\documentclass[aps,prd,twocolumn,superscriptaddress,nofootinbib,longbibliography,floatfix,amsmath,amssymb,10pt]{revtex4-2} \usepackage{mathptmx} \usepackage{amsthm} \usepackage{graphicx} \usepackage{booktabs} \usepackage{xcolor} \usepackage{enumitem} \usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black,bookmarksnumbered=true]{hyperref} \usepackage{orcidlink} \usepackage{microtype} \usepackage{xurl} \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} \begin{document} \title{Yang--Mills Mass Gap in 4D: Lattice SU($N$) and Every Compact Simple Group} \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, Heramba Chandra College, 23/49 Gariahat Road, Golpark, Kolkata 700029 (University of Calcutta, 87/1 College Street, Kolkata 700073), 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} \date{27 October 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, exponential clustering and an explicit mass gap at fixed lattice spacing, past the curvature limit of the ungauged theory, with the ranges $\beta_W\le0.345$ for SU(2) and $\beta_W\le0.76$ for SU(3). We also prove that one estimate of Dobrushin--Shlosman type in a cube of fixed physical size gives a mass gap in physical units for every continuum limit, and that this estimate, which holds at strong coupling and remains open along the continuum tuning, needs for SU($N$) a cube of side at least of order $(\beta_W/\log\beta_W)^{1/4}$. \end{abstract} \maketitle \begingroup \renewcommand\thefootnote{}\footnotesize \footnotetext{\footnotesize\emph{2020 Mathematics Subject Classification.} Primary 81T25, 82B20; Secondary 60K35, 81T13, 22E30, 58J50, 65G30. \emph{PACS.} 11.15.Ha, 05.50.+q, 02.50.Cw, 02.40.Ky. \emph{Keywords.} Lattice gauge theory; gauge fixing; Dobrushin uniqueness; mass gap.} \footnotetext{\footnotesize An earlier version, titled ``Mass Gap of Four-Dimensional Lattice Yang--Mills Theory in a Forest Gauge'', appeared as Hexagon preprint 2610.00341. The \LaTeX{} source, the checks and the Lean certificate are in \href{https://github.com/creelie/4Y-M}{github.com/creelie/4Y-M}, release v3.19.0.} \endgroup \addtocounter{footnote}{-2} \makeatletter \renewcommand\l@subsection[2]{} \renewcommand\l@subsubsection[2]{} \makeatother \tableofcontents \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.\footnote{The combination of a forest gauge fixing with the sharp quarter of~\cite{BB26a} was first assembled during a run of Anaxia, the reasoning engine of the BLLM system~\cite{Bha26}, release v3.0.5. In that run the engine combines proof steps along typed links (a step that produces a single-link Kantorovich bound may only feed a step that consumes one), and in every run with typed crossing it reached a composite accepting a lattice mass gap for four-dimensional $\SU(2)$ at $\beta_W<\frac13$, which no untyped or random run reached. The composite was first proposed by the language model inside the engine, and the engine's network reproduces it. The engine did not supply the line field, the free-boundary limit, the ceiling, the curvature bound or anything in the $\SU(3)$ part; the authors wrote and checked every proof in this paper. Release v3.1.0 of the engine~\cite{Bha26b} records its Yang--Mills runs in the failure log: the narrowing of the target from the continuum gap to explicit couplings in this gauge, the change from analytic one-link bounds to bounds certified box by box, and the change to blocks of several links. These records follow the course of the work and supply no step of the proofs.} 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 $0\le\beta_W\le0.345$. The number $\alpha(\beta_W)$ of~\eqref{eq:alphablock} is increasing, $\alpha(\frac13)\le0.9561$ and $\alpha(0.345)\le0.9973$. 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 $m\ge0.0224$. For $-0.345\le\beta_W<0$ 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. 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 much further. \begin{theorem}[$\SU(3)$]\label{thm:su3} Let $G=\SU(3)$ and $0\le\beta_W\le\frac34$. Then the conclusions of Theorem~\ref{thm:su2} hold, with decay at every rate below $-\log(1.33319\,\beta_W)$ and a transfer-matrix gap \[ m\ge-\log(1.33319\,\beta_W)>0 . \] \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 $b$. Its analytic part is a lower bound for the spectral gap of $\nu_b$ through Bochner's formula with a cross term (Lemma~\ref{lem:gap}), a corrector chosen box by box among polynomials of degree two in the matrix entries (Lemma~\ref{lem:corrector}), the reduction of $b$ to diagonal matrices (Lemma~\ref{lem:orbits}), the character expansion of the moments of $\nu_b$ (Lemma~\ref{lem:characters}) and an interpolation inequality on boxes of staple sums (Lemma~\ref{lem:boxes}). The remaining step evaluates the resulting bound on $2341$ boxes with rigorous error bounds; the theorem rests on that computation, which is described in full in Sec.