% % % % % \documentclass[11pt,letterpaper]{amsart} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{amsmath,amssymb,amsthm,mathrsfs} \usepackage[letterpaper,margin=1.1in]{geometry} \usepackage{enumitem} \usepackage{xcolor} % \usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref} \usepackage[capitalise,noabbrev]{cleveref} \crefname{openproblem}{Problem}{Problems} \Crefname{openproblem}{Problem}{Problems} \crefname{convention}{Convention}{Conventions} \Crefname{convention}{Convention}{Conventions} % \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{convention}[theorem]{Convention} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} % 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\providecommand{\RTc}{\ensuremath{{R_T}}} % \providecommand{\Wfr}{\ensuremath{{W^{\circ}}}} % \providecommand{\Zfree}{\ensuremath{{Z^{\free}}}} % \providecommand{\Ztor}{\ensuremath{{Z^{\tor}}}} % \providecommand{\sWt}{\ensuremath{{\sW^{\tor}}}} % \providecommand{\sbig}{\ensuremath{{s_{\mathrm b}}}} % \providecommand{\tScap}{{\tilde S^{\mathrm{cap}}}} % \providecommand{\tauz}{{\tau_0}} % \space% % % % % % % % % \newcounter{convention}[section] \renewcommand{\theconvention}{\thesection.\arabic{convention}} \makeatletter \ifcsname theHsection\endcsname \@namedef{theHconvention}{\theHsection.\arabic{convention}} \else \@namedef{theHconvention}{\arabic{section}.\arabic{convention}} \fi \makeatother \crefname{convention}{Convention}{Conventions} \Crefname{convention}{Convention}{Conventions} \space% % % % % \newcommand{\HPrefraw}[1]{\csname HPnum@#1\endcsname} \newcommand{\HPref}[1]{{\cite[\HPrefraw{#1}]{HP}}} % \expandafter\def\csname HPnum@ms:conv0\endcsname{Definition~A.1} \expandafter\def\csname HPnum@ms:conv\endcsname{Remark~A.2} \expandafter\def\csname HPnum@ms:fielddef\endcsname{Definition~A.3} \expandafter\def\csname HPnum@ms:hermite\endcsname{Lemma~A.4} \expandafter\def\csname HPnum@ms:borel\endcsname{Lemma~A.5} \expandafter\def\csname HPnum@ms:struct\endcsname{Lemma~A.6} \expandafter\def\csname HPnum@ms:density\endcsname{Lemma~A.7} \expandafter\def\csname HPnum@ms:carrier\endcsname{Lemma~A.8} \expandafter\def\csname HPnum@ms:unique\endcsname{Proposition~A.9} \expandafter\def\csname HPnum@ms:discharge\endcsname{Proposition~A.10} \expandafter\def\csname HPnum@ms:transfer\endcsname{Lemma~A.11} \expandafter\def\csname HPnum@ms:met\endcsname{Remark~A.12} \expandafter\def\csname HPnum@ms:x:detalg\endcsname{Remark~A.13} \expandafter\def\csname HPnum@ms:x:strongcont\endcsname{Lemma~A.14} \expandafter\def\csname HPnum@ms:x:proj\endcsname{Lemma~A.15} \expandafter\def\csname HPnum@ms:x:invloc\endcsname{Lemma~A.16} \expandafter\def\csname HPnum@ms:erg:rcp\endcsname{Lemma~A.17} \expandafter\def\csname HPnum@ms:erg:decomp\endcsname{Theorem~A.18} \expandafter\def\csname HPnum@ms:erg:transfer\endcsname{Proposition~A.19} \expandafter\def\csname HPnum@ms:erg:Ohdecomp\endcsname{Theorem~A.20} \expandafter\def\csname HPnum@ms:erg:growth\endcsname{Theorem~A.21} \expandafter\def\csname HPnum@ms:erg:growthX\endcsname{Corollary~A.22} \expandafter\def\csname HPnum@ms:recon\endcsname{Proposition~A.23} \expandafter\def\csname HPnum@ms:fixed\endcsname{Proposition~A.24} \expandafter\def\csname HPnum@ms:ladder\endcsname{Definition~A.25} \expandafter\def\csname HPnum@ms:mean\endcsname{Theorem~A.26} \expandafter\def\csname HPnum@ms:x:oneterm\endcsname{Lemma~A.27} \expandafter\def\csname HPnum@ms:diag\endcsname{Theorem~A.28} \expandafter\def\csname HPnum@ms:components\endcsname{Corollary~A.29} % \newcommand{\HPapp}{{\cite[Appendix~A]{HP}}} % \expandafter\def\csname HPnum@ms:erg:eq:hermitebound\endcsname{Theorem~A.21} \space% % \title[OS axioms for scalar QED$_3$ in the Coulomb phase]{Osterwalder--Schrader axioms for scalar QED$_3$ in the Coulomb phase} \hypersetup{urlcolor=blue} \DeclareRobustCommand{\repoline}{\texorpdfstring{\href{https://github.com/marikgoldstein/scalar-qed-existence}{\normalfont\small\ttfamily github.com/marikgoldstein/scalar-qed-existence}\textsuperscript{$\ast$}}{github.com/marikgoldstein/scalar-qed-existence}} \DeclareRobustCommand{\reposhort}{\texorpdfstring{{\normalfont\ttfamily github.com/marikgoldstein/scalar-qed-existence}}{github.com/marikgoldstein/scalar-qed-existence}} \author[OS axioms for scalar QED$_3$ in the Coulomb phase]{\repoline} % \thanks{\textsuperscript{$\ast$}This work and its companion are machine generated and have been contributed by M.~Goldstein and A.~Guevara, based on the results described in \url{https://github.com/marikgoldstein/scalar-qed-existence}.} \date{} % \theoremstyle{plain} \newtheorem{mainthm}{Theorem} \renewcommand{\themainthm}{\Alph{mainthm}} \crefname{mainthm}{Theorem}{Theorems} \Crefname{mainthm}{Theorem}{Theorems} \begin{document} \begin{abstract} King constructed the continuum, infinite-volume limit of three-dimensional scalar quantum electrodynamics, the $U(1)$ (abelian) Higgs model, along Neumann--Dirichlet cubes at block factor $5$ \cite{K1,K2}. For his state at fixed photon mass we prove the Osterwalder--Schrader axioms: regularity $\Ezz$ with absolute constants, invariance under the full Euclidean group, reflection positivity about every plane, symmetry, and the measure with its ergodic decomposition, on whose components the cluster axiom holds in all its forms. The existence of King's cube limits, the existence clause of his Theorem~2.1, is proved from his renormalisation-group analysis in a ratio form in which the vacuum energies cancel; rotation invariance is proved by comparing King's cubes with periodic tori carrying a smooth mass wall on two lattices whose orientations differ by $\arctan(3/4)$. With real scalar sources, the joint field of the curvature and the renormalised Wick square satisfies, on a tube of small Wick amplitudes, the axioms: regularity, invariance under translations and the lattice point group, Euclidean invariance of the joint field outside an exceptional set of couplings, reflection positivity for Wick amplitudes below a threshold, symmetry, and the measure with the cluster axiom in its weak form; above the threshold the Euclidean clause is open. The curvature results hold on King's whole coupling range; the Wick results hold under a one-sided sign clause of the Wick leg, which we prove on explicit coupling regions. The limit of vanishing photon mass is treated in a subsequent paper. \end{abstract} \maketitle \enlargethispage{3pt}% \setcounter{tocdepth}{1} \tableofcontents % % \section{Introduction}\label{sec:intro} \subsection{The model and the problem}\label{ssec:intro:model} The abelian Higgs model, scalar quantum electrodynamics, is the Euclidean field theory of a real vector field $A_\mu$ and a complex scalar field $\phi$ on $\R^3$ with formal action \begin{equation}\label{eq:intro:action} \begin{gathered} S(A,\phi)=\int_{\R^3}\Big[\tfrac14F_{\mu\nu}F_{\mu\nu}+\tfrac12\lmu^2A_\mu A_\mu+\tfrac12\abs{(\partial_\mu+\mathrm ieA_\mu)\phi}^2+\tfrac12m^2\abs{\phi}^2+\lambda\abs{\phi}^4\Big]\,dx,\\ F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu , \end{gathered} \end{equation} in the normalisation of King \cite[(2.1)--(2.4)]{K1}; on the lattice the photon is written in Feynman form, with the gradient terms of each of its components, and carries the mass $\lmu>0$ (Section~\ref{sec:s02}). In three dimensions $e^2$ and $\lambda$ have the dimension of a mass and the model is superrenormalisable: one mass counterterm $\dm_{\eps}$ for the scalar, one number per spacing, renormalises it. For $m^2<0$ the scalar condenses and the photon acquires a mass by the Higgs mechanism; for $m^2>0$ the scalar is massive and the photon is expected to stay massless. In the language of superconductivity (the abelian Higgs model is the relativistic field-theoretic version of its Ginzburg--Landau theory), the first regime, the \emph{Higgs phase}, is the superconducting state and the second, the \emph{Coulomb phase}, the normal state. The two phases are treated in two papers: the Higgs phase in the companion paper \cite{HP}, the Coulomb phase here, where the bare mass $m$ is positive throughout and the photon mass $\lmu>0$, the infrared regulator of \cite{Ba1,Ba2,Ba3,Ba4,K1,K2}, is kept fixed. None of our proofs uses the phase. One-loop perturbation theory predicts a fluctuation-induced first-order transition at very small quartic coupling \cite{CW73,HLM74}; no claim about the phase diagram is made here. The gauge field is non-compact: on the lattice $A$ is real-valued on the bonds and the gauge action is a quadratic form, as in Ba{\l}aban \cite{Ba1,Ba2,Ba3,Ba4} and King \cite{K1,K2}. For observables of integer charge on a finite lattice with free boundary conditions the non-compact model is the compact Villain model with every monopole removed \cite{BBIJ84}. The choice matters in the Coulomb phase: in the compact pure gauge theory in three dimensions monopoles produce a mass gap and confinement at every coupling \cite{GM82}, while the non-compact pure gauge theory is the free field, whose photon is massless. Masslessness of the photon as $\lmu\to0$ is the expectation for the model with matter; we work at fixed $\lmu$. The problem treated here is due to King \cite{K1,K2}: with the renormalisation-group method of Ba{\l}aban \cite{Ba1,Ba2,Ba3,Ba4} he constructed the continuum limit of the model in finite volume and then its infinite-volume limit, through cubes with Neumann conditions for $A$ and Dirichlet conditions for $\phi$, proved its translation invariance and reflection positivity, and addressed its rotation invariance through a comparison of two rotated lattices; the details, and the earlier work on the model, are recalled in Section~\ref{ssec:intro:inputs}. The present paper establishes the Osterwalder--Schrader axioms for his state, rotation invariance included, by a different route. The question is whether the Schwinger functions of the local gauge-invariant fields, the \emph{curvature} $F=dA$ and the renormalised \emph{Wick square} of $\phi$, can be constructed as limits of the lattice regularisation and satisfy the axioms of Osterwalder and Schrader \cite{OS73,OS75}: regularity (\OSz), Euclidean invariance (\OSo), reflection positivity (\OSt), symmetry (\OSth) and clustering (\OSf). Translations, the cubic group and reflection positivity are properties of the lattice model that have to be carried through the cube limits (King's cube itself is not reflection symmetric, Remark~\ref{rem:s07:R1}); invariance under rotations is a symmetry of no lattice theory and has to be recovered in the limit. This paper proves the Osterwalder--Schrader axioms, with full Euclidean invariance, for the state King constructs in \cite{K2}, at fixed photon mass, on all of King's coupling range and at his block factor $5$; and, with real scalar sources and under a one-sided sign clause on the couplings, for the joint field of the curvature and the renormalised Wick square on a tube of small Wick amplitudes, the axioms: regularity, invariance under translations and the lattice point group, Euclidean invariance of the joint field outside an exceptional set of couplings (Theorem~\ref{rot:thm:mainGS}), reflection positivity about the coordinate planes for Wick amplitudes below a threshold, symmetry, and the measure with its ergodic decomposition and the cluster axiom in its weak form, ergodicity of the components under translations. We call it \emph{King's state}, $\ZKinf$: the infinite-volume state of King's construction \cite{K1,K2}, taken on his own cube convention, as opposed to the comparison convention of Section~\ref{sec:s04}. The existence of King's cube limits, the existence clause of his Theorem~2.1, is proved here (Section~\ref{sec:k01}; with scalar sources, in the corrected form of Section~\ref{sec:k02}), by King's route in a form in which the vacuum energies cancel. The passage from the cube limits to the axioms in \cite{K2} rests on three steps that we carry out differently. First, King's Neumann--Dirichlet cube is not reflection symmetric about any plane, and its natural symmetrisation is not reflection positive at small charge (Remark~\ref{rem:s07:R1}); reflection positivity goes through a modified box state, whose positivity, King's Proposition~3.3, is proved here directly (Lemma~\ref{lem:s07:RPbox}). Second, the inequality by which \cite{K2} returns from the boxes to the cube is false for the observables the argument needs, and holds in the reverse direction (Lemma~\ref{lem:s07:R}); it is replaced by a two-sided sandwich between cube states (Lemmas~\ref{lem:s07:L} and~\ref{lem:s07:U}). Third, where King passes to subsequences of spacings, the subsequences are constructed here, and for King's own state the limits are taken along the full sequence (Remark~\ref{rem:s12:K9}). With scalar sources, King's Theorem~2.1 needs a vacuum counterterm that his Wick ordering does not contain; we work with a \emph{scheme} $a$ that provides it. \subsection{Main results}\label{ssec:intro:results} Sources are real $2$-forms $f_j\in C^\infty_c(\R^3;\Lambda^2)$ with complex coefficients $z_j$ and real scalar test functions $h_j\in C^\infty_c(\R^3)$ with coefficients $w_j$; $F(f)$ is the curvature paired with $f$, $W^a(h)$ the renormalised Wick square in the scheme $a$, $g(z)=\sum_j(\Re z_j)f_j$, and the norms $\norm{\cdot}_2$, $\fournorm{\cdot}$, $\wnorm{\cdot}$ and the tube $\Dtube$ are those of Definitions~\ref{def:s02:sources} and~\ref{def:s02:tube}. The Schwinger functions $S_n$, $S_{n,p}$ are the Taylor coefficients of the limit functionals at $0$. The limits are iterated on King's cubes $C_J=[-MJ,MJ)^3$: the spacing $\eps_k=5^{-k}\to0$ first, then $J\to\infty$; the limit objects do not depend on King's large-block integer $M$, only the tube of (6) may (Remark~\ref{rem:s02:Mindep}). \smallskip\noindent\textbf{(1) Existence of the cube limits.} For every finite family of real $2$-forms, every cube containing their supports at distance greater than $1$ from its complement, and every real coefficient vector, the generating functional of the curvature on King's cube converges as $k\to\infty$, geometrically, to a limit in $(0,\infty)$ (Theorem~\ref{thm:k01:EC}); this is the existence clause of \cite[Thm.~2.1]{K2}. With real scalar sources, for couplings satisfying the sign clause $\Tthree$ of (7), in the reference scheme and for coefficient vectors whose positive scalar part lies below the threshold of the Wick tube, the cube limits exist as well (Theorem~\ref{thm:k02:W1}). \smallskip\noindent\textbf{(2) Regularity.} King's state $\ZKinf(\sum_jz_jf_j)$ is an entire function of the coefficients and satisfies $\abs{\ZKinf(\sum_jz_jf_j)}\le e^{\norm{g(z)}_2^2}$, with no dependence on $\lmu$, $e$, $\lambda$, $m$ or $M$. For real test functions $\abs{S_n(f_1,\dots,f_n)}\le(\sqrt2e)^n(n!)^{1/2}\prod_j\norm{f_j}_2$; the $S_n$ are tempered distributions and satisfy the regularity axiom $\Ezz$ with $s=4$ (Theorem~\ref{thm:s05:CK}). \smallskip\noindent\textbf{(3) Euclidean invariance.} $\ZKinf$ is invariant under all translations of $\R^3$ and under the cubic group $\Oh$ about every point (Theorem~\ref{thm:s06:S1K}), and under the full orthogonal group, for all complex coefficients, $F$ transforming as a $2$-form; the $S_n$ are Euclidean covariant (Theorem~\ref{thm:s06:O3}). The rotations are Theorem~\ref{rot:thm:main}: King's state on curvature sources is $O(3)$-invariant at every point of King's range. \smallskip\noindent\textbf{(4) Reflection positivity and symmetry.} Reflection positivity about every coordinate plane holds for King's state, in matrix form for exponentials with arbitrary complex curvature coefficients (Theorem~\ref{thm:s08:OS2C}) and in polynomial form (Proposition~\ref{prop:s08:poly}); with (3) it holds about every plane $\{x\cdot n=c\}$. The $S_n$ are real and symmetric, and $S_n=0$ for odd $n$ (Proposition~\ref{prop:s08:OS3}). \smallskip\noindent\textbf{(5) The measure and clustering.} For every even $r\ge14$ there is exactly one Borel probability measure $\mu_F$ on $\cH^{-r}(\R^3;\Lambda^2)$ with moments $S_n$ and characteristic functional $\ZKinf(\mathrm i\,\cdot)$, and the localised lattice curvature fields converge in law to those of $\mu_F$. The measure is invariant under the cubic group about every point and its characteristic functional under the Euclidean group; it decomposes into translation-ergodic components, and almost every component is $\Euc$-invariant, reflection positive about every plane, clusters in the mean and under diagonal separation, has a unique joint vacuum in every direction and satisfies $\OSf$ in all its forms for every choice of the time direction (Theorem~\ref{thm:s09:S2K}). Whether $\mu_F$ is itself ergodic is not decided (Section~\ref{sec:s23}). \smallskip\noindent\textbf{(6) The Wick tube.} Let the couplings satisfy the sign clause $\Tthree$ of (7) and let $a$ be an admissible scheme (Definition~\ref{def:s03:adm}). For sources in a fixed box, the joint functional of $(F,W^a)$ on King's cubes converges on the tube $\Dtube$ of small Wick amplitudes to a holomorphic limit $G^K_\infty$, King's tube object, with $\abs{G^K_\infty(\zeta)}\le\exp(\norm g_2^2+\kappa\norm{k_+}_{L^1}+K_-\wnorm{k_-})$, $k_\pm$ the positive and negative parts of the scalar source and $\kappa$, $K_-$ constants depending only on the couplings, the scheme, $M$ and fixed auxiliary choices; the $S_{n,p}$ satisfy factorial bounds and $\Ezz$ with $s=4$; translations and $\Oh$ hold on the tube; reflection positivity about every coordinate plane holds for Wick amplitudes below $\sigma_+/8$; $S_{n,p}=0$ for odd $n$; and the measure on $\cH^{-r}(\R^3;\Lambda^2\oplus\R)$, with the weak form of $\OSf$ on its components, exists as in (5). The Euclidean clause of the Wick sector is proved on the tube outside the exceptional set $E_W$ (Theorem~\ref{rot:thm:mainGS}) and is open above the amplitude threshold; the constants of the Wick sector are not uniform in $\lmu$. The tube bound is proved on the comparison convention (Theorem~\ref{thm:s11:W}) and transferred to King's convention, on which the limits are taken (Theorem~\ref{thm:s13:TWK}, Proposition~\ref{prop:s13:WK}); the comparison family enters only through its limit states along continuum subsequences, which coincide with $G^K_\infty$ (Theorem~\ref{thm:s12:LIM}, Proposition~\ref{prop:s13:LIMK}). \smallskip\noindent\textbf{(7) The coupling range.} The curvature results (1)--(5) hold on all of King's range, $e>0$ and $\lambda,m,\lmu>0$ arbitrary: every smallness condition in \cite{K1,K2,Ba1,Ba2,Ba3,Ba4} is a cap on the running couplings at a stopping scale chosen after the couplings and the datum, and no relation between $\lmu$ and $m$ is used (Lemma~\ref{lem:s14:CW}, Remark~\ref{rem:s14:quantifier}). The only coupling condition in the paper is the one-sided sign clause $\Tthree$ of the Wick leg, $\sigma_1(M_1)>0$ for some auxiliary scalar mass $M_1\in(0,m)$, where $\sigma_1(M_1)$ is the lower limit, as the spacing goes to zero, of half the difference of King's free-boundary bare mass coefficients at the scalar masses $m$ and $M_1$. It fails exactly when $m$ minimises, over the scalar mass in $(0,m]$, the continuum limit of the difference of the bare mass coefficients at that mass and at $m$ (only this relative limit exists, Lemma~\ref{lem:s14:CT}); to one-loop order it holds if and only if $e^2$ lies below an explicit threshold $e_c^2(\lambda,m,\lmu)$ (equation~\eqref{eq:s14:ec2}); and beyond one loop it is proved on explicit coupling regions, at $\lmu=m$ on a region containing $0<\lambda,e^2\le0.239\,m$ and at every $\lmu\in(0,m]$ on explicit boxes (Proposition~\ref{prop:s14:T3}). The Wick results hold wherever $\Tthree$ holds. \smallskip The main theorem, Theorem~\ref{thm:s13:main}, states (1)--(7) with every clause in full; its proof assembles the sections between. \subsection{Methods}\label{ssec:intro:method} \emph{Neumann--Dirichlet cubes.} King's cube state keeps the photon gradient terms inside the cube (Neumann) and kills the scalar on a boundary layer (Dirichlet). Both features are used: by the correlation inequalities of \cite{BFS1} the cube states are monotone in $J$ on a cone of observables, which gives the infinite-volume limit, and the faces of the cube lie on lattice hyperplanes at every block scale, which is what King's renormalisation group needs; the cube of the existence theorem is the only cube on which his block-spin flow runs (Remark~\ref{rem:k01:cube}). Two cube conventions occur: King's own, on which his state lives, and a comparison convention with the scalar alive on the whole cube, on which the Wick tube and the lattice comparisons of the rotation part are written. The two are interlaced in $J$ and have the same infinite-volume limit (Lemma~\ref{lem:s04:G}, Corollary~\ref{cor:s04:G2}), while the existence of the fixed-cube limit does not transfer between them and is proved on King's convention. \emph{Existence in ratio form.} The existence proof follows King: the generating functional is approximated, up to a small-field remainder, by an integral of the effective action after $j$ renormalisation steps, and two spacings whose effective actions live on the same unit lattice are compared. We write the comparison as a Cauchy property of the normalised functional, in a ratio in which every field-independent constant of the effective action cancels; this removes from King's route the convergence of the vacuum energies, which is not available on the cube (Theorem~\ref{thm:k01:CA}). \emph{Reflection positivity through the modified box.} Since the natural symmetrisation of King's cube is not reflection positive at small charge, the lattice statement is proved for King's modified box (Lemma~\ref{lem:s07:RPbox}) and carried to the cube by a two-sided sandwich between cube states with a margin of order $\lmu^{-1}$ (Lemmas~\ref{lem:s07:L} and~\ref{lem:s07:U}). The limits $k\to\infty$ and $J\to\infty$ are taken through uniform local bounds at complex coefficients and Montel's theorem, for the curvature sector and the Wick tube alike (Section~\ref{sec:s07}, Theorem~\ref{thm:s08:OS2C}). \emph{The Wick tube.} The joint functional of the curvature and the Wick square is controlled on the tube by three results: reflection positivity and the chessboard estimate on King's periodic tori, the correlation inequalities, and an \emph{extensive bound} for constant positive scalar sources in the fixed-spacing volume limit, derived from Ba{\l}aban's stability bounds at two scalar masses and the sign clause $\Tthree$ (Proposition~\ref{prop:s10:U}). The positive part of the scalar source is bounded through the tori, the negative part through a tiling into cells of fixed size, each controlled by one instance of the existence theorem with sources (Theorem~\ref{thm:s11:W}). \emph{Rotations.} The ultraviolet analysis needs boundaries that are lattice hyperplanes, and no bounded region has that property for a lattice and its copy rotated by $\theta_0=\arctan(3/4)$, whose rotations generate a dense subgroup of $SO(3)$. The two-lattice comparison is therefore placed on periodic tori carrying a smooth \emph{mass wall}, a raise of the scalar mass away from the sources. On each lattice King's cube functional is bounded by walled-torus functionals; the walled tori on the two lattices have the same continuum limit (Theorem~\ref{rot:hyp:K}; an involution exchanging the two lattices reduces it to one-sided bounds at a single spacing); and King's state is the lower envelope of the walled tori (Theorem~\ref{rot:hyp:TIGHT}). The heavy-shell estimate behind the last statement, the passage from the generators of the dense subgroup to $O(3)$ and the exceptional set of scalar masses are the subject of Section~\ref{sec:rot} (the chain \eqref{rot:eq:chain}). Throughout the paper, routine verifications are left to the reader. \emph{What is not addressed.}\label{ssec:intro:notaddressed} The photon mass $\lmu$ is fixed throughout, and the constants of the Wick leg are not uniform in it. The limit of vanishing photon mass is treated in a subsequent paper. We do not address the non-degeneracy of the limit (that King's state is not the state of a constant field) or its non-triviality. We do not prove the Wick sector at arbitrary Wick amplitudes, nor its Euclidean clause above the threshold or at the points of $E_W$. We do not identify King's state with the limits of periodic-torus states, and we do not prove its uniqueness or ergodicity. These questions are collected in Section~\ref{sec:s23}. \subsection{Previous work}\label{ssec:intro:inputs} Reflection positivity of lattice gauge theories goes back to Osterwalder and Seiler \cite{OS78} for compact gauge groups and to \cite[Thm.~5.3]{BFS1} for the non-compact abelian field. In two dimensions the model was constructed by Brydges, Fr\"ohlich and Seiler \cite{BFS1,BFS2,BFS3}, whose correlation inequalities are used below. The ultraviolet problem in three dimensions was solved by Ba{\l}aban \cite{Ba1,Ba2,Ba3,Ba4}: two-sided bounds $e^{-E_-\abs T}\le Z^{\eps}(T)\le e^{E_+\abs T}$ on the partition function with a photon mass on a periodic torus $T$, with constants independent of the spacing $\eps$ and of $T$ \cite[(1.14)]{Ba1}, \cite[(1.3)]{Ba2}. King \cite{K1,K2} turned these bounds into limits: the generating functionals converge as $\eps\to0$ on tori and on cubes, the infinite-volume limit is taken through cubes with Neumann conditions for $A$ and Dirichlet conditions for $\phi$, monotone in the volume by the correlation inequalities, and the resulting Schwinger functions are translation invariant and reflection positive; he kept $\lmu>0$ throughout and listed ergodicity, rotation invariance and regularity as open \cite[p.~326]{K2}. For rotations he compared, on a fixed torus, the lattice with its copy rotated by $\theta_0=\arctan(3/4)$ \cite[Thm.~2.4]{K2}. The Higgs phase was treated on the unit lattice in \cite{BBIJ84} and in the continuum in \cite{BIJ88}. Dimock \cite{Di15} revisited the ultraviolet renormalisation of the model (and of its fermionic counterpart \cite{Di18}); continuum limits of the compact pure $U(1)$ theory are due to Gross \cite{Gr83} and Driver \cite{Dr87}; for the stochastic quantisation of gauge theories with matter in three dimensions see \cite{CCHS24}. The results of King and Ba{\l}aban are used in the form stated at their point of use, each with its source: Ba{\l}aban's stability bounds and the $c$-number lemma in Section~\ref{sec:s10}, his torus analysis on King's cube in Section~\ref{sec:k01}, and the statements of the rotation part in Sections~\ref{rot:ssec:inputsN}, \ref{rot:ssec:shellest} and~\ref{rot:sec:shellestB}. Every statement taken from the renormalisation-group analysis is stated, with its proof, in the section that uses it; Remark~\ref{rem:s03:extended} lists the statements of \cite{K1} and \cite{Ba1,Ba2,Ba3,Ba4} used beyond their printed order or case. \label{ssec:intro:companion}% The Higgs phase of the abelian Higgs model, in its Villain form, is treated in the companion paper \cite{HP}, which provides (\HPapp) the construction of the measure and the reconstruction in a coordinate direction used in Section~\ref{sec:s09}, by the same general scheme: gauge-invariant fields on a lattice regularisation, infinite-volume states through boxes with killed matter, and rotation invariance recovered in the limit. There the ultraviolet problem needs no renormalisation group and the rotations are obtained from reflection positivity about nine families of planes; here the rotations come from the walled tori of Section~\ref{sec:rot}. \subsection*{Added note} The v1 of this manuscript provides a more detailed account of the construction hosted in \url{https://github.com/marikgoldstein/scalar-qed-existence} on October 10th, 2026. This line of research is rapidly evolving including parallel and independent efforts by many groups, such as \cite{YMLLM,YM3T} which also approach three-dimensional QFTs. \subsection{Organisation}\label{ssec:intro:organisation} Section~\ref{sec:s02} fixes the model, King's cube and the comparison cube, the sources and schemes, the coupling region of the Wick leg and the standing lemmas. Section~\ref{sec:s03} outlines the proof: it states the three theorems on which the main theorem rests, the existence of King's cube limits without and with scalar sources (Theorems~\ref{hyp:s03:EC} and~\ref{hyp:s03:W1}) and the rotation theorem (Theorem~\ref{hyp:s03:RC}), each with the section where it is proved, and Table~\ref{tab:intro:roadmap} lists what is proved where. Section~\ref{sec:s04} interlaces the two cube conventions. The existence part proves King's existence theorem on the cube without (Section~\ref{sec:k01}) and with scalar sources (Section~\ref{sec:k02}). The curvature sector follows in the order of the axioms: regularity (Theorem~\ref{thm:s05:CK}), translations, the cubic group and the rotation clause (Theorem~\ref{thm:s06:O3}), reflection positivity on the modified box (Section~\ref{sec:s07}) and in the limit (Theorem~\ref{thm:s08:OS2C}), and the measure with its ergodic decomposition (Theorem~\ref{thm:s09:S2K}). The Wick tube occupies Sections~\ref{sec:s10} and~\ref{sec:s11}; continuum subsequences form Section~\ref{sec:s12}. The rotation part is Sections~\ref{sec:rot}--\ref{rot:sec:gensrc}, rows (R0)--(R4) of Table~\ref{tab:intro:roadmap}: the rotation theorem, Theorem~\ref{rot:thm:main}, is stated at its start, and the results it uses are stated where they are first needed, in Sections~\ref{rot:sec:K} and~\ref{rot:sec:shell}. Theorem~\ref{thm:s13:main} assembles the main theorem, Section~\ref{sec:s14} establishes the coupling range, and Section~\ref{sec:s23} collects the open problems. \space% % % % \section{The model, King's cube and the standing lemmas}\label{sec:s02} This section fixes the lattice model with King's action and his free-boundary counterterm, King's Neumann--Dirichlet cube and the comparison cube, the sources, schemes and generating functionals, and the standing lemmas: the correlation inequalities of \cite{BFS1}, with the two inequalities only sketched there proved in full, Gaussian domination, cutting and the elementary facts. Nothing here involves a limit. Every later section uses these definitions; the correlation inequalities drive the interlacing of Section~\ref{sec:s04}, the modified box of Section~\ref{sec:s07} and the comparisons of the rotation part. \subsection{Couplings, lattices and the action} \label{conv:v065d_s02_setting:1}Throughout, $e>0$ and $\lambda,m,\lmu>0$ are the charge, the quartic coupling, the bare scalar mass and the photon mass; no further condition on the couplings is imposed, since every smallness condition of \cite{K1,K2,Ba1,Ba2,Ba3,Ba4} is a cap on the running couplings at a stopping scale chosen after the point $(e,\lambda,m,\lmu)$ and after the datum (Section~\ref{sec:s14}). The block factor is $5$: the spacings are $\eps_k\defeq5^{-k}$, the lattices $\Lk\defeq\eps_k\Z^3$ with the standard frame $(e_\alpha)$, and $M$ is King's large-block integer, ``a large integer'' \cite[\S2.3]{K1}, the integer of \cite{Ba4}, with $M\in50\N$ and $M\ge M_{\min}$. Here $M_{\min}$ is the least integer such that every integer $M\ge M_{\min}$ satisfies the lower bounds placed on the large-block integer, or on the side of the big or of the large blocks, in the schedule of constants described in Lemma~\ref{lem:s14:CW} (the constants are fixed in an acyclic order, $L=5$, $\bar n$ and the decay and coercivity constants first, then $M\in50\N$, then $R$, then $b_0$ (with $c_1p_1(\beta)+p(5\beta)\le p(\beta)$), then $\beta$ below Ba{\l}aban's and King's caps, then $J_K$, each step imposing finitely many one-sided or divisibility conditions on one entry, so that every threshold depending on $M$ is imposed on $\beta$ after $M$; at $L=5$ every condition added to Ba{\l}aban's parameter list is a lower bound on $R$ ($R\ge D^\flat+3$, $R\ge10(M+1)$, $R\ge10s_b+10$, $R\ge52$) or on $b_0$, or a downward-closed cap on $\eps_0$, each depending only on data fixed earlier in his order, hence none on the couplings, none on $L$ and none on $M$), in the hypotheses of Ba{\l}aban's stability bounds \cite[(1.14)]{Ba1}, \cite[(1.3)]{Ba2} as used in \cref{hyp:s03:T1}, in the torus form of his propagator bounds (they hold on every torus with period lattice in $M\Z^d$ and sup-systole at least $3M$, for all $x,x'$ when $\Omega=T$, with his lattice constants $\delta_0,c_0,R_0$ independent of the period lattice; the torus-size condition (T.nw) of Theorem~\ref{cpi:thm:main} implies this), and in the conditions of Theorem~\ref{cpi:thm:main} (each a lower bound on $L$, satisfied by every integer $L\ge5$, on $M$ or on $b_0$, a fixed exponent constraint, or a cap on $\eps_0$ chosen after $b_0$; the order of choices is acyclic and $e$, $\lambda$, $m$, $\lmu$ stay free). Each of these bounds holds for all $M$ above a threshold that depends only on the couplings and on constants fixed before $M$, not on the spacing, the cube, the torus or the sources; so $M_{\min}<\infty$. (The symbols of this schedule are Ba{\l}aban's and King's parameters, recalled where they are used, in Section~\ref{sec:k01} and among the hypotheses of the rotation part, Section~\ref{rot:ssec:inputsN}; only the conclusion $M_{\min}<\infty$ is used before then.) Of the divisibility $50\mid M$, the rotation part uses $M=5M'$ with $M'$ even, for the grid offset at the box $H'$ of \cref{rot:def:shell} (\cref{rot:def:geom}). $M$ is fixed after the couplings. Where an argument follows \cite{K1,K2}, a \emph{large block} is King's block of $M^3$ sites of the lattice of its scale \cite[\S2.3]{K1}; where it follows \cite{Ba1,Ba2,Ba3,Ba4}, these cubes are called \emph{big blocks}, and a large block is a cube of $5M$ sites per side \cite[p.~607]{Ba1}. King's cubes are $\CJ\defeq[-MJ,MJ)^3$, $J\in\N$, and $\XJ\defeq\CJ\cap\Lk$ (the spacing understood from the context). A \emph{King torus} is $\R^3/(\ell\Z)^3$ with $\ell\in2M\N$; $\Lk$ descends to it, with $\ell\,5^k$ sites per side. The scalar field is $\phi(x)=r_xe^{\mathrm i\vartheta_x}\in\C$ and the photon field a real $A_\alpha(x)$ for every site $x$ and direction $\alpha$; the bond $\langle x,x+\eps e_\alpha\rangle$ carries $A_\alpha(x)$ and the reversed bond $-A_\alpha(x)$. King's action \cite[(2.5)]{K2} at spacing $\eps$ is \begin{equation*} \begin{aligned} S&=\half\sum_{\alpha,\beta}\sum_x\eps\,\big(A_\alpha(x+\eps e_\beta)-A_\alpha(x)\big)^2+\frac{\lmu^2}2\sum_x\eps^3|A(x)|^2\\ &\quad+\half\sum_\beta\sum_x\eps\,\big|e^{\mathrm ie\eps A_\beta(x)}\phi(x+\eps e_\beta)-\phi(x)\big|^2\\ &\quad+\sum_x\eps^3\Big[\half\big(m^2+\dm_\eps\big)r_x^2+\lambda r_x^4\Big], \end{aligned} \end{equation*} the photon in Feynman form with mass $\lmu$, and the hopping term of a bond $\langle x,y\rangle$, $y=x+\eps e_\beta$, equal to $\frac\eps2(r_x^2+r_y^2)-\eps r_xr_y\cos(\vartheta_x-e\eps A_\beta(x)-\vartheta_y)$: a \emph{diagonal part} and a \emph{cosine part}. The counterterm $\dm_\eps$ is King's free-boundary counterterm \cite[p.