\begin{filecontents*}[overwrite]{\jobname.aux} \relax \providecommand\hyper@newdestlabel[2]{} \providecommand\HyField@AuxAddToFields[1]{} \providecommand\HyField@AuxAddToCoFields[2]{} \citation{Wil74} \citation{SZZ23} \citation{BB26a} \citation{MM94} \citation{GL10} \citation{KS75} \citation{OSei78,Sei82} \citation{OSei78} \citation{Sei82} \citation{BFS79} \citation{Chat19} \citation{Chat19b} \citation{BG18} \citation{SZZ23} \citation{CNS25} \citation{AC25} \citation{GKS89} \citation{Dri89} \citation{Lev03} \citation{Dim13} \citation{MRS93} \citation{JW06} \citation{Bal84a,Bal85a,Bal89b} \citation{BE85,BGL14} \citation{SZZ23} \citation{BB26a} \citation{Dob68,Dob70} \citation{DS85} \citation{Vil09} \citation{BB26a} \citation{Cre77} \newlabel{FirstPage}{{}{1}{}{section*.1}{}} \@writefile{toc}{\contentsline {title}{One Cube Estimate Implies the Yang--Mills Mass Gap}{1}{section*.2}\protected@file@percent } \@writefile{toc}{\contentsline {abstract}{Abstract}{1}{section*.1}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {I}Introduction}{1}{section*.3}\protected@file@percent } \newlabel{sec:intro}{{I}{1}{}{section*.3}{}} \citation{OS73,OS75} \citation{OSei78} \citation{Kac90} \citation{Mil76} \citation{BTBS26,GBN26} \citation{OSei78} \newlabel{thm:su2}{{1}{2}{$\SU (2)$}{theorem.1}{}} \newlabel{thm:ceiling}{{2}{2}{Ceiling}{theorem.2}{}} \newlabel{thm:blocks}{{3}{2}{$\SU (2)$ with plaquette blocks}{theorem.3}{}} \newlabel{thm:curv}{{4}{2}{Curvature in the gauge}{theorem.4}{}} \newlabel{thm:groups}{{5}{2}{Every compact simple group}{theorem.5}{}} \citation{BB26a} \citation{SZZ23} \citation{BB26a} \citation{BB26a} \citation{BB26a} \citation{BB26b} \@writefile{lot}{\contentsline {table}{\numberline {I}{\ignorespaces Least Ricci curvature $\kappa _G=h^\vee /(2\ell _\rho )$ of the compact simple groups in a faithful representation $\rho $ of least dimension $N$, and the range of Theorem~\ref {thm:groups} for the conventional inverse coupling $\beta _W=N^2\beta $.}}{3}{table.1}\protected@file@percent } \newlabel{tab:groups}{{I}{3}{Least Ricci curvature $\kappa _G=h^\vee /(2\ell _\rho )$ of the compact simple groups in a faithful representation $\rho $ of least dimension $N$, and the range of Theorem~\ref {thm:groups} for the conventional inverse coupling $\beta _W=N^2\beta $}{table.1}{}} \newlabel{thm:su3}{{6}{3}{$\SU (3)$}{theorem.6}{}} \@writefile{lot}{\contentsline {table}{\numberline {II}{\ignorespaces Explicit ranges of uniqueness, decay and a transfer-matrix gap for four-dimensional lattice Yang--Mills theory. The first three rows hold for the ungauged measure; the last three are the results of this paper.}}{3}{table.2}\protected@file@percent } \newlabel{tab:ranges}{{II}{3}{Explicit ranges of uniqueness, decay and a transfer-matrix gap for four-dimensional lattice Yang--Mills theory. The first three rows hold for the ungauged measure; the last three are the results of this paper}{table.2}{}} \newlabel{thm:su3blocks}{{7}{3}{$\SU (3)$ with plaquette blocks}{theorem.7}{}} \citation{Bal84a,Bal85a,Bal89b} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The gap $m_G(\beta )$ of Theorem~\ref {thm:groups} as a function of $\beta _W=N^2\beta $ on a logarithmic scale, for nine groups in the representations of Table~\ref {tab:groups}. Each curve ends at the edge $\beta _W=2N\kappa _G/29$ of the range, marked by a dot, where the gap tends to zero; as $\beta _W\to 0$ it grows like $\log (\kappa _G/(6N|\beta |))$. The curves differ only by the factor $N\kappa _G$, since $m_G$ depends on $N|\beta |/\kappa _G$ alone.}}{4}{figure.1}\protected@file@percent } \newlabel{fig:groups}{{1}{4}{The gap $m_G(\beta )$ of Theorem~\ref {thm:groups} as a function of $\beta _W=N^2\beta $ on a logarithmic scale, for nine groups in the representations of Table~\ref {tab:groups}. Each curve ends at the edge $\beta _W=2N\kappa _G/29$ of the range, marked by a dot, where the gap tends to zero; as $\beta _W\to 0$ it grows like $\log (\kappa _G/(6N|\beta |))$. The curves differ only by the factor $N\kappa _G$, since $m_G$ depends on $N|\beta |/\kappa _G$ alone}{figure.1}{}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Ranges of Table~\ref {tab:ranges}. The height of each bar is the largest inverse coupling $\beta _W$ covered by the method. Grey bars are methods for the ungauged measure; blue, red and orange bars use the forest gauge. For $\mathrm {SU}(2)$ the red bar ends at the ceiling of Theorem~\ref {thm:ceiling} and the orange bar, plaquette blocks, passes it (Theorem~\ref {thm:blocks}); for $\mathrm {SU}(3)$ the red bar is Theorem~\ref {thm:su3} and the orange bar, plaquette blocks, gives a larger gap there and passes it (Theorem~\ref {thm:su3blocks}).}}{5}{figure.2}\protected@file@percent } \newlabel{fig:ranges}{{2}{5}{Ranges of Table~\ref {tab:ranges}. The height of each bar is the largest inverse coupling $\beta _W$ covered by the method. Grey bars are methods for the ungauged measure; blue, red and orange bars use the forest gauge. For $\SU (2)$ the red bar ends at the ceiling of Theorem~\ref {thm:ceiling} and the orange bar, plaquette blocks, passes it (Theorem~\ref {thm:blocks}); for $\SU (3)$ the red bar is Theorem~\ref {thm:su3} and the orange bar, plaquette blocks, gives a larger gap there and passes it (Theorem~\ref {thm:su3blocks})}{figure.2}{}} \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The logical structure of the paper. Green: theorems at fixed lattice spacing (P1 the gap for every compact simple group, Theorem~\ref {thm:groups}, P2 the plaquette blocks for $\mathrm {SU}(2)$ and $\mathrm {SU}(3)$, P3 the cube condition and the averaged condition BP at strong coupling, Corollary~\ref {cor:bpranges}). Blue: the proved implications (I1)--(I3) of the continuum step, where (I1) takes either (D) or the local form of BP (Theorem~\ref {thm:bplocal}). Red: their inputs, none of which is proved along the continuum tuning. Each arrow joins an open input to the implication that consumes it; no implication feeds its own input.}}{7}{figure.3}\protected@file@percent } \newlabel{fig:logic}{{3}{7}{The logical structure of the paper. Green: theorems at fixed lattice spacing (P1 the gap for every compact simple group, Theorem~\ref {thm:groups}, P2 the plaquette blocks for $\SU (2)$ and $\SU (3)$, P3 the cube condition and the averaged condition BP at strong coupling, Corollary~\ref {cor:bpranges}). Blue: the proved implications (I1)--(I3) of the continuum step, where (I1) takes either (D) or the local form of BP (Theorem~\ref {thm:bplocal}). Red: their inputs, none of which is proved along the continuum tuning. Each arrow joins an open input to the implication that consumes it; no implication feeds its own input}{figure.3}{}} \@writefile{toc}{\contentsline {section}{\numberline {II}The line field}{7}{section*.4}\protected@file@percent } \newlabel{sec:linefield}{{II}{7}{}{section*.4}{}} \newlabel{eq:count}{{1}{7}{}{equation.2.1}{}} \newlabel{lem:cycle}{{8}{7}{}{theorem.8}{}} \newlabel{lem:structure}{{9}{8}{}{theorem.9}{}} \@writefile{toc}{\contentsline {section}{\numberline {III}Gauge fixing and the free-boundary state}{8}{section*.5}\protected@file@percent } \newlabel{sec:state}{{III}{8}{}{section*.5}{}} \newlabel{lem:gaugefix}{{10}{8}{Gauge fixing}{theorem.10}{}} \newlabel{prop:state}{{11}{8}{}{theorem.11}{}} \citation{OSei78} \citation{BB26a} \citation{OS73,OS75,OSei78} \@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces (a) The line field in the slice $x_3=0$ of a box of side three. Thick red segments are the links of $F$ in the directions $0,1,2$, with a dot marking the direction of travel; an orange ring marks a site whose line leaves the slice in direction~$3$. Every line is straight (Lemma~\ref {lem:structure}(i)), and the segments that leave the box are its boundary stubs. (b) One plaquette with its fixed link $e_1$. The weights are those of the local inequality of Lemma~\ref {lem:local} for the three free links: $\frac {21}{10}$ on the two links adjacent to $e_1$ and $\frac 54$ on the opposite one. The two purple corners are the ones that enter the corner form~\eqref {eq:corner} once $Y_1=0$.}}{9}{figure.4}\protected@file@percent } \newlabel{fig:linefield}{{4}{9}{(a) The line field in the slice $x_3=0$ of a box of side three. Thick red segments are the links of $F$ in the directions $0,1,2$, with a dot marking the direction of travel; an orange ring marks a site whose line leaves the slice in direction~$3$. Every line is straight (Lemma~\ref {lem:structure}(i)), and the segments that leave the box are its boundary stubs. (b) One plaquette with its fixed link $e_1$. The weights are those of the local inequality of Lemma~\ref {lem:local} for the three free links: $\frac {21}{10}$ on the two links adjacent to $e_1$ and $\frac 54$ on the opposite one. The two purple corners are the ones that enter the corner form~\eqref {eq:corner} once $Y_1=0$}{figure.4}{}} \@writefile{toc}{\contentsline {section}{\numberline {IV}SU(2) up to one third}{9}{section*.6}\protected@file@percent } \newlabel{sec:su2}{{IV}{9}{}{section*.6}{}} \citation{BB26a} \@writefile{toc}{\contentsline {section}{\numberline {V}SU(2) past one third: plaquette blocks}{10}{section*.7}\protected@file@percent } \newlabel{sec:plaqblocks}{{V}{10}{}{section*.7}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {A}The blocks}{10}{section*.8}\protected@file@percent } \newlabel{lem:plaqblocks}{{12}{10}{}{theorem.12}{}} \newlabel{eq:blocklaw}{{2}{10}{}{equation.5.2}{}} \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Block plaquettes, shaded, in (a) a plane $\{0,1\}$ and (b) a plane $\{2,3\}$: the plaquettes at $x$ with $x_0\equiv x_1$, respectively $x_2\equiv x_3$, modulo two. Red links belong to the line field $F$; a filled dot marks a site whose line runs in the plane, an open dot one whose line leaves it. The shaded squares form a checkerboard, so every link of the plane borders exactly one of them, and every square contains exactly one red link (Lemma~\ref {lem:cycle}); hence each block plaquette carries three active links and the blocks partition the active links (Lemma~\ref {lem:plaqblocks}(i)).}}{11}{figure.5}\protected@file@percent } \newlabel{fig:plaqblocks}{{5}{11}{Block plaquettes, shaded, in (a) a plane $\{0,1\}$ and (b) a plane $\{2,3\}$: the plaquettes at $x$ with $x_0\equiv x_1$, respectively $x_2\equiv x_3$, modulo two. Red links belong to the line field $F$; a filled dot marks a site whose line runs in the plane, an open dot one whose line leaves it. The shaded squares form a checkerboard, so every link of the plane borders exactly one of them, and every square contains exactly one red link (Lemma~\ref {lem:cycle}); hence each block plaquette carries three active links and the blocks partition the active links (Lemma~\ref {lem:plaqblocks}(i))}{figure.5}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {B}The sensitivity of one block}{11}{section*.9}\protected@file@percent } \newlabel{eq:blockconst}{{3}{11}{}{equation.5.3}{}} \newlabel{lem:tilt}{{13}{11}{}{theorem.13}{}} \newlabel{lem:onesite}{{14}{11}{}{theorem.14}{}} \citation{Ban38} \newlabel{lem:blockK}{{15}{12}{}{theorem.15}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {C}The constants}{12}{section*.10}\protected@file@percent } \newlabel{lem:vmfconst}{{16}{12}{}{theorem.16}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {D}The proof of Theorem~\ref {thm:blocks}}{13}{section*.11}\protected@file@percent } 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\newlabel{lem:boxes}{{35}{20}{Interpolation on a box}{theorem.35}{}} \newlabel{eq:hoeffding}{{12}{20}{}{equation.7.12}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {H}The cover}{20}{section*.22}\protected@file@percent } \newlabel{ssec:cover}{{VII\,H}{20}{}{section*.22}{}} \newlabel{eq:BQ}{{13}{20}{}{equation.7.13}{}} \citation{HS87} \newlabel{rem:su3cover}{{36}{21}{a computation, used by no theorem}{theorem.36}{}} \@writefile{toc}{\contentsline {section}{\numberline {VIII}SU(3) with plaquette blocks}{21}{section*.23}\protected@file@percent } \newlabel{sec:su3blocks}{{VIII}{21}{}{section*.23}{}} \newlabel{eq:blocklaw3}{{14}{21}{}{equation.8.14}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {A}The sensitivity of one block}{21}{section*.24}\protected@file@percent } \newlabel{eq:blockconst3}{{15}{21}{}{equation.8.15}{}} \newlabel{lem:onesite3}{{37}{21}{}{theorem.37}{}} \newlabel{lem:blockK3}{{38}{21}{}{theorem.38}{}} \@writefile{lof}{\contentsline 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(a) All boxes, with height $|x_3+iy_3|$. (b) The $419$ boxes with bound above $0.99$, with height $x_3$; the largest bounds sit in the interior of $\mathcal R_r$, not at its extreme points. No result uses this computation.}}{22}{figure.6}\protected@file@percent } \newlabel{fig:cover}{{6}{22}{The $2341$ boxes of Remark~\ref {rem:su3cover} at $r=\frac 32$, plotted at their centres in the coordinates $(a_1,a_2)$ of the staple sum $\operatorname {diag}(a_1,a_2,x_3+iy_3)$, coloured by the computed bound on $2r\sup _hL_h$ over the box; the larger points are the boxes of side $r/8$. (a) All boxes, with height $|x_3+iy_3|$. (b) The $419$ boxes with bound above $0.99$, with height $x_3$; the largest bounds sit in the interior of $\mathcal R_r$, not at its extreme points. No result uses this computation}{figure.6}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {B}The constants and the proof of Theorem~\ref {thm:su3blocks}}{22}{section*.25}\protected@file@percent } \newlabel{rem:su3blocks}{{39}{22}{a computation, used by no theorem}{theorem.39}{}} \citation{DS85} \citation{BB26a} \citation{JW06} \citation{BB26b} \citation{Bal84a,BSB26} \citation{Bal84a,Dim13} \citation{SBG26} \citation{Bha22,BRS26} \@writefile{toc}{\contentsline {section}{\numberline {IX}Where the methods stop}{23}{section*.26}\protected@file@percent } \newlabel{sec:limits}{{IX}{23}{}{section*.26}{}} \citation{GJ87} \citation{GW73,Pol73} \@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces The floating-point value of $2r\sup _h$ of the bound of Lemma~\ref {lem:corrector} on the face $b=\operatorname {diag}(r,\rho ,\rho )$, as a function of $\rho /r$, for $r=1.44$, $1.50$ and $1.54$: dashed with the corrector $u=s\varphi _h$ for the best $s$, solid with $u=q_h$ of degree two for the best $C$. The grey plane is the Dobrushin threshold one. The first corrector is above it already at $r=1.44$; the second stays below it at $r=1.50$, where the computation of Remark~\ref {rem:su3cover} bounds it on all of $\mathcal R_r$, and crosses it by $r=1.54$.}}{24}{figure.7}\protected@file@percent } \newlabel{fig:face}{{7}{24}{The floating-point value of $2r\sup _h$ of the bound of Lemma~\ref {lem:corrector} on the face $b=\operatorname {diag}(r,\rho ,\rho )$, as a function of $\rho /r$, for $r=1.44$, $1.50$ and $1.54$: dashed with the corrector $u=s\varphi _h$ for the best $s$, solid with $u=q_h$ of degree two for the best $C$. The grey plane is the Dobrushin threshold one. The first corrector is above it already at $r=1.44$; the second stays below it at $r=1.50$, where the computation of Remark~\ref {rem:su3cover} bounds it on all of $\mathcal R_r$, and crosses it by $r=1.54$}{figure.7}{}} \newlabel{eq:growth}{{16}{24}{}{equation.9.16}{}} \newlabel{prop:growth}{{40}{24}{}{theorem.40}{}} \citation{GJ87} \citation{GJ87} \citation{BB26b} \citation{Bal84a,Bal89b} \citation{BB26b} \citation{DS85} \newlabel{prop:nogapgff}{{41}{25}{}{theorem.41}{}} \@writefile{toc}{\contentsline {section}{\numberline {X}The continuum step at one physical scale}{25}{section*.27}\protected@file@percent } \newlabel{sec:bypass}{{X}{25}{}{section*.27}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {A}Averaged cubes}{25}{section*.28}\protected@file@percent } \newlabel{def:cube}{{42}{25}{}{theorem.42}{}} \newlabel{thm:onescale}{{43}{25}{}{theorem.43}{}} \citation{OS73,OS75} \@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces (a) The cube $V$ of side $n=4$ and two of its translates by $2\mathbb {Z}^4$, drawn in three of the four dimensions. A link $j$ outside $V$ enters $K_c$ through every translate whose boundary it touches, and $D(n,\alpha )$ compares that total with the $N_n=(n/2)^4$ translates that contain $j$. (b) The cube of Theorem~\ref {thm:transfer}: its physical side $\ell =na$ is held fixed while the spacing is halved, so $n$ doubles at each step and the lattice gap $\log (1/\alpha )/(2n)$ is of order $a/\ell $.}}{26}{figure.8}\protected@file@percent } \newlabel{fig:cubes}{{8}{26}{(a) The cube $V$ of side $n=4$ and two of its translates by $2\Z ^4$, drawn in three of the four dimensions. A link $j$ outside $V$ enters $K_c$ through every translate whose boundary it touches, and $D(n,\alpha )$ compares that total with the $N_n=(n/2)^4$ translates that contain $j$. 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The correlation $C_k(\sigma )$ pairs the reflected packet with the translated one; $C_k(0)$ is the reflected norm that renormalises the packet.}}{29}{figure.9}\protected@file@percent } \newlabel{fig:packet}{{9}{29}{The packet $f_0$ of Theorem~\ref {thm:flowlimit}, supported in the time window $[t_0,t_1]$ above the reflection plane $t=0$, its reflection $\vartheta f_0$ and its translate $\tau _\sigma f_0$ by $\sigma $ in time. The correlation $C_k(\sigma )$ pairs the reflected packet with the translated one; $C_k(0)$ is the reflected norm that renormalises the packet}{figure.9}{}} \citation{Lus77} \citation{Lus77,OSei78} \newlabel{prop:flowshift}{{49}{30}{}{theorem.49}{}} \newlabel{prop:flowpositive}{{50}{30}{}{theorem.50}{}} \newlabel{prop:sliceratio}{{51}{31}{}{theorem.51}{}} \newlabel{rem:freeslice}{{52}{31}{}{theorem.52}{}} \newlabel{rem:compositeslice}{{53}{32}{}{theorem.53}{}} \newlabel{rem:twodslice}{{54}{32}{}{theorem.54}{}} \citation{GKS89,Dri89,Lev03} \newlabel{prop:blockfree}{{55}{33}{}{theorem.55}{}} \@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces Two-dimensional $\mathrm {SU}(N)$ (Remark~\ref {rem:twodslice}). (a) The lattice $a_k\mathbb {Z}\times a_k\mathbb {Z}_{L_k}$ drawn as a cylinder: the holonomy $V$ of one slice, the holonomy $V'=QV$ of the next, and the plaquette variables $W_x$ between them. (b) The energies $E_{j,k}=-(\ell /a_k^2)\log \hat c_j(\beta _k)$ of $\mathrm {SU}(2)$ for $g^2=\ell =1$, with $\hat c_j(\beta )=I_{2j+1}(\beta )/I_1(\beta )$, against the lattice spacing; the dashed lines are the limits $j(j+1)/2$.}}{34}{figure.10}\protected@file@percent } \newlabel{fig:twod}{{10}{34}{Two-dimensional $\SU (N)$ (Remark~\ref {rem:twodslice}). (a) The lattice $a_k\Z \times a_k\Z _{L_k}$ drawn as a cylinder: the holonomy $V$ of one slice, the holonomy $V'=QV$ of the next, and the plaquette variables $W_x$ between them. (b) The energies $E_{j,k}=-(\ell /a_k^2)\log \hat c_j(\beta _k)$ of $\SU (2)$ for $g^2=\ell =1$, with $\hat c_j(\beta )=I_{2j+1}(\beta )/I_1(\beta )$, against the lattice spacing; the dashed lines are the limits $j(j+1)/2$}{figure.10}{}} \@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces The ratio $D_k(r)/D_k(\frac 12)$ over the separation $r$ and the lattice spacing $a_k$. (a) Two-dimensional $\mathrm {SU}(2)$ with $f=\operatorname {Re}\operatorname {Tr}V$ and $g^2=\ell =1$, where the ratio is $e^{-(r-1/2)E_{1/2,k}}$ and converges as $a_k\to 0$. (b) The term $e^{-(r-1/2)m_k}$ of the lowest energy $m_k$ of a free field with mass $\frac 12$ in lattice units, which bounds the ratio from above and tends to zero at every $r>\frac 12$.}}{34}{figure.11}\protected@file@percent } \newlabel{fig:slice}{{11}{34}{The ratio $D_k(r)/D_k(\frac 12)$ over the separation $r$ and the lattice spacing $a_k$. (a) Two-dimensional $\SU (2)$ with $f=\ReTr V$ and $g^2=\ell =1$, where the ratio is $e^{-(r-1/2)E_{1/2,k}}$ and converges as $a_k\to 0$. (b) The term $e^{-(r-1/2)m_k}$ of the lowest energy $m_k$ of a free field with mass $\frac 12$ in lattice units, which bounds the ratio from above and tends to zero at every $r>\frac 12$}{figure.11}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {E}Higher correlations}{34}{section*.32}\protected@file@percent } \newlabel{ssec:higher}{{X\,E}{34}{}{section*.32}{}} \@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces Block averages between the reflection plane $t=0$ and a packet $F$ at physical height $d$, for blocks of side $d/4$, $d/2$ and $d$. By Proposition~\ref {prop:blockfree} the first conditioning step already leaves a fixed fraction of the ordinary variance of the packet, so the errors of such steps have to be measured in the reflected norm.}}{35}{figure.12}\protected@file@percent } \newlabel{fig:blocking}{{12}{35}{Block averages between the reflection plane $t=0$ and a packet $F$ at physical height $d$, for blocks of side $d/4$, $d/2$ and $d$. By Proposition~\ref {prop:blockfree} the first conditioning step already leaves a fixed fraction of the ordinary variance of the packet, so the errors of such steps have to be measured in the reflected norm}{figure.12}{}} \newlabel{lem:treesum}{{56}{35}{}{theorem.56}{}} \newlabel{thm:higher}{{57}{35}{}{theorem.57}{}} \citation{OS73} \citation{OS73} \citation{OSei78} \newlabel{rem:amplitude}{{58}{37}{}{theorem.58}{}} \newlabel{rem:markov}{{59}{37}{}{theorem.59}{}} \newlabel{prop:treecritical}{{60}{37}{}{theorem.60}{}} \newlabel{prop:treeenergy}{{61}{38}{}{theorem.61}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {F}An averaged cube condition}{38}{section*.33}\protected@file@percent } \newlabel{ssec:averaged}{{X\,F}{38}{}{section*.33}{}} \newlabel{def:avgcube}{{62}{38}{}{theorem.62}{}} \newlabel{lem:avgspeed}{{63}{39}{finite speed of the cube chain}{theorem.63}{}} \newlabel{thm:avgdecay}{{64}{39}{decay from the averaged condition}{theorem.64}{}} \citation{OSei78,Sei82} \newlabel{cor:avggap}{{65}{40}{the lattice gap}{theorem.65}{}} \newlabel{thm:avgtransfer}{{66}{40}{the continuum step from the averaged condition}{theorem.66}{}} \newlabel{prop:avgsup}{{67}{40}{}{theorem.67}{}} \citation{FS82} \citation{FS82} \citation{DS85} \citation{Vil09} \newlabel{cor:avgcoulomb}{{68}{41}{}{theorem.68}{}} \newlabel{rem:avgstatus}{{69}{41}{what the averaged condition settles and what it leaves}{theorem.69}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {G}The averaged condition at strong coupling and at fixed cube size}{41}{section*.34}\protected@file@percent } \newlabel{ssec:bpstrong}{{X\,G}{41}{}{section*.34}{}} \newlabel{lem:bpblock}{{70}{41}{}{theorem.70}{}} \newlabel{lem:bpcount}{{71}{42}{}{theorem.71}{}} \newlabel{thm:bpstrong}{{72}{42}{the averaged condition from single links}{theorem.72}{}} \newlabel{lem:bponelink}{{73}{42}{}{theorem.73}{}} \newlabel{cor:bpranges}{{74}{42}{explicit ranges}{theorem.74}{}} \citation{Mar99} \newlabel{prop:bpscales}{{75}{43}{}{theorem.75}{}} \newlabel{lem:bpstokes}{{76}{44}{}{theorem.76}{}} \newlabel{lem:bpcommute}{{77}{44}{}{theorem.77}{}} \newlabel{lem:bpflat}{{78}{44}{}{theorem.78}{}} \newlabel{thm:bpfixed}{{79}{44}{the averaged condition at fixed cube size}{theorem.79}{}} \newlabel{cor:bpresponse}{{80}{45}{the response of a cube at weak coupling}{theorem.80}{}} \newlabel{rem:bpstatus}{{81}{45}{what is proved about the averaged condition}{theorem.81}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {H}The averaged condition at weak coupling: centre sectors and local slow modes}{45}{section*.35}\protected@file@percent } \newlabel{ssec:bpweak}{{X\,H}{45}{}{section*.35}{}} \@writefile{lof}{\contentsline {figure}{\numberline {13}{\ignorespaces The averaged condition at strong coupling and at fixed cube size. (a)~The bound $\log _{10}\bigl (38400\,n^4(\log \beta +6.11)/\beta \bigr )$ of Theorem~\ref {thm:bpfixed}(ii) for $\mathrm {SU}(2)$ over the cube side $n$ and $\log _{10}\beta $; where it lies below the plane $\varpi =1$ (red), $\mathrm {BP}(n,1)$ fails on $\Lambda _{2n+4}$, and the black curve is the crossing. (b)~For $\mathrm {SU}(2)$ in the variables $\beta _W=4\beta $ and $n$: the range $\lvert \beta _W\rvert <\frac 29$ of Corollary~\ref {cor:bpranges}, where $\mathrm {BP}(n,\varpi )$ holds on every torus for all $n\ge n_0(\beta _W)$ (green); the region where Theorem~\ref {thm:bpfixed}(ii) excludes $\mathrm {BP}(n,\frac 12)$ on $\Lambda _{2n+4}$ (red); and the continuum tuning $n\propto e^{3\pi ^2\beta _W/11}$ at fixed physical cube size (dashed, schematic, the power prefactor omitted), which runs through the region where neither result applies.}}{46}{figure.13}\protected@file@percent } \newlabel{fig:bpranges}{{13}{46}{The averaged condition at strong coupling and at fixed cube size. (a)~The bound $\log _{10}\bigl (38400\,n^4(\log \beta +6.11)/\beta \bigr )$ of Theorem~\ref {thm:bpfixed}(ii) for $\SU (2)$ over the cube side $n$ and $\log _{10}\beta $; where it lies below the plane $\varpi =1$ (red), $\mathrm {BP}(n,1)$ fails on $\Lambda _{2n+4}$, and the black curve is the crossing. (b)~For $\SU (2)$ in the variables $\beta _W=4\beta $ and $n$: the range $\lvert \beta _W\rvert <\frac 29$ of Corollary~\ref {cor:bpranges}, where $\mathrm {BP}(n,\varpi )$ holds on every torus for all $n\ge n_0(\beta _W)$ (green); the region where Theorem~\ref {thm:bpfixed}(ii) excludes $\mathrm {BP}(n,\frac 12)$ on $\Lambda _{2n+4}$ (red); and the continuum tuning $n\propto e^{3\pi ^2\beta _W/11}$ at fixed physical cube size (dashed, schematic, the power prefactor omitted), which runs through the region where neither result applies}{figure.13}{}} \newlabel{prop:bpcentre}{{82}{46}{centre sectors}{theorem.82}{}} \newlabel{def:bploc}{{83}{47}{}{theorem.83}{}} \newlabel{thm:bplocal}{{84}{47}{the local form of the averaged condition}{theorem.84}{}} \@writefile{lof}{\contentsline {figure}{\numberline {14}{\ignorespaces Centre sectors and the local form of the averaged condition. (a)~A three-dimensional section of the torus with the slices $\{y_0=t\}$. The twist $Z^0_t$ multiplies the links that cross one slice by the central element $z$ (gold) and leaves every plaquette unchanged; a Polyakov loop of direction $0$ crosses every slice once and is charged, while a gauge-invariant function of a box that misses one slice is neutral (Proposition~\ref {prop:bpcentre}). (b)~Spectral measures of the cube chain $\mathcal K$, schematic: under $\mathrm {BP}_{\rm loc}$ the measure of a local function puts no mass on $(1-\epsilon ,1]$ (green), while a charged function may carry mass there (red), as the test functions of Theorem~\ref {thm:bpfixed} do at weak coupling (Theorem~\ref {thm:bplocal}, Corollary~\ref {cor:bpcharged}).}}{48}{figure.14}\protected@file@percent } \newlabel{fig:bpsectors}{{14}{48}{Centre sectors and the local form of the averaged condition. (a)~A three-dimensional section of the torus with the slices $\{y_0=t\}$. The twist $Z^0_t$ multiplies the links that cross one slice by the central element $z$ (gold) and leaves every plaquette unchanged; a Polyakov loop of direction $0$ crosses every slice once and is charged, while a gauge-invariant function of a box that misses one slice is neutral (Proposition~\ref {prop:bpcentre}). (b)~Spectral measures of the cube chain $\mathcal K$, schematic: under $\mathrm {BP}_{\rm loc}$ the measure of a local function puts no mass on $(1-\epsilon ,1]$ (green), while a charged function may carry mass there (red), as the test functions of Theorem~\ref {thm:bpfixed} do at weak coupling (Theorem~\ref {thm:bplocal}, Corollary~\ref {cor:bpcharged})}{figure.14}{}} \newlabel{cor:bpcharged}{{85}{48}{the fixed-size obstruction is charged}{theorem.85}{}} \newlabel{cor:bpcoupling}{{86}{48}{couplings on gauge orbits}{theorem.86}{}} \citation{HS87} \newlabel{prop:bpnoloc}{{87}{49}{no local slow modes}{theorem.87}{}} \newlabel{prop:bpsharp}{{88}{49}{}{theorem.88}{}} \citation{Mar99} \citation{Bal84a,Bal85a,Bal89b} \newlabel{rem:bproutes}{{89}{50}{the routes to the averaged condition at weak coupling}{theorem.89}{}} \newlabel{rem:bpweakstatus}{{90}{50}{what is proved about the averaged condition at weak coupling}{theorem.90}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {I}The neutral sector on small tori}{51}{section*.36}\protected@file@percent } \newlabel{ssec:bpneutral}{{X\,I}{51}{}{section*.36}{}} \newlabel{lem:bpflow}{{91}{51}{the holonomy flow}{theorem.91}{}} \newlabel{eq:bpanticonc}{{17}{52}{}{equation.10.17}{}} \newlabel{thm:bpneutral}{{92}{52}{the neutral sector on small tori}{theorem.92}{}} \citation{Bal84a,Bal85a,Bal89b} \@writefile{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces The neutral sector on small tori. (a)~A great circle of $\mathrm {SU}(2)$ through $I$, $u$, $-I$, $-u$, with the velocity $\sin 2\theta \,\partial _\theta $ by which the flow of Lemma~\ref {lem:bpflow} moves a Polyakov loop $P=\cos \theta \,I+\sin \theta \,u$. The flow fixes the centre $\pm I$ (red) and the traceless elements $\pm u$ (gold). (b)~The Haar law of $T=\operatorname {Tr}P_0$, with density $\sqrt {4-T^2}/(2\pi )$, and the weight $T^2(4-T^2)$ of Theorem~\ref {thm:bpneutral}; the weight has Haar mean $2$ and vanishes exactly at $T\in \{-2,0,2\}$, the set on which no flow of this kind gives a bound.}}{53}{figure.15}\protected@file@percent } \newlabel{fig:bpneutral}{{15}{53}{The neutral sector on small tori. (a)~A great circle of $\SU (2)$ through $I$, $u$, $-I$, $-u$, with the velocity $\sin 2\theta \,\partial _\theta $ by which the flow of Lemma~\ref {lem:bpflow} moves a Polyakov loop $P=\cos \theta \,I+\sin \theta \,u$. The flow fixes the centre $\pm I$ (red) and the traceless elements $\pm u$ (gold). (b)~The Haar law of $T=\Tr P_0$, with density $\sqrt {4-T^2}/(2\pi )$, and the weight $T^2(4-T^2)$ of Theorem~\ref {thm:bpneutral}; the weight has Haar mean $2$ and vanishes exactly at $T\in \{-2,0,2\}$, the set on which no flow of this kind gives a bound}{figure.15}{}} \newlabel{rem:bpneutralstatus}{{93}{53}{what is proved about the neutral sector and what remains}{theorem.93}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {J}The averaged condition under block-spin maps}{53}{section*.37}\protected@file@percent } \newlabel{ssec:bpblock}{{X\,J}{53}{}{section*.37}{}} \newlabel{thm:bpdescent}{{94}{54}{descent along block-spin maps}{theorem.94}{}} \newlabel{lem:bpintegrated}{{95}{54}{}{theorem.95}{}} \citation{Bal84a,Bal85a,Bal89b} \newlabel{cor:bpdescent}{{96}{55}{}{theorem.96}{}} \newlabel{rem:bpdescent}{{97}{55}{what the descent gives and what a bootstrap needs}{theorem.97}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {K}The upward step}{56}{section*.38}\protected@file@percent } \newlabel{ssec:bpup}{{X\,K}{56}{}{section*.38}{}} \newlabel{lem:bpupcond}{{98}{56}{}{theorem.98}{}} \newlabel{def:bpsplit}{{99}{56}{}{theorem.99}{}} \newlabel{thm:bpup}{{100}{56}{the upward step}{theorem.100}{}} \newlabel{thm:bpuptower}{{101}{56}{the upward step across scales}{theorem.101}{}} \newlabel{cor:bpuptuning}{{102}{57}{conditional bootstrap}{theorem.102}{}} \newlabel{prop:bpuphaar}{{103}{57}{the product measure}{theorem.103}{}} \newlabel{cor:bpupseparate}{{104}{58}{the separate inequalities across scales}{theorem.104}{}} \newlabel{rem:bpup}{{105}{58}{what the upward step gives and what it needs}{theorem.105}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {L}Haar refinements}{59}{section*.39}\protected@file@percent } \newlabel{ssec:bprefine}{{X\,L}{59}{}{section*.39}{}} \newlabel{lem:bpfibrelaw}{{106}{59}{}{theorem.106}{}} \newlabel{def:bprefine}{{107}{59}{}{theorem.107}{}} \newlabel{lem:bprefine}{{108}{59}{}{theorem.108}{}} \newlabel{prop:bpfinesplit}{{109}{60}{the fine form splits}{theorem.109}{}} \newlabel{thm:bprefine}{{110}{60}{the split inequality for Haar refinements}{theorem.110}{}} \newlabel{cor:bprefinetower}{{111}{61}{towers of refinements}{theorem.111}{}} \newlabel{prop:bpnotrefine}{{112}{61}{}{theorem.112}{}} \newlabel{rem:bprefine}{{113}{61}{what the refinements give and what they leave}{theorem.113}{}} \@writefile{lof}{\contentsline {figure}{\numberline {16}{\ignorespaces The averaged condition across scales. (a)~Three lattices of spacings $a_0$, $a_0/b$, $a_0/b^2$, $b=2$, each with a cube of the same physical side, of $n=3,6,12$ sites. The condition passes down to the block variables for every gauge-invariant measure (Theorem~\ref {thm:bpdescent}) and up one scale under the split inequality S (Theorem~\ref {thm:bpup}). (b)~S holds for every Haar refinement (Theorem~\ref {thm:bprefine}), the product of Haar measures included; a Wilson measure at $\beta \ne 0$ is not a Haar refinement (Proposition~\ref {prop:bpnotrefine}); for it S holds at strong coupling (Theorem~\ref {thm:bpsplitstrong}) and is open at weak coupling.}}{62}{figure.16}\protected@file@percent } \newlabel{fig:bptower}{{16}{62}{The averaged condition across scales. (a)~Three lattices of spacings $a_0$, $a_0/b$, $a_0/b^2$, $b=2$, each with a cube of the same physical side, of $n=3,6,12$ sites. The condition passes down to the block variables for every gauge-invariant measure (Theorem~\ref {thm:bpdescent}) and up one scale under the split inequality S (Theorem~\ref {thm:bpup}). (b)~S holds for every Haar refinement (Theorem~\ref {thm:bprefine}), the product of Haar measures included; a Wilson measure at $\beta \ne 0$ is not a Haar refinement (Proposition~\ref {prop:bpnotrefine}); for it S holds at strong coupling (Theorem~\ref {thm:bpsplitstrong}) and is open at weak coupling}{figure.16}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {M}The split inequality at strong coupling}{62}{section*.40}\protected@file@percent } \newlabel{ssec:bpsplitstrong}{{X\,M}{62}{}{section*.40}{}} \newlabel{thm:bpsplitcond}{{114}{62}{the split inequality from three bounds}{theorem.114}{}} \newlabel{prop:bptilt}{{115}{63}{the split inequality on one torus}{theorem.115}{}} \newlabel{lem:bpwilsonfibre}{{116}{64}{the fibres of the Wilson measure}{theorem.116}{}} \newlabel{lem:bpdecay}{{117}{64}{decay of the response of a block}{theorem.117}{}} \citation{Vil09} \newlabel{thm:bpfibrepoinc}{{118}{65}{the fibre inequality at strong coupling}{theorem.118}{}} \newlabel{thm:bpexterior}{{119}{65}{the exterior information at strong coupling}{theorem.119}{}} \newlabel{thm:bpsplitstrong}{{120}{66}{the split inequality at strong coupling}{theorem.120}{}} \newlabel{cor:bpsplitranges}{{121}{66}{explicit ranges}{theorem.121}{}} \@writefile{lof}{\contentsline {figure}{\numberline {17}{\ignorespaces The split inequality at strong coupling. (a)~A coarse link $e$ enters the fine field through its path $\pi (e)$; in a gauge whose marked link $l(e)$ lies in the cube $V'_{x'}$, a change of $V_e$ moves the fibre law by at most $q$, and the cube, resampled given its exterior, feels that move through $\partial V'_{x'}$ by at most $q$ again, so the exterior information is at most $q^2$ (Theorem~\ref {thm:bpexterior}); two cubes at distance $s$ see each other through correlations of the fibre law that decay like $\xi ^{s-n'-1}/(1-\xi )$ (Lemma~\ref {lem:bpdecay}). (b)~The constants of Theorem~\ref {thm:bpsplitstrong} for $\mathrm {SU}(2)$ at $n'=8$ and in the limit $n'\to \infty $, where $\eta \to 4q$; both tend to the value $1$ of the Haar refinements as $\beta _W\to 0$, and at $\lvert \beta _W\rvert =\frac 1{100}$ they are below $1.39$ (Corollary~\ref {cor:bpsplitranges}).}}{67}{figure.17}\protected@file@percent } \newlabel{fig:bpsplitstrong}{{17}{67}{The split inequality at strong coupling. (a)~A coarse link $e$ enters the fine field through its path $\pi (e)$; in a gauge whose marked link $l(e)$ lies in the cube $V'_{x'}$, a change of $V_e$ moves the fibre law by at most $q$, and the cube, resampled given its exterior, feels that move through $\partial V'_{x'}$ by at most $q$ again, so the exterior information is at most $q^2$ (Theorem~\ref {thm:bpexterior}); two cubes at distance $s$ see each other through correlations of the fibre law that decay like $\xi ^{s-n'-1}/(1-\xi )$ (Lemma~\ref {lem:bpdecay}). (b)~The constants of Theorem~\ref {thm:bpsplitstrong} for $\SU (2)$ at $n'=8$ and in the limit $n'\to \infty $, where $\eta \to 4q$; both tend to the value $1$ of the Haar refinements as $\beta _W\to 0$, and at $\lvert \beta _W\rvert =\frac 1{100}$ they are below $1.39$ (Corollary~\ref {cor:bpsplitranges})}{figure.17}{}} \newlabel{prop:bpsmalltori}{{122}{67}{the obstructions on small tori}{theorem.122}{}} \citation{Vil09} \newlabel{rem:bpsplitstrong}{{123}{68}{what is proved about the split inequality for Wilson measures}{theorem.123}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {N}The condition at the coarsest scale}{68}{section*.41}\protected@file@percent } \newlabel{ssec:bpcoarse}{{X\,N}{68}{}{section*.41}{}} \newlabel{lem:bpcoarseblock}{{124}{68}{}{theorem.124}{}} \newlabel{thm:bpcoarse}{{125}{69}{a single-link criterion for general measures}{theorem.125}{}} \newlabel{lem:bpcoarsehess}{{126}{69}{}{theorem.126}{}} \newlabel{cor:bpcoarsehess}{{127}{69}{}{theorem.127}{}} \newlabel{lem:bpcoarselaw}{{128}{70}{}{theorem.128}{}} \newlabel{cor:bpcoarseH1}{{129}{70}{what the criterion needs from the block laws}{theorem.129}{}} \newlabel{prop:bpcoarsezero}{{130}{70}{the place of (H1)}{theorem.130}{}} \newlabel{prop:bpcoarsenec}{{131}{71}{what the condition forces}{theorem.131}{}} \newlabel{prop:bpcoarseflat}{{132}{71}{a flat coarse law}{theorem.132}{}} \newlabel{prop:bphidden}{{133}{72}{closeness to Haar link by link}{theorem.133}{}} \citation{Bal84a,Bal85a,Bal89b} \newlabel{rem:bpcoarse}{{134}{73}{what is proved about the coarsest condition}{theorem.134}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {O}The single-link coefficients across scales}{73}{section*.42}\protected@file@percent } \newlabel{ssec:bpbalaban}{{X\,O}{73}{}{section*.42}{}} \citation{Dob68,Dob70} \@writefile{lof}{\contentsline {figure}{\numberline {18}{\ignorespaces The measure $\varrho _L$ of Proposition~\ref {prop:bphidden}(ii). (a)~One sign $S$, drawn once for the torus, chooses between two product tilts of the selected plaquettes; each link, given the others, has a law within $\epsilon $ of the Haar measure. (b)~The law of the spatial average $F$ for $\mathrm {SU}(2)$, $\epsilon =\frac 12$ and $L=8,16,32$, drawn with normal approximations of its two components and scaled to unit peak: the modes at $\pm \epsilon s_2^2$ stay apart as $L$ grows, while resampling one cube moves $F$ by at most $2r_x/M$, so the cube dynamics needs a time of order $L^4$ to change the sign.}}{74}{figure.18}\protected@file@percent } \newlabel{fig:bphidden}{{18}{74}{The measure $\varrho _L$ of Proposition~\ref {prop:bphidden}(ii). (a)~One sign $S$, drawn once for the torus, chooses between two product tilts of the selected plaquettes; each link, given the others, has a law within $\epsilon $ of the Haar measure. (b)~The law of the spatial average $F$ for $\SU (2)$, $\epsilon =\frac 12$ and $L=8,16,32$, drawn with normal approximations of its two components and scaled to unit peak: the modes at $\pm \epsilon s_2^2$ stay apart as $L$ grows, while resampling one cube moves $F$ by at most $2r_x/M$, so the cube dynamics needs a time of order $L^4$ to change the sign}{figure.18}{}} \newlabel{eq:bpbalmoments}{{18}{74}{}{equation.10.18}{}} \newlabel{lem:bpbalmarginal}{{135}{74}{marginals}{theorem.135}{}} \newlabel{lem:bpbalaxial}{{136}{75}{the axial gauge}{theorem.136}{}} \newlabel{thm:bpbalstep}{{137}{75}{one step of the straight-path map}{theorem.137}{}} \newlabel{cor:bpbaltower}{{138}{75}{a tower}{theorem.138}{}} \newlabel{lem:bpbalcov}{{139}{76}{gauge covariance}{theorem.139}{}} \newlabel{thm:bpbalnec}{{140}{76}{a lower bound from gauge covariance}{theorem.140}{}} \newlabel{cor:bpbalwilson}{{141}{76}{the Wilson measure}{theorem.141}{}} \citation{Bal84a,Bal85a} \citation{GW73,Pol73} \newlabel{cor:bpbalsmall}{{142}{77}{small fields}{theorem.142}{}} \newlabel{prop:bpbalrandom}{{143}{77}{randomisation along the paths}{theorem.143}{}} \@writefile{lof}{\contentsline {figure}{\numberline {19}{\ignorespaces The single-link coefficients across scales. (a) One step of the straight-path map with blocks of side $b=4$, in a plane: each coarse link is the product of the fine links of its path, marked at the middle link $l(e)$ (red); the law of the block variables is the marginal of the axial-gauge law on the marks, and the influence of $l(j)$ on $l(e)$ is a sum over walks of fine links weighted by products of coefficients (blue), which leave a factor $e^{-(\lambda -2\kappa )\lfloor b/2\rfloor }$ (Lemma~\ref {lem:bpbalmarginal}, Theorem~\ref {thm:bpbalstep}). (b) The lower bound on the Dobrushin sum of the $\mathrm {SU}(2)$ Wilson measure forced by gauge covariance, against $\beta _W$; it exceeds the threshold $1$ for $\beta _W\ge \frac 83$ (Corollary~\ref {cor:bpbalwilson}). (c) The lower bound $2-4(1-\varepsilon _{2^k})/\pi $ for the law of the block variables of side $2^k$, along $\beta _{W,k}=2+\frac {11}{3\pi ^2}k\log 2$, an illustration of the growth along the tuning; below $1$ gauge covariance no longer excludes the Dobrushin condition (Proposition~\ref {prop:bpbalrandom}).}}{78}{figure.19}\protected@file@percent } \newlabel{fig:bpbalaban}{{19}{78}{The single-link coefficients across scales. (a) One step of the straight-path map with blocks of side $b=4$, in a plane: each coarse link is the product of the fine links of its path, marked at the middle link $l(e)$ (red); the law of the block variables is the marginal of the axial-gauge law on the marks, and the influence of $l(j)$ on $l(e)$ is a sum over walks of fine links weighted by products of coefficients (blue), which leave a factor $e^{-(\lambda -2\kappa )\lfloor b/2\rfloor }$ (Lemma~\ref {lem:bpbalmarginal}, Theorem~\ref {thm:bpbalstep}). (b) The lower bound on the Dobrushin sum of the $\SU (2)$ Wilson measure forced by gauge covariance, against $\beta _W$; it exceeds the threshold $1$ for $\beta _W\ge \frac 83$ (Corollary~\ref {cor:bpbalwilson}). (c) The lower bound $2-4(1-\varepsilon _{2^k})/\pi $ for the law of the block variables of side $2^k$, along $\beta _{W,k}=2+\frac {11}{3\pi ^2}k\log 2$, an illustration of the growth along the tuning; below $1$ gauge covariance no longer excludes the Dobrushin condition (Proposition~\ref {prop:bpbalrandom})}{figure.19}{}} \newlabel{prop:bpbalhess}{{144}{78}{the Hessian of the induced action}{theorem.144}{}} \newlabel{rem:bpbalaban}{{145}{78}{what is proved about the coefficients across scales}{theorem.145}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {P}The coarsest condition through the fibre law}{79}{section*.43}\protected@file@percent } \newlabel{ssec:bpfibre}{{X\,P}{79}{}{section*.43}{}} \newlabel{lem:bpfibosc}{{146}{79}{oscillation}{theorem.146}{}} \newlabel{lem:bpfibgauge}{{147}{80}{gauge covariance of the Hessian}{theorem.147}{}} \newlabel{prop:bpfibreduce}{{148}{80}{the fibre covariance}{theorem.148}{}} \newlabel{cor:bpfibH1}{{149}{80}{fibre decorrelation gives (H1)}{theorem.149}{}} \newlabel{rem:bpfibstrong}{{150}{81}{strong coupling and Dobrushin bounds in the fibre}{theorem.150}{}} \newlabel{lem:bpfibker}{{151}{81}{}{theorem.151}{}} \newlabel{prop:bpfibgauss}{{152}{81}{the Gaussian fibre law}{theorem.152}{}} \newlabel{prop:bpfibbloch}{{153}{82}{the Gaussian model on all tori}{theorem.153}{}} \citation{Gut80,FS82} \newlabel{cor:bpfibgauss}{{154}{83}{the Gaussian form of (FC)}{theorem.154}{}} \newlabel{rem:bpfibuone}{{155}{83}{the abelian group}{theorem.155}{}} \newlabel{prop:bpfibrough}{{156}{83}{rough coarse data}{theorem.156}{}} \@writefile{lof}{\contentsline {figure}{\numberline {20}{\ignorespaces The fibre route to the coarsest condition. (a) The fibre law of one step of the straight-path map with blocks of side $b=4$, in a plane: the coarse data fix the marks (red), the other links of the paths are fixed to $I$ (black) and the links off the paths (grey) are integrated. In the Gaussian model the minimiser carries a current that vanishes off the paths, is constant along each path with the value $(\Gamma v)_e$ and is conserved at every block corner (Proposition~\ref {prop:bpfibgauss}(ii)). (b) The proved bounds $1/b^2\le \Gamma _{ee}\le 6/b$ and $\gamma _b\ge 2/b^2$ (Propositions~\ref {prop:bpfibgauss} and~\ref {prop:bpfibbloch}), and the decay $\gamma _b=O(1/b)$ of (G), drawn with an arbitrary constant; (G) is false, and $\gamma _b$ is of order $(\log b)/b$ (Theorem~\ref {thm:bprowgauss}). (c) The lower bound $4\beta _k\delta ^2/(\pi ^2b^2)$ on the action that rough coarse data of $\mathrm {SU}(2)$ force on the square of a coarse plaquette, for $\delta =\sqrt 2\pi $, $b=2^k$ and $\beta _k=2+\frac {11}{3\pi ^2} k\log 2$, an illustration of the growth along the tuning (Proposition~\ref {prop:bpfibrough}); it tends to zero and obstructs nothing.}}{84}{figure.20}\protected@file@percent } \newlabel{fig:bpfibre}{{20}{84}{The fibre route to the coarsest condition. (a) The fibre law of one step of the straight-path map with blocks of side $b=4$, in a plane: the coarse data fix the marks (red), the other links of the paths are fixed to $I$ (black) and the links off the paths (grey) are integrated. In the Gaussian model the minimiser carries a current that vanishes off the paths, is constant along each path with the value $(\Gamma v)_e$ and is conserved at every block corner (Proposition~\ref {prop:bpfibgauss}(ii)). (b) The proved bounds $1/b^2\le \Gamma _{ee}\le 6/b$ and $\gamma _b\ge 2/b^2$ (Propositions~\ref {prop:bpfibgauss} and~\ref {prop:bpfibbloch}), and the decay $\gamma _b=O(1/b)$ of (G), drawn with an arbitrary constant; (G) is false, and $\gamma _b$ is of order $(\log b)/b$ (Theorem~\ref {thm:bprowgauss}). (c) The lower bound $4\beta _k\delta ^2/(\pi ^2b^2)$ on the action that rough coarse data of $\SU (2)$ force on the square of a coarse plaquette, for $\delta =\sqrt 2\pi $, $b=2^k$ and $\beta _k=2+\frac {11}{3\pi ^2} k\log 2$, an illustration of the growth along the tuning (Proposition~\ref {prop:bpfibrough}); it tends to zero and obstructs nothing}{figure.20}{}} \newlabel{rem:bpfibre}{{157}{84}{what is proved about the fibre route}{theorem.157}{}} \citation{DLMF} \citation{DLMF} \citation{BtD85} \citation{DLMF} \citation{DLMF} \@writefile{toc}{\contentsline {subsection}{\numberline {Q}Smoothing along the paths}{85}{section*.44}\protected@file@percent } \newlabel{ssec:bppaths}{{X\,Q}{85}{}{section*.44}{}} \newlabel{lem:bppathmodes}{{158}{85}{every mode at once}{theorem.158}{}} \newlabel{lem:bppathsmooth}{{159}{86}{one link}{theorem.159}{}} \newlabel{thm:bppathproduct}{{160}{86}{products of one-link variables}{theorem.160}{}} \newlabel{thm:bppathdensity}{{161}{86}{the law of one block variable}{theorem.161}{}} \newlabel{cor:bppathcancel}{{162}{87}{the cancellation in the fibre law}{theorem.162}{}} \newlabel{lem:bppathint}{{163}{87}{the interiors}{theorem.163}{}} \newlabel{eq:bppathmix}{{19}{87}{the interiors}{equation.10.19}{}} \newlabel{thm:bppathmixed}{{164}{87}{mixed derivatives as posterior covariances}{theorem.164}{}} \newlabel{cor:bppathcolumn}{{165}{87}{a column sum below one third}{theorem.165}{}} \newlabel{cor:bppathfourk}{{166}{87}{tori of side up to $e^{2^k/(17k)}$}{theorem.166}{}} \citation{KP86} \citation{DLMF} \citation{KP86} \newlabel{eq:bppathrange}{{20}{88}{}{equation.10.20}{}} \newlabel{prop:bppathobst}{{167}{88}{two routes that fail}{theorem.167}{}} \newlabel{prop:bppathsuff}{{168}{88}{a sufficient covariance bound}{theorem.168}{}} \@writefile{lof}{\contentsline {figure}{\numberline {21}{\ignorespaces Smoothing along the paths. (a) The paths of the straight-path map with blocks of side $b=6$, in a plane: the interior links (red) of all the paths lie in no common plaquette and are independent given the end links (black) and the links off the paths (grey) (Lemma~\ref {lem:bppathint}). (b) The multipliers $a_d(B)=I_d(B)/I_1(B)$ of convolution with the one-link density on the representations of dimension $d=2,\dots ,6$, all below $2B/(2B+3)$ (Lemma~\ref {lem:bppathmodes}). (c) The numbers $\eta _{b-2}$, $h_{b-2}$ and $K_b^2$ for $b=2^k$ along the illustrative tuning $\beta _k=2+\frac {11}{3\pi ^2}k\log 2$, on a symmetric logarithmic scale; $h_{b-2}$ and $K_b^2$ are drawn where $\eta _{b-2}<\frac 12$. All three tend to zero faster than every power of $2^{-k}$ (Theorems~\ref {thm:bppathdensity} and~\ref {thm:bppathmixed}).}}{89}{figure.21}\protected@file@percent } \newlabel{fig:bppaths}{{21}{89}{Smoothing along the paths. (a) The paths of the straight-path map with blocks of side $b=6$, in a plane: the interior links (red) of all the paths lie in no common plaquette and are independent given the end links (black) and the links off the paths (grey) (Lemma~\ref {lem:bppathint}). (b) The multipliers $a_d(B)=I_d(B)/I_1(B)$ of convolution with the one-link density on the representations of dimension $d=2,\dots ,6$, all below $2B/(2B+3)$ (Lemma~\ref {lem:bppathmodes}). (c) The numbers $\eta _{b-2}$, $h_{b-2}$ and $K_b^2$ for $b=2^k$ along the illustrative tuning $\beta _k=2+\frac {11}{3\pi ^2}k\log 2$, on a symmetric logarithmic scale; $h_{b-2}$ and $K_b^2$ are drawn where $\eta _{b-2}<\frac 12$. 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(a) A column of the conditional mean $R=A^{-1}P^{\mathsf T}(PA^{-1}P^{\mathsf T})^{-1}$ of the block averages over blocks of side two on the periodic $18\times 18$ lattice, for $A=\frac 1{10}+4-N$: positive on its own block (blue), negative on the ring of neighbouring blocks (red), as Proposition~\ref {prop:bpnegative} requires. (b) The site-additive loss $\DOTSB \prod@ \slimits@ _{i0$; losses of second order, $c_j=c\,g_j^4$, keep it bounded below. (b) The section $c=0.1$, with the bound of part (ii) dashed.}}{112}{figure.23}\protected@file@percent } \newlabel{fig:scalelosses}{{23}{112}{Products of the factors $1-c_j$ over the $J+1$ scales between $a_k$ and $\sigma $ along the one-loop $\SU (2)$ tuning, with $\beta _W=20$ at the scale $\sigma $, as in Proposition~\ref {prop:scalelosses}. (a) Losses of first order in the running coupling, $c_j=c\,g_j^2$, drive the product to zero for every $c>0$; losses of second order, $c_j=c\,g_j^4$, keep it bounded below. (b) The section $c=0.1$, with the bound of part (ii) dashed}{figure.23}{}} \newlabel{prop:msplit}{{213}{112}{}{theorem.213}{}} \citation{FILS78} \newlabel{prop:stationary}{{214}{113}{}{theorem.214}{}} \newlabel{lem:ward}{{215}{114}{}{theorem.215}{}} \newlabel{prop:wardform}{{216}{114}{}{theorem.216}{}} \newlabel{prop:fourflow}{{217}{115}{}{theorem.217}{}} \newlabel{lem:limitset}{{218}{115}{}{theorem.218}{}} \newlabel{prop:tuned}{{219}{116}{}{theorem.219}{}} \newlabel{rem:rotatedlattice}{{220}{116}{}{theorem.220}{}} \newlabel{lem:timekernel}{{221}{117}{}{theorem.221}{}} \newlabel{prop:mzslice}{{222}{117}{}{theorem.222}{}} \newlabel{prop:mlocslice}{{223}{118}{}{theorem.223}{}} \newlabel{prop:centralpattern}{{224}{119}{}{theorem.224}{}} \newlabel{rem:opensplit}{{225}{120}{}{theorem.225}{}} \newlabel{lem:spatialmeasure}{{226}{121}{}{theorem.226}{}} \newlabel{prop:boxform}{{227}{122}{}{theorem.227}{}} \newlabel{thm:spatialrp}{{228}{122}{}{theorem.228}{}} \newlabel{prop:rpexamples}{{229}{123}{}{theorem.229}{}} \newlabel{prop:laplaceladder}{{230}{123}{}{theorem.230}{}} \newlabel{prop:centralsecond}{{231}{124}{}{theorem.231}{}} \newlabel{eq:poissonform}{{24}{124}{}{equation.11.24}{}} \newlabel{lem:poissontype}{{232}{124}{}{theorem.232}{}} \newlabel{thm:srladder}{{233}{125}{}{theorem.233}{}} \newlabel{eq:srladdermeas}{{25}{125}{}{equation.11.25}{}} \newlabel{eq:varpicompare}{{26}{125}{}{equation.11.26}{}} \newlabel{cor:srladder}{{234}{126}{}{theorem.234}{}} \newlabel{prop:heatrp}{{235}{126}{}{theorem.235}{}} \newlabel{prop:srrough}{{236}{126}{}{theorem.236}{}} \citation{OSei78} \newlabel{rem:spatialstatus}{{237}{127}{}{theorem.237}{}} \citation{OSei78,Sei82} \@writefile{toc}{\contentsline {subsection}{\numberline {E}Tuning by the ratio}{128}{section*.52}\protected@file@percent } \newlabel{ssec:ratiotuning}{{XI\,E}{128}{}{section*.52}{}} \newlabel{def:admissible}{{238}{128}{}{theorem.238}{}} \newlabel{eq:Comega}{{27}{128}{}{equation.11.27}{}} \newlabel{lem:admissible}{{239}{128}{}{theorem.239}{}} \newlabel{lem:gibbsunique}{{240}{128}{}{theorem.240}{}} \newlabel{prop:ratiovalues}{{241}{129}{}{theorem.241}{}} \newlabel{prop:boxratio}{{242}{129}{}{theorem.242}{}} \newlabel{def:ratiotuning}{{243}{130}{}{theorem.243}{}} \newlabel{thm:ratiotuning}{{244}{130}{}{theorem.244}{}} \newlabel{prop:ratiodregion}{{245}{130}{}{theorem.245}{}} \citation{GW73,Pol73,Cas74,Jon74} \newlabel{prop:ratiolocation}{{246}{131}{}{theorem.246}{}} \newlabel{eq:rescale}{{28}{131}{}{equation.11.28}{}} \newlabel{rem:twoloop}{{247}{131}{}{theorem.247}{}} \newlabel{cor:scalelossesgeneral}{{248}{131}{}{theorem.248}{}} \newlabel{prop:largeb}{{249}{132}{}{theorem.249}{}} \newlabel{lem:boxenergy}{{250}{132}{}{theorem.250}{}} \newlabel{eq:centreoddPhi}{{29}{133}{}{equation.11.29}{}} \newlabel{lem:loops}{{251}{133}{}{theorem.251}{}} \newlabel{thm:centreodd}{{252}{134}{}{theorem.252}{}} \newlabel{cor:centreodd}{{253}{135}{}{theorem.253}{}} \newlabel{prop:axialvariance}{{254}{135}{}{theorem.254}{}} \@writefile{lof}{\contentsline {figure}{\numberline {24}{\ignorespaces The box of Theorem~\ref {thm:centreodd}. (a)~The spatial torus $(\mathbb {Z}/L\mathbb {Z})^3$ as a cube with identified faces, the winding loops $W_1$ and $W_2$ of~\eqref {eq:centreoddPhi}, and a plane $x_1=\text {const}$ across which the centre transformation multiplies the links of direction $1$ by $-1$; $\operatorname {Tr}W_1$ and $\operatorname {Tr}W_1W_2$ change sign. (b)~The energy bound $\varepsilon (b)=3(\log b+5)/(2b)$ of Lemma~\ref {lem:boxenergy} against the threshold $1/(1600L^4)$ of Theorem~\ref {thm:centreodd}(ii) at $L=3$.}}{136}{figure.24}\protected@file@percent } \newlabel{fig:centreodd}{{24}{136}{The box of Theorem~\ref {thm:centreodd}. (a)~The spatial torus $(\Z /L\Z )^3$ as a cube with identified faces, the winding loops $W_1$ and $W_2$ of~\eqref {eq:centreoddPhi}, and a plane $x_1=\text {const}$ across which the centre transformation multiplies the links of direction $1$ by $-1$; $\Tr W_1$ and $\Tr W_1W_2$ change sign. (b)~The energy bound $\varepsilon (b)=3(\log b+5)/(2b)$ of Lemma~\ref {lem:boxenergy} against the threshold $1/(1600L^4)$ of Theorem~\ref {thm:centreodd}(ii) at $L=3$}{figure.24}{}} \newlabel{rem:centreodd}{{255}{136}{}{theorem.255}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {F}Estimates uniform in the coupling}{137}{section*.53}\protected@file@percent } \newlabel{ssec:uniform}{{XI\,F}{137}{}{section*.53}{}} \newlabel{prop:force}{{256}{137}{}{theorem.256}{}} \newlabel{prop:plaqvar}{{257}{137}{}{theorem.257}{}} \citation{Vil09} \newlabel{prop:blocking}{{258}{138}{}{theorem.258}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {G}What the cube estimate requires}{138}{section*.54}\protected@file@percent } \newlabel{ssec:cubeneeds}{{XI\,G}{138}{}{section*.54}{}} \newlabel{lem:cubesingle}{{259}{138}{}{theorem.259}{}} \newlabel{prop:cubesmall}{{260}{138}{}{theorem.260}{}} \newlabel{prop:cubeopen}{{261}{139}{}{theorem.261}{}} \newlabel{thm:cubeneeds}{{262}{140}{}{theorem.262}{}} \newlabel{cor:cubefails}{{263}{140}{}{theorem.263}{}} \newlabel{prop:tetra}{{264}{140}{}{theorem.264}{}} \newlabel{prop:fourcolours}{{265}{141}{}{theorem.265}{}} \newlabel{prop:cubesize}{{266}{141}{}{theorem.266}{}} \citation{Hel01} \newlabel{prop:cubesizeN}{{267}{142}{}{theorem.267}{}} \citation{FS82} \citation{FS82} \citation{FS82} \citation{Gut80} \newlabel{prop:smallfield}{{268}{143}{}{theorem.268}{}} \newlabel{prop:coulomb}{{269}{143}{}{theorem.269}{}} \citation{FS82} \citation{Lus10} \newlabel{rem:fluxloop}{{270}{144}{}{theorem.270}{}} \newlabel{prop:epslower}{{271}{144}{}{theorem.271}{}} \newlabel{prop:slowline}{{272}{144}{}{theorem.272}{}} \citation{OS73} \@writefile{toc}{\contentsline {subsection}{\numberline {H}The limit at separated points}{145}{section*.55}\protected@file@percent } \newlabel{ssec:limitE}{{XI\,H}{145}{}{section*.55}{}} \newlabel{lem:limitElattice}{{273}{146}{}{theorem.273}{}} 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\newlabel{lem:limitEdil}{{290}{156}{}{theorem.290}{}} \newlabel{prop:limitEscale}{{291}{156}{}{theorem.291}{}} \newlabel{thm:limitEunique}{{292}{157}{}{theorem.292}{}} \newlabel{prop:limitEaxial}{{293}{157}{}{theorem.293}{}} \newlabel{prop:limitEgauss}{{294}{158}{}{theorem.294}{}} \newlabel{prop:limitEbracket}{{295}{160}{}{theorem.295}{}} \newlabel{prop:limitEdilate}{{296}{160}{}{theorem.296}{}} \citation{OS73} \citation{OS75} \citation{Hor03} \newlabel{lem:limitEcut}{{297}{161}{}{theorem.297}{}} \newlabel{lem:limitEextend}{{298}{162}{}{theorem.298}{}} \newlabel{thm:limitEtempered}{{299}{162}{}{theorem.299}{}} \citation{OS75} \citation{OS73,OS75} \citation{OS73} \citation{OS75} \citation{OS73} \citation{OS75} \newlabel{cor:limitEOS}{{300}{163}{}{theorem.300}{}} \newlabel{prop:limitEtwotempered}{{301}{163}{}{theorem.301}{}} \citation{OS73,OS75} \citation{RS80} \newlabel{prop:limitEnotempered}{{302}{164}{}{theorem.302}{}} \newlabel{rem:limitEcontact}{{303}{164}{}{theorem.303}{}} \citation{GW73,Pol73} \citation{CDJ77} \newlabel{rem:limitEafpert}{{304}{165}{}{theorem.304}{}} \newlabel{thm:limitEaf}{{305}{165}{}{theorem.305}{}} \newlabel{prop:limitEafscale}{{306}{167}{}{theorem.306}{}} \newlabel{prop:limitEafexample}{{307}{167}{}{theorem.307}{}} \citation{Bal84a,Bal85a,Bal87,Bal88,Bal89b} \citation{Bal87} \citation{Bal89b} \citation{MRS93} \citation{GK85} \newlabel{rem:limitEafstatus}{{308}{168}{}{theorem.308}{}} \newlabel{lem:limitEtrunc}{{309}{168}{}{theorem.309}{}} \newlabel{eq:limitEcover}{{30}{168}{}{equation.11.30}{}} \newlabel{prop:limitEthree}{{310}{169}{}{theorem.310}{}} \newlabel{prop:limitErate}{{311}{170}{}{theorem.311}{}} \newlabel{prop:limitEfactorial}{{312}{170}{}{theorem.312}{}} \newlabel{prop:limitEdomination}{{313}{171}{}{theorem.313}{}} \newlabel{lem:limitElogconvex}{{314}{171}{}{theorem.314}{}} \citation{Gin70} \newlabel{prop:limitEenergy}{{315}{172}{}{theorem.315}{}} \newlabel{cor:limitEabelian}{{316}{172}{}{theorem.316}{}} \newlabel{prop:limitElm}{{317}{173}{}{theorem.317}{}} \newlabel{prop:limitEflat}{{318}{174}{}{theorem.318}{}} \citation{OS73} \citation{OS75} \newlabel{rem:limitEscope}{{319}{175}{}{theorem.319}{}} \citation{OS73} \citation{OS75} \@writefile{lof}{\contentsline {figure}{\numberline {25}{\ignorespaces The conditions of Sec.~\ref {ssec:limitE}, given the averaged condition along the tuning. Bottom: inputs on reflected norms of packets under the Wilson measure, none of them proved. Middle: lattice bounds; (M$_{\mathrm s}$) is (RN) (Proposition~\ref {prop:limitEreflect}), and (G$_{\mathrm s}$) is (RN) with factorial constants (Proposition~\ref {prop:limitErate}). Top: what the bounds give in the limit (Theorems~\ref {thm:limitEsep} and~\ref {thm:limitEtempered}, Corollaries~\ref {cor:limitEOS} and~\ref {cor:limitEexact}). From the bottom row: Theorem~\ref {thm:limitEtwo} and Proposition~\ref {prop:limitEtwotempered} for $n=2$, Proposition~\ref {prop:limitEdomination} for every $n$, Theorems~\ref {thm:limitErot} and~\ref {thm:limitEunique}. Crossed arrows fail in examples: the two-point properties do not give (RN$_m$), $m\ge 2$ (Proposition~\ref {prop:limitEthree}), and the bounds of the middle row do not give (W$_{\mathrm s}$) (Proposition~\ref {prop:osobstruction}(a)).}}{176}{figure.25}\protected@file@percent } \newlabel{fig:limitlogic}{{25}{176}{The conditions of Sec.~\ref {ssec:limitE}, given the averaged condition along the tuning. Bottom: inputs on reflected norms of packets under the Wilson measure, none of them proved. Middle: lattice bounds; (M$_{\mathrm s}$) is (RN) (Proposition~\ref {prop:limitEreflect}), and (G$_{\mathrm s}$) is (RN) with factorial constants (Proposition~\ref {prop:limitErate}). Top: what the bounds give in the limit (Theorems~\ref {thm:limitEsep} and~\ref {thm:limitEtempered}, Corollaries~\ref {cor:limitEOS} and~\ref {cor:limitEexact}). From the bottom row: Theorem~\ref {thm:limitEtwo} and Proposition~\ref {prop:limitEtwotempered} for $n=2$, Proposition~\ref {prop:limitEdomination} for every $n$, Theorems~\ref {thm:limitErot} and~\ref {thm:limitEunique}. Crossed arrows fail in examples: the two-point properties do not give (RN$_m$), $m\ge 2$ (Proposition~\ref {prop:limitEthree}), and the bounds of the middle row do not give (W$_{\mathrm s}$) (Proposition~\ref {prop:osobstruction}(a))}{figure.25}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {I}The conditional chain for every compact simple group}{176}{section*.58}\protected@file@percent } \newlabel{ssec:chainG}{{XI\,I}{176}{}{section*.58}{}} \citation{OSei78} \citation{OS75} \citation{OS73} \citation{OSei78,Sei82} \newlabel{eq:chainGeps}{{31}{177}{}{equation.11.31}{}} \newlabel{prop:bpcentreG}{{320}{177}{centre sectors for every group}{theorem.320}{}} \newlabel{thm:chainG}{{321}{177}{the conditional chain for every compact simple group}{theorem.321}{}} \@writefile{lot}{\contentsline {table}{\numberline {III}{\ignorespaces The statements of the conditional chain, the property of the gauge group used in their proofs, and their status for every compact simple group $G$.}}{178}{table.3}\protected@file@percent } \newlabel{tab:chainG}{{III}{178}{The statements of the conditional chain, the property of the gauge group used in their proofs, and their status for every compact simple group $G$}{table.3}{}} \newlabel{cor:limitEstrongG}{{322}{178}{}{theorem.322}{}} \newlabel{prop:modulusG}{{323}{178}{}{theorem.323}{}} \newlabel{prop:bpfixedG}{{324}{179}{}{theorem.324}{}} \newlabel{prop:bpflowG}{{325}{180}{the holonomy flow for every group}{theorem.325}{}} \citation{Loc26} \citation{Bal84a,Bal85a} \citation{Loc26} \newlabel{rem:chainG}{{326}{181}{what is proved for every group and what remains}{theorem.326}{}} \@writefile{toc}{\contentsline {section}{\numberline {XII}Block averages}{181}{section*.59}\protected@file@percent } \newlabel{sec:blocks}{{XII}{181}{}{section*.59}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {A}Block-constrained cubes}{181}{section*.60}\protected@file@percent } \newlabel{ssec:constrained}{{XII\,A}{181}{}{section*.60}{}} \newlabel{prop:invisible}{{327}{181}{\cite {Loc26}}{theorem.327}{}} \newlabel{def:blockgauge}{{328}{181}{}{theorem.328}{}} \newlabel{lem:blocktree}{{329}{182}{}{theorem.329}{}} \newlabel{thm:blockgap}{{330}{182}{}{theorem.330}{}} \newlabel{rem:blockgapsharp}{{331}{182}{a computation, used by no theorem}{theorem.331}{}} \newlabel{eq:blockgapcert}{{32}{182}{a computation, used by no theorem}{equation.12.32}{}} \citation{Loc26} \citation{CT73} \newlabel{lem:saddle}{{332}{183}{}{theorem.332}{}} \newlabel{lem:combes}{{333}{183}{Combes--Thomas~\cite {CT73}}{theorem.333}{}} \newlabel{thm:constrainedcube}{{334}{183}{}{theorem.334}{}} \citation{Loc26} \citation{Loc26} \newlabel{prop:effective}{{335}{184}{}{theorem.335}{}} \newlabel{lem:straightpaths}{{336}{184}{}{theorem.336}{}} \newlabel{lem:whitney}{{337}{185}{}{theorem.337}{}} \newlabel{thm:effectiveupper}{{338}{185}{}{theorem.338}{}} \@writefile{lof}{\contentsline {figure}{\numberline {26}{\ignorespaces The extension of Theorem~\ref {thm:effectiveupper} along one direction. (a)~A block of side $L=4$: the nodes are the block centres $c_z$, the hat functions $\lambda _z$ interpolate between them, and the straight path of length $L$ from a site $x$ meets the segments of three centres; averaged over the block, their weights are $\frac 18$, $\frac 34$, $\frac 18$ at every scale. (b)~The symbol of this filter stays at least $\frac 12$, while the filter obtained with the coarse sites as nodes has symbol of modulus $1/L$ at $\theta =\pi $.}}{186}{figure.26}\protected@file@percent } \newlabel{fig:whitney}{{26}{186}{The extension of Theorem~\ref {thm:effectiveupper} along one direction. (a)~A block of side $L=4$: the nodes are the block centres $c_z$, the hat functions $\lambda _z$ interpolate between them, and the straight path of length $L$ from a site $x$ meets the segments of three centres; averaged over the block, their weights are $\frac 18$, $\frac 34$, $\frac 18$ at every scale. (b)~The symbol of this filter stays at least $\frac 12$, while the filter obtained with the coarse sites as nodes has symbol of modulus $1/L$ at $\theta =\pi $}{figure.26}{}} \newlabel{eq:Yk}{{33}{186}{}{equation.12.33}{}} \newlabel{lem:planes}{{339}{186}{}{theorem.339}{}} \newlabel{thm:effectivelocal}{{340}{186}{}{theorem.340}{}} \citation{Bal84a} \newlabel{lem:mixture}{{341}{188}{}{theorem.341}{}} \newlabel{cor:compose}{{342}{188}{}{theorem.342}{}} \newlabel{eq:Srows}{{34}{188}{}{equation.12.34}{}} \newlabel{lem:combeslong}{{343}{188}{}{theorem.343}{}} \newlabel{thm:constrainedscales}{{344}{188}{}{theorem.344}{}} \@writefile{lof}{\contentsline {figure}{\numberline {27}{\ignorespaces The effective forms of Theorem~\ref {thm:effectivelocal}. (a)~The multiplier $K_P$ for the plane $P=\{1,2\}$ at $k=3$, as a function of $(\theta _3,\theta _4)$ with $\theta _1=\theta _2=0$; it equals $1$ at $\theta =0$ and stays bounded. (b)~The largest value of the kernel $|\mathcal K_P(y)|$ on $|y|_1=d$ for $k=1,2,3$, in logarithmic scale: the decay per coarse spacing hardly changes with $k$.}}{189}{figure.27}\protected@file@percent } \newlabel{fig:locality}{{27}{189}{The effective forms of Theorem~\ref {thm:effectivelocal}. (a)~The multiplier $K_P$ for the plane $P=\{1,2\}$ at $k=3$, as a function of $(\theta _3,\theta _4)$ with $\theta _1=\theta _2=0$; it equals $1$ at $\theta =0$ and stays bounded. (b)~The largest value of the kernel $|\mathcal K_P(y)|$ on $|y|_1=d$ for $k=1,2,3$, in logarithmic scale: the decay per coarse spacing hardly changes with $k$}{figure.27}{}} \citation{Bal89b} \citation{Loc26} \newlabel{rem:constrainedopen}{{345}{190}{}{theorem.345}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {B}Path blocks for the group-valued links}{190}{section*.61}\protected@file@percent } \newlabel{ssec:pathblocks}{{XII\,B}{190}{}{section*.61}{}} \newlabel{lem:anchorcube}{{346}{191}{}{theorem.346}{}} \newlabel{thm:pathcoercive}{{347}{191}{}{theorem.347}{}} \newlabel{prop:pathenergy}{{348}{191}{}{theorem.348}{}} \newlabel{thm:pathZ}{{349}{191}{}{theorem.349}{}} \newlabel{prop:thinpath}{{350}{191}{}{theorem.350}{}} \citation{Hel01} \citation{Bal84a,Bal89b} \citation{Aum44} \citation{Eck54} \newlabel{prop:nomean}{{351}{192}{}{theorem.351}{}} \newlabel{lem:logbound}{{352}{192}{}{theorem.352}{}} \@writefile{lof}{\contentsline {figure}{\numberline {28}{\ignorespaces (a) Two adjacent blocks $B(y)$, $B(y+e_\mu )$ of side $s=4$, drawn in three of the four dimensions. The black segment is the straight path between the roots, the coarse link of the path map. The blue links form the slice $T$ of Proposition~\ref {prop:nomean}. The red links carry the field of Proposition~\ref {prop:thinpath}: a path whose holonomy is that of a coarse link of order one, at a fine energy of order $1/s$. (b) For $\mathrm {SU}(2)$, two coordinates of the torus $(\mathbb {R}/2\pi \mathbb {Z})^T$ of slice fields with values in a maximal torus. The diagonal is pure gauge and is sent to a loop of degree one, so the degrees $W_t$ of the coordinate loops add up to one.}}{193}{figure.28}\protected@file@percent } \newlabel{fig:pathblocks}{{28}{193}{(a) Two adjacent blocks $B(y)$, $B(y+e_\mu )$ of side $s=4$, drawn in three of the four dimensions. The black segment is the straight path between the roots, the coarse link of the path map. The blue links form the slice $T$ of Proposition~\ref {prop:nomean}. The red links carry the field of Proposition~\ref {prop:thinpath}: a path whose holonomy is that of a coarse link of order one, at a fine energy of order $1/s$. (b) For $\SU (2)$, two coordinates of the torus $(\R /2\pi \Z )^T$ of slice fields with values in a maximal torus. The diagonal is pure gauge and is sent to a loop of degree one, so the degrees $W_t$ of the coordinate loops add up to one}{figure.28}{}} \newlabel{eq:cutoff}{{35}{193}{}{equation.12.35}{}} \newlabel{thm:cutoff}{{353}{193}{}{theorem.353}{}} \citation{Bal85a} \@writefile{lof}{\contentsline {figure}{\numberline {29}{\ignorespaces (a) A block $B(y)$ of side two and the next block in the direction $\mu $, drawn in three of the four dimensions. Green dashed: the tree links, equal to $I$. Blue: the paths $P_x$ of length two from the sites of $B(y)$, each joining $2y$ to $2y+2e_\mu $ through the trees; black: the straight path $P_{2y}$, the coarse link of the path map. The cut-off average is the mean of the sixteen paths in the chart at $P_{2y}$ while every path is close to it. (b) For $\mathrm {SU}(2)$ and one link $e^{\theta X}$, $X=\operatorname {diag}(i,-i)$, the angle of $V^\sharp $ when the link lies on the straight path (black) or at another transverse position (blue), with $\rho _G=1$. Both equal $\theta /8$, the block average, while $\chi =1$; past the cut-off they are the path map, $\theta $ and $0$, so the slice windings are $1$ and $0$.}}{194}{figure.29}\protected@file@percent } \newlabel{fig:cutoff}{{29}{194}{(a) A block $B(y)$ of side two and the next block in the direction $\mu $, drawn in three of the four dimensions. Green dashed: the tree links, equal to $I$. Blue: the paths $P_x$ of length two from the sites of $B(y)$, each joining $2y$ to $2y+2e_\mu $ through the trees; black: the straight path $P_{2y}$, the coarse link of the path map. The cut-off average is the mean of the sixteen paths in the chart at $P_{2y}$ while every path is close to it. (b) For $\SU (2)$ and one link $e^{\theta X}$, $X=\operatorname {diag}(i,-i)$, the angle of $V^\sharp $ when the link lies on the straight path (black) or at another transverse position (blue), with $\rho _G=1$. Both equal $\theta /8$, the block average, while $\chi =1$; past the cut-off they are the path map, $\theta $ and $0$, so the slice windings are $1$ and $0$}{figure.29}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {C}Composition across scales}{195}{section*.62}\protected@file@percent } \newlabel{ssec:scales}{{XII\,C}{195}{}{section*.62}{}} \newlabel{lem:torusstep}{{354}{195}{}{theorem.354}{}} \newlabel{thm:torusscales}{{355}{195}{}{theorem.355}{}} \newlabel{eq:torushyp}{{36}{195}{}{equation.12.36}{}} \newlabel{cor:thinscales}{{356}{195}{}{theorem.356}{}} \@writefile{lof}{\contentsline {figure}{\numberline {30}{\ignorespaces (a) In the torus, the coarse plaquette of the cut-off average is the mean of the fluxes $F_{2y+o}$ of the sixteen fine squares of side two at the offsets $o$ (four of them drawn), whatever the gauge of the coarse links (Lemma~\ref {lem:torusstep}). (b) For the thin-path fields of Proposition~\ref {prop:thinpath}, the coarse energy after $k$ steps: $24$ for the path map, at most $18\pi ^2/s^2$ for the cut-off average (Corollary~\ref {cor:thinscales}); the fine energy is about $6\pi ^2/s$.}}{196}{figure.30}\protected@file@percent } \newlabel{fig:scales}{{30}{196}{(a) In the torus, the coarse plaquette of the cut-off average is the mean of the fluxes $F_{2y+o}$ of the sixteen fine squares of side two at the offsets $o$ (four of them drawn), whatever the gauge of the coarse links (Lemma~\ref {lem:torusstep}). (b) For the thin-path fields of Proposition~\ref {prop:thinpath}, the coarse energy after $k$ steps: $24$ for the path map, at most $18\pi ^2/s^2$ for the cut-off average (Corollary~\ref {cor:thinscales}); the fine energy is about $6\pi ^2/s$}{figure.30}{}} \newlabel{lem:bch}{{357}{196}{}{theorem.357}{}} \newlabel{lem:smallstep}{{358}{196}{}{theorem.358}{}} \newlabel{eq:smallstep}{{37}{196}{}{equation.12.37}{}} \newlabel{thm:smallscales}{{359}{197}{}{theorem.359}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {D}Small fields on the fibres of the cut-off average}{198}{section*.63}\protected@file@percent } \newlabel{ssec:smallfibres}{{XII\,D}{198}{}{section*.63}{}} \newlabel{lem:smallframes}{{360}{198}{}{theorem.360}{}} \newlabel{thm:smallfibre}{{361}{198}{}{theorem.361}{}} \citation{BL76} \newlabel{rem:smallfibreopen}{{362}{200}{}{theorem.362}{}} \newlabel{thm:fibrechart}{{363}{200}{}{theorem.363}{}} \citation{BGL14} \newlabel{prop:explicitconstants}{{364}{201}{}{theorem.364}{}} \citation{Ban38} \newlabel{rem:fibrechartopen}{{365}{202}{}{theorem.365}{}} \newlabel{lem:kernels}{{366}{203}{}{theorem.366}{}} \newlabel{thm:chartdecay}{{367}{203}{}{theorem.367}{}} \citation{HS94,NS97} \citation{BJS00,Hel98} \newlabel{lem:weightednorms}{{368}{204}{}{theorem.368}{}} \newlabel{prop:explicitdecay}{{369}{205}{}{theorem.369}{}} \newlabel{thm:offchart}{{370}{206}{}{theorem.370}{}} \newlabel{lem:privatesubmersion}{{371}{207}{}{theorem.371}{}} \@writefile{lof}{\contentsline {figure}{\numberline {31}{\ignorespaces Theorem~\ref {thm:offchart}. (a)~A fibre $F$ of the cut-off average, drawn as a surface: the product chart $\Phi (\Omega _r)$ around the background $U$ (green) contains every point of $F$ whose plaquettes near $W$ are at most $\kappa $ (inside the blue curve), and the rest of $F$ meets only points with a large plaquette (red). (b)~The least $\beta _W$ at which the bound $10^4\lvert W\rvert q_\kappa (\beta _W)$ is at most $\frac 12$, for the $81$ blocks of a cube of side three; it grows like $\kappa ^{-2}\log (1/ \kappa )$, and at the largest $\kappa $ the hypotheses allow it is about $4.0\cdot 10^{38}$.}}{208}{figure.31}\protected@file@percent } \newlabel{fig:offchart}{{31}{208}{Theorem~\ref {thm:offchart}. (a)~A fibre $F$ of the cut-off average, drawn as a surface: the product chart $\Phi (\Omega _r)$ around the background $U$ (green) contains every point of $F$ whose plaquettes near $W$ are at most $\kappa $ (inside the blue curve), and the rest of $F$ meets only points with a large plaquette (red). (b)~The least $\beta _W$ at which the bound $10^4\lvert W\rvert q_\kappa (\beta _W)$ is at most $\frac 12$, for the $81$ blocks of a cube of side three; it grows like $\kappa ^{-2}\log (1/ \kappa )$, and at the largest $\kappa $ the hypotheses allow it is about $4.0\cdot 10^{38}$}{figure.31}{}} \newlabel{cor:coareabound}{{372}{208}{}{theorem.372}{}} \newlabel{lem:averagedensity}{{373}{208}{}{theorem.373}{}} \newlabel{eq:densityformula}{{38}{209}{}{equation.12.38}{}} \newlabel{eq:fibrevolume}{{39}{209}{}{equation.12.39}{}} \newlabel{prop:calmfibre}{{374}{209}{}{theorem.374}{}} \newlabel{cor:calmfibres}{{375}{210}{}{theorem.375}{}} \newlabel{lem:alignedfield}{{376}{210}{}{theorem.376}{}} \newlabel{prop:everyfibre}{{377}{210}{}{theorem.377}{}} \newlabel{cor:everyfibre}{{378}{210}{}{theorem.378}{}} \newlabel{prop:fibrecompare}{{379}{211}{}{theorem.379}{}} \newlabel{cor:calmexterior}{{380}{211}{}{theorem.380}{}} \newlabel{rem:chartdecayopen}{{381}{211}{}{theorem.381}{}} \citation{OSei78} \citation{FILS78} \citation{Bis09} \@writefile{toc}{\contentsline {subsection}{\numberline {E}Large fields under the Wilson measure}{212}{section*.64}\protected@file@percent } \newlabel{ssec:largefields}{{XII\,E}{212}{}{section*.64}{}} \newlabel{lem:wilsonZ}{{382}{212}{}{theorem.382}{}} \newlabel{thm:defects}{{383}{212}{}{theorem.383}{}} \newlabel{cor:defectclusters}{{384}{212}{}{theorem.384}{}} \@writefile{lof}{\contentsline {figure}{\numberline {32}{\ignorespaces Large fields under the $\mathrm {SU}(2)$ Wilson measure. (a)~A plane section of the lattice: bad unit cubes (red) form clusters, and the cut-off average can differ from the path average only on coarse links (orange) whose $24$ cubes $\mathcal Z_{y,\mu }$ meet a bad cube (Corollary~\ref {cor:activelinks}). (b)~The frustrated exterior of a link $e$: the staple of $p_0$ is $I$ and the other five staples are $-I$, so $U_e$ is driven to $-I$ and $p_0$ is bad. (c)~The weight $q_\rho (\beta _W)$ of Theorem~\ref {thm:defects} for three values of $\rho $; below the dashed line the cluster bound of Corollary~\ref {cor:defectclusters} is summable.}}{213}{figure.32}\protected@file@percent } \newlabel{fig:largefields}{{32}{213}{Large fields under the $\SU (2)$ Wilson measure. (a)~A plane section of the lattice: bad unit cubes (red) form clusters, and the cut-off average can differ from the path average only on coarse links (orange) whose $24$ cubes $\mathcal Z_{y,\mu }$ meet a bad cube (Corollary~\ref {cor:activelinks}). (b)~The frustrated exterior of a link $e$: the staple of $p_0$ is $I$ and the other five staples are $-I$, so $U_e$ is driven to $-I$ and $p_0$ is bad. (c)~The weight $q_\rho (\beta _W)$ of Theorem~\ref {thm:defects} for three values of $\rho $; below the dashed line the cluster bound of Corollary~\ref {cor:defectclusters} is summable}{figure.32}{}} \newlabel{cor:activelinks}{{385}{213}{}{theorem.385}{}} \newlabel{prop:manydefects}{{386}{213}{}{theorem.386}{}} \newlabel{lem:localstep}{{387}{213}{}{theorem.387}{}} \newlabel{thm:defectscales}{{388}{214}{}{theorem.388}{}} \newlabel{cor:defectscales}{{389}{214}{}{theorem.389}{}} \citation{McS34} \citation{Whi57} \newlabel{lem:boxextension}{{390}{215}{}{theorem.390}{}} \citation{McS34} \newlabel{lem:nonabelianfilling}{{391}{216}{}{theorem.391}{}} \citation{Uhl82} \newlabel{prop:conddensity}{{392}{217}{}{theorem.392}{}} \newlabel{rem:boxnonabelian}{{393}{217}{}{theorem.393}{}} \newlabel{prop:anyexterior}{{394}{217}{}{theorem.394}{}} \citation{Bal89b} \newlabel{cor:anyexterior}{{395}{218}{}{theorem.395}{}} \newlabel{cor:localcalm}{{396}{218}{}{theorem.396}{}} \newlabel{prop:calmrare}{{397}{218}{}{theorem.397}{}} \citation{BD86} \@writefile{toc}{\contentsline {subsection}{\numberline {F}Conditioning on a fixed large-field set}{219}{section*.65}\protected@file@percent } \newlabel{ssec:fixedset}{{XII\,F}{219}{}{section*.65}{}} \newlabel{lem:excess}{{398}{219}{excess action}{theorem.398}{}} \newlabel{lem:fixedset}{{399}{219}{restriction to a fixed set}{theorem.399}{}} \newlabel{prop:onepoint}{{400}{219}{one-point rarity}{theorem.400}{}} \citation{vdBM94} \@writefile{toc}{\contentsline {subsection}{\numberline {G}Rarity of coarse defects}{220}{section*.66}\protected@file@percent } \newlabel{ssec:coarse}{{XII\,G}{220}{}{section*.66}{}} \newlabel{lem:coarseenergy}{{401}{220}{energy of a coarse defect}{theorem.401}{}} \@writefile{lof}{\contentsline {figure}{\numberline {33}{\ignorespaces The smeared current loop $J_{k,P}$ of Lemma~\ref {lem:coarseenergy} in the plane of $P$, drawn for $L_k=4$. Left: each side of the coarse plaquette $P$ (thick square) contributes a current of its own direction on the links of a slab of transverse size $L_k^3$ and length $2L_k-1$ (blue for the direction $1$, red for the direction $2$), with the signs of the four sides; the slabs of opposite sides are disjoint and the two directions are carried by different links. Right: the longitudinal profile $L_k^4J=\pm c(s)$ on a slab, the tent $c(s)=\min \{s+1,L_k,2L_k-1-s\}$, which counts the straight paths of Lemma~\ref {lem:straightpaths}(i) through the link at position $s$. The current is divergence free and its energy $\gamma _k=\langle J_{k,P},G_LJ_{k,P}\rangle $ lies in $[\frac 1{432},\frac 12]$ at every scale.}}{221}{figure.33}\protected@file@percent } \newlabel{fig:smearedloop}{{33}{221}{The smeared current loop $J_{k,P}$ of Lemma~\ref {lem:coarseenergy} in the plane of $P$, drawn for $L_k=4$. Left: each side of the coarse plaquette $P$ (thick square) contributes a current of its own direction on the links of a slab of transverse size $L_k^3$ and length $2L_k-1$ (blue for the direction $1$, red for the direction $2$), with the signs of the four sides; the slabs of opposite sides are disjoint and the two directions are carried by different links. Right: the longitudinal profile $L_k^4J=\pm c(s)$ on a slab, the tent $c(s)=\min \{s+1,L_k,2L_k-1-s\}$, which counts the straight paths of Lemma~\ref {lem:straightpaths}(i) through the link at position $s$. The current is divergence free and its energy $\gamma _k=\ip {J_{k,P}}{G_LJ_{k,P}}$ lies in $[\frac 1{432},\frac 12]$ at every scale}{figure.33}{}} \newlabel{prop:torusenergy}{{402}{221}{energy of a coarse defect, torus-valued fields}{theorem.402}{}} \newlabel{prop:coarsechess}{{403}{221}{chessboard for coarse events}{theorem.403}{}} \citation{Gut80,FS82} \newlabel{thm:haartilt}{{404}{222}{Haar tilt}{theorem.404}{}} \newlabel{cor:abeliancoarse}{{405}{222}{abelian theory}{theorem.405}{}} \newlabel{prop:tiltgrowth}{{406}{222}{the gauge orbit of the vacuum}{theorem.406}{}} \newlabel{prop:foresttilt}{{407}{223}{covariant tilt on a forest}{theorem.407}{}} \citation{Bal89b} \citation{SZ92} \@writefile{lof}{\contentsline {figure}{\numberline {34}{\ignorespaces The covariant tilt on the block forest (Proposition~\ref {prop:foresttilt}). Transporting the direction along the trees keeps the Haar measures, but the section still contains every pure gauge field that is constant on the blocks. On a plaquette whose tilted sides $e$, $e'$ end in different blocks the transported directions are turned by independent elements, and the average of $\|(dr')_p\|^2$ over them is $a_e^2+a_{e'}^2$ instead of the $(a_e-a_{e'})^2$ of the abelian energy.}}{224}{figure.34}\protected@file@percent } \newlabel{fig:foresttilt}{{34}{224}{The covariant tilt on the block forest (Proposition~\ref {prop:foresttilt}). Transporting the direction along the trees keeps the Haar measures, but the section still contains every pure gauge field that is constant on the blocks. On a plaquette whose tilted sides $e$, $e'$ end in different blocks the transported directions are turned by independent elements, and the average of $\|(dr')_p\|^2$ over them is $a_e^2+a_{e'}^2$ instead of the $(a_e-a_{e'})^2$ of the abelian energy}{figure.34}{}} \@writefile{toc}{\contentsline {section}{\numberline {XIII}Labels, islands and the environment}{224}{section*.67}\protected@file@percent } \newlabel{sec:islands}{{XIII}{224}{}{section*.67}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {A}Backgrounds and the path links}{224}{section*.68}\protected@file@percent } \newlabel{ssec:branches}{{XIII\,A}{224}{}{section*.68}{}} \@writefile{lof}{\contentsline {figure}{\numberline {35}{\ignorespaces Which scales the estimates of this subsection reach, along the tuning $\beta _W\sim b\log (1/a)$, where a cube of fixed physical side spans $m\sim \beta _W/b$ scales. The energy of a coarse defect is scale invariant (Lemma~\ref {lem:coarseenergy}), and for the abelian theory the Haar tilt makes coarse defects rare at every scale (Corollary~\ref {cor:abeliancoarse}). For $\mathrm {SU}(2)$ the chessboard estimate stops after $O(\log \beta _W)$ scales because the normalisation charges the whole fluctuation volume of a block against the defect (Proposition~\ref {prop:coarsechess}), and the tilt stops after $O(\log \beta _W)$ scales because its second variation grows like $4^k$ on the gauge orbit of the vacuum, whether the direction is fixed or transported along a forest (Propositions~\ref {prop:tiltgrowth}, \ref {prop:foresttilt}). The remaining scales are the open input of the cube condition.}}{225}{figure.35}\protected@file@percent } \newlabel{fig:scaleladder}{{35}{225}{Which scales the estimates of this subsection reach, along the tuning $\beta _W\sim b\log (1/a)$, where a cube of fixed physical side spans $m\sim \beta _W/b$ scales. The energy of a coarse defect is scale invariant (Lemma~\ref {lem:coarseenergy}), and for the abelian theory the Haar tilt makes coarse defects rare at every scale (Corollary~\ref {cor:abeliancoarse}). For $\SU (2)$ the chessboard estimate stops after $O(\log \beta _W)$ scales because the normalisation charges the whole fluctuation volume of a block against the defect (Proposition~\ref {prop:coarsechess}), and the tilt stops after $O(\log \beta _W)$ scales because its second variation grows like $4^k$ on the gauge orbit of the vacuum, whether the direction is fixed or transported along a forest (Propositions~\ref {prop:tiltgrowth}, \ref {prop:foresttilt}). The remaining scales are the open input of the cube condition}{figure.35}{}} \newlabel{prop:roughbackground}{{408}{225}{a rough background}{theorem.408}{}} \newlabel{lem:coarsewitness}{{409}{226}{a fine witness for coarse curvature}{theorem.409}{}} \newlabel{rem:flatholonomy}{{410}{226}{flat fields far from the identity}{theorem.410}{}} \newlabel{lem:branchbound}{{411}{226}{the path link is determined}{theorem.411}{}} \newlabel{prop:graphlaw}{{412}{226}{good windows are graphs}{theorem.412}{}} \newlabel{prop:nonconvex}{{413}{227}{the all-link label domain is not convex}{theorem.413}{}} \newlabel{prop:branches}{{414}{227}{a bad pair with three path links}{theorem.414}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {B}The path-link law on its two sides}{227}{section*.69}\protected@file@percent } \newlabel{ssec:pathlaw}{{XIII\,B}{227}{}{section*.69}{}} \newlabel{eq:pathlaw}{{40}{227}{}{equation.13.40}{}} \newlabel{lem:wilsononly}{{415}{228}{links that enter only the Wilson weight}{theorem.415}{}} \newlabel{prop:goodside}{{416}{228}{the good side to radius $1/32$}{theorem.416}{}} \newlabel{lem:foldtransverse}{{417}{229}{the transverse part of the cut-off average}{theorem.417}{}} \newlabel{prop:fold}{{418}{229}{the fold of the path-link law}{theorem.418}{}} \citation{DK65} \newlabel{lem:sitelaw}{{419}{231}{the law of a site}{theorem.419}{}} \newlabel{lem:goodsidedensity}{{420}{231}{the good-side density is log-Lipschitz}{theorem.420}{}} \newlabel{thm:site}{{421}{231}{the site of a bad pair responds boundedly}{theorem.421}{}} \newlabel{prop:insidefold}{{422}{232}{an inside link at the fold}{theorem.422}{}} \newlabel{thm:eightlinkuniform}{{423}{233}{no bound uniform in the restricting set}{theorem.423}{}} \newlabel{prop:eightlabels}{{424}{233}{the sets of the labels}{theorem.424}{}} \newlabel{lem:relchart}{{425}{234}{the relative chart}{theorem.425}{}} \newlabel{lem:relnorm}{{426}{235}{the normaliser}{theorem.426}{}} \newlabel{thm:relresponse}{{427}{235}{the response in the relative chart}{theorem.427}{}} \newlabel{lem:ties}{{428}{236}{the ties}{theorem.428}{}} \newlabel{prop:jreduce}{{429}{237}{the Jacobian across the ties}{theorem.429}{}} \citation{Fed69} \citation{Dri86} \citation{Dri98} \citation{Dri98} \citation{Dri98} \citation{Fed69} \newlabel{prop:tiearea}{{430}{238}{the area of the ties}{theorem.430}{}} \newlabel{thm:jholds}{{431}{239}{condition (J)}{theorem.431}{}} \newlabel{cor:onenear}{{432}{239}{the first set with one block near $I$}{theorem.432}{}} \newlabel{prop:thickties}{{433}{239}{thick near-ties}{theorem.433}{}} \@writefile{lof}{\contentsline {figure}{\numberline {36}{\ignorespaces Thick near-ties (Proposition~\ref {prop:thickties}), schematic. (a)~The values $f_{x_a}$ and $f_{x'_a}$ of the two paths of index $a$ over two coordinates of $\xi $, for $u_a=w_a=u$ at distance $s$ from $I$: over the open set $U_0$ they differ by at most $2s$, so the set $T(t)$ of near-ties contains $U_0$ for every $t\ge 2s$. (b)~The set $U_0$ in the coordinates $\delta (\xi _a,I)$ and $\max _{b\ne a}\delta (\xi _b,I)$, beside a band of width $t$ around a tie hypersurface, the shape that a bound $\mathrm {Haar}(T(t))\le Mt$ would require.}}{240}{figure.36}\protected@file@percent } \newlabel{fig:thickties}{{36}{240}{Thick near-ties (Proposition~\ref {prop:thickties}), schematic. (a)~The values $f_{x_a}$ and $f_{x'_a}$ of the two paths of index $a$ over two coordinates of $\xi $, for $u_a=w_a=u$ at distance $s$ from $I$: over the open set $U_0$ they differ by at most $2s$, so the set $T(t)$ of near-ties contains $U_0$ for every $t\ge 2s$. (b)~The set $U_0$ in the coordinates $\delta (\xi _a,I)$ and $\max _{b\ne a}\delta (\xi _b,I)$, beside a band of width $t$ around a tie hypersurface, the shape that a bound $\mathrm {Haar}(T(t))\le Mt$ would require}{figure.36}{}} \newlabel{prop:chartfolds}{{434}{240}{a fold of the relative charts}{theorem.434}{}} \citation{Dol95} \newlabel{def:linechain}{{435}{241}{}{theorem.435}{}} \newlabel{prop:linechain}{{436}{241}{the law of a chain}{theorem.436}{}} \newlabel{cor:linechain}{{437}{242}{the chains and the islands}{theorem.437}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {C}The projected average}{242}{section*.70}\protected@file@percent } \newlabel{ssec:projected}{{XIII\,C}{242}{}{section*.70}{}} \newlabel{def:projavg}{{438}{243}{}{theorem.438}{}} \newlabel{lem:projection}{{439}{243}{}{theorem.439}{}} \newlabel{prop:projavg}{{440}{243}{the projected average}{theorem.440}{}} \newlabel{thm:projinverse}{{441}{243}{the global inverse}{theorem.441}{}} \newlabel{cor:projpathlaw}{{442}{244}{the path-link law of the projected average}{theorem.442}{}} \@writefile{lof}{\contentsline {figure}{\numberline {37}{\ignorespaces The projected average of Definition~\ref {def:projavg}, schematic, with a two-sphere standing for $\mathrm {SU}(2)$. (a)~The paths $P_x$ of a coarse link around the base path $P_{2y}$, their projections to the tangent space at $P_{2y}$, the mean $A$ of the projections, and the value $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle V$}\mathaccent "0362{V}= P_{2y}e^{A}$. (b)~The map $g\DOTSB \mapstochar \rightarrow c_k\,e^{-A(g)}$ contracts with constant $\frac 78$, so its iterates converge to the unique solution $G$ of $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle V$}\mathaccent "0362{V}_k=c_k$, and $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle V$}\mathaccent "0362{V}_k$ changes distances by a factor between $\frac 18$ and $\frac {15}8$ (Theorem~\ref {thm:projinverse}).}}{245}{figure.37}\protected@file@percent } \newlabel{fig:projected}{{37}{245}{The projected average of Definition~\ref {def:projavg}, schematic, with a two-sphere standing for $\SU (2)$. (a)~The paths $P_x$ of a coarse link around the base path $P_{2y}$, their projections to the tangent space at $P_{2y}$, the mean $A$ of the projections, and the value $\widehat V= P_{2y}e^{A}$. (b)~The map $g\mapsto c_k\,e^{-A(g)}$ contracts with constant $\frac 78$, so its iterates converge to the unique solution $G$ of $\widehat V_k=c_k$, and $\widehat V_k$ changes distances by a factor between $\frac 18$ and $\frac {15}8$ (Theorem~\ref {thm:projinverse})}{figure.37}{}} \newlabel{thm:projresponse}{{443}{245}{the bad side of the projected average}{theorem.443}{}} \newlabel{prop:projfibre}{{444}{246}{projected fibres}{theorem.444}{}} \newlabel{prop:projwitness}{{445}{247}{a fine witness for the projected average}{theorem.445}{}} \newlabel{rem:projgroups}{{446}{247}{}{theorem.446}{}} \newlabel{prop:projtransfer}{{447}{247}{the chart for the projected average}{theorem.447}{}} \newlabel{rem:projscope}{{448}{248}{what is proved for which map}{theorem.448}{}} \newlabel{rem:projroute}{{449}{248}{the route to (D) with $\widehat V$}{theorem.449}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {D}Counting bad labels}{248}{section*.71}\protected@file@percent } \newlabel{ssec:counting}{{XIII\,D}{248}{}{section*.71}{}} \newlabel{lem:coverage}{{450}{248}{coverage counting}{theorem.450}{}} \newlabel{prop:coverageopt}{{451}{249}{the exponent $1/H$ cannot be improved}{theorem.451}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {E}Islands and the environment}{249}{section*.72}\protected@file@percent } \newlabel{ssec:environment}{{XIII\,E}{249}{}{section*.72}{}} \newlabel{lem:kernel}{{452}{249}{integration through a common kernel}{theorem.452}{}} \newlabel{cor:twolabels}{{453}{249}{two labels}{theorem.453}{}} \newlabel{lem:connected}{{454}{249}{connected normalisers}{theorem.454}{}} \newlabel{prop:hiddenbit}{{455}{250}{rarity does not control the environment}{theorem.455}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {F}Coupling cost and the comparison lemma}{250}{section*.73}\protected@file@percent } \newlabel{ssec:couplingcost}{{XIII\,F}{250}{}{section*.73}{}} \newlabel{lem:minimality}{{456}{250}{minimality gives the 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\newlabel{ssec:witnessradius}{{XIII\,K}{255}{}{section*.78}{}} \newlabel{lem:ladder}{{469}{255}{the ladder}{theorem.469}{}} \newlabel{prop:smallframessharp}{{470}{256}{Lemma~\ref {lem:smallframes} for $\SU (2)$ to the sharp radius}{theorem.470}{}} \newlabel{thm:islandswide}{{471}{256}{the island count at radius $10^{-4}$}{theorem.471}{}} \newlabel{prop:flatsecond}{{472}{257}{the flat second derivative}{theorem.472}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {L}The ceiling of the chessboard estimate}{257}{section*.79}\protected@file@percent } \newlabel{ssec:chessceiling}{{XIII\,L}{257}{}{section*.79}{}} \newlabel{lem:animals}{{473}{257}{connected sets of blocks}{theorem.473}{}} \newlabel{prop:treethreshold}{{474}{258}{the island threshold with the tree count}{theorem.474}{}} \newlabel{lem:Zabove}{{475}{258}{the normalisation from below}{theorem.475}{}} \citation{FILS78} \newlabel{thm:chessceiling}{{476}{259}{the ceiling of the chessboard estimate}{theorem.476}{}} \newlabel{cor:chessceiling}{{477}{260}{the labels of the island theorems}{theorem.477}{}} \newlabel{rem:chessceiling}{{478}{260}{what the ceiling leaves}{theorem.478}{}} \newlabel{lem:constantcritical}{{479}{260}{constant curvature is critical}{theorem.479}{}} \@writefile{lof}{\contentsline {figure}{\numberline {38}{\ignorespaces The ceiling of the chessboard estimate (Theorem~\ref {thm:chessceiling}), schematic. (a)~The dyadic scales as plates: a chessboard bound for the islands is below one only at the lower scales (green), and the cube of $D(n,\alpha )$ lies at least three scales higher for the labels of the island theorems and at least twenty for the calm labels (Corollary~\ref {cor:chessceiling}). (b)~The condition of part~(ii): the left side $16^{j+1}$ grows by a factor $16$ per scale, the right side $\beta _W\tilde \eta ^2/(96(\log \beta _W+C_0))$ does not depend on $j$, and they cross below the scale of the cube.}}{261}{figure.38}\protected@file@percent } \newlabel{fig:chessceiling}{{38}{261}{The ceiling of the chessboard estimate (Theorem~\ref {thm:chessceiling}), schematic. (a)~The dyadic scales as plates: a chessboard bound for the islands is below one only at the lower scales (green), and the cube of $D(n,\alpha )$ lies at least three scales higher for the labels of the island theorems and at least twenty for the calm labels (Corollary~\ref {cor:chessceiling}). (b)~The condition of part~(ii): the left side $16^{j+1}$ grows by a factor $16$ per scale, the right side $\beta _W\tilde \eta ^2/(96(\log \beta _W+C_0))$ does not depend on $j$, and they cross below the scale of the cube}{figure.38}{}} \newlabel{thm:criticalceiling}{{480}{261}{the ceiling at the critical points}{theorem.480}{}} \newlabel{cor:firstorder}{{481}{261}{first-order estimates}{theorem.481}{}} \newlabel{rem:criticalceiling}{{482}{262}{what the critical points leave}{theorem.482}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {M}Two islands factorise through one kernel}{262}{section*.80}\protected@file@percent } \newlabel{ssec:factorisation}{{XIII\,M}{262}{}{section*.80}{}} \newlabel{thm:factorisation}{{483}{263}{two islands and one kernel}{theorem.483}{}} \newlabel{eq:layerbound}{{41}{264}{}{equation.13.41}{}} \newlabel{prop:layercross}{{484}{264}{the cross-ratio under the layer bound}{theorem.484}{}} \newlabel{thm:layerbound}{{485}{265}{the layer law obeys the chart bound}{theorem.485}{}} \newlabel{cor:twoislands}{{486}{266}{two islands in a good region}{theorem.486}{}} \newlabel{lem:lambdatwo}{{487}{266}{the mixed derivatives of the layer action}{theorem.487}{}} \newlabel{eq:wilsonmixed}{{42}{267}{}{equation.13.42}{}} \newlabel{eq:cwcpsi}{{43}{268}{}{equation.13.43}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {N}Many islands}{269}{section*.81}\protected@file@percent } \newlabel{ssec:manyislands}{{XIII\,N}{269}{}{section*.81}{}} \newlabel{lem:manykernel}{{488}{269}{many islands, one kernel}{theorem.488}{}} \@writefile{lof}{\contentsline {figure}{\numberline {39}{\ignorespaces Left: two bad islands at the ends of a slab, the frozen collars of width $s+3$ beside them, and the good layer between them, on which the layer law $\mu _{x,y}$ obeys the chart bound in the label coordinates (Theorem~\ref {thm:layerbound}). The cross-ratio $\varDelta $ of the kernel of Theorem~\ref {thm:factorisation} compares the two islands through the layer. Right: the factor $e^{-2\alpha _G(w-2s-8)}$ of Corollary~\ref {cor:twoislands} at $\alpha _G=2.5\cdot 10^{-4}$; the amplitude in front of it is $1920\Lambda _2^2r^2\sqrt {n_xn_y}/\beta _W$, of order $10^{-34}\Lambda _2^2r^2\sqrt {n_xn_y}$ at the threshold $\beta _W=\beta ^*$.}}{270}{figure.39}\protected@file@percent } \newlabel{fig:layer}{{39}{270}{Left: two bad islands at the ends of a slab, the frozen collars of width $s+3$ beside them, and the good layer between them, on which the layer law $\mu _{x,y}$ obeys the chart bound in the label coordinates (Theorem~\ref {thm:layerbound}). The cross-ratio $\varDelta $ of the kernel of Theorem~\ref {thm:factorisation} compares the two islands through the layer. Right: the factor $e^{-2\alpha _G(w-2s-8)}$ of Corollary~\ref {cor:twoislands} at $\alpha _G=2.5\cdot 10^{-4}$; the amplitude in front of it is $1920\Lambda _2^2r^2\sqrt {n_xn_y}/\beta _W$, of order $10^{-34}\Lambda _2^2r^2\sqrt {n_xn_y}$ at the threshold $\beta _W=\beta ^*$}{figure.39}{}} \newlabel{lem:manylayer}{{489}{270}{the layer between many islands}{theorem.489}{}} \newlabel{lem:islandcov}{{490}{270}{covariances over the islands}{theorem.490}{}} \citation{Pop35} \citation{Fol82} \citation{Dob68,Dob70} \newlabel{thm:manyislands}{{491}{271}{many islands}{theorem.491}{}} \newlabel{cor:islandchains}{{492}{271}{decay along chains of islands}{theorem.492}{}} \newlabel{prop:islandmass}{{493}{272}{the island sums in the mean}{theorem.493}{}} \newlabel{thm:islandwindow}{{494}{273}{the island sums in a window}{theorem.494}{}} \newlabel{eq:islandmassG}{{44}{273}{}{equation.13.44}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {O}The covariance of the labels}{275}{section*.82}\protected@file@percent } \newlabel{ssec:labelcov}{{XIII\,O}{275}{}{section*.82}{}} \newlabel{lem:groupcross}{{495}{275}{two groups of islands}{theorem.495}{}} \citation{Pop35} \@writefile{lof}{\contentsline {figure}{\numberline {40}{\ignorespaces (a) Islands of bad blocks (red: the two islands $I_a$, $I_b$ compared; blue: other islands $I_q$) in a window whose remaining blocks are good. The solid arc is the two-island bound $\varDelta _{ab}$ of Corollary~\ref {cor:twoislands}; the dashed arcs are the bounds $\varDelta _{uq}$ through which the other islands enter $\varDelta _*$ in Theorem~\ref {thm:manyislands}. (b) The bound $\varDelta $ for two islands against the width $w$ of the good layer between them, with $\Lambda _2$ from Lemma~\ref {lem:lambdatwo} at $\beta _W=1.13\cdot 10^{37}$; it falls below one at $w=99072$.}}{276}{figure.40}\protected@file@percent } \newlabel{fig:manyislands}{{40}{276}{(a) Islands of bad blocks (red: the two islands $I_a$, $I_b$ compared; blue: other islands $I_q$) in a window whose remaining blocks are good. The solid arc is the two-island bound $\varDelta _{ab}$ of Corollary~\ref {cor:twoislands}; the dashed arcs are the bounds $\varDelta _{uq}$ through which the other islands enter $\varDelta _*$ in Theorem~\ref {thm:manyislands}. (b) The bound $\varDelta $ for two islands against the width $w$ of the good layer between them, with $\Lambda _2$ from Lemma~\ref {lem:lambdatwo} at $\beta _W=1.13\cdot 10^{37}$; it falls below one at $w=99072$}{figure.40}{}} \newlabel{lem:labelchains}{{496}{277}{chains in two copies}{theorem.496}{}} \newlabel{eq:groupmean}{{45}{277}{}{equation.13.45}{}} \newlabel{thm:labelcov}{{497}{277}{the label covariance}{theorem.497}{}} \citation{vdBM94} \newlabel{thm:fibrecov}{{498}{278}{the fibre covariance}{theorem.498}{}} \newlabel{cor:fibrecovnumbers}{{499}{279}{the covariances in numbers}{theorem.499}{}} \citation{Fol82} \citation{Dob68,Dob70} \citation{Pop35} \newlabel{lem:sweep}{{500}{280}{the sweep}{theorem.500}{}} \newlabel{rem:labelcovopen}{{501}{280}{}{theorem.501}{}} \newlabel{lem:onelaw}{{502}{280}{one law}{theorem.502}{}} \newlabel{prop:fixedgood}{{503}{280}{fixed coarse fields}{theorem.503}{}} \newlabel{cor:labelstretched}{{504}{281}{the labels, stretched}{theorem.504}{}} \newlabel{thm:fixedobstruction}{{505}{281}{what rarity and the group cross-ratios give}{theorem.505}{}} \citation{KP86} \newlabel{lem:mobius}{{506}{282}{differences of higher order}{theorem.506}{}} \newlabel{rem:fixedcopen}{{507}{282}{}{theorem.507}{}} \newlabel{prop:pairsonly}{{508}{282}{pair bounds give the diameter form}{theorem.508}{}} \newlabel{prop:layerthird}{{509}{283}{the layer bound and third cumulants}{theorem.509}{}} \newlabel{rem:treeform}{{510}{283}{}{theorem.510}{}} \citation{HS94,NS97,Hel98} \newlabel{thm:thirdcumulant}{{511}{284}{third cumulants under the layer bound}{theorem.511}{}} \newlabel{cor:kthird}{{512}{285}{third differences of the kernel}{theorem.512}{}} \newlabel{prop:layerorders}{{513}{285}{the layer bound and cumulants of every order}{theorem.513}{}} \newlabel{lem:twoindex}{{514}{286}{two-index decay}{theorem.514}{}} \citation{BGL14} \newlabel{thm:thirdtree}{{515}{287}{third cumulants in tree form}{theorem.515}{}} \newlabel{cor:kthirdtree}{{516}{288}{third differences of the kernel at the full rate}{theorem.516}{}} \newlabel{rem:treeformstatus}{{517}{288}{}{theorem.517}{}} \citation{HS94,NS97,Hel98} \newlabel{thm:treeorders}{{518}{289}{cumulants of every order under finite range}{theorem.518}{}} \newlabel{eq:treewords}{{46}{289}{}{equation.13.46}{}} \newlabel{eq:treeenergy}{{47}{290}{}{equation.13.47}{}} \newlabel{rem:treeorderswalls}{{519}{291}{}{theorem.519}{}} \newlabel{lem:islandgas}{{520}{291}{derivatives through the gas of islands}{theorem.520}{}} \newlabel{prop:islandchain}{{521}{291}{a chain of strongly coupled islands}{theorem.521}{}} \newlabel{rem:treeordersstatus}{{522}{292}{}{theorem.522}{}} \newlabel{lem:condweights}{{523}{292}{conditioned weights}{theorem.523}{}} \newlabel{lem:coarsewitnesssharp}{{524}{292}{the sharp witness}{theorem.524}{}} \newlabel{prop:badchart}{{525}{293}{coarse curvature and the chart}{theorem.525}{}} \newlabel{thm:badcross}{{526}{293}{a bad component on the chart}{theorem.526}{}} \newlabel{rem:badcompopen}{{527}{294}{}{theorem.527}{}} \newlabel{lem:ballexit}{{528}{294}{leaving a ball}{theorem.528}{}} \newlabel{thm:offchartmass}{{529}{294}{the mass off the chart}{theorem.529}{}} \newlabel{cor:offchartmass}{{530}{295}{}{theorem.530}{}} \newlabel{prop:uniformflux}{{531}{295}{a uniform field strength}{theorem.531}{}} \@writefile{lof}{\contentsline {figure}{\numberline {41}{\ignorespaces The mass off the chart in a bad component $W$ (Theorem~\ref {thm:offchartmass}). (a)~The lower bound $1-\exp (-15\lvert W\rvert e^{-10^{-24}\beta _W})$ on $p$ over $\beta _W/10^{24}$ and $\log _{10}\lvert W\rvert $. (b)~The curve $\beta _W=10^{24}(4.40+\ln \lvert W\rvert )$, above which the bound of Theorem~\ref {thm:badcross} improves on the trivial one, and the region in which the lower bound on $p$ exceeds $\frac 12$.}}{296}{figure.41}\protected@file@percent } \newlabel{fig:offchartmass}{{41}{296}{The mass off the chart in a bad component $W$ (Theorem~\ref {thm:offchartmass}). (a)~The lower bound $1-\exp (-15\lvert W\rvert e^{-10^{-24}\beta _W})$ on $p$ over $\beta _W/10^{24}$ and $\log _{10}\lvert W\rvert $. (b)~The curve $\beta _W=10^{24}(4.40+\ln \lvert W\rvert )$, above which the bound of Theorem~\ref {thm:badcross} improves on the trivial one, and the region in which the lower bound on $p$ exceeds $\frac 12$}{figure.41}{}} \newlabel{prop:badpassage}{{532}{296}{the window under $\mu ^c$}{theorem.532}{}} \newlabel{cor:badcrossratio}{{533}{297}{}{theorem.533}{}} \newlabel{rem:badmassopen}{{534}{297}{}{theorem.534}{}} \newlabel{thm:localcharts}{{535}{298}{local charts}{theorem.535}{}} \newlabel{prop:spread}{{536}{298}{spreading a coarse plaquette}{theorem.536}{}} \@writefile{lot}{\contentsline {table}{\numberline {IV}{\ignorespaces The fields of Proposition~\ref {prop:spread}, in units of $\theta $, at the sites with coordinates $x_\mu $ and $x_\nu $: the angles of the links in the direction $\mu $ for $0\le x_\nu \le 3$, of the links in the direction $\nu $ at $x_\nu =1$, and the real sums of the plaquettes in the plane $\{\mu ,\nu \}$ with corner $x$. 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(a)~The least eigenvalue of $M_p(\varphi )$ over the torus of phases for the plaquette class with the smallest minimum, about $0.0057$ on a grid of $96^2$ phases (red dot); the exact computation of Remark~\ref {rem:certificate} finds it positive at every phase. (b)~The forty-eight totals in the definition of $h$, in increasing order, against $\frac {29}2$, the value $14.420001$ of the largest, the floor $14.384$ for certificates on the period-two cell, and the lower bound $9+\sqrt {17}$ for $h_F$. (c)~The constant $C$ in $\DOTSB \sum@ \slimits@ \phi ^2\le CQ$ for the block gauge: $52$ from Theorem~\ref {thm:blockgap}, $5000/537$ from the computation of Remark~\ref {rem:blockgapsharp}, and the bulk Bloch value $1/0.10749=9.3034$, the best constant when $W$ is the set of all blocks. The eigenvalues in (a) and the Bloch value are in floating point.}}{383}{figure.47}\protected@file@percent } \newlabel{fig:certificate}{{47}{383}{The certificate of Appendix~\ref {app:certificate}. 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\usepackage{orcidlink} \usepackage{microtype} \usepackage{xurl} \usepackage{tikz} \usetikzlibrary{arrows.meta,patterns} \makeatletter % Ragged-right paragraph cells without the array package, whose current % release is not recognised by revtex4-2. \newcommand*{\raggedcells}{% \let\YM@arrayparboxrestore\@arrayparboxrestore \def\@arrayparboxrestore{\YM@arrayparboxrestore\rightskip\z@\@plus1fil\relax}} \makeatother \makeatletter \AtBeginDocument{\def\NAT@bibsetup{\setlength{\labelwidth}{\z@}% \setlength{\labelsep}{0.5em}\setlength{\itemindent}{\z@}% \setlength{\listparindent}{\z@}\setlength{\topsep}{\z@}% \setlength{\parsep}{\z@}\NATx@bibsetup}} \let\write@bibliographystyle\relax \let\bibliography@latex\@gobble \makeatother \tolerance=2000 \emergencystretch=2em \hbadness=2000 \hfuzz=1pt \raggedbottom \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\C}{\mathbb{C}} \newcommand{\E}{\mathbb{E}} \newcommand{\SU}{\mathrm{SU}} \newcommand{\SO}{\mathrm{SO}} \newcommand{\U}{\mathrm{U}} \newcommand{\g}{\mathfrak{g}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\ReTr}{\operatorname{Re}\operatorname{Tr}} \newcommand{\ImTr}{\operatorname{Im}\operatorname{Tr}} \newcommand{\Hess}{\operatorname{Hess}} \newcommand{\Ric}{\operatorname{Ric}} \newcommand{\Cov}{\operatorname{Cov}} \newcommand{\Var}{\operatorname{Var}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\ip}[2]{\langle #1,#2\rangle} \setlength{\emergencystretch}{3em} % Preprints.org DOI of the amsart version: put it between the braces once % it is published, e.g. \def\preprintsDOI{10.20944/preprints2026xx.xxxx.v1} \def\preprintsDOI{} \begin{document} \title{One Cube Estimate Implies the Yang--Mills Mass Gap} \author{Deep Bhattacharjee$^{*}$\,\orcidlink{0000-0003-0466-750X}} \email{itsdeep@live.com} \thanks{$^{*}$\,Corresponding author. Secondary address: \nolinkurl{d.bhattacharjee@erl-forschung.de}; GitHub contact: \nolinkurl{egspl@proton.me}, \href{https://github.com/creelie}{github.com/creelie}.} \affiliation{Formerly, Electro-Gravitational Space Propulsion Laboratory (EGSPL), Bhubaneswar, Odisha 751030, India} \affiliation{Formerly, University of Calcutta (Heramba Chandra College), 23/49 Gariahat Road, Golpark, Kolkata 700029, West Bengal, India} \author{Ushashi Bhattacharya\,\orcidlink{0009-0002-2254-3914}} \email{rai.bhattacharya.in@gmail.com} \affiliation{Formerly, National Taiwan University, Taipei 106319, Taiwan} \author{James Montgomery Lockwood} \email{spectralityworkinggroup@gmail.com} \thanks{Secondary address: \nolinkurl{Blaze61029dogQ@proton.me}.} \affiliation{Independent Researcher, San Antonio, USA} \author{Tristan Romanov} \email{romanov360@gmail.com} \affiliation{Independent Researcher, San Francisco, CA, USA} \date{4 November 2026} \begin{abstract} \itshape We prove that four-dimensional lattice Yang--Mills theory with any compact simple gauge group has a unique infinite-volume state and a mass gap at strong coupling, and that one averaged estimate in a single cube of fixed physical size gives a mass gap to every continuum limit. For $\SU(2)$ the block variables approach the Haar measure in every representation as the coupling is tuned to the continuum, and this proves the estimate at the coarsest scale on every torus of side up to $e^{2^k/(17k)}$ after $k$ blocking steps. The estimate along the whole tuning remains open. \end{abstract} \maketitle \begingroup \renewcommand\thefootnote{}\footnotesize \footnotetext{\footnotesize\emph{2020 Mathematics Subject Classification.} Primary 81T25, 82B20; Secondary 81T13, 60K35, 82B26, 22E30, 43A77. \emph{PACS.} 11.15.Ha, 11.15.Tk, 05.50.+q, 05.10.Cc. \emph{Keywords.} Lattice gauge theory; mass gap; Dobrushin uniqueness; block-spin renormalisation; reflection positivity. \scriptsize An earlier version of this paper is available at \url{https://hexagonmath.org/2610.00341v6}.\ifx\preprintsDOI\empty\else{} The paper is also available on Preprints.org, \href{https://doi.org/\preprintsDOI}{doi:\preprintsDOI}.\fi{} Compared with it, this paper adds Sec.~\ref{sec:islands}, on the labels of the blocks, the islands of bad blocks, the bad side of a projected block average, the obstructions to the remaining routes and the effective action across one scale, Secs.~\ref{ssec:averaged}--\ref{ssec:bpmetrics}, on an averaged form of the cube condition, its local form, the couplings where it holds and fails, its neutral part on small tori, its passage to block variables, the upward step, the measures for which it holds, the split inequality at strong coupling, the condition at the coarsest scale, the coefficients across scales, the forces in the fibre law, the smoothing along the paths, the order of the row sums of the Gaussian model and the metrics that compose across scales, and Secs.~\ref{ssec:osaxioms}, \ref{ssec:limitE} and~\ref{ssec:chainG}, on the Osterwalder--Schrader axioms for the flowed fields, the limit at separated points of the fields without flow, with the inputs it needs and the examples that separate them, and the conditional chain for every compact simple group, together with the concluding Sec.~\ref{sec:conclusion}, and it proves by hand every constant that the earlier version took from computer certificates.} \endgroup \addtocounter{footnote}{-1} \section{Introduction}\label{sec:intro} Wilson's lattice gauge theory~\cite{Wil74} assigns an element $Q_e$ of a compact group $G\subset\U(N)$ to every link $e$ of $\Z^4$ and weighs a configuration by $\exp(N\beta\sum_p\ReTr Q_p)$, where $Q_p$ is the ordered product of the four link variables around the plaquette~$p$.\footnote{This is the normalisation of Shen, Zhu and Zhu~\cite{SZZ23} and of our earlier paper~\cite{BB26a}. The conventional inverse coupling is $\beta_W=N^2\beta$, so the weight is $\exp(\frac{\beta_W}N\sum_p\ReTr Q_p)$; for $\SU(2)$, identified with the unit quaternions, $\frac12\Tr Q_p=\operatorname{Re}Q_p$ and $\beta_W=4\beta$. Standard accounts of the lattice theory are the books of Montvay and M\"unster~\cite{MM94} and of Gattringer and Lang~\cite{GL10}; the Hamiltonian version is due to Kogut and Susskind~\cite{KS75}.} At small $|\beta|$ the theory is under control: it has a unique infinite-volume state, correlations decay exponentially, and the transfer matrix has a gap~\cite{OSei78,Sei82}.\footnote{Strong-coupling results of this kind go back to Osterwalder and Seiler~\cite{OSei78}, who proved exponential clustering and a gap by a convergent expansion, and to Seiler's monograph~\cite{Sei82}; the general framework for lattice gauge fields and their continuum limits is that of Brydges, Fr\"ohlich and Seiler~\cite{BFS79}. More recent work treats the strong-coupling regime by probabilistic methods: the survey of Chatterjee~\cite{Chat19}, his large-$N$ solution of $\SO(N)$ lattice gauge theory at strong coupling~\cite{Chat19b} and its planar extension by Basu and Ganguly~\cite{BG18}, the Langevin dynamics of Shen, Zhu and Zhu~\cite{SZZ23}, and the area law in an enlarged region by Cao, Nissim and Sheffield~\cite{CNS25}. For finite gauge groups Adhikari and Cao proved correlation decay at weak coupling~\cite{AC25}. In two dimensions the continuum Yang--Mills measure has been constructed by Gross, King and Sengupta~\cite{GKS89}, Driver~\cite{Dri89} and L\'evy~\cite{Lev03}; in four dimensions Ba{\l}aban's renormalisation group, as expounded by Dimock~\cite{Dim13}, gives ultraviolet stability in finite volume, and Magnen, Rivasseau and S\'en\'eor constructed the theory with an infrared cutoff~\cite{MRS93}. None of these results gives a mass gap in physical units.} The question is how far into the coupling range such statements can be proved with explicit constants, since the problem of the continuum limit~\cite{JW06} begins where strong coupling ends and Ba{\l}aban's weak-coupling analysis~\cite{Bal84a,Bal85a,Bal89b} takes over. Two methods give explicit ranges. The Bakry--\'Emery method~\cite{BE85,BGL14} of Shen, Zhu and Zhu~\cite{SZZ23} bounds the Hessian of the action; in~\cite{BB26a} we showed that the sharp bound is $4d=16$ and that no curvature argument for the ungauged measure goes beyond $|\beta|<1/32$. The Dobrushin method~\cite{Dob68,Dob70}\footnote{Dobrushin's theorem: if the conditional law of each variable given all others moves, in Kantorovich distance, by at most $C_{xy}$ times the distance by which the variable at $y$ moves, and $\sup_x\sum_yC_{xy}<1$, then the Gibbs measure is unique and correlations decay at the rate given by the Neumann series of the matrix $C$. Dobrushin and Shlosman~\cite{DS85} replaced single variables by finite boxes, which is the form used in Sec.~\ref{sec:bypass}.} bounds the influence of each link on the conditional law of another; for $\SU(2)$ the single-link conditional law is a von Mises--Fisher law on the three-sphere, the sharp Kantorovich modulus\footnote{The Kantorovich distance between two probability measures on a metric space is the least expected distance over their couplings, equivalently the largest difference of their integrals over $1$-Lipschitz functions~\cite{Vil09}. The modulus of a family of laws indexed by a metric space is the best Lipschitz constant of the map from the index to the law in that distance.} of that family is one quarter~\cite{BB26a} (Theorem~\ref{q-thm:quarter} in Appendix~\ref{app:sharp}), and every link has eighteen partners, which gives $\beta_W<2/9$ and nothing beyond, because the coefficients are attained. Both limits come from the same count. A link of $\Z^4$ lies in six plaquettes and shares each with three other links. Gauge invariance makes part of this redundant: in a box, the variables on a forest of links can be set to the identity without changing any gauge-invariant expectation~\cite{Cre77}.\footnote{Creutz used a maximal tree to remove the gauge freedom completely; the temporal gauge, with all links in one direction fixed, is the special case of a family of straight lines. In a maximal tree almost every link is fixed in some region and none in others, so the number of partners of a free link is not uniform, and the temporal gauge leaves fourteen partners per link. The point here is a pattern that fixes exactly one link in every plaquette.} The forest we use is a field of straight lines with one line through every site, chosen so that every plaquette contains exactly one fixed link. Then every free link has exactly twelve free partners, two in each of its plaquettes (Lemma~\ref{lem:structure}), and both methods improve. \begin{theorem}[$\SU(2)$]\label{thm:su2} Let $G=\SU(2)$ and $0\le\beta_W<\frac13$. For every gauge-invariant local observable $O$ the expectations $W_\Lambda(O)$ in boxes with free boundary converge, along every sequence of boxes that exhausts $\Z^4$, to a limit $\omega(O)$ that does not depend on the sequence. The state $\omega$ is invariant under translations and the hypercubic group and reflection positive;\footnote{That is, $\omega(\vartheta(\bar F)F)\ge0$ for every $F$ depending on the links on one side of a lattice hyperplane, $\vartheta$ being the reflection in it. It gives the Hilbert space and the transfer matrix of Osterwalder and Schrader~\cite{OS73,OS75} and of Osterwalder and Seiler~\cite{OSei78}, reconstructed in Appendix~\ref{app:transfer}.} truncated correlations of Lipschitz observables decay at every rate below $-\log(3\beta_W)$; and the transfer matrix has spectral gap \[ m\ge-\log(3\beta_W)>0 . \] For $-\frac13<\beta_W<0$ all statements except reflection positivity and the gap hold. \end{theorem} The step behind Theorem~\ref{thm:su2} is short once the line field is in place. The number $\frac13$ is also the end of the road for this kind of argument. \begin{theorem}[Ceiling]\label{thm:ceiling} Let $F'$ be any set of links of $\Z^4$ without cycles, and give two distinct links outside $F'$ that share a plaquette the coefficient $|\beta|$, the largest single-link Kantorovich coefficient of $\SU(2)$. Then the weighted Dobrushin condition with these coefficients fails for $\beta_W\ge\frac13$. For the line field the coefficients are attained, so no single-link Kantorovich argument in that gauge reaches $\beta_W=\frac13$. \end{theorem} The ceiling concerns coefficients between single links. Take instead as a block the three free links of one plaquette. The free links of the line field split into such triples with period two (Lemma~\ref{lem:plaqblocks}), the coupling inside the block plaquette is no longer paid for, and each link keeps ten outside partners. The price is the sensitivity of a law on three spheres instead of one, which we bound analytically by integrating out the middle link of the block. \begin{theorem}[$\SU(2)$ with plaquette blocks]\label{thm:blocks} Let $G=\SU(2)$ and $0\le\beta_W\le0.348$. The number $\alpha(\beta_W)$ of~\eqref{eq:alphablock} is increasing, $\alpha(\frac13)\le0.9456$, $\alpha(0.345)\le0.9837$ and $\alpha(0.348)\le0.9936$. The conclusions of Theorem~\ref{thm:su2} hold with decay at every rate below $-\frac12\log\alpha(\beta_W)$ and a transfer-matrix gap \[ m\ge-\tfrac12\log\alpha(\beta_W)>0 ; \] at $\beta_W=\frac13$ this gives $m\ge0.0279$. For $-0.348\le\beta_W<0$ all statements except reflection positivity and the gap hold. \end{theorem} At small coupling the rate of Theorem~\ref{thm:su2} is the better one, so both bounds stand side by side; Theorem~\ref{thm:blocks} matters past $\beta_W=\frac13$, where the gap of Theorem~\ref{thm:su2} has closed. The curvature method gains as well, and for every $N$. Let $\Hess S_0$ be the Hessian of $\sum_p\ReTr Q_p$ on $G^E$ and $F$ the set of fixed links. \begin{theorem}[Curvature in the gauge]\label{thm:curv} For every $N$, every field $Q$ and every tangent vector $v=(X_eQ_e)$ with $X_e=0$ for $e\in F$, \[ |\Hess S_0(v,v)|\le\tfrac{29}2|v|^2 , \] against the sharp constant $16$ without gauge fixing. Consequently, for $G=\SU(N)$ and $|\beta|<\frac1{29}$, or $G=\SO(N)$, $N\ge3$, and $|\beta|<\frac{N-2}{58N}$, the conclusions of Theorem~\ref{thm:su2} hold with the rate and gap of Theorem~\ref{thm:groups}. For $\SU(3)$ this is $\beta_W<\frac9{29}=0.310$. \end{theorem} The constant $\frac{29}2$ is a statement about $\U(N)$, so it holds on every closed subgroup, and the curvature of the group does the rest. This gives the theorem for every compact simple gauge group, the exceptional groups included, with an explicit gap. Let $\rho:G\to\U(N)$ be a faithful unitary representation of a compact connected Lie group $G$, weigh a configuration by $\exp(N\beta\sum_p\ReTr\rho(Q_p))$, and let $\kappa_G$ be the least Ricci\footnote{For a bi-invariant metric on a compact Lie group the Ricci curvature is $-\frac14\Tr((\operatorname{ad}X)^2)$ in the direction of $X$. It is positive for every $X\ne0$ exactly when the Lie algebra is semisimple, and $\kappa_G$ is its minimum over unit vectors.} curvature of $G$ for the bi-invariant metric $\ip XY=-\Tr(\rho(X)\rho(Y))$; we identify $G$ with $\rho(G)\subset\U(N)$. \begin{theorem}[Every compact simple group]\label{thm:groups} Suppose that the Lie algebra of $G$ is semisimple, so that $\kappa_G>0$, and that \[ N|\beta|<\tfrac2{29}\,\kappa_G . \] Then the conclusions of Theorem~\ref{thm:su2} hold, for $\beta<0$ all of them except reflection positivity and the gap, with decay at every rate below \[ m_G(\beta)=\operatorname{arccosh}\Bigl(1+\frac1{12}\Bigl(\frac{\kappa_G}{N|\beta|} -\frac{29}2\Bigr)\Bigr)>0 \] and a transfer-matrix gap $m\ge m_G(\beta)$. For a simple $G$, $\kappa_G=h^\vee/(2\ell_\rho)$, where $h^\vee$ is the dual Coxeter number and $\ell_\rho$ the Dynkin index of $\rho$; Table~\ref{tab:groups} lists the values. \end{theorem} For $\SU(N)$, $\kappa_G=\frac N2$, and for $\SO(N)$, $\kappa_G=\frac{N-2}4$, so Theorem~\ref{thm:groups} contains the ranges of Theorem~\ref{thm:curv} and supplies their rate.\footnote{The dual Coxeter number is $h^\vee=N$ for $\SU(N)$, $N-2$ for $\SO(N)$, $n+1$ for $\mathrm{Sp}(n)$, $4,9,12,18,30$ for $G_2,F_4,E_6,E_7,E_8$, and the Dynkin index of a representation is normalised to one on the defining representation of $\SU(N)$~\cite[Ch.~6]{Kac90}. The identity $\kappa_G=h^\vee/(2\ell_\rho)$ is the statement that the Ricci tensor of a compact simple group with a bi-invariant metric is one quarter of the negative Killing form, $\Ric(X,X)=-\frac14B(X,X)$~\cite{Mil76}, together with $B(X,X)=2h^\vee\Tr(\rho(X)^2)/\ell_\rho$. Lower Ricci bounds of the same kind, for special holonomy and for Riemannian submersions, are studied in~\cite{BTBS26,GBN26}.} For small $|\beta|$ the existence of a gap is classical for every compact group, with no explicit range~\cite[Thm.~3.6]{OSei78}; what is new is an explicit range for every group, past the range $N|\beta|<\kappa_G/16$ that the Hessian bound $16$ gives without gauge fixing, and an explicit gap.\footnote{The computation of Remark~\ref{rem:certificate}, from a certificate of T.~Romanov, gives $h_F\le14.420001$; with it in place of $\frac{29}2$, every range of Table~\ref{tab:groups} would grow by the factor $29/28.840002=1.0055$. No result of the paper uses it.} As $\beta\to0$ the gap grows like $\log\frac{\kappa_G}{6N|\beta|}$. \begin{table}[t] \caption{Least Ricci curvature $\kappa_G=h^\vee/(2\ell_\rho)$ of the compact simple groups in a faithful representation $\rho$ of least dimension $N$, and the range of Theorem~\ref{thm:groups} for the conventional inverse coupling $\beta_W=N^2\beta$.} \label{tab:groups} \begin{ruledtabular} \begin{tabular}{llcccc} Group&$\rho$&$h^\vee$&$\ell_\rho$&$\kappa_G$&$\beta_W<$\\ \colrule $\SU(N)$, $N\ge2$&$\C^N$&$N$&$1$&$\frac N2$&$\frac{N^2}{29}$\\ $\SO(N)$, $N\ge5$&$\R^N$&$N-2$&$2$&$\frac{N-2}4$&$\frac{N(N-2)}{58}$\\ $\SO(3)$&$\R^3$&$2$&$4$&$\frac14$&$\frac3{58}$\\ $\mathrm{Sp}(n)$, $n\ge2$&$\C^{2n}$&$n+1$&$1$&$\frac{n+1}2$&$\frac{2n(n+1)}{29}$\\ $G_2$&$7$&$4$&$2$&$1$&$\frac{14}{29}$\\ $F_4$&$26$&$9$&$6$&$\frac34$&$\frac{39}{29}$\\ $E_6$&$27$&$12$&$6$&$1$&$\frac{54}{29}$\\ $E_7$&$56$&$18$&$12$&$\frac34$&$\frac{84}{29}$\\ $E_8$&$248$&$30$&$60$&$\frac14$&$\frac{124}{29}$\\ \end{tabular} \end{ruledtabular} \end{table} \begin{figure*}[t] \centering \includegraphics[width=0.9\textwidth]{figures/fig_groups.png} \caption{The gap $m_G(\beta)$ of Theorem~\ref{thm:groups} as a function of $\beta_W=N^2\beta$ on a logarithmic scale, for nine groups in the representations of Table~\ref{tab:groups}. Each curve ends at the edge $\beta_W=2N\kappa_G/29$ of the range, marked by a dot, where the gap tends to zero; as $\beta_W\to0$ it grows like $\log(\kappa_G/(6N|\beta|))$. The curves differ only by the factor $N\kappa_G$, since $m_G$ depends on $N|\beta|/\kappa_G$ alone.} \label{fig:groups} \end{figure*} No Bakry--\'Emery argument for the ungauged measure can pass $|\beta|= \frac1{32}$~\cite[Thm.~C]{BB26a}; the gauge fixing changes the measure, not the metric, which is why it can. For $\SU(2)$ the curvature range $\beta_W<\frac4{29}$ lies inside that of Theorem~\ref{thm:su2}, and no curvature argument for the gauge-fixed $\SU(2)$ measure passes $\beta_W=\frac13$: when one active link $e$ is $-1$ and all others are $1$, the six plaquettes through $e$ have variable $-1$, and along a geodesic through $Q_e$ the Hessian of $-\frac{\beta_W}2\sum_p\ReTr Q_p$ is $-6\beta_W$ times the metric of the unit sphere $S^3$, whose Ricci curvature is $2$, so the Bakry--\'Emery tensor is negative there once $\beta_W>\frac13$. For $\SU(3)$ the Dobrushin route goes further. \begin{theorem}[$\SU(3)$]\label{thm:su3} Let $G=\SU(3)$ and $0\le\beta_W\le0.515$, put $r=2\beta_W$ and \[ \epsilon(r)=1-\frac{5\,e^{-r/5}}{9\,Z_*(r)},\qquad\alpha_1(\beta_W)= \frac{2r\sqrt{\epsilon(r)}}{\gamma(r)}, \] where $\gamma(r)=\frac{283}{128}-\frac r2$ and $Z_*(r)=\int_{\SU(3)} e^{r\ReTr U}dU$. The number $\alpha_1(\beta_W)$ is increasing, $\alpha_1(\frac12)\le0.9491$ and $\alpha_1(0.515)\le0.9924$. The conclusions of Theorem~\ref{thm:su2} hold with decay at every rate below $-\log\alpha_1(\beta_W)$ and a transfer-matrix gap \[ m\ge-\log\alpha_1(\beta_W)>0 ; \] at $\beta_W=\frac12$ this gives $m\ge0.052$. \end{theorem} The proof of Theorem~\ref{thm:su3} bounds the Kantorovich modulus of the one-link measure $\nu_b(dU)\propto e^{\ReTr(Ub)}dU$ on $\SU(3)$ uniformly over staple sums $b$. It uses a lower bound $\gamma(r)$ for the spectral gap of $\nu_b$ through Bochner's formula with a cross term (Lemma~\ref{lem:gap}), the Poincar\'e inequality twice, and a bound for the mean of $|\nabla\ReTr(Uh)|^2$ under $\nu_b$ through Jensen's inequality and Haar integrals of degree three (Lemma~\ref{lem:su3modulus}); $Z_*$ is a series in the moments of $\Tr U$, which are counts of standard tableaux (Lemma~\ref{lem:su3haar}). A sharper bound for the same modulus, evaluated by a computer on boxes of staple sums, would carry the range to $\beta_W=\frac34$; Remark~\ref{rem:su3cover} describes it, and no result uses it. The plaquette blocks of Theorem~\ref{thm:blocks} work for $\SU(3)$ as well. They use the same modulus of the one-link measure, but only through the three links of a block, so they need $\Lambda(2\beta_W)$ small and not $2r\Lambda(r)<1$: near the end of Theorem~\ref{thm:su3} they give a larger gap, and they go past it. \begin{theorem}[$\SU(3)$ with plaquette blocks]\label{thm:su3blocks} Let $G=\SU(3)$ and $0\le\beta_W\le0.52$, let $r$, $\epsilon(r)$, $\gamma(r)$ be as in Theorem~\ref{thm:su3}, and let $\alpha_3(\beta_W)$ be the number of~\eqref{eq:blockconst3} with $\Lambda(r)$, $\tau$, $\lambda_*$ and $M$ replaced by $\sqrt{\epsilon(r)}/\gamma(r)$, $\epsilon(r)/\gamma(r)$, $0$ and $2\sqrt3\,\epsilon(r)/\gamma(r)$. Then $\alpha_3(\beta_W)<1$, with $\alpha_3(0.515)\le0.9742$ and $\alpha_3(0.52)\le0.9927$. The conclusions of Theorem~\ref{thm:su2} hold with decay at every rate below $-\frac12\log\alpha_3(\beta_W)$ and a transfer-matrix gap \[ m\ge-\tfrac12\log\alpha_3(\beta_W)>0 ; \] at $\beta_W=0.515$ this gives $m\ge0.013$, against $0.0076$ from Theorem~\ref{thm:su3}, and at $\beta_W=0.52$, outside the range of Theorem~\ref{thm:su3}, $m\ge0.0036$. \end{theorem} The constants of Theorem~\ref{thm:su3blocks} are the bounds of Lemma~\ref{lem:su3modulus}; Remark~\ref{rem:su3blocks} records what computed constants would give. \begin{table}[t] \caption{Explicit ranges of uniqueness, decay and a transfer-matrix gap for four-dimensional lattice Yang--Mills theory. The first three rows hold for the ungauged measure; the last three are the results of this paper.} \label{tab:ranges} \begin{ruledtabular} \begin{tabular}{lll} Method&$\SU(2)$, $\beta_W$&$\SU(3)$, $\beta_W$\\ \colrule Curvature, Shen--Zhu--Zhu~\cite{SZZ23}&$<\frac1{12}$&$<\frac3{16}$\\ Curvature, sharp~\cite{BB26a}&$<\frac18$&$<\frac9{32}$\\ Dobrushin, sharp quarter~\cite{BB26a}&$<\frac29$&\\ Curvature, forest gauge&$<\frac4{29}$&$<\frac9{29}$\\ Dobrushin, forest gauge&$<\frac13$&$\le0.515$\\ Plaquette blocks, forest gauge&$\le0.348$&$\le0.52$\\ \end{tabular} \end{ruledtabular} \end{table} \begin{figure*}[t] \centering \includegraphics[width=0.86\textwidth]{figures/fig_ranges.png} \caption{Ranges of Table~\ref{tab:ranges}. The height of each bar is the largest inverse coupling $\beta_W$ covered by the method. Grey bars are methods for the ungauged measure; blue, red and orange bars use the forest gauge. For $\SU(2)$ the red bar ends at the ceiling of Theorem~\ref{thm:ceiling} and the orange bar, plaquette blocks, passes it (Theorem~\ref{thm:blocks}); for $\SU(3)$ the red bar is Theorem~\ref{thm:su3} and the orange bar, plaquette blocks, gives a larger gap there and passes it (Theorem~\ref{thm:su3blocks}).} \label{fig:ranges} \end{figure*} All seven theorems are statements at fixed lattice spacing. They do not touch the continuum limit, the crossover to Ba{\l}aban's regime, or a gap in physical units, which together are the Clay problem; Sec.~\ref{sec:limits} says where each method stops and what remains of the four open problems of~\cite{BB26a}. The obstruction proved in~\cite{BB26b}, that a lattice gap uniform in lattice units forces trivial continuum two-point functions, applies to all of them. \emph{The continuum step.} Sec.~\ref{sec:bypass} goes past that obstruction in one respect, through three implications, each proved. \begin{enumerate}[label=\textup{(I\arabic*)},leftmargin=*] \item A Dobrushin--Shlosman condition on one cube of fixed physical size, uniform along the continuum tuning, gives a mass gap in physical units for every continuum limit (Theorems~\ref{thm:onescale} and~\ref{thm:transfer}), and so does exponential decay with constants of the renormalised correlations of a spanning set of observables (Theorem~\ref{thm:clustering}); this holds also for observables smoothed by the spatial gradient flow (Proposition~\ref{prop:spatialflow}). A spectral gap of the dynamics that resamples one such cube at a time, uniform along the tuning and required only on local gauge-invariant functions, gives the same conclusion (Theorems~\ref{thm:avgtransfer} and~\ref{thm:bplocal}). \item The flowed observables, renormalised by one condition at a single physical separation, have reflection positive continuum limits along subsequences, and such a limit is nontrivial exactly when one ratio of lattice two-point functions stays away from zero (Theorem~\ref{thm:flowlimit}); that ratio is bounded above and below by ratios of correlations of one slice field at fixed physical separations (Proposition~\ref{prop:sliceratio}). \item Tree bounds on the truncated correlations give all the higher Schwinger functions as reflection positive distributions, imply the ratio bound, and carry the conditional mass gap to every polynomial in the smeared fields (Theorem~\ref{thm:higher}); for a smooth packet they give the ratio with an explicit rate and a reference vector of finite energy (Proposition~\ref{prop:treeenergy}). \end{enumerate} \emph{What is open.} The inputs of (I1)--(I3), namely the cube condition along the tuning, the slice ratio in four dimensions and the tree bounds, are not proved, and so the nontrivial existence of the continuum limit and the gap in physical units remain open. Figure~\ref{fig:logic} shows how the proved statements, the implications and the open inputs fit together. Sec.~\ref{sec:inputs} measures the open inputs. The cube condition holds at strong coupling, fails at every fixed cube size at weak coupling, forces the gauge-fixed variables to be disordered on the scale of the cube, and cannot be reached from estimates on single lines of links; for $\SU(N)$ it needs cubes of side at least of order $(\beta_W/\log\beta_W)^{1/4}$ (Propositions~\ref{prop:cubesize} and~\ref{prop:cubesizeN}). The spectral gap of the cube dynamics holds on every torus at strong coupling (Corollary~\ref{cor:bpranges}) and fails on the torus of side $2n+4$ when $n$ is small compared with $(\beta/\log\beta)^{1/4}$ (Theorem~\ref{thm:bpfixed}), but only through functions of nonzero centre charge, which its local form does not involve (Corollary~\ref{cor:bpcharged}). For a Gaussian field it holds at a fixed physical scale when the field has a mass and fails exactly, at every size, when it is massless (Proposition~\ref{prop:gauss}); it also fails for the quadratic approximation of the forest gauge itself (Proposition~\ref{prop:gaussforest}), and of every gauge fixed on an invariant forest with infinite trees (Proposition~\ref{prop:gaussevery}). The order-two cases of the tree bounds force the lattice gap below a fixed multiple of the lattice spacing, so no strong coupling expansion gives them (Proposition~\ref{prop:treecritical}). The slice ratio holds in two dimensions, where it is the exact solution of lattice gauge theory (Remark~\ref{rem:twodslice}), and fails for a field whose mass is fixed in lattice units; for the free field, conditioning an energy packet on averages over blocks leaves a fixed fraction of its ordinary variance at every lattice spacing, so a reduction to block averages has to be measured in the reflected norm (Proposition~\ref{prop:blockfree}). \emph{Across scales.} The spectral gap of the cube dynamics passes from the fine lattice to the law of the block variables of every local gauge-covariant block-spin map (Theorem~\ref{thm:bpdescent}). In the opposite direction it climbs one scale under a split inequality that bounds the variance of the fine field given the block variables together with the coarse Dirichlet form (Theorem~\ref{thm:bpup}), so that along the tuning it follows from the condition at the coarsest scale and the split inequality at every scale with constants bounded in product (Corollary~\ref{cor:bpuptuning}). The split inequality holds with the constants of the product measure for every Haar refinement (Theorem~\ref{thm:bprefine}), for the Wilson measure at strong coupling on every torus with constants within factors $1+O(\beta)$ of those (Theorem~\ref{thm:bpsplitstrong}), and on each torus for every measure with a continuous positive density, with constants that grow with the volume (Proposition~\ref{prop:bptilt}). The condition at the coarsest scale follows from a bound on the sums of the single-link coefficients of the law of the block variables (Theorem~\ref{thm:bpcoarse}), and the closeness of each single-link conditional law to the Haar measure does not suffice (Proposition~\ref{prop:bphidden}). One step of the straight-path map contracts a weighted Dobrushin sum of these coefficients, so that a bound at one level of the tower suffices (Theorem~\ref{thm:bpbalstep}, Corollary~\ref{cor:bpbaltower}); but gauge covariance bounds the sum below by two minus four times the normalised spread of the single-link conditional laws, which excludes the Wilson measure at weak coupling and every law of the block variables with concentrated plaquette variables (Theorem~\ref{thm:bpbalnec}, Corollaries~\ref{cor:bpbalwilson} and~\ref{cor:bpbalsmall}), while at the coarsest scale the conditional laws are spread out along the tuning and this obstruction is absent (Proposition~\ref{prop:bpbalrandom}). The coefficients at the coarsest scale are at most $D^2/4$ times the covariances of the forces on distinct marks in the fibre law, uniformly in the coarse data, so that the condition there follows from a decorrelation of these forces; in a Gaussian model they are given by an effective quadratic form of the straight-path map, whose row sums are of order $(\log b)/b$, which gives the decorrelation in that model along the tuning (Proposition~\ref{prop:bpfibreduce}, Corollaries~\ref{cor:bpfibH1} and~\ref{cor:bprowFC}, Theorem~\ref{thm:bprowgauss}). For $\SU(2)$ the single-link conditional laws of the block variables have densities within $48(1+6\beta_W)^2(4\beta_W/ (4\beta_W+1))^{b-2}$ of the Haar density, with their first two derivatives, because convolution with a one-link density contracts every nontrivial representation; along every tuning with $\beta_k=O(k)$ the single-link curvature of the induced action therefore tends to zero, and the mixed derivatives are covariances of the scores of the interior path products under one posterior law, each tending to zero, which gives the condition at the coarsest scale on every torus of side at most $e^{2^k/(17k)}$ (Theorems~\ref{thm:bppathdensity} and~\ref{thm:bppathmixed}, Corollary~\ref{cor:bppathfourk}); on tori of every size it needs these covariances to be summable uniformly in the torus (Proposition~\ref{prop:bppathsuff}). Along the tuning neither input is proved; the single-link bounds for the laws of the block variables are the effective-action problem of Ba{\l}aban.\footnote{Refs.~\cite{Bal84a,Bal85a,Bal89b} control the effective actions of the small-field region in a box of fixed physical size; the inputs here need them on tori of every size and for all fields, and in the small-field region itself the Dobrushin sums exceed one (Corollary~\ref{cor:bpbalsmall}).} Figure~\ref{fig:bptower} shows the route. \emph{Block averages.} Sec.~\ref{sec:blocks} poses the cube condition for block averages in a gauge whose fixed links are trees inside blocks of side two. Its quadratic approximation then holds at every coupling once the cube is large (Theorem~\ref{thm:constrainedcube}), and integrating out one scale reduces it to a cube condition for the averages, which carry the coarse quadratic form at every scale up to a factor between $1$ and $256$ that does not depend on the scale, with an exponentially decaying kernel whose constants do not depend on the scale either, and the constrained cube condition holds at every scale with one cube side (Corollary~\ref{cor:compose}, Proposition~\ref{prop:effective}, Theorems~\ref{thm:effectiveupper}, \ref{thm:effectivelocal} and~\ref{thm:constrainedscales}); the averages along the lines of the forest gauge, by contrast, leave one class of lines invisible and fail at every size (Proposition~\ref{prop:invisible}). For the group-valued links, the straight path between block roots gives a coarse link whose fibres satisfy a coercivity bound for every compact group and every field (Theorem~\ref{thm:pathcoercive}), but this map does not compose across scales (Proposition~\ref{prop:thinpath}), and for every compact connected group no continuous covariant block map treats the positions of a block alike (Proposition~\ref{prop:nomean}). A cut-off of Ba{\l}aban's averaging operation is continuous, covariant and coercive for every field and agrees with the block averages to first order; it breaks the symmetry only at large fields, as it must, and its constants are those of one step (Theorem~\ref{thm:cutoff}). On fields with values in a maximal torus whose curvature stays small at every scale it composes, with the energy nonincreasing and the thin paths contracted (Theorem~\ref{thm:torusscales}), and for every compact group it composes on fields whose curvatures at the successive scales have a small sum (Theorem~\ref{thm:smallscales}). Under the $\SU(2)$ Wilson measure the plaquettes that are not small lie in clusters whose physical size tends to zero in probability, and the cut-off acts only next to them (Theorem~\ref{thm:defects}, Corollaries~\ref{cor:defectclusters} and~\ref{cor:activelinks}); such plaquettes are nevertheless present along the one-loop tuning (Proposition~\ref{prop:manydefects}). The cut-off average composes through these clusters, which spread by at most three spacings of the current scale per step, up to a coarse spacing of order $(\beta_W/\log\beta_W)^{1/4}a$, the order of the cube side that the cube condition needs (Theorem~\ref{thm:defectscales} and Corollary~\ref{cor:defectscales}). Under every exterior, excess action in a box is rare, restricting one link to a fixed set leaves its Dobrushin response bounded, and one-point rarity of a bad set gives approximate uniqueness of the Gibbs measures but no decay (Lemmas~\ref{lem:excess} and~\ref{lem:fixedset}, Proposition~\ref{prop:onepoint}). The curvature of the block averages costs action independent of the scale (Lemma~\ref{lem:coarseenergy}, Proposition~\ref{prop:torusenergy}); the chessboard estimate applies to coarse events and is lost after $O(\log\beta_W)$ scales in the normalisation alone (Proposition~\ref{prop:coarsechess}); a right translation of the Haar measures bounds the tails of the force along any fixed direction by the second variation of the action, at every scale and under every exterior (Theorem~\ref{thm:haartilt}), which for the abelian theory makes coarse defects rare at the rate $\beta_Wx^2$ at every scale (Corollary~\ref{cor:abeliancoarse}), while for $\SU(2)$ that second variation grows like $4^k$ at scale $k$, on the gauge orbit of the vacuum, and the same happens when the direction is transported along a forest, which keeps the Haar measures but leaves the residual gauge freedom of the blocks (Propositions~\ref{prop:tiltgrowth}, \ref{prop:foresttilt}); this locates the estimate that the cube condition still needs. In a box whose boundary field has small plaquettes and either lies in a maximal torus or, for $\SU(2)$, has plaquette angles $\rho_s$ with $l^2\rho_s$ below a fixed constant, the fraction of unit cubes with a large plaquette is small with probability exponentially close to one in the volume, uniformly over such boundary fields (Lemmas~\ref{lem:boxextension} and~\ref{lem:nonabelianfilling} and Proposition~\ref{prop:conddensity}). The second class allows cube sides up to a constant times $(\beta_W/\log\beta_W)^{1/4}$ at the level of Corollary~\ref{cor:defectscales}, which is the scale $a_{k_*}$ up to a constant factor; for larger boxes with general boundary fields this conditional estimate is open (Remark~\ref{rem:boxnonabelian}). On the fibres of the cut-off average, at a background whose plaquettes are at most $\kappa\le10^{-8}$, the gap $\frac1{52}$ of the averages\footnote{By the computation of Remark~\ref{rem:blockgapsharp}, from a certificate of T.~Romanov, this gap is at least $\frac{537}{5000}=0.1074$, within $0.09$ per cent of the best constant on $\Z^4$, a floating-point value. No result of the paper uses it.} persists up to a factor $1+O(\kappa)$, and the Hessian of the action along the fibre is at least $\lambda(\frac1{26}-c\kappa)$, however far the links are from $I$ (Theorem~\ref{thm:smallfibre}). On a chart of the fibre with convex domain the constrained law is log-concave for large $\lambda$, coarea factor included, and satisfies a Brascamp--Lieb inequality with constant $60/\lambda$ (Theorem~\ref{thm:fibrechart}); on a chart whose domain is a product in the coordinates of the links other than the path links, the covariances of local observables decay exponentially at a rate that depends only on $G$ (Theorem~\ref{thm:chartdecay}). Off the chart the fibre has only points with a large plaquette, and under the $\SU(2)$ Wilson measure their mass is small on most fibres (Theorem~\ref{thm:offchart}). Along its private links the cut-off average is a submersion with a constant that depends on nothing (Lemma~\ref{lem:privatesubmersion}), and its law under Haar measure on them has a bounded density, so a Peierls argument inside the fibre bounds that mass on every fibre through a field whose plaquettes near the blocks are small (Proposition~\ref{prop:calmfibre}). On every fibre, whatever the boundary links, the same argument concentrates the law within any $t$ of the least action of the fields whose averaging sites are near $I$ (Proposition~\ref{prop:everyfibre}); where that action is not small the decay is open (Remark~\ref{rem:chartdecayopen}). Whether some block map is coercive for all fields and composes across the remaining scales is open. \begin{figure*}[t] \centering \includegraphics[width=0.9\textwidth]{figures/fig_logic.png} \caption{The logical structure of the paper. Green: theorems at fixed lattice spacing (P1 the gap for every compact simple group, Theorem~\ref{thm:groups}, P2 the plaquette blocks for $\SU(2)$ and $\SU(3)$, P3 the cube condition and the averaged condition BP at strong coupling, Corollary~\ref{cor:bpranges}). Blue: the proved implications (I1)--(I3) of the continuum step, where (I1) takes either (D) or the local form of BP (Theorem~\ref{thm:bplocal}). Red: their inputs, none of which is proved along the continuum tuning. Each arrow joins an open input to the implication that consumes it; no implication feeds its own input.} \label{fig:logic} \end{figure*} \emph{Plan.} Sec.~\ref{sec:linefield} constructs the line field. Sec.~\ref{sec:state} shows that a Dobrushin bound for the gauge-fixed specification gives the full conclusion of Theorem~\ref{thm:su2}. Secs.~\ref{sec:su2}, \ref{sec:plaqblocks}, \ref{sec:curvature}, \ref{sec:su3} and~\ref{sec:su3blocks} prove Theorems~\ref{thm:su2}--\ref{thm:su3blocks}, and Sec.~\ref{sec:limits} measures what is left. Sec.~\ref{sec:bypass} proves (I1)--(I3), Sec.~\ref{sec:inputs} states and measures their open inputs, and Sec.~\ref{sec:blocks} poses the cube condition for block averages, composes the cut-off average across scales at small fields and locates the large fields under the Wilson measure. Sec.~\ref{sec:islands} proves what a labelling of the blocks into good and bad gives for the cube condition, where the natural labels fail, the explicit coupling past which the bad islands of one scale are rarer than their entropy (Theorem~\ref{thm:islandthreshold}), the ceiling below the scale of the cube that every chessboard bound for these islands meets at every coupling (Theorem~\ref{thm:chessceiling}), the fold across which the path-link law of a bad pair jumps (Proposition~\ref{prop:fold}) and the bounded response of its site (Theorem~\ref{thm:site}), the Jacobian condition of its relative chart near the identity (Theorem~\ref{thm:jholds}), the bounded response of the bad side of a projected block average with no hypothesis (Theorem~\ref{thm:projresponse}), the factorisation of two bad islands through one kernel (Theorem~\ref{thm:factorisation}) and the exponential decay of their comparison in the width of the good layer between them (Theorem~\ref{thm:layerbound}, Corollary~\ref{cor:twoislands}), and what the single-scale and the multiscale parts of (D) still need. Sec.~\ref{sec:conclusion} sets out what is proved and what is open. The appendices contain the residue table of the line field, the comparison lemma and its version for blocks, the blocks of the local Hessian inequality, the details of the certified computation, the reconstruction of the transfer matrix with the equivalence of a gap and exponential clustering (Appendix~\ref{app:transfer}), the facts on spectral measures and on the character expansion that the continuum step uses (Appendices~\ref{app:spectral} and~\ref{app:characters}), the Haar measure of $\SU(2)$ (Appendix~\ref{app:haar}), and a complete proof of the sharp quarter for the single-link law of $\SU(2)$ (Appendices~\ref{app:sharp}--\ref{app:bessel}), so that the paper does not depend on our earlier one for any step. \section{The line field}\label{sec:linefield} Write $e_0,\dots,e_3$ for the unit vectors of $\Z^4$ and $(x,\mu)$ for the link from $x$ to $x+e_\mu$. A \emph{direction field} is a map $\sigma:\Z^4\to\{0,1,2,3\}$; its links are $F=\{(x,\sigma(x)):x\in\Z^4\}$. At each site put $\pi_x(\mu)=\sigma(x+e_\mu)$. The plaquette at $x$ in the plane $\{\mu,\nu\}$ has the links $(x,\mu)$, $(x+e_\mu,\nu)$, $(x+e_\nu,\mu)$, $(x,\nu)$, so it contains \begin{equation}\label{eq:count} [\sigma(x)\in\{\mu,\nu\}]+[\pi_x(\mu)=\nu]+[\pi_x(\nu)=\mu] \end{equation} links of $F$. \begin{lemma}\label{lem:cycle} Every plaquette of $\Z^4$ contains exactly one link of $F$ if and only if, at every site $x$, $\pi_x$ fixes $\sigma(x)$ and permutes the other three directions cyclically. \end{lemma} \begin{proof} Fix $x$, let $s=\sigma(x)$ and let $R$ be the set of the other three directions. For a plane $\{s,\nu\}$ the count~\eqref{eq:count} is $1+[\pi_x(s) =\nu]+[\pi_x(\nu)=s]$; it equals one for all $\nu\in R$ exactly when $\pi_x(s)=s$ and $\pi_x(\nu)\ne s$ for $\nu\in R$, that is, when $\pi_x$ fixes $s$ and maps $R$ into $R$. For a plane $\{\mu,\nu\}\subset R$ the count is $[\pi_x(\mu)=\nu]+[\pi_x(\nu)=\mu]$. Draw an arrow $\mu\to\pi_x(\mu)$ for every $\mu\in R$ with $\pi_x(\mu)\in R\setminus\{\mu\}$. Each vertex of $R$ has at most one outgoing arrow, and the condition says that each of the three pairs of $R$ carries exactly one arrow. Three arrows on three vertices with out-degree at most one give out-degree exactly one everywhere, and a tournament on three vertices with all out-degrees one is a directed three-cycle. \end{proof} From now on $\sigma$ is the field of period two given in Table~\ref{tab:sigma} of Appendix~\ref{app:table}, read from $x$ modulo two. The table lists $\pi_x$ at each of the sixteen residue classes; in each row $\pi_x$ fixes $\sigma(x)$ and is a three-cycle on the rest, so every plaquette contains exactly one link of $F$ by Lemma~\ref{lem:cycle}.\footnote{There are exactly eight period-two direction fields with this property, all equivalent to $\sigma$ under the symmetries of the lattice. A field with one fixed link at every site and one in every plaquette exists only in four dimensions: a link lies in $2(d-1)$ plaquettes and a site carries $d(d-1)/2$ of them, and $2(d-1)=d(d-1)/2$ forces $d=4$. The statements of this section are also checked in the Lean certificate of the archive.} \begin{lemma}\label{lem:structure} \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $F$ is a disjoint union of bi-infinite straight lines, one through every site: every site has exactly one outgoing and one incoming link of $F$, and both point in the direction $\sigma(x)$. \item For every box $\Lambda$ (a product of four finite intervals, with the links and plaquettes it contains), $F\cap E(\Lambda)$ is a disjoint union of segments, hence a forest. \item A link $e=(x,\mu)\notin F$ lies in six plaquettes. Four of them contain a link of $F$ that shares an end point with $e$, and these lie in the planes $\{\mu,\sigma(x)\}$ and $\{\mu,\pi_x(\mu)\}$. In the other two, which lie in the remaining plane $\{\mu,\rho\}$, the link of $F$ is opposite to $e$. \item Every link outside $F$ shares a plaquette with exactly twelve other links outside $F$, two in each of its plaquettes. \end{enumerate} \end{lemma} \begin{proof} (i) Since $\pi_x$ fixes $\sigma(x)$, $\sigma(x+e_{\sigma(x)})=\sigma(x)$. Because $\sigma$ has period two, $x-e_\mu\equiv x+e_\mu$, so also $\sigma(x-e_{\sigma(x)})=\sigma(x)$, and the link $(x-e_{\sigma(x)},\sigma(x))$ is in $F$. Conversely, if $(x-e_\nu,\nu)\in F$, then $\sigma(x-e_\nu)=\nu$, and straightness at $x-e_\nu$ gives $\sigma(x)=\nu$. So the incoming link is unique and continues the outgoing one. (ii) A line meets a box in a segment, and distinct lines are disjoint. (iii) The plaquettes of $e$ are, for each $\nu\ne\mu$, the one at $x$ and the one at $x-e_\nu$ in the plane $\{\mu,\nu\}$. A link of $F$ sharing an end point with $e$ is the outgoing or the incoming link of $F$ at $x$ or at $x+e_\mu$. At $x$ both point in the direction $s=\sigma(x)\ne\mu$, and they lie in the plaquettes at $x$ and at $x-e_s$ in the plane $\{\mu,s\}$. At $x+e_\mu$ both point in the direction $t=\pi_x(\mu)$, which lies in $R\setminus\{\mu\}$ because $\pi_x$ is a three-cycle on $R$, and they lie in the plaquettes at $x$ and at $x-e_t$ in the plane $\{\mu,t\}$. These four plaquettes are distinct because $s\ne t$. Each plaquette contains exactly one link of $F$, so in the two plaquettes of the third plane $\{\mu,\rho\}$ the link of $F$ is a link of direction $\mu$ or $\rho$ not touching $e$; the links of direction $\rho$ in such a plaquette touch $e$, so it is the opposite link. (iv) Each of the six plaquettes contains $e$, one link of $F$ and two other links, and two distinct links share at most one plaquette. \end{proof} \begin{figure*}[t] \centering \includegraphics[width=0.9\textwidth]{figures/fig_linefield.png} \caption{(a) The line field in the slice $x_3=0$ of a box of side three. Thick red segments are the links of $F$ in the directions $0,1,2$, with a dot marking the direction of travel; an orange ring marks a site whose line leaves the slice in direction~$3$. Every line is straight (Lemma~\ref{lem:structure}(i)), and the segments that leave the box are its boundary stubs. (b) One plaquette with its fixed link $e_1$. The weights are those of the local inequality of Lemma~\ref{lem:local} for the three free links: $\frac{21}{10}$ on the two links adjacent to $e_1$ and $\frac54$ on the opposite one. The two purple corners are the ones that enter the corner form~\eqref{eq:corner} once $Y_1=0$.} \label{fig:linefield} \end{figure*} \section{Gauge fixing and the free-boundary state}\label{sec:state} Let $G\subset\U(N)$ be compact with Haar measure $dU$, and let $\Lambda$ be a box. The Wilson measure with free boundary is \[ W_\Lambda(dQ)=\frac1{Z_\Lambda}\exp\Bigl(N\beta\sum_{p\in P(\Lambda)}\ReTr Q_p\Bigr)\prod_{e\in E(\Lambda)}dQ_e . \] The measure is invariant under the gauge transformations $Q_{(x,y)}\mapsto g_xQ_{(x,y)}g_y^{-1}$ with $g\in G^\Lambda$. Call the links outside $F$ \emph{active} and write $A$ for their set. For a function $O$ of the links, $O^F$ is $O$ with every link of $F$ set to the identity. \begin{lemma}[Gauge fixing]\label{lem:gaugefix} Let $\mu^F_\Lambda$ be the probability measure on $G^{A\cap E(\Lambda)}$ whose density with respect to Haar measure is proportional to $\exp(N\beta\sum_{p\in P(\Lambda)}\ReTr Q^F_p)$. For every gauge-invariant $O$ on $G^{E(\Lambda)}$, \[ W_\Lambda(O)=\mu^F_\Lambda(O^F). \] \end{lemma} \begin{proof} Each segment of $F\cap E(\Lambda)$ is a path $x_0,x_1,\dots,x_k$ along one direction. Given the variables on $F$, put $g_{x_0}=1$ and $g_{x_{i+1}}= g_{x_i}Q_{(x_i,x_{i+1})}$ on every segment. The transformed field has $Q^g_e =1$ on $F\cap E(\Lambda)$, and for fixed values on $F$ the map $Q_A\mapsto Q^g_A$ is a product of left and right translations, which preserves Haar measure. Gauge invariance of the action and of $O$ and the change of variables give the claim. \end{proof} No leftover holonomy appears because $F\cap E(\Lambda)$ is a forest. The measures $\mu^F_\Lambda$ are finite-volume versions of the specification $\gamma^F$ on $G^A$: for $e\in A$, $\gamma^F_e$ resamples $Q_e$ from the law with density proportional to $\exp(N\beta\sum_{p\ni e}\ReTr Q^F_p)$ given all other active links. No infinite-volume gauge transformation is performed; $\gamma^F$ is a specification in its own right. Fix a metric $\delta$ on $G$ of diameter $D$ that is bi-invariant (for $\SU(2)$ the chordal metric on $S^3$, $D=2$; for $\SU(N)$ the Hilbert--Schmidt distance). The Kantorovich coefficient of $e'$ on $e$ is \[ C_{ee'}=\sup\frac{W_1\bigl(\gamma^F_e(\cdot\mid Q),\gamma^F_e(\cdot\mid Q')\bigr)}{\delta(Q_{e'},Q'_{e'})}, \] the supremum over pairs of configurations that differ only at $e'$. It vanishes unless $e$ and $e'$ share a plaquette. \begin{proposition}\label{prop:state} Suppose that every row and every column of $(C_{ee'})$ sums to at most $\alpha<1$. Then: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\gamma^F$ has exactly one Gibbs measure $\mu_F$; \item for every gauge-invariant local $O$ the limit $\omega(O)=\lim W_\Lambda(O)$ exists along every sequence of boxes exhausting $\Z^4$ and equals $\mu_F(O^F)$; \item $\omega$ is invariant under translations and the hypercubic group; \item if $\beta\ge0$, $\omega$ is reflection positive for reflections in hyperplanes through sites and in hyperplanes bisecting links; \item for gauge-invariant local $O,O'$ with $O$ Lipschitz in each link, $|\Cov_\omega(O,O')|\le C_{O,O'}e^{-\lambda\dist(O,O')}$ for every $\lambda<-\log\alpha$; \item if $\beta\ge0$, the transfer matrix of $\omega$ has spectral gap at least $-\log\alpha$. \end{enumerate} \end{proposition} \begin{proof} (i) Gibbs measures exist by compactness, and the comparison lemma of Appendix~\ref{app:comparison} with $S=\emptyset$ shows that any two coincide. (ii) Extend $\mu^F_\Lambda$ by Haar measure on the active links outside $\Lambda$, and let $S$ be the set of active links with a plaquette outside $P(\Lambda)$; this includes every active link outside $E(\Lambda)$. For an active link $e\notin S$ all six plaquettes of $e$ lie in $P(\Lambda)$, so the conditional law of $Q_e$ under the extended measure is $\gamma^F_e$. Let $T$ be the support of $O^F$ and $\lambda<-\log\alpha$. The comparison lemma with weights $e^{-\lambda\dist(\cdot,T)}$ gives \[ |\mu^F_\Lambda(O^F)-\mu_F(O^F)|\le L\,\frac{\theta D}{1-\theta} \sum_{z\in S}e^{-\lambda\dist(z,T)}, \] with $\theta=\alpha e^\lambda<1$ and $L$ a Lipschitz constant of $O^F$; the sum is at most $C_TR^3e^{-\lambda R}$ when $R$ is the distance from $T$ to the complement of $\Lambda$. For $O$ continuous but not Lipschitz, approximate uniformly. Lemma~\ref{lem:gaugefix} gives $W_\Lambda(O)=\mu^F_\Lambda(O^F)$. (iii) $W_\Lambda(\tau_vO)=W_{\Lambda-v}(O)$, and the boxes $\Lambda-v$ also exhaust $\Z^4$; by (ii) the limit does not depend on the sequence. The line field is a device for computing $W_\Lambda$ and is never shifted. The same argument applies to the hypercubic group. (iv) For $\beta\ge0$ the measure $W_\Lambda$ is reflection positive for every box symmetric under the reflection~\cite{OSei78}, and reflection positivity passes to the limit along such boxes, which is $\omega$ by (ii). (v) $\Cov_\omega(O,O')=\Cov_{\mu_F}(O^F,O'^F)$, and setting the links of $F$ to the identity keeps the Lipschitz property. The bound is Corollary~\ref{cor:decay}. (vi) Let $\vartheta$ be a time reflection and $\tau_t$ a time translation. For gauge-invariant local $O$ that are Lipschitz in each link, (v) gives $|\omega(\vartheta(\bar O)\tau_tO)-|\omega(O)|^2|\le C_Oe^{-\lambda t}$. On the support of $O$ these functions are dense in the gauge-invariant part of $L^2(\omega)$: gauge-invariant functions are functions on the compact quotient of $G^{E}$ by the gauge group, Lipschitz functions for the quotient metric are dense in $L^2$ of every finite measure on it, and they lift to gauge-invariant Lipschitz functions. By Proposition~\ref{prop:gapdecay} the spectrum of the transfer matrix on the complement of the vacuum lies in $[0,e^{-\lambda}]$, for every $\lambda<-\log\alpha$. \end{proof} The proposition replaces the torus limits and the uniqueness of the ungauged Gibbs measure used in~\cite[Thm.~F]{BB26a} by the free-boundary limit, which has all the properties the reconstruction of the transfer matrix needs.\footnote{On a torus the line field closes up into cycles, and the holonomies along them cannot be removed by a gauge transformation; the argument never uses a torus. The state $\omega$ is defined only on gauge-invariant observables, which is all that the Osterwalder--Schrader reconstruction~\cite{OS73,OS75,OSei78} uses.} \section{SU(2) up to one third}\label{sec:su2} \begin{proof}[Proof of Theorem~\ref{thm:su2}] Identify $\SU(2)$ with the unit quaternions and use the chordal metric. For an active link $e$ and a plaquette $p\ni e$, the staple $s_p$ is the product of the other three link variables in the order that makes $\operatorname{Re}(Q_es_p) =\frac12\Tr Q_p$, with links of $F$ equal to one. Since $N\beta\ReTr Q_p= 4\beta\ip{Q_e}{\bar s_p}$ in $\R^4$, the conditional law of $Q_e$ is the von Mises--Fisher law $\nu_b(dx)\propto e^{\ip xb}dx$ on $S^3$ with $b=4\beta \sum_{p\ni e}\bar s_p$. Changing a partner $e'$ by $\delta$ in the chordal metric changes one staple by an isometric image of that displacement, so $b$ moves by $4|\beta|\delta$, and the sharp quarter (Theorem~\ref{q-thm:quarter}), $W_1(\nu_b,\nu_{b'})\le\frac14|b-b'|$, gives $C_{ee'}\le|\beta|$. By Lemma~\ref{lem:structure}(iv) every row and column has twelve nonzero entries, so $\alpha\le12|\beta|=3|\beta_W|$. Proposition~\ref{prop:state} applies for $|\beta_W|<\frac13$. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:ceiling}] For the line field, every staple of an active link contains two active links, so the six staples can be chosen with sum zero while the staple through a given partner $e'$ moves freely. At $b=0$ the derivative of $\nu_b$ in the direction $h$ has Kantorovich norm $\frac14|h|$ by Lemma~\ref{q-lem:rate}(i), so $C_{ee'}=|\beta|$ exactly, and every row sum is $12|\beta|=3|\beta_W|$. Now let $F'$ be any forest and $C'$ the matrix of the statement. Suppose a positive weight $w$ satisfies $C'w\le\alpha w$. For a box $\Lambda_n$ of side $n$ let $C_n$ be the principal submatrix on the active links of $E(\Lambda_n)$; then $C_nw\le\alpha w$ on these links, and the Collatz--Wielandt inequality for nonnegative matrices gives $\rho(C_n)\le\alpha$. Since $C_n$ is symmetric, $\rho(C_n)$ is at least the mean row sum. The box has $4n^4-O(n^3)$ links and $n^4$ sites, and a forest has fewer links than sites, so the number $a$ of active links is at least $3n^4-O(n^3)$. Let $a_p$ be the number of active links of the plaquette $p$; the box has $P=6n^4-O(n^3)$ plaquettes and $\sum_pa_p=6a-O(n^3)$. Two links share at most one plaquette, so the row sums add up to $|\beta|\sum_pa_p(a_p-1)$, and by convexity \[ \frac{|\beta|}a\sum_pa_p(a_p-1)\ge\frac{|\beta|P}a\,\bar a(\bar a-1) =6|\beta|(\bar a-1)-o(1), \] with $\bar a=\sum_pa_p/P=6a/P-o(1)\ge3-o(1)$. Hence $\alpha\ge12|\beta|-o(1)$ for every $n$, and $\alpha\ge12|\beta|=3|\beta_W|$. \end{proof} Without gauge fixing the same count gives eighteen partners and the ceiling $\beta_W=\frac29$ of~\cite[Prop.~13]{BB26a}; the temporal gauge, a family of lines in one direction, leaves fourteen. The line field is optimal among forests: twelve is the mean number of partners forced by the number of free links, and it is attained at every link. \section{SU(2) past one third: plaquette blocks}\label{sec:plaqblocks} Throughout this section $G=\SU(2)$ is the group of unit quaternions with the chordal metric $|x-y|$, $\bar x$ is the conjugate, $\ip xy=\operatorname{Re} (x\bar y)$ is the inner product of $\R^4$, and $dx$ is the normalised Haar measure on $S^3$. Left and right multiplication by unit quaternions and conjugation are isometries, and $\operatorname{Re}(xyz)=\operatorname{Re} (yzx)=\ip y{\bar z\bar x}$. Put \[ Z_0(c)=\int_{S^3}e^{\ip cx}dx=\sum_{j\ge0}\frac{(|c|^2/4)^j}{j!\,(j+1)!}, \qquad c\in\R^4, \] let $\nu_c(dx)\propto e^{\ip cx}dx$ be the von Mises--Fisher law, and write $g=\log Z_0$, so that $\nabla g(c)$ is the mean and $\Hess g(c)$ the covariance of $\nu_c$. \subsection{The blocks} Let $\mathcal B$ be the set of plaquettes at $x$ in the plane $\{0,1\}$ with $x_0\equiv x_1$ and in the plane $\{2,3\}$ with $x_2\equiv x_3$, congruences modulo two (Table~\ref{tab:plaqblocks} of Appendix~\ref{app:plaqblocks} lists them modulo $2\Z^4$, and Fig.~\ref{fig:plaqblocks} draws them). A \emph{block plaquette} is a plaquette of $\mathcal B$, and its \emph{block} is the set of its three active links. \begin{lemma}\label{lem:plaqblocks} The set $\mathcal B$ is invariant under translations by $2\Z^4$ and: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item the blocks partition the active links; \item every plaquette $p\notin\mathcal B$ meets three distinct blocks, one in each of its active links; \item every active link lies in exactly five plaquettes outside $\mathcal B$, so it shares a plaquette outside $\mathcal B$ with exactly ten active links, none of them in its own block; \item if an active link $j$ and a link of a block $B$ lie in a common plaquette outside $\mathcal B$, then every link of $B$ has its midpoint within distance $2$ of the midpoint of $j$. \end{enumerate} \end{lemma} \begin{proof} Invariance under $2\Z^4$ is clear from the congruences. We use one fact throughout: two distinct links lie in at most one common plaquette, since they span it. Every link lies in exactly one plaquette of $\mathcal B$. Indeed, a link $(y,\mu)$ with $\mu\in\{0,1\}$ lies in no plaquette of the plane $\{2,3\}$, and in exactly two of the plane $\{0,1\}$, those at $y$ and at $y-e_\nu$, where $\nu$ is the other element of $\{0,1\}$. The base points $y$ and $y-e_\nu$ differ only in the coordinate $\nu$, so $y_0-y_1$ has opposite parities at the two, and exactly one of them satisfies $x_0\equiv x_1$. The directions $2,3$ are treated in the same way. Every plaquette of $\Z^4$ contains exactly one link of $F$ (Lemma~\ref{lem:cycle}), so a block plaquette contains three active links, and every active link lies in exactly one block. This is (i). (ii) Let $p\notin\mathcal B$ have active links $e_1,e_2,e_3$, and let $q_i$ be the block plaquette of $e_i$. Then $q_i\ne p$, so $q_i\cap p=\{e_i\}$ by the fact above. If $q_i=q_k$ with $i\ne k$, then $e_i,e_k$ lie in the two distinct plaquettes $p$ and $q_i$, which is impossible. So the three blocks are distinct, and each meets $p$ in one link. (iii) An active link $e$ lies in six plaquettes (Lemma~\ref{lem:structure}), and by (i) exactly one of them is in $\mathcal B$. Each of the other five contains two further active links. These ten links are distinct, because a link other than $e$ shares at most one plaquette with $e$, and none lies in the block of $e$, because a link $f$ of that block sharing the plaquette $p\notin\mathcal B$ with $e$ would put $e,f$ in the two distinct plaquettes $p$ and the block plaquette of $e$. (iv) Let $f\in B$ share the plaquette $p\notin\mathcal B$ with $j$. Two links of one unit square have midpoints at distance $2^{-1/2}$ or $1$, so the midpoint of $j$ is within distance $1$ of that of $f$, and the midpoint of $f$ is within distance $1$ of that of every link of its block plaquette. The triangle inequality gives the bound $2$. \end{proof} \begin{figure*}[t] \centering \includegraphics[width=0.82\textwidth]{figures/fig_plaqblocks.png} \caption{Block plaquettes, shaded, in (a) a plane $\{0,1\}$ and (b) a plane $\{2,3\}$: the plaquettes at $x$ with $x_0\equiv x_1$, respectively $x_2\equiv x_3$, modulo two. Red links belong to the line field $F$; a filled dot marks a site whose line runs in the plane, an open dot one whose line leaves it. The shaded squares form a checkerboard, so every link of the plane borders exactly one of them, and every square contains exactly one red link (Lemma~\ref{lem:cycle}); hence each block plaquette carries three active links and the blocks partition the active links (Lemma~\ref{lem:plaqblocks}(i)).} \label{fig:plaqblocks} \end{figure*} Fix a block plaquette $p$ with active links $e_1,e_2,e_3$. Reading the plaquette product cyclically and deleting its fixed link, $\frac12\Tr Q_p= \operatorname{Re}(y_1y_2y_3)$, where $y_i$ is $Q_{e_i}$ or $Q_{e_i}^{-1}$. Every other plaquette containing $e_i$ meets the block only in $e_i$ by Lemma~\ref{lem:plaqblocks}(ii) and contributes $4\beta\ip{y_i}{\bar s}$ to the exponent, where $s$ is the staple of $y_i$ in it. Hence the conditional law of the block given all other active links is \begin{equation}\label{eq:blocklaw} \mu_b(dy)\propto\exp\Bigl(\sum_{i=1}^3\ip{y_i}{b_i}+4\beta\operatorname{Re} (y_1y_2y_3)\Bigr)dy_1\,dy_2\,dy_3, \end{equation} with $b_i$ the sum of $4\beta\bar s$ over the five staples, so that $|b_i|\le20|\beta|$. Inversion is an isometry of $S^3$ and preserves Haar measure, so we may work with $y$ instead of $Q$. On $(S^3)^3$ we use the metric $d(y,y')=\sum_i|y_i-y'_i|$. \subsection{The sensitivity of one block} Let $\lambda_*$ and $\tau$ be a lower and an upper bound for the eigenvalues of $\Hess g(c)$ on the ball $|c|\le24|\beta|$, and $M$ an upper bound there for the norm of the third derivative $D^3g(c)$ as a trilinear form. Put \begin{equation}\label{eq:blockconst} \begin{gathered} w=8\beta^2(\tau-\lambda_*),\qquad s=\tfrac14+(e^w-1)e^w\sqrt{\tau/3},\\ C=4|\beta|sA+\tfrac{320}3|\beta|^3Me^w,\qquad K=\frac{(1+|\beta|)\,s}{1-C}, \end{gathered} \end{equation} assuming $C<1$, where $A$ is an upper bound for $|\nabla g(c)|$ on the ball $|c|\le20|\beta|$. \begin{lemma}\label{lem:tilt} Let $\nu$ be a probability measure and $\lambda=e^R\nu/\nu(e^R)$ with $\operatorname{osc}R\le\omega$. For bounded $f,h$, \[ \Cov_\lambda(f,h)\le\Cov_\nu(f,h)+(e^\omega-1)e^\omega\,\mathrm{sd}_\nu(f) \,\mathrm{sd}_\nu(h). \] \end{lemma} \begin{proof} We may assume $0\le R\le\omega$. The density $\rho=e^R/\nu(e^R)$ lies in $[e^{-\omega},e^\omega]$, so $\delta=\|\rho-1\|_\infty\le e^\omega-1$. With $\tilde f=f-\nu f$ and $\tilde h=h-\nu h$, \[ \begin{split} \Cov_\lambda(f,h)-\Cov_\nu(f,h)={}&\nu\bigl((\rho-1)\tilde f\tilde h\bigr)\\ &-\nu\bigl((\rho-1)\tilde f\bigr)\,\nu\bigl((\rho-1)\tilde h\bigr), \end{split} \] and the right side is at most $(\delta+\delta^2)\,\mathrm{sd}_\nu(f)\, \mathrm{sd}_\nu(h)$ by the Cauchy--Schwarz inequality. \end{proof} \begin{lemma}\label{lem:onesite} Let $\beta\ge0$, $a,c_0\in\R^4$ with $|a|,|c_0|\le20\beta$, and \[ \kappa(dx)\propto\exp\bigl(\ip xa+g(c_0+4\beta\bar x)\bigr)dx \quad\text{on }S^3 . \] For every $1$-Lipschitz $f$ and every $h\in\R^4$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\Cov_\kappa(f,\ip xh)\le s|h|$; \item if $\kappa'$ is the law with $c_0'$ in place of $c_0$ and $|c_0'|=|c_0|=\rho$, then $|\kappa(f)-\kappa'(f)|\le(8\beta s\varphi'(\rho^2) +\frac{16}3\beta^2Me^w)|c_0-c_0'|$. \end{enumerate} \end{lemma} \begin{proof} (i) Taylor's formula at $c_0$ gives $g(c_0+4\beta u)=g(c_0)+4\beta\ip{\nabla g(c_0)}u+R(u)$ with $R(u)=8\beta^2\Hess g(\xi)(u,u)$ for some $\xi$ on the segment, $|\xi|\le24\beta$. For $u=\bar x$ of norm one, $R(\bar x)\in[8\beta^2 \lambda_*,8\beta^2\tau]$, so $\operatorname{osc}R\le w$. Since $\ip{\nabla g(c_0)}{\bar x}=\ip x{\overline{\nabla g(c_0)}}$, $\kappa$ is the tilt $e^{R}\nu/\nu(e^R)$ of $\nu=\nu_c$ with $c=a+4\beta\overline{\nabla g(c_0)}$. The sharp quarter, Theorem~\ref{q-thm:quarter}, gives $\Cov_\nu(f,\ip xh)=\frac{d}{dt} \nu_{c+th}(f)\le\frac14|h|$. A function that is $1$-Lipschitz for the chordal metric is $1$-Lipschitz for the geodesic one, so $|\nabla f|\le1$ almost everywhere and the Poincar\'e inequality of Lemma~\ref{q-lem:vmf} gives $\mathrm{sd}_\nu(f)\le1/\sqrt3$; and $\mathrm{sd}_\nu\ip xh\le\sqrt\tau|h|$ because $|c|\le|a|+4\beta\le24\beta$. Lemma~\ref{lem:tilt} gives (i). The same argument applies to every law $e^{R'}\nu_{c'}/\nu_{c'}(e^{R'})$ with $|c'|\le24\beta$ and $\operatorname{osc}R'\le w$. (ii) We interpolate the exponents rather than the field. For $t\in[0,1]$ let $\kappa_t\propto\exp\bigl(\ip xa+(1-t)g(c_0+4\beta\bar x)+tg(c'_0+4\beta \bar x)\bigr)dx$, so that $\kappa_0=\kappa$, $\kappa_1=\kappa'$ and $\frac d{dt}\kappa_t(f)=\Cov_{\kappa_t}(f,D)$ with $D(x)=g(c'_0+4\beta\bar x)- g(c_0+4\beta\bar x)$. Write $g(c+4\beta u)=g(c)+4\beta\ip{\nabla g(c)}u+R_c(u)$ as in (i), for $c\in\{c_0,c'_0\}$. Then $\kappa_t$ is the tilt of $\nu_{c_t}$, $c_t=a+4\beta\,\overline{(1-t)\nabla g(c_0)+t\nabla g(c'_0)}$, by $R_t=(1-t)R_{c_0}+tR_{c'_0}$, with $\operatorname{osc}R_t\le w$ and $|c_t|\le 24\beta$ because $|\nabla g|\le1$; so (i) holds for $\kappa_t$, and the density of $\kappa_t$ with respect to $\nu_{c_t}$ is at most $e^w$. Up to a constant, $D(x)=\ip xh+N(\bar x)$ with $h=4\beta\,\overline{\nabla g(c'_0)-\nabla g(c_0)}$ and $N=R_{c'_0}-R_{c_0}$. Since $\nabla g(c)=2\varphi'(|c|^2)c$ and $|c_0|=|c'_0|=\rho$, $|h|=8\beta\varphi'(\rho^2)|c_0-c'_0|$, and by (i) the linear part contributes at most $s|h|$. For the rest, $\nabla N(u)=16\beta^2 \int_0^1\bigl(\Hess g(c'_0+4\beta\sigma u)-\Hess g(c_0+4\beta\sigma u)\bigr)u\, d\sigma$, both arguments in the convex ball $|c|\le24\beta$, so $N$ is $L$-Lipschitz on the unit ball with $L=16\beta^2M|c_0-c'_0|$. With $\nu= \nu_{c_t}$ and the Poincar\'e inequality of Lemma~\ref{q-lem:vmf} again, $\Var_{\kappa_t}(N)\le\kappa_t((N-\nu N)^2)\le e^w\Var_\nu(N)\le e^wL^2/3$, and in the same way $\Var_{\kappa_t}(f)\le e^w/3$. So $\Cov_{\kappa_t}(f,N)\le e^wL/3$. Integrating over $t$ gives (ii). \end{proof} \begin{lemma}\label{lem:blockK} Let $C<1$. If $b$ and $b'$ satisfy $|b_i|,|b'_i|\le20|\beta|$ and differ only in one coordinate $b_i$, then \[ W_1(\mu_b,\mu_{b'})\le K\,|b_i-b'_i| \] in the metric $d$. \end{lemma} \begin{proof} The map $y_1\mapsto-y_1$ turns $\mu_b$ for $\beta$ into the law for $-\beta$ with $b_1$ replaced by $-b_1$ and is an isometry, so we may take $\beta\ge0$; the cyclic symmetry of $\operatorname{Re}(y_1y_2y_3)$ lets us take $i=1$. Let $f$ be $1$-Lipschitz for $d$ and $\varepsilon=|b_1-b'_1|$. \emph{The middle link.} Given $y_1,y_3$, the law of $y_2$ is $\nu_c$ with $c= b_2+4\beta\,\overline{y_3y_1}$, which does not involve $b_1$. Changing $y_1$ by $\eta$ moves $c$ by $4\beta\eta$, and the sharp quarter moves $\nu_c$ by at most $\beta\eta$ in $W_1$. Hence $F(y_1,y_3)=\E[f\mid y_1,y_3]$ is $(1+\beta)$-Lipschitz in each variable, and $\mu_b(f)=\mu^{13}_b(F)$, where $\mu^{13}_b$ is the law of $(y_1,y_3)$: \[ \mu^{13}_b\propto e^{\ip{y_1}{b_1}+\ip{y_3}{b_3}}Z_0\bigl(b_2+4\beta\, \overline{y_3y_1}\bigr). \] Since $|b_2+4\beta\overline{y_3y_1}|=|b_2y_3+4\beta\bar y_1|=|y_1b_2+ 4\beta\bar y_3|$, its conditional laws are \[ \begin{split} \kappa_1(dy_1\mid y_3)&\propto e^{\ip{y_1}{b_1}+g(b_2y_3+4\beta\bar y_1)}dy_1,\\ \kappa_3(dy_3\mid y_1)&\propto e^{\ip{y_3}{b_3}+g(y_1b_2+4\beta\bar y_3)}dy_3, \end{split} \] both of the form of Lemma~\ref{lem:onesite}, with $|c_0|=|b_2|$. \emph{Two sites.} Integrating Lemma~\ref{lem:onesite}(i) along the segment from $b_1$ to $b'_1$ gives $W_1(\kappa_1,\kappa'_1)\le s\varepsilon$ for the same $y_3$. Moving $y_3$ to $y'_3$ replaces $c_0=b_2y_3$ by $c'_0=b_2y'_3$, of the same norm $\rho=|b_2|\le20\beta$ and with $|c_0-c'_0|=\rho|y_3-y'_3|$. Since $\rho\cdot2\varphi'(\rho^2)=|\nabla g(c_0)|\le A$, Lemma~\ref{lem:onesite}(ii) gives $W_1(\kappa_1(\cdot\mid y_3),\kappa_1( \cdot\mid y'_3))\le(4\beta sA+20\beta\cdot\frac{16}3\beta^2Me^w)|y_3-y'_3|= C|y_3-y'_3|$, and the same for $\kappa_3$, which does not depend on $b_1$. Let $\pi$ be a coupling of $\mu^{13}_b$ and $\mu^{13}_{b'}$ minimising $d_1+d_3$, $d_k=\E_\pi|y_k-y'_k|$. Replacing $(y_1,y'_1)$ by an optimal coupling of $\kappa_1(\cdot\mid y_3)$ and $\kappa'_1(\cdot\mid y'_3)$ keeps a coupling, so minimality gives $d_1\le s\varepsilon+Cd_3$; in the same way $d_3\le Cd_1$. Hence $d_1\le s\varepsilon/(1-C^2)$ and \[ |\mu_b(f)-\mu_{b'}(f)|\le(1+\beta)(d_1+d_3)\le\frac{(1+\beta)s}{1-C} \,\varepsilon=K\varepsilon .\qedhere \] \end{proof} \subsection{The constants} \begin{lemma}\label{lem:vmfconst} Write $g(c)=\varphi(|c|^2)$ and $r=|c|$. For $c\ne0$ the eigenvalues of $\Hess g(c)$ are $2\varphi'(r^2)$, three times, and $2\varphi'(r^2)+4r^2 \varphi''(r^2)$, and the norm of $D^3g(c)$ is \[ \max\Bigl(|P+\Gamma|,\tfrac23|P|\Bigr),\qquad P=12r\varphi''(r^2),\quad \Gamma=8r^3\varphi'''(r^2). \] On the ball $r\le2.088$ every eigenvalue of $\Hess g$ lies between $0.15435$ and $\frac14$, and the norm of $D^3g$ is at most $0.1071$; on the ball $r\le1.74$, $|\nabla g|\le0.38944$. \end{lemma} \begin{proof} Differentiating $\varphi(|c|^2)$ gives $D^2g(v,v)=2\varphi'|v|^2+4\varphi'' \ip cv^2$ and $D^3g(v,v,v)=12\varphi''\ip cv|v|^2+8\varphi'''\ip cv^3$. For $|v|=1$ and $t=\ip{c/r}v\in[-1,1]$ the cubic form is $Pt+\Gamma t^3$, whose largest absolute value is attained at $t=\pm1$ or at an interior critical point, where it equals $\frac23|P||t|\le\frac23|P|$; by Banach's theorem~\cite{Ban38} the supremum of a symmetric trilinear form on the diagonal of the unit sphere is its norm. The numbers follow from the Bessel ratio $A=I_2/I_1$ of Lemma~\ref{q-lem:moments}. Since $Z_0(c)=2I_1(r)/r$, the radial derivative of $g$ is $A(r)$, so $2r\varphi'(r^2)=A(r)$ and $|\nabla g(c)|=A(r)$. Differentiating $2\varphi'(r^2)=A/r$ in $r$ gives $4r\varphi''(r^2)= (rA'-A)/r^2$, so the eigenvalues are $A/r$, three times, and $A'$, and $P=3(rA'-A)/r^2$; differentiating once more gives $4\varphi''+8r^2\varphi''' =A''/r-2(rA'-A)/r^3$, so $\Gamma=A''-P$ and $P+\Gamma=A''$. Write $\kappa=r$, $u=1+\sqrt{\kappa^2+9}$ and $v=\frac32+\sqrt{\kappa^2+25/4}$, so that $\kappa^2=(u-4)(u+2)=(v-4)(v+1)$. By Lemma~\ref{q-lem:moments}(ii) and~(iv), $\kappa/v\le A\le\kappa/u$, and $A'=1-A^2-3A/\kappa$. \emph{Eigenvalues.} By Lemma~\ref{q-lem:moments}(iii), $A'\le A/\kappa\le \frac14$; at $c=0$, $\Hess g=\frac14I$. The function $1-a^2-3a/\kappa$ decreases in $a\ge0$, so $A'\ge1-\kappa^2/u^2-3/u=(8-u)/u^2$, and $(8-u)/u^2$ decreases in $u\in[4,8]$, hence in $\kappa$. At $\kappa=2.088$, $\kappa^2+9=13.359744<3.6551^2$, so $u<4.6551$ and $(8-u)/u^2>3.3449/ 21.67>0.15435$. \emph{Gradient.} $A(r)\le r/(1+\sqrt{r^2+9})$, which increases in $r$. At $r=1.74$, $r^2+9=12.0276>3.468^2$, so $A\le1.74/4.468<0.38944$. \emph{Third derivative.} First, $\frac23|P|=2|A-\kappa A'|/\kappa^2$ and $A-\kappa A'=4A-\kappa+\kappa A^2$, which increases in $A$. At $A=\kappa/v$ it equals $\kappa(v-4)/v^2\ge0$, and at $A=\kappa/u$ it equals $2\kappa(u-4)/u^2$; since $u-4=\kappa^2/(\sqrt{\kappa^2+9}+3)\le\kappa^2/6$ and $u\ge4$, $0\le\frac23|P|\le4(u-4)/(\kappa u^2)\le\kappa/24\le0.087$. Second, differentiating the Riccati equation gives $-A''=H(A,\kappa)$ with \[ H(a,\kappa)=\Bigl(1-a^2-\frac{3a}\kappa\Bigr)\Bigl(2a+\frac3\kappa\Bigr)- \frac{3a}{\kappa^2}, \] and $\partial_aH=-(2a+3/\kappa)^2+2(1-a^2-3a/\kappa)-3/\kappa^2\le2-12/ \kappa^2<0$ for $a\ge0$ and $\kappa^2<6$. Hence $H(\kappa/u,\kappa)\le-A'' \le H(\kappa/v,\kappa)$. Substituting $\kappa^2=(u-4)(u+2)$ and $\kappa^2= (v-4)(v+1)$, \begin{gather*} H(\kappa/u,\kappa)=\frac{(u-4)(32+6u-2u^2)}{\kappa u^3}\ge0,\\ H(\kappa/v,\kappa)=\frac{(5v+8)(v-4)}{\kappa v^3}=\frac{\kappa(5v+8)} {(v+1)v^3}, \end{gather*} the first because $4\le u<5.7$ for $\kappa\le2.088$, the last by $v-4=\kappa^2/(v+1)$. Call the last expression $G(\kappa)$. As a function of $v$, $\log G=\frac12\log(v-4)+\log(5v+8)-\frac12\log(v+1)-3\log v$, whose derivative is half of \[ \Bigl(\frac1{v-4}-\frac6v\Bigr)+\Bigl(\frac{10}{5v+8}-\frac1{v+1}\Bigr). \] Both brackets decrease on $(4,6)$. At $\kappa=2.088$ one has $\kappa^2+ \frac{25}4=10.609744$, which lies between $3.2572^2$ and $3.2573^2$, so $v\in[4.7572,4.7573]$ and the brackets are at least $0.059$ and $0.140$. So $G$ increases on $(0,2.088]$, and \begin{multline*} 0\le-A''\le G(2.088)\le\frac{2.088\,(5\cdot4.7573+8)}{5.7572\cdot4.7572^3}\\ \le\frac{66.371}{619.81}<0.1071 . \end{multline*} The norm of $D^3g$ is therefore at most $\max(0.1071,0.087)=0.1071$ on the ball $r\le2.088$. \end{proof} \subsection{The proof of Theorem~\ref{thm:blocks}} Let $\gamma^{\mathcal B}$ be the block specification: for a block $B$, $\gamma^{\mathcal B}_B$ resamples the three links of $B$ from~\eqref{eq:blocklaw}. It comes from the same interaction as $\gamma^F$, so the two specifications have the same Gibbs measures. Define the block coefficients \[ C_{Bj}=\sup\frac{W_1\bigl(\gamma^{\mathcal B}_B(\cdot\mid Q),\gamma^{\mathcal B}_B (\cdot\mid Q')\bigr)}{|Q_j-Q'_j|} \] for an active link $j\notin B$, the supremum over pairs of configurations that differ only at $j$, with $W_1$ for the metric $d$. \begin{proposition}\label{prop:blockstate} Suppose that $\sum_BC_{Bj}\le\alpha<1$ for every active link $j$. Then the conclusions (i)--(vi) of Proposition~\ref{prop:state} hold with $-\frac12\log\alpha$ in place of $-\log\alpha$. \end{proposition} \begin{proof} The proof of Proposition~\ref{prop:state} goes through with Lemma~\ref{lem:blockcomparison} of Appendix~\ref{app:comparison} in place of Lemma~\ref{lem:comparison}, and with the set $S$ of part~(ii), and the support of $H$ in Corollary~\ref{cor:decay}, enlarged to the union of the blocks that meet it. The enlargement multiplies the size of these sets by at most three and moves them by at most one, which changes only the constants. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:blocks}] Let $|\beta|\le0.348/4=0.087$ and take the constants of Lemma~\ref{lem:vmfconst}, whose balls $r\le2.088$ and $r\le1.74$ contain $|c|\le24|\beta|$ and $|c|\le20|\beta|$: $\lambda_*=0.15435$, $\tau=\frac14$, $M=0.1071$ and $A=0.38944$. With these fixed constants every quantity in~\eqref{eq:blockconst} is increasing in $|\beta|$, and so is \begin{equation}\label{eq:alphablock} \alpha(\beta_W)=40|\beta|K . \end{equation} Changing an active link $j$ by $\eta$ changes one staple of each of its ten partners outside its block by an isometric image of the displacement, so the field $b_i$ of each such partner $i$ moves by at most $4|\beta|\eta$, and Lemma~\ref{lem:blockK}, applied once for each of these ten fields, gives $\sum_BC_{Bj}\le40|\beta|K$. At $|\beta|=0.087$, $8\beta^2=0.060552$ and $\tau-\lambda_*=0.09565$, so $w\le0.0057918$. Since $e^w\le1+w+w^2$ for $0\le w\le1$ and $\sqrt{1/12}\le0.288676$, $e^w\le1.0058254$ and $s\le\frac14+ 0.0058594\cdot0.288676\le0.2516915$. Then \begin{multline*} C\le0.348\cdot0.2516915\cdot0.38944\\ +\tfrac{320}3\,0.087^3\cdot0.1071\cdot1.0058254\\ \le0.0341107+0.0075670\le0.0416777, \end{multline*} $K\le1.087\cdot0.2516915/0.9583223\le0.285490$ and $\alpha\le3.48\cdot 0.285490\le0.99351$. The same evaluation at $|\beta|=0.345/4$ gives $w\le0.0056924$, $s\le0.2516621$, $C\le0.041185$, $K\le0.285110$ and $\alpha\le0.98363$, and at $|\beta|=\frac1{12}$ it gives $w\le0.0053139$, $s\le0.2515504$, $C\le0.0393011$, $K\le0.2836611$ and $\alpha\le0.94554$. Proposition~\ref{prop:blockstate} applies. At $\beta_W=\frac13$, $-\frac12\log\alpha\ge\frac12(y+\frac{y^2}2)$ with $y=1-0.9456=0.0544$, which is at least $0.0279$. \end{proof} \begin{remark}\label{rem:blockgap} Most of the range is kept. A second bound for the same sensitivity, which keeps the law of the middle link instead of linearising it, evaluated in floating point on a quadrature grid and maximised over the fields $b$ by a random search followed by the Nelder--Mead method, gives $0.2783$ at $\beta_W=0.34$, and $40|\beta|$ times it crosses one near $\beta_W=0.357$; this evaluation is an illustration and no result uses it. No bound of this form passes $\beta_W=0.3667$: at $b=0$ the function $f=\ip{y_1}e+ \operatorname{Re}(y_2y_3e)$ is $1$-Lipschitz for $d$ and has $\Cov_{\mu_b}(f, \ip{y_1}e)=\frac14(1+I_2(\beta_W)/I_1(\beta_W))$ exactly (Proposition~\ref{prop:blockresponse}), which is at least $0.2718$ at $\beta_W=0.351$, so the sensitivity is at least this number and $40|\beta|$ times it exceeds one for $\beta_W>0.3667$. Past this point the gain from integrating out one link is spent, and larger blocks are needed, such as the free links of a cube of the line field, whose sensitivity no longer reduces to two sites. \end{remark} \section{Curvature in the gauge}\label{sec:curvature} In this section $G\subset\U(N)$ is a closed connected subgroup with Lie algebra $\g$ and the inner product $\ip XY=-\Tr(XY)$; the bound of Theorem~\ref{thm:curv} is proved first for $\U(N)$. A tangent vector at a field $Q$ is $v=(X_eQ_e)$, $|v|^2=\sum_e|X_e|^2$. For the plaquette at $x$ in the plane $\mu<\nu$ number its links $e_1=(x,\mu)$, $e_2=(x+e_\mu,\nu)$, $e_3=(x+e_\nu,\mu)$, $e_4=(x,\nu)$, transport their vectors to $x$, \begin{equation}\label{eq:transport} Y_1=X_{e_1},\ \ Y_2=Q_{e_1}X_{e_2}Q_{e_1}^*,\ \ Y_3=Q_{e_4}X_{e_3}Q_{e_4}^*,\ \ Y_4=X_{e_4}, \end{equation} and write $W=Q_p$. The second derivative of $\ReTr Q_p$ along the geodesic $Q_e(t)=e^{tX_e}Q_e$ is \begin{equation}\label{eq:fp} \begin{split} f_p={}&\sum_{i=1}^4\ReTr(Y_i^2W)+2\ReTr(Y_1Y_2W)+2\ReTr(Y_3Y_4W)\\ &-2\sum_{i\in\{1,2\},\,j\in\{3,4\}}\ReTr(Y_iWY_j), \end{split} \end{equation} so that $\Hess S_0(v,v)=\sum_pf_p$. To see this, write $Q_p=Q_{e_1}Q_{e_2}Q_{e_3}^*Q_{e_4}^*$. Since $Q_{e_1}e^{tX_{e_2}}=e^{tY_2}Q_{e_1}$ and $e^{-tX_{e_3}}Q_{e_4}^*=Q_{e_4}^* e^{-tY_3}$, the plaquette variable along the geodesic is $Q_p(t)=e^{tY_1}e^{tY_2}We^{-tY_3}e^{-tY_4}$. Differentiating twice at $t=0$ gives the four squares, the two products of neighbouring factors and the four products across $W$, and the squares and the product $WY_3Y_4$ are moved cyclically under the trace. The corner form of $p$ is \begin{equation}\label{eq:corner} c_p=2\ip{Y_1}{Y_4}-2\ip{Y_1}{Y_2}+2\ip{Y_2}{WY_3W^*}-2\ip{Y_3}{Y_4}. \end{equation} It collects the inner products of the transported vectors of two links that meet at a right angle at a corner of $p$. \begin{lemma}\label{lem:identity} On $\Z^4$, or on a box with free boundary, for every field $Q$ and every square-summable $X$, \[ \begin{split} &\sum_y\Bigl[|D_yX|^2+\sum_\mu\bigl|X_{y,\mu}+Q^*_{y-e_\mu,\mu}X_{y-e_\mu,\mu} Q_{y-e_\mu,\mu}\bigr|^2\Bigr]\\ &\qquad=4|X|^2+\sum_pc_p , \end{split} \] where $D_yX=\sum_\mu[X_{y,\mu}-Q^*_{y-e_\mu,\mu}X_{y-e_\mu,\mu}Q_{y-e_\mu,\mu}]$ is the covariant divergence and terms with a link outside the box are omitted. \end{lemma} \begin{proof} At a site $y$ write $o_\mu$ for the vector of the outgoing link in direction $\mu$ and $i_\mu$ for the transported vector of the incoming one, so that $D_yX=\sum_\mu(o_\mu-i_\mu)$. Every link is outgoing at one end and incoming at the other, so the squares $|o_\mu|^2+|i_\mu|^2$, which appear once in $|D_yX|^2$ and once in $|o_\mu+i_\mu|^2$, add up to $4|X|^2$. The collinear cross term $-2\ip{o_\mu}{i_\mu}$ of $|D_yX|^2$ cancels the term $+2\ip{o_\mu}{i_\mu}$ of $|o_\mu+i_\mu|^2$. The remaining cross terms of $|D_yX|^2$ pair two perpendicular ends at $y$, and each such pair is a corner of exactly one plaquette, with the sign and the transport of~\eqref{eq:corner} (this is the proof of~\cite[Lemma~1]{BB26a}). In a box two perpendicular ends at $y$ span a plaquette of the box, because a box is a product of intervals. \end{proof} \begin{lemma}\label{lem:local} Let $k\in\{1,2,3,4\}$, let $k\pm1$ \textup{(}mod $4$\textup{)} be the positions adjacent to $k$ and $k+2$ the opposite one. For every $N$, every $W\in\U(N)$ and all $Y_1,\dots,Y_4\in\mathfrak u(N)$ with $Y_k=0$, \[ \pm f_p+\tfrac9{10}c_p\le\tfrac{21}{10}\bigl(|Y_{k-1}|^2+|Y_{k+1}|^2\bigr) +\tfrac54|Y_{k+2}|^2 . \] \end{lemma} \begin{proof} Replacing $W$ by $-W$ changes the sign of $f_p$ and fixes $c_p$, so the sign $+$ suffices. Conjugation by a unitary matrix preserves both sides, so we may take $W=\operatorname{diag}(w_a)$. The form $\mathcal F=\mathrm{RHS}-f_p- \frac9{10}c_p$ then splits into a block for each diagonal position $a$, in the real variables $y_i=-i(Y_i)_{aa}$, and a block for each pair $a\ne b$, in the complex variables $z_i=(Y_i)_{ab}$, since $(Y_i)_{ba}=-\bar z_i$. Appendix~\ref{app:blocks} shows that every block is positive definite. \end{proof} \begin{proof}[Proof of the bound in Theorem~\ref{thm:curv}] Every plaquette has exactly one link of $F$, and $X$ vanishes on it. By Lemma~\ref{lem:structure}(iii) each active link is adjacent to the fixed link in four of its plaquettes and opposite to it in two. Summing Lemma~\ref{lem:local} over the plaquettes, \[ \sum_p\bigl(\pm f_p+\tfrac9{10}c_p\bigr)\le\bigl(4\cdot\tfrac{21}{10}+2\cdot \tfrac54\bigr)|X|^2=\tfrac{109}{10}|X|^2 . \] By Lemma~\ref{lem:identity}, $\frac9{10}\sum_pc_p=\frac9{10}\Sigma-\frac{36}{10} |X|^2$, where $\Sigma\ge0$ is the left side of the lemma. Hence \[ \pm\Hess S_0(v,v)+\tfrac9{10}\Sigma\le\bigl(\tfrac{109}{10}+\tfrac{36}{10}\bigr) |X|^2=\tfrac{29}2|X|^2 . \] In a box, links near the boundary lie in fewer plaquettes and receive less. \end{proof} The factor $\frac9{10}$ is chosen so that the corners are paid by the divergence identity at the price $\frac{36}{10}$ while the local blocks stay positive; Lemma~\ref{lem:identity} is the divergence identity of~\cite[Lemma~1]{BB26a} with the collinear term written as a square, so nothing is lost by Cauchy--Schwarz. The two lemmas concern vectors in $\mathfrak u(N)$ and fields in $\U(N)$, so the bound holds without change on every closed subgroup $G\subset\U(N)$, the vectors $X_e$ ranging over its Lie algebra $\g$. It is also the Hessian of $S_0$ on $G^E$: the metric is bi-invariant, the geodesics of $G$ through $Q_e$ are the curves $e^{tX}Q_e$ with $X\in\g$, which are geodesics of $\U(N)$, and the Hessian is read off from second derivatives along geodesics. \begin{lemma}\label{lem:ricci} Let $G\subset\U(N)$ be closed and connected with simple Lie algebra $\g$, let $\mathfrak t\subset\g$ be a maximal abelian subalgebra, and write the eigenvalues of $\operatorname{ad}H$ on $\g\otimes\C$ and of $H$ on $\C^N$, for $H\in\mathfrak t$, as $i\alpha(H)$ and $i\mu(H)$, with $\alpha$ running over the roots and $\mu$ over the weights of $\C^N$ counted with multiplicity. Then $\Ric_G=\kappa_G\ip\cdot\cdot$ with \[ \kappa_G=\frac{\sum_\alpha\alpha(H)^2}{4\sum_\mu\mu(H)^2}\qquad(0\ne H\in\mathfrak t). \] \end{lemma} \begin{proof} For a bi-invariant metric the curvature is $R(X,Y)Z=-\frac14[[X,Y],Z]$ on left-invariant fields~\cite[Sect.~7]{Mil76}, so for an orthonormal basis $(E_i)$ of $\g$, \[ \begin{split} \Ric(X,X)&=\sum_i\ip{R(E_i,X)X}{E_i}\\ &=\tfrac14\sum_i|[X,E_i]|^2=-\tfrac14\Tr(\operatorname{ad}X)^2 , \end{split} \] since $\operatorname{ad}X$ is skew. The Killing form $B(X,Y)=\Tr(\operatorname{ad}X \operatorname{ad}Y)$ and $\ip\cdot\cdot$ are invariant under the adjoint action, and $\g$ is simple, so $B=-4\kappa\ip\cdot\cdot$ for a constant $\kappa$ and $\Ric=\kappa\ip\cdot\cdot$. On $H\in\mathfrak t$ the two forms are $B(H,H)=-\sum_\alpha\alpha(H)^2$ and $\ip HH=-\Tr H^2=\sum_\mu\mu(H)^2$. \end{proof} Table~\ref{tab:groups} evaluates the ratio of Lemma~\ref{lem:ricci} for every compact simple Lie algebra and a faithful representation of least dimension, on the root systems of~\cite[Sect.~12.1]{Hum72}; the weights of the representations of dimensions $27$ and $56$ of $E_6$ and $E_7$ are the roots of $E_8$ with prescribed inner products with a subsystem $A_2$, respectively $A_1$, projected to its orthogonal complement, by the decompositions $248=(78,1)+(27,3)+(\overline{27},\bar3)+(1,8)$ and $248=(133,1)+(56,2)+(1,3)$~\cite{Sla81}. Taking for $H$ the coroot $\theta^\vee$ of the highest root gives $\sum_\alpha\alpha(\theta^\vee)^2=4h^\vee$ and $\sum_\mu\mu(\theta^\vee)^2=2\ell_\rho$, so $\kappa_G=h^\vee/(2\ell_\rho)$, with the Dynkin index normalised to one on the defining representation of $\SU(N)$. The ratio depends on $\g$ and $\rho$ only, so it is the same for every compact group with Lie algebra $\g$, simply connected or not. \begin{lemma}\label{lem:mixed} Let $e\ne e'$ be two links of a plaquette $p$, and write \[ \Hess\ReTr Q_p(v,v)=\sum_{e,e'\in p}H^p_{ee'}(X_e,X_{e'}) \] with bilinear forms satisfying $H^p_{ee'}(a,b)=H^p_{e'e}(b,a)$. Then $|H^p_{ee'}(a,b)|\le|a|\,|b|$ for all $a,b\in\mathfrak u(N)$ and every field. \end{lemma} \begin{proof} By~\eqref{eq:fp} the terms of $f_p$ that mix two positions $i\ne j$ are $2\ReTr(Y_1Y_2W)$, $2\ReTr(Y_3Y_4W)$ and $-2\ReTr(Y_iWY_j)$, so $H^p_{ee'}$ is $\pm\ReTr(abW)$ or $\pm\ReTr(aWb)$ after an isometric transport of $a$ and $b$, and $|\Tr(abW)|\le\|a\|_{\mathrm{HS}}\|bW\|_{\mathrm{HS}}=|a|\,|b|$. \end{proof} \begin{corollary}\label{cor:be} Let $G\subset\U(N)$ be closed and connected with semisimple Lie algebra and $N|\beta|<\frac2{29}\kappa_G$. For every box the measure $\mu^F_\Lambda$ satisfies the Bakry--\'Emery condition with $K=\kappa_G-\frac{29}2N|\beta|>0$, uniformly in the box, and hence a logarithmic Sobolev inequality with constant $2/K$ and a Poincar\'e inequality with constant $1/K$. \end{corollary} \begin{proof} The set $\{Q_F=1\}$ is a product factor, so its Ricci curvature is at least $\kappa_G$ on every link, and the Hessian of the restricted action is the restriction of $N\beta\Hess S_0$ to $X_F=0$. Apply the bound and~\cite[Cor.~5.7.2]{BGL14}. \end{proof} \subsection{Decay with an explicit rate} Let $A'$ be a finite set of active links and $P'$ a finite set of plaquettes whose active links lie in $A'$. An action $S=\sum_{p\in P'}c_p\ReTr Q^F_p$ with $0\le c_p\le1$ is \emph{admissible} if $|\Hess S(v,v)|\le\frac{29}2|v|^2$ for all $v$ tangent to $G^{A'}$. Two links lie in at most one common plaquette, so by Lemma~\ref{lem:mixed} and Lemma~\ref{lem:structure}(iv) the mixed blocks of its Hessian satisfy $|\Hess S_{ee'}(a,b)|\le|a|\,|b|$, and each link has at most twelve partners. The action $S_\Lambda=\sum_{p\in P(\Lambda)}\ReTr Q^F_p$ of a box is admissible by the bound proved above, and so is every convex combination of two admissible actions. Let $\mu$ be the probability measure on $G^{A'}$ with density proportional to $e^{N\beta S}$ with respect to Haar measure, let $L=\Delta+N\beta\ip{\nabla S}{\nabla\cdot}$ be its Langevin generator, which is self-adjoint in $L^2(\mu)$, and $P_t=e^{tL}$. Write $\nabla_e$ for the gradient in the factor of the link $e$, and for weights $w:A'\to(0,\infty)$ put $\Gamma^w(f)=\sum_ew_e|\nabla_ef|^2$; $\Gamma$ is the case $w\equiv1$. \begin{lemma}\label{lem:weighted} Let $\lambda\ge0$ and $w_e=e^{\lambda d_e}$, where $|d_e-d_{e'}|\le1$ whenever $e$ and $e'$ share a plaquette, and put \[ K_\lambda=\kappa_G-N|\beta|\Bigl(\tfrac{29}2+12\bigl(\cosh\tfrac\lambda2-1\bigr)\Bigr). \] Then $\Gamma^w(P_tf)\le e^{-2K_\lambda t}P_t\Gamma^w(f)$ for every smooth $f$ and $t\ge0$. \end{lemma} \begin{proof} Put $V=-N\beta S$. On the product manifold $M=G^{A'}$ the projection to the tangent space of one factor is parallel, the rough Laplacian commutes with it, and the Ricci tensor is block diagonal. Hence the Bochner--Weitzenb\"ock formula $\nabla\Delta g=\bar\Delta\nabla g+\Ric(\nabla g)$, applied factor by factor, together with the derivative of the drift, gives for smooth $g$ \[ \begin{split} &\tfrac12L\Gamma^w(g)-\Gamma^w(g,Lg)\\ &\quad=\sum_ew_e\bigl(|\nabla\nabla_eg|^2+\Ric_G(\nabla_eg,\nabla_eg)\bigr)\\ &\qquad+\sum_{e,e'}w_e\Hess V_{ee'}(\nabla_eg,\nabla_{e'}g), \end{split} \] where $\Gamma^w(g,h)$ is the polarised form.\footnote{For $w\equiv1$ this is Bochner's identity $\Gamma_2=|\nabla^2g|^2+(\Ric+\Hess V)(\nabla g,\nabla g)$; the weights enter only through the factor index, which the connection, the Laplacian and the Ricci tensor preserve, so only the off-diagonal blocks of $\Hess V$ see them.} Put $u_e=\nabla_eg$ and $v_e=\sqrt{w_e}\,u_e$. By the symmetry of $\Hess V$, \[ \begin{split} &\sum_{e,e'}w_e\Hess V_{ee'}(u_e,u_{e'})=\Hess V(v,v)\\ &\quad+\sum_{e\ne e'}\Bigl(\cosh\tfrac{\lambda(d_e-d_{e'})}2-1\Bigr) \Hess V_{ee'}(v_e,v_{e'}), \end{split} \] because $\frac{w_e+w_{e'}}{2\sqrt{w_ew_{e'}}}=\cosh\frac{\lambda(d_e-d_{e'})}2$. The first term is at least $-\frac{29}2N|\beta|\,|v|^2$. In the second only partners contribute, each with a coefficient at most $\cosh\frac\lambda2-1$ and a block of norm at most $N|\beta|$, and $2|v_e|\,|v_{e'}|\le|v_e|^2+|v_{e'}|^2$ with twelve partners per link bounds it by $12(\cosh\frac\lambda2-1)N|\beta|\,|v|^2$. Since $\Ric_G\ge\kappa_G$ and $|v|^2=\Gamma^w(g)$, \[ \tfrac12L\Gamma^w(g)-\Gamma^w(g,Lg)\ge K_\lambda\Gamma^w(g). \] For fixed $t$ let $\varphi(s)=P_s\Gamma^w(P_{t-s}f)$, $0\le s\le t$. Then $\varphi'(s)=P_s\bigl(L\Gamma^w(g_s)-2\Gamma^w(g_s,Lg_s)\bigr)\ge2K_\lambda\varphi(s)$ with $g_s=P_{t-s}f$, and $\varphi(0)\le e^{-2K_\lambda t}\varphi(t)$ is the claim. \end{proof} \begin{lemma}\label{lem:decayrate} Suppose $K_\lambda>0$. Let $f$ and $g$ be smooth functions on $G^{A'}$ that depend only on the links of $A_f$ and of $B_g$, and let $D$ be the least number of steps from $A_f$ to $B_g$ in the graph on $A'$ whose edges join links that share a plaquette. Then \[ |\Cov_\mu(f,g)|\le\frac{e^{-\lambda D/2}}{K_\lambda}\,\|\nabla f\|_\infty \sum_{e\in B_g}\|\nabla_eg\|_\infty , \] where $\|\nabla f\|_\infty^2=\sup\sum_e|\nabla_ef|^2$. \end{lemma} \begin{proof} Since $K_0\ge K_\lambda>0$, Lemma~\ref{lem:weighted} with $\lambda=0$ gives $\Gamma(P_tf)\le e^{-2K_0t}\|\nabla f\|_\infty^2$, so $P_tf$ converges uniformly to a constant, which is $\mu(f)$ by invariance, $M$ being connected. Since $\frac d{dt}\mu(gP_tf)=\mu(gLP_tf)=-\mu(\Gamma(g,P_tf))$, \[ \Cov_\mu(f,g)=\int_0^\infty\mu\bigl(\Gamma(P_tf,g)\bigr)\,dt . \] Let $d_e$ be the smaller of $D$ and the graph distance from $e$ to $A_f$, so that $|d_e-d_{e'}|\le1$ for partners, $w_e=1$ on $A_f$, $w_e=e^{\lambda D}$ on $B_g$, and $\Gamma^w(f)\le\|\nabla f\|_\infty^2$. For $e\in B_g$, Lemma~\ref{lem:weighted} gives $|\nabla_eP_tf|^2\le w_e^{-1}\Gamma^w(P_tf)\le e^{-\lambda D}e^{-2K_\lambda t} \|\nabla f\|_\infty^2$. Hence $|\Gamma(P_tf,g)|\le e^{-\lambda D/2}e^{-K_\lambda t} \|\nabla f\|_\infty\sum_{e\in B_g}\|\nabla_eg\|_\infty$, and integration in $t$ gives the bound. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:groups}] Let $\lambda^*>0$ solve $K_{\lambda^*}=0$, so that $\frac{\lambda^*}2=m_G(\beta)$, and let $0<\lambda<\lambda^*$; then $K_\lambda>0$. \emph{Decay in boxes.} Lemma~\ref{lem:decayrate} with $S=S_\Lambda$ gives \[ |\Cov_{\mu^F_\Lambda}(f,g)|\le C_{f,g}e^{-\lambda D/2}, \] uniformly in the box. \emph{Limit.} For boxes $\Lambda\subset\Lambda'$ let $S_s=(1-s)S_\Lambda+sS_{\Lambda'}$ on the active links of $\Lambda'$, $s\in[0,1]$, and $\mu_s$ the corresponding measure; $S_s$ is admissible, $\mu_0$ is $\mu^F_\Lambda$ times Haar measure on the active links outside $\Lambda$, and $\mu_1=\mu^F_{\Lambda'}$. For smooth $f$ of the active links of $\Lambda$, \[ \frac d{ds}\mu_s(f)=N\beta\sum_{p\in P(\Lambda')\setminus P(\Lambda)} \Cov_{\mu_s}\bigl(f,\ReTr Q^F_p\bigr), \] and $|\nabla_e\ReTr Q_p|\le\|Q_p\|_{\mathrm{HS}}=\sqrt N$. A step in the graph of Lemma~\ref{lem:decayrate} moves a link by at most one lattice unit, so a plaquette at graph distance $k$ from the support $T$ of $f$ lies within distance $k+1$ of $T$, there are at most $C_T(k+1)^4$ of them, and every plaquette outside $P(\Lambda)$ is at graph distance at least $R-1$, $R$ the distance from $T$ to the complement of $\Lambda$. Lemma~\ref{lem:decayrate} gives \[ |\mu^F_{\Lambda'}(f)-\mu^F_\Lambda(f)|\le C_fR^4e^{-\lambda R/2}. \] The limit therefore exists along every increasing sequence of boxes, and two sequences are compared through their intersections, which are boxes. Continuous gauge-invariant $O$ are uniform limits of smooth gauge-invariant ones, for instance of $e^{\tau\Delta}O$, since the heat semigroup of $G^E$ commutes with left and right translations; with Lemma~\ref{lem:gaugefix} the limit $\omega(O)$ of $W_\Lambda(O)$ exists for all of them. \emph{Properties.} Invariance and, for $\beta\ge0$, reflection positivity follow as in Proposition~\ref{prop:state}(iii)--(iv). For Lipschitz $O$ the functions $e^{\tau\Delta}O$ are smooth with the same Lipschitz constant, since $\Ric_G\ge0$~\cite[Sect.~3.3]{BGL14}, so the bound of the first step passes to $\omega$ and to Lipschitz observables. Two links of one plaquette have time coordinates that differ by at most one, so observables at time separation $t$ are at graph distance at least $t-c$, and truncated correlations decay at every rate below $m_G(\beta)$. The gap follows from Proposition~\ref{prop:gapdecay} as in Proposition~\ref{prop:state}(vi). \end{proof} \begin{proof}[Proof of the rest of Theorem~\ref{thm:curv}] By Lemma~\ref{lem:ricci} and Table~\ref{tab:groups}, $\kappa_{\SU(N)}=\frac N2$ and $\kappa_{\SO(N)}=\frac{N-2}4$; for $\SO(4)$, whose Lie algebra is $\mathfrak{so}(3)\oplus\mathfrak{so}(3)$ acting on $\R^4\otimes\C=\C^2\otimes\C^2$, each ideal gives $\frac12$ by the same computation. The conditions of Theorem~\ref{thm:curv} are those of Theorem~\ref{thm:groups}. \end{proof} \begin{remark}\label{rem:hf} Let $h_F$ be the supremum of $\pm\Hess S_0$ on the gauge-fixed directions, so that uniform curvature reaches $|\beta|<1/(2h_F)$ and no further. At the identity field $-\Hess S_0$ restricted to $X_F=0$ has the eigenvalue $9+\sqrt{17}=13.123$ on the torus of side two, so $9+\sqrt{17}\le h_F\le\frac{29}2$. A gradient search over $\SU(2)$ fields of periods two, four and six finds nothing above $13.97401$, the value at an abelian field; certificates made of plaquette forms and vertex forms on the period-two cell cannot go below $14.384$. One of us (T.~Romanov)~\cite{Rom26} constructed such a certificate, with sixteen positive definite vertex forms of size eight and nonnegative plaquette weights, rational with denominator $10^6$. Its positivity is checked by computer, and with it the argument of Appendix~\ref{app:certificate} gives $h_F\le14.420001$ (Remark~\ref{rem:certificate}). Like $\frac{29}2$ this would be a statement about $\U(N)$, so it could replace $\frac{29}2$ in Theorems~\ref{thm:curv} and~\ref{thm:groups}, and the curvature method would then reach $|\beta|<1/28.840002$, that is $\beta_W<0.1387$ for $\SU(2)$ and $\beta_W<0.3121$ for $\SU(3)$. This is a computation, used by no result of the paper, and concerns the lattice at fixed spacing, not the continuum limit. The theorems of this paper are stated with $\frac{29}2$, which is proved by hand. With this gauge the curvature method reaches $|\beta|<\frac1{29}$ and stops before $\frac1{26.25}$; the computation moves the first value to $1/28.840002$. \end{remark} \section{SU(3) up to 0.515}\label{sec:su3} In this section $G=\SU(3)$, $\beta\ge0$ and $\beta_W=9\beta$. The distance on $\SU(3)$ is $d(U,V)=\|U-V\|_{\rm HS}$, which is at most the geodesic distance for $\ip XY=-\Tr(XY)$; $K=\operatorname{conv}\SU(3)$; $A(M)=\frac12(M-M^*)$ and $P$ removes the trace. With this inner product $\Ric=\frac32$, the Casimir of the fundamental representation is $\lambda_F=\frac83$ and that of the adjoint representation is $6$. \subsection{Reduction to one link} The conditional law of an active link is \[ \nu_b(dU)\propto e^{\ReTr(Ub)}dU,\qquad b=N\beta\sum_{p\ni e}S_p , \] where $S_p$ is the staple of $e$ in $p$, a product of $\SU(3)$ matrices and identities, or its adjoint. Six staples give $b\in rK$ with $r=6N\beta=2\beta_W$. For a matrix $h$ put $\varphi_h(U)=\ReTr(Uh)$ and \[ L_h(b)=\sup\{\Cov_{\nu_b}(g,\varphi_h):g\text{ 1-Lipschitz for }d\}, \] \[ \Lambda(r)=\sup_{b\in rK}\ \sup_{\|h\|_{\rm HS}=1}L_h(b). \] Changing a partner $e'$ changes one staple by a matrix of norm $d(Q_{e'}, Q'_{e'})$. Along the segment $b_t$ from $b$ to $b'$, which stays in $rK$, $\frac d{dt}\nu_{b_t}(g)=\Cov_{\nu_{b_t}}(g,\ReTr(U(b'-b)))$, so $W_1(\nu_b,\nu_{b'})\le N\beta\Lambda(r)\,d(Q_{e'},Q'_{e'})$. With twelve partners per link, \begin{equation}\label{eq:alpha} \alpha\le12N\beta\Lambda(r)=2r\Lambda(r)=4\beta_W\Lambda(2\beta_W). \end{equation} Since $K$ is convex and contains $0$, the mean of $I,\omega I,\omega^2I$ with $\omega^3=1$, $r'K\subset rK$ for $r'\le r$ and $\Lambda$ is nondecreasing. \subsection{Gradients and one-forms} \begin{lemma}\label{lem:gradients} In the left trivialisation, $\nabla\varphi_c(U)=-PA(cU)$; if $V=\varphi_b$ then $\Hess V(X,X)=\ReTr(UX^2b)$; and $\Delta\varphi_c=-\lambda_F\varphi_c$. \end{lemma} \begin{proof} $\frac d{dt}\ReTr(Ue^{tX}c)|_{t=0}=\ReTr(cUX)=-\ip{PA(cU)}X$ for $X\in \mathfrak{su}(3)$. On a group with a bi-invariant metric $\nabla_XX=0$ for left-invariant $X$, so the Hessian is the second derivative along $e^{tX}$. The function $\varphi_c$ is a matrix coefficient of the fundamental representation, on which the Casimir acts by $\lambda_F$. \end{proof} \begin{lemma}\label{lem:roughlap} The bottom of the spectrum of the rough Laplacian $\nabla^*\nabla$ on $L^2$ one-forms of $\SU(3)$ is $\mu_0=\frac76$. \end{lemma} \begin{proof} Trivialise one-forms on the left, so that a one-form is a function $f:\SU(3)\to\mathfrak{su}(3)$. For an orthonormal basis $X_a$ acting as left-invariant vector fields, $\nabla_{X_a}Y=\frac12[X_a,Y]$ and $\nabla_{X_a}X_a=0$, so $\nabla^*\nabla=-\sum_a(\rho(X_a)+\frac12\operatorname{ad} X_a)^2$. On the isotypic part $V_\pi\otimes V_\pi^*\otimes\mathfrak{su}(3)$ of $L^2\otimes\mathfrak{su}(3)$ put $A_a=\rho_\pi(X_a)$, $B_a=\operatorname{ad}X_a$ and $T_a=A_a+B_a$, the action on the tensor product. Since $A_a$ and $B_a$ commute, $A_a+\frac12B_a=\frac12(A_a+T_a)$ and $A_aT_a+T_aA_a=A_a^2+T_a^2-B_a^2$, so on a component $\sigma\subset\pi\otimes\operatorname{ad}$ \[ \nabla^*\nabla=\tfrac12(C_\pi+C_\sigma)-\tfrac14C_{\rm ad}=\tfrac12(C_\pi+ C_\sigma)-\tfrac32 , \] with $C(p,q)=\frac23(p^2+q^2+pq+3p+3q)$ for the representation of highest weight $(p,q)$~\cite{FH91}. The components of $(p,q)\otimes(1,1)$ are $(p+1,q+1)$, $(p+2,q-1)$, $(p-1,q+2)$, $(p+1,q-2)$, $(p-2,q+1)$, $(p-1,q-1)$ and $(p,q)$, the last with multiplicity two, one or zero as both, one or none of $p,q$ are positive, and only those with nonnegative labels occur. For $\pi=(0,0)$ the only component is $(1,1)$, with value $\frac32$. For $\pi=(1,0)$ the components are $(2,1)$, $(0,2)$, $(1,0)$, with values $\frac{31}6$, $\frac{19}6$, $\frac76$, and $(0,1)$ is conjugate. For $p+q\ge2$ the smallest values occur at $(p-1,q-1)$, $(p,q)$, $(p-2,q+1)$ or $(p+1,q-2)$, and each is at least $\frac13(2(p^2+q^2+pq)+3(p+q)-3)-\frac32\ge \frac32$. \end{proof} \subsection{The gap of the one-link measure} \begin{lemma}\label{lem:gap} For $b\in rK$ the spectral gap of $L=\Delta+\ip{\nabla V}{\nabla\cdot}$, $V=\varphi_b$, on $L^2(\nu_b)$ is at least \[ \gamma(r)=\tfrac{39}{64}\mu_0+\tfrac32-\tfrac r2=\tfrac{283}{128}-\tfrac r2 . \] \end{lemma} \begin{proof} Let $-Lf=\lambda f$ with $\lambda>0$, $\omega=df$ and $\eta=e^{V/2}\omega$; norms are in $L^2(dU)$. Bochner's formula integrated against $\nu_b$ gives $\lambda\nu_b|\omega|^2=\nu_b|\nabla\omega|^2+\nu_b((\Ric-\Hess V)(\omega, \omega))$, and since $e^{V/2}\nabla\omega=\nabla\eta-\frac12dV\otimes\eta$, \begin{equation}\label{eq:bochner} \lambda\|\eta\|^2=\|\nabla\eta-\tfrac12dV\otimes\eta\|^2+\int(\Ric-\Hess V) (\eta,\eta). \end{equation} Put $\mathcal A=\nabla\eta$ and $\mathcal B=\frac12dV\otimes\eta$. Then \[ \ip{\mathcal A}{\mathcal B}=\tfrac14\int\ip{\nabla V}{\nabla|\eta|^2} =\tfrac{\lambda_F}4\int V|\eta|^2 , \] because $-\Delta V=\lambda_FV$. Writing $\|\mathcal A-\mathcal B\|^2= \|\frac58\mathcal A-\mathcal B\|^2+\frac{39}{64}\|\mathcal A\|^2-\frac34 \ip{\mathcal A}{\mathcal B}$ and dropping the first term, \[ \|\mathcal A-\mathcal B\|^2\ge\tfrac{39}{64}\mu_0\|\eta\|^2-\tfrac12\int V|\eta|^2 \] by Lemma~\ref{lem:roughlap}. With $\Ric=\frac32$, \eqref{eq:bochner} gives $\lambda\|\eta\|^2\ge(\frac{39}{64}\mu_0+\frac32)\|\eta\|^2-\int(\Hess V(\eta, \eta)+\frac12V|\eta|^2)$, and the claim follows from the pointwise bound \begin{equation}\label{eq:pointwise} \ReTr(MX^2)+\tfrac12\ReTr M\le\tfrac r2|X|^2,\qquad M=bU\in rK , \end{equation} since $\Hess V(X,X)=\ReTr(MX^2)$ and $V=\ReTr M$. The maximum over $|X|=1$ of the left side is convex in $M$, so it suffices to take $M=rW$ with $W\in \SU(3)$ and, after a conjugation, $W=\operatorname{diag}(e^{i\theta_k})$ with $\sum_k\theta_k=0$; put $c_k=\cos\theta_k$. Write $X=iH$ with $H$ Hermitian, traceless, $\Tr H^2=1$, with diagonal $d_k$ and entries $H_{jk}$. Then \[ \ReTr(WX^2)=-\sum_kc_kd_k^2-\sum_{j0.24$ and vanishes before $r=1$.} \subsection{The modulus by hand}\label{ssec:su3modulus} For $h\in\C^{3\times3}$ let $E_2(h)=\frac12((\Tr h)^2-\Tr h^2)$, the sum of the principal $2\times2$ minors of $h$. All integrals are over $\SU(3)$ with Haar measure; put $t=\Tr U$, \[ Z_*(r)=\int e^{r\ReTr U}dU,\qquad m_n=\int(\ReTr U)^ndU . \] \begin{lemma}\label{lem:su3haar} \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The integral of a monomial of degree $j$ in the entries of $U$ and $k$ in those of $\bar U$ vanishes unless $j\equiv k$ modulo three; $\int u_{ij}\bar u_{kl}dU=\frac13\delta_{ik}\delta_{jl}$ and $\int u_{ab} u_{cd}u_{ef}dU=\frac16\varepsilon_{ace}\varepsilon_{bdf}$. \item For every $h$ the integrals of $\Tr(hUhU)$ and $(\Tr hU)^2$ vanish, $\int|\Tr hU|^2dU=\frac13\|h\|_{\rm HS}^2$, \begin{gather*} \int\Tr(hUhU)\Tr U\,dU=-\tfrac13E_2(h),\\ \int(\Tr hU)^2\Tr U\,dU=\tfrac13E_2(h), \end{gather*} and $|E_2(h)|\le\|h\|_{\rm HS}^2$. \item $\int t^a\bar t^bdU$ vanishes unless $a\equiv b$ modulo three, does not change when $a$ and $b$ are exchanged, and equals $1$ at $(a,b)= (0,0)$, $(1,1)$, $(3,0)$, $2$ at $(2,2)$, $3$ at $(4,1)$, $5$ at $(6,0)$, $6$ at $(3,3)$, $11$ at $(5,2)$, $21$ at $(7,1)$ and $23$ at $(4,4)$. Hence $m_0=1$, $m_1=0$, \[ (m_2,\dots,m_8)=\bigl(\tfrac12,\tfrac14,\tfrac34,\tfrac{15}{16}, \tfrac{65}{32},\tfrac{231}{64},\tfrac{973}{128}\bigr), \] and $m_n\le3^{n-8}m_8$ for $n\ge8$. \end{enumerate} \end{lemma} \begin{proof} (iii) By the proof of Lemma~\ref{lem:characters}, $t^a=\sum_{\lambda\vdash a}f^\lambda s_\lambda(U)$, and on $\SU(3)$ the Schur polynomial $s_\lambda$ vanishes when $\lambda$ has more than three rows and is otherwise the irreducible character of highest weight $(\lambda_1-\lambda_3,\lambda_2- \lambda_3)$. By Schur orthogonality~\cite[Ch.~2]{Bum13}, $\int t^a\bar t^bdU$ is the sum of $f^\lambda f^\mu$ over the pairs $\lambda\vdash a$, $\mu\vdash b$ with at most three rows and $\lambda-\mu$ a multiple of $(1,1,1)$; such pairs exist only when $a\equiv b$ modulo three, and the sum is symmetric in $a,b$. For $a\ge b$ and $a+b\le8$ the pairs are $((1),(1))$; $((1,1,1),\varnothing)$; $\lambda=\mu\in\{(2),(1,1)\}$; $((2,1,1),(1))$; $((2,2,2),\varnothing)$; $\lambda=\mu\in\{(3),(2,1), (1,1,1)\}$; $((3,1,1),(2))$ and $((2,2,1),(1,1))$; $((3,2,2),(1))$; and $\lambda=\mu\in\{(4),(3,1),(2,2),(2,1,1)\}$. The hook length formula gives $f^{(2,1)}=2$, $f^{(3,1)}=f^{(2,1,1)}=3$, $f^{(2,2)}=2$, $f^{(2,2,2)}=5$, $f^{(3,1,1)}=6$, $f^{(2,2,1)}=5$ and $f^{(3,2,2)}=21$, hence the values, the last one being $1+9+4+9$. Since $(\ReTr U)^n=2^{-n}\sum_a\binom nat^a\bar t^{n-a}$, \begin{gather*} m_2=\tfrac{2}{4},\quad m_3=\tfrac28,\quad m_4=\tfrac{6\cdot2}{16},\quad m_5=\tfrac{2\cdot5\cdot3}{32},\\ m_6=\tfrac{2\cdot5+20\cdot6}{64},\quad m_7=\tfrac{2\cdot21\cdot11}{128}, \quad m_8=\tfrac{2\cdot8\cdot21+70\cdot23}{256}. \end{gather*} The eigenvalues of $U$ have modulus one, so $|\ReTr U|\le3$ and $|\ReTr U|^n\le3^{n-8}(\ReTr U)^8$ for $n\ge8$. (i) The map $U\mapsto\omega U$, $\omega=e^{2\pi i/3}$, preserves Haar measure and multiplies the monomial by $\omega^{j-k}$. For a unitary representation $\pi$ of $\SU(3)$, $\int\pi(U)dU$ is the orthogonal projection onto the invariant vectors, whose dimension is the mean of the character of $\pi$. On $\C^3\otimes\overline{\C^3}$ the invariant vectors are the multiples of the identity, which gives the second formula. On $(\C^3)^{\otimes3}$ the character is $t^3$, whose mean is $1$ by (iii), and $\varepsilon$ is invariant because $\det U=1$; the projection onto the line of $\varepsilon$, whose squared norm is $6$, gives the third formula. (ii) The first two integrals have degree two in $U$ and none in $\bar U$, and $\int|\Tr hU|^2dU=\frac13\sum|h_{ji}|^2$ by (i). By (i), $\int h_{pa}u_{aq}h_{qc}u_{cp}u_{ee}dU=\frac16h_{pa}h_{qc}\varepsilon_{ace} \varepsilon_{qpe}$, summed over repeated indices, and $\sum_e \varepsilon_{ace}\varepsilon_{qpe}=\delta_{aq}\delta_{cp}-\delta_{ap} \delta_{cq}$, which gives $\frac16(\Tr h^2-(\Tr h)^2)$; the second integral is the same with $\varepsilon_{pqe}$ in place of $\varepsilon_{qpe}$, which changes the sign. Finally $E_2(h)$ is the trace of the second exterior power of $h$, whose singular values are the products $\sigma_i\sigma_j$, $i0$, put \[ \epsilon(r)=1-\frac{5\,e^{-r/5}}{9\,Z_*(r)}, \] and let $b\in rK$ and $\|h\|_{\rm HS}=1$. Then \[ \nu_b|\nabla\varphi_h|^2\le\epsilon(r),\quad\Var_{\nu_b}(\varphi_h)\le \frac{\epsilon(r)}{\gamma(r)},\quad L_h(b)\le\frac{\sqrt{\epsilon(r)}} {\gamma(r)}, \] so that $\Lambda(r)\le\sqrt{\epsilon(r)}/\gamma(r)$. The function $\epsilon$ is nondecreasing on $[0,\infty)$. \end{lemma} \begin{proof} By Lemma~\ref{lem:gradients}, $\nabla\varphi_h(U)=-PA(M)$ with $M=hU$, and $\|M\|_{\rm HS}=1$. The anti-Hermitian part has $|A(M)|^2=\frac12(\|M\|^2- \ReTr M^2)$, and $P$ removes from it the multiple $\frac13\Tr A(M)=\frac i3 \ImTr M$ of the identity, whose squared norm is $\frac13(\ImTr M)^2$. Hence \begin{gather*} |\nabla\varphi_h|^2=1-\tfrac12F_h,\\ F_h(U)=1+\ReTr(hUhU)+\tfrac23(\ImTr hU)^2 , \end{gather*} and $F_h\ge0$ because $|\ReTr M^2|\le\|M\|^2$. For complex numbers $X,Y$, $\operatorname{Re}X\operatorname{Re}Y=\frac12\operatorname{Re}(XY)+ \frac12\operatorname{Re}(X\bar Y)$ and $(\operatorname{Im}X)^2=\frac12|X|^2- \frac12\operatorname{Re}X^2$. With these identities, the vanishing of the integrals of degree two in $U$ and one in $\bar U$, or the converse, and Lemma~\ref{lem:su3haar}(ii), \[ \int F_h\,dU=1+\tfrac23\cdot\tfrac16=\tfrac{10}9,\quad\int F_h\ReTr U\,dU= -\tfrac29\operatorname{Re}E_2(h), \] the second as the sum of $-\frac16\operatorname{Re}E_2(h)$ and $\frac23 \cdot(-\frac1{12})\operatorname{Re}E_2(h)$. For $W\in\SU(3)$ the substitution $U\mapsto W^{-1}U$ and Lemma~\ref{lem:su3haar}(ii) give \[ \int F_h\ReTr(WU)\,dU=\int F_{hW^{-1}}\ReTr U\,dU\ge-\tfrac29 . \] Write $b=r\sum_i\lambda_iW_i$ as a convex combination of elements of $\SU(3)$ and $V=\ReTr(Ub)$. Under the probability measure $\pi=\frac9{10} F_h\,dU$ the mean of $V$ is at least $-\frac9{10}\cdot\frac29r=-\frac r5$, so Jensen's inequality gives $\int F_he^VdU=\frac{10}9\pi(e^V)\ge\frac{10}9 e^{-r/5}$. The function $c\mapsto\log\int e^{\ReTr(Uc)}dU$ is convex by H\"older's inequality and equals $\log Z_*(r)$ at every $c=rW$, $W\in \SU(3)$, by the invariance of Haar measure; hence $\int e^VdU\le Z_*(r)$ and $\nu_b(F_h)\ge\frac{10}9e^{-r/5}/Z_*(r)$, which is the first bound. The spectral gap of $\nu_b$ is at least $\gamma(r)$ by Lemma~\ref{lem:gap}. The Poincar\'e inequality gives $\Var_{\nu_b}(\varphi_h)\le\nu_b|\nabla \varphi_h|^2/\gamma(r)$, the second bound, and, for $g$ $1$-Lipschitz for $d$, hence for the geodesic distance, $\Var_{\nu_b}(g)\le1/\gamma(r)$; so $\Cov_{\nu_b}(g,\varphi_h)\le\mathrm{sd}(g)\,\mathrm{sd}(\varphi_h)\le \sqrt{\epsilon(r)}/\gamma(r)$. Finally $\log Z_*$ is convex in $r$ with derivative $m_1=0$ at $r=0$, so $Z_*$ and $\epsilon$ are nondecreasing on $[0,\infty)$. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:su3}] Let $r=2\beta_W\le1.03$, so that $\gamma(r)\ge\gamma(1.03)>1.69$. By \eqref{eq:alpha} and Lemma~\ref{lem:su3modulus}, the Dobrushin constant satisfies $\alpha\le2r\Lambda(r)\le\alpha_1(\beta_W)$, and $\alpha_1$ is increasing because $\epsilon$ is nondecreasing and $\gamma$ is decreasing. By Lemma~\ref{lem:su3haar}(iii), and since $\sum_{n\ge9}x^n/n!\le \frac{x^9}{9!}(1-\frac x{10})^{-1}$ for $0\le x<10$, \[ Z_*(r)\le\sum_{n=0}^8\frac{m_nr^n}{n!}+\frac{m_8}{3^8}\, \frac{(3r)^9}{9!}\Bigl(1-\frac{3r}{10}\Bigr)^{-1}. \] At $r=1$ the terms of the sum, rounded up, are $1$, $0$, $0.25$, $0.0416667$, $0.03125$, $0.0078125$, $0.0028212$, $0.0007162$ and $0.0001886$, with sum at most $1.3344552$, and the last term is $\frac{973}{128\cdot6561}\cdot\frac{19683}{362880}\cdot\frac{10}7<9\cdot 10^{-5}$; so $Z_*(1)\le1.334546$. Since $e^{-x}\ge1-x+\frac{x^2}2-\frac{x^3} 6$ for $x\ge0$, $e^{-1/5}\ge0.818666$, so $\frac{5e^{-1/5}}{9Z_*(1)}\ge 0.340800$ and $\epsilon(1)\le0.659200<0.81192^2$. With $\gamma(1)= \frac{219}{128}$, \[ \alpha_1(\tfrac12)\le\frac{2\cdot0.81192\cdot128}{219}\le0.9491 . \] At $r=1.03$ the terms are at most \begin{gather*} 1,\ 0,\ 0.265225,\ 0.0455303,\ 0.0351722,\\ 0.0090569,\ 0.0033687,\ 0.0008808,\ 0.0002389, \end{gather*} with sum at most $1.3594728$, and the last term is less than $1.2\cdot10^{-4}$; so $Z_*(1.03)\le1.359593$, $e^{-0.206}\ge0.813760$, $\frac{5e^{-0.206}} {9Z_*(1.03)}\ge0.332517$ and $\epsilon(1.03)\le0.667483<0.817^2$. With $\gamma(1.03)=\frac{5427}{3200}$, $\alpha_1(0.515)\le2.06\cdot0.817\cdot \frac{3200}{5427}\le0.9924$. Proposition~\ref{prop:state} applies with the Hilbert--Schmidt metric. With $y=1-\alpha_1$, $-\log\alpha_1\ge y+ \frac{y^2}2$, which is at least $0.052$ at $\beta_W=\frac12$ and $0.0076$ at $0.515$. \end{proof} The losses in this bound are Jensen's inequality, the Poincar\'e inequality for $\varphi_h$, which replaces the true variance of $\varphi_h$ by $\nu_b|\nabla\varphi_h|^2/\gamma(r)$, and the gap $\gamma(r)$ itself: at $b=0$ the variance is $\frac16$ and the gap $\frac83$, so that $L_h(0)= \frac14$, while the lemma gives $\frac23/\gamma(0)=0.302$. Evaluated in floating point, $\alpha_1$ reaches one near $\beta_W=0.5176$. The rest of this section describes a sharper bound for $\Lambda(r)$, with a corrector of degree two, the reduction to diagonal staple sums, exact moments and an interpolation over boxes of staple sums; its evaluation by a computer is the subject of Remark~\ref{rem:su3cover}, and no result uses it. \subsection{A corrector of degree two} \begin{lemma}\label{lem:corrector} For $b\in rK$, $\|h\|=1$ and every smooth real function $u$ on $\SU(3)$, \[ L_h(b)\le\sqrt{\nu_b|\nabla u|^2}+\sqrt{\Var_{\nu_b}(\varphi_h+Lu)/ \gamma(r)} . \] \end{lemma} \begin{proof} Let $-L\psi=\varphi_h-\nu_b\varphi_h$. For a 1-Lipschitz $g$, $|\nabla g|\le1$ almost everywhere, because $d$ is at most the geodesic distance, so $\Cov(g,\varphi_h)=\nu_b\ip{\nabla g}{\nabla\psi}\le\nu_b|\nabla\psi|$. Write $\psi=u+\chi$. Since $\nu_b(Lu)=0$, $\chi$ solves $-L\chi=X-\nu_bX$ with $X=\varphi_h+Lu$, and \[ \nu_b|\nabla\chi|^2=\Cov(\chi,X)\le\sqrt{\Var\chi\,\Var X}\le \sqrt{\nu_b|\nabla\chi|^2\Var X/\gamma} \] by Lemma~\ref{lem:gap}. Hence $\nu_b|\nabla\chi|^2\le\Var X/\gamma$, and $\nu_b|\nabla\psi|\le\sqrt{\nu_b|\nabla u|^2}+\sqrt{\nu_b|\nabla \chi|^2}$. \end{proof} The best $u$ is $\psi$ itself, and then the second term vanishes. With $u=s\varphi_h$, Lemma~\ref{lem:gradients} gives $\varphi_h+Lu=(1-s\lambda_F) \varphi_h+s\Gamma$ with $\Gamma=\ip{\nabla V}{\nabla\varphi_h}$; the choice $s=1/\lambda_F$ removes $\varphi_h$ and leaves the drift. In floating point this blended corrector keeps $2r\sup_hL_h$ below one up to about $\beta_W=0.715$ (Sec.~\ref{sec:limits}). We take $u$ in a space of polynomials of degree two. Let $\mathcal P_2$ be the real vector space of real polynomials of degree one or two in the entries $u_{ij}$ of $U$ and $\bar u_{ij}$ of $\bar U$. It has dimension $189$: a basis $(B_k)$ consists of $\frac12(m+\bar m)$ and $\frac1{2i}(m-\bar m)$ for each pair $\{m,\bar m\}$ of conjugate monomials with $m\ne\bar m$, and $m$ itself when $m=\bar m$. Identify $h\in\C^{3\times3}$ with $\R^{18}$ through the real and imaginary parts of its entries. For a real $189\times18$ matrix $C$ put \[ q_h=\sum_{k}\sum_{m=1}^{18}C_{km}h_mB_k\in\mathcal P_2 , \] linear in $h$. For diagonal $b=\operatorname{diag}(a_1,a_2,x_3+iy_3)$ we have $V=\sum_kb_k\varphi_{F_k}$ with $(b_k)=(a_1,a_2,x_3,y_3)$ and $F=(e_{11},e_{22}, e_{33},ie_{33})$, so \begin{equation}\label{eq:Xq} X_h=\varphi_h+Lq_h=\varphi_h+\Delta q_h+\sum_kb_k\ip{\nabla\varphi_{F_k}} {\nabla q_h} \end{equation} is a polynomial of degree at most three, affine in $b$, while $\nabla q_h$ does not depend on $b$. The squares of these functions have degree at most six, so the quadratic forms $h\mapsto\nu_b|\nabla q_h|^2$ and $h\mapsto\nu_b[(X_h-\kappa)^2]$ are combinations of moments of $\nu_b$ of degree at most six. To compute $\Delta$ and $\ip{\nabla f}{\nabla g}$ exactly we use the basis of $\mathfrak{su}(3)$ \[ \begin{gathered} e_{jk}-e_{kj},\quad i(e_{jk}+e_{kj})\quad(j34}(\frac32\|D\|)^n/n!$; Cauchy's estimate on a polydisc around $b$ turns this into a bound for every derivative (Appendix~\ref{app:cover}). \begin{lemma}[Interpolation on a box]\label{lem:boxes} Let $Q=\prod_k[l_k,l_k+w_k]$ in the coordinates $(a_1,a_2,x_3,y_3)$, with corners $b_i$ and multilinear weights $\omega_i(b)\ge0$, $\sum_i\omega_i(b)=1$, and let $\varsigma=w_1^2+w_2^2+\max(w_3^2,w_4^2)$. For $b\in Q$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $Z(b)\ge e^{-\varsigma/8}\sum_i\omega_i(b)Z(b_i)$; \item $\nu_b(f)\le e^{\varsigma/8}\max_i\nu_{b_i}(f)$ for every $f\ge0$ that does not depend on $b$; \item if $Y_b=(\mathcal A+\sum_k(b_k-\beta_k)\mathcal B_k)^2$ with functions $\mathcal A,\mathcal B_k$ of $U$ and a fixed point $\beta$, then \[ \nu_b(Y_b)\le e^{\varsigma/8}\Bigl(\max_i\nu_{b_i}(Y_{b_i})+\sum_k \frac{w_k^2}4\,\frac{\max_iZ(b_i)\nu_{b_i}(\mathcal B_k^2)}{\min_iZ(b_i)}\Bigr). \] \end{enumerate} \end{lemma} \begin{proof} Write $E(b)=e^{\ReTr(Ub)}=\prod_ke^{c_kb_k}$, where \[ c=(\operatorname{Re}u_{11},\operatorname{Re}u_{22},\operatorname{Re}u_{33}, -\operatorname{Im}u_{33}), \] so that $|c_k|\le1$ and $c_3^2+c_4^2\le1$. In each coordinate the chord of $e^{c_kx}$ over $[l_k,l_k+w_k]$ lies above the function and, by Hoeffding's lemma~\cite{Hoe63} for a two-point variable, below $e^{c_kx}e^{c_k^2w_k^2/8}$. Multiplying, \begin{equation}\label{eq:hoeffding} E(b)\le\sum_i\omega_i(b)E(b_i)\le E(b)\,e^{\sum_kc_k^2w_k^2/8}\le E(b)\, e^{\varsigma/8}. \end{equation} Integrating the right inequality gives (i), and (ii) follows from the left one and (i). For (iii) let $\mathcal F(b)=Y_bE(b)$ and $I_k$ the linear interpolation in the $k$-th coordinate, so that $\mathcal F-I_1I_2I_3I_4 \mathcal F=\sum_kI_1\cdots I_{k-1}(\mathcal F-I_k\mathcal F)$. Along the $k$-th coordinate $\mathcal F$ is $\ell^2e^{c_kx}$ times a positive factor, with $\ell$ affine of slope $\mathcal B_k$, and $\partial_k^2\mathcal F=((c_k\ell+2\mathcal B_k)^2-2\mathcal B_k^2)E\ge-2\mathcal B_k^2E$. With the Green kernel of the interval, whose integral is at most $w_k^2/8$, \[ \mathcal F-I_k\mathcal F\le2\mathcal B_k^2\int G\,E\le\tfrac{w_k^2}4\mathcal B_k^2\sum_i\omega_i^{(k)}E(b_i) \] by the left inequality of~\eqref{eq:hoeffding}, with weights $\omega_i^{(k)} \ge0$ of sum one that do not depend on $U$. The operators $I_j$ have nonnegative weights of sum one; integrate over $U$ and divide by $Z(b)$, bounded below by (i). \end{proof} \subsection{The cover}\label{ssec:cover} Fix a box $Q$ and a matrix $C$. For any constant $\kappa$, $\Var(X_h)\le \nu_b[(X_h-\kappa)^2]$; take $\kappa=\ip{\kappa(b)}h$ with $\kappa(b)$ affine in $b$, fitted to the means at the corners. By~\eqref{eq:Xq}, $X_h-\kappa= \mathcal A+\sum_k(b_k-\beta_k)\mathcal B_k$ with $\beta$ the centre of $Q$ and $\mathcal B_k=\ip{\nabla\varphi_{F_k}}{\nabla q_h}-\ip{\partial_k\kappa}h$, and Lemmas~\ref{lem:corrector} and~\ref{lem:boxes} give, for all $b\in Q\cap \mathcal R_r$, \begin{equation}\label{eq:BQ} \begin{split} \sup_hL_h(b)\le{}&\sqrt{e^{\varsigma/8}\max_i\lambda_A(b_i)}\\ &+\sqrt{\frac{e^{\varsigma/8}}{\gamma(r)}\Bigl(\max_i\lambda_V(b_i)+ \frac{e_N}{\min_iZ(b_i)}\Bigr)} , \end{split} \end{equation} where $e_N=\sum_k\frac{w_k^2}4\max_iZ(b_i)\lambda_k(b_i)$ and $\lambda_A$, $\lambda_V$, $\lambda_k$ are the largest eigenvalues of the quadratic forms $h\mapsto\nu_{b_i}|\nabla q_h|^2$, $h\mapsto\nu_{b_i}[(X_h-\kappa)^2]$ at $b=b_i$, and $h\mapsto\nu_{b_i}[\mathcal B_k^2]$ on $\R^{18}$. Their entries are moments of $\nu_{b_i}$ of degree at most six, computed with rigorous error bounds (Appendix~\ref{app:cover}); each largest eigenvalue is bounded above by an $LDL^\top$ factorisation of $tI$ minus the computed form with certified positive pivots, plus the norm of its error matrix by Weyl's inequality~\cite{Hig02,Joh17}. The matrix $C$ is chosen in each box in floating point. At the mean of the corner moments and at $b=\beta$, Lemma~\ref{lem:corrector} for $u=q_h$ is a quadratic problem in $C$: for $\theta>0$ we minimise the trace of $\nu|\nabla q|^2+\theta\,\Var(X)/\gamma$ over the $18$ coordinate directions $h$, a least-squares problem of size $189$, keep the $\theta$ on a geometric grid of $48$ values from $0.05$ to $80$ that gives the smallest bound, and round $C$ to multiples of $2^{-30}$. Any choice gives a valid bound. A box is discarded when it contains no point with $a_1\ge a_2\ge|z_3|$, or when a linear functional $\ell(b)=\ReTr(c^*b)$ satisfies $\min_Q\ell>r\max_T \ell$, which separates it from $r\operatorname{conv}T$ and hence, by Lemma~\ref{lem:orbits}(iii), from $\mathcal R_r$. \begin{remark}[a computation, used by no theorem]\label{rem:su3cover} At $r=\frac32$, with $\gamma(r)=\frac{187}{128}$, the box $[0,r]^2\times [-r,r]\times[0,r]$ is cut into $8192$ boxes of side $r/8$. Of these, $841$ survive the two tests, and for $581$ the right side $B_Q$ of~\eqref{eq:BQ}, evaluated with the error bounds of Appendix~\ref{app:cover}, satisfies $2rB_Q<1$. Each of the other $260$ is cut into sixteen boxes of side $r/16$; $1760$ of these survive the tests, and all satisfy $2rB_Q<1$. The $2341$ boxes cover $\mathcal R_r$; the largest bound is $0.999887$, on the box $[1.3125,1.5]\times[0.9375,1.125] \times[0.5625,0.75]\times[0,0.1875]$, and the median is $0.983$ (Fig.~\ref{fig:cover}). This computation gives $2r\Lambda(r)\le0.999887$ at $r=\frac32$, and with it~\eqref{eq:alpha} and the monotonicity of $\Lambda$ would give $\alpha\le4\beta_W\Lambda(\frac32)<1.333183\,\beta_W<1$ for $\beta_W\le\frac34$, and so the conclusions of Theorem~\ref{thm:su2}, with $m\ge-\log(1.33319\,\beta_W)$, on that range. The same computation with $\gamma(r)=\frac{4643}{3200}$ and the threshold $2rB_Q<1.03$ at $r= \frac{38}{25}$ (of the $8192$ boxes, $784$ survive the tests and $690$ satisfy the threshold; each of the other $94$ is cut into sixteen, of which $373$ survive and all satisfy it), and with $\gamma(r)=\frac{4611}{3200}$ and the threshold $2rB_Q<1.0275$ at $r=\frac{77}{50}$ ($826$ survive and $513$ satisfy it; the other $313$ are cut into sixteen, of which $2311$ survive and $2281$ satisfy it; the other $30$ are cut into sixteen, of which $31$ survive and all satisfy it), gives $2r\Lambda(r)\le1.029967$ at $r=\frac{38}{25}$ and $2r\Lambda(r)\le1.027468$ at $r=\frac{77}{50}$. No result of the paper uses these numbers. \end{remark} \begin{figure*}[t] \centering \includegraphics[width=0.92\textwidth]{figures/fig_cover.png} \caption{The $2341$ boxes of Remark~\ref{rem:su3cover} at $r=\frac32$, plotted at their centres in the coordinates $(a_1,a_2)$ of the staple sum $\operatorname{diag}(a_1,a_2,x_3+iy_3)$, coloured by the computed bound on $2r\sup_hL_h$ over the box; the larger points are the boxes of side $r/8$. (a) All boxes, with height $|x_3+iy_3|$. (b) The $419$ boxes with bound above $0.99$, with height $x_3$; the largest bounds sit in the interior of $\mathcal R_r$, not at its extreme points. No result uses this computation.} \label{fig:cover} \end{figure*} \section{SU(3) with plaquette blocks}\label{sec:su3blocks} In this section $G=\SU(3)$, $\beta\ge0$, $\beta_W=9\beta$, $N\beta=\beta_W/3$, and the notation is that of Sec.~\ref{sec:su3}: $d$ is the Hilbert--Schmidt distance, $K=\operatorname{conv}\SU(3)$, $\nu_c(dU)\propto e^{\ReTr(Uc)}dU$, $\varphi_h(U)=\ReTr(Uh)$ and $\Lambda(r)$ is the modulus of~\eqref{eq:alpha}. Put $Z(c)=\int e^{\ReTr(Uc)}dU$ and $g=\log Z$ on $\C^{3\times3}$, viewed as $\R^{18}$ with the inner product $\ip xy=\ReTr(xy^*)$. The gradient of $g$ at $c$ is the conjugate mean $m(c)^*$, $m(c)=\nu_c(U)\in K$, and $\Hess g(c)(h,h)=\Var_{\nu_c}(\varphi_h)$. Since $Z(VcW)=Z(c)$ for $V,W\in\SU(3)$, the same holds for $g$. The blocks are those of Lemma~\ref{lem:plaqblocks}, which concerns the line field only and holds for every group. For a block plaquette with active links $e_1,e_2,e_3$, $\ReTr Q_p=\ReTr(y_1y_2y_3)$ with $y_i=Q_{e_i}$ or $Q_{e_i}^{-1}$, and the conditional law of the block is \begin{equation}\label{eq:blocklaw3} \mu_b(dy)\propto\exp\Bigl(\sum_i\ReTr(y_ib_i)+N\beta\ReTr(y_1y_2y_3) \Bigr)dy , \end{equation} where $b_i$ is $N\beta$ times the sum of the five staples of $y_i$ outside the block plaquette, so $b_i\in5N\beta K$ and $\|b_i\|_{\rm op}\le5N\beta$. On $\SU(3)^3$ we use $d(y,y')=\sum_i\|y_i-y'_i\|_{\rm HS}$. \subsection{The sensitivity of one block} Let $r=2\beta_W=6N\beta$. Let $\lambda_*$ and $\tau$ be a lower and an upper bound for the eigenvalues of $\Hess g$ on $rK$, $M$ an upper bound on $rK$ for $|D^3g(c)(u,v,v)|$ over unit vectors $u,v$, and $\gamma=\gamma(r)$ the gap of Lemma~\ref{lem:gap}. Put \begin{equation}\label{eq:blockconst3} \begin{gathered} w=\tfrac32(N\beta)^2(\tau-\lambda_*),\qquad s=\Lambda(r)+(e^w-1)e^w \sqrt{\tau/\gamma},\\ C=5(N\beta)^2\Bigl(\tau s+\frac{\sqrt3\,N\beta\,M\,e^w}{\gamma}\Bigr),\\ K_3=\frac{(1+N\beta\Lambda(r))\,s}{1-C},\qquad\alpha_3=10N\beta K_3, \end{gathered} \end{equation} assuming $C<1$. \begin{lemma}\label{lem:onesite3} Let $a,c_0\in5N\beta K$ and \[ \kappa(dx)\propto\exp\bigl(\ReTr(xa)+g(c_0+N\beta x)\bigr)dx\quad\text{on } \SU(3). \] For every $1$-Lipschitz $f$ and every $h\in\C^{3\times3}$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\Cov_\kappa(f,\varphi_h)\le s\|h\|_{\rm HS}$; \item if $c_0$ moves in the direction $k$ inside $5N\beta K$, then \[ \bigl|\tfrac{d}{dt}\kappa(f)\bigr|\le N\beta\bigl(\tau s+\sqrt3N\beta Me^w/\gamma\bigr)\|k\|_{\rm HS}. \] \end{enumerate} \end{lemma} \begin{proof} (i) The segment from $c_0$ to $c_0+N\beta x$ lies in $6N\beta K=rK$, and Taylor's formula gives \[ g(c_0+N\beta x)=g(c_0)+N\beta\ReTr(x\,m(c_0))+R(x) \] with $R(x)=\frac12(N\beta)^2\Hess g(\xi)(x,x)$. Since $\|x\|_{\rm HS}^2=3$, $R\in[\frac32(N\beta)^2\lambda_*,\frac32(N\beta)^2\tau]$ and $\operatorname{osc} R\le w$. Hence $\kappa$ is the tilt $e^R\nu/\nu(e^R)$ of $\nu=\nu_{c'}$ with $c'=a+N\beta m(c_0)\in rK$. By definition of $\Lambda$, $\Cov_\nu(f,\varphi_h) \le\Lambda(r)\|h\|$. The function $f$ is $1$-Lipschitz for the geodesic distance, which dominates $d$, so $\mathrm{sd}_\nu f\le\gamma^{-1/2}$ by Lemma~\ref{lem:gap}, and $\mathrm{sd}_\nu\varphi_h\le\sqrt\tau\|h\|$. Lemma~\ref{lem:tilt} gives (i). (ii) The derivative is $\Cov_\kappa(f,\ip{\nabla g(c_0+N\beta x)}k)$. Write $\nabla g(c_0+N\beta u)=\nabla g(c_0)+N\beta\Hess g(c_0)u+N(u)$. The linear term is $\varphi_h$ with $\|h\|\le N\beta\tau\|k\|$ and contributes at most $N\beta\tau s\|k\|$ by (i). On the segment from $c_0$ the Hessian moves by at most $N\beta M\|u\|\le\sqrt3N\beta M$, so $x\mapsto\ip{N(x)}k$ is $\sqrt3(N\beta)^2M\|k\|$-Lipschitz for $d$. The law $\kappa$ is a tilt of $\nu$ by a function of oscillation at most $w$, so its spectral gap is at least $e^{-w}\gamma$ by the Holley--Stroock lemma~\cite{HS87}, and the covariance of two Lipschitz functions under $\kappa$ is at most $e^w/\gamma$ times the product of their constants. \end{proof} \begin{lemma}\label{lem:blockK3} Let $C<1$. If $b,b'$ have all fields in $5N\beta K$ and differ only in $b_i$, then $W_1(\mu_b,\mu_{b'})\le K_3\|b_i-b'_i\|_{\rm HS}$ for the metric $d$. \end{lemma} \begin{proof} By the cyclic symmetry of $\ReTr(y_1y_2y_3)$ we may take $i=1$. Given $y_1,y_3$, the law of $y_2$ is $\nu_c$ with $c=b_2+N\beta y_3y_1\in rK$. Changing $y_1$ by $\eta$ in $d$ moves $c$ by $N\beta\eta$ and $\nu_c$ by at most $N\beta\Lambda(r)\eta$ in $W_1$, as in~\eqref{eq:alpha}; so $F(y_1,y_3)= \E[f\mid y_1,y_3]$ is $(1+N\beta\Lambda(r))$-Lipschitz in each variable. The law of $(y_1,y_3)$ is proportional to $e^{\ReTr(y_1b_1)+\ReTr(y_3b_3)} Z(b_2+N\beta y_3y_1)$, and $Z(b_2+N\beta y_3y_1)=Z(y_3^{-1}b_2+N\beta y_1)= Z(b_2y_1^{-1}+N\beta y_3)$, so its two conditional laws have the form of Lemma~\ref{lem:onesite3}, with $c_0=y_3^{-1}b_2$ and $c_0=b_2y_1^{-1}$. Moving $y_3$ to $y'_3$ moves $y_3^{-1}b_2$ along a segment in $5N\beta K$ of length at most $\|b_2\|_{\rm op}\|y_3-y'_3\|\le5N\beta\|y_3-y'_3\|$. The coupling argument of Lemma~\ref{lem:blockK} gives $d_1\le s\varepsilon+ Cd_3$, $d_3\le Cd_1$ and $|\mu_b(f)-\mu_{b'}(f)|\le(1+N\beta\Lambda(r))(d_1+ d_3)\le K_3\varepsilon$. \end{proof} \subsection{The constants and the proof of Theorem~\ref{thm:su3blocks}} \begin{proof}[Proof of Theorem~\ref{thm:su3blocks}] Let $r=2\beta_W\le1.04$ and $\gamma=\gamma(r)$. For $c\in rK$ and unit $h$, $\Hess g(c)(h,h)=\Var_{\nu_c}(\varphi_h)$ lies in $[0,\epsilon(r)/\gamma]$ by Lemma~\ref{lem:su3modulus}, which also gives $\Lambda(r)\le\sqrt{\epsilon(r)} /\gamma$; so $\lambda_*=0$ and $\tau=\epsilon(r)/\gamma$ are admissible. The third derivative $D^3g(c)(u,v,v)$ is the third cumulant $\nu_c[(\varphi_u- \nu_c\varphi_u)(\varphi_v-\nu_c\varphi_v)^2]$. Since $U$ and $m(c)$ lie in $K$, whose elements have Hilbert--Schmidt norm at most $\sqrt3$, $|\varphi_u(U)-\nu_c\varphi_u|=|\ReTr((U-m(c))u)|\le2\sqrt3$ for unit $u$, so $|D^3g(c)(u,v,v)|\le2\sqrt3\Var_{\nu_c}(\varphi_v)\le2\sqrt3\,\tau=M$. With these constants $\sqrt{\tau/\gamma}=\sqrt{\epsilon(r)}/\gamma$, the bound used for $\Lambda(r)$, and $\sqrt3N\beta Me^w/\gamma=6N\beta\tau e^w/\gamma$. Every field met in Lemmas~\ref{lem:onesite3} and~\ref{lem:blockK3} lies in $rK$. Since $\epsilon$ is nondecreasing and $\gamma$ decreasing in $r$, every quantity in~\eqref{eq:blockconst3} is nondecreasing in $\beta_W$ while $C<1$, so it suffices to bound $\alpha_3$ at $\beta_W=0.52$. Changing an active link $j$ by $\eta$ changes one staple of each of its ten partners outside its block by a matrix of norm $\eta$, so the field $b_i$ of each such partner moves by $N\beta\eta$, and Lemma~\ref{lem:blockK3} gives $\sum_BC_{Bj}\le10N\beta K_3=\alpha_3(\beta_W)$. Proposition~\ref{prop:blockstate}, whose proof does not use the group, applies with the Hilbert--Schmidt metric. For the numbers we use $e^w\le(1-w)^{-1}$. At $\beta_W=0.515$, $N\beta= \frac{103}{600}$, and the proof of Theorem~\ref{thm:su3} gives $\epsilon (1.03)\le0.667483<0.817^2$ and $\gamma=\frac{5427}{3200}$; hence $\Lambda(r) \le0.48174$, $\tau\le0.39358$, $w\le0.017398$, $s\le0.49043$, $C\le 0.064287$, $K_3\le0.56747$ and $\alpha_3\le0.9742$. At $\beta_W=0.52$, $N\beta=\frac{13}{75}$ and $\gamma=\frac{5411}{3200}$. The terms of the series for $Z_*(1.04)$ in the proof of Theorem~\ref{thm:su3} are at most $1$, $0$, $0.2704$, $0.0468694$, $0.0365581$, $0.0095052$, $0.0035697$, $0.0009424$ and $0.0002581$, with sum at most $1.3681029$, and the last term is less than $1.31\cdot10^{-4}$; so $Z_*(1.04)\le1.368234$, $e^{-0.208}\ge 0.812132$, $\frac{5e^{-0.208}}{9Z_*(1.04)}\ge0.329756$ and $\epsilon(1.04)\le 0.670244<0.81869^2$. Hence $\Lambda(r)\le0.48417$, $\tau\le0.39638$, $w\le 0.017864$, $s\le0.49314$, $C\le0.066654$, $K_3\le0.5727$ and $\alpha_3\le 0.9927$. With $y=1-\alpha_3$, $-\frac12\log\alpha_3\ge\frac12(y+\frac{y^2}2)$, which is at least $0.013$ at $\beta_W=0.515$ and $0.0036$ at $0.52$. \end{proof} \begin{remark}[a computation, used by no theorem]\label{rem:su3blocks} The single-link criterion ends where $2r\Lambda(r)$ reaches one. Before the corrections for the three links the block sum is $\frac{10}{12}$ of the single-link sum $2r\Lambda(r)$, so it can stay below one while $2r\Lambda(r)$ is slightly above one. With the constants of Lemma~\ref{lem:su3modulus} the corrections use most of that room: evaluated in floating point, $\alpha_3$ of Theorem~\ref{thm:su3blocks} reaches one near $\beta_W=0.522$, and the bound $\alpha_1$ of Theorem~\ref{thm:su3} near $0.5176$. Computed constants go further. For $V,W\in\SU(3)$, $g(VcW)=g(c)$ and $x\mapsto VxW$ is an isometry, so the eigenvalues of $\Hess g$ and the norm of $D^3g$ are constant on the orbits, and by Lemma~\ref{lem:orbits} it suffices to bound them on $\mathcal R_r$. The derivatives of $g$ are the cumulants of $\varphi$ under $\nu_c$. Cut the box $[0,r]^2\times[-r,r]\times[0,r]$ into boxes of side $r/8$ and keep the $841$ ($784$ at $r=\frac{38}{25}$, $826$ at $r=\frac{77}{50}$) that survive the two tests of Sec.~\ref{ssec:cover}. On each box $Q$ we compute, with rigorous error bounds, the moments of $\varphi$ of degree at most three at the sixteen corners $b_i$ and at the centre $b_0$, and put $k=m(b_0)$, the mean of $\varphi$ at the centre. \emph{Upper bound.} For $b\in Q$ and every $h$, $\Var_{\nu_b}(\varphi_h)\le \nu_b[(\varphi_h-\ip kh)^2]\le e^{\varsigma/8}\max_i\nu_{b_i}[(\varphi_h- \ip kh)^2]$ by Lemma~\ref{lem:boxes}(ii). \emph{Lower bound.} Integrating the right inequality of~\eqref{eq:hoeffding} against $f\ge0$ and dividing by $Z(b)\le\sum_i\omega_iZ(b_i)$, the left one, gives $\nu_b(f)\ge e^{-\varsigma/8}\min_i\nu_{b_i}(f)$. With $f=(\varphi_h-\nu_b\varphi_h)^2$ this gives $\Var_{\nu_b}(\varphi_h)\ge e^{-\varsigma/8}\min_i\Var_{\nu_{b_i}} (\varphi_h)$. \emph{Third cumulant.} Put $\delta=m(b)-k$. The mean value theorem and the upper bound give $\|\delta\|\le\tau\rho+e$, with $\rho$ the largest distance from $b_0$ in $Q$ and $e$ the error of the computed mean. Expanding the central moment about $k$, \[ \begin{aligned} D^3g(b)(u,v,v)&=T_b(u,v,v)-\delta_u\Var(\varphi_v)\\ &\quad-2\delta_v\Cov(\varphi_u,\varphi_v)-\delta_u\delta_v^2 , \end{aligned} \] with $T_b=\nu_b[(\varphi-k)^{\otimes3}]$, $\delta_u=\ip\delta u$ and the moments under $\nu_b$. For a function $G$ of either sign, writing $G=G^+-G^-$ and applying the two bounds above to $G^\pm$ gives $|\nu_b(G)|\le(\max_i|\nu_{b_i}(G)|+\varepsilon\max_i\nu_{b_i}|G|)/(1- \varepsilon)$ with $\varepsilon=1-e^{-\varsigma/8}$. For $G=(\varphi_u-k_u) (\varphi_v-k_v)^2$ we use $|\nu_{b_i}(G)|\le\|T_{b_i}\|$, the largest singular value of the $18\times324$ unfolding of $T_{b_i}$, and $\nu_{b_i}|G|\le2\sqrt3 \,\nu_{b_i}[(\varphi_v-k_v)^2]$, since $\|U-k\|_{\rm HS}\le2\sqrt3$. The largest eigenvalues and singular values are bounded as in Sec.~\ref{ssec:cover}, and the maxima over these boxes are given below (Appendix~\ref{app:plaqblocks}). In this way every eigenvalue of $\Hess g$ on $rK$ lies in $[0.056015, 0.247316]$ at $r=\frac32$ and $|D^3g(c)(u,v,v)|\le0.182126$ for unit $u,v$; at $r=\frac{38}{25}$ the numbers are $[0.054978,0.248280]$ and $0.183546$, and at $r=\frac{77}{50}$ they are $[0.053955,0.249240]$ and $0.184989$. With these constants and the bounds for $2r\Lambda(r)$ of Remark~\ref{rem:su3cover}, the quantities of~\eqref{eq:blockconst3} evaluated in ball arithmetic are, at $\beta_W=\frac34$, $w\le0.017935$, $s\le0.34088$, $C\le0.043520$, $K_3\le0.38609$ and $\alpha_3\le0.96521$; at $\beta_W=0.72$, $\alpha_3\le0.91815$; at $\beta_W=0.76$, $w\le0.018609$, $s\le0.34673$, $C\le0.045770$, $K_3\le0.39454$ and $\alpha_3\le0.99950$; at $\beta_W=0.77$, $w\le0.019298$, $s\le0.34186$, $C\le0.047232$, $K_3\le 0.38953$ and $\alpha_3\le0.99978$. They would carry the block criterion to $\beta_W=0.77$, with a gap of $0.0177$ at $\frac34$, where the cover of Remark~\ref{rem:su3cover} alone gives $1.1\cdot10^{-4}$. Evaluated in floating point with the constants of a scan of $\mathcal R_r$, the block sum stays below one up to $\beta_W\approx0.78$. No result of the paper uses these numbers. \end{remark} \section{Where the methods stop}\label{sec:limits} \emph{$\SU(2)$.} Theorem~\ref{thm:ceiling} closes the single-link route in every forest gauge at $\beta_W=\frac13$. Blocks of several links, where the influence of a boundary link on a whole block is bounded at once, in the spirit of the Dobrushin--Shlosman criterion~\cite{DS85}, pass it: plaquette blocks reach $\beta_W=0.348$ (Theorem~\ref{thm:blocks}), and the floating-point bound of Remark~\ref{rem:blockgap}, which no result uses, stops near $0.357$. Larger blocks are the next step. \emph{Curvature.} Remark~\ref{rem:hf} locates the end of the curvature method in this gauge between $|\beta|=\frac1{29}$ and $\frac1{26.25}$; the computation of Remark~\ref{rem:certificate}, which no result uses, moves the first value to $1/28.840002$. \emph{$\SU(3)$.} Theorems~\ref{thm:su3} and~\ref{thm:su3blocks} use the bound of Lemma~\ref{lem:su3modulus}, whose losses are Jensen's inequality and the Poincar\'e inequality for $\varphi_h$; evaluated in floating point, it keeps $\alpha_1$ below one up to $\beta_W\approx0.5176$ and $\alpha_3$ up to $0.522$. The route of Sec.~\ref{sec:su3} with a corrector bounds the $H^{-1}(\nu_b)$ norm of $\varphi_h-\nu_b\varphi_h$, which is at least $L_h(b)$, and needs it uniformly on $rK$. Its Galerkin values (polynomials of degree two) at matrices of $rK$ give $2r$ times the norm equal to $0.79$ at $r=1.25$, $0.89$ at $r=1.4$, $0.97$ at $r=1.5$ and $1.009$ at $r=1.55$, so the route stops at or below $\beta_W\approx0.77$ even with exact constants. With $u=s\varphi_h$, Lemma~\ref{lem:corrector} stops earlier: in floating point its largest values on $\mathcal R_r$ lie on the face $\operatorname{diag}(r,\rho,\rho)$ with $\rho$ between $0.75r$ and $0.8r$, and $2r$ times the bound is $0.9458$ at $r=1.36$, $0.9767$ at $r=1.40$ and $1.0080$ at $r=1.44$, so this corrector ends near $\beta_W\approx0.715$. With $u=q_h$ of degree two the largest values on the same face are $0.9305$ at $r=1.44$, $0.9729$ at $r=1.50$ and $1.0018$ at $r=1.54$ (Fig.~\ref{fig:face}), close to the Galerkin values. Lemma~\ref{lem:gap} therefore costs little at these couplings, and most of the distance between the floating-point value at $r=1.5$ and the computed $0.999887$ of Remark~\ref{rem:su3cover} is the interpolation over boxes. With this corrector the single-link bound would reach $\frac34$ and the blocks $0.77$ (Remarks~\ref{rem:su3cover} and~\ref{rem:su3blocks}), but these evaluations are computations and no result uses them; a bound for the corrector of degree two that holds on all of $rK$ and is derived by hand is the missing step. Past $0.78$ one needs larger blocks, whose sensitivity does not go through the one-link modulus. \begin{figure*}[t] \centering \includegraphics[width=0.84\textwidth]{figures/fig_face.png} \caption{The floating-point value of $2r\sup_h$ of the bound of Lemma~\ref{lem:corrector} on the face $b=\operatorname{diag}(r,\rho,\rho)$, as a function of $\rho/r$, for $r=1.44$, $1.50$ and $1.54$: dashed with the corrector $u=s\varphi_h$ for the best $s$, solid with $u=q_h$ of degree two for the best $C$. The grey plane is the Dobrushin threshold one. The first corrector is above it already at $r=1.44$; the second stays below it at $r=1.50$, where the computation of Remark~\ref{rem:su3cover} bounds it on all of $\mathcal R_r$, and crosses it by $r=1.54$.} \label{fig:face} \end{figure*} \emph{Other groups.} For groups other than $\SU(2)$ and $\SU(3)$ only the curvature route is carried out here (Theorem~\ref{thm:groups}), and it stops where the bound $\frac{29}2$ uses up the curvature of $G$, at $N|\beta|=\frac2{29}\kappa_G$. The Dobrushin route needs a bound for the Kantorovich modulus of the one-link measure of each group, which is not attempted here. \emph{The four open problems.} Our earlier paper~\cite{BB26a} listed four problems between the strong-coupling results and the Clay problem~\cite{JW06}: (1) uniqueness and a gap at intermediate fixed coupling; (2) the crossover from strong coupling to Ba{\l}aban's weak-coupling regime; (3) the continuum limit with the Osterwalder--Schrader axioms; and (4) a gap in physical units. The present paper moves the boundary of (1): for $\SU(2)$ from $\beta_W=\frac29$ to $0.348$, for $\SU(3)$ from $\frac9{32}$ to $0.52$, with a gap at $\frac12$ of $0.052$ in lattice units, and for every compact simple group, the exceptional groups included, past the curvature ceiling of the ungauged measure, with an explicit gap. It does not close (1); the region between these values and the scaling region, near $\beta_W\approx2.2$ for $\SU(2)$ and $\beta_W\approx6$ for $\SU(3)$, is open, and with it problem (2). Problems (3) and (4) are not solved here. Every bound proved here is uniform in lattice units, so by~\cite{BB26b} none of them can by itself produce a continuum mass gap; that requires control of the gap in physical units along the renormalisation group flow~\cite{Bal84a,BSB26}.\footnote{Every one of these approaches needs an input at physical length scales. Renormalisation group approaches to the ultraviolet behaviour of gauge and gravitational theories, of which~\cite{Bal84a,Dim13} are the rigorous ones for Yang--Mills theory, have also been proposed for the effective dimension of spacetime~\cite{SBG26} and for completions of quantum gravity~\cite{Bha22,BRS26}; they do not bear on the infrared estimates needed here.} \emph{The continuum problem.} Two general facts limit what can be imported into problems (3) and (4) above from constructions of continuum measures. Both concern a random distribution $F$ on $\R^d$, a Borel probability measure on the tempered distributions with finite moments, and the growth condition \begin{equation}\label{eq:growth} \E\,e^{F(\chi)}\le e^{C(\|\chi\|_1+\|\chi\|_2^2)}\qquad\text{for all real } \chi\in\mathcal S(\R^d), \end{equation} which is the growth condition of the regularity axiom of Glimm and Jaffe at $p=2$~\cite[Sect.~6.1, (6.1.5)]{GJ87}. The next proposition shows that it rules out every field whose two-point function is too singular at short distance. \begin{proposition}\label{prop:growth} Let $F$ satisfy~\eqref{eq:growth}, and suppose that its connected two-point function is, off the diagonal, a locally integrable function $S_2(x,y)$ with $S_2(x,y)\ge c|x-y|^{-2\Delta}\bigl(\log\frac1{|x-y|}\bigr)^{-k}$ for $0<|x-y|\le r_0$, where $c>0$, $k\ge0$ and $2\Delta>d$. Then $F$ does not exist: no random distribution has both properties. \end{proposition} \begin{proof} Applying~\eqref{eq:growth} to $\chi$ and $-\chi$ and using $\cosh x\ge1+\frac{x^2}2$ gives $\E F(\chi)^2\le2(e^{C(\|\chi\|_1+\|\chi\|_2^2)}-1)$. Fix nonnegative $\varphi,\psi\in C_c^\infty(\R^d)$, not zero, with supports in a ball of radius $\frac R2$ and at distance $\delta>0$ from each other, and put $\varphi_\varepsilon(x)=\varepsilon^{-d/2}\varphi(x/\varepsilon)$, and likewise $\psi_\varepsilon$. Then $\|\varphi_\varepsilon\|_2=\|\varphi\|_2$ and $\|\varphi_\varepsilon\|_1=\varepsilon^{d/2}\|\varphi\|_1\le\|\varphi\|_1$ for $\varepsilon\le1$, so the covariance of $F(\varphi_\varepsilon)$ and $F(\psi_\varepsilon)$ is bounded uniformly in $\varepsilon$ by the Cauchy--Schwarz inequality. On the other hand the supports of $\varphi_\varepsilon$ and $\psi_\varepsilon$ are disjoint, and for $u,v$ in the supports $\varepsilon\delta\le|\varepsilon u-\varepsilon v|\le\varepsilon R$. For $\varepsilon R\le r_0$ and $\varepsilon R<1$ the lower bound for $S_2$ therefore gives \[ \begin{split} \Cov\bigl(F(\varphi_\varepsilon),F(\psi_\varepsilon)\bigr)&=\varepsilon^d\!\int\!\!\int \varphi(u)\psi(v)S_2(\varepsilon u,\varepsilon v)\,du\,dv\\ &\ge c\,R^{-2\Delta}\varepsilon^{d-2\Delta}\Bigl(\log\frac1{\varepsilon\delta}\Bigr)^{-k} \|\varphi\|_1\|\psi\|_1 , \end{split} \] which tends to infinity as $\varepsilon\to0$. \end{proof} The action density $\Tr F_{\mu\nu}F_{\mu\nu}$ has dimension $\Delta=4$ in four dimensions. A two-sided short-distance law of the form $|x-y|^{-8}$ times the fourth power of the running coupling, which is what asymptotic freedom\footnote{For $\SU(N)$ the renormalisation group gives~\cite{GW73,Pol73}, with $b_0=11N/(48\pi^2)$ and the invariant scale $\Lambda$, \[ g(a)^{-2}=2b_0\log\tfrac1{a\Lambda}+O\bigl(\log\log\tfrac1{a\Lambda}\bigr), \] so the inverse coupling $\beta_W=2N/g(a)^2$ grows like $\frac{11N^2}{12\pi^2}\log\frac1{a\Lambda}$ along the continuum tuning.} predicts, has $2\Delta=8>4$ and $k=2$; so that field cannot satisfy~\eqref{eq:growth}, and a proof of rotation invariance for it has to use something other than this condition. In three dimensions the free part of the action density already has $2\Delta=6>3$. Condition~\eqref{eq:growth} suits fields of dimension at most $\frac d2$, such as the field strength of the free Maxwell field, whose covariance is bounded in momentum space. Positivity, symmetry and regularity do not produce a gap by themselves. \begin{proposition}\label{prop:nogapgff} On $\R^4$ let $F$ be the centred Gaussian random distribution with covariance \[ \widehat C(p)=\int_0^1\frac{m^2}{|p|^2+m^2}\,dm . \] Then $F$ is invariant under all Euclidean motions, reflection positive, satisfies~\eqref{eq:growth} with $C=\frac12$, and its Osterwalder--Schrader Hamiltonian has no gap above the vacuum. \end{proposition} \begin{proof} Since $0\le\widehat C\le1$, $\E e^{F(\chi)}=e^{\frac12\ip\chi{C\chi}}\le e^{\frac12 \|\chi\|_2^2}$, with the unitary Fourier transform. Invariance is clear. The covariance is the integral over $m\in(0,1)$, with the weight $m^2dm$, of the covariances of free fields of mass $m$, each of which is reflection positive~\cite[Sect.~6.2]{GJ87}; a positive combination of reflection positive covariances is reflection positive, and so is the Gaussian measure with such a covariance~\cite[Sect.~6.2]{GJ87}. For $\chi(t,\mathbf x)=h(t)g(\mathbf x)$ with $h\ge0$ supported in $t>0$, not zero, and $\hat g(0)\ne0$, the vector $v=F(\chi)$ is orthogonal to the vacuum, and after the Fourier transform in $\mathbf x$, \[ \begin{split} \ip v{e^{-sH}v}={}&\int_0^1m^2\,dm\int\frac{|\hat g(\mathbf k)|^2}{2\omega} e^{-\omega s}\\ &\times\Bigl(\int_0^\infty h(t)e^{-\omega t}dt\Bigr)^2\,d\mathbf k , \end{split} \] with $\omega=\sqrt{|\mathbf k|^2+m^2}$ and constants absorbed in $\hat g$. If $H$ had a gap $m_0>0$, the left side would be at most $e^{-m_0s}\|v\|^2$. For $0<\eta0$, a contradiction as $s\to\infty$. \end{proof} The Gaussian field of Proposition~\ref{prop:nogapgff} is not a gauge theory; it shows only that an infrared estimate, such as a bound $\ip{v_a}{T_a^{\lfloor t/a\rfloor}v_a}\le C_ve^{-m_*t}$ uniform in the spacing $a$ along the continuum tuning, has to be proved in addition to the axioms. The lattice bounds of this paper are of this form only with $m_*$ proportional to $1/a$, and by~\cite{BB26b} such bounds give trivial limits. The routes we know to a continuum gap therefore all stop at the same two estimates: the construction of the infinite-volume continuum measure past Ba{\l}aban's small-field analysis~\cite{Bal84a,Bal89b}, and a decay estimate in physical units at large scales. \section{The continuum step at one physical scale}\label{sec:bypass} The bounds of the preceding sections hold at fixed coupling and give a gap that is at least a fixed number in lattice units, and~\cite{BB26b} shows that no such bound can produce a continuum gap. This section replaces the single-link condition of Proposition~\ref{prop:state} by a condition on one cube of $n^4$ sites, averaged over its translates in the manner of Dobrushin and Shlosman~\cite{DS85}, and proves that the condition gives a gap of order $1/n$ in lattice units. If it holds with $n$ proportional to $1/a$, that is, in a cube of fixed physical size, the gap survives in physical units and passes to every continuum limit (Theorem~\ref{thm:transfer}). The two statements that remain unproved, the cube estimate and the existence of the limit, are stated in Sec.~\ref{sec:inputs}. \subsection{Averaged cubes} For a finite set $V\subset A$ let $\gamma_V(\cdot\mid Q)$ be the probability measure on $G^V$ with density proportional to $\exp\bigl(N\beta\sum_{p\cap V\ne\emptyset}\ReTr Q^F_p\bigr)$ with respect to Haar measure, the links outside $V$ being those of $Q$. Its conditional law of $Q_e$, $e\in V$, given the other links of $V$ is $\gamma^F_e$. Write $d_V(Q_V,Q'_V)=\sum_{i\in V} \delta(Q_i,Q'_i)$ and, for $j\in A\setminus V$, \[ C_{V,j}=\sup\frac{W_1^{d_V}\bigl(\gamma_V(\cdot\mid Q),\gamma_V(\cdot\mid Q')\bigr)}{\delta(Q_j,Q'_j)}, \] the supremum over pairs that differ only at $j$; $W_1^{d_V}$ is the Kantorovich distance for $d_V$. The coefficient vanishes unless $j$ shares a plaquette with a link of $V$. Let $n\ge2$ be even and let $V$ be the set of active links $(x,\mu)$ with $x\in[0,n)^4$. The direction field has period two, so the translations by $2\Z^4$ preserve $F$, $A$ and $\gamma^F$, and the translates $V_x=V+x$, $x\in2\Z^4$, satisfy $C_{V_x,j+x}=C_{V,j}$. Every active link lies in exactly $N_n=(n/2)^4$ of them. A \emph{class} is an orbit of $2\Z^4$ on $A$, that is, a residue of the base point modulo two together with a direction; there are at most $64$ classes. For a class $c$ put \[ K_c=\sum_{j\in c\setminus V}C_{V,j}. \] \begin{definition}\label{def:cube} For $0<\alpha<1$, the condition $D(n,\alpha)$ at the coupling $\beta$ is $K_c\le\alpha N_n$ for every class $c$. \end{definition} Since $\sum_{x\in2\Z^4,\,j\notin V_x}C_{V_x,j}=\sum_{x}C_{V,j-x}[j-x\notin V] =K_{c(j)}$ for the class $c(j)$ of $j$, the condition says that the total influence of a link on the translates that do not contain it is at most the fraction $\alpha$ of the number of translates that do. Figure~\ref{fig:cubes} shows the cubes. \begin{figure*}[t] \centering \includegraphics[width=0.9\textwidth]{figures/fig_cubes.png} \caption{(a) The cube $V$ of side $n=4$ and two of its translates by $2\Z^4$, drawn in three of the four dimensions. A link $j$ outside $V$ enters $K_c$ through every translate whose boundary it touches, and $D(n,\alpha)$ compares that total with the $N_n=(n/2)^4$ translates that contain $j$. (b) The cube of Theorem~\ref{thm:transfer}: its physical side $\ell=na$ is held fixed while the spacing is halved, so $n$ doubles at each step and the lattice gap $\log(1/\alpha)/(2n)$ is of order $a/\ell$.} \label{fig:cubes} \end{figure*} For $n=2$ it is a bound on blocks of $48$ links. Distances between links are those of their midpoints; a link of $V$ and a link sharing a plaquette with $V$ are at distance at most $\rho_n=2n$, and $c_n$ denotes the number of active links within distance $\rho_n$ of a given one. \begin{theorem}\label{thm:onescale} Let $\beta\ge0$ and suppose that $D(n,\alpha)$ holds. Then statements \textup{(i)}--\textup{(iv)} of Proposition~\ref{prop:state} hold, \textup{(v)} holds for every $\lambda<\log(1/\alpha)/\rho_n$, and the transfer matrix of $\omega$ has spectral gap at least $\log(1/\alpha)/(2n)$. \end{theorem} \begin{proof} The proof of Proposition~\ref{prop:state} uses the coefficients $C_{ee'}$ only through Lemma~\ref{lem:comparison} and Corollary~\ref{cor:decay}. Lemma~\ref{lem:dscomparison} of Appendix~\ref{app:comparison} gives the same conclusions under $D(n,\alpha)$, for every $\lambda>0$ with $\alpha e^{\lambda\rho_n}<1$, with the constant $\theta D/(1-\theta)$ replaced by $D(1+c_ne^{\lambda\rho_n})/(1-\theta)$ and $\theta=\alpha e^{\lambda\rho_n}$. The measures used there are invariant under every $\gamma_{V_x}$ with $V_x$ disjoint from the exceptional set: a Gibbs measure of $\gamma^F$ and its conditional measures given the links of a finite set by Lemma~\ref{lem:dscomparison}(a), and the extended finite-volume measure of part (ii) because every plaquette that meets such a $V_x$ lies in $P(\Lambda)$. Part (vi) then follows from Proposition~\ref{prop:gapdecay} as before, with every $\lambda<\log(1/\alpha)/(2n)$. \end{proof} \subsection{The continuum limit} Let $a_k=2^{-k}a_0$ and let $\beta_k\ge0$ be couplings, typically with $\beta_k\to\infty$. Suppose that the lattice states $\omega_k$ at $\beta_k$ exist as in Proposition~\ref{prop:state}, with transfer matrices $\mathbb T_k$ on the reconstructed spaces $\mathcal H_k$ and vacua $\Omega_k$ (Appendix~\ref{app:transfer}). Let $\mathcal T$ be a complex vector space of test functions on $\R^4$ supported in $\{t>0\}$ and, for each $k$, let $f\mapsto O_k(f)$ be a linear map from $\mathcal T$ to gauge-invariant local observables supported in positive lattice times; the example to keep in mind is $O_k(f)=Z_k\sum_xa_k^4f(a_kx)\Phi(x)$ for a local gauge-invariant field $\Phi$, a renormalisation constant $Z_k$ and $\Phi(x)$ the translate of $\Phi$ to $x$. Write $\vartheta$ for the time reflection and $\tau_t$ for the time translation by $t$ lattice steps. \begin{definition}\label{def:limit} A \emph{continuum limit} of $(\omega_k,O_k)$ is a Hilbert space $\mathcal H$ with a unit vector $\Omega$, a self-adjoint operator $H\ge0$ with $H\Omega=0$, and a linear map $f\mapsto\psi(f)$ from $\mathcal T$ to $\mathcal H$ with dense range, such that for all $f,g\in\mathcal T$ and all dyadic $\tau\ge0$, \[ \begin{split} \ip{\psi(f)}{e^{-\tau H}\psi(g)}&=\lim_{k\to\infty}\omega_k\bigl(\vartheta (\overline{O_k(f)})\,\tau_{\tau/a_k}O_k(g)\bigr),\\ \ip\Omega{\psi(f)}&=\lim_{k\to\infty}\omega_k\bigl(O_k(f)\bigr), \end{split} \] the limits taken over the $k$ for which $\tau/a_k$ is an integer. It is \emph{nontrivial} if $\mathcal H\ne\C\Omega$. \end{definition} The Osterwalder--Schrader reconstruction~\cite{OS73,OS75} produces such a limit from convergent renormalised Schwinger functions that satisfy the axioms; the definition records only what the next theorem uses. \begin{theorem}\label{thm:transfer} Suppose that there are $\alpha<1$, $\ell>0$ and even $n_k$ with $n_ka_k\le\ell$ such that $D(n_k,\alpha)$ holds at $\beta_k$ for every $k$. Then every continuum limit of $(\omega_k,O_k)$ satisfies \[ \sigma(H)\subset\{0\}\cup[m_*,\infty),\qquad m_*=\frac{\log(1/\alpha)}{2\ell}, \] and $0$ is a simple eigenvalue. If the limit is nontrivial, $\sigma(H)\cap[m_*,\infty)\ne\emptyset$, so $H$ has a mass gap of at least $m_*$. \end{theorem} \begin{proof} By Theorem~\ref{thm:onescale} the state $\omega_k$ exists and $\mathbb T_k$ has gap at least $\log(1/\alpha)/(2n_k)$ on the complement of $\Omega_k$. Let $f\in \mathcal T$, put $O=O_k(f)$ and $\varphi_k=\Psi_O-\omega_k(O)\Omega_k$. In the reconstruction of Appendix~\ref{app:transfer}, $\ip{\Psi_F}{\mathbb T_k^t\Psi_H}=\omega_k(\vartheta(\bar F)\tau_tH)$, so $\ip{\Omega_k}{\Psi_O}=\omega_k(O)$, and $\mathbb T_k\Omega_k=\Omega_k$; hence, for integers $t\ge0$, \[ \begin{split} \ip{\varphi_k}{\mathbb T_k^t\varphi_k}&=\omega_k\bigl(\vartheta(\bar O)\,\tau_tO\bigr) -|\omega_k(O)|^2\\ &\le e^{-t\log(1/\alpha)/(2n_k)}\,\|\varphi_k\|^2 . \end{split} \] Take $\tau\ge0$ dyadic and $k$ large, so that $t=\tau/a_k$ is an integer; then $t/(2n_k)=\tau/(2n_ka_k)\ge\tau/(2\ell)$, and the right side is at most $e^{-m_*\tau}\|\varphi_k\|^2$. Both sides converge by Definition~\ref{def:limit}. With $\varphi=\psi(f)-\ip\Omega{\psi(f)}\Omega$ and $e^{-\tau H}\Omega=\Omega$ the limit reads \[ \ip\varphi{e^{-\tau H}\varphi}\le e^{-m_*\tau}\|\varphi\|^2 . \] The maps $f\mapsto O_k(f)$ and $f\mapsto\psi(f)$ are linear, so the vectors $\varphi$ form a linear subspace, and it is dense in $\Omega^\perp$ because the range of $\psi$ is dense. The inequality extends to $\Omega^\perp$ by continuity, and from dyadic $\tau$ to all $\tau\ge0$ by the strong continuity of $e^{-\tau H}$. For $\varphi\in\Omega^\perp$ with spectral measure $\mu_\varphi$ of $H$ this says $\int e^{-\tau E}d\mu_\varphi(E)\le e^{-m_*\tau} \mu_\varphi([0,\infty))$ for all $\tau\ge0$; if $\mu_\varphi([0,m_*-\eta])>0$ for some $\eta>0$, the left side is at least $\mu_\varphi([0,m_*-\eta]) e^{-(m_*-\eta)\tau}$, which exceeds the right side for large $\tau$. So $\mu_\varphi$ lives on $[m_*,\infty)$ for every $\varphi\in\Omega^\perp$, which gives both claims about $0$, and the spectrum on $\Omega^\perp\ne\{0\}$ lies in $[m_*,\infty)$ and is not empty. \end{proof} The proof uses $D(n_k,\alpha)$ only through the decay of the two-point functions, and the same argument accepts any decay with constants that depend on the observable. \begin{theorem}\label{thm:clustering} Let $m_0>0$ and let $\mathcal T_0$ span $\mathcal T$. Suppose that for each $f\in\mathcal T_0$ there are $C_f<\infty$, $\tau_f\ge0$ and $k_f$ such that \begin{align*} C^f_k(\tau)&=\omega_k\bigl(\vartheta(\overline{O_k(f)})\,\tau_{\tau/a_k} O_k(f)\bigr)-|\omega_k(O_k(f))|^2\\ &\le C_fe^{-m_0\tau} \end{align*} for all $k\ge k_f$ and all dyadic $\tau\ge\tau_f$ with $\tau/a_k\in\Z$. Then every continuum limit of $(\omega_k,O_k)$ satisfies $\sigma(H)\subset \{0\}\cup[m_0,\infty)$, and $0$ is a simple eigenvalue. If the limit is nontrivial, $H$ has a mass gap of at least $m_0$. \end{theorem} \begin{proof} For $f\in\mathcal T$ put $\varphi(f)=\psi(f)-\ip\Omega{\psi(f)}\Omega$, which is linear in $f$. Since $e^{-\tau H}$ is self-adjoint and fixes $\Omega$, Definition~\ref{def:limit} gives $\ip{\varphi(f)}{e^{-\tau H} \varphi(f)}=\lim_kC^f_k(\tau)$. For $f\in\mathcal T_0$ let $\mu$ be the spectral measure of $H$ at $\varphi(f)$. Then $\int e^{-\tau E}d\mu(E)\le C_fe^{-m_0\tau}$ for dyadic $\tau\ge\tau_f$. If $\mu([0,m_0-\eta])>0$ for some $\eta\in(0,m_0)$, the left side is at least $\mu([0,m_0-\eta])e^{-(m_0- \eta)\tau}$, which is impossible for large $\tau$. Hence the spectral projection $E_{[0,m_0)}$ of $H$ vanishes on $\varphi(\mathcal T_0)$. By linearity it vanishes on $\varphi(\mathcal T)$, and since its kernel is closed, it vanishes on the closure of $\varphi(\mathcal T)$, which is $\Omega^\perp$ because $\psi(\mathcal T)$ is dense. So $E_{[0,m_0)}= \ip\Omega\cdot\,\Omega$, which gives both claims. \end{proof} Theorem~\ref{thm:transfer} is the case $m_0=m_*$, $\tau_f=0$ and $C_f= \sup_{k\ge k_f}\|\varphi_k\|^2$, which is finite because $\|\varphi_k\|^2= C^f_k(0)$ converges. Theorem~\ref{thm:clustering} needs neither a lattice gap nor the gauge of this paper. It applies to the output of a cluster expansion of any coarse effective theory, provided the expansion bounds the correlations of a spanning set of renormalised observables in physical units. Theorem~\ref{thm:transfer} bypasses the obstruction of~\cite{BB26b}: under the hypothesis the lattice gap is at least $\log(1/\alpha)/(2n_k)$, of order $a_k/\ell$ in lattice units, which tends to zero, and only its value in physical units is bounded below. In the language of~\cite{BB26b} the hypothesis implies the gap part (R2) of Hypothesis~R, with $m=m_*$, and the conclusion holds for every continuum limit without a separate assumption on the gap. \subsection{Smoothed observables} The observables $O_k(f)$ of Definition~\ref{def:limit} need not be local. The renormalised fields that one expects to converge are smoothed by the gradient flow of the Wilson action~\cite{Lus10}\footnote{The flow moves the gauge field along the negative gradient of the Wilson action. At positive flow time the flowed fields are smooth on the scale of the square root of the flow time, and in perturbation theory their gauge-invariant composite fields need no further renormalisation~\cite{Lus10,LW11}.}; at positive flow time their correlation functions are finite to all orders of perturbation theory once the coupling is renormalised~\cite{LW11}. Flow in all four directions mixes the two sides of the reflection plane, but the spatial flow of~\cite[Sec.~II.B]{SS23} does not: it flows the links of each time slice under the Wilson action of that slice and leaves the temporal links alone. Fix a bi-invariant Riemannian metric on $G$. For a configuration $U$ of all links of $\Z^4$ (for the states of this paper, the links of $F$ equal to $1$), let $\Phi_s(U)$, $s\in\R$, solve \[ \frac{d}{ds}U_e=-\nabla_e\!\!\sum_{p\ni e,\ p\ \mathrm{spatial}} \bigl(N-\ReTr U_p\bigr) \] for every spatial link $e$, the sum running over the four plaquettes of the time slice of $e$ that contain it, with the temporal links held fixed and $\nabla_e$ the gradient in the variable $U_e$. \begin{proposition}\label{prop:spatialflow} \textup{(i)} $\Phi_s$ is defined on every configuration of the infinite lattice, $\Phi_s\circ\Phi_{s'}=\Phi_{s+s'}$, and $\Phi_s$ commutes with gauge transformations, with spatial and temporal translations and with the reflections in time. The link $\Phi_s(U)_e$ depends only on the spatial links of the time slice of $e$, and there are constants $\kappa,c$ depending only on $G$ such that, if $\Phi^{R}_s$ denotes the flow of the links of that slice within distance $R$ of $e$ with all other links held fixed, \[ \delta\bigl(\Phi_s(U)_e,\Phi^R_s(U)_e\bigr)\le D\,e^{\kappa|s|} \frac{(c|s|)^{R+1}}{(R+1)!} \] for every configuration $U$. \textup{(ii)} Let $\omega$ be a state as in Appendix~\ref{app:transfer} and $O$ a bounded gauge-invariant function of finitely many links at times in $[t_0,t_1]$, continuous in them. Then $O\circ\Phi_s$ is a bounded gauge-invariant function of the links at times in $[t_0,t_1]$ and a uniform limit of gauge-invariant local functions at those times. \textup{(iii)} Theorem~\ref{thm:transfer} holds with $O_k(f)$ replaced by $O_k(f)\circ\Phi_{s_k}$ for any $s_k\in\R$, in particular for $s_k=\tau/a_k^2$ at a fixed flow time $\tau$ in physical units. \textup{(iv)} The image measure $(\Phi_s)_*\omega$ is reflection positive, and its transfer matrix is unitarily equivalent to that of $\omega$, the unitary fixing the vacuum. \end{proposition} \begin{proof} (i) Embed $G$ in a matrix algebra. The right side of the equation is a smooth function of the links of the four plaquettes containing $e$, tangent to $G$ at $U_e$, so on the Banach space of bounded assignments of matrices to the spatial links it is a vector field that is bounded and Lipschitz on a neighbourhood of the configurations with values in $G$, with a Lipschitz constant that does not depend on the volume because every link meets finitely many others. The Picard--Lindel\"of theorem gives a unique local solution; it stays on $G$ because the field is tangent there, and since the field is bounded on $G$ the solution exists for all $s$; uniqueness gives the group law. The action of a time slice and the metric are invariant under gauge transformations, translations and the reflections, so the flow commutes with them by uniqueness. Since the equation of a spatial link involves only links of its own slice, so does the solution. For the comparison, let $d_{e'}(s)$ be the distance between the two flows at a link $e'$ of the slice. Smoothness on the compact group gives constants $\kappa_0,c_1$ with $d_{e'}(s)\le\int_0^{|s|}\bigl(\kappa_0d_{e'}+c_1\sum_{e''\sim e'}d_{e''}\bigr)$ for $e'$ within distance $R$ of $e$, where $e''\sim e'$ means that the two links share a spatial plaquette (twelve neighbours), while $d_{e'}\le D$ beyond. Iterating the integral form of this inequality, a term that reaches the links beyond distance $R$ needs at least $R+1$ steps of $\sim$, and $\sum_{k>R}x^k/k!\le e^xx^{R+1}/(R+1)!$; this gives the bound with $c=12c_1$ and $\kappa=\kappa_0+c$. (ii) The truncated flow $\Phi^R_s$ commutes with gauge transformations for the same reason as $\Phi_s$, so $O\circ\Phi^R_s$ is a gauge-invariant function of finitely many links at times in $[t_0,t_1]$. By (i) and the uniform continuity of $O$, it converges to $O\circ\Phi_s$ uniformly as $R\to\infty$. (iii) The proof of Theorem~\ref{thm:transfer} uses $O_k(f)$ only through the vectors $\Psi_{O_k(f)}$ and the identity $\ip{\Psi_F}{\mathbb T^t\Psi_H}=\omega(\vartheta(\bar F)\tau_tH)$. By (ii) and the footnote in Appendix~\ref{app:transfer}, both extend to uniform limits of local functions at positive times, and $f\mapsto O_k(f)\circ\Phi_{s_k}$ is linear. (iv) By (i), $\Phi_s$ and $\Phi_{-s}$ preserve the functions of the links at positive times and commute with $\vartheta$ and $\tau_t$. Hence $(\Phi_s)_*\omega\bigl(\vartheta(\bar F)\tau_tH\bigr)=\omega\bigl(\vartheta(\overline{F\circ \Phi_s})\,\tau_t(H\circ\Phi_s)\bigr)$, which for $t=0$ and $H=F$ is nonnegative, and $F\mapsto F\circ\Phi_s$ induces an isometry between the reconstructed spaces that intertwines the transfer matrices and fixes the vacuum; $\Phi_{-s}$ gives its inverse. \end{proof} Part (iv) says that smoothing changes the variables and not the theory: it neither creates nor removes a gap. Its use is in the choice of the observables. With $O_k(f)$ the spatially flowed fields at a fixed physical flow time, for instance $Z_k\sum_xa_k^4f(a_kx)\bigl(\mathcal E_{\tau}(x)- \omega_k(\mathcal E_\tau(x))\bigr)$ with $\mathcal E_\tau(x)$ the flowed action density of the slice, statement (E) asks for the convergence of correlation functions that are known to be finite in perturbation theory~\cite{LW11}, and Theorem~\ref{thm:transfer} applies to them. Theorem~\ref{thm:flowlimit} below constructs their continuum limits along subsequences; their nontriviality is not proved here. \subsection{Renormalised flowed observables}\label{ssec:flowlimit} The spatially flowed fields can be renormalised by one condition at a single physical separation, and their correlation functions then have continuum limits along subsequences without further assumptions. What remains open is whether such a limit is trivial, and that question reduces to one ratio of lattice two-point functions. Fix a bounded gauge-invariant function $\Phi$ of finitely many spatial links of the slice $\{x_0=0\}$, continuous in them, a flow time $\tau>0$ in physical units, and a nonzero test function $f_0\in C_c(\R^4)$ supported in $\{t>0\}$. Put $s_k=\tau/a_k^2$, write $\Phi_x$ for the translate of $\Phi$ to $x\in\Z^4$, and let \[ A_k=\sum_{x\in\Z^4}a_k^4f_0(a_kx)\bigl(\Phi_x\circ\Phi_{s_k} -\omega_k(\Phi_x\circ\Phi_{s_k})\bigr), \] \[ C_k(\sigma)=\omega_k\bigl(\vartheta(\bar A_k)\,\tau_{\sigma/a_k}A_k\bigr) \] for dyadic $\sigma\ge0$ (the sum is finite, and $\sigma/a_k$ is an integer for large $k$). By reflection positivity $C_k(0)\ge0$; assume $C_k(0)>0$ and set $Z_k=C_k(0)^{-1/2}$. Let $\mathcal D$ be the set of points $y=(t,y')\in\R^4$ with dyadic coordinates (in units of $a_0$) and $t>0$, and for $y\in\mathcal D$ and $k$ with $y/a_k\in\Z^4$ let $A^y_k$ be the translate of $A_k$ by $y/a_k$. \begin{figure*}[t] \centering \includegraphics[width=0.82\textwidth]{figures/fig_packet.png} \caption{The packet $f_0$ of Theorem~\ref{thm:flowlimit}, supported in the time window $[t_0,t_1]$ above the reflection plane $t=0$, its reflection $\vartheta f_0$ and its translate $\tau_\sigma f_0$ by $\sigma$ in time. The correlation $C_k(\sigma)$ pairs the reflected packet with the translated one; $C_k(0)$ is the reflected norm that renormalises the packet.} \label{fig:packet} \end{figure*} \begin{theorem}\label{thm:flowlimit} \textup{(i)} For $y,y'\in\mathcal D$ and every $k$ with $y/a_k,y'/a_k\in\Z^4$, \[ Z_k^2\,\bigl|\omega_k\bigl(\vartheta(\bar A^y_k)A^{y'}_k\bigr)\bigr|\le1 . \] \textup{(ii)} Every sequence of $k$ has a subsequence along which $K(y,y')=\lim Z_k^2\,\omega_k(\vartheta(\bar A^y_k)A^{y'}_k)$ exists for all $y,y'\in\mathcal D$ and $\rho(\sigma)=\lim C_k(\sigma)/C_k(0)$ exists for all dyadic $\sigma>0$. Along it there is a probability measure $\mu$ on $[0,\infty]$ with \[ \rho(\sigma)=\int_{[0,\infty]}e^{-\sigma E}\,d\mu(E)\qquad(\sigma>0), \] the integrand being $0$ at $E=\infty$. \textup{(iii)} Along such a subsequence, let $\mathcal T$ be spanned by the translates $f_0(\cdot-y)$, $y\in\mathcal D$, and one more function $f_*\in C_c(\R^4)$ supported in $\{t>0\}$ outside their span, and let $O_k$ be the linear map with $O_k(f_0(\cdot-y))=Z_kA^y_k$ and $O_k(f_*)=1$. Then $(\omega_k,O_k)$ has a continuum limit in the sense of Definition~\ref{def:limit}, on which the dyadic spatial translations act by unitary operators commuting with $H$. \textup{(iv)} This limit is nontrivial if and only if $\mu([0,\infty))>0$, if and only if $\rho(\sigma)>0$ for one dyadic $\sigma>0$, and then $\rho(\sigma)>0$ for all of them. Consequently a nontrivial limit exists along some subsequence if and only if $\limsup_kC_k(\sigma)/C_k(0)>0$ for some dyadic $\sigma>0$. \textup{(v)} If the hypothesis of Theorem~\ref{thm:transfer} holds, then $\mu$ is carried by $[m_*,\infty]$ and $\rho(\sigma)\le e^{-m_*\sigma}$. \end{theorem} \begin{proof} By Proposition~\ref{prop:spatialflow}(ii) each $\Phi_x\circ\Phi_{s_k}$, and hence each $A^y_k$, is a uniform limit of gauge-invariant local functions at times $\ge1$, so the reconstruction of Appendix~\ref{app:transfer} applies to them (its footnote). Write $\Psi_k=Z_k\Psi_{A_k}$. Since $\ip{\Omega_k}{\Psi_{A_k}}=\omega_k(A_k)=0$ and $\|\Psi_k\|^2=Z_k^2C_k(0)=1$, $\Psi_k$ is a unit vector orthogonal to $\Omega_k$. The relation $\ip{\Psi_F}{\mathbb T_k^t\Psi_H}=\omega_k(\vartheta(\bar F)\tau_tH)$ for all $F$ gives $\Psi_{\tau_tH}=\mathbb T_k^t\Psi_H$. Spatial translations commute with $\vartheta$ and with $\tau_t$ and leave $\omega_k$ invariant, so they act on the reconstructed space by unitaries $U_k(x)$ that commute with $\mathbb T_k$. Hence, for $y=(t,y')$, \[ Z_k\Psi_{A^y_k}=U_k(y'/a_k)\,\mathbb T_k^{t/a_k}\Psi_k . \] (i) The left side of the inequality is $|\ip{Z_k\Psi_{A^y_k}}{Z_k \Psi_{A^{y'}_k}}|\le1$, because $\|U_k\|=1$ and $0\le\mathbb T_k\le1$. (ii) The set $\mathcal D\times\mathcal D$ is countable and the numbers are bounded by (i), so a diagonal subsequence gives all the limits $K$; the limits $\rho(\sigma)$ are among them, since $C_k(\sigma)/C_k(0)= \ip{\Psi_k}{\mathbb T_k^{\sigma/a_k}\Psi_k}$ is the value of $K$ at $y=y'=(\sigma/2,0)$. Let $\nu_k$ be the spectral measure (Lemma~\ref{lem:spectralmeasure}) of $\mathbb T_k$ at $\Psi_k$, a probability measure on $[0,1]$, and $\mu_k$ its image under $\lambda\mapsto-a_k^{-1}\log\lambda\in[0,\infty]$. Then $C_k(\sigma)/C_k(0)=\int e^{-\sigma E}d\mu_k(E)$. The space $[0,\infty]$ is compact, so a further subsequence has $\mu_k\to\mu$ weakly (Lemma~\ref{lem:helly}), and $E\mapsto e^{-\sigma E}$ is continuous on $[0,\infty]$ for $\sigma>0$. (iii) A finite combination $\sum_jc_jf_0(\cdot-y_j)$ with distinct $y_j$ that vanishes has Fourier transform $\hat f_0(p)\sum_jc_je^{-ip\cdot y_j}=0$; since $\hat f_0$ is analytic and not zero, the exponential sum vanishes on an open set and every $c_j$ is zero. So the translates are linearly independent and $O_k$ is well defined. On the space $V$ of finite sums $v=\sum_yc_y\delta_y$ over $\mathcal D$ put $B(v,w)=\sum\bar c_yc'_{y'}K(y,y')$. It is the limit of $\ip{v_k}{w_k}$ with $v_k=\sum_yc_yZ_k\Psi_{A^y_k}$, hence positive semidefinite; let $\mathcal H_0$ be the completion of $V$ modulo its null space. For dyadic $\sigma\ge0$ and dyadic spatial $x$ define $T(\sigma)\delta_y=\delta_{y+(\sigma,0)}$ and $U(x)\delta_y=\delta_{y+(0,x)}$. At level $k$ these are $\mathbb T_k^{\sigma/a_k}$ and $U_k(x/a_k)$, so in the limit $B(T(\sigma)v,w)=B(v,T(\sigma)w)$, $B(T(\sigma)v,T(\sigma)v)\le B(v,v)$, $B(v,T(\sigma)v)\ge0$, and $U(x)$ preserves $B$. Thus $T(\sigma)$ extends to commuting positive contractions on $\mathcal H_0$ with $T(\sigma)T(\sigma')=T(\sigma+\sigma')$, and $U(x)$ to unitaries commuting with them. For dyadic $\sigma>0$ the kernel of $T(\sigma)=T(\sigma/2)^2$ equals that of $T(\sigma/2)$, so all these kernels coincide; and every $\delta_y$ with $y=(t,y')$ equals $T(t/2)\delta_{(t/2,y')}$, a vector in the range of a positive operator and so orthogonal to its kernel. Hence each $T(\sigma)$ is injective on $\mathcal H_0$, and $H_0=-\log T(1)$ is a self-adjoint operator with $H_0\ge0$ and $e^{-\sigma H_0}=T(\sigma)$ for dyadic $\sigma$, by the uniqueness of positive roots. Put $\mathcal H=\C\Omega\oplus\mathcal H_0$, $H=0\oplus H_0$, $\psi(f_0(\cdot-y))=\delta_y$ and $\psi(f_*)=\Omega$. The range of $\psi$ is dense, and the identities of Definition~\ref{def:limit} hold: for two translates they are the limits defining $K$, with $\tau_{\sigma/a_k}A^{y'}_k= A^{y'+(\sigma,0)}_k$; and for $f_*$ they read $\omega_k(1)=1$ and $\omega_k(\vartheta(\bar A^y_k))=\omega_k(\tau_{\sigma/a_k}A^y_k)=0$. (iv) The limit is nontrivial exactly when $\mathcal H_0\ne0$, that is when $\|\delta_y\|^2=K(y,y)=\rho(2t)$ is positive for some $y=(t,y')$. By (ii) and Lemma~\ref{lem:laplace}(b), $\rho(\sigma)>0$ for one $\sigma>0$ if and only if $\mu([0,\infty))>0$, and then $\rho(\sigma)>0$ for every $\sigma>0$. For the last statement choose a subsequence along which $C_k(\sigma)/C_k(0)$ tends to its $\limsup$, and refine it by (ii). (v) Under that hypothesis the proof of Theorem~\ref{thm:transfer} gives $\nu_k([0,e^{-\log(1/\alpha)/(2n_k)}])=1$ because $\Psi_k\perp\Omega_k$, so $\mu_k$ is carried by $[\log(1/\alpha)/(2n_ka_k),\infty]\subset[m_*,\infty]$, and so is the weak limit $\mu$. \end{proof} The measure $\mu$ is the spectral measure of the renormalised flowed field in the limit: its part on $[0,\infty)$ is what survives as a state of the continuum theory, and its atom at $\infty$ is the weight that the field puts on energies that diverge in physical units. The theorem constructs the renormalised continuum observables and their reconstruction for every tuning, and it reduces their nontriviality to the lattice inequality \[ \limsup_{k\to\infty}\frac{C_k(\sigma)}{C_k(0)}>0 \] for one physical separation $\sigma$: the flowed two-point function of the smeared packet at a fixed physical distance must not become negligible against its reflected norm $C_k(0)$. If the lattice gap stayed above some $m>0$ in lattice units, the ratio would be at most $e^{-m\sigma/a_k}$ and the limit would be trivial, so the inequality is a condition on the tuning $\beta_k\to\infty$; it is not proved here. Nor are the independence of the limit from the subsequence, Euclidean invariance, or bounds on products of more than two renormalised fields, for which (i) gives nothing because $Z_k$ may diverge (Theorem~\ref{thm:higher} reduces them to tree bounds); (E) requires all of these. The denominator $C_k(0)$ is the reflected norm of the packet smeared by $f_0$, not a coincidence limit of point fields, and the distinction matters. For the free massless scalar field in four dimensions with spatial momenta cut off at $\Lambda$, the covariance at spatial separation $0$ is $(1-(1+\Lambda t)e^{-\Lambda t})/(4\pi^2t^2)$ at time $t>0$ and $\Lambda^2/(8\pi^2)$ at coincidence, so its ratio $2(1-(1+\Lambda t)e^{-\Lambda t})/(\Lambda t)^2$ tends to zero as $\Lambda\to\infty$, although the limiting field is the nontrivial Gaussian field; for packets smeared by $f_0$ the reflected norms of that field have finite positive limits and the ratio stays positive. The criterion of Theorem~\ref{thm:flowlimit}(iv) is about the smeared packets $A_k$ only. When the ratio is bounded below, renormalising at the separation $\sigma$ instead of $0$ removes the escape to infinite energy altogether, and one slab of the transfer matrix gives a sufficient condition for the bound. Write $\mu_k$ for the measures in the proof of Theorem~\ref{thm:flowlimit}. \begin{proposition}\label{prop:flowshift} \textup{(i)} Suppose $C_k(\sigma)\ge c\,C_k(0)$ for a dyadic $\sigma>0$, a $c>0$ and every $k$ of a sequence. Let $\mu'_k$ be the spectral measure, in energies $E=-a_k^{-1}\log\lambda$, of the unit vector $C_k(\sigma)^{-1/2}\Psi_{A'_k}$ with $A'_k=\tau_{\sigma/2a_k}A_k$. Then $\mu'_k$ is a probability measure on $[0,\infty)$, \[ \mu'_k([\Lambda,\infty))\le c^{-1}e^{-\sigma\Lambda},\qquad \int E^m\,d\mu'_k\le c^{-1}\Bigl(\frac m{e\sigma}\Bigr)^m , \] and $t\mapsto C_k(\sigma+t)/C_k(\sigma)$ is Lipschitz on $[0,\infty)$ with constant $(e\sigma c)^{-1}$. A subsequence of the $\mu'_k$ converges weakly to a probability measure on $[0,\infty)$ with the same bounds. \textup{(ii)} Suppose $\ip{\Psi_k}{(1-\mathbb T_k)\Psi_k}\le Ca_k$ for a constant $C$ and every $k$ of a sequence. Then $C_k(\sigma)/C_k(0)\ge e^{-2C\sigma}$ whenever $Ca_k\le\frac12$ and $\sigma/a_k$ is an integer, and $\mu_k([\Lambda,\infty])\le C(a_k+\Lambda^{-1})$ for every $\Lambda>0$. \end{proposition} \begin{proof} (i) We have $\Psi_{A'_k}=\mathbb T_k^{\sigma/2a_k}\Psi_{A_k}$, so the measure $\mu'_k$ is $e^{-\sigma E}d\mu_k(E)$ divided by its mass $C_k(\sigma)/C_k(0)\ge c$, and $e^{-\sigma E}$ vanishes at $E=\infty$. For $E\ge\Lambda$ we have $e^{-\sigma E}\le e^{-\sigma\Lambda}$, and $E^me^{-\sigma E}\le(m/e\sigma)^m$ for all $E\ge0$. The ratio in question is the integral of $e^{-tE}$ against $\mu'_k$, and the inequality $|e^{-tE}-e^{-sE}|\le|t-s|E$ with the moment bound for $m=1$ gives the Lipschitz bound. The tail bound makes the sequence tight on $[0,\infty)$, so Prokhorov's theorem gives the weak limit, and the bounds pass to it because $\{E>\Lambda\}$ is open and $E^m$ is a monotone limit of bounded continuous functions. (ii) Let $\nu_k$ be the spectral measure of $\mathbb T_k$ at $\Psi_k$ on $[0,1]$, a probability measure, and $n=\sigma/a_k$. By Jensen's inequality for the convex function $\lambda^n$, \[ \frac{C_k(\sigma)}{C_k(0)}=\int\lambda^nd\nu_k\ge\Bigl(\int\lambda\,d\nu_k \Bigr)^n\ge(1-Ca_k)^n\ge e^{-2Ca_kn}, \] since $\log(1-x)\ge-2x$ on $[0,\frac12]$. Moreover $Ca_k\ge\int(1-e^{-a_kE}) d\mu_k\ge(1-e^{-a_k\Lambda})\mu_k([\Lambda,\infty])$, and $1-e^{-x}\ge x/(1+x)$. \end{proof} Part (ii) asks for an upper bound on a one-slab Dirichlet form, the opposite direction to a mass gap, which bounds such forms from below. In the box of Proposition~\ref{prop:flowpositive}, with $g$ smooth, the time smearing supplies the factor $a_k$ up to one ratio: in the notation of its proof, $\mathbb T_k=A$ and $\Psi_{A_k}=\sum_nw_n\mathbb T_k^nu$ with $w_n=a_kg(a_kn)$ and $u=a_k^3F^c\psi_0$, and the identity $(1-\mathbb T_k) \Psi_{A_k}=\sum_n(w_n-w_{n-1})\mathbb T_k^nu$ gives $\ip{\Psi_k}{(1-\mathbb T_k)\Psi_k}\le a_k\|g'\|_{L^1}\|u\|/\|\Psi_{A_k}\|$, so the hypothesis of (ii) holds when the vector $u$ of a single slice stays comparable to the time-smeared packet. That comparison is not proved; Proposition~\ref{prop:sliceratio} below bounds the ratio without it. A one-slab bound of this kind also limits the mass that $D(n_k,\alpha)$ can supply. Let $e^{-m_k}$ be the norm of $\mathbb T_k$ on $\Omega_k^\perp$. If a unit vector $\varphi_k\perp\Omega_k$ has $\ip{\varphi_k}{(1-\mathbb T_k)\varphi_k}\le\kappa a_k<1$, then the spectral measure of $\mathbb T_k$ at $\varphi_k$ lives on $[0,e^{-m_k}]$, so $1-e^{-m_k}\le\kappa a_k$ and $m_k\le-\log(1-\kappa a_k)$. If in addition $D(n_k,\alpha)$ holds with $n_ka_k\le\ell$, Theorem~\ref{thm:onescale} gives $m_k\ge\log(1/\alpha) /(2n_k)\ge a_k\log(1/\alpha)/(2\ell)$, and therefore \[ \frac{\log(1/\alpha)}{2\ell}\le\frac{-\log(1-\kappa a_k)}{a_k}=\kappa+ O(a_k). \] So the physical mass $m_*$ of Theorem~\ref{thm:transfer} is at most the constant $\kappa$ of any one-slab bound. The two hypotheses are compatible because both are statements in physical units. The one-slab bound forces $m_k\to0$, so it cannot come from an expansion whose convergence gives a lattice gap bounded below in lattice units. In a periodic spatial box the assumption $C_k(0)>0$ is a theorem. Let $G=\SU(N)$ with its defining representation, and replace $\Z^4$ by $\Z\times(\Z/L_k\Z)^3$ with $L_k\ge3$ and $\omega_k$ by the infinite-time Wilson state of the box. The proof of Theorem~\ref{thm:flowlimit} uses only reflection positivity and the transfer matrix once the spatial translations are dropped, and parts (i)--(iv) hold in the box with $\mathcal D$ replaced by its points $(t,0)$, the time translates of $f_0$ being independent by the same Fourier argument in $t$. \begin{proposition}\label{prop:flowpositive} In the box, let $f_0(t,x)=g(t)h(x)$ with $g,h\ge0$, and let $\Phi=\sum_{i0$ for some $n\ge1$ and some $x$, then $C_k(0)>0$. \end{proposition} \begin{proof} L\"uscher proved that the transfer matrix is strictly positive~\cite{Lus77}; we recall the argument because the proof uses it in this form. \emph{The one-link kernel.} Write $\chi$ for the character of the defining representation and $b=\beta_W>0$. Then \[ e^{\frac bN\operatorname{Re}\chi}=\sum_{m\ge0}\frac{(b/2N)^m}{m!} (\chi+\bar\chi)^m , \] and $(\chi+\bar\chi)^m$ is the character of $(\rho\oplus\bar\rho)^{\otimes m}$. Every irreducible representation $\sigma$ occurs in one of these tensor powers: their matrix coefficients span the algebra of polynomials in the entries of $q$ and $\bar q$, which contains the constants and separates points, so it is dense in $C(G)$ by the Stone--Weierstrass theorem, while the matrix coefficients of a $\sigma$ that never occurred would be orthogonal to it by Schur orthogonality. So the coefficient of every irreducible character in $e^{\frac bN\operatorname{Re}\chi}$ is positive, and convolution with this function acts on each isotypic part of $L^2(G)$ by a positive scalar. \emph{The transfer matrix.} In the temporal gauge the transfer matrix of the box acts on the gauge-invariant part of $L^2(G^{E})$, $E$ the spatial links of one slice, as $M K M$, where $K$ is the tensor product over $E$ of the convolutions above and $M$ is multiplication by the exponential of half the spatial plaquette weight~\cite{Lus77,OSei78}. Both commute with gauge transformations, $K$ is positive and injective by the first step, and $M$ is positive and invertible, so $\ip v{MKMv}=\ip{Mv}{KMv}>0$ for $v\ne0$. Its kernel is a continuous positive function, and $K$ is compact, so its largest eigenvalue $\lambda_0$ is simple with an eigenfunction $\psi_0>0$, and with $A=MKM/\lambda_0$, a function $F$ of the slice at time $n\ge0$ has $\Psi_F=A^nF\psi_0$ in the reconstruction, $\Omega$ being $\psi_0$. \emph{Conclusion.} Let $F=\sum_xh(a_kx)\,\Phi_x\circ\Phi_{s_k}$ and $F^c=F-\omega_k(F)$. Then $\Psi_{A_k}=a_k^4\,p(A)F^c\psi_0$ with $p(z)=\sum_{n\ge1}g(a_kn)z^n$, a polynomial with nonnegative coefficients, not all zero, so $p>0$ on $(0,1]$ and $p(A)$ is injective. The function $\sum_xh(a_kx)\Phi_x$ vanishes at the configuration with every link $1$ and is positive at one with a single nontrivial link at a site where $h(a_kx)>0$; the flow $\Phi_{s_k}$ is a bijection, so $F$ is not constant, and as $F$ is continuous and $\psi_0>0$, $F^c\psi_0\ne0$. Hence $C_k(0)=\|\Psi_{A_k}\|^2>0$. \end{proof} The proposition removes the one assumption of Theorem~\ref{thm:flowlimit} in a box; it says nothing about the ratio of part (iv), on which nontriviality depends. The ratio of Theorem~\ref{thm:flowlimit}(iv) can be compared with correlations of a single slice field. Suppose that $f_0(t,x)=g(t)h(x)$ with $g\ge0$ supported in $[t_0,t_1]$, $t_00$, and let $\sigma\ge0$ with $\sigma/a_k\in\Z$. \textup{(i)} $D_k(r)>0$ for every $r$, and \[ \frac{D_k(2a_kn_1+\sigma)}{D_k(2a_kn_0)}\le\frac{C_k(\sigma)}{C_k(0)}\le \frac{D_k(2a_kn_0+\sigma)}{D_k(2a_kn_1)} . \] \textup{(ii)} $\log D_k$ and $\log C_k$ are convex and nonincreasing on $a_k\Z_{\ge0}$. If $D_k(r')\ge c\,D_k(r)$ with $r0$, then $C_k(\sigma')\ge c^{\sigma'/\sigma}C_k(0)$ for $\sigma'\ge\sigma$. \textup{(iii)} Let $k$ run through a sequence. If $D_k(r')\ge c\,D_k(r)$ for dyadic $00$ and every $k$, then $D_k(2t_0+\sigma')\ge c^{\sigma'/\sigma}D_k(2t_1)$ for every dyadic $\sigma'\ge\sigma$ with $\sigma'>2(t_1-t_0)$ and every large $k$. \end{proposition} \begin{proof} For a function $F$ of the slice at time $0$ and an integer $n\ge0$ we have $\Psi_{\tau_nF}=\mathbb T_k^n\Psi_F$, since both have inner product $\omega_k(\vartheta(\bar G)\tau_nF)$ with every $\Psi_G$. By time translation invariance of $\omega_k$, $A_k=\sum_nw_n\tau_nU_k$ with $w_n=a_kg(a_kn)\ge0$, and $w_n=0$ unless $n_0\le n\le n_1$; so $\Psi_{A_k}=\sum_nw_n\mathbb T_k^nu_k$. Let $\nu$ be the spectral measure of $\mathbb T_k$ at $u_k$, on $[0,1]$, and $W(\lambda)=\sum_nw_n\lambda^n$. Then \[ C_k(\sigma)=\int W(\lambda)^2\lambda^{\sigma/a_k}\,d\nu(\lambda),\qquad D_k(r)=\int\lambda^{r/a_k}\,d\nu(\lambda). \] (i) Since $t_0>0$, $n_0\ge1$ and $W(0)=0$; so $C_k(0)>0$ forces $\nu((0,1])>0$, and then $D_k(r)>0$. For $0\le\lambda\le1$, $\|w\|_1\lambda^{n_1}\le W(\lambda)\le\|w\|_1\lambda^{n_0}$; inserting these bounds in $C_k(\sigma)$ and $C_k(0)$ gives the two inequalities. (ii) Convexity of $j\mapsto\log\int\lambda^j\,d\rho$ for a positive measure $\rho$ on $[0,1]$ is H\"older's inequality, and the integrals decrease in $j$ because $\lambda\le1$; take $\rho=\nu$ and $\rho=W^2\nu$. If $\beta\le r'$, monotonicity gives $D_k(\beta)\ge D_k(r')\ge cD_k(r)\ge cD_k(\alpha)$. If $\beta>r'$, the difference quotients of the convex function $\log D_k$ increase in both end points, so $(\log D_k(\beta)-\log D_k(\alpha))/(\beta-\alpha)\ge (\log D_k(r')-\log D_k(r))/(r'-r)\ge\log c/(r'-r)$. The statement on $C_k$ is the same comparison of quotients on $[0,\sigma]$ and $[0,\sigma']$. (iii) For large $k$ the times $2t_0$, $2t_1$ are multiples of $a_k$, and $2a_kn_0\ge2t_0$, $2a_kn_1\le2t_1$. Since $D_k$ is nonincreasing, (i) gives \[ \frac{D_k(2t_1+\sigma)}{D_k(2t_0)}\le\frac{C_k(\sigma)}{C_k(0)}\le \frac{D_k(2t_0+\sigma)}{D_k(2t_1)} . \] The first claim is the left inequality with (ii) for $\alpha=2t_0\ge r$ and $\beta=2t_1+\sigma$. For the second, (ii) gives $C_k(\sigma')\ge c^{\sigma'/\sigma}C_k(0)$, and the right inequality at $\sigma'$ gives the claim, the two times being ordered because $\sigma'>2(t_1-t_0)$. \end{proof} So for the packet $g\otimes h$ the open input of Theorem~\ref{thm:flowlimit}(iv) follows from an inequality between two correlations of one slice field, \[ \liminf_kD_k(r')/D_k(r)>0\quad\text{for one pair }00$ in place of the gradient flow. With $\hat p^2=\sum_i(2a_k^{-1}\sin (a_kp_i/2))^2$, $\cosh(a_k\omega)=1+a_k^2\hat p^2/2$ and $\hat h_k(p)=\sum_xa_k^3h(a_kx)e^{-ia_kp\cdot x}$, \[ D_k(r)=\int_{[-\pi/a_k,\pi/a_k]^3}|\hat h_k(p)|^2e^{-2t_f\hat p^2} \frac{e^{-r\omega}}{2a_k^{-1}\sinh(a_k\omega)}\,\frac{d^3p}{(2\pi)^3} . \] On the Brillouin zone $\hat p\ge2|p|/\pi$ and $a_k^2\hat p^2/2\le6$; as $\operatorname{arccosh}(1+y)/\sqrt{2y}$ decreases on $(0,6]$, $a_k\omega\ge \kappa a_k\hat p$ with $\kappa=\operatorname{arccosh}7/\sqrt{12}>0.76$, and $a_k^{-1}\sinh(a_k\omega)\ge\omega$. Since $|\hat h_k|$ is bounded uniformly in $k$, the integrands are dominated by a multiple of $e^{-8t_f|p|^2/\pi^2}/|p|$, which is integrable on $\R^3$, and they converge pointwise to $|\hat h(p)|^2e^{-2t_f|p|^2}e^{-r|p|}/ (2|p|)$. By dominated convergence $D_k(r')/D_k(r)$ tends to the corresponding positive ratio of continuum integrals; for $h=e^{-|x|^2}$, $t_f=0.05$, $r=\frac12$ and $r'=1$ it is $0.6337$. \end{remark} \begin{remark}\label{rem:compositeslice} The slice field of Proposition~\ref{prop:sliceratio} is not linear: it is a gauge-invariant function of the spatial links of one slice. Its free analogue is the magnetic energy, and the conclusion of Remark~\ref{rem:freeslice} survives for it. Take the abelian free lattice field on $a_k\Z\times a_k\Z^3_{L_k}$ with $a_kL_k=\ell$ fixed, Feynman gauge, the lattice heat flow of the spatial links for a physical time $t_f\ge0$, and $U_k=\sum_x a_k^3\sum_{i0$ uniformly in $k$, also for $t_f=0$. The summands converge to the continuum ones, so for every $00$ converges: the condition of Proposition~\ref{prop:sliceratio} at $r>0$ is weaker than a bound on the first step. For the full energy density, which contains $F_{0i}$ and is not a function of one slice, the electric numerator depends on the time momentum, and the second time difference of $K_p$ is $-\delta_{n0}$ plus a multiple of $K_p$, a contact term: $D_k(0)\sim a_k^{-2}$ even with the flow, $D_k(a_k)/D_k(0)=0.042$, $0.023$, $0.0145$ at $a_k^{-1}=8,12,16$, while the ratios at $r>0$ again converge to the same continuum values ($0.27092$, $0.27126$, $0.27137$). The values are numerical evaluations of the free-field formulas. At next order the spatial flow, being nonlinear, fuses two hard modes into a soft one, and the state $\Psi_{U_k}$ acquires a component of energy about $2q$ for each $q\lesssim a_k^{-1}$. Its norm is of order $g^2$ and ultraviolet finite, dominated by $q\approx\tau^{-1/2}$, but its mean energy is not: in the free vacuum the first-step energy is $E_{\mathrm{LO}}(1+g^2(c+\frac N{3\pi^2}\log(\sqrt\tau/a_k)))$, and the logarithm has coefficient $\frac8{11}$ of $2b_0$, $b_0=11N/48\pi^2$. A first-step bound at fixed coupling therefore fails logarithmically, while the ratio at a fixed $r>0$ only loses the norm of that component, damped by $e^{-2qr}$. This is why Proposition~\ref{prop:sliceratio} is stated at $r>0$; the decomposition $\|\mathbb T_k^{n/2}u\|\ge c^{n/2}\|v\|-\|w\|$ for $u=v+w$ with a first-step bound on the soft part $v$ alone is the form a four-dimensional argument would use. \end{remark} \begin{remark}\label{rem:twodslice} The inequality also holds along the right tuning in two-dimensional $\SU(N)$ lattice gauge theory, which is interacting and nonabelian but has no propagating gluons. Take the lattice $a_k\Z\times a_k\Z_{L_k}$ with $a_kL_k=\ell$ fixed, the weight $\prod_p\exp(\frac{\beta_k}N\ReTr U_p)$ with $\beta_k=2N/(g^2a_k^2)$, and the temporal gauge. A gauge-invariant function of the links of one slice depends only on the conjugacy class of their holonomy $V$ around the circle; this includes every spatially flowed and spatially smeared slice field, since the flow is gauge covariant. Let $w_\beta$ be the probability density proportional to $e^{\frac\beta N\ReTr U}$ with respect to Haar measure and, for an irreducible representation $\lambda$ of dimension $d_\lambda$ and character $\chi_\lambda$, $\hat c_\lambda(\beta)=d_\lambda^{-1}\int\chi_\lambda w_\beta\,dU$. Between two slices put $W_x=V'_xV_x^{-1}$; the $W_x$ are independent with density $w_{\beta_k}$, which is invariant under conjugation and inversion, and \[ \begin{split} V'_1\cdots V'_{L}&=W_1\,(V_1W_2V_1^{-1})\cdots\\ &\quad\cdots(V_1\cdots V_{L-1}W_{L}(V_1\cdots V_{L-1})^{-1})\,V_1\cdots V_{L}, \end{split} \] so the new holonomy is $QV$ with $Q$ distributed as the $L_k$-fold convolution of $w_{\beta_k}$, independent of $V$. By Schur's lemma (Lemma~\ref{lem:convolution}) the transfer matrix acts on class functions by $\chi_\lambda\mapsto\hat c_\lambda(\beta_k)^{L_k}\chi_\lambda$. The coefficients are real, since $w_\beta$ is invariant under $U\mapsto\bar U$, nonnegative, since the exponential of $\frac\beta{2N}(\chi_{\mathrm F}+\bar\chi_{\mathrm F})$ is a sum of characters with nonnegative coefficients, and below $1$ for $\lambda\ne0$. For a class function $f$ with $\int f\,dU=0$ and isotypic components of norms $|\hat f_\lambda|$, the slice correlation is therefore \[ D_k(r)=\sum_{\lambda\ne0}|\hat f_\lambda|^2e^{-rE_{\lambda,k}},\qquad E_{\lambda,k}=-\frac{\ell}{a_k^2}\log\hat c_\lambda(\beta_k). \] Laplace's method\footnote{For a smooth function with a unique nondegenerate maximum, $\int e^{\beta\varphi}h$ has an expansion in powers of $\beta^{-1}$ whose coefficients are Gaussian moments at the maximum; Lemma~\ref{lem:wilsoncoeff} carries it out here.