~\ref{ssec:cover} and Appendix~\ref{app:cover}. 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 much larger gap, and they go past it. \begin{theorem}[$\SU(3)$ with plaquette blocks]\label{thm:su3blocks} Let $G=\SU(3)$ and $0\le\beta_W\le\frac{19}{25}=0.76$. Then $\alpha_3(\beta_W)<1$ for the number of~\eqref{eq:blockconst3}, with $\alpha_3(0.72)\le0.9182$, $\alpha_3(\frac34)\le0.9653$ and $\alpha_3(0.76)\le0.99950$. 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 $\beta_W=\frac34$ this gives $m\ge0.0177$, against $1.1\cdot10^{-4}$ from Theorem~\ref{thm:su3}, and at $\beta_W=0.76$, where Theorem~\ref{thm:su3} no longer applies, $m\ge2.5\cdot10^{-4}$. \end{theorem} \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&$<\frac13$&$\le\frac34$\\ Plaquette blocks, forest gauge&$\le0.345$&$\le0.76$\\ \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}), also for observables smoothed by the spatial gradient flow (Proposition~\ref{prop:spatialflow}). \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. The open inputs are measured as follows. 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}). 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}). 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}). \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 at strong coupling). Blue: the proved implications (I1)--(I3) of the continuum step. 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) and measures the open inputs. 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}), 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 by \texttt{decide} in the Lean file \texttt{GaugeFix.lean} 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} A \emph{block plaquette} is a plaquette of the set $\mathcal B$ described in Table~\ref{tab:plaqblocks} of Appendix~\ref{app:plaqblocks}, 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} All four statements are invariant under $2\Z^4$ and involve plaquettes at distance at most two from one site, so they hold on $\Z^4$ as soon as they hold on the torus $(\Z/P\Z)^4$ with $P\ge6$, where no such configuration wraps around. They are checked on the tori of side four and six by the program described in Appendix~\ref{app:plaqblocks}; the side four already suffices for (i)--(iii). For (iii), an active link lies in six plaquettes, one of which is its block plaquette, and each of the other five contains two further active links, which lie in other blocks by (ii). \end{proof} 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,\\ C=20|\beta|\bigl(4|\beta|\tau s+16\beta^2M\bigr),\qquad K=\frac{(1+|\beta|)\,s}{1-C}, \end{gathered} \end{equation} assuming $C<1$. \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 $c_0$ moves in the direction $k$, then $|\frac{d}{dt}\kappa(f)|\le (4\beta\tau s+16\beta^2M)|k|$. \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 $1$-Lipschitz function on a set of diameter two has $\mathrm{sd}\le1$, and $\mathrm{sd}_\nu\ip xh\le\sqrt\tau|h|$ because $|c|\le|a|+4\beta\le24\beta$. Lemma~\ref{lem:tilt} gives (i). (ii) The derivative is $\Cov_\kappa(f,\ip{\nabla g(c_0+4\beta\bar x)}k)$. Write $\nabla g(c_0+4\beta u)=\nabla g(c_0)+4\beta\Hess g(c_0)u+N(u)$. The constant term does not contribute. The linear term is $\ip xh$ with $h=4\beta\,\overline{\Hess g(c_0)k}$, $|h|\le4\beta\tau|k|$, and contributes at most $4\beta\tau s|k|$ by (i). For $|u|,|u'|\le1$, $\|\Hess g(c_0+4\beta v)- \Hess g(c_0)\|\le4\beta M|v|\le4\beta M$ on the segment between them, so $|N(u)-N(u')|\le16\beta^2M|u-u'|$; the function $\ip{N(\bar x)}k$ is therefore $16\beta^2M|k|$-Lipschitz, its standard deviation is at most that constant, and its covariance with $f$ is at most $16\beta^2M|k|$. \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 $y'_3$ moves $c_0=b_2y_3$ along a segment of length $|b_2||y_3-y'_3|\le20\beta|y_3-y'_3|$ inside the ball $|c_0|\le20\beta$, so Lemma~\ref{lem:onesite}(ii) gives $W_1(\kappa_1(\cdot\mid y_3),\kappa_1( \cdot\mid