~330]{K2}, fixed by his two-point graphs of order at most four in $(e,\lambda)$ at zero momentum with the propagators of the infinite lattice $\eps\Z^3$: it is \emph{one number per spacing}, the same at every site of every box and torus. The Wick constant is the free one, $\sW_\eps\defeq(-\Delta_{\eps\Z^3}+m^2)^{-1}(0,0)$. This \emph{free convention} is used on cubes and tori alike unless said otherwise; King's own torus convention appears only inside the torus theorems of the Wick leg. For a scalar mass $\mathsf M>0$ (the sans-serif letter $\mathsf M$ denotes a scalar mass throughout, $M$ being King's large-block integer) let $\dm_\eps(\mathsf M)$ be the same counterterm computed with the bare scalar mass $\mathsf M$ in place of $m$, so that $\dm_\eps=\dm_\eps(m)$, and let \[ \alpha^\infty_k(\mathsf M)\defeq\mathsf M^2+\dm_k(\mathsf M),\qquad \dm_k(\mathsf M)\defeq\dm_{\eps_k}(\mathsf M), \] be King's free-boundary bare coefficient on $\Lk$ at scalar mass $\mathsf M$. For an auxiliary scalar mass $M_1\in(0,m)$ we put \[ s^\infty_k(M_1)\defeq\tfrac12\big(\alpha^\infty_k(m)-\alpha^\infty_k(M_1)\big),\qquad \sigma_1(M_1)\defeq\liminf_{k\to\infty}s^\infty_k(M_1). \] That $\sigma_1(M_1)$ is finite is Lemma~\ref{hyp:s03:T2} (Section~\ref{sec:s10}); each coefficient $\alpha^\infty_k(\mathsf M)$ alone diverges as $k\to\infty$, only the difference having a limit, given in closed form in Lemma~\ref{lem:s14:CT}. \begin{definition}[\Tthree: the coupling region]\label{hyp:s03:T3} The couplings $(e,\lambda,m,\lmu)$ \emph{satisfy the sign clause} \Tthree\ if $\sigma_1(M_1)>0$ for some $M_1\in(0,m)$, and the set of couplings satisfying it is the \emph{coupling region} of this paper. By Proposition~\ref{prop:s14:T3} the coupling region contains the explicit regions listed there: at $\lmu=m$ a region containing $0<\lambda,e^2\le0.239\,m$, and at every $\lmu\in(0,m]$ explicit boxes. \end{definition} The Wick leg of the main theorem is stated for couplings in the coupling region; the curvature leg carries no condition on the couplings. Membership of the coupling region is the only condition on the couplings anywhere in this paper, and it is one-sided: the set $E^<_W$ of couplings on which the sign clause fails (Lemma~\ref{lem:s14:CW}) and the explicit regions on which it holds (Proposition~\ref{prop:s14:T3}) are described in Section~\ref{sec:s14}. \subsection{The states}\label{conv:v065d_s02_setting:2} \begin{definition}[Neumann--Dirichlet boxes, killed sites, the comparison cube]\label{def:s02:acube} For a finite set $X\subset\Lk$ the \emph{Neumann--Dirichlet state} $\langle\cdot\rangle^{\ND}_{k,X}$ is the normalised measure $Z^{-1}e^{-S}\allowbreak\prod_{x\in X}d^2\phi(x)\allowbreak\prod_{x\in X,\alpha}dA_\alpha(x)$ with $\phi=0$ at every site outside $X$ (Dirichlet condition: a bond leaving $X$ contributes only its diagonal part $\frac\eps2r_x^2$ at its end $x\in X$), photon gradient terms only for pairs $x,x+\eps e_\beta$ both in $X$, and the photon mass term on $X$ (Neumann condition). For $U\subset X$ the state \emph{with the matter killed on $U$} replaces the single-site measure $d^2\phi(x)$ times its weight by the point mass at $\phi(x)=0$ for $x\in U$; every hopping term with an endpoint in $U$ then reduces to its diagonal part at the other endpoint, and the photon variables are unchanged. The \emph{comparison cube} is $\NDa_{k,J}\defeq\langle\cdot\rangle^{\ND}_{k,\XJ}$, written $\ava{\cdot}$: the scalar is alive on all of $\XJ$, with Dirichlet planes $x_\mu=-MJ-\eps_k$ and $x_\mu=MJ$. \end{definition} \begin{definition}[King's RG cube]\label{def:s02:Kcube} Let $\alive^K_J\defeq\{x\in\XJ:\ x_\mu0$, and the exponential moments are finite because $\lambda>0$. So is every measure obtained from such a state by multiplying some $c_{xy}$ or some photon gradient terms by $t\in[0,1]$, adding $s|\ell(A)|^2$ ($s\ge0$) to the action, multiplying single-site measures by $e^{\eps^3q(x)r^2}$ with real $q$, or killing sites. A BFS measure is \emph{of King type} if $c_{xy}\in[0,\eps]$ and each $\rho_x$ is a point mass at $0$ or has the density $r\,e^{-\eps^3[\frac12(m^2+\dm_\eps)r^2+\lambda r^4]-\eps^3w(x)r^2-\frac\eps2n_xr^2}$ with $w(x)\ge0$, $n_x\in\{0,\dots,6\}$; every state above, and every measure obtained from one by the operations just listed with $q\le0$, is of King type. Every state above is invariant under the \emph{charge conjugation} $(A,\vartheta)\mapsto(-A,-\vartheta)$, which changes the sign of every field strength and fixes $r$. \subsection{Sources, schemes and the objects} \begin{definition}[sources, schemes, classes, transport]\label{def:s02:sources} A \emph{curvature source} is a real $2$-form $f\in C^\infty_c(\R^3;\Lambda^2)$, a \emph{scalar source} a real $h\in C^\infty_c(\R^3)$. For the plaquette $p=(x;\alpha<\beta)$ with centre $c(p)$ let $F(p)\defeq\eps^{-1}(A_\alpha(x)+A_\beta(x+\eps e_\alpha)-A_\alpha(x+\eps e_\beta)-A_\beta(x))$ and \[ \begin{gathered} F_{\Lk}(f)\defeq\sum_p\eps^3f_{c(p)}(e_\alpha,e_\beta)F(p),\qquad \sS_{\Lk}(h)\defeq\sum_x\eps^3h(x),\\ \Wick{h}\defeq\sum_x\eps^3h(x)\big(|\phi(x)|^2-2\sW_\eps\big), \end{gathered} \] the sums over the plaquettes (corners in the box), respectively the sites, of the region. A \emph{scheme} is a real sequence $a=(a_k)$, $a_k$ depending only on $\eps_k$ and the couplings; $W^a(h)\defeq\sum_x\eps^3h(x)(|\phi(x)|^2-2\sW_\eps-a_k)$. For a finite family $\zeta=(z_j,f_j;w_j,h_j)_j$ with complex coefficients put $\Theta(\zeta)\defeq\exp(\sum_jz_jF(f_j)+\sum_jw_j\Wick{h_j})$ and, for a Neumann--Dirichlet state on a box $X$ (killed sites allowed), \[ Z^a_{k,X}(\zeta)\defeq\Big\langle\exp\Big(\sum_jz_jF(f_j)+\sum_jw_jW^a(h_j)\Big)\Big\rangle^{\ND}_{k,X} =e^{-\sum_jw_ja_k\sS_{\Lk}(h_j)}\big\langle\Theta(\zeta)\big\rangle^{\ND}_{k,X}; \] $Z^a_{k,J}\defeq Z^a_{k,\XJ}$ for the comparison cube, $Z^{a,K}_{k,J}$ for King's RG cube, and \[N^a_{k,X}(0,q)\defeq e^{-(a_k+2\sW_\eps)\sS(q)}\langle e^{\sum_x\eps^3q(x)r_x^2}\rangle^{\ND}_{k,X}\] for a real scalar source. At $h=0$ the multiplier is $1$ and $a$ is dropped: $Z_{k,J}(z)\defeq\ava{e^{\sum_jz_jF(f_j)}}$, $Z^K_{k,J}(z)\defeq\avK{e^{\sum_jz_jF(f_j)}}$ for $z\in\C^n$. The \emph{cone} is $\cK\defeq\{(f,g):f\text{ a real $2$-form},\ g\ge0\}$ with the observable $\Theta(\mathrm if,g)$ (\emph{class $\cI$}); \emph{class $\cD$} consists of the observables $\Theta(f,-g)$, $(f,g)\in\cK$. Transport: $(O\cdot h)(x)\defeq h(O^{-1}x)$, $(O\cdot f)_x(u,v)\defeq f_{O^{-1}x}(O^{-1}u,O^{-1}v)$ for $O\in\Oth$; $\tau_bh\defeq h(\cdot-b)$, $(\tau_bf)_x\defeq f_{x-b}$. For $\nu\in\{1,2,3\}$, $c\in\R$ let $\rho=\rho_{\nu,c}$, $x\mapsto x-2(x_\nu-c)e_\nu$; $\theta h\defeq h\circ\rho$, $\theta f\defeq\rho^*f$, $(\rho^*f)_x(u,v)\defeq f_{\rho x}(\rho u,\rho v)$ (the components $f_{\mu\nu}$, $\mu\ne\nu$, change sign); $\theta\zeta^*\defeq(\bar z_j,\rho^*f_j;\bar w_j,h_j\circ\rho)_j$, and $\zeta\sqcup\zeta'$ is the union of two families. The \emph{amplitude} is $\amp(\zeta)\defeq\sum_j|\Re w_j|\,\norm{h_j}_\infty$. The \emph{reference scheme} is $z=a^\sharp$, $a^\sharp_k\defeq e^2X_{\eps_k}$, $X_\eps$ the coefficient of $e^2$ in the perturbative expansion of $\langle|\phi(0)|^2\rangle$ on $\eps\Z^3$ at $\lambda=0$ with the order-$e^2$ part of $\dm_\eps$ inserted. The letter $z$ names this scheme only as a superscript, $Z^{z}_{k,J}$, or through its terms $z_k$; as an argument, $z=(z_j)$ is the vector of curvature coefficients. \end{definition} By charge conjugation $\langle\Theta(\mathrm if,g)\rangle=\langle\cos F(f)\,e^{\Wick{g}}\rangle$ is real for every state above (\cref{conv:v065d_s02_setting:2}), and $Z^a_{k,X}(-z,w)=Z^a_{k,X}(z,w)$. \begin{remark}[schemes]\label{rem:s02:scheme} The Wick leg is stated for the reference scheme $z$, or for an admissible scheme $a$ (\cref{def:s03:adm}). If the fixed-cube limits exist in $(0,\infty)$ for both $a$ and $z$ at a real scalar source, the scheme is admissible: for $0\le h\in C^\infty_c$ with $\int h>0$ supported well inside some $\CJ$, $Z^z_{k,J}(0,h)/Z^a_{k,J}(0,h)=e^{-(z_k-a_k)\sS_{\Lk}(h)}$ converges to a number in $(0,\infty)$, and $\sS_{\Lk}(h)\to\int h>0$. What the Wick leg uses is only $\sup_k|z_k-a_k|<\infty$. \end{remark} \begin{definition}[admissible schemes]\label{def:s03:adm} A scheme $a$ (Definition~\ref{def:s02:sources}) is \emph{admissible} if $a=z$ or $\lim_k(z_k-a_k)$ exists. For an admissible scheme $\sup_k|z_k-a_k|<\infty$. \end{definition} \begin{definition}[norms and the tube]\label{def:s02:tube} For a family with sources $f_1,\dots,f_n$, $h_1,\dots,h_p$ supported in $Q_R\defeq[-R,R]^3$ and $\zeta=(z,w)\in\C^{n+p}$ put $g(\zeta)\defeq\sum_i(\Re z_i)f_i$ and $k(\zeta)\defeq\sum_j(\Re w_j)h_j=k_+-k_-$ (pointwise positive and negative parts). Lattice norms: $\norm{g}_{2,k}^2\defeq\sum_p\eps_k^3g_{c(p)}(e_\alpha,e_\beta)^2$ over the plaquettes of $\Lk$, $\norm{q}_{1,k}\defeq\sum_{x\in\Lk}\eps_k^3|q(x)|$. Continuum norms: $\norm{g}_2^2\defeq\sum_{\alpha<\beta}\norm{g_{\alpha\beta}}_{L^2}^2$; $\wnorm{q}\defeq\sup_x(1+|x|^2)^2|q(x)|$; $\fournorm{q}\defeq\sup_{x,|\gamma|\le4}(1+|x|^2)^2|D^\gamma q(x)|$ and $\fournorm{g}\defeq(\sum_{\alpha<\beta}\fournorm{g_{\alpha\beta}}^2)^{1/2}$; on scalar functions in $\cS(\R^3)$, $\fournorm q=|q|_4$, where $|q|_p\defeq\sup_{x,|\gamma|\le p}|(1+|x|^2)^{p/2}D^\gamma q(x)|$ is the norm of \cite[(2.1), p.~284]{OS75}. For $\sigma,H>0$ the \emph{tube} is \[ \cD_{\sigma,H}\defeq\big\{\zeta\in\C^{n+p}:\ \norm{k_+(\zeta)}_\infty<\sigma/2,\ \ \norm{k_-(\zeta)}_\infty1$ from $\R^3\setminus\CJ$. King's \emph{cube limits} and his \emph{state} are \[ \ZKJ(z)\defeq\lim_{k\to\infty}Z^K_{k,J}(z)\quad(J\ge J_0(f)),\qquad \ZKinf\Big(\sum_jz_jf_j,0\Big)\defeq\lim_{J\to\infty}\ZKJ(z), \] whenever the limits exist: their existence at real $z$ is \cref{hyp:s03:EC}, and their existence at every $z\in\C^n$, entire in $z$, is derived from it in Section~\ref{sec:s05}. The object is scheme-free. The limit $\ZKinf$ is \emph{King's state}, the infinite-volume state of King's construction \cite{K1,K2}, on his own cube convention. For general sources in a scheme $a$ the \emph{tube object} of King's state is $G^K_J(\zeta)\defeq\lim_kZ^{a,K}_{k,J}(\zeta)$, $G^K_\infty(\zeta)\defeq\lim_JG^K_J(\zeta)$, $\zeta\in\Dtube$ (Sections~\ref{sec:s10}--\ref{sec:s13}; its existence is \cref{thm:s13:TWK}); the corresponding limits of the comparison family along a continuum subsequence $\mathbf{k}$ are written $Z^{a,\mathbf{k}}_J$, $Z^{a,\mathbf{k}}_\infty$ (\cref{sec:s12}), and they coincide with $G^K_\infty$ on the tube (\cref{prop:s13:LIMK}). \subsection{Elementary facts} \begin{lemma}[elementary facts]\label{lem:s02:E} \emph{(E1)} For $\eps\le1$ and any translate $\Gamma$ of $\eps\Z^3$: $\sum_{x\in\Gamma}\eps^3(1+|x|^2)^{-4}\le C_0\defeq(5/2)^4\pi^2/8<49$ and $\sum_{x\in\Gamma}\eps^3(1+|x|^2)^{-2}\le C_0'\defeq(5/2)^2\pi^2<62$. \emph{(E2)} $\norm{q}_{L^1}\le\pi^2\wnorm{q}$, $\norm{q}_{L^2}\le\frac{\pi}{2\sqrt2}\fournorm{q}$, $\norm{q}_\infty\le\wnorm{q}\le\fournorm{q}$, $\norm{g}_2\le\frac{\pi}{2\sqrt2}\fournorm{g}$; $\wnorm{\cdot}$ is a norm, so $\wnorm{k(\zeta)}\le\sum_j|\Re w_j|\wnorm{h_j}$ and $\wnorm{k_\pm}\le\wnorm{k}$. \emph{(E3)} Sampling is evaluation at points: it commutes with pointwise positive and negative parts, and a sampled sup is at most the continuum sup, so $\zeta\in\cD_{\sigma,H}$ for the continuum sources implies the same bounds for the sampled ones. For continuous compactly supported $g,q$: $\norm{g}_{2,k}\to\norm{g}_2$, $\norm{q}_{1,k}\to\norm{q}_{L^1}$, and for every $k$: $\norm{g}_{2,k}^2\le C_0\fournorm{g}^2$, $\norm{q}_{1,k}\le C_0'\wnorm{q}$. \emph{(E4)} $n^{n/2}\le e^{n/2}(n!)^{1/2}$. \end{lemma} \begin{proof} (E1) For $y$ in the cube of side $\eps$ centred at $x\in\Gamma$, $|x-y|^2\le\frac34$ and $|y|^2\le2|x|^2+2|x-y|^2\le2|x|^2+\frac32$, so $1+|y|^2\le\frac52(1+|x|^2)$ and $(1+|x|^2)^{-r}\le(5/2)^r(1+|y|^2)^{-r}$. Integrate over the cube and sum over the disjoint cubes: $\sum_{x\in\Gamma}\eps^3(1+|x|^2)^{-r}\le(5/2)^r\int_{\R^3}(1+|y|^2)^{-r}dy$, with $\int(1+|y|^2)^{-4}=4\pi\cdot\frac\pi{32}=\frac{\pi^2}8$ and $\int(1+|y|^2)^{-2}=4\pi\cdot\frac\pi4=\pi^2$. (E2) $\norm{q}_{L^1}\le\wnorm{q}\int(1+|y|^2)^{-2}$, $\norm{q}_{L^2}^2\le\fournorm{q}^2\int(1+|y|^2)^{-4}$; the rest is immediate. (E3) Riemann sums for the convergences; the plaquette centres of one orientation form a translate of $\eps_k\Z^3$, so (E1) bounds $\sum_{p:(\alpha,\beta)}\eps_k^3g_{c(p)}(e_\alpha,e_\beta)^2$ by $C_0\fournorm{g_{\alpha\beta}}^2$, and the three orientations sum to $C_0\fournorm{g}^2$; the last bound is (E1) with $r=2$. (E4) is $n^n\le e^nn!$. \end{proof} \paragraph{Lattice-to-continuum.} On a box $X$ let $\ip AB_k\defeq\sum_{x\in X}\sum_\alpha\eps^3A_\alpha(x)B_\alpha(x)$, and for a real $2$-form $g$ whose non-vanishing plaquettes have their corners in $X$ define $\dstar_kg$ by $F_{\Lk}(g)=\ip{\dstar_kg}{A}_k$. Each component $(\dstar_kg)_\gamma$ is a backward difference quotient of the sampled components $g_{\gamma\beta}$, $\beta\ne\gamma$; for smooth compactly supported $g$ it converges uniformly to $(\dstar g)_\gamma$, up to the orientation sign convention of the components, at the rate $O(\eps_k)$ of Taylor's theorem. Hence \begin{equation}\label{eq:s02:riemann} \norm{\dstar_kg}_{2,k}\to\norm{\dstar g}_{L^2},\qquad \sup_k\norm{\dstar_kg}_{2,k}<\infty,\qquad \norm{g}_{2,k}\to\norm{g}_2\qquad(k\to\infty). \end{equation} \subsection{Correlation inequalities} For a BFS measure, a real linear functional $F$ of $A$ and $g\ge0$ on $V$ put $\Theta_{F,g}\defeq e^{\mathrm iF}e^{\sum_xg_xr_x^2}$, $\Theta^-_{F,g}\defeq e^{F}e^{-\sum_xg_xr_x^2}$, and $\langle X;Y\rangle\defeq\langle XY\rangle-\langle X\rangle\langle Y\rangle$. Since $F(f)$ is a real linear functional of $A$ and $e^{\Wick{g}}$ a positive constant times $e^{\sum_x\eps^3g(x)r_x^2}$, class $\cI$ consists of multiples of observables $\Theta_{F,g}$ and class $\cD$ of multiples of observables $\Theta^-_{F,g}$. \begin{lemma}[correlation inequalities]\label{lem:s02:BFS} Let $\langle\cdot\rangle$ be a BFS measure, $F$ a real linear functional of $A$, $g\ge0$, $\Theta=\Theta_{F,g}$, $\Theta^-=\Theta^-_{F,g}$. Then $\langle\Theta\rangle$ is real, and \begin{enumerate}[label=\textup{(\alph*)},nosep] \item $\langle\Theta;|\ell(A)|^2\rangle\le0$ for every complex linear functional $\ell$ of $A$; hence adding $s|\ell(A)|^2$ to the action, $s\ge0$, does not decrease $\langle\Theta\rangle$, and deleting a photon gradient term does not increase it; \item $\langle\Theta;r_xr_y\cos(\vartheta_x-\vartheta_y-\ell_{xy}(A))\rangle\ge0$; hence decreasing $c_{xy}$ does not increase $\langle\Theta\rangle$; \item $\langle\Theta;r_x^2\rangle\ge0$; hence multiplying $\rho_x$ by $e^{-sr^2}$, $s\ge0$, does not increase $\langle\Theta\rangle$, and killing a set $U$ of sites does not increase it; \item for $\Theta^-$ the three inequalities hold with the opposite signs, and each monotonicity statement holds with the same hypothesis and the opposite conclusion (``does not increase'' and ``does not decrease'' interchanged); \item if the pairs in $B$ and the Gaussian $\gamma$ do not couple the fields at the sites of $X\subset V$ (and the photon variables attached to them) to the others, the measure is a product, and for an observable depending only on the fields in $X$ the expectation equals the expectation in the $X$-factor. \end{enumerate} \end{lemma} \begin{proof} (a)--(c) are \cite[Thms.~6.1, 6.2(1),(2)]{BFS1}, whose hypotheses are those of a BFS measure: an arbitrary finite lattice, $c_{xy}\ge0$, finite single-site measures on $[0,\infty)$ with $\int e^{\alpha r^2}d\rho_x<\infty$ for all $\alpha\ge0$ (a point mass at $0$ is one), an arbitrary centred Gaussian with covariance $C\ge0$, and $g\ge0$ (so $g=0$ is included). Theorem 6.1 states $\langle P(r)\cos(n\cdot\vartheta+f\cdot A);Q(r)\cos(m\cdot\vartheta+h\cdot A)\rangle\ge0$ for polynomials $P,Q$ in the $r_x$ with positive coefficients; (a), (b) follow as in the proof of Theorem~6.2(1),(2) given there: $e^{\sum g_xr_x^2}$ is a limit of such polynomials, $e^{\mathrm iF}$ may be replaced by $\cos F$ because the expectation and both observables are invariant under $(A,\vartheta)\mapsto(-A,-\vartheta)$ (which also gives the realness), and $|\ell(A)|^2=\ell_1(A)^2+\ell_2(A)^2$ with $\ell_r(A)^2=\lim_{t\downarrow0}t^{-2}[1-\cos(t\ell_r(A))]$; (c) is Theorem 6.1 with $Q=r_x^2$ and $m=0$, $h=0$. Each monotonicity statement follows by differentiating in the parameter, every interpolating measure being a BFS measure. For the killing statements of (c) and (d), let $x\in V$ and let $\langle\cdot\rangle_x$ be the measure with $\rho_x$ replaced by the point mass at $0$, in which every term $c_{xy}r_xr_y\cos(\cdot)$ vanishes. Decreasing $c_{xy}$ to $0$ for every pair $\{x,y\}\in B$ containing $x$ does not increase $\langle\Theta\rangle$, by (b), and does not decrease $\langle\Theta^-\rangle$, by (d). In the resulting measure $(r_x,\vartheta_x)$ is coupled to nothing, so by (e) it is the product of $\rho_x(dr_x)\,d\vartheta_x$ and of the law of the other variables, which is that of $\langle\cdot\rangle_x$; the expectations of $\Theta$ and $\Theta^-$ in it are $m^+_x\langle\Theta\rangle_x$ and $m^-_x\langle\Theta^-\rangle_x$ with $m^\pm_x\defeq\int e^{\pm g_xr^2}d\rho_x/\int d\rho_x$, so that $m^-_x\le1\le m^+_x$. Since $\Theta^->0$, $\langle\Theta^-\rangle_x\ge m^-_x\langle\Theta^-\rangle_x\ge\langle\Theta^-\rangle$; and $\langle\Theta\rangle_x\ge0$ (multiplying the single-site measures by $e^{g_yr_y^2}$ gives a BFS measure, in which the expectation of $e^{\mathrm iF}$ is non-negative by identity (b) of Lemma~\ref{lem:s02:GD}, whose proof does not use the present lemma), hence $\langle\Theta\rangle_x\le m^+_x\langle\Theta\rangle_x\le\langle\Theta\rangle$. Several sites are killed one at a time; no condition on $\rho_x$ beyond the definition of a BFS measure is used. (d) is stated in \cite[Thm.~6.2(3),(4)]{BFS1} for the photon and the hopping inequality with a sketched proof, and its $r^2$-inequality is \cite[(3.28)]{K2}, stated there without proof; we prove all three. Let $\mu$ be the unnormalised measure and $\omega'=(r',\vartheta',A')$ an independent copy of $\omega=(r,\vartheta,A)$; then $(\int d\mu)^2\langle\Theta^-;Y\rangle=\iint\Theta^-(\omega)(Y(\omega)-Y(\omega'))\,d\mu(\omega)d\mu(\omega')$. Write $r=u-v$, $r'=u+v$ (so $u_x\ge|v_x|$), $\vartheta=\vartheta_++\vartheta_-$, $\vartheta'=\vartheta_+-\vartheta_-$, $A=a_++a_-$, $A'=a_+-a_-$. Then $a_\pm=(A\pm A')/2$ are independent with the common centred Gaussian law of covariance $C/2$, and the radial law $\prod_xd\rho_x(u_x-v_x)d\rho_x(u_x+v_x)$ on $\{u\ge|v|\}$ is invariant under $v_x\mapsto-v_x$ at each site. With $\chi_{xy}\defeq\vartheta_{+,x}-\vartheta_{+,y}-\ell_{xy}(a_+)$ and $\psi_{xy}\defeq\vartheta_{-,x}-\vartheta_{-,y}-\ell_{xy}(a_-)$ the unprimed phase is $\chi+\psi$, the primed one $\chi-\psi$, and \begin{align*} r_xr_y\cos(\chi+\psi)+r'_xr'_y\cos(\chi-\psi)&=2(u_xu_y+v_xv_y)\cos\chi\cos\psi+2(u_xv_y+v_xu_y)\sin\chi\sin\psi,\\ r_xr_y\cos(\chi+\psi)-r'_xr'_y\cos(\chi-\psi)&=-2(u_xv_y+v_xu_y)\cos\chi\cos\psi-2(u_xu_y+v_xv_y)\sin\chi\sin\psi, \end{align*} $r_x^2-r_x'^2=-4u_xv_x$, $\ell(A)^2-\ell(A')^2=4\ell(a_+)\ell(a_-)$ for real $\ell$, and $\Theta^-(\omega)=e^{F(a_+)}e^{F(a_-)}\allowbreak e^{-\sum_xg_x(u_x^2+v_x^2)}\allowbreak e^{2\sum_xg_xu_xv_x}$. Expand $\exp(\sum_bc_b[\text{first line}])$ and $e^{2\sum g_xu_xv_x}$ in power series (absolutely convergent at fixed spacing). Every term of the integrand is a sign $\sigma$ times a monomial in $(u,v)$ with non-negative coefficient, times the even radial factor $e^{-\sum g_x(u_x^2+v_x^2)}$, times $V(\vartheta_+,a_+)V(\vartheta_-,a_-)$ with one real function $V$ (a product of $e^{F(\cdot)}$, of factors $\cos\chi_b$ or $\sin\chi_b$, and of $\ell(\cdot)$ for the photon slot). The photon integral of $V(\vartheta_+,a_+)V(\vartheta_-,a_-)$ factorises because $a_+,a_-$ are independent and identically distributed. The angular integral factorises by the following argument: per site $d\vartheta\,d\vartheta'=2\,d\vartheta_+d\vartheta_-$, and $[0,2\pi)^2$ in the $(\vartheta,\vartheta')$-variables is a fundamental domain of the lattice $L$ generated by $(\pi,\pi)$ and $(\pi,-\pi)$ in the $(\vartheta_+,\vartheta_-)$-plane, which contains $(2\pi\Z)^2$ with index $2$; $V(\vartheta_+,\cdot)V(\vartheta_-,\cdot)$ is $L$-invariant, because the simultaneous shift of $\vartheta_{+,x}$ and $\vartheta_{-,x}$ by $\pi$ at one site changes every $\chi_b$ and $\psi_b$ with $b\ni x$ by $\pi$ and leaves $\cos\chi_b\cos\psi_b$ and $\sin\chi_b\sin\psi_b$ unchanged; hence $\int_{[0,2\pi)^2}VV\,d\vartheta\,d\vartheta'=2\int_{L\text{-domain}}VV\,d\vartheta_+d\vartheta_-=\int_{[0,2\pi)^2}VV\,d\vartheta_+d\vartheta_-=(\int_0^{2\pi}V)^2$. The radial integral of a monomial vanishes if its degree in some $v_x$ is odd, by the $v\mapsto-v$ symmetry, and is non-negative otherwise, since $u_x\ge|v_x|$. So every term has the sign $\sigma$: $\sigma=-1$ for $Y=r_x^2$ (from $-4u_xv_x$) and for the hopping term (second line), $\sigma=+1$ for $Y=\ell(A)^2$ (for complex $\ell$, real and imaginary parts). Hence $\langle\Theta^-;r_x^2\rangle\le0$, $\langle\Theta^-;r_xr_y\cos(\cdot)\rangle\le0$, $\langle\Theta^-;|\ell(A)|^2\rangle\ge0$. Monotonicity and killing follow as in (a)--(c), with (d)'s hopping inequality in place of (b). (e) is clear. \end{proof} \begin{lemma}[current representation and Gaussian domination]\label{lem:s02:GD} Let $\langle\cdot\rangle$ be a BFS measure whose Gaussian has covariance $C$. Integrating out the matter at fixed $A$, the matter integral is $\Psi(A)=\sum_qw(q)\cos\ip{\kappa(q)}A$, $q$ running over the integer-valued divergence-free currents on $B$, $w(q)\ge0$, $\sum_qw(q)<\infty$, $\ip{\kappa(q)}A\defeq\sum_{xy}q_{xy}\ell_{xy}(A)$. Let $\E^\sharp$ be the probability measure on currents with weights proportional to $w(q)e^{-\ip{\kappa(q)}{C\kappa(q)}/2}$. Then for $v\in\R^m$, $s\in\R$ ($C$ is symmetric, so $\ip{Cv}\kappa=\ip v{C\kappa}$): \begin{align*} &\textup{(a)}\qquad\langle e^{s\ip vA}\rangle=e^{s^2\ip v{Cv}/2}\,\E^\sharp\cos\big(s\ip{Cv}{\kappa}\big)\le e^{s^2\ip v{Cv}/2},\\ &\textup{(b)}\qquad\langle e^{\mathrm i\ip vA}\rangle=e^{-\ip v{Cv}/2}\,\E^\sharp\cosh\big(\ip{Cv}{\kappa}\big)\ \ge\ e^{-\ip v{Cv}/2}>0. \end{align*} \end{lemma} \begin{proof} Each bond factor expands as $e^{c_{xy}r_xr_y\cos t}=\sum_{q\in\Z}I_q(c_{xy}r_xr_y)e^{\mathrm iqt}$ with $I_q\ge0$, $I_{-q}=I_q$, $\sum_qI_q(c)=e^c$; after the radial integrals the multiple series converges absolutely, dominated by $\prod_xe^{\frac12\sum_yc_{xy}r_x^2}$, integrable against the single-site measures. Integrating the angles forces $\sum_yq_{xy}=0$ at every site (at a killed site $r_x=0$ and $I_q(0)=\delta_{q0}$, so the currents vanish on its bonds); pairing $q$ with $-q$ gives $\Psi$ with $w(q)\ge0$. For a centred Gaussian $\gamma$ with covariance $C$ and $a\in\C^m$, $\int e^{\ip aA}d\gamma=e^{\ip a{Ca}/2}$; with $a=sv\pm\mathrm i\kappa$ and the average of the two signs, $\int e^{s\ip vA}\cos\ip\kappa A\,d\gamma=e^{s^2\ip v{Cv}/2-\ip\kappa{C\kappa}/2}\cos(s\ip v{C\kappa})$. Sum over $q$ with the weights $w(q)$, divide by the same sum at $s=0$ (positive), and use $|\cos|\le1$. For the imaginary exponent take $a=\mathrm iv\pm\mathrm i\kappa$: $\int e^{\mathrm i\ip vA}\cos\ip\kappa A\,d\gamma=\frac12\big[e^{-\ip{v+\kappa}{C(v+\kappa)}/2}+e^{-\ip{v-\kappa}{C(v-\kappa)}/2}\big]=e^{-\ip v{Cv}/2-\ip\kappa{C\kappa}/2}\cosh\ip v{C\kappa}$; the same sum over $q$ and the same division give the second identity, and $\cosh\ge1$ gives the lower bound. \end{proof} The lemma applies to every state of this paper at every $k$, $J$, $\lmu>0$, with or without a real scalar source absorbed in the single-site measures (the photon covariance is unchanged). \begin{lemma}[killing]\label{lem:s02:Kprime} Let $\langle\cdot\rangle$ be a BFS measure of King type, $U$ a set of alive sites, $\chi:U\to(0,\infty)$, and $\langle\cdot\rangle_{U,s}$ the measure with $\rho_x$ multiplied by $e^{-s\eps^3\chi(x)r_x^2}$ for $x\in U$. Let $\Theta$ be continuous with $|\Theta(A,\phi)|\le e^{\ell(A)}e^{C\sum_xr_x^2}$ for a real linear functional $\ell$ and $C\ge0$. Then $\langle\Theta\rangle_{U,s}\to\langle\Theta\rangle_U$ as $s\to\infty$, $\langle\cdot\rangle_U$ the measure with $\rho_x$ replaced by the point mass at $0$ for $x\in U$ (the state with the matter killed on $U$). \end{lemma} \begin{proof} The unnormalised density is \[ e^{-Q(A)/2}dA\prod_{x\text{ alive}}w_x(r_x)r_xdr_xd\vartheta_x\prod_{b}e^{c_br_xr_y\cos(\cdot)}\prod_{x\in U}e^{-s\eps^3\chi(x)r_x^2}, \] $Q$ positive definite, killed sites carrying no variables. Substitute $r_x=(s\eps^3\chi(x))^{-1/2}u_x$ for $x\in U$; the Jacobians cancel between numerator and denominator. \emph{Domination for $s\ge s_0$:} at $x\in U$, $w_x(r_x)\le e^{\eps^3|\dm_\eps|r_x^2/2}$ (the other exponents in $w_x$ are $\le0$), $|\Theta|$ contributes $e^{Cr_x^2}$, and each of the at most six bond weights at $x$ is at most $e^{\frac\eps2r_x^2+\frac\eps2r_y^2}$ ($0\le c_b\le\eps$); in the variable $u_x$ these factors together are $\le e^{u_x^2/2}$ once $s\ge s_0(\eps,m,\dm_\eps,C,\min\chi)$, so the $U$-factor is at most $e^{-u_x^2/2}u_x$, integrable on $[0,\infty)\times[0,2\pi)$; at an alive $y\notin U$ the factor is at most $w_y(r_y)e^{(C+3\eps)r_y^2}r_y$, integrable since $\lambda>0$; the photon factor $e^{\ell(A)}e^{-Q(A)/2}$ is integrable since $Q$ is positive definite. \emph{Pointwise limit:} at fixed $(u,\vartheta,A,r_{V\setminus U})$, $r_x\to0$ for $x\in U$, so $\Theta\to\Theta|_{\phi_U=0}$ by continuity, every bond weight with an endpoint in $U$ and every $w_x(r_x)$, $x\in U$, tends to $1$, and the $U$-factor tends to $e^{-u_x^2}u_x\,du_xd\vartheta_x$, whose integral cancels between numerator and denominator. By dominated convergence the ratio tends to $\int\Theta|_{\phi_U=0}d\mu_U/\int d\mu_U$, where $\mu_U$ has the cosines at $U$ removed, the weights $w_y$ (which already contain all diagonals at $y$) and the photon unchanged, and the point mass at $U$: that is $\langle\cdot\rangle_U$, with a positive denominator. \end{proof} \begin{lemma}[cutting]\label{lem:s02:cut} Let $\langle\cdot\rangle_Y$ be a Neumann--Dirichlet box state on $\Lk$, possibly with killed sites, or a periodic state on a King torus, and let $X$ be a set of sites of $Y$ such that $X$ and its nearest neighbours lie in one fundamental cell (a clause in torus language: for a box state it has no content beyond $X\subset Y$). Let $\langle\cdot\rangle^{\ND}_X$ be the Neumann--Dirichlet state on $X$ whose single-site measures (killed sites and the diagonal parts of the bonds leaving $X$ included) are those of $\langle\cdot\rangle_Y$. Let $(f,g)\in\cK$ be such that every plaquette and every site at which $f$, respectively $g$, does not vanish lies in $X$. Then, at every lattice spacing, \[ \langle\Theta(\mathrm if,g)\rangle^{\ND}_X\le\langle\Theta(\mathrm if,g)\rangle_Y,\qquad \langle\Theta(f,-g)\rangle^{\ND}_X\ge\langle\Theta(f,-g)\rangle_Y, \] and the same for $Z^a$. The first inequality uses Lemma~\ref{lem:s02:BFS}(a),(b),(e) only; the second uses (d),(e). If no site of $X$ is killed, $\langle\cdot\rangle^{\ND}_X$ is the state of Definition~\ref{def:s02:acube}. \end{lemma} \begin{proof} Delete from $Y$ every photon gradient term coupling a site of $X$ to a site outside $X$, and the cosine parts of all hopping terms between $X$ and its complement, keeping their diagonal parts $\frac\eps2(r_x^2+r_y^2)$. By Lemma~\ref{lem:s02:BFS}(a),(b) this does not increase the first expectation, by (d) it does not decrease the second, and every intermediate measure is a BFS measure. The result is a product (Lemma~\ref{lem:s02:BFS}(e)) whose $X$-factor is $\langle\cdot\rangle^{\ND}_X$: the diagonal parts of the cut bonds are exactly the Dirichlet terms and the remaining photon terms the Neumann ones of $X$; the observable depends on the $X$-variables only. The free Wick constant, $\dm_\eps$ and the scheme multiplier are the same on both sides. \end{proof} With $Y=X_{J+1}$, $X=\XJ$ this is King's monotonicity in the volume \cite[Thm.~3.2, (3.31)--(3.33)]{K2} between comparison cubes, in the form his proof proves: with one Wick constant for all $J$, the free subtraction King himself switches to at the start of that proof \cite[p.~333]{K2}. With the Dirichlet Wick constant of \cite[(2.13)]{K2}, which increases with $J$, the $c$-number factor moves against the monotone factor and the monotonicity does not follow; that form is not used. The box version with killed sites and the monotonicity of King's RG cubes are in Section~\ref{sec:s04}. \subsection{Algebra of the scheme, covariance, reflection, continuity} \begin{lemma}[algebra of the scheme]\label{lem:s02:alg} Let $a$ be a scheme and $Z^a$ the functional of a state above (\cref{conv:v065d_s02_setting:2}) on a box at spacing $\eps_k$. \begin{enumerate}[label=\textup{(\alph*)},nosep] \item For real coefficients $w$ the multiplier $e^{-\sum_jw_ja_k\sS(h_j)}$ is a positive constant depending only on $k$, $w$ and $\sS_{\Lk}(h_j)$. Hence the inequalities of Lemmas~\ref{lem:s02:BFS} and~\ref{lem:s02:cut} between two states on the same lattice with the same sources hold for $Z^a$ in the same direction. \item $|Z^a(\zeta)|\le Z^a(\Re\zeta)$, and $\ln Z^a$ is convex in the real coefficients. \item $Z^a(f,h)\,Z^a(-f,-h)\ge1$ for real $(f,h)$. \item For real $f$ and real $k_1,k_2\ge0$: $Z^a(f,k_1-k_2)\le Z^a(2f,-2k_2)^{1/2}\,Z^a(0,2k_1)^{1/2}$. \end{enumerate} \end{lemma} \begin{proof} (a) is immediate. (b) $|\Theta(\zeta)|=\Theta(\Re\zeta)$ and the multiplier has modulus $e^{-\sum_j(\Re w_j)a_k\sS(h_j)}$; convexity is H\"older's inequality. (c) $\langle e^{X}\rangle\langle e^{-X}\rangle\ge1$ by Cauchy--Schwarz, and the multipliers cancel. (d) Cauchy--Schwarz gives $\langle e^{F+\Wick{k_1}-\Wick{k_2}}\rangle\le\langle e^{2F-2\Wick{k_2}}\rangle^{1/2}\langle e^{2\Wick{k_1}}\rangle^{1/2}$, and $e^{-a_k\sS(k_1-k_2)}=(e^{2a_k\sS(k_2)})^{1/2}(e^{-2a_k\sS(k_1)})^{1/2}$. \end{proof} \begin{lemma}[passive covariance]\label{lem:s02:pass} Let $O\in\Oth$, $\Lambda$ a lattice of spacing $\eps$ with frame $(f_\alpha)$, and $X$ a box or a torus for $\Lambda$. Equip $O\Lambda$ with the frame $(Of_\alpha)$ and $OX$ with the transported boundary conditions. Then $\langle\Theta(z\,O\cdot f,w\,O\cdot h)\rangle_{O\Lambda,OX}=\langle\Theta(zf,wh)\rangle_{\Lambda,X}$ and $\sS_{O\Lambda}(O\cdot h)=\sS_\Lambda(h)$, likewise for families and for $Z^a$. On a torus the periodic model does not change if the frame $(f_\alpha)$ is replaced by $(\pm f_{\pi(\alpha)})$, $\pi$ a permutation. \end{lemma} \begin{proof} The map $\phi'(Ox)=\phi(x)$, $A'_\alpha(Ox)=A_\alpha(x)$, components taken in the frames $(f_\alpha)$ and $(Of_\alpha)$, is a bijection of the configuration spaces preserving Lebesgue measure; the action is written in the frame, and $\dm_\eps$, $\sW_\eps$, $a_k$ depend on $\eps$ only; plaquettes, centres and field strengths are transported, and $(O\cdot f)_{Oc}(Of_\alpha,Of_\beta)=f_c(f_\alpha,f_\beta)$. A permutation of the frame relabels the components; reversing $f_\alpha$ replaces $A_\alpha(x)$ by $-A_\alpha(x-\eps f_\alpha)$, a bijection of the torus variables preserving the action, which changes the sign of both $F(p)$ and $f_{c(p)}(f_\alpha,f_\beta)$ and preserves the set of plaquette centres. \end{proof} \begin{lemma}[the reflection sandwich at fixed spacing, comparison convention]\label{lem:s02:refl} Let $\sigma_3=\diag(1,1,-1)$ and let $(f,g)\in\cK$ be supported at $\ell^\infty$-distance $>1$ from $\R^3\setminus C_{J-1}$. Then for every $k\ge0$ \[ Z^a_{k,J-1}(\mathrm if,g)\le Z^a_{k,J}(\mathrm i\,\sigma_3\!\cdot\!f,\sigma_3\!\cdot\!g)\le Z^a_{k,J+1}(\mathrm if,g), \] and the reversed inequalities hold for $Z^a(f,-g)$. \end{lemma} \begin{proof} $\sigma_3$ preserves $\Lk$ and $\sS_{\Lk}$ and maps $\XJ$ onto $X_+\defeq(\CJ+\eps_ke_3)\cap\Lk$. By Lemma~\ref{lem:s02:pass}, $Z^a_{k,J}(\mathrm i\sigma_3\cdot f,\sigma_3\cdot g)$ is the functional with sources $(f,g)$ of the Neumann--Dirichlet state on $X_+$ in the frame $(e_1,e_2,-e_3)$. Written in the frame $(e_\alpha)$, with $A'_3(x)=-A_3(x-\eps e_3)$, this is the BFS measure $R$ in which $\phi,A_1,A_2$ live on $X_+$ and $A_3$ on $X_+-\eps e_3=\XJ$, with Dirichlet conditions for $\phi$ on $X_+$, photon gradient and mass terms within these sets, and the hopping term in direction $3$ the standard one of the bond $\langle y-\eps e_3,y\rangle$; the plaquette terms $(x;\alpha,3)$ pick up two sign changes and keep their centres, so the observable is $\Theta(\mathrm if,g)$ in the standard frame. $R$ is obtained from the Neumann--Dirichlet state on $X_{J+1}$ by deleting photon gradient terms and hopping cosines and factorising, since $X_+\cup\XJ\subset X_{J+1}$, and the state on $X_{J-1}$ is obtained from $R$ in the same way, since $X_{J-1}\subset\XJ\cap X_+$; the diagonal parts of the cut bonds are the Dirichlet terms of the smaller state, and the sources stay inside $X_{J-1}$. Lemma~\ref{lem:s02:BFS}(a),(b),(e), respectively (d),(e), give the directions, and Lemma~\ref{lem:s02:alg}(a) carries them to $Z^a$. \end{proof} \begin{lemma}[continuity of convex functions]\label{lem:s02:baire} Let $U$ be a non-empty open convex subset of a Fr\'echet space and $\Phi:U\to\R$ a convex Borel function. Then $\Phi$ is continuous on $U$. \end{lemma} \begin{proof} $U$ is a Baire space and $U=\bigcup_{j\in\N}\{\Phi\le j\}$, so some $S\defeq\{\Phi\le j\}$ is non-meagre in $U$; being Borel, $S$ has the Baire property \cite{Kec95}: $O\setminus\mathcal M\subset S$ for a non-empty open $O\subset U$ and a meagre $\mathcal M$. Pick $x_0\in O$ and a balanced open neighbourhood $W$ of $0$ with $x_0+W+W\subset O$. For $w\in W$ and $y\defeq x_0+w$, $y+W\subset O$, and the point reflection $\iota(x)\defeq2y-x$ maps $y+W$ onto itself and preserves meagre sets; so $S\cap(y+W)$ and $\iota(S\cap(y+W))$ are comeagre in $y+W$ and intersect: there is $u\in S$ with $2y-u\in S$, and $\Phi(y)\le\frac12(\Phi(u)+\Phi(2y-u))\le j$. Hence $\Phi\le j$ on $x_0+W$. A convex function on an open convex set bounded above near one point is bounded above near every point $y\in U$ (choose $z\in U$ with $y$ strictly between $x_0$ and $z$, and take the convex combinations of $x_0+W$ with $z$), and a convex function locally bounded above is continuous \cite{RS1}. \end{proof} \subsection{Results of Ba{\l}aban and King} Ba{\l}aban's torus analysis \cite{Ba1,Ba2,Ba3,Ba4} enters this paper through the torus theorems of Section~\ref{sec:s10} (his stability bounds, used as stated at two scalar masses, and a $c$-number computation), through the statements of Section~\ref{sec:k01} in the existence part, and through its re-runs for the hybrid flow in the rotation part, stated with their proofs in Sections~\ref{rot:ssec:inputsN}, \ref{rot:ssec:shellest} and~\ref{rot:sec:shellestB}. The following remark lists what is used beyond the printed order or case. \begin{remark}[statements of Ba{\l}aban and King used beyond their printed order or case]\label{rem:s03:extended} The following statements of \cite{K1} and \cite{Ba1,Ba2,Ba3,Ba4} are used in a form read from the printed argument rather than from a printed statement, beyond the order, the model or the case treated in print; each is stated in that form, with its proof, at the place indicated. (i) King's constant-background reduction \cite[pp.~661--662, (3.43)--(3.55)]{K1}: the identities (3.43)--(3.44), (3.47)--(3.48) and (3.50)--(3.53) hold for any background, and in (3.47)--(3.55) the non-constant part of the background enters the bounds only through (3.44)--(3.45); so the reduction runs at the backgrounds of the hybrid flow (Theorem~\ref{cpi:hyb:BD}(ii), Section~\ref{rot:ssec:inputsN}). (ii) King's renormalisation induction \cite[pp.~663--668, (3.66)--(3.70), (3.76)--(3.79)]{K1} with Ba{\l}aban's enumeration of the divergent families \cite[Sects.