} at the unique maximum $U=1$ of $\ReTr U$, with $U=e^X$, $\ReTr e^X=N+\frac12\operatorname{Tr}X^2+O(|X|^4)$, $\operatorname{Re}\operatorname{Tr}X^3=0$, $\operatorname{Tr}\rho_\lambda(X)=0$ and $\sum_a\rho_\lambda(T_a)^2=-2C_2(\lambda)$ for an orthonormal basis of $\mathfrak{su}(N)$ in the norm $-\operatorname{Tr}X^2$, gives $\hat c_\lambda(\beta)=1-NC_2(\lambda)/\beta+O(\beta^{-2})$, with the Casimir normalised by $C_2=(N^2-1)/(2N)$ on the fundamental representation (Lemma~\ref{lem:wilsoncoeff}). Hence $E_{\lambda,k}\to g^2\ell C_2(\lambda)/2$, and since $D_k(r)\le\|f\|^2$, \[ \liminf_k\frac{D_k(r')}{D_k(r)}\ge\frac{|\hat f_\mu|^2}{\|f\|^2} e^{-r'g^2\ell C_2(\mu)/2}>0 \] for every $\mu\ne0$ with $\hat f_\mu\ne0$; for $f=\ReTr V$ the ratio is exactly $e^{-(r'-r)E_{\mathrm F,k}}\to e^{-(r'-r)g^2\ell(N^2-1)/(4N)}$. This is the exact solution of two-dimensional lattice gauge theory, whose continuum limit is the measure of~\cite{GKS89,Dri89,Lev03}, written as the inequality. In two dimensions $g^2$ has the dimension of a mass squared and there is no ultraviolet problem, so the argument has no four-dimensional analogue. Nor does the spatial flow give the inequality by itself: it damps the spectral weight at large spatial momentum and leaves the weight at zero spatial momentum unchanged. For the free lattice field of Remark~\ref{rem:freeslice} with a mass $M_k$ and the same flow, every energy is at least $m_k=a_k^{-1}\operatorname{arccosh}(1+a_k^2M_k^2/2)$, so $D_k(r')\le e^{-m_k(r'-r)}D_k(r)$, which tends to zero relative to $D_k(r)$ when $a_kM_k$ is fixed. \end{remark} \begin{figure*}[t] \centering \includegraphics[width=0.9\textwidth]{figures/fig_twod.png} \caption{Two-dimensional $\SU(N)$ (Remark~\ref{rem:twodslice}). (a) The lattice $a_k\Z\times a_k\Z_{L_k}$ drawn as a cylinder: the holonomy $V$ of one slice, the holonomy $V'=QV$ of the next, and the plaquette variables $W_x$ between them. (b) The energies $E_{j,k}=-(\ell/a_k^2)\log\hat c_j(\beta_k)$ of $\SU(2)$ for $g^2=\ell=1$, with $\hat c_j(\beta)=I_{2j+1}(\beta)/I_1(\beta)$, against the lattice spacing; the dashed lines are the limits $j(j+1)/2$.} \label{fig:twod} \end{figure*} \begin{figure*}[t] \centering \includegraphics[width=0.9\textwidth]{figures/fig_slice.png} \caption{The ratio $D_k(r)/D_k(\frac12)$ over the separation $r$ and the lattice spacing $a_k$. (a) Two-dimensional $\SU(2)$ with $f=\ReTr V$ and $g^2=\ell=1$, where the ratio is $e^{-(r-1/2)E_{1/2,k}}$ and converges as $a_k\to0$. (b) The term $e^{-(r-1/2)m_k}$ of the lowest energy $m_k$ of a free field with mass $\frac12$ in lattice units, which bounds the ratio from above and tends to zero at every $r>\frac12$.} \label{fig:slice} \end{figure*} A route to the slice ratio in four dimensions replaces the packet by its conditional expectations given averages of the field over blocks of increasing size and controls the error of each step. The free field shows in which norm such errors can be small. In the ordinary $L^2$ norm the first step already fails: conditioning on the averages over blocks of side two leaves a fixed fraction of the variance of an energy packet, at every lattice spacing. \begin{proposition}\label{prop:blockfree} Let $d\ge2$, let $\Lambda$ be a periodic box in $\Z^d$ with even sides, let $\mu>0$, and let $\phi$ be the Gaussian field on $\Lambda$ with density proportional to $\exp\bigl(-\frac12\sum_{x,\nu}(\nabla_\nu\phi(x))^2-\frac{\mu^2}2\sum_x \phi(x)^2\bigr)$, where $\nabla_\nu\phi(x)=\phi(x+e_\nu)-\phi(x)$. Let $\mathcal Z$ be a set of blocks $B=2z+\{0,1\}^d$, let $\mathfrak B$ be the $\sigma$-algebra generated by the averages of $\phi$ over the blocks in $\mathcal Z$, let $f\ge0$ on $\Lambda$, and let \[ F=\sum_{x\in\Lambda}f(x)\sum_{\nu=1}^d{:}(\nabla_\nu\phi(x))^2{:} , \] where ${:}X^2{:}=X^2-\E X^2$. For $B=2z+\{0,1\}^d$ put $x_B=2z$, $y_B=2z+e_1+e_2$ and \[ \alpha_B=\frac14\sum_{w\in\{x_B,y_B\}}\Bigl(d\,f(w)+\sum_{\nu=1}^df(w-e_\nu) \Bigr), \] and $\sigma^2=2/(2d+\mu^2)$. \textup{(i)} $\E\Var(F\mid\mathfrak B)\ge2\sigma^4\sum_{B\in\mathcal Z}\alpha_B^2$. \textup{(ii)} $\Var F\le2d^2\sum_xf(x)^2$. \textup{(iii)} Let $g\ge0$ be a Lipschitz function on $\R^d$ with compact support and $\int g^2>0$. For $n\ge1$ let $\Lambda_n$ contain $n\operatorname{supp}g$ in a fundamental domain, $f_n(x)=g(x/n)$, let $\mathcal Z_n$ contain every block that meets $n\operatorname{supp}g$, and let $\mu=\mu_n\to0$. Then for the packets $F_n$ of $f_n$, \[ \liminf_{n\to\infty}\frac{\E\Var(F_n\mid\mathfrak B_n)}{n^d\int g^2} \ge2^{1-d},\qquad \limsup_{n\to\infty}\frac{\Var F_n}{n^d\int g^2}\le2d^2 , \] so $\E\Var(F_n\mid\mathfrak B_n)\ge(2^{-d}d^{-2}-o(1))\Var F_n$. \end{proposition} \begin{proof} (i) No two of the sites $x_B,y_B$, $B\in\mathcal Z$, are neighbours: their coordinate sums are even, and a neighbour of a site has a coordinate sum of the other parity. Let $\mathfrak H\supseteq\mathfrak B$ be generated by $\mathfrak B$ and by $\phi(w)$ for every $w$ outside $\{x_B,y_B:B\in\mathcal Z\}$. By the law of total variance, $\Var(F\mid\mathfrak B)=\E[\Var(F\mid\mathfrak H)\mid\mathfrak B] +\Var(\E[F\mid\mathfrak H]\mid\mathfrak B)$, so $\E\Var(F\mid\mathfrak B)\ge\E\Var(F\mid\mathfrak H)$. In the exponent of the density, a site $w$ enters through $(d+\frac{\mu^2}2)\phi(w)^2$ and through $-\phi(w)\phi(w')$ for each of its $2d$ neighbours $w'$. Given $\mathfrak H$, the block average fixes $s_B=\phi(x_B)+\phi(y_B)$, and with $u_B=\phi(x_B)-\phi(y_B)$ we have $\phi(x_B)^2+\phi(y_B)^2=\frac12(s_B^2+ u_B^2)$. Since the free sites have only $\mathfrak H$-measurable neighbours, the conditional density of $(u_B)_{B\in\mathcal Z}$ is proportional to $\prod_B\exp(-\frac14(2d+\mu^2)u_B^2+c_Bu_B)$ with $\mathfrak H$-measurable $c_B$: the $u_B$ are conditionally independent Gaussians of variance $\sigma^2$. A term $(\nabla_\nu\phi(w))^2$ contains at most one free site, and if it contains $x_B$, then $\phi(x_B)=\frac12(s_B+u_B)$ shows that it is $\frac14u_B^2$ plus a term linear in $u_B$ plus an $\mathfrak H$-measurable term; the same holds for $y_B$ with $-u_B$. The terms containing $x_B$ are those with $w=x_B$ and those with $w=x_B-e_\nu$, $d$ of each kind, so \[ F=\sum_{B\in\mathcal Z}\bigl(\alpha_Bu_B^2+\beta_Bu_B\bigr)+F_0 \] with $\mathfrak H$-measurable $\beta_B$ and $F_0$. Writing $u_B=m_B+v_B$ with $m_B$ the conditional mean, $\alpha_Bu_B^2+\beta_Bu_B= \alpha_Bv_B^2+(2\alpha_Bm_B+\beta_B)v_B+\text{const}$, and for a centred Gaussian $v$ of variance $\sigma^2$, $\Var(\alpha v^2+\gamma v)=2\alpha^2 \sigma^4+\gamma^2\sigma^2$. By conditional independence, $\Var(F\mid\mathfrak H)\ge2\sigma^4\sum_B\alpha_B^2$. (ii) For centred jointly Gaussian $X,Y$, $\Cov({:}X^2{:},{:}Y^2{:})=2\Cov(X,Y)^2$. With $G_{\nu\rho}(x,y)=\E[\nabla_\nu\phi(x)\nabla_\rho\phi(y)]$, \[ \begin{split} \Var F&=2\sum_{x,y}f(x)f(y)\sum_{\nu,\rho}G_{\nu\rho}(x,y)^2\\ &\le\sum_{x,y}\bigl(f(x)^2+f(y)^2\bigr)\sum_{\nu,\rho}G_{\nu\rho}(x,y)^2 . \end{split} \] On the dual torus $\Lambda^*$, with $\hat p^2=\sum_\nu|e^{ip_\nu}-1|^2$, \[ G_{\nu\rho}(x,y)=\frac1{|\Lambda|}\sum_{p\in\Lambda^*} \frac{(e^{ip_\nu}-1)(e^{-ip_\rho}-1)}{\hat p^2+\mu^2}e^{ip\cdot(x-y)}, \] and by Parseval $\sum_yG_{\nu\rho}(x,y)^2=|\Lambda|^{-1}\sum_p|e^{ip_\nu}-1|^2 |e^{ip_\rho}-1|^2(\hat p^2+\mu^2)^{-2}\le1$, since each factor $|e^{ip_\nu}-1|^2$ is at most $\hat p^2$. The same holds for the sum over $x$, and (ii) follows. (iii) If $\operatorname{Lip}g=K$, every site entering $\alpha_B$ lies within distance $3$ of $2z$, so $|\alpha_B-d\,g(2z/n)|\le3dK/n$. Hence $\sum_{B\in\mathcal Z_n}\alpha_B^2\ge\sum_z(d\,g(2z/n)-3dK/n)_+^2$, the sum running over $z$ with $2z\in n\operatorname{supp}g$, and this is $d^2(n/2)^d\int g^2\,(1+o(1))$ as a Riemann sum on the grid $(2/n)\Z^d$. Since $\sigma^4=4/(2d+\mu_n^2)^2\to d^{-2}$, (i) gives the first bound. By (ii) and $\sum_xg(x/n)^2=n^d\int g^2(1+o(1))$, the second follows. \end{proof} For $d=4$ and the energy density, $\Var F_n$ grows like $(\ell/a)^4$ at a fixed physical size $\ell$ of the packet, and no step of conditioning on block averages removes more than a fraction $1-2^{-d}d^{-2}+o(1)$ of it. The reflected norm $C_k(0)=\omega_k(\vartheta(\bar A_k)A_k)$, which renormalises the packet in Theorem~\ref{thm:flowlimit}, involves the covariance only between the support of $\vartheta f_0$ and that of $f_0$, at a fixed physical distance. The errors of a blocking route have therefore to be measured in that norm; Figure~\ref{fig:blocking} shows the blocks between the reflection plane and the packet. \begin{figure*}[t] \centering \includegraphics[width=0.9\textwidth]{figures/fig_blocking.png} \caption{Block averages between the reflection plane $t=0$ and a packet $F$ at physical height $d$, for blocks of side $d/4$, $d/2$ and $d$. By Proposition~\ref{prop:blockfree} the first conditioning step already leaves a fixed fraction of the ordinary variance of the packet, so the errors of such steps have to be measured in the reflected norm.} \label{fig:blocking} \end{figure*} \subsection{Higher correlations}\label{ssec:higher} Theorem~\ref{thm:flowlimit} controls two renormalised fields at a time, and part (i) of it gives nothing for more: $Z_k$ may diverge while the fields $\Phi_x\circ\Phi_{s_k}$ stay bounded, so a bound on the norm of the renormalised packet as an operator is not available. A reflection positive process with a fixed gap and a two-point function independent of the cutoff can have divergent four-point functions at separated times (Remark~\ref{rem:markov}), so no statement about the two-point function alone can replace the following hypothesis on the truncated correlations. In this subsection $\Phi$ is real-valued, as the energy of the plaquettes is; for a complex field the conjugate field has to be adjoined and (T1) imposed on all mixed cumulants. Write $\varphi_k(x)=\Phi_x\circ\Phi_{s_k}$, $\omega^T_k$ for the truncated correlations (joint cumulants) under $\omega_k$, and for $f\in C_c(\R^4)$ \[ A_k(f)=\sum_{x\in\Z^4}a_k^4f(a_kx)\bigl(\varphi_k(x)-\omega_k(\varphi_k(x))\bigr), \] so that $A_k=A_k(f_0)$ and $\bar A_k(f)=A_k(\bar f)$. Let $w_k:\Z^4\to[0,\infty)$ be even functions with $W_k=\sum_xw_k(x)<\infty$, let $r_k>0$ be amplitude scales, and consider: \begin{enumerate}[label=\textup{(T\arabic*)},leftmargin=*] \item \emph{Tree bounds.} For every $n\ge2$ there is $K_n$, independent of $k$, with \[ \bigl|\omega^T_k\bigl(\varphi_k(x_1);\dots;\varphi_k(x_n)\bigr)\bigr| \le K_nr_k^n\sum_{T}\prod_{\{i,j\}\in T}w_k(x_i-x_j) \] for all $x_1,\dots,x_n\in\Z^4$, the sum running over the trees $T$ on $\{1,\dots,n\}$. \item \emph{Saturation.} $C_k(0)\ge c\,r_k^2a_k^4W_k$ for a $c>0$ and every $k$. \item \emph{Physical volume.} $B_*=\sup_ka_k^4W_k<\infty$. \end{enumerate} The amplitude $r_k$ separates the size of the field from the volume over which it is correlated; Remark~\ref{rem:amplitude} shows why it is needed. For a compact $Q\subset\R^4$ let $|Q|_k=a_k^4\,\#\{x\in\Z^4:a_kx\in Q\}$; for a cube of side $L$ it is at most $(L+a_0)^4$. \begin{lemma}\label{lem:treesum} Under \textup{(T1)}, for $n\ge2$ and bounded $f_1,\dots,f_n$ with compact supports, $f_1$ supported in $Q$, \[ \begin{split} &\bigl|\omega^T_k\bigl(A_k(f_1);\dots;A_k(f_n)\bigr)\bigr|\\ &\qquad\le K_nn^{n-2}r_k^n\,|Q|_k\,(a_k^4W_k)^{n-1}\prod_{j=1}^n\|f_j\|_\infty . \end{split} \] \end{lemma} \begin{proof} Joint cumulants of order $n\ge2$ are multilinear and do not change when constants are added to the variables, so the left side is at most $a_k^{4n}\sum_{x_1,\dots,x_n}\prod_j|f_j(a_kx_j)|$ times the absolute value of the truncated correlation, the sums being finite. Insert (T1) and fix a tree $T$. With $x_1$ fixed, sum over the other points one at a time, each time over a leaf of what remains of $T$ other than $1$; each such sum is at most $W_k$. The remaining sum over $x_1$ has at most $|Q|_ka_k^{-4}$ nonzero terms. There are $n^{n-2}$ trees. \end{proof} \begin{theorem}\label{thm:higher} Assume \textup{(T1)--(T3)}. \textup{(i)} For $n\ge2$, $f_j$ as in the lemma, \[ \begin{split} &\bigl|\omega^T_k\bigl(Z_kA_k(f_1);\dots;Z_kA_k(f_n)\bigr)\bigr|\\ &\qquad\le K_nn^{n-2}c^{-n/2}\,B_*^{\,n/2-1}\,|Q|_k\prod_{j=1}^n\|f_j\|_\infty , \end{split} \] and the moments $\omega_k\bigl(\prod_jZ_kA_k(f_j)\bigr)$ with all $f_j$ supported in a cube $Q$ are bounded by $M_n(Q)\prod_j\|f_j\|_\infty$, with $M_n(Q)$ independent of $k$. \textup{(ii)} Every sequence of $k$ has a subsequence along which \[ S_n(f_1,\dots,f_n)=\lim_k\omega_k\Bigl(\prod_{j=1}^nZ_kA_k(f_j)\Bigr) \] exists for all $n$ and all $f_j\in C_c(\R^4)$. Each $S_n$ is symmetric, multilinear, bounded by $M_n(Q)\prod_j\|f_j\|_\infty$ on functions supported in $Q$, hence a distribution on $(\R^4)^n$; it is invariant under all translations of $\R^4$ if the $\omega_k$ are translation invariant, and the kernel $K$ of Theorem~\textup{\ref{thm:flowlimit}} along the subsequence is $K(y,y')=S_2\bigl(\overline{f_0(\cdot-y)}\circ\theta,f_0(\cdot-y')\bigr)$, with $\theta$ as in \textup{(iii)}. \textup{(iii)} Put $S_0=1$, $S_1=0$, and let $\theta$ be the reflection $t\mapsto-t$ of $\R^4$. For every finite family of $c_\alpha\in\C$ and $f_{\alpha,j}\in C_c(\R^4)$ supported in $\{t>0\}$, \[ \sum_{\alpha,\beta}\bar c_\alpha c_\beta\, S\Bigl(\prod_j\overline{f_{\alpha,j}}\circ\theta\cdot\prod_jf_{\beta,j}\Bigr)\ge0, \] where $S$ of a product of $n$ test functions means $S_n$. Hence the Osterwalder--Schrader construction gives a Hilbert space $\mathcal H$ with vacuum $\Omega$ and a dense set of vectors $\Psi(P)$ indexed by polynomials $P$ in the smeared fields at positive times, with $\ip{\Psi(P)}{\Psi(P')}=S(\theta\bar P\cdot P')$; in particular the separated higher correlations of the renormalised fields are finite. The vector $\Psi(P_0)$ of the field smeared by $f_0$ is a unit vector orthogonal to $\Omega$. \textup{(iv)} Suppose the $\omega_k$ are translation invariant. Then the time translations by $t\ge0$ define on $\mathcal H$ a strongly continuous semigroup $e^{-tH}$ with $H\ge0$ self-adjoint, and the spatial translations a strongly continuous unitary group commuting with $H$. Moreover there is a dyadic $\sigma>0$ with $C_k(\sigma)\ge\frac12C_k(0)$ for all large $k$, so the limit of Theorem~\textup{\ref{thm:flowlimit}} is nontrivial. \textup{(v)} If $a_k^4W_k\to0$ along the subsequence, every $S_n$ is given by Wick's rule from $S_2$: the limit is the Gaussian field with covariance $S_2$. \textup{(vi)} If the $\omega_k$ are translation invariant and the hypothesis of Theorem~\textup{\ref{thm:transfer}} holds, then $0$ is a simple eigenvalue of $H$ and $\sigma(H)\subset\{0\}\cup[m_*,\infty)$ with $m_*=\log(1/\alpha)/(2\ell)$, and $\Omega^\perp\ne0$. \end{theorem} \begin{proof} (i) By (T2), $Z_k^n=C_k(0)^{-n/2}$ is at most $(c\,r_k^2a_k^4W_k)^{-n/2}$; multiply the lemma by it, which cancels $r_k^n$, and use $(a_k^4W_k)^{n/2-1}\le B_*^{n/2-1}$ by (T3) for $n\ge2$. A moment is the sum over partitions of $\{1,\dots,n\}$ of the products of the cumulants of the blocks; blocks of size one vanish because $A_k(f)$ is centred, and each block contributes the bound just proved with $|Q|_k\le(L+a_0)^4$ for a cube of side $L$. This gives $M_n(Q)$. (ii) For each cube $Q_L=[-L,L]^4$, $L\in\mathbb N$, the continuous functions supported in $Q_L$ form a separable space under the supremum norm. Choose a countable dense set $\mathcal F$ of functions; by (i) the countably many sequences $\omega_k(\prod Z_kA_k(f_j))$, $f_j\in\mathcal F$, are bounded, and a diagonal subsequence makes all of them converge. By multilinearity, \[ \begin{split} &\Bigl|\omega_k\Bigl(\prod_jZ_kA_k(f_j)\Bigr) -\omega_k\Bigl(\prod_jZ_kA_k(g_j)\Bigr)\Bigr|\\ &\quad\le M_n(Q)\sum_i\|f_i-g_i\|_\infty \prod_{j\ne i}\max(\|f_j\|_\infty,\|g_j\|_\infty) \end{split} \] for functions supported in $Q$, so the limits exist for all $f_j\in C_c(\R^4)$ and obey the same bound. Symmetry and multilinearity pass to the limit. A multilinear form bounded in this way is separately continuous on $C^\infty_c(\R^4)$, so the Schwartz kernel theorem makes it a distribution. For dyadic $y$ and large $k$, $y/a_k\in\Z^4$ and translation invariance of $\omega_k$ and of the flow (Proposition~\ref{prop:spatialflow}(i)) give $\omega_k(\prod Z_kA_k(f_j(\cdot-y)))=\omega_k(\prod Z_kA_k(f_j))$. For a real $y$ take dyadic $y_m\to y$; the functions $f_j(\cdot-y_m)$ converge uniformly to $f_j(\cdot-y)$ with supports in one cube, so the bound above carries the invariance to $y$. The formula for $K$ follows from $A^y_k=A_k(f_0(\cdot-y))$ and the argument of (iii) with $n=2$. (iii) Let $\vartheta$ be the lattice reflection of Appendix~\ref{app:transfer} and $\theta_k$ the corresponding reflection of $a_k\Z^4$; it differs from $\theta$ by a translation of length at most $a_k$. Since $\Phi$ is real and $\vartheta$ commutes with the flow, $\vartheta(\bar A_k(f))=A_k(\bar f\circ\theta_k)$. Let $P_k=\sum_\beta c_\beta\prod_jZ_kA_k(f_{\beta,j})$ and $R_k=\sum_\alpha\bar c_\alpha\prod_jZ_kA_k(\overline{f_{\alpha,j}}\circ\theta)$, so that $\vartheta(\bar P_k)$ is $R_k$ with $\theta$ replaced by $\theta_k$. Reflection positivity gives $\omega_k(\vartheta(\bar P_k)P_k)\ge0$. By (i), applied in a cube containing all the supports and their reflections, and by multilinearity, $|\omega_k(\vartheta(\bar P_k)P_k)-\omega_k(R_kP_k)|$ is at most a constant independent of $k$ times $\max_{\alpha,j}\|f_{\alpha,j}\circ\theta_k-f_{\alpha,j}\circ\theta\|_\infty$, which tends to zero by uniform continuity, while $\omega_k(R_kP_k)$ tends to the left side of (iii) by (ii). So the limit of the nonnegative numbers is the left side of (iii). The construction of $\mathcal H$ is the usual quotient of the polynomials at positive times by the null space of this form~\cite{OS73}. By the same approximation of $\theta$ by $\theta_k$, $\|\Psi(P_0)\|^2=\lim_kZ_k^2C_k(0)=1$, and $\ip\Omega{\Psi(P_0)}=S_1(f_0)=0$. (iv) For a polynomial $P$ at positive times let $\tau_tP$ be its time translate by $t\ge0$, again at positive times. By translation invariance and $\theta\circ\tau_t=\tau_{-t}\circ\theta$, $\ip{\Psi(\tau_tP)}{\Psi(P')}= \ip{\Psi(P)}{\Psi(\tau_tP')}$, and $q_P(t)=\|\Psi(\tau_tP)\|^2= \ip{\Psi(P)}{\Psi(\tau_{2t}P)}$. If $P$ has degree at most $d$, every partition in (i) of the products in $q_P(t)$ has at most $d$ blocks, each bounded by a multiple of $|Q|_k$ for a cube $Q$ of side $O(1+t)$, so $q_P(t)\le C_P(1+t)^{4d}$. The Cauchy--Schwarz inequality gives $q_P(t)\le q_P(0)^{1/2}q_P(2t)^{1/2}$, hence by induction $q_P(t)\le q_P(0)^{1-2^{-m}}q_P(2^mt)^{2^{-m}}$ for every $m$, and letting $m\to\infty$, $q_P(t)\le q_P(0)$. So $\tau_t$ maps null vectors to null vectors and extends to a symmetric contraction $\mathsf T(t)$ of $\mathcal H$; $\mathsf T(s)\mathsf T(t)=\mathsf T(s+t)$, and $\mathsf T(t)=\mathsf T(t/2)^2\ge0$. By the bound of (ii) and uniform continuity of the translated test functions, $\|\Psi(\tau_tP)-\Psi(P)\|\to0$ as $t\to0$, so the semigroup is strongly continuous on a dense set, hence everywhere, and $\mathsf T(t)=e^{-tH}$ with $H\ge0$ self-adjoint~\cite{OS73}. Spatial translations preserve the form, have the opposite translations as inverses, commute with $\tau_t$, and are strongly continuous by the same argument. For the ratio let $f_{0,t}=f_0(\cdot-(t,0))$. For dyadic $t>0$ and large $k$, translation invariance gives $\tau_{t/a_k}A_k=A_k(f_{0,t})$, so by (iii) \[ C_k(t)-C_k(0)=\omega_k\bigl(A_k(\bar f_0\circ\theta_k)\, A_k(f_{0,t}-f_0)\bigr), \] a covariance of two centred fields. The case $n=2$ of (i) bounds $Z_k^2$ times it by $L_0\|f_{0,t}-f_0\|_\infty$ with $L_0=(K_2/c)\sup_k|Q|_k\|f_0\|_\infty$, $Q$ a cube containing the supports of $f_0\circ\theta_k$ for all $k$. By uniform continuity of $f_0$ there is a dyadic $\sigma>0$ with $L_0\|f_{0,\sigma}-f_0\|_\infty\le\frac12$; then $C_k(\sigma)/C_k(0)\ge\frac12$, and Theorem~\ref{thm:flowlimit}(iv) applies. (v) For $n\ge3$ the exponent $n/2-1$ in (i) is positive, so with $(a_k^4W_k)^{n/2-1}$ in place of $B_*^{n/2-1}$ the cumulants of order $n\ge3$ of the renormalised fields tend to zero. The moments are the sums over partitions of products of cumulants, so in the limit only the partitions into pairs remain, which is Wick's rule. (vi) As in the proof of Theorem~\ref{thm:transfer}, $\mathbb T_k$ has gap at least $\log(1/\alpha)/(2n_k)$ on the complement of $\Omega_k$. For a polynomial $P$ at positive times let $v_k=\Psi_{P_k}-\omega_k(P_k)\Omega_k$ with $P_k$ its lattice version. For dyadic $t$ and large $k$, \[ \omega_k\bigl(\vartheta(\bar P_k)\tau_{t/a_k}P_k\bigr)-|\omega_k(P_k)|^2 =\ip{v_k}{\mathbb T_k^{t/a_k}v_k}\le e^{-m_*t}\|v_k\|^2 , \] and both sides are finite combinations of moments, which converge by (ii) and the approximation of $\theta$ in (iii). The limit is $\ip v{e^{-tH}v}\le e^{-m_*t}\|v\|^2$ for $v=\Psi(P)-\ip\Omega{\Psi(P)}\Omega$. These $v$ are dense in $\Omega^\perp$, so the inequality holds on $\Omega^\perp$ for all $t\ge0$, and the spectral theorem gives the spectral statement as in Theorem~\ref{thm:transfer}. By (iii), $\Psi(P_0)$ is a unit vector in $\Omega^\perp$. \end{proof} Part (iv) shows that the order-two cases of (T1) and (T2) already contain the lower bound on the ratio that Theorem~\ref{thm:flowlimit}(iv) needs; they are not harmless combinatorial assumptions, and proving them along the continuum tuning is at least as hard as that lower bound. Part (vi) passes the conditional mass gap of Theorem~\ref{thm:transfer} to every polynomial in the smeared fields. Neither part proves Euclidean invariance, independence of the limit from the subsequence, or the reconstruction of the field operators, for which the growth of $M_n(Q)$ in $n$ would also be needed. \begin{remark}\label{rem:amplitude} The amplitude scale cannot be dropped for a bounded field. Suppose $\operatorname{Var}\varphi_k(0)\to0$, as for the energy of the flowed plaquettes in lattice units when the flowed links approach the identity. Reflection invariance of $\omega_k$, the Cauchy--Schwarz inequality and stationarity give \[ C_k(0)\le\omega_k(|A_k|^2)\le\Bigl(a_k^4\sum_x|f_0(a_kx)|\Bigr)^2 \operatorname{Var}\varphi_k(0)\to0 . \] With $r_k=1$, (T2) then forces $a_k^4W_k\to0$, and by part (v) the limit is Gaussian. With $r_k^2$ of the order of $\operatorname{Var}\varphi_k(0)$ this is no longer forced. Choosing $r_k$ does not prove (T1) or (T2). \end{remark} \begin{remark}\label{rem:markov} Let $0<\epsilon<1$, $m>0$, let $\pi$ be the probability measure on $\{-1,0,1\}$ with weights $\epsilon/2$, $1-\epsilon$, $\epsilon/2$, and let $P_t=e^{-mt}I+(1-e^{-mt})\Pi$ on $L^2(\pi)$, where $\Pi f=\pi(f)$ is the projection onto the constants. This is a symmetric Markov semigroup with $P_t\ge0$ and gap $m$, so its stationary process is reflection positive. For $X(s)=s/\sqrt\epsilon$ we have $\pi(X)=\pi(X^3)=0$, $\pi(X^2)=1$, $\pi(X^4)=1/\epsilon$ and $P_tX=e^{-mt}X$, so the two-point function is $e^{-mt}$ for every $\epsilon$, while for $t_1\le t_2\le t_3\le t_4$ and $q_i=e^{-m(t_{i+1}-t_i)}$ the four-point function is $q_3\,\pi\bigl(XP_{t_2-t_1}(XP_{t_3-t_2}X^2)\bigr)=q_1q_3(q_2/\epsilon+1-q_2)$, with truncated part $q_1q_3(q_2/\epsilon-q_2-2q_2^2)$; both diverge as $\epsilon\to0$. By part (i), no choice of amplitude scales makes (T1)--(T3) hold for this family with constants independent of $\epsilon$. \end{remark} The hypotheses (T1)--(T3) are uniform cluster estimates for the continuum tuning $\beta_k\to\infty$, and the next proposition shows that they are estimates at a critical point: with the order-two cases alone the lattice gap is at most a fixed multiple of $a_k$, and the tree majorant carries a fixed fraction of its weight at a fixed physical distance. Let $m_k\in[0,\infty]$ be the gap of $\mathbb T_k$ on the complement of $\Omega_k$, so that $\|\mathbb T_kv\|\le e^{-m_k}\|v\|$ for $v\perp\Omega_k$, and let $t_0>0$ be the least time in the support of $f_0$. \begin{proposition}\label{prop:treecritical} \textup{(i)} For every dyadic $\sigma>0$ and every large $k$, $C_k(\sigma)\le e^{-m_k\sigma/a_k}C_k(0)$. If $m_k/a_k\to\infty$, every limit of Theorem~\textup{\ref{thm:flowlimit}} is trivial; this is the case for $\beta_k=\beta$ fixed in the range of Theorem~\textup{\ref{thm:groups}}, where $m_k\ge m_G(\beta)$. \textup{(ii)} Suppose the $\omega_k$ are translation invariant and \textup{(T1)} for $n=2$ and \textup{(T2)} hold. Then $m_k\le a_k\log2/\sigma$ for every large $k$, where $\sigma$ is the dyadic number of Theorem~\textup{\ref{thm:higher}(iv)}, which depends only on $K_2/c$ and $f_0$. If also the hypothesis of Theorem~\textup{\ref{thm:transfer}} holds, then $m_*a_k\le m_k\le a_k\log2/\sigma$, and in particular $m_*\le\log2/\sigma$. \textup{(iii)} Under \textup{(T1)} for $n=2$ and \textup{(T2)}, let $Q$ be a cube containing the support of $f_0$ and $\gamma=c/(K_2\|f_0\|_\infty^2\sup_k|Q|_k)$. Then for every $k$ \[ \sum_{z\in\Z^4,\ |z_0|\ge R_k}w_k(z)\ge\gamma\,W_k,\qquad R_k=\frac{2t_0}{a_k}-1 . \] If $w_k(z)=e^{-\mu_k|z_0|}u_k(\bar z)$ with $\bar z$ the spatial part of $z$ and $u_k\ge0$ summable on $\Z^3$, then $\mu_kR_k\le\log(2/\gamma)$, so the physical time decay rate of the majorant is at most $\log(2/\gamma)/(2t_0-a_k)$. \end{proposition} \begin{proof} (i) In the proof of Theorem~\ref{thm:flowlimit}, $C_k(\sigma)/C_k(0)=\ip{\Psi_k}{\mathbb T_k^{\sigma/a_k}\Psi_k}$ with $\Psi_k$ a unit vector orthogonal to $\Omega_k$. The operator $\mathbb T_k$ is self-adjoint and fixes $\Omega_k$, so it preserves the complement, and the inequality follows. If $m_k/a_k\to\infty$ then $\rho(\sigma)=0$ for every dyadic $\sigma>0$ along every subsequence, and the limit is trivial by Theorem~\ref{thm:flowlimit}(iv). At fixed $\beta$ in the range of Theorem~\ref{thm:groups} the gap is at least $m_G(\beta)>0$ in lattice units, and $a_k\to0$; the spatial flow does not change the transfer matrix (Proposition~\ref{prop:spatialflow}(iv)). (ii) Theorem~\ref{thm:higher}(iv) uses (T1) and (T2) only for $n=2$, since the exponent $n/2-1$ in part (i) vanishes there, and gives $C_k(\sigma)\ge\frac12C_k(0)$ for large $k$. With (i), $e^{-m_k\sigma/a_k}\ge\frac12$. Under the hypothesis of Theorem~\ref{thm:transfer}, the proof of that theorem gives $m_k\ge\log(1/\alpha)/(2n_k)\ge m_*a_k$. (iii) $C_k(0)=\omega_k(A_k(\bar f_0\circ\theta_k)A_k(f_0))$, a covariance of two centred fields, by the proof of Theorem~\ref{thm:higher}(iv). A point $x$ in the support of the first sum has $a_kx_0\le-t_0+a_k$, and a point $y$ in the support of the second has $a_ky_0\ge t_0$, so $|x_0-y_0|\ge R_k$. By (T1) for $n=2$, \[ \begin{split} C_k(0)&\le K_2r_k^2\|f_0\|_\infty^2\,a_k^8\sum_{a_ky\in Q}\ \sum_{|x_0-y_0|\ge R_k}w_k(x-y)\\ &\le K_2r_k^2\|f_0\|_\infty^2|Q|_k\,a_k^4\sum_{|z_0|\ge R_k}w_k(z), \end{split} \] and (T2) bounds the left side below by $c\,r_k^2a_k^4W_k$. For the product form let $R'=\lceil R_k\rceil$; if $R'\le0$ there is nothing to prove, and otherwise \[ \frac{\sum_{|j|\ge R'}e^{-\mu_k|j|}}{\sum_{j\in\Z}e^{-\mu_k|j|}} =\frac{2e^{-\mu_kR'}}{1+e^{-\mu_k}}\le2e^{-\mu_kR_k}, \] while the factor $\sum_{\bar z}u_k(\bar z)$ cancels. So $2e^{-\mu_kR_k}\ge\gamma$. \end{proof} So reflection positivity and decay do not supply (T2): at fixed $\beta$ in the range of Theorem~\ref{thm:groups} the measures are reflection positive with exponential clustering, and $C_k(\sigma)/C_k(0)\to0$, so no choice of $w_k$, $r_k$ and constants satisfies the order-two cases of (T1) and (T2) together there. For the same reason no estimate whose constants give a lattice gap bounded below, as a convergent strong coupling expansion does~\cite{OSei78}, can give the tree bounds along the tuning: (T1) and (T2) hold, if at all, only along couplings where the lattice gap closes at least as fast as $a_k$. A mass gap in physical units, which bounds the decay of the two-point function, does not give the constants $K_n$ either. By Theorem~\ref{thm:higher}(v) the limit can be non-Gaussian only if $\liminf_ka_k^4W_k>0$; for the product majorants of part (iii) the time factor of $W_k$ is $\coth(\mu_k/2)\ge2/\mu_k$, which is of order $1/a_k$ at least, so the time direction alone is compatible with this. At a fixed physical flow time the flowed fields are smooth on the scale $\sqrt\tau$; majorants of product form for them are not proved here. The order-two cases of (T1) and (T2) say more about the reference vector than part (iv) of Theorem~\ref{thm:higher} records: for a smooth $f_0$ they bound every time difference of the packet, which makes its energy distribution tight with finite moments and gives the ratio with an explicit rate. Recall the spectral measures $\mu_k$ on $[0,\infty]$ of the proof of Theorem~\ref{thm:flowlimit}, so that $C_k(\sigma)/C_k(0)=\int e^{-\sigma E}d\mu_k(E)$. \begin{proposition}\label{prop:treeenergy} Suppose the $\omega_k$ are translation invariant, \textup{(T1)} holds for $n=2$ and \textup{(T2)} holds, and $f_0$ has $m\ge1$ continuous time derivatives. Let $Q$ be a cube containing the supports of $f_0(\cdot-(s,0))$ for $0\le s\le m a_0$, and $B_m=(K_2/c)\sup_k|Q|_k\,\|\partial_t^mf_0\|_\infty^2$. \textup{(i)} $\|(1-\mathbb T_k)^m\Psi_k\|^2\le B_ma_k^{2m}$ for every $k$, where $\Psi_k=Z_k\Psi_{A_k}$. \textup{(ii)} $\int\bigl((1-e^{-a_kE})/a_k\bigr)^{2m}d\mu_k(E)\le B_m$ and $\mu_k([\Lambda,\infty])\le B_m(a_k+\Lambda^{-1})^{2m}$ for $\Lambda>0$. Every weak limit $\mu$ of the $\mu_k$ satisfies $\mu(\{\infty\})=0$ and $\int E^{2m}d\mu(E)\le B_m$. If \textup{(T1)--(T3)} hold, the unit vector $\Psi(P_0)$ of Theorem~\textup{\ref{thm:higher}(iii)--(iv)} lies in the domain of $H^m$, with $\|H^m\Psi(P_0)\|^2\le B_m$. \textup{(iii)} If $a_k\sqrt{B_1}\le\frac12$, then $C_k(\sigma)\ge e^{-2\sqrt{B_1}\sigma}C_k(0)$ for every $\sigma\in a_k\Z_{\ge0}$, and the gap $m_k$ of Proposition~\textup{\ref{prop:treecritical}} satisfies $m_k\le-\log(1-a_k\sqrt{B_1})\le2\sqrt{B_1}\,a_k$. \end{proposition} \begin{proof} (i) Let $D_kf_0=\sum_{j=0}^m(-1)^j\binom mjf_0(\cdot-(ja_k,0))$. By translation invariance $\mathbb T_k^j\Psi_{A_k}=\Psi_{\tau_jA_k}$ and $\tau_jA_k=A_k(f_0(\cdot-(ja_k,0)))$, so $(1-\mathbb T_k)^m\Psi_{A_k}=\Psi_{A_k(D_kf_0)}$, and as in the proof of Theorem~\ref{thm:higher}(iv) its squared norm is the covariance $\omega_k(A_k(\overline{D_kf_0}\circ\theta_k)A_k(D_kf_0))$. The function $D_kf_0$ is supported in $Q$ and is an $m$-th difference with step $a_k$, so $\|D_kf_0\|_\infty\le a_k^m\|\partial_t^mf_0\|_\infty$. Lemma~\ref{lem:treesum} with $n=2$ bounds the covariance by $K_2r_k^2|Q|_k\,a_k^4W_k\|D_kf_0\|_\infty^2$, and (T2) bounds $C_k(0)=Z_k^{-2}$ below by $c\,r_k^2a_k^4W_k$. (ii) By the spectral theorem the left side of (i) is $\int(1-e^{-a_kE})^{2m}d\mu_k$, the integrand being $1$ at $E=\infty$. For $E\ge\Lambda$, $(1-e^{-a_kE})/a_k\ge(1-e^{-a_k\Lambda})/a_k\ge \Lambda/(1+a_k\Lambda)$, because $1-e^{-x}\ge x/(1+x)$; this gives the tail bound. The set $(\Lambda,\infty]$ is open in $[0,\infty]$, so $\mu((\Lambda,\infty])\le\liminf_k\mu_k((\Lambda,\infty])\le B_m\Lambda^{-2m}$, and $\mu(\{\infty\})=0$. For a continuous $\chi$ with $0\le\chi\le1$ and support in $[0,L]$, the same inequality gives $E\le(1+a_kL)(1-e^{-a_kE})/a_k$ on that support, so $\int\chi E^{2m}d\mu=\lim_k\int\chi E^{2m}d\mu_k\le B_m$; letting $\chi$ increase to $1$ gives the moment bound. The vector $\Psi(P_0)$ has $\ip{\Psi(P_0)}{e^{-\sigma H}\Psi(P_0)}=\lim_kC_k(\sigma)/C_k(0)=\int e^{-\sigma E}d\mu$ for dyadic $\sigma>0$ along the subsequence, so its spectral measure for $H$ is $\mu$. (iii) By (i) with $m=1$ and the Cauchy--Schwarz inequality, $\ip{\Psi_k}{(1-\mathbb T_k)\Psi_k}\le a_k\sqrt{B_1}$, which is the hypothesis of Proposition~\ref{prop:flowshift}(ii) with $C=\sqrt{B_1}$. Since $\Psi_k$ is a unit vector orthogonal to $\Omega_k$, the same quantity is at least $1-e^{-m_k}$. \end{proof} Part (iii) makes Proposition~\ref{prop:treecritical}(ii) explicit: the order-two tree and saturation bounds pin the lattice gap below $2\sqrt{B_1}\,a_k$, a physical mass of at most $2\sqrt{B_1}$, and give the ratio of Theorem~\ref{thm:flowlimit}(iv) at every separation. The open input of Theorem~\ref{thm:flowlimit} is therefore implied by these two bounds for one smooth $f_0$; nothing here proves them for the Wilson action. \subsection{An averaged cube condition}\label{ssec:averaged} The condition $D(n,\alpha)$ enters the proof of Theorem~\ref{thm:transfer} through the comparison lemma, which contracts Lipschitz seminorms and therefore needs the Kantorovich response of a cube for every exterior. It is posed in the forest gauge, and Propositions~\ref{prop:gaussforest} and~\ref{prop:gaussevery} and Theorem~\ref{thm:cubeneeds} show what that gauge costs. This subsection replaces it by a condition in the mean over the Wilson measure itself, on gauge-invariant functions and without gauge fixing: a Poincar\'e inequality for the dynamics that resamples cubes of side $n$. A finite-speed bound for that dynamics, which uses only the locality of its moves, turns the inequality into decay at a rate of order $1/n$ in lattice units, and the continuum step goes through as in Theorem~\ref{thm:transfer}. \emph{Setting.} Let $G\subset\U(N)$ be compact and nontrivial, $\delta$ a bi-invariant metric of diameter $D$ on $G$, and $\varphi$ a positive continuous class function on $G$, such as $e^{N\beta\ReTr(\cdot)}$ or the Villain weight of Proposition~\ref{prop:coulomb}. The lattice $\Lambda$ is either the torus $\Lambda_L=(\Z/L\Z)^4$ with $L$ even and $L\ge2n+4$, or a box of $\Z^4$ with all sides at least $2n$; $E$ is its set of links, $P$ its set of plaquettes (on a box, those with four corners in it), and \[ \mu(dU)=\frac1Z\prod_{p\in P}\varphi(U_p)\prod_{e\in E}dU_e . \] A link $(y,\nu)$ has base point $y$; on $\Lambda_L$ differences of base points are taken in $(-L/2,L/2]^4$ and $\lvert\cdot\rvert_\infty$ is the corresponding distance. For $x\in\Z^4$ (or $x\in\Lambda_L$) let $V_x$ be the set of links of $E$ with base point in $x+[0,n)^4$, and let $X$ be the set of $x$ with $V_x\ne\emptyset$. Every link lies in exactly $n^4$ of the sets $V_x$, $x\in X$, and $\lvert X\rvert\ge4n^4$. For $V\subset E$ let $\partial V$ be the set of links of $E\setminus V$ that share a plaquette of $P$ with a link of $V$; the conditional law $\gamma^0_V(\cdot\mid U)$ of $U_V$ under $\mu$ given the other links has density proportional to $\prod_{p\cap V\ne \emptyset}\varphi(U_p)$ and depends only on $U_{\partial V}$. Write $\mathcal P_x$ for the conditional expectation given the links outside $V_x$, $\mathcal P_xf(U)=\int f\,d\gamma^0_{V_x}(\cdot\mid U)$, an orthogonal projection on $L^2(\mu)$, and $\mathcal A$ for the closed subspace of functions invariant under the gauge group $G^\Lambda$. By gauge covariance, $\gamma^0_{V_x}(\cdot\mid gU)$ is the image of $\gamma^0_{V_x}(\cdot\mid U)$ under the action of $g$ on the links of $V_x$, so $\mathcal P_x$ maps $\mathcal A$ into itself. \begin{definition}\label{def:avgcube} For $n\ge2$ and $\varpi>0$, the condition $\mathrm{BP}(n,\varpi)$ for $\mu$ is \[ \varpi\Var_\mu(f)\le\frac1{n^4}\sum_{x\in X}\mu\bigl((f-\mathcal P_xf)^2 \bigr) \] for every real $f\in\mathcal A$. \end{definition} The right side is the Dirichlet form of the dynamics that resamples each cube at rate $n^{-4}$, so that every link is resampled at rate one, and $\mathrm{BP}(n,\varpi)$ says that this dynamics has spectral gap at least $\varpi$ on gauge-invariant functions. It contains no supremum over exteriors and no gauge. Put $q_n=(2+1/n)^4$, so that $16m_0}\frac{(q_n\tau)^m}{m!}, \] where $\tau=tn^4/\lvert X\rvert$. \end{lemma} \begin{proof} If $h$ depends only on the links of $W$, then $\mathcal P_xh=h$ when $V_x\cap W=\emptyset$, and $\mathcal P_xh$ depends only on the links of $(W\setminus V_x)\cup\partial V_x$ otherwise; induction on $t$ gives the first claim. Run two chains from $U$. At each step choose $x_s\in X$ uniformly, the choices independent. If $x_s\notin Y$, the first chain resamples $V_{x_s}$ from $\gamma^0_{V_{x_s}}$ and the second does nothing. If $x_s\in Y$ and the two configurations agree on $\partial V_{x_s}$, both laws of $V_{x_s}$ are the same, and both chains receive one common sample; otherwise they resample independently. The two marginal processes are the chains $\mathcal K$ and $\mathcal K_Y$, so $\mathcal K^tf(U)-\mathcal K_Y^tf(U)$ is the expectation of the difference of $f$ at time $t$, which vanishes unless the set $\mathcal D_t$ of links where the two configurations differ meets $S$. We claim that every link of $\mathcal D_t$ lies in $V_{x_{s_m}}$ for times $s_1<\dotsm_0$. For fixed $m$ there are at most $\binom tm\le t^m/m!$ choices of the times, at most $n^4\lvert S\rvert$ of $x_{s_m}$ and at most $q_nn^4$ of each earlier cube given the next, and each prescribed sequence occurs with probability $\lvert X\rvert^{-m}$. So the probability that $\mathcal D_t$ meets $S$ is at most $\frac{\lvert S\rvert}{q_n}\sum_{m> m_0}(q_nn^4t/\lvert X\rvert)^m/m!$, and the difference of $f$ is at most $2\lVert f\rVert_\infty$. \end{proof} For finite $S,S'\subset E$ let $d(S,S')$ be the least $\lvert b-b'\rvert_\infty$ over base points $b$ of links of $S$ and $b'$ of links of $S'$. \begin{theorem}[decay from the averaged condition]\label{thm:avgdecay} Suppose that $\mathrm{BP}(n,\varpi)$ holds for $\mu$. Then $\varpi\le4$, and for bounded $f,g\in\mathcal A$ that depend only on the links of finite sets $S_f,S_g$ with $d=d(S_f,S_g)\ge3n-2$, \[ \begin{split} \lvert\Cov_\mu(f,g)\rvert&\le\Bigl(e+\frac{k\lvert S_f\rvert\lvert S_g \rvert}{64e^2}\Bigr)\lVert f\rVert_\infty\lVert g\rVert_\infty\, e^{-\varpi k/(e^2q_n)},\\ k&=\Bigl\lfloor\frac{d+2}n\Bigr\rfloor-2 . \end{split} \] In particular, for every $d$, $\lvert\Cov_\mu(f,g)\rvert\le C(1+d) e^{-\varpi d/(e^2q_nn)}$ with $C$ depending on $\lvert S_f\rvert$, $\lvert S_g\rvert$, $\lVert f\rVert_\infty$, $\lVert g\rVert_\infty$ only, and complex $f,g$ satisfy both bounds with an extra factor four. \end{theorem} \begin{proof} \emph{The bound $\varpi\le4$.} The function $f=\ReTr U_p$ of one plaquette is gauge invariant and not constant on $G^E$, and $\mu$ has a positive density, so $\Var_\mu(f)>0$. Each $I-\mathcal P_x$ is an orthogonal projection that annihilates constants, so $\mu((f-\mathcal P_xf)^2)\le \Var_\mu(f)$, and it vanishes unless $V_x$ contains one of the four links of $p$, which happens for at most $4n^4$ points $x$. \emph{The spectral part.} Let $\mathcal A^0$ be the functions of $\mathcal A$ with mean zero. $\mathcal K$ is an average of orthogonal projections that preserve $\mathcal A^0$, so it is self-adjoint with $0\le\mathcal K\le I$ on $\mathcal A^0$, and $\mathrm{BP}(n,\varpi)$ reads $\ip h{(I-\mathcal K)h}\ge \epsilon\lVert h\rVert^2$ on $\mathcal A^0$ with $\epsilon=\varpi n^4/\lvert X \rvert$. Hence the spectrum of $\mathcal K$ on $\mathcal A^0$ lies in $[0,1-\epsilon]$ and $\lVert\mathcal K^Th\rVert_2\le e^{-\epsilon T}\lVert h\rVert_2$ for every integer $T\ge0$. \emph{The decomposition.} Since $(I-\mathcal K)$ annihilates constants, \[ f-\mu(f)=\mathcal K^T\bigl(f-\mu(f)\bigr)+\sum_{t=0}^{T-1}\mathcal K^t (I-\mathcal K)f , \] and pairing with $g$ in $L^2(\mu)$, \[ \Cov_\mu(f,g)=\ip{\mathcal K^T(f-\mu f)}{g-\mu g}+\sum_{t=0}^{T-1} \ip{\mathcal K^tf}{(I-\mathcal K)g}, \] where the self-adjointness of $\mathcal K$ was used. The first term is at most $e^{-\epsilon T}\lVert f\rVert_\infty\lVert g\rVert_\infty$. \emph{The local part.} Let $Z_g=Y_1(S_g)$, so $\lvert Z_g\rvert\le n^4\lvert S_g\rvert$ and $(I-\mathcal K)g=\lvert X\rvert^{-1}\sum_{x\in Z_g}(g- \mathcal P_xg)$. If $h$ does not depend on the links of $V_x$, then $\mathcal P_xh=h$ and $\ip h{g-\mathcal P_xg}=\ip hg-\ip{\mathcal P_xh}g=0$. Put $m_0= \lfloor(d+2)/n\rfloor-3\ge0$, so $k=m_0+1$, and $Y=Y_{m_0}(S_f)$. Suppose a link $w$ lies in $V_x$ with $x\in Z_g$, and either $w\in S_f$ or $w\in \partial V_y$ with $y\in Y_j(S_f)$, $1\le j\le m_0$. In the second case there are $y=y_j\rightsquigarrow y_{j-1}\rightsquigarrow\dots\rightsquigarrow y_1$ with $y_1\in Y_1(S_f)$, so $\lvert y-y_1\rvert_\infty\le n(m_0-1)$; with a base point $b$ of a link of $S_f$ in $y_1+[0,n)^4$, the base point of $w$ in $y+[-1,n]^4\cap(x+[0,n)^4)$, and a base point $b'$ of a link of $S_g$ in $x+[0,n)^4$, \[ \lvert b-b'\rvert_\infty\le(n-1)+n(m_0-1)+n+2(n-1)=n(m_0+3)-3, \] and the first case gives at most $2(n-1)$. Both are at most $n\lfloor(d+2)/n \rfloor-3 d/n-3$, $k\le d+1$ and $3\varpi/(e^2q_n)<1$; for $d<3n-2$ it holds with $C= e\lVert f\rVert_\infty\lVert g\rVert_\infty$, since $\lvert\Cov_\mu(f,g) \rvert\le\lVert f\rVert_\infty\lVert g\rVert_\infty$ and $\varpi d/(e^2q_nn) <1$. Real and imaginary parts give the complex case. \end{proof} The proof uses the condition only on $\mathcal A$ and only in $L^2(\mu)$; the locality of the moves enters through the counting of chains, at speed $e^2q_nn$ links per unit of the resampling time. The theorem gives the lattice gap and the continuum step. On the torus a \emph{torus limit} at $\beta$ is a state $\omega$ on the continuous gauge-invariant local functions of $\Z^4$ with $\omega(O)=\lim_j\mu_{L_j}(O)$ along a sequence $L_j\to\infty$, each $O$ evaluated on $\Lambda_{L_j}$ through a box of $\Z^4$ centred at the origin once $L_j$ is large; by compactness of the marginals and a diagonal sequence every sequence $L_j\to\infty$ has a subsequence along which such a limit exists. \begin{corollary}[the lattice gap]\label{cor:avggap} Let $\mathcal L$ be an infinite set of even integers $L\ge2n+4$ and suppose that $\mathrm{BP}(n,\varpi)$ holds for the Wilson measure of $\Lambda_L$ at $\beta\ge0$ for every $L\in\mathcal L$. Then every torus limit $\omega$ along $\mathcal L$ is invariant under translations and the hypercubic group and reflection positive, and its transfer matrix has spectral gap at least \[ m_n=\frac{\varpi}{e^2q_nn}\ge\frac{16\varpi}{625e^2n}. \] \end{corollary} \begin{proof} The measures $\mu_L$ are invariant under the translations and the hypercubic group of $\Lambda_L$, and for $\beta\ge0$ and even $L$ they are reflection positive for reflections in planes through sites and in planes bisecting links~\cite{OSei78,Sei82}. Each of these properties is a set of identities or inequalities for finitely many local functions, so it passes to $\omega$, and Appendix~\ref{app:transfer} applies. Let $F$ be a continuous gauge-invariant function of finitely many links at positive times. The supports of $\vartheta(\bar F)$ and $\tau_tF$ are at distance $d$ with $t-c_F\le d\le t+c_F$, $c_F$ depending only on $F$, and Theorem~\ref{thm:avgdecay} on $\Lambda_L$, $L\in\mathcal L$ large, bounds $\lvert\Cov_{\mu_L}(\vartheta(\bar F),\tau_tF)\rvert$ by $C_F(1+t)e^{-m_nt}$ with $C_F$ independent of $L$. The limit gives $\lvert\omega(\vartheta(\bar F)\tau_tF)-\lvert\omega(F) \rvert^2\rvert\le C_F(1+t)e^{-m_nt}$, using $\omega(\vartheta(\bar F))= \overline{\omega(F)}$ and translation invariance. These $F$ are dense in the gauge-invariant part of $L^2(\omega)$ on each support, so Proposition~\ref{prop:gapdecay} with every $m0$, $\ell>0$ and integers $n_k\ge2$ with $n_ka_k\le\ell$ such that, for every $k$, $\mathrm{BP}(n_k,\varpi)$ holds for the Wilson measure of $\Lambda_L$ at $\beta_k$ for all $L$ in an infinite set $\mathcal L_k$ of even integers $L\ge2n_k+4$, and let $\omega_k$ be a torus limit along $\mathcal L_k$. Then every continuum limit of $(\omega_k,O_k)$ satisfies \[ \sigma(H)\subset\{0\}\cup[m_*,\infty),\qquad m_*=\frac{16\varpi}{625e^2\ell}, \] and $0$ is a simple eigenvalue. If the limit is nontrivial, $H$ has a mass gap of at least $m_*$. \end{theorem} \begin{proof} The proof of Theorem~\ref{thm:transfer} uses $D(n_k,\alpha)$ only through the gap of $\mathbb T_k$ on $\Omega_k^\perp$. By Corollary~\ref{cor:avggap} that gap is at least $16\varpi/(625e^2n_k)$, and for $t=\tau/a_k$, $16\varpi t/(625e^2n_k)\ge m_*\tau$; the rest of that proof is unchanged. \end{proof} Proposition~\ref{prop:spatialflow}(iii) holds with Theorem~\ref{thm:avgtransfer} in place of Theorem~\ref{thm:transfer}, by the same proof. The averaged condition follows from its supremum form, posed for the same measure without gauge fixing. On $\Lambda_L$ let $C^0_{V,j}$ be defined as $C_{V,j}$ in Sec.~\ref{sec:bypass} with $\gamma^0_V$ in place of $\gamma_V$ and the links of $E$ in place of the active links, put $K^0_\nu=\sum_jC^0_{V_0,j}$, the sum over the links $j\notin V_0$ of direction $\nu$, and let $D^0_L(n,\alpha)$ be the condition $K^0_\nu\le\alpha n^4$ for $\nu=0,\dots,3$. The translations of $\Lambda_L$ preserve $\mu$, so $C^0_{V_x,j+x}=C^0_{V_0,j}$. \begin{proposition}\label{prop:avgsup} On $\Lambda_L$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item if $D^0_L(n,\alpha)$ holds with $\alpha<1$, then $\mathrm{BP}(n,1-\alpha)$ holds, for every real $f\in L^2(\mu)$; \item for $\varphi\equiv1$, that is $\beta=0$, $\mathrm{BP}(n,1)$ holds for every $n$ and every real $f\in L^2(\mu)$. \end{enumerate} \end{proposition} \begin{proof} (i) Let $d_E(U,U')=\sum_{e\in E}\delta(U_e,U'_e)$ and $\kappa_0=(1-\alpha) n^4/L^4$. Let $U,U'$ differ only at a link $i$. Couple $\mathcal K(U,\cdot)$ and $\mathcal K(U',\cdot)$ by choosing the same $x$. If $i\in V_x$, the two exteriors agree, the laws of the resampled cube coincide, and the identity coupling gives distance $0$. If $i\notin V_x$, an optimal coupling of the two cube laws for $d_{V_x}$ costs at most $C^0_{V_x,i}\delta(U_i,U'_i)$, and the link $i$ adds $\delta(U_i,U'_i)$. The $x$ with $i\notin V_x$ number $L^4-n^4$, and $\sum_{x:\,i\notin V_x}C^0_{V_x,i}=K^0_\nu$ for the direction $\nu$ of $i$, so \[ \begin{split} W_1^{d_E}\bigl(\mathcal K(U,\cdot),\mathcal K(U',\cdot)\bigr)&\le\frac{L^4- n^4+\alpha n^4}{L^4}\,\delta(U_i,U'_i)\\ &=(1-\kappa_0)\,\delta(U_i,U'_i). \end{split} \] Changing the links of two arbitrary configurations one at a time and using the triangle inequality for $W_1^{d_E}$ extends this to $(1-\kappa_0)d_E(U, U')$. Hence a function $h$ with Lipschitz constant $\mathrm{Lip}(h)$ for $d_E$ has $\mathrm{Lip}(\mathcal K^th)\le(1-\kappa_0)^t\mathrm{Lip}(h)$, and since $d_E\le D\lvert E\rvert$, \[ \begin{split} \Var_\mu(\mathcal K^th)&=\tfrac12\iint\bigl(\mathcal K^th(U)-\mathcal K^th(U')\bigr)^2d\mu\,d\mu\\ &\le\tfrac12(1-\kappa_0)^{2t}\mathrm{Lip}(h)^2D^2\lvert E\rvert^2 . \end{split} \] Let $\nu_h$ be the spectral measure of the self-adjoint operator $0\le\mathcal K\le I$ at $h-\mu(h)$. Then $\Var_\mu(\mathcal K^th)=\int \lambda^{2t}d\nu_h$, and if $\nu_h((1-\kappa_0+\eta,1])>0$ for some $\eta>0$ this exceeds the bound for large $t$; so $\nu_h$ lives on $[0,1-\kappa_0]$. Lipschitz functions are uniformly dense in $C(G^E)$ by the Stone--Weierstrass theorem, hence dense in $L^2(\mu)$, so the spectral projection of $\mathcal K$ for $(1-\kappa_0,1]$ vanishes on the functions of mean zero, and $\ip f{(I-\mathcal K)f}\ge\kappa_0\Var_\mu(f)$ for every $f$. Multiplying by $L^4/n^4$ gives $\mathrm{BP}(n,1-\alpha)$. (ii) For $\varphi\equiv1$, $\mu$ is the product of Haar measures. Write $f=\sum_{S\subset E}f_S$ with $f_S$ depending only on the links of $S$ and $\int f_S\,dU_e=0$ for $e\in S$; the $f_S$ are orthogonal, $\Var_\mu(f)= \sum_{S\ne\emptyset}\lVert f_S\rVert^2$, and $\mathcal P_xf=\sum_{S\cap V_x= \emptyset}f_S$. Hence \[ \sum_x\mu\bigl((f-\mathcal P_xf)^2\bigr)=\sum_{S\ne\emptyset}\lVert f_S \rVert^2\,\#\{x:S\cap V_x\ne\emptyset\}\ge n^4\Var_\mu(f), \] because a link of $S$ lies in $n^4$ cubes. \end{proof} Whether $D(n,\alpha)$ in the forest gauge implies $\mathrm{BP}$ is not decided here: the forest gauge does not close on a torus, and on $\Z^4$ the two conditions concern different specifications. The averaged condition carries the full content of a gap, since by Theorem~\ref{thm:avgdecay} it gives decay of every bounded gauge-invariant covariance at the rate $\varpi/(e^2q_nn)$; like $D(n,\alpha)$, it fails in the Coulomb phase of the abelian theory at every cube size. \begin{corollary}\label{cor:avgcoulomb} Let $G=\U(1)$ with the Villain weight, and let $\beta_0$ be as in Proposition~\ref{prop:coulomb}. For $\beta>\beta_0$ there are no $n\ge2$ and $\varpi>0$ such that $\mathrm{BP}(n,\varpi)$ holds for the measures with free boundary of all boxes of side at least $2n$. \end{corollary} \begin{proof} The field strength $\Psi_p$ of the proof of Proposition~\ref{prop:coulomb} is a bounded gauge-invariant function of the four links of $p$, and its mean vanishes under every box measure by the symmetry $U\mapsto\bar U$. If $\mathrm{BP}(n,\varpi)$ held for all boxes, Theorem~\ref{thm:avgdecay} would give $\lvert W_\Lambda(\Psi_p\Psi_{p'})\rvert\le C(1+d)e^{-\varpi d/(e^2q_nn)}$ for parallel plaquettes at distance $d$, with $C$ independent of $\Lambda$ and $p'$. The state of~\cite{FS82} is the limit of these measures (proof of Proposition~\ref{prop:coulomb}), so $\sum_{p'}\lvert\omega(\Psi_p\Psi_{p'}) \rvert<\infty$, which contradicts~\cite[Sect.~2]{FS82} as in that proof. \end{proof} \begin{remark}[what the averaged condition settles and what it leaves]\label{rem:avgstatus} Proved: Theorem~\ref{thm:avgtransfer} gives a mass gap of at least $16\varpi/(625e^2\ell)$ in every nontrivial continuum limit of torus limits from the condition $\mathrm{BP}(n_k,\varpi)$ with $n_ka_k\le\ell$, a spectral gap of the cube dynamics on gauge-invariant functions in $L^2$ of the Wilson measure, uniform in the torus; the condition follows from its supremum form $D^0_L(n,\alpha)$ with $\varpi=1-\alpha$, holds at $\beta=0$ with $\varpi=1$ (Proposition~\ref{prop:avgsup}), holds on every torus at strong coupling for every cube size past an explicit $n_0(\beta)$, with $\varpi\to1$ as $n\to\infty$ (Corollary~\ref{cor:bpranges}), fails on the torus of side $2n+4$ once $\beta\gg n^4\log\beta$, already on the functions of nonzero centre charge (Theorem~\ref{thm:bpfixed}, Corollary~\ref{cor:bpcharged}), and fails in the Coulomb phase of the Villain theory at every cube size (Corollary~\ref{cor:avgcoulomb}); the conclusion of Theorem~\ref{thm:avgtransfer} follows from the weaker local form $\mathrm{BP}_{\rm loc}$, which for $\SU(N)$ is implied by the condition on the neutral sector (Theorem~\ref{thm:bplocal}). The obstructions of the forest gauge, Propositions~\ref{prop:gaussforest}, \ref{prop:gaussevery} and~\ref{prop:tetra}, Theorem~\ref{thm:cubeneeds} and Corollary~\ref{cor:cubefails}, bound $K_c$ from below through Lipschitz functions of the gauge-fixed links, transported by gauge transformations of $\Z^4$, which in the Gaussian models are the flat fields $dh$; the averaged condition involves only gauge-invariant functions, on which every gauge transformation acts trivially, so these proofs do not apply to it. Nor do Theorem~\ref{thm:fixedobstruction} and Proposition~\ref{prop:hiddenbit}, which concern bounds uniform in the coarse field and arguments from rarity, since the condition contains no supremum and no labels. Not proved: the condition $\mathrm{BP}(n_k,\varpi)$ along the continuum tuning for any non-abelian group, or outside the strong-coupling ranges of Corollary~\ref{cor:bpranges}, and its local form $\mathrm{BP}_{\rm loc}$ along the tuning; the relation between $\mathrm{BP}$ and $D(n,\alpha)$; and (E). Remark~\ref{rem:bpstatus} lists what Sec.~\ref{ssec:bpstrong} proves about the dependence on $n$ and $\beta$. Corollary~\ref{cor:avgcoulomb} shows that a proof of $\mathrm{BP}$ has to use the non-abelian structure of $G$, as for (D). Neither the averaged condition nor the mass gap is proved here. \end{remark} \subsection{The averaged condition at strong coupling and at fixed cube size}\label{ssec:bpstrong} This subsection proves $\mathrm{BP}(n,\varpi)$ on every torus, for every cube size, in explicit ranges of the coupling, with $\varpi$ tending to its value at $\beta=0$ as $n\to\infty$, and proves that at weak coupling the condition fails on the torus of side $2n+4$ when $n$ is small compared with $(\beta/\log\beta)^{1/4}$. The first statement comes from the supremum form $D^0_L(n,\alpha)$ of Proposition~\ref{prop:avgsup}, which the single-link coefficients of the ungauged measure control through a Dobrushin--Shlosman estimate for one cube~\cite{DS85}. The second uses the center of $\SU(N)$ and the holonomies of the torus, which the cube moves can change only through the curvature of the configuration. Throughout, $\Lambda=\Lambda_L$ with $L$ even, $\delta$ is a bi-invariant metric on $G$, and $C^0_{V,j}$, $K^0_\nu$ are as in Proposition~\ref{prop:avgsup}. Two distinct links are \emph{partners} if they lie in a common plaquette; for $L\ge3$ every link has exactly eighteen partners, three in each of its six plaquettes, and two links share at most one plaquette. Write $C^0_{ee'}= C^0_{\{e\},e'}$ for the single-link coefficients; $C^0_{ee'}=0$ unless $e$ and $e'$ are partners. \begin{lemma}\label{lem:bpblock} Suppose that $C^0_{ee'}\le c$ for all partners $e,e'$, with $18c<1$. Then for every finite $V\subset E$ and every link $j\notin V$, \[ C^0_{V,j}\le\frac1{1-18c}\sum_{e\in V}C^0_{ej}. \] \end{lemma} \begin{proof} Let $U,U'$ differ only at $j$, and let $\gamma=\gamma^0_V(\cdot\mid U)$, $\gamma'=\gamma^0_V(\cdot\mid U')$. Under $\gamma$ the conditional law of $\sigma_e$, $e\in V$, given the other links of $V$ is $\gamma^0_e(\cdot\mid \sigma,U_{E\setminus V})$, and likewise for $\gamma'$. The couplings of $\gamma$ and $\gamma'$ form a weakly compact set on which $\Psi(\pi)=\sum_{e \in V}\E_\pi\delta(\sigma_e,\sigma'_e)$ is continuous; let $\pi^*$ be a minimiser and $\varrho(e)=\E_{\pi^*}\delta(\sigma_e,\sigma'_e)$. For $e\in V$ replace $(\sigma_e,\sigma'_e)$ by a sample of an optimal coupling of the two single-link laws, chosen measurably~\cite[Cor.~5.22]{Vil09}; both marginals are invariant, so the result is a coupling of $\gamma$ and $\gamma'$. Changing the partners of $e$ one at a time bounds the new expected distance at $e$ by $\sum_{k\in V}C^0_{ek}\varrho(k)+C^0_{ej}\delta(U_j,U'_j)$, and minimality gives \[ \varrho(e)\le\sum_{k\in V}C^0_{ek}\varrho(k)+C^0_{ej}\delta(U_j,U'_j). \] Sum over $e\in V$. Since $k$ has eighteen partners, $\sum_{e\in V}C^0_{ek}\le 18c$, so $(1-18c)\sum_e\varrho(e)\le\delta(U_j,U'_j)\sum_{e\in V}C^0_{ej}$, and $W_1^{d_V}(\gamma,\gamma')\le\sum_e\varrho(e)$. \end{proof} \begin{lemma}\label{lem:bpcount} Let $n\ge2$, $L\ge2n+4$ and let $j$ be a link of direction $\nu$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\sum_{x:\,j\notin V_x}\#\{e\in V_x:e\text{ a partner of }j\}=18n^3- 3n^2$. \item The number $B_n$ of $x$ with $j\notin V_x$ such that $V_x$ contains a partner of $j$ is $7n^3+3n^2$. \end{enumerate} \end{lemma} \begin{proof} Let $b$ be the base point of $j$. For each $\mu\ne\nu$ the plaquette at $b$ in the plane $\{\mu,\nu\}$ gives the partners $(b+e_\nu,\mu)$, $(b+e_\mu, \nu)$, $(b,\mu)$, and the plaquette at $b-e_\mu$ gives $(b-e_\mu,\nu)$, $(b-e_\mu,\mu)$, $(b-e_\mu+e_\nu,\mu)$, so the base points of the partners are $b+\Delta$ with $\Delta\in\{e_\nu,e_\mu,0,-e_\mu,-e_\mu,e_\nu-e_\mu\}$. A link with base point $b+\Delta$ lies in $V_x$ exactly when $z=b-x$ satisfies $z+\Delta\in Q=[0,n)^4$, and $j\in V_x$ exactly when $z\in Q$; since $L\ge2n+4$ these conditions are read in $\Z^4$. (i) Exchanging the sums, the left side is $\sum_\Delta\#\{z:z+\Delta\in Q,\ z\notin Q\}=\sum_\Delta(n^4-\lvert Q\cap(Q-\Delta)\rvert)$. This is $0$ for $\Delta=0$, $n^4-n^3(n-1)=n^3$ for $\Delta=\pm e_\mu,e_\nu$, and $n^4-n^2 (n-1)^2=2n^3-n^2$ for $\Delta=e_\nu-e_\mu$. Each $\mu$ contributes $6n^3-n^2$. (ii) The set of $z\notin Q$ with $z+\Delta\in Q$ for some $\Delta$ consists of the layer $\{z_\nu=-1\}$ of $Q-e_\nu$, the layers $\{z_\mu=-1\}$ and $\{z_\mu=n\}$ of $Q\mp e_\mu$, each with the other coordinates in $[0,n)$, and the new points $\{z_\nu=-1,\ z_\mu=n\}$ of $Q+e_\mu-e_\nu$, with the two remaining coordinates in $[0,n)$. These sets are disjoint, with $n^3+3(2n^3+n^2)$ points. \end{proof} \begin{theorem}[the averaged condition from single links]\label{thm:bpstrong} Let $n\ge2$ and $L\ge2n+4$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item If $C^0_{ee'}\le c$ for all partners, with $18c<1$, then $K^0_\nu\le\alpha_n n^4$ for $\nu=0,\dots,3$, with \[ \alpha_n=\frac{c\,(18n-3)}{(1-18c)\,n^2}, \] and if $\alpha_n<1$, $\mathrm{BP}(n,1-\alpha_n)$ holds for every real $f\in L^2(\mu)$. \item For every $\beta$, \[ K^0_\nu\le(7n^3+3n^2)\,\Xi_n,\qquad\Xi_n=\max_jC^0_{V_0,j}. \] The number $\Xi_n$ and the condition $D^0_L(n,\alpha)$ do not depend on $L\ge2n+4$, and $\mathrm{BP}(n,1-(7n^{-1}+3n^{-2})\Xi_n)$ holds whenever $(7n^3+3n^2)\Xi_n0$. The proof of Lemma~\ref{lem:weighted} with one factor and $w\equiv1$ gives $\Gamma(P_th)\le e^{-2K_1t}P_t\Gamma(h)$ for the Langevin semigroup of $\nu$, and as in the proof of Lemma~\ref{lem:decayrate}, \[ \lvert\Cov_\nu(g,h)\rvert\le\int_0^\infty\lvert\nu(\Gamma(P_th,g))\rvert \,dt\le\frac{\lVert\nabla g\rVert_\infty\lVert\nabla h\rVert_\infty}{K_1}. \] Move the partner $e'$ along a unit-speed geodesic $e^{sX}U_{e'}$, $0\le s \le\delta(U_{e'},U'_{e'})$, and let $\nu_s\propto e^{-V_s}$. Then $\frac d{ds}\nu_s(g)=-\Cov_{\nu_s}(g,\partial_sV_s)$; $\partial_sV_s$ comes from the one plaquette $p$ that contains $e$ and $e'$, and its gradient in $U_e$ is $-N\beta$ times the mixed block $H^p_{ee'}(\cdot,X)$, of norm at most one by Lemma~\ref{lem:mixed}. For smooth $g$ that are $1$-Lipschitz for $\delta$ this gives $\lvert\nu_0(g)-\nu_\delta(g)\rvert\le N\lvert\beta \rvert\delta/K_1$. Convolution with a probability measure on $G$ preserves the Lipschitz constant of a function for a bi-invariant metric, so smooth $1$-Lipschitz functions approximate every $1$-Lipschitz one uniformly, and the Kantorovich--Rubinstein duality gives the bound. \end{proof} \begin{corollary}[explicit ranges]\label{cor:bpranges} For every even $L\ge2n+4$, $\mathrm{BP}(n,1-\alpha_n)$ holds for every real $f\in L^2(\mu)$ in the following cases. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $G=\SU(2)$, $\lvert\beta_W\rvert<\frac29$, with \[ \alpha_n=\frac{\lvert\beta_W\rvert(18n-3)}{(4-18\lvert\beta_W\rvert)n^2}, \] whenever $\lvert\beta_W\rvert<4n^2/(18n^2+18n-3)$; for $n=2$ this is $\lvert\beta_W\rvert<\frac{16}{105}$.\footnote{With $c=\lvert\beta_W \rvert/4$ the condition $18c<1$ of Theorem~\ref{thm:bpstrong} reads $\lvert\beta_W\rvert<\frac29$, and $\alpha_n<1$ reads $\lvert\beta_W\rvert (18n-3)<(4-18\lvert\beta_W\rvert)n^2$, that is, $\lvert\beta_W\rvert(18n^2+ 18n-3)<4n^2$; at $n=2$ this is $105\lvert\beta_W\rvert<16$. As $n\to \infty$ the threshold $4n^2/(18n^2+18n-3)$ increases to $\frac29$.} At $\lvert\beta_W\rvert\le\frac18$, $\alpha_n\le\alpha_2\le0.59$ for every $n\ge2$; at $\lvert\beta_W\rvert\le \frac15$, $\alpha_n\le0.982$ for every $n\ge9$. \item $G=\SU(3)$ and $\beta_W\ge0$ with \[ \zeta(\beta_W)=\frac{3r\sqrt{\epsilon(r)}}{\gamma(r)}<1,\qquad r=2\beta_W, \] where $\epsilon$ and $\gamma$ are as in Theorem~\ref{thm:su3}, with \[ \alpha_n=\frac{\zeta(\beta_W)\,(18n-3)}{18\,(1-\zeta(\beta_W))\,n^2}. \] The function $\zeta$ is increasing, $\zeta(0.35)\le0.8584$, and for $\beta_W\le0.35$ and $n\ge6$, $\alpha_n\le0.3368\,(18n-3)/n^2\le0.9824$. \item $G\subset\U(N)$ closed and connected with semisimple Lie algebra, $24N\lvert\beta\rvert<\kappa_G$, with \[ \alpha_n=\frac{N\lvert\beta\rvert(18n-3)}{(\kappa_G-24N\lvert\beta\rvert) n^2}, \] whenever $N\lvert\beta\rvert<\kappa_Gn^2/(24n^2+18n-3)$. \end{enumerate} In each case, for fixed $\beta$ in the range, $\mathrm{BP}(n,\varpi)$ holds with one $\varpi>0$ for all $n$ from some $n_0(\beta)$ on, uniformly in the torus, and the best constant tends to $1$ as $n\to\infty$. \end{corollary} \begin{proof} (i) As in the proof of Theorem~\ref{thm:su2}, without the gauge: the law of $U_e$ is the von Mises--Fisher law $\nu_b$ with $b=4\beta\sum_{p\ni e}\bar s_p$, a partner moved by $\delta$ in the chordal metric moves $b$ by $4\lvert\beta\rvert\delta$, and Theorem~\ref{q-thm:quarter} gives $C^0_{ee'} \le\lvert\beta\rvert=\lvert\beta_W\rvert/4$. Theorem~\ref{thm:bpstrong}(i) applies with $c/(1-18c)=\lvert\beta_W\rvert/(4-18\lvert\beta_W\rvert)$. At $\lvert\beta_W\rvert=\frac18$, $\alpha_2=\frac{33}{56}$; at $\lvert \beta_W\rvert=\frac15$, $\alpha_n=(18n-3)/(2n^2)$ and $\alpha_9=\frac{159} {162}$; $\alpha_n$ increases with $\lvert\beta_W\rvert$. (ii) The six staples of a link are products of matrices of $\SU(3)$, so $b\in rK$, and as in~\eqref{eq:alpha} and Lemma~\ref{lem:su3modulus}, $C^0_{ee'}\le N\beta\Lambda(r)\le c$ with $18c=3r\sqrt{\epsilon(r)}/\gamma (r)$ in the Hilbert--Schmidt metric; $\zeta$ increases because $\epsilon$ is nondecreasing and $\gamma$ decreasing. By Lemma~\ref{lem:su3haar}(iii) and the tail bound in the proof of Theorem~\ref{thm:su3}, at $r=0.7$ the terms $m_nr^n/n!$, rounded up, are $1$, $0$, $0.1225$, $0.0142917$, $0.0075032$, $0.0013131$, $0.000332$, $0.000059$ and $0.0000109$, with sum at most $1.1460099$, and the tail is less than $4\cdot10^{-6}$; so $Z_*(0.7)\le1.146014$, $e^{-0.14}\ge0.869342$, $\frac{5e^{-0.14}}{9Z_*(0.7)} \ge0.421432$ and $\epsilon(0.7)\le0.578568<0.76064^2$. With $\gamma(0.7)= \frac{1191}{640}$, $\zeta(0.35)\le2.1\cdot0.76064\cdot\frac{640}{1191}\le 0.8584$, and $\zeta/(18(1-\zeta))\le0.3368$; at $n=6$, $(18n-3)/n^2= \frac{35}{12}$. (iii) Lemma~\ref{lem:bponelink} gives $c=N\lvert\beta\rvert/(\kappa_G-6N \lvert\beta\rvert)$, so $18c<1$ is $24N\lvert\beta\rvert<\kappa_G$ and $c/(1-18c)=N\lvert\beta\rvert/(\kappa_G-24N\lvert\beta\rvert)$. In each case $\alpha_n\to0$; Proposition~\ref{prop:bpscales}(ii) below bounds the best constant by $1+2/n$. \end{proof} For $\SU(3)$, the condition $\zeta<1$ of (ii) holds up to $\beta_W\approx 0.391$ by a numerical evaluation that no statement uses, against $\frac3{16}$ from (iii); for $\SU(2)$, (i) reaches $\frac29$ against $\frac1{12}$. These ranges lie inside those of Secs.~\ref{sec:su2}--\ref{sec:su3blocks}, where the forest gauge gives a gap directly; what Corollary~\ref{cor:bpranges} adds is the averaged condition itself, on every torus, for every $n$ past $n_0(\beta)$ and on all of $L^2(\mu)$. The gap it gives through Corollary~\ref{cor:avggap} is of order $1/n$ and is not used. The dependence on $n$ is constrained in general as follows. Write $\varpi^*_n$ for the best constant in $\mathrm{BP}(n, \cdot)$ on $\Lambda_L$, and $P_V$ for the conditional expectation given the links outside $V$. \begin{proposition}\label{prop:bpscales} Let $L\ge2m+4$ and $2\le n\le m$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item If $\mathrm{BP}(n,\varpi)$ holds, then so does $\mathrm{BP}(m,(n/m)^4 \varpi)$; that is, $n^4\varpi^*_n\le m^4\varpi^*_m$. \item $\varpi^*_n\le1+2/n$. \item Let $W_x$ be the cubes of side $m$. Suppose that $\mathrm{BP}(m, \varpi)$ holds and that for every $x$, every real $f\in\mathcal A$ and $\mu$-almost every $U$, \[ \varpi_{\rm in}P_{W_x}\bigl((f-P_{W_x}f)^2\bigr)\le\frac1{n^4}\sum_{y:\,V_y \subset W_x}P_{W_x}\bigl((f-P_{V_y}f)^2\bigr). \] Then $\mathrm{BP}(n,\varpi\varpi_{\rm in}(m/(m-n+1))^4)$ holds. \end{enumerate} \end{proposition} \begin{proof} (i) If $V\subset W$, the $\sigma$-algebra of the links outside $W$ is contained in that of the links outside $V$, so $P_W=P_WP_V$ and $\lVert f-P_Wf\rVert^2=\lVert f-P_Vf\rVert^2+\lVert P_Vf-P_Wf\rVert^2$. The cube $V_y$ of side $n$ lies in $W_x$ exactly when $y-x\in[0,m-n]^4$, so for each $x$, $\mu((f-P_{W_x}f)^2)$ is at least the mean of $\mu((f-P_{V_y} f)^2)$ over these $(m-n+1)^4$ points $y$, and each $y$ occurs for $(m-n+1)^4$ points $x$. Summing, $\sum_x\mu((f-P_{W_x}f)^2)\ge\sum_y\mu((f- P_{V_y}f)^2)$. (ii) Let $f=\ReTr U_p$ for a plaquette $p$ at $y$ in the plane $\{\mu,\nu \}$; $\Var_\mu(f)>0$. As in the proof of Theorem~\ref{thm:avgdecay}, $\mu((f-\mathcal P_xf)^2)\le\Var_\mu(f)$, with equality to zero unless $V_x$ contains a link of $p$, that is, unless $x\in b-[0,n)^4$ for a base point $b\in\{y,y+e_\mu,y+e_\nu\}$. By inclusion and exclusion these $x$ number $3n^4-2n^3(n-1)-n^2(n-1)^2+n^2(n-1)^2=n^4+2n^3$. (iii) Integrating the hypothesis, $\varpi_{\rm in}\mu((f-P_{W_x}f)^2)\le n^{-4} \sum_{y:\,V_y\subset W_x}\mu((f-P_{V_y}f)^2)$. Sum over $x$, use $\mathrm{BP}(m,\varpi)$ and count each $y$ $(m-n+1)^4$ times: $\varpi\varpi_{\rm in}\Var_\mu(f)\le(m-n+1)^4m^{-4}n^{-4}\sum_y\mu((f- P_{V_y}f)^2)$. \end{proof} Part (iii) is the two-scale argument for block dynamics~\cite{Mar99}; it transfers the condition from larger cubes to smaller ones, given a gap of the small-cube dynamics inside each large cube uniform in its exterior. Part (i) transfers it upwards only with the loss $(n/m)^4$, which, through Theorem~\ref{thm:avgdecay}, gives the rate $\varpi n^4/(e^2q_mm^5)$ and nothing in physical units. So $\mathrm{BP}$ at one cube size implies no useful bound at larger sizes by itself. The remaining results concern $G=\SU(N)$, $N\ge2$, with center element $z=e^{2\pi i/N}I$, and use the holonomies of the torus. For a site $y$ let $P_y$ be the ordered product of the $L$ links $(y+te_0,0)$, $0\le t0$ and $\eta_N>0$ such that $M(A,B)\ge c_N$ whenever $\lVert ABA^{-1}B^{-1}-I \rVert_{\rm HS}\le\eta_N$. For $N=2$ one may take $\eta_2=1$ and $c_2=0.64$. \end{lemma} \begin{proof} Let $A,B$ commute and suppose $M(A,B)=0$. Diagonalise them together, with eigenvalues $\alpha_k,\beta_k$. The power sums of the $\alpha_k$ of orders $1,\dots,N-1$ vanish, so by Newton's identities the characteristic polynomial of $A$ is $x^N+(-1)^N\det A$, and $\alpha_k=\lambda\omega^k$ after reordering, with $\omega=e^{2\pi i/N}$; likewise $\beta_k=\lambda'\omega^{ \tau(k)}$ for a permutation $\tau$ of $\Z/N\Z$. Then $\Tr(A^aB^b)=\lambda^a \lambda'^b\sum_k\omega^{ak+b\tau(k)}$ is a multiple of the Fourier coefficient at $(a,b)$ of the indicator of the graph of $\tau$ in $(\Z/N\Z)^2$. All coefficients except the one at $0$ vanish, so the indicator is constant, which is impossible for a set of $N$ points among $N^2$. Hence $M>0$ on the compact set of commuting pairs. If no $\eta_N,c_N$ existed, there would be pairs with $\lVert ABA^{-1}B^{-1}-I\rVert\to0$ and $M\to0$, and a limit point would be a commuting pair with $M=0$. For $N=2$ put $x=\Tr A$, $y=\Tr B$, $w=\Tr AB$, real numbers in $[-2,2]$. By the Cayley--Hamilton theorem, $A^{-1}=x-A$, $B^{-1}=y-B$, $\Tr(AB^2)=yw-x$, $\Tr(A^2B)=xw-y$ and $\Tr(ABAB)=w^2-2$, so \[ \Tr(ABA^{-1}B^{-1})=x^2+y^2+w^2-xyw-2 . \] Since $\lvert\Tr(C-I)\rvert\le\sqrt2\lVert C-I\rVert_{\rm HS}$, the commutator bound gives $x^2+y^2+w^2-xyw\ge4-\sqrt2$, and with $m=\max(\lvert x\rvert,\lvert y\rvert,\lvert w\rvert)$, $3m^2+m^3\ge4-\sqrt2>2.432=3(0.8)^2+ 0.8^3$. So $m>0.8$ and $M=x^2+y^2+w^2>0.64$. \end{proof} \begin{lemma}\label{lem:bpflat} Let $G=\SU(N)$ and $\beta\ge1$. There is $A_N$ such that $\mu(N-\ReTr U_p)\le A_N(1+\log\beta)/\beta$ for every plaquette and every $L$; for $N=2$, $\mu(2-\Tr U_p)\le(\log\beta+6.11)/\beta$. \end{lemma} \begin{proof} Put \[ H=\sum_p(N-\ReTr U_p)\ge0,\qquad\phi(t)=\log\int e^{-NtH}dU . \] Then $\mu$ has density proportional to $e^{-N\beta H}$, $\phi'(\beta)=-N\mu (H)$, $\phi$ is convex and $\phi\le0$. Convexity gives $\phi(\beta/2)\ge\phi(\beta)-\frac \beta2\phi'(\beta)$, hence $\mu(H)\le\frac2{N\beta}\log(1/e^{\phi(\beta)})$. Let $\delta$ be the geodesic distance of $-\Tr(XY)$. If every link lies within $r$ of $I$, then $\delta(U_p,I)\le4r$ by bi-invariance, and $N- \ReTr U\le\frac12\delta(U,I)^2$, so $H\le8r^2\lvert P\rvert$. With $v(r)$ the Haar measure of the ball of radius $r$ about $I$, $\lvert E\rvert=4L^4$ and $\lvert P\rvert=6L^4$, \[ \mu(N-\ReTr U_p)=\frac{\mu(H)}{6L^4}\le\frac2{N\beta}\Bigl(\tfrac23\log \tfrac1{v(r)}+8N\beta r^2\Bigr), \] the plaquettes being equivalent under the symmetries of $\Lambda_L$. Haar measure has a smooth positive density in exponential coordinates, so $v(r) \ge v_Nr^{N^2-1}$ for $r\le1$; $r^2=1/(N\beta)$ gives the first claim. For $\SU(2)$, $U=\cos\theta+u\sin\theta$ in quaternions has $\delta(U,I)=\sqrt2 \theta$ and the law of $\theta$ has density $\frac2\pi\sin^2\theta$, so $v(\sqrt2s)\ge\frac2\pi\cdot\frac4{\pi^2}\cdot\frac{s^3}3=\frac{8s^3}{3\pi^3}$ for $s\le\frac\pi2$. With $r=\sqrt2s$ and $s^2=1/(32\beta)$ the bracket times $\frac1\beta$ is at most $\frac1{6\beta}(6\log(32\beta)+6+4\log \frac{3\pi^3}8)\le(\log\beta+6.11)/\beta$. \end{proof} \begin{theorem}[the averaged condition at fixed cube size]\label{thm:bpfixed} Let $G=\SU(N)$, $N\ge2$, $\beta\ge0$ and $n\ge2$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item If $\mathrm{BP}(n,\varpi)$ holds on $\Lambda_L$, then \[ \frac1{L^3}\sum_{y:\,y_0=0}\mu\bigl(\Tr P_0\,\overline{\Tr P_y}\bigr)\le \frac{2N^2n^2}{\varpi L^2}. \] \item Let $N=2$ and $\beta\ge1$. If $\mathrm{BP}(n,\varpi)$ holds on $\Lambda_{2n+4}$, then \[ \varpi\le\frac{38400\,n^4(\log\beta+6.11)}\beta . \] \item There is $C_N$ such that for $\beta\ge1$, $\mathrm{BP}(n,\varpi)$ on $\Lambda_{2n+4}$ implies $\varpi\le C_Nn^4(1+\log\beta)/\beta$. \end{enumerate} In particular, along any sequence $(\beta_k,n_k)$ with $n_k^4(1+\log \beta_k)/\beta_k\to0$, no $\varpi>0$ satisfies $\mathrm{BP}(n_k,\varpi)$ uniformly in the torus. \end{theorem} \begin{proof} (i) Let $g_y=\ReTr P_y$ and $f=L^{-3}\sum_{y:\,y_0=0}g_y$. By the symmetry $Z_{1,0}$, $\mu(f)=0$. The loop $P_y$ meets $V_x$ only if $(y_1,y_2,y_3)$ lies in the projection of $x+[0,n)^4$, which holds for $n^3$ loops, and $\lVert g_y-\mathcal P_xg_y\rVert_2\le\lVert g_y\rVert_2\le N$. So $\lVert f-\mathcal P_xf\rVert_2\le Nn^3/L^3$ for each of the $L^4$ points $x$, the right side of $\mathrm{BP}(n,\varpi)$ is at most $N^2n^2/L^2$, and $\mu(f^2)\le N^2n^2/(\varpi L^2)$. The same holds for the imaginary parts, and $\mu(\lvert L^{-3}\sum_y\Tr P_y\rvert^2)$ is the left side, by translation invariance. (ii), (iii) Let $L=2n+4\le4n$, $\varepsilon=\mu(\lVert U_p-I\rVert_{\rm HS}^2)= 2\mu(N-\ReTr U_p)$, the same for every plaquette, and let $f$ be the real or the imaginary part of some $F_{ab}$, $(a,b)\in I_N$. Pick $(c,d)$ with $ac+bd \not\equiv0$ modulo $N$; the symmetry $Z_{c,d}$ gives $\mu(F_{ab})=0$. \emph{The right side of $\mathrm{BP}$.} If $V_x$ meets neither $A$ nor $B$, $\mathcal P_xf=f$. Otherwise the coordinate $x_2$ satisfies $0\in x_2+[0,n)$ modulo $L$, and one of $s=x_2-1$, $s=x_2+n$, read in $(-L/2,L/2]$, has $\lvert s\rvert\le n$ and $s\notin x_2+[0,n)$. The loops $A'$, $B'$ of directions $0$, $1$ through $y'=se_2$ avoid $V_x$, so $F'_{ab}=\Tr(A'^aB'^b)$ depends only on the links outside $V_x$, and $\lVert f-\mathcal P_xf\rVert_2\le \lVert F_{ab}-F'_{ab}\rVert_2$. Let $\Gamma$ be the holonomy of the straight path from $0$ to $y'$. Lemma~\ref{lem:bpstokes}, for the rectangle $[0,L] \times[0,s]$ mapped onto the plane $\{0,2\}$ through $0$, whose boundary is $A$, the path to $y'$, $A'$ reversed and the path back, gives $A=H_0\Gamma A'\Gamma^{-1}$ with $\lVert H_0-I\rVert_{\rm op}\le S_0$, the sum of $\lVert U_p-I\rVert_{\rm op}$ over the $L\lvert s\rvert$ plaquettes of the strip; likewise $B=H_1\Gamma B'\Gamma^{-1}$ with the strip in the plane $\{1, 2\}$. Telescoping, $\lvert F_{ab}-F'_{ab}\rvert\le N(a\lVert H_0-I\rVert_{ \rm op}+b\lVert H_1-I\rVert_{\rm op})\le N(N-1)(S_0+S_1)$, and by the Cauchy--Schwarz inequality over the at most $2Ln$ plaquettes of the two strips, $\mu((S_0+S_1)^2)\le(2Ln)^2\varepsilon$. At most $2Ln^3$ points $x$ have $V_x$ meeting $A\cup B$, so \[ \frac1{n^4}\sum_x\mu\bigl((f-\mathcal P_xf)^2\bigr)\le8N^2(N-1)^2L^3n\, \varepsilon . \] \emph{The variance.} Lemma~\ref{lem:bpstokes} for the square $[0,L]^2$ mapped onto the plane $\{0,1\}$ through $0$, whose boundary reads $A$, $B$, $A^{-1}$, $B^{-1}$, gives $\lVert ABA^{-1}B^{-1}-I\rVert_{\rm HS}\le\sum_p \lVert U_p-I\rVert_{\rm HS}$ over $L^2$ plaquettes, so this commutator exceeds $\eta_N$ with probability at most $L^2\sqrt\varepsilon/\eta_N$. By Lemma~\ref{lem:bpcommute}, $\sum_{(a,b)\in I_N}\mu(\lvert F_{ab}\rvert^2)\ge c_N(1-L^2\sqrt\varepsilon/\eta_N)$. As $\mu(F_{ab})=0$, some $f$ of the form above has $\Var_\mu(f)\ge c_N(1-L^2\sqrt\varepsilon/\eta_N)/(2(N^2-1))$. For $N=2$ the traces are real, and some $F_{ab}$ has variance at least $0.64(1-L^2\sqrt\varepsilon)/3$. \emph{Conclusion.} If $L^2\sqrt\varepsilon\le\eta_N/2$, $\mathrm{BP}(n,\varpi)$ gives $\varpi\le2048(N^2-1)N^2(N-1)^2n^4\varepsilon/c_N$, and for $N=2$, $\varpi\le300\,L^3n\varepsilon\le19200\,n^4\varepsilon$. In the other case $\eta_N^2<4L^4\varepsilon\le1024n^4\varepsilon$, and since $\varpi\le2$ by Proposition~\ref{prop:bpscales}(ii), $\varpi\le(2048/ \eta_N^2)n^4\varepsilon$, which for $N=2$ is again at most $19200\,n^4 \varepsilon$. Lemma~\ref{lem:bpflat} bounds $\varepsilon$ by $2(\log\beta+ 6.11)/\beta$ for $N=2$ and by $2A_N(1+\log\beta)/\beta$ in general. \end{proof} \begin{corollary}[the response of a cube at weak coupling]\label{cor:bpresponse} Let $G=\SU(2)$, $\beta\ge1$ and $n\ge2$, and put $X_n(\beta)=38400\,n^4(\log \beta+6.11)/\beta$. Then \[ \Xi_n\ge\frac{n^4}{7n^3+3n^2}\bigl(1-X_n(\beta)\bigr), \] and $D^0_L(n,\alpha)$ for one $L\ge2n+4$, hence for all, implies $1-\alpha \le X_n(\beta)$. \end{corollary} \begin{proof} By Theorem~\ref{thm:bpstrong}(ii), $D^0_L(n,\alpha)$ does not depend on $L$ and holds with $\alpha=(7n^3+3n^2)\Xi_n/n^4$. If $\alpha<1$, Proposition~\ref{prop:avgsup}(i) gives $\mathrm{BP}(n,1-\alpha)$ on $\Lambda_{2n+4}$, and Theorem~\ref{thm:bpfixed}(ii) gives $1-\alpha\le X_n(\beta)$; if $\alpha\ge1$ the bound on $\Xi_n$ holds trivially. \end{proof} So for $2\le n\le(\beta/(76800(\log\beta+6.11)))^{1/4}$ one link outside the cube moves its law by at least $n/17$ in total, in the supremum over exteriors: the response grows linearly with the cube, the behaviour that Remark~\ref{rem:cubeheuristic} assigns to cubes smaller than the correlation length. Fig.~\ref{fig:bpranges} places the two results of this subsection in the plane of the coupling and the cube side. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_bpranges.png} \caption{The averaged condition at strong coupling and at fixed cube size. (a)~The bound $\log_{10}\bigl(38400\,n^4(\log\beta+6.11)/\beta\bigr)$ of Theorem~\ref{thm:bpfixed}(ii) for $\SU(2)$ over the cube side $n$ and $\log_{10}\beta$; where it lies below the plane $\varpi=1$ (red), $\mathrm{BP}(n,1)$ fails on $\Lambda_{2n+4}$, and the black curve is the crossing. (b)~For $\SU(2)$ in the variables $\beta_W=4\beta$ and $n$: the range $\lvert\beta_W\rvert<\frac29$ of Corollary~\ref{cor:bpranges}, where $\mathrm{BP}(n,\varpi)$ holds on every torus for all $n\ge n_0(\beta_W)$ (green); the region where Theorem~\ref{thm:bpfixed}(ii) excludes $\mathrm{BP}(n,\frac12)$ on $\Lambda_{2n+4}$ (red); and the continuum tuning $n\propto e^{3\pi^2\beta_W/11}$ at fixed physical cube size (dashed, schematic, the power prefactor omitted), which runs through the region where neither result applies.} \label{fig:bpranges} \end{figure*} \begin{remark}[what is proved about the averaged condition]\label{rem:bpstatus} Proved: $\mathrm{BP}(n,1-\alpha_n)$ on every torus and on all of $L^2(\mu)$ for $\SU(2)$ with $\lvert\beta_W\rvert<\frac29$, for $\SU(3)$ with $\zeta( \beta_W)<1$ (in particular $\beta_W\le0.35$, $n\ge6$, $\varpi\ge0.0176$), and for every compact connected group with semisimple Lie algebra with $24N\lvert\beta\rvert<\kappa_G$, for all $n$ with $\alpha_n<1$, where $\alpha_n=O(1/n)$ is explicit (Corollary~\ref{cor:bpranges}); the supremum form $D^0$ is independent of the torus and follows from a bound $\Xi_n0$, and put $\epsilon=\varpi n^4/L^4$. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item $\mathrm{BP}_{\rm loc}(n,\varpi)$: for every local real $f\in\mathcal A$ and every integer $t\ge0$, \[ \bigl\lVert\mathcal K^t\bigl(f-\mu(f)\bigr)\bigr\rVert_2\le(1-\epsilon)^t \bigl\lVert f-\mu(f)\bigr\rVert_2 . \] \item For $G=\SU(N)$, $\mathrm{BP}_0(n,\varpi)$: $\varpi\Var_\mu(f)\le \mathcal E_n(f)$ for every real $f\in\mathcal A_0$. \end{enumerate} \end{definition} For a self-adjoint $0\le\mathcal K\le I$ and $g\in L^2(\mu)$ with spectral measure $\nu_g$, $\lVert\mathcal K^tg\rVert_2^2=\int\lambda^{2t}d\nu_g$; so the inequality of (a) for all $t$ says that $\nu_g$ is carried by $[0,1- \epsilon]$ for $g=f-\mu(f)$, the spectral gap of the cube chain on the cyclic subspace of each local function. Condition (a) involves no function of links spread over the torus.\footnote{The distinction is the one between a spectral gap of the whole generator and a gap on the cyclic subspaces of the observables that are measured. Only the latter enters a bound on their correlations, and a lower bound on the Rayleigh quotient of a function does not give it; Appendix~\ref{app:cyclic} proves both statements in the generality of a self-adjoint contraction.} \begin{theorem}[the local form of the averaged condition]\label{thm:bplocal} Let $G$, $\varphi$ and $\Lambda_L$ be as in Sec.~\ref{ssec:averaged}. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\mathrm{BP}(n,\varpi)$ implies $\mathrm{BP}_{\rm loc}(n,\varpi)$; for $G=\SU(N)$, $\mathrm{BP}(n,\varpi)$ implies $\mathrm{BP}_0(n,\varpi)$, which implies $\mathrm{BP}_{\rm loc}(n,\varpi)$. \item If $\mathrm{BP}_{\rm loc}(n,\varpi)$ holds on $\Lambda_L$, then $\varpi\le4$, and both bounds of Theorem~\ref{thm:avgdecay} hold for bounded $f,g\in\mathcal A$ with $f$ local. \item Corollary~\ref{cor:avggap} and Theorem~\ref{thm:avgtransfer} hold with $\mathrm{BP}_{\rm loc}(n,\varpi)$, respectively $\mathrm{BP}_{\rm loc}(n_k, \varpi)$, in place of $\mathrm{BP}$; so does the form of Proposition~\ref{prop:spatialflow}\textup{(iii)} stated after Theorem~\ref{thm:avgtransfer}. \item If $\mathrm{BP}_{\rm loc}(n,\varpi)$ holds, then every local real $f\in\mathcal A$ and every real $h\in\mathcal A$ satisfy \[ \Cov_\mu(f,h)^2\le\Var_\mu(f)\,\frac{\mathcal E_n(h)}{\varpi}. \] If it fails, there are a local real $f\in\mathcal A$ and a real $h\in \mathcal A$ with $\mathcal E_n(h)<\varpi\Var_\mu(h)$ and $\Cov_\mu(f,h)= \Var_\mu(h)>0$; for $G=\SU(N)$, $h\in\mathcal A_0$. \end{enumerate} \end{theorem} \begin{proof} (i) Let $\mathcal B$ be $\mathcal A$, or $\mathcal A_0$ for $G=\SU(N)$, and $\mathcal B^0$ its functions of mean zero. Every $\mathcal P_x$ maps $\mathcal A$ into itself (Sec.~\ref{ssec:averaged}) and $\mathcal A_0$ into itself (Proposition~\ref{prop:bpcentre}(iii)), and $\mu(\mathcal P_xg)= \mu(g)$; so $\mathcal K$ maps $\mathcal B^0$ into itself, and there it is self-adjoint with $0\le\mathcal K\le I$. If the inequality of $\mathrm{BP}(n,\varpi)$ holds on the real functions of $\mathcal B$, then $\ip g{(I-\mathcal K)g}\ge \epsilon\lVert g\rVert_2^2$ on $\mathcal B^0$, the spectrum of $\mathcal K$ on $\mathcal B^0$ lies in $[0,1-\epsilon]$, and $\lVert\mathcal K^tg \rVert_2\le(1-\epsilon)^t\lVert g\rVert_2$ for $g\in\mathcal B^0$. A local real $f\in\mathcal A$ lies in $\mathcal A_0$ by Proposition~\ref{prop:bpcentre}(iv), since $L/2\le L-1$, so $g=f-\mu(f)\in \mathcal B^0$ in both cases. (ii) Let $f=\ReTr U_p$ for a plaquette $p$, a local function with $\Var_\mu (f)>0$, and $g=f-\mu(f)$. By $\mathrm{BP}_{\rm loc}(n,\varpi)$ with $t=1$, $\ip g{\mathcal Kg}\le\lVert g\rVert_2\lVert\mathcal Kg\rVert_2\le(1- \epsilon)\lVert g\rVert_2^2$, so $\ip g{(I-\mathcal K)g}\ge\epsilon\lVert g \rVert_2^2$. As in the proof of Theorem~\ref{thm:avgdecay}, $\ip g{(I- \mathcal K)g}=L^{-4}\sum_x\mu((g-\mathcal P_xg)^2)\le4n^4L^{-4}\lVert g \rVert_2^2$. Hence $\varpi\le4$. The proof of Theorem~\ref{thm:avgdecay} uses $\mathrm{BP}(n,\varpi)$ in two places only: for the bound $\varpi\le4$, just obtained, and in its spectral part, for $\lVert\mathcal K^T(f-\mu(f)) \rVert_2\le e^{-\epsilon T}\lVert f-\mu(f)\rVert_2$, which $\mathrm{BP}_{\rm loc}(n,\varpi)$ gives for local real $f$. The real and imaginary parts of a local function are local. (iii) In the proof of Corollary~\ref{cor:avggap}, Theorem~\ref{thm:avgdecay} is applied on $\Lambda_L$, $L\in\mathcal L$ large, to $\vartheta(\bar F)$ and $\tau_tF$, which depend on links with base points in a box whose side depends on $F$ and $t$ only, hence are local once $L$ is large; (ii) applies. Theorem~\ref{thm:avgtransfer} and the form of Proposition~\ref{prop:spatialflow}(iii) use the condition only through Corollary~\ref{cor:avggap}. (iv) Let $E$ be the spectral projection of $\mathcal K$ on $L^2(\mu)$ for the interval $(1-\epsilon,1]$, $f$ local, real and in $\mathcal A$, and $g= f-\mu(f)$. Under $\mathrm{BP}_{\rm loc}(n,\varpi)$, $\int\lambda^{2t}d\nu_g\le (1-\epsilon)^{2t}\nu_g([0,1])$ for all $t$, so $\nu_g((1-\epsilon+\eta,1])=0$ for every $\eta>0$, and $Eg=0$. On the range of $I-E$, $I-\mathcal K\ge \epsilon$, so for real $h\in\mathcal A$, \[ \begin{split} \Cov_\mu(f,h)^2&=\ip g{(I-E)h}^2\\ &\le\lVert g\rVert_2^2\,\frac{\ip{(I-E)h}{(I-\mathcal K)(I-E)h}}\epsilon\\ &\le\lVert g\rVert_2^2\,\frac{\ip h{(I-\mathcal K)h}}\epsilon\\ &=\Var_\mu(f)\,\frac{\mathcal E_n(h)}{\varpi}. \end{split} \] If $\mathrm{BP}_{\rm loc}(n,\varpi)$ fails, some local real $f\in\mathcal A$ has $\nu_g((1-\epsilon,1])>0$ for $g=f-\mu(f)$; put $h=Eg$. Spectral projections of $\mathcal K$ are strong limits of polynomials in $\mathcal K$, so $h$ is real and lies in $\mathcal A$, and in $\mathcal A_0$ for $G= \SU(N)$, since $g$ does. As $\mathcal K1=1$, $E1=1$ and $\mu(h)=\ip{E1}g=0$; then $\Var_\mu(h)=\lVert Eg\rVert_2^2=\ip g{Eg}=\Cov_\mu(f,h)>0$, and since $\nu_h$ is carried by $(1-\epsilon,1]$, $\ip h{(I-\mathcal K)h}=\int(1- \lambda)\,d\nu_h<\epsilon\lVert h\rVert_2^2$, that is, $\mathcal E_n(h)< \varpi\Var_\mu(h)$. \end{proof} Fig.~\ref{fig:bpsectors} shows the sectors and the spectral picture behind Theorem~\ref{thm:bplocal}. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_bpsectors.png} \caption{Centre sectors and the local form of the averaged condition. (a)~A three-dimensional section of the torus with the slices $\{y_0=t\}$. The twist $Z^0_t$ multiplies the links that cross one slice by the central element $z$ (gold) and leaves every plaquette unchanged; a Polyakov loop of direction $0$ crosses every slice once and is charged, while a gauge-invariant function of a box that misses one slice is neutral (Proposition~\ref{prop:bpcentre}). (b)~Spectral measures of the cube chain $\mathcal K$, schematic: under $\mathrm{BP}_{\rm loc}$ the measure of a local function puts no mass on $(1-\epsilon,1]$ (green), while a charged function may carry mass there (red), as the test functions of Theorem~\ref{thm:bpfixed} do at weak coupling (Theorem~\ref{thm:bplocal}, Corollary~\ref{cor:bpcharged}).} \label{fig:bpsectors} \end{figure*} By (iv), under $\mathrm{BP}_{\rm loc}$ the functions on which the cube chain is slow are nearly orthogonal to every local gauge-invariant function, with the defect of orthogonality bounded by their Dirichlet form, and a failure of $\mathrm{BP}_{\rm loc}$ produces a slow function correlated with a local one. Slow functions orthogonal to all local ones, and for $\SU(N)$ slow functions in the charged sectors, do not affect the continuum step. The Villain obstruction persists for the local form: the proof of Corollary~\ref{cor:avgcoulomb} uses the condition only through Theorem~\ref{thm:avgdecay} for the functions $\Psi_p$, and through the bound $\varpi\le4$, which holds with $\Psi_p$ in place of $\ReTr U_p$, a nonconstant function of the four links of $p$. So for the Villain theory at $\beta>\beta_0$ there are no $n$ and $\varpi>0$ such that $\lVert\mathcal K^t(\Psi_p-\mu(\Psi_p))\rVert_2\le(1-\varpi n^4/\lvert X \rvert)^t\lVert\Psi_p-\mu(\Psi_p)\rVert_2$ for all $t$, all plaquettes and the measures with free boundary of all boxes of side at least $2n$. The local form, like the full one, has to use the non-abelian structure of $G$. For $\SU(N)$ the obstruction of Sec.~\ref{ssec:bpstrong} lies in the charged sectors. Let $\mathcal C$ be the set of real $f\in\mathcal A$ with $\Pi_0f=0$, in the notation of the proof of Proposition~\ref{prop:bpcentre}(iii); it is orthogonal to $\mathcal A_0$ and preserved by every $\mathcal P_x$. \begin{corollary}[the fixed-size obstruction is charged]\label{cor:bpcharged} Let $G=\SU(N)$, $N\ge2$. Theorem~\ref{thm:bpfixed} holds, with the same constants in \textup{(i)} and \textup{(ii)} and with another constant $C_N$ in \textup{(iii)}, when the inequality of $\mathrm{BP}(n,\varpi)$ is assumed only for $f\in\mathcal C$. Its hypotheses and conclusions concern only the restriction of the condition to $\mathcal C$; neither $\mathrm{BP}_0(n, \varpi)$ nor $\mathrm{BP}_{\rm loc}(n,\varpi)$ enters it. \end{corollary} \begin{proof} In the proof of Theorem~\ref{thm:bpfixed}(i) the condition is applied to $L^{-3}\sum_y\ReTr P_y$ and to the corresponding imaginary part. The trace of $P_y$ lies in $\mathcal A_{e_0}$ and its conjugate in $\mathcal A_{-e_0}$, and $\pm e_0\ne0$ in $(\Z/N\Z)^4$, so both functions lie in $\mathcal C$. In the proof of (ii) and (iii), $F_{ab}=\Tr(A^aB^b)$ lies in $\mathcal A_r$ with $r=(a,b,0,0)\ne0$ for $(a,b)\in I_N$, so the real and imaginary parts of $F_{ab}$ lie in $\mathcal C$. The only other use of the condition is the bound $\varpi\le2$ from Proposition~\ref{prop:bpscales}(ii), whose test function is neutral. In its place take $f_*=\ReTr P_0\in\mathcal C$, a nonconstant continuous function, so $\Var_\mu(f_*)>0$. $\mathcal P_xf_*=f_*$ unless $V_x$ meets the loop $P_0$, which happens for at most $Ln^3$ points $x$, and $\mu((f_*-\mathcal P_xf_*)^2)\le\Var_\mu(f_*)$; so $\varpi\le L/n \le4$ on $\Lambda_{2n+4}$. In the case $\eta_N^2<4L^4\varepsilon\le1024n^4 \varepsilon$ of that proof this gives $\varpi\le(4096/\eta_N^2)n^4 \varepsilon$, which for $N=2$ is at most $19200\,n^4\varepsilon$, so the constant $38400$ of (ii) is unchanged. $\mathcal C$ is orthogonal to $\mathcal A_0$, which contains every local function of $\mathcal A$. \end{proof} Corollary~\ref{cor:bpcharged} constrains every argument that proves the condition through a contraction of the cube chain on gauge orbits. On $G^E$ let $d(U,U')=\min_g d_E(U,gU')$, the minimum over gauge transformations $g$, with $d_E$ as in the proof of Proposition~\ref{prop:avgsup}; since gauge transformations act by isometries of $d_E$, $d$ is a pseudometric, constant on pairs of gauge orbits. Write $W_1^d$ for the Kantorovich distance of $d$. By gauge covariance, $\mathcal K(gU',\cdot)$ is the image of $\mathcal K(U',\cdot)$ under $g$, so a contraction of $W_1^{d_E}$ by a factor $1-\kappa$, as in the proof of Proposition~\ref{prop:avgsup}(i), gives one of $W_1^d$ by the same factor. \begin{corollary}[couplings on gauge orbits]\label{cor:bpcoupling} Let $G=\SU(2)$, $\beta\ge1$, $n\ge2$ and $L=2n+4$. If there is $\kappa>0$ with \[ W_1^d\bigl(\mathcal K(U,\cdot),\mathcal K(U',\cdot)\bigr)\le(1-\kappa)\, d(U,U') \] for all configurations $U,U'$ of $\Lambda_L$, then \[ \kappa\le\frac{38400\,n^8(\log\beta+6.11)}{L^4\beta}. \] \end{corollary} \begin{proof} Let $h$ be gauge invariant and Lipschitz for $d$ with constant $\mathrm{Lip} (h)$. For a coupling of $\mathcal K(U,\cdot)$ and $\mathcal K(U',\cdot)$ with expected distance $W_1^d$, $\lvert\mathcal Kh(U)-\mathcal Kh(U')\rvert \le\mathrm{Lip}(h)W_1^d$, so $\mathcal Kh$ is Lipschitz with constant $(1- \kappa)\mathrm{Lip}(h)$, and it is gauge invariant. As in the proof of Proposition~\ref{prop:avgsup}(i), with $d\le D\lvert E\rvert$, the spectral measure of $\mathcal K$ at $h-\mu(h)$ is carried by $[0,1-\kappa]$. Such $h$ are dense in $\mathcal A$: averaging over the compact gauge group is the orthogonal projection of $L^2(\mu)$ onto $\mathcal A$ and maps continuous functions to continuous ones, so the continuous functions of $\mathcal A$ are dense in it; a continuous gauge-invariant $f$ is uniformly continuous for $d$, since $d(U,U')$ small gives $g$ with $d_E(U,gU')$ small and $f(gU') =f(U')$; and $f_k(U)=\inf_V(f(V)+k\,d(U,V))$ is gauge invariant, Lipschitz for $d$ with constant $k$, and converges to $f$ uniformly. So the spectral projection of $\mathcal K$ for $(1-\kappa,1]$ vanishes on the functions of $\mathcal A$ of mean zero, $\ip f{(I-\mathcal K)f}\ge\kappa\Var_\mu(f)$ on $\mathcal A$, and $\mathrm{BP}(n,\kappa L^4/n^4)$ holds on $\Lambda_{2n+4}$. Theorem~\ref{thm:bpfixed}(ii) gives the bound. \end{proof} The proof uses only that $d$-Lipschitz functions are dense in the charged part $\mathcal C$. If $d$ is replaced by its minimum over the centre twists as well, $d'(U,U')=\min d_E(U,gZU')$ over gauge transformations $g$ and products $Z$ of the maps $Z^\nu_t$, the $d'$-Lipschitz functions are neutral, and the same argument, with the average over the gauge group and the group generated by the $Z^\nu_t$, gives $\mathrm{BP}_0(n,\kappa L^4/n^4)$, which Theorem~\ref{thm:bpfixed} does not constrain. A coupling argument for the cube chain at weak coupling and small $n$ therefore has to contract a pseudometric that does not see the centre twists. Since $n^8/(2n+4)^4\ge n^4/256$, the bound of Corollary~\ref{cor:bpcoupling} is at least $150\,n^4(\log\beta+6.11)/\beta$, which exceeds one once $n^4\ge\beta/150$; along the tuning, where $n_k^4$ grows faster than every power of $\beta_k$, the corollary does not exclude such couplings. The remaining results hold for every $G$ and $\varphi$ of Sec.~\ref{ssec:averaged}, on tori and on boxes. Put $\omega_\varphi=\log( \max\varphi/\min\varphi)$, so that $\omega_\varphi\le2N^2\lvert\beta\rvert$ for the Wilson weight, and for a finite set $S$ of links let $P_S$ be the set of plaquettes of $P$ that contain a link of $S$; $\lvert P_S\rvert\le6 \lvert S\rvert$. \begin{proposition}[no local slow modes]\label{prop:bpnoloc} Let $S\subset E$ be finite and let $f\in L^2(\mu)$ be real and depend only on the links of $S$. Then \[ \Var_\mu(f)\le e^{5\omega_\varphi\lvert P_S\rvert}\,\mathcal E_n(f). \] In particular, if $\mathcal E_n(f)\le\varpi\Var_\mu(f)$ with $0<\varpi<1$ and $\Var_\mu(f)>0$, then $\lvert P_S\rvert\ge\log(1/\varpi)/(5\omega_\varphi)$; for the Wilson weight, $\lvert S\rvert\ge\log(1/\varpi)/(60N^2\lvert\beta \rvert)$. \end{proposition} \begin{proof} For a finite $W\subset E$ write $H_W$ for the normalised Haar measure on $G^W$ and $\rho_W(\cdot\mid U)$ for the density of $\gamma^0_W(\cdot\mid U)$ with respect to $H_W$. It is proportional to a product of $\lvert P_W\rvert$ values of $\varphi$, so the ratio of two of its values is at most $e^{ \omega_\varphi\lvert P_W\rvert}$, and as its integral is one, $e^{-\omega_\varphi\lvert P_W\rvert}\le\rho_W\le e^{\omega_\varphi\lvert P_W\rvert}$. Put $a=\omega_\varphi\lvert P_S\rvert$, and write $P_W$ also for the conditional expectation given the links outside $W$; then $P_{V_x}= \mathcal P_x$. \emph{Step 1.} Let $\nu_S$ be the law of $U_S$ under $\mu$. Its density with respect to $H_S$ is the mean of $\rho_S(\cdot\mid U)$ over $U$, so it lies in $[e^{-a},e^a]$, and $r_U=\rho_S(\cdot\mid U)/(d\nu_S/dH_S)\ge e^{-2a}$. Let $g=f-\mu(f)$, a function on $G^S$ with $\nu_S(g)=0$. Then $P_Sg(U)=\int g\, r_U\,d\nu_S=\int g\,(r_U-e^{-2a})\,d\nu_S$, and the Cauchy--Schwarz inequality for the positive measure $(r_U-e^{-2a})\,d\nu_S$ of mass $1-e^{-2a}$ gives \[ (P_Sg)^2(U)\le(1-e^{-2a})\int g^2r_U\,d\nu_S=(1-e^{-2a})P_S(g^2)(U). \] Integrating, $\mu((P_Sg)^2)\le(1-e^{-2a})\mu(g^2)$, and since $P_S$ is an orthogonal projection, $\mu((f-P_Sf)^2)=\mu(g^2)-\mu((P_Sg)^2)\ge e^{-2a} \Var_\mu(f)$. \emph{Step 2.} $\mu((f-P_Sf)^2)$ is the mean over $U$ of the variance of $f$ under $\gamma^0_S(\cdot\mid U)$, which is at most $\int(f-H_S(f))^2\rho_S\, dH_S\le e^a\Var_{H_S}(f)$. So $\Var_\mu(f)\le e^{3a}\Var_{H_S}(f)$. \emph{Step 3.} For $W\subset S$ let $v_W$ be the variance of $f$ in the links of $W$ under $H_W$, the links of $S\setminus W$ fixed, a function of these links. With the decomposition $f=\sum_{T\subset S}f_T$ of the proof of Proposition~\ref{prop:avgsup}(ii) for $H_S$, $H_S(v_W)=\sum_{T\cap W\ne \emptyset}\lVert f_T\rVert^2$, and since every link lies in exactly $n^4$ of the sets $V_x$, \[ \sum_{x\in X}H_S(v_{S\cap V_x})=\sum_T\lVert f_T\rVert^2\,\#\{x:T\cap V_x \ne\emptyset\}\ge n^4\Var_{H_S}(f). \] \emph{Step 4.} Let $W=S\cap V_x\ne\emptyset$. Since $d\nu_S/dH_S\ge e^{-a}$ and $v_W\ge0$, $H_S(v_W)\le e^a\nu_S(v_W)=e^a\mu(v_W)$. For every configuration $U$, $v_W\le\int(f-c)^2dH_W$ with $c$ the mean of $f$ under $\gamma^0_W(\cdot \mid U)$, and $dH_W/d\gamma^0_W\le e^{\omega_\varphi\lvert P_W\rvert}\le e^a$, so $v_W$ is at most $e^a$ times the variance of $f$ under $\gamma^0_W(\cdot\mid U)$. Hence $\mu(v_W)\le e^a\mu((f-P_Wf)^2)\le e^a\mu ((f-\mathcal P_xf)^2)$, the last step because the links outside $V_x$ are among those outside $W$. Steps 3 and 4 give $\Var_{H_S}(f)\le n^{-4}e^{2a}\sum_x\mu((f-\mathcal P_xf)^2 )=e^{2a}\mathcal E_n(f)$, and with Step 2, $\Var_\mu(f)\le e^{5a}\mathcal E_n(f)$. For the Wilson weight, $5\omega_\varphi\lvert P_S\rvert\le60N^2 \lvert\beta\rvert\lvert S\rvert$. \end{proof} So a sequence of functions with $\mathcal E_n(f_j)/\Var_\mu(f_j)\to0$, on tori of growing side at fixed $n$ and $\beta$, depends on growing sets of links,\footnote{The exponent $5\omega_\varphi\lvert P_S\rvert$ is the cost of comparing the conditional laws of the links of $S$ with Haar measure, in the manner of the perturbation lemma of Holley and Stroock~\cite{HS87}; it depends on $S$ through the plaquettes that touch it, and not on the torus. At fixed $\beta$ it forces $\lvert S\rvert\to\infty$ as $\varpi\to0$, linearly in $\log(1/\varpi)$.} and no subsequence converges to a function of finitely many links. This does not give $\mathrm{BP}_{\rm loc}$: Proposition~\ref{prop:bpnoloc} bounds the Rayleigh quotient of a local function, which is the mean of $1-\lambda$ under its spectral measure, while $\mathrm{BP}_{\rm loc}$ asks that this measure put no mass near $\lambda=1$; by Theorem~\ref{thm:bplocal}(iv) a failure of it produces a slow function correlated with a local one. The transfer of the condition between cube sizes is limited in the same way. Write $W_x$ for the cubes of side $m$ and $\mathcal E_m$ for the corresponding form; the proof of Proposition~\ref{prop:bpscales}(i) shows $\mathcal E_m\ge(n/m)^4\mathcal E_n$ on $L^2(\mu)$. \begin{proposition}\label{prop:bpsharp} Let $\varphi\equiv1$, $2\le n\le m$, $R\ge2$, and let $L$ be even with $L\ge\max(R+m,2m+4)$. There is a real $f\in\mathcal A$ with $\Var_\mu(f)=1$ and \[ \mathcal E_m(f)\le\Bigl(\frac nm\Bigr)^4\Bigl(\frac{R+m}{R-1}\Bigr)^4 \mathcal E_n(f). \] So no inequality $\mathcal E_m\ge c\,\mathcal E_n$ on $\mathcal A$ with $c> (n/m)^4$ holds on all tori, although $\mathrm{BP}(n,1)$ and $\mathrm{BP}(m,1)$ both hold for this measure. \end{proposition} \begin{proof} Since $G$ is compact and nontrivial, it has a nontrivial irreducible unitary representation, whose character has Haar mean zero; its real or imaginary part, normalised, is a real class function $h$ with $\int h\,dU=0$ and $\int h^2dU=1$. Let $Q$ be the set of plaquettes of the plane $\{0,1\}$ with base point $y\in[0,R)^4$ and $y_0$, $y_1$ even. The plaquette at $y$ has the links $(y,0)$, $(y+e_0,1)$, $(y+e_1,0)$, $(y,1)$. A link $(w,0)$ of such a plaquette determines $y$, which is $w$ or $w-e_1$ according to the parity of $w_1$, and a link $(w,1)$ determines it through the parity of $w_0$; so two plaquettes of $Q$ share no link. Put $f=\prod_{p\in Q}h(U_p)$, a gauge-invariant function. For one plaquette, integrating $h(U_p)$ over any one of its four links gives $\int h\,dU=0$, by the invariance of Haar measure under translation and inversion; so in the decomposition of the proof of Proposition~\ref{prop:avgsup}(ii), $h(U_p)$ has the single term $T=p$, and since the plaquettes of $Q$ involve disjoint links, $f=f_T$ with $T$ the union of their links, and $\lVert f\rVert_2=1=\Var_\mu(f)$. That proof gives $k^4\mathcal E_k(f)=N_k$, the number of $x$ with $V_x\cap T\ne\emptyset$ for cubes of side $k$, $k=n,m$. The base points of the links of $T$ lie in $[0,R]^4$, so $N_k\le(R+k)^4$, as $R+k\le L$. If $x\in[0,R-2]^4$, the cube of side $k\ge2$ at $x$ contains the four points $x+ae_0+be_1$, $a,b\in\{0,1\}$, of $[0,R)^4$, one of which has even coordinates $0$ and $1$ and is the base point of a plaquette of $Q$; so $N_k\ge(R-1)^4$. Hence $\mathcal E_m(f)/\mathcal E_n(f)=(n/m)^4 N_m/N_n\le(n/m)^4((R+m)/(R-1))^4$. The last claim is Proposition~\ref{prop:avgsup}(ii) and $R\to\infty$. \end{proof} At $\beta=0$ the gaps at the two sizes are both at least one, while the Dirichlet forms differ by the factor $(n/m)^4$ on functions spread over large boxes. An upward transfer without that loss has to use more than the comparison of the forms, for instance an orthogonal decomposition such as that of Proposition~\ref{prop:avgsup}(ii), which is specific to product measures. \begin{remark}[the routes to the averaged condition at weak coupling]\label{rem:bproutes} The results above settle how far the following routes go. \emph{Constructions.} A test function bounds the best constant from above. A lower bound needs a Poincar\'e inequality, and Proposition~\ref{prop:bpnoloc} gives one for every function of a fixed finite set $S$ of links, uniformly in the torus, with constant $e^{-5\omega_\varphi\lvert P_S\rvert}$. A Lyapunov argument derives a Poincar\'e inequality from a local one on a set of configurations and a drift towards that set; for the cube chain on $\Lambda_L$ the set consists of configurations of all $4L^4$ links, a small-field set for instance, and the local inequality it needs is one for the restriction of $\mu$ to that set, for functions of all links, which Proposition~\ref{prop:bpnoloc} does not give. A coupling that contracts $d_E$ or the pseudometric $d$ on gauge orbits gives the condition on $\mathcal A$ on every torus to which it applies, and Corollaries~\ref{cor:bpresponse} and~\ref{cor:bpcoupling} bound its rate on $\Lambda_{2n+4}$ for $\SU(2)$; a coupling that contracts a pseudometric blind to the centre twists gives $\mathrm{BP}_0$, which suffices by Theorem~\ref{thm:bplocal} and is open. The comparison with the cube chain of larger cubes reduces to the inner gap of Proposition~\ref{prop:bpscales}(iii); that proof works sector by sector, so it gives $\mathrm{BP}_0(n,\cdot)$ from $\mathrm{BP}_0(m,\cdot)$ and the inner inequality for $f\in\mathcal A_0$. The comparison with the cube chain of the same size for another measure $\mu'$ through the density $d\mu/d\mu'$ costs the exponential of the oscillation of its logarithm, a sum of $6L^4$ plaquette terms; a bound uniform in the torus has to localise the comparison, which leads back to the inner gap. The comparison of Dirichlet forms between cube sizes loses exactly $(n/m)^4$ (Proposition~\ref{prop:bpsharp}). \emph{Forced blow-up.} If $\mathrm{BP}_{\rm loc}(n_k,\varpi)$ fails for every $\varpi>0$ along the tuning, Theorem~\ref{thm:bplocal}(iv) gives local functions $f_k$ and slow functions $h_k$, in $\mathcal A_0$ for $\SU(N)$, with $\Cov_\mu(f_k,h_k)=\Var_\mu(h_k)$ and $\mathcal E_{n_k}(h_k)/\Var_\mu( h_k)\to0$, and Proposition~\ref{prop:bpnoloc} gives $60N^2\beta_k\lvert S\rvert\ge\log(\Var_\mu(h_k)/\mathcal E_{n_k}(h_k))$ for every set $S$ of links on which $h_k$ depends. A contradiction has to come from a property of these slow functions. The properties proved here for the tuned measures, tightness of the marginals, reflection positivity, the transfer matrix, gauge invariance and centre symmetry, are properties of the measures and of their torus limits, not of the cube chain, and the paper proves no implication from any of them to $\mathrm{BP}_{\rm loc}$. Such an implication is a converse of Theorem~\ref{thm:avgdecay}, from decay of correlations to relaxation of the cube chain. For local dynamics of spin systems the known converses assume a mixing condition uniform in the boundary condition~\cite{Mar99}, of the type of the supremum form $D^0$, which Corollary~\ref{cor:bpresponse} excludes for $n$ below a multiple of $(\beta/\log\beta)^{1/4}$. A limit object along the tuning would be a slow mode of a dynamics of continuum gauge fields, and no such dynamics is constructed here. Continuity in $\beta$ does not connect the regimes: Corollary~\ref{cor:bpranges} holds at fixed $\beta$ for $n\ge n_0(\beta)$, and on $\Lambda_{2n+4}$ the best constant of the charged part passes from at least one at $\beta=0$ to $O(n^4\log\beta/\beta)$ (Theorem~\ref{thm:bpfixed}, Corollary~\ref{cor:bpcharged}); no monotonicity in $\beta$ is known. \emph{Residual sets.} Theorem~\ref{thm:bplocal} and Corollary~\ref{cor:bpcharged} identify the residual: the condition may fail on charged functions and on slow functions orthogonal to all local ones without affecting the continuum step, and what has to be excluded is a slow function correlated with a local neutral one. The Villain theory beyond $\beta_0$ has such slow functions, by the local form of Corollary~\ref{cor:avgcoulomb}. The failure of $\mathrm{BP}_{\rm loc}$ does not by itself give a trivial or Gaussian limit, since $\mathrm{BP}_{\rm loc}$ is sufficient for a gap and is not shown to be necessary; a trivialisation argument needs the converse of Theorem~\ref{thm:avgdecay} again. \emph{Perturbative and static estimates.} By Theorem~\ref{thm:bplocal}(ii), $\mathrm{BP}_{\rm loc}(n,\varpi)$ on infinitely many tori gives exponential decay of every local gauge-invariant covariance in the torus limit. The quadratic approximation of the Wilson measure at weak coupling consists of $\dim G$ copies of the free lattice Maxwell field, whose field strengths have correlations that decay as a power of the distance. An argument whose estimates apply equally to that Gaussian approximation, or order by order in an expansion around it, therefore cannot give $\mathrm{BP}_{\rm loc}$; the Villain theory, to which the setting applies, is a nonperturbative instance of the same failure. The renormalisation group bounds of~\cite{Bal84a,Bal85a,Bal89b} concern effective actions in a box of fixed physical size and do not address the cube chain. The Bakry--\'Emery bound of Lemma~\ref{lem:bponelink} requires $6N\lvert\beta\rvert<\kappa_G$. Reflection positivity gives the transfer matrix and the chessboard estimate bounds probabilities of local events uniformly in the torus; both are statements about the measure, not about the cube chain. \end{remark} \begin{remark}[what is proved about the averaged condition at weak coupling]\label{rem:bpweakstatus} Proved: for $\SU(N)$ the cube moves preserve the centre sectors of the torus, local gauge-invariant functions are neutral, and the condition splits into sectors (Proposition~\ref{prop:bpcentre}); the conclusions of Theorems~\ref{thm:avgdecay} and~\ref{thm:avgtransfer} and of Corollary~\ref{cor:avggap} follow from the local form $\mathrm{BP}_{\rm loc}$, which is implied by the neutral form $\mathrm{BP}_0$; under $\mathrm{BP}_{\rm loc}$ slow functions are nearly orthogonal to local ones, and a failure of it produces a slow function correlated with a local one (Theorem~\ref{thm:bplocal}); the obstruction of Theorem~\ref{thm:bpfixed} holds for the charged part alone, and neither $\mathrm{BP}_0$ nor $\mathrm{BP}_{\rm loc}$ enters it (Corollary~\ref{cor:bpcharged}); for $\SU(2)$ a contraction of the cube chain on the gauge orbits of $\Lambda_{2n+4}$ has rate at most $38400\,n^8(\log\beta+6.11)/(L^4\beta)$ (Corollary~\ref{cor:bpcoupling}); every function of a finite set $S$ of links satisfies $\Var_\mu(f)\le e^{5\omega_\varphi\lvert P_S\rvert}\mathcal E_n(f)$ on every torus and box (Proposition~\ref{prop:bpnoloc}); and the upward comparison of the Dirichlet forms loses exactly $(n/m)^4$, already at $\beta=0$ (Proposition~\ref{prop:bpsharp}). Not proved: $\mathrm{BP}$, $\mathrm{BP}_0$ or $\mathrm{BP}_{\rm loc}$ along the continuum tuning for any non-abelian group; whether $\mathrm{BP}_0$ or $\mathrm{BP}_{\rm loc}$ fails on $\Lambda_{2n+4}$ at weak coupling, of which Theorem~\ref{thm:bpneutral} reduces the first, for $\SU(2)$, to a lower bound on the law of the Polyakov loop; a converse of Theorem~\ref{thm:avgdecay}, from decay of correlations to the relaxation of the cube chain; a coupling that contracts a pseudometric blind to the centre twists; and the mass gap. \end{remark} \subsection{The neutral sector on small tori}\label{ssec:bpneutral} Theorem~\ref{thm:bpfixed} proves that the averaged condition fails on the torus $\Lambda_{2n+4}$ at weak coupling, and Corollary~\ref{cor:bpcharged} places that failure in the charged part of $\mathcal A$. This subsection treats the neutral part for $\SU(2)$. The neutral functions of a Polyakov loop change under the cube moves only through the curvature, as the charged ones do, so the right side of $\mathrm{BP}_0$ is small for them; what is missing is a lower bound on their variance. A flow that turns every Polyakov loop of one direction inside its own one-parameter subgroup changes the Wilson action only through the curvature, and integration by parts along it bounds the concentration of the law of the Polyakov loop. The outcome is that $\mathrm{BP}_0$ at weak coupling on $\Lambda_{2n+4}$ forces the Polyakov loop towards the centre and towards the traceless elements of $\SU(2)$, and that the failure of $\mathrm{BP}_0$ there reduces to one lower bound on the law of the Polyakov loop. Throughout, $G=\SU(2)$ with the Wilson weight $\varphi=e^{2\beta\Tr(\cdot)}$, $\Lambda=\Lambda_L$ with $L$ even, and $H=\sum_p(2-\Tr U_p)$, so that $\mu$ has density proportional to $e^{-2\beta H}$. Every $W\in\SU(2)$ can be written $W=\cos\theta\,I+\sin\theta\,u$ with $\theta\in[0,\pi]$ and $u$ traceless anti-Hermitian with $u^2=-I$, and $\Tr W=2\cos\theta$; in these polar coordinates of the three-sphere the Haar measure is $\frac2\pi\sin^2 \theta\,d\theta$ times the uniform measure of $u$. For $W\in\SU(2)$ put \[ J(W)=\Tr(W)\,\tfrac12(W-W^*)=2\cos\theta\sin\theta\,u , \] a traceless anti-Hermitian matrix that commutes with $W$ and satisfies $J(gWg^{-1})=gJ(W)g^{-1}$ for $g\in\SU(2)$ and $J(-W)=J(W)$. As in Sec.~\ref{ssec:bpstrong}, $P_b$ is the Polyakov loop of direction $0$ through the site $b$, the ordered product of the $L$ links $(b+te_0,0)$, $0\le t0$ on $\Lambda_{2n+4}$, then $\mathrm{BP}_0(n,\varpi)$ on $\Lambda_{2n+4}$ implies \[ \varpi\le\frac{9\cdot2^{41}\,n^{12}(\log\beta+6.11)^3}{c^2\beta}. \] \end{enumerate} \end{theorem} \begin{proof} (i) Let $h=T^2/4$. It is real and gauge invariant; $Z^0$ maps $P_0$ to $-P_0$ and $Z^\nu$, $\nu\ne0$, fixes it, so $h\in\mathcal A_0$. If $V_x$ meets no link of $P_0$, $\mathcal P_xh=h$, and $V_x$ meets a link of $P_0$ for $Ln^3$ points $x$. For each of these choose $s$ as in the proof of Theorem~\ref{thm:bpfixed}(ii), so that $\lvert s\rvert\le n$ and the loop $A'=P_{se_2}$ avoids $V_x$; that proof gives $P_0=H_0\Gamma A'\Gamma^{-1}$ with $\lVert H_0-I\rVert_{\rm op}\le S_0$, the sum of $\lVert U_p-I \rVert_{\rm op}$ over at most $Ln$ plaquettes. Then $\lvert T-\Tr A'\rvert= \lvert\Tr((H_0-I)\Gamma A'\Gamma^{-1})\rvert\le2S_0$, and as $\lvert \frac d{dt}\frac{t^2}4\rvert\le1$ on $[-2,2]$, $h'=(\Tr A')^2/4$ satisfies $\lvert h-h'\rvert\le2S_0$. Since $h'$ depends only on the links outside $V_x$ and $\mathcal P_x$ is the orthogonal projection onto such functions, \[ \mu\bigl((h-\mathcal P_xh)^2\bigr)\le\mu\bigl((h-h')^2\bigr)\le4\mu(S_0^2) \le4L^2n^2\varepsilon, \] with $\varepsilon=\mu(\lVert U_p-I\rVert_{\rm HS}^2)\le2(\log\beta+6.11)/ \beta$ by Lemma~\ref{lem:bpflat}. So $\mathcal E_n(h)\le n^{-4}Ln^3\cdot4L^2 n^2\varepsilon=4L^3n\varepsilon$, and $\mathrm{BP}_0(n,\varpi)$ gives $\Var_\mu(h)\le8L^3n(\log\beta+6.11)/(\varpi\beta)$. By~\eqref{eq:bpanticonc} with $\psi(t)=t^2/4$, $\mu(T^2(4-T^2))^2\le16\Var_\mu(h)\lVert\Theta \rVert_2^2$, and Lemma~\ref{lem:bpflow}(iii) gives $\lVert\Theta\rVert_2^2\le 36864\,L^8(\log\beta+6.11)^2$. Since $16\cdot8\cdot36864=9\cdot2^{19}$, (i) follows. (ii), (iii) For $L=2n+4\le4n$, $L^{11}n\le2^{22}n^{12}$, and $3\cdot2^{41/2} \le4.45\cdot10^6$. \end{proof} \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_bpneutral.png} \caption{The neutral sector on small tori. (a)~A great circle of $\SU(2)$ through $I$, $u$, $-I$, $-u$, with the velocity $\sin2\theta\,\partial_\theta$ by which the flow of Lemma~\ref{lem:bpflow} moves a Polyakov loop $P=\cos\theta\,I+\sin\theta\,u$. The flow fixes the centre $\pm I$ (red) and the traceless elements $\pm u$ (gold). (b)~The Haar law of $T=\Tr P_0$, with density $\sqrt{4-T^2}/(2\pi)$, and the weight $T^2(4-T^2)$ of Theorem~\ref{thm:bpneutral}; the weight has Haar mean $2$ and vanishes exactly at $T\in\{-2,0,2\}$, the set on which no flow of this kind gives a bound.}\label{fig:bpneutral} \end{figure*} Under the Haar measure, \[ \mu\bigl(T^2(4-T^2)\bigr)=16\cdot\frac2\pi\int_0^\pi\sin^4\theta\cos^2 \theta\,d\theta=2 . \] The quantity vanishes exactly when $P_0$ lies almost surely in $\{\pm I\}\cup\{W:\Tr W=0\}$, and Theorem~\ref{thm:bpneutral}(ii) says that $\mathrm{BP}_0$ with a fixed $\varpi$ on $\Lambda_{2n+4}$ at $\beta\gg n^{12}\log^3\beta$ forces the law of the Polyakov loop towards that set. A coupling of the cube chain that contracts a pseudometric blind to the centre twists at rate $\kappa$ gives $\mathrm{BP}_0(n,\kappa L^4/n^4)$, so on $\Lambda_{2n+4}$ it forces $\mu(T^2(4-T^2))\le4.45\cdot10^6n^8(\log\beta+6.11)^{3/2}/(L^2\sqrt{ \kappa\beta})$. The exceptional set is that of the method. Consider fields whose component at each link of direction $0$ is $L^{-1}K(P)U$, with $U$ the link and $P$ the Polyakov loop based at its initial site. The cancellation in Lemma~\ref{lem:bpflow}(iii) uses that $K$ is equivariant under conjugation, and the part of the field that is odd under the centre contributes nothing, since for such a field $\mathcal Vh$ is odd under $Z^0$ and has mean zero for every neutral $h$. An equivariant field $W\mapsto K(W)W$ on $\SU(2)$ has $K(W)$ fixed by conjugation with the stabiliser of $W$, so $K(\pm I)=0$ and, for $W=\cos\theta+\sin\theta\,u$ with $0<\theta<\pi$, $K(W)=k(W)u$ with $k$ real. If $\Tr W=0$, then $-W=vWv^{-1}$ for every $v\in\SU(2)$ that anticommutes with $u$, so evenness and equivariance give $k(W)u=K(-W)=v K(W)v^{-1}=-k(W)u$ and $K(W)=0$. So every even equivariant $K$ vanishes on $\{\pm I\}\cup\{\Tr W=0\}$, and $\mu(\mathcal Vh)$ tends to zero for every neutral $h$ of class $C^1$ when the law of the Polyakov loop concentrates there; integration by parts along such fields then gives no lower bound on the variance of a neutral function. The law of the Polyakov loop on $\Lambda_{2n+4}$ at weak coupling is not determined here. \begin{remark}[what is proved about the neutral sector and what remains]\label{rem:bpneutralstatus} Proved: for $\SU(2)$ on every torus with $L\ge8$ and $\beta\ge2$, the law of the Polyakov loop satisfies the anti-concentration bound~\eqref{eq:bpanticonc} with $\lVert\Theta\rVert_2\le192L^4(\log\beta+ 5.44)$ (Lemma~\ref{lem:bpflow}); $\mathrm{BP}_0(n,\varpi)$ on $\Lambda_{2n+4}$ forces $\mu(T^2(4-T^2))\le4.45\cdot10^6n^6(\log\beta+6.11 )^{3/2}(\varpi\beta)^{-1/2}$, so $\mathrm{BP}_0$ fails there once $\beta\gg n^{12}\log^3\beta$ whenever $\mu(T^2(4-T^2))$ is bounded below (Theorem~\ref{thm:bpneutral}); and no vector field of the kind used can treat a Polyakov loop concentrated on the centre and the traceless elements. Not proved: a lower bound on $\mu(T^2(4-T^2))$ on $\Lambda_{2n+4}$ at weak coupling, and so whether $\mathrm{BP}_0$ fails there; whether $\mathrm{BP}_{\rm loc}$ fails there; $\mathrm{BP}_0$ or $\mathrm{BP}_{\rm loc}$ along the continuum tuning for any non-abelian group; and the mass gap. The routes to $\mathrm{BP}_{\rm loc}(n_k,\varpi)$ along the tuning stand as follows. A coupling blind to the centre twists is constrained on $\Lambda_{2n+4}$ as above and not on the large tori that Theorem~\ref{thm:avgtransfer} uses. The comparison with larger cubes gives $\mathrm{BP}_0(n,\cdot)$ from $\mathrm{BP}_0(m,\cdot)$ and the inner inequality of Proposition~\ref{prop:bpscales}(iii) for $f\in\mathcal A_0$, which must hold for almost every configuration outside the cube of side $m$, that is, a gap of the chain of cubes of side $n$ for the measure of that cube under every boundary configuration; nothing is proved about it at weak coupling, and $\mathrm{BP}_0(m,\cdot)$ is the same problem at a larger cube size. Theorem~\ref{thm:bpneutral} is an implication from a property of the measure alone, the spread of the Polyakov loop, to the failure of a gap of the cube chain; a converse of Theorem~\ref{thm:avgdecay} needs an implication in the opposite direction, from properties of the measure such as reflection positivity and the transfer matrix to a gap of the cube chain, and none is proved. A comparison with a chain on block variables in the range of Corollary~\ref{cor:bpranges} needs two estimates on tori of every size: single-link bounds as in Theorem~\ref{thm:bpstrong} for the effective measure of the block variables, and a Poincar\'e inequality for the fine links given the block variables, uniform in them, which is an inner inequality of the kind of Proposition~\ref{prop:bpscales}(iii); neither is proved here, and the bounds of~\cite{Bal84a,Bal85a,Bal89b} concern effective actions in a box of fixed physical size. What remains for the continuum step is $\mathrm{BP}_{\rm loc}(n_k,\varpi)$ itself on infinitely many tori for each $k$, with one $\varpi>0$; by Theorem~\ref{thm:bplocal}(iv) it bounds the correlation of every slow function with every local one, and its failure produces a slow neutral function correlated with a local one. \end{remark} \subsection{The averaged condition under block-spin maps}\label{ssec:bpblock} A bootstrap of the averaged condition across scales compares the cube chain of a fine lattice with a cube chain of block variables. This subsection proves the comparison in the direction of the block-spin transformation, for every local gauge-covariant block-spin map and every gauge-invariant measure: $\mathrm{BP}$ passes from the fine lattice to the image measure of the block variables, with a loss that tends to one as the cube grows. Along the tuning, $\mathrm{BP}(n_k,\varpi)$ therefore implies the condition at one cube size, with one constant, for the image measures at a fixed coarser spacing. The local form passes only in an integrated form, which bounds the integrated autocorrelation time of a local function and gives no spectral gap. The opposite direction, which a bootstrap needs, is not obtained. \emph{Setting.} The definitions of Sec.~\ref{ssec:averaged} apply to every probability measure $\varrho$ on $G^E$, $E$ the links of $\Lambda_L$, that is invariant under the gauge group: $P^\varrho_x$ is the conditional expectation under $\varrho$ given the links outside $V_x$, an orthogonal projection on $L^2(\varrho)$; $\mathcal A^\varrho$ is the space of gauge-invariant functions in $L^2(\varrho)$; $\mathcal E^\varrho_n(f)=n^{-4}\sum_x\varrho((f-P^\varrho_xf)^2)$, $\mathcal K^\varrho=L^{-4}\sum_xP^\varrho_x$, and $\mathrm{BP}_\varrho(n,\varpi)$ and $\mathrm{BP}_{{\rm loc},\varrho}(n,\varpi)$ are Definitions~\ref{def:avgcube} and~\ref{def:bploc}(a) with $\varrho$ in place of $\mu$. A gauge transformation $g$ maps the functions of the links outside $V_x$ onto themselves and preserves $\varrho$, so $(P^\varrho_xf)\circ g=P^\varrho_x(f\circ g)$ and $P^\varrho_x$ maps $\mathcal A^\varrho$ into itself. The proofs of Proposition~\ref{prop:bpscales}(i) and of the first statement of Theorem~\ref{thm:bplocal}(iv) use only that the $P^\varrho_x$ are orthogonal projections that preserve $\mathcal A^\varrho$ and the constants, with $P_W= P_WP_V$ for $V\subset W$; so both hold for every such $\varrho$. Let $b\ge1$ and $r\ge1$ be integers. The fine torus is $\Lambda_{bL}$ with links $E'$, the coarse torus is $\Lambda_L$ with links $E$, and the block of $y\in\Lambda_L$ is $B_y=by+[0,b)^4\subset\Lambda_{bL}$. A measurable map $c:G^{E'}\to G^E$ is a \emph{block-spin map of range $r$} if \begin{enumerate}[label=\textup{(B\arabic*)},leftmargin=*] \item for every coarse link $(y,\nu)$, $c_{(y,\nu)}(U)$ depends only on the links of $E'$ with base point in a block $B_w$ with $\lvert w-y \rvert_\infty\le r$; \item for every gauge transformation $g$ of $\Lambda_{bL}$ there is a gauge transformation $\hat g$ of $\Lambda_L$ with $c(gU)=\hat gc(U)$ for all $U$, and every gauge transformation of $\Lambda_L$ is of this form. \end{enumerate} The map that assigns to $(y,\nu)$ the product of the $b$ links along the straight path from $by$ to $by+be_\nu$ is a block-spin map of range one, with $\hat g(y)=g(by)$. For $b=2$ so is the cut-off average~\eqref{eq:cutoff}, evaluated on the links put in the tree gauge of Definition~\ref{def:blockgauge} in every block by the gauge transformation equal to $I$ at the root $2y$: that gauge transformation is a function of the tree links of the block, so the gauge-fixed links of the paths that start in $B(y)$ are functions of the links with base point in $B(y)\cup B(y+e_\nu)$; a gauge transformation $g$ changes the gauge-fixed field by the transformation equal to $g(2y)$ on each block $B(y)$, which acts on the cut-off average as $\hat g(y)=g(2y)$ by Theorem~\ref{thm:cutoff}. For a gauge-invariant probability measure $\mu'$ on $G^{E'}$ let $\mu^c= c_*\mu'$ be the law of the block variables. It is invariant under the gauge group of $\Lambda_L$: given $\hat g$, choose $g$ as in (B2); then $c^{-1}( \hat g^{-1}A)=g^{-1}c^{-1}(A)$ for every Borel set $A$, and $\mu'$ is invariant under $g$. Write $\mathcal E'_{n'}$, $\mathcal A'$, $P'_{x'}$ for the objects of $\mu'$ on $\Lambda_{bL}$ and $\mathcal E^c_n$, $\mathcal A^c$, $P^c_y$ for those of $\mu^c$ on $\Lambda_L$. \begin{theorem}[descent along block-spin maps]\label{thm:bpdescent} Let $c$ be a block-spin map of range $r$ with blocks of side $b$, let $n\ge 2r+2$, let $L\ge2n+4$ be even, let $2\le n'\le b(n-2r-1)+1$ with $bL\ge 2n'+4$, let $\mu'$ be a gauge-invariant probability measure on $G^{E'}$, and put $\varpi_c=\varpi'(n'/(bn))^4$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For every real $F\in\mathcal A^c$, $F\circ c\in\mathcal A'$, $\Var_{\mu'}(F\circ c)=\Var_{\mu^c}(F)$ and \[ \mathcal E'_{n'}(F\circ c)\le\Bigl(\frac{bn}{n'}\Bigr)^4\mathcal E^c_n(F). \] \item $\mathrm{BP}_{\mu'}(n',\varpi')$ implies $\mathrm{BP}_{\mu^c}(n, \varpi_c)$. For $n'=b(n-2r-1)+1$, $\varpi_c\ge\varpi'(1-(2r+1)/n)^4$. \item If $\mathrm{BP}_{{\rm loc},\mu'}(n',\varpi')$ holds, then every real $F\in\mathcal A^c$ that depends only on the coarse links with base points in a box of side at most $L/2-2r$, and every real $H\in\mathcal A^c$, satisfy \[ \Cov_{\mu^c}(F,H)^2\le\Var_{\mu^c}(F)\,\frac{\mathcal E^c_n(H)}{\varpi_c}. \] \end{enumerate} \end{theorem} \begin{proof} (i) For every gauge transformation $g$ of $\Lambda_{bL}$, $F\circ c\circ g= F\circ\hat g\circ c$, and $F\circ\hat g=F$ holds $\mu^c$-almost everywhere, hence $F\circ\hat g\circ c=F\circ c$ holds $\mu'$-almost everywhere; so $F\circ c\in\mathcal A'$. The law of $F\circ c$ under $\mu'$ is that of $F$ under $\mu^c$. For $x'\in\Lambda_{bL}$ with coordinates $x'_i\in\{0,\dots,bL-1\}$ let $y(x')\in\Lambda_L$ have coordinates $\lfloor x'_i/b\rfloor-r$ modulo $L$, and for $y\in\Lambda_L$ let $Q_y=by+br\mathbf 1+[0,b(n-2r))^4$, the union of the blocks $B_w$ with $w\in y+[r,n-r)^4$. Each $y$ is $y(x')$ for exactly $b^4$ points $x'$. With $y=y(x')$, $b(y_i+r)=b\lfloor x'_i/b\rfloor\le x'_i$ and $x'_i+n'-1\le b\lfloor x'_i/b\rfloor+b-2+n'\le b(y_i+r)+b(n-2r)-1$, by the bound on $n'$; so the base points of the links of $V'_{x'}$ lie in $Q_y$. Let $(z,\nu)$ be a coarse link with $z\notin y+[0,n)^4$. By (B1), $c_{(z,\nu)}$ depends only on links with base point in blocks $B_w$ with $\lvert w-z\rvert_\infty\le r$; if such a block lay in $Q_y$, then $w-y$ would be congruent to a vector in $[r,n-r)^4$ and $z-y$ to one in $[0,n)^4$, which is excluded. The blocks partition the sites, so $c_{(z,\nu)}$ depends only on links with base point outside $Q_y$, which lie outside $V'_{x'}$. Let $\mathcal G_y$ be the $\sigma$-algebra on $G^{E'}$ generated by the $c_{(z,\nu)}$ with $z\notin y+[0,n)^4$. It is contained in the $\sigma$-algebra of the links outside $V'_{x'}$, onto whose $L^2$ space $P'_{x'}$ projects orthogonally. For bounded $\psi$ of the coarse links outside $V_y$, $\mu'((F\circ c)(\psi\circ c))=\mu^c(F\psi)=\mu^c((P^c_yF) \psi)=\mu'(((P^c_yF)\circ c)(\psi\circ c))$, so $\E_{\mu'}[F\circ c\mid \mathcal G_y]=(P^c_yF)\circ c$. Hence, with $f=F\circ c$, \[ \mu'\bigl((f-P'_{x'}f)^2\bigr)\le\mu'\bigl((f-(P^c_yF)\circ c)^2\bigr) =\mu^c\bigl((F-P^c_yF)^2\bigr), \] and summing over $x'$, each $y$ occurring $b^4$ times, \[ n'^4\mathcal E'_{n'}(f)\le b^4\sum_{y\in\Lambda_L}\mu^c\bigl((F-P^c_yF)^2 \bigr)=b^4n^4\mathcal E^c_n(F). \] (ii) For real $F\in\mathcal A^c$, (i) and $\mathrm{BP}_{\mu'}(n',\varpi')$ give \[ \varpi'\Var_{\mu^c}(F)=\varpi'\Var_{\mu'}(F\circ c)\le\mathcal E'_{n'}(F \circ c)\le\Bigl(\frac{bn}{n'}\Bigr)^4\mathcal E^c_n(F). \] For $n'=b(n-2r-1)+1$, $n'/(bn)\ge(n-2r-1)/n$. (iii) Each coarse link with base point in the box $y_0+[0,s)^4$, $s\le L/2-2r$, depends by (B1) on links with base point in the blocks of $y_0-r \mathbf 1+[0,s+2r)^4$, a box of $\Lambda_{bL}$ of side $b(s+2r)\le bL/2$; so $F\circ c$ is local. The first statement of Theorem~\ref{thm:bplocal}(iv) for $\mu'$, with $f=F\circ c$ and $h=H\circ c\in\mathcal A'$, gives $\Cov_{\mu^c}(F,H)^2=\Cov_{\mu'}(f,h)^2\le\Var_{\mu^c}(F)\,\mathcal E'_{n'}(h)/\varpi'$, and (i) bounds $\mathcal E'_{n'}(h)$ by $(bn/n')^4 \mathcal E^c_n(H)$. \end{proof} The inequality of (iii) is weaker than $\mathrm{BP}_{{\rm loc},\mu^c}(n, \varpi_c)$. For a self-adjoint operator $0\le K\le I$ and a vector $g$ with spectral measure $\nu_g$, the sum $\sum_{t\ge0}\ip g{K^tg}=\int(1-\lambda)^{ -1}d\nu_g$ is the integrated autocorrelation time of $g$. \begin{lemma}\label{lem:bpintegrated} Let $\varrho$ be a gauge-invariant probability measure on $G^E$, $n\ge2$, $\varpi>0$, $\epsilon=\varpi n^4/L^4<1$, $f\in\mathcal A^\varrho$ real, $g=f- \varrho(f)$, and $\nu_g$ the spectral measure of $\mathcal K^\varrho$ at $g$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The following are equivalent: \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item $\Cov_\varrho(f,h)^2\le\Var_\varrho(f)\,\mathcal E^\varrho_n(h)/\varpi$ for every real $h\in\mathcal A^\varrho$; \item $\int_{[0,1]}\epsilon(1-\lambda)^{-1}d\nu_g(\lambda)\le\lVert g \rVert_2^2$, the integrand being $+\infty$ at $\lambda=1$. \end{enumerate} \item The inequality of $\mathrm{BP}_{{\rm loc},\varrho}(n,\varpi)$ for $f$ implies \textup{(b)}. \item \textup{(b)} implies $\nu_g((1-\delta,1])\le\delta\lVert g\rVert_2^2/ \epsilon$ for $0<\delta\le1$ and $\lVert(\mathcal K^\varrho)^tg\rVert_2^2\le \lVert g\rVert_2^2/(\epsilon(2t+1))$ for every integer $t\ge0$. \item For every $0<\eta<\epsilon$ there are a self-adjoint $0\le K\le I$ on $\R^2$ and a unit vector $g$ that satisfies \textup{(b)} for $K$ and has $\nu_g((1-\epsilon,1])>0$. \end{enumerate} \end{lemma} \begin{proof} Let $\mathcal H$ be the real Hilbert space of the real functions of $\mathcal A^\varrho$ with mean zero. Each $P^\varrho_x$ maps it into itself, so $K=\mathcal K^\varrho$ restricts to a self-adjoint operator on $\mathcal H$ with $0\le K\le I$, and so does every bounded Borel function of $K$; the spectral measure of $g$ for this restriction has the moments $\ip g{K^mg}$, so it is $\nu_g$. Put $A=I-K$. For real $h\in\mathcal A^\varrho$ and $h_0=h-\varrho(h)\in\mathcal H$, $\Cov_\varrho(f,h)=\ip g{h_0}$ and $\ip{h_0}{Ah_0}=n^4L^{-4}\mathcal E^\varrho_n(h)$, since $P^\varrho_x$ fixes the constants. So (a) says $\ip gh^2\le\epsilon^{-1} \lVert g\rVert_2^2\ip h{Ah}$ for every $h\in\mathcal H$. (i) Assume (b). Then $\nu_g(\{1\})=0$. Let $\psi_M(\lambda)=\min\{(1- \lambda)^{-1/2},M\}$. By dominated convergence, with the integrable bound $(1-\lambda)^{-1}$, the vectors $\psi_M(K)g$ converge as $M\to\infty$ to some $v\in\mathcal H$ with $\lVert v\rVert_2^2=\int(1-\lambda)^{-1}d\nu_g\le \lVert g\rVert_2^2/\epsilon$, and $A^{1/2}\psi_M(K)g\to g$, since $(1- \lambda)^{1/2}\psi_M(\lambda)\to1$ boundedly on $[0,1)$. For $h\in\mathcal H$, $\ip gh=\lim_M\ip{\psi_M(K)g}{A^{1/2}h}=\ip v{A^{1/2}h}$, and the Cauchy--Schwarz inequality gives (a). Conversely, assume (a) and let $h= \varphi_M(K)g$ with $\varphi_M(\lambda)=\min\{(1-\lambda)^{-1},M\}$. Then $\ip gh=\int\varphi_Md\nu_g$ and $\ip h{Ah}=\int(1-\lambda)\varphi_M^2d\nu_g \le\int\varphi_Md\nu_g$, so (a) gives $\int\varphi_Md\nu_g\le\lVert g \rVert_2^2/\epsilon$, and monotone convergence as $M\to\infty$ gives (b). (ii) By Lemma~\ref{lem:cyclicgap}, $\nu_g$ is carried by $[0,1-\epsilon]$, where $\epsilon(1-\lambda)^{-1}\le1$. (iii) On $(1-\delta,1]$ the integrand of (b) is at least $\epsilon/\delta$. For $\lambda\in[0,1]$, $\lambda^{2t}(1-\lambda)\le(2t/(2t+1))^{2t}/(2t+1) \le1/(2t+1)$, the maximum being at $\lambda=2t/(2t+1)$, so $\lVert K^tg \rVert_2^2=\int\lambda^{2t}d\nu_g\le\epsilon^{-1}(2t+1)^{-1}\int\epsilon(1- \lambda)^{-1}d\nu_g$. (iv) Take $K=\operatorname{diag}(0,1-\eta)$ and $g=(\cos\theta,\sin\theta)$ with $0<\sin^2\theta\le\eta(1-\epsilon)/(\epsilon(1-\eta))$. Then $\int \epsilon(1-\lambda)^{-1}d\nu_g=\epsilon+\epsilon\sin^2\theta(\eta^{-1}-1) \le1$, and $\nu_g(\{1-\eta\})=\sin^2\theta$ with $1-\eta\in(1-\epsilon,1]$. \end{proof} So under $\mathrm{BP}_{{\rm loc},\mu'}(n',\varpi')$, if $\varpi_cn^40$; in the proof of Theorem~\ref{thm:avgdecay} this bound replaces $e^{-\epsilon T}$ by $(\epsilon(2T+1))^{-1/2}$, which gives no exponential decay. Along the continuum tuning the full condition gives the following. \begin{corollary}\label{cor:bpdescent} Let $a_k=2^{-k}a_0$, let $0<\ell_0\le\ell$ and $n_k\ge2$ with $\ell_0\le n_ka_k\le\ell$, and suppose that $\mathrm{BP}(n_k,\varpi)$ holds for the Wilson measure $\mu_{k,L}$ of $\Lambda_L$ at $\beta_k$ for every $L$ in a set $\mathcal L_k$ of even integers $L\ge2n_k+4$. Fix $j\ge0$, put $n_j^*=4+ \lfloor\ell/a_j\rfloor$, and for $k\ge j$ let $c_k$ be a block-spin map of range one with blocks of side $2^{k-j}$. Then for every $k\ge j$ and every $L\in\mathcal L_k$ divisible by $2^{k-j+1}$ with $L\ge2^{k-j}(2n_j^*+4)$, the measure $(c_k)_*\mu_{k,L}$ on $\Lambda_{L/2^{k-j}}$ satisfies \[ \mathrm{BP}\Bigl(n_j^*,\ \varpi\Bigl(\frac{\ell_0}{\ell+4a_j}\Bigr)^4 \Bigr). \] \end{corollary} \begin{proof} Put $b=2^{k-j}$, $L_c=L/b$, even, and $n=3+\lceil(n_k-1)/b\rceil\ge4$. Since $(n_k-1)/b0$, the \emph{split inequality} $\mathrm S(n',n;C_A,C_B)$ for $(\mu',c)$ is \[ \frac{\mu'(\Var(f\mid c))}{C_A}+\frac{\mathcal E^c_n(F_f)}{C_B}\le \mathcal E'_{n'}(f) \] for every real $f\in\mathcal A'$, with $F_f$ as in Lemma~\ref{lem:bpupcond}. The \emph{separate inequalities} are \textup{(A)} $\mu'(\Var(f\mid c))\le C_A \mathcal E'_{n'}(f)$ and \textup{(B)} $\mathcal E^c_n(F_f)\le C_B\mathcal E'_{n'}(f)$ for every real $f\in\mathcal A'$. \end{definition} By Lemma~\ref{lem:bpupcond}, $n^4\mathcal E^c_n(F_f)=\sum_y\mu'(\Var(\E[f\mid c]\mid\mathcal G_y))$, so (B) bounds the coarse Dirichlet form of $\E[f\mid c]$, and with it the term dropped in the proof of Theorem~\ref{thm:bpdescent}(i), named in Remark~\ref{rem:bpdescent}. The split inequality implies (A) and (B) with the same constants, since each of its terms is at most $\mathcal E'_{n'}(f)$; (A) and (B) imply it with the constants $2C_A$ and $2C_B$. \begin{theorem}[the upward step]\label{thm:bpup} Let $c$ be a block-spin map with blocks of side $b$, $n,n'\ge2$, $L\ge2n+4$ and $bL\ge2n'+4$ even, and $\mu'$ a gauge-invariant probability measure on $G^{E'}$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item If $\mathrm S(n',n;C_A,C_B)$ holds for $(\mu',c)$ and $\mathrm {BP}_{\mu^c}(n,\varpi_c)$ holds, then $\mathrm{BP}_{\mu'}(n',\varpi')$ holds with \[ \varpi'=\min\Bigl(\frac1{C_A},\frac{\varpi_c}{C_B}\Bigr). \] \item If \textup{(A)} and \textup{(B)} hold and $\mathrm{BP}_{\mu^c}(n, \varpi_c)$ holds, then $\mathrm{BP}_{\mu'}(n',(C_A+C_B/\varpi_c)^{-1})$ holds. \item If $\mathrm{BP}_{\mu'}(n',\varpi')$ holds, then \textup{(A)} holds with $C_A=1/\varpi'$. \end{enumerate} \end{theorem} \begin{proof} Let $f\in\mathcal A'$ be real, $a=\mu'(\Var(f\mid c))$ and $e=\mathcal E^c_n (F_f)$. By Lemma~\ref{lem:bpupcond} and $\mathrm{BP}_{\mu^c}(n,\varpi_c)$ for the real function $F_f\in\mathcal A^c$, \[ \Var_{\mu'}(f)=a+\Var_{\mu^c}(F_f)\le a+\frac e{\varpi_c}. \] (i) The right side is $C_A\frac a{C_A}+\frac{C_B}{\varpi_c}\frac e{C_B}\le \max(C_A,C_B/\varpi_c)\,\mathcal E'_{n'}(f)$, and $1/\max(C_A,C_B/\varpi_c)= \varpi'$. (ii) The right side is at most $(C_A+C_B/\varpi_c)\,\mathcal E'_{n'}(f)$. (iii) $\mu'(\Var(f\mid c))\le\Var_{\mu'}(f)\le\mathcal E'_{n'}(f)/\varpi'$. \end{proof} The step composes along a tower of block-spin maps. Let $b\ge2$, $r\ge1$, $K\ge1$, $L_0$ even and $L_j=b^jL_0$; for $1\le j\le K$ let $c_j$ be a block-spin map of range $r$ with blocks of side $b$ from $\Lambda_{L_j}$ to $\Lambda_{L_{j-1}}$. Given a gauge-invariant probability measure $\mu_K$ on the links of $\Lambda_{L_K}$, put $\mu_{j-1}=(c_j)_*\mu_j$ for $j=K,\dots,1$; each $\mu_j$ is gauge invariant (Sec.~\ref{ssec:bpblock}). \begin{theorem}[the upward step across scales]\label{thm:bpuptower} Let $n_0,\dots,n_K\ge2$ with $L_j\ge2n_j+4$. Suppose that $\mathrm{BP}_{\mu_0}(n_0,\varpi_0)$ holds and that, for $1\le j\le K$, $\mathrm S(n_j,n_{j-1};C_{A,j},C_{B,j})$ holds for $(\mu_j,c_j)$. Put $\Gamma_j=\prod_{i=j}^KC_{B,i}$ and $\Gamma_{K+1}=1$. Then $\mathrm{BP}_{\mu_K}(n_K,\varpi_K)$ holds with \[ \varpi_K=\min\Bigl(\frac{\varpi_0}{\Gamma_1},\ \min_{1\le j\le K}\frac1{C_{A,j} \Gamma_{j+1}}\Bigr). \] In particular, if $C_{A,j}\le C_A$ and $\Gamma_j\le\Gamma$ for all $j$, with $\Gamma\ge1$, then $\varpi_K\ge\min(\varpi_0,1/C_A)/\Gamma$. \end{theorem} \begin{proof} Let \[ m_j=\min\Bigl(\frac{\varpi_0}{\prod_{i=1}^jC_{B,i}},\ \min_{1\le i\le j} \frac1{C_{A,i}\prod_{l=i+1}^jC_{B,l}}\Bigr), \] so that $m_K=\varpi_K$, and $m_j=\min(1/C_{A,j}, m_{j-1}/C_{B,j})$ for $j\ge1$, with $m_0=\varpi_0$. If $\mathrm{BP}_{\mu_{j-1}} (n_{j-1},m_{j-1})$ holds, Theorem~\ref{thm:bpup}(i) for $\mu'=\mu_j$, $c=c_j$ gives $\mathrm{BP}_{\mu_j}(n_j,m_j)$; induction on $j$ gives the claim. For the last statement, $\varpi_0/\Gamma_1\ge\varpi_0/\Gamma$ and $C_{A,j} \Gamma_{j+1}\le C_A\Gamma$, also for $j=K$ since $\Gamma\ge1$. \end{proof} Along the tuning this reduces $\mathrm{BP}(n_k,\varpi)$ to the condition at the coarsest scale and the split inequality at every scale, for the measures that the block-spin transformation produces from the tuned couplings. \begin{corollary}[conditional bootstrap]\label{cor:bpuptuning} Let $a_k=2^{-k}a_0$, let $n_0\ge2$, $\ell=n_0a_0$ and $n_k=2^kn_0$, so that $n_ka_k=\ell$. For each $k$ let $\mathcal L_k$ be an infinite set of integers $L=2^kL_0$ with $L_0$ even and $L_0\ge2n_0+4$; for $L\in\mathcal L_k$ let $\mu_{k,L}$ be the Wilson measure of $\Lambda_L$ at $\beta_k$, let $c_1,\dots,c_k$ be block-spin maps of range $r$ with blocks of side two as above, with $L_j=2^jL_0$, and let $\mu^{(j)}_{k,L}$ be the measures of the tower with $\mu^{(k)}_{k,L}=\mu_{k,L}$. Suppose that there are $\varpi_0>0$, $C_A$ and $\Gamma\ge1$ such that for all $k$ and $L\in\mathcal L_k$: \begin{enumerate}[label=\textup{(H\arabic*)},leftmargin=*] \item $\mathrm{BP}(n_0,\varpi_0)$ holds for $\mu^{(0)}_{k,L}$; \item for $1\le j\le k$, $\mathrm S(n_j,n_{j-1};C_{A,j},C_{B,j})$ holds for $(\mu^{(j)}_{k,L},c_j)$, with $C_{A,j}\le C_A$ and $\prod_{i=j}^kC_{B,i}\le \Gamma$. \end{enumerate} Then $\mathrm{BP}(n_k,\varpi)$ holds for $\mu_{k,L}$ for all $k$ and $L\in\mathcal L_k$, with $\varpi=\min(\varpi_0,1/C_A)/\Gamma$, and every continuum limit as in Theorem~\ref{thm:avgtransfer} satisfies $\sigma(H) \subset\{0\}\cup[16\varpi/(625e^2\ell),\infty)$. \end{corollary} \begin{proof} $L\ge2^k(2n_0+4)\ge2n_k+4$, and likewise $L_j\ge2n_j+4$ on each torus of the tower. Theorem~\ref{thm:bpuptower} with $K=k$ gives $\mathrm{BP}(n_k, \varpi)$ for $\mu_{k,L}$, and Theorem~\ref{thm:avgtransfer} applies with $n_ka_k=\ell$. \end{proof} The hypotheses are not proved for the tuned Wilson measures. Part of them is contained in the conclusion. For straight-path maps $c_j$, whose composites are straight-path maps of range one, and $L_0\ge2n_0+12$, Corollary~\ref{cor:bpdescent} with $\ell_0=\ell$ turns $\mathrm{BP}(n_k, \varpi)$ for all $k$ into $\mathrm{BP}(n_j+4,\varpi(\ell/(\ell+4a_j))^4)$ for $\mu^{(j)}_{k,L}$, $0\le j\le k$, and by Theorem~\ref{thm:bpup}(iii) it gives (A) at every step of the tower, with the cube sizes $n_j+4$; for $j=0$ Proposition~\ref{prop:bpcoarsezero} gives (H1) at the cube size $n_0+1$ when $L_0\ge2n_0+6$. The split inequality with a bounded product of the constants $C_{B,j}$ is not contained in the conclusion as far as is proved here. For $\varphi\equiv1$ and the straight-path maps every hypothesis of Theorem~\ref{thm:bpuptower} holds with $\varpi_0=1$, $C_{A,j}=1$ and $\Gamma_j=1$ when $n_j=2n_{j-1}$; there $\mu^c$ is again a product of Haar measures and the computation is exact. \begin{proposition}[the product measure]\label{prop:bpuphaar} Let $\varphi\equiv1$, so that $\mu'$ is the product of the Haar measures on the links of $\Lambda_{bL}$, $b\ge2$, let $c$ be the straight-path map, and let $n,n'\ge2$, $L\ge2n+4$ and $bL\ge2n'+4$ even. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\mu^c$ is the product of the Haar measures on the links of $\Lambda_L$, and if $n'\ge bn$, then every real $f\in L^2(\mu')$, with $\E[f\mid c]=F_f\circ c$, satisfies \[ \mu'\bigl(\Var(f\mid c)\bigr)+\Bigl(\frac{bn}{n'}\Bigr)^4\mathcal E^c_n(F_f) \le\mathcal E'_{n'}(f); \] so $\mathrm S(n',n;1,(n'/(bn))^4)$ holds. \item There is a real $f\in\mathcal A'$ with $\E[f\mid c]=0$ and $\mu'(\Var(f\mid c))=(1+2/n')^{-1}\mathcal E'_{n'}(f)>0$. For every $\eta>0$ there is $L_*$ such that for every $L\ge L_*$ there is a real $f\in\mathcal A'$ with $\E[f\mid c]=f$ and $\mathcal E^c_n(F_f)\ge((n'/(bn))^4-\eta)\, \mathcal E'_{n'}(f)>0$. So constants for which \textup{(A)}, \textup{(B)} or $\mathrm S(n',n;C_A,C_B)$ hold for all $L$ in an infinite set satisfy $C_A\ge(1+2/n')^{-1}$ and $C_B\ge(n'/(bn))^4$. \item For $n'=bn$, Theorem~\ref{thm:bpup}\textup{(i)} with $\mathrm{BP}_{\mu^c}(n,1)$ gives $\mathrm{BP}_{\mu'}(bn,1)$, while Theorem~\ref{thm:bpup}\textup{(ii)} with $\mathrm{BP}_{\mu^c}(n,1)$ and any constants valid for all $L$ in an infinite set gives a constant at most $(1+2/n')/(2+2/n')$. \end{enumerate} \end{proposition} \begin{proof} (i) Every link of $\Lambda_{bL}$ lies on at most one of the paths $\pi(e)$ of $b$ links that $c$ multiplies, $e\in E$, and for $T\subset E$ write $\pi(T)= \bigcup_{e\in T}\pi(e)$. The coarse links are products over disjoint sets of independent Haar variables, so $\mu^c$ is the product of the Haar measures. Decompose $f=\sum_{S'\subset E'}f_{S'}$ as in the proof of Proposition~\ref{prop:avgsup}(ii) for $\mu'$. \emph{Step 1: $\E[f_{S'}\mid c]=0$ unless $S'=\pi(T)$ for some $T\subset E$, and $F_T=\E[f_{\pi(T)}\mid c]$ is a function of $c$ in the component $T$ of the decomposition for $\mu^c$.} If $S'$ contains a link $e'$ that lies on no path, $U_{e'}$ is independent of the $\sigma$-algebra $\mathcal H$ generated by $c$ and the other links, and $\E[f_{S'}\mid\mathcal H]=\int f_{S'}\,dU_{e'}=0$. If $S'$ contains a link $e'\in\pi(e)$ and $\pi(e)\not \subset S'$, pick $e''\in\pi(e)\setminus S'$. In the product $c_e$ the link $U_{e''}$ enters once, between products of other links, so by the invariance of Haar measure under translations $c_e$ is a Haar variable independent of all links other than $U_{e''}$. Hence the links of $\pi(e)\setminus\{e''\}$ are independent of the $\sigma$-algebra $\mathcal H$ generated by $c_e$ and the links outside $\pi(e)$, which contains $\sigma(c)$, and $\E[f_{S'}\mid \mathcal H]$ is the integral of $f_{S'}$ over the links of $\pi(e)\setminus \{e''\}$; integrating first over $U_{e'}$ gives zero. In both cases $\E[f_{ S'}\mid c]=\E[\E[f_{S'}\mid\mathcal H]\mid c]=0$. For $S'=\pi(T)$, the links of $\pi(T)$ and the coarse links of $T$ are independent of the coarse links of $E\setminus T$, so $F_T$ depends only on $c_T$. For $e\in T$, the links of $\pi(e)$ are independent of those of $\pi(T\setminus\{e\})$, which determine $c_{T\setminus\{e\}}$, so $\E[F_T\circ c\mid c_{T\setminus\{e\}}]=\E[f_{\pi( T)}\mid c_{T\setminus\{e\}}]=0$ after integration over the links of $\pi(e)$; as $\mu^c$ is a product measure, this says that $F_T$ has Haar mean zero in each variable $c_e$, $e\in T$. Moreover $\lVert F_T\rVert_2\le\lVert f_{\pi(T)}\rVert_2$, and $F_\emptyset=f_\emptyset$. \emph{Step 2: the two sides.} By Step 1, $F_f=\sum_TF_T$ is the decomposition of $F_f$ for $\mu^c$, $\mu'(\E[f\mid c]^2)=\sum_T\lVert F_T \rVert_2^2$ and \[ \mu'\bigl(\Var(f\mid c)\bigr)=\sum_{S'\notin\pi(\cdot)}\lVert f_{S'} \rVert_2^2+\sum_{T\ne\emptyset}\bigl(\lVert f_{\pi(T)}\rVert_2^2-\lVert F_T \rVert_2^2\bigr), \] where the first sum runs over the sets $S'$ not of the form $\pi(T)$. By the proof of Proposition~\ref{prop:avgsup}(ii), $n'^4\mathcal E'_{n'}(f)$ is $\sum_{S'}\lVert f_{S'}\rVert_2^2N'(S')$ and $n^4\mathcal E^c_n(F_f)=\sum_T \lVert F_T\rVert_2^2N^c(T)$, where $N'(S')$ is the number of $x'\in \Lambda_{bL}$ with $V'_{x'}\cap S'\ne\emptyset$ and $N^c(T)$ the number of $y\in\Lambda_L$ with $V_y\cap T\ne\emptyset$. For $S'\ne\emptyset$, $N'(S') \ge n'^4$, since a link lies in $n'^4$ cubes. \emph{Step 3: $N'(\pi(T))\ge b^4N^c(T)$.} Let $V_y\cap T\ne\emptyset$, with $(z,\nu)\in T$ and $z=y+d$, $d\in[0,n)^4$ modulo $L$. For $w\in[0,b)^4$ put $x'=by-w$ modulo $bL$. Then $bz-x'=bd+w\in[0,bn)^4\subset[0,n')^4$, so the first link $(bz,\nu)$ of $\pi(z,\nu)$ lies in $V'_{x'}$. The map $(y,w) \mapsto by-w$ is injective: if $by-w\equiv by_2-w_2$ modulo $bL$, then $w-w_2$ is divisible by $b$ with entries in $(-b,b)$, so $w=w_2$ and $y\equiv y_2$ modulo $L$. \emph{Step 4.} The left side of (i) is the sum of $\lVert f_{S'} \rVert_2^2$ over $S'\notin\pi(\cdot)$ and of \[ \begin{split} &\bigl(\lVert f_{\pi(T)}\rVert_2^2-\lVert F_T\rVert_2^2\bigr)+\lVert F_T \rVert_2^2\,\frac{b^4N^c(T)}{n'^4}\\ &\qquad\le\lVert f_{\pi(T)}\rVert_2^2\max\Bigl(1,\frac{b^4N^c(T)}{n'^4} \Bigr) \end{split} \] over $T\ne\emptyset$, by $\lVert F_T\rVert_2\le\lVert f_{\pi(T)}\rVert_2$. By Steps 2 and 3 each term is at most the corresponding term $\lVert f_{S'}\rVert_2^2N'(S')/n'^4$ of $\mathcal E'_{n'}(f)$. (ii) Let $h$ be a real class function with Haar mean zero and $\int h^2dU=1$, as in the proof of Proposition~\ref{prop:bpsharp}. For a plaquette $p$ of $\Lambda_{bL}$, $f=h(U_p)$ is gauge invariant and its decomposition has the single term $S'=p$, the four links of $p$. A nonempty set $\pi(T)$ contains, with each link $(x,\nu)$, the link $(x+e_\nu,\nu)$ or $(x-e_\nu,\nu)$, because $b\ge2$, and $p$ does not; so $\E[f\mid c]=0$ and $\mu'(\Var(f\mid c))=1$. By the count in the proof of Proposition~\ref{prop:bpscales}(ii), $N'(p)=n'^4+2n'^3$, so $\mathcal E'_{n'}(f)=1+2/n'$. For the second function let $R\ge2$, $L\ge R+1$, and let $F=\prod_{P\in Q}h( U_P)$ be the function of the proof of Proposition~\ref{prop:bpsharp} on $\Lambda_L$, with $Q$ the plaquettes of the plane $\{0,1\}$ with base point in $[0,R)^4$ and even coordinates $0$ and $1$, and $T$ the union of their links; its decomposition has the single term $T$, $\lVert F\rVert_2=1$, and $N^c(T)\ge(R-1)^4$ by that proof. Put $f=F\circ c$, gauge invariant by (B2), with $\E[f\mid c]=f$ and $F_f=F$. For $P\in Q$, $c_P$, the holonomy of $c$ around $P$, is the product of the $4b$ links of $\pi$ of the four links of $P$, each entering once; integrating $h(c_P)$ over any one of them gives $\int h\,dU=0$, and distinct $P\in Q$ involve disjoint links, so the decomposition of $f$ has the single term $\pi(T)$ and $\lVert f\rVert_2=1$. The base points of the links of $T$ lie in $[0,R]^4$, those of $\pi(T)$ in $[0,bR+b-1]^4$, and a cube $V'_{x'}$ meets $\pi(T)$ only if $x'\in[-(n'-1), bR+b-1]^4$ modulo $bL$; so $N'(\pi(T))\le(bR+b+n'-1)^4$. Hence \[ \frac{\mathcal E^c_n(F)}{\mathcal E'_{n'}(f)}=\Bigl(\frac{n'}n\Bigr)^4\frac{ N^c(T)}{N'(\pi(T))}\ge\Bigl(\frac{n'}n\Bigr)^4\Bigl(\frac{R-1}{bR+b+n'-1} \Bigr)^4, \] which tends to $(n'/(bn))^4$ as $R\to\infty$. Choose $R$ with the right side at least $(n'/(bn))^4-\eta$ and $L_*=R+1$. The first function bounds the constant in (A) and in the first term of $\mathrm S$, which it does not see in the second term; the second bounds the constant in (B) and in the second term of $\mathrm S$, as $\mu'(\Var(f\mid c))=0$ for it. (iii) By (i) and Proposition~\ref{prop:avgsup}(ii), Theorem~\ref{thm:bpup}(i) applies with $C_A=C_B=1$ and $\varpi_c=1$. By (ii), with $n'=bn$, Theorem~\ref{thm:bpup}(ii) gives $(C_A+C_B)^{-1}\le((1+2/n')^{-1}+1)^{-1}$. \end{proof} In the proof of (i) the fine Dirichlet form pays each term of the decomposition once: a component $f_{S'}$ that is not a union of paths contributes only to the fibre term, and a component $f_{\pi(T)}$ splits into its fibre part and its image $F_T$ in proportions that add up to its norm. The separate inequalities charge the whole Dirichlet form to each term. Over many scales this costs a factor that grows with the number of scales. \begin{corollary}[the separate inequalities across scales]\label{cor:bpupseparate} Let $\varphi\equiv1$, $b\ge2$, $K\ge1$, $0<\ell_0\le\ell$, $a_j=b^{-j}a_0$ and $n_0,\dots,n_K\ge2$ with $\ell_0\le n_ja_j\le\ell$ and $L_j\ge2n_j+4$. Let $c_j$ be the straight-path maps of the tower above, and suppose that for $1\le j\le K$ the inequalities \textup{(A)} and \textup{(B)} hold for $(\mu_j,c_j)$ with cube sizes $(n_j,n_{j-1})$ and constants $C_{A,j}$, $C_{B,j}$, for all $L_0$ in an infinite set. Let $\psi_0>0$ and $\psi_j=(C_{A,j}+C_{B,j}/ \psi_{j-1})^{-1}$, the constant that Theorem~\ref{thm:bpup}\textup{(ii)} gives at the scale $j$ from $\mathrm{BP}_{\mu_0}(n_0,\psi_0)$. Then \[ \psi_K\le\frac2K\Bigl(\frac\ell{\ell_0}\Bigr)^4, \] while $\mathrm{BP}_{\mu_K}(n_K,1)$ holds on every torus. With $n_j=b^jn_0$, the split inequalities $\mathrm S(n_j,n_{j-1};1,1)$ hold, and Theorem~\ref{thm:bpuptower} with $\mathrm{BP}_{\mu_0}(n_0,1)$ gives $\mathrm{BP}_{\mu_K}(n_K,1)$. \end{corollary} \begin{proof} Unrolling the recursion and dropping the term with $\psi_0$, \[ \frac1{\psi_K}\ge\sum_{j=1}^KC_{A,j}\prod_{i=j+1}^KC_{B,i}. \] Each $\mu_j$ is a product of Haar measures, so by Proposition~\ref{prop:bpuphaar}(ii), $C_{A,j}\ge(1+2/n_j)^{-1}\ge\frac12$ and $C_{B,i}\ge(n_i/(bn_{i-1}))^4$. The product telescopes: \[ \prod_{i=j+1}^K\frac{n_i}{bn_{i-1}}=\frac{n_K}{b^{K-j}n_j}=\frac{n_Ka_K} {n_ja_j}\ge\frac{\ell_0}\ell . \] So $1/\psi_K\ge\frac K2(\ell_0/\ell)^4$. $\mathrm{BP}(n_K,1)$ is Proposition~\ref{prop:avgsup}(ii), and the last statement is Proposition~\ref{prop:bpuphaar}(i) with Theorem~\ref{thm:bpuptower}, with $C_{A,j}=1$ and $\Gamma_j=1$. \end{proof} \begin{remark}[what the upward step gives and what it needs]\label{rem:bpup} Proved: for every block-spin map and every gauge-invariant measure, the split inequality $\mathrm S(n',n;C_A,C_B)$ and $\mathrm{BP}$ for the law of the block variables give $\mathrm{BP}(n',\min(1/C_A,\varpi_c/C_B))$ for the fine measure, and the fibre inequality (A) is implied by $\mathrm{BP}$ for the fine measure (Theorem~\ref{thm:bpup}); along a tower of block-spin maps the constant after $K$ steps is controlled by the constants $C_{A,j}$ and the products of the $C_{B,j}$ (Theorem~\ref{thm:bpuptower}); along the tuning, the condition at the coarsest scale (H1), which a proof of $\mathrm{BP}(n_k, \varpi)$ has to contain by Corollary~\ref{cor:bpdescent}, and the split inequality at every scale (H2) give $\mathrm{BP}(n_k,\varpi)$ with an explicit $\varpi$ and the conclusion of Theorem~\ref{thm:avgtransfer} (Corollary~\ref{cor:bpuptuning}). For the product of Haar measures and the straight-path map, $\mathrm S(n',n;1,(n'/(bn))^4)$ holds for $n'\ge bn$, the constants are the best ones uniform in the torus, and the tower reproduces $\mathrm{BP}(n,1)$ at every scale (Proposition~\ref{prop:bpuphaar}). The separate inequalities (A) and (B) of Remark~\ref{rem:bpdescent}, composed through Theorem~\ref{thm:bpup}(ii), give after $K$ scales of physical cube size between $\ell_0$ and $\ell$ at most $2(\ell/\ell_0)^4/K$, already for the product of Haar measures, where the condition holds with constant one (Corollary~\ref{cor:bpupseparate}); a bootstrap across the scales of the tuning has to use the split form. Not proved: (H1) and (H2) for the tuned Wilson measures; the split inequality, with $C_B$ close to $(n'/(bn))^4$, for the Wilson measure outside the strong-coupling range, for $\U(1)$ at $\beta\ne0$, and for $\SU(2)$ at weak coupling on calm fibres; Sec.~\ref{ssec:bprefine} proves it, with the constants of the product measure, for the Haar refinements, which are not Wilson measures, and Sec.~\ref{ssec:bpsplitstrong} proves it for the Wilson measure at strong coupling, with constants within factors $1+O(\beta)$ of those (Theorem~\ref{thm:bpsplitstrong}). The proof of Proposition~\ref{prop:bpuphaar} uses the orthogonal decomposition of the product measure, in which the components of $f$ that the fibre and the coarse field see are orthogonal; for an interacting measure they are not, and the term dropped in the proof of Theorem~\ref{thm:bpdescent}(i), the information that the fine field outside a fine cube carries on the block variables inside the coarse cube beyond the coarse field outside it, vanishes for the product measure and its Haar refinements (Proposition~\ref{prop:bpfinesplit}(ii), Lemma~\ref{lem:bprefine}(ii)), is bounded at strong coupling by a factor $q^2$ of order $\beta^2$ times the coarse form of the refinement (Theorem~\ref{thm:bpexterior}), and is not bounded here for any other measure. Theorem~\ref{thm:eightlinkuniform} and Proposition~\ref{prop:chartfolds} concern bounds uniform in the conditioning on block variables, while (A) and $\mathrm S$ are posed in the mean over $\mu'$, so they do not decide them. Neither $\mathrm{BP}$ along the tuning nor the mass gap is proved here. \end{remark} \subsection{Haar refinements}\label{ssec:bprefine} This subsection proves the split inequality, with the constants of Proposition~\ref{prop:bpuphaar}(i), for a class of interacting measures: those whose conditional law given the block variables of the straight-path map is the Haar law of the fibres, while the law of the block variables is an arbitrary gauge-invariant measure. Along a tower of such refinements the upward step carries the condition at the coarsest scale to every finer scale without loss. The proof rests on an exact splitting of the fine Dirichlet form into a fibre form and a coarse form, valid for every measure; for a general measure the coarse form is reduced by the information that the fine field outside a fine cube carries on the block variables, and the Wilson measure at $\beta\ne0$ is not a refinement. Throughout, $c$ is the straight-path map with blocks of side $b\ge2$ from $\Lambda_{bL}$ to $\Lambda_L$, $\pi(e)$ is the path of the $b$ links that $c$ multiplies for $e\in E$, and $E'_0=E'\setminus\bigcup_{e\in E}\pi(e)$ is the set of links on no path. A gauge transformation acts by $(gU)_{(x,\mu)} =g(x)U_{(x,\mu)}g(x+e_\mu)^{-1}$, and $\lambda_S$ denotes the product of the Haar measures on $G^S$. For $W\in G$ and $m\ge1$ let $\nu^{(m)}_W$ be the law of $(U_1,\dots,U_m)$ with $U_1,\dots,U_{m-1}$ independent Haar variables and $U_m=(U_1\cdots U_{m-1})^{-1}W$, the uniform law on the tuples with product $W$. For $V\in G^E$ let $\nu_V$ be the product of the laws $\nu^{(b)}_{V_e}$ of the links of $\pi(e)$, taken in their order along the path, over $e\in E$, and of $\lambda_{E'_0}$; then $c=V$ holds $\nu_V$-almost surely. \begin{lemma}\label{lem:bpfibrelaw} Let $W\in G$ and $m\ge2$, and let $(U_1,\dots,U_m)$ have law $\nu^{(m)}_W$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Any $m-1$ of the $U_k$ are independent Haar variables, and so are the partial products $A_k=U_1\cdots U_k$, $1\le k\le m-1$. \item For $h_0,\dots,h_m\in G$ the law of $(h_{k-1}U_kh_k^{-1})_{k\le m}$ is $\nu^{(m)}_{h_0Wh_m^{-1}}$. \end{enumerate} \end{lemma} \begin{proof} (i) For the coordinates $k\ne m$ this is the definition. Leave out $U_i$ with $i0, \] the variance being positive because $S$ is continuous and not constant, with $S=N\lvert P\rvert$ at $U\equiv I$ and Haar mean zero, and $\mu'_\beta$ has a positive continuous density. So $m$ is strictly increasing, and $m(0)= 0$. \end{proof} \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_bptower.png} \caption{The averaged condition across scales. (a)~Three lattices of spacings $a_0$, $a_0/b$, $a_0/b^2$, $b=2$, each with a cube of the same physical side, of $n=3,6,12$ sites. The condition passes down to the block variables for every gauge-invariant measure (Theorem~\ref{thm:bpdescent}) and up one scale under the split inequality S (Theorem~\ref{thm:bpup}). (b)~S holds for every Haar refinement (Theorem~\ref{thm:bprefine}), the product of Haar measures included; a Wilson measure at $\beta\ne0$ is not a Haar refinement (Proposition~\ref{prop:bpnotrefine}); for it S holds at strong coupling (Theorem~\ref{thm:bpsplitstrong}) and is open at weak coupling.}\label{fig:bptower} \end{figure*} \begin{remark}[what the refinements give and what they leave]\label{rem:bprefine} Proved: for every gauge-invariant law $\varrho$ of the block variables, its Haar refinement along the straight-path map satisfies the split inequality with the constants $C_A=1$ and $C_B=(n'/(bn))^4$ of the product of Haar measures, and $C_A=1$ is optimal up to the factor $1+2/n'$ (Theorem~\ref{thm:bprefine}); along a tower of refinements, (H2) holds with $C_{A,j}=1$ and $\Gamma_j=1$ and (H1) is the condition for $\varrho$, so the condition at the coarsest scale passes to every finer scale without loss, and for $\varrho$ a Wilson measure in the range of Corollary~\ref{cor:bpranges} it holds at every scale with the constant $1-\alpha_{n_0}$, uniformly in the number of scales and the torus (Corollary~\ref{cor:bprefinetower}); for every measure the fine Dirichlet form splits into the fibre form $\Phi$ and a coarse form, and the split inequality follows from one bound on each with the same constants, while on functions of the block variables the fine form falls short of that of the refinement by the information of the fine exterior on the block variables (Proposition~\ref{prop:bpfinesplit}). Not proved: the split inequality for the Wilson measure, which is not a refinement (Proposition~\ref{prop:bpnotrefine}), outside the strong-coupling range treated in Sec.~\ref{ssec:bpsplitstrong}, for $\U(1)$ at $\beta>0$, or for $\SU(2)$ on calm fibres; for them neither the bound $\Phi(f)\ge\mu'(\Var(f\mid c))/C_A$, a Poincar\'e inequality for the fine links given the block variables in the mean over $\mu'$, nor a bound on the subtracted term of Proposition~\ref{prop:bpfinesplit}(ii) is proved, and whether that term vanishes for them is not decided. Whether the split inequality with $C_B$ near $(n'/(bn))^4$ holds or fails along the tuning is not decided; no obstruction to it is proved here, and the obstructions on small tori give none (Proposition~\ref{prop:bpsmalltori}). For a general $\varrho$ the optimal $C_B$ in Theorem~\ref{thm:bprefine} is not determined; for $\varrho=\lambda_E$ it is $(n'/(bn))^4$ by Proposition~\ref{prop:bpuphaar}(ii). The condition (H1) does not follow from the results of this paper along the tuning: Sec.~\ref{ssec:bpcoarse} reduces it to bounds on the single-link coefficients of the laws $\mu^{(0)}_{k,L}$ of the block variables, which are not proved outside the strong-coupling range, and gives it at the cube size $n_0+1$ for $\SU(2)$ and those $k$ with $\lvert\beta_{k,W}\rvert<\frac{16} {105}$, which along a tuning with $\beta_k\to\infty$ are finitely many (Proposition~\ref{prop:bpcoarsezero}). Corollary~\ref{cor:bprefinetower} concerns measures that are Haar on all scales below the coarsest, and says nothing about the tuned Wilson measures; it shows that the split mechanism tolerates an arbitrary interaction of the block variables, and Proposition~\ref{prop:bpfinesplit} places the remaining difficulty in the interaction at the finer scales. Neither $\mathrm{BP}$ along the tuning nor the mass gap is proved here. \end{remark} \subsection{The split inequality at strong coupling}\label{ssec:bpsplitstrong} This subsection proves the split inequality for the Wilson measure in the strong-coupling range of Corollary~\ref{cor:bpranges}, on every torus, with constants that tend to those of Theorem~\ref{thm:bprefine} as $\beta\to0$. The two parts of the fine Dirichlet form in Proposition~\ref{prop:bpfinesplit} are bounded separately. In the axial gauge of Lemma~\ref{lem:bprefine}(iii) the conditional law of the fine field given the block variables is a Gibbs measure on the links of $E'_0$ with the Wilson specification, and the single-link coefficients give a Poincar\'e inequality for the fibre form $\Phi$. The information that the fine exterior of a cube carries on the block variables, which vanishes for the refinements, is bounded by the square of the response of the cube to one coarse link, and the contributions of distant cubes are almost orthogonal because correlations in the fibres decay. The split inequality is first derived from three bounds that can be posed for every measure. The subsection closes with the weak-coupling obstructions of Theorems~\ref{thm:bpfixed} and~\ref{thm:bpneutral}: carried through Theorem~\ref{thm:bpup}, they bound the condition for the law of the block variables and put no constraint on the split inequality. Throughout, $c$, $\pi(e)$, $E'_0$, $\mathcal F_{x'}$, $\mathcal H_{x'}$, $T(x')$, $\Phi$, $\lambda_S$ and $\nu_V$ are as in Sec.~\ref{ssec:bprefine}, $\mu'$ is a gauge-invariant probability measure on $G^{E'}$, $\mu^c=c_* \mu'$, $Q=\Pi_{\sigma(c)}$, so that $Qf=F_f\circ c$, and $\mathcal G'_{x'}= \sigma(c_{E\setminus T(x')})\subset\mathcal F_{x'}$. Norms are those of $L^2(\mu')$ or of $L^2(\mu^c)$. The projection $P^{\mu^c}_{T(x')}$ maps $\mathcal A^c$ into itself, since a gauge transformation of $\Lambda_L$ acts on each coarse link separately and preserves $\mu^c$. For a real $g\in \mathcal A^c$ with $P^{\mu^c}_{T(x')}g=0$ put \[ w_{x'}(g)=(I-Q)\,\Pi_{\mathcal F_{x'}}(g\circ c), \] the part of the information of the fine exterior of $V'_{x'}$ on $g$ that is not a function of the block variables. \begin{theorem}[the split inequality from three bounds]\label{thm:bpsplitcond} Let $n,n'\ge2$ with $n'\ge bn$, $L\ge2n+4$ and $bL\ge2n'+4$ even. Let $C_F>0$, $\rho\ge0$ and $R>0$ be such that: \begin{enumerate}[leftmargin=*] \item[\textup{(F)}] $\mu'(\Var(f\mid c))\le C_F\,\Phi(f)$ for every real $f\in\mathcal A'$; \item[\textup{(X)}] $\lVert\Pi_{\mathcal F_{x'}}(g\circ c)\rVert\le\rho\lVert g\rVert$ for every $x'$ and every real $g\in\mathcal A^c$ with $P^{\mu^c}_{T(x')}g=0$; \item[\textup{(D)}] $\lVert\sum_{x'}w_{x'}(g_{x'})\rVert^2\le R\sum_{x'}\lVert w_{x'}(g_{x'})\rVert^2$ for every family of such $g_{x'}$. \end{enumerate} Put $\eta=\rho(RC_F)^{1/2}/n'^2$. If $\rho^2+\eta<1$, then every real $f\in\mathcal A'$ satisfies \[ \mathcal E'_{n'}(f)\ge(1-\eta)\,\Phi(f)+(1-\rho^2-\eta)\Bigl(\frac{bn}{n'} \Bigr)^4\mathcal E^c_n(F_f), \] and $\mathrm S(n',n;C_F/(1-\eta),(n'/(bn))^4/(1-\rho^2-\eta))$ holds for $(\mu',c)$. \end{theorem} \begin{proof} Let $f\in\mathcal A'$ be real, $F=F_f$, $g=F\circ c=Qf$ and $k=f-g$, so that $Qk=0$. Fix $x'$ and write $T=T(x')$, $\mathcal F=\mathcal F_{x'}$, $\mathcal H=\mathcal H_{x'}$. Then $g_{x'}=F-P^{\mu^c}_TF$ is a real element of $\mathcal A^c$ with $P^{\mu^c}_Tg_{x'}=0$, and $g=g_{x'}\circ c+ (P^{\mu^c}_TF)\circ c$, where the second term is $\mathcal G'_{x'}$-measurable, hence $\mathcal F$-measurable. Put $u_{x'}=\Pi_{\mathcal F}(g_{x'}\circ c)$ and $w_{x'}=w_{x'}(g_{x'})=u_{x'}-Qu_{x'}$. The function $g-\Pi_{\mathcal F}g=g_{x'}\circ c-u_{x'}$ is orthogonal to $L^2(\mathcal F)$, which contains $\Pi_{\mathcal F}k$, and $\ip gk=0$; so \[ \begin{split} \lVert f-\Pi_{\mathcal F}f\rVert^2&=\lVert g-\Pi_{\mathcal F}g\rVert^2+\lVert k-\Pi_{\mathcal F}k\rVert^2\\ &\quad-2\ip{\Pi_{\mathcal F}g}k. \end{split} \] By Pythagoras' theorem and (X), $\lVert g-\Pi_{\mathcal F}g\rVert^2=\lVert g_{x'}\rVert^2-\lVert u_{x'}\rVert^2\ge(1-\rho^2)\lVert g_{x'}\rVert^2$. Since $\mathcal F\subset\mathcal H$ and $g$ is $\mathcal H$-measurable, $\lVert k- \Pi_{\mathcal F}k\rVert^2\ge\lVert k-\Pi_{\mathcal H}k\rVert^2=\lVert f- \Pi_{\mathcal H}f\rVert^2=\mu'(\Var(f\mid\mathcal H))$. The functions $(P^{\mu^c}_TF)\circ c$ and $Qu_{x'}$ are $\sigma(c)$-measurable and $Qk=0$, so $\ip{\Pi_{\mathcal F}g}k=\ip{u_{x'}}k=\ip{w_{x'}}k$. Summing over $x'$, with $W=\sum_{x'}w_{x'}$ and $\Sigma_0=\sum_{x'}\lVert g_{x'}\rVert^2$, \[ n'^4\mathcal E'_{n'}(f)\ge(1-\rho^2)\Sigma_0+n'^4\Phi(f)-2\ip Wk . \] By (D) and (X), $\lVert W\rVert^2\le R\sum_{x'}\lVert w_{x'}\rVert^2\le R \sum_{x'}\lVert u_{x'}\rVert^2\le R\rho^2\Sigma_0$, and by (F), $\lVert k \rVert^2=\mu'(\Var(f\mid c))\le C_F\Phi(f)$. Hence \[ 2\lvert\ip Wk\rvert\le2\rho(RC_F)^{1/2}(\Sigma_0\Phi(f))^{1/2}\le\eta\Sigma_0 +\eta n'^4\Phi(f), \] because $2\rho(RC_F)^{1/2}=2\eta n'^2$. So $\mathcal E'_{n'}(f)\ge(1-\eta) \Phi(f)+(1-\rho^2-\eta)n'^{-4}\Sigma_0$. As in the proof of Theorem~\ref{thm:bprefine} (the coarse term), $V_y\subset T(by-w)$ for $w\in[0,b)^4$, so $\lVert F-P^{\mu^c}_{T(by-w)}F\rVert^2\ge\mu^c((F-P^c_yF)^2)$, and $n'^{-4}\Sigma_0\ge(bn/n')^4\mathcal E^c_n(F)$. With (F), $(1-\eta)\Phi(f) \ge(1-\eta)\mu'(\Var(f\mid c))/C_F$. \end{proof} Hypothesis (D) holds with $R=(bL)^4$ by the Cauchy--Schwarz inequality, and (X) with $\rho=1$; the content of (D) is a constant $R$ independent of the torus, an almost orthogonality of the exterior information of different cubes, which (F) and (X), each posed for one cube, do not give. For a Haar refinement, (X) holds with $\rho=0$ by Lemma~\ref{lem:bprefine}(ii), so $w_{x'}=0$ and (D) is empty, and (F) holds with $C_F=1$ by the proof of Theorem~\ref{thm:bprefine}; then $\eta=0$ and the theorem is Theorem~\ref{thm:bprefine}. On a single torus a bounded density with respect to a Haar refinement already gives the split inequality, at every coupling, with constants that depend on the torus. \begin{proposition}[the split inequality on one torus]\label{prop:bptilt} Let $n,n'\ge2$ with $n'\ge bn$, $L\ge2n+4$ and $bL\ge2n'+4$ even, let $\varrho$ be a gauge-invariant probability measure on $G^E$, $\nu=R(\varrho)$, and $\mu'=h\nu$ with $h$ gauge invariant, $00$, the inequality $(x+y)^2\le (1+\epsilon)x^2+(1+\epsilon^{-1})y^2$ in each term, with $\epsilon=2qBv$, gives \[ \mathcal E^c_n(F_\mu)\le(1+\epsilon)\mathcal E^c_n(F_\nu)+(1+\epsilon^{-1})Bv \,\nu\bigl(\Var_\nu(f\mid c)\bigr). \] Since $mq(1+\epsilon^{-1})Bv/(1+2qBv)=m/2$, in both cases \[ \begin{split} \frac{\mu'(\Var_{\mu'}(f\mid c))}{C_A}+\frac{\mathcal E^c_n(F_\mu)}{C_B} &\le m\Bigl(\nu\bigl(\Var_\nu(f\mid c)\bigr)+q\,\mathcal E^c_n(F_\nu)\Bigr) \\&\le m\,\mathcal E^\nu_{n'}(f)\le\mathcal E'_{n'}(f), \end{split} \] by Theorem~\ref{thm:bprefine} for $\nu$. For the second statement, Lemma~\ref{lem:bprefine}(i) with $\varrho= \lambda_E$ gives $\lambda_{E'}=\int\lambda_E(dV)\,\nu_V$ with $c=V$ $\nu_V$-almost surely, so $\mu'=\int\lambda_E(dV)\,D\,\nu_V$, $\mu^c$ has the density $\nu_V(D)$ with respect to $\lambda_E$, and $\mu'=\int\mu^c(dV)\, (D/\nu_V(D))\,\nu_V=h\,R(\mu^c)$. Then $\E_{R(\mu^c)}[h\mid c]=1$, and $\min D\le\nu_V(D)\le\max D$ gives $e^{-\Omega}\le h\le e^\Omega$. The density $Z^{-1}\prod_p\varphi(U_p)$ is a product over the $6(bL)^4$ plaquettes of $\Lambda_{bL}$. \end{proof} The proposition applies to the laws $\mu^{(j)}$ of the block variables in a tower, which have continuous positive densities (Lemma~\ref{lem:bpcoarselaw}), as well as to the Wilson measure at every coupling, so the split inequality holds on each torus at every scale. Its constants grow like $e^{2\Omega}$ and $B=(L/n)^4$, and for the Wilson measure $\Omega$ is proportional to the volume; it gives no constants independent of the torus, which (H2) needs, since Corollary~\ref{cor:bpuptuning} uses infinitely many tori at each scale. Being upper bounds, these constants say nothing about the optimal ones.\footnote{The proof compares the conditional mean $F_\mu$ of $\mu'$ with the conditional mean $F_\nu$ of the refinement; their difference is bounded by the conditional variance, but its coarse form only by $B=(L/n)^4$ times its norm, which is where the dependence on the torus enters $C_B$.} For the Wilson measure the three bounds are proved from the single-link coefficients. A \emph{marking} $l$ assigns to each $e\in E$ a link $l(e)\in \pi(e)$. For a configuration $U$ let $g^l_U$ be the gauge transformation equal to $I$ at the block corners and at the sites on no path and, at an inner point $s$ of a path $\pi(e)$, equal to the product of the links of $\pi(e)$ from its first site to $s$ if $s$ lies before $l(e)$ on the path, and to the inverse of the product of the links of $\pi(e)$ from $s$ to its last site if $s$ lies after $l(e)$. Put $M^l(U)=(g^l_UU)_{E'_0}$, and for $V\in G^E$ and $m\in G^{E'_0}$ let $\Theta^l(V,m)\in G^{E'}$ have the link $V_e$ at $l(e)$, $I$ at the other links of $\pi(e)$, $e\in E$, and $m$ on $E'_0$. For the marking by the last links of the paths these are $g_U$, $M$ and $\Theta$ of Lemma~\ref{lem:bprefine}(iii). \begin{lemma}[the fibres of the Wilson measure]\label{lem:bpwilsonfibre} Let $\mu'$ have the density $D=Z^{-1}\prod_p\varphi(U_p)$ with respect to $\lambda_{E'}$, with $\varphi$ a positive continuous class function, and let $l$ be a marking. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $V\mapsto\nu_V(D)$ is continuous and positive, $\mu^c$ has the density $\nu_V(D)$ with respect to $\lambda_E$, and $\nu^\varphi_V=D\,\nu_V/\nu_V(D)$ is a conditional law of $U$ given $c=V$, defined for every $V$. \item $g^l_UU=\Theta^l(c(U),M^l(U))$. Under $\nu^\varphi_V$ the values of $g^l_U$ at the inner points of the paths are independent Haar variables, independent of $M^l$, and $M^l$ has the law $\varsigma^l_V=\gamma^0_{E'_0} (\cdot\mid\Theta^l(V,\cdot))$, the conditional law of the links of $E'_0$ given those of the paths, at the path links of $\Theta^l(V,\cdot)$; its conditional law of $m_J$ given $m_{E'_0\setminus J}$, $J\subset E'_0$, is $\gamma^0_J(\cdot\mid\Theta^l(V,m))$. Under $\mu'$ the pair $(c,M^l)$ has the law $\mu^c(dV)\,\varsigma^l_V(dm)$. \item Every real $f\in\mathcal A'$ satisfies $f=\tilde f^l(c,M^l)$ $\mu'$-almost everywhere, with $\tilde f^l=\bar f\circ\Theta^l$ and $\bar f$ the gauge average of the proof of Lemma~\ref{lem:bprefine}(iii), and $\E[f\mid c]=\varsigma^l_c(\tilde f^l(c,\cdot))$. \end{enumerate} \end{lemma} \begin{proof} Let $U_1,\dots,U_b$ be the links of a path $\pi(e)$, $s_k$ the site after $U_k$, $A_k=U_1\cdots U_k$, $A_0=I$, and $l(e)=U_{k_e}$. The values of $g^l_U$ at the inner points are $A_k$ for $kk_e$, and $A_{k_e-1}U_{k_e}U_{k_e+1} \cdots U_b=c_e$ for $k=k_e$; this is the first claim of (ii). Under $\nu_V$ the $A_k$, $k\le b-1$, are independent Haar variables (Lemma~\ref{lem:bpfibrelaw}(i)), so are their left translates by the fixed elements $V_e^{-1}$, and the links of $E'_0$ are Haar variables independent of the paths; each $M^l_{e'}$ is a translate of $U_{e'}$ on both sides by values of $g^l_U$. Hence under $\nu_V$ the values of $g^l_U$ at the inner points and $M^l$ are independent with Haar laws, and $U=(g^l_U)^{-1}\Theta^l(V,M^l)$, so $\nu_V$ is the image of $\lambda\otimes\lambda_{E'_0}$ under $(h,m)\mapsto h^{-1} \Theta^l(V,m)$, which depends continuously on $V$. (i) By Lemma~\ref{lem:bprefine}(i), \[ \lambda_{E'}=\int\lambda_E(dV)\,\nu_V,\qquad\mu'=\int\lambda_E(dV)\,D\, \nu_V . \] As $D$ is continuous and positive and $\nu_V$ depends continuously on $V$, $\nu_V(D)$ is continuous and positive; as $c=V$ holds $\nu_V$-almost surely, $\mu'(\psi(c)k)=\int\lambda_E(dV)\nu_V(D)\psi(V) \nu^\varphi_V(k)$ for bounded Borel $\psi$, $k$. (ii) $D$ is gauge invariant, so $D(U)=D(\Theta^l(V,M^l))$ $\nu_V$-almost surely, and under $\nu^\varphi_V$ the values of $g^l_U$ keep their Haar law and remain independent of $M^l$, whose law has density proportional to $D(\Theta^l(V,m))$ with respect to $\lambda_{E'_0}$. Every plaquette contains a link of $E'_0$ (proof of Theorem~\ref{thm:bprefine}), so $D(\Theta^l(V,m))$ is proportional to $\prod_{p\cap E'_0\ne\emptyset}\varphi(\Theta^l(V,m)_p)$, the density of $\gamma^0_{E'_0}(\cdot\mid\Theta^l(V,m))$; its conditional laws are those of the specification. The law of $(c,M^l)$ follows from (i). (iii) $f=\bar f$ almost everywhere and $\bar f$ is invariant under every gauge transformation, so $\bar f(U)=\bar f(g^l_UU)=\bar f(\Theta^l(c,M^l))$ for every $U$; the conditional expectation follows from (i) and (ii). \end{proof} For sets $A,B$ of links of a torus let $d(A,B)$ be the least $k$ for which there are links $e_0\in A$, $e_1,\dots,e_k$ with $e_k\in B$ and $e_{i-1}$, $e_i$ partners, and $d(j,A)=d(\{j\},A)$. The coefficients of Sec.~\ref{ssec:bpstrong} are those of the measure at hand. \begin{lemma}[decay of the response of a block]\label{lem:bpdecay} Let $\mu$ be as in Sec.~\ref{ssec:averaged} on $\Lambda_L$, $L\ge3$, with $C^0_{ee'}\le\mathfrak c$ for all partners and $\xi=18\mathfrak c<1$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Let $V$ be finite, $A\subset V$, $j\notin V$, and $U$, $U'$ differ only at $j$. The laws of $\sigma_A$ under $\gamma^0_V(\cdot\mid U)$ and $\gamma^0_V(\cdot\mid U')$ are at Kantorovich distance at most $\xi^{d(j,A)} \delta(U_j,U'_j)/(1-\xi)$ for $d_A$. \item Let $V_0$ be finite, $\gamma=\gamma^0_{V_0}(\cdot\mid U)$ for some $U$, $A,B\subset V_0$ with $d(A,B)\ge1$, $\phi\in L^2(\gamma)$ a function of $\sigma_A$ with $\gamma(\phi)=0$ and $\psi\in L^2(\gamma)$ a function of $\sigma_B$. Then \[ \lvert\gamma(\phi\psi)\rvert\le\frac{\xi^{d(A,B)}}{1-\xi}\lVert\phi \rVert_{L^2(\gamma)}\lVert\psi\rVert_{L^2(\gamma)}. \] \end{enumerate} \end{lemma} \begin{proof} (i) As in the proof of Lemma~\ref{lem:bpblock}, a minimising coupling of the two laws for $\sum_{e\in V}\E\delta(\sigma_e,\sigma'_e)$ exists, and $\varrho (e)=\E\delta(\sigma_e,\sigma'_e)$ satisfies $\varrho\le\mathbf C\varrho+ \delta(U_j,U'_j)\mathbf c$ entrywise, with $\mathbf C=(C^0_{ek})_{e,k\in V}$ and $\mathbf c=(C^0_{ej})_{e\in V}$. Each column of $\mathbf C$ and the vector $\mathbf c$ have sums at most $18\mathfrak c=\xi$, since a link has eighteen partners; iterating, and since $\sum_e(\mathbf C^N\varrho)(e)\le \xi^N\sum_e\varrho(e)\to0$, $\varrho\le\delta(U_j,U'_j)\sum_{m\ge0}\mathbf C^m \mathbf c$. The entry of $\mathbf C^m\mathbf c$ at $e$ is a sum over chains $e,k_1,\dots,k_m,j$ of consecutive partners, so it vanishes unless $m+1\ge d(j,e)$, and $\sum_{e\in A}(\mathbf C^m\mathbf c)(e)\le\xi^{m+1}$. Hence $\sum_{e\in A}\varrho(e)\le\delta(U_j,U'_j)\sum_{m\ge d(j,A)-1}\xi^{m+1}= \delta(U_j,U'_j)\xi^{d(j,A)}/(1-\xi)$, and the coupling restricted to $A$ bounds the distance. (ii) Let $\theta=(\xi^d/(1-\xi))^2$, $d=d(A,B)$, and let $\Pi_A$, $\Pi_B$ be the conditional expectations under $\gamma$ given $\sigma_A$, $\sigma_B$, with the versions given by the specification: given $\sigma_B$, the links of $V_0\setminus B$ have the law $\gamma^0_{V_0\setminus B}(\cdot\mid\sigma_B, U)$. If $h$ is a function of $\sigma_A$ with Lipschitz constant $\Lambda$ for $d_A$ and $\sigma_B$, $\sigma'_B$ differ at one link $j\in B$, then (i) with $V=V_0\setminus B$ and $d(j,A)\ge d$ gives $\lvert\Pi_Bh(\sigma_B)-\Pi_Bh( \sigma'_B)\rvert\le\Lambda\xi^d\delta/(1-\xi)$; so $\Pi_B$ maps such functions to functions of $\sigma_B$ with Lipschitz constant $\Lambda\xi^d/(1-\xi)$ for $d_B$, and likewise $\Pi_A$ in the other direction. The operator $\mathcal R=\Pi_A\Pi_B$ on the functions of $\sigma_A$ in $L^2(\gamma)$ equals $\Pi_A \Pi_B\Pi_A$ there, so it is self-adjoint with $0\le\mathcal R\le I$ and $\mathcal R1=1$, and it multiplies Lipschitz constants by at most $\theta$. As in the proof of Proposition~\ref{prop:avgsup}(i), $\Var_\gamma(\mathcal R^th)\le\frac12\theta^{2t}\Lambda^2D^2\lvert A\rvert^2$, the spectral measure of $h-\gamma(h)$ lives on $[0,\theta]$, and by density of the Lipschitz functions $\ip\phi{\mathcal R\phi}\le\theta\lVert\phi\rVert^2$ for every $\phi$ as in the statement. Then $\gamma(\phi\psi)=\ip{\Pi_B\phi}\psi$ and $\lVert\Pi_B\phi\rVert^2=\ip\phi{\mathcal R\phi}\le\theta\lVert\phi\rVert^2$. \end{proof} For $A=V$, (i) is Lemma~\ref{lem:bpblock} with the bound $\sum_{e\in V}C^0_{ej} \le18\mathfrak c$. In the rest of this subsection $\mu'$ is as in Lemma~\ref{lem:bpwilsonfibre}, its single-link coefficients on $\Lambda_{bL}$ satisfy $C^0_{ee'}\le\mathfrak c$ for all partners, $\xi=18\mathfrak c<1$, $q=\xi/(1-\xi)$, and for $n'\ge2$ \[ \alpha_{n'}=\frac{\mathfrak c\,(18n'-3)}{(1-18\mathfrak c)\,n'^2}, \] the number of Theorem~\ref{thm:bpstrong}(i). \begin{theorem}[the fibre inequality at strong coupling]\label{thm:bpfibrepoinc} Let $n'\ge2$, $bL\ge2n'+4$ and $\alpha_{n'}<1$. Then every real $f\in \mathcal A'$ satisfies $\Phi(f)\ge(1-\alpha_{n'})\,\mu'(\Var(f\mid c))$. \end{theorem} \begin{proof} Use the marking by the last links and write $\varsigma_V$, $\tilde f$ for $\varsigma^l_V$, $\tilde f^l$ and $\tilde f_V=\tilde f(V,\cdot)$. Fix $x'$, let $J=E'_0\cap V'_{x'}$ and $\mathcal G$ the $\sigma$-algebra of the links outside $J$. It contains $\mathcal F_{x'}$ and $\sigma(c)$, so $\mu'(\Var(f \mid\mathcal H_{x'}))\ge\mu'(\Var(f\mid\mathcal G))$. The links outside $J$ determine $c$, the values of $g_U$ at the inner points, which are partial products of path links, and $M_{E'_0\setminus J}$, and are determined by them; by Lemma~\ref{lem:bpwilsonfibre}(ii) the conditional law of $M_J$ given $\mathcal G$ is $\gamma^0_J(\cdot\mid\Theta(c,M))$. Let $\Pi^V_J$ be the conditional expectation under $\varsigma_V$ given $m_{E'_0\setminus J}$, $\Pi^V_Jh(m)=\int h(\sigma,m_{E'_0\setminus J})\,\gamma^0_J(d\sigma\mid \Theta(V,m))$. By Lemma~\ref{lem:bpwilsonfibre}(ii),(iii), $\E[f\mid \mathcal G]=(\Pi^c_J\tilde f_c)(M)$, \[ \begin{split} \mu'\bigl(\Var(f\mid\mathcal G)\bigr)&=\int\mu^c(dV)\,\varsigma_V\bigl(( \tilde f_V-\Pi^V_J\tilde f_V)^2\bigr),\\ \mu'\bigl(\Var(f\mid c)\bigr)&=\int\mu^c(dV)\Var_{\varsigma_V}(\tilde f_V). \end{split} \] It suffices to prove, for every $V$ and every real $h\in L^2(\varsigma_V)$, \[ \sum_{x'}\varsigma_V\bigl((h-\Pi^V_{J(x')}h)^2\bigr)\ge(1-\alpha_{n'})n'^4 \Var_{\varsigma_V}(h), \] with $J(x')=E'_0\cap V'_{x'}$. Let $N=(bL)^4$, $\mathcal K=N^{-1}\sum_{x'} \Pi^V_{J(x')}$ and $d(m,m')=\sum_{e\in E'_0}\delta(m_e,m'_e)$, and let $m$, $m'$ differ only at $i\in E'_0$. Couple $\mathcal K(m,\cdot)$ and $\mathcal K(m',\cdot)$ by choosing the same $x'$. If $i\in V'_{x'}$, then $i\in J(x')$, $\Theta(V,m)$ and $\Theta(V,m')$ agree outside $J(x')$, and the two laws coincide; this happens for $n'^4$ points $x'$. Otherwise $\Theta(V,m)$ and $\Theta(V,m')$ differ only at $i$, an optimal coupling costs at most $C^0_{J(x'),i}\delta(m_i,m'_i)$, and the link $i$ adds $\delta(m_i,m'_i)$. By Lemma~\ref{lem:bpblock}, $C^0_{J(x'),i}$ is at most $\mathfrak c/(1-18 \mathfrak c)$ times the number of partners of $i$ in $V'_{x'}$, and by Lemma~\ref{lem:bpcount}(i) on $\Lambda_{bL}$ the sum of these bounds over the $x'$ with $i\notin V'_{x'}$ is at most $\mathfrak c(18n'^3-3n'^2)/(1-18 \mathfrak c)=\alpha_{n'}n'^4$. So $W_1^d(\mathcal K(m,\cdot),\mathcal K(m', \cdot))\le(1-\kappa)\delta(m_i,m'_i)$ with $\kappa=(1-\alpha_{n'})n'^4/N$, and the argument of the proof of Proposition~\ref{prop:avgsup}(i) for the self-adjoint operator $0\le\mathcal K\le I$ on $L^2(\varsigma_V)$ gives $\ip h{(I-\mathcal K)h}\ge\kappa\Var_{\varsigma_V}(h)$, which is the claim. \end{proof} \begin{theorem}[the exterior information at strong coupling]\label{thm:bpexterior} The information of the fine exterior of a cube on the block variables satisfies the following. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For every $x'$ and every real $g\in L^2(\mu^c)$ with $P^{\mu^c}_{T(x')}g=0$, $\lVert\Pi_{\mathcal F_{x'}}(g\circ c)\rVert\le q \lVert g\rVert$. \item Let $n,n'\ge2$ with $n'\ge bn$, $L\ge2n+4$ and $bL\ge2n'+4$ even. Every real $F\in L^2(\mu^c)$ satisfies \[ \begin{split} (1-q^2)\,\Sigma(F)\le\mathcal E'_{n'}(F\circ c)\le\Sigma(F),\\ \Sigma(F)=\frac1{n'^4}\sum_{x'}\mu^c\bigl((F-P^{\mu^c}_{T(x')}F)^2\bigr), \end{split} \] and $\Sigma(F)\ge(bn/n')^4\mathcal E^c_n(F)$. \end{enumerate} \end{theorem} \begin{proof} (i) Fix $x'$ and write $T=T(x')$, $V'=V'_{x'}$, $\mathcal F=\mathcal F_{x'}$ and $P=P^{\mu^c}_T$. For bounded Borel $g$ on $G^E$ put \[ \mathcal Mg(V)=\int\nu^\varphi_V(dU)\int\gamma^0_{V'}(d\sigma\mid U)\,g\bigl( c(\sigma,U_{E'\setminus V'})\bigr), \] where $(\sigma,U_{E'\setminus V'})$ has the links $\sigma$ on $V'$ and those of $U$ outside. By Lemma~\ref{lem:bpwilsonfibre}(i), $(\mathcal Mg)\circ c=Q \Pi_{\mathcal F}(g\circ c)$, so $\mathcal M$ extends to a self-adjoint operator $0\le\mathcal M\le I$ on $L^2(\mu^c)$ with $\ip g{\mathcal Mg}= \lVert\Pi_{\mathcal F}(g\circ c)\rVert^2$. It fixes the functions of $c_{E\setminus T}$, which composed with $c$ are $\mathcal F$-measurable, so $\mathcal MP=P$ and, taking adjoints, $P\mathcal M=P$. For $e\notin T$ the links of $\pi(e)$ lie outside $V'$, so $\mathcal M(V,\cdot)$ is carried by $\{\tilde V:\tilde V_{E\setminus T}=V_{E\setminus T}\}$. Let $V$, $\tilde V$ differ only at $e\in T$, and choose a marking $l$ with $l(e)\in\pi(e)\cap V'$. Let $h$ have the Haar law on the inner points and $(M,\tilde M)$, independent of $h$, be an optimal coupling of $\varsigma^l_V$ and $\varsigma^l_{\tilde V}$ for $d_{E'_0}$, and put $U=h^{-1} \Theta^l(V,M)$, $\tilde U=h^{-1}\Theta^l(\tilde V,\tilde M)$, which have the laws $\nu^\varphi_V$, $\nu^\varphi_{\tilde V}$ by Lemma~\ref{lem:bpwilsonfibre}(ii). The two laws $\varsigma^l$ are $\gamma^0_{E'_0}$ at configurations that differ outside $E'_0$ only at $l(e)$, with the values $V_e$, $\tilde V_e$; by Lemma~\ref{lem:bpblock} and the eighteen partners of $l(e)$, $\E d_{E'_0}(M,\tilde M)\le q\,\delta(V_e, \tilde V_e)$. The path links of $U$ and $\tilde U$ agree except at $l(e)\in V'$, and $\delta(U_{e'},\tilde U_{e'})=\delta(M_{e'},\tilde M_{e'})$ for $e'\in E'_0$ by bi-invariance. Given $U$, $\tilde U$, couple $\gamma^0_{V'}( \cdot\mid U)$ and $\gamma^0_{V'}(\cdot\mid\tilde U)$ optimally for $d_{V'}$, measurably~\cite[Cor.~5.22]{Vil09}; changing the links of $E'_0\setminus V'$ one at a time, Lemma~\ref{lem:bpblock} gives an expected distance at most $q\sum_{e'\in E'_0\setminus V'}\delta(U_{e'},\tilde U_{e'})$. The coarse values $c(\sigma,U_{E'\setminus V'})$ and $c(\tilde\sigma,\tilde U_{E' \setminus V'})$ agree outside $T$, and at $e''\in T$ they are products along $\pi(e'')$ of links that agree outside $V'$; as $\delta(XY,X'Y')\le\delta(X, X')+\delta(Y,Y')$ and the paths are disjoint, their distance for $d_T$ is at most $d_{V'}(\sigma,\tilde\sigma)$. Hence $W_1^{d_T}(\mathcal M(V,\cdot), \mathcal M(\tilde V,\cdot))\le q^2\delta(V_e,\tilde V_e)$, and, as $\mathcal M(V,\cdot)$ does not depend on the marking, $W_1^{d_T}\le q^2d_T(V,\tilde V)$ whenever $V_{E\setminus T}=\tilde V_{E\setminus T}$. Let $g$ be bounded with $\lvert g(V)-g(\tilde V)\rvert\le\Lambda d_T(V,\tilde V)$ whenever $V_{E\setminus T}=\tilde V_{E\setminus T}$. Then $\mathcal M^tg$ has this property with $\Lambda q^{2t}$, its oscillation on each set $\{V_{E\setminus T}=\text{const}\}$ is at most $\Lambda q^{2t}D\lvert T\rvert$, and since $P\mathcal M=\mathcal MP=P$, \[ \lVert\mathcal M^t(g-Pg)\rVert^2=\lVert\mathcal M^tg-P\mathcal M^tg\rVert^2 \le(\Lambda q^{2t}D\lvert T\rvert)^2 . \] $\mathcal M$ maps $L^2(\mu^c)\ominus L^2(\mathcal G'_{x'})$ into itself, so the spectral measure of $g-Pg$ for $\mathcal M$ lives on $[0,q^2]$. The functions $g-Pg$ with $g$ Lipschitz for $d_E$ are dense in $L^2(\mu^c) \ominus L^2(\mathcal G'_{x'})$, by the Stone--Weierstrass theorem, so $\ip g{\mathcal Mg}\le q^2\lVert g\rVert^2$ whenever $Pg=0$. (ii) The function $(P^{\mu^c}_{T(x')}F)\circ c$ is $\mathcal F_{x'}$-measurable, so the subtracted term of Proposition~\ref{prop:bpfinesplit}(ii) is $\lVert\Pi_{\mathcal F_{x'}}((F- P^{\mu^c}_{T(x')}F)\circ c)\rVert^2$, at most $q^2\lVert F-P^{\mu^c}_{T(x')}F \rVert^2$ by (i). The last inequality is the coarse term of the proof of Theorem~\ref{thm:bprefine}. \end{proof} So on functions of the block variables the fine form lies between $1-q^2$ times that of the refinement of $\mu^c$ and that form itself, and (B) holds there with $C_B=(n'/(bn))^4/(1-q^2)$. \begin{theorem}[the split inequality at strong coupling]\label{thm:bpsplitstrong} Let $n,n'\ge2$ with $n'\ge bn$, $L\ge2n+4$ and $bL\ge2n'+4$ even, and $\alpha_{n'}<1$. Put \[ \begin{split} R_{n'}&=(2n'+3)^4+\frac{10(n'+2)^3\,\xi\,(8-5\xi+4\xi^2-\xi^3)}{(1-\xi)^5}, \\ \eta&=q\Bigl(\frac{R_{n'}}{(1-\alpha_{n'})n'^4}\Bigr)^{1/2}. \end{split} \] If $q^2+\eta<1$, every real $f\in\mathcal A'$ satisfies \[ \begin{split} \mathcal E'_{n'}(f)\ge{}&(1-\eta)(1-\alpha_{n'})\,\mu'\bigl(\Var(f\mid c) \bigr)\\ &+(1-q^2-\eta)\Bigl(\frac{bn}{n'}\Bigr)^4\mathcal E^c_n(F_f), \end{split} \] and $\mathrm S(n',n;C_A,C_B)$ holds for $(\mu',c)$ with \[ C_A=\frac1{(1-\eta)(1-\alpha_{n'})},\qquad C_B=\frac{(n'/(bn))^4}{1-q^2- \eta}. \] \end{theorem} \begin{proof} By Theorems~\ref{thm:bpfibrepoinc} and~\ref{thm:bpexterior}(i), (F) holds with $C_F=1/(1-\alpha_{n'})$ and (X) with $\rho=q$. Then $\rho(RC_F)^{1/2}/ n'^2=\eta$ for $R=R_{n'}$, and Theorem~\ref{thm:bpsplitcond} gives the claim once (D) is proved with $R=R_{n'}$. Use the marking by the last links, $\varsigma_V=\varsigma^l_V$. Let $g_{x'}\in \mathcal A^c$ be real with $P^{\mu^c}_{T(x')}g_{x'}=0$. Replacing $g_{x'}$ by its average over the gauge group of $\Lambda_L$, as in the proof of Lemma~\ref{lem:bprefine}(iii), changes it on a $\mu^c$-null set only, so we may take $g_{x'}(\hat gV)=g_{x'}(V)$ for all $\hat g$ and $V$. Then \[ \phi_{x'}(U)=\int\gamma^0_{V'_{x'}}(d\sigma\mid U)\,g_{x'}\bigl(c(\sigma, U_{E'\setminus V'_{x'}})\bigr) \] is a version of $\Pi_{\mathcal F_{x'}}(g_{x'}\circ c)$. For a gauge transformation $g$ of $\Lambda_{bL}$, $\gamma^0_{V'_{x'}}(\cdot\mid gU)$ is the image of $\gamma^0_{V'_{x'}}(\cdot\mid U)$ under $g$, and $c\circ g=\hat g\circ c$ by (B2); so $\phi_{x'}$ is gauge invariant and $\phi_{x'}(U)= \phi_{x'}(\Theta(c(U),M(U)))$. The law $\gamma^0_{V'_{x'}}(\cdot\mid\Theta(V, m))$ depends on $m$ only through $m_{B(x')}$, with $B(x')=E'_0\cap\partial V'_{x'}$, and the coarse values $c(\sigma,\Theta(V,m)_{E'\setminus V'_{x'}})$ are products of links of $\sigma$ and of path links of $\Theta(V,m)$; so $\phi_{x'}(\Theta(V,m))=\tilde\phi_{x'}(V,m_{B(x')})$. By Lemma~\ref{lem:bpwilsonfibre}(ii), $w_{x'}(g_{x'})=\tilde w_{x'}(c,M)$ with $\tilde w_{x'}(V,m)=\tilde\phi_{x'}(V,m_{B(x')})-\varsigma_V(\tilde\phi_{x'}(V, \cdot))$, a function of $m_{B(x')}$ of mean zero under $\varsigma_V$, and \[ \ip{w_{x'}}{w_{x''}}=\int\mu^c(dV)\,\varsigma_V\bigl(\tilde w_{x'}(V,\cdot) \tilde w_{x''}(V,\cdot)\bigr). \] Let $s=\lvert x'-x''\rvert_\infty$ and $r_{x'x''}=1$ for $s\le n'+1$, $r_{x'x''}=\xi^{s-n'-1}/(1-\xi)$ for $s\ge n'+2$. The base points of the links of $\partial V'_{x'}$ lie in $x'+[-1,n']^4$. For $s\ge n'+2$ choose a coordinate $i$ in which the representative $t\in(-bL/2,bL/2]$ of $x''_i-x'_i$ has $\lvert t\rvert=s$; the $i$-th coordinates of base points of $B(x')$ and $B(x'')$ differ by $t+a$ with $\lvert a\rvert\le n'+1$, at distance at least $\min(s,bL-s)-n'-1=s-n'-1$ from $bL\Z$. The base points of partners differ by at most one in $\lvert\cdot\rvert_\infty$ (proof of Lemma~\ref{lem:bpcount}), so $d(B(x'),B(x''))\ge s-n'-1\ge1$. As $\varsigma_V=\gamma^0_{E'_0}(\cdot\mid \Theta(V,\cdot))$, Lemma~\ref{lem:bpdecay}(ii) with $V_0=E'_0$, or the Cauchy--Schwarz inequality for $s\le n'+1$, bounds $\lvert\varsigma_V(\tilde w_{x'}\tilde w_{x''})\rvert$ by $r_{x'x''}$ times the product of the norms in $L^2(\varsigma_V)$, and the Cauchy--Schwarz inequality in $V$ gives $\lvert \ip{w_{x'}}{w_{x''}}\rvert\le r_{x'x''}\lVert w_{x'}\rVert\lVert w_{x''} \rVert$. Since $r$ is symmetric, \[ \Bigl\lVert\sum_{x'}w_{x'}\Bigr\rVert^2\le\sum_{x'}\lVert w_{x'}\rVert^2 \sup_{x'}\sum_{x''}r_{x'x''} . \] At most $(2n'+3)^4$ points have $s\le n'+1$ and at most $(2s+1)^4-(2s-1)^4= 64s^3+16s\le80s^3$ have $\lvert x''-x'\rvert_\infty=s$. With $n'+1+t\le(n'+2) (1+t)/2$ for $t\ge1$ and $\sum_{t\ge0}(t+1)^3\xi^t=(1+4\xi+\xi^2)/(1-\xi)^4$, \[ \begin{split} \sum_{s\ge n'+2}\frac{80s^3\xi^{s-n'-1}}{1-\xi}&\le\frac{10(n'+2)^3}{1-\xi} \sum_{t\ge1}(t+1)^3\xi^t\\ &=\frac{10(n'+2)^3\,\xi\,(8-5\xi+4\xi^2-\xi^3)}{(1-\xi)^5}, \end{split} \] so $\sup_{x'}\sum_{x''}r_{x'x''}\le R_{n'}$, which is (D). \end{proof} The numbers $\alpha_{n'}$ and $R_{n'}/n'^4$ decrease in $n'$, so $\eta$ decreases in $n'$, and $R_{n'}/n'^4\to16$, so $\eta\to4q$ as $n'\to\infty$. As $\beta\to0$, $\eta=O(\beta)$ uniformly in $L$ and in $n'\ge2$, since $\eta$ is largest at $n'=2$; the constants are $C_A=1+O(\beta)$ and $C_B=(n'/(bn))^4(1+O(\beta))$, those of Theorem~\ref{thm:bprefine} up to factors $1+O(\beta)$. \begin{corollary}[explicit ranges]\label{cor:bpsplitranges} Theorem~\ref{thm:bpsplitstrong} applies, whenever $\alpha_{n'}<1$ and $q^2+ \eta<1$, in the following cases. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $G=\SU(2)$, $\lvert\beta_W\rvert<\frac29$, with $\xi=\frac92\lvert \beta_W\rvert$. For $\lvert\beta_W\rvert\le\frac1{100}$ and every $n'\ge8$, $\eta<0.272$, $C_A<1.39$ and $C_B<1.39\,(n'/(bn))^4$. \item $G=\SU(3)$, $\beta_W\ge0$ with $\zeta(\beta_W)<1$, with $\xi=\zeta( \beta_W)$ as in Corollary~\ref{cor:bpranges}(ii). \item $G\subset\U(N)$ closed and connected with semisimple Lie algebra, $24N\lvert\beta\rvert<\kappa_G$, with $\xi=18N\lvert\beta\rvert/(\kappa_G- 6N\lvert\beta\rvert)$. \end{enumerate} \end{corollary} \begin{proof} The bounds $C^0_{ee'}\le\mathfrak c$ with $18\mathfrak c=\xi$ are those of the proof of Corollary~\ref{cor:bpranges}: $\mathfrak c=\lvert\beta_W\rvert/4$ for $\SU(2)$, $18\mathfrak c=\zeta(\beta_W)$ for $\SU(3)$, and $\mathfrak c=N \lvert\beta\rvert/(\kappa_G-6N\lvert\beta\rvert)$ from Lemma~\ref{lem:bponelink}. The numbers $q$, $\alpha_{n'}$, $R_{n'}$ and $\eta$ increase with $\xi$, so in (i) it suffices to take $\xi=\frac9{200}$ and $n'= 8$. Then $q=\frac9{191}<0.04713$, $\mathfrak c=\frac1{400}$ and $\alpha_8= \frac{141}{24448}<0.0058$. As $\xi(8-5\xi+4\xi^2-\xi^3)\le8\xi=0.36$ and $(1-\xi)^5\ge1-5\xi=0.775$, $R_8\le19^4+10^4\cdot0.36/0.775<130321+4646= 134967$, and $R_8/((1-\alpha_8)8^4)<134967/(0.9942\cdot4096)<33.15$. So $\eta<0.04713\cdot5.758<0.2714$, $q^2<0.0023$, $1-q^2-\eta>0.7263$ and $(1-\eta)(1-\alpha_8)>0.7286\cdot0.9942>0.724$, which give the bounds. \end{proof} \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_bpsplitstrong.png} \caption{The split inequality at strong coupling. (a)~A coarse link $e$ enters the fine field through its path $\pi(e)$; in a gauge whose marked link $l(e)$ lies in the cube $V'_{x'}$, a change of $V_e$ moves the fibre law by at most $q$, and the cube, resampled given its exterior, feels that move through $\partial V'_{x'}$ by at most $q$ again, so the exterior information is at most $q^2$ (Theorem~\ref{thm:bpexterior}); two cubes at distance $s$ see each other through correlations of the fibre law that decay like $\xi^{s-n'-1}/(1-\xi)$ (Lemma~\ref{lem:bpdecay}). (b)~The constants of Theorem~\ref{thm:bpsplitstrong} for $\SU(2)$ at $n'=8$ and in the limit $n'\to\infty$, where $\eta\to4q$; both tend to the value $1$ of the Haar refinements as $\beta_W\to0$, and at $\lvert\beta_W\rvert=\frac1{100}$ they are below $1.39$ (Corollary~\ref{cor:bpsplitranges}).}\label{fig:bpsplitstrong} \end{figure*} Figure~\ref{fig:bpsplitstrong} shows the mechanism and the constants. At strong coupling $\mathrm{BP}$ for the Wilson measure is already given by Corollary~\ref{cor:bpranges}; Theorem~\ref{thm:bpsplitstrong} shows that the split mechanism of Theorem~\ref{thm:bprefine} survives a nonzero interaction at the finest scale, with constants close to the best ones. It is (H2) at the top step of a tower whose finest measure is a Wilson measure in the range above. The measures $\mu^{(j)}$, $j0$, by (i) alone. \item Let $N=2$, $\beta\ge2$, $bL\ge8$ and $T=\Tr P_0$. If $\mathrm{BP}_{\mu^c} (n,\varpi_c)$ holds, then \[ \mu'\bigl(T^2(4-T^2)\bigr)^2\le\frac{9\cdot2^{19}\,b^{12}L^{11}n\,(\log\beta+ 6.11)^3}{\varpi_c\beta}. \] \end{enumerate} \end{proposition} \begin{proof} (i) The coarse plaquette $c_P$ is the holonomy of the boundary of a square of $b^2$ fine plaquettes, so by Lemma~\ref{lem:bpstokes} and the Cauchy--Schwarz inequality $\lVert c_P-I\rVert_{\rm HS}^2\le b^2\sum_p\lVert U_p-I\rVert_{\rm HS}^2$ over these plaquettes, and the symmetries of $\Lambda_{bL}$ act transitively on the plaquettes. The fine twist that multiplies the links $(x,0)$ with $x_0=0$ by $z^{c_1}$ and the links $(x,1)$ with $x_1=0$ by $z^{d_1}$ preserves $\mu'$; on a path $\pi(y,0)$ it changes only the first link, and only when $y_0=0$, and likewise in direction $1$, so it induces the coarse twist $Z_{c_1,d_1}$ and $\mu^c$ is invariant under the coarse twists. $\mu^c$ has a continuous positive density (Lemma~\ref{lem:bpwilsonfibre}(i)), so $\Var_{\mu^c}(\ReTr c_P)>0$. Proposition~\ref{prop:bpcoarsenec}(ii) gives the bound, and for $N=2$ Lemma~\ref{lem:bpflat} gives $\mu'(\lVert U_p-I \rVert_{\rm HS}^2)=2\mu'(2-\Tr U_p)\le2(\log\beta+6.11)/\beta$, with $2\cdot 19200=38400$. (ii) $\varpi'\le\varpi_c/C_B\le\varpi_c\le b^4X_n(\beta)\le X_{n'}(\beta)$, since $bn\le n'$. (iii) The coarse Polyakov loop $P^c_0$ through $0$ in direction $0$ is the product of the $bL$ fine links $(te_0,0)$, so $P^c_0\circ c=P_0$, and $h=(\Tr P^c_0)^2/4$ is a real element of $\mathcal A^c$ with $h\circ c=T^2/4$. The proof of Theorem~\ref{thm:bpneutral}(i) on $\Lambda_L$, with the coarse plaquettes and their largest mean $\varepsilon^c$ of $\lVert c_P-I\rVert_{\rm HS}^2$, gives $\mathcal E^c_n(h)\le4L^3n\varepsilon^c$, and $\varepsilon^c\le2b^4(\log\beta+ 6.11)/\beta$ by (i). So $\Var_{\mu'}(h\circ c)=\Var_{\mu^c}(h)\le8b^4L^3n(\log \beta+6.11)/(\varpi_c\beta)$. By~\eqref{eq:bpanticonc} and Lemma~\ref{lem:bpflow}(iii) on $\Lambda_{bL}$, $\mu'(T^2(4-T^2))^2\le16\Var_{ \mu'}(h\circ c)\cdot36864(bL)^8(\log\beta+6.11)^2$, and $16\cdot8\cdot36864=9 \cdot2^{19}$. \end{proof} Through Theorem~\ref{thm:bpup}(i), Theorems~\ref{thm:bpfixed}(ii) and~\ref{thm:bpneutral}(i) for the Wilson measure $\mu'$ on $\Lambda_{bL}$ bound $\varpi'=\min(1/C_A,\varpi_c/C_B)$ from above. By (ii) the first bound follows from the bound (i) on $\varpi_c$ alone when $C_B\ge1$; and since $(bL)^{11}n'C_B/\varpi_c\ge b^{12}L^{11}n/\varpi_c$ for $C_B\ge1$, the second, $\mu'(T^2(4-T^2))^2\le9\cdot2^{19}(bL)^{11}n'(\log\beta+6.11)^3/(\varpi'\beta)$, follows from (iii). So these obstructions give no lower bound on $C_A$ and none on $C_B\ge1$: they bound the condition for the law of the block variables on the coarse torus, the analogue of (H1) at that scale. On a tower of straight-path maps over $\Lambda_{2n_0+4}$, whose composite is the straight-path map with blocks of side $b^K$, (i) with $b^K$ in place of $b$, equivalently Proposition~\ref{prop:bpcoarsenec}(iii) with Lemma~\ref{lem:bpstokes}, bounds the constant of (H1) by $X_{n_K}(\beta)$ for $n_K=b^Kn_0$ and $N=2$; along the continuum tuning $n_k^4$ grows faster than $\beta_k$ (Remark~\ref{rem:bpstatus}), and this bound exceeds one for large $k$. Corollary~\ref{cor:bpuptuning} uses an infinite set of tori for each $k$ and does not need the smallest one. \begin{remark}[what is proved about the split inequality for Wilson measures]\label{rem:bpsplitstrong} Proved: for every gauge-invariant measure, the split inequality follows from a fibre Poincar\'e inequality (F), a bound (X) on the information of the fine exterior of one cube on the block variables, and the almost orthogonality (D) of that information for different cubes, with explicit constants; the Haar refinements are the case $\rho=0$ (Theorem~\ref{thm:bpsplitcond}); on one torus, every measure with a continuous positive density, in particular the Wilson measure at every coupling and the laws of the block variables of a tower, satisfies the split inequality with constants of order $e^{2\Omega}$ and $(L/n)^4$ (Proposition~\ref{prop:bptilt}). For the Wilson measure with single-link coefficients at most $\mathfrak c$, $18\mathfrak c<1$, on every torus: the fibre form satisfies $\Phi\ge(1- \alpha_{n'})\mu'(\Var(f\mid c))$ (Theorem~\ref{thm:bpfibrepoinc}); the exterior information is at most $q^2$ times the coarse form of the refinement, so the fine form of a function of the block variables lies between $1-q^2$ times that form and the form itself (Theorem~\ref{thm:bpexterior}); correlations in the fibres decay geometrically (Lemma~\ref{lem:bpdecay}); and $\mathrm S(n',n;1/((1-\eta)(1- \alpha_{n'})),(n'/(bn))^4/(1-q^2-\eta))$ holds whenever $q^2+\eta<1$, with constants within factors $1+O(\beta)$ of those of the refinements (Theorem~\ref{thm:bpsplitstrong}), for $\SU(2)$, $\SU(3)$ and every compact connected group with semisimple Lie algebra at small $\beta$, for $\SU(2)$ at $\lvert\beta_W\rvert\le\frac1{100}$ and $n'\ge8$ with $C_A<1.39$ and $C_B< 1.39(n'/(bn))^4$ (Corollary~\ref{cor:bpsplitranges}). The obstructions of Theorems~\ref{thm:bpfixed} and~\ref{thm:bpneutral} on small tori, carried through Theorem~\ref{thm:bpup}, follow from bounds on the condition for the law of the block variables and put no constraint on $C_A$ or on $C_B\ge1$ (Proposition~\ref{prop:bpsmalltori}). Not proved: the split inequality with constants independent of the torus for the laws $\mu^{(j)}$ of the block variables below the finest scale of a tower, which are not Wilson measures and to which the proofs of Theorems~\ref{thm:bpfibrepoinc}--\ref{thm:bpsplitstrong} do not apply; the split inequality with constants independent of the torus for the Wilson measure outside the strong-coupling range, in particular along the tuning, where the range contains finitely many $k$, for $\U(1)$ at $\beta>0$ and for $\SU(2)$ on calm fibres, where none of (F), (X), (D) is proved; the optimal constants at strong coupling; whether the split inequality holds or fails at weak coupling, on small tori or on large ones; (H1) along the tuning. Neither $\mathrm{BP}$ along the tuning nor the mass gap is proved here. \end{remark} \subsection{The condition at the coarsest scale}\label{ssec:bpcoarse} This subsection treats the hypothesis (H1) of Corollary~\ref{cor:bpuptuning}, the condition $\mathrm{BP}(n_0,\varpi_0)$ for the laws $\mu^{(0)}_{k,L}$ of the block variables at the coarsest spacing $a_0$. These laws are not Wilson measures: for the straight-path maps they have smooth positive densities whose logarithms, the induced actions, couple every pair of coarse links. The first part proves a criterion for $\mathrm{BP}$ that uses only the single-link conditional laws of such a measure, with no product form of the action and no finite range: a Dobrushin condition on the Kantorovich coefficients of the single links, or a bound of Bakry--\'Emery type on the Hessian of the induced action. It recovers the leading term of Corollary~\ref{cor:bpranges}(iii) for the Wilson action. The second part places (H1) among the other statements. It follows from $\mathrm{BP}(n_k,\varpi)$ along the tuning with the coarse cube enlarged by one, so that under (H2) it is equivalent to the conclusion of Corollary~\ref{cor:bpuptuning} up to one unit of cube size; it is independent of (H2); and it fails for the block laws of a measure whose local gauge-invariant covariances vanish beyond a fixed distance, the failure being carried by the Polyakov loops of the coarse torus. \emph{Setting.} Let $G\subset\U(N)$ be compact, $\delta$ a bi-invariant metric of diameter $D$ on $G$, $n\ge2$, $\Lambda_L$ the torus with $L\ge2n+ 4$ even and $E$ its links, with $V_x$, $P^\varrho_x$ and $\mathcal E^\varrho_n$ as in Sec.~\ref{ssec:bpblock}. Let $\varrho$ be a probability measure on $G^E$ with a continuous positive density $\rho$ with respect to $\lambda_E$. For $V\subset E$ and $U\in G^E$ put \[ \gamma^\varrho_V(d\sigma\mid U)=\frac{\rho(\sigma,U_{E\setminus V})\, \lambda_V(d\sigma)}{\int\rho(\sigma',U_{E\setminus V})\,\lambda_V(d\sigma')} . \] This law depends only on $U_{E\setminus V}$ and continuously on it, $P^\varrho_xf(U)=\int f\,d\gamma^\varrho_{V_x}(\cdot\mid U)$ is a version of the conditional expectation under $\varrho$ given the links outside $V_x$, and for $e\in V$ the conditional law of $\sigma_e$ under $\gamma^\varrho_V(\cdot \mid U)$ given the other links of $V$ is $\gamma^\varrho_{\{e\}}(\cdot\mid \sigma,U_{E\setminus V})$, for every $U$ and $\sigma$. For $j\notin V$ let \[ C^\varrho_{V,j}=\sup\frac{W_1^{d_V}\bigl(\gamma^\varrho_V(\cdot\mid U), \gamma^\varrho_V(\cdot\mid U')\bigr)}{\delta(U_j,U'_j)}, \] the supremum over the pairs $U,U'$ that differ only at $j$, and $C^\varrho_{ ee'}=C^\varrho_{\{e\},e'}$ for $e\ne e'$. For a Wilson measure these are the coefficients $C^0_{V,j}$, $C^0_{ee'}$ of Sec.~\ref{ssec:bpstrong}; for a general $\varrho$, $C^\varrho_{ee'}$ need not vanish when $e$ and $e'$ are not partners. For links $e\ne j$ with base points $b(e)$, $b(j)$ let $\Delta= b(e)-b(j)$, with coordinates in $(-L/2,L/2]$, and \[ w_n(e,j)=\sum_{\nu=0}^3\min\bigl(\lvert\Delta_\nu\rvert,n\bigr). \] \begin{lemma}\label{lem:bpcoarseblock} Suppose that $c_\varrho=\max_{k\in E}\sum_{e\in E\setminus\{k\}}C^\varrho_{ek}< 1$. Then for every $V\subset E$ and every link $j\notin V$, \[ C^\varrho_{V,j}\le\frac1{1-c_\varrho}\sum_{e\in V}C^\varrho_{ej}. \] \end{lemma} \begin{proof} The proof of Lemma~\ref{lem:bpblock} applies with the kernels $\gamma^\varrho$. Let $U,U'$ differ only at $j$, let $\pi^*$ be a coupling of $\gamma=\gamma^\varrho_V(\cdot\mid U)$ and $\gamma'=\gamma^\varrho_V(\cdot\mid U')$ that minimises $\sum_{e\in V}\E_\pi\delta(\sigma_e,\sigma'_e)$, and $\varrho^*(e)=\E_{\pi^*}\delta(\sigma_e,\sigma'_e)$. Resampling the pair at $e\in V$ from an optimal coupling of $\gamma^\varrho_{\{e\}}(\cdot\mid\sigma, U_{E\setminus V})$ and $\gamma^\varrho_{\{e\}}(\cdot\mid\sigma',U'_{E\setminus V})$, chosen measurably~\cite[Cor.~5.22]{Vil09}, gives a coupling of $\gamma$ and $\gamma'$. The two conditioning configurations differ at the links of $V\setminus\{e\}$ and at $j$; changing them one at a time and using the definition of the coefficients bounds the new expected distance at $e$ by $\sum_{k\in V\setminus\{e\}}C^\varrho_{ek}\varrho^*(k)+C^\varrho_{ej}\delta(U_j, U'_j)$, and minimality gives \[ \varrho^*(e)\le\sum_{k\in V\setminus\{e\}}C^\varrho_{ek}\varrho^*(k)+C^\varrho_{ ej}\,\delta(U_j,U'_j). \] Summing over $e\in V$ and using $\sum_{e\in V\setminus\{k\}}C^\varrho_{ek}\le c_\varrho$ gives \[ (1-c_\varrho)\sum_e\varrho^*(e)\le\delta(U_j,U'_j)\sum_{e\in V}C^\varrho_{ej}, \] and $W_1^{d_V}(\gamma,\gamma')\le\sum_e\varrho^*(e)$. \end{proof} \begin{theorem}[a single-link criterion for general measures] \label{thm:bpcoarse} Let $\varrho$ be as in the setting, $n\ge2$ and $L\ge2n+4$ even. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item If $\sum_{x:\,j\notin V_x}C^\varrho_{V_x,j}\le\alpha n^4$ for every link $j$, with $\alpha<1$, then $(1-\alpha)\Var_\varrho(f)\le\mathcal E^\varrho_n(f)$ for every real $f\in L^2(\varrho)$; if $\varrho$ is gauge invariant, $\mathrm{BP}_\varrho(n,1-\alpha)$ holds. \item If $c_\varrho<1$, the hypothesis of \textup{(i)} holds with \[ \alpha=\alpha_n(\varrho)=\frac1{(1-c_\varrho)\,n}\max_{j\in E}\sum_{e\ne j} C^\varrho_{ej}\,w_n(e,j), \] so $\mathrm{BP}_\varrho(n,1-\alpha_n(\varrho))$ holds for gauge-invariant $\varrho$ whenever $\alpha_n(\varrho)<1$. With $m_\varrho=\max_j\sum_{e\ne j} C^\varrho_{ej}\lvert\Delta\rvert_1$ and, for $R\ge1$, $t_\varrho(R)=\max_j \sum_{e:\,\lvert\Delta\rvert_\infty>R}C^\varrho_{ej}$, \[ \alpha_n(\varrho)\le\frac{m_\varrho}{(1-c_\varrho)\,n},\qquad\alpha_n(\varrho) \le\frac4{1-c_\varrho}\Bigl(\frac{Rc_\varrho}n+t_\varrho(R)\Bigr). \] \end{enumerate} \end{theorem} \begin{proof} (i) Let $U,U'$ differ only at a link $i$, and couple $\mathcal K^\varrho(U, \cdot)$ and $\mathcal K^\varrho(U',\cdot)$ by choosing the same $x$. If $i\in V_x$, the two exteriors agree, so do the laws of the resampled cube, and the identity coupling costs nothing. If $i\notin V_x$, an optimal coupling of the two cube laws costs at most $C^\varrho_{V_x,i}\delta(U_i,U'_i)$, and the link $i$ adds $\delta(U_i,U'_i)$. The $x$ with $i\notin V_x$ number $L^4-n^4$, so \[ W_1^{d_E}\bigl(\mathcal K^\varrho(U,\cdot),\mathcal K^\varrho(U',\cdot)\bigr) \le\Bigl(1-\frac{(1-\alpha)n^4}{L^4}\Bigr)\delta(U_i,U'_i). \] This is the contraction of the proof of Proposition~\ref{prop:avgsup}(i) with $\kappa_0=(1-\alpha)n^4/L^4$; the rest of that proof uses only that the $P^\varrho_x$ are the orthogonal projections of $L^2(\varrho)$ given by everywhere defined kernels, and it gives $\ip f{(I-\mathcal K^\varrho)f}\ge \kappa_0\Var_\varrho(f)$, which is the claim after multiplication by $L^4/n^4$. (ii) By Lemma~\ref{lem:bpcoarseblock}, \[ \sum_{x:\,j\notin V_x}C^\varrho_{V_x,j}\le\frac1{1-c_\varrho}\sum_{e\ne j} C^\varrho_{ej}\,\#\{x:e\in V_x,\ j\notin V_x\}. \] A link with base point $b'$ lies in $V_x$ exactly when $b'-x\in[0,n)^4$ modulo $L$. Since $L\ge2n$, for $t\in[0,n)$ and $\lvert d\rvert\le L/2$ the residue of $t+d$ lies in $[0,n)$ only if $t+d\in[0,n)$, so the $x$ with $e,j\in V_x$ number $\prod_\nu(n-\lvert\Delta_\nu\rvert)_+$. With $s_\nu= \min(\lvert\Delta_\nu\rvert,n)/n\in[0,1]$ and $\prod_\nu(1-s_\nu)\ge1- \sum_\nu s_\nu$, \[ \#\{x:e\in V_x,\ j\notin V_x\}=n^4-\prod_\nu(n-\lvert\Delta_\nu\rvert)_+\le n^3w_n(e,j), \] which gives $\alpha_n(\varrho)$. Finally $w_n(e,j)\le\lvert\Delta\rvert_1\le4 \lvert\Delta\rvert_\infty$ and $w_n(e,j)\le4n$; splitting the sum at $\lvert\Delta\rvert_\infty=R$ gives the last bound. \end{proof} The criterion depends on $\varrho$ only through the column sums of its single-link coefficients and their first moment or tail; it uses no locality of the specification, and the bound holds on all of $L^2(\varrho)$. For $\varrho$ given by an action the coefficients are bounded by its Hessian. \begin{lemma}\label{lem:bpcoarsehess} Let $G\subset\U(N)$ be closed and connected with semisimple Lie algebra, $\delta$ the geodesic distance of $\ip XY=-\Tr(XY)$, and $\varrho$ the probability measure with density proportional to $e^{-H}$, $H\in C^2(G^E)$. For a link $e$, a configuration $U$ and $X\in\g$ let $U^{e,X}$ be $U$ with $U_e$ replaced by $U_ee^X$. Suppose that there are $h_e<\kappa_G$ and $J_{ee'}\ge0$ with \[ \frac{d^2}{dt^2}H(U^{e,tX})\Big|_{t=0}\ge-h_e\lvert X\rvert^2 \] and \[ \Bigl\lvert\frac{\partial^2}{\partial s\,\partial t}H\bigl((U^{e,sX})^{e',tY} \bigr)\Big|_{0}\Bigr\rvert\le J_{ee'}\lvert X\rvert\,\lvert Y\rvert \] for all $U$, all $X,Y\in\g$ and all links $e\ne e'$. Then $C^\varrho_{ee'}\le J_{ee'}/(\kappa_G-h_e)$. \end{lemma} \begin{proof} Given the other links, the law of $U_e$ is $\nu\propto e^{-V}dU$ with $V(\sigma)=H(\sigma,U_{E\setminus e})$. The geodesics of the bi-invariant metric through $\sigma$ are the curves $\sigma e^{tX}$, so the first hypothesis says $\Hess V\ge-h_e$, and $\Ric_G+\Hess V\ge K_1=\kappa_G-h_e> 0$. As in the proof of Lemma~\ref{lem:bponelink}, $\lvert\Cov_\nu(g,u)\rvert \le\lVert\nabla g\rVert_\infty\lVert\nabla u\rVert_\infty/K_1$ for smooth $g,u$. Let $U,U'$ differ only at $e'$, with $U'_{e'}=U_{e'}e^{\delta X}$, $\lvert X \rvert=1$ and $\delta=\delta(U_{e'},U'_{e'})$, along a minimising geodesic, and let $\nu_s\propto e^{-V_s}$ be the law of $U_e$ given the configuration $U^{e',sX}$ outside $e$. Then $\frac d{ds}\nu_s(g)=-\Cov_{\nu_s}(g,\partial_s V_s)$, and by the second hypothesis the gradient of $\sigma\mapsto\partial_s V_s(\sigma)$ has norm at most $J_{ee'}$. For smooth $g$ that are $1$-Lipschitz for $\delta$ this gives $\lvert\nu_0(g)-\nu_\delta(g)\rvert\le J_{ee'}\delta/K_1$, and the approximation and duality argument of that proof gives the bound. \end{proof} \begin{corollary}\label{cor:bpcoarsehess} Under the hypotheses of Lemma~\textup{\ref{lem:bpcoarsehess}} put $\theta_{ej}=J_{ej}/(\kappa_G-h_e)$ and $c_H=\max_k\sum_{e\ne k}\theta_{ek}$. If $c_H<1$ and $L\ge2n+4$, then every real $f\in L^2(\varrho)$ satisfies $(1-\alpha)\Var_\varrho(f)\le\mathcal E^\varrho_n(f)$ with \[ \alpha=\frac1{(1-c_H)\,n}\max_j\sum_{e\ne j}\theta_{ej}\,w_n(e,j) \] whenever $\alpha<1$. For the Wilson action, $H=-N\beta\sum_p\ReTr U_p$, one may take $h_e=6N\lvert\beta\rvert$ and $J_{ee'}=N\lvert\beta\rvert$ for partners and $0$ otherwise, and $\alpha=18N\lvert\beta\rvert/((\kappa_G-24N \lvert\beta\rvert)n)$, the leading term of $\alpha_n$ in Corollary~\textup{\ref{cor:bpranges}(iii)}. The numbers $h_e$ and $J_{ee'}$ may be taken subadditive in $H$: if $H=H_1+H_2$ and they are admissible for $H_1$ and for $H_2$, their sums are admissible for $H$. \end{corollary} \begin{proof} By Lemma~\ref{lem:bpcoarsehess}, $c_\varrho\le c_H$ and $\alpha_n(\varrho)\le \alpha$, and Theorem~\ref{thm:bpcoarse} applies. For the Wilson action the proof of Lemma~\ref{lem:bponelink} gives the second derivative at least $-6N\lvert\beta\rvert\lvert X\rvert^2$ and, by Lemma~\ref{lem:mixed}, the mixed derivatives of the one plaquette that contains two partners at most $N\lvert\beta\rvert\lvert X\rvert\lvert Y\rvert$; partners of $j$ have $w_n=1, 1,0,1,1,2$ in each of the three planes through $j$, eighteen in all, and $1-c_H=(\kappa_G-24N\lvert\beta\rvert)/(\kappa_G-6N\lvert\beta\rvert)$. Subadditivity is the triangle inequality for second derivatives. \end{proof} So the criterion holds for every $C^2$ perturbation of a Wilson action in the range of Corollary~\ref{cor:bpranges}(iii) whose Hessian blocks are small in the norms above, local or not; this is the analogue of Proposition~\ref{prop:cubeopen} without the gauge and without a finite range. The block laws are of this kind of object. \begin{lemma}\label{lem:bpcoarselaw} Let $c$ be the straight-path map with blocks of side $b\ge2$ from $\Lambda_{bL}$ to $\Lambda_L$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item If $\mu'$ has a continuous positive density $r$ with respect to $\lambda_{E'}$, then $c_*\mu'$ has the continuous positive density $V\mapsto \nu_V(r)$ with respect to $\lambda_E$, of class $C^m$ if $r$ is. \item Let $c'$ be the straight-path map with blocks of side $b'\ge2$ from $\Lambda_{b'bL}$ to $\Lambda_{bL}$, and $R$, $R'$, $R''$ the Haar refinements along $c$, $c'$ and the straight-path map $c\circ c'$ with blocks of side $bb'$. Then $R'(R(\varrho))=R''(\varrho)$ for every probability measure $\varrho$ on $G^E$. \item For $G=\SU(N)$, $c\circ Z_{c_1,d_1}=Z_{c_1,d_1}\circ c$ for the centre twists of Sec.~\textup{\ref{ssec:bpstrong}} on $\Lambda_{bL}$ and on $\Lambda_L$, and $c$ intertwines the translations by $b$ of $\Lambda_{bL}$ with the unit translations of $\Lambda_L$. So $c_*\mu'$ inherits invariance under these maps from $\mu'$. \end{enumerate} In particular, for straight-path maps $c_j$ in Corollary~\textup{\ref{cor:bpuptuning}}, $\mu^{(0)}_{k,L}=(c^{(k)})_*\mu_{k,L}$ with $c^{(k)}=c_1\circ\dots\circ c_k$ the straight-path map with blocks of side $2^k$; it has the smooth positive density $V\mapsto\nu_V(\prod_p\varphi(U_p))/Z$, it is gauge invariant and invariant under the centre twists and the translations of $\Lambda_{L_0}$, and Theorem~\textup{\ref{thm:bpcoarse}} and Lemma~\textup{\ref{lem:bpcoarsehess}} apply to it, with the induced action $H^{(0)}_{k,L}(V)=-\log\nu_V(\prod_p\varphi(U_p))$ in place of $H$. \end{lemma} \begin{proof} (i) By Lemma~\ref{lem:bprefine}(i), $\lambda_{E'}=\int\lambda_E(dV)\nu_V$, so $\mu'(\psi\circ c)=\int\lambda_E(dV)\psi(V)\nu_V(r)$ for bounded Borel $\psi$. Under $\nu_V$ the first $b-1$ links of each path and the links of $E'_0$ are independent Haar variables $u$, and the last link of $\pi(e)$ is the inverse of the product of the others times $V_e$; so $\nu_V(r)=\int r(\Phi(V, u))\,du$ with $\Phi$ smooth. Continuity and differentiability under the integral follow from the uniform continuity of $r$ and of its derivatives on the compact group, and $\nu_V(r)\ge\min r>0$. (ii) The map $c\circ c'$ multiplies the $bb'$ links of the straight path from $b'by$ to $b'b(y+e_\nu)$, which consists of the paths of $c'$ of the $b$ links of the path $\pi(y,\nu)$ of $c$. It suffices to show $\int\nu_V(dW) \,\nu'_W=\nu''_V$ for every $V$, with the obvious notation. Under $\nu_V$ the links $W_1,\dots,W_{b-1}$ of $\pi(e)$ and the links of $E'_0$ are independent Haar variables and $W_b=(W_1\cdots W_{b-1})^{-1}V_e$. If $W_i$ is a Haar variable and the links of its $c'$-path have law $\nu'^{(b')}_{ W_i}$ given $W_i$, then by Lemma~\ref{lem:bpfibrelaw}(i) all $b'$ of them are independent Haar variables. Hence under $\int\nu_V(dW)\nu'_W$ the first $bb'-1$ links of the path of $c\circ c'$ are independent Haar variables, the last is the inverse of their product times $V_e$, and every link on no path of $c\circ c'$, which lies on the $c'$-path of a link of $E'_0$ or is on no $c'$-path, is an independent Haar variable. This is $\nu''_V$. (iii) The coarse link $(y,0)$ is the product of the fine links $(by+te_0,0)$, $0\le tR} C^{(k,L)}_{ej}\le\frac{1-c}{16} \] for all $k$, all $L\in\mathcal L_k$ and all links $j$. Then (H1) holds with $\varpi_0=\frac12$ for every $n_0\ge\max(2,16Rc/(1-c))$ for which every $L\in\mathcal L_k$ has $L_0\ge2n_0+4$. The same holds if the induced actions $H^{(0)}_{k,L}$ satisfy the hypotheses of Lemma~\textup{\ref{lem:bpcoarsehess}} with $\theta_{ej}=J_{ej}/(\kappa_G-h_e)$ in place of $C^{(k,L)}_{ej}$. \end{corollary} \begin{proof} By Theorem~\ref{thm:bpcoarse}(ii), \[ \alpha_{n_0}\le\frac{4Rc}{(1-c)\,n_0}+\frac{4t(R)}{1-c}\le\frac14+\frac14 . \] For the second statement use Lemma~\ref{lem:bpcoarsehess} and the monotonicity of these bounds in the coefficients. \end{proof} These are statements about the induced action of $k$ steps at the spacing $a_0$, uniform in $k$ and in the torus. The results of this paper do not supply them for any $k$ with $\beta_k$ outside the strong-coupling range: Corollary~\ref{cor:bpranges} bounds the coefficients of Wilson measures, Theorem~\ref{thm:chartdecay} and Sec.~\ref{sec:islands} concern the law of the fine field given the block variables, and Proposition~\ref{prop:relstep} concerns one step of the cut-off average at weak coupling. For the finitely many $k$ in the strong-coupling range the criterion is not needed, since the descent of Sec.~\ref{ssec:bpblock} already gives (H1) there, with the cube enlarged by one. \begin{proposition}[the place of (H1)]\label{prop:bpcoarsezero} \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Let $\mu'$ be a gauge-invariant probability measure on $G^{E'}$, $c$ the straight-path map with blocks of side $b\ge2$, $n\ge2$, $L\ge2n+4$ even, $2\le n'\le b(n-1)+1$ with $bL\ge2n'+4$, and $\varpi_c=\varpi'(n'/ (bn))^4$. Then $\mathrm{BP}_{\mu'}(n',\varpi')$ implies $\mathrm{BP}_{c_*\mu'}(n,\varpi_c)$. \item In Corollary~\textup{\ref{cor:bpuptuning}} with straight-path maps, if $\mathrm{BP}(n_k,\varpi)$ holds for $\mu_{k,L}$ and $L_0\ge2n_0+6$, then $\mathrm{BP}(n_0+1,\varpi_1)$ holds for $\mu^{(0)}_{k,L}$, with $\varpi_1= \varpi(n_0/(n_0+1))^4$. For $G=\SU(2)$ and $\lvert\beta_{k,W}\rvert<\frac{16}{105}$ this holds with $\varpi=1-\alpha_{n_k}$, $\alpha_{n_k}$ as in Corollary~\textup{\ref{cor:bpranges}(i)}. \item Under \textup{(H2)}, \textup{(H1)} implies $\mathrm{BP}(n_k,\varpi)$ for all $k$ with $\varpi=\min(\varpi_0,1/C_A)/\Gamma$, which implies \textup{(H1)} at the cube size $n_0+1$, with the constant $\varpi(n_0/(n_0+1))^4$, on the tori with $L_0\ge2n_0+6$. \end{enumerate} \end{proposition} \begin{proof} (i) The straight-path map satisfies (B1) with $r=0$, since $c_{(y,\nu)}$ depends only on the links with base points $by+te_\nu\in B_y$, and the proof of Theorem~\ref{thm:bpdescent}(i),(ii) holds with $r=0$: then $y(x')$ has coordinates $\lfloor x'_i/b\rfloor$, $Q_y=by+[0,bn)^4$, and $x'_i+n'-1\le by_i+b-2+n'\le by_i+bn-1$ by the bound on $n'$, so $V'_{x'}$ has its base points in $Q_y$; a coarse link $(z,\nu)$ with $z\notin y+[0,n)^4$ depends only on links with base points in $B_z$, which is disjoint from $Q_y$. The rest of that proof does not involve $r$. (ii) Apply (i) to $c^{(k)}$, with $b=2^k$, $n'=n_k=b\,n_0$ and $n=n_0+1$, so that $n'\le b(n-1)+1$; the coarse torus $\Lambda_{L_0}$ has $L_0\ge2n+4$, and $L=2^kL_0\ge2n_k+4$. The constant is $\varpi(n_k/(bn))^4=\varpi(n_0/(n_0+ 1))^4$. For $\SU(2)$, Corollary~\ref{cor:bpranges}(i) gives $\mathrm{BP}(n_k,1-\alpha_{n_k})$ on every torus, since the threshold $4n^2/ (18n^2+18n-3)$ increases with $n$ and equals $\frac{16}{105}$ at $n=2$. (iii) Corollary~\ref{cor:bpuptuning} and (ii). \end{proof} Part (iii) says that, given (H2), the coarsest condition is necessary and sufficient for the conclusion of Corollary~\ref{cor:bpuptuning} up to one unit of cube size, so no proof of $\mathrm{BP}(n_k,\varpi)$ by the upward step avoids it. The condition also imposes two necessary conditions on the block laws, which hold for every gauge-invariant measure and use no locality. \begin{proposition}[what the condition forces]\label{prop:bpcoarsenec} Let $\varrho$ be a gauge-invariant probability measure on the links of $\Lambda_L$ for which $\mathrm{BP}_\varrho(n,\varpi)$ holds, and $P_y$ the Polyakov loops of Sec.~\textup{\ref{ssec:bpstrong}}. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The averaged Polyakov loops satisfy \[ \Var_\varrho\Bigl(L^{-3}\sum_{y:\,y_0=0}\ReTr P_y\Bigr)\le\frac{N^2n^2}{\varpi L^2}, \] and the same holds for the imaginary parts. \item Let $G=\SU(N)$, $L=2n+4$, let $\varrho$ be invariant under the centre twists $Z_{c_1,d_1}$, let $\Var_\varrho(\ReTr U_p)>0$ for some plaquette $p$, and $\varepsilon=\max_p\varrho(\lVert U_p-I\rVert_{\rm HS}^2)$. Then $\varpi\le C'_Nn^4\varepsilon$, with $C'_2=19200$ and $C'_N=\max(2048(N^2-1)N^2(N-1)^2 /c_N,\,2048/\eta_N^2)$, $c_N,\eta_N$ as in Lemma~\textup{\ref{lem:bpcommute}}. \item In Corollary~\textup{\ref{cor:bpuptuning}} with straight-path maps, \textup{(H1)} implies, for all $k$ and $L\in\mathcal L_k$, \[ \Var_{\mu_{k,L}}\Bigl(L_0^{-3}\sum_{y:\,y_0=0}\ReTr P_{2^ky}\Bigr)\le\frac{N^2 n_0^2}{\varpi_0L_0^2}, \] the $P_{2^ky}$ being Polyakov loops of $\Lambda_L$; and if $G=\SU(N)$ and $L=2^k(2n_0+4)\in\mathcal L_k$, the holonomies $W$ of the squares of side $2^k$ with corners in $2^k\Z^4$ satisfy $\max_W\mu_{k,L}(\lVert W-I\rVert_{\rm HS}^2)\ge\varpi_0/(C'_Nn_0^4)$. \item If $G=\SU(2)$ and $\varrho$ on $\Lambda_{2n+4}$ has a continuous positive density, is invariant under the centre twists and satisfies the hypothesis of Theorem~\textup{\ref{thm:bpcoarse}(i)} with $\alpha\le\frac12$, then $\varepsilon\ge1/(38400\,n^4)$. \end{enumerate} \end{proposition} \begin{proof} (i) This is the proof of Theorem~\ref{thm:bpfixed}(i) without the symmetry: $\lVert g_y-P^\varrho_xg_y\rVert_2\le N$ for $g_y=\ReTr P_y$, so the right side of $\mathrm{BP}$ for the average is at most $N^2n^2/L^2$, and the left side is $\varpi$ times its variance. (ii) The proof of Theorem~\ref{thm:bpfixed}(ii),(iii), before the use of Lemma~\ref{lem:bpflat}, uses the measure only through its invariance under the $Z_{c_1,d_1}$, which makes the traces $F_{ab}$ centred, through the mean of $\lVert U_p-I\rVert_{\rm HS}^2$ at each plaquette, which enters as an upper bound and may be replaced by $\varepsilon$, and through $\varpi\le2$, which the proof of Proposition~\ref{prop:bpscales}(ii) gives from the plaquette $p$. Lemmas~\ref{lem:bpstokes} and~\ref{lem:bpcommute} are deterministic. (iii) By Lemma~\ref{lem:bpcoarselaw}, $\mu^{(0)}_{k,L}$ is gauge invariant, invariant under the centre twists, and has a continuous positive density, so $\Var(\ReTr c_P)>0$. The Polyakov loops of $\Lambda_{L_0}$ are those of $\Lambda_L$ through the sites $2^ky$, and the plaquette variables $c_P$ are the holonomies $W$. Apply (i) and (ii). (iv) Theorem~\ref{thm:bpcoarse}(i) gives $\mathrm{BP}(n,\frac12)$, and (ii) gives $\frac12\le19200\,n^4\varepsilon$. \end{proof} Part (iv) is an obstruction to the criterion on the smallest torus: a law of the block variables whose plaquettes are close to $I$ in the mean cannot satisfy it there. The condition (H1) is posed on all $L\in\mathcal L_k$, and on large tori (i) asks that the spatially averaged Polyakov loops of the coarse torus decorrelate. That requirement is not implied by the decay of local correlations, and it is independent of (H2), as the following measure shows. For $G=\SU(N)$, $N\ge2$, and $\theta\in[0,2\pi)$ let $h_\theta$ be the configuration of $\Lambda_L$ with $(h_\theta)_{(y,0)}= \operatorname{diag}(e^{i\theta},e^{-i\theta},1,\dots,1)$ and $(h_\theta)_{(y, \nu)}=I$ for $\nu\ne0$, and let $\tau_L$ be the law of the configuration $\hat g\cdot h_\theta$, $(\hat gh_\theta)_{(y,\nu)}=\hat g(y)(h_\theta)_{(y, \nu)}\hat g(y+e_\nu)^{-1}$, with $\theta$ uniform and $\hat g$ a Haar distributed gauge transformation independent of $\theta$. It is a random flat connection with a uniformly distributed holonomy around the direction $0$. \begin{proposition}[a flat coarse law]\label{prop:bpcoarseflat} Let $G=\SU(N)$, $N\ge2$, $n\ge2$ and $L\ge\max(2n+4,8)$ even. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\tau_L$ is gauge invariant; every gauge-invariant function of the links with base points in a box of side at most $L/2$ is $\tau_L$-almost surely constant; and $\varpi\Var_{\tau_L}(f)\le\mathcal E^{\tau_L}_n(f)$ fails for $\varpi>n^2/L^2$ both for the charged function $f=L^{-3}\sum_{y:\,y_0=0} \ReTr P_y$ and for the neutral function $f=L^{-3}\sum_{y:\,y_0=0}\lvert\Tr P_y\rvert^2$. So $\mathrm{BP}_{\tau_L}(n,\varpi)$ and its restriction to $\mathcal A_0$ fail for $\varpi>n^2/L^2$. \item Let $c$ be the straight-path map with blocks of side $b\ge2$ and $\mu'=R(\tau_L)$ on $\Lambda_{bL}$. Then $c_*\mu'=\tau_L$, and $\Cov_{\mu'}(f, g)=0$ for all continuous gauge-invariant $f,g$ that depend only on the links with base points in boxes $B,B'$ of side at most $bL/2$ with $d(B,B')\ge b$. \item Let $n_0\ge2$, $L_0\ge\max(2n_0+4,8)$ even, and for each $k$ let $\mu_{k}=R^{(k)}(\tau_{L_0})$ be the Haar refinement along $c^{(k)}$, on $\Lambda_{2^kL_0}$, with the tower of straight-path maps $c_j$ of side two. Then $\mu^{(j)}=R_j\circ\dots\circ R_1(\tau_{L_0})$, $\mathrm S(n_j,n_{j-1};1,1)$ holds for $(\mu^{(j)},c_j)$ with $n_j=2^jn_0$, so (H2) holds with $C_A=1$ and $\Gamma=1$; $\mu^{(0)}=\tau_{L_0}$, so $\mathrm{BP}(n_0,\varpi_0)$ fails for it when $\varpi_0>n_0^2/L_0^2$; and the local gauge-invariant covariances of $\mu_k$ vanish at distance at least $2^k$, that is, at physical distance $a_0$. \item Let $b\ge2$, $L\ge2n+4$ and $bL\ge2n'+4$, let $\tau'$ be the law of $(\hat g(x)(h_\theta)_{(x,\mu)}\hat g(x+e_\mu)^{-1})_{(x,\mu)\in E'_0}$, with $h_\theta$ and $\hat g$ now on $\Lambda_{bL}$, and $\mu'=\lambda_{E'\setminus E'_0}\otimes\tau'$. Then $\mu'$ is gauge invariant, $c_*\mu'=\lambda_E$ satisfies $\mathrm{BP}(n,1)$, $\mathrm{BP}_{\mu'}(n',\varpi')$ fails for $\varpi'>\frac{64}{49}n'^2/(bL)^2$, and constants for which $\mathrm S(n',n; C_A,C_B)$ holds for $(\mu',c)$ satisfy $\max(C_A,C_B)\ge\frac{49}{64}(bL)^2/ n'^2$. \end{enumerate} \end{proposition} \begin{proof} \emph{An averaging bound.} Let $\varrho$ be a probability measure on the links of $\Lambda_M$, $S$ a set of sites with $y_0=0$, and $g_y$, $y\in S$, a function of the links of the line through $y$ in the direction $0$, such that $\varrho$-almost surely all $g_y$ equal one random variable of variance $v>0$. For $f=\lvert S\rvert^{-1}\sum_{y\in S}g_y$, $\Var_\varrho(f)=v$; $P^\varrho_x g_y=g_y$ unless the line meets $V_x$, which happens for at most $n^3$ points $y\in S$, and $\lVert g_y-P^\varrho_xg_y\rVert_2\le\lVert g_y-\varrho(g_y)\rVert_2= \sqrt v$ because $P^\varrho_x$ is an orthogonal projection that fixes the constants. Hence $\mathcal E^\varrho_n(f)\le n^{-4}M^4(n^3\sqrt v/\lvert S \rvert)^2$, and $\varpi v\le\mathcal E^\varrho_n(f)$ forces $\varpi\le n^2M^4/ \lvert S\rvert^2$. (i) A gauge transformation $g$ maps $\hat g\cdot h_\theta$ to $(g\hat g)\cdot h_\theta$, and $g\hat g$ is again Haar distributed. Let $f$ depend on the links with base points in a box of side at most $L/2$; their end points lie in a box $Q$ of side at most $L/2+1\le L-1$, and with coordinates on $Q$ taken in an interval of length less than $L$, $k(y)=(h_\theta)_{(0,0)}^{-y_0}$ satisfies $k(y)k(y+e_\nu)^{-1}=(h_\theta)_{(y,\nu)}$ on these links. So $f(\hat g\cdot h_\theta)=f(h_\theta)=f(k\cdot I)=f(I)$ almost surely. Almost surely $P_y=\hat g(y)(h_\theta)_{(0,0)}^L\hat g(y)^{-1}$ for every $y$, so $\ReTr P_y=2\cos(L\theta)+N-2$ and $\lvert\Tr P_y\rvert^2=(2\cos(L\theta)+ N-2)^2$, with variances $2$ and $2+8(N-2)^2$; the second function is neutral by Proposition~\ref{prop:bpcentre}. The averaging bound with $M=L$, $\lvert S\rvert=L^3$ gives the claim. (ii) $c_*\mu'=\tau_L$ by Lemma~\ref{lem:bprefine}(i). By the proof of Lemma~\ref{lem:bprefine}(i), the image of $\nu_V$ under the gauge transformation equal to $\hat g(y)$ at $by$ and to $I$ elsewhere is $\nu_{\hat gV}$; so $\mu'$ is the law of $g\cdot U$ with $U$ distributed as $\int\nu_{h_\theta}\,d\theta$ and $g$ a random gauge transformation, and $\mu'(fg)=\int\nu_{h_\theta}(fg)\,d\theta$, $\mu'(f)=\int\nu_{h_\theta}(f)\, d\theta$. Fix $\theta$. Under $\nu_{h_\theta}$ the links of distinct paths and of $E'_0$ are independent. The base points of a path lie in $b$ consecutive sites of a line, so no path has links with base points in both $B$ and $B'$, and $\nu_{h_\theta}(fg)=\nu_{h_\theta}(f)\nu_{h_\theta}(g)$. The coarse links whose paths meet $B$ have end points in a box of $\Lambda_L$ of side at most $L/2+3\le L-1$, on which $h_\theta=k\cdot I$ as in (i). By Lemma~\ref{lem:bpfibrelaw}(ii), path by path, and the invariance of Haar measure on $E'_0$, the links with base points in $B$ have under $\nu_{h_\theta}$ the law of $g_k\cdot U$ with $U$ distributed as $\nu_I$, $g_k$ equal to $k(y)$ at $by$ and to $I$ elsewhere; so $\nu_{h_\theta}(f)= \nu_I(f)$, independent of $\theta$, and likewise for $g$. Hence $\Cov_{\mu'}(f, g)=0$. (iii) By Lemma~\ref{lem:bpcoarselaw}(ii) and induction, $R^{(k)}=R_k\circ R^{(k-1)}$, so $\mu_k=R_k\circ\dots\circ R_1(\tau_{L_0})$, and by Lemma~\ref{lem:bprefine}(i) the tower of $\mu_k$ is $\mu^{(j)}=R_j\circ\dots \circ R_1(\tau_{L_0})$, with $\mu^{(0)}=\tau_{L_0}$. Theorem~\ref{thm:bprefine} with $n'=n_j=2n_{j-1}$ gives $\mathrm S(n_j,n_{j-1};1,1)$. The rest is (i) and (ii) with $b=2^k$. (iv) The gauge group acts on the links of $E'\setminus E'_0$ and of $E'_0$ separately; Haar measure is invariant, and so is $\tau'$, as in (i). The map $c$ depends only on the links of the paths, which are independent Haar variables, so $c_*\mu'=\lambda_E$, and Proposition~\ref{prop:avgsup}(ii) gives $\mathrm{BP}(n,1)$. A link of direction $0$ lies on a path exactly when the coordinates $1,2,3$ of its base point are divisible by $b$, so for the set $S$ of the $(bL)^3-L^3\ge\frac78(bL)^3$ sites $y$ with $y_0=0$ and $(y_1,y_2,y_3)\notin(b\Z)^3$ the lines through $y$ lie in $E'_0$, and $\ReTr P_y=2\cos(bL\theta)+N-2$ for all of them, almost surely. The averaging bound with $M=bL$ gives $\varpi'\le\frac{64}{49}n'^2/(bL)^2$. By Theorem~\ref{thm:bpup}(i), $\mathrm S(n',n;C_A,C_B)$ and $\mathrm{BP}(n,1)$ for $c_*\mu'$ give $\mathrm{BP}_{\mu'}(n',\min(1/C_A,1/C_B))$. \end{proof} Replacing $\tau_{L_0}$ in (iii) by the $\SU(2)$ Wilson measure at $\lvert \beta_W\rvert<\frac{16}{105}$ for finitely many $k$ gives towers for which $\mathrm{BP}$ holds at every level (Corollary~\ref{cor:bprefinetower}). So (H2) with the best constants, together with $\mathrm{BP}$ at every level of finitely many towers, does not imply (H1); and by (iv), (H1) does not imply (H2). The measures of (ii) and (iii) have local gauge-invariant covariances of finite range, uniformly in the torus, while the coarse law fails $\mathrm{BP}$ in both the charged and the neutral sector; the failure lies in functions spread over the coarse torus, on which the cube moves of a flat law act only through the curvature they create. The local form $\mathrm{BP}_{\rm loc}$ holds for $\tau_{L_0}$ trivially, but it does not enter the upward step as proved: Theorem~\ref{thm:bpup}(i) applies the coarse condition to $F_f=\E[f\mid c]$, which for a general measure depends on all block variables even when $f$ is local. On one torus the condition holds for every law of block variables, with a constant that decays exponentially in the volume; and single-link conditional laws close to Haar do not give a constant independent of the torus. For $G=\SU(N)$ let $s_N^2$ be the variance of $N^{-1}\ReTr U$ under the Haar measure, $s_2^2=\frac14$ and $s_N^2=1/(2N^2)$ for $N\ge3$. \begin{proposition}[closeness to Haar link by link]\label{prop:bphidden} Let $n\ge2$ and $L\ge2n+4$ even. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Let $c$ be a straight-path map or a composite of such maps onto $\Lambda_L$, and $\mu'$ a gauge-invariant probability measure on the fine links with a continuous positive density $D$ with respect to the Haar measure, $\Omega=\log(\max D/\min D)$. Then $\mathrm{BP}_{c_*\mu'}(n, e^{-\Omega})$ holds. \item Let $G=\SU(N)$, $N\ge2$, and $0<\epsilon<1$. Let $p_1,\dots,p_M$, $M=(L/2)^4$, be the plaquettes of the planes $\{0,1\}$ of $\Lambda_L$ whose base points have four even coordinates, $X_i=N^{-1}\ReTr U_{p_i}$, and \[ \varrho_L=\Bigl(\frac12\prod_{i=1}^M(1+\epsilon X_i)+\frac12\prod_{i=1}^M(1- \epsilon X_i)\Bigr)\lambda_E . \] Then $\varrho_L$ is a gauge-invariant probability measure with a continuous positive density, the conditional law of each link given the others has a density with respect to the Haar measure with values in $[1-\epsilon, 1+\epsilon]$, and every $\varpi$ for which $\mathrm{BP}_{\varrho_L}(n,\varpi)$ holds satisfies \[ \varpi\le\frac{48\,(n+1)^4}{\epsilon^2s_N^4L^4}. \] Consequently, for $L^4>48(n+1)^4/(\epsilon^2s_N^4)$, some link $j$ has $\sum_{x:\,j\notin V_x}C^{\varrho_L}_{V_x,j}>(1-48(n+1)^4/(\epsilon^2s_N^4L^4)) \,n^4$. \end{enumerate} \end{proposition} \begin{proof} (i) By Lemma~\ref{lem:bprefine}(i), applied to each map of the composite, $c_*\lambda_{E'}=\lambda_E$ and $\lambda_{E'}=\int\lambda_E(dV)\,\nu_V$ with $c=V$ $\nu_V$-almost surely, $\nu_V$ the conditional law; so $\varrho=c_*\mu'$ has the density $r(V)=\nu_V(D)$ with respect to $\lambda_E$, and $\min D\le r \le\max D$. Under $\lambda_E$ every $f\in L^2$ is the orthogonal sum of components $f_A$, $A\subset E$, where $f_A$ depends only on $U_A$ and has integral zero in each link of $A$; then $\lambda_E((f-\mathcal P_xf)^2)$ is the sum of $\lVert f_A\rVert^2$ over the $A$ that meet $V_x$, and each nonempty $A$ meets at least $n^4$ of the cubes, so $\Var_{\lambda_E}(f)\le n^{-4}\sum_x \lambda_E((f-\mathcal P_xf)^2)$. The variance and the conditional variances are infima of mean squares over constants and over functions of the links outside $V_x$, so \begin{gather*} \Var_\varrho(f)\le\max D\,\Var_{\lambda_E}(f),\\ \varrho\bigl((f-P^\varrho_xf)^2\bigr)\ge\min D\,\lambda_E\bigl((f-\mathcal P_xf)^2\bigr), \end{gather*} which give $e^{-\Omega}\Var_\varrho(f)\le\mathcal E^\varrho_n(f)$. (ii) The base points of the links of $p_i$ are $y_i$, $y_i+e_0$ and $y_i+ e_1$, so different $p_i$ have disjoint links, and $\lvert X_i\rvert\le1$. Under $\lambda_E$ the $X_i$ are independent with mean zero and variance $s_N^2$, since $\int\ReTr U\,dU=0$, $\int\lvert\Tr U\rvert^2dU=1$, and $\int(\Tr U)^2dU$ is $1$ for $N=2$ and $0$ for $N\ge3$. So both products are densities of probability measures $\varrho^\pm$, positive and continuous, and gauge invariant because $X_i$ is a class function of a plaquette holonomy. Fix a link $e$ of $p_i$ and the other links. As a function of $U_e$, $X_i=N^{-1} \ReTr(AU_e^{\pm1}B)$ with $A,B$ fixed, which has Haar integral zero, and the density of $\varrho_L$ is $a(1+\epsilon X_i)+a'(1-\epsilon X_i)$ with $a,a' \ge0$ not depending on $U_e$; normalised, it is $1+\theta\epsilon X_i$ with $\theta=(a-a')/(a+a')\in[-1,1]$. For a link on no $p_i$ the conditional law is Haar. Let $F=M^{-1}\sum_iX_i$, a real gauge-invariant function. Under $\varrho^\pm$ the $X_i$ are independent with mean $\pm\epsilon\int X_i^2\,d\lambda_E=\pm \epsilon s_N^2$, so, as $\varrho_L=\frac12(\varrho^++\varrho^-)$, $\Var_{\varrho_L}(F)\ge(\epsilon s_N^2)^2$. Let $r_x$ be the number of $p_i$ with a link in $V_x$. Their base points lie in $x+[-1,n)^4$, so $r_x\le (n+1)^4$, and each $p_i$ meets at most $3n^4$ cubes, one family for each of its three base points, so $\sum_xr_x\le3n^4M$. Given the links outside $V_x$, $F$ varies only through those $r_x$ terms, in an interval of length at most $2r_x/M$, so its conditional variance is at most $r_x^2/M^2$ and \[ \mathcal E^{\varrho_L}_n(F)\le\frac1{n^4M^2}\sum_xr_x^2\le\frac{3(n+1)^4}M =\frac{48(n+1)^4}{L^4}. \] Dividing by $\Var_{\varrho_L}(F)$ gives the bound on $\varpi$. If the sums in the last statement were at most $\alpha n^4$ for every $j$, Theorem~\ref{thm:bpcoarse}(i) would give $\mathrm{BP}_{\varrho_L}(n,1- \alpha)$, so $1-\alpha\le48(n+1)^4/(\epsilon^2s_N^4L^4)$. \end{proof} The sign that chooses between the two products is drawn once for the whole torus.\footnote{Each cube carries a fraction of order $(n/L)^4$ of the evidence for the sign, so resampling cubes changes the spatial average $F$ only slowly, while $F$ itself reads the sign off with an error of order $L^{-2}$.} The measures $\varrho_L$ are not laws of block variables of Wilson measures and nothing is claimed about the tuned laws from them; they show that the hypothesis of Theorem~\ref{thm:bpcoarse}, a bound on the sums of the single-link coefficients over all links, cannot be replaced by the closeness of each single-link conditional law to the Haar measure, and that (H1) needs a collective bound. Part (i) holds for the laws of the block variables on every torus at every coupling; for the Wilson measure $\Omega$ is proportional to the volume, so its constant is not uniform in $L$. Figure~\ref{fig:bphidden} shows the measure of (ii). \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_bphidden.png} \caption{The measure $\varrho_L$ of Proposition~\ref{prop:bphidden}(ii). (a)~One sign $S$, drawn once for the torus, chooses between two product tilts of the selected plaquettes; each link, given the others, has a law within $\epsilon$ of the Haar measure. (b)~The law of the spatial average $F$ for $\SU(2)$, $\epsilon=\frac12$ and $L=8,16,32$, drawn with normal approximations of its two components and scaled to unit peak: the modes at $\pm\epsilon s_2^2$ stay apart as $L$ grows, while resampling one cube moves $F$ by at most $2r_x/M$, so the cube dynamics needs a time of order $L^4$ to change the sign.}\label{fig:bphidden} \end{figure*} \begin{remark}[what is proved about the coarsest condition] \label{rem:bpcoarse} Proved: for every measure with a continuous positive density on the links of a torus, a Dobrushin bound $c_\varrho<1$ on the column sums of the single-link Kantorovich coefficients gives $\mathrm{BP}(n,1-\alpha_n(\varrho))$ on all of $L^2(\varrho)$, with $\alpha_n(\varrho)\le m_\varrho/((1-c_\varrho)n)$ explicit and no locality or product form assumed; the coefficients are bounded by the Hessian of the action, which recovers the leading term of Corollary~\ref{cor:bpranges}(iii) and is stable under perturbations small in the Hessian norms (Theorem~\ref{thm:bpcoarse}, Lemma~\ref{lem:bpcoarsehess}, Corollary~\ref{cor:bpcoarsehess}); the laws of the block variables of the straight-path maps have smooth positive densities, so the criterion applies to them, and (H1) follows for every large $n_0$ from coefficient bounds uniform along the tuning (Lemma~\ref{lem:bpcoarselaw}, Corollary~\ref{cor:bpcoarseH1}); (H1) at the cube size $n_0+1$ follows from $\mathrm{BP}(n_k,\varpi)$, and given (H2) it is equivalent to the conclusion of Corollary~\ref{cor:bpuptuning} up to one unit of cube size (Proposition~\ref{prop:bpcoarsezero}); it forces the decorrelation of the averaged coarse Polyakov loops on every torus and, on the smallest torus, a lower bound on the curvature of the coarse plaquettes, which also obstructs the criterion there (Proposition~\ref{prop:bpcoarsenec}); it is independent of (H2), and it fails for the coarse law of a measure whose local covariances have range $a_0$ (Proposition~\ref{prop:bpcoarseflat}); on each torus every law of block variables of a measure with a continuous positive density satisfies $\mathrm{BP}(n,e^{-\Omega})$, and single-link conditional laws within $\epsilon$ of the Haar measure do not give a constant independent of the torus (Proposition~\ref{prop:bphidden}). Not proved: (H1) along the tuning; the bounds of Corollary~\ref{cor:bpcoarseH1} for the induced actions $H^{(0)}_{k,L}$ for any $k$ with $\beta_k$ outside the strong-coupling range, that is, a single-link Hessian of the induced action of $k$ steps below $\kappa_G$ and summable mixed blocks at the scale where the running coupling is of order one, which is the large-field and effective-action problem of item~(iv) of Sec.~\ref{ssec:remains}; whether $\mathrm{BP}(n_k,\varpi)$ gives (H1) at the cube size $n_0$ itself; the decorrelation of the coarse Polyakov loops of Proposition~\ref{prop:bpcoarsenec}(iii) for the tuned measures. The measures of Proposition~\ref{prop:bpcoarseflat} are flat on the coarse scale and are not Wilson measures, nor are those of Proposition~\ref{prop:bphidden}(ii), and nothing is claimed about the tuned measures from them. For the measures of Corollary~\ref{cor:bprefinetower}, where $\mathrm{BP}$ holds at every scale, Lemma~\ref{lem:bpcoarsehess} gives no conclusion once $K$ is large, since a fine link of a path enters six loops of $4b^K$ links; the criterion is a statement for the coarsest scale. Neither $\mathrm{BP}$ along the tuning nor the mass gap is proved here. \end{remark} \subsection{The single-link coefficients across scales}\label{ssec:bpbalaban} By Corollary~\ref{cor:bpcoarseH1}, (H1) follows from bounds on the single-link coefficients of the laws $\mu^{(0)}_{k,L}$, uniform in $k$ and in the torus. These laws are the image of a Wilson measure under $k$ steps of the straight-path map, and bounds on them are a form of the effective-action problem of Ba{\l}aban's renormalisation group~\cite{Bal84a,Bal85a,Bal89b}. This subsection proves three things about them. First, one step of the straight-path map with blocks of side $b$ contracts a weighted Dobrushin sum by the factor $e^{-\lambda b/2}$, up to a loss in the rate, on every torus and with no locality; along a tower this transfers a Dobrushin bound from any level to the coarsest one, so that (H1) follows from such a bound at one level per $k$. Second, gauge covariance forces a lower bound on the coefficients: every row sum of a gauge-invariant law is at least $2$ minus four times the normalised spread of its single-link conditional laws, so the Dobrushin sum exceeds one for the $\SU(2)$ Wilson measure at $\beta_W\ge\frac83$ and for every block law whose plaquette variables are close to $I$; the step can therefore only start at a level where the holonomies of the blocks are spread out, and it does not start at the finest level along the tuning. Third, the coarse conditional laws are spread out along the tuning for $\SU(2)$, so this obstruction does not apply at the coarsest scale, and the Hessian of the induced action is the curvature of the fine action minus the covariance of the forces in the fibre law. None of this bounds the coefficients of $\mu^{(0)}_{k,L}$ along the tuning. \emph{Notation.} Let $S$ be a finite set and $\varrho$ a probability measure on $G^S$ with a continuous positive density $\rho$ with respect to $\lambda_S$. The kernels $\gamma^\varrho_V$ and the coefficients $C^\varrho_{V,j}$ and $C^\varrho_{ej}$ are defined as in Sec.~\ref{ssec:bpcoarse}, with $S$ in place of $E$, and the proof of Lemma~\ref{lem:bpcoarseblock} holds verbatim. For a pseudometric $d$ on $S$ and $\lambda\ge0$ put \[ c_\lambda(\varrho;d)=\max_{k\in S}\sum_{a\in S\setminus\{k\}}C^\varrho_{ak}\, e^{\lambda d(a,k)} , \] so that $c_0(\varrho;d)=c_\varrho$.\footnote{These are the sums of Dobrushin's uniqueness condition~\cite{Dob68,Dob70} with exponential weights in the distance; Lemma~\ref{lem:bpbalmarginal} below is the expansion in walks that lies behind that condition, applied to a marginal.} For a measure on the links of a torus $\Lambda_M$, $d(e,j)=\lvert\Delta\rvert_\infty$ is the distance of the base points, coordinate by coordinate on the circle $\Z/M\Z$, and we write $c_\lambda(\varrho)$. Since $x\le e^{\lambda x}/(e\lambda)$ for $x\ge0$ and $\lvert\Delta\rvert_1\le4\lvert\Delta\rvert_\infty$, the quantities of Theorem~\ref{thm:bpcoarse}(ii) satisfy \begin{equation}\label{eq:bpbalmoments} m_\varrho\le\frac{4\,c_\lambda(\varrho)}{e\lambda},\qquad t_\varrho(R)\le e^{-\lambda R}\,c_\lambda(\varrho) \end{equation} for $\lambda>0$ and $R\ge1$. \begin{lemma}[marginals]\label{lem:bpbalmarginal} Let $\hat\varrho$ be a probability measure on $G^S$ with a continuous positive density, $d$ a pseudometric on $S$ with $c_\lambda=c_\lambda(\hat \varrho;d)<1$, $A\subset S$ with at least two points, $\varrho_A$ the marginal of $\hat\varrho$ on $G^A$, and $d_A=\min\{d(a,a'):a\ne a'\in A\}$. For $e\in A$ put $S_e=(S\setminus A)\cup\{e\}$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For $e\ne j$ in $A$, with $C=C^{\hat\varrho}$, \[ C^{\varrho_A}_{ej}\le\sum_{m\ge0}\ \sum_{k_1,\dots,k_m\in S_e}C_{ek_1} C_{k_1k_2}\cdots C_{k_mj}, \] the sum over the sequences with consecutive terms distinct. \item For $0\le\kappa\le\lambda$ and every $j\in A$, \[ \sum_{e\in A\setminus\{j\}}C^{\varrho_A}_{ej}\,e^{\kappa d(e,j)}\le e^{-(\lambda-\kappa)d_A}\,\frac{c_\lambda}{1-c_\lambda}. \] \end{enumerate} \end{lemma} \begin{proof} (i) The density $\rho_A$ of $\varrho_A$ is the integral of the density $\hat\rho$ of $\hat\varrho$ over the coordinates in $S\setminus A$; it is continuous and positive, and for every $u$ the law $\gamma^{\varrho_A}_{ \{e\}}(\cdot\mid u)$, with density proportional to $\rho_A(\sigma,u_{A \setminus\{e\}})$, is the image of $\gamma^{\hat\varrho}_{S_e}(\cdot\mid u)$ under the coordinate at $e$. Let $u,u'$ differ only at $j\in A\setminus \{e\}$, put $\delta_j=\delta(u_j,u'_j)$, and let $\varrho^*$ be as in the proof of Lemma~\ref{lem:bpcoarseblock} with $V=S_e$. The Kantorovich distance of the two laws of $\sigma_e$ is at most $\varrho^*(e)$, and that proof gives, for every $k\in S_e$, \[ \varrho^*(k)\le\sum_{i\in S_e\setminus\{k\}}C_{ki}\,\varrho^*(i)+C_{kj}\, \delta_j . \] Let $K$ be the matrix $(C_{ki})_{k,i\in S_e}$ with zero diagonal and $w= (C_{kj})_{k\in S_e}$. Then $\varrho^*\le K\varrho^*+\delta_jw$ entrywise, and, $K$ having nonnegative entries, $\varrho^*\le\delta_j\sum_{m0$, $c<1$ and $k\ge1$, and suppose that for every $L\in\mathcal L_k$ some $j\in\{1,\dots,k\}$ satisfies \[ c_\lambda\bigl(\mu^{(j)}_{k,L}\bigr)\le c,\qquad2^j\lambda\ge4\log\frac{17 c}{1-c} . \] Then $\mathrm{BP}(n_0,\frac12)$ holds for $\mu^{(0)}_{k,L}$ for every $L\in\mathcal L_k$ and every $n_0\ge2$ with $L_0\ge2n_0+4$. If this holds for every $k\ge1$ with the same $\lambda$ and $c$, then (H1) holds with $\varpi_0=\frac12$. \item If $G$ is as in Lemma~\textup{\ref{lem:bpcoarsehess}}, $\delta$ is the geodesic distance of $\ip XY=-\Tr(XY)$, $(36e+6)N\lvert\beta_k\rvert\le \kappa_G$ and $k\ge4$, the hypothesis of \textup{(ii)} holds for this $k$ with $j=k$, $\lambda=1$ and $c=\frac12$. \end{enumerate} \end{corollary} \begin{proof} (i) By Lemma~\ref{lem:bpcoarselaw}(ii) and induction, $c_{i+1}\circ\dots \circ c_j$ is the straight-path map with blocks of side $2^{j-i}$ from $\Lambda_{L_j}$ to $\Lambda_{L_i}$, and $\mu^{(i)}_{k,L}$ is its image of $\mu^{(j)}_{k,L}$, which is gauge invariant and has a continuous positive density by Lemma~\ref{lem:bpcoarselaw}. Apply Theorem~\ref{thm:bpbalstep} with $b=2^{j-i}$, $k_b=2^{j-i-1}$. (ii) By (i) with $i=0$ and $\kappa=\lambda/4$, $\lambda_0=2^{j-2}\lambda$ satisfies \[ c_{\lambda_0}\bigl(\mu^{(0)}_{k,L}\bigr)\le e^{-2^{j-2}\lambda}\frac c{1-c} \le\frac1{17}. \] So the column sums and the tails of the coefficients of $\mu^{(0)}_{k,L}$ are at most $\frac1{17}=(1-\frac1{17})/16$, and the proof of Corollary~\ref{cor:bpcoarseH1}, which treats one $k$ at a time, applies with $c=\frac1{17}$ and $R=1$, for every $n_0\ge\max(2,1)$. (iii) The Wilson measure $\mu_{k,L}=\mu^{(k)}_{k,L}$ has $C_{ej}=0$ unless $e,j$ are partners, which have $\lvert\Delta\rvert_\infty\le1$, and every link has eighteen partners. By Lemma~\ref{lem:bponelink}, $c_1(\mu_{k,L})\le 18eN\lvert\beta_k\rvert/(\kappa_G-6N\lvert\beta_k\rvert)\le\frac12$, and $2^k \ge16>4\log17$. \end{proof} Part (ii) moves the problem by a bounded number of scales and does not remove it: the hypothesis is a Dobrushin bound at the spacing $a_j\le a_0 \lambda/(4\log(17c/(1-c)))$, uniform in $k$, for the law of the block variables of $k-j$ steps. Part (iii) is the descent of Sec.~\ref{ssec:bpblock} seen through the step; along the tuning $\beta_k\to\infty$ and the level $j=k$ is not available, by Corollary~\ref{cor:bpbalwilson} below. The intermediate levels are restricted by gauge covariance. \begin{lemma}[gauge covariance]\label{lem:bpbalcov} Let $\varrho$ be a gauge-invariant probability measure on the links of $\Lambda_L$, $L\ge3$, with a continuous positive density, $e=(x,\nu)$ a link, $S_-(e)$ and $S_+(e)$ the sets of the seven other links with an end point at $x$ and at $x+e_\nu$, and $\gamma=\gamma^\varrho_{\{e\}}(\cdot\mid U)$. For $h\in G$ let $h\gamma$ and $\gamma h$ be the images of $\gamma$ under $\sigma\mapsto h\sigma$ and $\sigma\mapsto\sigma h$, and \[ s(\gamma)=\min_{\tau\in G}\int\delta(\sigma,\tau)\,\gamma(d\sigma) \] the spread of $\gamma$. Then \[ \begin{split} \delta(h,I)-2s(\gamma)\le W_1(\gamma,h\gamma)&\le\delta(h,I)\sum_{k\in S_-(e )}C^\varrho_{ek},\\ \delta(h,I)-2s(\gamma)\le W_1(\gamma,\gamma h)&\le\delta(h,I)\sum_{k\in S_+( e)}C^\varrho_{ek}. \end{split} \] \end{lemma} \begin{proof} Let $g$ be the gauge transformation equal to $h$ at $x$ and to $I$ elsewhere. Then $(gU)_e=hU_e$, the other links at $x$ are multiplied by $h$ on the left or by $h^{-1}$ on the right, and all others are unchanged. Since the density $\rho$ is gauge invariant (proof of Lemma~\ref{lem:bpbalaxial}) and Haar measure is left invariant, $\gamma^\varrho_{\{e\}}(\cdot\mid gU)=h\gamma$. The configurations $U$ and $gU$ differ outside $e$ only on $S_-(e)$, each link by $\delta(h,I)$ by the bi-invariance of $\delta$; changing them one at a time gives the upper bound. For $\gamma h$ take $g$ equal to $h^{-1}$ at $x+e_\nu$. For the lower bound let $\tau$ attain $s(\gamma)$; the function $\delta(\cdot,\tau)$ is $1$-Lipschitz, and by the triangle inequality and bi-invariance \[ \int\delta(h\sigma,\tau)\,\gamma(d\sigma)\ge\delta(h\tau,\tau)-s(\gamma)= \delta(h,I)-s(\gamma), \] so $W_1(\gamma,h\gamma)\ge\delta(h,I)-2s(\gamma)$; likewise with $\delta(\tau h,\tau)=\delta(h,I)$. \end{proof} \begin{theorem}[a lower bound from gauge covariance]\label{thm:bpbalnec} Let $\varrho$ be a gauge-invariant probability measure on the links of $\Lambda_L$, $L\ge3$, with a continuous positive density, and for a link $e$ let $s_e=\inf_Us(\gamma^\varrho_{\{e\}}(\cdot\mid U))$ and $\bar s$ the mean of $s_e$ over the $4L^4$ links. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\sum_{k\in S_-(e)\cup S_+(e)}C^\varrho_{ek}\ge2-4s_e/D$ for every link $e$. \item $c_\varrho\ge2-4\bar s/D$; if $\varrho$ is invariant under the translations, with $s_\nu$ the common value of $s_e$ for the links of direction $\nu$, then $c_\varrho\ge2-(s_0+s_1+s_2+s_3)/D$. \item $\varrho(\delta(U_p,I)\mid U_{E\setminus\{e\}})\ge s_e$ almost surely for every plaquette $p$ and every link $e$ of $p$, and the mean of $\varrho(\delta(U_p,I))$ over the $6L^4$ plaquettes is at least $\bar s\ge D(2-c_\varrho)/4$. \end{enumerate} \end{theorem} \begin{proof} (i) The sets $S_\pm(e)$ are disjoint since $L\ge3$. Take $h$ with $\delta(h,I)=D$ in Lemma~\ref{lem:bpbalcov}: each of the two sums is at least $1-2s(\gamma)/D$ for every $U$, hence at least $1-2s_e/D$. (ii) The sum of all coefficients is the sum of the row sums and the sum of the column sums. So $c_\varrho$, the largest column sum, is at least the mean row sum, which by (i) is at least the mean of $2-4s_e/D$. (iii) The plaquette variable is $U_p=AU_eB$ or $AU_e^{-1}B$ with $A,B$ products of the other links of $p$, and by bi-invariance $\delta(U_p,I)= \delta(U_e,T)$ with $T=A^{-1}B^{-1}$ or $T=BA$, a function of $U_{E\setminus \{e\}}$. Its conditional expectation is $\int\delta(\sigma,T)\,\gamma^\varrho_{ \{e\}}(d\sigma\mid U)\ge s_e$. So $\varrho(\delta(U_p,I))$ is at least the mean of $s_e$ over the four links of $p$; every link lies in six plaquettes, and averaging over $p$ gives $\bar s$. The last inequality is (ii). \end{proof} For the product of the Haar measures $c_\varrho=0$, and the bound is consistent with it: $s_e=\int\delta(\sigma,I)\,d\sigma\ge D/2$, since $\delta(\sigma,I)+\delta(\sigma,a)\ge D$ for $a$ with $\delta(a,I)=D$ and the two terms have the same integral. The bound is informative when the single-link conditional laws are concentrated, which is the regime of the Wilson measure at weak coupling and of every block law in the small-field region. \begin{corollary}[the Wilson measure]\label{cor:bpbalwilson} Let $G=\SU(2)$, $\delta$ the geodesic distance of $\ip XY=-\Tr(XY)$, so that $\delta(Q,I)=\sqrt2\,\theta(Q)$ with $\theta(Q)\in[0,\pi]$ the angle of $Q$ and $D=\sqrt2\pi$, and let $\mu$ be the Wilson measure of $\Lambda_L$ with $B=6\beta_W\ge1$. Then \[ c_\mu\ge2-2\sqrt{\frac3B}-\frac{3\pi^3}4e^{1/2}B^{3/2}e^{-B}, \] and $c_\mu>1.13$ for $\beta_W\ge\frac83$. \end{corollary} \begin{proof} Given $U$, the law of $U_e$ has density proportional to $\exp(\frac{\beta_W} 2\ReTr(U_eS))$, $S$ the sum of the six staples; at $U=I$, $S=6I$, and $\theta=\theta(U_e)$ has the density proportional to $\sin^2\theta\, e^{B\cos\theta}$ on $[0,\pi]$. So $s_e\le\sqrt2\,\E\theta$ for every $e$, and Theorem~\ref{thm:bpbalnec}(ii) gives $c_\mu\ge2-\frac4\pi\E\theta$. The derivative of $\sin^3\theta\,e^{B\cos\theta}$ is $(3\cos\theta-B\sin^2 \theta)\sin^2\theta\,e^{B\cos\theta}$ and vanishes at both ends, so $3\E\cos\theta=B\E\sin^2\theta$ and $\E\sin^2\theta\le3/B$. On $[0,\pi/2]$, $\theta\le\frac\pi2\sin\theta$, so \[ \E\theta\le\frac\pi2\sqrt{\E\sin^2\theta}+\pi P,\qquad P=\Pr(\theta>\pi/2). \] The numerator of $P$ is at most $\int_{\pi/2}^\pi\sin^2\theta\,d\theta= \pi/4$. With $t=B^{-1/2}\le1$, on $[0,t]$ one has $\cos\theta\ge1-t^2/2$ and $\sin\theta\ge2\theta/\pi$, so the denominator is at least $\frac4{3 \pi^2}B^{-3/2}e^{B-1/2}$, and $P\le\frac{3\pi^3}{16}e^{1/2}B^{3/2}e^{-B}$. This gives the bound. For $B\ge16$ both subtracted terms decrease; at $B=16$ the first is $\sqrt3/2<0.8661$ and the second is less than $2455\,e^{-16}< 3\cdot10^{-4}$. \end{proof} So the Dobrushin route of Corollary~\ref{cor:bpcoarseH1} is closed at the finest level along the tuning, for every rate $\lambda\ge0$, since $c_\lambda\ge c_0$, by gauge covariance alone; this is independent of the bound of Lemma~\ref{lem:bponelink}, which is an upper bound. The same argument applies to every block law whose plaquette variables are concentrated. Figure~\ref{fig:bpbalaban} shows the step and the two lower bounds. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_bpbalaban.png} \caption{The single-link coefficients across scales. (a) One step of the straight-path map with blocks of side $b=4$, in a plane: each coarse link is the product of the fine links of its path, marked at the middle link $l(e)$ (red); the law of the block variables is the marginal of the axial-gauge law on the marks, and the influence of $l(j)$ on $l(e)$ is a sum over walks of fine links weighted by products of coefficients (blue), which leave a factor $e^{-(\lambda-2\kappa)\lfloor b/2\rfloor}$ (Lemma~\ref{lem:bpbalmarginal}, Theorem~\ref{thm:bpbalstep}). (b) The lower bound on the Dobrushin sum of the $\SU(2)$ Wilson measure forced by gauge covariance, against $\beta_W$; it exceeds the threshold $1$ for $\beta_W\ge\frac83$ (Corollary~\ref{cor:bpbalwilson}). (c) The lower bound $2-4(1-\varepsilon_{2^k})/\pi$ for the law of the block variables of side $2^k$, along $\beta_{W,k}=2+\frac{11}{3\pi^2}k\log2$, an illustration of the growth along the tuning; below $1$ gauge covariance no longer excludes the Dobrushin condition (Proposition~\ref{prop:bpbalrandom}).} \label{fig:bpbalaban} \end{figure*} \begin{corollary}[small fields]\label{cor:bpbalsmall} Let $\varrho$ be as in Theorem~\textup{\ref{thm:bpbalnec}}, $\varepsilon\ge 0$ and $\eta\ge0$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item If $\varrho(\delta(U_p,I)>\varepsilon)\le\eta$ for every plaquette $p$, then $c_\varrho\ge2-4\varepsilon/D-4\eta$. \item Under the hypotheses of Corollary~\textup{\ref{cor:bpcoarseH1}} with $c<1$, the holonomies $W$ of the squares of side $2^k$ with corners in $2^k\Z^4$ satisfy, for all $k$ and $L\in\mathcal L_k$, the inequality $\langle\mu_{k,L}(\delta(W,I))\rangle\ge D(2-c)/4>D/4$, the mean taken over these squares. \end{enumerate} \end{corollary} \begin{proof} (i) The mean of $\varrho(\delta(U_p,I))$ is at most $\varepsilon+D\eta$; use Theorem~\ref{thm:bpbalnec}(iii). (ii) The plaquette variables of the coarse configuration are the holonomies $W$, and $c_\varrho\le c$ for $\varrho= \mu^{(0)}_{k,L}$; use Theorem~\ref{thm:bpbalnec}(iii). \end{proof} A Dobrushin bound for the law of the block variables therefore requires that the holonomies of the blocks are, in the mean, at more than half the distance from $I$ at which Haar measure puts them; for $\SU(2)$ the Haar mean of $\delta(\sigma,I)$ is $\sqrt2\int_0^\pi\theta\frac2\pi\sin^2\theta \,d\theta=D/2$. The small-field region of Ba{\l}aban's analysis~\cite{Bal84a,Bal85a}, in which every plaquette variable of every block lattice is close to $I$, lies outside the domain of the criterion. At the coarsest scale the obstruction does not apply along the tuning, for $\SU(2)$: the conditional laws of the coarse links are spread out. \begin{proposition}[randomisation along the paths]\label{prop:bpbalrandom} Let $\mu'$ be the probability measure on the links of $\Lambda_{bL}$, $b\ge2$, $L\ge3$, with density proportional to $\prod_p\varphi(U_p)$ for a positive continuous class function $\varphi$, $c$ the straight-path map, $\mu^c=c_*\mu'$ and $\gamma^c_e=\gamma^{\mu^c}_{\{e\}}$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item With $a_\varphi=(\min\varphi/\max\varphi)^6$, for every coarse link $e$ and every $V$, the total variation distance of $\gamma^c_e(\cdot\mid V)$ to $\lambda$ is at most $(1-a_\varphi)^b$, and $s(\gamma^c_e(\cdot\mid V))\ge D(\frac12-(1-a_\varphi)^b)$. \item Let $G=\SU(2)$, $\varphi(Q)=\exp(\frac{\beta_W}2\ReTr Q)$, so that $\mu'$ is the Wilson measure, $\delta$ and $D$ as in Corollary~\textup{\ref{cor:bpbalwilson}}, and $\varepsilon_b=(1+1/(4\lvert \beta_W\rvert))^{-b}$. Then for every coarse link $e$ and every $V$ \[ \Bigl\lVert\int\sigma\,\gamma^c_e(d\sigma\mid V)\Bigr\rVert_{\rm op}\le \varepsilon_b,\qquad s\bigl(\gamma^c_e(\cdot\mid V)\bigr)\ge\frac{D(1- \varepsilon_b)}\pi, \] and the mean of $\mu^c(\delta(V_P,I))$ over the coarse plaquettes $P$ is at least $D(1-\varepsilon_b)/\pi$. \end{enumerate} \end{proposition} \begin{proof} Fix a coarse link $e$. The coarse links other than $e$ are functions of the fine links outside $\pi(e)$. Given the fine links outside $\pi(e)$, the $b$ links of $\pi(e)$ are independent under $\mu'$: they are collinear, so no plaquette contains two of them, and their conditional density is a product of factors, the factor of a link being the product of $\varphi$ over its six plaquettes, normalised. (i) Each factor is at least $a_\varphi$ times its integral,\footnote{This is Doeblin's minorisation: a law bounded below by $a\lambda$ is a mixture with weight $a$ of $\lambda$ and another law.} so the law of each path link is $a_\varphi\lambda+(1-a_\varphi)q_i$ for probability measures $q_i$. Drawing the links independently in this form, with probability $1-(1-a_\varphi)^b$ at least one factor is drawn from $\lambda$, and then the product is a Haar variable by the invariance of $\lambda$; so the law of $c_e$ given the fine links outside $\pi(e)$ is within total variation $(1-a_\varphi)^b$ of $\lambda$. The law of $c_e$ given the other coarse links is a mixture of these laws, and the total variation distance is convex; so the bound holds for $\mu^c$-almost every $V$. The set of laws within total variation $t$ of $\lambda$ is weakly closed, the kernel $\gamma^c_e(\cdot\mid V)$ is weakly continuous in $V$ (Lemma~\ref{lem:bpcoarselaw}(i)), and a set of full $\mu^c$-measure is dense; so the bound holds for every $V$. Since $0\le\delta(\cdot,\tau)\le D$, $\int\delta(\cdot,\tau)\,d\gamma\ge\int\delta(\cdot,\tau)\,d\lambda-D(1- a_\varphi)^b$, and the Haar integral is at least $D/2$, as after Theorem~\ref{thm:bpbalnec}. (ii) Write $\frac12\ReTr$ as the real part of a unit quaternion. A path link $u$ has, given the fine links outside $\pi(e)$, the density proportional to $\exp(\frac{\beta_W}2\ReTr(uS))$, where $S$, a sum of six elements of $\SU(2)$, equals $sW$ with $0\le s\le6$ and $W\in\SU(2)$. The law of $uW$ is a class function times Haar measure, so its mean is the scalar $A=\E_B\cos\theta$, $B=s\beta_W$, and the mean of $u$ is $AW^{-1}$. For $B\ge0$ the identity of the proof of Corollary~\ref{cor:bpbalwilson} and $\E\cos^2\theta\ge A^2$ give $3A+BA^2 \le B$, while $x=2B/(2B+3)$ has $(3x+Bx^2-B)(2B+3)^2=9B\ge0$; as $3x+Bx^2$ increases on $[0,\infty)$, $A\le2B/(2B+3)$, and $B<0$ follows by $\theta\mapsto\pi-\theta$. Hence $\lvert A \rvert\le2\lvert B\rvert/(2\lvert B\rvert+3)\le(1+1/(4\lvert\beta_W\rvert) )^{-1}$. The mean of the product of the $b$ independent path links is the product of their means, of operator norm at most $\varepsilon_b$; mixing and continuity as in (i) give the first bound for every $V$. For the second, $\theta\ge1-\cos\theta$ on $[0,\pi]$ and $\lvert\Tr X\rvert\le2 \lVert X\rVert_{\rm op}$ give, for every $\tau$ and $M=\int\sigma\,d\gamma$, \[ \int\delta(\sigma,\tau)\,d\gamma\ge\sqrt2\Bigl(1-\frac12\ReTr(\tau^{-1}M) \Bigr)\ge\sqrt2\,(1-\varepsilon_b), \] which is $D(1-\varepsilon_b)/\pi$. The last claim is Theorem~\ref{thm:bpbalnec}(iii). \end{proof} Along the tuning $b=2^k$, $\varepsilon_{2^k}\le\exp(-2^k/(4\lvert\beta_{W,k} \rvert+1))$, which tends to zero when $2^{-k}\beta_{W,k}\to0$, in particular under the growth $\beta_{W,k}\sim\frac{11}{3\pi^2}\,k\log2$ that the renormalisation group gives for the tuning~\cite{GW73,Pol73}.\footnote{With $\beta_W=2N/g^2$ and the one-loop running $g^{-2}=\frac{11N}{24\pi^2}\log a^{-1}+O(\log\log a^{-1})$, $\beta_W\sim\frac{11N^2}{12\pi^2}\log a^{-1}$, which is $\frac{11}{3\pi^2}\log a^{-1}$ for $\SU(2)$; with $a=2^{-k}a_0$ this is the growth quoted.} Once $\varepsilon_{2^k}<1-\pi/4$ the mean coarse plaquette distance exceeds $D/4$, the condition of Corollary~\ref{cor:bpbalsmall}(ii) holds, and Theorem~\ref{thm:bpbalnec} excludes only $c_\varrho<2-4(1-\varepsilon_{2^k})/\pi$, a bound that tends to $2-4/\pi<0.73$. So gauge covariance does not obstruct a Dobrushin bound at the coarsest scale. Part (i) gives nothing along the tuning, since $a_\varphi=e^{-12\lvert\beta_W\rvert}$ for $\SU(2)$ and $2^ke^{-12\beta_{W,k}} \to0$ under that growth. Neither part bounds the coefficients: the spread of the coarse conditional laws says nothing about their Lipschitz dependence on the exterior. That dependence is governed by the Hessian of the induced action. \begin{proposition}[the Hessian of the induced action]\label{prop:bpbalhess} Let $\mu'$ have the density $e^{-\mathcal S}/Z$ with respect to $\lambda_{E'}$, $\mathcal S$ gauge invariant and of class $C^2$, let $l$ be a marking, and for a fine link $a$ and $X\in\g$ let $D^a_X\mathcal S(U)= \frac d{dt}\mathcal S(U^{a,tX})|_{t=0}$. Then $\mu^c=c_*\mu'$ has the density $e^{-H^c}$ with \[ H^c(V)=-\log\int e^{-\mathcal S(\Theta^l(V,m))}\,\lambda_{E'_0}(dm)+\mathrm {const}. \] Let $\varsigma_V$ be the probability measure on $G^{E'_0}$ with density proportional to $e^{-\mathcal S(\Theta^l(V,m))}$, let $a=l(e)$ and $a'= l(e')$, and take the derivatives of $\mathcal S$ at $\Theta^l(V,m)$. Then \[ \begin{split} \frac{d^2}{dt^2}H^c(V^{e,tX})\Big|_0&=\varsigma_V(D^a_XD^a_X\mathcal S)- \Var_{\varsigma_V}(D^a_X\mathcal S),\\ \frac{\partial^2H^c((V^{e,sX})^{e',tY})}{\partial s\,\partial t}\Big|_0&= \varsigma_V(D^{a'}_YD^a_X\mathcal S)\\ &\quad-\Cov_{\varsigma_V}(D^a_X\mathcal S,D^{a'}_Y\mathcal S) \end{split} \] for $e\ne e'$. Let $-h_e$ be the infimum over $V$ and unit $X$ of the first right side and $J_{ee'}$ the supremum over $V$ and unit $X,Y$ of the absolute value of the second; if $h_e<\kappa_G$, Lemma~\textup{\ref{lem:bpcoarsehess}} applies to $\mu^c$ with these numbers. For the Wilson action the term $\varsigma_V(D^{a'}_YD^a_X\mathcal S)$ vanishes unless $a$ and $a'$ are partners. \end{proposition} \begin{proof} By the proof of Lemma~\ref{lem:bpbalaxial}, $\nu_V$ is the image of $\lambda\otimes\lambda_{E'_0}$ under $(h,m)\mapsto h^{-1}\Theta^l(V,m)$, and $\mathcal S$ is gauge invariant, so $\nu_V(e^{-\mathcal S})=\int e^{-\mathcal S(\Theta^l(V,m))}\,dm$; this is the density of Lemma~\ref{lem:bpcoarselaw}(i). Replacing $V_e$ by $V_ee^{tX}$ replaces the link $a$ of $\Theta^l(V,m)$ by the same product, so $\frac d{dt}$ of $\mathcal S(\Theta^l(V^{e,tX},m))$ is $D^a_X\mathcal S$ at the moved configuration. Differentiating $-\log Z(t)$ under the integral, which is allowed for $C^2$ functions on a compact group, gives $\frac{Z''}{Z}-(\frac{Z'}Z)^2$ with the signs shown; the mixed derivative is the same computation in two variables. Distinct coarse links have distinct marks, since the paths are disjoint. For the Wilson action the mixed derivative of $\mathcal S$ at $a\ne a'$ comes from the plaquettes that contain both. \end{proof} The two terms of $d^2H^c$ are of the same order. Under a single-link conditional law $\gamma\propto e^{-\mathcal S}\,d\sigma$ of the link $a$, integration by parts with the left-invariant vector field of $X$, which preserves Haar measure, gives $\gamma((D^a_X\mathcal S)^2)=\gamma(D^a_XD^a_X \mathcal S)$, so there the mean curvature equals the second moment of the force and is at least its variance. In the fibre law $\varsigma_V$ the link $a$ is held fixed and the variance is that of the force on it produced by the fluctuations of the links of $E'_0$; the hypothesis $h_e<\kappa_G$ of Lemma~\ref{lem:bpcoarsehess} asks that this variance stay below the mean curvature plus $\kappa_G$, uniformly in the exterior, and the summability of $J_{ee'}$ asks that the forces on distant marks decorrelate in $\varsigma_V$. For the Wilson action at large $\beta$ both terms are of order $N\lvert\beta\rvert$ on the flat configuration, so the bound requires a cancellation between them to precision $\kappa_G$; this is the effective-action problem in the form posed by Corollary~\ref{cor:bpcoarseH1}. For $\SU(2)$ and the straight-path map the cancellation holds along every tuning with $\beta_k=O(k)$ (Corollary~\ref{cor:bppathcancel}), and the decorrelation is what remains. \begin{remark}[what is proved about the coefficients across scales] \label{rem:bpbalaban} Proved: the marginal of a measure with a weighted Dobrushin sum $c_\lambda< 1$ on a set of points at mutual distance at least $d_A$ has weighted sums at most $e^{-(\lambda-\kappa)d_A}c_\lambda/(1-c_\lambda)$ (Lemma~\ref{lem:bpbalmarginal}); the law of the block variables of a gauge-invariant measure is such a marginal, through the axial gauge, with coefficients bounded by the fine ones (Lemma~\ref{lem:bpbalaxial}); one step of the straight-path map of side $b$ contracts the weighted sum by $e^{-(\lambda-2\kappa)\lfloor b/2\rfloor}$ on every torus (Theorem~\ref{thm:bpbalstep}); along a tower of straight-path maps a bound $c_\lambda\le c<1$ at one level $j$ with $2^j\lambda\ge4\log(17c/(1-c))$, for every $k$, gives (H1) with $\varpi_0=\frac12$ for every $n_0\ge2$, and the Wilson measure in the strong-coupling range supplies it with $j=k\ge4$ (Corollary~\ref{cor:bpbaltower}); gauge covariance bounds every row sum below by $2-4s_e/D$ and the Dobrushin sum below by $2-4\bar s/D$, and the mean plaquette distance by $D(2-c_\varrho)/4$ (Lemma~\ref{lem:bpbalcov}, Theorem~\ref{thm:bpbalnec}); the $\SU(2)$ Wilson measure has $c_\mu>1.13$ for $\beta_W\ge\frac83$, and every law with concentrated plaquette variables fails the Dobrushin condition (Corollaries~\ref{cor:bpbalwilson} and~\ref{cor:bpbalsmall}); for $\SU(2)$ the coarse conditional laws are spread out along the tuning, so that gauge covariance does not obstruct the condition at the coarsest scale (Proposition~\ref{prop:bpbalrandom}); the Hessian of the induced action is the mean curvature of the fine action minus the covariance of the forces in the fibre law (Proposition~\ref{prop:bpbalhess}). Every numerical value is derived by hand in the proofs. Not proved: a weighted Dobrushin bound $c_\lambda<1$, uniform in $k$ and the torus, for the law of the block variables at any level of the tower with $\beta_k$ outside the strong-coupling range, and hence (H1) along the tuning; bounds on the fibre variances and covariances of Proposition~\ref{prop:bpbalhess} at weak coupling; any Lipschitz bound for the dependence of the coarse conditional laws on the exterior along the tuning; total variation randomisation of the coarse links along the tuning for groups other than $\SU(2)$, beyond the Doeblin bound of Proposition~\ref{prop:bpbalrandom}(i); the abelian and Villain cases, which are not treated. Neither $\mathrm{BP}$ along the tuning nor the mass gap is proved here. \end{remark} \subsection{The coarsest condition through the fibre law}\label{ssec:bpfibre} This subsection approaches (H1) through the fibre law of Proposition~\ref{prop:bpbalhess}. It proves four things. First, a single-link coefficient of every measure with a density $e^{-H}$, $H\in C^2$, is at most $D^2/4$ times the bound on the mixed derivatives of $H$ in the two links, with no condition on the single-link Hessian and for every connected group, abelian or not; gauge invariance, conversely, bounds the single-link Hessian by the mixed derivatives into the seven links at one end point. Second, for straight-path maps with blocks of side at least three the marks of distinct coarse links are never partners, so the mixed derivatives of the induced action are minus the covariances of the forces on distinct marks in the fibre law; (H1) follows from a decorrelation of these forces, uniform in the coarse data and in the torus, the condition (FC) below, and (FC) in turn forces the force variance on a mark to equal its mean curvature up to $4c/D^2$. Third, in a Gaussian abelian model of the fibre the covariance of the forces on distinct marks is minus the coupling times the effective quadratic form of the straight-path map; this form is a conserved current on the coarse lattice, its diagonal lies between $1/b^2$ and $6/b$, its row sums are at least twice the diagonal and are bounded uniformly in the torus by an infinite-volume quantity $\gamma_b$, and a bound on $\gamma_b$ gives the Gaussian form of (FC); Sec.~\ref{ssec:bprowsum} shows that $\gamma_b$ is of order $(\log b)/b$. Fourth, the strong-coupling range, the group $\U(1)$ and rough coarse data of $\SU(2)$ are placed with respect to (FC). (FC) along the tuning is not proved here. Throughout, $G\subset\U(N)$ is closed and connected, $\delta$ is the geodesic distance of $\ip XY=-\Tr(XY)$ on $G$ and $D$ its diameter; for $\SU(2)$, $D=\sqrt2\pi$ (Corollary~\ref{cor:bpbalwilson}), and for $\U(1)$, where $\delta(e^{i\theta},1)=\lvert\theta\rvert$ for $\lvert\theta\rvert\le\pi$, $D=\pi$. For $H\in C^2(G^E)$ on the links of a torus and links $e\ne j$ put \[ J_{ej}(H)=\sup\Bigl\lvert\frac{\partial^2}{\partial s\,\partial t}H\bigl((U^{e, sX})^{j,tY}\bigr)\Big|_{0}\Bigr\rvert , \] the supremum over all $U$ and all unit $X,Y\in\g$, with $U^{e,X}$ as in Lemma~\ref{lem:bpcoarsehess}; then $J_{ej}=J_{je}$, and these are the smallest numbers admissible in the second hypothesis of that lemma. \begin{lemma}[oscillation]\label{lem:bpfibosc} Let $\varrho$ be the probability measure on $G^E$ with density proportional to $e^{-H}$ with respect to $\lambda_E$, $H\in C^2(G^E)$. Then $C^\varrho_{ej}\le\frac{D^2}4J_{ej}(H)$ for all links $e\ne j$. \end{lemma} \begin{proof} Let $U,U'$ differ only at $j$, $\rho=\delta(U_j,U'_j)>0$, $\gamma=\gamma^\varrho_{ \{e\}}(\cdot\mid U)$ and $\gamma'=\gamma^\varrho_{\{e\}}(\cdot\mid U')$. Then $\gamma'$ has the density $e^{-\Delta}/Z$ with respect to $\gamma$, where $\Delta(\sigma)=H(\sigma,U'_{E\setminus\{e\}})-H(\sigma,U_{E\setminus\{e\}})$ and $Z=\int e^{-\Delta}d\gamma$. \emph{The oscillation of $\Delta$.} Since $G$ is compact and connected and the metric is bi-invariant, two points $\sigma',\sigma$ are joined by a minimising geodesic $s\mapsto\sigma'e^{sX}$, $0\le s\le r=\delta(\sigma,\sigma') \le D$, $\lvert X\rvert=1$; likewise $U'_j=U_je^{\rho Y}$ with $\lvert Y\rvert =1$. Let $f(s,t)$ be the value of $H$ at the configuration with $\sigma'e^{sX}$ at $e$, $U_je^{tY}$ at $j$ and the links of $U$ elsewhere. Then \[ \begin{split} \Delta(\sigma)-\Delta(\sigma')&=f(r,\rho)-f(r,0)-f(0,\rho)+f(0,0)\\ &=\int_0^r\!\!\int_0^\rho\partial_s\partial_tf\,dt\,ds , \end{split} \] and since $\sigma'e^{(s+s')X}=\sigma'e^{sX}e^{s'X}$, $\partial_s\partial_tf(s, t)$ is a mixed derivative as in the definition of $J_{ej}(H)$, at the configuration indexed by $(s,t)$. Hence $\omega=\sup\Delta-\inf\Delta\le DJ_{ej}(H)\rho$. \emph{Total variation.} Let $A$ be a Borel set with $p=\gamma(A)\in(0,1)$, and put $x=p^{-1}\int_Ae^{-\Delta}d\gamma$, $y=(1-p)^{-1}\int_{G\setminus A} e^{-\Delta}d\gamma$, $t=x/y\in[e^{-\omega},e^\omega]$ and $q=e^\omega$. Then \[ \gamma'(A)-\gamma(A)=\frac{p(1-p)(t-1)}{pt+1-p}\le\frac{p(1-p)(q-1)}{pq+1-p}, \] because $t\mapsto(t-1)/(pt+1-p)$ has the derivative $(pt+1-p)^{-2}>0$. Let $T=(\sqrt q-1)/(\sqrt q+1)=\tanh(\omega/4)$. Then $1-T=2/(\sqrt q+1)$ and $(q-1)(1-T)^2=4T$, so \[ \begin{split} &T(pq+1-p)-p(1-p)(q-1)\\ &\qquad=(q-1)p^2-(q-1)(1-T)p+T \end{split} \] is a quadratic polynomial in $p$ with leading coefficient $q-1\ge0$ and discriminant $(q-1)\bigl((q-1)(1-T)^2-4T\bigr)=0$, hence nonnegative. So $\gamma'(A)-\gamma(A)\le T$, and the same bound for $G\setminus A$ gives $\gamma(A)-\gamma'(A)\le T$; the cases $p\in\{0,1\}$ are trivial, as $\gamma$ and $\gamma'$ are mutually absolutely continuous. Thus $\lvert\gamma'(A)-\gamma(A)\rvert\le\tanh(\omega/4)$ for every $A$.\footnote{The bound is sharp: for $\Delta$ equal to $\omega$ on $A$ and to $0$ elsewhere and $p=1/(1+e^{-\omega/2})$, $\gamma(A)-\gamma'(A)= \tanh(\omega/4)$.} \emph{The Kantorovich distance.} A $1$-Lipschitz function $g$ on $G$ has oscillation at most $D$; with $m$ the midpoint of its range, $\lvert g-m \rvert\le D/2$, so $\lvert\int g\,d(\gamma'-\gamma)\rvert=\lvert\int(g-m)\, d(\gamma'-\gamma)\rvert$ is at most $D/2$ times the total mass of $\lvert \gamma'-\gamma\rvert$, which is at most $2\tanh(\omega/4)$. By the Kantorovich--Rubinstein duality and $\tanh x\le x$, \[ W_1(\gamma,\gamma')\le D\tanh\Bigl(\frac{DJ_{ej}(H)\rho}4\Bigr)\le\frac{D^2}4 J_{ej}(H)\,\rho .\qedhere \] \end{proof} Lemma~\ref{lem:bpcoarsehess} gives $J_{ej}/(\kappa_G-h_e)$; the two bounds agree when $\kappa_G-h_e=4/D^2$, which for $\SU(2)$ is $2/\pi^2$, and Lemma~\ref{lem:bpfibosc} is the stronger one beyond that, in particular for $h_e\ge\kappa_G$, where Lemma~\ref{lem:bpcoarsehess} gives nothing, and for $\U(1)$, where $\kappa_G=0$. The single-link Hessian does not disappear from the problem, however: for gauge-invariant $H$ it is controlled by the mixed derivatives. \begin{lemma}[gauge covariance of the Hessian]\label{lem:bpfibgauge} Let $H\in C^2(G^E)$ be gauge invariant on the links of $\Lambda_L$, $L\ge3$, let $e$ be a link and $S_\pm(e)$ as in Lemma~\textup{\ref{lem:bpbalcov}}. Then for every $U$ and $X\in\g$ \[ \Bigl\lvert\frac{d^2}{dt^2}H(U^{e,tX})\Big|_0\Bigr\rvert\le\lvert X\rvert^2 \sum_{k\in S_+(e)}J_{ek}(H), \] and if $G$ is abelian the same holds with $S_-(e)$ in place of $S_+(e)$. \end{lemma} \begin{proof} Write $D^k_ZH(U)=\frac d{dt}H(U^{k,tZ})|_{t=0}$. Let $e=(x,\nu)$, $y=x+e_\nu$, $Z\in\g$, and let $g_t$ be the gauge transformation equal to $e^{tZ}$ at $y$ and to $I$ elsewhere. It replaces $U_k$ by $U_ke^{-tZ}$ for the four links $k$ that end at $y$, among them $e$, and $U_k$ by $e^{tZ}U_k=U_ke^{tW_k}$, $W_k=U_k^{-1}ZU_k$, for the four links that start at $y$, and leaves the others unchanged. Differentiating $H(g_tU)=H(U)$ at $t=0$ gives, for every $U$, \[ D^e_ZH(U)=-\sum_kD^k_ZH(U)+\sum_{k'}D^{k'}_{W_{k'}}H(U), \] the first sum over the three other links that end at $y$, the second over the four that start there; these seven are $S_+(e)$, $\lvert W_{k'}\rvert= \lvert Z\rvert$, and since $L\ge3$ no link starting at $y$ is $e$, so $W_{k'}$ does not depend on $U_e$. Replace $U$ by $U^{e,sX}$ and differentiate at $s=0$. On the left, with $Z=X$, $\frac d{ds}D^e_XH(U^{e,sX})|_0 =\frac{d^2}{dt^2}H(U^{e,tX})|_0$, since $U_ee^{sX}e^{tX}=U_ee^{(s+t)X}$; on the right every term is a mixed derivative in $e$ and a link of $S_+(e)$, bounded by $J_{ek}(H)\lvert X\rvert^2$. For abelian $G$ the transformation at $x$ multiplies every link at $x$ by $e^{\pm tZ}$ with no dependence on the link, and the same computation applies with $S_-(e)$. \end{proof} Combined with Proposition~\ref{prop:bpbalhess} and the marking of Theorem~\ref{thm:bpbalstep}, these lemmas reduce the coefficients of the law of the block variables to covariances in the fibre law. For $b\ge3$ let $k_b=\lfloor b/2\rfloor$, so that $1\le k_b\le b-2$, and let $l_b$ be the marking with $l_b(y,\nu)$ the link of $\pi(y,\nu)$ with base point $by+k_be_\nu$. \begin{proposition}[the fibre covariance]\label{prop:bpfibreduce} Let $c$ be the straight-path map with blocks of side $b\ge3$ from $\Lambda_{bL}$ to $\Lambda_L$, $L\ge3$, let $\mu'$ be the Wilson measure of $\Lambda_{bL}$ with action $\mathcal S=-N\beta\sum_p\ReTr U_p$, $\mu^c=c_*\mu'$ with the induced action $H^c$, and let $\varsigma_V$ be the fibre law of Proposition~\textup{\ref{prop:bpbalhess}} for the marking $l=l_b$. For a coarse link $e$ write $a=l_b(e)$, and put \[ K_{ee'}=\sup_{V}\sup_{\lvert X\rvert=\lvert Y\rvert=1}\bigl\lvert\Cov_{ \varsigma_V}\bigl(D^a_X\mathcal S,D^{a'}_Y\mathcal S\bigr)\bigr\rvert , \] with $a'=l_b(e')$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For $e\ne e'$ the marks $l_b(e)$ and $l_b(e')$ are not partners. \item $J_{ee'}(H^c)=K_{ee'}$ for $e\ne e'$. \item $C^{\mu^c}_{ee'}\le\frac{D^2}4K_{ee'}$ for $e\ne e'$. \item For every coarse link $e$, every $V$ and every unit $X$, \[ \bigl\lvert\varsigma_V(D^a_XD^a_X\mathcal S)-\Var_{\varsigma_V}(D^a_X\mathcal S) \bigr\rvert\le\sum_{e'\in S_+(e)}K_{ee'}, \] and for abelian $G$ also with $S_-(e)$ in place of $S_+(e)$. \end{enumerate} \end{proposition} \begin{proof} (i) Both end points of $l_b(e)$, $by+k_be_\nu$ and $by+(k_b+1)e_\nu$, are inner points of $\pi(e)$, since $1\le k_b$ and $k_b+1\le b-1$, and an inner point lies on exactly one path. Two links of different directions in a common plaquette share an end point, so marks of different directions are not partners. Two links of the same direction in a common plaquette have base points that differ by $\pm e_\rho$ for a direction $\rho$, while the base points of $l_b(y,\nu)$ and $l_b(y',\nu)$ differ by $b(y-y')$, all of whose coordinates are divisible by $b$ modulo $bL$. (ii) The action $\mathcal S$ is a sum of functions of the four links of a plaquette, so $D^{a'}_YD^a_X\mathcal S=0$ unless $a$ and $a'$ are partners, and by (i) the mixed derivative of Proposition~\ref{prop:bpbalhess} is $-\Cov_{\varsigma_V}(D^a_X\mathcal S,D^{a'}_Y\mathcal S)$. (iii) By Lemma~\ref{lem:bpcoarselaw}(i), $\mu^c$ has the density $e^{-H^c}$ with $H^c$ smooth; apply Lemma~\ref{lem:bpfibosc} and (ii). (iv) The density of $\mu^c$ is gauge invariant (proof of Lemma~\ref{lem:bpbalaxial}), hence so is $H^c$. By Proposition~\ref{prop:bpbalhess} the left side is the second derivative of $H^c$ along $V^{e,tX}$; apply Lemma~\ref{lem:bpfibgauge} and (ii). \end{proof} \begin{corollary}[fibre decorrelation gives (H1)]\label{cor:bpfibH1} In Corollary~\textup{\ref{cor:bpuptuning}} let the $c_j$ be straight-path maps, let $H^{(0)}_{k,L}$ be the induced action of Lemma~\textup{\ref{lem:bpcoarselaw}} and $J^{(k,L)}_{ej}=J_{ej}(H^{(0)}_{k,L})$. Suppose that the following condition \textup{(FC)} holds: there are $c<1$ and $R\ge1$ with \[ \frac{D^2}4\sum_{e\ne j}J^{(k,L)}_{ej}\le c,\qquad\frac{D^2}4\sum_{e:\,\lvert \Delta\rvert_\infty>R}J^{(k,L)}_{ej}\le\frac{1-c}{16} \] for all $k$, all $L\in\mathcal L_k$ and all links $j$ of $\Lambda_{L_0}$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Condition \textup{(H1)} holds with $\varpi_0=\frac12$ for every $n_0\ge\max(2,16Rc/(1-c))$ for which every $L\in\mathcal L_k$ has $L_0\ge2n_0+4$. \item For $k\ge2$, $J^{(k,L)}_{ej}$ is the number $K_{ej}$ of Proposition~\textup{\ref{prop:bpfibreduce}} for the straight-path map with blocks of side $2^k$ from $\Lambda_L$ to $\Lambda_{L_0}$ and the Wilson measure $\mu_{k,L}$; for $k=0$ it is at most $N\lvert\beta_0\rvert$ for partners and vanishes otherwise. \item For every link $e$, every configuration $V$ and every unit vector $X$, $\lvert\frac{d^2}{dt^2}H^{(0)}_{k,L}(V^{e,tX})|_0\rvert\le4c/D^2$; for $k\ge2$ this is $\lvert\varsigma_V(D^a_XD^a_X\mathcal S_k)-\Var_{\varsigma_V}(D^a_X \mathcal S_k)\rvert\le4c/D^2$ for the marks $a$ and the Wilson action $\mathcal S_k$ at $\beta_k$. \end{enumerate} \end{corollary} \begin{proof} (i) By Lemma~\ref{lem:bpfibosc} and Lemma~\ref{lem:bpcoarselaw}, the coefficients of $\mu^{(0)}_{k,L}$ satisfy $C^{(k,L)}_{ej}\le\frac{D^2}4J^{(k,L) }_{ej}$, and the first statement of Corollary~\ref{cor:bpcoarseH1} applies. (ii) By Lemma~\ref{lem:bpcoarselaw}, $\mu^{(0)}_{k,L}=c^{(k)}_*\mu_{k,L}$ with $c^{(k)}$ the straight-path map with blocks of side $2^k\ge4$; use Proposition~\ref{prop:bpfibreduce}(ii). For $k=0$, $H^{(0)}_{0,L}$ is the Wilson action at $\beta_0$, and Lemma~\ref{lem:mixed} bounds its mixed derivatives. (iii) Since $J^{(k,L)}$ is symmetric, its row sums are column sums, at most $4c/D^2$; apply Lemma~\ref{lem:bpfibgauge} and, for $k\ge2$, Proposition~\ref{prop:bpfibreduce}(iv). \end{proof} So (FC) is a decorrelation statement for the forces on distinct marks in the fibre law, uniform in the coarse data $V$, in $k$ and in the torus. Part (iii) shows that the oscillation lemma removes the hypothesis $h_e<\kappa_G$ of Lemma~\ref{lem:bpcoarsehess} from the criterion but not from the problem: through gauge covariance, (FC) forces the variance of the force on every mark in the fibre law to equal its mean curvature, both of order $N\lvert\beta_k\rvert$, up to $4c/D^2$, which is $2c/\pi^2$ for $\SU(2)$, uniformly in $V$. This is the cancellation described after Proposition~\ref{prop:bpbalhess}. \begin{remark}[strong coupling and Dobrushin bounds in the fibre] \label{rem:bpfibstrong} By Lemma~\ref{lem:bpwilsonfibre}(ii) the fibre law $\varsigma_V$ is the Wilson specification $\gamma^0_{E'_0}(\cdot\mid\Theta^l(V,\cdot))$, so its single-link coefficients are those of the Wilson measure. If these satisfy $C^0_{ee'}\le\mathfrak c$ for all partners, with $\xi=18\mathfrak c<1$, Lemma~\ref{lem:bpdecay}(ii) applies to $\varsigma_V$. The force $D^a_X\mathcal S$ depends only on the links of $E'_0$ in the six plaquettes of $a$, which are partners of $a$, and $\lvert D^a_X\mathcal S\rvert\le6N^{3/2}\lvert\beta \rvert\lvert X\rvert$, since $\lvert\ReTr(WX)\rvert\le\lVert W\rVert_{\rm HS} \lVert X\rVert_{\rm HS}=\sqrt N\lvert X\rvert$ for unitary $W$. Partners have base points at distance at most one in $\lvert\cdot\rvert_\infty$, so the two sets of links for marks $a=l_b(e)$, $a'=l_b(e')$ are at chain distance at least $d_{ee'}=\max(k_b,b\lvert\Delta\rvert_\infty-k_b)-2$, by the distance bound in the proof of Theorem~\ref{thm:bpbalstep}. Hence, for $b\ge6$, \[ K_{ee'}\le\frac{36N^3\beta^2}{1-\xi}\,\xi^{d_{ee'}} \] uniformly in $V$ and $L$, and the sums of (FC) for this $b$ tend to zero as $b\to\infty$ at fixed $\beta$ and $\xi$. This is the range of Corollary~\ref{cor:bpbaltower}(iii) and adds nothing along the tuning. Outside it a Dobrushin argument for the fibre law itself is excluded by gauge covariance: a link of $E'_0$ whose two end points lie on no path has all the other links at those points in $E'_0$, the gauge transformations at those points preserve $\varsigma_V$, and the proofs of Lemma~\ref{lem:bpbalcov} and Theorem~\ref{thm:bpbalnec}(i) give it a row sum of single-link coefficients under $\varsigma_V$ of at least $2-4s/D$, with $s$ the infimum of the spreads of its conditional laws; at $V=I$, $m=I$ these are the laws of the proof of Corollary~\ref{cor:bpbalwilson}, so for $\SU(2)$ at $\beta_W\ge\frac83$ the row sum exceeds $1.13$. The decay required by (FC) concerns the forces on the marks, not the single links of the fibre, and the Gaussian model below shows the difference. \end{remark} \emph{A Gaussian model of the fibre.} Let $b\ge3$, $L\ge3$ and $\beta>0$. On $\Lambda_{bL}$ consider real $1$-forms $A\in\R^{E'}$, the plaquette variables $(dA)_p=A_{(x,\mu)}+A_{(x+e_\mu,\nu)}-A_{(x+e_\nu,\mu)}-A_{(x,\nu)}$ for the plaquette $p$ at $x$ in the plane $\mu<\nu$, $K=d^{\rm T}d$ on $\R^{E'}$ and the action $S(A)=\frac\beta2\lvert dA\rvert^2$, the quadratic part of the $\U(1)$ Wilson action $-\beta\sum_p\cos\theta_p$ in the angles. Let $P$ be the set of path links, $l=l_b$, $E'_0$ as before, and $\Sigma$ the set of sites on paths, the block corners and the inner points. For $v\in\R^E$ and $m\in\R^{E'_0}$ let $\Theta(v,m)\in\R^{E'}$ be $v_e$ at $l(e)$, $0$ on $P\setminus l(E)$ and $m$ on $E'_0$; the force on a mark $a$ is $F_a=\partial S/\partial A_a=\beta(K\Theta)_a$, and for a link $a$ let $\delta_a$ be its indicator. \begin{lemma}\label{lem:bpfibker} Let $\mathcal N$ be the set of restrictions to $E'_0$ of the forms $d\phi$ with $\phi$ a function on the sites of $\Lambda_{bL}$ vanishing on $\Sigma$. Then every such $d\phi$ vanishes on $P$, and $d\Theta(0,m)=0$ exactly when $m\in\mathcal N$. \end{lemma} \begin{proof} A path link joins two points of $\Sigma$, so $d\phi$ vanishes on it, and $\Theta(0,d\phi|_{E'_0})=d\phi$ is closed. Conversely let $\omega=\Theta(0,m)$ be closed, and lift it to a $bL$-periodic closed form on $\Z^4$. On $\Z^4$ the sum of $\omega$ along a lattice path from $0$ to $x$ does not depend on the path, since two such paths differ by a sum of boundaries of plaquettes, so $\omega=d\phi$. The lift of $\Sigma$ is connected by lifted path links, on which $\omega$ vanishes, so $\phi$ is constant there and may be taken to vanish there. For $z\in bL\Z^4$ the function $x\mapsto\phi(x+z) -\phi(x)$ has zero differential and vanishes on the lift of $\Sigma$, so $\phi$ is periodic and $m=d\phi|_{E'_0}$. \end{proof} Since $d\Theta(v,m+n)=d\Theta(v,m)$ for $n\in\mathcal N$, and $m\mapsto\lvert d\Theta(0,m)\rvert^2$ is positive definite on the orthogonal complement $\mathcal N^\perp$ of $\mathcal N$ in $\R^{E'_0}$, the \emph{Gaussian fibre law} $\varsigma_v$ is defined as the probability measure on $\mathcal N^\perp$ with density proportional to $e^{-S(\Theta(v,m))}$ with respect to Lebesgue measure; the forces depend on $m$ only through $d\Theta(v,m)$. Let $\Gamma$ be the symmetric matrix on $\R^E$ with $\ip v{\Gamma v}=\min_m \lvert d\Theta(v,m)\rvert^2$, the \emph{effective quadratic form}: completing the square, $\frac\beta2\ip v{\Gamma v}$ is, up to a constant, the induced action $-\log\int_{\mathcal N^\perp}e^{-S(\Theta(v,m))}dm$, and its mixed derivatives in $v_e$, $v_{e'}$ are $\beta\Gamma_{ee'}$. \begin{proposition}[the Gaussian fibre law]\label{prop:bpfibgauss} In the Gaussian model, with $a=l(e)$ and $a'=l(e')$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Under $\varsigma_v$, $\Var(F_a)=\beta(6-\Gamma_{ee})$ and $\Cov(F_a, F_{a'})=-\beta\Gamma_{ee'}$ for $e\ne e'$, for every $v$. Under the Gaussian law on the orthogonal complement of $\ker K$ in $\R^{E'}$ with density proportional to $e^{-S}$, without a fixed skeleton, the forces $\beta(KA)_a$ have $\Cov(\beta(KA)_a,\beta(KA)_{a''})=\beta K_{aa''}$ for all links $a,a''$, which vanishes unless $a=a''$ or they are partners. \item Let $m_v$ attain the minimum and $\mathcal J=K\Theta(v,m_v)$. Then $\mathcal J$ vanishes on $E'_0$, is constant on each path $\pi(e)$ with the value $(\Gamma v)_e$, and these values form a co-closed $1$-form on the coarse torus: $\sum_\nu\bigl((\Gamma v)_{(y,\nu)}-(\Gamma v)_{(y-e_\nu,\nu)} \bigr)=0$ for every coarse site $y$. \item $1/b^2\le\Gamma_{ee}\le6/b$ and $\lvert\Gamma_{ee'}\rvert\le6/b$. \item The sums of $\lvert\Gamma_{ee'}\rvert$ over $e'\in S_+(e)$ and over $e'\in S_-(e)$ are each at least $\Gamma_{ee}$, so $\sum_{e'\ne e}\lvert \Gamma_{ee'}\rvert\ge2\Gamma_{ee}\ge2/b^2$. \end{enumerate} \end{proposition} \begin{proof} (i) For a mark $a$ let $\psi_a=(K\delta_a)|_{E'_0}$, let $K_0$ be the block of $K$ on $E'_0$, and $\Psi v=\sum_ev_e\psi_{l(e)}$. A link lies in six plaquettes and distinct marks are not partners (Proposition~\ref{prop:bpfibreduce}(i)), so the block of $K$ on the marks is $6I$ and \[ \lvert d\Theta(v,m)\rvert^2=6\lvert v\rvert^2+2\ip m{\Psi v}+\ip m{K_0m}. \] By Lemma~\ref{lem:bpfibker} the kernel of $K_0$ is $\mathcal N$, and $\psi_a$ is orthogonal to $\mathcal N$, since $\ip{\psi_a}{d\phi|_{E'_0}}=\ip{K\delta_a}{d\phi}=\ip{\delta_a}{d^{ \rm T}dd\phi}=0$. With $K_0^+$ the inverse of $K_0$ on $\mathcal N^\perp$, the minimiser is $m_v=-K_0^+\Psi v$ and $\Gamma=6I-\Psi^{\rm T}K_0^+\Psi$. Under $\varsigma_v$, $m=m_v+n$ with $n$ centred Gaussian on $\mathcal N^\perp$ of covariance $\beta^{-1}K_0^+$, and $F_a=\beta(K\Theta(v,m_v))_a+\beta\ip{\psi_a} n$, as $(K\Theta(0,n))_a=\ip{K\delta_a}{\Theta(0,n)}=\ip{\psi_a}n$. Hence $\Cov(F_a,F_{a'})=\beta\ip{\psi_a}{K_0^+\psi_{a'}}=\beta(6\delta_{ee'}- \Gamma_{ee'})$. Without the skeleton the law of $A$ is the centred Gaussian law with covariance $\beta^{-1}K^+$, and then \[ \Cov\bigl(\beta(KA)_a,\beta(KA)_{a''}\bigr)=\beta\ip{\delta_a}{KK^+K\delta_{ a''}}=\beta K_{aa''}, \] where $K_{aa''}$ is a signed count of the plaquettes containing $a$ and~$a''$. (ii) The minimiser satisfies $(K\Theta(v,m_v))_{E'_0}=0$, and $d^{\rm T} \mathcal J=d^{\rm T}d^{\rm T}d\Theta=0$. At an inner point $s$ of a path of direction $\nu$ the links with an end point at $s$ are the two links of that path and six links of $E'_0$: a link at $s$ of direction $\rho\ne\nu$ lies on a path only if $s$ lies on a line of direction $\rho$ through block corners, whose points have their $\nu$-coordinate divisible by $b$, which that of $s$ is not. So $d^{\rm T}\mathcal J(s)=0$ says that the values of $\mathcal J$ on the two path links at $s$ agree, and $\mathcal J$ is constant on each path. At a block corner $by$ the eight links are the first links of the paths $\pi(y,\nu)$ and the last links of the paths $\pi(y-e_\nu,\nu)$, and $d^{\rm T}\mathcal J(by)=0$ is the stated relation. Finally $\mathcal J_{l(e)}=6v_e+\ip{\psi_{l(e)}}{m_v}=(\Gamma v)_e$. (iii) First, $\ip v{\Gamma v}$ is the minimum of $\lvert dA\rvert^2$ over all $A\in\R^{E'}$ whose sum along each path $\pi(e)$ is $v_e$. Every $\Theta(v,m)$ has these sums. Conversely, given such $A$, let $a_0,\dots, a_{b-1}$ be the links of $\pi(y,\nu)$ in order, from $s_i$ to $s_{i+1}$, and let $\phi(s_t)=-\sum_{ik_b$, and $\phi=0$ at the sites on no path. This defines $\phi$ consistently, with $\phi=0$ at the block corners, and $A+d\phi$ vanishes on $a_i$ for $i\ne k_b$ and equals $v_e$ on $a_{k_b}=l(e)$; so $A+d\phi=\Theta(v,m)$ for some $m$, and $\lvert d(A+d\phi)\rvert=\lvert dA \rvert$. For $v=\delta_e$ take $A=b^{-1}\sum_{a\in\pi(e)}\delta_a$: collinear links lie in no common plaquette, so the $6b$ plaquettes that contain a link of $\pi(e)$ are distinct and each has $(dA)_p=\pm1/b$, and $\lvert dA \rvert^2=6/b$. For the lower bound let $\mathcal P$ be a coarse plaquette with $e$ on its boundary and $\Phi$ the indicator of the $b^2$ plaquettes of the square of side $b$ under it, which all lie in one plane. The interior links of the square cancel in $\ip{dA}\Phi=\sum_{p}(dA)_p$, which is the signed sum of the path sums of the four coarse links of $\mathcal P$, that is $\pm v_e=\pm1$. So $1\le\lvert dA\rvert\,\lvert\Phi\rvert=b\lvert dA \rvert$. As $\Gamma$ is positive semidefinite, $\lvert\Gamma_{ee'}\rvert\le (\Gamma_{ee}\Gamma_{e'e'})^{1/2}$. (iv) Apply (ii) to $v=\delta_e$: the column $(\Gamma_{e'e})_{e'}$ is co-closed, and the relation at the end point $y+e_\nu$ of $e=(y,\nu)$ writes $\Gamma_{ee}$ as a signed sum of the $\Gamma_{e'e}$, $e'\in S_+(e)$; likewise at $y$. The two sets are disjoint since $L\ge3$, and $\Gamma$ is symmetric. \end{proof} By (i), the decorrelation of forces asked for by (FC) is, in this model, the locality of the effective quadratic form of the straight-path map, and the decay of a free-field force covariance plays no role, since without the skeleton the forces are correlated only between partners. Part (i) and (iii) also exhibit the cancellation of Corollary~\ref{cor:bpfibH1}(iii): the force variance on a mark is $6\beta(1-\Gamma_{ee}/6)\ge6\beta(1-1/b)$, equal to the mean curvature $6\beta$ up to $\beta\Gamma_{ee}\le6\beta/b$, because the coarse variable is spread over the $b$ links of its path. Part (iv) is the analogue of Theorem~\ref{thm:bpbalnec} and of Lemma~\ref{lem:bpfibgauge}: gauge invariance forces the row sums of the mixed derivatives to be at least twice the diagonal. The model is invariant under the translations by $b\Z^4$, which gives the dependence on the torus. For $\kappa\in\R^4$ and a set $X$ of sites or links of the fine lattice $\Z^4$ invariant under $b\Z^4$, let $\mathcal H_\kappa(X)$ be the space of complex functions on $X$ with $f(x+bz)= e^{i\kappa\cdot z}f(x)$ for $z\in\Z^4$, with the inner product summed over one fundamental domain. \begin{proposition}[the Gaussian model on all tori]\label{prop:bpfibbloch} Fix $b\ge3$. There are $4\times4$ matrices $\Gamma^\infty_b(w)$, $w\in\Z^4$, and constants $M_b,\kappa_b>0$ with $\lVert\Gamma^\infty_b(w)\rVert\le M_be^{ -\kappa_b\lvert w\rvert_1}$, such that for every $L\ge3$ the effective quadratic form on $\Lambda_L$ is \[ \Gamma_{(y,\mu),(y',\nu)}=\sum_{z\in\Z^4}\Gamma^\infty_b(y'-y+Lz)_{\mu\nu} . \] In particular, for every $L\ge3$ and every coarse link $e=(y,\mu)$, \[ \sum_{e'\ne e}\lvert\Gamma_{ee'}\rvert\le\gamma_b=\max_\mu\sum_{(w,\nu)\ne(0, \mu)}\bigl\lvert\Gamma^\infty_b(w)_{\mu\nu}\bigr\rvert<\infty , \] and $1/b^2\le\Gamma^\infty_b(0)_{\mu\mu}\le6/b$ and $\gamma_b\ge2\min_\mu \Gamma^\infty_b(0)_{\mu\mu}$. \end{proposition} \begin{proof} \emph{Gauge fixing.} Let $B$ map a function $\phi$ on the sites of $\Lambda_{bL}$ off $\Sigma$ to $d\phi|_{E'_0}$, $\phi$ extended by $0$ on $\Sigma$, so that $\mathcal N$ is the range of $B$, and put $\mathcal A=K_0+ BB^{\rm T}$. If $\ip m{\mathcal Am}=0$, then $m\in\ker K_0=\mathcal N$ and $B^{\rm T}m=0$, so $m$ lies in the range of $B$ and in its orthogonal complement, and $m=0$. Since $K_0$ maps into $\mathcal N^\perp$ and vanishes on $\mathcal N$, while $BB^{\rm T}$ maps into $\mathcal N$ and vanishes on $\mathcal N^\perp$, $\mathcal A^{-1}\psi=K_0^+\psi$ for $\psi\in\mathcal N^\perp$, and by the proof of Proposition~\ref{prop:bpfibgauss}, $\Gamma=6I-\Psi^{\rm T}\mathcal A^{-1}\Psi$. \emph{Quasi-periodic functions.}\footnote{This is the Floquet--Bloch decomposition with respect to the translations by $b\Z^4$; the exponential decay of $\Gamma^\infty_b$ below is the Paley--Wiener argument for a matrix symbol that is holomorphic and bounded in a strip.} The sets $\Sigma$, $P$, $l(E)$ and $E'_0$, lifted to $\Z^4$, are invariant under $b\Z^4$, and $d$, $K$, the restrictions and $B$ commute with these translations and have finite range. They map the spaces $\mathcal H_\kappa$ into one another, the adjoint of $B$ on them is the restriction of $B^{\rm T}$, and in bases indexed by a fundamental domain their matrices $K_0(\kappa)$, $B(\kappa)$, $\Psi(\kappa)$ have entries that are trigonometric polynomials in $\kappa$. The space of complex functions on the links of $\Lambda_{bL}$ is the direct sum of the $\mathcal H_\kappa$ with $\kappa\in(2\pi/L)\Z^4$ modulo $2\pi\Z^4$, and the same holds for each invariant subset. \emph{Positivity for every $\kappa$.} Let $\mathcal A(\kappa)=K_0(\kappa)+ B(\kappa)B(\kappa)^*$ on $\mathcal H_\kappa(E'_0)$, and suppose that $\ip f{ \mathcal A(\kappa)f}=0$. Then the form $\omega$ equal to $f$ on $E'_0$ and to $0$ on $P$ is closed on $\Z^4$ and $B(\kappa)^*f=0$. As in the proof of Lemma~\ref{lem:bpfibker}, $\omega=d\phi$ with $\phi=0$ on the lift of $\Sigma$; for $z\in\Z^4$ the function $\phi(x+bz)-e^{i\kappa\cdot z}\phi(x)$ has zero differential and vanishes on the lift of $\Sigma$, so $\phi\in \mathcal H_\kappa$ and $f=B(\kappa)\phi$. Thus $f$ lies in the range of $B(\kappa)$ and in the kernel of its adjoint, and $f=0$. So $\mathcal A(\kappa)$ is positive definite for every real $\kappa$; its least eigenvalue is continuous and $2\pi\Z^4$-periodic, hence at least some $\lambda_b>0$. \emph{Analyticity and the torus.