y'_3))\le C|y_3-y'_3|$, 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 \[ \max\Bigl(|A+\Gamma|,\tfrac23|A|\Bigr),\qquad A=12r\varphi''(r^2),\quad \Gamma=8r^3\varphi'''(r^2). \] On the ball $r\le2.07$ every eigenvalue of $\Hess g$ lies between $0.158491$ and $0.25000001$, and the norm of $D^3g$ is at most $0.0615586$. \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 $At+\Gamma t^3$, whose largest absolute value is attained at $t=\pm1$ or at an interior critical point, where it equals $\frac23|A||t|\le\frac23|A|$; by Banach's theorem~\cite{Ban38} the supremum of a symmetric trilinear form on the diagonal of the unit sphere is its norm. The numerical bounds are obtained in ball arithmetic~\cite{Joh17} from the series of $Z_0$, truncated after forty terms with a proved bound on the tail, on $2\cdot10^4$ subintervals of $[0,2.07]$; the program is described in Appendix~\ref{app:plaqblocks}. \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 $|\beta|\le0.345/4$ and take the constants of Lemma~\ref{lem:vmfconst}, whose ball $r\le2.07$ contains $|c|\le24|\beta|$. 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 $\sum_BC_{Bj}\le40|\beta|K$. Evaluating~\eqref{eq:blockconst} in ball arithmetic at $\beta_W=0.345$ gives $w\le0.0054459$, $s\le0.252746$, $C\le0.050243$, $K\le0.289069$ and $\alpha\le0.99729$; at $\beta_W=\frac13$ it gives $\alpha\le0.95602$. Proposition~\ref{prop:blockstate} applies. \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 $\beta_W=0.34$, against $0.2881$ from Lemma~\ref{lem:blockK}, and $40|\beta|$ times it crosses one near $\beta_W=0.357$. That computation is not certified and is not used in the proof. 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 gives $h_F=13.97401$, attained by an abelian field; certificates made of plaquette forms and vertex forms on the period-two cell cannot go below $14.384$. With this gauge the curvature method stops between $|\beta|=\frac1{29}$ and $\frac1{26.25}$. \end{remark} \section{SU(3) up to 3/4}\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{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. This blended corrector reaches $\beta_W=\frac{45}{64}$ and stops near $0.715$ (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 and the theorem}\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{proposition}\label{prop:cover} $2r\Lambda(r)\le0.999887$ at $r=\frac32$. \end{proposition} \begin{proof} 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 $581$ satisfy $2rB_Q<1$ for the right side $B_Q$ of~\eqref{eq:BQ}, with $\gamma(r)= \frac{187}{128}$. 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 $2341$ certified 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 $0.983$. Figure~\ref{fig:cover} shows the cover, and Appendix~\ref{app:cover} the error analysis. \end{proof} The same cover at a larger $r$ bounds $\Lambda(r)$ where $2r\Lambda(r)$ is no longer below one; the plaquette blocks of Section~\ref{sec:su3blocks} use that bound. \begin{proposition}\label{prop:cover2} $2r\Lambda(r)\le1.029967$ at $r=\frac{38}{25}$. \end{proposition} \begin{proof} As in the proof of Proposition~\ref{prop:cover}, with $\gamma(r)= \frac{4643}{3200}$ and the threshold $2rB_Q<1.03$: of the $8192$ boxes of side $r/8$, $784$ survive the two tests and $690$ satisfy the threshold; each of the other $94$ is cut into sixteen boxes of side $r/16$, of which $373$ survive the tests, and all satisfy it. The $1063$ certified boxes cover $\mathcal R_r$, and the largest bound is $1.029967$. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:su3}] Proposition~\ref{prop:cover} gives \[ \Lambda\bigl(\tfrac32\bigr)\le\frac{0.999887}3<0.333296 . \] For $\beta_W\le\frac34$ we have $2\beta_W\le\frac32$, and~\eqref{eq:alpha} with the monotonicity of $\Lambda$ gives \[ \alpha\le4\beta_W\Lambda\bigl(\tfrac32\bigr)<1.333183\,\beta_W\le \tfrac34\cdot1.333183<1 . \] Proposition~\ref{prop:state} applies with the Hilbert--Schmidt metric. \end{proof} \begin{figure*}[t] \centering \includegraphics[width=0.92\textwidth]{figures/fig_cover.png} \caption{The $2341$ certified boxes 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 certified 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.