~2--3]{Ba3}: the same induction is run for the shell model, with its face residues as vertices, and for the derivative of the effective action with one inserted vertex (the renormalisation bounds of the shell model, Theorems~\ref{cpi:thm:main} and~\ref{cpi:swc:main}, Sections~\ref{rot:ssec:shellest} and~\ref{rot:sec:shellestB}). (iii) Ba{\l}aban's combinatorial estimate \cite[p.~592, (3.41)--(3.42)]{Ba2}: the two $O(1)$ of its exponents are read as constants independent of the scale, the spacing, the torus and the family history, the constant condition of the shell model (Sections~\ref{rot:sec:shell} and~\ref{rot:sec:shellestB}). (iv) The Taylor step on the external legs \cite[p.~435, (3.10); pp.~440--442, (3.26)--(3.30)]{Ba3}: the vanishing of the zeroth and first moments, printed for the vertices (1.8) and (1.10) of Ba{\l}aban's model, is used for the seagull and block-averaging members of the shell model (Lemma~\ref{cpi:lem:ward}(a) and~(c)). (v) The consequences drawn in \cite[pp.~421--422]{Ba3} from its Proposition~1 for the lower and upper bounds: the same consequences, with the same constants, for the shell model, whose scalar legs obey the same small-field bound (Proposition~\ref{cpi:prop:TD}(b)--(d)). (vi) The rewriting of a photon two-point class \cite[p.~442, (3.30), with pp.~437--445]{Ba3}: the two coefficient clauses of (3.30) are printed; that after the rewriting the explicit block projections sit only inside composite kernels on which no later step acts is a reading of pp.~437--445 carried to every order (Lemma~\ref{cpi:uc:lem:QPKinf}). (vii) The regularity assumption \cite[p.~610, Proposition~2.1, (2.23)]{Ba1}, \cite[pp.~572--573, (1.7)]{Ba4} and \cite[p.~571, Lemma~2.3, (2.59)--(2.60)]{Ba2}: the assumption (1.7) is read as (2.23) with the rescaled charge, so that the three regularity senses coincide and (2.60) implies it once the scale is small; at block factor $5$ the bounds (2.59)--(2.60) hold with $O(p(L^k\eps))$ replaced by $O_L(p(L^k\eps))$ (Corollary~\ref{cpi:asa:B5Omega}). \end{remark} \space% % % % % % % % \section{Outline of the proof}\label{sec:s03} \ifcsname mainthm\endcsname\else \theoremstyle{plain} \newtheorem{mainthm}{Theorem} \renewcommand{\themainthm}{\Alph{mainthm}} \crefname{mainthm}{Theorem}{Theorems} \Crefname{mainthm}{Theorem}{Theorems} \fi The main theorem, Theorem~\ref{thm:s13:main} (\cref{sec:s13}), rests on three theorems of this paper, stated in this section and proved later: the existence of King's cube limits, without and with scalar sources (\cref{hyp:s03:EC,hyp:s03:W1}; the existence part, \cref{sec:k01,sec:k02}), and the rotation theorem (\cref{hyp:s03:RC}; the rotation part, Sections~\ref{sec:rot}--\ref{rot:sec:gensrc}, whose main theorem is \cref{rot:thm:main}). It rests further on the two torus theorems of the Wick leg, stated in \cref{sec:s10}, where they are used: Ba{\l}aban's stability bounds (Proposition~\ref{hyp:s03:T1}), that is \cite[(1.14)]{Ba1}, \cite[(1.3)]{Ba2} used as stated there, read at two scalar masses and in King's torus normalisation of the counterterm, and the $c$-number lemma (Lemma~\ref{hyp:s03:T2}), a computation with lattice Feynman integrals. One definition completes the list, the coupling region of the Wick leg (Definition~\ref{hyp:s03:T3}, \cref{sec:s02}): the set of couplings at which the sign clause \Tthree\ holds, which contains the explicit regions of \cref{prop:s14:T3}; the Wick leg of the main theorem is stated for couplings in that region, and the curvature leg carries no condition on the couplings. After the two cube conventions, on which every statement is posed, the three theorems are stated, each with the section where it is proved and with what is used of it; the section ends with the route from them to the main theorem. \smallskip\noindent\emph{The two cube conventions.} The two cube conventions of Definitions~\ref{def:s02:Kcube} and~\ref{def:s02:acube} are King's convention $\NDK$ (the scalar field killed on the three top face layers $\TJ$ of the cube $\CJ$, the convention of the infinite-volume state of King's construction \cite{K1,K2}) and the comparison convention $\NDa$ (the scalar alive on all of $\XJ$), on which the tube bound (Theorem~\ref{thm:s11:W}), the cell estimates of the Wick leg and the lattice comparisons of the rotation part are written. The two are interlaced by Lemma~\ref{lem:s04:G}. Which convention a statement is posed on is part of it. The existence theorems, \cref{hyp:s03:EC,hyp:s03:W1}, are posed on King's convention $\NDK$, on which King's renormalisation-group flow runs (\cref{rem:k01:cube}); by Remark~\ref{rem:s04:notfollow} the existence of the fixed-cube limits on one convention does not imply it on the other, and the comparison family enters the main theorem through its limit states along continuum subsequences (\cref{sec:s12}), which exist without any such clause and coincide with King's state (\cref{cor:s12:KK}, \cref{prop:s13:LIMK}). The rotation theorem, \cref{hyp:s03:RC}, is proved in the rotation part for King's state $\ZKinf$ on King's convention (\cref{rot:thm:main}) and is carried to the limit states of the comparison family through Corollary~\ref{cor:s04:G2}; the Euclidean clause of the Wick sector below the amplitude threshold and outside the exceptional set $E_W$ is Theorem~\ref{rot:thm:mainGS} (in the scheme $\Theta^\sharp$ and in every admissible scheme on the tube $\Om(h)$); above the threshold it is open (\cref{sec:s23}). The torus theorems and the sign clause \Tthree\ are statements about King's tori (Definition~\ref{def:s10:torus}) and about $c$-numbers. All limit statements of this paper are proved for King's state, whose fixed-cube limits exist by \cref{hyp:s03:EC,hyp:s03:W1}. Several of them are proved by arguments that use, of the cube functionals, only the existence of the fixed-cube limits at real coefficients, the cube monotonicity on the cone and the bounds established here; a remark in each of the sections concerned notes that these arguments apply verbatim to the comparison family at every family and cube where that family's fixed-cube limits exist. Whether they exist is not decided here (Remark~\ref{rem:s04:notfollow}, \cref{rem:k01:compfamily}), and no statement of this paper assumes it: the comparison family enters through its limit states along continuum subsequences (\cref{thm:s12:LIM}), which exist unconditionally. \begin{mainthm}[King's cube limits, curvature]\label{hyp:s03:EC} For every finite family $f_1,\dots,f_n$ of real $2$-forms in $C^\infty_c(\R^3;\Lambda^2)$, every $J\ge J_0(f)$, where $J_0(f)$ is the least $J$ such that every $\supp f_j$ lies at $\ell^\infty$-distance greater than $1$ from $\R^3\setminus\CJ$, and every $z\in\R^n$, the limit \[ \lim_{k\to\infty}\avK{e^{\sum_j z_j\Fcurv{f_j}}} \] exists and lies in $(0,\infty)$. \end{mainthm} Proved in \cref{sec:k01} (\cref{thm:k01:EC}), from King's renormalisation-group analysis of the cube. Of \cref{hyp:s03:EC} only the existence of the fixed-cube limit at real coefficients is used: no constants clause, no uniformity in $J$ and no locally uniform convergence in $z$ enter from it, and every bound used on the limit is proved here, with absolute constants, in Theorem~\ref{thm:s05:CK}. It is the existence clause of King's Theorem~2.1 \cite{K2} at zero scalar source; the mass clause of \cite[Prop.~3.1]{Ba2}, upstream of it, is discussed in Remark~\ref{rem:s14:BALm} and is not a condition of this paper. \begin{mainthm}[King's cube limits with scalar sources]\label{hyp:s03:W1} Let the couplings lie in the coupling region of Definition~\ref{hyp:s03:T3} (the sign clause \Tthree, under which Proposition~\ref{prop:s10:U} provides the extensive bound \Itwo\ used in the proof), let $z=a^\sharp$ be the reference scheme and let $s_W\defeq s_1$ be the amplitude of Proposition~\ref{prop:s10:U}. For every finite family of real $2$-forms $f_j\in C^\infty_c(\R^3;\Lambda^2)$ and real functions $h_i\in C^\infty_c(\R^3)$, every $J\ge J_0(f,h)$, the least $J$ such that every $\supp f_j$ and every $\supp h_i$ lies at $\ell^\infty$-distance greater than $1$ from $\R^3\setminus\CJ$, and every real coefficient vector $(z,w)$ whose positive scalar part $k_+\defeq\big(\sum_iw_ih_i\big)_+$ satisfies $2\norm{k_+}_\infty{\raggedright\arraybackslash}p{0.25\textwidth}>{\raggedright\arraybackslash}p{0.14\textwidth}>{\raggedright\arraybackslash}p{0.33\textwidth}>{\raggedright\arraybackslash}p{0.20\textwidth}} \caption{What is proved where. ``Uses'' names what each result uses from the preceding rows and from the literature; a result taken from the renormalisation-group analysis is stated, with its proof, in the section named with it.}\label{tab:intro:roadmap}\\ \toprule Result & Section & Uses & Where proved\\ \midrule\endfirsthead \toprule Result & Section & Uses & Where proved\\ \midrule\endhead \multicolumn{4}{l}{\emph{The existence part}}\\ \midrule \relax(1) Existence of King's cube limits, curvature sources & Section~\ref{sec:k01} & Ba{\l}aban's block-factor-$5$ estimates on tori \cite{Ba1,Ba2,Ba3,Ba4}, transferred to King's cube (statements and proofs in Section~\ref{sec:k01}); King's comparison of two spacings at a common block scale \cite[Sect.~3.1]{K2} & Theorem~\ref{thm:k01:EC}; ratio form Theorem~\ref{thm:k01:CA}\\ \relax(1$'$) Cube limits with scalar sources, reference scheme, below the tube threshold & Section~\ref{sec:k02} & as (1), with the scheme $c$-number carried along; the sign clause $\Tthree$ (through the extensive bound of (6a)); King's localisation bound \cite[(3.38)]{K1} & Theorem~\ref{thm:k02:W1}\\ Interlacing of the two cube conventions & Section~\ref{sec:s04} & correlation inequalities \cite[Thms.~6.1, 6.2]{BFS1}, Lemma~\ref{lem:s02:BFS}; cutting (Lemma~\ref{lem:s02:cut}) & Lemma~\ref{lem:s04:G}, Corollary~\ref{cor:s04:G2}\\ \midrule \multicolumn{4}{l}{\emph{The axioms, curvature sector}}\\ \midrule \relax(2) Regularity with absolute constants & Theorem~\ref{thm:s05:CK} & Gaussian domination (Lemma~\ref{lem:s02:GD}); (1) & Theorem~\ref{thm:s05:CK} (Theorem~\ref{thm:s05:C} on the comparison convention)\\ \relax(3a) Translations, cubic group & Theorem~\ref{thm:s06:S1K} & lattice symmetries; interlacing; (2) & Theorem~\ref{thm:s06:S1K}, Corollary~\ref{cor:s06:TK}\\ \relax(3b) $O(3)$ invariance & Theorem~\ref{thm:s06:O3} & rotation theorem (rows (R1)--(R4)); Corollary~\ref{cor:s04:G2}(iii) & Theorem~\ref{thm:s06:O3}\\ \relax(4a) Reflection positivity of the modified box; two-sided sandwich & Section~\ref{sec:s07} & \cite[Thm.~6.2(1),(2)]{BFS1}; cutting; Gaussian domination & Lemma~\ref{lem:s07:RPbox}; Lemmas~\ref{lem:s07:L}, \ref{lem:s07:U}\\ \relax(4b) Reflection positivity in the limit, $\OSth$, polynomial form & Theorem~\ref{thm:s08:OS2C} & (1); (2); (4a); Montel's theorem; Lemma~\ref{lem:s02:baire} & Theorem~\ref{thm:s08:OS2C}, Propositions~\ref{prop:s08:OS3}, \ref{prop:s08:poly}\\ \relax(5) Measure, ergodic decomposition, clustering, $\OSf$ on the components & Theorem~\ref{thm:s09:S2K} & (2), (3a), (4b); (3b) for the Euclidean clauses; measure theory on spaces of distributions, ergodic decomposition and reconstruction, taken from the companion paper \HPapp\ (see Section~\ref{ssec:intro:inputs}; hypotheses established in Section~\ref{sec:s09}); \cite{OS73,OS75}, \cite[Thm.~6.1.3]{GJ}, \cite{Kec95} & Theorem~\ref{thm:s09:S2K}\\ \midrule \multicolumn{4}{l}{\emph{The Wick tube}}\\ \midrule \relax(6a) Torus theorems and the extensive bound & Section~\ref{sec:s10} & Ba{\l}aban's stability bounds at two scalar masses (Proposition~\ref{hyp:s03:T1}); the $c$-number statement (Lemma~\ref{hyp:s03:T2}); sign clause $\Tthree$; chessboard estimate \cite[Thms.~4.1--4.2]{FILS78} & Propositions~\ref{prop:s10:RP}, \ref{prop:s10:U}\\ \relax(6b) Uniform tube bound, comparison convention & Theorem~\ref{thm:s11:W} & (6a); (1$'$) one cell at a time; correlation inequalities & Theorem~\ref{thm:s11:W}, Lemma~\ref{lem:s11:TWtube}\\ Continuum subsequences and the states they define & Section~\ref{sec:s12} & cell bound, scheme bound, independence of the subsequence (Proposition~\ref{prop:s12:ID}); (3b) & Theorem~\ref{thm:s12:LIM}, Corollary~\ref{cor:s12:KK}\\ \relax(6c) Wick tube on King's convention: bound, fixed-cube limits, axioms & Theorem~\ref{thm:s13:TWK} & (6b); interlacing; (1$'$) & Theorem~\ref{thm:s13:TWK}, Proposition~\ref{prop:s13:WK}\\ \midrule \multicolumn{4}{l}{\emph{The rotation part}}\\ \midrule \relax(R0) King's state on curvature sources (the object) & Section~\ref{rot:sec:wick} & (1); Lemma~\ref{lem:s04:G}; \cite[Thm.~3.2]{K2} in the form King's proof proves & Theorem~\ref{rot:thm:W}, Definition~\ref{rot:def:object}\\ \relax(R1) Torus comparison on walled tori, two lattices at one spacing & Section~\ref{rot:sec:K} & corrected form of \cite[Thm.~2.4]{K2}: one-sided forward bounds on each torus (statement and proof in Section~\ref{rot:sec:K}) with the large-field clause of King's upper bound; an involution exchanging the lattices & Theorem~\ref{rot:hyp:K}; Theorems~\ref{rot:thm:sandwich}, \ref{rot:thm:CT}\\ \relax(R2) Heavy-shell estimate & Section~\ref{rot:sec:shell} & four estimates on King's block-spin flow with the shell mass inside the matter covariance (statements and proofs in Section~\ref{rot:sec:shell}); Ba{\l}aban's bounds on his propagator domains & Theorem~\ref{rot:thm:shell}\\ \relax(R3) King's state as the lower envelope of the walled tori & Section~\ref{rot:sec:env} & (R2); (R1); replacement of the photon beyond a dead shell by King's Neumann photon & Theorem~\ref{rot:hyp:TIGHT}, Theorem~\ref{rot:thm:E}\\ \relax(R4) Invariance under the generators; closure to $O(3)$ and to the Euclidean group & Section~\ref{rot:sec:rotinv} & (R1), (R3); continuity lemma for convex functionals (Lemma~\ref{lem:s02:baire}); translations (3a); right-continuity in the scalar mass at the exceptional set & Theorems~\ref{rot:thm:B}, \ref{rot:thm:C}, \ref{rot:thm:curv}; Theorem~\ref{rot:thm:main}\\ King's state with scalar sources, Euclidean invariance on a tube & Section~\ref{rot:sec:gensrc} & cube limits with scalar sources (Section~\ref{sec:k02}); (R1)--(R4) at the scalar source & Theorem~\ref{rot:thm:mainGS} (not used by Theorem~\ref{thm:s13:main})\\ \midrule \multicolumn{4}{l}{\emph{The main theorem and the couplings}}\\ \midrule \relax(7) Coupling range: King's range for the curvature leg; $\Tthree$ on explicit regions & Section~\ref{sec:s14} & the $c$-number statement (Lemma~\ref{hyp:s03:T2}); two-loop majorants of the lattice mass counterterm (statements and proofs in Section~\ref{sec:s14}); Reisz's power counting \cite{Rei88,Rei88b} & Lemma~\ref{lem:s14:CW}, Proposition~\ref{prop:s14:T3}\\ Main theorem & Theorem~\ref{thm:s13:main} & rows (1)--(7) & Theorem~\ref{thm:s13:main}\\ \bottomrule \end{longtable} \end{small} \space% % % % \section{The two cube conventions: interlacing and margins}\label{sec:s04} King's state lives on King's convention $\NDK_{k,J}$ (\cref{def:s02:Kcube}); the rotation part and the Wick-tube estimates of this paper use the comparison convention $\NDa_{k,J}$ (\cref{def:s02:acube}). We show that the two families are interlaced in $J$ (\cref{lem:s04:G}), draw the consequences for the limits (\cref{cor:s04:G1,cor:s04:G2}), and show that the comparisons of a cube with a reflected cube, a larger box and a tiling by cells carry over to King's convention, with the margins of the comparison-convention statements unchanged (\cref{lem:s04:C,lem:s04:D,lem:s04:E}). \subsection{The two conventions and the support condition}\label{conv:v065_s04_interlacing:1} Throughout, $\eps=\eps_k=5^{-k}$, $\XJ=\CJ\cap\Lk=\{-MJ,\dots,MJ-\eps\}^3$, $\TJ=\{x\in\XJ:\ x_\mu=MJ-\eps\text{ for some }\mu\}$ and $\alive^K_J=\XJ\setminus\TJ=\{-MJ,\dots,MJ-2\eps\}^3$. Both states carry the same photon variables $A_\alpha(x)$, $x\in\XJ$, with the Neumann form of $\XJ$, so their photon Gaussians are identical and every plaquette observable is the same function of the shared variables. They differ exactly by the matter on $\TJ$: alive in the comparison convention (scalar Dirichlet planes $x_\mu=-MJ-\eps$, $x_\mu=MJ$), killed in King's (planes $x_\mu=-MJ-\eps$, $x_\mu=MJ-\eps$, $2MJ$ apart, a multiple of every block scale; cf.\ \cref{def:s02:Kcube}). A bond with one end alive and the other killed or outside the site set contributes its diagonal part $\frac\eps2r_x^2$ at the alive end, so an alive site carries six diagonals in either convention. In the proof of his Theorem~3.2 and in his box form (3.38), King \cite{K2} works in the comparison convention: every site of the region is alive there. For $(f,g)$ with $f$ a real $2$-form and $g\ge0$ a scalar, both in $C^\infty_c$, we write as in \cref{def:s02:sources} $\Theta(\mathrm if,g)=e^{\mathrm iF(f)+W(g)}$ (class $\cI$), $\Theta(f,-g)=e^{F(f)-W(g)}$ (class $\cD$), $W(g)=\sum_x\eps^3g(x)(|\phi(x)|^2-2\sW_\eps)$ (free Wick constant), and $Z^a_{k,J}(\mathrm if,g)=e^{-a_k\sS(g)}\av{\Theta(\mathrm if,g)}$, $Z^a_{k,J}(f,-g)=e^{a_k\sS(g)}\av{\Theta(f,-g)}$, $\sS(g)=\sum_x\eps^3g(x)$; a superscript $K$ ($Z^{a,K}_{k,J}$) marks King's convention. All these expectations are real. The sources lie inside $\XJ$ if \begin{equation}\label{eq:s04:S}\tag{S} \begin{gathered} \text{every plaquette $p=(x;\alpha,\beta)$ of $\Lk$ with $f_{c(p)}(e_\alpha,e_\beta)\ne0$ has its four corners in $\XJ$, and}\\ \text{every site $x\in\Lk$ with $g(x)\ne0$ lies in $\alive^K_J=\XJ\setminus\TJ$.} \end{gathered} \end{equation} If $\supp f\cup\supp g\subset\{y:\dist_\infty(y,\R^3\setminus\CJ)>1\}$, as every hypothesis of this paper demands of its sources, then \eqref{eq:s04:S} holds: a site with $g(x)\ne0$ has $-MJ+11$ from $\R^3\setminus\CJ$ and its corners within $\eps/2\le\frac12$ of it, hence in $\XJ$. Also \eqref{eq:s04:S} for $X_{J_0}$ implies \eqref{eq:s04:S} for every $J\ge J_0$. \subsection{Cut-outs and the interlacing lemma} \begin{lemma}[cut-outs from a box with killed sites]\label{lem:s04:G3} Let $\av{\cdot}_Y$, $Y\defeq(V,T)$, be the state of \cref{def:s02:acube} on a finite set $V\subset\Lk$ of sites with the matter killed on $T\subset V$, and $X\subset V$ any subset. Let $(f,g)$, $g\ge0$, satisfy \textup{(S$_X$)}: every plaquette with $f\ne0$ has its four corners in $X$ and every site with $g\ne0$ lies in $X$ ($g$ on $X\cap T$ is immaterial). Then deleting from $\av{\cdot}_Y$ the photon gradient terms coupling $X$ to $V\setminus X$ and the cosine parts of the hopping terms between $X$ and $V\setminus X$ yields a product whose $X$-factor is $\av{\cdot}^{(a),X\cap T}_X$, the state of \cref{def:s02:acube} on $X$ with the matter killed on $X\cap T$; consequently \[ \av{\Theta(\mathrm if,g)}^{(a),X\cap T}_X\ \le\ \av{\Theta(\mathrm if,g)}_Y,\qquad \av{\Theta(f,-g)}^{(a),X\cap T}_X\ \ge\ \av{\Theta(f,-g)}_Y, \] and the same for $Z^a$. In particular, \textup{(a)} if $X\cap T=\emptyset$ the factor is $\av{\cdot}^{(a)}_X$; \textup{(b)} if $X=\prod_\mu\{a_\mu,\dots,b_\mu\}$ is a box and $X\cap T=\{x\in X:\ x_\mu=b_\mu\text{ for some }\mu\}$, the factor is the state of King's convention on $X$. \end{lemma} \begin{proof} Each deletion is in the cone direction: deleting a photon gradient term does not increase $\av{\Theta_{F,g}}$ and does not decrease $\av{\Theta^-_{F,g}}$ (\cref{lem:s02:BFS}(a),(d)), and setting a hopping coefficient $c_{xy}$ to $0$ likewise (\cref{lem:s02:BFS}(b),(d)); every intermediate measure is a BFS measure. After all deletions nothing couples the $X$-variables to the rest, so the measure is a product (\cref{lem:s02:BFS}(e)). In the $X$-factor the photon is the Neumann photon of $X$; a site of $X\cap T$ carries its point mass; an alive $x\in X\setminus T$ carries the single-site weight times $\frac\eps2r_x^2$ for each of its six bonds, since in $\av{\cdot}_Y$ every bond at an alive $x$ contributes $\frac\eps2r_x^2$ at $x$ whether its other end is alive in $V$ (the diagonal part of $\frac\eps2|e^{\mathrm ie\eps A}\phi_y-\phi_x|^2$), killed, or outside $V$ ($\phi=0$ there), and the cut keeps all diagonals; the cosines are those of the bonds within $X$ between alive sites. That is the state of \cref{def:s02:acube} on $X$ killed on $X\cap T$. By (S$_X$) the observable is a function of the $X$-factor's variables (at $x\in X\cap T$ the factor $e^{\eps^3g(x)r_x^2}$ is $1$ in both states), and the $c$-numbers $e^{\mp2\sW_\eps\sS(g)}$, $e^{\mp a_k\sS(g)}$ are the same positive constants on both sides (\cref{lem:s02:alg}(a); the counterterm is site-independent). Hence $\av{\Theta}^{(a),X\cap T}_X$ is the expectation of $\Theta$ in the fully deleted measure, which is $\le\av{\Theta}_Y$ on $\cI$ and $\ge$ on $\cD$. \end{proof} The clause of \cref{lem:s02:cut} that $X$ and its neighbours lie in one fundamental cell is a torus clause; for a box it has no content, since a bond leaving $V$ already carries its Dirichlet diagonal in $\av{\cdot}_Y$. Every sandwich $C_{J-1}\subset\cdot\subset C_{J+1}$ of this paper, read in King's convention, is an instance of \cref{lem:s04:G3}. \begin{lemma}[interlacing of the two cube families]\label{lem:s04:G} Fix $k\ge0$, $J\ge1$ and $(f,g)$, $g\ge0$, satisfying \eqref{eq:s04:S} for $\XJ$. Then \[ \begin{gathered} \avK{\Theta(\mathrm if,g)}\ \le\ \ava{\Theta(\mathrm if,g)}\ \le\ \langle\Theta(\mathrm if,g)\rangle^{K}_{k,J+1},\\ \avK{\Theta(f,-g)}\ \ge\ \ava{\Theta(f,-g)}\ \ge\ \langle\Theta(f,-g)\rangle^{K}_{k,J+1}, \end{gathered} \] and the same for the scheme functionals $Z^a_{k,J}(\mathrm if,g)$, $Z^a_{k,J}(f,-g)$ of the three states. \end{lemma} \begin{proof} \emph{Step 0 (one observable, one multiplier).} By \eqref{eq:s04:S} the plaquettes with non-zero coefficient have all corners in $\XJ$, whose photon variables are present in all three states, so $F(f)$ is one real linear functional $F$ of the shared variables; the sites with $g\ne0$ lie in $\alive^K_J$, alive in all three states; $\sW_\eps$, $\dm_\eps$, $\sS(g)$ are the same for the three regions. Hence, with $g'_x:=\eps^3g(x)\ge0$, $\Theta(\mathrm if,g)=e^{-2\sW_\eps\sS(g)}\Theta_{F,g'}$ and $\Theta(f,-g)=e^{2\sW_\eps\sS(g)}\Theta^-_{F,g'}$, where $\Theta_{F,g'}=e^{\mathrm iF}e^{\sum_xg'_xr_x^2}$, $\Theta^-_{F,g'}=e^{F}e^{-\sum_xg'_xr_x^2}$ are the observables of \cref{lem:s02:BFS}, and the scheme multiplier $e^{\mp a_k\sS(g)}$ is one positive constant for all three states (\cref{lem:s02:alg}(a)). It suffices to treat $\av{\Theta_{F,g'}}$, $\av{\Theta^-_{F,g'}}$. \emph{Step 1 (kill step on $\cI$).} For $s\ge0$ let $\av{\cdot}_s$ be $\ava{\cdot}$ with the single-site measure at each $x\in\TJ$ multiplied by $e^{-s\eps^3r_x^2}$, a BFS measure, and $\varphi(s):=\av{\Theta_{F,g'}}_s=N(s)/D(s)$, $N(s)=\int\Theta\,e^{-s\eps^3\sum_{\TJ}r_x^2}d\mu^{(0)}$, $D(s)=\int e^{-s\eps^3\sum_{\TJ}r_x^2}d\mu^{(0)}$, $\mu^{(0)}$ the unnormalised measure at $s=0$. The $s$-derivative of either integrand is bounded in modulus by $\eps^3\sum_{\TJ}r_x^2$ times the integrand at $s=0$, integrable against the quartic weight $e^{-\lambda\eps^3r^4}$ and the Gaussian; so differentiation under the integral sign is justified, $D(s)>0$, and \[ \varphi'(s)=-\eps^3\sum_{x\in\TJ}\big(\av{\Theta\,r_x^2}_s-\av{\Theta}_s\av{r_x^2}_s\big)=-\eps^3\sum_{x\in\TJ}\av{\Theta;r_x^2}_s\ \le\ 0 \] by \cref{lem:s02:BFS}(c). So $\varphi(s)\le\varphi(0)=\ava{\Theta}$. Since $\Theta_{F,g'}$ is continuous with $|\Theta_{F,g'}|=e^{\sum_xg'_xr_x^2}$ and $\ava{\cdot}$ is of King type, \cref{lem:s02:Kprime} gives $\varphi(s)\to\av{\Theta}_{\TJ}$, the state with the matter killed on $\TJ$, which is $\avK{\cdot}$. Hence $\avK{\Theta}\le\ava{\Theta}$. \emph{Step 2 (cut step).} Apply \cref{lem:s04:G3} with $V:=X_{J+1}$, $T:=T_{J+1}$ (so $\av{\cdot}_Y=\langle\cdot\rangle^K_{k,J+1}$), $X:=\XJ$: $\XJ\subset X_{J+1}$, and $\XJ\cap T_{J+1}=\emptyset$ since every site of $T_{J+1}$ has a coordinate $M(J+1)-\eps\ge MJ>MJ-\eps$. (At $k=0$, $M=1$ the outward neighbours of $\XJ$ lie on $T_{J+1}$ and are killed in $\av{\cdot}_Y$; harmless, since the bond from an alive $x\in\XJ$ to such a neighbour already carries $\frac\eps2r_x^2$ at $x$, the Dirichlet diagonal of the comparison cube.) By \cref{lem:s04:G3}(a) the $\XJ$-factor is $\ava{\cdot}$, and \eqref{eq:s04:S} is (S$_{\XJ}$): $\ava{\Theta(\mathrm if,g)}\le\langle\Theta(\mathrm if,g)\rangle^K_{k,J+1}$, $\ava{\Theta(f,-g)}\ge\langle\Theta(f,-g)\rangle^K_{k,J+1}$. \emph{Step 3 (kill step on $\cD$).} With $\av{\cdot}_s$ as in Step~1 and $\varphi^-(s):=\av{\Theta^-_{F,g'}}_s$ (differentiation justified as before, now with $|\Theta^-|\le e^{F(A)}$ integrable against the Gaussian), $(\varphi^-)'(s)=-\eps^3\sum_{x\in\TJ}\allowbreak\av{\Theta^-;r_x^2}_s\ge0$ by \cref{lem:s02:BFS}(d), so $\varphi^-(s)\ge\varphi^-(0)=\ava{\Theta^-}$; as $|\Theta^-_{F,g'}|\le e^{F(A)}$ with $F$ real linear, \cref{lem:s02:Kprime} (with $\ell=F$, $C=0$) gives $\varphi^-(s)\to\avK{\Theta^-}$. Hence $\avK{\Theta(f,-g)}\ge\ava{\Theta(f,-g)}$. \emph{Step 4 ($Z^a$).} The multiplier of Step~0 is one positive constant. \end{proof} Nothing in the proof uses a reflection symmetry of either cube (neither has one), the counterterm beyond its site-independence, or the renormalisation group. Since in every kill step of this section the observable does not depend on the fields at the killed sites (its factor there is $1$), each such step also follows with no limit: set $c_{xy}=0$ on every bond at the set $U$ to be killed, one bond at a time (\cref{lem:s02:BFS}(b),(d)); the measure is then a product (\cref{lem:s02:BFS}(e)) whose $U$-factor cancels, and the other factor is the state with $U$ killed. \subsection{Monotonicity of the K family and the limits} \begin{corollary}[the K family is $J$-monotone]\label{cor:s04:G1} Under \eqref{eq:s04:S} for $\XJ$: $\avK{\Theta(\mathrm if,g)}\le\langle\Theta(\mathrm if,g)\rangle^K_{k,J+1}$ and $\avK{\Theta(f,-g)}\ge\langle\Theta(f,-g)\rangle^K_{k,J+1}$, and the same for $Z^{a,K}$; hence for all $J\ge J_0$ whenever the sources satisfy \eqref{eq:s04:S} for $X_{J_0}$. \end{corollary} \begin{proof} Combine the two inequalities of \cref{lem:s04:G}. (For the comparison family this is \cref{lem:s04:G3}, part (a), with $V=X_{J+1}$, $X=\XJ$, \ie the first inequality of \cref{lem:s02:cut}.) \end{proof} The proof of Theorem~3.2 of \cite{K2} first replaces the $J$-dependent Dirichlet Wick subtraction (2.13) by the free, $J$-independent one and then proves the monotonicity in $J$ in the comparison convention (\cref{lem:s04:G3}(a)); by \cref{cor:s04:G1} that statement holds between cubes of King's convention too, with the kill step (Step~1), which is absent there. The form with the Dirichlet constant follows in neither convention and is not used here; at $g=0$ no Wick constant enters. \begin{corollary}[the limits]\label{cor:s04:G2} Let $f=(f_1,\dots,f_N)$ be real $2$-forms in $C^\infty_c$ with $\dist_\infty(\supp f_j,\R^3\setminus C_{J_0})>1$; for $J\ge J_0$, $z\in\C^N$ put $a^K_{k,J}(z):=\avK{e^{\sum_jz_jF(f_j)}}$, $a^{(a)}_{k,J}(z):=\ava{e^{\sum_jz_jF(f_j)}}$, $g(z):=\sum_j(\Re z_j)f_j$. At the cone points $z=\mathrm it$, $t\in\R^N$, both are real with modulus $\le1$, and \eqref{eq:s04:S} holds for $\XJ$. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item \textup{(One $J$-limit at fixed $k$; no hypothesis.)} For every $k$ and $t$, $J\mapsto a^K_{k,J}(\mathrm it)$ and $J\mapsto a^{(a)}_{k,J}(\mathrm it)$ are nondecreasing on $J\ge J_0$, bounded by $1$ and interlaced; hence $\lim_Ja^K_{k,J}(\mathrm it)=\lim_Ja^{(a)}_{k,J}(\mathrm it)\in[-1,1]$. The same holds for $\av{\Theta(\mathrm if,g)}$, $g\ge0$, with a common limit in $(-\infty,\infty]$. \item \textup{(King's cube limits.)} By \cref{hyp:s03:EC}, posed on King's convention, for every such family, every $J\ge J_0$ and every $t\in\R^N$, $\lim_ka^K_{k,J}(t)$ exists in $(0,\infty)$. Then: ($\alpha$) $a^K_{k,J}\to\ZKJ$ locally uniformly on $\C^N$, $\ZKJ$ is entire and $|\ZKJ(z)|\le e^{\norm{g(z)}_2^2}$; ($\beta$) $\ZKJ\to\ZKinf$ locally uniformly on $\C^N$, $\ZKinf$ is entire and $|\ZKinf(z)|\le e^{\norm{g(z)}_2^2}$; ($\gamma$) for all $J\ge J_0$ and $t\in\R^N$, \[ \ZKJ(\mathrm it)\ \le\ \liminf_ka^{(a)}_{k,J}(\mathrm it)\ \le\ \limsup_ka^{(a)}_{k,J}(\mathrm it)\ \le\ Z^K_{J+1}(\mathrm it), \] hence $\lim_J\liminf_ka^{(a)}_{k,J}(\mathrm it)=\lim_J\limsup_ka^{(a)}_{k,J}(\mathrm it)=\ZKinf(\mathrm it)$. \item \textup{(The subsequential limits of the comparison family.)} Let $\mathbf{k}=(k_m)$ be an increasing sequence of spacings along which $a^{(a)}_{k_m,J}\to Z^{\mathbf{k}}_J$ locally uniformly on $\C^N$ for every $J\ge J_0$ and every such family (a continuum subsequence of the comparison family, \cref{def:s12:contsub}; they exist with no hypothesis, \cref{lem:s12:LIM0}). Then $Z^{\mathbf{k}}_J\to\ZKinf$ locally uniformly on $\C^N$ as $J\to\infty$: every subsequential fixed-cube limit of the comparison family, followed by $J\to\infty$, is King's state. In particular, if the fixed-cube limits of the comparison family exist along the full sequence, $\lim_J\lim_ka^{(a)}_{k,J}=\ZKinf$. \end{enumerate} \end{corollary} \begin{proof} (i) Monotonicity is \cref{cor:s04:G1} for the K family and \cref{lem:s04:G3}(a) for the comparison family; \cref{lem:s04:G} interlaces them; two nondecreasing sequences with $a_J\le b_J\le a_{J+1}$ have the same limit, finite or $+\infty$. (ii)($\alpha$) The Montel argument, run on the K family. (1) $a^K_{k,J}$ is entire (Gaussian decay in $A$, $e^{-\lambda\eps^3r^4}$ decay in $r$ at the alive sites). (2) The local bound of \cref{lem:s05:C2} holds for the K family with the same constant: $\Re\sum_jz_jF(f_j)=F(g(z))$, so $|a^K_{k,J}(z)|\le\avK{e^{F(g(z))}}$; \cref{lem:s02:GD}(a) with $s=1$ and $v:=d^*_kg(z)$ (defined by $F_{\Lk}(g(z))=\ip{v}{A}_k$) applies to any BFS measure whose Gaussian has covariance $C$ and gives $\av{e^{\ip{v}{A}}}\le e^{\ip{v}{Cv}/2}$; the K cube is a BFS measure whose Gaussian is the Neumann photon Gaussian of $\XJ$, the same as the comparison cube's, so its covariance is the $C$ of \cref{lem:s05:C1}, a statement about this $C$ and the plaquettes with corners in $\XJ$: $\ip{v}{Cv}_k\le2\norm{g(z)}_{2,k}^2$. Hence $|a^K_{k,J}(z)|\le e^{\norm{g(z)}_{2,k}^2}\le e^{3C_0\fournorm{g(z)}^2}$ uniformly in $k\ge0$, $J\ge J_0$, by (E3) of \cref{lem:s02:E}: the plaquette centres of each of the three orientations form a translate of $\eps_k\Z^3$, each contributing at most $C_0\fournorm{g}^2$ to $\norm{g}_{2,k}^2$ by (E1), and the three orientations give $3C_0$. Nothing in this argument sees the matter beyond the state being a BFS measure. (3) \Cref{hyp:s03:EC} gives convergence at every real point. (4) $\R^N$ is a set of uniqueness for entire functions on $\C^N$; by Montel's theorem every subsequence has a locally uniformly convergent sub-subsequence, all limits are entire and agree on $\R^N$, hence everywhere \cite{Hor90}. So $a^K_{k,J}\to\ZKJ$ locally uniformly, $\ZKJ$ is entire, and the bound passes to the limit, $|\ZKJ(z)|\le\lim_ke^{\norm{g(z)}_{2,k}^2}=e^{\norm{g(z)}_2^2}$ (\cref{lem:s02:E}), a $J$-free bound. ($\beta$) By \cref{cor:s04:G1} at every $k$ and ($\alpha$), $J\mapsto\ZKJ(\mathrm it)$ is nondecreasing and $\le1$, so $\ZKinf(\mathrm it):=\lim_J\ZKJ(\mathrm it)$ exists; $\{\ZKJ\}_{J\ge J_0}$ is locally uniformly bounded by the $J$-free bound of ($\alpha$); $(\mathrm i\R)^N$ is a maximal totally real subspace, hence a set of uniqueness for entire functions; Montel's theorem and the identity theorem give $\ZKJ\to\ZKinf$ locally uniformly, $\ZKinf$ entire, $|\ZKinf(z)|\le e^{\norm{g(z)}_2^2}$. (The same Montel argument as in ($\alpha$), with the uniqueness set $(\mathrm i\R)^N$ and the cube monotonicity of \cref{cor:s04:G1} supplying the convergence there; \cref{thm:s05:CK}(b) states it.) ($\gamma$) \cref{lem:s04:G} at every $k$, with $k\to\infty$ at the two ends (limits by ($\alpha$)) and $\liminf$, $\limsup$ in the middle, gives the sandwich; ($\beta$) gives the limits. (iii) \cref{lem:s04:G} along $\mathbf{k}$, with the full-sequence limits of ($\alpha$) at the ends, gives $\ZKJ(\mathrm it)\le Z^{\mathbf{k}}_J(\mathrm it)\le Z^K_{J+1}(\mathrm it)$, so $\lim_JZ^{\mathbf{k}}_J(\mathrm it)=\ZKinf(\mathrm it)$ by ($\beta$). $\{Z^{\mathbf{k}}_J\}_J$ is locally bounded by the $J$-free bound $e^{\norm{g(z)}_2^2}$ (\cref{lem:s05:C2} passed to the limit), so by Montel's theorem and the uniqueness set $(\mathrm i\R)^N$ it converges locally uniformly to an entire function (\cref{lem:s12:LIM1}), which agrees with $\ZKinf$ on $(\mathrm i\R)^N$, hence on $\C^N$. If the fixed-cube limits exist along the full sequence, that sequence is a continuum subsequence. \end{proof} Statements (ii)--(iii) also hold with the two conventions exchanged, for $J\ge J_0+1$, at every family and cube where the comparison family's fixed-cube limits exist at real coefficients (read \cref{lem:s04:G} as $a^{(a)}_{k,J-1}\le a^K_{k,J}\le a^{(a)}_{k,J}$; whether they exist is not decided here, see Remark~\ref{rem:s04:notfollow}); the exchanged (iii) needs \cref{lem:s12:LIM0,lem:s12:LIM1} for the K family, whose proofs use only the local bound and the photon-only bound $\av{F(u-u')^2}\le2\norm{u-u'}_{2,k}^2$, both transferred by the covariance identity in ($\alpha$). \begin{remark}[what does not follow]\label{rem:s04:notfollow} At fixed $J$, \cref{lem:s04:G} yields exactly ($\gamma$) and nothing on whether $\lim_ka^{(a)}_{k,J}(\mathrm it)$ exists: the bounds $\ZKJ(\mathrm it)$, $Z^K_{J+1}(\mathrm it)$ differ in general (the cubes differ by $M$, so the gap does not shrink with $k$), and nothing constrains the oscillation of $a^{(a)}_{k,J}(\mathrm it)$ in $k$ between them. So the existence of the fixed-cube limits on one convention does not imply it on the other by these results, and likewise with scalar sources: the existence theorem must be proved on the family on which the renormalisation-group construction is carried out; the other is a comparison family with the same infinite-volume state by (ii)--(iii). No counterexample is claimed. \end{remark} \subsection{Three margins in King's convention}\label{conv:v065_s04_interlacing:2} Let $\sigma_3=\diag(1,1,-1)$, $X:=\XJ$, $X_+:=\sigma_3X=(\CJ+\eps e_3)\cap\Lk$ (third coordinates $\{-MJ+\eps,\dots,MJ\}$), $\sigma_3\TJ=\{x\in X_+:\ x_1=MJ-\eps\text{ or }x_2=MJ-\eps\text{ or }x_3=-MJ+\eps\}$ and $A_\sigma:=\sigma_3(\alive^K_J)=\{x:\ x_1,x_2\in\{-MJ,\dots,MJ-2\eps\},\ x_3\in\{-MJ+2\eps,\dots,MJ\}\}$. \begin{lemma}[the reflection sandwich]\label{lem:s04:C} Let $(f,g)\in\cK$ be supported at $\ell^\infty$-distance $>1$ from $\R^3\setminus C_{J_0}$. \textup{(a)} If $M\ge2\eps_k$, which the standing choice $M\in50\N$ (\cref{conv:v065d_s02_setting:1}) guarantees for every $k$, then for every $J\ge J_0+1$ \[ Z^{a,K}_{k,J-1}(\mathrm if,g)\ \le\ Z^{a,K}_{k,J}(\mathrm i\,\sigma_3\!\cdot\!f,\sigma_3\!\cdot\!g)\ \le\ Z^{a,K}_{k,J+1}(\mathrm if,g). \] \textup{(b)} For every $k\ge0$, $M\ge1$ and $J\ge J_0+2$ the same holds with $J\mp2$ in place of $J\mp1$. \textup{(c)} Were $M=1$ allowed, at $k=0$ the one-step sandwich \textup{(a)} would not be obtainable by deletions and kills: $A_\sigma$ contains alive sites ($x_3=MJ$) killed in King's cube of index $J+1$, and $\alive^K_{J-1}$ contains sites ($x_3=-MJ+\eps$) killed in the reflected state; this is why the condition in \textup{(a)} is stated. \end{lemma} \begin{proof} \emph{Step 1 (transport).} By \cref{lem:s06:PK}, $Z^{a,K}_{k,J}(\mathrm i\sigma_3\!\cdot\!f,\sigma_3\!\cdot\!g)=e^{-a_k\sS(g)}\av{\Theta(\mathrm if,g)}_R$, with $R$ the state on $X_+$ for the frame $(e_1,e_2,-e_3)$ killed on $\sigma_3\TJ$ ($\sS(\sigma_3\cdot g)=\sS(g)$). In the frame $(e_\alpha)$, as in \cref{lem:s02:refl}, $R$ is the BFS measure with $A_1,A_2$ on $X_+$ and $A_3$ on $X_+-\eps e_3=X$ (gradient pairs within these sets, masses on them), matter alive exactly on $A_\sigma$ with point masses on $\sigma_3\TJ$ and $\phi=0$ outside $X_+$, all six diagonals at each alive site, and the standard cosine on every bond with both ends in $A_\sigma$ (the flipped bond with $A'_3(x)=-A_3(x-\eps e_3)$ gives the standard cosine of $\langle x-\eps e_3,x\rangle$); $F(f)$, $W(g)$ are the standard ones (the plaquettes $(x;\alpha,3)$ pick up two sign changes and keep their centres). $R$ is of King type. The sources satisfy \eqref{eq:s04:S} for $X_{J-1}$ (resp.