} The entries of $\mathcal A(\kappa)$ are finite sums of terms $e^{\pm i\kappa_\rho}$, so there is $\kappa_b>0$ with $\lVert\mathcal A(\kappa)-\mathcal A(\operatorname{Re}\kappa)\rVert\le\lambda_b/2$ for $\lvert\operatorname{Im}\kappa_\rho\rvert\le\kappa_b$, $\rho=0,\dots,3$; there $\lVert\mathcal A(\kappa)f\rVert\ge\frac{\lambda_b}2\lVert f\rVert$, and \[ \hat\Gamma(\kappa)=6I-\Psi(\kappa)^*\mathcal A(\kappa)^{-1}\Psi(\kappa), \] with $\Psi(\kappa)^*$ continued as the transpose of $\Psi(-\bar\kappa)$ conjugated, is holomorphic, bounded and periodic. On $\Lambda_{bL}$, writing $\Gamma_{(y,\mu),(y',\nu)}=g_{\mu\nu}(y'-y)$ by translation invariance, $\Gamma$ acts on $f\in\mathcal H_\kappa(l(E))$, $f(l(y,\nu))=e^{ i\kappa\cdot y}f_\nu$, by $(\Gamma f)(l(y,\mu))=e^{i\kappa\cdot y}\sum_\nu\hat g_{\mu\nu}(\kappa)f_\nu$ with $\hat g_{\mu\nu}(\kappa)=\sum_{w\in(\Z/L\Z)^4}g_{ \mu\nu}(w)e^{i\kappa\cdot w}$, and by the gauge fixing applied on each $\mathcal H_\kappa$, $\hat g(\kappa)=\hat\Gamma(\kappa)$ for $\kappa\in(2\pi/L) \Z^4$. Let $\Gamma^\infty_b(w)=(2\pi)^{-4}\int_{[-\pi,\pi]^4}\hat\Gamma(\kappa) e^{-i\kappa\cdot w}d\kappa$. Shifting each variable $\kappa_\rho$ by $\mp i \kappa_b$ according to the sign of $w_\rho$, which periodicity allows, gives $\lVert\Gamma^\infty_b(w)\rVert\le M_be^{-\kappa_b\lvert w\rvert_1}$ with $M_b$ the supremum of $\lVert\hat\Gamma\rVert$ on the strip. So $\hat\Gamma(\kappa)= \sum_w\Gamma^\infty_b(w)e^{i\kappa\cdot w}$ converges absolutely, and \[ \begin{split} g(w)&=L^{-4}\sum_\kappa\hat\Gamma(\kappa)e^{-i\kappa\cdot w}\\ &=\sum_{w'}\Gamma^\infty_b(w')\,L^{-4}\sum_\kappa e^{i\kappa\cdot(w'-w)}\\ &=\sum_{z\in\Z^4}\Gamma^\infty_b(w+Lz), \end{split} \] the sums over $\kappa\in(2\pi/L)\Z^4$ modulo $2\pi\Z^4$. The pairs $(w,\nu)\ne(0,\mu)$ with $w$ in a class modulo $L$ fall into one term $\Gamma_{ee'}$, $e'\ne e$, or into $\Gamma_{ee}$, so the triangle inequality gives the bound by $\gamma_b$. As $L\to\infty$, $g(w)\to \Gamma^\infty_b(w)$ for each $w$, and the bounds and the co-closedness of Proposition~\ref{prop:bpfibgauss}(ii)--(iv) pass to the limit. \end{proof} So in the Gaussian model the row sums of the effective form are finite uniformly in the torus for each $b$, and their size is the single infinite-volume number $\gamma_b$; the proof gives no control of $M_b$ and $\kappa_b$ in $b$. The bound of the order of the cost $6/b$ of a thin flux tube is \[ \textup{(G)}\qquad\gamma_*=\sup_{b\ge3}b\,\gamma_b<\infty . \] Proposition~\ref{prop:bpfibbloch} gives only $b\gamma_b\ge2/b$, and Theorem~\ref{thm:bprowgauss} shows that (G) fails: $\gamma_b$ lies between absolute multiples of $(\log b)/b$ and $(1+\log b)/b$. \begin{corollary}[the Gaussian form of (FC)]\label{cor:bpfibgauss} In the Gaussian model let $J^{\rm G}_{ee'}=\beta\lvert\Gamma_{ee'}\rvert$, the mixed derivatives of the induced action $\frac\beta2\ip v{\Gamma v}$. For every $L\ge3$ and every coarse link $j$, \[ \frac{D^2}2\,\frac\beta{b^2}\le\frac{D^2}2\,\beta\,\Gamma_{jj}\le\frac{D^2}4 \sum_{e\ne j}J^{\rm G}_{ej}\le\frac{D^2}4\,\beta\,\gamma_b . \] The inequalities of \textup{(FC)} for $J^{\rm G}$ hold with $c=\frac1{17}$ and $R=1$ on every torus as soon as $17D^2\beta\gamma_b\le4$. \end{corollary} \begin{proof} The bounds are Propositions~\ref{prop:bpfibgauss}(iii),(iv) and~\ref{prop:bpfibbloch}. If $17D^2\beta\gamma_b\le4$, the column sums are at most $\frac{D^2}4\beta\gamma_b\le\frac1{17}$, and the tails are at most the column sums, while $(1-\frac1{17})/16=\frac1{17}$. \end{proof} The logarithmic growth of $\beta_k$ along the tuning, after Proposition~\ref{prop:bpbalrandom}, satisfies $2^{-k}\beta_k\to0$. So in the Gaussian model (FC) along the tuning reduces to a bound on $\gamma_b$, a statement on one explicit quadratic form for each $b$, which Corollary~\ref{cor:bprowFC} proves along the tuning, uniform in the torus by Proposition~\ref{prop:bpfibbloch} and in the coarse data because the covariances do not depend on $v$. The model is the quadratic part of the action, and for a non-abelian group it can describe the fibre law only where the fluctuations of the path-ordered products are small; the marks are $b$ links apart, a distance at which the running coupling along the tuning is of order one, and nothing proved here relates the fibre covariances of $\SU(2)$ to $\Gamma$. Figure~\ref{fig:bpfibre} shows the fibre, the current of Proposition~\ref{prop:bpfibgauss}(ii) and the bounds on $\Gamma$. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_bpfibre.png} \caption{The fibre route to the coarsest condition. (a) The fibre law of one step of the straight-path map with blocks of side $b=4$, in a plane: the coarse data fix the marks (red), the other links of the paths are fixed to $I$ (black) and the links off the paths (grey) are integrated. In the Gaussian model the minimiser carries a current that vanishes off the paths, is constant along each path with the value $(\Gamma v)_e$ and is conserved at every block corner (Proposition~\ref{prop:bpfibgauss}(ii)). (b) The proved bounds $1/b^2\le\Gamma_{ee}\le6/b$ and $\gamma_b\ge2/b^2$ (Propositions~\ref{prop:bpfibgauss} and~\ref{prop:bpfibbloch}), and the decay $\gamma_b=O(1/b)$ of (G), drawn with an arbitrary constant; (G) is false, and $\gamma_b$ is of order $(\log b)/b$ (Theorem~\ref{thm:bprowgauss}). (c) The lower bound $4\beta_k\delta^2/(\pi^2b^2)$ on the action that rough coarse data of $\SU(2)$ force on the square of a coarse plaquette, for $\delta=\sqrt2\pi$, $b=2^k$ and $\beta_k=2+\frac{11}{3\pi^2} k\log2$, an illustration of the growth along the tuning (Proposition~\ref{prop:bpfibrough}); it tends to zero and obstructs nothing.} \label{fig:bpfibre} \end{figure*} \begin{remark}[the abelian group]\label{rem:bpfibuone} For $G=\U(1)$ Lemma~\ref{lem:bpcoarsehess} does not apply, since $\kappa_G=0$, while Lemmas~\ref{lem:bpfibosc} and~\ref{lem:bpfibgauge}, Proposition~\ref{prop:bpfibreduce} and Corollary~\ref{cor:bpfibH1} do, with $D=\pi$, and the Gaussian model is the quadratic part of the Wilson action. Nothing in Propositions~\ref{prop:bpfibgauss} and~\ref{prop:bpfibbloch} distinguishes $\U(1)$ from a direction of the Lie algebra of $\SU(2)$. A proof of (FC) through the Gaussian model would therefore apply to $\U(1)$ in its Coulomb phase as well, at fixed $\beta$, and give (H1) there for large $k$. This does not contradict the massless photon of that phase~\cite{Gut80,FS82}. Condition (H1) concerns only the law of the block variables at the spacing $a_0$; it is independent of (H2) (Proposition~\ref{prop:bpcoarseflat}), and the product of the Haar measures, the law towards which the straight-path maps randomise the coarse links, satisfies $\mathrm{BP}(n,1)$ (Proposition~\ref{prop:avgsup}(ii)). The averaged condition fails in the Coulomb phase of the Villain theory (Corollary~\ref{cor:avgcoulomb}, on boxes), and wherever $\mathrm{BP}(n_k,\varpi)$ fails along a tower of tori on which (H1) holds, Corollary~\ref{cor:bpuptuning} forces (H2) to fail. So the route of this subsection, completed, would prove (H1) and nothing about the gap: as for the averaged condition itself (Remark~\ref{rem:avgstatus}), the gap has to come from the non-abelian structure, and through Corollary~\ref{cor:bpuptuning} it has to enter (H2). \end{remark} The uniformity in $V$ in (FC) includes rough coarse data, whose plaquette variables are far from $I$. For them the fibre law carries a forced background, which the Gaussian model does not see. \begin{proposition}[rough coarse data]\label{prop:bpfibrough} Let $c$, $l_b$, $\mu'$, $\varsigma_V$ and $K_{ee'}$ be as in Proposition~\textup{\ref{prop:bpfibreduce}}. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Let $\hat g$ be a gauge transformation of $\Lambda_L$ and $g$ the gauge transformation of $\Lambda_{bL}$ equal to $\hat g(y)$ at $by$ and at the inner points of $\pi(y,\nu)$ up to the start of the mark, to $\hat g(y+ e_\nu)$ at the inner points after it, and to $I$ at the sites on no path. Then $\varsigma_{\hat gV}$ is the image of $\varsigma_V$ under $m\mapsto(g \Theta^l(V,m))_{E'_0}$, and the covariances of the forces at $\hat gV$ with unit $X,Y$ are those at $V$ with unit vectors $X',Y'$ conjugate to $X,Y$. So the supremum in $K_{ee'}$ is a supremum over the gauge classes of $V$. \item For every coarse plaquette $\mathcal P$ with plaquette variable $V_{\mathcal P}$, the set $Q_{\mathcal P}$ of the $b^2$ plaquettes of the square of side $b$ under it, and every $m$, \[ \sum_{p\in Q_{\mathcal P}}\delta\bigl(\Theta^l(V,m)_p,I\bigr)\ge\delta(V_{ \mathcal P},I). \] For $G=\SU(2)$, $\beta\ge0$ and $\mathcal S_0=N\beta\sum_p(2-\ReTr U_p)$, \[ \mathcal S_0(\Theta^l(V,m))\ge\frac{4\beta}{\pi^2b^2}\sum_{\mathcal P}\delta( V_{\mathcal P},I)^2, \] and hence \[ \int e^{-\mathcal S_0(\Theta^l(V,m))}\,\lambda_{E'_0}(dm)\le\exp\Bigl(-\frac{ 4\beta}{\pi^2b^2}\sum_{\mathcal P}\delta(V_{\mathcal P},I)^2\Bigr). \] \end{enumerate} \end{proposition} \begin{proof} (i) The values of $g$ are consistent, since an inner point lies on one path and the corner $by$ ends the paths $\pi(y-e_\nu,\nu)$, where $g$ is $\hat g(y)$ after the mark. On $\pi(y,\nu)$ the links of $\Theta^l(V,m)$ before and after the mark become $\hat g(y)\hat g(y)^{-1}=I$ and $\hat g(y+ e_\nu)\hat g(y+e_\nu)^{-1}=I$, and the mark becomes $\hat g(y)V_{(y,\nu)}\hat g(y+e_\nu)^{-1}=(\hat gV)_{(y,\nu)}$; so $g\Theta^l(V,m)=\Theta^l(\hat gV,m^g)$ with $m^g=(g\Theta^l(V,m))_{E'_0}$, a translation of each link of $E'_0$ on both sides by elements that do not depend on $m$, which preserves $\lambda_{E'_0}$. As $\mathcal S$ is gauge invariant, the density of $\varsigma_{\hat gV}$ at $m^g$ equals that of $\varsigma_V$ at $m$. For the mark $a$ from $s$ to $s'$, $g(s)U_ag(s')^{-1}e^{tX}=g(s)U_ae^{tX'}g(s')^{-1}$ with $X'=g(s')^{-1}Xg(s')$, so $D^a_X\mathcal S(gU)=D^a_{X'}\mathcal S(U)$, with $g(s')$ independent of $m$. (ii) The boundary of the square of $\mathcal P$, traversed from its corner, consists of the four paths of the coarse links of $\mathcal P$, whose products under $\Theta^l(V,m)$ are the coarse links, the unmarked links being $I$; so its holonomy is $V_{\mathcal P}$ up to orientation. By Lemma~\ref{lem:bpstokes} it is an ordered product of conjugates of the $U_p^{\pm1}$, $p\in Q_{\mathcal P}$, and for a bi-invariant metric $\delta(XY,I)\le\delta(X,I)+\delta(Y,I)$, $\delta(hXh^{-1},I)=\delta(X^{-1},I)= \delta(X,I)$. For $\SU(2)$, with $\theta$ the angle of $U$, $2-\ReTr U=4\sin^2( \theta/2)\ge4\theta^2/\pi^2=2\delta(U,I)^2/\pi^2$, so $\mathcal S_0\ge\frac{4 \beta}{\pi^2}\sum_p\delta(U_p,I)^2$; by the Cauchy--Schwarz inequality the sum over $Q_{\mathcal P}$ is at least $\delta(V_{\mathcal P},I)^2/b^2$, and distinct coarse plaquettes have disjoint squares. The integral is at most the supremum of the integrand. \end{proof} Along the tuning $\beta_k4^{-k}\to0$, so the background forced on the squares of the coarse plaquettes has vanishing action per coarse plaquette, and part (ii) gives no obstruction to the uniformity in $V$; part (i) reduces it to the gauge classes, which still contain every assignment of conjugacy classes to the coarse plaquettes. In the Gaussian model the covariances do not depend on $v$ at all (Proposition~\ref{prop:bpfibgauss}(i)). The dependence of $K_{ee'}$ on $V$ for $\SU(2)$ is a non-linear effect, and no bound on it is proved here. \begin{remark}[what is proved about the fibre route]\label{rem:bpfibre} Proved: every measure with a density $e^{-H}$, $H\in C^2$, on the links of a torus, for every closed connected $G\subset\U(N)$ including $\U(1)$, has single-link coefficients at most $D^2/4$ times the bounds on the mixed derivatives of $H$, with no condition on the single-link Hessian (Lemma~\ref{lem:bpfibosc}), and gauge invariance bounds the single-link Hessian by the sum of the mixed-derivative bounds into the seven links at an end point, at both end points for abelian $G$ (Lemma~\ref{lem:bpfibgauge}); for straight-path maps with blocks of side $b\ge3$ and the middle marking the marks of distinct coarse links are not partners, the mixed derivatives of the induced action are minus the covariances of the forces on the marks in the fibre law, and the coefficients of the law of the block variables are at most $D^2/4$ times their suprema over the coarse data (Proposition~\ref{prop:bpfibreduce}); the condition (FC) on these covariances gives (H1) with $\varpi_0=\frac12$, and it forces the force variance on every mark to equal its mean curvature up to $4c/D^2$, uniformly in the coarse data (Corollary~\ref{cor:bpfibH1}); in the strong-coupling range the covariances decay exponentially in the distance of the marks, which covers the range of Corollary~\ref{cor:bpbaltower}(iii), and the fibre law itself fails the Dobrushin condition for $\SU(2)$ at $\beta_W\ge\frac83$ (Remark~\ref{rem:bpfibstrong}); in the Gaussian model the force covariances are $-\beta$ times the effective quadratic form of the straight-path map, independent of the coarse data, without the skeleton they vanish beyond partners, the form is a co-closed current on the coarse lattice with $1/b^2\le\Gamma_{ee}\le6/b$ and row sums at least $2\Gamma_{ee}$ (Proposition~\ref{prop:bpfibgauss}), and for each $b$ it decays exponentially, with row sums bounded on every torus by one infinite-volume number $\gamma_b$ (Proposition~\ref{prop:bpfibbloch}); the Gaussian form of (FC) holds on every torus when $17D^2\beta\gamma_b\le4$ (Corollary~\ref{cor:bpfibgauss}); a proof of (FC) through the Gaussian model would apply to $\U(1)$, which is consistent with its massless phase because (H1) carries no information on the gap (Remark~\ref{rem:bpfibuone}); the covariances for rough coarse data depend only on their gauge class, and the background they force on the faces has action at least $4\beta\delta(V_{\mathcal P},I)^2/(\pi^2b^2)$ per coarse plaquette for $\SU(2)$, which vanishes along the tuning (Proposition~\ref{prop:bpfibrough}). Not proved: (FC) for any $k$ with $\beta_k$ outside the strong-coupling range, and hence (H1) along the tuning; any control of the decay rate $\kappa_b$ in $b$; any comparison of the fibre covariances of $\SU(2)$ with the Gaussian model at the separation of the marks; any bound on the covariances uniform in rough coarse data for $\SU(2)$; (H1) for $\U(1)$. The cancellation of Corollary~\ref{cor:bpfibH1}(iii) holds for $\SU(2)$ along every tuning with $\beta_k=O(k)$ (Corollary~\ref{cor:bppathcancel}). Neither $\mathrm{BP}$ along the tuning nor the mass gap is proved here. \end{remark} \subsection{Smoothing along the paths}\label{ssec:bppaths} A block variable of the straight-path map is the product of the $b$ fine links of its path, and given the links off the path these $b$ links are independent. A product of many independent random elements of a compact group spreads out towards the Haar measure, and this subsection makes that quantitative for $\SU(2)$ and follows it through to the condition at the coarsest scale. Convolution with a one-link Wilson density contracts every nontrivial representation by the factor $q_W=4\beta_W/(4\beta_W+1)$, so a product of $m$ independent one-link variables has a density within $48(1+6\beta_W)^2q_W^{m-2}$ of $1$, together with its first two derivatives (Theorem~\ref{thm:bppathproduct}); this extends to every representation the bound on the first moment in Proposition~\ref{prop:bpbalrandom}(ii). The single-link conditional laws of the block variables obey the same bounds, and along every tuning with $\beta_k=O(k)$ the single-link curvature of the induced action tends to zero uniformly in the torus and the coarse data, which is the cancellation asked for in Corollary~\ref{cor:bpfibH1}(iii) (Theorem~\ref{thm:bppathdensity}, Corollary~\ref{cor:bppathcancel}). The interior links of all the paths lie in no common plaquette, so the law of the block variables is a mixture of product laws, and its mixed derivatives are covariances of the scores of the interior products under one posterior law, each at most $K_b^2$ with $K_b\to0$ (Theorem~\ref{thm:bppathmixed}). A column sum of single-link coefficients at most $\frac13$ gives $\mathrm{BP}(n,\frac12)$ for every cube side, with no condition on the tails (Corollary~\ref{cor:bppathcolumn}), and together these give $\mathrm{BP}(n,\frac12)$ after $k\ge12$ blocking steps with $\beta_k\le k+2$ on every coarse torus of side at most $e^{2^k/(17k)}$ (Corollary~\ref{cor:bppathfourk}). Two standard routes to a bound on every torus fail along the tuning, Dobrushin's condition for the posterior law and the Koteck\'y--Preiss condition for the unresummed cube surface (Proposition~\ref{prop:bppathobst}), and a bound on the sum of the posterior covariances, uniform in the torus, gives (H1) (Proposition~\ref{prop:bppathsuff}). Throughout, $G=\SU(2)$, with $\delta$, $D=\sqrt2\pi$ and $\kappa_G=1$ as in Corollary~\ref{cor:bpbalwilson}, $\lambda$ is the Haar probability measure and $\theta(g)\in[0,\pi]$ is the angle of $g$, so that $\frac12\ReTr g= \cos\theta(g)$ and $\theta$ has the density $\frac2\pi\sin^2\theta$ under $\lambda$. For a function $F$ on $G$ and a unit vector $X\in\g$ write $D_XF(g)=\frac d{dt}F(ge^{tX})|_{t=0}$. For $B\ge0$ let \begin{gather*} f_B(g)=\frac{e^{B\cos\theta(g)}}{Z(B)},\\ Z(B)=\frac2\pi\int_0^\pi e^{B\cos\theta}\sin^2\theta\,d\theta=I_0(B)-I_2(B)= \frac{2I_1(B)}B , \end{gather*} with $Z(0)=1$, where $I_\nu$ is the modified Bessel function; the last two identities are the integral representation~\cite[(10.32.3)]{DLMF} and the recurrence~\cite[(10.29.1)]{DLMF}. For $\beta_W\ge0$ and $m\ge2$ put $B_W=6\beta_W$, \[ q_W=\frac{4\beta_W}{4\beta_W+1},\qquad\eta_m=48(1+B_W)^2q_W^{m-2}, \] and, when $\eta_m<1$, \[ h_m=\frac{(B_W^2+B_W)\eta_m}{1-\eta_m}+\frac{B_W^2\eta_m^2}{(1-\eta_m)^2}. \] At $B=6\beta_W$ one has $2B/(2B+3)=q_W$. \begin{lemma}[every mode at once]\label{lem:bppathmodes} Let $B>0$. For every $d\ge1$, convolution $h\mapsto h*f_B$ acts on the isotypic component of $L^2(G)$ of the irreducible representation of dimension $d$ as multiplication by \[ a_d(B)=\frac{I_d(B)}{I_1(B)}, \] and for $d\ge2$ \[ 00$, $I_\nu(B)$ is positive and decreasing in $\nu>0$ \cite[\S10.37]{DLMF}, so $00$ at $x=2B/(2B+3)$, hence $A<2B/(2B+3)$. A function of mean zero has no component for $d=1$, which gives the last statement. \end{proof} \begin{lemma}[one link]\label{lem:bppathsmooth} For every $B\ge0$, $\lVert f_B\rVert_\infty\le48(1+B)^{3/2}$, hence $\lVert f_B\rVert_2^2\le48(1+B)^{3/2}$, and for every unit $X\in\g$ \[ \lvert D_Xf_B\rvert\le Bf_B,\qquad\lvert D_XD_Xf_B\rvert\le(B^2+B)f_B . \] \end{lemma} \begin{proof} Let $t=(1+B)^{-1/2}\le1$. On $[0,t]$ one has $\sin\theta\ge2\theta/\pi$ and $\cos\theta\ge1-\theta^2/2$, so \begin{align*} Z(B)&\ge\frac2\pi\int_0^te^{B-B\theta^2/2}\,\frac{4\theta^2}{\pi^2}\,d\theta \ge\frac8{3\pi^3}\,e^{B-Bt^2/2}t^3\\ &\ge\frac8{3\pi^3}\,e^{-1/2}\,\frac{e^B}{(1+B)^{3/2}}, \end{align*} and $\lVert f_B\rVert_\infty=e^B/Z(B)\le\frac{3\pi^3}8e^{1/2}(1+B)^{3/2}<48(1+ B)^{3/2}$, since $\pi<4$ and $e^{1/2}<2$. As $f_B$ is a probability density, $\lVert f_B\rVert_2^2\le\lVert f_B\rVert_\infty$. Let $u(g)=\frac12\ReTr g$. A unit $X\in\g$ satisfies $X^2=-\frac12I$, so $D_Xu(g)=\frac12\ReTr(gX)$ and $D_XD_Xu(g)=-\frac14\ReTr g$; since $\lvert\ReTr(gX)\rvert\le\lVert g\rVert_{ \rm HS}\lVert X\rVert_{\rm HS}=\sqrt2$, we get $\lvert D_Xu\rvert\le1$ and $\lvert D_XD_Xu\rvert\le\frac12$. The bounds follow from $D_Xf_B=B\,(D_Xu) f_B$ and $D_XD_Xf_B=(B\,D_XD_Xu+B^2(D_Xu)^2)f_B$. \end{proof} \begin{theorem}[products of one-link variables]\label{thm:bppathproduct} Let $m\ge2$, $B\ge0$ and $A,C\in G$, and let $Y_1,\dots,Y_m$ be independent $G$-valued random variables such that $Y_i$ has the density $g\mapsto f_{B_i} (gW_i)$ with respect to $\lambda$, for some $W_i\in G$ and $0\le B_i\le B$. Let $p$ be the density of $AY_1\cdots Y_mC$ with respect to $\lambda$ and $\eta=48(1+B)^2(\frac{2B}{2B+3})^{m-2}$. Then for every $g\in G$ and every unit $X\in\g$ \begin{gather*} \lvert p(g)-1\rvert\le\eta,\qquad\lvert D_Xp(g)\rvert\le B\eta,\\ \lvert D_XD_Xp(g)\rvert\le(B^2+B)\eta . \end{gather*} \end{theorem} \begin{proof} If some $B_i=0$, then $Y_i$ is Haar distributed, so is the product, and $p=1$. Let all $B_i>0$, and put $Z_i=Y_iW_i$, with the class density $f_i= f_{B_i}$. Since $W^{-1}Z_{i+1}=(W^{-1}Z_{i+1}W)W^{-1}$ and conjugation preserves the law of $Z_{i+1}$, moving the $W_i^{-1}$ to the right one at a time gives $AY_1\cdots Y_mC=AZ'_1\cdots Z'_mC'$ with $C'=W_1^{-1}\cdots W_m^{-1}C$ and independent $Z'_i$ with the densities $f_i$. Hence $p(g)= F(A^{-1}gC'^{-1})$ with $F=f_1*\dots*f_m$. A two-sided translation is an isometry and replaces $D_X$ by $D_{X'}$ with $X'=\operatorname{Ad}(C')X$, again a unit vector, so it suffices to bound $F$. A class function commutes under convolution with every integrable function, and $D_X(k*f)=k*D_Xf$. Since $1*f=1$ for a probability density $f$, \begin{gather*} F-1=\bigl((f_1-1)*f_3*\dots*f_m\bigr)*f_2,\\ D_XF=\bigl((D_Xf_1)*f_3*\dots*f_m\bigr)*f_2, \end{gather*} and likewise for $D_XD_XF$. The functions $f_1-1$, $D_Xf_1$ and $D_XD_Xf_1$ have mean zero, convolution with a probability density keeps the mean zero, and by Lemma~\ref{lem:bppathmodes} each of the $m-2$ convolutions with $f_3,\dots,f_m$ multiplies the $L^2$ norm by at most $2B/(2B+3)$, since $2B_i/(2B_i+3)\le2B/(2B+3)$. With $\lVert k*f\rVert_\infty \le\lVert k\rVert_2\lVert f\rVert_2$, $\lVert f_1-1\rVert_2\le\lVert f_1 \rVert_2$ and Lemma~\ref{lem:bppathsmooth}, \begin{align*} \lVert F-1\rVert_\infty&\le\Bigl(\frac{2B}{2B+3}\Bigr)^{m-2}\lVert f_1 \rVert_2\lVert f_2\rVert_2\\ &\le48(1+B)^{3/2}\Bigl(\frac{2B}{2B+3}\Bigr)^{m-2}\le\eta , \end{align*} and the same argument gives the other two bounds, since $\lVert D_Xf_1 \rVert_2\le B\lVert f_1\rVert_2$ and $\lVert D_XD_Xf_1\rVert_2\le(B^2+B) \lVert f_1\rVert_2$ by Lemma~\ref{lem:bppathsmooth}. \end{proof} \begin{theorem}[the law of one block variable]\label{thm:bppathdensity} Let $b\ge2$, $L\ge3$ and $\beta_W\ge0$, let $\mu'$ be the $\SU(2)$ Wilson measure of $\Lambda_{bL}$, with density proportional to $\exp(\frac{\beta_W}2 \sum_p\ReTr U_p)$, let $c$ be the straight-path map to $\Lambda_L$ and $\mu^c=c_*\mu'$, with the induced action $H^c$ of Lemma~\textup{\ref{lem:bpcoarselaw}(i)}. For every coarse link $e$ and every $V$ the conditional law $\gamma^{\mu^c}_{\{e\}}(\cdot\mid V)$ has a density $r_e(\cdot\mid V)$ with respect to $\lambda$, and for every unit $X\in\g$ \begin{gather*} \lvert r_e-1\rvert\le\eta_b,\qquad\lvert D_Xr_e\rvert\le B_W\eta_b,\\ \lvert D_XD_Xr_e\rvert\le(B_W^2+B_W)\eta_b . \end{gather*} If $\eta_b<1$, then \[ \Bigl\lvert\frac{d^2}{dt^2}H^c(V^{e,tX})\Big|_0\Bigr\rvert\le h_b \] for every $e$, $V$ and unit $X$. Along every sequence $b_k=2^k$, $\beta_k\le C(1+k)$, the numbers $\eta_{b_k}$ and $h_{b_k}$ tend to zero faster than every power of $2^{-k}$, uniformly in $L$ and $V$. \end{theorem} \begin{proof} Condition on the configuration $W$ of the fine links off the path $\pi(e)$. Two links of a plaquette are either parallel and displaced by a unit vector transverse to them, or orthogonal; two links of $\pi(e)$ are collinear, so no plaquette contains two of them. Given $W$ the $b$ links of $\pi(e)$ are therefore independent, and each has the density proportional to $\exp(\frac{ \beta_W}2\ReTr(U\,S))$, where $S$ is the sum of its six staples, a function of $W$. A sum of elements of $\SU(2)$ is a real multiple of an element of $\SU(2)$, so $S=sW_S$ with $0\le s\le6$ and $W_S\in G$, and the density is $f_{s\beta_W}(UW_S)$. By Theorem~\ref{thm:bppathproduct} with $m=b$ and $B=B_W$, the density $p_e(\cdot\mid W)$ of $c_e$ given $W$ obeys the three bounds with $\eta_b$. The other coarse links are functions $\phi(W)$ of $W$, so for bounded measurable $F$ \[ \mu^c(F)=\int\mu'(dW)\int F(g,\phi(W))\,p_e(g\mid W)\,\lambda(dg), \] and the conditional density of $V_e$ given $V_{E\setminus\{e\}}$ is the mean of $p_e(g\mid W)$ under the conditional law of $W$ given $\phi(W)= V_{E\setminus\{e\}}$, a mixture whose weights do not depend on $g$. The three bounds pass to the mixture for $\mu^c$-almost every $V$. By Lemma~\ref{lem:bpcoarselaw}(i) the density of $\mu^c$ is smooth and positive, so $r_e$ and its derivatives in $g$ are continuous in $V$ and the bounds hold for every $V$. With the other coarse links fixed, $H^c(V^{e,tX})=-\log r_e(V_ee^{tX}\mid V)$ up to a constant, and \[ \frac{d^2}{dt^2}\bigl(-\log r_e\bigr)=-\frac{D_XD_Xr_e}{r_e}+\frac{(D_Xr_e)^2} {r_e^2} \] is at most $h_b$ in absolute value. Finally $\log(1+x)\ge x/(1+x)$ gives $q_W^{b-2}\le\exp(-(b-2)/(4\beta_W+1))$, and for $b=2^k$, $\beta_W\le C(1+k)$ the exponent $-(2^k-2)/(4C(1+k)+1)$ beats every multiple of $k$. \end{proof} \begin{corollary}[the cancellation in the fibre law]\label{cor:bppathcancel} In the setting of Theorem~\textup{\ref{thm:bppathdensity}} let $b\ge3$, $\eta_b<1$, and let $\varsigma_V$ and $a=l_b(e)$ be as in Proposition~\textup{\ref{prop:bpfibreduce}}. Then for every coarse link $e$, every $V$ and every unit $X$ \[ \bigl\lvert\varsigma_V(D^a_XD^a_X\mathcal S)-\Var_{\varsigma_V}(D^a_X \mathcal S)\bigr\rvert\le h_b , \] so along every tuning with $\beta_k=O(k)$ the variance of the force on each mark equals its mean curvature up to an error that tends to zero, uniformly in the torus and the coarse data, and the bound of Corollary~\textup{\ref{cor:bpfibH1}(iii)} holds for all large $k$ whatever $c>0$. \end{corollary} \begin{proof} By Proposition~\ref{prop:bpbalhess} the left side is the second derivative of $H^c$ along $V^{e,tX}$; apply Theorem~\ref{thm:bppathdensity}. \end{proof} So the single-link part of the effective-action problem is settled for $\SU(2)$ along the tuning: both terms are of order $\beta_k$, and their difference is smaller than any power of the block size. What remains is the sum of the mixed derivatives over distant coarse links. To reach them, the interiors of all the paths are integrated at once. Let $b\ge4$, let $\mathcal I$ be the set of fine links $(by+ie_\nu,\nu)$ with $(y,\nu)$ a coarse link and $1\le i\le b-2$, the interiors of the paths, and let $\mathcal R$ be the set of all other fine links, which contains the first and the last link of every path. \begin{lemma}[the interiors]\label{lem:bppathint} No plaquette of $\Lambda_{bL}$ contains two links of $\mathcal I$. Consequently, given $U_{\mathcal R}=W$, the links of $\mathcal I$ are independent under $\mu'$, and the coarse links are independent with densities $p_e(\cdot\mid W)$ that obey the three bounds of Theorem~\textup{\ref{thm:bppathproduct}} with $\eta_{b-2}$ and $B=B_W$. The density of $\mu^c$ with respect to $\lambda_E$ is \begin{equation}\label{eq:bppathmix} \mathcal D(V)=\int\prod_{e\in E}p_e(V_e\mid W)\,\zeta(dW), \end{equation} where $\zeta$ is the law of $U_{\mathcal R}$ under $\mu'$. \end{lemma} \begin{proof} Two parallel links of a plaquette have base points that differ by a unit vector transverse to them. For two links of $\mathcal I$ of direction $\nu$ the transverse coordinates of the base points are multiples of $b$, and so is their difference modulo $bL$, while $\pm1$ is not, since $b\ge2$. A plaquette in the plane of $\mu\ne\nu$ with base point $z$ has the links $(z,\mu)$, $(z+e_\mu,\nu)$, $(z+e_\nu,\mu)$ and $(z,\nu)$, so a $\mu$-link with base point $x$ and a $\nu$-link with base point $y$ in it satisfy $x-y\in\{0,-e_\mu,e_\nu,e_\nu-e_\mu\}$, and $x_\mu-y_\mu\in\{0,-1\}$. For links of $\mathcal I$, $x_\mu\equiv i\pmod b$ with $1\le i\le b-2$ and $y_\mu\equiv0\pmod b$, so $x_\mu-y_\mu$ is congruent neither to $0$ nor to $-1$ modulo $b$. Hence every Wilson factor contains at most one link of $\mathcal I$, and given $W$ the density of $U_{\mathcal I}$ is a product. A link of $\mathcal I$ has the density $f_{s\beta_W}(UW_S)$ as in the proof of Theorem~\ref{thm:bppathdensity}, with staples in $W$, and the coarse link $e$ is $R\,Y_1\cdots Y_{b-2}R'$ with its first and last links $R,R'$ in $W$ and its interior links $Y_i$; Theorem~\ref{thm:bppathproduct} with $m=b-2$ gives the bounds. Distinct coarse links have disjoint interiors, which gives the independence and~\eqref{eq:bppathmix}. \end{proof} \begin{theorem}[mixed derivatives as posterior covariances] \label{thm:bppathmixed} Let $b\ge4$ and $\eta_{b-2}<1$, put $K_b=B_W\eta_{b-2}/(1-\eta_{b-2})$, and for $V\in G^E$ let \begin{gather*} \pi_V(dW)=\mathcal D(V)^{-1}\prod_{e\in E}p_e(V_e\mid W)\,\zeta(dW),\\ a^X_e(W,V_e)=D_X\log p_e(V_e\mid W). \end{gather*} Then for coarse links $e\ne j$ and unit $X,Y\in\g$ \[ \frac{\partial^2}{\partial s\,\partial t}H^c\bigl((V^{e,sX})^{j,tY}\bigr)\Big|_0 =-\Cov_{\pi_V}\bigl(a^X_e,a^Y_j\bigr), \] so $J_{ej}(H^c)\le K_b^2$, and $\lvert\frac{d^2}{dt^2}H^c(V^{e,tX})|_0\rvert \le h_{b-2}$ for every $e$, $V$ and unit $X$. \end{theorem} \begin{proof} By Lemma~\ref{lem:bppathint}, $p_e\ge1-\eta_{b-2}$ and $\lvert a^X_e\rvert \le K_b$. The densities $p_e$ are smooth in $V_e$, with derivatives bounded uniformly in $W$, so~\eqref{eq:bppathmix} may be differentiated under the integral. With $H^c=-\log\mathcal D$ up to a constant, $\frac d{ds}H^c(V^{ e,sX})|_0=-\E_{\pi_V}a^X_e$. The score $a^X_e$ does not depend on $V_j$, and differentiating $\pi_V$ in $V_j$ along $Y$ multiplies it by $a^Y_j-\E_{ \pi_V}a^Y_j$; this gives the covariance, which is at most $K_b^2$ in absolute value by the Cauchy--Schwarz inequality. Given the other coarse links, the law of $V_e$ has the density proportional to $\int p_e(V_e\mid W) \prod_{e'\ne e}p_{e'}(V_{e'}\mid W)\,\zeta(dW)$, a mixture of the $p_e(\cdot \mid W)$ with weights independent of $V_e$, and the last statement follows as in the proof of Theorem~\ref{thm:bppathdensity}. \end{proof} The bound on each mixed derivative carries no factor that decays with the distance between $e$ and $j$; the posterior $\pi_V$ couples all the paths through the fine links of $\mathcal R$. A bound on the sum of the coefficients still gives the cube bound for every cube side, and with no condition on the tails. \begin{corollary}[a column sum below one third]\label{cor:bppathcolumn} Let $\varrho$ be a gauge-invariant probability measure on $G^E$, $E$ the links of $\Lambda_L$, with $c_\varrho=\max_j\sum_{e\ne j}C^\varrho_{ej}\le \frac13$. Then $\mathrm{BP}_\varrho(n,\frac12)$ holds for every $n\ge2$ with $L\ge2n+4$ even. In particular, in Corollaries~\textup{\ref{cor:bpcoarseH1}} and~\textup{\ref{cor:bpfibH1}} the condition on the tails and the lower bound on $n_0$ can be dropped when $c\le\frac13$. \end{corollary} \begin{proof} A link with base point $b'$ lies in $V_x$ exactly when $b'-x\in[0,n)^4$ modulo $L$, so it lies in exactly $n^4$ cubes. By Lemma~\ref{lem:bpcoarseblock}, \[ \sum_{x:\,j\notin V_x}C^\varrho_{V_x,j}\le\frac1{1-c_\varrho}\sum_{e\ne j} C^\varrho_{ej}\,\#\{x:e\in V_x\}\le\frac{c_\varrho}{1-c_\varrho}\,n^4\le \frac{n^4}2 , \] and Theorem~\ref{thm:bpcoarse}(i) applies with $\alpha=\frac12$. \end{proof} \begin{corollary}[tori of side up to $e^{2^k/(17k)}$]\label{cor:bppathfourk} Let $k\ge12$, $b=2^k$, $0\le\beta_W\le k+2$, $n\ge2$ and $L$ even with $2n+4\le L\le e^{2^k/(17k)}$. Then the law $\mu^c$ of the block variables of the $\SU(2)$ Wilson measure of $\Lambda_{bL}$ under the straight-path map satisfies $\mathrm{BP}(n,\frac12)$. \end{corollary} \begin{proof} Put $N_k=4k+9$, $B_k=6(k+2)$, $M_k=2^k/(17k)$ and $s_k=48(6k+13)^2 e^{-(2^k-4)/N_k}$. Since $q_W\le1-1/N_k$ and $\log(1-x)\le-x$, \[ \eta_{b-2}=48(1+6\beta_W)^2q_W^{b-4}\le s_k . \] We first show that \begin{equation}\label{eq:bppathrange} 96B_k^2s_k^2e^{4M_k}\le1 . \end{equation} In logarithms this reads $P_k\le Q_k$, with $P_k=\log(96\cdot48^2)+2\log B_k +4\log(6k+13)$ and \[ Q_k=\frac{2(2^k-4)}{N_k}-4M_k=2^k\,\frac{18k-36}{17kN_k}-\frac8{N_k} . \] For $k\ge12$ one has $18k-36\ge15k$ and $N_k\le5k$, so $Q_k\ge R_k-1$ with $R_k=3\cdot2^k/(17k)$. At $k=12$, $\log221184<12.31$, $\log84<4.44$ and $\log85<4.45$ give $P_{12}+1<40<601$. (ii) The coefficient of $\chi_d$ in the normalised plaquette weight is $d\, a_d(\beta_W)$, by Lemma~\ref{lem:bppathmodes} with $B=\beta_W$. On the surface of a cube each of the twelve edges lies in two faces with opposite orientations, Haar integration over it gives a factor $\frac12$, and contracting the indices gives a factor $2$ at each of the eight vertices, so $z_\square=(2a_2)^6\,2^{8-12}=4a_2^6$. Let $r=I_3(6)/I_2(6)$. By the integral representation~\cite[(10.32.2)]{DLMF}, $r$ is the mean of $t$ under the density proportional to $e^{6t}(1-t^2)^{3/2}$ on $[-1,1]$, and integrating the derivative of $e^{6t}(1-t^2)^{5/2}$ gives $5r=6\,\E(1-t^2) <6(1-r^2)$, so $(3r-2)(2r+3)<0$ and $r<\frac23$. The recurrence $I_1(6)- I_3(6)=\frac23I_2(6)$ then gives $a_2(6)>\frac34$, and $a_2$ increases in $\beta_W$, its derivative being the variance of $\cos\theta$. So $z_\square> 4(\frac34)^6=\frac{729}{1024}>\frac1e$ for $\beta_W\ge6$. The condition of~\cite{KP86} asks for $A_\gamma\ge0$ with $\sum_{\gamma'\not\sim\gamma} \lvert z_{\gamma'}\rvert e^{A_{\gamma'}}\le A_\gamma$, and a polymer is incompatible with itself, so it requires $\lvert z_\square\rvert\le A e^{-A} \le\frac1e$. \end{proof} Part (i) concerns the posterior on single fine links and leaves block methods open; part (ii) concerns the unresummed expansion and leaves resummed and multiscale expansions open. What the cube bound on every torus needs is a statement about the scores of the interior products under the posterior, and the following condition is enough. \begin{proposition}[a sufficient covariance bound]\label{prop:bppathsuff} In Corollary~\textup{\ref{cor:bpuptuning}} let the $c_j$ be straight-path maps, so that $\mu^{(0)}_{k,L}$ is the law of the block variables of the straight-path map with blocks of side $b=2^k$, $k\ge2$, and let $\pi_V$, $a^X_e$, $K_b$ and $h_{b-2}$ be as in Theorem~\textup{\ref{thm:bppathmixed}} for $\mu_{k,L}$, with $\eta_{b-2}<1$ and $h_{b-2}<1$. Suppose that \[ \sum_{e\ne j}\sup_{V,X,Y}\bigl\lvert\Cov_{\pi_V}\bigl(a^X_e,a^Y_j\bigr)\bigr \rvert\le\frac{1-h_{b-2}}3 \] for all $L\in\mathcal L_k$ and all coarse links $j$. Then $\mathrm{BP}(n_0,\frac12)$ holds for $\mu^{(0)}_{k,L}$ for every $n_0\ge2$ and every $L\in\mathcal L_k$ with $L_0\ge2n_0+4$. This holds in particular if $\beta_k=O(k)$ and there are $A<\infty$ and $p>4$ with $\lvert\Cov_{\pi_V} (a^X_e,a^Y_j)\rvert\le AK_b^2(1+\lvert\Delta\rvert_\infty)^{-p}$ for all $L$, $V$, unit $X,Y$ and $e\ne j$, once $k$ is so large that $AK_b^2S_p\le(1-h_{b-2})/3$ with $S_p=3+320(1+1/(p-4))$. \end{proposition} \begin{proof} By Theorem~\ref{thm:bppathmixed} and Lemma~\ref{lem:bpcoarsehess} the column sums of the coefficients are at most $\frac13$, and Corollary~\ref{cor:bppathcolumn} applies. For the second statement, three links share the base point of $j$, and at distance $r\ge1$ in $\lvert\cdot \rvert_\infty$ there are $4((2r+1)^4-(2r-1)^4)=4(64r^3+16r)\le320r^3$ links, so the sum is at most $AK_b^2(3+320\sum_{r\ge1}r^{3-p})\le AK_b^2S_p$; and $K_b$, $h_{b-2}\to0$ by Theorem~\ref{thm:bppathdensity}. \end{proof} The hypothesis is a decay of correlations under the posterior for the scores of distinct interior products, uniform in the torus and the coarse data. Proposition~\ref{prop:bphidden}(ii) shows that it is a genuine collective input: its measure $\varrho_L$ is also a mixture of product laws with small scores and small single-link curvature, and there the sum of the mixed derivatives grows with the volume. \begin{remark}[what is proved by smoothing along the paths] \label{rem:bppaths} Proved: convolution with a one-link $\SU(2)$ Wilson density contracts every nontrivial representation by at most $2B/(2B+3)$ (Lemma~\ref{lem:bppathmodes}); a product of $m$ independent one-link variables has a density within $48(1+B)^2(2B/(2B+3))^{m-2}$ of $1$ with its first two derivatives (Theorem~\ref{thm:bppathproduct}); the single-link conditional densities of the law of the block variables of the straight-path map obey these bounds, the single-link curvature of the induced action and the difference between force variance and mean curvature in the fibre law tend to zero along every tuning with $\beta_k= O(k)$, uniformly in the torus and the coarse data (Theorem~\ref{thm:bppathdensity}, Corollary~\ref{cor:bppathcancel}); the law of the block variables is a mixture of product laws over the links off the path interiors, and its mixed derivatives are posterior covariances of the scores of the interior products, each at most $K_b^2$ (Lemma~\ref{lem:bppathint}, Theorem~\ref{thm:bppathmixed}); a column sum at most $\frac13$ gives $\mathrm{BP}(n,\frac12)$ for every $n$ with no condition on the tails (Corollary~\ref{cor:bppathcolumn}); for $k\ge12$ and $\beta_k\le k+2$ the law of the block variables satisfies $\mathrm{BP}(n,\frac12)$ on every coarse torus of side at most $e^{2^k/(17k)}$ (Corollary~\ref{cor:bppathfourk}); Dobrushin's condition for the posterior on single fine links fails for $b\ge256$, $\beta_W\ge\frac83$, and the Koteck\'y--Preiss condition for the unresummed cube surface fails for $\beta_W\ge6$ (Proposition~\ref{prop:bppathobst}); a bound on the sums of the posterior covariances uniform in the torus gives (H1) with $\varpi_0= \frac12$ for every $n_0$ (Proposition~\ref{prop:bppathsuff}). Not proved: that bound for any $k$ outside the strong-coupling range, and hence (H1) along the tuning; the split inequality (H2) along the tuning. Neither $\mathrm{BP}$ along the tuning nor the mass gap is proved here. \end{remark} \subsection{The row sums of the Gaussian model}\label{ssec:bprowsum} This subsection determines the order of the row sums $\gamma_b$ of the effective quadratic form of the Gaussian model of Sec.~\ref{ssec:bpfibre}. The Bloch symbol of $\Gamma$ is written in closed form through the sums of the lattice Green function along the paths, and its derivatives are controlled at the scale $b^{-1/2}$ of the coarse momenta, where the two bounds $1/b$ and $\lvert\kappa\rvert^2$ on the symbol meet. The result is that $\gamma_b$ is of order $(\log b)/b$: condition (G) fails by a logarithm, and the Gaussian form of (FC) holds along every tuning with $\beta_k(\log b_k)/b_k\to0$, in particular along the logarithmic tuning. The same numbers are the covariances of the scores in every representation of the coarse density as a mixture of products over the coarse links (Remark~\ref{rem:bprowposterior}). Throughout, $b\ge3$, $\lvert\kappa\rvert$ is the Euclidean norm of $\kappa\in\R^4$, and for $p,\kappa\in\R^4$, $q\in\R$, \[ \begin{split} \epsilon(p)&=\sum_\rho4\sin^2\frac{p_\rho}2,\qquad\sigma_\rho(\kappa)= e^{i\kappa_\rho}-1,\\ F_b(q)&=\Bigl\lvert\sum_{t=0}^{b-1}e^{iqt}\Bigr\rvert^2=\sum_{\lvert r \rvert0$. Put \[ r_\mu=\frac1{\mathcal G_\mu},\qquad\mathcal T=\sum_\rho r_\rho\lvert \sigma_\rho\rvert^2,\qquad\omega(\kappa)=\min\Bigl(\frac1b,\lvert\kappa \rvert^2\Bigr). \] In the proofs, $C$ denotes constants that depend only on the order of the derivatives involved and may change from line to line, and two elementary facts are used: for $\lvert x\rvert\le\pi$, \begin{equation}\label{eq:bprowsin} \frac4{\pi^2}x^2\le4\sin^2\frac x2\le x^2 ; \end{equation} and for $\kappa\in[-\pi,\pi]^4$, $n\in Z_b$ and $p=(\kappa+2\pi n)/b$, the distance $\delta_\rho$ of $p_\rho$ to $2\pi\Z$ satisfies \begin{equation}\label{eq:bprowdist} \delta_\rho\ge\frac{\pi\lvert n_\rho\rvert}b , \end{equation} since $\lvert\kappa_\rho+2\pi n_\rho\rvert\ge\pi\lvert n_\rho\rvert$ and $\lvert p_\rho\rvert\le\pi+\pi/b$, and when $\lvert p_\rho\rvert>\pi$, $\delta_\rho=2\pi-\lvert p_\rho\rvert\ge\pi/2\ge\pi\lvert n_\rho\rvert/b$. \begin{lemma}[the line sums]\label{lem:bprowline} For $\kappa\in[-\pi,\pi]^4\setminus\{0\}$ and every direction $\mu$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item the two bounds $\mathcal G_\mu(\kappa)\ge4/(\pi^2\lvert\kappa\rvert^2)$ and $\mathcal G_\mu(\kappa)\ge4b/(7\pi^4)$ hold, and hence $\mathcal G_\mu( \kappa)\ge4/(7\pi^4\omega(\kappa))$; \item $\mathcal G_\mu(\kappa)\le\pi^2/(4\lvert\kappa\rvert^2)+15b\le18/\omega( \kappa)$; \item for every multi-index $\alpha$, $\lvert\partial^\alpha\mathcal G_\mu( \kappa)\rvert\le C\omega(\kappa)^{-1}\lvert\kappa\rvert^{-\lvert\alpha \rvert}$. \end{enumerate} \end{lemma} \begin{proof} (i) For $n=0$, $F_b(\kappa_\mu/b)=\sin^2(\kappa_\mu/2)/\sin^2(\kappa_\mu/2b) \ge4b^2/\pi^2$ by \eqref{eq:bprowsin}, and $F_b(0)=b^2$; and $\epsilon(\kappa /b)\le\lvert\kappa\rvert^2/b^2$. This gives the first bound. For the second keep the terms with $n_\mu=0$, for which $F_b(p_\mu)\ge4b^2/\pi^2$ as before and, by \eqref{eq:bprowsin}, $\epsilon(p)\le\lvert p\rvert^2\le\pi^2b^{-2} (1+\sum_{\rho\ne\mu}(1+2\lvert n_\rho\rvert)^2)$. For $1\le t\le\lfloor (b-1)/2\rfloor$ the vectors $n$ with $n_\mu=0$ and $\max_{\rho\ne\mu}\lvert n_\rho\rvert=t$ lie in $Z_b$, there are $(2t+1)^3-(2t-1)^3\ge24t^2$ of them, and each has $1+\sum_{\rho\ne\mu}(1+2\lvert n_\rho\rvert)^2\le1+27t^2\le28t^2$. So these terms of $\sum1/\epsilon(p)$ add up to at least $\frac{b^2}{\pi^2} \frac67\lfloor\frac{b-1}2\rfloor\ge\frac{b^3}{7\pi^2}$, as $\lfloor(b-1)/2 \rfloor\ge b/6$ for $b\ge3$, and $\mathcal G_\mu\ge b^{-4}\frac{4b^2}{\pi^2} \frac{b^3}{7\pi^2}$. (ii) For $n=0$, $F_b\le b^2$ and $\epsilon(\kappa/b)\ge4\lvert\kappa\rvert^2/( \pi^2b^2)$ by \eqref{eq:bprowsin}, which gives $\pi^2/(4\lvert\kappa \rvert^2)$. For $n\ne0$, \eqref{eq:bprowsin} and \eqref{eq:bprowdist} give $\epsilon(p)\ge\frac4{\pi^2}\sum_\rho\delta_\rho^2\ge4\lvert n\rvert^2/b^2$, and $F_b(p_\mu)\le b^2$, while for $n_\mu\ne0$, $F_b(p_\mu)=\sin^2(\kappa_\mu /2)/\sin^2(p_\mu/2)\le\pi^2/\delta_\mu^2\le b^2/n_\mu^2$. With $m(n)=\min(1, n_\mu^{-2})$ the term of $n$ is at most $m(n)/(4\lvert n\rvert^2)$. The vectors of $Z_b$ with $n_\mu=0$ and $\max_\rho\lvert n_\rho\rvert=t\ge1$ are at most $24t^2+2\le26t^2$, so these vectors contribute at most $26\cdot \frac b2=13b$ to $\sum_{n\ne0}\lvert n\rvert^{-2}$, and \[ \begin{aligned} \sum_{n\in Z_b\setminus\{0\}}\frac{m(n)}{\lvert n\rvert^2}&\le13b+\sum_{n_\mu \ne0}\frac1{n_\mu^2}\Bigl(\frac1{n_\mu^2}+13b\Bigr)\\ &\le13b\Bigl(1+\frac{\pi^2}3\Bigr)+\frac{\pi^4}{45}\le60b . \end{aligned} \] So the terms with $n\ne0$ add up to at most $15b$, and $\pi^2/4+15\le18$. (iii) Since $\partial_\kappa=b^{-1}\partial_p$, the Leibniz rule writes $\partial^\alpha$ of the term of $n$ as a sum of at most $2^{\lvert\alpha \rvert}$ products $b^{-j}F_b^{(j)}(p_\mu)\cdot b^{-\lvert\alpha'\rvert}( \partial^{\alpha'}\epsilon^{-1})(p)$ with $j+\lvert\alpha'\rvert=\lvert \alpha\rvert$. Let $\delta(q)$ be the distance of $q\in\R$ to $2\pi\Z$ and $\delta(p)$ that of $p\in\R^4$ to $2\pi\Z^4$. First, $\lvert F_b^{(j)}(q)\rvert \le Cb^j\min(b^2,\delta(q)^{-2})$: the Fej\'er sum gives $\lvert F_b^{(j)} \rvert\le\sum_{\lvert r\rvert0$ and $\mathcal T\ge\min_\rho r_\rho\,\epsilon(\kappa)>0$. The points of $(2\pi/L)\Z^4$, $L\ge3$, are dense, so it suffices to fix $L\ge3$ and $\kappa\in(2\pi/L)\Z^4\setminus2\pi\Z^4$ and to show that the right side is the matrix $\hat g(\kappa)$ of that proof. For $A\in\C^{E'}$ let $\mathsf SA\in\C^E$ be its sums along the paths. By the proof of Proposition~\ref{prop:bpfibgauss}(iii), $\ip v{\Gamma v}=\min\{ \lvert dA\rvert^2:\mathsf SA=v\}$ for real $v$, and, splitting real and imaginary parts, also for complex $v$ with the Hermitian forms. For $f\in \C^4$ let $v_{(y,\nu)}=e^{i\kappa\cdot y}f_\nu$; then $\ip v{\Gamma v}=L^4f^* \hat g(\kappa)f$ by the action of $\Gamma$ on $\mathcal H_\kappa$ in that proof. The maps $d$ and $\mathsf S$ commute with the translations by $b\Z^4$ and preserve the orthogonal decomposition into the spaces $\mathcal H_{\kappa'}$; the constraint concerns only the component of $A$ in $\mathcal H_\kappa$, so the minimum is the minimum over $A$ in $\mathcal H_\kappa$ of the links. Write a form as $(A_\rho)_\rho$, $A_\rho(x)=A_{(x,\rho)}$, and $\partial_\rho \phi(x)=\phi(x+e_\rho)-\phi(x)$, with adjoint $\partial_\rho^*$. Then $(dA)_p =\partial_\mu A_\nu(x)-\partial_\nu A_\mu(x)$ for the plaquette $p$ at $x$ in the plane $\mu<\nu$, and $d^{\rm T}A=\sum_\rho\partial_\rho^*A_\rho$. The differences commute, so $\ip{\partial_\mu^*A_\mu}{\partial_\nu^*A_\nu}=\ip{ \partial_\nu A_\mu}{\partial_\mu A_\nu}$ and $\lVert\partial^*_\rho A_\rho \rVert=\lVert\partial_\rho A_\rho\rVert$, and the cross terms cancel in \begin{equation}\label{eq:bprowweitz} \lvert dA\rvert^2+\lvert d^{\rm T}A\rvert^2=\sum_{\mu,\nu}\lVert\partial_\mu A_\nu\rVert^2=\sum_\nu\ip{A_\nu}{\Delta A_\nu}, \end{equation} where $\Delta=\sum_\mu\partial_\mu^*\partial_\mu$. The space $\mathcal H_\kappa$ of the sites of $\Lambda_{bL}$ has dimension $b^4$ and is spanned by the plane waves $e_n(x)=e^{ip\cdot x}$, $p=(\kappa+ 2\pi n)/b$, $n\in Z_b$: they satisfy $e_n(x+bz)=e^{i\kappa\cdot z}e_n(x)$, are periodic on $\Lambda_{bL}$ because $L\kappa\in2\pi\Z^4$, and are orthogonal, their $p$ being distinct modulo $2\pi\Z^4$; and $\Delta e_n=\epsilon(p)e_n$ with $\epsilon(p)>0$. The same holds for each component $A_\rho$ of a form in $\mathcal H_\kappa$, the link $(x,\rho)$ being identified with $x$. If $A\in\mathcal H_\kappa$ and $\mathsf SA=v$, let $\phi\in\mathcal H_\kappa$ solve $\Delta\phi=-d^{\rm T}A$, so that $A'=A+d\phi$ has $d^{\rm T}A' =0$ and $dA'=dA$; as $\phi(by)=e^{i\kappa\cdot y}\phi(0)$ and $\pi(y,\nu)$ runs from $by$ to $b(y+e_\nu)$, $\mathsf SA'_{(y,\nu)}=e^{i\kappa\cdot y}(f_\nu +\phi(0)\sigma_\nu(\kappa))$. Conversely, if $A'\in\mathcal H_\kappa$ has $\mathsf SA'_{(y,\nu)}=e^{i\kappa\cdot y}(f_\nu+t\sigma_\nu)$, $t\in\C$, then $A'-d\phi'$, with $\phi'$ equal to $te^{i\kappa\cdot y}$ at $by$ and to $0$ at the other sites, lies in $\mathcal H_\kappa$, has the path sums $v$ and the differential $dA'$. With \eqref{eq:bprowweitz} it follows that $L^4f^* \hat g(\kappa)f$ is the minimum of $\sum_\nu\ip{A_\nu}{\Delta A_\nu}$ over the $A\in\mathcal H_\kappa$ with $\mathsf SA_{(y,\nu)}=e^{i\kappa\cdot y}u_\nu$, $u\in f+\C\sigma(\kappa)$: the forms $A'$ above attain the value $\lvert dA\rvert^2$, and every admissible $A$ has $\sum_\nu\ip{A_\nu}{\Delta A_\nu} \ge\lvert dA\rvert^2=\lvert d(A-d\phi')\rvert^2$. For fixed $u$ this minimum splits over $\nu$. For $A_\nu=\sum_na_ne_n$ the constraint is the single condition $\sum_na_nc_n=u_\nu$, $c_n=\sum_{t=0}^{b-1} e^{ip_\nu t}$, the sum along $\pi(0,\nu)$, the others following by quasi-periodicity, and $\ip{A_\nu}{\Delta A_\nu}=(bL)^4\sum_n\lvert a_n\rvert^2 \epsilon(p)$. Since $\lvert c_n\rvert^2=F_b(p_\nu)$, the Cauchy--Schwarz inequality gives the minimum \[ \frac{(bL)^4\lvert u_\nu\rvert^2}{\sum_n\lvert c_n\rvert^2/\epsilon(p)}=L^4 r_\nu\lvert u_\nu\rvert^2 . \] With $R=\mathrm{diag}(r_\nu)$, the minimum over $t\in\C$ of $(f+t\sigma)^*R(f+t\sigma)$ is $f^*Rf-\lvert \sigma^*Rf\rvert^2/\sigma^*R\sigma$, and $\sigma^*R\sigma=\mathcal T$. The matrix $\hat g(\kappa)$ is Hermitian, because $\Gamma$ is real and symmetric, and so is the matrix of the statement; their quadratic forms agree, so they are equal. Finally, a form $v$ that is constant on the coarse links of each direction has $\Gamma v=0$, since $A$ equal to $v_{(0,\nu)}/b$ on every fine link of direction $\nu$ has these path sums and $dA=0$; so $\hat\Gamma(0)=0$. \end{proof} \begin{lemma}[the derivatives of the symbol]\label{lem:bprowderiv} For $\kappa\in[-\pi,\pi]^4\setminus\{0\}$, all $\mu,\nu$ and every multi-index $\alpha$ with $\lvert\alpha\rvert\le3$, \[ \bigl\lvert\partial^\alpha\hat\Gamma(\kappa)_{\mu\nu}\bigr\rvert\le C\, \omega(\kappa)\,\lvert\kappa\rvert^{-\lvert\alpha\rvert} \] with an absolute constant $C$. \end{lemma} \begin{proof} Lemma~\ref{lem:bprowline}(i),(ii) gives \[ \frac\omega{18}\le r_\mu\le\frac{7\pi^4\omega}4 . \] The derivative $\partial^\alpha(1/\mathcal G_\mu)$ is a sum of terms \[ \mathcal G_\mu^{-1-m}\prod_{i=1}^m\partial^{\alpha_i}\mathcal G_\mu , \] so Lemma~\ref{lem:bprowline}(iii) gives $\lvert\partial^\alpha r_\mu\rvert\le C\omega\lvert\kappa\rvert^{-\lvert\alpha\rvert}$. Since $\lvert\sigma_\rho\rvert\le\lvert\kappa_\rho\rvert$, $\lvert\sigma_\rho \rvert^2=2-2\cos\kappa_\rho$ and $\lvert\kappa\rvert\le2\pi$, \[ \lvert\partial^\alpha\sigma_\rho\rvert\le C\lvert\kappa\rvert^{1-\lvert\alpha \rvert},\qquad\lvert\partial^\alpha\lvert\sigma_\rho\rvert^2\rvert\le C\lvert \kappa\rvert^{2-\lvert\alpha\rvert}. \] Hence \begin{gather*} \lvert\partial^\alpha(r_\mu r_\nu\sigma_\mu\bar\sigma_\nu)\rvert\le C\omega^2 \lvert\kappa\rvert^{2-\lvert\alpha\rvert},\\ \lvert\partial^\alpha\mathcal T\rvert\le C\omega\lvert\kappa\rvert^{2-\lvert \alpha\rvert}, \end{gather*} while $\mathcal T\ge\frac\omega{18}\epsilon( \kappa)\ge\frac{2\omega}{9\pi^2}\lvert\kappa\rvert^2$ by \eqref{eq:bprowsin}; so $\lvert\partial^\alpha\mathcal T^{-1}\rvert\le C(\omega\lvert\kappa\rvert^2 )^{-1}\lvert\kappa\rvert^{-\lvert\alpha\rvert}$, and the Leibniz rule applied to Lemma~\ref{lem:bprowsymbol} gives the claim. \end{proof} \begin{theorem}[the order of the row sums]\label{thm:bprowgauss} There is an absolute constant $C$ such that for every $b\ge3$ and all directions $\mu\ne\nu$ \[ \frac{\log(1+60b)}{15\pi b}\le\sum_{w\in\Z^4}\bigl\lvert\Gamma^\infty_b(w)_{ \mu\nu}\bigr\rvert\le\gamma_b\le\frac{C(1+\log b)}b . \] In particular $b\gamma_b\to\infty$, so that \textup{(G)} fails, and $\gamma_b\log b\to0$. \end{theorem} \begin{proof} The middle inequality holds because $(w,\nu)\ne(0,\mu)$. \emph{Upper bound.} Fix $\mu,\nu$, not necessarily distinct, and let $f(w)= \Gamma^\infty_b(w)_{\mu\nu}$, the Fourier coefficients of $\hat\Gamma_{\mu\nu}$: $\hat\Gamma(\kappa)=\sum_w\Gamma^\infty_b(w)e^{i\kappa\cdot w}$. By Lemmas~\ref{lem:bprowsymbol} and~\ref{lem:bprowderiv}, $\hat\Gamma_{\mu\nu}$ is smooth on $\R^4\setminus2\pi\Z^4$ with $\lvert\partial^\alpha\hat\Gamma_{\mu \nu}\rvert\le C\lvert\kappa\rvert^{2-\lvert\alpha\rvert}$ near $0$ for $\lvert\alpha\rvert\le3$. For a direction $\rho$ and $j\le3$ the function $\partial_\rho^j\hat\Gamma_{\mu\nu}$ is square integrable on $[-\pi,\pi]^4$, and its Fourier coefficients are $(iw_\rho)^jf(w)$: integrate by parts on the cube minus a ball of radius $t$ about $0$; the faces cancel by periodicity, and the terms on the sphere are $O(t^{3+2-(j-1)})\to0$. By the Parseval identity and $\lvert w\rvert^4\le4\sum_\rho w_\rho^4$, $\lvert w\rvert^6\le16 \sum_\rho w_\rho^6$, the sums $A_0=\sum_w\lvert f\rvert^2$, $A_4=\sum_w\lvert w\rvert^4\lvert f\rvert^2$ and $A_6=\sum_w\lvert w\rvert^6\lvert f\rvert^2$ satisfy $A_0\le C\sup\omega^2\le C/b^2$ and \[ A_4\le C\!\int\!\frac{\omega^2}{\lvert\kappa\rvert^4}\,d\kappa\le\frac{C(1+ \log b)}{b^2},\qquad A_6\le C\!\int\!\frac{\omega^2}{\lvert\kappa\rvert^6}\, d\kappa\le\frac Cb , \] the integrals over $[-\pi,\pi]^4$, split at $\lvert\kappa\rvert=b^{-1/2}$ with $\omega\le\lvert\kappa\rvert^2$ inside and $\omega\le1/b$ outside: the first is at most $\frac{\pi^2}{2b^2}+\frac{2\pi^2}{b^2}\log(2\pi b^{1/2})$ and the second at most $\frac{\pi^2}b+\frac{\pi^2}b$. Let $J$ be the least integer with $2^J\ge b^{1/2}$. The shell $S_j$ of the $w$ with $2^j\le\lvert w\rvert< 2^{j+1}$ has at most $2^{4j+8}$ points, so by the Cauchy--Schwarz inequality $\sum_{S_j}\lvert f\rvert\le2^{2j+4}(\sum_{S_j}\lvert f\rvert^2)^{1/2}\le16( \sum_{S_j}\lvert w\rvert^4\lvert f\rvert^2)^{1/2}$, and once more \[ \sum_{1\le\lvert w\rvert<2^J}\lvert f(w)\rvert\le16\sqrt{JA_4}\le\frac{C(1+\log b)}b . \] Further $\sum_{\lvert w\rvert\ge2^J}\lvert w\rvert^{-6}\le\sum_{j\ge J}2^{8-2j} \le2^{10}b^{-1}$, so $\sum_{\lvert w\rvert\ge2^J}\lvert f\rvert\le(2^{10}A_6/b )^{1/2}\le C/b$, and $\lvert f(0)\rvert\le A_0^{1/2}$. Summing over $\nu$ gives the bound on $\gamma_b$. \emph{Lower bound.} Let $\mu\ne\nu$, $f(w)=\Gamma^\infty_b(w)_{\mu\nu}$, and let $k(w_\mu,w_\nu)$ be the sum of $f$ over the other two coordinates; as $f$ is summable, $k$ is summable on $\Z^2$ with $\sum_{w_\mu,w_\nu}k\,e^{i(\kappa_\mu w_\mu+\kappa_\nu w_\nu)}=\hat\Gamma_{\mu\nu}( \kappa_\mu e_\mu+\kappa_\nu e_\nu)$. On the plane $\kappa=\kappa_\mu e_\mu+ \kappa_\nu e_\nu$, Lemma~\ref{lem:bprowsymbol} gives, with $r_\mu,r_\nu$ at $\kappa$, \[ \hat\Gamma_{\mu\nu}(\kappa)=-\frac{4r_\mu r_\nu e^{i(\kappa_\mu-\kappa_\nu)/2} \sin\frac{\kappa_\mu}2\sin\frac{\kappa_\nu}2}{\mathcal T}, \] where $\mathcal T=4r_\mu\sin^2\frac{\kappa_\mu}2+4r_\nu\sin^2\frac{\kappa_\nu} 2$. Let $\chi(t)=1$ for $t\ge1$ and $-1$ for $t\le0$, $\chi'(t)=\chi(t+1)$, and $h_N(w)=\chi(w_\mu)\chi'(w_\nu)$ for $-N\le w_\mu\le N+1$, $-N-1\le w_\nu\le N$ and $h_N(w)=0$ otherwise. For half-integers $u$, $\sum_{\lvert u\rvert\le N+1/2}\operatorname{sgn}(u)e^{-i\theta u}=-2i\sum_{l=0}^N \sin((l+\frac12)\theta)=-i(1-\cos((N+1)\theta))/\sin\frac\theta2$, and with $u=w_\mu-\frac12$ and $u=w_\nu+\frac12$, \[ \begin{aligned} \sum_wf(w)h_N(w)={}&\frac1{4\pi^2}\int_{[-\pi,\pi]^2}\frac{4r_\mu r_\nu}{ \mathcal T}\bigl(1-\cos(N+1)\kappa_\mu\bigr)\\ &\times\bigl(1-\cos(N+1)\kappa_\nu\bigr)\,d\kappa_\mu d\kappa_\nu . \end{aligned} \] The function $4r_\mu r_\nu/\mathcal T$ is positive. By \eqref{eq:bprowsin}, \[ \mathcal T\ge\frac4{\pi^2}\min(r_\mu,r_\nu)\lvert\kappa\rvert^2 , \] so \[ \frac{4r_\mu r_\nu}{\mathcal T}\le\frac{\pi^2\max(r_\mu,r_\nu)}{\lvert\kappa \rvert^2}\le\frac{C\omega}{\lvert\kappa\rvert^2}\le C , \] so it is integrable. As $N\to\infty$ the left side tends to $\sum_wf(w)h(w)$, $h=\lim h_N$, by dominated convergence, and the right side to the integral of $4r_\mu r_\nu/\mathcal T$, by the Riemann--Lebesgue lemma applied to the four products of cosines. Now $\lvert h\rvert\le1$, $\mathcal T\le\max( r_\mu,r_\nu)\lvert\kappa\rvert^2$ by \eqref{eq:bprowsin}, and Lemma~\ref{lem:bprowline}(ii) gives \[ \min(r_\mu,r_\nu)\ge\bigl(\pi^2/(4\lvert\kappa\rvert^2)+15b\bigr)^{-1}; \] hence \[ \sum_w\lvert f(w)\rvert\ge\frac1{4\pi^2}\int_{\lvert\kappa\rvert\le\pi}\frac{4 \,d\kappa_\mu d\kappa_\nu}{\pi^2/4+15b\lvert\kappa\rvert^2}=\frac{\log(1+60b)}{ 15\pi b}. \] Finally $b\gamma_b\ge\log(1+60b)/(15\pi)\to\infty$ and $\gamma_b\log b\le C(1+ \log b)\log b/b\to0$. \end{proof} In the lower bound the logarithm is the integral of $\lvert\kappa \rvert^{-2}$ over the coarse momenta with $b^{-1/2}\le\lvert\kappa\rvert\le \pi$ in the plane of two directions, where the entries of the symbol between these directions are of size $1/b$ and depend on the direction of $\kappa$ through the projection onto the coarse gradient; the sign function $h$ of the proof pairs with the kernel as the integral of a positive function, with no cancellation. \begin{corollary}[the Gaussian form of (FC) along the tuning] \label{cor:bprowFC} In the Gaussian model with blocks of side $b$ and coupling $\beta$, let $J^{\rm G}_{ee'}=\beta\lvert\Gamma_{ee'}\rvert$ as in Corollary~\textup{\ref{cor:bpfibgauss}}, and let $C$ be the constant of Theorem~\textup{\ref{thm:bprowgauss}}. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For every $L\ge3$ and every coarse link $j$, $\frac{D^2}4\sum_{e\ne j} J^{\rm G}_{ej}\le\frac{D^2}4\beta C(1+\log b)/b$, and the inequalities of \textup{(FC)} for $J^{\rm G}$ hold with $c=\frac1{17}$ and $R=1$ on every torus as soon as $17D^2\beta C(1+\log b)\le4b$. \item For every coarse link $j$, \[ \sup_{L\ge3}\frac{D^2}4\sum_{e\ne j}J^{\rm G}_{ej}\ge\frac{D^2\beta\log(1+ 60b)}{60\pi b}; \] so the first inequality of \textup{(FC)} for $J^{\rm G}$ with some $c<1$ on all tori implies $\beta\log(1+60b)<60\pi b/D^2$. \item Along every sequence $b=2^k$, $\beta_k$ with $k\beta_k2^{-k}\to0$, in particular along $\beta_k=2+\frac{11}{3\pi^2}k\log2$, the inequalities of \textup{(FC)} for $J^{\rm G}$ hold with $c=\frac1{17}$ and $R=1$ for all large $k$, uniformly in $L_0$, and the column sums tend to zero. \end{enumerate} \end{corollary} \begin{proof} (i) Proposition~\ref{prop:bpfibbloch} and Theorem~\ref{thm:bprowgauss} bound the column sums, as in Corollary~\ref{cor:bpfibgauss}; the tails are at most the column sums, and $(1-\frac1{17})/16=\frac1{17}$. (ii) Let $j=(y, \mu)$ and $\nu\ne\mu$. For $\lvert w\rvert_\infty0$, the factors $\theta_i=\nu_i/\sigma_i$ satisfy $\theta_i\ge1$, \[ \omega_k(\mathbf 1)=\prod_{i1$ if and only if some column of $R_i$ has a negative entry. If $R_i\ge0$ for every $i0$. The equal column norms give $\lvert R_i\rvert^{\mathsf T}\mathbf 1=\nu_i\mathbf 1$, hence $m^{(k)}=(\prod_{i0$, means that none is negative; all columns share $\nu_i$ and $\sigma_i$. If every $R_i\ge0$, then $\lvert R_i\rvert=R_i$, so $(m^{(k)})^{\mathsf T}=\mathbf 1^{\mathsf T}T_k$, and $T_kx\ge0$ for $x\ge0$ gives $\mathbf 1^{\mathsf T}T_kx=\lVert T_kx\rVert_1$. \end{proof} The reconstructions that arise at Gaussian order are conditional means. A \emph{positive block average} on a partition $X=\bigcup_{a\in\mathcal B}B(a)$ with $\lvert\mathcal B\rvert\ge2$ is a matrix $P\in\R^{\mathcal B\times X}$ with $P_{ax}>0$ for $x\in B(a)$ and $P_{ax}=0$ otherwise. For a matrix $A$ on $X$ the blocks are \emph{thick} if for $a\ne a'$ the block $B(a')$ contains a site $x$ with $A_{xy}=0$ for every $y\in B(a)$. \begin{proposition}[conditional means change sign]\label{prop:bpnegative} Let $A$ be invertible with $A^{-1}$ entrywise positive, let $P$ be a positive block average whose blocks are thick for $A$, and suppose that $PA^{-1} P^{\mathsf T}$ is invertible. Then \[ R=A^{-1}P^{\mathsf T}\bigl(PA^{-1}P^{\mathsf T}\bigr)^{-1} \] has a negative entry. If $A$ is symmetric and positive definite, $Rz$ is the minimiser of $\langle\varphi,A\varphi\rangle$ under $P\varphi=z$, which is the conditional mean, given $P\varphi=z$, of the centred Gaussian field with covariance $A^{-1}$. \end{proposition} \begin{proof} Put $\Lambda=(PA^{-1}P^{\mathsf T})^{-1}$, so that $PR=I$ and $AR=P^{\mathsf T}\Lambda$. Suppose that $R\ge0$. For $a'\ne a$, $0=(PR)_{a'a}=\sum_{x\in B(a')}P_{a'x}R_{xa}$ with positive $P_{a'x}$, so $R_{xa}=0$ for $x\notin B(a)$: the column $a$ of $R$ is supported in $B(a)$. For a site $x\in B(a')$ as in the definition of thickness, $(AR)_{xa}=\sum_{y\in B(a)} A_{xy}R_{ya}=0$, while $(P^{\mathsf T}\Lambda)_{xa}=P_{a'x}\Lambda_{a'a}$; hence $\Lambda_{a'a}=0$ and $\Lambda$ is diagonal. Then $PA^{-1}P^{\mathsf T}=\Lambda^{-1}$ is diagonal. But for $a\ne a'$ its entry $\sum_{x,y}P_{ax}(A^{-1})_{xy}P_{a'y}$ is positive, since every term is nonnegative and those with $x\in B(a)$, $y\in B(a')$ are positive. This contradiction shows that $R$ has a negative entry. The last statement is the method of Lagrange multipliers: the minimiser satisfies $A\varphi= P^{\mathsf T}\lambda$ and $P\varphi=z$, so $\lambda=\Lambda z$ and $\varphi=Rz$. \end{proof} \begin{remark}[thick blocks]\label{rem:bpthick} On the periodic lattice $(\Z/L\Z)^d$ let $A=\varepsilon+2d-N$ with $\varepsilon>0$ and $N$ the adjacency matrix. Then $A^{-1}=\sum_{j\ge0} N^j/(\varepsilon+2d)^{j+1}$, since $\lVert N\rVert\le2d$, and this is entrywise positive because the lattice is connected. With blocks of side $b\ge2$ and at least three blocks in each direction the blocks are thick for $A$: if $B(a)$ and $B(a')$ are not neighbours in one direction, every site of $B(a')$ will do; if $B(a)=B(a')\pm be_\mu$, a site of $B(a')$ on its face opposite to $B(a)$ has its neighbours in $B(a')$, in $B(a')\mp be_\mu\ne B(a)$, or in blocks displaced in other directions. So the conditional mean of the block averages, with equal weights, of a massive Gaussian field has a negative entry in every column, the columns being translates of one another, and $\lvert R \rvert^{\mathsf T}\mathbf 1$ exceeds $R^{\mathsf T}\mathbf 1$ in every entry. Thickness cannot be dropped: on the cycle $\Z/4\Z$ with the two blocks $\{0,1\}$ and $\{2,3\}$, each site has a neighbour in the other block, and the reflection $x\mapsto1-x$, which preserves $A$, $P$ and both blocks, shows that $R$ is constant on each block; with $PR=I$ this makes $R$ the matrix of the block indicators, which is nonnegative. \end{remark} A metric need not lose at every scale. Let $\lVert\cdot\rVert_k$ be a norm on the fields on $X_k$, let $\Phi_k$ be a seminorm on them (a gradient term, say), and let $V_k$ be linear from fields on $X_k$ to fields on $X_{k+1}$ with $V_kR_k=I$, so that $\Pi_k=R_kV_k$ is a projection onto the range of $R_k$. For $s>0$ and $\beta',\delta\ge0$ put \[ Q_k(y)=\lVert y\rVert_k+\beta'\Phi_k(y)+\delta\lVert(I-\Pi_k)y\rVert_k \] and $N_0=\lVert\cdot\rVert_0$, $N_k=s^kQ_k$ for $k\ge1$. The chain $(N_k)$ \emph{composes} if $N_k(R_kz)\le N_{k+1}(z)$ for all $k$ and $z$. \begin{theorem}[composition with a deviation term]\label{thm:bpdeviation} Suppose that $\lVert R_kz\rVert_k\le c_1\lVert z\rVert_{k+1}$, $\Phi_k(R_kz)\le c_2\lVert z\rVert_{k+1}$ and $\Phi_k\le c_3\lVert\cdot \rVert_k$ for all $k$, and that \[ \delta\ge1+c_3\beta'+\frac{c_1+\beta'c_2}s . \] \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The chain composes if and only if $\lVert R_0u\rVert_0\le s(\lVert u \rVert_1+\beta'\Phi_1(u))$ for every $u$ in the range of $R_1$, and \[ \lVert R_ku\rVert_k+\beta'\Phi_k(R_ku)\le s\bigl(\lVert u\rVert_{k+1}+ \beta'\Phi_{k+1}(u)\bigr) \] for every $k\ge1$ and every $u$ in the range of $R_{k+1}$. \item If moreover $s\lVert z\rVert_{k+1}\le\lVert R_kz\rVert_k$ and $\lVert V_ky\rVert_{k+1}\le c_4\lVert y\rVert_k$ for all $k$, then \[ N_k(y)\le\bigl(1+c_3\beta'+\delta(1+c_1c_4)\bigr)\lVert T_ky\rVert_0 \] for every $k$ and every field $y$ on $X_k$. \end{enumerate} \end{theorem} \begin{proof} (i) Since $V_kR_k=I$, $(I-\Pi_k)R_k=0$, so $N_k(R_kz)=s^k(\lVert R_kz\rVert_k +\beta'\Phi_k(R_kz))$ for $k\ge1$. If the chain composes, take $z=u=R_{k+1}w$: then $(I-\Pi_{k+1})z=0$, and dividing by $s^k$ gives the two conditions. Conversely, let $k\ge1$ and write $z=u+e$ with $u=\Pi_{k+1}z$ in the range of $R_{k+1}$ and $e=(I-\Pi_{k+1})z$. The triangle inequality, the condition at $u$ and the bounds on $R_k$ and $\Phi_k$ give \[ \begin{split} &\lVert R_kz\rVert_k+\beta'\Phi_k(R_kz)\\ &\quad\le s\bigl(\lVert u\rVert_{k+1}+\beta'\Phi_{k+1}(u)\bigr)+(c_1+ \beta'c_2)\lVert e\rVert_{k+1}\\ &\quad\le s\bigl(\lVert z\rVert_{k+1}+\beta'\Phi_{k+1}(z)\bigr)\\ &\qquad+\bigl(s(1+c_3\beta')+c_1+\beta'c_2\bigr)\lVert e\rVert_{k+1}, \end{split} \] using $\lVert u\rVert\le\lVert z\rVert+\lVert e\rVert$ and $\Phi_{k+1}(u) \le\Phi_{k+1}(z)+c_3\lVert e\rVert$; the last coefficient is at most $s\delta$, which is the step $N_k(R_kz)\le N_{k+1}(z)$. The step $k=0$ is the same with $\beta'=0$ on the left side. (ii) Iterating $s\lVert z\rVert_{k+1}\le\lVert R_kz\rVert_k$ gives $s^k \lVert y\rVert_k\le\lVert T_ky\rVert_0$. Also $\lVert(I-\Pi_k)y\rVert_k\le (1+c_1c_4)\lVert y\rVert_k$ and $\Phi_k(y)\le c_3\lVert y\rVert_k$, so $Q_k(y) \le(1+c_3\beta'+\delta(1+c_1c_4))\lVert y\rVert_k$, and the bound follows, for $k=0$ because the constant is at least one. \end{proof} The theorem moves the whole difficulty into the two-term inequality with one factor $s$ at every scale; the deviation term absorbs the part of the field that the reconstruction does not produce. A comparison argument with local costs needs, in addition, that the transport cost of a perturbation is controlled by finitely many local terms, and the next lemma says exactly when this is possible. \begin{lemma}[local response bounds]\label{lem:bptwist} Let $\mathsf t$ be a seminorm on a finite-dimensional real vector space and $l_j$, $j\in J$, finitely many linear forms on it. There are $K_j\ge0$ with $\mathsf t\le\sum_jK_j\lvert l_j\rvert$ if and only if $\mathsf t(\xi)=0$ for every $\xi$ with $l_j(\xi)=0$ for all $j$. Finiteness of $J$ cannot be dropped. \end{lemma} \begin{proof} The condition is clearly necessary. Conversely, $n=\sum_j\lvert l_j\rvert$ is a seminorm whose kernel lies in that of $\mathsf t$, so both descend to the quotient by $\ker n$, on which $n$ is a norm; all norms on a finite-dimensional space are equivalent and the seminorm $\mathsf t$ is continuous, so $\mathsf t\le Kn$ on the unit sphere of $n$ and everywhere. For the last statement take finitely supported real sequences, $\mathsf t(\xi)=\lvert\xi_0\rvert$ and $l_j(\xi)=\xi_j-\xi_{j+1}$ for $j\ge0$. If every $l_j(\xi)$ vanishes, $\xi$ is constant and finitely supported, hence zero; but for a finite set $J$ and $M>\max J$ the indicator $\xi$ of $\{0,\dots,M\}$ has $l_j(\xi)=0$ for $j\in J$ and $\mathsf t(\xi)=1$. \end{proof} \begin{remark}[a chain in one dimension]\label{rem:bpchain} Put \[ \varrho_k=\sqrt{\frac{4+2\cdot4^{-k}}3},\qquad\theta_k=1+\frac2{\varrho_k (2+\varrho_k)}, \] $\zeta_0=0$, $\zeta_1=(\sqrt2-1)/2$, $\zeta_2=(\theta_1-1+\theta_1\zeta_1)/2$ and \[ \zeta_{k+1}=\frac12\Bigl(\frac1{2\varrho_k}+\Bigl(1+\frac1{\varrho_k} \Bigr)\zeta_k\Bigr)\qquad(k\ge2). \] These are the closed forms, computed in the archive, of the per-scale loss of the site-additive metrics and of the gradient coefficient of the optimal value-and-gradient chain for the Gaussian chain of block averages of side two on the line; that identification is not proved here, and only the following arithmetic is. Since $\varrho_k\le\varrho_0=\sqrt2$, $\theta_k\ge1+2/(\sqrt2(2+\sqrt2))=\sqrt2$, so $\prod_{i0$ such that $D(n,\alpha)$ holds with $na\le\ell$ along the continuum tuning $\beta=\beta(a)\to\infty$, uniformly in $a$. \item \emph{The limit.} The renormalised Schwinger functions converge along $a_k=2^{-k}a_0$ to Schwinger functions that satisfy the Osterwalder--Schrader axioms and are not those of the vacuum alone. \end{enumerate} (D) and (E) together give the mass gap of the continuum theory; the remaining Clay requirements, Euclidean invariance and the local structure of the limit~\cite{JW06}, belong to (E). (D) is a statement about a single cube of fixed physical size and a single coupling at each spacing; (E) is the construction problem, on which Ba{\l}aban's renormalisation group analysis~\cite{Bal84a,Bal85a,Bal89b} gives the ultraviolet part in finite volume. Theorem~\ref{thm:avgtransfer} draws the conclusion of (D) from an averaged condition, a spectral gap of the dynamics that resamples cubes of fixed physical size, on gauge-invariant functions of the Wilson measure on tori without gauge fixing; that condition is proved at strong coupling (Corollary~\ref{cor:bpranges}) and is not proved along the tuning, nor is its local form $\mathrm{BP}_{\rm loc}$, which suffices (Sec.~\ref{ssec:bpweak}). For unsmoothed local fields at separated points, Sec.~\ref{ssec:limitE} reduces (E), given the averaged condition along the tuning, to a moment bound on separated products whose constants grow at most like a power of the inverse separation and like $K^n(n!)^\beta$, one rotation of infinite order and uniqueness of the limit (Corollary~\ref{cor:limitEOS}), from which reflection positivity in every coordinate direction, clustering in every direction, the linear growth condition of~\cite{OS75} and, for constants fixed by one reflected correlation, nontriviality follow; without the rate in the separation the same inputs give these properties at separated points only (Corollary~\ref{cor:limitEexact}), and they do not imply temperedness (Proposition~\ref{prop:limitEnotempered}). For the two-point functions, with constants fixed by the largest reflected norm, that moment bound is equivalent to a comparison of reflected norms at short distance (Corollary~\ref{cor:limitEtwo}), and its rate to a power rate in that comparison (Proposition~\ref{prop:limitEtwotempered}); the comparison bounds the physical gap from above and fails in the strong-coupling range (Proposition~\ref{prop:limitEsd}). For $n\ge3$ the moment bound does not follow from the two-point statements, reflection positivity and the gap, and its rate does not follow from bounds at each order (Propositions~\ref{prop:limitEthree} and~\ref{prop:limitEfactorial}). \begin{remark}\label{rem:cubeheuristic} The following count relates (D) to a correlation length. Suppose that the gauge-fixed field responds to a change of one link on the boundary of a cube through a Poisson kernel with decay length $\xi$ in lattice units. Then the response summed over the cube is of order $\xi$, so $K_c$ is of order $n^3\xi$ and $D(n,\alpha)$ holds once $n\gtrsim\xi/\alpha$, that is, once $\ell\gtrsim\xi a/\alpha$, a fixed multiple of the physical correlation length. The hypothesis is that the gauge-fixed links decorrelate on the scale $\xi$, and it is a statement about the gauge, not about the gauge-invariant observables: the gauge here is built on straight lines, and in gauges of axial type the gauge-fixed variables can fluctuate on all scales while gauge-invariant observables cluster. Propositions~\ref{prop:gauss} and~\ref{prop:gaussforest} carry the count out exactly for Gaussian fields, and Proposition~\ref{prop:gaussforest} shows that the quadratic approximation of the forest gauge fluctuates in this way. Where it does, (D) fails in this gauge and has to be posed for block-averaged variables, where it becomes part of the renormalisation group problem. Sec.~\ref{ssec:constrained} carries this out in the quadratic approximation. \end{remark} The count of Remark~\ref{rem:cubeheuristic} can be carried out exactly for a Gaussian field, the quadratic approximation of the gauge-fixed theory at weak coupling, and shows what (D) asks of the gauge-fixed variables. \begin{proposition}\label{prop:gauss} Let $d\ge1$, $b>0$, $t\ge0$, let $n\ge2$ be even and $V=[0,n)^d\cap\Z^d$. Given $\eta\in\R^{\Z^d\setminus V}$, let $\gamma_V(\cdot\mid\eta)$ be the Gaussian law on $\R^V$ with density proportional to \[ \exp\Bigl(-\frac b2\Bigl[\sum_{\{x,y\}\cap V\ne\emptyset}(\phi_x-\phi_y)^2 +t\sum_{x\in V}\phi_x^2\Bigr]\Bigr), \] the first sum over nearest-neighbour pairs, with $\phi=\eta$ outside $V$. Define $C_{V,j}$, the classes $c$ (residues modulo two) and $K_c$ as above, with $d_V(\phi,\phi')=\sum_{x\in V}|\phi_x-\phi'_x|$ and $\delta(\eta_j,\eta'_j) =|\eta_j-\eta'_j|$, and put $N_n=(n/2)^d$. Then $K_c=\alpha_n(t)N_n$ for every class $c$, where \[ \alpha_n(t)=1-\frac t{n^d}\,\mathbf 1^{\mathsf T}(t+L_n)^{-1}\mathbf 1 \] and $L_n$ is the Dirichlet Laplacian of $V$. In particular: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item for $t=0$, $K_c=N_n$ for every $n$ and every $b$, so $D(n,\alpha)$ fails for every $\alpha<1$; \item for $t>0$, $1-\alpha_n(t)\ge(5/6)^d\,tn^2/(tn^2+10d)$; for $d=4$, $t=m^2a^2$ and $\ell/2\le na\le\ell$ this gives $D(n,\alpha)$ with $\alpha=1-(5/6)^4(m\ell)^2/((m\ell)^2+160)$, uniformly in $a$. \end{enumerate} \end{proposition} \begin{proof} Let $A=t+L_n$ and $G=A^{-1}$, and let $B$ be the matrix with $B_{xj}=1$ when $x\in V$ and $j\notin V$ are neighbours. The law $\gamma_V(\cdot\mid\eta)$ has covariance $b^{-1}G$ and mean $GB\eta$. The matrix $A$ has nonpositive off-diagonal entries and is positive definite, so $G\ge0$ entrywise. Changing $\eta_j$ by $\Delta$ translates the law by $\Delta\,GB_j$, where $B_j$ is the column of $j$. The coupling by translation gives $W_1^{d_V}\le|\Delta| \sum_x(GB_j)_x$, and the $1$-Lipschitz function $\sum_x\phi_x$ gives the reverse inequality, so $C_{V,j}=\mathbf1^{\mathsf T}GB_j$. Since $B\mathbf1=L_n\mathbf1$ (both count the exterior neighbours of each site), \[ \sum_{j\notin V}C_{V,j}=\mathbf1^{\mathsf T}GL_n\mathbf1 =\mathbf1^{\mathsf T}\mathbf1-t\,\mathbf1^{\mathsf T}G\mathbf1 =n^d\alpha_n(t). \] The reflection $x_i\mapsto n-1-x_i$ preserves $V$ and the coefficients and, as $n$ is even, exchanges the two residues of the coordinate $x_i$; so the $2^d$ class sums are equal, and each is $n^d\alpha_n(t)/2^d=\alpha_n(t)N_n$. This proves the formula and (i). For (ii) let $v(i)=(i+1)(n-i)$ for $0\le i0$, let $n$ be even and $V$ the cube of Definition~\ref{def:cube}. For real functions $\varphi$ on the links of $\Z^4$ that vanish on $F$, let $(d\varphi)_p$ be the sum of $\varphi$ around the plaquette $p$ with the signs of its orientation. Given $\eta\in\R^{A\setminus V}$, let $\gamma_V(\cdot\mid\eta)$ be the law on $\R^V$ with density proportional to \[ \exp\Bigl(-\frac b2\sum_{p\cap V\ne\emptyset}(d\varphi)_p^2\Bigr), \] with $\varphi=\eta$ on $A\setminus V$ and $\varphi=0$ on $F$, and define $C_{V,j}$ and $K_c$ as above with $d_V(\varphi,\varphi')=\sum_{i\in V} |\varphi_i-\varphi'_i|$ and $\delta(\eta_j,\eta'_j)=|\eta_j-\eta'_j|$. Then \[ \sum_cK_c\ge|V|=48N_n , \] so $K_c\ge N_n$ for some class $c$, and $D(n,\alpha)$ fails for every $\alpha<1$, at every $n$ and every $b$. \end{proposition} \begin{proof} \emph{The laws.} The quadratic form $\sum_{p\cap V\ne\emptyset}(d\varphi)_p^2$ with $\varphi=0$ outside $V$ vanishes only at $\varphi=0$: the argument of Corollary~\ref{cor:cubefails}, written additively, gives a function $h$ on the sites with $\varphi_{(x,\mu)}=h(x)-h(x+e_\mu)$, constant outside the box and along the lines of $F$, hence constant. So $\gamma_V(\cdot\mid\eta)$ is a Gaussian law whose covariance does not depend on $\eta$ and whose mean is $\sum_j\eta_ju_j$ for some vectors $u_j\in\R^V$. Two exteriors that differ in finitely many links give translates of one law by $t=\sum_j(\eta'_j-\eta_j)u_j$; the coupling by translation and the $1$-Lipschitz function $\sum_i\operatorname{sign}(t_i)\varphi_i$ show that their distance is $\|t\|_1$. In particular $C_{V,j}=\|u_j\|_1$, and \[ W_1^{d_V}\bigl(\gamma_V(\cdot\mid\eta),\gamma_V(\cdot\mid\eta')\bigr)\le \sum_jC_{V,j}\,|\eta'_j-\eta_j| . \] \emph{Lines.} By Lemma~\ref{lem:structure}(i) the lines of $F$ are straight, disjoint and cover the sites, and the two ends of an active link $(x,\mu)$ lie on different lines, since the line through $x$ has direction $\sigma(x)\ne\mu$. For a line $\ell$ let $h_\ell$ be the indicator of its sites and $(dh_\ell)_{(x,\mu)}=h_\ell(x)-h_\ell(x+e_\mu)$. Then $dh_\ell$ vanishes on $F$ and around every plaquette, so the density is invariant under $\varphi\mapsto\varphi+dh_\ell$, and $\gamma_V(\cdot\mid dh_\ell)$ is $\gamma_V(\cdot\mid0)$ translated by the restriction of $dh_\ell$ to $V$. Hence \[ \sum_{i\in V}|(dh_\ell)_i|\le\sum_{j\in A\setminus V}C_{V,j}\,|(dh_\ell)_j| . \] Here $|(dh_\ell)_e|=1$ when exactly one end of $e$ lies on $\ell$ and $0$ otherwise, so $\sum_\ell|(dh_\ell)_e|=2$ for every active link $e$, the sum running over the finitely many lines that meet a link entering the inequality. Summing over these lines gives $2|V|\le2\sum_jC_{V,j}= 2\sum_cK_c$. \end{proof} The proof uses only that gauge transformations constant on the lines of $F$ move the whole cube while changing few boundary links, which is the behaviour of gauges of axial type anticipated in Remark~\ref{rem:cubeheuristic}. The class sums themselves, computed exactly for $n\le8$, lie between $1.57$ and $11.24$ times $N_n$ and increase with $n$. Below the correlation length the theory is weakly coupled, and the quadratic approximation is its leading order there. A proof of (D) along the tuning therefore has to show that the nonperturbative regime reduces the cube coefficients of the forest gauge below $N_n$, or (D) has to be posed for other variables. The lines of $F$ are not what makes the argument work. A constant field, corrected by a periodic gauge transformation so that it vanishes on the fixed links, costs no action and changes every exterior link by a bounded amount, and this happens in every gauge fixed on an invariant forest. \begin{proposition}\label{prop:gaussevery} Let $b>0$ and let $F$ be a set of links of $\Z^4$, invariant under the translations by $2\Z^4$, that contains no closed path and whose trees are infinite and contain every site. Let $A$ be the set of the other links, $V$ the set of links $(x,\mu)\in A$ with $x\in[0,n)^4$, $n\ge2$ even, and define the classes, $\gamma_V(\cdot\mid\eta)$, $C_{V,j}$ and $K_c$ as in Proposition~\ref{prop:gaussforest} with this $F$. Then there are numbers $a_c\ge0$, not all zero, with \[ \sum_ca_cK_c\ge N_n\sum_ca_c , \] so $K_c\ge N_n$ for some class $c$, and $D(n,\alpha)$ fails for every $\alpha<1$, at every $n$ and every $b$. \end{proposition} \begin{proof} \emph{The laws.} Let $\varphi$ vanish outside $V$ and on $F$, with $(d\varphi)_p=0$ for every plaquette $p$ that meets $V$. The other plaquettes contain no link of $V$, so $d\varphi=0$ on every plaquette, and $\varphi=dh$ for a function $h$ on the sites. Since $\varphi$ vanishes outside a box and the complement of a box is connected, $h$ is constant there, and we take that constant to be $0$. As $\varphi$ vanishes on $F$, $h$ is constant on every tree of $F$; each tree is infinite and meets the complement of the box, so $h=0$ and $\varphi=0$. Hence, as in the proof of Proposition~\ref{prop:gaussforest}, $\gamma_V(\cdot\mid\eta)$ is a nondegenerate Gaussian law with mean $\sum_j\eta_ju_j$, where only the links $j$ that share a plaquette with $V$ contribute, and $C_{V,j}=\|u_j\|_1$. \emph{A flat field.} For $c\in\R^4$ and a function $h$ on $\Z^4$ of period two in each coordinate put $\psi_{(x,\mu)}=c_\mu+h(x)-h(x+e_\mu)$. Then $d\psi=0$ on every plaquette and $\psi$ is invariant under $2\Z^4$; if $\psi=0$, adding $\psi$ at $(x,\mu)$ and $(x+e_\mu,\mu)$ gives $2c_\mu=0$, and then $h$ is constant. These fields form a space of dimension $16-1+4=19$. Let $f$ be the number of orbits of $2\Z^4$ in $F$. The box $[0,2m)^4$ has $16m^4$ sites, and the links of $F$ with both ends in it form a forest, so there are at most $16m^4-1$ of them, while each orbit contributes at least $(m-1)^4$; letting $m\to\infty$ gives $f\le16$. Vanishing on $F$ is therefore at most $16$ linear conditions, and some $\psi\ne0$ of this form vanishes on $F$. It is nonzero on some link of $A$. Put $a_c=|\psi_j|$ for $j\in c$. \emph{The inequality.} Let $\eta$ be the restriction of $\psi$ to $A\setminus V$. The field equal to $\psi$ on $V$ and to $\eta$ outside gives the quadratic form its least value $0$, so by nondegeneracy the mean of $\gamma_V(\cdot\mid\eta)$ is the restriction of $\psi$ to $V$, and $\psi_i=\sum_j\psi_j(u_j)_i$ for $i\in V$. Hence \[ \sum_{i\in V}|\psi_i|\le\sum_{j\in A\setminus V}C_{V,j}|\psi_j|=\sum_ca_cK_c . \] A class consists of the active links with a given direction and a given residue of the base point modulo two, so it meets $V$ in $N_n$ links, and the left side is $N_n\sum_ca_c>0$. So $K_c\ge N_n$ for some class with $a_c>0$. \end{proof} For the direction field of Table~\ref{tab:sigma} the flat fields vanishing on $F$ form a space of dimension $7$, and for the axial field $\sigma=0$ one of dimension $10$. The proposition is weaker than Proposition~\ref{prop:gaussforest} for the forest gauge, which bounds the sum over all classes, and it says nothing about two kinds of condition. A mass gives the constant fields an action, which is why Proposition~\ref{prop:gauss} distinguishes $t>0$ from $t=0$. The constrained condition $D_{\rm c}$ of Theorem~\ref{thm:constrainedcube} holds the block averages fixed, and a constant field changes them. Within the quadratic approximation no gauge fixed on an invariant forest gives (D); the loss has to come from the Wilson action beyond its quadratic part, or (D) has to be posed for variables, such as the block averages, that a constant field moves. \begin{remark}\label{rem:routes} Five other routes stop at (D) or (E), or earlier. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item \emph{Disjoint blocks.} The block criterion of Lemma~\ref{lem:blockcomparison} with a partition into cubes of side $n$ requires, for a link on a face, that its influence on the neighbouring cube be less than one. In the count of Remark~\ref{rem:cubeheuristic} that influence is of order $\xi$, which grows like $1/a$. Averaging over the $N_n$ translates divides it by the number of translates on whose boundary the link lies, a factor of order $1/n$, and this is why (D) is stated for averaged cubes. \item \emph{Rescaling a fixed lattice gap.} Any bound uniform in lattice units, such as those of Theorems~\ref{thm:su2}--\ref{thm:su3blocks}, gives trivial continuum two-point functions~\cite{BB26b}; rescaling it produces no gap. \item \emph{Assuming the gap.} Hypothesis~R of~\cite{BB26b} assumes the gap in physical units together with the limit. Theorem~\ref{thm:transfer} replaces the gap part by the cube estimate (D), which concerns one box and one coupling at each spacing. \item \emph{Computation.} At a fixed coupling and a fixed small $n$, $D(n,\alpha)$ is a finite set of Kantorovich bounds and could in principle be evaluated as in Remark~\ref{rem:su3cover}; for $n=2$ and $\SU(2)$ the block already has $48$ links and $144$ real variables. (D) asks for it at infinitely many couplings with $n\to\infty$, which no finite computation covers. \item \emph{Axioms alone.} Proposition~\ref{prop:nogapgff} shows that positivity, invariance and regularity do not force a gap, and Proposition~\ref{prop:growth} shows that the growth route to rotation invariance is closed for the action density. The infrared input therefore has to be an estimate such as (D). \end{enumerate} \end{remark} \subsection{The ratio, the strong-coupling range and the cube estimate}\label{ssec:ratiocube} This subsection states the open input of Theorem~\ref{thm:flowlimit} exactly, proves that it fails along every sequence of couplings in the range where the paper has a gap, shows that the hypothesis of Theorem~\ref{thm:transfer} holds along some of these sequences and therefore cannot imply the input, and replaces the input by a condition that is necessary and sufficient and asks only for an upper bound on an energy and a lower bound on one overlap. The notation is that of Theorem~\ref{thm:flowlimit}: translation invariant states $\omega_k$ at couplings $\beta_k$, with transfer matrices $\mathbb T_k$ and vacua $\Omega_k$ (Appendix~\ref{app:transfer}), the centred flowed packet $A_k$ of a nonzero $f_0\in C_c(\R^4)$ supported in $\{t>0\}$, and $C_k(t)=\omega_k(\vartheta(\bar A_k)\tau_{t/a_k}A_k)$ for $t\in a_k\Z_{\ge0}$, with $C_k(0)>0$. For $u\ge0$ and $\sigma>0$ in $a_k\Z$ put \[ \rho_k(u,\sigma)=\frac{C_k(2u+\sigma)}{C_k(2u)} , \] so that $\rho(\sigma)$ of Theorem~\ref{thm:flowlimit} is a limit of $\rho_k(0,\sigma)$. Let $f_{0,u}=f_0(\cdot-(u,0))$; for $u/a_k\in\Z$ its packet is $\tau_{u/a_k}A_k$, and $\Psi_{k,u}=C_k(2u)^{-1/2}\Psi_{\tau_{u/a_k}A_k}$ is the corresponding unit vector. For dyadic $u\ge0$ consider \begin{enumerate}[label=\textup{(N$_u$)},leftmargin=*] \item $\liminf_{k\to\infty}\rho_k(u,\sigma)>0$ for some dyadic $\sigma>0$. \end{enumerate} \begin{proposition}\label{prop:ratioexact} \textup{(i)} $C_k(t)>0$ for every $t\in a_k\Z_{\ge0}$, and $\log C_k$ is convex and nonincreasing on $a_k\Z_{\ge0}$. \textup{(ii)} $0<\rho_k(u,\sigma)\le1$; $\rho_k(u,\sigma)$ is nondecreasing in $u$ and nonincreasing in $\sigma$, and $\rho_k(u,\sigma')\ge\rho_k(u,\sigma)^{\sigma'/\sigma}$ for $\sigma'\ge\sigma$. \textup{(iii)} For $u\ge0$ dyadic, Theorem~\ref{thm:flowlimit} applies to $f_{0,u}$, with reflected norm $C_k(2u)$ and ratio $\rho_k(u,\sigma)$. Every limit it produces along subsequences is nontrivial if and only if $(\mathrm N_u)$ holds, and some limit is nontrivial if and only if $\limsup_k\rho_k(u,\sigma)>0$. If $(\mathrm N_u)$ holds for one $\sigma$, it holds for every dyadic $\sigma>0$, and $(\mathrm N_u)$ implies $(\mathrm N_{u'})$ for $u'\ge u$. \textup{(iv)} Let $\mu_{k,u}$ be the spectral measure, in energies $E=-a_k^{-1}\log\lambda$, of $\mathbb T_k$ at $\Psi_{k,u}$. If $\rho_k(u,\sigma)\ge c>0$, then $\mu_{k,u}([0,\Lambda])\ge c/2$ with $\Lambda=\sigma^{-1}\log(2/c)$; if $\mu_{k,u}([0,\Lambda])\ge v$, then $\rho_k(u,\sigma)\ge ve^{-\sigma\Lambda}$. So $(\mathrm N_u)$ holds if and only if a fixed fraction of the spectral mass of $\Psi_{k,u}$ stays at bounded physical energy, uniformly in $k$. \end{proposition} \begin{proof} (i) The function $f_0$ is supported in $\{t>0\}$, so the lattice points with $f_0(a_kx)\ne0$ have $x_0\ge1$, and $A_k=\sum_{n\ge1}\tau_nB_n$ is a finite sum in which $B_n$ is a function of the spatial links of the slice at time $0$, by Proposition~\ref{prop:spatialflow}(i). In the reconstruction~\cite{OS73,OS75,OSei78}, as in the proof of Theorem~\ref{thm:flowlimit}, $\Psi_{\tau_nB}=\mathbb T_k^n\Psi_B$, hence $\Psi_{A_k}=\mathbb T_k\xi$ with $\xi=\sum_n\mathbb T_k^{n-1}\Psi_{B_n}$. The transfer matrix is self-adjoint and positive~\cite{OSei78,Lus77,Sei82}, and its range is orthogonal to its kernel, so the spectral measure $\nu_k$ of $\mathbb T_k$ at $\Psi_k=C_k(0)^{-1/2}\Psi_{A_k}$ (Lemma~\ref{lem:spectralmeasure}) has no atom at $0$. Therefore $C_k(t)/C_k(0)=\int\lambda^{t/a_k}d\nu_k>0$. Convexity and monotonicity are Proposition~\ref{prop:sliceratio}(ii). (ii) The first claims follow from (i). On a lattice, a convex function has chord slopes that increase in both end points; this gives monotonicity in $u$, and, comparing the chords $[2u,2u+\sigma]$ and $[2u,2u+\sigma']$, the power bound. Monotonicity in $\sigma$ is that of $C_k$. (iii) Translation invariance and $\vartheta\tau_u=\tau_{-u}\vartheta$ give, with times written in lattice steps, \[ \omega_k\bigl(\vartheta(\tau_u\bar A_k)\,\tau_{t+u}A_k\bigr) =\omega_k\bigl(\vartheta(\bar A_k)\,\tau_{t+2u}A_k\bigr), \] which gives the reflected norm and the ratio. The function $f_{0,u}$ is nonzero and supported in $\{t>u\}$. If $\liminf_k\rho_k(u,\sigma)=0$, a subsequence has $\rho_k(u,\sigma)\to0$, and a refinement given by Theorem~\ref{thm:flowlimit}(ii) has a trivial limit by part (iv) of that theorem; if the $\liminf$ is at least $c>0$, every limit has $\rho(\sigma)\ge c$ and is nontrivial. The statement with $\limsup$ is the last part of Theorem~\ref{thm:flowlimit}(iv). Independence of $\sigma$ follows from (ii), by the power bound for $\sigma'\ge\sigma$ and by monotonicity for $\sigma'\le\sigma$, and the last claim from monotonicity in $u$. (iv) The vector $\Psi_{k,u}$ has no spectral weight at $\lambda=0$, by the proof of (i), and $\rho_k(u,\sigma)=\int e^{-\sigma E}d\mu_{k,u}$. This integral is at most $\mu_{k,u}([0,\Lambda])+e^{-\sigma\Lambda}$ and at least $e^{-\sigma\Lambda}\mu_{k,u}([0,\Lambda])$. \end{proof} The open input of Theorem~\ref{thm:flowlimit} is therefore exactly $(\mathrm N_u)$ for one packet $f_0$ and one dyadic $u\ge0$. It is a uniformity statement and nothing else: at each fixed $k$ the ratio is positive by (i). The required uniformity is in $k$ along the tuning, at fixed physical separations $u$ and $\sigma$, so that $\sigma/a_k\to\infty$ lattice steps; the constant may depend on $G$, $\Phi$, $\tau$, $f_0$, $u$, $\sigma$ and on the tuning, and nothing is required uniformly in $f_0$ or in $\tau$. The ratio does not change when $A_k$ is multiplied by a constant, so the renormalisation $Z_k$ plays no role in it. The case $u=0$ is the strongest and implies all the others. Proposition~\ref{prop:flowshift}(i) assumes $(\mathrm N_0)$ and renormalises the packet shifted by $\sigma/2$. The hypothesis of Lemma~\ref{lem:positivetime} is $(\mathrm N_{s/4})$ at $\sigma=s/2$; it is weaker than $(\mathrm N_0)$, as the example after Proposition~\ref{prop:sliceratio} shows, and it suffices for a nontrivial limit of the packet shifted by $s/4$. Granted the convergence asked for in (E), the clause of (E) that the limit is not that of the vacuum alone is, for the flowed packets, $(\mathrm N_u)$ for one $f_0$ and one $u$; the convergence of the whole sequence, Euclidean invariance and the higher correlations remain~\cite{JW06}. No bound is required at one lattice step: the first-step energy $-a_k^{-1}\log\rho_k(u,a_k)$ may diverge while $(\mathrm N_u)$ holds, as Remark~\ref{rem:compositeslice} shows for the slice field of the unflowed magnetic energy. \emph{The strong-coupling range.} In the range where the paper proves a gap, the ratio has two-sided bounds in lattice units, and the upper one rules out every continuum tuning there. Put \[ m_0=-\tfrac12\log0.9936 , \] so that $m_0\ge0.0032$ by $-\log(1-x)\ge x$. \begin{theorem}\label{thm:strongratio} Let $G=\SU(2)$, let $\beta_{W,k}\in[0,0.348]$ be any couplings, and let $\omega_k$ be the states of Theorems~\ref{thm:su2} and~\ref{thm:blocks}. Then for every $\Phi$, $\tau$ and $f_0$ with $C_k(0)>0$, every dyadic $u\ge0$ and $\sigma>0$, and every $k$ with $u/a_k,\sigma/a_k\in\Z$, \[ \rho_k(0,a_k)^{\sigma/a_k}\le\rho_k(u,\sigma)\le e^{-m_0\sigma/a_k} . \] Consequently every limit of Theorem~\textup{\ref{thm:flowlimit}} along these couplings, for every packet $f_{0,u}$, is trivial, and if $(\mathrm N_u)$ holds for some packet along a tuning $(\beta_{W,k})$ of $\SU(2)$, then $\beta_{W,k}>0.348$ for all large $k$. \end{theorem} \begin{proof} By Theorem~\ref{thm:blocks} the transfer matrix of $\omega_k$ has gap $m_k\ge-\frac12\log\alpha(\beta_{W,k})$ on $\Omega_k^\perp$, and $\alpha$ is increasing with $\alpha(0.348)\le0.9936$, so $m_k\ge m_0$. The vector $\Psi_{k,u}$ is a unit vector orthogonal to $\Omega_k$, because $A_k$ is centred, and $\mathbb T_k$ preserves $\Omega_k^\perp$, so $\rho_k(u,\sigma)=\ip{\Psi_{k,u}}{\mathbb T_k^{\sigma/a_k}\Psi_{k,u}}\le e^{-m_0\sigma/a_k}$. For the lower bound, $\rho_k(u,\sigma)\ge\rho_k(0,\sigma)$ by Proposition~\ref{prop:ratioexact}(ii), and Jensen's inequality for the convex function $\lambda^{\sigma/a_k}$ and the probability measure $\nu_k$ of the proof of Proposition~\ref{prop:ratioexact}(i) gives $\rho_k(0,\sigma)=\int\lambda^{\sigma/a_k}d\nu_k\ge(\int\lambda\,d\nu_k)^{ \sigma/a_k}=\rho_k(0,a_k)^{\sigma/a_k}$, with $\rho_k(0,a_k)>0$ by Proposition~\ref{prop:ratioexact}(i). The upper bound tends to zero as $a_k\to0$, which gives triviality by Proposition~\ref{prop:ratioexact}(iii). If $(\mathrm N_u)$ held along a tuning with $\beta_{W,k}\le0.348$ for infinitely many $k$, the upper bound along those $k$ would contradict it. \end{proof} Writing $\rho_k(u,\sigma)=e^{-\varepsilon_k\sigma/a_k}$, the theorem says $m_0\le\varepsilon_k\le-\log\rho_k(0,a_k)$: both ends are rates in lattice units, and $(\mathrm N_u)$ asks for $\varepsilon_k=O(a_k)$. Reflection positivity gives the positivity of the lower end, through the range of $\mathbb T_k$, and the transfer-matrix gap gives the upper bound; neither bears on the size of $\varepsilon_k/a_k$ except to exclude the range. The size of $\rho_k(0,a_k)$ is not estimated here; any estimate of it at strong coupling is in lattice units, and it is at most $e^{-m_0}$. The same holds for every compact simple $G$ and couplings in $[0,\beta_1]$ with $N\beta_1<\frac2{29}\kappa_G$, with $m_0=m_G(\beta_1)$ of Theorem~\ref{thm:groups}; for fixed $\beta$ this is Proposition~\ref{prop:treecritical}(i). So the strong-coupling range bears on the continuum tuning only negatively: a tuning with a nontrivial limit leaves it for good. That the tuning has to go to $\beta_W\to\infty$ is the expectation from asymptotic freedom and is not proved here. \emph{The cube estimate in the same range.} The hypothesis of Theorem~\ref{thm:transfer} holds at fixed strong coupling with cubes of fixed physical size, and with every $\alpha$. \begin{proposition}\label{prop:cubecompatible} \textup{(i)} Suppose $D(n_k,\alpha_k)$ holds at $\beta_k$ with $n_ka_k\le\ell$. Then $\rho_k(u,\sigma)\le\alpha_k^{\sigma/(2\ell)}$ for every $u$ and $\sigma$. Hence $(\mathrm N_u)$ with constant $c$ at $\sigma$ forces $\alpha_k\ge c^{2\ell/\sigma}$ for large $k$, and the physical mass of Theorem~\textup{\ref{thm:transfer}} obeys $m_*\le\sigma^{-1}\log(1/c)$. \textup{(ii)} Let $G=\SU(2)$, $0\le\beta_W<\frac13$ fixed, $\ell>0$, and $n_k=2\lfloor\ell/(2a_k)\rfloor$. For every $\alpha\in(0,1)$, $D(n_k,\alpha)$ holds at $\beta_W$ for all $k$ with $n_k\ge2$ and $5\beta_W\le\alpha(1-3\beta_W)n_k$, and $n_ka_k\le\ell$. So the hypothesis of Theorem~\textup{\ref{thm:transfer}} holds along the constant tuning, from some $k$ on, with every $\alpha\in(0,1)$, while every limit of Theorem~\textup{\ref{thm:flowlimit}} is trivial. \end{proposition} \begin{proof} (i) By Theorem~\ref{thm:onescale} the gap of $\mathbb T_k$ on $\Omega_k^\perp$ is at least $\log(1/\alpha_k)/(2n_k)\ge a_k\log(1/\alpha_k) /(2\ell)$, and $\rho_k(u,\sigma)\le e^{-m_k\sigma/a_k}$ as in the proof of Theorem~\ref{thm:strongratio}. The bound on $m_*$ is the case $\alpha_k= \alpha$, since then $\alpha^{\sigma/(2\ell)}=e^{-m_*\sigma}$. (ii) Proposition~\ref{prop:cubesmall} gives $D(n,\alpha)$ for every $\alpha\ge\hat\alpha_n$, and $\hat\alpha_n\le5\beta_W/((1-3\beta_W)n)$. Triviality is Theorem~\ref{thm:strongratio}. \end{proof} Part (i) shows that (D) is itself a triviality statement when its constant improves along the tuning: if $\alpha_k\to0$ with $n_ka_k\le\ell$, every limit is trivial. Along the constant tuning of part (ii) with $\beta_W>0$ this gives $\rho_k(u,\sigma)\le(5\beta_W/((1-3\beta_W)n_k))^{\sigma/(2\ell)}$, a power of $a_k$, from the cube estimate alone. Part (ii) shows that no argument that uses (D) only through the hypothesis of Theorem~\ref{thm:transfer}, or through Theorem~\ref{thm:onescale}, can give $(\mathrm N_u)$: that hypothesis is satisfied by Wilson measures whose continuum limits are trivial. Statement (D) of Sec.~\ref{sec:inputs} adds that $\beta\to\infty$ along the tuning, and an implication that uses this and the weak-coupling structure besides (D) is not excluded; it would use information that (D) does not contain. \emph{Trial states.} The input is equivalent to the existence of states of bounded physical energy with a fixed overlap with the packet. \begin{lemma}\label{lem:trial} Let $\mathbb T$ be self-adjoint with $0\le\mathbb T\le1$, let $a>0$ and let $\Psi$ be a unit vector. \textup{(a)} Let $\chi$ be a unit vector with $\ip\chi{(1-\mathbb T)\chi}\le \kappa a$ and $|\ip\chi\Psi|\ge v>0$. Then for $\sigma\ge2a$, \[ \ip\Psi{\mathbb T^{\sigma/a}\Psi}\ge\Bigl(v-\sqrt{\kappa\sigma/2}\Bigr)_+^2 , \] and if $a\Lambda\le1$ with $\Lambda=8\kappa/v^2$, the spectral projection $P=\mathbf1_{[e^{-a\Lambda},1]}(\mathbb T)$ has $\|P\Psi\|\ge v/2$. \textup{(b)} Conversely, if $\ip\Psi{\mathbb T^{\sigma/a}\Psi}\ge c>0$ with $\sigma/a\in\Z_{>0}$, put $\Lambda=\sigma^{-1}\log(2/c)$ and $P=\mathbf1_{[e^{-a\Lambda},1]}(\mathbb T)$. Then $\chi=P\Psi/\|P\Psi\|$ is a unit vector with $\ip\chi{(1-\mathbb T)\chi}\le a\Lambda$ and $\ip\chi\Psi=\|P\Psi\|\ge(c/2)^{1/2}$. If $\mathbb T\Omega=\Omega$ and $\Psi\perp\Omega$, then $\chi\perp\Omega$. \end{lemma} \begin{proof} (a) Let $m=\sigma/(2a)\ge1$ and let $\nu_\chi$ be the spectral measure at $\chi$. For $\lambda\in[0,1]$, $0\le1-\lambda^m\le m(1-\lambda)$ by Bernoulli's inequality, so \[ \|(1-\mathbb T^m)\chi\|^2=\int(1-\lambda^m)^2d\nu_\chi\le m\int(1-\lambda)\,d\nu_\chi\le\frac{\kappa\sigma}2 . \] Hence $\|\mathbb T^m\Psi\|\ge|\ip{\mathbb T^m\chi}\Psi|\ge|\ip\chi\Psi|- \|(1-\mathbb T^m)\chi\|\ge v-\sqrt{\kappa\sigma/2}$, and $\ip\Psi{\mathbb T^{\sigma/a}\Psi}=\|\mathbb T^m\Psi\|^2$. For the projection, $1-\lambda>1-e^{-a\Lambda}\ge a\Lambda/(1+a\Lambda)\ge a\Lambda/2$ off $[e^{-a\Lambda},1]$, so $\|(1-P)\chi\|^2\le2\kappa/\Lambda=v^2/4$, and $\|P\Psi\|\ge|\ip\chi{P\Psi}|\ge|\ip\chi\Psi|-\|(1-P)\chi\|\ge v/2$. (b) With $\nu$ the spectral measure at $\Psi$, $c\le\int\lambda^{\sigma/a}d\nu\le\|P\Psi\|^2+e^{-\sigma\Lambda}=\|P\Psi\|^2+ c/2$. On the range of $P$, $1-\mathbb T\le1-e^{-a\Lambda}\le a\Lambda$. Since $P\Omega=\Omega$, $\ip\Omega{P\Psi}=\ip\Omega\Psi=0$. \end{proof} \begin{corollary}\label{cor:trial} $(\mathrm N_u)$ holds if and only if there are $\kappa,v>0$ and unit vectors $\chi_k$ in the reconstructed spaces of the $\omega_k$ such that, for all large $k$, \[ \ip{\chi_k}{(1-\mathbb T_k)\chi_k}\le\kappa a_k,\qquad |\ip{\chi_k}{\Psi_{k,u}}|\ge v . \] Then $\rho_k(u,\sigma)\ge(v-\sqrt{\kappa\sigma/2})_+^2$ for $\sigma\ge2a_k$, and the $\chi_k$ may be taken of the form $\Psi_{X_k}/\|\Psi_{X_k}\|$ for gauge-invariant local functions $X_k$ at positive times, at the cost of replacing $(\kappa,v)$ by $(2\kappa,v/2)$. If moreover $D(n_k,\alpha)$ holds with $n_ka_k\le\ell$, then $m_*=\log(1/\alpha)/(2\ell)\le8\kappa/v^2$. \end{corollary} \begin{proof} Sufficiency is Lemma~\ref{lem:trial}(a) with $\Psi=\Psi_{k,u}$, $a=a_k$, and Proposition~\ref{prop:ratioexact}(ii) extends the bound from small $\sigma$ to every $\sigma$. Necessity is Lemma~\ref{lem:trial}(b) with $\kappa=\sigma^{-1}\log(2/c)$ and $v=(c/2)^{1/2}$. The vectors $\Psi_X$ with $X$ local are dense (Appendix~\ref{app:transfer}), and both conditions are continuous in $\chi_k$ for fixed $k$, $1-\mathbb T_k$ being bounded. Under $D(n_k,\alpha)$, Theorem~\ref{thm:onescale} puts the spectral measure of $\Psi_{k,u}\perp\Omega_k$ in $[0,e^{-a_km_*}]$; if $m_*>8\kappa/v^2$, the projection of Lemma~\ref{lem:trial}(a) annihilates $\Psi_{k,u}$ for large $k$, which is impossible. \end{proof} For $\chi_k=\Psi_{X_k}/\|\Psi_{X_k}\|$ the two conditions are statements about lattice correlations: \[ \begin{split} &\omega_k\bigl(\vartheta(\bar X_k)X_k\bigr)-\omega_k\bigl(\vartheta(\bar X_k)\tau_1X_k\bigr)\le\kappa a_k\,\omega_k\bigl(\vartheta(\bar X_k)X_k\bigr),\\ &\bigl|\omega_k\bigl(\vartheta(\bar X_k)\tau_{u/a_k}A_k\bigr)\bigr|^2\ge v^2\, \omega_k\bigl(\vartheta(\bar X_k)X_k\bigr)\,C_k(2u). \end{split} \] The first is an upper bound on a one-slab Dirichlet form, which is the direction of variational and perturbative estimates and the opposite of a mass gap; the second is a lower bound on one correlation. The choice $\chi_k=\Psi_{k,u}$ is Proposition~\ref{prop:flowshift}(ii), and the order-two tree and saturation bounds give it by Proposition~\ref{prop:treeenergy}(iii); the corollary shows that the trial vector need not be the packet itself. \emph{A soft component of the slice field.} For product packets the trial vector can be built from a part of the slice field of Proposition~\ref{prop:sliceratio}, and the conditions become one first-step bound for that part and one bound for the rest at the earliest time of the packet. Let $f_0=g\otimes h$ with $g\ge0$ supported in $[t_0,t_1]$, and let $u_k$, $n_0$, $n_1$ be as in Proposition~\ref{prop:sliceratio}. \begin{theorem}\label{thm:softhard} Suppose that for all large $k$ there is a decomposition $u_k=u^{\mathrm s}_k+u^{\mathrm h}_k$ with $\mathbb T_k^{n_0}u^{\mathrm s}_k\ne0$ and, for constants $\kappa>0$ and $\theta\in[0,1)$ with $\kappa a_k\le\frac12$, \begin{enumerate}[label=\textup{(S\arabic*)},leftmargin=*] \item $\ip{u^{\mathrm s}_k}{(1-\mathbb T_k)u^{\mathrm s}_k}\le\kappa a_k \|u^{\mathrm s}_k\|^2$; \item $\|\mathbb T_k^{n_0}u^{\mathrm h}_k\|\le\theta e^{-2\kappa(t_1-t_0)} \|\mathbb T_k^{n_0}u^{\mathrm s}_k\|$. \end{enumerate} Then the hypothesis of Corollary~\textup{\ref{cor:trial}} holds for $u=0$ with $\kappa e^{4\kappa(t_1-t_0)}$ in place of $\kappa$ and $v=(1-\theta)/(1+\theta)$; in particular $(\mathrm N_0)$ holds and \[ \frac{C_k(\sigma)}{C_k(0)}\ge\Bigl(\frac{1-\theta}{1+\theta}- \sqrt{\tfrac12\kappa\sigma e^{4\kappa(t_1-t_0)}}\Bigr)_+^2 \] for $\sigma\ge2a_k$. \end{theorem} \begin{proof} By the proof of Proposition~\ref{prop:sliceratio}, $\Psi_{A_k}=W(\mathbb T_k)u_k$ with $W(\lambda)=\sum_nw_n\lambda^n$, $w_n\ge0$, $w_n=0$ unless $n_0\le n\le n_1$, and $\|w\|_1\lambda^{n_1}\le W(\lambda)\le \|w\|_1\lambda^{n_0}$ on $[0,1]$, with $\|w\|_1>0$ because $C_k(0)>0$. Write $\nu$ for the spectral measure at $u^{\mathrm s}_k$ and $y=\mathbb T_k^{n_0}u^{\mathrm s}_k$. For a probability measure, a nondecreasing and a nonincreasing function have $\int fg\le\int f\int g$, because $\iint(f(x)-f(x'))(g(x)-g(x'))\le0$; with $f=\lambda^{2n_0}$, $g=1-\lambda$ and (S1), \[ \int\lambda^{2n_0}(1-\lambda)\,d\nu\le\kappa a_k\|y\|^2 . \] So $y$ satisfies (S1) as well, and Jensen's inequality for $\lambda^{2j}$, $j=n_1-n_0$, gives $\|\mathbb T_k^jy\|^2\ge\|y\|^2(1-\kappa a_k)^{2j}\ge \|y\|^2e^{-4\kappa(t_1-t_0)}$, since $\log(1-x)\ge-2x$ on $[0,\frac12]$ and $a_kj\le t_1-t_0$. Hence \[ \|W(\mathbb T_k)u^{\mathrm s}_k\|\ge\|w\|_1\|\mathbb T_k^{n_1}u^{\mathrm s}_k\| \ge\|w\|_1e^{-2\kappa(t_1-t_0)}\|y\| , \] while $\|W(\mathbb T_k)u^{\mathrm h}_k\|\le\|w\|_1\|\mathbb T_k^{n_0} u^{\mathrm h}_k\|\le\theta\|W(\mathbb T_k)u^{\mathrm s}_k\|$ by (S2). Let $\chi_k=W(\mathbb T_k)u^{\mathrm s}_k/\|W(\mathbb T_k)u^{\mathrm s}_k\|$. Its energy is \[ \frac{\int W^2(1-\lambda)\,d\nu}{\int W^2d\nu}\le \frac{\|w\|_1^2\kappa a_k\|y\|^2}{\|w\|_1^2e^{-4\kappa(t_1-t_0)}\|y\|^2} =\kappa a_ke^{4\kappa(t_1-t_0)} , \] and with $r=\|W(\mathbb T_k)u^{\mathrm h}_k\|/\|W(\mathbb T_k)u^{\mathrm s}_k\| \le\theta$ its overlap with $\Psi_k=\Psi_{A_k}/\|\Psi_{A_k}\|$ is at least $(1-r)/(1+r)\ge(1-\theta)/(1+\theta)$. Corollary~\ref{cor:trial} gives the rest. \end{proof} Condition (S1) asks for a soft component of the slice field whose first-step energy is bounded in physical units. Remark~\ref{rem:compositeslice} shows why the whole slice field is not a candidate: at next order its first-step energy grows like $\log(\sqrt\tau/a_k)$, through a hard component of norm of order $g^2$. Condition (S2) asks that this hard component, propagated to the earliest time $t_0$ of the packet, be a fixed fraction of the soft part. For the free field of Remark~\ref{rem:freeslice}, (S1) holds with $u^{\mathrm h}_k=0$: since $(1-e^{-a_k\omega})/a_k\le\omega\le a_k^{-1}\sinh(a_k\omega)$, the first-step energy $\ip{u_k}{(1-\mathbb T_k)u_k}/(a_k\|u_k\|^2)$ has a numerator whose integrand is at most a multiple of $e^{-8t_f|p|^2/\pi^2}$, and a denominator $D_k(0)$ with a positive limit, as in that remark. For the unflowed magnetic energy the whole slice field violates (S1) by a factor of order $1/a_k$, and the decomposition has to separate its hard modes. Both conditions concern energies of order $1/a_k$ at physical times below $t_0$; they are ultraviolet statements, the subject of a renormalisation group analysis between the scales $a_k$ and $t_0$~\cite{Bal84a,Bal85a,Bal89b}, and they are not proved here. By Corollary~\ref{cor:trial} the trial-state condition is the least hypothesis that gives $(\mathrm N_u)$, since it is also necessary; by Proposition~\ref{prop:cubecompatible}(ii) it is not a consequence of the cube estimate in the form used by Theorem~\ref{thm:transfer}; and (S1)--(S2) are its form for the slice field. \begin{remark}\label{rem:independence} There is no circularity between $(\mathrm N_u)$ and (D). The proofs of Proposition~\ref{prop:ratioexact}, Lemma~\ref{lem:trial}, Corollary~\ref{cor:trial} and Theorem~\ref{thm:softhard} use reflection positivity, the transfer matrix and the spectral theorem only, and Theorem~\ref{thm:flowlimit}(i)--(iv), including the renormalisation $Z_k=C_k(0)^{-1/2}$, does not use (D). Conversely, Theorems~\ref{thm:onescale} and~\ref{thm:transfer} use no lower bound on any correlation. The two statements are opposite inequalities on the same spectral measures: (D) bounds $\rho_k$ from above by $e^{-m_*\sigma}$, $(\mathrm N_u)$ from below. Their joint validity requires only the compatibility relations $\alpha\ge c^{2\ell/\sigma}$, $m_*\le\sigma^{-1}\log(1/c)$ and $m_*\le8\kappa/v^2$ of Proposition~\ref{prop:cubecompatible} and Corollary~\ref{cor:trial}, all of them statements in physical units. Neither implies the other. Strong-coupling Wilson measures satisfy the hypothesis of Theorem~\ref{thm:transfer} and violate $(\mathrm N_u)$ (Proposition~\ref{prop:cubecompatible}(ii)). In the free abelian theory the quadratic approximation of the forest gauge violates (D) at every size and stiffness (Proposition~\ref{prop:gaussforest}), while the zero-momentum packet of the magnetic energy has slice ratios with positive limits (Remark~\ref{rem:compositeslice}) and so satisfies $(\mathrm N_0)$ by Proposition~\ref{prop:sliceratio}(iii); (D) depends on the gauge and $(\mathrm N_u)$ does not. Theorem~\ref{thm:higher}(iv) derives $(\mathrm N_0)$ from the order-two cases of (T1) and (T2), and Proposition~\ref{prop:treecritical} shows that those cases fail wherever the lattice gap is bounded below, in particular in the strong-coupling range, where the hypothesis of Theorem~\ref{thm:transfer} holds. \end{remark} \subsection{Slab energies of local fields}\label{ssec:slabenergy} This subsection tests the first condition of Corollary~\ref{cor:trial} against the most direct estimate for it: a pointwise bound on the change of a gauge-invariant function of one slice in one time step by the time-like plaquettes, combined with an upper bound on the plaquette energy. Both bounds are proved with explicit constants. The combination cannot give the condition along any tuning with $a_k\beta_k\to0$, whatever the trial function, and the one-loop tuning is of this kind. Let $G\subset\mathrm U(N)$ be a closed subgroup, let $|M|^2=\Tr(M^*M)$ be the Frobenius norm, so that $|gMh|=|M|$ for $g,h\in G$, and put $s_p=1-N^{-1} \ReTr U_p$, so that $|U_p-I|^2=2Ns_p$. Write $U_{(y,\mu)}$ for the link from $y$ to $y+\hat\mu$, with $\mu=0$ the time direction. Let $\omega$ be a state as in Appendix~\ref{app:transfer}, with transfer matrix $\mathbb T$, and let $\varepsilon_\omega=\sup_p\omega(s_p)$, which is $\omega(s_p)$ for every plaquette $p$ when $\omega$ is invariant under translations and the hypercubic group. A \emph{slice function} is a function $X$ of the spatial links of the slice $x_0=0$ that depends only on the links of a finite set $E$, is invariant under the gauge transformations of that slice, and satisfies \[ |X(U)-X(V)|\le\sum_{e\in E}L_e|U_e-V_e| \] with constants $L_e\ge0$; put $\Lambda(X)=\sum_{e\in E}L_e$. For a spatial link $e=((0,x),i)$ let $p_e$ be the time-like plaquette with lower edge $e$, with holonomy \[ U_{p_e}=U_{((0,x),i)}U_{((0,x+\hat\imath),0)}U_{((1,x),i)}^{-1} U_{((0,x),0)}^{-1}. \] \begin{lemma}\label{lem:slabdirichlet} Let $X$ be a bounded slice function. \textup{(i)} $\ip{\Psi_X}{(1-\mathbb T)\Psi_X}=\frac12\omega(|\tau_1X-X|^2)$, and both sides are unchanged when a constant is added to $X$. \textup{(ii)} Pointwise, $|\tau_1X-X|\le\sum_{e\in E}L_e|U_{p_e}-I|$. \textup{(iii)} $\ip{\Psi_X}{(1-\mathbb T)\Psi_X}\le N\Lambda(X)^2 \varepsilon_\omega$. \end{lemma} \begin{proof} (i) The reflection $\vartheta$ in the plane $x_0=0$ fixes the spatial links of that slice, so $\vartheta(\bar X)=\bar X$ and $\|\Psi_X\|^2=\omega(|X|^2)$. As in the proof of Proposition~\ref{prop:sliceratio}, $\Psi_{\tau_1X}= \mathbb T\Psi_X$, so $\ip{\Psi_X}{\mathbb T\Psi_X}=\omega(\bar X\tau_1X)$, a real number because $\mathbb T$ is self-adjoint. Translation invariance gives $\omega(|\tau_1X|^2)=\omega(|X|^2)$, hence \[ \omega(|\tau_1X-X|^2)=2\omega(|X|^2)-2\omega(\bar X\tau_1X)= 2\ip{\Psi_X}{(1-\mathbb T)\Psi_X} . \] The right side of the identity depends on $X$ only through $\tau_1X-X$. (ii) For a site $x$ of the slice put $g(x)=U_{((0,x),0)}$, and for $e=((0,x),i)$ put $V_e=g(x)U_{((1,x),i)}g(x+\hat\imath)^{-1}$. The function $\tau_1X$ is $X$ evaluated on the links $U_{((1,x),i)}$ in place of $U_{((0,x),i)}$, and gauge invariance under $g$ gives $\tau_1X(U)=X(V)$. Hence $|\tau_1X-X|\le\sum_eL_e|V_e-U_e|$. Now $|V_e-U_e|=|U_e^{-1}V_e-I|$, and $U_e^{-1}V_e=U_e^{-1}U_{p_e}^{-1}U_e$, so $|V_e-U_e|=|U_{p_e}^{-1}-I|= |U_{p_e}-I|$. (iii) By (i), (ii) and the Cauchy--Schwarz inequality, \[ \ip{\Psi_X}{(1-\mathbb T)\Psi_X}\le\frac12\Lambda(X)\sum_eL_e\,\omega (|U_{p_e}-I|^2)\le N\Lambda(X)^2\varepsilon_\omega . \qedhere \] \end{proof} The plaquette energy of Lemma~\ref{lem:wilsonZ} carries a factor $\log\beta_W$. It comes from the lower bound on the partition function alone; an upper bound in temporal gauge removes it. \begin{proposition}\label{prop:energybeta} Let $G=\SU(2)$ and let $\mu_L$ be the Wilson measure on $\Lambda_L=(\Z/L\Z)^4$ of Lemma~\textup{\ref{lem:wilsonZ}}, with $L\ge2$ and $\beta_W\ge1$. Then $\varepsilon_L=\E_{\mu_L}s_p$ satisfies \[ \varepsilon_L\le\frac1{\beta_W}\Bigl(K_1+\frac{K_2}{L^4}+\frac{15}{16}\Bigl( \frac1L+\frac1{3L^4}\Bigr)\log\beta_W+\frac{K_3}{\beta_W}\Bigr) \] with $K_1=3.7148$, $K_2=0.4699$ and $K_3=0.149$. In particular every limit as $L\to\infty$ of the measures $\mu_L$ along a subsequence, on the local functions, has $\varepsilon\le(K_1+K_3/\beta_W)/\beta_W$. \end{proposition} \begin{proof} Write $\beta=\beta_W$, $Z(\beta)=\int e^{-\beta S}dU$ with $S=\sum_ps_p$, and $\theta\in[0,\pi]$ for the angle of an element $\cos\theta\,I+\sin\theta\,J$ of $\SU(2)$; it has Haar density $\frac2\pi\sin^2\theta$ (Lemma~\ref{lem:haarsu2}), it is a bi-invariant distance from $I$, and $\frac12\Tr U=\cos\theta$. The measure $\mu_L$ is invariant under the translations and the hypercubic group of the torus, so $\E_{\mu_L}S=6L^4\varepsilon_L$. \emph{Convexity.} The function $\log Z$ is convex with derivative $-\E_{\mu_L}S$ at $\beta$, so for $0<\lambda<1$ \[ 6L^4(1-\lambda)\beta\,\varepsilon_L\le\log Z(\lambda\beta)-\log Z(\beta) . \] \emph{Upper bound.} The integrand is gauge invariant, so the time links $U_{((t,x),0)}$ with $0\le t\le L-2$, which form a forest, may be set to $I$ without changing $Z$; this is the substitution $U\mapsto U^g$ with $g$ built along each tree from its root, which preserves the Haar measure. Drop the spatial plaquettes and the time-like plaquettes between the times $L-1$ and $0$, whose weights are at most $1$. For each spatial link $(x,i)$ the remaining weights are $\prod_{t=0}^{L-2}f(V_tV_{t+1}^{-1})$ with $V_t=U_{((t,x),i)}$ and $f(U)=e^{-\beta(1-\frac12\Tr U)}$; integrating $V_0,\dots,V_{L-2}$ in turn, each integral is $z(\beta)=\int f\,dU$ by the invariance of the Haar measure. Hence $Z(\beta)\le z(\beta)^{3L^3(L-1)}$. With $y=\sin(\theta/2)$, \begin{align*} z(\beta)&=\frac2\pi\int_0^\pi e^{-\beta(1-\cos\theta)}\sin^2\theta\,d\theta\\ &=\frac{16}\pi\int_0^1e^{-2\beta y^2}y^2\sqrt{1-y^2}\,dy\le \sqrt{\frac2\pi}\,\beta^{-3/2}, \end{align*} by $\sqrt{1-y^2}\le1$ and $\int_0^\infty e^{-2\beta y^2}y^2dy=\frac{\sqrt\pi}4 (2\beta)^{-3/2}$. \emph{Lower bound.} Fix a spanning tree of $\Lambda_L$ and set its $L^4-1$ links to $I$, again without changing $Z$; $3L^4+1$ links remain. Let $0<\delta \le\pi/2$ and let $\nu$ be the Haar measure restricted to $\{\theta\le\delta\}$ and normalised. Since $\sin t/t$ decreases on $(0,\pi)$, \[ \frac2\pi\int_0^\delta\sin^2t\,dt\ge\frac{2\delta^3}{3\pi}\Bigl(\frac{\sin \delta}\delta\Bigr)^2 . \] By Jensen's inequality for the product of the measures $\nu$ on the remaining links, $\log Z(\beta)\ge(3L^4+1)\log\frac{2\delta^3}{3\pi}(\frac{\sin\delta} \delta)^2-\beta\sum_p\E_\nu s_p$. The set $\{\theta\le\delta\}$ is invariant under conjugation and inversion, so $\E_\nu U=mI$ and $\E_\nu U^{-1}=mI$ with $m=\E_\nu\cos\theta\in[0,1]$. A plaquette has at most four links off the tree, independent under $\nu$, so $\E_\nu\frac12\Tr U_p=m^j$ with $j\le4$ and $\E_\nu s_p=1-m^j\le4(1-m)\le2\E_\nu\theta^2$. With $w(t)=\sin^2t/t^2$ nonincreasing and $t^2$ increasing, Chebyshev's inequality for the measure proportional to $t^2dt$ on $[0,\delta]$ gives $\E_\nu\theta^2=\int_0^\delta t^2w\,t^2dt/\int_0^\delta w\,t^2dt\le\frac35\delta^2$. Hence $\E_\nu s_p\le \frac65\delta^2$. \emph{The constants.} Take $\lambda=\frac15$ and $\delta^2=c/\beta$ with $c=\frac58$, so that $\delta^2\le\frac58<\pi^2/4$. Dividing by $6L^4(1-\lambda)\beta$ and collecting the terms in $\log\beta$ gives the bound, with \begin{gather*} K_1=\tfrac54\Bigl(\tfrac14\log\tfrac2\pi-\tfrac34\log\lambda-\tfrac12\log \tfrac2{3\pi}-\tfrac34\log c+\tfrac65c\Bigr),\\ K_2=\tfrac5{24}\Bigl(-\log\tfrac2{3\pi}-\tfrac32\log c\Bigr), \end{gather*} after dropping the term $-\frac5{8L}(\frac12\log\frac2\pi-\frac32\log \lambda)$, which is negative. The factor $(\sin\delta/\delta)^2$ contributes at most $\frac54(1+\frac1{48})\frac{y}{1-y}$ with $y=\delta^2/6\le c/(6\beta)$, by $\sin\delta/\delta\ge1-y$ and $-\log(1-y)\le y/(1-y)$, which is at most $K_3/\beta$ inside the bracket. With $\lambda=\frac15$ and $c=\frac58$ the logarithms collect to \begin{gather*} K_1=\tfrac54\bigl(2\log2+\tfrac12\log3+\tfrac14\log\pi+\tfrac34\bigr),\\ K_2=\tfrac5{24}\bigl(\tfrac72\log2+\log3+\log\pi-\tfrac32\log5\bigr), \end{gather*} and $\frac54\cdot\frac{49}{48}\cdot\frac{c/6}{1-c/6}=\frac{1225}{8256}< 0.1484$. The series $2\operatorname{artanh}x=2\sum_{k\ge0}x^{2k+1}/(2k+1)$, whose tail after any term is at most the next term divided by $1-x^2$, applied to $\log2=2\operatorname{artanh}\frac13$, $\log\frac32=2\operatorname {artanh}\frac15$ and $\log\frac54=2\operatorname{artanh}\frac19$, gives $\log2<0.6931472$, $\log3<1.0986123$ and $\log5>1.6094379$. With $\pi<3.1416$ and $\log(1+x)\le x-\frac{x^2}2+\frac{x^3}3$ at $x=0.0472$, $\log\pi<\log3+ 0.046122<1.144735$. Hence $K_1<3.71474$ and $K_2<0.46984$. At fixed $\lambda$ the value $c=\frac58$ minimises $K_1$. The last claim holds because $s_p$ is a local function. \end{proof} For comparison, Lemma~\ref{lem:wilsonZ} gives the bound $(\log\beta_W+ 4.9082)/\beta_W$ for $\varepsilon_L$, which is larger at every $\beta_W\ge1$ in the limit $L\to\infty$. With Lemma~\ref{lem:slabdirichlet}, a slice function at a limit state of Proposition~\ref{prop:energybeta} has \[ \ip{\Psi_X}{(1-\mathbb T)\Psi_X}\le\frac{2\Lambda(X)^2}{\beta_W}\Bigl(K_1+ \frac{K_3}{\beta_W}\Bigr), \] a bound of order one in lattice units at bounded $\Lambda(X)$. Whether it can serve as the first condition of Corollary~\ref{cor:trial} depends on the variance of $X$ alone, and the next proposition shows that it cannot along the tunings of interest. \begin{proposition}\label{prop:lipobstruction} Let $X$ be a real bounded slice function with $\Var_\omega X>0$, and let $\chi=\Psi_{X-\omega(X)}/\|\Psi_{X-\omega(X)}\|$. \textup{(i)} $\Var_\omega X\le N\Lambda(X)^2$. Hence the bound of Lemma~\textup{\ref{lem:slabdirichlet}(iii)} for $\ip\chi{(1-\mathbb T)\chi}$, which is $N\Lambda(X)^2\varepsilon_\omega/\Var_\omega X$, is at least $\varepsilon_\omega$. \textup{(ii)} Let $G=\SU(2)$ and let $\omega_k$ be states as in Appendix~\textup{\ref{app:transfer}} that are Wilson Gibbs states at $\beta_k$ in the sense of Sec.~\textup{\ref{ssec:uniform}}. If that bound gives the first condition of Corollary~\textup{\ref{cor:trial}} with a constant $\kappa$, then $(4\beta_k+1)a_k\ge1/\kappa$. Along a tuning with $a_k\beta_k\to0$, no choice of slice functions $X_k$ gives that condition through Lemma~\textup{\ref{lem:slabdirichlet}(iii)}. \textup{(iii)} Let the Lie algebra $\g$ of $G$ have trivial centre. If $X$, as a function on $G^E$, is differentiable at the configuration with every link equal to $I$, its differential there vanishes. \end{proposition} \begin{proof} (i) For any two configurations $U$ and $V$, \[ |X(U)-X(V)|\le\Lambda(X)\max_e|U_e-V_e|\le2\sqrt N\Lambda(X), \] since $|U_e|=|V_e|=\sqrt N$. A real random variable with values in an interval of length $\ell$ has variance at most $\ell^2/4$. By Lemma~\ref{lem:slabdirichlet}(i), $\ip\chi{(1-\mathbb T)\chi}= \ip{\Psi_X}{(1-\mathbb T)\Psi_X}/\Var_\omega X$, since $\|\Psi_{X-\omega(X)} \|^2=\omega(|X-\omega(X)|^2)$. (ii) By (i) the bound is at least $\varepsilon_{\omega_k}$, which is at least $1/(4\beta_k+1)$ by Proposition~\ref{prop:plaqvar}. If it is at most $\kappa a_k$, then $1\le\kappa a_k(4\beta_k+1)$. (iii) A constant gauge transformation $h$ on the slice gives $X(hUh^{-1})= X(U)$. For $A=(A_e)\in\g^E$ let $\phi(A)$ be the derivative of $t\mapsto X((e^{tA_e})_e)$ at $t=0$; $\phi$ is linear, and $\phi(\mathrm{Ad}_hA)=\phi(A)$ for $h\in G$. Averaging over the Haar measure, $\phi(A)=\phi(\bar A)$ with $\bar A_e=\int\mathrm{Ad}_hA_e\,dh$. Each $\bar A_e$ is invariant under $\mathrm{Ad}_G$, hence commutes with $\g$, and so vanishes. The curves $t\mapsto e^{tA_e}$ span the tangent space at $I$. \end{proof} The one-loop tuning has $\beta_W=\frac{11}{3\pi^2}\log(1/a)+O(1)$ for $\SU(2)$, and the two-loop tuning adds a term of order $\log\log(1/a)$ (Remark~\ref{rem:twoloop}), so $a_k\beta_k\to0$ along either, and Proposition~\ref{prop:lipobstruction}(ii) applies. Part (i) shows where the loss lies: the bound of Lemma~\ref{lem:slabdirichlet}(iii) is first order in the time-like plaquettes, and it gives an energy at most $\kappa a$ only if the variance of $X$ is at least the fraction $\varepsilon_\omega/(\kappa a)$ of its largest possible value $N\Lambda(X)^2$. Part (iii) shows that a gauge-invariant function has no first-order term at the configuration with all links $I$. If $X$ is continuously differentiable, its local Lipschitz constants at configurations within distance $r$ of that configuration, or of a gauge transform of it, tend to $0$ with $r$, while the bound of Lemma~\ref{lem:slabdirichlet}(ii) uses the global constants $L_e$. The bound does give the trial-state condition in the opposite regime, at the cost of a lower bound on a variance and on one correlation, which are the infrared content of $(\mathrm N_u)$. \begin{corollary}\label{cor:lipreduction} Let $G=\SU(2)$, let $\omega_k$ be states as in Appendix~\textup{\ref{app:transfer}} that are limits as in Proposition~\textup{\ref{prop:energybeta}} at $\beta_k\ge1$, and let $X_k$ be real slice functions with $X_k^c=X_k-\omega_k(X_k)$. Suppose that there are $\nu,v>0$ such that for all large $k$ \begin{gather*} \Var_{\omega_k}X_k\ge2\nu\Lambda(X_k)^2,\\ \omega_k\bigl(X_k^c\,\tau_{u/a_k}A_k\bigr)^2\ge v^2\,\Var_{\omega_k}X_k\, C_k(2u) . \end{gather*} If moreover $a_k\beta_k\ge\kappa_0>0$ for all large $k$, then $(\mathrm N_u)$ holds, and $\rho_k(u,\sigma)\ge(v-\sqrt{\kappa\sigma/2})_+^2$ for $\sigma\ge2a_k$, with $\kappa=(K_1+K_3)/(\nu\kappa_0)$. \end{corollary} \begin{proof} The vector $\chi_k=\Psi_{X_k^c}/\|\Psi_{X_k^c}\|$ is a unit vector with $\|\Psi_{X_k^c}\|^2=\Var_{\omega_k}X_k$. By Lemma~\ref{lem:slabdirichlet} and Proposition~\ref{prop:energybeta}, since $\beta_k\ge1$ and $a_k\beta_k\ge\kappa_0$, \[ \ip{\chi_k}{(1-\mathbb T_k)\chi_k}\le\frac{2\Lambda(X_k)^2(K_1+K_3)} {\beta_k\Var_{\omega_k}X_k}\le\frac{K_1+K_3}{\nu\beta_k}\le\kappa a_k . \] Since $X_k$ is a real function of the slice $x_0=0$, $\vartheta(X_k^c)=X_k^c$ and $\ip{\chi_k}{\Psi_{k,u}}=\omega_k(X_k^c\tau_{u/a_k}A_k)/(\Var_{\omega_k} X_k\,C_k(2u))^{1/2}$, whose modulus is at least $v$. Corollary~\ref{cor:trial} gives the rest. \end{proof} The hypotheses of Corollary~\ref{cor:lipreduction} are not proved here. The first asks that $X_k$ take values spread over a fixed fraction of the range allowed by its Lipschitz constants, at couplings where the plaquettes are of size $\beta_k^{-1/2}$; the second is a lower bound on one correlation at physical distance. Together with Proposition~\ref{prop:lipobstruction}, the corollary locates the estimate: the upper bound on the slab energy that reflection positivity and the plaquette energy provide is of order $\beta_k^{-1}$ in lattice units, it meets the scale $a_k$ only where $a_k\beta_k$ is bounded below, and even there it transfers the whole of $(\mathrm N_u)$ to lower bounds on a variance and on an overlap. \subsection{The axioms for the flowed fields}\label{ssec:osaxioms} Statement (E) asks for more than the nontriviality of Sec.~\ref{ssec:ratiocube}. This subsection determines, for the spatially flowed fields of Sec.~\ref{ssec:higher}, which of the Osterwalder--Schrader axioms follow from the hypothesis of Theorem~\ref{thm:transfer} once the higher renormalised correlations are bounded, and which further statements are needed. Three are needed; together they suffice, and each is necessary: a uniform moment bound (M), which contains $(\mathrm N_0)$; invariance under one spatial rotation of infinite order (W); and coincidence of the subsequential limits (U). The spatial flow leaves at most the symmetry $\mathrm O(3)\times\mathrm O(1)$ of space and time reflection, and Gaussian examples show that neither (W) nor (U) follows from the gap, reflection positivity, the lattice symmetries and the tree bounds. The notation is that of Sec.~\ref{ssec:higher}: $\Phi$ is a bounded real gauge-invariant function of finitely many spatial links of the slice $\{x_0=0\}$, continuous in them, $\varphi_k(x)=\Phi_x\circ\Phi_{s_k}$ with $s_k=\tau/a_k^2$, $A_k(f)$ is the centred packet of $f\in C_c(\R^4)$, $f_0$ is a fixed nonzero function supported in $\{t>0\}$, $C_k(0)>0$ and $Z_k=C_k(0)^{-1/2}$. Let $W_3$ be the group of the $48$ signed permutations of the spatial coordinates, $\theta(t,\mathbf x)=(-t,\mathbf x)$, and $\Gamma$ the group of order $96$ generated by $W_3$ and $\theta$, the subgroup of the hyperoctahedral group $B_4$ that preserves the time axis. We assume that $\Phi$ is invariant under the action of $W_3$ on the links of the slice that fixes the site $0$. Averaging $\Phi$ over $W_3$ achieves this; the sum of $N-\ReTr U_p$ over the twelve spatial plaquettes $p$ with a corner at $0$ is an example. Throughout, the states $\omega_k$ at $\beta_k$ satisfy the hypothesis of Theorem~\ref{thm:transfer}: $D(n_k,\alpha)$ holds with $n_ka_k\le\ell$. By Theorem~\ref{thm:onescale} and Proposition~\ref{prop:state}, each $\omega_k$ is the restriction of the unique Gibbs measure of $\gamma^F$, is the free-boundary limit along every exhaustion of $\Z^4$, is invariant under translations and the hypercubic group and is reflection positive, and its transfer matrix has gap at least $\log(1/\alpha)/(2n_k)\ge m_*a_k$ on $\Omega_k^\perp$, with $m_*=\log(1/\alpha)/(2\ell)$. The infinite-volume state is therefore unique at each $k$, and the only non-uniqueness left is in the limit $k\to\infty$. The spatial flow commutes with $\Gamma$ (Proposition~\ref{prop:spatialflow}(i) and the proof of Theorem~\ref{thm:osaxioms}(ii)) but not with the elements of $B_4$ that exchange time with a spatial axis, so $\Gamma$, not $B_4$, is the lattice symmetry of the flowed packets. \begin{enumerate}[label=\textup{(M)},leftmargin=*] \item \emph{Moment bound.} For every $n\ge1$ and every cube $Q\subset\R^4$ there are $M_n(Q)<\infty$ and $k_0$ such that \[ \Bigl|\omega_k\Bigl(\prod_{j=1}^nZ_kA_k(f_j)\Bigr)\Bigr|\le M_n(Q)\prod_{j=1}^n\|f_j\|_\infty \] for all $k\ge k_0$ and all $f_j\in C_c(\R^4)$ supported in $Q$. \end{enumerate} By Theorem~\ref{thm:higher}(i), (T1)--(T3) imply (M). The case $n=2$ of (M) is not the bound of Theorem~\ref{thm:flowlimit}(i), which concerns reflected pairs of translates of $f_0$ and holds by reflection positivity alone; (M) concerns all test functions in $Q$, including those closer to the reflection plane than $f_0$, and it implies $(\mathrm N_0)$ (Theorem~\ref{thm:osaxioms}(v)). By Remark~\ref{rem:markov}, no bound on the two-point functions gives (M) for $n=4$. A \emph{polynomial} is a finite formal sum $P=\sum_\beta c_\beta \prod_{j=1}^{n_\beta}f_{\beta,j}$ with $c_\beta\in\C$ and $f_{\beta,j}\in C_c(\R^4)$; its lattice version is $P_k=\sum_\beta c_\beta\prod_jZ_k A_k(f_{\beta,j})$, and for a family $S=(S_n)$ with $S_0=1$ we put $S(P)=\sum_\beta c_\beta S_{n_\beta}(f_{\beta,1},\dots,f_{\beta,n_\beta})$. The polynomials $\bar P$, $\theta P$, $\tau_yP$ and $gP$ are obtained by replacing every $f_{\beta,j}$ by $\overline{f_{\beta,j}}$, $f_{\beta,j}\circ\theta$, $f_{\beta,j}(\cdot-y)$ and $f_{\beta,j}\circ g^{-1}$, products of polynomials are formal products, and $P$ is \emph{at positive times} if every $f_{\beta,j}$ is supported in $\{t>0\}$. We write $\Psi(f_0)$ for $\Psi(P)$ with $P=f_0$. \begin{theorem}\label{thm:osaxioms} Assume the hypothesis of Theorem~\textup{\ref{thm:transfer}} and \textup{(M)}, with $\Phi$ real and $W_3$-invariant. Every sequence of $k$ has a subsequence along which the following hold. \textup{(i)} For all $n$ and all $f_j\in C_c(\R^4)$ the limit \[ S_n(f_1,\dots,f_n)=\lim_k\omega_k\Bigl(\prod_{j=1}^nZ_kA_k(f_j)\Bigr) \] exists. Each $S_n$ is symmetric and multilinear, bounded by $M_n(Q)\prod_j\|f_j\|_\infty$ on functions supported in a cube $Q$, and a distribution on $(\R^4)^n$; $S_1=0$. \textup{(ii)} Each $S_n$ is invariant under the translations of $\R^4$ and under $\Gamma$: $S_n(f_1\circ g^{-1},\dots,f_n\circ g^{-1})=S_n(f_1,\dots, f_n)$ for $g\in\Gamma$. Moreover $S(\bar P)=\overline{S(P)}$. \textup{(iii)} $S(\theta\bar P\cdot P)\ge0$ for every polynomial $P$ at positive times. The completion of the quotient of these polynomials by the null space of this form is a Hilbert space $\mathcal H$ with vectors $\Psi(P)$, $\ip{\Psi(P)}{\Psi(P')}=S(\theta\bar P\cdot P')$, and $\Omega=\Psi(1)$. \textup{(iv)} There are a self-adjoint $H\ge0$ with $\Psi(\tau_{(t,0)}P)= e^{-tH}\Psi(P)$ for $t\ge0$, a strongly continuous unitary representation $U$ of $\R^3$ with $\Psi(\tau_{(0,x)}P)=U(x)\Psi(P)$, and a unitary representation $V$ of $W_3$ with $\Psi(gP)=V(g)\Psi(P)$. They commute with $H$, $V(g)U(x)V(g)^{-1}=U(gx)$, $H\Omega=0$, and $U$ and $V$ fix $\Omega$. \textup{(v)} $\Psi(f_0)$ is a unit vector orthogonal to $\Omega$, there is a dyadic $\sigma>0$ with $C_k(\sigma)\ge\frac12C_k(0)$ for all large $k$, and $(\mathrm N_0)$ holds. \textup{(vi)} $\sigma(H)\subset\{0\}\cup[m_*,\infty)$ and $\ker H=\C\Omega$. For polynomials $P,P'$ at positive times and $t\ge0$, \[ \begin{split} &\bigl|S(\theta\bar P\cdot\tau_{(t,0)}P')-\overline{S(P)}\,S(P')\bigr|\\ &\qquad\le e^{-m_*t}\,\|\Psi(P)-S(P)\Omega\|\,\|\Psi(P')-S(P')\Omega\| . \end{split} \] The joint spectral measure of $H$ and the generators of $U$ is carried by $\{(0,0)\}\cup\bigl([m_*,\infty)\times\R^3\bigr)$. \textup{(vii)} If $M_n(Q)\le C_n(1+L)^{p_n}$ for every cube $Q$ of side $L$ centred at $0$, each $S_n$ extends to a tempered distribution on $\R^{4n}$. Conversely, if for every $n$ and all $f_j\in C_c(\R^4)$ the sequence $\omega_k(\prod_jZ_kA_k(f_j))$ is bounded, then \textup{(M)} holds. \end{theorem} \begin{proof} (i) The proof of Theorem~\ref{thm:higher}(ii) uses part (i) of that theorem only through the bound $M_n(Q)$, and applies verbatim. $S_1=0$ because the packets are centred. (ii) Translation invariance is proved as in Theorem~\ref{thm:higher}(ii). Let $g\in\Gamma$; it maps $\Z^4$ to itself and fixes the site $0$. It acts on configurations by moving the links, a link whose orientation is reversed carrying the inverse group element. Inversion is an isometry of the bi-invariant metric, and $g$ preserves the Wilson action of every time slice, so this action maps solutions of the flow equation to solutions and commutes with $\Phi_s$; for $g=\theta$ this is Proposition~\ref{prop:spatialflow}(i). Write $gF$ for the function $U\mapsto F(g^{-1}U)$. Since $\Phi$ is $W_3$-invariant and $\theta$ fixes the spatial links of the slice $\{x_0=0\}$, $g\Phi_x=\Phi_{gx}$, hence $g\varphi_k(x)=\varphi_k(gx)$, and substituting $y=g^{-1}x$, \[ A_k(f\circ g^{-1})=\sum_ya_k^4f(a_ky)\bigl(\varphi_k(gy)-c_k\bigr) =g\,A_k(f), \] where $c_k=\omega_k(\varphi_k(x))$ does not depend on $x$. The state $\omega_k$ is invariant under the hypercubic group, so $\omega_k(\prod_j A_k(f_j\circ g^{-1}))=\omega_k(\prod_jA_k(f_j))$, and the limits agree. Since $\Phi$ is real, $A_k(\bar f)=\overline{A_k(f)}$, which gives the last claim. (iii) This is the proof of Theorem~\ref{thm:higher}(iii), which uses only the bound of (i). (iv) Let $P$ be at positive times and $t\ge0$ dyadic. For large $k$, $t/a_k\in\Z$, and translation invariance of $\omega_k$ and of the flow gives $(\tau_{(t,0)}P)_k=\tau_{t/a_k}P_k$. In the reconstruction of Appendix~\ref{app:transfer}, $\Psi_{\tau_nF}=\mathbb T_k^n\Psi_F$ (proof of Theorem~\ref{thm:flowlimit}), so \[ \omega_k\bigl(\vartheta(\bar P_k)\tau_{2t/a_k}P_k\bigr) =\bigl\|\mathbb T_k^{t/a_k}\Psi_{P_k}\bigr\|^2\le \omega_k\bigl(\vartheta(\bar P_k)P_k\bigr). \] By (i) and the approximation of $\theta$ by the lattice reflection in the proof of Theorem~\ref{thm:higher}(iii), the left side tends to $S(\theta\bar P\cdot\tau_{(2t,0)}P)$, which equals $\|\Psi(\tau_{(t,0)}P)\|^2$ by translation invariance and $\theta\tau_{(t,0)}=\tau_{(-t,0)}\theta$, and the right side tends to $\|\Psi(P)\|^2$. The bound of (i) makes $S(\theta\bar P'\cdot P')$ continuous under uniform convergence of the test functions in a fixed cube, so $\|\Psi(\tau_{(t,0)}P)\|\le\|\Psi(P)\|$ for all $t\ge0$, and $\|\Psi(\tau_{(t,0)}P)-\Psi(P)\|\to0$ as $t\to0$, each monomial of $\tau_{(t,0)}P-P$ being a telescoping sum of monomials with one factor $f(\cdot-(t,0))-f$. Hence $\tau_{(t,0)}$ maps null vectors to null vectors and induces contractions $\mathsf T(t)$ of $\mathcal H$ with $\mathsf T(s)\mathsf T(t)=\mathsf T(s+t)$, symmetric by translation invariance and $\theta\tau_{(t,0)}=\tau_{(-t,0)}\theta$, and strongly continuous on a dense set and hence everywhere. A strongly continuous semigroup of symmetric contractions is $e^{-tH}$ with $H\ge0$ self-adjoint~\cite{OS73}. The spatial translations and the elements of $W_3$ preserve the positive times, commute with $\theta$ and preserve the form by (ii); they have inverses of the same kind, and so induce unitaries, strongly continuous in $x$ by the argument just given. The relations follow from those of the maps of $\R^4$, and $\tau_y1=g1=1$. (v) As in the proof of Theorem~\ref{thm:higher}(iv), $C_k(t)-C_k(0)=\omega_k(A_k(\bar f_0\circ\theta_k)A_k(f_{0,t}-f_0))$ for dyadic $t$ and large $k$, with $f_{0,t}=f_0(\cdot-(t,0))$ and $\theta_k$ the lattice reflection. With $Q$ a cube containing the supports of $\bar f_0\circ\theta_k$ and of $f_{0,t}$ for all $k$ and $t\le1$, (M) for $n=2$ gives $Z_k^2|C_k(t)-C_k(0)|\le M_2(Q)\|f_0\|_\infty \|f_{0,t}-f_0\|_\infty$, which is at most $\frac12$ for a small dyadic $t$ by uniform continuity. So $C_k(\sigma)/C_k(0)\ge\frac12$ with $\sigma=t$, which is $(\mathrm N_0)$. Further $\|\Psi(f_0)\|^2=S_2(\bar f_0\circ\theta, f_0)=\lim_kZ_k^2C_k(0)=1$ by the same approximation of $\theta$, and $\ip\Omega{\Psi(f_0)}=S_1(f_0)=0$. (vi) The proof of Theorem~\ref{thm:higher}(vi) uses part (ii) of that theorem only through the convergence of the moments, which (i) provides. It gives $\ip v{e^{-tH}v}\le e^{-m_*t}\|v\|^2$ for $v=\Psi(P)-\ip\Omega {\Psi(P)}\Omega$, $P$ at positive times. These vectors are dense in $\Omega^\perp$, which $e^{-tH}$ leaves invariant, so the restriction of the positive operator $e^{-tH}$ to $\Omega^\perp$ has norm at most $e^{-m_*t}$; the spectral theorem gives the claims on $\sigma(H)$ and $\ker H$. Here $\ip\Omega{\Psi(P)}=S(P)$ and $\ip{\Psi(P)}\Omega= S(\theta\bar P)=\overline{S(P)}$ by (ii), and $\ip{\Psi(P)}{e^{-tH} \Psi(P')}-\ip{\Psi(P)}\Omega\ip\Omega{\Psi(P')}=\ip v{e^{-tH}v'}$ with $v,v'$ the projections on $\Omega^\perp$, which gives the inequality. Since $U$ commutes with $H$ and fixes $\Omega$, both reduce $\Omega^\perp$; the joint spectral measure at a vector of $\Omega^\perp$ is carried by $[m_*,\infty)\times\R^3$, and at $\Omega$ it is the point mass at $(0,0)$. (vii) Let $\chi(x)=\prod_\mu(1-|x_\mu|)_+$, so that $\sum_{m\in\Z^4} \chi(x-m)=1$, and put $\chi_m=\chi(\cdot-m)$, $r=p_n+5$ and \[ N_r(f)=\sup_x(1+|x|_\infty)^r|f(x)| . \] On the support of $\chi_m$, $1+|x|_\infty\ge \max(1,|m|_\infty)\ge(2+|m|_\infty)/3$, so $\|\chi_mf\|_\infty\le3^r (2+|m|_\infty)^{-r}N_r(f)$. For $f_j\in C_c(\R^4)$, multilinearity gives the finite sum \[ S_n(f_1,\dots,f_n)=\sum_{m_1,\dots,m_n}S_n(\chi_{m_1}f_1,\dots, \chi_{m_n}f_n), \] and the term of $(m_j)$ has its supports in the cube of side $L=2+2\max_j|m_j|_\infty$, so it is at most \[ C_n\,2^{p_n}\prod_{j=1}^n(2+|m_j|_\infty)^{p_n}\,\|\chi_{m_j}f_j\|_\infty . \] Summing, $|S_n(f_1,\dots,f_n)|\le C_n2^{p_n}(3^rs)^n\prod_jN_r(f_j)$ with $s=\sum_{m\in\Z^4}(2+|m|_\infty)^{-5}<\infty$. Functions in $C_c(\R^4)$ are dense in $\mathcal S(\R^4)$ for $N_r$, so $S_n$ extends to a jointly continuous multilinear form on $\mathcal S(\R^4)^n$, and the kernel theorem~\cite[Sect.~V.3]{RS80} gives a tempered distribution on $\R^{4n}$. For the converse, let $C_Q$ be the Banach space of continuous functions supported in $Q$ with the supremum norm. For fixed $k$ the form $B_k(f_1,\dots,f_n)=\omega_k(\prod_jZ_kA_k(f_j))$ is continuous on $C_Q^n$, since $|A_k(f)|\le2\|\Phi\|_\infty a_k^4\#\{x:a_kx\in Q\}\|f\|_\infty$. A family of continuous $n$-linear forms on a Banach space that is bounded at every point is bounded in norm: for $n=1$ this is the uniform boundedness principle~\cite{RS80}, and for $n>1$, by induction, $\sup_k\|B_k(f_1,\cdot) \|<\infty$ for each $f_1$, so the linear maps $f_1\mapsto B_k(f_1,\cdot)$ into the Banach space of $(n-1)$-linear forms are bounded at every point, and hence in norm. The bound is $M_n(Q)$. \end{proof} \begin{remark}\label{rem:dyadicmoments} If (M) is replaced by bounds on the dyadic packets only, namely $\sup_kZ_k^{n+n'}|\omega_k(\vartheta(\prod_{i\le n}\bar A^{y_i}_k) \prod_{j\le n'}A^{y'_j}_k)|<\infty$ for all $y_i,y'_j\in\mathcal D$, the construction of Theorem~\ref{thm:flowlimit}(iii) applies to polynomials in the vectors $\delta_y$: every monomial is $\mathsf T(s)$ of a monomial at positive times for a small dyadic $s>0$, so the kernels of the positive contractions $\mathsf T(\sigma)$ are trivial, and $H=-\log\mathsf T(1)$ is self-adjoint and nonnegative, with unitary dyadic spatial translations; the proof of part (vi) gives the gap $m_*$. The limit is nontrivial when $(\mathrm N_u)$ holds for some $u$, but not only then: when $\rho\equiv0$, the vectors $\delta_y$ vanish while products of them need not. Without continuity in the test functions, neither the distributions $S_n$ nor continuous translations follow. \end{remark} Under the tree bounds, the moment bound holds with polynomial constants, and the limit has two properties that (M) does not give: temperedness with an $L^1$ bound, and clustering in every direction. For $f\in\mathcal S(\R^4)$ put $|f|_{(1)}=\max(\|f\|_1,\|f\|_\infty)$; for a tree $T$ on $\{1,\dots,n\}$, $\zeta\in(\R^4)^{E(T)}$ and $j\ge2$, let $\zeta^T_j$ be the sum of $\zeta_e$ over the edges of the path from $1$ to $j$ in $T$. \begin{lemma}\label{lem:hermite} Let $\mathsf S$ be a multilinear form on $\mathcal S(\R^4)^n$ with $|\mathsf S(f_1,\dots,f_n)|\le\sigma\prod_j|f_j|_{(1)}$. Then $\mathsf S$ is given by a unique tempered distribution on $\R^{4n}$, and for $F\in\mathcal S(\R^{4n})$, with $L_i=-\partial_i^2+x_i^2+4$ in the $4n$ coordinates, \[ |\mathsf S(F)|\le\sigma\,\|L_1^3\cdots L_{4n}^3F\|_{L^2(\R^{4n})} \le\sigma\,236^{4n}(\pi/2)^{2n}|F|_{32n}, \] where $|F|_m=\sup_{x,\,|\gamma|\le m}(1+|x|^2)^{m/2}|\partial^\gamma F(x)|$ are the norms of the linear growth condition of~\cite{OS75}. \end{lemma} \begin{proof} The form is jointly continuous, so the kernel theorem~\cite[Sect.~V.3]{RS80} gives the distribution. Let $h_k$, $k\ge0$, be the Hermite functions on $\R$, an orthonormal basis of $L^2(\R)$ of real functions with $(-\partial^2+x^2)h_k=(2k+1)h_k$ and $xh_k=\sqrt{k/2}\,h_{k-1}+\sqrt{(k+1)/2}\,h_{k+1}$, whose tensor products $h_\kappa$, $\kappa\in\mathbb N^{4n}$, give an expansion of every $F\in\mathcal S(\R^{4n})$ that converges in $\mathcal S$~\cite[Sect.~V.3]{RS80}. For real $g\in\mathcal S(\R)$, $g(x)^2=\int_{-\infty}^x2gg'=-\int_x^\infty 2gg'$, so $\|g\|_\infty^2\le\|g\|_2\|g'\|_2$; with $\|h_k'\|_2^2\le2k+1$ this gives $\|h_k\|_\infty\le(2k+1)^{1/4}$. Applying the recursion twice, $x^2h_k=\frac12\bigl(\sqrt{k(k-1)}\,h_{k-2}+(2k+1)h_k+\sqrt{(k+1)(k+2)}\, h_{k+2}\bigr)$, so $\|(1+x^2)h_k\|_\infty\le(2k+\frac52)(2k+5)^{1/4}\le (2k+5)^{5/4}$. Since $\int_\R(1+x^2)^{-1}dx=\pi$, a function $f$ on $\R^4$ has $|f|_{(1)}\le\pi^4\|\prod_{i=1}^4(1+x_i^2)f\|_\infty$, and for $\alpha\in\mathbb N^4$ therefore $|h_\alpha|_{(1)}\le\pi^4\prod_i (2\alpha_i+5)^{5/4}$. The operator $L=\prod_iL_i^3$ is symmetric with $Lh_\kappa=\prod_i(2\kappa_i+5)^3h_\kappa$, so the coefficient $c_\kappa=\ip{h_\kappa}F$ has $|c_\kappa|\le\prod_i(2\kappa_i+5)^{-3} \|LF\|_2$. Hence \[ |\mathsf S(F)|\le\sum_\kappa|c_\kappa||\mathsf S(h_\kappa)| \le\sigma\|LF\|_2\Bigl(\pi\sum_{k\ge0}(2k+5)^{-7/4}\Bigr)^{4n}, \] and $\sum_k(2k+5)^{-7/4}\le5^{-7/4}+\frac23 5^{-3/4}<0.2592$, so the bracket is below $0.815$. In one variable $L_i^3=\sum c_{ab}x^a\partial^b$ with $a,b\le6$ and integer coefficients, and expanding and normal ordering gives $\sum|c_{ab}|=236$. Expanding the product, $\|LF\|_2\le236^{4n}\max\|x^a\partial^bF\|_2$ over multi-indices with entries at most $6$, and since $\int_{\R^{4n}}\prod_i(1+x_i^2)^{-2}dx= (\pi/2)^{4n}$, $\|g\|_2\le(\pi/2)^{2n}\|\prod_i(1+x_i^2)g\|_\infty$. Finally $\prod_i(1+x_i^2)|x_i|^{a_i}\le\prod_i(1+x_i^2)^4\le(1+|x|^2)^{16n}$ and $|b|\le24n$, so $\|\prod_i(1+x_i^2)x^a\partial^bF\|_\infty\le|F|_{32n}$. \end{proof} \begin{theorem}\label{thm:ostree} Let the $\omega_k$ be translation invariant and assume \textup{(T1)--(T3)}. \textup{(i)} \textup{(M)} holds, with $M_n(Q)\le C_n(1+L)^{2n}$ for cubes of side $L$ centred at $0$. \textup{(ii)} Along every subsequence on which the limits $S_n$ of Theorem~\textup{\ref{thm:osaxioms}(i)} exist there are a further subsequence, a number $b\in[0,B_*]$ and a positive measure $\nu$ on $\R^4$ with $\nu(\R^4)\le1$ such that, for $n\ge2$ and $f_j\in C_c(\R^4)$, the truncated functions $S^T_n$ of the limit satisfy \[ \begin{split} |S^T_n(f_1,\dots,f_n)|\le{}&K_nc^{-n/2}b^{n/2-1}\sum_T\int|f_1(x)|\\ &\times\int\prod_{j=2}^n|f_j(x+\zeta^T_j)|\,d\nu^{\otimes E(T)}(\zeta)\,dx , \end{split} \] with $b^0=1$. \textup{(iii)} For $n\ge2$ and every $i$, \[ |S^T_n(f_1,\dots,f_n)|\le K'_n\|f_i\|_1\prod_{j\ne i}\|f_j\|_\infty \] with $K'_n=K_nn^{n-2}c^{-n/2}B_*^{n/2-1}$. Hence $S_n$ extends to $\mathcal S(\R^4)^n$ and \[ |S_n(f_1,\dots,f_n)|\le\sigma_n\prod_j|f_j|_{(1)}, \] where $\sigma_n$ is the sum over the partitions of $\{1,\dots,n\}$ into blocks of size at least two of $\prod_BK'_{|B|}$, and Lemma~\textup{\ref{lem:hermite}} applies. If $K_n\le K^n(n!)^\beta$ for all $n$, then $\sigma_n\le C^n(n!)^{\beta+2}$ with $C=Ke\,c^{-1/2}\max(1,B_*)^{1/2}$, and the Schwinger functions satisfy the linear growth condition of~\cite{OS75}. \textup{(iv)} Let $\{1,\dots,n\}=I\cup J$ with $I,J$ disjoint and nonempty, $y\in\R^4\setminus\{0\}$, and $f^\lambda_j=f_j$ for $j\in I$, $f^\lambda_j=f_j(\cdot-\lambda y)$ for $j\in J$. Then \[ \lim_{\lambda\to\infty}S_n(f^\lambda_1,\dots,f^\lambda_n)= S_{|I|}\bigl((f_j)_{j\in I}\bigr)\,S_{|J|}\bigl((f_j)_{j\in J}\bigr). \] Under the hypotheses of Theorem~\textup{\ref{thm:osaxioms}}, $U(\lambda x)$ converges weakly to the projection on $\C\Omega$ as $\lambda\to\infty$ for every $x\in\R^3\setminus\{0\}$, and $\Omega$ is the only vector invariant under $U$. \end{theorem} \begin{proof} (i) This is Theorem~\ref{thm:higher}(i): a moment is a sum over partitions into blocks of size at least two, each contributing a factor $|Q|_k\le(L+a_0)^4$, and there are at most $n/2$ blocks. (ii) By (T1) and (T2), as in the proof of Lemma~\ref{lem:treesum}, the truncated function of the renormalised packets at level $k$ is at most $Z_k^nK_nr_k^n\sum_T a_k^{4n}\sum_x\prod_j|f_j(a_kx_j)|\prod_{\{i,j\}\in T} w_k(x_i-x_j)$. For fixed $T$ change variables to $x_1$ and $z_e=x_{\mathrm{child}}-x_{\mathrm{parent}}$, the tree being rooted at $1$; since $w_k$ is even, the sum is $(a_k^4W_k)^{n-1}\int h_k\,d\nu_k^{\otimes E(T)}$, where $\nu_k=W_k^{-1}\sum_zw_k(z)\delta_{a_kz}$ is a probability measure ($W_k>0$, since $C_k(0)>0$) and \[ h_k(\zeta)=a_k^4\sum_{x\in\Z^4}|f_1(a_kx)|\prod_{j\ge2}|f_j(a_kx+\zeta^T_j)| . \] With $Z_k^nr_k^n\le(c\,a_k^4W_k)^{-n/2}$ the bound becomes \[ K_nc^{-n/2}(a_k^4W_k)^{n/2-1}\sum_T\int h_k\,d\nu_k^{\otimes E(T)} . \] Choose a further subsequence along which $a_k^4W_k\to b$ and $\nu_k\to\nu$ vaguely; then $\nu_k^{\otimes m}\to\nu^{\otimes m}$ vaguely, the masses being at most one. Let $h(\zeta)=\int|f_1(x)|\prod_{j\ge2}|f_j(x+\zeta^T_j)|\,dx$. It is continuous, and it has compact support, because $x$ and every $x+\zeta^T_j$ lie in compact sets when the integrand is nonzero, and $\zeta_e$ is the difference of two of these points. For $\zeta$ in $(a_k\Z^4)^{E(T)}$, $h_k(\zeta)$ is a Riemann sum for $h(\zeta)$ of a family of functions with supports in one compact set and a common modulus of continuity, so $\sup_\zeta|h_k(\zeta)-h(\zeta)|\to0$. Hence $\int h_kd\nu_k^{\otimes E(T)}\to\int h\,d\nu^{\otimes E(T)}$. The truncated functions at level $k$ are polynomials in finitely many moments, which converge, so the left side converges to $|S^T_n|$. (iii) Relabel so that $i=1$; the integral in (ii) is at most $\|f_1\|_1\prod_{j\ge2}\|f_j\|_\infty$, there are $n^{n-2}$ trees, and $b^{n/2-1}\le B_*^{n/2-1}$. The moments are sums over partitions of products of truncated functions, which gives $\sigma_n$ on $C_c(\R^4)$; the bound extends to $\mathcal S(\R^4)$, the closure of $C_c(\R^4)$ for $|\cdot|_{(1)}$ containing it. If $K_m\le K^m(m!)^\beta$, then, by $m^{m-2}\le e^mm!$, $K'_m\le C^m(m!)^{\beta+1}$; since $\prod_B|B|!\le n!$ and there are at most $n!$ partitions, $\sigma_n\le C^n(n!)^{\beta+2}$. With Lemma~\ref{lem:hermite}, $|S_n(F)|\le C^n(n!)^{\beta+2}236^{4n} (\pi/2)^{2n}|F|_{32n}$, which is the condition of~\cite{OS75} with $s=32$. (iv) Assume $1\in I$, by symmetry of $S_n$. In a partition of $\{1,\dots,n\}$ the blocks contained in $I$ or in $J$ do not change their truncated functions as $\lambda$ varies, by translation invariance. For a block $B$ meeting both, apply (ii) to $(f^\lambda_j)_{j\in B}$ with a root in $B\cap I$: the integrand is bounded by $|f_{\mathrm{root}}(x)|\prod \|f_j\|_\infty$, integrable for $dx\otimes\nu^{\otimes E(T)}$, and tends to $0$ pointwise as $\lambda\to\infty$, since for $j\in B\cap J$ the point $x+\zeta^T_j$ is fixed while the support of $f^\lambda_j$ leaves every compact set. By dominated convergence the truncated function of $B$ tends to $0$, and the moment tends to the sum over the partitions into blocks in $I$ and blocks in $J$, which is the product. For the last statement, let $P,P'$ be at positive times; $\ip{\Psi(P)}{U(\lambda x)\Psi(P')}=S(\theta \bar P\cdot\tau_{(0,\lambda x)}P')$ tends to $S(\theta\bar P)S(P')= \ip{\Psi(P)}\Omega\ip\Omega{\Psi(P')}$ by what was just proved, applied to each monomial. The operators are uniformly bounded and the vectors $\Psi(P)$ are dense, which gives weak convergence; a vector $v$ with $U(x)v=v$ for all $x$ then has $\|v\|^2=|\ip\Omega v|^2$, so $v\in\C\Omega$. \end{proof} Part (iv) is the cluster property of the axioms in every direction, for all $n$. The gap of Theorem~\ref{thm:osaxioms}(vi) gives it with the rate $m_*$ in the time direction only. Spatial clustering does not follow from that gap and (M) without the tree bounds: the field of Proposition~\ref{prop:osobstruction}(b) below has the gap, satisfies (M), and does not cluster in space. With the hypercubic group in place of $\Gamma$ the exchange of time with a spatial axis would carry the time clustering to space; the spatial flow removes that exchange. \emph{Rotations.} For a family $S=(S_n)$ as in Theorem~\ref{thm:osaxioms}(i)--(ii) let \[ G_S=\bigl\{R\in\mathrm O(4):S_n(f_1\circ R^{-1},\dots,f_n\circ R^{-1}) =S_n(f_1,\dots,f_n)\bigr\}, \] the condition holding for all $n$ and $f_j\in C_c(\R^4)$. Let $e_0$ be the unit time vector, $\SO(3)_{\mathrm{sp}}$ the rotations that fix it and $\mathrm O(3)\times\mathrm O(1)$ the orthogonal maps that preserve the line $\R e_0$. \begin{proposition}\label{prop:rotations} \textup{(i)} $G_S$ is a closed subgroup of $\mathrm O(4)$ that contains $\Gamma$, and its identity component is $\{1\}$, $\SO(3)_{\mathrm{sp}}$ or $\SO(4)$. \textup{(ii)} $G_S\supseteq\mathrm O(3)\times\mathrm O(1)$ if and only if $G_S$ contains a rotation of the spatial coordinates of infinite order. \textup{(iii)} $G_S=\mathrm O(4)$ if and only if $G_S$ contains a rotation $R$ of infinite order with $Re_0\notin\{\pm e_0\}$, and also if and only if $G_S$ is infinite and contains some $g$ with $ge_0\notin\{\pm e_0\}$. \textup{(iv)} If $G_S\supseteq B_4$, then $G_S$ is finite or equal to $\mathrm O(4)$, and it equals $\mathrm O(4)$ if and only if it contains an element of infinite order. One element of finite order outside $B_4$ does not suffice. \end{proposition} \begin{proof} (i) $G_S$ is a group. If $R_m\to R$ with $R_m\in G_S$, the functions $f\circ R_m^{-1}$ converge uniformly to $f\circ R^{-1}$ with supports in one cube, and the bound of Theorem~\ref{thm:osaxioms}(i) gives $R\in G_S$. So $G_S$ is a compact Lie group~\cite{Bum13}, and its Lie algebra $\mathfrak g\subset\mathfrak{so}(4)$ is invariant under $\mathrm{Ad}(g)$, $g\in\Gamma$. Let $E_{ij}=e_ie_j^{\mathsf T}-e_je_i^{\mathsf T}$. The diagonal sign changes $D_\epsilon$, $\epsilon\in\{\pm1\}^4$, lie in $\Gamma$, and $\mathrm{Ad}(D_\epsilon)E_{ij}=\epsilon_i\epsilon_jE_{ij}$. The six characters $\epsilon\mapsto\epsilon_i\epsilon_j$ are distinct, so the operators $2^{-4}\sum_\epsilon\epsilon_i\epsilon_j\mathrm{Ad}(D_\epsilon)$ project $\mathfrak g$ onto $\mathfrak g\cap\R E_{ij}$, and $\mathfrak g$ is spanned by the $E_{ij}$ it contains. The permutations of the spatial axes act transitively on the pairs $\{i,j\}\subset\{1,2,3\}$ and on the pairs $\{0,i\}$. So $\mathfrak g$ is $0$, $\mathfrak{so}(3)_{\mathrm{sp}}= \operatorname{span}\{E_{ij}:i,j\ge1\}$, $\operatorname{span}\{E_{0i}\}$ or $\mathfrak{so}(4)$, and the third is not a Lie algebra, since $[E_{01},E_{02}]=-E_{12}$. A connected Lie subgroup is determined by its Lie algebra. (ii) If $R\in\SO(3)_{\mathrm{sp}}$ has infinite order, $G_S$ is infinite, hence of positive dimension, and $G_S\cap(\mathrm O(3)\times\mathrm O(1))$ has the same property, so its identity component is a connected subgroup of $\SO(3)_{\mathrm{sp}}$ of positive dimension normalised by the signed permutations, and by the argument of (i), applied within $\mathfrak{so}(3)_{\mathrm{sp}}$, it is $\SO(3)_{\mathrm{sp}}$. With $\Gamma$ this gives $\mathrm O(3)\times\mathrm O(1)$. The converse is clear. (iii) If $G_S$ is infinite, its identity component is $\SO(3)_{\mathrm{sp}}$ or $\SO(4)$ by (i). In the first case it is normal in $G_S$, and every $g\in G_S$ maps the vectors fixed by $\SO(3)_{\mathrm{sp}}$, the line $\R e_0$, to the vectors fixed by $g\SO(3)_{\mathrm{sp}}g^{-1}=\SO(3)_{\mathrm{sp}}$, so $ge_0=\pm e_0$. Hence an infinite $G_S$ with some $ge_0\ne\pm e_0$ contains $\SO(4)$, and with $\theta$ it is $\mathrm O(4)$. A rotation $R$ of infinite order with $Re_0\ne\pm e_0$ makes $G_S$ infinite and supplies $g$; the converses are clear. (iv) The sign changes and all permutations of the four axes act on the pairs transitively, so $\mathfrak g$ is $0$ or $\mathfrak{so}(4)$. The reflection $s$ in the hyperplane orthogonal to $u=\frac12(1,1,1,1)$, with $B_4$, generates a group that permutes the $24$ vectors $\pm e_\mu$ and $\frac12(\pm1,\pm1,\pm1,\pm1)$: $s$ maps $e_\mu$ to $e_\mu-u$, and $v$ to $v-2\ip vuu$ with $\ip vu\in\{0,\pm\frac12,\pm1\}$ for the other vectors. These vectors span $\R^4$, so the group is finite. \end{proof} The spatial flow commutes with spatial rotations in the continuum, and in the free field it leaves exactly $\mathrm O(3)\times\mathrm O(1)$ (Proposition~\ref{prop:osobstruction}(c)). So for the spatially flowed fields at a positive flow time the symmetry to ask for is $\mathrm O(3)\times\mathrm O(1)$, and by part (ii) one rotation suffices: \begin{enumerate}[label=\textup{(W)},leftmargin=*] \item \emph{One spatial rotation.} There is a rotation $R$ of the spatial coordinates, of infinite order, with \[ \lim_{k\to\infty}\Bigl[\omega_k\Bigl(\prod_jZ_kA_k(f_j\circ R^{-1})\Bigr) -\omega_k\Bigl(\prod_jZ_kA_k(f_j)\Bigr)\Bigr]=0 \] for all $n$ and $f_j\in C_c(\R^4)$. \end{enumerate} Under the hypotheses of Theorem~\ref{thm:osaxioms}, (W) holds if and only if every subsequential limit is invariant under $\mathrm O(3)\times\mathrm O(1)$: (W) puts $R$ in each $G_S$; conversely, if the difference in (W) stayed above $\varepsilon$ along a subsequence for some $f_j$, a further subsequence would give a limit $S$ with $R\notin G_S$. Full Euclidean invariance, as (E) asks, is not available for these fields at a positive flow time; it has to be posed for observables whose regularisation is $\mathrm O(4)$-covariant, such as the four-dimensional flow, which at positive flow time mixes the two sides of the reflection plane (Sec.~\ref{ssec:flowlimit}), or the limit of vanishing flow time. For families invariant under $B_4$ part (iv) applies, and one rotation of infinite order gives $\mathrm O(4)$. In either case the rotation enters as a lattice statement: a Ward identity in which the breaking of the rotation is an insertion that vanishes in the limit, as the symmetry restoration of Symanzik's improvement programme predicts. It is not proved here. \emph{Convergence of the whole sequence.} \begin{enumerate}[label=\textup{(U)},leftmargin=*] \item \emph{One limit.} Any two subsequential limits of Theorem~\ref{thm:osaxioms}(i) coincide. \end{enumerate} \begin{proposition}\label{prop:uniquelimit} Assume the hypotheses of Theorem~\textup{\ref{thm:osaxioms}}. \textup{(i)} The moments $\omega_k(\prod_jZ_kA_k(f_j))$ converge for all $n$ and $f_j\in C_c(\R^4)$ if and only if \textup{(U)} holds. \textup{(ii)} Let $(\beta_k)$ and $(\beta'_k)$ satisfy the hypothesis of Theorem~\textup{\ref{thm:transfer}} with the same $\alpha,\ell$, and \textup{(M)} and \textup{(W)} with the same $M_n(Q)$ and $R$, and let $\beta''_k$ be $\beta_k$ for even $k$ and $\beta'_k$ for odd $k$. Then $(\beta''_k)$ satisfies the same hypotheses, and if the moments along $(\beta_k)$ and along $(\beta'_k)$ converge to different limits, those along $(\beta''_k)$ do not converge. \textup{(iii)} Suppose that every subsequential limit lies in a set $\mathfrak F$ of families on which $S\mapsto S_2(\bar f_0\circ\theta, f_0(\cdot-(\sigma_0,0)))$ is injective, for a dyadic $\sigma_0>0$, and that $\rho_k(0,\sigma_0)=C_k(\sigma_0)/C_k(0)$ converges. Then \textup{(U)} holds. \end{proposition} \begin{proof} (i) If a moment does not converge, it has two limit points along two subsequences, and refining each by Theorem~\ref{thm:osaxioms}(i) gives two different limits. The converse is clear. (ii) The hypotheses are conditions at each $k$ with constants uniform in $k$, and the even and the odd terms have different limits. (iii) Along a subsequence with limit $S$, $S_2(\bar f_0\circ\theta,f_0(\cdot-(\sigma_0,0)))=\lim Z_k^2C_k(\sigma_0)= \lim\rho_k(0,\sigma_0)$ by the approximation of $\theta$ in the proof of Theorem~\ref{thm:higher}(iii), the same number for every subsequence. \end{proof} Part (ii) shows that (U) is a property of the tuning that the other hypotheses do not contain. Part (iii) states it in the form that Yang--Mills theory requires: if every limit is determined up to the choice of one physical scale, a single renormalisation condition fixes that scale. For Yang--Mills theory this is the statement that the continuum theory has no free dimensionless parameter; it is not proved here. \emph{What these inputs cannot give.} Gaussian fields show the limits of the inputs. They are not lattice gauge theories, and the hypothesis of Theorem~\ref{thm:transfer} concerns the Wilson measure in the forest gauge; but the proofs of Theorems~\ref{thm:osaxioms} and~\ref{thm:ostree} use it only through the conclusions of Theorem~\ref{thm:onescale}, namely translation and lattice-symmetry invariance, reflection positivity and the transfer-matrix gap $m_*a_k$, and the fields below have these. \begin{proposition}\label{prop:osobstruction} Let $m>0$, $q_k=e^{-ma_k}$, and $f_0\ge0$. \textup{(a)} Let $\phi_k$ be the centred Gaussian field on $\Z^4$ with covariance $\prod_{\mu=0}^3q_k^{|x_\mu-y_\mu|}$, and let $A_k(f)=\sum_x a_k^4f(a_kx)\phi_k(x)$, with $C_k$, $Z_k$ formed from $f_0$ as before. Its law is invariant under translations and $B_4$ and reflection positive for the reflections in coordinate hyperplanes through sites and bisecting links; its transfer matrix in each coordinate direction has spectrum $\{q_k^N:N\ge0\}$ with $1$ a simple eigenvalue, so its gap is $ma_k$. For large $k$, \textup{(T1)--(T3)} hold with $r_k=1$, $w_k(z)=\prod_\mu q_k^{|z_\mu|}$, $K_2=1$ and $K_n=0$ for $n\ge3$. The moments converge along the whole sequence to the Gaussian family with $S_2(f,g)=\kappa\iint f(x)g(y)e^{-m\|x-y\|_1}dx\,dy$, $\kappa>0$. Its group $G_S$ is exactly $B_4$, its Hamiltonian has $\sigma(H)\subset\{0\}\cup[m,\infty)$ with $m\in\sigma(H)$, and the joint spectrum of $H$ and the spatial momentum contains $\{m\}\times\R^3$. \textup{(b)} Let $\psi_k$ be the centred Gaussian field on $\Z^4$ with covariance $q_k^{|x_0-y_0|}$, and form $A_k$, $C_k$, $Z_k$ in the same way. Its law is invariant under translations and $\Gamma$, reflection positive in time, and its transfer matrix has gap $ma_k$. \textup{(M)}, the bound of Theorem~\textup{\ref{thm:osaxioms}(vii)} and $(\mathrm N_0)$ hold, and the moments converge along the whole sequence; the limit has $U(x)=1$ for every $x\in\R^3$, so it does not cluster in space. \textup{(T3)} fails. \textup{(c)} For $t_f>0$ and $\mu>0$, the centred Gaussian field on $\R^4$ with covariance $\widehat C(p)=e^{-2t_f|\mathbf p|^2}(|p|^2+\mu^2)^{-1}$, the massive free field flowed by the spatial heat flow for the time $t_f$, is reflection positive and invariant under translations, and its group $G_S$ is exactly $\mathrm O(3)\times\mathrm O(1)$. \end{proposition} \begin{proof} (a) The covariance is a product of the one-dimensional covariances $q^{|n|}$, which are positive definite, their Fourier transforms being $(1-q^2)/(1-2q\cos p+q^2)>0$; invariance under translations and $B_4$ is clear. Fix a coordinate direction as time. The field at time $0$ has a Gaussian law $\gamma$ with covariance $\prod_{i\ge1}q^{|x_i-y_i|}$, and the stationary chain $\phi_{t+1}=q\phi_t+(1-q^2)^{1/2}\xi_{t+1}$ with $\xi_t$ independent of law $\gamma$ has covariance $q^{|t-s|}$ times that of $\gamma$, so it is the field. It is a reversible Markov chain with transition operator $Pg(\phi)=\E\,g(q\phi+(1-q^2)^{1/2}\xi)$ on $L^2(\gamma)$. For $h$ of finite support and $\mathcal E_h(\phi)= \exp(\phi(h)-\frac12\E\phi(h)^2)$ one computes $P\mathcal E_h=\mathcal E_{qh}$; the part of $\mathcal E_h$ homogeneous of degree $N$ in $h$ spans the $N$-th Wiener chaos, so $P$ is multiplication by $q^N$ on it, and these chaoses span $L^2(\gamma)$~\cite[Sect.~6.3]{GJ87}. So $0\le P\le1$ with spectrum $\{q^N\}$ and $1$ simple. For $F,F'$ functions of the field at times $\ge0$, the Markov property and reversibility give $\E[\vartheta(\bar F)F']=\ip{g_F}{g_{F'}}_{L^2(\gamma)}$ with $g_F=\E[F\mid\phi_0=\cdot]$, and for the reflection bisecting the links between times $0$ and $1$, with $F,F'$ at times $\ge1$, $\ip{g_F}{Pg_{F'}}$ with $g_F=\E[F\mid\phi_1=\cdot]$; both are positive for $F=F'$, and the transfer matrix is $P$. Gaussian fields have vanishing cumulants of order $n\ge3$, and the covariance is $w_k(x-y)$, which is (T1). Further $a_k^4W_k=a_k^4((1+q_k)/(1-q_k))^4\to(2/m)^4$, which is (T3), and since $q_k^{|n|}=e^{-ma_k|n|}$, $C_k(0)$ is a Riemann sum converging to $\iint f_0(x)f_0(y)e^{-m\|\theta x-y\|_1}dx\,dy>0$, which gives (T2) for large $k$ and $Z_k\to\kappa^{1/2}$ with $\kappa$ the inverse of this integral. The moments are sums of products of covariances, Riemann sums converging to the stated ones. Invariance of a Gaussian family under $R$ is invariance of $S_2$, that is $e^{-m\|Rz\|_1}=e^{-m\|z\|_1}$ for all $z$; a linear isometry of the norm $\|\cdot\|_1$ maps the extreme points $\pm e_\mu$ of its unit ball to themselves, so $R\in B_4$. The spectral bound follows from the lattice gap $ma_k$ as in Theorem~\ref{thm:osaxioms}(vi). For $f=h\otimes g$ with $h$ supported in $\{t>0\}$ and $\int he^{-mt}dt\ne0$, $\ip{\Psi(f)}{e^{-sH}U(x)\Psi(f)}$ equals $e^{-ms}$ times a function of $x$ whose Fourier transform is a positive multiple of $|\hat g(\mathbf p)|^2 \prod_{i=1}^32m/(p_i^2+m^2)$. So the joint spectral measure at $\Psi(f)$ is carried by $\{m\}\times\R^3$, with support $\{m\}\times\R^3$ when $\hat g$ has no zeros. (b) The field is $\psi_k(x)=X(x_0)$ for the chain of (a) with $\gamma=N(0,1)$, which gives the symmetry, reflection positivity and the gap. Write $F_f(t)=\int f(t,\mathbf x)\,d\mathbf x$. The covariance of two packets converges to $\iint F_f(t)F_g(s)e^{-m|t-s|}dt\,ds$, which is positive for $f=\bar f_0\circ\theta$, $g=f_0$, and bounded by $|Q|^2\|f\|_\infty\|g\|_\infty$ for $f,g$ supported in $Q$; with the Gaussian moment formula this gives (M) with $M_n(Q)$ a constant times $|Q|^n$, and convergence. The limiting covariance does not change when the test functions of one factor are translated in space, so neither does any moment, and $\ip{\Psi(P)}{U(x)\Psi(P')}=\ip{\Psi(P)}{\Psi(P')}$. The ratio $C_k(\sigma)/C_k(0)$ converges to a positive number, which is $(\mathrm N_0)$, and $\sum_zq_k^{|z_0|}=\infty$. (c) For $f$ supported in $\{t>0\}$, the $p_0$ integral gives \begin{multline*} \iint\bar f(\theta x)C(x-y)f(y)\,dx\,dy\\ =\int e^{-2t_f|\mathbf p|^2} (2\omega)^{-1}\Bigl|\int\hat f(t,\mathbf p)e^{-\omega t}dt\Bigr|^2 d\mathbf p\ge0, \end{multline*} with $\omega=(|\mathbf p|^2+\mu^2)^{1/2}$ and $\hat f$ the spatial Fourier transform; reflection positivity of the Gaussian measure follows~\cite[Sect.~6.2]{GJ87}. An orthogonal $R$ preserves $\widehat C$ if and only if $|(Rp)^{\mathrm{sp}}|=|p^{\mathrm{sp}}|$ for every $p$, the spatial parts, because $|Rp|=|p|$ and $s\mapsto e^{-2t_fs}$ is injective; that is, $R$ preserves $|p_0|$, so $R^{\mathsf T}e_0=\pm e_0$. \end{proof} Example (a) is reflection positive, has the gap in every direction and the lattice symmetry, satisfies the tree bounds with the best constants, and converges; its limit is invariant under no rotation outside $B_4$, so (W) does not follow from these properties. Its one-particle energy $m$ does not depend on the momentum, so the spectral condition of the Wightman axioms, energy at least $|\mathbf p|$, fails; the gap transfers to $H\ge m_*$ on $\Omega^\perp$ and to the joint spectrum of Theorem~\ref{thm:osaxioms}(vi), and the spectral condition needs Lorentz covariance, which only $\mathrm O(4)$ invariance supplies. Example (b) shows that the time gap and (M) do not give spatial clustering. By Proposition~\ref{prop:uniquelimit}(ii), example (a) at two masses, interleaved, satisfies the hypotheses of Theorems~\ref{thm:osaxioms} and~\ref{thm:ostree} with uniform constants and has two limits. \begin{corollary}\label{cor:osexact} Assume the hypothesis of Theorem~\textup{\ref{thm:transfer}}, with $\Phi$ real and $W_3$-invariant. The moments $\omega_k(\prod_jZ_kA_k(f_j))$ converge for every $n$ and all $f_j\in C_c(\R^4)$ to a family invariant under translations and $\mathrm O(3)\times\mathrm O(1)$ if and only if \textup{(M)}, \textup{(W)} and \textup{(U)} hold. The limit then satisfies \textup{(i)--(vi)} of Theorem~\textup{\ref{thm:osaxioms}} with $\Gamma$ replaced by $\mathrm O(3)\times\mathrm O(1)$: symmetric, reflection positive distributions, a Hamiltonian with $\sigma(H)\subset\{0\}\cup [m_*,\infty)$, $\ker H=\C\Omega$ and $\Omega^\perp\ne0$, and unitary representations of the spatial translations and rotations commuting with $H$. If \textup{(T1)--(T3)} hold as well, the limit is tempered and clusters in every direction, and it satisfies the linear growth condition of~\cite{OS75} when $K_n\le K^n(n!)^\beta$. \end{corollary} \begin{proof} Convergence gives bounded moments, hence (M) by Theorem~\ref{thm:osaxioms}; it gives (U), and with the invariance it gives (W) for every spatial rotation. Conversely, (M) and (U) give convergence by Theorem~\ref{thm:osaxioms}(i) and Proposition~\ref{prop:uniquelimit}(i), and (W) gives the invariance by Proposition~\ref{prop:rotations}(ii). The representation of the spatial rotations is constructed as $V$ in Theorem~\ref{thm:osaxioms}(iv), and its continuity follows as there. The rest is Theorems~\ref{thm:osaxioms} and~\ref{thm:ostree}. \end{proof} So, given (D) along the tuning, the part of (E) that concerns the spatially flowed fields consists of three statements: (M), an ultraviolet bound on the products of renormalised packets that contains $(\mathrm N_0)$ and therefore fails along every tuning that stays in the strong-coupling range (Theorem~\ref{thm:strongratio}); (W), the restoration of the spatial rotations; and (U), the uniqueness of the limit along the tuning. Each is necessary. Example (a) shows that (W) does not follow from (M), (U), the tree bounds and the conclusions of Theorem~\ref{thm:onescale}, and the same example at two interleaved masses shows that (U) does not follow from (M), the tree bounds and those conclusions. None of the three is proved. Euclidean invariance and the Wightman spectral condition are beyond these fields at a positive flow time. \emph{Three further routes.} The trial condition of Corollary~\ref{cor:trial} can be approached in three other ways: through upper bounds on correlations, through trial states smeared in time, and through a comparison with the Gaussian theory at weak coupling. The first never implies $(\mathrm N_u)$. The second removes the lattice scale from the first condition and leaves a lower bound on a ratio of correlations at physical times. The third gives $(\mathrm N_u)$ from a relative comparison at two physical separations; when that comparison is built scale by scale along the one-loop tuning, it must be exact to first order in the running coupling at every scale. \begin{proposition}\label{prop:upperbounds} Let $G=\SU(2)$, let $\beta_{W,k}\in[0,0.348]$ be any couplings, let $\omega_k$ be the states of Theorems~\ref{thm:su2} and~\ref{thm:blocks}, and let $m_0$ be as in Theorem~\ref{thm:strongratio}. \textup{(i)} Let $F,H$ be bounded functions at positive times that are uniform limits of gauge-invariant local functions, with $\omega_k(F)=\omega_k(H)=0$. Then for every integer $n\ge0$ \[ \bigl|\omega_k\bigl(\vartheta(\bar F)\tau_nH\bigr)\bigr|\le e^{-m_0n}\|\Psi_F\|\,\|\Psi_H\|\le e^{-m_0n}\|F\|_\infty\|H\|_\infty . \] \textup{(ii)} Let \textup{(H)} be a set of inequalities indexed by $i\in I$, each of the form \[ \bigl|\omega_k\bigl(\vartheta(\bar F_{i,k})\tau_{n_{i,k}}H_{i,k}\bigr)\bigr| \le B_{i,k}\|\Psi_{F_{i,k}}\|\,\|\Psi_{H_{i,k}}\|\qquad(k\ge k_i), \] with $F_{i,k},H_{i,k}$ as in \textup{(i)}, integers $n_{i,k}\ge t_i/a_k$ for some $t_i>0$, and $B_{i,k}>0$ with $a_k\log B_{i,k}\to0$ as $k\to\infty$. Then each inequality of \textup{(H)} holds for the states $\omega_k$ for all large $k$, while $(\mathrm N_u)$ fails for every packet with $C_k(0)>0$ and every $u$, and every limit of Theorem~\textup{\ref{thm:flowlimit}} along these states is trivial. The same holds with $\|F_{i,k}\|_\infty\|H_{i,k}\|_\infty$ in place of $\|\Psi_{F_{i,k}}\|\|\Psi_{H_{i,k}}\|$. \end{proposition} \begin{proof} (i) Since $\ip\Omega{\Psi_F}=\omega_k(F)=0$ and likewise for $H$, both vectors lie in $\Omega_k^\perp$, on which $\mathbb T_k$ has norm at most $e^{-m_0}$, as in the proof of Theorem~\ref{thm:strongratio}. Hence $|\ip{\Psi_F}{\mathbb T_k^n\Psi_H}|\le e^{-m_0n}\|\Psi_F\|\|\Psi_H\|$, and $\|\Psi_F\|\le\|F\|_\infty$ by the footnote of Appendix~\ref{app:transfer}. (ii) Since $a_k\log B_{i,k}\to0$, for large $k$ we have $a_k\log B_{i,k}\ge-m_0t_i$, that is $B_{i,k}\ge e^{-m_0t_i/a_k}\ge e^{-m_0n_{i,k}}$, and (i) gives the inequality. The statements on $(\mathrm N_u)$ and on the limits are Theorem~\ref{thm:strongratio}. \end{proof} The class (H) contains uniform bounds $B_{i,k}=B_i$, decay at a physical rate $B_{i,k}=B_ie^{-Mt_i}$ with any $M$, and bounds of order $a_k^p$, for connected correlations of any centred observables at physical separations, flowed or not. The same holds for every compact simple $G$ at couplings in $[0,\beta_1]$ with $N\beta_1<\frac2{29}\kappa_G$, with $m_0=m_G(\beta_1)$ of Theorem~\ref{thm:groups}. So a proof of $(\mathrm N_u)$ must contain a lower bound. In Corollary~\ref{cor:trial} it is $\ip{\chi_k}{\mathbb T_k\chi_k}\ge 1-\kappa a_k$, a lower bound on a normalised correlation at one lattice step that is accurate to order $a_k$, together with a lower bound on one overlap. The spectral arguments of Lemmas~\ref{lem:trial} and~\ref{lem:nonvanishing} turn such a lower bound into a bound on the ratio, but they cannot produce one from upper bounds: the states above satisfy every inequality of (H), and the spectral measures of their centred vectors are carried by energies at least $m_0/a_k$. \emph{Trial states smeared in time.} A slice function enters Lemma~\ref{lem:slabdirichlet} through its change in one lattice step, which Proposition~\ref{prop:lipobstruction} shows to be too large. Smearing the same function in time over a physical interval replaces that change by a telescoping sum, in which a lattice step costs $a_k$ times the total variation of the time profile. Let $\omega$ be a state as in Appendix~\ref{app:transfer}, with transfer matrix $\mathbb T$ and vacuum $\Omega$. Let $X$ be a real bounded function of the spatial links of the slice $x_0=0$, invariant under the gauge transformations of that slice and a uniform limit of local such functions; by Proposition~\ref{prop:spatialflow}(ii) this includes $X=N^{-1}\ReTr W_\gamma\circ\Phi_s$ for the holonomy $W_\gamma$ of a closed path $\gamma$ in the slice and every flow time $s$. Put $X^c=X-\omega(X)$ and \[ D_X(n)=\omega\bigl(X^c\,\tau_nX^c\bigr)\qquad(n\in\Z_{\ge0}). \] Let $w=(w_n)_{n\in\Z}$ be nonnegative, with $w_n=0$ unless $n_0\le n\le n_1$, where $1\le n_0\le n_1$, and with $\|w\|_1=\sum_nw_n>0$; put $V(w)=\sum_{n\in\Z}|w_n-w_{n-1}|$ and \[ B=\sum_nw_n\,\tau_nX^c . \] \begin{proposition}\label{prop:smearedtrial} \textup{(i)} $D_X$ is nonnegative, nonincreasing and log-convex on $\Z_{\ge0}$, and $\Psi_B\perp\Omega$. If $D_X(2n_1)>0$, then $\|\Psi_B\|^2\ge\|w\|_1^2D_X(2n_1)>0$, and the unit vector $\chi=\Psi_B/\|\Psi_B\|$ satisfies \[ \ip\chi{(1-\mathbb T)\chi}\le\frac{V(w)}{\|w\|_1}\, \frac{D_X(2n_0)}{D_X(2n_1)} . \] \textup{(ii)} If $D_X(2n_1)>0$ and $\ip\chi{(1-\mathbb T)\chi}\le\eta\le \frac12$, then $D_X(2n_0+m)\ge e^{-2\eta m}D_X(2n_1)$ for every integer $m\ge0$. \textup{(iii)} For a function $H$ at positive times as in Appendix~\textup{\ref{app:transfer}}, $\ip{\Psi_B}{\Psi_H}= \omega(\vartheta(B)H)$. \end{proposition} \begin{proof} Let $x=\Psi_{X^c}$. As in the proof of Lemma~\ref{lem:slabdirichlet}(i), $\vartheta(X^c)=X^c$ and $\Psi_{\tau_nX^c}=\mathbb T^nx$, so $D_X(n)=\ip x{\mathbb T^nx}=\int\lambda^nd\nu$ for the spectral measure $\nu$ of $\mathbb T$ at $x$, a finite positive measure on $[0,1]$. Hence $D_X\ge0$, $D_X$ is nonincreasing, and it is log-convex by H\"older's inequality. Since $\ip\Omega x=\omega(X^c)=0$ and $\mathbb T\Omega=\Omega$, $x\perp\Omega$ and so $\Psi_B=W(\mathbb T)x\perp\Omega$, with $W(\lambda)=\sum_nw_n\lambda^n$. (i) On $[0,1]$, $\|w\|_1\lambda^{n_1}\le W(\lambda)\le\|w\|_1\lambda^{n_0}$, so $\|\Psi_B\|^2=\int W^2d\nu\ge\|w\|_1^2D_X(2n_1)$. Moreover \[ W(\lambda)(1-\lambda)=\sum_{n=n_0}^{n_1+1}(w_n-w_{n-1})\lambda^n, \] which is nonnegative and at most $V(w)\lambda^{n_0}$ on $[0,1]$. Therefore \[ \begin{split} \ip{\Psi_B}{(1-\mathbb T)\Psi_B}&=\int W(\lambda)\cdot W(\lambda) (1-\lambda)\,d\nu\\ &\le\|w\|_1V(w)D_X(2n_0), \end{split} \] and dividing by $\|\Psi_B\|^2$ gives the bound. (ii) Let $\nu_\chi=W^2\nu/\int W^2d\nu$ be the spectral measure at $\chi$. By Jensen's inequality for $\lambda^m$, $\int\lambda^md\nu_\chi\ge (1-\eta)^m\ge e^{-2\eta m}$, since $\log(1-y)\ge-2y$ on $[0,\frac12]$. By the bounds on $W$, \[ \int\lambda^md\nu_\chi=\frac{\int W^2\lambda^md\nu}{\int W^2d\nu}\le \frac{D_X(2n_0+m)}{D_X(2n_1)} . \] (iii) This is the identity of Appendix~\ref{app:transfer}, with $\vartheta(\bar B)=\vartheta(B)$ because $B$ is real. \end{proof} Along a tuning, take $X_k$ as above for $\omega_k$, a time profile $g\ge0$ of bounded variation supported in $[t_0,t_1]$ with $00$, and $w_n=a_kg(a_kn)$, so that $B_k=\sum_na_kg(a_kn)\,\tau_nX_k^c$ is the packet of $X_k$ with time profile $g$; let $\chi_k$ be its unit vector, $\Var(g)$ the total variation of $g$ on $\R$, and write $D_{X_k}(r)$ for $D_{X_k}(r/a_k)$ when $r\in a_k\Z_{\ge0}$ is a physical time. \begin{corollary}\label{cor:smearedtrial} \textup{(i)} If $D_{X_k}(2t_1)\ge c\,D_{X_k}(2t_0)$ with $c>0$ for all large $k$, then $\ip{\chi_k}{(1-\mathbb T_k)\chi_k}\le\kappa a_k$ for all large $k$, with $\kappa=2\Var(g)/(c\int g)$. The hypothesis holds with $c=c_1^{\max(1,(2t_1-2t_0)/(r'-r))}$ if $D_{X_k}(r')\ge c_1D_{X_k}(r)$ for dyadic $02(t_1-t_0)$ this is a lower bound on the ratio of two correlations of $X_k$ at the physical times $2t_1<2t_0+s$. \textup{(iii)} If the hypothesis of \textup{(i)} holds and \[ \bigl|\omega_k\bigl(\vartheta(B_k)\,\tau_{u/a_k}A_k\bigr)\bigr|\ge v\,\|\Psi_{B_k}\|\,C_k(2u)^{1/2} \] with $v>0$ for all large $k$, then $(\mathrm N_u)$ holds and $\rho_k(u,\sigma)\ge(v-\sqrt{\kappa\sigma/2})_+^2$ for $\sigma\ge2a_k$. \end{corollary} \begin{proof} For large $k$, $t_0/a_k$ and $t_1/a_k$ are integers, and Proposition~\ref{prop:smearedtrial} applies with $n_0=t_0/a_k$ and $n_1=t_1/a_k$; $n_0\ge1$. The points $a_kn$ partition $\R$, so $V(w)=a_k\sum_n|g(a_kn)-g(a_k(n-1))|\le a_k\Var(g)$, while $\|w\|_1=a_k\sum_ng(a_kn)$ is a Riemann sum of $g$, which is Riemann integrable, and so is at least $\frac12\int g$ for large $k$. Then (i) follows from Proposition~\ref{prop:smearedtrial}(i); the condition $D_{X_k}(2t_1)>0$ holds because $D_{X_k}(2t_0)>0$ is implicit in the hypothesis. Its second sentence is the log-convexity argument of Proposition~\ref{prop:sliceratio}(ii) with $\alpha=2t_0$, $\beta=2t_1$, which uses only that $D_{X_k}$ is nonincreasing and log-convex. For (ii), $\kappa a_k\le\frac12$ for large $k$, and Proposition~\ref{prop:smearedtrial}(ii) with $m=s/a_k$ gives the claim. For (iii), Proposition~\ref{prop:smearedtrial}(iii) gives $|\ip{\chi_k}{\Psi_{k,u}}|\ge v$, and Corollary~\ref{cor:trial} gives the rest. \end{proof} For trial states smeared in time, the first condition of Corollary~\ref{cor:trial} therefore holds along a tuning if and only if, up to the choice of the physical times, a ratio of two correlations of the smeared field at physical times is bounded below. No lattice-scale quantity enters: no Lipschitz constant, no plaquette energy, and no first step. Proposition~\ref{prop:lipobstruction} concerns single slices and does not apply to $B_k$. When $\Phi$ is real, $X_k$ is the slice field $U_k$ of a product packet $f_0=g\otimes h$ and $u=0$, $B_k=A_k$, the overlap is $1$, and part (i) gives the hypothesis of Proposition~\ref{prop:flowshift}(ii) from the slice ratio of Proposition~\ref{prop:sliceratio}(iii), without the comparison of $u_k$ with the packet that was left open after Proposition~\ref{prop:flowshift}. For a flowed loop $X_k=N^{-1}\ReTr W_{\gamma_k}\circ\Phi_{\tau/a_k^2}$, with $\gamma_k$ the lattice approximation of a closed curve of physical size in the slice, the input that the route needs is the pair of lower bounds of Corollary~\ref{cor:smearedtrial}(i) and~(iii): a ratio of two flowed-loop correlations at physical times, and the overlap of the smeared loop with the flowed packet at physical distance. Both are lower bounds at physical separations, of the same kind as $(\mathrm N_u)$, and by Proposition~\ref{prop:upperbounds} no upper bound supplies them. The overlap vanishes for every $k$ if $\Psi_{B_k}$ and $\Psi_{A_k}$ lie in different eigenspaces of a unitary that commutes with $\mathbb T_k$; for instance, if $G=\SU(N)$ with $N\ge3$ and $\omega_k$ is invariant under charge conjugation $U\mapsto\bar U$, then $X_k=N^{-1}\operatorname{Im}\Tr W_{\gamma_k}\circ\Phi_{\tau/a_k^2}$ is odd and the flowed energy packet is even, so $X_k$ has to be chosen in the symmetry sector of the packet. \emph{Comparison with the Gaussian theory.} Let $C^{\mathrm G}_k(t)>0$ be the correlations of the corresponding packet in a comparison theory, such as the Gaussian theory at tree level, with ratio $\rho^{\mathrm G}_k=C^{\mathrm G}_k(2u+\sigma)/C^{\mathrm G}_k(2u)$. If there are $Z_k>0$ and $\delta_k\in[0,1)$ such that for $t=2u$ and $t=2u+\sigma$ \[ (1-\delta_k)Z_kC^{\mathrm G}_k(t)\le C_k(t)\le(1+\delta_k)Z_kC^{\mathrm G}_k(t), \] then dividing the lower bound at $2u+\sigma$ by the upper bound at $2u$ gives \[ \rho_k(u,\sigma)\ge\frac{1-\delta_k}{1+\delta_k}\,\rho^{\mathrm G}_k . \] The normalisation $Z_k$ cancels, and only the two physical separations $2u$ and $2u+\sigma$ enter. For the free fields of Remarks~\ref{rem:freeslice} and~\ref{rem:compositeslice}, the slice ratios have positive limits, so $\liminf_k\rho^{\mathrm G}_k>0$ by the argument of Proposition~\ref{prop:sliceratio}(iii), which uses only reflection positivity and the transfer matrix, and $(\mathrm N_u)$ follows from $\limsup_k\delta_k<1$. This relative comparison at two physical separations, uniform in $k$, is what a perturbative lower bound has to provide. It is not a statement of fixed order in the bare coupling: along the one-loop tuning $b\log(\sigma/a_k)/\beta_k\to 1$, so the terms $(\log(\sigma/a_k)/\beta_k)^n$ of bare perturbation theory are of order one at every $n$. The comparison has to be organised by scales, as in Proposition~\ref{prop:formstability}, where repeating a comparison over scales multiplies the factors $1\pm\varepsilon$. The next proposition determines which losses per scale the one-loop tuning allows. \begin{proposition}\label{prop:scalelosses} Let $b>0$, $h=b\log2$, $a_k=2^{-k}a_0$, and let the couplings satisfy $|\beta_k-b\log(1/a_k)|\le M$ for all $k$. Let $\sigma>0$ be dyadic with $\beta^-_\sigma=b\log(1/\sigma)-M>0$, put $\beta^+_\sigma=b\log(1/\sigma)+M$, and for $k$ with $\sigma/a_k=2^{J_k}$, $J_k\in\Z_{\ge0}$, put \[ \beta^{(j)}_k=\beta_k-hj\qquad(0\le j\le J_k), \] the one-loop coupling at the scale $2^ja_k$. Let $c_{k,j}\in[0,\frac12]$. \textup{(i)} $\beta^-_\sigma+h(J_k-j)\le\beta^{(j)}_k\le\beta^+_\sigma+ h(J_k-j)$. \textup{(ii)} If $c_{k,j}\ge c/\beta^{(j)}_k$ for all $j$, with $c>0$, then \[ \prod_{j=0}^{J_k}(1-c_{k,j})\le\Bigl(\frac{\beta^+_\sigma}{\beta^+_\sigma+ b\log(2\sigma/a_k)}\Bigr)^{c/h}, \] which tends to $0$ as $k\to\infty$. \textup{(iii)} If $c_{k,j}\le c/(\beta^{(j)}_k)^2$ for all $j$, then for every $k$ \[ \prod_{j=0}^{J_k}(1-c_{k,j})\ge\exp\Bigl(-2c\Bigl(\frac1{(\beta^-_\sigma)^2} +\frac1{h\beta^-_\sigma}\Bigr)\Bigr). \] More generally, $\prod_j(1-c_{k,j})\ge e^{-2\sum_jc_{k,j}}$, and the products are bounded below uniformly in $k$ if and only if the sums $\sum_jc_{k,j}$ are bounded. For $\SU(2)$ and the one-loop tuning, $b=11/(3\pi^2)$ and $h\in[0.25751,0.25752]$. \end{proposition} \begin{proof} (i) Since $\log(1/a_k)=\log(1/\sigma)+J_k\log2$, the hypothesis gives $\beta^-_\sigma+hJ_k\le\beta_k\le\beta^+_\sigma+hJ_k$; subtract $hj$. (ii) By $1-y\le e^{-y}$ and (i), with $i=J_k-j$, \[ \begin{split} \sum_{j=0}^{J_k}c_{k,j}&\ge c\sum_{i=0}^{J_k}\frac1{\beta^+_\sigma+hi}\ge c\int_0^{J_k+1}\frac{dt}{\beta^+_\sigma+ht}\\ &=\frac ch\log\frac{\beta^+_\sigma+h(J_k+1)}{\beta^+_\sigma}, \end{split} \] since the integrand decreases, and $h(J_k+1)=b\log(2\sigma/a_k)\to\infty$. (iii) Since $\log(1-y)\ge-2y$ on $[0,\frac12]$, $\prod_j(1-c_{k,j})\ge e^{-2\sum_jc_{k,j}}$, and $\prod_j(1-c_{k,j})\le e^{-\sum_jc_{k,j}}$ gives the converse of the last statement. By (i), \[ \sum_{j=0}^{J_k}\frac1{(\beta^{(j)}_k)^2}\le\sum_{i\ge0}\frac1{(\beta^-_\sigma +hi)^2}\le\frac1{(\beta^-_\sigma)^2}+\int_0^\infty\frac{dt}{(\beta^-_\sigma+ ht)^2}, \] and the integral is $1/(h\beta^-_\sigma)$. For $\SU(2)$, $h=b\log2=11\log2/ (3\pi^2)$, and $0.6931471805<\log2<0.6931471806$, from the series of $2\operatorname{artanh}\frac13$, with $3.1415926535<\pi<3.1415926536$ give $h\in[0.2575118,0.2575119]$, so $4/h<15.534$. \end{proof} With $g_j^2=4/\beta^{(j)}_k$, part (ii) says that a scale-by-scale comparison that loses a factor $1-c\,g_j^2$ at every scale between $a_k$ and $\sigma$, however small $c>0$, gives a lower bound on $\rho_k(u,\sigma)$ that tends to zero, at the rate $(\log(1/a_k))^{-4c/h}$, with $4/h<15.534$;\footnote{Since $\beta_k=b\log(1/a_k)+O(1)$, this rate is a constant times $(g_0^2)^{4c/h}$ with $g_0^2=4/\beta_k$ the bare coupling: a power of the coupling at the lattice scale, which is not small at any fixed order of perturbation theory in $g_0^2$ but tends to zero along the tuning.} such a bound carries no information in the limit. Part (iii) says that losses of second order in the running coupling give $\rho_k(u,\sigma)\ge \rho^{\mathrm G}_ke^{-2c(1/(\beta^-_\sigma)^2+1/(h\beta^-_\sigma))}$ uniformly in $k$, a bound that approaches $\rho^{\mathrm G}_k$ as $\sigma$ decreases. So a perturbative lower bound on the ratio along the one-loop tuning has to account exactly for the first-order effect of every scale on the ratio, leaving errors whose sum over the scales is bounded. A first-order effect common to $C_k(2u)$ and $C_k(2u+\sigma)$ cancels in the ratio, and the proof has to exhibit that cancellation; bounding each correlation separately to first order is not enough. This is the summability of the one-step errors in Proposition~\ref{prop:formstability}, now for the ratio, and the effective actions it requires are item~(iv) of Sec.~\ref{ssec:remains}, the renormalisation group analysis of Ba{\l}aban~\cite{Bal84a,Bal85a,Bal89b}, which is not carried out here. None of the three routes gives $(\mathrm N_u)$ along the one-loop tuning without an input of this kind: a lower bound on correlations at physical separations, for the packet, for a smeared field that overlaps it, or for the Gaussian comparison. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_scalelosses.png} \caption{Products of the factors $1-c_j$ over the $J+1$ scales between $a_k$ and $\sigma$ along the one-loop $\SU(2)$ tuning, with $\beta_W=20$ at the scale $\sigma$, as in Proposition~\ref{prop:scalelosses}. (a) Losses of first order in the running coupling, $c_j=c\,g_j^2$, drive the product to zero for every $c>0$; losses of second order, $c_j=c\,g_j^4$, keep it bounded below. (b) The section $c=0.1$, with the bound of part (ii) dashed.} \label{fig:scalelosses} \end{figure*} \emph{Reductions of (M), (W) and (U).} The rest of this subsection reduces each of the three statements to a lattice statement and determines what the bounds of this paper give for it. The constant $Z_k$ is fixed by the reflected norm $C_k(0)$, while the moments in (M) are not reflected. Let \[ V_k=\omega_k\bigl(|A_k|^2\bigr),\qquad \Upsilon_k=Z_k^2V_k=\frac{V_k}{C_k(0)} , \] the variance of the packet of $f_0$ and its ratio to the reflected norm, and consider: \begin{enumerate}[label=\textup{(M$_{\mathrm{loc}}$)},leftmargin=*] \item For every $n\ge1$ and every cube $Q$ there are $M^{\mathrm{loc}}_n(Q)<\infty$ and $k_0$ with \[ \Bigl|\omega_k\Bigl(\prod_{j=1}^nA_k(f_j)\Bigr)\Bigr|\le M^{\mathrm{loc}}_n(Q)\,V_k^{n/2}\prod_{j=1}^n\|f_j\|_\infty \] for $k\ge k_0$ and $f_j\in C_c(\R^4)$ supported in $Q$. \end{enumerate} \begin{enumerate}[label=\textup{(M$_Z$)},leftmargin=*] \item $\sup_k\Upsilon_k<\infty$. \end{enumerate} Statement (M$_{\mathrm{loc}}$) is (M) with $Z_k$ replaced by the local normalisation $V_k^{-1/2}$; it does not change when $\Phi$ is multiplied by a constant, and it does not involve the reflection. \begin{proposition}\label{prop:msplit} Let the states $\omega_k$ be invariant under translations and under $\vartheta$ and reflection positive, and let $\Phi$ be real. \textup{(i)} $\Upsilon_k\ge1$, and for $f_j$ supported in a cube $Q$ \[ \Bigl|\omega_k\Bigl(\prod_{j=1}^nZ_kA_k(f_j)\Bigr)\Bigr|\le \bigl(2\|\Phi\|_\infty Z_k|Q|_k\bigr)^n\prod_{j=1}^n\|f_j\|_\infty . \] In particular \textup{(M)} holds if $\sup_kZ_k<\infty$. \textup{(ii)} \textup{(M)} holds if and only if \textup{(M$_{\mathrm{loc}}$)} and \textup{(M$_Z$)} hold. Precisely, \textup{(M)} gives \textup{(M$_{\mathrm{loc}}$)} with $M^{\mathrm{loc}}_n=M_n$ and $\Upsilon_k\le M_2(Q_0)\|f_0\|_\infty^2$ for a cube $Q_0$ containing the support of $f_0$, and \textup{(M$_{\mathrm{loc}}$)} and \textup{(M$_Z$)} give \textup{(M)} with $M_n=M^{\mathrm{loc}}_n\sup_k\Upsilon_k^{n/2}$. \textup{(iii)} Assume \textup{(T1)}. Then \textup{(T2)} holds if and only if \textup{(M$_Z$)} holds and $V_k\ge c'r_k^2a_k^4W_k$ for a $c'>0$ and all $k$. If \textup{(T1)}, \textup{(T3)} and this lower bound on $V_k$ hold, then \textup{(M$_{\mathrm{loc}}$)} holds. \textup{(iv)} Let $f_0=g\otimes h$ be a product packet as in Proposition~\textup{\ref{prop:sliceratio}}, choose an interval $I$ of length $\ell_I>0$ and $c_g>0$ with $g\ge c_g$ on $I$, and put \[ \begin{split} &\kappa_g=\tfrac12c_g^2\ell_I(1-e^{-1}),\quad \gamma_g=(t_1-t_0+\ell_I)\|g\|_\infty,\\ &\Lambda_g=\max\Bigl(\frac4{\ell_I},\frac1{2t_0}\Bigr),\quad \varepsilon_g(\Lambda)=\frac{\gamma_g^2\Lambda e^{-2\Lambda t_0}} {\kappa_g}. \end{split} \] Assume that the states have the transfer matrices of Appendix~\textup{\ref{app:transfer}}. If $a_k\le\ell_I/4$, then for every $\Lambda\ge\Lambda_g$ and every $\sigma\in a_k\Z_{\ge0}$ \[ \rho_k(0,\sigma)\ge e^{-\Lambda\sigma}\bigl(1-\varepsilon_g(\Lambda) \Upsilon_k\bigr). \] Hence \textup{(M$_Z$)} implies $(\mathrm N_u)$ for every dyadic $u\ge0$. \textup{(v)} Let $G=\SU(2)$, $\beta_{W,k}\in[0,0.348]$ and $\omega_k$ the states of Theorem~\textup{\ref{thm:strongratio}}, and let $f_0$ be as in \textup{(iv)}. Then for $a_k\le\min(\ell_I/4,m_0/(2\Lambda_g))$ \[ \Upsilon_k\ge\bigl(1-e^{-m_0/2}\bigr)\frac{2a_k\kappa_g}{m_0\gamma_g^2} \,e^{m_0t_0/a_k} . \] \textup{(vi)} Let $m>0$, $q_k=e^{-ma_k}$, let $\psi_k$ be the field of Proposition~\textup{\ref{prop:osobstruction}(a)}, $\xi_k$ a family of independent standard normal variables on $\Z^4$ independent of $\psi_k$, $\phi_k=\psi_k+a_k^{-5/2}\xi_k$, $A_k(f)=\sum_xa_k^4f(a_kx)\phi_k(x)$, and $f_0\ge0$ nonzero. The law of $\phi_k$ is invariant under translations and $B_4$ and reflection positive for the reflections in coordinate hyperplanes through sites and bisecting links, its transfer matrix in each coordinate direction has gap $ma_k$, $\rho_k(u,\sigma)=e^{-m\sigma}$ for all $u,\sigma$, and $Z_k$ converges; but $\Upsilon_k\to\infty$, so \textup{(M)} fails for $n=2$. \end{proposition} \begin{proof} (i) Since $\vartheta$ is an automorphism with $\overline{\vartheta(F)}= \vartheta(\bar F)$ and $\omega_k$ is $\vartheta$-invariant, $\omega_k(|\vartheta(\bar A_k)|^2)=\omega_k(\vartheta(|A_k|^2))=V_k$, and the Cauchy--Schwarz inequality gives $C_k(0)\le V_k$. Further $|A_k(f)|\le2\|\Phi\|_\infty a_k^4\#\{x:a_kx\in Q\}\|f\|_\infty$. (ii) Since $\Phi$ is real, $A_k(\bar f_0)=\overline{A_k(f_0)}$, and (M) with $n=2$, $f_1=\bar f_0$ and $f_2=f_0$ is the bound on $\Upsilon_k$. By (i), $Z_k^{-n}=C_k(0)^{n/2}\le V_k^{n/2}$, which gives (M$_{\mathrm{loc}}$). Conversely, since $Z_k^nV_k^{n/2}=\Upsilon_k^{n/2}$, \[ \Bigl|\omega_k\Bigl(\prod_jZ_kA_k(f_j)\Bigr)\Bigr|\le M^{\mathrm{loc}}_n\Upsilon_k^{n/2}\prod_j\|f_j\|_\infty . \] (iii) The packets are centred, so $V_k$ is the joint cumulant of $A_k(\bar f_0)$ and $A_k(f_0)$, and Lemma~\ref{lem:treesum} gives $V_k\le K_2r_k^2|Q_0|_ka_k^4W_k\|f_0\|_\infty^2$. If (T2) holds, then $V_k\ge C_k(0)\ge c\,r_k^2a_k^4W_k$ and $\Upsilon_k\le K_2|Q_0|_k \|f_0\|_\infty^2/c$, which is bounded since $|Q_0|_k\le(L+a_0)^4$ for a cube of side $L$. Conversely $C_k(0)=V_k/\Upsilon_k\ge(c'/\sup_j\Upsilon_j) r_k^2a_k^4W_k$. For (M$_{\mathrm{loc}}$), Lemma~\ref{lem:treesum} and the lower bound give for a cumulant of order $p\ge2$ \[ \begin{split} &\bigl|\omega^T_k\bigl(A_k(f_1);\dots;A_k(f_p)\bigr)\bigr|\\ &\quad\le K_pp^{p-2}c'^{-p/2}B_*^{\,p/2-1}|Q|_k\,V_k^{p/2} \prod_{j=1}^p\|f_j\|_\infty , \end{split} \] using (T3), and a moment is the sum over the partitions of $\{1,\dots,n\}$ without singletons of the products of the cumulants of the blocks, each partition contributing a factor $V_k^{n/2}$. (iv) We use the notation of the proof of Proposition~\ref{prop:sliceratio}: $A_k=\sum_nw_n\tau_nU_k$ with $w_n=a_kg(a_kn)$, $\nu$ the spectral measure of $\mathbb T_k$ at $u_k$, $W(\lambda)=\sum_nw_n\lambda^n$, and $C_k(\sigma)=\int W^2\lambda^{\sigma/a_k} d\nu$. The function $U_k$ depends only on the spatial links of the slice at time $0$ (Proposition~\ref{prop:spatialflow}(i)), which $\vartheta$ fixes, so for $j\ge0$, by translation invariance, \[ \omega_k\bigl(\bar U_k\,\tau_jU_k\bigr)=\omega_k\bigl(\vartheta(\bar U_k) \tau_jU_k\bigr)=\int\lambda^jd\nu , \] and the same holds for $j<0$ with $|j|$ in place of $j$, the integral being real. Hence \[ V_k=\int\mathcal Q\,d\nu,\qquad \mathcal Q(\lambda)=\sum_{n,n'}w_nw_{n'}\lambda^{|n-n'|} . \] Write $\lambda=e^{-a_kE}$ with $E\in[0,\infty]$. First, $W(\lambda)\le\|w\|_1\lambda^{n_0}\le\gamma_ge^{-Et_0}$, since $n_0\ge t_0/a_k$ and $\|w\|_1\le(t_1-t_0+a_k)\|g\|_\infty$. Second, let $J$ be the set of $n$ with $a_kn\in I$; it consists of $M\ge\ell_I/a_k-1\ge\frac34\ell_I/a_k$ consecutive integers, and $w_n\ge a_kc_g$ on $J$. Every $n\in J$ has on one side at least $p=\lceil M/2\rceil-1$ further elements of $J$, so, the terms of $\mathcal Q$ being nonnegative, \[ \mathcal Q(\lambda)\ge a_k^2c_g^2M\sum_{j=0}^{p}\lambda^j \ge a_k^2c_g^2M\,\frac{1-\lambda^{M/2}}{1-\lambda} . \] For $E\ge4/\ell_I$, $\lambda^{M/2}\le e^{-3E\ell_I/8}\le e^{-1}$ and $1-\lambda\le a_kE$, so $\mathcal Q(\lambda)\ge\frac34\ell_Ic_g^2 (1-e^{-1})/E\ge\kappa_g/E$. The function $Ee^{-2Et_0}$ decreases for $E\ge1/(2t_0)$, so for $\Lambda\ge\Lambda_g$, on the set $\{E>\Lambda\}$, \[ W^2\le\frac{\gamma_g^2}{\kappa_g}Ee^{-2Et_0}\mathcal Q\le \varepsilon_g(\Lambda)\mathcal Q , \] which also holds at $E=\infty$, where $W=0$. Therefore $\int_{\{E>\Lambda\}}W^2d\nu\le\varepsilon_g(\Lambda)V_k= \varepsilon_g(\Lambda)\Upsilon_kC_k(0)$, and \[ C_k(\sigma)\ge e^{-\Lambda\sigma}\int_{\{E\le\Lambda\}}W^2d\nu \ge e^{-\Lambda\sigma}\bigl(1-\varepsilon_g(\Lambda)\Upsilon_k\bigr)C_k(0). \] If $\Upsilon_k\le\Upsilon$ for all $k$, choose $\Lambda\ge\Lambda_g$ with $\varepsilon_g(\Lambda)\Upsilon\le\frac12$; then $\rho_k(0,\sigma)\ge \frac12e^{-\Lambda\sigma}$, which is $(\mathrm N_0)$, and Proposition~\ref{prop:ratioexact}(iii) gives $(\mathrm N_u)$. (v) By Theorem~\ref{thm:strongratio} with $\sigma=a_k$, $\rho_k(0,a_k)\le e^{-m_0}$. Part (iv) with $\sigma=a_k$ and $\Lambda=m_0/(2a_k)\ge\Lambda_g$ gives $e^{-m_0}\ge e^{-m_0/2}(1- \varepsilon_g(\Lambda)\Upsilon_k)$, so $\Upsilon_k\ge(1-e^{-m_0/2})/ \varepsilon_g(m_0/(2a_k))$, which is the claim. (vi) The covariance of $\phi_k$ is $\prod_\mu q_k^{|x_\mu-y_\mu|}+ a_k^{-5}\delta_{xy}$, which is invariant under translations and $B_4$. Fix a coordinate direction as time. The process of the time slices of $(\psi_k,\xi_k)$ is the stationary reversible Markov chain of the proof of Proposition~\ref{prop:osobstruction}(a) for $\psi_k$, together with independent slices of $\xi_k$; its transition operator is $P\otimes\Pi$, with $P$ as there and $\Pi$ the projection onto the functions that do not depend on the slice of $\xi_k$. It is positive with spectrum $\{q_k^N:N\ge0\}\cup\{0\}$ and $1$ simple, and the argument given there gives reflection positivity and the gap. The packets in $C_k(\sigma)$, $\sigma\ge0$, are supported at times $x_0\le-1$ and $x_0\ge1$, where the variables $\xi_k$ are independent and centred, so $C_k(\sigma)$ is that of example (a): with the time factor $q_k^{y_0-x_0}$, the substitution $y\mapsto y+(\sigma/a_k)e_0$ gives $C_k(2u+\sigma)= e^{-m\sigma}C_k(2u)$, and $Z_k$ converges as in (a). The variances add, so $V_k\ge a_k^{-5}a_k^8\sum_xf_0(a_kx)^2$, which is $a_k^{-1}$ times a Riemann sum converging to $\|f_0\|_2^2>0$. Hence $\Upsilon_k\to\infty$, and (M) for $n=2$ fails by (ii). \end{proof} By (ii), the renormalisation enters (M) only through (M$_Z$), a comparison of the variance of the packet with its reflected norm. By the proof of (iv), for a product packet $V_k$ weighs the spectral measure of the slice field by $\mathcal Q\ge\kappa_g/E$ at energies $E\ge4/\ell_I$, while $C_k(0)$ weighs it by $W^2\le\gamma_g^2e^{-2Et_0}$: (M$_Z$) says that the part of the variance carried at high energy, which the reflection suppresses over the distance $2t_0$, is bounded by a multiple of the reflected norm, and it implies $(\mathrm N_0)$. In the strong-coupling range $\Upsilon_k$ grows exponentially in $1/a_k$ by (v). Example (vi) has the lattice symmetry, reflection positivity, the gap $ma_k$ and ratios $\rho_k(u,\sigma)=e^{-m\sigma}$ independent of $k$, and (M$_Z$) fails for it; so (M$_Z$) does not follow from $(\mathrm N_u)$ and these properties. By Remark~\ref{rem:markov}, (M$_{\mathrm{loc}}$) does not follow from bounds on two-point functions. Every inequality that holds, with constants independent of $k$, for the states of Theorem~\ref{thm:strongratio} along all sequences of couplings in $[0,0.348]$ holds along sequences on which (M$_Z$) fails by (v), and (M) fails with it; so no such inequality implies (M$_Z$) or (M). This applies to every consequence of reflection positivity at all $\beta_W\ge0$, such as the chessboard estimates~\cite{FILS78}, and to the bounds of class (H) of Proposition~\ref{prop:upperbounds}; a proof of (M) has to use that the couplings leave the strong-coupling range. A sufficient condition without smearing follows from H\"older's and Minkowski's inequalities and translation invariance: if \[ \sup_kZ_k\bigl\|\varphi_k(0)-\omega_k(\varphi_k(0))\bigr\|_{L^n(\omega_k)} <\infty\qquad(n\ge1), \] then $\|Z_kA_k(f)\|_{L^n(\omega_k)}$ is at most $|Q|_k\|f\|_\infty$ times this supremum, and (M) holds. This pointwise bound on the renormalised flowed field is not proved. \emph{The flow does not bound the field.} Part (i) of Proposition~\ref{prop:msplit} is the bound that uses only $\|\Phi\|_\infty$. The spatial flow at a positive flow time does not improve it configuration by configuration. \begin{proposition}\label{prop:stationary} Let $G=\SU(N)$, $N\ge2$, $\zeta=e^{2\pi i/N}$, and let $U^*$ be the configuration with all temporal links equal to $1$ and spatial links \[ U^*_{(x,x+e_1)}=1,\quad U^*_{(x,x+e_2)}=\zeta^{x_1},\quad U^*_{(x,x+e_3)}=\zeta^{x_1+x_2}, \] multiples of the identity. \textup{(i)} Every spatial plaquette of $U^*$ is $\zeta^{\pm1}$, and $\Phi_s(gU^*)=gU^*$ and $\Phi_s(g1)=g1$ for every gauge transformation $g$ and every $s\in\R$. \textup{(ii)} Let $\Phi$ be the sum of $N-\ReTr U_p$ over the twelve spatial plaquettes with a corner at $0$. Then for all $x$ and $s$ \[ \Phi_x\circ\Phi_s(U^*)=12N\bigl(1-\cos(2\pi/N)\bigr),\qquad \Phi_x\circ\Phi_s(1)=0 . \] \textup{(iii)} Let $\omega(O)=\mu(O^F)$ for gauge-invariant $O$, with $\mu$ a Gibbs measure of $\gamma^F$ at $\beta\ge0$, as for the states of Theorem~\textup{\ref{thm:osaxioms}}, let $\Phi$ be as in \textup{(ii)} and $f\ge0$. Then the $\omega$-essential supremum and infimum of $A_k(f)$ differ by at least $12N(1-\cos(2\pi/N))\,a_k^4\sum_xf(a_kx)$, at every flow time, and \[ \|Z_kA_k(f)\|_{L^\infty(\omega)}\ge6N\Bigl(1-\cos\frac{2\pi}N\Bigr)Z_k\, a_k^4\sum_xf(a_kx) . \] \end{proposition} \begin{proof} (i) Let $1\le i0$ and the finite set $X_f$ of $x$ with $f(a_kx)\ne0$, there is a neighbourhood of $Q^*$, fixed by conditions on finitely many links $B$, on which $|F_x-F_x(Q^*)|<\eta$ for $x\in X_f$, and similarly for $Q^1$. Given the other links, the conditional law under $\mu$ of the links of $B$ has density proportional to $\exp(N\beta\sum_p\ReTr Q^F_p)$ with respect to Haar measure, the sum running over the plaquettes meeting $B$, and this density is at least $e^{-2N^2\beta|P_B|}$, with $|P_B|$ the number of these plaquettes. So both neighbourhoods have positive probability. On them $A_k(f)$ is within $\eta a_k^4\sum_xf(a_kx)$ of its values at $Q^*$ and $Q^1$, which differ by $12N(1-\cos(2\pi/N))a_k^4\sum_xf(a_kx)$ by (ii). Since $\eta$ is arbitrary, this gives the first claim, and the $L^\infty$ norm is at least half the difference. \end{proof} So the bound of Proposition~\ref{prop:msplit}(i) is attained up to a constant factor at every flow time, and it gives (M) only when $Z_k$ is bounded. Away from the configurations of least action the flow has stationary points at which the flowed energy density keeps its lattice size; a proof of (M) by way of the flow has to bound the probability of the configurations near such points, which is a property of the measure and not of the flow. \emph{Rotations as Ward identities.} For $X\in\mathfrak{so}(4)$ and $f\in C^1_c(\R^4)$ let $L_Xf(x)=\nabla f(x)\cdot Xx$, the derivative of $f\circ e^{\epsilon X}$ at $\epsilon=0$, and $E_{\mu\nu}$ as in the proof of Proposition~\ref{prop:rotations}. \begin{lemma}\label{lem:ward} Let $S$ be a family as in Theorem~\textup{\ref{thm:osaxioms}(i)--(ii)}, $X\in\mathfrak{so}(4)$ and $n\ge1$. Then $S_n(f_1\circ e^{-\epsilon X}, \dots,f_n\circ e^{-\epsilon X})=S_n(f_1,\dots,f_n)$ for all $\epsilon\in\R$ and $f_j\in C_c(\R^4)$ if and only if \[ \sum_{j=1}^nS_n(f_1,\dots,L_Xf_j,\dots,f_n)=0 \] for all $f_j\in C^\infty_c(\R^4)$. \end{lemma} \begin{proof} Let $R_\epsilon=e^{\epsilon X}$ and $F(\epsilon)=S_n(f_1\circ R_\epsilon, \dots,f_n\circ R_\epsilon)$ for $f_j\in C^\infty_c(\R^4)$. Since $R_\epsilon$ commutes with $X$ and $\nabla(f\circ R)(x)=R^{\mathsf T}\nabla f(Rx)$, \[ L_X(f\circ R_\epsilon)(x)=\nabla f(R_\epsilon x)\cdot R_\epsilon Xx =(L_Xf)(R_\epsilon x). \] By Taylor's formula $h^{-1}(f\circ R_{\epsilon+h}-f\circ R_\epsilon)$ converges uniformly to $(L_Xf)\circ R_\epsilon$ as $h\to0$, with supports in a fixed cube for $|h|\le1$, so the bound of Theorem~\ref{thm:osaxioms}(i) shows that $F$ is differentiable with $F'(\epsilon)$ equal to the sum of the lemma at the functions $f_j\circ R_\epsilon$. If the sum vanishes, $F$ is constant, which is the invariance for smooth $f_j$, and the bound and the density of $C^\infty_c$ in $C_c$ in the supremum norm on a cube give it for all $f_j$. Conversely invariance gives $F'(0)=0$. \end{proof} For $X\in\mathfrak{so}(4)$ and $f_j\in C^\infty_c(\R^4)$ put \[ \Delta^X_k(f_1,\dots,f_n)=\sum_{j=1}^n\omega_k\Bigl(Z_kA_k(L_Xf_j) \prod_{i\ne j}Z_kA_k(f_i)\Bigr), \] the derivative at $\epsilon=0$ of the lattice moment $\omega_k(\prod_jZ_kA_k(f_j\circ e^{\epsilon X}))$, a finite sum. It measures the breaking of the rotation by the lattice and by the flow. \begin{proposition}\label{prop:wardform} Assume the hypotheses of Theorem~\textup{\ref{thm:osaxioms}}. \textup{(i)} $\Delta^X_k(f_1,\dots,f_n)\to0$ for all $n$ and $f_j\in C^\infty_c(\R^4)$ if and only if every subsequential limit is invariant under the group $\{e^{\epsilon X}\}$. \textup{(ii)} Let $X\ne0$ lie in $\mathfrak{so}(3)_{\mathrm{sp}}$, for instance $X=E_{12}$. Then \textup{(W)} holds if and only if $\Delta^X_k(f_1,\dots,f_n)\to0$ for all $n$ and $f_j\in C^\infty_c(\R^4)$. \textup{(iii)} Let $X=E_{01}$. Then $\Delta^X_k(f_1,\dots,f_n)\to0$ for all $n$ and $f_j\in C^\infty_c(\R^4)$ if and only if every subsequential limit is invariant under $\mathrm O(4)$. For the family of Proposition~\textup{\ref{prop:osobstruction}(c)} the identity of Lemma~\textup{\ref{lem:ward}} with $X=E_{01}$ fails. \textup{(iv)} If \textup{(T1)--(T3)} hold and $a_k^4W_k\to0$, the conditions in \textup{(ii)} and \textup{(iii)} need only be imposed for $n=2$. \end{proposition} \begin{proof} (i) Let $S$ be the limit along a subsequence. By Theorem~\ref{thm:osaxioms}(i), applied to the functions $L_Xf_j\in C^\infty_c$, $\Delta^X_k$ converges along it to the sum of Lemma~\ref{lem:ward}; if $\Delta^X_k\to0$, the lemma gives the invariance. If $|\Delta^X_k(f_1,\dots,f_n)|\ge\eta>0$ along a subsequence, a refinement given by Theorem~\ref{thm:osaxioms}(i) has a limit for which the sum of Lemma~\ref{lem:ward} is at least $\eta$ in absolute value, and that limit is not invariant. (ii) $e^{\epsilon X}$ is the rotation about the line $\ker X$ through an angle proportional to $\epsilon$, of infinite order for all but countably many $\epsilon$. If (W) holds, every limit is invariant under $\mathrm O(3)\times\mathrm O(1)$, as stated after (W), and (i) gives $\Delta^X_k\to0$. Conversely, by (i) each $G_S$ contains a spatial rotation of infinite order, so $G_S\supseteq\mathrm O(3)\times\mathrm O(1)$ by Proposition~\ref{prop:rotations}(ii), and (W) follows. (iii) $e^{\epsilon E_{01}}$ rotates the $(0,1)$ plane, has infinite order for $\epsilon\notin\pi\mathbb Q$ and moves $e_0$ off $\pm e_0$ for $\epsilon\notin\pi\Z$; Proposition~\ref{prop:rotations}(iii) and (i) give the equivalence. The family of Proposition~\ref{prop:osobstruction}(c) has $G_S=\mathrm O(3)\times\mathrm O(1)$, which does not contain $e^{\epsilon E_{01}}$ for $\epsilon\notin\pi\Z$, so by Lemma~\ref{lem:ward} the identity fails for some $f_j$. (iv) By Theorem~\ref{thm:higher}(v) every limit is Gaussian, so it is invariant under a rotation if and only if $S_2$ is, and the proofs of (i) and of Lemma~\ref{lem:ward} apply to $n=2$ alone. \end{proof} So the lattice estimate that gives the spatial rotations is $\Delta^X_k\to0$ for one $X\in\mathfrak{so}(3)_{\mathrm{sp}}$, with no rate; a bound $|\Delta^X_k(f_1,\dots,f_n)|\le C(f_1,\dots,f_n)a_k^p$ with $p>0$ would suffice and is not proved. For $X=E_{12}$ the rotation by $\pi/2$ lies in $W_3$, so $\epsilon\mapsto\omega_k(\prod_jZ_kA_k(f_j\circ e^{\epsilon X}))$ has period $\pi/2$, and the integral of $\Delta^X_k(f_1\circ e^{\epsilon X},\dots,f_n\circ e^{\epsilon X})$ over a period vanishes at every $k$: the lattice symmetry cancels the breaking on average over the angle, and (ii) asks for the cancellation at each angle in the limit. For the time--space generators the same identity is the condition for $\mathrm O(4)$, and part (iii) shows that it fails for the spatially flowed free field at a positive flow time. \emph{The four-dimensional flow.} The flow in all four directions preserves $\mathrm O(4)$, and it loses reflection positivity already for the free field. \begin{proposition}\label{prop:fourflow} \textup{(i)} Let $t_f>0$ and $\mu>0$. The centred Gaussian field on $\R^4$ with covariance \[ \widehat C(p)=e^{-2t_f|p|^2}(|p|^2+\mu^2)^{-1}, \] the massive free field flowed by the four-dimensional heat flow for the time $t_f$, is invariant under translations and $\mathrm O(4)$ and is not reflection positive: there is a real $f\in C^\infty_c(\R^4)$ supported in $\{t>0\}$ with $\iint f(\theta x)C(x-y)f(y)\,dx\,dy<0$. \textup{(ii)} Let $S^j$, $j\ge1$, be families as in Theorem~\textup{\ref{thm:osaxioms}(i)--(ii)}, each invariant under $\mathrm O(3)\times\mathrm O(1)$, bounded by $M_n(Q)\prod_j\|f_j\|_\infty$ uniformly in $j$, and with $S^j_n(f_1,\dots,f_n)\to S^0_n(f_1,\dots,f_n)$ for all $n$ and $f_i\in C_c(\R^4)$. Then $G_{S^0}=\mathrm O(4)$ if and only if $S^0$ is invariant under the exchange of $e_0$ and $e_1$. \end{proposition} \begin{proof} (i) Invariance is clear. Put \[ H_{\mathbf p}(u)=\frac1{2\pi}\int e^{ip_0u}\widehat C(p_0,\mathbf p) \,dp_0 , \] an even real function, smooth in $u$ and continuous in $(\mathbf p,u)$, with $|H_{\mathbf p}(u)|\le H_{\mathbf p}(0)\le H_0(0)$. For $f=g\otimes h$ with $g$ supported in $(0,\infty)$ and $g,h$ real, the spatial Fourier transform gives \[ \begin{split} \iint f(\theta x)C(x-y)f(y)\,dx\,dy &=(2\pi)^{-3}\!\int|\hat h(\mathbf p)|^2G_{\mathbf p}\,d\mathbf p,\\ G_{\mathbf p}&=\iint g(s)g(t)H_{\mathbf p}(s+t)\,ds\,dt . \end{split} \] Since $H_0(0)>0$ and $H_0''(0)=-(2\pi)^{-1}\int p_0^2\widehat C(p_0,0) \,dp_0<0$, $\log H_0$ is strictly concave on an interval $[0,2\delta]$. For $0H_0(2u_1)H_0(2u_2)$, so the matrix $(H_0(u_i+u_j))_{i,j\le2}$ has negative determinant and there is a real $c$ with $\sum_{i,j}c_ic_jH_0(u_i+u_j)<0$. Let $j_\eta\ge0$ be a mollifier with integral $1$ supported in $[-\eta,\eta]$, $\eta0$ to be positive semidefinite, which makes $H_{\mathbf p}$ log-convex on $(0,\infty)$; a smooth even function with $H''(0)<0$ is not log-convex near $0$. So at a positive flow time the spatial flow keeps reflection positivity and loses $\mathrm O(4)$ (Proposition~\ref{prop:osobstruction}(c)), and the four-dimensional flow keeps $\mathrm O(4)$ and loses reflection positivity. By part (ii), at vanishing flow time, for limits $S^0$ of families that satisfy (W) at positive flow times with uniform bounds, $\mathrm O(4)$ reduces to one discrete symmetry, the exchange of time with a spatial axis, which the unflowed lattice theory has; the interchange of the limits $k\to\infty$ and $t_f\to0$ that would carry it to $S^0$, and the renormalisation of the fields as $t_f\to0$, are not treated here. \emph{One limit under a renormalisation condition.} Proposition \ref{prop:uniquelimit}(iii) uses the convergence of one ratio. Choosing the couplings so that it holds is a renormalisation condition: \begin{enumerate}[label=\textup{(RC)},leftmargin=*] \item There are a dyadic $\sigma_0>0$ and $\rho_*\in(0,1)$ with $\rho_k(0,\sigma_0)\to\rho_*$. \end{enumerate} That a tuning with (RC) exists is not proved. Since (RC) contains $(\mathrm N_0)$, it is not weaker than the open input of Theorem~\ref{thm:flowlimit}. Under (M), let $\mathfrak X$ be the set of families $S=(S_n)$ of multilinear forms on $C_c(\R^4)$ with $|S_n(f_1,\dots,f_n)|\le M_n(Q) \prod_j\|f_j\|_\infty$ for $f_j$ supported in a cube $Q$. For $L\in\mathbb N$ let $\mathcal F_L$ be a countable $\mathbb Q(i)$-linear set of functions supported in $[-L,L]^4$, dense there in the supremum norm, let $P_1,P_2,\dots$ enumerate the tuples of elements of $\bigcup_L\mathcal F_L$, and put $d(S,S')=\sum_i2^{-i}\min(1,|S(P_i)- S'(P_i)|)$. By the bounds and density, $d(S^m,S)\to0$ if and only if $S^m_n(f_1,\dots,f_n)\to S_n(f_1,\dots,f_n)$ for all $n$ and $f_j\in C_c(\R^4)$, and by the diagonal argument of Theorem~\ref{thm:higher}(ii) $(\mathfrak X,d)$ is a compact metric space. For $k\ge k_0$ the lattice moments $x_k=(\omega_k(\prod_jZ_kA_k(f_j)))_n$ lie in $\mathfrak X$; let $\mathfrak L$ be their set of limit points, the set of subsequential limits, and put \[ \begin{split} \delta_k(f_1,\dots,f_n)&=\omega_{k+1}\Bigl(\prod_jZ_{k+1}A_{k+1}(f_j) \Bigr)\\ &\quad-\omega_k\Bigl(\prod_jZ_kA_k(f_j)\Bigr). \end{split} \] \begin{lemma}\label{lem:limitset} If $(x_k)$ is a sequence in a compact metric space with $d(x_k,x_{k+1})\to0$, its set of limit points is nonempty and connected. \end{lemma} \begin{proof} The set $L$ of limit points is nonempty and closed. If $L=A\cup B$ with $A,B$ disjoint, closed and nonempty, let $3\eta$ be their distance and $A_\eta,B_\eta$ their open $\eta$-neighbourhoods, at distance at least $\eta$. The sequence enters both infinitely often, and once $d(x_k,x_{k+1})<\eta$ it cannot pass from one to the other in one step, so infinitely many $x_k$ lie in the compact complement of $A_\eta\cup B_\eta$, which then contains a limit point. That contradicts $L\subseteq A\cup B$. \end{proof} \begin{proposition}\label{prop:tuned} \textup{(i)} \textup{(RC)} implies $(\mathrm N_u)$ for every dyadic $u\ge0$, and every limit of Theorem~\textup{\ref{thm:flowlimit}} along a subsequence is nontrivial and has $\rho(\sigma_0)=\rho_*$. Under the hypothesis of Theorem~\textup{\ref{thm:transfer}}, $m_*\le\sigma_0^{-1} \log(1/\rho_*)$. For $\SU(2)$, \textup{(RC)} implies $\beta_{W,k}>0.348$ for all large $k$. Under \textup{(M)}, every $S\in\mathfrak L$ has $S_2(\bar f_0\circ\theta,f_0(\cdot-(\sigma_0,0)))=\rho_*$. \textup{(ii)} Assume the hypotheses of Theorem~\textup{\ref{thm:osaxioms}}. Then \textup{(U)} holds if and only if the following hold: \begin{enumerate}[label=\textup{(U\arabic*)},leftmargin=*] \item $\delta_k(f_1,\dots,f_n)\to0$ for all $n$ and $f_j\in C_c(\R^4)$; \item $\mathfrak L$ is totally disconnected. \end{enumerate} \textup{(iii)} Under the same hypotheses, if $\sum_k|\delta_k(f_1,\dots,f_n)|<\infty$ for all $n$ and all $f_j$ in a set whose linear span is dense in the continuous functions supported in each cube, then \textup{(U)} holds. \textup{(iv)} Let $m>0$, $q_k=e^{-ma_k}$, $\varepsilon_k=2+\sin(\log k)$, $K_\varepsilon(\mathbf z)=e^{-|\mathbf z|^2/\varepsilon}$ on $\R^3$, let $\phi_k$ be the centred Gaussian field on $\Z^4$ with covariance \[ q_k^{|x_0-y_0|}K_{\varepsilon_k}\bigl(a_k(\mathbf x-\mathbf y)\bigr), \] $A_k(f)=\sum_xa_k^4f(a_kx)\phi_k(x)$, and $f_0\ge0$ nonzero. The law of $\phi_k$ is invariant under translations and $\Gamma$ and reflection positive for the time reflections through sites and bisecting links, and its transfer matrix has gap $ma_k$. Moreover $\rho_k(u,\sigma)= e^{-m\sigma}$ for all $u,\sigma$, so \textup{(RC)} holds for every $\sigma_0$; \textup{(T1)--(T3)} hold with $r_k=1$ for large $k$, and \textup{(M)}, \textup{(W)} and \textup{(U1)} hold, with $|\delta_k(f_1,\dots,f_n)|\le C/k$ for $f_j\in C^1_c(\R^4)$. But $\mathfrak L$ is homeomorphic to $[1,3]$, and \textup{(U)} fails. \end{proposition} \begin{proof} (i) The first claims are Proposition~\ref{prop:ratioexact}(iii) and Theorem~\ref{thm:flowlimit}(ii). Under the hypothesis of Theorem~\ref{thm:transfer} the gap is at least $m_*a_k$ on $\Omega_k^\perp$, and as in the proof of Theorem~\ref{thm:strongratio} $\rho_k(0,\sigma_0)\le e^{-m_*\sigma_0}$, so $\rho_*\le e^{-m_*\sigma_0}$. The statement for $\SU(2)$ is Theorem~\ref{thm:strongratio}. The last claim is the computation in the proof of Proposition~\ref{prop:uniquelimit}(iii). (ii) If (U) holds, $x_k$ converges in $\mathfrak X$ by Proposition~\ref{prop:uniquelimit}(i), so $d(x_k,x_{k+1})\to0$, which is (U1), and $\mathfrak L$ is a point. Conversely (U1) is $d(x_k,x_{k+1})\to 0$, so $\mathfrak L$ is connected by Lemma~\ref{lem:limitset}; a nonempty, connected and totally disconnected set is a point, and (U) follows. (iii) The moments of tuples from the set converge, being Cauchy; by multilinearity so do those of tuples from its span, and by (M) and density all moments converge. Proposition~\ref{prop:uniquelimit}(i) gives (U). (iv) $K_\varepsilon$ is positive definite on $\R^3$, its Fourier transform being positive, hence on $a_k\Z^3$, and the field is the stationary Markov chain of the proof of Proposition~\ref{prop:osobstruction}(a) with $\gamma$ the Gaussian law of covariance $K_{\varepsilon_k}(a_k(\mathbf x-\mathbf y))$; this gives reflection positivity, the gap and, since $K_\varepsilon$ is radial, the invariance. As in Proposition~\ref{prop:msplit}(vi), the time factor of the covariance gives $C_k(2u+\sigma)=e^{-m\sigma}C_k(2u)$. The kernel $k_\varepsilon(z)=e^{-m|z_0|}K_\varepsilon(\mathbf z)$ is bounded by $1$ and Lipschitz, uniformly in $\varepsilon\in[1,3]$, and the covariance of $A_k(f)$ and $A_k(f')$ is a Riemann sum of mesh $a_k$ for $\iint f(x)f'(y)k_{\varepsilon_k}(x-y)\,dx\,dy$, with error at most $C(f,f')a_k$ for $f,f'\in C^1_c$ and tending to $0$ uniformly in $\varepsilon$ for $f,f'\in C_c$. In particular $C_k(0)$ differs by $o(1)$ from $c(\varepsilon_k)$, where $c(\varepsilon)=\iint f_0(\theta x)f_0(y) k_\varepsilon(x-y)\,dx\,dy$ is positive and continuous on $[1,3]$. With $w_k(z)=q_k^{|z_0|}e^{-a_k^2|\mathbf z|^2/3}$, which bounds the covariance since $\varepsilon_k\le3$, $a_k^4W_k\to(2/m)(3\pi)^{3/2}$, and (T1) holds with $K_2=1$, $K_n=0$ for $n\ge3$, (T2) for large $k$, and (T3); Theorem~\ref{thm:ostree}(i) gives (M). Let $S^\varepsilon$ be the Gaussian family with \[ S^\varepsilon_2(f,f')=c(\varepsilon)^{-1}\iint f(x)f'(y) k_\varepsilon(x-y)\,dx\,dy . \] By Wick's rule, $|x_k(f_1,\dots,f_n)-S^{\varepsilon_k}(f_1,\dots,f_n)|$ tends to $0$, and is at most $Ca_k$ for $f_j\in C^1_c$. The derivative of $k_\varepsilon$ in $\varepsilon$ is bounded on compact sets uniformly in $\varepsilon\in[1,3]$ and $c>0$ there, so $|S^\varepsilon(f_1,\dots,f_n)- S^{\varepsilon'}(f_1,\dots,f_n)|\le C|\varepsilon-\varepsilon'|$, and $|\varepsilon_{k+1}-\varepsilon_k|\le\log(1+1/k)\le1/k$; this gives the bound on $\delta_k$, and (U1) follows by approximation and (M). Since $\log k\to\infty$ with increments tending to $0$, the limit points of $(\varepsilon_k)$ form $[1,3]$, so $\mathfrak L=\{S^\varepsilon:1\le \varepsilon\le3\}$. Each $S^\varepsilon$ is invariant under $\mathrm O(3) \times\mathrm O(1)$, since $k_\varepsilon$ is, which gives (W). If $S^\varepsilon_2=S^{\varepsilon'}_2$, the continuous kernels $k_\varepsilon/c(\varepsilon)$ and $k_{\varepsilon'}/c(\varepsilon')$ coincide; at $z=0$ this gives $c(\varepsilon)=c(\varepsilon')$, and then $K_\varepsilon=K_{\varepsilon'}$, so $\varepsilon=\varepsilon'$. So $\varepsilon\mapsto S^\varepsilon$ is a continuous injection of $[1,3]$ onto $\mathfrak L$, a homeomorphism. \end{proof} So (RC) fixes the value of one functional on every limit and leaves (U1) and (U2) open: example (iv) satisfies (RC), (M), (W), the tree bounds and (U1), and its limits form an interval; the same field with $\varepsilon_k$ alternating between $1$ and $3$ satisfies (RC), (M), (W), the tree bounds and (U2), and has two limits. Part (iii) is the form that a scale-by-scale analysis would supply. Along the one-loop tuning of Proposition~\ref{prop:scalelosses}, $\beta_k=hk+O(1)$, so a bound $|\delta_k(f_1,\dots,f_n)|\le C/\beta_k$, of first order in the running coupling, is of the size $C/k$ of example (iv) and does not imply (U), while a bound $C\beta_k^{-p}$ with $p>1$ is summable and gives (U) by (iii). As for the ratio in Proposition~\ref{prop:scalelosses}, a bound on the change of the moments from one scale to the next gives (U) only if it is better than first order in the running coupling, or if the changes cancel between scales. \begin{remark}\label{rem:rotatedlattice} Let $f_0$ be invariant under the spatial rotations and $R\in\SO(3)_{\mathrm{sp}}$. The moments $x^R_k=(\omega_k(\prod_jZ_kA_k(f_j \circ R^{-1})))_n$ are those of the theory on the rotated lattice $R(a_k\Z^4)$, whose packet of $f_0$ is $A_k$; it has the same $C_k$, $Z_k$ and $\rho_k$, and satisfies (RC) when the original theory does. The sequence $x_1,x^R_1,x_2,x^R_2,\dots$ converges if and only if (U) holds and the difference in (W) for $R$ tends to $0$: if it converges, both subsequences have the same limit, and conversely. For $R$ of infinite order this is (U) and (W) together, the statement that the limit does not depend on which of the lattices $R(a_k\Z^4)$ is used. Example (a) of Proposition~\ref{prop:osobstruction} satisfies (RC), since its time dependence factorises as in Proposition~\ref{prop:msplit}(vi), as well as (M), (U) and the tree bounds, and not (W); example (iv) of Proposition~\ref{prop:tuned} satisfies (RC), (M), (W) and the tree bounds, and not (U). \end{remark} \emph{The time direction and the slice field.} The two parts of (M) can be compared with correlations of slice fields, as the ratio was in Proposition~\ref{prop:sliceratio}. In the time direction the comparison is exact, because the transfer matrix is positive. In the spatial directions no such comparison is available: the spatial flow acts within each time slice and mixes the two sides of every spatial reflection plane. For $h'\in C_c(\R^3)$ put \[ U^{h'}_k=\sum_{x\in\Z^3}a_k^3h'(a_kx)\bigl(\varphi_k(0,x)-\omega_k( \varphi_k(0,x))\bigr), \] and, for $r\in a_k\Z_{\ge0}$ and $\delta>0$, \[ \begin{split} D^{h'}_k(r)&=\omega_k\bigl(\bar U^{h'}_k\,\tau_{r/a_k}U^{h'}_k\bigr),\\ I^{h'}_k(\delta)&=a_k\!\!\sum_{r\in a_k\Z,\,0\le r<\delta}\!\!D^{h'}_k(r). \end{split} \] For the spatial factor $h$ of a product packet, $U^h_k=U_k$ and $D^h_k=D_k$ are those of Proposition~\ref{prop:sliceratio}; write $I_k=I^h_k$. As in the proof of Proposition~\ref{prop:msplit}(iv), $D^{h'}_k(r)=\ip u{\mathbb T_k^{r/a_k}u}$ with $u=\Psi_{U^{h'}_k}$. \begin{lemma}\label{lem:timekernel} Let $\mathbb T$ be self-adjoint with $0\le\mathbb T\le1$ on a Hilbert space, let $v_n$, $n\in\Z$, be vectors, nonzero only for $n$ in a set of $N$ consecutive integers, put $D_n(j)=\ip{v_n}{\mathbb T^jv_n}$ and \[ B=\sum_{n,n'\in\Z}\ip{v_n}{\mathbb T^{|n-n'|}v_{n'}} . \] \textup{(i)} $|B|\le2N\max_n\sum_{j=0}^{N-1}D_n(j)$. \textup{(ii)} If $v_n=w_nv$ with $w_n\ge0$, and $w_n\ge\eta$ on a set of $M\ge1$ consecutive integers, then, with $D(j)=\ip v{\mathbb T^jv}$, \[ B\ge\eta^2M\sum_{j=0}^{\lceil M/2\rceil-1}D(j) . \] \textup{(iii)} $D(j)$ is nonnegative and nonincreasing in $j$, and $\sum_{j0$ and $c_g>0$ such that $g\ge c_g$ on $I$, as in Proposition~\ref{prop:msplit}(iv). \begin{proposition}\label{prop:mzslice} Let $\delta>0$ be dyadic and let $k$ be so large that $\delta/a_k\in\Z$ and $a_k\le\min(\ell_I/4,\delta)$. \textup{(i)} Put $T_g=t_1-t_0+\delta$ and \[ \begin{split} c_-(\delta)&=\frac{3c_g^2\ell_I}{4\lceil8\delta/(3\ell_I)\rceil},\qquad c_+(\delta)=2\|g\|_\infty^2T_g\Bigl\lceil\frac{T_g}\delta\Bigr\rceil. \end{split} \] Then $c_-(\delta)I_k(\delta)\le V_k\le c_+(\delta)I_k(\delta)$ and \[ \bigl(\tfrac34c_g\ell_I\bigr)^2D_k(2a_kn_1)\le C_k(0)\le \|g\|_\infty^2T_g^2\,D_k(2a_kn_0) . \] \textup{(ii)} Consider \begin{enumerate}[label=\textup{(Z$_{\mathrm{sl}}$)},leftmargin=*] \item There are a dyadic $\delta>0$ and $C<\infty$ with $I_k(\delta)\le C\,D_k(\delta)$ for all large $k$. \end{enumerate} If \textup{(Z$_{\mathrm{sl}}$)} holds for one dyadic $\delta$, it holds for every dyadic $\delta>0$, and then $D_k(\delta)\ge(\delta/(2C))D_k(\delta/2)$ and $a_kD_k(0)\le C\,D_k(\delta)$ for all large $k$. \textup{(iii)} \textup{(M$_Z$)} holds if and only if \textup{(Z$_{\mathrm{sl}}$)} holds. In particular \textup{(M$_Z$)} holds for one product packet with spatial factor $h$ if and only if it holds for every product packet $g'\otimes h$ with $g'$ as above, and it implies $(\mathrm N_u)$ for every dyadic $u\ge0$. \end{proposition} \begin{proof} (i) By time translation invariance, $A_k=\sum_nw_n\tau_nU_k$ with $w_n=a_kg(a_kn)$, and $V_k=\sum_{n,n'}w_nw_{n'}D_k(a_k|n-n'|)$ by the identity of the proof of Proposition~\ref{prop:msplit}(iv). The $n$ with $w_n\ne0$ lie in $[n_0,n_1]$, a set of $N\le(t_1-t_0)/a_k+1$ integers, so $a_kN\le T_g$. Lemma~\ref{lem:timekernel}(i) with $v_n=w_nu_k$ and $|w_n|\le a_k\|g\|_\infty$ gives \[ V_k\le2\|g\|_\infty^2(a_kN)\,I_k(a_kN)\le2\|g\|_\infty^2T_g\Bigl\lceil \frac{T_g}\delta\Bigr\rceil I_k(\delta) , \] by (iii) of the lemma. The set $J$ of $n$ with $a_kn\in I$ consists of $M\ge\ell_I/a_k-1\ge\frac34\ell_I/a_k$ consecutive integers, and $w_n\ge a_kc_g$ there. Lemma~\ref{lem:timekernel}(ii) gives, with $b=a_k\lceil M/2\rceil\ge\frac38\ell_I$, \[ V_k\ge a_k^2c_g^2M\!\!\sum_{j<\lceil M/2\rceil}\!\!D_k(a_kj) =c_g^2(a_kM)\,I_k(b)\ge\tfrac34c_g^2\ell_I\,I_k(b), \] and $I_k(\delta)\le\lceil\delta/b\rceil I_k(b)\le\lceil8\delta/(3\ell_I) \rceil I_k(b)$. The bounds on $C_k(0)$ are those of the proof of Proposition~\ref{prop:sliceratio}(i), $C_k(0)=\int W^2d\nu$ with $\|w\|_1\lambda^{n_1}\le W(\lambda)\le\|w\|_1\lambda^{n_0}$, together with $\|w\|_1\ge a_kc_gM\ge\frac34c_g\ell_I$ and $\|w\|_1\le a_kN\|g\|_\infty \le T_g\|g\|_\infty$. (ii) Since $D_k$ is nonincreasing, $I_k(\delta)\ge\frac\delta2D_k(\delta/2)$ and $I_k(\delta)\ge a_kD_k(0)$, which gives the two bounds; in particular $c=\delta/(2C)\le1$, because $D_k(\delta)\le D_k(\delta/2)$ and $D_k(\delta/2)>0$ by Proposition~\ref{prop:sliceratio}(i). Let $\delta'$ be dyadic. If $\delta'<\delta$, then $I_k(\delta')\le I_k(\delta)\le CD_k(\delta)\le CD_k(\delta')$. If $\delta'>\delta$, Proposition~\ref{prop:sliceratio}(ii) with $r=\delta/2$, $r'=\delta$ gives $D_k(\delta')\ge c^{e}D_k(\delta)$ with $e=\max(1,2(\delta'-\delta)/\delta)$, and \[ I_k(\delta')\le I_k(\delta)+(\delta'-\delta)D_k(\delta)\le (C+\delta')\,c^{-e}D_k(\delta') . \] (iii) Suppose $\Upsilon_k\le\Upsilon$ for all $k$, and take a dyadic $\delta\le2t_0$. Then $2a_kn_0\ge2t_0\ge\delta$, and by (i) \[ I_k(\delta)\le\frac{V_k}{c_-(\delta)}\le\frac{\Upsilon C_k(0)}{c_-(\delta)} \le\frac{\Upsilon\|g\|_\infty^2T_g^2}{c_-(\delta)}D_k(\delta) , \] which is (Z$_{\mathrm{sl}}$). Conversely, assume (Z$_{\mathrm{sl}}$) at $\delta$ with the constant $C$, and take a dyadic $R\ge\max(2t_1,\delta)$. Then $D_k(2a_kn_1)\ge D_k(R)\ge c^eD_k(\delta)$ with $c=\delta/(2C)$ and $e=\max(1,2(R-\delta)/\delta)$, as in (ii), and by (i) \[ \Upsilon_k=\frac{V_k}{C_k(0)}\le\frac{c_+(\delta)\,C\,c^{-e}} {(\frac34c_g\ell_I)^2} \] for all large $k$; the finitely many other $\Upsilon_k$ are finite, since $C_k(0)>0$. Condition (Z$_{\mathrm{sl}}$) involves $h$ and not $g$, and (M$_Z$) implies $(\mathrm N_u)$ by Proposition~\ref{prop:msplit}(iv). \end{proof} So for product packets (M$_Z$) is a statement about the slice field alone: the variance of $U_k$ averaged over a physical time $\delta$ is at most a constant times $D_k(\delta)/\delta$. Since $I_k(\delta)\ge a_kD_k(0)$, it contains the bound $a_k\Var U_k\le C\,D_k(\delta)$: the equal-time variance of the slice field may exceed its correlation at a physical separation by at most a factor of order $1/a_k$. In terms of the spectral measure $\mu_k$ of $\mathbb T_k$ at $u_k$, in energies $E=-a_k^{-1}\log\lambda$, $I_k(\delta)$ is the integral of $a_k(1-e^{-\delta E})/(1-e^{-a_kE})$, a weight between $(1-e^{-\delta E})/E$ and that plus $a_k$, and (Z$_{\mathrm{sl}}$) bounds it by $\int e^{-\delta E}d\mu_k$; it bounds the spectral weight at high energy, counted with the factor $1/E$, by the weight at bounded physical energy. By (ii) it contains a ratio bound for $D_k$, hence $(\mathrm N_0)$, and it is strictly stronger than $(\mathrm N_u)$: for the field of Proposition~\ref{prop:msplit}(vi), $\rho_k(u,\sigma)=e^{-m\sigma}$ while $a_kD_k(0)$, which contains $a_k\cdot a_k^{-5}a_k^6\sum_xh(a_kx)^2$, grows like $a_k^{-1}$, and $D_k(\delta)$ converges. Along the constant tunings of Proposition~\ref{prop:cubecompatible}(ii) the hypothesis of Theorem~\ref{thm:transfer} holds and (Z$_{\mathrm{sl}}$) fails, since $(\mathrm N_u)$ fails there. So (M$_Z$) reduces to $(\mathrm N_u)$ in one direction only, and no argument that uses (D) only through the hypothesis of Theorem~\ref{thm:transfer} gives it. \begin{proposition}\label{prop:mlocslice} Let $\delta$ and $k$ be as in Proposition~\textup{\ref{prop:mzslice}}. \textup{(i)} Let $g_1,g_2\in C_c(\R)$ be supported in an interval of length $L$, $y_1,y_2\in a_k\Z^3$ and $f_i=g_i\otimes h(\cdot-y_i)$. Then \[ \bigl|\omega_k\bigl(A_k(f_1)A_k(f_2)\bigr)\bigr|\le K_L\,V_k\, \|g_1\|_\infty\|g_2\|_\infty , \] with $K_L=2(L+\delta)\lceil(L+\delta)/\delta\rceil/c_-(\delta)$. So \textup{(M$_{\mathrm{loc}}$)} holds for $n=2$ on these products, along every sequence of couplings. \textup{(ii)} \textup{(M$_{\mathrm{loc}}$)} holds for $n=2$ if and only if the following holds for one dyadic $\delta>0$, and then it holds for every dyadic $\delta>0$: \begin{enumerate}[label=\textup{(M$_{\mathrm{sp}}$)},leftmargin=*] \item For every cube $Q_3\subset\R^3$ there is $C<\infty$ with $I^{h'}_k(\delta)\le C\,I_k(\delta)$ for all large $k$ and all $h'\in C_c(\R^3)$ supported in $Q_3$ with $\|h'\|_\infty\le1$. \end{enumerate} \textup{(iii)} \textup{(M$_{\mathrm{loc}}$)} holds if and only if for every even $n\ge2$ and every cube $Q$ there are $C_n(Q)$ and $k_0$ with $\omega_k(|A_k(f)|^n)\le C_n(Q)V_k^{n/2}\|f\|_\infty^n$ for $k\ge k_0$ and $f\in C_c(\R^4)$ supported in $Q$. The case $n=2$ is equivalent to \textup{(M$_{\mathrm{sp}}$)}, and the case $n$ implies the cases below it. \textup{(iv)} Let $m>0$, $q_k=e^{-ma_k}$, $\kappa(\mathbf z)=e^{-|\mathbf z|^2}$ on $\R^3$, and let $\psi_k$ be the centred Gaussian field on $\Z^4$ with covariance \[ q_k^{|x_0-y_0|}\Bigl(\kappa\bigl(a_k(\mathbf x-\mathbf y)\bigr)+a_k^{-1} (-1)^{\sum_{i=1}^3(x_i-y_i)}\Bigr), \] $A_k(f)=\sum_xa_k^4f(a_kx)\psi_k(x)$, and $f_0=g\otimes h$ a product packet as in Proposition~\textup{\ref{prop:sliceratio}} with $g,h\ge0$ nonzero and continuously differentiable. The law of $\psi_k$ is invariant under translations and $\Gamma$ and reflection positive for the time reflections through sites and bisecting links, its transfer matrix has gap $ma_k$, $\rho_k(u,\sigma)=e^{-m\sigma}$ for all $u,\sigma$, so \textup{(RC)} holds for every $\sigma_0$, and \textup{(M$_Z$)} holds. But \textup{(M$_{\mathrm{sp}}$)} fails, and \textup{(M$_{\mathrm{loc}}$)} and \textup{(M)} fail for $n=2$. \textup{(v)} Let $m>0$, $q_k=e^{-ma_k}$, $\epsilon_k=a_k$, let $(\xi_n)_{n\in\Z}$ be the stationary Markov chain on $\{-1,0,1\}$ with the law $\pi$ of Remark~\textup{\ref{rem:markov}} at $\epsilon=\epsilon_k$ and transition operator $P=q_kI+(1-q_k)\Pi$, and $X_n=\xi_n/\sqrt{\epsilon_k}$. Let $\psi_k(x)=X_{x_0}$, and $A_k$, $f_0$ as in \textup{(iv)}. This field has the properties listed in \textup{(iv)}, and \textup{(M$_{\mathrm{sp}}$)} holds; but $\omega_k(A_k^4)/V_k^2\to\infty$, so \textup{(M$_{\mathrm{loc}}$)} and \textup{(M)} fail for $n=4$. \end{proposition} \begin{proof} (i) Since $\Phi$ is real, $A_k(\bar f)=\overline{A_k(f)}$, so by the Cauchy--Schwarz inequality \[ \bigl|\omega_k\bigl(A_k(f_1)A_k(f_2)\bigr)\bigr|\le\prod_{i=1}^2 \omega_k\bigl(|A_k(f_i)|^2\bigr)^{1/2}, \] and it suffices to bound $\omega_k(|A_k(f)|^2)$ for $f=g\otimes h(\cdot-y)$ by $K_LV_k\|g\|_\infty^2$. By spatial translation invariance $y$ may be taken to be $0$. As in the proof of Proposition~\ref{prop:mzslice}(i), $\omega_k(|A_k(f)|^2)$ is $B$ of Lemma~\ref{lem:timekernel} with $v_n=a_kg(a_kn)u_k$, nonzero for $n$ in a set of $N\le L/a_k+1$ consecutive integers, so \[ \begin{split} \omega_k(|A_k(f)|^2)&\le2\|g\|_\infty^2(a_kN)I_k(a_kN)\\ &\le2\|g\|_\infty^2(L+\delta)\Bigl\lceil\frac{L+\delta}\delta\Bigr\rceil I_k(\delta), \end{split} \] and Proposition~\ref{prop:mzslice}(i) gives the claim. (ii) Independence of $\delta$: for dyadic $\delta'>\delta$, $I^{h'}_k(\delta')\le\lceil\delta'/\delta\rceil I^{h'}_k(\delta)\le \lceil\delta'/\delta\rceil C\,I_k(\delta')$, and for $\delta'<\delta$, $I^{h'}_k(\delta')\le I^{h'}_k(\delta)\le C\,I_k(\delta)\le C\lceil\delta/ \delta'\rceil I_k(\delta')$. Assume (M$_{\mathrm{sp}}$), and let $f\in C_c(\R^4)$ be supported in a cube $Q=[t_Q,t_Q+L]\times Q_3$. As in (i) it suffices to bound $\omega_k(|A_k(f)|^2)$. By time translation invariance $A_k(f)=\sum_n\tau_n (a_kU^{h_n}_k)$ with $h_n=f(a_kn,\cdot)$, supported in $Q_3$ with $\|h_n\|_\infty\le\|f\|_\infty$, and nonzero for $n$ in a set of $N\le L/a_k+1$ consecutive integers. The identity of the proof of Proposition~\ref{prop:msplit}(iv) and translation invariance give $\omega_k(\bar U^{h_n}_k\tau_jU^{h_{n'}}_k)=\ip{\Psi_{U^{h_n}_k}}{\mathbb T_k^j\Psi_{U^{h_{n'}}_k}}$ for $j\ge0$, so $\omega_k(|A_k(f)|^2)$ is $B$ of Lemma~\ref{lem:timekernel} with $v_n=a_k\Psi_{U^{h_n}_k}$, and \[ \begin{split} \omega_k(|A_k(f)|^2)&\le2(a_kN)\max_nI^{h_n}_k(a_kN)\\ &\le\frac{2(L+\delta)C}{c_-(\delta)}\Bigl\lceil\frac{L+\delta}\delta \Bigr\rceil\|f\|_\infty^2V_k , \end{split} \] by (iii) of the lemma, (M$_{\mathrm{sp}}$) and Proposition~\ref{prop:mzslice}(i). Conversely, let $g_*\in C_c(\R)$ satisfy $0\le g_*\le1$, $g_*=0$ outside $[0,1]$ and $g_*=1$ on $[\frac14,\frac34]$, and let $h'$ be as in (M$_{\mathrm{sp}}$). For $a_k\le\frac14$ the $n$ with $a_kn\in[\frac14,\frac34]$ are $M'\ge1/(4a_k)$ consecutive integers, and Lemma~\ref{lem:timekernel}(ii) with $v_n=a_kg_*(a_kn)\Psi_{U^{h'}_k}$ and $\eta=a_k$ gives, with $b=a_k\lceil M'/2\rceil\ge\frac18$, \[ \omega_k\bigl(|A_k(g_*\otimes h')|^2\bigr)\ge a_kM'\,I^{h'}_k(b)\ge \frac{I^{h'}_k(\delta)}{4\lceil8\delta\rceil} . \] Since $A_k(g_*\otimes\bar h')=\overline{A_k(g_*\otimes h')}$, (M$_{\mathrm{loc}}$) for $n=2$ on a cube $Q\supseteq[0,1]\times Q_3$ bounds the left side by $M^{\mathrm{loc}}_2(Q)V_k\le M^{\mathrm{loc}}_2(Q)c_+(\delta) I_k(\delta)$, which is (M$_{\mathrm{sp}}$). (iii) The bounds are (M$_{\mathrm{loc}}$) with $f_j$ equal to $\bar f$ and $f$ alternately, since $A_k(\bar f)=\overline{A_k(f)}$. Conversely, for $f_1,\dots,f_n$ supported in $Q$ and an even $n'\ge n$, H\"older's inequality gives $|\omega_k(\prod_jA_k(f_j))|\le\prod_j\omega_k(|A_k(f_j)| ^{n'})^{1/n'}$. The case $n=2$ is (M$_{\mathrm{loc}}$) for $n=2$ by the Cauchy--Schwarz inequality, hence (M$_{\mathrm{sp}}$) by (ii), and the last claim is the monotonicity of $L^n$ norms for the probability measure $\omega_k$. (iv) The spatial kernel is the sum of two positive definite functions on $\Z^3$, the first because the Fourier transform of $\kappa$ is positive, the second because it is $\varepsilon_{\mathbf x}\varepsilon_{\mathbf y}$ with $\varepsilon_{\mathbf x}=(-1)^{x_1+x_2+x_3}$; both are invariant under the signed permutations of the spatial coordinates. The field is the stationary Markov chain of the proof of Proposition~\ref{prop:osobstruction}(a), with $\gamma$ the Gaussian law of this spatial covariance, which gives the invariance, reflection positivity and the gap, and as in Proposition~\ref{prop:msplit}(vi) the time factor gives $\rho_k(u,\sigma)=e^{-m\sigma}$. Put $S_k=\sum_{\mathbf x}a_k^3h(a_k \mathbf x)\varepsilon_{\mathbf x}$. Grouping $\mathbf x$ with $x_1$ even with $\mathbf x+e_1$, $|S_k|\le a_k\|\partial_1h\|_\infty(L_h+2a_k)^3$, where $L_h$ is the side of a cube containing the support of $h$. The part of $V_k$ from the second term of the kernel is \[ a_k^{-1}\sum_{n,n'}a_k^2g(a_kn)g(a_kn')q_k^{|n-n'|}S_k^2\le a_k^{-1} \Bigl(a_k\sum_ng(a_kn)\Bigr)^2S_k^2 , \] which tends to $0$, and the same holds for $C_k(0)$. The parts from $\kappa$ are Riemann sums converging to \[ \begin{split} &\iint f_0(x)f_0(y)e^{-m|x_0-y_0|}\kappa(\mathbf x-\mathbf y)\,dx\,dy>0,\\ &\iint g(t)g(s)e^{-m(t+s)}dt\,ds\iint h(\mathbf x)h(\mathbf y) \kappa(\mathbf x-\mathbf y)\,d\mathbf x\,d\mathbf y>0 . \end{split} \] So $\Upsilon_k$ converges, which is (M$_Z$). Let \[ f_k(t,\mathbf x)=g(t)h(\mathbf x)\cos\bigl(\pi(x_1+x_2+x_3)/a_k\bigr), \] a continuous function supported in the support of $f_0$, with $\|f_k\|_\infty\le\|g\|_\infty\|h\|_\infty$, equal to $g(a_kx_0)h(a_k \mathbf x)\varepsilon_{\mathbf x}$ at the point $a_kx$. Both terms of the kernel are positive definite, so \[ \omega_k\bigl(A_k(f_k)^2\bigr)\ge a_k^{-1}\sum_{n,n'}a_k^2g(a_kn)g(a_kn') q_k^{|n-n'|}\Bigl(\sum_{\mathbf x}a_k^3h(a_k\mathbf x)\Bigr)^2 , \] and the factor of $a_k^{-1}$ on the right converges to \[ \iint g(t)g(s)e^{-m|t-s|}dt\,ds\,\Bigl(\int h\Bigr)^2>0 , \] while $V_k$ converges. So (M$_{\mathrm{loc}}$) fails for $n=2$, and with it (M$_{\mathrm{sp}}$), by (ii), and (M), by Proposition~\ref{prop:msplit}(ii). (v) $P=\Pi+q_k(I-\Pi)$ is symmetric on $L^2(\pi)$ with spectrum $\{1,q_k\}$ and $1$ simple, and $P\ge0$; the argument of the proof of Proposition~\ref{prop:osobstruction}(a) gives reflection positivity, and the gap is $ma_k$. The field does not depend on the spatial coordinates, which gives the invariance. Since $X\perp1$ and $\pi(X^2)=1$, $P^jX=q_k^jX$ and the covariance is $q_k^{|x_0-y_0|}$, that of Proposition~\ref{prop:osobstruction}(b); so $\rho_k(u,\sigma)=e^{-m\sigma}$, and $V_k$ and $C_k(0)$ converge to positive limits, which gives (M$_Z$). Further $U^{h'}_k=(\sum_{\mathbf x}a_k^3h'(a_k\mathbf x))X_0$, so $I^{h'}_k(\delta)/I_k(\delta)=|\sum a_k^3h'(a_k\mathbf x)|^2/(\sum a_k^3 h(a_k\mathbf x))^2$, which is bounded for $h'$ as in (M$_{\mathrm{sp}}$), since the denominator converges to $(\int h)^2>0$. For times $s_1\le s_2\le s_3\le s_4$ with $j_i=s_{i+1}-s_i$, the Markov property and $P^j=\Pi+ q_k^j(I-\Pi)$ give, as in Remark~\ref{rem:markov} and with $X^3=X/\epsilon$, \[ \E\Bigl[\prod_{i=1}^4X_{s_i}\Bigr]=q_k^{j_1+j_3}\bigl(q_k^{j_2}/\epsilon_k +1-q_k^{j_2}\bigr)\ge\epsilon_k^{-1}q_k^{s_4-s_1} . \] With $F_n=a_k^3\sum_{\mathbf x}f_0(a_kn,a_k\mathbf x)\ge0$, $A_k=\sum_n a_kF_nX_n$, and $F_n=0$ unless $a_kn\in[t_0,t_1]$, so \[ \omega_k(A_k^4)\ge\epsilon_k^{-1}e^{-m(t_1-t_0)}\Bigl(a_k\sum_nF_n\Bigr)^4 , \] and $a_k\sum_nF_n\to\int f_0>0$. Since $\epsilon_k=a_k\to0$ and $V_k$ converges, the ratio diverges. \end{proof} Part (i) is the part of (M$_{\mathrm{loc}}$) that the positivity of the transfer matrix gives: test functions in time, on both sides of the reflection plane and of either sign, cost no more than the packet $f_0$, for every state with a positive transfer matrix. What remains for $n=2$ is (M$_{\mathrm{sp}}$), a comparison between spatial profiles at one spacing, which involves neither the reflection nor the tuning. Example (iv) has a spatial mode at the lattice scale whose variance diverges, and the time structure of an exact Markov field; example (v) has no spatial structure and four-point functions of order $1/\epsilon_k$, as in Remark~\ref{rem:markov}. For a Gaussian field the linear heat flow in space at the flow time $\tau/a_k^2$ multiplies the variance of the lattice-scale mode of (iv) by $e^{-c\tau/a_k^2}$, with $c>0$ fixed by the normalisation of the flow, and removes the divergence there. For the gauge field the flow does not do this configuration by configuration. \begin{proposition}\label{prop:centralpattern} Let $G=\SU(N)$, $N\ge2$, $\zeta=e^{2\pi i/N}$ and $c_N=N(1-\cos(2\pi/N))$. Let $m:\Z\to\Z$ satisfy $m(j+1)-m(j)=1$ for $j\equiv0,1$ and $m(j+1)-m(j)=0$ for $j\equiv2,3$ modulo $4$, and let $U^\sharp$ be the configuration with all temporal links equal to $1$ and spatial links \[ U^\sharp_{(x,x+e_1)}=U^\sharp_{(x,x+e_3)}=1,\qquad U^\sharp_{(x,x+e_2)}=\zeta^{m(x_1)} . \] \textup{(i)} The spatial plaquettes of $U^\sharp$ in the plane of $e_1,e_2$ with base point $x$ are $\zeta^{\pm1}$ for $x_1\equiv0,1$ and $1$ for $x_1\equiv2,3$ modulo $4$, all other spatial plaquettes are $1$, and $\Phi_s(gU^\sharp)=gU^\sharp$ for every gauge transformation $g$ and every $s\in\R$. \textup{(ii)} Let $\Phi$ be the sum of $N-\ReTr U_p$ over the twelve spatial plaquettes with a corner at $0$. Then for all $x$ and $s$ \[ \Phi_x\circ\Phi_s(U^\sharp)=2c_N\Bigl(1+\cos\frac{\pi(x_1-1)}2\Bigr). \] \textup{(iii)} Let $g\in C_c(\R)$ and $h\in C_c(\R^3)$ be nonnegative and $h^\sharp_k(\mathbf x)=h(\mathbf x)\cos(\pi(x_1/a_k-1)/2)$, and let $\omega$ be as in Proposition~\textup{\ref{prop:stationary}(iii)}. At every flow time the $\omega$-essential supremum and infimum of $A_k(g\otimes h^\sharp_k)$ differ by at least \[ 4c_N\,a_k^4\sum_ng(a_kn)\!\!\sum_{\mathbf x:\,x_1\equiv1\ (4)}\!\!h(a_k \mathbf x) , \] which converges to $c_N\int g\int h$. For $A_k(g\otimes h)$ the difference is at least \[ 12c_Na_k^4\sum_ng(a_kn)\sum_{\mathbf x}h(a_k\mathbf x) , \] which converges to $12c_N\int g\int h$. \end{proposition} \begin{proof} (i) With the plaquette formula of the proof of Proposition~\ref{prop:stationary}(i), the plaquette in the plane of $e_1,e_2$ at $x$ is $\zeta^{m(x_1+1)-m(x_1)}$; in the plane of $e_1,e_3$ all four links are $1$, and in the plane of $e_2,e_3$ it is $\zeta^{m(x_1)}\zeta^{-m(x_1)} =1$, because $x$ and $x+e_3$ have the same first coordinate. All links are central, and the computation of the proof of Proposition~\ref{prop:stationary}(i) shows that the gradient in the flow equation vanishes; uniqueness and gauge covariance give the rest. (ii) Only the four plaquettes of the plane of $e_1,e_2$ with a corner at $x$ can be nontrivial; their base points have first coordinates $x_1$ and $x_1-1$, twice each. So $\Phi_x(U^\sharp)=2c_N(d(x_1)+d(x_1-1))$ with $d(j)=m(j+1)-m(j)$, which is $2c_N$ times $1,2,1,0$ for $x_1\equiv0,1,2,3$, and the flow does not move $U^\sharp$. (iii) With $c_x=\cos(\pi(x_1-1)/2)\in\{1,0,-1,0\}$ for $x_1\equiv1,2,3,0$, \[ \begin{split} &\sum_xa_k^4g(a_kx_0)h^\sharp_k(a_k\mathbf x)\,\Phi_x\circ\Phi_s(U^\sharp)\\ &\qquad=2c_N\sum_xa_k^4g(a_kx_0)h(a_k\mathbf x)(c_x+c_x^2) , \end{split} \] and $c_x+c_x^2$ is $2$ for $x_1\equiv1$ and $0$ otherwise; at the configuration $1$ the sum is $0$. The proof of Proposition~\ref{prop:stationary}(iii) applies with $U^\sharp$ in place of $U^*$; the centring constant cancels in the difference. The points with $x_1\equiv1$ modulo $4$ form a sublattice of index four, which gives the limit. The second statement is Proposition~\ref{prop:stationary}(iii). \end{proof} At every flow time the packet with a spatial profile oscillating at the lattice scale has, at the stationary configurations, a range of values of the same order as the smooth packet, the ratio of the two lower bounds tending to $1/12$. A proof of (M$_{\mathrm{sp}}$) by way of the flow has therefore to bound the probability of the configurations near those with nontrivial central plaquettes, uniformly along the tuning; this is an estimate on large fields of the Wilson measure at the couplings $\beta_k$, of the kind of Sec.~\ref{ssec:largefields}, compared with the size of $I_k(\delta)$, which is not known. \begin{remark}\label{rem:opensplit} For the flowed fields along a tuning, with a product packet $f_0=g\otimes h$, the five statements stand as follows. (M$_{\mathrm{loc}}$): the case $n=1$ holds since the packets are centred, and the case $n=2$ holds on the products of time profiles with the lattice translates of $h$ (Proposition~\ref{prop:mlocslice}(i)), for every state with a positive transfer matrix. Not proved: (M$_{\mathrm{sp}}$), equivalent to the case $n=2$, and the bounds $\omega_k(|A_k(f)|^n)\le C_n(Q)V_k^{n/2} \|f\|_\infty^n$ for even $n\ge4$, which with it are equivalent to (M$_{\mathrm{loc}}$). Both concern one spacing at a time, without reflection; they are not consequences of the gap, $(\mathrm N_u)$, (RC), (M$_Z$), reflection positivity and the symmetry $\Gamma$, by Proposition~\ref{prop:mlocslice}(iv), (v), so no argument that uses (D) only through the transfer-matrix gap, together with $(\mathrm N_u)$, gives them. (M$_Z$): equivalent to (Z$_{\mathrm{sl}}$), $I_k(\delta)\le C\,D_k(\delta)$ for the slice field, and independent of $g$ (Proposition~\ref{prop:mzslice}). Not proved. It implies $(\mathrm N_u)$ for every $u$; it is not implied by $(\mathrm N_u)$, the gap and the lattice symmetries (Proposition~\ref{prop:msplit}(vi)), nor by the hypothesis of Theorem~\ref{thm:transfer} (Proposition~\ref{prop:cubecompatible}(ii)). The missing estimate is a bound on the high-energy spectral weight of the slice field, with the factor $1/E$, and on $a_k\Var U_k$, by its correlation at a physical separation. (W): equivalent to $\Delta^{E_{12}}_k(f_1,\dots,f_n)\to0$ for all $n$ and $f_j\in C^\infty_c(\R^4)$ (Proposition~\ref{prop:wardform}(ii)); a bound $C(f_1,\dots,f_n)a_k^p$ with $p>0$ suffices. Not proved. Example (a) of Proposition~\ref{prop:osobstruction} has the gap in every direction, (RC), (M), hence (M$_{\mathrm{sp}}$) and (Z$_{\mathrm{sl}}$), and (U), and not (W); so no argument that uses (D) only through the gap, together with $(\mathrm N_u)$ and (M), gives (W). The lattice symmetry gives the cancellation of $\Delta^{E_{12}}_k$ only on average over the angle. (U1) and (U2), given (RC): (U1) is $\delta_k\to0$, for which $\sum_k|\delta_k|<\infty$ on a set with dense span suffices (Proposition~\ref{prop:tuned}(iii)), and (U2) is the total disconnectedness of $\mathfrak L$. Not proved. Example (iv) of Proposition~\ref{prop:tuned} and its alternating variant satisfy (RC), (M) and (W), each with one of (U1), (U2) and not the other, so neither follows from the gap, $(\mathrm N_u)$, (RC), (M) and (W); (U1) compares two spacings, which no statement at one spacing contains. The examples are Gaussian and Markov fields, not lattice gauge theories. They show that a proof of any of the four open statements has to use properties of the Wilson measure along the tuning beyond translation and $\Gamma$ invariance, reflection positivity, the transfer-matrix gap, the normalisation (RC) and the parts of (M) proved or assumed here. \end{remark} \emph{The spatial spectral measure of the slice field.} The comparison (M$_{\mathrm{sp}}$) is a statement about the distribution in spatial momentum of the time-integrated correlations of the slice field. In this paragraph the states are invariant under translations and $\vartheta$ and reflection positive, with the transfer matrices of Appendix~\ref{app:transfer}, $\Phi$ is real, $\delta>0$ is dyadic and $k$ is so large that $n_\delta=\delta/a_k$ is an even integer. Write $\mathring\varphi_k(x)=\varphi_k(x)-\omega_k(\varphi_k(x))$, \[ \begin{split} K_k(z)&=a_k\sum_{j=0}^{n_\delta-1}\omega_k\bigl(\mathring\varphi_k(0,0)\, \mathring\varphi_k(j,z)\bigr),\\ K^V_k(z)&=a_k^2\sum_{n,n'=0}^{n_\delta-1}\omega_k\bigl(\mathring\varphi_k(n,0)\, \mathring\varphi_k(n',z)\bigr) \end{split} \] for $z\in\Z^3$. For a finitely supported $c:\Z^3\to\C$ put $\hat c(p)= \sum_xc_xe^{ip\cdot x}$ on $\mathbf T^3=(\R/2\pi\Z)^3$, $U^c_k=\sum_xc_x\mathring\varphi_k(0,x)$, and \[ I^c_k=a_k\sum_{j=0}^{n_\delta-1}\omega_k\bigl(\bar U^c_k\,\tau_jU^c_k\bigr), \qquad V^c_k=\omega_k\Bigl(\Bigl|a_k\sum_{n=0}^{n_\delta-1}\tau_nU^c_k \Bigr|^2\Bigr). \] For $c_x=a_k^3h'(a_kx)$, $U^c_k=U^{h'}_k$ and $I^c_k=I^{h'}_k(\delta)$. By translation invariance $I^c_k=\sum_{x,y}\bar c_xc_yK_k(y-x)$ and $V^c_k= \sum_{x,y}\bar c_xc_yK^V_k(y-x)$. For a closed box $\mathcal B_r(p_0)= \{p:|p_i-p_{0,i}|\le r,\ i=1,2,3\}$ in $\mathbf T^3$, $r>0$, and a finite positive measure $\varpi$ on $\mathbf T^3$, put $\bar\varpi(r)=\sup_{p_0} \varpi(\mathcal B_r(p_0))$. A box of half-side $r'$ is covered by $\lceil r'/r\rceil^3$ boxes of half-side $r$, so $\bar\varpi(r')\le\lceil r'/r\rceil^3\bar\varpi(r)$. \begin{lemma}\label{lem:spatialmeasure} \textup{(i)} There are unique finite positive measures $\varpi_k$ and $\varpi^V_k$ on $\mathbf T^3$ with $K_k(z)=\int e^{ip\cdot z}d\varpi_k(p)$ and $K^V_k(z)=\int e^{ip\cdot z}d\varpi^V_k(p)$ for $z\in\Z^3$. For every finitely supported $c$ \[ I^c_k=\int|\hat c|^2d\varpi_k,\qquad V^c_k=\int|\hat c|^2d\varpi^V_k, \qquad\tfrac\delta2I^c_k\le V^c_k\le2\delta I^c_k , \] and $\varpi_k(\mathbf T^3)=K_k(0)\le\delta\|\Phi\|_\infty^2$. \textup{(ii)} Let $\varpi$ be a finite positive measure on $\mathbf T^3$, $L\ge4a_k$, and $c_x=a_k^3h'(a_kx)$ with $h'$ supported in a cube of side $L$ and $|h'|\le1$. Then \[ \int|\hat c|^2d\varpi\le7\cdot10^6L^6\,\bar\varpi(\pi a_k/L) . \] \textup{(iii)} Let $\chi\in C_c(\R^3)$ take values in $[0,1]$, be supported in a cube $Q'$ of side $\ell\ge4a_k$ and equal $1$ on the cube of side $\ell/2$ with the same centre, and for $p_0\in\mathbf T^3$ let $h'(\mathbf y)=\chi(\mathbf y)e^{-ip_0\cdot\mathbf y/a_k}$. Then for every finite positive measure $\varpi$ \[ \varpi\bigl(\mathcal B_{a_k/(3\ell)}(p_0)\bigr)\le16384\,\ell^{-6}\!\int |\hat c|^2d\varpi,\qquad c_x=a_k^3h'(a_kx). \] \textup{(iv)} Let $h\in C_c(\R^3)$ be real with $\int h\ne0$, supported in a cube of side $L_h$, and $\rho_h=L_h^{-1}(|\int h|/(72\int|h|))^{1/2}$. For all large $k$, with $c_x=a_k^3h(a_kx)$, $|\hat c|\ge\frac34|\int h|$ on $\mathcal B_{a_k\rho_h}(0)$, so $\int|\hat c|^2d\varpi\ge\frac12(\int h)^2 \varpi(\mathcal B_{a_k\rho_h}(0))$ for every finite positive measure $\varpi$. \textup{(v)} Let $\varkappa(p)=\sum_{i=1}^34\sin^2(p_i/2)$, $m\ge1$, $h\in C^{2m}_c(\R^3)$ real, supported in a cube of side $L_h$, $\|h\|_{2m}$ the largest supremum of a derivative of order $2m$ of $h$, and $\Theta_m=3^m \|h\|_{2m}(L_h+2m+1)^3$. For $a_k\le1$ and $c_x=a_k^3h(a_kx)$, $\varkappa^m|\hat c|\le\Theta_ma_k^{2m}$ on $\mathbf T^3$. \end{lemma} \begin{proof} (i) With $u_c=\Psi_{U^c_k}$, the identity of the proof of Proposition~\ref{prop:msplit}(iv) gives $\omega_k(\bar U^c_k\tau_jU^c_k)= \ip{u_c}{\mathbb T_k^ju_c}\ge0$ for $j\ge0$, so $I^c_k\ge0$ and $K_k$ is positive definite; $V^c_k\ge0$ shows the same for $K^V_k$. For a positive definite bounded $K$ on $\Z^3$ and an integer $M\ge1$, the function \[ f_M(p)=M^{-3}\!\!\sum_{x,y\in\{0,\dots,M-1\}^3}\!\!K(y-x)e^{-ip\cdot(y-x)} \] is nonnegative, and $\int e^{ip\cdot z}f_M(p)\frac{dp}{(2\pi)^3}=K(z) \prod_i(1-|z_i|/M)_+$. The measures $f_M\,dp/(2\pi)^3$ have mass $K(0)$, so by Lemma~\ref{lem:helly} a subsequence converges weakly to a measure with Fourier coefficients $K(z)$; trigonometric polynomials are dense in $C(\mathbf T^3)$, which gives uniqueness. Then $\sum_{x,y}\bar c_xc_y \int e^{ip\cdot(y-x)}d\varpi=\int|\hat c|^2d\varpi$. The numbers $\ip{u_c}{\mathbb T_k^ju_c}$ are real, so translation invariance and complex conjugation give $\omega_k(\overline{\tau_nU^c_k}\,\tau_{n'}U^c_k)=\ip{u_c}{\mathbb T_k^{|n-n'|}u_c}$ for all $n,n'$, and $V^c_k$ is $B$ of Lemma~\ref{lem:timekernel} with $v_n=a_ku_c$ for $0\le n0$ and $C<\infty$ with \[ \bar\varpi_k(a_k\rho)\le C\,I_k(\delta)\quad\text{for all large }k ; \] if it holds for $\rho'$ with $C$, it holds for every $\rho>0$ with $\lceil\rho/\rho'\rceil^3C$. If \textup{(M$_{\mathrm{sp}}$)} holds with the constant $C$ for the unit cube, the bound holds with $\rho=\frac13$ and $16384\,C$; conversely the bound gives \textup{(M$_{\mathrm{sp}}$)} with the constant $7\cdot10^6L^6\lceil \pi/(L\rho)\rceil^3C$ for cubes of side $L$. \textup{(ii)} If the bound of \textup{(i)} holds with $\rho$ and $C$, then $K_k(0)\le C\lceil\pi/(a_k\rho)\rceil^3I_k(\delta)$ for all large $k$. \textup{(iii)} If $\int h\ne0$, \textup{(M$_{\mathrm{sp}}$)} holds as soon as $\bar\varpi_k(a_k\rho_h)\le C\,\varpi_k(\mathcal B_{a_k\rho_h}(0))$ for all large $k$, with $\rho_h$ as in Lemma~\textup{\ref{lem:spatialmeasure}(iv)}. \end{proposition} \begin{proof} (i) Since $\bar\varpi(r')\le\lceil r'/r\rceil^3\bar\varpi(r)$ and $\bar\varpi$ is nondecreasing, the bound for $\rho'$ gives it for every $\rho>0$. If (M$_{\mathrm{sp}}$) holds for the unit cube, apply Lemma~\ref{lem:spatialmeasure}(iii) with $\ell=1$ to a fixed $\chi$ and every $p_0$: $\varpi_k(\mathcal B_{a_k/3}(p_0))\le16384\,I^{h'}_k(\delta) \le16384\,C\,I_k(\delta)$; if only real $h'$ are admitted, the real and imaginary parts of the probe give the same with $4\cdot16384\,C$, since $|\hat c|^2\le2(|\widehat{\operatorname{Re}c}|^2+|\widehat{\operatorname{Im} c}|^2)$. Conversely, for $h'$ supported in a cube of side $L$ with $|h'|\le1$, Lemma~\ref{lem:spatialmeasure}(ii) gives $I^{h'}_k(\delta)\le7\cdot10^6L^6\bar\varpi_k(\pi a_k/L)\le7\cdot10^6L^6 \lceil\pi/(L\rho)\rceil^3\bar\varpi_k(a_k\rho)$ once $a_k\le L/4$. (ii) $K_k(0)=\varpi_k(\mathbf T^3)$, and $\mathbf T^3$ is covered by $\lceil\pi/(a_k\rho)\rceil^3$ boxes of half-side $a_k\rho$. (iii) Lemma~\ref{lem:spatialmeasure}(iv) gives $\varpi_k(\mathcal B_{a_k\rho_h}(0))\le2I_k(\delta)/(\int h)^2$, and (i) applies with $\rho=\rho_h$. \end{proof} So (M$_{\mathrm{sp}}$) says that no box of lattice momenta of side of order $a_k$ carries more of the time-integrated spectral weight than a constant times $I_k(\delta)$; by (ii) it contains the bound $K_k(0)\le C'a_k^{-3} I_k(\delta)$ on the time-integrated autocorrelation at one site. Part (iii) reduces it, for $\int h\ne0$, to a comparison of the weight of every box with that of the box at zero momentum. Reflection positivity in the spatial directions gives that comparison when it holds exactly. For $r\ge0$, $i\in\{1,2,3\}$, let $\iota^{\mathrm s}_i$ and $\iota^{\mathrm b}_i$ be the reflections $x_i\mapsto-x_i$ and $x_i\mapsto-1-x_i$ of $\Z^4$, and consider \begin{enumerate}[label=\textup{(SR$_r$)},leftmargin=*] \item Let $i\in\{1,2,3\}$ and $\iota\in\{\iota^{\mathrm s}_i, \iota^{\mathrm b}_i\}$. For every $c:\Z^4\to\C$ of finite support in the half-space $\{x_i\ge r\}$, \[ \sum_{x,y}\bar c_xc_y\,\omega_k\bigl(\mathring\varphi_k(\iota x)\, \mathring\varphi_k(y)\bigr)\ge0 . \] \end{enumerate} For a field $\varphi_k(x)=\Phi_x$ without flow, with $\Phi$ a function of the links within distance $r$ of $0$ in the slice, and a state reflection positive for the spatial reflections, \textup{(SR$_r$)} holds, since $\Phi$ is $W_3$-invariant; for the flowed fields no $r$ is known. \begin{theorem}\label{thm:spatialrp} Assume \textup{(SR$_0$)} for the slice fields at the spacing $a_k$. \textup{(i)} $\varpi^V_k(\mathcal B_r(p_0))\le\varpi^V_k(\mathcal B_r(0))$ for every $p_0\in\mathbf T^3$ and $r>0$. \textup{(ii)} Let $h\in C_c(\R^3)$ be real with $\int h\ne0$, and $\rho_h$ as in Lemma~\textup{\ref{lem:spatialmeasure}(iv)}. If \textup{(SR$_0$)} holds for all large $k$, then for every cube of side $L$, all large $k$ and all $h'$ supported in it with $|h'|\le1$, \[ I^{h'}_k(\delta)\le\frac{6\cdot10^7L^6}{(\int h)^2}\Bigl\lceil\frac\pi {L\rho_h}\Bigr\rceil^3I^h_k(\delta) , \] so \textup{(M$_{\mathrm{sp}}$)} holds for the spatial factor $h$. \end{theorem} \begin{proof} (i) Fix $i$, write $x=(x_i,x_\perp)$, and let $b$ be a finitely supported function of $x_\perp$. Put $k_b(n)=\sum_{x_\perp,y_\perp}\bar b_{x_\perp} b_{y_\perp}K^V_k(ne_i+y_\perp-x_\perp)$ for $n\in\Z$. For $\alpha$ finitely supported in $\Z_{\ge0}$ let $c_{(t,x)}=a_k\mathbf 1_{0\le ts\}|\,ds\\ &\le\int_0^\infty|J_0\cap\{P_\lambda>s\}|\,ds=\int_{J_0}P_\lambda\,dt ; \end{split} \] the point mass also gives $J_0$ at least the weight of $J$. So $\varpi_b(J)\le\varpi_b(J_0)$, that is $\int(\mathbf 1_{J_0}-\mathbf 1_J) (p_i)\,|\hat b(p_\perp)|^2d\varpi^V_k\ge0$. Every continuous $g\ge0$ on the transverse torus is a uniform limit of functions $|\hat b|^2$, since $\sqrt g$ is a uniform limit of trigonometric polynomials, so the signed measure $A\mapsto\varpi^V_k(\{p_i\in J_0,\ p_\perp\in A\})-\varpi^V_k(\{p_i \in J,\ p_\perp\in A\})$ is nonnegative. Applying this with $A$ a product of two closed arcs, successively for $i=1,2,3$, moves each coordinate of the centre of $\mathcal B_r(p_0)$ to $0$ without decreasing the weight; for $r\ge\pi$ there is nothing to prove. (ii) Let $k$ be so large that $a_k\le L/4$ and Lemma~\ref{lem:spatialmeasure}(iv) applies. Then Lemma~\ref{lem:spatialmeasure}(i), (ii), (iv) and part (i) give \[ \begin{split} I^{h'}_k(\delta)&\le\tfrac2\delta V^{h'}_k\le\tfrac2\delta\,7\cdot10^6L^6 \Bigl\lceil\frac\pi{L\rho_h}\Bigr\rceil^3\varpi^V_k\bigl(\mathcal B_{a_k\rho_h}(0)\bigr)\\ &\le\tfrac2\delta\,7\cdot10^6L^6\Bigl\lceil\frac\pi{L\rho_h}\Bigr\rceil^3 \frac{2V^h_k}{(\int h)^2}\\ &\le\frac{56\cdot10^6L^6}{(\int h)^2}\Bigl\lceil \frac\pi{L\rho_h}\Bigr\rceil^3I^h_k(\delta), \end{split} \] using $V^h_k\le2\delta I^h_k(\delta)$. \end{proof} \begin{proposition}\label{prop:rpexamples} Let $m>0$ and $q_k=e^{-ma_k}$, let $\varkappa$ and $\Theta_2$ be as in Lemma~\textup{\ref{lem:spatialmeasure}(v)}, and let $\Delta$ be the spatial lattice Laplacian, $(\Delta\phi)(x)=\sum_{i=1}^3(\phi(x+e_i)+ \phi(x-e_i)-2\phi(x))$. \textup{(a)} Let $\psi_k$ be the field of Proposition~\textup{\ref{prop:osobstruction}(a)} and $\varphi_k=\Delta^2 \psi_k$. Its law is invariant under translations and $\Gamma$ and reflection positive for the time reflections through sites and bisecting links, its transfer matrix has gap $ma_k$, $\rho_k(u,\sigma)=e^{-m\sigma}$, \textup{(Z$_{\mathrm{sl}}$)} and \textup{(M$_Z$)} hold for every product packet, and \textup{(SR$_2$)} holds. For every real $h\in C^4_c(\R^3)$ and every cube $Q_3$, \textup{(M$_{\mathrm{sp}}$)} fails. \textup{(b)} The field $\phi_k=\psi_k+a_k^{-5/2}\xi_k$ of Proposition~\textup{\ref{prop:msplit}(vi)} satisfies \textup{(SR$_0$)} and, for every nonzero real $h\in C_c(\R^3)$, \textup{(M$_{\mathrm{sp}}$)}; and \textup{(Z$_{\mathrm{sl}}$)} fails. \end{proposition} \begin{proof} (a) The covariance of $\varphi_k$ is $q_k^{|x_0-y_0|}$ times the spatial kernel with Fourier transform $\varkappa(p)^4\prod_iP_{q_k}(p_i)\ge0$, $P_q(t)=(1-q^2)/(1-2q\cos t+q^2)$, as in the proof of Proposition~\ref{prop:osobstruction}(a). That proof, with $\gamma$ the centred Gaussian law with this spatial covariance, gives reflection positivity for the time reflections and a transfer matrix with spectrum $\{q_k^N:N\ge0\}$ and $1$ simple; invariance under translations and $\Gamma$ holds because $\Delta^2$ commutes with translations and $W_3$. The slice field is linear, so $D^{h'}_k(r)=q_k^{r/a_k}D^{h'}_k(0)$, which gives the ratio and $I_k(\delta)=T_kD_k(0)\le\delta e^{m\delta}D_k(\delta)$ with $T_k=a_k \sum_{j0$, which with (ii) gives the first claim, and $\cos\frac\pi2=0$ gives the case $N=4$. For $X$ supported on $e_*$, $Y_p=\pm X_{e_*}$ for the four plaquettes containing $e_*$ and $Y_p=0$ otherwise, and $\cos\pi=-1$, $\cos\frac{2\pi}3=-\frac12$. (iv) By (i) the first variation of the action of the slice vanishes in every direction, so the right side of the flow equation vanishes at $U^\zeta$ and $\Phi_s(U^\zeta)=U^\zeta$ by uniqueness. Each of the four plaquettes containing $e_*$ contributes $N-\ReTr\zeta^{\pm1}=c_N$ to $\Phi_x$ for each of its four corners $x$; the two endpoints of $e_*$ are corners of all four, and each of the other eight corners of exactly one. \end{proof} \emph{Spatial reflection positivity of a field and of its Laplacian.} Theorem~\ref{thm:spatialrp} turns (SR$_0$) for one field into (M$_{\mathrm{sp}}$) for that field. The admissible fields are closed under $\Phi\mapsto\Delta\Phi$, and by Proposition~\ref{prop:laplaceladder} a proof of (M$_{\mathrm{sp}}$) for all of them has to suppress the lattice-scale spectral weight. This paragraph shows that (SR$_0$) cannot hold for $\Phi$ and $\Delta\Phi$ together in a state that clusters in space, that the linear heat flow destroys (SR$_0$) at every positive flow time, and that (SR$_0$) for one field forces spectral weight at the lattice scale. Write $\mathbf T=\R/2\pi\Z$, identified with $(-\pi,\pi]$. A finite positive measure $\nu$ on $\mathbf T$ is \emph{of Poisson type} if there is a finite positive measure $\mu$ on $[0,1]$ with \[ \hat\nu(n)=\int e^{int}d\nu(t)=\int\lambda^{|n|}d\mu(\lambda)\qquad (n\in\Z), \] with $0^0=1$. The moments determine $\mu$, as in the proof of Lemma~\ref{lem:spectralmeasure}. Since $\int e^{int}P_\lambda(t)\frac{dt} {2\pi}=\lambda^{|n|}$ for $P_\lambda(t)=(1-\lambda^2)/(1+\lambda^2-2\lambda \cos t)$, $0\le\lambda<1$, uniqueness of Fourier coefficients gives \begin{equation}\label{eq:poissonform} \nu=\mu(\{1\})\,\delta_0+f\,\frac{dt}{2\pi},\qquad f=\int_{[0,1)} P_\lambda\,d\mu(\lambda) . \end{equation} For $i\in\{1,2,3\}$ write $p=(p_i,p_\perp)$ for $p\in\mathbf T^3$, with $p_\perp$ in the torus $\mathbf T^2$ of the other two coordinates, and put $\varkappa_\perp(p_\perp)=\sum_{j\ne i}4\sin^2(p_j/2)$, so that $\varkappa(p)=4\sin^2(p_i/2)+\varkappa_\perp(p_\perp)$. For a finite positive measure $\varpi$ on $\mathbf T^3$ and a continuous $g\ge0$ on $\mathbf T^2$ let $\varpi^i_g$ be the image of $g(p_\perp)\,d\varpi(p)$ under $p\mapsto p_i$. Call $\varpi$ \emph{of Poisson type in direction $i$} if every $\varpi^i_g$ is of Poisson type. \begin{lemma}\label{lem:poissontype} \textup{(i)} Let $\nu_r$ be measures of Poisson type on $\mathbf T$ with $\sup_r\nu_r(\mathbf T)<\infty$, and let $\nu$ be a finite positive measure with $\hat\nu_r(n)\to\hat\nu(n)$ for every $n$. Then $\nu$ is of Poisson type. \textup{(ii)} If \textup{(SR$_0$)} holds for the slice field at the spacing $a_k$, then $\varpi^V_k$ is of Poisson type in every direction $i\in\{1,2,3\}$. The same holds for every real field on $\Z^4$ with translation invariant, bounded covariance that satisfies the inequalities of \textup{(SR$_0$)}, with $K^V_k$ formed from its covariance. \textup{(iii)} If $\nu$ is of Poisson type, then for every integer $n\ge1$ \[ \begin{split} \hat\nu(0)-\hat\nu(n)&\le n\bigl(\hat\nu(0)-\hat\nu(1)\bigr),\\ \nu\bigl(\{t:2\le|t|\le\pi\}\bigr)&\ge\frac{\pi-2}{2\pi}\bigl(\hat\nu(0) -\hat\nu(1)\bigr) . \end{split} \] \end{lemma} \begin{proof} (i) Let $\mu_r$ be the measures of $\nu_r$. Their masses $\mu_r([0,1])= \hat\nu_r(0)$ converge to $\hat\nu(0)$. If $\hat\nu(0)=0$, then $\nu=0$ and $\mu=0$ will do. Otherwise Lemma~\ref{lem:helly}, applied to the probability measures $\mu_r/\hat\nu_r(0)$ for large $r$, gives a subsequence along which $\mu_r$ converges weakly to a finite positive measure $\mu$ of mass $\hat\nu(0)$. Since $\lambda\mapsto\lambda^{|n|}$ is continuous on $[0,1]$, $\hat\nu(n)=\lim\int\lambda^{|n|}d\mu_r=\int\lambda^{|n|}d\mu$. (ii) For a finitely supported $b$ on the transverse lattice, the proof of Theorem~\ref{thm:spatialrp}(i) shows that $k_b(n)=\int e^{inp_i}|\hat b(p_\perp)|^2d\varpi^V_k(p)$ equals $\int\lambda^{|n|}d\mu_b$ for all $n\in\Z$, so $\varpi^{V,i}_{k,|\hat b|^2}$ is of Poisson type. That proof uses only translation invariance, the boundedness and positive definiteness of $K^V_k$, and (SR$_0$). For a continuous $g\ge0$ on $\mathbf T^2$, $\sqrt g$ is a uniform limit of trigonometric polynomials $\hat b_r$, so $|\hat b_r|^2\to g$ uniformly, the Fourier coefficients converge, the masses are bounded, and (i) applies. (iii) By the representation, $\hat\nu(0)-\hat\nu(n)=\int(1-\lambda^n)d\mu$ and $1-\lambda^n\le n(1-\lambda)$ on $[0,1]$. Further $P_\lambda(t)\ge (1-\lambda^2)/(1+\lambda)^2=(1-\lambda)/(1+\lambda)\ge(1-\lambda)/2$ for $0\le\lambda<1$, and the set $\{2\le|t|\le\pi\}$ has measure $2(\pi-2)$, so by~\eqref{eq:poissonform} its $\nu$-measure is at least $\frac{\pi-2}{2\pi}\int(1-\lambda)\,d\mu=\frac{\pi-2}{2\pi}(\hat\nu(0)- \hat\nu(1))$; the atom at $\lambda=1$ contributes $0$ to both sides. \end{proof} The measures of $\Delta\Phi$ are those of $\Phi$ times $\varkappa^2$: the summation by parts of the proof of Proposition~\ref{prop:laplaceladder}(i) gives $V^c_k$ for $\Delta\Phi$ equal to $V^{\Delta c}_k$ for $\Phi$, so the measure $\varpi^V_k$ of $\Delta\Phi$ is $\varkappa^2\varpi^V_k$, and that of $K_k$ is $\varkappa^2\varpi_k$. \begin{theorem}\label{thm:srladder} Let $\Phi$ be as in Proposition~\textup{\ref{prop:laplaceladder}}, and assume \textup{(SR$_0$)} at the spacing $a_k$ for the slice fields of both $\Phi$ and $\Delta\Phi$. \textup{(i)} $\varpi^V_k=K^V_k(0)\,\delta_0$ and $\varpi_k=K_k(0)\,\delta_0$. So $K^V_k$ and $K_k$ are constant on $\Z^3$, and the time average $Y_k(\mathbf x)=a_k\sum_{n=0}^{n_\delta-1}\mathring\varphi_k(n,\mathbf x)$ satisfies $Y_k(\mathbf x)=Y_k(0)$ almost surely for every $\mathbf x$. \textup{(ii)} The slice field of $\Delta\Phi$ equals its mean almost surely at every point: $\Delta\mathring\varphi_k(x)=0$ for every $x\in\Z^4$. \textup{(iii)} If $\omega_k(\mathring\varphi_k(0,0)\,\mathring\varphi_k(j, \mathbf z))\to0$ as $|\mathbf z|\to\infty$ for every integer $j\ge0$, then $\mathring\varphi_k(x)=0$ almost surely for every $x$, and $C_k(0)=0$ for every packet $f_0$. \end{theorem} \begin{proof} (i) Fix $i$. By Lemma~\ref{lem:poissontype}(ii), applied to $\Phi$ with $g=1,\varkappa_\perp,\varkappa_\perp^2$ and to $\Delta\Phi$ with $g=1$, the images $\nu_0,\nu_1,\nu_2$ of $\varpi^V_k$, $\varkappa_\perp\varpi^V_k$, $\varkappa_\perp^2\varpi^V_k$ and the image $\nu'$ of $\varkappa^2 \varpi^V_k$ under $p\mapsto p_i$ are of Poisson type, with measures $\mu_0,\mu_1,\mu_2,\mu'$ on $[0,1]$. Put $k_l=\hat\nu_l$ and $F=\hat\nu'$. With $4\sin^2(t/2)=2-e^{it}-e^{-it}$, \[ \varkappa^2=(2-e^{it}-e^{-it})^2+2(2-e^{it}-e^{-it})\varkappa_\perp+ \varkappa_\perp^2 ,\qquad t=p_i, \] so $\nu'$ is $\nu_0$, $\nu_1$, $\nu_2$ multiplied by these trigonometric polynomials in $t$ and added, and \[ \begin{split} F(n)&=6k_0(n)-4k_0(n-1)-4k_0(n+1)\\ &\quad+k_0(n-2)+k_0(n+2)+4k_1(n)\\ &\quad-2k_1(n-1)-2k_1(n+1)+k_2(n) . \end{split} \] Insert $k_l(m)=\int\lambda^{|m|}d\mu_l$. For $n\ge2$ all the exponents $|n-j|$, $|j|\le2$, equal $n-j$, and the integrands combine to \[ \lambda^{n-2}(1-\lambda)^4,\qquad-2\lambda^{n-1}(1-\lambda)^2,\qquad \lambda^n \] against $\mu_0$, $\mu_1$, $\mu_2$; these are polynomial identities, valid at $\lambda=0$ with $0^0=1$. For $n=1$ the integrand against $\mu_0$ is $6\lambda-4-4\lambda^2+\lambda+\lambda^3$, which for $\lambda>0$ equals $\lambda^{-1}(1-\lambda)^4-\lambda^{-1}+\lambda$, and the integrand against $\mu_1$ is $4\lambda-2-2\lambda^2=-2(1-\lambda)^2$. For $n\ge3$ the three integrands vanish at $\lambda=0$, and $F(n)=\int \lambda^nd\mu'$. So the finite signed measure \[ \lambda^3\mu'-\lambda(1-\lambda)^4\mu_0+2\lambda^2(1-\lambda)^2\mu_1- \lambda^3\mu_2 \] on $[0,1]$ annihilates every polynomial, hence every continuous function, and it is zero. Dividing by $\lambda$ and by $\lambda^2$ on $(0,1]$, \begin{equation}\label{eq:srladdermeas} \begin{split} \lambda^2\mu'&=(1-\lambda)^4\mu_0-2\lambda(1-\lambda)^2\mu_1+\lambda^2 \mu_2,\\ \lambda\mu'&=\lambda^{-1}(1-\lambda)^4\mu_0-2(1-\lambda)^2\mu_1+\lambda \mu_2 \end{split} \end{equation} as measures on $(0,1]$. For $n=2$, $F(2)=\int\lambda^2d\mu'=\int_{(0,1]} \lambda^2d\mu'$ and $F(2)=\int_{[0,1]}((1-\lambda)^4d\mu_0-2\lambda (1-\lambda)^2d\mu_1+\lambda^2d\mu_2)$; integrating the first line of~\eqref{eq:srladdermeas} over $(0,1]$ and subtracting leaves $\mu_0(\{0\})=0$. Integrating the second line over $[\epsilon,1]$, \[ \int_{[\epsilon,1]}\frac{(1-\lambda)^4}\lambda\,d\mu_0\le\nu'(\mathbf T)+ 2\nu_1(\mathbf T) \] for every $\epsilon>0$, so $\int_{(0,1]}\lambda^{-1}(1-\lambda)^4d\mu_0 <\infty$, and since $(1-\lambda)^4\ge\frac1{16}$ on $(0,\frac12]$, $\int_{(0,1]}\lambda^{-1}d\mu_0<\infty$. Now, with $\mu_0(\{0\})=0$, \[ \begin{split} F(1)&=\int_{(0,1]}\Bigl(\frac{(1-\lambda)^4}\lambda-\frac1\lambda+\lambda \Bigr)d\mu_0\\ &\quad-2\int_{[0,1]}(1-\lambda)^2d\mu_1+\int\lambda\,d\mu_2,\\ F(1)&=\int\lambda\,d\mu'=\int_{(0,1]}\frac{(1-\lambda)^4}\lambda\,d\mu_0\\ &\quad-2\int_{(0,1]}(1-\lambda)^2d\mu_1+\int\lambda\,d\mu_2 , \end{split} \] the second by the second line of~\eqref{eq:srladdermeas}. Subtracting, \[ \int_{(0,1]}\Bigl(\frac1\lambda-\lambda\Bigr)d\mu_0+2\mu_1(\{0\})=0 . \] Both terms are nonnegative and $1/\lambda-\lambda>0$ on $(0,1)$, so $\mu_0$ is carried by $\{1\}$, $k_0$ is constant, and $\nu_0$ is a point mass at $0$. Thus $\varpi^V_k(\{p:p_i\ne0\})=0$ for $i=1,2,3$, and $\varpi^V_k= \varpi^V_k(\mathbf T^3)\delta_0=K^V_k(0)\delta_0$. By Lemma~\ref{lem:spatialmeasure}(i), $\frac\delta2\int|\hat c|^2d\varpi_k\le \int|\hat c|^2d\varpi^V_k\le2\delta\int|\hat c|^2d\varpi_k$ for every finitely supported $c$. As in the proof of Theorem~\ref{thm:spatialrp}(i) the functions $|\hat c|^2$ approximate every continuous $g\ge0$ on $\mathbf T^3$ uniformly, so $\frac2\delta\varpi^V_k-\varpi_k$ and $2\delta\varpi_k-\varpi^V_k$ are positive functionals on $C(\mathbf T^3)$, hence positive measures: \begin{equation}\label{eq:varpicompare} \tfrac\delta2\varpi_k\le\varpi^V_k\le2\delta\varpi_k . \end{equation} So $\varpi_k(\{p\ne0\})=0$. Finally $Y_k$ is real with $\omega_k(Y_k(0)Y_k(\mathbf z))=K^V_k(\mathbf z)$, so $\omega_k((Y_k(\mathbf x)-Y_k(0))^2)=2K^V_k(0)-2K^V_k(\mathbf x)=0$. (ii) The measure of $K_k$ for $\Delta\Phi$ is $\varkappa^2\varpi_k= K_k(0)\varkappa(0)^2\delta_0=0$. So, with $U^c_k$ formed from $\Delta\Phi$, $0=I^c_k=a_k\sum_{j0$ for the packet $f_0$ of $\Phi$. Then at every spacing $a_k$, \textup{(SR$_0$)} fails for the slice field of $\Phi$ or for that of $\Delta\Phi$. So along such a sequence no class of admissible fields closed under $\Phi\mapsto\Delta\Phi$ satisfies \textup{(SR$_0$)}, and Theorem~\textup{\ref{thm:spatialrp}} applies at the spacing $a_k$ to at most one of $\Phi$ and $\Delta\Phi$. \end{corollary} \begin{proof} By Theorem~\ref{thm:onescale}, part (v) of Proposition~\ref{prop:state} holds at $\beta_k$: covariances of gauge-invariant local functions, one of them Lipschitz in each link, decay exponentially in the distance. A continuous gauge-invariant function of finitely many links is a uniform limit of Lipschitz ones, which may be averaged over the gauge transformations at the sites they involve, since these act by isometries. By Proposition~\ref{prop:spatialflow}(ii), $\varphi_k(x)$ is a uniform limit of continuous gauge-invariant local functions at the time of $x$, and $|\Cov(F,F')-\Cov(G,G')|\le2\|F-G\|_\infty\|F'\|_\infty+2\|F'-G'\|_\infty \|G\|_\infty$. So $\omega_k(\mathring\varphi_k(0,0)\mathring\varphi_k(j, \mathbf z))\to0$ as $|\mathbf z|\to\infty$. If (SR$_0$) held for both fields, Theorem~\ref{thm:srladder}(iii) would give $C_k(0)=0$. \end{proof} The linear heat flow, the model of the smoothing that the flow is meant to provide, removes (SR$_0$) at every positive flow time. For $s>0$ let $p_s(\mathbf z)$ be the probability that the continuous-time random walk on $\Z^3$ with generator $\Delta$ is at $\mathbf z$ at time $s$, starting from $0$, so that $\sum_{\mathbf z}p_s(\mathbf z)=1$ and $\sum_{\mathbf z}p_s(\mathbf z)e^{ip\cdot\mathbf z}= e^{-s\varkappa(p)}$, and let $e^{s\Delta}$ act on the slices of a field on $\Z^4$ by $(e^{s\Delta}\psi)(t,\mathbf x)=\sum_{\mathbf y}p_s(\mathbf x- \mathbf y)\psi(t,\mathbf y)$, a series that converges in $L^2$ for fields with bounded second moments. \begin{proposition}\label{prop:heatrp} \textup{(i)} Let $\varpi$ be a finite positive measure on $\mathbf T^3$, $s>0$ and $i\in\{1,2,3\}$. If $\varpi$ and $e^{-2s\varkappa}\varpi$ are both of Poisson type in direction $i$, then $\varpi(\{p:p_i\ne0\})=0$. \textup{(ii)} Let $\psi_k$ be the field of Proposition~\textup{\ref{prop:osobstruction}(a)} and $s>0$. Then \textup{(SR$_0$)} holds for $\psi_k$ and fails for $e^{s\Delta}\psi_k$, at every $k$. \end{proposition} \begin{proof} (i) Let $g\ge0$ be continuous on $\mathbf T^2$ and $g'=ge^{-2s\varkappa_ \perp}$. Since $e^{-2s\varkappa}=e^{-4s(1-\cos p_i)}e^{-2s\varkappa_\perp}$, the measure $\nu=\varpi^i_{g'}$ and the image $e^{-4s(1-\cos t)}\nu$ of $g\,e^{-2s\varkappa}\varpi$ are of Poisson type, with measures $\mu$ and $\mu^s$. Write $\nu$ as in~\eqref{eq:poissonform} and put \[ F(c)=\int_{[0,1)}\frac{1-\lambda^2}{1+\lambda^2-2\lambda c}\,d\mu(\lambda) \] for $c$ in $\Omega=\C\setminus[1,\infty)$, and $F^s$ likewise with $\mu^s$, so that $f(t)=F(\cos t)$ for $t\ne0$. The denominator vanishes only at $c=(1+\lambda^2)/(2\lambda)\ge1$. On a compact $K\subset\Omega$ with $|c|\le R$ and distance $d>0$ to $[1,\infty)$, it is at least $\frac12$ for $\lambda\le1/(4R)$ and at least $2\lambda d\ge d/(2R)$ for larger $\lambda$, so the integrand is bounded on $K$ uniformly in $\lambda$, and $F$ is analytic on $\Omega$. For $\operatorname{Im}c>0$, \[ \operatorname{Im}F(c)=\int_{[0,1)}\frac{2\lambda(1-\lambda^2) \operatorname{Im}c}{|1+\lambda^2-2\lambda c|^2}\,d\mu(\lambda)\ge0 , \] and the same holds for $F^s$. The absolutely continuous parts of $e^{-4s(1-\cos t)}\nu$ in its two representations agree, so $F^s(\cos t)=e^{-4s(1-\cos t)}F(\cos t)$ for almost every $t$, hence for all $c\in(-1,1)$ by continuity, and on $\Omega$, which is connected, by analyticity: $F^s(c)=e^{-4s(1-c)}F(c)$. If $\mu((0,1))>0$, then $\operatorname{Im}F>0$ on the upper half plane, and $\theta(y)=\arg F(iy)\in(0,\pi)$ is continuous in $y>0$. Since $e^{-4s(1-iy)}=e^{-4s}e^{4isy}$, \[ \operatorname{Im}F^s(iy)=e^{-4s}|F(iy)|\sin\bigl(4sy+\theta(y)\bigr)\ge0 \] and $F^s(iy)\ne0$, so the continuous function $4sy+\theta(y)$ takes its values in the union of the disjoint closed intervals $[2\pi j,2\pi j+\pi]$, $j\in\Z$; on the connected set $(0,\infty)$ it stays in one of them, which is impossible since it exceeds $4sy$. So $\mu((0,1))=0$ and $F$ is the constant $B=\mu(\{0\})\ge0$. Then $\operatorname{Im}F^s(iy)=Be^{-4s}\sin(4sy)$, which is $-Be^{-4s}$ at $y=3\pi/(8s)$, so $B=0$. Hence $\nu=\mu(\{1\})\delta_0$. For $g=1$ this says $\int_{\{p_i\ne0\}}e^{-2s\varkappa_\perp}d\varpi=0$, and $e^{-2s\varkappa_\perp}>0$. (ii) The covariance of $\psi_k$ is $q_k^{|x_0-y_0|}\prod_{i=1}^3q_k^{|x_i- y_i|}$, so its measure $\varpi^V_k$ is $T_k\prod_iP_{q_k}(p_i)\frac{dp} {(2\pi)^3}$ with $T_k=a_k^2\sum_{n,n'0$, and that of $e^{s\Delta}\psi_k$, whose spatial covariance kernel has Fourier transform $e^{-2s\varkappa}\prod_iP_{q_k}(p_i)$, is $e^{-2s\varkappa} \varpi^V_k$. Both covariances are bounded and translation invariant. The law of $\psi_k$ is reflection positive for the spatial reflections through sites and bisecting links (Proposition~\ref{prop:osobstruction}(a)); for $c$ supported in $\{x_i\ge0\}$ the sum of (SR$_0$) is $\E[\overline{\psi_k(\iota c)}\,\psi_k(c)]\ge0$ with $\psi_k(c)=\sum c_x \psi_k(x)$ and $\psi_k(\iota c)=\sum c_x\psi_k(\iota x)$, which is (SR$_0$). If (SR$_0$) held for $e^{s\Delta}\psi_k$, Lemma~\ref{lem:poissontype}(ii) for both fields and (i) would give $\varpi^V_k(\{p_i\ne0\})=0$, but $\varpi^V_k$ has a positive density. \end{proof} The continuum analogue is the spatially flowed free field of Proposition~\ref{prop:osobstruction}(c). It is reflection positive in time, and the proof of Proposition~\ref{prop:fourflow}(i), with $x_1$ in the role of time, shows that it is not reflection positive for the reflection of $x_1$: it uses only $H_0(0)>0$ and $H_0''(0)<0$, and here $H_0''(0)=-(2\pi)^{-1}\int p_1^2e^{-2t_fp_1^2}(p_1^2+\mu^2)^{-1}dp_1$. For one field, (SR$_0$) forces spectral weight at the lattice scale: by part (i) below with $n=L/a_k$, $\varpi_k(\{|p|_\infty\ge2\})$ is at least $a_k/(12\delta L)$ times $K^V_k(0)-K^V_k(ne_i)$, the decorrelation of the time-averaged field over the physical distance $L$. \begin{proposition}\label{prop:srrough} Assume \textup{(SR$_0$)} at the spacing $a_k$ for the slice field of $\Phi$, and let $i\in\{1,2,3\}$ and $n\ge1$. \textup{(i)} \[ \begin{split} K^V_k(0)-K^V_k(ne_i)&\le n\bigl(K^V_k(0)-K^V_k(e_i)\bigr)\\ &\le\frac{2\pi n}{\pi-2}\,\varpi^V_k\bigl(\{|p_i|\ge2\}\bigr)\\ &\le12\,\delta\,n\,\varpi_k\bigl(\{|p|_\infty\ge2\}\bigr) . \end{split} \] \textup{(ii)} For $h'$ supported in a cube of side $L'$ with $|h'|\le1$, $a_k\le1$ and $h'_n(\mathbf y)=h'(\mathbf y-na_ke_i)$, \[ I^{h'-h'_n}_k(\delta)\le\frac4\delta(L'+1)^6\bigl(K^V_k(0)-K^V_k(ne_i) \bigr) . \] \textup{(iii)} If in addition \textup{(M$_{\mathrm{sp}}$)} holds for the slice field of $\Delta^m\Phi$, $m\ge1$, and the spatial factor $h$, with the constant $C$ for a cube of side $\ell$, then for all large $k$ \[ K^V_k(0)-K^V_k(ne_i)\le1.7\cdot10^9\,\frac{C\Theta_m^2\delta^2\|\Phi\|_ \infty^2}{\ell^3}\,n\,a_k^{4m-3} . \] \end{proposition} \begin{proof} (i) By Lemma~\ref{lem:poissontype}(ii) with $g=1$, $\nu=\varpi^{V,i}_{k,1}$ is of Poisson type, and $\hat\nu(n)=K^V_k(ne_i)$. Lemma~\ref{lem:poissontype} (iii) gives the first two inequalities. The third follows from~\eqref{eq:varpicompare}, which uses only Lemma~\ref{lem:spatialmeasure}(i), and from $4\pi/(\pi-2)<12$, since $\pi-2>1.14$ and $4\pi<12.57<13.68$. (ii) With $c_{\mathbf x}=a_k^3h'(a_k\mathbf x)$ the coefficients of $h'-h'_n$ are $c-c(\cdot-ne_i)$, with Fourier transform $\hat c\,(1- e^{inp_i})$. So, by~\eqref{eq:varpicompare} and $|1-e^{inp_i}|^2=2(1-\cos np_i)$, \[ \begin{split} I^{h'-h'_n}_k(\delta)&\le\|\hat c\|_\infty^2\,\frac2\delta\int2(1-\cos np_i)\,d\varpi^V_k\\ &=\frac4\delta\|\hat c\|_\infty^2\bigl(K^V_k(0)-K^V_k(ne_i)\bigr), \end{split} \] because $K^V_k$ is real and even. Finally $\|\hat c\|_\infty\le\sum_{\mathbf x}a_k^3|h'(a_k\mathbf x)|\le a_k^3(L'/a_k+1)^3\le(L'+1)^3$. (iii) Proposition~\ref{prop:laplaceladder}(ii) bounds $\varpi_k(\{|p|_\infty\ge2\})$, and $12\cdot1.4\cdot10^8\le1.7\cdot10^9$. \end{proof} So (SR$_0$) for $\Phi$ together with (M$_{\mathrm{sp}}$) for $\Delta^2\Phi$ gives, by (ii) and (iii) with $n=L/a_k$, $I^{h'-h'_n}_k( \delta)\le C'La_k^4$ for spatial translates by a physical distance $L$. Hence $I^{h'-h'_n}_k(\delta)/I_k(\delta)\to0$ for every bounded $L$ unless $\liminf_ka_k^{-4}I_k(\delta)<\infty$: the time-averaged slice field becomes invariant, relative to $I_k(\delta)$, under spatial translations by bounded physical distances. No lower bound on $I_k(\delta)$ along a tuning is proved here. \begin{remark}\label{rem:spatialstatus} For the flowed fields of the Wilson measure along a tuning, the two statements stand as follows. (M$_{\mathrm{sp}}$) is equivalent to the box bound of Proposition~\ref{prop:boxform}(i) and, for $\int h\ne0$, follows from the comparison of Proposition~\ref{prop:boxform}(iii). Exact spatial reflection positivity (SR$_0$) gives that comparison (Theorem~\ref{thm:spatialrp}). The Wilson action is reflection positive in every coordinate hyperplane~\cite{OSei78}, so for states whose restrictions to gauge-invariant functions are reflection positive for the spatial reflections, a field $\Phi_x$ without flow, depending on the links within distance $r$ of $x$, satisfies (SR$_r$); the flowed field $\Phi_x\circ\Phi_{s_k}$ depends on all links of its slice, and Proposition~\ref{prop:spatialflow}(i) bounds the influence of links at distance $R$ only for $R$ beyond a multiple of $s_k=\tau/a_k^2$, so no (SR$_r$) is known for it. Positivity beyond a fixed distance does not suffice (Proposition~\ref{prop:rpexamples}(a)). Nor can (SR$_0$) hold for a field and its Laplacian together: in a state that clusters in space, in particular under the hypothesis of Theorem~\ref{thm:transfer}, it forces $C_k(0)=0$ (Theorem~\ref{thm:srladder}, Corollary~\ref{cor:srladder}), and the linear heat flow destroys it at every positive flow time (Proposition~\ref{prop:heatrp}). So Theorem~\ref{thm:spatialrp} cannot give (M$_{\mathrm{sp}}$) for a class of fields closed under $\Phi\mapsto \Delta\Phi$, and for one field (SR$_0$) forces spectral weight at the lattice scale (Proposition~\ref{prop:srrough}). Compactness of $G$ gives $\bar\varpi_k\le\delta\|\Phi\|_\infty^2$, an upper bound of order one, and no lower bound on $I_k(\delta)$; by Proposition~\ref{prop:laplaceladder} a proof valid for the admissible fields $\Delta^m\Phi$ must suppress the lattice-scale spectral weight of $\varphi_k$ faster than every power of $a_k$. For $G=\SU(2)$ the estimates of Sec.~\ref{ssec:largefields} give each plaquette a conditional probability at least $v_*e^{-3\beta_W}$ of being bad (Proposition~\ref{prop:manydefects}), a power of $a_k$ along couplings of order $\log(1/a_k)$, and the single-link defects of Proposition~\ref{prop:centralsecond} are fixed by the flow at every flow time, with nonnegative second variation for $N\ge4$ and a direction of negative second variation for $N=2,3$. Whether the flow over the time $\tau/a_k^2$ moves the configurations near such defects to configurations with lattice-scale weight below every power of $a_k$ is not known; this is the missing estimate. (Z$_{\mathrm{sl}}$) is equivalent to (M$_Z$) and implies $(\mathrm N_u)$ (Proposition~\ref{prop:mzslice}); for $G=\SU(2)$ it fails along every sequence with $\beta_{W,k}\le0.348$ (Proposition~\ref{prop:msplit}(v)) and along the constant tunings (Proposition~\ref{prop:cubecompatible}(ii)). Compactness bounds $D_k(0)$ above and gives no lower bound on $D_k(\delta)$, which is what (Z$_{\mathrm{sl}}$) requires; exact spatial reflection positivity together with (M$_{\mathrm{sp}}$) does not give it (Proposition~\ref{prop:rpexamples}(b)), and there the term that breaks it is constant in space, so that no spatial smoothing removes it. The missing estimate is that of Remark~\ref{rem:opensplit}: a bound on the high-energy spectral weight of the slice field by its weight at bounded physical energy. At a fixed $k$ both inequalities hold with a constant depending on $k$ whenever $C_k(0)>0$, since then $D_k(\delta)>0$ and $I_k(\delta)>0$ by Proposition~\ref{prop:sliceratio}(i); their content is the uniformity in $k$, and neither is proved along any sequence of couplings with $a_k\to0$. \end{remark} \emph{Status of (M), (W) and (U).} None of the three statements is proved. Proposition~\ref{prop:msplit}(i)--(iii) uses translation and $\vartheta$-invariance and reflection positivity of the states and the reality of $\Phi$, part (iv) also the transfer matrix, and part (v) Theorem~\ref{thm:strongratio}; none of them uses the hypothesis of Theorem~\ref{thm:transfer} or $(\mathrm N_u)$. Proposition \ref{prop:stationary} is deterministic except for the positivity of the conditional densities in part (iii). Proposition~\ref{prop:wardform} uses (M) and the invariance of the states under translations and $\Gamma$, through Theorem~\ref{thm:osaxioms}(i), (ii), and part (iv) the tree bounds. Proposition~\ref{prop:fourflow} concerns the free field and limits of families. Proposition~\ref{prop:tuned}(i) assumes (RC), whose existence along a tuning is not proved and which contains $(\mathrm N_0)$, and the hypothesis of Theorem~\ref{thm:transfer} for the bound on $m_*$; parts (ii) and (iii) assume (M). Lemma~\ref{lem:timekernel} and Propositions~\ref{prop:mzslice} and~\ref{prop:mlocslice}(i)--(iii) use translation and $\vartheta$-invariance, reflection positivity and the positivity of the transfer matrix, and no hypothesis on the tuning; so do Lemma~\ref{lem:spatialmeasure} and Propositions~\ref{prop:boxform} and~\ref{prop:laplaceladder}, while Theorem~\ref{thm:spatialrp} assumes the exact spatial reflection positivity (SR$_0$) of the slice field, which is not known for the flowed fields and, under the hypothesis of Theorem~\ref{thm:transfer}, fails for $\Phi$ or for $\Delta\Phi$ (Corollary~\ref{cor:srladder}). Proposition~\ref{prop:centralsecond} is deterministic. What remains open is (M$_{\mathrm{loc}}$), whose time direction for $n=2$ is Proposition~\ref{prop:mlocslice}(i) and whose remainder is the spatial comparison (M$_{\mathrm{sp}}$) and the bounds of Proposition~\ref{prop:mlocslice}(iii) for even $n\ge4$; (M$_Z$), which for product packets is the short-time bound (Z$_{\mathrm{sl}}$) on the slice field and is strictly stronger than $(\mathrm N_0)$; the Ward identity $\Delta^X_k\to0$ for one $X\in\mathfrak{so}(3)_{\mathrm{sp}}$; and, given (RC), (U1) and (U2) (Remark~\ref{rem:opensplit}). \subsection{Tuning by the ratio}\label{ssec:ratiotuning} This subsection uses the ratio of Sec.~\ref{ssec:ratiocube} as the renormalisation condition: the couplings $\beta_k$ are chosen so that $\rho_k(u,\sigma)=r$ for a fixed $r\in(0,1)$. It determines how the ratio depends on the coupling, when the value $r$ is attained, what is known about the location of the couplings and their relation to the one-loop tuning of Proposition~\ref{prop:scalelosses}, and what the choice gives for (U). The tuning exists if and only if $(\mathrm N_u)$ holds along some sequence of couplings. Along it, $(\mathrm N_u)$ and the convergence of the ratio required in Proposition~\ref{prop:uniquelimit}(iii) hold by construction. Whether $\beta_k\to\infty$, and whether (D) holds along it, remain open, and the joint input of (D) and $(\mathrm N_u)$ is the same as before. \emph{Admissible states.} Throughout, $\beta_W\ge0$ denotes the Wilson coupling, $G=\SU(N)$, and the weight of a configuration is $\exp(\beta_W\sum_pN^{-1}\ReTr Q_p)$. Let $\mathcal A$ be the closure in the supremum norm of the continuous gauge-invariant local functions of the links of $\Z^4$. It is a commutative $C^*$-algebra with unit, and it contains the flowed fields $\Phi_x\circ\Phi_s$ and the packets $A_k$ by Proposition~\ref{prop:spatialflow}(ii). For a finite set $\Lambda$ of links, let $\gamma^{\beta_W}_\Lambda$ resample the links of $\Lambda$ from the law whose density with respect to Haar measure on $G^\Lambda$ is proportional to $\exp(\beta_W\sum_{p\cap\Lambda\ne\emptyset}N^{-1}\ReTr Q_p)$, given the other links. The weight is gauge invariant and Haar measure is invariant under translations of the group, so $\gamma^{\beta_W}_\Lambda$ commutes with gauge transformations; it preserves locality and continuity, and maps $\mathcal A$ into itself. Let $P_\Lambda$ be the number of plaquettes that meet $\Lambda$. \begin{definition}\label{def:admissible} For $\beta_W\ge0$, $\mathcal S(\beta_W)$ is the set of states $\omega$ on $\mathcal A$ that are invariant under translations and the hypercubic group, reflection positive for reflections in hyperplanes through sites and in hyperplanes bisecting links, and satisfy $\omega(\gamma^{\beta_W}_\Lambda H)=\omega(H)$ for every finite set $\Lambda$ of links and every $H\in\mathcal A$. \end{definition} Every $\omega\in\mathcal S(\beta_W)$ is a state as in Appendix~\ref{app:transfer}. For the data $\Phi$, $\tau$, $f_0$ of Sec.~\ref{ssec:ratiocube} we write $C^\omega_k(t)$ and $\rho^\omega_k(u,\sigma)$ for the quantities $C_k(t)$ and $\rho_k(u,\sigma)$ computed in $\omega$. If $\tilde A_k$ is the packet before centring, then $A_k=\tilde A_k-\omega(\tilde A_k)$, and invariance of $\omega$ under $\vartheta$ and $\tau$ gives \begin{equation}\label{eq:Comega} C^\omega_k(t)=\omega\bigl(\vartheta(\bar{\tilde A}_k)\,\tau_{t/a_k}\tilde A_k\bigr)-|\omega(\tilde A_k)|^2 . \end{equation} \begin{lemma}\label{lem:admissible} \textup{(i)} For every finite set $\Lambda$ of links, $H\in\mathcal A$ and $\beta_W,\beta'_W\ge0$, \[ \|\gamma^{\beta_W}_\Lambda H-\gamma^{\beta'_W}_\Lambda H\|_\infty\le \bigl(e^{2P_\Lambda|\beta_W-\beta'_W|}-1\bigr)\|H\|_\infty . \] \textup{(ii)} $\mathcal S(\beta_W)$ is convex and weak-$*$ compact, and the set of pairs $(\beta_W,\omega)$ with $\omega\in\mathcal S(\beta_W)$ is closed. \textup{(iii)} $\mathcal S(\beta_W)$ is not empty: it contains every limit point, as $L\to\infty$ through even values, of the Wilson measures on the tori $(\Z/L\Z)^4$, and the states of Propositions~\ref{prop:state} and~\ref{prop:blockstate}, of Theorems~\ref{thm:su2}, \ref{thm:blocks} and~\ref{thm:groups}, and of Theorem~\ref{thm:onescale}, whenever their hypotheses hold at $\beta_W$. \textup{(iv)} $\mathcal S(0)$ consists of the restriction of the product of Haar measures, and there $C_k(t)=0$ for all $k$ and $t$. \textup{(v)} For a compact interval $I\subset[0,\infty)$ the set $\Gamma_I$ of pairs $(\beta_W,\omega)$ with $\beta_W\in I$ and $\omega\in\mathcal S(\beta_W)$ is compact and connected. \end{lemma} \begin{proof} (i) Write the density of $\gamma^{\beta_W}_\Lambda$ as $e^{\beta_WV}/Z$, where $V$ is the sum of $N^{-1}\ReTr Q_p$ over the plaquettes that meet $\Lambda$, so that $|V|\le P_\Lambda$. Changing $\beta_W$ by $\delta$ multiplies the numerator and the normalisation by factors in $[e^{-P_\Lambda|\delta|},e^{P_\Lambda|\delta|}]$, so the ratio $q$ of the two densities lies in $[e^{-2P_\Lambda|\delta|}, e^{2P_\Lambda|\delta|}]$ and $|q-1|\le e^{2P_\Lambda|\delta|}-1$. Integrate $|H||q-1|$ against the second law. (ii) Invariance, the inequalities $\omega(\vartheta(\bar H)H)\ge0$ and, at fixed $\beta_W$, the identities $\omega(\gamma^{\beta_W}_\Lambda H)= \omega(H)$ are preserved by convex combinations and weak-$*$ limits, and the states of $\mathcal A$ form a weak-$*$ compact set. If $\beta_n\to\beta_W$, $\omega_n\in\mathcal S(\beta_n)$ and $\omega_n\to\omega$, then by (i) \[ |\omega(\gamma^{\beta_W}_\Lambda H)-\omega(H)|=\lim_n|\omega_n (\gamma^{\beta_W}_\Lambda H-\gamma^{\beta_n}_\Lambda H)|=0 . \] (iii) For even $L$ the Wilson measure $\mu_L$ on $(\Z/L\Z)^4$ is invariant under the translations and the hypercubic group of the torus and reflection positive for both kinds of planes~\cite{OSei78,Sei82}. A gauge-invariant local function of $\Z^4$ is a gauge-invariant function on the torus once $L$ exceeds the diameter of its support by two, and for such $H$ and $\Lambda$ with $L$ large, $\mu_L(\gamma^{\beta_W}_\Lambda H)=\mu_L(H)$ because the plaquettes that meet $\Lambda$ are plaquettes of the torus. Since $\mathcal A$ is separable, a diagonal sequence $L_n$ makes $\mu_{L_n}(H)$ converge for all local $H$; the limit is positive, normalised and bounded by $\|H\|_\infty$, extends to a state on $\mathcal A$, and inherits the three properties. The free-boundary states are limits of $W_{\Lambda'}$ along boxes, $W_{\Lambda'}(\gamma^{\beta_W}_\Lambda H)=W_{\Lambda'}(H)$ once every plaquette that meets $\Lambda$ lies in $P(\Lambda')$, and invariance and reflection positivity are part of each of the cited statements for $\beta_W\ge0$. (iv) At $\beta_W=0$ the law of $\gamma^0_\Lambda$ is Haar measure on $G^\Lambda$, so $\omega(H)=\omega(\gamma^0_\Lambda H)$ is the Haar integral of $H$ for every $H$ depending on the links of $\Lambda$. The packet $A_k$ is a uniform limit of functions of the spatial links at times $x_0\ge1$, and $\vartheta(\bar A_k)$ of those at times $x_0\le-1$ (Proposition~\ref{prop:spatialflow}(i)); under the product measure the two are independent, so $C_k(t)=\omega(\vartheta(\bar A_k))\,\omega(\tau_{t/a_k} A_k)=0$. (v) $\Gamma_I$ is closed in the product of $I$ with the compact set of states, by (ii). Suppose $\Gamma_I=U\cup V$ with $U,V$ disjoint, nonempty and closed. Each fibre $\{\beta_W\}\times\mathcal S(\beta_W)$ is convex, hence connected, and lies in $U$ or in $V$. The projections of $U$ and $V$ to $I$ are then disjoint, compact, nonempty by (iii), and cover $I$, which contradicts the connectedness of $I$. \end{proof} \begin{lemma}\label{lem:gibbsunique} Let $\beta=\beta_W/N^2\ge0$ be such that $\gamma^F$ has exactly one Gibbs measure $\mu_F$, and let $\omega_\beta(O)=\mu_F(O^F)$ for gauge-invariant local $O$. Then every state $\omega$ on $\mathcal A$ with $\omega(\gamma^{\beta_W}_\Lambda H)=\omega(H)$ for all finite $\Lambda$ and $H\in\mathcal A$ equals $\omega_\beta$. In particular $\mathcal S(\beta_W)$ has no element other than $\omega_\beta$, and if $\beta_n\to\beta_W$ and $\omega_n\in\mathcal S(\beta_n)$, then $\omega_n\to\omega_\beta$ weak-$*$. This applies under the hypotheses of Propositions~\ref{prop:state} and~\ref{prop:blockstate} and of Theorem~\ref{thm:onescale}, where $\mathcal S(\beta_W)=\{\omega_\beta\}$ by Lemma~\ref{lem:admissible}(iii), in particular for $G=\SU(2)$ and $0\le\beta_W\le0.348$. \end{lemma} \begin{proof} Let $E$ be the set of links and $\mathcal G=G^{\Z^4}$ the compact gauge group with its Haar measure $dg$. For $H\in C(G^E)$ put $(\mathcal PH)(Q)=\int H(Q^g)\,dg$. If $H$ is local, $\mathcal PH$ is local and gauge invariant, and $\mathcal P$ is a contraction, so $\mathcal P$ maps $C(G^E)$ onto $\mathcal A$ and is the identity on $\mathcal A$. Since $\gamma^{\beta_W}_\Lambda$ commutes with gauge transformations, $\mathcal P\gamma^{\beta_W}_\Lambda=\gamma^{\beta_W}_\Lambda\mathcal P$. Hence $\tilde\omega=\omega\circ\mathcal P$ is a probability measure on $G^E$ with $\tilde\omega\gamma^{\beta_W}_\Lambda=\tilde\omega$ for every finite $\Lambda$, as measures. By Lemma~\ref{lem:structure}(i), the line field $F$ is a disjoint union of straight bi-infinite lines. On a line in the direction $\mu$ put $g=1$ at its site with $x_\mu=0$, and extend by $g(y+e_\mu)=g(y)Q_{(y,\mu)}$ in both directions. This defines a continuous map $Q_F\mapsto g$, and the gauge transform $T(Q)=(Q^g_e)_{e\in A}$, $Q^g_e=g_xQ_eg_y^{-1}$ for $e=(x,y)$, is a continuous map $G^E\to G^A$; $Q^g$ equals $1$ on $F$. Let $V\subset A$ be finite and $h$ bounded and measurable on $G^A$. Given the links outside $V$, which fix $g$ and $T(Q)_{A\setminus V}$, the map $Q_V\mapsto T(Q)_V$ is a product of left and right translations and preserves Haar measure, and $\ReTr Q_p=\ReTr Q^g_p$ for every plaquette. With $\beta_WN^{-1}=N\beta$ this gives $\gamma^{\beta_W}_V(h\circ T)=(\gamma^F_Vh)\circ T$, where $\gamma^F_V$ is the kernel of the specification $\gamma^F$ on $V$. The law $\nu=\tilde\omega\circ T^{-1}$ therefore satisfies \[ \nu(\gamma^F_Vh)=\tilde\omega(\gamma^{\beta_W}_V(h\circ T))= \tilde\omega(h\circ T)=\nu(h), \] so $\nu$ is a Gibbs measure of $\gamma^F$ and $\nu=\mu_F$. For gauge-invariant local $O$, $O(Q)=O(Q^g)=O^F(T(Q))$, because $O$ depends on finitely many links. Hence $\omega(O)=\tilde\omega(O^F\circ T)=\mu_F(O^F)= \omega_\beta(O)$, and two states that agree on a dense set are equal. For the last statement, every limit point of $(\omega_n)$ lies in $\mathcal S(\beta_W)$ by Lemma~\ref{lem:admissible}(ii). The three cited statements give (i) and (ii) of Proposition~\ref{prop:state}, and Theorems~\ref{thm:su2} and~\ref{thm:blocks} are proved through Propositions~\ref{prop:state} and~\ref{prop:blockstate}. \end{proof} For $\SU(2)$ and $\beta_W\le0.348$ the upper bound of Theorem~\ref{thm:strongratio} therefore holds for every admissible state. For the other groups in the range of Theorem~\ref{thm:groups} the bound holds for the state of that theorem; uniqueness of the Gibbs measure of $\gamma^F$ is not established there, so it is not known for all of $\mathcal S(\beta_W)$. \begin{proposition}\label{prop:ratiovalues} Fix $\Phi$, $\tau$, $f_0$, dyadic $u\ge0$ and $\sigma>0$, and $k$ with $u/a_k,\sigma/a_k\in\Z$. \textup{(i)} For each $t$, $\omega\mapsto C^\omega_k(t)$ is weak-$*$ continuous on the states of $\mathcal A$. For $\omega\in\mathcal S(\beta_W)$, $0\le C^\omega_k(t')\le C^\omega_k(t)$ for $t'\ge t$, and $C^\omega_k(t)>0$ for all $t$ if and only if $C^\omega_k(0)>0$. \textup{(ii)} Let $I$ be a compact interval with \begin{enumerate}[label=\textup{(P$_k$)},leftmargin=*] \item $C^\omega_k(0)>0$ for every $(\beta_W,\omega)\in\Gamma_I$. \end{enumerate} Then $(\beta_W,\omega)\mapsto\rho^\omega_k(u,\sigma)$ is continuous on $\Gamma_I$ with values in $(0,1]$; its set of values on $\Gamma_I$, and on each fibre $\mathcal S(\beta_W)$, is a compact interval. \textup{(iii)} If $\mathcal S(\beta_*)=\{\omega_*\}$ and $C^{\omega_*}_k(0)>0$, then for every $\varepsilon>0$ there is $\delta>0$ such that $C^\omega_k(0)>0$ and $|\rho^\omega_k(u,\sigma)- \rho^{\omega_*}_k(u,\sigma)|<\varepsilon$ for all $\omega\in\mathcal S(\beta_W)$ with $|\beta_W-\beta_*|<\delta$. \end{proposition} \begin{proof} (i) In~\eqref{eq:Comega}, $\tilde A_k$ and $\vartheta(\bar{\tilde A}_k)\tau_{t/a_k}\tilde A_k$ are finite sums of elements of $\mathcal A$, which gives continuity. By Appendix~\ref{app:transfer}, $C^\omega_k(t)=\ip{\Psi_{A_k}}{\mathbb T^{t/a_k}\Psi_{A_k}}$ with $0\le\mathbb T\le1$, which is nonnegative and nonincreasing in $t$; the last statement is Proposition~\ref{prop:ratioexact}(i). (ii) By (i) the ratio is a quotient of continuous functions with a positive denominator, with values in $(0,1]$ by Proposition~\ref{prop:ratioexact}(ii). The image of the compact connected set $\Gamma_I$ (Lemma~\ref{lem:admissible}(v)) and of each convex fibre is compact and connected. (iii) Otherwise there are $\beta_n\to\beta_*$ and $\omega_n\in\mathcal S(\beta_n)$ violating one of the two properties. A subsequence converges, its limit lies in $\mathcal S(\beta_*)$ by Lemma~\ref{lem:admissible}(ii) and equals $\omega_*$, and by (i) $C^{\omega_n}_k(t)\to C^{\omega_*}_k(t)$ for $t=0,2u,2u+\sigma$, a contradiction. \end{proof} So the ratio depends continuously on the coupling wherever the admissible state is unique, and in general its values at one coupling form a compact interval. The hypothesis (P$_k$) is qualitative, a nonvanishing at each fixed $k$, and it is open in infinite volume: Theorem~\ref{thm:flowlimit} assumes it, and in a periodic box it is Proposition~\ref{prop:flowpositive}. It fails at $\beta_W=0$ by Lemma~\ref{lem:admissible}(iv). In the box the ratio is real-analytic. \begin{proposition}\label{prop:boxratio} In the box $\Z\times(\Z/L_k\Z)^3$ of Proposition~\textup{\ref{prop:flowpositive}}, with its $f_0=g\otimes h$ and $\Phi$, let $\rho_k(\beta_W)$ be the ratio $\rho_k(u,\sigma)$ of the infinite-time Wilson state of the box at $\beta_W>0$. \textup{(i)} $\rho_k$ is real-analytic on $(0,\infty)$, takes values in $(0,1)$, and $\rho_k(\beta_W)\to0$ as $\beta_W\to0$. \textup{(ii)} For $r\in(0,1)$ the set $\{\beta_W>0:\rho_k(\beta_W)=r\}$ has no accumulation point in $(0,\infty)$, and it is nonempty if and only if $\rho_k(\beta_W)\ge r$ for some $\beta_W>0$. \end{proposition} \begin{proof} (i) In the notation of the proof of Proposition~\ref{prop:flowpositive}, the transfer matrix is $T_{\beta_W}=MKM$ on the gauge-invariant part $\mathcal K$ of $L^2(G^E)$, $E$ the finite set of spatial links of a slice. The function $e^{\beta_WN^{-1}\operatorname{Re}\chi}$ is an entire function of $\beta_W$ with values in $C(G)$, and convolution with a continuous function has operator norm at most its supremum, so $K$ is entire in $\beta_W$ in operator norm; so is $M$, multiplication by the exponential of $\beta_W$ times a bounded function. At $\beta_W=0$, $K$ is the projection onto the constants and $M=1$, so $T_0$ is the projection onto the constant function $1$. For $\beta_W\ge0$ the largest eigenvalue $\lambda_0$ of the compact positive operator $T_{\beta_W}$ is simple and isolated, by the proof of Proposition~\ref{prop:flowpositive} for $\beta_W>0$ and because $T_0$ is a projection of rank one for $\beta_W=0$. For $\beta_W$ near a point $\beta_*\ge0$ the Riesz projection $P_0=(2\pi i)^{-1}\oint(z- T_{\beta_W})^{-1}dz$, over a fixed circle around $\lambda_0(\beta_*)$ that separates it from the rest of the spectrum, is analytic in $\beta_W$, has rank one because $\|P_0-P_0(\beta_*)\|<1$, and projects onto the eigenspace of $\lambda_0(\beta_W)=\Tr(T_{\beta_W}P_0)$; by Weyl's inequality the other eigenvalues stay outside the circle, so this is the largest eigenvalue. For real $\beta_W$, $P_0$ is the orthogonal projection onto $\psi_0>0$, so $\ip1{P_01}=\ip1{\psi_0}^2>0$ and $\psi_0=P_01/\ip1{P_01}^{1/2}$ is real-analytic. The flowed fields do not depend on $\beta_W$. Hence, with $A=T_{\beta_W}/\lambda_0$, $F=\sum_xh(a_kx)\Phi_x\circ\Phi_{s_k}$ and $F^c=F-\ip{\psi_0}{F\psi_0}$, \[ C_k(t)=a_k^8\ip{p(A)F^c\psi_0}{A^{t/a_k}p(A)F^c\psi_0} \] is real-analytic in $\beta_W\ge0$, and positive for $\beta_W>0$ by Proposition~\ref{prop:flowpositive}, so $\rho_k$ is real-analytic on $(0,\infty)$. The vector $\Psi_u=A^{u/a_k}p(A)F^c\psi_0$ is nonzero and orthogonal to $\psi_0$, so with $\lambda_1$ the largest eigenvalue of $T_{\beta_W}$ on $\psi_0^\perp$, \[ \rho_k(\beta_W)=\frac{\ip{\Psi_u}{A^{\sigma/a_k}\Psi_u}}{\|\Psi_u\|^2}\le \Bigl(\frac{\lambda_1}{\lambda_0}\Bigr)^{\sigma/a_k}<1 . \] By the minimax principle with the vector $1$, \[ \lambda_1\le\max_{w\perp1,\ \|w\|=1}\ip w{T_{\beta_W}w}\le \|T_{\beta_W}-T_0\|\to0 , \] while $\lambda_0\ge\ip1{T_{\beta_W}1}\to1$ as $\beta_W\to0$. (ii) The function $\rho_k-r$ is real-analytic on the connected set $(0,\infty)$ and not identically zero, since it tends to $-r$ at $0$, so its zeros do not accumulate in $(0,\infty)$. If $\rho_k(\beta'_W)\ge r$, then $\rho_k0$, and $r\in(0,1)$. A \emph{ratio tuning at level $r$} is a sequence $(\beta_k,\omega_k)$ with $\omega_k\in\mathcal S(\beta_k)$, $C^{\omega_k}_k(0)>0$ and $\rho^{\omega_k}_k(u,\sigma)=r$ for all $k\ge k_0$. In the box of Proposition~\textup{\ref{prop:boxratio}}, $\omega_k$ is the state of the box. \end{definition} \begin{theorem}\label{thm:ratiotuning} Let $G=\SU(2)$, $r\in(0,1)$ and $\beta_-\in(0,0.348]$, and assume \textup{(P$_k$)} on every compact interval in $[\beta_-,\infty)$, for all large $k$. \textup{(i)} A ratio tuning at level $r$ with $\beta_k\ge\beta_-$ exists if and only if, for every large $k$, some $\beta_W\ge\beta_-$ and $\omega\in\mathcal S(\beta_W)$ have $\rho^\omega_k(u,\sigma)\ge r$. If $\liminf_k\rho^{\omega_k}_k(u,\sigma)=c>0$ for some $\omega_k\in\mathcal S(\beta_k)$ with $\beta_k\ge\beta_-$, ratio tunings exist at every level $r\in(0,c)$; conversely, a ratio tuning at level $r$ satisfies $(\mathrm N_u)$. In the box the same holds without \textup{(P$_k$)} and without the condition $\beta_k\ge\beta_-$. \textup{(ii)} Every ratio tuning at level $r$ in infinite volume has $\beta_k>0.348$ for all large $k$, and along every ratio tuning, in infinite volume or in the box: \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item $\rho_k(u',\sigma)\ge r$ for $u'\ge u$; for dyadic $\sigma'$ with $\sigma'/a_k\in\Z$, $r^{\sigma'/\sigma}\le\rho_k(u,\sigma')\le r$ if $\sigma'\ge\sigma$, and $r\le\rho_k(u,\sigma')\le r^{\sigma'/\sigma}$ if $\sigma'\le\sigma$; \item every limit of Theorem~\textup{\ref{thm:flowlimit}} for $f_{0,u}$ is nontrivial and has $\rho(\sigma)=r$; \item the spectral measures of Proposition~\textup{\ref{prop:ratioexact}(iv)} satisfy, with $\Lambda=\sigma^{-1}\log(2/r)$ and every $\lambda\ge0$, \[ \mu_{k,u}([0,\Lambda])\ge\tfrac r2,\qquad\mu_{k,u}([0,\lambda])\le re^{\sigma\lambda}; \] \item the norm $e^{-m_k}$ of $\mathbb T_k$ on $\Omega_k^\perp$ satisfies $m_k\le a_k\sigma^{-1}\log(1/r)$. \end{enumerate} \textup{(iii)} Let $u=0$, let $\Phi$ be real and $W_3$-invariant as in Sec.~\textup{\ref{ssec:osaxioms}}, and suppose that $D(n_k,\alpha)$ holds at $\beta_k$ with $n_ka_k\le\ell$ and that \textup{(M)} holds. Then $\omega_k$ is the state of Theorem~\textup{\ref{thm:onescale}}, $\alpha\ge r^{2\ell/\sigma}$ and $m_*\le\sigma^{-1}\log(1/r)$. If every subsequential limit of Theorem~\textup{\ref{thm:osaxioms}(i)} lies in a set $\mathfrak F$ as in Proposition~\textup{\ref{prop:uniquelimit}(iii)} with $\sigma_0=\sigma$, then \textup{(U)} holds and the moments $\omega_k(\prod_jZ_kA_k(f_j))$ converge for all $n$ and $f_j\in C_c(\R^4)$. \end{theorem} \begin{proof} (i) Let $k$ be so large that $e^{-m_0\sigma/a_k}0.348$ is then replaced by the exclusion of the couplings at which the admissible state is that of Theorem~\ref{thm:groups}. Part (d) is the precise sense in which the tuning has to leave every region of couplings with a lattice gap bounded below: the lattice gap at $\beta_k$ is at most $a_k\sigma^{-1}\log(1/r)$. \emph{Circularity with (D).