} \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} now 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{lemma}\label{lem:su3const} At $r=\frac32$ every eigenvalue of $\Hess g$ on $rK$ lies in $[0.056015,0.247316]$, and $|D^3g(c)(u,v,v)|\le0.182126$ for $c\in rK$ and unit vectors $u,v$. At $r=\frac{38}{25}$ the same holds with $[0.054978,0.248280]$ and $0.183546$. \end{lemma} \begin{proof} 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 $r=\frac{38}{25}$) 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 the numbers stated (Appendix~\ref{app:plaqblocks}). \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:su3blocks}] Let first $\beta_W\le\frac34$, so that $r=2\beta_W\le\frac32$. Every field met in Lemmas~\ref{lem:onesite3} and~\ref{lem:blockK3} lies in $\frac32K$, so the constants of Lemma~\ref{lem:su3const} at $\frac32$, $\Lambda(\frac32)\le\frac{0.999887}3$ from Proposition~\ref{prop:cover} and $\gamma=\gamma(\frac32)=\frac{187}{128}$ apply; with these fixed constants every quantity in~\eqref{eq:blockconst3} is increasing in $\beta_W$. For $\frac34<\beta_W\le\frac{19}{25}$ the fields lie in $\frac{38}{25}K$, and the constants of Lemma~\ref{lem:su3const} at $\frac{38}{25}$, $\Lambda(\frac{38}{25})\le\frac{1.029967}{3.04}$ from Proposition~\ref{prop:cover2} and $\gamma(\frac{38}{25})=\frac{4643}{3200}$ apply in the same way. 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 $\sum_BC_{Bj}\le10N\beta K_3=\alpha_3(\beta_W)$. In ball arithmetic, 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 $\beta_W=0.76$, with the second set of constants, $w\le0.018609$, $s\le0.34673$, $C\le0.045770$, $K_3\le0.39454$ and $\alpha_3\le0.99950$. Proposition~\ref{prop:blockstate}, whose proof does not use the group, applies with the Hilbert--Schmidt metric. \end{proof} \begin{remark}\label{rem:su3blocks} Theorem~\ref{thm:su3} ends where $2r\Lambda(r)$ reaches one. The blocks remove the factor $\frac{12}{10}$ of the single-link sum and keep most of it after paying for the three links: at $\beta_W=\frac34$ the gap bound rises from $1.1\cdot10^{-4}$ to $0.0177$, and the block bound is the better one for $\beta_W\ge0.715$. Before the corrections for the three links their 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; Proposition~\ref{prop:cover2} certifies $2r\Lambda(r)\le1.03$ at $r=1.52$, which is enough at $\beta_W=0.76$, with little room. Evaluated in floating point with the constants of a scan of $\mathcal R_r$, the block sum stays below one up to $\beta_W\approx0.78$; that computation is not certified and is not used. \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 $\beta_W=0.345$ (Theorem~\ref{thm:blocks}), and the sharper floating-point bound of Remark~\ref{rem:blockgap} 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 $\frac1{26.25}$. \emph{$\SU(3)$.} The route of Sec.~\ref{sec:su3} 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 certified $0.999887$ of Proposition~\ref{prop:cover} is the interpolation over boxes. Between $\beta_W=\frac34$ and $0.77$ there is room only for smaller boxes. The plaquette blocks of Theorem~\ref{thm:su3blocks} use $\Lambda(2\beta_W)$ as well, with a little more room: they reach $0.76$ with the certified cover of Proposition~\ref{prop:cover2} and near $0.78$ in floating point (Remark~\ref{rem:su3blocks}); past that 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 Proposition~\ref{prop:cover} certifies the bound 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 $0.345$, for $\SU(3)$ from $\frac9{32}$ to $0.76$, with a gap at $\frac34$ of $0.0177$ 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}. It is the hypothesis under which reflection positivity across coordinate and diagonal hyperplanes is shown in~\cite{Sus26a} to give rotation invariance of the Schwinger functions, a step proposed there for the continuum limit of lattice gauge theories. \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, 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 and what the conditional construction of~\cite{Sus26b} states for the smeared action density, has $2\Delta=8>4$ and $k=2$; so that field cannot satisfy~\eqref{eq:growth}, and rotation invariance for it has to come from another source, as in~\cite{Sus26b}, where it is reduced to a separate infrared hypothesis. 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}, which the conditional construction of~\cite{Sus26b} assumes clause by clause and which yields there a non-negative Hamiltonian without any statement on its spectrum, and a decay estimate in physical units at large scales. The finite-size criterion described in~\cite[Sect.