\ $X_{J-2}$), so by the inclusions below Step~0 of \cref{lem:s04:G} applies to all states compared. \emph{Step 2 (upper end).} If $A_\sigma\subset\alive^K_{J'}$ and $X_+\cup X\subset X_{J'}$: from $\langle\cdot\rangle^K_{k,J'}$ kill every site of $\alive^K_{J'}\setminus A_\sigma$ (\cref{lem:s02:BFS}(c), \cref{lem:s02:Kprime}, which allows the already killed $T_{J'}$), leaving exactly the cosines of bonds within $A_\sigma$, \ie $R$'s; then delete every photon gradient term except the pairs within $X_+$ for $A_1,A_2$ and within $X$ for $A_3$ (\cref{lem:s02:BFS}(a); these are gradient pairs of $X_{J'}$). The result is a product (\cref{lem:s02:BFS}(e)) whose factor on $R$'s variables is $R$ (photon, alive set with all six diagonals by the bookkeeping of \cref{lem:s04:G3}, cosines); the other factor cancels. Hence $\av{\Theta(\mathrm if,g)}_R\le\langle\Theta(\mathrm if,g)\rangle^K_{k,J'}$. The inclusions hold iff $MJ\le MJ'-2\eps$ (direction $3$; the rest is trivial): for $J'=J+1$ iff $M\ge2\eps$, for $J'=J+2$ always. \emph{Step 3 (lower end).} If $\alive^K_{J''}\subset A_\sigma$ and $X_{J''}\subset X\cap X_+$: from $R$ kill every site of $A_\sigma\setminus\alive^K_{J''}$ (\cref{lem:s02:BFS}(c), \cref{lem:s02:Kprime}) and delete every photon gradient term coupling an $X_{J''}$-variable to another variable of $R$ (\cref{lem:s02:BFS}(a)). The $X_{J''}$-factor has the Neumann photon of $X_{J''}$ (gradient pairs within $X_+\cap X_{J''}=X\cap X_{J''}=X_{J''}$) and the matter alive on $\alive^K_{J''}$ with all diagonals, killed on $T_{J''}$: it is $\langle\cdot\rangle^K_{k,J''}$; the rest cancels. Hence $\langle\Theta(\mathrm if,g)\rangle^K_{k,J''}\le\av{\Theta(\mathrm if,g)}_R$. The inclusions hold iff $-MJ''\ge-MJ+2\eps$: for $J''=J-1$ iff $M\ge2\eps$, for $J''=J-2$ always. \emph{Step 4.} Steps 2--3 with $J\pm1$ give (a), with $J\pm2$ give (b); the multipliers agree (\cref{lem:s02:alg}(a)). For (c): at $k=0$, $M=1$ the top of $A_\sigma$ in direction $3$ is $J$ while that of $\alive^K_{J+1}$ is $J-1$, and the bottom of $\alive^K_{J-1}$ is $-J+1=-MJ+\eps$, the killed layer of $R$; deletions and kills never revive a killed site. \end{proof} In the curvature sector ($g=0$) the limit sentences of \cref{thm:s06:S1K} run on (a) or on (b), for all $k$, with $k\to\infty$ by \cref{cor:s04:G2}(ii) and $J\to\infty$ by \cref{cor:s04:G1}; with scalar sources they need the K-family tube limits, established in \cref{thm:s13:TWK} (\cref{rem:s04:wick}). \begin{lemma}[the upper box needs no change]\label{lem:s04:D} Let $\Omega\subset B$ be lattice boxes at spacing $\eps$ such that $B$ contains every site within $\ell^\infty$-distance $N$ lattice steps of $\Omega$, $N\ge1$ (the hypothesis of \cref{lem:s07:U}, whose mass condition forces $N\ge1$), and let $\av{\cdot}^K_B$ be the state of King's convention on $B$ (killed on its top face layers $T_B$). Then \textup{(a)} $\Omega\cap T_B=\emptyset$; \textup{(b)} the second part of \cref{lem:s07:U} holds with $\av{\cdot}^K_B$ in place of the comparison state on $B$, with the same proof, and its first part is a photon statement, unchanged; \textup{(c)} consequently the sandwich in the proof of \cref{thm:s08:OS2C} holds for King's cubes with the cubes and margins of \eqref{eq:s07:margin}: for every $k$ and every $\zeta$ admitted there, $Z^{a,K}_{k,J}(\zeta)\le\Phi_k(\zeta)\le Z^{a,K}_{k,J+2n}(\zeta)$, $\Phi_k$ the functional of the box state of \cref{def:s07:box} on $\Omega_{J,k}$; the lower end is \cref{lem:s07:L} with Step~1 of \cref{lem:s04:G} prepended. (By \cref{cor:s04:G1} the cube of index $J+2n+1$ would also do; it is not needed.) \end{lemma} \begin{proof} (a) For $x\in\Omega$ and each $\mu$, $x+N\eps e_\mu\in B$, so $x_\mu<\max_Bx_\mu$ and $x\notin T_B$. (b) By (a) every site of $\Omega$ is alive, so once the hopping cosines touching $B\setminus\Omega$ are removed the matter on $\Omega$ is the Dirichlet matter of $\Omega$, and the photon law of $B$ is the same in both conventions, so the proof of \cref{lem:s07:U} goes through unchanged. (c) Lower end: \cref{lem:s07:L} gives $\langle Y\rangle^{(a)}_{C_J}\le\langle Y\rangle_{\Omega_{J,k}}$ on $\cI$ (its proof lands on the Dirichlet matter of $C_J$, the comparison convention), and Step~1 of \cref{lem:s04:G} gives $\langle Y\rangle^K_{C_J}\le\langle Y\rangle^{(a)}_{C_J}$, \eqref{eq:s04:S} holding since the sources of the enlarged family and their reflections are supported at $\ell^\infty$-distance $>1$ from $\R^3\setminus C_{J_0}$, $C_{J_0}\subset C_J$. Upper end: (b) with $B=X_{J+2n}$, whose hypothesis is the second margin of \eqref{eq:s07:margin}, a statement about photon sites. The multipliers agree (\cref{lem:s02:alg}(a)). \end{proof} For the cell bound we use the cells of \cref{def:s11:cells}: $\ell_c=2ML_c\ge10$; the cells $c=\prod_\mu[w_\mu,w_\mu+\ell_c)$ of the tiling $\cT_s$, $s\in\{0,1\}^3$, $w_\mu\in W_\mu(s_\mu)$, $W_\mu(0)=\ell_c\Z$, $W_\mu(1)=ML_c+\ell_c\Z$ (so $w_\mu,w_\mu+\ell_c,MJ\in\Z$); the good sets $G_s=\{x:\dist(x_\mu,W_\mu(s_\mu))>2\ \forall\mu\}$ and cutoffs $\chi_s$ with $\supp\chi_s\subset G_s$; $X_c:=c\cap\XJ$; $J_1(R)=\min\{J:MJ\ge R+\ell_c\}$ for sources supported in $Q_R=[-R,R]^3$; $B_{\rm cell}$, defined on the comparison cell $C_{L_c}$. $N^K_{k,C_J}$ is the normalised scalar-source functional of $\avK{\cdot}$ and $N^{(a),T}_{k,X}$ that of the comparison state on $X$ killed on $T\subset X$. \begin{lemma}[the cell lemmas hold as stated]\label{lem:s04:E} \textup{(a)} For a tiling $\cT_s$ and any lattice function $q\ge0$ on $\XJ$: $N^K_{k,C_J}(0,-q)\le\prod_iN^{(a),X_i\cap\TJ}_{k,X_i}(0,-q|_{X_i})$, over the site sets $X_i=c\cap\XJ$ of the cells meeting $\CJ$. \textup{(b)} For a full cell $c\subset\CJ$: $X_c\cap\TJ\ne\emptyset$ iff $w_\mu+\ell_c=MJ$ for some $\mu$. \textup{(c)} Let $q\ge0$ be any lattice function supported in $Q_R$, $J\ge J_1(R)$, $s\in\{0,1\}^3$ and $c\in\cT_s$ with $(\chi_sq)|_{X_c}\not\equiv0$. Then $-MJ2$ from $\R^3\setminus c$. \textup{(d)} Hence \cref{lem:s11:W4,lem:s11:W5,lem:s11:W67} are unchanged and \cref{prop:s11:W8} holds for $\avK{\cdot}$ as stated, under its hypotheses and with its constant $K_-$: $\ln N^K_{k,C_J}(0,-q)\le K_-\wnorm{q}$ for every $k$, every $J\ge J_1(R)$ and every lattice $0\le q\le H$ supported in $Q_R$. \textup{(e)} (Direction.) For a cell $c$ and $\emptyset\ne T\subset X_c$: $N^{(a),T}_{k,c}(0,-\tilde q)\ge N^{(a)}_{k,c}(0,-\tilde q)$ for $\tilde q\ge0$ vanishing on $T$. So $B_{\rm cell}$, a bound on a comparison cell, does not bound a source-carrying cell of King's type, and \textup{(c)} is essential. \end{lemma} \begin{proof} (a) The proof of \cref{lem:s11:W3} deletes the cross-cell photon gradient terms and hopping cosines, each in the $\cD$ direction (\cref{lem:s02:BFS}(d)); by \cref{lem:s02:BFS}(e) and the factor identification of \cref{lem:s04:G3} with $V=\XJ$, $T=\TJ$, $X=X_i$, the product has the factors $N^{(a),X_i\cap\TJ}_{k,X_i}$; the observable $e^{-\sum\eps^3qr^2}$ and the multiplier $e^{a_k\sS(q)}=\prod_ie^{a_k\sS(q|_{X_i})}$ factorise, and at the sites of $\TJ$ the observable factor is $1$ in every state. (b) The sites of a full cell have $x_\mu\in\{w_\mu,\dots,w_\mu+\ell_c-\eps\}$, and $x\in\TJ$ iff $w_\mu+\ell_c-\eps=MJ-\eps$ for some $\mu$. (c) Let $x\in X_c$ with $(\chi_sq)(x)\ne0$. Then $x\in G_s$; since $w_\mu,w_\mu+\ell_c\in W_\mu(s_\mu)$ and $w_\mu\le x_\mu-R-\ell_c+2\ge-MJ+2$. So $c\subset(-MJ,MJ)^3$, $w_\mu+\ell_c\ne MJ$, hence $X_c\cap\TJ=\emptyset$ by (b), and $\dist_\infty(x,\R^3\setminus c)=\min_\mu\min(x_\mu-w_\mu,w_\mu+\ell_c-x_\mu)>2$. (d) In the chain of inequalities of \cref{prop:s11:W8} the Hölder step of \cref{lem:s11:W67} is an inequality within one state, convention-free; then (a) with the tiling $\cT_s$ and $q_s:=8\chi_sq$. A factor with $q_s|_{X_i}\equiv0$ equals $1$ whatever is killed on $X_i$ (this covers the clipped cells at the faces of $\CJ$ and any cell flush with them); a factor with $q_s|_{X_i}\not\equiv0$ is, by (c), the comparison-cell functional $N^{(a)}_{k,c}(0,-q_s|_c)$ of a full cell $c\subset\CJ$ with $q_s|_c$ supported at distance $>2$ from $\R^3\setminus c$ and $0\le q_s\le8H=H'$: exactly the hypotheses of \cref{lem:s11:W5} (hence of \cref{lem:s11:W4}: translation to $C_{L_c}$, $B_{\rm cell}$, \cref{prop:s11:W1prime} on the comparison box), as stated; \cref{lem:s11:W67} sums the cells. (A cell flush with a face, $w_\mu+\ell_c=MJ$, can meet $\supp q$ but never $\supp(\chi_sq)$; (a) is applied to the cutoff sources, not to $q$.) (e) Step~3 of \cref{lem:s04:G} applied to the comparison cell with $U=T$: $s\mapsto\av{\Theta^-}_s$ is nondecreasing (\cref{lem:s02:BFS}(d)) and tends to the killed state (\cref{lem:s02:Kprime}). \end{proof} \begin{remark}[one-line kill steps]\label{rem:s04:F} \Cref{prop:s11:W2} (cube $\le$ torus on $\cI$), \cref{prop:s11:W1prime} (box $\le$ cube $\le$ torus), the lower end of \cref{lem:s07:L}, and the comparison of the cube with the box state of \cref{def:s07:box} on the cone (\cref{rem:s07:K8}) are each proved for the comparison cube; for King's cube prepend Step~1 of \cref{lem:s04:G} ($\avK{\cdot}\le\ava{\cdot}$ on $\cI$), and for the $\cD$ versions (cube $\ge$ torus for negative sources) Step~3. The same applies to a box of King's type (a killed face layer) in \cref{prop:s11:W1prime}. Nothing beyond \cref{lem:s04:G} is needed. \end{remark} \begin{remark}[the Wick sector]\label{rem:s04:wick} With scalar sources $g\ge0$, \cref{cor:s04:G2}(i) holds as stated. The analogues of \cref{cor:s04:G2}(ii)--(iii) on the Wick tube need the fixed-cube limits and the $(k,J)$-uniform bound \eqref{eq:s11:W} for both families. For King's family the bound is the sequence of estimates \eqref{eq:s11:W} carried out on $\NDK$, whose convention-sensitive steps are the cube-versus-torus comparison (\cref{cor:s11:W2prime}, via \cref{rem:s04:F}) and the cell bound (\cref{lem:s04:E}); that sequence of estimates and the existence of the K-family tube limits are the content of \cref{thm:s13:TWK}, on which the tube statements for King's state rest (\cref{rem:s11:Kconvention}, \cref{prop:s13:WK}). \end{remark} \space% % \part*{The existence part} % % % % % \providecommand{\Hrel}{\ensuremath{(\mathrm{H0rel})_{C}}} \providecommand{\Kpos}{\ensuremath{(\mathrm{K}\text{-}\mathrm{pos})_{N}}} \providecommand{\BALint}{\ensuremath{(\mathrm{BAL}_{5})[\mathrm{int}]}} \providecommand{\RENorb}{\ensuremath{(\mathrm{REN}_{\mathrm{orb}})}} \providecommand{\Kh}{\ensuremath{(\mathrm{K}_{h})}} \providecommand{\Uplus}{\ensuremath{(\mathrm{U}^{+})}} \providecommand{\Bh}{\ensuremath{(\mathrm{B}^{h})}} \providecommand{\Rzero}{\ensuremath{(\mathrm{R}_{0})}} \providecommand{\UBm}{\ensuremath{(\mathrm{UB}^{-})}} \providecommand{\NEGinf}{\ensuremath{(\mathrm{NEG}_{\infty})}} \providecommand{\Zk}{\ensuremath{Z_{k}}} \providecommand{\Ik}[1]{\ensuremath{I^{(#1)}_{k}}} \providecommand{\Ihat}[1]{\ensuremath{\hat I^{(#1)}_{k}}} \providecommand{\Cl}{\operatorname{Cl}} \section{King's existence theorem on the cube}\label{sec:k01} The existence theorem for King's cube limits (\cref{hyp:s03:EC}), and its form with scalar sources that the Wick leg uses (\cref{hyp:s03:W1}), are proved in this section and the next from estimates on King's renormalisation-group integrals on the cube $\CK$. The argument is the one King gives for his Theorem~2.1 \cite[Sect.~3.1]{K2} and \cite[Sect.~3.2]{K1}: the generating functional is approximated, up to a small-field remainder, by an integral of the effective action after $j$ renormalisation steps, and two spacings whose effective actions live on the same unit lattice are compared. We write it as a Cauchy property of the \emph{normalised} functional, in a ratio form in which every field-independent constant of the effective action cancels; this removes from King's route the convergence of the vacuum energies, which the argument of \cite{K2} uses and which is not available on the cube. The results we use are of two kinds. Estimates taken from Ba{\l}aban's block-factor-$5$ analysis \cite{Ba1,Ba2,Ba3,Ba4} on tori, and their transfer to King's cube, are stated as propositions or lemmas, each followed by a description of its content and of the course of its proof, and by the definitions and auxiliary statements that the proof uses. Three of these statements are used at several places and are therefore stated once, with a precise formulation, in \cref{ssec:k01:open} (\BALint, \RENorb) and \cref{ssec:k02:setting} (\Kh). Everything else in the two sections is proved in place. \subsection{Renormalisation-group integrals on King's cube}\label{ssec:k01:setting} Fix the couplings, a cube $\CJ$ and King's RG cube $\CK$ of \cref{def:s02:Kcube}. In this section and the next the index $k$ is the spacing index ($\eps_k=5^{-k}$, as throughout), and $j$ counts renormalisation steps, $0\le j\le k-J_*$ for a fixed $J_*$ (King's ``$J$'' of \cite[Thm.~3.1]{K1}); the block scale after $j$ steps is $\eta\defeq5^{j}\eps_k=5^{j-k}\le1$, and the unit lattice after $j$ steps is $\eta^{-1}(\CJ\cap\eta\Z^3)$. Since the side $2MJ$ of $\CJ$ is an integer, the faces of $\CJ$ lie on block planes at every scale. \paragraph{Sources.} A real source is $s=(u,h)$ with $u=u_x\defeq\sum_ix_if_i$ a real combination of $2$-forms $f_i\in C^\infty_c(\R^3;\Lambda^2)$ sampled at the plaquette centres in the lattice's own frame, and $h\in C^\infty_c(\R^3)$ real; all supports lie at $\ell^\infty$-distance $>1$ from $\R^3\setminus\CJ$ (so $J\ge J_0$). Put $X(s)\defeq F(u)+\Wick{h}$, with the free Wick constant of \cref{def:s02:sources}; $X$ is real and linear in $s$. The unnormalised functional is \[ \Zk(s)\ \defeq\ \int e^{-S}\,e^{X(s)}\ \prod dA\prod d^2\phi \] over the variables of $\CK$ at spacing $\eps_k$ (any fixed, $s$-independent normalisation of the measure; it cancels everywhere below), and $N_k(s)\defeq\Zk(s)/\Zk(0)=\avK{e^{X(s)}}$. At $h=0$ we write $a_k(x)\defeq N_k(u_x,0)$, which is the $a^K_{k,J}(x)$ of \cref{cor:s04:G2}. \begin{definition}[King's block spin at zero field on the infinite lattice]\label{def:k01:bulkflow} On $\Lk=\eps_k\Z^3$ run King's block-spin averaging at $A=0$ with block factor $5$: the uniform photon average, and for the scalar the weighted average \cite[(3.1)--(3.2)]{K2} with the Dirichlet weight $f_D$ of \cite[(3.8)]{K2} at the finest step and the uniform weight above it, where $f_D$ is $5\eps_k$-periodic, equal to $(5/4)^3$ on the sites of each $5\eps_k$-block whose coordinates relative to the block's corner lie in $\{0,1,2,3\}\eps_k$ and to $0$ on the sites with a residue $4\eps_k$ in some coordinate, so that its block averages are $1$ and $\norm{f_D}_\infty=(5/4)^3$. Let $Q_j$ be the composite average onto the blocks of side $\eta=5^j\eps_k$ and $Q_j^*$ its adjoint for the inner products $\sum\eps_k^3$, $\sum\eta^3$. The fluctuation covariance after $j$ steps is \[G_j\defeq\big(-\Delta_{\eps_k}+m^2+\mathfrak a_j\eta^{-2}Q_j^*Q_j\big)^{-1},\qquad\mathfrak a_j=\mathfrak a(1-5^{-2})(1-5^{-2j})^{-1}\] \cite[(2.13)]{K1}, \cite[(3.5)]{K2}, with $\mathfrak a>0$ the stiffness of King's block-spin kernel; the free propagator splits as $C\defeq(-\Delta_{\eps_k}+m^2)^{-1}=G_j+\Gamma_j$; the photon is the same with $m^2$ replaced by $\lmu^2$, the uniform weight and $A=0$, $D=D_j+\Gamma^{\mathrm{ph}}_j$ with $D_j=\sum_{i0$ on the block lattice of $\CJ$, with $\iota^*(-\Delta_{\eps_k}+m^2)\iota\,H_j(\CK)=(Q^X_j)^*\Delta^{(j)}(\CK)$ and the stiffness $\mathfrak a_j$ of \cref{def:k01:bulkflow}; the Dirichlet weight at the finest step does not change the composite stiffness. The Dirichlet planes of $\CK$ are $2MJ$ apart, a multiple of every block side, which is what makes this flow possible on $\CK$ and not on the comparison cube (\cref{rem:k01:cube}). Let $S^{(j)}_k(A_j,\phi_j;s)$ be King's effective action after $j$ steps \cite[(3.14)]{K1}, \cite[(3.12)]{K2}, with the source vertices \cite[(3.20)--(3.21)]{K1}; let $\chi_j$ be King's small-field indicator, \cite[(3.2)]{K1} times the face factor \cite[(5.41)]{K2}, a function of $(A_j,\phi_j)$ alone; and \[ \Ik{j}(s)\ \defeq\ \int dA_j\,d\phi_j\ \chi_j\ e^{-S^{(j)}_k(A_j,\phi_j;s)} . \] The \emph{vacuum constant} $v_{k,j}\defeq S^{(j)}_k(0,0;0)$ is finite; it contains King's $E_0$, $E_1$, the normalisation constants $\ln Z_j$, $\ln Z_j(0)$ of the block-spin kernels and every field- and source-free graph. The \emph{normalised action} is $S^{(j)}_{N,k}(\cdot;s)\defeq S^{(j)}_k(\cdot;s)-v_{k,j}$ for every $s$ (the field-free, source-dependent part is kept), and $\Ihat{j}(s)\defeq\int\chi_je^{-S^{(j)}_{N,k}(\cdot;s)}=e^{v_{k,j}}\Ik{j}(s)$. The flow is a device of the proof: the state $\avK{\cdot}$ does not depend on it. \begin{lemma}[Gaussian data of the flow on $\CK$]\label{lem:k01:cubeops} Let the block-spin kernel of step $i$ at $A=0$ be $\exp(-\frac{\gamma_i}2\|\psi_i-Q_{(i)}\psi_{i-1}\|^2_{\ell_i})$ per real component, up to a field-independent constant (the flow of \cite[Sect.~2]{K1} run on $\CK$ as above), with $Q_{(1)}$ the $f_D$-weighted and $Q_{(i)}$, $i\ge2$, the uniform average (for the photon the uniform average throughout), with the same stiffnesses $\gamma_i$ on $\CK$ and on $\Lk$, and let the Gaussian part of the action at $A=0$ be $\frac12\langle\phi,\iota^*(-\Delta_{\eps_k}+m^2)\iota\phi\rangle$ per real scalar component and $\frac12\langle A_\alpha,(-\Delta^N_{\XJ}+\lmu^2)A_\alpha\rangle$ per photon component (the photon quadratic form of \cref{def:s02:Kcube}), with $\iota$, $\ell_i$ and $Q^X_j=Q_j\iota$ as above. Then $\gamma_i=\mathfrak a\ell_i^{-2}$ with the stiffness $\mathfrak a$ of \cref{def:k01:bulkflow}; the composite of $j\ge1$ steps on $\CK$ is a field-independent constant times $\exp(-\frac{\mathfrak a_j\eta^{-2}}2\|\psi_j-Q^X_j\phi\|_\eta^2)$, $\eta=\ell_j$; and the formulas for $G_j(\CK)$, $H_j(\CK)$, $\Delta^{(j)}(\CK)$, $D_i(\CK)$ and $G^{\mathrm{ph}}_{(i)}(\CK)$ displayed above hold, with $\Delta^{(j)}(\CK)>0$, $G^{\mathrm{ph}}_{(i)}(\CK)\ge0$ and $\iota^*(-\Delta_{\eps_k}+m^2)\iota\,H_j(\CK)=(Q^X_j)^*\Delta^{(j)}(\CK)$. \end{lemma} \begin{proof} The proof is the standard Gaussian computation, based on the following facts. The $\ell_i$-blocks with corners in $-MJ+\ell_i\Z^3$ tile $\XJ$ for $i\le k$ and nest $125$-to-one, because the side $2MJ$ of $\CJ$ is an integer (\cref{def:s02:Kcube}, as noted at the head of this section); so $Q_{(i)}Q_{(i)}^*=\mathbf 1$ for $i\ge2$ and $Q_i^*Q_i=\mathsf P_{\XJ,\ell_i}$ for the uniform averages. Integrating out the intermediate block field in the product of two consecutive kernels gives, up to a field-independent constant, a kernel of the same form in the composite average, the stiffnesses adding harmonically, $\Gamma_{i+1}^{-1}=\Gamma_i^{-1}+\gamma_{i+1}^{-1}$ (this uses $Q_{(i+1)}Q_{(i+1)}^*=\mathbf 1$ only). The recursion holds on $\Lk$ blockwise, and \cref{def:k01:bulkflow} gives $\Gamma_j=\mathfrak a_j\eta^{-2}$ there for every $j$, whence $\gamma_1=\mathfrak a\ell_1^{-2}$ and $\gamma_j^{-1}=\Gamma_j^{-1}-\Gamma_{j-1}^{-1}=\ell_j^2/\mathfrak a$ for $j\ge2$; the same recursion on $\CK$, with the same $\gamma_i$, returns $\Gamma_j=\mathfrak a_j\eta^{-2}$, the Dirichlet weight at the finest step changing neither the recursion (no condition on $Q_{(1)}$ enters it) nor the composite stiffness. The weight $f_D$ vanishes at the sites with a residue $4$, hence on $\TJ$, so $Q^X_j=Q_j\iota$. The form $\iota^*(-\Delta_{\eps_k}+m^2)\iota$ is positive (a function whose Dirichlet gradient form vanishes is constant and vanishes with its values on $\TJ$, and $m^2>0$), and $-\Delta^N_{\XJ}+\lmu^2\ge\lmu^2>0$. The formulas then follow by completing the square, with $\mathsf A=\iota^*(-\Delta_{\eps_k}+m^2)\iota$, $Q=Q^X_j$ and $\Gamma=\mathfrak a_j\eta^{-2}$: the fluctuation covariance is $(\mathsf A+\Gamma Q^*Q)^{-1}=G_j(\CK)$; the minimiser of the exponent at the block field $\psi$ is $\Gamma G_j(\CK)Q^*\psi=H_j(\CK)\psi$; the quadratic form left in $\psi$ is $\frac12\langle\psi,\Gamma(\mathbf 1-\Gamma QG_j(\CK)Q^*)\psi\rangle=\frac12\langle\psi,\Delta^{(j)}(\CK)\psi\rangle$, and $\Delta^{(j)}(\CK)^{-1}=\Gamma^{-1}+Q\mathsf A^{-1}Q^*$ shows $\Delta^{(j)}(\CK)>0$; and $\mathsf A\,H_j(\CK)=\Gamma(\mathsf A+\Gamma Q^*Q-\Gamma Q^*Q)G_j(\CK)Q^*=Q^*\Delta^{(j)}(\CK)$ is the identity. For the photon, with $B=-\Delta^N_{\XJ}+\lmu^2$, $\mathsf P_i=\mathsf P_{\XJ,\ell_i}$ and $\Gamma_i=\mathfrak a_i\ell_i^{-2}$, which is strictly decreasing in $i$ by the recursion, the same computation gives the covariance $D_i(\CK)=(B+\Gamma_i\mathsf P_i)^{-1}$ after $i$ steps; since the ranges of the $\mathsf P_i$ decrease with $i$, $\Gamma_{i+1}\mathsf P_{i+1}\le\Gamma_i\mathsf P_i$, so the inverses satisfy $D_{i+1}(\CK)\ge D_i(\CK)$, which gives the slices $G^{\mathrm{ph}}_{(i)}(\CK)\ge0$ and, with $D_0(\CK)=0$, $\sum_{i0$ there are $t_0(\kappa)>0$ and $C_\kappa<\infty$, depending on $\kappa$, on Ba{\l}aban's parameters $(R,\varrho,b_0,\tilde p,\bar n,\alpha_0)$ (in his notation) and on the couplings only, such that for $t\le t_0(\kappa)$ the sum over all localisations of the absolute values of all terms containing at least one difference of propagators or of external fields is at most $C_\kappa\,t^{\kappa}\,|\Lambda^{(k)}_7|$. The constants do not depend on $k$, on the domains or on $\tilde B$, and $\tilde B=0$ is included. \end{lemma} The proof writes the difference of the two domains' class sums as a sum over cut skeletons with $(n,m)\ne(0,0)$ difference lines and bounds the total by $C_\kappa t^\kappa|\Lambda^{(k)}_7|$ for every $\kappa$; its course is the following. This is the interaction-term half of Ba{\l}aban's domain-change estimate \cite[Prop.~2.6]{Ba2}, at every background. The terms with exactly one difference line, grouped line by line, carry their counterterm partners but not the complete renormalised classes of the cut graph, and a lone divergent two-photon-leg subgraph is not small per graph; the proof expands every line and every external field in the difference and regroups the terms exactly into contractions of \emph{full} classes of the cut components, to which the renormalisation bound \cite[Prop.~1, (1.33)]{Ba3} applies with constants free of the domains and of the background. Every use of the upper stability bound \cite[(1.3)]{Ba2} and of \cite[Thm.~3.1, (3.4)]{K1} at block factor $5$ passes through this statement. % \begin{definition}[cut terms, cut graphs and orbits]\label{cpi:def:a07:cut} Let $\cR=\sum_{\Gamma\in\cG}c_\Gamma E(\Gamma)$ be a renormalised family (\cref{cpi:def:a07:class}). Fix a set of \emph{decomposed kinds} of objects of the family (for instance all lines and legs on a domain, or all restricted lines) and write every object $o$ of such a kind as $o=o'-\Delta o$ with $\Delta o$ a symmetric kernel, $\Delta o(x,y)=\Delta o(y,x)^{T}$, as is every difference of two symmetric resolvents. An \emph{oriented cut term} is a triple $\tau=(\Gamma,S,\theta)$ with $\Gamma\in\cG$, $S$ a set of decomposed objects of $\Gamma$ and $\theta$ an orientation of every line of $S$; its weight is $2^{-|S_L|}c_\Gamma$, $S_L$ the set of lines of $S$. Since the two orientations of a line add up to $\Delta o$ by its symmetry, multilinearity gives the exact expansion \begin{equation}\label{cpi:eq:a07:tel} \cR=\sum_{\tau}2^{-|S_L|}c_\Gamma\prod_{o\in S}(-\Delta o)_\theta\prod_{o\notin S}o'. \end{equation} The \emph{cut graph} $\Gamma\setminus S$ is $\Gamma$ with the lines of $S$ removed, their ends carrying the two arguments of $\Delta o$ (end $0$ and end $1$ according to $\theta$), each leg of $S$ carrying $\Delta o$, and the objects of $S$ labelled; this is Ba{\l}aban's ``we replace these lines by pairs of external legs'' \cite[p.~432]{Ba3}, with external fields in the form of functions of several variables \cite[p.~419]{Ba3}. A subgraph $H$ of $\Gamma$ \emph{avoids} $S$ if it contains no line of $S$. For a move $\mu$ at a subgraph $H$ avoiding $S$ put $\mu\cdot\tau\defeq(\mu\Gamma,s_\mu(S),s_\mu(\theta))$: the ends of the $S$-lines and the $S$-legs sitting at slots of $H$ are moved to the image slots, and everything else is fixed. The \emph{orbits} are the classes of the equivalence relation generated by $\tau\sim\mu\cdot\tau$. \end{definition} \begin{lemma}[properties of the orbits]\label{r13:def:orbitfacts} By the properties \textup{(A1)}--\textup{(A5)} of the moves (\cref{r13:def:moves}): a connected line set of $\Gamma$ avoids $S$ exactly when it is a connected line set of $\Gamma\setminus S$, with the same vertices, slots, slot types, order and degree in both, so the moves available at it and their results are the same in $\Gamma$ and in $\Gamma\setminus S$; when $\cR$ is closed under the moves, $\mu\cdot\tau$ is again an oriented cut term of $\cR$, $\mu^{-1}$ acts at $H^*$, which avoids $s_\mu(S)$, and $\mu^{-1}\cdot(\mu\cdot\tau)=\tau$, so that the orbits partition the cut terms, every orbit is finite and the number of $S$-objects of each kind is constant on it; and cutting commutes with the moves, $(\mu\Gamma)\setminus s_\mu(S)=\mu(\Gamma\setminus S)$, so that the cut graphs of one orbit form exactly one class of the (possibly disconnected) cut graph, the product of the classes of its connected components, each cut graph being reached from exactly $r_O\ge1$ cut terms of the orbit $O$, where $r_O$ is the order of the group of relabellings of identical $S$-objects (same kind, same kernel, same orientation pattern) that preserve the class. These three facts are what the orbit regrouping of \cref{hyp:k01:RENorb} rests on. \end{lemma} \space% % % \begin{definition}[renormalised families, labels, lines and legs]\label{cpi:def:a07:class}\emergencystretch=3em\relax A \emph{renormalised family} is a finite sum $\cR=\sum_{\Gamma\in\cG}c_\Gamma E(\Gamma)$ over a finite set $\cG$ of graphs built from the vertices \cite[(1.6)--(1.15)]{Ba3}, from the mass-counterterm vertices \cite[(1.7)]{Ba3} with the coefficients \cite[(1.23), (1.29)]{Ba3} and from source vertices; $c_\Gamma$ collects the couplings, the combinatorial weight and the counterterm values, every counterterm being a vertex of the action with a field- and domain-independent coefficient \cite[(1.29)--(1.31), p.~419]{Ba3}. Every vertex carries \emph{labels} \cite[p.~420]{Ba3}: each vector-field leg the support box of its partition function, each vertex of type (1.6)/(1.7) its unit cube, each block vertex (1.13)--(1.15) its block; the counterterm vertex that replaces a divergent two-scalar-leg subgraph $\gamma$ carries the labels of $\gamma$, so that a graph and its counterterm partner have the same label multiset \cite[pp.~417, 420, 432]{Ba3}. A \emph{line} is a pair of legs with a kernel type (scalar or photon, with its domain and background); an unpaired leg is an \emph{external leg} and carries an external field, possibly a function of several variables \cite[p.~419]{Ba3}. \end{definition} \begin{definition}[subgraphs, degrees and the divergent subgraphs]\label{r13:def:divsub}\emergencystretch=3em\relax A \emph{subgraph} $H$ of $\Gamma$ is a connected set of lines; its \emph{slots} are the legs of its vertices not on lines of $H$ (legs on other lines, external legs, cut ends), its \emph{type} is the number and kind of its slots with the differentiation of the scalar ones, its order is the sum of the orders of its vertices, and its degree $D(H)$ is that of \cite[(2.1)--(2.3)]{Ba3}, a counterterm vertex counting as the graph it replaces; type and degree depend only on the vertices of $H$, its lines and which legs are slots. The divergent subgraphs and their completions \cite[pp.~420, 429--432]{Ba3} are: a two-scalar-leg one-particle-irreducible $H$ with undifferentiated slots and $D(H)\le0$, completed by its \emph{counterterm partner}; a two-vector-leg $H$ with $D(H)\le0$, completed by the \emph{two-vector-leg class}, all connected graphs of the considered type, order and label multiset \cite[p.~431]{Ba3}; a $\phi\phi A$ subgraph with $D(H)=0$, one-particle-irreducible or reducible, completed by the \emph{permutation} of its two external scalar legs \cite[p.~429]{Ba3}. Graphs with a block vertex have $D>0$ \cite[Cor.~2.3]{Ba3}, so no divergent subgraph contains one. \end{definition} \begin{definition}[the moves]\label{r13:def:moves}\emergencystretch=3em\relax Each completion is a \emph{move} $\mu$ at a subgraph $H$ of its source $\Gamma$: it replaces $H$ by a graph $H^*$ together with a bijection $s_\mu$ from the slots of $H$ to those of $H^*$, and whatever sits at a slot of $H$ (an outer line end, an external field, a cut end) is re-attached at the image slot; nothing else changes. The moves are: \textup{(P)}, the counterterm partner, which replaces the vertices and lines of $H$ by one counterterm vertex carrying the labels of $H$, the two slots going to its two legs, and its inverse \textup{(P$'$)}, the expansion of a counterterm vertex; \textup{(W)}, the two-vector-leg completion $H\to H'$ for every $H'$ in the per-label two-vector-leg class of $H$ (all connected graphs with the label multiset and order of $H$ whose two free photon legs carry the two slot labels of $H$, with their counterterm members), the slots mapped by their vector-leg labels, applied at every divergent two-vector-leg subgraph, maximal or not; \textup{(X)}, the exchange of the attachments at the two scalar slots of a divergent $\phi\phi A$ subgraph (a line joining the two slots is mapped to itself with its ends exchanged; two external fields of one kind exchange their arguments). The moves have the five properties on which the regrouping rests: \textup{(A1)} a move changes only the lines and vertices of $H$ and the attachment of the slots, independently of what is attached there; \textup{(A2)} its availability at $H$ and its results $(H^*,s_\mu)$ depend only on $H$ with its slots, because $D(H)$, the slot types and the per-label two-vector-leg class are intrinsic to $H$; \textup{(A3)} every move has an inverse at $H^*$ with slot map $s_\mu^{-1}$; \textup{(A4)} a move keeps the label multiset, the kind of every slot and the kernel type of every surviving line; \textup{(A5)} $c_{\mu\Gamma}/c_\Gamma$ depends only on $(H,H^*,s_\mu)$. \end{definition} \begin{definition}[classes]\label{r13:def:classes}\emergencystretch=3em\relax The \emph{class} $\operatorname{Cl}(\Gamma)$ of a graph is its connected component under the moves, the moves at $\Gamma$ itself included when $\Gamma$ is of a primitively divergent type (its one-vertex counterterm partner, its permuted graph, all two-vector-leg graphs of its order). It is Ba{\l}aban's class $G_{\mathrm{ren}}$ of $\Gamma$: ``the smallest sets of graphs, symmetric with respect to permutation of external scalar field legs and with each divergent subgraph renormalized'' \cite[p.~429]{Ba3} are exactly the connected components, the moves being invertible. A \emph{per-label full class} is the set of all connected graphs of one type, one order and one label multiset, with their counterterm members \cite[p.~431]{Ba3}; it is closed under every move. A family is \emph{closed under the moves} if $\cG$ is a finite union of classes; Ba{\l}aban's renormalised families are. \end{definition} \space% % \begin{lemma}[\RENorb: orbit regrouping of cut renormalised families]\label{hyp:k01:RENorb} Let $\cR=\sum_{\Gamma\in\cG}c_\Gamma E(\Gamma)$ be a renormalised family of \cite[Sects.~1--3]{Ba3} with $\cG$ a finite union of renormalised classes (\cref{r13:def:classes}), in which the objects (lines or external legs) of certain kinds are decomposed as $o=o'-\Delta o$ with symmetric kernels $\Delta o$, and expand multilinearly, $\cR=\sum_\tau2^{-|S_L|}c_\Gamma\prod_{o\in S}(-\Delta o)\prod_{o\notin S}o'$ over the oriented cut terms $\tau=(\Gamma,S,\theta)$ of \cref{cpi:def:a07:cut}. Call two cut terms equivalent when one arises from the other by finitely many of Ba{\l}aban's three completions of \cite[pp.~420, 429--432]{Ba3} (the counterterm partner; the two-vector-leg class; the permutation of the external scalar legs of a divergent $\phi\phi A$ subgraph) applied at divergent subgraphs containing no line of $S$, the objects of $S$ carried along to the image slots. Then: \textup{(i)} the equivalence classes (orbits) partition the cut terms, and every move keeps the label multiset and the number and kinds of the objects of $S$; \textup{(ii)} a connected line set of $\Gamma$ avoiding $S$ is a subgraph of the cut graph $\Gamma\setminus S$ with the same vertices, slots, slot types, order and degree \cite[(2.1)--(2.3)]{Ba3}, and cutting commutes with the moves, so that the cut graphs of one orbit form exactly one renormalised class of the cut graph, the product of the classes of its connected components; \textup{(iii)} at fixed labels, after the sum over the separable indices of the cut kernels, the sum of the cut terms of an orbit $O$ is $(-1)^{|S|}2^{-|S_L|}r_O^{-1}c_*\prod_KE(\Cl(K);\text{its fields})$, the product over the components $K$ of the class sums with the cut kernels as external fields, $c_*$ the common value on the orbit of $c_{\Gamma'}/\prod_Kc(K')$, $c(K')$ the product of the vertex weights of the component (a constant by (A5) of \cref{r13:def:moves}), and $r_O\ge1$ the order of the group of relabellings of identical $S$-objects preserving the class (\cref{r13:def:orbitfacts}); \textup{(iv)} the sum over all orbits is the full expansion; \textup{(v)} if $\cG$ is a per-label full class, the cut terms with a given cut datum (components with labels and orders, stubs and fields at each component, incidence of the $S$-objects) correspond, with a multiplicity $s_\delta\ge1$ depending only on the cut datum $\delta$, to the tuples of members of the per-label full classes of the components, members without internal line and of order $0$ included; each such group is a union of orbits and is closed under the exchange of the scalar legs of every two-scalar-leg or $\phi\phi A$ subgraph. \end{lemma} The proof partitions the cut terms of \cref{cpi:def:a07:cut} of a move-closed family into the orbits of the three completions, every vertex carrying Ba{\l}aban's labels (support boxes, unit cube, block) and a counterterm vertex inheriting the labels of the subgraph it replaces, so that the regrouping acts within fixed label multisets; the sum over an orbit is then computed from the class sums of the components of the cut graph, and the count $s_\delta$ of \textup{(v)} from the per-label full classes of the components. This is Ba{\l}aban's regrouping step in the proof of \cite[Prop.~1, p.~432]{Ba3} (``if we remove a line in any graph of the class, then for the obtained graph we can find in this class all graphs necessary to form a new renormalized class''), made explicit for decomposed objects, with the sharper conclusion that one orbit gives exactly one class of the cut graph. It is needed wherever the renormalisation bound is applied to a family some of whose lines have been cut, in particular in the comparison of two propagator domains and in the telescoping of the cube's large-field remainders; at a nonzero background, where the bound of \cite[Prop.