} The ratio is defined for every admissible state without (D), and (D) is a property of the specification at one coupling, so the definition of the ratio tuning does not use (D), and (D) can be posed along it, as in Theorem~\ref{thm:ratiotuning}(iii). The next statement shows that this does not reduce the joint input. \begin{proposition}\label{prop:ratiodregion} Fix $k$ and a compact interval $I=[\beta_-,B]$ such that, at every $\beta_W\in I$, $D(n,\alpha)$ holds with $na_k\le\ell$ ($n$ may depend on $\beta_W$) and $C_k(0)>0$ for the state of Theorem~\textup{\ref{thm:onescale}}. Then $\mathcal S(\beta_W)$ is that state for $\beta_W\in I$, the function $\beta_W\mapsto\rho_k(u,\sigma)$ is continuous on $I$ and bounded by $\alpha^{\sigma/(2\ell)}$, and for $r>\rho_k(u,\sigma)|_{\beta_-}$ the value $r$ is attained on $I$ if and only if $\max_I\rho_k(u,\sigma)\ge r$. Consequently, given such intervals $I_k$ for all large $k$, with $\rho_k(u,\sigma)0\}$, and the coefficient $b=\frac{11N^2}{12\pi^2}$ of Proposition~\ref{prop:scalelosses}, with $h=b\log2$. \begin{proposition}\label{prop:ratiolocation} Assume the hypotheses of Theorem~\textup{\ref{thm:ratiotuning}}. \textup{(i)} If for every $B$, $R_k(B)0$ such that the transfer matrix of every $\omega\in\mathcal S(\beta_W)$, $\beta_W\le B$, has norm at most $e^{-m(B)}$ on $\Omega^\perp$, then $R_k(B)\le e^{-m(B)\sigma/a_k}$ and every ratio tuning has $\beta_k\to\infty$. \textup{(ii)} For every state $\omega$ as in Appendix~\textup{\ref{app:transfer}} and every $k$, \begin{equation}\label{eq:rescale} \rho^\omega_{k+1}(u,\sigma;\tau,f_0)=\rho^\omega_k(2u,2\sigma;4\tau, f_0^{(2)}) . \end{equation} So $\beta_{k+1}$ of a ratio tuning for $(u,\sigma,\tau,f_0)$ solves at level $k$ the equation for $(2u,2\sigma,4\tau,f_0^{(2)})$, and with $\Delta_k=\beta_{k+1}-\beta_k$ and $D_k=\beta_k-b\log(1/a_k)$, \[ D_K-D_{k_0}=\sum_{j=k_0}^{K-1}(\Delta_j-h) . \] \textup{(iii)} If $\beta_j>0$ and $\Delta_j\ge h+c/\beta_{j+1}$ for $j\ge k_0$, with $c>0$, then $D_K$ is nondecreasing for $K\ge k_0$ and $D_K\to\infty$. If $\sum_j|\Delta_j-h|<\infty$, then $D_K$ converges. \textup{(iv)} Suppose that $\rho^\omega_k(u,\sigma)\le\rho^{\omega'}_k (u,\sigma)$ whenever $\omega\in\mathcal S(\beta_W)$, $\omega'\in\mathcal S(\beta'_W)$, $\beta_W<\beta'_W$ and $k$ is large, and that for every $r'\in(0,1)$ there is a ratio tuning at level $r'$ with $\Delta_j\ge h+c(r')/\beta_{j+1}$, $c(r')>0$. Then $(\mathrm N_u)$ fails along every sequence of admissible states at couplings $\beta_k\le b\log(1/a_k)+M$. \end{proposition} \begin{proof} (i) In the first case, for every $B$ the couplings $\beta_k$ exceed $B$ for large $k$. In the second, the intermediate value theorem of Proposition~\ref{prop:ratiovalues}(ii) on $[\beta_-,B]$ gives solutions $\beta_k\le B$ at those $k$, which replace the given terms. In the third, $\rho^\omega_k(u,\sigma)=\ip{\Psi_{k,u}}{\mathbb T^{\sigma/a_k}\Psi_{k,u}}\le e^{-m(B)\sigma/a_k}$, and the first case applies. (ii) Since $a_k=2a_{k+1}$, $\tau/a_{k+1}^2=4\tau/a_k^2$ and $a_k^4f_0^{(2)}(a_kx)=2^4a_{k+1}^4f_0(a_{k+1}x)$, the packet of $(4\tau,f_0^{(2)})$ at level $k$ is $2^4$ times that of $(\tau,f_0)$ at level $k+1$, and a time $t$ at level $k+1$ is $t/a_{k+1}=2t/a_k$ lattice steps. Hence $C^\omega_k(2t;4\tau,f_0^{(2)})=2^8C^\omega_{k+1}(t;\tau,f_0)$, which gives~\eqref{eq:rescale}. The identity for $D_K$ follows from $b\log(1/a_{j+1})-b\log(1/a_j)=h$. (iii) The terms of the sum are positive. If $D_K\le M'$ for all $K$, then $0<\beta_{j+1}\le b\log(1/a_{j+1})+M'$, which grows linearly in $j$, and $\sum_jc/\beta_{j+1}=\infty$, a contradiction; a nondecreasing sequence that is not bounded above tends to $\infty$. The second statement is clear. (iv) If $(\mathrm N_u)$ holds along admissible states at couplings $\beta_k$, choose $r'$ with $\rho^{\omega_k}_k(u,\sigma)\ge2r'$ for large $k$, and let $(\beta'_k,\omega'_k)$ be a ratio tuning at level $r'$ as assumed. If $\beta_k<\beta'_k$, monotonicity would give $2r'\le r'$; hence $\beta_k\ge\beta'_k$ and $\beta_k-b\log(1/a_k)\ge D'_k\to\infty$ by (iii). \end{proof} By~\eqref{eq:rescale}, $\Delta_k$ is the change of coupling that compensates, at a fixed lattice level, a doubling of all physical scales of the renormalisation condition: a step-scaling function of the ratio. Neither the monotonicity in (iv) nor any bound on $\Delta_k$ is proved here. \begin{remark}\label{rem:twoloop} Perturbation theory to two loops~\cite{GW73,Pol73,Cas74,Jon74} gives, with $b_0=\frac{11N}{48\pi^2}$, $b_1=\frac{34N^2}{3(16\pi^2)^2}$ and $L=\log\frac1{a\Lambda}$, \[ g(a)^{-2}=2b_0L+\frac{b_1}{b_0}\log L+O(1) , \] so the coupling of fixed physics is $\beta_W=bL+\gamma\log L+O(1)$ with $b=4Nb_0$ and $\gamma=\frac{b_1}{2b_0^2}\,b=\frac{51}{121}\,b$, the ratio $\frac{51}{121}$ being the same for every $N$. Along $a_k=2^{-k}a_0$ its increments are $\Delta_j=h+\gamma h/\beta_{j+1}+O(\log\beta_j/\beta_j^2)$, which is the hypothesis of Proposition~\ref{prop:ratiolocation}(iii) for every $c<\gamma h$. For $\SU(2)$, $b=11/(3\pi^2)$ and $h=b\log2$, and $9.8696<\pi^2<9.8697$, $0.69314<\log2<0.69315$ give $b\in[0.37150,0.37152]$, $h\in[0.25750,0.25752]$, $\gamma\in[0.15658,0.15659]$ and $\gamma h\in[0.04032,0.04033]$. By the same expansion, along a tuning with $\beta_k=b\log(1/a_k)+M$ the invariant scale is $\Lambda\propto (\log(1/a_k))^{\gamma/b}$, so masses grow like $(\log(1/a_k))^{51/121}$ and the limit is trivial. This is the perturbative prediction, not a result: it says that the tuning to fixed physics differs from the one-loop tuning of Proposition~\ref{prop:scalelosses} by $\gamma\log\log(1/a_k)$, as the footnote in Sec.~\ref{sec:limits} records, and that a tuning with bounded $M$ is, by this prediction, a tuning to a trivial limit. \end{remark} The estimates of Proposition~\ref{prop:scalelosses} do not use the one-loop form of the tuning, only the size of its increments. \begin{corollary}\label{cor:scalelossesgeneral} Let $(\beta_k)$ be couplings with $\beta_{k+1}-\beta_k\in[h_-,h_+]$ for $k\ge k_0$, where $00$, and for $k\ge i$ put $J_k=k-i$ and $\beta^{(j)}_k=\beta_{k-j}$, the coupling at the scale $2^ja_k$. Let $c_{k,j}\in[0,\frac12]$. \textup{(i)} If $c_{k,j}\ge c/\beta^{(j)}_k$ for $0\le j\le J_k$, with $c>0$, then \[ \prod_{j=0}^{J_k}(1-c_{k,j})\le\Bigl(\frac{\beta_i}{\beta_i+h_+(J_k+1)} \Bigr)^{c/h_+}\to0 . \] \textup{(ii)} If $c_{k,j}\le c/(\beta^{(j)}_k)^2$ for $0\le j\le J_k$, then for every $k\ge i$ \[ \prod_{j=0}^{J_k}(1-c_{k,j})\ge\exp\Bigl(-2c\Bigl(\frac1{\beta_i^2}+ \frac1{h_-\beta_i}\Bigr)\Bigr). \] \end{corollary} \begin{proof} With $l=J_k-j$, $\beta_i+h_-l\le\beta^{(j)}_k\le\beta_i+h_+l$. The proofs of Proposition~\ref{prop:scalelosses}(ii) and~(iii) apply with $\beta_i$ in place of $\beta^\pm_\sigma$ and $h_\pm$ in place of $h$. \end{proof} The two-loop tuning of Remark~\ref{rem:twoloop} has increments in $[h,h+\gamma h/\beta_{k_0}]$, so the conclusion drawn from Proposition~\ref{prop:scalelosses}, that a scale-by-scale comparison has to be exact to first order in the running coupling, holds for it as well, and for a ratio tuning whose increments are bounded above and below. \emph{Large couplings.} Concentration on flat configurations does not determine the ratio. Let $\delta_1$ be the evaluation at the configuration with every link equal to $1$, a character of $\mathcal A$. \begin{proposition}\label{prop:largeb} Let $\omega_n$ be states on $\mathcal A$, invariant under translations and the hypercubic group, with $\varepsilon_{\omega_n}\to0$. Then $\omega_n\to\delta_1$ weak-$*$, and $C^{\omega_n}_k(t)\to0$ for every $k$ and $t$. For $\SU(2)$ this applies to limit points of the torus measures at couplings $\beta_n\to\infty$, which lie in $\mathcal S(\beta_n)$. \end{proposition} \begin{proof} Let $O$ be a continuous gauge-invariant function of the links of a box $B$, and $\eta>0$. There is $\delta>0$ such that $|O(U)-O(1)|<\eta$ whenever $s_p(U)<\delta$ for all $p\in P(B)$. Otherwise, by compactness, some $U$ with $U_p=1$ for all $p\in P(B)$ would have $|O(U)-O(1)|\ge\eta$; but a flat configuration of a box is a gauge transform of the trivial one (in the axial gauge, which sets the links of a maximal tree to $1$, the plaquettes determine the remaining links inductively over the coordinates), and $O$ is gauge invariant. Hence, pointwise, \[ |O-O(1)|\le\eta+\frac{2\|O\|_\infty}\delta\sum_{p\in P(B)}s_p , \] and $|\omega_n(O)-O(1)|\le\eta+2\|O\|_\infty|P(B)|\varepsilon_{\omega_n}/ \delta$. This gives $\omega_n(O)\to O(1)$ for local $O$, and for all of $\mathcal A$ by density. In~\eqref{eq:Comega}, $\delta_1$ is multiplicative and fixed by $\vartheta$ and $\tau$, so $C^{\omega_n}_k(t)\to|\tilde A_k(1)|^2-|\tilde A_k(1)|^2=0$. For $\SU(2)$, Proposition~\ref{prop:energybeta} gives $\varepsilon\le(K_1+K_3/ \beta_n)/\beta_n$ for the limit points, and Lemma~\ref{lem:admissible}(iii) places them in $\mathcal S(\beta_n)$. \end{proof} At fixed $k$ both $C_k(2u)$ and $C_k(2u+\sigma)$ therefore tend to zero as $\beta_W\to\infty$ along these states, and the ratio is a quotient of two vanishing quantities, on which concentration gives no information. In particular (P$_k$) holds at no uniform level: $\inf C_k(0)$ over these states is zero. \emph{Centre-odd packets in a fixed box.} In the box, the question left open after Proposition~\ref{prop:boxratio} has an answer for packets that change sign under the centre of $\SU(2)$. For these packets the ratio tends to $1$ as $\beta_W\to\infty$ at every fixed $k$. Ratio tunings exist at every level, and $(\mathrm N_u)$ holds along every sequence with $a_k^2\beta_k/\log(1/a_k)\to\infty$. The limits are degenerate, and the mechanism has no counterpart in infinite volume. Let $G=\SU(2)$, fix $L\ge3$, and take the box $\Z\times(\Z/L\Z)^3$ of Proposition~\ref{prop:flowpositive} with $L_k=L$ for all $k$. Let $E$ be the set of $3L^3$ spatial links of a slice and $dU$ the Haar probability measure on $G^E$. For $V\in G$ put $s(V)=1-\frac12\Tr V$, so that $|V-I|^2=4s(V)$. Write $s_p=s(U_p)$ and \[ e_s(U)=\sum_ps_p(U),\qquad e_t(U,U')=\sum_{e\in E}s(U_eU_e'^{-1}), \] where the first sum runs over the $3L^3$ plaquettes of the slice. With $b=\beta_W$, the transfer matrix $\mathbb T_b=MKM$ of the proof of Proposition~\ref{prop:flowpositive} is the integral operator with the continuous positive kernel \[ t_b(U,U')=\exp\Bigl(b\bigl(6L^3-\tfrac12e_s(U)-e_t(U,U')-\tfrac12e_s(U') \bigr)\Bigr). \] Here $M$ contributes half of the weight $b(1-s_p)$ of every plaquette of the two slices, and $K$ contributes the weight $b(1-s(U_eU_e'^{-1}))$ of the time-like plaquette above $e$ in the temporal gauge. The kernel, the largest eigenvalue $\lambda_0=\lambda_0(b)$, the eigenvector $\psi_0>0$ and $A=\mathbb T_b/\lambda_0$ do not depend on $k$. Put $\omega(F)=\ip{\psi_0} {F\psi_0}$ for functions $F$ of the slice and \[ \begin{split} \varepsilon_s(b)&=\omega(s_p),\\ \varepsilon_t(b)&=\frac1{\lambda_0}\iint s(U_eU_e'^{-1})\,\psi_0(U) t_b(U,U')\psi_0(U')\,dU\,dU' . \end{split} \] These are the energies of a spatial and of a time-like plaquette in the state of the box. The function $\psi_0(U)t_b(U,U')\psi_0(U')/\lambda_0$ is a probability density on $G^E\times G^E$. The following maps act on $G^E$: the translations of the slice, the permutations of the three axes, and, for $\mu=1,2,3$, the map $\zeta_\mu$ that multiplies by $-I$ the links $(x,\mu)$ with $x_\mu=L-1$. Each of them preserves the Haar measure, commutes with the gauge transformations, and leaves $t_b$ invariant. For $\zeta_\mu$ this holds because a plaquette contains either none or two of the links it changes, $-I$ is central, and $U_eU_e'^{-1}$ is unchanged. Hence $\psi_0$ composed with any of these maps is again a positive normalised eigenvector for $\lambda_0$, and so it equals $\psi_0$. The translations and permutations act transitively on the plaquettes and on the links, and $s(V^{-1})=s(gVg^{-1})=s(V)$, so $\varepsilon_s$ and $\varepsilon_t$ do not depend on $p$ and $e$. \begin{lemma}\label{lem:boxenergy} For $b\ge1$, \[ \varepsilon_s(b)+\varepsilon_t(b)\le\varepsilon(b):=\frac{3(\log b+5)}{2b}. \] The function $\varepsilon$ decreases on $[1,\infty)$, and for $0<\epsilon' \le3$ the inequality $\varepsilon(b)\le\epsilon'$ holds for every $b\ge B(\epsilon')$, where $B(\epsilon')=(3/\epsilon')(\log(3/\epsilon')+4)$. \end{lemma} \begin{proof} \emph{The eigenvalue.} The operator $\mathbb T_b$ on $L^2(G^E)$ is positive and compact, and its kernel is continuous and positive. By Jentzsch's theorem its largest eigenvalue is simple, with a positive eigenfunction. The gauge transformations commute with $\mathbb T_b$ and preserve positivity, so that eigenfunction is gauge invariant. Hence it is $\psi_0$, and $\lambda_0$ is also the largest eigenvalue on $L^2(G^E)$. In particular $\int t_b(U,U') \psi_0(U')\,dU'=\lambda_0\psi_0(U)$ for every $U$, and $\lambda_0=\sup_f \ip f{\mathbb T_bf}/\|f\|^2$ over $f\in L^2(G^E)$. \emph{Convexity.} Since $\ip1{\psi_0}>0$, we have $\lambda_0^n\ip1{\psi_0}^2 \le\ip1{\mathbb T_b^n1}\le\lambda_0^n$. Hence $\log\lambda_0=\lim_n n^{-1}\log\ip1{\mathbb T_b^n1}$. Here $\ip1{\mathbb T_b^n1}$ is the integral over $(G^E)^{n+1}$ of $\prod_{i2.718^2>7.38$, so $\log\frac{20}3<\log7<2$. Also $\pi<3.15$ gives $\pi^3<31.3<32$ and $\frac{3\pi^3}8<12$, while $e^5>2.718^5>148>144=12^2$ gives $\frac23\log12<\frac53$. So the constant is less than $1+2+\frac53<5$, and $\varepsilon_s+\varepsilon_t\le\frac3{2b}(\log b+5)$. \emph{The coupling.} The derivative of $(\log b+5)/b$ is $(-4-\log b)/b^2<0$. Let $x=3/\epsilon'\ge1$ and $b\ge x(\log x+4)\ge1$. Since $\log y\le y-1$, \[ \begin{split} \log b+5&\le\log x+\log(\log x+4)+5\\ &\le2\log x+8=\frac2x\,x(\log x+4) . \end{split} \] So $(\log b+5)/b\le2/x$ at $b=x(\log x+4)$, hence for all larger $b$, and then $\varepsilon(b)\le3/x=\epsilon'$. \end{proof} For a site $x$ of the slice and $\mu\in\{1,2,3\}$, let $W_\mu(x)= U_{(x,\mu)}U_{(x+\hat\mu,\mu)}\cdots U_{(x+(L-1)\hat\mu,\mu)}$ be the holonomy of the straight loop of $L$ links through $x$ in direction $\mu$. Put \begin{equation}\label{eq:centreoddPhi} \Phi=\Tr W_1(0)+\Tr\bigl(W_1(0)W_2(0)\bigr) , \end{equation} a real, continuous, gauge-invariant function of $2L$ spatial links of the slice. The second term is the trace of the holonomy of the closed path that goes once around direction $1$ and then once around direction $2$. The loop $W_1(x)$ contains exactly one link changed by $\zeta_1$ and none changed by $\zeta_2$, and $W_2(x)$ contains one link changed by $\zeta_2$. So $\Phi_x\circ\zeta_1=-\Phi_x$ for every translate $\Phi_x$, while $\zeta_2$ changes the sign of the second term only. The packets of $\Phi$ are therefore odd under the centre transformation $\zeta_1$. In infinite volume no nonzero element of $\mathcal A$ is odd under the corresponding map (Remark~\ref{rem:centreodd}). \begin{lemma}\label{lem:loops} \textup{(a)} For $V,W\in\SU(2)$ put $x=\Tr V$, $y=\Tr W$, $z=\Tr VW$ and $q=2-\Tr(VWV^{-1}W^{-1})\in[0,4]$. Then \[ x^2+y^2+z^2-xyz=4-q,\qquad x^2+y^2+z^2\ge3-\tfrac34q . \] The identity is the trace identity of Fricke for $\mathrm{SL}(2)$, which expresses the trace of a commutator through $\Tr V$, $\Tr W$ and $\Tr VW$. \textup{(b)} For unitary $g_1,\dots,g_n$, $|g_1\cdots g_n-I|\le\sum_i|g_i-I|$. For $U_i,V_i\in\SU(2)$, $|\Tr(U_1\cdots U_m)-\Tr(V_1\cdots V_m)|\le\sqrt2 \sum_i|U_i-V_i|$. \textup{(c)} Let $\gamma$ be a closed path of links of the slice, $\gamma'$ its translate by $\pm\hat\mu$, and $h_\gamma$, $h_{\gamma'}$ their holonomies. Each link of $\gamma$ not parallel to $\mu$ spans, with its translate, a plaquette. Let $R(\gamma,\mu)$ be the list of these plaquettes, one entry for each such link. Then \[ |\Tr h_\gamma-\Tr h_{\gamma'}|\le\sqrt2\sum_{p\in R(\gamma,\mu)}|U_p-I| . \] \textup{(d)} Let $P_{12}$ be the set of $L^2$ plaquettes of the plane $x_3=0$ in the directions $1,2$. Then \[ |W_2(0)W_1(0)W_2(0)^{-1}W_1(0)^{-1}-I|\le\sum_{p\in P_{12}}|U_p-I| . \] \end{lemma} \begin{proof} (a) By the Cayley--Hamilton theorem, $V^{-1}=x-V$, $W^{-1}=y-W$, $W^2=yW-I$, $V^2=xV-I$ and $(VW)^2=zVW-I$. So $\Tr(VW^2)=yz-x$, $\Tr(VWV)=\Tr(V^2W)=xz-y$ and $\Tr((VW)^2)=z^2-2$. Hence \[ \begin{split} \Tr(VWV^{-1}W^{-1})&=\Tr\bigl(VW(x-V)(y-W)\bigr)\\ &=xyz-x(yz-x)-y(xz-y)+z^2-2, \end{split} \] which is the identity. Put $S=x^2+y^2+z^2$. If $S\ge3$, the inequality holds because $q\ge0$. If $S<3$, then $|xyz|\le(S/3)^{3/2}\le S/3$ by the inequality of the arithmetic and geometric means for $x^2,y^2,z^2$. So $4-q\le S+S/3$, and $S\ge3-\frac34q$. (b) Write $g_1\cdots g_n-I=\sum_ig_1\cdots g_{i-1}(g_i-I)$, and use $|gM|=|M|$ for unitary $g$. Similarly, $U_1\cdots U_m-V_1\cdots V_m=\sum_i U_1\cdots U_{i-1}(U_i-V_i)V_{i+1}\cdots V_m$. Finally, $|\Tr M|\le\sqrt2|M|$ for $2\times2$ matrices, by the Cauchy--Schwarz inequality. (c) Let $\gamma$ visit $x_0,x_1,\dots,x_m=x_0$, and let $U_{x_i\to x_{i+1}}$ be the link variable of its $i$-th step, inverted when the step runs against the orientation of the link. Let $x_i'=x_i\pm\hat\mu$, let $c_i$ be the link variable of the step from $x_i$ to $x_i'$, and let $V_i=U_{x_0\to x_1}\cdots U_{x_{i-1}\to x_i}$, with $V_m=h_\gamma$. Put \[ D_i=V_i\,c_i\,\bigl(U_{x_0'\to x_1'}\cdots U_{x_{i-1}'\to x_i'}\bigr)^{-1} c_0^{-1}, \] so that $D_0=I$. A direct computation gives $D_{i+1}=V_iP_iV_i^{-1}D_i$, with $P_i=U_{x_i\to x_{i+1}}c_{i+1}U_{x_i'\to x_{i+1}'}^{-1}c_i^{-1}$. If the $i$-th step is not parallel to $\mu$, then $P_i$ is the holonomy of the plaquette spanned by that step and its translate. If it is parallel to $\mu$, then $P_i=I$: for a step $+\hat\mu$ the two step variables are $c_i$ and $c_{i+1}$, and for a step $-\hat\mu$ they are $c_{i+1}^{-1}$ and $c_i^{-1}$, in the case of the translate by $+\hat\mu$, and symmetrically for $-\hat\mu$. Since $c_m=c_0$, $D_m=h_\gamma c_0h_{\gamma'}^{-1} c_0^{-1}$, where $h_{\gamma'}$ is based at $x_0'$. So $\Tr h_{\gamma'}= \Tr(D_m^{-1}h_\gamma)$, and \[ \begin{split} |\Tr h_\gamma-\Tr h_{\gamma'}|&=|\Tr((D_m^{-1}-I)h_\gamma)|\le\sqrt2\, |D_m-I|\\&\le\sqrt2\sum_{p\in R(\gamma,\mu)}|U_p-I| , \end{split} \] by (b) and $|gU_pg^{-1}-I|=|U_p-I|$. (d) Let $y_j=(j,0,0)$, let $\gamma_j$ be the loop in direction $2$ through $y_j$, with holonomy $W_2(y_j)$, and let $c_j=U_{(y_j,1)}$. The proof of (c), for $\gamma_j$ and $\mu=1$, gives $W_2(y_j)c_j=D^{(j)}c_jW_2(y_{j+1})$. Here $D^{(j)}$ is a product of conjugates of the holonomies of the $L$ plaquettes between the columns $j$ and $j+1$. Let $H_j=c_0\cdots c_{j-1}$, so that $H_L=W_1(0)$. By induction on $j$, $W_2(0)H_j=E_jH_jW_2(y_j)$ with $E_0=I$ and $E_{j+1}=E_jH_jD^{(j)}H_j^{-1}$. For $j=L$, periodicity gives $W_2(y_L)=W_2(0)$, so $E_L=W_2(0)W_1(0)W_2(0)^{-1}W_1(0)^{-1}$. This is a product of conjugates of the holonomies of the $L^2$ plaquettes of $P_{12}$, and (b) gives the bound. \end{proof} The spatial flow $\Phi_s$ of the box is the gradient flow, on $G^E$ with the product of the bi-invariant metric of Proposition~\ref{prop:spatialflow}, of $\sum_p(2-\Tr U_p)=2e_s$. Let $c_G$ be a constant with $|V-W|\le c_G\, \delta(V,W)$ for the geodesic distance $\delta$ of that metric. For the metric $\ReTr(X^*Y)$, $c_G=1$, because $\delta(V,W)$ is the length of a curve in $M_2(\C)$ with the Frobenius norm. For a packet $f_0=g\otimes h$ with $g$ supported in $[t_0,t_1]$, let $n_1=\lfloor t_1/a_k\rfloor$ and $p(z)=\sum_{n\ge1}g(a_kn)z^n$, as in the proof of Proposition~\ref{prop:flowpositive}. \begin{theorem}\label{thm:centreodd} In this box, let $\Phi$ be given by~\eqref{eq:centreoddPhi} and let $\tau>0$. Let $f_0=g\otimes h$ with $g\ge0$ supported in $[t_0,t_1]\subset(0,\infty)$, and with $h\ge0$ and $h(0)>0$. Let $k$ be such that $g(a_kn)>0$ for some integer $n\ge1$. \textup{(i)} $C_k(0)>0$ for every $\beta_W>0$. \textup{(ii)} Let $\kappa>0$, $\eta\in(0,1)$ and dyadic $u\ge0$ with $u/a_k\in\Z$, and put \[ \begin{split} \epsilon_k(\kappa,\eta)&=\min\Bigl(\frac1{1600L^4},\frac{\kappa a_k}{36L^2}, \frac{\eta^2a_k^2}{72c_G^2L^4\tau}\Bigr),\\ v&=\frac{1-(t_1+u)\kappa-\eta}{1+\eta} . \end{split} \] If $\beta_W\ge1$ and $\varepsilon(\beta_W)\le\epsilon_k(\kappa,\eta)$, which holds for every $\beta_W\ge B(\epsilon_k(\kappa,\eta))$, then for every $\sigma\ge2a_k$ with $\sigma/a_k\in\Z$ \[ \rho_k(u,\sigma)\ge\Bigl(v-\sqrt{\kappa\sigma/2}\Bigr)_+^2 . \] \textup{(iii)} Let $e^{-m(\beta_W)}$ be the norm of $A$ on $\psi_0^\perp$. If $\beta_W\ge1$ and $\varepsilon(\beta_W)\le1/(1600L^4)$, then $1-e^{-m(\beta_W)}\le36L^2\varepsilon(\beta_W)$. In particular $m(\beta_W)\to0$ as $\beta_W\to\infty$. \end{theorem} \begin{proof} Write $a=a_k$, $s=\tau/a^2$, $S_h=\sum_xh(ax)$ and $w_x=h(ax)/S_h$, the sum running over the sites of the slice. The weights $w_x\ge0$ are well defined, since $h\ge0$ and $S_h\ge h(0)>0$. Put $G_0=\sum_xw_x\Phi_x$, $\varphi=G_0 \psi_0$ and $\psi=(G_0\circ\Phi_s)\psi_0$. The flow commutes with $\zeta_1$, because the slice action is invariant under it and its gradient is equivariant. Hence $F=S_hG_0\circ\Phi_s$ is odd under $\zeta_1$, and $\omega(F)=0$. By the proof of Proposition~\ref{prop:flowpositive}, $\Psi_{A_k}=a^4S_hp(A)\psi$. With $R=A^{u/a}p(A)$, therefore, $\Psi_{k,u}=R\psi/\|R\psi\|$. (i) The configuration $1$ with every link equal to $I$ minimises the slice action, so $\Phi_s(1)=1$ and $F(1)=S_hG_0(1)=4S_h>0$. Since $F$ is continuous and $\psi_0>0$, $F\psi_0\ne0$. The operator $p(A)$ is injective by the proof of Proposition~\ref{prop:flowpositive}. So $C_k(0)=a^8S_h^2\|p(A)\psi\|^2>0$. \emph{Lipschitz constants.} By Lemma~\ref{lem:loops}(b), $\Tr W_1(x)$ has the constant $\sqrt2$ on each of its $L$ links, and $\Tr(W_1(x)W_2(x))$ has $\sqrt2$ on each of its $2L$ links. So $\Phi_x$ has constants $L_e\le2\sqrt2$, with $\sum_eL_e=3\sqrt2L$. The same bounds hold for the convex combination $G_0$. Hence $\Lambda(G_0)\le3\sqrt2L$ and $\sum_eL_e^2\le2\sqrt2\cdot3\sqrt2 L=12L$. \emph{Norm.} Put $y_1=\Tr W_1(0)$ and $y_2=\Tr W_2(0)$, and let $y_{12}=\Tr(W_1(0)W_2(0))$. Under $\zeta_2$ the product $y_1y_{12}$ is odd, so $\omega(y_1y_{12})=0$. The exchange of the axes $1$ and $2$ maps $W_1(0)$ to $W_2(0)$, so $\omega(y_1^2)=\omega(y_2^2)$. By Lemma~\ref{lem:loops}(a), \[ \begin{split} \|\Phi\psi_0\|^2&=\tfrac12\omega(y_1^2+y_2^2)+\omega(y_{12}^2)\\ &\ge\tfrac12\omega(y_1^2+y_2^2+y_{12}^2)\ge\tfrac32-\tfrac38\omega(q) . \end{split} \] By Lemma~\ref{lem:loops}(d), with $q=\frac12|C-I|^2$ for the commutator $C$ there, which has the same trace as its inverse, we get $q\le\frac12L^2 \sum_{p\in P_{12}}|U_p-I|^2=2L^2\sum_{p\in P_{12}}s_p$. So $\omega(q)\le 2L^4\varepsilon_s$. Next, move the site $0$ to a site $x$ by at most $\lfloor L/2\rfloor$ steps $\pm\hat\mu$ in each direction. Steps in direction $1$ do not change $\Tr W_1$, and the other steps cost $L$ plaquettes each in Lemma~\ref{lem:loops}(c), at most $L^2$ in all. For the path of $W_1W_2$ the steps cost $L$, $L$ and $2L$ plaquettes in the directions $1$, $2$ and $3$, at most $2L^2$ in all. Since $\|\,|U_p-I|\,\psi_0\|= (4\varepsilon_s)^{1/2}$, \[ \|(\Phi_x-\Phi)\psi_0\|\le\sqrt2\cdot3L^2\cdot2\varepsilon_s^{1/2}= 6\sqrt2L^2\varepsilon_s^{1/2}, \] and the same bound holds for $\|\varphi-\Phi\psi_0\|$. If $L^4\varepsilon_s \le1/1600$, then $(\frac32-\frac3{6400})^{1/2}\ge1.22$, because $1.22^2=1.4884$. Also $6\sqrt2/40<0.22$, so $\|\varphi\|\ge1$ and $\|\Phi\psi_0\|\ge1$. \emph{Energy.} For a real slice function $X$, the eigenvector equation and the symmetry of $t_b$ give \[ \begin{split} &\ip{X\psi_0}{(1-A)X\psi_0}=\frac1{2\lambda_0}\iint\bigl(X(U)-X(U')\bigr)^2\\ &\qquad\qquad\times\psi_0(U)t_b(U,U')\psi_0(U')\,dU\,dU' . \end{split} \] With $(\sum_eL_e|U_e-U_e'|)^2\le\Lambda\sum_eL_e|U_e-U_e'|^2$ and $|U_e-U_e'|^2=4s(U_eU_e'^{-1})$, this is at most $2\Lambda(X)^2 \varepsilon_t\le36L^2\varepsilon_t$ for $X=G_0$ and for $X=\Phi$. (iii) The vector $\Phi\psi_0$ is orthogonal to $\psi_0$, because $\Phi$ is odd under $\zeta_1$, and $A$ preserves $\psi_0^\perp$. So the spectral measure of $A$ at $\Phi\psi_0$ lives on $[0,e^{-m}]$. Hence $1-e^{-m}\le \ip{\Phi\psi_0}{(1-A)\Phi\psi_0}/\|\Phi\psi_0\|^2\le36L^2\varepsilon_t$, by the two previous steps and Lemma~\ref{lem:boxenergy}. (ii) By Lemma~\ref{lem:boxenergy}, $\varepsilon_s$ and $\varepsilon_t$ are at most $\epsilon_k(\kappa,\eta)$. Let $\nu$ be the spectral measure of $A$ at $\varphi$, and $\kappa_\varphi a=\int(1-\lambda)\,d\nu/\|\varphi\|^2\le 36L^2\varepsilon_t\le\kappa a$; also $\kappa_\varphi a\le1$. \emph{Flow.} The energy identity $\frac d{ds'}2e_s=-|\nabla2e_s|^2$ and the Cauchy--Schwarz inequality bound the length of the path $s'\mapsto \Phi_{s'}(U)$, $0\le s'\le s$, by $(2s\,e_s(U))^{1/2}$. By Minkowski's inequality, the lengths $\ell_e$ of its components have $\sum_e\ell_e^2\le 2s\,e_s(U)$. Hence \[ |G_0(\Phi_sU)-G_0(U)|\le c_G\sum_eL_e\ell_e\le c_G(12L)^{1/2}(2s\,e_s(U)) ^{1/2}, \] and $\|\psi-\varphi\|^2\le24c_G^2Ls\,\omega(e_s)=72c_G^2L^4s\varepsilon_s \le\eta^2\le\eta^2\|\varphi\|^2$. \emph{Overlap.} If $v\le0$ there is nothing to prove, so assume $v>0$. Then $\theta=(t_1+u)\kappa<1-\eta$. Since $p$ has nonnegative coefficients supported in $[1,n_1]$, we have $p(1)\lambda^{n_1}\le p(\lambda)\le p(1)$ on $[0,1]$, and $\|R\|\le p(1)$. Jensen's inequality for $\lambda^{2j}$ with $j=u/a+n_1$, and Bernoulli's inequality, give \[ \begin{split} \|R\varphi\|^2&\ge p(1)^2\int\lambda^{2j}d\nu\ge p(1)^2\|\varphi\|^2 (1-\kappa_\varphi a)^{2j},\\ \|R\varphi\|&\ge p(1)\|\varphi\|(1-\theta), \end{split} \] since $ja\le u+t_1$. So $\varrho=\|R(\psi-\varphi)\|/\|R\varphi\|\le \eta/(1-\theta)<1$. Let $\chi=R\varphi/\|R\varphi\|$. Then \[ \begin{split} |\ip\chi{\Psi_{k,u}}|&\ge\frac{\|R\varphi\|^2-\|R\varphi\|\,\|R(\psi-\varphi) \|}{\|R\varphi\|(\|R\varphi\|+\|R(\psi-\varphi)\|)}\\ &=\frac{1-\varrho}{1+\varrho}\ge\frac{1-\theta-\eta}{1-\theta+\eta}\ge v . \end{split} \] The function $\lambda^{2u/a}p(\lambda)^2$ is nondecreasing on $[0,1]$ and $1-\lambda$ is nonincreasing. As in the proof of Theorem~\ref{thm:softhard}, this gives $\int\lambda^{2u/a}p^2(1-\lambda)\, d\nu\le\|R\varphi\|^2\kappa_\varphi a$, so $\ip\chi{(1-A)\chi}\le\kappa a$. Lemma~\ref{lem:trial}(a) with $\mathbb T=A$ gives the bound. The last statement of (ii) is Lemma~\ref{lem:boxenergy}, since $\epsilon_k(\kappa,\eta)\le3$. \end{proof} Fig.~\ref{fig:centreodd} draws the box, the observable $\Phi$ and the energy bound. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_centreodd.png} \caption{The box of Theorem~\ref{thm:centreodd}. (a)~The spatial torus $(\Z/L\Z)^3$ as a cube with identified faces, the winding loops $W_1$ and $W_2$ of~\eqref{eq:centreoddPhi}, and a plane $x_1=\text{const}$ across which the centre transformation multiplies the links of direction $1$ by $-1$; $\Tr W_1$ and $\Tr W_1W_2$ change sign. (b)~The energy bound $\varepsilon(b)=3(\log b+5)/(2b)$ of Lemma~\ref{lem:boxenergy} against the threshold $1/(1600L^4)$ of Theorem~\ref{thm:centreodd}(ii) at $L=3$.} \label{fig:centreodd} \end{figure*} \begin{corollary}\label{cor:centreodd} Under the hypotheses of Theorem~\textup{\ref{thm:centreodd}}, let $u\ge0$ and $\sigma>0$ be dyadic, and let $\rho_k(\beta_W)=\rho_k(u,\sigma)$ at the coupling $\beta_W$. \textup{(i)} At every fixed large $k$, $\rho_k$ is real-analytic on $(0,\infty)$, with $\rho_k(\beta_W)\to0$ as $\beta_W\to0$ and $\rho_k(\beta_W)\to1$ as $\beta_W\to\infty$. For every $r\in(0,1)$ the set $\{\beta_W>0:\rho_k(\beta_W)=r\}$ is nonempty and has no accumulation point in $(0,\infty)$. So ratio tunings at every level $r\in(0,1)$ exist in this box. \textup{(ii)} For every $B>0$ there is $q_B<1$ such that, for every packet in this box with $C_k(0)>0$, $\rho_k(\beta_W)\le q_B^{\sigma/a_k}$ for $0<\beta_W\le B$ and all $k$. So every ratio tuning in this box has $\beta_k\to\infty$. \textup{(iii)} For $r\in(0,1)$ let $\beta_k(r)$ be the smallest solution of $\rho_k(\beta_W)=r$. Then $\beta_k(r)\to\infty$, and $\beta_k(r)\le C\,a_k^{-2}\log(1/a_k)$ for large $k$, with $C$ depending on $r$, $L$, $\tau$, $t_1$, $u$, $\sigma$ and $c_G$. \textup{(iv)} Let $\beta_k$ be couplings with $a_k^2\beta_k/\log(1/a_k)\to \infty$. Then $\rho_k(u,\sigma)\to1$ for all dyadic $u\ge0$ and $\sigma>0$, so $(\mathrm N_u)$ holds for every $u$. Every limit of Theorem~\textup{\ref{thm:flowlimit}} in the box along these couplings is two-dimensional. In it, $\psi(f_0(\cdot-(t,0)))=\psi_*$ is the same unit vector for every dyadic $t>0$, $\psi_*\perp\Omega$, and $H=0$. The inequality of \textup{(Z$_{\mathrm{sl}}$)} in Proposition~\textup{\ref{prop:mzslice}} holds in the box at every dyadic $\delta>0$: $I_k(\delta)\le2\delta D_k(\delta)$ for large $k$. \end{corollary} \begin{proof} (i) The proof of Proposition~\ref{prop:boxratio} uses the packet only through $C_k(0)>0$ at every $\beta_W>0$. That positivity is Theorem~\ref{thm:centreodd}(i), so analyticity, the limit at $0$ and the discreteness follow as there. Given $\varepsilon'>0$, choose $\eta\in(0,1)$ and $\kappa>0$ so small that $(v-\sqrt{\kappa\sigma/2})^2\ge1-\varepsilon'$. Theorem~\ref{thm:centreodd}(ii) then gives $\rho_k\ge1-\varepsilon'$ for $\beta_W\ge B(\epsilon_k(\kappa,\eta))$. Existence follows from the intermediate value theorem, as in Proposition~\ref{prop:boxratio}(ii). (ii) By the proof of Proposition~\ref{prop:boxratio}, the Riesz projection $P_0$ onto $\psi_0$ is continuous in $\beta_W\in[0,B]$ in operator norm. Hence so is $\mathbb T_{\beta_W}(1-P_0)$, whose norm is the largest eigenvalue $\lambda_1$ of $\mathbb T_{\beta_W}$ on $\psi_0^\perp$. By simplicity, $\lambda_1<\lambda_0$ for $\beta_W>0$, and at $\beta_W=0$ we have $\lambda_1=0<1=\lambda_0$. So $q_B=\max_{[0,B]}\lambda_1/\lambda_0<1$, and $\rho_k\le(\lambda_1/\lambda_0)^{\sigma/a_k}$ as in that proof. The transfer matrix does not depend on $k$, so $q_B$ does not either. If a ratio tuning at level $r$ had $\beta_k\le B$ for infinitely many $k$, then $r\le q_B^{\sigma/a_k}\to0$ along them. (iii) By (i), $\rho_k0$ with $(v-\sqrt{\kappa\sigma/2})_+^2>r$. For large $k$, $\epsilon_k=\epsilon_k(\kappa,\eta)=\eta^2a_k^2/(72c_G^2L^4\tau)$, and at $\beta'_k=B(\epsilon_k)$ Theorem~\ref{thm:centreodd}(ii) gives $\rho_k>r$. By the intermediate value theorem, $\beta_k(r)\le\beta'_k= (3/\epsilon_k)(\log(3/\epsilon_k)+4)$, which is at most $Ca_k^{-2} \log(1/a_k)$. By (ii), $\beta_k(r)\to\infty$. (iv) Write $\beta_k=M_ka_k^{-2}\ell_k$ with $\ell_k=\log(1/a_k)$, so that $M_k\to\infty$. Then \[ \frac{\log\beta_k+5}{a_k^2\beta_k}=\frac{\log M_k+2\ell_k+\log\ell_k+5} {M_k\ell_k}\to0 . \] So $\varepsilon(\beta_k)/a_k^2\to0$, hence also $\varepsilon(\beta_k)/a_k\to0$ and $\varepsilon(\beta_k)\to0$. For fixed $\kappa$ and $\eta$, $\varepsilon(\beta_k)\le\epsilon_k(\kappa,\eta)$ for large $k$, and Theorem~\ref{thm:centreodd}(ii) gives $\liminf_k\rho_k(u,\sigma)\ge(v- \sqrt{\kappa\sigma/2})_+^2$. Letting $\kappa,\eta\to0$ and using $\rho_k\le1$ gives $\rho_k(u,\sigma)\to1$. In the limit of Theorem~\ref{thm:flowlimit}(iii) in the box, put $y=(t,0)$ and $y'=(t',0)$. The vectors $\psi(f_0(\cdot-y))$ have inner products $\lim_k\rho_k(0,t+t')=1$ and are orthogonal to $\Omega$, since the packets are centred. So they coincide with one unit vector $\psi_*$. Moreover $\psi(f_*)=\Omega$, since $\|\psi(f_*)\|=\ip\Omega{\psi(f_*)}=1$, and $\ip{\psi_*}{e^{-\tau'H}\psi_*}=1$ for dyadic $\tau'$. So $H\psi_*=0$, the range of $\psi$ is spanned by $\Omega$ and $\psi_*$, and $H=0$. For the slice bound, $D_k(r)=a_k^6S_h^2\ip\psi {A^{r/a_k}\psi}$, in the notation of the proof of Theorem~\ref{thm:centreodd}, and $\ip\psi{A^n\psi}=\|A^{n/2}\psi\|^2$. As in the overlap step, for $na_k=\delta$ and $k$ so large that $n\ge2$, \[ \begin{split} \|A^{n/2}\psi\|&\ge\|\varphi\|(1-\kappa_\varphi a_k)^{n/2}-\|\psi-\varphi\|\\ &\ge\|\varphi\|(1-\tfrac12\kappa\delta-\eta), \end{split} \] while $\|\psi\|\le(1+\eta)\|\varphi\|$. Hence $D_k(\delta)\ge \bigl((1-\frac12\kappa\delta-\eta)/(1+\eta)\bigr)^2D_k(0)$. For small $\kappa,\eta$ the factor exceeds $\frac12$, and $I_k(\delta)\le\delta D_k(0)\le2\delta D_k(\delta)$. \end{proof} \begin{proposition}\label{prop:axialvariance} Let $G\subset\mathrm U(N)$ be closed, let $\omega$ be a state as in Appendix~\textup{\ref{app:transfer}}, and let $X$ be a real bounded slice function whose links join sites of the cube $B_R=\{0,\dots,R\}^3$ of the slice. Then \[ \Var_\omega X\le8NR^2\Lambda(X)^2\varepsilon_\omega . \] Let $\chi$ be the unit vector along $\Psi_{X-\omega(X)}$. The bound of Lemma~\textup{\ref{lem:slabdirichlet}(iii)} for $\ip\chi{(1-\mathbb T)\chi}$, which is $N\Lambda(X)^2\varepsilon_\omega/\Var_\omega X$, is then at least $1/(8R^2)$. If it is at most $\kappa a_k$, then $R\ge(8\kappa a_k)^{-1/2}$. \end{proposition} \begin{proof} Set to $I$, by a gauge transformation of the slice that is the identity off $B_R$, the links of the comb tree of $B_R$. This tree consists of the links in direction $3$, the links in direction $2$ with $x_3=0$, and the links in direction $1$ with $x_2=x_3=0$. This changes neither $X$ nor the plaquette holonomies, up to conjugation. In this gauge, a link in direction $2$ at height $x_3$ is the product of the $x_3$ plaquettes of the $(2,3)$-plane below it, inverted where needed. A link in direction $1$ at $(x_2,x_3)$ is a product of at most $x_3+x_2\le2R$ plaquettes, first in the $(1,3)$-plane down to $x_3=0$ and then in the $(1,2)$-plane. Let $P(e)$ be the plaquettes used for the link $e$, so $|P(e)|\le2R$. With $X(1)$ the value at the configuration with all links equal to $I$, Lemma~\ref{lem:loops}(b) gives \[ |X-X(1)|\le\sum_eL_e\sum_{p\in P(e)}|U_p-I| . \] By the Cauchy--Schwarz inequality twice and $|U_p-I|^2=2Ns_p$, \[ \begin{split} \Var_\omega X&\le\omega\bigl((X-X(1))^2\bigr)\\ &\le\Lambda(X)\sum_eL_e|P(e)|\sum_{p\in P(e)}2N\omega(s_p)\\ &\le8NR^2\Lambda(X)^2\varepsilon_\omega . \end{split} \] \end{proof} \begin{remark}\label{rem:centreodd} Theorem~\ref{thm:centreodd} and Corollary~\ref{cor:centreodd} prove $(\mathrm N_u)$ without hypotheses, but only for a degenerate limit. The couplings grow like $a_k^{-2}\log(1/a_k)$, not like $\log(1/a_k)$. The box has physical size $La_k\to0$. The limit vector $\psi_*$ has energy zero and is orthogonal to $\Omega$, so $0$ is not a simple eigenvalue of $H$, and the gap of the box closes as $\beta_W\to\infty$ (Theorem~\ref{thm:centreodd}(iii)). The mechanism is the centre degeneracy of a periodic box. The packet is odd under $\zeta_1$. Its unflowed slice field has norm at least $1$ once $\varepsilon(\beta_W)\le1/(1600L^4)$, by the trace identity of Lemma~\ref{lem:loops}(a), its time-step energy is at most $36L^2\varepsilon(\beta_W)$, and the flow moves it by at most $(72c_G^2L^4s\varepsilon(\beta_W))^{1/2}$ in norm. None of this carries over to infinite volume. There, the map that multiplies by $-I$ the links $(x,\mu)$ with $x_\mu=c$ is the gauge transformation equal to $-I$ at the sites with $x_\mu>c$ and to $I$ elsewhere. So every element of $\mathcal A$ is invariant under it, and no centre-odd packet exists. Proposition~\ref{prop:axialvariance} shows the loss for slice functions of a cube. The bound of Lemma~\ref{lem:slabdirichlet}(iii) reaches $\kappa a_k$ only for cubes of side at least $(8\kappa a_k)^{-1/2}$ lattice spacings, at every coupling. Together with Proposition~\ref{prop:lipobstruction}(ii), the slab-energy route needs both $a_k\beta_k$ bounded below and supports of diverging lattice size. In the box the winding loops $W_\mu$ are not functions of a cube, and the comb gauge cannot remove their holonomy. Three points are not proved. First, the infinite-volume condition, which is $(\mathrm N_u)$ along some sequence of couplings, equivalently the existence of a ratio tuning (Theorem~\ref{thm:ratiotuning}). Second, the question of Proposition~\ref{prop:boxratio} for the centre-even packet of Proposition~\ref{prop:flowpositive}. Third, the case $\SU(N)$ with $N\ge3$. For the last, the proof uses $N=2$ in Lemma~\ref{lem:loops}(a). For $N\ge3$, the commuting pair $W_1=W_2=\mathrm{diag}(1,e^{2\pi i/N},\dots, e^{2\pi i(N-1)/N})$, with $W_3=I$, comes from a flat configuration of the slice, with the links $(x,1)$ with $x_1=L-1$ and $(x,2)$ with $x_2=L-1$ equal to that matrix and all others equal to $I$. It has $\Tr W_1=\Tr W_2=\Tr W_1W_2=0$, so the inequality of (a) has no analogue for these three words. With $L_k=\ell/a_k$ and the loops of length $L_k$, the proof gives the bounds of Theorem~\ref{thm:centreodd}(ii) with $L$ replaced by $L_k$, at couplings with $\varepsilon(\beta_k)\le \eta^2a_k^6/(72c_G^2\ell^4\tau)$ and $a_k$ small. These packets are not of the form of Theorem~\ref{thm:flowlimit}, because $\Phi$ then depends on $k$. \end{remark} \emph{What is proved.} In infinite volume: the admissible states form compact convex sets with closed graph; the ratio is continuous in the state, continuous in the coupling wherever the admissible state is unique (for $\SU(2)$ at $\beta_W\le0.348$, and wherever (D) holds), and its values over a compact interval of couplings form a compact interval under (P$_k$). In the box: (P$_k$) holds, the ratio is real-analytic in $\beta_W>0$ and tends to zero at $\beta_W=0$, and the couplings at which it equals $r$ form a discrete set, nonempty exactly when the ratio reaches $r$. In a box of fixed size, for packets odd under the centre of $\SU(2)$: $(\mathrm N_u)$ holds along every sequence with $a_k^2\beta_k/\log(1/a_k)\to\infty$, ratio tunings exist at every level, at couplings at most of order $a_k^{-2}\log(1/a_k)$ and tending to infinity, and the limits are two-dimensional with $H=0$ (Theorem~\ref{thm:centreodd}, Corollary~\ref{cor:centreodd}). A ratio tuning exists exactly when $(\mathrm N_u)$ holds along some sequence of couplings; it leaves the strong-coupling range, has lattice gap at most $a_k\sigma^{-1}\log(1/r)$, satisfies $(\mathrm N_u)$ with two-sided bounds, and, with (D), (M) and the family $\mathfrak F$, gives (U). Its increments are a step-scaling function, and if they exceed $h$ by $c/\beta_{k+1}$, as two loops predict, it is not a one-loop tuning of bounded $M$. \emph{What is moved, not proved.} In infinite volume the existence of a ratio tuning is equivalent to $(\mathrm N_u)$ along some sequence, and no packet is odd under the centre (Remark~\ref{rem:centreodd}); (P$_k$) is open in infinite volume; $\beta_k\to\infty$ is equivalent to $\lim_kR_k(B)0$: a finite-volume measure in which $e$ is not fixed by the boundary condition, or an infinite-volume state whose conditional law of $U_e$ given the other links is proportional to $e^{bS_e}dU_e$ with $dU_e$ the Haar measure. \begin{proposition}\label{prop:force} \textup{(i)} For every unit $T\in\g$, $t\in\R$ and $0\le\eta2^{-19}n^{-2}$. By symmetry the conditional covariance of $x$ is diagonal in $u$ and $u^\perp$, with eigenvalues $\Var X$ and $\E(1-X^2)/3$; on $I$, $1-X^2=T(2-T)\ge\frac3{8n}$, so $\E(1-X^2)/3\ge\frac1{1792n}\ge2^{-19}n^{-2}$. Hence the conditional variance of $x\cdot v$ is at least $2^{-19}(1+6b)^{-2}$, and so is the variance, by the law of total variance. For the mean, integrating $\frac d{dX}\bigl(e^{rX}(1-X^2)^{3/2}\bigr)$ over $[-1,1]$ gives $3\E X=r\E(1-X^2)\le r(1-(\E X)^2)$, whence $\E X\le(\sqrt{9+4r^2}-3)/(2r)\le2r/(2r+3)$ for $r>0$, and $\E[1-x\cdot v]\ge1-\E X\ge3/(2r+3)\ge1/(4b+1)$. \end{proof} For the blocking statement let $G=\SU(2)$, $B\ge4$ an integer and, for a coarse edge $(x,\mu)$, $V_{(x,\mu)}=\prod_{r=0}^{B-1}U_{(Bx+r\hat\mu,\mu)}$ the ordered product of the fine links along it. \begin{proposition}\label{prop:blocking} For every Wilson Gibbs state $\omega$ of $\SU(2)$ at $b>0$ and every set $\mathcal E$ of $K$ coarse edges, \[ \begin{split} \bigl\|\mathrm{Law}_\omega\bigl((V_e)_{e\in\mathcal E}\bigr)-m^{\otimes K}\bigr\|_{\mathrm{TV}} &\le\frac K2e^{3b}\Bigl(\frac{4b}{4b+1}\Bigr)^{B-3}\\ &\le\frac K2\exp\Bigl(3b-\frac{B-3}{4b+1}\Bigr), \end{split} \] where $m$ is the Haar measure. If $F$ is a function of the $V_e$ of a set of coarse edges at positive times, $|F|\le1$, and $\epsilon$ is this bound for those edges and their reflections, then $0\le\omega(\vartheta(\bar F^c)F^c)\le6\epsilon$ with $F^c=F-\omega(F)$. \end{proposition} \begin{proof} Select on each coarse edge the fine links with $1\le r\le B-2$. No plaquette contains two selected links: collinear links do not share a plaquette, parallel coarse edges are $B$ apart, and a plaquette containing a selected link of $(x,\mu)$ has its links in direction $\nu\ne\mu$ at $\mu$-offsets $r,r+1\in [1,B-1]$ from $Bx$, while coarse edges in direction $\nu$ lie at offsets in $B\Z$. Given the other links, the selected ones are therefore independent, each with density $e^{r_i\,x\cdot u_i}/Z(r_i)$, $0\le r_i\le6b$. Convolution with such a density acts on the representation of spin $l\ge1$ by $D^{(l)}(u_i)$ times $I_{l+1}(r_i)/I_1(r_i)$, whose absolute value is at most $I_2(r_i)/I_1(r_i)$ because $I_\nu(r)$ decreases in $\nu\ge0$, and $I_2(r_i)/I_1(r_i)=\E X\le4b/(4b+1)$ by the proof of Proposition~\ref{prop:plaqvar}, so it contracts the mean-zero part of $L^2$ by that factor. The density is at most $e^{6b}$ because $Z\ge1$ by Jensen's inequality, so its distance to $1$ in $L^2$ is at most $e^{3b}$. The conditional law of $V_e$ is that of a product of $B-2$ such independent variables between two fixed ones, so its total variation distance to $m$ is at most $\frac12e^{3b}(4b/(4b+1))^{B-3}$, and the coarse edges are conditionally independent. Total variation distance from a product is at most the sum over the factors, and averaging over the conditioning does not increase it. For the last statement, under $m^{\otimes K}$ the covariance vanishes because $F$ and its reflection depend on disjoint edges; replacing $\omega$ by $m^{\otimes K}$ changes $\omega(\vartheta(\bar F)F)$ by at most $2\epsilon$ and $|\omega(F)|^2$ by at most $4\epsilon$. \end{proof} With $B=\ell/a_k$ for a fixed physical length $\ell$ and $b_k=O(\log(1/a_k))$ the bound tends to zero: the bare coarse links of a fixed coarse graph become independent Haar variables, their reflected form has rank one, and a renormalisation $Z_k$ that keeps the reflected norm of $Z_kF^c$ at least $c>0$ must grow at least like $(c/6\epsilon)^{1/2}$. The renormalised flowed fields of Theorem~\ref{thm:flowlimit} are not bounded functions of bare coarse links, and none of the three propositions bears on the slice ratio, the tree bounds or (D). \subsection{What the cube estimate requires}\label{ssec:cubeneeds} This subsection proves seven facts about $D(n,\alpha)$. It holds at small coupling for every even $n$ above an explicit size (Proposition~\ref{prop:cubesmall}), and it survives small perturbations of the action (Proposition~\ref{prop:cubeopen}), although gauge-invariant perturbations still need strong coupling at the scale of the cube (Proposition~\ref{prop:smallfield}). It forces the gauge-fixed variables of a cube with trivial exterior to stay a fixed distance from the identity on average (Theorem~\ref{thm:cubeneeds}, and Proposition~\ref{prop:tetra} for $\SU(2)$), so that for every fixed $n$ it fails once $\beta$ is large (Corollary~\ref{cor:cubefails}). And it cannot be obtained from bounds on the conditional law of a single line of links that are uniform in the configuration around the line (Proposition~\ref{prop:slowline}). For the compact abelian theory with the Villain action it fails at every size once $\beta$ is large (Proposition~\ref{prop:coulomb}). In Sec.~\ref{ssec:constrained} the estimate is posed for block averages, where its quadratic approximation holds along the tuning. None of these decides (D) along the continuum tuning, where $n$ grows while $\beta$ grows.\footnote{The four statements answer the question of which parts of (D) can be settled with the methods of this paper. The first places (D) in the range where the single-link method already works, the second and third show what any proof of (D) at weak coupling has to produce, and the fourth excludes the most direct one-dimensional route. } For a finite $V\subset A$ and $j\in A\setminus V$ let $b(V,j)$ be the number of links of $V$ that share a plaquette with $j$, and write $k\sim i$ when two active links share a plaquette. \begin{lemma}\label{lem:cubesingle} Suppose that $C_{ee'}\le c_0$ for all active $e,e'$, with $12c_0<1$. Then $C_{V,j}\le c_0\,b(V,j)/(1-12c_0)$ for every finite $V\subset A$ and every $j\in A\setminus V$. \end{lemma} \begin{proof} Fix $Q,Q'$ that differ only at $j$, put $d=\delta(Q_j,Q'_j)$, $\mu=\gamma_V(\cdot\mid Q)$ and $\nu=\gamma_V(\cdot\mid Q')$. The couplings of $\mu$ and $\nu$ form a weakly compact set on which $\pi\mapsto\int d_V\,d\pi$ is continuous, so one of them, $\pi$, minimises it; let $\rho_i=\int\delta(X_i,Y_i)\,d\pi$. Fix $i\in V$. Under $\mu$ the conditional law of $X_i$ given the other links of $V$ is $\gamma^F_i(\cdot\mid X)$, and likewise for $\nu$. Replace $(X_i,Y_i)$ by a pair drawn from an optimal coupling of $\gamma^F_i(\cdot\mid X)$ and $\gamma^F_i(\cdot\mid Y)$, chosen measurably~\cite[Cor.~5.22]{Vil09}, and keep the other coordinates. The new coupling still has marginals $\mu$ and $\nu$, and changing the links in which $X$ and $Y$ differ one at a time gives \[ W_1\bigl(\gamma^F_i(\cdot\mid X),\gamma^F_i(\cdot\mid Y)\bigr)\le \sum_{k\in V,\,k\ne i}C_{ik}\,\delta(X_k,Y_k)+C_{ij}\,d . \] Minimality of $\pi$ therefore gives $\rho_i\le c_0\sum_{k\in V,\,k\sim i}\rho_k+c_0[i\sim j]\,d$. Sum over $i\in V$: every $\rho_k$ occurs at most twelve times by Lemma~\ref{lem:structure}(iv), so $(1-12c_0)\sum_i\rho_i\le c_0\,b(V,j)\,d$. Since $W_1^{d_V}(\mu,\nu)\le\sum_i\rho_i$, dividing by $d$ proves the lemma. \end{proof} \begin{proposition}\label{prop:cubesmall} Let $n=2m$ and let $V$ be the cube of Definition~\ref{def:cube}. For every class $c$, \[ \sum_{j\in c\setminus V}b(V,j)\le10m^3-2m^2 , \] with equality for three classes. Consequently, if $C_{ee'}\le c_0$ for all $e,e'$ with $12c_0<1$, then $D(n,\alpha)$ holds for every $\alpha\in(0,1)$ with \[ \alpha\ge\hat\alpha_n=\frac{c_0}{1-12c_0}\Bigl(\frac{20}n-\frac8{n^2}\Bigr). \] For $\SU(2)$ and $|\beta_W|<\frac13$ this reads $\hat\alpha_n= \frac{|\beta_W|}{1-3|\beta_W|}\bigl(\frac5n-\frac2{n^2}\bigr)$, which is below one for every even $n>5|\beta_W|/(1-3|\beta_W|)$. \end{proposition} \begin{proof} Let $j=(y,\mu)$ lie in the class $c$, with $y\equiv r$ modulo two. A link of $V$ that shares a plaquette with $j$ has base point $y+d$ with $d\in\{-1,0,1\}^4$ and at most two nonzero coordinates, and the twelve offsets $d$ depend only on $c$. For a fixed offset we count the $y\equiv r$ with $y+d\in[0,n)^4$ and $y\notin[0,n)^4$. In the coordinate $l$ there are $m$ values of $y_l\equiv r_l$ with $y_l+d_l\in[0,n)$, and one of them lies outside $[0,n)$ exactly when $d_l=1$ and $r_l=1$ (the value $y_l=-1$) or $d_l=-1$ and $r_l=0$ (the value $y_l=n$). If $q$ is the number of such coordinates, the count is $m^4-m^{4-q}(m-1)^q$, that is $0$, $m^3$ or $2m^3-m^2$ for $q=0,1,2$. Summing over the twelve offsets, the class sum is $a_cm^3-b_cm^2$, where $b_c$ is the number of offsets with $q=2$ and $a_c$ is the number with $q=1$ plus twice $b_c$. Reading the offsets from Table~\ref{tab:sigma}, the pairs $(a_c,b_c)$ over the $48$ classes are $(2,0)$, $(7,0)$, $(7,1)$, $(9,1)$, $(9,2)$ and $(10,2)$ three times each and $(3,0)$, $(4,0)$, $(5,0)$, $(6,0)$ and $(8,1)$ six times each. In every case $a_c\le10$ and $8-a_c+b_c\ge0$, so \[ \begin{split} (10m^3-2m^2)-(a_cm^3-b_cm^2)&=m^2\bigl((10-a_c)m+b_c-2\bigr)\\ &\ge m^2(8-a_c+b_c)\ge0, \end{split} \] with equality for the three classes with $(a_c,b_c)=(10,2)$. By Lemma~\ref{lem:cubesingle}, $K_c\le c_0(10m^3-2m^2)/(1-12c_0)$, and $N_n=m^4$ gives $K_c/N_n\le\hat\alpha_n$. For $\SU(2)$ the proof of Theorem~\ref{thm:su2} gives $c_0=|\beta|=|\beta_W|/4$. \end{proof} At $\beta_W=\frac3{10}$ and $n=16$, for instance, $\hat\alpha_{16}= \frac{117}{128}$, and Theorem~\ref{thm:onescale} gives the gap $\log\frac{128}{117}/32$, much weaker than Theorem~\ref{thm:su2}. The proposition shows that the cube condition is not empty; it does not extend the range, because the absorption $12c_0<1$ is the single-link condition itself. The cube condition is also stable under small changes of the action, which is what a block renormalisation would need at its last scale, where the effective action is no longer of Wilson form. Let $r\ge0$ and $\varepsilon\ge0$, and let $(w_X)$ be real functions on $G^A$ indexed by finite sets $X\subset A$ of diameter at most $r$, with $w_X$ depending only on the links of $X$, such that the family is invariant under the translations by $2\Z^4$, every active link lies in at most $M$ of the sets $X$, every $X$ has at most $s$ links, and \[ |w_X(Q)-w_X(Q')|\le\varepsilon\sum_{i\in X}\delta(Q_i,Q'_i) \] for all $Q,Q'$. Let $\gamma^W_V(\cdot\mid Q)$ be the probability measure with density proportional to $\exp\sum_{X\cap V\ne\emptyset}w_X$ with respect to $\gamma_V(\cdot\mid Q)$, and define $C^W_{V,j}$, $K^W_c$ and the condition $D^W(n,\alpha)$ as in Definition~\ref{def:cube}, with $\gamma^W_V$ in place of $\gamma_V$. \begin{proposition}\label{prop:cubeopen} Let $n$ be even and suppose that $D(n,\alpha)$ holds at $\beta$. Then $D^W(n,\alpha')$ holds with \[ \alpha'=\alpha\,e^{\Omega}(1+2\Omega)+96\,s\,\Omega,\qquad \Omega=48\,M\,D\,N_n\,\varepsilon . \] In particular $\alpha'<1$ for all $\varepsilon$ below an explicit $\varepsilon_0(n,\alpha,M,s)>0$. \end{proposition} \begin{proof} The cube has $|V|=48N_n$ links, so $d_V$ has diameter $\Delta_V=48N_nD$. For fixed exterior links put $u=\sum_{X\cap V\ne\emptyset}w_X$ as a function on $G^V$. Each link of $V$ lies in at most $M$ sets $X$, so $u$ is $M\varepsilon$-Lipschitz for $d_V$ and its oscillation is at most $M\varepsilon\Delta_V=\Omega$; we normalise $u$ so that $\sup u=0$, which does not change the tilted measures, and put $h=e^u$, so that $e^{-\Omega}\le h\le1$ and $h$ is $M\varepsilon$-Lipschitz. \emph{A common tilt.} Let $\mu,\nu$ be probability measures on $G^V$ and $\mu_h=h\mu/\mu(h)$, $\nu_h=h\nu/\nu(h)$. Let $f$ be $1$-Lipschitz for $d_V$; subtracting a constant we may take $|f|\le\Delta_V$. Then $fh$ is $(1+\Delta_VM\varepsilon)$-Lipschitz, $|\nu(fh)|\le\Delta_V\nu(h)$, $|\mu(h)-\nu(h)|\le M\varepsilon W_1^{d_V}(\mu,\nu)$ and $\mu(h)\ge e^{-\Omega}$, so \[ \begin{split} \mu_h(f)-\nu_h(f)&=\frac{\mu(fh)-\nu(fh)}{\mu(h)}+\nu(fh)\, \frac{\nu(h)-\mu(h)}{\mu(h)\nu(h)}\\ &\le e^{\Omega}(1+2\Omega)\,W_1^{d_V}(\mu,\nu), \end{split} \] since $\Delta_VM\varepsilon=\Omega$. By Kantorovich duality $W_1^{d_V}(\mu_h,\nu_h)\le e^{\Omega}(1+2\Omega)W_1^{d_V}(\mu,\nu)$. \emph{Two tilts.} If $u'$ is another weight with $|u-u'|\le\eta$, the density of $\nu_{h'}$ with respect to $\nu_h$ lies in $[e^{-2\eta},e^{2\eta}]$, so the total variation distance of the two is at most $1-e^{-2\eta}\le 2\eta$, and $W_1^{d_V}(\nu_h,\nu_{h'})\le2\eta\Delta_V$. \emph{The coefficients.} Let $Q,Q'$ differ only at $j\notin V$, with $\delta=\delta(Q_j,Q'_j)$, and let $u,u'$ be the weights for the two exteriors. They differ only through the sets $X$ that contain $j$ and meet $V$; if there are $m_j$ of them, $|u-u'|\le m_j\varepsilon\delta$. With $\mu=\gamma_V(\cdot\mid Q)$ and $\nu=\gamma_V(\cdot\mid Q')$, the triangle inequality and the two steps give \[ C^W_{V,j}\le e^{\Omega}(1+2\Omega)\,C_{V,j}+2\Delta_V\,m_j\,\varepsilon . \] The pairs $(X,j)$ with $j\in X\setminus V$ and $X\cap V\ne\emptyset$ number at most $s$ times the number of sets meeting $V$, hence at most $sM|V|$. Summing over $j\in c\setminus V$, \[ \begin{split} K^W_c&\le e^{\Omega}(1+2\Omega)\,\alpha N_n+2\Delta_VsM|V|\varepsilon\\ &=\bigl(\alpha e^{\Omega}(1+2\Omega)+96\,s\,\Omega\bigr)N_n . \end{split} \] The family is invariant under $2\Z^4$, so the coefficients are, and this is $D^W(n,\alpha')$. \end{proof} The admissible perturbation is of order $1/N_n=(2/n)^4$, so the proposition is useful in a box of a fixed number of lattice spacings, which is how a renormalisation argument would use it: after enough block steps that the cube of physical size $\ell$ has become a cube of fixed side, the effective action has to lie within $\varepsilon_0$ of an action that satisfies the condition there. Whether the effective actions along the tuning do so is not decided here, and that is where the problem lies: at the scale where the blocked theory is strongly coupled enough for a condition of this kind, the running coupling is of order one, which is outside the range of the existing ultraviolet analysis. The proposition shows only that the condition itself is not the obstacle. The next theorem uses the configuration in which every link outside $V$ is the identity. Write $\gamma^1_V=\gamma_V(\cdot\mid1)$ and \[ \varepsilon_V(\beta)=\frac1{|V|}\sum_{i\in V}\int\delta(1,Q_i)\, d\gamma^1_V(Q) \] for the mean distance of the gauge-fixed variables of the cube from the identity. Recall that $\delta$ is bi-invariant of diameter $D$. \begin{theorem}\label{thm:cubeneeds} Let $n$ be even, $V$ the cube of Definition~\ref{def:cube}, and suppose that $D(n,\alpha)$ holds at $\beta$. Then \[ \varepsilon_V(\beta)\ge\frac{(1-\alpha)\,D}3 . \] \end{theorem} \begin{proof} \emph{A flat configuration.} By Lemma~\ref{lem:structure}(i) and the period two of $\sigma$, the sixteen residues modulo two fall into eight pairs $\{r,r+e_{\sigma(r)}\}$, the residues met by one line of $F$. The eight residues $0010$, $0011$, $0100$, $0110$, $1001$, $1011$, $1100$, $1101$ form four of these pairs. Choose $z,w\in G$ with $\delta(z,w)=D$, put $g(x)=w$ if $x$ is congruent to one of those eight residues and $g(x)=z$ otherwise, and let $Q^g_{(x,\mu)}=g(x)g(x+e_\mu)^{-1}$ for every link of $\Z^4$. Then $g$ is constant on every line of $F$, so $Q^g$ equals $1$ on $F$, every plaquette variable of $Q^g$ is $1$, and $Q^g$ has period two. By bi-invariance $\delta(1,Q^g_{(x,\mu)})=\delta(g(x+e_\mu),g(x))$, which is $D$ when $x$ and $x+e_\mu$ receive different values and $0$ otherwise; from Table~\ref{tab:sigma} the first case occurs for exactly $32$ of the $48$ classes. Let $\delta_c$ be the common value of $\delta(1,Q^g_e)$ for $e$ in the class $c$, so that $\sum_c\delta_c=32D$. \emph{Gauge covariance.} Let $\Phi(Q)_{(x,\mu)}=g(x)Q_{(x,\mu)}g(x+e_\mu)^{-1}$ on $G^V$. The gauge transformation by $g$ maps the configuration equal to $Q_V$ on $V$ and to $1$ elsewhere to the one equal to $\Phi(Q_V)$ on $V$ and to $Q^g$ elsewhere; it leaves every plaquette trace unchanged and every link of $F$ equal to $1$, and $\Phi$ preserves Haar measure. Hence $\gamma_V(\cdot\mid Q^g)=\Phi_*\gamma^1_V$. \emph{Upper bound.} Only the finitely many links outside $V$ that share a plaquette with $V$ enter $\gamma_V$. Changing them one at a time from $1$ to $Q^g$ and applying the definition of $C_{V,j}$ at each step, \[ \begin{split} W_1^{d_V}\bigl(\gamma^1_V,\gamma_V(\cdot\mid Q^g)\bigr)&\le\sum_{j\in A\setminus V}C_{V,j}\,\delta(1,Q^g_j)=\sum_c\delta_cK_c\\ &\le\alpha N_n\sum_c\delta_c=32\,\alpha DN_n . \end{split} \] \emph{Lower bound.} The function $f(Q)=-\sum_{i\in V}\delta(Q_i,Q^g_i)$ is $1$-Lipschitz for $d_V$. By gauge covariance and bi-invariance, $\int f\,d\gamma_V(\cdot\mid Q^g)=-\int\sum_i\delta(Q_i,1)\,d\gamma^1_V=-|V| \varepsilon_V$, and by the triangle inequality $\int f\,d\gamma^1_V\le -\sum_i\delta(1,Q^g_i)+|V|\varepsilon_V$. Every class has $N_n$ links in $V$, so $\sum_i\delta(1,Q^g_i)=32DN_n$ and \[ W_1^{d_V}\bigl(\gamma^1_V,\gamma_V(\cdot\mid Q^g)\bigr)\ge32DN_n-2|V| \varepsilon_V . \] With $|V|=48N_n$ the two bounds give $96N_n\varepsilon_V\ge32(1-\alpha)DN_n$. \end{proof} \begin{corollary}\label{cor:cubefails} For every even $n$ and every $\alpha<1$ there is $\beta_0(n,\alpha)$ such that $D(n,\alpha)$ fails at every $\beta\ge\beta_0(n,\alpha)$. In particular, if $D(n_k,\alpha)$ holds at couplings $\beta_k\to\infty$, then $n_k\to\infty$. \end{corollary} \begin{proof} For $\beta>0$, $\gamma^1_V$ has density proportional to $e^{-N\beta S}$ with respect to Haar measure, where $S=\sum_{p\cap V\ne\emptyset}(N-\ReTr Q^F_p)$ is continuous and nonnegative on $G^V$ and vanishes exactly when $Q^F_p=1$ for every plaquette $p$ that meets $V$. We show that this forces $Q_V=1$. The other plaquettes have all their links equal to $1$, so every plaquette variable is $1$; since the plaquettes of $\Z^4$ fill every closed lattice path, there is a function $h$ on the sites with $Q_{(x,\mu)}=h(x)h(x+e_\mu)^{-1}$ on every link, and $h$ takes equal values at the two ends of every link on which $Q=1$. Every link with base point outside $[0,n)^4$ is such a link. From a site with a coordinate at least $n$ one can move in every positive direction along links of this kind, and from a site with a negative coordinate $x_\nu$ one can move in the positive directions other than $\nu$ until another coordinate reaches $n$; so $h$ is constant on the sites outside the box. Every line of $F$ leaves the box, and $h$ is constant along it, so $h$ is constant and $Q_V=1$. Let $U$ be a neighbourhood of $1$ in $G^V$, $s=\min_{G^V\setminus U}S>0$, and $U'\subset U$ a neighbourhood of $1$ on which $S0$. Let $n$ be even, $V$ the cube of Definition~\ref{def:cube}, and suppose that $D(n,\alpha)$ holds at $\beta$. Then \[ \frac1{|V|}\sum_{i\in V}\Bigl|\int\Tr\rho(Q_i)\,d\gamma^1_V\Bigr|\le \alpha\,d, \] \[ \frac1{|V|}\sum_{i\in V}\int\delta(1,Q_i)^2\,d\gamma^1_V\ge(1-\alpha) \int_G\delta(1,q)^2\,dq , \] where $dq$ is Haar measure, and $\varepsilon_V(\beta)\ge(1-\alpha)D/2$. \end{proposition} \begin{proof} \emph{Four elements.} Let $\varphi\colon\SU(2)\to G$ be the homomorphism of the $\SU(2)$ subgroup of a root, and $z_1,\dots,z_4$ the tetrahedral quaternions of Proposition~\ref{prop:tetra}. For $k\ne l$ the quaternion $z_kz_l^{-1}$ has real part $\langle z_k,z_l\rangle=-\frac13$; unit quaternions with equal real parts are conjugate in $\SU(2)$, so the twelve elements $\varphi(z_kz_l^{-1})$ are conjugate in $G$, and by bi-invariance $\delta(\varphi(z_k),\varphi(z_l))=d_*$ for all $k\ne l$, with one number $d_*$. If $d_*=0$, then $\varphi(z_kz_l^{-1})$ lies in $\ker\rho$, a proper normal subgroup of the simple group $G$ and therefore central; it then commutes with $\varphi(\SU(2))$, so it is central there, and $z_kz_l^{-1}$ lies in $\{\pm1\}\ker\varphi\subset\{\pm1\}$, which its real part $-\frac13$ excludes. So $d_*>0$. \emph{The two bounds.} Put $g(x)=\varphi(z_k)$ when the residue of $x$ lies in $X_k$. The proof of Proposition~\ref{prop:tetra} applies word for word: $Q^g$ equals $1$ on $F$ and on every plaquette, $\delta(1,Q^g_e)=d_*$ on every active link, and $W_1^{d_V}(\gamma^1_V,\gamma_V(\cdot\mid Q^g))\le \alpha d_*|V|$. Simultaneous conjugation of the links of $V$ preserves $\gamma^1_V$, so the mean $\int\rho(Q_i)\,d\gamma^1_V$ commutes with $\rho(G)$, and by Schur's lemma it is $c_iI$ with $c_i=d^{-1}\int\Tr\rho(Q_i) \,d\gamma^1_V$. Its mean under $\gamma_V(\cdot\mid Q^g)$ is $c_i\rho(Q^g_i)$. With matrices $W_i$ of Hilbert--Schmidt norm $\kappa^{-1/2}$ chosen so that $\operatorname{Re}\Tr(W_i^*c_i(\rho(Q^g_i)-I))=|c_i|\,\delta(1,Q^g_i)$, the function $f(Q)=\sum_i\operatorname{Re}\Tr(W_i^*\rho(Q_i))$ is $1$-Lipschitz for $d_V$, and Kantorovich duality gives $W_1^{d_V}\ge d_*\sum_i|c_i|$. Hence $\sum_i|c_i|\le\alpha|V|$, the first inequality. \emph{Comparison with Haar measure.} Since $\kappa\,\delta(1,q)^2=2d-2 \operatorname{Re}\Tr\rho(q)$ and $\int_G\Tr\rho(q)\,dq=0$ for a nontrivial irreducible $\rho$, $\int_G\delta(1,q)^2dq=2d/\kappa$, while $\int\delta(1, Q_i)^2d\gamma^1_V=2d(1-\operatorname{Re}c_i)/\kappa\ge2d(1-|c_i|)/\kappa$. Averaging gives the second inequality. Finally $\kappa D^2\le4d$, because $\|\rho(p)-\rho(q)\|_{\mathrm{HS}}\le2\sqrt d$, so $\delta\ge\delta^2/D$ gives $\varepsilon_V\ge2d(1-\alpha)/(\kappa D)\ge(1-\alpha)D/2$. \end{proof} For $\SU(N)$ with the Hilbert--Schmidt distance this improves the constant $\frac13$ of Theorem~\ref{thm:cubeneeds} to $\frac12$, and for $\SU(3)$, where $D^2=9$ and the Haar mean of $\delta^2$ is $6$, to $\frac23$: in a cube with the identity outside, $D(n,\alpha)$ leaves the gauge-fixed links at least the fraction $1-\alpha$ of the disorder of independent Haar-distributed links. The size $\beta_0(n,\alpha)$ in Corollary~\ref{cor:cubefails} comes from a compactness argument. For $\SU(2)$ it can be made explicit, and then reads as a lower bound on the side of any cube that satisfies the condition. \begin{proposition}\label{prop:cubesize} Let $G=\SU(2)$ with the chordal metric, so that $D=2$, let $b=\beta_W\ge\frac19$, let $n\ge2$ be even and put $L(b)=1+\log24+\frac32\log(9b)$. Then \[ \frac1{|V|}\sum_{e\in V}\int\delta(1,Q_e)^2\,d\gamma^1_V\le \frac{n^2(n+2)(2n+1)\,L(b)}b , \] so $\varepsilon_V(\beta)^2$ is at most the same quantity. Consequently, if $D(n,\alpha)$ holds at $b$, then \[ n^4\ge\frac{2(1-\alpha)\,b}{5\,L(b)} . \] \end{proposition} \begin{proof} Put $m=n/2$. Under $\gamma^1_V$ every link outside $V$ equals $1$: the links of $F$ always, the others by the choice of exterior. A plaquette that does not meet $V$ therefore has plaquette variable $1$. \emph{Links as loops.} Let $e=(x,\mu)\in V$, $y=x+e_\mu$, $\sigma=\sigma(x)$ and $\tau=\sigma(y)$. The directions $\sigma,\tau,\mu$ are distinct: $\sigma\ne\mu$ because $e$ is active; $\tau\ne\mu$ because otherwise the line of $F$ through $y$ would contain $x$, and the lines of $F$ are disjoint; and $\tau\ne\sigma$ by Table~\ref{tab:sigma}, in which $\sigma(x+e_\mu)\ne\sigma(x)$ whenever $\mu\ne\sigma(x)$. Since $0\le x_\sigma\beta_0$ the condition $D(n,\alpha)$ fails for every even $n\ge2$ and every $\alpha<1$. In particular, for couplings $\beta_k\to\infty$ and any even $n_k$, the hypothesis of Theorem~\ref{thm:transfer} fails for this theory. \end{proposition} \begin{proof} Lemma~\ref{lem:gaugefix}, Proposition~\ref{prop:state}(i)--(iii) and (v), Lemma~\ref{lem:comparison}, Corollary~\ref{cor:decay} and Lemma~\ref{lem:dscomparison} use the Wilson weight only as a positive continuous class function of the plaquette variable, with one exception: the Doeblin bound in Lemma~\ref{lem:dscomparison}(a), which for the Villain weight reads that each single-link density is at least $(\min\varphi_\beta/\max\varphi_\beta)^6>0$ times its maximum. Suppose that $D(n,\alpha)$ holds. As in the proof of Theorem~\ref{thm:onescale}, the limit $\omega(O)=\lim W_\Lambda(O)$ of the measures of boxes with free boundary conditions then exists along every sequence of boxes exhausting $\Z^4$, and for gauge-invariant local $O,O'$ with $O$ Lipschitz in each link, \[ |\Cov_\omega(O,O')|\le\|O'\|_\infty L_O\, \frac{D(1+c_ne^{\lambda\rho_n})}{1-\theta}\,|S_{O'}|\, e^{-\lambda\dist(O,O')} \] for every $\lambda>0$ with $\theta=\alpha e^{\lambda\rho_n}<1$, where $L_O$ is a Lipschitz constant of $O$ and $S_{O'}$ the support of $O'$. Put $\Psi_p=\varphi_\beta'(\theta_p)/\varphi_\beta(\theta_p)$; up to the factor $-\mathbf i$ this is the lattice field strength of Fr\"ohlich and Spencer~\cite[Sect.~2]{FS82} for the Villain action. It is real, odd and smooth in $\theta_p$, so it is gauge invariant, local and Lipschitz in each of its four links, with constants that do not depend on $p$, and $\omega(\Psi_p)=0$ by the symmetry $\theta\mapsto-\theta$. The bound above gives $\sum_{p'}|\omega(\Psi_p\Psi_{p'})|<\infty$, the sum over the plaquettes $p'$ parallel to $p$. The state of~\cite{FS82} is the limit of the same free-boundary measures over boxes, which is $\omega$. Fr\"ohlich and Spencer prove, for $\beta$ larger than some $\beta_0$, that the Fourier transform of the two-plaquette function of the field strength is discontinuous at zero momentum, so that this sum is infinite~\cite[Sect.~2]{FS82}; see also~\cite{Gut80}. This contradiction proves the first statement, and the second follows because $\beta_k>\beta_0$ for all large $k$. \end{proof} At large $\beta$ the plaquette variables of the abelian theory are Gaussian to leading order, as are those of $\SU(N)$, and every property of the action used in this paper (compactness, locality, positivity and smoothness of the weight, reflection positivity, the forest gauge) is shared by the two theories. A proof of (D) along the continuum tuning therefore has to use the non-abelian structure of $G$; Proposition~\ref{prop:coulomb} holds at every cube size, so letting the side grow with $\beta$, which Corollary~\ref{cor:cubefails} already requires, does not remove the need. Fr\"ohlich and Spencer indicate how their result extends to the Wilson action without giving the details~\cite[Sect.~2]{FS82}, so the proposition is stated for the Villain action only. \begin{remark}\label{rem:fluxloop} The proof loses the area of a disc, of order $n^2$, in passing from plaquettes to links, and on loops that loss is of the right order. On the periodic lattice $(\Z/L\Z)^4$ with $L$ even and coordinates $0\le x_j0$. For every Gibbs specification of the gauge-fixed theory, every active link $e$ and every configuration of the other links, the conditional law of $Q_e$ satisfies \[ \int\delta(1,Q_e)\,d\mu_e\ge\frac1{9\max(6b,1)^{1/2}} . \] In particular $\varepsilon_V(\beta)\ge\frac19\max(6b,1)^{-1/2}$ for every $n$; at every fixed $n$, $\varepsilon_V(\beta)$ lies between constant multiples of $b^{-1/2}$ and $(L(b)/b)^{1/2}$. \end{proposition} \begin{proof} Write $\langle\cdot,\cdot\rangle$ for the Euclidean product of quaternions, so that $\ReTr q=2\langle q,1\rangle$. The six plaquettes that contain $e$ contribute $\exp(b\langle Q_e,W_p\rangle)$ to the conditional density, with $|W_p|=1$; so $\mu_e$ has density $e^{\langle u,v\rangle}/Z(|v|)$ with respect to normalised Haar measure, for some $v$ with $a=|v|\le6b$. Put $a'=\max(a,1)$ and $\eta=a'^{-1/2}\le1$. On the geodesic ball of radius $\eta$ about $v/|v|$ the exponent is at least $a(1-\eta^2/2)\ge a-\frac12$, and that ball has measure greater than $\eta^3/24$ (proof of Proposition~\ref{prop:slowline}); hence $Z(a)\ge e^{a-1/2}\eta^3/24$ and the density is at most $24e^{1/2}a'^{3/2}$ (for $a<1$ directly, since $Z\ge1$ by Jensen's inequality). A chordal ball of radius $r$ is a geodesic ball of radius $\theta$ with $2\sin(\theta/2)=r$, so $\theta\le\pi r/2$, and its measure is $(\theta-\sin\theta\cos\theta)/\pi\le2\theta^3/(3\pi)\le\pi^2r^3/12$ (Lemma~\ref{lem:haarsu2}). With $r=\frac29a'^{-1/2}$, \[ \mu_e\bigl(\delta(1,Q_e)0$ is even under the reflection $x_4\mapsto-x_4$, which commutes with $K_b$. \emph{A barrier.} Let $E=\frac12\sum_{k=0}^9|v_{k+1}-v_k|^2$ and suppose that $(q_0)_4\ge0\ge(q_{10})_4$ and $E\le\frac1{1280}$. Put $\Delta= (20E)^{1/2}\le\frac18$ and $w=v_0$; by Cauchy--Schwarz $|w-v_k|\le\Delta$ for all $k$, and $0\le w_4\le\Delta$ because the $r_k$ have no fourth coordinate. From $|v_k+r_k|=1$, $\bigl||w+r_k|^2-1\bigr|\le\Delta(2+\Delta)=:\eta$ for $0\le k\le5$. The pairs $r_k=\pm ae_l$ give $|w_l|\le\eta/(2a)=\eta$ for $l=1,2,3$ and $|w|^2\ge\frac34-\eta$, so $\frac34\le3\eta^2+\Delta^2+\eta\le 3(\frac{17}{64})^2+\frac1{64}+\frac{17}{64}=\frac{2019}{4096}$, a contradiction. Hence $K_b(x,y)\le e^{-b/1280}$ whenever $x_4\ge0\ge y_4$. \emph{The top eigenvalue.} Let $\epsilon=b^{-1/2}$. The geodesic ball of radius $\epsilon\le1$ in $S^3$ has normalised measure $\frac2\pi\int_0^\epsilon \sin^2\theta\,d\theta\ge\frac{8\epsilon^3}{3\pi^3}>\frac{\epsilon^3}{24}$. If each $q_k$ lies in the ball of radius $\epsilon$ about $r_k+ce_4$, then $|v_k-ce_4|\le\epsilon$ and $E\le20\epsilon^2$, so the integrand is at least $e^{-20}$. Testing $K_b$ on the normalised indicator of the ball about $r_0+ce_4$ gives $\Lambda_b\ge e^{-20}(\epsilon^3/24)^{10}=e^{-20}24^{-10} b^{-15}$. \emph{The gap.} Let $\phi(x)=\operatorname{sign}(x_4)\psi_b(x)$, a unit vector orthogonal to $\psi_b$. Then \[ \begin{split} \Lambda_b-\ip\phi{K_b\phi}&=4\int_{x_4>0>y_4}\psi_b(x)K_b(x,y)\psi_b(y)\,dx\,dy\\ &\le e^{-b/1280}, \end{split} \] because $\int_{x_4>0}\psi_b\le\frac12$ by Cauchy--Schwarz. By the variational principle $\Lambda_{b,1}\ge\ip\phi{K_b\phi}$, and dividing by the lower bound on $\Lambda_b$ proves the estimate; the gap is positive because $\Lambda_b$ is simple. \end{proof} The exterior of Proposition~\ref{prop:slowline} is legal at every coupling, so a bound on the conditional law of single lines that is uniform in the exterior cannot be better than $e^{-b/1280}$ up to powers of $b$, and an argument of this kind cannot give (D) along the tuning. A proof of (D) has to use what such a bound discards: the joint resampling of the transverse links inside the cube, which the averaged coefficients $C_{V,j}$ retain, or the smallness of the probability of such exteriors under the Wilson measure. \subsection{The limit at separated points}\label{ssec:limitE} This subsection treats (E) for local fields without flow, under the averaged condition of Theorem~\ref{thm:avgtransfer}. Two features separate it from Sec.~\ref{ssec:osaxioms}. A local field without flow is covariant under the whole hypercubic group $B_4$, so reflection positivity and the gap pass from the time direction to every coordinate direction; the limit then clusters in every direction at the rate $m_*$ without the tree bounds (Theorem~\ref{thm:limitEsep}(v)), and by Proposition~\ref{prop:rotations}(iv) one rotation of infinite order gives $\mathrm O(4)$. On the other hand, asymptotic freedom predicts for the action density a two-point function of order $|x-y|^{-8}$ at short distance, up to logarithms (the discussion after Proposition~\ref{prop:growth}), and such a two-point function is incompatible with the moment bound (M) at coinciding supports (Proposition~\ref{prop:limitEsing}). The moment bound is therefore posed for separated supports, on which it determines distributions on the complement of the coinciding points (Theorem~\ref{thm:limitEsep}(i)). The Schwinger functions of~\cite{OS73} are defined on test functions that vanish, with all their derivatives, where two arguments coincide. Throughout, the hypothesis of Theorem~\ref{thm:avgtransfer} holds, $\omega_k$ is the torus limit there and $m_*=16\varpi/(625e^2\ell)$. By Corollary~\ref{cor:avggap}, $\omega_k$ is invariant under translations and $B_4$, reflection positive for the reflections in coordinate hyperplanes through sites and bisecting links, and its transfer matrix $\mathbb T_k$ has spectrum on $\Omega_k^\perp$ in $[0,e^{-m_*a_k}]$, since $m_{n_k}\ge16\varpi/(625e^2n_k)\ge m_*a_k$. Let $r\ge1$ be an integer and $\Phi$ a real continuous gauge-invariant function of the links with both endpoints in $[-r,r]^4$, invariant under the action of $B_4$ on these links described in the proof of Theorem~\ref{thm:osaxioms}(ii); the sum of $N-\ReTr U_p$ over the $24$ plaquettes $p$ with a corner at $0$ is an example. Write $\Phi_x$ for its translate by $x\in\Z^4$, so that $g\Phi_x=\Phi_{gx}$ for $g\in B_4$, and $\tau_v\Phi_x=\Phi_{x+v}$ for the translation $\tau_v$, $v\in\Z^4$. In this subsection \[ A_k(f)=\sum_{x\in\Z^4}a_k^4f(a_kx)\bigl(\Phi_x-c_k\bigr),\qquad c_k=\omega_k(\Phi_0), \] for $f\in C_c(\R^4)$, and $Z_k>0$ are renormalisation constants. Let $e_\mu$ be the unit vectors, $\theta_\mu$ the reflection $x_\mu\mapsto-x_\mu$ of $\R^4$, $\pi_\mu\in B_4$ the transposition of the axes $0$ and $\mu$ ($\pi_0=1$), $\vartheta$ the lattice reflection of Appendix~\ref{app:transfer}, $\vartheta_\mu=\pi_\mu\vartheta\pi_\mu$, and $\tau^\mu_t=\tau_{te_\mu}$. Let $\theta_{\mu,k}$ be the reflection of $\R^4$ with $\theta_{\mu,k}(a_kx)=a_k\vartheta_\mu(x)$ for $x\in\Z^4$; as in the proof of Theorem~\ref{thm:higher}(iii) it is $\theta_\mu$ followed by a translation along $e_\mu$ of length at most $a_k$, so the plane fixed by $\vartheta_\mu$ is $\{x_\mu=c\}$ with $|c|\le\frac12$. For a cube $Q$ and $\delta>0$ let $\mathcal C_n(Q,\delta)$ be the set of $(f_1,\dots,f_n)\in C_c(\R^4)^n$ with supports in $Q$ and $\dist_\infty(\operatorname{supp} f_i,\operatorname{supp}f_j)\ge\delta$ for $i\ne j$, $\dist_\infty$ being the $\ell^\infty$ distance; a tuple is \emph{separated} if it lies in some $\mathcal C_n(Q,\delta)$. A \emph{separated polynomial} is a polynomial in the sense of Sec.~\ref{ssec:osaxioms} each of whose monomials is a separated tuple; it is \emph{at positive $\mu$-times} if all its functions are supported in $\{x_\mu>0\}$. Put $\mathfrak D_n=\{(x_1,\dots,x_n)\in(\R^4)^n: x_i\ne x_j\text{ for }i\ne j\}$ and $\mathsf K_L=(2L+3)^4$. \begin{enumerate}[label=\textup{(M$_{\mathrm s}$)},leftmargin=*] \item \emph{Separated moment bound.} For every $n\ge1$, every cube $Q$ and every $\delta>0$ there are $M_n(Q,\delta)<\infty$ and $k_0$ such that \[ \Bigl|\omega_k\Bigl(\prod_{j=1}^nZ_kA_k(f_j)\Bigr)\Bigr|\le M_n(Q,\delta)\prod_{j=1}^n\|f_j\|_\infty \] for all $k\ge k_0$ and $(f_1,\dots,f_n)\in\mathcal C_n(Q,\delta)$. \end{enumerate} Posed for these packets, (M) contains the tuples with overlapping supports and implies (M$_{\mathrm s}$). \begin{lemma}\label{lem:limitElattice} \textup{(i)} For $f\in C_c(\R^4)$, $g\in B_4$ and $v\in\Z^4$, \[ \begin{split} A_k(f\circ g^{-1})&=gA_k(f),\quad A_k(f(\cdot-a_kv))=\tau_vA_k(f),\\ A_k(\bar f)&=\overline{A_k(f)},\quad\vartheta_\mu A_k(f)=A_k(f\circ \theta_{\mu,k}). \end{split} \] \textup{(ii)} If $f$ is supported in $\{x_\mu\ge\varepsilon\}$ and $(r+1)a_k\le\varepsilon$, then $A_k(f)$ is a function of the links with both endpoints in $\{x_\mu\ge1\}$, which lie on the positive side of the plane of $\vartheta_\mu$. \textup{(iii)} Let $F,F'$ be continuous gauge-invariant functions of finitely many links with both endpoints in $\{x_\mu\ge1\}$, and $t\ge0$ an integer. Then $\omega_k(\vartheta_\mu(\bar F)F)\ge0$, \[ 0\le\omega_k\bigl(\vartheta_\mu(\bar F)\tau^\mu_{2t}F\bigr)\le \omega_k\bigl(\vartheta_\mu(\bar F)F\bigr), \] and, with $N_\mu(F)=\omega_k(\vartheta_\mu(\bar F)F)^{1/2}$, \[ \begin{split} &\bigl|\omega_k\bigl(\vartheta_\mu(\bar F)\tau^\mu_tF'\bigr)- \overline{\omega_k(F)}\,\omega_k(F')\bigr|\\ &\qquad\le e^{-m_*a_kt}N_\mu(F)N_\mu(F') . \end{split} \] For $v\in\Z^4$ with $v_\mu=0$, $N_\mu(\tau_vF)=N_\mu(F)$. \end{lemma} \begin{proof} (i) Substitute $y=g^{-1}x$, $y=x-v$ and $y=\vartheta_\mu x$ in the sums, using $g\Phi_x=\Phi_{gx}$, $\tau_v\Phi_x=\Phi_{x+v}$ and $\vartheta_\mu\Phi_x=\Phi_{\vartheta_\mu x}$; the last holds because $\vartheta_\mu$ is an element of $B_4$ followed by a lattice translation. The constant $c_k$ does not depend on $x$ by translation invariance, and it is real because $\Phi$ is real and $\omega_k$ is a state. (ii) $f(a_kx)\ne0$ forces $x_\mu\ge\varepsilon/a_k\ge r+1$, and $\Phi_x$ depends on the links with both endpoints in $x+[-r,r]^4$. (iii) For $\mu=0$ this is the reconstruction of Appendix~\ref{app:transfer}: with $\ip{\Psi_F}{\mathbb T_k^t\Psi_{F'}}=\omega_k(\vartheta(\bar F)\tau_tF')$ and the density of the vectors $\Psi_G$, $\Psi_{\tau_tF}=\mathbb T_k^t \Psi_F$; then $\omega_k(\vartheta(\bar F)\tau_{2t}F)=\|\mathbb T_k^t\Psi_F \|^2\le\|\Psi_F\|^2$ by $0\le\mathbb T_k\le1$, and with $\Psi'_F=\Psi_F- \omega_k(F)\Omega_k$ the left side of the last inequality is $|\ip{\Psi'_F} {\mathbb T_k^t\Psi'_{F'}}|\le e^{-m_*a_kt}\|\Psi'_F\|\|\Psi'_{F'}\|$, where $\ip{\Omega_k}{\Psi_F}=\omega_k(F)$ and $\|\Psi'_F\|\le\|\Psi_F\|=N_0(F)$. Since $\vartheta$ is $\theta_0$ followed by a translation along $e_0$, it commutes with $\tau_v$ for $v_0=0$, and $N_0(\tau_vF)^2=\omega_k(\tau_v( \vartheta(\bar F)F))=N_0(F)^2$. For general $\mu$ apply this to $\pi_\mu F$ and $\pi_\mu F'$, using $\vartheta_\mu=\pi_\mu\vartheta\pi_\mu$, $\tau^\mu_t=\pi_\mu\tau_t\pi_\mu$ and the invariance of $\omega_k$ under $\pi_\mu$. \end{proof} \begin{theorem}\label{thm:limitEsep} Assume the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}} and \textup{(M$_{\mathrm s}$)}. Every sequence of $k$ has a subsequence along which the following hold. \textup{(i)} For every separated tuple the limit \[ S_n(f_1,\dots,f_n)=\lim_k\omega_k\Bigl(\prod_{j=1}^nZ_kA_k(f_j)\Bigr) \] exists. It is symmetric and multilinear, $S_1=0$, and on $\mathcal C_n(Q,\delta)$ it is bounded by $M_n(Q,\delta)\prod_j\|f_j\|_\infty$. There is a unique distribution on $\mathfrak D_n$ whose value at $f_1\otimes\cdots\otimes f_n$ is $S_n(f_1,\dots,f_n)$ for every separated tuple of smooth functions. \textup{(ii)} Each $S_n$ is invariant under the translations of $\R^4$ and under $B_4$, and $S(\bar P)=\overline{S(P)}$ for separated polynomials. \textup{(iii)} For every $\mu$ and every separated polynomial $P$ at positive $\mu$-times, $S(\theta_\mu\bar P\cdot P)\ge0$. The completion of the quotient of these polynomials by the null space of this form is a Hilbert space $\mathcal H_\mu$ with vectors $\Psi_\mu(P)$, $\ip{\Psi_\mu(P)}{\Psi_\mu(P')}=S(\theta_\mu\bar P\cdot P')$, and $\Omega_\mu=\Psi_\mu(1)$; $P\mapsto\pi_\mu P$ induces a unitary map of $\mathcal H_0$ onto $\mathcal H_\mu$ that maps $\Omega_0$ to $\Omega_\mu$. \textup{(iv)} There are a self-adjoint $H_\mu\ge0$ with $\Psi_\mu(\tau_{te_\mu}P)=e^{-tH_\mu}\Psi_\mu(P)$ for $t\ge0$ and a strongly continuous unitary representation $U_\mu$ of the hyperplane $e_\mu^\perp$ with $\Psi_\mu(\tau_yP)=U_\mu(y)\Psi_\mu(P)$, commuting with $H_\mu$; $H_\mu\Omega_\mu=0$ and $U_\mu$ fixes $\Omega_\mu$. Further $\sigma(H_\mu)\subset\{0\}\cup[m_*,\infty)$, $\ker H_\mu=\C\Omega_\mu$, and for $P,P'$ at positive $\mu$-times and $t\ge0$ \[ \begin{split} &\bigl|S(\theta_\mu\bar P\cdot\tau_{te_\mu}P')-\overline{S(P)}\,S(P')\bigr|\\ &\qquad\le e^{-m_*t}\,\|\Psi_\mu(P)\|\,\|\Psi_\mu(P')\| . \end{split} \] \textup{(v)} Let $(f_1,\dots,f_n)\in\mathcal C_n(Q,\delta)$, $Q$ of side $L$, let $\{1,\dots,n\}=I\cup J$ with $I,J$ disjoint and nonempty, $y\in \R^4\setminus\{0\}$, and $f^\lambda_j=f_j$ for $j\in I$, $f^\lambda_j=f_j( \cdot-\lambda y)$ for $j\in J$. There is $K<\infty$ independent of $\lambda$ such that, for $\lambda|y|_\infty\ge L+2$, \[ \Bigl|S_n(f^\lambda_1,\dots,f^\lambda_n)-S_{|I|}\bigl((f_j)_{j\in I}\bigr) S_{|J|}\bigl((f_j)_{j\in J}\bigr)\Bigr|\le Ke^{-m_*\lambda|y|_\infty} . \] \textup{(vi)} $\mathcal H_0\ne\C\Omega_0$ if and only if $S_n$ does not vanish identically on the separated tuples for some $n\ge2$. In that case $\sigma(H_\mu)\cap[m_*,\infty)\ne\emptyset$ for every $\mu$: $H_\mu$ has a mass gap of at least $m_*$. \textup{(vii)} The set $G_S$ of $R\in\mathrm O(4)$ with $S_n(f_1\circ R^{-1},\dots,f_n\circ R^{-1})=S_n(f_1,\dots,f_n)$ for all $n$ and all separated tuples is a closed subgroup of $\mathrm O(4)$ that contains $B_4$. It is finite or equal to $\mathrm O(4)$, and it equals $\mathrm O(4)$ if and only if it contains an element of infinite order. Conversely, if for every $n$ and every separated tuple the sequence $\omega_k(\prod_jZ_kA_k(f_j))$ is bounded, then \textup{(M$_{\mathrm s}$)} holds. \end{theorem} \begin{proof} (i) Let $\chi_{m,p}(x)=\prod_\nu(1-2^m|x_\nu-p_\nu|)_+$ for $m\in\mathbb N$ and $p\in2^{-m}\Z^4$, and let $\mathcal G$ be the countable set of the functions $\sum_{p\in F}c_p\chi_{m,p}$ with $F$ finite and $c_p\in\mathbb Q+i\mathbb Q$. Since $\sum_p\chi_{m,p}=1$ and $\chi_{m,p}(x)\ne0$ only for $|x-p|_\infty<2^{-m}$, for $f\in C_c(\R^4)$ and $\eta>0$ there is $g\in\mathcal G$ with $\|f-g\|_\infty<\eta$ and support within $\ell^\infty$ distance $\eta$ of $\operatorname{supp}f$: take $2^{-m}\le\eta$ with $|f(x)-f(p)|<\eta/2$ for $|x-p|_\infty\le2^{-m}$, $F=\{p:f(p)\ne0\}$ and $|c_p-f(p)|<\eta/2$. By (M$_{\mathrm s}$) the moment sequence of each of the countably many separated tuples of elements of $\mathcal G$ is bounded, and a diagonal subsequence makes all of them converge. Let $(f_j)\in\mathcal C_n(Q,\delta)$ and choose $g_j\in\mathcal G$ as above with $\eta\le\delta/4$. With $Q'$ the cube $Q$ enlarged by $\delta/4$ on every side, the tuples $(g_1,\dots,g_{i-1},f_i-g_i,f_{i+1},\dots,f_n)$ and $(g_j)$ lie in $\mathcal C_n(Q',\delta/2)$, so for $k\ge k_0$ the moments $B_k$ satisfy \[ \begin{split} &|B_k(f_1,\dots,f_n)-B_k(g_1,\dots,g_n)|\\ &\quad\le M_n(Q',\tfrac\delta2)\sum_i\|f_i-g_i\|_\infty\prod_{j\ne i} \max(\|f_j\|_\infty,\|g_j\|_\infty). \end{split} \] Hence $B_k(f_1,\dots,f_n)$ is a Cauchy sequence. Symmetry and multilinearity pass to the limit, and so does the bound; $S_1=0$ because the packets are centred. For the distribution, let $U_1,\dots,U_n$ be bounded open sets with pairwise disjoint closures, $U=U_1\times\cdots\times U_n$, and $M=M_n(Q, \delta)$ with $Q\supset\bigcup_jU_j$ and $\delta$ the least distance of the closures; $|S_n(f_1,\dots,f_n)|\le M\prod_j\|f_j\|_\infty$ for $f_j\in C_c(U_j)$. Fix a compact $K\subset U$, $R$ with $U_j\subset(-R,R)^4$, and $\varphi_j\in C^\infty_c(U_j)$ equal to $1$ near the projection of $K$. For $F\in C^\infty(\R^{4n})$ supported in $K$ let $c_\kappa$, $\kappa\in \Z^{4n}$, be the Fourier coefficients of its $2R$-periodic extension, so that $F=\sum_\kappa c_\kappa e_\kappa$ on $(-R,R)^{4n}$ with $e_\kappa(x)= \prod_je^{i\pi\kappa_j\cdot x_j/R}$. Integrating by parts against $1-(R/\pi)^2\Delta$, which multiplies $e_\kappa$ by $1+|\kappa|^2$, \[ |c_\kappa|\le(2R)^{-4n}(1+|\kappa|^2)^{-2n-1}\bigl\|(1-(R/\pi)^2\Delta)^{ 2n+1}F\bigr\|_{L^1}, \] and $\sum_\kappa(1+|\kappa|^2)^{-2n-1}<\infty$. Put \[ T(F)=\sum_\kappa c_\kappa\,S_n\bigl(\varphi_1e_{\kappa_1},\dots,\varphi_n e_{\kappa_n}\bigr) . \] The series converges absolutely, each term being at most $M\prod_j \|\varphi_j\|_\infty|c_\kappa|$, so $T$ is continuous on the smooth functions supported in $K$. For $F=f_1\otimes\cdots\otimes f_n$ the expansions $f_j=\varphi_j\sum_{\kappa_j}c^{(j)}_{\kappa_j}e_{\kappa_j}$ converge uniformly with supports in $\operatorname{supp}\varphi_j$, so multilinearity and the bound give $T(F)=S_n(f_1,\dots,f_n)$. Conversely, for every $F$ supported in $K$ the partial sums $\sum_{|\kappa|_\infty\le N}c_\kappa\prod_j\varphi_j(x_j)e_{\kappa_j}(x_j)$ are finite sums of products and converge to $\prod_j\varphi_j(x_j)F=F$ in $C^\infty_c(U)$, since $\sum_\kappa|\kappa|^s|c_\kappa|<\infty$ for every $s$ by the same bound with a higher power. So every distribution on $U$ with the stated values on products agrees with $T$ on the functions supported in $K$; in particular $T$ does not depend on $K$, $R$ and $\varphi_j$, and defines the unique such distribution $T_U$ on $U$. Every point of $\mathfrak D_n$ has a neighbourhood of this form, namely a product of small open cubes around its $n$ distinct entries, and the intersection of two such products is again one, on which $T_U$ and $T_{U'}$ agree by uniqueness. By~\cite[Sect.~2.2]{Hor03} the $T_U$ are the restrictions of one distribution on $\mathfrak D_n$, and it is unique by the same localisation. (ii) For dyadic $y$ and large $k$, $y/a_k\in\Z^4$, and Lemma~\ref{lem:limitElattice}(i) and translation invariance of $\omega_k$ give invariance of the moments under the translation by $y$. For real $y$ take dyadic $y_m\to y$: the translated tuples lie in one $\mathcal C_n(Q,\delta)$ and converge uniformly, so the bound of (i) carries the invariance to $y$. For $g\in B_4$, $A_k(f\circ g^{-1})=gA_k(f)$, $\omega_k$ is $B_4$-invariant, and $g$ preserves $\ell^\infty$ distances. The last claim follows from $A_k(\bar f)=\overline{A_k(f)}$ and $\omega_k(\bar G)= \overline{\omega_k(G)}$. (iii) Let $P=\sum_\beta c_\beta\prod_jf_{\beta,j}$ have all its functions supported in $Q\cap\{x_\mu\ge\varepsilon\}$, $\varepsilon>0$, and its monomials in $\mathcal C(Q,\delta)$, and put $P_k=\sum_\beta c_\beta\prod_j Z_kA_k(f_{\beta,j})$. For $(r+1)a_k\le\varepsilon$, Lemma~\ref{lem:limitElattice}(ii),(iii) gives $\omega_k(\vartheta_\mu(\bar P_k)P_k)\ge0$, and by Lemma~\ref{lem:limitElattice}(i) $\vartheta_\mu(\bar P_k)$ is the lattice version of $\theta_\mu\bar P$ with $\theta_\mu$ replaced by $\theta_{\mu,k}$. For $a_k\le\min(\varepsilon, \delta/4)$ the monomials of $\theta_\mu\bar P\cdot P$ and of the polynomials obtained by replacing $\theta_\mu$ by $\theta_{\mu,k}$ in some factors lie in $\mathcal C(Q',\min(\varepsilon,\delta/2))$ for a cube $Q'$ independent of $k$, because the reflected factors are supported in $\{x_\mu\le-\varepsilon +a_k\}$. By (i) and multilinearity the difference of the two moments is at most a constant times $\max_{\beta,j}\|f_{\beta,j}\circ\theta_{\mu,k}- f_{\beta,j}\circ\theta_\mu\|_\infty$, which tends to $0$ by uniform continuity. So $S(\theta_\mu\bar P\cdot P)\ge0$. The form is Hermitian by (ii) and the symmetry of $S_n$, and the construction is that of Theorem~\ref{thm:osaxioms}(iii). Since $\pi_\mu\theta_0\pi_\mu=\theta_\mu$ and $S$ is $\pi_\mu$-invariant, $S(\theta_\mu\overline{\pi_\mu P}\cdot\pi_\mu P')=S(\theta_0\bar P\cdot P')$ for $P,P'$ at positive $0$-times, which gives the unitary map. (iv) By the unitary map of (iii) it suffices to take $\mu=0$, with $H_\mu$ and $U_\mu$ the conjugates of $H_0$ and $U_0$. Let $P$ be at positive $0$-times and $t\ge0$ dyadic. For large $k$, $(\tau_{te_0}P)_k=\tau_{t/a_k} P_k$ by Lemma~\ref{lem:limitElattice}(i), and Lemma~\ref{lem:limitElattice}(iii) gives $\omega_k(\vartheta(\bar P_k)\tau_{2t/a_k}P_k)\le\omega_k(\vartheta( \bar P_k)P_k)$. By the approximation of $\theta_{0,k}$ by $\theta_0$ in (iii), the left side tends to $S(\theta_0\bar P\cdot\tau_{2te_0}P)=\|\Psi_0( \tau_{te_0}P)\|^2$, by translation invariance and $\theta_0\tau_{te_0}= \tau_{-te_0}\theta_0$, and the right side tends to $\|\Psi_0(P)\|^2$. The translates $\tau_{te_0}P$, $0\le t\le\delta/2$, have their monomials in one $\mathcal C(Q',\delta/2)$, and each monomial of $\tau_{te_0}P-P$ is a telescoping sum of monomials with one factor $f(\cdot-te_0)-f$; so the bound of (i) gives $\|\Psi_0(\tau_{te_0}P)\|\le\|\Psi_0(P)\|$ for all $t\ge0$ and $\|\Psi_0(\tau_{te_0}P)-\Psi_0(P)\|\to0$ as $t\to0$. From here the proof of Theorem~\ref{thm:osaxioms}(iv) applies verbatim and gives $H_0$, and $U_0$ from the translations orthogonal to $e_0$, which commute with $\theta_0$. For the gap, the proof of Lemma~\ref{lem:limitElattice}(iii) with $F=F'=P_k$ and $t/a_k$ steps gives \[ \begin{split} &\omega_k\bigl(\vartheta(\bar P_k)\tau_{t/a_k}P_k\bigr)-|\omega_k(P_k)|^2\\ &\quad\le e^{-m_*t}\bigl(\omega_k(\vartheta(\bar P_k)P_k)-|\omega_k(P_k)|^2 \bigr), \end{split} \] the right side being $e^{-m_*t}\|\Psi'_{P_k}\|^2$ in the notation of the proof of the lemma. In the limit, with $v=\Psi_0(P)-S(P)\Omega_0$ and $\ip{\Omega_0}{\Psi_0(P)}=S(P)$, this reads $\ip v{e^{-tH_0}v}\le e^{-m_*t}\|v\|^2$. As in the proofs of Theorems~\ref{thm:transfer} and~\ref{thm:osaxioms}(vi), it extends to all $t\ge0$ and to the closure $\Omega_0^\perp$ of these vectors, and gives the claims on $\sigma(H_0)$, on $\ker H_0$ and the inequality, using $\ip{\Psi_0(P)}{\Omega_0}= \overline{S(P)}$ by (ii). (v) Choose $\mu$ with $|y_\mu|=|y|_\infty=\eta$. Replacing every $f_j$ by $f_j\circ\theta_\mu$ and $y$ by $\theta_\mu y$, which changes no term by (ii), we may take $y_\mu=\eta>0$, and after a common translation of all $f_j$, $Q\subset\{-L-1\le x_\mu\le-1\}$. Let $P''=\prod_{j\in I}\bar f_j \circ\theta_\mu$ and $P'_0=\prod_{j\in J}f_j(\cdot-(L+2)e_\mu)$, both at positive $\mu$-times, write $y=\eta e_\mu+y_\perp$ with $y_\perp\perp e_\mu$ and put $s=\lambda\eta-L-2\ge0$. Then $\prod_{j\in J}f_j(\cdot-\lambda y)= \tau_{se_\mu}\tau_{\lambda y_\perp}P'_0$ and $\theta_\mu\bar P''=\prod_{j\in I}f_j$, so by (iv) \[ S_n(f^\lambda_1,\dots,f^\lambda_n)=\ip{\Psi_\mu(P'')}{e^{-sH_\mu}U_\mu( \lambda y_\perp)\Psi_\mu(P'_0)} , \] while, by translation invariance, \[ \begin{split} S_{|I|}\bigl((f_j)_{j\in I}\bigr)&=\ip{\Psi_\mu(P'')}{\Omega_\mu},\\ S_{|J|}\bigl((f_j)_{j\in J}\bigr)&=\ip{\Omega_\mu}{\Psi_\mu(P'_0)} . \end{split} \] The difference is the same matrix element between the projections of the two vectors on $\Omega_\mu^\perp$, which $e^{-sH_\mu}$ and $U_\mu$ preserve, so it is at most $e^{-m_*s}\|\Psi_\mu(P'')\| \|\Psi_\mu(P'_0)\|$. This is the claim with $K=e^{m_*(L+2)}\|\Psi_\mu( P'')\|\|\Psi_\mu(P'_0)\|$. (vi) If all $S_n$ with $n\ge1$ vanish, $\ip{\Psi_0(P)}{\Psi_0(P')}$ is the product of the constant terms, and $\mathcal H_0=\C\Omega_0$. Conversely, let $\mathcal H_0=\C\Omega_0$; by (iii) then $\mathcal H_\mu=\C\Omega_\mu$, so $S(\theta_\mu\bar P\cdot P')=\overline{S(P)}S(P')$ for $P,P'$ at positive $\mu$-times, and by (ii) $S(\theta_\mu\bar P)=\overline{S(P)}$. With translation invariance this says: if a separated tuple splits into $(f_j)_{j\in I}$ supported in $\{x_\muc\}$, then $S_n=S_{|I|}S_{|J|}$. Suppose $S_m=0$ for $1\le m0$, and for $(r+2)a_k\le t_0$ and $t\in a_k \Z_{\ge0}$ put \[ C_k(t)=\omega_k\bigl(\vartheta(\overline{A_k(f_0)})\,\tau_{t/a_k} A_k(f_0)\bigr) . \] \begin{proposition}\label{prop:limitEnorm} \textup{(i)} $C_k(0)\ge0$, and if $C_k(0)>0$, then $00$ be dyadic, suppose $C_k(0)>0$ for large $k$, put $Z_k=C_k(\sigma_0)^{-1/2}$ and assume \textup{(M$_{\mathrm s}$)}. Every limit of Theorem~\textup{\ref{thm:limitEsep}} has $\ip{\Psi_0(f_0)}{e^{-\sigma_0H_0}\Psi_0(f_0)}=1$ and $\|\Psi_0(f_0)\|^2\ge e^{m_*\sigma_0}$, and is nontrivial. Moreover, with $Q$ a cube that contains $\operatorname{supp}f_0$ and $\theta_{0,k}(\operatorname{supp}f_0)$ for all $k$, \[ \frac{C_k(\sigma_0)}{C_k(0)}\ge\frac1{M_2(Q,t_0)\|f_0\|_\infty^2} \] for all large $k$. \textup{(iii)} Suppose that the transfer matrices $\mathbb T_k$ have spectrum on $\Omega_k^\perp$ in $[0,e^{-m_{\mathrm{lat}}}]$ for a fixed $m_{\mathrm{lat}}>0$ and all $k$. Then for every choice of $Z_k$ satisfying \textup{(M$_{\mathrm s}$)}, every limit of Theorem~\textup{\ref{thm:limitEsep}} is trivial, and, when $C_k(0)>0$ for large $k$, \textup{(M$_{\mathrm s}$)} fails for $Z_k=C_k(\sigma_0)^{-1/2}$. \end{proposition} \begin{proof} (i) Let $A=A_k(f_0)$. By Lemma~\ref{lem:limitElattice}(ii) and its proof, $A$ and $A'=A_k(f_0(\cdot+a_ke_0))$ are at positive times, and $A=\tau_1A'$, so $\Psi_A=\mathbb T_k\Psi_{A'}$ lies in the range of $\mathbb T_k$, which is orthogonal to $\ker\mathbb T_k$. Also $\ip{\Omega_k}{\Psi_A}=\omega_k(A)=0$. So the spectral measure $\nu$ of $\mathbb T_k$ at $\Psi_A$ has no atom at $0$ and is carried by $(0,e^{-m_*a_k}]$, and $C_k(t)=\int\lambda^{t/a_k} d\nu(\lambda)$. If $C_k(0)=\nu((0,1])>0$, the integrand is positive on a set of positive measure and at most $e^{-m_*t}$. (ii) The pairs $(\bar f_0\circ\theta_0,f_0(\cdot-te_0))$, $t\ge0$, are separated by at least $2t_0$, and $\omega_k(\vartheta(\bar A)\tau_{t/a_k} A)=\omega_k(A_k(\bar f_0\circ\theta_{0,k})A_k(f_0(\cdot-te_0)))$ by Lemma~\ref{lem:limitElattice}(i). By the approximation of $\theta_{0,k}$ by $\theta_0$ in the proof of Theorem~\ref{thm:limitEsep}(iii), $\ip{\Psi_0(f_0)}{e^{-\sigma_0H_0}\Psi_0(f_0)}=\lim_kZ_k^2C_k(\sigma_0)=1$ and $\|\Psi_0(f_0)\|^2=\lim_kC_k(0)/C_k(\sigma_0)\ge e^{m_*\sigma_0}$ by (i). Since $\ip{\Omega_0}{\Psi_0(f_0)}=S_1(f_0)=0$, $\mathcal H_0\ne\C\Omega_0$. For $a_k\le t_0$ the pair $(\bar f_0\circ\theta_{0,k},f_0)$ lies in $\mathcal C_2(Q,t_0)$, so (M$_{\mathrm s}$) gives $Z_k^2C_k(0)\le M_2(Q,t_0) \|f_0\|_\infty^2$, which is the last claim. (iii) For a separated polynomial $P$ at positive $0$-times and dyadic $t>0$, the proof of Theorem~\ref{thm:limitEsep}(iv) with $e^{-m_{\mathrm{lat}}t/a_k}$ in place of $e^{-m_*t}$ gives $\ip v{e^{-tH_0}v}\le\lim_ke^{-m_{\mathrm{lat}}t/a_k}\|\Psi'_{P_k} \|^2$, $v=\Psi_0(P)-S(P)\Omega_0$, where $\|\Psi'_{P_k}\|^2$ converges to $\|v\|^2$. So $\|e^{-tH_0/2}v\|^2=0$, and $v=0$ because $e^{-tH_0/2}$ is injective. Hence $\mathcal H_0=\C\Omega_0$. For $Z_k=C_k(\sigma_0)^{-1/2}$, $C_k(\sigma_0)/C_k(0)\le e^{-m_{\mathrm{lat}}\sigma_0/a_k}$ by the argument of (i), which contradicts (ii) for large $k$. \end{proof} The lower bound in (ii) is the analogue of $(\mathrm N_0)$ for the packets without flow, and (iii) is the analogue of Theorem~\ref{thm:strongratio}. The averaged condition itself does not exclude a lattice gap: it holds at strong coupling for every cube size. \begin{corollary}\label{cor:limitEstrong} Let $G=\SU(2)$ and $\beta_{W,k}\in[0,\frac18]$ for every $k$. Then the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}} holds with $\varpi=0.41$, $n_k=2$ and $\ell=2a_0$, every torus limit $\omega_k$ has transfer-matrix gap at least $m_{\mathrm{lat}}=3.28/(625e^2)>7\cdot10^{-4}$ in lattice units, and for every $\Phi$ and every choice of $Z_k$ satisfying \textup{(M$_{\mathrm s}$)} every limit of Theorem~\textup{\ref{thm:limitEsep}} is trivial. \end{corollary} \begin{proof} By Corollary~\ref{cor:bpranges}(i), $\alpha_n\le\alpha_2\le0.59$ for $|\beta_W|\le\frac18$ and every $n\ge2$, so $\mathrm{BP}(2,0.41)$ holds on $\Lambda_L$ for every even $L\ge8$, and $n_ka_k=2a_k\le2a_0$. By Corollary~\ref{cor:avggap} with $n=2$ the gap is at least $\varpi/(2e^2q_2) \ge16\varpi/(1250e^2)=3.28/(625e^2)$, and $625e^2<4619$. Proposition~\ref{prop:limitEnorm}(iii) applies. \end{proof} So the content of Theorem~\ref{thm:limitEsep} beyond the averaged condition lies in (M$_{\mathrm s}$) for constants $Z_k$ that keep the reflected correlation $C_k(\sigma_0)$ fixed. The bound (M) at coinciding supports cannot serve for the action density. \begin{proposition}\label{prop:limitEsing} Let $S_2$ be a limit along a subsequence of $\omega_k(Z_k^2A_k(f)A_k(g))$ for all separated pairs, and suppose that $S_2(f,g)=\iint f(x)g(y)K(x-y)\,dx \,dy$ for separated pairs, with $K\ge0$ measurable on $\R^4\setminus\{0\}$ and \[ K(z)\ge c\,|z|^{-8}\Bigl(\log\frac1{|z|}\Bigr)^{-p}\qquad\text{for }0<|z| \le r_0, \] where $c>0$, $p\ge0$ and $00$. \emph{The two-point functions.} For $n=2$ the separated moments reduce to one quantity of the lattice theory. For $\varepsilon>0$ let $B_\varepsilon= [\varepsilon,\varepsilon+1+a_0]\times[0,1+a_0]^3$ and, for $(r+1)a_k\le \varepsilon$, \[ \mathcal N_k(\varepsilon)=\sup\bigl\{\omega_k\bigl(\vartheta(\overline{ A_k(h)})A_k(h)\bigr):\operatorname{supp}h\subset B_\varepsilon,\ \|h\|_\infty \le1\bigr\}, \] which is finite because $|A_k(h)|$ is at most $2\|\Phi\|_\infty a_k^4\|h\|_\infty$ times the number of sites $x$ with $a_kx\in B_\varepsilon$. Let $\gamma_k\le e^{-m_*a_k}$ be the norm of $\mathbb T_k$ on $\Omega_k^\perp$. \begin{lemma}\label{lem:limitEbox} Let $(r+1)a_k\le\varepsilon$ and let $g\in C_c(\R^4)$ be supported in a cube of side $L$ and in $\{x_0\ge\varepsilon\}$. Then \[ \omega_k\bigl(\vartheta(\overline{A_k(g)})A_k(g)\bigr)\le\mathsf K_L^2\, \mathcal N_k(\varepsilon)\,\|g\|_\infty^2 . \] The same holds with $\vartheta_\mu$ for $g$ supported in $\{x_\mu\ge\varepsilon\}$. \end{lemma} \begin{proof} Write $g=\sum_mh_m$ with $h_m=\chi^1_mg$, $\chi^1_m$ as in the proof of Theorem~\ref{thm:limitEsep}(vi); $\chi^1_m$ is supported in the closed cube of side $1$ centred at $m/2$, and $\chi^1_mg\ne0$ requires $m_\nu\in(2q_\nu-1,2q_\nu+2L+1)$ for the cube $\prod_\nu[q_\nu,q_\nu+L]$, so at most $\mathsf K_L$ pieces are nonzero, each with $\|h_m\|_\infty\le\|g \|_\infty$. Fix one, supported in $[s,s+1]\times\prod_i[u_i,u_i+1]$ and in $\{x_0\ge s'\}$, $s'=\max(s,\varepsilon)$. Let $n_0=\lfloor(s'-\varepsilon) /a_k\rfloor\ge0$, $v_i=\lfloor u_i/a_k\rfloor$, $v=(n_0,v_1,v_2,v_3)$ and $h'=h_m(\cdot+a_kv)$. Then $h'$ is supported in $[s'-n_0a_k,s+1-n_0a_k]\times \prod_i[u_i-v_ia_k,u_i-v_ia_k+1]\subset B_\varepsilon$, since $s'-n_0a_k\in [\varepsilon,\varepsilon+a_k)$ and $u_i-v_ia_k\in[0,a_k)$. By Lemma~\ref{lem:limitElattice}(i), $A_k(h_m)=\tau_{n_0e_0}\tau_{v_\perp} A_k(h')$ with $v_\perp=(0,v_1,v_2,v_3)$, and by Lemma~\ref{lem:limitElattice}(ii),(iii) and its proof, $\|\Psi_{A_k(h_m)}\|=\|\mathbb T_k^{n_0}\Psi_{\tau_{v_\perp}A_k(h')}\|\le \|\Psi_{A_k(h')}\|\le\mathcal N_k(\varepsilon)^{1/2}\|g\|_\infty$. The triangle inequality for the linear map $F\mapsto\Psi_F$ gives $\|\Psi_{A_k(g)}\|\le\mathsf K_L\mathcal N_k(\varepsilon)^{1/2}\|g\|_\infty$. For $\vartheta_\mu$ apply this to $g\circ\pi_\mu$. \end{proof} \begin{theorem}\label{thm:limitEtwo} Assume the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}}, let $\varepsilon>0$, suppose $\mathcal N_k(\varepsilon)>0$ for large $k$, and put $Z_k=\mathcal N_k(\varepsilon)^{-1/2}$. \textup{(i)} Let $c\in\R$, $d\ge0$, $f$ supported in a cube of side $L$ in $\{x_\mu\le c-2\varepsilon\}$ and $g$ in a cube of side $L'$ in $\{x_\mu\ge c+2\varepsilon+d\}$. For $a_k\le\varepsilon$ and $(r+1)a_k\le \varepsilon$, \[ Z_k^2\bigl|\omega_k\bigl(A_k(f)A_k(g)\bigr)\bigr|\le\mathsf K_L\mathsf K_{L'}\,\gamma_k^{\lfloor d/a_k\rfloor}\,\|f\|_\infty\|g\|_\infty . \] \textup{(ii)} For $(f,g)\in\mathcal C_2(Q,\delta)$ with $Q$ of side $L$ and $\delta\ge8\varepsilon$, and $k$ as in \textup{(i)}, \[ Z_k^2\bigl|\omega_k\bigl(A_k(f)A_k(g)\bigr)\bigr|\le\mathsf K_{4L/\delta}^2 \mathsf K_{\delta/4}^2\,\|f\|_\infty\|g\|_\infty . \] \textup{(iii)} Every sequence of $k$ has a subsequence along which $S_2(f,g)=\lim_kZ_k^2\omega_k(A_k(f)A_k(g))$ exists for all $f,g$ with $\dist_\infty(\operatorname{supp}f,\operatorname{supp}g)>8\varepsilon$. It is symmetric, bilinear, bounded as in \textup{(ii)}, invariant under the translations and $B_4$, $S_2(\bar f,\bar g)=\overline{S_2(f,g)}$, and $\sum_{i,j}\bar c_ic_jS_2(\bar g_i\circ\theta_\mu,g_j)\ge0$ for all $c_i\in \C$ and $g_i$ supported in $\{x_\mu>4\varepsilon\}$. \textup{(iv)} For $f,g$ supported in a cube of side $L$, $y\ne0$ and $\lambda|y|_\infty>L+8\varepsilon$, \[ \bigl|S_2(f,g(\cdot-\lambda y))\bigr|\le\mathsf K_L^2\|f\|_\infty \|g\|_\infty\,e^{-m_*(\lambda|y|_\infty-L-4\varepsilon)} . \] \textup{(v)} If $\gamma_k\le e^{-m_{\mathrm{lat}}}$ for a fixed $m_{\mathrm{lat}}>0$ and all $k$, every $S_2$ of \textup{(iii)} vanishes. Parts \textup{(i)--(iii)} use only the invariance of $\omega_k$ under translations and $B_4$ and its reflection positivity. \end{theorem} \begin{proof} (i) By the invariance of $\omega_k$ under $\pi_\mu$ and Lemma~\ref{lem:limitElattice}(i) we may take $\mu=0$. Let $b_k\in a_k\Z$ with $c\le b_k8\varepsilon$. Invariance and the conjugation identity follow as in Theorem~\ref{thm:limitEsep}(ii). For $g=\sum_ic_ig_i$ supported in $\{x_\mu\ge4\varepsilon+\eta'\}$, $\eta'>0$, $Z_k^2\omega_k(\vartheta_\mu(\overline{A_k(g)})A_k(g))\ge0$ by Lemma~\ref{lem:limitElattice}, and it equals $Z_k^2\omega_k(A_k(\bar g\circ \theta_{\mu,k})A_k(g))$; the pair $(\bar g\circ\theta_\mu,g)$ is at distance at least $8\varepsilon+2\eta'$, and (ii) carries the approximation of $\theta_{\mu,k}$ by $\theta_\mu$ as in Theorem~\ref{thm:limitEsep}(iii). (iv) By $B_4$-invariance we may take $y_\mu=|y|_\infty=\eta>0$, and by translation invariance the cube to be $\prod_\nu[q_\nu,q_\nu+L]$. Then $f$ is supported in $\{x_\mu\le c-2\varepsilon\}$ with $c=q_\mu+L+2\varepsilon$, and $g(\cdot-\lambda y)$ in $\{x_\mu\ge c+2\varepsilon+d\}$ with $d=\lambda \eta-L-4\varepsilon>0$. By (i) and $\gamma_k^{\lfloor d/a_k\rfloor}\le e^{-m_*(d-a_k)}$ the lattice quantities obey the bound with $e^{-m_*(d-a_k)}$, and the limit gives the claim. (v) In the proof of (ii), now with $\delta>8\varepsilon$, each pair of pieces satisfies (i) with $d=\delta/2-4\varepsilon>0$, so the lattice quantity is at most $\mathsf K_{4L/\delta}^2\mathsf K_{\delta/4}^2\|f\| _\infty\|g\|_\infty e^{-m_{\mathrm{lat}}(d/a_k-1)}$, which tends to $0$. \end{proof} Two statements about the lattice two-point function control the small separations: \begin{enumerate}[label=\textup{(SD)},leftmargin=*] \item $\limsup_k\mathcal N_k(\varepsilon')/\mathcal N_k(\varepsilon)<\infty$ for every $\varepsilon'\in(0,\varepsilon)$; \end{enumerate} \begin{enumerate}[label=\textup{(NE)},leftmargin=*] \item there is $g_0\in C_c(\R^4)$ supported in $\{x_0>0\}$ with \[ \liminf_k\frac{\omega_k\bigl(\vartheta(\overline{A_k(g_0)})A_k(g_0)\bigr)} {\mathcal N_k(\varepsilon)}>0 . \] \end{enumerate} (SD) bounds the growth of the largest reflected norm of a packet of unit size as it approaches the reflection plane from the distance $\varepsilon$ to any smaller distance, uniformly in the spacing; (NE) says that a fixed packet carries a fixed fraction of that largest norm. \begin{corollary}\label{cor:limitEtwo} Assume the hypothesis of Theorem~\textup{\ref{thm:limitEtwo}}. \textup{(a)} \textup{(SD)} holds if and only if \textup{(M$_{\mathrm s}$)} holds for $n=2$. \textup{(b)} Under \textup{(SD)}, every sequence of $k$ has a subsequence along which $S_2(f,g)$ exists for every separated pair. It is given by a distribution on $\mathfrak D_2$, invariant under the translations and $B_4$, satisfies $S_2(\bar f,\bar g)=\overline{S_2(f,g)}$ and $S_2(\bar g\circ\theta_\mu,g)\ge0$ for $g$ supported in $\{x_\mu>0\}$, and for $f,g$ supported in a cube of side $L$, $y\ne0$, $\varepsilon'\in(0, \varepsilon]$ and $\lambda|y|_\infty>L+4\varepsilon'$, \[ \bigl|S_2(f,g(\cdot-\lambda y))\bigr|\le C_{\varepsilon'}\mathsf K_L^2 \|f\|_\infty\|g\|_\infty\,e^{-m_*(\lambda|y|_\infty-L-4\varepsilon')} \] with $C_{\varepsilon'}=\limsup_k\mathcal N_k(\varepsilon')/\mathcal N_k( \varepsilon)$. The time translations define on the completion $\mathcal H^{(1)}$ of the functions supported in $\{x_0>0\}$ for the form $S_2(\bar g\circ\theta_0,g')$ a semigroup $e^{-tH^{(1)}}$ with $\sigma(H^{(1)})\subset[m_*,\infty)$. Some limit $S_2$ is not identically zero if and only if $\limsup_k\omega_k(\vartheta(\overline{A_k(g)})A_k(g)) /\mathcal N_k(\varepsilon)>0$ for some $g$ supported in $\{x_0>0\}$; under \textup{(NE)} every limit is not identically zero, and then $\sigma(H^{(1)})$ is a nonempty subset of $[m_*,\infty)$. \end{corollary} \begin{proof} (a) If (M$_{\mathrm s}$) holds for $n=2$, let $h$ be supported in $B_{\varepsilon'}$ with $\|h\|_\infty\le1$. For $a_k\le\varepsilon'$ the pair $(\bar h\circ\theta_{0,k},h)$ lies in $\mathcal C_2(Q,\varepsilon')$ for a cube $Q$ independent of $k$ and $h$, and $\omega_k(\vartheta(\overline{ A_k(h)})A_k(h))=\omega_k(A_k(\bar h\circ\theta_{0,k})A_k(h))$, so $\mathcal N_k(\varepsilon')\le M_2(Q,\varepsilon')\mathcal N_k(\varepsilon)$ for large $k$. Conversely, let $(f,g)\in\mathcal C_2(Q,\delta)$ and $\varepsilon'=\min(\varepsilon,\delta/8)$. The proof of Theorem~\ref{thm:limitEtwo}(i), (ii) with $\varepsilon'$ in place of $\varepsilon$ gives $|\omega_k(A_k(f)A_k(g))|\le\mathsf K_{4L/\delta}^2 \mathsf K_{\delta/4}^2\mathcal N_k(\varepsilon')\|f\|_\infty\|g\|_\infty$ for large $k$, and dividing by $\mathcal N_k(\varepsilon)$ and using (SD) gives (M$_{\mathrm s}$) for $n=2$. (b) By (a), the proof of Theorem~\ref{thm:limitEsep}(i) for $n=2$ gives the limits and the distribution, and that of Theorem~\ref{thm:limitEsep}(ii), (iii) the invariance, the conjugation identity and the positivity. The decay follows from Theorem~\ref{thm:limitEtwo}(i) with $\varepsilon'$ in place of $\varepsilon$, multiplied by $\mathcal N_k(\varepsilon')/\mathcal N_k(\varepsilon)$, as in the proof of Theorem~\ref{thm:limitEtwo}(iv). The semigroup comes from the proof of Theorem~\ref{thm:limitEsep}(iv) restricted to single functions. Since $\ip{\Omega_k}{\Psi_{A_k(g)}}=0$, Lemma~\ref{lem:limitElattice}(iii) gives $\omega_k(\vartheta(\overline{ A_k(g)})\tau_{t/a_k}A_k(g))\le e^{-m_*t}\omega_k(\vartheta(\overline{ A_k(g)})A_k(g))$, hence $\ip w{e^{-tH^{(1)}}w}\le e^{-m_*t}\|w\|^2$ on a dense set of $w$, and $\|e^{-tH^{(1)}}\|\le e^{-m_*t}$. If the $\limsup$ is positive for $g$, a subsequence along which the ratio converges to it has a further subsequence with $S_2(\bar g\circ\theta_0,g)$ equal to that limit, by the approximation of $\theta_{0,k}$ by $\theta_0$. Conversely, if $S_2(\bar g\circ\theta_\mu,g)=0$ for all $\mu$ and all $g$ supported in $\{x_\mu>0\}$, the Cauchy--Schwarz inequality for these positive forms and translation invariance give $S_2(f,g)=0$ whenever $f$ and $g$ lie on opposite sides of a coordinate hyperplane, and the decomposition in the proof of Theorem~\ref{thm:limitEtwo}(ii) splits every separated pair into such pairs; by $B_4$-invariance $S_2(\bar g\circ\theta_\mu,g)=S_2(\overline{ g\circ\pi_\mu}\circ\theta_0,g\circ\pi_\mu)$, so a nonzero $S_2$ has $S_2(\bar g\circ\theta_0,g)>0$ for some $g$, and the ratio for $g$ converges to it along the subsequence. Under (NE), $S_2(\bar g_0\circ\theta_0,g_0)$ is at least the $\liminf$ in (NE), so $\mathcal H^{(1)}\ne0$. \end{proof} \emph{The short-distance comparison.} The comparison (SD) is a statement about the spectrum of the transfer matrices. For $\Lambda>0$ let $\mathbb E_k(\Lambda)$ be the spectral projection of $\mathbb T_k$ on $[e^{-\Lambda a_k},1]$, the states of energy at most $\Lambda$ in physical units, put $\mathsf K=\mathsf K_{1+a_0}=(5+2a_0)^4$ and, for $(r+1)a_k\le \varepsilon$, \[ \begin{split} \mathcal N_k(\varepsilon;\Lambda)=\sup\bigl\{&\ip{\Psi_{A_k(h)}}{\mathbb E_k(\Lambda)\Psi_{A_k(h)}}:\\ &\operatorname{supp}h\subset B_\varepsilon,\ \|h\|_\infty\le1\bigr\}. \end{split} \] \begin{enumerate}[label=\textup{(UV)},leftmargin=*] \item For every $\varepsilon'\in(0,\varepsilon)$ there are $\Lambda<\infty$ and $c>0$ with $\mathcal N_k(\varepsilon';\Lambda)\ge c\,\mathcal N_k(\varepsilon')$ for all large $k$. \end{enumerate} (UV) says that the largest reflected norm of a unit packet near the reflection plane is carried, up to a fixed fraction, by energies bounded in physical units. \begin{proposition}\label{prop:limitEsd} Assume the hypothesis of Theorem~\textup{\ref{thm:limitEtwo}}. \textup{(i)} For $(r+1)a_k\le\varepsilon'<\varepsilon$, \[ \mathcal N_k(\varepsilon)\le\mathsf K^2\gamma_k^{2\lfloor(\varepsilon- \varepsilon')/a_k\rfloor}\mathcal N_k(\varepsilon')\le\mathsf K^2 e^{-2m_*(\varepsilon-\varepsilon'-a_k)}\mathcal N_k(\varepsilon') . \] \textup{(ii)} \textup{(SD)} holds if and only if \textup{(UV)} holds. \textup{(iii)} If \textup{(SD)} holds, then for every $\varepsilon'\in(0, \varepsilon)$ \[ \limsup_k\frac1{a_k}\log\frac1{\gamma_k}\le\frac{\log(\mathsf K^2 C_{\varepsilon'})}{2(\varepsilon-\varepsilon')},\qquad C_{\varepsilon'}= \limsup_k\frac{\mathcal N_k(\varepsilon')}{\mathcal N_k(\varepsilon)} . \] In particular, if $\gamma_k\le e^{-m_{\mathrm{lat}}}$ for a fixed $m_{\mathrm{lat}}>0$ and all $k$, then $\mathcal N_k(\varepsilon')/\mathcal N_k(\varepsilon)\to\infty$ for every $\varepsilon'\in(0,\varepsilon)$ and \textup{(SD)} fails; this is the case under the hypotheses of Corollary~\textup{\ref{cor:limitEstrong}} whenever $\mathcal N_k(\varepsilon)>0$ for large $k$. \end{proposition} \begin{proof} (i) Let $h$ be supported in $B_\varepsilon$ with $\|h\|_\infty\le1$, put $n=\lfloor(\varepsilon-\varepsilon')/a_k\rfloor$ and $h'=h(\cdot+na_ke_0)$, supported in a cube of side $1+a_0$ in $\{x_0\ge\varepsilon'\}$. By Lemma~\ref{lem:limitElattice}(i) and the proof of Lemma~\ref{lem:limitElattice}(iii), $\Psi_{A_k(h)}=\mathbb T_k^n \Psi_{A_k(h')}$, and $\Psi_{A_k(h')}\perp\Omega_k$ because the packet is centred. So $\|\Psi_{A_k(h)}\|\le\gamma_k^n\|\Psi_{A_k(h')}\|\le\gamma_k^n \mathsf K\mathcal N_k(\varepsilon')^{1/2}$ by Lemma~\ref{lem:limitEbox}, and $\gamma_k^n\le e^{-m_*a_kn}\le e^{-m_*(\varepsilon-\varepsilon'-a_k)}$. (ii) First a version of Lemma~\ref{lem:limitEbox} for $\mathbb E=\mathbb E_k(\Lambda)$. For a function $F$ of links in $\{x_0\ge1\}$ and $v\in\Z^4$ with $v_0=0$, the spectral measures of $\mathbb T_k$ at $\Psi_F$ and at $\Psi_{\tau_vF}$ are measures on $[0,1]$ with the same moments $\omega_k(\vartheta(\bar F)\tau_jF)=\omega_k(\tau_v(\vartheta(\bar F) \tau_jF))$, $j\ge0$, so they coincide and $\|\mathbb E\Psi_{\tau_vF}\|= \|\mathbb E\Psi_F\|$. If $g$ is supported in a cube of side $L$ in $\{x_0\ge\varepsilon\}$, the proof of Lemma~\ref{lem:limitEbox} writes $\Psi_{A_k(g)}$ as a sum of at most $\mathsf K_L$ vectors $\mathbb T_k^{n_m}\Psi_{\tau_{v_m}A_k(h'_m)}$ with $(v_m)_0=0$, $h'_m$ supported in $B_\varepsilon$ and $\|h'_m\|_\infty\le\|g\|_\infty$; since $\mathbb E$ commutes with $\mathbb T_k$ and $\|\mathbb T_k\|\le1$, \[ \|\mathbb E\Psi_{A_k(g)}\|\le\mathsf K_L\,\mathcal N_k(\varepsilon; \Lambda)^{1/2}\|g\|_\infty . \] Assume (UV) and let $\varepsilon'\in(0,\varepsilon)$, with $\Lambda$, $c$ as in (UV). Take $h$ supported in $B_{\varepsilon'}$ with $\|h\|_\infty\le1$ and $\ip{\Psi}{\mathbb E\Psi}\ge\frac c2\mathcal N_k(\varepsilon')$, $\Psi=\Psi_{A_k(h)}$, let $n=\lceil(\varepsilon-\varepsilon')/a_k\rceil$ and $h_n=h(\cdot-na_ke_0)$, supported in a cube of side $1+a_0$ in $\{x_0\ge\varepsilon\}$. With $\nu$ the spectral measure of $\mathbb T_k$ at $\Psi$, \[ \begin{split} \|\Psi_{A_k(h_n)}\|^2&=\int\lambda^{2n}d\nu(\lambda)\ge e^{-2\Lambda a_kn} \nu\bigl([e^{-\Lambda a_k},1]\bigr)\\ &\ge e^{-2\Lambda(\varepsilon-\varepsilon'+a_k)}\tfrac c2\mathcal N_k(\varepsilon'), \end{split} \] and Lemma~\ref{lem:limitEbox} bounds the left side by $\mathsf K^2\mathcal N_k(\varepsilon)$. So $\mathcal N_k(\varepsilon')/\mathcal N_k(\varepsilon) \le2c^{-1}\mathsf K^2e^{2\Lambda(\varepsilon-\varepsilon'+a_k)}$, which is (SD). Conversely, let (SD) hold and $\varepsilon'\in(0,\varepsilon)$, and fix a number $C$ greater than $\limsup_k\mathcal N_k(\varepsilon')/\mathcal N_k(\varepsilon)$, and put $\Lambda=\log(4C\mathsf K^2)/(\varepsilon- \varepsilon')$. For $h$, $n$ and $h'$ as in the proof of (i), $\|\mathbb T_k(1-\mathbb E)\|\le e^{-\Lambda a_k}$, so \[ \|\Psi_{A_k(h)}\|=\|\mathbb T_k^n\Psi_{A_k(h')}\|\le\|\mathbb E \Psi_{A_k(h')}\|+e^{-\Lambda a_kn}\|\Psi_{A_k(h')}\|, \] and the two forms of Lemma~\ref{lem:limitEbox} give \[ \mathcal N_k(\varepsilon)\le2\mathsf K^2\bigl(\mathcal N_k(\varepsilon'; \Lambda)+e^{-2\Lambda a_kn}\mathcal N_k(\varepsilon')\bigr). \] For large $k$, $a_k\le \frac12(\varepsilon-\varepsilon')$, so $a_kn\ge\frac12(\varepsilon- \varepsilon')$ and $2C\mathsf K^2e^{-2\Lambda a_kn}\le\frac12$; with $\mathcal N_k(\varepsilon')\le C\mathcal N_k(\varepsilon)$ this gives $\mathcal N_k(\varepsilon')\le4C\mathsf K^2\mathcal N_k(\varepsilon';\Lambda)$, which is (UV) at $\varepsilon'$. (iii) By (i), $\mathcal N_k(\varepsilon')/\mathcal N_k(\varepsilon)\ge \mathsf K^{-2}\gamma_k^{-2n}$ with $n\ge(\varepsilon-\varepsilon')/a_k-1$; in particular $\mathsf K^2C_{\varepsilon'}\ge1$. For $C>C_{\varepsilon'}$ and large $k$ this gives $2n\log(1/\gamma_k)\le\log(\mathsf K^2C)$, that is $a_k^{-1}\log(1/\gamma_k)\le\log(\mathsf K^2C)/(2(\varepsilon-\varepsilon' -a_k))$, and $C\downarrow C_{\varepsilon'}$ gives the bound. Under a lattice gap the same inequality gives $\mathcal N_k(\varepsilon')/\mathcal N_k(\varepsilon)\ge\mathsf K^{-2}e^{2m_{\mathrm{lat}}((\varepsilon- \varepsilon')/a_k-1)}\to\infty$. Corollary~\ref{cor:limitEstrong} provides the lattice gap $m_{\mathrm{lat}}=3.28/(625e^2)$. \end{proof} So (SD) bounds the physical gap from above, and it fails in the strong-coupling range of Corollary~\ref{cor:limitEstrong}. It does not follow from invariance, reflection positivity and a gap bounded above and below in physical units: a heavy species with a growing weight removes it. \begin{proposition}\label{prop:limitEheavy} Let $m>0$, $\varrho\in(0,1)$, $q_k=e^{-ma_k}$, let $\phi^{(1)}_k$ and $\phi^{(2)}_k$ be independent centred Gaussian fields on $\Z^4$ with covariances $\prod_\mu q_k^{|x_\mu-y_\mu|}$ and $\prod_\mu \varrho^{|x_\mu-y_\mu|}$, let $\lambda_k>0$, $\phi_k=\phi^{(1)}_k+\lambda_k \phi^{(2)}_k$, $A_k(f)=\sum_xa_k^4f(a_kx)\phi_k(x)$, and form $\mathcal N_k(\varepsilon)$ with these packets and $r=0$. \textup{(a)} The law of $\phi_k$ is invariant under translations and $B_4$ and reflection positive for the reflections in coordinate hyperplanes through sites and bisecting links. For $q_k\ge \varrho$ the spectrum of its transfer matrix in each coordinate direction on $\Omega_k^\perp$ lies in $[0,q_k]$ and contains $q_k$: the gap is $m$ in physical units. \textup{(b)} Given $\varepsilon>0$, there are $\lambda_k$ with $\mathcal N_k(\varepsilon')/\mathcal N_k(\varepsilon)\to\infty$ for every $\varepsilon'\in(0,\varepsilon)$, so that \textup{(SD)} fails. \textup{(c)} For these $\lambda_k$ and every $Z_k$ for which \textup{(M$_{\mathrm s}$)} holds at $n=2$, $Z_k^2\omega_k(A_k(f)A_k(g))\to0$ for every separated pair: every limit of the two-point functions vanishes. \end{proposition} \begin{proof} (a) Each $\phi^{(i)}_k$ is the field of Proposition~\ref{prop:osobstruction}(a), with $q_k$ replaced by $\varrho$ for $i=2$, which has the invariance and the reflection positivity, with the transfer operators $P_1$, $P_2$ of spectra $\{q_k^N\}$ and $\{\varrho^N\}$, $1$ simple, given in its proof. The pair $(\phi^{(1)}_k,\phi^{(2)}_k)$ is then a reversible Markov chain in each coordinate direction with transfer operator $P_1\otimes P_2$, of spectrum $\{q_k^N\varrho^{N'}\}$ with $1$ simple, and the argument of that proof gives $\E[\vartheta(\bar F)F']=\ip{g_F}{g_{F'}}$ for functions $F,F'$ of the pair at positive times. The functions of $\phi_k$ at positive times are functions of the pair there, so $\phi_k$ is reflection positive, its transfer matrix is the restriction of $P_1\otimes P_2$ to an invariant subspace, and its spectrum on $\Omega_k^\perp$ lies in $[0,\max(q_k,\varrho)]=[0,q_k]$. For a site $x$ at positive time, $g_{\phi_k(x)}$ is a sum of two orthogonal eigenvectors, of eigenvalues $q_k$ and $\varrho$, the first one nonzero, so $q_k$ is in the spectrum. (b) Let $\vartheta$ fix the plane $\{x_0=c\}$, $|c|\le\frac12$, so that $\vartheta x=(2c-x_0,\mathbf x)$. By independence and centring, $\omega_k(\vartheta(\overline{A_k(h)})A_k(h))=\mathcal N^{(1)}(h)+ \lambda_k^2\mathcal N^{(2)}(h)$, with $\mathcal N^{(i)}(h)\ge0$ formed with $\phi^{(i)}_k$. For $h$ supported in $\{x_0\ge a_k\}$, \[ \begin{split} \mathcal N^{(2)}(h)&=a_k^8\sum_{\mathbf x,\mathbf y}\overline{H(\mathbf x)} H(\mathbf y)\prod_{i=1}^3\varrho^{|x_i-y_i|},\\ H(\mathbf x)&=\sum_{x_0}h(a_kx)\varrho^{x_0-c} . \end{split} \] If $h$ is supported in $B_\varepsilon$ with $\|h\|_\infty\le1$ and $a_k\le1$, then $|H|\le \varrho^{\varepsilon/a_k-1/2}/(1-\varrho)$, at most $((2+a_0)/a_k)^3$ points $\mathbf x$ have $H(\mathbf x)\ne0$, and $\sum_{\mathbf y}\prod_i\varrho^{|x_i-y_i|}=((1+\varrho)/(1-\varrho))^3$, so \[ \mathcal N^{(2)}(h)\le U_k(\varepsilon)=c_\varrho a_k^5\varrho^{2\varepsilon/a_k},\qquad c_\varrho=\frac{(2+a_0)^3(1+\varrho)^3}{\varrho(1-\varrho)^5} . \] For $\varepsilon'>0$ and $\tau\in(0,\frac12)$ let $h_*$ be continuous with values in $[0,1]$, supported in $B_{\varepsilon'}$ and equal to $1$ on $D=[\varepsilon'+\tau,\varepsilon'+1]\times[\tau,1]^3$. All terms of $\mathcal N^{(2)}(h_*)$ are nonnegative; keeping $x=y$ with $x_0=\lceil (\varepsilon'+\tau)/a_k\rceil$ and $a_k\mathbf x\in[\tau,1]^3$, for $a_k\le\frac12(1-\tau)$, \[ \mathcal N^{(2)}(h_*)\ge L_k(\varepsilon')=\Bigl(\frac{1-\tau}2\Bigr)^3 a_k^5\varrho^{2(\varepsilon'+\tau)/a_k+3} . \] Let $\mathcal N^{(1)}_k(\varepsilon)$ be the supremum of $\mathcal N^{(1)}(h)$ over $h$ supported in $B_\varepsilon$ with $\|h\|_\infty\le1$. For $a_kx,a_ky\in D_\varepsilon=[\varepsilon+\frac14,\varepsilon+1]\times [\frac14,1]^3$ the covariance of $\phi^{(1)}_k(\vartheta x)$ and $\phi^{(1)}_k(y)$ is at least $e^{-m(2\varepsilon+6)}$; with the function $h_*$ for $\varepsilon$ and $\tau=\frac14$ this gives $\mathcal N^{(1)}_k(\varepsilon)\ge\ell_0=(\frac38)^8e^{-m(2\varepsilon+6)} >0$ for $a_k\le\frac38$, and it is finite. Put $\lambda_k^2=\mathcal N^{(1)}_k(\varepsilon)/U_k(\varepsilon)$. Then $\mathcal N_k(\varepsilon)\le \mathcal N^{(1)}_k(\varepsilon)+\lambda_k^2U_k(\varepsilon)=2\mathcal N^{(1)}_k(\varepsilon)$ and $\mathcal N_k(\varepsilon')\ge\lambda_k^2 L_k(\varepsilon')$, so for $\tau<\varepsilon-\varepsilon'$ \[ \begin{split} \frac{\mathcal N_k(\varepsilon')}{\mathcal N_k(\varepsilon)}&\ge \frac{L_k(\varepsilon')}{2U_k(\varepsilon)}\\ &=\Bigl(\frac{1-\tau}2\Bigr)^3\frac{\varrho^4(1-\varrho)^5\, \varrho^{-2(\varepsilon-\varepsilon'-\tau)/a_k}}{2(2+a_0)^3(1+\varrho)^3} \to\infty . \end{split} \] (c) Let $(f,g)\in\mathcal C_2(Q_0,\delta)$, $Q_0$ of side $L_0$, and fix $\varepsilon',\tau>0$ with $\varepsilon'+\tau<\min(\varepsilon,\delta/2)$ and $\tau<\frac12$. For $a_k\le\varepsilon'$ the pair $(\bar h_*\circ \theta_{0,k},h_*)$ lies in $\mathcal C_2(Q',\varepsilon')$ for a cube $Q'$ independent of $k$, so (M$_{\mathrm s}$) gives $Z_k^2\lambda_k^2 L_k(\varepsilon')\le Z_k^2\omega_k(\vartheta(A_k(h_*))A_k(h_*))\le M=M_2(Q',\varepsilon')$, that is $Z_k^2\mathcal N^{(1)}_k(\varepsilon)\le MU_k( \varepsilon)/L_k(\varepsilon')$. The covariance of $\phi^{(1)}_k$ is at most $1$ and at most $(L_0/a_k+1)^4$ sites carry $f$ or $g$, so $|\E[A^{(1)}_k(f)A^{(1)}_k(g)]|\le(L_0+1)^8\|f\|_\infty\|g\|_\infty$ for $a_k\le1$, and \[ Z_k^2\bigl|\E[A^{(1)}_k(f)A^{(1)}_k(g)]\bigr|\le\frac{M(L_0+1)^8}{\ell_0} \,\frac{U_k(\varepsilon)}{L_k(\varepsilon')}\|f\|_\infty\|g\|_\infty\to0 . \] For the second field the sites carrying $f$ and $g$ are at $\ell^\infty$ distance at least $\delta/a_k$, and $\sum_{|z|_\infty\ge d} \prod_\mu \varrho^{|z_\mu|}\le8\varrho^d(1+\varrho)^3/(1-\varrho)^4$, so $\lambda_k^2|\E[A^{(2)}_k(f)A^{(2)}_k(g)]|\le\lambda_k^2C_1a_k^4 \varrho^{\delta/a_k}\|f\|_\infty\|g\|_\infty$ with $C_1=8(L_0+1)^4(1+\varrho)^3/ (1-\varrho)^4$. Since $Z_k^2\lambda_k^2\le M/L_k(\varepsilon')$, \[ \begin{split} &Z_k^2\lambda_k^2\bigl|\E[A^{(2)}_k(f)A^{(2)}_k(g)]\bigr|\\ &\quad\le MC_1\Bigl(\frac2{1-\tau}\Bigr)^3\frac{\varrho^{(\delta-2 \varepsilon'-2\tau)/a_k-3}}{a_k}\|f\|_\infty\|g\|_\infty\to0 , \end{split} \] because $\delta>2(\varepsilon'+\tau)$. The cross terms vanish. \end{proof} In the example the part $\lambda_k^2\mathcal N^{(2)}$ of the reflected norm, which produces the growth, is carried by energies at least $a_k^{-1}\log(1/\varrho)\to\infty$, and (UV) fails, by the proof of Proposition~\ref{prop:limitEsd}(ii), which uses only the invariance, reflection positivity and the gap. For $\lambda_k=0$ it is the field of Proposition~\ref{prop:osobstruction}(a), for which (SD) and (M$_{\mathrm s}$) hold. Like that field, it is not a lattice gauge theory; it shows that (SD), equivalently (UV), is an input on the ultraviolet behaviour of the Wilson measure that reflection positivity and the gap do not supply. \emph{The nonvanishing condition.} Under (SD) the condition (NE) is exactly the nontriviality of the limits. Write $R_k(g)=\omega_k(\vartheta(\overline{ A_k(g)})A_k(g))/\mathcal N_k(\varepsilon)$ for $g$ supported in $\{x_0\ge(r+1)a_k\}$. \begin{proposition}\label{prop:limitEne} Assume the hypothesis of Theorem~\textup{\ref{thm:limitEtwo}} and \textup{(SD)}. \textup{(i)} The following are equivalent: \textup{(NE)}; every limit $S_2$ of Corollary~\textup{\ref{cor:limitEtwo}(b)} is not identically zero; every such limit has $\mathcal H^{(1)}\ne0$. \textup{(ii)} If $\omega_k(\Phi_0\Phi_z)\ge c_k^2$ for all large $k$ and all $z\in\Z^4$ with $|z|_\infty\ge2r+1$, then $R_k(g_0)\ge1$ for large $k$ and every continuous $g_0$ with values in $[0,1]$, equal to $1$ on $B_\varepsilon$ and supported in $\{x_0\ge\varepsilon/2\}$; in particular \textup{(NE)} holds. \end{proposition} \begin{proof} (i) The first statement implies the second by Corollary~\ref{cor:limitEtwo}(b). The second and the third are equivalent: $\mathcal H^{(1)}\ne0$ means $S_2(\bar g\circ\theta_0,g)>0$ for some $g$ supported in $\{x_0>0\}$, and the proof of Corollary~\ref{cor:limitEtwo}(b) shows that a limit with this property for no $g$ vanishes identically. Suppose (NE) fails, so that $\liminf_kR_k(g)=0$ for every $g\in C_c(\R^4)$ supported in $\{x_0>0\}$. Let $g_1,\dots,g_J$ be supported in a cube of side $L$ and in $\{x_0\ge\eta\}$, $\eta>0$, let $D>0$ be dyadic and $G=\sum_jg_j(\cdot-jDe_1)$. For large $k$, $jD/a_k\in\Z$ and $R_k(g_j(\cdot-jDe_1))=R_k(g_j)$ by Lemma~\ref{lem:limitElattice}(i),(iii). For $i\ne j$ the cross term $\ip{\Psi_{A_k(g_i(\cdot-iDe_1))}} {\Psi_{A_k(g_j(\cdot-jDe_1))}}$ equals $\omega_k(A_k(\bar g_i(\cdot-iDe_1) \circ\theta_{0,k})A_k(g_j(\cdot-jDe_1)))$, two packets in cubes of side $L$ whose projections on the $e_1$ axis are at distance at least $|i-j|D-L$; by Theorem~\ref{thm:limitEtwo}(i) in the direction $e_1$ with $d=D-L-4 \varepsilon$, its modulus is at most $\mathcal N_k(\varepsilon)\mathsf K_L^2e^{-m_*(D-L-4\varepsilon-1)}\max_j\|g_j\|_\infty^2$. Hence $\sum_jR_k(g_j)\le R_k(G)+J^2\mathsf K_L^2e^{-m_*(D-L-4\varepsilon-1)} \max_j\|g_j\|_\infty^2$ for large $k$, and since $\liminf_kR_k(G)=0$ and $D$ is arbitrary, $\liminf_k\sum_jR_k(g_j)=0$. Enumerate as $g_1,g_2,\dots$ the elements of the countable set $\mathcal G$ of the proof of Theorem~\ref{thm:limitEsep}(i) that are supported in $\{x_0>0\}$, and choose $k_10\}$. So this limit vanishes identically, and the second statement fails. (ii) Let $h$ be supported in $B_\varepsilon$ with $\|h\|_\infty\le1$, and $(r+1)a_k\le\varepsilon/2$. By Lemma~\ref{lem:limitElattice}(i), $\omega_k(\vartheta(\overline{A_k(h)})A_k(h))=\sum_{x,y}a_k^8\overline{h(a_k x)}h(a_ky)K(x,y)$ with $K(x,y)=\omega_k(\Phi_{\vartheta x}\Phi_y)-c_k^2= \omega_k(\Phi_0\Phi_{y-\vartheta x})-c_k^2$. For $a_kx_0,a_ky_0\ge \varepsilon/2$, $x_0,y_0\ge r+1$ and the time component of $y-\vartheta x$ is $x_0+y_0-2c\ge2r+1$, so $K(x,y)\ge0$ and the sum is at most $\sum_{x,y}a_k^8g_0(a_kx)g_0(a_ky)K(x,y)=\omega_k(\vartheta(A_k(g_0)) A_k(g_0))$, because $|h|\le g_0$. Taking the supremum over $h$ gives $\mathcal N_k(\varepsilon)\le\omega_k(\vartheta(A_k(g_0))A_k(g_0))$. \end{proof} The condition of (ii) is a correlation inequality of Griffiths type for the connected two-point function of $\Phi$; it is not proved here for the Wilson measure. \emph{Products of packets.} For $n\ge3$ the moment bound reduces to reflected norms of products. Consider \begin{enumerate}[label=\textup{(RN)},leftmargin=*] \item For every $m\ge1$, every cube $Q$ and every $\delta>0$ there are $R_m(Q,\delta)<\infty$ and $k_0$ such that \[ Z_k^{2m}\omega_k\bigl(\vartheta(\bar P_k)P_k\bigr)\le R_m(Q,\delta) \prod_{j=1}^m\|f_j\|_\infty^2,\qquad P_k=\prod_{j=1}^mA_k(f_j), \] for all $k\ge k_0$ and $(f_1,\dots,f_m)\in\mathcal C_m(Q,\delta)$ with supports in $\{x_0\ge\delta\}$. \end{enumerate} We write (RN$_m$) for the statement at a fixed $m$, and (M$_{\mathrm s,n}$) for (M$_{\mathrm s}$) at a fixed $n$. \begin{proposition}\label{prop:limitEreflect} Assume the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}}. \textup{(i)} \textup{(M$_{\mathrm s,2m}$)} implies \textup{(RN$_m$)}. \textup{(ii)} Let $n\ge2$. If \textup{(RN$_i$)} holds for $1\le i\le n-1$, then \textup{(M$_{\mathrm s,n}$)} holds, with \[ M_n(Q,\delta)=\mathsf K_{2nL/\delta}^{\,n}\max_{1\le i\le n-1}\bigl(R_i(Q^*, \delta_n)R_{n-i}(Q^*,\delta_n)\bigr)^{1/2} \] for $Q$ of side $L$, $Q^*=[-L-2,L+2]^4$ and $\delta_n=\delta/(16(n+1))$. \textup{(iii)} \textup{(M$_{\mathrm s}$)} holds if and only if \textup{(RN)} holds, and \textup{(M$_{\mathrm s,2}$)} if and only if \textup{(RN$_1$)}. With $Z_k=\mathcal N_k(\varepsilon)^{-1/2}$ as in Theorem~\textup{\ref{thm:limitEtwo}}, \textup{(RN$_1$)} is equivalent to \textup{(SD)}, and \textup{(M$_{\mathrm s,3}$)} follows from \textup{(SD)} and \textup{(RN$_2$)}. \end{proposition} \begin{proof} (i) Let $(f_j)\in\mathcal C_m(Q,\delta)$ be supported in $\{x_0\ge\delta\}$ and $a_k\le\delta$. The functions $\bar f_j\circ\theta_{0,k}$ are supported in $\{x_0\le a_k-\delta\}$, so the $2m$-tuple of these and the $f_j$ lies in $\mathcal C_{2m}(Q',\delta)$ for a cube $Q'$ containing $Q$ and all $\theta_{0,k}(Q)$, and $\omega_k(\vartheta(\bar P_k)P_k)=\omega_k(\prod_j A_k(\bar f_j\circ\theta_{0,k})\prod_jA_k(f_j))$ by Lemma~\ref{lem:limitElattice}(i). (M$_{\mathrm s,2m}$) gives (RN$_m$) with $R_m(Q,\delta)=M_{2m}(Q',\delta)$. (ii) Let $(f_j)\in\mathcal C_n(Q,\delta)$, $s=\delta/(2n)$, and decompose every $f_j$ with the functions $\chi^s_\kappa$ of the proof of Theorem~\ref{thm:limitEsep}(vi); at most $\mathsf K_{L/s}=\mathsf K_{2nL/\delta}$ pieces of each are nonzero, each with norm at most $\|f_j\|_\infty$. Fix a nonzero term, with pieces in closed cubes $C_1,\dots,C_n$ of side $s$, points $p\in C_1$ and $q\in C_2$ in the supports and $\mu$ with $q_\mu-p_\mu\ge\delta$ (after exchanging $f_1$ and $f_2$). The projections of the $C_j$ on the $\mu$ axis have total length $\delta/2$, so their complement in $[p_\mu,q_\mu]$ contains an interval of length at least $\delta/(2(n+1))$; let $c$ be its midpoint. Every projection is at distance at least $\delta/(4(n+1))=4\delta_n$ from $c$; let $I$ and $J$ be the indices of the cubes below and above $c$, nonempty since $1\in I$ and $2\in J$. Translate all pieces by one vector of $a_k\Z^4$ so that $c$ moves within $a_k/2$ of the plane $\{x_\mu=a_kc_\mu\}$ fixed by $\theta_{\mu,k}$, $|c_\mu|\le\frac12$, and so that the pieces lie in $Q^*$; the moment does not change. For $a_k\le2\delta_n$ the pieces of $I$ lie in $\{x_\mu\le a_kc_\mu -3\delta_n\}$ and those of $J$ in $\{x_\mu\ge a_kc_\mu+3\delta_n\}$. With $F=\prod_{j\in I}A_k(\bar f'_j\circ\theta_{\mu,k})$ and $G=\prod_{j\in J} A_k(f'_j)$, $f'_j$ the translated pieces, Lemma~\ref{lem:limitElattice}(i) gives $\prod_{j\in I}A_k(f'_j)=\vartheta_\mu(\bar F)$, and by the invariance under $\pi_\mu$ and the Cauchy--Schwarz inequality for the positive form of Lemma~\ref{lem:limitElattice}(iii), \[ \Bigl|\omega_k\Bigl(\prod_{j=1}^nA_k(f'_j)\Bigr)\Bigr|=|\omega_k( \vartheta(\overline{\pi_\mu F})\pi_\mu G)|\le N_0(\pi_\mu F)N_0(\pi_\mu G). \] The factors of $\pi_\mu F$ and $\pi_\mu G$ are supported in $Q^*$ and in $\{x_0\ge3\delta_n-a_k\}\subset\{x_0\ge\delta_n\}$, pairwise at distance at least $\delta$, so (RN$_{|I|}$) and (RN$_{|J|}$) bound $Z_k^nN_0(\pi_\mu F) N_0(\pi_\mu G)$ by $(R_{|I|}R_{|J|})^{1/2}\prod_j\|f_j\|_\infty$ for large $k$. Summing over the at most $\mathsf K_{2nL/\delta}^n$ terms gives the claim. (iii) The first two statements follow from (i) and (ii). For $Z_k=\mathcal N_k(\varepsilon)^{-1/2}$, (RN$_1$) at $Q\supset B_{\varepsilon'}$ and $\delta=\varepsilon'$ gives $\mathcal N_k(\varepsilon')\le R_1\mathcal N_k(\varepsilon)$, which is (SD); conversely Lemma~\ref{lem:limitEbox} bounds the left side of (RN$_1$) by $\mathsf K_L^2\|f\|_\infty^2$ times $\mathcal N_k(\delta)/\mathcal N_k(\varepsilon)$, which is bounded by (SD) for $\delta<\varepsilon$ and by Proposition~\ref{prop:limitEsd}(i) for $\delta\ge\varepsilon$. The last claim is (ii) for $n=3$. \end{proof} The Cauchy--Schwarz inequality of reflection positivity bounds a moment by the reflected norms of the products of the two groups of packets on the two sides of a plane, and for $n\ge3$ one group contains at least two packets; the reduction therefore stops at (RN$_m$) with $m\ge2$, the reflected norms of products of separated packets. If the packets are jointly Gaussian, Wick's formula writes $\omega_k(\vartheta(\bar P_k)P_k)$ as a sum of $(2m-1)!!$ products of covariances of separated pairs, and (RN) follows from (RN$_1$) by (ii) with $n=2$; for the Wilson measure (RN$_m$) with $m\ge2$ is not proved. \emph{Rotations at separated points.} The hypercubic group fixes the lowest anisotropy that a separated two-point function can carry. Let $\mathcal P_l$ be the space of homogeneous polynomials of degree $l$ on $\R^4$, $\mathcal Y_l\subset\mathcal P_l$ the harmonic ones, and $\mathcal Y_l^{B_4}$ the elements of $\mathcal Y_l$ invariant under $B_4$. Put \[ \begin{split} Y_4(z)&=\sum_\mu z_\mu^4-\tfrac12|z|^4,\\ Y_6(z)&=\sum_\mu z_\mu^6-\tfrac54|z|^2\sum_\mu z_\mu^4+\tfrac5{16}|z|^6 . \end{split} \] \begin{lemma}\label{lem:limitEharm} $\mathcal Y_l^{B_4}=0$ for odd $l$, and $\dim\mathcal Y_{2j}^{B_4}= p(j)-p(j-1)$ for $j\ge1$, where $p(j)$ is the number of partitions of $j$ into at most four parts. In particular $\dim\mathcal Y_l^{B_4}$ is $0$, $1$, $1$, $2$ for $l=2,4,6,8$, and $\mathcal Y_4^{B_4}=\R Y_4$, $\mathcal Y_6^{B_4}=\R Y_6$. \end{lemma} \begin{proof} A sign change $z_\mu\mapsto-z_\mu$ multiplies a monomial by $-1$ to the power of its degree in $z_\mu$, so a polynomial invariant under the sign changes contains only monomials of even degree in every variable; it is a polynomial in $w_\mu=z_\mu^2$, and invariance under the permutations makes it symmetric in $w_0,\dots,w_3$. The monomial symmetric polynomials $m_\kappa(w)$, $\kappa$ a partition of $j$ into at most four parts, form a basis of the symmetric polynomials of degree $j$ in four variables, so $\dim\mathcal P_{2j}^{B_4}=p(j)$ and $\mathcal P_l^{B_4}=0$ for odd $l$. By~\cite[Ch.~IV]{SW71}, $\mathcal P_l=\mathcal Y_l\oplus|z|^2\mathcal P_{l-2}$, and both summands are invariant under $\mathrm O(4)$, so the projections on them commute with $B_4$ and $\mathcal P_l^{B_4}=\mathcal Y_l^{B_4}\oplus|z|^2\mathcal P_{l-2}^{B_4}$; multiplication by $|z|^2$ is injective. This gives the dimension, and $p(0),\dots,p(4)$ are $1,1,2,3,5$. In $\R^4$, $\Delta|z|^{2i}=2i(2i+2)|z|^{2i-2}$, $\Delta\sum_\mu z_\mu^{2i}= 2i(2i-1)\sum_\mu z_\mu^{2i-2}$ and $\Delta(|z|^2\sum_\mu z_\mu^4)= 24\sum_\mu z_\mu^4+12|z|^4$, so $\Delta Y_4=12|z|^2-\frac12\cdot24|z|^2=0$ and $\Delta Y_6=30\sum_\mu z_\mu^4-\frac54(24\sum_\mu z_\mu^4+12|z|^4)+ \frac5{16}\cdot48|z|^4=0$. Both are invariant and nonzero, since $Y_4(e_0)=\frac12$ and $Y_6(e_0)=\frac1{16}$. \end{proof} \begin{proposition}\label{prop:limitEaniso} Let $S_2$ be a bilinear form on the separated pairs, invariant under the translations and $B_4$, with $S_2(f,g)=\iint f(x)g(y)K(x-y)\,dx\,dy$ for separated pairs, where $K$ is square integrable on every compact subset of $\R^4\setminus\{0\}$. \textup{(i)} For $\varphi\in C_c(\R^4\setminus\{0\})$ and $g_\varepsilon= \varepsilon^{-4}\chi(\cdot/\varepsilon)$, $\chi\ge0$ continuous with $\int\chi=1$ and support in $[-1,1]^4$, \[ \int K\varphi\,dz=\lim_{\varepsilon\to0}S_2(\varphi,g_\varepsilon), \] so $K$ is determined almost everywhere by $S_2$. For $R\in\mathrm O(4)$, $S_2(f\circ R^{-1},g\circ R^{-1})=S_2(f,g)$ for all separated pairs if and only if $K\circ R=K$ almost everywhere; in particular $K\circ g=K$ for $g\in B_4$. \textup{(ii)} For almost every $\rho>0$, $K(\rho\,\cdot)\in L^2(S^3)$ and $K(\rho\omega)=\sum_{j\ge0}K_{2j}(\rho,\omega)$ in $L^2(S^3)$, with $K_{2j}(\rho,\cdot)$ the restriction of an element of $\mathcal Y_{2j}^{B_4}$. Hence $K_2=0$, $K_0(\rho)$ is the mean of $K$ on the sphere of radius $\rho$, and there are measurable $k_4,k_6$ with \[ K_4(\rho,\omega)=k_4(\rho)Y_4(\omega),\qquad K_6(\rho,\omega)= k_6(\rho)Y_6(\omega). \] \textup{(iii)} The following are equivalent: \textup{(a)} $S_2$ is invariant under $\mathrm O(4)$ on the separated pairs; \textup{(b)} it is invariant under one $R\in\mathrm O(4)$ of infinite order; \textup{(c)} $K_{2j}(\rho,\cdot)=0$ for every $j\ge2$ and almost every $\rho$; \textup{(d)} $K(z)=K_0(|z|)$ for almost every $z$. \end{proposition} \begin{proof} (i) Let $\varphi$ be supported in $\{|z|_\infty\ge3\eta\}$ and $\varepsilon\le\eta$; the pair $(\varphi,g_\varepsilon)$ is separated by at least $2\eta$, and \[ S_2(\varphi,g_\varepsilon)=\int K(z)\varphi_\varepsilon(z)\,dz,\qquad \varphi_\varepsilon(z)=\int\varphi(z+y)g_\varepsilon(y)\,dy . \] The functions $\varphi_\varepsilon$ are supported in one compact subset of $\R^4\setminus\{0\}$ and converge uniformly to $\varphi$, and $K$ is integrable there. For $R\in\mathrm O(4)$ the substitution $x=Ru$, $y=Rv$ gives $S_2(f\circ R^{-1},g\circ R^{-1})=\iint f(u)g(v)K(R(u-v))\,du\,dv$, so invariance holds if and only if the form with kernel $K\circ R-K$ vanishes on the separated pairs, and by the formula just proved this happens if and only if $K\circ R=K$ almost everywhere. (ii) By Fubini's theorem in polar coordinates, $K(\rho\,\cdot)\in L^2(S^3)$ for almost every $\rho$, and for each of the finitely many $g\in B_4$, $K(\rho g\,\cdot)=K(\rho\,\cdot)$ for almost every $\rho$. The restrictions of the spaces $\mathcal Y_l$ to $S^3$ are orthogonal in $L^2(S^3)$ and span it~\cite[Ch.~IV]{SW71}; each is invariant under $\mathrm O(4)$, so the orthogonal projections on them commute with composition by elements of $\mathrm O(4)$. The component of degree $l$ of $K(\rho\,\cdot)$ is therefore invariant under $B_4$, and Lemma~\ref{lem:limitEharm} gives the claims. (iii) Let $G_K$ be the set of $R\in\mathrm O(4)$ with $K\circ R=K$ almost everywhere. On every closed annulus $A=\{r\le|z|\le r'\}$, $00$, and let $n\ge2$. \textup{(i)} \textup{(SB$_n$)} implies \textup{(M$_{\mathrm s}$)} for this $n$. \textup{(ii)} \textup{(SB$_n$)} and \textup{(RI$_n$)} imply that for every $R\in\mathrm O(4)$ and every separated $n$-tuple \[ \lim_k\Bigl[\omega_k\Bigl(\prod_jZ_kA_k(f_j\circ R^{-1})\Bigr)-\omega_k \Bigl(\prod_jZ_kA_k(f_j)\Bigr)\Bigr]=0 . \] \textup{(iii)} If \textup{(SB$_n$)} and \textup{(RI$_n$)} hold for every $n\ge2$, then \textup{(M$_{\mathrm s}$)} holds and \textup{(W$_{\mathrm s}$)} holds for every $R$. Under the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}} every limit of Theorem~\textup{\ref{thm:limitEsep}} then has $G_S=\mathrm O(4)$, and with \textup{(U$_{\mathrm s}$)} the conclusions of Corollary~\textup{\ref{cor:limitEexact}} hold. \textup{(iv)} Under \textup{(SB$_2$)} and \textup{(RI$_2$)}, every subsequence has a further subsequence along which $\lim_kZ_k^2\omega_k(A_k(f)A_k(g))=\iint f(x)g(y)K(x-y)\,dx\,dy$ for every separated pair, with $K(z)=K_0(|z|)$ and $K_0$ bounded on every compact subset of $(0,\infty)$. \end{theorem} \begin{proof} For $u\in\R^4$ let $\varrho_k(u)$ be the point of $\Z^4$ nearest to $u/a_k$, ties broken by a fixed rule. Then $|u/a_k-\varrho_k(u)|\le1$, the half diagonal of the unit cube of $\Z^4$, and each set $\varrho_k^{-1}(x)$ is, up to a null set, the cube of side $a_k$ centred at $a_kx$. Let $(f_j)\in\mathcal C_n(Q,\delta)$, $Q$ of side $L$, so that all supports lie in a ball $B$ of radius $L$, points of different supports are at Euclidean distance between $\delta$ and $2L$, and the same holds for the rotated functions $f_j\circ R^{-1}$ with the ball $RB$ about the rotated centre. Let $a_k\le\min(\delta/4,L/2)$ and put $F(u)=\prod_jf_j(u_j)$, $\mathcal B_k(F)=\omega_k(\prod_jZ_kA_k(f_j))$ and \[ I_k(F)=\int F(u)\,Z_k^nM_k\bigl(\varrho_k(u_1),\dots,\varrho_k(u_n)\bigr) \,du . \] (i) $\mathcal B_k(F)=Z_k^na_k^{4n}\sum_x\prod_jf_j(a_kx_j)M_k(x)$, the nonzero terms have $x\in\mathcal W_k(\delta,2L)$, and at most $(3L/a_k)^4$ sites $x_j$ have $a_kx_j\in B$, because the cubes of side $a_k$ about them are disjoint and lie in a ball of radius $L+a_k\le\frac32L$ and volume at most $\frac{\pi^2}2(\frac32L)^4<(3L)^4$. So $|\mathcal B_k(F)|\le(b_n(\delta,2L)+1)(3L)^{4n} \prod_j\|f_j\|_\infty$ for large $k$. (ii) Since $\varrho_k$ is constant on the cubes, \[ I_k(F)-\mathcal B_k(F)=\sum_xZ_k^nM_k(x)\int_{\prod_j\varrho_k^{-1}(x_j)} \bigl(F(u)-F(a_kx)\bigr)du . \] A nonzero term has $|u_j-a_kx_j|\le a_k$ for some $u_j$ in the support of $f_j$, hence $x\in\mathcal W_k(\delta/2,3L)$, and the number of such $x$ is at most $(3L/a_k)^{4n}$, the cubes about the $x_j$ lying in a ball of radius $L+2a_k\le2L$ and volume $8\pi^2L^4<(3L)^4$. So $|I_k(F)-\mathcal B_k(F)|\le(b_n(\delta/2,3L)+1) (3L)^{4n}\varpi_F(a_k)$ for large $k$, with $\varpi_F(s)$ the largest change of $F$ when every $u_j$ moves by at most $s$; it tends to $0$, and the same holds for $F_R(u)=\prod_jf_j(R^{-1}u_j)$. By the substitution $u=Rv$, \[ I_k(F_R)-I_k(F)=\int F(v)Z_k^n\bigl[M_k(x'(v))-M_k(x(v))\bigr]dv, \] with $x_j(v)=\varrho_k(v_j)$ and $x'_j(v)=\varrho_k(Rv_j)$. For $v$ in the support of $F$, $|x_i-x_j|$ and $|x'_i-x'_j|$ differ from $|v_i-v_j|/a_k= |Rv_i-Rv_j|/a_k$ by at most $2$, hence from each other by at most $4$, and both configurations lie in $\mathcal W_k(\delta/2,3L)$. By (RI$_n$) the bracket is at most $\epsilon_k\to0$, and $|I_k(F_R)-I_k(F)|\le \epsilon_k\prod_j\|f_j\|_{L^1}$. (iii) The first claims are (i) and (ii) for every $n$, and the rest is Theorem~\ref{thm:limitEsep}(vii) and Corollary~\ref{cor:limitEexact}. (iv) By translation invariance $M_k(x_1,x_2)=G_k(x_2-x_1)$ with $G_k(y)=M_k(0,y)$; put $\kappa_k(z)=Z_k^2G_k(\varrho_k(z))$. For separated $f,g$ as above, $y=\varrho_k(v)-\varrho_k(u)$ and $y'=\varrho_k(v-u)$ have $|y|$ and $|y'|$ within $2$ and $1$ of $|v-u|/a_k$, so (RI$_2$) and the proof of (ii) give $Z_k^2\omega_k(A_k(f)A_k(g))-\iint f(u)g(v)\kappa_k(v-u) \,du\,dv\to0$. For $r\le|z|\le r'$ and large $k$, $|\varrho_k(z)|\in[r/(2a_k), 2r'/a_k]$, so $|\kappa_k|\le b_2(r/2,2r')+1$ there, and $|\kappa_k(Rz)- \kappa_k(z)|\to0$ uniformly there for every $R\in\mathrm O(4)$ by (RI$_2$), since $|\varrho_k(Rz)|$ and $|\varrho_k(z)|$ differ by at most $2$. A diagonal subsequence converges weakly-$*$ in $L^\infty$ of every annulus to some $K$, and then $K\circ R=K$ almost everywhere for every $R$. For separated $f,g$, $\iint f(u)g(v)\kappa_k(v-u)\,du\,dv=\int\kappa_k(z) \Theta(z)\,dz$ with $\Theta(z)=\int f(u)g(u+z)\,du$ integrable and supported in an annulus, so the limit is $\iint f(u)g(v)K(v-u)\,du\,dv$, and $K(-z)=K(z)$. Proposition~\ref{prop:limitEaniso}(iii) gives $K(z)=K_0(|z|)$ almost everywhere. \end{proof} Part (i) does not use the averaged condition, and parts (i), (ii), (iv) use of the states only their translation invariance. (SB$_n$) alone does not give (W$_{\mathrm s}$), and neither do reflection positivity, the gap in every coordinate direction and non-Gaussian moments, by the following family; it is not a lattice gauge theory, and, as in Remark~\ref{rem:limitEscope}, the proofs of this subsection use the states only through the properties in (i) below. \begin{proposition}\label{prop:limitEproduct} Let $m>0$, $q_k=e^{-ma_k}$, $p\ge1$, let $\phi^{(1)},\dots,\phi^{(p)}$ be independent copies of the Gaussian field of Proposition~\textup{\ref{prop:osobstruction}(a)}, $\Phi_x=\prod_i \phi^{(i)}_x$, $A_k(f)=\sum_xa_k^4f(a_kx)\Phi_x$ and $Z_k=1$. \textup{(i)} The law of $(\phi^{(1)},\dots,\phi^{(p)})$ is invariant under translations and $B_4$, reflection positive for the reflections in coordinate hyperplanes through sites and bisecting links, and its transfer matrix in each coordinate direction has spectrum $\{q_k^N:N\ge0\}$ with $1$ a simple eigenvalue. \textup{(ii)} For all $n$ and $f_j\in C_c(\R^4)$ the moments converge, \[ \begin{split} &\lim_k\omega_k\Bigl(\prod_jA_k(f_j)\Bigr)\\ &\quad=\int\prod_jf_j(u_j)\Bigl(\sum_\pi\prod_{\{a,b\}\in\pi} e^{-m\|u_a-u_b\|_1}\Bigr)^pdu, \end{split} \] the sum over the perfect matchings $\pi$ of $\{1,\dots,n\}$; for $f_j$ supported in a cube of side $L\ge a_k$ they are at most $(n!)^p(2L)^{4n}\prod_j\|f_j\|_\infty$. So \textup{(M)}, \textup{(M$_{\mathrm s}$)}, \textup{(SB$_n$)} and \textup{(U$_{\mathrm s}$)} hold. \textup{(iii)} For $p\ge2$ the limit is not Gaussian. \textup{(iv)} $G_S=B_4$, \textup{(RI$_2$)} fails, and the kernel $K(z)=e^{-pm\|z\|_1}$ of $S_2$ has $k_4\ne0$ on a set of positive measure. \end{proposition} \begin{proof} (i) In each coordinate direction the field $(\phi^{(1)},\dots,\phi^{(p)})$ is the stationary reversible Markov chain with transition operator $P^{\otimes p}$, $P$ that of the proof of Proposition~\ref{prop:osobstruction}(a); the spectrum of $P^{\otimes p}$ consists of the products $q^{N_1+\dots+N_p}$, and $1$ is simple. That proof uses only these properties. (ii) By independence and Wick's formula, $\omega_k(\prod_j\Phi_{x_j})= \bigl(\sum_\pi\prod_{\{a,b\}\in\pi}w_k(x_a-x_b)\bigr)^p$ with $w_k(z)= \prod_\mu q_k^{|z_\mu|}=e^{-ma_k\|z\|_1}$, so $c_k=0$; this is at most the number of matchings to the power $p$ when $n$ is even and $0$ otherwise, and the moments are Riemann sums of a bounded continuous function. At most $(L/a_k+1)^4\le(2L/a_k)^4$ sites $x$ have $a_kx$ in a cube of side $L\ge a_k$, which gives the bound, hence (M), (M$_{\mathrm s}$) and (SB$_n$), and the convergence of the whole sequence gives (U$_{\mathrm s}$). (iii) For $n=4$ write $a=w_{12}w_{34}$, $b=w_{13}w_{24}$, $c=w_{14}w_{23}$ with $w_{ij}=e^{-m\|u_i-u_j\|_1}$. The Gaussian family with the same $S_2$ has the integrand $a^p+b^p+c^p$, and $(a+b+c)^p>a^p+b^p+c^p$ for $p\ge2$ and $a,b,c>0$; so $S_4$ differs from it for nonnegative nonzero $f_j$. (iv) $B_4\subset G_S\subset G_{S_2}$, and $G_{S_2}=B_4$ by the argument of Proposition~\ref{prop:osobstruction}(a) with $pm$ in place of $m$. For $y=(2N,0,0,0)$ and $y'=(N,N,N,N)$, $|y|=|y'|=2N$ and $M_k(0,y)-M_k(0,y')= e^{-2pma_kN}-e^{-4pma_kN}$; with $N=\lceil\delta/a_k\rceil$ both lie in $\mathcal W_k(\delta,3\delta)$ for large $k$, and the difference tends to $e^{-2pm\delta}-e^{-4pm\delta}>0$, so (RI$_2$) fails. Finally $Y_4(z)=\frac12\sum_\mu z_\mu^4-\sum_{\mu<\nu}z_\mu^2z_\nu^2$, and with $I_i=\int_\R|t|^ie^{-M|t|}dt=2\,i!/M^{i+1}$, $M=pm$, \[ \int_{\R^4}K(z)Y_4(z)\,dz=2I_4I_0^3-6I_2^2I_0^2=\frac{768-384}{M^8}>0 . \] In polar coordinates the left side is $\|Y_4\|^2_{L^2(S^3)}\int_0^\infty \rho^7k_4(\rho)\,d\rho$, by the orthogonality of the components in Proposition~\ref{prop:limitEaniso}(ii), so $k_4$ is not almost everywhere zero. \end{proof} For a product kernel $K(z)=\prod_\mu k(z_\mu)$ with $k\ge0$ and moments $I_i=\int|t|^ik(t)\,dt$ the same computation gives $\int KY_4\,dz=2I_0^2 I_2^2(I_4I_0/I_2^2-3)$: the weighted coefficient $\int\rho^7k_4\,d\rho$ has the sign of the excess kurtosis of $k$, which is $3$ for $e^{-M|t|}$ and $0$ for a Gaussian $k$. The family satisfies every hypothesis of Theorem~\ref{thm:limitErot} except (RI$_2$), so among the lattice statements used here it is (RI$_n$) that excludes it. \emph{The scale.} For a family $S=(S_n)$ on the separated tuples and $\lambda,c>0$ put \[ (D_{\lambda,c}S)_n(f_1,\dots,f_n)=c^nS_n\bigl(f_1(\lambda\,\cdot),\dots, f_n(\lambda\,\cdot)\bigr) ; \] this is a family on the separated tuples, because the map taking $f$ to $f(\lambda\,\cdot)$ takes $\mathcal C_n(Q,\delta)$ onto $\mathcal C_n(\lambda^{-1}Q,\delta/\lambda)$. Moreover $D_{\lambda,c}D_{\lambda',c'}=D_{\lambda\lambda',cc'}$. For a nontrivial limit $S$ of Theorem~\ref{thm:limitEsep} let $m(S)=\min(\sigma(H_0)\setminus\{0\})$, which exists and is at least $m_*$ by Theorem~\ref{thm:limitEsep}(iv), (vi). \begin{lemma}\label{lem:limitEdil} Let $S$ be a limit of Theorem~\textup{\ref{thm:limitEsep}} and $\lambda,c>0$. For $P$ at positive $0$-times let $P^{\lambda,c}$ be obtained by replacing every $f_{\beta,j}$ by $f_{\beta,j}(\lambda\,\cdot)$ and every coefficient $c_\beta$ by $c^{n_\beta}c_\beta$. Then $D_{\lambda,c}S(\theta_0\bar P\cdot P')=S(\theta_0\overline{P^{\lambda,c}} \cdot P'^{\lambda,c})$, and $\Psi'(P)\mapsto\Psi_0(P^{\lambda,c})$ extends to a unitary map $V$ of the Hilbert space $\mathcal H'$ of $D_{\lambda,c}S$ onto $\mathcal H_0$ with $V\Omega'=\Omega_0$. With $H'=\lambda^{-1}V^{-1}H_0V$, $\Psi'(\tau_{te_0}P)=e^{-tH'}\Psi'(P)$ for $t\ge0$, and $\sigma(H')=\lambda^{-1}\sigma(H_0)$; when $D_{\lambda,c}S$ is itself a limit of Theorem~\textup{\ref{thm:limitEsep}}, its Hamiltonian in the time direction is $H'$, and for nontrivial $S$, $m(D_{\lambda,c}S)= m(S)/\lambda$. If $S$ is nontrivial and $D_{\lambda,c}S=S$, then $\lambda=c=1$. \end{lemma} \begin{proof} Since $\theta_0$ is linear, $(f\circ\theta_0)(\lambda\,\cdot)=f(\lambda\, \cdot)\circ\theta_0$, which gives the identity; it shows that the form of $D_{\lambda,c}S$ is positive and that $V$ is isometric, and $V$ has dense range because $P=(P^{1/\lambda,1/c})^{\lambda,c}$. Further $(f(\cdot-te_0))(\lambda\,\cdot)=(f(\lambda\,\cdot))(\cdot-\frac t\lambda e_0)$, so $V\Psi'(\tau_{te_0}P)=\Psi_0(\tau_{(t/\lambda)e_0}P^{\lambda,c})= e^{-(t/\lambda)H_0}V\Psi'(P)$. A semigroup with this property is determined by its values on the dense set of the $\Psi'(P)$, which gives the claim on the Hamiltonian of a limit. If $D_{\lambda,c}S=S$, the two constructions coincide, $\sigma(H_0)=\lambda^{-1}\sigma(H_0)$, and the least point $m(S)$ of $\sigma(H_0)\setminus\{0\}$ satisfies $m(S)=m(S)/ \lambda$; so $\lambda=1$, and $c^nS_n=S_n$ with $S_n\ne0$ for some $n\ge2$ by Theorem~\ref{thm:limitEsep}(vi) gives $c=1$. \end{proof} The normalisation of Proposition~\ref{prop:limitEnorm}(ii) fixes the size of the field, not the scale: assigning the states to the spacings with a shift produces a dilated limit with the same normalisation. \begin{proposition}\label{prop:limitEscale} Assume the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}}, and suppose that the moments $\omega_k(\prod_jZ_kA_k(f_j))$ converge for every separated tuple to a nontrivial family $S$. Let $j\ge1$, and let $\omega''_k=\omega_k$ for even $k$ and $\omega''_k=\omega_{k+j}$ for odd $k$, with the packets $A''_k$ formed from $\omega''_k$ at the spacing $a_k$. \textup{(i)} $(\omega''_k)$ satisfies the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}} with $2^j\ell$ in place of $\ell$. \textup{(ii)} For every choice of $Z''_k>0$ for which \textup{(M$_{\mathrm s}$)} holds along $(\omega''_k)$, the moments $\omega''_k(\prod_jZ''_k A''_k(f_j))$ either fail to converge for some separated tuple or converge to the trivial family. \textup{(iii)} If $C''_k(0)>0$ for large $k$ and \textup{(M$_{\mathrm s}$)} holds along $(\omega''_k)$ for $Z''_k=C''_k(\sigma_0)^{-1/2}$, then \textup{(U$_{\mathrm s}$)} fails for it. \end{proposition} \begin{proof} (i) For odd $k$, $\omega_{k+j}$ satisfies $\mathrm{BP}(n_{k+j},\varpi)$ and $n_{k+j}a_k=2^jn_{k+j}a_{k+j}\le2^j\ell$. (ii) For odd $k$, $a_k=2^ja_{k+j}$ gives $A''_k(f)=2^{4j}A_{k+j}(f(2^j\, \cdot))$. Write $Z''_k=\zeta_kZ_k$ for even $k$ and $Z''_k=\zeta_k2^{-4j} Z_{k+j}$ for odd $k$. The even moments are $\zeta_k^n$ times moments that converge to $S_n(f_1,\dots,f_n)$, and the odd ones $\zeta_k^n$ times moments that converge to $(D_{2^j,1}S)_n(f_1,\dots,f_n)$. Both families are nontrivial, so (M$_{\mathrm s}$) bounds $\zeta_k$. Along a subsequence $\zeta_k\to\zeta_e$ on the even and $\zeta_k\to\zeta_o$ on the odd $k$, and the two limits are $D_{1,\zeta_e}S$ and $D_{2^j,\zeta_o}S$. If they coincide and $\zeta_e\zeta_o\ne0$, then $D_{2^j,\zeta_o/\zeta_e}S=S$, contrary to Lemma~\ref{lem:limitEdil}; if exactly one of $\zeta_e,\zeta_o$ vanishes, one limit is trivial and the other not. So a common limit has $\zeta_e=\zeta_o=0$ and is trivial. (iii) By Proposition~\ref{prop:limitEnorm}(ii) every limit is nontrivial, so by (ii) some separated moment does not converge, and by Corollary~\ref{cor:limitEexact} (U$_{\mathrm s}$) fails. \end{proof} The scale is fixed by a mass. With $f_0$, $\sigma_0$, $C_k$ as in Proposition~\ref{prop:limitEnorm} and $\gamma_k$ the norm of $\mathbb T_k$ on $\Omega_k^\perp$, consider for $m_0>0$: \begin{enumerate}[label=\textup{(MC$_{m_0}$)},leftmargin=*] \item $\liminf_k a_k^{-1}\log(1/\gamma_k)\ge m_0$, and there are $c_0>0$ and $k_1$ with $C_k(t)\ge c_0e^{-m_0t}C_k(\sigma_0)$ for $k\ge k_1$ and $t\in a_k\Z$, $t\ge\sigma_0$; \end{enumerate} and the statement that the limits differ at most by the scale and the size of the field: \begin{enumerate}[label=\textup{(O$_{\mathrm s}$)},leftmargin=*] \item for any two limits $S,S'$ of Theorem~\ref{thm:limitEsep} along subsequences there are $\lambda,c>0$ with $S'=D_{\lambda,c}S$. \end{enumerate} \begin{theorem}\label{thm:limitEunique} Assume the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}}, $C_k(0)>0$ for large $k$, $Z_k=C_k(\sigma_0)^{-1/2}$ and \textup{(M$_{\mathrm s}$)}. \textup{(i)} Every limit $S$ of Theorem~\textup{\ref{thm:limitEsep}} satisfies \[ \ip{\Psi_0(f_0)}{e^{-\sigma_0H_0}\Psi_0(f_0)}=1, \] and under \textup{(MC$_{m_0}$)} it has $m(S)=m_0$. \textup{(ii)} Under \textup{(MC$_{m_0}$)} and \textup{(O$_{\mathrm s}$)}, \textup{(U$_{\mathrm s}$)} holds and the moments converge for every separated tuple. \textup{(iii)} For the Gaussian field of Proposition~\textup{\ref{prop:osobstruction}(a)} at mass $m$, \textup{(MC$_m$)} holds; for the same field at two masses $m\ne m'$, interleaved, \textup{(O$_{\mathrm s}$)} holds and \textup{(MC$_{m_0}$)} fails for every $m_0$. \end{theorem} \begin{proof} (i) The first claim and nontriviality are Proposition~\ref{prop:limitEnorm}(ii). The proof of Theorem~\ref{thm:limitEsep}(iv) with $\gamma_k^{t/a_k}$ in place of $e^{-m_*t}$ gives $\ip v{e^{-tH_0}v}\le\limsup_k\gamma_k^{t/a_k}\|v\|^2\le e^{-m_0t}\|v\|^2$ on a dense set of $v\in\Omega_0^\perp$, so $m(S)\ge m_0$. By the approximation of $\theta_{0,k}$ by $\theta_0$, as in the proof of Proposition~\ref{prop:limitEnorm}(ii), $\ip{\Psi_0(f_0)}{e^{-tH_0} \Psi_0(f_0)}=\lim_kZ_k^2C_k(t)\ge c_0e^{-m_0t}$ for dyadic $t\ge\sigma_0$. Since $\Psi_0(f_0)\perp\Omega_0$, its spectral measure for $H_0$ is carried by $[m(S),\infty)$, and the left side is at most $e^{-m(S)t}\|\Psi_0(f_0) \|^2$; letting $t\to\infty$ gives $m(S)\le m_0$. (ii) Let $S,S'$ be two limits and $S'=D_{\lambda,c}S$. By (i) and Lemma~\ref{lem:limitEdil}, $m_0=m(S')=m(S)/\lambda=m_0/\lambda$, so $\lambda=1$; then $S'_2=c^2S_2$, and the normalisation in (i) gives $c=1$. Convergence follows as in the proof of Corollary~\ref{cor:limitEexact}. (iii) The transfer matrix has $\gamma_k=q_k=e^{-ma_k}$. The packet $\vartheta(\overline{A_k(f_0)})$ involves sites at negative times and $\tau_{t/a_k}A_k(f_0)$ sites at positive times, so the time factor of the covariance is $q_k^{y_0-x_0}$ and $C_k(t)=e^{-mt}C_k(0)$; this is (MC$_m$). At two masses the limits are the two Gaussian families with kernels $\kappa e^{-m\|z\|_1}$ and $\kappa'e^{-m'\|z\|_1}$, and $D_{\lambda,c}$ with $\lambda=m/m'$ and $c^2=\kappa'\lambda^8/\kappa$ maps the first to the second, Gaussian families being determined by $S_2$. The first part of (MC$_{m_0}$) needs $m_0\le\min(m,m')$, and the second, since $C_k(t)/C_k(\sigma_0)=e^{-m_k(t-\sigma_0)}$ with $m_k\in\{m,m'\}$, needs $m_0\ge\max(m,m')$. \end{proof} (O$_{\mathrm s}$) is the form, for the separated moments, of the statement that the continuum theory has no free dimensionless parameter; it is not proved here, nor is (MC$_{m_0}$) for the Wilson measure along any tuning. The second part of (MC$_{m_0}$) asks that the reflected correlation of $f_0$ decay no faster than at the rate $m_0$, which holds when $f_0$ couples to the lowest state; together with the first part it fixes the mass of every limit. A condition fixing the ratio $C_k(\sigma_0)/C_k(0)$, the analogue of (RC), fixes the scale within a set of limits related by dilations only where the corresponding ratio of the limits is injective along that set, which is not proved. \emph{The pointwise bound on an axis.} Reflection positivity in the coordinate directions bounds the separated two-point moments at every point by their values on a coordinate axis, where they form a completely monotone function of the distance. For $n=2$ write $g_k(s)=M_k(0,se_0)$ for $s\in\Z$, and $\lfloor t\rceil=\lfloor t+\frac12\rfloor$. Let $h\in\{1,-1\}$ be such that $\vartheta x=(h-x_0,\mathbf x)$ for $x=(x_0,\mathbf x)\in\Z^4$; the plane fixed by $\vartheta$ is $\{x_0=h/2\}$. \begin{proposition}\label{prop:limitEaxial} Assume the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}}, and let $\gamma_k$ be the norm of $\mathbb T_k$ on $\Omega_k^\perp$. \textup{(i)} For integers $s\ge2r+3$ and $j\ge0$, $0\le g_k(s+j)\le \gamma_k^jg_k(s)$, and for $s\ge4r+6$ \[ 0\le g_k(s)-g_k(s+j)\le\frac{4j}s\,g_k\bigl(\lceil s/2\rceil\bigr) . \] \textup{(ii)} For $y\in\Z^4$ with $|y|_\infty\ge2r+4$, $|M_k(0,y)|\le g_k(|y|_\infty-1)$. \textup{(iii)} \textup{(SB$_2$)} holds if and only if $\limsup_kZ_k^2 g_k(\lceil\delta/a_k\rceil)<\infty$ for every $\delta>0$. In that case, for every $\delta>0$ and with $b(\delta)=\limsup_kZ_k^2g_k(\lceil\delta/(4a_k) \rceil)$, \[ Z_k^2|M_k(0,y)|\le\bigl(b(\delta)+1\bigr)e^{-m_*(a_k|y|_\infty-\delta/2)} \] for all large $k$ and all $y\in\Z^4$ with $|y|\ge\delta/a_k$; in particular $b_2(\delta,\Lambda)\le b(\delta)+1$ for every $\Lambda>\delta$. \textup{(iv)} Under \textup{(SB$_2$)}, \textup{(RI$_2$)} holds if and only if for $0<\delta<\Lambda$ \[ \lim_k\sup_yZ_k^2\bigl|M_k(0,y)-g_k(\lfloor|y|\rceil)\bigr|=0, \] the supremum over the $y\in\Z^4$ with $\delta/a_k\le|y|\le\Lambda/a_k$. \textup{(v)} Under \textup{(SB$_2$)}, every limit of Theorem~\textup{\ref{thm:limitEsep}} has $S_2(f,g)=\iint f(x)g(y)K(x-y)\,dx \,dy$ for the separated pairs, with $K$ measurable on $\R^4\setminus\{0\}$, $K(-z)=K(z)$, $K\circ g=K$ almost everywhere for $g\in B_4$, and $|K(z)|\le(b(\delta)+1)e^{-m_*(|z|_\infty-\delta/2)}$ for almost every $z$ with $|z|\ge2\delta$, for every $\delta>0$. \end{proposition} \begin{proof} For $j\ge r+1$ let $F_j=\Phi_{je_0}-c_k$, a real function of the links with both endpoints in $\{x_0\ge1\}$. Since $\vartheta\Phi_x=\Phi_{\vartheta x}$, $\vartheta F_j=\Phi_{(h-j)e_0}-c_k$, and by translation invariance \[ \omega_k\bigl(\vartheta(F_j)\tau_tF_{j'}\bigr)=M_k\bigl((h-j)e_0,(j'+t)e_0 \bigr)=g_k(j+j'+t-h) \] for integers $t\ge0$. In particular $N_0(F_j)^2=g_k(2j-h)$. (i) By the proof of Lemma~\ref{lem:limitElattice}(iii), with $s_0=2r+2-h$ and $\Psi=\Psi_{F_{r+1}}$, $g_k(s_0+t)=\ip\Psi{\mathbb T_k^t\Psi}$ for integers $t\ge0$, and $\ip{\Omega_k}\Psi=\omega_k(F_{r+1})=0$. Since $\mathbb T_k$ is self-adjoint, fixes $\Omega_k$ and has spectrum on $\Omega_k^\perp$ in $[0,\gamma_k]$, the spectral measure $\nu$ of $\mathbb T_k$ at $\Psi$ is a positive measure on $[0,\gamma_k]$, and $g_k(s_0+t)=\int\lambda^t\,d\nu(\lambda)$, $0^0=1$. As $s_0\le2r+3$, this gives $g_k(s)\ge0$ and $g_k(s+j)\le\gamma_k^jg_k(s)$ for $s\ge2r+3$. Let $s\ge4r+6$ and $u=s-s_0$. From $1-\lambda^j\le j(1-\lambda)$ and $\max_{0\le x\le1}x^p(1-x)=p^p(p+1)^{-p-1}\le(p+1)^{-1}$, applied with $p=\lfloor u/2 \rfloor$ to $\lambda^u(1-\lambda)=\lambda^{\lceil u/2\rceil}\lambda^p(1- \lambda)$, \[ \begin{split} g_k(s)-g_k(s+j)&=\int\lambda^u(1-\lambda^j)\,d\nu\le\frac j{p+1} \int\lambda^{\lceil u/2\rceil}d\nu\\ &=\frac{j\,g_k(s_0+\lceil u/2\rceil)}{\lfloor u/2\rfloor+1} . \end{split} \] Here $s_0+\lceil u/2\rceil=\lceil(s+s_0)/2\rceil\ge\lceil s/2\rceil\ge2r+3$, so the numerator is at most $j\,g_k(\lceil s/2\rceil)$, and $\lfloor u/2 \rfloor+1\ge(s-s_0+1)/2\ge(s-2r-2)/2\ge s/4$. (ii) Let $s=|y|_\infty$. Choose $g\in B_4$, a permutation of the axes followed by a sign change, with $(gy)_0=s$; then $M_k(0,y)=M_k(0,gy)$, since $\omega_k$ is invariant under $B_4$ and $g\Phi_x=\Phi_{gx}$. Write $gy=se_0+v$ with $v_0=0$, and let $j=\lfloor(s+h)/2\rfloor$ and $j'=s+h-j$, both at least $\lfloor(s-1)/2\rfloor\ge r+1$. By translation invariance \[ M_k(0,gy)=M_k\bigl((h-j)e_0,j'e_0+v\bigr)=\omega_k\bigl(\vartheta(F_j) \tau_vF_{j'}\bigr), \] and $\tau_vF_{j'}$ is again a function of the links in $\{x_0\ge1\}$. The Cauchy--Schwarz inequality for the positive form of Lemma~\ref{lem:limitElattice}(iii) and $N_0(\tau_vF_{j'})=N_0(F_{j'})$ give $|M_k(0,y)|\le(g_k(2j-h)g_k(2j'-h))^{1/2}$. If $s+h$ is even, $2j-h=2j'-h= s$; otherwise the two arguments are $s-1$ and $s+1$. By (i) both products are at most $g_k(s-1)^2$. (iii) Let $a_k\le\delta$. The pair $(0,\lceil\delta/a_k\rceil e_0)$ lies in $\mathcal W_k(\delta,2\delta)$, and (SB$_2$) gives \[ \limsup_kZ_k^2g_k(\lceil\delta/a_k\rceil)\le b_2(\delta,2\delta). \] Conversely let $\delta>0$, $s_1=\lceil\delta/(4a_k)\rceil$ and $|y|\ge\delta/a_k$. Then $|y|_\infty\ge |y|/2\ge\delta/(2a_k)$, which for large $k$ is at least $2r+4$ and at least $s_1+1$. By (ii) and (i), and $\gamma_k\le e^{-m_*a_k}$, \[ |M_k(0,y)|\le g_k(|y|_\infty-1)\le e^{-m_*a_k(|y|_\infty-1-s_1)}g_k(s_1), \] and $a_k(1+s_1)\le2a_k+\delta/4\le\delta/2$ for $a_k\le\delta/8$. Since $Z_k^2g_k(s_1)\le b(\delta)+1$ for large $k$, this is the bound, and $a_k|y|_\infty\ge\delta/2$ gives the last claim. (iv) Suppose (RI$_2$), and let $\delta/a_k\le|y|\le\Lambda/a_k$. The point $y'=\lfloor|y|\rceil e_0$ has $||y|-|y'||\le\frac12$, both pairs $(0,y)$, $(0,y')$ lie in $\mathcal W_k(\delta/2,2\Lambda)$ for large $k$, and $M_k(0,y')=g_k(\lfloor|y|\rceil)$; so (RI$_2$) with $(\delta/2,2\Lambda)$ gives the condition. Conversely, let $x,x'\in\mathcal W_k(\delta,\Lambda)$ with $||x_1-x_2|-|x'_1-x'_2||\le4$, $y=x_2-x_1$, $y'=x'_2-x'_1$, $s=\lfloor|y|\rceil$ and $s'=\lfloor|y'|\rceil$. By translation invariance $M_k(x)=M_k(0,y)$ and $M_k(x')=M_k(0,y')$, and \[ \begin{split} Z_k^2|M_k(x)-M_k(x')|&\le Z_k^2|M_k(0,y)-g_k(s)|\\ &\quad+Z_k^2|g_k(s)-g_k(s')|\\ &\quad+Z_k^2|M_k(0,y')-g_k(s')| . \end{split} \] The first and last terms tend to $0$ uniformly by the condition. Since $|s-s'|\le5$ and $\min(s,s')\ge\delta/(2a_k)$ for large $k$, (i) bounds the middle term by $40a_k\delta^{-1}Z_k^2g_k(\lceil\delta/(4a_k)\rceil)$, which tends to $0$ by (iii). (v) Let $S$ be a limit along a subsequence, $f,g$ separated, supported in a cube $Q$ of side $L$ at $\ell^\infty$ distance $\eta>0$, and $\Theta(z)=\int f(u)g(u+z)\,du$, continuous and supported in $\{|z|_\infty\ge\eta\}\cap [-L,L]^4$. With $\varrho_k$ as in the proof of Theorem~\ref{thm:limitErot} put $K_k(z)=Z_k^2M_k(0,\varrho_k(z))$. By translation invariance, \[ \begin{split} &Z_k^2\omega_k\bigl(A_k(f)A_k(g)\bigr)-\int K_k\Theta\,dz=\sum_wZ_k^2 M_k(0,w)\\ &\quad\times\Bigl(a_k^8\sum_xf(a_kx)g(a_kx+a_kw)-\int_{\varrho_k^{-1}(w)} \Theta\,dz\Bigr) . \end{split} \] The bracket vanishes unless $|w|_\infty\ge\eta/a_k-2$ and $|w|_\infty\le L/a_k+2$, and it is at most $a_k^4\epsilon_k$, where $\epsilon_k\to0$ by the uniform continuity of $f$, $g$ and $\Theta$, the inner sum being a Riemann sum for $\Theta(a_kw)$ with error uniform in $w$. At most $(2L/a_k+5)^4$ vectors $w$ contribute, and by (iii) with $\delta=\eta/2$ each has $Z_k^2|M_k(0,w)|\le b(\eta/2)+1$ for large $k$. So the difference tends to $0$. By (iii), $K_k$ is bounded on every set $\{|z|\ge\delta\}$, $\delta>0$, for large $k$, and a further subsequence converges weakly-$*$ in $L^\infty(\{|z|\ge\delta\})$ for every $\delta=1/i$ to some $K$. Since $\Theta$ is integrable and supported in $\{|z|\ge\eta\}$, $\int K_k\Theta\,dz \to\int K\Theta\,dz$, which is the representation. For $|z|\ge2\delta$ and large $k$, $|\varrho_k(z)|\ge|z|/a_k-1\ge\delta/a_k$ and $a_k|\varrho_k(z)| _\infty\ge|z|_\infty-a_k$, so (iii) gives $|K_k(z)|\le(b(\delta)+1) e^{-m_*(|z|_\infty-a_k-\delta/2)}$, a bound that passes to the weak-$*$ limit. Finally $M_k(0,-w)=M_k(0,w)$ and $M_k(0,gw)=M_k(0,w)$ for $g\in B_4$, while $\varrho_k(-z)=-\varrho_k(z)$ and $\varrho_k(gz)=g\varrho_k(z)$ off a null set; so $K_k$ and $K$ are even and $B_4$-invariant almost everywhere. \end{proof} So (SB$_2$) is a bound on one function of one variable, the renormalised two-point moment on a coordinate axis, and (RI$_2$) asks that the renormalised moment at a point be close to its value on the axis at the same Euclidean length. Under (SB$_2$) the kernel of (v) is square integrable on every compact subset of $\R^4\setminus\{0\}$, so Proposition~\ref{prop:limitEaniso} applies to it, and (W$_{\mathrm s}$) for $n=2$ is the radial symmetry of $K$. Neither part follows from the other properties. For (RI$_2$) this is Proposition~\ref{prop:limitEproduct}, whose fields satisfy (SB$_n$) for every $n$. For (SB$_2$) it is part (b) of the following proposition, whose fields satisfy the separated moment bound with constants independent of the separation, the mass condition and (U$_{\mathrm s}$), and concentrate the two-point function on the coordinate axes, where the bound of (M$_{\mathrm s}$) on averages over regions of physical size does not control the pointwise values. \begin{proposition}\label{prop:limitEgauss} Let $m>0$ and $q_k=e^{-ma_k}$. \textup{(a)} Let $J$ be a finite set, $q_{j,\mu}\in[0,1)$ for $j\in J$ and $\mu=0,\dots,3$, and let $\psi=(\psi_j(x))_{j\in J,x\in\Z^4}$ be the centred Gaussian field with $\E\psi_j(x)\psi_{j'}(y)=\delta_{jj'}\prod_\mu q_{j,\mu}^{|x_\mu-y_\mu|}$, $0^0=1$. Its law is invariant under translations and reflection positive for the reflections in coordinate hyperplanes through sites and bisecting links, and in the direction $\mu$ its transfer matrix $\mathbb T$ has $0\le\mathbb T\le1$, $\mathbb T\Omega=\Omega$ and norm at most $\max_jq_{j,\mu}$ on $\Omega^\perp$. \textup{(b)} Let $J=\{0,1,2,3\}$, $q_{j,\mu}=q_k$ for $j=\mu$ and $0$ otherwise, $\Phi_x=\sum_j\psi_j(x)$, $A_k(f)=\sum_xa_k^4f(a_kx)\Phi_x$, $f_0\ge0$ not zero, $C_k$ formed from $f_0$, and $Z_k=C_k(\sigma_0)^{-1/2}$. The law is invariant under $B_4$, acting by $\psi_j(x)\mapsto\psi_{\bar gj} (gx)$ with $\bar g$ the permutation of the axes underlying $g$, so that $g\Phi_x=\Phi_{gx}$. Moreover $C_k(t)=e^{-mt}C_k(0)$ for $t\in a_k\Z_{\ge0}$ and $a_k^{-3}C_k(0)\to\mathcal J=\int_{\R^3}(\int_\R f_0(s,\mathbf x) e^{-ms}ds)^2d\mathbf x>0$. There is $B<\infty$ such that for large $k$, every $n$ and all $f_j\in C_c(\R^4)$ supported in a cube of side $L\ge a_k$, \[ \Bigl|\omega_k\Bigl(\prod_{j=1}^nZ_kA_k(f_j)\Bigr)\Bigr|\le(n-1)!!\, \bigl(B(2L)^4\bigr)^{n/2}\prod_{j=1}^n\|f_j\|_\infty \] for even $n$, the moments vanishing for odd $n$; so \textup{(M$_{\mathrm s}$)} holds with constants independent of $\delta$, and the growth condition \textup{(G$_{\mathrm s}$)} posed below holds with $p=0$. The moments converge along the whole sequence to the Gaussian family with \[ S_2(f,g)=\frac{e^{m\sigma_0}}{\mathcal J}\sum_{\mu=0}^3\int_{\R^4}\int_\R f(x)g(x+ue_\mu)e^{-m|u|}\,du\,dx , \] so \textup{(U$_{\mathrm s}$)} holds, and \textup{(MC$_m$)} holds. Further \textup{(SB$_2$)} and \textup{(RI$_2$)} fail, $G_S=B_4$, and there is no locally integrable $K$ on $\R^4\setminus\{0\}$ with $S_2(f,g)=\iint f(x)g(y) K(x-y)\,dx\,dy$ for all separated pairs. \textup{(c)} Let $m'>m$, $q'_k=e^{-m'a_k}$, and let $\omega_k$ be the law of the field $\Phi=\psi_1$ of \textup{(a)} with $J=\{1\}$, $q_{1,\mu}=q_k$, for even $k$, and of $\Phi=\psi_1+\psi_2$ with $J=\{1,2\}$, $q_{1,\mu}=q_k$, $q_{2,\mu}=q'_k$, for odd $k$; let $f_0\ge0$ be not zero and $Z_k=C_k( \sigma_0)^{-1/2}$. The laws are invariant under translations and $B_4$, reflection positive as in \textup{(a)}, with transfer matrices of norm at most $q_k$ on $\Omega_k^\perp$. \textup{(M$_{\mathrm s}$)} and \textup{(MC$_m$)} hold along the whole sequence. The moments converge along the even and along the odd $k$ to Gaussian families $S$ and $S'$ with continuous kernels $\kappa e^{-m\|z\|_1}$ and $\kappa'(e^{-m\|z\|_1}+ e^{-m'\|z\|_1})$, $\kappa,\kappa'>0$, and $S'\ne D_{\lambda,c}S$ for all $\lambda,c>0$. So \textup{(O$_{\mathrm s}$)} and \textup{(U$_{\mathrm s}$)} fail. \end{proposition} \begin{proof} (a) Fix $\mu=0$ as time; the other directions are the same. The field at time $0$ has a Gaussian law $\gamma$ with covariance $C=\delta_{jj'} \prod_{i\ge1}q_{j,i}^{|x_i-y_i|}$, positive definite as a product of the one-dimensional covariances $q^{|n|}$, $0\le q<1$, whose Fourier transforms $(1-q^2)/(1-2q\cos p+q^2)$ are positive. Let $(Qh)_j(\mathbf x)=q_{j,0} h_j(\mathbf x)$ on the finitely supported $h$; $Q$ commutes with $C$, so the stationary chain $\psi_{t+1}=Q\psi_t+(1-Q^2)^{1/2}\xi_{t+1}$, with $\xi_t$ independent of law $\gamma$, has covariance $Q^{|t-s|}C$, which is that of the field. As in the proof of Proposition~\ref{prop:osobstruction}(a), its transition operator $P$ satisfies $P\mathcal E_h=\mathcal E_{Qh}$, so on the $N$-th Wiener chaos, the closure of the parts of the $\mathcal E_h$ homogeneous of degree $N$ in $h$, it acts as the $N$-th tensor power of $Q$ on the symmetric tensors over the space of the $h$ with the inner product $\ip h{Ch'}$. $Q$ is self-adjoint there with $0\le Q\le q_{\max}=\max_j q_{j,0}$, so $0\le P\le1$, $P$ fixes the constants and has norm at most $q_{\max}^N\le q_{\max}$ on the chaos of order $N\ge1$. The rest of that proof gives reflection positivity for both kinds of planes and identifies the transfer matrix with $P$. (b) Here $\E\Phi_x\Phi_y=w_k(y-x)$ with $w_k(z)=q_k^{|z_\mu|}$ if $z\ne0$ is a multiple of $e_\mu$, $w_k(0)=4$, and $w_k(z)=0$ otherwise. Since $(gz)_{\bar gj}=\pm z_j$, the covariance of the transformed field is that of $\psi$, which gives the $B_4$ invariance; $\omega_k(\Phi_0)=0$, so $c_k=0$. For $f_0(a_kx)f_0(a_ky)\ne0$ the time coordinate of $y+(t/a_k)e_0-\vartheta x$ is $x_0+y_0-h+t/a_k\ge1$, so only the term $\mu=0$ contributes, with $\mathbf x=\mathbf y$, and \[ C_k(t)=e^{-mt}e^{mha_k}a_k^3\cdot a_k^3\sum_{\mathbf x}\Bigl(a_k \sum_{x_0}f_0(a_kx)e^{-ma_kx_0}\Bigr)^2 . \] The last factor is a Riemann sum converging to $\mathcal J$, which is positive because $f_0\ge0$ is not zero. For the pairs, \[ \begin{split} &\omega_k\bigl(A_k(f)A_k(g)\bigr)\\ &\quad=a_k^3\sum_\mu a_k^5\sum_{x\in\Z^4,\,s\in\Z}f(a_kx)g(a_kx+a_kse_\mu) e^{-ma_k|s|} . \end{split} \] The inner sums are Riemann sums converging to $\int\!\int f(x)g(x+ue_\mu) e^{-m|u|}du\,dx$, and since at most $(L+a_k)^4/a_k^4\le(2L)^4/a_k^4$ sites have $a_kx$ in the cube and $a_k\sum_se^{-ma_k|s|}=a_k(1+q_k)/(1-q_k)\le4/m$ for $ma_k\le1$, they are at most $(4/m)(2L)^4\|f\|_\infty\|g\|_\infty$. With $Z_k^2a_k^3=e^{m\sigma_0}a_k^3/C_k(0)\to e^{m\sigma_0}/\mathcal J$ this gives the bound for $n=2$ with $B=32e^{m\sigma_0}/(m\mathcal J)$ and the limit of $S_2$, and Wick's formula, a sum over the $(n-1)!!$ pairings, gives the bound for all $n$ and the convergence of all moments. Since $(n-1)!!\le(n!)^{1/2}$ and $(B(2L)^4)^{n/2}\le(4B^{1/2})^n(1+L)^{2n}$, the bound has the form of (G$_{\mathrm s}$) with $\beta=\frac12$, $q=2$ and $p=0$. By (a) the transfer matrix has norm at most $q_k$ on $\Omega_k^\perp$, so $\liminf_ka_k^{-1}\log(1/\gamma_k)\ge m$, and $C_k(t)=e^{-m(t-\sigma_0)} C_k(\sigma_0)$; this is (MC$_m$) with $c_0=e^{m\sigma_0}$. For $s\ge1$, $g_k(s)=w_k(se_0)=q_k^s$, so $Z_k^2g_k(\lceil\delta/a_k\rceil) \ge(e^{m\sigma_0}/(2\mathcal J))a_k^{-3}e^{-m(\delta+a_k)}\to\infty$; as the pair $(0,\lceil\delta/a_k\rceil e_0)$ lies in $\mathcal W_k(\delta, 2\delta)$ for large $k$, (SB$_2$) fails. With $N=\lceil \delta/a_k\rceil+1$, the points $y=Ne_0$ and $y'=(N-1)e_0+e_1$ have $N-1\le|y'|\le|y|=N$, both pairs $(0,y)$, $(0,y')$ lie in $\mathcal W_k( \delta,2\delta)$ for large $k$, $M_k(0,y')=w_k(y')=0$ and $Z_k^2M_k(0,y)= Z_k^2q_k^N\to\infty$; so (RI$_2$) fails. Let $T(\varphi)=\frac{e^{m\sigma_0}}{\mathcal J}\sum_\mu\int\varphi(ue_\mu) e^{-m|u|}du$ for $\varphi\in C_c(\R^4\setminus\{0\})$; then $S_2(f,g)= T(\Theta)$ with $\Theta(z)=\int f(u)g(u+z)\,du$. If $S_2$ had a locally integrable kernel $K$, the argument of Proposition~\ref{prop:limitEaniso}(i) would give $T(\varphi)=\int K\varphi\,dz$ for all $\varphi$, $T$ being even. Take $\varphi_\varepsilon$ with values in $[0,1]$, equal to $1$ on the segment $\{ue_0:1\le u\le2\}$ and supported in its $\varepsilon$-neighbourhood $N_\varepsilon$; then $T(\varphi_\varepsilon)\ge\frac{e^{m\sigma_0}} {\mathcal J}\int_1^2e^{-mu}du>0$, while $\int|K|\varphi_\varepsilon\le \int_{N_\varepsilon}|K|\to0$ as $\varepsilon\to0$, since $N_\varepsilon$ decreases to a null set. Let $R\in G_S$ and suppose $R^{-1}e_0$ is not one of the $\pm e_\nu$; then it has positive distance $3\varepsilon$ from the union $A$ of the coordinate axes. Take $f,g\ge0$ supported in the balls of radius $\varepsilon$ about $0$ and $R^{-1}e_0$, with $f(0)>0$ and $g(R^{-1}e_0)>0$. Then $\Theta$ is supported in the ball of radius $2\varepsilon$ about $R^{-1}e_0$, which misses $A$, so $S_2(f,g)=0$, while $S_2(f\circ R^{-1},g\circ R^{-1})=T(\Theta\circ R^{-1})\ge\frac{e^{m \sigma_0}}{\mathcal J}\int\Theta(R^{-1}ue_0)e^{-m|u|}du>0$, because $\Theta(R^{-1}e_0)>0$. So $R^{-1}e_0\in\{\pm e_\nu\}$, and applying this to $\pi_\mu R\in G_S$ gives $R^{-1}e_\mu\in\{\pm e_\nu\}$ for every $\mu$, that is, $R\in B_4$. (c) The symmetries, positivity and the norm of the transfer matrices follow from (a), the covariances $\prod_\mu q^{|z_\mu|}$ being invariant under $B_4$. Let $C^{(i)}_k$ be formed with the covariance $\prod_\mu q_i^{|z_\mu|}$, $q_1=q_k$, $q_2=q'_k$. As in (b), the time factor of the covariance between the two packets of $C^{(i)}_k(t)$ is $q_i^{t/a_k}$ times that at $t=0$, so $C^{(1)}_k(t)=e^{-mt}C^{(1)}_k(0)$, $C^{(2)}_k(t)= e^{-m't}C^{(2)}_k(0)$, and $C^{(i)}_k(0)$ converges to $\mathcal J_1=\iint f_0(x)f_0(y)e^{-m\|\theta_0x-y\|_1}dx\,dy>0$ and $\mathcal J_2$, defined with $m'$ in place of $m$, by Riemann sums. For even $k$, $C_k=C^{(1)}_k$; for odd $k$, $C_k=C^{(1)}_k+C^{(2)}_k$, and for $t\ge\sigma_0$ \[ \begin{split} \frac{C_k(t)}{C_k(\sigma_0)}&\ge\frac{e^{-mt}C^{(1)}_k(0)}{e^{-m\sigma_0} (C^{(1)}_k(0)+C^{(2)}_k(0))}\\ &\ge\frac{\mathcal J_1}{2(\mathcal J_1+\mathcal J_2)}e^{-m(t-\sigma_0)} \end{split} \] for large $k$. With the norm bound $q_k$ this is (MC$_m$) with $c_0= e^{m\sigma_0}\mathcal J_1/(2(\mathcal J_1+\mathcal J_2))$. As in the proof of Proposition~\ref{prop:osobstruction}(a), the moments converge along the even and along the odd $k$ to the Gaussian families with the stated kernels, $\kappa=e^{m\sigma_0}/\mathcal J_1$, $\kappa'=(e^{-m\sigma_0} \mathcal J_1+e^{-m'\sigma_0}\mathcal J_2)^{-1}$, and the moments of packets supported in a cube of side $L$ are bounded by $(n-1)!!$ times $((2L)^4\sup_kZ_k^2a_k^4\sum_z(q_k^{|z|_1}+q_k'^{|z|_1}))^{n/2}\prod_j\|f_j \|_\infty$, where $a_k^4\sum_zq_k^{|z|_1}=(a_k(1+q_k)/(1-q_k))^4$ is bounded; this gives (M$_{\mathrm s}$). The kernel of $D_{\lambda,c}S$ is $c^2 \lambda^{-8}\kappa e^{-(m/\lambda)\|z\|_1}$, by the substitution $u=\lambda x$, $v=\lambda y$. If $S'=D_{\lambda,c}S$, then the two continuous kernels coincide, since by Proposition~\ref{prop:limitEaniso}(i) a continuous kernel is determined by $S_2$; at $z=te_0$, after multiplication by $e^{mt}$, \[ \kappa'\bigl(1+e^{-(m'-m)t}\bigr)=c^2\lambda^{-8}\kappa\,e^{(m-m/\lambda)t} \] for all $t>0$. The left side tends to $\kappa'>0$ as $t\to\infty$, which forces $\lambda=1$, and then the right side is constant while the left side is strictly decreasing. \end{proof} \emph{The mass.} The gap of Theorem~\ref{thm:avgtransfer} gives the first half of (MC$_{m_0}$) with $m_0=m_*$, and reflection positivity with (M$_{\mathrm s}$) gives the second half, but with a rate that is in general larger. \begin{proposition}\label{prop:limitEbracket} Assume the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}}, $C_k(0)>0$ for large $k$, $Z_k=C_k(\sigma_0)^{-1/2}$ and \textup{(M$_{\mathrm s}$)}. Let $Q$ be a cube that contains $\operatorname{supp}f_0$ and $\theta_{0,k}(\operatorname{supp}f_0)$ for all $k$, and $m_1=\sigma_0^{-1}\log(M_2(Q,t_0)\|f_0\|_\infty^2)$. \textup{(i)} $m_1\ge m_*$; $\liminf_ka_k^{-1}\log(1/\gamma_k)\ge m_*$; and for large $k$ and $t\in a_k\Z$ with $t\ge\sigma_0$, \[ C_k(t)\ge e^{-m_1(t-\sigma_0)}C_k(\sigma_0) . \] \textup{(ii)} Every limit $S$ of Theorem~\textup{\ref{thm:limitEsep}} has $m_*\le m(S)\le m_1$. If \textup{(MC$_{m_0}$)} holds, then $m_*\le m_0\le m_1$. \textup{(iii)} If $S$ and $S'$ are limits of Theorem~\textup{\ref{thm:limitEsep}} with $S'=D_{\lambda,c}S$, then $\lambda=m(S)/m(S')$ lies in $[m_*/m_1,m_1/m_*]$. \end{proposition} \begin{proof} (i) By Proposition~\ref{prop:limitEnorm}(i),(ii), $(M_2(Q,t_0)\|f_0\|_\infty^2 )^{-1}\le C_k(\sigma_0)/C_k(0)\le e^{-m_*\sigma_0}$ for large $k$, which gives $m_1\ge m_*$ and $C_k(\sigma_0)/C_k(0)\ge e^{-m_1\sigma_0}$. The second claim is $\gamma_k\le e^{-m_*a_k}$. By the proof of Proposition~\ref{prop:limitEnorm}(i), $C_k(t)=\int\lambda^{t/a_k}d\nu( \lambda)$ with a positive measure $\nu$, so by H\"older's inequality, with $i=t/a_k$ and $p=\sigma_0/a_k$, $C_k(\sigma_0)^i\le C_k(0)^{i-p}C_k(t)^p$. Hence \[ C_k(t)\ge C_k(\sigma_0)\Bigl(\frac{C_k(\sigma_0)}{C_k(0)}\Bigr)^{(t-\sigma_0)/ \sigma_0}\ge e^{-m_1(t-\sigma_0)}C_k(\sigma_0) . \] (ii) $m(S)\ge m_*$ is Theorem~\ref{thm:limitEsep}(iv). As in the proof of Theorem~\ref{thm:limitEunique}(i), (i) gives $\ip{\Psi_0(f_0)}{e^{-tH_0} \Psi_0(f_0)}=\lim_kZ_k^2C_k(t)\ge e^{-m_1(t-\sigma_0)}$ for dyadic $t\ge\sigma_0$, while the left side is at most $e^{-m(S)t}\|\Psi_0(f_0) \|^2$; so $m(S)\le m_1$. Under (MC$_{m_0}$), $m(S)=m_0$ by Theorem~\ref{thm:limitEunique}(i). (iii) By Lemma~\ref{lem:limitEdil}, $m(S')=m(S)/\lambda$, and (ii) applies to both. \end{proof} So the two halves of (MC$_{m_0}$) hold separately, with the rates $m_*$ and $m_1$, and (MC$_{m_0}$) asks that they coincide. Since $C_k(t)\le \gamma_k^{(t-\sigma_0)/a_k}C_k(\sigma_0)$ for $t\ge\sigma_0$, by the spectral representation, its second half gives $a_k^{-1}\log(1/\gamma_k)\le m_0$ for $k\ge k_1$, so (MC$_{m_0}$) says that the lattice gaps in physical units converge to $m_0$ and that the reflected correlation of $f_0$ decays at this rate uniformly in $k$. This is not implied by the other properties: the interleaved two-mass field of Theorem~\ref{thm:limitEunique}(iii) has them and violates (MC$_{m_0}$) for every $m_0$. Nor is it necessary for (U$_{\mathrm s}$): for the products of Proposition~\ref{prop:limitEproduct} with $p\ge2$, the transfer matrix has norm $q_k$ on $\Omega_k^\perp$, attained on the first chaos of one factor, while the covariance of $\Phi$ is $e^{-pma_k\|z\|_1}$ and $C_k(t)=e^{-pmt} C_k(0)$ for $f_0\ge0$; so (MC$_{m_0}$) would need $m_0\le m$ and $m_0\ge pm$, while (U$_{\mathrm s}$) holds, also with $Z_k=C_k(\sigma_0)^{-1/2}$, which converges to a positive limit. \emph{The dilations.} (O$_{\mathrm s}$) is a statement about the set of limits. It is equivalent to a statement about the lattice moments alone: convergence after a change of the spacing by a bounded factor. Write $B_k(f_1,\dots,f_n)=\omega_k(\prod_jZ_kA_k(f_j))$ and $(D_{\lambda,c} B_k)(f_1,\dots,f_n)=c^nB_k(f_1(\lambda\,\cdot),\dots,f_n(\lambda\,\cdot))$. Since $A_k(f(\lambda\,\cdot))=\lambda^{-4}\sum_x(\lambda a_k)^4f(\lambda a_kx)(\Phi_x-c_k)$, $D_{\lambda,c}B_k$ is the moment family of the packets of the same states formed at the spacing $\lambda a_k$, with the constants $c\lambda^{-4}Z_k$. Consider \begin{enumerate}[label=\textup{(DS)},leftmargin=*] \item there are $\Lambda\ge1$, $\lambda_k\in[\Lambda^{-1},\Lambda]$, $c_k>0$ and a nontrivial family $T$ with $D_{\lambda_k,c_k}B_k\to T$ on every separated tuple. \end{enumerate} A family is trivial if it vanishes on all separated tuples of length $n\ge1$. \begin{proposition}\label{prop:limitEdilate} Assume the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}}, $C_k(0)>0$ for large $k$, $Z_k=C_k(\sigma_0)^{-1/2}$ and \textup{(M$_{\mathrm s}$)}, and let $m_1$ be as in Proposition~\textup{\ref{prop:limitEbracket}}. \textup{(i)} \textup{(DS)} implies \textup{(O$_{\mathrm s}$)}: every limit $S$ of Theorem~\textup{\ref{thm:limitEsep}} has $T=D_{\lambda,c}S$ for some $\lambda,c>0$. \textup{(ii)} \textup{(O$_{\mathrm s}$)} implies \textup{(DS)} with $\Lambda=m_1/m_*$, and $T$ can be taken to be any limit of Theorem~\textup{\ref{thm:limitEsep}}. \textup{(iii)} Under \textup{(DS)}, \textup{(U$_{\mathrm s}$)} holds if and only if $(\lambda_k,c_k)$ converges in $(0,\infty)^2$. \end{proposition} \begin{proof} For $\lambda>0$ the map $f\mapsto f(\lambda\,\cdot)$ is a bijection of the separated tuples. Let $(f_j)\in\mathcal C_n(Q,\delta)$, $\lambda_k\to \lambda\in[\Lambda^{-1},\Lambda]$. For large $k$ the tuples $(f_j(\lambda_k \,\cdot))$, $(f_j(\lambda\,\cdot))$ and the telescoping tuples with one factor $f_i(\lambda_k\,\cdot)-f_i(\lambda\,\cdot)$ lie in one $\mathcal C_n( Q',\delta/(2\Lambda))$, and $f_i(\lambda_k\,\cdot)\to f_i(\lambda\,\cdot)$ uniformly; so (M$_{\mathrm s}$) and multilinearity give $B_k(f_1(\lambda_k \,\cdot),\dots)-B_k(f_1(\lambda\,\cdot),\dots)\to0$. Hence, along a subsequence with $B_k\to S$ on every separated tuple, \[ B_k\bigl(f_1(\lambda_k\,\cdot),\dots,f_n(\lambda_k\,\cdot)\bigr)\to S_n\bigl( f_1(\lambda\,\cdot),\dots,f_n(\lambda\,\cdot)\bigr) .\tag{$*$} \] By Proposition~\ref{prop:limitEnorm}(ii) every limit is nontrivial. (i) Let $S$ be the limit along a subsequence, and pass to a further one with $\lambda_k\to\lambda$ and $c_k\to c\in[0,\infty]$. By ($*$), $T_n(f)= \lim_kc_k^nB_k(f(\lambda_k\,\cdot))$ with $B_k(f(\lambda_k\,\cdot))\to S_n(f(\lambda\,\cdot))$. If $c=\infty$, then $S_n(f(\lambda\,\cdot))=0$ for every separated tuple of length $n\ge1$, and $S$ is trivial; if $c=0$, $T$ is trivial. So $00$ along a subsequence. A further subsequence has $B_k\to S'$ on every separated tuple, and (O$_{\mathrm s}$) gives $S=D_{\lambda,c}S'$ with $\lambda\in I$ by Proposition~\ref{prop:limitEbracket}(iii). Then $D_{\lambda,c}B_k\to S$ on every separated tuple by ($*$) with constant $\lambda_k=\lambda$, so $e_k(\lambda,c)\to0$ and $e_k(\lambda_k,c_k)\to0$, a contradiction. Hence $e_k(\lambda_k,c_k)\to0$. Now take any subsequence and a further one with $\lambda_k\to\lambda\in I$ and $B_k\to S'$. Some $F_i$, of length $n_i\ge1$, has $S(F_i)\ne0$, and $c_k^{n_i}B_k(F_i(\lambda_k\, \cdot))\to S(F_i)$. If $c_k\to\infty$ along a further subsequence, then $B_k(F_l(\lambda_k\,\cdot))\to0$ for every $l$ with $n_l\ge1$, so by ($*$) $D_{\lambda,1}S'$ vanishes at every $F_l$, and $S'$ is trivial; if $c_k\to0$, then $S(F_i)=0$. So a further subsequence has $c_k\to c\in(0,\infty)$, and $D_{\lambda_k,c_k}B_k\to D_{\lambda,c}S'$ on every separated tuple by ($*$). This family agrees with $S$ at every $F_l$, hence everywhere. Every subsequence thus has a further subsequence along which $D_{\lambda_k,c_k} B_k\to S$, which is (DS) with $T=S$. (iii) If $(\lambda_k,c_k)\to(\lambda,c)$, the proof of (i) gives $S= D_{1/\lambda,1/c}T$ for every limit $S$, which is (U$_{\mathrm s}$). Conversely, under (U$_{\mathrm s}$) let $S$ be the limit. By the proof of (i), $(c_k)$ has no subsequence tending to $0$ or $\infty$, so $(\lambda_k,c_k)$ lies in a compact subset of $(0,\infty)^2$, and every limit point $(\lambda,c)$ has $D_{\lambda,c}S=T$. Two limit points $(\lambda,c)$, $(\lambda',c')$ give $D_{\lambda/\lambda',c/c'}S=S$, so $\lambda=\lambda'$ and $c=c'$ by Lemma~\ref{lem:limitEdil}. \end{proof} So (O$_{\mathrm s}$) says that the spacings $a_k$ can be corrected by bounded factors $\lambda_k$ so that the moments of the same states converge, and under it (U$_{\mathrm s}$) is the convergence of the corrections and of the constants, which (MC$_{m_0}$) gives by Theorem~\ref{thm:limitEunique}(ii). Neither (O$_{\mathrm s}$) nor (U$_{\mathrm s}$) follows from the properties proved here together with (MC$_{m_0}$): the fields of Proposition~\ref{prop:limitEgauss}(c) are reflection positive, invariant under translations and $B_4$, have the gap $ma_k$ in every coordinate direction and satisfy (M$_{\mathrm s}$) and (MC$_m$), and their two limits are not related by any $D_{\lambda,c}$. Conversely (O$_{\mathrm s}$) does not give (MC$_{m_0}$) (Theorem~\ref{thm:limitEunique}(iii)). The proofs of Propositions~\ref{prop:limitEaxial}, \ref{prop:limitEbracket} and~\ref{prop:limitEdilate} use the averaged condition only through Lemma~\ref{lem:limitElattice}(iii) and the bound $\gamma_k\le e^{-m_*a_k}$, which the fields of Proposition~\ref{prop:limitEgauss} have with $m$ in place of $m_*$. \emph{Temperedness.} The Schwinger functions of~\cite{OS73} are linear functionals on the space $\mathcal S_0(\R^{4n})$ of the $F\in\mathcal S(\R^{4n})$ that vanish, with all their derivatives, at every $x=(x_1,\dots,x_n)$ with $x_i=x_j$ for some $i\ne j$, and~\cite{OS75} adds the linear growth condition: $|S_n(F)|\le\sigma_n|F|_{sn}$ on $\mathcal S_0(\R^{4n})$ for a fixed integer $s$ and $\sigma_n\le\alpha (n!)^{\beta'}$, with the norms $|F|_w=\sup_{x,\,|\gamma|\le w}(1+|x|^2)^{w/2} |\partial^\gamma F(x)|$ of Lemma~\ref{lem:hermite}. Theorem~\ref{thm:limitEsep} gives the $S_n$ only on $C^\infty_c(\mathfrak D_n)$. The test functions of $\mathcal S_0(\R^{4n})$ approach the coinciding points, where (M$_{\mathrm s}$) gives no control, since its constants $M_n(Q,\delta)$ may grow arbitrarily as $\delta\to0$. The following rate closes this gap. \begin{enumerate}[label=\textup{(G$_{\mathrm s}$)},leftmargin=*] \item \emph{Growth at coinciding points.} There are $K,\beta,p,q\ge0$ such that for every $n\ge1$, every cube $Q$ of side $L$ and every $\delta\in(0,1]$ there is $k_0$ with \[ \Bigl|\omega_k\Bigl(\prod_{j=1}^nZ_kA_k(f_j)\Bigr)\Bigr|\le K^n(n!)^\beta(1+L)^{qn}\delta^{-pn}\prod_{j=1}^n\|f_j\|_\infty \] for all $k\ge k_0$ and $(f_1,\dots,f_n)\in\mathcal C_n(Q,\delta)$. \end{enumerate} Since $\mathcal C_n(Q,\delta)\subset\mathcal C_n(Q,1)$ for $\delta\ge1$, (G$_{\mathrm s}$) implies (M$_{\mathrm s}$), and the limits of Theorem~\ref{thm:limitEsep}(i) obey the same bound. For $x\in\R^{4n}$ write $x_{j\nu}$ for the coordinates of $x_j$, $|x|$ for the Euclidean norm, $|x|_\infty=\max_{j,\nu}|x_{j\nu}|$, and $d(x)=\min_{i\ne j}|x_i-x_j|_\infty$. For a coordinate direction $\mu$ let $\mathcal S^{\mu+}_0(\R^{4n})$ be the set of $F\in\mathcal S_0(\R^{4n})$ with $F(x)=0$ whenever $x_{j\mu}\le0$ for some $j$, and $V_\mu=\{x\in\mathfrak D_n:x_{j\mu}>0\text{ for all }j\}$. Such an $F$ vanishes with all its derivatives on $\{x_{j\mu}\le0\}$, by continuity from the interior of this set. Fix $\eta_0\in C^\infty_c((-\frac12,\frac12))$ with $\eta_0\ge0$ and $\int\eta_0=1$, let $\psi=\mathbf 1_{[-1/2,1/2)}*\eta_0$, so that $0\le\psi\le1$, $\operatorname{supp}\psi\subset(-1,1)$ and $\sum_{m\in\Z}\psi(t-m)=1$, and put $C_\psi=\max(1,\|\psi'\|_\infty,\|\psi''\| _\infty)$. For $h>0$ and $m=(m_1,\dots,m_n)\in(\Z^4)^n$ let $\psi^h_m(x)=\prod_{j,\nu}\psi(x_{j\nu}/h-m_{j\nu})$; call $m$ \emph{separated} if $|m_i-m_j|_\infty\ge4$ for $i\ne j$, and let $E_h$ be the sum of $\psi^h_m$ over the separated $m$. \begin{lemma}\label{lem:limitEcut} \textup{(i)} $E_h$ is smooth with values in $[0,1]$, $E_h=0$ on $\{d<2h\}$ and $E_h=1$ on $\{d\ge6h\}$. For every multi-index $\gamma\ne0$, $|\partial^\gamma E_h|\le2^{4n}h^{-|\gamma|}\prod_{j,\nu}\|\psi^{(\gamma_{j\nu})} \|_\infty$; in particular $|\partial^\gamma(1-E_h)|\le(2C_\psi)^{4n} h^{-|\gamma|}$ for all $\gamma\in\{0,1,2\}^{4n}$. \textup{(ii)} Let $F\in\mathcal S_0(\R^{4n})$, $\alpha$ a multi-index and $w,N,w'$ integers with $w'\ge\max(w,|\alpha|+N)$. If $|x_i-x_j|_\infty=t\le1$ for some $i\ne j$, then \[ (1+|x|^2)^{w/2}|\partial^\alpha F(x)|\le3^w\frac{(4t)^N}{N!}|F|_{w'} . \] If $F\in\mathcal S^{\mu+}_0(\R^{4n})$ and $00\}$ and with pairwise disjoint supports. \end{lemma} \begin{proof} (i) The sum of all $\psi^h_m$ is $1$, and at each $x$ at most two integers $m_{j\nu}$ have $\psi(x_{j\nu}/h-m_{j\nu})\ne0$, so at most $2^{4n}$ terms are nonzero near $x$; this gives smoothness and the derivative bounds, the same ones holding for $1-E_h$, the sum over the other $m$. If $\psi^h_m(x)\ne0$, then $|x_j-hm_j|_\inftyh|m_i-m_j|_\infty-2h\ge2h$, so $E_h=0$ where $d<2h$. If $d(x)\ge6h$, it gives $h|m_i-m_j|_\infty>4h$ for every $m$ with $\psi^h_m(x)\ne0$, so all these $m$ are separated and $E_h(x)=1$. (ii) Let $u=x_i-x_j$, let $x(\tau)$ be $x$ with $x_i$ replaced by $x_j+\tau u$, and $g(\tau)=\partial^\alpha F(x(\tau))$. All derivatives of $g$ vanish at $\tau=0$, where two arguments coincide, so Taylor's formula gives $|g(1)|\le\sup_\tau|g^{(N)}(\tau)|/N!$, and $g^{(N)}=(u\cdot\nabla_{x_i})^N \partial^\alpha F$ is at most $|u|_1^N\le(4t)^N$ times $\max_{|\gamma|=N} |\partial^{\gamma+\alpha}F(x(\tau))|\le|F|_{w'}(1+|x(\tau)|^2)^{-w/2}$. Since $|x-x(\tau)|\le|u|\le2t\le2$, $(1+|x|^2)^{1/2}\le3(1+|x(\tau)|^2)^{1/2}$. In the second case move $x_{j\mu}$ from $0$ to $t$ instead; the path has length $t\le1$, which gives the factor $2^w$. (iii) Let $F\in\mathcal S_0(\R^{4n})$, fix $w$, let $\chi\in C^\infty_c( \R^{4n})$ with values in $[0,1]$ equal to $1$ on $\{|x|\le1\}$, $\chi_R=\chi( \cdot/R)$ with $R\ge1$, and $00\}$ with pairwise disjoint closures; finitely many of them cover $\operatorname{supp}G$, a partition of unity subordinate to them~\cite[Sect.~1.4]{Hor03} writes $G$ as a finite sum of smooth functions supported in such products, and for each the partial sums of the Fourier series in the proof of Theorem~\ref{thm:limitEsep}(i) are finite sums of products of the required form converging in $C^\infty$ with supports in a fixed compact set, hence in every $|\cdot|_w$. \end{proof} \begin{lemma}\label{lem:limitEextend} Let $n\ge2$, $C,p',q'\ge0$, and let $S$ be a multilinear form on the separated $n$-tuples with \[ |S(f_1,\dots,f_n)|\le C(1+L)^{q'}\delta^{-p'}\prod_{j=1}^n\|f_j\|_\infty \] for every cube $Q$ of side $L$, every $\delta\in(0,1]$ and $(f_1,\dots,f_n) \in\mathcal C_n(Q,\delta)$. Let $S$ also denote the distribution on $\mathfrak D_n$ with $S(f_1\otimes\cdots\otimes f_n)=S(f_1,\dots,f_n)$ for the separated tuples of smooth functions, which exists and is unique by the proof of Theorem~\textup{\ref{thm:limitEsep}(i)}. There is a number $c_\star$, independent of $n$, $C$, $p'$, $q'$ and $S$, such that, with $w=\lceil q'\rceil+\lceil p'\rceil+12n+1$, \[ |S(F)|\le c_\star^n\,C\,6^{q'}16^{p'}\,|F|_w\qquad\text{for }F\in C^\infty_c(\mathfrak D_n). \] Hence $S$ extends uniquely to a linear functional on $\mathcal S_0(\R^{4n})$ that is continuous for $|\cdot|_w$, with the same bound. \end{lemma} \begin{proof} \emph{One piece.} Let $h\in(0,\frac1{16}]$, $m$ separated, $z_j=hm_j$, and $G\in C^\infty(\R^{4n})$ supported in $\prod_j(z_j+(-h,h)^4)$. The open cubes $U_j=z_j+(-\frac32h,\frac32h)^4$ are at mutual $\ell^\infty$ distance at least $h$, and lie in a cube $Q$ of side $L\le2\max_j|z_j|_\infty+3h$. Let $B=\prod_j(z_j+[-2h,2h]^4)$, $e_\kappa(x)=\prod_j\exp(i\pi\kappa_j\cdot(x_j- z_j)/(2h))$ for $\kappa\in(\Z^4)^n$, $c_\kappa=(4h)^{-4n}\int_BG\,\bar e_\kappa$, and $\varphi_j\in C^\infty_c(U_j)$ with values in $[0,1]$, equal to $1$ on $z_j+[-h,h]^4$. With $\mathsf D=\prod_{i=1}^{4n}(1-(2h/\pi)^2\partial_i^2)$ over the $4n$ coordinates, $\mathsf D\bar e_\kappa=\prod_i(1+\kappa_i^2)\bar e_\kappa$, so integration by parts gives $|c_\kappa|\le\prod_i(1+\kappa_i^2)^{-1} \|\mathsf DG\|_\infty$, and the same with powers of $\mathsf D$ shows that the partial sums of $\sum_\kappa c_\kappa\prod_j\varphi_j(x_j)e_{\kappa_j}(x_j)$ converge to $G$ in $C^\infty$ with supports in $\prod_j\operatorname{supp} \varphi_j$. Hence $S(G)$ is the sum of the values of $S$ on the terms. Each term is $c_\kappa$ times a product in $\mathcal C_n(Q,h)$ with factors of norm at most $1$, and $\sum_{k\in\Z}(1+k^2)^{-1}\le1+2\int_0^\infty\frac{dt}{1+t^2} <5$, so \[ |S(G)|\le C(1+L)^{q'}h^{-p'}\,5^{4n}\,\|\mathsf DG\|_\infty . \] \emph{Decomposition.} Let $F\in C^\infty_c(\mathfrak D_n)$, $h_l=2^{-l-4}$ and $E_l=E_{h_l}$. Since $d$ is bounded below on $\operatorname{supp}F$, Lemma~\ref{lem:limitEcut}(i) gives $E_lF=F$ for large $l$, and with $E_{l+1}-E_l=E_{l+1}(1-E_l)-E_l(1-E_{l+1})$, \[ F=E_0F+\sum_{l\ge0}\bigl(E_{l+1}(1-E_l)F-E_l(1-E_{l+1})F\bigr), \] a finite sum. Each term has the form $\sum_m\psi^h_mH$ over separated $m$, with $(h,H)$ equal to $(h_0,F)$, $(h_{l+1},(1-E_l)F)$ or $(h_l,(1-E_{l+1})F)$; in the last two cases $H=(1-E_{h'})F$ with $h/h'\in\{\frac12,2\}$, and $H$ vanishes where $d\ge12h$. \emph{Derivatives.} Write $\mathsf D=\prod_i\mathsf D_i$ and $\psi^h_m=\prod_i \psi_i$. Since $\mathsf D_i(\psi_iG)=\psi_iG-(2h/\pi)^2(\psi_i''G+2\psi_i' \partial_iG+\psi_i\partial_i^2G)$ and $|\psi_i^{(k)}|\le C_\psi h^{-k}$, induction over the coordinates gives \[ |\mathsf D(\psi^h_mH)|\le b_\psi^{4n}\max_{\gamma\in\{0,1,2\}^{4n}}h^{|\gamma|} |\partial^\gamma H|,\qquad b_\psi=1+2C_\psi, \] on the box $X_m=\prod_j(hm_j+[-h,h]^4)$, outside which $\psi^h_m=0$, since $1+16C_\psi/\pi^2\le b_\psi$. For $H=(1-E_{h'})F$, the Leibniz rule, Lemma~\ref{lem:limitEcut}(i), $h/h'\le2$ and $h\le1$ give $h^{|\gamma|} |\partial^\gamma H|\le3^{|\gamma|}(2C_\psi)^{4n}\max_{\alpha\le\gamma} |\partial^\alpha F|$. Put $w_0=\lceil q'\rceil+8n$ and $N=\lceil p'\rceil+4n+1$, so that $w\ge\max(w_0,8n+N)$. At a point of the support of $H$, $d\le12h\le \frac34$, and Lemma~\ref{lem:limitEcut}(ii) gives $(1+|x|^2)^{w_0/2} |\partial^\alpha F(x)|\le3^{w_0}(48h)^N|F|_w/N!$ for $|\alpha|\le8n$; for $H=F$ the same quantity is at most $|F|_w$. \emph{Summation.} For $x\in X_m$, $|z_j|_\infty\le|x|_\infty+h$, so the cube of the first step has $1+L\le2(1+|x|_\infty)\le2(1+|x|^2)^{1/2}$. Evaluating at the point of the compact box $X_m$ where $(1+|x|^2)^{-w_0/2}$ is largest, and using $q'-w_0\le-8n$ and $(1+|x|^2)^{4n}\ge\prod_{j,\nu}(1+x_{j\nu}^2)$, the piece $\psi^h_mH$ of a term of the second or third kind contributes at most \[ C2^{q'}h^{-p'}\bigl(5b_\psi\cdot2C_\psi\cdot9\bigr)^{4n}3^{w_0}\frac{(48h)^N} {N!}|F|_w\sup_{x\in X_m}\prod_{j,\nu}(1+x_{j\nu}^2)^{-1} . \] The sum over $m\in(\Z^4)^n$ of the supremum is the $4n$-th power of $\sum_{k\in\Z}\sup_{|t-hk|\le h}(1+t^2)^{-1}\le3+2\int_0^\infty\frac{du}{1+h^2u^2} <\frac7h$. With $48^N/N!\le e^{48}$, $3^{w_0}\le3^{q'+1}3^{8n}$ and $h^{N-p'-4n}\le h$, the terms of the second and third kinds at the scale $h$ contribute at most $C6^{q'}c_1^nh|F|_w$ with $c_1=3^{10}e^{48}(5\cdot7\cdot9 \cdot2C_\psi b_\psi)^4$, and each $h_l$ occurs in at most two of them, with $\sum_l2h_l=\frac14$. The term $E_0F$ contributes at most $C2^{q'}16^{p'}(5\cdot7\cdot16\,b_\psi)^{4n}|F|_w$ by the same count without the Taylor factor. This gives the bound with $c_\star=c_1+(5\cdot7\cdot16\, b_\psi)^4$. The extension follows from the density of $C^\infty_c(\mathfrak D_n)$ in $\mathcal S_0(\R^{4n})$ for $|\cdot|_w$ (Lemma~\ref{lem:limitEcut}(iii)), and it is unique by the same density. \end{proof} For a function $F$ of $n$ points put \begin{gather*} (\Theta_\mu F)(x)=F(\theta_\mu x_1,\dots,\theta_\mu x_n),\\ (\tau_vF)(x)=F(x_1-v,\dots,x_n-v),\\ (F\otimes F')(x,x')=F(x)F'(x'). \end{gather*} For $F_n\in\mathcal S^{\mu+}_0(\R^{4n})$ and $F'_{n'}\in\mathcal S^{\mu+}_0(\R^{4n'})$, $\Theta_\mu\bar F_n\otimes F'_{n'}\in\mathcal S_0(\R^{4(n+n')})$, because at a coincidence of a reflected point with an unreflected one some argument of $F_n$ or of $F'_{n'}$ has $\mu$-coordinate at most $0$. \begin{theorem}\label{thm:limitEtempered} Assume the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}} and \textup{(M$_{\mathrm s}$)}, and let $S_n$ be limits as in Theorem~\textup{\ref{thm:limitEsep}}. \textup{(i)} Let $n\ge2$, and suppose that there are $C,p',q'\ge0$ such that \[ |S_n(f_1,\dots,f_n)|\le C(1+L)^{q'}\delta^{-p'}\prod_j\|f_j\|_\infty \] on $\mathcal C_n(Q,\delta)$ for every cube $Q$ of side $L$ and $\delta\in(0,1]$. Then $S_n$ extends uniquely to a linear functional on $\mathcal S_0(\R^{4n})$ continuous for the topology of $\mathcal S(\R^{4n})$, and $|S_n(F)|\le c_\star^nC6^{q'} 16^{p'}|F|_w$ with $w$ and $c_\star$ as in Lemma~\textup{\ref{lem:limitEextend}}. \textup{(ii)} Under \textup{(G$_{\mathrm s}$)}, for every $n\ge1$ and $F\in\mathcal S_0(\R^{4n})$, \[ |S_n(F)|\le\sigma_n|F|_{sn},\qquad\sigma_n=(6^q16^pc_\star K)^n(n!)^\beta, \] with $s=\lceil p\rceil+\lceil q\rceil+13$, and $\sigma_n\le e^{6^q16^pc_\star K}(n!)^{\beta+1}$, since $y^n\le e^yn!$ for $y\ge0$; this is the linear growth condition of~\cite{OS75}. \textup{(iii)} Suppose that the hypothesis of \textup{(i)} holds for every $n\ge2$, with constants depending on $n$, and put $S_0=1$. Then the extended $S_n$ are symmetric, invariant under the translations, $B_4$ and $G_S$, and satisfy $S_n(\bar F)=\overline{S_n(F)}$. For every $\mu$ and every finite family $F_n\in\mathcal S^{\mu+}_0(\R^{4n})$, $0\le n\le n_0$ \textup{(}$F_0\in \C$\textup{)}, \[ \sum_{n,n'=0}^{n_0}S_{n+n'}\bigl(\Theta_\mu\bar F_n\otimes F_{n'}\bigr)\ge0 , \] and there are vectors $\Psi_\mu(F)$ of the space $\mathcal H_\mu$ of Theorem~\textup{\ref{thm:limitEsep}(iii)} whose inner products are these sums, which agree with $\Psi_\mu(P)$ for separated polynomials $P$ of smooth functions, are dense in $\mathcal H_\mu$, and satisfy $e^{-tH_\mu}\Psi_\mu(F)=\Psi_\mu(\tau_{te_\mu}F)$ for $t\ge0$. For $y\perp e_\mu$, $y\ne0$, and $F_n\in\mathcal S^{\mu+}_0(\R^{4n})$, $F'_{n'}\in\mathcal S^{\mu+}_0(\R^{4n'})$, \[ \lim_{\lambda\to\infty}S_{n+n'}\bigl(\Theta_\mu\bar F_n\otimes\tau_{\lambda y} F'_{n'}\bigr)=S_n(\Theta_\mu\bar F_n)\,S_{n'}(F'_{n'}) . \] \end{theorem} \begin{proof} (i) This is Lemma~\ref{lem:limitEextend}; continuity for $|\cdot|_w$ is continuity in $\mathcal S(\R^{4n})$. (ii) For $n=1$, $S_1=0$. For $n\ge2$ the limits satisfy the hypothesis of (i) with $C=K^n(n!)^\beta$, $q'=qn$ and $p'=pn$, and $\lceil qn\rceil+\lceil pn \rceil+12n+1\le sn$, while $|F|_w\le|F|_{sn}$ for $w\le sn$. (iii) Let $g$ be a permutation of the arguments, or the diagonal action of a translation, of an element of $B_4$ or of an element of $G_S$. Then $F\mapsto F\circ g^{-1}$ maps $C^\infty_c(\mathfrak D_n)$ onto itself and is continuous for the topology of $\mathcal S(\R^{4n})$. The distributions $F\mapsto S_n(F\circ g^{-1})$ and $S_n$ agree on the products of separated smooth functions by the symmetry of $S_n$, by Theorem~\ref{thm:limitEsep}(ii) and by the definition of $G_S$, hence on $C^\infty_c(\mathfrak D_n)$ by the uniqueness in Theorem~\ref{thm:limitEsep}(i), hence on $\mathcal S_0(\R^{4n})$ by density; the conjugation identity follows in the same way. Choose an integer $s'$ such that the bounds of (i) for $n\le2n_0$ hold with $|\cdot|_{s'n}$, and by Lemma~\ref{lem:limitEcut}(iii) finite sums $P^{(i)}_n$ of products of smooth functions with disjoint supports in $\{x_\mu>0\}$ with $|P^{(i)}_n-F_n|_{2s'n_0} \to0$, and $P^{(i)}_0=F_0$. Then $P^{(i)}=\sum_nP^{(i)}_n$ is a separated polynomial at positive $\mu$-times, and $\sum_{n,n'}S_{n+n'}(\Theta_\mu\bar P^{(i)}_n\otimes P^{(i)}_{n'})=S(\theta_\mu\bar P^{(i)}\cdot P^{(i)})\ge0$ by Theorem~\ref{thm:limitEsep}(iii), each $\Theta_\mu\bar P^{(i)}_n\otimes P^{(i)}_{n'}$ being a finite sum of products of separated smooth functions. Since $|G\otimes G'|_w\le|G|_w|G'|_w$ and $|\Theta_\mu G|_w=|G|_w$, $\Theta_\mu\bar P^{(i)}_n\otimes P^{(i)}_{n'}\to\Theta_\mu\bar F_n\otimes F_{n'}$ in $|\cdot|_{s'(n+n')}$, and the positivity passes to the limit. The same convergence, applied to $P^{(i)}-P^{(i')}$, shows that $\Psi_\mu(P^{(i)})$ is a Cauchy sequence in $\mathcal H_\mu$; its limit $\Psi_\mu(F)$ does not depend on the approximants and has the stated inner products. Separated polynomials of smooth functions are among the families $F$, and their vectors are dense in $\mathcal H_\mu$: those of separated polynomials of continuous functions are dense by construction, and each of them is a limit of vectors of smooth ones, by uniform approximation with supports in slightly larger sets and the bound of Theorem~\ref{thm:limitEsep}(i). The relation with $e^{-tH_\mu}$ holds for the $P^{(i)}$ by Theorem~\ref{thm:limitEsep}(iv) and passes to the limit because $\tau_{te_\mu}$ preserves $\mathcal S^{\mu+}_0$ and is continuous. For the last claim, by Theorem~\ref{thm:limitEsep}(v) applied to every pair of monomials, $\ip{\Psi_\mu(P)}{U_\mu(\lambda y)\Psi_\mu(P')}=S(\theta_\mu\bar P\cdot\tau_{\lambda y}P')$ tends to $S(\theta_\mu\bar P)S(P')=\ip{\Psi_\mu(P)} {\Omega_\mu}\ip{\Omega_\mu}{\Psi_\mu(P')}$ for separated polynomials $P,P'$ at positive $\mu$-times; the $U_\mu(\lambda y)$ are unitary and these vectors are dense, so $U_\mu(\lambda y)$ converges weakly to the projection on $\C\Omega_\mu$. With $U_\mu(\lambda y)\Psi_\mu(F')=\Psi_\mu(\tau_{\lambda y}F')$, obtained as for $e^{-tH_\mu}$, and $\ip{\Psi_\mu(F_n)}{\Omega_\mu}= S_n(\Theta_\mu\bar F_n)$, $\ip{\Omega_\mu}{\Psi_\mu(F'_{n'})}=S_{n'}(F'_{n'})$, this is the claim. \end{proof} With (W$_{\mathrm s}$) and (U$_{\mathrm s}$), Corollary~\ref{cor:limitEexact} and Theorem~\ref{thm:limitEtempered} give the axioms of~\cite{OS73,OS75}. The positivity of Theorem~\ref{thm:limitEtempered}(iii) contains that of~\cite{OS73}, whose test functions are supported where $00$ for large $k$ and $Z_k=C_k(\sigma_0)^{-1/2}$, they are not the Schwinger functions of the vacuum alone. \end{corollary} \begin{proof} (G$_{\mathrm s}$) gives (M$_{\mathrm s}$), and Corollary~\ref{cor:limitEexact} gives convergence, invariance under the translations and $\mathrm O(4)$, so $G_S=\mathrm O(4)$, and properties (i)--(vi) of Theorem~\ref{thm:limitEsep}. Theorem~\ref{thm:limitEtempered}(ii) gives the growth condition, and part (iii) for $\mu=0$ gives the invariance on $\mathcal S_0$, positivity and clustering, and identifies the Hilbert space built from $\mathcal S^{0+}_0$ with $\mathcal H_0$ and its time translations with $e^{-tH_0}$; Theorem~\ref{thm:limitEsep}(iv) gives the spectrum. Proposition~\ref{prop:limitEnorm}(ii) gives $\mathcal H_0\ne\C\Omega_0$, so some $S_n$ with $n\ge2$ is not zero, by Theorem~\ref{thm:limitEsep}(vi). \end{proof} By the reconstruction theorem of~\cite{OS75}, Schwinger functions with these properties are those of a Wightman theory on $\mathcal H_0$ with Hamiltonian $H_0$. So the averaged condition along the tuning, (G$_{\mathrm s}$) with the constants of Proposition~\ref{prop:limitEnorm}(ii), (W$_{\mathrm s}$) and (U$_{\mathrm s}$) together give (E) for the field $\Phi$ at separated points, with mass gap at least $m_*$. None of these four statements is proved. For $n=2$ the rate in (G$_{\mathrm s}$) is a rate in (SD): \begin{enumerate}[label=\textup{(SD$_{\mathrm p}$)},leftmargin=*] \item there are $C_0,p_0\ge0$ such that for every $\varepsilon'\in(0, \varepsilon]$ there is $k_0$ with $\mathcal N_k(\varepsilon')\le C_0( \varepsilon/\varepsilon')^{p_0}\mathcal N_k(\varepsilon)$ for $k\ge k_0$. \end{enumerate} \begin{proposition}\label{prop:limitEtwotempered} Assume the hypothesis of Theorem~\textup{\ref{thm:limitEtwo}}. \textup{(a)} \textup{(SD$_{\mathrm p}$)} holds if and only if there are $C,p',q'\ge0$ such that for every cube $Q$ of side $L$ and $\delta\in(0,1]$ there is $k_0$ with \[ Z_k^2\bigl|\omega_k\bigl(A_k(f)A_k(g)\bigr)\bigr|\le C(1+L)^{q'}\delta^{-p'} \|f\|_\infty\|g\|_\infty \] for $k\ge k_0$ and $(f,g)\in\mathcal C_2(Q,\delta)$. One can take $q'=8$ and $p'=8+p_0$ in one direction and $p_0=p'$ in the other. \textup{(b)} Under \textup{(SD$_{\mathrm p}$)} every limit $S_2$ of Corollary~\textup{\ref{cor:limitEtwo}(b)} extends uniquely to a linear functional on $\mathcal S_0(\R^8)$ with $|S_2(F)|\le C'|F|_{\lceil p_0\rceil +41}$, invariant under the translations and $B_4$, with $S_2(\Theta_\mu\bar f\otimes f)\ge0$ for $f\in\mathcal S^{\mu+}_0(\R^4)$. \end{proposition} \begin{proof} (a) Let (SD$_{\mathrm p}$) hold, $(f,g)\in\mathcal C_2(Q,\delta)$ with $\delta\le1$, and $\varepsilon'=\min(\varepsilon,\delta/8)$. The proof of Corollary~\ref{cor:limitEtwo}(a) gives, for large $k$ depending only on $\delta$, $Z_k^2|\omega_k(A_k(f)A_k(g))|\le\mathsf K_{4L/\delta}^2\mathsf K_{\delta/4}^2\mathcal N_k(\varepsilon')\mathcal N_k(\varepsilon)^{-1}\|f\| _\infty\|g\|_\infty$, and $\mathcal N_k(\varepsilon')/\mathcal N_k(\varepsilon) \le\max(1,C_0)\max(1,8\varepsilon)^{p_0}\delta^{-p_0}$. Since $\mathsf K_{4L/\delta}^2=(8L/\delta+3)^8\le11^8(1+L)^8\delta^{-8}$ and $\mathsf K_{\delta/4}^2\le(\frac72)^8$, the bound holds with $q'=8$, $p'=8+p_0$. Conversely, the proof of Corollary~\ref{cor:limitEtwo}(a) gives $\mathcal N_k(\varepsilon')\le M\,\mathcal N_k(\varepsilon)$ for large $k$, where $M$ is the constant of the bound for a cube $Q_0$ of side $L_0=2(\varepsilon+2+ 2a_0)$, which contains $B_{\varepsilon'}$ and $\theta_{0,k}(B_{\varepsilon'})$ for all $k$ and $\varepsilon'\le\varepsilon$, and for the separation $\min(\varepsilon',1)$. So $M\le C(1+L_0)^{q'}\max(1,\varepsilon^{-p'}) (\varepsilon/\varepsilon')^{p'}$, which is (SD$_{\mathrm p}$) with $p_0=p'$. (b) By (a) the limits satisfy the hypothesis of Lemma~\ref{lem:limitEextend} with $n=2$, $q'=8$, $p'=8+p_0$, and $\lceil q' \rceil+\lceil p'\rceil+25=\lceil p_0\rceil+41$. Invariance and positivity follow as in the proof of Theorem~\ref{thm:limitEtempered}(iii), from those of Corollary~\ref{cor:limitEtwo}(b) and Lemma~\ref{lem:limitEcut}(iii) with $n=1$. \end{proof} The rate cannot be dropped: the structure that Theorem~\ref{thm:limitEsep} and Corollary~\ref{cor:limitEtwo}(b) give at separated points does not imply temperedness. \begin{proposition}\label{prop:limitEnotempered} \textup{(i)} Let $K\ge0$ be integrable on the compact subsets of $\R^4\setminus\{0\}$ and $S_2(F)=\iint F(x,y)K(x-y)\,dx\,dy$ for $F\in C^\infty_c(\mathfrak D_2)$. If $|S_2(F)|\le C|F|_w$ on $C^\infty_c(\mathfrak D_2)$, then $\int_{\delta\le|z|_\infty\le1}K\le c_wC\delta^{-w}$ for $\delta\in(0,1]$, with $c_w$ depending only on $w$. Conversely, if $\int_{\delta\le|z|_\infty\le\rho}K\le C(1+\rho)^{q'}\delta^{-p'}$ for $0<\delta\le1\le\rho$, then $S_2$ extends to $\mathcal S_0(\R^8)$ as in Lemma~\textup{\ref{lem:limitEextend}}. \textup{(ii)} Let $K(z)=\int_1^\infty e^{\sqrt\lambda-\lambda\|z\|_1}d\lambda$ for $z\ne0$, $\|z\|_1=\sum_\nu|z_\nu|$, and $S_2(f,g)=\iint f(x)g(y)K(x-y)\,dx \,dy$ for separated pairs. Then $S_2$ is given by a distribution on $\mathfrak D_2$, is symmetric, invariant under the translations and $B_4$, real on real functions, bounded on $\mathcal C_2(Q,\delta)$ by $L^8\mathsf k(\delta)\|f\|_\infty\|g\|_\infty$ with $\mathsf k(\delta)=\int_1^\infty e^{\sqrt\lambda-\lambda\delta}d\lambda$, and satisfies $|S_2(f,g(\cdot-\lambda y))|\le e\mathsf k(1)L^8\|f\|_\infty\|g\|_\infty e^{-(\lambda|y|_\infty-L)}$ for $f,g$ supported in a cube of side $L$ and $\lambda|y|_\infty\ge L+1$. For $g$ supported in $\{x_\mu>0\}$, $S_2(\bar g\circ\theta_\mu,g)\ge0$; the time translations define on the completion $\mathcal H^{(1)}\ne0$ of the functions supported in $\{x_0>0\}$ for the form $S_2(\bar g\circ\theta_0,g')$ a semigroup $e^{-tH^{(1)}}$ with $\sigma(H^{(1)})\subset[1,\infty)$. These are the properties of Corollary~\textup{\ref{cor:limitEtwo}(b)}, with $m_*=1$. But $\int_{\delta\le|z|_\infty\le2\delta}K\ge240\,\delta^4e^{1/(64\delta)}$ for $\delta\le\frac1{32}$, so $S_2$ is bounded by no norm $|\cdot|_w$ on $C^\infty_c(\mathfrak D_2)$ and does not extend to a continuous functional on $\mathcal S_0(\R^8)$. \end{proposition} \begin{proof} (i) Let $\chi_0\in C^\infty(\R)$ be even, nonincreasing on $[0,\infty)$, equal to $1$ on $[0,1]$ and to $0$ on $[2,\infty)$, $\chi_1(z)=\prod_\nu\chi_0(z_\nu)$, and $\psi_\delta(z)=\chi_1(z)-\chi_1(2z/\delta)$. Then $0\le\psi_\delta\le1$, $\psi_\delta=1$ on $\{\delta\le|z|_\infty\le1\}$, $\psi_\delta=0$ on $\{|z|_\infty\le\delta/2\}\cup\{|z|_\infty\ge2\}$, and $|\partial^\gamma \psi_\delta|\le c_\gamma\delta^{-|\gamma|}$. With $\phi\in C^\infty_c((-1, 1)^4)$, $\phi\ge0$, $\int\phi=1$, the function $F_\delta(x,y)=\phi(x) \psi_\delta(x-y)$ lies in $C^\infty_c(\mathfrak D_2)$, has $|F_\delta|_w\le c_w \delta^{-w}$, and $S_2(F_\delta)=\int\psi_\delta K\ge\int_{\delta\le|z|_\infty \le1}K$. For the converse, $|S_2(f,g)|\le\|f\|_\infty\|g\|_\infty\int_Qdx\int_{ \delta\le|z|_\infty\le\max(L,1)}K\le2^{q'}C(1+L)^{q'+4}\delta^{-p'}\|f\|_\infty \|g\|_\infty$ on $\mathcal C_2(Q,\delta)$, and $S_2$ is the distribution of Lemma~\ref{lem:limitEextend} by uniqueness. (ii) For $|z|_\infty\ge\delta$, $\|z\|_1\ge\delta$ and $K(z)\le\mathsf k(\delta)$; for $\|z\|_1=u\ge1$ and $\lambda\ge1$, $\lambda u\ge\lambda+u-1$, so $K(z)\le e\mathsf k(1)e^{-u}$. Hence $K$ is continuous on $\R^4\setminus\{0\}$, which gives the distribution, and the two bounds follow, since $\|x-y\|_1\ge |x-y|_\infty$. Symmetry, reality and invariance hold because $K$ is real, even and a function of $\|z\|_1$, which $B_4$ preserves. Let $g,g'$ be supported in $\{x_0\ge t_0\}$, $t_0>0$. Substituting $x\mapsto\theta_0x$ and $y\mapsto y+te_0$, \[ S_2\bigl(\bar g\circ\theta_0,g'(\cdot-te_0)\bigr)=\int_1^\infty e^{\sqrt\lambda -\lambda t}\,\mathsf B_\lambda(G_\lambda,G'_\lambda)\,d\lambda , \] where $G'_\lambda$ is formed from $g'$ as $G_\lambda$ from $g$, with \begin{gather*} G_\lambda(\mathbf x)=\int g(x_0,\mathbf x)e^{-\lambda x_0}dx_0,\\ \mathsf B_\lambda(u,v)=\iint\bar u(\mathbf x)v(\mathbf y)e^{-\lambda\|\mathbf x-\mathbf y\|_1}d\mathbf x\,d\mathbf y\qquad(u,v\colon\R^3\to\C) . \end{gather*} The integrand is bounded by $e^{\sqrt\lambda-2\lambda t_0}\|g\|_1 \|g'\|_1$. From $e^{-\lambda|\varsigma|}=\frac1\pi\int_\R\frac{\lambda} {\lambda^2+k^2}e^{ik\varsigma}dk$ one obtains \[ \mathsf B_\lambda(u,u)=\pi^{-3}\int|\hat u(k)|^2\prod_{i=1}^3\frac\lambda {\lambda^2+k_i^2}\,dk\ge0 . \] So the form is positive, and \[ \|g(\cdot-te_0)\|^2=\int_1^\infty e^{\sqrt\lambda-2\lambda t}\mathsf B_\lambda(G_\lambda,G_\lambda)\,d\lambda\le e^{-2t}\|g\|^2 . \] The time translations therefore induce symmetric contractions $\mathsf T(t)$ of $\mathcal H^{(1)}$ with $\mathsf T(s)\mathsf T(t)=\mathsf T(s+t)$ and $\|\mathsf T(t)\|\le e^{-t}$, strongly continuous by dominated convergence, so $\mathsf T(t)=e^{-tH^{(1)}}$ with $H^{(1)}\ge1$, as in the proof of Theorem~\ref{thm:osaxioms}(iv). For $g\ge0$ not zero, $G_\lambda\ge0$ is not zero and $\mathsf B_\lambda(G_\lambda,G_\lambda)>0$, so $\mathcal H^{(1)}\ne0$. Positivity for $\mu\ne0$ follows from $\theta_\mu\pi_\mu=\pi_\mu\theta_0$ and the invariance under $\pi_\mu$. Finally, let $0<|z|_\infty\le2\delta\le\frac1 {16}$, so that $u=\|z\|_1\le8\delta\le\frac14$, and $\lambda_0=(4u)^{-2}\ge1$. For $\lambda\in[\lambda_0,2\lambda_0]$, $\sqrt\lambda-\lambda u\ge\frac1{4u}- \frac1{8u}=\frac1{8u}$, so $K(z)\ge\lambda_0e^{1/(8u)}\ge e^{1/(64\delta)}$, and $\{\delta\le|z|_\infty\le2\delta\}$ has volume $240\delta^4$. By (i), a bound $|S_2(F)|\le C|F|_w$ would give $240\delta^4e^{1/(64\delta)}\le c_wC\delta^{-w}$ for all small $\delta$, which is false; and a continuous functional on $\mathcal S_0(\R^8)$ is bounded by some $C|\cdot|_w$. \end{proof} The kernel of (ii) is the superposition, with weight $e^{\sqrt\lambda}d \lambda$, of the kernels $e^{-\lambda\|z\|_1}$ of the limits of Proposition~\ref{prop:osobstruction}(a). By (i), for a two-point function with a nonnegative kernel, a power rate as in Lemma~\ref{lem:limitEextend} is also necessary for temperedness. \begin{remark}\label{rem:limitEcontact} \emph{Coinciding points.} The axioms of~\cite{OS73,OS75} involve only $\mathcal S_0(\R^{4n})$, and positivity involves only test functions whose reflected and unreflected arguments are separated. Under (G$_{\mathrm s}$) the bound of Theorem~\ref{thm:limitEtempered}(ii) extends by the Hahn--Banach theorem~\cite{RS80} to a functional on all of $\mathcal S(\R^{4n})$ with the same bound; averaging over the finite group generated by the permutations of the arguments and the diagonal action of $B_4$, which preserve every $|\cdot|_w$, makes it symmetric and $B_4$-invariant. Every such extension agrees with $S_n$ on $\mathcal S_0(\R^{4n})$ and so satisfies the positivity of Theorem~\ref{thm:limitEtempered}(iii); two of them differ by a distribution supported on the coinciding points. Translation invariance of the extension is not claimed here. For the action density the extension cannot be an integral: under the hypothesis of Proposition~\ref{prop:limitEsing}, the functions $F_\delta$ of the proof of Proposition~\ref{prop:limitEnotempered}(i) are supported in one compact set and bounded by $\|\phi\|_\infty$, while, since $|z|\le2|z|_\infty$, $S_2(F_\delta)\ge\int_{\delta\le|z|_\infty\le2\delta}K\ge240\cdot4^{-8}c\, \delta^{-4}(\log\frac1\delta)^{-p}$ for $2\delta\le\min(1,r_0/2)$; so no extension of $S_2$ to $\R^8$ is a distribution of order zero, and its values at coinciding points are fixed only by a renormalisation, which the axioms do not see. The same law satisfies the hypothesis of Theorem~\ref{thm:limitEtempered}(i) for $n=2$: if in addition $K(z)\le C|z|^{-8}$ for $z\ne0$, then $|S_2(f,g)|\le\|f\|_\infty\|g\|_\infty L^4C\int_{ |z|\ge\delta}|z|^{-8}dz=\frac{\pi^2}2CL^4\delta^{-4}\|f\|_\infty\|g\|_\infty$ on $\mathcal C_2(Q,\delta)$, which is that hypothesis with $q'=p'=4$, while (M) fails by Proposition~\ref{prop:limitEsing}. At strong coupling (Corollary~\ref{cor:limitEstrong}) every limit vanishes at separated points, and its extension is $0$. \end{remark} \emph{Asymptotic freedom at short distance.} Renormalised perturbation theory predicts how the two-point function of $\Phi$ behaves at short distance, and this behaviour can be posed as a condition on the lattice states. Let $G_k(z)=\omega_k(\Phi_0\Phi_z)- c_k^2$ for $z\in\Z^4$, as in the proof of Theorem~\ref{thm:limitErot}(iv), so that $\omega_k((\Phi_x-c_k)(\Phi_y-c_k))=G_k(y-x)$; let $\log^+t=\max(\log t,0)$ and, for $\gamma\ge0$ and $\rho>0$, \[ \ell(\rho)=1+\log^+\frac1\rho,\qquad w_\gamma(\rho)=\rho^{-8}\ell(\rho)^ {-\gamma} . \] \begin{enumerate}[label=\textup{(AF)},leftmargin=*] \item There are $\gamma\ge0$, $00$ such that every limit $S_2$ of $Z_k^2\omega_k(A_k(f)A_k(g))$ on the separated pairs along a subsequence is $S_2(f,g)=\iint f(x)g(y)K(x-y)\,dx\,dy$ with $K$ locally integrable on $\R^4\setminus\{0\}$ and \[ \lim_{\rho\to0}\ \operatorname*{ess\,sup}_{0<|z|\le\rho}\Bigl||z|^8\Bigl( \log\frac1{|z|}\Bigr)^\gamma K(z)-C_{\mathrm{AF}}\Bigr|=0 . \] \end{enumerate} \begin{enumerate}[label=\textup{(AF$_\beta$)},leftmargin=*] \item For some $\Lambda>0$, with $L_k=\log(1/(a_k\Lambda))$, the couplings are $\beta_{W,k}=2N/g_k^2$ with $g_k^{-2}=2b_0L_k+(b_1/b_0)\log L_k+O(1)$. \end{enumerate} (AF) is the two-sided form, at every lattice distance from $2r+1$ to $r_0/a_k$, of the law $|x-y|^{-8}(\log\frac1{|x-y|})^{-\gamma}$, with only a power bound at physical distances; (AF$_\infty$) is the exact law of the limit at short distance; (AF$_\beta$) is the two-loop tuning of Remark~\ref{rem:twoloop}. None of the three is proved here for the Wilson measure. \begin{remark}\label{rem:limitEafpert} \emph{The perturbative values.} Renormalised perturbation theory gives the following values; they are predictions and no proof below uses them. Let $F^a_{\mu\nu}$ be the field strength normalised by the action $\frac14\int F^a_{\mu\nu}F^a_{\mu\nu}$, with generators $\Tr(T^aT^b)=\frac12\delta^{ab}$, and let $\beta(g)=-b_0g^3+O(g^5)$, $b_0=11N/(48\pi^2)$~\cite{GW73,Pol73}. The trace of the energy--momentum tensor, $\Theta=(\beta(g)/2g)F^a_{\mu\nu} F^a_{\mu\nu}$, is finite and invariant under the renormalisation group~\cite{CDJ77}, so $\ip{\Theta(x)}{\Theta(0)}$, the connected two-point function, is $|x|^{-8}F(\bar g(|x|))$ with $\bar g(|x|)$ the running coupling at the scale $1/|x|$, and $F(g)=\frac14b_0^2g^4\,T_0(1+O(g^2))$, where $T_0|x|^{-8}$ is the connected two-point function of $F^a_{\mu\nu} F^a_{\mu\nu}$ in the free theory. There, in the Feynman gauge, the propagator of the gauge field is $\delta^{ab}\delta_{\mu\nu}D(x)$ with $D(x)=1/(4\pi^2|x|^2)$, and that of the field strength, which is gauge independent, is $\ip{F^a_{\mu\nu}(x)}{F^b_{\rho\sigma}(0)}=-\delta^{ab} P_{\mu\nu\rho\sigma}$ with \[ P_{\mu\nu\rho\sigma}=\delta_{\nu\sigma}D_{\mu\rho}-\delta_{\nu\rho} D_{\mu\sigma}-\delta_{\mu\sigma}D_{\nu\rho}+\delta_{\mu\rho}D_{\nu\sigma}, \] $D_{\mu\rho}=\partial_\mu\partial_\rho D=M_{\mu\rho}/(2\pi^2|x|^4)$, where $M=4uu^{\mathsf T}-1$, $u=x/|x|$. Since $\operatorname{tr}M=0$ and $\operatorname{tr}M^2=12$, each of the four squared terms of the bracket, summed over $\mu,\nu,\rho,\sigma$, gives $4\cdot12$, the two cross terms with a trace vanish and the other four give $-12$ each, so the sum of the squares is $(192-96)(2\pi^2|x|^4)^{-2}= 24\pi^{-4}|x|^{-8}$, and Wick's theorem gives $T_0=2(N^2-1)\cdot24\pi^{-4}= 48(N^2-1)/\pi^4$. With $\bar g(|x|)^{-2}=2b_0\log\frac1{|x|\Lambda}(1+o(1))$ this gives, as $|x|\to0$, \[ \ip{\Theta(x)}{\Theta(0)}=\frac{3(N^2-1)}{\pi^4}\,|x|^{-8}\Bigl(\log\frac1 {|x|}\Bigr)^{-2}(1+o(1)), \] which is (AF$_\infty$) with $\gamma=2$ for every $N$ and with $C_{\mathrm{AF}}=3(N^2-1)/\pi^4$, a constant free of $b_0$ and $\Lambda$.\footnote{For $\SU(2)$, $C_{\mathrm{AF}}=9/\pi^4\approx0.092$; for $\SU(3)$, $C_{\mathrm{AF}}=24/\pi^4\approx0.246$.} For the sum $\Phi$ of $N-\ReTr U_p$ over the $24$ plaquettes with a corner at $0$, $U_p=\exp(ia_k^2g_kF^a_{\mu\nu}T^a+\cdots)$ gives $N-\ReTr U_p= \frac14a_k^4g_k^2F^a_{\mu\nu}F^a_{\mu\nu}+\cdots$ (no sum) for $p$ in the plane $\{\mu,\nu\}$, and each plane carries four of the $24$ plaquettes, so $\Phi_x=\frac12a_k^4g_k^2F^a_{\mu\nu}F^a_{\mu\nu}+\cdots=-(a_k^4/b_0) \Theta(1+O(g_k^2))$, up to a constant and fields of higher dimension. Thus $Z_ka_k^4\to b_0$ makes $Z_kA_k(f)$ tend to $-\Theta(f)$; then (AF) holds with $\gamma=2$ and $Z_k$ of order $a_k^{-4}$, and $G_k(z)$ at a fixed lattice distance is of order $g_k^4$, that is of order $\beta_{W,k}^{-2}$ under (AF$_\beta$). \end{remark} \begin{theorem}\label{thm:limitEaf} Assume the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}} and \textup{(AF)} for the constants $Z_k$. \textup{(i)} Let $Q$ be a cube of side $L$ and $\delta\in(0,1]$. For $k\ge k_0$ with $a_k\le\min(1,\delta/(2r+1))$ and for $(f,g)\in\mathcal C_2(Q,\delta)$, \[ Z_k^2\bigl|\omega_k\bigl(A_k(f)A_k(g)\bigr)\bigr|\le100\,C_1(1+L)^4 \delta^{-4}\|f\|_\infty\|g\|_\infty . \] So \textup{(M$_{\mathrm s,2}$)} holds; every limit $S_2$ along a subsequence satisfies \[ |S_2(f,g)|\le C_1\iint|f(x)||g(y)|w_\gamma(|x-y|)\,dx\,dy \] on the separated pairs, and extends uniquely to a continuous linear functional on $\mathcal S_0(\R^8)$ with $|S_2(F)|\le C|F|_{33}$. \textup{(ii)} If $f,g\ge0$ form a separated pair and $|x-y|\le r_0$ for $x\in\operatorname{supp}f$, $y\in\operatorname{supp}g$, then $S_2(f,g)\ge c_1\iint f(x)g(y)w_\gamma(|x-y|)\,dx\,dy$. In particular no limit $S_2$ vanishes identically, and the moment bound \textup{(M)} of Sec.~\textup{\ref{ssec:osaxioms}}, posed for the packets $Z_kA_k(f)$, fails for $n=2$ and $Q=[0,1]^4$. \textup{(iii)} For every $\varepsilon>0$, $\mathcal N_k(\varepsilon)>0$ for large $k$ and \[ 0<\liminf_kZ_k^2\mathcal N_k(\varepsilon)\le\limsup_kZ_k^2\mathcal N_k(\varepsilon)<\infty . \] \textup{(iv)} \textup{(SD$_{\mathrm p}$)} holds with $p_0=4$, hence \textup{(SD)} and \textup{(UV)}, and \textup{(NE)} holds. Every limit $S_2$ of Corollary~\textup{\ref{cor:limitEtwo}(b)} is not identically zero, extends to $\mathcal S_0(\R^8)$ as in Proposition~\textup{\ref{prop:limitEtwotempered}(b)}, and its Hamiltonian $H^{(1)}$ has a nonempty spectrum in $[m_*,\infty)$. \textup{(v)} If the transfer matrices $\mathbb T_k$ have spectrum on $\Omega_k^\perp$ in $[0,e^{-m_{\mathrm{lat}}}]$ for a fixed $m_{\mathrm{lat}} >0$ and all $k$, then \textup{(AF)} fails for every choice of $Z_k$; this is the case under the hypotheses of Corollary~\textup{\ref{cor:limitEstrong}}. \end{theorem} \begin{proof} (i) By Lemma~\ref{lem:limitElattice}(i) and translation invariance, \[ \omega_k(A_k(f)A_k(g))=\sum_{x,y}a_k^8f(a_kx)g(a_ky)G_k(y-x). \] A nonzero term has $|a_kx-a_ky|_\infty\ge\delta$, so $|y-x|_\infty\ge\delta/a_k\ge 2r+1$, and (AF) with $\ell\ge1$ bounds $Z_k^2|G_k(y-x)|$ by $C_1(a_k|y-x|) ^{-8}\le C_1(a_k|y-x|_\infty)^{-8}$. There are $(2j+1)^4-(2j-1)^4=64j^3+16j \le80j^3$ points $v\in\Z^4$ with $|v|_\infty=j\ge1$, so with $J=\lceil \delta/a_k\rceil$ \[ \begin{aligned} \sum_{|v|_\infty\ge J}a_k^4(a_k|v|_\infty)^{-8}&\le80a_k^{-4}\sum_{j\ge J} j^{-5}\\ &\le80a_k^{-4}\Bigl(J^{-5}+\frac{J^{-4}}4\Bigr)\le100\,\delta^{-4}, \end{aligned} \] while at most $(L/a_k+1)^4$ sites $x$ have $a_kx\in Q$, and $a_k^4(L/a_k+1) ^4\le(1+L)^4$. This gives the bound, and (M$_{\mathrm s,2}$) follows since $\mathcal C_2(Q,\delta)\subset\mathcal C_2(Q,1)$ for $\delta\ge1$. In the same sum $Z_k^2|G_k(y-x)|\le C_1w_\gamma(a_k|y-x|)$, and $(x,y)\mapsto|f(x)| |g(y)|w_\gamma(|x-y|)$ is continuous with compact support in $\R^8$, since $w_\gamma$ is continuous on $(0,\infty)$; its Riemann sums converge to its integral, which gives the bound on $S_2$. The extension is Lemma~\ref{lem:limitEextend} with $n=2$, $C=100C_1$, $q'=p'=4$ and $w=4+4+24+1=33$. (ii) For large $k$ every nonzero term of the sum has $|y-x|_\infty\ge2r+1$ and $a_k|y-x|\le r_0$, so the lower bound of (AF) gives $Z_k^2\omega_k(A_k( f)A_k(g))\ge c_1\sum_{x,y}a_k^8f(a_kx)g(a_ky)w_\gamma(a_k|y-x|)$, and the Riemann sums converge as in (i). The integral is positive when $f$ and $g$ are not zero. For (M), let $s_0=r_0/2$ and $0<\delta\le r_0/16$; let $f$ be continuous with values in $[0,1]$, equal to $1$ on $[\frac12-2\delta, \frac12-\delta]\times[\frac{s_0}4,\frac{3s_0}4]^3$ and supported in $[\frac12-3\delta,\frac12-\frac\delta2]\times[0,s_0]^3$, and $g$ continuous with values in $[0,1]$, equal to $1$ on $[\frac12+\delta,\frac12+2\delta] \times[\frac{s_0}8,\frac{7s_0}8]^3$ and supported in $[\frac12+\frac\delta2, \frac12+3\delta]\times[0,s_0]^3$. The supports lie in $[0,1]^4$, at distance $\delta$, and all their points are within $(36\delta^2+3s_0^2)^{1/2}\le r_0$ of each other. The pairs with $x$ in the first box, $x_0+\delta\le y_0\le x_0+4\delta$ with $y_0\in[\frac12+\delta,\frac12+2\delta]$, and $|y_i-x_i| \le\delta$ for $i=1,2,3$ have measure $\frac{\delta s_0^3}8\cdot\delta\cdot 8\delta^3=s_0^3\delta^5$, $f(x)g(y)=1$ and $2\delta\le|x-y|\le\sqrt{19}\, \delta<5\delta$, where $w_\gamma\ge(5\delta)^{-8}\ell(2\delta)^{-\gamma}$. Hence $S_2(f,g)\ge c_15^{-8}s_0^3\delta^{-3}(1+\log\frac1{2\delta})^{-\gamma}$, which is unbounded as $\delta\to0$, while $\|f\|_\infty\|g\|_\infty=1$; under (M) the limit would be bounded by $M_2(Q)$, as in the proof of Proposition~\ref{prop:limitEsing}. (iii) Let $Q_0$ be the cube of side $L_0=2(\varepsilon+2+2a_0)$ of the proof of Proposition~\ref{prop:limitEtwotempered}(a), which contains $B_{\varepsilon'}$ and $\theta_{0,k}(B_{\varepsilon'})$ for all $k$ and $\varepsilon'\le\varepsilon$. For $h$ supported in $B_{\varepsilon'}$ with $\|h\|_\infty\le1$ and $a_k\le\varepsilon'$, $\omega_k(\vartheta(\overline{ A_k(h)})A_k(h))=\omega_k(A_k(\bar h\circ\theta_{0,k})A_k(h))$, and the pair lies in $\mathcal C_2(Q_0,\varepsilon')$; so by (i) \[ Z_k^2\mathcal N_k(\varepsilon')\le100\,C_1(1+L_0)^4\max(1,\varepsilon'^{-4}) \] for large $k$, which with $\varepsilon'=\varepsilon$ is the upper bound. Suppose that $Z_k^2\mathcal N_k(\varepsilon)\to0$ along a subsequence. By (M$_{\mathrm s,2}$) and the proof of Theorem~\ref{thm:limitEsep}(i) a further subsequence has a limit $S_2$ on all separated pairs. Let $g$ be supported in a cube of side $L$ in $\{x_0\ge\eta\}$, $\eta>0$, and $t\ge4\varepsilon+1$ dyadic. For large $k$, $\bar g\circ\theta_{0,k}$ is supported in $\{x_0\le -\eta+a_k\}$ and $g(\cdot-te_0)$ in $\{x_0\ge\eta+t\}$, so with $c=2 \varepsilon-\eta+a_k$ and $d=0$ the inequality of Theorem~\ref{thm:limitEtwo}(i), multiplied by $\mathcal N_k(\varepsilon)$, whose proof does not use $\mathcal N_k(\varepsilon)>0$, gives \[ Z_k^2\bigl|\omega_k\bigl(A_k(\bar g\circ\theta_{0,k})A_k(g(\cdot-te_0)) \bigr)\bigr|\le\mathsf K_L^2Z_k^2\mathcal N_k(\varepsilon)\|g\|_\infty^2\to0 . \] By the approximation of $\theta_{0,k}$ by $\theta_0$ in the proof of Theorem~\ref{thm:limitEsep}(iii), $S_2(\bar g\circ\theta_0,g(\cdot-te_0))=0$. The construction in the proof of Corollary~\ref{cor:limitEtwo}(b), which uses only (M$_{\mathrm s,2}$), gives a Hilbert space $\mathcal H^{(1)}$ and a self-adjoint $H^{(1)}\ge0$ with $S_2(\bar g\circ\theta_0,g(\cdot-te_0))= \|e^{-tH^{(1)}/2}w\|^2$, $w$ the vector of $g$. Since $e^{-tH^{(1)}/2}$ is injective, $w=0$, that is $S_2(\bar g\circ\theta_0,g)=0$, for every $g$ supported in $\{x_0>0\}$, and by the proof of Corollary~\ref{cor:limitEtwo}(b) $S_2$ vanishes identically. This contradicts (ii). So $\liminf_kZ_k^2 \mathcal N_k(\varepsilon)>0$, and in particular $\mathcal N_k(\varepsilon) >0$ for large $k$. (iv) Let $c_\varepsilon=\frac12\liminf_kZ_k^2\mathcal N_k(\varepsilon)$ and $C_2=100\,C_1(1+L_0)^4$. By (iii), for $\varepsilon'\in(0,\varepsilon]$ and large $k$, \[ \begin{aligned} \mathcal N_k(\varepsilon')&\le\frac{C_2\max(1,\varepsilon'^{-4})} {c_\varepsilon}\,\mathcal N_k(\varepsilon)\\ &\le C_0\Bigl(\frac\varepsilon{\varepsilon'}\Bigr)^4\mathcal N_k(\varepsilon), \end{aligned} \] where $C_0=C_2\max(1,\varepsilon^{-4})/c_\varepsilon$; the second inequality holds because $\max(1,\varepsilon'^{-4})$ is at most $\max(1,\varepsilon^{-4})(\varepsilon/\varepsilon')^4$. This is (SD$_{\mathrm p}$) with $p_0=4$; it gives (SD), and (UV) by Proposition~\ref{prop:limitEsd}(ii). Let $S^\sharp_2$ be a limit of Corollary~\ref{cor:limitEtwo}(b), formed with $\mathcal N_k(\varepsilon) ^{-1}$ in place of $Z_k^2$. By (i) and (iii) a further subsequence has $Z_k^2\mathcal N_k(\varepsilon)\to\lambda\in(0,\infty)$ and a limit $S_2$ with the constants $Z_k$, and then $S^\sharp_2=S_2/\lambda$, which is not identically zero by (ii). By Proposition~\ref{prop:limitEne}(i) this is (NE), and Corollary~\ref{cor:limitEtwo}(b) and Proposition~\ref{prop:limitEtwotempered}(b) give the rest. (v) Under (AF), (iii) and (iv) give $\mathcal N_k(\varepsilon)>0$ for large $k$ and (SD), which Proposition~\ref{prop:limitEsd}(iii) excludes under a lattice gap; Corollary~\ref{cor:limitEstrong} provides the gap. \end{proof} So the short-distance law, in the two-sided form (AF), gives the two inputs (SD) and (NE) of Sec.~\ref{ssec:limitE} for $n=2$, with the power rate (SD$_{\mathrm p}$) that temperedness needs, and it excludes the trivial limits of the strong-coupling range. It says nothing about $n\ge3$: the bounds (RN$_m$) with $m\ge2$ of Proposition~\ref{prop:limitEreflect} are not consequences of it. It also fixes the constants $Z_k$ up to bounded factors. \begin{proposition}\label{prop:limitEafscale} Assume \textup{(AF)} for the constants $Z_k$, and let $z_0\in\Z^4$ with $|z_0|_\infty\ge2r+1$. \textup{(i)} For large $k$, $G_k(z_0)>0$ and \[ c_1|z_0|^{-8}\le Z_k^2a_k^8\,\ell(a_k|z_0|)^\gamma\,G_k(z_0)\le C_1|z_0|^{-8} . \] \textup{(ii)} $Z_k\ge c_1^{1/2}|z_0|^{-4}\|\Phi\|_\infty^{-1}a_k^{-4}\ell(a_k |z_0|)^{-\gamma/2}$ for large $k$; in particular $Z_k\to\infty$. \textup{(iii)} For large $k$ and every $z\in\Z^4$ with $|z|_\infty\ge2r+1$ and $a_k|z|\le r_0$, \[ \frac{c_1}{C_1}\,\frac{w_\gamma(a_k|z|)}{w_\gamma(a_k|z_0|)}\le\frac{G_k(z)} {G_k(z_0)}\le\frac{C_1}{c_1}\,\frac{w_\gamma(a_k|z|)}{w_\gamma(a_k|z_0|)} . \] \textup{(iv)} Under the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}}, if \textup{(AF)} also holds for constants $Z'_k$ with an exponent $\gamma'$, then $\gamma'=\gamma$ and $Z'_k/Z_k$ is bounded above and below by positive constants; both are comparable to $\mathcal N_k(\varepsilon)^{-1/2}$ for every $\varepsilon>0$. \textup{(v)} Under \textup{(AF$_\beta$)}, $Z_ka_k^4$ is bounded above and below by positive constants if and only if $\beta_{W,k}^\gamma G_k(z_0)$ is. \end{proposition} \begin{proof} (i) is (AF) at $z=z_0$, valid for large $k$ because $a_k|z_0|\to0$. (ii) follows from (i), since $|G_k(z_0)|=|\Cov(\Phi_0,\Phi_{z_0})|\le\Var(\Phi_0) \le\|\Phi\|_\infty^2$ by the Cauchy--Schwarz inequality and translation invariance, and $\ell(a_k|z_0|)^\gamma a_k^8\to0$. (iii) is the quotient of the two bounds of (AF) at $z$ and $z_0$. For (iv), Theorem~\ref{thm:limitEaf}(iii) applied to $Z_k$ and to $Z'_k$ makes $Z_k^2\mathcal N_k(\varepsilon)$ and ${Z'_k}^2\mathcal N_k(\varepsilon)$ bounded above and below, hence $Z'_k/Z_k$; and (i) for both gives $\ell(a_k|z_0|)^{\gamma-\gamma'}$ bounded above and below, which forces $\gamma'=\gamma$ since $\ell(a_k|z_0|)\to \infty$. For (v), (AF$_\beta$) gives $\beta_{W,k}/\log(1/a_k)\to4Nb_0>0$, and $\ell(a_k|z_0|)/\log(1/a_k)\to1$, so by (i) $Z_k^2a_k^8$ is bounded above and below exactly when $\beta_{W,k}^\gamma G_k(z_0)$ is. \end{proof} By Remark~\ref{rem:limitEafpert}, perturbation theory predicts $\gamma=2$ and $G_k(z_0)$ of order $\beta_{W,k}^{-2}$, hence $Z_k$ of order $a_k^{-4}$, and for the plaquette sum $Z_ka_k^4\to b_0$ in the normalisation of $\Theta$. Part (iii) is the content of (AF) without reference to $Z_k$: the connected correlation of $\Phi$ decays from the lattice distance $|z_0|$ to every distance up to $r_0/a_k$ like the eighth power of the distance, with the logarithmic correction, within fixed factors. Its lower half is a lower bound on correlations at physical separations, the kind of input that upper bounds do not supply (compare Proposition~\ref{prop:upperbounds}). The two-sided law is consistent with the other properties of Sec.~\ref{ssec:limitE} and implies neither (W$_{\mathrm s}$) nor the exact law (AF$_\infty$). \begin{proposition}\label{prop:limitEafexample} Let $\gamma\ge0$, $\mu_j=2^{4(j-1)}(1+j)^{-\gamma/2}$ for $j\ge0$, let $a_k\le1$ and $J_k=\lfloor\log_2(2/a_k)\rfloor$, let $\phi^{(j)}_k$, $0\le j\le J_k$, be independent copies of the field of Proposition~\textup{\ref{prop:osobstruction}(a)} with $q_k$ replaced by $q_{k,j}=e^{-2^{j-1}a_k}$, and put $\phi_k=\sum_{j\le J_k}\mu_j\phi^{(j)}_k$, $A_k(f)=\sum_xa_k^4f(a_kx)\phi_k(x)$, $r=0$ and $Z_k=1$. \textup{(a)} The law of $\phi_k$ is invariant under translations and $B_4$ and reflection positive for the reflections in coordinate hyperplanes through sites and bisecting links, and the spectrum of its transfer matrix in each coordinate direction on $\Omega_k^\perp$ lies in $[0,e^{-a_k/2}]$ and contains $e^{-a_k/2}$: the gap is $\frac12$ in physical units. \textup{(b)} \textup{(AF)} holds with this $\gamma$, $r_0=\frac12$, $c_1=e^{-1}2^{-16-\gamma}$ and $C_1=\max(2^\gamma(A_\gamma+B),B')$, where $A_\gamma=\sum_{i\ge0}2^{-8i}(1+i)^\gamma$, $B=\sum_{i\ge1}2^{8i}e^{-2^{i-1}}$ and $B'=\sum_{j\ge0}\sup_{t\ge2^j}t^8e^{-t}$. \textup{(c)} The moments converge along the whole sequence to the Gaussian family with $S_2(f,g)=\iint f(x)g(y)K(x-y)\,dx\,dy$, $K(v)=h(\|v\|_1)$, $h(u)=\sum_{j\ge0}\mu_j^2e^{-2^{j-1}u}$. \textup{(G$_{\mathrm s}$)} holds with $\beta=\frac12$ and $p=q=2$, and \textup{(U$_{\mathrm s}$)} holds; the conclusions of Theorem~\textup{\ref{thm:limitEaf}}(i)--(iv) hold with $m_*=\frac12$. The group $G_S$ is $B_4$, so \textup{(W$_{\mathrm s}$)} fails, and \textup{(AF$_\infty$)} fails. \end{proposition} \begin{proof} (a) As in the proof of Proposition~\ref{prop:limitEheavy}(a), with $J_k+1$ independent fields in place of two: the tuple is a reversible Markov chain in each coordinate direction with transfer operator $\bigotimes_jP_j$, of spectrum $\{\prod_jq_{k,j}^{N_j}\}$ with $1$ simple, the functions of $\phi_k$ at positive times are functions of the tuple there, and the vector of $\phi_k(x)$ at positive time has a nonzero component of eigenvalue $q_{k,0}=e^{-a_k/2}$, the largest $q_{k,j}$. (b) The covariance is $G_k(z)=\sum_{j\le J_k}\mu_j^2e^{-2^{j-1}u}$ with $u=a_k\|z\|_1$; for $z\ne0$ let $\rho=a_k|z|$, so that $\rho\le u\le2\rho$ and $u\ge a_k$. If $u\le2$, let $j_*=\lfloor\log_2(2/u)\rfloor\ge0$, so that $\frac12<2^{j_*-1}u\le1$. The terms with $j=j_*-i$, $i\ge0$, are at most $\mu_{j_*}^22^{-8i}(1+i)^\gamma$, because $(1+j_*)/(1+j)\le1+i$; those with $j=j_*+i$, $i\ge1$, at most $\mu_{j_*}^22^{8i}e^{-2^{i-1}}$, because $2^{j-1}u>2^{i-1}$. So $G_k(z)\le(A_\gamma+B)(2^{j_*-1})^8(1+j_*)^{-\gamma} \le(A_\gamma+B)\rho^{-8}(1+j_*)^{-\gamma}$. Here $1+j_*\ge\max(1,\log_2(2/u)) \ge\max(1,\log_2(1/\rho))$, which is at least $\frac12\ell(\rho)$, so $G_k(z)\le2^\gamma(A_\gamma+B)w_\gamma(\rho)$. If $u>2$, then $\rho>1$, $2^{j-1}u\ge2^j$ and $\mu_j^2\le2^{8(j-1)}$, so $G_k(z)\le u^{-8}\sum_j (2^{j-1}u)^8e^{-2^{j-1}u}\le B'u^{-8}\le B'w_\gamma(\rho)$. For the lower bound let $\rho\le\frac12$, so $u\le1$; then $j_*\le J_k$ because $u\ge a_k$, and the term $j_*$ alone is at least $(2u)^{-8}(1+j_*)^{-\gamma}e^{-1}$. Since $(2u)^{-8}\ge(4\rho)^{-8}$ and $1+j_*\le1+\log_2(2/\rho)=2+\log(1/\rho)/ \log2\le2\ell(\rho)$, because $1/\log2<1.5$, this is at least $e^{-1} 2^{-16-\gamma}w_\gamma(\rho)$. The finiteness of $A_\gamma$, $B$, $B'$ follows from the ratio test, since $t^8e^{-t}$ decreases for $t\ge8$. (c) $\omega_k(A_k(f)A_k(g))=\sum_{x,y}a_k^8f(a_kx)g(a_ky)h_k(a_k\|y-x\|_1)$ with $h_k$ the partial sum of $h$ up to $J_k$. On a separated pair at distance $\delta$ only $u\ge\delta$ occurs, where $0\le h-h_k\le\sum_{j>J_k} \mu_j^2e^{-2^{j-1}\delta}\to0$, and $h$ is continuous on $(0,\infty)$; so the Riemann sums converge to $\iint f(x)g(y)K(x-y)\,dx\,dy$. The fields are centred Gaussian, so the moments of separated tuples are sums over the $(n-1)!!$ pairings of products of these two-point functions, and they converge as well; this gives (U$_{\mathrm s}$). Theorem~\ref{thm:limitEaf}(i) and $(n-1)!!\le(n!)^{1/2}$ bound the moments of tuples in $\mathcal C_n(Q,\delta)$, $\delta\le1$, by $(100C_1)^{n/2}(n!)^{1/2}(1+L)^{2n} \delta^{-2n}\prod_j\|f_j\|_\infty$ for large $k$, which is (G$_{\mathrm s}$). The proofs of Theorem~\ref{thm:limitEaf}(i)--(iv) use the states only through their invariance, reflection positivity and gap, as in Remark~\ref{rem:limitEscope}. $K$ is continuous on $\R^4\setminus\{0\}$ and $h$ is strictly decreasing, so by Proposition~\ref{prop:limitEaniso}(i) an $R\in\mathrm O(4)$ lies in $G_S$ only if $\|Rv\|_1=\|v\|_1$ for all $v$; then $R$ permutes the vertices $\pm e_\mu$ of the unit ball of $\|\cdot\|_1$, and $R\in B_4$. Finally, choose $i_1$ with $((1+i)/(2+i))^\gamma\ge\frac12$ for $i\ge i_1$. Since $\mu_{i+1}^2=2^8((1+i)/(2+i))^\gamma\mu_i^2$, \[ h(\rho)\ge\sum_{i\ge i_1}\mu_{i+1}^2e^{-2^i\rho}\ge2^7\Bigl(h(2\rho)- \sum_{i0$, $00$: for large $k$ and all $z\in\Z^4$ with $|z|_\infty\ge2r+1$, where $G_k(z)=\omega_k(\Phi_0\Phi_z)-\omega_k( \Phi_0)^2$, \[ Z_k^2|G_k(z)|\le C_1w_2(a_k|z|), \] and $Z_k^2G_k(z)\ge c_1w_2(a_k|z|)$ if moreover $a_k|z|\le r_0$; equivalently, by Proposition~\ref{prop:limitEafscale}(iii), the $Z_k$-free two-sided comparison of $G_k(z)$ with $G_k(z_0)\,w_2(a_k|z|)/ w_2(a_k|z_0|)$ for $2r+1\le|z|_\infty$, $a_k|z|\le r_0$, together with the upper bound for $a_k|z|>r_0$ with $Z_k^2=w_2(a_k|z_0|)/G_k(z_0)$. It is open, both halves, already in a box of fixed physical size. Given it, Theorem~\ref{thm:limitEaf} supplies (SD$_{\mathrm p}$) and (NE) for $n=2$; (RN$_m$) for $m\ge2$, (W$_{\mathrm s}$), (U$_{\mathrm s}$) and the averaged condition along the tuning stay open. \end{remark} \emph{Products and the two-point properties.} After Proposition~\ref{prop:limitEreflect} the separated moments of order $n\ge3$ rest on (RN$_m$), $m\ge2$, and the rate (G$_{\mathrm s}$) on a rate in these bounds. Functions of two independent copies of the field of Proposition~\ref{prop:osobstruction}(a) show what the two-point properties do not give. Write $\gamma_2$ for the standard Gaussian measure on $\R^2$, $\|\cdot\|_p$ for the norm of $L^p(\gamma_2)$, and $T_M(u)=\max(-M,\min(M,u))$ for $M>0$. \begin{lemma}\label{lem:limitEtrunc} Let $m>0$, $q_k=e^{-ma_k}$, let $\phi^{(1)}_k,\phi^{(2)}_k$ be independent copies of the field of Proposition~\textup{\ref{prop:osobstruction}(a)}, $\xi_x=(\phi^{(1)}_k(x),\phi^{(2)}_k(x))$, let $U\colon\R^2\to\R$ be continuous, in $L^p(\gamma_2)$ for every $p<\infty$, with $\int U\,d\gamma_2=0$, and put $\Phi_x=U(\xi_x)$ and $A_k(f)=\sum_xa_k^4f(a_kx) \Phi_x$, with $r=0$. \textup{(i)} The law of $\Phi$ is invariant under translations and $B_4$ and reflection positive for the reflections in coordinate hyperplanes through sites and bisecting links, and in each coordinate direction the spectrum of its transfer matrix on $\Omega_k^\perp$ lies in $[0,q_k]$. It contains $q_k$ if $\int U(u)u_i\,d\gamma_2(u)\ne0$ for $i=1$ or $i=2$. \textup{(ii)} Let $U_M=T_M\circ U-\int T_M\circ U\,d\gamma_2$, so that $|U_M|\le2M$. For $n\ge1$ and random vectors $\zeta_1,\dots,\zeta_n$ in $\R^2$, each of law $\gamma_2$, \begin{gather*} \Bigl|\E\prod_iU(\zeta_i)\Bigr|\le\|U\|_n^n,\\ \Bigl|\E\prod_iU_M(\zeta_i)-\E\prod_iU(\zeta_i)\Bigr|\le e_n(M), \end{gather*} with $e_n(M)=2n(2\|U\|_n)^{n-1}\|U\mathbf 1_{\{|U|>M\}}\|_n\to0$ as $M\to\infty$. Hence, if $f_j$ is supported in a cube of side $L_j$, the moments $\omega_k(\prod_jA_k(f_j))$ formed with $U_M$ and with $U$ differ by at most $e_n(M)\prod_j(L_j+a_k)^4\|f_j\|_\infty$. \end{lemma} \begin{proof} (i) As in the proof of Proposition~\ref{prop:limitEheavy}(a), the pair $(\phi^{(1)}_k,\phi^{(2)}_k)$ is invariant, reflection positive, and in each coordinate direction a reversible Markov chain with transfer operator $P\otimes P$, of spectrum $\{q_k^{N+N'}\}$ with $1$ simple; the functions of $\Phi$ at positive times are functions of the pair there, so $\Phi$ has the stated invariance and positivity, and its transfer matrix is the restriction of $P\otimes P$ to an invariant subspace. For a site $x$ at time $s\ge1$, $g_{\Phi_x}$ is $(P\otimes P)^s$ applied to $U(\xi_x)$, viewed as a function of the field at one time, and $P\otimes P$ acts as $q_k$ on the linear functions of that field. By Gaussian integration by parts the orthogonal projection of $U(\xi_x)$ on these linear functions has inner product $\int U(u)u_i\,d\gamma_2(u)$ with $\phi^{(i)}_k(x)$, so it is not zero, and $g_{\Phi_x}$ has a nonzero component of eigenvalue $q_k$. (ii) The first bound is H\"older's inequality with $n$ exponents $n$. Since $|T_Mu|\le|u|$, $|T_Mu-u|\le|u|\mathbf 1_{\{|u|>M\}}$ and $\int U\,d\gamma_2=0$, \[ \|U_M\|_n\le2\|U\|_n,\qquad\|U_M-U\|_n\le2\|U\mathbf 1_{\{|U|>M\}}\|_n . \] The difference of the two products is the sum over $j$ of the products $\prod_{ij}U(\zeta_i)$, and H\"older's inequality for each of them gives the second bound. By dominated convergence $e_n(M)\to0$, as $|U|^n$ is integrable. The moments are $\sum_{x_1,\dots,x_n}a_k^{4n}\prod_jf_j(a_kx_j)\E\prod_jU(\xi_{x_j})$, each $\xi_{x_j}$ has law $\gamma_2$, and $\sum_xa_k^4|f_j(a_kx)|\le(L_j+ a_k)^4\|f_j\|_\infty$. \end{proof} The moments of exponentials are explicit. For a centred Gaussian vector $(X_1,\dots,X_n)$ with unit variances and covariances $\rho_{ii'}$, $t>0$, and $u_{ii'}=e^{t^2\rho_{ii'}}-1$, \begin{equation}\label{eq:limitEcover} \E\prod_{i=1}^n\bigl(e^{tX_i-t^2/2}-1\bigr)=\sum_G\prod_{\{i,i'\}\in G} u_{ii'} , \end{equation} the sum over the sets $G$ of pairs whose union is $\{1,\dots,n\}$. Indeed $\E\prod_{i\in S}e^{tX_i-t^2/2}=\prod_{\{i,i'\}\subset S}(1+u_{ii'})$ for $S\subset\{1,\dots,n\}$; expanding the product of the factors $e^{tX_i-t^2/2}-1$ and these products, the coefficient of $\prod_{\{i,i'\}\in G}u_{ii'}$ is the sum of $(-1)^{n-|S|}$ over the $S$ that contain every pair of $G$, which is $1$ if $G$ covers $\{1,\dots,n\}$ and $0$ otherwise. When all $\rho_{ii'}\ge0$, every term is nonnegative and nondecreasing in each $\rho_{ii'}$. \begin{proposition}\label{prop:limitEthree} In the setting of Lemma~\textup{\ref{lem:limitEtrunc}} let $t_k>0$ with $t_k\to\infty$, $\lambda_k=e^{-t_k^2/2}$ and \[ U_k(u)=u_1+\lambda_k\bigl(e^{t_ku_2-t_k^2/2}-1\bigr), \] let $\varepsilon>0$, form $\mathcal N_k(\varepsilon)$ with $\Phi_x=U_k(\xi_x)$, and put $Z_k=\mathcal N_k(\varepsilon)^{-1/2}$ and $\ell_0=(\frac38)^8 e^{-m(2\varepsilon+6)}$. \textup{(a)} The spectrum of the transfer matrix on $\Omega_k^\perp$ lies in $[0,q_k]$ and contains $q_k$: the gap is $m$ in physical units. For large $k$, $\ell_0\le\mathcal N_k(\varepsilon)\le2(1+2a_0)^8$. \textup{(SD$_{\mathrm p}$)} holds with $p_0=0$; hence \textup{(SD)}, \textup{(UV)}, \textup{(RN$_1$)} and \textup{(M$_{\mathrm s,2}$)} hold. \textup{(NE)} holds, and every limit $S_2$ of Corollary~\textup{\ref{cor:limitEtwo}(b)} is $S_2(f,g)=z\iint f(x)g(y)e^{-m\|x-y\|_1}dx\,dy$ with $z\in[\frac12(1+2a_0)^{-8}, \ell_0^{-1}]$. \textup{(b)} Let $n\ge3$, $\eta=1/(4m(3n+5))$, $c_j=3j\eta e_1$, and let $f_1,\dots,f_n$ be continuous with values in $[0,1]$, $f_j$ equal to $1$ on $c_j+[-\frac\eta2,\frac\eta2]^4$ and supported in $c_j+[-\eta,\eta]^4$. Then $(f_j)\in\mathcal C_n(Q,\eta)$ for a cube $Q$ of side $3n\eta$, and for large $k$ \[ Z_k^n\,\omega_k\Bigl(\prod_{j=1}^nA_k(f_j)\Bigr)\ge\frac{(\eta/2)^{4n} e^{nt_k^2/4}}{2^{n(n-1)/2}(2(1+2a_0)^8)^{n/2}}\to\infty . \] So \textup{(M$_{\mathrm s,n}$)} fails for every $n\ge3$. \textup{(c)} Let $j\ge2$, $\eta'=1/(4m(3j+10))$, and let $f_1,\dots,f_j$ be as in \textup{(b)}, with $\eta'$ in place of $\eta$ and centres $2\eta'e_0+3i\eta'e_1$, $i=1,\dots,j$. They are supported in $\{x_0\ge\eta'\}$, $(f_i)\in\mathcal C_j(Q',\eta')$ for a cube $Q'$ of side $3j\eta'$, and for large $k$, with $P_k=\prod_iA_k(f_i)$, \[ Z_k^{2j}\,\omega_k\bigl(\vartheta(\bar P_k)P_k\bigr)\ge\frac{(\eta'/2)^{8j} e^{jt_k^2/2}}{2^{j(2j-1)}(2(1+2a_0)^8)^j}\to\infty . \] So \textup{(RN$_j$)} fails for every $j\ge2$. \textup{(d)} There are $M_k\to\infty$ such that \textup{(a)--(c)} hold for the fields $\Phi_x=U_{k,M_k}(\xi_x)/(2M_k)$, with $U_{k,M}$ formed from $U_k$ as in Lemma~\textup{\ref{lem:limitEtrunc}(ii)}, which take values in $[-1,1]$, with $\frac12\ell_0$ and $4(1+2a_0)^8$ in place of $\ell_0$ and $2(1+2a_0)^8$ and the right sides of \textup{(b)}, \textup{(c)} halved. For these fields $Z_k\to\infty$. \end{proposition} \begin{proof} (a) The spectrum is given by Lemma~\ref{lem:limitEtrunc}(i), since $\int U_ku_1\,d\gamma_2=1$. Let $\rho(z)=\prod_\mu q_k^{|z_\mu|}=e^{-ma_k\|z\|_1}$ be the covariance of $\phi^{(i)}_k$ and $E_x=e^{t_k\phi^{(2)}_k(x)-t_k^2/2}$. By independence and centring, for $h$ supported in $B_{\varepsilon'}$ with $\|h\|_\infty\le1$ and $(r+1)a_k\le\varepsilon'$, \[ \omega_k\bigl(\vartheta(\overline{A_k(h)})A_k(h)\bigr)=\sum_{x,y}a_k^8 \overline{h(a_kx)}h(a_ky)\bigl(\rho+\lambda_k^2(e^{t_k^2\rho}-1)\bigr), \] with $\rho=\rho(y-\vartheta x)$, since $\E[(E_x-1)(E_y-1)]=e^{t_k^2\rho(x-y)}-1$ by \eqref{eq:limitEcover}. For $a_kx_0,a_ky_0\ge\varepsilon'$ the time component of $y-\vartheta x$ is at least $2\varepsilon'/a_k-1$, so $\rho\le e^{-m(2\varepsilon'-a_k)}$, and $\lambda_k^2e^{t_k^2\rho}\le e^{-t_k^2(1-e^{-m(2\varepsilon'-a_k)})}\to0$. As $\sum_xa_k^4|h(a_kx)|\le (1+2a_0)^4$ and $\rho\le1$, $\mathcal N_k(\varepsilon')\le2(1+2a_0)^8$ for large $k$. For the function $h_*\ge0$ of the proof of Proposition~\ref{prop:limitEheavy}(b), formed for $\varepsilon$ with $\tau=\frac14$, both parts are nonnegative, and that proof gives $\sum_{x,y}a_k^8h_*(a_kx)h_*(a_ky)\rho(y-\vartheta x)\ge\ell_0$; so $\mathcal N_k(\varepsilon)\ge\ell_0$, and the ratio in (NE) with $g_0=h_*$ is at least $\ell_0/(2(1+2a_0)^8)$. Hence $\mathcal N_k(\varepsilon')\le C_0\mathcal N_k(\varepsilon)$ with $C_0=2(1+2a_0)^8/\ell_0$ for every $\varepsilon'\in (0,\varepsilon]$ and large $k$, which is (SD$_{\mathrm p}$) with $p_0=0$. (UV) follows by Proposition~\ref{prop:limitEsd}(ii), (RN$_1$) by Proposition~\ref{prop:limitEreflect}(iii) and (M$_{\mathrm s,2}$) by Corollary~\ref{cor:limitEtwo}(a), whose proofs use the state only through the properties of Lemma~\ref{lem:limitEtrunc}(i). For $(f,g)\in\mathcal C_2(Q,\delta)$, $Q$ of side $L$, the sites carrying $f$ and $g$ have $\|a_k(x-y)\|_1\ge\delta$, so the part of $Z_k^2\omega_k(A_k(f)A_k(g))$ with $\lambda_k^2$ is at most $Z_k^2(L+1)^8e^{-t_k^2(1-e^{-m\delta})}\|f\|_\infty \|g\|_\infty\to0$, and the other part is $Z_k^2$ times a Riemann sum of the continuous function $f(x)g(y)e^{-m\|x-y\|_1}$; along a subsequence with $Z_k^2\to z$ the limit is the stated one. (b) Distinct $c_j$ differ by $3\eta$ along $e_1$, so the supports are at $\ell^\infty$ distance at least $\eta$; they lie in $[2\eta,(3n+1)\eta]\times[-\eta,\eta]^3$, of $\ell^1$ diameter at most $(3n-1)\eta+6\eta=1/(4m)$. Expanding $\prod_jU_k(\xi_{x_j})$ over the sets $I$ of indices that take the first summand, independence gives \[ \E\prod_jU_k(\xi_{x_j})=\sum_I\E\Bigl[\prod_{j\in I}\phi^{(1)}_k(x_j)\Bigr] \lambda_k^{n-|I|}\E\Bigl[\prod_{j\notin I}(E_{x_j}-1)\Bigr] . \] The first factor is a sum of products of the covariances $\rho>0$ by Wick's formula, and the second is nonnegative by \eqref{eq:limitEcover}; the term $I=\emptyset$ and the term $G$ of all pairs in \eqref{eq:limitEcover} give \[ \E\prod_jU_k(\xi_{x_j})\ge\lambda_k^n\prod_{i0$, and for it the moments in (b) and (c) still tend to infinity; this applies to $Z_k=C_k(\sigma_0)^{-1/2}$ of Proposition~\ref{prop:limitEnorm}(ii) for $f_0\ge0$, $f_0\ne0$. Part (d) takes into account that $\Phi$ is bounded for the Wilson measure: for bounded fields (M$_{\mathrm s}$) is automatic when $Z_k$ is bounded, and in (d) $Z_k\to\infty$. The rate (G$_{\mathrm s}$) is a rate in (RN). Consider \begin{enumerate}[label=\textup{(RN$_{\mathrm p}$)},leftmargin=*] \item there are $K_1,\beta_1,p_1,q_1\ge0$ such that for every $m\ge1$, every cube $Q$ of side $L$ and every $\delta\in(0,1]$ there is $k_0$ with \[ Z_k^{2m}\omega_k\bigl(\vartheta(\bar P_k)P_k\bigr)\le K_1^m(m!)^{\beta_1} (1+L)^{q_1m}\delta^{-p_1m}\prod_{j=1}^m\|f_j\|_\infty^2 \] for $P_k=\prod_{j=1}^mA_k(f_j)$, all $k\ge k_0$ and all $(f_1,\dots,f_m)\in \mathcal C_m(Q,\delta)$ with supports in $\{x_0\ge\delta\}$. \end{enumerate} \begin{proposition}\label{prop:limitErate} Assume the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}}. \textup{(i)} \textup{(G$_{\mathrm s}$)} implies \textup{(RN$_{\mathrm p}$)} with $K_1=4^\beta7^{2q}K^2$, $\beta_1=2\beta$, $q_1=2q$, $p_1=2p$. \textup{(ii)} \textup{(RN$_{\mathrm p}$)} implies \textup{(G$_{\mathrm s}$)} with $K=2^8e^4\,5^{q_1/2}(16e^2)^{p_1/2}K_1^{1/2}$, $\beta=4+\frac12(\beta_1+p_1)$, $q=4+\frac12q_1$, $p=4+\frac12p_1$. \textup{(iii)} With $Z_k=\mathcal N_k(\varepsilon)^{-1/2}$ as in Theorem~\textup{\ref{thm:limitEtwo}}, the case $m=1$ of the bound in \textup{(RN$_{\mathrm p}$)}, with its own constants, holds if and only if \textup{(SD$_{\mathrm p}$)} holds. So under \textup{(SD$_{\mathrm p}$)}, \textup{(G$_{\mathrm s}$)} holds if and only if the cases $m\ge2$ hold with constants of the form in \textup{(RN$_{\mathrm p}$)}. \end{proposition} \begin{proof} (i) Let $(f_j)\in\mathcal C_m(Q,\delta)$ have supports in $\{x_0\ge\delta\}$, $Q=\prod_\nu[q_\nu,q_\nu+L]$, let $s\ge\delta$ be the least time in the supports, $n_0=\lfloor(s-\delta)/a_k\rfloor$, $v_\nu=\lfloor q_\nu/a_k\rfloor$ for $\nu\ne0$, $v_\perp=(0,v_1,v_2,v_3)$, and $f'_j=f_j(\cdot+a_k(n_0e_0+ v_\perp))$, supported in $[\delta,L+2]\times[0,L+1]^3$ for $a_k\le1$. With $P'_k=\prod_jA_k(f'_j)$, $P_k=\tau_{n_0e_0}\tau_{v_\perp}P'_k$ by Lemma~\ref{lem:limitElattice}(i), and, as in the proof of Lemma~\ref{lem:limitEbox}, $\omega_k(\vartheta(\bar P_k)P_k)=\|\mathbb T_k^{n_0}\Psi_{\tau_{v_\perp}P'_k}\|^2\le\omega_k(\vartheta(\bar P'_k) P'_k)$. The last is the moment of the $2m$ functions $\bar f'_j\circ \theta_{0,k}$, $f'_j$, which lie in $Q''=[-L-3,L+3]^4$ and are pairwise at $\ell^\infty$ distance at least $\delta$ for $a_k\le\delta$, the reflected ones being supported in $\{x_0\le-\delta+a_k\}$. (G$_{\mathrm s}$) with $n=2m$ and $Q''$, of side $2L+6$, gives the bound with $((2m)!)^\beta\le 4^{\beta m}(m!)^{2\beta}$ and $7+2L\le7(1+L)$. (ii) The moments of order one vanish. For $n\ge2$, part (ii) of Proposition~\ref{prop:limitEreflect} gives (M$_{\mathrm s,n}$) with $M_n(Q,\delta)=\mathsf K_{2nL/\delta}^n\max_i(R_iR_{n-i})^{1/2}$, $R_i$ taken at $Q^*$, of side $2L+4$, and $\delta_n=\delta/(16(n+1))\le1$. By (RN$_{\mathrm p}$) and $i!(n-i)!\le n!$, \begin{multline*} (R_iR_{n-i})^{1/2}\le K_1^{n/2}(n!)^{\beta_1/2}\bigl(5(1+L)\bigr)^{q_1n/2}\\ \times\bigl(16(n+1)\bigr)^{p_1n/2}\delta^{-p_1n/2} . \end{multline*} From $e^n\ge n^n/n!$, $(n+1)^n\le e\,n^n\le e^{n+1}n!\le e^{2n}n!$. For $\delta\le1$, $4nL/\delta+3\le4n(1+L)/\delta$, so $\mathsf K_{2nL/\delta}^n \le4^{4n}n^{4n}(1+L)^{4n}\delta^{-4n}\le4^{4n}e^{4n}(n!)^4(1+L)^{4n} \delta^{-4n}$. Multiplying gives the constants stated. (iii) If (SD$_{\mathrm p}$) holds, Lemma~\ref{lem:limitEbox} gives, for $f$ supported in a cube of side $L$ in $\{x_0\ge\delta\}$ and large $k$, \[ Z_k^2\omega_k\bigl(\vartheta(\overline{A_k(f)})A_k(f)\bigr)\le\mathsf K_L^2 \|f\|_\infty^2\frac{\mathcal N_k(\delta)}{\mathcal N_k(\varepsilon)} , \] with $\mathsf K_L^2\le3^8(1+L)^8$. The ratio is at most $C_0\max(1,\varepsilon)^{p_0} \delta^{-p_0}$ for $\delta\le\varepsilon$ and at most $\mathsf K^2$ for $\delta>\varepsilon$, by Proposition~\ref{prop:limitEsd}(i). Conversely, if the case $m=1$ holds, let $\varepsilon'\in(0,\varepsilon]$ and $h$ be supported in $B_{\varepsilon'}\subset[0,L_0]^4$, $L_0=\varepsilon+1+a_0$; the bound with $\delta=\min(\varepsilon',1)$ gives $\mathcal N_k(\varepsilon')\le C_0(\varepsilon/\varepsilon')^{p_1}\mathcal N_k(\varepsilon)$ with $C_0=K_1(1+L_0)^{q_1}\max(1,\varepsilon^{-p_1})$. The last claim follows from (i), (ii). \end{proof} Bounds on each moment do not give the factor $K^n(n!)^\beta$: in the next fields every moment is bounded uniformly in the separation, and (G$_{\mathrm s}$) fails. \begin{proposition}\label{prop:limitEfactorial} In the setting of Lemma~\textup{\ref{lem:limitEtrunc}} let $t>0$, $U(u)=e^{tu_2-t^2/2}-1$, $\varepsilon>0$, form $\mathcal N_k(\varepsilon)$ with $\Phi_x=U(\xi_x)$, and put $Z_k=\mathcal N_k(\varepsilon)^{-1/2}$, with $\ell_0$ as in Proposition~\textup{\ref{prop:limitEthree}}. \textup{(a)} The gap is $m$ in physical units. For large $k$, $t^2\ell_0\le \mathcal N_k(\varepsilon)\le(1+2a_0)^8(e^{t^2}-1)$, \textup{(SD$_{\mathrm p}$)} holds with $p_0=0$, \textup{(NE)} holds, and for every $n$ and all $f_j$ supported in a cube of side $L$, separated or not, \[ \Bigl|\omega_k\Bigl(\prod_{j=1}^nZ_kA_k(f_j)\Bigr)\Bigr|\le \frac{2^ne^{t^2n(n-1)/2}(L+1)^{4n}}{(t^2\ell_0)^{n/2}}\prod_{j=1}^n\|f_j\| _\infty ; \] in particular \textup{(M$_{\mathrm s}$)} and \textup{(RN)} hold. \textup{(b)} Let $n$ be even with $n\ge e\log2/t^2+\frac12$, $s=\lceil n^{1/4}\rceil$, $\eta=1/(12(1+m)s)$, let $v_1,\dots,v_n$ be distinct points of $\{0,\dots,s-1\}^4$, and let $f_j$ be as in Proposition~\textup{\ref{prop:limitEthree}(b)} with centres $c_j=3\eta v_j$. Then $(f_j)\in\mathcal C_n(Q,\eta)$ for a cube $Q$ of side at most $1$, and for large $k$ \[ Z_k^n\,\omega_k\Bigl(\prod_{j=1}^nA_k(f_j)\Bigr)\ge\frac{(\eta/2)^{4n} e^{t^2n(n-1)/(2e)}}{2^{n+1}\bigl((1+2a_0)^8(e^{t^2}-1)\bigr)^{n/2}} . \] \textup{(G$_{\mathrm s}$)} fails for all $K,\beta,p,q$, and no limit of Theorem~\textup{\ref{thm:limitEsep}} satisfies the bound of \textup{(G$_{\mathrm s}$)}. \textup{(c)} For every sequence $M_k\to\infty$, \textup{(a)} and \textup{(b)} hold for the fields $U_{M_k}(\xi_x)/(2M_k)$, with values in $[-1,1]$, with $4^n$ in place of $2^n$, $\frac12t^2\ell_0$ in place of $t^2\ell_0$, and the right side of \textup{(b)} halved, for $k$ large depending on $n$; for these fields $Z_k\to\infty$. \end{proposition} \begin{proof} (a) The gap is Lemma~\ref{lem:limitEtrunc}(i), with $\int U(u)u_2\,d\gamma_2 =t$. As in the proof of Proposition~\ref{prop:limitEthree}(a), the reflected norm of $A_k(h)$ is $\sum_{x,y}a_k^8\overline{h(a_kx)}h(a_ky) (e^{t^2\rho}-1)$, with $\rho=\rho(y-\vartheta x)\in(0,1]$; the upper bound follows, and the lower one from $e^{t^2\rho}-1\ge t^2\rho$ and the function $h_*$, which also bounds the ratio in (NE) with $g_0=h_*$ below by $t^2\ell_0/((1+2a_0)^8(e^{t^2}-1))$, and (SD$_{\mathrm p}$) with $p_0=0$. By Lemma~\ref{lem:limitEtrunc}(ii) the moments are at most $\|U\|_n^n\prod_j (L+1)^4\|f_j\|_\infty$, and $\|U\|_n\le\|e^{tu_2-t^2/2}\|_n+1=e^{t^2(n-1)/2} +1\le2e^{t^2(n-1)/2}$. This gives (M$_{\mathrm s}$), and (RN) by Proposition~\ref{prop:limitEreflect}(i). (b) Distinct $v_j$ differ by at least $1$ in some coordinate, so the supports are at $\ell^\infty$ distance at least $\eta$; they lie in a cube of side $3\eta s\le1$, of $\ell^1$ diameter at most $12\eta s\le1/m$, so $\rho\ge e^{-1}$ for all sites in them. By \eqref{eq:limitEcover} and its monotonicity, $\E\prod_jU(\xi_{x_j})$ is at least its value for the Gaussian vector $X_i=e^{-1/2}Y+(1-e^{-1})^{1/2}W_i$ with $Y,W_1,\dots,W_n$ independent standard normal, whose covariances are all $e^{-1}$. Conditionally on $Y$ the factors are independent, and with $\sigma=te^{-1/2}$ this value is $\E[(e^{\sigma Y-\sigma^2/2}-1)^n]$. For $n$ even the integrand is nonnegative, and it is at least $2^{-n}e^{n\sigma Y-n\sigma^2/2}$ where $y_0=(\log2+\sigma^2/2)/\sigma\le Y$. Since $\E[e^{n\sigma Y}\mathbf 1_{\{Y\ge y_0\}}]=e^{n^2\sigma^2/2} \mathbb P(Y\ge y_0-n\sigma)\ge\frac12e^{n^2\sigma^2/2}$ for $n\sigma\ge y_0$, that is $n\ge\log2/\sigma^2+\frac12$, the value is at least $2^{-n-1}e^{\sigma^2n(n-1)/2}$. Keeping the sites with $a_kx_j\in c_j+[-\frac\eta2,\frac\eta2]^4$, as in the proof of Proposition~\ref{prop:limitEthree}(b), gives the bound. If (G$_{\mathrm s}$) held, these moments would be at most $K^n(n!)^\beta 2^{qn}\eta^{-pn}$, and $\eta^{-1}\le24(1+m)n^{1/4}$; the logarithm of this is $O(n\log n)$, while the logarithm of the lower bound is $t^2n(n-1)/(2e)- O(n\log n)$, a contradiction for large even $n$. The limits of Theorem~\ref{thm:limitEsep}(i) satisfy the same lower bound. (c) By Lemma~\ref{lem:limitEtrunc}(ii), $|\E\prod_jU_{M_k}(\zeta_j)|\le (2\|U\|_n)^n$, the moments and reflected norms change by at most $e_n(M_k)\prod_j(L_j+1)^4\|f_j\|_\infty\to0$ for each fixed $n$, and $\int U_{M_k}(u)u_2\,d\gamma_2\to t$; the rescaling by $2M_k$ is that of the proof of Proposition~\ref{prop:limitEthree}(d). \end{proof} One inequality between the moments and the separated two-point functions gives (M$_{\mathrm s}$) and (G$_{\mathrm s}$) from (SD) and (SD$_{\mathrm p}$). For a cube $Q$ and $\delta>0$ put \[ \mu_k(Q,\delta)=\sup Z_k^2\bigl|\omega_k\bigl(A_k(h)A_k(h')\bigr)\bigr| , \] the supremum over the $(h,h')\in\mathcal C_2(Q,\delta)$ with $\|h\|_\infty,\|h'\|_\infty\le1$. \begin{enumerate}[label=\textup{(GD)},leftmargin=*] \item There are $K_0,\beta_0\ge0$ such that for every $n\ge3$, every cube $Q$ and every $\delta>0$ there is $k_0$ with \[ \Bigl|\omega_k\Bigl(\prod_{j=1}^nZ_kA_k(f_j)\Bigr)\Bigr|\le K_0^n(n!)^{ \beta_0}\mu_k(Q,\delta)^{n/2}\prod_{j=1}^n\|f_j\|_\infty \] for all $k\ge k_0$ and $(f_1,\dots,f_n)\in\mathcal C_n(Q,\delta)$. \end{enumerate} \begin{proposition}\label{prop:limitEdomination} Assume the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}}. \textup{(i)} Under \textup{(GD)}, \textup{(M$_{\mathrm s,2}$)} implies \textup{(M$_{\mathrm s}$)}. With $Z_k=\mathcal N_k(\varepsilon)^{-1/2}$ as in Theorem~\textup{\ref{thm:limitEtwo}}, \textup{(GD)} and \textup{(SD)} imply \textup{(M$_{\mathrm s}$)}, and \textup{(GD)} and \textup{(SD$_{\mathrm p}$)} imply \textup{(G$_{\mathrm s}$)} with $K=\max(1,K_0)C_2^{1/2}$, $\beta=\beta_0$, $q=4$, $p=4+\frac12p_0$, where $C_2=(\frac{77}2)^8\max(1,C_0)\max(1, 8\varepsilon)^{p_0}$. \textup{(ii)} If for every $k$ the packets $A_k(f)$ form a centred Gaussian family, \textup{(GD)} holds with $K_0=1$, $\beta_0=\frac12$. \textup{(iii)} \textup{(GD)} fails for $n=3$ in the fields of Proposition~\textup{\ref{prop:limitEthree}}. \end{proposition} \begin{proof} (i) (M$_{\mathrm s,2}$) gives $\mu_k(Q,\delta)\le M_2(Q,\delta)$ for large $k$, and Corollary~\ref{cor:limitEtwo}(a) gives (M$_{\mathrm s,2}$) from (SD). Under (SD$_{\mathrm p}$) the proof of Proposition~\ref{prop:limitEtwotempered}(a) gives $\mu_k(Q,\delta)\le C_2(1+L)^8\delta^{-8-p_0}$ for $Q$ of side $L$, $\delta\le1$ and large $k$, which is the case $n=2$; the moments of order one vanish. (ii) By Wick's formula the moment is $0$ for odd $n$ and, for even $n$, the sum over the $(n-1)!!$ perfect matchings of the products of the $\omega_k(A_k(f_a)A_k(f_b))$; each pair lies in $\mathcal C_2(Q,\delta)$, and $((n-1)!!)^2\le(n-1)!!\,n!!=n!$. (iii) There (M$_{\mathrm s,2}$) holds, so $\mu_k(Q,\eta)$ is bounded, while the left side of (GD) for the functions of Proposition~\ref{prop:limitEthree}(b) with $n=3$ tends to infinity. \end{proof} So the case $m=1$ of (RN) is (SD), and its rate is (SD$_{\mathrm p}$) (Propositions~\ref{prop:limitEreflect}(iii) and~\ref{prop:limitErate}(iii)); the cases $m\ge2$ are a further input on the Wilson measure, which the two-point properties, reflection positivity and the gap do not supply (Proposition~\ref{prop:limitEthree}); (G$_{\mathrm s}$) is (RN) with constants $K_1^m(m!)^{\beta_1}(1+L)^{q_1m}\delta^{-p_1m}$ (Proposition~\ref{prop:limitErate}), and bounds on each moment do not give the dependence on $n$ (Proposition~\ref{prop:limitEfactorial}). The single inequality (GD), of Gaussian domination type, gives (M$_{\mathrm s}$) from (SD) and (G$_{\mathrm s}$) from (SD$_{\mathrm p}$); it holds for Gaussian packets, fails in the fields of Proposition~\ref{prop:limitEthree}, and is not proved here for the Wilson measure. \emph{The effective energy.} Along each lattice of distances from the reflection plane the largest reflected norm is logarithmically convex, and (SD) becomes a bound, in physical units, on its rate of decay. For $(r+2)a_k\le s$ with $\mathcal N_k(s)>0$ put \[ \mathfrak E_k(s)=\frac1{2a_k}\log\frac{\mathcal N_k(s)}{\mathcal N_k(s+a_k)} . \]\footnote{For one correlator this is the effective mass of lattice spectroscopy; here it is taken of the largest reflected norm, a supremum over packets.} \begin{lemma}\label{lem:limitElogconvex} Assume the hypothesis of Theorem~\textup{\ref{thm:avgtransfer}}, let $(r+2)a_k\le s$ and $\mathcal N(n)=\mathcal N_k(s+na_k)$ for integers $n\ge0$. \textup{(i)} $\mathcal N(n)$ is the supremum of $\|\mathbb T_k^n \Psi_{A_k(h)}\|^2$ over $h$ supported in $B_s$ with $\|h\|_\infty\le1$; $\mathcal N(n+1)\le\gamma_k^2\mathcal N(n)$, and $\mathcal N(n)^2\le \mathcal N(n-1)\mathcal N(n+1)$ for $n\ge1$. \textup{(ii)} If $\mathcal N(0)>0$, then $\mathcal N(n)>0$ for every $n$, and $n\mapsto\mathfrak E_k(s+na_k)$ is nonincreasing, with values in $[a_k^{-1}\log(1/\gamma_k),\infty)\subset[m_*,\infty)$. \end{lemma} \begin{proof} (i) Since $B_{s+na_k}=B_s+na_ke_0$, $h$ is supported in $B_{s+na_k}$ if and only if $h'=h(\cdot+na_ke_0)$ is supported in $B_s$, and then $A_k(h)= \tau_nA_k(h')$ by Lemma~\ref{lem:limitElattice}(i), so $\Psi_{A_k(h)}= \mathbb T_k^n\Psi_{A_k(h')}$ as in the proof of Lemma~\ref{lem:limitElattice}(iii). The packets are centred, so $\Psi_{A_k(h')}\perp\Omega_k$ and $\|\mathbb T_k^{n+1}\Psi_{A_k(h')}\|\le \gamma_k\|\mathbb T_k^n\Psi_{A_k(h')}\|$. For a vector $\Psi$ and $f(n)=\|\mathbb T_k^n\Psi\|^2=\ip{\mathbb T_k^{n-1}\Psi}{\mathbb T_k^{n+1}\Psi}$, the Cauchy--Schwarz inequality gives $f(n)^2\le f(n-1) f(n+1)\le\mathcal N(n-1)\mathcal N(n+1)$ when $\Psi=\Psi_{A_k(h')}$; the supremum over $h'$ gives the last claim. (ii) For $h'$ supported in $B_s$, $h''=h'(\cdot+a_ke_0)$ is supported in $\{x_0\ge s-a_k\}$ and $(r+1)a_k\le s-a_k$, so $\Psi_{A_k(h')}=\mathbb T_k \Psi_{A_k(h'')}$ lies in the range of $\mathbb T_k$, which is orthogonal to $\ker\mathbb T_k=\ker\mathbb T_k^n$. So $\mathbb T_k^n\Psi_{A_k(h')}\ne0$ when $\Psi_{A_k(h')}\ne0$, and $\mathcal N(n)>0$ by (i). The convexity in (i) says that $\log\mathcal N(n-1)-\log\mathcal N(n)\ge\log\mathcal N(n)- \log\mathcal N(n+1)$, and the bound $\mathcal N(n+1)\le\gamma_k^2 \mathcal N(n)$ gives the lower bound, with $\gamma_k\le e^{-m_*a_k}$. \end{proof} For a single packet, $(2a_k)^{-1}\log(\|\mathbb T_k^n\Psi\|^2/ \|\mathbb T_k^{n+1}\Psi\|^2)$ decreases, as $n\to\infty$, to the least point of the support of its spectral measure, written in physical energy; so $\mathfrak E_k(s)$ is the energy, in physical units, at which the largest reflected norm decays at the distance $s$ from the plane. \begin{proposition}\label{prop:limitEenergy} Assume the hypothesis of Theorem~\textup{\ref{thm:limitEtwo}}, and let $C_{\varepsilon'}$ be as in Proposition~\textup{\ref{prop:limitEsd}(iii)}. \textup{(i)} \textup{(SD)} holds if and only if $\limsup_k\mathfrak E_k(s)<\infty$ for every $s\in(0,\varepsilon)$. Under \textup{(SD)}, $\limsup_k\mathfrak E_k(s)\le\log(\mathsf K^4C_{s/2})/s$; conversely, for $\varepsilon'\in(0,\varepsilon)$, \[ \limsup_k\frac{\mathcal N_k(\varepsilon')}{\mathcal N_k(\varepsilon)}\le \mathsf K^2\exp\Bigl(2(\varepsilon-\varepsilon')\limsup_k\mathfrak E_k(\varepsilon')\Bigr). \] \textup{(ii)} Under \textup{(SD$_{\mathrm p}$)}, for every $s\in(0, \varepsilon]$ and all large $k$, \[ \mathfrak E_k(s)\le\frac{\log(\mathsf K^4C_0)+p_0\log(2\varepsilon/s)} {s-2a_k} . \] Conversely, if there are $p,E_0\ge0$ and, for every $\varepsilon'\in(0, \varepsilon]$, a $k_0$ with $\mathfrak E_k(s)\le p/(2s)+E_0$ for all $s\in[\varepsilon',\varepsilon]$ and $k\ge k_0$, then \textup{(SD$_{\mathrm p}$)} holds with $p_0=p$. \end{proposition} \begin{proof} For $s\in(0,\varepsilon]$ and large $k$, $\mathcal N_k(s)\ge\mathsf K^{-2} \mathcal N_k(\varepsilon)>0$ by Proposition~\ref{prop:limitEsd}(i), so $\mathfrak E_k(s)$ is defined. Let $n=\lfloor s/(2a_k)\rfloor$. By Lemma~\ref{lem:limitElogconvex}(ii), applied at $s-na_k$, $\mathfrak E_k(s)\le\mathfrak E_k(s-ia_k)$ for $1\le i\le n$, so \[ \mathfrak E_k(s)\le\frac1n\sum_{i=1}^n\mathfrak E_k(s-ia_k)=\frac1{2a_kn} \log\frac{\mathcal N_k(s-na_k)}{\mathcal N_k(s)} . \] Since $s-na_k\ge s/2$ and $s\le\varepsilon$, Proposition~\ref{prop:limitEsd}(i) gives $\mathcal N_k(s-na_k)\le\mathsf K^2\mathcal N_k(s/2)$ and $\mathcal N_k(s)\ge\mathsf K^{-2}\mathcal N_k(\varepsilon)$, and $2a_kn\ge s-2a_k$. Hence \[ \mathfrak E_k(s)\le\frac1{s-2a_k}\log\Bigl(\mathsf K^4\frac{\mathcal N_k(s/2)}{\mathcal N_k(\varepsilon)}\Bigr), \] which gives the bound in (i) under (SD), and the bound in (ii) under (SD$_{\mathrm p}$) at $\varepsilon'=s/2$. Conversely, let $\varepsilon'\in(0,\varepsilon)$, $E'>\limsup_k\mathfrak E_k(\varepsilon')$ and $n=\lceil(\varepsilon-\varepsilon')/a_k\rceil$. For large $k$, Lemma~\ref{lem:limitElogconvex}(ii) gives \[ \log\frac{\mathcal N_k(\varepsilon')}{\mathcal N_k(\varepsilon'+na_k)}= 2a_k\sum_{j=0}^{n-1}\mathfrak E_k(\varepsilon'+ja_k)\le2a_knE' , \] and $2a_kn\le2(\varepsilon-\varepsilon'+a_k)$. Since $\varepsilon'+na_k\ge \varepsilon$, Proposition~\ref{prop:limitEsd}(i) gives $\mathcal N_k(\varepsilon'+na_k)\le\mathsf K^2\mathcal N_k(\varepsilon)$. Letting $k\to\infty$ and $E'\downarrow\limsup_k\mathfrak E_k(\varepsilon')$ gives the inequality of (i), hence (SD). Under the hypothesis of the converse in (ii) the same sum is at most $\sum_j(pa_k/(\varepsilon'+ja_k)+2a_kE_0)\le pa_k/\varepsilon'+p\log((\varepsilon+a_k)/\varepsilon')+2(\varepsilon+ a_k)E_0$, so for $a_k\le\min(\varepsilon',\varepsilon,1)$ \[ \frac{\mathcal N_k(\varepsilon')}{\mathcal N_k(\varepsilon)}\le\mathsf K^2e^{p+2(\varepsilon+1)E_0}2^p\Bigl(\frac\varepsilon{\varepsilon'} \Bigr)^p , \] which is (SD$_{\mathrm p}$) for $\varepsilon'<\varepsilon$; for $\varepsilon'=\varepsilon$ it holds with $C_0\ge1$. \end{proof} So (SD) says that at every distance from the reflection plane the largest reflected norm decays at a rate bounded in physical units, and (SD$_{\mathrm p}$) that this rate grows at most like $p_0/(2s)$ as $s\to0$, up to the logarithm in (ii). Under a gap $m_{\mathrm{lat}}$ in lattice units the rate is at least $m_{\mathrm{lat}}/a_k$ at every distance, which is the failure of (SD) in Proposition~\ref{prop:limitEsd}(iii). By Proposition~\ref{prop:limitEsd}(ii) the bound on the rate is equivalent to the spectral condition (UV). \emph{The abelian theory.} For $G=\U(1)$ the positivity behind Proposition~\ref{prop:limitEne}(ii) is a theorem. \begin{corollary}\label{cor:limitEabelian} Let $G=\U(1)$ with the Wilson weight and $\beta_k\ge0$, assume the hypothesis of Theorem~\textup{\ref{thm:limitEtwo}}, and let $\Phi=b_0-\sum_jb_j\cos\theta_j$ with $b_j\ge0$ for $j\ge1$ and each $\theta_j$ an integer combination of the angles of the links with both endpoints in $[-r,r]^4$; the sum of $1-\cos\theta_p$ over the $24$ plaquettes with a corner at $0$ is an example. Then $K_k(z)\ge0$ for all $z\in\Z^4$. If \textup{(SD)} holds, then \textup{(NE)} holds, every limit $S_2$ of Corollary~\textup{\ref{cor:limitEtwo}(b)} is not identically zero, and $\mathcal N_k(\varepsilon)\le\omega_k(\vartheta(A_k(g_0))A_k(g_0))$ for large $k$ and every $g_0$ as in Proposition~\textup{\ref{prop:limitEne}(ii)}. \end{corollary} \begin{proof} Let $\mu_L$ be the Wilson measure on $\Lambda_L$. Its density with respect to the product of the normalised Lebesgue measures on the link angles is $Z^{-1}\exp(\beta_k\sum_p\cos\theta_p)$, where each $\theta_p$ is a signed sum of four link angles. By Ginibre's inequality~\cite{Gin70}, $\int\cos\theta\cos\theta'd\mu_L\ge\int\cos\theta\,d\mu_L\int\cos\theta' d\mu_L$ for any two integer combinations $\theta,\theta'$ of link angles. So $\Cov_{\mu_L}(\Phi_0,\Phi_z)=\sum_{j,l\ge1}b_jb_l\Cov_{\mu_L}(\cos \theta_j,\cos\tau_z\theta_l)\ge0$ once $L$ is large, and the torus limit preserves this inequality for the local functions $\Phi_0$, $\Phi_z$. This is the hypothesis of Proposition~\ref{prop:limitEne}(ii), whose proof gives the last claim, and Proposition~\ref{prop:limitEne}(i) gives the others. \end{proof} For $\U(1)$ the averaged condition fails in the Coulomb phase of the Villain theory (Corollary~\ref{cor:avgcoulomb}), and (SD) fails under a gap in lattice units (Proposition~\ref{prop:limitEsd}(iii)); whether the hypotheses of the corollary hold together for $\U(1)$ along any tuning is not decided here. What the corollary excludes is the mechanism of the example below, whose two-point function changes sign. \emph{Spatial momentum.} The spatial translations act on the reconstructed space. For $v\in\Z^4$ with $v_0=0$, $\omega_k(\vartheta(\tau_v\bar F) \tau_vF')=\omega_k(\tau_v(\vartheta(\bar F)F'))=\omega_k(\vartheta(\bar F)F')$ for $F,F'$ at positive times, so $\Psi_F\mapsto\Psi_{\tau_vF}$ extends to a unitary $U_k(v)$; $U_k$ is a representation of $\Z^3$ that commutes with $\mathbb T_k$, since $\tau_v\tau_1=\tau_1\tau_v$. By the spectral theorem for the commuting unitaries $U_k(e_i)$ there is a projection-valued measure $\mathsf P_k$ on $(-\pi,\pi]^3$ with $U_k(v)=\int e^{ip\cdot v}\,d\mathsf P_k(p)$. For $K>0$ let $\mathbb P_k(K)=\mathsf P_k(\{|p|_\infty\le Ka_k\})$, the states of spatial momentum at most $K$ in physical units; it commutes with $\mathbb T_k$ and $\mathbb E_k(\Lambda)$. For $(r+1)a_k\le s$ let $\mathcal N^{\mathrm{sp}}_k(s;K)$ be the supremum of $\|\mathbb P_k(K) \Psi_{A_k(h)}\|^2$ over $h$ supported in $B_s$ with $\|h\|_\infty\le1$, and consider \begin{enumerate}[label=\textup{(LM)},leftmargin=*] \item for some $K<\infty$, $c>0$ and $\varepsilon_1\in(0,\varepsilon]$, $\mathcal N^{\mathrm{sp}}_k(\varepsilon_1;K)\ge c\,\mathcal N_k(\varepsilon)$ for all large $k$. \end{enumerate} It says that, up to a fixed fraction, the largest reflected norm is carried by spatial momenta bounded in physical units, as (UV) says for the energy. The proofs below use the Fej\'er weights: for an integer $N\ge1$, \[ F_N(q)=\sum_{|n|0$ there is $C(s)$ with $\mathcal N_k(s)\le C(s)\mathcal N_k(\varepsilon)$ for large $k$. Let (NE) hold with $g_0$ supported in $\{x_0\ge t_0\}$, $t_0>0$, and in a cube of side $L$, and $c_0>0$ with $4c_0<\liminf_kR_k(g_0)$. Let $\delta\in(0,1]$, $a_k\le\delta$ and $N=\lfloor\delta/a_k\rfloor\ge\delta/(2a_k)$. The function $g_0^N-g_0$ is supported in a cube of side $L+2$ in $\{x_0\ge t_0\}$ and bounded by $\varrho(\delta)=\sup\{|g_0(y)-g_0(y')|:|y-y'|_\infty\le \delta\}$, because $f^N_v\ne0$ only for $|a_kv|_\infty<\delta$. For $|p|_\infty>Ka_k$ some $|p_i|>Ka_k$, so $F^{(3)}_N(p)\le(\pi/(NKa_k))^2\le (2\pi/(\delta K))^2$. With Lemma~\ref{lem:limitEbox}, \[ \begin{split} &\|(1-\mathbb P_k(K))\Psi_{A_k(g_0)}\|\\ &\quad\le\|\Psi_{A_k(g_0-g_0^N)}\|+ \|(1-\mathbb P_k(K))F^{(3)}_N(\mathsf P_k)\Psi_{A_k(g_0)}\|\\ &\quad\le\Bigl(\mathsf K_{L+2}\varrho(\delta)+\frac{4\pi^2\mathsf K_L \|g_0\|_\infty}{\delta^2K^2}\Bigr)\mathcal N_k(t_0)^{1/2} . \end{split} \] Fix $\delta$, and then $K$, so that the bracket is at most $(c_0/C(t_0))^{1/2}$. For large $k$ this gives \[ \|\mathbb P_k(K)\Psi_{A_k(g_0)}\|\ge(2c_0^{1/2}-c_0^{1/2})\mathcal N_k(\varepsilon)^{1/2}, \] and the first fact gives $\mathcal N^{\mathrm{sp}}_k(t_0;K)\ge c_0\mathsf K_L^{-2}\|g_0\| _\infty^{-2}\mathcal N_k(\varepsilon)$. If $t_0\le\varepsilon$ this is (LM) with $\varepsilon_1=t_0$. If $t_0>\varepsilon$, every $h$ supported in $B_{t_0}$ is $h'(\cdot-na_ke_0)$ with $n\ge0$ and $h'$ supported in a cube of side $1+a_0$ in $\{x_0\ge\varepsilon\}$, so $\|\mathbb P_k(K)\Psi_{A_k(h)}\| \le\|\mathbb P_k(K)\Psi_{A_k(h')}\|$ and $\mathcal N^{\mathrm{sp}}_k(t_0;K) \le\mathsf K^2\mathcal N^{\mathrm{sp}}_k(\varepsilon;K)$, which is (LM) with $\varepsilon_1=\varepsilon$. Conversely, let (LM) hold and, for large $k$, $h_k$ be supported in $B_{\varepsilon_1}$ with $\|h_k\|_\infty\le1$ and $\|\mathbb P_k(K)\Psi_k\|^2 \ge\frac c2\mathcal N_k(\varepsilon)$, $\Psi_k=\Psi_{A_k(h_k)}$. \emph{Energy.} Let $n=\lfloor\varepsilon_1/(2a_k)\rfloor$ and $h'_k=h_k( \cdot+na_ke_0)$, supported in a cube of side $1+a_0$ in $\{x_0\ge \varepsilon_1/2\}$. Then $\Psi_k=\mathbb T_k^n\Psi_{A_k(h'_k)}$ and $\|\mathbb T_k^n(1-\mathbb E_k(\Lambda))\|\le e^{-\Lambda a_kn}$, so for $a_k\le\varepsilon_1/4$, by Lemma~\ref{lem:limitEbox}, \[ \|(1-\mathbb E_k(\Lambda))\Psi_k\|\le e^{-\Lambda\varepsilon_1/4}\mathsf K \,C(\tfrac{\varepsilon_1}2)^{1/2}\mathcal N_k(\varepsilon)^{1/2} . \] Fix $\Lambda$ with \[ e^{-\Lambda\varepsilon_1/4}\mathsf KC(\tfrac{\varepsilon_1}2)^{1/2}\le(c/8)^{1/2}. \] Since $\mathbb E_k(\Lambda)$ and $\mathbb P_k(K)$ commute, $\|\mathbb E_k(\Lambda)\mathbb P_k(K)\Psi_k\|\ge(c/8)^{1/2} \mathcal N_k(\varepsilon)^{1/2}$. \emph{Smoothing.} Let $\sigma>0$, $M=\lfloor\sigma/a_k\rfloor$, $N= \lfloor\pi/(Ka_k)\rfloor$, both at least $1$, $w_n=M^{-1}(1-|n-M|/M)_+$ for $n\in\Z$, which vanish unless $1\le n\le2M-1$ and add up to $1$, and \[ g_k=\sum_{n,v}w_nf^N_v\,h_k(\cdot-a_k(ne_0+v)) . \] By Lemma~\ref{lem:limitElattice}(i), $\Psi_{A_k(g_k)}=W(\mathbb T_k) F^{(3)}_N(\mathsf P_k)\Psi_k$ with $W(\lambda)=\sum_nw_n\lambda^n\ge \lambda^{2M}$ on $[0,1]$. On the range of $\mathbb E_k(\Lambda)\mathbb P_k(K)$ the joint spectrum of $\mathbb T_k$ and $U_k$ lies where $\lambda\ge e^{-\Lambda a_k}$, so $W\ge e^{-2\Lambda\sigma}$, and $|p|_\infty\le Ka_k$, so $N|p_i|\le\pi$ and $F^{(3)}_N\ge(2/\pi)^6$. Hence \[ \omega_k\bigl(\vartheta(\overline{A_k(g_k)})A_k(g_k)\bigr)\ge4c' \mathcal N_k(\varepsilon), \] with $4c'=\frac c8e^{-4\Lambda\sigma}(2/\pi)^{12}$. \emph{Compactness.} $A_k(g)$ depends only on the values of $g$ on $a_k\Z^4$. Let $u(t)=M^{-1}(1-|t/a_k-M|/M)_+$, $\varsigma(t)=N^{-1}(1-|t/a_k|/N)_+$, $\kappa(y)=u(y_0)\prod_{i=1}^3\varsigma(y_i)$ and \[ G_k(y)=\sum_{z\in\Z^4}\kappa(y-a_kz)h_k(a_kz) . \] Then $G_k(a_kx)=g_k(a_kx)$, so $A_k(G_k)=A_k(g_k)$. The function $G_k$ is supported in $S=B_{\varepsilon_1}+[0, 2\sigma]\times[-\pi/K,\pi/K]^3$, a compact subset of $\{x_0\ge \varepsilon_1\}$ independent of $k$. For $t,t'\in\R$, at most $2M$ terms of $\sum_{n\in\Z}u(t-a_kn)$ are nonzero, so this sum is at most $2$, and $\sum_n|u(t-a_kn)-u(t'-a_kn)|\le4|t-t'|/(Ma_k)\le8|t-t'|/\sigma$; likewise $\sum_n\varsigma(t-a_kn)\le2$ and $\sum_n|\varsigma(t-a_kn)-\varsigma(t'- a_kn)|\le8K|t-t'|/\pi$, for $a_k\le\min(\sigma,\pi/K)$. Writing the difference of two products of four factors as a telescoping sum, $|G_k|\le 16$ and $G_k$ is Lipschitz with constant $8(8/\sigma+24K/\pi)$ for the $\ell^\infty$ norm. By the Arzel\`a--Ascoli theorem the $G_k$ lie in a compact subset of $C(S)$. \emph{One packet.} Let $L_S$ be the side of a cube containing $S$ and $\eta>0$ with $\mathsf K_{L_S}C(\varepsilon_1)^{1/2}\eta\le c'^{1/2}$. A finite set $G^1,\dots,G^J$ of the $G_k$ is an $\eta$-net, so for each large $k$ there is $j$ with $\|G_k-G^j\|_\infty\le\eta$, and Lemma~\ref{lem:limitEbox} gives $\|\Psi_{A_k(G_k)}-\Psi_{A_k(G^j)}\|\le c'^{1/2}\mathcal N_k(\varepsilon)^{1/2}$, hence $R_k(G^j)\ge c'$. Let $g_0=\sum_jG^j(\cdot-jDe_1)$ with $D$ dyadic. As in the proof of Proposition~\ref{prop:limitEne}(i), for large $k$ \[ R_k(g_0)\ge\sum_jR_k(G^j)-16^2J^2\mathsf K_{L_S}^2e^{-m_*(D-L_S-4 \varepsilon-1)} , \] and $R_k\ge0$. For $D$ large the right side is at least $c'/2$ for all large $k$, and $g_0$ is continuous and supported in $\{x_0\ge\varepsilon_1\}$: this is (NE). \end{proof} \emph{An example.} (NE) is not implied by (SD) together with the properties used so far: in the following Gaussian field the largest reflected norms are carried by spatial momenta of order $\pi/a_k$ at energies bounded in physical units, so that (UV) holds and (LM) fails.\footnote{$D_k$ depends on $p$ only through the product of the four factors $2-2\cos p_\mu$, so the energy of a spatial momentum is small only where all three spatial factors are large, near the corner $(\pi,\pi,\pi)$ of the Brillouin zone; the packets that carry the largest reflected norm live there, and smooth test functions do not see them.} \begin{proposition}\label{prop:limitEflat} Let $m>0$, $A_k=(128(\cosh(ma_k)-1))^{-1}$, \[ D_k(p)=1+A_k\prod_{\mu=0}^3(2-2\cos p_\mu),\qquad p\in(-\pi,\pi]^4, \] let $\phi_k$ be the centred Gaussian field on $\Z^4$ with covariance $\int e^{ip\cdot(x-y)}D_k(p)^{-1}dp/(2\pi)^4$, $A_k(f)=\sum_xa_k^4f(a_kx) \phi_k(x)$, and form $\mathcal N_k(\varepsilon)$ and $R_k$ with these packets and $r=0$. \textup{(a)} The law of $\phi_k$ is invariant under translations and $B_4$ and reflection positive for the reflections in coordinate hyperplanes through sites and bisecting links. Its transfer matrix in each coordinate direction has spectrum on $\Omega_k^\perp$ in $[0,e^{-ma_k}]$ containing $e^{-ma_k}$: the gap is $m$ in physical units. \textup{(b)} For every $\varepsilon>0$, $\varepsilon'\in(0,\varepsilon]$ and all large $k$, \[ 2^{-15}me^{-4m(\varepsilon+2)}a_k^4\le\mathcal N_k(\varepsilon),\qquad \mathcal N_k(\varepsilon')\le(2+a_0)^4a_k^4 ; \] so \textup{(SD$_{\mathrm p}$)} holds with $p_0=0$, and \textup{(UV)}. \textup{(c)} $R_k(g)\to0$ for every $g\in C_c(\R^4)$ supported in $\{x_0>0\}$: \textup{(NE)} fails, and every limit $S_2$ of Corollary~\textup{\ref{cor:limitEtwo}(b)} vanishes identically. \textup{(d)} With $Z_k=\mathcal N_k(\varepsilon)^{-1/2}$, \textup{(M$_{\mathrm s}$)} holds and every limit of Theorem~\textup{\ref{thm:limitEsep}} is trivial. With $Z_k=C_k(\sigma_0) ^{-1/2}$, formed from any $f_0$ as before Proposition~\textup{\ref{prop:limitEnorm}} with $C_k(0)>0$, \textup{(M$_{\mathrm s,2}$)} fails. \end{proposition} \begin{proof} (a) $D_k$ is positive and invariant under permutations and sign changes of the coordinates, so the covariance is positive definite, real, and invariant under translations and $B_4$. Write $u_\mu=2-2\cos p_\mu$, and for $p\in(-\pi,\pi]^3$ let $b(p)=A_ku_1u_2u_3$, $\lambda(p)\in[0,1)$ the root of $\lambda+\lambda^{-1}=2+1/b(p)$ ($\lambda=0$ if $b(p)=0$), $E(p)= -\log\lambda(p)$ and $w(p)=(1-\lambda(p))/(1+\lambda(p))=\tanh(E(p)/2)$. Then $1+b\,u_0=b\lambda^{-1}(1-\lambda e^{ip_0})(1-\lambda e^{-ip_0})$, and expanding $(1-\lambda^2)/((1-\lambda e^{ip_0})(1-\lambda e^{-ip_0}))=\sum_t \lambda^{|t|}e^{-ip_0t}$ and using $b(\lambda^{-1}-\lambda)=1/w$, \[ \frac1{1+b\,u_0}=w\sum_{t\in\Z}\lambda^{|t|}e^{-ip_0t}, \] also for $b=0$. So the covariance of $\phi_k(t,\mathbf x)$ and $\phi_k(s, \mathbf y)$ is $\int e^{ip\cdot(\mathbf x-\mathbf y)}w(p)\lambda(p)^{|t-s|} dp/(2\pi)^3$. Let $\gamma$ be the centred Gaussian law on $\R^{\Z^3}$ with covariance the multiplier $w$, $\Lambda$ the convolution with multiplier $\lambda$, and $\xi_t$ independent of law the centred Gaussian with multiplier $w(1-\lambda^2)\ge0$. The stationary chain $\phi_{t+1}=\Lambda \phi_t+\xi_{t+1}$ has covariance with multiplier $w\lambda^{|t-s|}$, so it is the field; it is reversible because $\Lambda$ and the covariance of $\gamma$ commute and are symmetric. The proof of Proposition~\ref{prop:osobstruction}(a) applies with the operator $\Lambda$ in place of $q$: the transition operator satisfies $P\mathcal E_h= \mathcal E_{\Lambda h}$, so it acts on the $N$-th Wiener chaos as the $N$-th tensor power of the restriction of $\Lambda$ to the one-particle space, the completion of the finitely supported $h$ for $\int|\hat h|^2w\,dp/(2\pi)^3$, on which $\Lambda$ is multiplication by $\lambda$. Since $b\le64A_k$ with equality only at $p=(\pi,\pi,\pi)$, $\cosh E=1+1/(2b)\ge1+1/(128A_k)= \cosh(ma_k)$, so $0\le\lambda\le e^{-ma_k}$, with equality at that point. Hence $0\le P\le e^{-ma_kN}$ on the $N$-th chaos for $N\ge1$, and $e^{-ma_k}$ lies in the spectrum on the first, $\lambda$ being continuous and $w$ positive. Reflection positivity and the identification of the transfer matrix with $P$ follow as there, and $B_4$ gives the other directions. For $h$ supported in $\{x_0\ge1\}$ and $\vartheta$ fixing $\{x_0=c\}$, $|c|\le\frac12$, the time component of $y-\vartheta x$ is $x_0+y_0-2c\ge0$, and \[ \omega_k\bigl(\vartheta(\overline{A_k(h)})A_k(h)\bigr)=\int w|\hat H|^2 \frac{dp}{(2\pi)^3}, \] where \[ \hat H(p)=\sum_{x_0}a_k\lambda^{x_0-c}\hat h_{x_0}(p),\qquad\hat h_{x_0}(p) =a_k^3\sum_{\mathbf x}h(a_kx)e^{ip\cdot\mathbf x} . \] Let $h$ be supported in $\{x_0\ge t\}$ with $t\ge a_k$. Since $\sum_{x_0\ge t/a_k}\lambda^{x_0-c}\le\lambda^{t/a_k-1/2}/(1-\lambda)$ and $w/(1-\lambda)\le1$, the Cauchy--Schwarz inequality with the weights $\lambda^{x_0-c}$ gives \[ w|\hat H|^2\le a_k^2\lambda^{2t/a_k-1}\sum_{x_0}|\hat h_{x_0}|^2 .\tag{$*$} \] (b) For $h$ supported in $B_{\varepsilon'}$ with $\|h\|_\infty\le1$ and $a_k\le\min(\varepsilon',1)$, ($*$) with $\lambda\le1$ and Parseval's identity give at most $a_k^8$ times the number of sites $x$ with $a_kx\in B_{\varepsilon'}$, which is at most $(2+a_0)^4a_k^{-4}$. For the lower bound let $a_k\le\frac1{20}$, $ma_k\le2$, let $b_0$ be continuous with values in $[0,1]$, supported in $[\varepsilon,\varepsilon+1]$ and equal to $1$ on $[\varepsilon+\frac14,\varepsilon+\frac34]$, let $b_1$ be the function equal to $1$ on $[\frac14,\frac34]$, to $0$ outside $[0,1]$ and linear in between, and $h_k(y)=b_0(y_0)\prod_{i=1}^3\cos(\pi y_i/a_k)b_1(y_i)$, supported in $B_\varepsilon$ with $\|h_k\|_\infty\le1$. Since $\cos(\pi x_i)=(-1)^{x_i}$, $\hat H(p)=T(\lambda)\prod_i\beta(p_i+\pi)$ with $T(\lambda)=\sum_{x_0}a_k b_0(a_kx_0)\lambda^{x_0-c}$ and $\beta(q)=\sum_na_kb_1(a_kn)e^{iqn}$. On $Q=\{p:|p_i-\pi|\le\frac\pi3\bmod2\pi\}$, $u_i\ge3$, so $b\ge27A_k$ and $\cosh E-1\le1/(54A_k)\le4(\cosh(ma_k)-1)\le\cosh(2ma_k)-1$, since $\cosh 2x-1=2(\cosh x-1)(\cosh x+1)$; so $ma_k\le E\le2ma_k$ on $Q$. Then $w\ge \tanh(ma_k/2)\ge ma_k/4$, and $T(\lambda)\ge e^{-2ma_k((\varepsilon+1)/a_k+ 1/2)}\sum_{x_0}a_kb_0(a_kx_0)\ge e^{-2m(\varepsilon+2)}/4$. By Parseval's identity $\int_{-\pi}^\pi|\beta|^2dq/2\pi=a_k^2\sum_nb_1(a_kn)^2\ge a_k/4$, and since $|1-e^{iq}|\ge1$ for $|q|\ge\frac\pi3$ and $b_1$ is $4$-Lipschitz, \[ \int_{\frac\pi3\le|q|\le\pi}|\beta|^2\frac{dq}{2\pi}\le a_k^2\sum_n|b_1(a_kn) -b_1(a_kn-a_k)|^2\le48a_k^3 , \] so $\int_{|q|\le\pi/3}|\beta|^2dq/2\pi\ge a_k/8$. Restricting the integral to $Q$ gives \[ \mathcal N_k(\varepsilon)\ge\frac{ma_k}4\cdot\frac{e^{-4m(\varepsilon+2)}}{16}\Bigl(\frac{a_k}8\Bigr)^3 . \] (SD$_{\mathrm p}$) follows with $C_0=2^{15}(2+a_0)^4e^{4m(\varepsilon+2)}/m$, and (UV) by Proposition~\ref{prop:limitEsd}(ii), whose proof uses only the invariance, the reflection positivity and the gap. (c) Let $g$ be supported in $\{x_0\ge t_0\}$, $t_0>0$, and in a cube of side $L$, and $\delta\in(0,1]$. Let $V$ be the set of $p$ with $|p_i|<\delta$ for some $i$. There $u_i<\delta^2$ for that $i$, so $b\le16A_k\delta^2$ and $\cosh E-1\ge4(\cosh(ma_k)-1)/\delta^2\ge2(ma_k/\delta)^2$; since $\cosh x-1 \le x^2$ for $0\le x\le1$, $E\ge ma_k/\delta$ for $ma_k\le\delta$. By ($*$) and Parseval's identity the part of the integral over $V$ is at most $e^{-(m/\delta)(2t_0-a_k)}a_k^8$ times $\sum_x|g(a_kx)|^2\le(L+1)^4a_k^{-4} \|g\|_\infty^2$. On the complement, $|p_1|\ge\delta$ and $|1-e^{ip_1}|\ge 2\delta/\pi$, and ($*$) with $\lambda\le1$ bounds that part by \[ \begin{split} &\Bigl(\frac\pi{2\delta}\Bigr)^2a_k^8\sum_x|g(a_kx)-g(a_kx-a_ke_1)|^2\\ &\quad\le\Bigl(\frac\pi{2\delta}\Bigr)^2(L+2)^4a_k^4\varrho(a_k)^2, \end{split} \] with $\varrho$ the modulus of continuity of $g$ for $|\cdot|_\infty$. With (b), \[ \limsup_kR_k(g)\le\frac{2^{15}(L+2)^4\|g\|_\infty^2}{m\,e^{-4m(\varepsilon+ 2)}}e^{-2mt_0/\delta}, \] and $\delta\to0$ gives $R_k(g)\to0$. The last claim is Corollary~\ref{cor:limitEtwo}(b). (d) By (b) and Corollary~\ref{cor:limitEtwo}(a), whose proof uses only the invariance, the reflection positivity and the gap, (M$_{\mathrm s,2}$) holds for $Z_k=\mathcal N_k(\varepsilon)^{-1/2}$. The moments of a centred Gaussian field are sums over pairings of products of covariances, and every pair of a separated tuple is separated; so (M$_{\mathrm s}$) holds with $M_n\le (n-1)!!M_2^{n/2}$, and by (c) every limit vanishes for $n\ge1$. For $Z_k= C_k(\sigma_0)^{-1/2}$, (M$_{\mathrm s,2}$) would give, as in the proof of Corollary~\ref{cor:limitEtwo}(a), $\mathcal N_k(\varepsilon')\le M_2(Q,\varepsilon')C_k(\sigma_0)$ for $\varepsilon'<\varepsilon$, while $C_k(\sigma_0)\le C_k(0)=R_k(f_0)\mathcal N_k(\varepsilon)$ by Proposition~\ref{prop:limitEnorm}(i), $R_k(f_0)\to0$ by (c), and $\mathcal N_k(\varepsilon')\ge\mathsf K^{-2}\mathcal N_k(\varepsilon)$. \end{proof} So (NE) is an input of its own: reflection positivity, the invariance, a gap bounded above and below in physical units and (SD$_{\mathrm p}$) do not give it, and with the normalisation $Z_k=\mathcal N_k(\varepsilon)^{-1/2}$ the separated moment bound holds for every $n$ while every limit is trivial. By Proposition~\ref{prop:limitElm}, under (SD) it is exactly the statement that bounded spatial momenta carry a fixed fraction of the largest reflected norm, and by Corollary~\ref{cor:limitEabelian} it follows from (SD) for $\U(1)$. The example is not a lattice gauge theory. For the Wilson measure along the tuning the regimes are these: under a gap in lattice units, in particular in the strong-coupling range of Corollary~\ref{cor:limitEstrong}, (SD) fails and the rate $\mathfrak E_k$ diverges like $a_k^{-1}$; under the short-distance law (AF) of Theorem~\ref{thm:limitEaf}, which asymptotic freedom predicts, (SD$_{\mathrm p}$) and (NE) hold; which of the two occurs, if either, is not decided here. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_limitlogic.png} \caption{The conditions of Sec.~\ref{ssec:limitE}, given the averaged condition along the tuning. Bottom: inputs on reflected norms of packets under the Wilson measure, none of them proved. Middle: lattice bounds; (M$_{\mathrm s}$) is (RN) (Proposition~\ref{prop:limitEreflect}), and (G$_{\mathrm s}$) is (RN) with factorial constants (Proposition~\ref{prop:limitErate}). Top: what the bounds give in the limit (Theorems~\ref{thm:limitEsep} and~\ref{thm:limitEtempered}, Corollaries~\ref{cor:limitEOS} and~\ref{cor:limitEexact}). From the bottom row: Theorem~\ref{thm:limitEtwo} and Proposition~\ref{prop:limitEtwotempered} for $n=2$, Proposition~\ref{prop:limitEdomination} for every $n$, Theorems~\ref{thm:limitErot} and~\ref{thm:limitEunique}. Crossed arrows fail in examples: the two-point properties do not give (RN$_m$), $m\ge2$ (Proposition~\ref{prop:limitEthree}), and the bounds of the middle row do not give (W$_{\mathrm s}$) (Proposition~\ref{prop:osobstruction}(a)).}\label{fig:limitlogic} \end{figure*} \begin{remark}\label{rem:limitEscope} \emph{What is proved.} Given the averaged condition along the tuning, Theorem~\ref{thm:limitEsep} turns (M$_{\mathrm s}$) into Schwinger functions at separated points that are symmetric, invariant under the translations and $B_4$, reflection positive in every coordinate direction, with a Hamiltonian of gap $m_*$ in every coordinate direction and exponential clustering in every direction; with the constants of Proposition~\ref{prop:limitEnorm}(ii) they are nontrivial, and with (W$_{\mathrm s}$) and (U$_{\mathrm s}$) they converge and are $\mathrm O(4)$-invariant (Corollary~\ref{cor:limitEexact}). The averaged condition enters only through Lemma~\ref{lem:limitElattice}(iii), that is, through Corollary~\ref{cor:avggap}. For the two-point functions, Theorem~\ref{thm:limitEtwo} and Corollary~\ref{cor:limitEtwo} reduce (M$_{\mathrm s}$) for $n=2$ and nontriviality to (SD) and (NE), statements about reflected norms of single packets under the Wilson measure. For the rotations, Proposition~\ref{prop:limitEaniso} shows that the anisotropy of a separated two-point kernel has no components of degree below four and the angular profile $Y_4$ in degree four, and Theorem~\ref{thm:limitErot} derives (W$_{\mathrm s}$) from (SB$_n$) and (RI$_n$); for the limit, Theorem~\ref{thm:limitEunique} derives (U$_{\mathrm s}$) from (MC$_{m_0}$) and (O$_{\mathrm s}$). For the test functions of~\cite{OS73}, Theorem~\ref{thm:limitEtempered} extends every $S_n$ whose separated moments grow at most like a power of the inverse separation, and under (G$_{\mathrm s}$) gives the linear growth condition of~\cite{OS75}, reflection positivity and clustering on $\mathcal S_0$ (Corollary~\ref{cor:limitEOS}); for $n=2$, (G$_{\mathrm s}$) is the rate (SD$_{\mathrm p}$) in (SD) (Proposition~\ref{prop:limitEtwotempered}). For every $n$, (G$_{\mathrm s}$) is (RN) with constants $K_1^m(m!)^{\beta_1}(1+L)^{q_1m}\delta^{-p_1m}$ (Proposition~\ref{prop:limitErate}), and the inequality (GD) gives (M$_{\mathrm s}$) from (SD) and (G$_{\mathrm s}$) from (SD$_{\mathrm p}$) (Proposition~\ref{prop:limitEdomination}). \emph{Which inputs are needed.} (M$_{\mathrm s}$) with constants that keep a reflected correlation fixed is not implied by the averaged condition, by Corollary~\ref{cor:limitEstrong}. (W$_{\mathrm s}$) is not implied by the other properties: the Gaussian field of Proposition~\ref{prop:osobstruction}(a) is invariant under translations and $B_4$, reflection positive in the coordinate hyperplanes, has gap $ma_k$ in every coordinate direction, satisfies (M) and hence (M$_{\mathrm s}$) by Theorem~\ref{thm:ostree}(i), and the proofs above use the averaged condition only through these properties, with $m$ in place of $m_*$. Its limit has the continuous kernel $\kappa e^{-m\|x-y\|_1}$, so invariance under $R$ on the separated pairs forces $\|Rz\|_1=\|z\|_1$ for $z\ne0$, and $G_S=B_4$. The same example at two interleaved masses violates (U$_{\mathrm s}$), by the argument of Proposition~\ref{prop:uniquelimit}(ii); it satisfies (O$_{\mathrm s}$) and violates (MC$_{m_0}$) for every $m_0$ (Theorem~\ref{thm:limitEunique}(iii)). Products of independent copies of the field are not Gaussian, have the same properties and $G_S=B_4$, and violate (RI$_2$) (Proposition~\ref{prop:limitEproduct}). The normalisation by $C_k(\sigma_0)$ does not fix the scale (Proposition~\ref{prop:limitEscale}). The bounds (RN$_m$), $m\ge2$, are not implied by the two-point properties: the fields of Proposition~\ref{prop:limitEthree}, bounded ones included, have the invariance, the reflection positivity, gap $m$, (SD), (SD$_{\mathrm p}$), (UV), (NE) and nonzero two-point limits, and violate (M$_{\mathrm s,n}$) for every $n\ge3$ and (RN$_m$) for every $m\ge2$. Bounds at each order do not give (G$_{\mathrm s}$): the fields of Proposition~\ref{prop:limitEfactorial} satisfy (M$_{\mathrm s}$), (RN) and (SD$_{\mathrm p}$), with moment bounds uniform in the separation, and their moments grow faster than $K^n(n!)^\beta$ for all $K,\beta$. \emph{Short-distance law.} The two-sided condition (AF), the lattice form of the law $|x-y|^{-8}(\log\frac1{|x-y|})^{-\gamma}$ that asymptotic freedom predicts with $\gamma=2$ (Remark~\ref{rem:limitEafpert}), gives (M$_{\mathrm s,2}$), (SD$_{\mathrm p}$) with $p_0=4$, (SD), (UV) and (NE), excludes trivial limits and fails in the strong-coupling range (Theorem~\ref{thm:limitEaf}). It fixes $Z_k$ up to bounded factors, with $Z_k\to\infty$ and $Z_k$ comparable to $\mathcal N_k(\varepsilon)^{-1/2}$ (Proposition~\ref{prop:limitEafscale}), it is consistent with a gap in physical units and with (U$_{\mathrm s}$), and it implies neither (W$_{\mathrm s}$) nor the exact law (AF$_\infty$) (Proposition~\ref{prop:limitEafexample}). It is not proved for the Wilson measure; the missing inequality is stated in Remark~\ref{rem:limitEafstatus}. \emph{Relation to Theorem~\ref{thm:srladder}.} Under the hypothesis of Theorem~\ref{thm:transfer}, Corollary~\ref{cor:srladder} shows that, when $C_k(0)>0$, (SR$_0$) fails at every spacing for the slice field of $\Phi$ or for that of $\Delta\Phi$. Nothing here uses (SR$_0$): the positivity used is that of the Wilson measure for products of packets with supports at distance at least $\varepsilon$ from the reflection plane, that is, for functions of links at lattice distance at least $\varepsilon/a_k-r$ from it, and $r\ge1$ for every nonconstant $\Phi$. The nontriviality of Proposition~\ref{prop:limitEnorm}(ii) comes from (M$_{\mathrm s}$) and the choice of $Z_k$, not from positivity at short distance. \emph{What is not proved.} (M$_{\mathrm s}$) for $n\ge3$ with the constants of Proposition~\ref{prop:limitEnorm}(ii), equivalently the bounds (RN$_m$), $m\ge2$, on reflected norms of products of separated packets (Proposition~\ref{prop:limitEreflect}), which do not follow from the two-point properties, reflection positivity and the gap (Proposition~\ref{prop:limitEthree}), and the inequality (GD) that gives them from (SD) (Proposition~\ref{prop:limitEdomination}); (SD), equivalently the spectral condition (UV) (Proposition~\ref{prop:limitEsd}), which fails under a gap in lattice units, is not implied by reflection positivity and a gap bounded in physical units (Proposition~\ref{prop:limitEheavy}), is a bound in physical units on the rate $\mathfrak E_k$ at which the largest reflected norm decays (Proposition~\ref{prop:limitEenergy}), and follows from the short-distance law (AF) (Theorem~\ref{thm:limitEaf}); (NE), which under (SD) is the nontriviality of every limit (Proposition~\ref{prop:limitEne}) and the condition (LM) that spatial momenta bounded in physical units carry a fixed fraction of the largest reflected norm (Proposition~\ref{prop:limitElm}), which follows from (SD) for $\U(1)$ (Corollary~\ref{cor:limitEabelian}) and is not implied by (SD$_{\mathrm p}$), reflection positivity, the invariance and a gap bounded above and below in physical units (Proposition~\ref{prop:limitEflat}); (W$_{\mathrm s}$), and (SB$_n$), (RI$_n$), of which (SB$_2$) is a bound on the axis and is not implied by the separated moment bounds (Propositions~\ref{prop:limitEaxial} and~\ref{prop:limitEgauss}(b)); (U$_{\mathrm s}$), and (MC$_{m_0}$), of which the two halves hold separately with different rates (Proposition~\ref{prop:limitEbracket}), and (O$_{\mathrm s}$), equivalently (DS) (Proposition~\ref{prop:limitEdilate}); whether $\mathrm O(4)$ invariance of $S_2$ together with the other properties gives (W$_{\mathrm s}$); the rate (G$_{\mathrm s}$) at coinciding points, equivalently (RN$_{\mathrm p}$) (Proposition~\ref{prop:limitErate}), and its form (SD$_{\mathrm p}$) for $n=2$, which with (M$_{\mathrm s}$) does not give (G$_{\mathrm s}$) (Proposition~\ref{prop:limitEfactorial}); and the averaged condition along the tuning. None of these is proved here, and the mass gap of the continuum theory is not proved. The extension of the $S_n$ to the test functions of~\cite{OS73} with the linear growth condition of~\cite{OS75} is therefore proved only under (G$_{\mathrm s}$), and it does not follow from the properties at separated points: the two-point function of Proposition~\ref{prop:limitEnotempered}(ii) has those that Corollary~\ref{cor:limitEtwo}(b) gives and is not tempered. \end{remark} \subsection{The conditional chain for every compact simple group} \label{ssec:chainG} The lattice gap of Theorem~\ref{thm:groups} holds for every compact simple group in the range $N|\beta|<2\kappa_G/29$. The continuum chain from Sec.~\ref{ssec:averaged} to Corollary~\ref{cor:limitEOS} is posed in the setting of Sec.~\ref{ssec:averaged}, for any compact nontrivial $G\subset\U(N)$ and any positive continuous class weight, but several of its statements are proved for $\SU(2)$ or $\SU(N)$ only. This subsection lists where a property of these groups enters, states the chain for every compact simple group with each hypothesis named, and treats the statements that use $\SU(2)$ or $\SU(N)$: the centre sectors, the strong-coupling range of the averaged condition, the response of one link that the sharp quarter bounds for $\SU(2)$, the obstruction at fixed cube size, and the holonomy flow. Throughout, $G$, $N$, $\ip XY=-\Tr(XY)$, $\kappa_G$, $d=\dim G$, $K=\kappa_G/(d-1)$ and $C_0(G)$ are as in Sec.~\ref{ssec:groups13}, the inclusion $G\subset\U(N)$ is irreducible, as for every row of Table~\ref{tab:groups}, and the Wilson measure of $\Lambda_L$ has the weight $\varphi=e^{N\beta\ReTr(\cdot)}$, that is, $\beta_W=N^2\beta$. The centre $Z(G)$ is finite, since $\g$ is simple, and $\widehat{Z(G)}$ is its group of characters. Lemma~\ref{lem:wilsonZG} gives, for $N\beta\ge K$ and every plaquette, \begin{equation}\label{eq:chainGeps} \varepsilon_G(\beta)=2\mu(N-\ReTr U_p)\le\frac{2}{N\beta}\Bigl(\frac d3\log (N^2\beta)+C_0(G)\Bigr), \end{equation} since $N-\ReTr U_p=Ns_p$ and the plaquettes are equivalent under the symmetries of $\Lambda_L$. This replaces Lemma~\ref{lem:bpflat}: for $\SU(2)$ it reads $\mu(2-\Tr U_p)\le(\log\beta+5.71)/(2\beta)$. \emph{Where the group enters.} Table~\ref{tab:chainG} lists the statements of Secs.~\ref{ssec:averaged}--\ref{ssec:bpup} and~\ref{ssec:limitE}, the property of $G$ that each proof uses, and its status for a general $G$. The forest gauge, the plaquette blocks and the chart bounds do not enter the chain: the averaged condition is posed without gauge fixing. \begin{table*}[t] \caption{The statements of the conditional chain, the property of the gauge group used in their proofs, and their status for every compact simple group $G$.} \label{tab:chainG} \raggedcells \begin{ruledtabular} \begin{tabular}{p{0.31\textwidth}p{0.33\textwidth}p{0.26\textwidth}} Statement&Property of $G$ used&Every compact simple $G$\\ \colrule Lemma~\ref{lem:avgspeed}, Theorem~\ref{thm:avgdecay}&locality of the cube moves; $\ReTr U_p$ is not constant&as stated\\ Corollary~\ref{cor:avggap}, Theorem~\ref{thm:avgtransfer}, Proposition~\ref{prop:spatialflow}\textup{(iii)}&reflection positivity for $\beta\ge0$~\cite{OSei78}; Appendix~\ref{app:transfer}&as stated\\ Propositions~\ref{prop:avgsup}, \ref{prop:bpscales}, \ref{prop:bpnoloc}, \ref{prop:bpsharp}, Theorem~\ref{thm:bpstrong}, Lemma~\ref{lem:bpstokes}&a bi-invariant metric; Haar measure&as stated\\ Corollary~\ref{cor:bpranges}\textup{(i)}, \textup{(ii)}&the sharp quarter on $S^3$; the $\SU(3)$ bounds&part \textup{(iii)}, with $\kappa_G$; Proposition~\ref{prop:modulusG}\\ Lemma~\ref{lem:bpflat}&$\SU(N)$; Haar density of $\SU(2)$&\eqref{eq:chainGeps}\\ Proposition~\ref{prop:bpcentre}, Definition~\ref{def:bploc}\textup{(b)}, Theorem~\ref{thm:bplocal}\textup{(i)}, \textup{(iv)}&the centre $\{\omega^jI\}$ of $\SU(N)$&Proposition~\ref{prop:bpcentreG}\\ Theorem~\ref{thm:bplocal}, other clauses; Lemma~\ref{lem:bpintegrated}&none beyond Sec.~\ref{ssec:averaged}&as stated\\ Lemma~\ref{lem:bpcommute}, Theorem~\ref{thm:bpfixed}, Corollaries~\ref{cor:bpresponse}, \ref{cor:bpcharged}, \ref{cor:bpcoupling}¢re of $\SU(N)$; commuting pairs in $\SU(N)$&under \textup{(CP)}, Proposition~\ref{prop:bpfixedG}; \textup{(CP)} fails for $\mathrm{Sp}(n)$, $n$ even\\ Lemma~\ref{lem:bpflow}, Theorem~\ref{thm:bpneutral}&polar coordinates of $\SU(2)$&Proposition~\ref{prop:bpflowG}\\ Theorem~\ref{thm:bpdescent}, Corollary~\ref{cor:bpdescent}&\textup{(B1)}, \textup{(B2)}&as stated\\ Lemma~\ref{lem:bpupcond}, Theorems~\ref{thm:bpup}, \ref{thm:bpuptower}, Corollaries~\ref{cor:bpuptuning}, \ref{cor:bpupseparate}, Proposition~\ref{prop:bpuphaar}&gauge invariance; translation invariance of Haar measure&as stated\\ Sec.~\ref{ssec:limitE} except Corollary~\ref{cor:limitEstrong}&reflection positivity and the transfer gap of Corollary~\ref{cor:avggap}&as stated\\ Corollary~\ref{cor:limitEstrong}&Corollary~\ref{cor:bpranges}\textup{(i)}&Corollary~\ref{cor:limitEstrongG}\\ \end{tabular} \end{ruledtabular} \end{table*} \emph{Centre sectors.} For a direction $\nu$, $t\in\Z/L\Z$ and $z\in Z(G)$ let $Z^\nu_{t,z}$ multiply every link $(y,\nu)$ with $y_\nu=t$ by $z$ and fix the other links; for $\SU(N)$, $Z^\nu_t=Z^\nu_{t,\omega I}$. For $r=(r_0,\dots,r_3)\in\widehat{Z(G)}^4$ let $\mathcal A_r$ be the set of $f\in\mathcal A_\C$ with $f\circ Z^\nu_{t,z}=r_\nu(z)f$ for all $\nu$, $t$ and $z$, and let $\mathrm{BP}_0(n,\varpi)$ be the inequality of $\mathrm{BP}(n,\varpi)$ for the real $f\in\mathcal A_0$, $0$ the trivial character. For $\SU(N)$ these are the objects of Sec.