~9.4]{Sus26b} reduces a gap to a threshold condition on one box, uniformly in the cutoff; as stated there, the condition fails in every small box and would have to be checked at large scales, which the construction does not reach. \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 are named at the end: the cube estimate and the existence of the limit. \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 theorem 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. 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: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{Estimates uniform in the coupling}\label{ssec:uniform} Some local quantities of the Wilson measure can be controlled uniformly in $\beta_W$, the volume and the boundary condition, by integrating by parts in a single link. Such estimates hold along every tuning, but they concern local fields in lattice units, and the three results below show how far they are from the open inputs: the force has bounded moments but its centred form is a null vector of the reflected form, the plaquette has fluctuations of order $\beta_W^{-2}$ but this is a single-site variance, and bare links over a fixed physical distance become independent Haar variables. Let $G=\SU(N)$, write $b=\beta_W$, so that the Wilson weight is $e^{bS}$ with $S=\sum_pN^{-1}\ReTr Q_p$, and give $\g$ the metric $\ip TR=-\Tr(TR)$. For a link $e$ let $S_e$ be the sum of the six plaquette terms containing $e$, $X_T$ the derivative along $U_e\mapsto e^{uT}U_e$ at $u=0$, $\nabla_e$ and $\Delta_e$ the gradient and Laplacian in $U_e$, $q=N^2-1$, $c_N=(N^2-1)/N$, and \[ E_e=b|\nabla_eS_e|^2,\qquad W_e=E_e-c_NS_e=b|\nabla_eS_e|^2+\Delta_eS_e . \] Below, $\omega$ is a Wilson Gibbs state at $b>0$: 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{The two residues} Theorem~\ref{thm:transfer} reduces a continuum mass gap to two statements, neither of which is proved here or, to our knowledge, anywhere else. \begin{enumerate}[label=\textup{(\Alph*)},leftmargin=*] \setcounter{enumi}{3} \item \emph{The cube estimate.} There are $\alpha<1$ and $\ell>0$ 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. \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. \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 in the archive for $n\le8$, lie between $1.5$ and $11$ 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. \begin{remark}\label{rem:routes} The other routes we tried 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 certified as in Propositions~\ref{prop:cover} and~\ref{prop:cover2}; 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{What the cube estimate requires}\label{ssec:cubeneeds} This subsection proves four facts about $D(n,\alpha)$. It holds at small coupling for every even $n$ above an explicit size (Proposition~\ref{prop:cubesmall}). 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}). None of the four 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. The finite counts used below, from Table~\ref{tab:sigma}, are rechecked by \texttt{cube\_requirements.py} in the archive.} 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 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 $0\le x_\sigma0$. 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 $(\theta-\sin\theta\cos\theta)/\pi\le2\theta^3/(3\pi)\le\pi^2r^3/12$. 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. \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 gauge fixing on a forest is Creutz's~\cite{Cre77}, and the curvature of Lie groups is Milnor's~\cite{Mil76}. 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}. The Yang--Mills manuscripts of Sussman and collaborators~\cite{Sus26a,Sus26b} were read in the preparation of Sec.~\ref{sec:limits}. \section*{Data and code availability} The analytic proofs are complete as written, except the evaluations in Propositions~\ref{prop:cover} and~\ref{prop:cover2} and Lemma~\ref{lem:su3const} and the numerical bounds of Lemma~\ref{lem:vmfconst}, which are computations with rigorous error bounds, and the finite check of Lemma~\ref{lem:plaqblocks}. The repository \texttt{creelie/4Y-M} contains the \LaTeX{} source of this paper; the programs of that computation, in Python with python-flint (FLINT and Arb) and a C kernel for products of Taylor polynomials, with their outputs and the list of certified boxes; the programs and outputs of the earlier certificate at $\beta_W=\frac{45}{64}$; independent checks in exact rational arithmetic of Lemmas~\ref{lem:local} and~\ref{lem:roughlap} and of~\eqref{eq:pointwise}; numerical checks of Lemma~\ref{lem:identity} on tori; the searches of Remark~\ref{rem:hf} and Sec.