~1]{Ba3} is available for per-label full classes, it is used in the form \textup{(v)}. \begin{remark}[the cube]\label{rem:k01:cube} The propositions below are stated for King's RG cube $\CK$, on which King's Dirichlet weight is compatible with the image group of the two Dirichlet planes (they are $2MJ$ apart). On the comparison cube $\NDa$ the planes are $2MJ+\eps_k$ apart, no partition into consecutive blocks is invariant under the image group, and the flow does not exist; this is why \cref{hyp:s03:EC} is posed on King's convention, and why the fixed-cube limits of the comparison family are reached only through \cref{cor:s04:G2}. The cube of the existence theorem is $\CK$ itself (Definition~\ref{def:s02:Kcube}). \end{remark} \subsection{The estimates at zero scalar source}\label{ssec:k01:inputs} \begin{proposition}[\Hrel: the small-field representation on King's cube, zero scalar source]\label{prop:k01:H0rel} Fix $J\ge J_0$, a family $f_1,\dots,f_N$ and $x\in\R^N$, and let $s\in\{u_x,0\}$. There are $\eta_1>0$, $k_0$ and $C,c,\sigma>0$, depending on the couplings, on $J$, on $f$ and $x$, and on nothing else (not on $k$, $j$), such that for all $k\ge k_0$ and all $j$ with $J_*\le j\le k-J_*$ and $\eta=5^{j-k}\le\eta_1$: \begin{equation}\label{eq:k01:H0rel} e^{-C\eta^{\sigma}}\,\Ik{j}(s)\ \le\ \Zk(s)\ \le\ e^{C\eta^{\sigma}}\,\Ik{j}(s)+\rho(\eta)\,\Zk(0),\qquad \rho(\eta)\defeq e^{-c\,p(\eta)^2+C}, \end{equation} where $p(\eta)=b_0(1+\ln\eta^{-1})^{6}$ is Ba{\l}aban's small-field threshold (the exponent $6$ is compatible with the conditions \cite[(3.54)]{Ba1}). Moreover, at $s=0$ the two terms of the upper line are the two parts of Ba{\l}aban's decomposition: $\int P\,d\mu_{k,0}\le e^{C\eta^\sigma}\Ik{j}(0)$ and $\int(1-P)\,d\mu_{k,0}\le\rho(\eta)\Zk(0)$ \textup{(\Rzero)}. \end{proposition} The proof is given below: Lemmas~\ref{cpi:a01:WQ}--\ref{cpi:a01:FPC} provide the four modifications of the torus argument, Proposition~\ref{cpi:a01:lines} the two lines and Theorem~\ref{cpi:a01:R0} the clause \Rzero. This is \cite[Thm.~3.1]{K1} on the cube, which \cite[Sect.~5.2]{K2} states (``the proof follows by modifying suitably the material in [Ba~1] and [Ba~2]''), in the form in which the remainder is measured against $\Zk(0)$ rather than against an absolute constant, and with constants uniform in the step $j$ (the form the diagonal pairs need, since $\eta_k\to0$). The lower line is the weaker of the two readings of the sign in \cite[(3.3)]{K1}. The statement contains no lower bound on $\Zk(0)$; the proof uses Ba{\l}aban's lower bound only to pass from the absolute remainder to the ratio form. Of the statements of \cref{ssec:k01:open}, \BALint\ (through Ba{\l}aban's torus bounds) and \RENorb\ (through the comparison of propagator domains and the telescoping of the large-field remainders) enter. % \subsubsection*{Proof of Proposition~\ref{prop:k01:H0rel}} The proof is King's proof of \cite[Thm.~3.1]{K1}, that is Ba{\l}aban's lower and upper bounds \cite{Ba1,Ba2} with the modifications of \cite[Appendix]{K1}, run on King's cube $\CK$ with King's block-spin flow of Section~\ref{ssec:k01:setting}. Two things change relative to the torus. First, the scalar field is killed on the Dirichlet planes of the cube and averaged with King's Dirichlet weight; every place where the torus arguments use that the block average is uniform ($QQ^*=1$, $\mathsf P$ a projector, the constant reproduced by the average) is replaced by the corresponding fact for the weighted average (Lemma~\ref{cpi:a01:WQ}), and every place where the torus has no boundary acquires a Dirichlet face, at which the scalar background vanishes linearly (Lemma~\ref{cpi:a01:DV}). Second, the photon has Neumann conditions; every photon object of the cube is the even restriction of the same object on a doubled torus, so every photon estimate of the torus analysis holds on the cube with its constant multiplied by at most $8$ (Lemma~\ref{cpi:a01:PE}). The large-field analysis near the Dirichlet faces uses King's face test \cite[(5.41)]{K2} as a large-field family at every scale, paid by a Dirichlet-face positivity valid for every gauge field (Lemma~\ref{cpi:a01:FPC}). Throughout, $d=3$, $L=5$, the couplings are fixed, $J\ge J_0$, $s\in\{u_x,0\}$, and constants $C,c,\delta$ never depend on the spacing index $k$, the step $j$, the field or the domain unless this is said. We work in unit-$j$ units, in which the lattice after $j$ steps has unit spacing, and $\eps_j\defeq5^{-j}$ denotes the fine spacing. \emph{The weighted average.} Let $X_{\mathrm{live}}$ be the live scalar sites of the cube at the current scale and $\mathsf K$ the killed sites (the Dirichlet planes). A \emph{block weight} is a function $w\ge0$ on the fine sites of a unit block with (W1) $\sum_{x\in B}\eps_j^3w(x)=1$, (W2) $w\le W$, (W3) $w=0$ on every killed site of the block, (W4) the same $w$ on every block up to translation; King's Dirichlet weight, which vanishes on the top layers of his block, is in the class with $W=125/64$, and the uniform weight satisfies (W1), (W2), (W4) with $W=1$ but not (W3) (on a block whose first site plane is a Dirichlet plane its live sites carry total weight at most $1-\eps_j$), so it is used only on lattices without killed sites: the coarse steps on the block lattice and the photon. The weighted average is $(Q(A)u)(y):=\sum_{x\in B(y)}\eps_j^3w(x)U(A(\Gamma_{y,x}))^{-1}u(x)$ with King's hierarchical contours $\Gamma_{y,x}$ \cite[(2.11)--(2.12)]{K1}, which start at the live lower corner of the block and stay off the killed set; $\mathsf P:=Q(A)^*Q(A)$ and $M:=-\Delta^D_A+V+\mathfrak a_j\mathsf P$ is the cube operator, with the Dirichlet Laplacian $-\Delta^D_A$ of the killed planes and the composite stiffness $\mathfrak a_j$ of \cref{def:k01:bulkflow} (in unit-$j$ units; $\mathfrak a$ denotes the bare stiffness). \begin{lemma}[the weighted average]\label{cpi:a01:WQ} For every weight in the class and every gauge field $A$, with $W_2:=\sum_{x\in B}\eps_j^3w(x)^2\in[1,W]$ ($W_2=W$ for King's weight): (Q1) $Q(0)\mathbf 1=\mathbf 1$ on every block, and $Q(A)$ maps the transported constant to the constant; (Q2) $Q(A)Q(A)^*=W_2\mathbf 1$; (Q3) $\mathsf P=W_2\Pi$ with $\Pi$ an orthogonal projector, $0\le\mathsf P\le W_2$ and $\|1-\mathsf P\|=\max(1,|1-W_2|)\le1$ since $W_2<2$; (Q4) $|(Qu)(y)|\le W_2^{1/2}\|\mathbf 1_{B(y)}u\|_2$ and $\|Q^*e_y\|_2=W_2^{1/2}$; (Q5) $|(Qu)(y)|\le\sup_{B(y)}|u|$; (Q6) $|Q^*b|_\infty\le W|b|_\infty$, $\|\mathsf P\|_{\infty\to\infty}\le W$ and the Schur norm of $\mathsf P$ is at most $W$; (Q7) $\sum_x\eps_j^3|(\mathsf Pu)(x)|=\sum_y|(Qu)(y)|$; (Q8) $\mathsf P(x,x')=0$ unless $x,x'$ lie in one block, and $\mathsf P$ vanishes on killed sites, so $\mathsf P$ never couples a live site to a killed one and every union of blocks is invariant. Every use of $QQ^*=1$, of $\mathsf P$ being a projector and of the reproduction of the constant in \cite{Ba1,Ba2,K1} is replaced by (Q1)--(Q3), the factor $W_2$ entering the constants only; in particular the block-spin minimiser identity $\langle\phi,\Delta^{(j)}\phi\rangle=\min_\Phi[\langle\Phi,(-\Delta^D_A+V_j)\Phi\rangle+\mathfrak a_j\|\phi-Q\Phi\|^2]$ and the exact renormalisation group hold verbatim. \end{lemma} \emph{The photon sector.} Let $2X$ be the torus of period $2N_\eta$ in each direction, $N_\eta\defeq2MJ\cdot5^{k-j}$ the side of the cube in unit lengths (physical period $4MJ$), the cube embedded as $\{0,\dots,N_\eta-\eps_j\}^3$, and $\mathcal W\cong(\Z_2)^3$ the group generated by the reflections in the photon planes $x_\mu=-\eps_j/2$. Since $4MJ$ is a multiple of every block side, the blocks of the cube are blocks of the periodic tiling of $2X$ and $\mathcal W$ maps blocks to blocks. For a photon component (a site field) let $\mathsf E$ be the even extension to $2X$, and $\mathsf E_c$ the same on the coarse lattices. \begin{lemma}[photon objects are even restrictions]\label{cpi:a01:PE} For every photon object $O$ of Ba{\l}aban's and King's construction built from the Neumann form $-\Delta^N+\mu_0^2\eta^2$ and the uniform block averages on the cube, and its counterpart $O_{2X}$ on $2X$: (a) the propagators $G_j$, the block forms $\Delta^{(j)}$, the fluctuation covariances $C^{(j)}$, the background map $A_j\mapsto\mathfrak a_jG_jQ_j^*A_j$ and the slices satisfy $O_{2X}\mathsf E=\mathsf E\,O$; (b) $\langle\mathsf EA,\Delta^{(j)}_{2X}\mathsf EA\rangle=8\langle A,\Delta^{(j)}A\rangle$, and likewise for every positive form; (c) for every coarse set $\Lambda$ and every fine union of blocks $\Omega$ the Dirichlet-conditioned compressed operators on $\Lambda$ and the Neumann operators on $\Omega$ intertwine with $\mathsf E$ and the $\mathcal W$-invariant sets $\mathcal W\Lambda$, $\mathcal W\Omega$; (d) with Ba{\l}aban's cut-off kernel $\zeta$ of \cite[(2.44)]{Ba2} applied image-wise on $2X$, the cut-off objects of the cube are even restrictions of those of $2X$ with the torus constants. Consequently every photon estimate of the torus analysis (decay, regularity, the cut-off and localisation lemmas) holds on the cube with its constant multiplied by at most $|\mathcal W|=8$, and the regularity of the photon background is preserved. \end{lemma} \begin{proof} (a) The Neumann Laplacian of the cube is the periodic Laplacian of $2X$ on the $\mathcal W$-even fields, the reflection planes lying between sites: every plane-crossing bond of $2X$ joins a site to its mirror image, where an even field has zero difference, and every bond of the cube appears $8$ times. The uniform block averages commute with $\mathsf E$ (blocks go to blocks) and $Q^*_{j,2X}\mathsf E_c=\mathsf EQ_j^*$ (the adjoint of a $\mathcal W$-covariant uniform average is the uniform block extension). Hence $(-\Delta^N_{2X}+\mu_0^2\eta^2+\mathfrak a_jQ_j^*Q_j)\mathsf E=\mathsf E(-\Delta^N+\mu_0^2\eta^2+\mathfrak a_jQ_j^*Q_j)$; the torus operator commutes with $\mathcal W$, preserves the even subspace and is invertible on it, so $G_{j,2X}\mathsf E=\mathsf EG_j$, and the block forms, the fluctuation covariances, the background map and the slices follow from the algebra of \cite[(2.13)--(2.17)]{K1}, which is the same on both sides. (b) $\mathsf E^*\mathsf E=8$ on the cube, so by (a) $\langle\mathsf EA,\Delta^{(j)}_{2X}\mathsf EA\rangle=\langle\mathsf EA,\mathsf E\Delta^{(j)}A\rangle=8\langle A,\Delta^{(j)}A\rangle$, and likewise for every positive form. (c) Compression to a $\mathcal W$-invariant set commutes with $\mathcal W$, and $P_{\mathcal W\Lambda}\mathsf E=\mathsf EP_\Lambda$ with $P_\Lambda$ the projection onto the functions supported on $\Lambda$; then (a) on the invariant subspace. For Neumann conditions on $\Omega$, the bonds of $\mathcal W\Omega$ are the images of those of $\Omega$ plus plane-crossing bonds, on which even fields have zero difference. (d) $\zeta(x,gy)$ depends on $|x-gy|$ only; the image-wise kernel $K(x,y)\defeq\sum_{g\in\mathcal W}\zeta(x,gy)K_{2X}(x,gy)$ commutes with $\mathcal W$ by construction and restricts to the cube kernel by definition; the support and size statements are statements about $\zeta(x,\cdot)$ on $2X$, and $|x-gy|\ge|x-y|$ for $x,y$ in the cube keeps every support statement (the cut is a device inside the definition of Ba{\l}aban's background, so this choice is free). Finally, (a) and (b) give $O=\frac18\mathsf E^*O_{2X}\mathsf E$, that is, since $O_{2X}$ commutes with $\mathcal W$, $O(x,y)=\sum_{g\in\mathcal W}O_{2X}(x,gy)$ for the kernels: a sum of $|\mathcal W|=8$ torus kernels at the distances $|x-gy|\ge|x-y|$, so every decay, regularity, cut-off and localisation estimate of the torus analysis holds on the cube with its constant multiplied by at most $8$; every positivity bound of the photon sector (the photon part of \cite[(3.29), Prop.~3.1]{Ba2}, the last-step mass bound of \cite[(3.11)]{Ba2}, the photon exponents of \cite[(3.26)]{Ba2}) holds with the constant of $2X$, because by (b) the cube form of an even field is $\frac18$ of the torus form and the right-hand sides (gradient sums, mass sums, block sums) scale by the same factor, plane-crossing gradients of even fields vanishing; and the regularity of a photon configuration is preserved by $\mathsf E$, since $\mathsf EA=A\circ F$ with $F$ the coordinatewise folding map of $2X$ onto the cube, which is $1$-Lipschitz, lattice differences across a plane vanishing and the others being copies. \end{proof} \emph{Dirichlet-face operations.} For a sub-domain $\Omega$ of the cube (a union of blocks, with the Dirichlet condition of the cube at the killed planes and Neumann cuts elsewhere) and block data $b$ on its blocks, the \emph{leg} of $b$ is $u:=M^{-1}\mathfrak a_jQ^*b$, the background of the data; $t(x)$ denotes the distance of $x$ to the Dirichlet planes. \begin{lemma}[linear vanishing at the Dirichlet face]\label{cpi:a01:DV} For every $\Omega$, every $A$, block data $b$ and every live $x\in\Omega$: $|u(x)|\le C_{\mathrm{DV}}|b|_\infty\min(1,t(x))$, and the same for $u=M^{-1}f$ with bounded $f$. Moreover, with $\delta_5>0$, the leg $l(\cdot,v)$ of the unit datum on a block $v$ satisfies $|l(x,v)|\le C(\eps_j+t_\mu(x))e^{-\delta_5\dist(x,B(v))}$ for $x$ in the first-layer slab of the face $\mu$, so that $|u(x)|\le C|b|_\infty(\eps_j+t(x))e^{-\delta_5\dist(x,\supp b)/2}$ there. \end{lemma} \begin{proof} Fix the face $\mu$ with $t_\mu(x)=t(x)\le1$. Let $F_0:=\mathfrak a_jW(1+C)|b|_\infty$ bound $|\mathfrak a_jQ^*(b-Qu)|$ by (Q6) and the sup bound on $u$, let $S_0$ be that sup bound, and put $\rho(s):=(S_0+F_0)s-F_0s^2/2$ on $[0,1+\eps_j]$, increasing, with $\rho(0)=0$, $\rho(1)\ge S_0$ and $-\Delta_s\rho=F_0$ exactly. On the first-layer slab $\mathcal S_1:=\Omega\cap X_{\mathrm{live}}\cap\{0\sigma_i\}$ at the face blocks $y'\in L_1^{(i)}$ inside the current small-field region (for $i=0$ on the first live layer itself, where $\phi_0$ is the fine field), with $\sigma_i:=(2/(\theta_2\gamma_D^{(i)}))^{1/2}p_i\le C_8p_i$, $\gamma_D^{(0)}:=1$ and $C_8\defeq\sup_i(2/(\theta_2\gamma_D^{(i)}))^{1/2}$, finite because the stiffnesses $\mathfrak a_i$ are bounded below; at the stopping scale it is King's test $\lambda_j$, \cite[(5.41)]{K2} with $C=\sigma_j/p_j$. A violation at scale $i$ is paid by the factor $e^{-p_i^2/4}$ from the $\theta_2$-share of the scale-$i$ form (Lemma~\ref{cpi:a01:FPC}; at $i=0$ from the face-bond term itself), and violating blocks of different scales are disjoint, so the payments add. On the complement of the face and gradient families, a point within one corridor $r(\ell_i)$ of the face obeys $|\phi_i(x)|\le\sigma_i+r(\ell_i)p_{i-1}\le Cp_i^2$ along a block path to the face layer, and the fluctuation field obeys the same bound; beyond one corridor the Dirichlet image effect on the background is at most $Ce^{-\delta_1r(\ell_i)}M_i\le p_i$ by Lemma~\ref{cpi:a01:DV}, so \cite[(2.113)]{Ba2} holds there in Ba{\l}aban's form; and within distance $R_0$ of a face a block with $|\phi_i|\ge M_i$ forces either a gradient violation or a face-family element along a shortest block path. \emph{The two lines and the assembly.} With $\eta=5^{j-k}$, $|\CJ|=(2MJ)^3$ the volume of the cube $\CJ=[-MJ,MJ)^3$ and $p(\eta)=b_0(1+\ln\eta^{-1})^6$: \begin{proposition}[lower and upper lines]\label{cpi:a01:lines} For $\eta\le\eta_1$ and $J_*\le j\le k-J_*$, in the normalisation of Ba{\l}aban's block-spin representation, \begin{gather*} e^{-a}\Ik j(s)\le\Zk(s)\le e^{a}\Ik j(s)+e^{-\frac12c_0'p(\eta)^2+E_+|\CJ|},\\ \Zk(0)\ge e^{-E_-|\CJ|},\qquad a=\sum_{i0$, $c_0'$ the constant of the large-field payment, and $E_\pm(M)$ depending on $\eps_0^{\mathrm{Ba}}$ and King's data only; all constants are independent of $k$ and $j$. \end{proposition} \begin{proof} The lower line is Ba{\l}aban's lower-bound induction \cite[\S\S3--4]{Ba1} with King's Appendix class, carried through the scales with a background-gradient clause and a block-mismatch clause in place of King's Laplacian clause; the sum $\sum_{i0$ and $C_3<\infty$, depending on the couplings, $J$, $f$, $x$ only, such that for all $k\ge k_0$, all $J_*\le j\le k-J_*$ with $\eta=5^{j-k}\le\eta_1$, and all $(A_j,\phi_j)$ in the support of $\chi_j$, \begin{equation}\label{eq:k01:Kpos} \big|S^{(j)}_{N,k}(A_j,\phi_j;s)-S^{(j+1)}_{N,k+1}(A_j,\phi_j;s)\big|\ \le\ C_3\big(5^{-\gamma j}\eta^{-\beta}+\eta^{\sigma}\big). \end{equation} \end{proposition} The proof compares the $k$-model after $j$ steps and the $(k+1)$-model after $j+1$ steps on one unit lattice, every rate across the pair being $5^{-\gamma j}$ with the exponent $\gamma$ set by the line comparisons; its course is the following. This is \cite[Thm.~3.4]{K1} at $n=1$ on the cube, for the field- and source-dependent part of the effective action: the two members of the pair $(k,j)$, $(k+1,j+1)$ live on one unit lattice with one indicator $\chi_j$, and the statement compares them there. It is assembled from six pieces: the photon quadratic form (an even-restriction identity reduces the cube to the doubled torus, where \cite[Prop.~3.10]{K1} applies); the scalar quadratic form at zero background (the Dirichlet symbol) and with background (a two-leg tree graph); the background determinant $B_3\defeq-\ln[Z_j(A^{(j)})/Z_j(0)]$ (King's interpolation \cite[(3.94)--(3.98)]{K1}, of which the middle line of (3.98) is false as stated in \cite{K1}: at a constant block field the order-two traces of the two members differ by an amount growing like $5^{j}$ times the volume, so no bound order by order at a fixed background exists and only the sum over the orders converges; it is replaced by the scale decomposition of the determinant, $\ln\det G_j(A)-\ln\det G_j(0)=\sum_{i0$, is the rate at which the coarse operators of the two members converge and the two unmatched finest steps cost $\eps_k$ each; summed over the sites and the scales, with $\sum_{i3$; \cref{cpi:thm:hrs:NP3} is the same bound on the torus); the graphs with curvature vertices; the source-free field-dependent graphs away from the faces and near them (the photon face terms and the scalar boundary remainders of \cite[pp.~331--332]{K2}); and the block-vertex classes. For the curvature-vertex and the source-free field-dependent classes, the localisations whose tree length $R$ exceeds the small-field threshold $p$ are bounded by cutting the localised graph into clusters at mutual distance $\ge R_{\mathrm{cl}}=3\sqrt3$ (the cut by clusters, \cref{r13:def:clusters} below) and reducing each cluster in its own box of side $O(p)$, with the cut lines as external lines of both clusters (the degree count row by row at every tuple, which gives King's (3.38) and (3.42) per cluster, and the position sum, which costs $\eta^{-3}|\CJ|$ and in the branch $R>p$ leaves the factor $e^{-(\delta-\delta')p(\eta)/2}\le C_N\eta^N$; the zone-B tails by \cref{cpi:thm:a02:Bsc}(a) and \cref{r13:thm:BVPpair}(d)). The vacuum constant $v_{k,j}$ is not part of the statement. \begin{lemma}[bounds for free]\label{lem:k01:free} Put $\beta_f\defeq C_0\sum_i\fournorm{f_i}^2$ with $C_0$ as in \cref{lem:s02:E}. For all $k\ge0$, $J\ge J_0$ and $x\in\R^N$, \[ e^{-\beta_f|x|^2}\ \le\ a_k(x)\ \le\ e^{\beta_f|x|^2}. \] \end{lemma} \begin{proof} By \cref{thm:s05:CK}(a) (a finite-$k$ Gaussian bound, proved without this section), $a_k(x)\le\exp(\norm{u_x}^2_{2,k})$, and $\norm{u_x}_{2,k}\le\sum_i|x_i|\norm{f_i}_{2,k}\le|x|(\sum_i\norm{f_i}^2_{2,k})^{1/2}\le|x|\,\beta_f^{1/2}$ by \cref{lem:s02:E}(E3). The lower bound is $a_k(x)a_k(-x)\ge1$, \cref{lem:s02:alg}(c). \end{proof} % \begin{definition}[units, averaging operators and propagators of depth $i$]\label{cpi:def:a02:setting} Let $k$ be the spacing index, $j$ the number of renormalisation steps and $\eta=5^{j-k}$ the block scale; the two members of the pair at $(k,j)$ are the $k$-model after $j$ steps and the $(k+1)$-model after $j+1$ steps (\cref{ssec:k01:setting}), and quantities of the second member carry a prime. For a depth $0\le i\le j$ we work in \emph{unit units of depth $i$}: the lattice $X_i$ of the $i$-th block variables has spacing $1$, the physical unit of length is $l_i\defeq5^i\eps_k$ (so $l_j=\eta$), the fine lattice has spacing $5^{-i}$, and \[ e_i\defeq e\,l_i^{1/2},\qquad m_i^2\defeq m^2l_i^2,\qquad \mu_i^2\defeq\lmu^2l_i^2,\qquad A_i(x)\defeq l_i^{1/2}A(l_ix), \] so that a unit charge sees the transporter $U(A_b)=\exp(e_i5^{-i}qA_{i,b})$, $q$ the real antisymmetric generator with $q^2=-1$. At depth $j$ the slice of index $i\le j$ has the scale $\rho_i\defeq5^{i-j}$ ($\rho_j=1$). The $(k+1)$-model at depth $i+1$ lives on the same lattice $X_i$, with fine spacing $5^{-i-1}$. On the cube $\CK$ the scalar field carries the Dirichlet condition of \cref{def:s02:Kcube}, the bonds to the Dirichlet planes being kept; $H_A\defeq D_A^*D_A$ is the covariant Laplacian and $M_0(A)\defeq H_A+m_i^2$. The \emph{averaging operator} $Q_i(A)$ averages the fine field over the $i$-th blocks along the hierarchical staircases of \cite[(2.10)--(2.12)]{K1}, paths of length at most $3$ in unit units inside the block, with the weights $w(x)=5^{-3i}f(x)$, where $f=1$ above the finest step and $f=f_D$ at the finest step: the \emph{Dirichlet weight} $f_D$ equals $(5/4)^3$ on the sites of a block all of whose coordinates have residues in $\{0,1,2,3\}$ relative to the block's corner and $0$ on the others, those with a residue $4$ in some coordinate (for a block whose top face lies on a Dirichlet plane these include the layer on that plane), so that $\sum_xw(x)=1$ and the density of $w$ is at most $\varrho\defeq(5/3)^3$; every bound below holds for every weight of density $\le\varrho$. With the stiffnesses $\mathfrak a$ and $\mathfrak a_i$ of \cref{def:k01:bulkflow} (the bare one and the composite one after $i$ steps, in the units of depth $i$) we set \begin{gather*} M(A)\defeq M_0(A)+\mathfrak a_iQ_i(A)^*Q_i(A),\qquad G_i(A)\defeq M(A)^{-1},\\ H_i(A)\defeq \mathfrak a_iG_i(A)Q_i(A)^*,\qquad \Delta^{(i)}(A)\defeq \mathfrak a_i-\mathfrak a_iQ_i(A)H_i(A), \end{gather*} and, with $Q$ the plain one-step block average $X_i\to X_{i+1}$, \[ K_i(A)\defeq\Delta^{(i)}(A)+\mathfrak a\,5^{-2}Q(A)^*Q(A),\qquad C^{(i)}(A)\defeq K_i(A)^{-1}, \] the \emph{coarse factor} of the slice of index $i$: the slice itself is $G_{(i)}=\mathfrak a_i^2\rho_i^{-4}\,(G_iQ_i^*)\,C^{(i)}\,(Q_iG_i)$ \cite[(2.17)]{K1}, an outer factor on each side of the coarse factor. The photon operators are the same with $f=1$, $m$ replaced by $\lmu$, the Neumann condition of \cref{def:s02:Kcube} and blocks tiling $\CJ$; the photon background after $j$ steps is $A^{(j)}\defeq H^{\mathrm{ph}}_jA_j$. On the support of the small-field indicator $\chi_j$ (\cite[(3.2)]{K1} with the face factor of \cite[(5.41)]{K2}) the fields $|A_j|$, $|dA_j|$ and $|\phi_j|$ are bounded by powers of $p=p(\eta)$, and $\varpi_i\defeq e_i\sup|dA^{(j)}_i|$ is the smallness parameter of the expansion at depth $i$. All statements of this kind hold at every point $(e,\lambda,m^2,\lmu)$ of King's range; the constants depend on it, on $J$, on $b_0$ and $p$ and on the order $\bar n=13$ of the expansion, and on nothing else unless said. The field- and source-free parts of the effective action \cite[(3.90)]{K1} belong to the vacuum constant $v_{k,j}$ and are dropped wherever they occur. \end{definition} \space% % % \begin{definition}[localisation, boxes and zones at depth $j$]\label{cpi:def:a02:zones} From here on everything is in the unit units of depth $j$ (\cref{cpi:def:a02:setting}): the \emph{live sites} are $T_\eta\defeq X_j$, the fine spacing is $\vartheta\defeq5^{-j}$ (the $\eps_j$ of the proof of \cref{prop:k01:H0rel}; not the phase of $\phi$), $\mathsf F$ denotes the set of scalar Dirichlet planes of $\partial T_\eta$, and $t(x)\defeq\dist(x,\mathsf F)\ge\vartheta$ for a live site. Every graph of the expansion is localised as in \cite[p.~661]{K1}, by centres: a \emph{tuple} $\tau$ of a graph assigns to each vertex a localisation weight $\chi_z$ supported within $\frac23$ of its centre $z$ in each coordinate, $P(\tau)$ is the set of centres, $n_\tau\defeq|P(\tau)|\le N_*$ with $N_*$ depending on the order $\bar n$ of the expansion only, and the \emph{tree length} $R=R(\tau)\defeq\dist(P(\tau))$ is the length of a shortest tree joining the centres (the points of the external fields included, \cref{cpi:lem:a02:Z}); $\square$ is the smallest union of large blocks (side $M$) containing its vertices, $\square''\subset\square'$ are the smallest boxes containing the large blocks within $p$, resp.\ $2p$, of $\square$, truncated at $\partial T_\eta$, and $\Omega\defeq\square'\cap T_\eta$ is a rectangular box with faces on block planes at every level (this is where the integer side $2MJ$ of the cube enters). A localisation is in \emph{zone~I} if $\square'\cap\mathsf F=\emptyset$ and in \emph{zone~B} otherwise; the faces of $\Omega$ not on $\mathsf F$ lie at distance $\ge2p$ from $\square$. The range of $j$ is $0\le j\le k-J_*$, with the step threshold $J_*$ of Section~\ref{ssec:k01:setting} (King's ``$J$'' of \cite[Thm.~3.1]{K1}) taken so large that $5^{-J_*}\le\delta_*$, where $\delta_*^{-1}>3p(\delta_*)+4M+1$; so $\eta=5^{j-k}\le\delta_*$ throughout. \end{definition} \begin{definition}[the lines on a box and the Dirichlet convention]\label{r13:def:boxlines} The internal scalar lines are the slices $G_{(i)}(\square',0)$ of the $f_D$ flow with the Dirichlet condition on all of $\partial\Omega$, the photon slices carry the Neumann condition, and the lines $\delta G_j(\square',A^{(j)})$ of the exact rewriting $G=G(\square')+\delta G(\square')$ are external in King's sense. The Dirichlet condition is taken in the following convention: if $\Omega$ occupies $\prod_\mu[c^-_\mu,c^+_\mu)$ with $c^\pm_\mu\in\Z$, its Dirichlet planes are $x_\mu=c^\mp_\mu-\vartheta$, its live sites are the sites of $T_\eta$ in $\prod_\mu[c^-_\mu,c^+_\mu-\vartheta)$, and the matter is killed on the top face layers $x_\mu=c^+_\mu-\vartheta$; on the faces lying on $\mathsf F$ this is the convention of $T_\eta$ itself (\cref{def:s02:Kcube}). On every such box the lines obey King's slice bounds \cite[(3.63)--(3.65)]{K1} up to and including the faces, the scalar and the photon slices being image sums of free kernels over the reflection group of the box, and the external objects obey \cite[(5.2)--(5.3)]{K2}, King's Prop.~5.1 on boxes whose faces lie on block planes, which follows from the image sums and a bootstrap on the sup of $|u|+|D_Au|$ with the commutator localisation of \cite[Lemma~5.3]{K2}, with no a priori sup bound. \end{definition} \begin{definition}[the cut by clusters]\label{r13:def:clusters} Put $c_{\mathrm{cl}}\defeq\frac{13}3\sqrt3$ and $R_{\mathrm{cl}}\defeq3\sqrt3$, the \emph{cluster distance}. The \emph{clusters} $K$ of a tuple $\tau$ are the components of the $c_{\mathrm{cl}}$-proximity graph on $P(\tau)$ (two centres adjacent when at distance $\le c_{\mathrm{cl}}$), with $P_K\defeq P(\tau)\cap K$; every line of the graph whose ends lie in different clusters is \emph{cut} and kept whole as an external line of both cluster graphs $H_K$, and $H_K$ is reduced with its own boxes $\square_K\subset\square''_K\subset\square'_K$, built from the vertices of $K$ as above, and with the gauge $B_K\defeq A_j(x^K_0)$, $x^K_0$ the centre of $\square'_K$; put $\Omega_K\defeq\square'_K\cap T_\eta$, with the Dirichlet convention above. A cluster is in zone~I if $\Omega_K\cap\mathsf F=\emptyset$ (then $\square'_K$ is not truncated and $\Omega_K=\square'_K$) and in zone~B otherwise. The geometry is elementary: two vertices in different clusters are at distance $>c_{\mathrm{cl}}-\frac43\sqrt3=R_{\mathrm{cl}}$, so every cut line has length $\ge R_{\mathrm{cl}}$; $\operatorname{diam}P_K\le c_{\mathrm{cl}}(|P_K|-1)$; $|\square'_K|\le C'p^3$; the quotient graph with the clusters as nodes and the cut lines as edges is connected, and $\dist(P(\tau))\le\sum_{\ell\in T}|\ell|+c_{\mathrm{cl}}n_\tau$ over any spanning tree $T$ of it; and the partition, the boxes, the gauges and the set of cut lines are the same for every member of every family and every position. The two members of the pair live on the same unit lattice and carry the same cut. \end{definition} \begin{definition}[the range]\label{r13:def:range} Put $C_{\mathrm{box}}\defeq\max(c_{\mathrm{cl}}N_*+6M+4,\,4M+7)$; it bounds, in units of $p$, the side of every cluster box $\square'_K$ and of the one box $\square'$ of a localisation with $R\le p$ (an elementary count, \cref{lem:k02:Zh}(e)). With $M(A)$ the operator of \cref{cpi:def:a02:setting} at depth $j$ on a box, $\tilde A_K\defeq A^{(j)}-B_K$ the cluster field, $V_j(\tilde A_K)\defeq M(\tilde A_K)-M(0)$ and $G_0(\Omega_K)$ the zero-background propagator on $\Omega_K$, let $C_\flat$ collect $\sqrt3\,C_{\mathrm{box}}$, the constant of the difference bound \cite[(3.45)]{K1} and the constant of the operator bound for the expansion of the reduced propagator in $\tilde A_K$, so that $\norm{V_j(\tilde A_K)G_0(\Omega_K)}_{\infty\to\infty}\le C_\flat\,e\,\eta^{1/2}p^{2}$ on every cluster box, from $|\tilde A_K|,|\partial\tilde A_K|\le C'p^2$ on $\Omega_K$ (the same with $\Omega_K$ replaced by the one box $\Omega$). Let $\eta_2\le\eta_1$ be so small that for $\eta\le\eta_2$, with $p=p(\eta)$, \begin{equation}\label{eq:k02:range} \eta^{-1}\ \ge\ 4p+8M+2c_{\mathrm{cl}}N_*^2+8,\qquad p\ \ge\ c_{\mathrm{cl}}N_*^2,\qquad C_\flat\,e\,\eta^{1/2}\,p^{2}\ \le\ \tfrac12; \end{equation} all three hold for $\eta$ small because $p(\eta)$ grows slower than every power of $\eta^{-1}$ and faster than every logarithm. The first condition contains the condition $\delta_*^{-1}>3p(\delta_*)+4M+1$ on the range of $j$ above; where the background is expanded in $\tilde A$ on one box, $\eta$ lies moreover in the range \eqref{eq:k02:range}, whose third condition is the convergence condition of that expansion on every cluster box $\square'_K$ and on the box $\square'$ of every localisation with $R\le p$, both of side $\le C_{\mathrm{box}}p$. \end{definition} \begin{lemma}[zones]\label{cpi:lem:a02:Z} Let a localisation have the points $\{y\}$ of its external fields and the centres $\{z\}$ of its vertices, and put $R\defeq\dist(\{y\},\{z\})$, the length of a shortest tree joining them. \textup{(i)} Every localisation with $R\le p$ that contains a source vertex lies in zone~I. \textup{(ii)} Every zone-B localisation containing a source vertex has $R\ge\eta^{-1}-2p-4M\ge\eta^{-1}/2$. \textup{(iii)} Every source vertex $s$ has $t(s)\ge\eta^{-1}/2\ge1$ and $\dist(s,\partial\square'\setminus\mathsf F)\ge2p-\sqrt3$. \end{lemma} \begin{proof} The sources are supported at continuum $\ell^\infty$-distance $>1$ from $\R^3\setminus\CJ$, that is at $\ge\eta^{-1}$ unit lengths from $\partial T_\eta$, and the summed curvature vertex sits on bonds whose weight vanishes within $1-\eps$ of the faces. In (i) every vertex lies within $p$ of the source vertex, $\square$ within $p+M$ and $\square'$ within $3p+3M<\eta^{-1}$ of it, by the first condition of \eqref{eq:k02:range}; (ii) and (iii) are the same arithmetic. \end{proof} \space% % % \begin{definition}[cube rewriting and remainder vertices]\label{cpi:def:a02:RC} Let a renormalised family be localised in zone~B (\cref{cpi:def:a02:zones}), its internal lines the cube slices with the Dirichlet condition on $\partial\Omega$ and its vertex positions ranging over the live sites of $\Omega$. Apply to it, step by step, the \emph{free rewriting} of its divergent source-free subgraphs, that is Ba{\l}aban's renormalisation procedure \cite[Sect.~3]{Ba3} in the form of \cite[(3.77)--(3.78)]{K1}: Taylor transport of the legs to a reference vertex, H\"older splits of the remainders, resummation over the orderings up to the lowest external index $j_0$, and evaluation of the resummed coefficients. Each evaluation of a resummed coefficient $c_S(x)$ of a divergent subgraph $S$ rooted at the live site $x$ is a \emph{C-site}. At a C-site write \[ c^{C}_S(x)=c^{F}_S(x)+\Delta_S(x), \] where $c^F_S(x)$ is the value of the same coefficient with the free $f_D$ lines and with the positions summed over $5^{-j}\Z^3$, so that $c^F_S$ is the coefficient of the free rewriting, which vanishes or is a local polynomial exactly as on the torus, and the \emph{remainder} $\Delta_S(x)$ is attached at $x$ as a generalised \emph{remainder vertex} $v_S$ carrying the legs of the evaluated term. Every graph carrying a remainder is \emph{halted}: no later step of the rewriting acts on a stage that contains a remainder vertex. The \emph{moment degree} of $S$ is $D'(S)\defeq D(S)$ when the zeroth moment is evaluated and $D'(S)\defeq D(S)+1$ for a first moment, $D(S)$ the degree of $S$ \cite[(2.1)--(2.3)]{Ba3}. The remainder vertices with $D'\in\{0,-1\}$ are the layered vertices of \cref{cpi:thm:a02:Bsc}, bounded there by face weights; the free parts $c^F_S$ are governed by the degree count on the free $f_D$ flow (\cref{cpi:thm:a02:T}). \end{definition} \space% % % \begin{lemma}[regularised cut lines]\label{cpi:lem:a02:XL} Fix a radial $\bar\psi\in C^\infty(\R^3)$ with $0\le\bar\psi\le1$, $\bar\psi(w)=0$ for $|w|\le\frac12$ and $\bar\psi(w)=1$ for $|w|\ge1$. For a cut line $\ell$ of the cut by clusters (\cref{r13:def:clusters}) between an end $u$ in the cluster $K$ and an end $v$ in the cluster $K'$ put \[ \ell^\flat(u,v)\defeq\bar\psi(u-v)\,\ell(u,v) \] (for an external line from $u$ to a point $y$ of the external field, $\bar\psi(u-y)\ell(u,y)$), and let \[ \ell^{\flat,\mathrm{ph}}\defeq U\big(B_K(\Gamma_{x_0^K,u})\big)^{-1}\,\ell^\flat\,U\big(B_{K'}(\Gamma_{x_0^{K'},v})\big) \] be its \emph{phase-dressed} form, the transporters along the staircases from the centres $x_0^K$, $x_0^{K'}$ of the two cluster boxes in the gauges $B_K$, $B_{K'}$ of the two clusters, which are the unitary factors that King's reduction \cite[(3.46)]{K1} attaches to the ends of a line leaving the box. Then: \begin{itemize}[nosep] \item[(a)] Replacing every cut line by $\ell^\flat$ leaves $E^{(j)}(H;\tau)$ unchanged, term by term: for every member of every family, every ordering and every position configuration, each pair $(u',v')$ of evaluation points at which a cut line enters, the lattice differences of the vertex functions at its ends included, satisfies $|u'-v'|\ge R_{\mathrm{cl}}-4\cdot5^{-j}>1$, where $\bar\psi=1$; here $R_{\mathrm{cl}}=3\sqrt3$ is the cluster distance (\cref{r13:def:clusters}), the four fine steps account for the differences at the two ends, and when an end lies on a block product of a block vertex the margin of the block product gives $|u'-v'|>1$ as well. \item[(b)] $\ell^{\flat,\mathrm{ph}}$ is $C^{1,\alpha_{\mathrm H}}\otimes C^{1,\alpha_{\mathrm H}}$ at unit scale in both arguments, where $\alpha_{\mathrm H}\in(0,1)$ denotes throughout the H\"older exponent of the line bounds, with the decay $e^{-\delta_1|u-v|}$, at all positions and with no distance condition: for all operations $O_u$, $O_v$ at its two ends of order at most $1+\alpha_{\mathrm H}$ (the identity, a lattice first difference, or a H\"older quotient of a first difference), \[ |O_uO_v\ell^{\flat,\mathrm{ph}}(u,v)|\le C\,e^{-\delta_1|u-v|}, \] the constant depending on $\bar n$, $d$, $L$ and $\alpha_{\mathrm H}$ only, times $(1+k-j)^5$ for a dressed line, the size of the coarse mass-counterterm blob it carries. This is the line bound \cite[(3.1)]{Ba3} in the form of a cut line with the endpoint localisations: the cut-off $\bar\psi$ removes the short-distance singularity, so the bound holds with one constant at every distance. \end{itemize} \end{lemma} \space% % % \paragraph{The degree count.} Ba{\l}aban's renormalised degree count \cite[Sect.