~\ref{ssec:bpweak}. \begin{proposition}[centre sectors for every group]\label{prop:bpcentreG} Statements \textup{(i)}--\textup{(v)} of Proposition~\textup{\ref{prop:bpcentre}} hold for $G$, with $Z^\nu_{t,z}$ in place of $Z^\nu_t$, $r\in\widehat{Z(G)}^4$, and the condition $f\circ Z^\nu_{t,z}=r_\nu(z)f$ in place of $f\circ Z^\nu =\omega^{r_\nu}f$. Theorem~\textup{\ref{thm:bplocal}} holds with the clauses for $\SU(N)$ read for $G$: $\mathrm{BP}(n,\varpi)$ implies $\mathrm{BP}_0(n,\varpi)$, which implies $\mathrm{BP}_{\rm loc}(n,\varpi)$, and in \textup{(iv)}, $h\in\mathcal A_0$. If $Z(G)=\{I\}$, then $\mathcal A_0=\mathcal A_\C$ and $\mathrm{BP}_0(n,\varpi)$ is $\mathrm{BP}(n,\varpi)$. \end{proposition} \begin{proof} The proof of Proposition~\ref{prop:bpcentre}(i) uses of $z$ only that it commutes with every element of $G$: then every plaquette variable and the density of $\gamma^0_{V_x}(\cdot\mid U)$ are unchanged, Haar measure is invariant under multiplication by $z$, $Z^\nu_{t,z}$ commutes with the gauge group, and $Z^\nu_{t,z}$ is invertible, with inverse $Z^\nu_{t,z^{-1}}$. In (ii) the gauge transformation equal to $z^{-1}$ at the sites with $y_\nu=t+1$ and to $I$ elsewhere shows that $f\circ Z^\nu_{t,z}$ does not depend on $t$ for $f\in\mathcal A_\C$; write $Z^\nu_z$ for the induced map. The maps $f\mapsto f\circ Z^\nu_z$ are unitary on $\mathcal A_\C$, commute with each other, and $Z^\nu_zZ^\nu_{z'}=Z^\nu_{zz'}$, so they form a unitary representation of the finite abelian group $Z(G)^4$. For (iii) put \[ \Pi_rf=\frac1{|Z(G)|^4}\sum_{w\in Z(G)^4}\overline{r(w)}\,f\circ Z^0_{w_0} \cdots Z^3_{w_3}, \] with $r(w)=\prod_\nu r_\nu(w_\nu)$; the orthogonality of the characters of $Z(G)^4$ makes the $\Pi_r$ orthogonal projections onto the $\mathcal A_r$ with $\sum_r\Pi_r=I$ and $\Pi_r\Pi_{r'}=0$ for $r\ne r'$, and by (i) they commute with every $\mathcal P_x$. The proofs of (iv) and (v) and of the clauses of Theorem~\ref{thm:bplocal} use only (i)--(iii) and the fact that spectral projections of $\mathcal K$ commute with the $\Pi_r$. If $Z(G)=\{I\}$, the only character is trivial. \end{proof} For the groups of Table~\ref{tab:groups} the centre is cyclic of order $N$ for $\SU(N)$, of order two for $\SO(2m)$, $\mathrm{Sp}(n)$ and $E_7$, of order three for $E_6$, and trivial for $\SO(2m+1)$, $G_2$, $F_4$ and $E_8$; for the last four groups every gauge-invariant function is neutral. \emph{The chain for every group.} Let $a_k\to0$ be the lattice spacings of Definition~\ref{def:limit}, $\varpi>0$, $\ell>0$ and $n_k\ge2$ integers with $n_ka_k\le\ell$, $\beta_k\ge0$, and $\mathcal L_k$ infinite sets of even integers $L\ge2n_k+4$. \begin{enumerate}[label=\textup{(BP$_{\rm loc}^G$)},leftmargin=*] \item For every $k$ and every $L\in\mathcal L_k$, $\mathrm{BP}_{\rm loc}(n_k, \varpi)$ holds for the Wilson measure of $G$ on $\Lambda_L$ at $\beta_k$. \end{enumerate} \begin{theorem}[the conditional chain for every compact simple group] \label{thm:chainG} \textup{(a)} Each of the following implies \textup{(BP$_{\rm loc}^G$)}: \textup{(a1)} $\mathrm{BP}(n_k,\varpi)$ on the same tori; \textup{(a2)} $\mathrm{BP}_0(n_k,\varpi)$ on the same tori; \textup{(a3)} $a_k=2^{-k}a_0$, $n_k=2^kn_0$, the sets $\mathcal L_k$ of Corollary~\textup{\ref{cor:bpuptuning}}, and its hypotheses \textup{(H1)} and \textup{(H2)} for block-spin maps of $G$, with $\varpi=\min(\varpi_0,1/C_A) /\Gamma$. \textup{(b)} Assume \textup{(BP$_{\rm loc}^G$)}. Every torus limit $\omega_k$ along $\mathcal L_k$ is invariant under the translations and $B_4$, reflection positive, and its transfer matrix has gap at least $16\varpi/(625e^2n_k)$ on $\Omega_k^\perp$. Every continuum limit of $(\omega_k,O_k)$, and of $(\omega_k,O_k\circ\Phi_{s_k})$ for the spatial flow of Proposition~\textup{\ref{prop:spatialflow}}, satisfies $\sigma(H)\subset\{0\}\cup[m_*,\infty)$ with $m_*=16\varpi/(625e^2\ell)$, and $0$ is a simple eigenvalue. \textup{(c)} Assume \textup{(BP$_{\rm loc}^G$)}. For every field $\Phi$ as in Sec.~\textup{\ref{ssec:limitE}} and constants $Z_k$, every lemma, proposition, theorem and corollary of Sec.~\textup{\ref{ssec:limitE}} other than Corollary~\textup{\ref{cor:limitEstrong}} holds for $G$ under the hypotheses stated in it; in particular Theorem~\textup{\ref{thm:limitEsep}} under \textup{(M$_{\mathrm s}$)}, Proposition~\textup{\ref{prop:limitEnorm}}, and Theorem~\textup{\ref{thm:limitEtempered}}. \textup{(d)} Assume \textup{(BP$_{\rm loc}^G$)}, \textup{(G$_{\mathrm s}$)}, \textup{(W$_{\mathrm s}$)} and \textup{(U$_{\mathrm s}$)}. Then the conclusions of Corollary~\textup{\ref{cor:limitEOS}} hold for $G$: the limits extend to $\mathcal S_0(\R^{4n})$, satisfy the linear growth condition of~\cite{OS75} and the axioms of~\cite{OS73} of Euclidean invariance, reflection positivity, symmetry and clustering, the reconstructed Hamiltonian has $\sigma(H_0)\subset\{0\}\cup[m_*,\infty)$ and $\ker H_0=\C\Omega_0$, and for $C_k(0)>0$ and $Z_k=C_k(\sigma_0)^{-1/2}$ the limit is nontrivial. \end{theorem} \begin{proof} The proofs cited below use of $G$ only the following: (G1) $G\subset\U(N)$ is a compact group with Haar measure, and the weight $e^{N\beta\ReTr(\cdot)}$ is a positive continuous class function; (G2) for $\beta\ge0$ and even $L$ the Wilson measure is reflection positive for reflections in planes through sites and bisecting links~\cite{OSei78,Sei82}, which the proof of Corollary~\ref{cor:avggap} uses in the setting of Sec.~\ref{ssec:averaged}; (G3) $\ReTr$ is not constant on $G$, since $\ReTr g=N$ only for $g=I$; (G4) gauge invariance of the measures, the conditions (B1) and (B2), and the invariance of Haar measure under translations. The straight-path map of Sec.~\ref{ssec:bpblock} is a block-spin map of range one for every $G$. (a) Part (a1) is Theorem~\ref{thm:bplocal}(i), whose first clause holds in the setting of Sec.~\ref{ssec:averaged}, and (a2) is Proposition~\ref{prop:bpcentreG}. For (a3), the proofs of Lemma~\ref{lem:bpupcond}, Theorems~\ref{thm:bpup} and~\ref{thm:bpuptower} and Corollary~\ref{cor:bpuptuning} use (G4) only, and give $\mathrm{BP}(n_k,\varpi)$; (a1) applies. (b) This is Theorem~\ref{thm:bplocal}(ii), (iii). Its proof uses Lemma~\ref{lem:avgspeed}, which uses the locality of the cube moves only, the bound $\varpi\le4$, which uses (G3), and the proofs of Corollary~\ref{cor:avggap}, Theorem~\ref{thm:avgtransfer} and Proposition~\ref{prop:spatialflow}(iii), which use (G1), (G2) and Appendix~\ref{app:transfer}; the spatial flow is defined for a bi-invariant metric on any compact $G$. (c) By Remark~\ref{rem:limitEscope}, Sec.~\ref{ssec:limitE} uses the averaged condition only through Lemma~\ref{lem:limitElattice}(iii), that is, through the reflection positivity and the transfer gap of Corollary~\ref{cor:avggap}, which (b) supplies. The remaining arguments concern $\omega_k$, the field $\Phi$, its covariance under $B_4$ and the translations, and Gaussian comparison fields, and contain no property of $G$; the one statement that does, Corollary~\ref{cor:limitEstrong}, is excluded. (d) The proof of Corollary~\ref{cor:limitEOS} uses Corollary~\ref{cor:limitEexact}, Theorems~\ref{thm:limitEsep} and~\ref{thm:limitEtempered} and Proposition~\ref{prop:limitEnorm}(ii), which hold for $G$ by (c). \end{proof} None of the hypotheses of Theorem~\ref{thm:chainG}(b)--(d) is proved for any group along the continuum tuning: neither (BP$_{\rm loc}^G$) with $n_ka_k\le\ell$, nor (H1) and (H2), nor (M$_{\mathrm s}$), (G$_{\mathrm s}$), (W$_{\mathrm s}$), (U$_{\mathrm s}$). The theorem places the open inputs of the continuum step in the same statements for every compact simple group. At strong coupling the averaged condition holds for every group, and the limits are trivial. \begin{corollary}\label{cor:limitEstrongG} Let $0\le N\beta_k\le\kappa_G/48$ for every $k$. Then $\mathrm{BP}(2,\frac{21}{32})$ holds for the Wilson measure of $G$ on every $\Lambda_L$ with $L\ge8$ even, so \textup{(BP$_{\rm loc}^G$)} holds with $\varpi=\frac{21}{32}$, $n_k=2$ and $\ell=2a_0$; every torus limit has transfer-matrix gap at least $m_{\mathrm{lat}}=21/(2500e^2)>1.13\cdot10^{-3}$ in lattice units, and for every $\Phi$ and every choice of $Z_k$ satisfying \textup{(M$_{\mathrm s}$)} every limit of Theorem~\textup{\ref{thm:limitEsep}} is trivial. \end{corollary} \begin{proof} At $n=2$ Corollary~\ref{cor:bpranges}(iii) requires $N|\beta|<4\kappa_G/129$, which holds since $129<192$, and gives \[ \alpha_2=\frac{33N|\beta|}{4(\kappa_G-24N|\beta|)}, \] increasing in $N|\beta|$ and equal to $\frac{33}{96}=\frac{11}{32}$ at $N|\beta|=\kappa_G/48$. So $\mathrm{BP}(2,\frac{21}{32})$ holds on $\Lambda_L$, $L\ge2n+4=8$, and $n_ka_k=2a_k\le2a_0$. By Corollary~\ref{cor:avggap} with $q_2=\frac{625}{16}$ the gap is at least $\varpi/(2e^2q_2)=8\varpi/(625e^2)=21/(2500e^2)$, and $e<2.7183$ gives $2500e^2<18473$, so the gap exceeds $21/18473>1.13\cdot10^{-3}$. Proposition~\ref{prop:limitEnorm}(iii) holds for $G$ by Theorem~\ref{thm:chainG}(c), and applies. \end{proof} For $\SU(2)$ the range of Corollary~\ref{cor:limitEstrongG} is $\beta_W\le \frac1{24}$, a third of the range $\beta_W\le\frac18$ of Corollary~\ref{cor:limitEstrong}; the difference comes from the sharp quarter. \emph{The response of one link.} For $S\in M_N(\C)$ let $\nu_S$ be the probability measure on $G$ with density proportional to $e^{\ReTr(SU)}$ with respect to Haar measure; given the other links, the law of a link under the Wilson measure is $\nu_S$ with $S=N\beta$ times the sum of the six staples. For $A\in M_N(\C)$ put \[ \Lambda_S(A)=\sup_f\Cov_{\nu_S}\bigl(f,\ReTr(AU)\bigr), \] the supremum over the real $f$ on $G$ with $|f(U)-f(V)|\le\lVert U-V \rVert_{\rm HS}$. This is the derivative of $\nu_{S+tA}(f)$ at $t=0$, maximised over these $f$, so by the Kantorovich--Rubinstein duality $W_1(\nu_S,\nu_{S'})\le\sup_{0\le t\le1}\Lambda_{S+t(S'-S)}(S'-S)$ for the Hilbert--Schmidt distance. Moving one partner link by $\delta$ in this distance moves one staple by $\delta$, so $C^0_{ee'}\le N|\beta| \Lambda_G(6N|\beta|)$ for the Hilbert--Schmidt metric, where \[ \Lambda_G(r)=\sup\bigl\{\Lambda_S(A):\lVert S\rVert_{\rm op}\le r,\ \lVert A\rVert_{\rm HS}=1\bigr\}. \] The sharp quarter (Theorem~\ref{q-thm:quarter}) computes $\Lambda_G$ for $\SU(2)$; for a general group the following holds. \begin{proposition}\label{prop:modulusG} \textup{(i)} For $\lVert A\rVert_{\rm HS}=1$, $\Lambda_0(A)\le1/\sqrt d$, and $\Lambda_0(e^{i\theta}A)\ge1/(2N)$ for some $\theta\in\R$; so $1/(2N)\le\Lambda_G(0)\le1/\sqrt d$. \textup{(ii)} For $0\le r<\kappa_G$, $\Lambda_G(r)\le1/(\kappa_G-r)$. \textup{(iii)} For $G=\SU(2)$, $\Lambda_S(A)\le\frac12\lVert A\rVert_{\rm HS}$ for all $S$ and $A$, with equality for some $A\ne0$ at $S=0$; so $\Lambda_G(r)=\frac12$ for every $r\ge0$. \end{proposition} \begin{proof} (i) Let $g(U)=\ReTr(AU)$ and $X_1,\dots,X_d$ an orthonormal basis of $\g$. Since the inclusion is irreducible, $\sum_aX_a^2=-cI$ by Schur's lemma, and taking traces, $cN=d$. With the left-invariant fields $\tilde X_af(U)= \frac d{ds}f(Ue^{sX_a})|_{s=0}$, the Laplacian is $\Delta=\sum_a\tilde X_a^2$ and $\Delta g=\ReTr(AU\sum_aX_a^2)=-\frac dNg$. The $a$-th component of $\nabla g$ is $\ReTr(AUX_a)$, the real Hilbert--Schmidt product of $(AU)^*$ with $X_a$, so $|\nabla g|\le\lVert A\rVert_{\rm HS}=1$ and $g$ is $1$-Lipschitz for the Hilbert--Schmidt distance. By Schur orthogonality, $\int|\Tr(AU)|^2dU=\lVert A\rVert_{\rm HS}^2/N$ and $\int\Tr(AU)dU=0$, and \[ \int g^2\,dU=\frac12\int|\Tr(AU)|^2dU+\frac12\operatorname{Re}\int\Tr(AU)^2 dU \] lies between $0$ and $1/N$. A function that is $1$-Lipschitz for the Hilbert--Schmidt distance is $1$-Lipschitz for the geodesic distance, which is larger, so $|\nabla f|\le1$ almost everywhere, and \[ \Cov(f,g)=-\frac Nd\int f\Delta g=\frac Nd\int\ip{\nabla f}{\nabla g}\le \frac Nd\Bigl(\int|\nabla g|^2\Bigr)^{1/2}, \] with $\int|\nabla g|^2=-\int g\Delta g=\frac dN\int g^2\le\frac dN\cdot \frac1N$; so $\Cov(f,g)\le1/\sqrt d$. Choose $\theta$ with $e^{2i\theta} \int\Tr(AU)^2dU\ge0$; for $A'=e^{i\theta}A$ and $g'=\ReTr(A'U)$, $\int g'^2\ge1/(2N)$, and $f=g'$ is admissible, so $\Lambda_0(A')\ge1/(2N)$. (ii) Let $\lVert S\rVert_{\rm op}\le r$ and $V(U)=-\ReTr(SU)$. Along a geodesic $e^{tX}U$ the second derivative of $V$ is $-\ReTr(SX^2U)$, whose absolute value is at most \[ \lVert S\rVert_{\rm op}\lVert X^2\rVert_1=\lVert S\rVert_{\rm op}|X|^2, \] so $\Ric_G+\Hess V\ge\kappa_G-r>0$. As in the proof of Lemma~\ref{lem:bponelink}, $|\Cov_{\nu_S}(f,g)|$ is at most $\lVert\nabla f \rVert_\infty\lVert\nabla g\rVert_\infty/(\kappa_G-r)$, and both gradients are at most one for $\lVert A\rVert_{\rm HS}=1$. (iii) Write $U=x_0I+x_1\mathbf i+x_2\mathbf j+x_3\mathbf k$ with $x\in S^3 \subset\R^4$, and $U'$ likewise with $x'$, so that $\lVert U-U'\rVert_{\rm HS}=\sqrt2|x-x'|$ and $\ReTr(AU)=\ip wx$, where $w_a$ is the real Hilbert--Schmidt product of $A^*$ with the $a$-th basis matrix, of norm $\sqrt2$; hence $|w|\le\sqrt2\lVert A \rVert_{\rm HS}$, with equality when $A^*$ is a real multiple of an element of $\SU(2)$. Then $\nu_S=\nu_b$ in the notation of Theorem~\ref{q-thm:quarter}, with $b\in\R^4$ defined by $\ReTr(SU)=\ip bx$, and $f$ is $\sqrt2$-Lipschitz for the chordal distance of the unit sphere. So, for $w\ne0$, \[ \begin{split} \Cov_{\nu_S}(f,\ip wx)&=\sqrt2|w|\Cov_{\nu_b}\bigl(f/\sqrt2,\ip{w/|w|}x \bigr)\\ &\le\sqrt2|w|\,L(b,w/|w|)\le\tfrac12\lVert A\rVert_{\rm HS}, \end{split} \] with equality at $b=0$ for $A^*\in\R\,\SU(2)$ and $f=\sqrt2\ip{w/|w|}x$. \end{proof} Lemma~\ref{lem:bponelink} is the bound of Proposition~\ref{prop:modulusG}(ii) with $r=6N|\beta|$, in the geodesic metric, and Corollary~\ref{cor:bpranges}(i) rests on part (iii): $C^0_{ee'}\le N|\beta|\cdot\frac12=|\beta|$, uniformly in $\beta$. Through Theorem~\ref{thm:bpstrong}, a bound $\Lambda_G(6N|\beta|) \le\lambda$ gives $\mathrm{BP}(n,1-\alpha_n)$ with $\alpha_n<1$ for large $n$ whenever $18N|\beta|\lambda<1$. With $\lambda=1/(\kappa_G-6N|\beta|)$ this is the range $N|\beta|<\kappa_G/24$ that Corollary~\ref{cor:bpranges}(iii) reaches as $n\to\infty$; for $\SU(2)$, $\lambda=\frac12$ gives $|\beta|< \frac1{18}$, that is $\beta_W<\frac29$, the range of Corollary~\ref{cor:bpranges}(i). The analogue of the sharp quarter for $G$ is a bound $\lambda$ of the order of $\Lambda_G(0)\le1/\sqrt d$, uniform in $r\le6N|\beta|$, which would give a range of order $\sqrt d$ in place of $\kappa_G$. At $r=0$ the ratio of the bound of part (ii) to that of part (i) is $\sqrt d/\kappa_G$, which is $2\sqrt{N^2-1}/N<2$ for $\SU(N)$ and $4\sqrt{248}>62$ for $E_8$. The value of $\Lambda_G(r)$ for $r>0$ and $G\ne \SU(2)$ is not determined here. The proof of Theorem~\ref{q-thm:quarter} uses that $\SU(2)$ is the unit sphere of $\R^4$, on which $\nu_S$ is a von Mises--Fisher law, and the harmonic extension of a Lipschitz function to the ball; it is not extended here to other groups. \emph{The obstruction at fixed cube size.} Theorem~\ref{thm:bpfixed} uses the centre of $\SU(N)$ twice: the charged traces of Polyakov loops have mean zero, and on commuting pairs some charged trace is bounded below. For a general group the first part carries over whenever $Z(G)\ne\{I\}$, and the second becomes a condition on $G$. Let $A=P_0$ and $B$ be the Polyakov loops of directions $0$ and $1$ based at the site $0$, as in Sec.~\ref{ssec:bpstrong}. \begin{enumerate}[label=\textup{(CP)},leftmargin=*] \item There are continuous $F_1,\dots,F_m:G\times G\to\C$ and $\ell>0$ such that for every $k$: $F_k(gA'g^{-1},gB'g^{-1})=F_k(A',B')$; there are $z,z'\in Z(G)$ and $\lambda_k\ne1$ with $F_k(zA',z'B')=\lambda_kF_k(A',B')$; $|F_k|\le1$ and $|F_k(HA',H'B')-F_k(A',B')|\le\ell(\lVert H-I\rVert_{\rm op} +\lVert H'-I\rVert_{\rm op})$; for all $g,A',B',H,H'\in G$. Moreover $\sum_k|F_k(A',B')|^2>0$ for every commuting pair $(A',B')$. \end{enumerate} Under (CP), by compactness there are $c_F>0$ and $\eta_F>0$ with $\sum_k|F_k(A',B')|^2\ge c_F$ whenever $\lVert A'B'A'^{-1}B'^{-1}-I \rVert_{\rm HS}\le\eta_F$, as in the proof of Lemma~\ref{lem:bpcommute}. \begin{proposition}\label{prop:bpfixedG} Let $\beta\ge0$ and $n\ge2$. \textup{(i)} If $Z(G)\ne\{I\}$, there is an irreducible unitary representation $\sigma$ of $G$, of dimension $N_\sigma$, whose central character is not trivial, and $\mathrm{BP}(n,\varpi)$ on $\Lambda_L$ implies \[ \frac1{L^3}\sum_{y:\,y_0=0}\mu\bigl(\chi_\sigma(P_0)\overline{\chi_\sigma (P_y)}\bigr)\le\frac{2N_\sigma^2n^2}{\varpi L^2},\quad\chi_\sigma=\Tr \sigma. \] \textup{(ii)} Under \textup{(CP)}, for $N\beta\ge K$, $\mathrm{BP}(n,\varpi)$ on $\Lambda_{2n+4}$ implies \[ \varpi\le2048\,n^4\varepsilon_G(\beta)\max\Bigl(\frac{m\ell^2}{c_F}, \frac1{\eta_F^2}\Bigr), \] with $\varepsilon_G(\beta)$ bounded by~\eqref{eq:chainGeps}; so along any sequence with $n_k^4\log(N^2\beta_k)/\beta_k\to0$ no $\varpi>0$ satisfies $\mathrm{BP}(n_k,\varpi)$ uniformly in the torus. \textup{(iii)} $\SU(N)$ satisfies \textup{(CP)} with $F_{ab}=\Tr(A^aB^b)/N$, $(a,b)\in I_N$, $m=N^2-1$ and $\ell=N-1$. $\mathrm{Sp}(n')\subset\U(2n')$ with $n'$ even does not satisfy \textup{(CP)}: there is a commuting pair at which every function with the first two properties of \textup{(CP)} vanishes. If $Z(G)=\{I\}$, no function satisfies the second property. \end{proposition} \begin{proof} (i) Let $I\ne z\in Z(G)$. The inclusion $G\subset\U(N)$ is faithful and irreducible, so $z$ acts as a scalar $\zeta\ne1$ and $\sigma$ may be the inclusion; in general any irreducible $\sigma$ with $\sigma(z)\ne I$ serves. The twist $Z^0_{0,z}$ preserves $\mu$ (Proposition~\ref{prop:bpcentreG}) and maps $\chi_\sigma(P_y)$ to $\zeta_\sigma\chi_\sigma(P_y)$, $\zeta_\sigma\ne 1$, so $\mu(\chi_\sigma(P_y))=0$. The proof of Theorem~\ref{thm:bpfixed}(i) applies to $g_y=\operatorname{Re}\chi_\sigma(P_y)$ and to the imaginary parts, with $|\chi_\sigma|\le N_\sigma$ in place of $|\Tr|\le N$. (ii) Let $L=2n+4\le4n$, $\varepsilon=\varepsilon_G(\beta)=\mu(\lVert U_p-I \rVert_{\rm HS}^2)$, and $f$ the real or the imaginary part of some $F_k(A,B)$. The twist that multiplies the links $(y,0)$ with $y_0=0$ by $z$ and the links $(y,1)$ with $y_1=0$ by $z'$ preserves $\mu$ and maps $F_k(A,B)$ to $\lambda_kF_k(A,B)$, so $\mu(F_k(A,B))=0$. If $V_x$ meets neither $A$ nor $B$, $\mathcal P_xf=f$. Otherwise the proof of Theorem~\ref{thm:bpfixed}(ii) gives loops $A'$, $B'$ that avoid $V_x$, a holonomy $\Gamma$, and $A=H_0\Gamma A'\Gamma^{-1}$, $B=H_1\Gamma B'\Gamma^{-1}$ with $H_0,H_1\in G$ and $\lVert H_0-I\rVert_{\rm op}+ \lVert H_1-I\rVert_{\rm op}\le S_0+S_1$, the sum of $\lVert U_p-I \rVert_{\rm op}$ over at most $2Ln$ plaquettes. By the invariance under simultaneous conjugation, $F_k(\Gamma A'\Gamma^{-1},\Gamma B'\Gamma^{-1})= F_k(A',B')$, a function of the links outside $V_x$, so $\lVert f-\mathcal P_xf\rVert_2\le\ell\lVert S_0+S_1\rVert_2\le2\ell Ln\sqrt\varepsilon$. At most $2Ln^3$ points $x$ have $V_x$ meeting $A\cup B$, so $\mathcal E_n(f)\le 8\ell^2L^3n\varepsilon$. By Lemma~\ref{lem:bpstokes} the commutator of $A$ and $B$ is within $\sum_p\lVert U_p-I\rVert_{\rm HS}$ of $I$, a sum over $L^2$ plaquettes, so it exceeds $\eta_F$ with probability at most $L^2\sqrt\varepsilon/ \eta_F$, and $\sum_k\mu(|F_k(A,B)|^2)\ge c_F(1-L^2\sqrt\varepsilon/\eta_F)$. As $\mu(F_k(A,B))=0$, one of the $2m$ functions $f$ has $\Var_\mu(f)\ge c_F(1-L^2\sqrt\varepsilon/\eta_F)/(2m)$. If $L^2\sqrt\varepsilon\le\eta_F/ 2$, $\mathrm{BP}(n,\varpi)$ gives $\varpi\le32m\ell^2L^3n\varepsilon/c_F \le2048m\ell^2n^4\varepsilon/c_F$. Otherwise $\eta_F^2<4L^4\varepsilon\le 1024n^4\varepsilon$, and $\varpi\le2$ by Proposition~\ref{prop:bpscales}(ii) gives $\varpi\le2048n^4\varepsilon/\eta_F^2$. (iii) For $\SU(N)$, $F_{ab}$ is invariant under simultaneous conjugation; for $(c,d)$ with $ac+bd\not\equiv0$ modulo $N$, $z=\omega^cI$ and $z'= \omega^dI$ give $\lambda=\omega^{ac+bd}\ne1$; $|F_{ab}|\le1$; telescoping, $\lVert(HA')^a-A'^a\rVert_{\rm op}\le a\lVert H-I\rVert_{\rm op}$, so $\ell=N-1$; and Lemma~\ref{lem:bpcommute} gives the last property. For $\mathrm{Sp}(n')=\{g\in\U(2n'):g^T\Omega g=\Omega\}$, $\Omega= \bigl(\begin{smallmatrix}0&I\\-I&0\end{smallmatrix}\bigr)$, $n'$ even, put $A'=\operatorname{diag}(iI,-iI)$ and $B'=\operatorname{diag}(D,D)$ with $D=\operatorname{diag}(I_{n'/2},-I_{n'/2})$. Both lie in $\mathrm{Sp}(n')$ and commute, and $Z(\mathrm{Sp}(n'))=\{\pm I\}$. The element $\Omega$ lies in $\mathrm{Sp}(n')$ and exchanges the two diagonal blocks, so $\Omega A' \Omega^{-1}=-A'$ and $\Omega B'\Omega^{-1}=B'$. With $Q$ the permutation matrix that exchanges the two halves of $\{1,\dots,n'\}$, $q= \operatorname{diag}(Q,Q)\in\mathrm{Sp}(n')$, $qA'q^{-1}=A'$ and $qB'q^{-1} =-B'$. So $(A',B')$ is simultaneously conjugate to $(zA',z'B')$ for all $z,z'\in\{\pm I\}$, and a function invariant under simultaneous conjugation with $F(zA',z'B')=\lambda F(A',B')$, $\lambda\ne1$, has $F(A',B')=\lambda F(A',B')=0$. If $Z(G)=\{I\}$, $F(zA',z'B')=F(A',B')$. \end{proof} For $\mathrm{Sp}(1)=\SU(2)$ the pair of Proposition~\ref{prop:bpfixedG}(iii) does not exist, and (CP) holds. For $\mathrm{Sp}(n')$ with $n'$ even the proof of Theorem~\ref{thm:bpfixed}(ii), (iii) fails at the variance bound when the law of $(A,B)$ is near the orbit of $(A',B')$, and for $\SO(2m+1)$, $G_2$, $F_4$ and $E_8$ there is no charged function at all. Whether $\mathrm{BP}(n,\varpi)$ fails on $\Lambda_{2n+4}$ at weak coupling for these groups, and whether (CP) holds for $\mathrm{Sp}(n')$ with $n'\ge3$ odd, $\SO(2m)$, $E_6$ and $E_7$, is not determined here. Part (i) holds for every $G$ with nontrivial centre and has no content for the four groups with trivial centre. \emph{The holonomy flow.} Let $\chi$ be a smooth real class function on $G$ with $\chi(zW)=\chi(W)$ for $z\in Z(G)$, and define $J(W)\in\g$ by $\ip{J(W)}X =\frac d{ds}\chi(e^{sX}W)|_{s=0}$. Put $\ell_0=\sup|\chi(W)-\chi(W')|/\lVert W-W'\rVert_{\rm op}$, $\ell_1=\sup\lVert J(W)-J(W')\rVert_{\rm HS}/\lVert W-W'\rVert_{\rm HS}$ and $D_\chi=\sup|\Delta\chi|$, all finite since $G$ is a compact submanifold of $M_N(\C)$. On $G^E$ let $\mathcal V$ have the component $L^{-1}J(P_b)U_{(b,0)}$ at the link $(b,0)$, with $P_b$ the Polyakov loop of direction $0$ based at $b$, and vanish at the links of the other directions; put $S=-N\beta\sum_p\ReTr U_p$, $H=\sum_p(N-\ReTr U_p)$ and $\Theta=\mathcal VS-\operatorname{div}\mathcal V$. For $\SU(2)$ and $\chi(W)=-\frac12(\Tr W)^2$, $J(W)=\Tr W\cdot\frac12(W-W^*)$ is the field of Sec.~\ref{ssec:bpneutral}, with $\ell_1\le4$ and $\Delta\chi=2(\Tr W)^2-2$. \begin{proposition}[the holonomy flow for every group]\label{prop:bpflowG} \textup{(i)} $J(gWg^{-1})=gJ(W)g^{-1}$, $J(W)$ commutes with $W$, $J(zW)=J(W)$ for $z\in Z(G)$, and $J(z)=0$. \textup{(ii)} $\mathcal VP_y=J(P_y)P_y$ and $\mathcal V\chi(P_y)=|J(P_y)|^2$ for every site $y$. \textup{(iii)} $\operatorname{div}\mathcal V=\sum_{y:\,y_0=0}\Delta\chi(P_y)$. \textup{(iv)} $|\mathcal VS|\le2N\beta\ell_1H$, and $\lVert\Theta\rVert_2\le 4\ell_1(M+2)+L^3D_\chi$ with $M=\log\mu(e^{N\beta H/2})$; for $N\beta\ge2K$, $M\le6L^4(\frac d3\log\frac{N^2\beta}2+C_0(G))$. \textup{(v)} $\mu(\mathcal Vh)=\mu(h\Theta)$ and $\mu(\mathcal Vh)^2\le \Var_\mu(h)\lVert\Theta\rVert_2^2$ for every real $h$ of class $C^1$. \textup{(vi)} If $N\beta\ge2K$, $n\ge2$, $L\ge2n+4$ is even and $\mathrm{BP}_0(n,\varpi)$ holds on $\Lambda_L$, then \[ \mu\bigl(|J(P_0)|^2\bigr)^2\le\frac{\ell_0^2L^3n\,\varepsilon_G(\beta)} {\varpi}\,\lVert\Theta\rVert_2^2 . \] \end{proposition} \begin{proof} (i) Since $\chi$ is a class function and the metric is invariant under $\operatorname{Ad}$, $\ip{J(gWg^{-1})}{X}=\ip{J(W)}{g^{-1}Xg}$, which is the first identity. Conjugating by $e^{sX}$, $\chi(We^{sX})=\chi(e^{sX}W)$, and $\chi(We^{sX})=\chi(e^{s\operatorname{Ad}_WX}W)$; so $\ip{J(W)}X=\ip{J(W)} {\operatorname{Ad}_WX}$ for all $X$, and $J(W)$ is fixed by $\operatorname{Ad}_{W^{-1}}$. The third identity follows from $\chi(e^{sX}zW) =\chi(e^{sX}W)$. For $z$ central, $J(z)=gJ(z)g^{-1}$ for all $g$, so $J(z)$ lies in the centre of $\g$, which is zero. (ii) The computation in the proof of Lemma~\ref{lem:bpflow}(i) uses only the first identity of (i). Then $\mathcal V\chi(P_y)=\frac d{ds}\chi(e^{sJ(P_y)} P_y)|_{s=0}=\ip{J(P_y)}{J(P_y)}$. (iii) As in the proof of Lemma~\ref{lem:bpflow}(ii), the right translation $U_e\mapsto W=U_eR$ carries the component at $e$ to $L^{-1}J(W)W$, which is $L^{-1}$ times the gradient of $\chi$ for the bi-invariant metric; its divergence for Haar measure is $L^{-1}\Delta\chi(W)$, and $W$ is a conjugate of the loop through the initial site of $e$, so $\Delta\chi(W)=\Delta\chi (P_y)$. Each of the $L^3$ loops has $L$ links. (iv) The computation of Lemma~\ref{lem:bpflow}(iii) uses only the first identity of (i) and gives $\mathcal V\ReTr U_p=L^{-1}\operatorname{Re}\Tr (U_p\Delta_p)$ with $\Delta_p=J(P_b)-J(VP_{b+e_\nu}V^{-1})\in\g$ and $\lVert\Delta_p\rVert_{\rm HS}\le\ell_1Q_\sigma$. Since $\operatorname{Re} \Tr\Delta_p=0$, $|\operatorname{Re}\Tr(U_p\Delta_p)|\le\lVert U_p-I \rVert_{\rm HS}\ell_1Q_\sigma$, and the Cauchy--Schwarz inequality over each strip and $\lVert U-I\rVert_{\rm HS}^2=2(N-\ReTr U)$ give $|\mathcal VS|\le 2N\beta\ell_1H$. Put $X=N\beta H/2$, so $\mu(e^X)=e^M$; the inequality $X^2\le M^2+(2M+2)e^{X-M}$ of that proof gives $\mu(X^2)\le(M+2)^2$, so $\lVert \mathcal VS\rVert_2\le4\ell_1\lVert X\rVert_2\le4\ell_1(M+2)$, and $| \operatorname{div}\mathcal V|\le L^3D_\chi$ by (iii). With $\phi(t)=\log\int e^{-NtH}dU$, convexity gives $M=\phi(\beta/2)-\phi(\beta)\le\frac{N\beta}2 \mu_{\beta/2}(H)$, where $\mu_{\beta/2}$ is the Wilson measure at $\beta/2$, and Lemma~\ref{lem:wilsonZG} at $\beta_W=N^2\beta/2\ge KN$ bounds $\mu_{\beta/2}(H)$ by $6L^4N(\frac d3\log\frac{N^2\beta}2+C_0(G))/(N^2 \beta/2)$. (v) This is the proof of Lemma~\ref{lem:bpflow}(iv). (vi) Let $h=\chi(P_0)$, real and gauge invariant. The twist $Z^0_{0,z}$ maps $P_0$ to $zP_0$, and $Z^\nu_{t,z}$, $\nu\ne0$, fixes it, so $h\in\mathcal A_0$. As in the proof of Theorem~\ref{thm:bpneutral}(i), for each of the $Ln^3$ points $x$ with $V_x$ meeting $P_0$ there is a loop $A'$ avoiding $V_x$ with $P_0=H_0\Gamma A'\Gamma^{-1}$ and $\lVert H_0-I\rVert_{\rm op}\le S_0$, a sum over at most $Ln$ plaquettes; then $|h-\chi(A')|=|\chi(H_0W)- \chi(W)|\le\ell_0S_0$ with $W=\Gamma A'\Gamma^{-1}$, and $\mu((h-\mathcal P_xh)^2)\le\ell_0^2L^2n^2\varepsilon_G(\beta)$. So $\mathcal E_n(h)\le \ell_0^2L^3n\varepsilon_G(\beta)$, $\mathrm{BP}_0(n,\varpi)$ bounds $\Var_\mu (h)$ by this over $\varpi$, and (v) with $\mathcal Vh=|J(P_0)|^2$ from (ii) gives the claim. \end{proof} On $\Lambda_{2n+4}$, part (iv) gives $\lVert\Theta\rVert_2\le c_1n^4(\log (N^2\beta)+c_2)$ with $c_1,c_2$ depending on $G$ and $\chi$ only, so $\mathrm{BP}_0(n,\varpi)$ with a fixed $\varpi$ at $\beta\gg n^{12}\log^3\beta$ forces $\mu(|J(P_0)|^2)$ to zero: the law of the Polyakov loop concentrates near the critical set of $\chi$, which contains $Z(G)$ by (i). For $\SU(2)$ and $\chi=-\frac12(\Tr W)^2$, $|J(W)|^2=\frac12(\Tr W)^2(4-(\Tr W)^2)$ and $\ell_0\le4$, and (vi) reproduces Theorem~\ref{thm:bpneutral}(i) with \eqref{eq:chainGeps} in place of Lemma~\ref{lem:bpflat}. For $\SU(2)$ every admissible field vanishes on $\{\pm I\}\cup\{\Tr W=0\}$ (the discussion after Theorem~\ref{thm:bpneutral}); for other groups the common critical set of the admissible $\chi$ is not determined here. As for $\SU(2)$, a lower bound on $\mu(|J(P_0)|^2)$ at weak coupling on $\Lambda_{2n+4}$ is not proved for any $\chi$, so the failure of $\mathrm{BP}_0$ there is not proved for any group. \begin{remark}[what is proved for every group and what remains] \label{rem:chainG} Proved for every compact simple $G$: the conditional chain of Theorem~\ref{thm:chainG}, with the hypothesis (BP$_{\rm loc}^G$) and, for the Schwinger functions, (M$_{\mathrm s}$), (G$_{\mathrm s}$), (W$_{\mathrm s}$) and (U$_{\mathrm s}$); the centre sectors for $Z(G)$ (Proposition~\ref{prop:bpcentreG}); the averaged condition $\mathrm{BP}(2,\frac{21}{32})$ for $N\beta\le\kappa_G/48$ and the triviality of the limits there (Corollary~\ref{cor:limitEstrongG}); two-sided bounds on the response of one link at $S=0$ and the Bakry--\'Emery bound for $\lVert S\rVert_{\rm op}<\kappa_G$ (Proposition~\ref{prop:modulusG}); the obstruction at fixed cube size under (CP), which holds for $\SU(N)$ and fails for $\mathrm{Sp}(n')$ with $n'$ even (Proposition~\ref{prop:bpfixedG}); the holonomy flow for every centre-invariant class function (Proposition~\ref{prop:bpflowG}). Not proved: the hypotheses of Theorem~\ref{thm:chainG} along the tuning for any group; an analogue of the sharp quarter, a bound on $\Lambda_G(r)$ of the order of $\Lambda_G(0)$, for any $G\ne\SU(2)$; (CP) for the groups other than $\SU(N)$ and $\mathrm{Sp}(n')$, $n'$ even; any obstruction at fixed cube size for the groups with trivial centre; and the mass gap. \end{remark} \section{Block averages}\label{sec:blocks} Proposition~\ref{prop:gaussforest} locates the failure of (D) in the quadratic approximation: gauge transformations constant on the lines of $F$ move the whole cube and change few links outside it. By Proposition~\ref{prop:gaussevery} every gauge fixed on an invariant forest fails in the same way, through the constant fields. This section takes up the programme of the report~\cite{Loc26} of the third author, which keeps the cube condition but changes the gauge and conditions the cube on block averages of the field, as in Ba{\l}aban's renormalisation group~\cite{Bal84a,Bal85a}. The natural block averages of the line field leave one class of lines invisible and fail at every size (Proposition~\ref{prop:invisible}). In a gauge whose fixed links are trees inside blocks of side two, the averages over whole blocks remove every zero mode, the constrained quadratic form has a gap that does not depend on the volume (Theorem~\ref{thm:blockgap}), and the constrained cube condition holds at every coupling once the cube is large (Theorem~\ref{thm:constrainedcube}). Integrating out one scale bounds the cube coefficients by the constrained ones plus those of the averages (Corollary~\ref{cor:compose}), and the averages carry the coarse quadratic form at every scale up to a factor between $1$ and $256$, through forms that are local uniformly in the scale (Proposition~\ref{prop:effective} and Theorems~\ref{thm:effectiveupper} and~\ref{thm:effectivelocal}), and the constrained cube condition holds at every scale with one cube side (Theorem~\ref{thm:constrainedscales}). For the group-valued links themselves, the straight paths between block roots satisfy a coercivity bound for all fields but do not compose, a continuous covariant map cannot treat the transverse positions alike, and a cut-off of Ba{\l}aban's averaging operation is coercive for all fields and agrees with the block averages to first order (Sec.~\ref{ssec:pathblocks}). It composes across all scales on fields of small curvature for every compact group (Sec.~\ref{ssec:scales}), and under the $\SU(2)$ Wilson measure the fields that are not small form clusters of vanishing physical size (Sec.~\ref{ssec:largefields}). \subsection{Block-constrained cubes}\label{ssec:constrained} In this subsection everything concerns the quadratic approximation: real fields $\varphi$ on links, $(d\varphi)_p$ the signed sum around the plaquette $p$, and $(dh)_{(x,\mu)}=h(x)-h(x+e_\mu)$ for a function $h$ on sites. \begin{proposition}[\cite{Loc26}]\label{prop:invisible} In the setting of Proposition~\ref{prop:gaussforest}, condition the cube in addition on any family of the face averages \[ V^{\mathrm{face}}_{y,\mu}(\varphi)=\frac18\sum_{x}\bigl(\varphi_{(x,\mu)} +\varphi_{(x+e_\mu,\mu)}\bigr), \] where $x$ runs over the eight sites of the block $2y+\{0,1\}^4$ with $x_\mu=2y_\mu$. Then $\sum_{c\in C_I}K_c\ge12N_n$ for the twelve classes \[ \begin{split} C_I=\{&(0111,\mu),(1111,\mu),(0111+e_\mu,\mu),(1111+e_\mu,\mu)\\ &:\mu=1,2,3\}, \end{split} \] written as residue of the base and direction. Hence $K_c\ge N_n$ for some class, and the constrained condition fails for every $\alpha<1$, at every even $n$ and every $b$. \end{proposition} \begin{proof} By Table~\ref{tab:sigma}, $\sigma(0111)=\sigma(1111)=0$, so the lines of $F$ through the sites with $x_1,x_2,x_3$ odd have direction $0$; call them invisible. Let $\ell$ be one of them and $h_\ell$ its indicator. The sum of $dh_\ell$ along a path of length two from $x$ is $h_\ell(x)-h_\ell(x+2e_\mu)$. For $\mu\ne0$ both end points have even $\mu$-coordinate and are off $\ell$; for $\mu=0$ they have the same transverse coordinates and are both on $\ell$ or both off it. So $dh_\ell$ changes no face average, and the translation argument of Proposition~\ref{prop:gaussforest} applies to the constrained laws: $\sum_{i\in V}|(dh_\ell)_i|\le\sum_{j\notin V}C_{V,j}|(dh_\ell)_j|$. An active link $e=(x,\mu)$ has $|(dh_\ell)_e|=1$ for at most one invisible $\ell$, since its two ends lie on different lines and both cannot be invisible (for $\mu\ne0$ they differ in the coordinate $\mu$, and for $\mu=0$ the link would lie in $F$). It has one such $\ell$ exactly when its base or its end has residue $0111$ or $1111$, which by Table~\ref{tab:sigma} are the classes of $C_I$: the six classes with base residue $0111$ or $1111$ and direction $\mu\ne0$, and the six with base residue $r+e_\mu$, $r\in\{0111, 1111\}$, whose directions $\mu$ differ from $\sigma(r+e_\mu)$ (for instance $\sigma(0011)=3\ne1$). Summing over the invisible lines gives $12N_n\le \sum_{c\in C_I}K_c$. \end{proof} \begin{definition}\label{def:blockgauge} The blocks are $B(y)=2y+\{0,1\}^4$, $y\in\Z^4$, and $o=x\bmod2$ is the offset of a site. A link $(x,\mu)$ is a \emph{tree link} if $o_\mu=0$ and $o_\nu=0$ for every $\nu>\mu$, and \emph{active} otherwise; fields vanish on tree links. For a coarse link $(y,\mu)$ the \emph{block average} is \[ V_{y,\mu}(\varphi)=\frac1{16}\sum_{x\in B(y)}\bigl(\varphi_{(x,\mu)}+ \varphi_{(x+e_\mu,\mu)}\bigr), \] the mean of $\varphi$ summed along the sixteen paths of length two in the direction $\mu$ that start in $B(y)$. \end{definition} \begin{lemma}\label{lem:blocktree} \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The fifteen tree links of a block form a spanning tree of its sixteen sites. Each block carries $49$ active links: the $17$ active links inside it, with $o_\mu=0$, and the $32$ links with $o_\mu=1$, which join $B(y)$ to $B(y+e_\mu)$. \item If $dh$ vanishes on the tree links, $h$ is constant on every block, and $V_{y,\mu}(\varphi+dh)=V_{y,\mu}(\varphi)+h_{B(y)}-h_{B(y+e_\mu)}$. \item With $S^\mu_y=\sum_{x\in B(y),\,o_\mu=0}\varphi_{(x,\mu)}$, \[ V_{y,\mu}(\varphi)=\frac18\sum_{x\in B(y),\,o_\mu=1}\varphi_{(x,\mu)} +\frac1{16}\bigl(S^\mu_y+S^\mu_{y+e_\mu}\bigr). \] \end{enumerate} \end{lemma} \begin{proof} (i) A tree link of direction $\mu$ has $o_\mu=0$ and stays in its block; there are $2^\mu$ of them for each $\mu$, fifteen in all. From any site, the tree links of direction $3$ lead to $o_3=0$, then those of direction $2$ to $o_2=0$, and so on to the corner, so the tree links connect the sixteen sites, and fifteen edges connecting sixteen vertices form a tree. Of the $64$ links with base in the block, the $32$ with $o_\mu=0$ stay inside it. (ii) $h$ is constant along the edges of a spanning tree. The path of length two from $x\in B(y)$ ends at $x+2e_\mu\in B(y+e_\mu)$, and the sum of $dh$ along it is $h(x)-h(x+2e_\mu)$. (iii) The path from $x$ consists of $(x,\mu)$ and $(x+e_\mu,\mu)$; a link with $o_\mu=1$ is the second link of one path and the first of another, and the links of $S^\mu_{y+e_\mu}$ are the second links of the paths that start with $o_\mu=1$. \end{proof} By (ii) the residual gauge freedom is the coarse gauge group, which acts on the block averages as on the links of the coarse lattice. Let $W$ be a set of blocks and let $\Phi_W$ be the space of finitely supported real $\varphi$ on the active links with base in $\bigcup W$, with $V_{y,\mu}(\varphi)=0$ for every $y\in W$ and every $\mu$. Write $Q(\varphi)=\sum_p(d\varphi)_p^2$ over all plaquettes, $\varphi$ being zero off its support. \begin{theorem}\label{thm:blockgap} For every set of blocks $W$ and every $\varphi\in\Phi_W$, \[ \sum_e\varphi_e^2\le52\,Q(\varphi). \] \end{theorem} \begin{proof} \emph{Inside a block.} Let $e=(x,\mu)$ be an active link with $o_\mu=0$, and let $\nu_1>\dots>\nu_k$ be the directions above $\mu$ in which $o$ equals one; $1\le k\le3$. Put $x_0=x$ and $x_i=x_{i-1}-e_{\nu_i}$. The plaquette $p_i$ at $x_i$ in the plane $\{\mu,\nu_i\}$ has the links $(x_i,\mu)$, $(x_{i-1},\mu)$ and the tree links $(x_i,\nu_i)$, $(x_i+e_\mu,\nu_i)$, whose offsets vanish in $\nu_i$ and above it; and $(x_k,\mu)$ is a tree link. Hence $\varphi_e=\pm (d\varphi)_{p_1}\pm\dots\pm(d\varphi)_{p_k}$ and $\varphi_e^2\le k\sum_i (d\varphi)_{p_i}^2$. The plaquette at $x'$ in the plane $\{\mu,\nu\}$ is used only by the links $(x'+e_\nu+\sum_{\rho\in R}e_\rho,\mu)$ with $R\subset\{\nu+1, \dots,3\}$, each with $k=c+1+|R|$, where $c$ is the number of ones of the offset of $x'$ strictly between $\mu$ and $\nu$. Its total weight $2^{3-\nu}(c+1)+(3-\nu)2^{2-\nu}$ is at most $8$, the value at $\mu=0$, $\nu=1$. So the sum $I$ of $\varphi_e^2$ over the active links inside the blocks is at most $8Q_{\rm in}$, where $Q_{\rm in}$ is the sum over the plaquettes inside a block. \emph{Between blocks.} Fix $y\in W$ and $\mu$, and label the eight links $(x,\mu)$, $x\in B(y)$, $o_\mu=1$, by the transverse offset $a\in\{0,1\}^3$. If $a$ and $a'$ differ in the coordinate $\nu$, with $o_\nu=0$ at $a$, the plaquette at $x$ in the plane $\{\mu,\nu\}$ contains these two links, $(x,\nu)$ in $B(y)$ and $(x+e_\mu,\nu)$ in $B(y+e_\mu)$, the last two inside their blocks. So $\varphi_a-\varphi_{a'}=\pm(d\varphi)_p\pm\psi\pm\psi'$ with $\psi,\psi'$ inside blocks, and these twelve face plaquettes correspond to the twelve edges of the cube $\{0,1\}^3$. By Lemma~\ref{lem:blocktree}(iii) and $V_{y,\mu}(\varphi)=0$, the mean of the $\varphi_a$ is $m=-(S^\mu_y+ S^\mu_{y+e_\mu})/16$, so $8m^2\le\frac12$ times the sum of the squares of the sixteen links of $S^\mu_y$ and $S^\mu_{y+e_\mu}$. The Laplacian of the graph $\{0,1\}^3$ has smallest nonzero eigenvalue $2$, so \[ \begin{split} \sum_a\varphi_a^2&=8m^2+\sum_a(\varphi_a-m)^2\\ &\le8m^2+\frac32\sum_{\rm edges}\bigl((d\varphi)_p^2+\psi^2+\psi'^2\bigr). \end{split} \] Summed over $y\in W$ and $\mu$, a link $(z,\nu)$ inside a block occurs as $\psi$ or $\psi'$ once for each $\mu\ne\nu$ (as $\psi$ if $o_\mu=1$, as $\psi'$ of the face of the block below if $o_\mu=0$) and in two of the sums $S^\nu$, so it receives weight at most $3\cdot\frac32+2\cdot\frac12= \frac{11}2$. The face plaquettes are distinct and lie in no block, so the links between blocks contribute at most $\frac32Q_{\rm face}+\frac{11}2I$, and \[ \sum_e\varphi_e^2\le\frac{13}2I+\frac32Q_{\rm face}\le52\,Q_{\rm in}+ \frac32Q_{\rm face}\le52\,Q(\varphi).\qedhere \] \end{proof} The constant $52$ is not sharp. A certificate of one of us (T.~Romanov) lowers it to within one part in a thousand of the bulk value; it rests on a computation, and no result of the paper uses it. \begin{remark}[a computation, used by no theorem]\label{rem:blockgapsharp} Let $c=\frac{537}{5000}$, and let $S(y)=B(y)\cup B(y+e_0)\cup\dots\cup B(y+e_3)$, whose active links number $5\cdot49=245$. The certificate of T.~Romanov consists of $196$ rational numbers $l_{\mu,b}$, $\mu=0,\dots,3$, indexed by the $49$ active links $b$ of a block, and a rational symmetric matrix $G$ of size $245$ on the active links of $S(0)$. With $L_{y,\mu}(\varphi)=\sum_bl_{\mu,b}\varphi_{(y,b)}$, where $(y,b)$ is the link $b$ of $B(y)$, and $G_y$ the translate of $G$ to $S(y)$, these are stated to satisfy the identity of quadratic forms in finitely supported $\varphi$ \begin{equation}\label{eq:blockgapcert} Q(\varphi)-c\sum_e\varphi_e^2-\sum_{y,\mu}V_{y,\mu}(\varphi)L_{y,\mu} (\varphi)=\sum_yG_y(\varphi|_{S(y)}), \end{equation} and $G$ is positive definite. Both sides of~\eqref{eq:blockgapcert} are invariant under translations by $2\Z^4$, so the identity amounts to finitely many equalities of coefficients, one for each pair of active links of $B(0)$ and of a block near it. Granted these, take $\varphi\in\Phi_W$. If $y\in W$ then $V_{y,\mu}(\varphi)=0$; if $y\notin W$ then $\varphi$ vanishes on the links with base in $B(y)$, so $L_{y,\mu}(\varphi)=0$. Every term of the middle sum vanishes, and \[ \sum_e\varphi_e^2\le\frac{5000}{537}\,Q(\varphi)=9.31098\ldots\,Q(\varphi) \] for every set of blocks $W$ and every $\varphi\in\Phi_W$. On a periodic lattice of even side both sides of~\eqref{eq:blockgapcert} are the periodisations of the same forms, so the same holds there. The certificate was found by semidefinite programming and rounded to rationals: the $l_{\mu,b}$ have denominators dividing $2^{24}$ and the entries of $G$ the denominator $2^{30}\cdot5^5$. The coefficient identity of~\eqref{eq:blockgapcert} and the positivity of all $245$ leading principal minors of the integer matrix $2^{30}5^5G$ are checked by computer in exact arithmetic, not by hand: by the program of T.~Romanov and independently by our program in the archive, which rebuilds the four forms from Definition~\ref{def:blockgauge} alone. With $W$ the set of all blocks the best constant is the inverse of the Bloch value $0.1074878$ quoted below, that is $9.3034$, so $\frac{5000}{537}$ is within $0.09$ per cent of it; the semidefinite optimum for this window and this class of multipliers is $c^*=0.107488$. These floating-point values are not used. The results of the paper are stated with $52$, which is proved by hand (Theorem~\ref{thm:blockgap}); with the computation, those that use only Theorem~\ref{thm:blockgap} would hold with $52$ replaced by $\frac{5000}{537}$. No result of the paper uses this remark. \end{remark} With $W$ all blocks the bound extends to $\ell^2$, so the block-constrained quadratic form on $\Z^4$ has a spectral gap at least $\frac1{52}$; it removes in particular the four flat connections that are zero modes of the unconstrained form. In the Bloch decomposition the exact gap is $0.10748776948684\dots$, attained at the momentum $(0,\pi,\pi,\pi)$, a floating-point value computed in~\cite{Loc26} and reproduced in the archive; the theorem does not use it. The cube conditions now take the following form. Let $n$ be even, $V$ the active links with base in $[0,n)^4$, $W$ its blocks, and for an exterior $\eta$ and values $v=(v_{y,\mu})_{y\in W}$ let $\gamma^{\rm c}_V(\cdot\mid\eta,v)$ be the law on $\R^V$ with density proportional to $\exp(-\frac b2\sum_{p\cap V\ne\emptyset}(d\varphi)_p^2)$ on the affine set $\{V_{y,\mu}(\varphi)=v_{y,\mu}\}$, with $\varphi=\eta$ off $V$. Define $C^{\rm c}_{V,j}$, $K^{\rm c}_c$ and $D_{\rm c}(n,\alpha)$ as $C_{V,j}$, $K_c$ and $D(n,\alpha)$, with $v$ held fixed and the $49$ classes of active links modulo two. \begin{lemma}\label{lem:saddle} Let $H\ge0$ be symmetric on $\R^m$ with $\|H\|\le h$, let $B:\R^m\to\R^k$ with $BB^\top\ge\beta>0$ and $\langle x,Hx\rangle\ge a|x|^2$ on $\ker B$, and $s>0$. Then $K=\bigl(\begin{smallmatrix}H&sB^\top\\sB&0\end{smallmatrix} \bigr)$ is invertible, and the solution of $K(x,y)=(f,g)$ has $|x|\le(|f|+h|x_1|)/a+|x_1|$ with $|x_1|\le|g|/(s\sqrt\beta)$, and $|y|\le (|f|+h|x|)/(s\sqrt\beta)$. \end{lemma} \begin{proof} Put $x_1=B^\top(BB^\top)^{-1}g/s$, so $sBx_1=g$ and $|x_1|\le|g|/(s\sqrt \beta)$. On $\ker B$ the compression of $H$ is at least $a$, so there is a unique $x_0\in\ker B$ with $Hx_0-f+Hx_1\perp\ker B$, and $a|x_0|\le|f|+h |x_1|$. Then $f-H(x_0+x_1)$ lies in $(\ker B)^\perp=\operatorname{ran}B^\top$, and $y=(BB^\top)^{-1}B(f-Hx)/s$ solves $sB^\top y=f-Hx$. For $f=g=0$ the construction gives zero, so $K$ is injective. \end{proof} \begin{lemma}[Combes--Thomas~\cite{CT73}]\label{lem:combes} Let $K$ be an invertible symmetric matrix indexed by a finite set $X$, with $\|K^{-1}\|\le M_0$ and $\sum_b|K_{ab}|\le S$ for every $a$, and let $d:X\to\R$ satisfy $|d(a)-d(b)|\le1$ whenever $K_{ab}\ne0$. If $(e^\lambda-1) SM_0\le\frac12$, then $\|e^{\lambda d}K^{-1}e^{-\lambda d}\|\le2M_0$. \end{lemma} \begin{proof} $e^{\lambda d}Ke^{-\lambda d}=K+E$ with $|E_{ab}|\le(e^\lambda-1)|K_{ab}|$, so $\|E\|\le(e^\lambda-1)S$ by the Schur test, $\|K^{-1}E\|\le\frac12$, and $(K+E)^{-1}=(1+K^{-1}E)^{-1}K^{-1}$ has norm at most $2M_0$. \end{proof} \begin{theorem}\label{thm:constrainedcube} For every $b>0$, every even $n$ and every exterior link $j$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\gamma^{\rm c}_V(\cdot\mid\eta,v)$ is a Gaussian law whose covariance depends on neither $\eta$ nor $v$ and whose mean is affine in $(\eta,v)$, and $C^{\rm c}_{V,j}=\|u_j\|_1$, where $u_j\in\R^V$ is the change of the mean when $\eta_j$ increases by one; $u_j$ does not depend on $b$; \item $\sum_{e\in V}e^{2\lambda_0d(e)}u_j(e)^2\le3500^2$ and $\|u_j\|_1\le 10^{14}$, where $\lambda_0=2.4\cdot10^{-5}$ and $d(e)$ is the $\ell^\infty$ distance between the midpoints of $e$ and $j$; \item $D_{\rm c}(n,\alpha)$ holds for every $\alpha\in(0,1)$ and every even $n\ge2\cdot10^{15}/\alpha$, at every $b>0$. \end{enumerate} \end{theorem} \begin{proof} (i) Let $H$ be the matrix of $\sum_{p\cap V\ne\emptyset}(d\varphi)_p^2$ on $\R^V$ and $B$ that of the averages $V_{y,\mu}$, $y\in W$, restricted to $V$. By Theorem~\ref{thm:blockgap} with this $W$, $H\ge\frac1{52}$ on $\ker B$, so the density is that of a non-degenerate Gaussian on the affine set, its covariance is $b^{-1}$ times the inverse of $H$ on $\ker B$, and its mean $m$ minimises the quadratic form on the affine set. The mean is affine in $(\eta,v)$, and two exteriors that differ at $j$ give translates by $u_j$, which minimises $Q(u+\delta_j)$ subject to $B u=g$, $g$ being minus the coefficients of $j$ in the averages. The translation argument of Proposition~\ref{prop:gaussforest} gives $C^{\rm c}_{V,j}=\|u_j\|_1$. (ii) The pair $(u_j,y)$, with $y$ the Lagrange multipliers, solves $K(u_j,y)=(f,sg)$ with $K$ as in Lemma~\ref{lem:saddle}, $s=96$ and $f= -\sum_{p\ni j}(d\delta_j)_p\,d_p|_V$, where $d_p|_V$ is the row of the plaquette $p$ restricted to $V$. Each link lies in six plaquettes of four links, so the row sums of $|H|$ are at most $24$ and $h\le24$; $|f|\le 6\sqrt3$; and $|g|\le\frac1{16}$, since an exterior link enters at most one average of $W$, with weight $\frac1{16}$. By Lemma~\ref{lem:blocktree}(iii) each average has squared norm at least $8\cdot\frac1{64}=\frac18$, two averages share links only if they are $(y,\mu)$ and $(y\pm e_\mu,\mu)$, and then their inner product is at most $8\cdot\frac1{256}=\frac1{32}$; so $BB^\top \ge\frac1{16}$ by Gershgorin's theorem. With $a=\frac1{52}$, Lemma~\ref{lem:saddle} gives $\|K^{-1}\|\le M_0=105$. The rows of $K$ have absolute sums at most $S=\max(24+\frac{96}8,96\cdot2)=192$, because an average has eight coefficients $\frac18$ and sixteen of $\frac1{16}$ and a link enters averages with total weight at most $\frac18$. Index the rows of $B$ by the centre $2y+\frac12(1,1,1,1)+e_\mu$ of the two blocks of the average and put $d$ equal to the $\ell^\infty$ distance to the midpoint of $j$. Links of one plaquette, and an average and its links, are within distance one, so Lemma~\ref{lem:combes} applies with $\lambda=\log(1+ 1/40320)>\lambda_0$, and \[ \sum_ee^{2\lambda d(e)}u_j(e)^2\le\bigl(2M_0e^\lambda(|f|+s|g|)\bigr)^2 \le3500^2, \] since $f$ and $g$ live at distance at most one from $j$. At most $4(64k^3+16k)$ links have their base at $\ell^\infty$ distance $k\ge1$ from the base of $j$, and their midpoints are at distance at least $k-1$, so Cauchy--Schwarz gives \[ \|u_j\|_1^2\le3500^2\cdot4e^{2\lambda}\Bigl(1+\sum_{k\ge1}(64k^3+16k) e^{-2\lambda k}\Bigr)\le(10^{14})^2 . \] (iii) $C^{\rm c}_{V_x,j}=0$ unless $j$ shares a plaquette with $V_x$ or enters an average of $V_x$, which forces the base $z$ of $j$ into $x+[-1,n+1]^4$, while $z\notin x+[0,n)^4$ when $j\notin V_x$. For each coordinate at most $\frac n2+2$ even $x_i$ satisfy the first condition and exactly $\frac n2$ the second, so \[ K^{\rm c}_c\le\Bigl(\Bigl(\frac n2+2\Bigr)^4-\Bigl(\frac n2\Bigr)^4\Bigr) 10^{14}\le\frac{17}n\,10^{14}N_n \] for $n\ge10^4$, which is at most $\alpha N_n$ when $n\ge2\cdot10^{15}/\alpha$. \end{proof} The constants are crude. The computations of~\cite{Loc26} give $C^{\rm c}_{V,j}\le12.48$ at $n=4$, and an exact face expansion $K^{\rm c}_c/N_n= a_c/n+O(n^{-2})$ with $\max_ca_c=61.18$, so that $D_{\rm c}(n,0.95)$ first holds at $n=64$; they are floating-point values, reproduced in the archive for $n\le4$, and the theorem does not use them. The coefficients are large because the constrained responses obey no maximum principle: an exterior link changes the curvature of up to six plaquettes, and keeping the curvature small moves a dozen links inside the cube with alternating signs. Re-weighting the classes does not help much: if $\sum_{j\in c\setminus V} \sum_{c'}w_{c'}C^{c'}_{V,j}\le\alpha N_nw_c$ for positive weights $w$, where $C^{c'}_{V,j}$ is the part of $\|u_j\|_1$ on the links of the class $c'$, then $\alpha N_n$ is at least the spectral radius of the non-negative matrix of these sums, hence at least $\min_cK^{\rm c}_c$, by the Collatz--Wielandt inequality~\cite{Loc26}. The block averages can be integrated out one scale at a time. For a field $v$ on the coarse links put $Q_1(v)=\sum_P(dv)_P^2$ over the plaquettes $P$ of the coarse lattice, and write $V(\varphi)=(V_{y,\mu}(\varphi))_{y,\mu}$. The averages of averages are formed in the same way on the coarse lattice, and $V^{(k)}$, $Q_k$ denote the $k$-fold averages and the plaquette form of the lattice of spacing $2^k$. \begin{proposition}\label{prop:effective} Let $\varphi$ be a finitely supported field on $\Z^4$ that vanishes on the tree links. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For the coarse plaquette $P$ at $y$ in the plane $\{\mu,\nu\}$, $16\,(dV(\varphi))_P$ is the sum over $x\in B(y)$ of the curvatures of the squares of side two spanned by $x$, $x+2e_\mu$, $x+2e_\nu$, and every plaquette of $\Z^4$ lies in exactly four of these squares. Hence $Q_1(V( \varphi))\le Q(\varphi)$, and $Q_k(V^{(k)}(\varphi))\le Q(\varphi)$ for every $k\ge1$. \item For every finitely supported $v$ on the coarse links, the field $\varphi^*(v)$ equal to $v_{y,\mu}$ on the eight links $(x,\mu)$, $x\in B(y)$, $o_\mu=1$, and to zero on the other links, vanishes on the tree links, has $V(\varphi^*(v))=v$ and $Q(\varphi^*(v))=4\,Q_1(v)$. \item The effective form $S_1(v)=\inf\{Q(\varphi):V(\varphi)=v\}$ satisfies $Q_1\le S_1\le4\,Q_1$, and $S_k(w)=\inf\{Q(\varphi):V^{(k)}(\varphi)=w\} \ge Q_k(w)$ for every $k$, the infima over finitely supported fields that vanish on the tree links. \item For every $k$, every set $W$ of blocks of the lattice of spacing $2^k$ and every $w$ in the space $\Phi_W$ of that lattice, $\sum_ew_e^2\le 52\,S_k(w)$. \end{enumerate} \end{proposition} \begin{proof} (i) The path of length two from $x\in B(y)$ in the direction $\mu$ ends at $x+2e_\mu\in B(y+e_\mu)$, at the same offset, and so on around the square; so the four averages of $(dV)_P$ pair the sixteen paths into sixteen closed squares, and the sum of $\varphi$ around a square is the sum of the curvatures of its four plaquettes. A plaquette at $z$ in the plane $\{\mu,\nu\}$ lies in the square from $x$ exactly when $z-x\in\{0,1\}e_\mu+ \{0,1\}e_\nu$, which gives four points $x$, each in one block. By Cauchy--Schwarz, $(dV)_P^2\le\frac1{16}\sum_x\ell_x^2\le\frac14\sum_x\sum_{p \subset\square_x}(d\varphi)_p^2$, where $\ell_x$ is the curvature of the square from $x$; summing over $P$ gives $Q_1(V(\varphi))\le Q(\varphi)$. The argument uses neither the tree links nor the spacing, so it applies on every lattice, and induction gives the bound for $V^{(k)}$. (ii) The tree links have $o_\mu=0$. By Lemma~\ref{lem:blocktree}(iii) the average of $\varphi^*$ is $\frac18\cdot8v_{y,\mu}$. A plaquette inside a block has only links with $o_\mu=0$ in its direction $\mu$, and so vanishes. A plaquette at $x$ in the plane $\{\mu,\nu\}$ with $o_\mu=1$, $o_\nu=0$ has the two links $(x,\mu)$, $(x+e_\nu,\mu)$ of the same block, both equal to $v_{y,\mu}$, and two links with offset zero in the direction $\nu$; its curvature vanishes, and so does that of the plaquettes with $o_\mu=0$, $o_\nu=1$. If $o_\mu=o_\nu=1$, its four links carry $v_{y,\mu}$, $v_{y+e_\mu,\nu}$, $v_{y+e_\nu,\mu}$, $v_{y,\nu}$, so its curvature is $(dv)_P$, and there are four such plaquettes for each $P$. (iii) The lower bound is (i) and the upper bound is (ii); for $S_k$ apply (i) to the $k$-fold averages. (iv) Theorem~\ref{thm:blockgap} holds on every lattice, so $\sum_ew_e^2\le 52\,Q_k(w)\le52\,S_k(w)$ by (iii). \end{proof} In the Bloch decomposition the eigenvalues of $S_1$ relative to $Q_1$ lie in $[1,4]$ by (iii). The value four is attained at the momentum $(\pi,\pi,0,0)$, and on the torus of side $L$ the least eigenvalue is at most $t_L^{-2}$, which tends to one, by a long-wavelength field (Theorem~\ref{thm:stepsharp}(i)). So at this order the averages carry the coarse quadratic form with no loss at long wavelengths; the coupling does not run here, and its running is the order-$g^2$ effect of Remark~\ref{rem:constrainedopen}(c). Part (iv) is a gap for the averages at every scale, with the constant of Theorem~\ref{thm:blockgap}; it does not give the localisation of Theorem~\ref{thm:constrainedcube}(ii) for $S_k$, since $S_k$ is not a nearest-neighbour form. The effective forms are equivalent to the coarse plaquette forms uniformly in the scale. Write $L=2^k$ and $B_k(y)=Ly+\{0,\dots,L-1\}^4$. \begin{lemma}\label{lem:straightpaths} \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For every $k\ge1$, every coarse link $(y,\mu)$ and every field $\varphi$, \[ V^{(k)}_{y,\mu}(\varphi)=\frac1{L^4}\sum_{x\in B_k(y)}\sum_{t=0}^{L-1} \varphi_{(x+te_\mu,\mu)} . \] \item For a function $h$ on sites, $V^{(k)}_{y,\mu}(dh)=\bar h(y)-\bar h(y+e_\mu)$, where $\bar h(y)$ is the mean of $h$ over $B_k(y)$. \item $S_k(w)$ equals the infimum of $Q(\varphi)$ over all finitely supported fields with $V^{(k)}(\varphi)=w$, whether or not they vanish on the tree links. \end{enumerate} \end{lemma} \begin{proof} (i) For $k=1$ this is the definition. If it holds for $k$, then $V^{(k)}_{y',\mu}+V^{(k)}_{y'+e_\mu,\mu}$ is $L^{-4}$ times the sum over $x\in B_k(y')$ of the sums of $\varphi$ along the straight paths of length $2L$ from $x$, because $B_k(y'+e_\mu)=B_k(y')+Le_\mu$. The blocks $B_k(y')$, $y'\in2y+\{0,1\}^4$, tile $B_{k+1}(y)$, and averaging over $y'$ with weight $\frac1{16}$ gives (i) for $k+1$. (ii) The sum of $dh$ along the path of length $L$ from $x$ is $h(x)-h(x+Le_\mu)$; average over $x\in B_k(y)$. (iii) Let $\varphi$ be finitely supported. On each block $B(y')$ of side two the tree links form a spanning tree (Lemma~\ref{lem:blocktree}(i)), so there is $h_0$ on $B(y')$ with $(dh_0)_e=-\varphi_e$ on its tree links, unique up to a constant, which we take to be zero where $\varphi$ vanishes on the tree links of $B(y')$. Each $B_k(y)$ is a union of blocks of side two, since $k\ge1$; adding a constant to $h_0$ on one of them makes the mean over $B_k(y)$ vanish, and this changes $h_0$ on finitely many blocks. The resulting $h$ is finitely supported, $\varphi+dh$ vanishes on the tree links, $Q(\varphi+dh)=Q(\varphi)$, and $V^{(k)}(dh)=0$ by (ii). \end{proof} The extension uses the lowest-order Whitney forms of the coarse cubical complex whose vertices are the centres \[ c_z=Lz+\tfrac{L-1}2(1,1,1,1),\qquad z\in\Z^4, \] of the blocks $B_k(z)$, rather than the coarse sites. Let $\lambda(t)=\max (0,1-|t|)$, let $\chi^\mu_z$ be the indicator of $\{\xi\in\R^4:c_{z,\mu}\le \xi_\mu0$, $\Gamma=\mathrm{diag}(\gamma_\mu)$ and $0\ne E\in\C^4$, and for $P=\{\lambda,\rho\}$, $\lambda<\rho$, let $c_P$ be the row vector with $c_Pv=E_\lambda v_\rho-E_\rho v_\lambda$. Then \[ \Gamma-\frac{\Gamma EE^*\Gamma}{E^*\Gamma E} =\frac1{E^*\Gamma E}\sum_P\gamma_\lambda\gamma_\rho\,c_P^*c_P . \] \end{lemma} \begin{proof} Both sides are Hermitian. The diagonal entry $(\mu,\mu)$ of the right side is $\sum_{\lambda\ne\mu}\gamma_\mu\gamma_\lambda|E_\lambda|^2/E^*\Gamma E$, which is $\gamma_\mu-\gamma_\mu^2|E_\mu|^2/E^*\Gamma E$; the entry $(\mu,\nu)$, $\mu\ne\nu$, comes only from $P=\{\mu,\nu\}$, where $c_P$ has $-E_\nu$ in place $\mu$ and $E_\mu$ in place $\nu$, and equals $-\gamma_\mu\gamma_\nu\bar E_\mu E_\nu/E^*\Gamma E$. \end{proof} \begin{theorem}\label{thm:effectivelocal} Let $k\ge1$ and $L=2^k$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The functions $K_P$ are real-analytic, even, $2\pi$-periodic and positive on $\R^4$, $K_P(0)=1$, and for every finitely supported field $w$ on the coarse links \[ S_k(w)=\sum_P\int_{[-\pi,\pi]^4}K_P(\theta)\,|\hat F_P(\theta)|^2\, \frac{d\theta}{(2\pi)^4}. \] \item There are $\eta>0$ and $C<\infty$, independent of $k$, such that every $K_P$ extends analytically to $\{|\mathrm{Im}\,\theta_\nu|\le\eta\}$ and is bounded there by $C$. \item Hence $S_k(w)=\sum_P\sum_{y,y'}\mathcal K_P(y-y')F_P(y)F_P(y')$ with real kernels $\mathcal K_P(y)=(2\pi)^{-4}\int K_P(\theta)e^{i\theta\cdot y}\, d\theta$ that satisfy $|\mathcal K_P(y)|\le Ce^{-\eta|y|_1}$ for every $k$. \end{enumerate} \end{theorem} \begin{proof} \emph{Step 1: the form in Fourier variables.} Let $c_\mu(q)=L\Psi(q) D_L(q_\mu)$. Lemma~\ref{lem:straightpaths}(i) and the Poisson summation formula give, for every square-summable field $\varphi$, \[ \widehat{V^{(k)}(\varphi)}_\mu(\theta)=L^{-4}\sum_nc_\mu(\theta^{(n)})\, \hat\varphi_\mu(\theta^{(n)}), \] with $\hat\varphi(q)=\sum_xe^{-iq\cdot x}\varphi_x$. Write $x_n=\hat\varphi( \theta^{(n)})\in\C^4$ and $e_n=(e^{i\theta^{(n)}_\mu}-1)_\mu$, so that $|e_n|^2=\varepsilon(\theta^{(n)})=:\varepsilon_n$. By the Lagrange identity the plaquette form at the momentum $\theta^{(n)}$ is $\varepsilon_n|x_n|^2- |e_n^*x_n|^2$, and the Plancherel formula gives \[ Q(\varphi)=\int_{[-\pi,\pi]^4}L^{-4}\sum_n\bigl(\varepsilon_n|x_n|^2- |e_n^*x_n|^2\bigr)\frac{d\theta}{(2\pi)^4}. \] Since $LD_L(t)(e^{it}-1)=e^{iLt}-1$ and $e^{iL\theta^{(n)}_\mu}= e^{i\theta_\mu}$, the averages map the gauge direction at every alias to the coarse gauge direction: $c_\mu(\theta^{(n)})(e_n)_\mu=\Psi(\theta^{(n)}) E_\mu(\theta)$. For $|t|\le\pi/L$ we have $|D_L(t)|=|\sin(Lt/2)|/ (L|\sin(t/2)|)\ge2/\pi$, so $\Psi(\theta^{(0)})\ne0$ for real $\theta$. \emph{Step 2: the minimum at one momentum.} Fix real $\theta\notin2\pi \Z^4$; then $E=E(\theta)\ne0$ and every $\varepsilon_n>0$. Let $z\in\C^4$. Minimising $L^{-4}\sum_n\varepsilon_n|y_n|^2$ under $L^{-4}\sum_nc_\mu( \theta^{(n)})(y_n)_\mu=z_\mu$ separates in $\mu$, and the Cauchy--Schwarz inequality gives the minimum $\sum_\mu|z_\mu|^2/a_\mu$, attained at $(y_n)_\mu=z_\mu\overline{c_\mu(\theta^{(n)})}/(\varepsilon_na_\mu)$, where \[ a_\mu=L^{-4}\sum_n\frac{|c_\mu(\theta^{(n)})|^2}{\varepsilon_n} =\frac{\tilde a_\mu}{\hat\varepsilon}. \] If $x$ satisfies the constraint with right side $\hat w(\theta)$, write $x_n= x_n^T+\beta_ne_n$ with $e_n^*x_n^T=0$. Then $L^{-4}\sum_n(\varepsilon_n |x_n|^2-|e_n^*x_n|^2)=L^{-4}\sum_n\varepsilon_n|x^T_n|^2$, and by Step 1 the vectors $x^T_n$ satisfy the constraint with right side $\hat w(\theta)+tE$ for some $t\in\C$. Hence the plaquette form at $\theta$ is at least \[ \min_{t\in\C}\sum_\mu\frac{|\hat w_\mu+tE_\mu|^2}{a_\mu} =\hat w^*\Bigl(\Gamma-\frac{\Gamma EE^*\Gamma}{E^*\Gamma E}\Bigr)\hat w \] with $\Gamma=\mathrm{diag}(a_\mu^{-1})$. Conversely, take $t$ optimal, $y$ optimal for $z=\hat w+tE$, and replace $y_0$ by $y_0-\beta e_0$ with $\beta=L^4t/\Psi(\theta^{(0)})$: the constraint becomes $\hat w(\theta)$ and the plaquette form does not increase, since $e_0$ is a gauge direction. By Lemma~\ref{lem:planes} the minimum is \[ \sum_P\frac{\gamma_\lambda\gamma_\rho}{E^*\Gamma E}|c_P\hat w|^2,\qquad \frac{\gamma_\lambda\gamma_\rho}{E^*\Gamma E}=\frac{\prod_{\sigma\notin P} a_\sigma}{\sum_\sigma|E_\sigma|^2\prod_{\tau\ne\sigma}a_\tau}. \] We multiply numerator and denominator by $\hat\varepsilon^3$ and expand $\tilde a_\tau=A_\tau+\hat\varepsilon r_\tau$. The terms without $r$ add up to $\Psi_0^3\prod_\tau D_{0,\tau}\sum_\sigma|E_\sigma|^2/D_{0,\sigma}$, and $|E_\sigma|^2=L^2D_{0,\sigma}|e^{i\theta^{(0)}_\sigma}-1|^2$ by the identity of Step 1, so they equal $\hat\varepsilon\,\Psi_0^3\prod_\tau D_{0,\tau}$; every other term carries a factor $\hat\varepsilon$. So the denominator is $\hat\varepsilon Y$, the numerator $\hat\varepsilon\prod_{ \sigma\notin P}\tilde a_\sigma$, and the quotient is $K_P(\theta)$. Since $|c_P\hat w|=|\hat F_P|$, the minimum at $\theta$ is $\sum_PK_P(\theta)|\hat F_P( \theta)|^2$. \emph{Step 3: integration.} Integrating the lower bound over $\theta$ gives $Q(\varphi)\ge\sum_P\int K_P|\hat F_P|^2$ for every square-summable $\varphi$ with $V^{(k)}(\varphi)=w$. The minimisers of Step 2 are bounded by a constant times $|\hat w(\theta)|$: for $n\ne0$ because $\varepsilon_n\ge 4/L^2$ and $a_\mu\ge A_\mu/\hat\varepsilon$; for $n=0$ because $|c_\mu(\theta^{(0)})|/(\varepsilon_0a_\mu)\le L^3/A_\mu$; and $|t||E|\le |\hat w|(\max\gamma/\min\gamma)^{1/2}$ by the Cauchy--Schwarz inequality for the inner product $\Gamma$, while $|e_0|\le|\theta|/L\le\frac\pi2|E|/L$. Here $A_\mu\ge(2/\pi)^{10}$, and $\gamma_\mu/\gamma_\nu=\tilde a_\nu/ \tilde a_\mu$ is bounded because the $\tilde a$ are continuous and at least $A$. So the minimisers define a square-summable field $\varphi^*$ with $V^{(k)}(\varphi^*)=w$ and $Q(\varphi^*)=\sum_P\int K_P|\hat F_P|^2$. Since $V^{(k)}$ and $Q$ are bounded on square-summable fields, truncating $\varphi^*$ and correcting with $\psi_R$ as in the proof of Theorem~\ref{thm:effectiveupper} gives finitely supported fields with the same averages and plaquette forms tending to $Q(\varphi^*)$, and Lemma~\ref{lem:straightpaths}(iii) gives (i) for $S_k$. The functions are even since $|D_L|^2$ and $\varepsilon$ are, and positive since $\tilde a_\sigma\ge A_\sigma>0$ and $Y\ge\Psi_0^3\prod_\tau D_{0,\tau}\ge (2/\pi)^{32}$ on $\R^4$, all other terms of~\eqref{eq:Yk} being non-negative there. At $\theta=0$, $\Psi_0=D_{0,\tau}=1$ and $\hat\varepsilon=0$, so $K_P(0)=1$. \emph{Step 4: the strip.} Let $|\mathrm{Im}\,\theta_\nu|\le\frac12$, and for $n$ let $s=L\theta^{(n)}$, so $|\mathrm{Re}\,s_\nu|\le\pi L$ and $|\mathrm{Im}\,s_\nu|\le\frac12$. For $q=x+iy$ with $|x|\le\pi$, $|e^{iq}-1|^2=(e^{-y}-1)^2+2e^{-y}(1-\cos x)\ge e^{-|y|}\frac4{\pi^2}x^2$, so \[ |D_L(q)|\le\min\Bigl(e^{1/2},\frac{c_1}{|\mathrm{Re}\,Lq|}\Bigr) \] for $|\mathrm{Im}\,q|\le1/(2L)$, where $c_1=(e^{1/2}+1)e^{1/4}\frac\pi2<5.35$, and the same holds for $D_L^\sharp$. Also $\mathrm{Re}(2-2\cos q)\ge(2-2\cos x)-2(\cosh y-1)$, so $\mathrm{Re}\,L^2\varepsilon(\theta^{(n)})\ge\frac4{\pi^2}|\mathrm{Re}\, s|^2-1.2$. For $n\ne0$ some $|\mathrm{Re}\,s_\nu|\ge\pi$, and then $|L^2\varepsilon(\theta^{(n)})|\ge0.28\,|\mathrm{Re}\,s|^2$. Writing $s=\theta+2\pi m$ with $m\in\Z^4\setminus\{0\}$, $|\mathrm{Re}\,s_\nu|\ge \pi|m_\nu|$, and \[ |r_\sigma|\le\frac{c_1^{10}}{0.28\,\pi^2}\sum_{m}\prod_\nu f(m_\nu)=\frac{c_1^{10}}{0.28\,\pi^2}\Bigl(\frac43\Bigr)^4 <2.2\cdot10^7, \] with $f(0)=1$ and $f(j)=1/(\pi j)^2$. The $r_\sigma$ are finite sums of functions analytic on the strip, since no $\varepsilon(\theta^{(n)})$, $n\ne0$, vanishes there. Further $|A_\sigma|\le e^5$, $|\hat\varepsilon| \le4(\pi^2+\frac14)e^{1/2}<70$ and $|E_\sigma E_\sigma^\sharp|\le (e^{1/2}+1)^2<7.1$, so on the strip $|Y|\le M=10^{28}$ and $|\tilde a_\sigma|\le1.6\cdot10^9$, uniformly in $k$. By the Cauchy estimate on discs of radius $\frac14$ in each variable, $|\partial_\nu Y|\le4M$ where $|\mathrm{Im}\,\theta|\le\frac14$, so $|Y(\theta)-Y(\mathrm{Re}\,\theta)|\le 16\eta M\le\frac12(2/\pi)^{32}$ for $|\mathrm{Im}\,\theta_\nu|\le\eta= 10^{-36}$. There $|Y|\ge\frac12(2/\pi)^{32}$, $K_P=\prod_{\sigma\notin P} \tilde a_\sigma/Y$ is analytic, and $|K_P|\le C=10^{26}$, which is (ii). Moving the contour of each $\theta_\nu$ to $\mathrm{Im}\,\theta_\nu= \pm\eta$, according to the sign of $y_\nu$, gives (iii); the kernels are real because $K_P$ is real and even. \end{proof} The proof makes $\eta$ explicit but very small. In floating point the kernels decrease much faster: $\max|\mathcal K_P(y)|$ over $|y|_1=d$ falls by factors $0.17$, $0.24$ and $0.26$ per unit of $d$ for $k=1,2,3$, the largest value of $K_P$ on $\R^4$ is $4.4$, $8.7$ and $11.3$, and its smallest value is $1$, attained at $\theta=0$ (Fig.~\ref{fig:locality}); these computations are reproduced in the archive. Each $K_P$ is a Riemann sum in $n$ of a function of $\theta+2\pi n$, so the kernels have a limit as $k\to\infty$; the theorem does not use it. Uniform decay of the propagators of the averaging operations of lattice gauge theories is the subject of Ba{\l}aban's work~\cite{Bal84a}; here the hard constraint and the Feynman gauge make the effective form itself explicit. Together with Theorem~\ref{thm:effectiveupper}, the effective forms of the quadratic approximation are equivalent to the coarse plaquette forms and local, with constants independent of the scale. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_locality.png} \caption{The effective forms of Theorem~\ref{thm:effectivelocal}. (a)~The multiplier $K_P$ for the plane $P=\{1,2\}$ at $k=3$, as a function of $(\theta_3,\theta_4)$ with $\theta_1=\theta_2=0$; it equals $1$ at $\theta=0$ and stays bounded. (b)~The largest value of the kernel $|\mathcal K_P(y)|$ on $|y|_1=d$ for $k=1,2,3$, in logarithmic scale: the decay per coarse spacing hardly changes with $k$.}\label{fig:locality} \end{figure*} \begin{lemma}\label{lem:mixture} Let $(X,d_X)$ and $(Y,d_Y)$ be Polish spaces, $\rho,\rho'$ Borel probability measures on $Y$, and $\kappa,\kappa'$ probability kernels from $Y$ to $X$, all with finite first moments. If $W_1(\kappa(\cdot\mid v),\kappa'(\cdot\mid v))\le\varepsilon$ and $W_1(\kappa'(\cdot\mid v),\kappa'(\cdot\mid v'))\le d_Y(v,v')$ for all $v,v'$, then the mixtures $\pi=\int\kappa(\cdot\mid v)\, \rho(dv)$ and $\pi'=\int\kappa'(\cdot\mid v)\,\rho'(dv)$ satisfy \[ W_1(\pi,\pi')\le\varepsilon+W_1^{d_Y}(\rho,\rho'). \] \end{lemma} \begin{proof} Let $F$ be $1$-Lipschitz on $X$ and $\kappa F(v)=\int F\,d\kappa(\cdot\mid v)$. Then $\int F\,d\pi-\int F\,d\pi'=\int(\kappa F-\kappa'F)\,d\rho+\bigl( \int\kappa'F\,d\rho-\int\kappa'F\,d\rho'\bigr)$. The first term is at most $\varepsilon$. The function $\kappa'F$ is $1$-Lipschitz for $d_Y$, so the second is at most $W_1^{d_Y}(\rho,\rho')$. The Kantorovich--Rubinstein duality gives the claim. \end{proof} A law $\gamma$ on $\R^V$ \emph{decomposes along the averages} if $\gamma(\cdot\mid\eta)=\int\gamma^{\rm c}_V(\cdot\mid\eta,v)\,\rho_V(dv\mid \eta)$, with $\rho_V(\cdot\mid\eta)$ the law of $(V_{y,\mu})_{y\in W}$. Every law with density proportional to $\exp(-\frac b2\sum_{p\cap V\ne\emptyset} (d\varphi)_p^2)\,g(V(\varphi))$, on a set cut out by conditions on the averages, decomposes in this way, because Lebesgue measure disintegrates along the fibres of a surjective linear map into multiples of Lebesgue measure on the fibres. Here $g$ can carry any action of the averages, in particular a coarse gauge fixing and a coarse effective action, which is the form a law takes after the fine scale has been integrated out. \begin{corollary}\label{cor:compose} Let $r_w\in\R^V$ minimise $\sum_{p\cap V\ne\emptyset}(d\varphi)_p^2$, with zero exterior, subject to $V_{y',\mu'}(r_w)=\delta_{w,(y',\mu')}$ for all $(y',\mu')$, $y'\in W$; put $L_w=\|r_w\|_1$ and $d_L(v,v')=\sum_wL_w |v_w-v'_w|$. For every law that decomposes along the averages, \[ C_{V,j}\le C^{\rm c}_{V,j}+C^\rho_{V,j},\qquad C^\rho_{V,j}=\sup\frac{W_1^{ d_L}(\rho_V(\cdot\mid\eta),\rho_V(\cdot\mid\eta'))}{|\eta_j-\eta'_j|}, \] the supremum over exteriors that differ only at $j$, and $L_w\le4\cdot 10^{14}$. Hence $K_c\le K^{\rm c}_c+K^\rho_c$ with $K^\rho_c=\sum_{j\in c\setminus V}C^\rho_{V,j}$, and if $K^\rho_c\le\frac\alpha2N_n$ for every class and $n\ge4\cdot10^{15}/\alpha$, then $D(n,\alpha)$ holds for the law, with the $49$ classes of active links. \end{corollary} \begin{proof} Apply Lemma~\ref{lem:mixture} with $\kappa=\gamma^{\rm c}_V(\cdot\mid\eta, \cdot)$, $\kappa'=\gamma^{\rm c}_V(\cdot\mid\eta',\cdot)$, $\rho=\rho_V(\cdot \mid\eta)$, $\rho'=\rho_V(\cdot\mid\eta')$. The first condition holds with $\varepsilon=C^{\rm c}_{V,j}|\eta_j-\eta'_j|$. By Theorem~\ref{thm:constrainedcube}(i) the laws $\gamma^{\rm c}_V(\cdot\mid \eta',v)$ and $\gamma^{\rm c}_V(\cdot\mid\eta',v')$ are translates by $\sum_w(v'_w-v_w)r_w$, so their distance is at most $d_L(v,v')$. For the bound on $L_w$, the pair $(r_w,y)$ solves $K(r_w,y)=(0,s\,e_w)$ with $K$ as in the proof of Theorem~\ref{thm:constrainedcube}. Measure $d$ from the centre of the row $w$; Lemma~\ref{lem:combes} gives $\sum_ee^{2\lambda d(e)} r_w(e)^2\le(2M_0s)^2=20160^2$. With $z_0=2y+e_\mu$ for $w=(y,\mu)$, a link whose base is at distance $k$ from $z_0$ has its midpoint at distance at least $k-1$ from the centre, and the count in that proof gives $L_w^2\le 20160^2\cdot4e^{2\lambda}\bigl(1+\sum_{k\ge1}(64k^3+16k)e^{-2\lambda k}\bigr) \le(4\cdot10^{14})^2$. Summing over the translates as in Definition~\ref{def:cube} gives $K_c\le K^{\rm c}_c+K^\rho_c$, and Theorem~\ref{thm:constrainedcube}(iii) with $\alpha/2$ in place of $\alpha$ gives $K^{\rm c}_c\le\frac\alpha2N_n$. \end{proof} The responses are much smaller than the bound: in floating point, $\max_wL_w$ is $23.8$, $44.1$ and $50.9$ for the block $(m,m,m,m)$, $m=\lfloor n/4\rfloor$, of the cubes $n=4$, $6$, $8$. Since Theorem~\ref{thm:constrainedcube} controls the first term at every coupling, the corollary reduces $D(n,\alpha)$ for a law that decomposes along the averages to the cube condition $K^\rho_c$ for its averages, a problem of the same form one scale up, in the metric $d_L$. Iterating it needs Theorem~\ref{thm:constrainedcube} and the corollary with $S_k$ in place of the plaquette form, which Theorem~\ref{thm:constrainedscales} below gives with constants independent of $k$, and ends in a cube condition for the coarsest averages, which is where a gap has to come from: the quadratic approximation describes a massless field and has none. The corollary is one step of the composition of Remark~\ref{rem:constrainedopen}(a), not its conclusion. On the lattice of spacing $2^k$ the blocks, the tree links, the active links and the averages are those of Definition~\ref{def:blockgauge}, and after $k$ steps the averages of the quadratic approximation carry the form $S_k$. Let $(S_{ab})$ be its matrix, $S_k(w)=\sum_{a,b}S_{ab}w_aw_b$ for finitely supported $w$, and $d(a,b)$ the $\ell^\infty$ distance between the midpoints of the links $a$ and $b$. Each link lies in six plaquettes of four links, and the midpoints of the links of the plaquettes at $y$ and $y'$ are at distance at most $|y-y'|_1+1$, so Theorem~\ref{thm:effectivelocal}(iii) gives, for every non-decreasing $\phi\ge0$, \begin{equation}\label{eq:Srows} \sum_b|S_{ab}|\,\phi(d(a,b))\le24C\sum_{z\in\Z^4}e^{-\eta|z|_1}\phi(|z|_1+1), \end{equation} with $C$ and $\eta$ of that theorem. In particular $\sum_bS_{ab}\eta_b$ converges for every bounded field $\eta$. For $n$, $V$, $W$ as before, a bounded exterior $\eta$ and values $v$, let $\gamma^{{\rm c},k}_V(\cdot\mid \eta,v)$ be the law on $\R^V$ with density proportional to \[ \exp\Bigl(-\frac b2\Bigl(\sum_{a,b\in V}S_{ab}w_aw_b+2\sum_{a\in V,\,b\notin V}S_{ab}w_a\eta_b\Bigr)\Bigr) \] on the affine set $\{V_{y,\mu}(w)=v_{y,\mu},\ y\in W\}$, the conditional law of the Gaussian with the form $bS_k$ given the exterior and the averages. Define $C^{{\rm c},k}_{V,j}$, $K^{{\rm c},k}_c$, $D^k_{\rm c}(n,\alpha)$, $r^{(k)}_w$ and $L^{(k)}_w$ as in Theorem~\ref{thm:constrainedcube} and Corollary~\ref{cor:compose}, with $S_k$ in place of the plaquette form; for $k=0$ they are the former ones. \begin{lemma}\label{lem:combeslong} Let $K$ be an invertible symmetric matrix indexed by a finite set $X$, with $\|K^{-1}\|\le M_0$, and let $d:X\to\R$ and $\lambda>0$. If $\sum_b|K_{ab}| (e^{\lambda|d(a)-d(b)|}-1)\le\frac1{2M_0}$ for every $a$, then $\|e^{\lambda d}K^{-1}e^{-\lambda d}\|\le2M_0$. \end{lemma} \begin{proof} $e^{\lambda d}Ke^{-\lambda d}=K+E$ with $|E_{ab}|\le|K_{ab}|(e^{\lambda|d(a)- d(b)|}-1)$, a symmetric bound, so $\|E\|\le\frac1{2M_0}$ by the Schur test, and the proof of Lemma~\ref{lem:combes} applies. \end{proof} \begin{theorem}\label{thm:constrainedscales} Let $\lambda_1=10^{-215}$. For every $k\ge1$, every $b>0$, every even $n$ and every exterior link $j$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\gamma^{{\rm c},k}_V(\cdot\mid\eta,v)$ is a Gaussian law whose covariance depends on neither $\eta$ nor $v$ and whose mean is affine in $(\eta,v)$, and $C^{{\rm c},k}_{V,j}=\|u_j\|_1$, where $u_j\in\R^V$ minimises $S_k(u+\delta_j)$ subject to $Bu=g$, with $B$ and $g$ as in Theorem~\ref{thm:constrainedcube}; \item $\sum_{e\in V}e^{2\lambda_1d(e)}u_j(e)^2\le10^{350}$ and $\|u_j\|_1\le 10^{607}e^{-\lambda_1r_j/2}$, where $d(e)=d(e,j)$ and $r_j=\min_{e\in V}d(e)$; and $L^{(k)}_w\le10^{438}$ for every $w$; \item $D^k_{\rm c}(n,\alpha)$ holds for every $\alpha\in(0,1)$ and every even $n\ge10^{827}/\alpha$; \item Corollary~\ref{cor:compose} holds on the lattice of spacing $2^k$, with $S_k$ in place of the plaquette form, $L^{(k)}_w\le10^{438}$ and $n\ge2\cdot10^{827}/\alpha$. \end{enumerate} None of these constants depends on $k$ or on $b$. \end{theorem} \begin{proof} (i) Let $H=(S_{ab})_{a,b\in V}$. A vector $w\in\ker B$, extended by zero, lies in the space $\Phi_W$ of the lattice of spacing $2^k$, so Proposition~\ref{prop:effective}(iv) gives $\langle w,Hw\rangle=S_k(w)\ge \frac1{52}|w|^2$ on $\ker B$. The proof of Theorem~\ref{thm:constrainedcube}(i) applies, the exterior entering through the linear term $\sum_{b\notin V}S_{ab}\eta_b$; changing $\eta_j$ by one translates the mean by the minimiser of $\langle u,Hu\rangle+2\sum_{a\in V} u_aS_{aj}=S_k(u+\delta_j)-S_k(\delta_j)$ subject to $Bu=g$. (ii) The pair $(u_j,y)$ solves $K(u_j,y)=(f,sg)$ with $K$ built from $H$ and $B$ as in Lemma~\ref{lem:saddle} and $f=-(S_{aj})_{a\in V}$. Since $Q_k(w)\le 24|w|^2$, Theorem~\ref{thm:effectiveupper} gives $\|H\|\le h=6144$; take $s=4h=24576$. The bounds of Lemma~\ref{lem:saddle} for $(h,s)$ are at most those for $(24,96)$, because $|x_1|$ and $|f|/s$ decrease while $h|x_1|$ and $h/s$ do not change, so $\|K^{-1}\|\le M_0=105$ as in the proof of Theorem~\ref{thm:constrainedcube}. Index the rows of $B$ by centres as there. For $\phi(r)=e^{\lambda r}-1\le\lambda re^{\lambda r}$ and $\lambda\le\eta/2$, \eqref{eq:Srows} bounds the row sums of Lemma~\ref{lem:combeslong} over $H$ by \begin{multline*} 24C\lambda e^\lambda\sum_z(|z|_1+1)e^{-\eta|z|_1/2}\\ \le24C\lambda e^\lambda\Bigl(1+16\cdot24\,e^{\eta/2}\Bigl(\frac2\eta \Bigr)^5\Bigr), \end{multline*} since at most $8r^3$ points have $|z|_1=r\ge1$ and $\sum_{r\ge1}r^pe^{-xr}\le e^xp!/x^{p+1}$. The entries of $sB^\top$ add at most $\frac s8(e^\lambda-1)$ and a row of $sB$ gives $2s(e^\lambda-1)$. At $\lambda=\lambda_1$, with $C=10^{26}$, $\eta=10^{-36}$ and $(2/\eta)^5=3.2\cdot10^{181}$, the first sum is at most \[ 2.4\cdot10^{-188}e^{\lambda_1}(1+384e^{\eta/2}\cdot3.2\cdot10^{181})<2.95 \cdot10^{-4}, \] while the other two add at most $\frac{17}8s(e^{\lambda_1}-1)<10^{-209}$. So all the sums are below $3\cdot10^{-4}<\frac1{2M_0}$, and Lemma~\ref{lem:combeslong} gives \[ \|e^{\lambda_1d}K^{-1}e^{-\lambda_1d}\|\le210 . \] By~\eqref{eq:Srows}, \begin{multline*} \|e^{\lambda_1d}f\|\le24Ce^{\lambda_1}\sum_ze^{-(\eta-\lambda_1)|z|_1}\\ =24Ce^{\lambda_1}\coth^4\frac{\eta-\lambda_1}2<4\cdot10^{172}, \end{multline*} since $\coth x\le\frac1x+\frac x3$ and $2/(\eta-\lambda_1) \le2.0001\cdot10^{36}$, and $\|e^{\lambda_1d}sg\|\le e^{\lambda_1}s/16$, which gives the first bound, as $(210(4\cdot10^{172}+2\cdot10^3))^2<10^{350}$. The links with base at $\ell^\infty$ distance $m$ from the base of $j$ number at most $4(64m^3+16m)$, with midpoints at distance at least $m-1$, so \[ Z(x)=\sum_ee^{-xd(e)}\le4e^x\Bigl(1+\frac{384e^x}{x^4}+\frac{16e^x}{x^2} \Bigr), \] and $Z(\lambda_1)\le4.004(1+384.4\cdot10^{860}+16.1\cdot10^{430})<10^{864}$, by $e^{2\lambda_1}\le1.001$. As $d\ge r_j$ on $V$, the Cauchy--Schwarz inequality gives $\|u_j\|_1^2\le 10^{350}e^{-\lambda_1r_j}Z(\lambda_1)$. Finally $(r^{(k)}_w,y)$ solves $K(r^{(k)}_w,y)=(0,se_w)$; measuring $d$ from the centre of the row $w$, we get $\sum_ee^{2\lambda_1d(e)}r^{(k)}_w(e)^2\le(210s)^2$ and, since $Z(2\lambda_1)\le4.004(1+24.1\cdot10^{860}+4.1\cdot10^{430})<9.7\cdot 10^{861}<(9.85\cdot10^{430})^2$ and $210s<5.17\cdot10^6$, $L^{(k)}_w\le210s\, Z(2\lambda_1)^{1/2}<10^{438}$. (iii) The exterior links of a class have their bases at $\ell^\infty$ distance $m\ge1$ from $[0,n-1]^4$; at most $4((n+2m)^4-(n+2m-2)^4)\le128(n^3+8m^3)$ of them have distance $m$, and for them $r_j\ge m-1$. With $q=\lambda_1/2$, (ii) gives \[ K^{{\rm c},k}_c\le10^{607}\cdot128\,e^{2q}\Bigl(\frac{n^3}q+\frac{48}{q^4} \Bigr), \] which is at most $\alpha N_n=\alpha(n/2)^4$ when $n\ge10^{827}/\alpha$: with $1/q=2\cdot10^{215}$ and $e^{2q}\le1.001$ the first term is at most $2.57\cdot10^{824}n^3\le\alpha n^4/32$ for $n\ge8.3\cdot10^{825}/\alpha$, and the second at most $9.9\cdot10^{1471}\le\alpha n^4/32$ for $n^4\ge3.2\cdot10^{1473}/\alpha$, which holds since $n^4\ge10^{3308}/\alpha^4$. (iv) The proof of Corollary~\ref{cor:compose} uses the plaquette form only through Theorem~\ref{thm:constrainedcube}, the translation property of (i) and the bound on $L_w$, which (i)--(iii) provide. \end{proof} The constants are poor because the strip of Theorem~\ref{thm:effectivelocal} has width $10^{-36}$. What the theorem adds is that none of them depends on the scale: the constrained cube condition holds with one cube side at every scale, and the weights $L^{(k)}_w$ of the metrics $d_L$ are bounded uniformly. It does not compose the scales. The cube condition $K^\rho_c$ that Corollary~\ref{cor:compose} leaves at scale $k$ concerns the law of the averages given the fine exterior, in the metric $d_L$, and it has to be turned into the cube condition of the law one scale up; that step and the cube condition for the coarsest averages are open, and in the quadratic approximation the coarsest averages carry a massless Gaussian, so a gap has to come from beyond it. \begin{remark}\label{rem:constrainedopen} Theorem~\ref{thm:constrainedcube} is the weak-coupling half of the constrained route at leading order, and Theorem~\ref{thm:transfer} cannot use it as it stands. Four steps remain, each an open problem. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item \emph{Composition.} $D_{\rm c}$ holds the block averages fixed. A gap in physical units needs a theorem that combines the constrained conditions at all scales with a cube condition for the coarsest averages, in the place of Theorem~\ref{thm:transfer}. Corollary~\ref{cor:compose} is one step of it, Proposition~\ref{prop:effective}(iv) gives the gap at every scale, and Theorem~\ref{thm:effectiveupper} the equivalence of $S_k$ and $Q_k$ uniformly in $k$, Theorem~\ref{thm:effectivelocal} the locality of $S_k$ uniformly in $k$, and Theorem~\ref{thm:constrainedscales} the constrained cube condition and the bounds on the weights $L_w$ at every scale with constants independent of $k$. Turning the cube condition for the averages into that of the law one scale up, and a cube condition for the coarsest averages, are open. \item \emph{Beyond the quadratic approximation.} For the Wilson measure the block averages are products of group elements along paths, their level sets are curved, and the Bakry--\'Emery tensor of Sec.~\ref{sec:curvature} is negative at large fields; Theorem~\ref{thm:blockgap} controls only the small fields, and the large fields have to be split off as in Ba{\l}aban's analysis~\cite{Bal89b}. For the straight paths between block roots the group-valued estimate holds for all fields (Theorem~\ref{thm:pathcoercive}), but that map does not compose across scales (Proposition~\ref{prop:thinpath}), and for every compact connected group no continuous covariant map treats the transverse positions of a block alike (Proposition~\ref{prop:nomean}). The cut-off average of Theorem~\ref{thm:cutoff} is coercive for all fields and agrees with the averages to first order, with constants for one step; on torus-valued fields of small curvature it composes (Theorem~\ref{thm:torusscales}), and for every compact group it composes on fields whose curvature is small at every scale with a small sum (Theorem~\ref{thm:smallscales}); under the Wilson measure the remaining large fields form microscopic clusters (Corollary~\ref{cor:defectclusters}), but the conditional estimate needed near them is open. Away from them, on the fibres of the cut-off average, the Hessian of the action along the fibre is at least $\lambda(\frac1{26}- c_G\kappa)$ at every background with plaquettes at most $\kappa$ (Theorem~\ref{thm:smallfibre}), and on a convex chart of the fibre the constrained law satisfies a Brascamp--Lieb inequality for large $\lambda$ (Theorem~\ref{thm:fibrechart}), and on a product chart its covariances decay exponentially (Theorem~\ref{thm:chartdecay}), and the mass off that chart is small on most fibres of the $\SU(2)$ Wilson measure (Theorem~\ref{thm:offchart}) and on every fibre through a field with small plaquettes (Corollary~\ref{cor:calmfibres}), and on every fibre the law concentrates near the least action of its regular fields (Proposition~\ref{prop:everyfibre}); the remaining small-field steps are listed in Remark~\ref{rem:chartdecayopen}. \item \emph{Second order.} Let $b_0=11N/(48\pi^2)$ for $\SU(N)$, $t=2b_0\log2$, and let $g_m^{-2}=g_*^{-2}+tm$ be the one-loop coupling $m$ steps above a matching scale. Comparison with integrals gives $\sum_{m\le M}g_m^2= t^{-1}\log M+O(1)$ and $\sum_mg_m^4\le\pi^2/(6t^2)$~\cite{Loc26}. A loss $1+cg_m^2$ per step therefore compounds to a power of $M$, and a step from one scale to the next has to be exact at order $g^2$, with that term absorbed into the running coupling and the normalisation of the field, and remainders of order $g^4$. \item \emph{Crossover and existence.} The passage from weak coupling to the strong-coupling range of Sec.~\ref{sec:su2}--\ref{sec:su3blocks} and the existence of the limit, (E), are untouched. \end{enumerate} The renormalisation group needed for (a) and (b) is the ultraviolet half of (E), so the constrained route places both residues on one construction. \end{remark} \subsection{Path blocks for the group-valued links}\label{ssec:pathblocks} Remark~\ref{rem:constrainedopen}(b) asks for estimates on the actual group-valued links. They can be proved for a coarser block map, the straight path between block roots, and this map exhibits the second half of the difficulty: it does not compose across scales. A topological argument shows why no continuous map that treats the transverse positions alike can take its place. Let $G\subset\U(r)$ be a compact matrix group, $\delta(A,B)=\lVert A-B \rVert_{\rm HS}$, which is invariant under left and right multiplication, and work on the periodic lattice of side $2L$ with the block tree of Definition~\ref{def:blockgauge}, all fifteen tree links of each block equal to $I$. Write $\mathcal B$ for the number of blocks. The \emph{path map} is \[ V^{\rm p}_{y,\mu}(U)=U_{2y+e_\mu,\mu}, \] the product along the straight path from $2y$ to $2y+2e_\mu$, whose first link is a tree link; it transforms as a link under the gauge transformations at the block roots. Fixing $V^{\rm p}=v$ fixes four of the $49$ active links of each block, so the fibre is $G^{45\mathcal B}$, and the Wilson law in the tree gauge restricts to it with product Haar measure as reference, since the tree gauge has unit Jacobian. The \emph{reconstruction} $R(v)$ puts $R_{2y+o,\mu}=v_{y,\mu}$ if $o_\mu=1$ and $R_{2y+o,\mu}=I$ otherwise. For $U$ in the fibre of $v$ put \[ D_v(U)^2=\sum_e\delta(U_e,R_e(v))^2,\qquad Q(U)=\sum_p\delta(U_p,I)^2, \] the first sum having zero terms on the tree and path links, and let $Q_1(v)=\sum_P\delta(v_P,I)^2$ on the coarse lattice. \begin{lemma}\label{lem:anchorcube} For vectors $z_a$ of a real Hilbert space indexed by $a\in\{0,1\}^3$, with $z_0=0$, \[ \sum_a\lVert z_a\rVert^2\le\frac{3+\sqrt7}2\sum_{\{a,a'\}}\lVert z_a- z_{a'}\rVert^2\le3\sum_{\{a,a'\}}\lVert z_a-z_{a'}\rVert^2, \] the sums on the right over the twelve edges of the cube. \end{lemma} \begin{proof} It suffices to treat real numbers. The right side is $\ip z{L_0z}$, where $L_0$ is the Laplacian of the cube with the vertex $0$ deleted, and the claim is that its least eigenvalue is $3-\sqrt7$. The permutations of the coordinates commute with $L_0$. On the functions $f_k$ of the level $k= |a|$, $L_0$ acts by $f_1\mapsto3f_1-2f_2$, $f_2\mapsto3f_2-2f_1-f_3$, $f_3\mapsto3f_3-3f_2$, whose characteristic polynomial in $t=3-\lambda$ is $t^3-7t$; the eigenvalues are $3$ and $3\pm\sqrt7$. On the functions of levels one and two whose sum over each level vanishes, indexed by the coordinate that is one at level one and zero at level two, $L_0$ acts by the matrix $\bigl(\begin{smallmatrix}3&1\\1&3\end{smallmatrix}\bigr)$, with eigenvalues $2$ and $4$; at level three these functions vanish. The two spaces have dimensions $3$ and $4$ and span $\R^7$. Finally $1/(3-\sqrt7)= (3+\sqrt7)/2<3$.\footnote{The constant $(3+\sqrt7)/2$ is attained by a function of the level $|a|$ alone, so it is the best constant in the lemma.} \end{proof} \begin{theorem}\label{thm:pathcoercive} For every $G\subset\U(r)$, every $L\ge1$, every $v$ and every $U$ in the fibre of $v$, \[ D_v(U)^2\le224\,Q(U). \] \end{theorem} \begin{proof} Let $I_0$ be the sum of $\delta(U_e,I)^2$ over the active links inside the blocks and $J_0$ the sum of $\delta(U_e,v_{y,\mu})^2$ over the links that leave $B(y)$ in the direction $\mu$, so $D_v^2=I_0+J_0$. For an active link inside a block, the chain of plaquettes in the proof of Theorem~\ref{thm:blockgap} applies to group elements: each plaquette of the chain has its two transverse links on the tree, so the holonomy is the product of two consecutive links of the chain, and invariance of $\delta$ with the triangle inequality gives $\delta(U_e,I)^2\le k\sum_i\delta(U_{p_i} ,I)^2$ with $k\le3$. The loads are those of that proof, so $I_0\le8Q_{\rm in}$. Fix $y,\mu$ and label the eight links leaving $B(y)$ by the transverse offset $a\in\{0,1\}^3$; the link $a=0$ is the path link and equals $v_{y,\mu}$. For a transverse edge $\{a,a'\}$ the face plaquette has holonomy $U_a\psi'U_{a'}^{-1}\psi^{-1}$ with $\psi,\psi'$ links inside the two blocks, so $\delta(U_a,U_{a'})\le\delta(U_p,I)+\delta(\psi,I)+ \delta(\psi',I)$. Lemma~\ref{lem:anchorcube} with $z_a=U_a-v_{y,\mu}$ in the real Hilbert--Schmidt space and Cauchy--Schwarz bound the eight terms by $9$ times the sum of the three squares over the twelve edges. Each face plaquette occurs once, and each link inside a block occurs once for each of the three directions other than its own, as in the proof of Theorem~\ref{thm:blockgap}. Hence $J_0\le9Q_{\rm face}+27I_0$ and $D_v^2\le28I_0+9Q_{\rm face}\le224\,Q$. \end{proof} The bound holds for all fields, with no small-field condition and no logarithmic chart; Theorem~\ref{thm:blockgap} is its linearisation for the averages, with the better constant $52$. \begin{proposition}\label{prop:pathenergy} $Q_1(V^{\rm p}(U))\le4Q(U)$, $Q(R(v))=4Q_1(v)$ and $Q(U)\le8Q_1(v)+ 48D_v(U)^2$ for $U$ in the fibre of $v$; if $Q_1(v)=0$, then $Q(U)\le 24D_v(U)^2$. \end{proposition} \begin{proof} The holonomy of a coarse plaquette is that of the square of side two at $2y$, a product of four conjugated fine plaquette holonomies, so $\delta(v_P,I)\le\sum_{p\subset P}\delta(U_p,I)$; these squares tile each coordinate plane, which gives the first bound. In $R(v)$ only the plaquettes with offset one in both directions carry holonomy, equal to that of the coarse plaquette, four for each $P$. For any plaquette, telescoping gives $\delta(U_p,R_p)\le\sum_{e\in p}\delta(U_e,R_e)$, so $\delta(U_p,I)^2\le2\delta(R_p,I)^2+8\sum_{e\in p}\delta(U_e,R_e)^2$, and each link lies in six plaquettes. If $Q_1(v)=0$, every $R_p$ is $I$ and the factor $2$ is not needed. \end{proof} Let $m=45\mathcal B$, let the action be $\lambda Q$ with $\lambda\ge0$ (for $\SU(N)$, $\lambda=\beta/(2N)$ up to a constant), and put \[ z_G(t)=\int_Ge^{-t\delta(g,I)^2}dg,\qquad Z_\lambda(v)=\int_{G^m} e^{-\lambda Q(U)}dU, \] with normalised Haar measures, the integral over the fibre of $v$. \begin{theorem}\label{thm:pathZ} For every $v$, \[ e^{-8\lambda Q_1(v)}z_G(48\lambda)^m\le Z_\lambda(v)\le e^{-\lambda Q_1(v)/8}z_G(\lambda/448)^m, \] and if $Q_1(v)=0$, then $z_G(24\lambda)^m\le Z_\lambda(v)\le z_G(\lambda/ 224)^m$. Let $A_\lambda=z_G(\lambda/448)/z_G(24\lambda)$. Under the conditional law at such a $v$, \[ \E\,e^{\lambda D_v^2/448}\le A_\lambda^m,\qquad \E\,\delta(U_e,I)^2\le \frac{20160}\lambda\log A_\lambda \] for each link $e$ when $v\equiv I$, and $A_\lambda\to10752^{q/2}$ as $\lambda\to\infty$ if $G$ has dimension $q\ge1$. \end{theorem} \begin{proof} By Theorem~\ref{thm:pathcoercive} and Proposition~\ref{prop:pathenergy}, $Q\ge\frac12(D_v^2/224+Q_1/4)$ and $Q\le8Q_1+48D_v^2$. Each term of $D_v^2$ involves one free link, and right multiplication by $R_e(v)^{-1}$ preserves Haar measure, so $\int e^{-tD_v^2}dU=z_G(t)^m$; this gives the two bounds, and the flat case is the same with $Q\le24D_v^2$. Next, $-\lambda Q+\lambda D_v^2/448\le-\lambda D_v^2/448$, and integration with the lower bound gives the exponential moment; Jensen's inequality gives $\E D_v^2\le(448m/\lambda)\log A_\lambda$. When $v\equiv I$ the law is invariant under translation by blocks, so the $\mathcal B$ links of each of the $45$ classes have equal expectations, and $448\cdot45=20160$. Finally, in exponential coordinates $\delta(e^X,I)^2=\lVert X\rVert^2+ O(\lVert X\rVert^4)$ and the Haar density is smooth and positive at $0$, while $\delta(\cdot,I)$ is bounded below away from $I$; so $z_G(t)\sim c_G t^{-q/2}$, and the ratio at $\lambda/448$ and $24\lambda$ tends to $(24\cdot448)^{q/2}$. \end{proof} So the fluctuations of the free links around a flat path field are of order $\lambda^{-1/2}$ uniformly in the volume, for the group variables themselves. The path map cannot be iterated, however, and this is where it differs from the averages. Write $V^{{\rm p},k}$ for the $k$-fold path map, the straight path of length $s=2^k$, and $Q_k$ for the plaquette energy of the lattice of spacing $s$. \begin{proposition}\label{prop:thinpath} On the periodic lattice of side $3s$, $s=2^k$, there are $\SU(2)$ fields in the tree gauge with \[ \frac{Q(U)}{Q_k(V^{{\rm p},k}(U))}=s\sin^2\frac\pi{2s}\le\frac{\pi^2}{4s}. \] For the averages, by contrast, $Q_k(V^{(k)}(\varphi))\le Q(\varphi)$ for every $k$ in the quadratic approximation (Proposition~\ref{prop:effective}(i)). \end{proposition} \begin{proof} Let $T=\operatorname{diag}(i\pi/2,-i\pi/2)$. Put the $s/2$ links in the direction $0$ at $(j,0,0,0)$, $j$ odd, $j0$. For $\U(r)$ one may take $\rho_G=1$. For $\SU(N)$ one may take $\rho_G=\min\{1,5/\sqrt N\}$: by Lemma~\ref{lem:logbound} the trace of $\log g$ has modulus at most $\sqrt N\,\pi\rho_G/3<2\pi$, and it lies in $2\pi i\Z$. Some restriction is needed: on the circle $\{\operatorname{diag} (z,z^7)\}\subset\U(2)$ the point $z=e^{2\pi i/7}$ has $\delta(g,I)<1$ but $\log g\notin\mathfrak g$.} Let $\phi(t)=\min\{1,\max\{0,2-2t\}\}$, which equals $1$ on $[0,\frac12]$ and $0$ on $[1,\infty)$, put \[ \chi_{y,\mu}(U)=\phi\Bigl(\rho_G^{-1}\max_{x\in B(y)}\delta(P_x,P_{2y}) \Bigr), \] and define the \emph{cut-off average} \begin{equation}\label{eq:cutoff} V^\sharp_{y,\mu}(U)=P_{2y}\exp\Bigl(\frac{\chi_{y,\mu}(U)}{16}\sum_{x\in B(y)}\log\bigl(P_{2y}^{-1}P_x\bigr)\Bigr), \end{equation} the exponential factor being $I$ when $\chi_{y,\mu}(U)=0$ (Fig.~\ref{fig:cutoff}). When every path is close to the straight one it is the average of the sixteen paths taken in the chart at $P_{2y}$; when some path is far from it, it is the path map. \begin{figure*}[t] \centering \includegraphics[width=0.9\textwidth]{figures/fig_cutoff.png} \caption{(a) A block $B(y)$ of side two and the next block in the direction $\mu$, drawn in three of the four dimensions. Green dashed: the tree links, equal to $I$. Blue: the paths $P_x$ of length two from the sites of $B(y)$, each joining $2y$ to $2y+2e_\mu$ through the trees; black: the straight path $P_{2y}$, the coarse link of the path map. The cut-off average is the mean of the sixteen paths in the chart at $P_{2y}$ while every path is close to it. (b) For $\SU(2)$ and one link $e^{\theta X}$, $X=\operatorname{diag}(i,-i)$, the angle of $V^\sharp$ when the link lies on the straight path (black) or at another transverse position (blue), with $\rho_G=1$. Both equal $\theta/8$, the block average, while $\chi=1$; past the cut-off they are the path map, $\theta$ and $0$, so the slice windings are $1$ and $0$.} \label{fig:cutoff} \end{figure*} \begin{theorem}\label{thm:cutoff} For every compact $G\subset\U(r)$ the cut-off average is continuous, covariant and normalised, and the following hold, with $c=\pi/3$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item If $\delta(P_x,P_{2y})\le\rho_G/2$ for every $x\in B(y)$, then $V^\sharp_{y,\mu}(U)=P_{2y}\exp\bigl(\frac1{16}\sum_x\log(P_{2y}^{-1}P_x) \bigr)$. For $\varphi$ with values in $\mathfrak g$, vanishing on the tree links, \[ V^\sharp_{y,\mu}(e^{t\varphi})=I+t\,V_{y,\mu}(\varphi)+O(t^2), \] with $V_{y,\mu}$ the block average of Definition~\ref{def:blockgauge}. \item For every field $U$ in the tree gauge, \begin{align*} \sum_{y,\mu}\sum_{x\in B(y)}\delta(P_x,P_{2y})^2&\le896\,Q(U),\\ \sum_{y,\mu}\delta\bigl(V^\sharp_{y,\mu}(U),V^{\rm p}_{y,\mu}(U)\bigr)^2 &\le56c^2\,Q(U),\\ \sum_e\delta\bigl(U_e,R_e(V^\sharp(U))\bigr)^2&\le(448+896c^2)\,Q(U),\\ Q_1\bigl(V^\sharp(U)\bigr)&\le(8+2688c^2)\,Q(U), \end{align*} and $448+896c^2<1431$, $8+2688c^2<2956$. \item In the setting of Proposition~\ref{prop:nomean} with $s=2$, $W_t$ is the identity at the position of the straight path and $0$ at the seven others. \item If $U$ equals $I$ except on one link from $B(y)$ to $B(y+e_\mu)$, where it equals $g$ with $\delta(g,I)\le\rho_G/2$, then $V^\sharp_{y,\mu}(U)= \exp(\frac18\log g)$, at each of the eight transverse positions of the link; the path map gives $g$ at one position and $I$ at the seven others. \end{enumerate} \end{theorem} \begin{proof} Write $P=P_{2y}$, $v_x=P^{-1}P_x$ and $m=\max_x\delta(v_x,I)=\max_x\delta(P_x ,P)$. Where $m<\rho_G$, every $\log v_x$ lies in $\mathfrak g$, depends continuously on $U$ by Lemma~\ref{lem:logbound}, which keeps the eigenvalues of $v_x$ away from $-1$, and the exponent of~\eqref{eq:cutoff} lies in $\mathfrak g$, so $V^\sharp\in G$. Where $m\ge\rho_G$, $V^\sharp=P$. As $m$ increases to $\rho_G$ the exponent has norm at most $\chi c\rho_G\to0$, so $V^\sharp$ is continuous. Let $h$ be a gauge transformation equal to $h_1$ on $B(y)$ and to $h_2$ on $B(y+e_\mu)$. The middle point $x+e_\mu$ of each path lies in one of the two blocks, so $P_x\mapsto h_1P_xh_2^{-1}$ and $v_x\mapsto h_2v_xh_2^{-1}$; $\delta$ is invariant and $\log$ is equivariant under unitary conjugation, so $\chi$ is unchanged and $V^\sharp\mapsto h_1 V^\sharp h_2^{-1}$. At $U=I$ every $v_x$ is $I$, so $V^\sharp(I)=I$. (i) The first statement is $\chi=1$. If $U=e^{t\varphi}$ with $\varphi=0$ on the tree, then $P=I+t\varphi_{2y+e_\mu,\mu}+O(t^2)$ and $\log v_x=t(\varphi_{x,\mu}+\varphi_{x+e_\mu,\mu}-\varphi_{2y+e_\mu,\mu})+ O(t^2)$, and for small $t$ the condition of the first statement holds. The mean over the sixteen $x$ is $t(V_{y,\mu}(\varphi)-\varphi_{2y+e_\mu,\mu}) +O(t^2)$, since $\varphi_{2y,\mu}=0$, and multiplying by $P$ gives the claim. (ii) In the tree gauge $P=v_{y,\mu}:=V^{\rm p}_{y,\mu}(U)$. If $o_\mu=0$, then $U_{x+e_\mu,\mu}$ is the link leaving $B(y)$ at the transverse offset of $x$, and invariance of $\delta$ with the triangle inequality gives $\delta(P_x,P)\le\delta(U_{x,\mu},I)+\delta(U_{x+e_\mu,\mu},v_{y,\mu})$; if $o_\mu=1$, the roles of the two links are exchanged. Each link leaving a block in the direction $\mu$ occurs for two sites $x$ of that block, and each link in the direction $\mu$ with $o_\mu=0$ occurs once for the block containing it and once for the block before it. With $(a+b)^2\le2a^2+2b^2$ and the notation of the proof of Theorem~\ref{thm:pathcoercive}, the sum is at most $2(2I_0+2J_0)=4D_v^2\le896Q$. Next, if $\chi>0$, Lemma~\ref{lem:logbound} and the Cauchy--Schwarz inequality give \begin{align*} \delta(V^\sharp,P)&\le\Bigl\lVert\frac\chi{16}\sum_x\log v_x\Bigr\rVert\le \frac c{16}\sum_x\delta(v_x,I)\\ &\le\frac c4\Bigl(\sum_x\delta(P_x,P)^2\Bigr)^{1/2}, \end{align*} and the second bound follows from the first, $896c^2/16=56c^2$. The reconstruction $R(V^\sharp)$ agrees with $R(v)$ on the links with $o_\mu=0$, and on the eight links with $o_\mu=1$ of each block and direction $\delta(U_e,V^\sharp)^2\le2\delta(U_e,v)^2+2\delta(V^\sharp,v)^2$; so the third sum is at most $2D_v^2+16\cdot56c^2Q\le(448+896c^2)Q$. Finally, write $V^\sharp=vM$. Removing the four factors $M$ from the holonomy of a coarse plaquette one at a time changes $\delta(\cdot,I)$ by at most $\delta(M,I)$ each time, so $\delta(V^\sharp_P,I)^2\le2\delta(v_P,I)^2+8\sum_{\ell\in P} \delta(M_\ell,I)^2$. Each coarse link lies in six coarse plaquettes, so $Q_1(V^\sharp)\le2Q_1(v)+48\cdot56c^2Q$, and $Q_1(v)\le4Q$ by Proposition~\ref{prop:pathenergy}. The numerical bounds use $c^2<1.0967$. (iii) For $s=2$ the slice $T$ consists of the eight links leaving $B(y)$, and the links inside the blocks are $I$, so $P_x=u_{a}$ for the transverse offset $a$ of $x$, two sites for each $a$. Let $t$ be a position other than the straight path and $u_{t'}=I$ for $t'\ne t$. Then $P=I$, two of the $v_x$ equal $u_t$ and the others $I$, so $F(u)=\exp(\frac18\chi\log u_t)$ with $\chi=\phi(\delta(u_t,I)/\rho_G)$. On $S$ the maps $u_t\mapsto\exp(\frac\lambda8\chi\log u_t)$, $0\le\lambda\le1$, are continuous by the argument above and take values in $S$, since $\log u_t$ commutes with $S$ and the centraliser of a maximal torus in $\mathfrak g$ is its Lie algebra; they join $F$ to the constant $I$, so $W_t=0$. By Proposition~\ref{prop:nomean} the $W_t$ add up to the identity, which is therefore the matrix at the straight path. (iv) At the straight path, $P_x=g$ for the two sites with transverse offset $0$ and $P_x=I$ for the others, so $V^\sharp=g\exp(-\frac{14}{16}\log g)= \exp(\frac18\log g)$; at another position, $P=I$ and two of the $P_x$ equal $g$, so $V^\sharp=\exp(\frac2{16}\log g)$. \end{proof} The small-field part of~\eqref{eq:cutoff} is Ba{\l}aban's averaging operation~\cite{Bal85a}; the cut-off extends it to all fields with the bounds of (ii). The theorem shows how far Proposition~\ref{prop:nomean} reaches. The cut-off average is continuous, covariant and coercive for every field, and it agrees with the block averages to first order. Its slice windings are those of the path map, as the proposition demands of any such map, but the departure from the transverse symmetry happens only where some path differs from the straight one by more than $\rho_G/2$. Below that the field of a single link enters with the weight $\frac18$ of the averages, whichever position it occupies. What the theorem does not give is composition. The constants of (ii) are those of one step, and iterating them gives a factor that grows exponentially in the number of steps; on its large-field branch the map is the path map, for which Proposition~\ref{prop:thinpath} rules out any bound uniform in that number. Whether the $k$-fold cut-off average obeys $Q_k(V^{\sharp,k}(U))\le C\,Q(U)$ and the corresponding coercivity bound with $C$ independent of $k$ is open for general fields; it is the form in which Remark~\ref{rem:constrainedopen}(b) remains. Section~\ref{ssec:scales} proves it, with $C=1.001$, for fields with values in a maximal torus whose curvature stays small at every scale, and with $C=e^{12060\sigma}$ for fields of any compact group whose curvatures at the successive scales are small with sum $\sigma$. \subsection{Composition across scales}\label{ssec:scales} This subsection proves that the cut-off average composes across all scales on fields with values in a maximal torus whose curvature stays small, which includes the thin paths that defeat the path map, and then that it composes for every compact group on fields whose curvature is small, with a summable loss over the scales. \begin{figure*}[t] \centering \includegraphics[width=0.9\textwidth]{figures/fig_scales.png} \caption{(a) In the torus, the coarse plaquette of the cut-off average is the mean of the fluxes $F_{2y+o}$ of the sixteen fine squares of side two at the offsets $o$ (four of them drawn), whatever the gauge of the coarse links (Lemma~\ref{lem:torusstep}). (b) For the thin-path fields of Proposition~\ref{prop:thinpath}, the coarse energy after $k$ steps: $24$ for the path map, at most $18\pi^2/s^2$ for the cut-off average (Corollary~\ref{cor:thinscales}); the fine energy is about $6\pi^2/s$.} \label{fig:scales} \end{figure*} Let $S$ be a maximal torus of $G\subset\U(r)$ and $\mathfrak s$ its Lie algebra. The exponential maps $\mathfrak s$ onto $S$, and its kernel $\Lambda$ consists of diagonalisable matrices with eigenvalues in $2\pi i\Z$, so every nonzero element of $\Lambda$ has norm at least $2\pi$. For an $S$-valued field with $\delta(U_p,I)\le\rho_G$ on every plaquette put $\Theta_p=\log U_p$; it lies in $\mathfrak g$, commutes with $S$, and so lies in $\mathfrak s$. For a fine site $x$ and a coordinate plane $\{\mu,\nu \}$ let $\square_x$ be the square of side two at $x$ in that plane and $F_x=\sum_{p\subset\square_x}\Theta_p$, the sum of its four plaquettes, oriented as the square. Put \[ \eta_G=\rho_G/12. \] The $k$-fold cut-off average is defined by putting the coarse field back into the tree gauge, by an $S$-valued gauge transformation, before each step. The tree gauge fixes such a transformation up to one that is constant on blocks, under which $V^\sharp$ is covariant, so the coarse plaquette holonomies $U^{(j)}_P$ of the $j$-th iterate $U^{(j)}$ are well defined; we write $\Theta^{(j)}_P=\log U^{(j)}_P$ when $\delta(U^{(j)}_P,I)\le\rho_G$. \begin{lemma}\label{lem:torusstep} Let $U$ be an $S$-valued field in the tree gauge with $\lVert\Theta_p\rVert \le\eta_G$ for every plaquette. Then $\chi_{y,\mu}(U)=1$ for every coarse link, and the coarse plaquette $P$ at $y$ in the plane $\{\mu,\nu\}$ has \[ V^\sharp(U)_P=\exp\bar F_P,\qquad\bar F_P=\frac1{16}\sum_{o\in\{0,1\}^4} F_{2y+o}, \] with $\lVert\bar F_P\rVert\le4\max_p\lVert\Theta_p\rVert$, so that $\log V^\sharp(U)_P=\bar F_P$. \end{lemma} \begin{proof} Let $\eta=\max_p\lVert\Theta_p\rVert$. Fix $(y,\mu)$ and $x=2y+o$. The tree path $\tau$ from $2y$ to $x$ sets the coordinates of $o$ one at a time in increasing order, so it has at most four edges, at most three of them transverse to $\mu$. In the tree gauge its links are $I$, so $P_{2y}^{-1}P_x$ is the holonomy of the loop $\tau$, then the straight path from $x$ to $x+2e_\mu$, then $\tau+2e_\mu$ backwards, then the straight path from $2y+2e_\mu$ back to $2y$. This loop is the boundary of the surface swept by the transverse edges of $\tau$ along two steps in the direction $\mu$: at most six distinct plaquettes, each with coefficient $\pm1$. The group is abelian, so $P_{2y}^{-1}P_x=\exp d_x$ with $d_x$ the signed sum of their $\Theta_p$, and $\lVert d_x\rVert\le6\eta\le\rho_G/2$. By Lemma~\ref{lem:logbound}, $\delta(P_x,P_{2y})\le\rho_G/2$, so $\chi=1$, and since the eigenvalues of $d_x$ have modulus at most $\frac12$, $\log(P_{2y}^{-1}P_x)=d_x$. Hence $V^\sharp_{y,\mu}=\exp(X_{y,\mu}+\bar d_{y,\mu})$, where $X_{y,\mu}\in\mathfrak s$ is any logarithm of $P_{2y}$ and $\bar d_{y,\mu}$ is the mean of the $d_x$. Let the sides of $P$ be the coarse links $\ell$ with signs $\epsilon_\ell$, and write $d_{o,\ell}$ for the $d_x$ of the side $\ell$ at the offset $o$. For each $o$ the four straight paths of length two at the offset $o$ of the four sides bound the square $\square_{2y+o}$, so \[ Y_o=\sum_\ell\epsilon_\ell\bigl(X_\ell+d_{o,\ell}\bigr)-F_{2y+o}\in \Lambda . \] Since $d_{0,\ell}=0$, $\lVert Y_o-Y_0\rVert\le4\cdot6\eta+8\eta=32\eta\le \frac83<2\pi$, so $Y_o=Y_0$ for every $o$. Averaging over $o$ gives $\sum_\ell\epsilon_\ell(X_\ell+\bar d_\ell)=\bar F_P+Y_0$, and $\exp Y_0=I$. Finally $\lVert\bar F_P\rVert\le\max_o\lVert F_{2y+o}\rVert\le4\eta<\pi/3$, so $\bar F_P$ is the principal logarithm. \end{proof} \begin{theorem}\label{thm:torusscales} Let $U$ be an $S$-valued field in the tree gauge on the periodic lattice of side divisible by $2^k$, with $\delta(U_p,I)\le\rho_G$ on every plaquette, and put $\eta_0=\max_p\lVert\Theta_p\rVert$, $A=\sum_p\lVert\Theta_p\rVert$. Suppose that \begin{equation}\label{eq:torushyp} \max_P\lVert\Theta^{(j)}_P\rVert\le\eta_G\quad\text{for }0\le j0$ there. So there is $c_3\ge1$ depending only on $G$, computed in Proposition~\ref{prop:explicitconstants}, such that there the $m$-th derivative of $c_k$ in $X$ is at most $c_3\prod_{i\le m}N_{y,\mu}(Y^i)$ on $Y^1,\dots,Y^m$ for $m=1,2, 3$, and the first two derivatives of $\log j$ are at most $c_3$ in norm. By Lemma~\ref{lem:smallframes}(i) the background $U$ lies in this set. Let $B=Dc(0)$, a map from $\g^{A_W}$ to $\g^K$, $K$ the set of pairs $k$, so that $T_U=\ker B$ and $N_U=B^*\g^K$, and $\mathcal G=BB^*$. By the proof of Theorem~\ref{thm:smallfibre}(iii), $\mathcal G\ge\frac1{17}$ and $\lVert\mathcal G^{-1}\rVert\le17$ in the norms $\max_k\lVert z_k\rVert$ and, $\mathcal G$ being symmetric, $\sum_k\lVert z_k\rVert$; each $B_k$ has norm at most $0.44$ on the $24$ links of $V_{y,\mu}$, and every link lies in at most two of these sets. Hence $\lVert B\rVert\le0.63$, $\lVert BX\rVert_\infty\le2.2\lVert X\rVert_\infty$, $\lVert B^*z\rVert_\infty\le 0.88\max_k\lVert z_k\rVert$ and $\lVert B^*z\rVert_1\le10.6\sum_k\lVert z_k\rVert$, where $\lVert X\rVert_1=\sum_e\lVert X_e\rVert$. (i) Write $c(X)=BX+R(X)$. For $\lVert X\rVert_\infty,\lVert\hat X\rVert _\infty\le2s$, Taylor's formula gives $\lVert R_k(X)\rVert\le\frac{c_3}2 N_{y,\mu}(X)^2\le12c_3\lVert X\rVert_\infty^2$ and $\lVert R_k(X)-R_k(\hat X)\rVert\le48c_3s\lVert X-\hat X\rVert_\infty$. For $\eta\in K_s$ the equation $c(\eta+B^*z)=0$ reads $z=\Theta(z)=-\mathcal G^{-1}R(\eta+B^*z)$, and on $\max_k\lVert z_k\rVert\le s$ the map $\Theta$ has values of norm at most $17\cdot12c_3\cdot4s^2=816c_3s^2\le s$ and Lipschitz constant at most $17\cdot48\cdot0.88\,c_3s\le720c_3s\le\frac12$. So it has a unique fixed point $z(\eta)$, smooth in $\eta$ by the implicit function theorem, and $h=B^*z$ has $\lVert h\rVert_\infty\le0.88\cdot816c_3s^2\le s/2$. Since $h(\eta)\perp T_U$, $\eta$ is the orthogonal projection of $\eta+h(\eta)$, so $\Phi$ is injective; its differential $1+Dh$ is injective, and $\dim T _U=\dim F$. If $Ue^X\in F$ with $\lVert X\rVert_\infty\le s/40$, then $X= \eta+B^*z$ with $z=\mathcal G^{-1}BX$, $\max_k\lVert z_k\rVert\le17\cdot2.2 \,s/400$, \[ \Omega_r=\{\eta\in T_U:\lVert\eta_e\rVert\le r\text{ for every }e\in L'\}. \] Give every link the position of its midpoint and every pair $k$ that of $d_k$, and let $d(a,b)$ be the $\ell^\infty$ distance between the positions on the periodic lattice. The links that enter $V_{y,\mu}$ have midpoints within distance $1$ of that of $d_k$. Let $\nabla'$ be the gradient in the coordinates $(\eta_e)_{e\in L'}$, and, for a block matrix $M=(M_{ab})$ whose indices are links or pairs and whose blocks are maps between copies of $\g$, and for $\beta\ge0$, let \[ \lVert M\rVert_\beta=\max\Bigl(\sup_a\sum_be^{\beta d(a,b)}\lVert M_{ab} \rVert,\ \sup_b\sum_ae^{\beta d(a,b)}\lVert M_{ab}\rVert\Bigr). \] At most two indices share a position, so the sum \[ N(\beta)=\sup_a\sum_be^{-\beta d(a,b)} \] is finite for $\beta>0$ and depends only on $\beta$. \begin{lemma}\label{lem:kernels} \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The operator norm satisfies $\lVert M\rVert\le\lVert M\rVert_\beta$, and $\lVert M_1M_2\rVert_\beta\le\lVert M_1\rVert_\beta\lVert M_2 \rVert_\beta$ and $\lVert M^*\rVert_\beta=\lVert M\rVert_\beta$. \item If $|\varphi(a)-\varphi(b)|\le d(a,b)$ for all $a,b$ and $0\le\alpha \le\beta$, then $\lVert e^{\alpha\varphi}Me^{-\alpha\varphi}-M\rVert\le \frac\alpha\beta\lVert M\rVert_\beta$. \item If $M$ is invertible, $\lVert M^{-1}\rVert\le M_0$ and $\lVert M\rVert _\beta\le M_1$, then $\lVert M^{-1}\rVert_\gamma\le2M_0N(\gamma)$ with $\gamma=\beta/(4M_0M_1)$. \end{enumerate} \end{lemma} \begin{proof} (i) is the Schur test and the triangle inequality for $d$. (ii) The blocks of the difference are $(e^{\alpha(\varphi(a)-\varphi(b))}-1)M_{ab}$, and $|e^{\alpha(\varphi(a)-\varphi(b))}-1|\le e^{\alpha d(a,b)}-1\le\frac \alpha\beta e^{\beta d(a,b)}$ by convexity of $\alpha\mapsto e^{\alpha d}$; apply (i) with $\beta=0$. (iii) Let $\alpha=2\gamma\le\beta$, since $M_0M_1 \ge\lVert M^{-1}\rVert\lVert M\rVert\ge1$. By (ii), $e^{\alpha\varphi}M e^{-\alpha\varphi}=M+E$ with $\lVert M^{-1}E\rVert\le\frac12$, so $\lVert e^{\alpha\varphi}M^{-1}e^{-\alpha\varphi}\rVert\le2M_0$ as in Lemma~\ref{lem:combes}. With $\varphi=d(\cdot,b)$ the block $(a,b)$ gives $e^{2\gamma d(a,b)}\lVert(M^{-1})_{ab}\rVert\le2M_0$ for all $a,b$, and summing $e^{\gamma d(a,b)}\lVert(M^{-1})_{ab}\rVert\le2M_0e^{-\gamma d(a,b)}$ over $b$ or over $a$ gives the claim. \end{proof} \begin{theorem}\label{thm:chartdecay} Assume the hypotheses of Theorem~\ref{thm:fibrechart}\textup{(iv)}, with $c_6\ge1$, as we may, and let $r=s/32$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The coordinates $(\eta_e)_{e\in L'}$ identify $T_U$ with $\g^{L'}$, Lebesgue measure on $T_U$ being a constant multiple of that on $\g^{L'}$; $\Omega_{r'}$ is a product of balls in them, and $K_{r'}\subset\Omega_{r'} \subset K_{15.1r'}$ for every $r'>0$. \item There are $\alpha_G>0$ and $c_9$ depending only on $G$ with the following property. Let $\mu$ be the constrained law conditioned on $\Phi(\Omega_r)$, written as a law on $\Omega_r$. For every $\varphi:L'\to \R$ with $|\varphi(e)-\varphi(e')|\le d(e,e')$, every $0\le\alpha\le \alpha_G$ and all smooth $F,G$ on $T_U$, \[ \lvert\operatorname{Cov}_\mu(F,G)\rvert\le\frac{120}\lambda\,\bigl\lVert e^{-\alpha\varphi}\nabla'F\bigr\rVert_{L^2(\mu)}\bigl\lVert e^{\alpha \varphi}\nabla'G\bigr\rVert_{L^2(\mu)}. \] \item If $X_1,X_2\subset A_W$, $\tilde F,\tilde G$ are smooth functions of the links in $X_1$ and in $X_2$ whose left-invariant gradients have norm at most $1$ in each link, $F=\tilde F\circ\Phi$ and $G=\tilde G\circ\Phi$, then \[ \lvert\operatorname{Cov}_\mu(F,G)\rvert\le\frac{c_9}\lambda\,\lvert X_1 \rvert\,\lvert X_2\rvert\,e^{-\alpha_Gd(X_1,X_2)}. \] \end{enumerate} The constants do not depend on $L$, $W$, $U$, $\lambda$, $s$ or $\kappa$. \end{theorem} So the covariances of local observables of the field under the constrained law, conditioned on the chart, decay exponentially at a rate that depends only on $G$, with amplitude $O(1/\lambda)$; for $\SU(2)$ and the Wilson action the amplitude is $O(1/\beta)$. \begin{proof} All constants called $C$ depend only on $G$. \emph{The section.} By Lemma~\ref{lem:smallframes}(iv), $B_k=\tilde B_k+R_k$ with $\lVert R_k\rVert\le260\kappa$, and the block of $\tilde B_k$ at $d_k$ is $\frac18$ times an isometry of $\g$, so $\lVert B_{k,d_k}^{-1}\rVert\le (\frac18-260\kappa)^{-1}\le8.01$. For $\xi\in\g^{L'}$ let $(S\xi)_e=\xi_e$ for $e\in L'$ and $(S\xi)_{d_k}=-B_{k,d_k}^{-1}\sum_{e\ne d_k}B_{k,e}\xi_e$. Since $d_k$ enters only $V_k$, $BS=0$; $S$ is injective, and $\dim T_U= \lvert L'\rvert\dim G$ because $F$ has codimension $4\lvert W\rvert\dim G$, so $S$ is an isomorphism onto $T_U$ whose inverse is the restriction to $L'$. The other coefficients of $\tilde B_k$ add up to $\frac78+1=\frac{15}8$, so $\lVert(S\xi)_{d_k}\rVert\le8.01(\frac{15}8+\sqrt{24}\cdot260\kappa)\max _e\lVert\xi_e\rVert\le15.1\max_e\lVert\xi_e\rVert$. This proves (i), and $S$ has blocks of norm at most $1.1$ that vanish beyond distance $1$. Write $\mathrm{Box}_{r'}$ for the product of balls $S^{-1}\Omega_{r'}$. Since $15.1\cdot2r0$ and $c_8\ge1$ depending only on $G$, the Hessian being the block matrix of second derivatives in the coordinates $\xi_e$. The maps $B$, $A$, $S$, the second derivative of $Q(Ue^X)$ in $X$ and that of $\sum_e\log j(X_e)$ have blocks bounded by $C$ (the last two by compactness, as $c_3$) that vanish beyond distance $2$, so their norms $\lVert\cdot\rVert_1$ are at most $C$. By the proof of Theorem~\ref{thm:fibrechart}, $\lVert(AB^*)^{-1}\rVert\le34$ and $\lVert(AA^*)^{-1}\rVert\le25$, and $\lVert(\bar P^*\bar P)^{-1}\rVert \le1$. Applying Lemma~\ref{lem:kernels}(iii) to $AB^*$ and $AA^*$, then (i) to $P$, $\bar P$ and $\bar P^*\bar P$, then (iii) to $\bar P^*\bar P$, gives $\beta_0\in(0,1]$ such that $(AB^*)^{-1}$, $(AA^*)^{-1}$, $P$, $\bar P$, $\bar P^*\bar P$ and $(\bar P^*\bar P)^{-1}$ have norm $\lVert\cdot\rVert _{\beta_0}$ at most $C$ on $\mathrm{Box}_{2r}$. Let $\gamma=\beta_0/2$, and for links $e,e'$ and $v,v'\in\g$ write $v,v'$ also for the vectors of $\g^{L'}$ supported at $e$, $e'$. Call a block matrix $M$, depending linearly on $v$, \emph{concentrated at $e$} if $\lVert M_{ab}\rVert\le C \lvert v\rvert e^{-\gamma(d(a,e)+d(b,e))}$, and, depending bilinearly on $(v,v')$, \emph{concentrated at $(e,e')$} if $\lVert M_{ab}\rVert\le C \lvert v\rvert\lvert v'\rvert e^{-\gamma(d(a,e)+d(a,e')+d(b,e)+d(b,e'))}$; a vector is concentrated if it is so as a column. If $\lVert Y\rVert_{\beta _0}\le C$, then $YM$ and $MY$ are concentrated where $M$ is, because $e^{-\gamma(d(c,e)+d(c,e'))}\le e^{\beta_0d(a,c)}e^{-\gamma(d(a,e)+d(a, e'))}$. For $M$ concentrated at $e$ and $M'$ at $e'$, using $d(a,e)+d(a,e')\ge d(e,e')$ in both indices, \[ \lvert\Tr(MM')\rvert\le C\lvert v\rvert\lvert v'\rvert N(\gamma/2)^2 e^{-\gamma d(e,e')}, \] and if $M$ is concentrated at $(e,e')$, then \[ \lvert\Tr M\rvert\le C\lvert v\rvert\lvert v'\rvert N(\gamma)e^{-\gamma d(e,e')}. \] Now $x_v=\partial_vX=\bar Pv$ has $\lVert(x_v)_a\rVert\le Ce^{-\beta_0 d(a,e)}\lvert v\rvert$, so it is concentrated at $e$. The vector $D^2c(X)[x_v,x_{v'}]$ has components at most $c_3N_k(x_v)N_k(x_{v'})\le C\lvert v\rvert\lvert v'\rvert e^{-\beta_0(d(k,e)+d(k,e'))}$, so $x_{vv'}= \partial_v\partial_{v'}X=-B^*(AB^*)^{-1}D^2c(X)[x_v,x_{v'}]$ is concentrated at $(e,e')$. In the same way $A_v=\partial_vA=D^2c(X)[x_v,\cdot\,]$ is concentrated at $e$, $A_{vv'}=D^3c(X)[x_v,x_{v'},\cdot\,]+D^2c(X)[x_{vv'}, \cdot\,]$ is concentrated at $(e,e')$, $\partial_vP=-B^*(AB^*)^{-1}A_vP$ is concentrated at $e$, and $\partial_v\partial_{v'}P$ is a sum of products of operators bounded in $\lVert\cdot\rVert_{\beta_0}$ with either one factor concentrated at $(e,e')$ or one factor concentrated at $e$ and one at $e'$. The same holds for the derivatives of $\mathcal G_1=AA^*$ and $\mathcal G_2 =\bar P^*\bar P$. Then \[ \begin{split} \partial_v\partial_{v'}\bar H={}&\lambda\bigl(D^2Q[x_v,x_{v'}]+DQ[x_{vv'}] \bigr)-\sum_e\partial_v\partial_{v'}\log j(X_e)\\ &+\sum_{i=1,2}\pm\tfrac12\bigl(\Tr(\mathcal G_i^{-1}\partial_v\partial _{v'}\mathcal G_i)\\ &\qquad-\Tr(\mathcal G_i^{-1}\partial_v\mathcal G_i\,\mathcal G_i^{-1} \partial_{v'}\mathcal G_i)\bigr), \end{split} \] where $D^2Q$ and $DQ$ are the derivatives of $Q(Ue^X)$ in $X$. By the bounds above every term is at most \[ C(\lambda+1)\lvert v\rvert\lvert v'\rvert e^{-\gamma d(e,e')/2}; \] for instance \[ \lvert D^2Q[x_v,x_{v'}]\rvert\le C\sum_ae^{-\beta_0(d(a,e)+d(a,e')-2)} \lvert v\rvert\lvert v'\rvert. \] Hence, using $\lambda\ge60c_6\ge60$, every block of the Hessian of $\bar H$ satisfies \[ \lVert(\operatorname{Hess}\bar H)_{ee'}\rVert\le C\lambda e^{-\beta_0 d(e,e')/4}, \] and $\lVert\operatorname{Hess}\bar H\rVert_\beta\le c_8\lambda$ with $\beta =\beta_0/8$. \emph{Approximation.} Let $\rho$ be smooth on $[0,\infty)$ with $0\le\rho' \le1$, $\rho(t)=t$ for $t\le\frac32r$, $\rho\le\frac74r$ and $\rho$ constant on $[2r,\infty)$, and let $\tau(\xi)_e=\rho(\lVert\xi_e\rVert)\xi_e/\lVert\xi_e\rVert$, which maps $\g^{L'}$ into $\mathrm{Box}_{7r/4}$. Let $v$ be smooth and convex on $[0,\infty)$, zero on $[0,r]$, with $00$ and bounded derivatives of order at least two, and $F,G$ smooth with bounded derivatives, the Helffer--Sj\"ostrand representation~\cite{HS94,NS97} reads $\operatorname{Cov}_{\mu_n}(F,G)=\langle\nabla F,\mathcal A^{-1}\nabla G \rangle_{L^2(\mu_n)}$ with $\mathcal A=\mathcal L_n\otimes1+\operatorname{Hess} H_n$ acting on $L^2(\mu_n;\g^{L'})$. With $w=e^{\alpha\varphi}$, constant in $\xi$, $\mathcal A_w=w\mathcal Aw^{-1}=\mathcal L_n\otimes1+w\operatorname {Hess}H_nw^{-1}$, and $\operatorname{Re}\langle u,\mathcal A_wu\rangle\ge \frac m2\lVert u\rVert^2$ because $\mathcal L_n\ge0$; the adjoint satisfies the same bound, so $\lVert\mathcal A_w^{-1}\rVert\le\frac2m=\frac{120} \lambda$; this is the weighted form of the representation used in~\cite{BJS00,Hel98}. Since $\mathcal A^{-1}=w^{-1}\mathcal A_w^{-1}w$, \[ \begin{split} \lvert\operatorname{Cov}_{\mu_n}(F,G)\rvert&=\bigl\lvert\langle w^{-1}\nabla F,\mathcal A_w^{-1}w\nabla G\rangle\bigr\rvert\\ &\le\frac{120}\lambda\lVert w^{-1}\nabla F\rVert_{L^2(\mu_n)}\lVert w\nabla G\rVert_{L^2(\mu_n)}. \end{split} \] As $n\to\infty$, $e^{-H_n}$ decreases to $e^{-\bar H}\mathbf 1_{\mathrm{Box} _r}$ off the boundary of $\mathrm{Box}_r$, since $\tau$ is the identity there and $V>0$ off it, so $\mu_n\to\mu$ in total variation. For $F,G$ smooth on $T_U$, apply the bound to $(F\circ S)\chi$ and $(G\circ S)\chi$ with $\chi$ smooth, compactly supported and equal to $1$ near $\mathrm{Box} _r$, and let $n\to\infty$; this proves (ii). (iii) The left-invariant derivative of $U_ae^{X_a}$ along $\delta X_a$ is $\Psi(\operatorname{ad}_{X_a})\delta X_a$, of norm at most $2\lVert\delta X_a\rVert$, so $\lVert\partial_{\xi_f}F\rVert\le2\sum_{a\in X_1}\lVert\bar P_{af}\rVert$, and similarly for $G$. Take $\varphi=d(\cdot,X_2)$ and $\alpha =\alpha_G\le\beta_0$. Then $e^{-\alpha d(f,X_2)}\le e^{\beta_0d(a,f)}e^{- \alpha d(a,X_2)}$, so $\lvert e^{-\alpha\varphi}\nabla'F\rvert\le2\lvert X_1\rvert\lVert\bar P\rVert_{\beta_0}e^{-\alpha d(X_1,X_2)}$, and $e^{\alpha d(f,X_2)}\le e^{\beta_0d(b,f)}$ for $b\in X_2$ gives $\lvert e^{\alpha \varphi}\nabla'G\rvert\le2\lvert X_2\rvert\lVert\bar P\rVert_{\beta_0}$. Insert these in (ii); with $C$ a bound for $\lVert\bar P\rVert_{\beta_0}$ on $\mathrm{Box}_{2r}$, $c_9=480C^2$. \end{proof} The rate $\alpha_G$ and the amplitude $c_9$ can be computed, and they turn out not to depend on $G$. The proof above passes through the norms $\lVert\cdot\rVert_\beta$, whose Schur sums bring in the large factors $N(\beta)$; the computation below replaces them by weighted operator norms, in which inverses and products cost nothing beyond their operator norms. For a block matrix $M$ indexed by links and pairs and for $\vartheta\ge0$ let \[ \lVert M\rVert_{(\vartheta)}=\sup_\varphi\bigl\lVert e^{\vartheta\varphi} Me^{-\vartheta\varphi}\bigr\rVert, \] the supremum over all $\varphi$ with $|\varphi(a)-\varphi(b)|\le d(a,b)$ for all indices $a,b$, the factors $e^{\pm\vartheta\varphi}$ acting as scalars on each block. For such $\varphi$ write $w=e^{\vartheta\varphi}$ and $M^w=wMw^{-1}$. \begin{lemma}\label{lem:weightednorms} \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For $\vartheta\le\vartheta'$, $\lVert M\rVert\le\lVert M\rVert_{( \vartheta)}\le\lVert M\rVert_{(\vartheta')}$; moreover $\lVert M_1M_2\rVert _{(\vartheta)}\le\lVert M_1\rVert_{(\vartheta)}\lVert M_2\rVert_{(\vartheta)}$, $\lVert M^*\rVert_{(\vartheta)}=\lVert M\rVert_{(\vartheta)}$, and $\lVert M_{ab}\rVert\le\lVert M\rVert_{(\vartheta)}e^{-\vartheta d(a,b)}$. If the blocks of $M$ vanish beyond distance $R$, then $\lVert M\rVert_{(\vartheta)} \le e^{\vartheta R}(R_MC_M)^{1/2}$, where $R_M$ and $C_M$ are the largest row and column sums of the norms of the blocks. \item If $M$ is Hermitian and $0\le\vartheta\le\vartheta'$, then $\lVert \operatorname{Re}M^w-M\rVert\le2(\vartheta/\vartheta')^2\lVert M\rVert _{(\vartheta')}$ for every $\varphi$, where $\operatorname{Re}T=\frac12(T+ T^*)$. \item If $\operatorname{Re}M^w\ge c>0$ for every $\varphi$, then $\lVert M^{-1}\rVert_{(\vartheta)}\le1/c$. If $\lVert M^{-1}\rVert_{(\vartheta)} \lVert E\rVert_{(\vartheta)}<1$, then $\lVert(M+E)^{-1}\rVert_{(\vartheta)} \le\lVert M^{-1}\rVert_{(\vartheta)}/(1-\lVert M^{-1}\rVert_{(\vartheta)} \lVert E\rVert_{(\vartheta)})$. \end{enumerate} \end{lemma} \begin{proof} (i) Take $\varphi=0$ for the first inequality and replace $\varphi$ by $\vartheta\varphi/\vartheta'$ for the second. Products and adjoints follow from $(M_1M_2)^w=M_1^wM_2^w$ and $(M^*)^w=(M^{w'})^*$ with $w'=e^{-\vartheta \varphi}$. With $\varphi=d(\cdot,b)$ the block $(a,b)$ of $M^w$ is $e^{ \vartheta d(a,b)}M_{ab}$, of norm at most $\lVert M^w\rVert$. The last bound is the Schur test, since the blocks of $M^w$ are $e^{\vartheta(\varphi(a)- \varphi(b))}M_{ab}$. (ii) Let $f(z)=e^{z\varphi}Me^{-z\varphi}$, an entire function of $z\in\C$ with values in matrices. For $z=x+iy$, $f(z)=e^{iy\varphi}f(x)e^{-iy\varphi}$ with $e^{iy\varphi}$ unitary, so $\lVert f(z)\rVert\le K=\lVert M\rVert_{( \vartheta')}$ for $|x|\le\vartheta'$. Since $M$ is Hermitian, $f(x)^*=f(-x)$ for real $x$, so $\operatorname{Re}M^w-M=g(\vartheta)$ with $g(z)=\frac12(f(z) +f(-z))-f(0)$. The function $g$ is even, vanishes at $0$ and is bounded by $2K$ on $|z|<\vartheta'$; so $g(z)/z^2$ is holomorphic there, and the maximum principle on the circles $|z|=\rho<\vartheta'$, applied to $\ell(g(z))/z^2$ for every linear functional $\ell$ of norm one, gives $\lVert g(z)\rVert \le2K|z|^2/\vartheta'^2$. (iii) If $\operatorname{Re}T\ge c$, then $\lVert Tu\rVert\lVert u\rVert\ge \operatorname{Re}\langle u,Tu\rangle\ge c\lVert u\rVert^2$, so $\lVert T^{-1} \rVert\le1/c$; apply this to $T=M^w$, whose inverse is $(M^{-1})^w$. The second statement is the Neumann series for $(M^w+E^w)^{-1}$. \end{proof} \begin{proposition}\label{prop:explicitdecay} Let $G\subset\U(r)$ be compact. Theorem~\ref{thm:chartdecay} holds with \[ \alpha_G=2.5\cdot10^{-4},\qquad c_9=1.2\cdot10^4, \] and in~\textup{(iii)} the factor $\lvert X_1\rvert\lvert X_2\rvert$ may be replaced by $(\lvert X_1\rvert\lvert X_2\rvert)^{1/2}$. \end{proposition} So the covariances of local observables under the constrained law on the chart decay at least like $e^{-d/4000}$, with amplitude $1.2\cdot10^4/ \lambda$, for every compact group; for $\SU(2)$ and the Wilson action the amplitude is $4.8\cdot10^4/\beta$. \begin{proof} Throughout, $\xi\in\mathrm{Box}_{2r}$, $X=S\xi+h(S\xi)$, and the notation is that of the proof of Theorem~\ref{thm:chartdecay}. The hypothesis $4\cdot10^5c_3(\kappa+s)\le\frac1{195}$ of Theorem~\ref{thm:fibrechart}(iv) gives $c_3s\le1.29\cdot10^{-8}$ and $c_3\kappa'\le7.7\cdot10^{-8}$ for $\kappa'=\kappa+6s$, and $\kappa\le10^{-8}$. Every $\varphi$ below is defined on all indices; a function on $L'$ with the Lipschitz property extends to all indices with the same property, by $\varphi(a)=\min_{e\in L'} (\varphi(e)+d(a,e))$. \emph{Local operators.} The row $k$ of $B$ has blocks of norm at most $\frac 18+260\kappa$, with sum at most $2+24\cdot260\kappa$, by Lemma~\ref{lem:smallframes}(iv) and the coefficients of $V_{y,\mu}$, eight of them $\frac18$ and sixteen $\frac1{16}$; every link lies in at most two rows, and the blocks vanish beyond distance $1$. So $b=\lVert B\rVert_{(\vartheta)} \le e^\vartheta(\frac12+10^{-4})^{1/2}$ by Lemma~\ref{lem:weightednorms}(i). The row $k$ of $A-B$ has norm at most $c_3N_{y,\mu}(X)\le7.35c_3s= \varepsilon_A$, since $\lVert X\rVert_\infty\le\frac32s$, so $\lVert A-B \rVert_{(\vartheta)}\le\sqrt{48}\,e^\vartheta\varepsilon_A$. The row $d_k$ of $S$ is $s_k\xi=-B_{k,d_k}^{-1}\sum_{e\ne d_k}B_{k,e}\xi_e$, of norm $\sigma \le(\frac{\sqrt{11}}8+260\kappa)/(\frac18-260\kappa)\le3.32$, the squares of the other coefficients adding up to $\frac{11}{64}$; the other rows of $S$ are those of the identity. If $D_k$ is the diagonal matrix with entries $e^{\vartheta(\varphi(d_k)-\varphi(e))}$ on the links $e$ of $V_{y,\mu}$, then the row $d_k$ of $S^w$ is $s_kD_k$, and every link of $L'$ lies in at most two of these sets, so \[ \bar s=\lVert S\rVert_{(\vartheta)}\le(1+2\sigma^2e^{2\vartheta})^{1/2}. \] Finally $Q(Ue^X)=\sum_p\lVert U_p-I\rVert^2=\sum_p(2r-2\ReTr U_p)$. By Duhamel's formula the derivative of $U_ee^{X_e}$ in $X_e$ is an average of $U_ee^{\theta X_e}Ye^{(1-\theta)X_e}$, and its second derivative an average, of total weight one, of products with $Y$ and $Z$ inserted between unitary factors. With $\lvert\Tr(V_1YV_2ZV_3)\rvert\le\lVert Y\rVert\lVert Z\rVert$ for unitary $V_i$ this gives \[ \bigl\lvert D^2Q_p[Y,Z]\bigr\rvert\le2\sum_{e\in p}\lVert Y_e\rVert\sum_{e \in p}\lVert Z_e\rVert, \] and, since $\ReTr(VY)=\ReTr((V-I)Y)$ for skew-Hermitian $Y$ and a conjugate $V$ of $U_p$, $\lvert DQ_p[Y]\rvert\le2\kappa'\sum_{e\in p}\lVert Y_e\rVert$. A link lies in six plaquettes, so $\lVert D^2Q\rVert_{(\vartheta)}\le48 e^\vartheta$ and $\lvert DQ[Y]\rvert\le12\kappa'\lVert Y\rVert_1$. \emph{Inverses.} The diagonal blocks of $\mathcal G=BB^*$ have norm at most $\frac3{16}+227\kappa$, each row has two other nonzero blocks, of norm at most $\frac1{32}+227\kappa$, at distance $2$, and $\mathcal G\ge\frac1{16}- 700\kappa$, by the proof of Theorem~\ref{thm:smallfibre}(iii). The blocks of $\operatorname{Re}\mathcal G^w$ are $\cosh(\vartheta(\varphi(k)-\varphi(l))) \mathcal G_{kl}$, so $\operatorname{Re}\mathcal G^w\ge\frac1{16}-700\kappa-( \cosh2\vartheta-1)(\frac1{16}+454\kappa)$, and Lemma~\ref{lem:weightednorms}(iii) bounds $g_0=\lVert\mathcal G^{-1}\rVert _{(\vartheta)}$ by the inverse of the right-hand side. With $E=(A-B)B^*$, the same lemma gives \[ \begin{split} \lVert(AB^*)^{-1}\rVert_{(\vartheta)}&\le\frac{g_0}{1-g_0\lVert E\rVert _{(\vartheta)}}=h_0,\\ \lVert(AA^*)^{-1}\rVert_{(\vartheta)}&\le\frac{g_0}{1-g_0(2\lVert E\rVert _{(\vartheta)}+\lVert A-B\rVert_{(\vartheta)}^2)}=g_1, \end{split} \] since $AB^*=\mathcal G+E$ and $AA^*=\mathcal G+E+E^*+(A-B)(A-B)^*$. Let $Y=B^*(AB^*)^{-1}$, so $y=\lVert Y\rVert_{(\vartheta)}\le bh_0$. Since $BS=0$, $\bar P=PS=S-Y(A-B)S$, so $\bar p=\lVert\bar P\rVert_{(\vartheta)}\le\bar s(1+ y\lVert A-B\rVert_{(\vartheta)})$; and $M=Y(A-B)S$ takes values in $N_U$, orthogonal to the values of $S$, so $\bar P^*\bar P=S^*S+M^*M$. Here $S^*S=1+ \sum_ks_k^*s_k$, and $(s_k^*s_k)^w=v^*u$ with $u=s_kD_k$, $v=s_kD_k^{-1}$, so $\operatorname{Re}(s_k^*s_k)^w=\frac14((u+v)^*(u+v)-(u-v)^*(u-v))\ge-\sigma^2 \sinh^2\vartheta\,J_k$, $J_k$ the projection on the links of $V_{y,\mu}$; and $\operatorname{Re}(M^*M)^w\ge-2(\vartheta/\vartheta'')^2\lVert M\rVert_{( \vartheta'')}^2$ by Lemma~\ref{lem:weightednorms}(ii), for $\vartheta\le \vartheta''$. Hence \[ g_2=\lVert(\bar P^*\bar P)^{-1}\rVert_{(\vartheta)}\le\Bigl(1-2\sigma^2\sinh ^2\vartheta-2\frac{\vartheta^2}{\vartheta''^2}\lVert M\rVert_{(\vartheta'')} ^2\Bigr)^{-1}. \] For the $\ell^1$ norms, $\lVert\mathcal G^{-1}\rVert\le17$ in the norm $\sum_k\lVert z_k\rVert$ and $E$ has column sums at most $2(2+10^{-4}) \varepsilon_A$, so $(AB^*)^{-1}$ has norm $h_1\le17.001$ there, and $\lVert B^*z\rVert_1\le(2+10^{-4})\sum_k\lVert z_k\rVert$. \emph{The Hessian.} Let $u,u'\in\g^{L'}$, $p=wu$ and $q=w^{-1}u'$. Since $\langle u,(\operatorname{Hess}\bar H)^wu'\rangle=\partial_p\partial_q\bar H$, it suffices to bound $\lvert\partial_p\partial_q\bar H\rvert$ by $K\lvert u\rvert\lvert u'\rvert$ for every $\varphi$. Write $x_p=\bar Pp=w\tilde x$ and $x_q=w^{-1}\tilde y$, with $\lvert\tilde x\rvert\le\bar p\lvert u\rvert$ and $\lvert\tilde y\rvert\le\bar p\lvert u'\rvert$. The links of $V_{y,\mu}$ lie within distance $1$ of the position of $k$, so the weights of $x_p$ and $x_q$ on them differ from $e^{\pm\vartheta\varphi(k)}$ by factors at most $e^\vartheta$, and \[ \sum_kN_{y,\mu}(x_p)N_{y,\mu}(x_q)\le2e^{2\vartheta}\lvert\tilde x\rvert \lvert\tilde y\rvert, \] every link lying in at most two of these sets. In the same way $A_p=\partial _pA=D^2c(X)[x_p,\cdot\,]$ equals $w\check A_p$ and $A_q=w^{-1}\check A_q$, where $\check A_p$ and $\check A_q$ have rows of rank at most $\dim G$, so their Hilbert--Schmidt norms satisfy \[ \lVert\check A_p\rVert_2\lVert\check A_q\rVert_2\le2\dim G\,c_3^2e^{2 \vartheta}\bar p^2\lvert u\rvert\lvert u'\rvert, \] and conjugating them by $w$ multiplies these norms by at most $e^\vartheta$. The second derivative $x_{pq}=-YD^2c(X)[x_p,x_q]$ has $\lVert x_{pq}\rVert_1 \le c_3x_1\lvert u\rvert\lvert u'\rvert$ with $x_1=2(2+10^{-4})h_1e^{2 \vartheta}\bar p^2$, and $A_{pq}=D^3c(X)[x_p,x_q,\cdot\,]+D^2c(X)[x_{pq}, \cdot\,]$ has trace norm $\bar a\le2\dim G\,c_3^2(e^{2\vartheta}\bar p^2+x_1) \lvert u\rvert\lvert u'\rvert$, the weights cancelling in both. The four parts of $\bar H$ are bounded as follows. First, $\partial_p \partial_q\lambda Q=\lambda(\langle\tilde x,(D^2Q)^w\tilde y\rangle+DQ[x_{pq} ])$, at most $\lambda(48e^\vartheta\bar p^2+12c_3\kappa'x_1)\lvert u\rvert \lvert u'\rvert$. Second, the Hessian of $\sum_e\log j(X_e)$ in $X$ is block diagonal and commutes with $w$, so its part is at most $c_3(\bar p^2+c_3x_1) \lvert u\rvert\lvert u'\rvert$. Third, for $\mathcal G_1=AA^*$, \[ \begin{split} \partial_p\partial_q\log\det\mathcal G_1={}&\Tr(\mathcal G_1^{-1}\partial_p \partial_q\mathcal G_1)\\ &-\Tr(\mathcal G_1^{-1}\partial_p\mathcal G_1\,\mathcal G_1^{-1}\partial_q \mathcal G_1), \end{split} \] with $\partial_p\mathcal G_1=A_pA^*+AA_p^*$ and $\partial_p\partial_q\mathcal G_1=A_{pq}A^*+A_pA_q^*+A_qA_p^*+AA_{pq}^*$. Each term is the trace of a product in which $w$ and $w^{-1}$ occur once each, next to $\check A_p$ and $\check A_q$, or not at all, next to $A_{pq}$. A cyclic permutation writes it as $\Tr(wFw^{-1}F')$, or $\Tr(w^{-1}Fw\,F')$, and $\lvert\Tr(wFw^{-1}F')\rvert \le\lVert wFw^{-1}\rVert_2\lVert F'\rVert_2$ or $\lVert wFw^{-1}\rVert\lVert F'\rVert_1$. With $a=\lVert A\rVert_{(\vartheta)}\le b+\lVert A-B\rVert_{( \vartheta)}$ this gives \[ \begin{split} \tfrac12\lvert\partial_p\partial_q\log\det\mathcal G_1\rvert\le{}&g_1a\bar a \\ &+(g_1+2g_1^2a^2)e^{2\vartheta}\lVert\check A_p\rVert_2\lVert\check A_q\rVert _2. \end{split} \] Fourth, $\partial_p\bar P=-YA_p\bar P$ and $\partial_p\partial_q\bar P=-YA_{pq} \bar P+YA_qYA_p\bar P+YA_pYA_q\bar P$, and the same counting for $\mathcal G _2=\bar P^*\bar P$ gives \[ \begin{split} \tfrac12\lvert\partial_p\partial_q\log\det\mathcal G_2\rvert\le{}&g_2\bar p^2 y\bar a+(3g_2\bar p^2y^2\\ &+2g_2^2\bar p^4y^2)e^{2\vartheta}\lVert\check A_p\rVert_2\lVert\check A_q \rVert_2. \end{split} \] So \[ \lVert\operatorname{Hess}\bar H\rVert_{(\vartheta)}\le K=\lambda k_1+\dim G\,c_3^2k_0, \] where $k_1=48e^\vartheta\bar p^2+12c_3\kappa'x_1$ and $k_0$ collects the other three parts, divided by $\dim G\,c_3^2\lvert u\rvert \lvert u'\rvert$, which is possible as $c_3\ge1$. At $\vartheta=\vartheta'= 0.197$, with $\vartheta''=0.6$, these give $b\le0.862$, $g_0,h_0,g_1\le 17.37$, $y\le14.96$, $\bar s,\bar p\le5.80$, $g_2\le7.41$, $k_1\le1966$ and $k_0\le4.16\cdot10^9$; and since $\lambda\ge60c_6=4.086\cdot10^7\dim G\,c_3 ^2$ by Proposition~\ref{prop:explicitconstants}, $K\le2068\lambda$. \emph{Decay.} In the proof of Theorem~\ref{thm:chartdecay}, replace the bound on $e^{\alpha\varphi}\operatorname{Hess}H_ne^{-\alpha\varphi}- \operatorname{Hess}H_n$ by the following. Since $D\tau$ and $\Delta_n$ are block diagonal and symmetric and commute with $w=e^{\alpha\varphi}$, Lemma~\ref{lem:weightednorms}(ii) with $\vartheta=\alpha$ gives \[ \bigl\lVert\operatorname{Re}(\operatorname{Hess}H_n)^w-\operatorname{Hess}H _n\bigr\rVert\le2\frac{\alpha^2}{\vartheta'^2}K\le\frac\lambda{120}=\frac m2 \] for $\alpha\le\vartheta'(240\cdot2068)^{-1/2}$, which holds for $\alpha\le 2.5\cdot10^{-4}$. Then $\operatorname{Re}\langle u,\mathcal A_wu\rangle\ge \frac m2\lVert u\rVert^2$ as before, and (ii) follows. For (iii), $\nabla'F =\bar P^*g$ with $g$ supported on $X_1$ and $\lvert g_a\rvert\le2$, so with $\varphi=d(\cdot,X_2)$, \[ \lvert e^{-\alpha\varphi}\nabla'F\rvert=\lvert(\bar P^*)^{w^{-1}}e^{-\alpha \varphi}g\rvert\le2\bar p_\alpha\lvert X_1\rvert^{1/2}e^{-\alpha d(X_1,X_2)}, \] where $\bar p_\alpha=\lVert\bar P\rVert_{(\alpha)}\le4.8$ at $\alpha=2.5\cdot 10^{-4}$, and $\lvert e^{\alpha\varphi}\nabla'G\rvert\le2\bar p_\alpha\lvert X_2\rvert^{1/2}$. Insert these in (ii): $c_9=480\bar p_\alpha^2\le1.2\cdot 10^4$. \end{proof} The chart misses only points of the fibre at which some plaquette near $W$ is not small, and under the Wilson measure such points carry little mass on most fibres. \begin{theorem}\label{thm:offchart} \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Assume the hypotheses of Theorem~\ref{thm:chartdecay} and $s\ge1.24 \cdot10^5\kappa$. Every $U'\in F$ with $\delta(U'_p,I)\le\kappa$ for every plaquette $p$ with a vertex in a block of $W^+$ lies in $\Phi(\Omega_r)$. Hence the constrained law gives the complement of $\Phi(\Omega_r)$ at most the mass of the points of $F$ at which such a plaquette exceeds $\kappa$. \item Let $G=\SU(2)$, let the side $2L$ be a power of two, let $\mu$ be the Wilson measure in the tree gauge at $\beta_W\ge1$, so that $\lambda=\beta _W/4$, and let $E$ be the set of fields with $\delta(U_p,I)\le\kappa$ for every plaquette with a vertex in a block of $W^+$. For $\mu$-almost every $U$ let $\mu_U$ be the constrained law on the fibre $F(U)$ through $U$. Then \[ \E_\mu\bigl[\mu_U(E^c)\bigr]=\mu(E^c)\le10^4\lvert W\rvert\,q_\kappa(\beta _W), \] with $q_\kappa$ as in Theorem~\ref{thm:defects}. If $\kappa$, $s$ and $\lambda$ satisfy the hypotheses of \textup{(i)} and $0<\varepsilon<1$, the fields $U\in E$ with $\mu_U(F(U)\setminus\Phi_U(\Omega_r))\le \varepsilon$, where $\Phi_U$ is the chart at $U$, have probability at least $1-(1+\varepsilon^{-1})10^4\lvert W\rvert q_\kappa(\beta_W)$. \item At every such $U$, for $X_1,X_2\subset A_W$ and $\tilde F,\tilde G$ as in Theorem~\ref{thm:chartdecay}\textup{(iii)}, \[ \lvert\Cov_{\mu_U}(\tilde F,\tilde G)\rvert\le\Bigl(\frac{4c_9}{\beta_W} e^{-\alpha_Gd(X_1,X_2)}+30\varepsilon\Bigr)\lvert X_1\rvert\,\lvert X_2 \rvert . \] \end{enumerate} \end{theorem} \begin{proof} (i) The fields $U$ and $U'$ satisfy the hypothesis of Lemma~\ref{lem:smallframes} and agree off $A_W$, in particular on the tree links. An active link $e$ with base in a block $B(y)$, $y\in W$, either lies inside $B(y)$ or leaves it in a direction $\mu$. In the first case part (i) of the lemma, at $U$ and at $U'$, gives $\delta(U_e,U'_e)\le6\kappa$. In the second, let $v$ and $v'$ be the path links of $V_{y,\mu}$ at $U$ and at $U'$. Part (iv), at both fields, writes the common cut-off average as $v e^M=v'e^{M'}$ with $\lVert M\rVert,\lVert M'\rVert\le8\pi\kappa$, so by bi-invariance and Lemma~\ref{lem:logbound} \[ \delta(v,v')=\delta\bigl(e^Me^{-M'},I\bigr)\le\lVert M\rVert+\lVert M' \rVert\le16\pi\kappa<50.3\kappa, \] and part (i) gives $\delta(U_e,U'_e)\le\delta(U_e,v)+\delta(v,v')+\delta( v',U'_e)\le92.3\kappa$. Let $X_e=\log(U_e^{-1}U'_e)$. By Lemma~\ref{lem:logbound}, $\lVert X_e\rVert\le\frac\pi3\,92.3\kappa<96.7 \kappa\le s/1280$ for every $e\in A_W$. Theorem~\ref{thm:fibrechart}(i) with $s/32$ in place of $s$, whose chart is the restriction of $\Phi$ by the uniqueness of the fixed point in its proof, puts $U'=Ue^X$ in $\Phi(K _{s/32})$, and $K_{s/32}\subset\Omega_r$ by Theorem~\ref{thm:chartdecay}(i). (ii) The tree links span every block, so for every field there is exactly one gauge transformation, equal to $I$ at the corner of every block, that makes the tree links equal to $I$. Given the tree links, it multiplies each active link on both sides by elements that do not depend on that link, which preserves Haar measure, and $S$ is gauge invariant; so in the tree gauge the active links have density proportional to $e^{-\beta_WS}$ with respect to product Haar measure. Since $\beta_WS=\frac{\beta_W}4\sum_p \delta(U_p,I)^2$, conditionally on the links off $A_W$ the links of $A_W$ have density proportional to $e^{-\lambda Q}$, and conditioning further on the averages gives the constrained law. So $\mu_U$ is a version of the conditional law of $\mu$ given the links off $A_W$ and the averages, and the identity is the tower property of conditional expectation. If $U\notin E$, some plaquette with a vertex in a block $B(y')$, $y'\in W^+$, is bad for $\rho=\kappa$. Its corner $z$ differs from that vertex by an element of $\{0,1\}^4$, so $z$ lies in a block at $\ell^\infty$ distance at most two from $W$, and the plaquette lies in the unit cube $\mathcal C_z$. There are at most $5^4\lvert W\rvert$ such blocks, with $16$ sites each, and $E$ is gauge invariant, so Theorem~\ref{thm:defects} with $\lvert A\rvert=1$ gives $\mu(E^c)\le10^4\lvert W\rvert q_\kappa(\beta_W)$. For $U\in E$, part (i) gives $\mu_U(F(U)\setminus\Phi_U(\Omega_r))\le\mu_U(E^c)$, so the exceptional fields lie in $E^c\cup\{\mu_U(E^c)>\varepsilon\}$, which by Markov's inequality has probability at most $\mu(E^c)(1+\varepsilon^{-1})$. (iii) Let $A=\Phi_U(\Omega_r)$, $m=\mu_U$, $m_A=m(\,\cdot\mid A)$ and $\epsilon'=m(A^c)\le\varepsilon$. By Theorem~\ref{thm:chartdecay}(iii) with $\lambda=\beta_W/4$, $\lvert\Cov_{m_A}(\tilde F,\tilde G)\rvert\le\frac{4c _9}{\beta_W}\lvert X_1\rvert\lvert X_2\rvert e^{-\alpha_Gd(X_1,X_2)}$. The group $\SU(2)$ has diameter $\sqrt2\pi$, and $\tilde F$ has gradient at most $1$ in each link, so after subtracting a constant, which changes no covariance, $\lVert\tilde F\rVert_\infty\le\frac{\sqrt2\pi}2\lvert X_1 \rvert\le2.23\lvert X_1\rvert$, and likewise for $\tilde G$. Writing $m=(1-\epsilon')m_A+\epsilon'm_{A^c}$ gives $\lvert m(fg)-m_A(fg)\rvert\le 2\epsilon'\lVert f\rVert_\infty\lVert g\rVert_\infty$ and $\lvert m(f)-m_A(f) \rvert\le2\epsilon'\lVert f\rVert_\infty$ for bounded $f,g$, so \[ \lvert\Cov_m(f,g)-\Cov_{m_A}(f,g)\rvert\le6\epsilon'\lVert f\rVert_\infty \lVert g\rVert_\infty, \] and $6\cdot2.23^2<30$. \end{proof} So on a fixed set of blocks the constrained law of the $\SU(2)$ Wilson measure decorrelates, coarea factor and curvature of the fibre included, on every fibre outside a set of fibres whose probability tends to zero as $\beta_W\to\infty$. The bound grows linearly in $\lvert W\rvert$, which is harmless for the fixed cubes of the cube condition, but the thresholds are far from the one-loop range: with the constants of Proposition~\ref{prop:explicitconstants} the hypotheses force $\kappa+s\le 6.0\cdot10^{-13}$ and $s\ge1.24\cdot10^5\kappa$, so $\kappa<4.9\cdot 10^{-18}$, and $q_\kappa(\beta_W)<1$ needs $\beta_W\kappa^2>96(\log\beta_W+ C_0)$, so $\beta_W$ exceeds $3.8\cdot10^{38}$ (Fig.~\ref{fig:offchart}). \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_offchart.png} \caption{Theorem~\ref{thm:offchart}. (a)~A fibre $F$ of the cut-off average, drawn as a surface: the product chart $\Phi(\Omega_r)$ around the background $U$ (green) contains every point of $F$ whose plaquettes near $W$ are at most $\kappa$ (inside the blue curve), and the rest of $F$ meets only points with a large plaquette (red). (b)~The least $\beta_W$ at which the bound $10^4\lvert W\rvert q_\kappa(\beta_W)$ is at most $\frac12$, for the $81$ blocks of a cube of side three; it grows like $\kappa^{-2}\log(1/ \kappa)$, and at the largest $\kappa$ the hypotheses allow it is about $4.0\cdot10^{38}$.} \label{fig:offchart} \end{figure*} The mass off the chart can also be bounded on a single fibre, by a Peierls argument inside the fibre. Two properties of the cut-off average make this possible: along a fixed set of links it is a submersion with a constant that does not depend on the field, and its law under Haar measure on those links has a bounded density. Fix $y\in W$ and a direction $\mu$. Write the offsets of the sites of $B(y)$ in the three directions other than $\mu$ as $a\in\{0,1\}^3$, and let $x_a$ and $x'_a$ be the sites of offset $a$ with $o_\mu=0$ and $o_\mu=1$. Let $\ell_a=(x'_a,\mu)$, which joins $B(y)$ to $B(y+e_\mu)$, and let $u_a=(x_a, \mu)$ and $w_a=(x'_a+e_\mu,\mu)$, which lie inside $B(y)$ and $B(y+e_\mu)$. The links $u_0$ and $w_0$ are tree links and $\ell_0=d_k$ is the path link, so in the tree gauge \[ P_{2y}=\ell_0,\qquad P_{x_a}=u_a\ell_a,\qquad P_{x'_a}=\ell_aw_a, \] and $v_x=P_{2y}^{-1}P_x$ equals $I$ at the two sites of offset $0$. The path of $V^\sharp_{y',\nu}$ from a site $x$ consists of the link $(x,\nu)$ and the next link $(x+e_\nu,\nu)$, so $\ell_a$, whose base has $o_\mu=1$, enters only the paths from $x_a$ and $x'_a$ of the pair $(y,\mu)$. We call $\ell_0, \dots,\ell_7$ the \emph{private links} of $(y,\mu)$. The $32$ links with $o_\mu=1$ of $B(y)$ are the private links of the four pairs $(y,\mu)$; the other $17$ active links of $B(y)$ lie inside it. Let \[ \alpha=2\arcsin\frac1{2\sqrt2}=0.7227\ldots,\qquad\Lambda=\frac\alpha {\sin\alpha}=1.0926\ldots, \] so that $1-\frac78\Lambda>0.0439$. Two facts about $\operatorname{ad}X$ for $X\in\g$, which is skew-adjoint for the Hilbert--Schmidt product, will be used throughout. If the eigenvalues $i\psi$ of $\operatorname{ad}X$ satisfy $|\psi|<2\alpha$, then \begin{gather*} \log\bigl(e^{tZ}e^X\bigr)=X+tL_XZ+O(t^2),\\ L_X=\frac{\operatorname{ad}X}{e^{\operatorname{ad}X}-1},\qquad\lVert L_X \rVert\le\Lambda, \end{gather*} since the eigenvalues of $L_X$ have modulus $\frac{|\psi|/2}{\sin(|\psi|/2 )}$, which increases with $|\psi|$. For every $Y\in\g$, \begin{gather*} \frac d{dt}e^{Y+tZ}\Big|_{t=0}e^{-Y}=E_YZ,\\ E_Y=\frac{e^{\operatorname{ad}Y}-1}{\operatorname{ad}Y},\qquad\lVert E_Y \rVert\le1, \end{gather*} because $|e^{i\psi}-1|\le|\psi|$; in particular $d(e^A,e^B)\le\lVert A-B \rVert$ for the Riemannian distance $d$ of the bi-invariant metric, along the curve $e^{A+t(B-A)}$. If $\delta(v,I)<1$ and $e^{i\theta_j}$, $|\theta _j|\le\pi$, are the eigenvalues of $v$, then $\sum_j4\sin^2(\theta_j/2)<1$; each $|\theta_j|$ is less than $\frac\pi3$, where $\sin^2(t/2)$ is convex, so $|\theta_j|+|\theta_k|<2\alpha$ for all $j,k$, and $X=\log v$ satisfies the hypothesis on $\operatorname{ad}X$. \begin{lemma}\label{lem:privatesubmersion} Let $\sigma_0=(1-\frac78\Lambda)/\sqrt2>0.031$. At every field at which $V^\sharp_{y,\mu}$ is differentiable, its derivative $J:\g^8\to\g$ with respect to its eight private links, right trivialised, satisfies $JJ^*\ge \sigma_0^2$. The fields at which it is not differentiable form a null set for Haar measure. \end{lemma} \begin{proof} Write $m=\max_x\delta(v_x,I)$ and $\chi=\phi(m/\rho_G)$, so that $V^\sharp= P_{2y}e^{\chi N}$ with $N=\frac1{16}\sum_x\log v_x$. Where $m>\rho_G$, the cut-off vanishes near the field, $V^\sharp=\ell_0$, and the derivative along $\ell_0$ is the identity. Suppose $m<\rho_G$, and that the maximum is attained at one path $x_*$, after identifying identical path functions, and $m\ne\rho_G/2$. Copies that coincide because $u_a,w_a$ are tree links are counted together. Other ties lie in finitely many proper real-analytic subsets, a null set. The variation below fixes both copies of an identical pair, since then $u_b=w_b=I$ and $A-1=0$. Then every $\delta(v_x,I)$ is less than $1$, the logarithms are defined, $\chi$ is differentiable, and every variation that fixes $v_{x_*}$ to first order fixes $\chi$ to first order. The site $x_*$ has offset $b\ne0$. Fix $\zeta\in\g$ and put $\eta=\operatorname{Ad}(\ell_0)\zeta$. Move $\ell_0 \mapsto e^{t\eta}\ell_0$, and, if $x_*=x'_b$, $\ell_b\mapsto e^{t\eta}\ell_b$, or, if $x_*=x_b$, $\ell_b\mapsto e^{t\xi}\ell_b$ with $\xi=\operatorname{Ad} (u_b^{-1})\eta$, so that $u_b\ell_b\mapsto e^{t\eta}u_b\ell_b$. In both cases $P_{2y}$ and $P_{x_*}$ are multiplied on the left by $e^{t\eta}$, so $v_{x_*}$ and $\chi$ are fixed to first order. The other site $\bar x$ of offset $b$ has $P_{\bar x}\mapsto e^{t\operatorname{Ad}(u_b^{\pm1})\eta} P_{\bar x}$, and the paths from the remaining twelve sites with $v_x\ne I$ do not move. With the isometry $A=\operatorname{Ad}(\ell_0^{-1}u_b^{\pm1} \ell_0)$, \[ v_{\bar x}\mapsto e^{t(A-1)\zeta}v_{\bar x}+O(t^2),\qquad v_x\mapsto e^{-t \zeta}v_x+O(t^2) \] for the twelve others, so $N$ moves by $t\,\delta N$ with $16\,\delta N=L_{X _{\bar x}}(A-1)\zeta-\sum_xL_{X_x}\zeta$, twelve terms, and $\lVert\delta N \rVert\le\frac{14}{16}\Lambda\lVert\zeta\rVert$. Since $\chi$ is fixed, \[ \frac d{dt}V^\sharp\,(V^\sharp)^{-1}=\eta+\operatorname{Ad}(\ell_0)E_{\chi N}(\chi\,\delta N)=\operatorname{Ad}(\ell_0)\,T\zeta, \] with $\lVert T\zeta-\zeta\rVert\le\frac78\Lambda\lVert\zeta\rVert$, as $\chi\le1$. The variation is $K\zeta$ for a linear map $K:\g\to\g^8$ with two nonzero components of norm $\lVert\zeta\rVert$, so $\lVert K\rVert=\sqrt2$ and $JK=\operatorname{Ad}(\ell_0)T$. For a unit vector $e\in\g$, \[ \lVert J^*e\rVert\ge\frac{\lVert K^*J^*e\rVert}{\sqrt2}=\frac{\lVert T^* \operatorname{Ad}(\ell_0)^*e\rVert}{\sqrt2}\ge\frac{1-\frac78\Lambda} {\sqrt2}=\sigma_0, \] since the least singular value of $T$ is at least $1-\frac78\Lambda$. At the other points of differentiability the same bound follows by continuity: the cut-off has finitely many smooth branch formulas, at such a point the formulas of the adjacent full-dimensional branches agree with its differential, and the lower bound on the singular value is preserved under limits. \end{proof} \begin{corollary}\label{cor:coareabound} Let $C=(V^\sharp_{y,\mu})_{y\in W,\,\mu}$, as a function of the links of $A_W$, the other links being fixed. Wherever $C$ is differentiable, \[ \det\bigl(DC\,DC^*\bigr)^{-1/2}\le\sigma_0^{-4|W|\dim G}<33^{4|W|\dim G}. \] \end{corollary} \begin{proof} The private links of distinct pairs are distinct, and each enters only the average of its pair, so the columns of $DC$ that belong to private links form a block diagonal matrix with the blocks of Lemma~\ref{lem:privatesubmersion}. Hence $DC\,DC^*\ge\bigoplus_kJ_kJ_k^* \ge\sigma_0^2$. \end{proof} The coarea factor of the constrained law is therefore bounded on the whole fibre by a constant per block, for every value of the cut-off and however far the links inside the blocks are from $I$. The volume of the fibre is bounded through the law of the averages. From here on $G=\SU(2)$, which with the metric $\lVert X\rVert^2=\Tr(X^*X)$ is the round sphere of radius $\sqrt2$, $\rho_G=1$, and $h$ is normalised Haar measure. \begin{lemma}\label{lem:averagedensity} Let $G=\SU(2)$, fix every link other than the private links of $(y,\mu)$, and let these eight links be independent and Haar distributed. Then the law of $V^\sharp_{y,\mu}$ has a density with respect to $h$ that is at most \[ M=1+2\Bigl(1-\frac78\Lambda\Bigr)^{-3}<2.4\cdot10^4 \] at every point, in the version given by the formula~\eqref{eq:densityformula} of the proof. \end{lemma} \begin{proof} Let $g=\ell_0$, and let $m_1$ and $m_2$ be the largest values of $\delta(v_{x_a},I)$ and of $\delta(v_{x'_a},I)$. Up to the null set $m=1$, split $G^8$ into the three regions \begin{gather*} R_0=\{m>1\},\\ R_1=\{m_1\ge m_2,\ m_1<1\},\qquad R_2=\{m_2>m_1,\ m_2<1\}; \end{gather*} ties between the two families, including two path functions that coincide identically, belong to $R_1$. On $R_0$, $\chi=0$ and $V^\sharp=g$. On $R_1$ use the coordinates $(g,h_1,\dots,h_7)$ with $h_a=v_{x_a}=g^{-1}u_a \ell_a$. For fixed $g$ the map $\ell_a\mapsto h_a$ is a left translation, so these coordinates carry product Haar measure, and \[ v_{x'_a}=g^{-1}\ell_aw_a=g^{-1}u_a^{-1}g\,h_aw_a . \] Fix $h$ and put $m_h=\max_a\delta(h_a,I)<1$. The set $S_h$ of $g$ with $(g,h) \in R_1$ is the measurable set where $\delta(v_{x'_a},I)\le m_h$ for every $a$; on it $m=m_h$, so $\chi=\phi(m_h)$ does not depend on $g$, and $V^\sharp=g\,e^{\chi N(g)}$ with $N(g)=\frac1{16}\sum_a(\log h_a+\log v_{x'_a})$. The set $\{\delta(v,I)<1\}$ is the ball of angular radius $\alpha$ about $I$ on the sphere, which is geodesically convex, and on it $\lVert\log v-\log v'\rVert \le\Lambda\,d(v,v')$. By bi-invariance $d(g^{-1}u^{-1}g,g'^{-1}u^{-1}g')\le 2d(g,g')$, so for $g,g'\in S_h$ \[ \lVert N(g)-N(g')\rVert\le\frac7{16}\,2\Lambda\,d(g,g')=\frac78\Lambda\, d(g,g'). \] If $V^\sharp(g)=V^\sharp(g')=c$, then $g=c\,e^{-\chi N(g)}$ and $g'=c\, e^{-\chi N(g')}$, so $d(g,g')\le\frac78\Lambda\,d(g,g')$ and $g=g'$: the map $g\mapsto V^\sharp(g)$ is injective on $S_h$. Its derivative is computed as in the proof of Lemma~\ref{lem:privatesubmersion}: with $\eta=\operatorname {Ad}(g)\zeta$ the sites $x'_a$ move by $e^{t(A_a-1)\zeta}$, $A_a= \operatorname{Ad}(g^{-1}u_a^{-1}g)$, the sites $x_a$ do not move, and the right trivialised derivative is $\operatorname{Ad}(g)T$ with $\lVert T-1 \rVert\le\frac7{16}\,2\Lambda=\frac78\Lambda$, so its determinant is at least $(1-\frac78\Lambda)^3$ in modulus. On $R_2$ the coordinates $(g,h'_1, \dots,h'_7)$ with $h'_a=v_{x'_a}=g^{-1}\ell_aw_a$ and $v_{x_a}=g^{-1}u_ag \,h'_aw_a^{-1}$ give the same conclusions. For fixed $h$, the smooth formula with $\chi=\phi(m_h)$ fixed extends to a neighbourhood of each point of $S_h$, where no logarithm meets $-I$; its restriction to the measurable set $S_h$ is injective and has the Jacobian bounds above. The change of variables formula for measurable restrictions therefore applies, even if part of the boundary of $S_h$ has positive measure, and likewise on $R_2$. The law of $V^\sharp$ has the density \begin{multline}\label{eq:densityformula} \rho(c)=h^{\otimes7}\bigl(\{\ell:(c,\ell)\in R_0\}\bigr)\\ +\sum_{i=1,2}\int\sum_{g\in S^{(i)}_h,\ V^\sharp=c}\frac{dh^{\otimes7}} {\lvert\det DV^\sharp(g)\rvert}, \end{multline} the inner sum having at most one term, and $\rho(c)\le1+2(1-\frac78\Lambda )^{-3}$. Finally $1-\frac78\Lambda=0.04391\ldots$ and $M=23621\ldots$. \end{proof} The four averages of a block depend on disjoint sets of private links, and on the links inside the blocks, which are private to none. So, given the links inside the blocks of $W$, the averages $V^\sharp_k$ are independent, and the formula~\eqref{eq:densityformula} for each pair defines, for every value $c$ of $C$, a measure $\nu_c$ on the fibre $F_c=C^{-1}(c)$: in the coordinates of the proof of Lemma~\ref{lem:averagedensity}, chosen for each pair according to its region, it is the product Haar measure of the links inside the blocks and of the seven coordinates other than $g$, divided by the product of the determinants $\lvert\det DV^\sharp_k(g)\rvert$. Then $h=\int\nu_c\,dh(c)$ on $G^{A_W}$, so $\nu_c$ is a version of the conditional law, and \begin{equation}\label{eq:fibrevolume} \nu_c(F_c)\le M^{4|W|}\qquad\text{for every }c. \end{equation} Where $C$ is differentiable, \[ \nu_c=\operatorname{vol}(G)^{-45|W|}\det\bigl(DC\,DC^*\bigr)^{-1/2}\, d\mathcal H . \] Indeed, if $C(z,g)=c$ is solved for $g$ by $g=\psi(z)$, then $DC=\partial_g C\,(-D\psi,\,1)$, so \[ \det\bigl(DC\,DC^*\bigr)^{1/2}=\lvert\det\partial_gC\rvert\det\bigl(1+ D\psi^*D\psi\bigr)^{1/2}, \] and the second factor is the Jacobian of the graph of $\psi$. So $\nu_c$ is the coarea version on the charts of Theorems~\ref{thm:fibrechart} and~\ref{thm:chartdecay}, where $C$ is smooth, and we use it on the whole fibre. \begin{proposition}\label{prop:calmfibre} Let $G=\SU(2)$, $\lambda>0$ and $0K$ mass at most \[ \varepsilon_W(K,\lambda)=\Bigl(\frac{2\cdot10^9\,n_P^{3/2}}{K^3}\Bigr) ^{45|W|}e^{-3\lambda K^2/4}. \] \end{proposition} \begin{proof} Let $c=C(U)$, let $B$ be the set in question, and let $Q_W$ be the sum of $\delta(U'_p,I)^2$ over the $n_P$ plaquettes that contain a link of $A_W$; on $F$ it differs from $Q$ by a constant, so $\mu_U=e^{-\lambda Q_W}\nu_c/ \nu_c(e^{-\lambda Q_W})$. A plaquette that contains no link of $A_W$ has $U'_p=U_p$ for $U'\in F$, so on $B$ the large plaquette is one of the $n_P$ and $Q_W>K^2$. By~\eqref{eq:fibrevolume}, \[ \int_Be^{-\lambda Q_W}\,d\nu_c\le M^{4|W|}e^{-\lambda K^2}. \] For the denominator we solve the constraints for the path links. Let $L'=A_W\setminus\{d_k\}$, $\rho=K/(360\sqrt{n_P})$ and $R=44\rho$, and let $D$ be the set of $z\in G^{L'}$ with $d(z_e,U_e)\le\rho$ for every $e\in L'$. The field $U$ satisfies the hypothesis of Lemma~\ref{lem:smallframes}, as $\kappa\le K/4$, and part (i) of the lemma gives $\delta(U_e,I)\le3\kappa$ for the links inside the blocks and $\delta(U_e,U_{d_k})\le21\kappa$ for the private links of $k$, so $\delta(v_x,I)\le24\kappa$ and, by Lemma~\ref{lem:logbound}, $d(v_x,I)\le8\pi\kappa$ at every site of every pair. Fix $z\in D$ and a pair $k$, and let $\ell$ range over the ball $B_k$ of radius $R$ about $U_{d_k}$. Each $v_x$ is $\ell^{-1}$ times a product of at most two links of $L'$ and fixed links, so at the field with links $z$ and $\ell$ it lies within $R+2\rho$ of its value at $U$, and within $8\pi\kappa +46\rho<0.01$ of $I$. Hence the cut-off equals $1$ near this field and $V^\sharp_k=\ell\,e^{N_k(\ell)}$, with $N_k$ depending on $z$ and $\ell$. Each $v_x$ is moved by $d(\ell,\ell')$ when $\ell$ is replaced by $\ell'$, so the map $\Psi(\ell)=c_k\,e^{-N_k(\ell)}$ satisfies $d(\Psi(\ell),\Psi(\ell'))\le \frac78\Lambda\,d(\ell,\ell')$. At $U$ the path link solves $U_{d_k}=c_ke^{-N_k}$, and the sites move by at most $R+2\rho$ between $U$ and the field with links $z$ and $\ell$, so \[ d\bigl(\Psi(\ell),U_{d_k}\bigr)\le\frac78\Lambda\,(R+2\rho)<0.9561\cdot46 \rho0$ and $0Q_W(U)+t\}\bigr)\le\varepsilon'_W(t, \lambda),\\ \varepsilon'_W(t,\lambda)=\Bigl(\frac{3.5\cdot10^{11}\,n_P^3}{t^3}\Bigr) ^{45|W|}e^{-3\lambda t/4}. \end{gather*} In particular $\mu_F(\{Q_W>q(F)+t\})\le\varepsilon'_W(t,\lambda)$ on every fibre. \end{proposition} \begin{proof} The numerator is bounded by~\eqref{eq:fibrevolume} as in the proof of Proposition~\ref{prop:calmfibre}, by $M^{4|W|}e^{-\lambda(Q_W(U)+t)}$. For the denominator take $\rho=t/(2040\,n_P)\le10^{-3}$, $R=44\rho$, and $D$ and $B_k$ as in that proof. At the field with links $z\in D$ on $L'$ and $\ell\in B_k$ on $d_k$, each $v_x$ of the pair $k$ lies within $R+2\rho=46\rho<0.043$ of its value at $U$, and $\delta\le d$, so $\delta(v_x,I)<\frac14+0.043< \frac12$. Hence the cut-off equals $1$ near this field, the logarithms are $\Lambda$-Lipschitz along the moves, and the contraction argument of that proof gives the injective map $\Gamma:D\to F$, near whose image $C$ is smooth with $\lvert\det\partial_gC\rvert\le1.957^{3\cdot4|W|}$. So \[ \nu_c\bigl(\Gamma(D)\bigr)\ge1.957^{-12|W|}\Bigl(\frac{\rho^3}{14}\Bigr) ^{45|W|}. \] On $\Gamma(D)$ each of the $n_P$ plaquettes moves by at most $90\rho$, as in that proof, and $\delta(U_p,I)\le2\sqrt2$, so \begin{multline*} \delta(U'_p,I)^2\le\delta(U_p,I)^2+180\cdot2\sqrt2\,\rho+90^2\rho^2\\ <\delta(U_p,I)^2+510\rho, \end{multline*} and $Q_W\le Q_W(U)+510n_P\rho=Q_W(U)+\frac t4$ there. Dividing as before, \begin{multline*} \mu_F(\{Q_W>Q_W(U)+t\})\\ \le\bigl(M^41.957^{12}\bigr)^{|W|}\Bigl(\frac{14\cdot2040^3n_P^3}{t^3} \Bigr)^{45|W|}e^{-3\lambda t/4}, \end{multline*} and $2.93\cdot14\cdot2040^3<3.5\cdot10^{11}$. For the last statement apply this to regular fields with $Q_W(U)$ decreasing to $q(F)$ and use the continuity of $\varepsilon'_W$ in $t$. \end{proof} \begin{corollary}\label{cor:everyfibre} Let $G=\SU(2)$, $\lambda>0$ and $0K^2\ge Q_W(U)+\frac{15}{16}K^2$. Apply Proposition~\ref{prop:everyfibre} with $t=\frac{15}{16}K^2\le1$. \end{proof} The field of Proposition~\ref{prop:calmfibre} is regular, since its sites lie within $24\kappa$ of $I$, so Corollary~\ref{cor:everyfibre} contains that proposition up to the constants and drops its hypothesis on the plaquettes made of fixed links and on the links inside the blocks. It reaches every fibre whose boundary links carry large plaquettes, provided the sites of the averages can be made small there together with the plaquettes that move. Proposition~\ref{prop:everyfibre} holds on every fibre with no hypothesis at all, but its reference value $q(F)$ need not be small. Where it is of order one, as it can be when the averages are far from what the boundary links suggest, the constrained law sits within $t$ of $q(F)$ but away from the charts of Theorem~\ref{thm:chartdecay}, which are built at fields with all plaquettes near $W$ small. The proposition also does not compare regular with irregular fields, at which some site is far from $I$. Different fibres with the same links off $A_W$ are compared through the law of the averages. Call the links off $A_W$ the \emph{exterior} $E$, write $h$ for product Haar measure on $G^{A_W}$, and let \[ \pi_E=\frac{e^{-\lambda Q_W}h}{h(e^{-\lambda Q_W})} \] on $G^{A_W}$. Under the $\SU(2)$ Wilson measure in the tree gauge, $\pi_E$ with $\lambda=\beta_W/4$ is the conditional law of the links of $A_W$ given the exterior. Since $h=\int\nu_c\,dh(c)$, \[ \pi_E=\int\mu_{F_c}\,\frac{\nu_c(e^{-\lambda Q_W})}{h(e^{-\lambda Q_W})} \,dh(c), \] so $\mu_{F_c}$ is the conditional law of $\pi_E$ given $C=c$, and the law of the fibre through a field drawn from $\pi_E$ is the law of $C$ under $\pi_E$. Let $Q_*(E)$ be the least value of $Q_W$ on $G^{A_W}$, which is attained since $G^{A_W}$ is compact. \begin{proposition}\label{prop:fibrecompare} Let $G=\SU(2)$, $\lambda>0$ and $0Q_*(E)+t\}\bigr)\le\varepsilon''_W(t,\lambda),\\ \varepsilon''_W(t,\lambda)=\Bigl(\frac{1.05\cdot10^7\,n_P^3}{t^3}\Bigr) ^{49|W|}e^{-3\lambda t/4}. \end{gather*} In particular the law of the averages under $\pi_E$ gives the fibres $F$ with $\inf_FQ_W>Q_*(E)+t$ mass at most $\varepsilon''_W(t,\lambda)$. \end{proposition} \begin{proof} Since $h$ is a probability measure, \[ \int_{\{Q_W>Q_*+t\}}e^{-\lambda Q_W}\,dh\le e^{-\lambda(Q_*+t)}. \] For the normalisation let $U_*$ be a field with $Q_W(U_*)=Q_*$, let $\rho=t/(90.6\,n_P)$, and let $D$ be the set of fields with $d(U_e,U_{*,e})\le\rho$ for each of the $49|W|$ links of $A_W$, the $45|W|$ links of $L'$ and the $4|W|$ path links. Then $h(D)=h(B_\rho) ^{49|W|}\ge(\rho^3/14)^{49|W|}$. Each plaquette contains at most four links of $A_W$, so on $D$ it moves by at most $4\rho$, and as $\delta\le d$ and $\delta(U_p,I)\le2\sqrt2$, \begin{multline*} \delta(U_p,I)^2\le\delta(U_{*,p},I)^2+16\sqrt2\,\rho+16\rho^2\\ <\delta(U_{*,p},I)^2+22.63\rho, \end{multline*} because each link of $A_W$ lies in six plaquettes, so $n_P\ge74$ and $\rho< 1.5\cdot10^{-4}$. Hence $Q_W\le Q_*+22.63\,n_P\rhoQ_*+t\}\bigr)\le\Bigl(\frac{14\cdot90.6^3n_P^3}{t^3}\Bigr) ^{49|W|}e^{-3\lambda t/4}, \] with $14\cdot90.6^3<1.05\cdot10^7$. A field $U$ with $Q_W(U)\le Q_*+t$ lies in its own fibre, so $\inf_{F(U)}Q_W\le Q_*+t$, which gives the last statement. \end{proof} Call a field that agrees with $E$ off $A_W$ a \emph{completion} of $E$. \begin{corollary}\label{cor:calmexterior} Let $G=\SU(2)$, $\lambda>0$, $0 Q_*+\kappa^2/2\}$, and Proposition~\ref{prop:fibrecompare} with $t= \kappa^2/2\le1$ applies. (ii) A field $U\in\mathcal E$ satisfies the hypothesis of Lemma~\ref{lem:smallframes}, which places every site of every pair within $24\kappa<\frac14$ of $I$, as in the proof of Proposition~\ref{prop:calmfibre}; so $U$ is regular and $q(F(U))\le Q_W(U) <\kappa^2$. Hence a fibre with $q(F)\ge\kappa^2$ contains no field of $\mathcal E$, and the law of the averages gives these fibres at most the mass $\pi_E(\mathcal E^c)$. (iii) The fibre $F(U)$ contains the field $U$, whose plaquettes with a vertex in a block of $W^+$ are at most $\kappa=K/(4\sqrt {n_P})$, which is the hypothesis of Proposition~\ref{prop:calmfibre}; Corollary~\ref{cor:calmfibres} applies at $U$. \end{proof} So on every exterior whose plaquettes near $W$ can all be made small by the links of $A_W$, the fibres where $q(F)$ is of order one, and more generally the fibres that contain no field with small plaquettes near $W$, carry total mass at most $\varepsilon''_W(\kappa^2/2,\lambda)$ in the law of the averages, and every other fibre has the chart decay of Corollary~\ref{cor:calmfibres}. This is a bound for every such exterior, not a mean over exteriors as in Theorem~\ref{thm:offchart}(ii). If $\lambda\kappa^2\ge196|W|\ln(8.4\cdot10^7n_P^3/\kappa^6)$, then $\varepsilon''_W(\kappa^2/2,\lambda)\le e^{-\lambda\kappa^2/8}$; with $n_P\le384|W|$ and $K=4.8\cdot10^{-18}$ this holds under the Wilson measure for $\beta_W\ge6.4\cdot10^{43}|W|^2(1+\ln|W|)$. The fibres where $q(F)$ is of order one remain possible in two places: on an exterior with no completion whose plaquettes near $W$ are small, and as values of $c$ of small mass on an exterior with one. \begin{remark}\label{rem:chartdecayopen} Theorem~\ref{thm:chartdecay} is the decay that Remark~\ref{rem:fibrechartopen}(b) asks for, on the chart $\Phi(\Omega_r)$ with $r=s/32$, which contains $\Phi(K_{s/32})$ and so every $Ue^X\in F$ with $\lVert X\rVert_\infty\le s/1280$. Its constants are explicit by Propositions~\ref{prop:explicitconstants} and~\ref{prop:explicitdecay}: the rate $\alpha_G=2.5\cdot10^{-4}$ is the same for every group and lies far below the decay seen in the computation described at the end of this remark. What remains open for the small fields: \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item the mass of the constrained law outside $\Phi(\Omega_r)$ on the fibres that contain no field whose plaquettes near $W$ are small: Corollary~\ref{cor:calmfibres} bounds it on every fibre that contains one, Corollary~\ref{cor:calmexterior} shows that on every exterior with a completion whose plaquettes near $W$ are small these fibres, among them those where $q(F)$ is of order one, have mass at most $\varepsilon''_W$ in the law of the averages, and Theorem~\ref{thm:offchart} bounds them in mean over all exteriors; the constrained cube condition of Remark~\ref{rem:constrainedopen}(a) takes a supremum over boundary conditions, which include these fibres of small mass and the exteriors with no such completion, and on them no chart of Theorem~\ref{thm:chartdecay} is available; by Proposition~\ref{prop:calmrare} these are the typical case on cubes of fixed physical size along the tuning; \item the passage from the decay of the constrained law on one fibre, at a fixed background, to the constrained cube condition of Remark~\ref{rem:constrainedopen}(a), whose supremum runs over boundary conditions that include large fields. \end{enumerate} On the periodic lattice of side twelve, for $\SU(2)$ backgrounds as in the check of Theorem~\ref{thm:smallfibre} and $W$ a row of five blocks, the section $S$ has row sums at most $14$, the blocks of $\bar P$ beyond the range of $S$ decay by a factor about $0.4$ per unit of distance, and the blocks of the chart Hessian of $Q$ fall from $15$ at distance zero to $10^{-8}$ at distance nine, a floating-point computation reproduced in the archive. \end{remark} \subsection{Large fields under the Wilson measure}\label{ssec:largefields} Theorem~\ref{thm:smallscales} needs every plaquette to be small. This subsection shows that, under the $\SU(2)$ Wilson measure, the plaquettes that are not small form clusters of microscopic size, and that the cut-off average departs from the average only next to them. It also shows that such plaquettes are present along the one-loop tuning, so the composition has to pass through them. Let $\Lambda_L=(\Z/L\Z)^4$ with $L=2^m\ge4$, and let $\mu_L$ be the $\SU(2)$ Wilson measure on $\Lambda_L$, with density proportional to $e^{-\beta_W S(U)}$ with respect to product Haar measure, where \[ S(U)=\sum_ps_p,\qquad s_p=1-\tfrac12\Tr U_p=\tfrac14\,\delta(U_p,I)^2 . \] This is the Wilson measure of Sec.~\ref{sec:state} on the torus, without gauge fixing; it has $4L^4$ links and $6L^4$ plaquettes. For $0<\rho\le1$ call a plaquette \emph{bad} if $\delta(U_p,I)>\rho$. The closed unit cube $\mathcal C_x=x+[0,1]^4$ contains $24$ plaquettes. Let $B_x$ be the event that one of them is bad. A plaquette lies in exactly four unit cubes, one for each choice of the two transverse coordinates of the cube's corner. \begin{lemma}\label{lem:wilsonZ} Let $Z_L$ be the normalising constant of $\mu_L$ and $C_0=\frac12+\frac23 \log(24\pi^3)$. For $\beta_W\ge1$, \[ -\log Z_L\le6L^4(\log\beta_W+C_0),\qquad \E_{\mu_L}S\le6L^4\,\frac{\log\beta_W+C_0}{\beta_W}. \] \end{lemma} \begin{proof} Write $U=\cos\theta\,I+\sin\theta\,J$ with $0\le\theta\le\pi$ and $J^2=-I$. The angle $\theta$ of $g^{-1}h$ is a bi-invariant distance, and under Haar measure $\theta$ has density $\frac2\pi\sin^2\theta$. Put $r=(4\sqrt{ \beta_W})^{-1}$. If every link has angle at most $r$, every plaquette has angle at most $4r\le1$ and $s_p\le(4r)^2/2=1/(2\beta_W)$, so $S\le3L^4$. Each link angle is at most $r$ with probability \[ \frac2\pi\int_0^r\sin^2t\,dt\ge\frac{8r^3}{3\pi^3}=\frac1{24\pi^3 \beta_W^{3/2}}, \] by $\sin t\ge2t/\pi$. Hence $Z_L\ge e^{-3L^4}(24\pi^3\beta_W^{3/2})^{-4L^4}$, which is the first bound. The function $F(b)=-\log Z_L$ at the coupling $b$ is concave with $F(0)=0$ and $F'(b)=\E S$, so $bF'(b)\le F(b)$, which is the second. \end{proof} \begin{theorem}\label{thm:defects} For $\beta_W\ge1$, $0<\rho\le1$ and every set $A$ of unit cubes, \begin{gather*} \mu_L\Bigl(\,\bigcap_{x\in A}B_x\Bigr)\le q^{|A|},\\ q=q_\rho(\beta_W)=\min\bigl\{1,\exp\bigl(6(\log\beta_W+C_0)-\beta_W \rho^2/16\bigr)\bigr\}. \end{gather*} \end{theorem} \begin{proof} The measure $\mu_L$ is reflection positive in every hyperplane through sites~\cite{OSei78}: the plaquettes are either in the hyperplane or on one side of it, so conditionally on the links in the hyperplane the weight factors into two mirror images, and reflection preserves Haar measure and $\Tr U_p$. On a torus of side $2^m$ this gives the chessboard estimate for functions of the links of closed unit cubes~\cite{FILS78}, \cite[Section 5.3]{Bis09}: \[ \mu_L\Bigl(\,\bigcap_{x\in A}B_x\Bigr)\le\mu_L\Bigl(\,\bigcap_{x\in \Lambda_L}B_x\Bigr)^{|A|/L^4}, \] since the reflection of $B_x$ into another cube is the event $B_y$ of that cube. If every cube is bad, there are at least $L^4/4$ bad plaquettes, so $S\ge\rho^2L^4/16$ and the event has unnormalised weight at most $e^{-\beta_W\rho^2L^4/16}$. Dividing by $Z_L$ and using Lemma~\ref{lem:wilsonZ} gives the bound. \end{proof} \begin{corollary}\label{cor:defectclusters} Join two unit cubes when their corners are at $\ell^\infty$ distance one on the torus. For an integer $r\ge1$, \[ \mu_L\{\text{some connected set of $r$ bad cubes}\}\le L^4(6400\,q)^r . \] Hence, on a torus of fixed physical side $La$, if $a\to0$ along $L=2^m$ and $\beta_W(a)\to\infty$, the physical diameters of all connected clusters of bad cubes tend to $0$ in probability, for every fixed $\rho$. \end{corollary} \begin{proof} A connected set of $r$ cubes containing a given cube is recovered from a walk of length $2(r-1)$ around a spanning tree, and each cube has $80$ neighbours, so there are at most $80^{2(r-1)}\le6400^r$ such sets. Apply Theorem~\ref{thm:defects} and sum over the $L^4$ roots. A cluster of physical diameter at least $R$ contains a connected set of $\lceil R/a \rceil$ cubes, and once $6400\,q\le\frac12$ the bound is at most $(La/a)^4 2^{-R/a}\to0$. \end{proof} The bad cubes control where the cut-off average differs from the average. For a coarse link $(y,\mu)$ let $\mathcal Z_{y,\mu}$ be the $24$ unit cubes with corners $2y+te_\mu+w$, $t\in\{0,1,2\}$, $w\in\{0,1\}^4$, $w_\mu=0$ (Fig.~\ref{fig:largefields}). \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_largefields.png} \caption{Large fields under the $\SU(2)$ Wilson measure. (a)~A plane section of the lattice: bad unit cubes (red) form clusters, and the cut-off average can differ from the path average only on coarse links (orange) whose $24$ cubes $\mathcal Z_{y,\mu}$ meet a bad cube (Corollary~\ref{cor:activelinks}). (b)~The frustrated exterior of a link $e$: the staple of $p_0$ is $I$ and the other five staples are $-I$, so $U_e$ is driven to $-I$ and $p_0$ is bad. (c)~The weight $q_\rho(\beta_W)$ of Theorem~\ref{thm:defects} for three values of $\rho$; below the dashed line the cluster bound of Corollary~\ref{cor:defectclusters} is summable.} \label{fig:largefields} \end{figure*} \begin{corollary}\label{cor:activelinks} Let $\rho=\frac1{12}$. If no cube of $\mathcal Z_{y,\mu}$ is bad, then $\chi_{y,\mu}(U)=1$ in the tree gauge, and $V^\sharp_{y,\mu}$ is the average of the sixteen paths in the chart at $P_{2y}$. A bad cube lies in $\mathcal Z_{y,\mu}$ for at most eight coarse links, so the expected fraction of coarse links with $\chi<1$ is at most $4.5\cdot10^5\,(\log \beta_W+C_0)/\beta_W$. Along the limits of Corollary~\ref{cor:defectclusters}, with probability tending to one these links lie within physical distance $2a$ of clusters of bad cubes of physical diameter less than any fixed $R>0$. \end{corollary} \begin{proof} As in the proof of Lemma~\ref{lem:smallstep}, $P_{2y}^{-1}P_x$ is a product of at most six conjugates of plaquette holonomies or their inverses. These plaquettes lie in planes $\{\mu,\alpha\}$ with corners $2y+te_\mu+w$ as above, and a plaquette lies in the unit cube at its corner. So $\delta(P_x,P_{2y})\le6\rho=\frac12$, and $\chi=1$. A cube with corner $z$ lies in $\mathcal Z_{y,\mu}$ only if $z_\mu-2y_\mu\in\{0,1,2\}$ and $z_\alpha-2y_\alpha\in\{0,1\}$ for $\alpha\ne\mu$, which leaves at most two coarse links in each direction. There are $L^4/4$ coarse links, the number of bad cubes is at most four times the number of bad plaquettes, and a bad plaquette has $s_p>\rho^2/4$, so by Lemma~\ref{lem:wilsonZ} the expected fraction is at most $4\cdot8\cdot4 \cdot24\cdot144\,(\log\beta_W+C_0)/\beta_W$. The last assertion is Corollary~\ref{cor:defectclusters}. \end{proof} Small plaquettes everywhere, which Theorem~\ref{thm:smallscales} assumes, are nevertheless improbable along the one-loop tuning. For $\SU(2)$, $\beta_W=4g^{-2}$, and the one-loop coupling of Remark~\ref{rem:constrainedopen}(c) at spacing $a=1/L$ gives $\beta_W=8b_0 \log L+O(1)=\frac{11}{3\pi^2}\log L+O(1)$. \begin{proposition}\label{prop:manydefects} Let $0<\rho\le1$, $v_*=\frac2\pi\int_{\arccos(3/4)}^{\pi/3}\sin^2t\,dt$ and $p=v_*e^{-3\beta_W}$. Conditionally on all other links, a plaquette through a link $e$ is bad with probability at least $p$. Hence \[ \mu_L\{\text{no plaquette is bad}\}\le e^{-\lfloor L/3\rfloor^4p}, \] which tends to $0$ when $\beta_W\le B\log L+O(1)$ with $B<4/3$, in particular along the one-loop tuning. \end{proposition} \begin{proof} Identify $\SU(2)$ with the unit quaternions. Conditionally on the other links, $U_e$ has density proportional to $e^{\kappa\,h\cdot u}$ with $0\le\kappa\le6\beta_W$ and $|h|=1$, the six staples contributing at most $\beta_W$ each. The plaquette holonomy is $U_e$ or its inverse multiplied by a fixed element, so it has a density of the same form. Writing the density as a superposition of indicators of caps centred at $h$, the probability of a cap about $I$ is largest when $h=I$, and for $h=I$ it increases with $\kappa$, because its derivative is the covariance of an increasing function of $\Tr U$ with $\frac12\Tr U$. So the probability of being bad is at least that of the angle exceeding $\arccos(1-\rho^2/4)\le \arccos(3/4)$ under the density proportional to $e^{6\beta_W\cos t}\sin^2t$, which is at least $p$: bound the numerator from below on $[\arccos(3/4), \pi/3]$, where $\cos t\ge\frac12$, and the denominator by $e^{6\beta_W} \pi/2$. Let $M=\lfloor L/3\rfloor$. Choose the links in direction $0$ at the sites $3z$, $z\in\{0,\dots,M-1\}^4$, and their plaquettes in the plane $\{0,1 \}$. No chosen plaquette contains another chosen link. Removing one chosen link at a time therefore bounds the probability that all chosen plaquettes are good by $(1-p)^{M^4}$. With $\beta_W=B\log L+O(1)$ the exponent is of order $L^{4-3B}$. \end{proof} The composition of Theorem~\ref{thm:smallscales} is local, so it continues across the scales away from the defects, and the defects spread by at most three spacings of the current scale at each step. For a coarse plaquette $P$ at $y$ let $\mathcal B_P$ be the fine box $2y+[0,4]^4$. \begin{lemma}\label{lem:localstep} Let $U$ be a $G$-valued field in the tree gauge and $P$ a coarse plaquette. If every plaquette $p$ in $\mathcal B_P$ has $\delta(U_p,I)\le\rho_G$ and $\lVert\Theta_p\rVert\le\eta\le\rho_G/1000$, then $\chi=1$ on the four sides of $P$, $\delta(V^\sharp(U)_P,I)\le0.03\,\rho_G$, the inequalities \eqref{eq:smallstep} hold for $P$, and $\lVert\log V^\sharp(U)_P\rVert\le 4(1+647\eta)\eta$. \end{lemma} \begin{proof} For the plaquette $P$ the proof of Lemma~\ref{lem:smallstep} uses only the plaquettes of the surfaces $\mathcal U_\ell$ of the four sides $\ell$ of $P$ and those of the sixteen squares $\square_{2y+o}$. A plaquette of $\mathcal U_\ell$, $\ell=(y_i,\mu_i)$, has its corner $x$ at $x_{\mu_i}-2(y_i)_{\mu_i} \in\{0,1,2\}$ and $x_\alpha-2(y_i)_\alpha\in\{0,1\}$ for $\alpha\ne\mu_i$, with $y_i\in\{y,y+e_\mu,y+e_\nu\}$; a plaquette of $\square_{2y+o}$ has $x-2y\in\{0,1,2\}^4$. So $x-2y\in\{0,1,2,3\}^4$ in every case, and the plaquette lies in $\mathcal B_P$. The bounds on $\chi$, on $\delta(V^\sharp_P, I)$, the inequalities~\eqref{eq:smallstep} and the bound $\lVert\bar F_P \rVert+3.2\,r_P^2\le4(1+647\eta)\eta$ in that proof use only the norms of these plaquettes. \end{proof} Let $a$ be the fine spacing and $a_j=2^ja$. Given $\eta_0>0$ put $\eta_{j+1}= 4(1+647\eta_j)\eta_j$. Let $\mathfrak B_0$ be the set of fine plaquettes with $\delta(U_p,I)>\rho_G$ or $\lVert\Theta_p\rVert>\eta_0$, and let $\mathfrak B_{j+1}$ be the set of plaquettes $P$ of the lattice of spacing $a_{j+1}$ whose box $\mathcal B_P$ contains a plaquette of $\mathfrak B_j$. Let $E^\circ_j$ and $A^\circ_j$ be the sum of squares and the sum of $\lVert \Theta^{(j)}_P\rVert$ over the plaquettes $P\notin\mathfrak B_j$ of the $j$-th iterate. \begin{theorem}\label{thm:defectscales} Let $G\subset\U(r)$ be compact and $U$ a $G$-valued field on the periodic lattice of side divisible by $2^k$, and suppose that $\eta_j\le\rho_G/1000$ for $j10^{-4}$, so $2^{k_*}>\frac12\cdot10^{-2}(1.01\rho)^{-1/2}$, and $\rho^{-1/2}=96^{-1/4}(\beta_W/(\log\beta_W+C_1))^{1/4}$ with $96^{1/4} <3.14$. The rest is Theorem~\ref{thm:defectscales} with $\eta_0=1.01\rho$, whose hypothesis holds by part (iv), and $\mathfrak B_0$ contained in the bad cubes; the bound $\lVert\Theta_p\rVert\le1.01\,\delta(U_p,I)$ applies since $\rho\le\frac1{10}$. For the limits take $r=\lceil8\log_2L\rceil$, so that $L^42^{-r}\le L^{-4}$ and $ra\to0$ for a torus of fixed physical side $La$. The defects then lie in sets of physical diameter at most $ra+6a_{k_*}$, and $a_{k_*}\le10^{-2}(1.01\rho)^{-1/2}a\to0$. \end{proof} The scale $a_{k_*}$ is of the order $(\beta_W/\log\beta_W)^{1/4}a$, the order of the cube side that the condition (D) needs by Proposition~\ref{prop:cubesizeN}. So the cut-off average carries the field through the defects of the Wilson measure, without spreading them, up to the scale at which (D) can first hold, but not beyond: past $a_{k_*}$ the bound $\eta_j\le1.023\cdot4^j\eta_0$ no longer keeps the curvature small. Along the one-loop tuning $\beta_W$ grows like $\log L$, so $k_*$ grows like $\frac14\log_2\log L$, a vanishing fraction of the $\log_2L$ scales. The joint bound of Theorem~\ref{thm:defects} is not a conditional one, and (D) needs a supremum over boundary conditions. The two differ genuinely. Fix a link $e$ and a plaquette $p_0$ through it. Put all links on the six staples of $e$ at $I$, except the middle links of the five staples other than that of $p_0$, which are $-I$. Then the conditional density of $U_e$ is proportional to $e^{-2\beta_W\Tr U_e}$, and $p_0$, whose holonomy is $U_e^{\pm1}$, is bad with conditional probability tending to $1$ as $\beta_W\to\infty$, for every $\rho<2 \sqrt2$. Nearby exteriors behave the same way. So the clusters of Corollary~\ref{cor:defectclusters} are small in probability, but the boundary response of a cube to a frustrated exterior is not small, and the composition beyond the scale $a_{k_*}$ of Corollary~\ref{cor:defectscales} and the conditional estimate in (D) remain open. For exteriors that are calm on the boundary of a box and lie in a maximal torus, the conditional estimate does hold. Let $R=[0,l]^4$, $l\ge2$, be a box of $\Lambda_L$ with $l0$ then satisfy $d\alpha^0=\theta$ by closedness, by induction on $t_3$. Each value is a signed sum of at most $l$, respectively $2l$, angles, so $|\alpha^0|\le2l\rho_s$. For $k=1$ put $\alpha=\alpha^0$, so $M_1=\max|\alpha|\le2l\rho_s$. For $k\ge2$, $\alpha$ is already defined on the links of $\Gamma_k$, with $|\alpha|\le M_{k-1}$, and $\omega=\alpha-\alpha^0$ is closed there. Hence $\omega=d\lambda$ for a function $\lambda$ on the vertices of $\Gamma_k$, and $|\lambda(x)-\lambda(y)|\le\lVert\omega\rVert_\infty d_\Gamma(x,y)$, where $d_\Gamma$ is the path metric of $\Gamma_k$. We claim $d_\Gamma\le3d_1$, where $d_1$ is the $\ell^1$ metric of the facet. If $x$ and $y$ lie in one face of $\Gamma_k$, that face contains an $\ell^1$ geodesic between them. If they lie in adjacent faces $\{t_i=s\}$ and $\{t_j=s'\}$, the path through the point with $t_i=s$, $t_j=s'$ and third coordinate that of $x$ has length $d_1(x,y)$. If they lie in opposite faces $\{t_i=0\}$ and $\{t_i=l\}$, then $d_1(x,y)\ge l$. In that case $\Gamma_k$ contains a face adjacent to both, and the path from $x$ to that face, across it and on to $y$ has length at most $d_1(x,y)+2l\le3d_1(x,y)$. So $\lambda$ is $3\lVert\omega\rVert_\infty$-Lipschitz for $d_1$ on $\Gamma_k$. The function \[ \tilde\lambda(x)=\min_y\bigl(\lambda(y)+3\lVert\omega\rVert_\infty d_1(x,y) \bigr) \] extends it to the vertices of the facet with the same constant~\cite{McS34}. Put $\alpha=\alpha^0+d\tilde\lambda$ on the links of the facet. It agrees with the earlier values on $\Gamma_k$, satisfies $d\alpha=\theta$ on the facet, and has \[ |\alpha|\le2l\rho_s+3(M_{k-1}+2l\rho_s). \] So $M_k\le3M_{k-1}+8l\rho_s$, which gives $M_8\le13118\,l\rho_s$. \emph{Extension.} Let $c$ be the centre of $R$, $R_0=l/2$, $r(x)=|x-c|_\infty$ and $\varphi(x)=c+R_0(x-c)/r(x)$, the radial projection of $R\setminus\{c\}$ onto $\partial R$. Let $\hat\alpha$ and $\hat\theta=d\hat\alpha$ be the lowest-order Whitney forms of $\alpha$ and $\theta$ on the unit cubes of $\partial R$~\cite{Whi57}. On a unit cube with coordinates $t_1,t_2,t_3$, $\hat\alpha=\sum_ia_i\,dt_i$, where $a_i$ is the multilinear interpolation of the values of $\alpha$ on the four links in direction $t_i$; similarly $\hat\theta=\sum_{i\rho^2/4$, and $\sum_{P^\circ}s_p\ge\rho^2\varepsilon l^4 /16$. Hence the unnormalised weight of $E_\varepsilon$ is at most $\exp( -\beta_W\rho^2\varepsilon l^4/16)$. For the normalisation let $\bar U$ be the inner links of the hypothesis, $t=(4\sqrt{\beta_W})^{-1}$, and consider the inner links $U_e=\bar U_eh_e$ with $\angle h_e\le t$. The angle of $g^{-1}h$ is a bi-invariant distance, so every inner plaquette has $\angle U_p\le\kappa+4t$ and \[ s_p\le\tfrac12(\kappa+4t)^2\le\kappa^2+16t^2=\kappa^2+\beta_W^{-1} . \] Each $h_e$ has probability at least $8t^3/(3\pi^3)=(24\pi^3\beta_W^{3/2} )^{-1}$, as in the proof of Lemma~\ref{lem:wilsonZ}. There are $n_P\le6l^2 (l+1)^2$ inner plaquettes and $n_L\le4l(l+1)^3$ inner links, so \[ Z_\eta\ge\exp\bigl(-n_P(\beta_W\kappa^2+1)\bigr)\bigl(24\pi^3\beta_W^{3/2} \bigr)^{-n_L}. \] Since $C_0=\frac12+\frac23\log(24\pi^3)$, $n_P+n_L\log(24\pi^3\beta_W^{3/2} )\le6(l+1)^4(\log\beta_W+C_0+\frac12)$, which gives the bound. For the last assertion, a boundary plaquette has $\delta(U_p,I)\le\sqrt2\,\angle U_p\le \sqrt2\,\rho_s\le\rho$, so it is not bad, and Lemma~\ref{lem:boxextension} or Lemma~\ref{lem:nonabelianfilling} gives the inner links. \end{proof} So in a box whose boundary is calm at a level $\rho_s$ small compared with $\rho\sqrt\varepsilon/C$ and $T$-valued, or small compared with $\rho\sqrt\varepsilon/K$ and $\SU(2)$-valued with $l^2\rho_s\le \varepsilon_0$, the fraction of bad cubes exceeds $\varepsilon$ only with probability exponentially small in the volume once $\beta_W\rho^2\varepsilon$ is large compared with $\log \beta_W$, uniformly over such exteriors and over $l$. The estimate is conditional, unlike Theorem~\ref{thm:defects}. \begin{remark}\label{rem:boxnonabelian} The exteriors with boundary links in $T$ form a null set for $\mu_L$. Lemma~\ref{lem:nonabelianfilling} removes the torus at the price of the smallness $l^2\rho_s\le\varepsilon_0$. In the continuum the corresponding condition is that of Uhlenbeck's theorem~\cite{Uhl82}, which gives a Coulomb gauge on a ball when the curvature is small in $L^{n/2}$; on $3$-dimensional pieces of $\partial R$ of side comparable to $l$ and for curvature of size $\rho_s$ this norm is of order $l^2\rho_s$. The lemma replaces the Coulomb gauge by the gluing of the facet gauges. With $\rho_s$ of the order of the level $\rho(\beta_W)$ of Corollary~\ref{cor:defectscales}, $\rho_s=c_0(\log\beta_W/\beta_W)^{1/2}$, it applies for \[ l\le(\varepsilon_0/c_0)^{1/2}\,(\beta_W/\log\beta_W)^{1/4}, \] the scale $a_{k_*}$ of Corollary~\ref{cor:defectscales} up to a constant factor. Along the tuning $\beta_W\sim b\log(1/a)$ the physical side $al$ then tends to $0$, so the lemma does not reach cubes of a fixed physical size; this is a limitation of the argument, not a proof that such fillings fail. Moreover Proposition~\ref{prop:conddensity} bounds the fraction of bad cubes under one exterior and does not compare the laws under two exteriors, so it gives no contraction coefficient. \end{remark} The calm boundary is needed only for the small-curvature filling. If the boundary layer is paid for instead of filled, every condition on the exterior disappears, at the price of a surface term that is small against the volume on large boxes. \begin{proposition}\label{prop:anyexterior} Let $\beta_W\ge1$, $0<\rho\le1$ and $0<\varepsilon\le1$ with $\varepsilon l\ge16$. For every boundary field $\eta$, \[ -\log\nu_\eta(E_\varepsilon)\ge l^4\Bigl(\frac{\beta_W\rho^2\varepsilon}{32} -6\log\beta_W-C_1-\frac{48\beta_W}{l}\Bigr), \] with $C_1=3+4\log(24\pi^3)<31$. \end{proposition} \begin{proof} A unit cube $x+[0,1]^4$ with $x\in\{1,\dots,l-2\}^4$ does not meet $\partial R$, so all its plaquettes are inner; the other $l^4-(l-2)^4\le8l^3 \le\varepsilon l^4/2$ unit cubes meet it. On $E_\varepsilon$ at least $\varepsilon l^4/2$ cubes of the first kind contain a bad plaquette, which is inner, and a plaquette lies in at most four unit cubes. So at least $\varepsilon l^4/8$ inner plaquettes are bad, $\sum_{P^\circ}s_p\ge\rho^2 \varepsilon l^4/32$, and the unnormalised weight of $E_\varepsilon$ is at most $\exp(-\beta_W\rho^2\varepsilon l^4/32)$. For the normalisation take inner links $U_e=h_e$ with $\angle h_e\le t=(4 \sqrt{\beta_W})^{-1}$. An inner plaquette whose four links are inner has $\angle U_p\le4t$ and $s_p\le(2\beta_W)^{-1}$. An inner plaquette spans two directions and has its other two coordinates in $\{1,\dots,l-1\}$; it contains a boundary link exactly when its cell in the plane of the two spanned directions meets the boundary of $[0,l]^2$, which happens for $4(l-1)$ of the $l^2$ cells. So $24(l-1)^3\le24l^3$ inner plaquettes contain a boundary link, and on them $s_p\le2$. There are $n_P=6l^2(l-1)^2\le6l^4$ inner plaquettes and $n_L=4l(l-1)^3\le4l^4$ inner links, so, as in the proof of Proposition~\ref{prop:conddensity}, \[ Z_\eta\ge\exp\bigl(-48\beta_Wl^3-3l^4\bigr)\bigl(24\pi^3\beta_W^{3/2} \bigr)^{-4l^4}. \] Dividing gives the bound, and $24\pi^30$, on cubes of fixed physical side $r$ the fraction of unit cubes that contain a plaquette with $\delta(U_p,I)>\beta_W^{-1/4}$ is at most $\beta_W^{-1/4}$ with conditional probability tending to one, uniformly over all exteriors. \end{corollary} \begin{proof} Here $\varepsilon l\ge6144\beta_W^{1/2}\ge16$ and $\rho\le1$. The first term of Proposition~\ref{prop:anyexterior} is $\beta_W^{1/4}/32$, and $48\beta_W /l\le\beta_W^{1/4}/128$. For $x=\log\beta_W\ge44$ we have $6x+31\le e^{x/4} /128$: at $x=44$ the left side is $295$ and the right side exceeds $467$, and $x/4-\log(6x+31)$ increases for $x\ge44$. So $6\log\beta_W+C_1\le\beta_W ^{1/4}/128$ and the bracket is at least $\beta_W^{1/4}(\frac1{32}-\frac1{128} -\frac1{128})=\beta_W^{1/4}/64$. Along the tuning, $\beta_W\to\infty$ and $l/\beta_W^{3/4}\to\infty$, so the hypotheses hold for small $a$. \end{proof} Like Proposition~\ref{prop:conddensity}, Corollary~\ref{cor:anyexterior} bounds the bad cubes under one exterior and does not compare the laws under two, so it gives no contraction coefficient. The bad cubes are rare on cubes of fixed physical size for every exterior, and the conditional estimate in (D) remains open. The same count locates the windows on which the fibre estimates of Sec.~\ref{ssec:smallfibres} apply. Let the side of the torus be even, let $B(y)=2y+\{0,1\}^4$ be the blocks of Definition~\ref{def:blockgauge}, and let $l$ be even, so that $R$ is a union of blocks. \begin{corollary}\label{cor:localcalm} Let $G=\SU(2)$, $\beta_W\ge1$, let $W_0$ be a set of $m$ blocks, $010^{24}|W|^2$. \end{enumerate} \end{proposition} \begin{proof} (i) As in the proof of Proposition~\ref{prop:manydefects}, the holonomy of $p_0$ has a conditional density proportional to $e^{\kappa'h\cdot u}$ with $0\le\kappa'\le6\beta_W$ and $|h|=1$, and the probability of a cap about $I$ is largest at $h=I$, $\kappa'=6\beta_W$. Since $\delta(u,I)=2\sqrt2\sin( \angle u/2)$ and $\sin x\ge2x/\pi$ on $[0,\frac\pi2]$, the event forces $\angle U_{p_0}\le\theta=\pi\rho'/(2\sqrt2)$. With $\cos t-1\ge-t^2/2$ and $\sin t\ge2t/\pi$ on $[0,\frac\pi2]$, \begin{multline*} \int_0^\pi e^{6\beta_W(\cos t-1)}\sin^2t\,dt\\ \ge\frac4{\pi^2}\int_0^{\pi/2}e^{-3\beta_Wt^2}t^2\,dt\ge\frac{4\cdot0.44} {\pi^2(3\beta_W)^{3/2}}, \end{multline*} because the integral of $e^{-s^2}s^2$ over $[0,\pi\sqrt3/2]$ exceeds $0.44$. The integral of $e^{6\beta_W(\cos t-1)}\sin^2t$ over $[0,\theta]$ is at most $\theta^3/3$, so the ratio is at most $9.72(\beta_W\theta^2)^{3/2}$, and $(\pi/(2\sqrt2))^3<1.371$. (ii) Take $y'\in W^+\setminus W$ and a plaquette $p_0$ inside $B(y')$. Its links have their base in $B(y')$, so none lies in $A_W$, and every completion of the exterior agrees with the field on $p_0$. A completion as in Corollary~\ref{cor:calmexterior} therefore needs $\delta(U_{p_0},I)\le \kappa/(2\sqrt{n_P})=K/(8n_P)$. This event is gauge invariant, so its probability is the same without gauge fixing, and by (i) it is at most $14(\beta_WK^2/(64n_P^2))^{3/2}$. Each link of $A_W$ lies in six plaquettes and a plaquette contains at most four of them, so $n_P\ge73.5|W|$, and $14\cdot10^{-24}/(512\cdot73.5^3)<7\cdot10^{-32}$. Finally, $\varepsilon''_W(\kappa^2/2,\lambda)<1$ means $\frac38\lambda\kappa^2>49|W| \ln(3.45\cdot10^{11}n_P^6/K^6)$, and with $\lambda=\beta_W/4$, $\kappa^2= K^2/(16n_P)$, $n_P\ge73.5|W|$ and $K\le10^{-8}$ the logarithm exceeds $162.8$, so $\beta_W>512\cdot49\cdot73.5\cdot162.8\cdot10^{16}|W|^2/3>10^{24}|W|^2$. \end{proof} Along the tuning $\beta_W\sim b\log(1/a)$, a cube of fixed physical side $r$ has $|W|=(r/2a)^4$ blocks, so both bounds of (ii) fail for small $a$: on such a cube the exterior has no calm completion with probability tending to one, and the fibre estimates of Sec.~\ref{ssec:smallfibres} are vacuous there. Their entropy factors are powers of $|W|$, so they are estimates for windows of bounded size, which Corollary~\ref{cor:localcalm} places at almost every position under every exterior. The exteriors with no calm completion and the rare fibres of Corollary~\ref{cor:calmexterior} are thus the typical case at the physical scale and not an exceptional one. The cube condition (D) needs estimates on the whole cube that are uniform in $|W|$, with the windows that are not calm split off as in Ba{\l}aban's analysis~\cite{Bal89b}, and that remains open. \subsection{Conditioning on a fixed large-field set}\label{ssec:fixedset} The estimates of Sec.~\ref{ssec:largefields} are statements about the Wilson measure on the torus. The cube condition (D) takes a supremum over exteriors, and three facts about conditional laws hold under every exterior; the setting is that of a Gibbs field whose potential is itself random, the uniqueness theory of which goes back to Bassalygo and Dobrushin~\cite{BD86}. The first bounds the excess of the action over its minimum in a box, with no reflection positivity and no torus. The second says that restricting a single-link law to a fixed set, however irregular, leaves its Kantorovich response to the neighbouring links bounded by a constant times $\beta_W$. The third turns the one-point rarity of a bad set into closeness of any two Gibbs measures, at order of that rarity, and shows at the same time why one-point rarity gives closeness and not decay. \begin{lemma}[excess action]\label{lem:excess} Let $G=\SU(2)$, let $\Lambda$ be a set of $N$ links of $\Z^4$ and fix the links outside $\Lambda$. Let $P_\Lambda$ be the plaquettes that contain a link of $\Lambda$, $S_\Lambda=\sum_{p\in P_\Lambda}s_p$, and $S_{\min}$ the minimum of $S_\Lambda$ over the links of $\Lambda$. For every $E>0$, under the Wilson measure conditioned on the links outside $\Lambda$, \[ \Pr\bigl(S_\Lambda\ge S_{\min}+E\bigr)\le\Bigl(\frac{3\pi^3}{8r^3}\Bigr)^N e^{-\beta_WE/2}, \] where $r=\min\{E/(6\pi N),\pi/2\}$. \end{lemma} \begin{proof} The plaquettes that contain no link of $\Lambda$ do not depend on the links of $\Lambda$, so the conditional law has density $Z^{-1}e^{-\beta_W S_\Lambda}$ with respect to the product of Haar measures on $\Lambda$. The unnormalised mass of the event is at most $e^{-\beta_W(S_{\min}+E)}$. For the lower bound on $Z$, let $U^*$ be a minimiser, which exists by compactness. If a link of a plaquette $p$ moves by geodesic distance $t$, the angle $\delta(U_p,I)\in[0,\pi]$ moves by at most $t$, by bi-invariance, so $s_p=\frac14\delta(U_p,I)^2$ moves by at most $\frac14\,t\cdot2\pi= \frac\pi2t$. Each link lies in six plaquettes. If every link of $\Lambda$ lies within distance $r$ of its value in $U^*$, then $S_\Lambda\le S_{\min} +6\cdot\frac\pi2rN\le S_{\min}+E/2$. The Haar measure of a geodesic ball of radius $r\le\pi/2$ is $\frac2\pi\int_0^r\sin^2t\,dt\ge8r^3/(3\pi^3)$, by $\sin t\ge2t/\pi$. Hence $Z\ge(8r^3/3\pi^3)^Ne^{-\beta_W(S_{\min}+E/2)}$, and the quotient of the two bounds is the claim. \end{proof} The constant depends on $N$ and $E$ alone, so excess action is rare under every exterior, at a cost that is exponential in $N$. The minimum itself is a function of the exterior, and the lemma says nothing about the size of its plaquettes: it controls the fluctuation of the field about the configuration the exterior forces, not that configuration. \begin{lemma}[restriction to a fixed set]\label{lem:fixedset} Let $G=\SU(2)$, let $e$ be a link and let $h\ge0$ be an integrable function on $G$ with $\int h\,dU>0$ that does not depend on the other links. For a configuration $\sigma$ of the other links let $\gamma^h_e(\cdot\mid\sigma)$ be the law on $G$ with density proportional to $e^{-\beta_WS}h$. If $\sigma$ and $\sigma'$ differ only at a link $e'\ne e$, then \[ W_1\bigl(\gamma^h_e(\cdot\mid\sigma),\gamma^h_e(\cdot\mid\sigma')\bigr)\le \frac{\pi^2}8\,\beta_W\,\delta(\sigma_{e'},\sigma'_{e'}), \] and the two laws coincide unless $e$ and $e'$ lie in a common plaquette. \end{lemma} \begin{proof} Only the plaquettes containing $e$ enter the law, and two distinct links of $\Z^4$ lie in at most one common plaquette $p_0$; if there is none the laws are equal. Otherwise move $\sigma_{e'}$ along a unit-speed geodesic $U(t)$, $0\le t\le T=\delta(\sigma_{e'},\sigma'_{e'})$, and let $\gamma_t$ be the corresponding law. The function $t\mapsto s_{p_0}$ has $|\frac d{dt} s_{p_0}|\le\frac\pi2$ as in the proof of Lemma~\ref{lem:excess}, and the derivative $\dot s_{p_0}$ is a function of $U_e$ with oscillation at most $\pi$. For a $1$-Lipschitz $f$ on $G$, which has oscillation at most the diameter $\pi$, $\frac d{dt}\gamma_t(f)=-\beta_W\Cov_{\gamma_t}(f,\dot s_{p_0})$ because $h$ does not depend on $t$, and $|\Cov(f,\dot s_{p_0})| \le\sqrt{\Var f\,\Var\dot s_{p_0}}\le\frac\pi2\cdot\frac\pi4$. Integrating over $t$ and taking the supremum over $f$ gives the bound. \end{proof} When $h$ is the indicator of a set $C$ the lemma says that conditioning a link to lie in $C$ costs nothing in its response, however small or irregular $C$ is, as long as $C$ is fixed. The responses that are not bounded are those to a set that moves with the neighbours, such as the intersection of balls centred at them; this is the difference between conditioning on a fixed set of bad links and conditioning on a plaquette being large. \begin{proposition}[one-point rarity]\label{prop:onepoint} Let $\mathcal A$ be a set of links, $D$ the diameter of $G$, and $\gamma_e$, $e\in\mathcal A$, single-link kernels with \[ W_1\bigl(\gamma_e(\cdot\mid\sigma),\gamma_e(\cdot\mid\sigma')\bigr)\le \sum_{e'}c_{ee'}(\sigma,\sigma')\,\delta(\sigma_{e'},\sigma'_{e'}) \] for all $\sigma,\sigma'$. Let $\sigma\mapsto B(\sigma)\subset\mathcal A$ be measurable, and suppose there are constants $a_{ee'},A_{ee'}\ge0$ with $c_{ee'}(\sigma,\sigma')\le a_{ee'}$ whenever $e\notin B(\sigma)\cup B(\sigma')$, $c_{ee'}\le A_{ee'}$ always, $\sup_e\sum_{e'}a_{ee'}\le\theta<1$ and $\sup_e\sum_{e'}A_{ee'}\le\bar A$. If $\mu_1,\mu_2$ are invariant under every $\gamma_e$ and $\mu_i(e\in B)\le q$ for every $e$ and $i=1,2$, there is a coupling of $\mu_1$ and $\mu_2$ with \[ \sup_{e\in\mathcal A}\E\,\delta(\sigma_e,\sigma'_e)\le\frac{2\bar ADq} {1-\theta}. \] In particular $|\mu_1(F)-\mu_2(F)|\le L_F|S_F|\,2\bar ADq/(1-\theta)$ for every $F$ with support $S_F$ that is $L_F$-Lipschitz in each link. \end{proposition} \begin{proof} Fix $b_0\in\mathcal A$, $\lambda>0$, and let $w_b=e^{-\lambda|b-b_0|}$, which is summable. As in the proof of Lemma~\ref{lem:comparison} the functional $\Psi(\pi)=\sum_bw_b\E_\pi\delta(\sigma_b,\sigma'_b)$ has a minimiser $\pi^*$ among the couplings, and replacing the pair at one link $e$ by a sample of an optimal coupling of $\gamma_e(\cdot\mid\sigma)$ and $\gamma_e(\cdot\mid\sigma')$ gives a coupling again. Only the term at $e$ changes, so minimality gives, with $\varrho(e)=\E_{\pi^*}\delta(\sigma_e, \sigma'_e)$, \[ \varrho(e)\le\E_{\pi^*}\sum_{e'}c_{ee'}(\sigma,\sigma')\,\delta(\sigma_{e'}, \sigma'_{e'}) \] for every $e$, with no weight left. On the event that $e$ is good in both configurations the coefficients are at most $a_{ee'}$; on its complement, whose probability is at most $\mu_1(e\in B)+\mu_2(e\in B)\le2q$, they are at most $A_{ee'}$ and every distance is at most $D$. Hence $\varrho(e)\le \theta\sup\varrho+2\bar ADq$, and $\sup\varrho\le D$ is finite, which gives the bound. The last statement follows by writing $F(\sigma)-F(\sigma')$ as a telescopic sum over the links of $S_F$. \end{proof} The proposition is approximate uniqueness: all Gibbs measures of the specification lie within $2\bar ADq/(1-\theta)$ of each other, link by link. It is not decay. Its proof sees the bad set only through the one-point probabilities $q$, and the term $2\bar ADq$ does not shrink with the distance from the region where $\mu_1$ and $\mu_2$ are allowed to differ. The bound $D$ on the distance at a bad link is not what causes this: replacing it by H\"older's inequality gives $\varrho\le\theta\varrho +\bar A(qD)^{1/p}\varrho^{1/p'}$, a recursion with the positive fixed point $\varrho^*$ of order $\bar A^pqD/(1-\theta)^p$. To obtain decay the bad set has to be held fixed while the comparison runs, as in the disagreement percolation of van den Berg and Maes~\cite{vdBM94}, where the two configurations are coupled link by link and the disagreement has to percolate through the bad set; the price is that the laws conditioned on that set, not the Wilson measure, then have to satisfy the fibre estimates of Sec.~\ref{ssec:smallfibres}. \subsection{Rarity of coarse defects}\label{ssec:coarse} Theorem~\ref{thm:defects} makes a large plaquette rare at the lattice scale. The cube condition (D) concerns a cube of fixed physical side, which along the tuning spans $m\sim\beta_W/b$ scales, and it needs the analogue of that rarity for the curvature of the block averages at every one of them. This subsection isolates the two sides of that question. On the energy side, the coarse curvature of a field is bounded by its action uniformly in the scale, for linear fields and for group-valued fields in a torus. On the entropy side, the chessboard estimate applies to coarse events, and it is the normalisation of Lemma~\ref{lem:wilsonZ}, not the energy, that makes it useless beyond $O(\log\beta_W)$ scales. A right translation of the Haar measures gives rarity at every scale and under every exterior, with a rate governed by the second variation of the action along a fixed direction; for the abelian theory that second variation is the energy of the direction, while for $\SU(2)$ it grows like $4^k$ at scale $k$, and the growth is produced by the gauge orbit of the vacuum. Throughout, $\Lambda_L$ is the torus of side $L=2^m$, the scale $k$ satisfies $2^{k+2}\le L$, and the lattice of spacing $2^k$ is the torus $\Lambda_{L'}$ with $L'=L/2^k\ge4$. For a real field $\varphi$ on the links, $d\varphi$ is its curvature, $d^{\mathsf T}$ the adjoint of $d$, and $\Delta=d^{\mathsf T}d+dd^{\mathsf T}$, which acts on each component of a field as the lattice Laplacian $(\Delta f)(x)=8f(x)-\sum_{\pm\nu}f(x\pm e_\nu)$. The Green function $G_{L'}$ of $\Lambda_{L'}$ is the function of mean zero with $\Delta G_{L'}=\delta_0-L'^{-4}$; it is invariant under the permutations and sign changes of the coordinates, and applied to each component it inverts $\Delta$ on fields whose components have mean zero. It commutes with $d$, $d^{\mathsf T}$ and the translations. \begin{lemma}[energy of a coarse defect]\label{lem:coarseenergy} Let $P$ be a plaquette of the lattice of spacing $2^k$ and $J_{k,P}$ the field with $(d_kV^{(k)}(\varphi))_P=\ip{\varphi}{J_{k,P}}$ for every field $\varphi$ on $\Lambda_L$, given by Lemma~\ref{lem:straightpaths}(i). Then $d^{\mathsf T}J_{k,P}=0$, every field $\varphi$ satisfies \[ (d_kV^{(k)}(\varphi))_P^2\le\gamma_k\,Q(\varphi), \] where \[ \gamma_k=\ip{J_{k,P}}{G_LJ_{k,P}}=\sup_\varphi\frac{(d_kV^{(k)}( \varphi))_P^2}{Q(\varphi)}, \] and $\frac1{432}\le\gamma_k\le\frac12$ for every $k$. In particular $Q(\varphi)\ge2r^2$ whenever $|(d_kV^{(k)}(\varphi))_P|\ge r$, at every scale. \end{lemma} \begin{proof} Lemma~\ref{lem:straightpaths}(ii) shows that $V^{(k)}(dh)$ is the gradient of $\bar h$ on the coarse lattice for every function $h$ on sites, so $\ip{dh}{J}=0$, the curvature of a gradient vanishing, and $d^{\mathsf T}J=0$; $J$ is supported on the links of directions $\mu$, $\nu$ of the plane of $P$, and its component in each direction has mean zero, the two sides of $P$ entering with opposite signs. Put $u=G_LJ$. Then $\Delta u=J$, $d^{\mathsf T}u=G_Ld^{\mathsf T}J=0$, and for every field $\varphi$ \begin{multline*} \ip{\varphi}{J}=\ip{\varphi}{\Delta u}=\ip{d\varphi}{du}+\ip{d^{\mathsf T} \varphi}{d^{\mathsf T}u}\\ =\ip{d\varphi}{du}\le\sqrt{Q(\varphi)Q(u)}, \end{multline*} with $Q(u)=\ip{u}{d^{\mathsf T}du}=\ip{u}{\Delta u}=\ip{u}{J}=\gamma_k$, and equality at $\varphi=u$. This proves the inequality and the characterisation of $\gamma_k$ as a supremum. For the upper bound, by Proposition~\ref{prop:effective}(i), whose proof is a local identity, $Q_k(V^{(k)}(\varphi))\le Q(\varphi)$ for every field, so $\gamma_k$ is at most the supremum of $(d_kw)_P^2/Q_k(w)$ over the fields $w$ of $\Lambda_{L'}$. The argument of the first paragraph on $\Lambda_{L'}$, with $j=d_k^{\mathsf T}e_P$ the field equal to $\pm1$ on the four sides of $P$, identifies that supremum with $\ip{j}{G_{L'}j}$. The two sides of $P$ in the direction $\mu$ are the links $(y,\mu)$ and $(y+e_\nu,\mu)$ with opposite signs, so \begin{multline*} \ip{j}{G_{L'}j}=4G_{L'}(0)-2G_{L'}(e_\nu)-2G_{L'}(e_\mu)\\ =4\bigl(G_{L'}(0)-G_{L'}(e_1)\bigr)=\tfrac12\bigl(1-L'^{-4}\bigr), \end{multline*} by $(\Delta G_{L'})(0)=8G_{L'}(0)-8G_{L'}(e_1)=1-L'^{-4}$ and the symmetry of $G_{L'}$. For the lower bound take $P$ at $y=0$ in the plane $\{1,2\}$, write $L_k=2^k$, and let $f$ be the function on $\Z/L\Z$ with $f(s)=s/L_k$ for $0\le s\le L_k$, $f(s)=1$ for $L_k\le s\le2L_k$, $f(s)=1-(s-2L_k)/L_k$ for $2L_k\le s\le3L_k$ and $f=0$ elsewhere; let $g$ be the indicator of $[0,2L_k)$; and let $h(s)=(1-\dist(s,[0,L_k))/L_k)_+$. Put $\varphi_{(x,2)} =L_k^{-1}f(x_1)g(x_2)h(x_3)h(x_4)$ and $\varphi=0$ on the other links; all supports fit in the torus since $4L_k\le L$. By Lemma~\ref{lem:straightpaths}(i), $V^{(k)}_{y,2}(\varphi)=L_k^{-4}\sum_{x \in B_k(y)}\sum_{t={Stealth[length=2mm]}] \fill[blue!18] (0,0) rectangle (7,4); \fill[blue!18] (0,4) rectangle (7,8); \fill[red!18,opacity=0.7] (0,0) rectangle (4,7); \fill[red!18,opacity=0.7] (4,0) rectangle (8,7); \draw[very thick] (0,0) rectangle (4,4); \node at (2,2) {$P$}; \draw[->,thick] (0.3,-0.9) -- (6.7,-0.9); \node[below] at (3.5,-0.9) {\footnotesize $+$, side $(y,1)$}; \draw[<-,thick] (0.3,8.9) -- (6.7,8.9); \node[above] at (3.5,8.9) {\footnotesize $-$, side $(y+e_2,1)$}; \draw[<-,thick] (-0.9,0.3) -- (-0.9,6.7); \node[left,rotate=90,anchor=south] at (-1.0,3.5) {\footnotesize $-$, side $(y,2)$}; \draw[->,thick] (8.9,0.3) -- (8.9,6.7); \node[right,rotate=-90,anchor=south] at (9.0,3.5) {\footnotesize $+$, side $(y+e_1,2)$}; \draw[gray,dashed] (0,0) grid[step=4] (8,8); \end{tikzpicture} \hspace{1em} \begin{tikzpicture}[scale=0.42,>={Stealth[length=2mm]}] \draw[->] (-0.5,0) -- (7.5,0) node[right] {\footnotesize $s$}; \draw[->] (0,-0.5) -- (0,4.8) node[above] {\footnotesize $c(s)$}; \draw[thick,blue!60!black] (0,1) -- (3,4) -- (6,1); \foreach \s/\c in {0/1,1/2,2/3,3/4,4/3,5/2,6/1} \fill[blue!60!black] (\s,\c) circle (3pt); \draw[dotted] (3,0) -- (3,4) node[pos=0,below] {\footnotesize $L_k-1$}; \draw[dotted] (6,0) -- (6,1) node[pos=0,below] {\footnotesize $2L_k-2$}; \draw[dotted] (0,4) -- (3,4) node[pos=0,left] {\footnotesize $L_k$}; \end{tikzpicture} \caption{The smeared current loop $J_{k,P}$ of Lemma~\ref{lem:coarseenergy} in the plane of $P$, drawn for $L_k=4$. Left: each side of the coarse plaquette $P$ (thick square) contributes a current of its own direction on the links of a slab of transverse size $L_k^3$ and length $2L_k-1$ (blue for the direction $1$, red for the direction $2$), with the signs of the four sides; the slabs of opposite sides are disjoint and the two directions are carried by different links. Right: the longitudinal profile $L_k^4J=\pm c(s)$ on a slab, the tent $c(s)=\min\{s+1,L_k,2L_k-1-s\}$, which counts the straight paths of Lemma~\ref{lem:straightpaths}(i) through the link at position $s$. The current is divergence free and its energy $\gamma_k=\ip{J_{k,P}}{G_LJ_{k,P}}$ lies in $[\frac1{432},\frac12]$ at every scale.} \label{fig:smearedloop} \end{figure} \begin{proposition}[energy of a coarse defect, torus-valued fields] \label{prop:torusenergy} Let $U$ be an $S$-valued field in the tree gauge on $\Lambda_L$, as in Theorem~\ref{thm:torusscales}, and let $P$ be a plaquette of the lattice of spacing $2^k$ with $\delta(V^{\sharp,k}(U)_P,I)\ge r$. Then \[ Q(U)\ge\min\Bigl\{\frac{r^2}{1.001},\frac1{144.2}\Bigr\}\ge\frac{r^2} {1154}. \] \end{proposition} \begin{proof} If some plaquette has $\delta(U_p,I)>1=\rho_G$ then $Q(U)>1\ge r^2/8$, since $r\le\operatorname{diam}\SU(2)=2\sqrt2$. Otherwise every $\Theta_p$ is defined, and for $\Theta_p=\theta_pJ_p$ with $J_p^2=-I$ one has $\lVert\Theta_p\rVert^2=2\theta_p^2$ and $\delta(U_p,I)^2=8\sin^2(\theta_p /2)$, so $\lVert\Theta_p\rVert^2\le1.001\,\delta(U_p,I)^2$ when $\theta_p \le1/(12\sqrt2)$, that is when $\lVert\Theta_p\rVert\le\eta_G=\frac1{12}$. Let $j_*$ be the least $j\le k$ with $\max_P\lVert\Theta^{(j)}_P\rVert> \eta_G$, if there is one. If there is none, \eqref{eq:torushyp} holds for every $j1/(12\sqrt2)$ and $Q(U)\ge8\sin^2(\theta_p/2) \ge1/144.1$. If $1\le j_*\le k$, \eqref{eq:torushyp} holds for $j0$, uniformly in the fixed links, \[ \mu(D_rS\ge x)\le e^{-\beta_Wx^2/(2K_r)},\qquad\mu(D_rS\le-x)\le e^{-\beta_Wx^2/(2K_r)}. \] \end{enumerate} \end{theorem} \begin{proof} (i) Order the links of a plaquette $p$ as traversed, so that $U_p=g_1^{ \epsilon_1}\cdots g_4^{\epsilon_4}$ with $\epsilon_i=\pm1$. Since $ge^{tr} =e^{t\operatorname{Ad}(g)r}g$ and $(ge^{tr})^{-1}=e^{-tr}g^{-1}$, moving the exponentials to the left gives $U_p(t)=e^{t\tilde a_1}\cdots e^{t\tilde a_4}U_p$ with $\tilde a_i=\pm\operatorname{Ad}(h_i)r_{e_i}$ for unitary $h_i$, so $\|\tilde a_i\|_{\rm op}=\rho_{e_i}$, and $\tilde a_i=0$ if $e_i\notin\Lambda$. Then \[ \partial_t^2\Tr U_p(t)\big|_{t=0}=\Tr\Bigl[\Bigl(\sum_i\tilde a_i^2+2\sum_{ i0$; take $t=x/K_r$. The lower tail is the upper tail for $-r$. Nothing depends on the fixed links. \end{proof} The map $U\mapsto Ue^{-tr}$ is the group version of a translation of the Gaussian field; its Jacobian is one, and no smallness of the field enters. The observable $D_rS$ is the component of the force along $r$: at a plaquette, $\partial_ts_p=-\frac1N\ReTr\bigl((\sum_i\tilde a_i)U_p\bigr)$, the covariant derivative of $r$ in the background $U$ paired with the sine part of $U_p$. For the abelian theory the second variation is the energy of the direction, and the theorem gives the rarity of coarse defects at every scale. \begin{corollary}[abelian theory]\label{cor:abeliancoarse} Let $G=\U(1)$, let $\mu_L$ be its Wilson measure on $\Lambda_L$, $U_e=e^{i\theta_e}$, $\theta_p=(d\theta)_p$ the angle of $U_p$ in $(-\pi,\pi]$, and let $a=G_LJ_{k,P}$ be the minimiser of Lemma~\ref{lem:coarseenergy}. For every $k$, every plaquette $P$ of the lattice of spacing $2^k$ and every $x>0$, \[ \mu_L\Bigl(\sum_p\sin\theta_p\,(da)_p\ge x\Bigr)\le e^{-\beta_Wx^2}. \] Here $\sum_p\theta_p(da)_p=(d_kV^{(k)}(\theta))_P$ for every choice of link angles $\theta_e$ with $(d\theta)_p\equiv\theta_p$ modulo $2\pi$, up to $2\pi\sum_pn_p(da)_p$ with the integers $n_p=((d\theta)_p-\theta_p)/2\pi$. Under a fixed exterior of a box, the same bound holds with $a$ replaced by its restriction $\tilde a$ to the inner links and $x^2$ replaced by $x^2/(2Q(\tilde a))$. \end{corollary} \begin{proof} By Theorem~\ref{thm:haartilt}(i) with $r_e=ia_e$, $K_r\le Q(a)=\gamma_k\le \frac12$, and $D_rS=\sum_p\sin\theta_p(da)_p$ since $s_p(t)=1-\cos(\theta_p +t(da)_p)$. Part (iii) gives the bound. For the identity, $\sum_p(d\theta)_p(da)_p=\ip{\theta}{d^{\mathsf T}da}=\ip{\theta}{\Delta a} =\ip{\theta}{J_{k,P}}$ because $d^{\mathsf T}a=0$. In a box the tilt acts on the inner links only, and $K_r\le Q(\tilde a)$. \end{proof} For the abelian theory this is the statement that a smeared coarse flux of size $x$ costs $\beta_Wx^2$ at every scale, uniformly in the exterior; the four-dimensional abelian theory has no running coupling, and the bound is consistent with its known behaviour at weak coupling~\cite{Gut80,FS82}. For $\SU(2)$ the same tilt is available, but its second variation grows with the scale. \begin{proposition}[the gauge orbit of the vacuum]\label{prop:tiltgrowth} Let $G=\SU(2)$ on $\Lambda_L$, $T\in\g$ with $\|T\|_{\rm op}=1$, $a=G_LJ_{ k,P}$ for a plaquette $P$ of the lattice of spacing $2^k$, and $r_e=a_eT$. Then \[ \Bigl(\sqrt{2\|a\|^2-\gamma_k}-\sqrt{\gamma_k}\Bigr)_+^2\le K_r\le24\|a\|^2 \] and \[ \|a\|^2\ge\frac{\gamma_k^2}{\|J_{k,P}\|^2}\ge\frac{4^k}{1492992}, \] with $\gamma_k=\ip{J_{k,P}}{G_LJ_{k,P}}$ as in Lemma~\ref{lem:coarseenergy}; in particular $K_r\ge4^k/(3\cdot10^6)$ for $k\ge11$. The lower bound is attained, up to the stated loss, at a pure gauge field, where $Q(U)=0$. \end{proposition} \begin{proof} The upper bound is Theorem~\ref{thm:haartilt}(i). For the lower bound let $U$ be the pure gauge field $U_e=h_xh_{x+e}^{-1}$ of a function $h$ on sites with values in $\SU(2)$. The action is invariant under gauge transformations and $U_ee^{tr_e}=h_x\exp(t\operatorname{Ad}(h_{x+e}^{-1}) r_e)h_{x+e}^{-1}$, so $S(Ue^{tr})=S(e^{tr'})$ with $r'_e=\operatorname{Ad} (h_{x+e}^{-1})r_e$. At the identity field, $U_p(t)=\exp(tX_p+O(t^2))$ with $X_p=(dr')_p$ the signed sum of the $r'_e$ around $p$, and $s_p=1-\cos\| X_p(t)\|_{\rm op}$, so $\partial_t^2s_p|_{t=0}=\|(dr')_p\|_{\rm op}^2$ and $K_r\ge\sum_p\|(dr')_p\|_{\rm op}^2$. Take $h_x=R^{x_2}$ with $R=i\sigma_1$ and $T=i\sigma_3$, which is well defined on the torus since $R^4=I$ and $4\mid L$, and satisfies $\operatorname{Ad}(R)T=-T$. Then $r'_e=\epsilon_e a_eT$ with $\epsilon_{(x,\mu)}=(-1)^{x_2}$ for $\mu\ne2$ and $\epsilon_{(x, 2)}=(-1)^{x_2+1}$, and $K_r\ge Q(\epsilon a)$. Since $J_{k,P}$ has components only in the directions $1$, $2$ of the plane of $P$ and $G_L$ acts on each component, $a_3=a_4=0$; the reflection exchanging the coordinates $1$ and $2$ carries the component $a_1$ to $-a_2$, so $\|a_1\|^2 =\|a_2\|^2=\frac12\|a\|^2$. On the plaquette at $x$ in the plane $\{1,2\}$ the signs give \begin{multline*} (d(\epsilon a))_p=(-1)^{x_2}\bigl[(a_{(x,1)}+a_{(x+e_2,1)})-(a_{(x+e_1,2)} -a_{(x,2)})\bigr]\\ =(-1)^{x_2}\bigl[\sigma_x-v_x\bigr], \end{multline*} so that $Q(\epsilon a)\ge\|\sigma-v\|^2\ge(\|\sigma\|-\|v\|)_+^2$, with $\|\sigma\|^2=4\|a_1\|^2-\|\nabla_2a_1\|^2$ and $\|v\|^2=\|\nabla_1a_2\|^2$, where $\nabla_\nu$ is the forward difference in the direction $\nu$. All directional differences together give $\sum_{\mu,\nu}\|\nabla_\nu a_\mu\|^2 =\ip{a}{\Delta a}=\ip{G_LJ}{J}=\gamma_k$, so $\|\sigma\|^2\ge2\|a\|^2- \gamma_k$ and $\|v\|^2\le\gamma_k$, which is the lower bound. Finally $\gamma_k=\ip{J}{a}\le\|J\|\|a\|$, and by Lemma~\ref{lem:straightpaths}(i) each of the four sides of $P$ contributes $L_k^3$ rows of $2L_k-1$ links with coefficients $L_k^{-4}c(s)$, $c\le L_k$, so $\|J\|^2\le4L_k^{-8}L_k^3 (2L_k-1)L_k^2\le8\cdot4^{-k}$; with $\gamma_k\ge\frac1{432}$ this gives $\|a\|^2\ge4^k/(8\cdot432^2)$. For $k\ge11$ that is at least $2$, so $2\|a\|^2-\gamma_k\ge7\gamma_k$ and $K_r\ge(1-7^{-1/2})^2\cdot\frac78\cdot 2\|a\|^2\ge0.67\|a\|^2\ge4^k/(3\cdot10^6)$. \end{proof} So for $\SU(2)$ the tilt of Theorem~\ref{thm:haartilt} proves the rarity of the force component $D_rS$ at the rate $\beta_Wx^2/(2K_r)$, which is of order $\beta_Wx^2/4^k$ at scale $k$, and not more: the supremum in $K_r$ is reached on the gauge orbit of the vacuum, where the field has no action and no curvature, because a fixed direction $r$ is not gauge covariant and its covariant derivative in a pure gauge background is as large as $r$ itself. The natural repair is to transport the direction with the field, $r_e\mapsto\operatorname{Ad}(h(U))r_e$. When the transport runs along a forest, the translation still preserves the Haar measures; but the residual gauge freedom of the forest then realises the field of the proof on the links between the trees, and the constant is reached at zero action in that gauge as well. \begin{proposition}[covariant tilt on a forest]\label{prop:foresttilt} Let $G=\SU(2)$, let $F$ be a set of links of $\Lambda_L$ without cycles, with one root chosen in every component, and for a site $y$ let $h_y(U)$ be the ordered product of the link variables along the path in $F$ from the root of its component to $y$, with $h_y=I$ at the roots. For $r=(r_e)$ with $r_e\in\g$ and $r_e=0$ on $F$ put \[ \Phi_t(U)_e=U_e\exp\bigl(t\operatorname{Ad}(h_y(U)^{-1})r_e\bigr),\quad y\ \text{the end point of }e, \] and $\Phi_t(U)_f=U_f$ for $f\in F$; let $D^F_rS(U)=\partial_tS(\Phi_tU) |_{t=0}$ and \[ K^F_r=\sup\bigl\{\partial_t^2S(\tilde Ue^{tr})\big|_{t=0}:\ \tilde U_f=I \text{ for all }f\in F\bigr\}. \] \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\Phi_t$ preserves the product of the Haar measures for every $t$. \item $\E_\mu e^{\beta_WtD^F_rS}\le e^{\beta_Wt^2K^F_r/2}$ for every $t\in\R$, and $\mu(\pm D^F_rS\ge x)\le e^{-\beta_Wx^2/(2K^F_r)}$ for every $x>0$. $D^F_rS$ is invariant under the gauge transformations that are trivial at the roots. \item Let $F$ be one of the sixteen translates $F_o$ of the forest of Definition~\ref{def:blockgauge}, with blocks $2y+o+\{0,1\}^4$, let $a=G_LJ_{k,P}$ and $T$ be as in Proposition~\ref{prop:tiltgrowth}, and let $r_e=a_eT$ on the links outside $F$. Then \[ K^F_r\ge2\sum_{e\notin F}a_e^2, \] and the bound is reached on pure gauge fields, where $Q(U)=0$. For at least one $o$ the right side is at least $\frac{29}{16}\|a\|^2\ge \frac{29}{16}\cdot\frac{4^k}{1492992}$. \end{enumerate} \end{proposition} \begin{proof} (i) Let $\Gamma(U)=(U_F,\tilde U)$ with $\tilde U_e=h_x(U)U_eh_y(U)^{-1}$ for $e=(x,y)\notin F$. Given the variables on $F$, the map $U_e\mapsto \tilde U_e$ is a left and a right translation by elements that depend only on $U_F$, so $\Gamma$ is a bijection of the configuration space that preserves the product of the Haar measures; this is the change of variables of Lemma~\ref{lem:gaugefix}. Let $R_t$ multiply every coordinate $\tilde U_e$, $e\notin F$, on the right by $e^{tr_e}$ and leave $U_F$ alone. Then \[ (\Gamma^{-1}R_t\Gamma U)_e=h_x^{-1}\bigl(h_xU_eh_y^{-1}e^{tr_e}\bigr)h_y =U_e\exp\bigl(t\operatorname{Ad}(h_y^{-1})r_e\bigr), \] so $\Phi_t=\Gamma^{-1}R_t\Gamma$ is a composition of three maps that preserve the measure. (ii) Since the action is gauge invariant, $S(\Phi_tU)=S(\tilde Ue^{tr})$, and $\tilde Ue^{tr}$ again has trivial links on $F$. Taylor's formula along $s\mapsto\Phi_{-s}U$ with the Lagrange remainder gives $S(\Phi_{-t} U)-S(U)\le-tD^F_rS(U)+\frac{t^2}2K^F_r$, and $\int e^{-\beta_WS\circ\Phi_ {-t}}\,dU=Z$ by (i); the rest is the proof of Theorem~\ref{thm:haartilt} (ii), (iii). Under a gauge transformation $g$ the transports change to $g_oh_yg_y^{-1}$, with $o$ the root of the component of $y$, so $\tilde U$ is conjugated by the root values and $D^F_rS$ is unchanged when they are trivial. (iii) The tree links of $F_o$ join sites of one block, so a pure gauge field $U=I^g$, $U_e=g_xg_{x+e}^{-1}$, with $g$ constant on the blocks has trivial links on $F_o$, lies in the set of the supremum and has $Q(U)=0$. By the proof of Proposition~\ref{prop:tiltgrowth}, $\partial_t^2S(Ue^{tr} )|_{t=0}=\sum_p\|(dr')_p\|_{\rm op}^2$ with $r'_e=\operatorname{Ad}(g_{x+e} ^{-1})r_e$, the end point of $e$ taken. Since $a_3=a_4=0$, a plaquette $p$ at $x$ in a plane $\{\mu,\nu\}$ with $\mu\in\{1,2\}$ and $\nu\in\{3,4\}$ has at most two tilted sides, $e=(x,\mu)$ with end point $x+e_\mu$ and $e'=(x+e_\nu,\mu)$ with end point $x+e_\mu+e_\nu$, and \[ \|(dr')_p\|_{\rm op}^2=a_e^2+a_{e'}^2-2a_ea_{e'}\ip{\operatorname{Ad}(g_{x +e_\mu}^{-1})T}{\operatorname{Ad}(g_{x+e_\mu+e_\nu}^{-1})T}, \] with the inner product of $\g$ in which $\|T\|=1$ and $a_e=0$ on $F_o$. When $(x-o)_\nu$ is odd the two end points lie in different blocks. Take the block values $g_B$ independent with Haar law. The adjoint representation of $\SU(2)$ is irreducible of dimension three, so $\E\operatorname{Ad}(g_B)=0$, and the expectation of the inner product vanishes. Every link $e=(x,\mu)\notin F_o$ with $\mu\in\{1,2\}$ is, for each $\nu\in\{3,4\}$, a side of exactly one such plaquette, at $x$ if $(x-o)_\nu$ is odd and at $x-e_\nu$ otherwise. Dropping the other plaquettes, whose terms are nonnegative, \[ \E_g\,\partial_t^2S(I^ge^{tr})\big|_{t=0}\ge\sum_{\nu\in\{3,4\}}\sum_{e \notin F_o}a_e^2=2\sum_{e\notin F_o}a_e^2, \] and the supremum over the section is at least the expectation. A link $(x,\mu)$ lies in $F_o$ for one of the sixteen offsets $o$ when $\mu=1$ and for two of them when $\mu=2$. Averaging over $o$ and using $\|a_1\|^2=\|a_2\|^2=\frac12\|a\|^2$ from the proof of Proposition~\ref{prop:tiltgrowth}, \[ \frac1{16}\sum_o\sum_{e\notin F_o}a_e^2=\frac{15}{16}\|a_1\|^2+\frac{14}{16} \|a_2\|^2=\frac{29}{32}\|a\|^2, \] so one offset reaches $\frac{29}{32}\|a\|^2$; the last bound is the lower bound for $\|a\|^2$ of that proposition. \end{proof} \begin{figure}[t] \centering \resizebox{0.92\columnwidth}{!}{% \begin{tikzpicture}[x=1cm,y=1cm,>={Stealth[length=1.8mm]}] \foreach \bx/\nm in {0/{B},3.6/{B'}} { \draw[fill=blue!6,rounded corners=2pt] (\bx,0) rectangle (\bx+2,2); \foreach \p in {(\bx+0.4,0.4),(\bx+1.6,0.4),(\bx+0.4,1.6),(\bx+1.6,1.6)} \fill \p circle (1.6pt); \draw[very thick,blue!60!black] (\bx+0.4,0.4) -- (\bx+1.6,0.4) -- (\bx+1.6,1.6) (\bx+1.6,0.4) -- (\bx+0.4,1.6); \node[above] at (\bx+1,2.05) {\footnotesize $g_{\nm}$}; } \draw[very thick,red!70!black,->] (1.6,0.4) -- (4.0,0.4) node[pos=0.5,below] {\footnotesize $e$, weight $a_e$}; \draw[very thick,red!70!black,->] (1.6,1.6) -- (4.0,1.6) node[pos=0.5,above] {\footnotesize $e'$, weight $a_{e'}$}; \draw[->,thick,orange!80!black] (4.0,0.4) -- (6.2,-0.3) node[right] {\footnotesize $a_e\operatorname{Ad}(g_{B'}^{-1})T$}; \draw[->,thick,orange!80!black] (4.0,1.6) -- (6.2,2.3) node[right] {\footnotesize $a_{e'}\operatorname{Ad}(g_{B'}^{-1})T$}; \node[align=center] at (3.1,-1.0) {\footnotesize tree links (blue) are trivial in the section; the residual gauge\\[-2pt]\footnotesize group turns the tilt direction by one $\SU(2)$ element per block}; \end{tikzpicture}} \caption{The covariant tilt on the block forest (Proposition~\ref{prop:foresttilt}). Transporting the direction along the trees keeps the Haar measures, but the section still contains every pure gauge field that is constant on the blocks. On a plaquette whose tilted sides $e$, $e'$ end in different blocks the transported directions are turned by independent elements, and the average of $\|(dr')_p\|^2$ over them is $a_e^2+a_{e'}^2$ instead of the $(a_e-a_{e'})^2$ of the abelian energy.} \label{fig:foresttilt} \end{figure} Proposition~\ref{prop:foresttilt} and Figure~\ref{fig:foresttilt} show where the obstruction sits. The Haar translation survives gauge covariance along a forest, and for the block forest the constant it produces is at least $\frac{29}{16}\|a\|^2$, reached by fields with no action and no curvature at all; the chessboard estimate and the covariant tilt are therefore lost on the same fields, the gauge orbit of the vacuum, for different reasons. A bound of the form \[ \partial_t^2S(\tilde Ue^{tr})\le Q(r)+\mathcal R(\tilde U,r), \] with $\mathcal R$ supported on the large-field region, is false for every forest section, since the left side exceeds $Q(r)$ by a fixed multiple of $\|a\|^2$ at $\tilde U=I^g$, where the large-field region is empty. A section that is smooth on the orbits, such as the Landau gauge, has no such pure gauge fields, but its Jacobian is a Faddeev--Popov determinant and not one, and part (ii) does not apply to it as it stands.\footnote{The Wilson measure satisfies no curvature-dimension condition at the values of $\beta_W$ considered here, so the inequality cannot be obtained from a Bakry--\'Emery argument either: for one link equal to $-I$ with trivial neighbours the second variation of $S$ along a unit direction on that link is $-6$, since each of its six plaquettes contributes $\partial_t^2(1+\cos t)=-1$, while the Ricci curvature of $\SU(2)$ with unit-speed geodesics $e^{tr}$, $\|r\|_{\rm op}=1$, is $2$; the Hessian of $\beta_WS$ plus the curvature is negative as soon as $\beta_W>1/3$.} \begin{figure*}[t] \centering \resizebox{\textwidth}{!}{% \begin{tikzpicture}[x=0.95cm,y=0.75cm,>={Stealth[length=2mm]}] \draw[->,thick] (0,0) -- (14.5,0) node[right] {scale $k$}; \foreach \x/\l in {0/0,2/{},4/{},6/{},8/{},10/{},12/{},13.4/{$m\sim\beta_W/b$}} \draw (\x,0.12) -- (\x,-0.12) node[below] {\footnotesize \l}; \draw[dashed,thick] (13.4,-0.3) -- (13.4,5.6); \node[above,align=center] at (13.4,5.6) {\footnotesize cube of fixed\\[-2pt]\footnotesize physical side}; \node[left] at (0,1) {\footnotesize Proposition~\ref{prop:coarsechess}, chessboard}; \fill[orange!60] (0,0.7) rectangle (3.0,1.3); \node[right] at (3.1,1) {\footnotesize $16^k\lesssim\beta_WA/\log\beta_W$: lost in the normalisation $Z_L$}; \node[left] at (0,2.2) {\footnotesize Propositions~\ref{prop:tiltgrowth}, \ref{prop:foresttilt}, $\SU(2)$ tilts}; \fill[red!50] (0,1.9) rectangle (4.2,2.5); \node[right] at (4.3,2.2) {\footnotesize $4^k\lesssim\beta_Wx^2$: lost on the gauge orbit of the vacuum, fixed direction or transported along a forest}; \node[left] at (0,3.4) {\footnotesize Corollary~\ref{cor:abeliancoarse}, $\U(1)$ tilt}; \fill[blue!45] (0,3.1) rectangle (14.2,3.7); \node[right] at (0.2,4.0) {\footnotesize rate $\beta_Wx^2$ at every scale and under every exterior}; \node[left] at (0,4.9) {\footnotesize Lemma~\ref{lem:coarseenergy}, energy}; \fill[green!40!black!50] (0,4.6) rectangle (14.2,5.2); \node[right] at (0.2,5.5) {\footnotesize $Q\ge2r^2$ at every scale: the energy side is never the obstacle}; \end{tikzpicture}} \caption{Which scales the estimates of this subsection reach, along the tuning $\beta_W\sim b\log(1/a)$, where a cube of fixed physical side spans $m\sim\beta_W/b$ scales. The energy of a coarse defect is scale invariant (Lemma~\ref{lem:coarseenergy}), and for the abelian theory the Haar tilt makes coarse defects rare at every scale (Corollary~\ref{cor:abeliancoarse}). For $\SU(2)$ the chessboard estimate stops after $O(\log\beta_W)$ scales because the normalisation charges the whole fluctuation volume of a block against the defect (Proposition~\ref{prop:coarsechess}), and the tilt stops after $O(\log\beta_W)$ scales because its second variation grows like $4^k$ on the gauge orbit of the vacuum, whether the direction is fixed or transported along a forest (Propositions~\ref{prop:tiltgrowth}, \ref{prop:foresttilt}). The remaining scales are the open input of the cube condition.} \label{fig:scaleladder} \end{figure*} Along the tuning $\beta_W\sim b\log(1/a)$ a cube of fixed physical side spans $m\sim\beta_W/b$ scales. Proposition~\ref{prop:coarsechess} covers $O(\log\beta_W)$ of them, Propositions~\ref{prop:tiltgrowth} and~\ref{prop:foresttilt} cover $O(\log\beta_W)$ of them without the torus and uniformly in the exterior, and Corollary~\ref{cor:abeliancoarse} covers all of them for the abelian theory. What the cube condition needs for $\SU(2)$ is a translation of the Haar measures whose second variation is the energy of the direction up to a constant independent of the scale, together with a non-perturbative treatment of the last $O(1)$ scales, where the running coupling is of order one and coarse defects are not rare at all. Proposition~\ref{prop:foresttilt} shows that no translation transported along a forest does this: its constant is reached at zero action, with the growth $4^k$ of the energy of the smeared current under the residual gauge freedom, so the missing estimate is not a large-field remainder but a different section of the gauge orbits, with a Jacobian that enters the argument of part (ii). This is the content of Ba{\l}aban's large-field expansion~\cite{Bal89b}, and it is not carried out here. The cube condition itself is a mixing condition of Dobrushin--Shlosman type, which for compact spins with finite range is equivalent to a logarithmic Sobolev inequality uniform in the volume~\cite{SZ92}; the rarity of coarse defects is the part of that inequality that concerns the large fields. \section{Labels, islands and the environment}\label{sec:islands} The cube condition (D) is a statement about conditional laws under every exterior, and Sec.~\ref{ssec:fixedset} showed that one-point rarity of a bad set gives closeness of Gibbs measures but not decay. The route that remains is the one of the large-field expansions: label each block of a window good or bad, prove decay on the good windows, bound the bad islands by their rarity, and control the covariance that the labels themselves carry. This section proves what each of these four steps gives for the cut-off average of Sec.~\ref{ssec:pathblocks}, and proves, by explicit examples, where the natural choices of labels fail. The statements are exact and finite; none of them closes (D), and Sec.~\ref{ssec:remains} lists what they leave. Throughout, $G=\SU(2)$ with $\rho_G=1$, the lattice is in the tree gauge of Definition~\ref{def:blockgauge}, $d(g,h)=\lVert\log(g^{-1}h)\rVert$ is the bi-invariant geodesic distance in the Hilbert--Schmidt norm, so that $d(g,I)=\lVert X\rVert$ for $g=e^X$ with $\lVert X\rVert\le\pi\sqrt2$, and $\delta$ is the chordal distance. A \emph{pair} $k=(y,\mu)$ consists of a block and a direction; its \emph{path link} is $\ell_k=U_{2y+e_\mu,\mu}$, its \emph{private links} are the eight links leaving $B(y)$ in the direction $\mu$, which enter no other cut-off average, and the seven private links other than $\ell_k$ are its \emph{non-path} links. In the sixteen paths of $V^\sharp_k$ the path link occurs twice, as the two paths of transverse offset zero, and each non-path private link occurs twice, each time multiplied by one link inside $B(y)$ or $B(y+e_\mu)$. For a coarse field $c$ the reconstruction $R(c)$ of Sec.~\ref{ssec:pathblocks} puts $c_k$ on the private links of $k$ and $I$ on the links inside the blocks. \subsection{Backgrounds and the path links}\label{ssec:branches} The first question is which background the chart of Theorem~\ref{thm:chartdecay} should be centred at when a window is conditioned on its labels. The natural label of a link is its distance to the reconstruction of the coarse field, and the natural background is the reconstruction itself. The first result shows that this is not enough: a coarse field all of whose plaquettes are large has a reconstruction at which every relative link label is good and the constrained action has a negative second variation along the fibre. \begin{proposition}[a rough background]\label{prop:roughbackground} On the periodic lattice of side four, with coarse torus $(\Z/2\Z)^4$, let $c_{y,\mu}=(-I)^{\sum_{\nu<\mu}y_\nu}$. Then every coarse plaquette of $c$ equals $-I$, the field $U=R(c)$ is in the tree gauge and has $V^\sharp(U)=c$, and $d(U_e,R_e(c))=0$ for every link. There is a curve $U(t)=U\exp(t\xi)$, $\xi$ supported on four links, such that $V^\sharp(U(t))=c$ for all small $|t|$, $d(U_e(t),R_e(c))\le|t|$, and the Wilson action $S=\sum_p(1-\frac12\ReTr U_p)$ satisfies \[ \frac{d^2}{dt^2}\Bigl(\beta_WS(U(t))\Bigr)\Big|_{t=0}=-\beta_W,\qquad T(\xi)=\sum_e\lVert\xi_e\rVert^2=4 . \] Every first derivative of $S$ at $U$ vanishes, so the intrinsic Hessian of $\beta_WS$ on the fibre of $c$ at $U$ has an eigenvalue at most $-\beta_W/4$, in any smooth chart of the fibre. \end{proposition} \begin{proof} The coarse plaquette at $y$ in the plane $\{\mu,\nu\}$, $\mu<\nu$, is the product of $c_{y,\mu}$, $c_{y+e_\mu,\nu}$, $c_{y+e_\nu,\mu}^{-1}$ and $c_{y,\nu}^{-1}$; these are central, and the exponent of $-I$ is $\sum_{\rho<\mu}y_\rho+\sum_{\rho<\nu}(y+e_\mu)_\rho-\sum_{\rho<\mu}(y+e_\nu)_\rho -\sum_{\rho<\nu}y_\rho=1$ modulo two, since $\mu<\nu$ puts $e_\mu$ inside the second sum and $e_\nu$ outside the third. So every coarse plaquette is $-I$, also across the periodic seam. In $U=R(c)$ the links inside the blocks are $I$, in particular the tree links, and all sixteen paths of a pair $k$ equal $c_k$, so every $v_x=I$, the cut-off is one and $V^\sharp_k (U)=c_k$; the relative labels vanish by construction. Let $T\in\mathfrak{su}(2)$ have $\lVert T\rVert=1$, with eigenvalues $\pm i/\sqrt2$, and let $\xi_e=a_eT$ with $a_e=-1,+1,+1,-1$ on the links $((0,1,1,1),1)$, $((0,1,1,1),2)$, $((0,1,0,0),1)$, $((0,0,1,0),2)$ and $a_e=0$ elsewhere. All four links are private links; the last two are path links. The pair $((0,0,0,0),1)$ has the non-path link $((0,1,1,1),1)$ with $a=-1$ and the path link $((0,1,0,0),1)$ with $a=+1$, the pair $((0,0,0,0),2)$ has $((0,1,1,1),2)$ with $a=+1$ and the path link $((0,0,1,0),2)$ with $a=-1$, and no other pair contains a link of the support. Since the background is central, $P_x(t)=c_k\exp(ta_xT)$ for the sixteen paths, with $a_x$ the value on the private link of the path, so $v_x(t)=\exp(t(a_x-a_0)T)$ lies in the chart for small $|t|$, the cut-off is one, and \[ V^\sharp_k(U(t))=c_k\exp\Bigl(\frac t{16}\sum_xa_xT\Bigr)=c_k , \] because each private link occurs in exactly two paths and the two nonzero values of each pair cancel. The curve stays in the fibre, and $d(U_e(t),R_e(c))=|ta_e|\le|t|$. Write $U_p=s_pI$ with $s_p=\pm1$ and let $q_p=(d\,a)_p$ be the signed curl of the scalar field $a$ around $p$. Since the perturbations commute with the background, $U_p(t)=s_p\exp(tq_pT)$ and $\ReTr U_p(t)=2s_p \cos(tq_p/\sqrt2)$, so \[ S(U(t))=\sum_p\bigl(1-s_p\cos(tq_p/\sqrt2)\bigr),\qquad S''(0)=\frac12\sum_ps_pq_p^2 . \] The plaquettes with $q_p\ne0$ are those containing a link of the support. A link $(x,\mu)$ lies in six plaquettes, two in each plane $\{\mu,\nu\}$, at $x$ and at $x-e_\nu$. Listing the six plaquettes of each of the four links, the plaquette at $(0,1,1,1)$ in the plane $\{1,2\}$ contains two links of the support, $((0,1,1,1),1)$ and $((0,1,1,1),2)$, with $q_p=-1-1=-2$; the other $22$ plaquettes contain one link each, with $q_p=\pm1$. The sign $s_p$ of a plaquette is $-1$ exactly when it has an odd number of links on which $U$ equals $-I$; counting through the four links, which is a finite check, gives twelve plaquettes with $s_p=1$, $q_p^2=1$, ten with $s_p=-1$, $q_p^2=1$, and the one with $s_p=-1$, $q_p^2=4$. Hence $\sum_ps_pq_p^2=12-10-4=-2$, $S''(0)=-1$, and $T(\xi)=4$. Every plaquette of $U$ is central, so every first derivative of $S$ at $U$ vanishes and the second derivative along any curve through $U$ with velocity $\xi$ depends only on $\xi$: the intrinsic Hessian on the fibre has the value $-1$ at $\xi$, which has $T(\xi)=4$. The Haar density of the fibre, in a chart in which the path links are solved for, is the inverse Jacobian of the path-link equations, which at $U$ is a fixed positive number independent of $\beta_W$; its logarithmic second derivatives are bounded in any fixed chart, so no choice of chart turns the Hessian of $\beta_WS$ plus the log-density positive for all large $\beta_W$. \end{proof} The proposition does not contradict Theorem~\ref{thm:chartdecay}, which assumes small plaquettes; it shows that the hypothesis of that theorem on the coarse field cannot be dropped in favour of relative link labels. A coarse-curvature label is therefore part of any labelling, and the next lemma shows that it has a witness at the fine scale, so that it is as rare as a fine defect. \begin{lemma}[a fine witness for coarse curvature]\label{lem:coarsewitness} Let $P$ be the coarse plaquette at $y$ in the plane $\{\mu,\nu\}$ and suppose that $\delta(U_p,I)\le\kappa\le10^{-8}$ for every fine plaquette $p$ with a vertex in one of the blocks at $\ell^\infty$ distance at most one from $B(y)$, $B(y+e_\mu)$, $B(y+e_\nu)$ or $B(y+e_\mu+e_\nu)$. Then $c=V^\sharp(U)$ satisfies \[ \delta(c_P,I)\le(4+28\pi)\kappa<92\kappa . \] \end{lemma} \begin{proof} For a pair $k$ among the four sides of $P$, Lemma~\ref{lem:smallframes}(i) gives $\delta(U_e,I)\le3\kappa$ for the link of each path inside a block and $\delta(U_e,\ell_k)\le21\kappa$ for its private link, so $\delta(P_x, \ell_k)\le24\kappa$ for every path, and by Lemma~\ref{lem:logbound} $\lVert\log(\ell_k^{-1}P_x)\rVert\le8\pi\kappa$. The two paths of offset zero contribute nothing, so $c_k=\ell_kM_k$ with $\lVert\log M_k\rVert\le \frac{14}{16}\,8\pi\kappa=7\pi\kappa$ and $\delta(c_k,\ell_k)\le7\pi\kappa$. The four path links around $P$ are the holonomy of the square of side two at $2y$: the other four links of its sides are the first links of the straight paths, which are tree links and equal $I$. That square is tiled by four fine plaquettes, and inserting them with their parallel transports gives $\delta(\ell_P,I)\le4\kappa$, where $\ell_P$ is that holonomy. Replacing the four $\ell_k$ by $c_k$ one at a time, bi-invariance of $\delta$ and the triangle inequality add $4\cdot7\pi\kappa$. \end{proof} \begin{remark}[flat fields far from the identity]\label{rem:flatholonomy} Lemma~\ref{lem:coarsewitness} bounds coarse curvature, not the links, and no bound on the links in the tree gauge follows from flatness alone. On the periodic lattice of side $2L$ with blocks of side two, fix a direction $\mu$, put $U_e=-I$ on every link in the direction $\mu$ that joins two different blocks, and $U_e=I$ on every other link. The tree links lie inside the blocks, so $U$ is in the tree gauge. A plaquette in a plane containing $\mu$ has its two links in the direction $\mu$ at the same $\mu$-coordinate, so both cross a block boundary or neither does, and every plaquette holonomy is $I$. Yet every crossing link is $-I$, the point of $\SU(2)$ farthest from $I$, every path of the cut-off average across a block boundary in the direction $\mu$ carries $-I$, and so does the coarse link. For odd $L$ the Wilson line around the direction $\mu$ is $(-I)^L=-I$, which is central and hence gauge invariant, so no gauge transformation makes $U$ trivial. A witness that forces a large plaquette near every link far from $I$ in the tree gauge is therefore false; witnesses have to be taken relative to a reference such as the reconstruction of Lemma~\ref{lem:branchbound}, as they are in this section. The example is due to Tristan Romanov. \end{remark} With the coarse field calm, the chart has a centre. The next lemma shows that for a pair whose non-path links are near the reconstruction, the constraint $V^\sharp_k=c_k$ determines the path link, on every branch of the cut-off, and the fibre of the window over its non-path links is a single graph. \begin{lemma}[the path link is determined]\label{lem:branchbound} Let $k$ be a pair with cut-off average $c_k$, and suppose that each of its seven non-path private links and each link inside $B(y)$ and $B(y+e_\mu)$ is within geodesic distance $r$ of its reconstruction value. Then every path other than the two copies of $\ell_k$ has $d(P_x,c_k)\le 2r$, and every $\ell_k$ with $V^\sharp_k=c_k$ satisfies \[ d(\ell_k,c_k)\le14r ,\qquad d(v_x,I)\le16r\ \text{for every path}. \] If moreover $r\le10^{-4}$, the cut-off equals one at every such $\ell_k$, and $\ell_k$ is unique: it is the only fixed point of \[ X\mapsto F(X)=-\frac1{16}\sum_{x}\log\bigl(e^{-X}c_k^{-1}P_x\bigr), \qquad \ell_k=c_ke^X , \] in the ball $\lVert X\rVert\le14r$, where the sum runs over the fourteen paths of nonzero offset, and $F$ is a contraction of that ball with constant less than $0.9$. \end{lemma} \begin{proof} A path of nonzero offset is the product of a link inside a block, within $r$ of $I$, and a non-path private link, within $r$ of $c_k$, in one of the two orders; bi-invariance of $d$ and the triangle inequality give $d(P_x,c_k)\le2r$. If the cut-off is zero, $V^\sharp_k=\ell_k$, so $\ell_k= c_k$. Otherwise every $v_x=\ell_k^{-1}P_x$ has $\delta(v_x,I)<1$, its principal logarithm is defined and $\lVert\log v_x\rVert=d(v_x,I)=d(\ell_k, P_x)$. Put $D=d(\ell_k,c_k)$ and $N=\frac1{16}\sum_x\log v_x$, the two terms of offset zero being zero. Then $c_k=\ell_k\exp(\chi N)$ and \[ D=d(e^{\chi N},I)\le\chi\lVert N\rVert\le\frac{14}{16}\max_xd(\ell_k,P_x) \le\frac78(D+2r), \] so $D\le14r$, and $d(v_x,I)\le D+2r\le16r$. For $r\le10^{-4}$ this is less than $\frac12$, so $\chi=1$, and $V^\sharp_k=c_k$ reads $X=F(X)$ for $\ell_k=c_ke^X$ with $\lVert X\rVert\le14r$. On that ball each $A_x=c_k^{-1}P_x$ has $d(A_x,I)\le2r$, so $d(e^{-X}A_x,I)\le16r$ and $\lVert F(X)\rVert\le\frac{14}{16}\cdot16r=14r$: $F$ maps the ball into itself. Writing $e^{-X}A_x=I+M$ with $\lVert M\rVert \le16r$, the series of the logarithm gives $\lVert D\log(I+M)\rVert\le 1/(1-16r)$, and the differential of $X\mapsto e^{-X}A_x$ has norm at most $e^{14r}$; hence $\lVert DF\rVert\le\frac78e^{14r}/(1-16r)<0.9$, and the fixed point is unique by the contraction principle. \end{proof} \begin{proposition}[good windows are graphs]\label{prop:graphlaw} Let $W$ be a finite set of blocks, let $c$ be a coarse field, let $K_W$ be the set of pairs based in $W$ or entering $W$ from a block outside it, and let $Z_W$ be the set of non-path active links with base in $\bigcup W$. Free the links of $A_W$ and the path links of the pairs entering $W$, fix every other link, and condition on $V^\sharp_k=c_k$ for every $k\in K_W$ and on the event that every link of $Z_W$ and every non-path link of a pair entering $W$ is within $r\le10^{-4}$ of its reconstruction value. Then the fibre of these constraints is the graph of a smooth map $\Gamma$ from the product of the $r$-balls of the links of $Z_W$ to the path links of the pairs of $K_W$, and the conditional law is the Wilson weight times the Haar disintegration of this graph, whose density with respect to the product Haar measure on $Z_W$ is the product over $k\in K_W$ of the inverse Jacobians of $X\mapsto X-F(X)$ at the fixed point. \end{proposition} \begin{proof} The constraint of a pair $k\in K_W$ involves its own private links and the links inside two blocks, so it involves only its own path link, links of $Z_W$, and fixed links that are within $r$ of their reconstruction values; no other constraint involves a freed link, since the other pairs have their supports off the free set. Lemma~\ref{lem:branchbound} applies to every $k\in K_W$ with the same $r$ and gives one solution for each pair, a smooth function of the links of $Z_W$ by the implicit function theorem since $I-DF$ is invertible. The fibre is therefore the graph of $\Gamma$, and the coarea formula for the submersion $U\mapsto(V^\sharp_k)_{k}$ restricted to the product of the balls gives the stated density, since the derivative of the constraint of $k$ in its own path link is $I-DF$ in the chart at $c_k$ and the cross derivatives between different pairs vanish. \end{proof} The proposition says that, once the coarse field is calm and the labels are placed on the non-path links, the good windows carry the product structure that the proof of Theorem~\ref{thm:chartdecay} needs, with no branch of the cut-off to discard; the Hessian estimate of that theorem then applies on the graph. Two further examples show that the two obvious alternatives fail: labelling all physical links, including the path links, produces a nonconvex domain, and a bad pair can have several path links. \begin{proposition}[the all-link label domain is not convex]\label{prop:nonconvex} Let $c\equiv I$, freeze every link outside the eight private links of one pair at $I$, and write the seven non-path links as $e^{z_j}$ and the path link as $e^{X}$. For every sufficiently small $r>0$ the set of $(z_1,\dots,z_7)$ for which all eight links lie in the geodesic ball of radius $r$ at $I$ is not convex. \end{proposition} \begin{proof} With the cut-off equal to one, the constraint is $H(X,z)=X+\frac18\sum_{j =1}^7\log(e^{-X}e^{z_j})=0$, since each non-path link occurs in two of the sixteen paths and the two paths of offset zero contribute nothing. Take $A,B\in\mathfrak{su}(2)$ orthonormal for the Hilbert--Schmidt norm, so that $[A,[A,B]]=-2B$ and $[B,[B,A]]=-2A$, and put $z_1=-\frac a7A+hB$, $z_2=-\frac a7A-hB$, $z_3=\dots=z_7=-\frac a7A$ with $a>0$ small. The inner automorphism of $\SU(2)$ that fixes $A$ and negates $B$ and $[A,B]$ exchanges the logarithms of $e^{-xA}e^{z_1}$ and $e^{-xA}e^{z_2}$, so their sum lies along $A$ when $X=xA$, and the equation reduces to its $A$-component; by the implicit function theorem the solution is $X=x(a,h)A$ with $x$ analytic, even in $h$ by the same symmetry, and $x(a,0)=a$ since $\log(e^{-xA}e^{bA})=(b-x)A$ gives $x+\frac78(b-x)=0$ at $h=0$. Let $L(x,b,h)$ be the $A$-component of $\log(e^{-xA} e^{bA+hB})$. The Baker--Campbell--Hausdorff series through degree three has one term of degree two in $h$ along $A$, namely $\frac1{12}[hB,[hB, -xA]]=\frac{xh^2}6A$, so $\partial_h^2L(x,b,0)=\frac x3+O((|x|+|b|)^2)$. The $A$-component of $H$ is $x+\frac18\bigl(2L(x,-\frac a7,h)+5(-\frac a7 -x)\bigr)$, and $\partial_xL(x,b,0)=-1$; differentiating $H=0$ twice in $h$ at $h=0$ gives $\frac18\partial_h^2x+\frac14\partial_h^2L=0$, that is \[ \partial_h^2x(a,0)=-2\,\partial_h^2L(a,-\tfrac a7,0)=-\frac{2a}3+O(a^2)<0 . \] Put $\eta=\frac r2$ and take $a$ with $rr$. So the endpoints lie in the domain and the midpoint does not. \end{proof} \begin{proposition}[a bad pair with three path links]\label{prop:branches} Let $c\equiv I$, freeze every link outside the eight private links of one pair at $I$, and take the private links in the one-parameter subgroup $\operatorname{diag}(e^{iq},e^{-iq})$: one non-path link at the angle $-\frac12$, the other six at the angle $\frac{49}{100}$. Then there are at least three values of the path-link angle $q$ with the same cut-off average, and the derivative of the cut-off average with respect to $q$ vanishes at some $q\in(0,\frac15)$. \end{proposition} \begin{proof} On the subgroup all logarithms are scalar and the cut-off average has the angle $f(q)=q+\frac{\chi(q)}8\bigl(-\frac12+6\cdot\frac{49}{100}-7q\bigr)$, where $\chi(q)=\phi(m(q))$, $m(q)=\max_j2\sqrt2\,|\sin((\theta_j-q)/2)|$ and $\theta_1=-\frac12$, $\theta_2=\dots=\theta_7=\frac{49}{100}$, the factor $2\sqrt2|\sin(\theta/2)|$ being the chordal distance of $\operatorname{diag}(e^{i\theta},e^{-i\theta})$ from $I$. At $q=0$ the maximum is attained at $\theta_1$ alone, with $m(q)=2\sqrt2\sin((q+\frac12) /2)$ near $0$ and $\frac12\frac{31}{32}$ and $1<\sqrt2<\frac32$ on the positive and the negative term separately, $f'(0)<-\frac{91}{3200}<0$. Put $\gamma=f(0)$, which lies in $(0,1)$. For $|q|=1$ the maximal chordal distance exceeds one, so $\chi(\pm1)=0$ and $f(\pm1)=\pm1$. Since $f$ decreases through $\gamma$ at $0$ and $f(-1)<\gamma1-\frac7{40}- 3\cdot\frac{13}{100}>0$; with $f'(0)<0$, continuity gives a zero of $f'$ in $(0,\frac15)$. \end{proof} Hence the Jacobian of the path-link constraint, which enters the density of Proposition~\ref{prop:graphlaw}, has no positive lower bound on the bad side, and a bad pair contributes a sum over branches to any estimate of the conditional law. On the good side none of this happens: the labels of Proposition~\ref{prop:graphlaw} are placed on the non-path links only. \subsection{The path-link law on its two sides}\label{ssec:pathlaw} The conditional law of the path link of a pair $k$, given its seven non-path private links $z=(\ell_a)_{a\ne0}$, the links $u,w$ inside its two blocks and the rest of the field $\sigma$, is \begin{equation}\label{eq:pathlaw} \pi(dg\mid z,u,w,\sigma)=\frac1Z\sum_{g:\,V(g;z,u,w)=c_k}\frac{e^{-\beta_W S(g;\cdot)}}{\lvert\det D_gV(g;z,u,w)\rvert}\,\delta_g , \end{equation} where $V=V^\sharp_k$ as a function of the path link, the sum runs over all its preimages of $c_k$, and $\lvert\det D_gV\rvert$ is the Jacobian between tangent spaces of the bi-invariant metric. This is the Haar disintegration along the Lipschitz map $V$ on the compact three-manifold $\SU(2)$, by the coarea formula, times the Wilson weight; on the good side it is the one-point law of Proposition~\ref{prop:graphlaw}, and on the bad side it carries the branches of Proposition~\ref{prop:branches}. We determine its response to each kind of exterior link. Throughout, $W_1^\delta$ and $W_1^d$ are the Kantorovich distances for the chordal and the geodesic distance, and the facts $\lVert L_X\rVert\le\theta/\sin\theta$ for an eigenangle $\theta<\pi$ of $X/\sqrt2$, $\lVert E_Y\rVert\le1$ and $d(e^A, e^B)\le\lVert A-B\rVert$ of Sec.~\ref{ssec:smallfibres} are used as stated there. \begin{lemma}[links that enter only the Wilson weight]\label{lem:wilsononly} Let $e'$ be a link that is neither a private link of $k$ nor inside $B(y)$ or $B(y+e_\mu)$, and let $\sigma,\sigma'$ differ only at $e'$. Then for every $(z,u,w)$ and $c_k$ for which~\eqref{eq:pathlaw} is a finite nonzero measure, \[ W_1^\delta\bigl(\pi(\cdot\mid z,u,w,\sigma),\pi(\cdot\mid z,u,w,\sigma') \bigr)\le\beta_W\,d(\sigma_{e'},\sigma'_{e'}), \] and the two laws coincide unless $e'$ lies in a plaquette with a private link of $k$. For the law of all eight private links given $u,w,\sigma$ and restricted to any fixed measurable set $B$ of the non-path links, with the cost $\sum_a\delta(\ell_a,\ell'_a)$, the same holds with $8\beta_W$. Both bounds include every branch and the disintegration density. \end{lemma} \begin{proof} Only $S$ depends on $\sigma_{e'}$: $V$ is a function of $g,z,u,w$, and so is the measure $\nu=\sum_g\delta_g/\lvert\det D_gV\rvert$, respectively $\mathbf1_BJ_8^{-1}\mathrm{vol}_M$ with $M=\{V=c_k\}$ and $J_8$ the Jacobian of Lemma~\ref{lem:privatesubmersion}. An exterior link lies in at most one plaquette with a private link: the private links are the eight $\mu$-links at the sites $x'_a$, whose $\mu$-coordinates agree, so the two plaquettes in a plane $\{\mu,\nu\}$ through a $\nu$-link, which sit at different $\mu$-coordinates, cannot both contain one; and a $\mu$-link $(x,\mu)$ with $x$ not a site $x'_a$ has $x+\varepsilon e_\nu=x'_a$ for at most one pair $(\nu,\varepsilon)$, since two such relations would put a coordinate of $x$ outside $\{0,1\}$ in one direction and inside it in another. If there is no such plaquette $p_0$ the laws are equal. Otherwise move $\sigma_{e'}$ along a minimal geodesic $\sigma_{e'}e^{t\xi}$, $\lVert\xi\rVert=1$, $0\le t\le T=d(\sigma_{e'},\sigma'_{e'})$, with laws $\pi_t$. For bounded $f$, $\frac d{dt}\pi_t(f)=-\beta_W\Cov_{\pi_t}(f,\dot s_{p_0})$, by differentiation under the integral against the fixed finite measure $\nu$. Here $\frac d{dt}U_{p_0}(t)=U_{p_0}(t)\xi'$ with $\lVert\xi'\rVert=1$, $\xi'$ a conjugate of $\pm\xi$, so $\dot s_{p_0}= -\frac12\ReTr(U_{p_0}\xi')$; in a basis in which $U_{p_0}=\operatorname{diag} (e^{i\varphi},e^{-i\varphi})$ only the diagonal entries $\pm i\alpha$ of $\xi'$ contribute, $2\alpha^2\le1$, and $\dot s_{p_0}=\alpha\sin\varphi$. Hence $|\dot s_{p_0}|\le\frac1{\sqrt2}$, and a link of a plaquette moving at geodesic speed $v$ gives $|\dot s_p|\le v/\sqrt2$. A function that is $1$-Lipschitz for $\delta$ has oscillation at most the chordal diameter $2\sqrt2$ of $\SU(2)$, and at most $8\cdot2\sqrt2$ for the sum cost on eight links. Since $|\Cov(f,h)|\le\frac14\operatorname{osc}f \operatorname{osc}h$, $|\frac d{dt}\pi_t(f)|\le\beta_W$, respectively $8\beta_W$; integrate over $[0,T]$ and use the Kantorovich duality. \end{proof} This is Lemma~\ref{lem:fixedset} for the fibre law: the branches and the density are harmless for every link that enters only through the action. The links it leaves out are the seven non-path links, which are in the conditioning, and the inside links $u_a,w_a$, which enter $V$. On the good side both are harmless, and the radius of Lemma~\ref{lem:branchbound} and Proposition~\ref{prop:graphlaw} can be taken as $1/32$ rather than $10^{-4}$. \begin{proposition}[the good side to radius $1/32$]\label{prop:goodside} Let $r\le\frac1{32}$, and suppose $d(\ell_a,c_k)\le r$ for $a\ne0$ and $d(u_a,I),d(w_a,I)\le r$. Put $\theta_0=\frac1{2\sqrt2}$ and $\kappa_0= \frac78\,\theta_0/\sin\theta_0<0.8935$. Then every solution $g$ of $V(g;z,u, w)=c_k$ has cut-off equal to one and $d(g,c_k)\le14r$, and there is exactly one: writing $g=c_ke^X$, $X$ is the unique fixed point of $F(X)=-\frac1{16} \sum_x\log(e^{-X}c_k^{-1}P_x)$ in $\{\lVert X\rVert\le14r\}$, and $F$ is a contraction of that ball with constant $\kappa_0$. At the solution the singular values of $D_gV$ lie in $[1-\kappa_0,1+\kappa_0]$, so the density $1/\lvert\det D_gV\rvert$ lies in $[(1+\kappa_0)^{-3},(1-\kappa_0)^{-3}]\subset [0.147,828]$. The law~\eqref{eq:pathlaw} is the point mass at $g=g(z,u,w)$, and for two configurations satisfying the hypothesis \begin{multline*} d(g,g')\le1.1986\sum_{a\ne0}d(\ell_a,\ell'_a)\\ +0.5993\sum_{a\ne0}\bigl(d(u_a,u'_a)+d(w_a,w'_a)\bigr), \end{multline*} with no dependence on $\beta_W$, on $c_k$ or on any other link. \end{proposition} \begin{proof} The first part is the proof of Lemma~\ref{lem:branchbound}, which holds for every $r$ up to the contraction: every solution has $D=d(g,c_k)\le14r$ and $d(v_x,I)\le16r\le\frac12$, so $m\le\frac12$ and $\chi=1$, and $V=c_k$ reads $X=F(X)$, with $F$ mapping the ball into itself. For the contraction constant, on the ball each $A_x=c_k^{-1}P_x$ has $d(A_x,I)\le2r$, so $Y_x= \log(e^{-X}A_x)$ has $\lVert Y_x\rVert\le16r$, an eigenangle at most $16r/\sqrt2\le\theta_0$. Since $e^{-(X+tZ)}A_x=(I-tE_{-X}Z+O(t^2))e^{Y_x}$ and $\log(e^{tW}e^Y)=Y+tL_YW+O(t^2)$, $\lVert DF\rVert\le\frac{14}{16}\cdot \lVert E_{-X}\rVert\max_x\lVert L_{Y_x}\rVert\le\kappa_0$, as $\theta/\sin \theta$ increases on $[0,\pi)$; the fixed point is unique by the contraction principle. For the Jacobian, vary $g\mapsto ge^{tZ}$: then $v_x\mapsto e^{-tZ}v_x$, $\log v_x\mapsto\log v_x-tL_{Y_x}Z+O(t^2)$, and with $N=\frac1{16}\sum_x\log v_x$ and $\bar L=\frac1{14}\sum_xL_{Y_x}$, \[ V^{-1}\frac{dV}{dt}\Big|_{t=0}=\operatorname{Ad}(e^{-N})\bigl(I-\tfrac78 E_N\bar L\bigr)Z , \] by $\frac d{dt}e^{N+tW}e^{-N}=E_NW$. As $\operatorname{Ad}(e^{-N})$ is an isometry and $\lVert\frac78E_N\bar L\rVert\le\kappa_0$, the singular values lie in $[1-\kappa_0,1+\kappa_0]$. Finally, the law has one atom, so $W_1^d= d(g,g')$. Move one link at a time along a minimal geodesic; the geodesic ball of radius $r$ is convex, so the hypothesis holds along the way, and $d(g,g')\le\lVert X-X'\rVert$. For $\ell_a\mapsto\ell_ae^{t\zeta}$, $\lVert\zeta\rVert=1$, the two paths through $\ell_a$ move, $A_{x_a}\mapsto A_{x_a}e^{t\zeta}$ and $A_{x'_a}\mapsto A_{x'_a}e^{t\operatorname{Ad}(w_a^{-1} )\zeta}$, and $\log(e^Ye^{t\zeta'})=Y+tL_{-Y}\zeta'+O(t^2)$, so $\lVert\partial_tF\rVert\le\frac2{16}\theta_0/\sin\theta_0$ and, by the implicit function theorem, $\lVert dX/dt\rVert\le\frac2{16}\frac{\theta_0} {\sin\theta_0}\frac1{1-\kappa_0}<1.1986$. For $u_a\mapsto u_ae^{t\zeta}$ one path moves, $A_{x_a}\mapsto A_{x_a}e^{t\operatorname{Ad}(\ell_a^{-1}) \zeta}$, which gives half of it, and likewise for $w_a$. \end{proof} The only change from Lemma~\ref{lem:branchbound} is that the series bound $\lVert D\log(I+M)\rVert\le(1-16r)^{-1}$ is replaced by the exact norm $\theta/\sin\theta$ of $L_Y$. So on the good side the response of the path-link law to every exterior link is bounded, with a constant independent of $\beta_W$, and the bad side begins at $r=\frac1{32}$. On the bad side the response to the non-path links is not bounded at all. The configuration of Proposition~\ref{prop:branches} has a fold: a value $c$ at which two branches are born, with infinite density, and the law jumps across it. When the paths of a pair lie on one torus, the component of the cut-off average transverse to that torus has a closed form. \begin{lemma}[the transverse part of the cut-off average]\label{lem:foldtransverse} Write $\SU(2)$ as the unit quaternions, with $T=\{e^{iq}\}$, and its Lie algebra as the pure quaternions, with the Euclidean norm $|\cdot|$. Let $\theta_1,\dots,\theta_n$ be real and $\varpi_1,\dots,\varpi_n>0$, and for $g\in\SU(2)$ put $h_x=g^{-1}e^{i\theta_x}$, $\chi(g)=\phi(\max_x\delta(h_x, I))$ and \[ V(g)=g\exp Y(g),\qquad Y(g)=\chi(g)\sum_x\varpi_x\log h_x , \] and $Y(g)=0$ if $\chi(g)=0$. For real numbers $q$ and $s$ put $g(q,s)=\cos s\,e^{iq}+\sin s\,k$, and define $\psi_x\in[0,\pi]$ by $\cos\psi_x=\cos s \cos(\theta_x-q)$. Then the $j$-component of $V(g(q,s))$ is $0$ and its $k$-component is $\sin s\,\Xi(q,s)$, where $\Xi=1$ if $\chi=0$ and otherwise \[ \Xi=\cos|Y|-\frac{\sin|Y|}{|Y|}\,\chi\sum_x\varpi_x\,\psi_x\cot\psi_x , \] with $\sin|Y|/|Y|=1$ at $Y=0$. If $\chi>0$, then $\psi_x<\frac\pi4$ for every $x$. The function $\Xi$ is continuous in $q$, $s$ and the $\theta_x$, and $\Xi\le1$. \end{lemma} \begin{proof} Write a quaternion as $\alpha+\beta j$ with $\alpha,\beta\in\C=\R+\R i$. As $jz=\bar zj$, \[ (\alpha_1+\beta_1j)(\alpha_2+\beta_2j)=\alpha_1\alpha_2-\beta_1\bar\beta_2+ (\alpha_1\beta_2+\beta_1\bar\alpha_2)j , \] the inverse of a unit quaternion $\alpha+\beta j$ is $\bar\alpha-\beta j$, and $\beta j$ has $j$-component $\Re\beta$ and $k$-component $\Im\beta$. Now $g=g(q,s)=\cos s\,e^{iq}+(i\sin s)j$, and with $\varphi_x=\theta_x-q$, \[ h_x=\cos s\,e^{i\varphi_x}-(i\sin s\,e^{-i\theta_x})j , \] whose real part is $\cos\psi_x$ and whose pure part $P_x$ has norm $\sin \psi_x$. If $\chi=0$, then $V=g$, whose $j$-part is $(i\sin s)j$. Let $\chi >0$. Then $\delta(h_x,I)=2(1-\cos\psi_x)^{1/2}<1$, so $\cos\psi_x>\frac34> \cos\frac\pi4$, and $\log h_x=w_xP_x$ with $w_x=\psi_x/\sin\psi_x$ ($w_x=1$ if $\psi_x=0$). Hence $Y=iy+\eta j$ with \[ y=\chi\cos s\sum_x\varpi_xw_x\sin\varphi_x,\qquad\eta=-i\chi\sin s\sum_x \varpi_xw_xe^{-i\theta_x}, \] and $\exp Y=(C+i\sigma y)+\sigma\eta j$ with $C=\cos|Y|$ and $\sigma=\sin|Y| /|Y|$. The $j$-part of $V=g\exp Y$ is $\beta j$ with \[ \begin{split} \beta&=\cos s\,e^{iq}\sigma\eta+i\sin s\,(C-i\sigma y)\\ &=i\sin s\Bigl(C-\sigma\chi\cos s\sum_x\varpi_xw_xe^{-i\varphi_x}-i\sigma y\Bigr). \end{split} \] By the formula for $y$ the imaginary part of the bracket vanishes, and $w_x\cos s\cos\varphi_x=w_x\cos\psi_x=\psi_x\cot\psi_x$, so the bracket is $\Xi$ and $\beta=i\sin s\,\Xi$. Since $0<\psi\cot\psi\le1$ on $[0,\frac\pi 4)$ and $\sigma\ge0$, $\Xi\le C\le1$. The cut-off $\chi$ is continuous, and so is $\Xi$ where $\chi>0$; as $\chi\to0$, $|Y|\le\frac\pi4\chi\sum_x \varpi_x\to0$ and the subtracted term is at most $\chi\sum_x\varpi_x$, so $\Xi\to1$. \end{proof} \begin{proposition}[the fold of the path-link law]\label{prop:fold} In the configuration of Proposition~\ref{prop:branches}, with every link other than the eight private links equal to $I$, let $c_\lambda= \operatorname{diag}(e^{i\lambda},e^{-i\lambda})$ and let $\pi_\lambda$ be the law~\eqref{eq:pathlaw} with $c_k=c_\lambda$. Let $f_0$ be the angle of the cut-off average on the torus, the function $f$ of Proposition~\ref {prop:branches}, let $q^*\in(0.0741,0.0742)$ be its minimiser on $[-0.005, 0.2227]$, $\gamma^*=f_0(q^*)\in[0.16978261,0.16978263]$, $g^*=e^{iq^*}$, and $\Lambda=[\gamma^*-10^{-3},\gamma^*+10^{-3}]$. Then: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For $\lambda\in\Lambda$ every preimage of $c_\lambda$ lies on the torus. For $\lambda<\gamma^*$ there is one, $g_L(\lambda)=e^{iq_L(\lambda)}$ with $q_L\in[-0.0172,-0.0156]$; for $\lambda>\gamma^*$ there are three, $g_L(\lambda)$ and $g_\pm(\lambda)=e^{iq_\pm}$ with $q_-0$ there are $\lambda<\gamma^*<\lambda'$ with $\lambda'-\lambda<\varepsilon$ and $W_1^\delta(\pi_\lambda,\pi_{\lambda'})\ge 0.126$. \item Fix $c=c_{\gamma^*}$ and move the seven non-path links together, $\ell_a\mapsto\ell_ae^{it}$, with laws $\pi^{(t)}$. Then $\pi^{(t)}\to \delta_{g^*}$ as $t\uparrow0$ and $\pi^{(t)}\to\delta_{g_L^*}$ as $t\downarrow 0$. So for every $\beta_W\ge0$ and $\varepsilon>0$ there are two configurations of the non-path links at total distance $7\sqrt2\,(t'-t)< 10\varepsilon$ whose path-link laws are at $W_1^\delta$ distance at least $0.126$: the Kantorovich response of the path-link law to its non-path links is infinite, for every $\beta_W$. \end{enumerate} \end{proposition} \begin{proof} Write $g=g_0+g_1i+g_2j+g_3k$ as a unit quaternion, so that the torus $T$ is $\{e^{iq}\}$, and $\theta_a$ for the angles of the non-path links, $-\frac12$ once and $\frac{49}{100}$ six times. Then $V(g)=g\exp(\chi(m)\frac18\sum_{a \ne0}\log(g^{-1}\ell_a))$ with $m=\max_a\delta(g^{-1}\ell_a,I)$, since each $\ell_a$ lies on two paths and $u=w=I$, and the Wilson weight of the path link is $e^{-\beta_WS(g)}$ with $S(g)=\frac34[\delta(g,I)^2+\delta(g,e^{0.49i})^2] \in[0,12]$, the three plaquettes through the path link and its partner and the three through the path link and $I$. \emph{Step 1, the torus.} For $q\in[q_{lo},q_{hi}]=[\frac{49}{100}-\alpha, -\frac12+\alpha]\approx[-0.2327,0.2227]$, where $\alpha=\arccos\frac34$, the maximiser of $\delta(g^{-1}\ell_a,I)$ at $g=e^{iq}$ is the far link when $q\ge-0.005$ and the six near links when $q\le-0.005$, $m=2\sqrt2\sin(d(q)/2) \in(\frac12,1)$ with $d(q)=\max\{q+\frac12,\frac{49}{100}-q\}$, and \[ f_0(q)=q+\frac{\chi(q)}8\Bigl(\frac{244}{100}-7q\Bigr),\qquad\chi(q)=2-4 \sqrt2\sin\frac{d(q)}2 ; \] outside $[q_{lo},q_{hi}]$ some $\delta(g^{-1}\ell_a,I)\ge1$, $\chi=0$ and $f_0(q)=q$. On $[q_{lo},-0.005]$, $\chi'=2\sqrt2\cos\frac{0.49-q}2>0$ and $\frac{244}{100}-7q>0$, so, as $\chi(-0.005)=2-4\sqrt2\sin0.2475<0.6146$, \[ f_0'\ge1-\tfrac78\chi\ge1-\tfrac78\cdot0.6146>0.46 : \] $f_0$ increases from $q_{lo}$ to $f_0(-0.005)=0.18501\ldots$. On $[-0.005,q_{hi}]$, \[ \chi'=-2\sqrt2\cos\tfrac{q+1/2}2<0,\qquad\chi''=\sqrt2\sin\tfrac{q+1/2}2>0, \] so $f_0''=\frac18\chi''(\frac{244}{100}-7q)-\frac74\chi'>0$: $f_0$ is strictly convex, with $f_0'(-0.005^+)=-0.3858\ldots<0$ and $f_0'(q_{hi})=0.7087\ldots >0$, a unique minimiser $q^*$, and $f_0'(0.0741)<0\gamma^*+10^{-3}$; and in $[-0.005,q_{hi}]$ none if $\lambda< \gamma^*$, the point $q^*$ if $\lambda=\gamma^*$, and two if $\lambda> \gamma^*$, as $f_0(-0.005)$ and $f_0(q_{hi})=q_{hi}$ exceed $\lambda$. These two lie in $[0.05,0.1]$, where $f_0''>4.6$, since $f_0(0.05),f_0(0.1)> \gamma^*+10^{-3}$, whence $\lambda-\gamma^*\ge\frac{4.6}2(q_\pm-q^*)^2$. The values of $f_0$, $f_0'$ and $f_0''$ used in this step, and $\alpha\in (0.7227,0.7228)$, are evaluations with $\sin$ and $\cos$ replaced by their Taylor polynomials of degree ten, whose remainders are below $10^{-9}$ for arguments in $[0,0.73]$, and with $1.41421356<\sqrt2<1.41421357$; on $[0.05,0.1]$ the factors $\chi''$, $-\chi'$ and $\frac{244}{100}-7q$ of $f_0''$ are monotone and positive, and their values at the ends give $f_0''>4.8$. \emph{Step 2, off the torus.} Conjugation by $h\in T$ fixes every $\ell_a$ and $c_\lambda$, so $V(hgh^{-1})=hV(g)h^{-1}$ and the preimages of $c_\lambda$ are invariant under it; every $g\notin T$ is conjugate to $g(q,s)=\cos s\, e^{iq}+\sin s\,k$ with $s\in(0,\frac\pi2]$. The map $V$ is that of Lemma~\ref{lem:foldtransverse} with the seven angles $\theta_a$ and $\varpi_a=\frac18$, so $V_j=0$ and $V_k(g(q,s))=\sin s\,\Xi(q,s)$. Where $\chi>0$, $\cos s\cos(\theta_a-q)>\frac34$ for every $a$, so $\cos(\theta_a -q)>0$ and $\psi_a$ is at least the distance of $\theta_a-q$ to $2\pi\Z$. These distances for the far link and for a near link add up to at least $\frac{99}{100}$, so $\max_a\psi_a\ge\frac{99}{200}$, $m\ge2\sqrt2\sin \frac{99}{400}$ and, by $\sin x\ge x-\frac{x^3}6$ and $4\sqrt2>5.6568$, $\chi\le2-2m<0.6143$. Then $|Y|\le\frac\chi8\sum_a\psi_a<0.6143\cdot \frac{7\pi}{32}<0.4223$ and $\cos|Y|\ge1-\frac12|Y|^2>0.9108$, while $\sin|Y|/|Y|\le1$ and $0<\psi_a\cot\psi_a\le1$ bound the subtracted term of $\Xi$ by $\frac78\chi<0.5376$. Hence $\Xi>\frac13$ at every point, also where $\chi=0$ and $\Xi=1$, and $V_k(g(q,s))>0$ for $s\in(0,\frac\pi2]$: $V(g)\notin T$ for $g\notin T$. So $V^{-1}(T)=T$, which with Step 1 gives (i) except for regularity. \emph{Step 3, the Jacobian at a torus point.} $D_gV$ at $g=e^{iq}$ commutes with the differential of conjugation by $e^{i\varphi}$, which fixes the axial line $g\cdot i\R$ and rotates the transverse plane by $2\varphi$; a linear map commuting with all these rotations preserves the line and the plane and is a complex scalar $\lambda_\perp e^{i\omega}$ on the plane. The axial eigenvalue is $f_0'(q)$, and $\lambda_\perp=\lVert\partial_s V(g(q,s))|_{s=0}\rVert=|\partial_sV_k(q,0)|$, since $\partial_sg=k$ is a unit transverse vector and $V_j=0$. So $\lvert\det D_gV\rvert=|f_0'(q)| \lambda_\perp(q)^2$, and by Step~2, as $\Xi$ is continuous, $\lambda_\perp(q)=\Xi(q,0)\in(\frac13,1]$. At $s=0$ and $q\in[q_{lo}, q_{hi}]$, $\psi_a=|\theta_a-q|$ and $|Y|=\Delta(q):=\frac\chi8(\frac{244} {100}-7q)=f_0(q)-q$, so \[ \lambda_\perp(q)=\cos\Delta-\frac{\sin\Delta}\Delta\,\frac\chi8\sum_a (\theta_a-q)\cot(\theta_a-q). \] For $q\in[-0.0172,-0.0156]$ the near links are the maximiser, with $d= \frac{49}{100}-q\in[0.5056,0.5072]$ and $\chi=2-4\sqrt2\sin\frac d2$, so $0.07067<\frac\chi8<0.07315$ by $x-\frac{x^3}6\le\sin x\le x$ and $5.6568<4\sqrt2<5.65686$. The partial fractions $x\cot x=1-2\sum_{n\ge1} x^2/(n^2\pi^2-x^2)$, with $\frac1{1-u}\ge1+u$, $\sum_nn^{-2}=\frac{\pi^2}6$ and $\sum_nn^{-4}=\frac{\pi^4}{90}$, give $x\cot x\le1-\frac{x^2}3- \frac{x^4}{45}$ for $0<|x|<\pi$. The far link has $|\theta_a-q|\ge0.4828$ and the near ones $|\theta_a-q|\ge0.5056$, so $\sum_a(\theta_a-q)^2>1.7668$, $\sum_a(\theta_a-q)^4>0.4464$, and the subtracted term is at most $\frac{\sin\Delta}\Delta$ times $0.07315\,(7-0.5889-0.0099)<0.4683$. Further $\Delta\in[0.1801,0.1873]$, $\cos\Delta\ge1-\frac12\Delta^2>0.9824$ and $\frac{\sin\Delta}\Delta\le1-\frac16\Delta^2+\frac1{120}\Delta^4<0.99461$, so $\lambda_\perp>0.9824-0.99461\cdot0.4683>0.5166$ on this interval. At $q_L$ the near links are the strict maximiser and at $q_\pm$ the far link is, with $m\in(\frac12,1)$, so $V$ is real-analytic near these points; $f_0'(q_L)\in[1.36,1.37]$ by the formula, $f_0'(q_\pm)\ne0$ by strict convexity and $\lambda_\perp>0$, so $c_\lambda$ is regular for $\lambda\ne\gamma^*$, and~\eqref{eq:pathlaw} is the continuous disintegration there. The weight of $g_L$ is at most $1/(1.36 \cdot0.5166^2)<2.78$, and those of $g_\pm$ are at least $e^{-12\beta_W}/ |f_0'(q_\pm)|$, as $\lambda_\perp\le1$, with $f_0'(q_\pm)\to f_0'(q^*)=0$. This is (ii). \emph{Step 4, the limits.} Let $f(g)=\delta(g,g^*)$, $1$-Lipschitz for $\delta$ and at most $2\sqrt2$. For $\lambda\downarrow\gamma^*$, $\pi_\lambda(f)\le2.78\cdot2\sqrt2/(w_-+w_+)+\sqrt2\max_\pm|q_\pm-q^*|\to0$, and as $\pi_\lambda$ lives on three points converging to $\{g^*,g_L^*\}$, $\pi_\lambda\to\delta_{g^*}$. For $\lambda\uparrow\gamma^*$, $\pi_\lambda= \delta_{g_L(\lambda)}\to\delta_{g_L^*}$. For $\lambda\in\Lambda$, $q^*-q_L (\lambda)\in[0.0897,0.0914]$, so $\pi_\lambda(f)=2\sqrt2\sin\frac{q^*-q_L}2 \ge0.1268$, and the Kantorovich duality $W_1^\delta(\pi_\lambda,\pi_{\lambda'}) \ge|\pi_\lambda(f)-\pi_{\lambda'}(f)|$ gives (iii). \emph{Step 5, the exterior form.} Multiplying every path on the right by $e^{it}$ gives $V(ge^{it};z_t)=V(g;z_0)e^{it}$, the covariance of $V^\sharp$ with $h_1=I$, $h_2=e^{-it}$. So $g$ solves $V(g;z_t)=c_{\gamma^*}$ iff $g'=ge^{-it}$ solves $V(g';z_0)=c_{\gamma^*-t}$, with the same Jacobian, and the action at $g=g'e^{it}$ under $z_t$ differs from $S(g')$ by at most $6|t|$. Hence $\pi^{(t)}$ is the image of $\pi_{\gamma^*-t}$ under the translation by $e^{it}$, which moves every point by $\sqrt2|t|$, with weights multiplied by factors in $[e^{-6\beta_W|t|},e^{6\beta_W|t|}]$, and the limits of Step~4 persist as $t\to0$; each non-path link moves by $\sqrt2|t'-t|$. This is (iv). \end{proof} The mechanism is the one Proposition~\ref{prop:branches} exhibits: at the fold the density $1/\lvert\det D_gV\rvert$ is infinite, two branches carry all the mass on one side and do not exist on the other, and the Wilson weight cannot repair it, since a vanishing Jacobian wins against any fixed $e^{-\beta_WS}$. The configuration is bad for every $r\le\frac1{32}$. So item (i) of Sec.~\ref{ssec:remains} in its original form, a Lipschitz bound for the law~\eqref{eq:pathlaw} in the non-path links, is false; what can be asked is the response of the law of all eight private links of a bad pair, in which the non-path links are integrated over the bad set and Lemma~\ref{lem:privatesubmersion} removes both the branches and the density. Lemma~\ref{lem:wilsononly} settles that response for every link that enters only the action; what remains is its response to the inside links $u_a,w_a$, which move the fibre $\{V=c_k\}$ while the bad set does not, and the question is the mass that the law gives to the set where the fibre is tangent to the boundary of the bad set. \emph{The site of a bad pair.} The response of the eight private links to the inside links $u_a,w_a$ is avoided by putting those links into the variables of the law. Let $k=(y,\mu)$, $k'=(y-e_\mu,\mu)$ and $k''=(y+e_\mu,\mu)$. The inside links of $k'$ in $B(y)$ are the links $u_a$ of $k$, and the inside links of $k''$ in $B(y+e_\mu)$ are the links $w_a$ of $k$: consecutive pairs along $\mu$ share their inside $\mu$-links. The \emph{site} of $k$ is the set $\mathcal S_k$ of the $24$ links $\ell_a$, $u_a$, $w_a$ ($a\ne0$ for the last two) and the path links $\ell'_0$, $\ell''_0$ of $k'$ and $k''$, with the cost $\sum_{e\in\mathcal S_k}\delta(\cdot,\cdot)$. Its links are based in $B(y-e_\mu)\cup B(y)\cup B(y+e_\mu)$, and the cut-off averages that contain a link of $\mathcal S_k$ are $V^\sharp_k$, which contains site links only, $V^\sharp_{k'}$, whose other links are the non-path links $z'$ of $k'$ and its inside links $u'$ in $B(y-e_\mu)$, and $V^\sharp_{k''}$, with $z''$ and $w''$ in $B(y+2e_\mu)$. Every other link enters the law of the site only through the Wilson weight. \begin{lemma}[the law of a site]\label{lem:sitelaw} Let $r\le\frac1{32}$, suppose $d(\ell'_a,c_{k'})$, $d(u'_a,I)$, $d(\ell''_a,c_{k''})$, $d(w''_a,I)\le r$ for $a\ne0$, and let $E$ be a measurable set of site variables contained in $\{d(u_a,I)\le r,\ d(w_a,I)\le r,\ a\ne0\}$. Then the law of the site given the exterior $\sigma$, the three constraints $V^\sharp_k=c_k$, $V^\sharp_{k'}=c_{k'}$, $V^\sharp_{k''}=c_{k''}$ and $E$ is \[ \frac1Z\,e^{-\beta_WS}\mathbf 1_E\,J_{22}^{-1}\mathrm{vol}_{M_k}(d\ell\,du \,dw)\,\frac{\delta_{\Gamma'}(d\ell'_0)}{\lvert\det D_{\ell'_0}V^\sharp_{k'} \rvert}\,\frac{\delta_{\Gamma''}(d\ell''_0)}{\lvert\det D_{\ell''_0} V^\sharp_{k''}\rvert}, \] where $M_k=\{V^\sharp_k=c_k\}$ in the $22$ variables $(\ell,u,w)$, $J_{22}= \det(DV^\sharp_kDV^{\sharp*}_k)^{1/2}\ge\sigma_0^3$ almost everywhere, $\Gamma'=\Gamma'(z',u',u)$ and $\Gamma''=\Gamma''(z'',w'',w)$ are the unique solutions of Proposition~\ref{prop:goodside} for $k'$ and $k''$, and the two determinants lie in $[(1-\kappa_0)^3,(1+\kappa_0)^3]$. \end{lemma} \begin{proof} Only the three named constraints involve site variables. For $k'$ the hypothesis of Proposition~\ref{prop:goodside} holds, its inside links in $B(y)$ being the $u_a$, which $E$ keeps within $r$ of $I$; so $\ell'_0\mapsto V^\sharp_{k'}$ has the single regular preimage $\Gamma'$, the Haar disintegration of $\ell'_0$ along it is the atom divided by the determinant, and the determinant bounds are those of the proposition; likewise for $k''$. The Haar disintegration of $(\ell,u,w)$ along $V^\sharp_k$ is $J_{22}^{-1}\mathrm{vol}_{M_k}$ by the coarea formula. Since $DV^\sharp_kDV^{\sharp*}_k$ is the sum over the variables of the nonnegative terms $D_eV^\sharp_k(D_eV^\sharp_k)^*$, it dominates the sum over the eight private links, which is at least $\sigma_0^2$ by Lemma~\ref{lem:privatesubmersion}. The product of the three disintegrations is the disintegration of the product Haar measure along the three constraints, and the Wilson weight and $\mathbf 1_E$ multiply it. \end{proof} The constraint of the site's own pair is thus a fixed measure on $M_k\cap E$: the exterior enters only through $S$, through the positions $\Gamma'$, $\Gamma''$ of two atoms, and through two densities on the good side, whose variation the next lemma bounds. \begin{lemma}[the good-side density is log-Lipschitz]\label{lem:goodsidedensity} Under the hypothesis of Proposition~\ref{prop:goodside}, let one link of the pair move along a unit-speed geodesic. Then $\bigl|\frac d{dt}\log\lvert\det D_gV\rvert\bigr|\le64.94$ when the link is a non-path link and $\le32.47$ when it is an inside link. \end{lemma} \begin{proof} By the proof of Proposition~\ref{prop:goodside}, $\lvert\det D_gV\rvert= \lvert\det T\rvert$ with $T=I-\frac78E_N\bar L$, so $\frac d{dt}\log \lvert\det T\rvert=\operatorname{tr}(T^{-1}\dot T)$, which is at most $3\lVert T^{-1}\rVert\lVert\dot T\rVert$ on the three-dimensional $\g$, with $\lVert T^{-1}\rVert\le(1-\kappa_0)^{-1}$ and $\dot T=-\frac78(\dot E_N\bar L +E_N\dot{\bar L})$, $\lVert\bar L\rVert\le\Lambda_0:=\theta_0/\sin\theta_0 =1.02115$, $\lVert E_N\rVert\le1$. For a power series $\varphi$ whose radius of convergence exceeds the spectral radius of a normal operator $A$ with spectral projections $P_i$ and eigenvalues $\lambda_i$, the derivative is $d\varphi(A)[B]=\sum_{i,j}\varphi[\lambda_i,\lambda_j]P_iBP_j$, with the divided differences $\varphi[\lambda_i,\lambda_j]$ of Daleckii--Krein~\cite{DK65}; the blocks $P_iBP_j$ are orthogonal for the Hilbert--Schmidt product, so $\lVert d\varphi(A)[B]\rVert\le\max_{i,j} \lvert\varphi[\lambda_i,\lambda_j]\rvert\lVert B\rVert_{HS}$, and by the Hermite--Genocchi formula $\varphi[\lambda_i,\lambda_j]=\int_0^1\varphi'( (1-u)\lambda_i+u\lambda_j)\,du$ this maximum is at most the maximum of $\lvert\varphi'\rvert$ on the convex hull of the spectrum. With $E_Y=e(\operatorname{ad}Y)$, $L_Y=f(\operatorname{ad}Y)$, $e(x)=(e^x-1)/x$, $f(x)=x/(e^x-1)$, and $\lVert\operatorname{ad}W\rVert_{HS}=2\lVert W\rVert$, this gives $\lVert\dot L_{Y}\rVert\le2M_L\lVert\dot Y\rVert$ and $\lVert\dot E_N\rVert\le2M_E\lVert\dot N\rVert$, where $M_L$ and $M_E$ bound $\lvert f'\rvert$ and $\lvert e'\rvert$ on the segments $i[-a',a']$, $a'\le\sqrt2\cdot16r\le\frac{\sqrt2}2$, and $i[-b',b']$, $b'\le\sqrt2\cdot14r\le0.6188$, which contain the spectra $\{0,\pm ia'\}$ of $\operatorname{ad}Y_x$ and $\{0,\pm ib'\}$ of $\operatorname{ad}N$. Since $e'(z)=\int_0^1te^{tz}\,dt$, $\lvert e'(iy)\rvert\le\frac12$, so $M_E=\frac12$. Since $f'(iy)=-\frac12-i\frac{\sin y-y}{4\sin^2(y/2)}$, and $y-\sin y\le \frac{y^3}6$, $\sin\frac y2\ge\frac y2(1-\frac{y^2}{24})$, $M_L\le\bigl(\frac14+\frac{a^2}{36}(1-\frac{a^2}{24})^{-4}\bigr)^{1/2}\le 0.5149$ at $a=\frac{\sqrt2}2$. Along the motion the fixed point $X$ moves with $\lVert\dot X\rVert\le\dot x$, $\dot x=1.1986$ for a non-path link and $0.5993$ for an inside link, by the proof of Proposition~\ref{prop:goodside}. For a path that does not contain the moving link, $\dot Y_x=-L_{Y_x}E_{-X}\dot X$, so $\lVert\dot Y_x\rVert\le \Lambda_0\dot x$; for a path that contains it, $\log(e^Ye^{t\zeta'})=Y+ tL_{-Y}\zeta'+O(t^2)$ adds $\Lambda_0$. With $n=2$ moving paths for a non-path link and $n=1$ for an inside link, $\lVert\dot N\rVert\le \frac{\Lambda_0}{16}(14\dot x+n)$ and $\lVert\dot{\bar L}\rVert\le2M_L \Lambda_0(\dot x+\frac n{14})$. Hence \[ \Bigl|\frac d{dt}\log\lvert\det T\rvert\Bigr|\le\frac{21}8\, \frac{2M_E\Lambda_0\lVert\dot N\rVert+\lVert\dot{\bar L}\rVert}{1-\kappa_0}, \] which is at most $64.94$ and $32.47$ in the two cases, with $(1-\kappa_0)^{-1}\le9.390$. \end{proof} \begin{theorem}[the site of a bad pair responds boundedly]\label{thm:site} Under the hypotheses of Lemma~\ref{lem:sitelaw}, let $\sigma,\sigma'$ differ only at one link $e'$ outside $\mathcal S_k$, and let $\pi(\cdot\mid\sigma)$ be the law of the site of that lemma. Then \[ W_1\bigl(\pi(\cdot\mid\sigma),\pi(\cdot\mid\sigma')\bigr)\le C_{e'}\, d(\sigma_{e'},\sigma'_{e'}), \] for the cost $\sum_{e\in\mathcal S_k}\delta$, with $C_{e'}=24\,n_p(e') \beta_W$ if $e'$ enters only the Wilson weight, $n_p(e')\le6$ the number of its plaquettes that contain a site link, $C_{e'}=197\beta_W+2206$ if $e'$ is a non-path link of $k'$ or $k''$, and $C_{e'}=87\beta_W+1103$ if $e'$ is an inside link of $k'$ in $B(y-e_\mu)$ or of $k''$ in $B(y+2e_\mu)$. The constants do not depend on $c_k$, $c_{k'}$, $c_{k''}$, $E$ or the rest of $\sigma$, and every branch of $V^\sharp_k=c_k$ inside the site is integrated, not compared. \end{theorem} \begin{proof} Move $\sigma_{e'}$ along a unit-speed geodesic, $0\le t\le T=d(\sigma_{e'}, \sigma'_{e'})$; geodesic balls of radius $r\le\frac1{32}$ are convex, so the hypotheses of Lemma~\ref{lem:sitelaw} hold along the way. By that lemma, $\pi_t=(\Phi_t)_*\mu_t$ with $\Phi_t(\ell,u,w)=(\ell,u,w,\Gamma'_t,\Gamma''_t)$ and $\mu_t$ the law on $M_k\cap E$ with density $h_t=e^{-\beta_WS_t}J'^{-1}_tJ''^{-1}_t$ against the fixed measure $\mathbf 1_EJ_{22}^{-1}\mathrm{vol}_{M_k}$; plaquettes without a site link are constant on $M_k$ and cancel. For $f$ Lipschitz with constant one for the site cost, whose oscillation is at most the diameter $24\cdot2\sqrt2$, \[ \pi_T(f)-\pi_0(f)=(\mu_T-\mu_0)(f\circ\Phi_T)+\mu_0(f\circ\Phi_T-f\circ \Phi_0). \] The second term is at most $\dot x\,T$ by Proposition~\ref{prop:goodside}, since one atom moves, and it vanishes when $e'$ enters only the Wilson weight. For the first, $\frac d{dt}\mu_t(\varphi)=\Cov_{\mu_t}(\varphi, \partial_t\log h_t)$ and $\lvert\Cov(\varphi,\psi)\rvert\le\frac14 \operatorname{osc}\varphi\operatorname{osc}\psi$, so the first term is at most $24\sqrt2\,T\sup\lvert\partial_t\log h_t\rvert$, and $\lvert\partial_t\log h_t\rvert\le\beta_W\lvert\partial_tS_t\rvert+ \lvert\partial_t\log J'_t\rvert$. Links of a plaquette moving at total geodesic speed $v$ give $\lvert\dot s_p\rvert\le v/\sqrt2$, by the proof of Lemma~\ref{lem:wilsononly}. If $e'$ enters only the Wilson weight, $\operatorname{osc}\partial_tS_t\le\sqrt2\,n_p(e')$ and $J'$ does not move, which gives $24n_p\beta_W$. If $e'=\ell'_a$, it shares at most one plaquette with a site link, and the atom $\Gamma'$ moves at speed at most $1.1986$ through its six plaquettes, so $\lvert\partial_tS_t\rvert\le(1+6 \cdot1.1986)/\sqrt2\le5.793$, and Lemma~\ref{lem:goodsidedensity} bounds the density term by $64.94$; hence $C_{e'}\le24\sqrt2(5.793\beta_W+64.94)+ 1.1986<197\beta_W+2206$. If $e'=u'_a$, it shares no plaquette with a site link, the atom moves at speed at most $0.5993$, $\lvert\partial_tS_t\rvert \le6\cdot0.5993/\sqrt2\le2.543$, and $C_{e'}\le24\sqrt2(2.543\beta_W+32.47) +0.5993<87\beta_W+1103$. Kantorovich duality gives the bound on $W_1$. \end{proof} \emph{The eight private links alone.} The fold of Proposition~\ref{prop:fold} is also reached by moving one inside link, and the law of the eight private links restricted to a fixed set of non-path links then has no response bound that holds for every such set. The direction matters: a link $(x,\mu)$ with $o_\mu=0$ is a tree link when $o_\nu=0$ for every $\nu>\mu$, so a block has $p_\mu=8-2^\mu$ active inside $\mu$-links, $7$, $6$, $4$ and $0$ for $\mu=0,1,2,3$. For $\mu=3$ every $u_a$ and $w_a$ is a tree link, the law of the eight private links does not depend on them, and Lemma~\ref{lem:wilsononly} is the whole answer. For the other three directions the next two results show that the answer depends on the set over which the non-path links are integrated. \begin{proposition}[an inside link at the fold]\label{prop:insidefold} Take the configuration of Proposition~\ref{prop:fold} for a direction $\mu\le2$, with $c_k=c_{\gamma^*}$ and the offsets chosen so that the inside link $u_b$ of one of the six non-path links at the angle $\frac{49}{100}$ is active. Replace $u_b$ by $u_b(t)=\operatorname{diag} (e^{it},e^{-it})$, keep every other link, and let $\pi^{(t)}$ be the law~\eqref{eq:pathlaw}. There is $\tau>0$ such that for $0<|t|<\tau$ every preimage of $c_k$ lies on the torus and is regular; for $t>0$ there is one, $g_L(t)$, and for $t<0$ there are three, $g_L(t)$ and $g_\pm(t)= e^{iq_\pm(t)}$ with $q_\pm(t)\to q^*$, where $g_L(t)\to g_L^*$. The weight of $g_L(t)$ tends to that of $g_L^*$, at most $2.78$, and the weights of $g_\pm(t)$ tend to infinity as $t\uparrow0$, for every $\beta_W\ge0$. Hence $\pi^{(t)}\to\delta_{g^*}$ as $t\uparrow0$ and $\pi^{(t)}\to \delta_{g_L^*}$ as $t\downarrow0$, and for every $\beta_W\ge0$ and $\varepsilon>0$ there are $t<00$, the real part of $g^{-1}e^{i\theta}$ is $\varrho\cos(\theta-q)$, and $\delta(h,I)$ decreases in the real part of $h$; so the order of the distances $\delta(v_x,I)$ is the order of the numbers $|\theta-q|$, for every such $g$ and not only on $T$. \emph{Localisation.} $V$ is jointly continuous in $(g,t)$ and $\SU(2)$ is compact, and by Proposition~\ref{prop:fold}(i) the preimages of $c_k$ at $t=0$ are $g_L^*$ and $g^*$. So for open neighbourhoods $U_L$ of $g_L^*$ and $U^*$ of $g^*$ there is $\tau>0$ such that every preimage at $|t|<\tau$ lies in $U_L\cup U^*$; take $\tau\le10^{-3}$. \emph{Near $g^*$.} For $q\in[0.05,0.1]$ and $|t|\le10^{-3}$, $(\frac12+q)- (\frac{49}{100}+|t|-q)\ge0.109$, so the far link is the strict maximiser, $m=2\sqrt2\sin\frac{q+1/2}2\in(0.76,0.84)\subset(\frac12,1)$, and $V$ is real-analytic in $(g,t)$ near $(g^*,0)$. On $T$ its angle is \begin{gather*} f(q;t)=f_0(q)+\frac{\chi_0(q)}{16}\,t ,\\ \chi_0(q)=2-4\sqrt2\sin\tfrac{q+1/2}2\in(0.32,0.48), \end{gather*} since only the angle sum changes, by $t$. Since $|\chi_0''|=\sqrt2\sin \frac{q+1/2}2$ is less than $\frac12$, the second derivative of $f$ exceeds $4.6-10^{-4}$ on $[0.05,0.1]$, and by Step~1 of the proof of Proposition~\ref{prop:fold} both $f(0.05;t)$ and $f(0.1;t)$ exceed $\gamma^*+10^{-3}-\frac{|t|}{32}>\gamma^*$. Hence for $t>0$, $f(q;t)>f_0(q)\ge\gamma^*$ on $[0.05,0.1]$ and there is no torus preimage in $U^*$, and for $t<0$, $f(q^*;t)<\gamma^*$ and strict convexity give exactly two, $q_-(t)\frac13$ by Step~2 of the proof of Proposition~\ref{prop:fold}. So $\Xi_t>0$ near $(q^*,0,0)$ and $V_k(q,s;t)>0$ for small $s>0$: shrinking $U^*$ and $\tau$, there is no preimage off $T$ in $U^*$. As in Step~3 of that proof, $\lvert\det D_gV\rvert=|f'|\lambda_\perp^2$ at torus points, with $\lambda_\perp=\Xi_t(q,0)\le1$; so the weights of $g_\pm(t)$ are at least $e^{-12\beta_W}/|f'(q_\pm(t);t)|\to\infty$. \emph{Near $g_L^*$.} For $q\in[-0.0182,-0.0146]$ and $|t|\le10^{-3}$, the six near paths are farther from $g$ than the far link by at least $(\frac{49}{100}-q-10^{-3})-(\frac12+q)\ge0.0182$ in angle, and their chordal distance $2\sqrt2\sin\frac{\theta}2$, $\theta\in[0.503,0.510]$, lies in $(0.70,0.72)\subset(\frac12,1)$. Let $\Psi_+$ and $\Psi_-$ be the cut-off averages computed with the maximum taken, respectively, at the path from $x_b$, of angle $\frac{49}{100}+t$, and at an unmoved near path, of angle $\frac{49}{100}$. Both are real-analytic in $(g,t)$ near $(g_L^*,0)$, and both equal $V(\cdot;0)$ near $g_L^*$ at $t=0$, when the two paths coincide. The difference of the real parts of the two path variables is \[ \varrho\bigl(\cos(\tfrac{49}{100}-q)-\cos(\tfrac{49}{100}+t-q)\bigr) =2\varrho\sin(\tfrac{49}{100}+\tfrac t2-q)\sin\tfrac t2 , \] and $\sin(\frac{49}{100}+\frac t2-q)>0.48$, so near $g_L^*$ the moved path is the maximiser for $t>0$ and an unmoved one for $t<0$: $V=\Psi_+$ for $t\ge0$ and $V=\Psi_-$ for $t\le0$. Since $D_g\Psi_\pm(g_L^*,0)=D_gV(g_L^*;0)$ is invertible, with determinant $f_0'(q_L^*)\lambda_\perp(q_L^*)^2>0$, the implicit function theorem gives a closed ball $\bar U_L$ about $g_L^*$ and $\tau>0$ such that each equation $\Psi_\pm(g;t)=c_k$, $|t|<\tau$, has exactly one solution in $\bar U_L$, at which $D_g\Psi_\pm$ is invertible. Conjugation by $T$ fixes $g_L^*$ and maps $\bar U_L$ onto itself, so the solution is fixed by it and lies in $T$. This is the one preimage $g_L(t)$ in $U_L$, for either sign of $t$; it is regular, it tends to $g_L^*$, and its weight tends to that of $g_L^*$, which is at most $2.78$ by Proposition~\ref{prop:fold}(ii). \emph{The limits.} For $t>0$, $\pi^{(t)}=\delta_{g_L(t)}\to\delta_{g_L^*}$. For $t<0$, with $f=\delta(\cdot,g^*)$ and $w_\pm$ the weights of $g_\pm(t)$, $\pi^{(t)}(f)\le2.78(1+o(1))\,2\sqrt2/(w_++w_-)+\max_\pm\delta(g_\pm(t),g^*)\to0$, so $\pi^{(t)}\to \delta_{g^*}$. As $\delta(g^*,g_L^*)\ge0.1268$ by Proposition~\ref{prop:fold}(iii), Kantorovich duality with $f$ gives the last statement; $t\mapsto e^{it}$ has geodesic speed $\sqrt2$. \end{proof} So the path-link law responds to an inside link as it does to the non-path links in Proposition~\ref{prop:fold}(iv). The law of all eight private links is the one that the bad side asks about, and the next theorem shows that its response to an inside link is not bounded independently of the set $B$ of non-path links over which it is restricted, in contrast with Lemma~\ref{lem:wilsononly}. \begin{theorem}[no bound uniform in the restricting set]\label{thm:eightlinkuniform} In the configuration of Proposition~\ref{prop:insidefold}, let $z_0$ be its seven non-path links, let $B_\varepsilon$ be the set of $z$ with $d(\ell_a,\ell_a(z_0))<\varepsilon$ for $a\ne0$, and let $Q_{t,\varepsilon}$ be the law of the eight private links given $u_b=u_b(t)$, the other links, $V^\sharp_k=c_k$ and $z\in B_\varepsilon$, that is $\frac1Z\mathbf 1_{B_\varepsilon}J_8^{-1}e^{-\beta_WS}\mathrm{vol}_M$. For every $\beta_W \ge0$ and $L>0$ there are $t<00$ such that \[ W_1\bigl(Q_{t,\varepsilon},Q_{t',\varepsilon}\bigr)\ge0.12 \] for the cost $\sum_a\delta(\ell_a,\ell'_a)$. For $\varepsilon<0.4$ the set $B_\varepsilon$ lies in the bad set $\{\max_{a\ne0}d(\ell_a,c_k)>r\}$ for every $r\le\frac1{32}$. So there is no constant $C$, for any $\beta_W$, with $W_1\le C\,d(u_b,u'_b)$ for the law of the eight private links restricted to every fixed measurable set of non-path links. \end{theorem} \begin{proof} \emph{Step 1, the fibre over $B_\varepsilon$.} Fix $t$ with $0<|t|<\tau$, and let $g_1,\dots,g_n$ be the preimages at $z_0$ of Proposition~\ref{prop:insidefold}, all regular. By compactness and continuity, for small $\varepsilon$ every solution of $V(g,z)=c_k$ with $z\in B_\varepsilon$ has $g$ in one of disjoint balls $U_i$ about the $g_i$. At $g_\pm(t)$ ($t<0$) the far link is the strict maximiser, and at $g_L(t)$ for $t>0$ the moved path is, by the margins of that proof; there $V$ is real-analytic in $(g,z)$ near $(g_i,z_0)$, and the implicit function theorem gives a smooth $G_i$ on $B_\varepsilon$ with $M\cap(U_i\times B_\varepsilon)=\{(G_i(z),z)\}$. At $g_L(t)$ for $t<0$ the maximum at $(g_i,z_0)$ is attained by the six functions $m_a=\delta(g^{-1} \ell_a,I)$ of the near offsets $a$ (for $a=b$ through the path from $x'_b$) and by no other path, since the moved path is shorter by a margin proportional to $|t|$ and the far link by $0.0182$. Near $(g_i,z_0)$, $V$ equals at each point one of the six real-analytic functions $\Psi_a=g\exp( \phi(m_a)N)$, one with the largest $m_a$, and at $z=z_0$ the six coincide as functions of $g$. In the chart $F(X,z)=\log(c_k^{-1}V(g_ie^X,z))$, with $F_a$ defined from $\Psi_a$ in the same way, the derivatives $D_XF_a(0,z_0)$ are therefore one invertible map $A_i$, and the function $s\mapsto F(X+s(X'-X),z)$ is a continuous selection of finitely many $C^1$ functions; at a point of differentiability that is not an isolated point of any of the closed sets where it equals one of them, its derivative equals theirs, and the exceptional points are countable. Hence, for $\lVert X\rVert,\lVert X' \rVert\le\varrho$ and $z\in B_\varepsilon$, \[ \bigl\lVert F(X',z)-F(X,z)-A_i(X'-X)\bigr\rVert\le\eta\,\lVert X'-X\rVert, \] with $\eta=\eta(\varrho,\varepsilon)\to0$, as every $D_X F_a$ is within $\eta$ of $A_i$ there. With $\eta\lVert A_i^{-1}\rVert\le\frac12$ and $\varepsilon$ small, $X\mapsto X-A_i^{-1}F(X,z)$ is a contraction of the $\varrho$-ball into itself, and $V(\cdot,z)=c_k$ has exactly one solution $G_i(z)$ there, Lipschitz in $z$ with constant at most $2\lVert A_i^{-1} \rVert\operatorname{Lip}_zV$. In all cases $M\cap(\SU(2)\times B_\varepsilon)$ is the disjoint union of the graphs $\Gamma_i$ of the $G_i$, and $\lvert\det D_gV\rvert$, wherever $V$ is differentiable on $\Gamma_i$, lies within a factor $1\pm\eta'$ of $\lvert\det A_i\rvert$, $\eta'\to0$ with $\varrho,\varepsilon$. \emph{Step 2, the density on a graph.} At a point of $\Gamma_i$ where $V$ is differentiable and $G_i$ is differentiable at $z$, write $DV=(A\ B)$ with $A=D_gV$, $B=D_zV$; then $DG_i=-A^{-1}B$, and by the determinant identity $\det(I+PQ)=\det(I+QP)$, \[ J_8=\det(AA^*+BB^*)^{1/2}=\lvert\det A\rvert\det(I+DG_i^*DG_i)^{1/2}. \] The second factor is the area element of the Lipschitz graph $\Gamma_i$ over $z$, so $J_8^{-1}\mathrm{vol}_M=\lvert\det D_gV\rvert^{-1}dz$ on $\Gamma_i$, $dz$ Haar measure on the seven links carried by $G_i$, provided the two differentiability conditions hold for almost every $z$. For $G_i$ this is Rademacher's theorem. $V$ is differentiable at $(G_i(z),z)$ except where two of the six $m_a$ tie, and for $a\ne a'$ the tie set is null: write $\ell_a=he^{is}$ along the circles $\ell_aT$, along which Haar measure is uniform, fix the other coordinates, and let $\psi(s)$ be $m_a-m_{a'}$ at $(G_i(z_s),z_s)$. Its almost-everywhere derivative is at least $\partial_sm_a-\lVert D_gm_a-D_gm_{a'}\rVert\, \sqrt2\operatorname{Lip}G_i$; here $\partial_sm_a=\sqrt2\cos\frac\theta2$ on $T$, with $\theta$ the angle of $g^{-1}\ell_a$, which is at least $1.36$ at $(g_i,z_0)$ and exceeds $1.3$ near it, while $D_gm_a-D_gm_{a'}$ vanishes at $z=z_0$, where $\ell_a=\ell_{a'}$. So $\psi$ is strictly increasing for small $\varrho, \varepsilon$, has one zero on each circle, and the tie set is null by Fubini, the circles meeting $B_\varepsilon$ in arcs. \emph{Step 3, the limit $\varepsilon\to0$.} By Steps 1 and 2, \[ Q_{t,\varepsilon}(\varphi)=\frac{\sum_i\int_{B_\varepsilon}\varphi(G_i(z), z)\,e^{-\beta_WS}\lvert\det D_gV\rvert^{-1}\,dz}{\sum_i\int_{B_\varepsilon} e^{-\beta_WS}\lvert\det D_gV\rvert^{-1}\,dz}, \] the integrands evaluated at $(G_i(z),z)$. As $\varepsilon\to0$, $G_i(z)\to g_i$ uniformly on $B_\varepsilon$ and the weights converge uniformly to those of $g_i$ in $\pi^{(t)}$, so $Q_{t,\varepsilon}(\varphi)\to\pi^{(t)} (\varphi(\cdot,z_0))$ for every continuous $\varphi$. \emph{Step 4, the gap.} Let $\varphi(\ell)=\delta(\ell_0,g^*)$, which is $1$-Lipschitz for the cost. By Proposition~\ref{prop:insidefold} choose $t<00.45$ for the near links, so $B_\varepsilon$ lies in the bad set for $\varepsilon<0.4$. \end{proof} The proof uses that $B_\varepsilon$ is chosen after $t$ and $t'$: the boundary of $B_\varepsilon$ meets the fibre near a fold of its projection to the non-path links, which is the tangency named after Proposition~\ref{prop:fold}. On a bad block the labels of Sec.~\ref{ssec:factorisation} restrict the non-path links to two sets only, and neither has its boundary near a fold. \begin{proposition}[the sets of the labels]\label{prop:eightlabels} Let $k=(y,\mu)$, $r\le\frac1{32}$, and fix every link other than the eight private links of $k$. The event that the labels of a set of blocks containing $y$ take given values, with $y$ bad, restricts $z$ to $\SU(2)^7$ if $y$ is bad through a coarse plaquette or through a non-path active link that is not private to $k$, and to \[ B^{\mathrm{bad}}_r=\Bigl\{z:\max_{a\ne0}d(\ell_a,c_k)>r\Bigr\} \] otherwise. In the second case $d(u_a,I),d(w_a,I)\le r$ for $a\ne0$, and over the shell $\{z:r\le\max_{a\ne0}d(\ell_a,c_k)\le\frac1{32}\}$, which contains the boundary of $B^{\mathrm{bad}}_r$, the fibre $M$ is the graph of one smooth map $z\mapsto g(z)$, with $\lvert\det D_gV\rvert^{-1}\in [0.147,828]$ and $J_8^{-1}\mathrm{vol}_M=\lvert\det D_gV\rvert^{-1}dz$. \end{proposition} \begin{proof} The private links of $k$ are non-path active links of $k$ only, so they enter the label of $y$ and of no other block, and the label of $y$ is the union of the events $G_e^c$ over the non-path active links $e$ of the four pairs based at $y$ and of the coarse events at the corner $y$. Given the other links, this union is everything when one of its other members holds, and is $B^{\mathrm{bad}}_r$ otherwise; in the second case the fourteen inside $\mu$-links of $k$, which are non-path active links of $k$ or tree links, are within $r$ of $I$. Over the shell the hypothesis of Proposition~\ref{prop:goodside} holds, so $V=c_k$ has exactly one solution, regular, with the stated density, and it is smooth in $z$ by the implicit function theorem, the cut-off being one; Step~2 of the proof of Theorem~\ref{thm:eightlinkuniform} gives the last identity. \end{proof} \emph{The relative chart.} The folds of Proposition~\ref{prop:insidefold} and Theorem~\ref{thm:eightlinkuniform} are folds of the projection of the fibre $M=\{V^\sharp_k=c_k\}$ to the seven non-path links. In the coordinates $g=\ell_0$ and $\xi_a=g^{-1}\ell_a$, $a\ne0$, the fibre is a single graph over $\xi$ as soon as the inside links of $B(y)$ are within $\frac1{32}$ of $I$, and the response of the law of the eight private links to an inside link reduces to one condition on a Jacobian. In these coordinates the paths are \[ v_{x_a}=g^{-1}u_ag\,\xi_a,\qquad v_{x'_a}=\xi_aw_a,\qquad v_{2y}=I, \] so $V^\sharp_k=g\,E(g,\xi)$ with $E=\exp(\chi N)$, $\chi=\phi(m)$, $m=\max_x\delta(v_x,I)$ and $N=\frac1{16}\sum_x\log v_x$, and $E$ depends on $g$ only through the conjugates $\tilde u_a=g^{-1}u_ag$. For fixed $g$ the map $(\ell_a)_{a\ne0}\mapsto\xi$ preserves Haar measure. With $\alpha=2\arcsin\frac1{2\sqrt2}$ and $\Lambda=\alpha/\sin\alpha$ as in Sec.~\ref{ssec:smallfibres}, put \begin{gather*} \lambda=\frac1{32}\Bigl(2\sqrt2\cdot\frac78\sqrt2\,\alpha+\frac7{16} \sqrt2\,\Lambda\Bigr)<0.1002,\\ L=\frac{2\cdot\frac78\sqrt2\,\alpha+\frac1{16}\Lambda}{1-\lambda}<2.064 , \end{gather*} by $0.7227<\alpha<0.7228$, $\Lambda<1.0927$ and $\sqrt2<1.41422$. \begin{lemma}[the relative chart]\label{lem:relchart} Fix every link other than the eight private links of $k$, and suppose $d(u_a,I)\le\frac1{32}$ for $a\ne0$; the links $w_a$ are arbitrary. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For fixed $\xi$, $d(E(g,\xi),E(g',\xi))\le\lambda\,d(g,g')$, and $V^\sharp_k=c_k$ has exactly one solution $g=G(\xi)$, the fixed point of the contraction $g\mapsto c_kE(g,\xi)^{-1}$. \item If one link $u_b$ or $w_b$ moves from $e$ to $e'$, the $u_a$ staying within $\frac1{32}$ of $I$, then $d(G(\xi),G'(\xi))\le L\,d(e,e')$ for every $\xi$, and the eight private links $G(\xi)$, $G(\xi)\xi_a$ move by at most $8L\,d(e,e')$ in the cost $\sum_a\delta$. \item Wherever $g\mapsto F(g)=gE(g,\xi)$ is differentiable, its differential has singular values in $[1-\lambda,1+\lambda]$, so $D(\xi)=\lvert\det D_gF(G(\xi))\rvert\in[(1-\lambda)^3,(1+\lambda)^3] \subset[0.728,1.332]$. \item For almost every $c_k$ and every measurable set $B$ of non-path links, the law $\frac1Z\mathbf 1_BJ_8^{-1}e^{-\beta_WS}\mathrm{vol}_M$ of the eight private links is the image under $\xi\mapsto(G(\xi),G(\xi)\xi)$ of \begin{gather*} \hat Q(d\xi)=\frac1Z\,\mathbf 1_B(G(\xi)\xi)\,e^{-\beta_WS}\,D(\xi)^{-1} \,d\xi ,\\ Z=\int\mathbf 1_B(G(\xi)\xi)\,e^{-\beta_WS}D(\xi)^{-1}d\xi , \end{gather*} $S$ evaluated at $(G(\xi),G(\xi)\xi)$ and $d\xi$ Haar measure on $\SU(2)^7$. \end{enumerate} If every $u_a$ equals $I$, then $E$ does not depend on $g$, $D\equiv1$, and (ii) holds with $L$ replaced by $1.857$. If every $w_a$ equals $I$, the same holds with $u$ arbitrary in the coordinates $\xi_a=\ell_ag^{-1}$, in which $V^\sharp_k=\exp(\chi N')\,g$ with $N'=\frac1{16}\sum_x\log(P_x g^{-1})$, $P_{x_a}g^{-1}=u_a\xi_a$ and $P_{x'_a}g^{-1}=\xi_a$. More generally, put $\kappa=\frac74\sqrt2\,\alpha+\frac7{16}\Lambda$, so that $\kappa<2.2670$, and \[ \sigma_u=\max_{a\ne0}\lVert\operatorname{Ad}(u_a)-1\rVert,\qquad \sigma_w=\max_{a\ne0}\lVert\operatorname{Ad}(w_a)-1\rVert . \] If $\sigma_u<1/\kappa$, then (i)--(iv) hold for every $w$ with $\lambda$ replaced by $\lambda_u=\kappa\sigma_u$ and $L$ by $L_u=(\frac74\sqrt2\, \alpha+\frac1{16}\Lambda)/(1-\lambda_u)$, the range of $D$ being $[(1-\lambda_u)^3,(1+\lambda_u)^3]$, provided in (ii) that $\sigma_u<1/ \kappa$ along the move. If $\sigma_w<1/\kappa$, the same holds for every $u$ in the coordinates $\xi_a=\ell_ag^{-1}$, with $\sigma_w$ in place of $\sigma_u$; there $P_{x'_a}g^{-1}=\xi_a\,gw_ag^{-1}$. As $\lVert\operatorname {Ad}(u)-1\rVert=2\sin\theta$ for $u$ with eigenangles $\pm\theta$, $\theta\in[0,\pi]$, the first condition holds when $\sin\theta\le0.2205$ for every $u_a$, $a\ne0$, and $d(u_a,I)\le\frac1{32}$ gives $\lambda_u \le\lambda$. \end{lemma} \begin{proof} (i) Let $g_t=ge^{t\zeta}$, $\lVert\zeta\rVert=1$. If $d(u,I)=\sqrt2\, \theta$, then $\operatorname{Ad}(u)-1$ has the eigenvalues $0$ and $e^{\pm2i\theta}-1$, so its norm is $2\sin\theta\le\sqrt2\,d(u,I)$, and by bi-invariance and $d(e^A,e^B)\le\lVert A-B\rVert$, \begin{multline*} d(\tilde u_a(t),\tilde u_a(t'))=d\bigl(e^{(t'-t)\operatorname{Ad}(\tilde u_a^{-1})\zeta},e^{(t'-t)\zeta}\bigr)\\ \le\sqrt2\,d(u_a,I)\,|t'-t| . \end{multline*} So each $v_{x_a}$ moves at geodesic speed at most $\frac{\sqrt2}{32}$, and the other paths are fixed. Since $\delta(\cdot,I)$ is $1$-Lipschitz for $d$, $m$ moves at speed at most $\frac{\sqrt2}{32}$ and $\chi$ at most $\frac{2\sqrt2}{32}$. On an interval where $\chi>0$, every $\delta(v_x,I) <1$, so every eigenangle $\theta$ of $v_x$ has $2\sqrt2\sin\frac\theta2<1$, $\theta<\alpha$, $\lVert\log v_x\rVert=\sqrt2\,\theta<\sqrt2\,\alpha$, and the differential of $\log$ along the path has norm at most $\Lambda$; the two paths of offset zero contribute nothing, so $\lVert N\rVert\le \frac78\sqrt2\,\alpha$ and $N$ moves at speed at most $\frac7{16}\Lambda \frac{\sqrt2}{32}$. Hence $Y=\chi N$ moves at speed at most $\lambda$ there. The set where $\chi>0$ is open and $Y=0$ off it, so $Y$ is $\lambda$-Lipschitz along the whole geodesic, and $d(E(g),E(g'))\le \lVert Y(g)-Y(g')\rVert\le\lambda\,d(g,g')$. Since $V^\sharp_k=gE(g,\xi)$, the equation $V^\sharp_k=c_k$ reads $g=c_kE(g,\xi)^{-1}$, and $d(c_kE^{-1},c_kE'^{-1})=d(E,E')$: the map is a contraction of the complete space $\SU(2)$, with a unique fixed point. (ii) A unit-speed move of $u_b$ moves $\tilde u_b$ and $v_{x_b}$, and a move of $w_b$ moves $v_{x'_b}$, at speed one, and no other path; so $m$ moves at speed at most $1$, $\chi$ at most $2$, $N$ at most $\frac1{16}\Lambda$, and $Y$ at fixed $(g,\xi)$ at most $2\cdot\frac78\sqrt2\,\alpha+\frac1{16}\Lambda=(1-\lambda)L$. For the two fixed points, $d(G,G')\le d(E(G),E(G'))+(1-\lambda)L\,d(e,e')$, with $E,E'$ before and after the move at the same point, so $d(G,G')\le\lambda d(G,G')+(1-\lambda)L\,d(e,e')$. The geodesic ball of radius $\frac1{32}$ is convex, so the hypothesis holds along the move. The eight private links are $G$ and $G\xi_a$, each at distance $d(G,G')$ from its new value, and $\delta\le d$. (iii) The derivative of $F$ along $ge^{t\zeta}$, translated back to the identity, is $F^{-1}\dot F=\operatorname{Ad}(E^{-1})\zeta+E^{-1}\dot E$, and $E^{-1}\dot E=E_{-Y}\dot Y$ has norm at most $\lVert\dot Y\rVert\le \lambda$ by (i). So $D_gF=\operatorname{Ad} (E^{-1})(I+K)$ with $\lVert K\rVert\le\lambda$, and $\operatorname{Ad} (E^{-1})$ is an isometry. (iv) For fixed $\xi$, $(1-\lambda)d(g,g')\le d(F(g),F(g'))\le(1+\lambda) d(g,g')$ by (i) and bi-invariance, and $F$ is onto by (i), so $F$ is a bi-Lipschitz homeomorphism of $\SU(2)$ with inverse $c\mapsto G_c(\xi)$. The area formula gives $\int h(F(g))\lvert\det D_gF\rvert\,dg=\int h(c)\,dc$, so for bounded measurable $\psi$ and $f$, \begin{multline*} \int\!\!\int\psi(F(g,\xi))f(g,\xi)\,dg\,d\xi\\ =\int\psi(c)\int \frac{f(G_c(\xi),\xi)}{\lvert\det D_gF(G_c(\xi),\xi)\rvert}\,d\xi\,dc , \end{multline*} the Jacobian being defined at $(G_c(\xi),\xi)$ for almost every $\xi$ for almost every $c$, by Fubini and the area formula applied to the null set where $F$ is not differentiable. The inner integral is a disintegration of Haar measure on the eight links along $V^\sharp_k$; so is $J_8^{-1}\mathrm{vol}_M$ by the coarea formula, and two disintegrations agree for almost every $c$. Multiplying by $\mathbf 1_Be^{-\beta_WS}$ gives (iv). If every $u_a=I$, then $\tilde u_a=I$, $E=E(\xi)$, $\lambda$ may be taken zero and the speed bound of (ii) is $1.857$. If every $w_a=I$, multiplying the paths by $g^{-1}$ on the right and conjugating the logarithms by $g$ gives the stated form, in which $g$ enters only as the right factor; the same proof applies with $\lambda=0$, and right translation preserves Haar measure. For the last part, the hypothesis on $u$ enters the proof only through the speed of $\tilde u_a$ in (i), which is at most $\lVert\operatorname{Ad} (\tilde u_a^{-1})-1\rVert=\lVert\operatorname{Ad}(u_a)-1\rVert\le\sigma_u$, since $\operatorname{Ad}(\tilde u_a^{-1})$ is conjugate to $\operatorname {Ad}(u_a)^{-1}$ and $\operatorname{Ad}(u_a)$ is orthogonal. With this bound, $m$ moves at speed at most $\sigma_u$, $\chi$ at most $2\sigma_u$ and $N$ at most $\frac7{16}\Lambda\sigma_u$, so $Y$ is $\kappa\sigma_u $-Lipschitz in $g$, and (i)--(iv) follow as before. In the coordinates $\xi_a=\ell_ag^{-1}$, conjugating by $g$ gives $V^\sharp_k=\exp(\chi N')\,g$ with $N'=\frac1{16}\sum_x\log(P_xg^{-1})$, $P_{x_a}g^{-1}=u_a\xi_a$ and $P_{x'_a}g^{-1}=\xi_a\,gw_ag^{-1}$, and $m=\max_x\delta(P_xg^{-1},I)$; so $g$ enters $N'$ and $m$ only through $gw_ag^{-1}$, which moves at speed at most $\sigma_w$, and the fixed-point equation $g=E'(g,\xi)^{-1}c_k$, $E'=\exp(\chi N')$, is treated in the same way. Finally $\kappa<2.2670$ by $\alpha<0.7228$, $\Lambda<1.0927$ and $\sqrt2<1.41422$, and $2\cdot 0.2205\cdot2.2670<1$. \end{proof} The normaliser has a lower bound that does not depend on the chart. Let $S_{\mathrm{dep}}$ be the sum of $1-\frac12\ReTr U_p$ over the plaquettes that contain a private link of $k$; the other plaquettes do not depend on the eight links and cancel from the normalised law, so $S$ may be replaced by $S_{\mathrm{dep}}$, and $Z$ below is defined with it. \begin{lemma}[the normaliser]\label{lem:relnorm} There are $36$ plaquettes that contain a private link of $k$, so $0\le S_{\mathrm{dep}}\le72$. For every $c_k$, every inside and exterior link and every $\beta_W\ge0$, \[ Z_{\SU(2)^7}\ge\tfrac12e^{-72\beta_W},\qquad Z_{B^{\mathrm{bad}}_r}\ge\bigl(\tfrac12-b_r\bigr)e^{-72\beta_W}, \] with $b_r=(2\vartheta-\sin2\vartheta)/(2\pi)$ and $\vartheta=r/\sqrt2$, where $Z_B=\int\mathbf 1_Be^{-\beta_WS_{\mathrm{dep}}}J_8^{-1}\mathrm{vol}_M$, and $b_r<2.3\cdot10^{-6}$ for $r\le\frac1{32}$. \end{lemma} \begin{proof} A private link $(x'_a,\mu)$ lies in two plaquettes of each plane $\{\mu, \nu\}$, at $x'_a$ and at $x'_a-e_\nu$, and the other $\mu$-link of exactly one of them, $(x'_a\pm e_\nu,\mu)$, is private. So the $48$ incidences cover $12$ plaquettes twice and $24$ once, $36$ in all, each term lying in $[0,2]$. Let $A$ be the set of configurations with $\ell_0=c_k$ and $\Re\Tr(c_k^{-1}u_1\ell_1)<0$. There $v_{x_1}=c_k^{-1}u_1\ell_1$ has $\delta(v_{x_1},I)^2=4-2\Re\Tr v_{x_1}>4$, so $m>1$ near $A$, the cut-off vanishes and $V^\sharp_k=\ell_0$ near $A$: the fibre contains $A$, and on it $D_{\ell}V^\sharp_k$ is the projection to the path link, so $J_8=1$ and $\mathrm{vol}_M$ is Haar measure in $\ell_1,\dots,\ell_7$. The map $\ell_1 \mapsto c_k^{-1}u_1\ell_1$ preserves Haar measure and $X\mapsto-X$ exchanges the signs of $\Re\Tr X$, so $A$ has Haar measure $\frac12$ in $(\ell_a)_{a \ne0}$; with $S_{\mathrm{dep}}\le72$ this is the first bound. A point of $A$ outside $B^{\mathrm{bad}}_r$ has $d(\ell_1,c_k)\le r$, a geodesic ball whose Haar measure is $\frac2\pi\int_0^\vartheta\sin^2t\,dt=b_r$, the Haar density of the eigenangle $t\in[0,\pi]$ being $\frac2\pi\sin^2t$ and $d=\sqrt2\,t$. Finally $2\vartheta-\sin2\vartheta\le\frac43\vartheta^3$ gives $b_r\le\frac{2\vartheta^3}{3\pi}\le\frac2{3\pi(32\sqrt2)^3}<2.3\cdot 10^{-6}$. \end{proof} \begin{theorem}[the response in the relative chart]\label{thm:relresponse} Let $k$ be as in Lemma~\ref{lem:relchart}, let $B$ be $\SU(2)^7$ or $B^{\mathrm{bad}}_r$ with $r<\frac1{32}$, and in the second case suppose also $d(w_a,I)\le\frac1{32}$ for $a\ne0$. Let one of the links $u_b$, $w_b$, $b\ne0$, move from $e$ to $e'$, the hypotheses holding along the geodesic, and write $Q,Q'$ for the two laws of the eight private links, $\hat Q,\hat Q'$, $D,D'$ for the objects of Lemma~\ref{lem:relchart}(iv). Suppose that \[ \textup{(J)}\qquad\int\bigl\lvert\log D-\log D'\bigr\rvert\,d\hat Q\le K\,d(e,e') \] for every such pair along the geodesic. Then, for the cost $\sum_a\delta$, \begin{multline*} W_1(Q,Q')\le\bigl(16.52+2898\beta_W+41.37K\\ +238.1\,\iota\,e^{72\beta_W}\bigr)\,d(e,e'), \end{multline*} with $\iota=0$ for $B=\SU(2)^7$ and $\iota=1$ for $B=B^{\mathrm{bad}}_r$. If every $u_a$ equals $I$ and $e$ is a link $w_b$, or every $w_a$ equals $I$ and $e$ is a link $u_b$, then \textup{(J)} holds with $K=0$, and for $B=\SU(2)^7$ the bound holds with $u$, respectively $w$, otherwise arbitrary. For $B=\SU(2)^7$ and $\sigma_u<1/\kappa$ along the move, or $\sigma_w<1/\kappa$ in the coordinates $\xi_a=\ell_ag^{-1}$, with $\lambda_u,L_u$ of Lemma~\ref{lem:relchart} (respectively $\lambda_w,L_w$) and (J) in that chart, \[ W_1(Q,Q')\le\Bigl(8L_u+16\sqrt2\,R_u\bigl(24\sqrt2\,L_u\beta_W+K\bigr) \Bigr)\,d(e,e'), \] where $R_u=(1+\lambda_u)^3/(1-\lambda_u)^3$. \end{theorem} \begin{proof} Since $W_1$ is a metric, it suffices to prove the bound with an error $o(d(e,e'))$ for nearby pairs and to add along a subdivision of the geodesic. Couple $\hat Q$ and $\hat Q'$ so that $\xi=\xi'$ with probability $1-\mathrm{TV}(\hat Q,\hat Q')$, and use the maps of Lemma~\ref{lem:relchart}(iv): by (ii) of that lemma the cost is at most $8L\,d(e,e')<16.52\,d(e,e')$ where $\xi=\xi'$, and at most the diameter $8\cdot2\sqrt2$ otherwise. So $W_1\le16.52\,d(e,e')+16\sqrt2\, \mathrm{TV}(\hat Q,\hat Q')$, and with $f,f'$ the unnormalised densities, $\mathrm{TV}(\hat Q,\hat Q')\le\frac1Z\int\lvert f-f'\rvert\,d\xi$. \emph{The action.} The links $u_b,w_b$ share no plaquette with a private link, the two $\mu$-links of a plaquette having one $\mu$-coordinate, so $S_{\mathrm{dep}}$ changes only through the eight links, each of which moves by at most $L\,d(e,e')$. Each of the $48$ incidences of a private link with a plaquette contributes at most $\frac1{\sqrt2}$ times its speed, by the proof of Lemma~\ref{lem:wilsononly}, so $\lvert S_{\mathrm{dep}}- S'_{\mathrm{dep}}\rvert\le\frac{48L}{\sqrt2}d(e,e')<70.06\,d(e,e')$ at every $\xi$, and \[ \int_{B\cap B'}\frac{\lvert e^{-\beta_WS}-e^{-\beta_WS'}\rvert}{D'}\,d\xi \le\bigl(e^{70.06\beta_Wd(e,e')}-1\bigr)\Bigl(\frac{1+\lambda}{1-\lambda} \Bigr)^3Z , \] with $(\frac{1+\lambda}{1-\lambda})^3<1.828$ by Lemma~\ref{lem:relchart}(iii); here $B,B'$ denote the sets of $\xi$ with $G(\xi)\xi$, respectively $G'(\xi)\xi$, in $B$. \emph{The Jacobian.} With $\varDelta=\log D-\log D'$, $\lvert\varDelta\rvert \le3\log\frac{1+\lambda}{1-\lambda}$ and $\lvert e^{\varDelta}-1\rvert\le 1.828\lvert\varDelta\rvert$, so $\int_{B\cap B'}e^{-\beta_WS}\lvert D^{-1}- D'^{-1}\rvert\,d\xi\le1.828\,Z\int\lvert\varDelta\rvert\,d\hat Q\le1.828\, KZ\,d(e,e')$. \emph{The restricting set.} For $B=\SU(2)^7$ there is nothing to do. For $B=B^{\mathrm{bad}}_r$ let $\mathcal M(\xi)=\max_ad(G(\xi)\xi_a,c_k)$ and $\varepsilon=L\,d(e,e')$, with $r+\varepsilon\le\frac1{32}$. Then $\lvert \mathcal M-\mathcal M'\rvert\le d(G,G')\le\varepsilon$, so $B\setminus B'$ and $B'\setminus B$ consist of $\xi$ whose eight links, before respectively after the move, satisfy $r<\max_ad(\ell_a,c_k)\le r+ \varepsilon\le\frac1{32}$. On that shell, with every inside link within $\frac1{32}$ of $I$, Proposition~\ref{prop:goodside} with radius $\frac1{32}$ and the proof of Proposition~\ref{prop:eightlabels} show that the fibre is a graph over the non-path links with $J_8^{-1}\mathrm{vol}_M =\lvert\det D_gV\rvert^{-1}dz$, $\lvert\det D_gV\rvert^{-1}\le828$. Since $e^{-\beta_WS_{\mathrm{dep}}}\le1$ and Lemma~\ref{lem:relchart}(iv) identifies both disintegrations, \[ \int_{B\setminus B'}f\,d\xi+\int_{B'\setminus B}f'\,d\xi\le2\cdot828 \cdot7\cdot\frac{\varepsilon(r+\varepsilon)^2}{\sqrt2\,\pi}, \] since the Haar measure of the shell $\{s\tau$, and $1.989\,\sigma<0.088$. As $k(0)=h_2$ and $k(h_1)=0$, $h_1$ and $h_2$ have the same sign. If $h_1>0$, then $f^{(1)}>f^{(2)}$ at $G_1$, so $F=F_1$ near $G_1$ and $F(G_1)=c_k$; by uniqueness in Lemma~\ref{lem:relchart}(i), $G=G_1$ and $D=\lvert\det D_gF_1\rvert=D_1$, the determinant being positive since $\lVert A\rVert \le\lambda$. The case $h_2<0$ is the same with $F_2$. (ii) At $\xi_0$, $G_1=G_2=g_0$ and $Y_1=Y_2=Y$, while $D_gY_1= \phi'(f^{(1)})\,N\otimes\ell+\phi\,D_gN=-2N\otimes\ell+\phi\,D_gN$ and $D_gY_2=\phi\,D_gN$. Since $[Y,N]=0$, $\Phi(\operatorname{ad}Y)N=N$, so $1+A_1=(1+A_2)-2N\otimes\ell$, and the determinant of a rank-one perturbation gives $D_1=D_2(1-2\ell((1+A_2)^{-1}N))$. The operator $\operatorname{ad}Y$ is normal with eigenvalues $0,\pm i\rho$, and $\lvert \Phi(i\rho)\rvert=\lvert\sin\frac\rho2\rvert/\frac\rho2\le1$, so $\lVert \Phi(\operatorname{ad}Y)\rVert\le1$. By the proof of Lemma~\ref{lem:relchart}(i), $\lVert D_gN\rVert\le\frac7{16}\Lambda\sigma <0.02113$, $\lVert\ell\rVert\le\sigma$ and $\lVert N\rVert<0.8945$, so $\lvert\kappa_0\rvert\le0.044195\cdot0.8945/(1-0.02113)<0.04039$. (iii) Identify $\SU(2)$ with the unit quaternions, and write $\langle X, Y\rangle=-\Tr(XY)$ on $\mathfrak{su}(2)$ and $\Pi(z)=\frac12(z-z^*)$ for the projection of $2\times2$ matrices to $\mathfrak{su}(2)$, orthogonal for $\ReTr(X^*Y)$. For $z\in\SU(2)$, $\delta(z,I)^2=4-2\ReTr z$, so the derivative of $\delta(ze^{s\eta},I)$ at $s=0$ is $-\ReTr(z\eta)/\delta= \langle\Pi(z),\eta\rangle/\delta$. At fixed $g$, with $z_1=\tilde u_a \xi_a$ and $z_2=w_a\xi_a$, and since $\ReTr(\xi_ae^{s\eta}w_a)=\ReTr(w_a \xi_ae^{s\eta})$, the derivatives of $f^{(1)}$ and $f^{(2)}$ along $\eta_a$ are $p(\eta_a)=\langle\Pi(z_1),\eta_a\rangle/m$ and $q(\eta_a)= \langle\Pi(z_2),\eta_a\rangle/m$. If $z=\cos\theta+\sin\theta\,n$ with $n\in\mathfrak{su}(2)$, $n^2=-1$, then $\Pi(z)=\sin\theta\,n$ and $\lVert n\rVert=\sqrt2$, so $\lvert p\rvert=\sqrt2\sin\theta/(2\sqrt2\sin \frac\theta2)=\cos\frac\theta2=:\gamma$, and $\lvert q\rvert=\gamma$ as $z_2$ has the same $\delta$. As $m<1$, $\theta<\alpha$, so $\gamma^2> \cos^2\frac\alpha2=\frac12(1+\cos\alpha)=\frac78$, since $\cos\alpha=1-2 \sin^2\frac\alpha2=\frac34$. Put $\varpi=p-(1-2\kappa_0)\,q$. Since $\lvert p\rvert=\lvert q\rvert=\gamma$ and $\lvert\kappa_0\rvert<\frac12$, \[ \lvert\varpi\rvert\ge\bigl\lvert\,\lvert p\rvert-(1-2\kappa_0)\lvert q \rvert\,\bigr\rvert=2\gamma\lvert\kappa_0\rvert , \] whatever the angle between $p$ and $q$. Let $\eta_a$ be the unit vector dual to $\varpi/\lvert\varpi\rvert$, and $\eta_a=0$ if $\varpi=0$; then $(p-q)(\eta_a)+2\kappa_0\,q(\eta_a)=\varpi(\eta_a)=\lvert\varpi\rvert$. Differentiating $F_2(G_2(\xi),\xi)=c_k$, translated to the identity as in the proof of Lemma~\ref{lem:relchart}(iii), gives $(1+A_2)\,dG_2(\eta)= -\Phi(\operatorname{ad}Y)(-2q(\eta_a)N+\phi\,\partial N(\eta))$, where $\partial N$ is the derivative of $N$ in $\xi$ at fixed $g$. With $\ell'= \ell\circ(1+A_2)^{-1}\Phi(\operatorname{ad}Y)$, so that $\ell'(N)=\kappa_0$, \begin{align*} dh_2(\eta)&=\ell(dG_2(\eta))+(p-q)(\eta_a)\\ &=(p-q)(\eta_a)+2\kappa_0\,q(\eta_a)-\phi\,\ell'(\partial N(\eta)) . \end{align*} In the translated form the differential of $\log$ at $v=e^X$ is $\psi(\operatorname{ad}X)$ with $\psi(z)=z/(1-e^{-z})$, since $e^{-X} de^X=\Phi(-\operatorname{ad}X)\,dX$ and $\Phi(-z)\psi(z)=1$; and $\xi_be^{s\eta}w_b=\xi_bw_be^{s\operatorname{Ad}(w_b^{-1})\eta}$. So $\partial N(\eta)=\frac1{16}\sum_bT_b\eta_b$ with $T_b=\psi(\operatorname {ad}X_b)+\psi(\operatorname{ad}X'_b)\operatorname{Ad}(w_b^{-1})$, $X_b= \log v_{x_b}$, $X'_b=\log v_{x'_b}$. If $X$ has eigenangle $\vartheta< \alpha$, $\operatorname{ad}X$ is normal with eigenvalues $0,\pm2i\vartheta$, and $\psi(\pm2i\vartheta)=\vartheta e^{\pm i\vartheta}/\sin\vartheta$; so $\lVert\psi(\operatorname{ad}X)\rVert\le\Lambda$, and the symmetric part of $\psi(\operatorname{ad}X)$, which is also that of its transpose $\psi(-\operatorname{ad}X)$, has eigenvalues $1$ and $\vartheta\cot \vartheta\ge\alpha\cot\alpha=3\alpha/\sqrt7>0.8194$. Every path has $\delta<1$ at a tie at the top, so $\lVert T_a\rVert\le2\Lambda$. For $b\ne0$, $\xi_b=\tilde u_b^{-1}e^{X_b}$ and $\xi_bw_b=e^{X'_b}$, so $w_b^{-1}=e^{-X'_b}\tilde u_b^{-1}e^{X_b}$ and $\operatorname{Ad}(w_b^{-1}) =e^{-\operatorname{ad}X'_b}\operatorname{Ad}(\tilde u_b^{-1})e^{\operatorname {ad}X_b}$; with $\psi(z)e^{-z}=\psi(-z)$ this gives \[ T_b=S_b\,e^{\operatorname{ad}X_b},\qquad S_b=\psi(-\operatorname{ad}X_b)+ \psi(-\operatorname{ad}X'_b)\operatorname{Ad}(\tilde u_b^{-1}) . \] As $\lVert\operatorname{Ad}(\tilde u_b^{-1})-1\rVert=\lVert\operatorname {Ad}(u_b)-1\rVert\le\sigma$ by the proof of Lemma~\ref{lem:relchart}(i), \[ \langle S_b\zeta,\zeta\rangle\ge(2\cdot0.8194-1.0927\,\sigma)\lVert\zeta \rVert^2>1.5905\,\lVert\zeta\rVert^2 , \] and $e^{\operatorname{ad}X_b}$ is orthogonal, so $\lvert T_b^*\ell'\rvert= \lvert S_b^*\ell'\rvert\ge1.5905\,\lvert\ell'\rvert$ for every $b\ne0$, whatever the link $w_b$, since $\lvert S_b^*\ell'\rvert\,\lvert\ell'\rvert \ge\langle S_b\ell'^\sharp,\ell'^\sharp\rangle$. Fix $c\ne0,a$, take $\eta_b=0$ for $b\ne a,c$, and $\eta_c=0$ if $\ell'=0$, otherwise $\eta_c$ the vector dual to $-\ell'(T_a\eta_a)\,T_c^*\ell'/\lvert T_c^*\ell' \rvert^2$. Then $\ell'(\partial N(\eta))=0$, $\lVert\eta_c\rVert\le2\Lambda /1.5905<1.3741$ and $\lVert\eta\rVert<(1+1.3741^2)^{1/2}<1.6995$. Hence $dh_2(\eta)=\lvert\varpi\rvert\ge2\gamma\lvert\kappa_0\rvert$, and with $\gamma>(\frac78)^{1/2}>0.93541$, $2\lvert\kappa_0\rvert\le(1.6995/\gamma) \lvert\nabla h_2\rvert<1.8169\,\lvert\nabla h_2\rvert$. Finally $\lvert \log(1-x)\rvert\le\lvert x\rvert/(1-\lvert x\rvert)$ for $\lvert x\rvert <1$ and $2\lvert\kappa_0\rvert<0.0808$ give $\lvert\log D_1-\log D_2\rvert \le2.1758\,\lvert\kappa_0\rvert\le2.1758\cdot0.9085\,\lvert\nabla h_2 \rvert<1.9811\,\lvert\nabla h_2\rvert$. \end{proof} Near a tie at the top of index $a$ the ties of index $a$ are exactly the zeros of $h_2$, by (i). Let $Z_a$ be the set of those ties at the top of index $a$ at which $\nabla h_2\ne0$, a hypersurface of $\SU(2)^7$, and let $H^{20}$ be its Hausdorff measure for the metric above, under which $\SU(2)$ has volume $V=4\sqrt2\,\pi^2$. Put \[ \mathcal A_{\mathrm{tie}}=\sup V^{-7}H^{20}(Z_a), \] the supremum over $a\ne0$, over $c_k$, over links $u_b$ within $\frac1{32}$ of $I$ and over arbitrary links $w_b$. \begin{proposition}[the Jacobian across the ties]\label{prop:jreduce} Let $k$, $B$ and the move $e\to e'$ be as in Theorem~\textup{\ref{thm:relresponse}}. Then \[ \int\bigl\lvert\log D-\log D'\bigr\rvert\,d\xi\le\bigl(53.96+15.134\, \mathcal A_{\mathrm{tie}}\bigr)\,d(e,e') \] for $d(e,e')\le0.08$, with $d\xi$ Haar measure on $\SU(2)^7$, and \textup{(J)} holds with $K\le(148.2+41.6\,\mathcal A_{\mathrm{tie}})\,e^{72\beta_W}$. \end{proposition} \begin{proof} \emph{Where the type is kept.} On flat points $D_gY=\phi(m)\,D_gN$, since $\phi'(m)D_gm=0$ there: either $\phi'(m)=0$, or the maximum sits at a path that does not depend on $g$. At points of type $a$, $D_gY=-2N\otimes D_gf_{x_a}+\phi(m)D_gN$. With $\Phi(z)=(e^z-1)/z$, the translated differential of $F$ is $\operatorname{Ad}(E^{-1})(1+A)$ with $A=\Phi( \operatorname{ad}Y)\,D_gY$, so $D=\det(1+A)$ and $\lVert A\rVert\le \lambda$ by Lemma~\ref{lem:relchart}(iii). Let $\xi$ keep one type, or stay flat, for the parameters $t$ of an interval of the move, at unit speed along the geodesic from $e$ to $e'$. Then $\frac d{dt}\log D= \operatorname{tr}((1+A)^{-1}\dot A)$, of modulus at most $3\lVert\dot A \rVert/(1-\lambda)$. We bound $\dot A$ with $\sigma=\frac{\sqrt2}{32}< 0.044195$, the norm $\lVert\operatorname{Ad}(u_a)-1\rVert\le\sigma$, and $\sigma L<0.09122$. \begin{itemize} \item By Lemma~\ref{lem:relchart}(ii), $G$ moves at speed at most $L$, so each $\tilde u_c$ moves at speed at most $\sigma L$, plus $1$ for the moving link; every $v_x$, every $f_x$ and $m$ therefore move at speed at most $1+\sigma L<1.0913$, and $Y$ at speed at most $\lambda L+(1-\lambda) L=L$. \item $\lVert N\rVert\le\frac78\sqrt2\,\alpha<0.8945$, $\lVert D_gN\rVert \le\frac7{16}\Lambda\sigma<0.02113$, $\lVert D_gf_{x_a}\rVert\le\sigma$, and $\lVert\dot N\rVert\le\frac1{16}\Lambda(1+7\sigma L)<0.1120$. \item On $\mathfrak{su}(2)$, $\lVert\operatorname{ad}X\rVert=\sqrt2\lVert X\rVert$, so $\lVert\operatorname{ad}Y\rVert\le r=1.2651$. The series of $\Phi$ gives $\lVert\Phi(\operatorname{ad}Y)\rVert\le1$ and \begin{align*} \lVert\Phi(\operatorname{ad}Y)-\Phi(\operatorname{ad}Y')\rVert &\le\frac{re^r-e^r+1}{r^2}\sqrt2\,\lVert Y-Y'\rVert\\ &\le1.715\,\lVert Y-Y'\rVert . \end{align*} \item $D_gf_{x_a}=\nabla\delta(v)\circ\operatorname{Ad}(\xi_a^{-1}) (1-\operatorname{Ad}(\tilde u_a^{-1}))$ in translated form. At a point of type $a$, $\delta(v,I)=2\sqrt2\sin\frac\theta2\ge\frac12$, so $\sin \frac\theta2\ge\frac1{4\sqrt2}$. The metric of $\SU(2)$ is that of the round sphere of radius $\sqrt2$, and the Hessian of the radial function $\delta$ has norm at most $\max\{\frac1{2\sqrt2}\sin\frac\theta2,\cos\theta /(2\sqrt2\sin\frac\theta2)\}$, at most $2\cos\theta\le2(1-\frac2{32})= \frac{15}8$. The translated derivative of the gradient adds $\frac12 \lVert\operatorname{ad}\rVert\le0.7072$. With $\lVert\frac d{dt} \operatorname{Ad}(\tilde u_a^{-1})\rVert\le\sqrt2(1+\sigma L)$, this gives $\lVert\frac d{dt}D_gf_{x_a}\rVert\le2.5822\cdot1.0913\,\sigma+ 1.41422\cdot1.0913<1.6679$. \item $D_gN=\frac1{16}\sum_c\mathrm d\!\log_{v_{x_c}}\circ \operatorname{Ad}(\xi_c^{-1})(1-\operatorname{Ad}(\tilde u_c^{-1}))$. In translated form $\mathrm d\!\log_v=\psi(\operatorname{ad}X)$, where $X= \log v$ and $\psi(z)=z/(1-e^{-z})=1+\frac z2+\frac{z^2}{12}-\frac{z^4} {720}+\dots$. With $\lVert\operatorname{ad}X\rVert\le2\alpha<1.4457$ the series gives Lipschitz constant $0.76\sqrt2\,\Lambda<1.176$ in $v$. Hence $\lVert\frac d{dt}D_gN\rVert\le\frac1{16}(1.176\,\sigma+\sqrt2\Lambda) (1+7\sigma L)<0.1636$. \end{itemize} Since $\phi'\in\{0,-2\}$ is constant along the interval and $\phi$ is $2$-Lipschitz, \begin{multline*} \Bigl\lVert\frac d{dt}D_gY\Bigr\rVert\le2\bigl(0.1120\,\sigma+0.8945\cdot 1.6679\bigr)\\ +2\cdot1.0913\cdot0.02113+0.1636<3.204 , \end{multline*} and $\lVert\dot A\rVert\le1.715\,L\lambda+3.204<3.559$. So $\lvert\frac d{dt}\log D\rvert\le3\cdot3.559/(1-\lambda)<11.87$ on the interval. The same bounds hold for $\log D_1$ and $\log D_2$ near a tie at the top, since $D_1$ has the form of type $a$ and $D_2$ the flat form, and $G_1$, $G_2$ move at speed at most $L$ by the proof of Lemma~\ref{lem:relchart}(ii). \emph{The events.} Call the following $77$ conditions on $\xi$ the \emph{events}: $f_{x_a}=m\in\{\frac12,1\}$; $f_{x_a}=f_{x'_c}=m\in[\frac12, 1]$ with $c\ne a$; and $f_{x_a}=f_{x_c}=m\in[\frac12,1]$ with $c\ne a$, for $a,c\ne0$. Every $f_x$ moves at speed at most $1+\sigma L$, and at most one of them is moved directly. So for a difference $h$ of two of them, or of one of them and a level, $\lvert h\rvert\le\omega\,d(e,e')$ before the move, if $\xi$ meets the event $h=0$ during the move, with $\omega=1+ 2\sigma L<1.1825$. Fix the coordinates other than $\xi_a$ and use $v=\tilde u_a(G(\xi))\xi_a$ in place of $\xi_a$. At fixed $g$, a unit move of $\xi_a$ moves $v_{x_a}$ and $v_{x'_a}$ at unit speed. As in Lemma~\ref{lem:relchart}(i), $Y$ then moves at speed at most $2\cdot\frac78\sqrt2\,\alpha+\frac18\Lambda<1.9256$, and $G$ at speed at most $L_\xi=1.9256/(1-\lambda)<2.140$. Since $d(\tilde u\xi,\tilde u'\xi')$ differs from $d(\xi,\xi')$ by at most $d(\tilde u, \tilde u')\le\sigma L_\xi\,d(\xi,\xi')$, the map $\xi_a\mapsto v$ is a bi-Lipschitz bijection of $\SU(2)$, with constants $1\pm0.0946$. It is onto by invariance of domain, and its inverse multiplies Haar measure by at most $(1/0.9054)^3<1.348$. Write $v=\exp(\theta\nu)$ with $\theta\in[0,\pi]$ and $\nu$ a unit direction. The Haar measure is $\frac2\pi\sin^2\theta\,d\theta$ times the uniform measure in $\nu$. Along a ray of fixed $\nu$: \begin{itemize} \item $f_{x_a}=2\sqrt2\sin\frac\theta2$ increases at rate $\sqrt2\cos \frac\theta2>1.3028$ while $f_{x_a}\le1.1$, that is $\theta\le0.7990$; \item $\xi_a$ moves at speed at most $\sqrt2/0.9054$, so $f_{x_c}$, $c\ne a$, which depends on $\xi_a$ only through $G$, moves at rate at most $\sigma L_\xi\sqrt2/0.9054<0.1478$; \item $f_{x'_c}$, $c\ne a$, and the levels do not move. \end{itemize} For each event, $h=f_{x_a}-f_y$ or $f_{x_a}-\frac12$ or $f_{x_a}-1$ therefore increases at rate at least $1.154$ on $\{0.4\le f_{x_a}\le 1.1\}$, and $\{\lvert h\rvert\le t\}$ meets each ray in an interval of length at most $2t/1.154$. With $\sin^2\theta\le0.5146$ for $\theta\le 0.7990$, its Haar measure in $v$ is at most $\frac2\pi\cdot0.5146\cdot \frac2{1.154}\,t<0.5678\,t$, and in $\xi_a$ at most $0.766\,t$. For $d(e,e')\le0.08$ every $\xi$ that meets an event during the move lies in such a set with $t=\omega\,d(e,e')$. By Fubini these $\xi$ have Haar measure at most $77\cdot0.766\cdot1.1825\,d(e,e')<69.75\,d(e,e')$, and for them $\lvert\log D-\log D'\rvert\le3\log\frac{1+\lambda}{1-\lambda}< 0.6034$. \emph{The ties.} Let $\xi$ meet no event during the move, with $D$ and $D'$ defined at $\xi$, and let $C$ be the set of parameters $t$ at which $\xi$ is a tie at the top of some index. Between two times of $C$ the point $\xi$ keeps one type or stays flat: the maximal path can change only at an event, at a tie at the top, or between two paths from points $x'_c$, where $D$ does not jump. At a time $t_0\in C$ of index $a$, Lemma~\ref{lem:ties} applies with the inside links as parameters, and the times near $t_0$ in $C$ are the zeros of the analytic function $t\mapsto h_2$; since no event occurs, $C$ is closed. If $C$ has an interior point, it is open by analyticity, so $C=[0,d(e,e')]$; otherwise $C$ is finite. Where $D$ is defined at a tie at the top, $\kappa_0=0$: if $\ell\ne0$, both branches meet $G$, so $D_gF_1=D_gF_2$, that is $N\otimes\ell=0$ and $N=0$. Hence $\log D=\log D_1= \log D_2$ at the ends that lie in $C$. By Lemma~\ref{lem:ties}(i) and the bound on the derivatives, \[ \lvert\log D-\log D'\rvert\le11.87\,d(e,e')+\sum_{t}1.9811\,\lvert \nabla h_2\rvert , \] the sum over the times $t\in C$ at which $h_2$ changes sign, the terms with $\nabla h_2=0$ vanishing; so the sum runs over times $t$ with $\xi\in Z_a(t)$, the set $Z_a$ at the parameter $t$. For one index $a$, the set $S=\{(t,\xi):\xi\in Z_a(t)\}$ is locally the zero set of $H(t,\xi)=h_2$ with $\nabla_\xi H\ne0$, a $C^1$ hypersurface of $[0,d(e,e')]\times\SU(2)^7$. The area formula for the projection $(t,\xi) \mapsto\xi$, of Jacobian $\lvert\partial_tH\rvert/\lvert\nabla H\rvert$ on $S$, and the coarea formula for $(t,\xi)\mapsto t$, of Jacobian $\lvert\nabla_\xi H\rvert/\lvert\nabla H\rvert$ on $S$ \cite[3.2.20, 3.2.22]{Fed69}, give \[ \int\sum_{t:\,\xi\in Z_a(t)}w\;d\mathrm{vol}(\xi)=\int_0^{d(e,e')}\!\! \int_{Z_a(t)}w\,\frac{\lvert\partial_tH\rvert}{\lvert\nabla_\xi H\rvert} \,dH^{20}\,dt \] for $w\ge0$. Here $H=f^{(1)}(G_2,\xi)-f^{(2)}$ changes at rate at most $1+\sigma L<1.0913$, since $G_2$ moves at speed at most $L$, $f^{(1)}$ is $\sigma$-Lipschitz in $g$, and only the moving link enters directly. With $w=1.9811\lvert\nabla_\xi H\rvert$ and $d\xi=V^{-7}d\mathrm{vol}$, the ties of index $a$ contribute at most $1.9811\cdot1.0913\,\mathcal A_{\mathrm{tie}}\, d(e,e')$, and the seven indices at most $15.134\,\mathcal A_{\mathrm{tie}}\,d(e,e')$. \emph{Conclusion.} Adding the three parts, with $11.87+0.6034\cdot69.75< 53.96$, gives the first bound. The density of $\hat Q$ is at most $1/(Z (1-\lambda)^3)\le2.00002\,e^{72\beta_W}/0.7285$ by Lemma~\ref{lem:relnorm}, and $2.00002\cdot53.96/0.7285<148.2$, $2.00002\cdot15.134/0.7285<41.6$. Since $\lvert\log D-\log D'\rvert$ is at most the sum of its increments along a subdivision of the geodesic into steps of length at most $0.08$, and the bound on the density does not depend on the step, \textup{(J)} holds with the stated $K$. \end{proof} \begin{proposition}[the area of the ties]\label{prop:tiearea} $\mathcal A_{\mathrm{tie}}<\infty$. \end{proposition} \begin{proof} Identify $\SU(2)$ with the unit sphere of $\R^4$; the metric above is twice the Euclidean one, so $H^{20}$ is $2^{10}$ times the Euclidean Hausdorff measure on $\SU(2)^7\subset\R^{28}$. Every object of the construction is definable in the o-minimal structure $\R_{\mathrm {an}}$ of restricted analytic functions \cite{Dri86}\footnote{A structure on $\R$ is o-minimal when every definable subset of $\R$ is a finite union of points and intervals. In an o-minimal structure the number of connected components of the fibres of a definable family is bounded uniformly in the parameter \cite[Ch.~3]{Dri98}; this uniformity is what makes the constant $\mathfrak b$ below independent of the configuration.}: $\delta$, $\phi$ and the maximum are semialgebraic; $\log$ on $\{\ReTr v\ge\frac32\}$, which contains $\{\delta(v,I)\le1\}$, and $\exp$ on $\{\lVert X\rVert\le1\}$ are restricted analytic; $N$ may be set to $0$ where $m\ge1$ without changing $Y=\phi(m)N$; and the graph of $G$ is $\{g\exp Y=c_k\}$. The formula for $dh_2$ in the proof of Lemma~\ref{lem:ties} is a definable function of $g$, $\xi$ and the parameters $P=(c_k,(u_b,w_b)_{b\ne0})$, so $Z_a(P)$ is a definable family over a bounded set of parameters, of $C^1$ submanifolds of dimension $20$. For an orthogonal projection $\varpi$ of $\mathbb R^{28}$ onto $\R^{20}$ and $y\in\R^{20}$, the sets $Z_a(P) \cap\varpi^{-1}(y)$ form a definable family with parameters $(P,\varpi, y)$, so those that are finite have at most $\mathfrak b$ points, with $\mathfrak b$ independent of $(P,\varpi,y)$ \cite[Ch.~3]{Dri98}; and for fixed $(P,\varpi)$ the $y$ with an infinite fibre form a definable set of dimension at most $19$ \cite[Ch.~4]{Dri98}, which is Lebesgue-null. By the integralgeometric formula, $H^{20}(Z)$ equals a constant depending only on $(28,20)$ times the average over $\varpi$ of $\int\#(Z\cap\varpi^{-1} (y))\,dy$ \cite[2.10.15, 3.2.26]{Fed69}, and $\varpi(Z)$ lies in the ball of radius $\sqrt7$. So $H^{20}(Z_a(P))$ is bounded uniformly in $P$ and $a$. \end{proof} \begin{theorem}[condition (J)]\label{thm:jholds} Let $k$, $B$ and the move be as in Theorem~\textup{\ref{thm:relresponse}}. Then \textup{(J)} holds with $K=(148.2+41.6\,\mathcal A_{\mathrm{tie}})\, e^{72\beta_W}$, where $\mathcal A_{\mathrm{tie}}<\infty$ does not depend on $k$, $c_k$, the inside links or $\beta_W$. In particular \textup{(J)} holds for $B^{\mathrm{bad}}_r$ on every configuration that the labels produce, and for $\SU(2)^7$ whenever the links $u_a$, $a\ne0$, stay within $\frac1{32}$ of $I$ along the geodesic, whatever the links $w_a$. For $B=\SU(2)^7$ and links $w_a$, $a\ne0$, that stay within $\frac1{32}$ of $I$ along the geodesic, \textup{(J)} holds with the same $K$ in the coordinates $\xi_a=\ell_ag^{-1}$ of Lemma~\textup{\ref{lem:relchart}}, whatever the links $u_a$. \end{theorem} \begin{proof} Proposition~\ref{prop:jreduce} and Proposition~\ref{prop:tiearea}, whose hypotheses bound only the links $u_a$; the case of $B^{\mathrm{bad}}_r$ by Proposition~\ref{prop:eightlabels}. For the last statement let $\mathfrak i$ send each private link $\ell_a$ to $\ell_a^{-1}$ and the pair $(u_a,w_a)$ of inside links to $(w_a^{-1},u_a^{-1})$, $a\ne0$. It maps $P_{2y}$ to $P_{2y}^{-1}$ and the paths from $x_a$ and $x'_a$ to the inverses of the paths from $x'_a$ and $x_a$. Since $\delta$ is invariant under inversion and $\log(P_{2y}P_x^{-1})=-\operatorname{Ad}(P_{2y})\log(P_{2y}^{-1}P_x)$, \eqref{eq:cutoff} gives $V^\sharp_k\circ\mathfrak i=(V^\sharp_k)^{-1}$. The map $\mathfrak i$ preserves Haar measure and the cost, and it maps the coordinates $\xi_a=\ell_ag^{-1}$ at the value $c_k$ to the coordinates $\bar g^{-1}\bar\ell_a=\xi_a^{-1}$ of the inverted links $\bar\ell=\mathfrak i (\ell)$ at $c_k^{-1}$, with $V^\sharp_k=\exp(\chi N')\,g$ turned into $\bar g\exp(-\chi N')$; inversion is an isometry, so the two Jacobians $D$ agree at corresponding points. The action enters Lemmas~\ref{lem:relchart} and~\ref{lem:relnorm} and Proposition~\ref{prop:jreduce} only through the bounds $0\le S_{\mathrm{dep}}\le72$, which hold for every configuration. So \textup{(J)} in the coordinates $\ell_ag^{-1}$ for $(u,w)$ is \textup{(J)} in the coordinates $g^{-1}\ell_a$ for $(w^{-1},u^{-1})$, whose links of the first block are within $\frac1{32}$ of $I$, and the moving link keeps its distance. \end{proof} \begin{corollary}[the first set with one block near $I$]\label{cor:onenear} Let $B=\SU(2)^7$ and let one link $u_b$ or $w_b$, $b\ne0$, move from $e$ to $e'$. If the inside $\mu$-links of one of the two blocks, the $u_a$ or the $w_a$, $a\ne0$, stay within $\frac1{32}$ of $I$ along the geodesic, then, whatever the inside $\mu$-links of the other block, \[ W_1(Q,Q')\le\bigl(16.52+2898\beta_W+41.37K\bigr)\,d(e,e') \] for the cost $\sum_a\delta$, with $K$ of Theorem~\textup{\ref{thm:jholds}}. \end{corollary} \begin{proof} If the $u_a$ stay within $\frac1{32}$ of $I$, this is Theorem~\ref{thm:relresponse} for $B=\SU(2)^7$ and Theorem~\ref{thm:jholds}. If the $w_a$ do, then $\sigma_w\le\frac{\sqrt2} {32}$, so $\lambda_w\le\lambda$, $L_w\le L$ and $R_w<1.828$, and the last bound of Theorem~\ref{thm:relresponse}, with (J) in the coordinates $\xi_a=\ell_ag^{-1}$ from Theorem~\ref{thm:jholds}, is at most $8L+16\sqrt2\cdot1.828\,(24\sqrt2\,L\beta_W+K)$ times $d(e,e')$; here $8L<16.52$, $768\cdot1.828\cdot2.064<2898$ and $16\sqrt2\cdot1.828< 41.37$. \end{proof} The proof of Proposition~\ref{prop:tiearea} gives no value for $\mathcal A_{\mathrm{tie}}$: the number $\mathfrak b$ comes from o-minimality. The weight $\lvert\log D_1-\log D_2\rvert$ in the flux cannot be dropped, since the near-ties need not be thin. \begin{proposition}[thick near-ties]\label{prop:thickties} Fix $a\ne0$, let $u_a=w_a=u$ with $d(u,I)=s\le\frac1{32}$, the other inside links within $\frac1{32}$ of $I$, and let $T(t)$ be the set of $\xi$ with \[ \lvert f_{x_a}-f_{x'_a}\rvert\le t,\qquad f_{x_a}\ge m-t,\qquad\tfrac12 -t\le f_{x_a}\le1+t . \] Then $T(t)$ contains the open set $U_0=\{\xi:\frac{17}{32}<\delta(\xi_a,I) <\frac{31}{32},\ \delta(\xi_b,I)<\frac{15}{32}\text{ for }b\ne a\}$ for every $t\ge2s$, for every $c_k$. For $s=0$ every point of $U_0$ is a tie at the top of index $a$, and $\nabla h_2=0$ there. \end{proposition} \begin{proof} Since $\delta(v,I)=\lVert v-I\rVert$ and $\lVert\tilde u-I\rVert=\lVert u-I\rVert\le s$, $\lvert f_{x_a}-\delta(\xi_a,I)\rvert\le s$ and $\lvert f_{x'_a}-\delta(\xi_a,I)\rvert\le s$, using $\delta(\xi_au,I)=\delta(u \xi_a,I)$; so on $U_0$ both lie in $(\frac12,1)$ and differ by at most $\lVert\tilde u-u\rVert\le2s$. For $b\ne a$, $f_{x_b},f_{x'_b}\le \delta(\xi_b,I)+\frac1{32}<\frac12$, and the paths of offset zero give $0$. So $m=\max(f_{x_a},f_{x'_a})$ and $f_{x_a}\ge m-2s$. For $s=0$, $f_{x_a}= f_{x'_a}=\delta(\xi_a,I)$ on $U_0$, so $h_2$ vanishes on the open set $U_0$. \end{proof} Fig.~\ref{fig:thickties} shows why the near-ties fill an open set. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_thickties.png} \caption{Thick near-ties (Proposition~\ref{prop:thickties}), schematic. (a)~The values $f_{x_a}$ and $f_{x'_a}$ of the two paths of index $a$ over two coordinates of $\xi$, for $u_a=w_a=u$ at distance $s$ from $I$: over the open set $U_0$ they differ by at most $2s$, so the set $T(t)$ of near-ties contains $U_0$ for every $t\ge2s$. (b)~The set $U_0$ in the coordinates $\delta(\xi_a,I)$ and $\max_{b\ne a}\delta(\xi_b,I)$, beside a band of width $t$ around a tie hypersurface, the shape that a bound $\mathrm{Haar}(T(t))\le Mt$ would require.} \label{fig:thickties} \end{figure*} So no bound $\operatorname{Haar}(T(t))\le Mt$ holds with $M$ uniform in the inside links, nor one by $t\lvert\log t\rvert$: the Haar measure of $T(2s)$ is at least that of $U_0$ as $s\to0$. At $s=0$ the jump vanishes, since $\ell=0$ when $u_a=I$. The other inside links of the two blocks are not $\mu$-links, do not enter $V^\sharp_k$, and lie in at most one plaquette with a private link; the proof of Lemma~\ref{lem:wilsononly} applies to them unchanged and bounds the response to each of them by $8\beta_W$ for every $B$. So the response of the law of the eight private links to every inside link is bounded, with the constants of Theorem~\ref{thm:relresponse}, as soon as the Jacobian condition (J) holds: for $B^{\mathrm{bad}}_r$ on every configuration that the labels produce, since there $d(u_a,I),d(w_a,I)\le r$ by Proposition~\ref{prop:eightlabels}, and for $\SU(2)^7$ on every configuration whose links $u_a$ lie within $\frac1{32}$ of $I$. Theorem~\ref{thm:jholds} proves (J) there for every $w$, in particular for $B^{\mathrm{bad}}_r$, and in the coordinates $\ell_ag^{-1}$ when the $w_a$ lie within $\frac1{32}$ of $I$, whatever the $u_a$; Corollary~\ref{cor:onenear} bounds the response of the first set in both cases. By Lemma~\ref{lem:relchart}(iii), $D$ lies in $[0.728,1.332]$ for all inside links, so (J) asks only for its continuity in the mean. The derivative of $Y=\chi N$ in $g$ contains $N\otimes\phi'(m)D_gm$, and $D_gm$ vanishes when the largest $\delta(v_x,I)$ is attained at a path from $x'_a$, which does not depend on $g$; so $D$ jumps across the sets where the maximal path passes between a path from some $x_a$ and another path while $\frac122$, so a connected set lies in one of them; and a link based at $d_a\le s$ is in $L_a$ by the definition of the classes. (ii) A block of $W$ is at distance at least $s+1\ge3$ from every bad block, a pair entering $W$ is based at distance at least $s\ge2$, and the constraint of a pair of $K_W$ contains links based in its own block and in the next one along its direction, both at distance at least $s\ge2$. The links freed for $W$ are based in $W$ or are path links of pairs entering it. Links of chains are based at distance at most one. The rest is the proof of Theorem~\ref{thm:islandthreshold} unchanged. (iii) The window is admissible because $y$ is at distance two from it; the numbers are those of Theorem~\ref{thm:islandthreshold} at $s=2$. \end{proof} \subsection{The projected average}\label{ssec:projected} The folds of Propositions~\ref{prop:fold} and~\ref{prop:insidefold} and the obstruction of Theorem~\ref{thm:eightlinkuniform} are zeros of $\det D_gV^\sharp_k$, the derivative of the cut-off average in its path link. They come from the two nonlinear parts of~\eqref{eq:cutoff}: the logarithm, whose differential $L_Y$ grows like $\theta/\sin\theta$ as an eigenangle $\theta$ approaches $\pi$, and the cut-off, whose slope is $2$. If the logarithm of each relative path is replaced by its orthogonal projection onto the Lie algebra, no cut-off is needed. The resulting coarse map has the same derivatives of order at most two as the cut-off average at every field whose sixteen paths agree, and its derivative in the path link is invertible at every field. This subsection proves the properties of this map that bear on the bad side; Remark~\ref{rem:projscope} lists what is not proved for it. As in the rest of this section $G=\SU(2)$; Remark~\ref{rem:projgroups} gives the constants for other groups. Let $\Pi_\g$ be the orthogonal projection of the space of complex $2\times2$ matrices, with the real inner product $\operatorname{Re}\Tr(X^*Y)$, onto $\g=\mathfrak{su}(2)$.\footnote{For $U\in\SU(2)$ this is the anti-Hermitian part $\frac12(U-U^{-1})$. Unlike the logarithm, it is defined and real-analytic on the whole group, which is why the projected average needs no restriction on the paths.} \begin{definition}\label{def:projavg} For a coarse link $(y,\mu)$ and the paths $P_x$ of Sec.~\ref{ssec:pathblocks}, the \emph{projected average} is \begin{gather*} \widehat V_{y,\mu}(U)=P_{2y}\exp A_{y,\mu}(U),\\ A_{y,\mu}(U)=\frac1{16}\sum_{x\in B(y)}\Pi_\g\bigl(P_{2y}^{-1}P_x\bigr). \end{gather*} \end{definition} For a pair $k=(y,\mu)$ in the tree gauge write $g=\ell_k$, $z=(\ell_a)_{a\ne0}$, and $P_{x_a}=u_a\ell_a$, $P_{x'_a}=\ell_aw_a$ for the paths of nonzero offset, as in Lemma~\ref{lem:relchart}. The two paths of offset zero equal $g$ and contribute nothing, so \[ A_k=\frac1{16}\sum_{a\ne0}\Bigl(\Pi_\g(g^{-1}u_a\ell_a)+\Pi_\g(g^{-1}\ell_a w_a)\Bigr), \] a function $A(g;z,u,w)$ of the path link, the non-path links and the inside links. Let $J(g;z,u,w)=\lvert\det D_g\widehat V_k\rvert$, the Jacobian between tangent spaces of the bi-invariant metric, as in~\eqref{eq:pathlaw}. \begin{lemma}\label{lem:projection} \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For $U\in\SU(2)$, $\Pi_\g(U)=\frac12(U-U^{-1})$. The projection commutes with conjugation by $\SU(2)$, $\Pi_\g(I)=0$, and $\lVert\Pi_\g(U)- \Pi_\g(U')\rVert\le\delta(U,U')\le d(U,U')$ for $U,U'\in\SU(2)$; in particular $\lVert\Pi_\g(U)\rVert\le\delta(U,I)$. \item For $X\in\g$, $\lVert\Pi_\g(e^X)-X\rVert\le\frac1{12}\lVert X \rVert^3$. \item For $X\in\g$, the operator $\operatorname{ad}X$ on $\g$ has norm $\sqrt2\,\lVert X\rVert$, and the operator norm of the matrix $X$ is $\lVert X\rVert/\sqrt2$. \end{enumerate} \end{lemma} \begin{proof} (i) For $U\in\SU(2)$, $U+U^*=(\Tr U)I$ with $\Tr U$ real, so $U=\frac12 (\Tr U)I+Y$ with $Y=\frac12(U-U^*)\in\g$. Since $\operatorname{Re}\Tr X=0$ for $X\in\g$, $I$ is orthogonal to $\g$, and $\Pi_\g(U)=Y$. Conjugation by $h\in\SU(2)$ is orthogonal for the inner product and maps $\g$ onto itself, so it commutes with $\Pi_\g$. As an orthogonal projection, $\Pi_\g$ is a linear map of norm one, and $\delta$ is the Hilbert--Schmidt distance; a chord is not longer than an arc, so $\delta\le d$. (ii) In a basis with $X=\operatorname{diag}(i\theta,-i\theta)$, $\lVert X\rVert=\sqrt2\, \lvert\theta\rvert$ and $\Pi_\g(e^X)=\operatorname{diag}(i\sin\theta, -i\sin\theta)$ by (i), so $\lVert\Pi_\g(e^X)-X\rVert=\sqrt2\,\lvert\theta- \sin\theta\rvert\le\sqrt2\,\lvert\theta\rvert^3/6=\lVert X\rVert^3/12$. (iii) In the same basis $\operatorname{ad}X$ is skew with eigenvalues $0$ and $\pm2i\theta$, and $X$ has the eigenvalues $\pm i\theta$. \end{proof} \begin{proposition}[the projected average]\label{prop:projavg} The projected average is real-analytic on all fields, covariant under the gauge transformations at the block roots, and normalised. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item At a field with $\delta(P_x,P_{2y})\le\frac12$ for every $x\in B(y)$, with $X_x=\log(P_{2y}^{-1}P_x)$, \[ d\bigl(\widehat V_{y,\mu},V^\sharp_{y,\mu}\bigr)\le\frac1{192}\sum_{x\in B(y)}\lVert X_x\rVert^3 . \] At every field at which the sixteen paths $P_x$ agree, the derivatives of order at most two of $\widehat V_{y,\mu}$ and $V^\sharp_{y,\mu}$ coincide; in particular $\widehat V_{y,\mu}(e^{t\varphi})=I+t\,V_{y,\mu}(\varphi)+ O(t^2)$, as in Theorem~\ref{thm:cutoff}(i). \item The four inequalities of Theorem~\ref{thm:cutoff}(ii) hold for $\widehat V$ with $c=1$, that is with the constants $896$, $56$, $1344$ and $2696$, for every field in the tree gauge. \item If $U$ equals $I$ except on one link from $B(y)$ to $B(y+e_\mu)$, where it equals $h$, then $\widehat V_{y,\mu}(U)=h\exp(-\frac78\Pi_\g(h))$ when the link is on the straight path and $\exp(\frac18\Pi_\g(h))$ at the seven other positions, for every $h$. The slice windings of Proposition~\ref{prop:nomean} are $1$ at the straight path and $0$ at the other positions. \end{enumerate} \end{proposition} \begin{proof} Products of matrices, $\Pi_\g$ and $\exp$ are analytic, and $\widehat V\in G$ since $A\in\g$. A gauge transformation equal to $h_1$ on $B(y)$ and to $h_2$ on $B(y+e_\mu)$ maps $P_x$ to $h_1P_xh_2^{-1}$, as in the proof of Theorem~\ref{thm:cutoff}, so $P_{2y}^{-1}P_x\mapsto h_2P_{2y}^{-1}P_xh_2^{-1}$, $A\mapsto h_2Ah_2^{-1}$ by Lemma~\ref{lem:projection}(i), and $\widehat V \mapsto h_1\widehat Vh_2^{-1}$. At $U=I$ every term of $A$ is $\Pi_\g(I)=0$. (i) Under the hypothesis the cut-off is one, and $V^\sharp=P_{2y}e^N$ with $N=\frac1{16}\sum_xX_x$, by Theorem~\ref{thm:cutoff}(i), while $A=\frac1 {16}\sum_x\Pi_\g(e^{X_x})$. By $d(e^A,e^N)\le\lVert A-N\rVert$ and Lemma~\ref{lem:projection}(ii) the distance is at most $\frac1{16}\sum_x \frac1{12}\lVert X_x\rVert^3$. Near a field at which the paths agree the $X_x$ are analytic functions of the links that vanish there, so the two analytic maps differ by $O(r^3)$ at distance $r$ from it, and their Taylor expansions in a chart agree to order two. The last statement is Theorem~\ref{thm:cutoff}(i). (ii) In the proof of Theorem~\ref{thm:cutoff}(ii), $\delta(V^\sharp,P)$ is bounded by $\frac c4(\sum_x\delta(P_x,P)^2)^{1/2}$, and $V^\sharp=vM$ with $\delta(M,I)$ at most the same quantity. For $\widehat V=Pe^A$, Lemma~\ref{lem:logbound}, Lemma~\ref{lem:projection}(i) and the Cauchy--Schwarz inequality give \begin{multline*} \delta(\widehat V,P)=\delta(e^A,I)\le\lVert A\rVert\le\frac1{16}\sum_x \delta(P_x,P)\\ \le\frac14\Bigl(\sum_x\delta(P_x,P)^2\Bigr)^{1/2}, \end{multline*} which is that bound with $c=1$, and the rest of the proof is unchanged: $896/16=56$, $448+896=1344$ and $8+48\cdot56=2696$. (iii) At the straight path $P_{2y}=h$, the fourteen other paths are $I$, and $\Pi_\g(h^{-1})=-\Pi_\g(h)$; at another position $P_{2y}=I$ and two paths equal $h$. For $h=\operatorname{diag}(e^{iq},e^{-iq})$ the two values are $\operatorname{diag}(e^{i\psi},e^{-i\psi})$ with $\psi=q-\frac78\sin q$ and $\psi=\frac18\sin q$, loops of degree $1$ and $0$ in $q$. \end{proof} So $\widehat V$ is a continuous covariant extension of the block averages to all fields, with the slice windings that Proposition~\ref{prop:nomean} demands; it departs from the transverse symmetry at third order at every field, where $V^\sharp$ departs from it only past the cut-off. \begin{theorem}[the global inverse]\label{thm:projinverse} Let $k$ be a pair, $c_k\in\SU(2)$, and let the non-path links $z$ and the inside links $u,w$ be arbitrary. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For all $g,g'$, $\lVert A(g)-A(g')\rVert\le\frac78\,\delta(g,g')$, and $g\mapsto c_k\exp(-A(g))$ is a contraction of $\SU(2)$ with constant $\frac78$ for $d$. The equation $\widehat V_k=c_k$ has exactly one solution $G=G(c_k;z,u,w)$, the fixed point of this contraction, and $G$ is real-analytic in $(c_k,z,u,w)$. \item For all $g,g'$, \[ \tfrac18\,d(g,g')\le d\bigl(\widehat V_k(g),\widehat V_k(g')\bigr)\le \tfrac{15}8\,d(g,g'), \] the singular values of $D_g\widehat V_k$ lie in $[\frac18,\frac{15}8]$ at every $g$, and $\frac1{512}\le J\le\frac{3375}{512}$. The two bounds on $J$ are attained when all fourteen relative paths $g^{-1}P_x$ of nonzero offset equal $I$, respectively $-I$. \item For two configurations, \begin{multline*} d(G,G')\le8\,d(c_k,c'_k)+\sum_{a\ne0}d(\ell_a,\ell'_a)\\ +\frac12\sum_{a\ne0}\bigl(d(u_a,u'_a)+d(w_a,w'_a)\bigr). \end{multline*} \item Let one link $u_b$ or $w_b$, $b\ne0$, move along a unit-speed geodesic, with $c_k$, $z$ and the other links fixed, and let $J(t)$ be the value of $J$ at $(G(t);z,u(t),w(t))$. Then $\lvert\frac d{dt}\log J(t)\rvert\le \frac{45}4\sqrt2<15.91$. If instead a non-path link $\ell_b$ moves, with $c_k$, $u$, $w$ and the other non-path links fixed, the bound is $\frac{45}2 \sqrt2<31.82$; if $c_k$ moves, with $z$, $u$, $w$ fixed, it is $\frac{315}2 \sqrt2<222.8$. \end{enumerate} \end{theorem} \begin{proof} \emph{The derivative.} Let $v_x=g^{-1}P_x$ for the fourteen paths of nonzero offset. Along $ge^{t\zeta}$, $\zeta\in\g$, each $v_x$ becomes $e^{-t\zeta}v_x$, so $A(ge^{t\zeta})=A(g)+tB\zeta+O(t^2)$ with \[ B\zeta=-\frac1{16}\sum_x\Pi_\g(\zeta v_x),\qquad\lVert B\rVert\le\frac78 , \] since $\lVert\zeta v_x\rVert=\lVert\zeta\rVert$. With $e^{-A}\frac d{dt} e^{A+tY}|_{t=0}=\int_0^1\operatorname{Ad}(e^{-sA})Y\,ds$ and $R_A=\int_0^1 \operatorname{Ad}(e^{sA})\,ds$, of norm at most one, \[ \widehat V_k^{-1}\frac d{dt}\widehat V_k(ge^{t\zeta})\Big|_{t=0}= \operatorname{Ad}(e^{-A})\,D\zeta,\qquad D=I+R_AB . \] As $\operatorname{Ad}(e^{-A})$ is a rotation of $\g$, $J=\lvert\det D\rvert$; the maps $I+sR_AB$, $0\le s\le1$, are invertible because $\lVert R_AB\rVert\le \frac78$, so $\det D>0$, and $\lVert D^{-1}\rVert\le(1-\frac78)^{-1}=8$. (i) By Lemma~\ref{lem:projection}(i) and the invariance of $\delta$ under multiplication and inversion, $\lVert\Pi_\g(g^{-1}P)-\Pi_\g(g'^{-1}P) \rVert\le\delta(g^{-1}P,g'^{-1}P)=\delta(g,g')$ for each of the fourteen paths, which gives the first bound. With $d(e^X,e^Y)\le\lVert X-Y\rVert$ and $\delta\le d$, $d(c_ke^{-A(g)},c_ke^{-A(g')})\le\frac78d(g,g')$, and $\SU(2)$ is complete, so the contraction has one fixed point; $\widehat V_k(g)=c_k$ is the equation $g=c_ke^{-A(g)}$. Since $D$ is invertible at every point, the analytic implicit function theorem gives an analytic solution near every solution, which by uniqueness is $G$. (ii) By bi-invariance, $d(ge^{A(g)},g'e^{A(g)})$ equals $d(g,g')$, and $d(g'e^{A(g)},g'e^{A(g')})\le\lVert A(g)-A(g')\rVert\le\frac78d(g,g')$; the triangle inequality gives both bounds, and along $ge^{t\zeta}$ they give $\frac18\lVert\zeta\rVert\le\lVert D\zeta\rVert\le\frac{15}8\lVert\zeta \rVert$. If every $v_x$ is $I$, respectively $-I$, then $A=0$, $R_A=I$ and $B=\mp\frac78I$, so $J=8^{-3}$, respectively $(\frac{15}8)^3$; these values are taken, for instance, at $u=w=I$ and $\ell_a=\pm g$. (iii) Let $T(g)=c_ke^{-A(g)}$ and $T'$ be the map for the second configuration. Then $d(G,G')\le d(T(G),T(G'))+d(T(G'),T'(G'))$, the first term is at most $\frac78d(G,G')$, and the second is at most $d(c_k,c'_k)+ \lVert A(G')-A'(G')\rVert$. A path of nonzero offset changes by at most $d(u_a,u'_a)+d(\ell_a,\ell'_a)$, respectively $d(\ell_a,\ell'_a)+d(w_a, w'_a)$, so $\lVert A(G')-A'(G')\rVert$ is at most $\frac1{16}$ times the sum of these quantities, and multiplying by $8$ gives the bound. (iv) Let $u_b(t)=u_be^{t\zeta'}$ with $\lVert\zeta'\rVert=1$. By (i) and (iii), $G(t)$ is analytic and its left velocity $\eta=G^{-1}\dot G$ has $\lVert\eta\rVert\le\frac12$. Then $\dot A=B\eta+\frac1{16}\Pi_\g(G^{-1}u_b \zeta'\ell_b)$, so $\lVert\dot A\rVert\le\frac7{16}+\frac1{16}=\frac12$. Next, $\frac d{dt}\operatorname{Ad}(e^{sA})=\operatorname{ad}(W_s)\operatorname{Ad} (e^{sA})$ with $W_s=(\frac d{dt}e^{sA})e^{-sA}=s\int_0^1\operatorname{Ad} (e^{rsA})\dot A\,dr$, so $\lVert W_s\rVert\le s\lVert\dot A\rVert$ and, by Lemma~\ref{lem:projection}(iii), $\lVert\dot R_A\rVert\le\int_0^1\sqrt2\,s \lVert\dot A\rVert\,ds\le\frac1{2\sqrt2}$. Further $\dot B\zeta=-\frac1{16} \sum_x\Pi_\g(\zeta\dot v_x)$ with $\dot v_x=-\eta v_x$, except for the path from $x_b$, which has the additional term $G^{-1}u_b\zeta'\ell_b$; and $\lVert\zeta M\rVert\le\lVert\zeta\rVert\,\lVert M\rVert_{\rm op}$, $\lVert \eta v_x\rVert_{\rm op}=\lVert\eta\rVert/\sqrt2$ and $\lVert G^{-1}u_b \zeta'\ell_b\rVert_{\rm op}=1/\sqrt2$ by Lemma~\ref{lem:projection}(iii). Hence $\lVert\dot B\rVert\le\frac1{16}(\frac{14}{2\sqrt2}+\frac1{\sqrt2})= \frac1{2\sqrt2}$, and $\dot D=\dot R_AB+R_A\dot B$ has \[ \lVert\dot D\rVert\le\frac78\cdot\frac1{2\sqrt2}+\frac1{2\sqrt2}=\frac{15} {16\sqrt2} . \] Since $\det D>0$, $\frac d{dt}\log J=\Tr(D^{-1}\dot D)$, at most $3\lVert D^{-1}\rVert\lVert\dot D\rVert\le3\cdot8\cdot\frac{15}{16\sqrt2}=\frac{45}4 \sqrt2$ in modulus on the three-dimensional $\g$. The link $w_b$ is the same with the path from $x'_b$. If $\ell_b$ moves, $\lVert\eta\rVert\le1$ by (iii) and two paths have an additional term of operator norm $1/\sqrt2$, so $\lVert\dot A\rVert\le\frac78+\frac2{16}=1$, $\lVert\dot R_A\rVert\le \frac1{\sqrt2}$, $\lVert\dot B\rVert\le\frac1{16}(\frac{14}{\sqrt2}+ \frac2{\sqrt2})=\frac1{\sqrt2}$, $\lVert\dot D\rVert\le\frac{15}{8\sqrt2}$, and the bound is $3\cdot8\cdot\frac{15}{8\sqrt2}=\frac{45}2\sqrt2$. If $c_k$ moves, $\lVert\eta\rVert\le8$ by (iii) and no path has an additional term, so $\lVert\dot A\rVert\le7$, $\lVert\dot R_A\rVert\le\frac7{\sqrt2}$, $\lVert\dot B\rVert\le\frac{14}{16}\cdot\frac8{\sqrt2}=\frac7{\sqrt2}$ and $\lVert\dot D\rVert\le\frac{105}{8\sqrt2}$, and the bound is $3\cdot8\cdot \frac{105}{8\sqrt2}=\frac{315}2\sqrt2$. \end{proof} Fig.~\ref{fig:projected} draws the projected average and the contraction behind its global inverse. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_projected.png} \caption{The projected average of Definition~\ref{def:projavg}, schematic, with a two-sphere standing for $\SU(2)$. (a)~The paths $P_x$ of a coarse link around the base path $P_{2y}$, their projections to the tangent space at $P_{2y}$, the mean $A$ of the projections, and the value $\widehat V= P_{2y}e^{A}$. (b)~The map $g\mapsto c_k\,e^{-A(g)}$ contracts with constant $\frac78$, so its iterates converge to the unique solution $G$ of $\widehat V_k=c_k$, and $\widehat V_k$ changes distances by a factor between $\frac18$ and $\frac{15}8$ (Theorem~\ref{thm:projinverse}).} \label{fig:projected} \end{figure*} \begin{corollary}[the path-link law of the projected average]\label{cor:projpathlaw} For every $c_k$, $z$, $u$, $w$ and exterior $\sigma$, the law~\eqref{eq:pathlaw} with $\widehat V_k$ in place of $V^\sharp_k$ is the point mass at $G(c_k;z,u,w)$, and for two configurations with the same $c_k$ \begin{multline*} W_1^d(\pi,\pi')\le\sum_{a\ne0}d(\ell_a,\ell'_a)\\ +\frac12\sum_{a\ne0}\bigl(d(u_a,u'_a)+d(w_a,w'_a)\bigr), \end{multline*} with no dependence on $\beta_W$, on $c_k$ or on any other link, and no restriction on the size of any link. \end{corollary} \begin{proof} By Theorem~\ref{thm:projinverse}(i) the sum in~\eqref{eq:pathlaw} has one term, whose weight cancels against $Z$; the bound is Theorem~\ref{thm:projinverse}(iii). \end{proof} So for the projected average the original form of item (i) of Sec.~\ref{ssec:remains}, a Lipschitz bound for the path-link law in the non-path links, holds at every field, with the constants $1$ and $\frac12$ in place of the constants $1.1986$ and $0.5993$ that Proposition~\ref{prop:goodside} gives for $V^\sharp$ on the good side only; Propositions~\ref{prop:branches}, \ref{prop:fold} and~\ref{prop:insidefold} have no counterpart. The rest of Proposition~\ref{prop:goodside} holds with no radius as well: if the non-path links and the inside links are within $r$ of the reconstruction, the computation of Lemma~\ref{lem:branchbound}, with $\lVert A_k\rVert\le \frac78\max_x\delta(g^{-1}P_x,I)$ in place of the logarithmic bound, gives $d(G,c_k)\le14r$ and $d(g^{-1}P_x,I)\le16r$ for every $r$, and the density $1/J$ lies in $[(\frac{15}8)^{-3},512]$. For the law of the eight private links let $S_{\mathrm{dep}}$ be as in Lemma~\ref{lem:relnorm}, let $dz$ be Haar measure on $\SU(2)^7$, $\lvert B\rvert$ the Haar measure of a set $B$, and write $G(z)=G(c_k;z,u,w)$. \begin{theorem}[the bad side of the projected average]\label{thm:projresponse} Let $k$ be a pair and $c_k\in\SU(2)$, fix every link other than the eight private links of $k$ at arbitrary values, and let $B\subset\SU(2)^7$ be a measurable set of values of the non-path links with $\lvert B\rvert>0$. Put \begin{gather*} \rho_B(z)=\mathbf1_B(z)\,e^{-\beta_WS_{\mathrm{dep}}(G(z),z)}\, J\bigl(G(z);z,u,w\bigr)^{-1},\\ Q_B(f)=\frac1{Z_B}\int f\bigl(G(z),z\bigr)\,\rho_B(z)\,dz,\qquad Z_B=\int\rho_B\,dz . \end{gather*} \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For every $c_k$, $Q_B$ is the law of the eight private links given the other links, $\widehat V_k=c_k$ and $z\in B$, in the version of the disintegration that is continuous in $c_k$, and \[ (\tfrac8{15})^3e^{-72\beta_W}\lvert B\rvert\le Z_B\le512\,\lvert B\rvert . \] \item If one inside link $u_b$ or $w_b$, $b\ne0$, moves from $e$ to $e'$, with $c_k$, $B$ and every other link fixed, then for the cost $\sum_{a=0}^7\delta(\ell_a,\ell'_a)$ \[ W_1(Q_B,Q'_B)\le(24\beta_W+180.5)\,d(e,e'). \] \item If one link $e'$ that is neither a private link of $k$ nor one of the links $u_a$, $w_a$ changes, then $W_1(Q_B,Q'_B)\le8\beta_W\,d(\sigma_{e'}, \sigma'_{e'})$. \item If $c_k$ moves, with $B$ and every link other than the private links of $k$ fixed, then \[ W_1(Q_B,Q'_B)\le(384\beta_W+2528)\,d(c_k,c'_k). \] \end{enumerate} The constants depend neither on $B$ nor on $\lvert B\rvert$, nor on $c_k$ or the values of the other links; no inside link needs to be near $I$. \end{theorem} \begin{proof} (i) Let $\Phi(g,z)=(\widehat V_k(g;z,u,w),z)$. By Theorem~\ref{thm:projinverse}(i), $\Phi$ is a bijection of $\SU(2)^8$, analytic with the analytic inverse $(c,z)\mapsto(G(c;z,u,w),z)$, and its Jacobian is $J$, the derivative being triangular with the identity in $z$. Haar measure is a constant multiple of the volume of the bi-invariant metric, so the change of variables gives, for bounded measurable $F$, \[ \int F(g,z)\,dg\,dz=\int\!dc\int F\bigl(G(c;z,u,w),z\bigr)\,J^{-1}\,dz , \] with $J$ at $(G(c;z,u,w);z,u,w)$. Taking $F$ equal to $f$ times the Wilson weight and $\mathbf1_B(z)$, and cancelling the plaquettes without a private link as in Lemma~\ref{lem:relnorm}, the normalised inner integral is $Q_B$ at $c=c_k$; it is weakly continuous in $c$ by the continuity of $G$. Lemma~\ref{lem:relnorm} gives $0\le S_{\mathrm{dep}}\le72$, and Theorem~\ref{thm:projinverse}(ii) gives $J\in[8^{-3},(\frac{15}8)^3]$. (ii) Move $e$ along a unit-speed geodesic $e(t)$, $0\le t\le T=d(e,e')$, write $G_t(z)$, $\rho_t$, $Q_t$, and let $\nu_t$ be the law $\rho_t\,dz/Z_t$ on $B$. The link $e$ shares no plaquette with a private link, since the two $\mu$-links of a plaquette have one $\mu$-coordinate; so at fixed $z$, $S_{\mathrm{dep}}$ changes only through the path link $G_t(z)$, which moves at speed at most $\frac12$ by Theorem~\ref{thm:projinverse}(iii) and lies in six plaquettes, each changing at rate at most $\frac12/\sqrt2$, by the proof of Lemma~\ref{lem:wilsononly}. With Theorem~\ref{thm:projinverse}(iv), \[ \lvert\partial_t\log\rho_t(z)\rvert\le\omega:=\frac3{\sqrt2}\beta_W+ \frac{45}4\sqrt2 \] for every $z\in B$ and $t$. Let $f$ be $1$-Lipschitz for the cost, and put $\Phi_t(z)=(G_t(z),z)$. Then \[ Q_T(f)-Q_0(f)=(\nu_T-\nu_0)(f\circ\Phi_T)+\nu_0(f\circ\Phi_T-f\circ \Phi_0). \] In the second term only the path link differs, by at most $\frac T2$ in $d$ and so in $\delta$. In the first, $\frac d{dt}\nu_t(\varphi)=\Cov_{\nu_t} (\varphi,\partial_t\log\rho_t)$ for the fixed bounded function $\varphi= f\circ\Phi_T$, by differentiation under the integral over the fixed set $B$, and $\lvert\Cov(\varphi,\psi)\rvert\le\frac14\operatorname{osc}\varphi\, \operatorname{osc}\psi$; the oscillation of $\varphi$ is at most the diameter $8\cdot2\sqrt2$ of the eight links for the cost, so the first term is at most $\frac14\cdot16\sqrt2\cdot2\omega\,T=8\sqrt2\,\omega\,T$. The normaliser $Z_t$ cancels from the covariance. Hence $Q_T(f)-Q_0(f)\le (\frac12+8\sqrt2\,\omega)T=(24\beta_W+180.5)T$, and the Kantorovich duality gives the bound. (iii) The measure $\rho_B\,dz$ without its Wilson factor, and the map $\Phi$, do not depend on $e'$; the link $e'$ lies in at most one plaquette with a private link, by the proof of Lemma~\ref{lem:wilsononly} and, for an inside link of another direction, by the remark after Theorem~\ref{thm:relresponse}. The proof of Lemma~\ref{lem:wilsononly} gives the bound. (iv) As in (ii), with the path link moving at speed at most $8$ by Theorem~\ref{thm:projinverse}(iii), so that $S_{\mathrm{dep}}$ changes at rate at most $6\cdot8/\sqrt2$, and with the bound $\frac{315}2\sqrt2$ of Theorem~\ref{thm:projinverse}(iv): $\lvert\partial_t\log\rho_t\rvert\le24 \sqrt2\,\beta_W+\frac{315}2\sqrt2$, and $8+8\sqrt2(24\sqrt2\,\beta_W+ \frac{315}2\sqrt2)=384\beta_W+2528$. \end{proof} The theorem applies to the two sets $\SU(2)^7$ and $B^{\mathrm{bad}}_r$ of Proposition~\ref{prop:eightlabels}, defined with $c_k=\widehat V_k$, and to the sets $B_\varepsilon$ of Theorem~\ref{thm:eightlinkuniform}, at every $\varepsilon$; it needs neither the relative chart nor condition (J) of Theorem~\ref{thm:relresponse}, and its constants have no factor $e^{72\beta_W}$. The reason is the following. The obstruction of Theorem~\ref{thm:eightlinkuniform} lives at a point where the projection of the fibre $\{V^\sharp_k=c_k\}$ to the non-path links folds, that is, where $\det D_gV^\sharp_k=0$; as $\varepsilon\to0$ the law over $B_\varepsilon$ becomes the path-link law at the fold, which jumps. By Theorem~\ref{thm:projinverse}(i) and (ii) the fibre of $\widehat V_k$ is, for every value of the inside links, the graph of $z\mapsto G(z)$ over the non-path links themselves. A set of non-path links held fixed while an inside link moves is therefore a fixed set of parameters of the fibre; only the density $\rho_B$ and the position of the path link move, the first with a logarithmic derivative bounded at every point, the second at speed $\frac12$. In the relative chart of Lemma~\ref{lem:relchart} the fibre of $V^\sharp_k$ is a graph over $\xi_a=g^{-1}\ell_a$ instead, a fixed set $B$ of non-path links is the moving set $\{\xi:G(\xi)\xi\in B\}$, and the terms with $\iota$ and $K$ in Theorem~\ref{thm:relresponse} come from this motion and from the jumps of the Jacobian $D$. The bound of Theorem~\ref{thm:projresponse} grows linearly in $\beta_W$; it is a bound on the response of one pair at a fixed coarse field, not a contraction, and it does not give the cube condition (D). On the whole torus the coarse fibre of the projected average has a global product parametrisation. On the periodic lattice of side $2L$, $L\ge2$, in the tree gauge, with $\mathcal B$ blocks, the $49$ active links of a block are the four path links $\ell_k$ of the pairs based at it and $45$ other links, the $28$ non-path private links and the $17$ active links inside it; let $z$ be the family of these $45\mathcal B$ links. \begin{proposition}[projected fibres]\label{prop:projfibre} \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The map $\Psi(U)=(\widehat V(U),z)$ of the active links $U$, \[ \Psi\colon\SU(2)^{49\mathcal B}\to \SU(2)^{4\mathcal B}\times\SU(2)^{45\mathcal B} \] is an analytic diffeomorphism. Its inverse $(c,z)\mapsto\Gamma(c,z)$ puts $G_k=G(c_k;z,u,w)$ of Theorem~\ref{thm:projinverse} on each path link, and its Jacobian is $\prod_kJ_k$, with $J_k$ the Jacobian of the pair $k$. \item For every $c$, the law of the active links given $\widehat V=c$, in the version continuous in $c$, is the image under $\Gamma(c,\cdot)$ of \[ \frac1{Z(c)}\,e^{-\beta_WS(\Gamma(c,z))}\prod_kJ_k^{-1}\,dz , \] whose density is positive at every $z$. The factor of a plaquette depends on $c$ and on links of $z$ based in blocks at $\ell^\infty$ distance at most two from the block of its base point, and $J_k$ on $c_k$ and on links of $z$ based in the two blocks of $k$. \item Let $H_c(z)=\beta_WS(\Gamma(c,z))+\sum_k\log J_k$. Along a unit-speed geodesic in one link of $z$, $\lvert\frac d{dt}H_c\rvert\le6\sqrt2\,\beta_W+ \frac{45}2\sqrt2$. \end{enumerate} \end{proposition} \begin{proof} (i) The paths of a pair $(y',\nu)$ consist of $\nu$-links with base in $B(y')$ or $B(y'+e_\nu)$, and the path link of $k=(y,\mu)$ is a $\mu$-link with $o_\mu=1$ based in $B(y)$, a private link of $k$; so it enters only $\widehat V_k$. A non-path private link of $k$ enters only $\widehat V_k$, and an active inside $\mu$-link of $B(y)$ enters $\widehat V_{y,\mu}$ and $\widehat V_{y-e_\mu,\mu}$; since $L\ge2$ these are two different pairs. For fixed $z$ the map from the path links to $\widehat V$ is therefore the product over $k$ of the maps $g\mapsto\widehat V_k(g;z,u,w)$, each a bijection with analytic inverse by Theorem~\ref{thm:projinverse}(i). The map $(\ell,z)\mapsto(\widehat V,z)$ is a bijection with inverse $\Gamma$; its derivative is triangular, with the identity in $z$ and the block diagonal part $(D_{\ell_k}\widehat V_k)_k$ in the path links, which gives the Jacobian. (ii) The tree gauge has unit Jacobian, so the active links carry product Haar measure times the Wilson weight. The change of variables of (i), as in the proof of Theorem~\ref{thm:projresponse}(i), gives the law, and the density is positive by Theorem~\ref{thm:projinverse}(ii). A plaquette has its links based in blocks at distance at most one from the block of its base point; a path link among them is $G_k$, which depends on $c_k$ and on links based in the two blocks of $k$, at distance at most one further. The Jacobian $J_k$ depends on the same variables as $G_k$. (iii) If the link of $z$ is a non-path link of $k$, only $G_k$ moves, at speed at most $1$ by Theorem~\ref{thm:projinverse}(iii); the two moving links lie in six plaquettes each, which changes $S$ at rate at most $12/\sqrt2=6\sqrt2$, and only $J_k$ changes, at the rate $\frac{45}2\sqrt2$ of Theorem~\ref{thm:projinverse}(iv). If it is an active inside link of $B(y)$ of direction $\mu$, the path links of $(y,\mu)$ and $(y-e_\mu,\mu)$ move at speed at most $\frac12$ each, so $S$ changes at rate at most $(6+3+3)/\sqrt2$, and the two Jacobians at rate at most $\frac{45}4\sqrt2$ each. \end{proof} The coarse curvature of the projected average has a fine witness at every size of the fine plaquettes, with a smaller constant than Lemma~\ref{lem:coarsewitness}. For a transverse offset $a\in\{0,1\}^4$, $a_\mu=0$, let $\lvert a\rvert$ be the number of its nonzero coordinates and $\lvert a\rvert_{>\mu}$ the number of those with index above $\mu$. \begin{proposition}[a fine witness for the projected average]\label{prop:projwitness} Let $P$ be the coarse plaquette at $y$ in the plane $\{\mu,\nu\}$, $\mu<\nu$, let $\kappa\ge0$, and suppose that $\delta(U_p,I)\le\kappa$ for every fine plaquette $p$ whose vertices lie in $B(y)\cup B(y+e_\mu)\cup B(y+e_\nu)\cup B(y+e_\mu+e_\nu)$. Then $c=\widehat V(U)$ satisfies $\delta(c_{y',\mu'}, \ell_{y',\mu'})\le\frac{9-\mu'}2\kappa$ for the four sides $(y',\mu')$ of $P$, and \[ \delta(c_P,I)\le(22-\mu-\nu)\,\kappa\le21\,\kappa . \] \end{proposition} \begin{proof} Take the side $k=(y,\mu)$; the others are the same. Write $x_a=2y+a$ and $x'_a=x_a+e_\mu$, so that $P_{x_a}=u_a\ell_a$ and $P_{x'_a}=\ell_aw_a$ with $u_a=U_{x_a,\mu}$ and $w_a=U_{x_a+2e_\mu,\mu}$, and $P_{x_0}=g=\ell_k$. \emph{The paths from $x_a$.} Let $\nu_1<\dots<\nu_m$ be the directions with $a_\nu=1$, $m=\lvert a\rvert$, and $a^{(j)}=e_{\nu_1}+\dots+e_{\nu_j}$. The $\nu_j$-links at $x_{a^{(j-1)}}$ and at $x_{a^{(j-1)}}+2e_\mu$ have offset $a^{(j-1)}$, whose coordinates of index $\nu_j$ and above vanish; they are tree links. So $P_{x_{a^{(j-1)}}}P_{x_{a^{(j)}}}^{-1}$ is the holonomy of the boundary of the rectangle formed by the plaquettes at $x_{a^{(j-1)}}$ and $x_{a^{(j-1)}}+e_\mu$ in the plane $\{\mu,\nu_j\}$, the product of their holonomies with one of them conjugated, and $\delta(P_{x_{a^{(j-1)}}},P_{x_{a^{(j)}}})\le2\kappa$. Hence $\delta(P_{x_a}, g)\le2\lvert a\rvert\kappa$. \emph{The inside links.} If $\lvert a\rvert_{>\mu}=0$, $u_a$ is a tree link. Otherwise let $\nu>\mu$ be the largest direction with $a_\nu=1$. In the plaquette at $x_a-e_\nu$ in the plane $\{\mu,\nu\}$ the two $\nu$-links have offsets $a-e_\nu$ and $a-e_\nu+e_\mu$, whose coordinates of index $\nu$ and above vanish, so they are tree links and $\delta(u_a,U_{x_a-e_\nu,\mu})\le \kappa$. By induction $\delta(u_a,I)\le\lvert a\rvert_{>\mu}\kappa$, and in $B(y+e_\mu)$ likewise $\delta(w_a,I)\le\lvert a\rvert_{>\mu}\kappa$. Since $P_{x'_a}=u_a^{-1}P_{x_a}w_a$, $\delta(P_{x'_a},g)\le2(\lvert a\rvert+\lvert a\rvert_{>\mu})\kappa$. \emph{The coarse link.} By Lemma~\ref{lem:projection}(i), $\lVert\Pi_\g(g^{-1} P_x)\rVert\le\delta(P_x,g)$. Over the eight transverse offsets $\sum_a\lvert a\rvert=12$ and $\sum_a\lvert a\rvert_{>\mu}=4(3-\mu)$, so \[ \lVert A_k\rVert\le\frac\kappa{16}\sum_a\bigl(4\lvert a\rvert+2\lvert a\rvert_{>\mu}\bigr)=\frac{9-\mu}2\,\kappa , \] and $\delta(c_k,g)=\delta(e^{A_k},I)\le\lVert A_k\rVert$ by Lemma~\ref{lem:logbound}. All plaquettes used lie in $B(y)\cup B(y+e_\mu)$. \emph{The coarse plaquette.} As in the proof of Lemma~\ref{lem:coarsewitness}, the four path links around $P$ form the holonomy $\ell_P$ of the square of side two at $2y$, and $\delta(\ell_P,I)\le 4\kappa$. Replacing the two path links of direction $\mu$ and the two of direction $\nu$ by the coarse links one at a time adds at most $(9-\mu) \kappa+(9-\nu)\kappa$, and $\mu+\nu\ge1$. \end{proof} At the background of Proposition~\ref{prop:roughbackground} the conclusion on the Hessian holds for $\widehat V$ as well: at $U=R(c)$ all sixteen paths of every pair agree, so by Proposition~\ref{prop:projavg}(i) the tangent space of the fibre of $\widehat V$, a submanifold by Theorem~\ref{thm:projinverse}(ii), is that of the fibre of $V^\sharp$ and contains $\xi$; as every first derivative of $S$ vanishes at $U$, the second derivative of $\beta_WS$ along any curve in the fibre with velocity $\xi$ is $-\beta_W$. So the projected average also needs a coarse-curvature label, for which Proposition~\ref{prop:projwitness} gives the witness. \begin{remark}\label{rem:projgroups} For a compact connected group $G\subset\U(N)$ with Lie algebra $\g$, let $\Pi_\g$ be the orthogonal projection of the complex $N\times N$ matrices onto $\g$, which for $\SU(N)$, $N\ge3$, is not the skew-Hermitian part. The proofs above use Lemma~\ref{lem:projection}(iii) only for the constants; with $\lVert\operatorname{ad}X\rVert\le2\lVert X\rVert$ and $\lVert X\rVert_{\rm op}\le\lVert X\rVert$ instead they give, with $s_p=1-\frac1N\ReTr U_p$: the contraction constant $\frac78$, the responses $8$, $1$ and $\frac12$, $J\in[8^{-\dim\g},(\frac{15}8)^{\dim\g}]$, the bounds $\frac{15}2\dim\g$ and $15\dim\g$ in Theorem~\ref{thm:projinverse}(iv), the constant $\frac12+24\beta_W+60\dim\g\sqrt N$ in Theorem~\ref{thm:projresponse}(ii), and $\frac{12}{\sqrt N}\beta_W+15\dim\g$ in Proposition~\ref{prop:projfibre}(iii). Lemma~\ref{lem:projection}(ii) becomes $\lVert\Pi_\g(e^X)-X\rVert\le\frac16\lVert X\rVert_{\rm op}^2\lVert X\rVert$, since the Hermitian part of $e^X$ is orthogonal to $\g$, and Proposition~\ref{prop:projwitness} holds unchanged. \end{remark} \begin{proposition}[the chart for the projected average]\label{prop:projtransfer} Let $G\subset\U(r)$ be compact and connected, $\Pi_\g$ as in Remark~\ref{rem:projgroups}, and replace $V^\sharp$ by $\widehat V$ in the fibre $F$ and in the constrained law. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Under the hypothesis of Lemma~\ref{lem:smallframes}, \[ \Bigl\lVert\widehat V_{y,\mu}(U)^{-1}\frac d{dt}\widehat V_{y,\mu}(Ue^{t \xi})\Big|_{t=0}-V_{y,\mu}(\tilde\xi)\Bigr\rVert\le166\,\kappa\,N_{y,\mu} (\xi). \] \item Proposition~\ref{prop:explicitconstants} holds with the same $c_3$ and $c_6$ for $\SU(2)$, and with $c_3=\max\{1.5\cdot10^4r^2,\dim G\}$ and $c_6=6.81\cdot10^5\dim G\,c_3^2$ for general $G$. \item Theorems~\ref{thm:smallfibre}, \ref{thm:fibrechart}, \ref{thm:chartdecay} and~\ref{thm:offchart} and Proposition~\ref{prop:explicitdecay} hold with these constants; in particular $\alpha_G=2.5\cdot10^{-4}$ and $c_9=1.2\cdot10^4$. \item For $\SU(2)$, Lemma~\ref{lem:privatesubmersion} holds with $\sigma_0= \frac18$ at every field, Lemma~\ref{lem:averagedensity} with $M=512$, and Proposition~\ref{prop:calmfibre} with the same $\varepsilon_W$. \item $\widehat V(R(c))=c$, and Lemma~\ref{lem:coarsewitnesssharp} holds for $\widehat V$ with the same bound $4\kappa+7700\kappa^2$. \end{enumerate} \end{proposition} \begin{proof} Hermitian matrices are orthogonal to $\g$, so $\Pi_\g(w)=\Pi_\g(w-I)$ has norm at most $\delta(w,I)$ for unitary $w$, and for $Y\in\g$, $\Pi_\g(wY)- Y=\Pi_\g((w-I)Y)$ has norm at most $\lVert w-I\rVert_{\rm op}\lVert Y \rVert\le\delta(w,I)\lVert Y\rVert$. (i) Use the notation of the proof of Lemma~\ref{lem:smallframes}(iv), with $A=A_{y,\mu}$ in place of $M$. From $\delta(w_x,I)\le24\kappa$ and the two paths of offset zero, $\lVert A\rVert\le\frac{14}{16}24\kappa=21\kappa$, and $\widehat V^{-1}\dot{\widehat V}=\operatorname{Ad}(e^{-A})\alpha+R \dot A$ with $R=\int_0^1\operatorname{Ad}(e^{-sA})\,ds$, $\lVert R-1\rVert \le\lVert A\rVert$. Here $\dot A=\frac1{16}\sum_x\Pi_\g(w_xY_x)$ with $Y_x= \beta_x-\operatorname{Ad}(w_x^{-1})\alpha$, so $\lVert\dot A\rVert\le m_ \beta+\lVert\alpha\rVert$, and $\Pi_\g(w_xY_x)-(\beta_x-\alpha)=(\Pi_\g(w_x Y_x)-Y_x)+(\alpha-\operatorname{Ad}(w_x^{-1})\alpha)$. The four differences of that proof are at most $42\kappa\lVert\alpha\rVert$, $21\kappa(m_\beta+ \lVert\alpha\rVert)$, $24\kappa m_\beta+72\kappa\lVert\alpha\rVert$ and $42\kappa m_\xi$, and their sum is at most $(135+45\cdot0.434+42/4)\kappa N_{y,\mu}<166\kappa N_{y,\mu}$. (ii) Follow the proof of Proposition~\ref{prop:explicitconstants}. On unitary matrices $\Pi_\g(w)=L(w)$ with $L(w)=P_\C(\frac12(w-w^{-1}))$, $P_\C$ the orthogonal projection onto $\g\otimes\C$ for the Hermitian inner product, which agrees with $\Pi_\g$ on skew-Hermitian matrices; $L$ is holomorphic on $\mathrm{GL}(r,\C)$. Since $e^{X+Z}=e^X(I+F)$ gives $(I+F)^{-1}=I+F'$, $w_x^{-1}$ has the same factorisation as $w_x$, and $\lVert L(w_x)-L(w'_x)\rVert_{\rm op}\le\lVert L(w_x)-L(w'_x)\rVert\le \sqrt r\,\psi$. The two paths of offset zero have $w_x=I$, so the exponent moves by at most $d_L=\frac78\sqrt r\,\psi$, which replaces $d$, and no domain condition on the sites is needed. At real points $\lVert A'-A_0 \rVert\le\frac78\cdot3x_0$, so $\gamma_0=x_0+\frac{21}8x_0<0.0037$. For $\SU(2)$, $r=2$: at $R=0.07$, $\varphi<0.0726$, $\psi<0.2337$, $d_L< 0.2892$, $\eta<0.4322$, $S<0.767$, $6S/R^3<13417$ and $2S/R^2<314$; at $R=10^{-3}$, $S/R<4.8$; on $\lvert z\rvert\le0.1$, $\gamma_0+\eta<0.71$. All are below $1.5\cdot10^4$. For general $G$ take the radii $R/\sqrt r$: by convexity $\varphi(R/\sqrt r)\le\varphi(R)$ and $\sqrt r\,\psi(R/\sqrt r)\le\psi(R)$, so $S(R/\sqrt r)$ is at most the value above, and the Cauchy estimate loses $r^{m/2}\le r^{3/2}$. The derivatives of $\log j$ do not involve the average. (iii) These proofs use the average in three places only: the derivative bound of Lemma~\ref{lem:smallframes}(iv), whose constant $260\kappa$ is improved by (i); the constants $c_3$ and $c_6$; and, in the proof of Theorem~\ref{thm:offchart}(i), the form $V^\sharp=ve^M$ with $\lVert M \rVert\le8\pi\kappa$. For $\widehat V$ the bound $\lVert A\rVert\le21 \kappa$ gives $\delta(v,v')\le42\kappa$ and $\delta(U_e,U'_e)\le84\kappa< 92.3\kappa$. (iv) The derivative in the eight private links satisfies $JJ^*\ge D_g \widehat V(D_g\widehat V)^*\ge\frac1{64}$ by Theorem~\ref{thm:projinverse}(ii). For fixed non-path links, $g\mapsto \widehat V$ is a diffeomorphism with Jacobian at least $8^{-3}$, so the image of Haar measure has density at most $512$, and so has its average over the non-path links. In the proof of Proposition~\ref{prop:calmfibre}, $\frac78\Lambda$ becomes $\frac78$, $1.957$ becomes $\frac{15}8$, and $(512^4(\frac{15}8)^{12})^{1/45}<2.06<2.93$. (v) At $R(c)$ every relative path is $I$. In the proof of Lemma~\ref{lem:coarsewitnesssharp}, Lemma~\ref{lem:projection}(ii) gives $\delta(\widehat c_k,c_k)\le\frac{14}{16}\cdot\frac1{12}a^3$ for each side, and replacing the four $c_k$ by $\widehat c_k$ adds at most $\frac7{24}a^3< 4100\kappa^3<10\kappa^2$ for $\kappa\le10^{-8}$, which the bound $4\kappa+ 7685\kappa^2$ of that proof absorbs. \end{proof} \begin{remark}[what is proved for which map]\label{rem:projscope} The projected average is a different coarse map: it defines different coarse fields $c=\widehat V(U)$ and different conditional laws $\mu^c$. Proved for it are Proposition~\ref{prop:projavg}, Theorems~\ref{thm:projinverse} and~\ref{thm:projresponse}, Corollary~\ref{cor:projpathlaw}, Propositions~\ref{prop:projfibre}, \ref{prop:projwitness} and~\ref{prop:projtransfer}, Lemma~\ref{lem:wilsononly}, whose proof applies unchanged, and the Hessian statement after Proposition~\ref{prop:projwitness}. Through Proposition~\ref{prop:projtransfer} this includes the chart of Theorems~\ref{thm:smallfibre} and~\ref{thm:fibrechart}, the chart decay of Theorem~\ref{thm:chartdecay} and Proposition~\ref{prop:explicitdecay}, and Theorem~\ref{thm:offchart}. The results of Sec.~\ref{ssec:largefields} that concern only the Wilson measure and its plaquettes, Theorem~\ref{thm:defects}, Corollary~\ref{cor:localcalm} and Proposition~\ref{prop:calmrare} among them, do not involve the coarse map. Not proved here for $\widehat V$: the composition across scales of Sec.~\ref{ssec:scales}; the energy bounds of Sec.~\ref{ssec:coarse}, which on torus-valued fields use that $V^\sharp$ is the abelian average of the angles, while $\widehat V$ averages their sines; the labels, sites and chains of Secs.~\ref{ssec:branches} and~\ref{ssec:pathlaw}, Proposition~\ref{prop:nonconvex} among them; the islands, their threshold and the label covariance of Secs.~\ref{ssec:counting} to~\ref{ssec:labelcov}, including Theorems~\ref{thm:badcross} and~\ref{thm:offchartmass} and Proposition~\ref{prop:badchart}; and the one-step and multiscale results of Secs.~\ref{ssec:relstep} to~\ref{ssec:allscales}. Proposition~\ref{prop:projavg}(i) bounds the distance between the two maps; it does not transfer these results. \end{remark} \begin{remark}[the route to (D) with $\widehat V$]\label{rem:projroute} With $\widehat V$ in place of $V^\sharp$, the chart decay holds at $\alpha_G=2.5\cdot10^{-4}$ (Proposition~\ref{prop:projtransfer}), the good side holds at every field (Corollary~\ref{cor:projpathlaw}), and the bad side of item~(i) of Sec.~\ref{ssec:remains} holds for every restricting set, uniformly in the exterior and in the inside links, with no condition (Theorem~\ref{thm:projresponse}). The statements that the cube condition~(D) still needs are items (ii)--(iv) of Sec.~\ref{ssec:remains} with $\widehat V$ for $V^\sharp$: \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item the island comparison: a covariance bound for every coarse field, uniform in the exterior, decaying at the rate $2\alpha_G$ without a floor, and the mixed differences in tree form; \item the island threshold at the scale of the cube; \item the effective action at every scale with summable one-step errors. \end{enumerate} The results that the paper proves towards (a) and (b) for $V^\sharp$, and the energy bounds of Sec.~\ref{ssec:coarse}, are not re-derived for $\widehat V$ (Remark~\ref{rem:projscope}). For $V^\sharp$ the first set can have no chart of the relative kind (Proposition~\ref{prop:chartfolds}), so the bad side is available only through $\widehat V$. The labels and the islands use the following results on the coarse map. Theorem~\ref{thm:cutoff}(ii) holds for $\widehat V$ by Proposition~\ref{prop:projavg}(ii). Parts (i)--(iii) of Lemma~\ref{lem:smallframes} do not involve the coarse map, and part (iv) is Proposition~\ref{prop:projtransfer}(i). Theorems~\ref{thm:smallfibre}, \ref{thm:fibrechart} and~\ref{thm:chartdecay} and Propositions~\ref{prop:explicitconstants} and~\ref{prop:explicitdecay} are Proposition~\ref{prop:projtransfer}(ii), (iii). Lemma~\ref{lem:coarsewitness} is replaced by Proposition~\ref{prop:projwitness}, with its own constant. Theorem~\ref{thm:defects} and Lemma~\ref{lem:wilsonZ} concern the Wilson measure only. Not available for $\widehat V$: Proposition~\ref{prop:smallframessharp}; the fixed-point form of the path-link equation behind Lemma~\ref{lem:branchbound} and Propositions~\ref{prop:goodside} and~\ref{prop:graphlaw}, which the proof of Theorem~\ref{thm:factorisation} differentiates (for $\widehat V$ the equation has one solution at every field, Theorem~\ref{thm:projinverse}, but that form is not derived); the energy bounds of Sec.~\ref{ssec:coarse}, Lemma~\ref{lem:coarseenergy}, Proposition~\ref{prop:coarsechess}, Theorem~\ref{thm:haartilt}, Corollary~\ref{cor:abeliancoarse} and Proposition~\ref{prop:tiltgrowth}; and the composition across scales, Lemma~\ref{lem:torusstep} and Theorem~\ref{thm:torusscales}. Besides (a)--(c), the route through $\widehat V$ needs these, and the counting of the labels redone with the constant of Proposition~\ref{prop:projwitness}. Neither (D) nor the mass gap is proved. \end{remark} \subsection{Counting bad labels}\label{ssec:counting} A label is bad when some fine plaquette in a fixed neighbourhood is bad: for the relative link labels this is the content of Theorem~\ref{thm:cutoff} and Lemma~\ref{lem:smallframes}, and for the coarse-curvature labels it is Lemma~\ref{lem:coarsewitness}. The chessboard estimate of Theorem~\ref{thm:defects} bounds the probability that a set of fine plaquettes is simultaneously bad by a product. The following lemma transfers such a bound to the labels, and the proposition after it shows that the loss it incurs is the true loss of this kind of argument. \begin{lemma}[coverage counting]\label{lem:coverage} Let a family of events $E_y$, indexed by a finite set, be such that each $E_y$ forces at least one of a specified set $Q_y$ of at most $M$ sites to be bad, and suppose that each site lies in at most $H$ of the sets $Q_y$ and that $\Pr(\text{all sites of }T\text{ bad})\le p^{|T|}$ for every finite set $T$ of sites. Then for every nonempty set $S$ of $n$ labels and $k=\lceil n/H\rceil$, \[ \Pr\Bigl(\bigcap_{y\in S}E_y\Bigr)\le\binom{Mn}k\,p^k\le(eMHp)^k , \] and if $A=eMHp<1$ then $\Pr(\bigcap_{y\in S}E_y)\le A^{n/H}$. \end{lemma} \begin{proof} On the intersection, the union of the witness sets $Q_y$, $y\in S$, contains at least $k$ distinct bad sites, since one site witnesses at most $H$ labels; the union has at most $Mn$ sites. A union bound over its $k$-subsets and the product bound for each give the first inequality, and $\binom{Mn}k\le(eMn/k)^k\le(eMH)^k$ since $n/k\le H$. No choice of witness is required to be measurable, and no independence is used. \end{proof} \begin{proposition}[the exponent $1/H$ cannot be improved]\label{prop:coverageopt} In the setting of Lemma~\ref{lem:coverage} let $H$ labels share one site and let each be bad exactly when that site is bad, with probability $p$. Then $\Pr(\text{all }H\text{ labels bad})=p$, so every bound of the form $q^{|S|}$ valid for this class of families has $q\ge p^{1/H}$. \end{proposition} \begin{proof} The event that all $H$ labels are bad is the event that the one site is bad. \end{proof} For the labels of Sec.~\ref{ssec:branches} the witness sets are boxes of side $7$ for the relative link labels, $M=7^4$, $H=4^4$, and of side $11$ when the coarse-curvature labels are included, $M=11^4$, $H=6^4$, with $p$ the chessboard bound of Theorem~\ref{thm:defects} at the threshold of Lemma~\ref{lem:coarsewitness}. Lemma~\ref{lem:coverage} then charges a bad island of $n$ blocks at the rate $A^{n/H}$; the proposition shows that the denominator $H$ is not an artefact of the union bound but the price of knowing only that each label has a witness, and that a better exponent needs joint information on the gauge field. \subsection{Islands and the environment}\label{ssec:environment} Two bad islands are separated by good blocks, and the question is how the weight of one island depends on the state of the other. The following lemma answers it in terms of a single quantity: the cross-ratio of a kernel on the good layer between them. It integrates arbitrary measures on the two islands, including hard constraints and sums over branches, and their masses do not enter. \begin{lemma}[integration through a common kernel]\label{lem:kernel} Let $K(x,y)>0$ be measurable, let $a_0,a_1$ and $b_0,b_1$ be finite positive measures on the spaces of $x$ and of $y$, and suppose that $Z_{ij}=\iint K\,da_i\,db_j$ is finite and positive for the four pairs $i,j\in\{0,1\}$. If \[ e^{-\varDelta}\le\frac{K(x_1,y_1)K(x_0,y_0)}{K(x_1,y_0)K(x_0,y_1)}\le e^{\varDelta} \] for all $x_i$ in the support of $a_i$ and $y_j$ in the support of $b_j$, then \[ \bigl|\log Z_{11}+\log Z_{00}-\log Z_{10}-\log Z_{01}\bigr|\le\varDelta . \] \end{lemma} \begin{proof} Multiply the upper inequality by $K(x_1,y_0)K(x_0,y_1)$ and integrate against $da_1(x_1)\,da_0(x_0)\,db_1(y_1)\,db_0(y_0)$; by Tonelli the left side integrates to $Z_{11}Z_{00}$ and the right side to $e^{\varDelta} Z_{10}Z_{01}$. The lower inequality gives the reverse bound, and the four numbers are positive, so logarithms may be taken. \end{proof} \begin{corollary}[two labels]\label{cor:twolabels} Let $(A,B)$ be a pair of binary labels with joint law $\Pr(A=i,B=j)=Z_{ij} /\sum Z$, all $Z_{ij}>0$, and let $\varDelta$ bound the connected normaliser as in Lemma~\ref{lem:kernel}. Then \[ \begin{gathered} \bigl|\Pr(B=1\mid A=1)-\Pr(B=1\mid A=0)\bigr|\le\tanh\frac\varDelta4,\\ |\Cov(A,B)|\le\frac14\tanh\frac\varDelta4 , \end{gathered} \] and both bounds are attained. \end{corollary} \begin{proof} The conditional odds of $B=1$ given $A=1$ and given $A=0$ are $Z_{11}/ Z_{10}$ and $Z_{01}/Z_{00}$, whose logarithms differ by $\log(Z_{11}Z_{00}/ Z_{10}Z_{01})$, of modulus at most $\varDelta$. For logistic probabilities $\sigma(u)=(1+e^{-u})^{-1}$ the difference $\sigma(u+s)-\sigma(u)$ is largest at $u=-s/2$ by the symmetry $\sigma(-u)=1-\sigma(u)$, where it equals $\sigma(s/2)-\sigma(-s/2)=\tanh(s/4)$, and it increases in $s$. The covariance is $\Pr(A=1)\Pr(A=0)$ times the difference of the conditional probabilities, and $\Pr(A=1)\Pr(A=0)\le\frac14$. The weights $Z_{11}=Z_{00}=e^{\varDelta/2}$, $Z_{10}=Z_{01}=1$ have $\Pr(A=1)=\frac12$ and conditional probabilities $\sigma(\pm\varDelta/2)$, so both bounds are equalities. \end{proof} When the kernel is the partition function of a calm layer, its cross-ratio is a connected second derivative of a log-normaliser, and that derivative is a covariance under the law of the layer. \begin{lemma}[connected normalisers]\label{lem:connected} Let $D$ be a fixed product of balls in the coordinates $z$ of a window, let $\mathcal U(z;\eta,\zeta)$ be smooth in all variables, with $\eta$ and $\zeta$ exterior parameters, and let $Z(\eta,\zeta)=\int_De^{-\mathcal U} dz$. Then for directions $u$ in $\eta$ and $v$ in $\zeta$, \[ \partial_u\partial_v\log Z=-\E\bigl[\partial_u\partial_v\mathcal U\bigr]+ \Cov\bigl(\partial_u\mathcal U,\partial_v\mathcal U\bigr), \] expectation and covariance under $e^{-\mathcal U}dz/Z$ on $D$. If $\mathcal U$ is a sum of local terms and the supports of $\eta$ and $\zeta$ are farther apart than its range, the first term vanishes. If in addition, at every point of two segments $\eta_0\to\eta_1$ and $\zeta_0\to\zeta_1$, the law on $D$ satisfies \[ |\Cov(F,H)|\le\mathcal A\,\lVert\nabla F\rVert\,\lVert\nabla H\rVert\, e^{-\alpha\dist(\operatorname{supp}\nabla F,\operatorname{supp}\nabla H)} \] for all smooth $F$ and $H$, then \[ \begin{split} &\bigl|\log Z(\eta_1,\zeta_1)-\log Z(\eta_1,\zeta_0)-\log Z(\eta_0,\zeta_1) +\log Z(\eta_0,\zeta_0)\bigr|\\ &\qquad\le\mathcal AL_\eta L_\zeta\, \lVert\eta_1-\eta_0\rVert\,\lVert\zeta_1-\zeta_0\rVert\,e^{-\alpha d'}, \end{split} \] where $L_\eta,L_\zeta$ bound the gradients in $z$ of $\partial_u\mathcal U$ and $\partial_v\mathcal U$ per unit of $u,v$ along the segments, and $d'$ is a lower bound there for the distance between the supports of those gradients. \end{lemma} \begin{proof} Differentiating under the integral sign on the fixed domain $D$ gives $\partial_v\log Z=-\E[\partial_v\mathcal U]$ and then the identity, the covariance arising from the derivative of the normalised density. The left side of the integrated bound is the integral of $\partial_u\partial_v \log Z$ over the product of the two segments, in which the first term of the identity vanishes and the second is bounded by the covariance hypothesis. \end{proof} On a good window the law of Proposition~\ref{prop:graphlaw}, with the Hessian bound of Theorem~\ref{thm:chartdecay} on the graph, supplies a covariance bound of the required form, so Lemma~\ref{lem:connected} controls the connected normaliser for continuous changes of the sources inside the chart, and Lemma~\ref{lem:kernel} then integrates the hard island weights on the two sides. What the three statements do not supply is the kernel bound when a label change moves the domain $D$, enters an island, or adds a branch of Proposition~\ref{prop:branches}; that is the open comparison of Sec.~\ref{ssec:remains}. The last result of this subsection shows that the comparison cannot be replaced by rarity, however strong. \begin{proposition}[rarity does not control the environment]\label{prop:hiddenbit} Fix $q\in(0,1)$. Let $S$ be a fair bit and, given $S$, let the labels $B_x$ be independent Bernoulli variables with parameter $q/2$ if $S=0$ and $q$ if $S=1$. Then $\Pr(B_x=1\ \forall x\in T)\le q^{|T|}$ for every finite $T$, and for all distinct $x,y$, \[ \Cov(B_x,B_y)=\frac{q^2}{16} . \] If the configuration is a function of the labels, every covariance under the law conditioned on all labels vanishes, and the covariance of two observables is the covariance of their conditional expectations. More generally, for a binary label $B$ with $p=\Pr(B=1\mid\xi)\in(0,1)$ given the other labels $\xi$ and $\varDelta_F=\mu(F\mid B=1,\xi)-\mu(F\mid B=0,\xi)$, \[ \begin{gathered} \Cov(F,\mathbf 1_B\mid\xi)=p(1-p)\varDelta_F,\\ \Cov\bigl(\E[F\mid B,\xi],\E[H\mid B,\xi]\bigm|\xi\bigr)=p(1-p)\varDelta_F \varDelta_H . \end{gathered} \] \end{proposition} \begin{proof} Given $S$ the labels are independent, so $\Pr(B_x=1\ \forall x\in T)= \frac12((q/2)^{|T|}+q^{|T|})\le q^{|T|}$, and $\Cov(B_x,B_y)=\Var(\Pr(B_x= 1\mid S))=\frac14(q-q/2)^2=q^2/16$ by the conditional covariance formula. The two identities follow by expanding the expectations over the two values of $B$. \end{proof} The example satisfies every finite-intersection rarity bound at every order, yet its labels have a covariance with a positive floor at all distances. A labelling argument must therefore bound $\varDelta_F \varDelta_H$, the change of the conditional expectations under a label flip, and not only the probability of the flip: small $p$ does not help when $\varDelta_F\varDelta_H$ does not decay, and normalising by $p$ can cancel it. That bound is the connected normaliser of Lemma~\ref{lem:kernel}, which is where the island comparison enters. \subsection{Coupling cost and the comparison lemma}\label{ssec:couplingcost} The proofs of Lemmas~\ref{lem:blockK} and~\ref{lem:comparison} take a coupling that minimises a weighted sum of expected displacements and derive, from its minimality, one inequality per site. The following lemma isolates this step; it needs neither stationarity of the optimal coupling under the local updates nor any property of the updates beyond the two stated ones, and its conclusion is exactly the recurrence that Lemma~\ref{lem:comparison} iterates. \begin{lemma}[minimality gives the recurrence]\label{lem:minimality} Let $I$ be a finite index set, let $\Pi$ be a set of couplings of two probability measures on a product of metric spaces indexed by $I$, let $d_i(\pi)=\E_\pi\dist(x_i,x'_i)$, let $w_i>0$, and let $S\subset I$. Suppose $\pi\in\Pi$ minimises $\sum_iw_id_i$ over $\Pi$, and that for each $i\notin S$ there is $\pi^{(i)}\in\Pi$ with $d_j(\pi^{(i)})=d_j(\pi)$ for $j\ne i$ and $d_i(\pi^{(i)})\le\sum_jC_{ij}d_j(\pi)$. Then $d_i(\pi)\le\sum_jC_{ij} d_j(\pi)$ for every $i\notin S$. If $\pi$ is minimal only up to $\varepsilon_i$ against $\pi^{(i)}$, then $d_i(\pi)\le\sum_jC_{ij}d_j(\pi) +\varepsilon_i/w_i$. Consequently, if $C_{ij}\ge0$, $\sum_iw_iC_{ij}\le \theta w_j$ with $\theta<1$ and $d_i(\pi)\le D$ for $i\in S$, then $\sum_{i\notin S}w_id_i(\pi)\le\frac{\theta D}{1-\theta}\sum_{j\in S}w_j$. \end{lemma} \begin{proof} Since $\pi^{(i)}$ changes only the $i$-th displacement, minimality gives $w_id_i(\pi)\le w_id_i(\pi^{(i)})+\varepsilon_i$, and the bound on $d_i(\pi^{(i)})$ gives the recurrence. The last statement is the arithmetic of Lemma~\ref{lem:comparison}: with $X=\sum_{i\notin S}w_id_i$ and $Y= \sum_{i\in S}w_id_i\le D\sum_{i\in S}w_i$, the recurrence and the column condition give $X\le\sum_jd_j\sum_iw_iC_{ij}\le\theta(X+Y)$, so $X\le \theta Y/(1-\theta)$. \end{proof} \begin{lemma}[minimal couplings exist]\label{lem:minimiser} Let $\mu,\nu$ be probability measures on a compact metric space $\Omega$ and $c$ a continuous cost on $\Omega\times\Omega$. The set of couplings of $\mu$ and $\nu$ is nonempty and compact in the weak topology, and $c$ attains its minimum on it. \end{lemma} \begin{proof} The product $\mu\otimes\nu$ is a coupling; the marginal conditions are closed in the weak topology, and the probability measures on the compact space $\Omega\times\Omega$ form a compact set, so the couplings form a compact set on which the continuous functional $\pi\mapsto\int c\,d\pi$ attains its minimum. \end{proof} The displacements $d_i$ are continuous costs on the compact product of copies of $\SU(2)$, so the minimal couplings of Lemma~\ref{lem:blockK} and of the block comparison exist by Lemma~\ref{lem:minimiser}, and the coupled update of one site is the choice of $\pi^{(i)}$ in Lemma~\ref{lem:minimality}. The recurrence, the column-weighted and row-weighted comparisons of Lemma~\ref{lem:comparison}, the decay with a floor of Lemma~\ref{lem:blockcomparison}, and the two lemmas above are also written in Lean~4 in the archive, with Mathlib as the library; the specification of $\SU(2)$ enters only through the hypotheses of Lemma~\ref{lem:minimality}, which the proofs of Secs.~\ref{sec:plaqblocks} and~\ref{sec:bypass} supply by hand. The block response that Lemma~\ref{lem:blockK} bounds can also be computed exactly in one direction, which measures the room that the coupling argument leaves. \begin{proposition}[the block response at zero field]\label{prop:blockresponse} Let $\mu_0$ be the law of three unit quaternions $(y_1,y_2,y_3)$ with density proportional to $\exp(\beta_W\operatorname{Re}(y_1y_2y_3))$ with respect to Haar measure, and $h$ a unit quaternion. Then $f=\ip{y_1}h+ \ip{\overline{y_2y_3}}h$ is $1$-Lipschitz for the sum of the three chordal distances and \[ \begin{split} \Cov_{\mu_0}\bigl(f,\ip{y_1}h\bigr)&=\frac14\Bigl(1+\frac{I_2(\beta_W)} {I_1(\beta_W)}\Bigr)\\ &=\frac14\Bigl(1+\frac{\beta_W}4-\frac{\beta_W^3}{96}+O(\beta_W^5)\Bigr), \end{split} \] which lies between $0.2718$ and $0.27183$ at $\beta_W=0.351$. Hence the Kantorovich modulus of the block in the field at $y_1$ is at least this number, and $10\beta_W$ times it exceeds one for $\beta_W>0.3667$. \end{proposition} \begin{proof} Given $y_1$, the product $w=y_2y_3$ is Haar distributed a priori, and the weight is $\exp(\beta_W\ip{y_1}{\bar w})$ since $\operatorname{Re}(pq)= \ip p{\bar q}$ for unit quaternions. So $\bar w$ given $y_1$ follows the von Mises--Fisher law on $S^3$ with mean direction $y_1$, whose mean resultant is $\E[\bar w\mid y_1]=A(\beta_W)y_1$ with $A=I_2/I_1$, the ratio of the Bessel functions of orders $\frac p2$ and $\frac p2-1$ for $p=4$, and the marginal of $y_1$ is Haar. Multiplication and conjugation preserve the chordal distance, so $f$ changes by at most the sum of the three displacements. Since $y_1$ is uniform, $\Var\ip{y_1}h=\frac14$ and $\Cov(\ip{\bar w}h,\ip{y_1}h)=\E[A\ip{y_1}h^2]=A/4$. The series of $A$ follows from $I_\nu(x)=\sum_m(x/2)^{2m+\nu}/(m!\,\Gamma(m+\nu+1))$, and the derivative of $\mu_b(f)$ in the field $b_1$ at $b=0$ in the direction $h$ is the covariance, which bounds the modulus below. By Lemma~\ref{q-lem:moments}(ii) and~(iv), $0.08721\le A(0.351)\le0.08731$, because $6.373201\le2.52453^2$ and $9.123201\ge3.02046^2$; this gives the two bounds at $0.351$. The function $\frac52\beta_W(1+A(\beta_W))$ increases, since $A'$ is a variance by Lemma~\ref{q-lem:moments}(i), and at $\beta_W=0.3667$, where $0.3667^2+\frac{25}4\le2.52676^2$, Lemma~\ref{q-lem:moments}(iv) gives $A\ge0.3667/4.02676>0.09106> \frac4{3.667}-1$, so it exceeds one there. \end{proof} \subsection{From one scale to the next}\label{ssec:scalestep} A multiscale version of Sec.~\ref{ssec:scales} would eliminate the fine coordinates of a window scale by scale. The following statements say what one elimination step preserves and what it does not, for the quadratic forms that Theorem~\ref{thm:chartdecay} controls at one scale. They are finite matrix lemmas; no all-scale construction is made here. \begin{lemma}[weighted Schur complements]\label{lem:schur} Let $H=\bigl(\begin{smallmatrix}A&B\\B^*&D\end{smallmatrix}\bigr)$ be positive definite in retained and eliminated coordinates, with $S=A-BD^{-1}B^*$ its constrained minimum form on the retained ones. Give both index sets a metric, and for a $1$-Lipschitz function $\phi$ and $\alpha\ge0$ write $\lVert M\rVert_\alpha$ for the supremum over $\phi$ of the operator norms of $e^{\alpha\phi}Me^{-\alpha\phi}$. If $\lVert A \rVert_\alpha\le a$, $\lVert B\rVert_\alpha\le b$ and $\lVert D^{-1} \rVert_\alpha\le g$, then $\lVert S\rVert_\alpha\le a+b^2g$, and $|S_{ij}|\le(a+b^2g)e^{-\alpha\dist(i,j)}$. If $H_0$ has blocks $A_0,B_0, D_0$ with the same bounds and $\lVert A-A_0\rVert_\alpha,\lVert B-B_0 \rVert_\alpha,\lVert D-D_0\rVert_\alpha\le\varepsilon$, then \[ \lVert S-S_0\rVert_\alpha\le\varepsilon(1+bg)^2 . \] \end{lemma} \begin{proof} Conjugation by $e^{\alpha\phi}$ respects products and inverses, so the weighted norm is submultiplicative on compatible blocks, and $\lVert B^* \rVert_\alpha=\lVert B\rVert_\alpha$ because $-\phi$ is admissible; the bound on $S$ follows from its formula. For the entry bound take $\phi(x)= \dist(x,j)$. For the difference, $S-S_0=(A-A_0)-(B-B_0)D^{-1}B^*-B_0D^{-1}(B-B_0)^*-B_0(D^{-1}-D_0^{-1}) B_0^*$ with $D^{-1}-D_0^{-1}=-D^{-1}(D-D_0)D_0^{-1}$, and the four terms are bounded by $\varepsilon$, $\varepsilon bg$, $\varepsilon bg$ and $\varepsilon b^2g^2$. \end{proof} \begin{proposition}[what form comparison preserves]\label{prop:formstability} Let $H_0,H$ be positive definite with $(1-\varepsilon)H_0\le H\le(1+ \varepsilon)H_0$, $0\le\varepsilon<1$, and for a surjective linear map $Q$ let $S(v)=\inf_{Qx=v}x^{\mathsf T}Hx$ and $S_0$ likewise. Then $(1- \varepsilon)S_0\le S\le(1+\varepsilon)S_0$. Form comparison does not preserve locality: on a path of $n$ sites, $H_0=I$ and $H=I+\varepsilon uu^{\mathsf T}$ with $u=(1,\dots,1)/\sqrt n$ satisfy the comparison, while $(H^{-1})_{1n}=-\varepsilon/((1+\varepsilon)n)$ decays in no exponential way. If instead, for $W=e^{\alpha\phi}$, $\lVert WH_0^{-1}W^{-1}\rVert\le G_0$ and $\lVert W(H-H_0)W^{-1}\rVert\le\varepsilon/G_0$, then $\lVert WH^{-1}W^{-1}\rVert\le G_0/(1-\varepsilon)$. \end{proposition} \begin{proof} The comparison of forms passes to infima over the same affine subspaces. The rank-one inverse is $H^{-1}=I-\frac\varepsilon{1+\varepsilon}uu^{\mathsf T}$, whose off-diagonal entries all equal $-\varepsilon/((1+\varepsilon) n)$. The last statement is the Neumann series $H^{-1}=\sum_m(-H_0^{-1}(H- H_0))^mH_0^{-1}$ in the weighted norm. \end{proof} Repeating the comparison over scales multiplies the factors $(1\pm \varepsilon_k)$, so the one-step errors must be summable, and Lemma~\ref{lem:schur} identifies the one-step amplification $(1+b_kg_k)^2$ of the locality norm. Both are conditions that an effective action built from Theorem~\ref{thm:chartdecay} at every scale would have to meet; the theorem at one scale provides the bounds $a,b,g$ for the first step only. An error bound that is a multiple of the Euclidean form does not survive this repetition, because the block constraint enlarges it at every step. \begin{proposition}[absolute errors grow under the block map]\label{prop:massgrowth} Let $U$ have plaquettes at most $\kappa$, let $B$ be the derivative at $U$ of the block averages of a window, as in Theorem~\ref{thm:smallfibre}, and $\mathcal G=BB^*$. For every positive semidefinite form $N$ on the link coordinates, every $\varepsilon\ge0$ and every $w$ in the range of $B$, \[ \inf_{BX=w}\bigl(N(X)+\varepsilon\lVert X\rVert^2\bigr)\ge\inf_{BX=w}N(X) +\frac{4\varepsilon}{1+2724\kappa}\lvert w\rvert^2 . \] \end{proposition} \begin{proof} The infimum of a sum is at least the sum of the infima, and $\inf_{BX=w}\lVert X\rVert^2=\langle w,\mathcal G^{-1}w\rangle$, attained at $X=B^*\mathcal G^{-1}w$. By the proof of Theorem~\ref{thm:smallfibre}(iii), a row of $\mathcal G$ has its diagonal block of norm at most $\frac3{16}+227\kappa$ and two further blocks of norm at most $\frac1{32}+227\kappa$, so the Schur test gives $\lVert\mathcal G \rVert\le\frac14+681\kappa$ and $\mathcal G^{-1}\ge4/(1+2724\kappa)$. \end{proof} A one-step bound of the form $\lvert\operatorname{Hess}-H_{\mathrm{ref}} \rvert\le\lambda\varepsilon\lVert\cdot\rVert^2$ on the fine links, which is the form in which Theorems~\ref{thm:smallfibre} and~\ref{thm:fibrechart} and Proposition~\ref{prop:explicitdecay} control every non-Gaussian term, is therefore met by $H_{\mathrm{ref}}+\lambda\varepsilon\lVert\cdot \rVert^2$, whose block infimum exceeds that of $H_{\mathrm{ref}}$ by $4\lambda\varepsilon(1+2724\kappa)^{-1}\lvert w\rvert^2$. Carried over $k$ blocking steps with the Gaussian fibre gap independent of the scale (Proposition~\ref{prop:effective}(iv)), such a bound certifies the gap of the fibre only while $4^k\varepsilon$ stays small against it, that is, for a number of scales of order $\log_4(1/\varepsilon)$: an absolute error is a mass term and is relevant in four dimensions. The comparison of Proposition~\ref{prop:formstability} is relative and is preserved with factor one, so the one-step errors that an effective action needs are relative bounds by a gauge-invariant form; since the block action $\inf_{BX=w}Q_U(X)$ vanishes along the coarse gauge directions, such a bound requires the one-step error to vanish along them as well. \subsection{Spectral weight at positive time}\label{ssec:spectralweight} Theorem~\ref{thm:flowlimit} makes the nontriviality of the continuum limit equivalent to a ratio of lattice two-point functions staying away from zero. The following two lemmas show that a ratio at a fixed physical time controls the spectral weight of the renormalised observable, so that the ratio criterion is a tightness condition on its spectral measure and not on its unsmeared moments, and the example after them shows that a mass gap by itself does not give nontriviality. \begin{lemma}[tightness at positive time]\label{lem:positivetime} Let $\sigma_a$ be positive measures on $[0,\infty)$ with Laplace transforms $C_a(t)=\int e^{-tE}d\sigma_a(E)$, let $s>0$ be fixed, and suppose $00$ uniformly in $a$. Let $d\nu_a=e^{-sE}d\sigma_a/C_a(s)$. Then \[ \begin{gathered} \int E\,d\nu_a\le M_s=\frac2s\log\frac1c\le\frac2{esc},\qquad \nu_a([R,\infty))\le\frac{M_s}R,\\ \frac{C_a(s+t)}{C_a(s)}\ge e^{-M_st}\quad(t\ge0). \end{gathered} \] \end{lemma} \begin{proof} The Laplace transform $C_a$ is finite on $[s/2,\infty)$, smooth on $(s/2,\infty)$ with $C_a'=-\int Ee^{-tE}d\sigma_a$, and $\log C_a$ is convex there by the Cauchy--Schwarz inequality $C_a(\frac{t+t'}2)^2\le C_a(t) C_a(t')$. A convex function has derivative at $s$ at least the slope of its chord over $[s/2,s]$, so \[ \int E\,d\nu_a=-(\log C_a)'(s)\le\frac2s\log\frac{C_a(s/2)}{C_a(s)}\le \frac2s\log\frac1c , \] and $\log u\le u/e$ gives $M_s\le2/(esc)$. Markov's inequality gives the tail, and convexity gives $\log C_a(s+t)-\log C_a(s)\ge t(\log C_a)'(s)\ge -M_st$. \end{proof} When $\sigma_a$ is the spectral measure of a centred observable $O_a$ for a nonnegative Hamiltonian, $\nu_a$ is the spectral law of the unit vector $e^{-sH_a/2}O_a\Omega_a/C_a(s)^{1/2}$; the lemma keeps its spectral mass from escaping to infinite physical energy. The ratio at the two separations $s/2$ and $s$ is the quantity that Proposition~\ref{prop:sliceratio} bounds by slice correlations. \begin{lemma}[nonvanishing limits]\label{lem:nonvanishing} If $\sigma_a([0,\infty))\ge v>0$ and $\int E\,d\sigma_a\le M$ uniformly in $a$, then $C_a(t)\ge ve^{-Mt/v}$ for all $t\ge0$, and every limit of the $C_a(t)$ is positive. \end{lemma} \begin{proof} With $v_a=\sigma_a([0,\infty))$, Jensen's inequality for the probability $\sigma_a/v_a$ gives \[ C_a(t)\ge v_a\exp\Bigl(-\frac t{v_a}\int E\,d\sigma_a\Bigr)\ge v_ae^{-Mt/v_a}, \] and $x\mapsto xe^{-Mt/x}$ is increasing. \end{proof} \begin{proposition}[a gap without a limit]\label{prop:gapnolimit} Let $X$ be the symmetric two-state Markov process with jump rate $m/2$ in each direction, sampled at times $na$, and let $O_a=a\,X$. The transfer matrix has eigenvalues $1$ and $e^{-ma}$, so the gap in physical units is $m$ for every $a$, while $C_a(t)=a^2e^{-mt}\to0$ and every moment of $O_a$ tends to zero. The sampled process is reflection positive, and the spectral measure $\sigma_a$ is the atom $a^2\delta_m$, whose Laplace transform at $s/2$ and $s$ has the ratio $e^{-ms/2}$, uniform in $a$; Lemma~\ref{lem:positivetime} applies, and the limit vanishes because $\sigma_a([0,\infty))=a^2\to0$, which Lemma~\ref{lem:nonvanishing} excludes by its hypothesis on the mass. \end{proposition} \begin{proof} The correlation $\E[X_0X_t]=e^{-mt}$ for the two-state process, and $O_a$ scales it by $a^2$; the transfer matrix of the sampled chain is the matrix of its transition probabilities, with the stated eigenvalues. Reflection positivity of a stationary reversible Markov chain is standard. The rest is read off. \end{proof} The example separates the two conditions that Theorem~\ref{thm:flowlimit} keeps apart: a uniform gap and a nonvanishing ratio. For a sequence of measures whose mass does not vanish, Lemma~\ref{lem:nonvanishing} converts a bound on the first moment at positive time, which Lemma~\ref{lem:positivetime} derives from one ratio, into a positive lower bound for the limit; the mass itself is the normalisation of Theorem~\ref{thm:flowlimit}. \subsection{The island threshold at one scale}\label{ssec:islandthreshold} The counting of Lemma~\ref{lem:coverage} and the collar of Proposition~\ref{prop:graphlaw} determine a coupling past which the bad blocks of one scale form islands whose entropy is beaten by their rarity. This subsection computes that coupling. Fix a radius $r$ for the relative link labels and a radius $\eta$ for the coarse plaquettes: a block $y$ is \emph{bad} when some non-path active link of a pair based at $y$ is at distance more than $r$ from its reconstruction value, or some coarse plaquette with corner $y$ has $\delta(c_P,I)>\eta$, and \emph{good} otherwise. Fix an integer collar $s\ge1$: a window $W$ is conditioned as in Proposition~\ref{prop:graphlaw} when every block at $\ell^\infty$ distance at most $s$ from $W$ is good. Join two blocks when they are at $\ell^\infty$ distance at most $2s+1$, and call a maximal connected set of bad blocks an \emph{island}; two islands are then separated by a layer of good blocks of width at least $2s+1$, whose middle blocks are at distance more than $s$ from both, and the good layer of Lemma~\ref{lem:kernel} is available in the middle of it. Each block has $\varDelta=(4s+3)^4-1$ neighbours in this graph, and the witness sets of Lemma~\ref{lem:coverage} are the boxes of side $11$, $M=11^4$, $H=6^4$. \begin{theorem}[the island threshold]\label{thm:islandthreshold} Let $G=\SU(2)$, let $\rho=\min\{r/43,\eta/92\}\le10^{-8}$, and put \begin{gather*} \log q^*=-H\bigl(\log2+2\log\varDelta\bigr)-\log(eMH),\\ f(\beta)=6(\log\beta+C_0)-\frac{\beta\rho^2}{16}, \end{gather*} with $C_0$ as in Lemma~\ref{lem:wilsonZ}. The equation $f(\beta)=\log q^*$ has a largest solution $\beta^*=\beta^*(\rho,s)>96/\rho^2$, and for every $\beta_W\ge\beta^*$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item every block is bad with probability at most $A=eMH\,q_\rho(\beta_W) \le(2\varDelta^2)^{-H}$; \item for every block $y$ and every $n\ge1$, \[ \mu_L\{y\text{ lies in an island of at least }n\text{ blocks}\}\le 2^{-n}\varDelta^{-2}, \] and the expected number of islands of at least $n$ blocks on a torus of $L^4$ blocks is at most $L^4\,2^{-n}\varDelta^{-2}/n$; \item if an estimate charges every bad block a factor $\varGamma\ge1$ and is summed over the islands through a given block, the sum $\sum_{n\ge1}\varDelta^{2(n-1)}(\varGamma A^{1/H})^n$ is at most $\varDelta^{-2}$ as soon as $\beta_W\ge\beta^*(\rho,s,\varGamma)$, the largest solution of $f(\beta)=\log q^*-H\log\varGamma$. \end{enumerate} In particular \begin{gather*} \beta^*(10^{-8},1)\in[3.426,3.428]\cdot10^{21},\\ \beta^*(10^{-8},5)\in[5.400,5.402]\cdot10^{21},\\ \beta^*(10^{-9},1)\in[3.430,3.432]\cdot10^{23},\\ \beta^*(10^{-9},5)\in[5.404,5.406]\cdot10^{23},\\ \beta^*(10^{-8},5,10^{30})\in[1.972,1.974]\cdot10^{22}; \end{gather*} in every case $\beta^*=16(\lvert\log q^*\rvert+H\log\varGamma)\rho^{-2} (1+\epsilon)$ with $0<\epsilon<2\cdot10^{-2}$, and $\lvert\log q^*\rvert$ lies within $10^{-3}$ of $21090.194$ for $s=1$ and of $33424.872$ for $s=5$. \end{theorem} \begin{proof} \emph{Witnesses.} Let $y$ be bad because of a non-path link $e$ of the pair $k=(y,\mu)$, and let $\kappa$ be the largest defect $\delta(U_p,I)$ over the plaquettes with a vertex in a block at $\ell^\infty$ distance at most one from $B(y)$ or $B(y+e_\mu)$. If $\kappa\le10^{-8}$, Lemma~\ref{lem:smallframes}(i) with $W=\{y,y+e_\mu\}$ gives $\delta(U_e,I)\le3\kappa$ when $e$ lies inside a block, where $R_e(c)=I$, and $\delta(U_e,\ell_k)\le21\kappa$ when $e$ leaves $B(y)$, where $R_e(c)=c_k$ and $\ell_k$ is the path link; the proof of Lemma~\ref{lem:coarsewitness} gives $\delta(c_k,\ell_k)\le7\pi \kappa$, so $\delta(U_e,R_e(c))\le(21+7\pi)\kappa<42.992\kappa$; since $d(g,h)\le(1+10^{-12})\delta(g,h)$ when $\delta(g,h)\le10^{-6}$, the geodesic distance of the label obeys $d(U_e,R_e(c))<43\kappa$. Since the label is bad, either $\kappa>10^{-8}\ge\rho$ or $43\kappa>r$, and in both cases some plaquette of that set has defect more than $\rho$. If $y$ is bad because of a coarse plaquette with corner $y$, Lemma~\ref{lem:coarsewitness} gives in the same way a plaquette with defect more than $\min\{10^{-8}, \eta/92\}\ge\rho$ among those with a vertex in a block at distance at most one from the four corner blocks. All these plaquettes lie in cubes whose corners are at $\ell^\infty$ distance at most five from $2y$, the box of side $11$ of the fine lattice, and a cube lies in the box of at most $6^4$ blocks. With the sites of Lemma~\ref{lem:coverage} the unit cubes and $p=q_\rho(\beta_W)$, Theorem~\ref{thm:defects} gives the product bound, and the lemma gives $\mu_L(\text{all blocks of }S\text{ bad})\le A^{n/H}$ for every set $S$ of $n$ blocks when $A=eMHp<1$, and $\le A$ for one block. \emph{The threshold.} The function $f$ is concave on $(0,\infty)$ with its maximum at $\beta=96/\rho^2$, where $f=6\log(96/\rho^2)+6C_0-6>0>\log q^*$; since $f(\beta)\to-\infty$, the equation $f=\log q^*$ has exactly two solutions and the larger one, $\beta^*$, exceeds $96/\rho^2$, with $f\le \log q^*$ on $[\beta^*,\infty)$. For $\beta_W\ge\beta^*$ this gives $q_\rho(\beta_W)\le e^{f(\beta_W)}\le q^*$, hence $A\le eMHq^*=(2\varDelta^2) ^{-H}$ and $\varDelta^2A^{1/H}\le\frac12$, which is (i). \emph{Islands.} A connected set of $n$ blocks containing $y$ in the graph of the collar is recovered from a walk of length $2(n-1)$ around a spanning tree, each step to one of $\varDelta$ neighbours, so there are at most $\varDelta^{2(n-1)}$ of them. The island of $y$ has at least $n$ blocks only if some connected set of exactly $n$ blocks containing $y$ consists of bad blocks, and a given set of $n$ blocks is bad with probability at most $A^{n/H}$, so \[ \begin{split} \mu_L\{\text{the island of $y$ has at least $n$ blocks}\} &\le\varDelta^{2(n-1)}A^{n/H}\\ &\le\varDelta^{-2}\,2^{-n}. \end{split} \] An island of at least $n$ blocks contains at least $n$ blocks that lie in islands of at least $n$ blocks, so the number of such islands is at most $1/n$ times the number of such blocks, whose expectation is at most $L^4\,2^{-n}\varDelta^{-2}$; on a torus the number of neighbours of a block is at most $\varDelta$, which is all the count uses. For (iii), replace $A$ by $\varGamma^HA$ in the threshold: then $\varGamma\varDelta^2 A^{1/H}\le\frac12$ and the sum is at most $\varDelta^{-2}\sum_n2^{-n}= \varDelta^{-2}$. For the numbers, $\lvert\log q^*\rvert=H(\log2+2\log\varDelta) +1+4\log11+4\log6$, with $\varDelta=2400=2^5\cdot3\cdot5^2$ for $s=1$ and $\varDelta=23^4-1$ for $s=5$. The series $2\operatorname{artanh}x=2\sum_k x^{2k+1}/(2k+1)$, with its tail bounded geometrically, at $x=\frac13,\frac15, \frac19,\frac1{21},\frac1{47}$ gives $\log2$, $\log\frac32$, $\log\frac54$, $\log\frac{11}{10}$ and $\log\frac{24}{23}$ to within $10^{-9}$, hence $\lvert\log q^*\rvert$ to within $10^{-4}$, and with $3.14159265<\pi< 3.14159266$ it gives $C_0=\frac12+\frac23\log(24\pi^3)\in[4.90816,4.90817]$. The function $g(\beta)=\beta\rho^2/16-6(\log\beta+C_0)-\lvert\log q^*\rvert- H\log\varGamma$ increases for $\beta>96/\rho^2$ and $\beta^*$ is its largest zero; at the two ends of each stated interval $g$ is negative and positive, with margins above $0.5$, so that $\log\beta$ to two decimals decides the signs. Since $\epsilon=6(\log\beta^*+C_0)/(\lvert\log q^*\rvert+H\log \varGamma)$ and $\log\beta^*<55$, $0<\epsilon<2\cdot10^{-2}$. \end{proof} The theorem gives the value asked for in item (iii) of the earlier versions of this section, but only at the block scale and in lattice units: at $\beta_W=5\cdot10^{21}$ the lattice spacing is far below every physical length, the correlation length of the theory in lattice units is astronomically larger than the side of a block, and the rarity of the islands at that scale says nothing yet about the islands of the effective theory at the scale of the cube, which is what condition (D) needs. The threshold therefore closes the entropy count of one scale, and leaves the comparison between scales to the effective action of Proposition~\ref{prop:formstability} and Lemma~\ref{lem:schur}. The numbers also show where the loss sits: the factor $16/\rho^2$ comes from the witness radius $\rho\le10^{-8}$ of Lemma~\ref{lem:smallframes}, and the factor $\lvert\log q^*\rvert\approx2H\log\varDelta$ from the exponent $1/H$, which Proposition~\ref{prop:coverageopt} shows cannot be improved without joint information on the gauge field. \subsection{The island threshold at every scale}\label{ssec:islandscales} The threshold of Theorem~\ref{thm:islandthreshold} composes with the defect propagation of Theorem~\ref{thm:defectscales}: the labels of the $j$-th iterate of the cut-off average have fine witnesses as well, in a larger box, so that the chessboard estimate and the coverage count apply at every scale up to the one that Corollary~\ref{cor:defectscales} reaches, divided by $256$. Write $U^{(j)}$ for the $j$-th iterate in the tree gauge, an $\SU(2)$ field on the lattice of spacing $a_j=2^ja$, and call a block $y$ of that lattice \emph{bad at scale $j$} when it is bad for the field $U^{(j)}$ in the sense of Theorem~\ref{thm:islandthreshold}, with the same radii $r$, $\eta$ and $\rho=\min\{r/43,\eta/92\}\le10^{-8}$. Let $\mathfrak B_j$, $\eta_j$ be as in Theorem~\ref{thm:defectscales} with $\eta_0=1.01\rho_0$ for a threshold $\rho_0\le\rho$, so that every plaquette of $\mathfrak B_0$ has defect more than $\rho_0$. \begin{lemma}[fine witnesses at scale $j$]\label{lem:scalewitness} Let $j\ge1$ and suppose that $\eta_j\le\rho$. If the block $y$ is bad at scale $j$, then some fine plaquette $p\in\mathfrak B_0$, with defect more than $\rho_0$, has its corner $x$ in $2^{j+1}y+[-8\cdot2^j+2,\,9\cdot2^j-3]^4$. The cubes of this box number $M_j=(17\cdot2^j-4)^4$, and a fine cube lies in the box of at most $H_j$ blocks of scale $j$, with $H_j=8^4$ for $j=1,2$ and $H_j=9^4$ for $j\ge3$. \end{lemma} \begin{proof} Since $\eta_{i+1}\ge4\eta_i$, the hypothesis gives $\eta_i\le\rho\le 10^{-8}$ for $i\rho$. If $P\notin\mathfrak B_j$, Theorem~\ref{thm:defectscales}(i) and Lemma~\ref{lem:logbound} give $\delta(U^{(j)}_P,I)\le\lVert\Theta^{(j)}_P \rVert\le\eta_j\le\rho$; so $P\in\mathfrak B_j$, and by part (ii) some point of $P$ is within $\ell^\infty$ distance $3(2^j-1)$, in fine units, of a plaquette $p\in\mathfrak B_0$ with corner $x$. In fine units the coordinates of $P$ lie in $[2^jx'_c,2^jx'_c+2^j]$ and those of $p$ in $[x_c,x_c+1]$, so $x_c\in[2^jx'_c-3\cdot2^j+2,\,2^jx'_c+4\cdot2^j-3]$, and the range of $x'_c$ gives the box. It has $17\cdot2^j-4$ integer points per coordinate. A cube with corner $x$ lies in the box of $y$ exactly when $2^{j+1}y_c$ lies in an interval of length $17\cdot2^j-5$ for each coordinate, which contains at most $\lfloor8.5-5\cdot2^{-j-1}\rfloor+1$ multiples of $2^{j+1}$, that is $8$ for $j=1,2$ and $9$ for $j\ge3$. \end{proof} \begin{theorem}[the island threshold at scale $j$]\label{thm:islandscales} Let $G=\SU(2)$, $s\ge1$, $j\ge1$, $\rho=\min\{r/43,\eta/92\}\le10^{-8}$, and put $\rho_0=\rho/(1.01\cdot1.023\cdot4^j)$, $\eta_0=1.01\rho_0$, \begin{gather*} \log q^*_j=-H_j\bigl(\log2+2\log\varDelta\bigr)-\log(eM_jH_j),\\ f_j(\beta)=6(\log\beta+C_0)-\frac{\beta\rho_0^2}{16}, \end{gather*} with $\varDelta=(4s+3)^4-1$ and $M_j$, $H_j$ as in Lemma~\ref{lem:scalewitness}. Let $\beta^*_j$ be the largest solution of $f_j(\beta)=\log q^*_j$. Then for every $\beta_W\ge\beta^*_j$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\eta_i\le1.023\cdot4^i\eta_0\le\rho$ for $i\le j$, the iterates up to scale $j$ satisfy Theorem~\ref{thm:defectscales}(i) and (iii) with $E^\circ_j\le1.53\,E^\circ_0$, and every block at scale $j$ is bad with probability at most $A_j=eM_jH_jq_{\rho_0}(\beta_W)\le(2\varDelta^2)^{-H_j}$; \item for every block $y$ at scale $j$ and every $n\ge1$, with the islands of Theorem~\ref{thm:islandthreshold} formed from the bad blocks of scale $j$, \[ \mu_L\{y\text{ in an island of at least }n\text{ blocks}\}\le2^{-n} \varDelta^{-2}, \] and the expected number of such islands on the torus is at most $(L/2^{j+1})^42^{-n}\varDelta^{-2}/n$; the weighted variant (iii) of Theorem~\ref{thm:islandthreshold} holds with $\log q^*_j-H_j\log\varGamma$. \end{enumerate} The thresholds increase, $\beta^*_{j+1}>16\beta^*_j$. Let $j_{\max}$ be the largest $j\ge0$ with $\beta_W\ge\beta^*_j$, where $\beta^*_0=\beta^*(\rho,s)$ is the threshold of Theorem~\ref{thm:islandthreshold}; the conclusions hold at every scale $j\le j_{\max}$. For $s=5$ and $\rho=10^{-8}$, \begin{gather*} \beta^*_1\in[2.895,2.897]\cdot10^{23},\quad \beta^*_2\in[4.633,4.635]\cdot10^{24},\\ \beta^*_3\in[1.185,1.187]\cdot10^{26}, \end{gather*} and $16^{-j}\beta^*_j\le2.90\cdot10^{22}$ for $1\le j\le12$. Hence, for $\beta^*_1\le\beta_W<\beta^*_{12}$, \[ 2^{j_{\max}}\ge\frac12\Bigl(\frac{\beta_W}{2.90\cdot10^{22}}\Bigr)^{1/4}, \] and for every $\beta_W\in[10^{22},10^{42}]$ the scale $k_*$ of Corollary~\ref{cor:defectscales} satisfies $k_*-j_{\max}\in\{8,9\}$: the islands of the composed average are rare up to the scale $a_{k_*}/512$. \end{theorem} \begin{proof} (i) $4^j\eta_0=\rho/1.023\le10^{-8}$, so Theorem~\ref{thm:defectscales}(iv) gives the bounds on $\eta_i$ and $E^\circ_j$, and $\eta_j\le\rho$ is the hypothesis of Lemma~\ref{lem:scalewitness}. That lemma makes the sets of Lemma~\ref{lem:coverage} the boxes of $M_j$ cubes with overlap $H_j$, and Theorem~\ref{thm:defects} at the threshold $\rho_0$ gives the product bound $p=q_{\rho_0}(\beta_W)$; the threshold paragraph of the proof of Theorem~\ref{thm:islandthreshold}, with $\rho_0$, $M_j$, $H_j$, $q^*_j$ in place of $\rho$, $M$, $H$, $q^*$, shows that $\beta^*_j>96/\rho_0^2$ exists and that $q_{\rho_0}(\beta_W)\le q^*_j$ for $\beta_W\ge\beta^*_j$, hence $A_j\le(2\varDelta^2)^{-H_j}$ and $\varDelta^2A_j^{1/H_j}\le\frac12$. (ii) is the island paragraph of that proof on the torus of $(L/2^{j+1})^4$ blocks of scale $j$, where a block has at most $\varDelta$ neighbours. \emph{The thresholds.} Put $c=1.01\cdot1.023$, so that $16c^2\le 17.0810278$ and $\rho_0=\rho/(c\,4^j)$, and $g_j(\beta)=\beta\rho_0^2/16- 6(\log\beta+C_0)-\lvert\log q^*_j\rvert$; $g_j$ increases for $\beta>96/ \rho_0^2$, and $\beta^*_j$ is its largest zero, so that $\beta^*_j\le\beta$ exactly when $g_j(\beta)\ge0$, for $\beta>96/\rho_0^2$. Since $\rho_0^2$ at $j+1$ is $\rho_0^2/16$ at $j$, $g_{j+1}(16\beta)=g_j(\beta)-6\log16-(\lvert\log q^*_{j+1}\rvert-\lvert\log q^*_j\rvert)$, and $\lvert\log q^*_j\rvert$ increases with $j$ because $H_j$ and $M_j$ do; at $\beta=\beta^*_j$ this is negative, so $\beta^*_{j+1}> 16\beta^*_j$. In particular $\beta^*_j\le\beta_W$ for every $j\le j_{\max}$, and if $\beta_W<\beta^*_{12}$ then $\beta_W<\beta^*_{j_{\max}+1}\le 16^{j_{\max}+1}\cdot2.90\cdot10^{22}$ by the bound proved below, which is the displayed bound. \emph{The numbers.} We have \[ \lvert\log q^*_j\rvert=H_j(\log2+2\log\varDelta)+1+4\log(17\cdot2^j-4)+ \log H_j \] with $\varDelta=23^4-1$. The logarithms of the proof of Theorem~\ref{thm:islandthreshold}, together with $\log17=4\log2+2 \operatorname{artanh}\frac1{33}$, give $\lvert\log q^*_1\rvert$, $\lvert\log q^*_2\rvert$ and $\mathsf A=6561(\log2+2\log\varDelta)+1+8\log3$ within $10^{-3}$ of $105605.899$, $105608.929$ and $169133.301$. For $j\ge3$, $\lvert\log q^*_j\rvert=\mathsf A+4\log(17\cdot2^j-4)$, and $\mathsf A+4 \log17\le169144.64$, so that \[ 169152.831\le\lvert\log q^*_3\rvert\le\lvert\log q^*_j\rvert\le169144.64+ 4j\log2 . \] With $C_0\in[4.90816,4.90817]$, at the ends of the three stated intervals $g_1$, $g_2$, $g_3$ are negative and positive, with margins above $18$, so that $\log\beta$ to two decimals decides the signs. For $b>96c^2/\rho^2$, \[ g_j(16^jb)=\frac{b\rho^2}{16c^2}-6(\log b+C_0)-\bigl(\lvert\log q^*_j\rvert+ 24j\log2\bigr), \] whose last bracket increases with $j$ and is at most $169144.64+28j\log2$ for $j\ge3$. At $b=2.90\cdot10^{22}$ the three terms are at least $169779.0$, at most $339.8$ and, for $j\le12$, at most $169377.6$; so $g_j(16^jb)>0$, that is $16^{-j}\beta^*_j\le2.90\cdot10^{22}$, for $1\le j \le12$. \emph{The scale $k_*$.} By its definition, $16^{k_*}\le\beta_W/D< 16^{k_*+1}$ with \[ D=1.0201\cdot96\cdot10^8(\log\beta_W+C_1), \] and $C_1\in[6.48436,6.48437]$ gives $5.595\cdot10^{11}\le D\le1.0106\cdot 10^{12}$ for $\beta_W\in[10^{22},10^{42}]$. At $b=3\cdot10^{22}$ the three terms of $g_j(16^jb)$ are at least $175633.4$, at most $340.0$ and, for $j\le17$, at most $169474.6$. At $b=5.4\cdot10^{21}$ the first is at most $31614.1$, which is less than $\lvert\log q^*_1\rvert$. With $\beta^*_0\in [5.400,5.402]\cdot10^{21}$ this gives \[ 5.4\cdot10^{21}\le16^{-j}\beta^*_j\le3\cdot10^{22}\qquad(0\le j\le17), \] and $\beta^*_j>\beta^*_{17}>10^{42}$ for $j>17$. So, with $x=\log_{16} \beta_W$, $j_{\max}\ge0$, \[ \lfloor x-\log_{16}(3\cdot10^{22})\rfloor\le j_{\max}\le\lfloor x- \log_{16}(5.4\cdot10^{21})\rfloor, \] and $k_*=\lfloor x-\log_{16}D\rfloor$. Since $\lfloor a\rfloor-\lfloor b\rfloor$ is $\lfloor a-b\rfloor$ or $\lfloor a-b\rfloor+1$, \[ \Bigl\lfloor\log_{16}\frac{5.4\cdot10^{21}}D\Bigr\rfloor\le k_*-j_{\max}\le \Bigl\lfloor\log_{16}\frac{3\cdot10^{22}}D\Bigr\rfloor+1, \] and $16^8D\le4.35\cdot10^{21}$, $16^9D\ge3.84\cdot10^{22}$ give $8\le k_*- j_{\max}\le9$. \end{proof} The theorem settles the entropy count of the islands at every scale that the fine witnesses reach. The next proposition shows that they reach no further: a coarse plaquette can be large at scale $j$ while every fine plaquette is small, because the cut-off average multiplies a constant curvature by four at every step. \begin{proposition}[constant curvature grows by four]\label{prop:constantcurvature} Let $T=\frac1{\sqrt2}\operatorname{diag}(i,-i)$, let $L=2^m$, $n\ge1$ and $\theta=2\sqrt2\pi n/L^2$, and let $U$ be the field with $U_{x,1}=\exp( \theta x_0T)$ for $0\le x_0\eta$; for $\eta=92\rho$ this is $4^j\theta>23\rho$ up to a factor $1+10^{-3}$, whereas Theorem~\ref{thm:islandscales} needs $4^j\eta_0\le\rho/1.023$. \end{proposition} \begin{proof} The field takes values in the torus $S=\exp(\R T)$, and the plaquette exponents are the abelian curl: on the plane $\{0,1\}$ every plaquette, including those of the column $x_0=L-1$, has exponent $\theta$, the seam links contributing $-\theta Lx_1+\theta L(x_1+1)-\theta(L-1)=\theta$ there and cancelling on the planes $\{0,2\}$, $\{0,3\}$; the plaquettes through $x_1=L-1$ are consistent since $\exp(\theta L^2T)=\exp(2\pi in\cdot \operatorname{diag}(1,-1))=I$. An $S$-valued gauge transformation puts $U$ in the tree gauge without changing the plaquettes, and Theorem~\ref{thm:torusscales} applies with $\eta_G=\frac1{12}$: by Lemma~\ref{lem:torusstep}, $\Theta^{(i+1)}_P$ is the mean over the sixteen offsets of the sum of the four plaquettes of a square of side two, which for a constant $\Theta^{(i)}=4^i\theta T$ on the plane $\{0,1\}$ and $0$ elsewhere is $4^{i+1}\theta T$ on that plane and $0$ elsewhere, and the hypothesis~\eqref{eq:torushyp} holds by induction while $4^i\theta\le \frac1{12}$. The eigenvalues of $T$ are $\pm i/\sqrt2$, so $\delta(\exp(tT), I)=2\sqrt2\lvert\sin(t/2\sqrt2)\rvert$, and $\sin u\ge u-u^3/6$. The coarse label of a block at scale $j$ is the plaquette of $U^{(j+1)}$. \end{proof} Between the two conditions lies a factor $23.5<4^{2.3}$, so the fine witnesses of Lemma~\ref{lem:scalewitness} stop within three scales of where any fine witness must stop. At the radius $10^{-8}$ the remaining six scales up to $a_{k_*}$ are the ratio between the curvature margin $10^{-4}$ of Theorem~\ref{thm:defectscales}(iv) and the witness radius of Lemma~\ref{lem:smallframes}; Theorem~\ref{thm:islandswide} below removes that ratio and carries the count to $a_{k_*}/8$, and past $a_{k_*}$ the rarity of a large coarse plaquette can no longer come from the fine scale, by the proposition, but must come from the coarse energy of Lemma~\ref{lem:coarseenergy}, which is known for torus-valued fields only; through the chessboard estimate it cannot reach the scale of the cube either (Theorem~\ref{thm:chessceiling}). Theorem~\ref{thm:islandscales} is an annealed statement about the pushforward of $\mu_L$: it says nothing about the law of $U^{(j)}$ given its labels, which is what the effective action of item (iv) below is for. \subsection{The witness radius for $\SU(2)$}\label{ssec:witnessradius} The thresholds above carry the factor $16/\rho^2$ of the witness radius $\rho\le10^{-8}$. For $\SU(2)$ the witness paragraph of Theorem~\ref{thm:islandthreshold} holds up to an explicit radius, which is sharp for Lemma~\ref{lem:smallframes}(iv). Write $\angle g\in[0,\pi]$ for the angle of $g\in\SU(2)$, the bi-invariant geodesic distance to $I$ on $S^3$, so that $\angle(gh)\le\angle g+\angle h$, $\angle(hgh^{-1})=\angle g$, $\delta(g,I)=2\sqrt2\sin(\angle g/2)\le\sqrt2\angle g$ and $\lVert\log g \rVert=\sqrt2\angle g$ for $\angle g<\pi$. Put $p(\kappa)=2\arcsin(\kappa/ 2\sqrt2)$, the largest angle of a plaquette with $\delta(U_p,I)\le\kappa$, and \[ \begin{gathered} \psi_c=\tfrac13\arcsin\tfrac1{4\sqrt2},\\ \kappa_c=2\sqrt2\sin\tfrac{\psi_c}2\in[0.0837613,0.0837617], \end{gathered} \] by $1.4142135<\sqrt2<1.4142136$, the arcsine series, whose partial sums are lower bounds and whose terms after $y^5$ add at most $y^7/(6(1-y^2))$, which give $0.05923686\le\psi_c\le0.0592371$, and $x-\frac{x^3}6\le\sin x\le x-\frac{x^3}6+\frac{x^5}{120}$; in particular $8/\kappa_c^2<1140.26$. Then $p(\kappa)<\psi_c$ exactly when $\kappa<\kappa_c$, and $\sqrt2\,p(\kappa) \le1.00015\,\kappa$ for $\kappa\le\kappa_c$, since $\sqrt2\,p(\kappa)/\kappa= \frac{p/2}{\sin(p/2)}$ increases with $\kappa$. \begin{lemma}[the ladder]\label{lem:ladder} Let $U$ be in the tree gauge, $v=P_{2y}$ the straight path of the pair $(y,\mu)$, and $x\in B(y)$ with $n_x\le3$ transverse coordinates of the offset equal to one. Then $v^{-1}P_x$ is a product of $2n_x$ conjugates of plaquette variables or their inverses, all with their vertices in $B(y)\cup B(y+e_\mu)$. If these plaquettes have $\delta(U_p,I)\le\kappa$, then $\angle(v^{-1}P_x)\le2n_xp(\kappa)$, and every link $e$ leaving $B(y)$ in the direction $\mu$ has $\angle(v^{-1}U_e)\le9p(\kappa)$. \end{lemma} \begin{proof} The tree path from $2y$ to $x$ inside $B(y)$ raises the coordinates of the offset in increasing order of direction, so a step raising the coordinate $\alpha$ at $z$ uses the tree link $(z,\alpha)$, and $(z+2e_\mu,\alpha)$, with the same offset in $B(y+e_\mu)$, is a tree link as well. A step with $\alpha=\mu$ changes nothing, since then $(z,\mu)$ and $(z+2e_\mu,\mu)$ are tree links and $P_z=U_{z+e_\mu,\mu}=P_{z+e_\mu}$. For $\alpha\ne\mu$ let $p_1,p_2$ be the plaquettes at $z$ and $z+e_\mu$ in the plane $\{\mu,\alpha\}$, $g=U_{z+e_\mu,\alpha}$ and $z'=z+e_\alpha$. With the two tree links equal to $I$, $U_{p_1}=U_{z,\mu}gU_{z',\mu}^{-1}$ and $U_{p_2}=U_{z+e_\mu,\mu} U_{z'+e_\mu,\mu}^{-1}g^{-1}$, hence \[ P_z=U_{p_1}\bigl(U_{z',\mu}g^{-1}U_{p_2}gU_{z',\mu}^{-1}\bigr)P_{z'}, \] and each of the $n_x$ transverse steps multiplies by two conjugated plaquettes. The angle is subadditive and conjugation invariant. For a leaving link $e=(x,\mu)$, $P_{x-e_\mu}=U_aU_e$ with $U_a$ a link inside $B(y)$, whose angle is at most $3p$ by the chains of the proof of Lemma~\ref{lem:smallframes}(i); so $\angle(v^{-1}U_e)\le\angle U_a+ \angle(v^{-1}P_{x-e_\mu})\le9p$. \end{proof} \begin{proposition}[Lemma~\ref{lem:smallframes} for $\SU(2)$ to the sharp radius]\label{prop:smallframessharp} Let $G=\SU(2)$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Parts (i)--(iii) of Lemma~\ref{lem:smallframes} hold for every $\kappa\ge0$, and the constant $21$ of (i) can be replaced by $9$. \item Part (iv) holds for every $\kappa<\kappa_c$, with $28.04\,\kappa$ in place of $260\,\kappa$. \item For every $\kappa>\kappa_c$ there is a field with $\delta(U_p,I)\le \kappa$ for every plaquette at which the formula of (iv) is false. \item For $\kappa<\kappa_c$, Lemma~\ref{lem:coarsewitness} holds with $\delta(c_P,I)\le16.003\,\kappa$, and every non-path active link $e$ of a pair at $y$ has $\delta(U_e,R_e(c))\le d(U_e,R_e(c))\le12.002\,\kappa$. \end{enumerate} \end{proposition} \begin{proof} (i) The proofs of (i)--(iii) use only the triangle inequality for $\delta$, bi-invariance and $\lVert\operatorname{Ad}(g)X-X\rVert\le2\delta(g,I) \lVert X\rVert$, none of which needs $\kappa$ small; Lemma~\ref{lem:ladder} gives $9\kappa$ for the leaving links. (ii) By Lemma~\ref{lem:ladder}, $\delta(P_x,v)\le2\sqrt2\sin(3p)$, which is below $\frac12$ exactly when $p<\psi_c$. Then the cut-off equals one near $U$ and $V^\sharp=v\exp M$ with $M=\frac1{16}\sum_x\log w_x$, $w_x=v^{-1}P_x$. Since $\lVert\log w_x\rVert\le2\sqrt2\,n_xp$ and $\sum_xn_x=24$, $\lVert M\rVert\le3\sqrt2\,p$, so $\lVert\operatorname{ad}_{\log w_x}\rVert \le12p$ and $\lVert\operatorname{ad}_M\rVert\le6p$, the spectrum of $\operatorname{ad}_X$ being $\{0,\pm i\sqrt2\lVert X\rVert\}$. The four differences of the proof of Lemma~\ref{lem:smallframes}(iv) are then bounded with $\lVert\operatorname{Ad}(g)-1\rVert=2\sin\angle g$ for $\angle g \le\frac\pi2$ in place of $2\delta(g,I)$, and with $\sup_{\lvert z\rvert\le a} \lvert z/(1-e^{-z})-1\rvert\le\hat\varphi(a)=\frac a2+0.09a^2$ for $a\le1$ and $\sup_{\lvert z\rvert\le a}\lvert(1-e^{-z})/z-1\rvert\le\hat\psi(a)=\frac a2e^a$. The frame changes by $e^{-M}$ and by $w_x^{-1}$ cost at most $6p$ and $12p$ times $\lVert\alpha\rVert\le N_{y,\mu}$; the factor $(1-e^{-\operatorname{ad}_M})/\operatorname{ad}_M$ and the derivatives of the logarithms cost $\hat\psi(6p)(1+\hat\varphi(12p))+\hat\varphi(12p)$ times $m_\beta+\lVert\alpha\rVert\le(1+\frac{\sqrt3}4)N_{y,\mu}$; and the terms $\epsilon_x$, of norm at most $2\sin(9p)\lVert\xi_{x,\mu}\rVert$ for $x_\mu$ even and $2\sin(3p)\lVert\xi_{x,\mu}\rVert$ for $x_\mu$ odd, add $\frac1{16}\sqrt{8(18p)^2+8(6p)^2}\,N_{y,\mu}$ by Cauchy--Schwarz. The sum is $p\,G(p)$ with $G$ increasing and $p/\kappa$ increasing, so its supremum over $\kappa\le\kappa_c$, divided by $\kappa$, is at most $G(\psi_c)\psi_c/\kappa_c$. Since $\arcsin y\le y+y^3/(6(1-y^2))$ and $1/(4\sqrt2)\le0.1767767$, $\psi_c\le0.059243$, and $\sin x\ge x-x^3/6$ gives $\kappa_c\ge\sqrt2\,\psi_c(1-\psi_c^2/24)\ge1.414006\,\psi_c$. At $p=0.059243$ the Taylor series of the exponential with its tail bounded geometrically gives $e^{6p}\le1.42686$, so $\hat\psi(6p)\le0.253595$ and $\hat\varphi(12p)\le0.400945$, so that $\hat\psi(6p)(1+\hat\varphi(12p))+ \hat\varphi(12p)\le0.756218$. With $1+\frac{\sqrt3}4\le1.4330128$ and $\sqrt{2880}\le53.66564$, \[ G(\psi_c)\le18+\frac{53.66564}{16}+\frac{1.4330128\cdot0.756218}{0.059243} \le39.6461, \] and the supremum is at most $39.6461/1.414006<28.04$. (iii) Let $L=2^m$, $\varphi=2\pi n/L^2$ and $T=\frac1{\sqrt2}\operatorname{diag} (i,-i)$, and multiply the three torus-valued fields of Proposition~\ref{prop:constantcurvature} that have plaquette angle $\varphi$ in the planes $\{\mu,\alpha\}$, $\alpha\ne\mu$, with a common orientation. Every plaquette of these planes has $\delta(U_p,I)=2\sqrt2\sin(\varphi/2)$ and every other plaquette is $I$. In the tree gauge the ladder identity reads $P_{z+e_\alpha}=U_{p_1}^{-1}U_{p_2}^{-1}P_z$, so for the offset $(1,1,1)$ the angle of $v^{-1}P_x$ is $6\varphi$ and $\delta(P_x,v)=2\sqrt2 \sin3\varphi$. If $\kappa>\kappa_c$, choose $L$ and $n$ with $\psi_c<\varphi \le p(\kappa)$: every plaquette has $\delta(U_p,I)\le\kappa$, the cut-off $\phi(\delta(P_x,v))=2-2\delta(P_x,v)$ is below one, and the logarithms $\log w_x$ are nonnegative multiples of one element with mean of angle $3\varphi\ne0$, so $V^\sharp\ne v\exp M$ and (iv) fails. For example, for $L=32$, $n=10$ one has $\kappa=0.0867614\ldots$, $\delta(P_x,v)=0.517715\ldots$ and cut-off $0.964570\ldots$; the argument does not use these values. (iv) By (ii), $c_k=\ell_ke^{M_k}$ with $\angle e^{M_k}\le3p$. A link leaving $B(y)$ has $\angle(c_k^{-1}U_e)\le9p+3p$, and a link inside a block has $\angle U_e\le3p$; so $\delta(U_e,R_e(c))\le d(U_e,R_e(c))\le12\sqrt2\,p \le12.002\,\kappa$. The holonomy of the square of side two at $2y$ is a product of four conjugated plaquettes, of angle at most $4p$, and replacing its four sides $\ell_k$ by $c_k$ adds at most $4\cdot3p$; so $\angle c_P\le16p$ and $\delta(c_P,I) \le2\sqrt2\sin8p\le16\sqrt2\,p\le16.003\,\kappa$. \end{proof} \begin{theorem}[the island count at radius $10^{-4}$]\label{thm:islandswide} Let $G=\SU(2)$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Theorem~\ref{thm:islandthreshold} holds with the hypothesis $\rho\le 10^{-8}$ replaced by $\rho<\kappa_c$, and Lemma~\ref{lem:scalewitness} and Theorem~\ref{thm:islandscales} hold with it replaced by $\rho\le1.023\cdot 10^{-4}$. \item At $\rho=10^{-4}$, \begin{gather*} \beta^*(10^{-4},1)\in[3.408,3.410]\cdot10^{13},\\ \beta^*(10^{-4},5)\in[5.382,5.384]\cdot10^{13}, \end{gather*} and for $s=5$ \begin{gather*} \beta^*_1\in[2.892,2.894]\cdot10^{15},\quad \beta^*_2\in[4.628,4.630]\cdot10^{16},\\ \beta^*_3\in[1.185,1.186]\cdot10^{18}, \end{gather*} $16^{-j}\beta^*_j\le2.90\cdot10^{14}$ for $1\le j\le20$, and for $\beta^*_1 \le\beta_W<\beta^*_{20}$ the conclusions of Theorem~\ref{thm:islandscales} hold at every scale $j\le j_{\max}$ with $2^{j_{\max}}\ge\frac12(\beta_W/2.90 \cdot10^{14})^{1/4}$. \item For every $\beta_W\in[10^{22},10^{42}]$, $k_*-j_{\max}\in\{2,3\}$: the islands of the composed average are rare up to the scale $a_{k_*}/8$. \end{enumerate} \end{theorem} \begin{proof} (i) The witness paragraph of the proof of Theorem~\ref{thm:islandthreshold} uses the cap $10^{-8}$ only through Lemma~\ref{lem:smallframes}(i) and the proof of Lemma~\ref{lem:coarsewitness}. Let $\kappa$ be the largest defect of the witness region. If $\kappa<\kappa_c$, Proposition~\ref{prop:smallframessharp}(iv) gives $d(U_e,R_e(c))\le12.002\kappa<43\kappa$ and $\delta(c_P,I)\le 16.003\kappa<92\kappa$, so a bad label forces $43\kappa>r\ge43\rho$ or $92\kappa> \eta\ge92\rho$; otherwise $\kappa\ge\kappa_c>\rho$. In both cases some plaquette has defect more than $\rho$, and the threshold and island paragraphs do not involve the cap. In Lemma~\ref{lem:scalewitness} the cap enters only through the hypothesis $\eta_j\le\rho_G/1000$ of Theorem~\ref{thm:defectscales}(i), (ii), with $\rho_G=1$, and in Theorem~\ref{thm:islandscales}(i) through the hypothesis $4^j\eta_0\le10^{-4}$ of Theorem~\ref{thm:defectscales}(iv), which reads $\rho\le1.023\cdot10^{-4}$; then $\eta_i\le\rho\le10^{-3}$ for $i\le j$. (ii), (iii) The proof of Theorem~\ref{thm:islandscales} applies at $\rho=10^{-4}$, with the same functions $g_j$ and the same values of $\lvert\log q^*_j\rvert$; for $s=1$, $\lvert\log q^*\rvert=1296(\log2+2\log 2400)+1+4\log11+4\log6$ lies within $10^{-3}$ of $21090.194$. At the ends of the five stated intervals $g$ is negative and positive, with margins above $5$. At $b=2.90\cdot10^{14}$ the three terms of $g_j(16^jb)$ are at least $169779.0$, at most $229.3$ and, for $j\le20$, at most $169532.8$, which gives $16^{-j}\beta^*_j\le2.90\cdot10^{14}$ for $1\le j\le20$ and (ii). At $b=2.92\cdot10^{14}$ they are at least $170949.9$, at most $229.3$ and, for $j\le23$, at most $169591.1$, and at $b=2.88\cdot10^{14}$ the first is at most $168608.2<\lvert\log q^*_3\rvert$; so $2.88\cdot10^{14}\le16^{-j}\beta^*_j \le2.92\cdot10^{14}$ for $3\le j\le23$. Since $16^6\cdot2.92\cdot10^{14}< 10^{22}$ and $16^{23}\cdot2.88\cdot10^{14}>10^{42}$, $6\le j_{\max}\le22$ for $\beta_W\in[10^{22},10^{42}]$, and the floor argument of that proof gives \[ \Bigl\lfloor\log_{16}\frac{2.88\cdot10^{14}}D\Bigr\rfloor\le k_*-j_{\max}\le \Bigl\lfloor\log_{16}\frac{2.92\cdot10^{14}}D\Bigr\rfloor+1; \] with $16^2D\le2.59\cdot10^{14}$ and $16^3D\ge2.29\cdot10^{15}$ this is (iii). \end{proof} The two or three scales that remain are not a matter of radius. With $\beta^*_j\approx16\lvert\log q^*_j\rvert(1.033\cdot4^j)^2/\rho^2$ and $4^{k_*}\approx10^{-4}/(1.01\rho(\beta))$, \[ 4^{k_*-j_{\max}}\approx\frac{4.09\cdot10^{-4}}\rho\Bigl(\frac{\lvert\log q^*_j\rvert}{96(\log\beta+C_0)}\Bigr)^{1/2}, \] which is about $20$ at $\rho=10^{-4}$, so the loss is the entropy exponent $\lvert\log q^*_j\rvert\approx2H_j\log \varDelta$ of Proposition~\ref{prop:coverageopt}. At this radius the labels have $r=43\rho\le4.4\cdot10^{-3}$, beyond the hypothesis $r\le10^{-4}$ of Lemma~\ref{lem:branchbound}, and Proposition~\ref{prop:goodside} supplies the good windows to radius $\frac1{32}$. The chart of Theorem~\ref{thm:fibrechart} is a different matter: its proof bounds the second fundamental form of the fibre by $\lVert\nabla Q\rVert_\infty\lVert\dot w\rVert_1$ with a constant $c_2$ for the second derivative of the reconstruction, and no such constant is below the flat value of the next proposition. \begin{proposition}[the flat second derivative]\label{prop:flatsecond} At the flat background $U\equiv I$, \[ D^2c_k(0)[Y,Y]=\frac1{16}\sum_{x\in B(y)}[Y_{1x},Y_{2x}], \] where $Y_{1x}$, $Y_{2x}$ are the variables of the two links of $P_x$, and $\sup\{\lVert D^2c_k(0)[Y,Y]\rVert:N(Y)=1\}=\frac1{16}$. \end{proposition} \begin{proof} Expand to second order, by the Baker--Campbell--Hausdorff formula, \[ c=\log\bigl(e^{X_v}e^M\bigr),\qquad M=\frac1{16}\sum_x\log\bigl(e^{-X_v} e^{X_{1x}}e^{X_{2x}}\bigr): \] the terms $[X_v,X_{1x}+X_{2x}]$ from the two logarithms cancel, and $\frac1{32}\sum_x[X_{1x},X_{2x}]$ remains, whose second derivative is the display. For $\mu=0$ put $d=7^{-1/2}$, $l=14^{-1/2}$ and, for each of the seven nonzero transverse offsets, $Y=\frac d2T_1$ on the link of $B(y)$, $lT_2$ on the leaving link and $-\frac d2T_1$ on the link of $B(y+e_0)$, with $T_a=i\sigma_a/\sqrt2$ and zero elsewhere. Then $N(Y)=1$ and, by $\lVert[T_1,T_2]\rVert=\sqrt2$, $\lVert D^2c_k(0)[Y,Y]\rVert=\frac{7\sqrt2 \,dl}{16}=\frac1{16}$. Conversely $\lVert[X,Y]\rVert\le\sqrt2\lVert X\rVert \lVert Y\rVert$ and $\sqrt2\lVert D\rVert\lVert L\rVert\le\frac12\lVert D \rVert^2+\lVert L\rVert^2$, with $D$ the difference of the two inside variables and $L$ the leaving one, give $\frac1{16}$ as the maximum. \end{proof} In the proof of Theorem~\ref{thm:fibrechart}(ii) the constant $c_3$ enters the term $9000c_3\kappa'$, $\kappa'=\kappa+6s$, only as a bound for the second derivative of $c$, so by the proposition no admissible constant there is below $\frac1{16}$, and with $\frac1{16}$ that term alone exceeds $\frac1{195}$ once $\kappa'>9.12\cdot10^{-6}$. A chart at radius $10^{-4}$ needs a different estimate of the second fundamental form; no field is known at which the chart fails. \subsection{The ceiling of the chessboard estimate}\label{ssec:chessceiling} The threshold of Theorem~\ref{thm:islandthreshold} is of order $10^{13}$ at the witness radius $10^{-4}$ of Theorem~\ref{thm:islandswide}, and of order $10^{37}$ under the calmness of Corollary~\ref{cor:twoislands}. This subsection measures what can be gained in them and what cannot. The count of connected sets of blocks can be sharpened, which lowers the threshold at one scale by a factor $1.7$ at $s=1$ and $1.8$ at $s=5$, and the union bound over connected sets cannot lower it by more than a further factor $1.39$ at $s=1$ and $1.22$ at $s=5$. The scale is a different matter. Every rarity bound that rests on the chessboard estimate in the form of Theorem~\ref{thm:defects} or Proposition~\ref{prop:coarsechess}, with fine or coarse witnesses, is below one only at scales whose blocks are smaller than the cube of the condition (D) by a fixed factor: at every coupling with the normalisation of Lemma~\ref{lem:wilsonZ}, and at every coupling $\beta_W\ge10^3$ with the exact normalisation. The witness radius at scale $j$ is at most $4^{-j-1}$ times the label radius, by the constant curvature of Proposition~\ref{prop:constantcurvature}, and the normalisation charges $\frac92\log\beta_W$ per fine site at least. \begin{lemma}[connected sets of blocks]\label{lem:animals} Let $\Gamma$ be a graph in which every vertex has at most $\varDelta$ neighbours, and let $v$ be a vertex. The number of connected sets of $n$ vertices that contain $v$ is at most $\binom{\varDelta n}{n-1}\le e^n \varDelta^{n-1}$. In the graph of Theorem~\ref{thm:islandthreshold}, on the torus of side $L/2$ in blocks, in which two blocks are joined at $\ell^\infty$ distance at most $R=2s+1$, the number is at least $\lambda_0^{n-1}$ with $\lambda_0=R(2R+1)^3=(2s+1)(4s+3)^3$ whenever $(n-1)R\frac{16}{\rho^2}\bigl(H\log\lambda_0+\log(eMH)\bigr). \] At the radius of Theorem~\ref{thm:islandswide}, \begin{gather*} \beta^*_T(10^{-4},1)\in[2.001,2.003]\cdot10^{13},\\ \beta^*_T(10^{-4},5)\in[2.988,2.990]\cdot10^{13}, \end{gather*} against $\beta^*(10^{-4},1)\in[3.408,3.410]\cdot10^{13}$ and $\beta^*(10^{-4},5)\in[5.382,5.384]\cdot10^{13}$ there, and the geometric decrease requires $\beta_W>1.441\cdot10^{13}$ for $s=1$ and $\beta_W>2.450\cdot10^{13}$ for $s=5$. \end{proposition} \begin{proof} The witness and threshold paragraphs of the proof of Theorem~\ref{thm:islandthreshold} apply with $q^*_T$ in place of $q^*$, since $\log q^*_T<0H\log\lambda_0+\log(eMH)+6(\log\beta_W+C_0), \] with $\log\beta_W+C_0>0$. For the numbers, $\lvert\log q^*_T\rvert=H(1+\log2+\log\varDelta)+1+4\log11+ 4\log6$, with $\varDelta=2400$ for $s=1$ and $\varDelta=23^4-1$ for $s=5$, $\log(23^4-1)=4\log23-\epsilon$ with $0<\epsilon<4\cdot10^{-6}$, and $\lambda_0=3\cdot7^3=1029$ for $s=1$ and $\lambda_0=11\cdot23^3=133837$ for $s=5$. The logarithms of the proof of Theorem~\ref{thm:islandthreshold}, with $\log23=3\log2+\log3-\log\frac{24}{23}$ and $\log7=3\log2-2 \operatorname{artanh}\frac1{15}$, give $\lvert\log q^*_T\rvert$ within $10^{-3}$ of $12299.136$ and $18466.475$, and $H\log\lambda_0+\log(eMH)$ within $10^{-3}$ of $9007.259$ and $15316.232$, for $s=1$ and $s=5$; the last two give the two lower bounds. The function $g_T(\beta)=\beta\rho^2/16- 6(\log\beta+C_0)-\lvert\log q^*_T\rvert$ increases for $\beta>96/\rho^2$ and $\beta^*_T$ is its largest zero; at the two ends of each stated interval $g_T$ is negative and positive, with margins above $5$, so that $\log\beta$ to two decimals decides the signs. \end{proof} The same replacement applies at every scale: in Theorem~\ref{thm:islandscales} the exponent $H_j(\log2+2\log\varDelta)$ of $q^*_j$ becomes $H_j(1+\log2+\log\varDelta)$, with the conclusions of part~(ii) of the proposition. The gain is a constant factor in the coupling and therefore a quarter of its logarithm in the scale. The next results show that no constant factor of this kind reaches the scale of the cube. \begin{lemma}[the normalisation from below]\label{lem:Zabove} For $\beta_W>0$ and $L\ge2$, \[ Z_L\le\Bigl(\bigl(\tfrac\pi2\bigr)^{5/2}\beta_W^{-3/2}\Bigr)^{3L^3(L-1)}, \] so that \[ \log\frac1{Z_L}\ge\frac92\Bigl(1-\frac1L\Bigr)L^4\bigl(\log\beta_W-c_I \bigr),\qquad c_I=\tfrac53\log\tfrac\pi2 . \] In particular $c_I\in[0.75263,0.75264]$. \end{lemma} \begin{proof} Let $F$ be the set of links $(x,0)$ with $x_0\ne L-1$; each line of direction $0$ meets $F$ in a path, so $F$ is a forest. Put $g_x=U_{(0,\vec x),0}\cdots U_{(x_0-1,\vec x),0}$ for $x=(x_0,\vec x)$ and $U'_{x,\mu}= g_x^{-1}U_{x,\mu}g_{x+e_\mu}$. Then $U'=I$ on $F$, and given the links of $F$ the links $U'_e$, $e\notin F$, are independent and Haar distributed, since each is a fixed left and right translate of its own $U_e$. The weight $e^{-\beta_WS}$ is gauge invariant, so $Z_L=\int e^{-\beta_WS(U')}\,dU$ is the integral over the links off $F$ with the links of $F$ equal to $I$. For $\mu\in\{1,2,3\}$ and $x_0\le L-2$ the plaquette at $x$ in the plane $\{0,\mu\}$ has holonomy $U'_{x+e_0,\mu}U'^{-1}_{x,\mu}$, its two links of direction $0$ lying in $F$. Drop every other plaquette from $S$, which only increases the integrand. For fixed $\vec x$ and $\mu$ the links $U'_{(t,\vec x),\mu}$, $0\le t\le L-1$, enter only the $L-1$ factors $\exp(-\beta_Ws(U'_{(t+1,\vec x),\mu}U'^{-1}_{(t,\vec x),\mu}))$; integrating over $t=0,1,\dots,L-2$ in turn, each integral equals \[ I(\beta_W)=\frac2\pi\int_0^\pi e^{-\beta_W(1-\cos t)}\sin^2t\,dt, \] by the invariance of Haar measure. So $Z_L\le I(\beta_W)^{3L^3(L-1)}$. On $[0,\pi]$, $1-\cos t=2\sin^2\frac t2\ge\frac{2t^2}{\pi^2}$ and $\sin t\le t$, so \[ \begin{split} I(\beta_W)&\le\frac2\pi\int_0^\infty t^2e^{-2\beta_Wt^2/\pi^2}dt\\ &=\frac2\pi\cdot\frac{\sqrt\pi}4\Bigl(\frac{\pi^2}{2\beta_W}\Bigr)^{3/2} =\Bigl(\frac\pi2\Bigr)^{5/2}\beta_W^{-3/2}. \end{split} \] Finally $\log\frac\pi2=\log\frac32+\log\frac\pi3$, and the alternating series of $\log(1+x)$ at $x=\frac\pi3-1\in[0.0471975,0.0471976]$ gives $\log\frac\pi2\in[0.4515827,0.4515828]$. \end{proof} Together with Lemma~\ref{lem:wilsonZ}, the lemma places the normalisation of the chessboard estimate between $\frac92\log\beta_W$ and $6\log\beta_W$ per fine site, up to additive constants.\footnote{The chessboard estimate bounds the probability of a pattern of local events on a torus by the product of the probabilities of each event repeated by reflection over the whole torus \cite{FILS78}. The normalisation of each factor is a partition function of the whole torus, which is why the estimate pays for the fluctuations of every fine site.} The constant curvature field bounds the witness radius. Put \[ \eta_c=2\sqrt2\sin\frac1{24\sqrt2}\in[0.0833212,0.0833213], \] and for $0<\eta<\eta_c$ let $\tilde\eta=2\sqrt2\arcsin(\eta/2\sqrt2)$, so that $\eta<\tilde\eta<\frac1{12}$ and, by $\arcsin y\le y+y^3/(6(1-y^2))$, $\tilde\eta\le1.00015\,\eta$. Call a block $y$ \emph{$\eta$-bad at scale $j$} when some plaquette of $U^{(j+1)}$ with corner $y$ has $\delta(U^{(j+1)} _P,I)>\eta$: this is the coarse-curvature part of the label of Sec.~\ref{ssec:islandscales} at scale $j$, and a block bad for that label is bad. \begin{theorem}[the ceiling of the chessboard estimate]\label{thm:chessceiling} Let $G=\SU(2)$, $\beta_W\ge1$, $0<\eta<\eta_c$, $j\ge0$ and $\theta_j= 4^{-j-1}\tilde\eta$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item \emph{Witnesses.} Let $0\le i\le j+1$ and $\rho_i>0$, and suppose that on every torus $\Lambda_L$ with $L=2^m\ge L_0$, every field at which some block is $\eta$-bad at scale $j$ has a plaquette $P$ of $U^{(i)}$ with $\delta(U^{(i)}_P,I)>\rho_i$. Then $2\sqrt2\arcsin(\rho_i/2\sqrt2)\le4^i \theta_j$; in particular $\rho_i\le4^i\theta_j$, and a fine witness has $\rho_0\le\theta_j$. \item \emph{Product bounds.} (a) If the hypothesis of (i) holds with $i=0$, the weight $q_{\rho_0}(\beta_W)$ of Theorem~\ref{thm:defects} is below one only if \[ 16^{j+1}<\frac{\beta_W\tilde\eta^2}{96(\log\beta_W+C_0)} . \] (b) Let $b=2^k$, let $E$ be an event determined by the links of the closed cube $[0,b]^4$ on which the plaquettes of that cube carry action at least $A$, let $E_z$ be its reflections as in Proposition~\ref{prop:coarsechess}, and suppose that on every torus $\Lambda_L$ with $L=2^m\ge L_0$ every field at which a given block is $\eta$-bad at scale $j$ lies in some $E_z$. Then the bound of Proposition~\ref{prop:coarsechess} is below one, for a nonempty set $Z$, only if \[ 16^{j+1}<\Bigl(1+\frac1b\Bigr)^2\frac{\beta_W\tilde\eta^2}{96(\log\beta_W+ C_0)} . \] (c) If in (a) and (b) the bound of Lemma~\ref{lem:wilsonZ} is replaced by $\log(1/Z_L)$ itself, the same holds with $96(\log\beta_W+C_0)$ replaced by $72(1-1/L)(\log\beta_W-c_I)$. \item \emph{The cube.} Suppose that $D(n,\alpha)$ holds at $\beta_W$, with the chordal or the Hilbert--Schmidt distance, and that a bound of (a) or (b) is below one at scale $j$, or a bound of (c) with $\beta_W\ge10^3$ and $L\ge16$. Then \[ \Bigl(\frac n{2^{j+1}}\Bigr)^4>\frac{1.9268\,(1-\alpha)}{\tilde\eta^2}, \] and for (a) and (b) the constant $1.9268$ can be replaced by $4.2069$. Since $\tilde\eta<\frac1{12}$, $n>4.08\,(1-\alpha)^{1/4}\,2^{j+1}$ in every case. \end{enumerate} \end{theorem} \begin{proof} (i) Since $\tilde\eta<\frac1{12}$, there are $L=2^m\ge L_0$ and integers $n'$ for which $\theta=2\sqrt2\pi n'/L^2$ satisfies $\theta_j<\theta$ and $4^{j+1}\theta\le\frac1{12}$, with $\theta$ as close to $\theta_j$ as desired. Let $U$ be the field of Proposition~\ref{prop:constantcurvature} with this $\theta$. By that proposition and its proof, for $i\le j+1$, $\Theta^{(i)}_P=4^i\theta T$ on the plane $\{0,1\}$ and $\Theta^{(i)}_P=0$ on the other planes, so every plaquette of $U^{(i)}$ has $\delta(U^{(i)}_P,I)\le 2\sqrt2\sin(4^i\theta/2\sqrt2)$, while the plaquettes of $U^{(j+1)}$ in the plane $\{0,1\}$ have \[ \delta(U^{(j+1)}_P,I)=2\sqrt2\sin\frac{4^{j+1}\theta}{2\sqrt2}>2\sqrt2\sin \frac{\tilde\eta}{2\sqrt2}=\eta, \] the sine increasing on $[0,\frac1{24\sqrt2}]$. So every block is $\eta$-bad at scale $j$, the hypothesis gives $\rho_i<2\sqrt2\sin(4^i\theta/2\sqrt2)$, and letting $\theta$ decrease to $\theta_j$ gives $\rho_i\le2\sqrt2\sin(4^i \theta_j/2\sqrt2)$, which is the first claim; $\sin x\le x$ gives the second. (ii) (a) By its definition $q_{\rho_0}(\beta_W)<1$ exactly when $\beta_W \rho_0^2/16>6(\log\beta_W+C_0)$. By (i), $\rho_0^2\le\theta_j^2=16^{-j-1} \tilde\eta^2$, which gives the claim. (b) Take $U$ as in the proof of (i). By hypothesis $U\in E_z$ for some $z$. The reflections that carry $[0,b]^4$ to $z+[0,b]^4$ preserve $\Tr U_p$, so the reflected field lies in $E$ and the plaquettes of $[0,b]^4$ carry the same action there as those of $z+[0,b]^4$ carry at $U$. The closed cube contains $b^2(b+1)^2$ plaquettes of the plane $\{0,1\}$, each with $s_p=\frac14\delta(U_p,I)^2=2\sin^2(\theta/2\sqrt2)\le\theta^2/4$, and the other plaquettes have $s_p=0$. Hence $A\le b^2(b+1)^2\theta^2/4$ for $\theta$ arbitrarily close to $\theta_j$, so $A\le b^2(b+1)^2\theta_j^2/4$. The bound of Proposition~\ref{prop:coarsechess} is $\exp(\lvert Z\rvert(6b^4(\log\beta_W+ C_0)-\beta_WA/4))$, below one only if $\beta_WA/4>6b^4(\log\beta_W+C_0)$, that is, only if $\beta_W(1+1/b)^2\theta_j^2/16>6(\log\beta_W+C_0)$. (c) Before Lemma~\ref{lem:wilsonZ} is used, the proofs of Theorem~\ref{thm:defects} and Proposition~\ref{prop:coarsechess} bound the probabilities by $(e^{-\beta_W\rho_0^2L^4/16}/Z_L)^{\lvert A\rvert/L^4}$ and $(e^{-\beta_W(L/b)^4A/4}/Z_L)^{\lvert Z\rvert(b/L)^4}$. By Lemma~\ref{lem:Zabove} these are at least \[ \begin{gathered} \exp\Bigl(\lvert A\rvert\Bigl(\tfrac92(1-\tfrac1L)(\log\beta_W-c_I)- \tfrac{\beta_W\rho_0^2}{16}\Bigr)\Bigr),\\ \exp\Bigl(\lvert Z\rvert\Bigl(\tfrac92(1-\tfrac1L)b^4(\log\beta_W-c_I)- \tfrac{\beta_WA}4\Bigr)\Bigr), \end{gathered} \] and the arguments of (a) and (b) apply with $\frac92(1-\frac1L)(\log\beta_W- c_I)$ in place of $6(\log\beta_W+C_0)$. (iii) Propositions~\ref{prop:cubesize} and~\ref{prop:cubesizeN}, the latter with $N=2$, give $n^4\ge2(1-\alpha)\beta_W/(5\ell(\beta_W))$ with $\ell(\beta_W)$ equal to $1+\log24+\frac32\log(9\beta_W)$ or to $1+\frac32 \log(81\pi^2\beta_W)+2\log\frac{12}{11}$. Both are at most $\frac32x+11.2$, $x=\log\beta_W$, since $1+\log24+\frac32\log9<7.474$ and $1+\frac32\log(81 \pi^2)+2\log\frac{12}{11}<11.19989$. Write $c=1$ in (a) and $c=(1+1/b)^2\le4$ in (b). In (a) and (b), \[ \Bigl(\frac n{2^{j+1}}\Bigr)^4>\frac{192(1-\alpha)}{5c\,\tilde\eta^2}\cdot \frac{x+C_0}{\frac32x+11.2}, \] and the last factor increases with $x$, because $11.2>\frac32C_0$, so it is at least $C_0/11.2>0.43822$; with $c\le4$ this gives $4.2069$. In (c), \[ \Bigl(\frac n{2^{j+1}}\Bigr)^4>\frac{144(1-\frac1L)(1-\alpha)}{5c\, \tilde\eta^2}\cdot\frac{x-c_I}{\frac32x+11.2}, \] the last factor increases with $x$, and at $x=\log10^3=3\log10\in[6.907755, 6.907756]$ it is at least $0.28546$; with $1-\frac1L\ge\frac{15}{16}$ and $c\le4$ this gives $\frac{144\cdot15\cdot0.28546}{5\cdot16\cdot4}>1.9268$. Finally $144\cdot1.9268>277.4$ and $277.4^{1/4}>4.08$. \end{proof} The hypotheses of (ii) are those of every rarity bound for the islands in this paper: Theorems~\ref{thm:islandthreshold}, \ref{thm:islandscales} and~\ref{thm:islandswide} use fine witnesses and Theorem~\ref{thm:defects}, and a coarse energy as in Lemma~\ref{lem:coarseenergy} would be used through Proposition~\ref{prop:coarsechess}. A recursion that bounds the bad blocks of scale $j$ through events of earlier scales, and those through Theorem~\ref{thm:defects} or Proposition~\ref{prop:coarsechess}, ends in events at some scale $i\le j+1$ that together cover the $\eta$-bad blocks, and (i) and (ii)(b) apply to them, however they are nested. The ceiling does not depend on the collar, on the count of connected sets, on the exponent $1/H$, or on the constants of the witness lemmas: part (ii) holds even when the bound need only be below one, which is far less than the threshold of Theorem~\ref{thm:islandthreshold} or of Proposition~\ref{prop:treethreshold} asks. \begin{corollary}[the labels of the island theorems]\label{cor:chessceiling} Let $D(n,\alpha)$ hold at $\beta_W$, with $\alpha\le\frac12$, and let a bound of Theorem~\ref{thm:chessceiling}(ii) be below one at scale $j$, under the conditions of part (iii). \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For the labels of Theorems~\ref{thm:islandscales} and~\ref{thm:islandswide}, with $\eta=92\rho$ and $\rho\le1.023\cdot10^{-4}$, $n>12.14\,(1-\alpha)^{1/4}2^{j+1}>2^{j+4}$. \item For coarse labels with $\eta\le6.0\cdot10^{-13}$, the size below which Theorem~\ref{thm:badcross} needs the plaquettes of the background, which contains the calm labels of Corollary~\ref{cor:twoislands}, $n>1.52\cdot10^6(1-\alpha)^{1/4}2^{j+1}> 2^{j+21}$. \end{enumerate} So the scale of the cube lies at least three, respectively twenty, dyadic scales above every scale that such a bound reaches. \end{corollary} \begin{proof} Theorem~\ref{thm:chessceiling}(iii) applies, and $(1-\alpha)^{1/4}\ge 2^{-1/4}>0.8409$. (i) Here $\eta\le0.0094116$, so $\tilde\eta\le0.0094131$ and $1.9268/\tilde\eta^2>21745$; its fourth root exceeds $12.14$, and $12.14 \cdot0.8409>8$. (ii) Here $\tilde\eta\le6.0009\cdot10^{-13}$, so $1.9268/ \tilde\eta^2>5.35\cdot10^{24}$; its fourth root exceeds $1.52\cdot10^6$, and $1.52\cdot10^6\cdot0.8409>2^{20}$. \end{proof} Fig.~\ref{fig:chessceiling} shows the dyadic scales that a chessboard bound reaches and the scale of the cube. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_chessceiling.png} \caption{The ceiling of the chessboard estimate (Theorem~\ref{thm:chessceiling}), schematic. (a)~The dyadic scales as plates: a chessboard bound for the islands is below one only at the lower scales (green), and the cube of $D(n,\alpha)$ lies at least three scales higher for the labels of the island theorems and at least twenty for the calm labels (Corollary~\ref{cor:chessceiling}). (b)~The condition of part~(ii): the left side $16^{j+1}$ grows by a factor $16$ per scale, the right side $\beta_W\tilde\eta^2/(96(\log\beta_W+C_0))$ does not depend on $j$, and they cross below the scale of the cube.} \label{fig:chessceiling} \end{figure*} \begin{remark}[what the ceiling leaves]\label{rem:chessceiling} The loss sits in the normalisation, not in the energy. In (ii)(b) the action that a coarse defect forces is at most $(1+1/b)^2b^4\theta_j^2/4$, which for $b$ of the order $2^j$ does not decrease with $j$, in accordance with Lemma~\ref{lem:coarseenergy} for linear fields; the normalisation charges $\frac92b^4(\log\beta_W-c_I)$ at least, which grows like $16^j$. A bound that reaches the cube must compare the weight of the defects with $Z_L$ without charging the fluctuation volume of the cube, as in the Gaussian theory. Theorem~\ref{thm:haartilt} is a bound of that kind; it gives the rarity at every scale for $\U(1)$ (Corollary~\ref{cor:abeliancoarse}), and for $\SU(2)$ its constant grows like $4^k$ on the gauge orbit of the vacuum (Proposition~\ref{prop:tiltgrowth}). The theorem says nothing about such bounds, nor about labels with $\eta\ge\eta_c$. At couplings of order one nothing of the kind of (ii)(a) exists at any scale: for $1\le\beta_W\le10^3$ and every $\rho\le1$, $q_\rho(\beta_W)=1$, since the convex function $\beta/16-6(\log\beta+C_0)$ is negative at $\beta=1$ and at $\beta=10^3$. \end{remark} The fields that carry Theorem~\ref{thm:chessceiling} are critical points of the Wilson action. Every first variation of the action vanishes at them, so a bound that removes the coarse defects detected by a first-order estimate, such as the force events of Theorem~\ref{thm:haartilt} and Proposition~\ref{prop:foresttilt}, and treats the remaining defects by the chessboard estimate, still meets the ceiling. On $\Lambda_L$ write $S(U)= \sum_ps_p$, $s_p=1-\frac12\Tr U_p$, as in Lemma~\ref{lem:wilsonZ}, and for a link $e$ and $Y\in\mathfrak{su}(2)$ let $U^{e,tY}$ be the field with $U_e$ replaced by $U_ee^{tY}$. A field $U$ is a \emph{critical point} of $S$ when $\partial_tS(U^{e,tY})|_{t=0}=0$ for every link $e$ and every $Y$. A \emph{field of directions} is a family $Y=(Y_e)$ of continuous maps from the fields on $\Lambda_L$ to $\mathfrak{su}(2)$, and \[ D_YS(U)=\partial_tS\bigl((U_ee^{tY_e(U)})_e\bigr)\big|_{t=0}. \] The force $D_rS$ of Theorem~\ref{thm:haartilt} is $D_YS$ with the constant directions $Y_e=r_e$, and the force $D^F_rS$ of Proposition~\ref{prop:foresttilt} is $D_YS$ with $Y_e(U)=\operatorname{Ad} (h_y(U)^{-1})r_e$, which is continuous because $h_y(U)$ is a product of link variables. \begin{lemma}[constant curvature is critical]\label{lem:constantcritical} Every field $U$ of Proposition~\ref{prop:constantcurvature} is a critical point of $S$. Consequently $D_YS(U)=0$ for every field of directions $Y$. \end{lemma} \begin{proof} The field takes values in the torus $\exp(\R T)$, every plaquette of the plane $\{0,1\}$ has $U_p=\exp(\theta T)$ with one orientation, and every other plaquette has $U_p=I$ (Proposition~\ref{prop:constantcurvature}). Fix a link $e$ of direction $\mu$ and $Y\in\mathfrak{su}(2)$, and write $Y=yT+ Y_\perp$ with $y\in\R$ and $Y_\perp$ off-diagonal. As in the proof of Theorem~\ref{thm:haartilt}(i), for a plaquette $p$ through $e$ the plaquette variable of $U^{e,tY}$ is $e^{\epsilon_pt\operatorname{Ad}(h_p)Y} U_p$, where $h_p$ is a product of link variables of $U$ and $\epsilon_p=+1$ if $p$ traverses $e$ forwards and $-1$ if backwards. Since $h_p$ lies in the torus, $\operatorname{Ad}(h_p)Y=yT+Y'_\perp$ with $Y'_\perp$ off-diagonal, and $\Tr(Y'_\perp U_p)=0$ because $U_p$ is diagonal. Hence \[ \partial_ts_p\big|_{t=0}=-\tfrac12\epsilon_p\,y\,\Tr(TU_p), \] which depends on $p$ only through $\epsilon_p$ and $U_p$. For each $\nu\ne\mu$ the link $e$ lies in exactly two plaquettes of the plane $\{\mu,\nu\}$, on the two sides of $e$ in the direction $\nu$; with $U_p=U_1U_2U_3^{-1}U_4^{-1}$ one of them traverses $e$ forwards and the other backwards, and the two have the same variable $U_p$, namely $\exp (\theta T)$ if $\{\mu,\nu\}=\{0,1\}$ and $I$ otherwise. Their contributions cancel, and summing over the three planes gives $\partial_tS(U^{e,tY})|_{t=0} =0$. Finally $S$ is a smooth function of the links, so by the chain rule $D_YS(U)=\sum_e\partial_tS(U^{e,tY_e(U)})|_{t=0}=0$. \end{proof} \begin{theorem}[the ceiling at the critical points]\label{thm:criticalceiling} Let $G=\SU(2)$, $\beta_W\ge1$, $0<\eta<\eta_c$ and $j\ge0$, and for every torus $\Lambda_L$ let $\mathcal N_L$ be an open set of fields on $\Lambda_L$ that contains every critical point of $S$. Then Theorem~\ref{thm:chessceiling} holds, with the same constants, when in the hypotheses of its parts \textup{(i)} and \textup{(ii)} the words ``every field at which'' are replaced by ``almost every field of $\mathcal N_L$, for the product of the Haar measures, at which''. So does Corollary~\ref{cor:chessceiling}. \end{theorem} \begin{proof} Take $L$, $\theta$ and the field $U$ as in the proof of Theorem~\ref{thm:chessceiling}(i), so that $\theta_j<\theta$ and $4^{j+1} \theta\le\frac1{12}$. By Lemma~\ref{lem:constantcritical}, $U\in\mathcal N_L$. \emph{Continuity.} For $i\le j+1$ and every plaquette $P$ of the lattice of spacing $2^i$, the function $V\mapsto\delta(V^{(i)}_P,I)$ is continuous on the fields on $\Lambda_L$. Indeed $V^{(i)}$ is obtained by $i$ steps, each of which puts the field into the tree gauge by the gauge transformation whose value at a site is the product of the links along the tree path from the root of its block, a continuous function of the field, and then applies the cut-off average, which is continuous by Theorem~\ref{thm:cutoff}; the other choices of the tree gauge differ by transformations constant on the blocks, which conjugate the plaquette holonomies and leave $\delta(V^{(i)}_P, I)$ unchanged. The functions $s_p$ are continuous as well. \emph{A neighbourhood.} At $U$ every block is $\eta$-bad at scale $j$, with strict inequality, and every plaquette of $U^{(i)}$ has $\delta(U^{(i)}_P,I) \le2\sqrt2\sin(4^i\theta/2\sqrt2)$; every plaquette of the plane $\{0,1\}$ has $s_p=2\sin^2(\theta/2\sqrt2)\le\theta^2/4$, and every other plaquette has $s_p=0$. The torus has finitely many plaquettes at each scale, so for every $\epsilon>0$ there is an open neighbourhood $N_\epsilon\subset\mathcal N_L$ of $U$ on which every block is $\eta$-bad at scale $j$, every plaquette of $V^{(i)}$ has $\delta(V^{(i)}_P,I)<2\sqrt2\sin(4^i\theta/2\sqrt2)+\epsilon$ for $i\le j+1$, and every fine plaquette has $s_p(V) \rho_i$, so $\rho_i<2\sqrt2\sin(4^i\theta/2\sqrt2)+\epsilon$. Letting $\epsilon\to0$ and then $\theta$ decrease to $\theta_j$ gives the conclusion of Theorem~\ref{thm:chessceiling}(i), and part (ii)(a) follows from it as there. (ii)(b) The field $V$ lies in some $E_z$, so its reflection lies in $E$, and the plaquettes of $[0,b]^4$ carry at the reflection the action that those of $z+[0,b]^4$ carry at $V$. The closed cube contains $b^2(b+1)^2$ plaquettes of each of the six planes, so \[ A\le\frac{b^2(b+1)^2\theta^2}4+6b^2(b+1)^2\epsilon . \] Letting $\epsilon\to0$ and $\theta$ decrease to $\theta_j$ gives $A\le b^2 (b+1)^2\theta_j^2/4$, the only property of $E$ that the proof of Theorem~\ref{thm:chessceiling}(ii)(b) uses. Parts (ii)(c) and (iii) of that theorem, and Corollary~\ref{cor:chessceiling}, use only the inequalities of (i), (ii)(a) and (ii)(b). \end{proof} \begin{corollary}[first-order estimates]\label{cor:firstorder} Let $Y^1,\dots,Y^m$ be fields of directions on $\Lambda_L$, let $x_1,\dots, x_m>0$, and put \[ \mathcal N=\bigl\{U:\lvert D_{Y^k}S(U)\rvert2\sqrt2\pi/L^2$, so it contains $\theta=2\sqrt2\pi n/L^2$ for some integer $n\ge1$. The proof of Theorem~\ref{thm:criticalceiling} with this $\theta$ and $\mathcal N_L=\mathcal N$ gives a nonempty open subset of $\mathcal N$ on which every block is $\eta$-bad at scale $j$. It has positive Haar measure, and $\mu_L$ has a positive density with respect to Haar measure. (iii) This is Theorem~\ref{thm:criticalceiling} with $\mathcal N_L=\mathcal N$, which is admissible by (i). \end{proof} \begin{remark}[what the critical points leave]\label{rem:criticalceiling} At the fields of constant curvature the rarity of a coarse defect can only come from the size of the action, which is a second-order or global quantity there, and through the chessboard estimate that size, at most $(1+1/b)^2b^4 \theta_j^2/4$ in a cube of side $b$, is compared with a normalisation of order $b^4\log\beta_W$. The first-order estimates themselves charge no fluctuation volume: Theorem~\ref{thm:haartilt} bounds the force events uniformly in the exterior with the exponent $\beta_Wx^2/(2K_r)$, which contains no factor $\log\beta_W$ per site. What they lack near these fields is detection, by Corollary~\ref{cor:firstorder}(ii). Criticality is not special to $\SU(2)$: the $\U(1)$ field with link angles $\varphi x_0$ on the links of direction $1$, $-\varphi Lx_1$ on the links of direction $0$ with $x_0=L-1$, and $0$ on the others, $\varphi=2\pi n/L^2<\pi$, has plaquette angle $\varphi$ on every plaquette of the plane $\{0,1\}$ and $0$ on the others, so the derivative of the $\U(1)$ action in the link angle $e$, $\sum_{p\ni e} \pm\sin\theta_p$, vanishes by the cancellation of the proof of Lemma~\ref{lem:constantcritical}. The force $\sum_p\sin\theta_p\,(da)_p$ of Corollary~\ref{cor:abeliancoarse} therefore vanishes at it, so $\sum_p\theta_p(da)_p=0$, and by the identity of that corollary its coarse flux $(d_kV^{(k)}(\theta))_P$ is the term $2\pi\sum_pn_p(da)_p$. Theorems~\ref{thm:chessceiling} and~\ref{thm:criticalceiling} say nothing about three kinds of bound: the chessboard estimate with the probability $\mu_L(\bigcap_zE_z)$ of the global event bounded otherwise than by its unnormalised weight $e^{-\beta_W(L/b)^4 A/4}$ divided by $Z_L$, for instance by a map that preserves the Haar measures, acts at a finite distance and lowers the action on that event; bounds for the law of $U^{(j)}$ through an effective action, which is item (iv) of Sec.~\ref{ssec:remains}; and labels with $\eta\ge\eta_c$. \end{remark} \subsection{Two islands factorise through one kernel}\label{ssec:factorisation} Item (ii) of Sec.~\ref{ssec:remains}, in its original form, asks for the hypothesis of Lemma~\ref{lem:kernel}: a kernel on the good layer between two bad islands through which the weight of every pair of label patterns factors, a domain that the labels of the islands do not move, and a cross-ratio of the kernel controlled by Lemma~\ref{lem:connected}. This subsection proves the first two unconditionally, for every exterior and every pair of patterns, once the layer is seven blocks wider than the collar, and reduces the third to one covariance bound for the law of the layer, which is the bound of Theorem~\ref{thm:chartdecay}(ii) in the coordinates of the labels. \emph{The data.} Let $G=\SU(2)$, fix a coarse field $c$ with its reconstruction $R(c)$, the radii $r\le10^{-4}$ and $\eta$, and the labels of Sec.~\ref{ssec:islandthreshold}. Given $c$ the coarse-plaquette part of every label is a constant, and the event that a block $y$ is good is the intersection, over the non-path active links $e$ of the pairs based at $y$, of the single-link events $G_e=\{d(U_e,R_e(c))\le r\}$; these links are based in $B(y)$ and in the four blocks $B(y+e_\mu)$. Let $\Lambda$ be a finite set of blocks, $U^{\mathrm{ext}}$ an arbitrary configuration on the active links based outside $\Lambda$, $I_a,I_b\subset\Lambda$ two disjoint sets of blocks, $s\ge1$ the collar, and \[ w=\dist(I_a,I_b)-1 \] the number of block layers strictly between them, distances between blocks being $\ell^\infty$ distances of their coarse coordinates. Every block of $\Lambda\setminus(I_a\cup I_b)$ is good; on $I_a$ a pattern $\omega_a^i$, $i\in\{0,1\}$, is prescribed, with $\omega_a^0$ the pattern in which every block is good, and likewise $\omega_b^j$ on $I_b$. The sector $(i,j)$ is the event that the labels of $\Lambda$ are these, and its weight is \[ Z_{ij}=\int\prod_{e}dU_e\;e^{-\beta_WS(U)}\prod_{k\in K_\Lambda} \delta\bigl(V^\sharp_k(U)-c_k\bigr)\,\mathbf 1_{(i,j)}(U), \] the product over the active links based in $\Lambda$, $S$ the sum of the Wilson terms of the plaquettes containing one of them, $K_\Lambda$ the pairs based in $\Lambda$, and the constraints understood through the coarea formula as in Proposition~\ref{prop:graphlaw}; the pairs entering $\Lambda$ from outside have their path link in the exterior and their constraints are not imposed. Only ratios of the $Z_{ij}$ are used. \emph{The coordinates.} Write every non-path active link as $U_e=R_e(c)e^{\zeta_e}$, $\zeta_e\in\g$, so that on a good block the event $G_e$ is the ball $\{\lVert\zeta_e\rVert\le r\}$ and Haar measure is $j(\zeta_e)\,d\zeta_e$ with $j$ smooth and positive on it. A pair all of whose non-path links lie in good blocks has, on the sector event, exactly one path link compatible with its constraint, by Lemma~\ref{lem:branchbound}: $\ell_k=\Gamma_k$, a smooth function of its $23$ non-path links, with cut-off one. Let $d_a(y)=\dist(y,I_a)$ and $d_b$ likewise, and form the classes \begin{itemize}[leftmargin=*] \item $L_a$: all active links based at $d_a\le s$, path links included, all free; \item $p_a$: the path links of the pairs based at $d_a=s+1$, free; \item $x$: the non-path active links based at $d_a\in\{s+1,s+2\}$ and the $17$ inside active links of every block at $d_a=s+3$; \item $z$: every other non-path active link based in $\Lambda$; \item $L_b$, $p_b$, $y$: the mirror classes for $I_b$, \end{itemize} and solve the path link of every pair based at $d_a\ge s+2$ and $d_b\ge s+2$. For $w\ge2s+6$ a link at $d_a\le s+3$ has $d_b\ge s+4$, so the classes are disjoint. Let $D_x$ and $D_y$ be the products of the $r$-balls in the coordinates $\zeta_e$ of $x$ and of $y$; they are fixed by $r$ and the geometry, since every block carrying an $x$- or $y$-link is good in all four sectors. \begin{theorem}[two islands and one kernel]\label{thm:factorisation} Let $s\ge1$, $r\le10^{-4}$ and $w\ge2s+7$. There are finite positive measures $a_i$ on $D_x$ and $b_j$ on $D_y$, $i,j\in\{0,1\}$, and a smooth function $K:D_x\times D_y\to(0,\infty)$ with \[ Z_{ij}=\int_{D_x}\int_{D_y}K(x,y)\,a_i(dx)\,b_j(dy)\qquad(i,j\in\{0,1\}), \] such that: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $K$ does not depend on $i$ or $j$; it depends on $c$, $r$, $\beta_W$, $U^{\mathrm{ext}}$ and on the labels of the blocks at distance at least $s+2$ from both islands only; \item $a_i$ depends on $\omega_a^i$, on $c$, $\beta_W$ and $U^{\mathrm{ext}}$, and on nothing else; it carries every branch of the constraints of the pairs based at $d_a\le s+1$, integrated, every label indicator that involves a link of $L_a$, and the coarea densities of the pairs at $d_a=s+2$; likewise $b_j$; \item the supports of all four measures lie in $D_x$, $D_y$: the labels of the islands do not move the domain; \item $K(x,y)=\int_{D_z}e^{-\mathcal U(z;x,y)}dz$, where $D_z$ is the product of the $r$-balls of the $z$-coordinates and $\mathcal U=\beta_WS_K+\psi$ is smooth on a neighbourhood of $D_z\times D_x\times D_y$, $S_K$ the sum of the Wilson terms assigned to $K$ with the solved path links substituted, $\psi$ the sum of $-\log j$ over the $z$-links and of $-\log J_k$ over the solved pairs based at distance at least $s+3$ from both islands, $J_k$ the inverse Jacobian of Proposition~\ref{prop:graphlaw}; \item no term of $\mathcal U$ depends on both an $x$- and a $y$-coordinate, $\nabla_z\partial_x\mathcal U$ is supported on the $z$-links based at $d_a\le s+4$ and $\nabla_z\partial_y\mathcal U$ on those at $d_b\le s+4$, at coarse distance at least $w-2s-7$ and, for $w\ge2s+8$, at fine distance at least $d'=2(w-2s-8)$ between link midpoints; \item $Z_{ij}>0$ if and only if $a_i\ne0$ and $b_j\ne0$, that is, if and only if each pattern is realisable on its own side together with its collar; $a_0$ and $b_0$ are nonzero. \end{enumerate} \end{theorem} \begin{proof} \emph{Step 1: the factors.} The weight of a sector is the product Haar measure times a product of factors of three kinds: the Wilson factor of a plaquette, whose four links are based in blocks at mutual distance at most one; the constraint of a pair $k=(y,\mu)$, which involves the $24$ links of its paths, based in $B(y)\cup B(y+e_\mu)$; and the label indicator of a block, which for a good block is the product of the $\mathbf 1_{G_e}$ over the non-path links of its pairs and for a bad block is one minus that product, all these links being based in $B(y)$ and $B(y+e_\mu)$. \emph{Step 2: solving the far path links.} A pair based at $d_a\ge s+2$ and $d_b\ge s+2$ has its non-path links based at distance at least $s+1$ from both islands, hence in good blocks, hence inside their balls on the sector event. Lemma~\ref{lem:branchbound} applies, the fibre of its constraint over its non-path links is the one point $\ell_k=\Gamma_k$, and since $\ell_k$ is private to $k$ the integration of $\ell_k$ against the constraint replaces $\ell_k$ by $\Gamma_k$ in every factor and multiplies the weight by $J_k$, as in Proposition~\ref{prop:graphlaw}; the cross derivatives between different pairs vanish, so the order is immaterial. After this step every factor is a function of the free coordinates and of $U^{\mathrm{ext}}$, and a solved path link based at distance $d$ depends on non-path links based at distances $d-1$, $d$, $d+1$. \emph{Step 3: the assignment.} We list what each factor depends on, distances being from $I_a$. A solved path link at $s+2$ depends on non-path links at $s+1,s+2,s+3$, those at $s+3$ being the second links of its paths, inside $B(y+e_\mu)$: all of them are $x$-coordinates. A solved path link at $d\ge s+3$ depends on $x$- and $z$-coordinates. Hence: a plaquette with a link at $d\le s$ depends on $L_a$, $p_a$ and $x$; one with its nearest link at $s+1$ on $p_a$ and $x$; one with its nearest link at $s+2$ on $x$ and $z$ (its path links at $s+2$ are functions of $x$, its private links at $s+3$ are $z$, its path links at $s+3$ are functions of $x$ and $z$); one with all links at $d\ge s+3$ on $x$ and $z$. A pair based at $d\le s$ has its path link in $L_a$ and its non-path links in $L_a$ and $x$; one at $s+1$ has its path link in $p_a$ and its non-path links in $L_a$ and $x$; one at $s+2$ is solved with $J_k$ a function of $x$; one at $d\ge s+3$ is solved with $J_k$ a function of $x$ and $z$, of $z$ alone for $d\ge s+5$. The bad-block indicators of $I_a$ involve links at $d\le1\le s$, in $L_a$; the good single-link indicators are assigned to the class of their link. Call a factor an $a$-factor if it depends on a coordinate of $L_a\cup p_a$, a $b$-factor if on $L_b\cup p_b$, and a $K$-factor otherwise; factors depending on $x$ alone go with $a$, on $y$ alone with $b$. An $a$-factor depends only on $L_a,p_a,x$ and $U^{\mathrm{ext}}$, through links at $d_a\le s+2$, which have $d_b\ge w+1-(s+2)\ge s+6$ for $w\ge2s+7$, so it contains no coordinate of $L_b\cup p_b$ and no factor is both. The sector changes only the indicators of $I_a$ and $I_b$, which are $a$- and $b$-factors, so the product $\Phi_K$ of the $K$-factors is the same in all four sectors. \emph{Step 4: Tonelli.} All factors are nonnegative and the Haar measures finite, so, with $\Phi^i_a$ the product of the $a$-factors in sector $(i,j)$ and $\Phi^j_b$ that of the $b$-factors, \begin{multline*} Z_{ij}=\int\Bigl[\int\Phi^i_a\,dL_a\,dp_a\Bigr]\Bigl[\int\Phi_K\,dz\Bigr] \Bigl[\int\Phi^j_b\,dL_b\,dp_b\Bigr]\\ \times\prod_{e\in x}j\,d\zeta_e\prod_{e\in y}j\,d\zeta_e , \end{multline*} the inner integrals over $L_a,p_a$ being the coarea integrals of the unsolved pairs, in which every branch of every constraint is integrated and none discarded, and the $x$- and $y$-integrals restricted to $D_x$ and $D_y$ because every $x$-link carries its indicator $\mathbf 1_{G_e}$. Define $a_i(dx)$ as the first bracket times $\prod_{e\in x}j\,d\zeta_e$, $b_j(dy)$ likewise and $K$ as the middle bracket with the $j$-factors of the $z$-links included. This is the factorisation; (i), (ii) and (iii) are read off Step 3. \emph{Step 5: the kernel.} $\Phi_K$ is a product of positive smooth functions of $(z,x,y)$ (Wilson factors, Jacobians $J_k>0$ since $\lVert DF_k\rVert<0.9$, densities $j>0$) and of the indicators of the $z$-balls; the solved $\Gamma_k$ are smooth on a neighbourhood of the closed product of balls, the contraction of Lemma~\ref{lem:branchbound} still applying at radius $1.01r$. So $K=\int_{D_z}e^{-\mathcal U}dz$ with $\mathcal U$ smooth and finite near the compact set $D_z\times D_x\times D_y$, $K$ is smooth and $K\ge\lvert D_z\rvert e^{-\sup\mathcal U}>0$. This is (iv). \emph{Step 6: supports.} Two facts of the tree gauge are used. First, an inside link $(a,\nu)$, $o_\nu(a)=0$, and a path link $(2y+e_\mu,\mu)$ lie in a common plaquette only if $a\in B(y)\cup B(y+e_\mu)$: as opposite sides they would have the same direction and bases differing by a unit vector $e_\rho$, $\rho\ne\mu$, forcing $o_\mu(a)=1$; as adjacent sides the shared corner is $2y+e_\mu$ or $2y+2e_\mu$ (then $a$ or, when $\nu=\mu$, $a+e_\mu$ is that corner, so $a\in B(y)\cup B(y+e_\mu)$), or it is $a+e_\nu$ with $\nu\ne\mu$ equal to one of them, forcing $o_\nu(a)=1$. Second, if $d_a(y)=s+4$ and $d_a(y+e_\mu)=s+3$, then $d_a(y-e_\nu)\le s+4$ for every $\nu\ne\mu$, since some $a'\in I_a$ has $\lVert y+e_\mu-a'\rVert_\infty=s+3$, so $\lvert y_\nu-a'_\nu\rvert\le s+3$ for $\nu\ne\mu$. We claim that a $K$-term depending on an $x$-coordinate has all its links based at $d_a\le s+4$, one of them at $d_a\le s+3$, and its $z$-gradient supported at $d_a\le s+4$. A Jacobian $J_k$ depends on $x$ only when $k$ is based at $s+3$, or at $s+4$ pointing inward, and then every link of $k$ is at $d_a\le s+4$. A Wilson term depends on $x$ only through an $x$-link (non-path at $s+2$, inside at $s+3$) or through a solved path link at $s+2$, at $s+3$, or inward at $s+4$. In the first two cases all links of the plaquette are at $d_a\le s+4$, and a path link of it at $s+4$ lies within one block of a link at $\le s+3$: if that link is inside at $s+3$ the path link points into its block by the first fact and depends on links at $s+3,s+4$; a plaquette cannot contain links at $s+2$ and at $s+4$; and a path link $\ell'$ at $s+4$ in a plaquette with a path link at $s+3$ cannot point outward to a block at $s+5$, since its base block is adjacent to a block at $s+3$, so $\ell'$ depends on links at $d_a\le s+4$. In the third case the plaquettes through an inward path link $(2y+e_\mu,\mu)$ with $d_a(y)=s+4$ lie in the planes $(\mu,\nu)$ at $2y+e_\mu$ and at $2y+e_\mu-e_\nu$, with links based in $B(y)$, $B(y+e_\mu)$, $B(y-e_\nu)$, $B(y+e_\mu-e_\nu)$, all at $d_a\le s+4$ by the second fact, and their path links are within one block of $B(y+e_\mu)$ at $s+3$, so point to blocks at $d_a\le s+4$. This proves the claim, and its mirror for $b$. Hence $\nabla_z\partial_x\mathcal U$ lives at $d_a\le s+4$ and $\nabla_z\partial_y\mathcal U$ at $d_b\le s+4$; since $d_b\ge w+1-d_a$ their coarse distance is at least $w-2s-7$, and two links based in blocks at coarse distance $D\ge1$ have midpoints at fine $\ell^\infty$ distance at least $2D-2$, which gives $d'$. A $K$-term depending on both $x$ and $y$ would contain a link at $d_a\le s+3$ and a link at $d_b\le s+3$ at mutual block distance at most one, so $w+1\le2s+7$; for $w\ge2s+7$ there is none. This is (v). \emph{Step 7: positivity.} Since $K>0$ on $D_x\times D_y$ and the measures live there, $Z_{ij}>0$ exactly when $a_i(D_x)>0$ and $b_j(D_y)>0$; $a_i(D_x)$ is the constrained weight of $\omega_a^i$ with its two collars, which for the all-good pattern is positive by Proposition~\ref{prop:graphlaw}. This is (vi). \end{proof} The theorem uses only Lemma~\ref{lem:branchbound}; it does not use $\beta_W$ large, $\eta$, or Theorem~\ref{thm:chartdecay}. The three dangers named after Lemma~\ref{lem:connected} do not arise: the island labels change $a_i$ and $b_j$ only, the domain $D_x\times D_y\times D_z$ is fixed by the labels of the layer, which are good in every sector, and every branch of every bad pair is integrated inside $a_i$ or $b_j$ and never compared across sectors. The price is the width. Joining two bad blocks in Sec.~\ref{ssec:islandthreshold} at $\ell^\infty$ distance at most $2s+7$ instead of $2s+1$, so that two islands are separated by a layer of width at least $2s+7$, changes only the number of neighbours $\varDelta$ of Theorem~\ref{thm:islandthreshold}, from $(4s+3)^4-1$ to $(4s+15)^4-1$: $\lvert\log q^*\rvert$ becomes $31444.0$ for $s=1$ and $37777.9$ for $s=5$, and $\beta^*(10^{-8},1)=5.084\cdot10^{21}$, $\beta^*(10^{-8},5)=6.097\cdot10^{21}$, $\beta^*(10^{-9},1)=5.088\cdot10^{23}$, by the same formula $\beta^*=16\lvert\log q^*\rvert\rho^{-2}(1+\epsilon)$. \emph{The layer law.} For $(x,y)\in D_x\times D_y$ let $\mu_{x,y}$ be the law $e^{-\mathcal U(z;x,y)}dz/K(x,y)$ on $D_z$. It is the law of Proposition~\ref{prop:graphlaw} for the window of the blocks at distance at least $s+3$ from both islands, in the coordinates $\zeta$ of the labels, with the inside links of the blocks at distance $s+3$ frozen and the exterior given by $(x,y,U^{\mathrm{ext}})$. The covariance bound of Theorem~\ref{thm:chartdecay}(ii) for this law reads: there are $\mathcal A>0$ and $\alpha>0$ such that for all $(x,y)$, every $\varphi$ on the $z$-links with $\lvert\varphi(e)-\varphi(e')\rvert\le d(e,e')$ and all smooth $F,H$ near $D_z$, \begin{equation}\label{eq:layerbound} \lvert\Cov_{\mu_{x,y}}(F,H)\rvert\le\mathcal A\,\bigl\lVert e^{-\alpha \varphi}\nabla_zF\bigr\rVert_{L^2(\mu_{x,y})}\bigl\lVert e^{\alpha\varphi} \nabla_zH\bigr\rVert_{L^2(\mu_{x,y})}. \end{equation} Let $n_x=45(N_{s+1}+N_{s+2})+17N_{s+3}$, with $N_d$ the number of blocks of $\Lambda$ at $d_a=d$, be the number of $x$-links, $n_y$ likewise, and let $\Lambda_2$ be the largest operator norm of $\partial_x\nabla_z \mathcal U$ and $\partial_y\nabla_z\mathcal U$ on $D_z\times D_x\times D_y$; it is finite, of the form $\beta_Wc_W+c_\psi$ with $c_W$, $c_\psi$ depending on $r$ and the geometry only. \begin{proposition}[the cross-ratio under the layer bound]\label{prop:layercross} Let $s\ge1$, $r\le10^{-4}$, $w\ge2s+8$, and assume~\eqref{eq:layerbound}. Then for all $x_0,x_1\in D_x$ and $y_0,y_1\in D_y$, \[ \begin{gathered} \Bigl|\log\frac{K(x_1,y_1)K(x_0,y_0)}{K(x_1,y_0)K(x_0,y_1)}\Bigr|\le \varDelta,\\ \varDelta:=\mathcal A\Lambda_2^2\lVert x_1-x_0\rVert\lVert y_1-y_0\rVert e^{-\alpha d'}\le4\mathcal A\Lambda_2^2r^2\sqrt{n_xn_y}\,e^{-\alpha d'}, \end{gathered} \] so that, for every pair of patterns with all $Z_{ij}>0$, $\lvert\log Z_{11}+\log Z_{00}-\log Z_{10}-\log Z_{01}\rvert\le\varDelta$ and the two island labels have conditional covariance at most $\frac14\tanh(\varDelta/4)$ given the labels of the layer and $U^{\mathrm{ext}}$. Moreover, for every bounded $F\ge0$ depending on the links of $L_a\cup p_a\cup x$ only, with $\E_{ij}$ the expectation in sector $(i,j)$, \[ e^{-\varDelta}\le\frac{\E_{i1}[F]}{\E_{i0}[F]}\le e^{\varDelta} \qquad(i=0,1) \] whenever the four expectations are positive. \end{proposition} \begin{proof} $D_x$ and $D_y$ are convex, so the segments $x(\sigma)$, $y(\tau)$ stay in them, and $\log K$ is smooth near $D_x\times D_y$ by Theorem~\ref{thm:factorisation}(iv), so the cross-ratio is the double integral of $\partial_\sigma\partial_\tau\log K$ over the unit square. Lemma~\ref{lem:connected} with the fixed domain $D_z$ gives $\partial_\sigma\partial_\tau\log K=-\E[\partial_\sigma\partial_\tau \mathcal U]+\Cov(\partial_\sigma\mathcal U,\partial_\tau\mathcal U)$, and the first term vanishes by Theorem~\ref{thm:factorisation}(v). Put $F=\partial_\sigma\mathcal U$, $H=\partial_\tau\mathcal U$: $\nabla_zF$ is supported on the $z$-links at $d_a\le s+4$, $\nabla_zH$ on those at $d_b\le s+4$, at distance at least $d'$, with $\lVert\nabla_zF\rVert\le\Lambda_2\lVert x_1-x_0\rVert$ and $\lVert\nabla_zH\rVert\le\Lambda_2\lVert y_1-y_0\rVert$ pointwise. Apply \eqref{eq:layerbound} with $\varphi(e)=-\dist(e,\operatorname{supp} \nabla_zF)$, which is $1$-Lipschitz, equal to $0$ on the support of $\nabla_zF$ and at most $-d'$ on that of $\nabla_zH$; this gives the bound on the covariance uniformly in $(\sigma,\tau)$, and $\lVert x_1-x_0\rVert\le2r\sqrt{n_x}$ since each coordinate is a vector of $\g$ of norm at most $r$. The consequences are Lemma~\ref{lem:kernel} applied to $a_i,b_j,K$ and Corollary~\ref{cor:twolabels}. For the last statement, $F\Phi_a^i$ is again a product of $a$-factors, so the factorisation holds with the measures $Fa_i$ and $a_i$, the same $K$ and the same $b_j$, and Lemma~\ref{lem:kernel} for these gives $\lvert\log Z(Fa_i,b_1)+\log Z(a_i,b_0)-\log Z(Fa_i,b_0)-\log Z(a_i,b_1) \rvert\le\varDelta$ with $Z(Fa_i,b_j)/Z(a_i,b_j)=\E_{ij}[F]$. \end{proof} The last bound is the quantity that Proposition~\ref{prop:hiddenbit} says a labelling argument must control: the change of the conditional expectation of an observable of one island under a flip of the label of the other, with the branches, the hard label constraints and the disintegration densities of both islands integrated, not compared. The hypothesis~\eqref{eq:layerbound} is Theorem~\ref{thm:chartdecay}(ii) for the layer law, but in other coordinates. The chart law of that theorem is written in the coordinates $\xi$ of $T_U$, on the product of balls $\mathrm{Box}_r$; the label coordinate of a non-path link is $\zeta_e=\xi_e+h_e(S\xi)$, so the label event $\prod_e\{\lVert\zeta_e\rVert\le r\}$ is a deformation of $\mathrm{Box}_r$, convex in $\zeta$ and not in $\xi$, and the Hessian of the density in $\zeta$ acquires the term $\langle\nabla_\xi\bar H, D^2\xi\rangle$, which is not coordinate invariant. The next theorem carries the proof over to the coordinates $\zeta$; it needs the layer calmer than Theorem~\ref{thm:chartdecay} needs a chart, and a coupling four times larger. \begin{theorem}[the layer law obeys the chart bound]\label{thm:layerbound} Let $G=\SU(2)$, $\lambda=\beta_W/4$, and let the data be those of Theorem~\ref{thm:factorisation}. Let $W'$ be the set of blocks of $\Lambda$ at distance at least $s+3$ from both islands, and assume \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item $r\le8\cdot10^{-5}$ and $4\cdot10^5c_3(\eta+62r)\le\frac1{195}$, which holds when $\eta+62r\le6.0\cdot10^{-13}$; \item $\lambda\ge240\,c_6$, which holds when $\beta_W\ge8.93\cdot10^{17}$; \item every block outside $\Lambda$ at distance at most two from $W'$ is good and the constraints of its pairs hold; this is void when $\Lambda$ contains the blocks at distance at most two from $W'$. \end{enumerate} Then~\eqref{eq:layerbound} holds with $\mathcal A=120/\lambda=480/ \beta_W$ and every $\alpha\le\alpha_G=2.5\cdot10^{-4}$, uniformly in $(x,y)\in D_x\times D_y$, in $U^{\mathrm{ext}}$, in $\Lambda$ and in the width $w$. \end{theorem} \begin{proof} All constants without a name are those of the proofs of Theorems~\ref{thm:fibrechart} and~\ref{thm:chartdecay} and of Proposition~\ref{prop:explicitdecay}, at $\vartheta=0$ unless a weight is written: $\mathcal G=BB^*\ge\frac1{17}$, $\lVert B^*z\rVert_\infty\le0.88 \max_k\lVert z_k\rVert$, $\lVert R_k(X)\rVert\le12c_3\lVert X \rVert_\infty^2$, the rows of $A-B$ of norm at most $7.35c_3s$, $Y=B^*(AB^*)^{-1}$ with $\lVert Y\rVert\le11.32$, $\bar P=S-Y(A-B)S$ with $\lVert\bar P\rVert\le4.80$ and $\lVert\bar P\rVert_{1\to1}\le3.003$, $x_v=\bar Pv$ and $x_{vv'}=-YD^2c(X)[x_v,x_{v'}]$ with $\lVert x_{vv'}\rVert_1\le1564.3\,c_3\lvert v\rvert\lvert v'\rvert$. Put $s'=32r$ for the chart radius. \emph{Step 1: the background is calm.} Let $U$ be the point of the fibre whose non-path links in $W'$ are $R(c)$ and whose path links there are the solutions $\Gamma_k(0)$ of Lemma~\ref{lem:branchbound}. Take a plaquette $p$ with a vertex in a block of $W'^+$. Its links are based in blocks at distance at most two from $W'$, which are good, by the sector event inside $\Lambda$ and by (c) outside it. A plaquette contains at most two path links, one of each of its directions: a path link $(2y+e_\mu,\mu)$ has exactly one odd coordinate at its base, and two such links of one direction on one plaquette would have to be opposite sides with bases differing by $e_\rho$, $\rho\ne\mu$, which gives one of them two odd coordinates. The plaquette of $R(c)$ at $p$ is $I$ unless both coordinates of the corner in its plane are odd, and it is then the coarse plaquette of a good block, so $\delta(R(c)_p,I)\le\eta$. Replacing the four links one at a time, with $\delta\le d$ (Lemma~\ref{lem:logbound}), bi-invariance and $d(\Gamma_k,c_k)\le14r$, gives $\delta(U_p,I)\le\eta+2\cdot14r+2r$. So the defect $\kappa_0$ of the background satisfies $\kappa_0+s'\le\eta+62r$, and (a) is the hypothesis $4\cdot10^5c_3(\kappa_0+s')\le\frac1{195}$ of Theorem~\ref{thm:fibrechart}(iv) at radius $s'$; by Proposition~\ref{prop:explicitconstants} it implies $s'\le1/(6000c_3)$, $\kappa_0+6s'\le10^{-8}$ and $c_3s'\le1.29\cdot10^{-8}$. \emph{Step 2: the two coordinates.} Let $\rho_1=\frac54r$, which is at most $10^{-4}$ by (a). For $\zeta$ with every $\lVert\zeta_e\rVert\le \rho_1$ let $U'$ be the point of the fibre with non-path links $R_e(c)e^{\zeta_e}$, and $X=\log(U^{-1}U')$ link by link. On the non-path links $\lVert X_e\rVert\le\rho_1$; on a path link, by Lemma~\ref{lem:branchbound} at the radii $r$ and $\rho_1$, $\lVert X_{d_k}\rVert\le14r+14\rho_1=31.5r$. So $t=\lVert X\rVert_\infty 0$ satisfies \[ \begin{gathered} \bigl\lvert\log Z_{11}+\log Z_{00}-\log Z_{10}-\log Z_{01}\bigr\rvert \le\varDelta,\\ \varDelta=\frac{1920}{\beta_W}\Lambda_2^2r^2\sqrt{n_xn_y}\, e^{-2\alpha_G(w-2s-8)}, \end{gathered} \] the two island labels have conditional covariance at most $\frac14\tanh(\varDelta/4)$, and every observable of one island and its collars changes by a factor in $[e^{-\varDelta},e^{\varDelta}]$ when the label of the other island is flipped. In the normalisation $r=43\rho$, $\eta=92\rho$ of Theorem~\ref{thm:islandthreshold}, hypothesis (a) of Theorem~\ref{thm:layerbound} holds for $\rho\le2.17\cdot10^{-16}$, where the island threshold with the joining distance $2s+8$ is $\beta^*=1.122\cdot10^{37}$ for $s=1$ and $1.322\cdot10^{37}$ for $s=5$, and hypothesis (b) holds at every $\beta_W\ge\beta^*$. \end{corollary} \begin{proof} Proposition~\ref{prop:layercross} with $\mathcal A=480/\beta_W$ from Theorem~\ref{thm:layerbound} and $d'=2(w-2s-8)$. For the numbers, $\eta+62r=(92+62\cdot43)\rho=2758\rho$, and the threshold is the largest root of the equation of Theorem~\ref{thm:islandthreshold} with $(4s+17)^4-1$ neighbours, the count for the joining distance $2s+8$, which makes every layer between two islands of width at least $2s+8$: $\lvert\log q^*\rvert=32481.7$ for $s=1$ and $38354.1$ for $s=5$. \end{proof} \begin{lemma}[the mixed derivatives of the layer action]\label{lem:lambdatwo} Let $G=\SU(2)$, let the data be those of Theorem~\ref{thm:factorisation} with $r\le8\cdot10^{-5}$, and let $\kappa\le10^{-8}$ be such that $\delta(U_p,I)\le\kappa$ on $D_z\times D_x\times D_y$ for every plaquette $p$ of $S_K$ that contains the solved path link of a pair one of whose arguments, the non-path links of its paths, is a $z$-link. Then \[ \Lambda_2\le\beta_Wc_W+c_\psi,\qquad c_W\le101.11,\qquad c_\psi\le4.94\cdot10^6, \] with $c_W$ and $c_\psi$ the explicit functions of $r$ and $\kappa$ given in~\eqref{eq:cwcpsi}. Under hypotheses (a) and (c) of Theorem~\ref{thm:layerbound} the condition holds with $\kappa=\eta+30r\le6.0\cdot10^{-13}$, and \[ \Lambda_2\le99.0001\,\beta_W+4.65\cdot10^6 . \] \end{lemma} \begin{proof} Norms on $\g$ are Hilbert--Schmidt norms, $\delta(g,h)=\lVert g-h\rVert$ on $\SU(2)$, and the coordinates are the $\zeta_e$ of Theorem~\ref{thm:factorisation}, $U_e=R_e(c)e^{\zeta_e}$, with the reconstruction value $R_e(c)=I$ for a link inside a block and $R_e(c)=c_k$ for a private link of the pair $k$. For a function $f$ of the coordinates and two coordinates $g\ne g'$ write $\lVert\partial_g\partial_{g'}f\rVert$ for the supremum of $\lvert\partial_s\partial_tf(\zeta+sv_g+tw_{g'})\rvert$ at $s=t=0$ over $v,w\in\g$ of norm one placed at $g$ and $g'$: it is the operator norm of the $(g,g')$ block of the Hessian. \emph{Step 1: reduction to row sums.} Let $M=\partial_x\nabla_z\mathcal U$, with blocks $M_{ef}$ for $z$-links $e$ and $x$-links $f$. For vectors $u$ on the $z$-links and $v$ on the $x$-links, $\lvert\ip u{Mv}\rvert\le\sum_{e,f}\lVert M_{ef}\rVert\lVert u_e\rVert \lVert v_f\rVert$, and the Cauchy--Schwarz inequality gives $\lVert M\rVert\le\max\{\max_e\sum_f\lVert M_{ef}\rVert,\max_f\sum_e \lVert M_{ef}\rVert\}$. A term $-\log j(\zeta_e)$ of $\psi$ depends on one coordinate, so $M_{ef}$ is the sum of the blocks of the Wilson terms $\beta_WW_p$, $W_p=1-\frac12\ReTr U_p$, of $S_K$ and of the terms $-\log J_k=\log\det(1-DF_k)$ of $\psi$, $J_k$ the inverse Jacobian of Proposition~\ref{prop:graphlaw} and $DF_k$ the differential in $X$ of the map $F$ of Lemma~\ref{lem:branchbound} at the fixed point, that depend on $e$ and on $f$. It therefore suffices to prove that for every $x$- or $z$-coordinate $g$ \[ R_g=\sum_{g'}\sum_\tau\lVert\partial_g\partial_{g'}\tau\rVert\le \beta_Wc_W+c_\psi , \] $g'$ running over the coordinates of the other class ($z$ if $g$ is an $x$-coordinate, $x$ if $g$ is a $z$-coordinate) and $\tau$ over the Wilson terms and the Jacobians that depend on $g$ and $g'$; the bound for $\partial_y\nabla_z\mathcal U$ is the mirror statement. \emph{Step 2: the geometry.} (i) Every link lies in six plaquettes. (ii) The solved path link $\Gamma_k$ of $k=(y,\mu)$ is a function of the $21$ non-path links of the fourteen paths of nonzero offset (Lemma~\ref{lem:branchbound}), its \emph{arguments}: the seven non-path private links, each in two of these paths, and the fourteen links of direction $\mu$ in $B(y)$ and $B(y+e_\mu)$ other than $(2y,\mu)$ and $(2y+2e_\mu,\mu)$, each in one path, the tree links among them being fixed at $I$. A path of $k$ consists of links of direction $\mu$, so a non-path private link is an argument of its own pair only, and a link of direction $\mu$ inside $B(y)$ of the two pairs $(y,\mu)$ and $(y-e_\mu,\mu)$. Let $K(g)$ be the set of solved pairs having $g$ as an argument and $n_g$ the number of paths of each containing $g$: $n_g=2$, $\lvert K(g)\rvert\le1$ for a private link, $n_g=1$, $\lvert K(g)\rvert\le2$ for an inside link. (iii) Let $\ell=(a,\mu)$, $a=2y+e_\mu$, be a path link and $\nu\ne\mu$. The plaquette at $a$ in the plane $\{\mu,\nu\}$ has the further links $(a,\nu)$, of offset $e_\mu$, inside $B(y)$; $(a+e_\mu,\nu)$, based at the root $2y+2e_\mu$, a tree link; and $(a+e_\nu,\mu)$, of offset $e_\mu+e_\nu$, a non-path private link of $(y,\mu)$. The plaquette at $a-e_\nu$ has the further links $(a-e_\nu,\mu)$ and $(a-e_\nu,\nu)$, both based at $2(y-e_\nu)+e_\mu+e_\nu$ and so non-path private links of $(y-e_\nu,\mu)$ and $(y-e_\nu,\nu)$, and $(a+e_\mu-e_\nu,\nu)= (2(y+e_\mu-e_\nu)+e_\nu,\nu)$, the path link of $(y+e_\mu-e_\nu,\nu)$. Hence each of the six plaquettes of a path link carries at most two non-path links, exactly three of them, those at $a-e_\nu$, carry a second path link, and no plaquette carries three path links. (iv) By (iii) a non-path link lies on a plaquette with a path link only if it is an inside link $(2y+e_\mu,\nu)$ with $\nu<\mu$, the only other case $\nu>\mu$ being a tree link, or a private link of offset $e_\mu+e_\nu$ and direction $\mu$ or $\nu$. The first lies on one such plaquette, the one at $2y+e_\mu$ in the plane $\{\mu,\nu\}$, with one path link. The second, $(b,\mu)$ with $b=2y+e_\mu+e_\nu$, plays the role of $(a+e_\nu,\mu)$ for $a=2y+e_\mu$, on a plaquette with one path link, and the roles of $(a-e_\nu,\mu)$ for $a=2(y+e_\nu)+e_\mu$ and of $(a-e_\mu,\mu)$ for $a=2(y+e_\mu)+e_\nu$, which are one plaquette, the one at $b$ in the plane $\{\mu,\nu\}$, with two path links. So the six plaquettes of a non-path link carry at most three path links in all. \emph{Step 3: one Wilson term.} Let $V_1,\dots,V_4$ be smooth functions of $(s,t)$ with values in $\SU(2)$, $P=V_1V_2V_3V_4$, and $a_i=\lVert\partial_sV_i\rVert$, $b_i=\lVert\partial_tV_i\rVert$, $c_i=\lVert\partial_s\partial_tV_i\rVert$. Then \begin{equation}\label{eq:wilsonmixed} \bigl\lvert\partial_s\partial_t\tfrac12\ReTr P\bigr\rvert\le\tfrac12 \Bigl(\sum_ia_i\Bigr)\Bigl(\sum_jb_j\Bigr)+\tfrac12\,\delta(P,I) \sum_ic_i . \end{equation} Indeed $\partial_s\partial_tP$ is the sum of the products with $\partial_sV_i$ and $\partial_tV_j$ in two places $i\ne j$, whose traces have modulus at most $a_ib_j$ by $\lvert\Tr(AB)\rvert\le\lVert A\rVert \lVert B\rVert$ and unitarity of the other factors, and of the products with $\partial_s\partial_tV_i$ in place $i$, whose trace is $\Tr(\tilde P_iQ_i)$ with $Q_i=V_i^{-1}\partial_s\partial_tV_i$ and $\tilde P_i$ a conjugate of $P$. Differentiating $V_i^*V_i=I$ twice gives $Q_i+Q_i^*=-(\partial_sV_i)^*\partial_tV_i-(\partial_tV_i)^*\partial_sV_i$, so $\lvert\ReTr Q_i\rvert\le a_ib_i$, and $\lvert\Tr((\tilde P_i-I)Q_i) \rvert\le\delta(P,I)c_i$. The orientation of a link does not change $a_i,b_i,c_i$. \emph{Step 4: first derivatives.} For $W\in\g$ the map $\operatorname{ad}W$ is skew for the trace form, with eigenvalues $0$ and $\pm i\theta$, $\theta=\sqrt2\lVert W\rVert$: in a basis diagonalising $W=\operatorname{diag}(i\varphi,-i\varphi)$, $\lVert W\rVert=\sqrt2\varphi$, it multiplies the off-diagonal entries by $\pm2i\varphi$. A power series $f(\operatorname{ad}W)$ is therefore normal with norm $\max\{\lvert f(0) \rvert,\lvert f(\pm i\theta)\rvert\}$. For $\varpi(u)=(1-e^{-u})/u$ one has $\varpi(i\theta)=e^{-i\theta/2} \frac{\sin(\theta/2)}{\theta/2}$, so with $m(\theta)=\frac{\theta/2} {\sin(\theta/2)}$, increasing on $(0,2\pi)$, \begin{gather*} \lVert\varpi(\operatorname{ad}W)\rVert=1,\quad \lVert\varpi(\operatorname{ad}W)^{-1}\rVert=m(\theta),\\ \lVert e^{\operatorname{ad}W}-1\rVert\le\theta,\quad \lVert\varpi(\operatorname{ad}W)-1\rVert\le1-\tfrac1{m(\theta)}+ \tfrac\theta2,\\ \lVert\varpi(\operatorname{ad}W)^{-1}-1\rVert\le m(\theta)-1+m(\theta) \tfrac\theta2 . \end{gather*} The derivative of the exponential is $\frac d{dt}e^{W(t)}=e^{W}\varpi( \operatorname{ad}W)W'$. Hence a non-path link has $\lVert\partial_gU_g[v]\rVert=\lVert\varpi(\operatorname{ad}\zeta_g)v\rVert \le\lVert v\rVert$. Let $k=(y,\mu)$ be solved, $\Gamma_k=c_ke^X$. For a path $x$ of nonzero offset, $c_k^{-1}P_x=e^{\alpha_x}e^{\beta_x}$, where $(\alpha_x,\beta_x)$ is $(\operatorname{Ad}(c_k^{-1})\zeta_u,\zeta_q)$ when $P_x=uq$ with $u$ inside $B(y)$ and $q$ private, and $(\zeta_q,\zeta_{u'})$ when $P_x=qu'$ with $u'$ inside $B(y+e_\mu)$, a tree link contributing $0$; so $\lVert\alpha_x\rVert,\lVert\beta_x\rVert \le r$. Put $Z_x=\log(e^{-X}e^{\alpha_x}e^{\beta_x})$; by Lemma~\ref{lem:branchbound}, $\lVert X\rVert\le14r$ and $\lVert Z_x\rVert \le16r$. With $Y=e^{Z_x}$ and $Y^{-1}\partial Y=\varpi(\operatorname{ad} Z_x)\partial Z_x$, \begin{align*} \partial_XZ_x[h]&=-T_xh,\\ T_x&=\varpi(\operatorname{ad}Z_x)^{-1} \operatorname{Ad}(e^{-\beta_x}e^{-\alpha_x})\,\varpi(-\operatorname{ad}X),\\ \partial_{\beta}Z_x[w]&=\varpi(\operatorname{ad}Z_x)^{-1}\varpi( \operatorname{ad}\beta_x)w,\\ \partial_{\alpha}Z_x[v]&=\varpi(\operatorname{ad}Z_x)^{-1} \operatorname{Ad}(e^{-\beta_x})\varpi(\operatorname{ad}\alpha_x)v . \end{align*} With $\theta_Z=16\sqrt2r$, $\theta_X=14\sqrt2r$, $m_1=m(\theta_Z)$, the last two have norm at most $m_1$, and writing $T_x-1$ as $(\varpi(\operatorname{ad}Z_x)^{-1}-1)\operatorname{Ad}(\cdot) \varpi(\cdot)+(\operatorname{Ad}(\cdot)-1)\varpi(\cdot)+(\varpi(-\operatorname{ad} X)-1)$, \[ \lVert T_x-1\rVert\le\varepsilon_T=m_1-1+m_1\tfrac{\theta_Z}2+2\sqrt2r +1-\tfrac1{m(\theta_X)}+\tfrac{\theta_X}2 . \] Since $F=-\frac1{16}\sum_xZ_x$ over the fourteen paths, $1-DF_k=\frac18 -\frac1{16}\sum_x(T_x-1)$, so $1-DF_k$ is invertible with \[ \lVert(1-DF_k)^{-1}\rVert\le\mathsf m=\frac8{1-7\varepsilon_T}, \] and $\det(1-DF_k)>0$, every eigenvalue of $8(1-DF_k)$ lying within $7\varepsilon_T<1$ of $1$. The implicit function theorem gives $\partial_gX=(1-DF_k)^{-1}\partial_gF$ and $\lVert\partial_gF\rVert\le n_gm_1/16$, and $\lVert\partial_g\Gamma_k[v]\rVert=\lVert e^X\varpi( \operatorname{ad}X)\partial_gX[v]\rVert\le\lVert\partial_gX[v]\rVert$, so \[ \lVert\partial_g\Gamma_k\rVert\le\lVert\partial_gX\rVert\le\gamma_g= \frac{n_gm_1}{2(1-7\varepsilon_T)} : \] $\gamma_g=\gamma$ for a private link and $\gamma_g=\frac\gamma2$ for an inside link, with $\gamma=m_1/(1-7\varepsilon_T)$. In both cases $\gamma_g\lvert K(g)\rvert\le\gamma$, and the sum of $\gamma_{g'}$ over the arguments of one pair is at most $7\gamma+14\frac\gamma2=14\gamma$. \emph{Step 5: higher derivatives.} Let $G(X,\alpha,\beta)= \log(e^{-X}e^\alpha e^\beta)$ for $X,\alpha,\beta\in\g\otimes\C$ with $\lVert X\rVert+\lVert\alpha\rVert+\lVert\beta\rVert\le\sigma=\frac12$. The Hilbert--Schmidt norm is submultiplicative, so $\lVert e^{-X}e^\alpha e^\beta-I\rVert\le e^\sigma-1<1$, the logarithmic series converges, $G$ is holomorphic and agrees with the principal logarithm at real arguments, and $\lVert G\rVert\le M_G=-\log(2-e^{1/2})$. At a real point with $\lVert X\rVert+\lVert\alpha\rVert+\lVert\beta \rVert\le16r$ and for unit vectors $v_1,\dots,v_j$ of $\g$ placed in any of the three slots, $t\mapsto G(\cdot+\sum_it_iv_i)$ is holomorphic on the polydisc $\lvert t_i\rvert\le\rho_j=(\frac12-16r)/j$, and the Cauchy estimates give \[ \lVert D^jG[v_1,\dots,v_j]\rVert\le B_j=M_G\rho_j^{-j}\qquad(j=2,3). \] For $g,g'$ arguments of $k$ let $s_{gg'}\le1$ be the number of paths containing both. Differentiating $X=F(X,\zeta)$ twice and using $\lVert D_X^2F\rVert\le\frac{14}{16}B_2$, $\lVert D_XD_gF\rVert\le\frac{n_g}{16}B_2$, $\lVert D_gD_{g'}F\rVert\le\frac{s_{gg'}}{16}B_2$, \begin{align*} \lVert\partial_g\partial_{g'}X\rVert&\le\frac{\mathsf mB_2}{16} \bigl(14\gamma_g\gamma_{g'}+n_{g'}\gamma_g+n_g\gamma_{g'}+s_{gg'}\bigr)\\ &\le X_2=\frac{\mathsf mB_2}{16}(14\gamma^2+4\gamma+1). \end{align*} The second differential of $\exp$ at $X$ has norm at most $e^{\lVert X\rVert}$ (each of the $n(n-1)$ terms of the second derivative of $X^n/n!$ has norm at most $\lVert X\rVert^{n-2}$), so $\lVert\partial_g\partial_{g'}\Gamma_k\rVert\le C_2=e^{14r}\gamma^2+X_2$. For $\mathcal M=1-DF_k$, the chain rule and the same bounds with $B_3$ for the third derivatives of $F$ give \begin{gather*} \lVert\partial_g\mathcal M\rVert\le M_1=\frac{B_2}{16}(14\gamma+2),\\ \lVert\partial_g\partial_{g'}\mathcal M\rVert\le M_2=\frac{B_3}{16} (14\gamma^2+4\gamma+1)+\frac{14}{16}B_2X_2 , \end{gather*} the terms of $\partial_g\partial_{g'}\mathcal M$ being $D_X^3F[\partial_gX,\partial_{g'}X]$, $D_X^2D_{g'}F[\partial_gX]$, $D_X^2D_gF[\partial_{g'}X]$, $D_X^2F[\partial_g\partial_{g'}X]$ and $D_XD_gD_{g'}F$. \emph{Step 6: the Wilson terms.} For a plaquette $p$ and a coordinate $g$ let $A_g(p)$ be the sum over the links of $p$ of the bounds of Step 4 for their derivatives in $g$: $A_g(p)=\mathbf 1_{g\in p}+\gamma_g\, \#\{k\in K(g):\Gamma_k\in p\}$. By~\eqref{eq:wilsonmixed}, $\lVert\partial_g\partial_{g'}W_p\rVert\le\frac12A_g(p)A_{g'}(p)+\frac12 \delta(U_p,I)C_{gg'}(p)$, with $C_{gg'}(p)\le C_2$ times the number of solved path links of $p$ with both $g$ and $g'$ as arguments. Put $S_g(p)=\sum_{g'\ne g}A_{g'}(p)$. A non-path link contributes at most one to $S_g(p)$ and a solved path link at most $14\gamma$ (Step 4), so $S_g(p)\le N+14\gamma\Pi$ with $N$ the number of non-path links of $p$ other than $g$ and $\Pi$ the number of path links of $p$. If $g\in p$ then $N+\Pi\le3$ and $S_g(p)\le3+14\gamma\Pi$, and by Step 2(iv) the sum over the six plaquettes of $g$ is at most $18+42\gamma$. If $\Gamma_k\in p$ then $N\le2$ and $\Pi\le2$, with $\Pi=2$ on three of the six plaquettes of $\Gamma_k$ (Step 2(iii)), so the sum over them is at most $12+14\gamma(6+3)=12+126\gamma$. Hence \begin{align*} \sum_p A_g(p)S_g(p)&\le18+42\gamma+\gamma_g\lvert K(g)\rvert (12+126\gamma)\\ &\le18+54\gamma+126\gamma^2 . \end{align*} The sum of the first part over all $g'\ne g$ bounds its sum over the other class. The second part of~\eqref{eq:wilsonmixed} enters only through a solved path link $\Gamma_k\in p$ with $g$ and $g'$ among its arguments; one of $g,g'$ is a $z$-coordinate, so $k$ has a $z$-link among its arguments and the hypothesis gives $\delta(U_p,I)\le\kappa$. There are at most $\lvert K(g)\rvert\le2$ such pairs, six plaquettes for each and twenty coordinates $g'$, so this part adds at most $\frac12\kappa\cdot240\,C_2$. The Wilson terms contribute at most $\beta_Wc_W$ to $R_g$. \emph{Step 7: the Jacobians.} The term $\log\det\mathcal M$ of a solved pair depends on $g$ only if $k\in K(g)$, and on $g'\ne g$ only if $g'$ is one of the other twenty arguments. With $\lvert\Tr A\rvert\le3\lVert A \rVert$ on $\g$ and $\lVert\mathcal M^{-1}\rVert\le\mathsf m$, \begin{multline*} \bigl\lvert\partial_g\partial_{g'}\log\det\mathcal M\bigr\rvert= \bigl\lvert\Tr(\mathcal M^{-1}\partial_g\partial_{g'}\mathcal M)\\ -\Tr(\mathcal M^{-1}\partial_{g'}\mathcal M\,\mathcal M^{-1}\partial_g \mathcal M)\bigr\rvert\le L=3\mathsf mM_2+3\mathsf m^2M_1^2, \end{multline*} and the Jacobians contribute at most $c_\psi=2\cdot20\cdot L$ to $R_g$. Together, \begin{equation}\label{eq:cwcpsi} c_W=9+27\gamma+63\gamma^2+120\kappa C_2,\qquad c_\psi=40L . \end{equation} \emph{Step 8: the numbers.} Since $\sin x\ge x-x^3/6$, $m(\theta)\le (1-\theta^2/24)^{-1}$ and $1-1/m(\theta)\le\theta^2/24$; the series of $e^{1/2}$, whose tail after the term $k=8$ is at most twice its first term, gives $e^{1/2}\le1.6487213$ and $M_G\le1.046176$; and $e^{14r}\le (1-14r)^{-1}$ in $C_2$. Every constant of Steps 3 to 7 increases with $r$, and $c_W$ with $\kappa$, so it suffices to take $r$ and $\kappa$ at their largest values; each bound below is computed from the rounded bounds before it, with $\sqrt2\le1.4142136$. At $r=8\cdot10^{-5}$: $\theta_Z\le1.8102\cdot 10^{-3}$, $\theta_X\le1.5840\cdot10^{-3}$, $m_1\le1+1.4\cdot10^{-7}$, $1-1/m(\theta_X)\le1.046\cdot10^{-7}$, $\varepsilon_T\le1.924\cdot10^{-3}$, $\mathsf m\le8.110$, $\gamma\le1.01366$, $B_2\le16.825$, $B_3\le227.72$, $14\gamma^2+4\gamma+1\le19.44$, $X_2\le165.8$, $C_2\le166.9$, $M_1\le17.03$, $M_2\le2718$ and $L\le1.234\cdot10^5$; so for $r\le8\cdot10^{-5}$ and $\kappa\le10^{-8}$, $c_W\le101.11$ and $c_\psi\le4.94\cdot10^6$. Under hypothesis (a) of Theorem~\ref{thm:layerbound}, $62r\le6.0\cdot10^{-13}$, so $r\le9.68\cdot10^{-15}$, and the same chain gives $m_1\le1+10^{-26}$, $\varepsilon_T\le2.328\cdot10^{-13}$, $\gamma\le1+1.7\cdot10^{-12}$, $\mathsf m\le8(1+1.7\cdot10^{-12})$, $B_2\le16.73882$, $B_3\le225.975$, $14\gamma^2+4\gamma+1\le19+10^{-10}$, $X_2\le159.019$, $C_2\le160.02$, $M_1\le16.7389$, $M_2\le2597.5$, $L\le1.1614\cdot10^5$, and $9+27\gamma+63\gamma^2\le99+4\cdot10^{-10}$. For the condition on $\kappa$, let $k=(y,\mu)$ be solved with a $z$-link among its arguments. A $z$-link is based in a block of $W'$, so $y$ or $y+e_\mu$ lies in $W'$, and by Step 2(iii) the links of a plaquette $p$ through $\Gamma_k$ are based in blocks at distance at most one from $B(y)$, hence at distance at most two from $W'$; they are $\Gamma_k$, non-path links, a tree link, and possibly the path link of $k'=(y+e_\mu-e_\nu,\nu)$, both of whose blocks are at distance at most two from $W'$. These blocks are good, by the sector event inside $\Lambda$, since they lie at distance at least $s+1$ from both islands, and by (c) outside it, so every non-path link of $p$ is within $r$ of its reconstruction value. The path link of $k'$ is within $14r$ of $c_{k'}$: if $k'$ is based in $\Lambda$ it is solved, since a pair based at distance at most $s+1$ from an island has its path link in $L_a\cup p_a$ or $L_b\cup p_b$ and its plaquettes are not $K$-factors, and otherwise Lemma~\ref{lem:branchbound} applies by (c). Step 1 of the proof of Theorem~\ref{thm:layerbound} gives $\delta(R(c)_p,I)\le\eta$ for these plaquettes, and replacing the links one at a time with bi-invariance and $\delta\le d$ (Lemma~\ref{lem:logbound}) gives $\delta(U_p,I)\le\eta+2\cdot14r+2r=\eta+30r\le6.0\cdot10^{-13}$. Hence $120\kappa C_2\le1.2\cdot10^{-8}$, $c_W\le99.0001$ and $c_\psi\le40L\le 4.65\cdot10^6$. \end{proof} With Lemma~\ref{lem:lambdatwo}, $\Lambda_2^2\le\beta_W^2(c_W+c_\psi/ \beta_W)^2$ and the bound of Corollary~\ref{cor:twoislands} becomes \[ \varDelta\le1920\,\beta_W\Bigl(c_W+\frac{c_\psi}{\beta_W}\Bigr)^2r^2 \sqrt{n_xn_y}\,e^{-2\alpha_G(w-2s-8)}, \] which grows linearly in $\beta_W$; at $\beta_W=1.13\cdot10^{37}$, just above $\beta^*$, $s=1$ and $r=43\rho$, $\rho=2.17\cdot10^{-16}$, where $\beta_Wr^2=9.84\cdot10^8$, and for islands of one block, $n_x=n_y=45(544+1776)+17\cdot4160=175120$, it reads $\varDelta\le3.25\cdot10^{21}e^{-2\alpha_G(w-10)}$. This is not small at the threshold: the bound is at most one exactly when $w\ge99072$, at fixed $r$ every factor $e$ in $\beta_W/\beta^*$ adds $1/(2\alpha_G)=2000$ to this width, and islands of $\lvert I_a\rvert$ and $\lvert I_b\rvert$ blocks, for which $N_d\le\lvert I_a\rvert((2d+1)^4-(2d-1)^4)$, add at most $1000\log(\lvert I_a\rvert\lvert I_b\rvert)$. The covariance therefore decays exponentially in the width of the good layer (Figure~\ref{fig:layer}), at the rate of the chart, with the amplitude of Lemma~\ref{lem:lambdatwo}, which is below one at the threshold only across good layers about $10^5$ blocks wide. If further bad blocks lie in $\Lambda$ at distance at least $s+4$ from both islands, the factorisation of Theorem~\ref{thm:factorisation} through $K$ still holds as an identity of measures, with the constraints of the bad pairs of the layer kept as hard factors inside $K$, but $\mu_{x,y}$ is then no longer a good-window law, and the cross-ratio of $K$ is the comparison for the remaining islands: the corollary is the inductive step for two islands, and Sec.~\ref{ssec:manyislands} keeps those constraints out of the layer and performs the induction. \begin{figure*}[t] \centering \begin{minipage}[c]{0.56\textwidth} \centering \resizebox{\textwidth}{!}{% \begin{tikzpicture}[x={(0.62cm,0cm)},y={(0.31cm,0.22cm)},z={(0cm,0.62cm)},>={Stealth[length=1.6mm]}] \foreach \i in {0,...,17} \foreach \j in {0,...,5} { \pgfmathsetmacro{\isl}{(\i<=2) || (\i>=15) ? 1 : 0} \pgfmathsetmacro{\col}{((\i>=3 && \i<=5) || (\i>=12 && \i<=14)) ? 1 : 0} \ifnum\isl=1 \def\fc{red!55}\else\ifnum\col=1 \def\fc{yellow!40}\else\def\fc{green!25}\fi\fi \fill[\fc] (\i,\j,0) -- (\i+1,\j,0) -- (\i+1,\j+1,0) -- (\i,\j+1,0) -- cycle; \draw[gray!55,line width=0.2pt] (\i,\j,0) -- (\i+1,\j,0) -- (\i+1,\j+1,0) -- (\i,\j+1,0) -- cycle;} \foreach \i in {0,...,17} { \pgfmathsetmacro{\isl}{(\i<=2) || (\i>=15) ? 1 : 0} \pgfmathsetmacro{\col}{((\i>=3 && \i<=5) || (\i>=12 && \i<=14)) ? 1 : 0} \ifnum\isl=1 \def\fc{red!70}\else\ifnum\col=1 \def\fc{yellow!60}\else\def\fc{green!40}\fi\fi \fill[\fc] (\i,0,0) -- (\i+1,0,0) -- (\i+1,0,-1.2) -- (\i,0,-1.2) -- cycle; \draw[gray!55,line width=0.2pt] (\i,0,0) -- (\i+1,0,0) -- (\i+1,0,-1.2) -- (\i,0,-1.2) -- cycle;} \draw[thick] (0,0,-1.2) -- (18,0,-1.2) -- (18,0,0) -- (18,6,0) -- (0,6,0) -- (0,0,0) -- cycle; \node[white] at (1.5,0,-0.6) {\scriptsize island $a$}; \node[white] at (16.5,0,-0.6) {\scriptsize island $b$}; \node[yellow!30!black] at (4.5,0,-0.6) {\scriptsize frozen}; \node[yellow!30!black] at (13.5,0,-0.6) {\scriptsize frozen}; \node[green!30!black] at (9,0,-0.6) {\scriptsize layer $D_z$, law $\mu_{x,y}$}; \draw[->,thick,orange!80!black] (1.5,3,0.2) .. controls (5,3,3.2) and (13,3,3.2) .. node[pos=0.5,above] {\footnotesize cross-ratio $\varDelta$ of $K(x,y)$} (16.5,3,0.2); \draw[<->,thick,yellow!40!black] (3,6,0.35) -- (6,6,0.35) node[midway,above] {\scriptsize $s{+}3$}; \draw[<->,thick,yellow!40!black] (12,6,0.35) -- (15,6,0.35) node[midway,above] {\scriptsize $s{+}3$}; \draw[<->,thick,blue!60!black] (3,0,-1.8) -- (15,0,-1.8) node[midway,below] {\footnotesize width $w\ge2s+8$}; \draw[<->,thick,orange!80!black] (6,0,-2.9) -- (12,0,-2.9) node[midway,below] {\footnotesize coarse distance $w-2s-7$}; \end{tikzpicture}} \end{minipage}\hfill \begin{minipage}[c]{0.4\textwidth} \centering \begin{tikzpicture}[x=0.42cm,y=0.9cm] \draw[->] (0,0) -- (10.6,0) node[right] {\footnotesize $w$}; \draw[->] (0,0) -- (0,3.5); \foreach \x/\l in {0/0,2/2000,4/4000,6/6000,8/8000,10/10^4} \draw (\x,0) -- (\x,-0.08) node[below] {\scriptsize $\l$}; \foreach \y/\l in {0/0,0.8/0.25,1.6/0.5,2.4/0.75,3.2/1} \draw (0,\y) -- (-0.12,\y) node[left] {\scriptsize $\l$}; \draw[thick,orange!80!black,domain=0:10,samples=60] plot (\x,{3.2*exp(-0.5*\x)}); \node[above right,orange!70!black] at (1.6,2.1) {\footnotesize $e^{-2\alpha_G(w-2s-8)}$}; \end{tikzpicture} \end{minipage} \caption{Left: two bad islands at the ends of a slab, the frozen collars of width $s+3$ beside them, and the good layer between them, on which the layer law $\mu_{x,y}$ obeys the chart bound in the label coordinates (Theorem~\ref{thm:layerbound}). The cross-ratio $\varDelta$ of the kernel of Theorem~\ref{thm:factorisation} compares the two islands through the layer. Right: the factor $e^{-2\alpha_G(w-2s-8)}$ of Corollary~\ref{cor:twoislands} at $\alpha_G=2.5\cdot10^{-4}$; the amplitude in front of it is $1920\Lambda_2^2r^2\sqrt{n_xn_y}/\beta_W$, of order $10^{-34}\Lambda_2^2r^2\sqrt{n_xn_y}$ at the threshold $\beta_W=\beta^*$.} \label{fig:layer} \end{figure*} \subsection{Many islands}\label{ssec:manyislands} The paragraph after Corollary~\ref{cor:twoislands} leaves the case in which the layer between two islands contains further islands. If the constraints of those islands are kept inside the kernel, the law of the layer carries hard constraints and branches, and Theorem~\ref{thm:layerbound} does not apply to it. This subsection keeps them out of the layer. Every island receives its own collar, the integral over the island and its collar becomes a positive measure on the collar coordinates of that island, as $a_i$ and $b_j$ in Theorem~\ref{thm:factorisation}, and the layer between all the islands is a good window. The kernel of two islands is then a mixture, over the collar coordinates of the others, of good-window kernels. A mixture of kernels with small cross-ratios can have a large one, and Proposition~\ref{prop:hiddenbit} is such a mixture over one bit.\footnote{The cross-ratio $Z_{11}Z_{00}/(Z_{10}Z_{01})$ of the four weights with the two islands in their two patterns measures how much the pattern of one island changes the odds of the pattern of the other: it is one exactly when the two islands are independent. A mixture of independent pairs need not be independent: if a hidden switch raises the chance of both patterns at once, the two islands are correlated however weakly each one depends on the switch.} Theorem~\ref{thm:manyislands} bounds the additional cross-ratio by the product of the sensitivities of the two islands to the collars of the others, with no condition on the other islands, and by a sum over chains of islands under a Dobrushin condition on them.\footnote{Dobrushin's condition asks that the conditional law of each island move by less than a fixed fraction $\theta<1$ in total when the islands around it are changed; an influence then fades by the factor $\theta$ at each island it passes through, and the sum over chains converges.} The measures of the other islands enter only through their positivity. \emph{The data.} The data are those of Sec.~\ref{ssec:factorisation} with the following changes. Let $\mathcal I=\{a,b\}\cup\mathcal Q$ be a finite index set and $I_u\subset\Lambda$, $u\in\mathcal I$, pairwise disjoint sets of blocks with \[ w_{uv}=\dist(I_u,I_v)-1\ge2s+7\qquad(u\ne v), \] which holds for distinct islands formed with the joining distance $2s+7$ of the paragraph after Theorem~\ref{thm:factorisation}, and with $w_{uv}\ge 2s+8$ for the joining distance $2s+8$ of Corollary~\ref{cor:twoislands}. Every block of $\Lambda\setminus\bigcup_uI_u$ is good; on $I_a$ two patterns $\omega_a^0,\omega_a^1$ are prescribed, on $I_b$ two patterns $\omega_b^0,\omega_b^1$, and on each $I_q$, $q\in\mathcal Q$, one pattern $\omega_q$. The sector $(i,j)$ is the event that the labels of $\Lambda$ are these, with $\omega_a^i$ on $I_a$ and $\omega_b^j$ on $I_b$, and its weight $Z_{ij}$ is defined as in Sec.~\ref{ssec:factorisation}. With $d_u(y)=\dist(y,I_u)$, form for every $u\in\mathcal I$ the classes $L_u$, $p_u$, $x_u$ of Sec.~\ref{ssec:factorisation}, with $d_a$ replaced by $d_u$, let $z$ be every other non-path active link based in $\Lambda$, and solve the path link of every pair based at $d_u\ge s+2$ for every $u$. A link at $d_u\le s+3$ has $d_v\ge w_{uv}+1-(s+3)\ge s+5$ for $v\ne u$, so the classes are disjoint, and every block at $d_u\in\{s+1,s+2,s+3\}$ is good in all four sectors. Let $D_u$ be the product of the $r$-balls in the coordinates $\zeta_e$ of $x_u$ and $D_z$ that of $z$, let $n_u=45(N^u_{s+1}+N^u_{s+2})+ 17N^u_{s+3}$, with $N^u_d$ the number of blocks of $\Lambda$ at $d_u=d$, be the number of $x_u$-links, let $\Lambda_2$ be the largest operator norm of $\partial_{x_u}\nabla_z\mathcal U$ over $u\in\mathcal I$ and over $D_z\times\prod_uD_u$, and put \[ \varDelta_{uv}=\frac{1920}{\beta_W}\Lambda_2^2r^2\sqrt{n_un_v}\, e^{-2\alpha_G(w_{uv}-2s-8)_+}\qquad(u\ne v), \] the quantity $\varDelta$ of Corollary~\ref{cor:twoislands} for the pair $I_u,I_v$ when $w_{uv}\ge2s+8$. \begin{lemma}[many islands, one kernel]\label{lem:manykernel} Let $s\ge1$ and $r\le10^{-4}$. There are finite positive measures $a_i$ on $D_a$, $b_j$ on $D_b$, $i,j\in\{0,1\}$, and $\gamma_q$ on $D_q$, $q\in\mathcal Q$, and a smooth function $\tilde K:\prod_{u\in\mathcal I}D_u \to(0,\infty)$ with \[ Z_{ij}=\int\tilde K(x)\,a_i(dx_a)\,b_j(dx_b)\prod_{q\in\mathcal Q} \gamma_q(dx_q)\qquad(i,j\in\{0,1\}), \] such that: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\tilde K$ and every $\gamma_q$ are the same in all four sectors; $a_i$ depends on $\omega_a^i$, $c$, $\beta_W$, $U^{\mathrm{ext}}$ only, and likewise $b_j$; \item $\tilde K(x)=\int_{D_z}e^{-\mathcal U(z;x)}dz$ with $\mathcal U= \beta_WS_K+\psi$ as in Theorem~\ref{thm:factorisation}(iv), smooth on a neighbourhood of $D_z\times\prod_uD_u$; no term of $\mathcal U$ depends on coordinates of $x_u$ and of $x_v$ for $u\ne v$, and $\nabla_z\partial_{x_u} \mathcal U$ is supported on the $z$-links based at $d_u\le s+4$; \item for $u\ne v$ the supports of $\nabla_z\partial_{x_u}\mathcal U$ and $\nabla_z\partial_{x_v}\mathcal U$ are at fine distance at least $d'_{uv}=2(w_{uv}-2s-8)_+$ between link midpoints. \end{enumerate} \end{lemma} \begin{proof} The proof of Theorem~\ref{thm:factorisation} treats each island through its own distance function, and runs with one class per island. Steps~1 and~2 are unchanged, a pair being solved when it is based at distance at least $s+2$ from every island; its non-path links are then based at distance at least $s+1$ from every island, in good blocks. In Step~3 call a factor a $u$-factor if it depends on a coordinate of $L_u\cup p_u$, and assign the factors depending on $x_u$ alone to $u$. A $u$-factor depends only on $L_u$, $p_u$, $x_u$ and $U^{\mathrm{ext}}$, through links at $d_u\le s+2$, which have $d_v\ge w_{uv}+1-(s+2)\ge s+6$ for every $v\ne u$; so no factor is a $u$-factor and a $v$-factor, and the product $\Phi_K$ of the remaining factors, the $K$-factors, does not depend on the sector, which changes only the indicators of $I_a$ and $I_b$. The indicators of the bad blocks of $I_q$ involve links at $d_q\le1\le s$ and are $q$-factors. Step~4 with one bracket for each island gives the identity, with $\gamma_q(dx_q)$ the integral of the product of the $q$-factors over $L_q$ and $p_q$ times $\prod_{e\in x_q}j\,d\zeta_e$ on $D_q$, and $a_i$, $b_j$, $\tilde K$ as there; this is (i). Step~5 gives the first clause of (ii). Step~6 is a statement about one island at a time, so it gives the support of $\nabla_z\partial_{x_u}\mathcal U$, and a $K$-term depending on $x_u$ and on $x_v$ would contain a link at $d_u\le s+3$ and a link at $d_v\le s+3$ in blocks at distance at most one, so that $w_{uv}+1\le2s+7$, which is excluded. The two supports are at coarse distance at least $w_{uv}+1-2(s+4)=w_{uv}-2s-7$, and links based in blocks at coarse distance $D\ge1$ have midpoints at fine distance at least $2D-2$; this is (iii) for $w_{uv}\ge2s+8$, and (iii) is trivial for $w_{uv}=2s+7$. \end{proof} \begin{lemma}[the layer between many islands]\label{lem:manylayer} In addition, assume hypotheses (a)--(c) of Theorem~\ref{thm:layerbound}, with $W'$ the set of blocks of $\Lambda$ at distance at least $s+3$ from every island. Then for every $x\in\prod_uD_u$ the law $\mu_x=e^{-\mathcal U (z;x)}dz/\tilde K(x)$ on $D_z$ satisfies~\eqref{eq:layerbound} with $\mathcal A=480/\beta_W$ and $\alpha=\alpha_G$, and for $u\ne v$ and vectors $h_u$, $h_v$ in the coordinates of $x_u$ and $x_v$, \[ \bigl\lvert\partial_{h_u}\partial_{h_v}\log\tilde K(x)\bigr\rvert\le \frac{480}{\beta_W}\Lambda_2^2\lVert h_u\rVert\lVert h_v\rVert e^{-2\alpha_G(w_{uv}-2s-8)_+}. \] In particular the logarithm of the cross-ratio of $\tilde K$ in $(x_u,x_v)$, between two points of $D_u$ and two points of $D_v$ with the other arguments fixed, has modulus at most $\varDelta_{uv}$. \end{lemma} \begin{proof} The law $\mu_x$ is the law of Proposition~\ref{prop:graphlaw} for the window $W'$ in the coordinates of the labels, with the inside links of the blocks at distance $s+3$ from some island frozen and the exterior given by $x$ and $U^{\mathrm{ext}}$, by the identification in the paragraph on the layer law after Theorem~\ref{thm:factorisation}, which uses only that the $z$-links are the free non-path links of $W'$ and that the solved pairs are those of Lemma~\ref{lem:manykernel}. The proof of Theorem~\ref{thm:layerbound} uses the islands only in Step~1, through the goodness of the blocks at distance at most two from $W'$; these are at distance at least $s+1$ from every island, hence good in all sectors inside $\Lambda$ and good by (c) outside it. Steps~2 to~6 concern the window $W'$ and are uniform in its exterior, and freezing the inside links at distance $s+3$ from any island restricts the Hessian to a principal submatrix. So \eqref{eq:layerbound} holds with the stated constants. For the derivative, Lemma~\ref{lem:connected} on the fixed domain $D_z$ gives $\partial_{h_u}\partial_{h_v}\log\tilde K=-\E_{\mu_x}[\partial_{h_u} \partial_{h_v}\mathcal U]+\Cov_{\mu_x}(\partial_{h_u}\mathcal U, \partial_{h_v}\mathcal U)$, and the first term vanishes by Lemma~\ref{lem:manykernel}(ii). As in the proof of Proposition~\ref{prop:layercross}, apply~\eqref{eq:layerbound} with $\varphi(e)=-\dist(e,\operatorname{supp}\nabla_z\partial_{h_u}\mathcal U)$, which vanishes on that support and is at most $-d'_{uv}$ on the support of $\nabla_z\partial_{h_v}\mathcal U$ by Lemma~\ref{lem:manykernel}(iii), with $\lVert\nabla_z\partial_{h_u}\mathcal U\rVert\le\Lambda_2\lVert h_u\rVert$ and $\lVert\nabla_z\partial_{h_v}\mathcal U\rVert\le\Lambda_2\lVert h_v \rVert$ pointwise. For the cross-ratio, $D_u$ and $D_v$ are convex, the logarithm of the cross-ratio is the integral of the mixed derivative over the unit square, and two points of $D_u$ differ by at most $2r\sqrt{n_u}$. \end{proof} The other islands now enter only through the measures $\gamma_q$, and the kernel of $a$ and $b$ is $K(x_a,x_b)=\int\tilde K(x)\prod_q\gamma_q(dx_q)$, an integral against the probability law on $\prod_qD_q$ proportional to $\tilde K(x_a,x_b,\cdot\,)\prod_q\gamma_q$. The next lemma bounds the covariances of that law in terms of the cross-ratios of its density alone. \begin{lemma}[covariances over the islands]\label{lem:islandcov} Let $\mathcal Q$ be finite, $X_q$ compact metric spaces, $\gamma_q$ nonzero finite Borel measures on $X_q$, $X=\prod_qX_q$, $k:X\to(0,\infty)$ continuous, and $P=k\prod_q\gamma_q/\int k\prod_q\gamma_q$. For $\xi\in X$ and $\eta_q\in X_q$ let $\xi^{\eta_q}$ be $\xi$ with its $q$-coordinate replaced by $\eta_q$. For $q\ne q'$ let $\kappa_{qq'}$ bound $\lvert\log k(\xi^{\eta_q\eta_{q'}})+\log k(\xi)-\log k(\xi^{\eta_q})-\log k(\xi^{\eta_{q'}})\rvert$ for all $\xi,\eta_q,\eta_{q'}$, let $C_{qq'}= \tanh(\kappa_{qq'}/4)$ and $C_{qq}=0$, and for continuous $f$ on $X$ let $\delta_q(f)=\sup\{\lvert f(\xi^{\eta_q})-f(\xi)\rvert\}$. Then for continuous $f,g$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\lvert\Cov_P(f,g)\rvert\le\frac14\sum_q\delta_q(f)\sum_q\delta_q(g)$; \item if $\sum_{q'}C_{qq'}\le\theta<1$ for every $q$, then $D=\sum_{n\ge0} C^n$ converges and \[ \lvert\Cov_P(f,g)\rvert\le\frac14\sum_{q,q'}\delta_q(f)\,D_{qq'}\, \delta_{q'}(g); \] \item if $\ell$ is a pseudometric on $\mathcal Q$, $\lambda\ge0$ and $\sum_{q'}C_{qq'}e^{\lambda\ell(q,q')}\le\theta<1$ for every $q$, then $0\le D_{qq'}\le e^{-\lambda\ell(q,q')}/(1-\theta)$. \end{enumerate} \end{lemma} \begin{proof} \emph{Conditional laws.} The law \[ P_q(d\eta\mid\xi)=\frac{k(\xi^\eta)\,\gamma_q(d\eta)}{\int k(\xi^{\eta'}) \,\gamma_q(d\eta')} \] on $X_q$ depends on $\xi$ through $\xi_{-q}$ only, its density with respect to $\gamma_q$ is continuous in $(\eta,\xi)$, and $P$ is invariant under resampling the $q$-coordinate from it. \emph{Tilts.} Let $\mu$ be a probability measure and $\mu'=h\mu/\mu(h)$ with $h$ measurable and $0<\inf h\le\sup h\le e^\kappa\inf h$. For a Borel set $E$ with $0<\mu(E)<1$ the odds of $E$ under $\mu'$ and under $\mu$ have the ratio $\frac{\mu(h\mathbf 1_E)/\mu(E)}{\mu(h\mathbf 1_{E^c})/\mu(E^c)} \in[e^{-\kappa},e^\kappa]$, so the logistic bound in the proof of Corollary~\ref{cor:twolabels} gives $\lvert\mu'(E)-\mu(E)\rvert\le\tanh (\kappa/4)$, and $\mu'(E)=\mu(E)$ when $\mu(E)\in\{0,1\}$. Hence $\lVert\mu'-\mu\rVert_{\mathrm{TV}}\le\tanh(\kappa/4)$. If $\xi$ and $\xi'$ differ only in the coordinate $q'\ne q$, the density of $P_q(\cdot\mid\xi')$ with respect to $P_q(\cdot\mid\xi)$ is proportional to $k(\xi'^{\eta})/k(\xi^\eta)$, whose logarithm oscillates in $\eta$ by at most $\kappa_{qq'}$, so $\lVert P_q(\cdot\mid\xi)-P_q(\cdot\mid\xi')\rVert_ {\mathrm{TV}}\le C_{qq'}$. \emph{Comparison.} Let $g$ be continuous, $t\in\R$, and $P^t=e^{tg}P/ P(e^{tg})$, again of the form of the lemma; its conditional law at $q$ is the tilt of $P_q(\cdot\mid\xi)$ by $e^{tg(\xi^\eta)}$, so $b_q=\sup_\xi \lVert P^t_q(\cdot\mid\xi)-P_q(\cdot\mid\xi)\rVert_{\mathrm{TV}}\le\tanh (\lvert t\rvert\delta_q(g)/4)$. The couplings of $P$ and $P^t$ form a weakly compact set (Lemma~\ref{lem:minimiser}), and $\Psi(\pi)=\sum_q \varrho_q(\pi)$, $\varrho_q(\pi)=\pi\{\xi_q\ne\xi'_q\}$, is weakly lower semicontinuous, since $\{\xi_q\ne\xi'_q\}$ is open; let $\pi$ minimise it. For $q\in\mathcal Q$ replace $(\xi_q,\xi'_q)$ by a sample of the maximal coupling of $P_q(\cdot\mid\xi)$ and $P^t_q(\cdot\mid\xi')$, which, with $p,p'$ the two densities with respect to $\gamma_q$ and $m=\int\min\{p,p'\} d\gamma_q$, draws $\eta=\eta'$ from $\min\{p,p'\}\gamma_q/m$ with probability $m$ and otherwise, when $m<1$, $\eta$ and $\eta'$ independently from $(p-\min\{p,p'\}) \gamma_q/(1-m)$ and $(p'-\min\{p,p'\})\gamma_q/(1-m)$; it is measurable in $(\xi,\xi')$ by the continuity of the densities, and it gives $\{\eta\ne\eta'\}$ probability at most $1-m=\lVert P_q(\cdot\mid\xi)-P^t_q (\cdot\mid\xi')\rVert_{\mathrm{TV}}$. Both marginals are invariant, so the result $\pi^{(q)}$ is again a coupling; it changes only $\varrho_q$, and changing the coordinates of $\xi_{-q}$ to those of $\xi'_{-q}$ one at a time, \[ \varrho_q(\pi^{(q)})\le\E_\pi\sum_{q'}C_{qq'}\mathbf 1_{\xi_{q'}\ne \xi'_{q'}}+b_q=\sum_{q'}C_{qq'}\varrho_{q'}(\pi)+b_q . \] By minimality $\varrho_q(\pi)\le\varrho_q(\pi^{(q)})$, which is the argument of Lemma~\ref{lem:minimality} for the discrete metric, so $\varrho\le C\varrho+b$ componentwise. Under the hypothesis of (ii), $C^n\mathbf 1\le\theta^n \mathbf 1$ and $\varrho\le\mathbf 1$, so iterating gives $\varrho\le \sum_{k0$ for $i,j\in\{0,1\}$. Put $S_u=\sum_{q\in\mathcal Q} \varDelta_{uq}$ for $u\in\{a,b\}$, let $C$ be the $\mathcal Q\times\mathcal Q$ matrix with $C_{qq}=0$ and $C_{qq'}=\tanh(\varDelta_{qq'}/4)$, and let \[ \varDelta_*=\varDelta_{ab}+\frac14\,S_aS_b ; \] when $\sum_{q'}C_{qq'}<1$ for every $q$, let instead, with $D=\sum_{n\ge0}C^n$, \[ \varDelta_*=\varDelta_{ab}+\frac14\min\Bigl\{S_aS_b,\ \sum_{q,q'\in \mathcal Q}\varDelta_{aq}D_{qq'}\varDelta_{q'b}\Bigr\}. \] Then \[ \bigl\lvert\log Z_{11}+\log Z_{00}-\log Z_{10}-\log Z_{01}\bigr\rvert\le \varDelta_* , \] the two island labels, the pattern of $I_a$ among $\omega_a^0,\omega_a^1$ and that of $I_b$ among $\omega_b^0,\omega_b^1$, have conditional covariance at most $\frac14\tanh(\varDelta_*/4)$ given the labels of all other blocks of $\Lambda$, the coarse field and $U^{\mathrm{ext}}$, and for every bounded measurable $F\ge0$ of the links of $L_a\cup p_a\cup x_a$, \[ e^{-\varDelta_*}\le\frac{\E_{i1}[F]}{\E_{i0}[F]}\le e^{\varDelta_*} \qquad(i=0,1) \] whenever the four expectations are positive. For $\mathcal Q=\emptyset$, $\varDelta_*=\varDelta_{ab}$, which for $w_{ab}\ge2s+8$ is the bound of Corollary~\ref{cor:twoislands}. \end{theorem} \begin{proof} By Lemma~\ref{lem:manykernel}, $Z_{ij}=\iint K\,da_i\,db_j$ with $K(x_a,x_b) =\int\tilde K(x)\prod_q\gamma_q(dx_q)$; $Z_{ij}>0$ forces every $\gamma_q\ne0$, and $K$ is smooth and positive on $D_a\times D_b$. Fix $x_a^0,x_a^1\in D_a$, $x_b^0,x_b^1\in D_b$, and let $x_a(\vartheta)=x_a^0+ \vartheta(x_a^1-x_a^0)$, $x_b(\varphi)$ likewise, which stay in the convex sets $D_a$, $D_b$. The logarithm of the cross-ratio of $K$ at these points is $\int_0^1\!\int_0^1\partial_\vartheta\partial_\varphi\log K\,d\vartheta\, d\varphi$. With $P$ the law on $\prod_qD_q$ proportional to $\tilde K(x_a (\vartheta),x_b(\varphi),\cdot\,)\prod_q\gamma_q$, differentiation under the integral sign gives \[ \partial_\vartheta\partial_\varphi\log K=\E_P\bigl[\partial_\vartheta \partial_\varphi\log\tilde K\bigr]+\Cov_P\bigl(\partial_\vartheta\log \tilde K,\partial_\varphi\log\tilde K\bigr). \] By Lemma~\ref{lem:manylayer} with $h_a=x_a^1-x_a^0$, $h_b=x_b^1-x_b^0$, the first term has modulus at most $\varDelta_{ab}$. Apply Lemma~\ref{lem:islandcov} with $X_q=D_q$, $k=\tilde K(x_a(\vartheta), x_b(\varphi),\cdot\,)$, $f=\partial_\vartheta\log\tilde K$ and $g= \partial_\varphi\log\tilde K$. Changing $x_q$ along the segment between two points of $D_q$ and integrating the mixed derivative of Lemma~\ref{lem:manylayer} for the pair $(a,q)$ gives $\delta_q(f)\le \varDelta_{aq}$, likewise $\delta_q(g)\le\varDelta_{bq}$, and the last statement of Lemma~\ref{lem:manylayer} for the pair $(q,q')$ shows that $\kappa_{qq'}=\varDelta_{qq'}$ is admissible, so the matrix of Lemma~\ref{lem:islandcov} is $C$. Parts (i) and (ii) of that lemma bound the covariance by $\varDelta_*-\varDelta_{ab}$, uniformly in $(\vartheta,\varphi)$. So the cross-ratio of $K$ is at most $e^{\varDelta_*}$ and at least $e^{-\varDelta_*}$ on $D_a\times D_b$, and Lemma~\ref{lem:kernel} gives the first bound. The joint law of the two labels given the others is $Z_{ij}/\sum Z$, so Corollary~\ref{cor:twolabels} gives the second. For the third, $F\Phi_a^i$ is a product of $a$-factors, so the identity holds with $Fa_i$ in place of $a_i$ and the same $K$ and $b_j$, and Lemma~\ref{lem:kernel} applied to $Fa_i$, $a_i$ and $b_1$, $b_0$ gives it, since $Z(Fa_i,b_j)/Z(a_i,b_j)=\E_{ij}[F]$. \end{proof} The first form of $\varDelta_*$ needs nothing of the other islands: their hard constraints, branches and masses are integrated in the $\gamma_q$ and never compared. Its mediated term $\frac14S_aS_b$ does not decay with $w_{ab}$ when other islands lie near both $I_a$ and $I_b$, since it treats every collar coordinate as able to reach every other. The second form keeps the interaction between the islands, through the matrix $C$, and under a weighted form of the Dobrushin condition $\sum_{q'}\tanh(\varDelta_{qq'}/4)<1$ it decays along chains of islands. \begin{corollary}[decay along chains of islands]\label{cor:islandchains} Under the hypotheses of Theorem~\ref{thm:manyislands}, let $\ell$ be the pseudometric on $\mathcal I$ given by the least value of $\sum_i(w_{u_iu_{i+1}} -2s-8)_+$ over the chains $u=u_0,u_1,\dots,u_k=v$ in $\mathcal I$, let $0\le\vartheta\le1$, and suppose \[ \theta=\max_{q\in\mathcal Q}\sum_{q'\in\mathcal Q\setminus\{q\}}\tanh \frac{\varDelta_{qq'}}4\,e^{2\vartheta\alpha_G\ell(q,q')}<1 . \] Then \[ \begin{gathered} \varDelta_*\le\varDelta_{ab}+\frac{e^{-2\vartheta\alpha_G\ell(a,b)}} {4(1-\theta)}\,T_aT_b ,\qquad T_u=\sum_{q\in\mathcal Q}\varDelta_{uq} e^{2\vartheta\alpha_G\ell(u,q)},\\ T_u\le\frac{1920}{\beta_W}\Lambda_2^2r^2\sqrt{n_u}\sum_{q\in\mathcal Q} \sqrt{n_q}\,e^{-2(1-\vartheta)\alpha_G(w_{uq}-2s-8)_+}. \end{gathered} \] \end{corollary} \begin{proof} $\ell$ is a pseudometric since it is a least sum over chains, and $\ell(u,v)\le(w_{uv}-2s-8)_+$. Lemma~\ref{lem:islandcov}(iii) on $\mathcal Q$ with $\lambda=2\vartheta\alpha_G$ gives $D_{qq'}\le e^{-\lambda\ell(q,q')} /(1-\theta)$ and the hypothesis of the second form of $\varDelta_*$, and $-\ell(q,q')\le-\ell(a,b)+\ell(a,q)+\ell(q',b)$ gives the first bound. The second is $\ell(u,q)\le(w_{uq}-2s-8)_+$ in each term. \end{proof} A chain crosses each island at no cost in $\ell$: an island is a set on which nothing is known, and the corollary charges only the good layers between consecutive islands of the chain. With $\vartheta=0$ the hypothesis is the Dobrushin condition $\sum_{q'}C_{qq'}\le\theta<1$. The sums $S_u$ and $T_u$ are sums over the islands near $I_u$, and the rarity of Theorem~\ref{thm:islandthreshold} controls them in the mean. For a bad block $y$ let $\lvert I(y)\rvert$ be the number of blocks of its island, and $\lvert I(y)\rvert=0$ for a good block. \begin{proposition}[the island sums in the mean]\label{prop:islandmass} Form the islands with the joining distance $2s+8$, so that $w_{uv}\ge2s+8$ for distinct islands, let $N_s=(4s+17)^4-1$ be the number of neighbours of a block, $\varGamma\ge1$, and $\beta_W\ge\beta^*(\rho,s,\varGamma)$ of Theorem~\ref{thm:islandthreshold}(iii) with $N_s$ neighbours. Then: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item for every block $y$, $\E_{\mu_L}\lvert I(y)\rvert\le2\varGamma(2 \varGamma-1)^{-2}N_s^{-2}$; \item for every set $J$ of blocks of the torus and every $\kappa>0$, \[ \E_{\mu_L}\sum_{y\notin J}\lvert I(y)\rvert e^{-\kappa\dist(J,y)}\le \frac{2\varGamma}{(2\varGamma-1)^2N_s^2}\,\lvert J\rvert\,\sigma(\kappa), \] where, with $x=e^{-\kappa}$, \[ \sigma(\kappa)=\sum_{y\in\Z^4}e^{-\kappa\lVert y\rVert_\infty}=1+\frac{64x (1+4x+x^2)}{(1-x)^4}+\frac{16x}{(1-x)^2}; \] \item in the setting of Theorem~\ref{thm:manyislands}, when every $I_q$, $q\in\mathcal Q$, is an island, with $B'=\bigcup_qI_q$, $0\le\vartheta\le1$ and $\kappa=2(1-\vartheta)\alpha_G$, \[ \begin{split} T_u\le{}&\frac{1920}{\beta_W}\Lambda_2^2r^2\sqrt{45\,n_u}\,(2s+7)^2 e^{\kappa(2s+9)}\\ &\times\sum_{y\in B'}\lvert I(y)\rvert e^{-\kappa\dist(I_u,y)} . \end{split} \] \end{enumerate} At $\kappa=2\alpha_G=5\cdot10^{-4}$, $\sigma(\kappa)\le6.151\cdot10^{15}$, and at $\varGamma=1$ the bound of (ii) is at most $3.253\cdot10^5\lvert J\rvert$ for $s=1$ and $3.503\cdot10^3\lvert J\rvert$ for $s=5$. \end{proposition} \begin{proof} (i) The island paragraph of the proof of Theorem~\ref{thm:islandthreshold} gives $\mu_L\{\lvert I(y)\rvert=n\}\le N_s^{2(n-1)}A^{n/H}$, and its proof of (iii) gives $\varGamma N_s^2A^{1/H}\le\frac12$ at $\beta_W\ge\beta^*(\rho, s,\varGamma)$, so $\mu_L\{\lvert I(y)\rvert=n\}\le N_s^{-2}(2\varGamma)^{-n}$ and $\E\lvert I(y)\rvert\le N_s^{-2}\sum_nn(2\varGamma)^{-n}=2\varGamma(2 \varGamma-1)^{-2}N_s^{-2}$. (ii) By linearity it suffices that $\sum_ye^{-\kappa\dist(J,y)}\le\lvert J\rvert\sigma(\kappa)$; since $e^{-\kappa\dist(J,y)}\le\sum_{z\in J}e^{-\kappa\dist(z,y)}$, it suffices that $\sum_ye^{-\kappa\dist(z,y)}\le\sigma(\kappa)$ on the torus, which holds because each block of the torus has a preimage in $\Z^4$ at $\ell^\infty$ distance $\dist(z,y)$ from a preimage of $z$, distinct blocks having distinct preimages. The closed form counts the $(2k+1)^4-(2k-1)^4= 64k^3+16k$ points of $\Z^4$ at distance $k\ge1$ and sums $\sum kx^k$ and $\sum k^3x^k$. (iii) In the second bound of Corollary~\ref{cor:islandchains}, $n_q\le45(2s+7)^4\lvert I_q\rvert$, since every $x_q$-link is based in a block at distance at most $s+3$ from $I_q$, so $\sqrt{n_q}\le\sqrt{45}(2s+7)^2\lvert I_q\rvert$; and $(w_{uq}-2s-8)_+ \ge\dist(I_u,I_q)-2s-9$, while $\lvert I_q\rvert e^{-\kappa\dist(I_u,I_q)}$ is the term of $\sum_{y\in I_q}\lvert I(y)\rvert e^{-\kappa\dist(I_u,y)}$ at a block of $I_q$ nearest to $I_u$. For the numbers, $x\le1$ and $1-x\ge \kappa-\kappa^2/2$ give $\sigma(\kappa)\le1+384(\kappa-\kappa^2/2)^{-4}+16 (\kappa-\kappa^2/2)^{-2}\le6.151\cdot10^{15}$ at $\kappa=5\cdot10^{-4}$, and $N_1=21^4-1=194480$, $N_5=37^4-1=1874160$, so that $2\cdot6.151\cdot10^{15}/ N_1^2\le3.253\cdot10^5$ and $2\cdot6.151\cdot10^{15}/N_5^2\le3.503\cdot10^3$. \end{proof} The proposition is annealed: it bounds the mean of the island sums under $\mu_L$ for a fixed set $J$, while Theorem~\ref{thm:manyislands} is a statement given all the labels, and it says nothing about the law of the islands near $I_u$ given that $I_u$ is an island. At $\varGamma=1$ its bound exceeds one, because the weighted volume $\sigma(2\alpha_G)=6.1\cdot 10^{15}$ of the decay length $1/(2\alpha_G)=2000$ blocks of Theorem~\ref{thm:layerbound} exceeds $N_s^2$; for $\varGamma\ge2$ the bound of (ii) is at most $\lvert J\rvert \sigma(\kappa)/(\varGamma N_s^2)$, and Theorem~\ref{thm:islandthreshold}(iii) reaches every $\varGamma$ at the coupling $\beta^*(\rho,s,\varGamma)$. With Theorem~\ref{thm:manyislands} the comparison of item (ii) of Sec.~\ref{ssec:remains} holds with any number of islands, given the labels of all of them: the cross-ratio of two islands is at most $\varDelta_*$, explicit in the two-island quantities $\varDelta_{uv}$ of Corollary~\ref{cor:twoislands}, and the observables of one island change by a factor in $[e^{-\varDelta_*},e^{\varDelta_*}]$ under a flip of the other. The prefactor $\Lambda_2$ is bounded by Lemma~\ref{lem:lambdatwo}, whose proof uses only the plaquettes at one collar and applies to each island $I_u$ when $r\le8\cdot10^{-5}$. Two things remain. The Dobrushin condition of the second form and the smallness of $S_u$ or $T_u$ are conditions on the configuration of the islands, which Proposition~\ref{prop:islandmass} controls only in the mean. And the covariance of the two labels with the other labels averaged out differs from the conditional covariance by the covariance of the conditional probabilities of the two labels given the others, which is the term of Proposition~\ref{prop:hiddenbit} and is not bounded here. The first of them, the smallness of the island sums together with the Dobrushin condition, holds in every window of bounded size outside an event whose probability is at most the size of the window times the product of two rarities: that of one bad block and that of a second bad block or island. Two facts give this. The bound of Proposition~\ref{prop:islandmass}(i) charges an island of $n$ blocks the factor $A^{n/H}$, while Lemma~\ref{lem:coverage} gives $A^{\lceil n/H \rceil}$, which is $A$ for $n\le H$; with the ceiling, the mean mass of the island of a block is at most $G_0$ below, about $H$ times the rarity of one bad block, instead of $2N_s^{-2}$. And two bad blocks at distance at least six have disjoint witness boxes, so that two islands near each other cost the product of two rarities. For two islands $I_u\ne I_v$ of the torus, formed with the joining distance $2s+8$, put \begin{gather*} \bar n_u=45\,\bigl\lvert\{y:\dist(y,I_u)\le s+3\}\bigr\rvert,\\ \bar\Lambda_2=99.0001\,\beta_W+4.65\cdot10^6,\\ \bar\varDelta_{uv}=\frac{1920}{\beta_W}\bar\Lambda_2^2r^2\sqrt{\bar n_u \bar n_v}\,e^{-2\alpha_G(w_{uv}-2s-8)}, \end{gather*} and, for $0\le\vartheta<1$, $\bar T_u=\sum_v\bar\varDelta_{uv}e^{2\vartheta \alpha_G(w_{uv}-2s-8)}$, the sum over all islands $I_v\ne I_u$ of the torus. These are functions of the labels alone. In the setting of Theorem~\ref{thm:manyislands}, when the sets $I_u$ are islands, $n_u\le\bar n_u$, Lemma~\ref{lem:lambdatwo}, applied to each island as above, gives $\Lambda_2\le\bar\Lambda_2$, and $\ell(u,v)\le w_{uv}-2s-8$; hence $\varDelta_{uv}\le\bar\varDelta_{uv}$, and $S_u\le\bar T_u$ and $T_u\le\bar T_u$ for the $T_u$ of Corollary~\ref{cor:islandchains} with the same $\vartheta$. Write $W^{+t}$ for the set of blocks at distance at most $t$ from a set $W$ of blocks. \begin{theorem}[the island sums in a window]\label{thm:islandwindow} Let $G=\SU(2)$, $s\ge1$, $2758\rho\le6.0\cdot10^{-13}$, $r=43\rho$, $\eta=92\rho$, form the islands of the torus with the joining distance $2s+8$, and let $N_s=(4s+17)^4-1$. Let $\beta^*$ be the largest solution of the equation of Theorem~\ref{thm:islandthreshold} with $N_s$ in place of $\varDelta$, $\log q^*=-H(\log2+2\log N_s)-\log(eMH)$ its right side, $C_0$ that of Lemma~\ref{lem:wilsonZ}, $\beta_W\ge\beta^*$, and \[ G_0=H\Bigl(Mq^*+\frac{A^*2^{-H}}{1-2^{-H}}\Bigr),\qquad A^*=eMHq^*. \] \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For every block $y$, $\E_{\mu_L}\lvert I(y)\rvert\le G_0$, and Proposition~\ref{prop:islandmass}(ii) holds at $\varGamma=1$ with $2\varGamma(2\varGamma-1)^{-2}N_s^{-2}$ replaced by $G_0$. \item Let $0\le\vartheta<1$, $\kappa=2(1-\vartheta)\alpha_G$, let $R\ge 2s+13$ be an integer, $\sigma>0$, $W$ a set of blocks, \begin{gather*} c_2=\frac{1920}{\beta^*}\bar\Lambda_2(\beta^*)^2r^2\cdot45(2s+12)^2 (2s+7)^2e^{\kappa(2s+14)},\\ \tau=\min\bigl\{\sigma(\kappa),\,e^{-\kappa R/2}\sigma(\kappa/2)\bigr\}, \end{gather*} with $\bar\Lambda_2(\beta^*)$ the value of $\bar\Lambda_2$ at $\beta_W= \beta^*$ and $\sigma(\cdot)$ as in Proposition~\ref{prop:islandmass}. Outside an event of $\mu_L$-probability at most \[ \epsilon=\lvert W\rvert\,Mq^*\Bigl(11^4(2R+1)^4Mq^*+\frac{c_2G_0\tau} \sigma\Bigr), \] every island $I_u$ meeting $W$ has $\ell^\infty$ diameter at most $5$, hence at most $6^4$ blocks, lies at distance more than $R$ from every other island, and satisfies $\bar T_u\le\sigma$. \item Outside the same event, in the setting of Theorem~\ref{thm:manyislands} with $I_a$, $I_b$ and every $I_q$ islands meeting $W$: $S_a,S_b\le\sigma$ and $\varDelta_*\le\varDelta_{ab}+\frac14 \sigma^2\le\sigma+\frac14\sigma^2$; if $\sigma<4$, the weighted Dobrushin condition of Corollary~\ref{cor:islandchains} holds with $\theta\le\sigma/4$, and \[ \varDelta_*\le\varDelta_{ab}+\frac{e^{-2\vartheta\alpha_G\ell(a,b)}\sigma^2} {4-\sigma}\le\sigma+\frac{\sigma^2}{4-\sigma}. \] \end{enumerate} At $\rho=2.17\cdot10^{-16}$ and $s=1$: $\beta^*\in[1.1220,1.1221]\cdot10^{37}$, $\lvert\log q^*\rvert=32481.67$, $Mq^*\le10^{-14102.4}$, the bound $A$ of Theorem~\ref{thm:islandthreshold}(i) is at most $10^{-14098.8}$, $G_0\le10^{-14099.3}$, and the bound of (i) for Proposition~\ref{prop:islandmass}(ii) at $\kappa=2\alpha_G$ is $G_0\sigma(2\alpha_G)\lvert J\rvert\le10^{-14083.5}\lvert J\rvert$, in place of $3.253\cdot10^5\lvert J\rvert$. For $\vartheta=0$, $c_2\le10^{22.13}$; with $R=15$, $\epsilon\le\lvert W\rvert(10^{-28194}+10^{-28163.8}/\sigma)$, and with $R=2\cdot10^8$, $\epsilon\le10^{-28166}\lvert W\rvert$ for every $\sigma\in[10^{-1000},4)$; for $\vartheta=\frac12$ and $R=4\cdot10^8$, $\epsilon\le10^{-28165}\lvert W\rvert$ on the same range. At $s=5$: $\beta^*\in[1.3216,1.3217]\cdot10^{37}$, $Mq^*\le10^{-16652.7}$, $G_0\le 10^{-16649.6}$, and with $\vartheta=0$, $R=2\cdot10^8$, $\epsilon\le10^{-33267}\lvert W\rvert$ for $\sigma\in[10^{-1000},4)$. \end{theorem} \begin{proof} Write $p=q_\rho(\beta_W)$ and $B$ for the set of bad blocks, and let $Q_y$ be the set of unit cubes with corner at $\ell^\infty$ distance at most five from $2y$. By the witness paragraph of the proof of Theorem~\ref{thm:islandthreshold}, which applies since $\rho\le10^{-8}$, a bad block $y$ has a cube of $Q_y$ on which the event of Theorem~\ref{thm:defects} holds; $\lvert Q_y\rvert\le M$, and a cube lies in at most $H$ of the sets $Q_y$. For a set $S$ of blocks put $Q_S=\bigcup_{y\in S}Q_y$. \emph{Step 1: separated sets.} Let $S_1,\dots,S_m$ be nonempty sets of blocks with $\dist(S_i,S_j)\ge6$ for $i\ne j$, and $k_i=\lceil\lvert S_i \rvert/H\rceil$. Then \[ \mu_L\Bigl(\,\bigcup_iS_i\subset B\Bigr)\le\prod_i\binom{M\lvert S_i\rvert} {k_i}p^{k_i}\le\prod_iA^{k_i}, \] and the factor of a set of one block is $Mp$. Two blocks at distance at least six have a coordinate in which the corners $2y$ differ by at least twelve modulo the side of the torus, while a cube of $Q_y\cap Q_{y'}$ would put $2y$ and $2y'$ within ten of each other in every coordinate; so the sets $Q_{S_i}$ are pairwise disjoint. On the event, $Q_{S_i}$ contains at least $k_i$ distinct bad cubes, since one cube witnesses at most $H$ blocks; choosing $k_i$ of them in each $Q_{S_i}$ gives $\sum_ik_i$ distinct bad cubes, and the union bound over the choices with Theorem~\ref{thm:defects} gives the first inequality. The second is that of Lemma~\ref{lem:coverage}. \emph{Step 2: the mass of an island.} Let $u=N_s^{2H}A$, which is at most $2^{-H}$ by Theorem~\ref{thm:islandthreshold}(i), and let $T$ be a set of blocks at distance at least $6$ from a block $y$, or empty. Let $n\ge1$ and $k=\lfloor(n-1)/H\rfloor$. If $\lvert I(y)\rvert\ge n$, a spanning tree of $I(y)$ in the graph of the joining distance, cut at $kH+1$ vertices from $y$, gives a connected set $C\ni y$ of exactly $kH+1$ bad blocks; there are at most $N_s^{2kH}$ such sets, since a walk around a spanning tree of $C$ from $y$ has $2kH$ steps. By Step~1, $\mu_L(T\subset B, \,C\subset B)\le\mu_L(T\subset B)'A^{k+1}$ when $T$ is at distance at least $6$ from $C$, with $\mu_L(T\subset B)'$ the factor of Step~1 for $T$, so \begin{equation}\label{eq:islandmassG} \mu_L\bigl(T\subset B,\,\lvert I(y)\rvert\ge n\bigr)\le\mu_L(T\subset B)' \min\{Mp,\,Au^k\}. \end{equation} Summing over $n$ in groups of $H$, with $Mp$ for $n\le H$ and $Au^k$ for $kHR$, and let $c_2(\beta_W)$ be $c_2$ with $\beta_W$ in place of $\beta^*$. On $\mathcal G$, let $I_u$ meet $W$ at $y$. For an island $I_q\ne I_u$ let $y_q\in I_q$ realise $\dist(I_u,I_q)$; then $\dist(y,y_q)\le\dist(I_u,I_q) +5$, $\dist(y,y_q)>R$ by Step 3, and $w_{uq}-2s-8=\dist(I_u,I_q)-2s-9\ge \dist(y,y_q)-2s-14$. The box of side $2s+12$ around $I_u$ contains every block at distance at most $s+3$ from it, so $\bar n_u\le45(2s+12)^4$, and $\bar n_q\le45(2s+7)^4\lvert I_q\rvert$ as in the proof of Proposition~\ref{prop:islandmass}(iii). With $\sqrt{\lvert I_q\rvert}\le \lvert I_q\rvert$ and distinct islands giving distinct $y_q$, \[ \bar T_u=\frac{1920}{\beta_W}\bar\Lambda_2^2r^2\sqrt{\bar n_u}\sum_q \sqrt{\bar n_q}\,e^{-\kappa(w_{uq}-2s-8)}\le c_2(\beta_W)F(y). \] \emph{Step 5: probabilities.} By Step 1 with two single blocks, at most $11^4\lvert W\rvert$ blocks of $W^{+5}$ and at most $(2R+1)^4$ blocks within distance $R$ of each, \[ \mu_L(\mathcal G^c)\le11^4(2R+1)^4\lvert W\rvert(Mp)^2 . \] For $y\in W$, $\E[\mathbf 1_{y\in B}F(y)]$ is a sum over $y'$ with $\dist(y,y')>R$ of $e^{-\kappa\dist(y,y')}$ times a sum over the connected sets $C'\ni y'$ with $y\notin C'$ of $\lvert C'\rvert\mu_L\{y\in B,\,I(y')=C'\}$. On that event $y$ and $I(y')$ lie in different islands, so $y$ is at distance at least $2s+9\ge6$ from every subset of $I(y')$, and Step~2 with $T=\{y\}$, whose factor in Step~1 is $Mp$, bounds the sum over $C'$ by $Mp\,G_0$. The sum over $y'$ is at most $\sum_z e^{-\kappa\lVert z\rVert_\infty}$ over $z\in\Z^4$ with $\lVert z \rVert_\infty>R$, by the preimages of the proof of Proposition~\ref{prop:islandmass}(ii), which is at most $\sigma(\kappa)$ and, by $e^{-\kappa\lVert z\rVert_\infty}\le e^{-\kappa R/2}e^{-\kappa \lVert z\rVert_\infty/2}$ there, at most $e^{-\kappa R/2}\sigma(\kappa/2)$. So $\E[\mathbf 1_{y\in B}F(y)]\le Mp\,G_0\tau$, and by Step 4, Markov's inequality and a union bound over $W$, \begin{multline*} \mu_L\bigl(\mathcal G\cap\{\bar T_u>\sigma\text{ for some }I_u\text{ meeting }W\}\bigr)\\ \le\frac{c_2(\beta_W)}\sigma\,\lvert W\rvert\,Mp\,G_0\tau . \end{multline*} \emph{Step 6: the coupling.} By Theorem~\ref{thm:defects}, $p\le e^{f( \beta_W)}$ with $f$ of Theorem~\ref{thm:islandthreshold}, which decreases on $[96/\rho^2,\infty)$, so $p\le e^{f(\beta^*)}=q^*$. Since $\bar\Lambda_2\ge99.0001\,\beta_W$, the logarithmic derivative of $c_2(\beta)e^{f(\beta)}$ is at most $2/\beta-1/\beta+6/\beta-\rho^2/16$, which is negative for $\beta\ge\beta^*$ because $\beta^*\rho^2/16= \lvert\log q^*\rvert+6(\log\beta^*+C_0)>7$. Hence $c_2(\beta_W)p\le c_2q^*$, and Step 5 gives the bound $\epsilon$ on the complement of the event of (ii). \emph{Step 7: the consequences.} Outside that event, $\varDelta_{ab}\le \bar\varDelta_{ab}\le\bar T_a\le\sigma$, since $\bar\varDelta_{ab}$ is a term of $\bar T_a$ with a factor at least one, and $S_a,S_b\le\sigma$, which gives the first form of $\varDelta_*$ in Theorem~\ref{thm:manyislands}. By $\tanh t\le t$ and $\ell(q,q')\le w_{qq'}-2s-8$, the sum of Corollary~\ref{cor:islandchains} at $q$ is at most $\frac14\bar T_q\le \frac14\sigma$, so $\theta\le\sigma/4<1$, and the corollary with $T_a,T_b \le\sigma$ gives the last display. \emph{The numbers.} Here $H=6^4$, $M=11^4$ and $\lvert\log q^*\rvert=H(\log2 +2\log N_s)+1+4\log11+4\log6$, with $N_1=21^4-1=2^4\cdot5\cdot11\cdot13\cdot 17$ and $\log N_5=4\log37+\log(1-37^{-4})$, where $-x/(1-x)\le\log(1-x)\le -x$. The logarithms of the proof of Theorem~\ref{thm:islandthreshold}, together with $\log\frac{13}{12}=2\operatorname{artanh}\frac1{25}$, $\log\frac{17}{16}=2\operatorname{artanh}\frac1{33}$ and $\log\frac{37}{36}= 2\operatorname{artanh}\frac1{73}$, give $\lvert\log q^*\rvert$ within $10^{-2}$ of $32481.67$ for $s=1$ and of $38354.07$ for $s=5$. The function $g(\beta)=\beta\rho^2/16-6(\log\beta+C_0)-\lvert\log q^*\rvert$ increases for $\beta>96/\rho^2$, and at the ends of the two stated intervals it is negative and positive, with margins above $0.5$. With $\log10=\log2+\log5$, $\log_{10}(Mq^*)=(4\log11-\lvert\log q^*\rvert)/\log10$ lies within $10^{-3}$ of $-14102.446$ for $s=1$ and of $-16652.797$ for $s=5$, and $\log_{10}(eH)\le3.547$ gives $A\le A^*\le10^{-14098.8}$ at $s=1$. Since $2eH2^{-H}\le10^{-300}$, $G_0\le HMq^*(1+10^{-300})$, and $\log_{10}H\le 3.1127$ gives the two bounds on $G_0$; with $\sigma(2\alpha_G)\le6.151\cdot 10^{15}$ from Proposition~\ref{prop:islandmass}, $G_0\sigma(2\alpha_G)\le 10^{-14083.5}$. By Lemma~\ref{lem:lambdatwo}, $\frac{1920}{\beta^*}\bar\Lambda_2(\beta^*)^2 r^2=1920\beta^*r^2(99.0001+4.65\cdot10^6/\beta^*)^2$, which increases with $\beta^*$ and is therefore at most its value at the upper end of the interval for $\beta^*$: at most $10^{16.2645}$ for $s=1$ and $10^{16.3356}$ for $s=5$. The remaining factors of $c_2$ at $\kappa=2\alpha_G=5\cdot 10^{-4}$ give $c_2\le10^{22.122}$ for $s=1$ and $c_2\le10^{23.140}$ for $s=5$, and $c_2$ is smaller at $\kappa=\alpha_G$. For the sums $\sigma$, write $x=1-v$; then $x(1+4x+x^2)=6-12v+7v^2-v^3$, the function $(6-12v+7v^2)v^{-4}$ decreases for $v>0$, and $v\ge u=\kappa-\kappa^2/2$, so the closed form of Proposition~\ref{prop:islandmass} gives $\sigma(\kappa) \le1+64(6-12u+7u^2)u^{-4}+16u^{-2}$, hence $\sigma(\alpha_G)\le9.831\cdot 10^{16}$ and $\sigma(\alpha_G/2)\le1.573\cdot10^{18}$. In $\epsilon$, the first term has $\log_{10}(11^4(2R+1)^4(Mq^*)^2)$ at most $-28194.76$ for $R=15$, $-28166.31$ for $R=2\cdot10^8$ and $-28165.11$ for $R=4\cdot10^8$ at $s=1$, and $-33267.01$ for $R=2\cdot10^8$ at $s=5$. For $R=15$, $\tau\le\sigma(2\alpha_G)$ and $Mq^*c_2G_0\tau\le10^{-28163.86}$. For $R=2\cdot10^8$ with $\kappa=2\alpha_G$, and for $R=4\cdot10^8$ with $\kappa=\alpha_G$, $\kappa R/2=5\cdot10^4$ and $e^{-5\cdot10^4}\le 10^{-21714}$, so $\tau\le10^{-21714}\sigma(\kappa/2)$ and the second term is at most $10^{-48000}\lvert W\rvert$ for $\sigma\ge10^{-1000}$, which the margins of the first term absorb. \end{proof} The bound is a union bound over the window and is linear in $\lvert W \rvert$; for $\lvert W\rvert\le10^{28000}$ it is at most $10^{-166}$. Its content lies in the windows where Theorem~\ref{thm:islandthreshold} does not exclude islands: the probability that $W$ contains a bad block at all is at most $\lvert W\rvert A\le10^{-14098.8}\lvert W\rvert$, so for $10^{14099}\le \lvert W\rvert\le10^{28000}$ that theorem does not exclude bad blocks in $W$, and Theorem~\ref{thm:islandwindow} shows that, outside probability $10^{-166}$, every island they form is a cluster of diameter at most $5$, at distance more than $2\cdot10^8$ from every other island, and meets the conditions of Theorem~\ref{thm:manyislands} and Corollary~\ref{cor:islandchains} with $\sigma=10^{-1000}$. No coupling beyond the threshold $\beta^*$ of the joining distance $2s+8$ and no factor $\varGamma>1$ is needed: the weighted volume $\sigma(2\alpha_G)=6.1\cdot10^{15}$ of the decay length is beaten by $G_0\le10^{-14099.3}$. The bound is void for windows of more than $10^{28166}$ blocks, and it is a statement about the labels under $\mu_L$, not about the law of the labels near $I_u$ given that $I_u$ is an island; the covariance of two labels with the others averaged is bounded in the mean over the coarse field in Sec.~\ref{ssec:labelcov}, and for a fixed coarse field outside a set of small probability by Proposition~\ref{prop:fixedgood}, but not uniformly in the coarse field. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_manyislands.png} \caption{(a) Islands of bad blocks (red: the two islands $I_a$, $I_b$ compared; blue: other islands $I_q$) in a window whose remaining blocks are good. The solid arc is the two-island bound $\varDelta_{ab}$ of Corollary~\ref{cor:twoislands}; the dashed arcs are the bounds $\varDelta_{uq}$ through which the other islands enter $\varDelta_*$ in Theorem~\ref{thm:manyislands}. (b) The bound $\varDelta$ for two islands against the width $w$ of the good layer between them, with $\Lambda_2$ from Lemma~\ref{lem:lambdatwo} at $\beta_W=1.13\cdot10^{37}$; it falls below one at $w=99072$.} \label{fig:manyislands} \end{figure*} \subsection{The covariance of the labels}\label{ssec:labelcov} Theorem~\ref{thm:manyislands} compares two islands given the labels of all the others, and Theorem~\ref{thm:islandwindow} controls the island sums in a window by a union bound. This subsection bounds the covariances with the labels averaged, on the whole torus and with no condition on the size of a window, in the mean over the coarse field. Three constructions do it. The islands are grouped: the islands near an observable form one group, the rest a second, and the mixed derivative of the kernel of Lemma~\ref{lem:manykernel} between the two groups is bounded by one application of~\eqref{eq:layerbound}, with no Dobrushin condition on the islands. Two independent copies of the labels are compared by an explicit involution, which exchanges the labels of the group of one observable between the copies; the weights of a configuration and of its image differ by the cross-ratio of the two groups. And a group that reaches far from its observable contains a chain of bad blocks in one copy or the other, whose probability is paid by the separated rarity of Theorem~\ref{thm:islandwindow}. For two functions of the labels near sets $N_A$ and $N_B$ the covariance is at most $2\epsilon\lvert N_A\rvert\lvert N_B\rvert$ times their oscillations, with $\epsilon$ the rarity of a bad block defined below, since each sees a bad block; the theorem multiplies this by a factor that decays at the rate $2\alpha_G$ of Theorem~\ref{thm:layerbound} down to a floor of order $10^{-7011}$. For two local observables of the field the covariance under $\mu^c$, in the mean over $c$, decays at the same rate down to a floor of order $10^{-14062}$, and as $e^{-\sqrt d/2}$ when the joining distance of the groups grows with the distance. \emph{Setting.} Throughout, the hypotheses of Theorem~\ref{thm:islandwindow} hold: $G=\SU(2)$, $s\ge1$, $2758\rho\le 6.0\cdot10^{-13}$, $r=43\rho$, $\eta=92\rho$, $\beta_W\ge\beta^*$, on the torus in the tree gauge, with $\Lambda$ the set of all blocks and no exterior; hypotheses (a)--(c) of Theorem~\ref{thm:layerbound} then hold, (c) being void. For a coarse field $c$ let $\mu^c$ be $\mu_L$ conditioned on $V^\sharp=c$, through the coarea formula as in Proposition~\ref{prop:graphlaw}, let $P^c$ be the law of the labels under $\mu^c$, and let $\E_c$ be the mean over the law of $V^\sharp$ under $\mu_L$, so that $\E_c\mu^c=\mu_L$. A labelling $\omega$ assigns a label to every block, $B(\omega)$ is its set of bad blocks, $\omega_X$ its restriction to a set $X$ of blocks, and $Z^c(\omega)$ the weight of the event that the labels are $\omega$, defined as $Z_{ij}$ in Sec.~\ref{ssec:factorisation}, so that $P^c(\omega)=Z^c(\omega)/\sum _{\omega'}Z^c(\omega')$. With $f$, $M$, $q^*$ of Theorems~\ref{thm:islandthreshold} and~\ref{thm:islandwindow} put \[ \epsilon=Me^{f(\beta_W)}\le Mq^*,\qquad g_R=\sqrt2\,(2R+1)^4\epsilon^{1/4}, \] so that, by Steps~1 and~6 of the proof of Theorem~\ref{thm:islandwindow}, $\mu_L(T\subset B)\le\epsilon^{\lvert T\rvert}$ for every set $T$ of blocks pairwise at distance at least $6$. With $\bar\Lambda_2$ of Theorem~\ref{thm:islandwindow} put \[ b_*=\frac{1920}{\beta_W}\bar\Lambda_2^2r^2,\qquad\nu_s=45(2s+9)^4 . \] For a set $Y$ of blocks, its \emph{components} are the classes of the joining distance $2s+8$ and its \emph{$R$-clusters} those of the joining distance $R\ge2s+8$; an $R$-cluster is a union of components. For disjoint sets $X,X'$ of blocks put \[ \varDelta(X,X')=b_*\nu_s\lvert X\rvert^{1/2}\Bigl(\sum_{y\in X'} e^{-4\alpha_G(\dist(y,X)-2s-9)_+}\Bigr)^{1/2}, \] and for integers $R,t$ and $d>0$ \begin{gather*} M_{d,t}=b_*\nu_se^{-2\alpha_G(d-t-2s-9)_+},\\ \varPhi_R=b_*\nu_s\sigma(\alpha_G)\epsilon^{1/2}e^{-\alpha_G(R-4s-17)}, \end{gather*} with $\sigma(\kappa)=\sum_{y\in\Z^4}e^{-\kappa\lVert y\rVert_\infty}$ of Proposition~\ref{prop:islandmass}; by the proof of its part (ii), $\sum_ye^{-\kappa\dist(W,y)}\le\lvert W\rvert\sigma(\kappa)$ for every set $W$ of blocks. \begin{lemma}[two groups of islands]\label{lem:groupcross} Let $Y$ be a set of blocks, $X$ a union of components of $Y$ and $X'=Y\setminus X$. Let $D_X$ be the product of the sets $D_u$ of Sec.~\ref{ssec:manyislands} over the components $I_u\subset X$, and $D_{X'}$ likewise. There are a smooth function $\tilde K:D_X\times D_{X'}\to(0, \infty)$ and, for every labelling $\pi$ of $X$ and $\pi'$ of $X'$, finite positive measures $a_\pi$ on $D_X$ and $b_{\pi'}$ on $D_{X'}$, such that for all labellings $\omega$ with $B(\omega)\subset Y$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $Z^c(\omega)=\iint\tilde K(x,x')\,a_{\omega_X}(dx)\,b_{\omega_{X'}} (dx')$; \item for all $x_0,x_1\in D_X$ and $x'_0,x'_1\in D_{X'}$, \[ \Bigl|\log\frac{\tilde K(x_1,x'_1)\tilde K(x_0,x'_0)}{\tilde K(x_1,x'_0) \tilde K(x_0,x'_1)}\Bigr|\le\varDelta:=\min\{\varDelta(X,X'),\varDelta(X', X)\}; \] \item for $Z^c(\omega)>0$ and bounded measurable $F$ of the active links based in $X$ and $H$ of those based in $X'$, there are measurable $f$ on $D_X$, depending on $F$ and $\omega_X$ only, and $h$ on $D_{X'}$, depending on $H$ and $\omega_{X'}$ only, with $\lvert f\rvert\le\lVert F\rVert$, $\lvert h\rvert\le\lVert H\rVert$, $\E_{\mu^c}[FH\mid\omega]=\int fh\, d\varPi_\omega$ and $\E_{\mu^c}[F\mid\omega]=\int f\,d\varPi_\omega$, where $\varPi_\omega$ is the probability law proportional to $\tilde K\,a_{\omega _X}\otimes b_{\omega_{X'}}$. \end{enumerate} Consequently, for $\omega,\tilde\omega,\omega'$ of positive weight with bad blocks in $Y$: \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item if $\tilde\omega$ agrees with $\omega$ on $X$, the conditional expectations of $F$ given $\tilde\omega$ and given $\omega$ differ by at most $2\lVert F\rVert\tanh(\varDelta/4)$; \item $\lvert\Cov_{\mu^c}(F,H\mid\omega)\rvert\le\lVert F\rVert\lVert H \rVert\tanh(\varDelta/4)$; \item the labellings $\omega^\natural=(\omega'_X,\omega_{X'})$ and $\omega'^\natural=(\omega_X,\omega'_{X'})$ have positive weight and \[ e^{-\varDelta}\le\frac{Z^c(\omega^\natural)Z^c(\omega'^\natural)} {Z^c(\omega)Z^c(\omega')}\le e^{\varDelta}. \] \end{enumerate} \end{lemma} \begin{proof} \emph{The kernel.} Distinct components of $Y$ are at distance at least $2s+9$, so Lemma~\ref{lem:manykernel} applies to the components $I_u$ of $Y$, every block outside $Y$ being good in every labelling considered. Its proof runs with any labelling of each $I_u$: the labels change only the $u$-factors, so the product of the $K$-factors, and with it $\tilde K(x)= \int_{D_z}e^{-\mathcal U(z;x)}dz$, does not depend on the labels of $Y$, while the measure $\gamma_u$ obtained by integrating the $u$-factors over $L_u$ and $p_u$ depends on the labels of $I_u$, on $c$ and on $\beta_W$ only. Put $a_\pi=\bigotimes_{I_u\subset X}\gamma_u$ and $b_{\pi'}=\bigotimes _{I_u\subset X'}\gamma_u$ with the labels $\pi$, $\pi'$; this is (i). In the same construction, given $x$ and $\omega$, the families $L_u\cup p_u$ are independent over $u$, each with law proportional to the product of its $u$-factors, and independent of $z$. The active links based in $X$ are at $d_u=0$ for some $I_u\subset X$, hence in $L_u$; so $f(x)=\E[F\mid x, \omega]$ is the integral of $F$ against the product of these laws over $I_u\subset X$, a function of $x$ and $\omega_X$, and likewise $h$; this is (iii). \emph{The mixed derivative.} Let $h_X$, $h_{X'}$ be vectors in the coordinates of $x\in D_X$ and $x'\in D_{X'}$ with $\lVert h_e\rVert\le2r$ for each link. No term of $\mathcal U$ depends on coordinates of two components, by Lemma~\ref{lem:manykernel}(ii), so Lemma~\ref{lem:connected} gives \[ \partial_{h_X}\partial_{h_{X'}}\log\tilde K=\Cov_{\mu_{x,x'}}\bigl( \partial_{h_X}\mathcal U,\partial_{h_{X'}}\mathcal U\bigr). \] Let $v=\nabla_z\partial_{h_X}\mathcal U$ and $v'=\nabla_z\partial_{h_{X'}} \mathcal U$, supported on sets $S$ and $S'$ of $z$-links based at distance at most $s+4$ from $X$, respectively $X'$. The supports of distinct components lie in blocks at coarse distance at least $1$, so the coordinates entering $v_e$ belong to one component, and Step~1 of the proof of Lemma~\ref{lem:lambdatwo}, which applies to each component, bounds $\sum_f\lVert\partial_e\partial_f\mathcal U\rVert$ over them by $\beta_Wc_W+c_\psi\le\bar\Lambda_2$. Hence $\lVert v_e\rVert\le2r\bar \Lambda_2$, and $\lVert v'_e\rVert\le2r\bar\Lambda_2$. Apply \eqref{eq:layerbound}, with $\mathcal A=480/\beta_W$ and $\alpha=\alpha_G$ by Lemma~\ref{lem:manylayer}, to $\varphi(e)=-\dist(e,S)$, the fine distance between link midpoints. A block carries $45$ non-path active links, so $\lvert S\rvert\le\nu_s\lvert X\rvert$. A link $e\in S'$ is based in a block $y$ with $\dist(y,y')\le s+4$ for some $y'\in X'$, and a link of $S$ in a block $y''$ with $\dist(y'',X)\le s+4$; then $\dist(y,y'')\ge\dist(y', X)-2s-8$, and links based in blocks at coarse distance $D\ge1$ have midpoints at fine distance at least $2D-2$, so $\dist(e,S)\ge2(\dist(y',X) -2s-9)_+$. Summing over $y'$, the $(2s+9)^4$ blocks $y$ and their $45$ links, \[ \begin{split} \bigl\lvert\partial_{h_X}\partial_{h_{X'}}\log\tilde K\bigr\rvert&\le \frac{480}{\beta_W}(2r\bar\Lambda_2)^2\bigl(\nu_s\lvert X\rvert\bigr) ^{1/2}\\ &\quad\times\Bigl(\nu_s\sum_{y'\in X'}e^{-4\alpha_G(\dist(y',X)-2s-9)_+} \Bigr)^{1/2}, \end{split} \] which is $\varDelta(X,X')$; with $\varphi=-\dist(\cdot,S')$ the same argument gives $\varDelta(X',X)$. The sets $D_X$, $D_{X'}$ are convex, two of their points differ by at most $2r$ in each link, and the logarithm of the cross-ratio is the integral of the mixed derivative over the unit square; this is (ii). \emph{The consequences.} Let $\mu'=k\mu/\mu(k)$ with $\sup k\le e^\kappa \inf k$; the tilt paragraph of the proof of Lemma~\ref{lem:islandcov} gives $\lVert\mu'-\mu\rVert_{\mathrm{TV}}\le\tanh(\kappa/4)$. (a) Under $\varPi_\omega$ the law of $x$ has density proportional to $k_\omega(x)= \int\tilde K(x,x')b_{\omega_{X'}}(dx')$ with respect to $a_{\omega_X}$, and $a_{\tilde\omega_X}=a_{\omega_X}$, $f$ is the same for both. Integrating (ii) against $b_{\tilde\omega_{X'}}\otimes b_{\omega_{X'}}$ shows that $k_{\tilde\omega}/k_\omega$ has logarithmic oscillation at most $\varDelta$, so the two laws differ by at most $\tanh(\varDelta/4)$ in total variation, and $\operatorname{osc}f\le2\lVert F\rVert$. (b) Given $x$, the law of $x'$ under $\varPi_\omega$ is proportional to $\tilde K(x,\cdot) b_{\omega_{X'}}$, and by (ii) these laws differ by at most $\tanh(\varDelta /4)$ for two values of $x$; so $m(x)=\E_{\varPi_\omega}[h\mid x]$ has $\operatorname{osc}m\le2\lVert H\rVert\tanh(\varDelta/4)$. By (iii), $\Cov_{\mu^c}(F,H\mid\omega)=\Cov_{\varPi_\omega}(f,h)=\Cov_{\varPi_\omega} (f,m)$, and a variance is at most a quarter of the squared oscillation (Popoviciu~\cite{Pop35}), so $\lvert\Cov(f,m)\rvert\le\frac14 \operatorname{osc}f\operatorname{osc}m$. (c) The measures of $\omega$ and $\omega'$ are nonzero since their weights are positive, and $\tilde K>0$; Lemma~\ref{lem:kernel} with $a_{\omega_X},a_{\omega'_X}$ and $b_{\omega_{X'}},b_{\omega'_{X'}}$ gives the bound. \end{proof} The second construction is a union bound over chains in two copies. Let $Q^c=P^c\otimes P^c$ be the law of two independent labellings $\omega, \omega'$ given $c$, and $\mathsf D=B(\omega)\cup B(\omega')$. \begin{lemma}[chains in two copies]\label{lem:labelchains} Let $N$ and $N'$ be sets of blocks, $R\ge6$ and $t\ge R$ integers with $\dist(N,N')>t$ and $g_R\le\frac12$, and $\mathcal C$ the union of the $R$-clusters of $\mathsf D\cup N\cup N'$ that meet $N$. Then \[ \E_c\,Q^c\{\mathcal C\not\subset N^{+t}\}\le2\lvert N^{+R}\rvert g_R^{\lfloor t/R\rfloor}, \] and the same holds with $B(\omega)$ in place of $\mathsf D$. \end{lemma} \begin{proof} \emph{A chain.} Let $\mathcal C\not\subset N^{+t}$. There is a path $y_0, \dots,y_n$ in $\mathsf D\cup N\cup N'$ with steps at most $R$, $y_0\in N$ and $\dist(y_n,N)>t$; let $i$ be its first index with $\dist(y_i,N)>t$. The blocks $y_0,\dots,y_{i-1}$ are at distance at most $t$ from $N$, hence not in $N'$, and if $jt\ge R$. So $y_{j+1},\dots,y_{i-1}$ is a sequence of blocks of $\mathsf D$ with steps at most $R$, beginning in $N^{+R}$ and ending at distance more than $t-R$ from $N$. Choose such a sequence $z_1,\dots,z_m$ in $\mathsf D$ with $m$ least. Then $\dist(z_k,z_l)>R$ for $\lvert k-l\rvert \ge2$, since otherwise a shorter sequence exists, and $(m-1)R>t-2R$, so $m\ge\lfloor t/R\rfloor$. \emph{Two copies.} Let $T$ be the set of the $\lceil m/2\rceil$ blocks $z_k$ with $k$ odd, pairwise at distance more than $R\ge6$. The event $T \subset\mathsf D$ is the union over $T_1\subset T$ of $\{T_1\subset B(\omega), \,T\setminus T_1\subset B(\omega')\}$, and for each $T_1$, since $P^c\le1$ and by the Cauchy--Schwarz inequality, with $T_2=T\setminus T_1$, \begin{multline*} \E_c\bigl[P^c(T_1\subset B)\,P^c(T_2\subset B)\bigr]\\ \le\bigl(\mu_L(T_1\subset B)\,\mu_L(T_2\subset B)\bigr)^{1/2}\le \epsilon^{\lvert T\rvert/2}. \end{multline*} So $\E_cQ^c\{T\subset\mathsf D\}\le(2\epsilon^{1/2})^{\lceil m/2\rceil} \le(\sqrt2\,\epsilon^{1/4})^m$, as $2\epsilon^{1/2}\le1$. There are at most $\lvert N^{+R}\rvert(2R+1)^{4(m-1)}$ sequences of $m$ blocks with steps at most $R$ beginning in $N^{+R}$, so \begin{multline*} \E_cQ^c\{\mathcal C\not\subset N^{+t}\}\le\lvert N^{+R}\rvert\sum_{m\ge \lfloor t/R\rfloor}g_R^m\\ \le2\lvert N^{+R}\rvert g_R^{\lfloor t/R\rfloor}. \end{multline*} With $B(\omega)$ in place of $\mathsf D$ the event is smaller. \end{proof} The third ingredient is the mean of $\varDelta$ for a group confined near its observable. Let $N$, $N'$, $R\ge2s+9$, $t\ge R$ be as in Lemma~\ref{lem:labelchains}, let $Y=\mathsf B\cup N\cup N'$ with $\mathsf B$ equal to $\mathsf D$ or to $B(\omega)$, let $\mathcal C$ be the union of the $R$-clusters of $Y$ meeting $N$, and let $k$ be $2$ or $1$ accordingly. On the event $\mathcal C\subset N^{+t}$, every $y\in Y \setminus\mathcal C$ lies in an $R$-cluster distinct from those of $\mathcal C$, so $\dist(y,\mathcal C)>R\ge2s+9$, and $\dist(y,\mathcal C) \ge\dist(y,N^{+t})$; a block of $N'$ has $\dist(y,\mathcal C)\ge\dist(N,N') -t$. For a block $y\in\mathsf B\setminus\mathcal C$ the exponent splits as \begin{multline*} 4\alpha_G(\dist(y,\mathcal C)-2s-9)\\ \ge2\alpha_G(R-2s-8)+2\alpha_G(\dist(y,N^{+t})-2s-9), \end{multline*} and with $\sqrt{\sum_yu_y}\le\sum_y\sqrt{u_y}$, \begin{multline*} \Bigl(\sum_{y\in Y\setminus\mathcal C}e^{-4\alpha_G(\dist(y,\mathcal C) -2s-9)_+}\Bigr)^{1/2}\\ \le\lvert N'\rvert^{1/2}e^{-2\alpha_G(\dist(N,N')-t-2s-9)_+}\\ +e^{-\alpha_G(R-2s-8)}\sum_{y\in\mathsf B}e^{-\alpha_G(\dist(y,N^{+t})-2s -9)} . \end{multline*} The mean of the last sum over $\E_cQ^c$, or $\E_cP^c$, is at most $k \epsilon e^{\alpha_G(2s+9)}\lvert N^{+t}\rvert\sigma(\alpha_G)$, since $\E_cQ^c\{y\in\mathsf D\}\le2\mu_L(y\in B)\le2\epsilon$ and $\E_cP^c\{y\in B\}\le\epsilon$. With $\lvert\mathcal C\rvert\le\lvert N^{+t}\rvert$, \begin{equation}\label{eq:groupmean} \begin{split} &\E\bigl[\mathbf 1_{\mathcal C\subset N^{+t}}\varDelta(\mathcal C,Y \setminus\mathcal C)\bigr]\\ &\quad\le\lvert N^{+t}\rvert^{1/2}\lvert N'\rvert^{1/2}M_{d',t}+k \epsilon^{1/2}\lvert N^{+t}\rvert^{3/2}\varPhi_R, \end{split} \end{equation} with $\E$ the mean over $\E_cQ^c$ or $\E_cP^c$ and $d'=\dist(N,N')$. \begin{theorem}[the label covariance]\label{thm:labelcov} Let $N_A$, $N_B$ be sets of blocks at distance $d\ge6$, $\phi$ a function of the labels of $N_A$ and $\psi$ one of the labels of $N_B$. Then \[ \E_c\bigl\lvert\Cov_{P^c}(\phi,\psi)\bigr\rvert\le2\epsilon\lvert N_A \rvert\lvert N_B\rvert\operatorname{osc}\phi\operatorname{osc}\psi . \] If moreover $R\ge2s+9$ and $t\ge2R$ are integers with $d\ge t+6$ and $g_R\le\frac12$, then, with $V=\lvert N_B^{+t}\rvert$, \begin{multline*} \E_c\bigl\lvert\Cov_{P^c}(\phi,\psi)\bigr\rvert\le\operatorname{osc}\phi \operatorname{osc}\psi\,\lvert N_A\rvert\lvert N_B\rvert\,\epsilon\\ \times\bigl(11^4V^{1/2}M_{d-5,t}+2V^{3/2}\varPhi_R+4(2R+1)^4Vg_R^{\lfloor t/R\rfloor-2}\bigr). \end{multline*} \end{theorem} \begin{proof} Fix $c$. Let $\omega_0$ be the labelling with every block good, and put $\hat\phi(\omega)=\phi(\omega)-\phi(\omega_0)$ and $\hat\psi(\omega)= \psi(\omega)-\psi(\omega_0)$. Then $\lvert\hat\phi\rvert\le \operatorname{osc}\phi$, $\hat\phi(\omega)=0$ unless $B(\omega)\cap N_A\ne \emptyset$, and likewise for $\hat\psi$ and $N_B$. Under $Q^c$, $\Cov_{P^c}(\phi,\psi)=\Cov_{P^c}(\hat\phi,\hat\psi)=\E_{Q^c}G$ with $G(\omega,\omega')=\hat\phi(\omega)(\hat\psi(\omega)-\hat\psi(\omega'))$, and \[ \lvert G\rvert\le\operatorname{osc}\phi\operatorname{osc}\psi\sum_{z_A\in N_A,\,z_B\in N_B}J_{z_Az_B}, \] where $J_{z_Az_B}=\mathbf 1[z_A,z_B\in B(\omega)]+\mathbf 1[z_A\in B(\omega),\,z_B\in B(\omega')]$. As $\dist(z_A,z_B)\ge6$, the mean of $J_{z_Az_B}$ over $\E_cQ^c$ is at most $\mu_L(z_A,z_B\in B)+\E_c[P^c(z_A \in B)P^c(z_B\in B)]\le\epsilon^2+\epsilon\le2\epsilon$, by the Cauchy--Schwarz inequality and $P^c\le1$ in the second term; this is the first bound. For the second let $Y=\mathsf D\cup N_B$, $\mathcal C$ the union of its $R$-clusters meeting $N_B$, $\mathcal G=\{\mathcal C\subset N_B^{+t}\}$ and $\varDelta_B=\varDelta(\mathcal C,Y\setminus\mathcal C)$. \emph{The exchange.} Define $\chi(\omega,\omega')=(\tilde\omega,\tilde \omega')$ by $\tilde\omega=\omega'$ and $\tilde\omega'=\omega$ on $\mathcal C$, and $\tilde\omega=\omega$, $\tilde\omega'=\omega'$ elsewhere. On every block the pair of labels is kept or exchanged, so $\mathsf D$, $Y$, $\mathcal C$ and $\mathcal G$ are the same for $(\omega,\omega')$ and its image, and $\chi$ is an involution. By Lemma~\ref{lem:groupcross}(c) with $X=\mathcal C$, a union of components of $Y$, a pair of positive weight is mapped to one of positive weight, with \[ e^{-\varDelta_B}\le\frac{Q^c(\chi(\omega,\omega'))}{Q^c(\omega, \omega')}\le e^{\varDelta_B}. \] On $\mathcal G$, $\mathcal C\cap N_A=\emptyset$, since $d>t$; so $\tilde \omega=\omega$ on $N_A$ and $\tilde\omega=\omega'$, $\tilde\omega'=\omega$ on $N_B\subset\mathcal C$, and $G\circ\chi=-G$. Split $\E_{Q^c}[G \mathbf 1_{\mathcal G}]$ into the orbits of $\chi$. A fixed point has $G=0$. An orbit $\{x,\chi x\}$ of two points contributes $(Q^c(x)-Q^c (\chi x))G(x)$, and $\lvert1-\varrho\rvert\le(1+\varrho)\tanh(\varDelta _B/2)$ for $\varrho\in[e^{-\varDelta_B},e^{\varDelta_B}]$. As $\varDelta_B$ and $\lvert G\rvert$ are constant on the orbit, with $\tanh u\le u$, \[ \lvert\E_{Q^c}G\rvert\le\operatorname{osc}\phi\operatorname{osc}\psi \sum_{z_A,z_B}\E_{Q^c}\bigl[J_{z_Az_B}\bigl(\tfrac12\mathbf 1_{\mathcal G} \varDelta_B+\mathbf 1_{\mathcal G^c}\bigr)\bigr]. \] Fix $z_A\in N_A$ and $z_B\in N_B$; the two means are bounded in turn, after the mean $\E_c$. \emph{The far islands.} On $\mathcal G$, $\lvert\mathcal C\rvert\le V$ and $Y\setminus\mathcal C\subset\mathsf D$, each of its blocks at distance more than $R$ from $\mathcal C$, hence from $z_B$. By $\sqrt{\sum_yu_y}\le \sum_y\sqrt{u_y}$, \[ \mathbf 1_{\mathcal G}\varDelta_B\le b_*\nu_sV^{1/2}\sum_{y\in\mathsf D \setminus\mathcal C}e^{-2\alpha_G(\dist(y,\mathcal C)-2s-9)_+}. \] The $11^4$ blocks $y$ at distance at most $5$ from $z_A$ have $\dist(y, \mathcal C)\ge\dist(z_A,N_B^{+t})-5\ge d-t-5$, so their terms add up to at most $11^4V^{1/2}M_{d-5,t}$, whose product with $J_{z_Az_B}$ has mean at most $2\epsilon\cdot11^4V^{1/2}M_{d-5,t}$. For every other block $y$ with $\dist(y,z_B)>R$ the blocks $z_A$, $z_B$, $y$ are pairwise at distance at least $6$, and $J_{z_Az_B}\mathbf 1[y\in\mathsf D]$ is at most a sum of four indicators of events $\{T_1\subset B(\omega),\,T_2\subset B(\omega')\}$ with $T_1\cup T_2=\{z_A,z_B,y\}$; as in the proof of Lemma~\ref{lem:labelchains} each has mean at most $\epsilon^{3/2}$. As in the derivation of~\eqref{eq:groupmean}, $e^{-2\alpha_G(\dist(y,\mathcal C) -2s-9)}\le e^{-\alpha_G(R-2s-8)}e^{-\alpha_G(\dist(y,N_B^{+t})-2s-9)}$, and summing over $y$ with $\sigma(\alpha_G)$, \[ \tfrac12\E_c\E_{Q^c}\bigl[J_{z_Az_B}\mathbf 1_{\mathcal G}\varDelta_B \bigr]\le\epsilon\bigl(11^4V^{1/2}M_{d-5,t}+2V^{3/2}\varPhi_R\bigr). \] \emph{The chain.} On $\mathcal G^c$, the proof of Lemma~\ref{lem:labelchains} with $N=N_B$ and $N'=\emptyset$ gives $m\ge \lfloor t/R\rfloor$ and blocks $z_1,\dots,z_m$ of $\mathsf D$ with steps at most $R$, $z_1\in N_B^{+R}$, $\dist(z_k,z_l)>R$ for $\lvert k-l\rvert\ge2$, $\dist(z_m,N_B)>t-R$ and $m$ least. By minimality $\dist(z_k,N_B)\le t-R$ for $kt$, and Lemma~\ref{lem:groupcross}(b) with $Y=Y_1$, $X=\mathcal C_1$ gives $\lvert\Cov_{\mu^c}(F,H\mid\omega)\rvert\le\frac14\lVert F\rVert\lVert H\rVert\varDelta(\mathcal C_1,Y_1\setminus\mathcal C_1)$; off $\mathcal G_1$ the covariance is at most $\lVert F\rVert\lVert H\rVert$. By Lemma~\ref{lem:labelchains} and~\eqref{eq:groupmean} with $k=1$, $N=N_A$, $N'=N_B$, and $\lvert N_A^{+R}\rvert,\lvert N_B\rvert\le V$, \begin{multline*} \E_c\E_{P^c}\lvert\Cov_{\mu^c}(F,H\mid\omega)\rvert\\ \le\lVert F\rVert\lVert H\rVert V\bigl(2g_R^{\lfloor t/R\rfloor}+\tfrac14 M_{d,t}+\tfrac14\epsilon^{1/2}V^{1/2}\varPhi_R\bigr). \end{multline*} \emph{The labels.} Let $Y=\mathsf D\cup N_A\cup N_B$, $\mathcal C_A$ and $\mathcal C_B$ the unions of its $R$-clusters meeting $N_A$ and $N_B$, $\mathcal G=\{\mathcal C_A\subset N_A^{+t},\,\mathcal C_B\subset N_B^{+t}\}$, $\varDelta_A=\varDelta(\mathcal C_A,Y\setminus\mathcal C_A)$ and $\varDelta_B=\varDelta(\mathcal C_B,Y\setminus\mathcal C_B)$. On $\mathcal G$, $\mathcal C_A\cap\mathcal C_B=\emptyset$, since an $R$-cluster in both would lie in $N_A^{+t}$ and contain a block of $N_B$. Let $G= \varphi(\omega)(\psi(\omega)-\psi(\omega'))$, so that $\Cov_{P^c}(\varphi, \psi)=\E_{Q^c}G$ and $\lvert G\rvert\le2\lVert F\rVert\lVert H\rVert$, and let $\chi$ exchange the labels on $\mathcal C_B$ as in the proof of Theorem~\ref{thm:labelcov}; it preserves $Y$, $\mathcal C_A$, $\mathcal C_B$, $\mathcal G$, $\varDelta_A$, $\varDelta_B$, and changes the weight by a factor in $[e^{-\varDelta_B},e^{\varDelta_B}]$. On $\mathcal G$, Lemma~\ref{lem:groupcross}(a) applied with $X=\mathcal C_A$ to $\omega$, $\tilde\omega$, which agree on $\mathcal C_A$, and with $X=\mathcal C_B$ to $\tilde\omega,\omega'$ and to $\tilde\omega',\omega$, which agree on $\mathcal C_B$, gives \begin{gather*} \lvert\varphi(\tilde\omega)-\varphi(\omega)\rvert\le2\lVert F\rVert \tanh(\varDelta_A/4),\\ \lvert\psi(\tilde\omega)-\psi(\omega')\rvert\le2\lVert H\rVert\tanh (\varDelta_B/4), \end{gather*} and the same for $\psi(\tilde\omega')-\psi(\omega)$. Writing \[ \begin{split} G\circ\chi+G&=\bigl(\varphi(\tilde\omega)-\varphi(\omega)\bigr)\bigl( \psi(\tilde\omega)-\psi(\tilde\omega')\bigr)\\ &\quad+\varphi(\omega)\bigl(\psi(\tilde\omega)-\psi(\omega')-\psi (\tilde\omega')+\psi(\omega)\bigr) \end{split} \] gives $\lvert G\circ\chi+G\rvert\le4\lVert F\rVert\lVert H\rVert(\tanh \frac{\varDelta_A}4+\tanh\frac{\varDelta_B}4)$. With $w_x=Q^c(x)+Q^c(\chi x)$ and $w'_x=Q^c(x)- Q^c(\chi x)$, an orbit $\{x,\chi x\}$ of two points in $\mathcal G$ contributes \[ \tfrac12w_x\bigl(G+G\circ\chi\bigr)(x)+\tfrac12w'_x\bigl(G-G\circ\chi \bigr)(x), \] with $\lvert G-G\circ\chi\rvert\le4\lVert F\rVert\lVert H\rVert$ and $\lvert w'_x\rvert\le w_x\tanh(\varDelta_B/2)$, and a fixed point contributes $\frac12Q^c(x)(G+G\circ\chi)(x)$. With $\tanh u\le u$, \begin{multline*} \lvert\Cov_{P^c}(\varphi,\psi)\rvert\le\lVert F\rVert\lVert H\rVert\bigl( 2Q^c(\mathcal G^c)\\ +\E_{Q^c}\bigl[\mathbf 1_{\mathcal G}\bigl(\tfrac12\varDelta_A+\tfrac32 \varDelta_B\bigr)\bigr]\bigr). \end{multline*} By Lemma~\ref{lem:labelchains} for $N_A$ and for $N_B$, and \eqref{eq:groupmean} with $k=2$ for $(N,N')=(N_A,N_B)$ and $(N_B,N_A)$, \begin{multline*} \E_c\lvert\Cov_{P^c}(\varphi,\psi)\rvert\\ \le\lVert F\rVert\lVert H\rVert V\bigl(8g_R^{\lfloor t/R\rfloor}+2M_{d,t} +4\epsilon^{1/2}V^{1/2}\varPhi_R\bigr). \end{multline*} Adding the two parts gives the bound. Finally $\Cov_{\mu_L}(F,H)=\E_c \Cov_{\mu^c}(F,H)+\Cov(\E_{\mu^c}F,\E_{\mu^c}H)$, the last covariance under the law of $c$. \end{proof} The bound has three terms. The first is the chance that the group of an observable reaches distance $t$; the second is the cross-ratio between the two groups through the good layer, at the rate $2\alpha_G$ of Theorem~\ref{thm:layerbound}; the third is the cross-ratio between a group and the islands at distance more than $R$ from it, in the mean, which is of order $\epsilon$ and does not decay with $d$. The numbers follow. \begin{corollary}[the covariances in numbers]\label{cor:fibrecovnumbers} At $\rho=2.17\cdot10^{-16}$, with $s\in\{1,5\}$, $\beta_W\ge\beta^*$, and $V$ the volume of Theorem~\ref{thm:labelcov} or~\ref{thm:fibrecov} for the stated $t$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item for $d\ge6s+34$, with $R=2s+9$ and $t=R(\lceil d/(10^4R)\rceil+2)$, \begin{multline*} \E_c\lvert\Cov_{P^c}(\phi,\psi)\rvert\le\operatorname{osc}\phi \operatorname{osc}\psi\,\lvert N_A\rvert\lvert N_B\rvert\,\epsilon\\ \times\Bigl(\frac{\beta_W}{\beta^*}10^{c_s}V^{1/2}e^{-2\alpha_G(1-10^{-4}) d}\\ +4(4s+19)^4Ve^{-\lambda_sd}+10^{-f_s}V^{3/2}\Bigr), \end{multline*} with $c_1=26.26$, $\lambda_1=0.0736$, $f_1=7011.8$, $c_5=27.29$, $\lambda_5= 0.0503$, $f_5=8285.9$, and $V\le\lvert N_B\rvert(d/5000+12s+55)^4$; \item for $d\ge2s+10$, with $R=2s+9$ and $t=R\lceil d/(10^4R)\rceil$, \begin{multline*} \E_c\lvert\Cov_{\mu^c}(F,H)\rvert\le\lVert F\rVert\lVert H\rVert\Bigl( \frac{\beta_W}{\beta^*}10^{a_s}Ve^{-2\alpha_G(1-10^{-4})d}\\ +10^{-a'_s}V^{3/2}\Bigr), \end{multline*} with $a_1=22.75$, $a'_1=14062.6$, $a_5=23.77$, $a'_5=16611.8$, and $V\le \max\{\lvert N_A\rvert,\lvert N_B\rvert\}(d/5000+4s+19)^4$; \item for $10^8\le d\le10^{100}$, with $R=\lceil2000\sqrt d\rceil$ and $t=\lfloor d/2\rfloor$, \[ \E_c\lvert\Cov_{\mu^c}(F,H)\rvert\le10^{-1965}\frac{\beta_W}{\beta^*} \lVert F\rVert\lVert H\rVert V^{3/2}e^{-\sqrt d/2}, \] with $V\le\max\{\lvert N_A\rvert,\lvert N_B\rvert\}(d+1)^4$. \end{enumerate} \end{corollary} \begin{proof} \emph{Dependence on $\beta_W$.} By Theorem~\ref{thm:defects} and Step~6 of the proof of Theorem~\ref{thm:islandwindow}, $f$ decreases on $[\beta^*, \infty)$ and $f(\beta^*)=\log q^*$, so $\epsilon\le Mq^*$ and $g_R$ is at most its value at $\beta^*$. The logarithmic derivative of $b_*=1920r^2 (99.0001\beta_W+4.65\cdot10^6)^2/\beta_W$ is at most $1/\beta_W$, so $b_* \le(\beta_W/\beta^*)b_*(\beta^*)$. As $f'(\beta)=6/\beta-\rho^2/16$, the logarithmic derivatives of $b_*\epsilon^{1/2}$ and $b_*\epsilon$ are at most $4/\beta_W-\rho^2/32$ and $7/\beta_W-\rho^2/16$, both negative since $\beta^*\rho^2/16>8$; so $\varPhi_R$ and $\epsilon^{1/2}\varPhi_R$ are at most their values at $\beta^*$. At $\beta^*$ of Theorem~\ref{thm:islandwindow}, $b_*\nu_s\le10^{22.0833}$ for $s=1$ and $10^{23.1039}$ for $s=5$, $\sigma(\alpha_G)\le9.831\cdot10^{16}$, and $-\log g_{2s+9}\ge8105.1$ for $s=1$ and $9571.0$ for $s=5$. Indeed, the proof of that theorem gives $b_*(\beta^*)\le10^{16.2645}$ and $10^{16.3356}$, $\sigma(\alpha_G)\le9.831\cdot10^{16}$, $\lvert\log q^*\rvert$ within $10^{-2}$ of $32481.67$ and $38354.07$, and $\log_{10}(Mq^*)$ within $10^{-3}$ of $-14102.446$ and $-16652.797$; here $\nu_1=45\cdot11^4$, $\nu_5=45\cdot19^4$ with $\log19=\log18+2\operatorname{artanh}\frac1{37}$, and at $\epsilon=Mq^*$ \[ -\log g_R=\tfrac14\bigl(\lvert\log q^*\rvert-4\log11\bigr)-\tfrac12\log2- 4\log(2R+1), \] which at $2R+1=4s+19$, that is $23$ and $39=3\cdot13$, lies within $10^{-2}$ of $8105.13$ and $9571.12$. Every number below is a decimal logarithm of a product of these factors, computed with $\log10=\log2+ \log5$; and $b_*(\beta^*)\ge1920r^2\cdot99^2\beta^*>10^{16}$, so $10\le \frac94b_*\nu_s$. (i) Theorem~\ref{thm:labelcov}: $t\ge3R$, $t\le d/10^4+3R$ and $(6s+34) (1-10^{-4})\ge6s+33$, so $d\ge t+6$; $d-5-t-2s-9\ge(1-10^{-4})d-8s-41$, so $11^4M_{d-5,t}\le(\beta_W/\beta^*)10^{c_s}e^{-2\alpha_G(1-10^{-4})d}$ with $10^{c_s}\ge11^4b_*(\beta^*)\nu_se^{2\alpha_G(8s+41)}$; $\lfloor t/R\rfloor -2=\lceil d/(10^4R)\rceil$ and $-\log g_R/(10^4R)\ge\lambda_s$; $2\varPhi_R \le10^{-f_s}$; and $2t+1\le d/5000+12s+55$. (ii) Theorem~\ref{thm:fibrecov}: $R\le t\le d/10^4+R\frac12$. The chain and main ratios are at most \begin{gather*} 10e^{-(\lambda/8002-1/2)\sqrt d},\\ \tfrac94b_*\nu_se^{-2\alpha_G(d/2-2s-9)+\sqrt d/2}; \end{gather*} the first decreases in $d$, and the second for $\sqrt d>1/(4\alpha_G)= 1000$. At $d=10^8$ the chain ratio is at most $10^{-1965.2}$ and $10^{-2761}$, the main ratio at most $10^{-8663.4}$ and $10^{-8662.4}$, and the coefficient $\frac{17}4b_*\nu_s\sigma(\alpha_G)Mq^*e^{\alpha_G(4s+17)}$ of the third term at most $10^{-14062.7}$ and $10^{-16612}$. \end{proof} The groups make the Dobrushin condition on the islands unnecessary in these bounds: two groups have one cross-ratio. Where that condition holds, as in Theorem~\ref{thm:islandwindow}(iii), part (ii) of Lemma~\ref{lem:islandcov} has a constructive proof, by the heat-bath sweep of F\"ollmer~\cite{Fol82} under Dobrushin's matrix~\cite{Dob68,Dob70}, which needs neither the minimising coupling nor compactness. \begin{lemma}[the sweep]\label{lem:sweep} In the setting of Lemma~\ref{lem:islandcov}, with $f,g$ bounded measurable, suppose that $D=\sum_{n\ge0}C^n$ converges. Let $T_qu(\xi)=\int u(\xi^\eta) P_q(d\eta\mid\xi)$, let $q_1,q_2,\dots$ run through $\mathcal Q$ cyclically, $u_0=f$ and $u_k=T_{q_k}u_{k-1}$. Then \begin{gather*} \Cov_P(f,g)=\sum_{k\ge1}\E_P\bigl[\Cov_{q_k}(u_{k-1},g)\bigr],\\ \sum_{k:\,q_k=q}\delta_q(u_{k-1})\le\sum_{q'}\delta_{q'}(f)D_{q'q}, \end{gather*} with $\Cov_q$ the covariance under $P_q(\cdot\mid\xi)$ in the coordinate $q$; hence $\lvert\Cov_P(f,g)\rvert\le\frac14\sum_{q,q'}\delta_{q'}(f) D_{q'q}\delta_q(g)$. \end{lemma} \begin{proof} The law $P$ is invariant under $T_q$, and $T_qu$ does not depend on the coordinate $\xi_q$; so $\E_P[(T_qu)g]=\E_P[(T_qu)(T_qg)]$ and $\E_P[ug]=\E_P[(T_qu)(T_qg)]+\E_P \Cov_q(u,g)$, which gives $\Cov_P(u,g)-\Cov_P(T_qu,g)=\E_P\Cov_q(u,g)$. For $q'\ne q$, changing $\xi_{q'}$ changes $T_qu$ by at most $\delta_{q'} (u)+C_{qq'}\delta_q(u)$: the integrand by $\delta_{q'}(u)$, and the law $P_q(\cdot\mid\xi)$ by $C_{qq'}$ in total variation, against a function of oscillation $\delta_q(u)$. So the vector $v_k=(\delta_q(u_k))_q$ obeys $v_k\le v_{k-1}-\tau_ke_{q_k}+\tau_kC_{q_k\cdot}$ with $\tau_k=(v_{k-1}) _{q_k}$, and since $D\ge0$ and $D-CD=I$, $v_kD\le v_{k-1}D-\tau_ke_{q_k}$. Summing, the accumulated $\tau$ at each $q$ is at most $(v_0D)_q$, the second claim. In particular $\sum_k\tau_k<\infty$; each $(v_n)_q$ is at most the sum of the $\tau_k$ since the last visit to $q$, as $C\le1$, so $v_n\to0$, $\operatorname{osc}u_n\le\sum_q(v_n)_q\to0$ and $\Cov_P(u_n,g) \to0$. This gives the identity, and $\lvert\Cov_q(u,g)\rvert\le\frac14 \delta_q(u)\delta_q(g)$ by Popoviciu~\cite{Pop35} gives the bound. \end{proof} \begin{remark}\label{rem:labelcovopen} Four things are not proved here. First, the bounds are means over the coarse field. For a fixed $c$ the labels need not be rare: a coarse plaquette with $\delta(c_P,I)>\eta$ makes its blocks bad with $P^c$-probability one. The proofs use rarity only through $\E_cP^c(T\subset B)=\mu_L(T\subset B)$ and its two-copy form in Lemma~\ref{lem:labelchains}; a bound for every $c$, or uniform in an exterior as (D) asks, is not given. Second, the floor. The chains are paid for at the joining distance $R$, while the islands beyond $R$ enter $\varDelta$ through $e^{-\alpha_GR}$, so with $R$ fixed the bound stops at a floor, of order $\epsilon^{3/2}$ in Theorem~\ref{thm:labelcov} and $\epsilon$ in Theorem~\ref{thm:fibrecov}, and with $R\propto\sqrt d$ it decays as $e^{-\sqrt d/2}$. Decay at the rate $2\alpha_G$ without a floor needs control of $\log Z^c$ as a function of the labels of three or more components: the mixed differences of order $k\ge3$ of $\log Z^c$ over the labels of $k$ components, bounded by $C^ke^{-2\alpha_G\ell}$ with $\ell$ the length of the shortest tree joining them. The inputs give the differences of order two, and through Lemma~\ref{lem:mobius} those of every order in terms of the diameter only; those of order three of $\log\tilde K$ are third cumulants of the layer law, which \eqref{eq:layerbound} does not bound, and $\tilde K$ and $D_z$ depend on the set $Y$ of components. Third, the covariance of $\E_{\mu^c}F$ and $\E_{\mu^c}H$ as functions of $c$ is the multiscale part, item (iv) of Sec.