~\ref{sec:limits}; and a Lean~4 certificate of the line field (Lemmas~\ref{lem:cycle} and~\ref{lem:structure} for the table of Appendix~\ref{app:table}, by \texttt{decide}); and the programs of Appendix~\ref{app:plaqblocks} for the plaquette blocks, with the floating-point search of Remark~\ref{rem:blockgap}; and an exact computation of the curvatures of Table~\ref{tab:groups} from the root systems; and exact checks of the finite counts, of the discs and of the Gaussian benchmark of Sec.~\ref{ssec:cubeneeds}, with Monte Carlo estimates of the influence coefficients and of $\varepsilon_V$ that the proofs do not use. This paper accompanies release v3.18.0 of the archive, \url{https://github.com/creelie/4y-m/releases/tag/v3.18.0}. The PDF of this paper is at \url{https://github.com/creelie/4Y-M/blob/main/preprintYangMillsForestGauge/YangMillsForestGauge.pdf}. The computations and the Lean, Python and shell checks were run with AI assistance (Claude Opus~5.5, Claude Fable~5.1 with Ultracode, GPT-6.1 Sol Ultra and GPT-6 Astra Ultra) under the direction of Deep Bhattacharjee. The first composite behind Theorem~\ref{thm:su2} was assembled by the Anaxia engine of the BLLM system~\cite{Bha26}, release v3.0.5, as described in the footnote to that theorem; release v3.1.0~\cite{Bha26b} records the engine's Yang--Mills runs in its failure log, whose last Yang--Mills entry is the change from single links to blocks of several links; the plaquette blocks of Sec.~\ref{sec:plaqblocks}, the bound of Lemma~\ref{lem:blockK}, their $\SU(3)$ version in Sec.~\ref{sec:su3blocks} and the certified constants were found and checked by the authors. All three authors approved the final text. \section*{Declarations} \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} The program \texttt{cover.py} in the folder \nolinkurl{strong-coupling/gauge-fixed/blocks/} of the archive finds this set by an exact-cover search on the torus of side two and checks Lemma~\ref{lem:plaqblocks} on the tori of side four and six. The program \texttt{constants.py} evaluates the bounds of Lemma~\ref{lem:vmfconst} in ball arithmetic with python-flint at $120$ bits: the series of $Z_0$ and of its first three derivatives in $s=r^2$ is summed to forty terms, and the tail is bounded for $s\le5$ by twice the first omitted term $(5/4)^{40}40^3/(40!\,41!)$, because consecutive terms decrease by a factor below one half. The program \texttt{alpha.py} evaluates the constants of~\eqref{eq:blockconst} and $\alpha$ at a given $\beta_W$, read exactly. For $\SU(3)$ the folder \nolinkurl{strong-coupling/gauge-fixed/blocks/su3/} contains \texttt{consts.py}, which evaluates the bounds of Lemma~\ref{lem:su3const} on the $841$ boxes of side $\frac3{16}$ at $r=\frac32$ and the $784$ boxes of side $\frac{19}{100}$ at $r=\frac{38}{25}$ with the moments of Appendix~\ref{app:cover}, and \texttt{alpha3.py}, which reads these bounds and the cover of Proposition~\ref{prop:cover} or Proposition~\ref{prop:cover2} and evaluates the constants of~\eqref{eq:blockconst3} and $\alpha_3$ in ball arithmetic at $120$ bits; its outputs at $\beta_W=\frac34$ and $0.76$ are listed with the bounds of each box. The cover of Proposition~\ref{prop:cover2} is computed by the program of Appendix~\ref{app:cover} with \texttt{--r 38/25 --thr 1.03}. \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} \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 $10^{-12}$. \emph{Running time.} The moments at one corner take about one and a half seconds and the bound on one box about two and a half seconds on one core. The cover runs in about an hour on four cores and writes every certified box with its $\theta$ and its bound. \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} \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. \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{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} \def\bibsection{\par\vskip10pt\noindent\hfil\rule{0.5\columnwidth}{0.4pt}\hfil\par\vskip8pt} \begin{thebibliography}{99}\raggedright\advance\leftskip\labelwidth \bibitem{AC25} A.~Adhikari and S.~Cao, Correlation decay for finite lattice gauge theories at weak coupling, Ann. 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