~3, (3.5)]{Ba3}, in the form King uses on the torus \cite[(3.77)--(3.78)]{K1}, says the following. After the rewriting of the divergent subgraphs (Taylor transport of the legs, H\"older splits, resummation over the orderings up to the lowest external index $j_0$, and evaluation of the resummed coefficients $c_S$ on the free lattice, where they vanish by the counterterm, by sign pairing, by translation invariance or parity, or by the Ward identity of the vacuum polarisation), every ordered ultraviolet stage of a source-free renormalised family has positive degree, and the infrared counterterm lines, which lie in the mass-counterterm units and are attached to the rest of the graph at one vertex, obey \cite[(3.78)]{K1}: the degree of the whole graph is at most that of every prefix containing the ultraviolet lines. We call this the \emph{degree count} (DEG). In both references it is stated for the flow with the weight $f=1$; the lines of the zone-I graphs are those of the $f_D$ flow (\cref{cpi:def:a02:setting}), and the transfer rests on the following lemma. \begin{lemma}[the two free flows differ by a top-scale kernel]\label{cpi:lem:a02:RSB} Let $\Delta G_i\defeq G^{f_D}_i-G^{1}_i$ on $5^{-i}\Z^3$ at $A=0$, in the units of depth $i$. For $a,b\in\{0,1\}$ and every $\gamma<1$, \[ |\partial^a_x\partial^b_y\Delta G_i(x,y)|\le C_\gamma5^{-\gamma i}e^{-\kappa'|x-y|}, \] and $\partial_x\Delta G_i(\cdot,y)$ is $\alpha'$-H\"older with the constant $C_{\alpha'}5^{-si}$, $s=(1-\alpha')/(1+\alpha')$, times the same decay. In the units of depth $j$ the kernel of level $i$ is the rescaled one, so $\Delta G_i$ obeys the slice bounds \cite[(3.63)--(3.65)]{K1} of the index $i$ times $5^{-\gamma i}$: it is a line without short-distance singularity. \end{lemma} \begin{proof} Both operators have the same $C^{-1}=-\Delta+m_i^2$, so $\Delta G_i=G^{f_D}RG^1$ with $R(u,u')=\mathfrak a_i[1-w(u)w(u')]\mathbf 1[B(u)=B(u')]$, $B(u)$ the block of $u$. Writing $1-w(u)w(u')=(1-w(u))+w(u)(1-w(u'))$ and using $\sum_{u\in B(y)}5^{-3i}G^{f_D}(x,u)w(u)=H^{f_D}(x,y)/\mathfrak a_i$ we get \[ \Delta G_i(x,x')=\mathfrak a_i\sum_yD_0(x,y)g_1(y,x')+\sum_yH^{f_D}(x,y)g_2(y,x'), \] with $D_0(x,y)\defeq\sum_{u\in B(y)}5^{-3i}(1-w(u))G^{f_D}(x,u)$, $g_1(y,x')\defeq\sum_{u'\in B(y)}5^{-3i}G^1(u',x')$ and $g_2(y,x')\defeq\sum_{u'\in B(y)}5^{-3i}(1-w(u'))G^1(u',x')$. The point is a \emph{dipole property} of the weight: every unit block is a union of finest cells $c$ of side $5^{1-i}$ on each of which $\sum_{u\in c}(1-w(u))=0$, because $\sum_{u\in c}w=\prod_\mu\sum_{r=0}^4w_1(r)=5^3=|c|$. Hence $\sum_{u\in B}5^{-3i}(1-w(u))h(u)=\sum_c\sum_{u\in c}5^{-3i}(1-w(u))[h(u)-h(u_c)]$ is at most $C\sum_c|c|\operatorname{osc}_ch$, and for $|\partial h|\le C(|x-v|+5^{-i})^{-\mathsf p}$ this is $\le C5^{-i}$ when $\mathsf p=2$ and $\le C5^{-i}(1+i)$ when $\mathsf p=3$. The free kernel bounds $|\partial^aG|\le C(|x-u|+5^{-i})^{-1-a}e^{-\kappa|x-u|}$ then give $|D_0|,|g_2|\le C5^{-i}e^{-\kappa\dist}$, $|\partial D_0|,|\partial g_2|\le C5^{-i}(1+i)e^{-\kappa\dist}$, $|g_1|,|\partial g_1|\le Ce^{-\kappa\dist}$ and $|\partial^aH^{f_D}|\le Ce^{-\kappa|x-y|}$; every product contains one dipole factor and the sums over $y$ converge. For the H\"older clause interpolate the cell oscillation of $h(u)=\partial_xG(x,u)-\partial_xG(\tilde x,u)$ between the H\"older bound $C|x-\tilde x|^{\alpha''}r_c^{-2-\alpha''}$ and $C5^{1-i}r_c^{-3}$, with $\alpha''\defeq(1+\alpha')/2$. \end{proof} \begin{theorem}[the degree count on the free $f_D$ flow]\label{cpi:thm:a02:T} Every source-free renormalised prefix component $S$ of the free $f_D$ flow, at zero background, admits the rewriting of \cite[Sect.~3]{Ba3} with: \textup{(i)} every resummed coefficient that vanishes exactly for $f=1$ vanishes exactly for $f_D$; \textup{(ii)} $c_a^{f_D}=c_a^1+c_a(M_S)$, where $|c_a^1|\le C\rho_{j_0}^{D(S)+|a|}$ is the moment of the $f=1$ flow and $|c_a(M_S)|\le C_\gamma5^{-\gamma i_0}\rho_{j_0}^{D(S)+|a|}$, $i_0$ the depth of the lowest external index $j_0$ of $S$; \textup{(iii)} the remainders have the same shapes as for $f=1$. Hence \textup{(DEG)} holds for the $f_D$ flow with the degree of every stage unchanged, and its infrared half \cite[(3.78)]{K1} holds with the mass-counterterm blob $U^{f_D}$ differing from $U^1$ by $C(\eps_k/\eta)^\gamma\eta^{1-\varsigma}$ in the units of \cite[(3.78)]{K1}. The $f=1$ statement of \textup{(ii)} on the rows whose coefficient vanishes by the zero-momentum gauge covariance (the Ward rows) is King's reduction \cite[(3.39)--(3.55)]{K1}, run on the cube with the gauge $B=A_j(x_0)$ and, for a cluster of the cut by clusters (\cref{r13:def:clusters}), inside the cluster with its own gauge $B_K$; on every other row it is the member-wise identity (charge parity, the derivative at a line end, reflection, the $\eta$-suppressed tadpole, or the counterterm matching between partners of identical vertex structure). \end{theorem} \begin{proof} Resum the lines of $S$ to $G^{f_D}_{j_0}$, write $G^{f_D}=G^1+\Delta G$ on every line and expand multilinearly: $F^{f_D}_S=F^1_S+M_S$, where $F^1_S$ is the inside family of the same class with $f=1$ lines (the members are the same graphs, since every vertex of a divergent class is weight-free; the counterterms $\dm$ and $E_1$ have the same value in both flows; a counterterm piece all of whose lines are $G^1$ is the $f=1$ decomposition of the same counterterm) and $M_S$ is the sum over the members and over the non-empty sets $T$ of lines carrying $\Delta G$. \emph{(i).} The exact-zero mechanisms (the counterterm per scale assignment, pairing, parity, the telescoping of a summable kernel) are identities for every charge-diagonal line kernel when the partners carry the same lines, so they hold for the $f_D$ lines directly. \emph{(ii).} In a term of $M_S$ treat each $\Delta G$ line as one line of the fixed index $j_0$; by \cref{cpi:lem:a02:RSB} it obeys the slice bounds of the index $j_0$ times $5^{-\gamma i_0}$. In any ordering the first $\Delta G$ line forces the index $j_0$ on every later line, so the free index sums run over $G^1$ slices only, and the prefixes containing a $\Delta G$ line enter no geometric sum. Each prefix component $\gamma$ with $D(\gamma)\le0$ before the first $\Delta G$ line is a source-free, block-free $f=1$ subgraph whose complete inside family, partners included, is present at fixed outside, because replacing $\gamma$ by another member of its class does not touch $T$; renormalise it by the $f=1$ rewriting, finest first, with the $\Delta G$ ends as legs at the scale $\rho_{j_0}$ (Lipschitz, with an $\alpha'$-H\"older first difference, which is what the transport and Taylor steps use), the debits landing on a $\Delta G$ line being paid at $\rho_{j_0}$. The doubly differentiated single line between two current vertices whose other ends carry $\Delta G$ is rewritten by the free-lattice identity $\sum D_xD_yG^1(\cdot)=0$ against a constant, whose transport remainder carries $|\Delta G(\cdot,v_2)-\Delta G(\cdot,v_1)|\le C|v_1-v_2|5^{-\gamma i_0}\rho_{j_0}^{-2}$. After this every prefix with a non-trivial index sum has positive degree, the lines obey their slice bounds, the moment factors $|x_i-x|^{a_i}$ are distributed along paths of lines (costing $\rho_{j_0}^{a_i}$ on a $\Delta G$ line), and the multiscale bound \cref{cpi:thm:a02:S3}(a) with the root held gives $|c_a(M_S)|\le C\rho_{j_0}^{D(S)+|a|}5^{-\gamma i_0|T|}$. \emph{(iii)} is the same argument. \emph{The infrared half.} The prefix condition of \cite[(3.78)]{K1} is a statement about the lines and vertices of the blob and is the same for both flows; its proof uses of the lines only their dimensions, the shapes (i)--(iii) of the rewriting at the indices below the infrared ones and the counterterm identities, so it holds for $f=1$ and for $f_D$ alike. For the value of the blob, $U^{f_D}=U^1-(\text{the terms with at least one soft line }\Delta G_j)$, and each such term is a skeleton term of the $f=1$ argument in which the soft line obeys the soft-line bounds with the constant $C(\eps_k/\eta)^\gamma$ (\cref{cpi:lem:a02:RSB} in physical units), which gives the stated difference. The remaining estimates are routine given the slice bounds of the lines and \cref{cpi:thm:a02:S3}(a). \end{proof} \space% % % \begin{theorem}[the split degree governs the multiscale bound]\label{cpi:thm:a02:S3} Let $\mathcal R$ be a renormalised family of King's expansion of order $\le\bar n$ (\cref{cpi:def:a02:setting}), written as a finite sum of graphs $G$ with the source vertices admitted. Write $G^{\mathrm{sp}}$ for $G$ with every block vertex \cite[(3.25)--(3.26)]{K1} split into its \emph{points}, each point carrying one scalar leg and the staircase fields of its own staircases, so that the staircase line of a field is a line of its own, the two summation points of a two-point block product being joined by a \emph{$\kappa$-line}; weight a vertex $v$ of a connected stage $H$ of $G^{\mathrm{sp}}$ by the \emph{split count} \[ w_v(H)=3+a_v-\tfrac12N_v+\tfrac12n^{\mathrm{ext}}_v(H)-\nu_v(H)+c_v(H),\qquad D^{\mathrm{sp}}(H)\defeq\sum_{v\in H}w_v(H)-3, \] where $a_v$ is the explicit power of the vertex, $N_v$ the number of all its legs, $n^{\mathrm{ext}}_v(H)$ the number of those which are ends of no line of $H$, $\nu_v(H)$ the number of derivatives on the legs which are ends of lines of $H$, and the \emph{staircase credit} $c_v(H)$ is the number of photon fields of $v$ whose line lies in $H$ if $v$ is a block point and $0$ otherwise; so a leg external to $H$ contributes $0$, an undifferentiated end of a line of $H$ contributes $-\frac12$ and a differentiated one $-\frac32$. Assume that, for every member that does not vanish identically at zero background, taken at zero background, and for every term of the expansion in the background field of such a member at a regular background on a box whose faces lie on block planes ($\CK$ among them), for each admitted ordering of the lines of $G^{\mathrm{sp}}$: every component of every ultraviolet prefix has current degree $D'>0$ after the rewriting (the degree count of \cref{cpi:thm:a02:T} in the split count); the infrared lines, all on the counterterm lines of the $\dm$ units, obey \cite[(3.78)]{K1}; and the lines obey the slice bounds \cite[(3.63)--(3.65)]{K1} up to the faces of the box (for a term at a regular background, its zero-field lines). Let $\alpha_0>0$ be the minimum of these degrees. Then, on the torus and on $\CK$, for every such graph $G$: \begin{itemize}[nosep] \item[(a)] the sum over orderings, compatible indices and positions of $|E^{(j)}(G)|$ is at most \[ C\eta^{\rho_0}p^{n_1}(1+k-j)^{p_0}\exp[-\delta\dist(\text{external vertices})], \] $\rho_0$ the coupling power of $G$ \cite[p.~665]{K1}; the same holds with one element weakened by $\theta<\alpha_0$ (a line with $c_l\to c_l\rho_{i_l}^{-\theta}$, a vertex with an explicit power $a_v\ge\theta$, or an external field); \item[(b)] at zero background, \[ |E'^{(j+1)}(G)-E^{(j)}(G)|\le C\eta^{\rho'_0}5^{-\gamma j}\exp[-\delta\dist] \] for $0<\gamma<\min(\alpha_0,\gamma_{\mathrm c},\alpha_{\mathrm H},1-\alpha_{\mathrm H})$, where $\alpha_{\mathrm H}\in(0,1)$ is the H\"older exponent of the line bounds (\cref{cpi:lem:a02:XL}) and $\gamma_{\mathrm c}>0$, depending on $\bar n$ and $\alpha_{\mathrm H}$ only, is the rate at which the resummed local coefficients of the rewritten units of the two members agree; \item[(c)] every rewritten subgraph is source-free, point-free and a component of a prefix of $G^{\mathrm{sp}}$, and $D'(\gamma)\ge D^{\mathrm{sp}}(\gamma)$ for every component $\gamma$ of every prefix; \item[(d)] the exponent of every stage of the ordered bound is $D'(H_t)$ in the split count, the credit for a staircase field being booked only when its staircase line lies in the stage, unlike King's per-vertex ``$+m$'' \cite[(3.30)]{K1}, which credits all $m$ fields of \cite[(3.26)]{K1} whether or not their lines are present. \end{itemize} At a regular background only (a), (c) and, with ``is at least $D'(H_t)$'' in place of ``is $D'(H_t)$'', (d) are claimed, on the boxes named above and not on the torus. The constants depend on $\bar n$, on the point of King's range and on the line constants, not on $j$, $k$, $\eta$. \end{theorem} \begin{proof} \emph{(a).} Apply the multiscale lemma of \cite[(3.66)--(3.70)]{K1}, a line-by-line induction over the ordering in which each step shrinks one line and sums one of its endpoints, the index sums being geometric by the positive degree, to the connected graph $G^{\mathrm{sp}}$ with the slice bounds of its lines. The degree of $G^{\mathrm{sp}}$ is $D^{\mathrm{sp}}$. A stage containing $\kappa$-lines is not a stage of the split count: if $c_1,\dots,c_r$ are its components after the $\kappa$-lines are deleted, each containing a point, its degree is $\sum_iD^{\mathrm{sp}}(c_i)+3(r-1)\ge2$, since an isolated point has $D^{\mathrm{sp}}=0$ and a component with a point and a line has $D^{\mathrm{sp}}\ge1$ (a block point weighs $3-\frac s2+\frac p2\ge\frac52$, $s\in\{0,1\}$ and $p\ge0$ the numbers of ends of lines at its scalar leg and at its staircase fields). Infrared stage: points lie in no counterterm unit, so no infrared line ends at a point; the $\kappa$-line has index $j\le$ every infrared index, and shrinking it first transfers only the external lines at one summation point to the other and gives $\sum_{x'}5^{-3j}\kappa\le1$, so that the infrared differences of \cite[(3.80)]{K1} are unchanged. Fixed lines, remainders and the position sum are as in \cite[pp.~663--664]{K1}, and the weakened element is handled by the same induction with $D'-\theta\ge\alpha_0-\theta>0$. On $\CK$, (a) holds verbatim with the cube line bounds (image sums of free kernels), position sums over $\CJ$ being bounded by sums over $\Z^3$; at a regular background the induction runs on the majorant of the expansion in the background field with halved decay rates, which gives (a), (c) and the weak form of (d). \emph{(b).} This is \cite[pp.~663--665]{K1} on $G^{\mathrm{sp}}$. The finer slices of the finer member give $5^{-\gamma j}$ through the finest index sum; otherwise one factor is replaced at a time, the external lines by \cite[(3.71)]{K1}, the sources by sampling ($|h(x')-h(x)|\le\sqrt3\,\eps_k\norm{\nabla h}_\infty$), the phase by \cite[(3.72)]{K1}, the plain lines by \cite[(3.73)]{K1}, the staircase lines by \cite[(3.74)]{K1}, the H\"older lines by \cite[(3.75)]{K1}, the local coefficients of the rewritten units at the rate $\gamma_{\mathrm c}$, and an explicit power of the spacing at a vertex by its own smallness, each error being the graph with one element weakened by $\gamma$ and the prefactor $5^{-\gamma j}$. Then the position sums of the two members are paired: the finer member's cell totals are exact, $\sum_{x'\in B^1(x)}5^{-3(j+1)}f'_D(x')=5^{-3j}$ at every alive coarse site $x$ (on $\CK$ the sum runs over the cell inside the box, $f'_D$ vanishing at the finer plane sites), and the other internal vertices are paired over the alive coarse sites. On $\CK$ three further terms arise. A term with one non-point vertex of the finer member restricted to the \emph{top layer}, the sites of the finer lattice in the last fine layer below a Dirichlet plane, which have no coarse partner (one such vertex per term, by the union bound), gives at the step at which that vertex is summed through a line of index $i\ge0$ the factor $C5^{-\gamma j}\rho_i^{3-\gamma}$ in place of $C\rho_i^3$, that is the graph with that line weakened by $\gamma$ and the prefactor $5^{-\gamma j}$; a held vertex of the finer member in the top layer has no partner and gives the factor $\frac{12}5\,5^{-j}$ per half-open unit cube; and the coarser member's weight $f_D$ differs from the uniform measure by a sum of differences of the graph at points of one level-$1$ block \cite[(3.13)]{K2}, each $C5^{-\gamma j}$ times the graph with one element at the point weakened by $\gamma$, by the H\"older clauses of the cube lines on the closed box (no point is a source vertex, so the sources are not involved). Finally $\kappa'=\kappa$ up to the block parameters, which differ by $|\mathfrak a'_{j+1}-\mathfrak a_j|\le C5^{-2j}$ (\cref{def:k01:bulkflow}). At $j=0$, where $5^{-\gamma j}=1$, (b) follows from (a) for the two members. For the comparison bounds the cube lines obey the pair clauses of the two-end lines, of the staircase lines and of their H\"older forms, transferred to the photon staircase lines on the photon box, and the phase of a staircase of length $\le3$ is compared through $|e^{ia}-e^{ib}|\le|a-b|$. \emph{(c).} A rewritten subgraph is divergent, so it contains no point (a component with a point and a line has $D^{\mathrm{sp}}\ge1$) and is one of Ba{\l}aban's divergent classes, which contain no block vertex \cite[pp.~428--431]{Ba3}; debits land on legs of the rewritten subgraph $S$, and if a leg lies in $\gamma$ then $S\subset\gamma$, so the net change per component is $\ge0$. \emph{(d)} is the booking of the dimensions: the credit $+1$ for a staircase field of a point is the factor $c_l$ in the dimension of its staircase line, booked at the point for the lines of the stage only, so it enters exactly the stages containing that line. The remaining estimates are routine given the slice bounds and the degree count. \end{proof} \space% % % \begin{definition}[the setting of the zone-B photon pieces]\label{r13:def:BVPset} Let a member family of the cube expansion be localised in zone~B (\cref{cpi:def:a02:zones}), every member carrying a source vertex or an external field (so that no member is a field-free vacuum graph), every photon two-point prefix component $S'$ rewritten by the extraction of the zeroth and first moments of \cite[(3.26)--(3.31)]{Ba3} with the lines $G_{j_0}=C^\xi_D+G_{j_0}(1-m^2_{j_0}-\mathfrak a_{j_0}P_{j_0})C^\xi_D$ (in the units of depth $j_0$), $C^\xi_D$ the Dirichlet covariance of the box, and every other divergent source-free stage rewritten as in \cref{cpi:def:a02:RC}. Inside each collapsed piece write every cube line as the free line plus its image part and every internal sum over $\Omega$ as the sum over $5^{-j}\Z^3$ minus the sum outside $\Omega$, and rewrite the free piece by the free rules. \end{definition} \begin{theorem}[the zone-B photon pieces]\label{cpi:thm:a02:BVP} Let a member family be as in \cref{r13:def:BVPset}. Then: \begin{itemize}[nosep] \item[(a)] The cube is compared with the free lattice only inside the collapsed pieces and at the two cancellation sites of the bulk structure, the free counterterm against the scalar local parts and the reflection symmetry for the first moment, piece by piece: for the one-loop class the zeroth moment of the $C^\xi_D$-part vanishes on both sides (an identity for the full kernel of the class, not a per-ordering one), leaving the residual $\Delta c^{S'}$; for a class with $D(S')=0$ the whole difference $c^\Omega-c^F$ of the zeroth moment is one piece; the \emph{face piece} $M$ comes from the first moment of the one-loop bubble with one leg differentiated; and there are the remaining differences. \item[(b)] Each difference piece is a generalised vertex at the root $x$ of $S'$, at the stage $j_0(S')$, bounded by its free power count times $\omega(x)+\omega_p$, with $\omega$ a layered weight of class $\mathcal L(\rho_{j_0},m,\theta)$ (\cref{cpi:def:a02:lay}) and $\omega_p\le Ce^{-\delta p/\rho_{j_0}}$ a term supported away from $\mathsf F$, of any prescribed degree $N$ at the cost $C_Ne^{-\delta p/2}$: the face piece $M$ is of class $\mathcal L(\rho_{j_0},1,0)$ and of the degree $D(S')+1$ of the bulk first-moment vertex; the residual satisfies $|\Delta c^{S'}(x)|\le C\rho_{j_0}^{D(S')}[e^{-\delta t(x)/\rho_{j_0}}+e^{-\delta p/\rho_{j_0}}]$ times the bounds of the legs, one line being replaced by its image part, for which $\sum_w5^{-3j}|\nabla G^{\mathrm{img}}_{j_0}(x,w)|\le C\rho_{j_0}e^{-\delta't(x)/\rho_{j_0}}$; it is of class $\mathcal L(\rho_{j_0},0,0)$ and degree $D(S')$, and for $D(S')=0$ it is entered together with $c^{(C)\Omega}-c^{(C)F}$ as the whole difference $c^\Omega-c^F$, of class $\mathcal L(\rho_{j_0},1,0)$; the difference of the first moment of the replaced-line part is of class $\mathcal L(\rho_{j_0},0,0)$ and degree $D(S')+1$; and the differences of the Taylor remainders are of class $\mathcal L(\rho_{j_0},0,0)$ and degree $D(S')+1+\alpha_{\mathrm H}$ when both moments are extracted ($D(S')=-1$) and $D(S')+\alpha_{\mathrm H}$ when only the zeroth moment is ($D(S')=0$), in both cases $\ge\alpha_{\mathrm H}$. The members with a block vertex produce no difference piece: they are not collapsed, and every stage containing one of their points has split degree $\ge1$ (\cref{cpi:thm:a02:S3}). No piece has the weight $t^{-1}$. \end{itemize} \end{theorem} \begin{proof} \emph{(a).} The image representations of the cube lines are exact and the rewriting rules are identities on each side, so the difference of two rewritten pieces is the sum of the differences of their parts; the two cancellation sites are the only places where the free structure of the coefficients is used. For the one-loop class the zeroth moment of the $C^\xi_D$-part vanishes on both sides by the gauge invariance of the scalar functional under constant shifts of the photon field on the simply connected cube. \emph{(b).} The component $S'$ is connected, has exactly two photon legs $\Phi_1$ at the root bond $b=(x,\mu)$, of index $j_0$, and $\Phi_2$ at $b'=(x',\nu)$, of index $j_2\ge j_0$, contains no source vertex and no block point, and has $D(S')\in\{-1,0\}$, with $D(S')=-1$ only for the one-loop class (the bubble and the one-vertex tadpoles). Resum its internal lines to the cumulative line $G_{j_0}$, which on $\Omega$ is the Dirichlet resolvent $(-\Delta^\Omega_D+\rho^{-2}\tilde m^2+\rho^{-2}\tilde{\mathfrak a}P_{j_0})^{-1}$, $\rho=\rho_{j_0}$ ($\tilde m^2=m^2_{j_0}$ and $\tilde{\mathfrak a}=\mathfrak a_{j_0}$ the mass and the stiffness of depth $j_0$, the factor $\rho^{-2}$ converting them to the units of depth $j$), and split it as Ba{\l}aban does \cite[p.~437]{Ba3}: $G_{j_0}=C^\xi_D+R_{j_0}$ with $C^\xi_D=(-\Delta^\Omega_D+\rho^{-2})^{-1}$ and $R_{j_0}=G_{j_0}K_{j_0}C^\xi_D$, $K_{j_0}=\rho^{-2}(1-\tilde m^2-\tilde{\mathfrak a}P_{j_0})$; the kernel $\Pi(b,b')$ of the class, the sum over the inside family with the positions summed over $\Omega$, splits multilinearly as $\Pi^{(C)}+\Pi^{(R)}$, the second collecting the terms with at least one $R$-line. The rewriting at the root Taylor-expands $\Phi_2$ at $b$: $\sum_{b'}\Pi(b,b')\Phi_2(b')=\Phi_2(b)c(x)+\partial_s\Phi_2(b)m_s(x)+\sum_{b'}\Pi(b,b')\tau_2(b,b')$ with the zeroth moment $c(x)=\sum_{b'}\Pi(b,b')$, the first moment $m_s(x)=\sum_{b'}\Pi(b,b')(x'-x)_s$ and $|\tau_2|\le C|x'-x|^{1+\alpha_{\mathrm H}}\rho_{j_2}^{-1-\alpha_{\mathrm H}}\bar\Phi_2$ (for $D(S')=0$ only the zeroth moment is extracted and the remainder is of order $\alpha_{\mathrm H}$). Each moment and remainder splits into its $(C)$- and $(R)$-parts; a quantity computed on the cube carries the superscript $\Omega$, the same expression with free lines and positions over $5^{-j}\Z^3$ the superscript $F$, and $\Delta=(\cdot)^\Omega-(\cdot)^F$. For the one-loop class $c^{(C)\Omega}=c^{(C)F}=0$; for $D(S')=0$ the difference $c^{(C)\Omega}-c^{(C)F}$ is kept and entered together with the residual; by the reflection symmetry \cite[(3.27)--(3.28)]{Ba3} $m^{(C)F}=0$, while $m^{(C)\Omega}=M$ is the face piece, the first moment of the bubble with the lines $C^\xi_D=\sum_g\operatorname{sgn}(g)C^\xi(\cdot,g\,\cdot)$ whose non-identity images have length $\ge t(x)$, which gives the weight of class $\mathcal L(\rho,1,0)$, the logarithm coming from the sum over the slices. So the difference pieces are $M$ and $\Delta m^{(R)}$ (for $D(S')=-1$), $\Delta c^{S'}$ with $\Delta c^{(C)}$ when $D(S')=0$, and the remainder differences $\Delta r^{(C)}$, $\Delta r^{(R)}$; each contains an image line of length $\ge t(x)$ or a position sum outside $\Omega$, at distance $\ge p$ from the vertices, which gives the exponentials $e^{-\delta t(x)/\rho}$ and $e^{-\delta p/\rho}$, and the free power count gives the prefactor $\rho^{D(S')}$ or $\rho^{D(S')+1}$. A piece is read as a generalised vertex $v$ at $x$ with $a_v$ its moment degree and $u_v=3+a_v-n_v/2-\epsilon_v$, $\epsilon_v$ the number of its differentiated legs internal to the stage; a remainder is read, as in \cite[Prop.~2.2, p.~428]{Ba3}, as a vertex of explicit power $\rho^{D'}$ joined to $x'$ by a generalised line of index $j_0$ whose kernel sums to $C$ over $x'$ at the scale $\rho$, equivalently as the graph $S'$ with a credit $\rho_{i_l}^{\theta_l}$ on its lines, $\sum_l\theta_l=1+\alpha_{\mathrm H}$ (resp.\ $\alpha_{\mathrm H}$), and the H\"older debit on the leg $\Phi_2$. The members with a block vertex are not part of the C-site and are treated separately: their stages have split degree $\ge1$ and need no layer (\cref{cpi:uc:lem:UC}). The image-part bound in (b) holds because $|x-gw|\ge\max(|x-w|,t(x))$ for every non-trivial reflection $g$ of the box. That the rewriting with the block part left uncollapsed is exact, and that no layered vertex then contains a block vertex or a block point, is the \emph{no-collapse lemma} for the block part, \cref{cpi:uc:lem:UC} below. \end{proof} \begin{theorem}[the zone-B photon pieces across the pair and at the root]\label{r13:thm:BVPpair} Let a member family be as in \cref{r13:def:BVPset}, with the difference pieces of \cref{cpi:thm:a02:BVP}(b). Then: \begin{itemize}[nosep] \item[(c)] The degree condition with layered vertices of \cref{cpi:thm:a02:Bsc} holds for every difference graph, and for $0<\gamma<\min(\alpha_0^B,1-\theta_{\max},\gamma_{\mathrm c})$, $\gamma_{\mathrm c}\le\min(\alpha_{\mathrm H},1-\alpha_{\mathrm H})$ the rate of \cref{cpi:thm:a02:S3}(b), every difference piece obeys the pair bound $5^{-\gamma j}\rho_{j_0}^{-\gamma}\times$(its bound), its weight weakened to the class $\theta+\gamma$; so, given the pair clauses of the lines and legs, their H\"older quotients included, the ordered bound with layered vertices applies and gives the pair bound with $5^{-\gamma j}$ and the root weight $\Lambda_\gamma$ of \cref{cpi:thm:a02:Bsc}(b). For a member with a line $\delta G_j(\square',\cdot)$, bounded in each model separately, the bound is in the two-term form of \cref{cpi:thm:a02:Bsc}(b); for these members and for the members with block points the pair rate is proved by the \emph{joint boundary and block-vertex statement}, \cref{cpi:uc:thm:J} below, which proves the root-weighted bound and the pair rate together for $\gamma$ in its own range, with which the range stated here is intersected. \item[(d)] In a host rooted at an interior source vertex ($t(x_0)\ge\eta^{-1}/2$), $\Lambda(x_0)\le C$ and $\Lambda_\gamma(x_0)\le C$: the zone-B photon pieces cost a constant. For the source-free field-dependent classes, root-summed, $\sum_{x_0}5^{-3j}\Lambda(x_0)\le C$ and $\sum_{x_0}5^{-3j}\Lambda_\gamma(x_0)\le C$ per unit face cell, the pair difference carries $5^{-\gamma j}$ at the localisations with $R\le p$ (in the two-term form for the members with a line $\delta G_j(\square',\cdot)$, by \cref{cpi:uc:thm:J}(J-b); at $R>p$ each member is bounded separately), and the zone-B total is $\le C|\partial T^{(j)}|\le C\eta|T^{(j)}|$ relative to the bulk form. \end{itemize} \end{theorem} \begin{proof} \emph{(c).} For every remainder graph of \cref{cpi:thm:a02:Bsc} and every difference graph here, the hypotheses of the ordered bound with layered vertices hold, the layered vertices being the $D'=0$ remainder vertices, the charged $D'=-1$ vertices after borrowing and the difference pieces of \cref{cpi:thm:a02:BVP}(b), the residual of every one-loop class included, all of classes with $m\le1$; every component of a stage that contains a difference piece contains a line at it (so the piece is never a component of its own), and $\alpha_0^B\ge\alpha_*>0$ with $\alpha_*$ depending on $\bar n$ and $\alpha_{\mathrm H}$ only. The pair hypothesis of the ordered bound holds piece by piece: the residual of a one-loop class by the two-cutoff comparison of the one-loop kernel across the pair; the face piece together with the first moment of the replaced-line part by the comparison of the Dirichlet image sums of the two members; the remainders, and for $D(S')=0$ the whole zeroth moment $c^\Omega-c^F$, by the comparison of the Taylor remainders and of the cube line differences; in each case the difference is $5^{-\gamma j}\rho^{-\gamma}$ times a weight of the class with $\theta$ raised by $\gamma$, for $\gamma<\gamma_{\mathrm c}$. The residuals do not change $\theta_{\max}$ or $\Lambda$, and the index $j_0$ of a residual is the index of its first line and is summed with it. \emph{(d).} The root weight $\Lambda$ of \cref{cpi:thm:a02:Bsc} is $\le C$ at $t(x_0)\ge1$, and $\sum_x5^{-3j}\omega(x)\le C\rho_w$ for every layered weight of width $\rho_w$ and class $\theta<1$, summed over the face cells; across the pair the root weight is $\Lambda_\gamma$, which is $\le C$ at $t(x_0)\ge1$ and root-summable since $\theta_{\max}+\gamma<1$. This replaces the per-ordering photon boundary step of \cite[pp.~331--332]{K2}: the cube is never compared with the torus ordering by ordering; the comparison is made once, inside the collapsed pieces, and every remainder is a layered vertex bounded by the ordered bound with layered vertices. The remaining estimates are routine given the slice bounds of the cube lines up to the faces and their pair clauses. \end{proof} \space% % % \begin{definition}[layered weights]\label{cpi:def:a02:lay} A weight $\omega$ on $\Omega$ is \emph{layered} of class $\mathcal L(\rho,m,\theta)$, where $5^{-j}\le\rho\le1$, $m\le\bar n$, $0\le\theta<1$ and $\delta_b>0$, if \[ |\omega(x)|\le\big(1+\ln_+(\rho/t(x))\big)^m(\rho/t(x))^\theta e^{-\delta_bt(x)/\rho}\qquad\text{for every live site $x$}, \] $t(x)=\dist(x,\mathsf F)$ as in \cref{cpi:def:a02:zones}. A vertex $v$ is \emph{layered} if its vertex function is a bounded function times $\omega_v(x_v)$ with $\omega_v\in\mathcal L(\rho_v,m,\theta)$ and $\rho_v\le\rho_{j(v)}$, $j(v)$ the lowest index of the lines at $v$. \end{definition} \begin{definition}[the setting of the zone-B scalar classes]\label{r13:def:Bscset} Let $\mathcal R$ be a renormalised family of order $\le\bar n$ of King's expansion of $P^{(j),1}$ on $\CK$ at zero background, localised in zone~B, with at least one external field or at least one source vertex, rewritten by \cref{cpi:def:a02:RC} (its vacuum-polarisation prefix components by \cref{cpi:thm:a02:BVP}), and reduced on one box: the box $\square'$ of a localisation with $R\le p$, or the box $\square'_K$ of a zone-B cluster of a tuple with several clusters, the cut lines being external fields of index $j$. Fix $0<\alpha_{\mathrm H}<1$ (the H\"older exponent of the line bounds, \cref{cpi:lem:a02:XL}), $0<\beta_{\mathrm H}<\alpha_{\mathrm H}/2$ (the \emph{borrowing exponent} of step (2) of the proof of \cref{cpi:thm:a02:Bsc}) and $0<\gamma<\min(\alpha_0^B,\beta_{\mathrm H},\gamma_{\mathrm c},1-\alpha_{\mathrm H})$, where $\alpha_0^B$ is the minimum positive stage degree over the finitely many graphs and orderings and $\gamma_{\mathrm c}$ the rate of \cref{cpi:thm:a02:S3}(b). \end{definition} \begin{theorem}[the zone-B scalar classes]\label{cpi:thm:a02:Bsc} Let $\mathcal R$, $\alpha_{\mathrm H}$, $\beta_{\mathrm H}$ and $\gamma$ be as in \cref{r13:def:Bscset}. Then: \begin{itemize}[nosep] \item[(a)] with one vertex $x_0$ held (or none), the sum over orderings, compatible indices and positions of $|E^{(j)}(G)|$ over the members of $\mathcal R$ is at most \[ C\eta^{\rho_0}p^{n_1}(1+k-j)^{p_0}\exp[-\delta\dist(V_*)]\,\Lambda(x_0), \] $V_*$ the vertices with external legs, the held vertex and the source vertices, with the \emph{face weight} \[ \Lambda(x_0)\le C\big(1+\ln_+(1/t(x_0))\big)^{\bar n+2}\,t(x_0)^{-(1-\beta_{\mathrm H})}\,e^{-\delta_\Lambda t(x_0)}, \] $\Lambda=1$ if no vertex is held, $\Lambda(x_0)\le C$ if $t(x_0)\ge1$ (in particular at every source vertex), and $\sum_{x_0\in Q}5^{-3j}\Lambda(x_0)\le C$ for every unit cube $Q$; \item[(b)] across the pair, on the box $\square'$ of a localisation with $R\le p$ (on a cluster box only (a) is claimed), $|E'^{(j+1)}-E^{(j)}|\le C5^{-\gamma j}\times$(the bound (a) with $t^{-(1-\beta_{\mathrm H})}$ replaced by $t^{-(1-\beta_{\mathrm H}+\gamma)}$) for $0<\gamma<\gamma_b$, with the threshold $\gamma_b\defeq\min(\alpha_0^B,\beta_{\mathrm H},\gamma_{\mathrm c},1-\alpha_{\mathrm H},\alpha_0,\alpha',\gamma_{F3}/2)$ for a fixed $\alpha'\in(0,1)$, independent of $j$, $k$, $\eta$, $J$, $x_0$ (with $\alpha_0$ and $\gamma_{F3}$ as in \cref{cpi:uc:thm:J}); for every member with a line $\delta G_j(\square',\cdot)$ (the remainders of \cite[(3.52)--(3.54), p.~665]{K1}, in every class) the right side is $(C5^{-\gamma j}+C_N\eta^N)\times$(the bound (a)) for every $N$, which is of the first form on every cone $j\le(1-c)k$, $c>0$, for $N\ge\gamma(1-c)/c$; \item[(c)] summed over the zone-B localisations with $R\le p$, the zone-B part of the source-free field-dependent effective-action difference is at most $(C5^{-\gamma j}+C_N\eta^N)\eta^{\rho_0}p^{n_1}(1+k-j)^{p_0}|\partial T^{(j)}|\le C'5^{-\gamma'j}\eta^{-\beta'}|\CJ|+C'_N\eta^{N+\rho_0-2-\epsilon}|\CJ|$ for every $\gamma'\le\gamma$ with $\gamma<\gamma_b$ as in (b), every $\beta'>\max(2-\rho_0,0)$, every $\epsilon>0$ and every $N$; the second term, from the members with a line $\delta G_j(\square',\cdot)$, is absent on every cone $j\le(1-c)k$ for $N$ large, and for $N>2-\rho_0+\epsilon$ is of the second form of \eqref{eq:k01:Kpos}; the zone-B localisations with $R>p$, among them every zone-B localisation of a source host, are bounded in each member separately by (a) in each zone-B cluster with no vertex held, with no factor $5^{-\gamma j}$. \end{itemize} The constants depend on the point of King's range, $\bar n$, $\alpha_{\mathrm H}$, $\beta_{\mathrm H}$, $\alpha'$, $\gamma$, the exponents $\beta'$, $\epsilon$, the order $N$, the cone parameter $c$ and the line constants, not on $j$, $k$, $\eta$, $J$, $x_0$. \end{theorem} \begin{proof} The free rewriting of \cref{cpi:def:a02:RC} splits $\mathcal R$ exactly into \emph{bulk-type graphs}, in which every resummed coefficient is the free one $c^F_S$, and \emph{remainder graphs}, which carry at least one remainder vertex $v_S$ with the coefficient $\Delta_S(x)=c^C_S(x)-c^F_S(x)$. The bulk-type graphs obey the degree count of \cref{cpi:thm:a02:T} with the cube slice bounds up to the faces, so \cref{cpi:thm:a02:S3} gives (a) for them with $\Lambda\equiv1$ and (b) with the cube line differences. The content of the theorem is the treatment of the remainder graphs, in four steps. The joint statement named in (b) is \cref{cpi:uc:thm:J} below, stated after the no-collapse lemma, \cref{cpi:uc:lem:UC}, whose rewriting it uses: of the threshold $\gamma_b$ of (b), the first four exponents are the range of \cref{r13:def:Bscset}, on which the proof below covers the point-free members; the last three complete the range of the \emph{joint boundary and block-vertex statement}, \cref{cpi:uc:thm:J}, which proves the root-weighted bound and the pair rate together for the zone-B members with block points, with or without layered vertices. \emph{(1) The layer weight of a remainder.