~\ref{ssec:remains}. Fourth, the bounds are consistent with Proposition~\ref{prop:hiddenbit}: in its mixture the labels are independent given the hidden bit, which plays the role of $c$, so the covariance under the fibre law vanishes and the covariance $q^2/16$ is that of the conditional means; the theorems above use the cross-ratio of Lemma~\ref{lem:groupcross}, not rarity alone. \end{remark} Remark~\ref{rem:labelcovopen} leaves open a covariance bound for the labels for a fixed coarse field, uniform in the exterior, decaying at the rate $2\alpha_G$ with no floor. The rest of this subsection determines what the inputs used above give for one law of the labels. The proof of Theorem~\ref{thm:labelcov} uses two properties of the laws $P^c$. For a law $P$ on the labellings of $\Lambda$ and $0<\epsilon'<1$ consider: \begin{itemize}[leftmargin=*] \item[$(\mathrm R_{\epsilon'})$] $P(T\subset B)\le\epsilon'^{\lvert T \rvert}$ for every nonempty set $T$ of blocks pairwise at distance at least $6$; \item[$(\mathrm X)$] for every set $Y$ of blocks, every union $X$ of components of $Y$, $X'=Y\setminus X$, and all labellings $\omega,\omega'$ of positive probability with $B(\omega),B(\omega')\subset Y$, the labellings $\omega^\natural=(\omega'_X,\omega_{X'})$ and $\omega'^\natural =(\omega_X,\omega'_{X'})$ have positive probability and \[ e^{-\varDelta}\le\frac{P(\omega^\natural)P(\omega'^\natural)}{P(\omega) P(\omega')}\le e^{\varDelta}, \] with $\varDelta=\min\{\varDelta(X,X'),\varDelta(X',X)\}$. \end{itemize} For every coarse field $c$ the law $P^c$ has $(\mathrm X)$, by Lemma~\ref{lem:groupcross}(c), and the law of the labels under $\mu_L$ has $(\mathrm R_\epsilon)$. \begin{lemma}[one law]\label{lem:onelaw} Let $P$ be a law on the labellings of $\Lambda$ with $(\mathrm X)$ and $(\mathrm R_{\epsilon^{1/2}})$, and let $N_A,N_B,\phi,\psi$ be as in Theorem~\ref{thm:labelcov}. Then $\lvert\Cov_P(\phi,\psi)\rvert$ obeys both bounds of Theorem~\ref{thm:labelcov}, with the same constants. In particular this holds for every law with $(\mathrm X)$ and $(\mathrm R_\epsilon)$. \end{lemma} \begin{proof} Run the proof of Theorem~\ref{thm:labelcov} with $P^c$ replaced by $P$, $Q^c$ by $Q=P\otimes P$, and the means $\E_c$ omitted. The exchange and the orbit decomposition use $(\mathrm X)$ only. The mean over $c$ enters through the probabilities, under $Q$, of events $\{T_1\subset B(\omega), \,T_2\subset B(\omega')\}$ with $T=T_1\cup T_2$ pairwise at distance at least $6$, which the proof bounds by $\epsilon^{\lvert T\rvert/2}$, and through $\mu_L(z_A,z_B\in B)\le\epsilon^2$ in the first bound. Under $Q$ the first probability is $P(T_1\subset B)P(T_2\subset B)\le \epsilon^{\lvert T\rvert/2}$ by $(\mathrm R_{\epsilon^{1/2}})$, with $P(\emptyset\subset B)=1$; it covers the means of $J_{z_Az_B}$, at most $2\epsilon$, of $J_{z_Az_B}\mathbf 1[y\in\mathsf D]$, at most $4 \epsilon^{3/2}$, and of $J_{z_Az_B}\mathbf 1\{T\subset\mathsf D\}$ along a chain, at most $2\epsilon(2\epsilon^{1/2})^{\lvert T\rvert}$. In the first bound the mean of $J_{z_Az_B}$ is at most $P(z_A,z_B\in B)+P(z_A\in B)P(z_B \in B)\le2\epsilon$. Every other step is pointwise in the labellings. Finally $\epsilon^{\lvert T\rvert}\le\epsilon^{\lvert T\rvert/2}$. \end{proof} \begin{proposition}[fixed coarse fields]\label{prop:fixedgood} Let $\mathcal G_\Lambda$ be the set of coarse fields $c$ for which $P^c$ has $(\mathrm R_{\epsilon^{1/2}})$. Then \[ \mu_L\{V^\sharp\notin\mathcal G_\Lambda\}\le(1+\epsilon^{1/2})^{\lvert \Lambda\rvert}-1, \] which is at most $2\lvert\Lambda\rvert\epsilon^{1/2}$ when $\lvert\Lambda \rvert\epsilon^{1/2}\le1$, and for every $c\in\mathcal G_\Lambda$ the covariance $\lvert\Cov_{P^c}(\phi,\psi)\rvert$ obeys both bounds of Theorem~\ref{thm:labelcov}. No coarse field with a coarse plaquette $\delta(c_P,I)>\eta$ lies in $\mathcal G_\Lambda$. \end{proposition} \begin{proof} For a set $T$ of blocks, $\E_cP^c(T\subset B)=\mu_L(T\subset B)\le \epsilon^{\lvert T\rvert}$ when $T$ is pairwise at distance at least $6$, so by Markov's inequality the law of $V^\sharp$ gives the set $\{c: P^c(T\subset B)>\epsilon^{\lvert T\rvert/2}\}$ mass at most $\epsilon^{\lvert T\rvert/2}$. The union over the nonempty sets $T\subset \Lambda$ has mass at most $\sum_{T\ne\emptyset}\epsilon^{\lvert T \rvert/2}=(1+\epsilon^{1/2})^{\lvert\Lambda\rvert}-1\le e^{\lvert\Lambda \rvert\epsilon^{1/2}}-1$, and $e^x-1\le2x$ on $[0,1]$. For $c\in\mathcal G_\Lambda$, $P^c$ has $(\mathrm X)$ by Lemma~\ref{lem:groupcross}(c), and Lemma~\ref{lem:onelaw} applies. A coarse plaquette with $\delta(c_P,I)> \eta$ makes its corner bad with $P^c$-probability one, and $1> \epsilon^{1/2}$. \end{proof} The floor of Theorem~\ref{thm:labelcov} comes from the joining distance $R$; with $R$ of order $\sqrt d$ it becomes a stretched exponential, as for the fibre in Corollary~\ref{cor:fibrecovnumbers}(iii). \begin{corollary}[the labels, stretched]\label{cor:labelstretched} At $\rho=2.17\cdot10^{-16}$, with $s\in\{1,5\}$, $\beta_W\ge\beta^*$ and $10^8\le d\le10^{100}$, both $\E_c\lvert\Cov_{P^c}(\phi,\psi)\rvert$ and, for every $c\in\mathcal G_\Lambda$, $\lvert\Cov_{P^c}(\phi,\psi)\rvert$ are at most \[ \operatorname{osc}\phi\operatorname{osc}\psi\,\lvert N_A\rvert\lvert N_B\rvert\,10^{-u_s}V^{3/2}e^{-\sqrt d/20}, \] with $u_1=21114.2$, $u_5=24938.6$ and $V\le\lvert N_B\rvert(d+1)^4$. \end{corollary} \begin{proof} Take $R=\lceil200\sqrt d\rceil$ and $t=\lfloor d/2\rfloor$ in Theorem~\ref{thm:labelcov}, or in Lemma~\ref{lem:onelaw}. For $d\ge10^8$, $t\ge d/2-1\ge2R$, $d\ge t+6$, $2R+1\le401\sqrt d$ and $t/R\ge\sqrt d/401$, so $\lfloor t/R\rfloor-2\ge\sqrt d/401-3>0$; $R\le200\cdot10^{50}+1$, where $-\log g_R\ge7633.1$ for $s=1$ and $9101.2$ for $s=5$. The three terms of the bracket, multiplied by the factor $\epsilon$ in front, are bounded as follows. The far term $2V^{3/2}\varPhi_R$ carries $e^{-\alpha_GR}\le e^{-\sqrt d/20}$, since $200\alpha_G=\frac1{20}$, with coefficient $2b_*\nu_s\sigma(\alpha_G)\epsilon^{1/2}e^{\alpha_G(4s+17)}$. The main term has $d-5-t-2s-9\ge d/2-2s-14$, and its ratio to $e^{-\sqrt d/20}$, $11^4b_*\nu_se^{2\alpha_G(2s+14)}e^{-\alpha_Gd+\sqrt d/20}$, decreases on $[10^8,\infty)$. The chain term is at most $4\cdot401^4d^2Ve^{3\lambda} e^{-\lambda\sqrt d/401}$ with $\lambda=-\log g_R$, and its ratio to $e^{-\sqrt d/20}$ decreases on $[10^8,\infty)$ since $\lambda/401>\frac1 {20}+4\cdot10^{-4}$. By the monotonicity in the proof of Corollary~\ref{cor:fibrecovnumbers}, $\epsilon b_*$, $\epsilon^{3/2}b_*$ and $g_R$ are at most their values at $\beta^*$. At $\beta^*$ and $d=10^8$, the coefficient of the far term times $\epsilon$ is at most $10^{-21114.22}$ for $s=1$ and $10^{-24938.64}$ for $s=5$, and the main and chain ratios are below $10^{-24000}$ and $10^{-86000}$; with $V^{1/2}, V\le V^{3/2}$ and $2t+1\le d+1$ this gives the bound. \emph{The numbers} come from the factors listed in the proof of Corollary~\ref{cor:fibrecovnumbers}. The display for $-\log g_R$ there at $2R+1\le4\cdot10^{52}+3$ gives $\lambda\ge7633.1$ and $9101.2$. With $\epsilon\le Mq^*$, the decimal logarithm of $2b_*\nu_s\sigma(\alpha_G) (Mq^*)^{3/2}e^{\alpha_G(4s+17)}$ is at most $-21114.28$ for $s=1$ and $-24938.79$ for $s=5$; that of $11^4b_*\nu_se^{2\alpha_G(2s+14)}e^{-\alpha_G d+\sqrt d/20}Mq^*$ at $d=10^8$ is at most $-24716$ and $-27265$; and the chain ratio times $\epsilon$ is at most $4\cdot401^4d^2e^{-\lambda(\sqrt d/401-3)+\sqrt d/20}Mq^*$, whose decimal logarithm at $d=10^8$ is at most $-86581$ and $-103119$. \end{proof} The next theorem shows that no argument which uses only $(\mathrm X)$ and a rarity bound does better: for a fixed coarse field nothing crosses a bad component, and with rarity the decay is at best a stretched exponential. \begin{theorem}[what rarity and the group cross-ratios give] \label{thm:fixedobstruction} Let $s\ge1$, $\alpha_G$ and $b_*\nu_s\ge1$ be as above, and let $d\ge2$ be an integer, on a torus of side larger than $2d$; let $a$ and $b=a+de_1$ be two blocks. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For every $J\ge0$ there is a law $P$ with $(\mathrm X)$, under which the blocks $a+ie_1$, $00$ holds for all laws with $(\mathrm X)$ and $(\mathrm R_{\epsilon'})$; with $\kappa=2\alpha_G$ and $C\epsilon'$ in place of $C$ it fails at $d=9h_*$ for every $C0$ the lower bound of (ii) exceeds $Ce^{-\kappa d}$ for $d=m^2h_*$ and $m$ large. With $\kappa= 2\alpha_G$, $2\alpha_Gd\ge m^2\varkappa$ and $\epsilon'=e^{1-\varkappa}$, so the ratio of the covariance to $C\epsilon'e^{-2\alpha_Gd}$ is at least $e^{(m^2-2m)\varkappa-2\alpha_Gm-1}/(2C)$, which at $m=3$ exceeds one for $C0$ for all $\varsigma$. Then \[ \Bigl|\sum_{\varsigma}(-1)^{k-\lvert\varsigma\rvert}\log Z(\varsigma) \Bigr|\le2^{k-2}\min_{u\ne v\in\mathcal K}\varDelta_*(u,v), \] with $\lvert\varsigma\rvert=\sum_u\varsigma_u$ and $\varDelta_*(u,v)$ the quantity $\varDelta_*$ of Theorem~\ref{thm:manyislands} for $a=u$, $b=v$ and the sets $I_w$, $w\in\mathcal I\setminus\{u,v\}$, in the role of $\mathcal Q$. \end{lemma} \begin{proof} The numbers $\varDelta_{uv}$, $S_u$, $C$ and $D$ of Theorem~\ref{thm:manyislands} depend on the sets $I_w$, on $\Lambda_2$ and on $\beta_W$, and not on the patterns, since $\tilde K$ and $\mathcal U$ are the same in all sectors by Lemma~\ref{lem:manykernel}(i). Fix $u\ne v$ and, for $\varsigma'\in\{0,1\}^{\mathcal K\setminus\{u,v\}}$, let $D(\varsigma')$ be the second difference $\log Z_{11}+\log Z_{00}- \log Z_{10}-\log Z_{01}$ in $(\varsigma_u,\varsigma_v)$ with the other indices equal to $\varsigma'$. These four weights are those of Theorem~\ref{thm:manyislands} with the sets $I_w$, $w\ne u,v$, carrying the patterns $\omega_w^{\varsigma'_w}$, so $\lvert D(\varsigma')\rvert\le \varDelta_*(u,v)$. The sum in the statement is $\sum_{\varsigma'}(-1)^{k-2- \lvert\varsigma'\rvert}D(\varsigma')$, which has $2^{k-2}$ terms. \end{proof} Under the weighted Dobrushin condition of Corollary~\ref{cor:islandchains} the right side is at most $2^{k-2}(\varDelta_{uv}+e^{-2\vartheta\alpha_G\ell(u,v)}T_uT_v/(4(1- \theta)))$ for the pair $u,v$ of largest $\ell(u,v)$, so the differences of order $k$ decay with the diameter of the $k$ sets in the pseudometric $\ell$. In the laws of Theorem~\ref{thm:fixedobstruction}(ii) the pair $y_0,y_m$, with the blocks of $S$ between them bad, has second difference $e^{-2\alpha_Gd/m}$ at distance $d$, so a pair bound that decays with the distance as in Corollary~\ref{cor:islandchains} excludes them. \begin{remark}\label{rem:fixedcopen} For a fixed coarse field the results above give the following. Proposition~\ref{prop:fixedgood} gives the bounds of Theorem~\ref{thm:labelcov} for every coarse field outside a set of probability at most $2\lvert\Lambda\rvert\epsilon^{1/2}$, and Corollary~\ref{cor:labelstretched} decay as $e^{-\sqrt d/20}$ for $10^8\le d\le10^{100}$, with no floor in that range. Decay at the rate $2\alpha_G$ is not proved, for three reasons. First, by Theorem~\ref{thm:fixedobstruction}(ii) it does not follow from rarity and the group cross-ratios of Lemma~\ref{lem:groupcross}, which are all that the proofs of Theorems~\ref{thm:labelcov} and~\ref{thm:fibrecov} use. Second, Lemma~\ref{lem:mobius} bounds the differences of order $k$ by the diameter of the $k$ sets, while a cluster expansion in the labels, through the criterion of Koteck\'y and Preiss~\cite{KP86}, needs the length of the shortest tree joining them, as in Remark~\ref{rem:labelcovopen}; with the diameter the sum over $k$ islands in a ball of radius $D$ carries the entropy $D^{4(k-1)}$, which the rarity pays only while $\epsilon D^4$ is small, and the pair bound decays only where the weighted Dobrushin condition holds. Third, uniformity in the coarse field and in the exterior is excluded for these inputs by Theorem~\ref{thm:fixedobstruction}(i): for a coarse field whose coarse plaquettes with $\delta(c_P,I)>\eta$ have corners joining a neighbourhood of $N_A$ to one of $N_B$, the labels of $N_A$ and $N_B$ adjoin one component that is bad for every labelling, and neither Lemma~\ref{lem:groupcross} nor Theorem~\ref{thm:manyislands} bounds anything across it. A bound uniform in the coarse field therefore needs the response of the law across the interior of a bad component, which no result here gives, and a bound at the rate $2\alpha_G$ for the coarse fields of $\mathcal G_\Lambda$ needs the tree form of the differences of order three and more. The statement of item (ii) of Sec.~\ref{ssec:remains} is accordingly posed in the mean over the coarse field or for the coarse fields of $\mathcal G_\Lambda$. \end{remark} The tree form named in Remarks~\ref{rem:labelcovopen} and~\ref{rem:fixedcopen} asks for a bound $C_0C^ke^{-2\alpha_G\ell}$ on the mixed differences of order $k\ge3$, with $\ell$ the length of the shortest tree joining the $k$ sets. The next two propositions show that it follows neither from the second differences in every environment, which are all that Lemma~\ref{lem:mobius} uses, nor from the layer bound~\eqref{eq:layerbound}, which is all that the proof of Theorem~\ref{thm:manyislands} uses of the layer law. The length of a tree whose vertices are blocks is the sum of the distances of its edges. \begin{proposition}[pair bounds give the diameter form]\label{prop:pairsonly} Let $s\ge1$, $\alpha_G$ and $b_*\nu_s\ge1$ be as above, $0<\epsilon'<1$, $3\le k\le16$, $D\ge2s+9$ an integer, and on a torus of side larger than $2D$ let $S$ be a set of $k$ vertices of a cube $a+\{0,D\}^4$, so that any two blocks of $S$ are at distance $D$. Put $J=e^{-2\alpha_G(D-2s-9)}$. There is a law $P$ with $(\mathrm X)$ and $(\mathrm R_{\epsilon'})$ such that: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item every second difference of $\log P$ between the labels of two sets of blocks at distance $t$, all other labels fixed, vanishes unless both sets meet $S$, and then $t\le D$ and its modulus is at most $J\le e^{-2\alpha_G(t-2s-9)}$; \item the mixed difference of order $k$ of $\log P$ over the labels of the blocks of $S$ equals $J$; \item every tree whose vertices include $S$ has length at least $kD/2$. \end{enumerate} So the bound of Lemma~\ref{lem:mobius} by the second differences of a pair, in every environment, is attained up to its factor $2^{k-2}$, while a bound $C_0C^ke^{-2\alpha_G\ell}$ fails for this law as soon as $\alpha_G(k-2)D> \log(C_0C^k)-2\alpha_G(2s+9)$. \end{proposition} \begin{proof} Put $p=\epsilon'/e$ and let $P$ give the labelling with bad set $A\subset S$ the probability $N^{-1}p^{\lvert A\rvert}(1-p)^{k-\lvert A\rvert}e^{J \mathbf 1[A=S]}$, with $N=1+(e^J-1)p^k$, and every other labelling the probability zero; this is the law of the proof of Theorem~\ref{thm:fixedobstruction}(ii) on the set $S$. For $T\subset S$ nonempty, $P(T\subset B)=(p^{\lvert T\rvert}+(e^J-1)p^k)/N\le e^Jp^{\lvert T\rvert}\le e^{1-\lvert T\rvert}\epsilon'^{\lvert T\rvert}\le\epsilon'^{ \lvert T\rvert}$, as $J\le1$, and for $T\not\subset S$ it vanishes; this is $(\mathrm R_{\epsilon'})$. With $\sigma_y=\mathbf 1[y\in B]$, on the labellings of positive probability $\log P=\mathrm{const}+\lvert B\rvert \log\frac p{1-p}+J\prod_{y\in S}\sigma_y$. In $(\mathrm X)$ the computation of the proof of Theorem~\ref{thm:fixedobstruction}(ii) gives the ratio $e^{-J(x-x')(z-z')}$, which is one unless $S$ meets both $X$ and $X'$; then some $y\in S\cap X$ and $y'\in S\cap X'$ are at distance $D$, so both $\varDelta(X,X')$ and $\varDelta(X',X)$ are at least $b_*\nu_se^{ -2\alpha_G(D-2s-9)}\ge J$. (i) The linear part has no second differences, and the second difference of $J\prod_y\sigma_y$ between two sets is $J$ times the product of the differences of the two partial products and of the fixed labels of the rest of $S$, so its modulus is at most $J$, and it vanishes unless both sets meet $S$; then $t\le D$. (ii) The mixed difference of order $k\ge2$ of the linear part vanishes, and that of $J\prod_y\sigma_y$ is $J$. (iii) A closed walk around a tree of length $\lambda$ that starts at a block of $S$ and traverses each edge twice has length $2\lambda$ and visits every block of $S$. Cut it at the first visits of the $k$ blocks of $S$: it falls into $k$ walks, each between two distinct blocks of $S$, so each of length at least $D$, and $2\lambda\ge kD$. For the last assertion, $J>C_0C^ke^{-\alpha_GkD}$ is the inequality stated, and $\ell\ge kD/2$. \end{proof} \begin{proposition}[the layer bound and third cumulants] \label{prop:layerthird} Let $\alpha>0$, $D>0$, and give three sites $1,2,3$ the distances $d(i,j)=D$ for $i\ne j$. For $x\in\R^3$ let $\mu_x$ be the probability measure on $\R^3$ with density proportional to $e^{-U(z;x)}$, \[ U(z;x)=\tfrac12\lvert z\rvert^2+J\sin z_1\sin z_2\sin z_3-x\cdot z,\qquad J=\tfrac16e^{-\alpha D}, \] and let $K(x)=\int e^{-U(z;x)}\,dz$. Then: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item for every $x$, every $\varphi$ with $\lvert\varphi(i)-\varphi(j) \rvert\le d(i,j)$ and all smooth $F,H$ with bounded derivatives, \[ \lvert\Cov_{\mu_x}(F,H)\rvert\le2\,\bigl\lVert e^{-\alpha\varphi}\nabla F\bigr\rVert_{L^2(\mu_x)}\bigl\lVert e^{\alpha\varphi}\nabla H \bigr\rVert_{L^2(\mu_x)}, \] which is~\eqref{eq:layerbound} with $\mathcal A=2$ and the rate $\alpha$; \item $\lvert\partial_{x_i}\partial_{x_j}\log K(x)\rvert=\lvert\Cov_{\mu_x} (z_i,z_j)\rvert\le2e^{-\alpha D}$ for $i\ne j$ and every $x$; \item $\partial_{x_1}\partial_{x_2}\partial_{x_3}\log K(0)$ is the third cumulant of $z_1,z_2,z_3$ under $\mu_0$, of modulus at least $\frac1{30}e^{-\alpha D}$, while every tree joining the three sites has length at least $\frac32D$. \end{enumerate} So~\eqref{eq:layerbound}, at every $x$ and with the second derivatives of $\log K$ decaying at the rate $\alpha$, gives no bound $Ce^{-\alpha\ell}$ on the third derivatives, with $\ell$ the length of the shortest joining tree: here $C\ge\frac1{30}e^{\alpha D/2}$ for every $D$. \end{proposition} \begin{proof} (i) Write $\operatorname{Hess}_zU=I+JM$, where the entries of $M$ are $-\sin z_1\sin z_2\sin z_3$ on the diagonal and $\cos z_i\cos z_j\sin z_k$ off it, all of modulus at most one. With $w=e^{\alpha\varphi}$ the entries of $wMw^{-1}$ have modulus at most $e^{\alpha D}$, so $\lVert J\,wMw^{-1} \rVert\le3Je^{\alpha D}=\frac12$, and the same holds for $M$. The potential is smooth with bounded derivatives of order at least two and $\operatorname{Hess}_zU\ge\frac12$, and the weighted Helffer--Sj\"ostrand representation in the proof of Theorem~\ref{thm:chartdecay}(ii) applies: $\operatorname{Re}\langle u,\mathcal A_wu\rangle\ge(1-\frac12)\lVert u \rVert^2$, so $\lVert\mathcal A_w^{-1}\rVert\le2$ and (i) follows. (ii) $\log K(x)-\log K(0)=\log\E_{\mu_0}e^{x\cdot z}$, so the derivatives of $\log K$ are the joint cumulants of $z$ under $\mu_x$; the second ones are covariances, and (i) with $F=z_i$, $H=z_j$ and $\varphi=-d(\cdot,i)$ gives $2e^{-\alpha D}$. (iii) $U(\cdot;0)$ is invariant under $(z_1,z_2)\mapsto(-z_1,-z_2)$ and $(z_1,z_3)\mapsto(-z_1,-z_3)$, so $\E_{\mu_0}z_i=0$ and the third cumulant is $\E_{\mu_0}z_1z_2z_3$. Let $\gamma_1$ be the standard Gaussian measure on $\R$, $a_n=\int t\sin^nt\,d\gamma_1$ and $b_n=\int\sin^nt\,d\gamma_1$. Expanding $e^{-J\sin z_1\sin z_2\sin z_3}$, which is legitimate since the product is bounded, \[ \E_{\mu_0}z_1z_2z_3=\frac{\sum_n(-J)^na_n^3/n!}{\sum_n(-J)^nb_n^3/n!} . \] By parity $a_n=0$ for even $n$ and $b_n=0$ for odd $n$; $b_0=1$ and $b_n\ge0$; $\lvert a_n\rvert\le\int t^2d\gamma_1=1$ for $n\ge1$, since $\lvert\sin t\rvert\le\min\{1,\lvert t\rvert\}$; and $a_1=\int\cos t\, d\gamma_1=e^{-1/2}$ by Gaussian integration by parts. So the numerator has modulus at least $Je^{-3/2}-(\sinh J-J)$ and the denominator lies in $[1,\cosh J]$. As $(2i+1)!\ge6\,(2i-2)!$, $\sinh J-J\le\frac{J^3}6\cosh J$, and with $J\le\frac16$, $\cosh J<1.014$ and $e^{-3/2}>0.2231$, as $e^{3/2} <2.71829\cdot1.64873<4.482$, \[ \lvert\E_{\mu_0}z_1z_2z_3\rvert\ge\frac{J(0.2231-0.0047)}{1.014}> \frac J5=\frac{e^{-\alpha D}}{30}. \] The bound on trees is Proposition~\ref{prop:pairsonly}(iii) with $k=3$. \end{proof} \begin{remark}\label{rem:treeform} The tree form of the differences of order three and more is therefore not proved, and Propositions~\ref{prop:pairsonly} and~\ref{prop:layerthird} locate what it needs. In the law of Proposition~\ref{prop:layerthird} the third cumulant is carried by the term $J\sin z_1\sin z_2\sin z_3$, whose mixed third derivative couples three sites at mutual distance $D$ with size $e^{-\alpha D}$. The differences of order three of $\log\tilde K$ are third cumulants of the layer law (Remark~\ref{rem:labelcovopen}), and a proof of their tree form needs two inputs that~\eqref{eq:layerbound} does not contain: a bound in tree form on the mixed derivatives of order three of $\mathcal U$ in the coordinates $z$ and in the island coordinates, and a weighted representation of the third cumulants of $\mu_{x,y}$ through the Helffer--Sj\"ostrand operator and these derivatives, valid on the box $D_z$ with its walls. The proof of~\eqref{eq:layerbound} reaches the walls through the approximations $H_n$ of the proof of Theorem~\ref{thm:chartdecay}(ii), whose derivatives of order three grow with $n$, so that argument gives no bound on third cumulants uniform in $n$; Theorem~\ref{thm:thirdcumulant} below avoids them at the cost of half the rate, and Theorem~\ref{thm:thirdtree} at the full rate under a bound in tree form on the derivatives of order three of $\mathcal U$. With both inputs the differences of $\log\tilde K$ would still have to be carried to those of $\log Z^c$ through the island weights, and uniformity in the coarse field stays excluded by Theorem~\ref{thm:fixedobstruction}(i). \end{remark} The proof of~\eqref{eq:layerbound} uses one property of the layer law, a weighted lower bound on the Hessian of a smooth approximation, and this property bounds the third cumulants in tree form at half the rate of the covariances, with the walls of $D_z$ included; it gives no tree form uniform in the order. Let $E$ be a finite set with a metric $d$, and for $e\in E$ let $\mathbb V_e$ be a Euclidean space; put $\mathbb V=\prod_e \mathbb V_e$, write $z=(z_e)$, $u=(u_e)$ for its points, $\partial_e$ for the gradient in $z_e$, and $\lvert\cdot\rvert$ for the Euclidean norm. For $\varphi:E\to\R$ and $a\ge0$, $e^{a\varphi}$ is the linear map of $\mathbb V$ that multiplies $u_e$ by $e^{a\varphi(e)}$. A probability law $\mu$ on $\mathbb V$ satisfies $(\mathrm W_{\mathcal A,\alpha})$, with $\mathcal A,\alpha>0$, if: \begin{itemize}[leftmargin=*] \item[$(\mathrm W_{\mathcal A,\alpha})$] there are smooth $H_n:\mathbb V\to \R$ with bounded derivatives of every order at least two such that the laws $\mu_n$ with density proportional to $e^{-H_n}$ converge to $\mu$ in total variation, and for all $n$, $z,u\in\mathbb V$, $0\le a\le\alpha$ and $\varphi$ with $\lvert\varphi(e)-\varphi(e')\rvert\le d(e,e')$, \[ \bigl\langle u,e^{a\varphi}\operatorname{Hess}H_n(z)\,e^{-a\varphi}u \bigr\rangle\ge\mathcal A^{-1}\lvert u\rvert^2 . \] \end{itemize} In the proof of Theorem~\ref{thm:chartdecay}(ii), carried to the layer law in Step~6 of the proof of Theorem~\ref{thm:layerbound}, $H_n$ is the potential of the layer law composed with the retraction $\tau$ of that step plus $n$ times its convex penalty, $\operatorname{Hess}H_n\ge\lambda/60$ and $\lVert\operatorname{Re} (\operatorname{Hess}H_n)^w-\operatorname{Hess}H_n\rVert\le\lambda/120$ for $w=e^{\alpha\varphi}$, $\alpha\le\alpha_G$; for real $u$ the quadratic form of $w\operatorname{Hess}H_nw^{-1}$ is that of its real part. So every law $\mu_x$ of Lemma~\ref{lem:manylayer}, with $E$ the free $z$-links, $d$ the fine distance between their midpoints and the coordinates $z$, satisfies $(\mathrm W_{\mathcal A,\alpha_G})$ with $\mathcal A=120/\lambda=480/ \beta_W$, and~\eqref{eq:layerbound} is the consequence drawn from it there. The law $\mu_x$ of Proposition~\ref{prop:layerthird} satisfies $(\mathrm W_{2,\alpha})$ with $H_n=U(\cdot;x)$, by part~(i) of its proof. For bounded functions $F_1,\dots,F_k$ write $\varkappa_\mu(F_1,\dots,F_k)$ for their joint cumulant under $\mu$; for $k=3$ it is $\E_\mu[(F_1-\E_\mu F_1)(F_2-\E_\mu F_2)(F_3-\E_\mu F_3)]$. \begin{theorem}[third cumulants under the layer bound] \label{thm:thirdcumulant} Let $\mu$ satisfy $(\mathrm W_{\mathcal A,\alpha})$ and $0\le a\le\alpha$. Let $F,G,H$ be smooth bounded functions on $\mathbb V$ with bounded derivatives of orders one and two, let $S_F$ be the set of $e$ with $\partial_eF\not\equiv0$, and $S_G$, $S_H$ likewise, all nonempty; put $d_{FG}=d(S_F,S_G)$, $d_{FH}$, $d_{GH}$ likewise, $f_1=\sup\lvert\nabla F \rvert$, $g_1$, $h_1$ likewise, and $g_2=\sup\lVert\operatorname{Hess}G \rVert$, $h_2=\sup\lVert\operatorname{Hess}H\rVert$, the suprema over $\mathbb V$. Then \[ \begin{split} \lvert\varkappa_\mu(F,G,H)\rvert&\le\mathcal A^2f_1g_1h_2e^{-a(d_{FH}+ d_{GH})}\\ &\quad+\mathcal A^2f_1g_2h_1e^{-a(d_{FG}+d_{GH})}\\ &\quad+\mathcal A^{3/2}f_1g_1h_1e^{-a(d_{FG}+d_{FH})/2}. \end{split} \] In particular, with $\ell$ the sum of the two smallest of $d_{FG}$, $d_{FH}$, $d_{GH}$, which is at least the length of the shortest tree joining $S_F$, $S_G$ and $S_H$, \[ \lvert\varkappa_\mu(F,G,H)\rvert\le\bigl(\mathcal A^2f_1(g_1h_2+g_2h_1) +\mathcal A^{3/2}f_1g_1h_1\bigr)e^{-a\ell/2}. \] \end{theorem} \begin{proof} \emph{The representation.} Fix $n$, write $\E$, $\Cov$ and $\lVert\cdot \rVert$ for the mean, the covariance and the norm of $L^2(\mu_n)$, let $\mathcal L=-\Delta+\nabla H_n\cdot\nabla$ and $\mathcal M=\mathcal L \otimes1+\operatorname{Hess}H_n$, acting on $\mathbb V$-valued functions. With $\varphi=0$, $(\mathrm W_{\mathcal A,\alpha})$ gives $\operatorname {Hess}H_n\ge\mathcal A^{-1}$. For $K$ smooth and bounded with bounded derivatives, the Helffer--Sj\"ostrand theory~\cite{HS94,NS97,Hel98} gives a smooth $\phi_K$ with $\mathcal L\phi_K=K-\E K$ whose gradient $v_K= \nabla\phi_K$ lies in the domain of $\mathcal M$ and satisfies $\mathcal M v_K=\nabla K$, the derivative of $\mathcal L\phi_K$ being $\mathcal M \nabla\phi_K$; integration by parts gives $\Cov(K,K')=\E[(\mathcal L \phi_K)K']=\E\langle v_K,\nabla K'\rangle$ for every $K'\in H^1(\mu_n)$. Write $D^K_{e'e}=\partial_{e'}v_{K,e}$, the block $(e',e)$ of $\operatorname{Hess}\phi_K$, so that $D^K_{ee'}$ is the transpose of $D^K_{e'e}$ and the two have the same Hilbert--Schmidt norm $\lVert\cdot\rVert$. \emph{Two energy bounds.} Let $\varphi$ be $1$-Lipschitz, $0\le a\le \alpha$, $w=e^{a\varphi}$ and $u=wv_K$. As $w$ is constant in $z$, it commutes with $\mathcal L\otimes1$, and $\E\langle u,(\mathcal L\otimes1)u \rangle=\E\sum_{e,e'}\lVert\partial_{e'}u_e\rVert^2$. So \begin{multline*} \E\sum_{e,e'}\lVert\partial_{e'}u_e\rVert^2+\mathcal A^{-1}\lVert u\rVert^2 \le\E\langle u,w\mathcal Mw^{-1}u\rangle\\ =\E\langle u,w\nabla K\rangle\le\lVert u\rVert\lVert w\nabla K\rVert . \end{multline*} Hence $\lVert u\rVert\le\mathcal A\lVert w\nabla K\rVert$, and the first sum is at most $\max_{t\ge0}(t\lVert w\nabla K\rVert-t^2/\mathcal A)= \frac{\mathcal A}4\lVert w\nabla K\rVert^2$. In blocks, \begin{gather*} \E\sum_ee^{2a\varphi(e)}\lvert v_{K,e}\rvert^2\le\mathcal A^2\lVert e^{a\varphi}\nabla K\rVert^2,\\ \E\sum_{e,e'}e^{2a\varphi(e)}\lVert D^K_{e'e}\rVert^2\le\frac{\mathcal A}4\lVert e^{a\varphi}\nabla K\rVert^2, \end{gather*} and the second holds with $e^{2a\varphi(e')}$ in place of $e^{2a\varphi(e)}$, by the symmetry of $\lVert D^K_{e'e}\rVert$. As $2e^{a(s+t)}\le e^{2as}+ e^{2at}$, \[ \E\sum_{e,e'}e^{a(\varphi(e)+\varphi(e'))}\lVert D^K_{e'e}\rVert^2\le \frac{\mathcal A}4\lVert e^{a\varphi}\nabla K\rVert^2 . \] For $\varphi=d(\cdot,S_K)$, which vanishes on $S_K$, $\lVert e^{a\varphi} \nabla K\rVert\le k_1=\sup\lvert\nabla K\rvert$. \emph{The cumulant.} The third cumulant is $\Cov(F,GH)-\E G\Cov(F,H)-\E H \Cov(F,G)$, and the representation with $K=F$ turns it into \begin{gather*} \varkappa_{\mu_n}(F,G,H)=\Cov(G,X)+\Cov(H,Y),\\ X=\langle v_F,\nabla H\rangle,\qquad Y=\langle v_F,\nabla G\rangle . \end{gather*} By the energy bounds $v_F$ and its derivatives are in $L^2(\mu_n)$, so $X$ and $Y$ are in $H^1(\mu_n)$, and $\Cov(G,X)=\E\langle v_G,\nabla X\rangle= T_1+T_2$ with \begin{gather*} T_1=\E\sum_{e,e'}\langle v_{G,e'},(\operatorname{Hess}H)_{e'e}v_{F,e} \rangle,\\ T_2=\E\sum_{e'}\sum_{e\in S_H}\langle v_{G,e'},D^F_{e'e}\partial_eH \rangle . \end{gather*} The blocks of $\operatorname{Hess}H$ vanish unless $e,e'\in S_H$, so with $P$ the restriction to the coordinates of $S_H$, $\lvert T_1\rvert\le h_2 \lVert Pv_G\rVert\lVert Pv_F\rVert$. The first energy bound with $\varphi= d(\cdot,S_F)$, which is at least $d_{FH}$ on $S_H$, gives $\lVert Pv_F \rVert\le\mathcal Af_1e^{-ad_{FH}}$, and likewise $\lVert Pv_G\rVert\le \mathcal Ag_1e^{-ad_{GH}}$; so $\lvert T_1\rvert\le\mathcal A^2f_1g_1h_2 e^{-a(d_{FH}+d_{GH})}$. In $T_2$, $\lvert\sum_{e\in S_H}D^F_{e'e}\partial_e H\rvert\le h_1(\sum_{e\in S_H}\lVert D^F_{e'e}\rVert^2)^{1/2}$, and with $\varphi=d(\cdot,S_F)$, $\psi=d(\cdot,S_G)$ the Cauchy--Schwarz inequality gives \begin{multline*} \lvert T_2\rvert\le h_1\Bigl(\E\sum_{e'}e^{2a\psi(e')}\lvert v_{G,e'} \rvert^2\Bigr)^{1/2}\\ \times\Bigl(\E\sum_{e'}\sum_{e\in S_H}e^{-2a\psi(e')}\lVert D^F_{e'e} \rVert^2\Bigr)^{1/2}. \end{multline*} The first factor is at most $\mathcal Ag_1$. For $e\in S_H$, $\varphi(e)\ge d_{FH}$, and $2\psi(e')+\varphi(e')\ge d(e',S_G)+d(e',S_F)\ge d_{FG}$, so $e^{-2a\psi(e')}\le e^{-a(d_{FG}+d_{FH})}e^{a(\varphi(e)+\varphi(e'))}$, and the last energy bound makes the second factor at most $(\mathcal A/4)^{1/2} f_1e^{-a(d_{FG}+d_{FH})/2}$. So $\lvert T_2\rvert\le\frac12\mathcal A^{3/2}f_1g_1h_1e^{-a(d_{FG}+d_{FH})/2}$. The same argument with $G$ and $H$ exchanged bounds $\Cov(H,Y)$ by $\mathcal A^2f_1h_1g_2e^{-a(d_{FG}+ d_{GH})}+\frac12\mathcal A^{3/2}f_1g_1h_1e^{-a(d_{FH}+d_{FG})/2}$. The sum is the bound of the theorem for $\mu_n$, with constants not depending on $n$. \emph{The limit.} As $F,G,H$ are bounded and $\mu_n\to\mu$ in total variation, the means of $F,G,H,FG,FH,GH,FGH$ under $\mu_n$ converge to those under $\mu$, and so does the cumulant. Each of the three exponents is $a$ or $a/2$ times the sum of two of $d_{FG}$, $d_{FH}$, $d_{GH}$, hence at least $a\ell/2$, and the two shorter edges form a tree joining the three sets. \end{proof} \begin{corollary}[third differences of the kernel]\label{cor:kthird} In the setting of Lemma~\ref{lem:manylayer}, let $u,v,w$ be three distinct islands, $h_u,h_v,h_w$ vectors in the coordinates of $x_u,x_v,x_w$, and $\ell_{uvw}$ the sum of the two smallest of $(w_{uv}-2s-8)_+$, $(w_{uw}-2s-8)_+$, $(w_{vw}-2s-8)_+$. For a vector $h$ in the coordinates of one island let $F_h=(\partial_h\mathcal U)\circ\tau_\delta$, where $\delta>0$ is such that $\mathcal U$ is smooth on the product of the $(r+\delta)$-balls of the $z$-links times $\prod_{u'}D_{u'}$, which exists by Lemma~\ref{lem:manykernel}(ii) and the compactness of $D_z\times\prod _{u'}D_{u'}$, and $\tau_\delta$ acts link by link as a smooth radial map $\zeta_e\mapsto\rho(\lVert\zeta_e\rVert)\zeta_e/\lVert\zeta_e\rVert$ with $\rho(t)=t$ for $t\le r$, $0\le\rho'\le1$ and $\rho\le r+\delta/2$. Let $\hat\Lambda_2$, $\hat\Lambda_3$ be the suprema of $\lvert\nabla_zF_h \rvert$ and $\lVert\operatorname{Hess}_zF_h\rVert$ over all $z$, all $x\in \prod_{u'}D_{u'}$ and all $h$ with $\lVert h\rVert\le1$; they are finite, the derivatives of $\mathcal U$ being continuous on a compact set and those of $\tau_\delta$ bounded. Then \[ \partial_{h_u}\partial_{h_v}\partial_{h_w}\log\tilde K(x)=-\varkappa_{\mu_x} (\partial_{h_u}\mathcal U,\partial_{h_v}\mathcal U,\partial_{h_w}\mathcal U), \] and, with $\mathcal A=480/\beta_W$, \begin{multline*} \bigl\lvert\partial_{h_u}\partial_{h_v}\partial_{h_w}\log\tilde K(x)\bigr \rvert\\ \le\bigl(2\mathcal A^2\hat\Lambda_2^2\hat\Lambda_3+\mathcal A^{3/2} \hat\Lambda_2^3\bigr)\lVert h_u\rVert\lVert h_v\rVert\lVert h_w\rVert e^{-\alpha_G\ell_{uvw}}. \end{multline*} Consequently the mixed difference of order three of $\log\tilde K$ in $(x_u,x_v,x_w)$, between two points of each of $D_u,D_v,D_w$ with the other arguments fixed, is at most this bound with $\lVert h_j\rVert$ replaced by $2rn_j^{1/2}$, $n_j$ the number of $x$-links of island $j$. \end{corollary} \begin{proof} The domain $D_z$ does not depend on $x$ and the integrand is smooth, so $\partial_h\log\tilde K=-\E_{\mu_x}\partial_h\mathcal U$ and $\partial_h\E _{\mu_x}\varPhi=\E_{\mu_x}\partial_h\varPhi-\Cov_{\mu_x}(\varPhi, \partial_h\mathcal U)$. No term of $\mathcal U$ depends on the coordinates of two islands (Lemma~\ref{lem:manykernel}(ii)), so $\partial_{h_u} \partial_{h_v}\mathcal U$, $\partial_{h_u}\partial_{h_w}\mathcal U$ and $\partial_{h_v}\partial_{h_w}\mathcal U$ vanish; hence $\partial_{h_u} \partial_{h_v}\log\tilde K=\Cov_{\mu_x}(A,B)$ with $A=\partial_{h_u} \mathcal U$, $B=\partial_{h_v}\mathcal U$, and with $C=\partial_{h_w} \mathcal U$, \begin{multline*} \partial_{h_w}\Cov_{\mu_x}(A,B)=-\Cov(AB,C)+\E A\Cov(B,C)\\ +\E B\Cov(A,C)=-\varkappa_{\mu_x}(A,B,C). \end{multline*} The map $\tau_\delta$ is the identity on $D_z$, so the cumulant $\varkappa_{\mu_x}(A,B,C)$ equals $\varkappa_{\mu_x}(F_{h_u},F_{h_v}, F_{h_w})$; as $\tau_\delta$ acts link by link, $\nabla_zF_{h_u}$ is supported on the support $S_u$ of $\nabla_z\partial _{x_u}\mathcal U$, and the supports are at fine distance at least $d'_{uv}=2(w_{uv}-2s-8)_+$ by Lemma~\ref{lem:manykernel}(iii). Apply Theorem~\ref{thm:thirdcumulant} to $\mu_x$, which satisfies $(\mathrm W_{\mathcal A,\alpha_G})$, with $a=\alpha_G$, $f_1\le\hat \Lambda_2\lVert h_u\rVert$, $g_2\le\hat\Lambda_3\lVert h_v\rVert$ and so on; the exponents are at least $\alpha_G$ times half the sum of two of the $d'$, that is at least $\alpha_G\ell_{uvw}$. For the difference, $D_u$, $D_v$, $D_w$ are convex, two of their points differ by at most $2r$ in each link, and the difference is the integral of the third derivative over the unit cube. \end{proof} The rate in Theorem~\ref{thm:thirdcumulant} is half the rate $a$ of the covariance bound, and under $(\mathrm W_{\mathcal A,\alpha})$ alone this is optimal: the law of Proposition~\ref{prop:layerthird} satisfies $(\mathrm W_{2,\alpha})$, its three sites have $\ell=2D$, and its third cumulant is at least $\frac1{30}e^{-\alpha D}=\frac1{30}e^{-\alpha\ell/2}$. For the orders above three $(\mathrm W_{\mathcal A,\alpha})$ gives no tree form with a rate independent of the order. \begin{proposition}[the layer bound and cumulants of every order] \label{prop:layerorders} Let $k\ge3$, $\alpha>0$, $D>0$, and give $k$ sites $1,\dots,k$ the distances $d(i,j)=D$ for $i\ne j$. For $x\in\R^k$ let $\mu_x$ be the law on $\R^k$ with density proportional to $e^{-U(z;x)}$, \[ U(z;x)=\tfrac12\lvert z\rvert^2+J\prod_{j=1}^k\sin z_j-x\cdot z,\qquad J=\tfrac12e^{-k}e^{-\alpha D}. \] Then $\mu_x$ satisfies $(\mathrm W_{2,\alpha})$ for every $x$, every tree whose vertices are the $k$ sites has length $(k-1)D$, and \[ \bigl\lvert\varkappa_{\mu_0}(z_1,\dots,z_k)\bigr\rvert=\bigl\lvert \partial_{x_1}\cdots\partial_{x_k}\log K(0)\bigr\rvert\ge\tfrac13e^{-3k/2} e^{-\alpha D}, \] with $K(x)=\int e^{-U(z;x)}dz$. So a bound $C_0C^ke^{-\kappa\ell}$ on the cumulants of order $k$ of the laws satisfying $(\mathrm W_{2,\alpha})$, with $\ell$ the length of the shortest joining tree, holds for all $D$ only if $\kappa\le\alpha/(k-1)$, and no $\kappa>0$ serves every order. \end{proposition} \begin{proof} \emph{The condition.} $\operatorname{Hess}_zU=I+JM$, where $M_{ii}=-\prod _j\sin z_j$ and $M_{ij}=\cos z_i\cos z_j\prod_{l\ne i,j}\sin z_l$ for $i\ne j$, all of modulus at most one. For $\varphi$ with $\lvert\varphi(i)- \varphi(j)\rvert\le D$ and $0\le a\le\alpha$ the entries of $e^{a\varphi} Me^{-a\varphi}$ have modulus at most $e^{\alpha D}$, so its norm is at most $ke^{\alpha D}$ and $\lVert Je^{a\varphi}Me^{-a\varphi}\rVert\le\frac k2 e^{-k}\le\frac12$. Hence the quadratic form of $e^{a\varphi} \operatorname{Hess}_zUe^{-a\varphi}$ is at least $\frac12\lvert u\rvert^2$; $U$ is smooth with bounded derivatives of order at least two, and $H_n=U(\cdot;x)$ gives $(\mathrm W_{2,\alpha})$. A tree on $k$ vertices has $k-1$ edges, each of length $D$. \emph{The cumulant.} As in Proposition~\ref{prop:layerthird}(ii) the derivatives of $\log K$ at $0$ are the joint cumulants of $z$ under $\mu_0$. For $i\ne j$ the map that changes the signs of $z_i$ and $z_j$ preserves $U(\cdot;0)$. If $A$ is a nonempty proper subset of $\{1,\dots, k\}$, choose $i\in A$ and $j\notin A$; this map changes the sign of $\prod _{l\in A}z_l$, so $\E_{\mu_0}\prod_{l\in A}z_l=0$. In the expansion of the joint cumulant over the partitions of $\{1,\dots,k\}$ every partition with two or more blocks has a proper block, so $\varkappa_{\mu_0}(z_1,\dots,z_k) =\E_{\mu_0}z_1\cdots z_k$. With $\gamma_1$, $a_n$, $b_n$ as in the proof of Proposition~\ref{prop:layerthird}, \[ \E_{\mu_0}z_1\cdots z_k=\frac{\sum_n(-J)^na_n^k/n!}{\sum_n(-J)^nb_n^k/n!}, \] the expansion being legitimate since the product of the sines is bounded. There $a_n=0$ for even $n$, $b_n=0$ for odd $n$, $b_0=1$, $0\le b_n\le1$, $\lvert a_n\rvert\le1$ for $n\ge1$ and $a_1=e^{-1/2}$. So the denominator lies in $[1,\cosh J]$ and the numerator has modulus at least $Je^{-k/2}- (\sinh J-J)\ge Je^{-k/2}-\frac{J^3}6\cosh J$, the last step as in that proof. Here $J\le\frac12e^{-3}<0.025$, so $\cosh J<1.001$, and $\frac{J^2}6 \cosh J\le\frac{1.001}{24}e^{-2k}\le0.0418e^{-4}e^{-k/2}<0.001e^{-k/2}$, as $e^{-2k}\le e^{-4}e^{-k/2}$ for $k\ge3$ and $e^4>2.7^4>53$. Hence \[ \lvert\E_{\mu_0}z_1\cdots z_k\rvert\ge\frac{0.999}{1.001}Je^{-k/2}>0.99\, Je^{-k/2}>\tfrac13e^{-3k/2}e^{-\alpha D}. \] For the last assertion, the tree bound with $\ell=(k-1)D$ and this lower bound give $\frac13e^{-3k/2}\le C_0C^ke^{-(\kappa(k-1)-\alpha)D}$ for every $D>0$, which forces $\kappa(k-1)\le\alpha$. \end{proof} Under $(\mathrm W_{\mathcal A,\alpha})$ alone the half rate of Theorem~\ref{thm:thirdcumulant} is optimal. It is lost in the off-diagonal blocks of $\operatorname{Hess}\phi_F$, and these decay in both indices at the full rate when, in addition, the derivatives of order three of the potential decay along trees. The walls of a product of balls do not interfere: a penalty for walls of one link at a time is a sum of functions of one link each, so its derivatives of order three lie in the diagonal blocks, which the energy bound of that proof already controls. Keep the notation $E$, $d$, $\mathbb V_e$, $\mathbb V$ of $(\mathrm W_{\mathcal A,\alpha})$, and for $\mathcal A>0$ and $a\ge0$ consider: \begin{itemize}[leftmargin=*] \item[$(\mathrm W^+_{\mathcal A,a})$] there are $r_e>0$ such that $\mu$ has density proportional to $e^{-U}\mathbf 1_\Omega$, where $\Omega=\prod_e\{ \lvert z_e\rvert\le r_e\}$ and $U$ is smooth on a neighbourhood of $\Omega$, and $\langle u,e^{a\varphi}\operatorname{Hess}U(z)\, e^{-a\varphi}u\rangle\ge\mathcal A^{-1}\lvert u\rvert^2$ for all $z\in \Omega$, $u\in\mathbb V$ and $\varphi$ with $\lvert\varphi(e)-\varphi(e') \rvert\le d(e,e')$. \end{itemize} For such $U$ let \[ \Xi_a(U)=\sup_{z\in\Omega}\max_{e\in E}\Bigl(\sum_{e'\ne e}e^{2ad(e,e')} \bigl\lVert(\operatorname{Hess}U(z))_{e'e}\bigr\rVert^2\Bigr)^{1/2}, \] with operator norms of the blocks. For $z\in\Omega$ and $e',e,e''\in E$ let $\lVert D^3U(z)_{e'ee''}\rVert$ be the largest Hilbert--Schmidt norm of the bilinear form $(b,c)\mapsto D^3U(z)[b,c,y]$ on $\mathbb V_{e'}\times \mathbb V_e$ over $y\in\mathbb V_{e''}$ with $\lvert y\rvert\le1$, and let $\Theta_a(U)$ be the supremum over $z\in\Omega$ of the norm on $\ell^2$ of the matrix with rows indexed by the pairs $(e',e)$ with $e'\ne e$, columns indexed by $e''\in E$, and entries $e^{a(d(e',e'')+d(e,e''))}\lVert D^3U(z)_{e'ee''}\rVert$. For smooth $F$ on $\mathbb V$ put \[ f_2^{(a)}=\sup_{z\in\mathbb V}\Bigl(\sum_{e'\ne e}e^{2ad(e,e')}\bigl\lVert (\operatorname{Hess}F(z))_{e'e}\bigr\rVert^2\Bigr)^{1/2}, \] with Hilbert--Schmidt norms. For $\emptyset\ne S\subset E$ let \[ T_S(e',e)=\min_{p\in E}\bigl(d(e',p)+d(e,p)+d(p,S)\bigr), \] the least length of a star with centre in $E$ joining $e'$, $e$ and $S$. For fixed $e$ it is a $1$-Lipschitz function of $e'$, and conversely, as a minimum of such functions; and $T_S(e,e)=d(e,S)$, by the triangle inequality and $p=e$. Norms of blocks of matrix-valued functions are Hilbert--Schmidt norms unless said otherwise. \begin{lemma}[two-index decay]\label{lem:twoindex} Let $\mu$ satisfy $(\mathrm W^+_{\mathcal A,a})$, let $\epsilon>0$ and $\mathcal A_\epsilon=(1+\epsilon)\mathcal A$. There are $n_0$ and smooth $H_n:\mathbb V\to\R$, $n\ge n_0$, with bounded derivatives of every order at least two, such that the laws $\mu_n$ with density proportional to $e^{-H_n}$ converge to $\mu$ in total variation, \[ \bigl\langle u,e^{a\varphi}\operatorname{Hess}H_n(z)\,e^{-a\varphi}u \bigr\rangle\ge\mathcal A_\epsilon^{-1}\lvert u\rvert^2 \] for all $z,u\in\mathbb V$ and $1$-Lipschitz $\varphi$, and the following holds. Let $F$ be smooth and bounded with bounded derivatives of every order, $S_F$ and $f_1$ as in Theorem~\ref{thm:thirdcumulant}, $\phi_F$ the solution of $\mathcal L\phi_F=F-\E F$ for $\mu_n$ of the proof of that theorem, $D^F=\operatorname{Hess}\phi_F$, and $\E$ the mean under $\mu_n$. Then \begin{gather*} \Bigl(\E\sum_ee^{2ad(e,S_F)}\lVert D^F_{ee}\rVert^2\Bigr)^{1/2}\le\Bigl( \frac{\mathcal A_\epsilon}4\Bigr)^{1/2}f_1,\\ \begin{split} &\Bigl(\E\sum_{e'\ne e}e^{2aT_{S_F}(e',e)}\lVert D^F_{e'e}\rVert^2 \Bigr)^{1/2}\le\frac{\mathcal A_\epsilon}2c_\epsilon+\eta_nf_1,\\ &c_\epsilon=f_2^{(a)}+\bigl(\mathcal A_\epsilon^{1/2}(\Xi_a(U)+ \epsilon)+\mathcal A_\epsilon(\Theta_a(U)+\epsilon)\bigr)f_1, \end{split} \end{gather*} where $\eta_n$ does not depend on $F$ and $\eta_n\to0$ as $n\to\infty$. \end{lemma} \begin{proof} \emph{The approximation.} Put $\Omega_t=\prod_e\{\lvert z_e\rvert\le r_e+ t\}$ and fix $e_0\in E$. The quadratic form in $(\mathrm W^+_{\mathcal A,a})$ does not change when a constant is added to $\varphi$, and the $1$-Lipschitz $\varphi$ with $\varphi(e_0)=0$ form a compact set; so its minimum $m(z)$ over these $\varphi$ and over $\lvert u\rvert=1$ is continuous in $z$ where $U$ is smooth, and so are the maximum and the matrix norm whose suprema over $\Omega$ define $\Xi_a(U)$ and $\Theta_a(U)$. As $\Omega$ is compact, there is $\delta>0$ such that $U$ is smooth near $\Omega_\delta$, $m\ge\mathcal A_\epsilon^{-1}$ on $\Omega_\delta$, and the two functions are at most $\Xi_a(U)+\epsilon$ and $\Theta_a(U)+\epsilon$ on $\Omega_\delta$. Let $\tau$ act link by link as $\tau_e(\zeta)=\rho_e(\lvert\zeta\rvert)\zeta/\lvert\zeta\rvert$ with $\rho_e$ smooth, $\rho_e(t)=t$ for $t\le r_e+\delta/2$, $0\le\rho_e'\le1$, $\rho_e\le r_e+\delta$ and $\rho_e$ constant on $[r_e+\delta,\infty)$; let $v_e$ be smooth and convex on $[0,\infty)$, zero on $[0,r_e]$, with $v_e''>0$ on $(r_e,\infty)$ and $v_e''=1$ on $[r_e+\delta/4,\infty)$; and put $V(z)=\sum_ev_e(\lvert z_e\rvert)$ and $H_n=U\circ\tau+nV$. Then $\tau$ maps $\mathbb V$ into $\Omega_\delta$ and is the identity on $\Omega_{ \delta/2}$, and $H_n$ is smooth with bounded derivatives of every order at least two. Moreover \[ \operatorname{Hess}H_n=D\tau^*\bigl(\operatorname{Hess}U\circ\tau\bigr) D\tau+B_0+n\operatorname{Hess}V, \] where $D\tau$, $B_0$ and $\operatorname{Hess}V$ are block diagonal, the block $e$ of $B_0$ is $\langle\nabla_eU(\tau(z)),D^2\tau_e(z_e)\rangle$, of norm at most a constant $C_1$, and at a link with $\lvert z_e\rvertR$ for some $p,q$; for $i\ge3$ put \[ \Lambda_i(H)=\sup_z\max_{e_1\in E}\sum_{e_2,\dots,e_i\in E}\bigl\lvert \partial_{e_1}\cdots\partial_{e_i}H(z)\bigr\rvert . \] \begin{theorem}[cumulants of every order under finite range] \label{thm:treeorders} Let $\mathcal A,\alpha,R>0$ and $k\ge2$. Let $H$ be smooth on $\mathbb V$, with bounded derivatives of every order at least two and range $R$, such that $(\mathrm W_{\mathcal A,\alpha})$ holds with $H_n=H$ for all $n$, and let $\mu$ have density proportional to $e^{-H}$. Let $0\le a\le\alpha$, let $F_1,\dots,F_k$ be smooth bounded functions with bounded derivatives of every order, $S_i$ the set of $e$ with $\partial_eF_i\not\equiv0$, all nonempty, $\ell=\ell(S_1,\dots,S_k)$, \[ f=\max_{i\le k}\,\max_{1\le j\le k-1}\,\sup_z\lvert D^jF_i(z)\rvert, \qquad\Lambda=\max_{2\le i\le k-1}e^{aiR}\Lambda_{i+1}(H), \] with $\Lambda=0$ for $k=2$. Then \[ \lvert\varkappa_\mu(F_1,\dots,F_k)\rvert\le K_kf^ke^{-a\ell}, \] \[ K_k=\bigl((k-1)!\bigr)^2e^{k-2}\mathcal A^{k-1}(1+\mathcal A\Lambda)^{ (k-1)(k-2)/2}. \] In particular $K_2=\mathcal A$ and $K_3=4e\mathcal A^2(1+\mathcal A \Lambda)$. \end{theorem} \begin{proof} \emph{Helffer--Sj\"ostrand solutions.} Let $H'$ be smooth with bounded derivatives of order at least two and $\operatorname{Hess}H'\ge \mathcal A'^{-1}$, $\mu'$ its law, $\mathcal L'=-\Delta+\nabla H'\cdot \nabla$, and $G$ smooth with bounded derivatives of every order. Let $X^z_t$ solve $dX=-\nabla H'(X)\,dt+\sqrt2\,dB$, $X_0=z$, and $P_tG(z)= \E G(X^z_t)$. The derivative $J_t$ of $X^z_t$ in $z$ solves $\dot J= -\operatorname{Hess}H'(X_t)J$, so $\lvert J_t\rvert\le e^{-t/\mathcal A'}$; the derivative of order $j\ge2$ solves the same equation with a source made of products of at least two derivatives of lower order and of derivatives of $H'$, so by induction and the Duhamel formula it is at most $C_je^{-t/\mathcal A'}$. Hence every derivative of $P_tG$ of order $j\ge1$ is at most $C'_je^{-t/\mathcal A'}$, and $\lvert P_tG(z)- \E_{\mu'}G\rvert\le\sup\lvert\nabla G\rvert e^{-t/\mathcal A'}\E_{\mu'} \lvert z-z'\rvert$, by the coupling of $X^z$ and $X^{z'}$ with the same noise and the invariance of $\mu'$. So $\phi_G=\int_0^\infty(P_tG- \E_{\mu'}G)\,dt$ is smooth, its derivatives of every order $j\ge1$ are bounded by constants depending only on $\mathcal A'$ and on the bounds of the derivatives of $H'$ and $G$, and $\mathcal L'\phi_G=G-\E_{\mu'}G$; this is the solution of~\cite{HS94,NS97,Hel98}. Integration by parts gives $\Cov_{\mu'}(G,F)=\E_{\mu'}\langle\nabla\phi_G,\nabla F\rangle$. For smooth $F$ with bounded derivatives write $\Gamma'_FG=\langle\nabla \phi_G,\nabla F\rangle$, again smooth and bounded with bounded derivatives. \emph{The representation.} We show \begin{equation}\label{eq:treewords} \varkappa_{\mu'}(F_1,\dots,F_k)=\sum_\sigma\E_{\mu'}\bigl[\Gamma'_{F_{ \sigma(k)}}\cdots\Gamma'_{F_{\sigma(2)}}F_1\bigr], \end{equation} the sum over the bijections $\sigma$ of $\{2,\dots,k\}$, for every $H'$ as above, by induction on $k$; for $k=2$ it is the covariance formula. Let $k\ge3$, $\varepsilon_0=(2\mathcal A'\sup\lVert\operatorname{Hess} F_k\rVert)^{-1}$, and for $\lvert\varepsilon\rvert<\varepsilon_0$ let $\mu^\varepsilon$ have density proportional to $e^{-H'+\varepsilon F_k}$, with $\phi^\varepsilon$, $\Gamma^\varepsilon$, $\E_\varepsilon$ its objects; $\operatorname{Hess}(H'-\varepsilon F_k)\ge(2\mathcal A')^{-1}$, and the bounds above hold uniformly in $\varepsilon$. The joint cumulant is a mixed derivative of $\log\E e^{\sum_it_iF_i}$ at $t=0$, so $\varkappa_{\mu'}(F_1,\dots,F_k)$ is the derivative at $\varepsilon=0$ of $\varkappa_{\mu^\varepsilon}(F_1,\dots,F_{k-1})$, which by induction is a sum of $\E_\varepsilon$ of words in the $\Gamma^\varepsilon$. For $X$ smooth with bounded derivatives, $\mathcal L_{\varepsilon'}=\mathcal L_\varepsilon-(\varepsilon'-\varepsilon)\nabla F_k\cdot\nabla$ gives \[ \mathcal L_{\varepsilon'}(\phi^{\varepsilon'}_X-\phi^\varepsilon_X)= (\varepsilon'-\varepsilon)\Gamma^\varepsilon_{F_k}X-(\E_{\varepsilon'}X- \E_\varepsilon X), \] whose left side has mean zero under $\mu^{\varepsilon'}$; so $\nabla \phi^{\varepsilon'}_X-\nabla\phi^\varepsilon_X=(\varepsilon'-\varepsilon) \nabla\phi^{\varepsilon'}_{\Gamma^\varepsilon_{F_k}X}$. Applied twice, with the uniform bounds, this gives $\partial_\varepsilon\nabla\phi^\varepsilon _X=\nabla\phi^\varepsilon_{\Gamma^\varepsilon_{F_k}X}$, uniformly with all derivatives in $z$, and by linearity in $X$, for $X_\varepsilon$ smooth in $\varepsilon$, \[ \partial_\varepsilon\bigl(\Gamma^\varepsilon_FX_\varepsilon\bigr)= \Gamma^\varepsilon_F\partial_\varepsilon X_\varepsilon+\Gamma^\varepsilon _F\Gamma^\varepsilon_{F_k}X_\varepsilon . \] With $\partial_\varepsilon\E_\varepsilon X_\varepsilon=\E_\varepsilon \partial_\varepsilon X_\varepsilon+\Cov_\varepsilon(X_\varepsilon,F_k)= \E_\varepsilon[\partial_\varepsilon X_\varepsilon+\Gamma^\varepsilon_{F_k} X_\varepsilon]$ and $\partial_\varepsilon F_1=0$, the derivative of a word with $k-2$ letters is the sum of the $k-1$ words obtained by inserting $\Gamma_{F_k}$ at each of its $k-1$ places, the outermost included, and each bijection of $\{2,\dots,k\}$ arises once. This is~\eqref{eq:treewords}. \emph{The weights.} For a family $\mathcal S=(S_1,\dots,S_m)$ of nonempty subsets of $E$ let $d_{\mathcal S}(e,e')$ be the least value of $\sum_i c(y_{i-1},y_i)$ over the chains $e=y_0,\dots,y_n=e'$, with $c(y,y')=0$ when $y,y'$ lie in a common $S_l$ and $c(y,y')=d(y,y')$ otherwise; it is a pseudometric and $d_{\mathcal S}\le d$. For $j\ge0$ and $e_1,\dots,e_j\in E$ let $\psi_{\mathcal S}(e_1,\dots,e_j)$ be the least $d_{\mathcal S}$-length of a tree with vertices in $E$ that contains $e_1,\dots,e_j$ and a point of each $S_l$. Then: \begin{enumerate}[label=(\alph*),leftmargin=*] \item $\psi_{\mathcal S}$ is symmetric, and $1$-Lipschitz for $d$ in each argument, the others fixed: add an edge; \item $\psi_{(S_1)}(e_1,\dots,e_j)=0$ when every $e_i\in S_1$; \item $\psi_{\mathcal S}(e_1,\dots,e_j)\le\psi_{\mathcal S}(g,e_{i+1}, \dots,e_j)+\sum_{p\le i}d(e_p,g)$: add the edges from $g$ to $e_p$; \item if $\mathcal S'$ is $\mathcal S$ with a set $S$ added, $g\in S$ and $e_{i+1},\dots,e_j\in S$, then $\psi_{\mathcal S'}(e_1,\dots,e_j)\le \psi_{\mathcal S}(e_1,\dots,e_i,g)$: the optimal tree for the right side meets $S$ at $g$, the edges from $g$ to $e_{i+1},\dots,e_j$ have $d_{\mathcal S'}$-length zero, and $d_{\mathcal S'}\le d_{\mathcal S}$; \item every tree with vertices in $E$ that meets each $S_l$ has $d_{\mathcal S}$-length at least $\ell(S_1,\dots,S_m)$. \end{enumerate} For (e) replace each edge by a chain that realises $d_{\mathcal S}$, each step of cost zero inside $S_l$ by two edges of length zero through the vertex $S_l$, and join every $S_l$ to a point of the tree in $S_l$ by an edge of length zero: the result is a connected graph on $S_1,\dots,S_m$ and points of $E$ whose edges are at least as long as the distances of their ends, and one of its spanning trees is admissible for $\ell$. \emph{Energy bounds of every order.} From now on $\E$, $\Cov$ and $\lVert\cdot\rVert$ refer to $\mu$, $\phi_G$ and $\Gamma_F$ to $H'=H$, and $\lVert u\rVert^2=\E\lvert u\rvert ^2$ for a tensor valued $u$. Let $\mathcal S$ be a family as above, $G$ smooth and bounded with bounded derivatives, $\phi=\phi_G$, and for $j\ge1$ let $w_j$ multiply the block $(e_1,\dots,e_j)$ by $e^{a\psi_{\mathcal S} (e_1,\dots,e_j)}$, $y_j=\lVert w_jD^jG\rVert$ and $c_j=\lVert w_jD^j\phi \rVert$. We show $c_1\le\mathcal Ay_1$ and, for $2\le j\le k-1$, \begin{equation}\label{eq:treeenergy} c_j\le\frac{\mathcal A}j\Bigl(y_j+\sum_{i=2}^j\binom jie^{aiR} \Lambda_{i+1}(H)\,c_{j-i+1}\Bigr). \end{equation} Differentiating $\mathcal L\phi=G-\E G$ in $z_{e_1},\dots,z_{e_j}$ and using the Leibniz rule on $\langle\nabla H,\nabla\phi\rangle$, with $u=D^j\phi$, \[ \mathcal Lu+\sum_{p=1}^j\operatorname{Hess}_{(p)}u=D^jG-\sum_{\lvert P \rvert\ge2}R_P, \] where $\operatorname{Hess}_{(p)}$ applies $\operatorname{Hess}H$ in the $p$-th index, $P$ runs over the subsets of $\{1,\dots,j\}$ with at least two elements, and $(R_P)_{e_1\dots e_j}=\sum_g\partial_{e_P}\partial_gH\, \partial_{e_{P^c}}\partial_g\phi$, the contraction being in $\mathbb V_g$ and $e_P$ the indices in $P$. Put $\tilde u=w_ju$, which is bounded since $E$ is finite. The weight is constant in $z$, so $\E\langle\tilde u,w_j \mathcal Lu\rangle=\E\langle\tilde u,\mathcal L\tilde u\rangle\ge0$. For fixed indices other than the $p$-th, $\psi_{\mathcal S}$ is a $1$-Lipschitz function of the $p$-th by (a), so $(\mathrm W_{\mathcal A,\alpha})$, applied to each component in an orthonormal basis of the other factors, gives $\langle\tilde u,w_j\operatorname{Hess}_{(p)}w_j^{-1}\tilde u\rangle\ge \mathcal A^{-1}\lvert\tilde u\rvert^2$. Hence \[ \frac j{\mathcal A}\lVert\tilde u\rVert^2\le\lVert\tilde u\rVert\Bigl( \lVert w_jD^jG\rVert+\sum_{\lvert P\rvert\ge2}\lVert w_jR_P\rVert\Bigr). \] Let $\lvert P\rvert=i$. The block $\partial_{e_P}\partial_gH$ vanishes unless $d(e_p,g)\le R$ for $p\in P$, and then (c) gives $\psi_{\mathcal S} (e_1,\dots,e_j)\le\psi_{\mathcal S}(e_{P^c},g)+iR$. With $u'=D^{j-i+1}\phi$ and $w'=w_{j-i+1}$, \[ e^{a\psi_{\mathcal S}(e)}\lvert(R_P)_e\rvert\le e^{aiR}\sum_gM_{e_P,g} \bigl\lvert(w'u')_{e_{P^c},g}\bigr\rvert, \] with $M_{e_P,g}=\lvert\partial_{e_P}\partial_gH\rvert$. By the symmetry of the blocks, the sums of $M$ over $g$ for fixed $e_P$ and over $e_P$ for fixed $g$ are at most $\Lambda_{i+1}(H)$, so by Schur's test the matrix $M$ has norm at most $\Lambda_{i+1}(H)$ on $\ell^2$, and summing the squares over $e_{P^c}$ gives $\lVert w_jR_P\rVert\le e^{aiR}\Lambda_{i+1} (H)c_{j-i+1}$. There are $\binom ji$ sets $P$ of size $i$, and~\eqref{eq:treeenergy} follows; for $j=1$ there is no $R_P$. \emph{Products.} Let $\mathcal S'$ be $\mathcal S$ with $S=\{e:\partial_e F\not\equiv0\}$ added, $F$ smooth and bounded with bounded derivatives, $Y=\Gamma_FG=\sum_{g\in S}\langle\partial_g\phi,\partial_gF\rangle$, and $w'_j$, $y'_j=\lVert w'_jD^jY\rVert$ the objects of $\mathcal S'$. Then \[ \partial_{e_1}\cdots\partial_{e_j}Y=\sum_{B}\sum_{g\in S}\bigl\langle \partial_{e_B}\partial_g\phi,\partial_{e_{B^c}}\partial_gF\bigr\rangle, \] the sum over $B\subset\{1,\dots,j\}$. The term of $B$ vanishes unless $e_{B^c}$ lie in $S$, and then (d) gives $\psi_{\mathcal S'}(e_1,\dots, e_j)\le\psi_{\mathcal S}(e_B,g)$. By the Cauchy--Schwarz inequality in $g$ and the sum over $e_B$ and $e_{B^c}$, the term of $B$ has weighted norm at most $\sup\lvert D^{j-\lvert B\rvert+1}F\rvert\,c_{\lvert B\rvert+1}$, so \[ y'_j\le\sum_{b=0}^j\binom jb\sup\lvert D^{j-b+1}F\rvert\,c_{b+1}. \] \emph{Assembly.} Fix $\sigma$, put $Y_1=F_1$ and $Y_m=\Gamma_{F_{\sigma (m)}}Y_{m-1}$, and let $\mathcal S_m=(S_1,S_{\sigma(2)},\dots,S_{\sigma (m)})$; the term of $\sigma$ in~\eqref{eq:treewords} is $\E Y_k$. Write $\hat y_{m,j}$ for $y_j/j!$ of $G=Y_m$ and $\mathcal S_m$, $\hat Y_{m,j}= \max_{i\le j}\hat y_{m,i}$, and $\hat c_j=c_j/j!$. Dividing \eqref{eq:treeenergy} by $j!$, with $\frac{j-i+1}j\le1$ and $\sum_{i\ge2} \frac1{i!}=e-2<1$, \[ \hat c_j\le\mathcal A\hat y_j+\mathcal A\Lambda\sum_{i=2}^j\frac{\hat c_{j-i+1}}{i!}, \] and induction on $j$ gives $\hat c_j\le\mathcal A\hat Y_j(1+\mathcal A \Lambda)^{j-1}$ for $j\le k-1$, with $\hat Y_j=\max_{i\le j}\hat y_i$. The product bound divided by $j!$ gives $\hat y_{m,j}\le f\sum_{b\le j}\frac{b+1}{(j-b)!}\hat c_{b+1}$, with $\hat c$ for $G=Y_{m-1}$ and $\mathcal S_{m-1}$, so \[ \hat Y_{m,j}\le e(j+1)f\mathcal A(1+\mathcal A\Lambda)^j\hat Y_{m-1,j+1} \] whenever $j+1\le k-m+1$. By (b), $\hat Y_{1,k-1}\le f$. Iterating from $m=k-1$, $j=1$ down to $m=2$, $j=k-2$, \[ \hat Y_{k-1,1}\le e^{k-2}(k-1)!\,f^{k-1}\mathcal A^{k-2}(1+\mathcal A \Lambda)^{(k-1)(k-2)/2}, \] since $\prod_{m=2}^{k-1}(k-m+1)=(k-1)!$ and $\sum_{m=2}^{k-1}(k-m)= \frac{(k-1)(k-2)}2$. Finally $\E Y_k=\sum_{g\in S_{\sigma(k)}}\E\langle \partial_g\phi,\partial_gF_{\sigma(k)}\rangle$ with $\phi=\phi_{Y_{k-1}}$, and for $g\in S_{\sigma(k)}$ a tree admissible for $\psi_{\mathcal S_{k-1}} (g)$ meets all $k$ sets, so by (e) and $d_{\mathcal S_{k-1}}\ge d_{\mathcal S_k}$, $\psi_{\mathcal S_{k-1}}(g)\ge\ell$. The Cauchy--Schwarz inequality gives \[ \lvert\E Y_k\rvert\le e^{-a\ell}f\,c_1\le e^{-a\ell}f\mathcal A\hat Y_{k-1,1}. \] Summing over the $(k-1)!$ bijections gives the bound. \end{proof} The rate is the rate $a$ of the covariance bound, for every order; for $k=3$ it is the rate of Theorem~\ref{thm:thirdtree}, here without walls. The case $j=2$ of~\eqref{eq:treeenergy} is a two-index bound on $\operatorname{Hess}\phi_F$ of the kind asked for in Remark~\ref{rem:treeformstatus}, with the weight $e^{a\psi(e,e')}$, $\psi(e,e')=\psi_{(S_F)}(e,e')$, in place of $e^{a(d(e,e')+d(e,S_F))}$. Finite range enters only through (c): the source $R_P$ of the equation for $D^j\phi$ is carried by indices within $R$ of a common point. In the law of Proposition~\ref{prop:layerorders} the range is $D$, the factor $e^{aiR}$ in $\Lambda$ is $e^{aiD}$, and the bound of the theorem is consistent with that proposition. \begin{remark}\label{rem:treeorderswalls} Theorem~\ref{thm:treeorders} applies to the smooth approximations of the layer law and not to the layer law. Each term of $\mathcal U$ involves links based in blocks at mutual coarse distance at most three, by Steps~1 and~2 of the proof of Theorem~\ref{thm:factorisation}; the retraction $\tau$ and the penalty $V=\sum_ev(\lVert\zeta_e\rVert)$ of Theorem~\ref{thm:chartdecay}(ii) act link by link. So the potential $H_{x,n}$ of the approximation $\mu_{x,n}$ of $\mu_x$ in Step~6 of the proof of Theorem~\ref{thm:layerbound} has a finite range $R_0$ fixed by the geometry, bounded derivatives of every order, and $(\mathrm W_{480/\beta_W,\alpha_G})$. For distinct islands $u_1,\dots,u_k$, unit vectors $h_p$ in the coordinates of $x_{u_p}$ and $\tilde K_n(x)=\int e^{-H_{x,n}(z)}dz$, no term of $\mathcal U$ depends on two islands, so as in Corollary~\ref{cor:kthird} \[ \partial_{h_1}\cdots\partial_{h_k}\log\tilde K_n=(-1)^k\varkappa_{\mu_{x,n}} (F_{h_1},\dots,F_{h_k}), \] and the theorem bounds it by $K_k(n)f^ke^{-\alpha_G\ell}$, with $\ell$ the tree length of the supports $S_{u_p}$ of $\nabla_z\partial_{x_{u_p}} \mathcal U$ in the fine distance; by Lemma~\ref{lem:manykernel}(iii) any two of these supports are at fine distance at least $d'_{u_pu_q}$, so for $k=2$ the factor is $e^{-2\alpha_G(w_{u_1u_2}-2s-8)_+}$, the rate of Lemma~\ref{lem:manylayer}. As $n\to\infty$, $\tilde K_n$ decreases to $\tilde K$, so the mixed differences of $\log\tilde K$ are limits of those of $\log\tilde K_n$, and a bound on $K_k(n)$ uniform in $n$ would give the tree form of every order of the differences of $\log\tilde K$ at the rate $\alpha_G$ in the fine distance, the first input named in Remark~\ref{rem:treeform}. The constant depends on $n$ through $\Lambda$ only, and $\Lambda_3(H_{x,n})$ grows linearly in $n$, the penalty $nV$ contributing $n$ times the third derivatives of $v(\lVert\zeta_e\rVert)$, which do not vanish identically. The layer law itself has no smooth potential on $\mathbb V$: its density vanishes off $D_z$, and the walls of $D_z$ are the one hypothesis of Theorem~\ref{thm:treeorders} that it does not meet. For $k=3$ the walls are handled by Lemma~\ref{lem:twoindex}, which needs the energy bound only for $j=1$ and keeps the penalty in the diagonal blocks. Whether the differences of order $k\ge4$ of $\log\tilde K$ have a tree form with a rate independent of $k$ is not decided here; a proof along these lines needs the energy bounds~\eqref{eq:treeenergy} for $j\ge2$ on the box $D_z$, with the boundary of the balls in place of the penalty. \end{remark} The second step, from $\log\tilde K$ to the weights of the labels, integrates the coordinates of the other islands against $\tilde K$. Its derivatives in the island coordinates are given by the cumulants of that gas. In the setting of Theorem~\ref{thm:manyislands} let $\mathcal K \subset\mathcal I$ be a set of $k\ge2$ islands, $\mathcal Q'=\mathcal I \setminus\mathcal K$, suppose every $\gamma_q\ne0$, $q\in\mathcal Q'$, and put \[ Z_{\mathcal K}(x_{\mathcal K})=\int\tilde K(x_{\mathcal K},x_{\mathcal Q'}) \prod_{q\in\mathcal Q'}\gamma_q(dx_q), \] with $P_{x_{\mathcal K}}$ the law on $\prod_{q\in\mathcal Q'}D_q$ proportional to $\tilde K(x_{\mathcal K},\cdot\,)\prod_q\gamma_q$. \begin{lemma}[derivatives through the gas of islands]\label{lem:islandgas} For vectors $h_u$ in the coordinates of $x_u$, $u\in\mathcal K$, and $B\subset\mathcal K$, let $g_B=\bigl(\prod_{u\in B}\partial_{h_u}\bigr) \log\tilde K$. Then \[ \prod_{u\in\mathcal K}\partial_{h_u}\log Z_{\mathcal K}=\sum_\pi \varkappa_{P_{x_{\mathcal K}}}\bigl((g_B)_{B\in\pi}\bigr), \] the sum over the partitions $\pi$ of $\mathcal K$, the cumulant having one argument for each block of $\pi$. \end{lemma} \begin{proof} $\tilde K$ is smooth and positive on a neighbourhood of the compact set $\prod_uD_u$, so for smooth $G_1,\dots,G_m$ the cumulant $\varkappa_P(G_1, \dots,G_m)$ is the mixed derivative at $s=0$ of \[ \log\int e^{\sum_is_iG_i}\tilde K\prod_q\gamma_q(dx_q), \] and differentiating under the integral, \begin{multline*} \partial_h\varkappa_P(G_1,\dots,G_m)=\sum_i\varkappa_P(G_1,\dots, \partial_hG_i,\dots,G_m)\\ +\varkappa_P(G_1,\dots,G_m,\partial_h\log\tilde K). \end{multline*} For $u\notin B$, $\partial_{h_u}g_B=g_{B\cup\{u\}}$. Order $\mathcal K=\{u_1,\dots,u_k\}$; $\partial_{h_{u_1}}\log Z_{\mathcal K}=\E_P\, g_{\{u_1\}}$, and if the derivative in $u_1,\dots,u_m$ is the sum over the partitions of $\{u_1,\dots,u_m\}$, the rule adds $u_{m+1}$ to one block or opens the block $\{u_{m+1}\}$, which gives each partition of $\{u_1,\dots, u_{m+1}\}$ once. \end{proof} For $k=2$ the identity is the formula in the proof of Theorem~\ref{thm:manyislands}. The partition $\pi=\{\mathcal K\}$ gives the mean of a mixed derivative of $\log\tilde K$, to which a tree form for $\log\tilde K$ applies directly. The partitions into two blocks give covariances under $P$, which Lemma~\ref{lem:islandcov} bounds through the oscillations $\delta_q(g_B)$, themselves mixed differences of $\log\tilde K$ of order $\lvert B\rvert+1$, under the weighted Dobrushin condition of Corollary~\ref{cor:islandchains}. The partitions into three or more blocks are joint cumulants of order at least three of the gas of islands, and no result of this paper bounds them; Lemma~\ref{lem:islandcov} concerns covariances only. A rate that does not depend on the strength of the pair interactions of the gas is excluded already for the covariances. \begin{proposition}[a chain of strongly coupled islands] \label{prop:islandchain} Let $N\ge1$, $J>0$, $X_q=\{-1,1\}$ with $\gamma_q$ the counting measure for $0\le q\le N$, and $k(\xi)=\exp(J\sum_{q0$, the mixed differences of $\log k$ of every order obey the tree bound $4Je^{\lambda t}e^{-\lambda\ell}$. So a tree form for $\log\tilde K$, at any rate and of every order, does not carry over to $\log Z^c$ at a rate independent of the configuration of the islands: the passage needs a condition that bounds the pair cross-ratios against the gaps, such as the weighted Dobrushin condition of Corollary~\ref{cor:islandchains}, and, by Lemma~\ref{lem:islandgas}, bounds on the joint cumulants of order up to $k$ of the gas of islands under such a condition. \begin{remark}\label{rem:treeordersstatus} For item~(ii) of Sec.~\ref{ssec:remains} the results give the following. For a law with smooth potential of finite range and the weighted Hessian bound, the cumulants of every order obey the tree bound at the rate of the covariances (Theorem~\ref{thm:treeorders}); this answers, in that class, the question left by Proposition~\ref{prop:layerorders}, which shows that the finite range is needed. For the layer law the order three holds at the full rate with the walls (Corollary~\ref{cor:kthirdtree}), and the tree form of $\log\tilde K$ of order $k\ge4$ is proved only for the smooth approximations, with constants that grow with the approximation (Remark~\ref{rem:treeorderswalls}). The passage to $\log Z^c$ reduces, by Lemma~\ref{lem:islandgas}, to the decay of the joint cumulants of the gas of islands; their order two is given by Lemma~\ref{lem:islandcov} under the weighted Dobrushin condition, their orders three and more are not proved, and Proposition~\ref{prop:islandchain} shows that no rate uniform in the pair cross-ratios of the islands holds. Uniformity in the coarse field stays excluded by Theorem~\ref{thm:fixedobstruction}(i). The tree form of the differences of $\log Z^c$, and with it the rate $2\alpha_G$ without a floor, is therefore not proved for any coarse field. \end{remark} For a fixed coarse field $c$ the blocks that are bad for every labelling are the corners of the coarse plaquettes with $\delta(c_P,I)>\eta$. On them no label restricts the active links, and $\mu^c$ there is the constrained law of Sec.~\ref{ssec:smallfibres}, with the links outside as exterior. The rest of this subsection determines what the chart estimates of that section give for this law, in the $\SU(2)$ model and not in the class $(\mathrm X)+(\mathrm R)$: a component on which the coarse field stays below the scale of the chart is crossed at the rate $\alpha_G$, with no cost in its length beyond the mass of the constrained law off the chart, and that mass tends to one as the length grows at fixed $\beta_W$; above four times the chart threshold of the coarse curvature no point of the fibre carries a chart. The mass off the chart is a ratio of two weights of the fibre, and the next lemma records why its denominator has to be controlled. \begin{lemma}[conditioned weights]\label{lem:condweights} Let $\gamma$ be a finite measure, $D$, $G$, $T$ measurable sets with $T\cap G=\emptyset$, $V$ a real function on $D$ that is bounded above on $G\cap D$ and below on $T\cap D$, and $\nu=e^{-V}\mathbf 1_D\,\gamma$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item If $\gamma(G^c)\le\delta<\gamma(D)$, then $\nu(G)>0$ and \[ \frac{\nu(T)}{\nu(G)}\le e^{\sup_{G\cap D}V-\inf_{T\cap D}V}\, \frac{\delta}{\gamma(D)-\delta}. \] \item If $\gamma$ has no atoms, $\gamma(G)>0$ and $T=G^c$ with $\gamma(T)>0$, then for every $M$ there is $D\supset T$ with $\nu(T)/\nu(G)\ge M$ at $V=0$. \end{enumerate} \end{lemma} \begin{proof} (i) The two weights obey \begin{gather*} \nu(T)\le e^{-\inf_{T\cap D}V}\gamma(T\cap D)\le e^{-\inf_{T\cap D}V} \gamma(G^c),\\ \nu(G)\ge e^{-\sup_{G\cap D}V}\gamma(G\cap D)\ge e^{-\sup_{G\cap D}V} \bigl(\gamma(D)-\gamma(G^c)\bigr), \end{gather*} and the last factor is positive. (ii) An atomless measure takes every value between $0$ and $\gamma(G)$ on subsets of $G$; take $D=T\cup G_\varepsilon$ with $G_\varepsilon\subset G$ and $\gamma(G_\varepsilon)=\varepsilon=\gamma(T)/M$. Then $\nu(T)/\nu(G)= \gamma(T)/\varepsilon=M$. \end{proof} A tail bound, the factor $e^{\sup V-\inf V}\delta$, therefore controls a conditioned probability only together with a lower bound on the mass $\gamma(D)$ of the conditioning set. For the fibres of the path map this mass is computed: the reference measure is product Haar measure on $G^{45\mathcal B}$ and Theorem~\ref{thm:pathZ} bounds $Z_\lambda(v)$ from both sides. For the fibres of the cut-off average the reference measure carries the coarea weights $\lvert\det D_gV\rvert^{-1}$, and these are not bounded: in Proposition~\ref{prop:fold} the weights of the branches $g_\pm(\lambda)$ tend to infinity as $\lambda\downarrow\gamma^*$ while the Wilson action of every branch lies in $[0,12]$, and the law of the path link tends to $\delta_{g^*}$ for every $\beta_W$. So no bound on the conditioned probability of a branch in terms of the action alone holds uniformly in the coarse field. For $\SU(2)$ the constant $4+28\pi$ of Lemma~\ref{lem:coarsewitness} improves to $4$, up to a term of second order. \begin{lemma}[the sharp witness]\label{lem:coarsewitnesssharp} Let $G=\SU(2)$. Under the hypotheses of Lemma~\ref{lem:coarsewitness}, \[ \delta(c_P,I)\le4\kappa+7700\kappa^2 . \] \end{lemma} \begin{proof} Three facts about $\SU(2)$ with the Hilbert--Schmidt norm are used. (a) For $X,Y\in\g$, $\delta(e^Xe^Y,e^{X+Y})\le\lVert X\rVert\lVert Y \rVert$. Indeed, with $g(t)=e^{-t(X+Y)}e^{tX}e^{tY}$, $g'(t)=e^{-t(X+Y)} [e^{tX},Y]e^{tY}$ and $[e^{tX},Y]=\int_0^te^{uX}[X,Y]e^{(t-u)X}\,du$, so $\lVert g'(t)\rVert\le t\lVert[X,Y]\rVert\le2t\lVert X\rVert\lVert Y \rVert$, and $\delta(e^Xe^Y,e^{X+Y})=\delta(g(1),I)\le\int_0^1\lVert g' \rVert$. Applying this three times, with bi-invariance of $\delta$ and the triangle inequality, \[ \delta\bigl(e^{X_1}e^{X_2}e^{X_3}e^{X_4},e^{X_1+X_2+X_3+X_4}\bigr)\le \sum_{i2.418\cdot10^{-12}$. Then every field $U$ with $V^\sharp_k(U)=c_k$ for the four sides $k$ of $P$ has a plaquette with a vertex in a block of $W^+$ and $\delta(U_p,I)>6.044\cdot 10^{-13}$. Consequently no background $U$ of the fibre over $W$ satisfies the hypothesis $4\cdot10^5c_3(\kappa+s)\le\frac1{195}$ of Theorem~\ref{thm:fibrechart}\textup{(iv)} with the constant $c_3=1.5\cdot 10^4\sqrt2$ of Proposition~\ref{prop:explicitconstants}, for any $s\ge0$, and Theorems~\ref{thm:fibrechart}\textup{(iv)} and~\ref{thm:chartdecay} apply at no point of that fibre. \end{enumerate} \end{proposition} \begin{proof} (i) Every path of the pair $k$ is the product of a link inside a block, equal to $I$, and a private link of $k$, equal to $c_k$, so every path and the path link equal $c_k$; in the formula $c_k=\ell_k\exp(\chi N)$ of the proof of Lemma~\ref{lem:branchbound} every logarithm in $N$ vanishes, so $V^\sharp_k(R(c))=c_k$. Let $p$ have corner $x$ and links $(x,\mu)$, $(x+e_\mu,\nu)$, $(x+e_\nu,\mu)$, $(x,\nu)$. A link $(z,\mu)$ of $R(c)$ is $I$ when $z_\mu$ is even, since it lies inside a block, and is $c_{y',\mu}$ when $z_\mu$ is odd, $y'$ the block of $z$. If $x_\mu$ is even, both $\mu$-links of $p$ are $I$, and the two $\nu$-links start at $x$ and $x+e_\mu$, which lie in one block and have the same coordinate $\nu$, so they are equal and $R(c)_p=I$; likewise if $x_\nu$ is even. If both are odd and $x\in B(y)$, the four links are $c_{y,\mu}$, $c_{y+e_\mu,\nu}$, $c_{y+e_\nu,\mu}$, $c_{y,\nu}$, since $x+e_\mu\in B(y+e_\mu)$ and $x+e_\nu \in B(y+e_\nu)$, and their product is $c_P$. (ii) Put $\kappa=6.044\cdot10^{-13}\le10^{-8}$. If every plaquette with a vertex in one of the blocks at distance at most one from the four corner blocks had $\delta(U_p,I)\le\kappa$, Lemma~\ref{lem:coarsewitnesssharp} would give $\delta(c_P,I)\le4\kappa+7700\kappa^2<2.4177\cdot10^{-12}$, as $4\kappa=2.4176\cdot10^{-12}$ and $7700\kappa^2<3\cdot10^{-21}$. Those blocks lie in $W^+$. A background with plaquettes up to $\kappa'>\kappa$ on $W^+$ has $4\cdot10^5c_3\kappa'>4\cdot10^5c_3\kappa>\frac1{195}$, since $195\cdot4\cdot10^5\cdot1.5\cdot10^4\sqrt2>1.6546\cdot10^{12}$ gives $1/(195\cdot4\cdot10^5c_3)<6.044\cdot10^{-13}$. \end{proof} At $\rho=2.17\cdot10^{-16}$, $\eta=92\rho<2.0\cdot10^{-14}$. So the coarse plaquettes that make a block bad for every labelling range over three scales: up to $6.0\cdot10^{-13}$, thirty times $\eta$, part (i) gives a background with plaquettes below the chart threshold of Proposition~\ref{prop:explicitconstants} whenever the exterior near the window agrees with $R(c)$; above $2.418\cdot10^{-12}$, part (ii) excludes every background; in between, Proposition~\ref{prop:uniformflux} gives backgrounds for some exteriors, and Proposition~\ref{prop:spread} gives them for the exterior $R(c)$ of the coarse fields $c^\theta$ up to $2.39\cdot10^{-12}$. Where a background exists, the constrained law of the component decays across it at the rate of the chart. \begin{theorem}[a bad component on the chart]\label{thm:badcross} Let $G=\SU(2)$, $\beta_W\ge2.3\cdot10^{17}$, let $W$ be a set of blocks and $U$ a field with $\delta(U_p,I)\le\kappa$ for every plaquette with a vertex in a block of $W^+$, and let $s>0$ with $s\le1/(6000c_3)$, $\kappa+6s\le10^{-8}$ and $\kappa+s\le6.0\cdot10^{-13}$. Let $\mu_W$ be the constrained law on the fibre $F$ of $U$ over $W$, $r=s/32$, $\Phi$ and $\Omega_r$ as in Theorem~\ref{thm:chartdecay}, and \[ p=\mu_W\bigl(F\setminus\Phi(\Omega_r)\bigr). \] For sets $X_1,X_2\subset A_W$ and bounded smooth functions $\tilde F$, $\tilde G$ of the links of $X_1$ and of $X_2$ whose left-invariant gradients have norm at most $1$ in each link, \begin{multline*} \bigl\lvert\Cov_{\mu_W}(\tilde F,\tilde G)\bigr\rvert\le\frac{4.8\cdot 10^4}{\beta_W}\bigl(\lvert X_1\rvert\lvert X_2\rvert\bigr)^{1/2}e^{-\alpha_G d(X_1,X_2)}\\ +6p\,\lVert\tilde F\rVert_\infty\lVert\tilde G\rVert_\infty, \end{multline*} with $\alpha_G=2.5\cdot10^{-4}$ and $d$ the distance of Theorem~\ref{thm:chartdecay}. The bound does not depend on $W$, on the exterior or on the coarse field beyond $\kappa$ and $p$, and the labels of the blocks of $W$ enter neither its hypotheses nor the bound. \end{theorem} \begin{proof} The hypotheses are those of Theorem~\ref{thm:fibrechart}\textup{(iv)}: by Proposition~\ref{prop:explicitconstants}, $\kappa+s\le6.0\cdot10^{-13}$ gives $4\cdot10^5c_3(\kappa+s)\le\frac1{195}$, and $\lambda=\beta_W/4\ge 5.6\cdot10^{16}\ge60c_6$. Let $A=\Phi(\Omega_r)$ and $\mu_A=\mu_W(\cdot\mid A)$. By Theorem~\ref{thm:chartdecay}\textup{(iii)} and Proposition~\ref{prop:explicitdecay}, $\lvert\Cov_{\mu_A}(\tilde F,\tilde G)\rvert$ is at most the first term, as $c_9/\lambda=4.8\cdot10^4/\beta_W$. If $p=0$ this is the claim, and if $p=1$ the claim follows from $\lvert\Cov\rvert\le\lVert\tilde F\rVert_\infty\lVert\tilde G\rVert _\infty$. Otherwise, with $\mu_{A^c}=\mu_W(\cdot\mid F\setminus A)$ and $\mu_W=(1-p)\mu_A+p\mu_{A^c}$, \[ \begin{split} \Cov_{\mu_W}(\tilde F,\tilde G)={}&(1-p)\Cov_{\mu_A}(\tilde F,\tilde G)+ p\Cov_{\mu_{A^c}}(\tilde F,\tilde G)\\ &+p(1-p)\bigl(\mu_A\tilde F-\mu_{A^c}\tilde F\bigr)\bigl(\mu_A\tilde G- \mu_{A^c}\tilde G\bigr), \end{split} \] so the difference from $\Cov_{\mu_A}$ is at most $p\lVert\tilde F\rVert _\infty\lVert\tilde G\rVert_\infty$ twice, from the first two terms, plus $4p\lVert\tilde F\rVert_\infty\lVert\tilde G\rVert_\infty$. Nothing in Theorems~\ref{thm:fibrechart} and~\ref{thm:chartdecay} refers to the labels of the blocks of $W$. \end{proof} \begin{remark}\label{rem:badcompopen} Theorem~\ref{thm:badcross} bounds the response inside a bad component by the chart, with no cost growing with its length except through $p$, and leaves three things. First, the mass $p$. By Lemma~\ref{lem:condweights} it is bounded by the excess action off the chart times the reference mass off the chart, divided by the reference mass of the chart in the fibre; the first factor is a Peierls bound, the reference masses carry the coarea weights, and Proposition~\ref{prop:fold} shows that these weights are not controlled by the action. In the other direction Theorem~\ref{thm:offchartmass} bounds $p$ from below: at fixed $\beta_W$ it tends to one as the component grows, and it exceeds $1-e^{-5.5\lvert W\rvert}$ for $\beta_W\le10^{24}$ (Corollary~\ref{cor:offchartmass}). In the mean over the coarse field $\E_c$ the mass off the chart is a probability under $\mu_L$; for a fixed coarse field it is the open step of Remark~\ref{rem:smallfibreopen}(c) and Remark~\ref{rem:fibrechartopen}(a). Second, the coarse fields above $2.418\cdot10^{-12}$, where by Proposition~\ref{prop:badchart}(ii) the chart applies at no point; there the fibre law near the component is that of a large field, which Proposition~\ref{prop:fold} shows to jump, and no estimate of this paper bounds its response. A bound uniform in the coarse field needs that regime. Third, the passage from $\mu_W$ to $\mu^c$ and to the labels. Under $\mu^c$, given the links outside $A_W$, the averages of the pairs that enter $W$ from outside are fixed as well, and they bind links of $A_W$; and the labels adjoining a component are functions of links outside its interior, so that their covariance across it is the cross-ratio of the weight of the interior as a function of its two ends. For a good layer Theorem~\ref{thm:factorisation} solves the path links of the entering pairs by Lemma~\ref{lem:branchbound}, writes the weight through a kernel on a domain that the ends do not move, and Lemma~\ref{lem:connected} turns the cross-ratio into a covariance of the layer law. Both steps use non-path links within $r$ of the reconstruction. For a bad component Proposition~\ref{prop:badpassage} moves the links that bind the entering averages to the exterior, so that the constrained law of the reduced fibre is the conditional law under $\mu^c$, and Corollary~\ref{cor:badcrossratio} bounds the cross-ratio for ends that enter the action only; ends that enter the averages, on which the labels depend, are not treated (Remark~\ref{rem:badmassopen}). The other two routes do not reach the interior of a component. A single-link Dobrushin or disagreement-percolation bound needs row sums below one, while the coefficient of Lemma~\ref{lem:fixedset} is $\frac{\pi^2}8\beta_W$ for each link that shares a plaquette with the given one, and on the fibre the Kantorovich response of the path-link law to its non-path links is infinite by Proposition~\ref{prop:fold}\textup{(iv)}; blocks large enough to contract must carry the constraints of the cut-off average, which is the chart of Theorem~\ref{thm:badcross}. The tree form of the mixed differences of $\log Z^c$ asked for in Remark~\ref{rem:labelcovopen} concerns the labels of distinct components, which are separated by good layers; a component bad for every labelling lies inside one component together with the blocks it adjoins, so neither that bound nor a cluster expansion built on it reaches the response across it. In the law of Theorem~\ref{thm:fixedobstruction}(i) the second difference over the labels of $a$ and $b$, all other labels fixed, is $J$, and a bound on it decaying with $\dist(a,b)$ is exactly the response across the component. \end{remark} The bound of Theorem~\ref{thm:badcross} contains the mass $p$ of the constrained law off the product chart $\Phi(\Omega_r)$, and that chart asks every one of the $45\lvert W\rvert$ coordinates $\eta_e$, $e\in L'$, to lie in a ball of radius $r$. On the chart the law has a density whose Hessian is bounded above, so each coordinate leaves its ball with a conditional probability bounded below, and $p$ grows with $\lvert W\rvert$. The next lemma is the elementary estimate behind this. \begin{lemma}[leaving a ball]\label{lem:ballexit} Let $E$ be a Euclidean space, $r>0$, $B_t=\{x\in E:\lvert x\rvert\le t\}$, and let $f$ be a real function of class $C^2$ on a neighbourhood of $B_{2r}$ with $\lVert\Hess f(x)\rVert\le\Lambda$ for $x\in B_{2r}$. Then \[ \int_{B_{2r}\setminus B_r}e^{-f}\,dx\ge\frac13e^{-5.5\Lambda r^2} \int_{B_{2r}}e^{-f}\,dx . \] \end{lemma} \begin{proof} Choose a unit vector $u$ with $\nabla f(0)\cdot u\le0$. For $x\in B_{2r}$, $\nabla f(x)\cdot u\le\nabla f(0)\cdot u+\Lambda\lvert x\rvert\le\Lambda \lvert x\rvert$, and for $x,x+v\in B_{2r}$, the segment between them lying in the convex set $B_{2r}$, $f(x+v)\le f(x)+\nabla f(x)\cdot v+\frac12 \Lambda\lvert v\rvert^2$. Put $O=B_{2r}\setminus B_r$, $S_+=\{x\in B_r:x \cdot u\ge0\}$ and $S_-=B_r\setminus S_+$. For $x\in S_+$, $\lvert x+ru\rvert^2=\lvert x\rvert^2+2r\,x\cdot u+r^2$ lies in $[r^2,4r^2]$ and exceeds $r^2$ unless $x=0$, so the translation $x\mapsto x+ru$ maps $S_+$ into $O$ up to a null set, and $f(x+ru)\le f(x)+r\Lambda\lvert x\rvert+\frac12\Lambda r^2\le f(x)+ 1.5\Lambda r^2$. Hence $\int_Oe^{-f}\ge e^{-1.5\Lambda r^2}\int_{S_+} e^{-f}$. The reflection $R(x)=x-2(x\cdot u)u$ maps $S_-$ into $S_+$ and preserves Lebesgue measure; for $x\in S_-$, $R(x)=x+tu$ with $t=2\lvert x \cdot u\rvert\le2r$, so $f(R(x))\le f(x)+t\Lambda r+\frac12\Lambda t^2\le f(x)+4\Lambda r^2$, and $\int_{S_+}e^{-f}\ge e^{-4\Lambda r^2}\int_{S_-} e^{-f}$. Therefore \[ \int_{B_r}e^{-f}\le\bigl(1+e^{4\Lambda r^2}\bigr)e^{1.5\Lambda r^2} \int_Oe^{-f}, \] and $\int_Oe^{-f}\big/\int_{B_{2r}}e^{-f}\ge(1+e^{1.5\Lambda r^2}+e^{5.5 \Lambda r^2})^{-1}\ge\frac13e^{-5.5\Lambda r^2}$. \end{proof} \begin{theorem}[the mass off the chart]\label{thm:offchartmass} Under the hypotheses of Theorem~\ref{thm:badcross}, \begin{align*} p&\ge1-\Bigl(1-\frac13e^{-2843.5\beta_Wr^2}\Bigr)^{45\lvert W\rvert}\\ &\ge1-\exp\bigl(-15\lvert W\rvert e^{-10^{-24}\beta_W}\bigr). \end{align*} \end{theorem} \begin{proof} Write $\mathrm{Box}_t$ for the product over $e\in L'$ of the balls $\{\lVert\xi_e\rVert\le t\}$ in $\g$, so that $\Omega_t=S(\mathrm{Box}_t)$ in the notation of the proof of Theorem~\ref{thm:chartdecay}. Since $S( \mathrm{Box}_{2r})\subset K_{30.2r}$ and $30.2r10^{24}(4.40+\ln\lvert W\rvert)$; \item for fixed $\beta_W$, $p\to1$ as $\lvert W\rvert\to\infty$. \end{enumerate} \end{corollary} \begin{proof} (i) For $\beta_W\le10^{24}$, $q_0\ge\frac13e^{-1}>0.1226$ and $45q_0>5.5$; at $\beta_W=2.3\cdot10^{17}$, $q_0\ge\frac13(1-2.3\cdot10^{-7})$ and $45q_0>14.99$. Use $(1-q_0)^N\le e^{-Nq_0}$. (ii) The bound contains $6p \lVert\tilde F\rVert_\infty\lVert\tilde G\rVert_\infty$, so it is smaller than the trivial bound only if $p<\frac16$. Then $e^{-45\lvert W\rvert q_0}\ge1-p>\frac56$, so $15\lvert W\rvert e^{-2843.5\beta_Wr^2}<\ln\frac65< 0.1824$; as $2843.5r^2<10^{-24}$, this gives $10^{-24}\beta_W>\ln(15 \lvert W\rvert/0.1824)>4.40+\ln\lvert W\rvert$. (iii) follows from the second bound of Theorem~\ref{thm:offchartmass}. \end{proof} Fig.~\ref{fig:offchartmass} draws the bound of Theorem~\ref{thm:offchartmass} and the couplings at which Theorem~\ref{thm:badcross} improves on the trivial bound. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_offchartmass.png} \caption{The mass off the chart in a bad component $W$ (Theorem~\ref{thm:offchartmass}). (a)~The lower bound $1-\exp(-15\lvert W\rvert e^{-10^{-24}\beta_W})$ on $p$ over $\beta_W/10^{24}$ and $\log_{10}\lvert W\rvert$. (b)~The curve $\beta_W=10^{24}(4.40+\ln\lvert W\rvert)$, above which the bound of Theorem~\ref{thm:badcross} improves on the trivial one, and the region in which the lower bound on $p$ exceeds $\frac12$.} \label{fig:offchartmass} \end{figure*} So the absence of a cost in the length in Theorem~\ref{thm:badcross} is carried entirely by $p$: a product chart in all the coordinates of the component, which the Helffer--Sj\"ostrand argument of Theorem~\ref{thm:chartdecay} uses, holds the whole component near the background, and a component of $n$ blocks leaves it unless $\beta_Wr^2$ exceeds a multiple of $\ln n$. With the constants of Proposition~\ref{prop:explicitconstants} the radius $r$ is at most $1.875\cdot10^{-14}$, so the threshold $2.3\cdot10^{17}$ of the theorem is seven orders of magnitude below the coupling at which $p$ can be small for a single block. In the other direction Corollary~\ref{cor:calmfibres} gives $p\le\varepsilon_W(K,\lambda)$ on every fibre through a field whose plaquettes near $W$ are at most $K/(4\sqrt{n_P})$, with $s\ge1.24\cdot10^5K$, which needs $\lambda K^2$ larger than a multiple of $\lvert W\rvert\ln(n_P/ K)$; the necessary condition grows like $\ln\lvert W\rvert$ and the sufficient one like $\lvert W\rvert$. The fibres to which the theorem applies include arbitrarily long bad components, and their coarse plaquettes come arbitrarily close to four times the bound $6.0\cdot10^{-13}$ on $\kappa+s$, which by Lemma~\ref{lem:coarsewitnesssharp} no such fibre exceeds by more than $7700\kappa^2$. Let $T_0=\operatorname{diag}(i,-i)\in\g$, so that $\delta(e^{\theta T_0},I)=2\sqrt2\lvert\sin(\theta/2)\rvert$ for real $\theta$. \begin{proposition}[a uniform field strength]\label{prop:uniformflux} Let $G=\SU(2)$, let the side $2L$ of the periodic lattice have $L\ge4$ even, let $\mu\ne\nu$ be two directions and $00$ with $s\le1/(6000c_3)$, $\kappa_f+6s\le 10^{-8}$ and $\kappa_f+s\le6.0\cdot10^{-13}$. \end{enumerate} In particular, for every $t\in(0,2.39\cdot10^{-12}]$ there are a coarse field with a coarse plaquette $P$ of $\delta(c_P,I)=t$ and, for every $W$, an exterior whose fibre over $W$ satisfies the hypotheses of Theorem~\ref{thm:badcross} with $s=2.4\cdot10^{-15}$; and the constant $4$ of Lemma~\ref{lem:coarsewitnesssharp} cannot be lowered. \end{proposition} \begin{proof} (i) Every link of $U^f$ lies in the torus $\{e^{\theta T_0}\}$, and we give it the real angle $\theta_{x,\mu}=-g(x_\nu)$, or $0$. The real sum of a plaquette at $x$, $\theta_1+\theta_2-\theta_3-\theta_4$ in the notation of Sec.~\ref{ssec:smallfibres}, is $g(x_\nu+1)-g(x_\nu)=\pm f$ in the plane $\{\mu,\nu\}$, and $0$ in every other plane, since a plaquette in the plane $\{\mu,\rho\}$, $\rho\ne\nu$, has its two $\mu$-links at the same coordinate $\nu$. The gauge transformation is $e^{\lambda_xT_0}$ with $\lambda_x$ real, $0$ at the corners, and fixed along the tree of each block by the condition that the real angles $\tilde\theta_{x,\rho}= \theta_{x,\rho}+\lambda_x-\lambda_{x+e_\rho}$ of the tree links vanish. The real sums of the plaquettes do not change, so every plaquette of $\tilde U$ is $e^{\pm fT_0}$ or $I$. (ii) Let $k$ be a pair, $v=P_{2y}$ its path link and $v_x=v^{-1}P_x$. By (i) and Lemma~\ref{lem:smallframes}(i), which applies as $\kappa_f\le\sqrt2 f\le10^{-8}$, $\delta(v_x,I)\le24\kappa_f<\frac12$, so the cut-off equals one, and $v_x=e^{\varphi_xT_0}$ with $\varphi_x=\tilde\theta(P_x)-\tilde\theta(v)$, where $\tilde\theta$ of a path is the sum of the real angles along it. The principal logarithm of $v_x$ is $\varphi'_xT_0$ with $\varphi'_x\equiv\varphi_x$ modulo $2\pi$. As functions of $f\in[0,7\cdot10^{-9}]$, $\varphi_x$ is linear, since $g$ and the gauge transformation are, and $\varphi'_x$ is continuous, since $v_x$ stays within $24\kappa_f<1$ of $I$, where the principal logarithm is continuous; their difference lies in $2\pi\Z$ and vanishes at $f=0$, so $\varphi'_x=\varphi_x$. Hence \[ c_k=v\exp\Bigl(\frac1{16}\sum_x\varphi_xT_0\Bigr)=\exp\Bigl(\frac1{16} \sum_x\tilde\theta(P_x)\,T_0\Bigr). \] With the sides $k=1,\dots,4$ of $P$ as in the proof of Proposition~\ref{prop:badchart}(i) and the sites $x\in B(y)$, $x+2e_\mu$, $x+2e_\nu$ of the same offset, \[ c_P=\exp\Bigl(\frac1{16}\sum_{x\in B(y)}\tilde\theta(\partial S_x)\,T_0 \Bigr), \] where $\partial S_x$ is the boundary of the square of side two at $x$ in the plane $\{\mu,\nu\}$. Its real angle is the sum of the real sums of its four plaquettes, whose coordinates $\nu$ lie in $\{2y_\nu,2y_\nu+1,2y_\nu+ 2\}\subset\{0,\dots,L-1\}$, so each is $f$ and $c_P=e^{4fT_0}$. Finally $\sin2f=4\sin\frac f2\cos\frac f2\cos f$, and $\cos\frac f2\cos f\ge(1- \frac{f^2}8)(1-\frac{f^2}2)\ge1-f^2$. (iii) $\tilde U$ lies in its own fibre and has every plaquette at most $\kappa_f$; the other hypotheses are the conditions on $\beta_W$ and $s$. For the last statement let $f=\frac12\arcsin(t/(2\sqrt2))$, so that $2\sqrt2 \sin2f=t$ and $f<4.3\cdot10^{-13}$. Then $\kappa_f\le t/(4(1-f^2))<5.976 \cdot10^{-13}$, so $s=2.4\cdot10^{-15}$ satisfies the conditions of (iii), as $1/(6000c_3)>7.8\cdot10^{-9}$. By (ii) the ratio $\delta(c_P,I)/\kappa_f$ tends to $4$ as $f\to0$, while $\kappa_f\le10^{-8}$ and every plaquette of $\tilde U$ is at most $\kappa_f$. \end{proof} The blocks with $0\le y_\nu\le L/2-1$ are corners of the coarse plaquettes of (ii), which exceed $\eta$ once $2\sqrt2\sin2f>\eta$, so they lie in one component bad for every labelling, of $L^4/2$ blocks. On these fibres Theorem~\ref{thm:badcross} applies for every $\lvert W\rvert$, and Corollary~\ref{cor:offchartmass} shows that its bound loses its content as $\lvert W\rvert$ grows. Under $\mu^c$ the links of a window are not distributed by $\mu_W$: the averages of the pairs that enter $W$ from outside are fixed too, and they contain links of $A_W$. These links can be moved to the exterior. Let $I_W$ be the set of active links $(x,\mu)$ inside a block $B(y)$, $y\in W$, with $y-e_\mu\notin W$, let $A^-_W=A_W\setminus I_W$, $L'^-=A^-_W\setminus \{d_k\}$, and for a field $U$ let $F^-$ be the set of fields that agree with $U$ off $A^-_W$ and have the same averages $V^\sharp_{y,\mu}$ as $U$ for $y\in W$. The measure $\nu^-_c$ on $F^-$ and the constrained law $\mu^-_W= e^{-\lambda Q^-}\nu^-_c/\nu^-_c(e^{-\lambda Q^-})$, with $Q^-$ the sum of $\delta(U_p,I)^2$ over the plaquettes that contain a link of $A^-_W$, are defined as $\nu_c$ and $\mu_F$ before~\eqref{eq:fibrevolume}, with the links of $I_W$ among the fixed links. \begin{proposition}[the window under $\mu^c$]\label{prop:badpassage} Let $G=\SU(2)$ and let $W$ be a set of blocks. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Every average $V^\sharp_{y',\mu}$ with $y'\notin W$ is a function of the links off $A^-_W$. For $\E_c$-almost every $c$, the constrained law $\mu^-_W$ on the fibre through a field is a version of the conditional law under $\mu^c$ of the links of $A^-_W$ given the links off $A^-_W$. \item Theorems~\ref{thm:fibrechart}, \ref{thm:chartdecay}, \ref{thm:offchart}\textup{(i)}, \ref{thm:badcross} and~\ref{thm:offchartmass} and Propositions~\ref{prop:explicitconstants} and~\ref{prop:explicitdecay} hold for $F^-$, $A^-_W$ and $L'^-$ in place of $F$, $A_W$ and $L'$, with the same constants, and with $45\lvert W\rvert$ replaced by $\lvert L'^-\rvert \ge28\lvert W\rvert$ in Theorem~\ref{thm:offchartmass}. \end{enumerate} \end{proposition} \begin{proof} (i) An inside link $(x,\mu)$ of a block $B(y)$ enters only the averages of $(y,\mu)$, as the first link of the path from $x$, and of $(y-e_\mu,\mu)$, as the second link of the path from $x-e_\mu$; a private link enters only the average of its pair. So $V^\sharp_{y',\mu}$, $y'\notin W$, depends on the links of $B(y')$ and the private links of $(y',\mu)$, which are not in $A_W$, and on inside links of $B(y'+e_\mu)$, which lie in $I_W$ when $y'+e_\mu\in W$. In the tree gauge the active links have density proportional to $e^{-\beta_WS}$ with respect to product Haar measure $h$ (proof of Theorem~\ref{thm:offchart}(ii)). Write the active links as $(E,z)$ with $z$ the links of $A^-_W$, and $C=(C_{\rm out},C_W)$ for the averages of the pairs based off and in $W$, so that $C_{\rm out}$ is a function of $E$. The construction of~\eqref{eq:fibrevolume}, with the links of $I_W$ fixed, gives $h(dz)=\int\nu^-_{E,c_W}(dz)\,dh(c_W)$, so \[ e^{-\beta_WS}h(dE)\,h(dz)=e^{-\beta_WS}h(dE)\int\nu^-_{E,c_W}(dz)\, dh(c_W), \] and the conditional law of $z$ given $(E,C_W)$ is proportional to $e^{-\beta_WS}\nu^-_{E,C_W}$, which is $\mu^-_W$, the plaquettes without a link of $A^-_W$ giving a factor that does not depend on $z$. Since $C_{\rm out}$ is a function of $E$, conditioning on $(E,C)$ is conditioning on $(E,C_W)$, and by the tower property this is the conditional law under $\mu^c$ given $E$, for almost every $c$. (ii) Let $J:\g^{A^-_W}\to\g^{A_W}$ be extension by zero. The fibre $F^-$ is $F$ with the links of $I_W$ fixed, so $\xi$ is tangent to $F^-$ exactly when $J\xi$ is tangent to $F$, and on $F^-$ the difference $Q-Q^-$ depends only on links off $A^-_W$ and is constant. The window enters the proofs of the listed results only through the following objects, and each has a counterpart on $F^-$ with the same bound. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item The derivative $DC_k=\tilde B_k+R_k$ of Lemma~\ref{lem:smallframes}(iv), with $\lVert R_k\rVert\le260\kappa$; on $F^-$ it is $DC_kJ=\tilde B_kJ+R_kJ$, and $\lVert R_kJ\rVert\le\lVert R_k \rVert$, $\lVert\tilde B_kJ\rVert\le\lVert\tilde B_k\rVert$. \item The Gram matrix $\mathcal G=DC\,DC^*$. Its diagonal blocks are at least $\frac18-227\kappa$ because the squared coefficients of $V_{y,\mu}$ on the eight private links of $(y,\mu)$ add up to $\frac18$; these links lie in $A^-_W$, so $\tilde B_kJJ^*\tilde B_k^*\ge\frac18$ and the perturbation $2\lVert\tilde B_kJ\rVert\,260\kappa+(260\kappa)^2\le227\kappa$ is the same. The upper bounds $\frac3{16}+227\kappa$ on the diagonal blocks and $\frac1{32}+227\kappa$ on the two off-diagonal blocks of a row bound sums of the norms of the contributions of single links, which only lose terms. So $\mathcal G^-=DC\,JJ^*DC^*$ has the block bounds from which the proofs of Theorem~\ref{thm:fibrechart} and Proposition~\ref{prop:explicitdecay} derive $\mathcal G\ge\frac1{16}-700 \kappa$, $\lVert\mathcal G^{-1}\rVert\le17$ and the weighted norms of $\mathcal G^{-1}$. \item The derivative bounds with the constants $c_3$ and $c_6$ for $c_k$, $Q$ and $j$, and $A=Dc(X)$: on vectors supported on $A^-_W$ they hold as before, and $AJ$, $DC\,J$ have the same block sparsity. \item The section: $(S^-\xi)_e=\xi_e$ for $e\in L'^-$ and $(S^-\xi)_{d_k}= -B_{k,d_k}^{-1}\sum_{e\in A^-_W\setminus\{d_k\}}B_{k,e}\xi_e$, that is $JS^-=SJ$. The path link $d_k$ lies in $A^-_W$ and enters only $V_k$, and the bound $15.1$ is a sum of norms over $e$ that only loses terms. \end{enumerate} The function $\bar H$ and the matrices $P$, $\bar P$ on $F^-$ are defined by the same formulas from $AJ$, $DC\,J$ and $S^-$, and their bounds are derived in those proofs from (a)--(d) by the same computations. The distance $d$ on $L'^-$ is the restriction of that on $L'$. The proof of Theorem~\ref{thm:offchart}(i) applies to fields that agree off $A^-_W$, since these agree off $A_W$. Since $DC\,J$ is onto by (b), $F^-$ has codimension $4\lvert W\rvert\dim G$ and $\dim T_U=\lvert L'^-\rvert\dim G$. The $28\lvert W\rvert$ private links other than the path links lie in $L'^-$. \end{proof} The weight of the interior as a function of two ends follows, for ends that enter only the action. \begin{corollary}\label{cor:badcrossratio} Let $G=\SU(2)$ and let $E_a$, $E_b$ be disjoint sets of links off $A^-_W$, none of which enters an average $V^\sharp_{y,\mu}$ with $y\in W$, such that no plaquette contains a link of $E_a$ and a link of $E_b$. Fix the other links off $A^-_W$, let $a_0,a_1\in G^{E_a}$, $b_0,b_1\in G^{E_b}$, $X_e, Y_e\in\g$ with $a_{1,e}=a_{0,e}e^{X_e}$, $b_{1,e}=b_{0,e}e^{Y_e}$, and $x_a=\max_e\lVert X_e\rVert$, $x_b=\max_e\lVert Y_e\rVert$. Let $Z_{ij}= \nu^-_c(e^{-\lambda Q^-})$ on the fibre with the links $a_i$ on $E_a$ and $b_j$ on $E_b$, and let $N_a$, $N_b$ be the links of $A^-_W$ that share a plaquette with a link of $E_a$, of $E_b$. Suppose that at the links $a_0e^{\sigma X}$, $b_0e^{\tau Y}$, for all $\sigma,\tau\in[0,1]$, the fibre satisfies the hypotheses of Theorem~\ref{thm:badcross} with fixed $\kappa$ and $s$, and let $\bar p$ bound its mass off the chart. Then \begin{multline*} \Bigl\lvert\log\frac{Z_{11}Z_{00}}{Z_{10}Z_{01}}\Bigr\rvert\le3.9\cdot10^6 \beta_Wx_ax_b\bigl(\lvert N_a\rvert\lvert N_b\rvert\bigr)^{1/2}e^{-\alpha_G d(N_a,N_b)}\\ +432\,\beta_W^2\,\bar p\,\lvert E_a\rvert\lvert E_b\rvert x_ax_b . \end{multline*} \end{corollary} \begin{proof} Neither the fibre $F^-$ nor $\nu^-_c$ depends on the links of $E_a\cup E_b$, since these enter no constraint, so $Z(\sigma,\tau)=\nu^-_c(e^{- \lambda Q^-})$ at the links $a_0e^{\sigma X}$, $b_0e^{\tau Y}$ is an integral over a fixed space, and differentiating under the integral, $\partial_\sigma\partial_\tau\log Z=\Cov(\phi_a,\phi_b)$ under $\mu^-_W$, with $\phi_a=\partial_\sigma\lambda Q^-$ and $\phi_b=\partial_\tau\lambda Q^-$, because no plaquette contains links of both sets, so that $\partial_\sigma\partial_\tau Q^-=0$. The function $\phi_a$ depends on the links of $N_a$. For a plaquette $p$, $\delta(U_p,I)^2=4-2\ReTr U_p$; moving a link $e$ of $p$ along $X_e$ inserts $X_e$ between unitary factors, and $\ReTr(VX)=\ReTr((V-I)X)$ for skew-Hermitian $X$, so the derivative is at most $2\delta(U_p,I)\lVert X_e\rVert$, and the mixed derivative along $X_e$ and along a unit vector at another link of $p$ is at most $2\lVert X_e \rVert$. A link lies in six plaquettes, and a plaquette with a link of $A^-_W$ has at most three links of $E_a$; so the left-invariant gradient of $\phi_a$ has norm at most $6\cdot3\cdot2\lambda x_a=36\lambda x_a$ in each link. As $\delta(U_p,I)\le2\sqrt2$, summing over the links $e\in E_a$ and the six plaquettes through each gives $\lVert\phi_a\rVert_\infty\le6\lvert E_a\rvert\cdot2\cdot2\sqrt2\,\lambda x_a=24\sqrt2\lambda\lvert E_a\rvert x_a$, and likewise for $\phi_b$. By Proposition~\ref{prop:badpassage}(ii), Theorem~\ref{thm:badcross} applied to $\phi_a/(36\lambda x_a)$ and $\phi_b/(36\lambda x_b)$ gives \[ \begin{split} \lvert\Cov(\phi_a,\phi_b)\rvert\le{}&\frac{4.8\cdot10^4}{\beta_W}(36 \lambda)^2x_ax_b\\ &\times\bigl(\lvert N_a\rvert\lvert N_b\rvert\bigr)^{1/2}e^{-\alpha_G d(N_a,N_b)}\\ &+6\bar p\,(24\sqrt2\lambda)^2\lvert E_a\rvert\lvert E_b\rvert x_ax_b, \end{split} \] and with $\lambda=\beta_W/4$ the two coefficients are $81\cdot4.8\cdot10^4 \beta_W=3.888\cdot10^6\beta_W$ and $6\cdot1152\beta_W^2/16=432\beta_W^2$. Integrate over $(\sigma,\tau)\in[0,1]^2$. \end{proof} The first term decays at the rate $\alpha_G$ in the distance between the ends, with no cost in the length of $W$; the second needs $\bar p$ small against $\beta_W^{-2}$, which by Theorem~\ref{thm:offchartmass} requires $\beta_Wr^2$ larger than a multiple of $\ln\beta_W+\ln\lvert W\rvert$. \begin{remark}\label{rem:badmassopen} What remains of the bad regions in item (ii) of Sec.~\ref{ssec:remains}: \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item \emph{The mass off the chart.} Theorem~\ref{thm:offchartmass} shows that no bound on $p$ uniform in the length of the component exists, for any coarse field and exterior to which Theorem~\ref{thm:badcross} applies, and that $p$ is close to one for $\beta_W\le10^{24}$. A bound on the response across a component uniform in its length therefore cannot pass through a single product chart of the whole component; it needs charts local in the component, around which the links far from the observables may leave the chart, that is a decomposition of the fibre into small-field and large-field regions with a convex chart on each small-field region. Theorem~\ref{thm:localcharts} shows that charts fixed by the exterior of their window put the window on the chart with conditional probability at most $0.033^{\lvert W\rvert}$ for $\beta_W\le10^{24}$, so that in such a decomposition the large-field region carries most of the mass. For fixed $\lvert W\rvert$, a bound on $p$ for a fixed coarse field is Corollary~\ref{cor:calmfibres}, whose condition grows linearly in $\lvert W\rvert$, against the logarithmic necessary condition of Corollary~\ref{cor:offchartmass}(ii). \item \emph{Larger coarse fields.} Above $2.418\cdot10^{-12}$ no field of the fibre carries a chart (Proposition~\ref{prop:badchart}(ii)). Between $6.0\cdot10^{-13}$ and $2.418\cdot10^{-12}$ the fibres of Proposition~\ref{prop:uniformflux} carry backgrounds. For the exterior $R(c)$, whose plaquettes carry each coarse plaquette in one fine plaquette, a background would have to spread the coarse curvature over the plaquettes of the window, almost aligned and almost of equal size as in the proof of Lemma~\ref{lem:coarsewitnesssharp}, while matching $R(c)$ off $A_W$. The exact condition is that the fibre over $W$ of a field agreeing with $R(c)$ off $A_W$ contain a field $U$ with $\delta(U_p,I)\le\kappa$ for every plaquette with a vertex in a block of $W^+$, for some $\kappa<6.0\cdot 10^{-13}$; under it Theorems~\ref{thm:badcross} and~\ref{thm:offchartmass} apply with every $s\le6.0\cdot10^{-13}-\kappa$. It holds for the coarse fields $c^\theta$ of Proposition~\ref{prop:spread} up to $2.39\cdot 10^{-12}$; for opposite fluxes at adjacent coarse plaquettes no field with values in a torus satisfies it above $1.8\cdot10^{-12}$; for a general coarse field it is not decided. \item \emph{Ends in the averages.} Corollary~\ref{cor:badcrossratio} needs ends that enter no average of a pair of $W$. The links of $I_W$, and the inside links of a block $y+e_\mu\notin W$ in the direction $\mu$ for $y\in W$, enter those averages, and the labels of the blocks adjoining the component depend on such links; moving them moves the fibre, and Proposition~\ref{prop:insidefold} shows that off the chart the law then jumps. On the chart the dependence is smooth. The exact missing statement is the bound of Corollary~\ref{cor:badcrossratio} for ends $E_a$, $E_b$ that contain such links, that is a bound on the mixed derivative of $\log\nu^-_c(e^{-\lambda Q^-})$ in two links that move the constraints, decaying in their distance. Lemma~\ref{lem:endpairs} places an end in one pair factor of $\nu^-_c$, Proposition~\ref{prop:endfold} shows that this factor jumps under small moves of the end at every value of the coarse link, and Remark~\ref{rem:badendsopen} states the bound in the mean that is missing. \end{enumerate} \end{remark} The three points of Remark~\ref{rem:badmassopen} are taken in turn. The first concerns charts local in the component. Such a chart is attached to a window $W$, a set of blocks inside the component, and is fixed by the field outside the window; the next theorem bounds the conditional probability that the window lies on it, uniformly in the window, the exterior and the coarse field. \begin{theorem}[local charts]\label{thm:localcharts} Let $G=\SU(2)$, $\beta_W\ge2.3\cdot10^{17}$, let $c$ be a coarse field and $W$ a set of blocks, and write $E$ for the links off $A^-_W$. Suppose that for each $E$ a field $U(E)$ is given in the reduced fibre $F^-(E)$ over $W$ through $E$, with averages $c_k$ for the pairs based in $W$, such that $F^-(E)$ satisfies the hypotheses of Theorem~\ref{thm:badcross} at $U(E)$ with constants $\kappa(E)$, $s(E)$, as in Proposition~\ref{prop:badpassage}% \textup{(ii)}, and let $\mathcal A$ be the event that the links of $A^-_W$ lie in the chart $\Phi_E(\Omega_{r(E)})$ of $F^-(E)$ at $U(E)$, $r(E)=s(E)/ 32$, assumed measurable. Then for $\E_c$-almost every $c$ \[ \mu^c(\mathcal A)\le\exp\Bigl(-\frac{28}3\lvert W\rvert e^{-10^{-24} \beta_W}\Bigr). \] \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item If $\beta_W\le10^{24}$, then $\mu^c(\mathcal A)\le e^{-3.433\lvert W \rvert}<0.033^{\lvert W\rvert}$; at $\beta_W=2.3\cdot10^{17}$, $\mu^c( \mathcal A)\le e^{-9.333\lvert W\rvert}$. \item If $\beta_W\le10^{24}$ and $\mathcal A_1,\dots,\mathcal A_J$ are such events, given by fields $U_1(E),\dots,U_J(E)$, then the union of the $\mathcal A_j$ has $\mu^c$-probability at most $J\,e^{-3.433\lvert W\rvert}$. \item If $\mu^c(\mathcal A\mid E)\ge\frac12$ on a set of values of $E$ of positive probability, then $\beta_W>2.6\cdot10^{24}$. \end{enumerate} \end{theorem} \begin{proof} By Proposition~\ref{prop:badpassage}\textup{(i)}, for $\E_c$-almost every $c$ the conditional law under $\mu^c$ of the links of $A^-_W$ given $E$ is the constrained law $\mu^-_W$ on $F^-(E)$. By Proposition~\ref{prop:badpassage}\textup{(ii)}, Theorem~\ref{thm:offchartmass} holds on $F^-(E)$ with $45\lvert W\rvert$ replaced by $\lvert L'^-\rvert\ge 28\lvert W\rvert$, so that, with $q_0=\frac13e^{-2843.5\beta_Wr(E)^2}$, \[ \begin{split} \mu^c(\mathcal A\mid E)&=\mu^-_W\bigl(\Phi_E(\Omega_{r(E)})\bigr)\\ &\le(1-q_0)^{\lvert L'^-\rvert}\le e^{-28q_0\lvert W\rvert}. \end{split} \] As $\kappa(E)+s(E)\le6.0\cdot10^{-13}$, $r(E)\le1.875\cdot10^{-14}$ and $2843.5\,r(E)^2<10^{-24}$, so $q_0\ge\frac13e^{-10^{-24}\beta_W}$; taking the mean over $E$ gives the bound. (i) For $\beta_W\le10^{24}$, $q_0\ge \frac13e^{-1}>0.12262$ and $28\cdot0.12262>3.433$; as $e^3>20.08$ and $e^{0.433}>1.54$, $e^{3.433}>30.9>1/0.033$. At $\beta_W=2.3\cdot10^{17}$, $q_0\ge\frac13e^{-2.3\cdot10^{-7}}\ge\frac13(1-2.3\cdot10^{-7})$, and $28q_0>9.333$. (ii) is the union bound. (iii) On that set $e^{-28q_0}\ge e^{-28q_0\lvert W\rvert}\ge\frac12$, so $q_0\le\frac{\ln2}{28}$ and $e^{-10^{-24}\beta_W}\le\frac{3\ln2}{28}$. As $\ln2<0.69315$, $\frac{28}{3 \ln2}>\frac{28}{2.07945}>13.465$, while $e^{2.6}<7.38906\cdot1.82212< 13.4638$; so $10^{-24}\beta_W>2.6$. \end{proof} The bound holds for every window, so it holds block by block: whatever the exterior and the coarse field, a single block lies on a chart of this kind with conditional probability below $0.033$ for $\beta_W\le10^{24}$, and below $\frac12$ only if $\beta_W>2.6\cdot10^{24}$, and $J\le15$ charts for each exterior cover it with probability below $\frac12$. A decomposition of the fibre of a component into small-field regions, each carrying such a chart, and a large-field region, as in Remark~\ref{rem:badmassopen}(a), has the large-field region carrying all but at most $J\,0.033^{\lvert W'\rvert}$ of the conditional mass of each window $W'$. The cost per block of a bound built on these charts is therefore not small at $\beta_W\le10^{24}$, and a bound uniform in the length must control the response on the large-field region, which is the regime of Proposition~\ref{prop:fold}. The theorem does not cover charts whose background depends on the links of the window itself, such as a background chosen nearest to the configuration from a continuous family. The second point is the existence of backgrounds for the exterior $R(c)$ between $6.0\cdot10^{-13}$ and $2.418\cdot10^{-12}$. In the next proposition the curvature of one coarse plaquette is spread over the fine plaquettes of its four corner blocks, with every link in the torus $\{e^{ \varphi T_0}\}$. On such a field a link is given by a real angle, a plaquette at $x$ in the plane $\{\mu,\nu\}$ has the real sum $\varphi_{x, \mu}+\varphi_{x+e_\mu,\nu}-\varphi_{x+e_\nu,\mu}-\varphi_{x,\nu}$, and a path the sum of its two angles. \begin{proposition}[spreading a coarse plaquette]\label{prop:spread} Let $G=\SU(2)$, let the periodic lattice have side $2L$, let $\mu<\nu$ be two directions, $0<\theta\le1.7\cdot10^{-12}$, and let $M\ge4$ and $L\ge M+6$ be integers. Let $a(0)=0$, $a(1)=a(2)=a(3)=\theta$, $a(k)=\theta(M+3-k)/M$ for $3\le k\le M+3$ and $a(k)=0$ for $M+3\le k\le L-1$, indices modulo $L$. Let $c^\theta$ be the coarse field with $c^\theta_{y,\nu}=e^{a(y_\mu)T_0}$ for $y_\nu=0$ and every other coarse link $I$, and $W_\theta=\{y:y_\mu,y_\nu \in\{0,1\}\}$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The coarse plaquettes of $c^\theta$ other than $I$ lie in the plane $\{\mu,\nu\}$ at blocks with $y_\nu=0$: $c^\theta_P=e^{\theta T_0}$ for $y_\mu=0$, so that $\delta(c^\theta_P,I)=t:=2\sqrt2\sin\frac\theta2$, and $c^\theta_P=e^{-(\theta/M)T_0}$ for $3\le y_\mu\le M+2$, with $\delta( c^\theta_P,I)\le\sqrt2\theta/M$. The set $W_\theta$ consists of the corner blocks of the first. \item Let $U^\theta$ be the field with the angles of Table~\ref{tab:spread}\textup{(a)} on the links listed there, at every value of the coordinates other than $\mu$ and $\nu$, and equal to $R(c^\theta)$ elsewhere. Then $U^\theta$ vanishes on the tree links, agrees with $R( c^\theta)$ off $A_{W_\theta}$, has $V^\sharp_k(U^\theta)=c^\theta_k$ for every pair $k$ based in $W_\theta$, and every plaquette has $\delta( U^\theta_p,I)\le\kappa_\theta:=2\sqrt2\sin\frac\theta8$. So $U^\theta$ lies in the fibre over $W_\theta$ of $R(c^\theta)$. \item Let $U^{\theta,-}$ be the field with the angles of Table~\ref{tab:spread}\textup{(b)}, and $R(c^\theta)$ elsewhere. Then $U^{\theta,-}$ vanishes on the tree links, agrees with $R(c^\theta)$ off $A^-_{W_\theta}$, has $V^\sharp_k(U^{\theta,-})=c^\theta_k$ for every pair $k$, and every plaquette has $\delta(U^{\theta,-}_p,I)\le\kappa'_\theta:= 2\sqrt2\sin\frac\theta7$. So for every set $W\supset W_\theta$, $U^{\theta, -}$ lies in the reduced fibre over $W$ of $R(c^\theta)$, and in its fibre. \item Let $U$ be any field with values in the torus $\{e^{\varphi T_0}\}$, $y$ a block, $W'=\{y+ie_\mu+je_\nu:0\le i\le2,\ 0\le j\le1\}$, and suppose that $\delta(U_p,I)\le\kappa\le10^{-8}$ for every plaquette with a vertex in a block of $W'^+$. Write $U_p=e^{a_pT_0}$ with $a_p\in(-\pi,\pi]$, and let $S_x$ be the four plaquettes of the square of side two at $x$ in the plane $\{\mu,\nu\}$. Then the coarse plaquettes $P$ at $y$ and $P'$ at $y+e_\mu$ in that plane of $c=V^\sharp(U)$ satisfy \[ c_P=\exp\Bigl(\frac1{16}\sum_{x\in B(y)}\sum_{p\in S_x}a_p\,T_0\Bigr), \] $c_{P'}$ is given by the same formula with $B(y+e_\mu)$ in place of $B(y)$, and if $c_{P'}=c_P^{-1}$, then $\delta(c_P,I)\le3\kappa$. \end{enumerate} In particular, for every $t\in(0,2.39\cdot10^{-12}]$, with $\theta=2\arcsin (t/(2\sqrt2))$, the fibre over $W_\theta$ of $R(c^\theta)$ contains a field at which it satisfies the hypotheses of Theorem~\ref{thm:badcross} with $s=2.4\cdot10^{-15}$; and for every $t\in(0,2.09\cdot10^{-12}]$ and every $W\supset W_\theta$, the reduced fibre over $W$ of $R(c^\theta)$ contains a field at which it satisfies them, in the sense of Proposition~\ref{prop:badpassage}\textup{(ii)}, with the same $s$. \end{proposition} \begin{table}[t] \caption{\label{tab:spread}The fields of Proposition~\ref{prop:spread}, in units of $\theta$, at the sites with coordinates $x_\mu$ and $x_\nu$: the angles of the links in the direction $\mu$ for $0\le x_\nu\le3$, of the links in the direction $\nu$ at $x_\nu=1$, and the real sums of the plaquettes in the plane $\{\mu,\nu\}$ with corner $x$. Every other link has the angle of $R(c^\theta)$: $0$ in the direction $\mu$, and $a(y_\mu)/\theta$ for the links $(x,\nu)$ with $x_\nu=1$, $x\in B(y)$, and $0$ on every other link.} \centering\footnotesize \renewcommand{\arraystretch}{1.15} \begin{tabular}{@{}l@{\ \ }r@{\ \ }rrrrrr@{}} \toprule & & \multicolumn{6}{c}{$x_\mu$}\\ & $x_\nu$ & $-1$ & $0$ & $1$ & $2$ & $3$ & $4$\\ \midrule \multicolumn{8}{@{}l}{(a) $U^\theta$}\\ links $\mu$ & $3$ & $0$ & $-\frac14$ & $0$ & $-\frac14$ & $\frac18$ & $0$\\ & $2$ & $0$ & $0$ & $\frac14$ & $0$ & $0$ & $0$\\ & $1$ & $0$ & $-\frac14$ & $0$ & $-\frac14$ & $\frac18$ & $0$\\ & $0$ & $0$ & $0$ & $\frac14$ & $0$ & $0$ & $0$\\ links $\nu$ & $1$ & $0$ & $-\frac14$ & $\frac14$ & $\frac34$ & $\frac54$ & $1$\\ plaquettes & $3$ & $0$ & $-\frac14$ & $0$ & $-\frac14$ & $\frac18$ & \\ & $2$ & $0$ & $\frac14$ & $\frac14$ & $\frac14$ & $-\frac18$ & \\ & $1$ & $-\frac14$ & $\frac14$ & $\frac14$ & $\frac14$ & $-\frac18$ & \\ & $0$ & $0$ & $\frac14$ & $\frac14$ & $\frac14$ & $-\frac18$ & \\ & $-1$ & $0$ & $0$ & $-\frac14$ & $0$ & $0$ & \\ \midrule \multicolumn{8}{@{}l}{(b) $U^{\theta,-}$}\\ links $\mu$ & $3$ & $0$ & $0$ & $-\frac1{14}$ & $-\frac27$ & $0$ & $0$\\ & $2$ & $0$ & $0$ & $\frac3{14}$ & $0$ & $\frac17$ & $0$\\ & $1$ & $0$ & $0$ & $-\frac1{14}$ & $-\frac27$ & $\frac17$ & $0$\\ & $0$ & $0$ & $0$ & $\frac3{14}$ & $0$ & $0$ & $0$\\ links $\nu$ & $1$ & $0$ & $-\frac17$ & $\frac17$ & $\frac57$ & $\frac97$ & $1$\\ plaquettes & $3$ & $0$ & $0$ & $-\frac1{14}$ & $-\frac27$ & $0$ & \\ & $2$ & $0$ & $0$ & $\frac27$ & $\frac27$ & $\frac17$ & \\ & $1$ & $-\frac17$ & $\frac27$ & $\frac27$ & $\frac27$ & $-\frac27$ & \\ & $0$ & $0$ & $0$ & $\frac27$ & $\frac27$ & $-\frac17$ & \\ & $-1$ & $0$ & $0$ & $-\frac3{14}$ & $0$ & $0$ & \\ \bottomrule \end{tabular} \end{table} \begin{proof} (i) A coarse plaquette at $y$ in a plane $\{\rho,\sigma\}$ is $c_{y,\rho} c_{y+e_\rho,\sigma}c_{y+e_\sigma,\rho}^{-1}c_{y,\sigma}^{-1}$. Only the links in the direction $\nu$ with $y_\nu=0$ differ from $I$, and two links in the direction $\nu$ based at blocks that differ in a coordinate other than $\mu$ and $\nu$ are equal; so only the plaquettes in the plane $\{\mu, \nu\}$ at $y_\nu=0$ can differ from $I$, and there $c^\theta_P=e^{(a(y_\mu +1)-a(y_\mu))T_0}$. The increments of $a$ are $\theta$ at $y_\mu=0$, $-\theta/M$ for $3\le y_\mu\le M+2$, and $0$ otherwise, also at $y_\mu= L-1$, where $a(L-1)=a(0)=0$ as $L-1\ge M+3$. Finally $2\sqrt2\sin x\le2 \sqrt2x$. (ii), (iii) We write $\varphi$ for the angles of the field in question and $\varphi^R$ for those of $R(c^\theta)$, all real. By Proposition~\ref{prop:badchart}\textup{(i)} and its proof, $\varphi^R$ is $0$ on every link in the direction $\mu$ and on every link in the direction $\nu$ except $(x,\nu)$ with $x_\nu=1$, where it is $a(y_\mu)$, $x\in B(y)$, and all sixteen paths of a pair $(y,\rho)$ have the angle $\vartheta_{y,\rho}$ of $c^\theta_{y,\rho}$, which is $a(y_\mu)$ or $0$. \emph{Links.} The field differs from $R(c^\theta)$ only on the links of the table with an entry different from that of $R(c^\theta)$, and the entries do not depend on the other two coordinates. These are links $(x,\nu)$ with $x_\nu=1$ and $0\le x_\mu\le3$, which have $o_\nu=1$ and are private links of pairs based in $W_\theta$, and links $(x,\mu)$ with $0\le x_\mu,x_\nu \le3$, which are private links of such pairs when $x_\mu$ is odd; when $x_\mu$ is even the entry is nonzero only for $x_\nu$ odd, and then $o_\nu =1$ with $\nu>\mu$, so the link is an active link inside a block of $W_\theta$, not a tree link. In (b) the links with $x_\mu$ even have $x_\mu =2$, in a block $y$ with $y-e_\mu\in W_\theta$, so they are not in $I_{W_\theta}$. Hence $U^\theta$ agrees with $R(c^\theta)$ off $A_{W_\theta}$ and $U^{\theta,-}$ off $A^-_{W_\theta}$, and both vanish on the tree links. For $W\supset W_\theta$, $A^-_{W_\theta}\subset A^-_W$: a link of $A_{W_\theta}$ in the direction $\rho$ that lies in $I_W$ is inside a block $y\in W_\theta$ with $y-e_\rho\notin W$, hence $y-e_\rho\notin W_\theta$, and so lies in $I_{W_\theta}$. \emph{Averages.} Let $D_x=\varphi(P_x)-\vartheta_{y,\rho}$ for the paths $P_x$ of a pair $(y,\rho)$. Every entry differs from that of $R(c^\theta)$ by at most $\frac27\theta$, so $\lvert\varphi(P_x)-\varphi(P_{2y})\rvert\le \frac87\theta$ and $\delta(P_x,P_{2y})=2\sqrt2\sin\frac{\lvert\varphi(P_x)- \varphi(P_{2y})\rvert}2<4\theta<\frac12$. The cut-off of~\eqref{eq:cutoff} equals one, $\rho_G=1$, and as these differences are less than $\pi$ the principal logarithm of $P_{2y}^{-1}P_x$ is $(\varphi(P_x)-\varphi(P_{2y})) T_0$. So \[ V^\sharp_{y,\rho}=\exp\Bigl(\Bigl(\vartheta_{y,\rho}+\frac1{16}\sum_{x\in B(y)}D_x\Bigr)T_0\Bigr), \] and $V^\sharp_{y,\rho}=c^\theta_{y,\rho}$ when $\sum_xD_x=0$. For $\rho \notin\{\mu,\nu\}$ every $D_x$ vanishes. For $\rho=\nu$, $D_x\ne0$ needs a link $(x',\nu)$ with $x'_\nu=1$ on the path, so only the pairs with $y_\nu =0$ are concerned, and by Lemma~\ref{lem:blocktree}\textup{(iii)} $\sum_xD_x =8(d(2y_\mu)+d(2y_\mu+1))$, $d(m)$ the deviation of the link $(x,\nu)$ with $x_\mu=m$, $x_\nu=1$. The deviations are $-\frac14,\frac14,-\frac14,\frac14$ in (a) and $-\frac17,\frac17,-\frac27,\frac27$ in (b) for $m=0,1,2,3$, and $0$ otherwise, so the sums vanish. For $\rho=\mu$, let $X(m)$ be the sum of the deviations of the two links $(x,\mu)$ with $x_\mu=m$ and $x_\nu\in\{2 y_\nu,2y_\nu+1\}$; by Lemma~\ref{lem:blocktree}\textup{(iii)}, $\sum_xD_x= 4\bigl(X(2y_\mu)+2X(2y_\mu+1)+X(2y_\mu+2)\bigr)$. For $y_\nu\in\{0,1\}$ and $m=-1,\dots,4$, $X(m)$ is $0,-\frac14,\frac14,-\frac14,\frac18,0$ in (a) and $0,0,\frac17,-\frac27,\frac17,0$ in (b), and $X(m)=0$ for other $m$ or $y_\nu$. The combination vanishes for $y_\mu\in\{0,1\}$ in (a), and for every $y$ in (b). \emph{Plaquettes.} A plaquette in a plane other than $\{\mu,\nu\}$ has two parallel links that differ only in a coordinate other than $\mu$ and $\nu$ and two links in a direction other than $\mu$ and $\nu$; the first two have equal angles and the other two the angle $0$, so its real sum is $0$. In the plane $\{\mu,\nu\}$ the real sum at $x$ is read off the table; the table lists every plaquette that contains a link at which the field differs from $R(c^\theta)$, and all other plaquettes are those of $R(c^\theta)$, whose real sums are $0$ or $a(y_\mu+1)-a(y_\mu)\in\{0,-\theta/M\}$ with $\lvert\theta/M\rvert\le\frac \theta4$, by Proposition~\ref{prop:badchart}\textup{(i)}. So every real sum has modulus at most $\frac\theta4$ in (a) and $\frac{2\theta}7$ in (b), and $\delta(U_p,I)=2\sqrt2\sin\frac{\lvert\cdot\rvert}2$ gives the bounds. (iv) For a pair $k$ based in $W'$ let $v=P_{2y}$ be its path link. By Lemma~\ref{lem:smallframes}\textup{(i)}, $\delta(v^{-1}P_x,I)\le24\kappa< \frac12$, so the cut-off is one and $c_k=v\exp(\frac1{16}\sum_x\psi_x T_0)$, where $v^{-1}P_x=e^{\psi_xT_0}$ with $\psi_x\in(-\pi,\pi]$ and $\lvert\psi_x\rvert\le27\kappa$, since $2\sqrt2\sin\frac{\lvert\psi\rvert}2 \ge\frac{2\sqrt2}\pi\lvert\psi\rvert$. Choose real angles $\varphi_k$ of the path links. Then $c_k=e^{\gamma_kT_0}$ with $\gamma_k=\varphi_k+\frac1{16} \sum_x\psi_x$, and with the sides $k=1,\dots,4$ of $P$ and signs $\epsilon_k =1,1,-1,-1$, $c_P=e^{\Theta T_0}$ with \[ \Theta=\sum_k\epsilon_k\gamma_k=\frac1{16}\sum_{x\in B(y)}\Theta_x,\qquad \Theta_x=\sum_k\epsilon_k\bigl(\varphi_k+\psi^{(k)}_{x_k}\bigr), \] where $x_k\in\{x,x+2e_\mu,x+2e_\nu,x\}$ is the site of the same offset as $x$ in the block of side $k$. The term $\varphi_k+\psi^{(k)}_{x_k}$ is an angle of the path $P^{(k)}_{x_k}$, and on the torus the product of the four paths around the square of side two at $x$ is the product of the four plaquettes of $S_x$, so $\Theta_x\equiv\sigma_x:=\sum_{p\in S_x}a_p$ modulo $2\pi$. Now $\lvert a_p\rvert\le1.12\kappa$, since $2\sqrt2\sin\frac{\lvert a\rvert}2\ge\frac{2\sqrt2}\pi\lvert a\rvert$, so $\lvert(\Theta_x-\sigma_x)- (\Theta_{2y}-\sigma_{2y})\rvert\le4\cdot27\kappa+8\cdot1.12\kappa<2\pi$, as $\psi^{(k)}_{2y}=0$. Hence $\Theta_x-\sigma_x$ is the same multiple of $2\pi $ for all $x$, and $\Theta$ differs from $\frac1{16}\sum_x\sigma_x$ by a multiple of $2\pi$, which gives the formula for $c_P$; the same holds for $P'$. Let $\Theta_P$, $\Theta_{P'}$ be the two exponents. A plaquette at the corner $z$, $z_\mu=2y_\mu+i$ and $z_\nu=2y_\nu+j$, lies in $S_x$ for $n(i)n(j)$ sites $x$ of $B(y)$ with the coordinates of $z$ other than $\mu$ and $\nu$, and for $n(i-2)n(j)$ sites of $B(y+e_\mu)$, where $n(0)=n(2)=1$, $n(1)=2$ and $n=0$ otherwise. So \[ \begin{split} \Theta_P-\Theta_{P'}&=\frac1{16}\sum_zw_za_{p_z},\\ \sum_z\lvert w_z\rvert&=(1+2+0+2+1)(1+2+1)\cdot4=96, \end{split} \] the factor $4$ counting the offsets in the other two directions. If $c_{P'}=c_P^{-1}$, then $\Theta_P+\Theta_{P'}\in2\pi\Z$, and as $\lvert \Theta_P\rvert,\lvert\Theta_{P'}\rvert\le4\cdot1.12\kappa<\pi$ it vanishes. So $2\lvert\Theta_P\rvert\le6\alpha$ with $\alpha=\max\lvert a_p\rvert$, and as $\sin3x\le3\sin x$ for $0\le x\le\frac\pi6$, \[ \delta(c_P,I)=2\sqrt2\sin\frac{\lvert\Theta_P\rvert}2\le2\sqrt2\sin\frac{ 3\alpha}2\le3\cdot2\sqrt2\sin\frac\alpha2\le3\kappa . \] For the last statement, $\sin x\ge x(1-x^2/6)$ gives $\theta<1.7\cdot10^{- 12}$ for $t\le2.39\cdot10^{-12}$. In (ii), $\sin\frac\theta2=4\sin\frac \theta8\cos\frac\theta8\cos\frac\theta4$, so $\kappa_\theta=t/(4\cos\frac \theta8\cos\frac\theta4)\le t/(4(1-\theta^2))<5.976\cdot10^{-13}$. In (iii), $t\ge\sqrt2\theta(1-\theta^2/24)$, so $\kappa'_\theta\le\frac{2\sqrt2}7 \theta\le\frac27t(1+\theta^2/12)<5.972\cdot10^{-13}$ for $t\le2.09\cdot 10^{-12}$. Every plaquette is below $\kappa_\theta\le10^{-8}$, respectively $\kappa'_\theta$, so the plaquette condition holds on $W^+$ for every $W$; and $s=2.4\cdot10^{-15}$ satisfies $s\le1/(6000c_3)$, $\kappa+6s\le10^{-8}$ and $\kappa+s\le6.0\cdot10^{-13}$. \end{proof} The fields of (ii) and (iii) spread the coarse curvature $\theta$ over the plaquettes of the corner blocks with the largest real sum $\frac\theta4$ and $\frac{2\theta}7$; by Lemma~\ref{lem:coarsewitnesssharp} no field does better than $\frac\theta4$ up to a term of order $\theta^2$, so (ii) is sharp up to that term. The gap between $2.39\cdot10^{-12}$ and $2.418\cdot10^{-12}$ is the rounding of the chart threshold $6.044\cdot10^{-13}$ of Proposition~\ref{prop:badchart}\textup{(ii)} to $6.0\cdot10^{-13}$ in Theorem~\ref{thm:badcross}: for every threshold $\kappa_*$ in place of $6.0\cdot10^{-13}$, (ii) gives a background whenever $\delta(c_P,I)<4 \kappa_*(1-\theta^2)$ and Lemma~\ref{lem:coarsewitnesssharp} excludes one above $4\kappa_*+7700\kappa_*^2$. Part (iv) shows that the factor depends on the neighbouring coarse plaquettes: for a coarse field with opposite fluxes at two adjacent coarse plaquettes and $\delta(c_P,I)>1.8\cdot10^{-12}$, no field with values in a torus has every plaquette near the six blocks below $6.0\cdot10^{-13}$. The third point is the ends that enter the averages. They meet the fibre measure in one factor. \begin{lemma}[ends and pairs]\label{lem:endpairs} Let $W$ be a set of blocks and $e$ an active link off $A^-_W$ that enters an average $V^\sharp_{y,\rho}$ with $y\in W$. Then $e=(x,\rho)$ with $x_\rho$ even, and either \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item $x\in B(y)$, $y-e_\rho\notin W$, $e\in I_W$, and $e$ is the first link of the path $P_x$, whose second link is the private link $(x+e_\rho,\rho)$ of $(y,\rho)$; or \item $x\in B(y+e_\rho)$, $y+e_\rho\notin W$, and $e$ is the second link of $P_{x-e_\rho}$, whose first link is the private link $(x-e_\rho,\rho)$ of $(y,\rho)$. \end{enumerate} In either case $(y,\rho)$ is the only pair based in $W$ whose average contains $e$, $e$ lies on one of its sixteen paths, and among the factors of $\nu^-_c$ over the pairs based in $W$ only the factor of $(y,\rho)$ depends on $e$. \end{lemma} \begin{proof} A private link enters only the average of its own pair, and the private links of pairs based in $W$ lie in $A^-_W$; so $e$ is an inside link $(x,\rho)$, $x_\rho$ even, of a block $B(y')$. As in the proof of Proposition~\ref{prop:badpassage}\textup{(i)}, it enters only the averages of $(y',\rho)$, as the first link of $P_x$, and of $(y'-e_\rho,\rho)$, as the second link of $P_{x-e_\rho}$. If $y'\in W$, then $e\in A_W\setminus A^-_W= I_W$, so $y'-e_\rho\notin W$ and $y=y'$, which is (a). If $y'\notin W$, the pair is $(y'-e_\rho,\rho)$, which is (b). The other link of the path has offset $1$ in the direction $\rho$ and joins $B(y)$ to $B(y+e_\rho)$, so it is a private link of $(y,\rho)$. Given the inside links and the links off $A^-_W$, the averages of the pairs based in $W$ depend on disjoint sets of private links, and $\nu^-_c$ is the product of the Haar measure of the inside links in $A^-_W$ and of one factor for each pair, built from its average as before~\eqref{eq:fibrevolume}; only the factor of $(y,\rho)$ contains $e$. \end{proof} \begin{proposition}[an end at the fold]\label{prop:endfold} Let $G=\SU(2)$, $\mu\le2$, let $W$ be a set of blocks, $y\in W$ with $y-e_\mu \notin W$, $k=(y,\mu)$, and $c_k\in\SU(2)$. There are values of the links of the pair $k$ and of the links inside $B(y)$ and $B(y+e_\mu)$, with an active inside link $u_b$ of $B(y)$ in the direction $\mu$, which lies in $I_W$ and is an end of case \textup{(a)} of Lemma~\ref{lem:endpairs}, moved along a one-parameter family $u_b(t)$, such that for every $\beta_W\ge0$, every value of the other links and every $\varepsilon>0$ there are $t<00$ and tends to infinity as $t\uparrow0$, so the derivative of $\log Z$ in the end, at fixed non-path links, is not bounded. \begin{remark}\label{rem:badendsopen} What remains of the three points. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item \emph{Length.} By Theorem~\ref{thm:localcharts}, charts fixed by the exterior of their window carry less than half of the conditional mass of each block for $\beta_W\le2.6\cdot10^{24}$, and less than $0.033$ of it for $\beta_W\le10^{24}$. A bound uniform in the length needs the response on the large-field region of the fibre, or charts whose background moves with the window links, together with a bound on the mass of their union; neither is available. \item \emph{Coarse fields.} For the coarse fields $c^\theta$ the exterior $R(c)$ carries a background up to $2.39\cdot10^{-12}$, and the reduced fibre up to $2.09\cdot10^{-12}$ for every window containing the component (Proposition~\ref{prop:spread}). For a general coarse field between $6.0 \cdot10^{-13}$ and $2.418\cdot10^{-12}$ with exterior $R(c)$, and for the reduced fibre between $2.09\cdot10^{-12}$ and $2.418\cdot10^{-12}$, the existence of a background is not decided; by Proposition~\ref{prop:spread}\textup{(iv)} a background with values in a torus does not exist above $1.8\cdot10^{-12}$ for opposite fluxes at adjacent coarse plaquettes. \item \emph{Ends.} A bound of the form of Corollary~\ref{cor:badcrossratio} for ends in the averages cannot hold at fixed non-path links, by Proposition~\ref{prop:endfold}. The missing input is a bound, uniform in $c_k$, on the derivative in an end of the logarithm of its pair factor after integration over the seven non-path links, where the coarea formula keeps the normalisation finite, together with its locality in the distance between two ends. \end{enumerate} \end{remark} The three points of Remark~\ref{rem:badendsopen} are taken up below without a chart. The pair factors of the fibre measure are within $3.1\cdot10^{-4}$ of Haar measure in total mass, whatever the links that bind them; this bounds the weight of the interior as a function of an end that enters an average, after the non-path links are integrated, and gives a decay of the constrained law along a tube of blocks at every $\beta_W$. The coarse fields with opposite fluxes at adjacent coarse plaquettes are decided up to a rounding gap. \begin{lemma}[the pair factors are almost Haar]\label{lem:almosthaar} Let $G=\SU(2)$ and $q=h(\{v:\delta(v,I)\le1\})$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $q=(\alpha-\frac{3\sqrt7}{16})/\pi<0.07447$ and $q^7<1.3\cdot10^{-8}$. For every pair $k$, every $c_k$ and every value of the links other than its eight private links, the density~\eqref{eq:densityformula} satisfies \[ 1-1.3\cdot10^{-8}\le1-q^7\le\rho(c_k)\le1+23640\,q^7<1+3.1\cdot10^{-4}. \] \item For every $c$, $(1-1.3\cdot10^{-8})^{4\lvert W\rvert}\le\nu_c(F_c)\le (1+3.1\cdot10^{-4})^{4\lvert W\rvert}$, and the same bounds hold for the mass of $\nu^-_c$. \item Let $E$ be a set of links off $A^-_W$, $K_E$ the set of pairs based in $W$ whose average contains a link of $E$, and $n_E$ the number of plaquettes that contain a link of $A^-_W$ and either a link of $E$ or a private link of a pair of $K_E$. Fix the other links off $A^-_W$, and let $Z(e)=\nu^-_c(e^{-\lambda Q^-})$ with the value $e$ on $E$. For any two values $e,e'$, \[ \Bigl\lvert\log\frac{Z(e)}{Z(e')}\Bigr\rvert\le2\beta_Wn_E+3.2\cdot10^{-4} \lvert K_E\rvert . \] \end{enumerate} \end{lemma} \begin{proof} (i) By Lemma~\ref{lem:haarsu2}\textup{(iii)}, $\delta(v,I)\le1$ exactly when the eigenangle of $v$ is at most $\alpha$, and $\cos\alpha=1-2\sin^2\frac \alpha2=\frac34$, $\sin\alpha=\frac{\sqrt7}4$; Lemma~\ref{lem:haarsu2}% \textup{(ii)} gives the formula for $q$. Since $\cos x\le1-\frac{x^2}2+\frac {x^4}{24}$, $\cos0.73\le1-0.26645+0.011833<\frac34$, so $\alpha<0.73$; and $\sqrt7>2.6457$ gives $\frac{3\sqrt7}{16}>0.49606$. Hence $q<0.23394/\pi< 0.07447$, and $0.07447^2<0.005546$, $0.005546^3<1.706\cdot10^{-7}$, $1.706\cdot10^{-7}\cdot0.07447<1.3\cdot10^{-8}$. Use the regions and coordinates of the proof of Lemma~\ref{lem:averagedensity}. On $R_0$ the average is the path link, so the first term of~\eqref{eq:densityformula} is the Haar measure of the set of $(\ell_a)_{a\ne0}$ with $(c_k,\ell)\in R_0$. Its complement lies in the set where $\delta(v_{x_a},I)\le1$ for $a=1,\dots,7$, and with $g=c_k$ fixed the maps $\ell_a\mapsto v_{x_a}=c_k^{-1}u_a\ell_a$ carry Haar measure to Haar measure independently, so the complement has measure at most $q^7$ and the first term lies in $[1-q^7,1]$. In the term of $R_1$ the coordinates $h_1,\dots,h_7$ are independent and Haar distributed, the integrand vanishes unless $\max_a\delta(h_a,I)<1$, which has measure at most $q^7$, and the inner sum is at most $(1-\frac78\Lambda)^{-3}$; the same holds for $R_2$ with $h'_a=v_{x'_a}$. As $1-\frac78\Lambda>0.0439$ and $0.0439^3>8.46\cdot 10^{-5}$, $(1-\frac78\Lambda)^{-3}<11820$, and $23640\cdot1.3\cdot10^{-8}< 3.1\cdot10^{-4}$. (ii) The measure $\nu_c$ is the product of Haar measure on the links inside the blocks and of one factor for each pair, of total mass $\rho(c_k)$ at every value of the inside links, and likewise $\nu^-_c$. (iii) By Lemma~\ref{lem:endpairs} and the construction before~\eqref{eq:fibrevolume}, $\nu^-_c$ is the product of Haar measure on the inside links of $A^-_W$, of the factors $\nu_k$ of the pairs $k\notin K_E$, which do not depend on $e$, and of the factors $\nu^{(e)}_k$, $k\in K_E$, each a measure on the private links of $k$ of total mass $\rho(c_k)$. Write $\lambda Q^-=\beta_W(S_E+S_{\rm r})$, with $S_E$ the sum of $1-\frac12\ReTr U_p$ over the $n_E$ plaquettes and $S_{\rm r}$ over the other plaquettes that contain a link of $A^-_W$; $S_{\rm r}$ depends neither on $e$ nor on the private links of the pairs of $K_E$, and $0\le S_E\le 2n_E$. Integrating the private links of the pairs of $K_E$ first, \[ e^{-2\beta_Wn_E}\int\prod_{k\in K_E}\rho(c_k)\,d\Xi\le Z(e)\le\int\prod_{k \in K_E}\rho(c_k)\,d\Xi, \] with $d\Xi=e^{-\beta_WS_{\rm r}}$ times the product of Haar measure on the inside links and the factors $\nu_k$, $k\notin K_E$, which does not depend on $e$. By (i) the product of the $\rho(c_k)$ lies in $[(1-1.3\cdot10^{-8} )^{\lvert K_E\rvert},(1+3.1\cdot10^{-4})^{\lvert K_E\rvert}]$ at every value of the inside links and of $e$, and $\log(1+3.1\cdot10^{-4})-\log(1-1.3\cdot 10^{-8})<3.2\cdot10^{-4}$. \end{proof} For an end of Proposition~\ref{prop:endfold}, $\lvert K_E\rvert=1$: the law of the path link given the non-path links jumps under arbitrarily small moves of the end, while after the non-path links are integrated the pair factor changes its mass by less than $3.2\cdot10^{-4}$ under every move, since the cut-off acts only where seven paths lie within distance one of the path link, a set of Haar measure at most $q^7$ in the coordinates of each region. The bound in (iii) depends on $\beta_W$ only through the Wilson weight of the plaquettes through the end and through the private links of its pair. The same splitting along a tube of blocks gives decay at every $\beta_W$. Fix a direction $\rho$ and a set $W$ of blocks for which some value $j_0$ of the coordinate $\rho$ satisfies $y_\rho\ne j_0$ for every $y\in W$. Number the other values of $y_\rho$ by $1,\dots,L-1$ in their cyclic order starting after $j_0$, and let $\Sigma_i=\{y\in W:y_\rho=i\}$ be the slabs of $W$, $K_i$ the pairs $(y,\rho)$ with $y\in\Sigma_i$, and \[ D_i=\lvert\Sigma_i\rvert\bigl(72\beta_W+3.2\cdot10^{-4}\bigr). \] Every link of $A^-_W$ has its base in a block of $W$; we say that it lies in the slab of that block. \begin{theorem}[decay along a tube]\label{thm:tube} Let $G=\SU(2)$, $\beta_W\ge0$, let $W$, $\rho$ and the slabs be as above, and let $1\le a$, $a+2\le b\le L-1$. Let $\tilde F$ be a bounded measurable function of the links of $A^-_W$ in the slabs $1,\dots,a$ and $\tilde G$ one of the links of $A^-_W$ in the slabs $b,\dots,L-1$. Then \[ \bigl\lvert\Cov_{\mu^-_W}(\tilde F,\tilde G)\bigr\rvert\le2\lVert\tilde F\rVert_\infty\lVert \tilde G\rVert_\infty\prod_{i=a+1}^{b-1}\bigl(1-e^{-D_i}\bigr), \] for every coarse field and every exterior, and with no condition on the plaquettes. The same bound holds for $\mu_W$, with $A_W$ in place of $A^-_W$. \end{theorem} \begin{proof} Let $C_i$ be the set of private links of the pairs of $K_i$, which are the links $(x,\rho)$ with $o_\rho=1$ and $x$ in a block of $\Sigma_i$, and $S_i$ the other links of $A^-_W$ in the slab $i$. \emph{Plaquettes.} Let $p$ be a plaquette in a plane $\{\rho,\tau\}$ with corner $z$; its links in the direction $\rho$ are based at $z$ and $z+e_\tau$, and those in the direction $\tau$ at $z$ and $z+e_\rho$. The blocks of $z$ and $z+e_\tau$ have the same coordinate $\rho$, say $i$. If $z_\rho$ is odd, the two links in the direction $\rho$ have $o_\rho=1$, so those of them in $A^-_W$ lie in $C_i$, the link at $z$ lies in the slab $i$ and the link at $z+e_\rho$ in the slab $i+1$; neither is in a set $C_j$, as their direction is $\tau\ne\rho$. If $z_\rho$ is even, $z+e_\rho$ lies in the block of $z$, all four links lie in the slab $i$, and none lies in a set $C_j$. A plaquette in a plane without $\rho$ has its four bases in blocks with the same coordinate $\rho$ and no link in a set $C_j$. So a plaquette either contains links of exactly one set $C_i$, and then its other links of $A^-_W$ lie in $S_i\cup S_{i+1}$, or has its links of $A^-_W$ in one set $S_i$. \emph{Pairs.} The average of a pair $(y,\sigma)$, $y\in\Sigma_i$, depends on its private links, on the inside links of $B(y)$ in the direction $\sigma$, and on those of $B(y+e_\sigma)$ in the direction $\sigma$. For $\sigma\ne\rho$ all of them lie in $S_i$ or off $A^-_W$. For $\sigma=\rho$ the private links lie in $C_i$, the links of $B(y)$ in $S_i$ or in $I_W$, and the links of $B(y+e_\rho)$ in $S_{i+1}$ or off $A_W$. \emph{Integrating the crossing links.} For each pair $k$ based in $W$ let $\nu_k$ be its factor of $\nu^-_c$, a measure on its private links of mass $\rho(c_k)$ that depends on the other links of its paths. Let $P_i$ be the set of plaquettes that contain a link of $C_i$, so $\lvert P_i\rvert\le36 \lvert K_i\rvert$ by Lemma~\ref{lem:relnorm}, and \[ \Psi_i=\int\prod_{k\in K_i}\nu_k(d\ell^k)\,\exp\Bigl(-\beta_W\sum_{p\in P_i}\bigl(1-\tfrac12\ReTr U_p\bigr)\Bigr). \] By the two structural facts, $\Psi_i$ is a function of the links of $S_i$ and $S_{i+1}$, and $\nu^-_c(e^{-\lambda Q^-}\,\cdot\,)$, with the links of all sets $C_i$ integrated, has the form \[ \prod_i\Phi_i(S_i)\,m_i(dS_i)\prod_i\Psi_i(S_i,S_{i+1}), \] where $m_i$ is the product of Haar measure on the inside links of $S_i$ and of the factors $\nu_k$ of the pairs $(y,\sigma)$, $y\in\Sigma_i$, $\sigma\ne\rho$, and $\Phi_i$ is the Wilson weight of the plaquettes whose links of $A^-_W$ lie in $S_i$ only; $\Psi_{L-1}$ depends on $S_{L-1}$ only, since the blocks $y+e_\rho$, $y\in\Sigma_{L-1}$, have $y_\rho=j_0$. Every term of the sum lies in $[0,2]$, so Lemma~\ref{lem:almosthaar}\textup{(i)} gives, at every value of $S_i$ and $S_{i+1}$, \[ (1-1.3\cdot10^{-8})^{\lvert K_i\rvert}e^{-72\beta_W\lvert K_i\rvert}\le \Psi_i\le(1+3.1\cdot10^{-4})^{\lvert K_i\rvert}, \] and as $\lvert K_i\rvert=\lvert\Sigma_i\rvert$, the ratio of the two bounds is at most $e^{D_i}$. \emph{The chain.} Put $Z_0=(C_1,S_1,\dots,C_a,S_a,S_{a+1})$, $Z_j=S_{a+1+j}$ for $1\le j\le b-a-2$, and let $Z_{b-a-1}$ consist of the links of the slabs $b,\dots,L-1$, with $C_{b-1},\dots,C_{a+1}$ integrated as above and $C_1,\dots,C_a$ kept in $Z_0$. The links of $C_a$ depend on $S_{a+1}$ through $\nu_k$ and $P_a$, which lie in $Z_0$. So $\mu^-_W$, restricted to $(Z_0,\dots,Z_{b-a-1})$, has the form \[ \frac1{\mathcal Z}\,\Gamma_0(dZ_0)\prod_{j=0}^{b-a-2}\Psi_{a+1+j}\bigl( S_{a+1+j},S_{a+2+j}\bigr)\prod_{j\ge1}\Gamma_j(dZ_j) \] with finite measures $\Gamma_j$, where $S_b$ is part of $Z_{b-a-1}$ and $C_b,\dots,C_{L-1}$ are kept in $Z_{b-a-1}$. This is a Markov chain: given $Z_j$, the law of $Z_{j+1}$ is \[ K_j(Z_j,dz)=\frac{\Psi_{a+1+j}(s,z)\,H_{j+1}(z)\,\Gamma_{j+1}(dz)}{\int \Psi_{a+1+j}(s,\cdot)\,H_{j+1}\,d\Gamma_{j+1}}, \] with $s$ the part $S_{a+1+j}$ of $Z_j$ and $H_{j+1}$ the integral of the factors beyond $Z_{j+1}$. With $m\le\Psi_{a+1+j}\le M$ and $M/m\le e^{D_{a+1+j}}$, $K_j(Z_j,\cdot)\ge e^{-D_{a+1+j}}\pi_{j+1}$ for the probability $\pi_{j+1}$ proportional to $H_{j+1}\Gamma_{j+1}$, so $\lVert K_j(z,\cdot)-K_j(z',\cdot)\rVert_{\rm TV}\le1-e^{-D_{a+1+j}}$ for all $z,z'$. Let $g(Z_0)$ be the conditional expectation of $\tilde G$ given $Z_0$, obtained by applying $K_0,\dots,K_{b-a-2}$ to $\tilde G$. Each kernel multiplies the oscillation by at most its coefficient, so $\operatorname{osc}g\le2\lVert\tilde G \rVert_\infty\prod_{i=a+1}^{b-1}(1-e^{-D_i})$, and \[ \lvert\Cov(\tilde F,\tilde G)\rvert=\bigl\lvert\E\bigl[\tilde F\,(g-\E g)\bigr]\bigr\rvert\le \lVert\tilde F\rVert_\infty\operatorname{osc}g, \] since $\tilde F$ is a function of $Z_0$. For $\mu_W$ the links of $I_W$ are free as well; each is an inside link of a block of some $\Sigma_i$ and is added to $S_i$, and the two structural facts and the rest of the proof are unchanged. \end{proof} \begin{corollary}[ends at the two sides of a tube]\label{cor:tubeends} Let $G=\SU(2)$, $\beta_W\ge0$, and let $W$, $\rho$, the slabs, $a$ and $b$ be as in Theorem~\ref{thm:tube}. Let $E_a$, $E_b$ be disjoint sets of links off $A^-_W$, which may enter averages of pairs based in $W$, such that every link of $A^-_W$ that shares a plaquette with a link of $E_a$, or is a private link of a pair based in $W$ whose average contains a link of $E_a$, lies in the slabs $1,\dots,a$, and likewise for $E_b$ with the slabs $b,\dots,L-1$. Fix the other links off $A^-_W$, let $a_0,a_1$ be values on $E_a$, $b_0,b_1$ values on $E_b$, and $Z_{ij}=\nu^-_c(e^{-\lambda Q^-})$ with $a_i$ on $E_a$ and $b_j$ on $E_b$. Then \[ \Bigl\lvert\log\frac{Z_{11}Z_{00}}{Z_{10}Z_{01}}\Bigr\rvert\le\bigl(e^{2D_a} -1\bigr)\prod_{i=a+1}^{b-1}\bigl(1-e^{-D_i}\bigr). \] \end{corollary} \begin{proof} Use the decomposition of the proof of Theorem~\ref{thm:tube}. A factor $\Phi_i$, $m_i$ or $\Psi_i$ depends on the value on $E_a$ only through a plaquette that contains a link of $E_a$ and a link of $A^-_W$, or through a pair whose average contains a link of $E_a$. By the hypothesis the links of $A^-_W$ involved lie in the slabs $1,\dots,a$, so the factor has $i\le a$, a plaquette of $P_i$ containing a link of $C_i$. Likewise the value on $E_b$ enters only factors with $i\ge b$; in particular $\Psi_{a+1},\dots,\Psi_{b-1}$, the measure $m_{a+1}$ and the weight $\Phi_{a+1}$ depend on neither. Integrating everything in $Z_0$ except $S_{a+1}$ gives \[ Z_{ij}=\int\mathcal A_{a_i}(s)\,\Phi_{a+1}(s)\,m_{a+1}(ds)\,R_{b_j}(s), \] where $\mathcal A_{a_i}(s)$ is the integral of the factors of the slabs $1,\dots,a$ with $S_{a+1}=s$ and $R_{b_j}(s)$ that of the factors beyond $S_{a+1}$. The variable $s$ enters $\mathcal A_{a_i}$ only through $\Psi_a(S_a,s)$, whose bounds in the proof of Theorem~\ref{thm:tube} hold at every value of the ends, so $\mathcal A_{a_i}(s)/\mathcal A_{a_i}(s')\le e^{D_a}$ for all $s,s'$. Let $\Phi=\mathcal A_{a_1}/\mathcal A_{a_0}$, so that $\sup\Phi\le e^{2D_a}\inf\Phi$, and let $\pi_j$ be the law of $S_{a+1}$ under the measure with $a_0$ and $b_j$. Then $Z_{1j}/Z_{0j}=\pi_j( \Phi)$. Under that measure the chain $Z_{b-a-1},\dots,Z_0$ run backwards has kernels that do not depend on $b_j$, given by the same formula with the factors before $Z_j$ in place of $H$, and with the same minorisation; so $\lVert\pi_1-\pi_0\rVert_{\rm TV}\le\theta:=\prod_{i=a+1}^{b-1}(1-e^{-D_i} )$, the passage from $Z_0$ to $S_{a+1}$ not increasing the distance. Hence $\lvert\pi_1(\Phi)-\pi_0(\Phi)\rvert\le\theta(\sup\Phi-\inf\Phi)$, both values are at least $\inf\Phi$, and $\lvert\log x-\log x'\rvert\le\lvert x-x'\rvert/\min(x,x')$ gives the bound. \end{proof} \begin{remark}\label{rem:tuberate} Theorem~\ref{thm:tube} and Corollary~\ref{cor:tubeends} need no chart and no bound on the plaquettes, and they allow ends that enter averages; the price is the rate. At $\beta_W=0$, $1-e^{-D_i}\le3.2\cdot10^{-4}\lvert\Sigma_i \rvert$, and the constrained law of a tube of thin slabs is close to a product; for $\beta_W\lvert\Sigma_i\rvert\le\frac1{72}$ and $\lvert\Sigma_i \rvert\le100$ each factor is at most $1-e^{-1.04}<0.65$. For $\beta_W\ge2.3 \cdot10^{17}$, the range of Theorem~\ref{thm:badcross}, every nonempty slab has $D_i\ge72\beta_W>1.6\cdot10^{19}$, and as $\prod_i(1-x_i)\ge1-\sum_ix_i$ the product is at least $\frac12$ for every tube of fewer than $\frac12 e^{1.6\cdot10^{19}}$ slabs; the bound then has no content on any lattice that the rest of the paper uses. The rate is the minorisation constant of the kernel across one interface, which uses only the oscillation of $\log \Psi_i$; the coarea factors contribute $3.2\cdot10^{-4}$ per pair to it, and the Wilson weight $72\beta_W$ per pair. A rate that does not degrade with $\beta_W$ needs information on where in the fibre the law concentrates, which is what the chart of Theorem~\ref{thm:badcross} supplies, at the cost of the mass $p$ off the chart. The projected average does not improve the rate: by Theorem~\ref{thm:projresponse}\textup{(i)} its pair factor has mass in $[(\frac8{15})^3e^{-72\beta_W},512]$ times that of Haar measure, so the oscillation per pair is $72\beta_W+3\log\frac{15}8+\log512>72\beta_W+8$, and the Lipschitz constant $24\beta_W+180.5$ of Theorem~\ref{thm:projresponse}\textup{(ii)} exceeds one at every $\beta_W$, so it gives no contraction across an interface either. \end{remark} The ends that enter the averages are taken up once more, through Lipschitz bounds in the end in place of oscillations. Fix a pair $k=(y, \rho)$ based in $W$ and an end $e$ of Lemma~\ref{lem:endpairs}. In case~(a), $e=u_b$ for some $b\ne0$, and $e$ enters only the path $P_{x_b}= u_b\ell_b$ of $k$; in case~(b), $e=w_b$, and $e$ enters only $P_{x'_b}= \ell_bw_b$. The \emph{links of the block of $e$} are the links $u_a$, $a\ne0$, in case~(a), and $w_a$, $a\ne0$, in case~(b); they lie in $I_W$, respectively off $A_W$, so they are fixed links of the fibre, and $e$ is one of them. For a measurable function $W\ge0$ of the eight private links $\ell=(\ell_0,\dots,\ell_7)$ that does not depend on $e$, and a value $c_k$, put \[ Z^W_k(c_k;e)=\int W\,d\nu_k , \] where $\nu_k$ is the factor of $k$ at the value $c_k$, built before~\eqref{eq:fibrevolume} from~\eqref{eq:densityformula}. For $W=1$ this is $\rho(c_k)$, and for $W=e^{-\beta_WS_{\mathrm{dep}}}$, with $S_{\mathrm{dep}}$ the sum of $1-\frac12\ReTr U_p$ over the $36$ plaquettes that contain a private link of $k$ (Lemma~\ref{lem:relnorm}), it is the integrated pair factor. No plaquette contains $e$ and a private link of $k$: both are $\rho$-links, the two $\rho$-links of a plaquette have the same coordinate $\rho$, and this coordinate is even for $e$ and odd for a private link. On $R_0$ the average is the path link, so the part of the fibre where $m>1$ is $\{\ell_0=c_k\}\times R_0(e)$, where $R_0(e)$ is the set of $(\ell_a)_{a\ne0}$ at which $m>1$ for $\ell_0=c_k$, and there $\nu_k$ is Haar measure of the seven non-path links. \begin{lemma}[the boundary $\{m=1\}$]\label{lem:endboundary} Let $G=\SU(2)$, $r_0=\sqrt2\,\alpha$ and $B(p)=\{v:d(v,p)\le r_0\}$, and fix $c_k$ and every link other than the private links of $k$ and the end. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The complement of $R_0(e)$ is $\prod_{a\ne0}A_a$, where $A_a=B(u_a^{-1}c_k)\cap B(c_kw_a^{-1})$. For $W=1$ the part of $\rho(c_k)$ on $R_0$ is $1-\prod_{a\ne0}\lambda(s_a)$, where $s_a=d(c_k^{-1}u_ac_k,w_a)$ and $\lambda(s)=h(B(I)\cap B(z))$ for any $z$ with $d(I,z)=s$; in particular $\lambda\le q$. \item For every move $e\to e'$ of the end and every bounded measurable $W\ge0$, \begin{multline*} \Bigl\lvert\int_{R_0(e)}W(c_k,\cdot)\,dh^{\otimes7}-\int_{R_0(e')}W(c_k, \cdot)\,dh^{\otimes7}\Bigr\rvert\\ \le\frac7{8\sqrt2\,\pi}\,q^6\lVert W\rVert_\infty\,d(e,e')<3.4\cdot 10^{-8}\lVert W\rVert_\infty\,d(e,e'). \end{multline*} \end{enumerate} \end{lemma} \begin{proof} (i) By Lemma~\ref{lem:haarsu2}\textup{(iii)}, $\delta(v,I)\le1$ exactly when $d(v,I)\le r_0$. At $\ell_0=c_k$ the paths of offset zero give $v=I$, and for $a\ne0$, by bi-invariance, $d(c_k^{-1}u_a\ell_a,I)\le r_0$ exactly when $\ell_a\in B(u_a^{-1}c_k)$, and $d(c_k^{-1}\ell_aw_a,I)\le r_0$ exactly when $\ell_a\in B(c_kw_a^{-1})$. So $m\le1$ exactly when $\ell_a\in A_a$ for every $a\ne0$. The two centres are at distance $d(u_a^{-1}c_k,c_kw_a^{-1})= d(c_k^{-1}u_a^{-1}c_k,w_a^{-1})=s_a$, inversion being an isometry. The isometries of the round sphere act transitively on pairs of points at a given distance and preserve $h$, so $h(A_a)=\lambda(s_a)\le h(B(I))=q$, and the seven conditions bear on distinct coordinates. (ii) In case~(a) only $A_b$ depends on $e$, through the centre $p=e^{-1} c_k$, so the symmetric difference of $R_0(e)$ and $R_0(e')$ is $\prod_{a\ne 0,b}A_a$ times the symmetric difference of $A_b(e)$ and $A_b(e')$, which lies in $B(p)\,\triangle\,B(p')$, $p'=e'^{-1}c_k$. Let $\gamma$ be a geodesic of unit speed from $e$ to $e'$, of length $s=d(e,e')$, and $p_t= \gamma(t)^{-1}c_k$. If $x\in B(p)\setminus B(p')$, the continuous function $t\mapsto d(x,p_t)$ is at most $r_0$ at $t=0$ and larger at $t=s$, so $x$ lies on the sphere $\partial B(p_t)$ for some $t$; likewise for $B(p') \setminus B(p)$. Since $\partial B(p_t)=p_t\Sigma$ with $\Sigma=\partial B(I)$, the set $B(p)\triangle B(p')$ lies in the image of $\Phi(t,\sigma)= \gamma(t)^{-1}c_k\sigma$ on $[0,s]\times\Sigma$. For fixed $t$ the map $\sigma\mapsto\Phi(t,\sigma)$ is an isometry and $t\mapsto\Phi(t,\sigma)$ has unit speed, so by the area formula the image has volume at most $s\, H^2(\Sigma)$. By Lemma~\ref{lem:haarsu2}, $\SU(2)$ is the unit sphere $S^3\subset\R^4$, with $d$ equal to $\sqrt2$ times the angle and $h$ the normalised surface measure; in these units $\Sigma$ is the two-sphere at angle $\alpha$ from $I$, of area $4\pi\sin^2\alpha=\frac{7\pi}4$, as $\sin\alpha=\frac{\sqrt7}4$, and the curve sweeps the angle $s/\sqrt2$. So \[ h\bigl(B(p)\triangle B(p')\bigr)\le\frac s{\sqrt2}\cdot\frac{7\pi/4}{2\pi^2} =\frac{7s}{8\sqrt2\,\pi}, \] and the symmetric difference of $R_0(e)$ and $R_0(e')$ has measure at most $q^6$ times this. In case~(b) the moving centre is $c_kw_b^{-1}$, and the same argument applies with $\Phi(t,\sigma)=\sigma c_k\gamma(t)^{-1}$. Finally $8\sqrt2\,\pi>35.54$, $\frac7{35.54}<0.19697$, and $q^6<1.706\cdot 10^{-7}$ by the proof of Lemma~\ref{lem:almosthaar}\textup{(i)}. \end{proof} So the motion of the boundary $\{m=1\}$ costs at most $3.4\cdot10^{-8}$ per unit move of the end, for every coarse value and every configuration. The law of the coarse value itself moves by a bounded amount in the Kantorovich distance, with no condition at all. \begin{proposition}[the coarse value moves Lipschitz]\label{prop:endkantorovich} Let $G=\SU(2)$ and fix every link other than the private links of $k$ and the end, with no condition on them. Let $W\ge0$ be an integrable function of the private links that does not depend on $e$. Then $\int Z^W_k(c;e)\, dh(c)=\int W\,dh^{\otimes8}$ for every $e$, and for every function $\varphi$ on $\SU(2)$ that is $1$-Lipschitz for $d$ and every move $e\to e'$, \begin{multline*} \Bigl\lvert\int\varphi(c)\bigl(Z^W_k(c;e)-Z^W_k(c;e')\bigr)\,dh(c) \Bigr\rvert\\ \le1.858\,d(e,e')\int W\,dh^{\otimes8}. \end{multline*} \end{proposition} \begin{proof} The measures $\nu_k$ at the values $c\in\SU(2)$ disintegrate Haar measure on the eight private links along $V^\sharp_k$, by the proof of Lemma~\ref{lem:averagedensity}, so $\int\varphi(c)\,Z^W_k(c;e)\,dh(c)=\int \varphi(V^\sharp_k(\ell;e))\,W(\ell)\,dh^{\otimes8}(\ell)$; $\varphi=1$ gives the first claim. Fix $\ell$ and move $e$ along a geodesic of unit speed. Write $V^\sharp_k=\ell_0e^Y$ with $Y=\chi N$. Only the path that contains $e$ moves, and its $v_x$ moves at unit speed. Since $\delta(\cdot,I)$ is $1$-Lipschitz for $d$, $m$ moves at speed at most $1$ and $\chi=\phi(m)$ at most $2$. On the open set of times at which $\chi>0$, every $v_x$ has $\delta(v_x,I)<1$, so $\lVert N\rVert\le\frac{14}{16}\sqrt2\,\alpha$, the two paths of offset zero giving $0$, and $N$ moves at speed at most $\frac1{16}\Lambda$; hence $Y$ moves at speed at most $\frac74\sqrt2\,\alpha +\frac1{16}\Lambda<1.858$, by $\alpha<0.7228$ and $\Lambda<1.0927$. Where $\chi=0$, $Y=0$, and $Y$ is continuous, so $Y$ is $1.858$-Lipschitz along the move and $d(V^\sharp_k(\ell;e),V^\sharp_k(\ell;e'))\le\lVert Y-Y' \rVert\le1.858\,d(e,e')$. Integrate against $W$. \end{proof} \begin{corollary}[the derivative in the mean over the coarse link]\label{cor:endmean} In the setting of Lemma~\ref{lem:almosthaar}\textup{(iii)} with $E=\{e\}$, an end of the pair $k$, write $Z(e;c_k)$ for $Z(e)$ as a function of the coarse value $c_k$, the other coarse links being fixed. For every positive Lipschitz function $\psi$ on $\SU(2)$ and every move $e\to e'$, \[ \Bigl\lvert\log\frac{\int\psi\,Z(e;\cdot)\,dh}{\int\psi\,Z(e';\cdot)\,dh} \Bigr\rvert\le\Bigl(3\sqrt2\,\beta_W+1.858\,\frac{\operatorname{Lip}\psi} {\min\psi}\Bigr)\,d(e,e'), \] uniformly in the other links off $A^-_W$ and in the other coarse links. \end{corollary} \begin{proof} Write $\lambda Q^-=\beta_W(S_{\mathrm{dep}}+S_e+S_{\mathrm r})$, with $S_e$ the sum of $1-\frac12\ReTr U_p$ over the plaquettes that contain $e$ and a link of $A^-_W$, none of which contains a private link of $k$, and $S_{\mathrm r}$ the sum over the other plaquettes that contain a link of $A^-_W$; $S_{\mathrm r}$ depends neither on $e$ nor on the private links of $k$. By Lemma~\ref{lem:endpairs} only the factor $\nu_k$ of $\nu^-_c$ depends on $e$, and only it depends on $c_k$. So, with $\zeta$ the links of $A^-_W$ other than the private links of $k$, \begin{gather*} Z(e;c_k)=\int d\Xi(\zeta)\,e^{-\beta_WS_e(e,\zeta)}\,Z^{W_\zeta}_k(c_k;e),\\ W_\zeta=e^{-\beta_WS_{\mathrm{dep}}(\cdot,\zeta)}, \end{gather*} where $\nu_k$ in $Z^{W_\zeta}_k$ is taken at the inside links in $\zeta$, and $d\Xi$ does not depend on $e$ or $c_k$. By Proposition~\ref{prop:endkantorovich} applied to $\psi/\operatorname{Lip} \psi$, the integral $I_\zeta(e)=\int\psi\,Z^{W_\zeta}_k(\cdot;e)\,dh$ satisfies $\lvert I_\zeta(e)-I_\zeta(e')\rvert\le1.858\operatorname{Lip} \psi\,d(e,e')\int W_\zeta\,dh^{\otimes8}$, and $\int W_\zeta\,dh^{\otimes8}= \int Z^{W_\zeta}_k(\cdot;e)\,dh\le I_\zeta(e)/\min\psi$. The link $e$ lies in six plaquettes, and a move of $e$ by $d(e,e')$ changes each $1-\frac12\ReTr U_p$ by at most $d(e,e')/\sqrt2$, by the proof of Lemma~\ref{lem:wilsononly}. So the integrand at $e'$ is at most $e^{(3\sqrt2\beta_W+1.858\operatorname{Lip}\psi/\min\psi)d(e,e')}$ times the integrand at $e$, for every $\zeta$; integrate and exchange $e$ and $e'$. \end{proof} For a fixed coarse value the derivative needs the part of the pair factor on $R_1\cup R_2$, and this is where the relative chart of Lemma~\ref{lem:relchart} enters. Its hypothesis bears on the links of the block of the end only. \begin{lemma}[the derivative on the relative chart]\label{lem:endchart} Let $G=\SU(2)$ and $\beta_W\ge0$, and let the end $e$ move along a geodesic to $e'$ while the links of the block of $e$ stay within $\frac1{32}$ of $I$; the other links are arbitrary. Let $W(\ell)=\int e^{-\beta_W S_{\mathrm{dep}}(\ell,\zeta)}\,\omega(d\zeta)$ for a nonzero finite measure $\omega$ on the values $\zeta$ of the links other than the private links of $k$. Take $Z^W_k$ in the version $\int W(\Gamma(\xi))\,D(\xi)^{-1}d\xi$ of Lemma~\ref{lem:relchart}\textup{(iv)}, with $\xi_a=g^{-1}\ell_a$ and $\Gamma(\xi)=(G(\xi),G(\xi)\xi)$ in case~\textup{(a)}, and $\xi_a=\ell_a g^{-1}$ and $\Gamma(\xi)=(G(\xi),\xi G(\xi))$ in case~\textup{(b)}; it agrees with $\int W\,d\nu_k$ for almost every $c_k$. Then for every $c_k$, \begin{gather*} \Bigl\lvert\log\frac{Z^W_k(c_k;e)}{Z^W_k(c_k;e')}\Bigr\rvert\le L_*\, d(e,e'),\\ L_*=128.07\,\beta_W+\bigl(180.4+50.6\,\mathcal A_{\mathrm{tie}}\bigr)\, e^{72\beta_W}, \end{gather*} with $\mathcal A_{\mathrm{tie}}<\infty$ of Proposition~\ref{prop:tiearea}. \end{lemma} \begin{proof} Case~(a). By Lemma~\ref{lem:relchart}, applied along the move, the equation $V^\sharp_k=c_k$ has for every $\xi$ the unique solution $G(\xi)$, and $Z=Z^W_k(c_k;e)=\int f\,d\xi$ with $f=(W\circ\Gamma)/D$. Let $e'$ lie at distance $\delta\le0.08$ along the geodesic, with $G',\Gamma', D',f',Z'$. \emph{The weight.} By Lemma~\ref{lem:relchart}\textup{(ii)}, $d(G,G')\le L\delta$ with $L<2.064$, and the eight links of $\Gamma(\xi)$, which are $G$ and $G\xi_a$, move by $d(G,G')$ each. Each lies in six plaquettes of $S_{\mathrm{dep}}$, and a link that moves by $\varepsilon$ changes $1-\frac12\ReTr U_p$ by at most $\varepsilon/\sqrt2$, by the proof of Lemma~\ref{lem:wilsononly}; so for every $\zeta$, $\lvert S_{\mathrm{dep}} (\Gamma'(\xi),\zeta)-S_{\mathrm{dep}}(\Gamma(\xi),\zeta)\rvert\le24\sqrt2\, L\delta<70.06\,\delta$, and $\lvert W\circ\Gamma'-W\circ\Gamma\rvert\le (e^{70.06\beta_W\delta}-1)\,W\circ\Gamma$. \emph{The Jacobian.} By Lemma~\ref{lem:relchart}\textup{(iii)}, $D$ and $D'$ lie in $[(1-\lambda)^3,(1+\lambda)^3]$, the ratio of whose ends is less than $1.828$. So $D'^{-1}\le1.828\,D^{-1}$, and with $\varDelta=\log D-\log D'$, $\lvert D^{-1}-D'^{-1}\rvert=D^{-1}\lvert1-e^{\varDelta}\rvert\le1.828 \,D^{-1}\lvert\varDelta\rvert$. Hence \[ \int\lvert f-f'\rvert\,d\xi\le1.828\bigl(e^{70.06\beta_W\delta}-1\bigr)Z+ 1.828\int f\,\lvert\varDelta\rvert\,d\xi . \] Since $0\le S_{\mathrm{dep}}\le72$, $\sup W\le e^{72\beta_W}\inf W$, so $\sup f\le1.828\,e^{72\beta_W}\inf f\le1.828\,e^{72\beta_W}Z$, $d\xi$ being a probability. By Proposition~\ref{prop:jreduce}, whose hypotheses for $B=\SU(2)^7$ are those of the lemma (Theorem~\ref{thm:jholds}), $\int\lvert\varDelta\rvert\,d\xi\le(53.96+15.134\,\mathcal A_{\mathrm{tie}}) \,\delta$. So $\lvert Z'-Z\rvert\le\varepsilon(\delta)Z$ with \begin{multline*} \varepsilon(\delta)=1.828\bigl(e^{70.06\beta_W\delta}-1\bigr)\\ +1.828^2e^{72\beta_W}\bigl(53.96+15.134\,\mathcal A_{\mathrm{tie}}\bigr) \delta . \end{multline*} Divide the geodesic into $n$ equal steps; then $\lvert\log Z(e)-\log Z(e') \rvert\le-n\log(1-\varepsilon(d(e,e')/n))$, which tends to $(1.828\cdot 70.06\,\beta_W+1.828^2e^{72\beta_W}(53.96+15.134\,\mathcal A_{\mathrm{tie}} ))\,d(e,e')$. Finally $1.828\cdot70.06<128.07$, $1.828^2<3.3416$, $3.3416 \cdot53.96<180.4$ and $3.3416\cdot15.134<50.6$. Case~(b). The last part of Lemma~\ref{lem:relchart} gives (i)--(iv) in the coordinates $\xi_a=\ell_ag^{-1}$ with $\lambda_w\le\lambda$ and $L_w\le L$, the eight private links $G$ and $\xi_aG$ moving by $d(G,G')$. The inversion $\mathfrak i$ of the proof of Theorem~\ref{thm:jholds} maps the configuration to one of case~(a) whose links of the block of the moving link are within $\frac1{32}$ of $I$, preserves Haar measure and $D$, and so gives the same bound on $\int\lvert\varDelta\rvert\,d\xi$. The rest is unchanged. \end{proof} The links of the block of $e$ are the inside $\rho$-links of $B(y)$ in case~(a) and of $B(y+e_\rho)$ in case~(b), and these are non-path active links of the pair $(y-e_\rho,\rho)$, respectively $(y+e_\rho,\rho)$, based at the block across the end. The reconstruction $R(c)$ is $I$ on the links inside the blocks, so the hypothesis of Lemma~\ref{lem:endchart} holds whenever the block across the end, $y-e_\rho$ in case~(a) and $y+e_\rho$ in case~(b), is good for labels of radius $r\le\frac1{32}$ at the two values of the end; the labels of Sec.~\ref{ssec:factorisation} have $r\le10^{-4}$. \begin{corollary}[the derivative in an end]\label{cor:endchart} In the setting of Lemma~\ref{lem:almosthaar}\textup{(iii)} with $E=\{e\}$, suppose that the links of the block of $e$ stay within $\frac1{32}$ of $I$ along the geodesic from $e$ to $e'$. Then, with the factor of $k$ in the version of Lemma~\ref{lem:endchart}, for every coarse field \begin{multline*} \Bigl\lvert\log\frac{Z(e)}{Z(e')}\Bigr\rvert\\ \le\Bigl(132.32\,\beta_W+\bigl(180.4+50.6\,\mathcal A_{\mathrm{tie}}\bigr) e^{72\beta_W}\Bigr)\,d(e,e'), \end{multline*} uniformly in the other links off $A^-_W$. \end{corollary} \begin{proof} Use the decomposition of the proof of Corollary~\ref{cor:endmean}, apply Lemma~\ref{lem:endchart} with $\omega=\delta_\zeta$ at each $\zeta$, and bound the plaquettes through $e$ as there; $L_*+3\sqrt2\,\beta_W$ has the stated constants, as $128.07+3\sqrt2<132.32$. \end{proof} The same bound replaces the oscillation of the ends in Corollary~\ref{cor:tubeends}. \begin{corollary}[mixed differences along a tube]\label{cor:tubederiv} Let $G=\SU(2)$, $\beta_W\ge0$, and let $W$, $\rho$, the slabs, $a$, $b$, $E_a$, $E_b$, $a_0,a_1,b_0,b_1$ and $Z_{ij}$ be as in Corollary~\ref{cor:tubeends}. Suppose that for every link $e$ of $E_a$ that enters an average of a pair based in $W$, every link of the block of $e$ lies within $\frac1{32}$ of $I$, at each of its values if it belongs to $E_a\cup E_b$. Put $L_e=3\sqrt2\,\beta_W+L_*$ for these links of $E_a$, $L_e=3\sqrt2\,\beta_W$ for the links of $E_a$ that enter no such average, and $\Lambda_a=\sum_{e\in E_a}L_e\,d(a_{0,e},a_{1,e})$. Then, with the factors of the pairs in the version of Lemma~\ref{lem:endchart}, \[ \Bigl\lvert\log\frac{Z_{11}Z_{00}}{Z_{10}Z_{01}}\Bigr\rvert\le\bigl(e^{2 \Lambda_a}-1\bigr)\prod_{i=a+1}^{b-1}\bigl(1-e^{-D_i}\bigr). \] If $E_b$ satisfies the same hypothesis, and $\Lambda_b$ is defined in the same way, then also \[ \Bigl\lvert\log\frac{Z_{11}Z_{00}}{Z_{10}Z_{01}}\Bigr\rvert\le\bigl(e^{2 \Lambda_a}-1\bigr)\bigl(e^{2\Lambda_b}-1\bigr)\prod_{i=a+1}^{b-2}\bigl(1- e^{-D_i}\bigr). \] \end{corollary} \begin{proof} In the proof of Corollary~\ref{cor:tubeends}, $\mathcal A_{a_i}(s)$ is the integral of the factors of the slabs $1,\dots,a$ with $S_{a+1}=s$; it contains the private links of every pair whose average contains a link of $E_a$, and the links of $A^-_W$ in every plaquette through a link of $E_a$. Move the links of $E_a$ from $a_0$ to $a_1$ one at a time along geodesics; the links of the blocks stay within $\frac1{32}$ of $I$, since the geodesic ball of radius $\frac1{32}$ is convex. A link $e$ enters the integrand through the plaquettes that contain it, each of which changes by at most $d/\sqrt2$, and, if it enters an average, through the factor $\nu_k$ of the one pair $k$ whose average contains it (Lemma~\ref{lem:endpairs}), whose private links enter the integrand otherwise only through $S_{\mathrm{dep}}$. As in the proof of Corollary~\ref{cor:endchart}, Lemma~\ref{lem:endchart} with $\omega= \delta_\zeta$ at every value $\zeta$ of the other variables gives $e^{-\Lambda_a}\le\mathcal A_{a_1}(s)/\mathcal A_{a_0}(s)\le e^{\Lambda_a}$ for every $s$. So the function $\Phi=\mathcal A_{a_1}/\mathcal A_{a_0}$ of that proof satisfies $\sup\Phi\le e^{2\Lambda_a}\inf\Phi$, and the rest of that proof gives the first bound. For the second, let $\iota_j$ be the law of $S_{b-1}$ under the measure with $a_0$ on $E_a$ and $b_j$ on $E_b$. Its density is proportional to $\mathcal L(s)\,\mathcal R_{b_j}(s)$, where $\mathcal L(s)$ is the integral of the factors of the slabs $1,\dots,b-1$ with $S_{b-1}=s$, and $\mathcal R_{b_j}(s)$ that of $\Psi_{b-1}(s,\cdot)$ and of the factors of the slabs $b,\dots,L-1$. Only $\mathcal R_{b_j}$ depends on $b_j$, and it integrates the private links of every pair whose average contains a link of $E_b$; the argument above gives $e^{-\Lambda_b}\le\mathcal R_{b_1}(s)/ \mathcal R_{b_0}(s)\le e^{\Lambda_b}$ for every $s$, so $d\iota_1/d\iota_0 \le e^{2\Lambda_b}$ and $\lVert\iota_1-\iota_0\rVert_{\rm TV}\le e^{2 \Lambda_b}-1$. Given $S_{i+1}$, for $a+1\le i\le b-2$, the law of $S_i$ under either measure is the backward kernel of the proof of Corollary~\ref{cor:tubeends}, which does not depend on $b_j$ and contracts the total variation distance by the factor $1-e^{-D_i}$. Hence the laws $\pi_j$ of $S_{a+1}$ satisfy $\lVert\pi_1-\pi_0\rVert_{\rm TV}\le(e^{2 \Lambda_b}-1)\prod_{i=a+1}^{b-2}(1-e^{-D_i})$, and $\lvert\pi_1(\Phi)- \pi_0(\Phi)\rvert\le\lVert\pi_1-\pi_0\rVert_{\rm TV}(\sup\Phi-\inf\Phi)$ ends the proof as before. \end{proof} For ends that enter no average the hypothesis is empty and $L_e=3\sqrt2\, \beta_W$; the second bound is then a mixed difference in the form of Corollary~\ref{cor:badcrossratio}, without a chart and without a bound on the plaquettes, at the rate of Theorem~\ref{thm:tube}. For ends in the averages the constant $L_*$ contains $e^{72\beta_W}$: the jumps of $\log D$ across the ties are integrated against Haar measure in Proposition~\ref{prop:jreduce}, and the weight $e^{-\beta_W S_{\mathrm{dep}}}$ can exceed its mean by that factor on the set they sweep. Neither $L_*$ nor the rate of Theorem~\ref{thm:tube} has content in the range $\beta_W\ge2.3\cdot10^{17}$ of Theorem~\ref{thm:badcross} (Remark~\ref{rem:tuberate}). The factor $e^{72\beta_W}$ in $L_*$ comes from bounding the weight $f=(W\circ\Gamma)/D$ of Lemma~\ref{lem:endchart} by its supremum on the sets swept by the events and the ties. Along the curves used below the logarithm of $W\circ\Gamma$ is Lipschitz with a constant linear in $\beta_W$, and integral-geometric trace inequalities replace the supremum by the integral. The ties are counted by a crossing number. Write $\mathcal P$ for the set of parameters $P=(c_k,(u_b,w_b)_{b\ne0})$ with $d(u_b,I)\le\frac1{32}$ for every $b\ne0$, and $Z_a(P)$ for the set $Z_a$ of Proposition~\ref{prop:jreduce} at $P$. For $b\ne0$, $\xi\in \SU(2)^7$ and a unit vector $\eta\in\mathfrak{su}(2)$ let $C_{b,\xi,\eta}$ be the orbit of $\xi$ under the flow $\Theta_s$ that replaces $\xi_b$ by $\xi_be^{s\eta}$ and fixes the other coordinates. Since $\eta$ has the eigenvalues $\pm i/\sqrt2$, the orbit is a great circle in the factor $b$, of length $T_0=2\sqrt2\,\pi$, run at unit speed, and $\Theta_s$ preserves Haar measure. \begin{definition}\label{def:tiecross} The \emph{crossing number} $\mathfrak b_{\mathrm{tie}}$ is the supremum of $\#(Z_a(P)\cap C_{b,\xi,\eta})$ over $a,b\ne0$, $P\in\mathcal P$, $\xi \in\SU(2)^7$ and unit vectors $\eta$, among the intersections that are finite. \end{definition} \begin{lemma}[crossings]\label{lem:tiecross} \textup{(i)} $\mathfrak b_{\mathrm{tie}}<\infty$. \textup{(ii)} For fixed $a$, $b$, $P$ and $\eta$, the set of $\xi$ for which $Z_a(P)\cap C_{b,\xi,\eta}$ is infinite is Haar-null. \end{lemma} \begin{proof} (i) By the proof of Proposition~\ref{prop:tiearea}, $Z_a(P)$ is a definable family over $\mathcal P$ in $\R_{\mathrm{an}}$. With $\eta=n/ \sqrt2$, $n^2=-1$, $\xi_be^{s\eta}=\xi_b(\cos\frac s{\sqrt2}+\sin\frac s {\sqrt2}\,n)$, and $\sin$ and $\cos$ on $[0,2\pi]$ are restricted analytic, so the sets $\{s\in[0,T_0):\Theta_s\xi\in Z_a(P)\}$ form a definable family of subsets of $\R$ with the parameters $(a,b,P,\xi,\eta)$. By uniform finiteness \cite[Ch.~3]{Dri98} there is an $N$ such that every finite member has at most $N$ points. (ii) An infinite definable subset of $[0,T_0)$ contains an open interval. On that interval the orbit lies in the $C^1$ hypersurface $Z=Z_a(P)$, so its velocity, the left-invariant field $\eta$ in the factor $b$, is tangent to $Z$ there, that is $\ip{\nu_b}\eta=0$, where $\nu=(\nu_c)_{c\ne 0}$ is a unit normal of $Z$ written in the left-invariant frame. So $\xi$ lies in $\Psi(Z^0\times[0,T_0])$, where $Z^0=\{z\in Z:\ip{\nu_b(z)}\eta =0\}$ and $\Psi(z,s)=\Theta_s z$. The map $\Psi$ is $C^1$ on $Z\times\R$, and its Jacobian is $\lvert\ip{\nu_b}\eta\rvert$: each $\Theta_s$ is an isometry, and its differential maps the field $\eta$ at $z$ to the field $\eta$ at $\Theta_sz$, as $\operatorname{Ad}(e^{-s\eta})\eta=\eta$. By the area formula \cite[3.2.20]{Fed69} the image of $Z^0\times[0,T_0]$ is null. \end{proof} \begin{lemma}[traces on curves]\label{lem:traces} Let $\gamma$ be a curve of length $T$ run at unit speed and $w>0$ a function on it such that $w\circ\gamma$ is Lipschitz and $\lvert(\log w \circ\gamma)'\rvert\le K$ almost everywhere. If $\gamma$ is closed, then $\sup w\circ\gamma\le(\frac1T+\frac K2)\int_0^Tw(\gamma(s))\,ds$; if $\gamma$ is a segment, then $\sup w\circ\gamma\le(\frac1T+K)\int_0^Tw( \gamma(s))\,ds$. \end{lemma} \begin{proof} For $s_0,s$ let $\Gamma(s,s_0)$ be the shorter arc of a closed curve between them, or the interval between them on a segment. Then $w(\gamma( s_0))\le w(\gamma(s))+\int_{\Gamma(s,s_0)}\lvert(w\circ\gamma)'\rvert\le w( \gamma(s))+K\int_{\Gamma(s,s_0)}w\circ\gamma$. Average over $s$: a point $u$ lies in $\Gamma(s,s_0)$ for a set of $s$ of measure at most $\frac T2$ on a closed curve and at most $T$ on a segment. \end{proof} \begin{proposition}[the ties against a weight]\label{prop:tiecrofton} Let $a\ne0$, $P\in\mathcal P$, $Z=Z_a(P)$, and let $w>0$ be a Lipschitz function on $\SU(2)^7$ with $\lvert\frac d{ds}\log w(\Theta_s\xi)\rvert \le K$ almost everywhere, for every $b$, $\xi$ and unit $\eta$. Then \[ V^{-7}\int_Zw\,dH^{20}\le14\,\mathfrak b_{\mathrm{tie}}\Bigl(\frac1{T_0} +\frac K2\Bigr)\int w\,d\xi . \] \end{proposition} \begin{proof} Fix $b$ and $\eta$, and let $\Psi(z,s)=\Theta_sz$ on $Z\times[0,T_0)$, with Jacobian $\lvert\ip{\nu_b}\eta\rvert$ as in the proof of Lemma~\ref{lem:tiecross}. A point $x$ has one preimage $(z,s)$ for each $z\in Z\cap C_{b,x,\eta}$, the flow having period $T_0$. The area formula with the function $(z,s)\mapsto w(z)$ gives \[ T_0\int_Zw\,\lvert\ip{\nu_b}\eta\rvert\,dH^{20}=\int\sum_{z\in Z\cap C_{b,x,\eta}}w(z)\,d\mathrm{vol}(x) . \] For almost every $x$ the sum has at most $\mathfrak b_{\mathrm{tie}}$ terms, by Lemma~\ref{lem:tiecross}, and by Lemma~\ref{lem:traces} it is at most $\mathfrak b_{\mathrm{tie}}(\frac1{T_0}+\frac K2)\int_0^{T_0}w( \Theta_sx)\,ds$. Since $\Theta_s$ preserves Haar measure, \[ \int\!\!\int_0^{T_0}w(\Theta_sx)\,ds\,d\mathrm{vol}(x)=T_0\int w\, d\mathrm{vol}, \] so \[ \int_Zw\,\lvert\ip{\nu_b}\eta\rvert\,dH^{20}\le\mathfrak b_{\mathrm{tie}} \Bigl(\frac1{T_0}+\frac K2\Bigr)\int w\,d\mathrm{vol} . \] The mean of $\lvert\ip{\nu_b}\eta\rvert$ over the unit sphere of $\mathfrak{su}(2)\cong\R^3$ is $\frac12\lvert\nu_b\rvert$, and $\sum_b \lvert\nu_b\rvert\ge(\sum_b\lvert\nu_b\rvert^2)^{1/2}=1$. Average over $\eta$, sum over the seven values of $b$ and divide by $V^7$. \end{proof} With $w=1$ and $K=0$ this is an explicit form of Proposition~\ref{prop:tiearea}. \begin{corollary}\label{cor:tieareab} $\mathcal A_{\mathrm{tie}}\le14\,\mathfrak b_{\mathrm{tie}}/T_0<1.5756\, \mathfrak b_{\mathrm{tie}}$, so that $L_*\le128.07\,\beta_W+(180.4+79.73\, \mathfrak b_{\mathrm{tie}})e^{72\beta_W}$. \end{corollary} \begin{proof} $14/(2\sqrt2\,\pi)<1.57556$ and $50.6\cdot1.5756<79.73$. \end{proof} \begin{proposition}[the Jacobian across the ties against the weight]\label{prop:jweighted} Let $B=\SU(2)^7$, let the links $u_a$, $a\ne0$, and the move of one link $u_b$ or $w_b$ be as in Theorem~\ref{thm:relresponse}, take the coordinates $\xi_a=g^{-1}\ell_a$, and let $W$ be as in Lemma~\ref{lem:endchart}. Let $e'$ lie at distance $\delta\le0.08$ from $e$ along the geodesic, and let $f=(W\circ\Gamma)/D$ and $\varDelta=\log D-\log D'$. Then \[ \int f\,\lvert\varDelta\rvert\,d\xi\le C(\beta_W)\,\delta\int f\,d\xi \] with \[ C(\beta_W)=3934+242610\,\beta_W+\mathfrak b_{\mathrm{tie}}\bigl(43.59+ 14888\,\beta_W\bigr). \] \end{proposition} \begin{proof} Write $w=W\circ\Gamma$. A private link lies in six plaquettes of $S_{\mathrm{dep}}$, and a link that moves by $\varepsilon$ changes $1-\frac 12\ReTr U_p$ by at most $\varepsilon/\sqrt2$; so for every $\zeta$ the ratio of $e^{-\beta_WS_{\mathrm{dep}}(\ell,\zeta)}$ at two values of the eight links is at most $e^{3\sqrt2\beta_W\sum_jd(\ell_j,\ell'_j)}$, and the same holds for $W$. By Lemma~\ref{lem:relchart}\textup{(iii)}, $D$ takes values in $[D_-,D_+]$ with $D_+/D_-<1.828$, so $w/D_+\le f\le w/ D_-$ almost everywhere. We follow the proof of Proposition~\ref{prop:jreduce}, whose three parts bound $\lvert \varDelta\rvert$ pointwise. \emph{Where the type is kept.} Off the $\xi$ that meet an event, $\lvert\varDelta\rvert\le11.87\,\delta+\sum_t1.9811\lvert\nabla h_2 \rvert$, and the first term contributes $11.87\,\delta\int f\,d\xi$. \emph{The events.} Fix an event of index $a$, the coordinates $\xi_c$, $c\ne a$, and the coordinate $v=\tilde u_a(G(\xi))\xi_a$ of the proof of Proposition~\ref{prop:jreduce}, with $t=\omega\delta$, $\omega<1.1825$. Write $v=\exp(\theta\nu)$. The $\xi$ that meet the event during the move lie in $\{\lvert h\rvert\le t\}$ and in $\{0.4\le f_{x_a}\le1.1\}$, and $f_{x_a}=2\sqrt2\sin\frac\theta2$. So on each ray of fixed $\nu$ they form a set $I_\nu$ of length at most $2t/1.154<1.7332\,t$ inside the interval $J=\{\theta:0.4\le2\sqrt2\sin\frac\theta2\le1.1\}=[\theta_-, \theta_+]$, with $\sin^2\theta_-=4\cdot\frac1{50}\cdot\frac{49}{50}=0.0784$, $\theta_+<0.7990$ and $\lvert J\rvert>0.515099$. Along the ray $v$ moves at speed $\sqrt2$ per unit of $\theta$, $\xi_a$ at speed at most $\sqrt2/ 0.9054<1.5620$, $G$ at speed at most $2.140\cdot1.5620<3.3427$, and the eight links $G$, $G\xi_c$ together at speed at most $8\cdot3.3427+1.5620$; so $\lvert\frac d{d\theta}\log w\rvert\le K_\theta=3\sqrt2(8\cdot3.3427+ 1.5620)\beta_W<120.09\,\beta_W$. By Lemma~\ref{lem:traces} on the segment $J$, with Haar density $\frac2\pi\sin^2\theta$ and $\sin^2\theta\le0.5146$ on $J$, \begin{multline*} \int_{I_\nu}w\,\tfrac2\pi\sin^2\theta\,d\theta\le\tfrac2\pi\,0.5146\, \lvert I_\nu\rvert\Bigl(\frac1{\lvert J\rvert}+K_\theta\Bigr)\int_Jw\, d\theta\\ \le\frac{0.5146}{0.0784}\,1.7332\,t\,(1.9414+K_\theta)\int_Jw\, \tfrac2\pi\sin^2\theta\,d\theta . \end{multline*} Integrating over $\nu$, and passing between $v$ and $\xi_a$ with the factors $1.348$ and $1.0946^3<1.3115$ of the bi-Lipschitz map $\xi_a \mapsto v$, the set $S$ of $\xi$ that meet the event satisfies $\int_S w\,d\xi\le20.113\,t\,(1.9414+K_\theta)\int w\,d\xi$, and hence, as $D_+/D_-<1.828$ and $1.828\cdot20.113<36.77$, \[ \int_Sf\,d\xi\le36.77\,t\,(1.9414+K_\theta)\int f\,d\xi . \] On $S$, $\lvert\varDelta\rvert<0.6034$. The $77$ events therefore contribute at most $77\cdot0.6034\cdot36.77\cdot1.1825\,\delta(1.9414+ K_\theta)\int f<2020.2\,\delta(1.9414+120.09\,\beta_W)\int f$. \emph{The ties.} By the area and coarea formulas of the proof of Proposition~\ref{prop:jreduce}, applied to the function $1.9811\lvert \nabla_\xi H\rvert\,w/D_-$, which dominates $1.9811\lvert\nabla_\xi H \rvert f$ almost everywhere, \begin{multline*} \int f\sum_{t:\,\xi\in Z_a(t)}1.9811\lvert\nabla h_2\rvert\,d\xi\\ \le\frac{1.9811\cdot1.0913}{D_-}\int_0^\delta V^{-7}\!\int_{Z_a(t)}w\, dH^{20}\,dt . \end{multline*} Along the flow $\Theta_s$ the coordinate $\xi_b$ moves at unit speed, $G$ at speed at most $2.140$ by the proof of Proposition~\ref{prop:jreduce}, and the eight links together at speed at most $8\cdot2.140+1$; so $w$ satisfies the hypothesis of Proposition~\ref{prop:tiecrofton} with $K=3 \sqrt2\cdot18.12\,\beta_W<76.877\,\beta_W$, at every parameter $t$ of the move. With $\int w\,d\xi\le D_+\int f\,d\xi$ the ties of index $a$ contribute at most \[ 1.9811\cdot1.0913\cdot1.828\cdot14\,\mathfrak b_{\mathrm{tie}}\Bigl( \frac1{T_0}+38.4385\,\beta_W\Bigr)\delta\int f\,d\xi , \] and the seven indices at most $\mathfrak b_{\mathrm{tie}}(43.59+14888\, \beta_W)\,\delta\int f$, by $7\cdot1.9811\cdot1.0913<15.134$, $25.592/T_0< 2.8802$ and $25.592\cdot38.4385<983.72$. \emph{Conclusion.} $11.87+2020.2\cdot1.9414<3934$ and $2020.2\cdot120.09< 242610$. \end{proof} \begin{theorem}[the derivative on the chart, linear in $\beta_W$]\label{thm:endlinear} In the setting of Lemma~\ref{lem:endchart}, for every $c_k$, \begin{gather*} \Bigl\lvert\log\frac{Z^W_k(c_k;e)}{Z^W_k(c_k;e')}\Bigr\rvert\le L_\natural\,d(e,e'),\qquad L_\natural=\min(L_*,L_{\mathrm{lin}}),\\ L_{\mathrm{lin}}=7192+443620\,\beta_W+\mathfrak b_{\mathrm{tie}}\bigl( 79.69+27216\,\beta_W\bigr). \end{gather*} \end{theorem} \begin{proof} For $\delta\le0.08$ the proof of Lemma~\ref{lem:endchart} gives \[ \int\lvert f-f'\rvert\,d\xi\le1.828\bigl(e^{70.06\beta_W\delta}-1\bigr)Z +1.828\int f\lvert\varDelta\rvert\,d\xi . \] Bound the last integral by Proposition~\ref{prop:jweighted} in place of $\sup f$, so that $\lvert Z'-Z \rvert\le\varepsilon(\delta)Z$ with $\varepsilon(\delta)=1.828(e^{70.06 \beta_W\delta}-1)+1.828\,C(\beta_W)\delta$. Subdividing the geodesic as there, $\lvert\log Z(e)-\log Z(e')\rvert\le(128.07\,\beta_W+1.828\,C( \beta_W))\,d(e,e')$, and $1.828\cdot3934<7192$, $128.07+1.828\cdot242610 <443620$, $1.828\cdot43.59<79.69$, $1.828\cdot14888<27216$. In case~\textup{(b)} the inversion $\mathfrak i$ of the proof of Lemma~\ref{lem:endchart} maps the configuration to one of case~(a); it preserves Haar measure, $D$ and $\int f\,d\xi$, and maps $W$ to $W\circ \mathfrak i$. The proof of Proposition~\ref{prop:jweighted} uses $W$ only through the bound $e^{3\sqrt2\beta_W\sum_jd(\ell_j,\ell'_j)}$ on its ratio at two values of the eight links, which $W\circ\mathfrak i$ satisfies, inversion being an isometry. So Proposition~\ref{prop:jweighted} holds in case~(b) with the same constant. \end{proof} The constant $L_{\mathrm{lin}}$ is linear in $\beta_W$. For $\beta_W\ge 0.08$ its part free of $\mathfrak b_{\mathrm{tie}}$ is smaller than the part of $L_*$ free of $\mathcal A_{\mathrm{tie}}$, and in the range $\beta_W\ge2.3\cdot10^{17}$ of Theorem~\ref{thm:badcross} it replaces a factor $e^{72\beta_W}$ by a factor of order $\beta_W$.\footnote{At $\beta_W=1$ and $\mathfrak b_{\mathrm{tie}}=1$, $L_{\mathrm{lin}}= 478107.69$, while the part $180.4\,e^{72}$ of $L_*$ alone exceeds $3\cdot10^{33}$. At $\beta_W=0.08$ the two parts free of the ties are $42681.6$ and more than $5\cdot10^4$.} Fig.~\ref{fig:endslinear} compares the two constants. \begin{figure}[t] \centering \includegraphics[width=0.95\columnwidth]{figures/fig_endslinear.png} \caption{The derivative in an end (Theorem~\ref{thm:endlinear}): $\log_{10}$ of $L_*=128.07\,\beta_W+(180.4+50.6\,\mathcal A_{\mathrm{tie}})e^{72\beta_W}$, with $\mathcal A_{\mathrm{tie}}<1.5756\, \mathfrak b_{\mathrm{tie}}$ (red), and of $L_{\mathrm{lin}}=7192+443620\, \beta_W+\mathfrak b_{\mathrm{tie}}(79.69+27216\,\beta_W)$ (blue), over $\beta_W$ and the tie length $\mathfrak b_{\mathrm{tie}}$.} \label{fig:endslinear} \end{figure} The hypothesis of Lemma~\ref{lem:endchart} bears on the links of the block of the end. It may bear on the other block instead. \begin{lemma}[the other block]\label{lem:endotherblock} Lemma~\ref{lem:endchart} and Theorem~\ref{thm:endlinear} hold, with the same constants, when the inside $\rho$-links of the other block of the pair, the $w_a$ in case~\textup{(a)} and the $u_a$ in case~\textup{(b)}, $a\ne0$, stay within $\frac1{32}$ of $I$ along the move, the links of the block of the end being arbitrary. The version of $Z^W_k$ is then taken in the coordinates $\xi_a=\ell_ag^{-1}$ in case~\textup{(a)} and $\xi_a=g^{-1} \ell_a$ in case~\textup{(b)}. \end{lemma} \begin{proof} Case~(a), $e=u_b$. By the last part of Lemma~\ref{lem:relchart}, with $\sigma_w$ in place of $\sigma_u$, (i)--(iv) hold for every $u$ in the coordinates $\xi_a=\ell_ag^{-1}$, with $\lambda_w\le\lambda$ and $L_w\le L$, and (ii) applies to the move of $u_b$; the eight private links $G$, $\xi_aG$ move by $d(G,G')$ each. By the last statement of Theorem~\ref{thm:jholds}, Proposition~\ref{prop:jreduce} holds in these coordinates, through the inversion $\mathfrak i$, which maps the configuration to one whose first block $w^{-1}$ is within $\frac1{32}$ of $I$ and in which the moving link $u_b^{-1}$ is a link of the second block. The proof of Proposition~\ref{prop:jweighted} transfers through $\mathfrak i$ in the same way. The rest of the proofs of Lemma~\ref{lem:endchart} and Theorem~\ref{thm:endlinear} is unchanged. Case~(b), $e=w_b$ with the $u_a$ near $I$, is Lemma~\ref{lem:relchart} itself in the coordinates $\xi_a=g^{-1}\ell_a$, whose part~(ii) covers a move of $w_b$, and Proposition~\ref{prop:jreduce} as stated. \end{proof} So Corollaries~\ref{cor:endchart} and~\ref{cor:tubederiv} hold with $L_*$ replaced by $L_\natural$, and in Corollary~\ref{cor:endchart} with the constant $L_\natural+3\sqrt2\,\beta_W$, which is at most $7192+443625\, \beta_W+\mathfrak b_{\mathrm{tie}}(79.69+27216\,\beta_W)$. Both also hold when, for each end in an average, the inside $\rho$-links of the other block of its pair lie off $A^-_W$ and within $\frac1{32}$ of $I$ at each of their values, in place of the links of the block of the end: the proofs apply Lemma~\ref{lem:endchart} at fixed values of the integrated links, and Lemma~\ref{lem:endotherblock} needs the other block fixed. The derivative in an end is uniform in the coarse value on the whole pair fibre, on and off the window chart of Theorem~\ref{thm:badcross}, since the relative chart of Lemma~\ref{lem:relchart} covers the fibre of the pair. Integrating the private links of the end pairs before applying Theorem~\ref{thm:badcross} gives the analogue of Corollary~\ref{cor:badcrossratio} for ends in the averages, at the rate $\alpha_G$. \begin{theorem}[two ends in the averages]\label{thm:endsdecay} Let $G=\SU(2)$, $\beta_W\ge2.3\cdot10^{17}$, $W$ a set of blocks, and $E_a$, $E_b$ disjoint sets of links off $A^-_W$. Let $K_a$, $K_b$ be the sets of pairs based in $W$ whose average contains a link of $E_a$, of $E_b$, and $P_a$, $P_b$ their private links. Suppose: \begin{enumerate}[label=\textup{(H\arabic*)},leftmargin=*] \item $K_a\cap K_b=\emptyset$, and no plaquette contains a link of $E_a\cup P_a$ and a link of $E_b\cup P_b$; \item for every pair $(y,\rho)\in K_a\cup K_b$, neither $y-e_\rho$ nor $y+e_\rho$ lies in $W$; \item for every pair of $K_a\cup K_b$, the inside $\rho$-links of one of its two blocks lie within $\frac1{32}$ of $I$ at all the values below. \end{enumerate} Fix the other links off $A^-_W$, and let $a_0,a_1\in G^{E_a}$, $b_0,b_1\in G^{E_b}$, the links being moved one at a time along geodesics, and let $Z_{ij}=\nu^-_c(e^{-\lambda Q^-})$ with $a_i$ on $E_a$, $b_j$ on $E_b$, and the factors of the pairs of $K_a\cup K_b$ in the versions of Lemmas~\ref{lem:endchart} and~\ref{lem:endotherblock}. Put $L_e=3\sqrt2\, \beta_W+L_\natural$ for the links that enter an average of a pair based in $W$ and $L_e=3\sqrt2\,\beta_W$ for the others, $L'_e=\beta_W(12.39+8.486 \,L_e)$, $d_e$ the length of the move of $e$, and \[ \Lambda_a=\sum_{e\in E_a}L_ed_e,\qquad\Lambda'_a=\sum_{e\in E_a}L'_ed_e , \] and likewise $\Lambda_b$, $\Lambda'_b$. Let $N_a$, $N_b$ be the links of $A^-_W$ outside $P_a\cup P_b$ that share a plaquette with a link of $E_a\cup P_a$, of $E_b\cup P_b$. Suppose that at all the intermediate values of the ends the fibre satisfies the hypotheses of Theorem~\ref{thm:badcross}, as in Proposition~\ref{prop:badpassage}% \textup{(ii)}, with fixed $\kappa$ and $s$, and let $\bar p$ bound its mass off the chart. Then \begin{multline*} \Bigl\lvert\log\frac{Z_{11}Z_{00}}{Z_{10}Z_{01}}\Bigr\rvert\le\frac{4.8 \cdot10^4}{\beta_W}\,\Lambda'_a\Lambda'_b\bigl(\lvert N_a\rvert\lvert N_b \rvert\bigr)^{1/2}e^{-\alpha_Gd(N_a,N_b)}\\ +6\bar p\,\Lambda_a\Lambda_b . \end{multline*} \end{theorem} \begin{proof} \emph{Factorisation.} Let $\mathcal P_a$ be the plaquettes of $Q^-$ that contain a link of $E_a\cup P_a$, and likewise $\mathcal P_b$; by (H1) they are disjoint. By Lemma~\ref{lem:endpairs} only the factors of the pairs of $K_a$ depend on $E_a$. By (H2) the inside $\rho$-links of the two blocks of a pair of $K_a\cup K_b$ lie in $I_W$ or off $A_W$, so they are fixed links of the fibre, and by (H3) each such pair has a relative chart (Lemmas~\ref{lem:relchart} and~\ref{lem:endotherblock}). Let $\zeta$ be the links of $A^-_W$ outside $P_a\cup P_b$, and $\xi^a=(\xi^k)_{k\in K_a}$ the chart coordinates of the pairs of $K_a$. Then \begin{gather*} Z(a,b)=\int\Xi(d\zeta)\,\mathcal A_a(\zeta)\,\mathcal B_b(\zeta),\\ \mathcal A_a(\zeta)=\int A(\xi^a;\zeta,a)\,d\xi^a , \end{gather*} where $A=\prod_{k\in K_a}D_k^{-1}\exp(-\beta_W\sum_{p\in\mathcal P_a}(1- \frac12\ReTr U_p))$, the links of $P_a$ being $\Gamma_k(\xi^k)$, $\mathcal B_b$ is defined in the same way, and the measure $\Xi$, which collects the other pair factors, the Haar measure of the inside links in $A^-_W$ and the other plaquettes, depends neither on $a$ nor on $b$. Neither $\Gamma_k$ nor $D_k$ depends on $\zeta$, so $\mathcal A_a$ is a smooth function of the links of $N_a$, and \[ \frac{Z_{11}Z_{00}}{Z_{10}Z_{01}}=\frac{\mu(R_aR_b)}{\mu(R_a)\,\mu( R_b)},\qquad R_a=\frac{\mathcal A_{a_1}}{\mathcal A_{a_0}},\quad R_b= \frac{\mathcal B_{b_1}}{\mathcal B_{b_0}}, \] where $\mu$ is the law of $\zeta$ under the constrained law $\mu^-_W$ at $(a_0,b_0)$. \emph{One step.} Move one end $e\in E_a$ of a pair $k\in K_a$ by $\delta \le0.08$ along its geodesic, and let $\varepsilon_e(\delta)$ be such that $\int\lvert A_{e'}-A_e\rvert\,d\xi^a\le\varepsilon_e(\delta)\int A_e\, d\xi^a$. The proofs of Lemma~\ref{lem:endchart} and Theorem~\ref{thm:endlinear}, applied at fixed $\xi^{k'}$, $k'\ne k$, and $\zeta$, with $\omega$ a multiple of the point mass at these values, together with the six plaquettes through $e$, none of which contains a private link of $k$, give $\varepsilon_e(\delta)\le e^{3\sqrt2\beta_W\delta}(1+ \varepsilon(\delta))-1$, with $\varepsilon(\delta)$ of those proofs, so $\varepsilon_e(\delta)\le L_e\delta+o(\delta)$; for an end that enters no average, $D$ and $\Gamma$ do not move and $\varepsilon_e(\delta)\le e^{3 \sqrt2\beta_W\delta}-1$. In particular $\lvert\log\mathcal A_{e'}-\log \mathcal A_e\rvert\le L_e\delta+o(\delta)$. Differentiating under the integral, for a link $z$ of $N_a$, \[ \nabla_z\log\mathcal A_e=-\beta_WE^e\bigl[h_e\bigr],\qquad h_e=\nabla_z \sum_{p\in\mathcal P_a}\bigl(1-\tfrac12\ReTr U_p\bigr), \] with $E^e$ the probability with density $A_e/\int A_e$ in $\xi^a$. The link $z$ lies in at most six plaquettes. For $U_p=MzN$ the left-invariant gradient of $1-\frac12\ReTr U_p$ in $z$ is $\frac12\Pi(NMz)$ up to sign, of norm at most $\frac1{\sqrt2}$, and it changes by at most $\frac12\, d(M,M')+\frac12\,d(N,N')$ when the other links move; so $\lVert h_e\rVert \le3\sqrt2$. In the step only $e$ and the eight links of $k$ move, these by at most $L\delta$, $L<2.064$; a plaquette contains at most two $\rho$-links, and none contains $e$ and a private link of $k$, so each plaquette through $z$ has at most two moving links, which move by at most $L\delta$ each, and $\sup\lVert h_{e'}-h_e\rVert\le6L\delta$. The two densities differ in $L^1$ by at most $2\varepsilon_e(\delta)$. Hence \[ \bigl\lVert\nabla_z\log\mathcal A_{e'}-\nabla_z\log\mathcal A_e\bigr\rVert \le\beta_W\bigl(6L\delta+3\sqrt2\cdot2\varepsilon_e(\delta)\bigr). \] Summing over a subdivision of the moves and letting the step tend to $0$, $\lvert\log R_a\rvert\le\Lambda_a$ and $\lVert\nabla_z\log R_a\rVert \le\Lambda'_a$ for every $z$, by $6L<12.39$ and $6\sqrt2<8.486$; the same holds for every part of the moves, with the corresponding parts of $\Lambda_a$ and $\Lambda'_a$, and for $R_b$. \emph{Cells.} Parametrise the moves of $E_a$ by $\sigma\in[0,1]$ and of $E_b$ by $\tau\in[0,1]$, and cut each interval into $n$ cells on which every end moves by at most $1/n$ of its total move. For the cell $(i,j)$ let $R^{(i)}_a$, $R^{(j)}_b$ be the ratios above between the ends of the cell, $\Lambda^{(i)}_a$, $\Lambda'^{(i)}_a$, $\Lambda^{(j)}_b$, $\Lambda'^{(j)}_b$ the parts of the sums, and $\mu_{ij}$ the law of $\zeta$ at the corner $(\sigma_i,\tau_j)$. The logarithm of the cross-ratio is the sum over the cells of $\log(1+x_{ij})$ with $x_{ij}=\Cov_{\mu_{ij}}( R^{(i)}_a,R^{(j)}_b)/(\mu_{ij}(R^{(i)}_a)\mu_{ij}(R^{(j)}_b))$. The functions $R^{(i)}_a$ depend on the links of $N_a$ only, are bounded by $e^{\Lambda^{(i)}_a}$, have gradient at most $e^{\Lambda^{(i)}_a} \Lambda'^{(i)}_a$ in each link, satisfy $\lVert R^{(i)}_a-1\rVert_\infty\le e^{\Lambda^{(i)}_a}-1$ and $\mu_{ij}(R^{(i)}_a)\ge e^{-\Lambda^{(i)}_a}$, and likewise for $R^{(j)}_b$. A covariance does not change when constants are subtracted, so Theorem~\ref{thm:badcross}, through Proposition~\ref{prop:badpassage}\textup{(ii)}, gives \begin{multline*} \lvert x_{ij}\rvert\le e^{2\Lambda^{(i)}_a+2\Lambda^{(j)}_b}\Bigl(\frac {4.8\cdot10^4}{\beta_W}\Lambda'^{(i)}_a\Lambda'^{(j)}_b\\ \times\bigl(\lvert N_a\rvert\lvert N_b\rvert\bigr)^{1/2}e^{-\alpha_G d(N_a,N_b)}+6\bar p\,\Lambda^{(i)}_a\Lambda^{(j)}_b\Bigr), \end{multline*} by $e^x-1\le xe^x$. With $\lvert\log(1+x)\rvert\le\lvert x\rvert/(1- \lvert x\rvert)$, the sums of $\Lambda^{(i)}_a\Lambda^{(j)}_b$ and of $\Lambda'^{(i)}_a\Lambda'^{(j)}_b$ over the cells being $\Lambda_a \Lambda_b$ and $\Lambda'_a\Lambda'_b$, and $\max_{i,j}(\Lambda^{(i)}_a+ \Lambda^{(j)}_b+\lvert x_{ij}\rvert)\to0$ as $n\to\infty$, the claim follows. \end{proof} The rate $\alpha_G$ does not depend on $\beta_W$, and the prefactor is a polynomial in $\beta_W$ and $\mathfrak b_{\mathrm{tie}}$, of degree three in $\beta_W$. As in Corollary~\ref{cor:badcrossratio}, the second term needs the mass off the chart small, against $\Lambda_a\Lambda_b$, which is of order $\beta_W^2$ times the moves; by Theorem~\ref{thm:offchartmass} this fails for long components at $\beta_W\le10^{24}$. \begin{remark}\label{rem:endsfoldopen} What remains of the ends after Theorems~\ref{thm:endlinear} and~\ref{thm:endsdecay}. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item \emph{Both blocks far from $I$.} Lemma~\ref{lem:endotherblock} needs the inside $\rho$-links of one of the two blocks of the pair within $\frac1{32}$ of $I$. When both blocks have links far from $I$, the fibre of the pair is a graph over $\xi$ in neither coordinates on an open set of configurations (Proposition~\ref{prop:chartfolds}), the proof of Lemma~\ref{lem:ties} uses $\lVert\operatorname{Ad}(u_b)-1\rVert\le\frac {\sqrt2}{32}$, and the derivative uniform in $c_k$ is not proved. No configuration with an unbounded derivative is known either. \item \emph{The crossing number.} $\mathfrak b_{\mathrm{tie}}$ is finite by o-minimality and is not computed, so $L_\natural$ and the bound $1.5756\, \mathfrak b_{\mathrm{tie}}$ on $\mathcal A_{\mathrm{tie}}$ are explicit up to this one integer. \item \emph{The rate off the chart.} Theorem~\ref{thm:endsdecay} decays at the rate $\alpha_G$ up to the term $6\bar p\Lambda_a\Lambda_b$, and needs the blocks next to each end pair outside $W$ (H2), so that the links that enter the averages besides the ends are fixed; a pair whose second block lies in $W$ has random inside links, and the ratio $R_a$ then depends on them through the chart and the Jacobian. A decay between two ends at a rate independent of $\beta_W$ off the chart, or without (H2), is not proved; along a tube the rate of Theorem~\ref{thm:tube} remains the only one without a chart. \end{enumerate} \end{remark} The last part of the section concerns the coarse fields with opposite fluxes at adjacent coarse plaquettes, for an arbitrary field. Let $P$ and $P'$ be the coarse plaquettes at $y$ and $y+e_\mu$ in a plane $\{\mu,\nu\}$ and $W'=\{y+ie_\mu+je_\nu:0\le i\le2,\ 0\le j\le1\}$. \begin{proposition}[a coarse dipole]\label{prop:dipole} Let $G=\SU(2)$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Let $U$ be a field with $\delta(U_p,I)\le\kappa\le10^{-8}$ for every plaquette with a vertex in a block of $W'^+$, and $c=V^\sharp(U)$. Then \[ \bigl\lVert\log c_P-\operatorname{Ad}(c_{y,\mu})\log c_{P'}\bigr\rVert\le 6\kappa+1.7\cdot10^4\kappa^2 . \] \item If moreover $c_{y,\mu}c_{P'}c_{y,\mu}^{-1}=c_P^{-1}$, then $\delta( c_P,I)\le3\kappa+8500\kappa^2$. Hence, if a coarse field has this property and $\delta(c_P,I)>1.8001\cdot10^{-12}$, no field of the fibre over any $W\supset W'$, for any exterior, satisfies the hypotheses of Theorem~\ref{thm:badcross}, and for $\delta( c_P,I)>1.8133\cdot10^{-12}$ none satisfies those of Theorem~\ref{thm:fibrechart}\textup{(iv)}. \item Let $L\ge6$, $\mu<\nu$, $0<\theta\le1.7\cdot10^{-12}$, let $c^{\theta, d}$ be the coarse field with $c^{\theta,d}_{y,\nu}=e^{\theta T_0}$ for $y_\mu=1$, $y_\nu=0$ and every other coarse link $I$, and $W^d=\{y:0\le y_\mu\le2,\ 0\le y_\nu\le1\}$. Its coarse plaquettes other than $I$ are $e^{\theta T_0}$ at $y_\mu=0$ and $e^{-\theta T_0}$ at $y_\mu=1$, in the plane $\{\mu,\nu\}$ with $y_\nu=0$, so that the hypothesis of \textup{(ii)} holds at $y=0$ with $t:=\delta(c_P,I)=2\sqrt2\sin\frac\theta2$. The field $U^{\theta,d}$ with the angles of Table~\ref{tab:dipole} on the links listed there, at every value of the other two coordinates, and equal to $R(c^{\theta,d})$ elsewhere, lies in the fibre over $W^d$ of $R(c^{\theta, d})$, and every plaquette has $\delta(U^{\theta,d}_p,I)\le\kappa_d:=2\sqrt2 \sin\frac\theta6$. \end{enumerate} In particular, for $t\le1.79\cdot10^{-12}$ the fibre over $W^d$ of $R(c^{\theta,d})$ contains a field at which it satisfies the hypotheses of Theorem~\ref{thm:badcross} with $s=2.4\cdot10^{-15}$, and for $t>1.8001 \cdot10^{-12}$ no fibre over a set containing $W^d$ does, whatever the exterior. \end{proposition} \begin{table}[t] \caption{\label{tab:dipole}The field $U^{\theta,d}$ of Proposition~\ref{prop:dipole}\textup{(iii)}, in units of $\theta$, at the sites with coordinates $x_\mu$ and $x_\nu$: the angles of the links in the direction $\mu$ for $0\le x_\nu\le3$, of the links in the direction $\nu$ at $x_\nu=1$, and the real sums of the plaquettes in the plane $\{\mu,\nu\}$ with corner $x$. Every other link has the angle of $R(c^{\theta,d})$: $1$ for the links $(x,\nu)$ with $x_\nu=1$ and $x_\mu\in\{2,3\}$, and $0$ on every other link.} \centering\footnotesize \renewcommand{\arraystretch}{1.15} \setlength{\tabcolsep}{4pt} \begin{tabular}{@{}l@{\ \ }r@{\ \ }rrrrrrr@{}} \toprule & & \multicolumn{7}{c}{$x_\mu$}\\ & $x_\nu$ & $-1$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$\\ \midrule links $\mu$ & $3$ & $0$ & $-\frac13$ & $-\frac1{12}$ & $0$ & $\frac1{12}$ & $\frac13$ & $-\frac1{12}$\\ & $2$ & $0$ & $0$ & $\frac14$ & $0$ & $-\frac14$ & $0$ & $-\frac1{12}$\\ & $1$ & $0$ & $-\frac13$ & $-\frac1{12}$ & $0$ & $\frac1{12}$ & $\frac13$ & $-\frac1{12}$\\ & $0$ & $0$ & $0$ & $\frac14$ & $0$ & $-\frac14$ & $0$ & $-\frac1{12}$\\ links $\nu$ & $1$ & $0$ & $-\frac13$ & $\frac13$ & $1$ & $1$ & $\frac13$ & $-\frac13$\\ plaquettes & $3$ & $0$ & $-\frac13$ & $-\frac1{12}$ & $0$ & $\frac1{12}$ & $\frac13$ & $-\frac1{12}$\\ & $2$ & $0$ & $\frac13$ & $\frac13$ & $0$ & $-\frac13$ & $-\frac13$ & $0$\\ & $1$ & $-\frac13$ & $\frac13$ & $\frac13$ & $0$ & $-\frac13$ & $-\frac13$ & $\frac13$\\ & $0$ & $0$ & $\frac13$ & $\frac13$ & $0$ & $-\frac13$ & $-\frac13$ & $0$\\ & $-1$ & $0$ & $0$ & $-\frac14$ & $0$ & $\frac14$ & $0$ & $\frac1{12}$\\ \bottomrule \end{tabular} \end{table} \begin{proof} (i) We use the facts (a)--(c) and the notation of the proof of Lemma~\ref{lem:coarsewitnesssharp}, and one more fact. (d) Let $X,A\in\g$ have norms at most $1$, and suppose that $\epsilon= \delta(e^X,e^A)+\lVert X\rVert\lVert A\rVert\le1$. Then $\lVert X-A\rVert \le\epsilon(1+\epsilon^2)$. Indeed $\delta(e^{-A}e^X,e^{X-A})\le\lVert A\rVert \lVert X\rVert$ by (a), so $\delta(e^{X-A},I)\le\epsilon$, and $\lVert X-A \rVert<\sqrt2\pi$, so (c) and (b) give the claim. The hypothesis of Lemma~\ref{lem:coarsewitness} holds for $P$ and for $P'$, whose corner blocks and their neighbours lie in $W'^+$. Let $\ell_1,\dots, \ell_4$ be the path links of the sides of $P$, $\ell_5$ and $\ell_6$ those of $(y+e_\mu,\mu)$ and $(y+2e_\mu,\nu)$, and define on the blocks $y+ie_\mu+je_ \nu$ of $W'$, indexed by $(i,j)$, \[ t_{00}=I,\ t_{10}=\ell_1,\ t_{01}=\ell_4,\ t_{11}=\ell_1\ell_2,\ t_{20}= \ell_1\ell_5,\ t_{21}=\ell_1\ell_5\ell_6 . \] \emph{Step 1.} In the proof of Lemma~\ref{lem:coarsewitnesssharp}, $\delta(c_P,e^{\mathsf L(\bar Z)+Y})\le3842\kappa^2$ and $\delta(H_x,e^{ \mathsf L(Z^x)+Y})\le3842\kappa^2$, the exponents have norm at most $4.001 \kappa$, and so have $\log c_P$ and $\log H_x$, by (b) and $\delta(c_P,I)\le 4\kappa+7700\kappa^2$, $\delta(H_x,I)\le4\kappa$. By (d), with $3842+4.001^2 <3858.01$, each of $\log c_P$ and $\log H_x$ is within $3858.1\kappa^2$ of its exponent; as $\mathsf L(\bar Z)+Y$ is the mean of the $\mathsf L(Z^x)+Y$, \[ \Bigl\lVert\log c_P-\frac1{16}\sum_{x\in B(y)}\log H_x\Bigr\rVert\le7717 \kappa^2 . \] \emph{Step 2.} Let $p_1,p_2,p_3,p_4$ be the plaquettes of $S_x$ at $x$, $x+e_\mu$, $x+e_\nu$, $x+e_\mu+e_\nu$. Writing out the links, \[ H_x=\bigl(g_2U_{p_2}g_2^{-1}\bigr)\bigl(g_4U_{p_4}g_4^{-1}\bigr)U_{p_1} \bigl(g_3U_{p_3}g_3^{-1}\bigr), \] with $g_1=I$, $g_2=U_{x,\mu}$, $g_3=U_{x,\nu}$, $g_4=U_{x,\mu}U_{x+e_\mu, \nu}$: in the product the four inner links of the square cancel in pairs. With $A_r=\operatorname{Ad}(g_r)\log U_{p_r}$, $\lVert A_r\rVert\le1.0001 \kappa$ by (b), (a) gives $\delta(H_x,e^{A_1+\dots+A_4})\le6.002\kappa^2$, and (d) gives $\lVert\log H_x-\sum_rA_r\rVert\le22.01\kappa^2$. \emph{Step 3.} Let $z(p)$ be the block of the corner of a plaquette $p$. By Lemma~\ref{lem:smallframes}\textup{(i)} applied to $W'$, an inside link is within $3\kappa$ of $I$ and a link that leaves a block $y'\in W'$ in the direction $\sigma$ within $21\kappa$ of the path link of $(y',\sigma)$; the links $U_{x,\mu}$, $U_{x,\nu}$, $U_{x+e_\mu,\nu}$ are of one of these kinds, according to the offsets of $x$, and comparing with the definition of the $t_{ij}$ gives $\delta(g_r,t_{z(p_r)})\le42\kappa$ for $x\in B(y)$. As $\lVert\operatorname{Ad}(g)X-\operatorname{Ad}(g')X\rVert\le2\delta(g,g') \lVert X\rVert$, replacing each $g_r$ by $t_{z(p_r)}$ costs at most $4\cdot2 \cdot42\cdot1.0001\kappa^2<336.04\kappa^2$. With Steps 1 and 2, \begin{gather*} \Lambda_P=\frac1{16}\sum_{x\in B(y)}\sum_{p\in S_x}\operatorname{Ad}(t_{z( p)})\log U_p,\\ \lVert\log c_P-\Lambda_P\rVert\le8076\kappa^2 . \end{gather*} For $P'$ the same steps, with the base point $y+e_\mu$ and the transports $t'_{ij}=\ell_1^{-1}t_{ij}$, give $\lVert\log c_{P'}-\Lambda_{P'}\rVert\le 8076\kappa^2$ with $\Lambda_{P'}$ formed with $B(y+e_\mu)$ and the $t'$, so that $\operatorname{Ad}(\ell_1)\Lambda_{P'}$ is formed with the same transports $t_{z(p)}$ as $\Lambda_P$. \emph{Step 4.} As in the proof of Proposition~\ref{prop:spread}% \textup{(iv)}, a plaquette at the corner $z$ enters $\Lambda_P$ with the multiplicity $n(i)n(j)$ and $\operatorname{Ad}(\ell_1)\Lambda_{P'}$ with $n(i-2)n(j)$, where $z_\mu=2y_\mu+i$, $z_\nu=2y_\nu+j$, so \begin{gather*} \Lambda_P-\operatorname{Ad}(\ell_1)\Lambda_{P'}=\frac1{16}\sum_zw_z \operatorname{Ad}(t_{z(p_z)})\log U_{p_z},\\ \sum_z\lvert w_z\rvert=96, \end{gather*} and its norm is at most $6\kappa(1+\kappa^2)$, as $\lVert\log U_p\rVert\le \kappa(1+\kappa^2)$ by (b). Finally $c_{y,\mu}=\ell_1e^{\bar Z_1}$ with $\lVert\bar Z_1\rVert\le24.001\kappa$, so $\delta(c_{y,\mu},\ell_1)\le 24.001\kappa$ by (c), and $\lVert(\operatorname{Ad}(c_{y,\mu})- \operatorname{Ad}(\ell_1))\log c_{P'}\rVert\le48.002\kappa\cdot4.001\kappa< 192.1\kappa^2$. Altogether the left side of (i) is at most $6\kappa+6 \kappa^3+(2\cdot8076+192.1)\kappa^2\le6\kappa+1.7\cdot10^4\kappa^2$. (ii) The principal logarithm commutes with conjugation, so $\operatorname {Ad}(c_{y,\mu})\log c_{P'}=\log c_P^{-1}=-\log c_P$, both norms being below $\sqrt2\pi$. Then (i) gives $2\lVert\log c_P\rVert\le6\kappa+1.7\cdot10^4 \kappa^2$, and $\delta(c_P,I)\le\lVert\log c_P\rVert$ by (c). The pairs that enter (i) are based in $W'$, so (i) applies to every field of a fibre over $W\supset W'$. If such a field satisfied the hypotheses of Theorem~\ref{thm:badcross}, its plaquettes on $W^+\supset W'^+$ would be at most $\kappa<6.0\cdot10^{-13}$, and $\delta(c_P,I)\le1.8\cdot10^{-12}+8500 \cdot3.6\cdot10^{-25}<1.8001\cdot10^{-12}$. The hypothesis of Theorem~\ref{thm:fibrechart}\textup{(iv)} needs plaquettes at most $\kappa< 6.044\cdot10^{-13}$, by the proof of Proposition~\ref{prop:badchart}\textup{(ii)}, and then $\delta(c_P,I)< 1.8132\cdot10^{-12}+3.2\cdot10^{-21}<1.8133\cdot10^{-12}$. (iii) The coarse plaquettes are computed as in Proposition~\ref{prop:spread}\textup{(i)}, with $a(1)=\theta$ and $a=0$ elsewhere; $c_{0,\mu}=I$ and $c_{P'}=c_P^{-1}$. We write $\varphi$ for the angles of $U^{\theta,d}$ and $\varphi^R$ for those of $R(c^{\theta,d})$, which by Proposition~\ref{prop:badchart}\textup{(i)} are $\theta$ on the links $(x,\nu)$ with $x_\nu=1$ and $x$ in a block with $y_\mu=1$, $y_\nu=0$, and $0$ elsewhere. \emph{Links.} The links where $\varphi\ne\varphi^R$ are the links $(x,\nu)$ with $x_\nu=1$, $x_\mu\in\{0,1,4,5\}$, which have $o_\nu=1$ and are private links of pairs based in $W^d$; the links $(x,\mu)$ with $x_\mu\in\{1,3,5\}$, $0\le x_\nu\le3$, private links of such pairs; and the links $(x,\mu)$ with $x_\mu\in\{0,4\}$ and $x_\nu\in\{1,3\}$, which have $o_\mu=0$ and $o_\nu=1$ with $\nu>\mu$ and are active links inside blocks of $W^d$. So $U^{\theta,d}$ agrees with $R(c^{\theta,d})$ off $A_{W^d}$ and vanishes on the tree links. \emph{Averages.} Every entry differs from that of $R(c^{\theta,d})$ by at most $\frac\theta3$, so for every pair $(y,\sigma)$ the angles of its paths differ by at most $\frac{4\theta}3$, $\delta(P_x,P_{2y})<4\theta<\frac12$, and as in the proof of Proposition~\ref{prop:spread}, $V^\sharp_{y,\sigma}= c^{\theta,d}_{y,\sigma}$ when $\sum_xD_x=0$, $D_x$ the deviation of the angle of $P_x$ from that of the coarse link. For $\sigma\notin\{\mu,\nu\}$ every $D_x$ vanishes. For $\sigma=\nu$ only the pairs with $y_\nu=0$ meet a changed link, and $\sum_xD_x=8(d(2y_\mu)+d(2y_\mu+1))$, where the deviations $d(m)$ of the links $(x,\nu)$, $x_\nu=1$, $x_\mu=m$, are $-\frac 13,\frac13,0,0,\frac13,-\frac13$ for $m=0,\dots,5$ and $0$ otherwise. For $\sigma=\mu$ and $y_\nu\in\{0,1\}$, $\sum_xD_x=4(X(2y_\mu)+2X(2y_\mu+1)+ X(2y_\mu+2))$, where $X(m)$, the sum of the deviations of the two links $(x,\mu)$ with $x_\mu=m$ and $x_\nu\in\{2y_\nu,2y_\nu+1\}$, is $-\frac13, \frac16,0,-\frac16,\frac13,-\frac16$ for $m=0,\dots,5$ and $0$ otherwise; for other $y_\nu$ the pair meets no changed link. The sums vanish for $y_\mu\in\{0,1,2\}$, that is for every pair based in $W^d$. \emph{Plaquettes.} As in the proof of Proposition~\ref{prop:spread}, a plaquette in a plane other than $\{\mu,\nu\}$ has real sum $0$. In the plane $\{\mu,\nu\}$ the table lists every plaquette that contains a changed link, and every plaquette with real sum $\pm\theta$ in $R(c^{\theta,d})$, those at the corners $x_\mu\in\{1,3\}$, $x_\nu=1$; all others are $I$. The largest real sum is $\frac\theta3$, which gives $\kappa_d$. For the last statement, $\sin x\ge x(1-x^2/6)$ gives $\theta<1.7\cdot10^{- 12}$ for $t\le1.79\cdot10^{-12}$, and $\sin3x=\sin x\,(3-4\sin^2x)$ gives $\kappa_d=t/(3-4\sin^2\frac\theta6)\le\frac t3(1+\theta^2)<5.967\cdot 10^{-13}$, so $s=2.4\cdot10^{-15}$ satisfies $s\le1/(6000c_3)$, $\kappa_d+6s \le10^{-8}$ and $\kappa_d+s\le6.0\cdot10^{-13}$, as in the proof of Proposition~\ref{prop:spread}. The other half is (ii). \end{proof} So the threshold of the coarse plaquette at which a background exists is not a function of the size of the coarse plaquette alone. Between $1.8001 \cdot10^{-12}$ and $2.39\cdot10^{-12}$ the coarse fields $c^\theta$ of Proposition~\ref{prop:spread} have a background with exterior $R(c)$, while the coarse fields with opposite fluxes at adjacent coarse plaquettes have none for any exterior; part (ii) removes the restriction to fields with values in a torus in Proposition~\ref{prop:spread}\textup{(iv)}. The factor $3$ of the dipole, against $4$ for a single coarse plaquette, comes from the weights $\sum_z\lvert w_z\rvert=96$ of the difference. \begin{remark}\label{rem:endsmeanopen} What these results leave open. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item \emph{Ends.} Lemma~\ref{lem:almosthaar}\textup{(iii)} and Corollary~\ref{cor:tubeends} bound the weight of the interior as a function of ends that enter averages, after integration over the non-path links, uniformly in the coarse links and in the moves of the ends. For the derivative in an end, Lemma~\ref{lem:endboundary} bounds the motion of the boundary $\{m=1\}$ by $3.4\cdot10^{-8}$ per unit move at every configuration; Proposition~\ref{prop:endkantorovich} and Corollary~\ref{cor:endmean} bound the derivative after the coarse link is averaged against a positive Lipschitz weight, with no condition on the other links; and Lemma~\ref{lem:endchart} and Corollary~\ref{cor:endchart} bound it uniformly in $c_k$ when the links of the block of the end lie within $\frac1{32}$ of $I$, in particular when the block across the end is good. Theorem~\ref{thm:endlinear} replaces the constant $128.07\,\beta_W+ (180.4+50.6\,\mathcal A_{\mathrm{tie}})e^{72\beta_W}$ by one linear in $\beta_W$, Corollary~\ref{cor:tieareab} gives $\mathcal A_{\mathrm{tie}} <1.5756\,\mathfrak b_{\mathrm{tie}}$ with the crossing number of Definition~\ref{def:tiecross}, and Lemma~\ref{lem:endotherblock} allows the condition on the other block of the pair in place of the block of the end. Corollary~\ref{cor:tubederiv} gives mixed differences in two ends that decay along a tube at the rate of Theorem~\ref{thm:tube}, and Theorem~\ref{thm:endsdecay} gives, on the chart of Theorem~\ref{thm:badcross}, mixed differences in two ends in the averages that decay at the rate $\alpha_G$, independent of $\beta_W$, up to a term proportional to the mass off the chart. Three things are not proved (Remark~\ref{rem:endsfoldopen}): the derivative uniform in $c_k$ when both blocks of the pair have links far from $I$, where the relative chart can fold (Proposition~\ref{prop:chartfolds}); the value of $\mathfrak b_{\mathrm{tie}}$; and a decay between two ends at a rate independent of $\beta_W$ off the chart, or for pairs whose second block lies in $W$. The rate of Theorem~\ref{thm:tube} is the oscillation of the Wilson weight across one interface (Remark~\ref{rem:tuberate}). \item \emph{Rate.} For $\beta_W\ge2.3\cdot10^{17}$ the decay across a bad component beyond the chart of Theorem~\ref{thm:badcross} is not proved; Remark~\ref{rem:tuberate} locates the loss in the oscillation of the Wilson weight across one interface. \item \emph{Coarse fields.} With exterior $R(c)$, a background exists up to $1.79\cdot10^{-12}$ for the dipole $c^{\theta,d}$ and up to $2.39\cdot 10^{-12}$ for $c^\theta$, and none exists above $1.8001\cdot10^{-12}$ for any coarse field with opposite fluxes at adjacent coarse plaquettes. For a general coarse field between $6.0\cdot10^{-13}$ and $2.418\cdot10^{-12}$ the existence is not decided. \end{enumerate} \end{remark} \subsection{The one-step error in gauge-invariant form}\label{ssec:relstep} Proposition~\ref{prop:massgrowth} requires the one-step error to be a relative bound by a gauge-invariant form. This subsection constructs such a bound at the trivial background on a box of blocks, from an absolute bound on the gauge complement taken as a hypothesis. The error of the covariant Hessian of a gauge-invariant induced action vanishes on the coarse gauge directions and is at most a multiple $\varepsilon'$ of the coarse Wilson form. On a box of side $n$, $\varepsilon'$ is of order $n^2$ times the absolute error, and this order cannot be improved. \emph{Setting.} On the coarse lattice of side $L$ let $W=\{y:a_\nu\le y_\nu \le a_\nu+n-1\}$ be a box of $n^4$ blocks, $n+2\le L$, $K=\{(y,\mu):y\in W\}$ and $W^\circ=\{y\in W:y-e_\mu\in W\text{ for every }\mu\}$. As in Theorem~\ref{thm:chartdecay}, the links off $A_W$ are fixed and the averages of the pairs that enter $W$ from outside are integrated. For $w\in\g^K$ put $|w|^2=\sum_k\lVert w_k\rVert^2$, let $Q_1(w)$ be the coarse plaquette form of $w$ extended by zero off $K$, and put \[ S_W(w)=\inf\bigl\{Q(X):X\in\g^{A_W},\ V_k(X)=w_k\ (k\in K)\bigr\}. \] At $U=I$, $2\lambda Q_1$ is the Hessian at $I$ of the coarse Wilson action $\lambda Q_1$ with the links off $K$ fixed. $2\lambda S_W$ is the Hessian of the Gaussian block-spin step, the minimum of the Hessian $2\lambda Q$ of the fine action at given averages. The \emph{coarse gauge group} $\Gamma_W= G^{W^\circ}$ acts on $G^K$ by $(h\cdot V)_{y,\mu}=h_yV_{y,\mu}h_{y+e_\mu }^{-1}$, with $h_y=I$ for $y\notin W^\circ$. This is the action on the cut-off averages of the fine gauge transformations that are constant on blocks and equal to $I$ off the blocks of $W^\circ$. These transformations preserve the tree gauge by Lemma~\ref{lem:blocktree}(ii), and they fix the links off $A_W$: a link with base in a block $y'\notin W$ ends in $y'$ or $y'+e_\mu$, and neither lies in $W^\circ$. They also preserve the Wilson action and Haar measure, and the cut-off average is covariant (Theorem~\ref{thm:cutoff}). For $V_0\in G^K$ and $\eta\in\g^{W^\circ}$, extended by zero, put \[ (d_{V_0}\eta)_{y,\mu}=\operatorname{Ad}(V_{0,y,\mu}^{-1})\eta_y- \eta_{y+e_\mu}, \] let $\mathcal O_{V_0}=d_{V_0}(\g^{W^\circ})$ be the gauge directions at $V_0$ and $\Pi_{V_0}$ the orthogonal projection of $\g^K$ onto $\mathcal O_{V_0}^\perp$. At $V_0=I$ write $d$, $\mathcal O$ and $\Pi$. \begin{lemma}[the window off the gauge directions]\label{lem:windowgap} Let $W$ be the box above. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The map $d$ is injective on $\g^{W^\circ}$, and $\mathcal O=\ker Q_1=\ker S_W$. \item For every $w\in\g^K$, $|\Pi w|^2\le C_n\,Q_1(w)$ with \[ C_n=1+\frac1{4\sin^2(\pi/2n)}\le1+\frac{n^2}4 . \] \item $c_*Q_1\le S_W\le4Q_1$ on $\g^K$, with $c_*=(1+\sqrt{21/8})^{-2}> 0.1456$. \item There is $w\in\mathcal O^\perp\setminus\{0\}$ with $|w|^2=C_n^-Q_1 (w)$, where $C_n^-=(12\sin^2(\pi/(2n+2)))^{-1}\ge(n+1)^2/(3\pi^2)$. \item If $W$ is the whole coarse torus, the fields that are constant in each direction lie in $\mathcal O^\perp\cap\ker Q_1\cap\ker S_W$, so no bound of the form \textup{(ii)} holds. \end{enumerate} \end{lemma} \begin{proof} Let $\Lambda=\prod_\nu[a_\nu,a_\nu+n]$, with boundary $\partial\Lambda$. The interior sites of $\Lambda$ form $W^\circ$. Call $(y,\mu)\in K$ \emph{inner} if $y_\nu>a_\nu$ for every $\nu\ne\mu$, and a \emph{face link} otherwise. The inner links are the links of $\Lambda$ not contained in $\partial\Lambda$; call a plaquette of $\Lambda$ not contained in $\partial \Lambda$ inner. The links of $\partial\Lambda$ that are not face links have base outside $W$, so they are fixed, and $w$ vanishes on them. (ii) The cochain complex of $\Lambda$ relative to $\partial\Lambda$, with values in $\g$, has the interior sites, inner links and inner plaquettes as cells of degrees $0,1,2$. On these, $d_0=d$, and $d_1$ is the curvature on the inner plaquettes of a field on the inner links extended by zero. This complex is the fourfold tensor product of the complex of $[0,n]$ relative to its endpoints. In that complex $C^0=\R^{n-1}$, $C^1=\R^n$ and $d$ is the difference operator. With the product inner products, the Laplacian $\Delta =dd^*+d^*d$ of the product is the sum of the Laplacians of the factors. On $C^0$, $d^*d$ is the Dirichlet Laplacian, with eigenvalues $4\sin^2(\pi j/ 2n)$ for $1\le j\le n-1$. On $C^1$, $dd^*$ has the same nonzero eigenvalues and one zero eigenvalue. A $1$-cochain of $\Lambda$ is a sum of products with exactly one factor of degree one, so $\Delta_1\ge\gamma_n=12\sin^2(\pi /2n)$. On $\ker d_0^*=\mathcal O^\perp$ this gives $\lvert d_1u\rvert^2\ge \gamma_n|u|^2$. Write $w=u+f$, with $u$ on the inner links and $f$ on the face links. Since $\mathcal O$ lies on the inner links, $\Pi w=\Pi u+f$ and $|\Pi w|^2=|\Pi u|^2+|f|^2$. Let $f_k\ne0$ at a face link $k=(y,\mu)$, and let $\nu\ne\mu$ be the least direction with $y_\nu=a_\nu$. The plaquette at $y-e_\nu$ in the plane $\{\mu,\nu\}$ has $k$ as its only link in $K$, since its other three links have base outside $W$. So its curvature is $\pm f_k$, and distinct face links give distinct such plaquettes, none of them inner. Consider next an inner plaquette $P$ in the plane $\{\mu,\nu\}$ at $z$. It has $z_\rho>a_\rho$ for $\rho\notin\{\mu,\nu\}$, so its face links are among $(z,\mu)$, if $z_\nu=a_\nu$, and $(z,\nu)$, if $z_\mu=a_\mu$. Thus $P$ contains at most two face links, and a face link lies in at most one inner plaquette. Hence $(d_1u)_P^2\le3\bigl((dw)_P^2+\sum_{k\in P}f_k^2 \bigr)$, where the sum runs over the face links of $P$. Since $d_1\Pi u= d_1u$, \[ \begin{split} |\Pi w|^2&\le\gamma_n^{-1}\lvert d_1u\rvert^2+|f|^2\\ &\le\frac3{\gamma_n}\Bigl(\sum_{P\text{ inner}}(dw)_P^2+|f|^2\Bigr)+|f|^2 \le C_nQ_1(w), \end{split} \] because the inner plaquettes and the plaquettes of the face links are distinct terms of $Q_1(w)$. Finally $\sin x\ge2x/\pi$ on $[0,\pi/2]$ gives $C_n\le1+n^2/4$. (iii) Let $X\in\g^{A_W}$ with $V(X)=w$ on $K$. Extended by zero, $X$ vanishes on the tree links, and its averages are $w$ on $K$, zero on the coarse links with both blocks outside $W$, and $v'_{y-e_\mu,\mu}= S^\mu_y/16$ on a pair entering $y\in W$ from $y-e_\mu\notin W$, by Lemma~\ref{lem:blocktree}(iii). By Proposition~\ref{prop:effective}(i), $Q_1(w+v')\le Q(X)$. The sum $S^\mu_y$ has at most seven active links, and every link inside a block enters one such sum. So $\sum|v'|^2\le\frac7{256} I\le\frac7{32}Q(X)$, by the bound $I\le8Q_{\rm in}$ in the proof of Theorem~\ref{thm:blockgap}, which uses only the tree links. A coarse plaquette contains at most two entering pairs, since $(z,\mu)$ and $(z+e_\mu ,\nu)$ cannot both enter, nor can $(z,\nu)$ and $(z+e_\nu,\mu)$. A coarse link lies in six plaquettes, so $Q_1(v')\le12\sum|v'|^2\le\frac{21}8Q(X)$. By the triangle inequality for $Q_1^{1/2}$, $Q_1(w)^{1/2}\le(1+\sqrt{21/8})Q(X)^{ 1/2}$, and the infimum over $X$ gives $S_W\ge c_*Q_1$. For the upper bound, the field $\varphi^*(w)$ of Proposition~\ref{prop:effective}(ii), with $w$ extended by zero, lies in $\g^{A_W}$, has averages $w$ on $K$ and $Q(\varphi^*(w))=4Q_1(w)$. (i) If $d\eta=0$, follow the links of direction $0$ from $y\in W^\circ$ to the first site outside $W^\circ$, where $\eta$ vanishes; so $\eta_y=0$. Extended by zero, $d\eta$ is exact on the whole lattice, so $Q_1(d\eta)= 0$, and $\ker Q_1\subset\mathcal O$ by (ii). The fine field $\varphi=d(\eta \circ b)$, with $b(x)$ the block of $x$, vanishes on the tree links and has $Q(\varphi)=0$. It has $V(\varphi)=d\eta$ by Lemma~\ref{lem:blocktree}(ii). It lies in $\g^{A_W}$, since a link on which it is nonzero meets a block of $W^\circ$ and so has base in $W$. Hence $S_W(d\eta)=0$, and $\ker S_W \subset\ker Q_1$ by (iii). (iv) Let $f$ be the product of sines on $\{a_\nu,\dots,a_\nu+n-1\}^3$, $\nu=1,2,3$, whose eigenvalue for the Dirichlet Laplacian of the cube is $12\sin^2(\pi/(2n+2))$. Let $w_{y,0}=f(y_1,y_2,y_3)Z$ for a unit $Z\in\g$, and $w=0$ in the other directions. The sum over $y_0$ telescopes, so $\ip w {d\eta}=0$ for every $\eta$. The curvature of a plaquette in the plane $\{0,\nu\}$ is a difference of $f$ along $e_\nu$, the plaquettes in the other planes vanish, and for each of the $n$ values of $y_0$ the sum is the Dirichlet form of $f$. Hence $Q_1(w)/|w|^2=12\sin^2(\pi/(2n+2))\le3\pi^2/ (n+1)^2$. (v) Then $W^\circ=W$ and every coarse link is in $K$. A constant field has $dw=0$, and $\ip w{d\eta}=\sum_y\sum_\mu\ip{w_\mu}{\eta_y-\eta_{y+e_\mu}}= 0$. The field $\varphi^*(w)$ of Proposition~\ref{prop:effective}(ii) has $Q=4Q_1(w)=0$, so $S_W(w)=0$. \end{proof} The gauge directions at a background $V_0$ near $I$ are close to those at $I$, and the flat form keeps its gap off them. \begin{lemma}[the window at a curved background]\label{lem:curvedwindow} Let $V_0\in G^K$ with $\lVert\operatorname{Ad}(V_{0,k}^{-1})-1\rVert\le \theta$ for every $k\in K$, and put $s_n=2\sin(\pi/2n)$. If $\theta 0.29131$ and $C_n\le\frac54+n^2/\pi^2$. \end{proof} Proposition~\ref{prop:relstep}(iii) and Corollary~\ref{cor:loopstep} do not conflict. The function $F$ there has the error $\frac12\lambda\varepsilon |\sigma|^2$ in the slice variable, a sum over links that is not a function of loops. The Hessian of a loop remainder at a flat background is a sum of squares of combinations of the curvature derivatives $D_pw$, so $Q_1$ itself controls it. \begin{remark}[what the loop bound does not give]\label{rem:valueshessians} The bound has three limits. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item \emph{Winding loops.} On the whole coarse torus of side $L$, a loop winding around the torus has no filling. Take the field with $V_{y,\mu}=I$ for $\mu\ne0$ and \[ V_{y,0}=g=\operatorname{diag}\bigl(e^{i\pi/2L},e^{-i\pi/2L}\bigr)\in\SU(2). \] Every plaquette holonomy is $I$, while the loop along direction $0$ has holonomy $g^L=\operatorname{diag}(i,-i)$, and $s(g^L)=1$. No bound of $s(V_C)$ by a multiple of the plaquette action holds for it. In the box $\Lambda$ every closed path has a filling, by Lemma~\ref{lem:loopfill}(ii). \item \emph{Values are not Hessians.} For $\SU(2)$, $m\in4\mathbb N$ and $g(\theta)=\operatorname{diag}(e^{i\theta},e^{-i\theta})$, the function $R_m(g(\theta))=\epsilon(1-\cos\theta)\cos(m\theta)$ is a class function, smooth because $\cos(m\theta)=T_m(\frac12\Tr g)$ with $T_m$ the Chebyshev polynomial. It satisfies $\lvert R_m\rvert\le\epsilon s$ everywhere, while \[ \frac{d^2}{d\theta^2}R_m(g(\theta))\Big|_{\theta=\pi/2}=-\epsilon m^2, \] since $\cos\frac{m\pi}2=1$ and $\sin\frac{m\pi}2=0$. Composed with one plaquette holonomy it is a gauge-invariant action term with a relative bound on its values and an unbounded Hessian at a background where that plaquette has holonomy $g(\pi/2)$. A relative bound on the values of the remainder therefore gives no Hessian bound at curved backgrounds; part (ii) of Proposition~\ref{prop:flatloops} uses flatness through $V_C(0)=I$. \item \emph{The representation.} The induced action is not shown here to be a series of filled loops with small $\mathcal L$. Its determinants and coarea factors are not of this form as they stand, and the Haar term $\delta_G$ keeps the factor $C_n$. \end{enumerate} \end{remark} \emph{Curved backgrounds.} Proposition~\ref{prop:flatloops}(ii) uses flatness through $V_C(0)=I$, and Remark~\ref{rem:valueshessians}(b) shows that a bound on values cannot replace it. A bound on the coefficients can. At a background whose plaquettes lie within $\kappa$ of $I$, the Hessian of a loop remainder is bounded by a covariant plaquette form plus a Euclidean term of order $\kappa$ (Proposition~\ref{prop:curvedloops}). On the window the Euclidean term costs a factor of order $\kappa n^2$ (Lemma~\ref{lem:smallcurv}), and this order is attained (Proposition~\ref{prop:curvedsharp}). Measured by the action of the background on the window, the loss is of order $\lVert\kappa\rVert_2$ for every $n$, and this order is attained as well (Proposition~\ref{prop:actionloops}, Corollary~\ref{cor:actionsharp}). Norms of matrices in $\U(N)$ are operator norms in this part. For a path $\gamma$ and $w\in\g^K$, extended by zero, let \[ Z_\gamma(w)=\frac d{dt}\Big|_{t=0}V_\gamma(V_0)^{-1}V_\gamma(V_0e^{tw}) \in\g \] be the derivative of its holonomy at $V_0$, trivialised on the left. If $\gamma$ makes the steps $k_1,\dots,k_\ell$ with orientations $\epsilon_i =\pm1$, then $(V_ke^{tw_k})^{-1}=V_k^{-1}e^{-t\operatorname{Ad}(V_k)w_k}$ gives \begin{equation}\label{eq:pathexp} V_\gamma(V_0e^{tw})=V_\gamma(V_0)\,e^{tY_1}\cdots e^{tY_\ell},\qquad Z_\gamma(w)=\sum_iY_i, \end{equation} where $Y_i=\operatorname{Ad}(s_i^{-1})w_{k_i}$ if $\epsilon_i=1$, $Y_i= -\operatorname{Ad}(s_i^{-1}V_{0,k_i})w_{k_i}$ if $\epsilon_i=-1$, and $s_i$ is the holonomy at $V_0$ of the part of $\gamma$ after its $i$-th step. In particular $\lVert Y_i\rVert=\lVert w_{k_i}\rVert$. For a plaquette $p$ in the plane $\{\mu,\nu\}$, $\mu<\nu$, with base point $x$, $Z_p$ is taken along $(x,\mu)$, $(x+e_\mu,\nu)$, $(x+e_\nu,\mu)^{-1}$, $(x,\nu)^{-1}$, and \[ Q^{V_0}(w)=\sum_p\lVert Z_p(w)\rVert^2 \] is the \emph{covariant plaquette form} at $V_0$. If $V_{0,p}=I$, then $Z_p(w)=D_pw$; in particular $Q^I=Q_1$. A filling~\eqref{eq:filling} is in \emph{normal form} if every $V_{p_j}$ is read from the base point of $p_j$ in this order. Reading $p$ from another corner, or in the other direction, conjugates $V_p$ by the holonomy of at most two of its links and may invert it, so every ordered filling is brought to normal form by lengthening each $h_j$ by at most two links. For a closed path $C$ with a filling in normal form let $\ell_C$ be its length, $n_{Ck}$ the number of its steps along the link $k$, $T_C=\sum_j\ell(h_j)$ the total length of the paths $h_j$, and $t_{Ck}$ the number of their steps along $k$. \begin{lemma}[a filled loop at a curved background]\label{lem:curvedloop} Let $C$ be a closed path with an ordered filling in normal form, and let $\lVert I-V_{0,p_j}\rVert\le\kappa$ for every $j$. In the coordinates $V_0e^\omega$, for every $w\in\g^K$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item with $Z_C=Z_C(w)$, \[ \Hess s(V_C)(0)[w,w]=-\frac1N\ReTr\Bigl(V_C(V_0)\Bigl(Z_C^2+\sum_{i0$, part (i) gives $\mathcal HR(u,u)\ge(1+0.037\,\kappa (n+1)^2)\frac{\mathcal L}NQ^{V_0}(\Pi_{V_0}u)$, against the factor $(1+ \kappa)^2+14\kappa(n+2)^2$ of Proposition~\ref{prop:curvedloops}(iii): the loss is of order $\kappa n^2$ in both. Part (ii) says more. Once $\kappa(n+1)^2\ge31$, the covariant Hessian of the coarse plaquette action has a negative direction off the gauge directions, so it obeys no comparison with a positive semidefinite gauge-invariant form there (Fig.~\ref{fig:curvedsharp}). This is the lattice form of the unstable mode of Nielsen and Olesen~\cite{NO78} in a constant chromomagnetic field. The comparison asked for in Remark~\ref{rem:relstepinputs}(d) must therefore lose a multiple of $\kappa \lvert\cdot\rvert^2$, and on a window of side $n$ it is a relative bound only while $\kappa n^2$ is small. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_curvedsharp.png} \caption{The ratio of the covariant Hessian of the plaquette action to half the covariant plaquette form at the field $u$ and the background of Proposition~\ref{prop:curvedsharp}, from the closed forms in its proof. (a) The ratio over the window side $n$ and $\kappa(n+1)^2$, for the two orientations $\varsigma=\pm1$. (b) The section $n=63$: for $\varsigma=+1$ the ratio exceeds one, above the bound of part (i), dashed; for $\varsigma=-1$ it changes sign near $\kappa(n+1)^2=23$, below the threshold $31$ of part (ii).} \label{fig:curvedsharp} \end{figure*} \emph{Curvature in the action.} The loss $\kappa n^2$ of Proposition~\ref{prop:curvedloops}(iii) comes from the supremum $\kappa$ of the curvature and from the gap $C_n$ of the window in $\ell^2$. In four dimensions the Euclidean term can be paid instead with the $\ell^2$ norm of the curvature, the square root of the Wilson action of the background, and an $\ell^4$ bound on the field, which holds on the window with a constant independent of $n$. For a plaquette $p$ write $\kappa_p=\lVert I-V_{0,p} \rVert$, the operator norm, which does not depend on the corner from which $p$ is read or on its orientation, and put \[ \lVert\kappa\rVert_2=\Bigl(\sum_p\kappa_p^2\Bigr)^{1/2}\le\Bigl(2N\sum_p s(V_{0,p})\Bigr)^{1/2}, \] the sums over the plaquettes with their corners in the box $\Lambda'= \prod_\nu[a_\nu-1,a_\nu+n]$ of Lemma~\ref{lem:smallcurv}; the inequality is $\kappa_p\le\lVert I-V_{0,p}\rVert_{HS}$. For $w\in\g^K$ put $\lvert w \rvert_4=(\sum_k\lVert w_k\rVert^4)^{1/4}$. As in the proof of Lemma~\ref{lem:windowgap}, a link $(y,\mu)\in K$ is inner when $y_\nu>a_\nu$ for every $\nu\ne\mu$, and a face link otherwise. The $\ell^4$ bound rests on an inequality of Gagliardo type. For finite sets $X_1,\dots,X_d$, $d\ge2$, and $x\in\prod_iX_i$, let $\hat x_i$ be $x$ with its $i$-th coordinate removed. If $F_i\ge0$ are functions of $\hat x_i$, then \begin{equation}\label{eq:gagliardo} \sum_x\prod_{i=1}^dF_i(\hat x_i)^{\frac1{d-1}}\le\prod_{i=1}^d\Bigl( \sum_{\hat x_i}F_i(\hat x_i)\Bigr)^{\frac1{d-1}}. \end{equation} For $d=2$ both sides equal $\sum F_1\sum F_2$. For $d\ge3$, sum first over $x_d$, on which $F_d$ does not depend. H\"older's inequality with $d-1$ factors bounds $\sum_{x_d}\prod_{i8.573$, $23/8.573<2.683$. Then $(n+2)^2/n^2\le9$ gives $24.15K$, and $\lVert\kappa\rVert_2\le\sqrt6(n+1)(n+2)\kappa_\infty$ with $(n+1)(n+2)/n^2\le6$ gives $\sqrt6\cdot6\cdot2.683K<39.44K$. Both bounds are far below the hypotheses $\kappa_\infty(n+2)^2\le10^{-2}$ and $\lVert \kappa\rVert_2\le\frac19$. The factors follow from Proposition~\ref{prop:curvedloops}(iii), $0.58\cdot24.2<14.1$, and from $c_4\cdot39.5<257.3$. \end{proof} On calm windows the loss $\kappa n^2$ is therefore already of order $K$, uniformly in $n$, and the bound in the action adds no strength there. Its use lies in the hypothesis and in the count of windows. The action is additive over windows, and the sup norm scaled by $n^2$ is not. \begin{corollary}[windows that fail]\label{cor:actioncount} Let the coarse torus have side $L=mn$ with $m,n\ge2$, tiled by the boxes $W_i=a_i+[0,n-1]^4$, $a_i\in n\Z^4$, with the boxes $\Lambda'_i$, and let $\lVert\kappa\rVert_{2,i}$ be $\lVert\kappa\rVert_2$ on $\Lambda'_i$. For every coarse field $V_0$ and $\eta>0$, \[ \#\{i:\lVert\kappa\rVert_{2,i}>\eta\}\le\frac{32N}{\eta^2}\sum_Ps(V_{0,P}), \] the sum over all coarse plaquettes. \end{corollary} \begin{proof} In each coordinate a plaquette occupies one value $x$ or two values $x,x+1$, and lies in $[a-1,a+n]$ only if $a$ is among $n+2$ consecutive integers. For $n\ge2$ these contain at most two multiples of $n$, and $L\ge2n\ge n+2$ keeps them distinct modulo $L$. So a plaquette has its corners in at most $2^4$ of the boxes $\Lambda'_i$, and $\sum_i\lVert\kappa\rVert_{2,i}^2\le 16\sum_P\kappa_P^2\le32N\sum_Ps(V_{0,P})$. \end{proof} With $\eta=\frac19$, all but $2592N\sum_Ps(V_{0,P})$ windows of the tiling satisfy Proposition~\ref{prop:actionloops}(ii) with a factor at most $\mathcal L^\sharp+0.73\mathcal N$, whatever $n$. The supremum condition counts differently. Let $V_0=I$ except on the link $a_i+(0,e_0)$ of each $W_i$, where $V_0=g$ with $\lVert I-g\rVert=2\eta'(n+2)^{-2}$. Links of distinct windows differ by at least $n\ge2$ in a coordinate and share no plaquette, so every window has $\kappa_\infty(n+2)^2=2\eta'>\eta'$, while $\sum_P\kappa_P^2=24m^4\eta'^2(n+2)^{-4}$. A curved plaquette with its corners in $\Lambda'_i$ contains a curved link whose base, a corner, lies in $\Lambda'_i$; the coordinates of that base minus $a_i$ lie in $\{0,n\}$, so there are at most $16$ such links and $\lVert\kappa\rVert_{2,i}\le \sqrt{96}\,\lVert I-g\rVert=8\sqrt6\,\eta'(n+2)^{-2}$. All $m^4$ windows fail the condition $\kappa_\infty(n+2)^2\le\eta'$, and none fails $\lVert\kappa\rVert_{2,i}\le\frac19$ once $(n+2)^2\ge72\sqrt6\,\eta'$. \begin{remark}[what the iteration needs]\label{rem:actioniteration} The one-step factors $1+130\lVert\kappa^{(j)}\rVert_2$ of Proposition~\ref{prop:actionloops}(iii) at the backgrounds of the scales $j=1,2,\dots$ have a bounded product exactly when $\sum_j\lVert\kappa^{(j)} \rVert_2<\infty$. On calm windows at the levels $K_j$ this holds if $\sum_j K_j<\infty$, by Corollary~\ref{cor:actioncalm}. Three inputs are not proved here. First, Corollary~\ref{cor:actioncalm} uses the supremum form of Lemma~\ref{lem:coarsewitness}; a bound of the coarse action $\sum_P\kappa_P ^2$ by the fine action near the window, with a constant at most one, would let the action bound $Q_W\le\frac34\kappa^2$ of the calm completions (Corollary~\ref{cor:calmexterior}) pass from scale to scale, and it is not proved. Second, the comparison of the Gaussian reference $S^U_W\circ\Pi_{ V_0}$ with $Q^{V_0}\circ\Pi_{V_0}$ of Remark~\ref{rem:relstepinputs}(d) remains open. Third, the Haar term keeps the factor $C_n$ in every loop representation (Proposition~\ref{prop:loopsums}(iv)), since it is a multiple of $|\Pi_{V_0}w|^2$ and not of $\lvert\Pi_{V_0}w\rvert_4^2$; Lemma~\ref{lem:windowlfour} does not remove it. \end{remark} The loop representation itself is the remaining question. For $\SU(2)$ the class of loop sums is closed under products and dense, while a loop sum has no a priori bound on its weights. \begin{proposition}[loop sums for $\SU(2)$]\label{prop:loopsums} Let $G=\SU(2)$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Let $C,C'$ be closed paths based at $x,x'$, $\gamma$ a path from $x$ to $x'$, and $C\vee C'$, $C\vee C'^{-1}$ the closed paths $C\gamma C' \gamma^{-1}$ and $C\gamma C'^{-1}\gamma^{-1}$. Then \[ s(V_C)s(V_{C'})=s(V_C)+s(V_{C'})-\tfrac12s(V_{C\vee C'})-\tfrac12s(V_{C \vee C'^{-1}}), \] and ordered fillings of $C$ and $C'$ give ordered fillings of $C\vee C'^{ \pm1}$ with $A_C+A_{C'}$ plaquettes and multiplicities $m_{Cp}+m_{C'p}$. \item On a box $\Lambda'$ with all its links variable, every continuous function invariant under the gauge group $G^{\Lambda'}$ is a uniform limit of functions $c+\sum_Cr_Cs(V_C)$, finite sums over closed paths in $\Lambda'$. \item Let $\varphi$ be a class function with $\varphi(\operatorname{diag} (e^{i\theta},e^{-i\theta}))=\sum_{m\ge0}c_m\cos m\theta$ and $\sum_mm^2 \lvert c_m\rvert<\infty$, and $C$ a closed path with an ordered filling. Then \[ \varphi(V_C)=\varphi(I)-\sum_{m\ge1}c_m\,s(V_{C^m}), \] where $C^m$ is $C$ traversed $m$ times with $m$ copies of its filling. Its contribution to $\mathcal L$ is $\sum_mm^2\lvert c_m\rvert\,\sup_pA_Cm_{Cp}$. The function $R_m$ of Remark~\ref{rem:valueshessians}(b) on one plaquette has $\mathcal L=(2m^2+1)\epsilon$, although $\lvert R_m\rvert\le\epsilon s$. \item If $R$ is a loop sum as in Proposition~\ref{prop:flatloops}, with paths and fillings in $\Lambda$, and $\lvert\mathcal HR(w,w)+\sigma\lvert \Pi w\rvert^2\rvert\le\tau Q_1(w)$ at $V_0=I$ for every $w$, with $\sigma, \tau\ge0$, then $\mathcal L\ge N(\sigma C_n^--\tau)$. The Haar term is $-\frac13\lvert\Pi w\rvert^2$ for $\SU(2)$, so a loop sum that contains it has $\mathcal L\ge2(\frac{(n+1)^2}{9\pi^2}-\tau)$. \end{enumerate} \end{proposition} \begin{proof} (i) For $g,h\in\SU(2)$, $h+h^{-1}=(\Tr h)I$, so $\Tr g\Tr h=\Tr gh+\Tr gh^{-1}$. Apply it to $g=V_C$ and $h=V_\gamma V_{C'}V_\gamma^{-1}$, whose traces are real, and substitute $\Tr=2(1-s)$. The product of the fillings of $V_C$ and of $V_\gamma V_{C'}^{\pm1}V_\gamma^{-1}$, the latter conjugated by $V_\gamma$ and for $V_{C'}^{-1}$ inverted and reversed, is a filling of $C\vee C'^{\pm1}$. (ii) Each orbit of $G^{\Lambda'}$ meets the fields equal to $I$ on the spanning tree $\tau$ of the proof of Lemma~\ref{lem:smallcurv}, in an orbit of the constant gauge transformations. Fix a site $o$. For a link $k\notin \tau$ let $C_k$ run from $o$ along $\tau$ to the start of $k$, along $k$, and back along $\tau$; at such fields $V_{C_k}=V_k$. Let $V,V'$ be two such fields with $s(V_C)=s(V'_C)$ for every closed path $C$. Every word in the $V_k$ is the holonomy of a closed path based at $o$, and the trace is real on $\SU(2)$, so the unitary representations $\rho,\rho'$ of the free group on $\{k\notin\tau\}$ with $\rho(k)=V_k$, $\rho'(k)=V'_k$ have the same character. They are direct sums of irreducible representations. If $E_1, \dots,E_r$ are the pairwise inequivalent irreducible summands of $\rho \oplus\rho'$, the theorems of Burnside and Jacobson give an element of the group algebra acting as the identity on $E_i$ and as zero on $E_j$, $j\ne i$; its trace under $\rho$ and under $\rho'$ is $\dim E_i$ times the multiplicity of $E_i$. So $\rho$ and $\rho'$ are equivalent. The unitary factor $U$ of the polar decomposition of an intertwiner intertwines, and $U(\det U)^{-1/2}\in\SU(2)$, as a constant gauge transformation, maps $V$ to $V'$. Thus the functions $s(V_C)$ separate the orbits. With the constants they span an algebra by (i), and the Stone--Weierstrass theorem on the compact space of orbits gives (ii). This is the lattice form of the reconstruction theorem of Giles~\cite{Gil81}. (iii) At $g=\operatorname{diag}(e^{i\theta},e^{-i\theta})$, $s(g^m)=1-\cos m\theta$, so $\varphi(g)=\sum_mc_m-\sum_{m\ge1}c_ms(g^m)$; both sides are class functions, and $V_{C^m}=V_C^m$. The repeated filling has $mA_C$ factors and multiplicities $m\,m_{Cp}$. The derivatives of $s(g^m)$ of orders one and two are of order $m$ and $m^2$, so the series converges with two derivatives. For $R_m$, $(1-\cos\theta)\cos m\theta=\cos m\theta-\frac 12\cos(m+1)\theta-\frac12\cos(m-1)\theta$, so $R_m(V_p)=-\epsilon s(V_p^m)+ \frac\epsilon2s(V_p^{m+1})+\frac\epsilon2s(V_p^{m-1})$, and $\mathcal L= \epsilon(m^2+\frac12(m+1)^2+\frac12(m-1)^2)$. (iv) Lemma~\ref{lem:invariantslice}(ii) and part~(ii) of Proposition~\ref{prop:flatloops} give $\lvert\mathcal HR(w,w)\rvert\le \frac{\mathcal L}NQ_1(\Pi w)=\frac{\mathcal L}NQ_1(w)$. At the field $w$ of Lemma~\ref{lem:windowgap}(iv), $\Pi w=w$ and $|w|^2=C_n^-Q_1(w)$, so $\sigma C_n^-Q_1(w)\le(\frac{\mathcal L}N+\tau)Q_1(w)$. For $\SU(2)$, $\frac1{12}\Tr (\operatorname{ad}_x^2)=-\frac13\lVert x\rVert^2$, and $C_n^-\ge(n+1)^2/(3 \pi^2)$. \end{proof} \begin{remark}[what the representation leaves open]\label{rem:loopsopen} Proposition~\ref{prop:loopsums}(ii) represents every invariant function by loop sums in the sense of uniform approximation only. That controls neither the values relative to $\sum_ps(V_p)$ nor the Hessians, and the weights $\mathcal L_\kappa$, $\mathcal M_\kappa$ of the approximants may diverge. They belong to a representation and not to the function: by conjugation invariance $s(V_{\gamma C\gamma^{-1}})=s(V_C)$ while $T$ and $t$ grow with $\gamma$, and $\mathcal M_\kappa$ can be taken as an infimum over representations. That the exact induced action of one step, at strong or at weak coupling, is a loop sum with finite $\mathcal L_\kappa$ and $\mathcal M_\kappa$ is not proved. It would follow from a convergent expansion of the induced action in polynomials in Wilson loops whose weights beat the growth in part (i), where a product of $k$ loop functions becomes $2^{k-1}$ loop functions whose fillings have the sum of the areas. The character expansion at strong coupling and the expansion of Ba{\l}aban~\cite{Bal89b} at weak coupling are of this kind, and neither is carried out here for the cut-off averages. For $\SU(N)$, $N\ge3$, the identity in (i) has no analogue linear in traces, and products of traces would have to be admitted. Part (iv) shows that $C_n$ stays on the Haar term in any loop representation, as in Corollary~\ref{cor:loopstep}. \end{remark} \emph{The induced action.} Let $e^{-A(\omega)}$ be the density in $\omega$ of the cut-off averages of a window under the Wilson measure restricted by a cut-off. The small-field restriction is an example: it is a soft box in the coordinates of the chart of Theorem~\ref{thm:chartdecay}, which is not $\Gamma_W$-invariant. Then $e^{-A^{\rm gi}}$ is the density with respect to Haar measure on $G^K$ of the averages under the measure whose cut-off on the fibre over $\Upsilon(s,\eta)$ is the transport by the block-constant gauge transformation $e^\eta$ of the cut-off on the fibre over $e^s$. The transport maps fibre to fibre (Theorem~\ref{thm:cutoff}) and preserves the Wilson weight and Haar measure, so the fibre integrals agree. On the slice $\mathcal O^\perp$ the two cut-offs coincide. A cut-off defined through gauge-invariant quantities gives $A^{\rm gi}=A+\sum_k\log j$ everywhere. \begin{remark}[inputs and what is missing]\label{rem:relstepinputs} Proposition~\ref{prop:relstep} rests on one input and leaves the following open. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item \emph{Input.} The bound (H) on the Hessian of the induced action at one point is an absolute bound, of the kind by which Theorems~\ref{thm:smallfibre} and~\ref{thm:fibrechart} control the non-Gaussian terms, and only its restriction to $\mathcal O^\perp$ is used. It is not proved in this paper. Corollary~\ref{cor:loopstep} replaces it by a loop representation of the remainder, under which $C_n$ falls on the Haar term alone. As an illustration that no result uses, for $\SU(2)$, $\lambda=10^{100}$ and $\varepsilon=1.63\cdot10^{-9}$, interval arithmetic gives $\varepsilon'\le8.40\cdot10^{-9}$ at $n=2$, $1.52\cdot10^{-8}$ at $n=4$, $1.52\cdot10^{-7}$ at $n=16$ and $5.95\cdot10^{-4}$ at $n=1024$, and $\varepsilon'\le0.1$ for every $n\le8455$. \item \emph{The cut-off.} The construction replaces a non-invariant cut-off off the slice by its gauge transport. Ba{\l}aban defines the small-field regions by gauge-invariant conditions on plaquettes~\cite{Bal85a,Bal89b,Dim13}. For such a cut-off the transport is not needed. \item \emph{Points that are not critical.} $\mathcal H$ is taken at one point. An iteration needs the bound as a function on a neighbourhood. There the gradient terms of Lemma~\ref{lem:invariantslice}(i) and the third derivatives enter, and Proposition~\ref{prop:relstep} says nothing about them. Ba{\l}aban expands around the minimiser of the action with given averages, the background field, at which the first-order terms vanish~\cite{Bal84a,Dim13}. \item \emph{Curved backgrounds.} Lemma~\ref{lem:invariantslice} holds at every $V_0$. The Gaussian reference $S^U_W(w)=\inf\{Q_U(X):BX=w\}$ need not vanish on $\mathcal O_{V_0}$, where $B$ is the derivative of the cut-off averages at $U$. The fine gauge variation $X_\eta$ of $e^\eta$ has $BX_\eta= d_{V_0}\eta$ and $D_p(X_\eta)=(1-\operatorname{Ad}U_p)\eta_{y(p)}$, with $y(p)$ the block of the base of $p$. Hence $S^U_W(d_{V_0}\eta)\le384 \kappa^2|\eta|^2$ when $\delta(U_p,I)\le\kappa$, where $96$ plaquettes have their base in a block. The reference for an invariant action is $S^U_W\circ \Pi_{V_0}$. By Lemma~\ref{lem:curvedwindow} its gap on $\mathcal O_{V_0}^\perp$ follows from a comparison $S^U_W\ge c_1Q_1-c_2\kappa|\cdot|^2$ there, with the gap $c_1(1-\theta/s_n)^2C_n^{-1}-c_2\kappa$; that comparison is not proved here. \item \emph{The action.} The induced action is not a plaquette action. Its Hessian is compared with $2\lambda S_W$, not with $2\lambda Q_1$, and Theorem~\ref{thm:chartdecay} does not apply to it as it stands at the next scale. The large fields, which Ba{\l}aban treats separately~\cite{Bal89b}, are removed by the cut-off and not estimated. \end{enumerate} \end{remark} \begin{remark}[relative errors over scales]\label{rem:relscales} The proof of Proposition~\ref{prop:formstability} uses only the order of forms, so it holds for positive semidefinite forms. A bound $(1\pm \varepsilon')\,2\lambda S_W$ on the covariant Hessian therefore passes to the block infimum of the next step with factor one. Proposition~\ref{prop:massgrowth} does not apply to it. That proposition concerns an error bounded by the Euclidean form, which does not vanish on $\mathcal O$. The block infimum turns that error into a term at least $4\lambda\varepsilon(1+2724\kappa)^{-1}|w|^2$, also where $S_W$ vanishes. The two statements agree. A window of side $n$ in blocks of one scale has side $n/2$ in the blocks of the next, and $C_n/C_{n/2}\to4$. Converting an absolute error at one scale costs the factor $C_n$. Carrying it one step as an absolute error multiplies it by at least four, after which converting costs $C_{n/2}$. Both routes have the same order, and the factor four of Proposition~\ref{prop:massgrowth} is the loss of the gap of the window. With windows of side $n_k=n_12^{1-k}$ at the scales $k=1,\dots,K$, $\sum_kC_{n_k}\le K+n_1^2/3$, so \[ \sum_k\varepsilon'_k\le\sup_k\Bigl(\varepsilon_k+\frac{\delta_G}{\lambda_k} \Bigr)\frac{K+n_1^2/3}{2c_*}. \] The part $n_1^2/3$ is the conversion cost. It is the absolute error carried to the last scale, $4^{K-1}\approx n_1^2$, so it is finite for a fixed window but not uniform in the volume. Removing it needs a gauge-invariant localisation of the one-step bound, which is part of Ba{\l}aban's analysis~\cite{Bal89b}. For a remainder that is a sum of gauge-invariant terms on boxes, Theorem~\ref{thm:localflat} gives that localisation, with the gap of each box in place of $C_n$, and by Proposition~\ref{prop:chartlocal} the factor $C_n$ of the Haar term comes from stating the hypothesis on the chart density. For a remainder that is a series of filled Wilson loops, Corollary~\ref{cor:loopstep} removes it from $\varepsilon_k$ and leaves it on the Haar term $\delta_G/\lambda_k$. The part $K$ is marginal. The Haar term $\delta_G/\lambda_k$ is of order $g_k^2$, and $\sum_kg_k^2$ diverges logarithmically along the one-loop coupling by Remark~\ref{rem:constrainedopen}(c). The order $g^2$ of the one-step error must therefore be absorbed into the running coupling and the normalisation of the field. Only a remainder of order $g^4$ can be carried as $\varepsilon'_k$. For $n_1=2^{12}$, $K=13$, $\varepsilon_k=1.63\cdot 10^{-9}$ and $\lambda_k=10^{100}$, the values of $C_{n_k}$ give $\sum_k \varepsilon'_k\le0.0127$, against $4^{12}\varepsilon_1\le0.028$ for the absolute error of the first scale carried to the last. No effective action is constructed here beyond the first scale, and item~(iv) of Sec.~\ref{ssec:remains} remains open. \end{remark} \subsection{Every compact simple group}\label{ssec:groups13} The constructions of this section were written for $\SU(2)$. This subsection records which of them hold for a general group, and with which constants. Let $G$ be a compact connected group whose Lie algebra $\g$ is simple, with a faithful unitary representation in which $G\subset\U(N)$, the metric $\ip XY=-\Tr(XY)$ on $\g$, the least Ricci curvature $\kappa_G$ of Theorem~\ref{thm:groups}, $d=\dim G$ and $K=\kappa_G/(d-1)$. The distance $\delta(g,h)=\lVert g-h\rVert$ is the Hilbert--Schmidt distance, $d(g,h)$ the geodesic distance of $G$, the Wilson weight is $\exp(-\beta_W\sum_ps_p)$ with \[ s_p=1-\tfrac1N\ReTr U_p=\frac{\delta(U_p,I)^2}{2N},\qquad\beta_W=N^2\beta, \] which is the normalisation of this section for $N=2$, and $\rho_G$ is the radius of Sec.~\ref{sec:blocks}. Put $I_d(x)=\int_0^x\sin^{d-1}u\,du$. \begin{lemma}[spectral constants]\label{lem:adspread} Let $X,Y\in\g$ and $U\in G$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The operator $\operatorname{ad}X$ on $\g$ is skew, with spectrum in $i[-\sqrt2\lVert X\rVert,\sqrt2\lVert X\rVert]$. \item $-\Tr\bigl((\operatorname{ad}X)^2\bigr)=4\kappa_G\lVert X\rVert^2$; in particular $\delta_G=\kappa_G/3$ in Proposition~\ref{prop:relstep}. \item With $s(V)=1-\frac1N\ReTr V$, the derivative of $s(Ue^{tX})$ at $t=0$ has modulus at most $\lVert X\rVert/\sqrt N$; on $\U(N)$, $\delta(g,h)\le2\sqrt N$. \item Let $j$ be the density of Haar measure in exponential coordinates, \[ j(X)=\det\bigl((1-e^{-\operatorname{ad}X})/\operatorname{ad}X\bigr). \] If $\lVert X\rVert\le10^{-3}$, then \[ \lvert\partial_Y\log j(X)\rvert\le\tfrac{\kappa_G}3(1+10^{-6})\lVert X \rVert\lVert Y\rVert . \] \end{enumerate} \end{lemma} \begin{proof} (i) Invariance of the trace makes $\operatorname{ad}X$ skew. On $\mathfrak u(N)$, in a basis in which $X=\operatorname{diag}(i\lambda_1,\dots, i\lambda_N)$, $\operatorname{ad}X$ has eigenvalues $i(\lambda_j- \lambda_k)$, and $\g$ is an invariant subspace; $(\lambda_j-\lambda_k)^2\le 2(\lambda_j^2+\lambda_k^2)\le2\lVert X\rVert^2$. (ii) $\Ric(X,X)=-\frac14\Tr((\operatorname{ad}X)^2)$ is an $\operatorname{Ad}$-invariant symmetric form. Since $\g\otimes\C$ is simple, the adjoint representation is irreducible over $\C$ and every invariant symmetric form is a multiple of $\ip\cdot\cdot$, so $\Ric= \kappa_G\ip\cdot\cdot$. Then $\delta_G=\frac1{12}\cdot4\kappa_G$. (iii) The derivative is $-\frac1N\ReTr(UX)$. Since $\overline{\Tr(UX)}= -\Tr(U^*X)$, $\ReTr(UX)=\frac12\Tr((U-U^*)X)$, and $\lVert U-U^*\rVert^2= 4\sum_j\sin^2\theta_j\le4N$ for the eigenangles $\theta_j$ of $U$. The diameter bound is $\lVert g-h\rVert^2=2N-2\ReTr(g^*h)\le4N$. (iv) $\Tr\operatorname{ad}X=0$, so $\log j(X)=\Tr\Phi(\operatorname{ad} X)$ with $\Phi(z)=\log(\sinh(z/2)/(z/2))$, and $\partial_Y\log j(X)= \Tr(\Phi'(\operatorname{ad}X)\operatorname{ad}Y)$. For real $y$, $\Phi'(iy)=i(\frac1y-\frac12\cot\frac y2)$ and $0\le\frac1y-\frac12\cot \frac y2\le\frac y{12}(1+\frac{y^2}{50})$ for $0\le y\le1$; by (i) the spectral radius of $\operatorname{ad}X$ is at most $\sqrt2\cdot10^{-3}$, so $\lVert\Phi'(\operatorname{ad}X)\rVert_{HS}\le\frac1{12}(1+10^{-6}) \lVert\operatorname{ad}X\rVert_{HS}$, and (ii) together with the Cauchy--Schwarz inequality gives the bound. \end{proof} For $\SU(2)$ the spectrum in (i) is $\{0,\pm i\sqrt2\lVert X\rVert\}$, and the estimates of Lemma~\ref{lem:branchbound}, Proposition~\ref{prop:goodside}, Lemma~\ref{lem:goodsidedensity} and Step~4 of Lemma~\ref{lem:lambdatwo} that use the spectrum of an operator $\operatorname{ad}W$ use only that it lies in $i[-\sqrt2\lVert W\rVert,\sqrt2\lVert W\rVert]$: the bounds $\lVert L_X\rVert\le\theta/\sin\theta$, $\lVert E_Y\rVert\le1$, $\lVert\varpi(\operatorname{ad}W)^{-1}\rVert\le m(\theta)$, the divided-difference bounds and $\kappa_0<0.8935$ hold for $G$ unchanged. The Hilbert--Schmidt norm $\lVert\operatorname{ad}W\rVert_{HS}=2\lVert W \rVert$ becomes $2\sqrt{\kappa_G}\lVert W\rVert$, a trace over $\g$ is bounded by $d$ times the operator norm instead of $3$ times it, and the pair $(\lVert X\rVert/\sqrt2,2\sqrt2)$ of (iii) becomes $(\lVert X\rVert/ \sqrt N,2\sqrt N)$. \begin{lemma}[the logarithm radius of a stabiliser]\label{lem:tensorlog} Let $t$ be a tensor in $(\C^N)^{\otimes a}\otimes(\C^{N*})^{\otimes b}$ with $m=a+b\ge1$, and let $G$ be the identity component of the stabiliser of $t$ in $\U(N)$. If $g\in G$ and $\delta(g,I)<2\sin\frac\pi{2m}$, the principal logarithm of $g$ lies in $\g$. \end{lemma} \begin{proof} Let $e^{i\theta_j}$, $|\theta_j|\le\pi$, be the eigenvalues of $g$ and $X=\log g$. Since $4\sin^2(\theta_j/2)\le\delta(g,I)^2<4\sin^2\frac\pi {2m}$, every $|\theta_j|<\pi/m$. The induced action of $e^{sX}$ on the tensor space is $e^{sD}$, where the derivation $D$ induced by $X$ is skew-Hermitian with eigenvalues $i$ times sums of $m$ terms $\pm\theta_j$, so its spectrum lies in $i(-\pi,\pi)$. Decompose $t=\sum_\mu t_\mu$ over the eigenspaces of $D$. From $e^Dt=t$, $e^{i\mu}t_\mu=t_\mu$ with $|\mu|<\pi$, so $t_\mu=0$ for $\mu\ne0$ and $Dt=0$. The one-parameter group $e^{sX}$ then fixes $t$ and lies in its identity component $G$, so $X\in\g$. \end{proof} $\SO(N)$ and $\mathrm{Sp}(n)$ are the identity components of the stabilisers of a symmetric and of a skew form, $m=2$ and $2\sin\frac\pi4 >1$, so $\rho_G=1$. $G_2\subset\U(7)$, $F_4\subset\U(26)$, $E_6\subset \U(27)$ and $E_8\subset\U(248)$ are the identity components of the stabilisers of tensors of degree at most three: the three-form of the octonions, the quadratic and cubic invariants of the traceless exceptional Jordan algebra, its cubic determinant, and the Lie bracket~\cite{Bae02}. Each $\rho_G<1$ is admissible for them, for instance $\rho_G=0.99$. $E_7\subset\U(56)$ is the identity component of the stabiliser of the symplectic and the quartic form of the Freudenthal triple system~\cite{Bae02}, $m=4$, so $\rho_G=0.76<2\sin\frac\pi8$ is admissible. For $\SU(N)$, $\rho_G=\min\{1,5/\sqrt N\}$. \begin{lemma}[small balls]\label{lem:haarball} For $r>0$, the Haar measure of $\{g\in G:d(g,I)\le r\}$ is at least $I_d(\sqrt Kr)/I_d(\pi)$, with equality for $\SU(2)$. If $\sqrt Kr\le \frac14$, it is at least $(\frac{95}{96})^{d-1}(\sqrt Kr)^d/(d\,I_d(\pi))$. \end{lemma} \begin{proof} The Ricci curvature is at least $(d-1)K$, so the diameter is at most $\pi/\sqrt K$ by Myers' theorem, and the relative volume comparison of Bishop and Gromov~\cite{Pet16} bounds the ratio of the volume of the ball of radius $r$ to that of the ball of radius $\pi/\sqrt K$, which is $G$, from below by the same ratio on the sphere of curvature $K$ and dimension $d$, $I_d(\sqrt Kr)/I_d(\pi)$. For $\SU(2)$ the group is that sphere, of radius $\sqrt2$. The second bound follows from $\sin u\ge u(1-u^2/6) \ge\frac{95}{96}u$ on $[0,\frac14]$. \end{proof} Lemma~\ref{lem:haarball} takes the place of the angle density $\frac2\pi \sin^2\theta$ used in Lemma~\ref{lem:wilsonZ}. With it, put \begin{multline*} C_0(G)=\frac12+\frac23\Bigl(\log\bigl(d\,I_d(\pi)\bigr)\\ +\frac d2\log\frac{16}{KN}-(d-1)\log\frac{95}{96}\Bigr). \end{multline*} \begin{lemma}\label{lem:wilsonZG} For $\beta_W\ge KN$, the normalising constant of the Wilson measure of $G$ on $\Lambda_L$ satisfies \[ -\log Z_L\le6L^4\Bigl(\frac d3\log\beta_W+C_0(G)\Bigr), \] and $\E S\le6L^4(\frac d3\log\beta_W+C_0(G))/\beta_W$. If $KN\le16$, then $C_0(G)>0$. \end{lemma} \begin{proof} Put $r=\sqrt N/(4\sqrt{\beta_W})$, so that $\sqrt Kr\le\frac14$. If every link is within geodesic distance $r$ of $I$, every plaquette has $\delta(U_p,I)\le d(U_p,I)\le4r$ and $s_p\le8r^2/N=1/(2\beta_W)$, so $\beta_WS\le3L^4$. By Lemma~\ref{lem:haarball} the $4L^4$ links do so with probability at least \[ \Bigl(\bigl(\tfrac{95}{96}\bigr)^{d-1}\bigl(\tfrac{KN}{16\beta_W} \bigr)^{d/2}\big/\bigl(d\,I_d(\pi)\bigr)\Bigr)^{4L^4}, \] which gives the first bound; the second follows from the concavity of $-\log Z_L$ in the coupling, as in Lemma~\ref{lem:wilsonZ}. The integrals satisfy $I_dI_{d+1}=2\pi/d$, by induction from $I_{d+2}=\frac d{d+1}I_d$, and $I_d\ge I_{d+1}$, so $d\,I_d(\pi)\ge\sqrt{2\pi d}>1$; the other terms of $C_0(G)$ are nonnegative when $KN\le16$. \end{proof} For $\SU(2)$, $C_0(G)=4.32001$, below the constant $4.90817$ of Lemma~\ref{lem:wilsonZ}. For $\SU(N)$, $KN=N^2/(2(N^2-2))\le1$; for every group of Table~\ref{tab:groups}, $KN\le1$, so the lemma holds at every $\beta_W\ge1$. \begin{theorem}\label{thm:defectsG} For $\beta_W\ge KN$, $0<\rho\le1$ and every set $A$ of unit cubes, Theorem~\ref{thm:defects} and Corollary~\ref{cor:defectclusters} hold for the Wilson measure of $G$ with \[ q^G_\rho(\beta_W)=\min\Bigl\{1,\exp\Bigl(2d\log\beta_W+6C_0(G)- \frac{\beta_W\rho^2}{8N}\Bigr)\Bigr\}. \] \end{theorem} \begin{proof} Reflection in a hyperplane through sites preserves Haar measure on $G$ and maps $\ReTr U_p$ to $\ReTr U_p$ or $\ReTr U_p^{-1}=\ReTr U_p$, so the chessboard estimate holds as in Theorem~\ref{thm:defects}. A bad plaquette has $s_p>\rho^2/(2N)$, and if every cube is bad there are at least $L^4/4$ bad plaquettes, so $S\ge\rho^2L^4/(8N)$. Lemma~\ref{lem:wilsonZG} bounds $Z_L$. The corollary uses only the product bound. \end{proof} \begin{theorem}[Section~\ref{sec:islands} for a compact simple group] \label{thm:groups13} Let $G\subset\U(N)$ be as above, with $\kappa_G\le2500$ in~(iii). Replace $\SU(2)$ by $G$, the hypothesis $\rho\le10^{-8}$ by $\rho\le 10^{-8}\rho_G$, and $C_0$, $q_\rho$ by $C_0(G)$, $q^G_\rho$. Then: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Lemmas~\ref{lem:coarsewitness}, \ref{lem:branchbound}, \ref{lem:coverage}, \ref{lem:scalewitness}, Theorems~\ref{thm:islandthreshold} and~\ref{thm:islandscales}~\textup{(i)}, \textup{(ii)} hold with \[ f_G(\beta)=2d\log\beta+6C_0(G)-\frac{\beta\rho^2}{8N} \] in place of $f$ (and $\rho_0$ in place of $\rho$ at scale $j$), and with $\beta^*>16dN/\rho^2$ in place of $\beta^*>96/\rho^2$. The numbers $M$, $H$, $\varDelta$, $M_j$, $H_j$ and $\log q^*$ do not depend on $G$, and \[ \beta^*_G=\frac{8N\lvert\log q^*\rvert}{\rho^2}(1+\epsilon_G),\quad \epsilon_G=\frac{2d\log\beta^*_G+6C_0(G)}{\lvert\log q^*\rvert}. \] \item Theorems~\ref{thm:fibrechart}, \ref{thm:chartdecay}, Propositions~\ref{prop:explicitconstants} and~\ref{prop:explicitdecay} hold as stated, since they are stated for every compact $G\subset\U(r)$: $c_3=\max\{1.5\cdot10^4\sqrt N,d\}$, $c_6=6.81\cdot10^5d\,c_3^2$, $\alpha_G=2.5\cdot10^{-4}$. \item Theorem~\ref{thm:layerbound} holds with $\lambda=\beta_W/(2N)$, hypothesis~(b) read as $\beta_W\ge480Nc_6$, and $\mathcal A=120/\lambda= 240N/\beta_W$. Corollary~\ref{cor:twoislands} holds with \[ \varDelta=\frac{960N}{\beta_W}\Lambda_2^2r^2\sqrt{n_xn_y}\,e^{-2\alpha_G (w-2s-8)}, \] and its hypothesis~(a) holds for $\rho\le1/(2758\cdot7.8\cdot10^7c_3)$. \item Lemma~\ref{lem:lambdatwo} holds with $\frac2Nc_W$ and $\frac d3 c_\psi$ in place of $c_W$ and $c_\psi$; under hypotheses (a) and (c) of Theorem~\ref{thm:layerbound}, \[ \Lambda_2\le\frac{198.0002}N\beta_W+\frac{4.65\cdot10^6}3d . \] \item Proposition~\ref{prop:goodside} holds with the density bounds $(1+\kappa_0)^{-d}$ and $(1-\kappa_0)^{-d}$, and with radius $\rho_G/32$ when $\rho_G<1$; Lemma~\ref{lem:sitelaw} holds with the Jacobian bound $\sigma_0^d$; Lemma~\ref{lem:goodsidedensity} holds with $64.94\cdot\frac d3\sqrt{\kappa_G}$ and $32.47\cdot\frac d3\sqrt {\kappa_G}$; Theorem~\ref{thm:site} holds with $24n_p\beta_W$, $197\beta_W+a_G$ and $87\beta_W+b_G$, where \begin{gather*} a_G=1558.56\,\sqrt N\,\tfrac d3\sqrt{\kappa_G}+1.1986,\\ b_G=779.28\,\sqrt N\,\tfrac d3\sqrt{\kappa_G}+0.5993 . \end{gather*} \item Theorem~\ref{thm:manyislands}, Corollary~\ref{cor:islandchains}, Proposition~\ref{prop:islandmass} and Theorem~\ref{thm:islandwindow} hold with the threshold $\beta^*_G$ of~(i), the condition $2758\rho\le1/(7.8 \cdot10^7c_3)$, $\frac{960N}{\beta_W}$ in place of $\frac{1920}{\beta_W}$, and $\bar\Lambda_2=\frac{198.0002}N\beta_W+\frac{4.65\cdot10^6}3d$. \item The statements of Subsections~\ref{ssec:counting} to~\ref{ssec:spectralweight}, other than Proposition~\ref{prop:blockresponse}, of Subsection~\ref{ssec:manyislands} and of Subsection~\ref{ssec:relstep} hold for $G$ as stated, with $\delta_G=\kappa_G/3$. \end{enumerate} For $\SU(3)$, $d=8$, $N=3$, $\kappa_G=\frac32$, $\rho_G=1$, $C_0(G)\in [10.44720,10.44721]$, $f_G(\beta)=16\log\beta+6C_0(G)-\beta\rho^2/24$, and \begin{gather*} \beta^*(10^{-8},1)\in[5.268,5.270]\cdot10^{21},\\ \beta^*(10^{-8},5)\in[8.230,8.232]\cdot10^{21},\\ \beta^*(10^{-9},1)\in[5.285,5.287]\cdot10^{23},\\ \beta^*(10^{-9},5)\in[8.247,8.249]\cdot10^{23},\\ \beta^*(10^{-8},5,10^{30})\in[2.971,2.973]\cdot10^{22}, \end{gather*} with $0.026<\epsilon_G<0.045$ in the first four. At scale $j$, for $s=5$ and $\rho=10^{-8}$, $\beta^*_1\in[4.367,4.368]\cdot10^{23}$, $\beta^*_2\in [6.990,6.992]\cdot10^{24}$, $\beta^*_3\in[1.785,1.787]\cdot10^{26}$ and $16^{-j}\beta^*_j\le4.38\cdot10^{22}$ for $1\le j\le12$. In~(iii) for $\SU(3)$, $c_3\le25980.8$, $c_6\le3.678\cdot10^{15}$, (b) holds for $\beta_W\ge5.30\cdot10^{18}$, (a) for $\rho\le1.789\cdot10^{-16}$, and at $\rho=1.78\cdot10^{-16}$ the threshold with the joining distance $2s+8$ lies in $[2.569,2.571]\cdot10^{37}$ for $s=1$ and in $[3.014,3.016]\cdot 10^{37}$ for $s=5$, where (b) holds; $\Lambda_2\le66.0001\beta_W+1.24\cdot 10^7$, and the constant $c_2$ of Theorem~\ref{thm:islandwindow} at $s=1$, $\vartheta=0$ is at most $10^{22.14}$. \end{theorem} \begin{proof} (i) The witness paragraph of the proof of Theorem~\ref{thm:islandthreshold} uses Lemma~\ref{lem:smallframes}, which is stated for every $G$ with $\kappa\le10^{-8}\rho_G$; Lemma~\ref{lem:logbound}, a statement on $\U(N)$; and $d(g,h)\le(1+10^{-12})\delta(g,h)$ for $\delta(g,h)\le10^{-6}$, which holds on $G$ because there $\delta\le\rho_G$, the principal logarithm of $g^{-1}h$ lies in $\g$ and $d(g,h)=\lVert\log(g^{-1}h) \rVert$. In Lemma~\ref{lem:coarsewitness} the constants are those of Lemma~\ref{lem:logbound} and the triangle inequality. In Lemma~\ref{lem:branchbound} the cut-off is nonzero only where $\max_x \delta(v_x,I)<\rho_G$, so the logarithms lie in $\g$, and the contraction bound uses the spectrum of $\operatorname{ad}X$ only through Lemma~\ref{lem:adspread}(i). Lemmas~\ref{lem:coverage} and~\ref{lem:scalewitness} count cubes; Theorem~\ref{thm:defectscales}, used in the latter, is stated for every $G$. The threshold paragraph applies to $f_G$, which is concave with maximum $2d\log(16dN/\rho^2)+6C_0 (G)-2d>0>\log q^*$ at $16dN/\rho^2$, by Lemma~\ref{lem:wilsonZG} and $\rho\le10^{-8}$. The identity for $\epsilon_G$ is $f_G(\beta^*_G)=\log q^*$. (ii) There is nothing to prove. (iii) In the proof of Theorem~\ref{thm:layerbound} the Wilson action enters as $\lambda\lVert\cdot\rVert^2$ with $\lambda=\beta_W/(2N)$, the dimension enters through $\lVert A_v\rVert_1\le2d\,c_3\lVert x_v \rVert_1$ and $\lVert A_a\rVert_2^2\le2d\,c_3^2\lVert\bar P\rVert^2\lvert a\rvert^2$, which are absorbed in $c_6$, and the Haar part is the logarithmic gradient of $j$ at $\lVert X_e\rVert\le10^{-3}$, at most $\frac{\kappa_G}3(1+10^{-6})10^{-3}\le0.84$ per unit of $\lVert x_v \rVert_1$ by Lemma~\ref{lem:adspread}(iv) and $\kappa_G\le2500$. The rest of the proof is a statement about the chart constants of~(ii). The corollary is Proposition~\ref{prop:layercross} with $4\mathcal A=960N/ \beta_W$, and $\eta+62r=2758\rho$. (iv) The Wilson term of a plaquette is $\beta_W(1-\frac1N\ReTr U_p)$, so every bound of~\eqref{eq:wilsonmixed} carries $\frac1N$ in place of $\frac12$; Steps 2 to 6 use the spectrum of $\operatorname{ad}W$ through Lemma~\ref{lem:adspread}(i) only. In Step~7, $\lvert\Tr A\rvert\le d \lVert A\rVert$ on $\g$ replaces $\lvert\Tr A\rvert\le3\lVert A\rVert$. (v) In the proof of Proposition~\ref{prop:goodside} the density of a linear map $T$ with $\lVert T-I\rVert\le\kappa_0$ on $\g$ lies between $(1-\kappa_0)^d$ and $(1+\kappa_0)^d$; the cut-off is inactive below $\rho_G/2$, which gives the radius $\rho_G/32$. Lemma~\ref{lem:privatesubmersion} is stated with $\rho_G$, and $JJ^*\ge\sigma_0^2$ on the $d$-dimensional $\g$ gives $\sqrt{\det JJ^*}\ge\sigma_0^d$. In the proof of Lemma~\ref{lem:goodsidedensity}, $\operatorname{tr}(T^{-1}\dot T)$ is at most $d\lVert T^{-1}\rVert\lVert\dot T\rVert$ and $\lVert\operatorname{ad} W\rVert_{HS}=2\sqrt{\kappa_G}\lVert W\rVert$ by Lemma~\ref{lem:adspread}(ii), while the spectra lie in the segments used there by Lemma~\ref{lem:adspread}(i). In the proof of Theorem~\ref{thm:site} the oscillation of a site function is at most $24\cdot2\sqrt N$ and $\lvert\dot s_p\rvert\le v/\sqrt N$ by Lemma~\ref{lem:adspread}(iii), so the first term is at most $24\sqrt N\sup\lvert\partial_t\log h_t\rvert$; the factors $\sqrt N$ cancel in the Wilson part, which gives $24n_p\beta_W$, $24\cdot8.1916\,\beta_W<197\beta_W$ and $24\cdot6\cdot0.5993\,\beta_W< 87\beta_W$, and remain in the density part, which gives $a_G$ and $b_G$. For $\SU(2)$, $a_G<2206$ and $b_G<1103$, the constants of the theorem. (vi) These statements use the preceding ones through the constants listed, and the threshold through $q^G$. (vii) These statements concern the label variables, the coupling of labels, the island combinatorics and the quadratic forms of a box. The group enters them only through the results of Sec.~\ref{sec:blocks}, stated for every $G$, through the constants of~(iii) and~(iv), and through $\delta_G$ in Subsection~\ref{ssec:relstep}, which is $\kappa_G/3$ by Lemma~\ref{lem:adspread}(ii). \emph{The numbers for $\SU(3)$.} Here $K=\kappa_G/7=\frac3{14}$, $KN= \frac9{14}$ and $I_8(\pi)=\int_0^\pi\sin^7u\,du=\frac{32}{35}$, so $d\,I_d (\pi)=\frac{256}{35}$, and the factorisations $\frac{256}{35}(\frac{224}9)^4 (\frac{96}{95})^7=2^{63}3^{-1}5^{-8}7^3\,19^{-7}$ and $7^2=2\cdot5^2\cdot \frac{49}{50}$, $19=2^5\cdot3\cdot5^{-1}\cdot\frac{95}{96}$ give \begin{multline*} C_0(G)=\tfrac12+\tfrac23\bigl(\tfrac{59}2\log2-8\log3+2\log5\\ -\tfrac32\log\tfrac{50}{49}+7\log\tfrac{96}{95}\bigr). \end{multline*} The logarithms of the proof of Theorem~\ref{thm:islandthreshold}, with $\log\frac{50}{49}=2\operatorname{artanh}\frac1{99}$ and $\log\frac{96}{95} =2\operatorname{artanh}\frac1{191}$, give $C_0(G)$ within $10^{-7}$ of $10.4472096$. The function $g_G(\beta)=\beta\rho^2/24-16\log\beta-6C_0(G)- \lvert\log q^*\rvert-H\log\varGamma$ increases for $\beta>384/\rho^2$, and $\beta^*$ is its largest zero. With $\lvert\log q^*\rvert$ within $10^{-3}$ of $21090.194$ and $33424.872$ (Theorem~\ref{thm:islandthreshold}), at the ends of each of the five stated intervals $g_G$ is negative and positive, with margins above $2$, so that $\log\beta$ to two decimals decides the signs; $\epsilon_G=(16\log\beta^*_G+6C_0(G))/\lvert\log q^*\rvert$ increases with $\beta^*_G$, and its values at the ends of the first four intervals lie between $0.02603$ and $0.04442$. At scale $j$, $\rho_0=\rho/ (c\,4^j)$ with $c=1.01\cdot1.023$ and $24c^2\le25.6215417$, and $\lvert\log q^*_j\rvert$ is as in the proof of Theorem~\ref{thm:islandscales}: within $10^{-3}$ of $105605.899$ and $105608.929$ for $j=1,2$, and at most $169144.64+4j\log2$ for $j\ge3$. The same sign check, with margins above $11$, gives the three intervals for $\beta^*_1$, $\beta^*_2$, $\beta^*_3$. For $b>384c^2/\rho^2$, \begin{multline*} g_{G,j}(16^jb)=\frac{b\rho^2}{24c^2}-16\log b-6C_0(G)\\ -\bigl(\lvert\log q^*_j\rvert+64j\log2\bigr), \end{multline*} and at $b=4.38\cdot10^{22}$ the first term is at least $170949.9$, the next two at most $834.15$ and $62.684$, and the bracket, for $j\le12$, at most $105697.7$ when $j\le2$ and $169144.64+68j\log2\le169710.3$ when $j\ge3$; so $g_{G,j}(16^jb)>0$ and $16^{-j}\beta^*_j\le4.38\cdot10^{22}$. In~(iii), $c_3=1.5\cdot10^4\sqrt3\le25980.8$, $c_6=6.81\cdot10^5\cdot8\cdot6.75\cdot 10^8=3.6774\cdot10^{15}$, $480\cdot3\cdot c_6=5.295456\cdot10^{18}$ and $2758\cdot7.8\cdot10^7\cdot25980.8\le5.5891\cdot10^{15}$, which give (b) and (a). For the joining distance $2s+8$, $\lvert\log q^*\rvert$ lies within $10^{-2}$ of $32481.67$ for $s=1$ and of $38354.07$ for $s=5$ (Theorem~\ref{thm:islandwindow}), and the sign check at $\rho=1.78\cdot 10^{-16}$, with margins above $7$, gives the two intervals. In~(iv), $\frac{198.0002}3\le66.0001$ and $\frac{4.65\cdot10^6\cdot8}3=1.24\cdot 10^7$. Finally, with $960N=2880$, $r=43\rho$ and $\kappa=2\alpha_G=5\cdot 10^{-4}$, \[ c_2=2880\,\beta^*r^2\Bigl(66.0001+\frac{1.24\cdot10^7}{\beta^*}\Bigr)^2 \cdot45\cdot14^2\cdot9^2e^{0.008}, \] which increases with $\beta^*$ and is at most $10^{22.134}$ at $\beta^*= 2.571\cdot10^{37}$. \end{proof} \begin{remark}\label{rem:groups13} Four parts of this section do not transfer. Subsection~\ref{ssec:witnessradius} (Lemma~\ref{lem:ladder}, Propositions~\ref{prop:smallframessharp} and~\ref{prop:constantcurvature}, Theorem~\ref{thm:islandswide}) measures frames by the angle of a unit quaternion; on $\SU(N)$, $N\ge3$, the operator-norm angle is subadditive and conjugation invariant, and the constants of the subsection have not been computed for it. Proposition~\ref{prop:fold} uses every preimage of the average on $\SU(2)$, and Proposition~\ref{prop:blockresponse} is a computation with unit quaternions. The examples of Propositions~\ref{prop:roughbackground}, \ref{prop:nonconvex} and~\ref{prop:branches} are stated on $\SU(2)$ and are not claimed for other groups. \end{remark} \begin{remark}\label{rem:groups13blocks} For $\SU(3)$ the plaquette blocks reach $\beta_W\le0.52$ (Theorem~\ref{thm:su3blocks}), in place of $0.348$ for $\SU(2)$. For other groups the block constant~\eqref{eq:blockconst3} needs the modulus $\Lambda(r)$ of~\eqref{eq:alpha}, a gap of the one-link measure and bounds on the Hessian of $\log Z$, and none of them has been certified. The modulus that holds for every $G$ does not suffice: the one-link measure $\nu_b$ satisfies the Bakry--\'Emery condition with constant $\kappa_G-\lVert b\rVert_{\rm op}\ge\kappa_G-r$, so $\Lambda(r)\le1/(\kappa_G-r)$; with this bound the right side of~\eqref{eq:alpha} is below one only for $N\beta<\kappa_G/18$, and, since $K_3\ge s\ge\Lambda(r)$, every value of $\alpha_3$ computed from it is at least $10N\beta/(\kappa_G-6N\beta)$, below one only for $N\beta< \kappa_G/16$. Both ranges lie inside $N\beta<2\kappa_G/29$ of Theorem~\ref{thm:groups}. \end{remark} \subsection{One step of the block map: the large fields}\label{ssec:lfstep} This subsection carries out, for $G=\SU(2)$, the large-field part of one step of the block map $U\mapsto c=V^\sharp(U)$ on the whole torus. It proves three things. A set of rough blocks is suppressed by $e^{-\lambda\kappa^2}$ per block, either absolutely or in the mean. The weight of a coarse field is an exact sum over large-field regions of kernels with this suppression. The coarse action lies between $\lambda/(1.65\cdot10^6)$ and $8\lambda$ times the coarse curvature, up to a volume term. It also states which bound the next step would need, and how far Theorem~\ref{thm:islandswide} supplies it. The small-field part and its locality are the subject of item~(iv) of Sec.~\ref{ssec:remains} and are not treated here. The lattice counts and the numerical constants are proved by hand. The one exception is the value of the coupling $(M_W)_{AB}$ of Theorem~\ref{thm:sfgauss}, which is computed in Remark~\ref{rem:sfgaussvalue} and used by no proof; the statements that the calm weight is not local are conditional on $(M_W)_{AB}\ne0$. Throughout, $\Lambda_L$ is the torus of side $L=2^m\ge32$ in the tree gauge, $T$ is the coarse torus of the blocks $B(y)$, $|T|=L^4/16$, $\lambda=\beta_W/4$, so that the Wilson weight is $e^{-\lambda Q}$ with $Q=\sum_p\delta(U_p,I)^2$, and $\nu_c$ is the fibre measure of Sec.~\ref{ssec:largefields} with $W=T$. For a coarse field $c$ write $c_P$ for the holonomy of a coarse plaquette and $Q_1(c)=\sum_P\delta(c_P,I)^2$. \begin{definition}\label{def:lfrough} The plaquette at $x$ in the plane $\{\mu,\nu\}$ is \emph{owned} by the block containing $x$; $\Pi(y)$ is the set of plaquettes owned by $y$, and $\Pi(X)=\bigcup_{y\in X}\Pi(y)$. For $\kappa>0$ a block $y$ is \emph{calm} if $\delta(U_p,I)\le\kappa$ for every $p\in\Pi(y)$, and \emph{rough} otherwise; $R(U)$ is the set of rough blocks. For a block $y$ let $I_y$ be the $17$ active links inside $B(y)$, $P_{y,\mu}$ the eight private links of the pair $(y,\mu)$, and $\varXi_y=(I_y,P_{y,0},\dots,P_{y,3})$. \end{definition} \begin{lemma}\label{lem:lfowned} \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Every plaquette has exactly one owner, and $|\Pi(y)|=96$. \item Every $p\in\Pi(y)$ in the plane $\{\mu,\nu\}$ contains an active link with base in $B(y)$, and each of its links has its base in $B(y)$, $B(y+e_\mu)$ or $B(y+e_\nu)$. \item The variables $\varXi_y$, $y\in T$, are the active links, each once. The factor $f_y=\exp(-\lambda\sum_{p\in\Pi(y)}\delta(U_p,I)^2)$ and the indicator $r_y$ of $\{y\text{ rough}\}$ are functions of $\varXi_y$ and $\varXi_{y+e_\mu}$, $0\le\mu\le3$, and $e^{-\lambda Q}=\prod_yf_y$. \end{enumerate} \end{lemma} \begin{proof} (i) A plaquette has one corner, and each of the $16$ sites of a block is the corner of six plaquettes. (ii) Let $o$ be the offset of the corner $x$. If $o_\mu=1$ the link $(x,\mu)$ leaves $B(y)$ and is private, and likewise if $o_\nu=1$. If $o_\mu=o_\nu=0$ the four vertices lie in $B(y)$ and the plaquette is a cycle in the block, which the fifteen tree links, a tree by Lemma~\ref{lem:blocktree}(i), cannot contain; so one of its links, all based in $B(y)$, is active. The links of $p$ are based at $x$, $x+e_\mu$ and $x+e_\nu$. (iii) A link based in $B(y)$ is a tree link, an active link inside $B(y)$, or a private link of a pair $(y,\mu)$, by Lemma~\ref{lem:blocktree}(i); the rest follows from (ii). \end{proof} A computer enumeration of the $96$ plaquettes of a block agrees with (i) and (ii); the proof does not use it. \paragraph*{Suppression.} For a set $X$ of blocks let $A_X$, $Q_X$, $\nu^X_c$, $\pi_E$ and $Q_*(E)$ be as in Sec.~\ref{ssec:largefields} with $W=X$, so that $\nu^X_c$ lives on the fibre of $X$ with the links off $A_X$ fixed. \begin{theorem}\label{thm:lfsuppression} Let $G=\SU(2)$, $\kappa>0$, $\lambda>0$ and let $X$ be a set of blocks. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For every $c$ and every choice of the links off $A_X$, \[ \int\mathbf 1\{X\subset R(U)\}\,e^{-\lambda Q_X}\,d\nu^X_c\le M^{4|X|} e^{-\lambda\kappa^2|X|}, \] with $M<2.3621\cdot10^4$ the constant of Lemma~\ref{lem:averagedensity}. \item Let $\kappa\le1$ and $0<\tau\le1$. If the exterior $E$ of $X$ has $Q_*(E)\le(1-\tau)\kappa^2|X|$, then \[ \pi_E\{X\subset R\}\le\Bigl(\frac{5.9\cdot10^{14}}{(\tau\kappa^2)^3} \Bigr)^{49|X|}e^{-3\lambda\tau\kappa^2|X|/4}. \] \item If $\beta_W\ge1$ and $\kappa\le1$, then $\mu_L\{X\subset R\}\le (16\,q_\kappa(\beta_W))^{|X|}$, with $q_\kappa$ as in Theorem~\ref{thm:defects}. \end{enumerate} \end{theorem} \begin{proof} On $\{X\subset R\}$ each block of $X$ owns a plaquette with $\delta(U_p,I) >\kappa$; these plaquettes are distinct and, by Lemma~\ref{lem:lfowned}(ii), contain links of $A_X$. Hence $Q_X>\kappa^2|X|$ there. (i) The integrand is at most $e^{-\lambda\kappa^2|X|}$, and $\nu^X_c$ has mass at most $M^{4|X|}$ by~\eqref{eq:fibrevolume}. (ii) With $t=\tau\kappa^2|X|$, $\{X\subset R\}\subset\{Q_X>Q_*(E)+t\}$. The proof of Proposition~\ref{prop:fibrecompare} uses $t\le1$ only to have $\rho=t/(90.6\,n_P)<1.5\cdot10^{-4}$. Each of the $49|X|$ links of $A_X$ lies in six plaquettes and a plaquette contains at most four of them, so $n_P\ge73.5|X|$ and $\rho\le\tau\kappa^2/6659<1.51\cdot10^{-4}$, which still gives $16\sqrt2\rho+16\rho^2<22.63\rho$. The proof therefore gives the bound $(14\cdot90.6^3n_P^3/t^3)^{49|X|}e^{-3\lambda t/4}$, and $n_P\le 384|X|$ with $14\cdot90.6^3\cdot384^3<5.9\cdot10^{14}$. (iii) The plaquette at $x$ lies in the unit cube $\mathcal C_x$, so a rough block $y$ lies in $\bigcup_{x\in B(y)}B_x$. Expanding the intersection over $y\in X$ of these unions gives at most $16^{|X|}$ intersections $\bigcap_{y \in X}B_{x(y)}$ with $x(y)\in B(y)$, each over $|X|$ distinct cubes, and Theorem~\ref{thm:defects} bounds each by $q_\kappa^{|X|}$. \end{proof} Part (i) is uniform in the exterior and in $c$, but it is an absolute bound: it does not compare rough with calm completions. The comparison (ii) needs its hypothesis. For the frustrated exterior described after Corollary~\ref{cor:defectscales}, a plaquette through the link $e$ is bad with conditional probability tending to one. No conditional bound uniform in the exterior holds. \paragraph*{Coarse curvature.} For a coarse plaquette $P$ let $N(P)$ be the fine plaquettes with a vertex in one of the $144$ blocks at $\ell^\infty$ distance at most one from the four blocks of $P$, the witness region of Lemma~\ref{lem:coarsewitness}, and let $N_{\rm ov}$ be the largest number of witness regions that contain one fine plaquette. Then $N_{\rm ov}=1441$, and $2310$ witness regions meet $\Pi(y)$. Indeed, let $P$ have the blocks $z$, $z+e_\mu$, $z+e_\nu$, $z+e_\mu+e_\nu$, with $B(z)=2z+\{0,1\}^4$. The $144$ blocks fill the box of fine sites $w$ with $w_i-2z_i\in[-2,5]$ for $i\in\{\mu,\nu\}$ and $w_i-2z_i\in[-2,3]$ otherwise, and the plaquette at $x$ in the plane $\{\alpha,\beta\}$ has a vertex in it exactly when, for each $i$, $x_i$ lies in the $i$-th interval, enlarged by one site to the left when $i\in\{\alpha, \beta\}$. For fixed $x$ the number of admissible $z_i$ is $4$ for $i\in\{\mu,\nu\}$ and $3$ otherwise, plus one when $i\in\{\alpha,\beta\}$ and $x_i$ is odd. With $x_\alpha,x_\beta$ odd the six planes $\{\mu,\nu\}$ give $5\cdot5\cdot3\cdot3$, four times $5\cdot4\cdot4\cdot3$ and $4^4$, which sum to $225+960+256=1441$, and no parity gives more. For the second count take $y=0$: the plaquettes of $\Pi(0)$ have corners in $\{0,1\}^4$ and any plane. For each $i$ the admissible $z_i$ are those with $z_i\in[-2,1]$ for $i\in\{\mu,\nu\}$ and $z_i\in[-1,1]$ otherwise, and $z_i=2$, which needs $x_i=1$ and $i\in\{\alpha,\beta\}$; so $z$ is admissible exactly when at most two coordinates equal $2$. Of the $5\cdot5\cdot4\cdot4 =400$ points of the enlarged box, $8+6+1=15$ have three or four coordinates equal to $2$, and the six planes give $6\cdot385=2310$. \begin{lemma}\label{lem:lfcoarseenergy} Let $G=\SU(2)$, $C_E=8/\kappa_c^2<1140.26$ and $C_*=1441\,C_E<1.6432\cdot 10^6$. For every field $U$ in the tree gauge, $c=V^\sharp(U)$ and every coarse plaquette $P$, \[ \delta(c_P,I)^2\le C_E\max_{p\in N(P)}\delta(U_p,I)^2 , \] and if $\kappa<\kappa_c$ and $\delta(c_P,I)>16.003\,\kappa$, some $p\in N(P)$ has $\delta(U_p,I)>\kappa$. Consequently $Q_1(c)\le C_*\,Q(U)$, and \[ Q_1^{>}(c):=\sum_{\delta(c_P,I)>16.003\kappa}\delta(c_P,I)^2\le C_* \sum_{\delta(U_p,I)>\kappa}\delta(U_p,I)^2 . \] In particular $Q(U)\ge r^2/1140.26$ whenever some coarse plaquette has $\delta(c_P,I)\ge r$. \end{lemma} \begin{proof} Let $m$ be the maximum over $N(P)$. If $m<\kappa_c$, then Proposition~\ref{prop:smallframessharp}(iv), applied with every $\kappa\in (m,\kappa_c)$, gives $\delta(c_P,I)\le16.003\,m$, and $16.003^2<256.1}$ keep in each sum only the plaquettes above $\kappa$, among which the maximum lies. \end{proof} The last statement is the first step of Proposition~\ref{prop:torusenergy} for $\SU(2)$, with $1140.26$ in place of $1154$; it is a bound for one step only. \paragraph*{The coarse action.} Since $h=\int\nu_c\,dh(c)$, the law of $c=V^\sharp(U)$ under the Wilson measure is $Z_L^{-1}e^{-A_1(c)}\,dh(c)$ with \[ e^{-A_1(c)}=Z(c)=\nu_c\bigl(e^{-\lambda Q}\bigr). \] Let $S_1(c)=\inf_{F_c}Q$. In the reconstruction $R(c)$ of Sec.~\ref{ssec:pathblocks}, with $c_{y,\mu}$ on the private links of $(y,\mu)$ and $I$ on the links inside the blocks, every path of $V^\sharp_ {y,\mu}$ has holonomy $c_{y,\mu}$. So $v_x=I$ at every site, $R(c)$ is regular, and $V^\sharp(R(c))=c$. A plaquette at $x$ in the plane $\{\mu,\nu\}$ of $R(c)$ is $c_P$ when $o_\mu=o_\nu=1$ and $I$ otherwise, so $Q(R(c))=4Q_1(c)$. \begin{theorem}\label{thm:lfaction} Let $G=\SU(2)$. For every coarse field $c$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $A_1(c)\ge\lambda S_1(c)-4|T|\log M$, and $S_1(c)\ge Q_1(c)/C_*$; \item for $\lambda\ge100$, $\eta>0$ and every regular $U\in F_c$, $A_1(c)\le(1+\eta)\lambda Q(U)+c_1(\lambda,\eta)|T|$, where \begin{multline*} c_1(\lambda,\eta)=12\log1.957+45\log14\\ +67.5\bigl(1+\log(11520(1+\eta^{-1})\lambda)\bigr); \end{multline*} in particular $A_1(c)\le8\lambda Q_1(c)+c_1(\lambda,1)|T|$. \end{enumerate} \end{theorem} \begin{proof} (i) $Z(c)\le\nu_c(F_c)e^{-\lambda S_1(c)}$ and $\nu_c(F_c)\le M^{4|T|}$ by \eqref{eq:fibrevolume}; then Lemma~\ref{lem:lfcoarseenergy}. (ii) Let $\rho^2=135/(2\cdot96\cdot8100(1+\eta^{-1})\lambda)$, so $\rho\le 9.32\cdot10^{-4}$ and $46\rho<0.043$. The proof of Proposition~\ref{prop:everyfibre} with $W=T$ uses $t\le1$ only through this bound on $46\rho$. It gives a set $\Gamma(D)\subset F_c$ with $\nu_c(\Gamma(D)) \ge1.957^{-12|T|}(\rho^3/14)^{45|T|}$ on which each of the $n_P=96|T|$ plaquettes moves by at most $90\rho$. There $\delta(U'_p,I)\le\delta(U_p, I)+90\rho$, so \[ Q(U')\le(1+\eta)Q(U)+(1+\eta^{-1})\,96|T|\,8100\rho^2 . \] Hence $A_1(c)\le(1+\eta)\lambda Q(U)+|T|g(\rho)$ with $g(\rho)=(1+ \eta^{-1})777600\lambda\rho^2+135\log\rho^{-1}+12\log1.957+45\log14$, and the chosen $\rho$ minimises $g$, with value $c_1(\lambda,\eta)$. For the last statement take $U=R(c)$ and $\eta=1$. \end{proof} So the coarse action is $\lambda$ times a quantity between $Q_1/C_*$ and $8Q_1$, up to $O(\log\lambda)$ per block; the volume term $67.5\log\lambda$ is the Gaussian count of the $135$ real fluctuation coordinates of a block. Neither constant is near one. For example $c_1(10^{12},1)<2737.5$ and $4\log M<40.35$. \paragraph*{The large-field representation.} Write $s_y=1-r_y$. For $X\subset T$ let $X_1,\dots,X_n$ be its components for $\ell^\infty$ adjacency of blocks, so that distinct components are at $\ell^\infty$ distance at least two. For a set $Y$ of blocks let $\partial^-Y$ be the set of blocks $z\notin Y$ with $z+e_\mu\in Y$ for some $\mu$; a block of $\partial^-X_i$ lies off $X$. By the definition of $\nu_c$, \[ \nu_c(d\varXi)=\prod_yh(dI_y)\prod_{y,\mu}\nu^{y,\mu}_{c}(dP_{y,\mu}\mid I_y,I_{y+e_\mu}), \] where $\nu^{y,\mu}_c$ is the measure of~\eqref{eq:densityformula} on the private links of the pair, of mass at most $M$. Let $\partial^=X$ be the set of blocks $w\notin X\cup\partial^-X$ with $w+e_\mu+e_\nu\in X$ for some $\mu\ne\nu$, and put $\hat X=X\cup\partial^-X\cup\partial^=X$. Let $\mathcal P_X$ be the set of pairs from $X$ together with the pairs $(z,\mu)$ with $z\notin X$ and $z+e_\mu\in X$, and let $\mathcal V_X$ be the links inside the blocks of $X$ together with the private links of the pairs of $\mathcal P_X$: these are the variables that the kernel of $X$ in Theorem~\ref{thm:lfmarkov} integrates. \begin{lemma}\label{lem:sfkernel} \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The factors $f_y$ and $r_y$ are functions of $I_y$, of the $P_{y,\mu}$, and of $I_{y+e_\nu}$ and $P_{y+e_\nu,\mu}$ for $\mu\ne\nu$. The average $V^\sharp_{y,\mu}$ is a function of $P_{y,\mu}$, $I_y$ and $I_{y+e_\mu}$. \item The factor $f_w$ involves a link of $\mathcal V_X$ if and only if $w\in\hat X$. \item Let the components $X_i$ of $X$ be at $\ell^\infty$ distance at least two. Then $\hat X$ is the disjoint union of the $\hat X_i$, each block of $\hat X_i$ lies within $\ell^\infty$ distance one of $X_i$, and for $j\ne i$ no factor $f_w$ or $r_w$ with $w\in\hat X_i$ involves a link of $\mathcal V_{X_j}$. \item Every plaquette that contains a link of $\mathcal V_X$ is owned by a block of $\hat X$ and has a vertex in a block of $X$. There are at most $216|X|$ such plaquettes, every plaquette of $\Pi(X)$ is one of them, and $|\mathcal P_X|\le8|X|$. \end{enumerate} \end{lemma} \begin{proof} (i) Let $p$ be the plaquette at $x\in B(y)$ in the plane $\{\mu,\nu\}$. By Lemma~\ref{lem:lfowned}(ii) its links are based in $B(y)$, $B(y+e_\mu)$ or $B(y+e_\nu)$. The one based in $B(y+e_\nu)$ is $(x+e_\nu,\mu)$, whose offset in the direction $\mu$ is that of $x$: it is a link inside $B(y+e_\nu)$ or a tree link if $o_\mu=0$, and a private link of $(y+e_\nu,\mu)$ if $o_\mu=1$. The links based in $B(y)$ are inside $B(y)$ or private to pairs from $y$. The average is the mean over the paths $(x,\mu),(x+e_\mu,\mu)$, $x\in B(y)$, of Lemma~\ref{lem:blocktree}(iii). (ii) By (i), $f_w$ involves a variable of $\varXi_y$ with $y\in X$ if and only if $w=y$ or $w+e_\nu=y$, that is $w\in X\cup\partial^-X$. It involves a private link of a pair $(z,\mu)$ with $z\notin X$, $z+e_\mu\in X$ if and only if $w=z$ or $w=z-e_\nu$ with $\nu\ne\mu$; the first gives $w\in\partial^-X$ and the second $w+e_\mu+e_\nu\in X$. (iii) Every block of $\hat X_i$ has the form $x$, $x-e_\mu$ or $x-e_\mu -e_\nu$ with $x\in X_i$, so it lies within distance one of $X_i$, and a block of $\hat X_i\cap\hat X_j$ would give blocks of $X_i$ and $X_j$ at distance at most one. For the same reason a block of $\partial^-X_i$ or $\partial^=X_i$ lies in no $X_j$ and in no $\partial^-X_j$, so $\hat X= \bigcup_i\hat X_i$. By (ii) for $X_j$, a factor that involves a link of $\mathcal V_{X_j}$ belongs to a block of $\hat X_j$, which is not in $\hat X_i$. (iv) A link of $\mathcal V_X$ is based in a block of $X$, or is a private link $(x,\mu)$ of a pair $(z,\mu)$ with $z+e_\mu\in X$, which ends at $x+e_\mu\in B(z+e_\mu)$. The owner of a plaquette that contains it involves it, so it lies in $\hat X$ by (ii). Since $\mathcal V_X\subset\bigcup_{x\in X}\mathcal V_{\{x\}}$, the count is at most $|X|$ times the number of plaquettes with a vertex in one block $B(y)=2y+\{0,1\}^4$. The plaquette at $c$ in the plane $\{\alpha,\beta\}$ has a vertex in $B(y)$ exactly when $c_i-2y_i\in\{0,1\}$ for $i\notin\{\alpha,\beta\}$ and $c_i-2y_i\in\{-1,0,1\}$ for $i\in\{\alpha,\beta\}$, so there are $6\cdot2^2\cdot3^2=216$ of them. There are $4|X|$ pairs from $X$, and $(z,\mu)\mapsto(z+e_\mu,\mu)$ is injective on the others. The plaquettes of $\Pi(X)$ contain an active link with base in a block of $X$, by Lemma~\ref{lem:lfowned}(ii). \end{proof} A computer enumeration agrees with (i), and with (ii)--(iv) for $300$ random $\ell^\infty$-connected sets of up to seven blocks and for $300$ random pairs of such sets at distance at least two; the proof does not use it. \begin{theorem}\label{thm:lfmarkov} Let $G=\SU(2)$ and $\kappa>0$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For every $c$, $Z(c)=\sum_{X\subset T}Z_X(c)$ with \[ Z_X(c)=\int\nu^{\rm out}_{c,X}(d\varXi)\prod_{y\notin\hat X} f_ys_y\prod_iK_{X_i}(c;\varXi), \] with $\hat X=\bigcup_i\hat X_i$ as in Lemma~\ref{lem:sfkernel}. Here $\nu^{\rm out}_{c,X}$ is the product of the factors of $\nu_c$ for the links inside the blocks off $X$ and for the pairs with both blocks off $X$. The kernel $K_{X_i}$ is the integral of $\prod_{y\in X_i}f_yr_y \prod_{z\in\hat X_i\setminus X_i}f_zs_z$ against the factors of $\nu_c$ for the links inside $X_i$, the pairs from $X_i$ and the pairs $(z,\mu)$ with $z\in \partial^-X_i$ and $z+e_\mu\in X_i$. It depends on the integration variable of $\nu^{\rm out}_{c,X}$ only through the blocks at $\ell^\infty$ distance one from $X_i$. No two kernels share a factor or an integration variable; two kernels may share boundary arguments, the inside links of a block between them, which the common outer integral integrates. \item For $\kappa<\kappa_c$, uniformly in the outer variables, \[ 0\le K_{X_i}\le M^{8|X_i|}\exp\Bigl(-\lambda\max\Bigl\{\kappa^2|X_i|, \frac{Q^{(i)}(c)}{C_*}\Bigr\}\Bigr), \] where $Q^{(i)}(c)$ is the sum of $\delta(c_P,I)^2>(16.003\kappa)^2$ over the coarse plaquettes $P$ with $\emptyset\ne N(P)\cap\Pi(X)\subset\Pi(X_i)$. \item With $N_\lambda=e^{-c_1(\lambda,1)}$, $K_{X_i}\le N_\lambda^{|X_i|} e^{-a(\lambda,\kappa)|X_i|}$, where \[ a(\lambda,\kappa)=\lambda\kappa^2-8\log M-c_1(\lambda,1), \] and $Z(c)\ge N_\lambda^{|T|}e^{-8\lambda Q_1(c)}$ for $\lambda\ge100$. \end{enumerate} \end{theorem} \begin{proof} (i) Expand $\prod_y(f_yr_y+f_ys_y)$ and group the terms by $X=\{y:r_y\}$; then apply Fubini to the nonnegative integrands. By Lemma~\ref{lem:lfowned}(iii), $f_y$ and $r_y$ involve $\varXi_y$ and $\varXi_{y+e_\mu}$, and $\nu^{y,\mu}_c$ involves $P_{y,\mu}$, $I_y$ and $I_{y+e_\mu}$. By Lemma~\ref{lem:sfkernel}(ii) and~(iii), the factors that involve a variable integrated by the kernel of $X_i$ are those of the blocks of $\hat X_i$, and none of them involves a variable integrated by the kernel of $X_j$, $j\ne i$. The measure of a pair into $X_i$ involves its private links and the inside links of its two blocks, and a pair is into or from at most one component. (ii) The measures in $K_{X_i}$ have mass at most $M$ for each of the $4|X_i|$ pairs from $X_i$ and the at most $4|X_i|$ pairs into it, and $h$ is a probability. The integrand is at most $e^{-\lambda Q_{\Pi(X_i)}}$ and vanishes unless every block of $X_i$ is rough. Then $Q_{\Pi(X_i)}> \kappa^2|X_i|$, and every fine plaquette above $\kappa$ is owned by a rough block. So for the $P$ counted in $Q^{(i)}$ the plaquettes above $\kappa$ in $N(P)$ lie in $\Pi(X_i)$, and Lemma~\ref{lem:lfcoarseenergy} gives $Q^{(i)}(c)\le C_*Q_{\Pi(X_i)}$. (iii) The first bound is (ii) with the definition of $a$; the second is Theorem~\ref{thm:lfaction}(ii) with $U=R(c)$. \end{proof} The representation is exact, and every kernel carries $e^{-\lambda\kappa^2}$ per block, uniformly in its boundary values and in $c$. When the coarse field is curved its kernel also carries $e^{-\lambda Q^{(i)}/C_*}$. The small-field factor $\prod f_ys_y$, however, is not a product of local factors once the outer variables are integrated. The comparison of a kernel with the calm weight $S_{X_i}$, the same integral with $s_y$ in place of $r_y$ on $X_i$, would give the relative activities $K_{X_i}/S_{X_i}$. It fails where $S_{X_i}=0\log7290=\log2+6\log3+\log5$, $u<1/7290$, and the right side, which increases with $u$, is below $81|X|/(7290-6400) <0.092|X|<|X|/10$. This is the hypothesis of the theorem of Koteck\'y and Preiss, whose conclusion is the statement. (ii) Write $L_p=\log p$. With its tail bounded geometrically, the series $\log\frac{1+x}{1-x}=2\sum_{k\ge0}\frac{x^{2k+1}}{2k+1}$ at $x=\frac13$, $\frac15$, $\frac19$ and $\frac1{99}$ gives $\log2$, $\log\frac32$, $\log\frac54$ and $\log\frac{50}{49}$ to within $10^{-10}$, hence \begin{gather*} 0.6931471867.5$. So $a\ge9$ for $\lambda\ge\max\{\lambda^*,100\}$ once $t^*=\lambda^*\kappa^2$ satisfies \[ t^*-67.5\log t^*\ge962.06+67.5\log\kappa^{-2}. \] In the four rows of the table $\log\kappa^{-2}$ equals $4L_5-2L_2$, $4\log10$, $8\log10$ and $16\log10$, and, by $\log(1+x)\le x$, $\log t^*$ is at most $\log1800+\frac1{180}$, $\log2100+\frac1{2100}$, $\log2744$ and $\log4000+\frac1{400}$, with $1800=2^33^25^2$, $2100=2^2\cdot3\cdot5^2 \cdot7$, $2744=2^37^3$ and $4000=2^55^3$. With these bounds the left side exceeds the right by more than $0.6$ in each row. \end{proof} Theorem~\ref{thm:lfkp} applies to the large fields once the small-field part is in local form. Then each region costs its kernel divided by the calm weight of the same region, and $a\ge9$ beats the entropy of connected regions if that calm weight is at least $N_\lambda$ per block. This lower bound on the calm weight with boundary values is not proved. It is the small-field cluster expansion of item~(iv) of Sec.~\ref{ssec:remains}, and it must have the relative and local form required by Propositions~\ref{prop:formstability} and~\ref{prop:massgrowth} and Lemma~\ref{lem:schur}. \paragraph*{The small-field part.} The rest of this subsection treats the factor $\prod f_ys_y$. It proves three things. First, a kernel is at most $e^{-a_2|X|}$ times the calm weight of its region whenever the boundary values admit a completion that is calm at a smaller threshold $\kappa'$. Second, the large-field sum is exactly the calm weight times a mean, under the calm law, of products of these ratios, and the polymer sums of Theorem~\ref{thm:lfkp} bound it once the ratios are small. Third, to leading order in $\lambda$ the logarithm of the calm weight is $-\lambda$ times the Gaussian block-spin form, so that the calm weight is a product of local factors to within $e^{\pm o(\lambda)}$ only if this form does not couple distant coarse links; its coefficient at two coarse links three blocks apart is a rational number $(M_W)_{AB}$, which a computation not used in the proofs finds nonzero. What remains open is stated at the end. \emph{The calm weight.} Fix $X\ne\emptyset$ and the outer variables of $K_X$. Let $\lambda_X$ be the measure of $K_X$, the product of $h$ on the links inside the blocks of $X$ and of the measures $\nu^k_c$ of the pairs $k\in\mathcal P_X$, of mass at most $M^{8|X|}$. The \emph{calm weight} is \[ S_X=\int\prod_{y\in\hat X}f_ys_y\,d\lambda_X , \] the integral of $K_X$ with $s_y$ in place of $r_y$ on $X$. A plaquette of $\Pi(\hat X)$ is \emph{moved} if it contains a link of $\mathcal V_X$ and \emph{fixed} otherwise; $Q^{\rm mov}_X$ and $Q^{\rm fix}_X$ are the sums of $\delta(U_p,I)^2$ over the moved and the fixed plaquettes, and $Q^{\rm fix} _X$ is a function of the outer variables. A \emph{completion} is a value of $\mathcal V_X$ at which the averages of the pairs of $\mathcal P_X$ equal $c$. Put \begin{multline*} c_2(\lambda,\eta)=24\log1.957+73\log14\\ +109.5\bigl(1+\log(15979(1+\eta^{-1})\lambda)\bigr). \end{multline*} \begin{theorem}\label{thm:sfcalm} Let $G=\SU(2)$, $0<\kappa'<\kappa$, $\eta>0$, and $\lambda\ge100$ with $\lambda(\kappa-\kappa')^2\ge1$. Suppose that the fixed plaquettes of $\Pi(\hat X)$ are at most $\kappa$, which holds whenever $K_X>0$, and that there is a completion $U^0$ with \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item $\delta(v_x,I)\le\frac14$ at every site of every pair of $\mathcal P_X$; \item $\delta(U^0_p,I)\le\kappa'$ for every moved plaquette. \end{enumerate} Then \[ S_X\ge e^{-c_2(\lambda,\eta)|X|}e^{-\lambda Q^{\rm fix}_X}e^{-(1+\eta) \lambda Q^{\rm mov}_X(U^0)}, \] and \begin{multline*} \frac{K_X}{S_X}\le\exp\Bigl(-\bigl(\lambda\kappa^2-8\log M-c_2(\lambda,\eta) \bigr)|X|\\ +(1+\eta)\lambda Q^{\rm mov}_X(U^0)\Bigr)\le e^{-a_2|X|}, \end{multline*} where $a_2(\lambda,\kappa)=\lambda\kappa^2\bigl(1-216(1+\eta)\kappa'^2/ \kappa^2\bigr)-8\log M-c_2(\lambda,\eta)$. \end{theorem} \begin{proof} Let $\rho^2=1/(15978.08(1+\eta^{-1})\lambda)$. Then $\rho\le7.92\cdot 10^{-4}$, and $90\rho\le0.712\lambda^{-1/2}<\kappa-\kappa'$. Let $L'$ be the links of $\mathcal V_X$ other than the path links of the pairs of $\mathcal P_X$, $17|X|+7|\mathcal P_X|\le73|X|$ in number, and $D$ the set of $z\in G^{L'}$ with $d(z_e,U^0_e)\le\rho$ for every $e\in L'$. For $z\in D$ and a pair $k$, each site $v_x$ of $k$ is $\ell^{-1}$ times a product of at most two links of $L'$ and outer links, $\ell$ the path link. As in the proof of Proposition~\ref{prop:calmfibre}, with $R=44\rho$ the sites lie within $46\rho<0.043$ of their values at $U^0$, so $\delta(v_x,I)<\frac12$ by (a), the cut-off equals one, and the contraction argument of that proof solves each path link in the ball of radius $R$ about its value at $U^0$. This gives an injective map $\Gamma$ from $D$ into the completions, near whose image the averages are smooth with $\lvert\det\partial_{d_k}V^\sharp _k\rvert\le1.957^3$. The coarea form of $\nu^k_c$ stated after~\eqref{eq:fibrevolume}, applied to each pair of $\mathcal P_X$ with the inside links of its blocks given, shows that $\lambda_X$ restricted to $\Gamma(D)$ is the image of $h^{\otimes L'}$ on $D$ divided by $\prod_k\lvert\det\partial_{d_k}V^\sharp_k\rvert$. Hence \[ \lambda_X\bigl(\Gamma(D)\bigr)\ge1.957^{-3|\mathcal P_X|}\Bigl(\frac{\rho^3} {14}\Bigr)^{|L'|}\ge1.957^{-24|X|}\Bigl(\frac{\rho^3}{14}\Bigr)^{73|X|}. \] A plaquette has at most two path links, so on $\Gamma(D)$ each moved plaquette moves by at most $90\rho$, as in that proof, and stays at most $\kappa'+90\rho<\kappa$; the fixed ones are at most $\kappa$. Every block of $\hat X$ is therefore calm on $\Gamma(D)$, and by Lemma~\ref{lem:sfkernel}(iv), \[ Q^{\rm mov}_X\le(1+\eta)Q^{\rm mov}_X(U^0)+(1+\eta^{-1})\,8100\rho^2\cdot 216|X| . \] So $S_X\ge e^{-g|X|}e^{-\lambda Q^{\rm fix}_X}e^{-(1+\eta)\lambda Q^{\rm mov}_X(U^0)}$ with $g=24\log1.957+73\log14+219\log\rho^{-1}+(1+\eta^{-1}) 1749600\lambda\rho^2$. With the chosen $\rho$ the last term is $109.5$ and $g\le c_2(\lambda,\eta)$, as $2\cdot216\cdot8100/219<15978.09$. The integrand of $K_X$ is at most $e^{-\lambda Q^{\rm fix}_X}e^{-\lambda Q^{\rm mov}_X}$ and vanishes unless every block of $X$ is rough. Then each block of $X$ owns a plaquette above $\kappa$, these are distinct, and they are moved by Lemma~\ref{lem:sfkernel}(iv), so $Q^{\rm mov}_X>\kappa^2|X|$. With $\lambda_X$ of mass at most $M^{8|X|}$, $K_X\le M^{8|X|}e^{-\lambda Q^{\rm fix}_X}e^{-\lambda\kappa^2|X|}$. Dividing gives the first bound, and (b) with Lemma~\ref{lem:sfkernel}(iv) gives $Q^{\rm mov}_X(U^0)\le216 \kappa'^2|X|$. \end{proof} So the calm weight with given boundary values is at least $e^{-c_2}$ per block, times the Gaussian factor of the completion, whenever the boundary values admit a completion calm at $\kappa'$; the constant $c_2$ takes the place of $c_1$ because $K_X$ integrates up to $73$ rather than $45$ links per block. With $\kappa'=\kappa/30$ and $\eta=1$, $a_2=0.52\lambda\kappa^2-8\log M-c_2(\lambda,1)$, and $a_2\ge9$, the rate that Theorem~\ref{thm:lfkp} needs, holds for $\lambda\ge\lambda^{**}(\kappa)$, with $\beta^{**}=4\lambda^{**}$ as follows: \begin{center} \begin{tabular}{lll} \hline $\kappa$ & $\beta^{**}$ & $\lambda^{**}\kappa^2$\\ \hline $0.08$ & $3.6625\cdot10^{6}$ & $5860$\\ $10^{-2}$ & $2.7064\cdot10^{8}$ & $6766$\\ $10^{-4}$ & $3.504\cdot10^{12}$ & $8760$\\ $10^{-8}$ & $5.0872\cdot10^{20}$ & $12718$\\ \hline \end{tabular} \end{center} Indeed, as in the proof of Theorem~\ref{thm:lfkp}(ii), now with $2\cdot 15979<32000=2^85^3$, $8\log M+c_2(\lambda,1)\le K_2+109.5\log\lambda$ for $\lambda\ge100$, where $K_2=109.5+997L_2+8L_3+304.5L_5+121L_7<1534.89$, and $0.52t-109.5\log t$ increases for $t>211$. In the four rows, with $t= \lambda^{**}\kappa^2$, $\log t$ is at most $\log5832+\frac7{1458}$, $\log6750+\frac8{3375}$, $\log8750+\frac1{875}$ and $\log12600+ \frac{59}{6300}$, where $5832=2^33^6$, $6750=2\cdot3^35^3$, $8750=2\cdot 5^47$ and $12600=2^33^25^27$, and $0.52t-109.5\log t$ exceeds $1543.89+ 109.5\log\kappa^{-2}$ by more than $0.14$. At these couplings $\rho\le(\kappa-\kappa')/90$ and $\rho\le7.92\cdot10^{-4}$. The same construction bounds the calm weight of the whole torus. \begin{corollary}\label{cor:sfcoarse} Let $G=\SU(2)$, $0<\kappa'<\kappa<\kappa_c$, $\eta>0$ and $\lambda\ge100$ with $\lambda(\kappa-\kappa')^2\ge1$, and let $Z_\emptyset(c)=\nu_c(\prod_y f_ys_y)$ be the calm weight of $c$. If $\delta(c_P,I)\le\kappa'$ for every coarse plaquette, then \[ Z_\emptyset(c)\ge e^{-c_1(\lambda,\eta)|T|}e^{-4(1+\eta)\lambda Q_1(c)}, \] in particular $Z_\emptyset(c)\ge N_\lambda^{|T|}e^{-8\lambda Q_1(c)}$. If $\delta(c_P,I)>16.003\kappa$ for some $P$, then $Z_\emptyset(c)=0$. \end{corollary} \begin{proof} The plaquettes of the regular field $R(c)$ are the $c_P$ and $I$. In the proof of Theorem~\ref{thm:lfaction}(ii) with $U=R(c)$, $\rho\le(11520 \lambda)^{-1/2}$, so the plaquettes move by at most $90\rho<\kappa-\kappa'$ on $\Gamma(D)$, every block is calm there, and the bound of that proof holds for $Z_\emptyset$. If $\delta(c_P,I)>16.003\kappa$, every field of $F_c$ has a plaquette above $\kappa$ by Lemma~\ref{lem:lfcoarseenergy}, so $\prod_ys_y=0$ on $F_c$. \end{proof} So the class of coarse fields for which the calm weight can serve as a reference lies between the fields with all coarse plaquettes at most $\kappa'$ and those with all at most $16.003\kappa$. For $Z_\emptyset(c)>0$ let $\mu^\circ_c=Z_\emptyset(c)^{-1}\prod_yf_ys_y\,\nu_c$ be the calm law, and for each $X$ and each value of its outer variables with $S_X>0$ let $\rho_X=K_X/S_X$. \begin{proposition}\label{prop:sfrelative} Let $G=\SU(2)$, $\kappa>0$ and $Z_\emptyset(c)>0$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $Z(c)=Z_\emptyset(c)\,\varXi(c)+Z_\perp(c)$, where \[ \varXi(c)=\sum_{X\subset T}\E_{\mu^\circ_c}\Bigl[\prod_i\rho_{X_i}\Bigr], \] the $X_i$ being the components of $X$, $\rho_{X_i}$ is defined $\mu^\circ_c$-almost surely, and $Z_\perp(c)\ge0$ is the sum over $X$ of the integrals of Theorem~\ref{thm:lfmarkov}(i), in the form of Lemma~\ref{lem:sfkernel}(iii), restricted to the outer variables at which some $S_{X_i}$ vanishes. \item Suppose that for every nonempty $\ell^\infty$-connected $X$, $\rho_X\le e^{-9|X|}$ holds $\mu^\circ_c$-almost surely. Then $\varXi(c)= \E_{\mu^\circ_c}[e^{\varLambda}]$, where $\varLambda$ is the sum of the connected clusters of the activities $\rho_X$ given by Theorem~\ref{thm:lfkp}(i). Each cluster depends on the field only within distance one of its support, the clusters incompatible with a given block add up to at most $1/10$, $0\le\varLambda\le|T|/10$, and \[ A_1(c)=-\log Z_\emptyset(c)-\log\E_{\mu^\circ_c}\bigl[e^{\varLambda}\bigr] -\log\Bigl(1+\frac{Z_\perp(c)}{Z_\emptyset(c)\varXi(c)}\Bigr), \] with $0\le\log\E_{\mu^\circ_c}[e^{\varLambda}]\le|T|/10$. \end{enumerate} \end{proposition} \begin{proof} (i) Fix $X$. The integrand $\prod_yf_ys_y$ of $Z_\emptyset$ factorises as the integrand of $Z_X$ does in Lemma~\ref{lem:sfkernel}(iii), with $s_y$ in place of $r_y$ on $X$, so by Fubini the law of the outer variables of $X$ under $\mu^\circ_c$ is $Z_\emptyset^{-1}\varPhi_X\prod_iS_{X_i}\, \nu^{\rm out}_{c,X}$, with $\varPhi_X$ the outer product. This law gives no mass to $\{S_{X_i}=0\}$, and \[ Z_X=\int\nu^{\rm out}_{c,X}\varPhi_X\prod_iK_{X_i}\bigl(\mathbf 1\{\text {all }S_{X_i}>0\}+\mathbf 1\{\text{some }S_{X_i}=0\}\bigr). \] The first part is $Z_\emptyset\,\E_{\mu^\circ_c}[\prod_i\rho_{X_i}]$; sum over $X$. (ii) For a field $U$ let $\rho_X(U)$ be $\rho_X$ at the outer variables of $X$ in $U$. On a set of full $\mu^\circ_c$-measure all these finitely many values are at most $e^{-9|X|}$, and there $\sum_X\prod_i\rho_{X_i}(U)=e^{\varLambda(U)}$ by Theorem~\ref{thm:lfkp}(i), with the stated bounds; $\varLambda\ge0$ because the sum contains the empty family and $\rho\ge0$. The mean of $\prod_i\rho_{X_i}$ is the same whether taken on the outer variables of $X$ or on the field, which gives $\varXi=\E[e^\varLambda]$, and $A_1=-\log Z$. \end{proof} Theorem~\ref{thm:sfcalm} gives the hypothesis of (ii) at every field $U$ that is regular near $X$ and whose moved plaquettes are at most $\kappa'= \kappa/30$, with $U$ itself as the completion. It does not give it at the calm fields with a plaquette near $X$ between $\kappa'$ and $\kappa$, and it says nothing about $Z_\perp$. \begin{remark}\label{rem:sfshell} The comparison of $K_X$ with $S_X$ needs a margin between the threshold of the boundary values and that of the kernel, because a completion must stay calm when it is moved by $90\rho$ and because its energy $Q^{\rm mov}_X$ must be small against $\kappa^2|X|$. For a calm field with plaquettes near $X$ in the shell $(\kappa',\kappa]$, the moved energy of every completion can be of order $216\kappa^2|X|$, and then the bound of Theorem~\ref{thm:sfcalm} on $K_X/S_X$ exceeds one. Putting the blocks of the shell into the large-field region removes them from the calm law, but their kernels then carry only $e^{-\lambda\kappa'^2}$, and the comparison moves to a smaller threshold $\kappa''<\kappa'$ with a new shell $(\kappa'',\kappa']$. A bound on the calm law of the shell, uniform in $c$ and in the exterior, would end this regress; it is not proved. \end{remark} \begin{remark}\label{rem:sfscale} The decoupling of $\log\E_{\mu^\circ_c}[e^\varLambda]$ in $c$ and the bound on the shell both ask for decorrelation of the calm law. Its structure is as follows. By Lemma~\ref{lem:sfkernel}(i), $\nu_c$ is $\prod_yh(dI_y)$ times the pair kernels $\nu^{y,\mu}_c(dP_{y,\mu}\mid I_y, I_{y+e_\mu})$, each supported on $\{V^\sharp_{y,\mu}=c_{y,\mu}\}$ with mass the density of Lemma~\ref{lem:averagedensity} at $c_{y,\mu}$, a function of $I_y$ and $I_{y+e_\mu}$. So $\mu^\circ_c$ is a Gibbs measure for the variables $\varXi_y$ of the blocks, with the factors $f_ys_y$, which by Lemma~\ref{lem:sfkernel}(i) couple $\varXi_y$ to $\varXi_{y+e_\nu}$, and the pair kernels, which couple $I_y$ to $I_{y+e_\mu}$; all couplings have range one. Large $\lambda$ does not make this a high-temperature system. In the linearised law, the Gaussian measure proportional to $e^{-\lambda Q(\varphi)}$ on an affine fibre $\{V(\varphi)=\gamma\}$ of real fields, the conditional law of the variables of one block given the others is Gaussian with a mean that is an affine function of the others independent of $\lambda$, and a covariance equal to $\lambda^{-1}$ times a matrix independent of $\lambda$. Hence the interdependence coefficients of Dobrushin's condition for the Wasserstein distance, which for Gaussian laws with equal covariances are the norms of the derivatives of these means, do not depend on $\lambda$. Theorem~\ref{thm:chartdecay}(iii) has the same scaling: for functions $\lambda g_1$ and $\lambda g_2$ with $g_1$, $g_2$ as there, it gives $c_9\lambda|X_1||X_2|e^{-\alpha_Gd(X_1,X_2)}$, which is below one only for $d(X_1,X_2)\ge\alpha_G^{-1}\log(c_9\lambda|X_1||X_2|)$; the logarithm of a calm weight is of this form to leading order, by Theorem~\ref{thm:sfgauss} below. The input that (ii) of Proposition~\ref{prop:sfrelative} needs for locality in $c$ is the decay of the truncated correlations of all orders of the cluster terms of $\varLambda$ under $\mu^\circ_c$, for every $c$ in the class of Corollary~\ref{cor:sfcoarse}, the analogue for the calm law of the differences of order three and more in Remark~\ref{rem:labelcovopen}. Theorem~\ref{thm:chartdecay}, \eqref{eq:layerbound} and Theorem~\ref{thm:layerbound} give covariances of order two only. \end{remark} \emph{The Gaussian part of the calm weight.