} Expanding the internal lines of a scalar C-site $S$ into slices of index $\le j_0$ (the lowest index of its legs, $\rho=\rho_{j_0}$) and comparing the cube slices with the free ones, the difference is a sum of image terms, each carrying one reflected line of length $\ge t(x)$; this gives \[ |\Delta_S(x)|\le C\rho^{D'}\big[w_{D'}(t(x)/\rho)+e^{-\delta p/\rho}\big],\qquad w_0(s)=(1+\ln_+(1/s))e^{-\delta s},\quad w_{-1}(s)=s^{-1}e^{-\delta s}, \] and across the pair $|\Delta'_S(x)-\Delta_S(x)|\le C5^{-\gamma j}\rho^{-\gamma}\times$(the same with the exponent $D'$ lowered by $\gamma$). So a $D'=0$ remainder vertex is layered of class $\mathcal L(\rho,1,0)$; a $D'=-1$ remainder vertex has the weight $t^{-1}$, which is not summable over the root, and is treated by borrowing. \emph{(2) Borrowing and the charging rule.} Every scalar leg ending at a live site $x$ at distance $t$ from the faces vanishes on $\mathsf F$ and is H\"older up to the faces, so it carries the factor $m_{\alpha_{\mathrm H}}(x;\rho_l)\defeq\min(1,t/\rho_l)^{\alpha_{\mathrm H}}$, $\rho_l$ the scale of the leg (half the exponent for a pair difference). In a non-vanishing remainder graph one can choose for each $D'=-1$ remainder vertex one of its two scalar legs so that no line is charged twice (a cut line between two zone-B clusters excepted, which is then charged at both ends and bounded by the two-ended face bound for cut lines, \cref{lem:k02:face}(b), proved in Section~\ref{ssec:k02:Khproof} without this theorem): the multigraph on the $D'=-1$ vertices with the shared legs as edges has no component with more edges than vertices, since a cycle would be a closed component made of remainder vertices in a connected graph with an external field or a source vertex. Moving the factor $m_{\beta_{\mathrm H}}(x;\rho_l)$, $\beta_{\mathrm H}<\alpha_{\mathrm H}/2$, from the charged leg to $v_S$ leaves the leg its standard bound and gives $v_S$ the weight $\rho|\Delta_S(x)|m_{\beta_{\mathrm H}}(x;\rho_l)\le C[(\rho/t)^{1-\beta_{\mathrm H}}e^{-\delta t/\rho}+e^{-\delta p/(2\rho)}]$, of class $\mathcal L(\rho,0,1-\beta_{\mathrm H})$ plus a term of any prescribed degree supported away from $\mathsf F$; across the pair the difference is $5^{-\gamma j}\rho^{-\gamma}$ times a weight of class $\mathcal L(\rho,0,1-\beta_{\mathrm H}+\gamma)$, which is why $\gamma<\beta_{\mathrm H}$ is required. \emph{(3) The degree condition with layered vertices.} The layered vertices are the $D'=0$ remainder vertices, the $D'=-1$ ones after borrowing, and the difference pieces of \cref{cpi:thm:a02:BVP}(b), all of classes with $m\le1$ and degrees $\ge-1$. In every admitted ordering every connected component $H$ of every stage has $D(H)\ge0$, and $D(H)=0$ only if $H$ contains a layered vertex: the stages containing a scalar remainder vertex by the census of the scalar remainders (a C-site is the zeroth moment of a scalar self-energy of order $2$ or $4$, the first moment of the order-$2$ one-photon exchange, a one-leg-differentiated two-point class or a $\phi\phi A$ class, and a stage through its remainder vertex with an external leg has $D\ge0$); the stages containing a vacuum-polarisation piece by the integrality of the photon two-point degrees and the presence of an external field or a source vertex; the stages containing a block point by the split count (a block point weighs $3-\frac s2+\frac p2\ge\frac52$, so every such stage has $D\ge1$); the \emph{mixed stages}, which contain a source vertex and a layered vertex, by re-expanding the generalised vertices, after which the source vertex and the standard vertices of the layered vertex's class give $D\ge u_s-2\ge0$, $u_s=2$ for the scalar source and $\frac52$ for the summed curvature vertex (no C-site contains a source vertex, and reserving half of the decay of every line of such a stage gives moreover a factor $C_N\rho^N$ at its first appearance); and all other stages are bulk-type stages with $D>0$ by \cref{cpi:thm:a02:T}. Let $\alpha_0^B$ be the minimum positive degree. \emph{(4) The ordered bound with layered vertices.} Run the multiscale induction of \cref{cpi:thm:a02:S3}(a) with one change: when the line being shrunk, of scale $\rho_i$, has a layered endpoint of width $\rho_w\le\rho_i$ that is not held, sum that endpoint against the line, using half of the line's exponential to transfer the layer weight: the factor is $C\rho_i^3(\rho_w/\rho_i)$ instead of $C\rho_i^3$, so an active layered composite gains $+1$ in the exponent at every step until it is merged into the held vertex $x_0$ or into the final volume sum, where $\sum_x5^{-3j}\omega(x)\le C\rho_w$ for $\omega\in\mathcal L(\rho_w,m,\theta)$, $\theta<1$. Hence every stage has exponent $\ge\min(\alpha_0^B,1)>0$ except the steps at which a $D=0$ component contains $x_0$ and an inactive layered composite: there the factor $e^{-\delta't(x_0)/\rho_*}$ forces the merge scale $\rho_*\ge ct(x_0)$ up to a summable tail, the at most $\bar n$ such steps and the merge index give the logarithms $(1+\ln_+(1/t(x_0)))^{\bar n+2}$ and the factor $e^{-\delta_\Lambda t(x_0)}$, and a layered vertex held at $x_0$ adds its own $t(x_0)^{-\theta}$ with $\theta\le\theta_{\max}=1-\beta_{\mathrm H}$: this is $\Lambda(x_0)$, root-summable because $\theta_{\max}<1$. The infrared stage, which contains no remainder, no source and no layered vertex, gives the factor $(1+k-j)^{p_0}$ as in \cref{cpi:thm:a02:S3}(a). This proves (a). For (b) the same induction is run for the difference of the two members, every line, coefficient and layered weight obeying a pair bound $5^{-\gamma j}\rho^{-\gamma}\times$(its bound), a weakened layered weight being of class $\theta+\gamma<1$, the positions of the two members paired cell by cell as in \cref{cpi:thm:a02:S3}(b), the finer member's held vertices in the top layer, which have no partner, bounded by (a) for the finer member and entering only the sum of (c), where they cost $C5^{-\gamma j}$ per unit cube; the root weight becomes $\Lambda_\gamma$, with $t^{-(1-\beta_{\mathrm H}+\gamma)}$, still root-summable as $1-\beta_{\mathrm H}+\gamma<1$. A member with a line $\delta G_j(\square',\cdot)$ is bounded in each model separately, which gives its term $C_N\eta^N\times$(the bound (a)); for these members and for the members with block points the pair rate is \cref{cpi:uc:thm:J}(J-b). (c) is (a), (b) and \cref{r13:thm:BVPpair}(d) summed over the root with $\sum_{x_0}5^{-3j}\Lambda(x_0)\le C$ per unit face cell, $|\partial T^{(j)}|\le C\eta^{-2}|\CJ|$ and $p^{n_1}(1+k-j)^{p_0}\le C_\epsilon\eta^{-\epsilon}$. The remaining estimates are routine given the slice bounds of the cube lines up to the faces and their pair clauses. \end{proof} \space% % % \begin{theorem}[the curvature classes and the source-free classes in zone~I]\label{cpi:thm:a02:B4} Let $H$ be a graph of King's expansion at step $j$ on $\CK$ at $h=0$ that is field-dependent, or contains a curvature vertex, and let it be localised in zone~I. Then \cite[(3.39)]{K1} holds for it, and \cite[(3.38)]{K1} with the extra factor $(1+k-j)^{p_0}$; summed over positions, the zone-I contribution to $|S^{(j)}_{N,k}-S^{(j+1)}_{N,k+1}|$ is $\le C[5^{-\gamma'j}\eta^{-\beta'}+\eta^{\sigma'}]|\CJ|$, a bound of the form of \eqref{eq:k01:Kpos}, with $C$ depending on the curvature sources through $\max\norm{f_{\mu\nu}}_{C^{2,\alpha}}$ only. \end{theorem} \begin{proof} At the localisations with $R>p$ the two members are bounded separately by the cut by clusters (Definition~\ref{r13:def:clusters}, as in Proposition~\ref{prop:k02:KhR}) and enter the second term only; at those with $R\le p$ it suffices to treat the free version $\tilde H^{\mathrm{free}}$ of the reduced graph on the free $f_D$ flow, with an error of the form of \cite[(3.39)]{K1}, since the free-boundary counterterm renormalises the free graphs exactly. For the free flow one runs King's proof of \cite[Thm.~3.5]{K1} and of the step bound \cite[Prop.~3.6]{K1}: the census of ordered subgraph components (every component containing a source vertex has degree $\ge\frac12$, except the field-independent source-linear family, which is absent at $h=0$ and vanishes for the curvature source at $A_j=\phi_j=0$ by charge parity, and the single-curvature-vertex tadpole, which vanishes by $\operatorname{Tr}q=0$, $q$ the antisymmetric charge matrix of the current vertex); the ordered multiscale bound, Theorem~\ref{cpi:thm:a02:S3}(a) with the degree count of Theorem~\ref{cpi:thm:a02:T} for the source-free components; the one-factor-at-a-time replacement of \cite[pp.~663--665]{K1}, with the comparison of the cube lines with the free lines and, for the curvature vertex, the comparison of the summed vertex weights across the pair of spacings, $|w_{\eps_{k+1}}(b')-w_{\eps_k}(b)|\le(2\sqrt3+1)\,\eps_k\max\norm{f_{\mu\nu}}_{C^2}$ for bonds $b'$ at spacing $\eps_{k+1}$ and $b$ at spacing $\eps_k$ in the same direction with $|m_{b'}-m_b|\le\sqrt3\,\eps_k$ (the mean value theorem), which carries the gain $\eps_k\le5^{-\gamma j}$ with a bounded summed vertex; and the identities \cite[(3.43)--(3.55)]{K1}, under which the curvature vertex is invariant. The curvature source enters only through the sup and lattice $(1,\alpha)$-norms of the summed vertex weight $w$, the bond function with $\sum_pg^\eps(p)F(p)=\sum_bw(b)A_b$ exactly for every bond field $A$ on $\CJ$ ($f$ a real $2$-form of the family, $g^\eps$ its centre sampling and $F(p)$ the plaquette curvature), $w(b)=W^\eps_\mu(m_b)$ with $W^\eps_\mu(x)=\sum_{\nu\ne\mu}\eps^{-1}[f_{\mu\nu}(x+\tfrac\eps2e_\nu)-f_{\mu\nu}(x-\tfrac\eps2e_\nu)]$ and $m_b$ the midpoint of the bond $b$; these norms are $\le C\max\norm{f_{\mu\nu}}_{C^{2,\alpha}}$ uniformly in $\eps$. \end{proof} \space% % % % % % \paragraph{The split degree count for the classes with block vertices.} King's expansion of the renormalisation transformation \cite[(3.24)--(3.26)]{K1} produces, besides the local vertices \cite[(3.18)--(3.23)]{K1}, the \emph{block products}: a vertex at the block $y\in T^{(k)}$ that is the product of two of the expressions \begin{gather*} \phi_k(y),\qquad -\big(Q_k(A^{(k)})(\phi'+\phi^{(k)})\big)(y),\\ -\frac{(e(L^k\eps)^{2-d/2})^m}{m!}\sum_{x\in B^k(y)}\eta^d\,U\big(A^{(k)}(\Gamma^{(k)}_{y,x})\big)\big(e'A'(\Gamma^{(k)}_{y,x})\big)^m q^m(\phi'+\phi^{(k)})(x), \end{gather*} one of which is of the third kind, multiplied by $-a_k$ or $-a_k/2$ and summed over $y$. Each factor of the second or third kind is a sum over a \emph{point} $x$ of the block with the measure $\eta^d$, and the $m$ fluctuation fields of the third kind live on the contour $\Gamma^{(k)}_{y,x}$; by \cite[(2.19)]{K1} the scale-$j$ field of the point $x$ enters only through the sub-contour $\Gamma^{(j+1)}_{x_{j+1},x}$, so a photon line from the contour at slice $j$ is the \emph{contour kernel} $g^{(j)}(x,b)=\sum_{b'\in\Gamma^{(j+1)}_{x_{j+1},x}}\eta\,G_{(j)}(b',b)$ of \cite[(3.27)]{K1}. A vertex of the expansion that is a block product is a \emph{block vertex}. King's degree \cite[(3.30)--(3.31)]{K1} credits a block vertex with $+m$ and merges its two summation points into one; Ba{\l}aban's degree \cite[(2.1)--(2.2)]{Ba3} represents a block product by two vertices and credits a vector leg of a block vertex only when it lies on an internal line of the graph, the factor $L^{j_l}\eta$ of such a leg being booked at the scale of its line. On whole graphs the two agree; on subgraphs King's per-vertex form over-credits a block vertex whose contour fields have their lines outside the subgraph, and the count that governs the bounds is Ba{\l}aban's, stated here as the split count. \begin{definition}[the split graph]\label{cpi:ir:split} Let $\tilde H$ be a connected graph of King's expansion after the constant-background reduction \cite[(3.43)--(3.55)]{K1} at zero background, on the torus or on a cube with the slices of the Dirichlet flow, its internal lines sliced as in \cite[(3.58)]{K1} (fluctuation slices $0\le j\le k-1$, the finer model's slices $-n\le j\le k-1$, the counterterm lines $C^\eta$ of the mass-counterterm units with $0\le j\le K$). The \emph{split graph} $G^{\mathrm{sp}}$ (that of \cref{cpi:thm:a02:S3}, recalled here) has as vertices every vertex of $\tilde H$ other than a block product and, for each block product, its summation points $x$ and (when present) $x'$, each with the measure $\eta^d$; the fields $\phi_k(y)$, the contour fields of the background and the field $\phi^{(k)}$ are external at the point. Its scale-summed lines are the internal fluctuation lines with their slice index, a photon field on the contour of a point entering through the kernel $g^{(j)}$, and, for each two-point product, the block kernel $\kappa_y(x,x'):=\mathbf 1[x,x'\in B^k(y)]$ times $-a_k$ or $-a_k/2$, read as a line with the single index $k$. A scale-summed line $l$ has dimension $d_l:=2-d-\#\{\text{derivatives on }l\}+c_l$, $c_l\in\{0,1,2\}$ the number of its ends on contours (a H\"older factor $|x-y|^\alpha$ adds $\alpha$), the block kernel has $d_\kappa:=0$, and a vertex carries the weight $a_v$ equal to the number of explicit factors $\eta$ at $v$ beyond the measure ($m-1$ for a vertex (3.22) of order $m$, $m-2$ for (3.23), and $a_X:=0$ at every point). For a connected subgraph $H$ of $G^{\mathrm{sp}}$ with all lines scale-summed, the \emph{degree of the multiscale bound} is $D^{\mathrm{MS}}(H):=\sum_{l\in H}d_l+\sum_{v}a_v+d(|V(H)|-1)$. For a connected component $c$ of the graph $\gamma^{\mathrm{sp}}$ obtained from a subgraph $\gamma$ of $\tilde H$ by the same splitting (the block kernel is not a line of it), the \emph{split degree} is \begin{gather*} D^{\mathrm{sp}}(c):=\sum_{v\in c}w_v-d,\\ w_v:=d+a_v-\tfrac{d-2}2\#\{\text{undifferentiated ends of }c\text{-lines at }v\}-\tfrac d2\#\{\text{differentiated ends}\}, \end{gather*} where at a point $X$ the weight $a_X$ is replaced by $m_{\mathrm{int}}(X)$, the number of photon fields on the contour of $X$ whose lines are lines of $c$; and $D^{\mathrm{nat}}(\gamma):=\sum_cD^{\mathrm{sp}}(c)$. For $\gamma$ without block vertex $D^{\mathrm{nat}}(\gamma)$ is King's degree. For a point $X$ with at least one end of a $c$-line, $w_X=3+\frac12m_{\mathrm{int}}(X)-\frac12s_{\mathrm{int}}(X)\ge\frac52$, $s_{\mathrm{int}}(X)\in\{0,1\}$ recording whether the scalar slot of $X$ is internal, and $D^{\mathrm{sp}}(c)$ is an integer. If $c_1,\dots,c_r$ are the components of $H$ with its $\kappa$-lines removed, then $D^{\mathrm{MS}}(H)=\sum_iD^{\mathrm{sp}}(c_i)+d(r-1)$. \end{definition} On King's vertex list at zero background the weights take the values: a standard vertex other than the source vertices has $w_v\ge\frac12$, with equality only for a vertex (3.22) of order one with its derivative internal and all three ends internal; a vertex of a mass-counterterm unit has $w\ge0$, with $w=0$ only for its base vertex with all legs and its line inside the subgraph; the scalar-source vertex $h$ has $w=3-e/2\ge2$ ($e$ its number of internal ends; the shell or insertion vertex of the shell model is of this type); the curvature vertex $g$ has $w=\frac32$ with its leg internal or $\frac52$ in its summed-by-parts form. \begin{lemma}[positivity of block-point components]\label{cpi:ir:BLKnat} Let $\gamma$ be any connected subgraph of King's expansion at zero background, on the torus or on a cube, with any external legs and with mass-counterterm units of any depth, and let $c$ be a connected component of $\gamma^{\mathrm{sp}}$ containing a point $X$ of a block vertex and at least one line. Then (i) $D^{\mathrm{sp}}(c)\ge1$; (ii) $D^{\mathrm{sp}}(c)\ge2$ if $c$ contains two points, an $h$-type vertex or a curvature vertex. Consequently $D^{\mathrm{nat}}(\gamma)\ge1$ for every connected $\gamma$ containing a block vertex, and $\ge2$ if both points of some block vertex of $\gamma$ carry $\gamma$-lines. \end{lemma} \begin{proof} The lemma is a finite case analysis of the weights: for a unit-free component, $D^{\mathrm{sp}}(c)=w_X+\sum_{v\ne X}w_v-3$ with $w_X\ge\frac52$, and the values $\frac12$ and $1$ of $\sum_{v\ne X}w_v$ are excluded by the pairing of the scalar and photon ends (a lone vertex (3.22) of order one has one scalar and one photon end left over, two such vertices leave three scalar ends); a component with mass-counterterm units is reduced to a unit-free one by replacing each unit met by a point vertex, each one-loop unit costing one and crediting two. No parity, Furry, Ward or reflection argument and no properness is used. \end{proof} \space% % % % \begin{lemma}[no collapse of the block part]\label{cpi:uc:lem:UC} Let $\mathcal R$ be a renormalised family of King's expansion of order $\le\bar n$, localised in zone~B (\cref{cpi:def:a02:zones}), at zero background and zero scalar source, with the lines of the $f_D$ flow of \cref{cpi:def:a02:setting}, in either member of the pair. For a photon two-point prefix component $S'$ split its members as $S'=S'_{\mathrm{hop}}+S'_{\mathrm{blk}}$, where $S'_{\mathrm{blk}}$ consists of the members containing at least one block product (of total multiplicity $\ge1$) and $S'_{\mathrm{hop}}$ of the others. Let $\mathcal R^{C,\mathrm{uc}}$ be the rewriting of \cref{cpi:def:a02:RC} and \cref{cpi:thm:a02:BVP}(a) with one change: no $S'_{\mathrm{blk}}$ is collapsed. For every photon two-point prefix component $S'$, at every nesting depth (a photon two-point prefix component may itself lie inside a larger rewritten prefix component), the members of $S'_{\mathrm{blk}}$ stay subgraphs of the \emph{host member}, the member of $\mathcal R$ of which $S'$ is a prefix component (their lines in slice form, their internal vertices \emph{host vertices}, that is, vertices of the host member carrying its localisation); only $S'_{\mathrm{hop}}$ is rewritten and compared, with its pieces. Then: \textup{(i)} $\mathcal R^{C,\mathrm{uc}}$ is an exact rewriting of $\mathcal R$; \textup{(ii)} every member of $\mathcal R^{C,\mathrm{uc}}$ is a generalised graph on the split graph $G^{\mathrm{sp}}$ of \cref{cpi:thm:a02:S3} whose layered vertices (\cref{cpi:def:a02:lay}) are scalar remainder vertices and $S'_{\mathrm{hop}}$ difference pieces only, no layered vertex contains a block vertex or a block point, and every block point is a host vertex; \textup{(iii)} every component of every stage of $G^{\mathrm{sp}}$ has $D\ge0$, with $D=0$ only if it contains a layered vertex: $D^{\mathrm{sp}}\ge1$ with a block point, $D\ge0$ with a scalar remainder vertex (the census of the scalar remainders, step~(3) of the proof of \cref{cpi:thm:a02:Bsc}), $D\ge0$ and layered with an $S'_{\mathrm{hop}}$ piece (the census of the vacuum-polarisation pieces, same step), $D>0$ otherwise (the degree count of the free $f_D$ flow, \cref{cpi:thm:a02:T}); \textup{(iv)} hence no member of $\mathcal R^{C,\mathrm{uc}}$ contains a collapsed block piece, \cref{cpi:thm:a02:BVP}(b) and \cref{r13:thm:BVPpair}(c)--(d) hold with the vacuum-polarisation pieces restricted to $S'_{\mathrm{hop}}$ pieces, and the $S'_{\mathrm{blk}}$ members are bounded on $\CK$ by the ordered bound with layered vertices (step~(4) of the proof of \cref{cpi:thm:a02:Bsc}) on $G^{\mathrm{sp}}$. \end{lemma} \begin{proof} \textup{(i)} The $S'_{\mathrm{blk}}$ members are never rewritten: every connected subgraph with a block vertex has $D^{\mathrm{nat}}\ge1$ (\cref{cpi:ir:BLKnat}: a component of the split graph containing a point of a block vertex and a line has $D^{\mathrm{sp}}\ge1$, a block point weighing $3-\frac s2+\frac p2\ge\frac52$ in the split count, step~(3) of the proof of \cref{cpi:thm:a02:Bsc}), so it is not divergent, and no evaluation, no moment extraction (the extraction of the zeroth and first moments of \cite[(3.26)--(3.31)]{Ba3}, \cref{cpi:thm:a02:BVP}) and no resummation acts on an $S'_{\mathrm{blk}}$ member or on a prefix component containing a point. The split $S'=S'_{\mathrm{hop}}+S'_{\mathrm{blk}}$ is a partition of the members, and the rewriting of $S'_{\mathrm{hop}}$ uses no $S'_{\mathrm{blk}}$ member: the counterterm units of $S'_{\mathrm{hop}}$ are summed over its own lines, the bounds of the coefficients $c^{S'_{\mathrm{hop}}}$ use $S'_{\mathrm{hop}}$ alone, and the total identity involving $S'_{\mathrm{blk}}$ is kept as an identity which no bound uses. In \cref{cpi:thm:a02:BVP}(a) the $S'_{\mathrm{blk}}$ part of the comparison is the identity $S'^{\,C}_{\mathrm{blk}}=S'^{\,\mathrm{free}}_{\mathrm{blk}}+\Delta S'_{\mathrm{blk}}$ followed by a regrouping, and omitting an identity changes no value. Without the collapse the internal scales of the $S'_{\mathrm{blk}}$ members are not resummed (undoing the resummation regroups finite sums back into the host's sum over orderings), and their internal vertices carry the host's localisation, as every unrewritten convergent subgraph does \cite[p.~661]{K1}, \cite[p.~330]{K2}. \textup{(ii)} By construction, since a C-site is the evaluation of a divergent subgraph and no divergent subgraph contains a block vertex (\cref{cpi:ir:BLKnat}), so that no C-site contains a block point: the census of the C-sites in step~(3) of the proof of \cref{cpi:thm:a02:Bsc} (a C-site is the zeroth moment of a scalar self-energy of order $2$ or $4$, the first moment of the order-$2$ one-photon exchange, a one-leg-differentiated two-point class or a $\phi\phi A$ class) contains no class with a block vertex. \textup{(iii)} As listed; a stage containing an uncollapsed $S'_{\mathrm{blk}}$ member contains its points. \textup{(iv)} From \textup{(ii)}, \textup{(iii)} and the ordered bound with layered vertices run on $G^{\mathrm{sp}}$, on which the $\kappa$-lines and the staircase lines of the points are lines of the graph and every stage exponent is the split-count degree (\cref{cpi:thm:a02:S3}(c), (d)), so that the contour and block-kernel lines of the points are shrunk as ordinary lines. \end{proof} \space% % % % \begin{lemma}[composite kernels]\label{cpi:uc:lem:QPK} In the setting of \cref{cpi:uc:lem:UC}, with $\vartheta=5^{-j}$ the spacing of the coarser member of the pair and $\vartheta'=\vartheta/5$ that of the finer one, let $\rho\defeq\rho_{j''}=5^{j''}\vartheta$ with $1\le j''\le j$, and $K=A\,M_P\,B$, where $A$, $B$ are cumulative (index $\le j''$) lines of the $f_D$ flow on the cube, $C^\xi_D$ (the Dirichlet covariance of \cref{cpi:thm:a02:BVP}) or their image parts, each slice obeying \cite[(3.63)--(3.65)]{K1} on the cube with $\mathsf q_A,\mathsf q_B\le1$ derivatives at the outer ends $u$, $v$ and none at the $P$-ends, and $M_P$ is $\rho^{-2}P_{j''}$, with $P_{j''}=Q_{j''}^*Q_{j''}$ the averaging kernel of \cite[(3.2)]{K2} carrying the weight $f_D$ at the finest step \cite[(A.1)--(A.3)]{K2}, or the dipole $\mathrm i[\Lambda,\rho^{-2}P_{j''}]$ ($\Lambda$ the multiplication by a linear function $\lambda$ with $B=d\lambda$ a constant field). Put $\bar K\defeq C\rho^{-1-\mathsf q_A-\mathsf q_B}e^{-\delta|u-v|/(2\rho)}$, times $|B|\rho$ for the dipole. Let $\gamma_{F3}$ be the rate of the parts of the lines specific to the weight $f_D$: any $\gamma_{F3}<\min(\alpha_0,s)$, with $\alpha_0$ the minimum positive stage degree of \cref{cpi:thm:a02:S3} and $s=(1-\alpha')/(1+\alpha')$ the H\"older rate of \cref{cpi:lem:a02:RSB}. Then (a) $|K|\le\bar K$; (b) across the pair, with $K'$ the composite of the finer member, $K'-K=(\text{line terms})+(\mathrm I_P)+(\mathrm{ZM}_P)$ with $|(\text{line terms})+(\mathrm I_P)|\le C5^{-\gamma j}\rho^{-\gamma}\bar K$ and $|(\mathrm{ZM}_P)|\le C(\vartheta/\rho)^{\alpha'}\bar K$, for every $\alpha'<1$ and every $\gamma<\gamma_{F3}$, and $\gamma<\gamma_{F3}/2$ when image parts occur; so $|K'-K|\le C5^{-\gamma j}\rho^{-\gamma}\bar K$ for $\gamma\le\alpha'$ in that range. The bound $\bar K$ is the bound of the part $M_P=\rho^{-2}P$; the replaced line $G(1-m^2-aP)C^\xi_D$ obeys the corresponding bound, and with derivatives at both outer ends its $(1-m^2)GC$ part has the integrable singularity $\rho^{-2}|u-v|^{-1}$ in three dimensions. \end{lemma} \begin{proof} \emph{(a).} $P(w,\tilde w)=\sum_yQ(y;w)Q(y;\tilde w)$ with $Q(y;w)=c\rho^{-3}f_*(w)1[w\in B^{j''}(y)]$ in the measure $\vartheta^3$, where $f_*$ is the finest-step weight of \cite[(3.8)]{K2}, equal to $(5/4)^3$ on $4^3$ sites of each $5\vartheta$-block and $0$ on the remaining layer, so that $\sum_{x\in B(z)}f_*(x)=5^3$ on every $5\vartheta$-block, the averaging above the finest step being uniform \cite[(A.1)--(A.3)]{K2}; use $\sum_w\vartheta^3|A(u,w)|\le C\rho^{2-\mathsf q_A}$, the same for $B$, and $|w-\tilde w|\le c\rho$ on the support. \emph{(b).} Telescope $K'-K=(A'-A)M'_PB'+A(M'_P-M_P)B'+AM_P(B'-B)$. The outer terms are cumulative line differences, bounded per factor by $C5^{-\gamma j}\sum_{i\le j''}\rho_i^{-1-\mathsf q-\gamma}e^{-\delta\dist/\rho_i}$ (the two-member comparison of the cube slices used in \cref{cpi:thm:a02:S3}(b), with the $f_D$-specific parts of \cref{cpi:lem:a02:RSB}; for $C^\xi_D$ the lattice Green's function difference), integrated against the other factor and the block kernel; since $2-\mathsf q-\gamma>0$ the sums over $i$ are geometric. For the middle term, the two summation points $w$, $\tilde w$ of $P$ enter only through their weights $\vartheta^3f_*$, through functions constant on each $5\vartheta$-block and through the factors $A(u,\cdot)$, $B(\cdot,v)$, undifferentiated there; the point-sum identities $\sum_{w'\in B^1(z)}\vartheta'^3f'_*(w')=\vartheta^3$ \cite[(3.13)]{K2} and $\sum_{z\in b}(1-f_*(z))=5^3-4^3(5/4)^3=0$ on every $5\vartheta$-block $b$ write the difference of the two members' point sums, at $w$ and then at $\tilde w$, as $(\mathrm I_P)+(\mathrm{ZM}_P)$: $(\mathrm I_P)$ is the identification $w'\to z$ of the finer sites with the coarser ones, a line-level difference, and $(\mathrm{ZM}_P)$ differences $A(u,\cdot)$ at $w$ between $z$ and a reference site $z_b$ of its block, $|z-z_b|\le\sqrt3\cdot5\vartheta$, by the H\"older clause of the slice bounds in the $P$-end variable (the mixed clause when $\mathsf q_A=1$), with $\sum_w\vartheta^3|A(u,z)-A(u,z_b)|\le C\sum_{i\le j''}(\vartheta/\rho_i)^{\alpha'}\rho_i^{2-\mathsf q_A}\le C(\vartheta/\rho)^{\alpha'}\rho^{2-\mathsf q_A}$, a geometric sum since $2-\mathsf q_A-\alpha'\ge1-\alpha'>0$ (this is where the absence of a derivative at the $P$-end is used). The dipole factor gives $|\lambda(z)-\lambda(z_b)|\le|B|\sqrt3\cdot5\vartheta$ against $|B|c\rho$. Finally $(\vartheta/\rho)^{\alpha'}=5^{-\alpha'j}\rho^{-\alpha'}\le5^{-\gamma j}\rho^{-\gamma}$ for $\gamma\le\alpha'$. Image parts keep their decay by $\min(a,b)\le a^{1/2}b^{1/2}$, which halves the admissible rate. \end{proof} \begin{lemma}[composite kernels at every order]\label{cpi:uc:lem:QPKinf} Let $X$ be a cube-side piece carrying an explicit $f_D$-weighted $P$ of a zone-B photon two-point prefix component $S'$ of order $\le\bar n$: the residual $\Delta c^{S'_{\mathrm{hop}}}$ or the replaced-line piece (the difference of the first moment of the replaced-line part) of \cref{cpi:thm:a02:BVP}(b), in the representation with the lines $G_{j_0}=C^\xi_D+G_{j_0}(1-m^2_{j_0}-\mathfrak a_{j_0}P_{j_0})C^\xi_D$, or an inner resummed coefficient computed with cube lines inside any vacuum-polarisation piece. Let the pair be as in \cref{cpi:uc:lem:UC}, $0<\alpha'<1$, and $0<\gamma<\min(\alpha_0^B,\alpha_0,1-\theta_{\max},\alpha',\gamma_{F3}/2)$, with $\alpha_0^B$ the minimum positive stage degree of \cref{cpi:thm:a02:Bsc}, $\theta_{\max}=1-\beta_{\mathrm H}$ the largest class exponent of the layered weights (step~(4) of the proof of \cref{cpi:thm:a02:Bsc}) and $\gamma_{F3}$ as in \cref{cpi:uc:lem:QPK}. Then $X$ has the \emph{composite-kernel structure} (a), (b) and obeys (c): \begin{itemize}[nosep] \item[(a)] every explicit $P$ in $X$ sits in exactly one composite $K=A\,M_P\,B$ created by the splitting of a resummed coefficient, with $A=d^{\mathsf q_A}G_{j''}(\cdot,0)$ (a cube cumulative line or its image part), $B=C^{\xi''}_Dd^{*\mathsf q_B}$, $\xi''=5^{-j''}$, $1\le j''\le j_0$; each outer end carries at most one derivative and at most one H\"older quotient, of exponent $\alpha_B\in(0,1)$ (at most one derivative at the classes of \cite[pp.~437--445]{Ba3}), the $P$-ends carry no derivative and no other factor, and the $P$-points enter $X$ only through $K$; \item[(b)] no step applied after the splitting has a $P$-point of $K$ as a vertex, a localisation point, a Taylor or H\"older base point, a contour end, a summation-by-parts site, or a slice or ordering variable. \item[(c)] $|X'(x)-X(x)|\le C5^{-\gamma j}\rho_{j_0}^{-\gamma}\times(\text{the bound of }X)\times\omega_\gamma(x)$ across the pair, $X'$ the piece of the finer member, with $\omega_\gamma$ a layered weight of class $\mathcal L(\rho_{j_0},m,\gamma)$ (\cref{cpi:def:a02:lay}), $m$ the multiplicity of the bound of $X$, provided $\alpha_B+\alpha'<1$ in the point step; the same for the $(j,k)$-member against the $(j+n,k+n)$-member, with the constant multiplied by $(1-5^{-\gamma})^{-1}$. \end{itemize} At every order $\le\bar n$, (a) and (b) hold by the argument of \cite[pp.~437--445]{Ba3}, carried to every order: the rewriting is a finite sequence of steps of four types (the resolvent splitting inside a resummed coefficient, the steps on the lines internal to the coefficient, the steps on the outer legs, and the $f$-free vector averaging), the splitting at a resummed coefficient is the only step that produces an explicit $P$, and no later step has a $P$-point as its target. \end{lemma} \begin{proof} \emph{(a), (b).} For the classes treated explicitly in \cite[pp.~436--445]{Ba3} they are read off Ba{\l}aban's rewriting: the explicit $P$'s outside lines are created only by the resolvent splitting inside a resummed coefficient \cite[p.~437, (3.16)]{Ba3}, the steps after the splitting act on lines internal to the coefficient and on the outer legs \cite[pp.~436--445]{Ba3}, and the vector averaging is $f$-free; on the cube the splitting is an identity on the space with the Dirichlet condition, and $P_{j''}$ at zero background is \cite[(3.2)]{K2} with $f_D$ at the finest step \cite[(A.1)--(A.3)]{K2}, so that for $j''\ge1$ the block indicator $1[w\in B^{j''}(y)]$ is constant on $5\vartheta$-blocks and $w$ is a summation point with exactly one undifferentiated line factor $A(u,w)$ (and $B(\tilde w,v)$ at $\tilde w$); an order-$(1+\alpha_B)$ decoration of an outer end arises when a leg of an enclosing graph carries a derivative of that order \cite[p.~668]{K1} and never reaches a $P$-end, and the dipole type occurs only at one loop. \emph{(c): pairing.} $X$ is built by the same graphs, orderings and steps in both members, except for the finest index $i=-1$ of the finer member and its composite at the lattice scale (absent in the coarser member, which has no $P_0$); these are bounded at their own scale by (a) and \cref{cpi:uc:lem:QPK}(a), where $(\vartheta/\rho)^\gamma\ge c$, and absorbed by $D-\gamma>0$. Otherwise $X'-X$ telescopes factor by factor, the $P$-points being differenced only inside their composites by (b), and the plain internal summation points are identified by $\sum_{w'\in B^1(z)}\vartheta'^3=\vartheta^3$ with the cube two-member line comparison. \emph{Composites.} \cref{cpi:uc:lem:QPK}(b), with the outer terms by the line comparison and the lattice Green's function difference, $(\mathrm I_P)$ a line-level difference, and $(\mathrm{ZM}_P)$ by the H\"older clause of $A$ in its $P$-end variable: the sum $\sum_{i\le j''}(\vartheta/\rho_i)^{\alpha'}\rho_i^{2-(1+\alpha_B)}$ is geometric if and only if $\alpha_B+\alpha'<1$, which is where that condition enters. The scalar constants of $M_P$ at one physical scale: $|\mathfrak a_{j''+1}-\mathfrak a_{j''}|\le C5^{-2j''}=C(\vartheta/\rho_{j''})^2$, and the mass $m^2_{j''}$ is the rescaled mass of the Gaussian part, the same in both members at the same physical scale (the mass counterterms are vertices \cite[(2.2)--(2.3)]{Ba3}). \emph{Inside the coefficient.} A resummed coefficient is bounded by absolute values of its factors \cite[p.~437]{Ba3}; replacing a composite by its difference keeps the form of the bound ($\bar K$), and replacing a line adds a factor $|r|^{-\gamma}$ at scale $\le\rho_{j''}$, integrable inside the coefficient for $\gamma<\alpha_0$, the replaced coefficient being convergent with positive room \cite[p.~441]{Ba3} (with derivatives at both outer ends the integrand is $\sim|r|^{-1}$, still integrable with room $2>\gamma$). Nested coefficients are weakened elements of the outer generalised graph, absorbed at their own scale since every ordered stage has $D>0$. \emph{Layer.} The image part of a line or of a composite's line survives every replacement, by $\min(a,b)\le a^{1/2}b^{1/2}$ in the outer terms and $(\mathrm I_P)$ and by $|z_b-gu|\ge|z-gu|-\sqrt3\cdot5\vartheta$ in $(\mathrm{ZM}_P)$; King's per-scale sum \cite[(3.17)--(3.18)]{K2} gives $\omega_\gamma\in\mathcal L(\rho_{j_0},m,\gamma)$ (one internal $D=0$ sub-stage adds $1$ to $m$). For the members at distance $n$ telescope over the $n$ steps. \end{proof} \space% % % % \smallskip\noindent Throughout \cref{cpi:uc:thm:J} and its proof, $0<\alpha_{\mathrm H}<1$ and $0<\beta_{\mathrm H}<\alpha_{\mathrm H}/2$ are the H\"older and the borrowing exponents of \cref{cpi:thm:a02:Bsc}, $\alpha_0^B$ is the minimum positive stage degree of \cref{cpi:thm:a02:Bsc}, $\alpha_0$ that of \cref{cpi:thm:a02:S3}, and for $0<\alpha'<1$, $\gamma_{F3}$ is the rate of the parts specific to the weight $f_D$: any $\gamma_{F3}<\min(\alpha_0,s)$, with $s=(1-\alpha')/(1+\alpha')$ the H\"older rate of \cref{cpi:lem:a02:RSB} (as in \cref{cpi:uc:lem:QPK}). The $S'_{\mathrm{hop}}$ pieces carrying an explicit averaging kernel $P_{j_0}=Q_{j_0}^*Q_{j_0}$ outside a line are the residual $\Delta c^{S'_{\mathrm{hop}}}$, the replaced-line piece of \cref{cpi:thm:a02:BVP}(b) and the inner resummed coefficients of the vacuum-polarisation pieces; the \emph{composite-kernel structure} (a), (b) of \cref{cpi:uc:lem:QPKinf} of such a piece is this: every explicit $P$ sits in exactly one composite $A\,M_P\,B$ created by the splitting of a resummed coefficient, whose outer ends are cube cumulative lines (or their image parts) and Dirichlet covariances carrying at most one derivative and at most one H\"older quotient each, of exponent $\alpha_B\in(0,1)$, whose $P$-ends carry no derivative and no other factor, and no step applied after the splitting has a $P$-point as a vertex, a localisation point, a Taylor or H\"older base point, a contour end, a summation-by-parts site, or a slice or ordering variable. \begin{theorem}[the boundary bound and the block-vertex difference clause, jointly]\label{cpi:uc:thm:J} In the setting of \cref{cpi:uc:lem:UC} let $\mathcal R$ be a renormalised family of King's expansion of order $\le\bar n$ with at least one external field, localised in zone~B (so source-free, \cref{cpi:lem:a02:Z}), rewritten by $\mathcal R^{C,\mathrm{uc}}$ (\cref{cpi:uc:lem:UC}) and written on the split graph $G^{\mathrm{sp}}$ of \cref{cpi:thm:a02:S3} with $b\ge0$ block points and $r\ge0$ layered vertices. Let $0<\alpha'<1$ and $0<\gamma<\min(\alpha_0^B,\beta_{\mathrm H},\alpha_0,\alpha',\gamma_{F3}/2)$. Every $S'_{\mathrm{hop}}$ piece of the family carrying an explicit averaging kernel $P_{j_0}=Q_{j_0}^*Q_{j_0}$ outside a line has the composite-kernel structure (a), (b) of \cref{cpi:uc:lem:QPKinf}; for the classes treated explicitly in \cite[pp.~436--445]{Ba3} this is Ba{\l}aban's rewriting, and at every order $\le\bar n$ it holds by the argument of \cite[pp.