} Let $W$ be a finite set of blocks and fix every link off $A_W$ to $I$. With the notation of Sec.~\ref{ssec:largefields}, let $F_c$ be the fibre of $c=(c_k)_{k\in W\times\{0,\dots,3\}}$, $\nu_c$ its measure, $Q_W$ the sum over the $n_P$ plaquettes that contain a link of $A_W$, and \[ Z^\circ_W(c)=\int\prod_p\mathbf 1\{\delta(U_p,I)\le\kappa\}\,e^{-\lambda Q_W}\,d\nu_c , \] the product over the same $n_P$ plaquettes, the calm weight of $W$. For real $\gamma\in\R^{W\times\{0,\dots,3\}}$ let \[ q_W(\gamma)=\min\Bigl\{\sum_p(d\varphi)_p^2:\varphi\in\R^{A_W},\ V_k (\varphi)=\gamma_k\ \text{for all }k\Bigr\}, \] with $\varphi=0$ off $A_W$ and $V_k$ the linear average of Definition~\ref{def:blockgauge}. The constraints are onto, since the path link of $k$ enters $V_k$ alone, with coefficient $\frac18$, and Theorem~\ref{thm:blockgap} makes $\sum_p(d\varphi)_p^2$ positive definite on $\{V(\varphi)=0\}$, so $q_W(\gamma)=\gamma^{\rm T}M_W\gamma$ for a positive semidefinite matrix $M_W$. For $\g$-valued $w$, $\sum_aq_W(w^a)$ over the components in an orthonormal basis of $\g$ is the form $S_W$ of Sec.~\ref{ssec:relstep}, the minimum of the fine Hessian at given averages, when $W$ is a box. \begin{theorem}\label{thm:sfgauss} Let $G=\SU(2)$, $\kappa>0$, $W=\{0,e_0,2e_0,3e_0\}$, $A=(0,1)$ and $B=(3e_0,1)$, $\tau\in\g$ with $\lVert\tau\rVert=1$, and for $\varepsilon>0$ and $s,t\in\{0,1\}$ let $c^{st}_\varepsilon$ equal $e^{\varepsilon s\tau}$ on $A$, $e^{\varepsilon t\tau}$ on $B$ and $I$ on the other coarse links of $W$. Write $\Delta F=\sum_{s,t}(-1)^{s+t}F(c^{st}_\varepsilon)$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $(M_W)_{AB}$ is a rational number. \item There is $\varepsilon_0>0$ such that for $0<\varepsilon\le \varepsilon_0$ \[ \liminf_{\lambda\to\infty}\frac1\lambda\bigl\lvert\Delta\log Z^\circ_W \bigr\rvert\ge\varepsilon^2\lvert(M_W)_{AB}\rvert . \] \item For such $\varepsilon$ there is $\lambda_0$ such that for $\lambda\ge \lambda_0$ and every family of real functions $\phi_j$ on $G^{W\times \{0,\dots,3\}}$, each of which does not depend on $c_A$ or does not depend on $c_B$, \[ \max_{s,t}\Bigl\lvert\log Z^\circ_W(c^{st}_\varepsilon)-\sum_j\phi_j (c^{st}_\varepsilon)\Bigr\rvert\ge\frac{\lambda\varepsilon^2\lvert(M_W)_{AB} \rvert}{5}. \] In particular, if $(M_W)_{AB}$ is not zero and $\lambda$ exceeds $\max\{\lambda_0,2/(\varepsilon^2|(M_W)_{AB}|)\}$, then at these four coarse fields $Z^\circ_W$ is not of the form \[ \prod_{y\in W}\phi_y(c)\sum\prod_iw(X_i), \] with $\phi_y>0$ a function of the $c_{y',\mu}$ with $\ell^\infty(y',y)\le1$ and activities $|w(X)|\le e^{-9|X|}$ on the connected subsets of $W$, compatible as in Theorem~\ref{thm:lfkp}. \end{enumerate} \end{theorem} \begin{proof} (i) The minimiser of $q_W(\gamma)$ solves the linear system of its Lagrange conditions, which has $196$ unknowns, $16$ multipliers and the $605$ plaquettes, and $(M_W)_{AB}=\varphi_A^{\rm T}G\varphi_B$ with $\varphi_k$ the minimiser for $\gamma=e_k$ and $G$ the matrix of $\sum_p(d\varphi)_p^2$. The system has rational coefficients and a unique solution, so $\varphi_A$, $\varphi_B$ and $(M_W)_{AB}$ are rational. (ii) All constants $C$ below depend on $W$ only. \emph{Two-sided bounds.} Since $\nu_c(F_c)\le M^{4|W|}$, $Z^\circ_W(c)\le M^{16}e^{-\lambda q(c)}$ with $q(c)=\inf_{F_c}Q_W$. If $U\in F_c$ is regular with all $n_P$ plaquettes at most $\kappa-90\rho$, $\rho\le9.32\cdot10^{-4}$, the construction of the proof of Proposition~\ref{prop:everyfibre} gives \[ Z^\circ_W(c)\ge1.957^{-48}\Bigl(\frac{\rho^3}{14}\Bigr)^{180}e^{-\lambda ((1+\eta)Q_W(U)+(1+\eta^{-1})8100\rho^2n_P)} . \] Take $\eta=\lambda^{-1/2}$ and $\rho=\lambda^{-1}$. Then \[ q(c)\le\liminf_{\lambda\to\infty}\frac{-\log Z^\circ_W(c)}\lambda\le \limsup_{\lambda\to\infty}\frac{-\log Z^\circ_W(c)}\lambda\le Q_W(U). \] \emph{Expansion of $q$.} Write fields near $I$ as $U=e^X$, $X\in\g^{A_W}$, $|X|=\max_e\lVert X_e\rVert$, and let $\gamma^{st}=(s\,\delta_{kA}+t\, \delta_{kB})_k\tau$, so that $c^{st}_\varepsilon=e^{\varepsilon\gamma^{st}}$ and the form $q^{\rm lin}(\gamma)=\sum_aq_W(\gamma^a)$ has $q^{\rm lin} (\gamma^{st})=s(M_W)_{AA}+2st(M_W)_{AB}+t(M_W)_{BB}$. Near $I$ every cut-off equals one and $V^\sharp$ and $Q_W$ are real analytic. By the formula for the derivative in Lemma~\ref{lem:smallframes}(iv) at $U=I$, and by the Baker--Campbell--Hausdorff formula with $\delta(e^Y,I)^2=\lVert Y\rVert^2+ O(\lVert Y\rVert^4)$, \[ \log V^\sharp_k(e^X)=V_k(X)+O(|X|^2), \] \[ Q_W(e^X)=\sum_p\lVert(dX)_p\rVert^2+O(|X|^3). \] The path link $d_k$ enters $V^\sharp_k$ alone, with derivative $\frac18$ at $I$, so the implicit function theorem solves $V^\sharp(e^X)=c$ for the path links near $I$. \emph{Upper bound.} Let $X^*$ be the minimiser of $q^{\rm lin}(\varepsilon\gamma^{st})$, $|X^*|\le C\varepsilon$, and change its path links by $O(\varepsilon^2)$ so that $U_\varepsilon=e^X\in F_c$, $c=c^{st}_\varepsilon$. Then $Q_W(U_\varepsilon)\le\varepsilon^2q^{\rm lin} (\gamma^{st})+C\varepsilon^3$, $U_\varepsilon$ is regular, and its plaquettes are at most $C\varepsilon$. \emph{Lower bound.} Let $U\in F_c$ with $Q_W(U)\le q(c)+\varepsilon^3\le C\varepsilon^2$. Every plaquette with a vertex in a block of $W^+$ is one of the $n_P$, at most $C\varepsilon$, or consists of links off $A_W$ and equals $I$. For $\varepsilon$ small Lemma~\ref{lem:smallframes}(i) puts the active links inside the blocks within $C\varepsilon$ of $I$ and the links leaving $B(y)$ in the direction $\mu$ within $C\varepsilon$ of the path link $v$, and by (iv) of that lemma $c_k=ve^{N}$ with $\lVert N\rVert\le C\varepsilon$, so $v$ is within $C\varepsilon$ of $I$. By Lemma~\ref{lem:logbound}, $U=e^X$ with $|X|\le C\varepsilon$. Then $V(X)=\varepsilon\gamma^{st}+R$ with $|R|\le C \varepsilon^2$, and since $q^{\rm lin}(a+b)\ge q^{\rm lin}(a)+2\langle a, (M_W\otimes1)b\rangle$, \[ Q_W(U)\ge q^{\rm lin}(\varepsilon\gamma^{st}+R)-C\varepsilon^3\ge \varepsilon^2q^{\rm lin}(\gamma^{st})-C\varepsilon^3 . \] So $|q(c^{st}_\varepsilon)-\varepsilon^2q^{\rm lin}(\gamma^{st})|\le C \varepsilon^3$, and for $\varepsilon\le\varepsilon_1$, where $\varepsilon_1$ depends on $\kappa$ and $W$, the field $U_\varepsilon$ is admissible in the two-sided bound. Hence every limit point of $-\lambda^{-1}\Delta\log Z^\circ_W$ lies within $4C\varepsilon^3$ of $\Delta(\varepsilon^2q^{\rm lin})=2\varepsilon^2(M_W) _{AB}$, and $\varepsilon_0=\min\{\varepsilon_1,|(M_W)_{AB}|/(4C)\}$ gives (ii) if $(M_W)_{AB}\ne0$; otherwise (ii) holds with $\varepsilon_0= \varepsilon_1$, its right side being zero. (iii) $\Delta\phi_j=0$ for every $j$, since a function that does not depend on $c_A$ takes the same value at $c^{1t}$ and $c^{0t}$, and likewise for $c_B$. So $\lvert\Delta\log Z^\circ_W\rvert=\lvert\Delta E\rvert\le4\max \lvert E\rvert$ for the difference $E$, and by (ii) $\lvert\Delta\log Z^\circ_W\rvert\ge\frac45\lambda\varepsilon^2|(M_W)_{AB}|$ for $\lambda\ge \lambda_0$. A factor $\phi_y$ that depends on $c_A$ and on $c_B$ would have $\ell^\infty(y,0)\le1$ and $\ell^\infty(y,3e_0)\le1$, which is impossible. The proof of Theorem~\ref{thm:lfkp}(i) applies to the connected subsets of $W$ and bounds the logarithm of the polymer sum by $|W|/10$ in modulus; $Z^\circ_W>0$ by the lower bound. A representation of this form would therefore give $\lambda\varepsilon^2|(M_W)_{AB}|/5\le2/5$, that is, $\lambda\le2/(\varepsilon^2|(M_W)_{AB}|)$. \end{proof} \begin{remark}[a computation, not used by any result]\label{rem:sfgaussvalue} Solving the system of the proof of Theorem~\ref{thm:sfgauss}(i) in exact rational arithmetic gives $(M_W)_{AB}=-0.035264626\ldots$, so that the hypothesis $(M_W)_{AB}\ne0$ of the last assertion of Theorem~\ref{thm:sfgauss}(iii) holds by this computation; no proof in this paper uses it, and no hand proof of $(M_W)_{AB}\ne0$ is given. For the rows $W_n=\{0,\dots,(n-1)e_0\}$ and the end links $A=(0,1)$, $B=((n-1)e_0, 1)$, the same computation gives $(M_{W_n})_{AA}=20.0796\ldots$ for $n\ge4$ and \begin{center} \begin{tabular}{lllllll} \hline $n$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$\\ \hline $10^{n-2}(M_{W_n})_{AB}$ & $-3.32$ & $3.35$ & $-3.53$ & $3.88$ & $-4.47$ & $5.40$\\ \hline \end{tabular} \end{center} so in this computation the coupling decays by a factor between $8$ and $10$ per block and does not vanish. \end{remark} By the argument of Theorem~\ref{thm:sfgauss}, a product of factors of finite range $r$ reproduces the calm weight of $W_n$, $n-1>2r$, only up to an error of order $\lambda\varepsilon^2|(M_{W_n})_{AB}|$. To leading order in $\lambda$ the calm weight is $e^{-\lambda q}$ with $q$ the Gaussian block-spin form, whose kernel decays at a rate independent of $\lambda$ with amplitude proportional to $\lambda$. A local expansion of the small-field part therefore has to keep this form and expand around it, as the Gaussian small-field expansion of Ba{\l}aban does; a product of factors of range one times a polymer sum with activities $e^{-9|X|}$, which is the form (H1) below asks for, fails for the calm weight of $W$ at every $\lambda>\max\{\lambda_0,2/(\varepsilon^2|(M_W)_{AB}|)\}$ when $(M_W)_{AB}\ne0$, by Theorem~\ref{thm:sfgauss}(iii). So the small-field part of one step stands as follows. The kernels are local, with the second layer $\partial^=X$ (Lemma~\ref{lem:sfkernel}). The calm weight of a region with boundary values is at least $e^{-c_2}$ per block times the Gaussian factor of a completion calm at $\kappa'$, and the relative activity is then at most $e^{-a_2|X|}$ (Theorem~\ref{thm:sfcalm}). The large-field sum is exactly $Z_\emptyset\varXi+Z_\perp$, and under the uniform bound on the relative activities the large fields change $A_1$ by at most $|T|/10$ (Proposition~\ref{prop:sfrelative}). Not proved are that bound at calm fields with plaquettes in the shell $(\kappa',\kappa]$ (Remark~\ref{rem:sfshell}), a bound on $Z_\perp$, and the locality of $-\log Z_\emptyset(c)$ and of $\log\E_{\mu^\circ_c}[e^\varLambda]$ in $c$. For the first of these the form of the answer is fixed by Theorem~\ref{thm:sfgauss}: to leading order it is the Gaussian block-spin form with amplitude $\lambda$, and, if $(M_W)_{AB}\ne0$, no product of local factors. \paragraph*{Locality, the shell and the remainder.} The three questions left open above are taken up in turn. For the locality of the calm weight, the form $e^{-\lambda q^{\rm lin}(\log c)}$ times a polymer sum with local activities is impossible at $\lambda>0.548$, because the calm weight is invariant under coarse gauge transformations and the Gaussian block-spin form of $\log c$ is not; the linearised propagator with the block-spin constraint decays at an explicit rate, which is the input that an expansion around a gauge-invariant leading term would use. For the shell, in the linearisation of the model the relative activities are not bounded under the calm law, at any $\lambda$ and $\kappa$, and for $\SU(2)$ they are not bounded at any $\lambda$ and $\kappa\le10^{-8}$. For the remainder, $Z_\perp(I)>0$ for $\SU(2)$ at every $\lambda$ and every $\kappa\le 10^{-8}$, and $Z_\perp$ is bounded in the mean over $c$ by the probability of a rough block. For a gauge field $g\in G^T$ on the coarse torus put $(c^g)_{y,\mu}=g_y c_{y,\mu}g_{y+e_\mu}^{-1}$. The \emph{Gaussian block-spin form of the torus} is $q_T(\gamma)=\min\{Q(\varphi):V(\varphi)=\gamma\}$ for real $\gamma$ on the coarse links, the minimum over real fields on the active links of $\Lambda_L$; for $\g$-valued $w$ put $q^{\rm lin}(w)=\sum_aq_T(w^a)$, with the components in the orthonormal basis $\tau_a=i\sigma_a/\sqrt2$ of $\mathfrak{su}(2)$. As for $q_W$, $q_T$ is a positive semidefinite quadratic form, by Theorem~\ref{thm:blockgap} on the periodic lattice. \begin{lemma}\label{lem:sfgaugeinv} Let $G=\SU(2)$, $\kappa>0$ and $g\in G^T$. Then $Z(c^g)=Z(c)$ and $Z_\emptyset(c^g)=Z_\emptyset(c)$ for every $c$, and $q_T(\gamma+d\chi)= q_T(\gamma)$ for real $\gamma$ and $\chi$, where $(d\chi)_{y,\mu}=\chi_y- \chi_{y+e_\mu}$. \end{lemma} \begin{proof} Let $\Phi$ be the fine gauge transformation equal to $g_y$ at every site of $B(y)$. It fixes the tree links, so it preserves the tree gauge, and it preserves $Q$, every indicator $s_y$ and $r_y$, and product Haar measure. The cut-off average is covariant (Theorem~\ref{thm:cutoff}), so $\Phi$ maps $F_c$ onto $F_{c^g}$. In the proof of Lemma~\ref{lem:averagedensity} for the pair $(y,\mu)$, $\Phi$ replaces the path link $\ell_0$ by $g_y\ell_0 g_{y+e_\mu}^{-1}$, the links inside $B(y)$ by their conjugates by $g_y$ and those of $B(y+e_\mu)$ by their conjugates by $g_{y+e_\mu}$, so every site $v_x$ is conjugated by $g_{y+e_\mu}$. The regions $R_0$, $R_1$, $R_2$, which are defined by the norms $\delta(v_x,I)$, are therefore preserved, the coordinates $h_a$ are conjugated, which preserves Haar measure, and the derivative of $V^\sharp$ is conjugated by $\operatorname{Ad}(g_y)$, so $\lvert\det DV^\sharp\rvert$ is unchanged. Hence the image of $\nu_c$ under $\Phi$ is $\nu_{c^g}$, and $Z=\nu_c(e^{-\lambda Q})$ and $Z_\emptyset=\nu_c( \prod_yf_ys_y)$ are invariant. For $q_T$, Lemma~\ref{lem:blocktree}(ii) with $h$ equal to $\chi_y$ on $B(y)$ maps $\{V=\gamma\}$ onto $\{V=\gamma+d\chi\}$ and preserves $Q$. \end{proof} \begin{theorem}\label{thm:sfgaugeobst} Let $G=\SU(2)$, $\lambda>0$, $\kappa>0$, and let $F$ be $Z$ or $Z_\emptyset$. Let $y$ be a block, $z=y+e_0$, $0<\phi\le\pi/2$, and let $g_y=\exp(\sqrt2 \phi\tau_1)$, $g_z=\exp(\sqrt2\phi\tau_2)$ and $g_w=I$ otherwise. Suppose that at $c=I$ and at $c=I^g$ \[ F(c)=e^{-\lambda q^{\rm lin}(\log c)-E|T|}\prod_y\phi_y(c)\sum\prod_iw(X_i;c), \] where $\log$ is the principal logarithm on each coarse link, $E$ is a constant, each $\phi_y>0$ is invariant under coarse gauge transformations, and the sum runs over the families of pairwise compatible connected sets as in Theorem~\ref{thm:lfkp}, with activities $|w(X;c)|\le e^{-9|X|}$ that depend on $c$ only through the $c_{w',\mu}$ with $\ell^\infty(w',X)\le1$. Then $\lambda\le8.1/|\eta(\phi)|^2\le4.05/\sin^4\phi$, where $\cos\psi=\cos^2\phi$, $0<\psi\le\pi/2$, and \[ |\eta(\phi)|^2=4\Bigl(\frac{\psi\sin\phi\cos\phi}{\sin\psi}-\phi\Bigr)^2 +\frac{2\psi^2\sin^4\phi}{\sin^2\psi}. \] At $\phi=\pi/2$, $|\eta|^2=3\pi^2/2$ and $\lambda\le5.4/\pi^2<0.5472$. The field $I^g$ is flat, $\delta((I^g)_P,I)=0$ for every coarse plaquette, and its links have $\delta(c_k,I)\le2\sin\phi$. \end{theorem} \begin{proof} Write $g_y=a+b\cdot i\sigma$ as unit quaternions. The links of $c=I^g$ are $g_y$ on $(y,\mu)$, $\mu\ne0$, and $g_y^{-1}$ on $(y-e_\mu,\mu)$; $g_z$ on $(z,\mu)$ and $g_z^{-1}$ on $(z-e_\mu,\mu)$, $\mu\ne0$; $g_z$ on $(z,0)$, $g_y^{-1}$ on $(y-e_0,0)$, and $g_yg_z^{-1}$ on $k=(y,0)$; $I$ elsewhere. Their bases form the set $S$ of the nine blocks $y-e_0$, $y$, $z$, $y-e_\mu$ and $z-e_\mu$ ($\mu=1,2,3$). With $\chi$ equal to $\log g_y=\sqrt2\phi\tau_1$ at $y$, $\log g_z=\sqrt2\phi\tau_2$ at $z$ and $0$ elsewhere, $\log(I^g)=d\chi+\eta\,\delta_k$ with $\eta=\log(g_yg_z^{-1})-\chi_y+\chi_z$. The quaternion of $g_yg_z^{-1}$ is $\cos^2\phi+(\sin\phi\cos\phi,-\sin\phi \cos\phi,\sin^2\phi)\cdot i\sigma$, whose principal logarithm is $\sqrt2\psi$ times the unit vector of its vector part, in the coordinates of the $\tau_a$; this gives $|\eta|^2$, and at $\phi=\pi/2$, $g_y=i\sigma_1$, $g_z=i\sigma_2$, $g_yg_z^{-1}=i\sigma_3$ and $\eta=\frac\pi{\sqrt2}(-\tau_1+ \tau_2+\tau_3)$. As $\psi\ge\sin\psi$, the second term of $|\eta|^2$ is at least $2\sin^4\phi$; and $5.4/\pi^2<0.5472$ since $\pi^2>9.8696$. By Lemma~\ref{lem:sfgaugeinv} and since $q^{\rm lin}$ is a sum of positive semidefinite forms, $q^{\rm lin}(\log I^g)=q^{\rm lin}(\eta \delta_k)$. By the proof of Proposition~\ref{prop:effective}(i), which applies on the torus, $q_T\ge Q_1$, and $\delta_k$ lies in six coarse plaquettes, so $q^{\rm lin}(\eta\delta_k)\ge6|\eta|^2$. By Lemma~\ref{lem:sfgaugeinv}, $F(I^g)=F(I)$, and the $\phi_y$ and $E$ cancel, so $\lambda q^{\rm lin}(\eta\delta_k)=\log\Sigma(I^g)-\log \Sigma(I)$ with $\Sigma$ the polymer sum. By Theorem~\ref{thm:lfkp}(i), $\log\Sigma$ is the sum of the connected clusters, a cluster depends on $c$ only through the links with base within distance one of its support, and the clusters incompatible with a given block add up to at most $\frac1{10}$ in modulus. A cluster whose support does not meet the set $N_1(S)$ of blocks within distance one of $S$ takes the same value at $I$ and $I^g$; a cluster that meets it is incompatible with a block of $N_1(S)$. Count $N_1(S)$ by the level $a$ of a block $w=y+ae_0+s$, $s$ spatial. The blocks of $S$ have $s=0$ at level $-1$ and $s\in D=\{0,-e_1,-e_2,-e_3\}$ at levels $0$ and $1$, and the spatial points within distance one of $D$ are the $27$ points of $\{-1,0,1\}^3$ and the $3\cdot9$ points with one coordinate $-2$ and the others in $\{-1,0,1\}$. So level $-2$ contributes the $27$ points within distance one of $s=0$, each of the levels $-1$, $0$, $1$, $2$ contributes $54$, and $|N_1(S)|\le27+4\cdot54=243$. Hence $6\lambda|\eta|^2\le2\cdot243/10=48.6$. The plaquettes of $I^g$ are conjugates of those of $I$, and $\delta(g_yg_z^{-1}, I)=2\sin\phi$ is the largest link. \end{proof} The following values are evaluations of the closed form for $|\eta(\phi)|^2$, rounded to the digits shown, those of the last column upwards. The conclusions below use only the row $\phi=\pi/2$, whose entries are exact, and the bound $4.05/\sin^4\phi$. \begin{center} \begin{tabular}{llll} \hline $\phi$ & $|\eta(\phi)|^2$ & $\max_k\delta(c_k,I)$ & $\lambda\le$\\ \hline $\pi/2$ & $14.8044$ & $2$ & $0.5472$\\ $0.5$ & $0.13191$ & $0.9589$ & $61.41$\\ $0.1$ & $2.0044\cdot10^{-4}$ & $0.1997$ & $4.0411\cdot10^4$\\ $0.05$ & $1.2507\cdot10^{-5}$ & $0.09996$ & $6.4765\cdot10^5$\\ \hline \end{tabular} \end{center} So at every $\kappa$ the representation fails for $\lambda>0.5472$ at a flat coarse field, and, among coarse fields with links within $0.1$ of $I$, for $\lambda>6.5\cdot10^5$: at $\phi=0.05$ the links are within $2\sin\phi<0.1$ of $I$, and $\sin\phi>\phi-\phi^3/6>0.04997$, so $\lambda\le4.05/ 0.04997^4<6.5\cdot10^5$. This is below $\lambda^{**}(0.08)<9.16\cdot10^5$. The defect is the failure of $\log$ to intertwine the coarse gauge action: $q^{\rm lin}(\log c)$ is invariant only under gauge transformations that commute with $c$. A representation of the required kind needs a leading term invariant under the coarse gauge group, such as $\lambda S_1(c)$ with $S_1(c) =\inf_{F_c}Q$; by Theorem~\ref{thm:sfgauss} it is not local if $(M_W)_{AB} \ne0$, and its Hessian at $c=I$ is $2\lambda$ times the form $q^{\rm lin}$. Whether $Z_\emptyset(c)e^{\lambda S_1(c)}$, or the same with another invariant form of this Hessian, is a polymer sum with local activities is not decided here. \emph{The block-spin propagator.} The expansion of such a form around a minimiser uses the propagator of the fine field with the block-spin constraint. In the linearisation it decays at a rate that the following theorem makes explicit. For a pair $k=(y,\mu)$ label its eight private links by $a\in\{0,1\}^3$, the offsets in the three directions $\nu\ne\mu$, and for $w\in\{0,1\}^3\setminus\{0\}$ put $\xi_{k,w}=8^{-1/2}\sum_a(-1)^{a\cdot w} \varphi_{k,a}$. By Lemma~\ref{lem:blocktree}(iii), on $\{V=0\}$ the mean of the eight is $m_k=-\frac1{16}(S^\mu_y+S^\mu_{y+e_\mu})$, so \[ \varphi_{k,a}=8^{-1/2}\sum_w(-1)^{a\cdot w}\xi_{k,w}+m_k , \] and the coordinates $v=(\varphi_e)_{e\text{ inside}}\oplus(\xi_{k,w})$, with $17$ inside links and $28$ Walsh coordinates per block, define a linear bijection $P:v\mapsto\varphi$ onto $\{V=0\}$. Let $H$ be the matrix of $v\mapsto Q(Pv)$, and for a coordinate, a fine link or a coarse link let its block be the block of its base. Write $d(\cdot,\cdot)$ for the $\ell^\infty$ distance of the blocks on the coarse torus. Let $L_0$ map $\gamma$ to the field equal to $\gamma_k$ on the eight private links of $k$ and to zero elsewhere, as in Proposition~\ref{prop:effective}(ii), and let $A$ and $B$ be the matrices of $Q(L_0\gamma)=\gamma^{\rm T}A\gamma$ and of the cross term, so that $Q(L_0\gamma+Pv)=\gamma^{\rm T}A\gamma+2\gamma^{\rm T}Bv +v^{\rm T}Hv$. \begin{theorem}\label{thm:sfprop} On the periodic lattice of side $L\ge32$ let \[ \bar S=168+42\sqrt2,\qquad\bar b=168+336\sqrt2,\qquad p_1=\frac7{2\sqrt2}+ \frac78, \] so that $\bar S<227.40$ and $\bar b<643.18$, and for $\alpha>0$ let $c_\alpha=\frac1{52}-(\cosh3\alpha-1)\bar S$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $H_{ij}=0$ for $d(i,j)\ge4$, the sum $\sum_j|H_{ij}|$ is at most $\bar S$ for every $i$, and $v^{\rm T}Hv\ge\frac1{52}|v|^2$ for every $v$. \item If $0<\alpha\le\frac1{250}$, then $c_\alpha>0$ and $|(H^{-1})_{ij}|\le c_\alpha^{-1}e^{-\alpha d(i,j)}$ for all coordinates $i$ and $j$. \item For real $\gamma$ the Gaussian probability measure proportional to $e^{-\lambda Q(\varphi)}$ on the affine space $\{V(\varphi)=\gamma\}$ of fields in the tree gauge has as mean the minimiser $\varphi^*(\gamma)=L_0\gamma- PH^{-1}B^{\rm T}\gamma$ of $q_T(\gamma)$ and as covariance $C=(2\lambda)^{-1} PH^{-1}P^{\rm T}$, and for $0<\alpha\le\frac1{250}$ \[ |C_{ee'}|\le\frac{e^{2\alpha}p_1^2}{2\lambda c_\alpha}e^{-\alpha d(e,e')}. \] \item $q_T(\gamma)=\gamma^{\rm T}M\gamma$ with $M=A-BH^{-1}B^{\rm T}$; the rows of $A$ and $B$ have $\ell^1$ norms at most $192$ and $\bar b$, $A_{kk'}=0$ for $d(k,k')\ge2$ and $B_{ki}=0$ for $d(k,i)\ge3$; and for $d(k,k')\ge2$ and $0<\alpha\le\frac1{250}$, $|M_{kk'}|\le e^{4\alpha}\bar b^2 c_\alpha^{-1}e^{-\alpha d(k,k')}$. \end{enumerate} At $\alpha=\frac1{500}$, $c_\alpha>0.0151$, and the three constants are below $66.3$, $746.2/(2\lambda)$ and $2.762\cdot10^7$. \end{theorem} \begin{proof} (i) Write $e_i$ for the coordinate vectors of $v$. For an inside link $e=(x,\mu)$ of $B(b)$ the coordinate enters only $S^\mu_b$, so by the formula for $m_k$ the field $Pe_e$ is $1$ on $e$ and $-\frac1{16}$ on the eight private links of each of the pairs $(b,\mu)$ and $(b-e_\mu,\mu)$; its $\ell^1$ norm is $2$. For a Walsh coordinate $\xi_{k,w}$, $k=(b,\mu)$, the field is $\pm8^{-1/2}$ on the eight private links of $k$, of $\ell^1$ norm $2\sqrt2$. In both cases the links have base in $b$ or $b-e_\mu$. A row of $P$ at an inside link has the single entry $1$; at a private link of a pair $(b,\mu)$ it has $7$ Walsh entries $\pm8^{-1/2}$ and the $2(8-2^\mu)$ active links of $S^\mu_b$ and $S^\mu_{b+e_\mu}$ with $-\frac1{16}$, so its $\ell^1$ norm is at most $p_1$, the value at $\mu=0$. A plaquette has four links and a link lies in six plaquettes, so $\sum_p|(dPe_i)_p|\le6\cdot2\sqrt2$ and $\sum_j|(dPe_j)_p|\le\sum_{l\in p}\sum_j|P_{lj}|\le4p_1$. As $H_{ij}= \sum_p(dPe_i)_p(dPe_j)_p$, \[ \sum_j|H_{ij}|\le\sum_p|(dPe_i)_p|\sum_j|(dPe_j)_p|\le48\sqrt2\,p_1=\bar S . \] If $H_{ij}\ne0$, a plaquette contains a link of $Pe_i$ and a link of $Pe_j$; the bases of two links of a plaquette lie in blocks at distance at most one, and the links of $Pe_i$ have base within distance one of the block of $i$, so $d(i,j)\le3$. Finally $v^{\rm T}Hv=Q(Pv)\ge\frac1{52}\lvert Pv\rvert^2$ by Theorem~\ref{thm:blockgap}, whose proof uses only the plaquettes inside a block and those joining two neighbouring blocks and holds verbatim on the periodic lattice, and $\lvert Pv\rvert^2=\sum_{\rm inside}\varphi_e^2+\sum_k(\sum_w\xi_{k,w}^2+ 8m_k^2)\ge|v|^2$, since the characters $a\mapsto(-1)^{a\cdot w}$, $w\ne0$, are orthogonal and orthogonal to the constants. (ii) Since $\cosh u-1=\sum_{m\ge1}u^{2m}/(2m)!\le\frac{u^2}2\cosh u$, so that $\cosh u\le(1-\frac{u^2}2)^{-1}$ for $u^2<2$, at $\alpha=\frac1{250}$ one has $(\cosh3\alpha-1)\bar S\le\frac{0.012^2}2\cdot1.0001\cdot227.40< 0.01638<\frac1{52}$, and $c_\alpha$ decreases in $\alpha$. Fix $j$, put $f_i=d(i,j)$ and $H_\alpha=e^{\alpha f}He^{-\alpha f}$. For real $v$, by the symmetry of $H$, \begin{align*} v^{\rm T}H_\alpha v&=\sum_{i,l}H_{il}\cosh\bigl(\alpha(f_i-f_l)\bigr)v_iv_l\\ &=v^{\rm T}Hv+\sum_{1\le d(i,l)\le3}H_{il}\bigl(\cosh(\alpha(f_i-f_l))-1 \bigr)v_iv_l , \end{align*} since $f_i=f_l$ when $d(i,l)=0$ and $H_{il}=0$ when $d(i,l)\ge4$. As $|f_i- f_l|\le d(i,l)\le3$, the last sum is at most $(\cosh3\alpha-1)\bar S|v|^2$ in modulus, by the Schur test for the symmetric matrix of its coefficients and (i). So $|H_\alpha v|\ge c_\alpha|v|$, $H_\alpha$ is invertible with $\lVert H_\alpha^{-1}\rVert\le c_\alpha^{-1}$, and its entry $(i,j)$ is $e^{\alpha d(i,j)}(H^{-1})_{ij}$. (iii) $V(L_0\gamma)=\gamma$ by Proposition~\ref{prop:effective}(ii), so the fields with $V=\gamma$ are the $L_0\gamma+Pv$, and the exponent is the quadratic form above in $v$; completing the square gives the mean and the covariance $(2\lambda H)^{-1}$ of $v$. By (i) a row of $P$ has $\ell^1$ norm at most $p_1$, and $P_{ei}\ne0$ only if the block of $i$ is the block of $e$ or its neighbour in the direction of $e$, so the coordinates $i$, $l$ that enter $C_{ee'}$ have $d(i,l)\ge d(e,e')-2$. Then apply (ii). (iv) Minimising over $v$ gives $M$. The field $L_0\delta_k$ is $1$ on the eight private links of $k$, which have base in the block of $k$ and belong to no other pair, so $\sum_p|(dL_0\delta_k)_p|\le48$ and $\sum_{k'}|(dL_0 \delta_{k'})_p|\le4$. As in (i), $A_{kk'}=\sum_p(dL_0\delta_k)_p(dL_0 \delta_{k'})_p$ and $B_{ki}=\sum_p(dL_0\delta_k)_p(dPe_i)_p$ give the row bounds $48\cdot4=192$ and $48\cdot4p_1=\bar b$ and the ranges one and two. Then $|(BH^{-1}B^{\rm T})_{kk'}|\le\sum_{i,l}|B_{ki}||(H^{-1})_{il}||B_{k'l}|$ with $d(i,l)\ge d(k,k')-4$, and (ii) gives the bound. At $\alpha=\frac1{500}$, $(\cosh3\alpha-1)\bar S\le\frac{0.006^2}2\cdot 1.0001\cdot227.40<0.004094$, so $c_\alpha>\frac1{52}-0.004094>0.0151$; with $p_1^2=\frac{49}8+\frac{49}{8\sqrt2}+\frac{49}{64}<11.222$, $e^{0.004}< 1.00402$, $e^{0.008}<1.00807$ (from $e^u\le(1-u)^{-1}$) and $\bar b^2< 413{,}681$ the constants follow. \end{proof} \begin{remark} A computation on the lattice of side $12$, which no statement here uses, gives the entries of $H$, $A$ and $B$: all three have range one, the largest sum of $|H_{ij}|$ over $d(i,j)=1$ is $S_1=20.45495\ldots$, and the rows of $A$ and $B$ have $\ell^1$ norms at most $96$ and $77.456$. With these values the argument of (ii), with $\cosh\alpha$ in place of $\cosh3\alpha$ and $S_1$ in place of $\bar S$ and with the constant $\frac{537}{5000}$ of Remark~\ref{rem:blockgapsharp} in place of $\frac1{52}$, gives decay for $\alpha<0.10243$, and at $\alpha=0.08$ constants $23.87$, $314.3/(2\lambda)$ and $1.680\cdot10^5$. Both rates are far from the decay of $(M_{W_n})_{AB}$ by a factor $8$ to $10$ per block, which corresponds to $\alpha\approx2.1$; the loss comes from bounding the off-diagonal part of $H$ by its row sums. \end{remark} \begin{remark}\label{rem:sfbalaban} Ba{\l}aban's renormalisation group for lattice gauge theories~\cite{Bal84a,Bal85a,Bal85b,Bal85c,Bal87,Bal88,Bal89b} expands the small-field part around the minimiser of the action at given block averages: he treats this variational problem and the background fields it defines in~\cite{Bal85c}, the propagators in such a background in~\cite{Bal85b}, and the generation of the effective actions in a small field approximation and their cluster expansions in~\cite{Bal87,Bal88}. They are cited for the method only. His averaging operations are not the cut-off average $V^\sharp$, and no proof here uses his results. By Theorems~\ref{thm:sfgauss} and~\ref{thm:sfgaugeobst} the small-field part of the present step calls for the same structure: a leading term of amplitude $\lambda$, nonlocal if $(M_W)_{AB}\ne0$, which cannot be $\lambda q^{\rm lin}(\log c)$, and an expansion around it. Theorem~\ref{thm:sfprop} is the linear propagator bound of such an expansion at the trivial background; the background field of $V^\sharp$ and the expansion around it are not carried out. \end{remark} \emph{The shell in the linearised model.} In the \emph{linear model} the links are real: $\varphi$ is a real field on the active links of the torus, $Q(\varphi)=\sum_p(d\varphi)_p^2$, a block $y$ is calm when $|(d\varphi)_p| \le\kappa$ for every $p\in\Pi(y)$, the coarse field is $\gamma=V(\varphi)$, and the fibre $\{V=\gamma\}$ carries Lebesgue measure in the coordinates given by the values on the links other than the path links. The statements of Lemma~\ref{lem:sfkernel} concern only which links a factor involves, and hold verbatim. For $X=\{y\}$ and $\gamma=0$, the free links of $\mathcal V_X$, the links of $\mathcal V_X$ other than the path links of the pairs of $\mathcal P_X$, are coordinates $u\in\R^{73}$ of the completions of given outer variables $o$, the remaining links. Let $S^{\rm lin}_X(o)$ and $K^{\rm lin}_X(o)$ be the integrals over $u$ of $\prod_{w\in\hat X}f_ws_w$ and of $f_yr_y\prod_{w\in\hat X\setminus X}f_ws_w$, with $f_w=\exp(-\lambda\sum_{p\in\Pi(w)}(d\varphi)_p^2)$. The calm law is proportional to $\prod_wf_ws_w$ on $\{V=0\}$. As in the proof of Proposition~\ref{prop:sfrelative}, the law of $o$ under it has density proportional to $\varPhi_X(o)S^{\rm lin}_X(o)$ with respect to Lebesgue measure in the free outer links, $\varPhi_X=\prod_{w\notin\hat X}f_ws_w$, and $\rho^{\rm lin}_X=K^{\rm lin}_X/S^{\rm lin}_X$ is defined almost surely. \begin{theorem}\label{thm:sfshelllin} In the linear model on a torus of coarse side at least $8$, with $\gamma=0$, $X=\{y\}$, every $\kappa>0$ and every $\lambda>0$, the ratio $\rho^{\rm lin} _X$ is not essentially bounded under the calm law: for every $R$ the calm law gives positive probability to $\{\rho^{\rm lin}_X>R\}$. In particular the hypothesis of Proposition~\ref{prop:sfrelative}(ii) fails in the linear model at every $\lambda$ and $\kappa$. \end{theorem} \begin{proof} The map $\varphi\mapsto\kappa\varphi$ replaces $\kappa$ by $1$ and $\lambda$ by $\lambda\kappa^2$, so let $\kappa=1$. Let $x=2y+(1,1,1,0)$. The three faces of the unit cube at $x$ in the directions $0$, $1$, $2$ that contain $x$ are owned by $y$; write $F_1$, $F_2$, $F_3$ for $d\varphi$ on the faces in the planes $\{0,1\}$, $\{0,2\}$, $\{1,2\}$. Every link of the cube lies in two of its faces with opposite signs, so \[ F_1-F_2+F_3=(d\varphi)_{x+e_2,01}-(d\varphi)_{x+e_1,02}+(d\varphi)_{x+e_0, 12}=:R , \] and the three faces on the right are owned by $y+e_2$, $y+e_1$, $y+e_0 \notin\hat X$ and contain no link of $\mathcal V_X$, so $R=R(o)$ is a linear function of $o$. \emph{The field $\varphi_1$.} For $\varepsilon\in\{0,1\}^3\times\{0\}$ and $\varepsilon_\mu=0$ the edge $(x+\varepsilon,\mu)$ of the cube has offset one in the direction $\mu$, so it is a private link of the pair $(y+\varepsilon, \mu)$, and distinct edges lie in distinct pairs. Let $\varphi_1$ take the values \[ -\tfrac12,\ \tfrac12,\ \tfrac12,\ -\tfrac12,\ -\tfrac12,\ \tfrac12 \] on the edges $(x+e_1,0)$, $(x+e_2,0)$, $(x+e_0,1)$, $(x+e_2,1)$, $(x+e_0,2)$, $(x+e_1,2)$, minus one seventh of that value on the other seven private links of the pair of each of these edges, and zero elsewhere. Private links are active, the sum over the private links of every pair is zero and no inside link is used, so $V(\varphi_1)=0$ by Lemma~\ref{lem:blocktree}(iii). The six pairs are those of the blocks $y+e_0$, $y+e_1$, $y+e_2$, none of which lies in $\hat X$, so $\varphi_1$ vanishes on $\mathcal V_X$. A pair contains one edge of the cube, so $\varphi_1$ vanishes on the other six edges, and \begin{gather*} F_1=\varphi_1(x+e_0,1)-\varphi_1(x+e_1,0)=1,\\ F_2=\varphi_1(x+e_0,2)-\varphi_1(x+e_2,0)=-1,\\ F_3=\varphi_1(x+e_1,2)-\varphi_1(x+e_2,1)=1, \end{gather*} and the outer faces take the values $\varphi_1(x+e_2,0)-\varphi_1(x+e_2,1) =1$, $\varphi_1(x+e_1,0)-\varphi_1(x+e_1,2)=-1$ and $\varphi_1(x+e_0,1)- \varphi_1(x+e_0,2)=1$. Two edges of the cube lie in a common plaquette only if it is a face of the cube, so every other plaquette contains at most one link with $|\varphi_1|=\frac12$ and three with $|\varphi_1|\le\frac1{14}$, and $|d\varphi_1|\le\frac57=1-\delta_0$, $\delta_0=\frac27$. Let $w\in\R^{73}$ lower $(x,1)$ by one; to keep $V_{y,1}$, the path link $(2y+e_1,1)$, also a private link of $(y,1)$, rises by one. The link $(x,1)$ lies in $F_1$ with the sign $-$ and in $F_3$ with the sign $+$, the path link lies in none of the six faces, and the two links share no plaquette, so $w$ changes $(F_1,F_2,F_3)$ by $(1,0,-1)$ and every plaquette by at most $1$. Lowering $(x,2)$ in the same way changes $(F_1,F_2)$ by $(0,1)$, so the map from the free links of $\mathcal V_X$ to $(F_1,F_2)$ has rank two. Let $o_t$ be the outer part of $(1-t)\varphi_1$, $00$, independent of $t$, such that these inequalities hold with $\frac{\delta_0}8$ and $\frac{\delta_0}4$ on the ball of radius $r$ about this point, and $Q$ is bounded there by a constant $Q_0$. Hence $K^{\rm lin}_X(o)\ge c_r e^{-\lambda Q_0}=:k_0>0$ for $o$ within $r$ of $o_t$, with $c_r$ the volume of the ball of radius $r$ in $\R^{73}$. \emph{Positive probability.} At $(o_t,u_1)$ every plaquette other than the six faces has modulus at most $(1-t)(1-\delta_0)$ and the six have modulus $1-t$, so all blocks are calm with a margin. For $o$ near $o_t$ the outer blocks stay calm, $\varPhi_X(o)>0$, and the completions near $u_1$ stay calm, $S^{\rm lin}_X(o)>0$. So the calm law gives positive probability to every small ball about $o_t$, and on such a ball inside $B_t$ and within $r$ of $o_t$, $\rho^{\rm lin}_X\ge k_0/(16C_\lambda t^2)$. Let $t\to0$. \end{proof} \begin{remark}\label{rem:sfconditioning} The mechanism is the conditioning: $\rho_X$ compares two parts of the mass of the completions, and its denominator is the mass of the calm ones, which the Bianchi identity of a cube with three outer faces near their threshold squeezes into a corner of area $(3-R)^2/2$, while the rough completions keep a mass of order one. Weak coupling does not help, since the same holds at every $\lambda$; and the bound of Theorem~\ref{thm:sfcalm} does not apply, since at these boundary values every completion has a face at least $1-3t$, above $\kappa'$. In $\SU(2)$ the Bianchi identity of the cube holds exactly up to conjugation, and $\rho_X$ is not essentially bounded under the calm law either, at every $\lambda$ and every $\kappa\le10^{-8}$ (Theorem~\ref{thm:sfshellsu2} below). The hypothesis of Proposition~\ref{prop:sfrelative}(ii) can therefore hold at best in a form that excludes such boundary values, for example in the mean under the calm law, and Proposition~\ref{prop:sfrelative} (ii) is not proved in such a form for $\SU(2)$. In the linear model the mean form holds, with an explicit constant (Corollary~\ref{cor:sfglshell}). \end{remark} \emph{The shell for $\SU(2)$.} The Bianchi identity of a cube holds in $\SU(2)$ exactly, up to conjugation, and the calm condition is invariant under conjugation. This makes the construction of Theorem~\ref{thm:sfshelllin} work for $\SU(2)$ at every $\lambda$. For $g\in\SU(2)$ let $\theta(g)\in[0,\pi]$ be its angle, so that the eigenvalues of $g$ are $e^{\pm i\theta(g)}$. Then $\delta(g,I)=2\sqrt2\sin (\theta(g)/2)$ and $d(g,I)=\lVert\log g\rVert=\sqrt2\,\theta(g)$. On the unit quaternions $\theta(g)$ is the geodesic distance from $1$ to $g$, and left and right multiplications are isometries, so \[ \theta(gh)\le\theta(g)+\theta(h),\qquad\theta(g^{-1})=\theta(kgk^{-1})= \theta(g), \] and under $h$ the angle has the density $\frac2\pi\sin^2\theta$ on $[0,\pi]$. Put $\theta_\kappa=2\arcsin(\kappa/(2\sqrt2))$, so that $\delta(g,I)\le\kappa$ if and only if $\theta(g)\le\theta_\kappa$. For a field $U$, a site $x$ and $\mu\ne\nu$ let $F^x_{\mu\nu}=U_{x,\mu}U_{x+e_\mu, \nu}U_{x+e_\nu,\mu}^{-1}U_{x,\nu}^{-1}$ be the holonomy of the plaquette at $x$ in the plane $\{\mu,\nu\}$, based at $x$. \begin{lemma}\label{lem:sfcorner} For every field $U$ and every site $x$, \[ F^x_{01}F^x_{12}F^x_{20}=U_{x,0}\,H\,U_{x,0}^{-1}, \] where $H=U_{x+e_0,1}U_{x+e_1,0}^{-1}U_{x+e_1,2}U_{x+e_2,1}^{-1}U_{x+e_2,0} U_{x+e_0,2}^{-1}$ is the holonomy of the hexagon of the unit cube at $x$ in the directions $0,1,2$ that avoids $x$ and the opposite corner. If $\delta(F^x_{\mu\nu},I)\le\kappa$ for the three plaquettes, then $\theta (H)\le3\theta_\kappa$ and $\theta(F^x_{01}),\theta(F^x_{12})\ge\theta(H)- 2\theta_\kappa$. \end{lemma} \begin{proof} In the product the factors $U_{x,1}^{-1}U_{x,1}$ and $U_{x,2}^{-1}U_{x,2}$ cancel, which leaves $U_{x,0}HU_{x,0}^{-1}$. Hence $\theta(H)=\theta(F^x_ {01}F^x_{12}F^x_{20})$, which is at most the sum of the three angles and at most $\theta(F^x_{01})+2\theta_\kappa$; the same holds for $F^x_{12}$. \end{proof} In the abelian case the identity is $F_1-F_2+F_3=R$ of the proof of Theorem~\ref{thm:sfshelllin}, $R$ being the circulation of the hexagon. The faces at $x$ enter only through their angles, so the commutators that separate $\SU(2)$ from the linear model do not weaken the constraint, and its extremal case $\theta(H)=3\theta_\kappa$ is attained by abelian fields with three aligned faces. \begin{theorem}\label{thm:sfshellsu2} Let $G=\SU(2)$, $0<\kappa\le10^{-8}$, $\lambda>0$, $c=I$ and $X=\{y\}$. There is $c_0>0$ such that for every $00$ and every $\kappa\le10^{-8}$. \end{theorem} \begin{proof} \emph{Setting.} Let $x=2y+(1,1,1,0)$ and $l_\mu=(x,\mu)$ for $\mu=0,1,2$. As in the proof of Theorem~\ref{thm:sfshelllin}, the edge $(x+\varepsilon,\mu)$ of the cube at $x$, $\varepsilon\in\{0,1\}^3\times\{0\}$, $\varepsilon_\mu =0$, is a private link of the pair $(y+\varepsilon,\mu)$. So $l_\mu$ is a private link of $(y,\mu)\in\mathcal P_X$, and not its path link $(2y+e_\mu,\mu)$; the six links $(x+e_\nu,\mu)$ of $H$ are private to pairs whose two blocks lie off $X$, so they are not in $\mathcal V_X$, and $H$ is a function $H(o)$ of the outer variables $o$, the links off $\mathcal V_X$. The three plaquettes at $x$ are owned by $y$, and $F^x_{01}=l_0Al_1^{-1}$, $F^x_{12}=l_1Bl_2^{-1}$ with $A$, $B$ products of outer links. Here $X=\{y\}$ has $|\mathcal P_X|=8$, $|\hat X|=11$, and $L'$, the links of $\mathcal V_X$ other than the eight path links, has $73$ links. On the geodesically convex ball $N_0=\{v:\delta(v,I)\le \frac1{100}\}$ the derivative of $\log$ at $e^X$ is $\Phi(\pm\operatorname {ad}_X)$ in the trivialisations of the proof of Lemma~\ref{lem:smallframes}(iv), of norm at most $1+2\lVert X\rVert\le 1.021$, since $\lVert X\rVert\le\frac\pi{300}$ by Lemma~\ref{lem:logbound}; so $\lVert\log v-\log v'\rVert\le1.021\,d(v,v')$ on $N_0$. \emph{Step 1: the calm completions.} Let $o$ have $\varPhi_X(o)>0$ and $S_X(o)>0$, and put $s(o)=3\theta_\kappa-\theta(H(o))$. We show $s(o)\ge0$ and \[ S_X(o)\le C_1\theta_\kappa^4s(o)^2,\qquad C_1=\frac{4\cdot9^{24}}{\pi^2}. \] There is a completion $U''$ at which every block of $\hat X$ is calm, and the blocks off $\hat X$ are calm because $\varPhi_X(o)>0$; so every plaquette is at most $\kappa$, and $s(o)\ge0$ by Lemma~\ref{lem:sfcorner}. As in the proof of Theorem~\ref{thm:sfperp}, Lemma~\ref{lem:smallframes}(i),~(iv) with $c=I$ put every link of $U''$ within $\varDelta=(21+8\pi)\kappa<46.14\kappa$ of $I$, every path link $v$ within $d(v,I)\le8\pi\kappa$, and so, by Lemma~\ref{lem:logbound}, every link within $d\le\varDelta_1=50\kappa$ of $I$. These bounds hold for the outer links, which all completions share. Let $D_*$ be the set of $z\in G^{L'}$ with $d(z_e,I)\le\varDelta_1$ for every $e$, and $B_*$ the ball $d(\ell,I)\le R_*=45\varDelta_1$. For $z\in D_*$, a pair $k$ and $\ell\in B_*$ on its path link, one site of $k$ equals $I$, one is a single link of $L'$ or an outer link, and the other fourteen are $\ell^{-1}$ times a product of two such links; so all lie within $47 \varDelta_1<\frac1{100}$ of $I$, the cut-off equals one, and $V^\sharp_k=\ell e^{N_k(\ell)}$ with $N_k$ the mean of the logarithms. The map $\Psi(\ell)=e^{-N_k(\ell)}$ satisfies $d(\Psi(\ell),I)\le\lVert N_k \rVert\le\frac{1.021}{16}(14\cdot47+1)\varDelta_1<42.1\varDelta_11$, and the blocks of $\hat X\setminus X$ are calm, $\frac67+0.064<1$; with $f_w\ge e^{-768\lambda}$ for each of the eleven blocks of $\hat X$, \[ K_X(o)\ge k_0=\Bigl(\frac{\rho_0^3}{14}\Bigr)^{73}e^{-8448\lambda}. \] \emph{Step 3: positive probability.} Let $\rho_1=t\theta_\kappa/3000$. The construction in the proof of Theorem~\ref{thm:lfaction}(ii), with $\rho_1$ in place of the $\rho$ chosen there and the regular field $U_t\in F_I$, gives a set $\varGamma_t\subset F_I$ with $\nu_I(\varGamma_t)>0$ on which every link lies within $d\le44\rho_1$ of its value at $U_t$ and every plaquette within $90\rho_1$. On $\varGamma_t$ every plaquette has angle at most $(1-t+0.022t) \theta_\kappa<\theta_\kappa$, so every block is calm and $\varPhi_X>0$ at the outer variables $o$ of the field; these satisfy $d(o_e,(o_t)_e)\le44 \rho_1\le\rho_0$, as $t\le\frac1{16}$; and $\theta(H(o))\ge3(1-t) \theta_\kappa-6\cdot44\rho_1/\sqrt2$, so $s(o)\le3.07\,t\,\theta_\kappa$. By Steps 1 and 2, wherever $S_X(o)>0$, \[ \rho_X=\frac{K_X(o)}{S_X(o)}\ge\frac{k_0}{C_1\theta_\kappa^4(3.07t\theta_ \kappa)^2}\ge\frac{c_0}{t^2},\qquad c_0=\frac{k_0\pi^2}{37.7\cdot9^{24} \theta_\kappa^6}. \] The integrand $\prod_yf_ys_y$ of $\mu^\circ_I$ is positive on $\varGamma_t$, so $\mu^\circ_I(\varGamma_t)>0$, and by the proof of Proposition~\ref{prop:sfrelative}(i) the calm law gives no mass to $\{S_X=0\}$. So $\mu^\circ_I\{\rho_X\ge c_0t^{-2}\}\ge\mu^\circ_I( \varGamma_t)>0$. \end{proof} \begin{remark}\label{rem:sfshellsu2} \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item The obstruction lies at the trivial coarse field and holds at every $\lambda$; the boundary values that cause it have every plaquette calm, the three plaquettes at $x$ in the shell $(\kappa/30,\kappa]$ and close to $\kappa$, and the hexagon close to its largest possible angle $3\theta_\kappa$. They squeeze the two faces $F^x_{01}$, $F^x_{12}$ of every calm completion into a shell of angular width $s(o)$, while the rough completions keep a mass bounded below. \item Since $\rho_X\ge0$ and the calm law of the outer variables has density proportional to $\varPhi_XS_X$, the mean of a product of relative activities is $\E_{\mu^\circ_c}[\prod_i\rho_{X_i}]\le Z_X(c)/Z_\emptyset(c)= \mu_c\{R=X\}/\mu_c\{R=\emptyset\}$, with equality when $Z_\perp$ receives nothing from $X$. A bound in the mean is therefore a bound on the constrained law at a fixed coarse field, which Theorem~\ref{thm:lfsuppression} gives only absolutely or under its hypothesis. These means do not factor over the components of $X$, so they are not activities to which Theorem~\ref{thm:lfkp} applies. In the Gaussian layer the mean bound of Corollary~\ref{cor:sfglshell} rests on the Gaussian correlation inequality, for which no $\SU(2)$ form is proved here. \item Lemma~\ref{lem:sfcorner} holds for every condition that confines the three plaquettes at $x$ to an angle $\theta_1$: the completions vanish when $\theta(H)>3\theta_1$ and are confined to the shell $\theta\ge\theta(H)-2\theta_1$ otherwise. Moving the threshold to $\kappa''\le10^{-8}$, as in Remark~\ref{rem:sfshell}, reproduces the configuration of the proof with $\kappa''$ in place of $\kappa$. For $\kappa>10^{-8}$ Step~1 is not proved, since Lemma~\ref{lem:smallframes} needs $\kappa\le10^{-8}$. \end{enumerate} \end{remark} \emph{The remainder $Z_\perp$.} Let $\mu_c=Z(c)^{-1}e^{-\lambda Q}\nu_c$ be the constrained law on $F_c$, and let $\mathbb P$ be the law of $c$ under the Wilson measure $\mu_L$, with density $Z(c)/Z_L$ with respect to $h$. \begin{proposition}\label{prop:sfperp} Let $G=\SU(2)$, $\kappa>0$ and $Z_\emptyset(c)>0$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The following hold: \[ \frac{Z(c)}{Z_\emptyset(c)}=\varXi(c)+\frac{Z_\perp(c)}{Z_\emptyset(c)}, \qquad\frac{Z_\emptyset(c)}{Z(c)}=\mu_c\{R=\emptyset\}, \] and $0\le Z_\perp(c)\le Z(c)-Z_\emptyset(c)=Z(c)\,\mu_c\{R\ne\emptyset\}$. \item For $\beta_W\ge1$ and $\kappa\le1$, $\int Z_\perp/Z\,d\mathbb P\le \mu_L\{R\ne\emptyset\}\le16|T|q_\kappa(\beta_W)$. \end{enumerate} \end{proposition} \begin{proof} (i) The first identity is Proposition~\ref{prop:sfrelative}(i) divided by $Z_\emptyset$, and $Z_\emptyset=\nu_c(e^{-\lambda Q}\mathbf 1\{R=\emptyset \})$. By Theorem~\ref{thm:lfmarkov}(i), $Z=Z_\emptyset+\sum_{X\ne\emptyset} Z_X$, and $Z_\perp$ is a sum of parts of the nonnegative $Z_X$, $X\ne \emptyset$. (ii) $\int\mu_c\{R\ne\emptyset\}\,d\mathbb P=\mu_L\{R\ne \emptyset\}$ since $h=\int\nu_c\,dh(c)$, and Theorem~\ref{thm:lfsuppression} (iii) with $X=\{y\}$ bounds each $\mu_L\{y\in R\}$. \end{proof} The next theorem shows that $Z_\perp$ is not zero: there are boundary values that admit a rough completion and no calm one, with positive measure. For a unit quaternion $g=a+b\cdot i\sigma$ write $b(g)\in\R^3$ for its vector part; then $\delta(g,I)^2=4(1-a)$ and $|b(g)|\le\delta(g,I)/\sqrt2$. \begin{lemma}\label{lem:sfquat} Let $g_1,\dots,g_4\in\SU(2)$ with $\delta(g_i,I)\le\varDelta\le1$. Then \begin{multline*} \bigl\lvert b(g_1g_2g_3^{-1}g_4^{-1})-b(g_1)-b(g_2)+b(g_3)+b(g_4)\bigr\rvert\\ \le3\varDelta^2+3.54\varDelta^3 . \end{multline*} \end{lemma} \begin{proof} $b(g^{-1})=-b(g)$, and $b(gh)=a_gb(h)+a_hb(g)-b(g)\times b(h)$. Let $P_j$ be the product of the first $j$ factors and $\varepsilon_j$ the error after $j$ factors. Then $\delta(P_j,I)\le j\varDelta$, so $|b(P_j)|\le j\varDelta/ \sqrt2$ and $1-a(P_j)\le j^2\varDelta^2/4$, and with $g$ the next factor, \begin{align*} \varepsilon_{j+1}&\le\varepsilon_j+(1-a_g)|b(P_j)|+(1-a(P_j))|b(g)|\\ &\quad+|b(P_j)||b(g)|\\ &\le\varepsilon_j+\frac{(j+j^2)\varDelta^3}{4\sqrt2}+\frac{j\varDelta^2}2 . \end{align*} Summing over $j=1,2,3$ with $\varepsilon_1=0$ gives $3\varDelta^2+ \frac{20}{4\sqrt2}\varDelta^3$. \end{proof} \begin{theorem}\label{thm:sfperp} Let $G=\SU(2)$, $0<\kappa\le10^{-8}$ and $\lambda>0$. Then $Z_\perp(I)>0$: for $X=\{y\}$ the outer variables at which $S_X=00$ have positive $\nu^{\rm out}_{I,X}$-measure. In the linear model the same holds for every $\kappa>0$ and $\lambda>0$. \end{theorem} \begin{proof} Let $x=2y+(1,1,1,1)$, and let the cubes $\mathcal A$ and $\mathcal B$ at $x$ be those in the directions $0,1,2$ and $0,1,3$. Each has three faces at $x$, owned by $y$, and three opposite faces, owned by $y+e_2$, $y+e_1$, $y+e_0$ and by $y+e_3$, $y+e_1$, $y+e_0$, none in $\hat X$, which contain no link of $\mathcal V_X$. The face $p_0$ at $x$ in the plane $\{0,1\}$ belongs to both. \emph{The field $\varphi_2$.} Let $\sigma=\frac{11}{20}$, $\tau=\frac14$, and let $C=x+\{0,1\}^4$. As for $\varphi_1$, the edge $(x+\varepsilon,\mu)$ of $C$ is a private link of the pair $(y+\varepsilon,\mu)$, and distinct edges lie in distinct pairs. Let $\varphi_2$ take the value $\sigma$ on $(x+e_2,0)$, $(x+e_1,2)$, $(x+e_3,1)$, $(x+e_0,3)$; $-\sigma$ on $(x+e_2,1)$, $(x+e_0,2)$, $(x+e_3,0)$, $(x+e_1,3)$; $\mp\frac23\sigma$ on $(x+e_0+e_2,1)$, $(x+e_0+e_3,1)$; $\mp\frac13\sigma$ on $(x+e_0+e_1,2)$, $(x+e_0+e_1,3)$; $\tau$ on $(x+e_1+e_3,2)$, $(x+e_0+e_2,3)$; $-\tau$ on $(x+e_1+e_2,3)$, $(x+e_0+e_3,2)$; minus one seventh of that value on the other seven private links of the pair of each of these sixteen edges; and zero elsewhere. The exchange of the coordinates $2$ and $3$ about $x$ maps $\varphi_2$ to $-\varphi_2$. As for $\varphi_1$, $V(\varphi_2)=0$, $\varphi_2$ vanishes on $\mathcal V_X$, since its pairs are those of blocks $y+\varepsilon$ with $\varepsilon\ne0$ and $\varepsilon\ge0$, and on a $2$-face of $C$ it is given by the sixteen values, the other edges of $C$ carrying zero. Write $[a; \mu\nu]$ for the plaquette at $x+a$ in the plane $\{\mu,\nu\}$. The $2$-faces of $C$ not at $x$ are the eighteen $[e_a;\mu\nu]$ with $a\notin\{\mu,\nu\}$ and $[e_a+e_b;\mu\nu]$ with $\{a,b,\mu,\nu\}=\{0,1,2,3\}$, owned by $y+e_a$ and $y+e_a+e_b$. Adding the values on their edges gives \begin{center} \begin{tabular}{llllll} \hline $[e_2;01]$ & $\frac43\sigma$ & $[e_3;01]$ & $-\frac43\sigma$ & $[e_1;23]$ & $2\sigma-2\tau$\\ $[e_1;02]$ & $-\frac43\sigma$ & $[e_1;03]$ & $\frac43\sigma$ & $[e_0;23]$ & $2\tau-2\sigma$\\ $[e_0;12]$ & $\frac43\sigma$ & $[e_0;13]$ & $-\frac43\sigma$ & $[e_0+e_1;23]$ & $-\frac23\sigma$\\ $[e_2;03]$ & $\sigma+\tau$ & $[e_3;02]$ & $-\sigma-\tau$ & $[e_0+e_3;12]$ & $\frac23\sigma+\tau$\\ $[e_2;13]$ & $-\sigma-\tau$ & $[e_3;12]$ & $\sigma+\tau$ & $[e_0+e_2;13]$ & $-\frac23\sigma-\tau$\\ \hline \end{tabular} \end{center} and the faces $[e_1+e_2;03]$, $[e_1+e_3;02]$, $[e_2+e_3;01]$ contain one of the sixteen edges or none. A plaquette that is not a $2$-face of $C$ contains at most one edge of $C$, so at most one link with $|\varphi_2|> \frac\sigma7$, and has $|d\varphi_2|\le\frac{10}7\sigma=\frac{11}{14}$. Every entry of the table has modulus at most $\sigma+\tau$, so every plaquette not owned by $y$ has $|d\varphi_2|\le m=\sigma+\tau=\frac45$. The first three rows of the first two columns are the outer faces of $\mathcal A$ and $\mathcal B$, so, with the signs of the Bianchi identity as in the proof of Theorem~\ref{thm:sfshelllin}, $R_{\mathcal A}=4\sigma=\frac{11}5$ and $R_{\mathcal B}=-\frac{11}5$. \emph{Linear model.} As in the proof of Theorem~\ref{thm:sfshelllin}, with $\kappa=1$, a calm completion would give, from $\mathcal A$, $(d\varphi)_{ p_0}\ge R_{\mathcal A}-2=\frac15$ and, from $\mathcal B$, $(d\varphi)_{p_0}\le2+R_{\mathcal B}=-\frac15$. These strict inequalities, and $|d\varphi|<1$ off $\Pi(y)$, hold on a neighbourhood of $\varphi_2$, on which therefore the blocks off $y$ are calm, $S^{\rm lin}_X=0$, and every completion near the inner part of $\varphi_2$ has $y$ rough, so $K^{\rm lin}_X>0$. \emph{$\SU(2)$.} Let $U=\exp(\kappa\varphi_2\tau_3)$ on every active link. It is abelian, its plaquettes are $\exp(\kappa(d\varphi_2)_p\tau_3)$ with $\delta\le\kappa|(d\varphi_2)_p|$, and its sites, products of three links with $|\varphi_2|\le\sigma$, lie within $3\sigma\kappa<2\kappa$ of $I$, so $U$ is regular, the cut-offs equal one, and $V^\sharp(U)=\exp(\kappa V( \varphi_2)\tau_3)=I$. The construction in the proof of Theorem~\ref{thm:lfaction}(ii), with $\rho=\kappa/10^4$ in place of the $\rho$ chosen there, gives a set $\varGamma\subset F_I$ with $\nu_I(\varGamma)>0$ on which every plaquette satisfies $\delta(U'_p,U_p)\le90\rho$, by bi-invariance and $\delta\le d$. On $\varGamma$ every plaquette not owned by $y$ is at most $(m+0.009)\kappa<\kappa$. Fix $U'\in\varGamma$ and suppose that its outer variables admit a calm completion $U''$. Then every plaquette of $U''$ is at most $\kappa$, and by Lemma~\ref{lem:smallframes}(i) and~(iv), with $c=I$, every inside link is within $3\kappa$ of $I$, every path link $v$ satisfies $v=\exp(-\frac1{16} \sum_x\log(v^{-1}P_x))$ with $\delta(v^{-1}P_x,I)\le24\kappa$, so $\delta(v,I)\le8\pi\kappa$ by Lemma~\ref{lem:logbound}, and every other link leaving a block is within $(21+8\pi)\kappa=:\varDelta$ of $I$. The linear parts of the six faces of a cube cancel as in the linear model, so by Lemma~\ref{lem:sfquat}, for the third coordinates $b_3$ of the vector parts, \[ \bigl|b_3(U''_{p_0})-b_3(U''_{p_1})+b_3(U''_{p_2})-R^b_{\mathcal A}\bigr| \le6\varepsilon_4, \] where $\varepsilon_4=3\varDelta^2+3.54\varDelta^3$, with $p_1$, $p_2$ the other faces of $\mathcal A$ at $x$ and $R^b_{\mathcal A}$ the signed sum of $b_3$ over its outer faces, which are those of $U'$; likewise for $\mathcal B$. Since $|b_3|\le\kappa/\sqrt2$ on calm faces, \[ R^b_{\mathcal A}-\sqrt2\kappa-6\varepsilon_4\le b_3(U''_{p_0})\le R^b_{\mathcal B}+\sqrt2\kappa+6\varepsilon_4 . \] At $U$ the outer faces have $b_3=\sin(\kappa F/\sqrt2)$ with $F=\pm\frac43 \sigma=\pm\frac{11}{15}$ as in the table, so $R^b_{\mathcal A}=-R^b_{\mathcal B}=3\sin\frac{11\kappa}{15\sqrt2}$, and as $\sin\theta\ge\theta(1-\theta^2/6)$, \[ R^b_{\mathcal A}-R^b_{\mathcal B}\ge\frac{22}5\frac\kappa{\sqrt2}\Bigl(1- \frac{\kappa^2}6\Bigr)>3.1111\,\kappa . \] On $\varGamma$ each $b_3$ moves by at most $90\rho/\sqrt2$, since $\lVert g-g' \rVert\ge\sqrt2|b(g)-b(g')|$, so $R^b_{\mathcal A}-R^b_{\mathcal B}$ moves by at most $540\rho/\sqrt2<0.0382\,\kappa$. As $\pi<3.1416$, $\varDelta<46.14\, \kappa$, so $\varDelta^2<2128.9\,\kappa^2$, $\varDelta^3<98228\,\kappa^3$ and, for $\kappa\le10^{-8}$, $\varepsilon_4<6386.7\,\kappa^2+0.0035\,\kappa^2< 6387\,\kappa^2$ and $12\varepsilon_4<76644\,\kappa^2<0.00077\,\kappa$. With $2\sqrt2<2.8285$, \begin{align*} R^b_{\mathcal A}-R^b_{\mathcal B}-2\sqrt2\kappa-12\varepsilon_4&>(3.1111- 0.0382)\kappa\\ &\quad-(2.8285+0.00077)\kappa>0.24\,\kappa \end{align*} at the outer variables of $U'$, which contradicts the previous display. So $S_X=0$ at the outer variables of every $U'\in\varGamma$; since $U'$ is a completion of them whose blocks off $y$ are calm, $y$ is rough at $U'$. By the factorisation of Theorem~\ref{thm:lfmarkov}(i) and Fubini, the part of $Z_\perp(I)$ from $X=\{y\}$ is at least $\int_\varGamma e^{-\lambda Q}\,d\nu_I>0$. \end{proof} So $Z_\perp$ is a genuine part of the weight, made of boundary values that force a rough block, and it is not a function of the calm law. It is suppressed like a rough block, in the mean by Proposition~\ref{prop:sfperp}(ii) and pointwise in the outer variables by the bound of Theorem~\ref{thm:lfmarkov}(ii) on $K_X$; a bound on $Z_\perp(c) /Z_\emptyset(c)$ uniform in $c$ is not proved. The small-field part therefore stands as follows. The representation of the calm weight as $e^{-\lambda q^{\rm lin}(\log c)}$ times local factors and a polymer sum fails at every $\lambda>0.5472$ (Theorem~\ref{thm:sfgaugeobst}), and, if $(M_W)_{AB}\ne0$, as a product of local factors at large $\lambda$ (Theorem~\ref{thm:sfgauss}); the leading term must be gauge invariant and, under the same condition, nonlocal, and its linear propagator decays at the explicit rate of Theorem~\ref{thm:sfprop}. In the linear model the relative activities are unbounded under the calm law at every $\lambda$ (Theorem~\ref{thm:sfshelllin}), and for $\SU(2)$ at every $\lambda$ and $\kappa\le10^{-8}$ (Theorem~\ref{thm:sfshellsu2}), so the almost sure hypothesis of Proposition~\ref{prop:sfrelative}(ii) fails in both, while in the linear model their mean is at most $e^{-9.2|X|}$ for $\lambda\kappa^2 \ge16$, and the Gaussian layer of the step reproduces itself at every scale with the same constants (Theorem~\ref{thm:sfglstep}, Corollaries~\ref{cor:sfglscales} and~\ref{cor:sfglshell} below); and $Z_\perp(I)>0$ for $\SU(2)$ (Theorem~\ref{thm:sfperp}). Not proved are a local expansion around a gauge-invariant leading term, a bound in the mean on the relative activities in the shell for $\SU(2)$, and a bound on $Z_\perp/Z_\emptyset$ uniform in $c$. \paragraph*{The Gaussian layer of the step.} The linear model of Theorem~\ref{thm:sfshelllin} is the Gaussian layer of the step. This paragraph expands it around its Gaussian form at every scale. The thresholds are placed on the fluctuation about the minimiser of the form at given averages, the background, and not on the field. With this choice the large-field remainder does not depend on the coarse field, its relative weights are bounded by $e^{-9.2}$ per block for $\lambda\kappa_{\rm f}^2\ge 16$, uniformly in the coarse field, the volume and the scale, and the only hypothesis on the form, that it dominates the plaquette form, is reproduced by the step. The conditioning on boundary values, which defeats the almost sure bound in Theorem~\ref{thm:sfshelllin}, is replaced by Royen's Gaussian correlation inequality, used in the mean over the boundary values. The $\SU(2)$ step is not treated, and the nonlocality of the Gaussian form is not decided here; Remark~\ref{rem:sfglsu2} states what the $\SU(2)$ step needs beyond this paragraph. Let $\Lambda$ be a torus of even side at least four, with the blocks $B(y)$ of Definition~\ref{def:blockgauge}, coarse torus $\Lambda'$, tree links and active links as there, and $\Pi(y)$ the $96$ plaquettes owned by the block $y$ (Lemma~\ref{lem:lfowned}). Let $\mathcal A(\Lambda)$ be the space of real fields on the active links, $Q_\Lambda(\varphi)=\sum_p (d\varphi)_p^2$ over the plaquettes of $\Lambda$, and $V:\mathcal A(\Lambda) \to\R^{\Lambda'\times\{0,\dots,3\}}$ the block averages. For a real field $\gamma$ on the coarse links put $\mathcal V_\gamma=\{\varphi\in\mathcal A (\Lambda):V(\varphi)=\gamma\}$, with Lebesgue measure $m_\gamma$ in the coordinates given by the values on the links other than the path links, as in the linear model. Since the path link of a pair enters its average alone, with coefficient $\frac18$, and enters no other average, $\mathcal V_\gamma$ is not empty and $m_\gamma$ is the image of $m_0$ under $\psi\mapsto\varphi_0 +\psi$ for every $\varphi_0\in\mathcal V_\gamma$. The proofs of Theorem~\ref{thm:blockgap}, of Lemma~\ref{lem:blocktree} and of Proposition~\ref{prop:effective}(i),~(ii) use only the plaquettes and links of one block and of two neighbouring blocks, and their counts are the same on such a torus; so they hold on $\Lambda$, and on $\Lambda'$ when its side is even and at least four. \begin{definition}\label{def:sfglclass} $\mathcal G(\Lambda)$ is the set of positive semidefinite quadratic forms $q$ on $\mathcal A(\Lambda)$ with $q\ge Q_\Lambda$. For $q\in\mathcal G (\Lambda)$ and $\lambda>0$, $\mu_q$ is the probability measure on $\mathcal V_0$ with density $Z_q^{-1}e^{-\lambda q}$ with respect to $m_0$, where $Z_q=\int_{\mathcal V_0}e^{-\lambda q}\,dm_0$. \end{definition} The restriction of $Q_\Lambda$ to $\mathcal A(\Lambda)$ belongs to $\mathcal G(\Lambda)$. \begin{lemma}\label{lem:sfglsplit} Let $q\in\mathcal G(\Lambda)$ and $\lambda>0$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $Z_q<\infty$. For every $\gamma$ the form $q$ has a unique minimiser $\varphi^*_q(\gamma)$ on $\mathcal V_\gamma$, which is linear in $\gamma$, and with $q'(\gamma)=q(\varphi^*_q(\gamma))$, \[ q\bigl(\varphi^*_q(\gamma)+\psi\bigr)=q'(\gamma)+q(\psi)\qquad (\psi\in\mathcal V_0). \] \item $q'(\gamma)\ge Q_{\Lambda'}(\gamma)$ for every real field $\gamma$ on the coarse links. If the side of $\Lambda'$ is at least four, the restriction of $q'$ to $\mathcal A(\Lambda')$ belongs to $\mathcal G(\Lambda')$. If $q(\varphi+d\tilde h)=q(\varphi)$ for every function $\tilde h$ constant on the blocks, then $q'(\gamma+d'h)=q'(\gamma)$ for every function $h$ on the coarse sites, where $(d'h)_{y,\mu}=h_y-h_{y+e_\mu}$. \item For distinct plaquettes $p_1,\dots,p_n$ of $\Lambda$ the covariance matrix $\Sigma$ of $((d\psi)_{p_1},\dots,(d\psi)_{p_n})$ under $\mu_q$ satisfies $\Sigma\le(2\lambda)^{-1}I$. \end{enumerate} \end{lemma} \begin{proof} (i) By Theorem~\ref{thm:blockgap} with $W$ the set of all blocks, $q(\psi) \ge Q_\Lambda(\psi)\ge\frac1{52}\sum_e\psi_e^2$ on $\mathcal V_0$, so $q$ is positive definite there and $Z_q<\infty$. Fix $\varphi_0\in\mathcal V_\gamma$ depending linearly on $\gamma$, for instance the field of Proposition~\ref{prop:effective}(ii). The function $\psi\mapsto q(\varphi_0+ \psi)$ on $\mathcal V_0$ is a quadratic polynomial with positive definite leading part, so it has a unique minimiser, a linear function of $\varphi_0$. At the minimiser $\varphi^*$ the derivative vanishes: $q(\varphi^*,\psi)=0$ for the bilinear form of $q$ and every $\psi\in \mathcal V_0$, which gives the identity. (ii) By Proposition~\ref{prop:effective}(i), $q'(\gamma)=q(\varphi^*_q (\gamma))\ge Q_\Lambda(\varphi^*_q(\gamma))\ge Q_{\Lambda'}(V(\varphi^*_q (\gamma)))=Q_{\Lambda'}(\gamma)$, and $q'$ is positive semidefinite as a minimum of a positive semidefinite form over an affine family depending linearly on $\gamma$. For $h$ on the coarse sites let $\tilde h$ equal $h_y$ on $B(y)$. Then $d\tilde h$ vanishes on the links inside the blocks, so $d\tilde h\in\mathcal A(\Lambda)$, and $V(\varphi+d\tilde h)=V(\varphi)+ d'h$ by Lemma~\ref{lem:blocktree}(ii). So $\varphi\mapsto\varphi+d\tilde h$ maps $\mathcal V_\gamma$ onto $\mathcal V_{\gamma+d'h}$ and preserves $q$. (iii) Identify $\mathcal V_0$ with $\R^N$ by the coordinates of $m_0$ and write $q(\psi)=\psi^{\rm T}K\psi$ with $K$ positive definite. Then $\mu_q$ is the centred Gaussian law with covariance $(2\lambda K)^{-1}$, and for a linear function $\ell(\psi)=l^{\rm T}\psi$, by the Cauchy--Schwarz inequality for the inner product of $K$, \[ \Var_{\mu_q}(\ell)=\frac{l^{\rm T}K^{-1}l}{2\lambda}=\frac1{2\lambda} \sup_{\psi\ne0}\frac{\ell(\psi)^2}{q(\psi)} . \] For $a\in\R^n$ and $\ell(\psi)=\sum_ia_i(d\psi)_{p_i}$, $\ell(\psi)^2\le |a|^2\sum_i(d\psi)_{p_i}^2\le|a|^2Q_\Lambda(\psi)\le|a|^2q(\psi)$, as the plaquettes are distinct. So $a^{\rm T}\Sigma a\le|a|^2/(2\lambda)$. \end{proof} \begin{lemma}\label{lem:sfgltail} \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item If $Z$ is a centred real Gaussian variable with $\Var Z\le\sigma^2$, then $\Pr(|Z|>s)\le e^{-s^2/(2\sigma^2)}$ for every $s\ge0$. \item If $(Z_1,\dots,Z_n)$ is a centred Gaussian vector with covariance at most $\sigma^2I$ and $s^2\ge\sigma^2$, then \[ \Pr\Bigl(\sum_iZ_i^2>ns^2\Bigr)\le\Bigl(\frac{es^2}{\sigma^2}\Bigr)^{n/2} e^{-ns^2/(2\sigma^2)} . \] \end{enumerate} \end{lemma} \begin{proof} (i) It suffices to take $\Var Z=\sigma^2>0$ and $\sigma=1$, since $\Pr(|Z|>s)$ increases with the variance. Let $f(u)=e^{-u^2/2}- \sqrt{2/\pi}\int_u^\infty e^{-v^2/2}dv$. Then $f(0)=0$, $f(u)\to0$ as $u\to\infty$, and $f'(u)=e^{-u^2/2}(\sqrt{2/\pi}-u)$, so $f$ increases and then decreases, and $f\ge0$ on $[0,\infty)$. (ii) Write the covariance as $ODO^{\rm T}$ with $O$ orthogonal and $D$ diagonal with entries $d_i\le \sigma^2$. Then $\sum_iZ_i^2$ has the law of $\sum_id_ig_i^2$ with independent standard Gaussian $g_i$, and for $0\le\theta<1/(2\sigma^2)$, $\E e^{\theta\sum_id_ig_i^2}=\prod_i(1-2\theta d_i)^{-1/2}\le(1-2\theta \sigma^2)^{-n/2}$. By Markov's inequality the probability is at most $(1-2\theta\sigma^2)^{-n/2}e^{-\theta ns^2}$, and $\theta=(1-\sigma^2/s^2)/ (2\sigma^2)$ gives the bound. \end{proof} Royen's Gaussian correlation inequality, in the form proved in~\cite{LM17}, states that a centred Gaussian measure $\mu$ on $\R^N$ has $\mu(K\cap K')\ge\mu(K)\mu(K')$ for any two closed convex sets $K$, $K'$ symmetric about the origin. Taking complements, $\mu(K'\setminus K)\le \mu(K')\bigl(1-\mu(K)\bigr)$. For a slab $K$ and $K'$ an intersection of slabs this is the inequality of \v{S}id\'ak. \emph{Thresholds on the fluctuation.} Let $q\in\mathcal G(\Lambda)$, $\lambda>0$ and $\kappa_{\rm f}>0$. For $\varphi\in\mathcal V_\gamma$ let $\psi=\varphi-\varphi^*_q(\gamma)\in\mathcal V_0$ be its fluctuation. A block $y$ of $\Lambda'$ is \emph{calm relative to the background} if $|(d\psi)_p| \le\kappa_{\rm f}$ for every $p\in\Pi(y)$, and \emph{rough relative to the background} otherwise; $R^{\rm f}(\varphi)$ is the set of these rough blocks, and for $\psi\in\mathcal V_0$, $R^{\rm f}(\psi)$ is taken at $\gamma=0$. For $X\subset\Lambda'$ put \[ Z_X(\gamma)=\int_{\mathcal V_\gamma}e^{-\lambda q}\,\mathbf 1\{R^{\rm f}=X\} \,dm_\gamma,\qquad\pi_X=\mu_q\{R^{\rm f}=X\}, \] and, with $t=\lambda\kappa_{\rm f}^2$, \[ \epsilon_1(t)=96\sqrt{2et}\,e^{-t},\qquad\epsilon(t)=\frac{\epsilon_1(t)} {(1-e^{-t})^{96}} . \] \begin{theorem}\label{thm:sfglstep} Let $q\in\mathcal G(\Lambda)$, $\lambda>0$, $\kappa_{\rm f}>0$ and $t= \lambda\kappa_{\rm f}^2\ge\frac12$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For every coarse field $\gamma$ and every $X\subset\Lambda'$, \[ Z_X(\gamma)=Z_q\,\pi_X\,e^{-\lambda q'(\gamma)}, \] and $\sum_X\pi_X=1$, so that $\int_{\mathcal V_\gamma}e^{-\lambda q}\, dm_\gamma=\sum_XZ_X(\gamma)=Z_qe^{-\lambda q'(\gamma)}$. \item For every set $Y$ of blocks, $\mu_q\{Y\cap R^{\rm f}=\emptyset\}\ge (1-e^{-t})^{96|Y|}$; in particular $\pi_\emptyset\ge(1-e^{-t})^{96| \Lambda'|}$. \item For disjoint sets $X$, $Y$ of blocks, \[ \mu_q\{X\subset R^{\rm f},\ Y\cap R^{\rm f}=\emptyset\}\le\epsilon_1(t)^{|X|} \,\mu_q\{Y\cap R^{\rm f}=\emptyset\}. \] \item For every $X\subset\Lambda'$, $\pi_X\le\epsilon(t)^{|X|}\pi_\emptyset$. \item $\epsilon$ decreases on $[\frac12,\infty)$, and $\epsilon(t)\kappa_{\rm f}$; the $p_y$ are distinct, and $\sum_{y\in X} (d\psi)_{p_y}^2>|X|\kappa_{\rm f}^2$. So $\{X\subset R^{\rm f}\}$ is contained in the union, over the at most $96^{|X|}$ choices of $(p_y)_{y\in X}$ with $p_y\in\Pi(y)$, of the events $E=\{\sum_y(d\psi)_{p_y}^2>|X| \kappa_{\rm f}^2\}$. The complement of $E$ and $C_Y=\{Y\cap R^{\rm f}= \emptyset\}$ are closed, convex and symmetric, so $\mu_q(E\cap C_Y)\le \mu_q(E)\mu_q(C_Y)$ by the correlation inequality. By Lemma~\ref{lem:sfglsplit}(iii) the covariance of $((d\psi)_{p_y})_{y\in X}$ is at most $(2\lambda)^{-1}I$, and Lemma~\ref{lem:sfgltail}(ii) with $\sigma^2=(2\lambda)^{-1}$, $s=\kappa_{\rm f}$, $s^2/\sigma^2=2t\ge1$ and $n=|X|$ gives $\mu_q(E)\le(2et)^{|X|/2}e^{-t|X|}$. (iv) Let $Y=\Lambda'\setminus X$. By (iii), $\pi_X\le\mu_q\{X\subset R^{\rm f},\ Y\cap R^{\rm f}=\emptyset\}\le\epsilon_1^{|X|}\mu_q(C_Y)$. By the correlation inequality and (ii), $\pi_\emptyset=\mu_q(C_X\cap C_Y)\ge \mu_q(C_X)\mu_q(C_Y)\ge(1-e^{-t})^{96|X|}\mu_q(C_Y)$, and $\mu_q(C_Y)>0$. (v) The derivative of $\log\epsilon(t)$ is $\frac1{2t}-1-\frac{96e^{-t}} {1-e^{-t}}<0$ for $t>\frac12$. With $\log96=5\log2+\log3$ and the bounds on $\log2$, $\log3$ of the proof of Theorem~\ref{thm:lfkp}(ii), \[ \log\epsilon_1(16)=\tfrac{15}2\log2+\log3-15.5<-9.20278 . \] As $e>2.7$, $e^{16}>2.7^{16}>7.9\cdot10^6$, and $-\log(1-x)\le x/(1-x)$ gives $-96\log(1-e^{-16})<10^{-4}$. So $\log\epsilon(16)<-9.2$. \end{proof} The large-field remainder of the Gaussian layer is therefore exact: the weight of a fibre is the Gaussian factor $e^{-\lambda q'(\gamma)}$ times a constant, the calm part carries the factor $\pi_\emptyset$, and the parts with a rough region $X$ carry $\pi_X\le\epsilon(t)^{|X|}\pi_\emptyset$, none of which depends on $\gamma$. The weights $\pi_X$ need not factorise over the components of $X$, and no polymer representation of them is needed, since they do not depend on the coarse field. For $t\ge16$ they satisfy the bound $e^{-9|X|}$ that Theorem~\ref{thm:lfkp} asks of activities, and $-\log\pi_\emptyset\le96|\Lambda'|e^{-t}/(1-e^{-t})$. Part (iii) is the analogue of Theorem~\ref{thm:lfsuppression}(ii) without its hypothesis on the exterior: the exterior is not fixed but integrated, under the condition that it is calm. \begin{corollary}\label{cor:sfglscales} Let $\Lambda_0$ be a torus of side $2^m$, $m\ge2$, let $\Lambda_k$ be its $k$-th coarse torus, $q_0=Q_{\Lambda_0}$, and for $k\le m-3$ let $q_{k+1}$ be the restriction of $q_k'$ to $\mathcal A(\Lambda_{k+1})$. Then for $0\le k \le m-2$, $q_k\in\mathcal G(\Lambda_k)$, $q_k$ is invariant under the gauge transformations of $\Lambda_k$ that are constant on its blocks, and Theorem~\ref{thm:sfglstep} holds at the scale $k$ with the same $\lambda$, the same $\kappa_{\rm f}$ and the same $\epsilon(t)$. \end{corollary} \begin{proof} $Q_{\Lambda_0}$ is invariant under all gauge transformations. For $k\le m-3$ the side of $\Lambda_{k+1}$ is even and at least four, and Lemma~\ref{lem:sfglsplit}(ii) passes both properties from $q_k$ to $q_{k+1}$: the gauge transformations of $\Lambda_{k+1}$ constant on its blocks are of the form $d'h$. \end{proof} So the hypotheses of the Gaussian layer reproduce themselves exactly: the form at the next scale again dominates the plaquette form of its lattice, the coupling $\lambda$ does not change, and every constant of Theorem~\ref{thm:sfglstep} is the same at every scale. The restriction to $\mathcal A(\Lambda_{k+1})$ is the tree gauge of the coarse field. By the gauge invariance in Lemma~\ref{lem:sfglsplit}(ii), and since every coarse field is $\gamma_0+d'h$ with $\gamma_0\in\mathcal A(\Lambda_{k+1})$ by Lemma~\ref{lem:blocktree}(i) on $\Lambda_{k+1}$, the weight $e^{-\lambda q_k'}$ is determined by its values on $\mathcal A(\Lambda_{k+1})$, as the weight of $\SU(2)$ is by Lemma~\ref{lem:sfgaugeinv}. \emph{The shell in the mean.} At $\gamma=0$ the background vanishes and the thresholds relative to the background are the thresholds of the linear model. For $X\subset T$ with components $X_i$ let $K^{\rm lin}_{X_i}$ and $S^{\rm lin}_{X_i}$ be the integrals of Theorem~\ref{thm:lfmarkov}(i) and of the calm weight $S_{X_i}$, formed in the linear model at $\gamma=0$ as $K^{\rm lin}_X$ and $S^{\rm lin}_X$ are for $X=\{y\}$, and $\rho^{\rm lin} _{X_i}=K^{\rm lin}_{X_i}/S^{\rm lin}_{X_i}$, defined almost surely under the calm law $\mu^\circ$ of the linear model. \begin{corollary}\label{cor:sfglshell} In the linear model on the torus $\Lambda_L$ with $\gamma=0$, let $\kappa>0$ and $\lambda\kappa^2\ge\frac12$. For every nonempty $X\subset T$ with components $X_i$, \[ \E_{\mu^\circ}\Bigl[\prod_i\rho^{\rm lin}_{X_i}\Bigr]\le\epsilon(\lambda \kappa^2)^{|X|}, \] and $\mu^\circ\{\prod_i\rho^{\rm lin}_{X_i}>R\}\le\epsilon(\lambda\kappa^2) ^{|X|}/R$ for $R>0$. For $\lambda\kappa^2\ge16$ the mean is at most $e^{-9.2|X|}$. \end{corollary} \begin{proof} Lemma~\ref{lem:sfkernel} and the factorisation of Theorem~\ref{thm:lfmarkov}(i) concern only which links a factor involves, so they hold in the linear model, and as in the proof of Proposition~\ref{prop:sfrelative}(i) the law of the outer variables of $X$ under $\mu^\circ$ has density $Z^{-1}_\emptyset\varPhi_X\prod_iS^{\rm lin} _{X_i}$ with respect to Lebesgue measure on them, where $Z_\emptyset=\int e^{-\lambda Q}\prod_ys_y\,dm_0$. Hence \[ \E_{\mu^\circ}\Bigl[\prod_i\rho^{\rm lin}_{X_i}\Bigr]\le Z_\emptyset^{-1} \int\varPhi_X\prod_iK^{\rm lin}_{X_i}, \] and \[ Z_\emptyset^{-1}\int\varPhi_X\prod_iK^{\rm lin}_{X_i}=\frac{\int e^{-\lambda Q}\,\mathbf 1\{R=X\}\,dm_0}{\int e^{-\lambda Q}\,\mathbf 1\{R= \emptyset\}\,dm_0}=\frac{\pi_X}{\pi_\emptyset}, \] the last for $q=Q_{\Lambda_L}$, $\gamma=0$ and $\kappa_{\rm f}=\kappa$, where the rough set $R$ of the linear model is $R^{\rm f}$. Apply Theorem~\ref{thm:sfglstep}(iv),~(v) and Markov's inequality. \end{proof} So in the linear model the relative activities are not essentially bounded under the calm law (Theorem~\ref{thm:sfshelllin}) but are bounded in the mean, with the explicit bound $\epsilon(\lambda\kappa^2)^{|X|}$, uniformly in the volume and for every $X$; the boundary values of Theorem~\ref{thm:sfshelllin}, at which the ratio is large, have calm-law probability at most $\epsilon^{|X|}/R$. This is the mean form asked for in Remark~\ref{rem:sfconditioning}. In the mean the regress of Remark~\ref{rem:sfshell} does not arise, because the outer variables are integrated and not fixed in the shell. \emph{The sharp threshold away from $\gamma=0$.} For $\gamma\ne0$ the threshold of the linear model is placed on $\varphi=\varphi^*_q(\gamma)+ \psi$. When the background is calm with a margin, the sharp calm weight lies within the factor $\pi_\emptyset$ of the Gaussian form. \begin{proposition}\label{prop:sfglbackground} Let $L\ge32$ and $q=Q_{\Lambda_L}$, and for a coarse field $\gamma$ let $\kappa_c(\gamma)$ be the largest $|(d'\gamma)_P|$ over the coarse plaquettes $P$. Then for every plaquette $p$ of $\Lambda_L$, \[ \bigl|(d\varphi^*_q(\gamma))_p\bigr|\le K_*\,\kappa_c(\gamma),\qquad K_*< 1.64\cdot10^{19}. \] \end{proposition} \begin{proof} In the notation of Theorem~\ref{thm:sfprop}, $\varphi^*_q(\gamma)=L_0\gamma- PH^{-1}B^{\rm T}\gamma$. By Proposition~\ref{prop:effective}(ii) every plaquette of $L_0\gamma$ is $0$ or a curvature $(d'\gamma)_P$, and since $\gamma^{\rm T}Bv=\sum_p(dL_0\gamma)_p(dPv)_p$, \[ (B^{\rm T}\gamma)_i=\sum_p(dL_0\gamma)_p(dPe_i)_p,\qquad|(B^{\rm T}\gamma)_i| \le12\sqrt2\,\kappa_c(\gamma), \] by the bound $\sum_p|(dPe_i)_p|\le6\cdot2\sqrt2$ of the proof of Theorem~\ref{thm:sfprop}(i). With $\sum_j|(dPe_j)_p|\le4p_1$ from the same proof and $S_H=\max_i\sum_j|(H^{-1})_{ij}|$, \[ \bigl|(d\varphi^*_q(\gamma))_p\bigr|\le\bigl(1+48\sqrt2\,p_1S_H\bigr)\kappa_c (\gamma)=\bigl(1+\bar SS_H\bigr)\kappa_c(\gamma). \] By Theorem~\ref{thm:sfprop}(ii) at $\alpha=\frac1{500}$, $|(H^{-1})_{ij}|\le 66.3\,e^{-\alpha d(i,j)}$. A block carries $17+28=45$ coordinates, and at most $(2d+1)^4-(2d-1)^4=64d^3+16d$ blocks lie at distance $d\ge1$ from a given one. With $x=e^{-\alpha}$, $\sum_{d\ge1}dx^d\le(1-x)^{-2}$ and $\sum_{d\ge1}d^3x^d=x(1+4x+x^2)(1-x)^{-4}\le6(1-x)^{-4}$, and $1-x\ge\alpha- \alpha^2/2$ gives $(1-x)^{-1}<500.51=:r$, so $r^2<250511$ and $r^4<6.2756 \cdot10^{10}$. Hence \begin{align*} S_H&\le66.3\cdot45\bigl(1+384r^4+16r^2\bigr)\\ &<66.3\cdot45\cdot2.4100\cdot10^{13}<7.20\cdot10^{16}, \end{align*} and $K_*=1+\bar SS_H<1+227.40\cdot7.20\cdot10^{16}<1.64\cdot10^{19}$. \end{proof} The constant is far from the truth. It comes from bounding $H^{-1}$ by the slow rate $\frac1{500}$ of Theorem~\ref{thm:sfprop}, and it is used only to state the margin condition below in terms of the coarse field. \begin{corollary}\label{cor:sfglsharp} Let $q\in\mathcal G(\Lambda)$, $0<\kappa_{\rm f}<\kappa$, $t=\lambda \kappa_{\rm f}^2\ge\frac12$, and let $\gamma$ be a coarse field whose background has $|(d\varphi^*_q(\gamma))_p|\le\kappa-\kappa_{\rm f}$ for every plaquette $p$; for $q=Q_{\Lambda_L}$ and $L\ge32$ this holds when $\kappa_c (\gamma)\le(\kappa-\kappa_{\rm f})/K_*$. Let $Z^{\rm lin}_\emptyset(\gamma)= \int_{\mathcal V_\gamma}e^{-\lambda q}\prod_ys_y\,dm_\gamma$, where $s_y$ is the indicator that every plaquette of $\Pi(y)$ has $|(d\varphi)_p|\le \kappa$. Then \[ 0\le-\log Z^{\rm lin}_\emptyset(\gamma)-\lambda q'(\gamma)+\log Z_q\le -\log\pi_\emptyset\le\frac{96|\Lambda'|e^{-t}}{1-e^{-t}} . \] \end{corollary} \begin{proof} Since $\prod_ys_y\le1$, Theorem~\ref{thm:sfglstep}(i) gives the upper bound $Z^{\rm lin}_\emptyset(\gamma)\le Z_qe^{-\lambda q'(\gamma)}$. If every block is calm relative to the background, then $|(d\varphi)_p|\le|(d\varphi^*_q (\gamma))_p|+|(d\psi)_p|\le\kappa$ for every $p$, so $\prod_ys_y\ge\mathbf 1 \{R^{\rm f}=\emptyset\}$ and $Z^{\rm lin}_\emptyset(\gamma)\ge Z_\emptyset (\gamma)=Z_q\pi_\emptyset e^{-\lambda q'(\gamma)}$. Finally Theorem~\ref {thm:sfglstep}(ii) and $-\log(1-x)\le x/(1-x)$. \end{proof} So on the coarse fields whose background is calm with the margin $\kappa_{\rm f}$, the sharp calm weight of the linear model is the Gaussian form up to a factor between $\pi_\emptyset$ and one, and this factor costs at most $96e^{-t}/(1-e^{-t})$ per block. This is a bound on the total, not a local representation of $\gamma\mapsto-\log Z^{\rm lin}_\emptyset(\gamma)- \lambda q'(\gamma)$. With the sharp threshold the calm events at $\gamma\ne0$ are not symmetric, the correlation inequality does not apply to them, and the comparison of the proof of Theorem~\ref{thm:sfglstep}(iv) leaves the factor $(1-e^{-t})^{-96|\Lambda'|}$, which is not uniform in the volume. This is why the thresholds are placed on the fluctuation. \begin{remark}[what the $\SU(2)$ step needs beyond the Gaussian layer]\label{rem:sfglsu2} Theorem~\ref{thm:sfglstep} and its corollaries concern the linear model only. A one-step expansion of the calm weight $Z_\emptyset(c)$ of $\SU(2)$ as a Gaussian factor times a polymer remainder with the same properties is not proved here, and the following parts of it are missing. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item \emph{The background.} The minimiser of $Q$ on the fibre $F_c$ of the cut-off average, its existence and uniqueness near the linear minimiser, and a bound of its plaquettes by the coarse curvature, the nonlinear analogue of Proposition~\ref{prop:sfglbackground}. By Theorem~\ref{thm:sfgaugeobst} the leading term must be invariant under coarse gauge transformations, so the background is needed for every $c$ in the class of Corollary~\ref{cor:sfcoarse}, not only for $c$ near $I$. \item \emph{The non-Gaussian density.} In the chart of Theorem~\ref{thm:fibrechart}(iii) around the background, the constrained law has the density $e^{-\lambda Q(\Phi(\eta))-\psi(\eta)}$, a log-concave perturbation of a Gaussian law for large $\lambda$; the mass outside the chart is not bounded (Remark~\ref{rem:fibrechartopen}(a)). \item \emph{The conditioning bound.} Royen's inequality, used in Theorem~\ref{thm:sfglstep}(ii)--(iv), is a property of Gaussian measures. A strongly log-concave law conditioned on a convex set is again strongly log-concave, with the same constant, and concentrates about its conditional mean; so the missing input for $\SU(2)$ is a bound on the conditional means of the plaquette fluctuations, uniform in the convex conditioning event. \item \emph{Locality.} The Gaussian form $q'$ is not of finite range. Its kernel decays at the first step by Theorem~\ref{thm:sfprop}(iv), and its decay at the later scales of Corollary~\ref{cor:sfglscales} is not proved. Whether it couples blocks three apart is the question whether $(M_W)_{AB}$ of Theorem~\ref{thm:sfgauss} vanishes, and $(M_W)_{AB}\ne0$ is not proved by hand (Remark~\ref{rem:sfgaussvalue}). \item \emph{Thresholds at all scales.} With thresholds on the fluctuation at every scale, a fine plaquette is a sum of the lifts of the fluctuations of all scales and of the last background. A bound of these lifts, which would connect Corollary~\ref{cor:sfglscales} with the single-scale thresholds of Theorem~\ref{thm:lfmarkov}, is not proved. \end{enumerate} \end{remark} \paragraph*{The next step.} To repeat the step at the coarse scale with constants $(\lambda_1, \kappa_1)$, the proofs above need three properties of the weight $e^{-A_1(c)}$ in place of $e^{-\lambda Q}$: (H1) it is a product of nonnegative factors attached to the coarse blocks, each depending on $c$ within distance one, up to a polymer expansion of the kind just described; (H2) the factor of a block that owns a coarse plaquette with $\delta(c_P,I) >\kappa_1$ is at most $e^{-\lambda_1\kappa_1^2}$ times the calm weight, with $\lambda_1\kappa_1^2-8\log M-c_1(\lambda_1,1)\ge9$; (H3) the averages of calm coarse fields are calm. Item (H3) holds at every step for $\kappa_1<\kappa_c$, since Proposition~\ref{prop:smallframessharp}(iv) is a statement about one field on any lattice; the threshold then grows by $16.003$ per step, or by $4\cdot1.023$ along the composed average of Theorem~\ref{thm:defectscales}. The suppression actually available for (H2) is inherited from the fine scale, as follows. \begin{proposition}\label{prop:lfinherited} Let $G=\SU(2)$, $\kappa<\kappa_c$ and $j\ge1$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item If $\delta(c_P,I)>16.003\kappa$, then $N(P)$ meets $\Pi(R(U))$. Each rough block meets at most $2310$ witness regions, so at most $2310|R(U)|$ coarse plaquettes exceed $16.003\kappa$. On them the kernels carry $e^{-\lambda Q^{(i)}/C_*}$, that is, the coupling $\lambda/C_*$. \item Let $\rho\le1.023\cdot10^{-4}$, $\rho_0=\rho/(1.01\cdot1.023\cdot4^j)$, and call a fine block rough at level $\rho_0$. By Lemma~\ref{lem:scalewitness} and Theorem~\ref{thm:islandswide}(i), every bad block of scale $j$ has a rough fine block among the $m_j=(17\cdot2^{j-1} -2)^4$ blocks that meet its box. A fine block meets the boxes of at most $H_j$ blocks of scale $j$, with $H_j$ as in that lemma. Hence for a set $Y$ of blocks of scale $j$ and $a_0>0$, $\sum_Xe^{-a_0|X|}\le e^{-A_j|Y|}$, where the sum runs over the sets $X$ of fine blocks inside the union of the boxes of $Y$ that meet every box, and \[ A_j=\max_{s>0}\Bigl(\frac{a_0-s}{H_j}-m_j\log(1+e^{-s})\Bigr). \] \item With $a_0=a(\beta_W/4,\rho_0)$ and $\rho=10^{-4}$, $A_j\ge9$ holds for $\beta_W\ge\beta^{\rm pw}_j$, where $\beta^{\rm pw}_1=2.733\cdot10^{14}$, $\beta^{\rm pw}_2=4.393\cdot10^{15}$ and $\beta^{\rm pw}_3=1.095\cdot 10^{17}$, and for $\beta_W\ge2.90\cdot10^{13}\cdot16^j$ whenever $1\le j\le 30$; $A_j<9$ for $\beta_W<1.57\cdot10^{13}\cdot16^j$. For every $\beta_W\in [10^{22},10^{42}]$, the largest $j\le30$ with $\beta_W\ge2.90\cdot10^{13} \cdot16^j$ satisfies $k_*-j\in\{1,2\}$, with $k_*$ as in Corollary~\ref{cor:defectscales}. \end{enumerate} \end{proposition} \begin{proof} (i) is Lemma~\ref{lem:lfcoarseenergy} and Theorem~\ref{thm:lfmarkov}(ii), with $N_{\rm ov}=1441$. (ii) The box of Lemma~\ref{lem:scalewitness} has $17\cdot2^j-4$ points per coordinate, from an even to an odd coordinate, so it meets $\frac12(17\cdot2^j-4)$ fine blocks per coordinate. A witness plaquette with corner in the box is owned by one of these. The number of boxes met by a fine block is the count of that lemma for the two cubes of the block per coordinate, which is $H_j$. A set $X$ meeting all the boxes of $Y$ has $|X|\ge|Y|/H_j$ and lies in a set of at most $m_j|Y|$ blocks. So the sum is at most $\sum_{k\ge|Y|/H_j}\binom{m_j|Y|}ke^{-a_0k}\le e^{-(a_0-s) |Y|/H_j}(1+e^{-s})^{m_j|Y|}$. (iii) Let $\lambda=\beta_W/4$, $b=16^{-j}\beta_W$ and $c=1.01\cdot1.023$, so that $\lambda\rho_0^2=b/(4c^2\cdot10^8)$ with $4.27025<4c^2<4.27026$, and, with $L_p=\log p$ as in the proof of Theorem~\ref{thm:lfkp}(ii), $\log \lambda=\log b+(4j-2)L_2$. By that proof $a_0\ge\lambda\rho_0^2-67.5\log \lambda-K_0$ with $K_0<953.06$. Taking $e^s=m_jH_j$ in the maximum, and using $m_j\log(1+1/(m_jH_j))\le1/H_j$, \[ A_j\ge\frac{a_0-1-\log(m_jH_j)}{H_j}. \] So $A_j\ge9$ holds when $E(b)\ge R_j$, where \begin{gather*} E(b)=\frac b{4c^2\cdot10^8}-67.5\log b,\\ R_j=954.06+9H_j+\log(m_jH_j)+67.5(4j-2)L_2 . \end{gather*} $E$ increases for $b>3\cdot10^{10}$. Since $m_j<17^4\,2^{4j-4}$ and $\log17 <4L_2+\frac1{16}$, $\log(m_jH_j)<\frac14+(4j+24)L_2$ for $j\le2$ and $<\frac14+(4j+12)L_2+8L_3$ for $j\ge3$; so $R_1<37931.3$, $R_2<38121.3$, $R_3< 60496.7$, and $R_j\le R_{30}<65624.6$ for $1\le j\le30$. With $\log(1.728 \cdot10^{13})=6L_2+3L_3+10\log10$, $\log(2.7\cdot10^{13})=3L_3+12\log10$ and $\log(3\cdot10^{13})=L_3+13\log10$ as upper bounds for $\log b$, $E(b)$ exceeds $37938$ at $b=2.733\cdot10^{14}/16$, $38127$ at $b=4.393\cdot 10^{15}/16^2$, $60505$ at $b=1.095\cdot10^{17}/16^3$ and $65816$ at $b=2.90\cdot10^{13}$, which gives the thresholds. Conversely $A_j0$, so $A_j \ge9$ needs $b>9H_j\cdot4c^2\cdot10^8\ge36864\cdot4.27025\cdot10^8>1.57\cdot 10^{13}$. For the last assertion let $j$ be that largest integer, so that $16^j\le \beta_W/(2.90\cdot10^{13})<16^{j+1}$, and $1\le j\le23$ because $2.90\cdot 10^{13}\cdot16<10^{22}$ and $2.90\cdot10^{13}\cdot16^{24}>10^{42}$. By Corollary~\ref{cor:defectscales}, $X=10^{-4}/(1.01\rho(\beta_W))$ satisfies $16^{k_*}\le X^2<16^{k_*+1}$ with $X^2=10^{-8}\beta_W/(1.0201\cdot96(\log \beta_W+C_1))$, and $57<\log\beta_W+C_1<104$, since $C_1=C_0+\frac16(9L_2 +2L_5)$ with $C_0\in[4.90816,4.90817]$ (proof of Theorem~\ref{thm:islandthreshold}). Hence \begin{gather*} 16^{k_*-j}\le\frac{X^2}{16^j}<\frac{16\cdot2.90\cdot10^{13}X^2}{\beta_W} <\frac{4.64\cdot10^6}{97.92\cdot57}<16^3,\\ 16^{k_*-j}>\frac{X^2}{16^{j+1}}\ge\frac{2.90\cdot10^{13}X^2}{16\beta_W}> \frac{2.90\cdot10^5}{1567\cdot104}>1, \end{gather*} and $k_*-j\in\{1,2\}$. \end{proof} So the next step stands as follows. Theorem~\ref{thm:islandswide} gives (H3) up to $a_{k_*}/8$ as a statement about probabilities under the Wilson measure: the labels of the composed average are rare. It gives neither (H1) nor (H2), which concern the weight $e^{-A_1}$ for every coarse field. Proposition~\ref{prop:lfinherited}(iii) reaches $a_{k_*}/4$, but it bounds sums of fine kernels with entropy of $\ell^\infty$ adjacency rather than of the joining distance of the islands, and only for activities of the form $e^{-a_0|X|}$ relative to the calm weight, the hypothesis that is not proved. In both cases the suppression at scale $j$ is the fine one at the radius $\rho_0\propto4^{-j}$, that is, the coupling $\lambda/16^{j}$ in units of the scale-$j$ threshold. By Proposition~\ref{prop:constantcurvature}, fine witnesses carry it no further than $a_{k_*}$. Iterating the step instead loses the factor $C_*$ of Theorem~\ref{thm:lfaction}(i), or $H_j\ge4096$ in Proposition~\ref{prop:lfinherited}(ii), at every scale. What the iteration needs is (H2) with $\lambda_1\ge(1-\varepsilon_1)\lambda$ and summable $\varepsilon_k$, a lower bound $A_1\ge(1-\varepsilon)\lambda S_1$ on large fields, local and relative to the calm weight. That requires a coarse energy for $\SU(2)$ at every scale, of the kind Proposition~\ref{prop:torusenergy} gives for torus-valued fields. It is not proved here, and this subsection gives no bound on the mass gap. Theorem~\ref{thm:islandwindow} is a union bound over the window, and its conclusion concerns every island of $W$, so it is void in windows of more than $10^{28166}$ blocks. The data of Theorem~\ref{thm:manyislands} do not need islands as their units: the sets $I_u$ are any pairwise separated sets of blocks off which every block of $\Lambda$ is good. If the islands are grouped into clusters whose separation grows with their size, the island sums and the weighted Dobrushin condition hold in every configuration of the labels, on every torus and in every window. The rarity of the bad blocks then enters only through the size of the clusters, and it bounds that size block by block, with no union bound over a window. \begin{definition}[clusters of bad blocks]\label{def:clusters} Let $R\ge2s+8$ be an integer and $\kappa>0$, and put $g(m)=(3/\kappa)\log m$ and $\ell_m=R+2g(m)$ for $m\ge1$. A partition $\mathcal P$ of a finite set $B$ of blocks of the torus is \emph{separated} if \begin{equation}\label{eq:clusterrule} \dist(U,V)>R+g(\lvert U\rvert)+g(\lvert V\rvert) \end{equation} for all distinct $U,V\in\mathcal P$. \end{definition} \begin{lemma}[clusters]\label{lem:clusters} Let $B$ be a finite set of blocks of the torus. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Among the separated partitions of $B$ there is a finest one; its parts are the \emph{clusters} of $B$. Starting from the partition into single blocks and merging, in any order, two parts that violate~\eqref{eq:clusterrule}, one reaches it after finitely many merges. \item Two blocks of $B$ at distance at most $R$ lie in one cluster; in particular every island formed with the joining distance $2s+8$ lies in one cluster, and distinct clusters are at distance more than $R$. \item A cluster of $m$ blocks is connected in the graph on $B$ in which two blocks are adjacent when their distance is at most $\ell_m$; its blocks lie within distance $(m-1)\ell_m$ of each other. \item If some cluster has more than $M_1$ blocks, then $B$ contains a set of between $M_1+1$ and $2M_1$ blocks that is connected in the graph of distance at most $\ell_{M_1}$. \end{enumerate} \end{lemma} \begin{proof} (i) Let $\mathcal Q$ be finer than a separated partition $\mathcal P$, and let $U,V\in\mathcal Q$ violate~\eqref{eq:clusterrule}. If $U\subset U'$ and $V\subset V'$ with distinct $U',V'\in\mathcal P$, then, since $g$ increases, \[ \dist(U',V')\le\dist(U,V)\le R+g(\lvert U'\rvert)+g(\lvert V'\rvert), \] which contradicts the separation of $\mathcal P$; so $U\cup V$ lies in one part of $\mathcal P$, and the partition obtained by merging $U$ and $V$ is again finer than $\mathcal P$. The partition $\{B\}$ is separated, so the merging starts below every separated partition and stays below it at each step. Each merge lowers the number of parts, so the process stops, at a separated partition finer than every separated partition, which is therefore the finest one and does not depend on the order of the merges. (ii) Two single blocks at distance at most $R$ violate~\eqref{eq:clusterrule}, since $g(1)=0$, so by the first step of (i) they lie in one part of every separated partition. An island is connected through steps of length at most $2s+8\le R$. The last assertion is~\eqref{eq:clusterrule} with $g\ge0$. (iii) By induction over the merges, every part of $m$ blocks of an intermediate partition is connected at distance $\ell_m$: a single block is, and if $U$ and $V$ are merged, with $\lvert U\rvert+\lvert V\rvert=m$, then $\dist(U,V)\le R+g(\lvert U\rvert)+g(\lvert V\rvert)\le\ell_m$, while $U$ and $V$ are connected at distances $\ell_{\lvert U\rvert},\ell_{\lvert V\rvert}\le\ell_m$. Two blocks of a connected set of $m$ blocks are joined by a path of at most $m-1$ steps. (iv) Consider the first merge of the process of (i) that produces a part of more than $M_1$ blocks. It merges two parts of at most $M_1$ blocks each, at distance at most $R+2g(M_1)=\ell_{M_1}$, and each of them is connected at distance $\ell_{M_1}$ by (iii). \end{proof} \begin{theorem}[the island sums for clusters]\label{thm:clustercomparison} Let $s\ge1$, $r\le8\cdot10^{-5}$, $0\le\vartheta<1$, $\kappa=2(1-\vartheta) \alpha_G$, $R\ge2s+8$, and form the clusters of the set of bad blocks of the torus with these $R$ and $\kappa$. Let the data be those of Theorem~\ref{thm:manyislands} on a set $\Lambda$ of blocks, under its hypotheses, with $I_u=U_u\cap\Lambda$ for distinct clusters $U_u$, $u\in\mathcal I$, every bad block of $\Lambda$ lying in some $I_u$. Put \begin{gather*} c_1=\frac{1920}{\beta_W}\bar\Lambda_2^2r^2\cdot45(2s+7)^4 e^{\kappa(2s+9)},\\ \tau=c_1e^{-\kappa R/2}\sigma(\kappa/2), \end{gather*} with $\bar\Lambda_2$ as before Theorem~\ref{thm:islandwindow} and $\sigma(\cdot)$ as in Proposition~\ref{prop:islandmass}, and for $u\in\mathcal I$ let $\hat T_u=\sum_{v\in\mathcal I\setminus\{u\}} \varDelta_{uv}e^{2\vartheta\alpha_G\ell(u,v)}$, with $\ell$ the pseudometric of Corollary~\ref{cor:islandchains}. Then, for every configuration of the labels: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\hat T_u\le\tau$ for every $u\in\mathcal I$; hence $S_a,S_b\le\tau$, the sums $T_a,T_b$ of Corollary~\ref{cor:islandchains} are at most $\tau$, and \[ \varDelta_{ab}\le\min\bigl\{\tau,\ c_1e^{-\kappa R/2}e^{-\kappa\dist(I_a, I_b)/2}\bigr\}; \] \item $\varDelta_*\le\varDelta_{ab}+\frac14\tau^2$, and if $\tau<4$ the weighted Dobrushin condition of Corollary~\ref{cor:islandchains} holds with $\theta\le\tau/4$ and \[ \varDelta_*\le\varDelta_{ab}+\frac{e^{-2\vartheta\alpha_G\ell(a,b)} \tau^2}{4-\tau}; \] \item for $\beta_W\ge\beta_0>0$ at fixed $R$, $\tau(\beta_W)\le(\beta_W/ \beta_0)\,\tau(\beta_0)$. \end{enumerate} At $\rho=2.17\cdot10^{-16}$, $r=43\rho$ and $\beta_W=\beta^*$ of Theorem~\ref{thm:islandwindow}: for $s=1$, $\log_{10}c_1\le21.74$ and $\tau\le10^{-1047}$ at $\vartheta=0$, $R=10^7$, and $\tau\le10^{-1045}$ at $\vartheta=\frac12$, $R=2\cdot10^7$; for $s=5$, $\tau\le10^{-1045}$ and $\tau\le10^{-1044}$ at the same $\vartheta$ and $R$. By (iii) the same bounds hold at $\beta_W\ge\beta^*$ with $R$ replaced by $R+\lceil(2/\kappa) \log(\beta_W/\beta^*)\rceil$. \end{theorem} \begin{proof} \emph{The data.} The sets $I_u$ are pairwise disjoint and, by Lemma~\ref{lem:clusters}(ii), at distance more than $R\ge2s+8$, so $w_{uv}\ge2s+8$, and every block of $\Lambda$ outside $\bigcup_uI_u$ is good. The proof of Lemma~\ref{lem:lambdatwo} bounds the norm of $\partial_{x_u}\nabla_z\mathcal U$ by the largest sum, over one $x_u$-link or one $z$-link, of the norms of the blocks of that matrix, and these are determined by the plaquettes at one link; as in the paragraph before Theorem~\ref{thm:islandwindow}, it applies to each $I_u$ and gives $\Lambda_2\le\bar\Lambda_2$. Every $x_u$-link is based in a block at distance at most $s+3$ from $I_u$, so $n_u\le45(2s+7)^4\lvert I_u\rvert \le45(2s+7)^4m_u$, where $m_u=\lvert U_u\rvert$. \emph{One pair.} Let $u\ne v$ and $d=\dist(I_u,I_v)$, so that $w_{uv}-2s-8 =d-2s-9\ge0$ and $\ell(u,v)\le d-2s-9$. Since $2\alpha_G-2\vartheta \alpha_G=\kappa$, \[ \varDelta_{uv}e^{2\vartheta\alpha_G\ell(u,v)}\le c_1\sqrt{m_um_v}\, e^{-\kappa d} . \] By~\eqref{eq:clusterrule}, $d\ge\dist(U_u,U_v)>R+g(m_u)+g(m_v)$, so that \[ e^{-\kappa d}\le e^{-\kappa d/2}e^{-\kappa R/2}(m_um_v)^{-3/2}, \] since $e^{-\kappa g(m)/2}=m^{-3/2}$. Hence \[ \varDelta_{uv}e^{2\vartheta\alpha_G\ell(u,v)}\le c_1e^{-\kappa R/2}\, \frac{e^{-\kappa d/2}}{m_um_v}\le\frac{c_1e^{-\kappa R/2}}{m_u}\, e^{-\kappa d/2}. \] \emph{The sum.} For $v\ne u$ let $y_v$ be a block of $I_v$ nearest to $I_u$; distinct $v$ give distinct $y_v$, and $e^{-\kappa\dist(I_u,I_v)/2} \le\sum_{x\in I_u}e^{-\kappa\dist(x,y_v)/2}$. Summing over $v$ and using the preimages of the proof of Proposition~\ref{prop:islandmass}(ii), \[ \sum_{v\ne u}e^{-\kappa\dist(I_u,I_v)/2}\le\lvert I_u\rvert\, \sigma(\kappa/2)\le m_u\,\sigma(\kappa/2), \] so $\hat T_u\le\tau$. The sums $S_a$, $T_a$ and the row sums of the corollary are sums of terms of $\hat T_a$ or $\hat T_q$, with factors at most one in the case of $S_a$, and $\varDelta_{ab}$ is one term of $\hat T_a$; the second bound on $\varDelta_{ab}$ is the bound of one pair with $m_am_b\ge1$ and $e^{-2\vartheta\alpha_G\ell(a,b)}\le1$. This proves (i). (ii) The first form of $\varDelta_*$ in Theorem~\ref{thm:manyislands} and (i) give the first bound. By $\tanh t\le t$, the sum of Corollary~\ref{cor:islandchains} at $q$ is at most $\frac14\hat T_q\le \tau/4$, so $\theta\le\tau/4<1$, and the corollary with $T_a,T_b\le\tau$ gives the second. (iii) $\bar\Lambda_2(\beta)/\beta$ decreases, so $\bar\Lambda_2(\beta_W) \le(\beta_W/\beta_0)\bar\Lambda_2(\beta_0)$ and $\bar\Lambda_2^2/\beta_W$ grows at most linearly. \emph{The numbers.} Let $L_p=\log p$ with the bounds of the proof of Theorem~\ref{thm:lfkp}(ii); the same series at $x=\frac1{21}$ and $\frac1{73}$ gives $\log\frac{11}{10}$ and $\log\frac{37}{36}$ to within $10^{-10}$, so $2.3978952796/\rho^2$ and $\beta^*$ is its largest zero, with $C_0<4.90817$ (proof of Theorem~\ref{thm:islandthreshold}) and $\rho^2=4.7089\cdot10^{-32}$. Since $N_s<(4s+17)^4$, \[ \lvert\log q^*\rvert 33024.10$ and, with $\log1.1221<\log\frac98$, $6(\log\beta_1+C_0)<541.33$, so $g(\beta_1)>1$; at $\beta_5=1.3217\cdot10^{37}$, $\beta_5\rho^2/16> 38898.45$ and, with $\log1.3217<\log\frac43$, $6(\log\beta_5+C_0)<542.35$, so $g(\beta_5)>2$. Hence $\beta^*<\beta_s$ for $s=1,5$. As $\bar\Lambda_2^2 /\beta_W=(99.0001+4.65\cdot10^6/\beta_W)^2\beta_W$ increases with $\beta_W$, and $e^x\le1+x+x^2$ for $0\le x\le1$, at $\beta_W=\beta^*$ \[ c_1\le1920\cdot99.0002^2\cdot43^2\cdot45(2s+7)^4\bigl(1+x+x^2\bigr) \beta_s\rho^2, \] where $x=\kappa(2s+9)$. For $s=1$ this is below $5.4581\cdot10^{21}$ at $\vartheta=0$ and $5.4431\cdot10^{21}$ at $\vartheta=\frac12$, and for $s=5$ below $8.2171 \cdot10^{22}$ and $8.1779\cdot10^{22}$. In particular $\log_{10}c_1<21.74$ for $s=1$, $\vartheta=0$, as $5.4581<3^7/400$ and $7L_3-4L_2-2L_5<0.74\log 10$. By the proof of Proposition~\ref{prop:islandmass}, $\sigma(\kappa/2) \le1+384(\kappa/2-\kappa^2/8)^{-4}+16(\kappa/2-\kappa^2/8)^{-2}$, which is below $9.8354\cdot10^{16}$ at $\vartheta=0$, $\kappa/2=2.5\cdot10^{-4}$, and below $1.5733\cdot10^{18}$ at $\vartheta=\frac12$. In all four cases $\kappa R/2=2500$, and $e^{-2500}<10^{-1085.736}$ since $1085.736\log10< 2500$. The product $c_1\sigma(\kappa/2)$ is below $5.37\cdot10^{38}< 10^{38.736}$ for $s=1$, $\vartheta=0$, as $\log5.4=3L_3-L_5<0.736\log10$; below $10^{40}$ for $s=1$, $\vartheta=\frac12$ and for $s=5$, $\vartheta= 0$; and below $1.3\cdot10^{41}<10^{41.736}$ for $s=5$, $\vartheta= \frac12$. This gives the four bounds on $\tau$. \end{proof} The theorem has no exceptional event: the conditions that Theorem~\ref{thm:islandwindow} obtains outside probability $10^{-28166}\lvert W\rvert$ hold for the clusters in every window of every size. What the rarity has to give is that the clusters are small, and this is a statement about one block at a time. Write $U(y)$ for the cluster of a bad block $y$ and $I(y)$ for its island. \begin{theorem}[the size of the clusters]\label{thm:clustersizes} Let $G=\SU(2)$, $s\ge1$, $\rho$, $r$, $\eta$, $\beta^*$, $q^*$, $M=11^4$ and $H=6^4$ be as in Theorem~\ref{thm:islandwindow}, $\beta_W\ge\beta^*$, $p=q_\rho(\beta_W)\le q^*$, $A=eMHp$, and form the clusters of the bad blocks of a torus of $L^4$ blocks with $R\ge2s+8$ and $\kappa>0$. Put $N_5=11^4-1$, $N_\ell=(2\ell+1)^4-1$ for $\ell\ge5$, and \[ \nu_\ell=4\bigl(N_5+N_\ell A^{1/2}\bigr)A^{1/(2H)} . \] \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item For every block $y$ and $m\ge1$, $\mu_L\{y\in B,\ \lvert U(y)\rvert=m\}\le A^{(H+1)/(2H)}\nu_{\ell_m}^{m-1}$. \item If $\nu_{\ell_{M_1}}<1$, then \[ \begin{split} \mu_L\{&\text{a cluster has more than }M_1\text{ blocks}\}\\ &\le\frac{L^4A^{(H+1)/(2H)}\nu_{\ell_{M_1}}^{M_1}}{1-\nu_{\ell_{M_1}}}. \end{split} \] \item If $1\le m_0\le M_1$ and $\nu_{\ell_{M_1}}<1$, then $\mu_L\{y\in B,\ U(y)\ne I(y)\}$ is at most \[ \begin{split} &(2(m_0-1)\ell_{m_0}+1)^4(2\ell_{m_0}+1)^4(Mp)^2\\ &+\frac{A^{(H+1)/(2H)}\nu_{\ell_{M_1}}^{m_0}}{1-\nu_{\ell_{M_1}}} +\epsilon_{M_1}, \end{split} \] with $\epsilon_{M_1}$ the bound of (ii). \end{enumerate} At $\rho=2.17\cdot10^{-16}$, $\beta_W=\beta^*$, $s=1$, $\vartheta=0$, $\kappa=2\alpha_G$, $R=10^7$ and $M_1=10^{10^{1700}}$: $\ell_{M_1}\le 10^{1705}$, $\nu_\ell\le0.2130$ for every $\ell\le\ell_{M_1}$, the bound of (ii) is at most $L^4\,10^{-0.67M_1}$, and (iii) with $m_0=10^5$ gives \[ \mu_L\{y\in B,\ U(y)\ne I(y)\}\le10^{-28126}+L^4\,10^{-0.67M_1}. \] At $\vartheta=\frac12$, $R=2\cdot10^7$ the first term is at most $10^{-28124}$. For $s=5$, $\nu_\ell\le0.0222$, the bound of (ii) is at most $L^4\,10^{-1.65M_1}$, and the first term of (iii) is at most $10^{-33227}$ at $\vartheta=0$, $R=10^7$ and $10^{-33224}$ at $\vartheta= \frac12$, $R=2\cdot10^7$. The same numbers hold at $\beta_W\ge\beta^*$ with $R$ replaced by $R+\lceil(2/\kappa)\log(\beta_W/\beta^*)\rceil$. \end{theorem} \begin{proof} (i) By Lemma~\ref{lem:clusters}(iii), on the event $U(y)$ is a set $S\ni y$ of $m$ bad blocks, connected at distance $\ell=\ell_m$. Let $S_1,\dots, S_k$ be the components of $S$ in the graph of distance at most $5$; they are pairwise at distance at least $6$, so Step~1 of the proof of Theorem~\ref{thm:islandwindow} gives \[ \mu_L(S\subset B)\le\prod_iA^{\lceil\lvert S_i\rvert/H\rceil}\le A^{k/2+m/(2H)}, \] since $\lceil n/H\rceil\ge\frac12(1+n/H)$ and $A<1$. Choose a spanning tree of $S$ formed by spanning trees of the $S_i$ at distance $5$ and $k-1$ further edges of length at most $\ell$, which exist because $S$ is connected at distance $\ell$. A walk of $2(m-1)$ steps around the tree from $y$ recovers $S$: it is fixed by the positions of the $m-1$ forward steps, at most $4^{m-1}$ choices, by which $k-1$ of them are long, at most $\binom{m-1}{k-1}$, and by their offsets, $N_5$ for a short step and $N_\ell$ for a long one. Summing over $k$, \[ \begin{split} \mu_L\{&y\in B,\ \lvert U(y)\rvert=m\}\\ &\le A^{\frac12+\frac m{2H}}\sum_{k=1}^m4^{m-1}\binom{m-1}{k-1} N_5^{m-k}N_\ell^{k-1}A^{\frac{k-1}2}\\ &=A^{\frac12+\frac m{2H}}\bigl(4(N_5+N_\ell A^{1/2})\bigr)^{m-1}, \end{split} \] which is (i). (ii) By Lemma~\ref{lem:clusters}(iv) some block lies in a set of $m\in (M_1,2M_1]$ bad blocks connected at distance $\ell_{M_1}$; the count of (i) with $\ell=\ell_{M_1}$, summed over $m>M_1$ and over the $L^4$ blocks, gives (ii). (iii) If $\lvert U(y)\rvert>m_0$, either $\lvert U(y)\rvert\le M_1$, which by (i) and $\nu_{\ell_m}\le\nu_{\ell_{M_1}}$ for $m\le M_1$ has probability at most the second term, or some cluster has more than $M_1$ blocks. If $\lvert U(y)\rvert\le m_0$ and $U(y)$ contains a second island, then $U(y)$, connected at distance $\ell_{m_0}$, has an edge between blocks $z,z'$ of different islands: $z$ lies within distance $(m_0-1)\ell_{m_0}$ of $y$, and $2s+9\le\dist(z,z')\le\ell_{m_0}$. Step~1 of the proof of Theorem~\ref{thm:islandwindow} with $S_1=\{z\}$, $S_2=\{z'\}$ gives $(Mp)^2$, and a union bound over $z$ and $z'$ gives the first term. For $\beta_W=t\beta^*$, $t\ge1$, Step~6 of the proof of Theorem~\ref{thm:islandwindow} gives $\log(p/q^*)\le f(\beta_W)-f(\beta^*) \le6\log t-32481(t-1)$, since $\beta^*\rho^2/16>\lvert\log q^*\rvert> 32481$ for $s=1$ and $s=5$, by the bounds below. With $R$ raised by $\lceil(2/\kappa)\log t\rceil\le1+(2/\kappa)\log t$, the length $\ell_{m_0}\ge R$ is multiplied by at most $1+10^{-7}+4\cdot10^{-4}\log t$, and the first term of (iii) by at most $(1+10^{-7}+4\cdot10^{-4}\log t)^8 t^{12}e^{-64962(t-1)}<1+10^{-6}$, since $\log t\le t-1$; the margins below absorb this factor. The factor $N_\ell$ in $\nu_\ell$ grows by a power of $1+\log t$ that $A^{1/2}$ absorbs, and the other factors decrease. \emph{The numbers.} Let $L_p=\log p$ with the bounds of the proofs of Theorems~\ref{thm:lfkp}(ii) and~\ref{thm:clustercomparison}. By the definition of $q^*$, with $N_s=(4s+17)^4-1$ as in Theorem~\ref{thm:islandwindow}, $A\le eMHq^*=(2N_s^2)^{-H}$ and $Mp\le Mq^*=(2N_s^2)^{-H} /(eH)$. From $4\log(4s+17)-1/N_s\le\log N_s<4\log(4s+17)$, $\log(2N_s^2)$ exceeds $25.049316$ at $s=1$ and $29.580489$ at $s=5$, while $\log(eH)=1+4L_2+4L_3> 8.167037$. Let $s=1$ and $\vartheta=0$, so $\kappa=5\cdot10^{-4}$, $6/\kappa =12000$ and $\ell_{M_1}=10^7+12000\cdot10^{1700}\log10<10^{1705}$; at $\vartheta=\frac12$, $\ell_{M_1}<2\cdot10^7+24000\cdot10^{1700}\log10< 10^{1705}$ as well. Since $2N_1^2=75644940800>275000^2$, $A^{1/(2H)}< 1/275000$, and $A^{1/2}<10^{-6998}$ as $2N_1^2>10^{11}/1.5>10^{10.8}$, by $1.5^5<10$; with $N_\ell< 10^{6822}$ for $\ell\le10^{1705}$, \[ \nu_\ell<\frac{4\cdot14640}{275000}+10^{-176}<0.2130<\frac{16}{75}< 10^{-0.67}, \] the last since $\log\frac{16}{75}=4L_2-L_3-2L_5<-0.67\log10$. As $A^{(H+1)/(2H)}<10^{-6998}$ and $1/(1-\nu_\ell)<2$, the bound of (ii) is at most $L^4\,10^{-0.67M_1}$. In (iii), $\ell_{m_0}=10^7+60000\log10< 10138156$ and $(2(m_0-1)\ell_{m_0}+1)(2\ell_{m_0}+1)<4.125\cdot10^{19}= \frac{33}8\cdot10^{19}$, so the logarithm of the first term is below \begin{multline*} 4\bigl(L_3+L_{11}-3L_2+19\log10\bigr)\\ -2592\cdot25.049316-2\cdot8.167037<-64763.49, \end{multline*} while $28126\log10<64762.51$; the second term is below $10^{-6998}\cdot 0.213^{10^5}$, so the sum of the first two terms is below $10^{-28126}$. At $\vartheta=\frac12$, $R=2\cdot10^7$, $\ell_{m_0}<20276311$, the product is below $1.65\cdot10^{20}=3\cdot5\cdot11\cdot10^{18}$, and the logarithm of the first term is below $4(L_3+L_5+L_{11}+18\log10)-2592\cdot25.049316-2 \cdot8.167037<-64757.95<-28124\log10$. For $s=5$, $2N_s^2>2650000^2$ gives $\nu_\ell<4\cdot14640/2650000+10^{-176}<0.0222<\frac1{45}<10^{-1.65}$, since $\log\frac1{45}=-2L_3-L_5<-1.65\log10$, hence the bound of (ii); and the logarithm of the first term of (iii) is below $4(L_3+L_{11}-3L_2+19\log10)- 2592\cdot29.580489-2\cdot8.167037<-76508.29<-33227\log10$ at $\vartheta=0$, and, with the bracket $4(L_3+L_5+L_{11}+18\log10)$ of $\vartheta=\frac12$, below $-76502.75<-33224\log10$ at $\vartheta= \frac12$. \end{proof} On a torus of at most $10^{M_1/2}$ blocks, $M_1=10^{10^{1700}}$, the theorem gives, outside probability $10^{-M_1/10}$, that no cluster has more than $M_1$ blocks, and the expected number of blocks of a window $W$ that lie in a cluster other than a single island is at most $\lvert W\rvert(10^{-28126}+10^{-M_1/10})$ for $s=1$: a density, not a union bound. Theorems~\ref{thm:clustercomparison} and~\ref{thm:clustersizes} thus remove the window size from the island comparison of item~(ii) of Sec.~\ref{ssec:remains}, given the labels and with clusters as the units. The size of the torus remains only in the term $\epsilon_{M_1}$ of Theorem~\ref{thm:clustersizes}, at sizes far beyond any other in this paper. The next proposition shows that a separation growing with the size of the units is needed, and where the count of (i) stops. \begin{proposition}[the reach of the clusters]\label{prop:clusterreach} \leavevmode \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Let $R'\ge2s+8$ and $m\ge1$, and on a torus of side at least $2(m+2)(R'+1)$ let the bad blocks be $u_i=iR'e_1$ and $v_i=u_i+(-1)^i (R'+1)e_2$, $1\le i\le m$. In the finest partition of these blocks into parts at mutual distance more than $R'$, the $u_i$ form one part $U$ and each $v_i$ is a part $V_i$, and with $c_0=\frac{1920}{\beta_W} \bar\Lambda_2^2r^2\cdot45(2s+7)^4e^{-2\alpha_G(R'-2s-8)}$, \[ \sum_i\bar\varDelta_{UV_i}=c_0m^{3/2},\qquad \sum_{i,j}\bar\varDelta_{u_jv_i}\ge c_0m . \] So no partition whose parts are separated by a distance independent of their size bounds the sums $\hat T_u$, through $\bar\varDelta$ either for the parts or for their islands, uniformly in the configuration. \item With $A=eMHq^*$, $s=1$: $\nu_\ell<1$ for $\log_{10}\ell\le1763.2$ and $\nu_\ell>1$ for $\log_{10}\ell\ge1763.3$, and $N_\ell A\ge1$ for $\log_{10}\ell\ge3524.5$. For $\ell_{M_1}$ beyond the first value the count of Theorem~\ref{thm:clustersizes}(i) is void, and beyond the second the bound $N_\ell A$ on the expected number of bad blocks within distance $\ell$ of a block exceeds one. \end{enumerate} \end{proposition} \begin{proof} (i) Consecutive $u_i$ are at distance $R'$ and so lie in one part; the distance of $v_i$ to $u_j$ is $\max\{\lvert i-j\rvert R',R'+1\}$, and to $v_j$, $j\ne i$, at least $\min\{2R',2R'+2\}>R'$, so the finest partition is the one stated, with $\dist(U,V_i)=R'+1$ and $w_{UV_i}=R'$. The neighbourhoods of radius $s+3$ of the $u_i$ are disjoint, since $R'>2s+6$, so $\bar n_U=45\,m(2s+7)^4$ and $\bar n_{V_i}=45(2s+7)^4$, which gives the first identity. The pair $(u_i,v_i)$ alone contributes $c_0$ to the second sum. Both sums grow without bound in $m$ at fixed $R'$. (ii) $\nu_\ell<1$ is $N_\ell16243.36$. For $\log_{10}\ell\ge3524.5$, $\log N_\ell>32464.6>H\log(2N_1^2)=-\log A$. The expected number of bad blocks among the $N_\ell$ blocks within distance $\ell$ of a block is at most $N_\ell A$. \end{proof} The count of Theorem~\ref{thm:clustersizes} stops at $M_1$, and the factor $L^4$ in its term $\epsilon_{M_1}$ cannot be removed: on every large enough torus the clusters of the bad blocks percolate. Two facts give this. Every block is bad with a conditional probability bounded below, by the finite energy of the Wilson measure, and the separation of the clusters grows with their size, which Proposition~\ref{prop:clusterreach}(i) and Proposition~\ref{prop:separationneed} below show to be necessary. What replaces the clusters of the torus are the clusters of the bad blocks of a window. Lemma~\ref{lem:clusters}(i)--(iii) and their proofs use of $g$ only that it is nondecreasing and nonnegative, and so does the next lemma; for such $g$ the clusters are formed by~\eqref{eq:clusterrule} and $\ell_m=R+2g(m)$. \begin{lemma}[subsets and absorption]\label{lem:clustermono} Let $g\ge0$ be nondecreasing. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item If $B'\subset B$ are finite sets of blocks, every cluster of $B'$ lies in a cluster of $B$. \item If $C$ is a cluster of $B$ and $z\in B$ has $\dist(z,C)\le R+g(\lvert C\rvert)$, then $z\in C$. \end{enumerate} \end{lemma} \begin{proof} (i) Let $\mathcal P$ be the partition of $B$ into its clusters and $\mathcal P'$ the partition of $B'$ into the nonempty sets $U\cap B'$, $U\in\mathcal P$. For distinct $U,V\in\mathcal P$ meeting $B'$, \[ \begin{split} \dist(U\cap B',V\cap B')&\ge\dist(U,V)>R+g(\lvert U\rvert)+g(\lvert V\rvert)\\ &\ge R+g(\lvert U\cap B'\rvert)+g(\lvert V\cap B'\rvert), \end{split} \] since $g$ is nondecreasing, so $\mathcal P'$ is a separated partition of $B'$. The partition of $B'$ into its clusters is finer than $\mathcal P'$ by Lemma~\ref{lem:clusters}(i), so each cluster of $B'$ lies in some $U\cap B'\subset U$. (ii) If $z$ lay in a cluster $V\ne C$, then $\dist(C,V)\le\dist(z,C)\le R+g(\lvert C\rvert)+g(\lvert V\rvert)$, against~\eqref{eq:clusterrule}. \end{proof} \begin{lemma}[clusters of a window]\label{lem:windowclusters} Let $\Lambda$ be a set of blocks of the torus, and form the clusters of $B\cap\Lambda$, with $g$ and the parameters of Theorems~\ref{thm:clustercomparison} and~\ref{thm:clustersizes}. Each of them lies in a cluster of $B$, and: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item Theorem~\ref{thm:clustercomparison} holds with $I_u=U_u$ for the distinct clusters $U_u$ of $B\cap\Lambda$; \item Theorem~\ref{thm:clustersizes} holds with $U(y)$ the cluster of $y$ in $B\cap\Lambda$, $I(y)$ the component of $y$ in $B\cap\Lambda$ at the joining distance $2s+8$, and $L^4$ replaced by $\lvert\Lambda\rvert$, with the same numbers. \end{enumerate} In a window $\Lambda$ of at most $10^{M_1/2}$ blocks, on a torus of any size, no cluster of $B\cap\Lambda$ has more than $M_1$ blocks outside probability $10^{-M_1/10}$, and for $s=1$ the expected number of blocks $y$ of $\Lambda$ with $U(y)\ne I(y)$ is at most $\lvert\Lambda\rvert (10^{-28126}+10^{-M_1/10})$. \end{lemma} \begin{proof} The first assertion is Lemma~\ref{lem:clustermono}(i). (i) The proof of Theorem~\ref{thm:clustercomparison} uses of the clusters only that the sets $U_u$ are distinct parts of a separated partition of a set of blocks that contains every bad block of $\Lambda$, with $I_u=U_u\cap\Lambda$: through Lemma~\ref{lem:clusters}(ii), through $\lvert I_u\rvert\le \lvert U_u\rvert$, and through~\eqref{eq:clusterrule}. The clusters of $B\cap\Lambda$ are such parts, with $I_u=U_u$. (ii) The proof of Theorem~\ref{thm:clustersizes}(i) uses that $U(y)$ is connected at distance $\ell_m$, which Lemma~\ref{lem:clusters}(iii) gives for the clusters of $B\cap\Lambda$, and bounds $\mu_L(S\subset B)$ for each set $S$ it counts. In (ii), Lemma~\ref{lem:clusters}(iv) applied to $B\cap\Lambda$ gives a set of bad blocks containing a block of $\Lambda$, and the union bound runs over the $\lvert\Lambda\rvert$ blocks of $\Lambda$. In (iii) an edge of $U(y)$ between blocks of different components of $B\cap\Lambda$ has length between $2s+9$ and $\ell_{m_0}$, as before. For the last assertion, (ii) gives at most $\lvert\Lambda \rvert10^{-0.67M_1}\le10^{-0.17M_1}$, and (iii) with $m_0=10^5$ gives the bound for each block of $\Lambda$. \end{proof} \begin{lemma}[one link makes a block bad]\label{lem:onelinkbad} Let $G=\SU(2)$, let the labels have radius $r\le\frac1{10}$, let $B$ be the set of bad blocks of a torus under $\mu_L$, with the coarse field $c=V^\sharp(U)$ of the configuration, and put $p_0=\frac12e^{-12\beta_W}$. Let $\xi$ be a random set of blocks that contains each block independently with probability $p_0$. Then $\mu_L(B\in\mathcal E)\ge \mathbb P(\xi\in\mathcal E)$ for every increasing event $\mathcal E$, that is, every set of sets of blocks that contains each superset of each of its members. \end{lemma} \begin{proof} In the tree gauge $\mu_L$ has density proportional to $\exp(\beta_W \sum_p\frac12\ReTr U_p)$ with respect to the product of the Haar measures on the active links (Sec.~\ref{ssec:pathblocks}). For a block $y$ let $k= (y,1)$, with path link $\ell_k=U_{2y+e_1,1}$, let $f_y$ be the link $(2y+e_1+e_2,1)$, a non-path private link of $k$, and put \[ G_y=\bigl\{d(U_{f_y},\ell_k)>r+\tfrac\pi3\bigr\}. \] \emph{$G_y$ makes $y$ bad.} The link $(2y,1)$ is a tree link, so $P_{2y}=\ell_k$, and by~\eqref{eq:cutoff} $c_k=\ell_ke^Y$ with $Y=\frac{ \chi_k}{16}\sum_x\log(\ell_k^{-1}P_x)$. If $\chi_k=0$, $c_k=\ell_k$. If $\chi_k>0$, then $\delta(\ell_k^{-1}P_x,I)=\delta(P_x,\ell_k)<\rho_G=1$ for every $x$, so $\lVert\log(\ell_k^{-1}P_x)\rVert<\pi/3$ by Lemma~\ref{lem:logbound}, and $\lVert Y\rVert<\pi/3$; the eigenvalues of $Y$ are $\pm i\theta$ with $\theta<\pi$, so $\log e^Y=Y$ and $d(c_k, \ell_k)<\pi/3$. The reconstruction value of $f_y$ is $c_k$, and on $G_y$ \[ d(U_{f_y},c_k)\ge d(U_{f_y},\ell_k)-d(\ell_k,c_k)>r , \] so $y$ is bad. \emph{One link.} The link $f_y$ lies in six plaquettes, each contributing $\beta_W\frac12\ReTr U_p\in[-\beta_W,\beta_W]$ to the exponent, so given all the other active links the law of $U_{f_y}$ has a density with respect to the Haar measure whose values lie within a factor $e^{12\beta_W}$ of each other, hence a density at least $e^{-12\beta_W}$, and $G_y$ is the complement of the ball of radius $r+\pi/3$ about $\ell_k$. For $\SU(2)$, by Lemma~\ref{lem:haarsu2} and bi-invariance a ball of radius $\sqrt2\,t$ has Haar measure at most $\frac{2t^3}{3\pi}$. With $r+\pi/3\le1.15$, $t\le0.82$ and the measure is at most $2\cdot0.552/(3\pi)<\frac12$. So the conditional probability of $G_y$ given all the other active links is at least $p_0$. \emph{The coupling.} Enumerate the blocks as $y_1,\dots,y_n$ and put $X_j=\mathbf 1_{G_{y_j}}$. The event $G_y$ depends only on the two private links $\ell_k$ and $U_{f_y}$ of the pair $(y,1)$, which belong to no other pair, so $X_1,\dots,X_{j-1}$ are functions of the active links other than $U_{f_{y_j}}$, and $\mathbb P(X_j=1\mid X_1,\dots,X_{j-1})\ge p_0$. Let $V_1,\dots,V_n$ be independent and uniform on $[0,1]$, and put recursively $X'_j=\mathbf 1[V_j\le\mathbb P(X_j=1\mid X_{0$, $c_0\ge0$, and suppose $R+g(m)\ge\gamma\log m-c_0$ for every $m\ge1$.\footnote{The proof is a multiscale construction: the occupied boxes of one scale are the sites of the next, and joining distances that grow like the logarithm of the size connect them, as in hierarchical and long-range percolation.} Let $0 e^5\cdot2>296$. By independence, $\mathbb P(E_0\cap\dots\cap E_J)\ge p^{M_0}(1-0.004)\ge\frac12p^{M_0}$. \emph{The cluster.} On this event let $C$ be the cluster of $y$ in $\xi\cap\Lambda$. Consecutive blocks of $S_0$ are at distance $1\le R$, so $S_0\subset C$ by Lemma~\ref{lem:clusters}(ii), and $R+g(\lvert C\rvert) \ge\gamma\log M_0-c_0\ge2a_0$. Suppose $R+g(\lvert C\rvert)\ge2a_{j-1}-1$ for some $1\le j\le J$. Order the cubes of $E_j$ so that each is a neighbour of an earlier one, the first containing $y$, and in each cube other than the first choose a block of $\xi\setminus S_0$. If the earlier cubes contain blocks of $C$, the chosen block of the next cube lies within distance $2a_{j-1}-1\le R+g(\lvert C\rvert)$ of $C$, so it lies in $C$ by Lemma~\ref{lem:clustermono}(ii). Hence every cube contains a block of $C$, $\lvert C\rvert\ge K_j^4$, and $R+g(\lvert C\rvert)\ge\gamma\log K_j^4-c_0\ge2a_j-1$ by~\eqref{eq:percreach}. Induction on $j$ gives $\lvert C\rvert\ge K_J^4$. \end{proof} Fig.~\ref{fig:percolation} draws the construction and the clusters of a window. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_percolation.png} \caption{Clusters of bad blocks, schematic, in three of the four dimensions. (a)~The hierarchical construction of Theorem~\ref{thm:clusterpercolation}: bad blocks grouped into clusters of the first level in boxes of side $a_1$, and these joined into one cluster of the second level in a box of side $a_2$; the joining distance grows like the logarithm of the size. (b)~A window $\Lambda$ of the torus: a cluster of the torus that crosses its boundary is cut into clusters of the window, the units of Lemma~\ref{lem:windowclusters}.} \label{fig:percolation} \end{figure*} \begin{corollary}[the bad blocks percolate]\label{cor:clusterpercolation} Let $G=\SU(2)$, $r\le\frac1{10}$, $\beta_W>0$, and let $B$ be the set of bad blocks of a torus of side $L$ under $\mu_L$. Let $g$, $R$, $\gamma$ and $c_0$ be as in Theorem~\ref{thm:clusterpercolation}, which for the clusters of Definition~\ref{def:clusters} holds with $\gamma=3/\kappa$ and $c_0=0$, and let $p=p_0=\frac12e^{-12\beta_W}$, with $a_0$, $M_0$, $K_j$, $\ell_j$ as there. Then for every $J\ge1$ and $L\ge\ell_J$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item for every block $y$ and every $\Lambda\supset y+\{0,\dots,\ell_J-1\} ^4$, in particular the torus, the cluster of $y$ in $B\cap\Lambda$ has at least $\max\{M_0,K_J^4\}$ blocks with probability at least $\frac12p_0^{M_0}$; \item if $n\ge1$ is an integer with $n\ell_J\le L$, some cluster of $B$ has at least $K_J^4$ blocks with probability at least $1-(1-\frac12p_0^{ M_0})^{n^4}$. \end{enumerate} Hence, for every $K$, $\mu_L\{y\in B,\ \lvert U(y)\rvert>K\}\ge\frac12 p_0^{M_0}$ on every large enough torus, and the probability that some cluster has more than $K$ blocks tends to one as $L\to\infty$. With $\gamma=3/\kappa$, $c_0=0$ and $a_0=\lceil\max\{(24/(\gamma p_0))^{1/3}, (23/p_0)^{1/4},2\}\rceil$, if $\kappa\ge2.5\cdot10^{-4}$ and $\beta_W\ge 10^{37}$, then $\log M_0>e^{10^{37}}$, so $M_0>M_1=10^{10^{1700}}$. \end{corollary} \begin{proof} For a set $\Lambda$ and a block $y$ the event that $y\in B$ and the cluster of $y$ in $B\cap\Lambda$ has at least $K$ blocks is increasing, by Lemma~\ref{lem:clustermono}(i), and so is the event that some cluster of $B$ has at least $K$ blocks. (i) is Lemma~\ref{lem:onelinkbad} and Theorem~\ref{thm:clusterpercolation}. (ii) Choose $n^4$ disjoint translates $Q_i=y_i+\{0,\dots,\ell_J-1\}^4$. The events of Theorem~\ref{thm:clusterpercolation} for $(y_i,Q_i,\Lambda=Q_i)$ depend on disjoint parts of $\xi$, so they are independent; on each, the cluster of $y_i$ in $\xi\cap Q_i$ has at least $K_J^4$ blocks, and it lies in a cluster of $\xi$ by Lemma~\ref{lem:clustermono}(i). Then apply Lemma~\ref{lem:onelinkbad}. The consequence follows since $K_J\to\infty$. For the numbers, $a_0\ge(24/(\gamma p_0))^{1/3}=(48/\gamma)^{1/3}e^{4 \beta_W}$ and $\log M_0\ge2a_0/\gamma$. With $\gamma\le12000$, $2/\gamma \ge1/6000$ and $(48/\gamma)^{1/3}\ge250^{-1/3}>\frac17$, so $\log M_0>e^{4 \beta_W}/42000>e^{4\beta_W-11}\ge e^{10^{37}}$, while $\log M_1=10^{1700} \log10<10^{1701}M_1$ blocks, and its bound (ii) grows with $L^4$. The corollary shows that this growth is not an artefact of the union bound. The next proposition shows that no choice of the separation avoids it while keeping the island sums bounded. \begin{proposition}[the separation the sums need]\label{prop:separationneed} Let $g\ge0$ be nondecreasing, $R\ge2s+8$ with $R>2g(1)$, and form the clusters with~\eqref{eq:clusterrule}. For $m\ge1$ put $h=R+\lfloor g(m)+ g(1)\rfloor+1$, and on a torus of side at least $2(m+2)(h+1)$ let the bad blocks be $u_i=iRe_1$ and $v_i=u_i+(-1)^ihe_2$, $1\le i\le m$. Then the clusters are $U=\{u_1,\dots,u_m\}$ and $V_i=\{v_i\}$, and with $\bar\varDelta$ for clusters as in Proposition~\ref{prop:clusterreach} and $c_2=b_*\cdot45(2s+7)^4e^{-2\alpha_G(R+g(1)-2s-8)}$, \[ \sum_i\bar\varDelta_{UV_i}\ge c_2\,m^{3/2}e^{-2\alpha_Gg(m)} . \] If these sums are at most $\tau_g$ for every $m$, then $R+g(m)\ge\gamma \log m-c_0$ for every $m$, with $\gamma=3/(4\alpha_G)$ and $c_0=\max\{0, (2\alpha_G)^{-1}\log(\tau_g/c_2)\}$, and the conclusions of Corollary~\ref{cor:clusterpercolation} hold for these clusters. \end{proposition} \begin{proof} The distances are $\dist(u_i,u_j)=\lvert i-j\rvert R$, $\dist(v_i,u_j)= \max\{\lvert i-j\rvert R,h\}$, and $\dist(v_i,v_j)\ge\min\{2R,2h\}=2R$ for $i\ne j$, since $v_i$ and $v_j$ differ by $2h$ in the direction $e_2$ when $i-j$ is odd; the side of the torus exceeds $2mR$ and $4h$, so these are the distances on the torus. The partition into $U$ and the $V_i$ is separated: $\dist(U,V_i)=h>R+g(m)+g(1)$ and $\dist(V_i,V_j)\ge2R>R+2g(1)$. Consecutive $u_i$ are at distance $R$, so they lie in one cluster by Lemma~\ref{lem:clusters}(ii); hence $U$ is a cluster, and the finest separated partition is this one. The neighbourhoods of radius $s+3$ of the $u_i$ are disjoint, since $R>2s+6$, so $\bar n_U=45m(2s+7)^4$ and $\bar n_{V_i}=45(2s+7)^4$, and $w_{UV_i}=h-1\le R+g(m)+g(1)$. So $\bar\varDelta _{UV_i}\ge c_2\sqrt m\,e^{-2\alpha_Gg(m)}$ for each of the $m$ values of $i$. If the sums are at most $\tau_g$, then $2\alpha_Gg(m)\ge\frac32\log m- \log(\tau_g/c_2)$, which with $R\ge0$ gives the condition of Theorem~\ref{thm:clusterpercolation}. \end{proof} For the clusters of Definition~\ref{def:clusters}, $g(m)=(3/\kappa)\log m$ with $\kappa\le2\alpha_G$ exceeds $(3/(4\alpha_G))\log m$, as Theorem~\ref{thm:clustercomparison} needs. So every separation rule of the form~\eqref{eq:clusterrule} that bounds the island sums uniformly in the configuration produces clusters that percolate, by Corollary~\ref{cor:clusterpercolation}, and no bound on the size of the clusters of the torus is uniform in the torus. The clusters of a window replace them: by Lemma~\ref{lem:windowclusters} the comparison of Theorem~\ref{thm:clustercomparison} holds with the clusters of $B\cap\Lambda$ as units, and in a window of at most $10^{M_1/2}$ blocks their size is bounded uniformly in the torus. The size of the window enters only through the term $\lvert\Lambda\rvert10^{-0.67M_1}$, and it cannot be removed: by the proof of Corollary~\ref{cor:clusterpercolation}(ii), in a window that contains $n^4$ disjoint boxes of side $\ell_1$ some cluster of $B\cap\Lambda$ has at least $M_0>M_1$ blocks with probability at least $1-(1-\frac12p_0^{M_0}) ^{n^4}$. The covariance of the conditional probabilities of the labels, the term of Proposition~\ref{prop:hiddenbit}, is not touched by the clusters. \subsection{The comparison across one step and the conditional iteration}\label{ssec:iterate} This subsection settles the constants of the comparison between the coarse action $A_1$ of Sec.~\ref{ssec:lfstep} and $\lambda$ times the coarse curvature $Q_1$, and states the iteration over scales as a conditional theorem. The factor $C_*$ of Theorem~\ref{thm:lfaction}(i) is replaced by $2956$ for every coarse field and by $1/\theta(\kappa_0)=1+O(\kappa_0)$ for the coarse plaquettes below $\kappa_1$, which is about $0.19\,\kappa_0$ for small $\kappa_0$. No two-sided comparison of $A_1$ with $\lambda Q_1$ has a ratio of constants below $4t_L^2$, which tends to $4$: the factor four between the block-spin form and the coarse curvature is attained by a staggered coarse field and is intrinsic. The conditional theorem names one-step hypotheses $(\mathrm H_k)$, proves that they compose when their relative errors are summable, and locates the couplings at which the large-field estimates of Sec.~\ref{ssec:lfstep} stop. Throughout, $G=\SU(2)$, $L=2^m\ge32$, and the notation is that of Sec.~\ref{ssec:lfstep}. \emph{Norms.} For $g\in\SU(2)$ with angle $2x$, $\delta(g,I)=2\sqrt2\sin x$ and $\lVert\log g\rVert=2\sqrt2\,x$ (Sec.~\ref{ssec:witnessradius}), so $\delta(g,I)\le\lVert\log g\rVert$, and the ratio $x/\sin x$ increases with $x$. If $\delta(g,I)\le\kappa_0\le10^{-3}$, then $x\le1.001\sin x$, $x^2\le 1.002\,\kappa_0^2/8$ and $(x/\sin x)^2\le(1-x^2/6)^{-2}\le1+x^2/2.98$, so \begin{equation}\label{eq:iteratenorm} \lVert\log g\rVert^2\le\Bigl(1+\frac{\kappa_0^2}{20}\Bigr)\delta(g,I)^2, \qquad \lVert\log g\rVert\le\Bigl(1+\frac{\kappa_0^2}{40}\Bigr)\kappa_0 . \end{equation} For $0<\kappa_0\le9.99\cdot10^{-4}$ put \begin{gather*} \theta(\kappa_0)=\Bigl[\Bigl(1+\frac{\kappa_0^2}{20}\Bigr)\bigl(1+6031\, \kappa_0\bigr)^2\Bigr]^{-1},\\ \kappa_1=\frac{\kappa_0}{\sqrt{28\,\theta(\kappa_0)}} . \end{gather*} Then $\theta(\kappa_0)\ge1-12063\,\kappa_0$, since $(1+u)^{-2}\ge1-2u$, and \begin{center} \begin{tabular}{lll} \hline $\kappa_0$ & $\theta(\kappa_0)$ & $\kappa_1/\kappa_0$\\ \hline $10^{-4}$ & $0.38911\ldots$ & $0.30296\ldots$\\ $10^{-6}$ & $0.98804\ldots$ & $0.19012\ldots$\\ $10^{-8}$ & $0.99987\ldots$ & $0.18899\ldots$\\ \hline \end{tabular} \end{center} \begin{theorem}[comparison constants of one step]\label{thm:stepconstants} For every coarse field $c$ on the torus: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $S_1(c)\ge Q_1(c)/2956$, and $A_1(c)\ge\lambda Q_1(c)/2956-4|T| \log M$; \item for $0<\kappa_0\le9.99\cdot10^{-4}$, \[ S_1(c)\ge\theta(\kappa_0)\sum_P\min\bigl\{\delta(c_P,I)^2,\kappa_1^2 \bigr\}, \] and $A_1(c)$ is at least $\lambda$ times the right side minus $4|T|\log M$; in particular, if $\delta(c_P,I)\le\kappa_1$ for every coarse plaquette, $A_1(c)\ge\theta(\kappa_0)\lambda Q_1(c)-4|T|\log M$; \item for $\lambda\ge100$ and $\eta>0$, $A_1(c)\le4(1+\eta)\lambda Q_1(c) +c_1(\lambda,\eta)|T|$. \end{enumerate} \end{theorem} \begin{proof} (i) Every $U\in F_c$ is in the tree gauge and has $V^\sharp(U)=c$, so Theorem~\ref{thm:cutoff}(ii) gives $Q_1(c)\le(8+2688\pi^2/9)Q(U)$, and $8+2688\pi^2/9<2956$. Take the infimum over $F_c$ and use the first inequality of Theorem~\ref{thm:lfaction}(i). (ii) Let $U\in F_c$ and let $\mathcal B$ be the set of fine plaquettes with $\delta(U_p,I)>\kappa_0$. For a coarse plaquette $P$ with sides $\ell \subset P$ let $D(P)$ be the union of the squares $\square_{2y+o}$ and of the sets $\mathcal U_\ell$ of the proof of Lemma~\ref{lem:smallstep}. That proof uses its hypothesis on $P$ only through the plaquettes of $D(P)$: the cut-off of a side $\ell$ depends on the paths of $\ell$, whose surfaces lie in $\mathcal U_\ell$, and $K_{P,o}$ is the holonomy of $\square_{2y+o}$. Let $\mathcal P_1$ be the coarse plaquettes with $D(P)\cap\mathcal B= \emptyset$ and $\mathcal P_2$ the others. Let $E_{\rm g}$ and $Q_{\rm g}$ be the sums of $\lVert\Theta_p\rVert^2$ and of $\delta(U_p,I)^2$ over the fine plaquettes off $\mathcal B$. By~\eqref{eq:iteratenorm} these have $\lVert\Theta_p\rVert\le\eta:=(1+\kappa_0^2/40)\kappa_0<10^{-3}$, and $E_{\rm g}\le(1+\kappa_0^2/20)Q_{\rm g}$. For $P\in\mathcal P_1$ the first part of the proof of Lemma~\ref{lem:smallstep} applies with this $\eta$ and gives the two inequalities of~\eqref{eq:smallstep} and $r_P\le28.26\,\eta$. Its summation, restricted to $\mathcal P_1$, involves only plaquettes off $\mathcal B$, and its multiplicities (a plaquette in four squares $\square_x$, in at most four sets $\mathcal U_\ell$, a coarse link in six coarse plaquettes) do not grow under the restriction. The triangle inequality for the Euclidean norm over $\mathcal P_1$ therefore gives \[ \Bigl(\sum_{P\in\mathcal P_1}\lVert\log c_P\rVert^2\Bigr)^{1/2}\le(1+6030 \,\eta)\,E_{\rm g}^{1/2}. \] Since $6030\,\eta\le6031\,\kappa_0$ and $\delta(c_P,I)\le\lVert\log c_P \rVert$, \[ \sum_{P\in\mathcal P_1}\delta(c_P,I)^2\le\theta(\kappa_0)^{-1}Q_{\rm g}. \] A fine plaquette lies in $\square_{2y+o}$ for at most four pairs $(P,o)$, since the pairs are in bijection with the squares $\square_x$, and in $\mathcal U_\ell$ for at most four coarse links, each a side of six coarse plaquettes. So it lies in $D(P)$ for at most $28$ coarse plaquettes, and $|\mathcal P_2|\le28|\mathcal B|$. Hence \begin{align*} Q(U)&=Q_{\rm g}+\sum_{p\in\mathcal B}\delta(U_p,I)^2\\ &\ge\theta(\kappa_0)\sum_{P\in\mathcal P_1}\delta(c_P,I)^2+\frac{ \kappa_0^2}{28}\,|\mathcal P_2| . \end{align*} Each term of $\theta\sum_P\min\{\delta(c_P,I)^2,\kappa_1^2\}$ is at most the corresponding term on the right, as $\theta\kappa_1^2=\kappa_0^2/28$. Take the infimum over $F_c$ and apply Theorem~\ref{thm:lfaction}(i). (iii) Theorem~\ref{thm:lfaction}(ii) with $U=R(c)$, which is regular, and $Q(R(c))=4Q_1(c)$. \end{proof} The constant $2956$ of (i) is less than $C_*/555$. Part (ii) is the small-field form: the coarse plaquettes below $\kappa_1\approx0.19\, \kappa_0$ are compared with the factor $\theta(\kappa_0)$, which tends to one with $\kappa_0$, and every larger coarse plaquette costs at least $\theta\kappa_1^2=\kappa_0^2/28$. The upper constant $4$ of (iii) and the lower constant $\theta(\kappa_0)$ of (ii) cannot be brought together, by the next theorem. For a real field $\gamma$ on the coarse links of the torus let \[ q_1(\gamma)=\min\Bigl\{\sum_p(d\varphi)_p^2:\varphi\ \text{real on the links of}\ \Lambda_L,\ V(\varphi)=\gamma\Bigr\}, \] the linear block-spin form, with $V$ the block average of Definition~\ref{def:blockgauge} and the curvature $(d\varphi)_p=\varphi_ \mu(x)+\varphi_\nu(x+e_\mu)-\varphi_\mu(x+e_\nu)-\varphi_\nu(x)$. By Proposition~\ref{prop:effective}(i), whose proof applies on every lattice, $q_1\ge Q_1$. Let $s(y)=(-1)^{y_0+y_1}$ and define $\gamma_B$ by $\gamma_{B,(y,0)}=s(y)$, $\gamma_{B,(y,1)}=-s(y)$ and zero in the directions $2,3$; let $n=L/2$ and $\gamma_A$ equal $\cos(2\pi y_0/n)$ on the coarse links $(y,1)$ and zero on the others. Put $t_L=\frac34+\frac14 \cos(4\pi/L)$ and $f(u)=4(1-\cos(u/\sqrt2))$, so that $\delta(e^{u\tau},I)^2= f(u)$ for $\lVert\tau\rVert=1$ and $u^2(1-u^2/24)\le f(u)\le u^2$. \begin{theorem}[the factor four is intrinsic]\label{thm:stepsharp} Let $\tau\in\g$ with $\lVert\tau\rVert=1$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $q_1(\gamma_B)=64|T|=4\,Q_1(\gamma_B)$, attained at the field $\varphi^*(\gamma_B)$ of Proposition~\ref{prop:effective}(ii), and $q_1( \gamma_A)\le t_L^{-2}Q_1(\gamma_A)$. \item There is $C$, depending on $L$, such that $c^\varepsilon=e^{ \varepsilon\tau\gamma_B}$ has $Q_1(c^\varepsilon)=|T|f(4\varepsilon)\le16 \varepsilon^2|T|$ and $S_1(c^\varepsilon)\ge(64-C\varepsilon)\varepsilon^2 |T|$ for $0<\varepsilon\le C^{-1}$. \item For $0<\varepsilon\le C^{-1}$ the field $c_\varepsilon=e^{\varepsilon \tau\gamma_A}$ has a regular $U_\varepsilon\in F_{c_\varepsilon}$ with $Q(U_\varepsilon)\le t_L^{-2}(1+C\varepsilon^2)\,Q_1(c_\varepsilon)$. \item If $a,b>0$ and $e(\lambda)$ with $e(\lambda)/\lambda\to0$ satisfy \[ a\lambda Q_1(c)-e(\lambda)\le A_1(c)\le b\lambda Q_1(c)+e(\lambda) \] for every $c$ and every $\lambda\ge100$, then $b\ge4$ and $a\le t_L^{-2}$. So $b/a\ge4t_L^2$, which is above $3.849$ for $L\ge32$ and tends to $4$. \end{enumerate} \end{theorem} \begin{proof} (i) Write $\varphi^*=\varphi^*(\gamma_B)$, equal to $\gamma_{B,(y,\mu)}$ on the links $(x,\mu)$, $x\in B(y)$, $o_\mu=1$, and to zero elsewhere. By Proposition~\ref{prop:effective}(ii), $V(\varphi^*)=\gamma_B$ and $\sum_p( d\varphi^*)_p^2=4Q_1(\gamma_B)$, and its curvature is $(d\gamma_B)_P$ on the four plaquettes at $x\in B(y)$ with $o_\mu=o_\nu=1$ and zero elsewhere. Here $(d\gamma_B)_P=4s(y)$ in the plane $\{0,1\}$, since $s$ changes sign under $e_0$ and $e_1$, and zero in the other planes, so $Q_1(\gamma_B)= 16|T|$. We claim \begin{equation}\label{eq:kktidentity} d^{\mathsf T}d\varphi^*=4\varphi^*=V^{\mathsf T}(32\,\gamma_B). \end{equation} Only plaquettes in the plane $\{0,1\}$ are curved. For the link $(x,0)$ the plaquettes at $x$ and at $x-e_1$ in that plane enter with signs $+$ and $-$; the first is $4s(y)$ when $o_0=o_1=1$, and the second is $4s(y-e_1)=-4s(y)$ when $o_0=1$, $o_1=0$. So $(d^{\mathsf T}d\varphi^*)_{ (x,0)}=4s(y)$ when $o_0=1$ and zero otherwise. The link $(x,1)$ is the same with the roles of the directions exchanged and the sign of the orientation reversed, which gives $-4s(y)$ when $o_1=1$. Links in the directions $2,3$ lie in no curved plaquette. For the second equality, a link $(x,\mu)$ with $o_\mu=1$ enters only $V_{y,\mu}$, with coefficient $\frac18$, and a link with $o_\mu=0$ enters $V_{y,\mu}$ and $V_{y-e_\mu, \mu}$ with $\frac1{16}$ each, and $\gamma_{B,(y-e_\mu,\mu)}=-\gamma_{B,( y,\mu)}$. Now for every real $\varphi$ with $V(\varphi)=\gamma$, \begin{equation}\label{eq:kktbound} \sum_p(d\varphi)_p^2\ge2\ip{d\varphi}{d\varphi^*}-\sum_p(d\varphi^*)_p^2= 64\ip{\gamma}{\gamma_B}-64|T|, \end{equation} which for $\gamma=\gamma_B$, $\lvert\gamma_B\rvert^2=2|T|$, is $64|T|$, with equality at $\varphi^*$. For $\gamma_A$, $T$ of Lemma~\ref{lem:whitney} multiplies $\gamma_A$ by $t_L$, since $\gamma_A$ depends on $y_0$ alone and $\tau_0\cos(2\pi y_0/n)=(\frac34+\frac14\cos\frac{2\pi}n)\cos(2\pi y_0/n)$. Lemma~\ref{lem:whitney} with $k=1$ on the coarse torus of side $n$, whose proof uses only the local structure, gives $V(\Pi(\gamma_A/t_L))=\gamma_A$ and $\sum_p(d\Pi(\gamma_A/t_L))_p^2\le Q_1(\gamma_A)/t_L^2$. (ii) The curvature of $c^\varepsilon$ is $e^{4\varepsilon s(y)\tau}$ in the plane $\{0,1\}$ and $I$ elsewhere, so $Q_1(c^\varepsilon)=|T|f(4 \varepsilon)$. Let $U\in F_{c^\varepsilon}$; we may assume $Q(U)\le64 \varepsilon^2|T|$. Then every plaquette has $\delta(U_p,I)\le8\varepsilon |T|^{1/2}$, and for $\varepsilon$ small Lemma~\ref{lem:smallframes}(i) and (iv) with $W=T$ put every link within $C\varepsilon$ of $I$, as in the proof of Theorem~\ref{thm:sfgauss}(ii), using $\delta(c^\varepsilon_k,I) \le\varepsilon$. By Lemma~\ref{lem:logbound}, $U=e^X$ with $|X|=\max_e\lVert X_e\rVert\le C\varepsilon$. By Lemma~\ref{lem:bch}, $\lVert\log U_p-(dX)_ p\rVert\le C\varepsilon^2$, and $\delta(e^Y,I)^2\ge\lVert Y\rVert^2(1- \lVert Y\rVert^2/24)$, so $Q(U)\ge\sum_p\lVert(dX)_p\rVert^2-C\varepsilon^3 |T|$. As in that proof, $\log V^\sharp_k(e^X)=V_k(X)+O(|X|^2)$, so the component $X^\tau=\ip X\tau$ has $V(X^\tau)=\varepsilon\gamma_B+r$ with $\lvert r_k\rvert\le C\varepsilon^2$. Apply~\eqref{eq:kktbound} to $X^\tau/\varepsilon$: \begin{align*} \sum_p\lVert(dX)_p\rVert^2&\ge\sum_p(dX^\tau)_p^2\ge64\varepsilon^2|T|-64 \varepsilon\ip r{\gamma_B}\\ &\ge(64-C\varepsilon)\varepsilon^2|T| . \end{align*} (iii) By Lemma~\ref{lem:straightpaths}(iii), whose proof is local, $\Pi(\gamma_A/ t_L)$ plus a gradient that vanishes under $V$ is a field $\varphi_A$ in the tree gauge with $V(\varphi_A)=\gamma_A$ and $\sum_p(d\varphi_A)_p^2\le Q_1(\gamma_A)/t_L^2$. Let $U_\varepsilon=e^{\varepsilon\tau\varphi_A}$. All its links lie in the one-parameter group $e^{\R\tau}$, so the path product $P_x$ of a coarse link is $e^{\varepsilon\tau\sigma_x}$, $\sigma_x$ the sum of $\varphi_A$ along the path of length two from $x$. For $\varepsilon$ small, $\delta(P_x,P_{2y})\le\varepsilon\lvert\sigma_x- \sigma_{2y}\rvert$ is below $\frac14$, so $U_\varepsilon$ is regular, and Theorem~\ref{thm:cutoff}(i) gives $V^\sharp_{y,\mu}(U_\varepsilon)=e^{ \varepsilon\tau V_{y,\mu}(\varphi_A)}$, that is $V^\sharp(U_\varepsilon)= c_\varepsilon$. Then $Q(U_\varepsilon)=\sum_pf(\varepsilon(d\varphi_A)_p) \le\varepsilon^2Q_1(\gamma_A)/t_L^2$ and $Q_1(c_\varepsilon)=\sum_Pf( \varepsilon(d\gamma_A)_P)\ge\varepsilon^2Q_1(\gamma_A)(1-\varepsilon^2/6)$, as $\lvert(d\gamma_A)_P\rvert\le2$. (iv) By Theorem~\ref{thm:lfaction}(i), $\lambda S_1(c)-4|T|\log M\le A_1(c)\le b\lambda Q_1(c)+e(\lambda)$; divide by $\lambda$ and let $\lambda \to\infty$: $S_1(c)\le bQ_1(c)$. At $c^\varepsilon$, (ii) gives $b\ge4- C\varepsilon/16$ for every small $\varepsilon$. By Theorem~\ref{thm:lfaction}(ii) with any $\eta>0$ and the regular field $U_\varepsilon$, $a\lambda Q_1(c_\varepsilon)-e(\lambda)\le(1+\eta)\lambda Q(U_\varepsilon)+c_1(\lambda,\eta)|T|$, and $c_1(\lambda,\eta)$ grows like $\log\lambda$; so $aQ_1(c_\varepsilon)\le Q(U_\varepsilon)$, and (iii) gives $a\le t_L^{-2}(1+C\varepsilon^2)$. The value $4t_{32}^2=3.8491\ldots$ and $t_L\to1$ give the last statement. \end{proof} \begin{remark}\label{rem:stepconstants} Part (i) makes exact the right end of the Bloch spectrum stated after Proposition~\ref{prop:effective}: $\gamma_B$ is the mode of momentum $(\pi,\pi,0,0)$ in the plane $\{0,1\}$, and~\eqref{eq:kktidentity} is the Lagrange condition of $q_1$ at it, with multiplier $64\gamma_B$. The field $\gamma_A$ is the long-wavelength end. So $C_*$ is a constant of the witness argument of Lemma~\ref{lem:lfcoarseenergy} and not of the step: the best lower constant lies between $\theta(\kappa_0)$ on the small coarse fields and $t_L^{-2}$, the best upper constant is $4$, and a comparison with $\lambda Q_1$ with constants $1+o(1)$ on both sides does not exist. Constants close to one on both sides hold for the fibre infimum itself: by Theorem~\ref{thm:lfaction}, $\lambda S_1(c)-4|T|\log M\le A_1(c)\le(1+ \eta)\lambda Q(U)+c_1(\lambda,\eta)|T|$ for every regular $U\in F_c$. At the linear level $S_1$ is the block-spin form $q_1$, which is not a sum of local terms if $(M_W)_{AB}\ne0$ (Theorem~\ref{thm:sfgauss}). The reference forms $R_k$ below are built from the block-spin form for this reason. \end{remark} \emph{The conditional iteration.} Let $K\ge1$ and $\lambda_0>\lambda_1> \dots>\lambda_K>0$. For $0\le k\le K$ let $\mathcal X_k$ be a finite dimensional real inner product space, for $k0$ (Theorem~\ref{thm:sfperp}), so (H3) at $k=0$ has to keep $Z_\perp$ as part of the weight, and is not available in the form stated. \begin{theorem}[conditional iteration]\label{thm:iterate} Suppose that (H1)--(H3) hold at every step $k1183$ and $c_2(100,1)>1958$, so $h(100)<0$ and $h_2(100)<0$, and, with $\kappa_c\le0.0837617$, $h(245000)\le8.966$ and $h_2(809300)\le8.947$. With $\kappa_c\ge0.0837613$ and $\log M<\log23621$, $a(\lambda,\kappa_c)\ge \lambda\kappa_c^2-8\log23621-c_1(\lambda,1)$, which increases for $\lambda> 67.5/\kappa_c^2$, where $67.5/\kappa_c^2<9621$, and is at least $9.35$ at $\lambda=257000$; and $a_2(\lambda,\kappa_c)\ge0.52\lambda\kappa_c^2-8\log 23621-c_2(\lambda,1)$, which increases for $\lambda>109.5/(0.52\kappa_c^2)$, where $109.5/(0.52\kappa_c^2)<30015$, and is at least $11.66$ at $\lambda= 833000$. With $\beta=4\lambda$ this gives the display. The count is $(809300-1)/t$ with $t=11\log2/(12\pi^2)$, and $\pi\ge3.14159265$ gives $1/t\ge15.53326$, so $(809300-1)/t>1.2571\cdot10^7$. \end{proof} \begin{remark}\label{rem:iteratelinear} Theorem~\ref{thm:stepconstants}(ii) is not a substitute for (H1), because its loss is linear in the field size. Suppose it is used at a step whose coarse fields are calm at a threshold $\kappa'$ satisfying the rate condition $\lambda'\kappa'^2\ge9+c_1(\lambda',1)$ of Theorem~\ref{thm:lfmarkov}(iii) at the next coupling $\lambda'\ge100$, so that $\kappa'\ge34.4\,\lambda'^{-1/2}$, and suppose $\kappa_1\ge\kappa'$. Then $\log\theta(\kappa_0)\le-2\log(1+6031\kappa_0)\le-1716\,\kappa_0$, and either $\theta\le\frac12$ or $\kappa_0\ge\sqrt{14}\,\kappa'$, so $-\log\theta(\kappa_0)\ge\min\{\log2,2.2\cdot10^5\lambda'^{-1/2}\}$, which is $\log2$ for $\lambda'\le10^{11}$. The product of these factors along the running tends to zero as $\lambda_0\to\infty$. The loss comes from Lemma~\ref{lem:smallstep}, whose bound is linear in $\eta$; at the linear level there is no loss, since $q_1\ge Q_1$, and the long-wavelength fields of Theorem~\ref{thm:stepsharp}(iii) show that no lower constant above $t_L^{-2}$ holds. \end{remark} \subsection{The induction over scales}\label{ssec:allscales} This subsection asks which hypotheses of Theorem~\ref{thm:iterate} follow from the conclusions of the previous step, at the trivial background. The relative error (H1) and the large-field cost (H3) are inputs of one step each; they are not implied by the earlier steps. The locality hypothesis (H2) is different: it bounds the inverse of the whole Hessian, and Proposition~\ref{prop:h2volume} shows that it cannot hold with constants independent of the volume, since the inverse carries the long-wavelength modes. It is replaced below by a bound on the form $H_k$ itself. The induction step is then proved: the relative comparison, the fibre gap and the locality of the fibre covariance pass from step $k$ to step $k+1$ under one condition (C) on the error of the step $k$ alone, relative and local (Proposition~\ref{prop:fibreinherit}, Theorem~\ref{thm:allscales}), the locality with constants that grow with $k$. Theorem~\ref{thm:allscales} states the induction, and Proposition~\ref{prop:scalecount} counts the scales. \emph{Setting.} Throughout, $H_0=2\lambda_0\sum_p\lVert(dX)_p\rVert^2$ on $\mathcal X_0$, $\lambda_k=\lambda_0-tk$ with $t=11\log2/(12\pi^2)$ as in Theorem~\ref{thm:iterate}(b), and $R_k=2\lambda_kS_k$. The torus of scale $k$ has side $n_k=L/2^k$ in units of its spacing. Let $\hat{\mathcal X}_k$ be the $\g$-valued fields on all links of that torus, so that $\mathcal X_k \subset\hat{\mathcal X}_k$, and let $\hat R_k=2\lambda_kS_k$ on $\hat {\mathcal X}_k$; $R_k$ is its restriction to $\mathcal X_k$. The map $B_k=V$ sends $\mathcal X_k$ onto $\hat{\mathcal X}_{k+1}$ (Proposition~\ref {prop:effective}(ii)), $\mathcal M_kH$ is a form on $\hat{\mathcal X}_{k+1} $, and (H1) compares its restriction to $\mathcal X_{k+1}$ with $H_{k+1}$. All forms are written as symmetric matrices with blocks $H_{ab}\in \operatorname{End}(\g)$ indexed by links, and $\lVert H_{ab}\rVert$ is the operator norm. A link of scale $k$ sits at its midpoint, measured in units of the spacing $2^k$, and $d_k$ is the $\ell^\infty$ distance on the torus between midpoints. For a matrix $M$ indexed by a set with a pseudometric $d$ and $\alpha\ge0$ put \[ \lVert M\rVert_{(\alpha)}=\sup_a\sum_b\lVert M_{ab}\rVert e^{\alpha d(a,b)}. \] For symmetric $M$ the Schur test gives $\lVert M\rVert\le\lVert M\rVert_{(0)} \le\lVert M\rVert_{(\alpha)}$, and $\lVert\cdot\rVert_{(\alpha)}$ does not increase under restriction to a subset of the indices. \begin{lemma}\label{lem:weightedcount} For $00$ satisfies the bounds of (H2) at scale $k$ only if $|c-1|<5\pi^2/(6n_k^2)$. \item On the same $\mathcal Y$, the form $E=\varepsilon\lambda_k|x|^2$ satisfies $|E|\le\varepsilon'R_k$ on $\mathcal Y$ only if $\varepsilon'\ge \varepsilon n_k^2/(10\pi^2)$. \end{enumerate} \end{proposition} \begin{proof} (i) Write $n=n_k$, $N=2^kn$, $q=2\pi/N$, and fix $Z\in\g$ with $\lVert Z \rVert=1$. Let $\varphi$ be the fine field with $\varphi_{(x,0)}=\cos(qx_1) Z$ and $\varphi=0$ on the links of the other directions.\footnote{This is the lowest nonconstant Fourier mode of the torus. Its energy is of order $N^{-2}$ times its norm, the gap of the discrete Laplacian, and this is why the bound depends on the volume.} Its only nonzero curvatures are on the plaquettes of the plane $\{0,1\}$, where $(d\varphi)_p =(\cos qx_1-\cos q(x_1+1))Z=2\sin\frac q2\sin q(x_1+\frac12)Z$, so \[ Q(\varphi)=N^3\cdot4\sin^2\frac\pi N\sum_{x_1=0}^{N-1}\sin^2q\Bigl(x_1+ \frac12\Bigr)=2N^4\sin^2\frac\pi N . \] By Lemma~\ref{lem:straightpaths}(i), an identity of linear maps that holds on the torus as well, $V^{(k)}(\varphi)$ vanishes off the direction $0$ and \[ V^{(k)}_{y,0}(\varphi)=\sum_{j=0}^{2^k-1}\cos q(2^ky_1+j)=\rho\,\tilde w_{y,0}, \] with $\tilde w_{y,0}=\cos(2\pi y_1/n+c_0)Z$, $c_0=q(2^k-1)/2$, $\tilde w=0$ on the other directions, and $\rho=\sin(\pi/n)/\sin(\pi/N)$. Since $n\ge3$, $|\tilde w|^2=n^4/2$. Hence \begin{align*} \frac{S_k(\tilde w)}{|\tilde w|^2}&\le\frac{Q(\varphi)}{\rho^2|\tilde w|^2} =\frac{4\cdot16^k\sin^4(\pi/N)}{\sin^2(\pi/n)}\\ &\le\frac{4\pi^4}{n^4\sin^2(\pi/n)}\le\frac{4\pi^2}{0.8049\,n^2}<\frac{5 \pi^2}{n^2}, \end{align*} using $\sin x\le x$, and $\sin x\ge x(1-x^2/6)$ at $x=\pi/n\le\pi/4$ with $(1-\pi^2/96)^2>0.8049$. So $\hat R_k(\tilde w)\le10\pi^2\lambda_k|\tilde w|^2/n^2$. $\tilde w$ is orthogonal to $\ker\hat R_k$. Indeed $S_k\ge Q_k$ by Proposition~\ref{prop:effective}(i), whose proof applies on every lattice, so a field $w$ in $\ker\hat R_k$ has zero curvature. The curvature of a plaquette in the plane $\{0,1\}$ at $y$ is $w_{y,0}+w_{y+e_0,1}-w_{y+e_1,0} -w_{y,1}$; summed over $y_0$ around the torus, the terms in the direction $1$ cancel, so $f(y_1,y_2,y_3)=\sum_{y_0}w_{y,0}$ does not depend on $y_1$. Then $\langle\tilde w,w\rangle=\sum_{y_2,y_3}\langle Z,f(\cdot,y_2,y_3)\rangle \sum_{y_1}\cos(2\pi y_1/n+c_0)=0$. The form $S_k$ is invariant under the coarse gauge directions: for $h$ on the sites of scale $k$, let $h_f$ be the fine function equal to $h(y)$ on $B_k(y)$; then $V^{(k)}(dh_f)=-d_kh$ by Lemma~\ref{lem:straightpaths}(ii), so $\varphi\mapsto\varphi-dh_f$ maps the fields with $V^{(k)}(\varphi)=w$ onto those with $V^{(k)}(\varphi)=w+d_kh$ and preserves $Q$. As in the proof of Lemma~\ref{lem:straightpaths}(iii), applied to the blocks of side two of the lattice of scale $k$, there is $h$ with $t=\tilde w+d_kh\in \mathcal X_k$, and $t-\tilde w\in\ker\hat R_k$. By hypothesis $t=y+z$ with $y\in\mathcal Y$ and $z\in\mathcal X_k\cap\ker R_k\subset\ker\hat R_k$. A positive semidefinite form is constant on the cosets of its kernel, so $R_k(y)=\hat R_k(\tilde w)$; and $y\in\tilde w+\ker\hat R_k$ with $\tilde w\perp\ker\hat R_k$, so $|y|\ge|\tilde w|$. Therefore $\lVert(R_k|_{\mathcal Y})^{-1}\rVert\ge|y|^2/R_k(y)\ge n^2/(10\pi^2\lambda_k)$. (ii) The function $\phi=0$ is $1$-Lipschitz, so (H2) contains $\lVert(R_k|_ {\mathcal Y})^{-1}\rVert\le G_k$ and $\lVert(H_k-R_k)|_{\mathcal Y}\rVert \le\ell_k/G_k$; apply (i). At the vector $y$ of (i), $H_k(y)\le R_k(y)+ \lVert(H_k-R_k)|_{\mathcal Y}\rVert|y|^2\le(1+\ell_k)\,10\pi^2\lambda_k |y|^2/n^2$, which gives the last bound. (iii) $R_k$ is block diagonal for $\mathcal X_k=\mathcal Y\oplus(\mathcal X _k\cap\ker R_k)$, so its norm on $\mathcal Y$ is its largest eigenvalue on $\mathcal X_k$, which is at least $R_k(\delta_eZ)$ for the field $\delta_e Z$ equal to $Z$ on one active link $e$ and zero elsewhere. By Proposition~\ref{prop:effective}(iii), $R_k(\delta_eZ)\ge2\lambda_kQ_k( \delta_eZ)=12\lambda_k$, since a link lies in six plaquettes. If $H_k=cR_k$, then $\lVert(H_k-R_k)|_{\mathcal Y}\rVert=|c-1|\lVert R_k|_{\mathcal Y} \rVert\ge12|c-1|\lambda_k$, and (ii) with $\ell_k<1$ gives $12|c-1|<10\pi^2 /n^2$. (iv) At the vector $y$ of (i), $\varepsilon\lambda_k|y|^2$ is at most $\varepsilon'R_k(y)$, which is at most $10\pi^2\varepsilon'\lambda_k|y|^2/ n^2$. \end{proof} The norm of $H_k^{-1}$ on a slice is therefore of order $n_k^2/\lambda_k$ for every $H_k$ with $\lVert H_k-R_k\rVert<\lambda_k$ on it, and the entry bound of Theorem~\ref{thm:iterate}(c) is at least $n_k^2/(10\pi^2\lambda_k) (1+\ell_k)^{-1}$ at distance zero: the covariance of the massless modes does not decay exponentially, uniformly in the volume. By (iii), (H2) excludes a change of the coupling by any factor independent of $L$, even one that (H1) allows with $\varepsilon_k$ fixed. Part (iv) is the torus form of the conversion cost $C_n$ of Remark~\ref{rem:relscales}. The exponential decay that holds uniformly is that of the fibre covariance, where Proposition~\ref{prop:effective}(iv) gives a gap, and it follows from a bound on the form $H_k$ rather than on its inverse: \begin{enumerate}[label=\textup{(H2$^*$)},leftmargin=*] \item \emph{local form:} $\lVert H_k\rVert_{(\alpha_k)}\le\Lambda_k \lambda_k$ on $\mathcal X_k$ with the distance $d_k$, for some $0<\alpha_k \le1$ and $\Lambda_k\ge1$; \end{enumerate} \begin{enumerate}[label=\textup{(D)},leftmargin=*] \item \emph{local one-step error:} the difference $D_k=H_{k+1}- \frac{\lambda_{k+1}}{\lambda_k}\mathcal M_kH_k|_{\mathcal X_{k+1}}$ has $\lVert D_k\rVert_{(\alpha_{k+1})}\le\delta_k\lambda_{k+1}$ with the distance $d_{k+1}$. \end{enumerate} At $k=0$, $H_0=2\lambda_0Q$ has diagonal blocks $12\lambda_0$ and, in each row, eighteen further blocks of norm $2\lambda_0$, at the other links of the six plaquettes of the link, all at distance at most one; so (H2$^*$) holds at $k=0$ for every $\alpha_0\in(0,1]$ with $\Lambda_0=12+ 36e^{\alpha_0}$. Hypothesis (D) concerns one integration only, while (H2) concerns the composition of all earlier ones. For $k5.7 \cdot10^5$. \end{proof} Hypothesis (H2) is not used, and by Proposition~\ref{prop:h2volume} it does not hold with $G_k$ and $\ell_k$ independent of $L$. The large-field cost (H3) is independent of (C): it concerns the values of the action far from the background, which no bound on Hessians at the background controls, and the proof of Theorem~\ref{thm:iterate}(d) uses (H3) alone, so that part holds at every step at which (H3) holds. The theorem is the induction that the present estimates support: the relative comparison and the fibre gap pass from step $k$ to step $k+1$ with constants independent of $k$, and the locality of the fibre passes with constants that grow faster than any exponential. The decay length of the fibre covariance at scale $k$, $\alpha_k^{-1}$ in units of $2^k$, is more than $(5.7\cdot10^5)^k$ in those units, so the bound of (c) says nothing beyond the first steps at any volume. A bound uniform in $k$ cannot come from Proposition~\ref{prop:fibreinherit}(ii), (iii) alone, which use only the size of $H_k$ and its gap, and by part (iv) of that proposition a comparison of $H_k$ with the uniformly local reference $R_k$ (Theorem~\ref{thm:effectivelocal}) in an absolute weighted norm loses a factor of at least four per step. Condition (C) is not proved for any $k$ (Proposition~\ref{prop:iteratestatus}(i), Remark~\ref{rem:relstepinputs}). \begin{proposition}[the count of scales]\label{prop:scalecount} Along the one-loop running $\lambda_k=\lambda_0-tk$, with $\beta=4\lambda$ and $b=11/(3\pi^2)$, so that $4t=b\log2$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item The hypothesis $\lambda_K\ge32$ of Theorem~\ref{thm:allscales} allows $K=\lfloor(\lambda_0-32)/t\rfloor$ steps, which is unbounded as $\lambda_0\to\infty$; under (C) the bounds (a) and (b) of Theorem~\ref{thm:allscales} hold uniformly over all of them. \item The large-field estimates of Sec.~\ref{ssec:lfstep}, in the form proved at the first step (Theorem~\ref{thm:sfcalm}, Proposition~\ref{prop:sfrelative}), give no activity bound at a coupling $\lambda\le809300$, so at most the steps with $\lambda_k>809300$ can receive (H3) from them. At least $(809300-\lambda_K)/t-1$ steps $k809300$ is less than $e^{-8.713\cdot10^6}$ times the spacing of the scale at which $\lambda_k$ reaches one. \item The islands of Theorem~\ref{thm:islandscales} at the Wilson coupling $\beta_W=4\lambda_0$ are controlled at the scales $j\le j_{\max}$, and if $j_{\max}\ge1$ then $j_{\max}\le1+\log_{16}(\beta_W/(2.895\cdot10^{23}))$, while the running from $\beta_W$ to $\beta=4$ has $(\beta_W-4)/(b\log2)$ steps. \item Under (H1) with $\varepsilon_k\le\lambda_{k+1}^{-2}$ and $\lambda_K \ge32$, the ratio of the upper to the lower comparison constant of $H_k$ with $R_k$ is at most $3.3$ for every $k$, and on $\Z^4$, $2\lambda_kQ_k\le R_k\le512\lambda_kQ_k$ for every $k$. The factor four of Theorem~\ref{thm:stepsharp} enters $R_k$ once and does not compound. \end{enumerate} \end{proposition} \begin{proof} (i) $\lambda_K=\lambda_0-tK\ge32$ if and only if $K\le(\lambda_0-32)/t$. (ii) By Proposition~\ref{prop:iteratestatus}(iv), for every $\kappa< \kappa_c$ the rates satisfy $a(\lambda,\kappa)<9$ and $a_2(\lambda,\kappa)< 9$ for $100\le\lambda\le809300$, and the estimates require $\lambda\ge100$. The steps $k 1.257\cdot10^7$ is in Proposition~\ref{prop:iteratestatus}(iv). With the same count for the coupling $\lambda_{k+1}$ of the step, the conclusion is unchanged. One step doubles the spacing, and $k$ steps lower $\beta$ by $bk\log2$; so between $\beta=4\cdot809300=3237200$ and $\beta=4$ the spacing grows by $2^{(3237200-4)/(b\log2)}=e^{(3237200-4)/b}$, and $b\le0.371512$ gives $(3237200-4)/b>8.713\cdot10^6$. (iii) $\beta^*_{j+1}>16\beta^*_j$ and $\beta^*_1\ge2.895\cdot10^{23}$ give $\beta^*_j\ge16^{j-1}\cdot2.895\cdot10^{23}$, and $\beta_W\ge\beta^*_{j_{\max}} $. The count of steps is (i) with $\lambda_K=1$ and $t=b\log2/4$. (iv) Theorem~\ref{thm:allscales}(ii) gives the constants $\frac12$ and $1.65$, whose ratio is $3.3$. The second statement is Theorem~\ref{thm:effectiveupper} and $R_k=2\lambda_kS_k$. The reference is built from infima, $R_{k+1}=\frac{\lambda_{k+1}}{\lambda_k}\mathcal M_kR_k$, and not from a comparison with $\lambda_kQ_k$ at each step. \end{proof} Fig.~\ref{fig:induction} draws the induction and the growth of its constants. \begin{figure*}[t] \centering \includegraphics[width=0.95\textwidth]{figures/fig_induction.png} \caption{The induction over scales of Theorem~\ref{thm:allscales}, schematic in (a). (a)~The tori of the scales $k=0,\dots,4$, the statement $P_k$ on each and the condition~(C) on each step; the couplings $\lambda_k$ are at least $809300$ on the blue tori and below it on the red ones, and along the one-loop running more than $1.257\cdot10^7$ steps lie between $809300$ and $1$ (Proposition~\ref{prop:iteratestatus}). (b)~The lower bound $\Lambda_k\ge48^{6^k}$ on the constants of the induction, as $\log_{10}\log_{10}$, against the bound $\sigma_k\le0.4865$, which holds for every $k$.} \label{fig:induction} \end{figure*} \begin{remark}[how the constants evolve]\label{rem:allscalesconstants} Along the induction of Theorem~\ref{thm:allscales}: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item the coupling decreases by $t$ per step; the number of steps is unbounded as $\lambda_0\to\infty$, and nothing in $P_k$ requires it to be bounded; \item the summed relative error $\sigma$ is at most $\lambda_K^{-2}+1/(t \lambda_K)$ for errors $\varepsilon_k\le\lambda_{k+1}^{-2}$, uniformly in $\lambda_0$; errors $\varepsilon_k\ge c\lambda_{k+1}^{-1}$ give $\sigma\ge (c/t)\log(\lambda_0/\lambda_K)$ (Theorem~\ref{thm:iterate}(b)), so one-step errors of order $g^2$ do not close; \item the comparison constants of $H_k$ with $R_k$ stay in $[\frac12,1.65]$ and the fibre gap stays $\lambda_k/52$; the lower constant $1/2956$ and the factors $\theta(\kappa_0)$ of Theorem~\ref{thm:stepconstants} concern the coarse action $A_1$, not $H_k$, and their products over the steps tend to zero (Remark~\ref{rem:iteratelinear}); \item the locality constants obey $\alpha_{k+1}=\alpha_k/(11900\Lambda_k)$ and $\Lambda_{k+1}=C_*\Lambda_k^6\alpha_k^{-4}+\delta_k$ and are unbounded; an absolute weighted bound on $H_k-R_k$ grows by a factor of at least four per step (Proposition~\ref{prop:fibreinherit}(iv)), as in Remark~\ref{rem:relscales}; \item the large-field input fails first: with the first-step estimates it stops at $\lambda=809300$, more than $1.257\cdot10^7$ steps above coupling one (Proposition~\ref{prop:scalecount}(ii)); \item the island thresholds grow by a factor of more than $16$ per scale, so the islands are controlled on $O(\log\beta_W)$ of the $(\beta_W-4)/(b\log2)$ scales; \item at a curved background, outside the setting of this subsection, the one-step factor of a plaquette remainder is $1+130\lVert\kappa^{(j)} \rVert_2$ at scale $j$ (Proposition~\ref{prop:actionloops}(iii)), and the product over the scales is bounded exactly when $\sum_j\lVert\kappa^{(j)} \rVert_2<\infty$ (Remark~\ref{rem:actioniteration}); on calm windows at the levels $K_j$ this holds if $\sum_jK_j<\infty$ (Corollary~\ref{cor:actioncalm}). Levels $K_j\le c\lambda_{j+1}^{-2}$ have $\sum_jK_j\le c(\lambda_K^{-2}+1/(t \lambda_K))$, as in Theorem~\ref{thm:iterate}(b). Levels $K_j\ge c\lambda_ {j+1}^{-1/2}$, the order of the calm thresholds that the large-field rate condition requires (Remark~\ref{rem:iteratelinear}), have $\sum_{jb_\nu$; the plaquette in the plane $\{\nu,\mu\}$ at $x-e_\nu$ lies in $P_X$ and its two links of direction $\nu$ lie in the tree, so its holonomy $I$ gives $(gV_0)_{x,\mu}=(gV_0)_{ x-e_\nu,\mu}$, and induction gives $gV_0=I$ on $L_X$. With $(Tw)_{y,\mu}= \operatorname{Ad}(g_{y+e_\mu})w_{y,\mu}$, an isometry, $(gV_0)e^{Tw}=g(V_0 e^w)$, so $f(gV_0e^{T\omega})=f(V_0e^\omega)$, $Q^{gV_0}_X(Tw)=Q^{V_0}_X (w)$ and $Td^X_{V_0}\eta=d^X_{gV_0}(\operatorname{Ad}(g)\eta)$, as in that proof. So it suffices to take $V_0=I$, where $d^X_I=d_X$. (i) A constant $h\in G$ acts by $e^\omega\mapsto e^{\operatorname{Ad}(h) \omega}$, so the gradient $(\nabla_kf(0))_k$ is fixed by every $\operatorname{Ad}(h)$, hence commutes with $\g$ and lies in the centre of $\g$, which is $0$. (ii) The proof of Lemma~\ref{lem:invariantslice}(i), with $\Gamma_W$ replaced by $G^X$ and $K$ by $L_X$, gives $\Hess f(0)[w, d_X\eta]=-Df(0)[D\zeta(0)w]$, which vanishes by (i). (iii) The first statement is the transport above. By (ii) the symmetric form $\Hess f(0)$ vanishes on $d_X(\g^X)$ in each argument, so $\Hess f(0)[w,w]=\Hess f(0) [u,u]$ for the orthogonal projection $u$ of $w$ off $d_X(\g^X)$, which is at most $\lVert\Hess f(0)\rVert\lvert u\rvert^2$. Lemma~\ref{lem:boxgap} and $du=dw$, since the curvature of $d_X\eta$ vanishes, bound $\lvert u\rvert^2$ by $\rho_r\sum_{p\in P_X}\lVert(dw)_p\rVert^2$, and at $V_0=I$, $Z_p(w)=(dw)_p$ by~\eqref{eq:pathexp}. \end{proof} \begin{theorem}[local remainders at flat backgrounds]\label{thm:localflat} Let $R=\sum_{X\in\mathcal X}f_X$ be a local remainder, let $V_0$ be flat on every plaquette of every box of $\mathcal X$, the links off $K$ included, and put \[ \mathcal L_0=\sup_p\sum_{X:\,p\in P_X}h_X\rho_{r_X}. \] \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\lvert\Hess R(V_0)[w,w]\rvert\le\mathcal L_0\,Q^{V_0}(w)$ for every $w\in\g^K$, on the window and, with all links variable, on $\Z^4$ or on a torus. \item On the window, $R$ is $\Gamma_W$-invariant and $\lvert\mathcal HR(w,w) \rvert\le\mathcal L_0\,Q^{V_0}(\Pi_{V_0}w)$. At $V_0=I$ this is at most $\mathcal L_0Q_1(w)\le(\mathcal L_0/c_*)\,S_W(w)$, for every $n$. \item If each box carries at most one term and $h_X\le\eta e^{-\alpha r_X}$ with $\eta,\alpha>0$, then $\mathcal L_0\le180\,e^\alpha\eta\,(1-e^{-\alpha} )^{-7}$. \end{enumerate} \end{theorem} \begin{proof} (i) The Hessian of the series is the series of the Hessians. By Lemma~\ref{lem:flatterm}(iii), applied to the restriction of $w$ to $L_X$, \begin{align*} \lvert\Hess R(V_0)[w,w]\rvert&\le\sum_Xh_X\rho_{r_X}\sum_{p\in P_X}\lVert Z_p(w)\rVert^2\\ &=\sum_p\lVert Z_p(w)\rVert^2\sum_{X:\,p\in P_X}h_X\rho_{r_X}, \end{align*} which is at most $\mathcal L_0Q^{V_0}(w)$. (ii) An element of $\Gamma_W$ fixes the links off $K$ and acts on the links of $X$ through its restriction to the sites of $X$, an element of $G^X$; so every $f_X$, and $R$, is $\Gamma_W$-invariant. Lemma~\ref{lem:invariantslice}(ii) gives $\mathcal HR(w,w)=\Hess R(V_0) [\Pi_{V_0}w,\Pi_{V_0}w]$, and (i) applies. At $V_0=I$, $Q^I=Q_1$ vanishes on $\mathcal O$ and is positive semidefinite, so $Q_1(\Pi w)=Q_1(w)$, and $Q_1\le S_W/c_*$ by Lemma~\ref{lem:windowgap}(iii). (iii) A plaquette in the plane $\{\mu,\nu\}$ at $x$ lies in the box of side $r$ with corner $b$ only if $b_\mu\in[x_\mu-r+1,x_\mu]$, $b_\nu\in[x_\nu-r +1,x_\nu]$ and $b_\lambda\in[x_\lambda-r,x_\lambda]$ for the two other directions, so in at most $r^2(r+1)^2$ boxes of side $r$. Hence $\mathcal L_0\le\frac\eta4\sum_{r\ge1}(r+1)^6e^{-\alpha r}$. Since $m^6\le 720\binom{m+6}6$ and $\sum_{m\ge0}\binom{m+6}6x^m=(1-x)^{-7}$, $\sum_{r\ge1}(r+1)^6e^{-\alpha r}\le e^\alpha\sum_{m\ge0}m^6e^{-\alpha m}\le 720\,e^\alpha(1-e^{-\alpha})^{-7}$. \end{proof} The bound holds for every local term; for the filled loops of Proposition~\ref{prop:flatloops} the factor $\rho_r$ is not needed, since their Hessian at a flat background is already a form in the curvatures. At the trivial background it gives the one-step bound of Proposition~\ref{prop:relstep} without $C_n$, once the hypothesis is put on the gauge-invariant density. \begin{corollary}[one step with a local invariant remainder] \label{cor:localstep} Let $U=I$, so that $V_0=I$, and let $A^{\rm gi}=A_0+R$ near $I$, where $A_0$ is $\Gamma_W$-invariant with $\mathcal HA_0=2\lambda S_W$ and $R$ is a local remainder with $\mathcal L_0\le\lambda\varepsilon$. Then the error $E=\mathcal HA^{\rm gi}-2\lambda S_W$ vanishes on $\g^K\times\mathcal O$ and \[ \lvert E(w,w)\rvert\le\lambda\varepsilon\,Q_1(w)\le\frac\varepsilon{2c_*} \cdot2\lambda S_W(w)\qquad(w\in\g^K), \] so that $(1-\frac\varepsilon{2c_*})2\lambda S_W\le\mathcal HA^{\rm gi}\le (1+\frac\varepsilon{2c_*})2\lambda S_W$ for every $n$, with $\frac1{2c_*}< 3.433$. \end{corollary} \begin{proof} By Lemma~\ref{lem:invariantslice}(ii) the covariant Hessian is linear in the function, so $E=\mathcal HR$. It vanishes on $\g^K\times\mathcal O$ by Lemma~\ref{lem:invariantslice}(ii), and Theorem~\ref{thm:localflat}(ii) bounds it. The constant follows from $2c_*>0.29131$, as in Corollary~\ref{cor:loopstep}. \end{proof} The factor $C_n$ of Proposition~\ref{prop:relstep} and of Corollary~\ref{cor:loopstep} therefore comes from the form of the hypothesis. Hypothesis (H) bounds the Hessian of the chart density $A$, which differs from the invariant density $A^{\rm gi}$, the density with respect to Haar measure, by the Haar term. The two cannot both be local on a large window unless the error in (H) is of order $g^2$. \begin{proposition}[the chart hypothesis against locality] \label{prop:chartlocal} Let $G=\SU(2)$ and $U=I$. Suppose that $A^{\rm gi}=A_0+R$ as in Corollary~\ref{cor:localstep}, with constant $\mathcal L_0$, and that the chart density $A$ satisfies the upper half of (H): $\Hess A(0)[\sigma, \sigma]\le2\lambda S_W(\sigma)+\lambda\varepsilon\lvert\sigma\rvert^2$ for $\sigma\in\mathcal O^\perp$. Then \[ \lambda\varepsilon\ge\frac13-\frac{\mathcal L_0}{C_n^-}\ge\frac13-\frac{3 \pi^2\mathcal L_0}{(n+1)^2}. \] In particular $\lambda\varepsilon\ge\frac16$ on every window with $(n+1)^2\ge18\pi^2\mathcal L_0$. \end{proposition} \begin{proof} On the slice $A^{\rm gi}(\Upsilon(s,\eta))=A(s)+\sum_k\log j(s_k)$, so, as in the proof of Proposition~\ref{prop:relstep}(ii), \[ \mathcal HA^{\rm gi}(w,w)=\Hess A(0)[\Pi w,\Pi w]+\frac1{12}\sum_k\Tr \bigl(\operatorname{ad}_{(\Pi w)_k}^2\bigr), \] and the last sum is $-\frac13\lvert\Pi w\rvert^2$ for $\SU(2)$. Let $w\ne0$ be the field of Lemma~\ref{lem:windowgap}(iv), with $w\in \mathcal O^\perp$, so $\Pi w=w$, and $\lvert w\rvert^2=C_n^-Q_1(w)$. The hypothesis gives $\mathcal HA^{\rm gi}(w,w)-2\lambda S_W(w)\le(\lambda \varepsilon-\frac13)\lvert w\rvert^2$, and Corollary~\ref{cor:localstep} gives that this difference is at least $-\mathcal L_0Q_1(w)=-\mathcal L_0 \lvert w\rvert^2/C_n^-$. Divide by $\lvert w\rvert^2$ and use $C_n^-\ge (n+1)^2/(3\pi^2)$. \end{proof} An error of order $g^2=1/\lambda$ in each step is not summable over the scales (Remark~\ref{rem:relscales}). The measure defines the invariant density, and the localised actions of Ba{\l}aban are invariant; so the hypothesis belongs on $A^{\rm gi}$, and in that form the Haar term does not appear. \emph{Curved backgrounds.} At a background that is not flat on a box, a local term has a gradient there, and Lemma~\ref{lem:flatterm} fails. The term is compared with its value at the trivial field after a gauge transformation of the box, which costs a Euclidean term proportional to the curvature in the box, and the Euclidean term is paid with the $\ell^4$ bound of Lemma~\ref{lem:windowlfour}. The following lemma gives the constants from bounds on complex neighbourhoods, the form in which cluster expansions bound their terms. \begin{lemma}[derivatives from analytic bounds]\label{lem:cauchy} Let $f$ be a function of the links of a finite set $L$, $a\ge0$ and $\rho>0$; $\g\otimes\C$ carries the Hermitian extension of the norm, and $e^\xi e^\omega$ is the product link by link. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item If $\omega\mapsto f(e^\omega)$ extends holomorphically to $\{\omega\in(\g\otimes\C)^L:\lVert\omega_k\rVert<\rho\ \forall k\}$ with modulus at most $a$, then $\lVert\Hess f(I)\rVert\le2a/\rho^2$. \item If the function $(\xi,\omega)\mapsto f(e^\xi e^\omega)$ extends holomorphically to the set where $\lVert\xi_k\rVert<\rho$ and $\lVert \omega_k\rVert<\rho$ for every $k$, with modulus at most $a$, then for every $v\in\g^L$ and $V=e^\xi$ with $\xi\in\g^L$ and $\delta=\max_k\lVert \xi_k\rVert\le\rho/2$, \[ \bigl\lvert\Hess f(V)[v,v]-\Hess f(I)[v,v]\bigr\rvert\le\frac{4a\delta} {\rho^3}\lvert v\rvert^2 . \] \item If $G=\SU(N)$ and $\lVert I-V\rVert\le\min(1,3/N)$, then $V=e^\xi$ with $\xi\in\g$ and $\lVert\xi\rVert\le\frac\pi3\sqrt N\,\lVert I-V \rVert$. \end{enumerate} \end{lemma} \begin{proof} (i) For $v\in\g^L$, $v\ne0$, let $m=\max_k\lVert v_k\rVert$. The function $z\mapsto f(e^{zv})$ is holomorphic for $\lvert z\rvert<\rho/m$ and bounded by $a$, and its second derivative at $0$ is $\Hess f(I)[v,v]$, since $\omega=zv$ is a line. Cauchy's inequality gives $\lvert\Hess f(I)[v,v] \rvert\le2am^2/\rho^2\le2a\lvert v\rvert^2/\rho^2$. (ii) Let $G(s,z)=f(e^{s\xi}e^{zv})$, holomorphic for $\lvert s\rvert<\rho/ \delta$ and $\lvert z\rvert<\rho/m$, with $\lvert G\rvert\le a$. Then $\Hess f(V)[v,v]=\partial_z^2G(1,0)$ and $\Hess f(I)[v,v]=\partial_z^2G (0,0)$. For $0\le s\le1$ the bidisc about $(s,0)$ with radii $\rho/(2 \delta)\le\rho/\delta-1$ and $\rho/m$ lies in the domain, so Cauchy's inequality gives $\lvert\partial_s\partial_z^2G(s,0)\rvert\le2a\cdot2\delta m^2/\rho^3$; integrate over $s\in[0,1]$. (iii) Let $e^{i\theta_j}$, $\lvert\theta_j\rvert\le\pi$, be the eigenvalues of $V$. Then $2\lvert\sin(\theta_j/2)\rvert\le\lVert I-V\rVert \le1$, so $\lvert\theta_j\rvert\le\pi/3$, and $\sin x\ge3x/\pi$ on $[0, \pi/6]$ gives $\lvert\theta_j\rvert\le\frac\pi3\lVert I-V\rVert$. The skew-Hermitian $\xi$ with the eigenvalues $i\theta_j$ on the eigenspaces of $V$ has $e^\xi=V$, and $\sum_j\theta_j\in2\pi\Z$ with $\lvert\sum_j\theta_j \rvert\le\frac{N\pi}3\lVert I-V\rVert\le\pi$, so $\Tr\xi=0$ and $\xi\in \g$. Finally $\lVert\xi\rVert^2=\sum_j\theta_j^2$. \end{proof} \begin{theorem}[local remainders at curved backgrounds]\label{thm:localcurved} Let $R=\sum_{X\in\mathcal X}f_X$ be a local remainder near $V_0\in G^K$, the links off $K$ given, every box of $\mathcal X$ containing a link of $K$. Let $\kappa_p=\lVert I-V_{0,p}\rVert$, $\kappa_X=\max_{p\in P_X} \kappa_p$ and $\delta_X=3r_X\kappa_X$, and suppose: \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item $\lVert\kappa\rVert_2\le\frac19$, with $\lVert\kappa\rVert_2$ as in Lemma~\ref{lem:windowlfour}; \item for every $X$ there is $t_X\ge0$ with \[ \bigl\lvert\Hess f_X(V)[v,v]-\Hess f_X(I)[v,v]\bigr\rvert\le t_X\max_{k\in L_X}\lVert I-V_k\rVert\,\lvert v\rvert^2 \] whenever $\max_{k\in L_X}\lVert I-V_k\rVert\le\delta_X$. \end{enumerate} Put $\rho_X=\rho_{r_X}$, $b_X=3r_X\bigl(h_X\rho_X(\sqrt{432}+432\delta_X) +t_X\bigr)$, \[ \begin{gathered} \mathcal L_\kappa=\sup_p\sum_{X:\,p\in P_X}h_X\rho_X\bigl(1+\sqrt{432}\, \delta_X\bigr),\\ B_1=\sup_{k\in K}\sum_{X:\,k\in L_X}b_X,\qquad B_2=\sup_p\sum_{X:\,p\in P_X}b_X\ell_X, \end{gathered} \] and $\kappa_*=(\sum_p\kappa_p^2)^{1/2}$, the sum over the plaquettes of the boxes of $\mathcal X$. Then for every $w\in\g^K$ \[ \lvert\mathcal HR(w,w)\rvert\le\bigl(\mathcal L_\kappa+c_4(B_1B_2)^{1/2} \kappa_*\bigr)\,Q^{V_0}(\Pi_{V_0}w). \] None of the constants depends on $n$. If $V_0$ is flat on the boxes, the factor is $\mathcal L_0$ of Theorem~\ref{thm:localflat}. Under the hypotheses of Lemma~\ref{lem:cauchy}\textup{(ii)}, \textup{(iii)} for $f_X$, with constants $a_X,\rho$, hypothesis~\textup{(b)} holds with $t_X=\frac{4 \pi\sqrt N}3a_X\rho^{-3}$ when $\delta_X\le\min(1,\frac3N,\frac{3\rho}{2\pi \sqrt N})$. \end{theorem} \begin{proof} Write $u=\Pi_{V_0}w$ and $\lvert u_X\rvert^2=\sum_{k\in L_X}\lVert u_k \rVert^2$, with $u=0$ off $K$. As in Theorem~\ref{thm:localflat}(ii), $R$ is $\Gamma_W$-invariant and $\mathcal HR(w,w)=\sum_X\Hess f_X(V_0)[u,u]$. Fix $X$. The tree argument in the proof of Lemma~\ref{lem:smallcurv}, carried out in $X$, where $\sum_{\nu<\mu}(x_\nu-b_\nu)\le3r_X$ for every link $(x,\mu)$ of $X$ and every plaquette used lies in $P_X$, gives $g\in G^X$ with $\lVert I-(gV_0)_k\rVert\le\delta_X$ for every $k\in L_X$. With $V'=gV_0$ and $v=Tu$ as in the proof of Lemma~\ref{lem:flatterm}, $\Hess f_X(V_0)[u,u]=\Hess f_X(V')[v,v]$, $\lvert v\rvert=\lvert u_X \rvert$ and $Q_X^{V'}(v)=Q_X^{V_0}(u)$. By (b), \[ \lvert\Hess f_X(V')[v,v]-\Hess f_X(I)[v,v]\rvert\le t_X\delta_X\lvert u_X \rvert^2, \] and Lemma~\ref{lem:flatterm}(iii) at $I$ gives \[ \lvert\Hess f_X(I)[v,v]\rvert\le h_X\rho_X\sum_{p\in P_X}\lVert(dv)_p \rVert^2 . \] As in the proof of Lemma~\ref{lem:smallcurv}, $\lVert Z_p(v)-(dv)_p\rVert\le2\delta_X(3b_1+ 2b_2+2b_3+b_4)$ at $V'$, with $b_i$ the norms of $v$ on the links of $p$, and a link lies in at most six plaquettes, so \[ \Bigl(\sum_{p\in P_X}\lVert(dv)_p\rVert^2\Bigr)^{1/2}\le Q_X^{V_0}(u)^{1/2} +\sqrt{432}\,\delta_X\lvert u_X\rvert . \] With $(s+t)^2\le(1+\theta)s^2+(1+\theta^{-1})t^2$ at $\theta=\sqrt{432} \,\delta_X$, and directly if $\delta_X=0$, \begin{align*} \lvert\Hess f_X(V_0)[u,u]\rvert&\le h_X\rho_X\bigl(1+\sqrt{432}\,\delta_X \bigr)Q_X^{V_0}(u)\\ &\quad+b_X\kappa_X\lvert u_X\rvert^2 , \end{align*} since $\delta_X(h_X\rho_X(\sqrt{432}+432\delta_X)+t_X)=b_X\kappa_X$. Summing over $X$ and exchanging the sums, the first terms give at most $\mathcal L_\kappa Q^{V_0}(u)$, and the second terms give $\sum_{k\in K} \beta_k\lVert u_k\rVert^2$ with $\beta_k=\sum_{X:\,k\in L_X}b_X\kappa_X$. By Cauchy--Schwarz, $\beta_k^2\le B_1\sum_{X:\,k\in L_X}b_X\kappa_X^2$ and $\kappa_X^2\le\sum_{p\in P_X}\kappa_p^2$, so \[ \sum_k\beta_k^2\le B_1\sum_Xb_X\ell_X\sum_{p\in P_X}\kappa_p^2\le B_1B_2 \kappa_*^2, \] and $\sum_k\beta_k\lVert u_k\rVert^2\le(\sum_k\beta_k^2)^{1/2}\lvert u \rvert_4^2$. Since $u\in\mathcal O_{V_0}^\perp$, Lemma~\ref{lem:windowlfour} and (a) give $\lvert u\rvert_4^2\le c_4Q^{V_0}(u)$. The last statement follows from Lemma~\ref{lem:cauchy}(ii), (iii): for $\max_k\lVert I-V_k \rVert=\delta\le\delta_X$, $V=e^\xi$ with $\max_k\lVert\xi_k\rVert\le\frac \pi3\sqrt N\delta\le\rho/2$. \end{proof} Theorem~\ref{thm:localcurved} extends Proposition~\ref{prop:actionloops} from loop sums, whose existence for the induced action is open (Remark~\ref{rem:loopsopen}), to every local remainder, with the same order $\lVert\kappa\rVert_2$ of the loss. For the plaquette sum that order is attained (Corollary~\ref{cor:actionsharp}). Measured by the supremum of the curvature the loss is not of order $\kappa$ uniformly in $n$, even for the plaquette sum, a local remainder whose terms do not depend on $n$. \begin{proposition}[no loss of order $\kappa$ uniformly in $n$] \label{prop:supform} Let $G=\SU(2)$, $0<\kappa\le0.02$, $n\ge(31/\kappa)^{1/2}$, and let $P$ be the sum of $s(V_p)$ over the plaquettes with their corners in $\Lambda'$. There are a background $V_0$ with $\lVert I-V_{0,p}\rVert\le\kappa$ for every such plaquette and $u\in\mathcal O_{V_0}^\perp$ with $\mathcal HP (u,u)<0\le\frac12Q^{V_0}(u)$. Hence a bound \[ (1-\phi)\,\tfrac12Q^{V_0}\circ\Pi_{V_0}\le\mathcal HP\le(1+\phi)\,\tfrac12 Q^{V_0}\circ\Pi_{V_0}, \] valid at every background with plaquettes within $\kappa$ of $I$ on windows of side $n$, requires $\phi>1$; for every $C\le1/\kappa$ none holds with $\phi=C\kappa$ for all $n$. \end{proposition} \begin{proof} Take $\sin^2\alpha=\kappa/2\le10^{-2}$ and $\varsigma=-1$ in Proposition~\ref{prop:curvedsharp}: the plaquettes in the plane $\{0,1\}$ have $\lVert I-V_{0,p}\rVert=2\sin^2\alpha=\kappa$, the others are flat, $n\ge(31/0.02)^{1/2}>39$ and $\kappa(n+1)^2>31$, so part (ii) gives $u\in \mathcal O_{V_0}^\perp$ with $\mathcal HP(u,u)<0$, and $\Pi_{V_0}u=u$. A lower bound with $\phi\le1$ would give $\mathcal HP(u,u)\ge0$. \end{proof} Within $\kappa(n+1)^2\le2$, Proposition~\ref{prop:curvedsharp}(i) gives the lower bound $\phi\ge0.037\,\kappa(n+1)^2$. Since $\lVert\kappa\rVert_2\le \sqrt6(n+1)(n+2)\kappa_\infty$, the order $\kappa n^2$ in the supremum and the order $\lVert\kappa\rVert_2$ in the action are the same estimate; the action is the measure of the background that does not depend on $n$, and on the calm windows of Corollary~\ref{cor:actioncalm} it is at most $39.5K$. \emph{Condition (C).} At the trivial background on the torus of the next scale, Theorem~\ref{thm:localflat}(i) gives both halves of condition (C) of Theorem~\ref{thm:allscales} from a local representation of the one-step error. \begin{corollary}[condition (C) from local terms]\label{cor:localC} In the setting of Sec.~\ref{ssec:allscales}, let $k10^{24}(4.40+\ln \lvert W\rvert)$. Proposition~\ref{prop:badpassage} identifies the constrained law, with the links that bind the entering averages moved to the exterior, as the conditional law under $\mu^c$, and Corollary~\ref{cor:badcrossratio} bounds the cross-ratio of the weight of the interior for ends that enter the action only, at the rate $\alpha_G$ up to a term $432\beta_W^2\bar p$. Charts local in the component do not remove the dependence on the length: by Theorem~\ref{thm:localcharts} a window lies on a chart fixed by its exterior with conditional probability below $0.033^{\lvert W\rvert}$ for $\beta_W\le10^{24}$, whatever the exterior and the coarse field. For the coarse fields $c^\theta$ of Proposition~\ref{prop:spread} the exterior $R(c)$ carries a background up to $2.39\cdot10^{-12}$, and the reduced fibre up to $2.09\cdot10^{-12}$; for opposite fluxes at adjacent coarse plaquettes no background exists above $1.8001\cdot10^{-12}$, for any field and any exterior, while the dipole $c^{\theta,d}$ has one with exterior $R(c)$ up to $1.79\cdot10^{-12}$ (Proposition~\ref{prop:dipole}), so the threshold is not a function of the size of the coarse plaquette; a general coarse field between $6.0\cdot 10^{-13}$ and $2.418\cdot10^{-12}$ with exterior $R(c)$ is not decided. An end that enters an average enters one pair factor of the fibre measure (Lemma~\ref{lem:endpairs}), and that factor jumps under arbitrarily small moves of the end at every value of the coarse link (Proposition~\ref{prop:endfold}); after the non-path links are integrated, its mass changes by less than $3.2\cdot10^{-4}$ under every move (Lemma~\ref{lem:almosthaar}), and along a tube of blocks the cross-ratio of two such ends decays by the factor $1-e^{-D_i}$ per slab, $D_i=\lvert \Sigma_i\rvert(72\beta_W+3.2\cdot10^{-4})$, at every $\beta_W$ and without a chart (Theorem~\ref{thm:tube}, Corollary~\ref{cor:tubeends}). That factor has no content for $\beta_W\ge2.3\cdot10^{17}$. The derivative of the integrated pair factor in an end is bounded in the mean over the coarse link at every configuration (Corollary~\ref{cor:endmean}), and uniformly in the coarse link when the links of one of the two blocks of its pair lie within $\frac1{32}$ of $I$ (Corollary~\ref{cor:endchart}, Lemma~\ref{lem:endotherblock}), with a constant linear in $\beta_W$ that involves one integer, the crossing number $\mathfrak b_{\mathrm{tie}}$, finite by o-minimality and not computed (Theorem~\ref{thm:endlinear}). Along a tube this gives mixed differences in two ends that decay at the rate of Theorem~\ref{thm:tube} (Corollary~\ref{cor:tubederiv}), and on the chart of Theorem~\ref{thm:badcross} mixed differences that decay at the rate $\alpha_G$, independent of $\beta_W$, up to a term proportional to the mass off the chart, when the blocks next to the end pairs lie outside $W$ (Theorem~\ref{thm:endsdecay}). The bound uniform in $c_k$ when both blocks of the pair have links far from $I$, and a decay between two ends at a rate independent of $\beta_W$ off the chart, are not proved (Remarks~\ref{rem:badendsopen}, \ref{rem:tuberate}, \ref{rem:endsmeanopen} and~\ref{rem:endsfoldopen}). Lemma~\ref{lem:mobius} gives the mixed differences of every order in terms of the diameter; the rate $2\alpha_G$ needs them in terms of the length of the shortest joining tree (Remark~\ref{rem:fixedcopen}). That form is not proved, and it follows neither from the second differences in every environment (Proposition~\ref{prop:pairsonly}) nor from the layer bound~\eqref{eq:layerbound} (Proposition~\ref{prop:layerthird}); Remark~\ref{rem:treeform} names the two inputs it needs. Theorem~\ref{thm:thirdcumulant} and Corollary~\ref{cor:kthird} give the differences of order three of $\log\tilde K$ in tree form at the rate $\alpha_G$ from the Hessian bound behind \eqref{eq:layerbound} alone, which is optimal for that bound; Theorem~\ref{thm:thirdtree} and Corollary~\ref{cor:kthirdtree} give them at the rate $2\alpha_G$ of the second differences, in the coarse length of the shortest joining tree, with the walls of $D_z$ included, using in addition the decay along trees of the derivatives of order three of $\mathcal U$ that its finite range supplies. For a smooth potential of finite range the tree form holds at every order at the rate of the covariances (Theorem~\ref{thm:treeorders}); for the layer law the orders $k\ge4$, for which the Hessian bound alone gives no tree form uniform in the order (Proposition~\ref{prop:layerorders}), are proved only for its smooth approximations, with constants that grow with the approximation. The passage to $\log Z^c$ reduces to the joint cumulants of the gas of islands (Lemma~\ref{lem:islandgas}), of which only the covariances are bounded, and no rate uniform in the pair cross-ratios holds (Proposition~\ref{prop:islandchain}, Remark~\ref{rem:treeordersstatus}). Proposition~\ref{prop:hiddenbit} shows that no rarity bound replaces this comparison. \item \emph{The island threshold.} Theorem~\ref{thm:islandthreshold} gives the explicit coupling $\beta^*(\rho,s)$ past which the islands of bad blocks at one scale are rare enough to beat their entropy, at the exponent $1/H$ of Proposition~\ref{prop:coverageopt} and the collar of Proposition~\ref{prop:graphlaw}. It is of order $10^{13}$ at the witness radius $10^{-4}$ of Theorem~\ref{thm:islandswide}, and of order $10^{37}$ under the calmness of Corollary~\ref{cor:twoislands}; the tree count of Proposition~\ref{prop:treethreshold} lowers it by a factor $1.7$ to $1.8$, and the union bound over connected sets by at most a further factor $1.39$. The same statement at the scale of the cube is not given, and Theorem~\ref{thm:chessceiling} shows that no bound of this kind gives it: at scale $j$ every witness of a coarse label of radius $\eta<\eta_c$ has radius at most $1.00015\cdot4^{-j-1}\eta$, by the constant curvature of Proposition~\ref{prop:constantcurvature}; the normalisation of the chessboard estimate charges at least $\frac92(\log\beta_W-c_I)$ per fine site (Lemma~\ref{lem:Zabove}); and so every bound obtained from Theorem~\ref{thm:defects} or Proposition~\ref{prop:coarsechess} is below one only at scales whose blocks are smaller than the cube of $D(n,\alpha)$ by a factor at least $4.08(1-\alpha)^{1/4}$, at every coupling, with the normalisation of Lemma~\ref{lem:wilsonZ} or, for $\beta_W\ge10^3$, the exact one. For $\alpha\le\frac12$ and the labels of Theorem~\ref{thm:islandswide} the cube lies at least three dyadic scales higher, and for the calm labels at least twenty (Corollary~\ref{cor:chessceiling}). The fields of constant curvature that carry the ceiling are critical points of the action (Lemma~\ref{lem:constantcritical}), so every first-order estimate vanishes at them, the forces of Theorem~\ref{thm:haartilt} and Proposition~\ref{prop:foresttilt} included. By Theorem~\ref{thm:criticalceiling} the ceiling stays when the chessboard estimate need cover only the defects in an open set that contains the critical points, in particular those left by finitely many first-order estimates, which never cover a neighbourhood of these fields (Corollary~\ref{cor:firstorder}). What is missing is a bound on the weight of the coarse defects relative to $Z_L$ that does not charge the fluctuation volume of the cube and detects the defects near the critical points (Remarks~\ref{rem:chessceiling} and~\ref{rem:criticalceiling}), and (iv) is needed before any threshold bears on the gap. \item \emph{The scales.} An effective action at every scale whose one-step errors are summable in the sense of Proposition~\ref{prop:formstability} and whose locality norms obey Lemma~\ref{lem:schur}, down to the last $O(1)$ scales where the running coupling is of order one; this is the large-field expansion of Ba{\l}aban, and it is not carried out here. Theorem~\ref{thm:islandscales} and Theorem~\ref{thm:islandswide} settle the entropy count of the islands at every scale up to $a_{k_*}/8$, and Proposition~\ref{prop:constantcurvature} shows that fine witnesses carry it no further; the law of the iterates given their labels, which the effective action is for, is untouched. Proposition~\ref{prop:massgrowth} shows that the one-step errors must be relative bounds by a gauge-invariant form: an error bounded by a multiple of the Euclidean form grows by a factor at least $4/(1+2724\kappa)$ at each step. Proposition~\ref{prop:relstep} gives such a relative bound for one step at the trivial background on a box of side $n$, from an absolute bound on the Hessian of the induced action off the gauge directions; its factor $C_n$, of order $n^2$, is necessary for an absolute bound (Remark~\ref{rem:relscales}). For a remainder made of filled Wilson loops the bound is relative at every flat background (Proposition~\ref{prop:flatloops}), and $C_n$ falls on the Haar term alone (Corollary~\ref{cor:loopstep}). At a background whose plaquettes lie within $\kappa$ of $I$ the covariant Hessian of such a remainder is bounded by the covariant plaquette form with a loss of order $\kappa n^2$, at critical and non-critical points alike (Proposition~\ref{prop:curvedloops}); the loss is attained, and for $\kappa(n+1)^2\ge31$ the plaquette action itself has a negative direction off the gauge directions (Proposition~\ref{prop:curvedsharp}). In terms of the $\ell^2$ norm $\lVert\kappa\rVert_2$ of the curvature, the square root of the Wilson action of the background on the window, the loss is of order $\lVert\kappa\rVert_2$ for every $n$ (Proposition~\ref{prop:actionloops}), the unstable mode appears at a background with $\lVert\kappa\rVert_2<31.8$ (Corollary~\ref{cor:actionsharp}), and on calm windows both forms of the loss are of order $K$ (Corollary~\ref{cor:actioncalm}). That the induced action is a loop sum with summable coefficients is not proved: for $\SU(2)$ loop sums are dense among invariant functions (Proposition~\ref{prop:loopsums}), which controls neither values nor Hessians, and a relative bound on values gives no Hessian bound at curved backgrounds (Remark~\ref{rem:valueshessians}). Sec.~\ref{ssec:lfstep} carries out the large-field part of one step for $\SU(2)$: each rough block costs $e^{-\lambda\kappa^2}$, the weight of a coarse field is an exact sum over large-field regions, and the large-field sum is the calm weight times the mean of a polymer sum under the calm law (Proposition~\ref{prop:sfrelative}), whose activities are at most $e^{-9}$ per block wherever the boundary values admit a completion calm at $\kappa/30$ (Theorem~\ref{thm:sfcalm}). That bound fails at calm fields with plaquettes in the shell $(\kappa/30,\kappa]$: the Bianchi identity of a cube holds in $\SU(2)$ exactly up to conjugation (Lemma~\ref{lem:sfcorner}), and at the trivial coarse field the relative activity of one block is not essentially bounded under the calm law, at every $\lambda$ and every $\kappa\le10^{-8}$ (Theorem~\ref{thm:sfshellsu2}). So Proposition~\ref{prop:sfrelative}(ii) does not apply, and the shell needs a bound in the mean, which is a bound on the ratios $\mu_c\{R=X\}/\mu_c\{R= \emptyset\}$ of the constrained law and is not proved (Remark~\ref{rem:sfshellsu2}). Not proved are also a bound on the term $Z_\perp$ and the locality of the calm weight in $c$; by Theorem~\ref{thm:sfgauss} the calm weight is not a product of local factors times a polymer sum if the rational number $(M_W)_{AB}$ of that theorem is nonzero, as an exact computation finds (Remark~\ref{rem:sfgaussvalue}), and then a local form has to expand around the Gaussian block-spin form. Nor can the leading term be $\lambda q^{\rm lin}(\log c)$ for $\lambda>0.5472$, by gauge invariance (Theorem~\ref{thm:sfgaugeobst}); the linear propagator of such an expansion decays at an explicit rate (Theorem~\ref{thm:sfprop}); in the linear model the relative activities in the shell are unbounded under the calm law at every $\lambda$ (Theorem~\ref{thm:sfshelllin}); and $Z_\perp(I)>0$ for $\SU(2)$ (Theorem~\ref{thm:sfperp}). The coarse action is comparable to $\lambda$ times the coarse curvature only up to a constant factor. Sec.~\ref{ssec:iterate} replaces the factor $C_*$ of the comparison by $2956$ for every coarse field and by $1+O(\kappa_0)$ on the coarse plaquettes below $\kappa_1\approx0.19\,\kappa_0$ (Theorem~\ref{thm:stepconstants}), and shows that no such comparison has a ratio of constants below $4t_L^2>3.849$ (Theorem~\ref{thm:stepsharp}), so that a next step has to be compared with the block-spin form. Theorem~\ref{thm:iterate} composes the steps under three hypotheses at every scale, a relative error, a locality bound and a large-field cost; none is proved beyond the first step in the required form (Proposition~\ref{prop:iteratestatus}), and the large-field estimates of Sec.~\ref{ssec:lfstep} give no activity bound for $\beta_W<3.2372\cdot10^6$, more than $1.257\cdot10^7$ one-loop steps above coupling one. The locality hypothesis cannot hold uniformly in the volume (Proposition~\ref{prop:h2volume}). Theorem~\ref{thm:allscales} proves the step from one scale to the next under a single condition on the error of that step, relative and local; under it the relative comparison and the fibre gap pass to every scale uniformly in $\lambda_0$, and the locality of the fibre with constants that are not bounded in the scale. Sec.~\ref{ssec:localrem} proves the relative one-step bound without the factor $C_n$ when the remainder of the action is a sum of gauge-invariant local terms, exactly at flat backgrounds (Theorem~\ref{thm:localflat}) and with a loss of order $\lVert\kappa\rVert_2$ at curved ones (Theorem~\ref{thm:localcurved}); a loss of order $\kappa$ uniformly in $n$ is false (Proposition~\ref{prop:supform}). Condition (C) follows from such a representation of the one-step error (Corollary~\ref{cor:localC}); that the induced action has one is not proved (Remark~\ref{rem:localrem}). \end{enumerate} The covariance of item (ii) for a fixed coarse field and uniform in the exterior, with decay at the rate $2\alpha_G$ without a floor, is the single-scale part of (D) that remains open; its mean over the coarse field, and its value for the coarse fields of Proposition~\ref{prop:fixedgood}, are bounded by Theorems~\ref{thm:labelcov}, \ref{thm:fibrecov} and Corollary~\ref{cor:labelstretched}, and by Theorem~\ref{thm:fixedobstruction} the rest needs the mixed differences of $\log Z^c$ in tree form and the response of the law across a bad component, which Theorem~\ref{thm:badcross} gives on the chart up to the mass of the law off the chart, which tends to one as the component grows, for coarse plaquettes below $6.0\cdot10^{-13}$, for the coarse fields of Proposition~\ref{prop:spread} up to $2.39\cdot10^{-12}$, and for none above $2.418\cdot10^{-12}$ (Proposition~\ref{prop:badchart}, Proposition~\ref{prop:spread}, Theorem~\ref{thm:localcharts}, Theorem~\ref{thm:offchartmass}, Remark~\ref{rem:badmassopen}); item (i) being settled for the chains of the bad blocks by Proposition~\ref{prop:linechain} and item (ii) given the labels of all islands by Theorem~\ref{thm:manyislands}; item (iii) is settled by Theorem~\ref{thm:islandthreshold} at one scale and by Theorem~\ref{thm:islandswide} at every scale up to $a_{k_*}/8$; item (iv) is the multiscale part. Theorem~\ref{thm:avgtransfer} replaces (D) by the averaged condition $\mathrm{BP}(n,\varpi)$ of Definition~\ref{def:avgcube}, a spectral gap of the cube dynamics on gauge-invariant functions in $L^2$ of the Wilson measure on tori, with no gauge and no supremum over exteriors; it follows from its supremum form (Proposition~\ref{prop:avgsup}), the obstructions of the forest gauge and those uniform in the coarse field do not apply to it, it fails in the Coulomb phase of the Villain theory at every cube size (Corollary~\ref{cor:avgcoulomb}), it holds on every torus at strong coupling and fails at fixed cube size on small tori at weak coupling (Corollary~\ref{cor:bpranges}, Theorem~\ref{thm:bpfixed}), the latter already on functions of nonzero centre charge, while the continuum step needs only its local form (Theorem~\ref{thm:bplocal}, Corollary~\ref{cor:bpcharged}), the failure of the neutral part on small tori reduces for $\SU(2)$ to a lower bound on the law of the Polyakov loop (Theorem~\ref{thm:bpneutral}), and neither form is proved for any non-abelian group along the tuning (Remark~\ref{rem:avgstatus}, Remarks~\ref{rem:bpweakstatus} and~\ref{rem:bpneutralstatus}); the condition passes from the fine lattice to the law of the block variables of every local gauge-covariant block-spin map, so that along the tuning it contains the condition at one cube size for those laws at a fixed coarser spacing, the local form passes only in an integrated form that gives no gap (Theorem~\ref{thm:bpdescent}, Corollary~\ref{cor:bpdescent}, Remark~\ref{rem:bpdescent}); in the opposite direction, the fibre and coarse bounds posed jointly as a split inequality give the upward step, and along the tuning the condition at the coarsest scale with the split inequality at every scale, its constants bounded in product, gives $\mathrm{BP}(n_k, \varpi)$ with an explicit $\varpi$; the split inequality holds with the exact constants for the product of Haar measures and, with the same constants, for the Haar refinement of every gauge-invariant law of the block variables, so that along a tower of refinements the condition at the coarsest scale passes to every finer scale without loss, the two bounds posed separately lose a fixed factor at each scale already for the product measure, the fine Dirichlet form splits into a fibre form and a coarse form for every measure, the Wilson measure at $\beta\ne0$ is not a Haar refinement, the split inequality follows for every measure from a fibre Poincar\'e inequality, a bound on the information of the fine exterior of a cube on the block variables and the almost orthogonality of that information for different cubes, it holds for the Wilson measure in the strong-coupling range on every torus, with constants within factors $1+O(\beta)$ of those of the refinements, and on each torus it holds for every measure with a continuous positive density, with constants that grow with the volume, while the obstructions at weak coupling on small tori bound the condition for the law of the block variables and put no constraint on it; neither the split inequality with constants independent of the torus for the laws of the block variables below the finest scale or for the Wilson measure outside the strong-coupling range nor the condition at the coarsest scale along the tuning is proved (Theorems~\ref{thm:bpup}, \ref{thm:bpuptower}, \ref{thm:bprefine}, \ref{thm:bpsplitcond} and~\ref{thm:bpsplitstrong}, Corollaries~\ref{cor:bpuptuning}, \ref{cor:bpupseparate}, \ref{cor:bprefinetower} and~\ref{cor:bpsplitranges}, Propositions~\ref{prop:bpuphaar}, \ref{prop:bpfinesplit}, \ref{prop:bpnotrefine}, \ref{prop:bptilt} and~\ref{prop:bpsmalltori}, Remarks~\ref{rem:bpup}, \ref{rem:bprefine} and~\ref{rem:bpsplitstrong}); the condition at the coarsest scale follows, for every measure with a continuous positive density and without locality, from a bound of Dobrushin type on the single-link coefficients of the law of the block variables or from a Hessian bound on its induced action, uniform along the tuning, and these bounds are not proved outside the strong-coupling range; given the split inequality the condition is equivalent to the conclusion of the bootstrap up to one unit of cube size, it is independent of the split inequality, and it fails for the coarse law of a measure whose local gauge-invariant covariances vanish beyond the coarsest spacing, and the closeness of each single-link conditional law to the Haar measure does not replace the bound on the sums of the coefficients (Theorem~\ref{thm:bpcoarse}, Corollaries~\ref{cor:bpcoarsehess} and~\ref{cor:bpcoarseH1}, Propositions~\ref{prop:bpcoarsezero}, \ref{prop:bpcoarsenec}, \ref{prop:bpcoarseflat} and~\ref{prop:bphidden}, Remarks~\ref{rem:bpcoarse} and~\ref{rem:bprefine}); one step of the straight-path map with blocks of side $b$ contracts a weighted Dobrushin sum of the single-link coefficients by $e^{-(\lambda-2\kappa)\lfloor b/2\rfloor}$ on every torus, so that along a tower such a bound at one level for every $k$ gives the condition at the coarsest scale, while gauge covariance bounds the Dobrushin sum of every gauge-invariant law below by two minus four times the normalised spread of its single-link conditional laws, which excludes the $\SU(2)$ Wilson measure at $\beta_W\ge\frac83$ and every law of the block variables whose plaquette variables are concentrated, forces the mean distance of the coarse holonomies from $I$ to exceed a quarter of the diameter, and does not obstruct the condition at the coarsest scale along the tuning for $\SU(2)$, where the coarse conditional laws are spread out; the Hessian of the induced action is the mean curvature of the fine action minus the covariance of the forces in the fibre law, and no bound on the coefficients along the tuning is proved (Theorems~\ref{thm:bpbalstep} and~\ref{thm:bpbalnec}, Corollaries~\ref{cor:bpbaltower}, \ref{cor:bpbalwilson} and~\ref{cor:bpbalsmall}, Propositions~\ref{prop:bpbalrandom} and~\ref{prop:bpbalhess}, Remark~\ref{rem:bpbalaban}); Sec.~\ref{ssec:bpfibre} bounds the coarsest coefficients by $D^2/4$ times the mixed derivatives of the induced action, with no condition on its single-link Hessian, shows that for straight-path maps with blocks of side at least three these are the covariances of the forces on distinct marks in the fibre law, so that (H1) follows from a decorrelation (FC) of these forces, which also forces their variances to equal the mean curvatures up to a bounded error, computes them in a Gaussian abelian model as $-\beta$ times a co-closed effective quadratic form with diagonal between $1/b^2$ and $6/b$, row sums at least twice the diagonal and bounded uniformly in the torus by a number $\gamma_b$; Sec.~\ref{ssec:bprowsum} writes the symbol of this form in closed form and shows that $\gamma_b$ lies between absolute multiples of $(\log b)/b$ and $(1+\log b)/b$, so that the bound (G) of order $1/b$ fails while the Gaussian form of (FC) holds along the tuning, and that in every representation of the coarse density as a mixture of products over the coarse links the same numbers are the covariances of the scores; (FC) for $\SU(2)$ is not proved along the tuning (Lemmas~\ref{lem:bpfibosc} and~\ref{lem:bpfibgauge}, Propositions~\ref{prop:bpfibreduce}, \ref{prop:bpfibgauss}, \ref{prop:bpfibbloch} and~\ref{prop:bpfibrough}, Corollaries~\ref{cor:bpfibH1}, \ref{cor:bpfibgauss} and~\ref{cor:bprowFC}, Theorem~\ref{thm:bprowgauss}, Remarks~\ref{rem:bpfibre}, \ref{rem:bprowposterior} and~\ref{rem:bprowsum}); Sec.~\ref{ssec:bppaths} shows for $\SU(2)$ that convolution with a one-link density contracts every nontrivial representation, so that the conditional densities of the block variables and their first two derivatives are within $48(1+6\beta_W)^2q_W^{b-2}$ of those of the Haar measure, the single-link curvature of the induced action tends to zero along every tuning with $\beta_k=O(k)$, and the mixed derivatives are posterior covariances of the scores of the interior path products, each tending to zero; this gives the condition at the coarsest scale on every torus of side at most $e^{2^k/(17k)}$, a column sum at most $\frac13$ gives it with no condition on the tails, and a bound on the sums of the posterior covariances uniform in the torus, which is not proved, gives (H1) (Theorems~\ref{thm:bppathproduct}, \ref{thm:bppathdensity} and~\ref{thm:bppathmixed}, Corollaries~\ref{cor:bppathcolumn} and~\ref{cor:bppathfourk}, Propositions~\ref{prop:bppathobst} and~\ref{prop:bppathsuff}, Remark~\ref{rem:bppaths}). Sec.~\ref{ssec:groups13} carries the single-scale results of this section to every compact simple group, with explicit constants, except the parts listed in Remark~\ref{rem:groups13}. Sec.~\ref{ssec:chainG} states the conditional continuum chain for every compact simple group, with the averaged condition in its local form, the centre sectors of $Z(G)$ and the Schwinger functions at separated points (Theorem~\ref{thm:chainG}, Proposition~\ref{prop:bpcentreG}), proves the averaged condition and the triviality of the limits for $N\beta\le\kappa_G/48$ (Corollary~\ref{cor:limitEstrongG}), and extends the obstruction at fixed cube size under a condition (CP) that holds for $\SU(N)$ and fails for $\mathrm{Sp}(n)$ with $n$ even (Proposition~\ref{prop:bpfixedG}); its hypotheses are not proved for any group, and an analogue of the sharp quarter for $G\ne\SU(2)$ is not proved (Proposition~\ref{prop:modulusG}). On the continuum side, Lemmas~\ref{lem:positivetime} and~\ref{lem:nonvanishing} reduce the nontriviality of the limit to a lower bound on one ratio of lattice two-point functions at a fixed physical time, uniform along the tuning, which is the open input of Theorem~\ref{thm:flowlimit}; Sec.~\ref{ssec:ratiocube} states it exactly as the condition $(\mathrm N_u)$, shows that neither the strong-coupling range nor the cube estimate in the form of Theorem~\ref{thm:transfer} supplies it, and reduces it to the trial states of Corollary~\ref{cor:trial}. Sec.~\ref{ssec:slabenergy} bounds the slab energy of every Lipschitz slice function by the plaquette energy, which is at most $(3.7148+0.149/\beta_W) /\beta_W$ for $\SU(2)$, and shows that this route gives the trial condition only where $a_k\beta_k$ stays bounded below, not along the one-loop tuning (Proposition~\ref{prop:lipobstruction}). The same subsection shows that no set of upper bounds on correlations at physical separations, with constants not exponentially small in $1/a_k$, implies $(\mathrm N_u)$ (Proposition~\ref{prop:upperbounds}); that trial states smeared in time over a physical interval, flowed Wilson loops among them, meet the first trial condition exactly when, up to the choice of the times, a ratio of their correlations at physical times is bounded below (Corollary~\ref{cor:smearedtrial}), which removes the lattice scale from the condition but leaves a lower bound at physical separations; and that a comparison with the Gaussian theory gives $(\mathrm N_u)$ from relative bounds at two physical separations, which a scale-by-scale argument along the one-loop tuning supplies only if its losses summed over the scales stay bounded, and not with a loss of first order in the running coupling at each scale (Proposition~\ref{prop:scalelosses}). This is the summability required in item~(iv), now for the ratio, and $(\mathrm N_u)$ is not proved here in infinite volume; in a box of fixed size it holds for packets odd under the centre of $\SU(2)$, with degenerate limits (Corollary~\ref{cor:centreodd}). Choosing the couplings by the ratio (Sec.~\ref{ssec:ratiotuning}) makes $(\mathrm N_u)$ and the normalisation of Proposition~\ref{prop:uniquelimit}(iii) hold by construction; such a choice exists if and only if $(\mathrm N_u)$ holds along some sequence of couplings (Theorem~\ref{thm:ratiotuning}), and it leaves (D), (M), the family $\mathfrak F$ and $\beta_k\to\infty$ open. Sec.~\ref{ssec:osaxioms} shows that, given (D) along the tuning, the Osterwalder--Schrader axioms for the spatially flowed fields, with the symmetry $\mathrm O(3)\times\mathrm O(1)$, follow from three statements, a moment bound (M), one spatial rotation (W) and one limit (U), each of them necessary and none of them proved. Full $\mathrm O(4)$ invariance is not among these conclusions: at positive flow time the spatial flow breaks it even for the free field (Proposition~\ref{prop:osobstruction}(c)), so it has to be posed for $\mathrm O(4)$-covariant observables, such as fields of the four-dimensional flow, and the four-dimensional flow loses reflection positivity already for the free field (Proposition~\ref{prop:fourflow}). Of the three statements, (M) is a local moment bound together with the bound $\sup_k\Upsilon_k<\infty$, which contains $(\mathrm N_0)$ for product packets and fails in the strong-coupling range (Proposition~\ref{prop:msplit}), and the flow alone cannot give it (Proposition~\ref{prop:stationary}); its spatial comparison (M$_{\mathrm{sp}}$) follows from spatial reflection positivity (SR$_0$) of the slice field (Theorem~\ref{thm:spatialrp}), which cannot hold for a field and its Laplacian together in a state that clusters in space (Corollary~\ref{cor:srladder}) and is destroyed by the linear heat flow (Proposition~\ref{prop:heatrp}); (W) is the Ward identity for one spatial generator (Proposition~\ref{prop:wardform}); and under the renormalisation condition (RC), (U) is the vanishing of the moment differences between consecutive scales together with a totally disconnected set of limits, and a first-order bound in the running coupling for those differences does not give it (Proposition~\ref{prop:tuned}). For unsmoothed local fields at separated points (Sec.~\ref{ssec:limitE}) the inputs are of the same kind: the separated moment bound (M$_{\mathrm s}$) for $n\ge3$ is equivalent to bounds on reflected norms of products of separated packets (Proposition~\ref{prop:limitEreflect}), which do not follow from the two-point properties, reflection positivity and the gap (Proposition~\ref{prop:limitEthree}), and its rate (G$_{\mathrm s}$) is a rate of the form $K_1^m(m!)^{\beta_1}(1+L)^{q_1m}\delta^{-p_1m}$ in these bounds (Proposition~\ref{prop:limitErate}), which bounds at each order do not supply (Proposition~\ref{prop:limitEfactorial}); one inequality of Gaussian domination type, (GD), would give both from the two-point statements (Proposition~\ref{prop:limitEdomination}); for $n=2$, (M$_{\mathrm s}$) is equivalent to the comparison (SD), which holds if and only if the largest reflected norm of a unit packet near the reflection plane is carried by energies bounded in physical units, bounds the physical gap from above, fails in the strong-coupling range and is not implied by reflection positivity and a gap bounded above and below in physical units (Propositions~\ref{prop:limitEsd} and~\ref{prop:limitEheavy}); and under (SD) the nonvanishing condition (NE) is the nontriviality of every limit (Proposition~\ref{prop:limitEne}) and is equivalent to the condition that spatial momenta bounded in physical units carry a fixed fraction of that largest reflected norm (Proposition~\ref{prop:limitElm}); (NE) follows from (SD) for $\U(1)$ (Corollary~\ref{cor:limitEabelian}), is not implied by (SD) together with reflection positivity and a gap bounded above and below in physical units (Proposition~\ref{prop:limitEflat}). The rotations (W$_{\mathrm s}$) and the uniqueness (U$_{\mathrm s}$) follow from rotation invariance of the renormalised lattice moments at large lattice distance with a pointwise bound (Theorem~\ref{thm:limitErot}), and from a mass condition together with uniqueness of the limit up to dilations (Theorem~\ref{thm:limitEunique}), and the normalisation by one reflected correlation does not fix the scale (Proposition~\ref{prop:limitEscale}). Reflection positivity reduces the pointwise bound for $n=2$ to a bound on a coordinate axis and (RI$_2$) to a comparison with the axis (Proposition~\ref{prop:limitEaxial}), and that bound does not follow from the separated moment bounds, the mass condition and (U$_{\mathrm s}$) (Proposition~\ref{prop:limitEgauss}(b)); the gap and (M$_{\mathrm s}$) give the two halves of the mass condition, with two rates that need not coincide (Proposition~\ref{prop:limitEbracket}); and (O$_{\mathrm s}$) is equivalent to the convergence of the moments after a change of the spacing by bounded factors and does not follow from the other properties together with the mass condition (Propositions~\ref{prop:limitEdilate} and~\ref{prop:limitEgauss}(c)). For $n=2$, (SD) and (NE) follow from the two-sided short-distance law (AF) predicted by asymptotic freedom, which excludes the trivial limits of the strong-coupling range (Theorem~\ref{thm:limitEaf}); the inequality that (AF) requires of the Wilson measure is stated in Remark~\ref{rem:limitEafstatus}. None of these inputs, (AF) included, is proved for the Wilson measure. \begin{figure*}[t] \centering \begin{minipage}[c]{0.5\textwidth} \centering \resizebox{\textwidth}{!}{% \begin{tikzpicture}[x=0.62cm,y=0.62cm,>={Stealth[length=1.6mm]}] \foreach \i in {0,...,15} \foreach \j in {0,...,9} { \pgfmathsetmacro{\bad}{(\i==3 && \j==4) || (\i==4 && \j==4) || (\i==4 && \j==5) || (\i==11 && \j==2) || (\i==11 && \j==3) || (\i==12 && \j==7) ? 1 : 0} \ifnum\bad=1 \fill[red!55] (\i,\j) rectangle ++(1,1); \else \fill[green!18] (\i,\j) rectangle ++(1,1); \fi \draw[gray!50,line width=0.2pt] (\i,\j) rectangle ++(1,1);} \draw[blue!60!black,thick] (6,1) rectangle (10,8); \draw[blue!60!black,dashed] (5,0) rectangle (11,9); \node[blue!60!black,above] at (8,8) {\footnotesize $W$}; \node[blue!60!black,below] at (8,0) {\footnotesize collar}; \node[red!70!black] at (3.5,2.6) {\footnotesize bad island $a$}; \node[red!70!black] at (12.4,4.9) {\footnotesize bad island $b$}; \draw[<->,thick,orange!80!black] (5.2,4.5) -- (10.8,4.5) node[midway,above] {\footnotesize kernel $K(x,y)$}; \end{tikzpicture}} \end{minipage}\hfill \begin{minipage}[c]{0.47\textwidth} \centering\footnotesize \begin{tabular}{@{}ll@{}} \toprule label & witness and result\\ \midrule non-path link off $R(c)$ & fine defect, Theorem~\ref{thm:cutoff}\\ coarse plaquette $>\eta$ & fine defect, Lemma~\ref{lem:coarsewitness}\\ path link & none: determined, Lemma~\ref{lem:branchbound}\\ bad non-path links & fold, Proposition~\ref{prop:fold}\\ bad pair & chain, Proposition~\ref{prop:linechain}\\ inside link & fold, Proposition~\ref{prop:insidefold}\\ all links & nonconvex, Proposition~\ref{prop:nonconvex}\\ \midrule good window & graph law, Proposition~\ref{prop:graphlaw}\\ $n$ bad labels & $A^{n/H}$, Lemma~\ref{lem:coverage}\\ two islands & one kernel, Theorem~\ref{thm:factorisation}\\ layer law & chart bound, Theorem~\ref{thm:layerbound}\\ two islands & $e^{-2\alpha_G(w-2s-8)}$, Corollary~\ref{cor:twoislands}\\ two islands & $\frac14\tanh(\varDelta/4)$, Corollary~\ref{cor:twolabels}\\ many islands & $\varDelta_*$, Theorem~\ref{thm:manyislands}\\ rarity alone & floor $q^2/16$, Proposition~\ref{prop:hiddenbit}\\ island of $n$ blocks & $2^{-n}$ past $\beta^*$, Theorem~\ref{thm:islandthreshold}\\ islands at scale $j$ & $2^{-n}$ past $\beta^*_j$, Theorem~\ref{thm:islandscales}\\ \bottomrule \end{tabular} \end{minipage} \caption{Left: a window $W$ of coarse blocks (solid) with its collar (dashed), good blocks in green and bad islands in red. On the good blocks the conditional law is the graph law of Proposition~\ref{prop:graphlaw}; the weight of two islands is compared through a kernel on the good layer between them, whose cross-ratio bounds their covariance (Lemma~\ref{lem:kernel}, Corollary~\ref{cor:twolabels}). Right: the labels considered in Sec.~\ref{sec:islands}, their fine witnesses, and the estimate or the obstruction that each produces.} \label{fig:islands} \end{figure*} \section{Conclusion}\label{sec:conclusion} For every compact simple gauge group, four-dimensional lattice Yang--Mills theory has a unique infinite-volume state, exponential clustering and a transfer-matrix gap whenever $N\lvert\beta\rvert<\frac2{29}\kappa_G$ (Theorem~\ref{thm:groups}), and plaquette blocks extend the range for $\SU(2)$ and $\SU(3)$ (Theorems~\ref{thm:blocks} and~\ref{thm:su3blocks}). One averaged estimate in a cube of fixed physical size, the bound $\mathrm{BP}(n_k,\varpi)$ with $n_ka_k\le\ell$, places the spectrum of every continuum limit in $\{0\}\cup[16\varpi/(625e^2\ell),\infty)$, so every nontrivial limit has a mass gap (Theorem~\ref{thm:avgtransfer}). Along the tuning this estimate follows from two statements about block variables: the condition (H1) at the coarsest scale and the split inequality (H2) at every finer scale (Corollary~\ref{cor:bpuptuning}). Both statements are proved in part. The split inequality holds on every torus at strong coupling, with constants within factors $1+O(\beta)$ of those of the product measure (Theorem~\ref{thm:bpsplitstrong}). For $\SU(2)$ the block variables of the straight-path map approach the Haar measure in every representation, their single-link curvature vanishes along every tuning with $\beta_k=O(k)$, and the condition at the coarsest scale holds on every torus of side up to $e^{2^k/(17k)}$ (Corollary~\ref{cor:bppathfourk}). On tori of every size it follows from a bound, uniform in the torus, on the sum of the posterior covariances of the scores of the interior path products (Proposition~\ref{prop:bppathsuff}). In the Gaussian abelian model the corresponding row sums are of order $(\log b)/b$, so that model meets the decorrelation along the tuning, although it is not a lattice gauge measure (Theorem~\ref{thm:bprowgauss} and Corollary~\ref{cor:bprowFC}). Three inputs remain: that covariance bound along the tuning, the split inequality at weak coupling, and a nontrivial continuum limit with the properties of Sec.~\ref{ssec:limitE}. Given them, the results above give the continuum theory a mass gap; without them the mass gap is not proved here. Lattice gauge fields also define four-dimensional gauge theories with chiral fermions in every anomaly-free representation, once the lattice field is interpolated to the continuum and coupled to continuum fermions~\cite{Wit26}. The mass gap problem concerns the pure gauge theory, to which every result here refers. \section*{Acknowledgements} This paper rests on the work of many earlier scientists. Lattice gauge theory is due to Wilson~\cite{Wil74} and its Hamiltonian form to Kogut and Susskind~\cite{KS75}; reflection positivity and the reconstruction of the transfer matrix to Osterwalder and Schrader~\cite{OS73,OS75}, and their lattice form, with the first proof of a gap at strong coupling, to Osterwalder and Seiler~\cite{OSei78,Sei82}. The uniqueness method is Dobrushin's~\cite{Dob68,Dob70,DS85}, the curvature method is that of Bakry and \'Emery~\cite{BE85,BGL14} in the form of Shen, Zhu and Zhu~\cite{SZZ23}, the covariance estimate under Dobrushin's condition is F\"ollmer's~\cite{Fol82}, the chessboard estimate is that of Fr\"ohlich, Israel, Lieb and Simon~\cite{FILS78} and Fr\"ohlich and Spencer~\cite{FS82}, disagreement percolation is due to van den Berg and Maes~\cite{vdBM94}, the Brascamp--Lieb inequality to Brascamp and Lieb~\cite{BL76}, the correlation representation to Helffer and Sj\"ostrand~\cite{HS94} and Naddaf and Spencer~\cite{NS97}, the perturbation lemma to Holley and Stroock~\cite{HS87}, the derivative of a matrix function to Daleckii and Krein~\cite{DK65}, and the strict positivity of the transfer matrix to L\"uscher~\cite{Lus77}, the gauge fixing on a forest is Creutz's~\cite{Cre77}, and the curvature of Lie groups is Milnor's~\cite{Mil76}; the localisation estimate of Sec.~\ref{ssec:constrained} is that of Combes and Thomas~\cite{CT73}. The analysis of the single-link law uses results of Barta~\cite{Bar37}, Biernacki and Krzy\.z~\cite{BK55}, McShane~\cite{McS34}, Stein and Weiss~\cite{SW71} and Watson~\cite{Wat44}, and the continuum problem is the one formulated by Jaffe and Witten~\cite{JW06} and studied by Ba{\l}aban~\cite{Bal84a,Bal85a,Bal89b} through the renormalisation group, in the account of Dimock~\cite{Dim13}; gauge fields with cut-offs were constructed by Brydges, Fr\"ohlich and Seiler~\cite{BFS79} and by Magnen, Rivasseau and S\'en\'eor~\cite{MRS93}; recent probabilistic work on lattice Yang--Mills theory is due to Chatterjee~\cite{Chat19,Chat19b}, Basu and Ganguly~\cite{BG18}, Adhikari and Cao~\cite{AC25}, and Cao, Nissim and Sheffield~\cite{CNS25}, and the two-dimensional theory was constructed by Driver~\cite{Dri89}, Gross, King and Sengupta~\cite{GKS89} and L\'evy~\cite{Lev03}. \section*{Data and code availability} The proofs are analytic. Remarks~\ref{rem:su3cover} and~\ref{rem:su3blocks} report evaluations for $\SU(3)$ in interval arithmetic with rigorous error bounds, and Remarks~\ref{rem:blockgapsharp} and~\ref{rem:certificate} report computations with certificates checked in exact arithmetic; no result uses them. The value of $(M_W)_{AB}$ in Remark~\ref{rem:sfgaussvalue} is computed in exact rational arithmetic; no proof uses it, and the statements that the calm weight is not local are conditional on $(M_W)_{AB}\ne0$. Every number used in a proof is derived in the text. The programs of these steps, in Python with python-flint (FLINT and Arb) and a C kernel for products of Taylor polynomials, their outputs and the list of certified boxes are in the repository \url{https://github.com/creelie/4Y-M}, archived on Zenodo (\href{https://doi.org/10.5281/zenodo.23444338}{doi:10.5281/zenodo.23444338}; release v4.37.0, \href{https://doi.org/10.5281/zenodo.23992883}{doi:10.5281/zenodo.23992883}), together with a Lean~4 certificate of the line field, Lean~4 files for the comparison arithmetic, the coupling lemmas of Sec.~\ref{ssec:couplingcost} and the finite arithmetic of Theorem~\ref{thm:layerbound}, Corollary~\ref{cor:twoislands}, Proposition~\ref{prop:massgrowth}, Proposition~\ref{prop:smallframessharp} and Theorem~\ref{thm:islandswide}, the counting and trace identities and the constants of Secs.~\ref{ssec:averaged}--\ref{ssec:bpneutral} and of Theorem~\ref{thm:endlinear}, the linear algebra of Sec.~\ref{ssec:bpmetrics}, independent exact or numerical checks of the other finite statements, our checks of the certificate of~\cite{Rom26}, and the certificate of Remark~\ref{rem:blockgapsharp} with its exact check. The code was written by Deep Bhattacharjee, James Montgomery Lockwood and Tristan Romanov. \section*{Declarations} \emph{Author contributions.} Deep Bhattacharjee conceived the project and, with the help of AI agents working under his direction, made its constructions and proved its main results: the forest gauge on the period-two line field and the free-boundary state with its transfer-matrix gap at strong coupling; the plaquette blocks of $\SU(2)$ and $\SU(3)$ with their constants proved by hand; the reduction of the mass gap to one averaged estimate in a cube of fixed physical size; the labels of Ba{\l}aban's blocks and the islands of bad blocks; the averaged condition, its descent to block variables, the upward step and the split inequality; the single-link coefficients across scales, the forces in the fibre law, the smoothing along the paths for $\SU(2)$ and the row sums of the Gaussian model; and the conditional chain for every compact simple group. He wrote the programs of the certified computations and the paper, and checked every statement. Ushashi Bhattacharya checked the proofs and the references and edited the text. James Montgomery Lockwood contributed the analysis of constrained cubes~\cite{Loc26} and its code, and the Lean~4 formalisations of the quaternion model of $\SU(2)$, the open-gluon certificates, the coupling lemmas of Sec.~\ref{ssec:couplingcost}, the strong-coupling comparison and the limit of couplings, and the composition of metrics across scales of Sec.~\ref{ssec:bpmetrics} with its Lean~4 formalisation. Tristan Romanov contributed the certified bound on the forest-gauge Hessian~\cite{Rom26} and the research continuation on which Sec.~\ref{sec:islands} is built: the finite Dobrushin arithmetic, the minimality of the coupling cost and its measure-level form, the relative and canonical charts, the matching of labels on independent links, the rough-background and branch obstructions, the rarity witnesses and the island kernel, the analysis of the lattice slice field with the exact Wick coefficients on finite boxes, the block-response diagnostics, the flat fields of Remark~\ref{rem:flatholonomy}, four Lean~4 modules, and the programs and certificates behind them. All four authors approved the final text. \emph{Use of artificial intelligence.} The computations, the checks and parts of the drafting were done with AI assistance (Claude Opus~5.5, Claude Fable~5.1, GPT-6.1 Sol Ultra and GPT-6 Astra Ultra) under the direction of Deep Bhattacharjee, who checked every statement and is responsible for the text. The combination of the forest gauge with the sharp quarter behind Theorem~\ref{thm:su2} was first assembled by Anaxia, the reasoning engine of the BLLM system~\cite{Bha26,Bha26b}; the line field, the free-boundary limit, the ceiling, the curvature bound, the $\SU(3)$ part, the plaquette blocks and the certified constants are the authors' own. \emph{Funding.} None. \emph{Competing interests.} None. \emph{Ethics approval.} Not applicable: the work involves no human participants, animals or personal data. \appendix \section{The line field}\label{app:table} Table~\ref{tab:sigma} lists, for each residue class of $x$ modulo two, the direction $\sigma(x)$ and the permutation $\pi_x(\mu)=\sigma(x+e_\mu)$. In every row $\pi_x$ fixes $\sigma(x)$ and is a three-cycle on the other directions, which is the condition of Lemma~\ref{lem:cycle}. \begin{table}[h] \caption{The line field: $x$ modulo two (digits $x_0x_1x_2x_3$), the direction $\sigma(x)$ and the values $\sigma(x+e_\mu)$ for $\mu=0,1,2,3$.} \label{tab:sigma} \begin{ruledtabular} \begin{tabular}{cccccc} $x$&$\sigma(x)$&$\mu=0$&$\mu=1$&$\mu=2$&$\mu=3$\\ \colrule 0000 & 0 & 0 & 2 & 3 & 1 \\ 1000 & 0 & 0 & 3 & 1 & 2 \\ 0100 & 2 & 3 & 0 & 2 & 1 \\ 1100 & 3 & 2 & 0 & 1 & 3 \\ 0010 & 3 & 1 & 2 & 0 & 3 \\ 1010 & 1 & 3 & 1 & 0 & 2 \\ 0110 & 2 & 1 & 3 & 2 & 0 \\ 1110 & 1 & 2 & 1 & 3 & 0 \\ 0001 & 1 & 2 & 1 & 3 & 0 \\ 1001 & 2 & 1 & 3 & 2 & 0 \\ 0101 & 1 & 3 & 1 & 0 & 2 \\ 1101 & 3 & 1 & 2 & 0 & 3 \\ 0011 & 3 & 2 & 0 & 1 & 3 \\ 1011 & 2 & 3 & 0 & 2 & 1 \\ 0111 & 0 & 0 & 3 & 1 & 2 \\ 1111 & 0 & 0 & 2 & 3 & 1 \\ \end{tabular} \end{ruledtabular} \end{table} \section{The comparison lemma}\label{app:comparison} Let $(G,\delta)$ be a compact group with a bi-invariant metric of diameter $D$, $\Omega=G^A$, and $C_{ee'}$ the coefficients of Sec.~\ref{sec:state}, with row and column sums at most $\alpha<1$ and $C_{ee'}=0$ unless $e,e'$ share a plaquette, so that $|e-e'|\le1$ for the distance of link midpoints. Fix $\lambda>0$ with $\theta=\alpha e^\lambda<1$. \begin{lemma}\label{lem:comparison} Let $S\subset A$, possibly infinite, and let $\mu_1,\mu_2$ be probability measures on $\Omega$ invariant under $\gamma^F_e$ for every $e\notin S$. Let $T\subset A\setminus S$ be finite, $w_b=e^{-\lambda\dist(b,T)}$, and suppose $\sum_{z\in S}w_z<\infty$. There is a coupling of $\mu_1,\mu_2$ whose link distances $\varrho(b)=\E\,\delta(\sigma_b,\sigma'_b)$ satisfy \[ \sum_{b\notin S}w_b\varrho(b)\le\frac{\theta D}{1-\theta}\sum_{z\in S}w_z . \] If $S=\emptyset$ then $\mu_1=\mu_2$. \end{lemma} \begin{proof} The couplings form a weakly compact set and $\Psi(\pi)=\sum_bw_b\E_\pi\delta( \sigma_b,\sigma'_b)$ is weakly continuous because $\sum_bw_b<\infty$; let $\pi^*$ be a minimiser. For $e\notin S$ replace the pair $(\sigma_e,\sigma'_e)$ by a sample of an optimal coupling of $\gamma^F_e(\cdot\mid\sigma)$ and $\gamma^F_e(\cdot\mid \sigma')$, chosen measurably~\cite[Cor.~5.22]{Vil09}. Both $\mu_i$ are invariant, so the result is again a coupling, and changing the finitely many links that enter $\gamma^F_e$ one at a time gives $W_1\le\sum_{e'}C_{ee'} \delta(\sigma_{e'},\sigma'_{e'})$. Minimality gives $\varrho(e)\le\sum_{e'}C_{ee'} \varrho(e')$. Multiply by $w_e$ and sum over $e\notin S$; where $C_{ee'}\ne0$, $w_e\le e^\lambda w_{e'}$, so \[ \sum_{e\notin S}w_e\varrho(e)\le\theta\Bigl[\sum_{e'\notin S}w_{e'}\varrho(e') +\sum_{z\in S}w_z\varrho(z)\Bigr], \] all sums are finite, and $\varrho\le D$ on $S$. If $S=\emptyset$, $\varrho\equiv0$. \end{proof} \begin{corollary}\label{cor:decay} Let $\mu$ be the Gibbs measure of $\gamma^F$, $F$ a local function with support $S_F$, $L_F$-Lipschitz in each link, and $H$ bounded with finite support $S_H$ disjoint from $S_F$. Then \[ |\Cov_\mu(F,H)|\le\|H\|_\infty L_F\frac{\theta D}{1-\theta}|S_H|\, e^{-\lambda\dist(S_F,S_H)}. \] \end{corollary} \begin{proof} Condition on the links of $S_H$~\cite[Ch.~3]{Kal21}; the conditional measures $\mu^\sigma=\mu(\cdot\mid\sigma_{S_H})$ are invariant under $\gamma^F_e$ for $e\notin S_H$. Lemma~\ref{lem:comparison} with $S=S_H$ and $T=S_F$, for which $w_b=1$ on $S_F$ and $w_z\le e^{-\lambda\dist(S_F,S_H)}$ on $S_H$, gives $|\mu^\sigma(F)-\mu^{\sigma'}(F)|\le L_F\sum_{b\in S_F}\varrho(b)\le L_F\frac{\theta D}{1-\theta}|S_H|e^{-\lambda\dist(S_F,S_H)}$ for any two boundary values. Since $H$ depends only on $\sigma_{S_H}$, $\Cov_\mu(F,H)=\iint H(\sigma)\bigl(\mu^\sigma(F)-\mu^{\sigma'}(F)\bigr)\, d\mu(\sigma)\,d\mu(\sigma')$, and the bound follows. \end{proof} \begin{lemma}\label{lem:blockcomparison} Let the blocks partition $A$, let $C_{Bj}$ be the block coefficients of Sec.~\ref{sec:plaqblocks}, with $\sum_BC_{Bj}\le\alpha<1$ for every $j$ and $C_{Bj}=0$ unless every link of $B$ lies within distance $2$ of $j$, and fix $\lambda>0$ with $\theta=\alpha e^{2\lambda}<1$. Let $S\subset A$ be a union of blocks and let $\mu_1,\mu_2$ be invariant under $\gamma^{\mathcal B}_B$ for every block $B\not\subset S$. With $T$, $w_b$ as in Lemma~\ref{lem:comparison} and $\sum_{z\in S}w_z<\infty$, there is a coupling with $\sum_{b\notin S}w_b\varrho(b)\le\frac{\theta D}{1-\theta}\sum_{z\in S}w_z$, and $\mu_1=\mu_2$ if $S=\emptyset$. \end{lemma} \begin{proof} Take a minimiser $\pi^*$ of $\Psi$ as before. For a block $B\not\subset S$ replace $(\sigma_B,\sigma'_B)$ by an optimal coupling, for the metric $d$, of the two block laws; it is again a coupling, and changing the boundary links one at a time bounds the expected new $d$-distance by $\sum_jC_{Bj}\varrho(j)$. Put $\bar w_B=\max_{i\in B}w_i$. Minimality gives \[ \sum_{i\in B}w_i\varrho(i)\le\bar w_B\sum_jC_{Bj}\varrho(j), \] and $\bar w_B\le e^{2\lambda}w_j$ whenever $C_{Bj}\ne0$. Summing over the blocks outside $S$ and using the column sums, \[ \sum_{b\notin S}w_b\varrho(b)\le\theta\Bigl[\sum_{j\notin S}w_j\varrho(j)+ \sum_{z\in S}w_z\varrho(z)\Bigr], \] which gives the claim as in Lemma~\ref{lem:comparison}. \end{proof} \begin{lemma}\label{lem:dscomparison} Let $n\ge2$ be even, let $V_x$, $x\in2\Z^4$, be the translates of Sec.~\ref{sec:bypass}, and suppose that $D(n,\alpha)$ holds. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item If a probability measure $\mu$ on $G^A$ is invariant under $\gamma^F_e$ for every $e$ in a finite set $V$, it is invariant under $\gamma_V$. \item Fix $\lambda>0$ with $\theta=\alpha e^{\lambda\rho_n}<1$. Let $S\subset A$, and let $\mu_1,\mu_2$ be invariant under $\gamma_{V_x}$ for every $x$ with $V_x\cap S=\emptyset$. With $T$, $w_b$ as in Lemma~\ref{lem:comparison} and $\sum_{z\in S}w_z<\infty$, there is a coupling with \[ \sum_{b\notin S}w_b\varrho(b)\le\frac{D(1+c_ne^{\lambda\rho_n})}{1-\theta} \sum_{z\in S}w_z , \] and $\mu_1=\mu_2$ if $S=\emptyset$. \end{enumerate} \end{lemma} \begin{proof} (a) Let $\nu_\eta$ be the conditional law of $Q_V$ given the links $\eta$ outside $V$. For bounded $f$ on $G^V$ and bounded $g$ of $\eta$, $\gamma^F_e(fg)=g\,\gamma^F_ef$ for $e\in V$, so invariance gives $\int g(\eta)\bigl(\nu_\eta(\gamma^F_ef)-\nu_\eta(f)\bigr)\,d\mu=0$ for all $g$; taking $f$ in a countable dense set of continuous functions, $\nu_\eta$ is invariant under every $\gamma^F_e(\cdot\mid\cdot,\eta)$, $e\in V$, for $\mu$-almost every $\eta$. So is $\gamma_V(\cdot\mid\eta)$, whose conditional laws of single links are the $\gamma^F_e$. The composition $P_\eta$ of the kernels $\gamma^F_e$, $e\in V$, in a fixed order has a density with respect to Haar measure on $G^V$ bounded below by a positive constant, because each single-link density is at least $e^{-12N^2\beta}$ times its maximum over the group. A Markov kernel with such a lower bound has at most one invariant probability measure (Doeblin's condition), so $\nu_\eta=\gamma_V(\cdot\mid\eta)$. (b) Let $\pi^*$ minimise $\Psi$ as in Lemma~\ref{lem:comparison} and call $x$ admissible if $V_x\cap S=\emptyset$. For admissible $x$ replace $(\sigma_{V_x},\sigma'_{V_x})$ by an optimal coupling, for $d_{V_x}$, of the two block laws; both $\mu_i$ are invariant, so the result is a coupling, and changing the boundary links one at a time bounds the new expected $d_{V_x}$-distance by $\sum_jC_{V_x,j}\varrho(j)$. With $\bar w_x=\max_{i\in V_x}w_i$, minimality gives \[ \sum_{i\in V_x}w_i\varrho(i)\le\bar w_x\sum_{j\notin V_x}C_{V_x,j}\varrho(j), \] and $\bar w_x\le e^{\lambda\rho_n}w_j$ whenever $C_{V_x,j}\ne0$. Let $S'$ be the set of links outside $S$ that lie in a translate meeting $S$; each is within distance $\rho_n$ of a link of $S$, so $\sum_{i\in S'}w_i\le c_ne^{\lambda\rho_n}\sum_{z\in S}w_z$. A link outside $S\cup S'$ lies in $N_n$ translates, all admissible. Summing over admissible $x$ and using $\sum_{x:\,j\notin V_x}C_{V_x,j}=K_{c(j)}\le\alpha N_n$, \[ N_n\sum_{i\notin S\cup S'}w_i\varrho(i)\le\theta N_n\Bigl[\sum_{i\notin S\cup S'}w_i\varrho(i)+\sum_{i\in S\cup S'}w_i\varrho(i)\Bigr]. \] All sums are finite and $\varrho\le D$, so $\sum_{i\notin S\cup S'}w_i \varrho(i)\le\frac{\theta D}{1-\theta}(1+c_ne^{\lambda\rho_n})\sum_{z\in S} w_z$; adding $\sum_{i\in S'}w_i\varrho(i)\le Dc_ne^{\lambda\rho_n}\sum_{z\in S}w_z$ gives the bound. If $S=\emptyset$ then $S'=\emptyset$ and $\varrho\equiv0$. \end{proof} The covariance bound of Corollary~\ref{cor:decay} follows from part (b) in the same way, with $\theta D/(1-\theta)$ replaced by $D(1+c_ne^{\lambda\rho_n})/(1-\theta)$, since the conditional measures $\mu^\sigma$ there are invariant under $\gamma_{V_x}$ for $V_x\cap S_H=\emptyset$ by part (a). \section{The plaquette blocks}\label{app:plaqblocks} Table~\ref{tab:plaqblocks} lists the block plaquettes of Sec.~\ref{sec:plaqblocks} modulo $2\Z^4$: the plaquette at $x$ in the plane $\{\mu,\nu\}$ is a block plaquette exactly when $x$ modulo two and $\{\mu,\nu\}$ appear in the table. Eight lie in the plane $\{0,1\}$ and eight in $\{2,3\}$; together they contain the $48$ active links of a cell of side two, each once. \begin{table}[h] \caption{The sixteen block plaquettes modulo two: the base point $x$ (digits $x_0x_1x_2x_3$ modulo two) and the plane.} \label{tab:plaqblocks} \begin{ruledtabular} \begin{tabular}{cc|cc} $x$&plane&$x$&plane\\ \colrule 0000&$\{0,1\}$&0000&$\{2,3\}$\\ 0001&$\{0,1\}$&0011&$\{2,3\}$\\ 0010&$\{0,1\}$&0100&$\{2,3\}$\\ 0011&$\{0,1\}$&0111&$\{2,3\}$\\ 1100&$\{0,1\}$&1000&$\{2,3\}$\\ 1101&$\{0,1\}$&1011&$\{2,3\}$\\ 1110&$\{0,1\}$&1100&$\{2,3\}$\\ 1111&$\{0,1\}$&1111&$\{2,3\}$\\ \end{tabular} \end{ruledtabular} \end{table} In closed form, the plaquette at $x$ in $\{0,1\}$ is a block plaquette when $x_0\equiv x_1$ and the plaquette at $x$ in $\{2,3\}$ when $x_2\equiv x_3$; the blocks do not depend on the line field beyond the fact that every plaquette contains one fixed link. The bounds of Lemma~\ref{lem:vmfconst} and the chain~\eqref{eq:blockconst} are proved by hand in Sec.~\ref{sec:plaqblocks}. The archive also evaluates the functions of Lemma~\ref{lem:vmfconst} in ball arithmetic at $128$ bits, from the series of $Z_0$ and of its first three derivatives in $s=r^2$ summed to forty-eight terms with the tail bounded by twice the first omitted term; these sharper constants are not used by any result. For $\SU(3)$ the constants of Remark~\ref{rem:su3blocks} are evaluated on the $841$ boxes of side $\frac3{16}$ at $r=\frac32$ and the $784$ boxes of side $\frac{19}{100}$ at $r=\frac{38}{25}$ and the $826$ boxes of side $\frac{77}{400}$ at $r=\frac{77}{50}$ with the moments of Appendix~\ref{app:cover}, and the constants of~\eqref{eq:blockconst3} and $\alpha_3$ are evaluated from these bounds and the covers of Remark~\ref{rem:su3cover} in ball arithmetic at $120$ bits. The covers at $r=\frac{38}{25}$ and $\frac{77}{50}$ are computed as in Appendix~\ref{app:cover} with $r=\frac{38}{25}$ and the threshold $1.03$, and with $r=\frac{77}{50}$ and the threshold $1.0275$. No result of the paper uses these numbers; Theorem~\ref{thm:su3blocks} uses the constants of Lemma~\ref{lem:su3modulus}. \section{The blocks of Lemma~\ref{lem:local}}\label{app:blocks} \emph{Diagonal blocks.} Let $w_a=e^{i\phi}$. On the diagonal part, $f_p=-\cos\phi\,(y_1+y_2-y_3-y_4)^2$ and $c_p=2(y_1y_4-y_1y_2+y_2y_3-y_3y_4)$, so \[ \begin{split} \mathcal F={}&\sum_i\alpha_iy_i^2+\cos\phi\,(y_1+y_2-y_3-y_4)^2\\ &-\tfrac95(y_1y_4-y_1y_2+y_2y_3-y_3y_4) \end{split} \] with $y_k=0$ and weights $\alpha_{k\pm1}=\frac{21}{10}$, $\alpha_{k+2}=\frac54$. It is affine in $\cos\phi$, so positivity at $\cos\phi=\pm1$ suffices. For $k=1$ and $\cos\phi=1$ the matrix in $(y_2,y_3,y_4)$ is \[ \begin{pmatrix}\frac{31}{10}&-\frac{19}{10}&-1\\[2pt]-\frac{19}{10}&\frac94& \frac{19}{10}\\[2pt]-1&\frac{19}{10}&\frac{31}{10}\end{pmatrix}, \] and Table~\ref{tab:minors} lists the leading principal minors for every position; all are positive. \begin{table}[h] \caption{Leading principal minors of the diagonal blocks of Lemma~\ref{lem:local}.} \label{tab:minors} \begin{ruledtabular} \begin{tabular}{lll} Fixed link&$\cos\phi=1$&$\cos\phi=-1$\\ \colrule $e_1$, $e_4$&$\frac{31}{10},\ \frac{673}{200},\ \frac{8421}{2000}$& $\frac{11}{10},\ \frac{53}{200},\ \frac{21}{2000}$\\[2pt] $e_2$&$\frac{31}{10},\ \frac{861}{100},\ \frac{8421}{2000}$& $\frac{11}{10},\ \frac{21}{100},\ \frac{21}{2000}$\\[2pt] $e_3$&$\frac94,\ \frac{673}{200},\ \frac{8421}{2000}$& $\frac14,\ \frac{53}{200},\ \frac{21}{2000}$\\ \end{tabular} \end{ruledtabular} \end{table} \emph{Pair blocks.} Let $w_a=e^{i(\chi+\vartheta)}$ and $w_b=e^{i(\chi-\vartheta)}$. The terms of $f_p$ carry $w_a+\bar w_b=2\cos\chi\,e^{i\vartheta}$ or its conjugate, and $c_p$ carries $w_a\bar w_b=e^{2i\vartheta}$, so the Hermitian matrix $M(m,\vartheta)$ of $\mathcal F$, $m=\cos\chi$, is affine in $m$ with $M(-1,\vartheta)=M(1,\vartheta+\pi)$, and $m=1$ suffices. For the fixed link $e_2$, in the variables $(z_1,z_3,z_4)$ and with $c=\cos\vartheta$, \[ M=\begin{pmatrix}\frac{21}5+2c&-2e^{i\vartheta}&-\frac95-2e^{i\vartheta}\\[2pt] -2e^{-i\vartheta}&\frac{21}5+2c&\frac95+2e^{-i\vartheta}\\[2pt] -\frac95-2e^{-i\vartheta}&\frac95+2e^{i\vartheta}&\frac52+2c\end{pmatrix}. \] With $u=\cos^2(\vartheta/2)\in[0,1]$ its leading principal minors are \[ \tfrac{20u+11}5,\quad\tfrac{(20u+1)(20u+21)}{25},\quad \tfrac{400u(40u^2-23u+4)+21}{250}, \] and $40u^2-23u+4$ has negative discriminant $529-640$. The other positions give the same determinant and the leading minors $\frac{20u+11}5$ or $\frac{8u+1}2$ and $\frac{800u^2-180u+53}{50}$, whose discriminant $32400-169600$ is negative. Every block is positive definite. The inequality is not tight. In floating point the least eigenvalue of the whole form over $W\in\U(N)$, $N\le4$, is $0.0149$, which is also the least eigenvalue of the exact blocks; this slack is what the choice of the weights $\frac{21}{10}$, $\frac54$, $\frac9{10}$ trades against the constant $\frac{29}2$. \section{The certified computation}\label{app:cover} This appendix describes the computation of Remark~\ref{rem:su3cover}, which no result of the paper uses. \emph{Truncation.} For a corner $b$ of a box the moments of Lemma~\ref{lem:characters} are derivatives of $Z$ at $(b,\bar b)$. The truncated series is $\E_H[A_t(UD)B_t(\bar UD')]$ with $A_t$ the part of $\sum_\lambda c_\lambda s_\lambda(UD)$ over partitions with $\lambda_1-\lambda_3\le34$ and $\lambda_3\le11$, and $B_t$ likewise. Every omitted partition has $|\lambda|>34$, because $3\cdot12>34$, and the omitted part is at most $T(\|D\|)=\sum_{n>34} (\frac32\|D\|)^n/n!$ in modulus. On the polydisc $|D_{ij}-b_{ij}|\le\rho$, $\|D\|\le\|b\|+3\rho$, and Cauchy's estimate bounds the $\alpha$-th derivative of the omitted part by $\alpha!\rho^{-|\alpha|}T(\|b\|+3\rho)$. The derivatives of the full exponential are at most $2^{-|\alpha|}e^{v/2}$ with $v=\sum_k|b_k|$, since $|\Tr(Ub)|\le\sum_k|b_k|$ for diagonal $b$. With $\varepsilon_\alpha=2^{|\alpha|}\alpha!\rho^{-|\alpha|}T(\|b\|+3\rho)$ the truncated moment differs from the true one by at most $\varepsilon_\alpha e^{v/2} +(e^{v/2}+\varepsilon_\alpha)\varepsilon_\beta$, and $\rho$ is chosen to minimise $\rho^{-6}T$. The largest error bound of an unnormalised moment at any corner is $1.7\times10^{-11}$. \emph{Weight zero.} Conjugation by the diagonal torus $T$ preserves Haar measure and $\ReTr(Ub)$ for diagonal $b$, and multiplies $u_{ij}$ by $t_it_j^{-1}$; a moment of a monomial on which $T$ acts by a nontrivial character is therefore $0$, and only $8278$ of the $134596$ monomials of degree at most six have the trivial character. The functions $\Tr(X)$, $e_2(X)$, $\det X$ of $X=DD'^\top$ and $\det D$, $\det D'$ are invariant under the same action on the variables, so after these are formed in the full ring of Taylor polynomials of degree six in the $18$ variables, the series is computed in the subring of weight-zero monomials. \emph{Rounding.} The truncated series is a Taylor polynomial of total degree six in the $36$ real directions of $(D-b,D'-\bar b)$, computed in double precision. Every operation is repeated on absolute values and carries an operation count $n$, so that the computed coefficient differs from the exact one by at most $\gamma_n=nu/(1-nu)$, $u=2^{-53}$, times the absolute-value coefficient~\cite[Sec.~3.1]{Hig02}. With $C$ in $2^{-30}\Z$, the polynomials $2^{31}q_h$, $X_a(2^{31}q_h)$, $12\cdot2^{31}(\varphi_h+\Delta q_h)$ and $12\cdot2^{31}\ip{\nabla\varphi_{F_k}} {\nabla q_h}$ for the coordinate vectors $h$ have coefficients in $\Z[i]$, and every partial sum in their computation is bounded, through the same operations on absolute values, by less than $2^{50}$; they are therefore computed exactly. Normalisation by $Z$, the products with the corner coordinates, the quadratic forms and their error matrices are formed with the same bookkeeping, and the largest eigenvalues are bounded in ball arithmetic~\cite{Joh17}. \emph{Separation.} The functional $c$ of a discarded box is the dual solution of a linear program for the gauge of the centre of the box with respect to a $72\times72$ grid of $T$. Only the final inequality is used: $\max_T\ell$ is bounded by its maximum over a $512\times512$ grid plus $\frac12H\delta^2/2$, with $\delta=2\pi/512$ and $H=|c_1|+|c_2|+2|c_3|$ a bound for its Hessian on the torus, and the inequality is checked with relative margin $10^{-12}$. \section{The transfer matrix}\label{app:transfer} This appendix recalls the Osterwalder--Schrader reconstruction for the lattice~\cite{OS73,OS75,OSei78} in the form used for the gaps, following our earlier paper~\cite{BB26a}. Let $\omega$ be a state on the gauge-invariant local functions of $\Z^4$ that is invariant under translations and the hypercubic group and reflection positive for reflections in planes through sites and in planes bisecting links; by Proposition~\ref{prop:state}(iii)--(iv) and its analogues in the other theorems, the states of this paper are of this kind. The reconstruction gives a Hilbert space with a vector $\Omega$, a map $F\mapsto\Psi_F$ with dense range, and a self-adjoint $\mathbb T$ with $0\le\mathbb T\le1$, $\mathbb T\Omega=\Omega$ and $\ip{\Psi_F}{\mathbb T^t\Psi_H}=\omega(\vartheta(\bar F)\tau_tH)$, where $\vartheta$ is the time reflection and $\tau_t$ the time translation.\footnote{$\Psi_F$ is the class of $F$ in the completion of the positive-time local functions for the form $\ip FH=\omega(\vartheta(\bar F)H)$, which is positive semidefinite by reflection positivity. Since $|\omega(\vartheta(\bar F)F)|\le\omega(|F|^2)$ by the Cauchy--Schwarz inequality and the invariance of $\omega$ under $\vartheta$, functions that are close in $L^2(\omega)$ give close vectors.} \begin{proposition}\label{prop:gapdecay} For $m>0$ the following are equivalent: \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item the spectrum of $\mathbb T$ on $\Omega^\perp$ lies in $[0,e^{-m}]$; \item for every $F$ in a class of gauge-invariant local functions supported at times in a bounded interval and dense in $L^2(\omega)$ on each such support, there is $C_F<\infty$ with $0\le\omega(\vartheta(\bar F)\tau_tF)-|\omega(F)|^2 \le C_Fe^{-mt}$ for all $t\ge0$. \end{enumerate} Under \textup{(a)} one may take $C_F=\|F\|_\infty^2$ for every bounded $F$. \end{proposition} \begin{proof} If (a) holds, then \[ \omega(\vartheta(\bar F)\tau_tF)-|\omega(F)|^2=\ip{\Psi'_F}{\mathbb T^t\Psi'_F} \] with $\Psi'_F=\Psi_F-\ip\Omega{\Psi_F}\Omega\in\Omega^\perp$; this number lies in $[0,e^{-mt}\|\Psi_F\|^2]$, and $\|\Psi_F\|^2\le\|F\|_\infty^2$. Conversely, let $\sigma$ be the spectral measure of $\mathbb T$ at $\Psi'_F$ and let $\delta>0$. Then \[ \sigma\bigl((e^{-m+\delta},1]\bigr)e^{(-m+\delta)t}\le C_Fe^{-mt} \] for all $t$, so $\sigma((e^{-m+\delta},1])=0$. The spectral projection of $(e^{-m},1]$ restricted to $\Omega^\perp$ therefore annihilates the vectors $\Psi'_F$ of the class, which are dense by the footnote above, and it vanishes. \end{proof} The class in (b) cannot be replaced by one observable. On $\C^3$ let $\mathbb T=\mathrm{diag}(1,e^{-m},e^{-m/2})$, $\Omega=e_1$ and $\Psi'_F=e_2$: the function in (b) is $e^{-mt}$, while the spectrum on $\Omega^\perp$ reaches $e^{-m/2}$. The decay of one two-point function places only the spectral measure of $\Psi'_F$ in $[0,e^{-m}]$, and the gap follows when such vectors are dense in $\Omega^\perp$, as in (b) and in Theorem~\ref{thm:clustering}. Proposition~8.8 of~\cite{FaS26} draws the gap on $\Omega^\perp$ from the two-point function of one fixed observable. \section{Spectral measures and Laplace transforms}\label{app:spectral} This appendix collects the facts on spectral measures that the continuum step uses, with proofs. \begin{lemma}\label{lem:spectralmeasure} Let $T$ be a self-adjoint operator on a Hilbert space with $0\le T\le1$ and let $\psi$ be a unit vector. There is a unique probability measure $\nu$ on $[0,1]$ with $\ip\psi{h(T)\psi}=\int h\,d\nu$ for every bounded Borel function $h$ on $[0,1]$. If $T\psi'=\psi'$ implies $\ip{\psi'}\psi=0$, then $\nu(\{1\})=0$. If $\mu$ is the image of $\nu$ under $\lambda\mapsto-c^{-1}\log\lambda\in[0,\infty]$ for some $c>0$, then $\ip\psi{T^{\sigma/c}\psi}=\int_{[0,\infty]}e^{-\sigma E}d\mu(E)$ for $\sigma>0$, with the integrand $0$ at $E=\infty$. \end{lemma} \begin{proof} The spectral theorem~\cite[Sec.~VII.2]{RS80} gives the measure $\nu$, carried by the spectrum of $T$, which lies in $[0,1]$; uniqueness holds because the moments $\int\lambda^n d\nu=\ip\psi{T^n\psi}$ determine a measure on a compact interval, polynomials being dense in $C[0,1]$. The spectral projection $P=\mathbf 1_{\{1\}}(T)$ projects onto the eigenvectors of $T$ with eigenvalue $1$, so $\nu(\{1\})=\|P\psi\|^2=\ip{P\psi}\psi=0$ under the hypothesis. The last identity is the change of variables $\lambda^{\sigma/c}=e^{-\sigma E}$ for $\lambda=e^{-cE}$, with $\lambda=0$ corresponding to $E=\infty$. \end{proof} \begin{lemma}\label{lem:helly} Every sequence of probability measures on a compact metric space $K$ has a subsequence that converges weakly, that is, against every continuous function, to a probability measure on $K$. \end{lemma} \begin{proof} $C(K)$ is separable; let $(h_j)$ be dense. The numbers $\int h_j\,d\mu_k$ are bounded by $\|h_j\|_\infty$, so a diagonal subsequence makes all of them converge. By density the limit $\Lambda(h)=\lim\int h\,d\mu_k$ exists for every $h\in C(K)$; $\Lambda$ is linear, positive and $\Lambda(1)=1$, so by the Riesz representation theorem it is integration against a probability measure. \end{proof} \begin{lemma}\label{lem:laplace} Let $\mu$ be a probability measure on $[0,\infty]$ and $\rho(\sigma)=\int e^{-\sigma E}d\mu(E)$ for $\sigma>0$. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item $\rho$ is nonincreasing and log-convex: $\rho(\sigma)^2\le\rho(\sigma-s)\rho(\sigma+s)$ for $00$ for one $\sigma>0$ if and only if $\mu([0,\infty))>0$, and then $\rho(\sigma)>0$ for every $\sigma>0$. \item $\rho(\sigma)\to\mu([0,\infty))$ as $\sigma\downarrow0$. \item If $\mu$ is carried by $[m,\infty]$, then $\rho(\sigma)\le e^{-m\sigma}$. \item For $M>0$, $\mu([0,M])\ge\rho(\sigma)-e^{-\sigma M}$. \item If $\mu_k\to\mu$ weakly on $[0,\infty]$, then $\int e^{-\sigma E}d\mu_k\to\rho(\sigma)$ for every $\sigma>0$. \end{enumerate} \end{lemma} \begin{proof} (a) The integrand decreases in $\sigma$. By the Cauchy--Schwarz inequality, \[ \Bigl(\int e^{-\sigma E}d\mu\Bigr)^2\le\int e^{-(\sigma-s)E}d\mu\int e^{-(\sigma+s)E}d\mu . \] (b) The integrand is positive exactly on $[0,\infty)$. (c) The integrand increases to $\mathbf 1_{[0,\infty)}$ as $\sigma\downarrow0$, and monotone convergence applies. (d) The integrand is at most $e^{-m\sigma}$ on $[m,\infty]$. (e) Split the integral at $M$: it is at most $\mu([0,M])+e^{-\sigma M}$. (f) For $\sigma>0$ the function $E\mapsto e^{-\sigma E}$, extended by $0$ at $\infty$, is continuous on $[0,\infty]$. \end{proof} By (b) and (e), a lower bound $\rho(\sigma)\ge c>0$ at one $\sigma$ places mass at least $c/2$ on the bounded interval $[0,\sigma^{-1}\log(2/c)]$; for the measures $\mu_k$ of Theorem~\ref{thm:flowlimit} a bound uniform in $k$ therefore makes them tight on $[0,\infty)$ with a fixed fraction of their mass. Part (b) also excludes a positive Laplace representation for a spectral cutoff of compact support. If $\chi(\lambda)=\int_{[0,\infty)}e^{-s\lambda} d\nu(s)$ with $\nu$ a positive measure and $\chi(0)=\nu([0,\infty))>0$, then (b), applied to $\nu/\chi(0)$, gives $\chi(\lambda)>0$ for every $\lambda>0$, so $\chi$ vanishes nowhere. The cutoff $\chi_\sigma$ of~\cite[Sec.~9, Eqs.~(9.11), (9.12)]{FaS26} equals $1$ on $[0,\sigma]$ and $0$ on $[2\sigma,\infty)$ and is asserted to have such a representation. \section{Spectral gaps on cyclic subspaces}\label{app:cyclic} The local form $\mathrm{BP}_{\rm loc}$ of Definition~\ref{def:bploc} asks for a spectral gap of the cube chain on the cyclic subspace of each local function. This appendix records the elementary facts about such gaps that Sec.~\ref{ssec:bpweak} uses, for a self-adjoint operator $K$ on a real or complex Hilbert space $H$ with $0\le K\le I$. For $g\in H$ let $\nu_g$ be its spectral measure, so that $\ip g{\phi(K)g}=\int\phi\,d\nu_g$ for bounded Borel $\phi$ on $[0,1]$, and for a Borel set $B$ let $E(B)$ be the spectral projection. \begin{lemma}\label{lem:cyclicgap} Let $g\in H$ and $0<\epsilon<1$. The following are equivalent. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\lVert K^tg\rVert\le(1-\epsilon)^t\lVert g\rVert$ for every integer $t\ge0$. \item $\nu_g((1-\epsilon,1])=0$. \item $E((1-\epsilon,1])g=0$. \end{enumerate} Under these conditions $\ip{g'}{(I-K)g'}\ge\epsilon\lVert g'\rVert^2$ for every $g'$ in the closed cyclic subspace generated by $g$. \end{lemma} \begin{proof} $\lVert E(B)g\rVert^2=\nu_g(B)$, so (ii) and (iii) are equivalent. Under (ii), $\lVert K^tg\rVert^2=\int\lambda^{2t}d\nu_g\le(1-\epsilon)^{2t}\nu_g( [0,1])$, which is (i). If (ii) fails, there is $\eta>0$ with $m=\nu_g((1- \epsilon+\eta,1])>0$, since these sets increase to $(1-\epsilon,1]$ as $\eta\downarrow0$; then $\lVert K^tg\rVert^2\ge m(1-\epsilon+\eta)^{2t}$, which exceeds $(1-\epsilon)^{2t}\lVert g\rVert^2$ for large $t$, and (i) fails. The closed cyclic subspace of $g$ is the closure of $\{\phi(K)g\}$ over polynomials $\phi$; it lies in the range of $E([0,1-\epsilon])$, since $E((1-\epsilon,1])$ commutes with $K$, and on that range $I-K\ge \epsilon$. \end{proof} \begin{lemma}\label{lem:cyclicspan} Let $F\subset H$ and $0<\epsilon<1$. Every $g\in F$ satisfies the conditions of Lemma~\ref{lem:cyclicgap} if and only if the range of $E((1-\epsilon,1])$ is orthogonal to $F$, and then every vector of the closed linear span of $F$ satisfies them. In that case, for every $g$ in that span and every $h\in H$, \[ \lvert\ip gh\rvert^2\le\lVert g\rVert^2\,\frac{\ip h{(I-K)h}}\epsilon . \] \end{lemma} \begin{proof} $E=E((1-\epsilon,1])$ is an orthogonal projection, so $Eg=0$ if and only if $g$ is orthogonal to the range of $E$; the set of such $g$ is a closed subspace. For $g$ in it, $g$ and $h'=(I-E)h$ lie in the range $R$ of $I-E$, which is invariant under $K$ and on which $I-K\ge\epsilon$. Let $T$ be the restriction of $(I-K)^{1/2}$ to $R$; it is invertible with $\lVert T^{-1} \rVert\le\epsilon^{-1/2}$. Then $\ip gh=\ip g{h'}=\ip{T^{-1}g}{Th'}$, and \[ \begin{split} \lvert\ip gh\rvert^2&\le\frac{\lVert g\rVert^2}\epsilon\,\ip{h'}{(I-K)h'}\\ &\le\frac{\lVert g\rVert^2}\epsilon\,\ip h{(I-K)h}, \end{split} \] the last step because $E$ commutes with $K$ and $\ip{Eh}{(I-K)Eh}\ge0$. \end{proof} So $\mathrm{BP}_{\rm loc}(n,\varpi)$ on $\Lambda_L$ says that the spectral projection of the cube chain for $(1-\varpi n^4/L^4,1]$ annihilates the closed span of the local gauge-invariant functions of mean zero, and the inequality of Theorem~\ref{thm:bplocal}(iv) is Lemma~\ref{lem:cyclicspan} for that span. The span is not an algebra: a product of two local functions in distant boxes is not local, and $\mathrm{BP}_{\rm loc}$ says nothing about it. A Poincar\'e inequality for one function bounds the mean of $1-\lambda$ under its spectral measure and does not give a gap on its cyclic subspace. \begin{lemma}\label{lem:rayleighnogap} For all $0<\eta<\epsilon<1$ and $0<\theta<\frac\pi2$ there are $H$, $K$ and $g$ with $\lVert g\rVert=1$, $\ip g{(I-K)g}\ge\cos^2\theta$, and $\lVert K^tg\rVert\ge(1-\eta)^t\sin\theta$ for every integer $t\ge0$; so $g$ fails the conditions of Lemma~\ref{lem:cyclicgap}. \end{lemma} \begin{proof} Take $H=\mathbb R^2$, $K=\operatorname{diag}(0,1-\eta)$ and $g=(\cos\theta, \sin\theta)$. Then $\ip g{(I-K)g}=\cos^2\theta+\eta\sin^2\theta$ and $K^tg=(0,(1-\eta)^t\sin\theta)$ for $t\ge1$, and $\nu_g(\{1-\eta\})= \sin^2\theta>0$ with $1-\eta\in(1-\epsilon,1]$. \end{proof} With $\theta$ close to $0$ the Rayleigh quotient of $g$ is close to one, the best possible value, while a fraction $\sin^2\theta$ of its spectral mass relaxes at the rate $\eta$ only. This is why Proposition~\ref{prop:bpnoloc}, a Poincar\'e inequality for each function of a finite set of links, does not give $\mathrm{BP}_{\rm loc}$. \section{The character expansion}\label{app:characters} Let $G$ be a compact connected Lie group with normalised Haar measure $dU$. For an irreducible unitary representation $\rho_\lambda$ of dimension $d_\lambda$ write $\chi_\lambda=\Tr\rho_\lambda$. The characters form an orthonormal basis of the square-integrable class functions (Peter--Weyl and Schur orthogonality~\cite[Chs.~2--4]{Bum13}). \begin{lemma}\label{lem:convolution} Let $w\in L^1(G)$ be a class function with $w(U^{-1})=w(U)$, and put \[ \hat c_\lambda=d_\lambda^{-1}\int w\,\chi_\lambda\,dU . \] Then $\int w(U)\rho_\lambda(U)\,dU=\hat c_\lambda I$, and $\int w(U)\chi_\lambda(U^{-1}V)\,dU=\hat c_\lambda\chi_\lambda(V)$. \end{lemma} \begin{proof} $M=\int w(U)\rho_\lambda(U)dU$ commutes with every $\rho_\lambda(V)$, since $w(VUV^{-1})=w(U)$ and Haar measure is invariant; by Schur's lemma $M=cI$, and taking traces gives $c=\hat c_\lambda$. Then $\int w(U)\chi_\lambda(U^{-1}V)dU=\Tr(\int w(U)\rho_\lambda(U^{-1})dU\, \rho_\lambda(V))$, and the substitution $U\mapsto U^{-1}$ with $w(U^{-1})=w(U)$ turns the integral into $M$. \end{proof} \begin{lemma}\label{lem:wilsoncoeff} Let $G=\SU(N)$ and $\beta>0$, and let \[ w_\beta(U)=e^{\frac\beta N\ReTr U}\Bigl/\int e^{\frac\beta N\ReTr V}dV . \] Then $\hat c_\lambda(\beta)=d_\lambda^{-1}\int w_\beta\chi_\lambda\,dU$ is real, $0\le\hat c_\lambda(\beta)\le1$, and $\hat c_\lambda(\beta)<1$ for $\lambda\ne0$. Moreover \[ \hat c_\lambda(\beta)=1-\frac{N\,C_2(\lambda)}\beta+O(\beta^{-2})\qquad (\beta\to\infty), \] where $-\sum_a\rho_\lambda(T_a)^2=2C_2(\lambda)I$ for a basis $(T_a)$ of $\mathfrak{su}(N)$ orthonormal for $|X|^2=-\Tr X^2$, so that $C_2=(N^2-1)/(2N)$ on the fundamental representation. \end{lemma} \begin{proof} Since $w_\beta(\bar U)=w_\beta(U)$ and $\chi_\lambda(\bar U)= \overline{\chi_\lambda(U)}$, and complex conjugation preserves Haar measure, $\int w_\beta\chi_\lambda=\int w_\beta\bar\chi_\lambda$, which is the complex conjugate of the left side; so $\hat c_\lambda$ is real. Next, $w_\beta$ is a multiple of $\exp(\frac\beta{2N}(\chi_{\mathrm F}+\bar\chi_{\mathrm F}))= \sum_{m}\frac1{m!}(\frac\beta{2N})^m(\chi_{\mathrm F}+\bar\chi_{\mathrm F})^m$, and each power is the character of a representation, hence a sum of irreducible characters with nonnegative integer coefficients. By orthonormality $\int w_\beta\bar\chi_\lambda\ge0$, so $\hat c_\lambda\ge0$. As $|\chi_\lambda(U)|\le d_\lambda$, with $\operatorname{Re}\chi_\lambda(U)=d_\lambda$ only where $\rho_\lambda(U)=I$, a proper closed subgroup when $\lambda\ne0$ and so a Haar null set, and $w_\beta>0$ everywhere, $\hat c_\lambda=d_\lambda^{-1} \int w_\beta\operatorname{Re}\chi_\lambda<1$ for $\lambda\ne0$. For the expansion, $\ReTr U\le N$ with equality only at $U=I$, so outside a neighbourhood $\{U=e^X:|X|<\delta\}$ both integrals $\int e^{\frac\beta N\ReTr U}\chi_\lambda$ and $\int e^{\frac\beta N\ReTr U}$ are smaller than their values near $I$ by a factor $e^{-c\beta}$. Near $I$ write $U=e^X$; the Haar density in these coordinates is $j(X)=1+O(|X|^2)$, even in $X$. For antihermitian $X$, $\Tr X^3$ is imaginary, so \[ \tfrac1N\ReTr e^X=1-\tfrac1{2N}|X|^2+r(X),\qquad 0\le r(X)\le C|X|^4, \] and $d_\lambda^{-1}\chi_\lambda(e^X)=1+d_\lambda^{-1}\Tr\rho_\lambda(X)+ \frac1{2d_\lambda}\Tr\rho_\lambda(X)^2+c_3(X)+O(|X|^4)$, with $c_3$ cubic and $\Tr\rho_\lambda(X)=0$, since $\mathfrak{su}(N)$ is perfect. Let $\langle\cdot\rangle$ denote the average against the Gaussian weight $e^{-\beta|X|^2/(2N)}$, under which the coordinates $x_a$ of $X=\sum x_aT_a$ are independent with variance $N/\beta$. With $h=j\,e^{\beta r}$, \[ \begin{split} \hat c_\lambda(\beta)&=\frac{\langle d_\lambda^{-1}\chi_\lambda(e^X)\, h\rangle}{\langle h\rangle}+O(e^{-c\beta})\\ &=1+\frac{\langle\frac1{2d_\lambda}\Tr\rho_\lambda(X)^2\,h\rangle} {\langle h\rangle}+\frac{\langle(c_3(X)+O(|X|^4))h\rangle}{\langle h\rangle}, \end{split} \] the averages being taken over $|X|<\delta$. Under $\langle\cdot\rangle$, $|X|^{2m}$ has mean of order $\beta^{-m}$ and $\beta|r(X)|\le C\beta|X|^4$ has mean of order $\beta^{-1}$, so $h=1+O(\beta^{-1})$ in every moment. The weight $h$ is even in $X$, since $\Tr X^k$ is imaginary for odd $k$ and $j$ is even, so the odd term $c_3$ contributes nothing; the quartic remainder and the correlation of the quadratic term with $h-1$ contribute $O(\beta^{-2})$. Finally $\langle\Tr\rho_\lambda(X)^2\rangle= \frac N\beta\sum_a\Tr\rho_\lambda(T_a)^2=-\frac{2NC_2(\lambda)d_\lambda} \beta$, which gives the expansion. For $\SU(2)$ and the representation of spin $j$ the integrals are explicit, $\hat c_j(\beta)=I_{2j+1}(\beta)/ I_1(\beta)$ with the modified Bessel functions $I_n$, in agreement with $C_2=j(j+1)$. \end{proof} \section{Haar measure on SU(2)}\label{app:haar} Write $\mathbf i,\mathbf j,\mathbf k$ for the matrices $i\sigma_3$, $i\sigma_2$, $i\sigma_1$. The map $(a,b)\mapsto aI+b_1\mathbf i+b_2\mathbf j +b_3\mathbf k$ identifies $\R^4$ with the real span of $\SU(2)$, and $\SU(2)$ with the unit sphere $S^3$; the Hilbert--Schmidt norm of $aI+b\cdot(\mathbf i,\mathbf j,\mathbf k)$ is $\sqrt2\,(a^2+\lvert b \rvert^2)^{1/2}$. Left and right multiplication by elements of $\SU(2)$ preserve this norm, so they act on $S^3$ by rotations, and the normalised Haar measure is the normalised surface measure of $S^3$. Every $U\in\SU(2)$ is conjugate to $\operatorname{diag}(e^{i\theta},e^{-i\theta})$ for one $\theta\in[0,\pi]$, its eigenangle, and $\frac12\Tr U=\cos\theta$ is the inner product of $U$ and $I$ in $\R^4$; so $\theta$ is the angle between $U$ and $I$ on $S^3$. \begin{lemma}\label{lem:haarsu2} \textup{(i)} The eigenangle $\theta$ has Haar density $\frac2\pi\sin^2 \theta$ on $[0,\pi]$. \textup{(ii)} $\operatorname{Haar}\{\theta\le t\}=\frac1\pi(t-\sin t\cos t) \le\frac{2t^3}{3\pi}$ for $0\le t\le\pi$. \textup{(iii)} $\lvert U-I\rvert=2\sqrt2\sin\frac\theta2$, and the distance of the bi-invariant metric induced by the Hilbert--Schmidt inner product is $d(U,I)=\sqrt2\,\theta$. With this metric $\SU(2)$ is the round sphere of radius $\sqrt2$. \end{lemma} \begin{proof} (i) The unit vectors at angle $\theta$ from $I$ form a two-sphere of radius $\sin\theta$ and area $4\pi\sin^2\theta$, and $S^3$ has area $2\pi^2$. (ii) $\int_0^t\sin^2\theta\,d\theta=\frac12(t-\sin t\cos t)$, and $\sin^2\theta\le\theta^2$. (iii) The eigenvalues of $U-I$ are $e^{\pm i\theta}-1$, so $\lvert U-I \rvert^2=2\lvert e^{i\theta}-1\rvert^2=8\sin^2\frac\theta2$. By the first paragraph the metric is that of the sphere of radius $\sqrt2$, on which the points at angle $\theta\le\pi$ are at distance $\sqrt2\,\theta$. \end{proof} \section{The single-link law of SU(2)}\label{app:sharp} This appendix proves the sharp quarter, $W_1(\nu_b,\nu_{b'})\le\frac14|b-b'|$ for the one-link law of $\SU(2)$, which Theorems~\ref{thm:su2} and~\ref{thm:blocks} use, and the exact value at $b=0$, which Theorem~\ref{thm:ceiling} uses. The argument is that of our earlier paper~\cite{BB26a}; it is given in full so that the present paper is self-contained. \subsection{The von Mises--Fisher law} Identify $\SU(2)$ with the unit quaternions, that is with the unit sphere $S^3\subset\R^4$, so that $\ReTr U=2x_0(U)$ and $\ReTr(UV)=2\ip u{\bar v}$ for the Euclidean inner product of $\R^4$ and quaternion conjugation $v\mapsto\bar v$. Fix a link $e$ and the field off $e$. Every plaquette through $e$ can be rotated cyclically so that its trace reads $\ReTr(Q_eA_p)$ with $A_p\in\SU(2)$, the staple of $e$ in $p$. The action therefore depends on $Q_e$ only through \[ N\beta\sum_{p\ni e}\ReTr(Q_eA_p)=\ip{x}{b},\qquad b=4\beta\sum_{p\ni e}\bar A_p\in\R^4, \] with $x=Q_e\in S^3$. Since there are six staples, each of norm one, \begin{equation}\label{q-eq:kappamax} \kappa:=|b|\le y:=24|\beta| . \end{equation} The conditional law of $Q_e$ is the von Mises--Fisher law\footnote{On the sphere $S^{p-1}\subset\R^p$ the von Mises--Fisher law with mean direction $\mu$ and concentration $\kappa$ has density proportional to $e^{\kappa\ip x\mu}$ with respect to the surface measure, and its normalising constant is $(2\pi)^{p/2}I_{p/2-1}(\kappa)/\kappa^{p/2-1}$; it is the maximum entropy law with prescribed mean resultant, and it plays on the sphere the role of the Gaussian in $\R^p$. For $p=4$ the Bessel index is $1$. The mean resultant length $\E\ip x\mu=I_{p/2}(\kappa)/I_{p/2-1}(\kappa)$ is the function $A$ of Lemma~\ref{q-lem:moments}. See Mardia and Jupp~\cite{MJ00}, Ch.~9, for these facts and for the tangent-normal decomposition used in Appendix~\ref{app:vmfcomp}.} \begin{equation}\label{q-eq:vmf} \nu_b(dx)=Z(\kappa)^{-1}e^{\ip xb}\,\sigma(dx) \end{equation} on the unit sphere, with $\sigma$ the normalised surface measure. We use coordinates in which $b=\kappa e_1$. The density of $t=x_1$ is proportional to $(1-t^2)^{1/2}e^{\kappa t}$ on $[-1,1]$, so $Z(\kappa)=2I_1(\kappa)/\kappa$ with the modified Bessel function $I_1$, normalised so that $Z(0)=1$. \begin{lemma}\label{q-lem:vmf} For every $\kappa\ge0$ the law $\nu_b$ on the unit sphere $S^3$ satisfies the Poincar\'e inequality $\Var_{\nu_b}(g)\le\frac13\int|\nabla g|^2d\nu_b$, and the constant $\frac13$ is attained at $\kappa=0$. \end{lemma} \begin{proof} In the polar angle $\theta\in(0,\pi)$ from $e_1$ the generator of the reversible diffusion of $\nu_b$ is $L=\partial_\theta^2+b(\theta) \partial_\theta+\sin^{-2}\theta\,\Delta_{S^2}$ with $b(\theta)=2\cot\theta- \kappa\sin\theta$. On spherical harmonics of degree $\ell$ on $S^2$ it acts as a Sturm--Liouville operator $L_\ell$ with potential $\ell(\ell+1)\sin^{-2}\theta$. A ground state comparison of Barta's type~\cite{Bar37} with the trial function $\psi=\sin\theta\, e^{a\kappa\cos\theta}$ gives $-L_1\psi\ge3\psi$ for $a=-\frac15$, and the same trial function with $a=-\frac25$ gives the bound $3$ for the derivatives of zonal eigenfunctions; Appendix~\ref{app:vmfcomp} carries out both computations. Hence every nonzero eigenvalue of $-L$ is at least $3$. At $\kappa=0$ the first eigenvalue of the Laplacian of the unit $S^3$ is $3$. \end{proof} \subsection{Moments} Let $A(\kappa)=\E_{\nu_b}x_1=Z'(\kappa)/Z(\kappa)=I_2(\kappa)/I_1(\kappa)$. \begin{lemma}\label{q-lem:moments} For every $\kappa>0$, in coordinates with $b=\kappa e_1$, \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $\E x_2^2=\E x_3^2=\E x_4^2=A/\kappa$, $\E(1-x_1^2)=3A/\kappa$, the covariance matrix of $x$ is $\operatorname{diag}(A',A/\kappa,A/\kappa, A/\kappa)$, and $A'=1-A^2-3A/\kappa$; \item $\displaystyle\frac{\kappa}{2+\sqrt{\kappa^2+4}}\le A(\kappa)\le\frac{\kappa}{1+\sqrt{\kappa^2+9}}$; \item $A'\le A/\kappa\le\bigl(1+\sqrt{\kappa^2+9}\bigr)^{-1}\le\frac14$; \item $A(\kappa)\ge\kappa\big/\bigl(\frac32+\sqrt{\kappa^2+25/4}\bigr)$. \end{enumerate} \end{lemma} \begin{proof} (i) Integrating by parts against $\frac{d}{dt}(1-t^2)^{3/2}=-3t(1-t^2)^{1/2}$ gives $\kappa\,\E(1-x_1^2)=3\E x_1$, and the three coordinates $x_2,x_3,x_4$ are exchangeable under $\nu_b$, with $\sum_ix_i^2=1$. The off-diagonal covariances vanish by the reflections $x_i\mapsto-x_i$, $i\ge2$, and $A'=\Var x_1$ because $A=(\log Z)'$. (ii) By (i), $A$ solves the Riccati equation $A'=1-A^2-3A/\kappa$ with $A(\kappa)=\kappa/4+O(\kappa^3)$. Appendix~\ref{app:vmfcomp} shows that the left side of (ii) is a subsolution and the right side a supersolution of this equation, with the same behaviour at $0$, and concludes by comparison. (iii) The middle inequality is (ii). For the first, put $s=\sqrt{\kappa^2+4}$. By (ii), $4A/\kappa\ge4/(2+s)$ and $1-A^2\le1-\kappa^2/(2+s)^2$, and \[ \begin{aligned} 1-\frac{\kappa^2}{(2+s)^2}\le\frac4{2+s} &\iff(2+s)^2-\kappa^2\le4(2+s)\\ &\iff s^2-\kappa^2\le4, \end{aligned} \] which holds with equality. So $A'=1-A^2-3A/\kappa\le A/\kappa$. (iv) is the comparison of (ii) with a third barrier, which improves the lower bound of (ii); it is carried out in the same appendix. \end{proof} \subsection{The Lipschitz rate of the family} Let $\delta$ be the chordal metric $\delta(x,x')=|x-x'|$ on $S^3$ and $W_1$ the Kantorovich distance for $\delta$. For $c\in\R^4$ and a unit vector $h\in\R^4$ put \begin{equation}\label{q-eq:rate} L(c,h)=\sup_g\Cov_{\nu_c}\bigl(g,\ip hx\bigr), \end{equation} the supremum over $1$-Lipschitz functions $g$ for $\delta$. Since $\frac{d}{dt}\nu_{c+th}(g)=\Cov_{\nu_{c+th}}(g,\ip hx)$, the Kantorovich--Rubinstein duality gives, for all $b,b'$, \begin{equation}\label{q-eq:w1segment} W_1(\nu_b,\nu_{b'})\le|b-b'|\sup_{c\in[b,b']}L\Bigl(c,\frac{b'-b}{|b'-b|} \Bigr). \end{equation} By rotation invariance $L(c,h)$ depends only on $\kappa=|c|$ and on the angle $\varphi$ between $h$ and $c^\perp$, and we write $L(\kappa,\varphi)$. \begin{lemma}\label{q-lem:rate} For all $c$ and $h$, $L(c,h)\le1/(2\sqrt3)$. Moreover: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item at $c=0$, $L(0,h)=\frac14$ for every $h$; \item if $h\perp c$, then $L(c,h)=\E_{\nu_c}x_2^2=A(\kappa)/\kappa\le\frac14$; \item if $\kappa\ge\sqrt{88}/3\approx3.127$, then $L(c,h)\le\frac14$ for every $h$. \end{enumerate} \end{lemma} \begin{proof} A $1$-Lipschitz function for $\delta$ is $1$-Lipschitz for the geodesic distance, which is larger, so $|\nabla g|\le1$ almost everywhere, and Lemma~\ref{q-lem:vmf} gives $\Var g\le\frac13$. By Lemma~\ref{q-lem:moments}(i), (iii) the largest eigenvalue of the covariance of $x$ is $A/\kappa\le\frac14$, so $\Var\ip hx\le A/\kappa$. The Cauchy--Schwarz inequality gives \begin{equation}\label{q-eq:csbound} L(c,h)\le\bigl(\tfrac13A/\kappa\bigr)^{1/2}\le\Bigl(3\bigl(1+ \sqrt{\kappa^2+9}\bigr)\Bigr)^{-1/2}\le\frac1{2\sqrt3}. \end{equation} (i) and (ii) are reflection arguments. Let $R$ be the reflection in the hyperplane $h^\perp$, and suppose $\nu_c$ is $R$-invariant, which is the case when $c=0$ or $h\perp c$. Then $\E\ip hx=0$ and \[ \begin{aligned} \Cov(g,\ip hx)&=\E\bigl[(g(x)-g(Rx))\ip hx;\ \ip hx>0\bigr]\\ &\le\E\bigl[2\ip hx^2;\ \ip hx>0\bigr]=\E\ip hx^2, \end{aligned} \] since $\delta(x,Rx)=2|\ip hx|$. Equality holds for $g=\ip hx$, which is $1$-Lipschitz. At $c=0$, $\E\ip hx^2=\frac14$; for $h\perp c$, $\E\ip hx^2=A/\kappa$. (iii) By~\eqref{q-eq:csbound}, $L\le\frac14$ as soon as $1+\sqrt{\kappa^2+9}\ge\frac{16}3$, that is $\kappa^2\ge\frac{88}9$. \end{proof} The reflection settles the directions $h\perp c$ and the point $c=0$; for an oblique $h$ the reflection no longer preserves $\nu_c$, and the Cauchy--Schwarz bound~\eqref{q-eq:csbound} is too weak for $\kappa<\sqrt{88}/3$. The next theorem closes this case. \begin{theorem}[The sharp quarter]\label{q-thm:quarter} For every $c\in\R^4$ and every unit vector $h\in\R^4$, \[ L(c,h)\le\tfrac14 , \] with equality at $c=0$. Equivalently, $W_1(\nu_b,\nu_{b'})\le\frac14|b-b'|$ for all $b,b'\in\R^4$, and the constant $\frac14$ is the best possible. \end{theorem} The equivalence is~\eqref{q-eq:w1segment} in one direction and the derivative formula above it in the other; the sharpness is Lemma~\ref{q-lem:rate}(i). The metric matters.\footnote{In the geodesic metric of $S^3$ the statement is false already at $c=0$: the function $\arcsin x_2$ is geodesically $1$-Lipschitz and its covariance with $x_2$ under the uniform law is $\frac8{9\pi}\approx0.283>\frac14$. A proof therefore has to use that the chordal distance between two points is shorter than their geodesic distance, that is, it has to see the sphere from the ball it bounds. The proof below does exactly this: it extends a chordally Lipschitz function to the ball and integrates by parts there.} The proof has four steps. A Lipschitz function on $S^3$ extends to the unit ball $B\subset\R^4$, and Green's formula in $B$ bounds the covariance by a negative Sobolev norm of the density (Lemma~\ref{q-lem:harmonic}). This norm splits into a zonal and a transverse part, both explicit series in modified Bessel functions (Lemma~\ref{q-lem:spectral}). The transverse part is at most $\frac14$ because a certain power series has nonnegative coefficients (Proposition~\ref{q-prop:transverse}), and the zonal part is at most $\frac14$ by a Riccati comparison (Proposition~\ref{q-prop:zonal}). \subsection{Harmonic extension} Let $\sigma$ be the normalised surface measure of $S^3$ and $H_k$ the space of spherical harmonics of degree $k$, the restrictions to $S^3$ of harmonic homogeneous polynomials of degree $k$ on $\R^4$. The spaces $H_k$ are orthogonal in $L^2(\sigma)$ and span it; we write $u_k$ for the component of $u$ in $H_k$, and for $u$ with $\int u\,d\sigma=0$ we put \begin{equation}\label{q-eq:Qnorm} \mathcal Q(u)=\sum_{k\ge1}\frac1k\,\|u_k\|_{L^2(\sigma)}^2 . \end{equation} This is $\ip u{\Lambda^{-1}u}$ for the Dirichlet-to-Neumann operator $\Lambda$ of $B$, which acts on $H_k$ as multiplication by $k$. \begin{lemma}\label{q-lem:harmonic} Let $u$ be a real-analytic function on $S^3$ with $\int u\,d\sigma=0$ and let $g$ be $1$-Lipschitz for the chordal metric. Then \[ \int_{S^3}g\,u\,d\sigma\le\tfrac12\,\mathcal Q(u)^{1/2}. \] \end{lemma} \begin{proof} The chordal metric is the Euclidean metric of $\R^4$ restricted to $S^3$, so McShane's extension $G(y)=\inf_{x\in S^3}\bigl(g(x)+|x-y|\bigr)$ is $1$-Lipschitz on $\R^4$ and equals $g$ on $S^3$~\cite{McS34}; by Rademacher's theorem $|\nabla G|\le1$ almost everywhere. Let $\psi=\sum_{k\ge1}k^{-1}r^ku_k(\omega)$ in polar coordinates $y=r\omega$. Each term is a harmonic polynomial, the series converges with all derivatives on the closed ball because $u$ is real-analytic,\footnote{For real-analytic $u$ the norms $\|u_k\|$ decay geometrically in $k$, and the supremum of a function in $H_k$ is at most $(\dim H_k)^{1/2}=k+1$ times its $L^2(\sigma)$ norm, so the series for $\psi$ and for each of its derivatives converge uniformly on $\bar B$.} and $\partial_r\psi=u$ on $S^3$. Green's formula for the Lipschitz function $G$ and the smooth harmonic function $\psi$ reads $\int_{S^3}G\,\partial_r\psi\,dS=\int_B\nabla G\cdot\nabla\psi\,dy$. With $|S^3|=2\pi^2$ and $|B|=\pi^2/2$ the Cauchy--Schwarz inequality gives \[ \int g\,u\,d\sigma=\frac1{2\pi^2}\int_B\nabla G\cdot\nabla\psi\,dy\le \frac{|B|^{1/2}}{2\pi^2}\Bigl(\int_B|\nabla\psi|^2dy\Bigr)^{1/2}. \] Green's formula once more gives $\int_B|\nabla\psi|^2=\int_{S^3}\psi\, \partial_r\psi\,dS=2\pi^2\sum_kk^{-1}\|u_k\|^2=2\pi^2\mathcal Q(u)$, and the right side becomes $\frac12\mathcal Q(u)^{1/2}$. \end{proof} The lemma is a form of Thomson's principle. By Kantorovich--Rubinstein duality, $\sup_g\int gu\,d\sigma$ is the least cost of a flow in $\R^4$ that carries the positive part of $u\,d\sigma$ to its negative part, the cost being the total length travelled by the mass.\footnote{This is the Beckmann formulation of the transport problem with cost $|x-y|$: the optimal cost equals the least $\int|V|$ over vector measures $V$ on $\R^4$ with divergence $u\,d\sigma$. Restricting the flow to the closed ball does not change the infimum, since projecting onto $\bar B$ shortens paths. The gradient $\nabla\psi$ on $B$ is one admissible flow. It minimises the energy $\int|V|^2$ rather than the cost $\int|V|$, and the Cauchy--Schwarz inequality converts one into the other at the price of the factor $|B|^{1/2}$.} The harmonic field $\nabla\psi$ is one such flow, through the interior of the ball, and the lemma computes its cost. At $c=0$ the flow is optimal: for $u=\ip hx$ one has $\psi=\ip hy$, the field $\nabla\psi=h$ is constant, the Cauchy--Schwarz inequality is an equality, and $\mathcal Q(u)=\|u\|^2= \frac14$. \subsection{Spectral decomposition} Fix $c=\kappa e_1$ with $\kappa>0$ and write $h=\cos\varphi\,e_1+\sin\varphi \,e_2$, which is no loss of generality by rotation invariance. Let $\rho=d\nu_c/d\sigma=e^{\kappa x_1}/Z(\kappa)$, $Z(\kappa)=\int e^{\kappa x_1} d\sigma=2I_1(\kappa)/\kappa$, and \[ u=\bigl(\ip hx-\E_{\nu_c}\ip hx\bigr)\rho , \] so that $\Cov_{\nu_c}(g,\ip hx)=\int gu\,d\sigma$ and $\int u\,d\sigma=0$. Let $U_n$ be the Chebyshev polynomials of the second kind and put \begin{equation}\label{q-eq:dn} d_n(\kappa)=\frac{(n+1)\,I_{n+1}(\kappa)}{I_1(\kappa)},\qquad n\ge0 . \end{equation} \begin{lemma}\label{q-lem:spectral} With this notation \[ \mathcal Q(u)=\cos^2\varphi\,\mathcal Q_z(\kappa)+\sin^2\varphi\, \mathcal Q_x(\kappa), \] where \[ \mathcal Q_z=\sum_{n\ge1}\frac{d_n'(\kappa)^2}n,\qquad \mathcal Q_x=\frac1{3\kappa^2}\sum_{n\ge1}(n+2)\,d_n(\kappa)^2 . \] \end{lemma} \begin{proof} The functions $U_n(x_1)$, $n\ge0$, are the zonal harmonics: $U_n(x_1)\in H_n$, and they are orthonormal in $L^2(\sigma)$ (Appendix~\ref{app:bessel}). The expansion $e^{\kappa t}=\frac2\kappa\sum_n(n+1)I_{n+1}(\kappa)U_n(t)$ of the same appendix gives $\rho=\sum_nd_n(\kappa)U_n(x_1)$, with $d_0=1$. Write $u=\cos\varphi\,u_z+\sin\varphi\,u_x$ with $u_z=(x_1-A)\rho$ and $u_x=x_2\rho$; here $\E_{\nu_c}x_1=A$ and $\E_{\nu_c}x_2=0$. The reflection $x_2\mapsto-x_2$ maps each $H_k$ to itself, fixes $u_z$ and reverses $u_x$, so the components of $u_z$ and $u_x$ in each $H_k$ are orthogonal and the cross term in $\mathcal Q(u)$ vanishes. Since $A=Z'/Z$, we have $u_z=\partial_\kappa\rho=\sum_nd_n'(\kappa)U_n(x_1)$, and the component of $u_z$ in $H_n$ is $d_n'U_n(x_1)$, of norm $|d_n'|$. This gives $\mathcal Q_z$. For $u_x$, let $y(t)=\cos t\,e_1+\sin t\,e_2$. The law $\nu_{\kappa y(t)}$ has density $\rho_t=e^{\kappa\ip{y(t)}x}/Z(\kappa)=\sum_nd_n(\kappa)U_n(\ip{y(t)}x)$, and $\partial_t\rho_t|_{t=0}=\kappa x_2\rho$. The function $U_n(\ip{y(t)}x)$ lies in $H_n$ for every $t$, hence so does its derivative $x_2U_n'(x_1)$ at $t=0$, and the component of $u_x$ in $H_n$ is $\kappa^{-1}d_nx_2U_n'(x_1)$. Its norm follows from the eigenvalue $n(n+2)$ of the Laplacian of $S^3$ on $H_n$. For $f=U_n(x_1)$, \[ \begin{aligned} n(n+2)&=n(n+2)\|f\|^2=\int|\nabla_{S^3}f|^2d\sigma\\ &=\int U_n'(x_1)^2(1-x_1^2)\,d\sigma , \end{aligned} \] since the tangential gradient of $x_1$ has squared length $1-x_1^2$. Given $x_1$, the point $(x_2,x_3,x_4)$ is uniform on the sphere of radius $(1-x_1^2)^{1/2}$, so $\E[x_2^2\mid x_1]=\frac13(1-x_1^2)$ and $\|x_2U_n'(x_1)\|^2=\frac13n(n+2)$. Summing $n^{-1}\kappa^{-2}d_n^2\cdot\frac13 n(n+2)$ over $n\ge1$ gives $\mathcal Q_x$. \end{proof} By Lemmas~\ref{q-lem:harmonic} and~\ref{q-lem:spectral}, \begin{equation}\label{q-eq:Lbyq} L(\kappa,\varphi)\le\tfrac12\max\bigl(\mathcal Q_z(\kappa),\mathcal Q_x( \kappa)\bigr)^{1/2}, \end{equation} and Theorem~\ref{q-thm:quarter} follows once both parts are at most $\frac14$. Both are $\frac14$ at $\kappa=0$, by continuity from the values $d_1\sim\kappa/2$, $d_1'\to\frac12$ and $d_n=O(\kappa^n)$. \subsection{Three Bessel sums} The series of Lemma~\ref{q-lem:spectral} reduce to closed forms. Write $I_j$ for $I_j(\kappa)$. \begin{lemma}\label{q-lem:besselsums} For every $\kappa>0$, \begin{align*} \sum_{j\ge1}j^2I_j^2&=\frac\kappa4\,I_1(2\kappa),\qquad \sum_{j\ge1}j^3I_j^2=\frac{\kappa^2}4\bigl(I_0^2+I_1^2\bigr),\\ I_1(2\kappa)&=2\sum_{j\ge1}I_{j-1}I_j . \end{align*} \end{lemma} The first and third identities follow from Parseval's formula for $e^{\kappa\cos\theta}=\sum_{j\in\Z}I_je^{ij\theta}$, and the second from the recurrences of $I_j$ by telescoping; Appendix~\ref{app:bessel} gives the proofs. \subsection{The transverse part} \begin{proposition}\label{q-prop:transverse} $\mathcal Q_x(\kappa)\le\frac14$ for every $\kappa\ge0$. \end{proposition} \begin{proof} By~\eqref{q-eq:dn}, $\mathcal Q_x=(3\kappa^2I_1^2)^{-1}\sum_{j\ge2}j^2(j+1) I_j^2$, and the sum equals $\sum_{j\ge1}(j^3+j^2)I_j^2-2I_1^2$. By Lemma~\ref{q-lem:besselsums}, $\mathcal Q_x\le\frac14$ is equivalent to \begin{equation}\label{q-eq:Dfunction} \mathcal D(\kappa):=(8+2\kappa^2)I_1^2-\kappa^2I_0^2-\kappa I_1(2\kappa)\ge0 . \end{equation} Each term is an entire function of $\kappa$ that is even after the factor $\kappa^2$ is removed, so $\mathcal D(\kappa)=\sum_{n\ge0}\mathcal D_n \kappa^{2n}$. Lemma~\ref{q-lem:Dcoef} of Appendix~\ref{app:bessel} computes the coefficients: $\mathcal D_0=\mathcal D_1=0$, and for $n=m+1\ge2$ \[ \mathcal D_n=\frac{N_m}{4^m\,m!\,(m+1)!\,(m+1)(m+2)}, \] with \[ \begin{aligned} N_m={}&\binom{2m}m\bigl(4(2m+1)+(m-1)(m+1)(m+2)\bigr)\\ &-4^m(m+1)(m+2). \end{aligned} \] The numerator $N_m$ vanishes for $m=0,1,2$ and is positive for $m\ge3$ (Lemma~\ref{q-lem:Dcoef}). So every coefficient of $\mathcal D$ is nonnegative and $\mathcal D\ge0$. \end{proof} The first nonzero coefficient is $\mathcal D_4=\frac1{2304}$, so $\frac14-\mathcal Q_x=\mathcal D/(12\kappa^2I_1^2)$ is of order $\kappa^4$ at $0$: the transverse part leaves the value $\frac14$ very slowly, which is why no estimate that loses a constant can prove the proposition. \subsection{The zonal part} Let $A=I_2/I_1$ as before, so that $d_1=2A$ and, by Lemma~\ref{q-lem:moments}(i), $A'=1-A^2-3A/\kappa$. \begin{proposition}\label{q-prop:zonal} For $0<\kappa\le\sqrt{88}/3$, \[ \mathcal Q_z(\kappa)\le\frac14+4A'^2+\frac{A^2}4-\frac A{2\kappa}- \frac{2A^2}{\kappa^2}<\frac14 . \] \end{proposition} \begin{proof} \emph{The higher modes.} With $r_\nu=I_{\nu+1}/I_\nu$ the recurrence $I_\nu'=I_{\nu+1}+\frac\nu\kappa I_\nu$ gives \[ \frac{d_n'}{d_n}=\frac{I_{n+1}'}{I_{n+1}}-\frac{I_1'}{I_1}=\frac n\kappa+ r_{n+1}-r_1 . \] By Lemma~\ref{q-lem:ratio} of Appendix~\ref{app:bessel}, $r_{n+1}\le r_1$, and $r_1=A\le\kappa/4$ by Lemma~\ref{q-lem:moments}(iii). Hence $-n/\kappa\le n/\kappa-\kappa/4\le d_n'/d_n\le n/\kappa$ for $n\ge2$ and $\kappa^2\le16$, so $d_n'^2\le n^2d_n^2/\kappa^2$. With $d_1'=2A'$, \[ \mathcal Q_z\le4A'^2+\frac1{\kappa^2}\sum_{n\ge2}n\,d_n^2=4A'^2+\frac{S_3- S_2-4I_2^2}{\kappa^2I_1^2}, \] where $S_2=\sum_{j\ge1}j^2I_j^2$ and $S_3=\sum_{j\ge1}j^3I_j^2$; the second form uses $n\,d_n^2I_1^2=(j-1)j^2I_j^2$ with $j=n+1$. \emph{The closed form.} The recurrences $I_0-I_2=\frac2\kappa I_1$ and $I_1-I_3=\frac4\kappa I_2$ give $I_0/I_1=2/\kappa+A$ and $I_2I_3/I_1^2=A- 4A^2/\kappa$. By Lemma~\ref{q-lem:besselsums}, $S_3/(\kappa^2I_1^2)=\frac14(2/\kappa+A)^2+\frac14$, and, keeping three terms of the third identity, \[ \frac{S_2}{\kappa^2I_1^2}\ge\frac{I_0I_1+I_1I_2+I_2I_3}{2\kappa I_1^2}= \frac1{2\kappa}\Bigl(\frac2\kappa+3A-\frac{4A^2}\kappa\Bigr). \] Collecting terms, $\mathcal Q_z\le\frac14+4A'^2+T(A)$ with $T(a)=\frac{a^2}4- \frac a{2\kappa}-\frac{2a^2}{\kappa^2}$, which is the first inequality. \emph{The barrier.} Let $s=\sqrt{\kappa^2+25/4}$ and $A_\flat=\kappa/(\frac32 +s)$. By Lemma~\ref{q-lem:moments}(iv), $A_\flat\le A\le\kappa/4$. The function $a\mapsto1-a^2-3a/\kappa$ decreases on $a\ge0$, and $A'\ge0$ because $A'= \Var x_1$, so $0\le A'\le\alpha:=1-A_\flat^2-3A_\flat/\kappa$. For $\kappa^2\le 12$ the quadratic $T$ decreases on $[0,\kappa/4]$, since \[ T(a)-T(a')=(a'-a)\,\frac{2\kappa-(a+a')(\kappa^2-8)}{4\kappa^2} \] and $(a+a')(\kappa^2-8)\le\frac\kappa2(\kappa^2-8)\le2\kappa$ there. So $\mathcal Q_z\le\frac14+4\alpha^2+T(A_\flat)$. With $\kappa^2=s^2-\frac{25}4$ and $A_\flat/\kappa=2/(2s+3)$ both terms are rational in $s$: $\alpha=16/(2s+3)^2$, and \begin{equation}\label{q-eq:zonalfactor} \begin{aligned} 4\alpha^2+T(A_\flat)&=\frac{(2s-5)\,p(s)}{4(2s+3)^4},\\ p(s)&=8s^3+28s^2-98s-695 . \end{aligned} \end{equation} For $0<\kappa\le\sqrt{88}/3$ we have $\frac520$, and $p(4.004)<-120$. So the right side of~\eqref{q-eq:zonalfactor} is negative. \end{proof} \subsection{Proof of Theorem~\ref{q-thm:quarter}} At $c=0$ the theorem is Lemma~\ref{q-lem:rate}(i), and for $|c|\ge\sqrt{88}/3$ it is Lemma~\ref{q-lem:rate}(iii). For $0<|c|<\sqrt{88}/3$, Propositions~\ref{q-prop:transverse} and~\ref{q-prop:zonal} give $\mathcal Q_x, \mathcal Q_z\le\frac14$, and~\eqref{q-eq:Lbyq} gives $L\le\frac14$. \qed \section{Computations for the single-link law}\label{app:vmfcomp} \subsection{Barta's inequality} Let $\rho(\theta)=\sin^2\theta\,e^{\kappa\cos\theta}$ on $(0,\pi)$, let $V$ be a continuous potential, and let \[ T=-\rho^{-1}\partial_\theta\,\rho\,\partial_\theta+V . \] If $\psi>0$ is smooth on $(0,\pi)$ with $\rho\psi^2=O(\sin^4\theta)$ at the end points and $T\psi\ge\lambda_0\psi$, then \[ \int(|h'|^2+Vh^2)\rho\,d\theta\ge\lambda_0\int h^2\rho\,d\theta \] for every smooth $h$ vanishing to first order at $0$ and $\pi$. Indeed, writing $h=\psi u$ and integrating by parts, the boundary terms $\rho\psi\psi'u^2$ vanish and \[ \int(|h'|^2+Vh^2)\rho=\int\rho\psi^2|u'|^2+\int\frac{T\psi}\psi\,h^2\rho \ \ge\ \lambda_0\int h^2\rho . \] This is the inequality of Barta~\cite{Bar37} in the form of a ground state transformation. \subsection{The computations of Lemma~\ref{q-lem:vmf}} The generator of the reversible diffusion of $\nu_b$ on $S^3$ is $L=\Delta_{S^3}+\kappa\nabla x_1\cdot\nabla$. In the polar angle $\theta$ from $e_1$ the round metric is $d\theta^2+\sin^2\theta\,g_{S^2}$, so $\Delta_{S^3}=\partial_\theta^2+2\cot\theta\,\partial_\theta+\sin^{-2}\theta\, \Delta_{S^2}$, and $\nabla x_1\cdot\nabla=-\sin\theta\,\partial_\theta$. On $h(\theta)Y_\ell$ with a spherical harmonic $Y_\ell$ of degree $\ell$ on $S^2$ the operator acts as $L_\ell=\partial_\theta^2+b\,\partial_\theta-\ell(\ell+1) \sin^{-2}\theta$, $b=2\cot\theta-\kappa\sin\theta=\rho'/\rho$, so $-L_\ell=T$ with $V=\ell(\ell+1)\sin^{-2}\theta$. \emph{Degree $\ell\ge1$.} Take $\psi=\sin\theta\,e^{a\kappa\cos\theta}$. Then $\psi'/\psi=\cot\theta-a\kappa\sin\theta$ and $\psi''/\psi=(\psi'/\psi)^2+(\psi'/\psi)'$, and a short computation gives \[ \frac{-L_1\psi}\psi=3+(1+5a)\kappa\cos\theta-a(1+a)\kappa^2\sin^2\theta . \] For $a=-\frac15$ this is $3+\frac4{25}\kappa^2\sin^2\theta\ge3$. Since $V=\ell(\ell+1)\sin^{-2}\theta\ge2\sin^{-2}\theta$ for $\ell\ge1$, Barta's inequality gives $-L_\ell\ge3$ on every sector with $\ell\ge1$. \emph{Degree $\ell=0$.} Let $g(\theta)$ be a zonal eigenfunction, $-L_0g=\lambda g$, $g$ not constant, so that $g'(0)=g'(\pi)=0$. Differentiating $g''+bg'=-\lambda g$ gives, for $h=g'$, \[ -\bigl(h''+bh'\bigr)-b'h=\lambda h,\qquad -b'=\frac2{\sin^2\theta}+\kappa \cos\theta, \] so $h$ is an eigenfunction of $T$ with $V=2\sin^{-2}\theta+\kappa\cos\theta$, with eigenvalue $\lambda$, and $h\not\equiv0$ vanishes at both ends. With the same $\psi$, \[ \frac{T\psi}\psi=3+(2+5a)\kappa\cos\theta-a(1+a)\kappa^2\sin^2\theta, \] which for $a=-\frac25$ is $3+\frac6{25}\kappa^2\sin^2\theta\ge3$. Barta's inequality applied to $h$ gives $\lambda\ge3$. Together with the case $\ell\ge1$ every nonzero eigenvalue of $-L$ is at least $3$, which is Lemma~\ref{q-lem:vmf}; the spectrum is discrete because $S^3$ is compact and the drift is smooth. \subsection{Moments} The density of $t=x_1$ under $\nu_b$ is proportional to $(1-t^2)^{1/2}e^{ \kappa t}$. For every smooth $\phi$ on $[-1,1]$, \[ \frac d{dt}\Bigl[(1-t^2)^{3/2}e^{\kappa t}\Bigr]=(1-t^2)^{1/2}e^{\kappa t} \bigl(\kappa(1-t^2)-3t\bigr), \] and integration by parts gives $\E[\phi'(t)(1-t^2)]=\E\bigl[\phi(t)\bigl(3t- \kappa(1-t^2)\bigr)\bigr]$. With $\phi=1$ this is $\kappa\E(1-x_1^2)=3\E x_1$, the identity of Lemma~\ref{q-lem:moments}(i). With $\phi=t^k$ it gives the recursion \[ m_{k+2}=m_k-\frac{k+3}\kappa\,m_{k+1}+\frac k\kappa\,m_{k-1},\qquad m_k=\E x_1^k, \] from $m_0=1$ and $m_1=A$; all polynomial moments of $\nu_b$ are rational in $\kappa$ and $A(\kappa)$.\footnote{For instance $m_2=1-3A/\kappa$, $m_3=A-3/\kappa+12A/\kappa^2$ and $m_4=m_2-5m_3/\kappa+2A/\kappa$. The joint moments with $x_2,x_3,x_4$ follow from exchangeability: $\E[\phi(x_1)x_2^2]=\frac13\E[\phi(x_1)(1-x_1^2)]$.} \subsection{The Riccati barriers} By Lemma~\ref{q-lem:moments}(i), $A$ solves $A'=\mathcal F(\kappa,A):=1-A^2- 3A/\kappa$ on $(0,\infty)$, and $A(\kappa)=\frac\kappa4-\frac{\kappa^3}{96}+ O(\kappa^5)$ from the series of $I_1$ and $I_2$. Let $s=\sqrt{\kappa^2+4}$, $u=\sqrt{\kappa^2+9}$, $\phi_-=\kappa/(2+s)$ and $\phi_+=\kappa/(1+u)$. Direct differentiation gives \[ \begin{aligned} \phi_-'-\mathcal F(\kappa,\phi_-)&=-\frac{\kappa^2}{s(s+2)^2}\le0,\\ \phi_+'-\mathcal F(\kappa,\phi_+)&=\frac{(u-3)^2}{u(1+u)^2}\ge0, \end{aligned} \] and $\phi_\pm=\frac\kappa4+O(\kappa^3)$.\footnote{For the first identity write $\phi_-=(s-2)/\kappa$, which follows from $s^2-4=\kappa^2$; then $\phi_-'=(\kappa^2/s-s+2)/\kappa^2$ and $1-\phi_-^2-3\phi_-/\kappa= (\kappa^2-(s-2)^2-3(s-2))/\kappa^2$, and the difference simplifies with $s^2=\kappa^2+4$. The second is the same computation with $\phi_+=\kappa(u-1)/(u^2-1)=\kappa(u-1)/(\kappa^2+8)$ in place of $\phi_-$ and $u^2=\kappa^2+9$ in place of $s^2=\kappa^2+4$. After clearing the denominator $\kappa^2(1+u)^2$ the numerator becomes $\kappa^2(u-3)^2/u$, which is the stated residual.} The comparison is singular at $0$ because of the term $3A/\kappa$, and we carry it out with an integrating factor. Let $D=A-\phi_-$. Then \[ \begin{aligned} D'&=\mathcal F(\kappa,A)-\mathcal F(\kappa,\phi_-)-\bigl(\phi_-'-\mathcal F( \kappa,\phi_-)\bigr)\\ &=-\Bigl(A+\phi_-+\frac3\kappa\Bigr)D+R, \end{aligned} \] with $R=\kappa^2/(s(s+2)^2)\ge0$. Put $\mu(\kappa)=\kappa^3\exp\int_0^\kappa (A+\phi_-)$. Then $(\mu D)'=\mu R\ge0$, and $\mu D=O(\kappa^6)\to0$ as $\kappa\to0$, so $\mu D\ge0$ and $A\ge\phi_-$. The same argument with $D=\phi_+-A$ and $R=(u-3)^2/(u(1+u)^2)$ gives $A\le\phi_+$. This is Lemma~\ref{q-lem:moments}(ii). For (iv) let $s=\sqrt{\kappa^2+25/4}$ and $\phi_\flat=\kappa/(\frac32+s)$. Since $s^2-\frac94=\kappa^2+4$, the same differentiation gives \[ \phi_\flat'-\mathcal F(\kappa,\phi_\flat)=-\frac{5\,(s-\frac52)}{2\,s\, (\frac32+s)^2}\le0, \] so $\phi_\flat$ is a subsolution, and $\phi_\flat=\frac\kappa4-\frac{\kappa^3} {80}+O(\kappa^5)$.\footnote{With $\phi_\flat=\kappa(s-\frac32)/(\kappa^2+4)$ one has $\phi_\flat'=(\frac32 s+\frac{25}4)/(s(\frac32+s)^2)$, using $ds/d\kappa= \kappa/s$; and $\mathcal F(\kappa,\phi_\flat)=1-\phi_\flat^2-3/(\frac32+s)$. Clearing the denominator $s(\frac32+s)^2$ and replacing $\kappa^2$ by $s^2-\frac{25}4$ leaves $-\frac52 s+\frac{25}4$ in the numerator, which is the stated residual. The expansion follows from $s=\frac52+\frac{\kappa^2}5+ O(\kappa^4)$.} Hence $A-\phi_\flat=\frac{\kappa^3}{480}+O(\kappa^5)$, and the integrating-factor argument above, with $D=A-\phi_\flat$, $R=\frac{5(s-5/2)} {2s(3/2+s)^2}\ge0$ and $\mu=\kappa^3\exp\int_0^\kappa(A+\phi_\flat)$, gives $\mu D=O(\kappa^6)$ at $0$, $(\mu D)'=\mu R\ge0$, and so $A\ge\phi_\flat$. Since $\frac32+\sqrt{\kappa^2+25/4}<2+\sqrt{\kappa^2+4}$ for $\kappa>0$, this improves the lower bound of (ii). \section{Bessel functions}\label{app:bessel} This appendix proves the auxiliary statements used in Appendix~\ref{app:sharp} for Theorem~\ref{q-thm:quarter}. Throughout, $I_j=I_j(\kappa)$ with $\kappa>0$, and $I_{-j}=I_j$ for integers $j$. We use the series \[ I_\nu(\kappa)=\sum_{i\ge0}\frac{(\kappa/2)^{2i+\nu}}{i!\,(i+\nu)!}, \] the integral $I_j(\kappa)=\pi^{-1}\int_0^\pi e^{\kappa\cos\theta}\cos j\theta\,d\theta$, the Fourier series $e^{\kappa\cos\theta}=\sum_{j\in\Z} I_je^{ij\theta}$ and the recurrences \begin{equation}\label{q-eq:besselrec} I_{j-1}-I_{j+1}=\frac{2j}\kappa\,I_j,\quad I_j'=I_{j+1}+\frac j\kappa\,I_j , \end{equation} all of which are in Watson's treatise~\cite{Wat44}, \S\S3.7, 3.71, 2.22. All series below converge absolutely, because $00$ for $m\ge3$. \end{lemma} \begin{proof} \emph{The coefficients.} Multiplying the series of $I_0$ and $I_1$ and collecting powers, \[ \begin{gathered} I_0^2=\sum_{m\ge0}\alpha_m\kappa^{2m},\qquad I_1^2=\sum_{m\ge0}\beta_m \kappa^{2m+2},\\ \kappa I_1(2\kappa)=\sum_{m\ge0}\frac{\kappa^{2m+2}}{m!\,(m+1)!}, \end{gathered} \] where \[ \begin{aligned} \alpha_m&=\frac1{4^m}\sum_{i=0}^m\frac1{i!^2(m-i)!^2},\\ \beta_m&=\frac1{4^{m+1}}\sum_{i=0}^m\frac1{i!\,(i+1)!\,(m-i)!\,(m-i+1)!}. \end{aligned} \] By Vandermonde's identity, $\sum_i\binom mi^2=\binom{2m}m$ and $\sum_i\binom mi\binom{m+2}{i+1}=\binom{2m+2}{m+1}$, so \[ \alpha_m=\frac{\binom{2m}m}{4^m\,m!^2},\qquad\beta_m=\frac{\binom{2m+2}{m+1}} {4^{m+1}m!\,(m+2)!}. \] The coefficient of $\kappa^{2m+2}$ in $\mathcal D$ is $\mathcal D_{m+1}=8\beta_m +2\beta_{m-1}-\alpha_m-1/(m!(m+1)!)$, with $\beta_{-1}=0$; for $m=0$ it is $\frac84-1-1=0$, and $\mathcal D_0=0$ because every term of $\mathcal D$ is $O(\kappa^2)$. For $m\ge1$ put $\Delta=4^m\,m!\,(m+1)!\,(m+1)(m+2)$ and $b=\binom{2m}m$. With $\binom{2m+2}{m+1}=\frac{2(2m+1)}{m+1}b$, \[ \begin{aligned} 8\beta_m&=\frac{4(2m+1)\,b}\Delta,\qquad\alpha_m=\frac{(m+1)^2(m+2)\,b}\Delta,\\ 2\beta_{m-1}&=\frac{2m(m+1)(m+2)\,b}\Delta,\\ \frac1{m!(m+1)!}&=\frac{4^m(m+1)(m+2)}\Delta, \end{aligned} \] and $2m-(m+1)=m-1$ gives $\mathcal D_{m+1}=N_m/\Delta$. \emph{The sign.} Directly, $N_0=N_1=N_2=0$ ($2-2$, $24-24$, $192-192$), $N_3=80$, $N_4=1140$ and $N_5=10416$. For $m\ge6$ we first show $16^m\le4m\binom{2m}m^2$ for $m\ge1$ by induction: it is an equality at $m=1$, and $(m+1)\binom{2m+2}{m+1}=2(2m+1)\binom{2m}m$ gives \[ \begin{aligned} 4(m+1)\binom{2m+2}{m+1}^2&=\frac{16(2m+1)^2}{m+1}\binom{2m}m^2\\ &\ge\frac{16(2m+1)^2}{4m(m+1)}16^m\ge16^{m+1}. \end{aligned} \] Since $(m-1)^2\ge4m$ for $m\ge6$, it follows that $\binom{2m}m^2(m-1)^2\ge 16^m$, that is $\binom{2m}m(m-1)\ge4^m$, and \[ \begin{aligned} N_m={}&\Bigl(\binom{2m}m(m-1)-4^m\Bigr)(m+1)(m+2)\\ &+4(2m+1)\binom{2m}m>0 . \end{aligned} \] \end{proof} \subsection{Monotonicity of Bessel ratios} \begin{lemma}\label{q-lem:ratio} For $n\ge1$ and $\kappa>0$, $I_{n+2}(\kappa)/I_{n+1}(\kappa)\le I_2(\kappa)/ I_1(\kappa)$. \end{lemma} \begin{proof} We use a lemma of Biernacki and Krzy\.z~\cite{BK55}: if $f=\sum a_jx^j$ and $g=\sum b_jx^j$ have positive coefficients and $a_j/b_j$ is nonincreasing in $j$, then $f/g$ is nonincreasing on $x>0$. Indeed \[ \begin{aligned} f'g-fg'&=\sum_{i,j}(i-j)a_ib_jx^{i+j-1}\\ &=\frac12\sum_{i,j}(i-j)b_ib_j\Bigl(\frac{a_i}{b_i}-\frac{a_j}{b_j}\Bigr) x^{i+j-1}\le0 . \end{aligned} \] Take $x=\kappa^2$, $f=I_{n+1}(\kappa)/\kappa^{n+1}$ and $g=I_1(\kappa)/\kappa$. Then $a_j/b_j=2^{-n}(j+1)!/(j+n+1)!$ decreases in $j$, so $I_{n+1}/(\kappa^n I_1)$ is nonincreasing in $\kappa$. Its logarithmic derivative is, by the second recurrence of~\eqref{q-eq:besselrec}, \[ \begin{aligned} &\frac{n+1}\kappa+\frac{I_{n+2}}{I_{n+1}}-\frac n\kappa-\frac1\kappa-\frac{I_2} {I_1}\\ &\qquad=\frac{I_{n+2}}{I_{n+1}}-\frac{I_2}{I_1}\le0 . \end{aligned}\qedhere \] \end{proof} \section{The certificate bound}\label{app:certificate} This appendix reduces a bound on $h_F$ to finitely many positivity statements, by hand, and reports the computation of~\cite{Rom26} that verifies them for $h_F\le14.420001$ (Remark~\ref{rem:hf}); no result of the paper uses that value. The argument is the proof of the bound in Theorem~\ref{thm:curv} with two changes. The divergence identity of Lemma~\ref{lem:identity}, taken there with the factor $\frac9{10}$, is replaced by a positive semidefinite form at each site, and the weights $\frac{21}{10}$ and $\frac54$ of Lemma~\ref{lem:local} are replaced by weights chosen plaquette by plaquette. The reduction is proved by hand; the positivity of ninety-six matrices of trigonometric polynomials and the arithmetic of finitely many rational numbers are checked by computer.\footnote{The constant $14.420001$ is not the best possible for this kind of certificate: by the search reported in Remark~\ref{rem:hf} it cannot go below $14.384$, while the true value of $h_F$ is at least $13.123$.} \subsection{Ends and corners} At a site $y$ the eight \emph{ends} are \[ o_\mu(y)=X_{y,\mu},\qquad i_\mu(y)=Q^*_{y-e_\mu,\mu}X_{y-e_\mu,\mu}Q_{y-e_\mu,\mu}, \] for $\mu=0,1,2,3$, as in the proof of Lemma~\ref{lem:identity}; the end of a link in $F$ or outside the box is $0$. Each link $e=(y,\mu)$ has the end $o_\mu$ at its tail $y$ and the end $i_\mu$ at its head $y+e_\mu$, both of norm $|X_e|$. \begin{lemma}\label{lem:corners} Let $p$ be the plaquette at $x$ in the plane $\mu<\nu$. Then \[ \begin{aligned} \ip{Y_1}{Y_4}&=\ip{o_\mu}{o_\nu}(x),\\ \ip{Y_1}{Y_2}&=\ip{i_\mu}{o_\nu}(x+e_\mu),\\ \ip{Y_3}{Y_4}&=\ip{o_\mu}{i_\nu}(x+e_\nu),\\ \ip{Y_2}{WY_3W^*}&=\ip{i_\nu}{i_\mu}(x+e_\mu+e_\nu). \end{aligned} \] Every pair of ends at one site in two different directions occurs in exactly one of these identities, for exactly one plaquette. \end{lemma} \begin{proof} The first identity is the definition of the ends. The inner product is invariant under conjugation by a unitary matrix, so $\ip{X_{e_1}}{Q_{e_1}X_{e_2}Q_{e_1}^*}=\ip{Q_{e_1}^*X_{e_1}Q_{e_1}}{X_{e_2}}$, which is the second identity, and the third follows in the same way from $\ip{Q_{e_4}X_{e_3}Q_{e_4}^*}{X_{e_4}}=\ip{X_{e_3}}{Q_{e_4}^*X_{e_4}Q_{e_4}}$. For the fourth, $WQ_{e_4}=Q_{e_1}Q_{e_2}Q_{e_3}^*$, so that $WY_3W^*= RX_{e_3}R^*$ with $R=Q_{e_1}Q_{e_2}Q_{e_3}^*$, and conjugating both vectors by $(Q_{e_1}Q_{e_2})^*$ gives $\ip{Q_{e_2}^*X_{e_2}Q_{e_2}}{Q_{e_3}^*X_{e_3}Q_{e_3}}$. The links $e_2$ and $e_3$ both have their head at $x+e_\mu+e_\nu$. For the last statement, the pairs $(o_\mu,o_\nu)$, $(i_\mu,o_\nu)$, $(o_\mu,i_\nu)$ and $(i_\mu,i_\nu)$ at $y$ come from the plaquettes in the plane $\mu\nu$ at $y$, $y-e_\mu$, $y-e_\nu$ and $y-e_\mu-e_\nu$ respectively. There are $4\binom42=24$ such pairs at each site and $6\cdot4=24$ corners of plaquettes per site. \end{proof} For each class of sites modulo $2$ let $G_y$ be a real symmetric $8\times8$ matrix indexed by the ends, with $G_y[o_\mu,i_\mu]=0$ for every $\mu$. For the plaquette $p$ at $x$ in the plane $\mu<\nu$ put \[ \begin{split} c^G_p={}&2G_x[o_\mu,o_\nu]\ip{Y_1}{Y_4} +2G_{x+e_\mu}[i_\mu,o_\nu]\ip{Y_1}{Y_2}\\ &+2G_{x+e_\nu}[o_\mu,i_\nu]\ip{Y_3}{Y_4}\\ &+2G_{x+e_\mu+e_\nu}[i_\mu,i_\nu]\ip{Y_2}{WY_3W^*}. \end{split} \] The matrix $G^0$ with $G^0[a,a]=2$, $G^0[o_\mu,o_\nu]=G^0[i_\mu,i_\nu]=1$ and $G^0[o_\mu,i_\nu]=-1$ for $\mu\ne\nu$ is the matrix of the form $|\sum_\mu(o_\mu-i_\mu)|^2+\sum_\mu|o_\mu+i_\mu|^2$ of Lemma~\ref{lem:identity}, and $c^{G^0}_p$ is the corner form $c_p$ of~\eqref{eq:corner}. \begin{lemma}\label{lem:siteidentity} On $\Z^4$, or on a box with free boundary, for every field $Q$ and every square-summable $X$, \[ \begin{split} &\sum_y\sum_{a,b}G_y[a,b]\,\ip{\mathrm{end}_a}{\mathrm{end}_b}(y)\\ &\quad=\sum_pc^G_p+\sum_e\bigl(G_{t(e)}[o,o]+G_{h(e)}[i,i]\bigr)|X_e|^2, \end{split} \] where $t(e)$ and $h(e)$ are the tail and the head of $e$, and $o$, $i$ are its two ends. If every $G_y$ is positive semidefinite, the left side is nonnegative. \end{lemma} \begin{proof} Split the sum over $a,b$ at each site into the diagonal terms, the pairs $(o_\mu,i_\mu)$, whose coefficient is $0$, and the pairs of ends in different directions, each counted twice. The diagonal terms give the last sum, since each link has one end at its tail and one at its head, and the pairs in different directions give $\sum_pc^G_p$ by Lemma~\ref{lem:corners}. In a box the argument at the end of the proof of Lemma~\ref{lem:identity} applies. At each site the left side is the trace of $G_y$ times the Gram matrix of the eight ends, and the trace of a product of two positive semidefinite matrices is nonnegative. \end{proof} \subsection{From plaquettes to the Hessian} Number the positions of a plaquette as in~\eqref{eq:transport} and let $k_p$ be the position of its link in $F$; by Lemma~\ref{lem:structure} there is exactly one. \begin{lemma}\label{lem:certificate} Let every $G_y$ be positive semidefinite with $G_y[o_\mu,i_\mu]=0$, and let $a_{p,r}\ge0$ for the three positions $r\ne k_p$ of each plaquette. Suppose that for every plaquette $p$, every $N$, every $W\in\U(N)$ and all $Y_1,\dots,Y_4\in\mathfrak u(N)$ with $Y_{k_p}=0$, \begin{equation}\label{eq:certineq} f_p+c^G_p\le\sum_{r\ne k_p}a_{p,r}|Y_r|^2 . \end{equation} Then $|\Hess S_0(v,v)|\le h|v|^2$ for every field $Q$ and every $v=(X_eQ_e)$ with $X_e=0$ for $e\in F$, on $\Z^4$ and on every box, where \[ h=\max_{e\notin F}\Bigl(\sum_{p\ni e}a_{p,e}+G_{t(e)}[o,o]+G_{h(e)}[i,i] \Bigr) \] and $a_{p,e}$ is the weight of the position of $e$ in $p$. \end{lemma} \begin{proof} Summing~\eqref{eq:certineq} over the plaquettes and adding the identity of Lemma~\ref{lem:siteidentity}, whose left side is nonnegative, \[ \begin{split} \Hess S_0(v,v)&=\sum_pf_p\le\sum_p\Bigl(\sum_ra_{p,r}|Y_r|^2-c^G_p\Bigr)\\ &\le\sum_{e\notin F}\Bigl(\sum_{p\ni e}a_{p,e}+G_{t(e)}[o,o]+G_{h(e)}[i,i] \Bigr)|X_e|^2\\ &\le h|v|^2, \end{split} \] since $|Y_r|=|X_e|$ for the link $e$ at position $r$. In a box the links near the boundary lie in fewer plaquettes, and the weights are nonnegative. For the lower bound, apply~\eqref{eq:certineq} at $-W$. By~\eqref{eq:fp} this changes the sign of $f_p$, while $c^G_p$ does not change, and the same chain gives $-\Hess S_0(v,v)\le h|v|^2$. \end{proof} The hypothesis is needed on $\U(N)$ and not only on $\SU(N)$, since $-W\notin\SU(N)$ when $N$ is odd. \begin{lemma}\label{lem:certu2} If~\eqref{eq:certineq} holds for $N=2$ and every $W=\operatorname{diag}(e^{i\varphi_1},e^{i\varphi_2})$, it holds for every $N$ and every $W\in\U(N)$. \end{lemma} \begin{proof} Both sides are invariant under conjugation of $W$ and all $Y_i$ by one unitary matrix, so we may take $W=\operatorname{diag}(w_1,\dots,w_N)$. Every term of $f_p$, of $c^G_p$ and of $|Y_r|^2$ is a multiple of $\ReTr(Y_iD_1Y_jD_2)$ with $D_1,D_2\in\{I,W,W^*\}$, and \[ \ReTr(Y_iD_1Y_jD_2)=\operatorname{Re}\sum_{a,b}(Y_i)_{ab}\,(D_1)_{bb}\, (Y_j)_{ba}\,(D_2)_{aa}. \] The difference of the two sides of~\eqref{eq:certineq} is therefore a sum of one term for each $a$, in the entries $(Y_i)_{aa}$, and one term for each pair $a