~437--445]{Ba3}, carried to every order (\cref{cpi:uc:lem:QPKinf}). Assume $\alpha_B+\alpha'<1$. Then for every member: \begin{itemize}[nosep] \item[(J-a)] the bound of \cref{cpi:thm:a02:Bsc}(a) holds for $G^{\mathrm{sp}}$ with the root weight $\Lambda$; the same holds with one element weakened by $\theta$, either a line or vertex in a stage of positive degree with $\theta<\alpha_0^B$ or a layered weight $\omega_v$ of class $\mathcal L(\rho_v,m,\theta_v)$ (\cref{cpi:def:a02:lay}), $\theta_v$ its class exponent, weakened to the class exponent $\theta_v+\theta<1$, or a factor at a block point with $\theta=\alpha'<1$; \item[(J-b)] $|E'^{(j+1)}-E^{(j)}|\le C5^{-\gamma j}\times$(the bound of (J-a) with $\Lambda$ replaced by $\Lambda_\gamma$); for a member with a line $\delta G_j(\square',A^{(j)})$ the right side is $(C5^{-\gamma j}+C_N\eta^N)\times$(the bound of (J-a)) for every $N$, which is of the first form on every cone $j\le(1-c)k$, $c>0$, for $N\ge\gamma(1-c)/c$, in particular along the diagonal pairs of \cref{ssec:k01:setting}; \item[(J-c)] summed over the localisations, the zone-B part is $\le C5^{-\gamma j}\eta^{\rho_0}p^{n_1}(1+k-j)^{p_0}|\partial T^{(j)}|$, and the block-vertex part has the form $\le C(5^{-\gamma j}\eta^{-\beta_1}+\eta^{\sigma})|\CJ|$ for some $\beta_1,\sigma>0$, the form of the one-step bound \eqref{eq:k01:Kpos}; here $\eta=5^{j-k}$ is as in (J-b), and the spacing $\vartheta=5^{-j}$ of the pair (\cref{cpi:uc:lem:QPK}) obeys $\vartheta^{-1}=5^{j}\le\eta^{-(1-c)/c}$ on the cone $j\le(1-c)k$. \end{itemize} In particular \cref{cpi:thm:a02:Bsc}(b) for the members with block points and the pair rate of \cref{cpi:thm:a02:S3}(b) for the block-vertex families in zone~B hold, and neither proof uses the other as a theorem. The constants depend on the point of King's range, $\bar n$, $\alpha_{\mathrm H}$, $\beta_{\mathrm H}$, $\alpha'$, $\alpha_B$, $\gamma$, the order $N$, the cone parameter $c$ and the line constants, not on $j$, $k$, $\eta$, $J$, $x_0$. \end{theorem} \begin{proof} The proof rests on unit-level statements only, none of them a bound on a whole graph: the slice bounds of the cube lines up to the faces and their pair clauses, with the $f_D$-specific parts of \cref{cpi:lem:a02:RSB} and \cref{cpi:thm:a02:T}(ii); the layered vertices, their weights and pair clauses (steps (1)--(2) of the proof of \cref{cpi:thm:a02:Bsc}, \cref{cpi:thm:a02:BVP}(b), \cref{r13:thm:BVPpair}(c)), and, for the $S'_{\mathrm{hop}}$ pieces carrying an explicit averaging kernel $P_{j_0}$, \cref{cpi:uc:lem:QPK} and \cref{cpi:uc:lem:QPKinf}(c) under the composite-kernel structure (a), (b) of \cref{cpi:uc:lem:QPKinf}, which holds at every order $\le\bar n$ by the argument of \cite[pp.~437--445]{Ba3}, carried to every order (their range is contained in the one stated, since $1-\theta_{\max}=\beta_{\mathrm H}$, and $\alpha_B+\alpha'<1$ is a hypothesis); the block points, through the split count of \cref{cpi:thm:a02:S3} and the comparison of the two members' point sums; and the \emph{separation of the two mechanisms}: no scalar remainder vertex and no $S'_{\mathrm{hop}}$ piece contains a block point (\cref{cpi:uc:lem:UC}(ii)), no C-site contains a block point (the census of the C-sites, step~(3) of the proof of \cref{cpi:thm:a02:Bsc}), block points lie in no divergent or remainder unit and no infrared slice ends at one (\cref{cpi:ir:BLKnat}), every connected stage containing a block point has $D\ge1$ in the split count, also when it contains layered vertices (\cref{cpi:ir:BLKnat} with \cref{cpi:thm:a02:S3}(c), (d): a block point weighs $\ge\frac52$, step~(3) of the proof of \cref{cpi:thm:a02:Bsc}), so the root-weight stages (degree $0$, containing $x_0$ and an inactive layered composite) contain no block point, and every stage containing a line with an end at a block point contains the point, a line lying in a stage only with both ends, hence has $D\ge1$. \emph{(J-a).} Run the ordered bound with layered vertices (step~(4) of the proof of \cref{cpi:thm:a02:Bsc}) on $G^{\mathrm{sp}}$: its lines are slices, contour and block-kernel lines, lines of the split graph of \cref{cpi:thm:a02:S3}(c), (d), on which the multiscale induction runs with the split-count exponents; its layered vertices are the scalar remainder vertices and the $S'_{\mathrm{hop}}$ pieces, of widths $\rho_v\le\rho_{j(v)}$, none of them a block point (\cref{cpi:uc:lem:UC}(ii)); the degree condition is \cref{cpi:uc:lem:UC}(iii). The induction with layered endpoints acts on layered endpoints only; host block points are never layered endpoints, and the points of former $S'_{\mathrm{blk}}$ members are host points (\cref{cpi:uc:lem:UC}); a point's contour and block-kernel lines are shrunk as ordinary lines, and the infrared stage is unchanged. Weakened forms: a weakened line in a stage of positive degree costs $\theta<\alpha_0^B$; a weakened layered weight stays admissible since $\theta_v+\theta<1$; a factor weakened by $\alpha'$ at a block point lies only in stages containing the point, where $D\ge1>\alpha'$. The root-weight steps occur only at stages of degree $0$ containing $x_0$ and an inactive layered composite, which contain no block point. \emph{(J-b).} Write the difference of the two members as a telescoping sum over the factors of $G^{\mathrm{sp}}$ plus the point step, $E'-E=(\mathrm I)+(\mathrm{ZM})$. Each term of $(\mathrm I)$ has exactly one differenced factor and is bounded by that factor's difference bound times the weakened form of (J-a): an internal slice line by the cube two-member line comparison weakened by $\gamma$, with its H\"older clause if the line is charged; a contour line by the pair clause of the staircase lines, with the H\"older clause of the slice bounds on each slice of the contour sum if its far end is charged; a layered coefficient or weight by its pair clause (the rate form of the borrowed weight, of class $\mathcal L(\rho,0,1-\beta_{\mathrm H}+\gamma)$, admissible since $\gamma<\beta_{\mathrm H}$; \cref{r13:thm:BVPpair}(c); \cref{cpi:uc:lem:QPK} and \cref{cpi:uc:lem:QPKinf}(c) for the $S'_{\mathrm{hop}}$ pieces carrying an explicit $P_{j_0}$); external lines and fields by the line comparison, the sources and $\chi_j$ by sampling, $\kappa'=\kappa$, no phase at $B=0$; the finer member's slices of negative index at their own scale (exponents $D-\gamma>0$ off layered stages, $D+1-\gamma$ for active layered ones, $D-\gamma\ge1-\gamma$ on point stages). The point step $(\mathrm{ZM})$ at a block point $w$: $w$ enters only through its weight $\vartheta^3f_*(w)$, with $\vartheta=5^{-j}$ the spacing and $f_*$ the finest-step weight of \cite[(3.8)]{K2} (equal to $(5/4)^3$ on $4^3$ sites of each $5\vartheta$-block and $0$ on the remaining layer), through functions constant on each $5\vartheta$-block (block indicators, $\kappa$, block corners of contours) and through at most $\bar n$ factors: undifferentiated slices of cube lines, free lines or image parts, contour lines, charged lines (the leg of a remainder vertex of degree $-1$ carrying the borrowed factor, the borrowing of step~(3) of the proof of \cref{cpi:thm:a02:Bsc}) and external fields through contours. The finer member's point sum is identified block by block with the coarser lattice, $\sum_{w'\in B^1(z)}\vartheta'^3f'_*(w')=\vartheta^3$ \cite[(3.13)]{K2} (this is the term $(\mathrm I)$), and the coarser member's weight is removed block by block by $\sum_{z\in b}(1-f_*(z))=5^3-4^3(5/4)^3=0$, which leaves $(\mathrm{ZM})=\vartheta^3\sum_b\sum_{z\in b}(1-f_*(z))[\prod_iF_i(z)-\prod_iF_i(z_b)]$ with a reference site $z_b$ in each block $b$; telescoping over the factors, each term is $G^{\mathrm{sp}}$ with one factor at the point weakened by $\alpha'$, times $C5^{-\alpha'j}$: a slice or contour line loses $\rho_i^{-\alpha'}$ by the H\"older clause of the slice bounds in the point variable (for $|z-z_b|\le\sqrt3\cdot5\vartheta$ this is $C(\vartheta/\rho_i)^{\alpha'}=C5^{-\alpha'j}\rho_i^{-\alpha'}$); a charged line $G(v,\cdot)$ with its $v$-end at a remainder vertex of degree $-1$ is differenced along a lattice path of at most $12$ bonds inside the block, each bond by the H\"older clause with the lattice derivative at the far end, which keeps the vanishing factor $\min(1,t(v)/\rho_i)^{\alpha_{\mathrm H}}$ of the leg and gives $(|z-z_b|/\rho_i)^{\alpha'}$ (for the external-line and $\delta G_j$ leg types the geometric mean of the vanishing bound and of the H\"older bound in the point variable gives the exponents $\alpha_{\mathrm H}/2$, $\alpha'/2$, enough since the borrowing uses $\beta_{\mathrm H}<\alpha_{\mathrm H}/2$); an external field is differenced by its gradient clause; the undifferenced factors keep their bounds up to $e^{5\sqrt3\,\delta}$, every decay rate being halved. Every stage containing the weakened factor contains the point, where $D\ge1>\alpha'$, so each term is $\le C5^{-\alpha'j}\times$(J-a). No layered vertex is a host block point (scalar remainder vertices are point-free, $S'_{\mathrm{hop}}$ pieces contain no block vertex, and there is no collapsed block piece, \cref{cpi:uc:lem:UC}), so $(\mathrm{ZM})$ never differences a layered weight, and differencing at a host point the end of a line that leads to a piece is a slice or contour line factor. A member with a line $\delta G_j(\square',A^{(j)})$, an external object \cite[p.~661]{K1}, is bounded in each model separately: by the exponential decay of $\delta G_j$ and since every vertex lies at distance $\ge2p$ from $\partial\Omega\setminus\mathsf F$, it is $\le Ce^{-2\delta_0p}\times$(J-a) with $p=p(\eta)$, and $e^{-2\delta_0p(\eta)}\le C_N\eta^N$ for every $N$, as $p>1$ \cite[(3.41)]{K1}; the remainders \cite[(3.48), (3.52)]{K1} likewise \cite[p.~665]{K1}. On a cone $j\le(1-c)k$ one has $\eta=5^{j-k}\le5^{-cj/(1-c)}$, so this is $\le C5^{-\gamma j}\times$(J-a) once $N\ge\gamma(1-c)/c$. Summing the finitely many terms gives (J-b) for every $\gamma$ in the stated range. \emph{(J-c).} Zone~B: the layer sum per unit face cell of \cref{r13:thm:BVPpair}(d), as in \cref{cpi:thm:a02:Bsc}(c). The block-vertex form: the position sums of the multiscale induction over $\CK$ cost a power of $\vartheta^{-1}=5^{j}$, and on the cone $j\le(1-c)k$ (or along the diagonal pairs) $5^{j}\le\eta^{-(1-c)/c}$, which gives the factor $\eta^{-\beta_1}$ in front of the rate $5^{-\gamma j}$ of (J-b); the terms $C_N\eta^N\times$(J-a) of the members with a line $\delta G_j(\square',A^{(j)})$, summed in the same way, give the second term, with $\sigma=N-\beta_1>0$ for $N>\beta_1$. The induction runs over the ranks (unit statements) $\to$ (J-a) $\to$ (J-b) $\to$ (J-c); inside (J-a) and (J-b) it is the ordered shrinking of the multiscale induction over finitely many lines and prefix components. At no rank is a graph-level rate statement, the boundary bound (b) or the difference clause, used, and the unit statements use neither. For $b=0$, (J-a)--(J-c) are the boundary bound for the point-free members, and for $r=0$ they are the difference clause of the block-vertex families. Two further points. The vacuum-polarisation rewriting of \cref{cpi:thm:a02:BVP}(a) with every $S'_{\mathrm{blk}}$ collapsed into one difference piece is not used: a member of $S'_{\mathrm{blk}}$ with two or more two-point block products joined only through the block kernel $\kappa$ is not one stage-$j_0$ object of $G^{\mathrm{sp}}$ and cannot be collapsed into a layered vertex of width $\rho_{j_0}$; under \cref{cpi:uc:lem:UC} these members are host members with block points, treated by (J-a), (J-b). And the pair rate of the members with block points, left to this statement by \cref{r13:thm:BVPpair}(c), is (J-b). \end{proof} \space% % \subsection{The Cauchy property in the spacing}\label{ssec:k01:CA} \begin{theorem}[the Cauchy property, ratio form]\label{thm:k01:CA} Fix $J\ge J_0$, $f$, $x\in\R^N$, and let $\eta_1$, $k_0$, $C$, $c$, $\sigma$ and $\gamma$, $\beta$, $C_3$ be the constants of \eqref{eq:k01:H0rel} and \eqref{eq:k01:Kpos} for $s\in\{u_x,0\}$ (\cref{prop:k01:H0rel,prop:k01:Kpos}). Choose $\alpha\in(0,\min\{\frac12,\gamma/(\gamma+\beta)\})$ and the diagonal pairs $j(k)=\lfloor(1-\alpha)k\rfloor$, $\eta_k=5^{j(k)-k}$. Put \[ \begin{gathered} b_k\defeq C\eta_k^{\sigma},\qquad \theta_k\defeq C_3\big(5^{-\gamma j(k)}\eta_k^{-\beta}+\eta_k^{\sigma}\big),\\ \rho_k\defeq\rho(\eta_k),\qquad r_k(0)\defeq2\rho_k,\qquad r_k(u_x)\defeq2\rho_ke^{\beta_f|x|^2}. \end{gathered} \] Then there is $k_1$ such that for all $k\ge k_1$ \begin{equation}\label{eq:k01:CA} \big|\ln a_{k+1}(x)-\ln a_k(x)\big|\ \le\ 4b_k+2\theta_k+r_k(u_x)+r_k(0). \end{equation} The right side is summable in $k$, and its tail from $k$ on is at most $C'5^{-\kappa k}$ for some $\kappa>0$. Hence $a_k(x)$ converges, and its limit lies in $[e^{-\beta_f|x|^2},e^{\beta_f|x|^2}]$. \end{theorem} \begin{proof} Fix $k\ge k_0$ so large that $j(k)\ge J_*$, $k-j(k)\ge J_*$, $\eta_k\le\eta_1$ and $r_k(u_x)\le1$ (possible by Step~6, since $k-j(k)\ge\alpha k$), and put $j\defeq j(k)$, $b\defeq b_k$, $\theta\defeq\theta_k$. For $s\in\{u_x,0\}$ write $Z\defeq\Zk(s)$, $I\defeq\Ik{j}(s)$, $\hat I\defeq\Ihat{j}(s)$ for the $k$-model after $j$ steps, and $Z'\defeq Z_{k+1}(s)$, $I'\defeq I^{(j+1)}_{k+1}(s)$, $\hat I'\defeq\hat I^{(j+1)}_{k+1}(s)$ for the $(k+1)$-model after $j+1$ steps. Both members have the block scale $\eta_k$ and the indicator $\chi_j$, and \eqref{eq:k01:H0rel} applies to both, since $j+1\le(k+1)-J_*$; their remainders are $\mathcal E\defeq\rho_k\Zk(0)$ and $\mathcal E'\defeq\rho_kZ_{k+1}(0)$. \emph{Step 1 (the remainders relative to $Z$).} For $s=0$, $\mathcal E/Z=\rho_k=r_k(0)/2$ and $\mathcal E'/Z'=\rho_k$. For $s=u_x$, $Z=a_k(x)\Zk(0)\ge e^{-\beta_f|x|^2}\Zk(0)$ by \cref{lem:k01:free}, so $\mathcal E/Z\le\rho_ke^{\beta_f|x|^2}=r_k(u_x)/2$, and likewise $\mathcal E'/Z'\le r_k(u_x)/2$. In both cases $\mathcal E/Z,\ \mathcal E'/Z'\le r_k(s)/2\le\frac12$. No lower bound on $\Zk(0)$ and no bound on $\Zk(u_x)$ beyond \cref{lem:k01:free} is used. \emph{Step 2 (one model).} From $e^{-b}I\le Z\le e^{b}I+\mathcal E$: $I\le e^{b}Z$ and $I\ge e^{-b}(Z-\mathcal E)=e^{-b}Z(1-\mathcal E/Z)$. So $I\in[e^{-b}Z(1-\mathcal E/Z),\,e^{b}Z]$, and in the same way $I'\in[e^{-b}Z'(1-\mathcal E'/Z'),\,e^{b}Z']$. In particular $I,I'>0$. \emph{Step 3 (across the pair).} On the support of $\chi_j$, \eqref{eq:k01:Kpos} gives $|S^{(j)}_{N,k}(\cdot;s)-S^{(j+1)}_{N,k+1}(\cdot;s)|\le\theta$. The two integrals $\hat I$, $\hat I'$ run over the same variables with the same $\chi_j$ and have non-negative integrands, so $\hat I'/\hat I\in[e^{-\theta},e^{\theta}]$. Since $I=e^{-v_{k,j}}\hat I$ and $I'=e^{-v_{k+1,j+1}}\hat I'$, \[ I'/I\ \in\ e^{\Delta}\,[e^{-\theta},e^{\theta}],\qquad \Delta\defeq v_{k,j}-v_{k+1,j+1}, \] and $\Delta$ does not depend on $s$: the vacuum constant is the same number in the numerator and in the denominator of $a_k$. \emph{Step 4 (ratio of partition functions).} By Steps 2 and 3, \[ \frac{Z'}{Z}=\frac{Z'}{I'}\cdot\frac{I'}{I}\cdot\frac{I}{Z}\ \in\ e^{\Delta}\Big[e^{-2b-\theta}\big(1-\tfrac{\mathcal E}{Z}\big),\ e^{2b+\theta}\big(1-\tfrac{\mathcal E'}{Z'}\big)^{-1}\Big]. \] With $|\ln(1-t)|\le2t$ for $0\le t\le\frac12$ and Step~1, $|\ln(1-\mathcal E/Z)|,\,|\ln(1-\mathcal E'/Z')|\le r_k(s)$, hence \[ \big|\ln(Z'/Z)-\Delta\big|\ \le\ 2b+\theta+r_k(s). \] \emph{Step 5 (ratio of ratios).} $\ln\dfrac{a_{k+1}(x)}{a_k(x)}=\Big[\ln\dfrac{Z_{k+1}(u_x)}{\Zk(u_x)}-\Delta\Big]-\Big[\ln\dfrac{Z_{k+1}(0)}{\Zk(0)}-\Delta\Big]$, and Step~4 at $s=u_x$ and at $s=0$ gives \eqref{eq:k01:CA}. (If the constants of \eqref{eq:k01:H0rel}, \eqref{eq:k01:Kpos} differ between the two sources, take the larger ones.) \emph{Step 6 (summability).} Since $k-j(k)\ge\alpha k$, $\eta_k\le5^{-\alpha k}$, so $b_k$ and $C_3\eta_k^\sigma$ decay geometrically. Since $j(k)\ge(1-\alpha)k-1$ and $k-j(k)\le\alpha k+1$, \[ 5^{-\gamma j(k)}\eta_k^{-\beta}\ \le\ 5^{\gamma+\beta}\,5^{-(\gamma(1-\alpha)-\beta\alpha)k}, \] which decays geometrically because $\alpha<\gamma/(\gamma+\beta)$. Finally $p(\eta_k)^2\ge b_0^2(1+\alpha k\ln5)^{12}\ge c_1k^2$, so $\rho_k\le e^{-cc_1k^2+C}$; thus $r_k(u_x)\le1$ for large $k$, $\sum_kr_k(s)<\infty$, and the tail of $\sum_k\rho_k$ is eventually below any geometric tail. The series in \eqref{eq:k01:CA} converges with a geometric tail, so $(\ln a_k(x))_k$ is Cauchy, $a_k(x)$ converges, and the limit lies in the closed interval of \cref{lem:k01:free}. \end{proof} \begin{remark}[what the argument does not use]\label{rem:k01:notused} (i) No convergence, and no bound, of the vacuum constants $v_{k,j}$ (King's $E_0$, $E_1$, the kernel normalisations, the face vacuum energies) enters: they cancel in Step~3. King's route \cite[p.~329]{K2} goes through the convergence of $\Zk$ itself, which requires exactly those constants to converge. (ii) No upper stability bound, no lower bound on $\Zk(0)$ and no bound on $\Zk(u_x)$ beyond \cref{lem:k01:free} is used; the remainder of \eqref{eq:k01:H0rel} is compared with $\Zk(0)$ only. (iii) No reflection positivity, no Ward identity, and no subsequence. (iv) The all-$n$ form of \cite[(3.13)]{K1} (comparing the $k$-model with the $(k+n)$-model at a fixed stopping scale $\eta$ for every $n$) also gives the Cauchy property, provided the constants of \eqref{eq:k01:H0rel} and \eqref{eq:k01:Kpos} are taken at fixed $\eta$ with no growth in $k$: then $|\ln a_{k+n}-\ln a_k|\le4b(\eta)+2\theta_k(\eta)+4\rho(\eta)(e^{\beta_f|x|^2}+1)$ for every $n$, and one lets $k\to\infty$ and then $\eta\to0$. The diagonal form above uses $n=1$ only, at the price of constants uniform in $j$; the two forms use the underlying estimates differently, and neither implies the other. \end{remark} \begin{theorem}[King's existence theorem on the cube: the curvature field]\label{thm:k01:EC} For every finite family $f_1,\dots,f_N$ of real $2$-forms in $C^\infty_c(\R^3;\Lambda^2)$, every $J\ge J_0(f)$ and every $x\in\R^N$ the limit \[ \ZKJ(x)=\lim_{k\to\infty}\avK{e^{\sum_ix_iF(f_i)}} \] exists and lies in $[e^{-\beta_f|x|^2},e^{\beta_f|x|^2}]\subset(0,\infty)$, with $\beta_f$ as in \cref{lem:k01:free}; the convergence is geometric. That is, \cref{hyp:s03:EC} holds. Consequently $\ZKJ$ and $\ZKinf$ exist locally uniformly on $\C^N$ and are entire, with the bounds and the monotonicity of \cref{thm:s05:CK}(b). \end{theorem} \begin{proof} \Cref{prop:k01:H0rel,prop:k01:Kpos} provide the two estimates \eqref{eq:k01:H0rel}, \eqref{eq:k01:Kpos} that \cref{thm:k01:CA} uses, at every $x$; the theorem gives the limit and its bounds; \cref{thm:s05:CK}(b) (that is, \cref{cor:s04:G2}(ii)) gives the rest. The constants $k_0$, $\eta_1$ and the rates may depend on $x$ and on $f$; existence needs no uniformity in $x$. \end{proof} \begin{remark}[the comparison family]\label{rem:k01:compfamily} \Cref{thm:k01:EC} says nothing about the fixed-cube limits of the comparison family $\NDa$ (\cref{rem:k01:cube}); for that family \cref{cor:s04:G2}(ii)($\gamma$) squeezes the limits inferior and superior at fixed $J$ between $\ZKJ$ and $Z^K_{J+1}$, and \cref{cor:s04:G2}(iii) identifies every subsequential limit, followed by $J\to\infty$, with $\ZKinf$. At fixed $J$ every inequality available for the comparison family bounds its oscillation in $k$ only by the difference of two distinct limits, which tends to zero only as $J\to\infty$; a Cauchy statement at fixed $J$ for that family would need the small-field analysis at its faces, where no flow of the kind used above exists. \end{remark} \space% % % % % % \section{King's existence theorem on the cube with scalar sources}\label{sec:k02} This section proves \cref{hyp:s03:W1}, the existence theorem with scalar sources in the form in which the Wick leg uses it: in the reference scheme $z$, the fixed-cube limits of the generating functional with curvature and scalar sources exist, locally uniformly and holomorphically, on the domain of coefficient vectors whose positive scalar part is below a threshold, and they are positive at real points; the negative scalar part is unrestricted. The route is the Cauchy argument of \cref{sec:k01} with the scheme $c$-number carried along (\cref{thm:k02:CAh}), applied at real points with a small non-negative scalar source, followed by a normal-family argument on the domain. The domain is bounded by Hölder's inequality from three one-sided estimates: the curvature bound of \cref{lem:k01:free}, a positive-source bound (\cref{prop:k02:Wsharp}), and a negative-source bound (\cref{thm:k02:UB}), the last obtained from positive-source statements at a heavier scalar mass by an exact mass shift and a correlation inequality. The notation is that of \cref{ssec:k01:setting}. \subsection{Setting; the source-dependent graphs}\label{ssec:k02:setting} Throughout, the scheme is the reference scheme $z=a^\sharp$ of \cref{def:s02:sources}; write $z_k$ for its $k$-th term. Put $\sS_k(h)\defeq\sum_{x\in\Lk\cap\CJ}\eps_k^3h(x)$ and $c_k(h)\defeq z_k\sS_k(h)$, which is linear in $h$. For a real source $s=(u,h)$ the normalised functional in the scheme is \[ n_k(s)\ \defeq\ e^{-c_k(h)}\,N_k(s)\ =\ e^{-z_k\sS_k(h)}\,\avK{e^{F(u)+\Wick{h}}}\ =\ Z^{z,K}_{k,J}(s), \] the object of \cref{hyp:s03:W1}. For a finite family $f_1,\dots,f_n$ of real $2$-forms and $h_1,\dots,h_p$ of real functions, all supported at $\ell^\infty$-distance $>1$ from $\R^3\setminus\CJ$, and $\zeta=(z,w)\in\C^{n+p}$, $n_k(\zeta)\defeq e^{-\sum_jw_jc_k(h_j)}\avK{e^{\sum_iz_iF(f_i)+\sum_jw_j\Wick{h_j}}}$ (here $z_i$ are coefficients and $z_k$ the scheme term; no confusion will arise). Every $n_k$ is entire in $\zeta$, $|n_k(\zeta)|\le n_k(\Re\zeta)$, and $\ln n_k$ is convex on real points (\cref{lem:s02:alg}(b)): a log-Laplace transform minus a linear function. A second scalar mass $M>0$ will be used with the \emph{same} $e$, $\lambda$, $\lmu$, lattice, cube and boundary conditions; $\langle\cdot\rangle^{(M)}$ denotes King's cube state at mass $M$, $\alpha_k(M)\defeq M^2+\dm_{\eps_k}(M)$ King's free-boundary bare coefficient at mass $M$ (\cref{conv:v065d_s02_setting:1}), $u_k(M)\defeq\frac12(\alpha_k(M)-\alpha_k(m))$, and $s_-(M)\defeq\liminf_ku_k(M)$. Every statement of \cref{sec:k01} and of this section holds at every point of King's range with point-dependent constants, in particular at the point $(e,\lambda,M,\lmu)$; we use this at $M=m$ and at one heavier $M_2$. \begin{lemma}[\Kh: one-step convergence of the source-dependent graphs on King's cube]\label{hyp:k02:Kh} Let $P^{(j),1}_k$ be the interaction part of King's effective action after $j$ steps on $\CK$ \cite[(3.17)]{K1}, written as a sum over connected renormalised graphs $H$, and let $\mathcal N_h$ be the family of its members that depend on the scalar source $h$: the field-dependent part of every graph with an $h$-vertex, and every field-independent $h$-dependent graph except the $h$-linear ones of the three lowest orders $(0,0)$, $(0,1)$, $(2,0)$ in $(e^2,\lambda)$ (the family $\mathcal T_h$ of \cref{cpi:def:families}). Then for every fixed real $h\in C^\infty_c$ supported at distance $>1$ from $\R^3\setminus\CJ$ and every member of $\mathcal N_h$, for $J_*\le j\le k-J_*$: the localisation bound \cite[(3.38)]{K1}, with the factor $(1+k-j)^{p_0}$ (King's power of $\ln(L^k\eps)^{-1}$, \cite[p.~666]{K1}, a choice of his $n_1$), holds for its localisations at distance from the faces larger than the small-field scale $2p(\eta)$ of the localisation boxes (zone~I of \cref{cpi:def:a02:zones}); the bound \cite[(3.38)]{K1} with the restricted degree holds with the factor $(1+k-j)^{p_0}$, summed over the localisations sharing the cube of an $h$-vertex, and for each single localisation at distance $>1$ from the faces. Moreover, there are exponents $\gamma',\beta',\sigma'>0$, depending neither on $h$ nor on the member, such that for every $k$ and every $j$ with $J_*\le j\le k-J_*$ the sum over positions of the difference between the member at $(k,j)$ and the member at $(k+1,j+1)$, on the common unit lattice and on the support of $\chi_j$, is bounded by $C\big[5^{-\gamma'j}\eta^{-\beta'}+\eta^{\sigma'}\big]\,|\CJ|$, $\eta=5^{j-k}$, with $C$ independent of $j$ and $k$. \end{lemma} This is the cube form, on $\CK$ with King's Dirichlet weight, of the corresponding statement on King's tori; the $h$-free graphs, those with a curvature vertex and no $h$-vertex and the source-free ones, are covered on the cube by \cref{prop:k01:Kpos}. The localisations of a member of $\mathcal N_h$ with tree length $R\le p$ lie at distance $>1$ from the faces, where the bounds reduce to those of the free flow on the torus (\cref{prop:k02:KhI}); a localisation with $R>p$ carries the factor $e^{-\delta R}$ once the localised graph is cut into clusters at mutual distance $\ge R_{\mathrm{cl}}=3\sqrt3$ (the cut by clusters of \cref{r13:def:clusters}), each reduced in its own box of side $O(p)$, the clusters containing the source vertex lying away from the faces and the clusters meeting the Dirichlet faces being source-free (\cref{prop:k02:KhR}). The third clause holds at every step of the range; the proof of \cref{prop:k02:Bh} uses it along the diagonal pairs of \cref{ssec:k01:setting}. The proof is given in \cref{ssec:k02:Khproof}. % % % % \subsection{Proof of \texorpdfstring{\cref{hyp:k02:Kh}}{the one-step convergence of the source-dependent graphs}}\label{ssec:k02:Khproof} We work in the unit units of depth $j$ of \cref{cpi:def:a02:zones}: the live sites are $T_\eta$, $\mathsf F$ denotes the set of scalar Dirichlet planes, $t(x)\defeq\dist(x,\mathsf F)$, and a localised graph $H$ carries King's boxes $\square\subset\square''\subset\square'$ and $\Omega=\square'\cap T_\eta$, in zone~I if $\square'\cap\mathsf F=\emptyset$ and in zone~B otherwise. The localisation is by centres, as in the proof of \cref{prop:k01:Kpos}: a tuple $\tau$ has the point set $P(\tau)$, $n_\tau=|P(\tau)|\le N_*$, and the tree length $R=R(\tau)=\dist(P(\tau))$ (\cref{cpi:def:a02:zones}). The cut by clusters, its constants $c_{\rm cl}=\frac{13}3\sqrt3$ and $R_{\rm cl}=3\sqrt3$, the clusters $K$, the cluster graphs $H_K$ with their boxes $\square_K\subset\square''_K\subset\square'_K$, gauges $B_K=A_j(x^K_0)$ and $\Omega_K=\square'_K\cap T_\eta$, and the zone of a cluster are those of \cref{r13:def:clusters}; we use its elementary geometry (cut lines between clusters have length $\ge R_{\rm cl}$, $\operatorname{diam}P_K\le c_{\rm cl}(|P_K|-1)$, $|\square'_K|\le C'p^3$, the quotient graph of clusters and cut lines is connected with $\dist(P(\tau))\le\sum_{\ell\in T}|\ell|+c_{\rm cl}n_\tau$ over any spanning tree $T$, and the partition, the boxes, the gauges and the set of cut lines are the same for every member of every family and every position). The two members of the pair, the $k$-model at depth $j$ and the $(k+1)$-model at depth $j+1$, live on the same unit lattice; every bound below holds for each of them separately with the same constants, and the two are compared only in zone~I. The $h$-vertex is King's vertex \cite[(3.21)]{K1}: the smooth function $h$ sampled at the points of the lattice's own frame, the coupling $\theta\eta^{2}$ in the units of depth $j$, and at most two fields $\phi^{(j)}$. In King's count of the coupling power $\rho(H)=\sum_v\rho_v$ of a graph \cite[p.~665]{K1}, every vertex has $\rho_v-s_v/4\ge0$, $s_v$ its number of external scalar fields, and the $h$-vertex has $\rho_h-s_h/4\ge2-\frac12=\frac32$; hence \begin{equation}\label{eq:k02:margin} \rho(H)-\tfrac14s(H)\ \ge\ \tfrac32\qquad\text{for every member of $\mathcal N_h$ with external fields,} \end{equation} $s(H)$ its number of external scalar legs, and $\rho(H)\ge0$ for the field-independent members. The Wick constant of \cite[(3.21)]{K1} is field-free and $h$-linear of the lowest order, so it enters only the excluded family $\mathcal T_h$ of Definition~\ref{cpi:def:families} and no member of $\mathcal N_h$ depends on it. % % \begin{lemma}[pointwise bounds at a regular background]\label{cpi:b7:lem:PW} In the unit units of depth $j$ (\cref{cpi:def:a02:zones}: live sites $T_\eta$, fine spacing $\vartheta$, scalar Dirichlet planes $\mathsf F$, $t(x)=\dist(x,\mathsf F)$) let $A$ be a photon field on the bonds of $T_\eta$ with $\theta_j\defeq e_j\sup_b|\partial A(b)|$, where $\partial A(b)\defeq(\partial_\nu A_\mu(b))_{\nu=1,2,3}$ is the vector of the lattice derivatives of the bond variable $A(b)=A_\mu(b)$, in the Euclidean norm, and let $G_A\defeq H_A^{-1}$, $H_A\defeq-\Delta_A+\mu+\mathfrak a_jQ_j(A)^*Q_j(A)$, $\mu\defeq m^2\eta^2$, be the cube propagator at the background $A$, on $\R^N$-valued functions on the live sites, with the Dirichlet condition on $\mathsf F$ and the weight $f_D$ in $Q_j$, its kernel taken with respect to the measure $\vartheta^3$ and extended by zero on $\mathsf F$ ($H_A$ is real symmetric, so $G_A(x,z)=G_A(z,x)^T$). An \emph{operation} $O_x$ at a point $x$ is the identity (\emph{order} $o=0$), a covariant bond difference $D_{A,b}$ along a bond $b$ at $x$ of the closed lattice, face bonds included ($o=1$), or an $\alpha$-H\"older quotient $|x-x''|^{-\alpha}[D_{A,b}-U(A(\Gamma_{x,x''}))D_{A,b''}]$ of two parallel bond differences at points $x,x''$ with $0<|x-x''|\le\frac18$, transported along an axis path $\Gamma_{x,x''}$ ($o=1+\alpha$); its evaluation set $E(O_x)$ has diameter $\le\frac18+2\vartheta$. A H\"older pair with $|x-x''|>\frac18$ is bounded by $8^\alpha$ times two differences and needs no clause of its own; an operation in the second variable acts through the transposed transport. There are $\theta_{\mathrm{PW}}>0$, $C$, $\delta_P>0$ and $s_*\defeq4+2\alpha$, depending on the block factor $5$, the point of King's range and $\alpha$ only, such that if $\theta_j\le\theta_{\mathrm{PW}}$, then for all operations $O_x$, $O'_z$ and all $x,z$: \begin{itemize}[nosep] \item[(PW1)] $|O_xG_A(x,z)|\le C(r+\vartheta)^{-1-o}e^{-\delta_Pr}$ with $r\defeq\dist(E(O_x),z)$, and the same for an operation in the second variable with $r\defeq\dist(x,E(O'_z))$; \item[(PW2)] $|O_xO'_zG_A(x,z)|\le C\min(1,r)^{-s_*}e^{-\delta_Pr}$ whenever $r\defeq\dist(E(O_x),E(O'_z))>0$; \item[(PW3)] if $t(x)\le r/2$ with $r\defeq\dist(x,E(O'_z))$ ($O'_z$ possibly the identity), then $|O'_zG_A(x,z)|\le C\min(1,t(x))\min(1,r/2)^{-s_*}e^{-\delta_Pr/2}$. \end{itemize} At $A=0$ and at a constant $A$ these hold with no condition on $\theta_j$. For the background $A=A^{(j)}$, the interpolation \cite[(2.18), (3.43)--(3.44)]{K1} of the block field $A_j$, on $\supp\chi_j$: $|dA^{(j)}|_\infty\le C_Fp$, where $C_F$ is the constant of the interpolation bound, which follows from King's small-field bound $|\partial_\nu A_{j,\mu}(b)|\le p$ on the block bonds \cite[(3.2)]{K1} through the interpolation kernel, which reproduces constants and decays exponentially \cite[(3.7)]{K1}, so that $C_F$ depends on the point only through the photon kernel constants of \cite[(3.7)]{K1}; hence $\theta_j\le C_{\mathrm{PW}}e_jp\le C_{\mathrm{PW}}\,e\,\eta^{1/2}p(\eta)$ with $C_{\mathrm{PW}}\defeq\sqrt3\,C_F$, and the condition $\theta_j\le\theta_{\mathrm{PW}}$ holds on $\supp\chi_j$ once $C_{\mathrm{PW}}\,e\,\eta^{1/2}p(\eta)\le\theta_{\mathrm{PW}}$, a smallness condition of the kind of \eqref{eq:k02:range}, which is added to the range; the cluster fields $\tilde A_K=A^{(j)}-B_K$ of \cref{r13:def:range} have the same lattice derivatives, hence the same $\theta_j$. \end{lemma} \begin{proof} \emph{Constant background.} A constant field $B$ on the simply connected cube is pure gauge, $B=d\lambda$, and $G_B=U_\lambda G_0U_\lambda^*$ with $U_\lambda$ the pointwise rotation by $\exp(e_j\vartheta q\lambda)$, so $B$-covariant operations correspond to ordinary ones. At $A=0$ the propagator is the odd image sum of the free $f_D$ propagator over the group generated by the reflections in the planes $\mathsf F$ (the image sums of \cref{r13:def:boxlines}), the images being lattice isometries with $|x-gw|\ge|x-w|$ coordinatewise and at most $C(1+r)^3$ of them within distance $r$; each slice factorises as $G_{(i)}=\mathfrak a_i^2\rho_i^{-4}(G_iQ_i^*)C^{(i)}(Q_iG_i)$ \cite[(2.17)]{K1}, an operation at each end acting on its own outer factor, and the profile bounds of the free slices give $|O_xO'_wG_{(i)}(x,w)|\le C\rho_i^{-1-o-o'}e^{-\kappa|x-w|/\rho_i}$; the sum over the slices, $\sum_i\rho_i^{-a}e^{-\kappa r/\rho_i}\le C(r+\vartheta)^{-a}e^{-\kappa r/2}$, gives (PW1) and (PW2) at a constant background with the rate $\delta_1\defeq\kappa/4$. \emph{Comparison at each point.} For a live site $x$ let $B_x$ be the constant field equal to $A$ on the bonds at $x$ and $V_x\defeq H_{B_x}-H_A$. Writing the hopping terms through covariant differences, $V_x=W_0+\sum_{\pm\mu}W_{\pm\mu}D_{B_x,\pm\mu}+V_{\mathrm{blk}}$ with $|W_0(w)|,|W_{\pm\mu}(w)|,|\partial W_{\pm\mu}(w)|\le C\theta_j(1+|w-x|)^2$ and a block-local $V_{\mathrm{blk}}$ with $|V_{\mathrm{blk}}(w,w')|\le C\theta_j(1+|w-x|)$, because $e_j|A(b)-B_x|\le\theta_j(1+|w-x|)$ on a bond $b$ at $w$ (integrate $\partial A$ along an axis path from $x$); the resolvent identity $G_A=G_{B_x}+G_{B_x}V_xG_A$ is exact on the finite lattice, and an $A$-covariant operation at $x$ differs from the $B_x$-covariant one by at most $C\theta_j(|f|+|Df|)$ on $E(O_x)$. \emph{Bootstrap.} Let $\Phi$ be the supremum over operations of order $\le1$ and over $x,z$ of $e^{\delta r}\min(1,r+\vartheta)^{1+o}|O_xG_A(x,z)|$, $\delta\defeq\delta_1/2$, finite on the finite lattice. Inserting the resolvent identity, the middle term is a convolution of the constant-background kernel with $V_xG_A$, which the definition of $\Phi$ bounds by $C\theta_j(1+|w-x|)^3\Phi\min(1,|w-z|+\vartheta)^{-2}e^{-\delta|w-z|}$; the convolution bound $\sum_w\vartheta^3(|x-w|+\vartheta)^{-a}(|w-z|+\vartheta)^{-b}e^{-c|x-w|-c|w-z|}\le C(r+\vartheta)^{3-a-b}(1+|\ln(r+\vartheta)|)e^{-cr}$ for $a+b\ge3$, the exponential $e^{-\delta_1|x-w|}$ of the constant-background kernel absorbing the polynomial growth $(1+|w-x|)^3$, gives $\Phi\le C_0+C_1\theta_j\Phi$, hence $\Phi\le2C_0$ once $\theta_j\le\min(1,1/(2C_1))$; the polynomial weight at large $r$ is recovered from the rate, $\delta_P\defeq\delta_1/4$. An operation of order $1+\alpha$ needs no bootstrap, the right side of its identity involving $\Phi$ only. \emph{The mixed bound (PW2).} Let $\Phi'$ be the maximum over pairs of operations and over the pairs $(x,z)$ with $r\defeq\dist(E(O_x),E(O'_z))>0$ of $e^{\delta_Pr}\min(1,r)^{s_*}|O_xO'_zG_A(x,z)|$, finite on the finite lattice. Lattice-scale pairs, $r\le8\vartheta$, need no bootstrap: a difference at $x$ costs $\le2\vartheta^{-1}$ and a H\"older quotient $\le2\vartheta^{-\alpha}$ times differences, so (PW1) in its second-variable form gives $|O_xO'_zG_A(x,z)|\le C\Phi\vartheta^{-3-2\alpha}$, and $\min(1,r)^{s_*}\le(8\vartheta)^{4+2\alpha}$ for $\vartheta\le\frac18$ (for $\frac18<\vartheta\le\frac15$, $\vartheta^{-3-2\alpha}\le5^{3+2\alpha}$): this is where $s_*=4+2\alpha$ enters. For $r>8\vartheta$ insert the resolvent identity at $x$ and split the middle term by a cutoff $\chi_z(w)\defeq\chi(\dist(w,E(O'_z))/\varrho)$ at the scale $\varrho\defeq\min(\max(\frac1{16},2\vartheta),r/4)$, so that $2\vartheta\le\varrho\le r/4$ and $\varrho\ge\min(1,r)/64$. The far part (the terms of $V_x$ with a difference on $G_A$, times $1-\chi_z$) is bounded through $\Phi'$: the bond of the difference has distance $\ge\varrho/2\ge\min(1,r)/128$ from $E(O'_z)$, which gives $\le C\theta_j(128^{s_*}\Phi'+\Phi)\min(1,r)^{-s_*}e^{-\delta_Pr}$. The near part (the same terms times $\chi_z$) is summed by parts in $w$, exactly and with no boundary term, since $g\defeq O'_zG_A(\cdot,z)$ vanishes on $\mathsf F$ and $\chi_z$ has compact support: the difference moves onto $[O^{(B_x)}_xG_{B_x}](x,\cdot)\chi_zW$, whose adjoint difference is $\le C\theta_j\min(1,r)^{-2-o}e^{-\delta_1r/4}$ on $\supp\chi_z$ (there $\dist(E(O_x),w)\ge r/2$), while $\sum_{w\in\supp\chi_z}\vartheta^3|g(w)|\le C\Phi$ by (PW1), the singularity $(\dist(w,E(O'_z))+\vartheta)^{-1-o'}$ with $1+o'\le2+\alpha<3$ being integrable over a set of diameter $\le1$; so the near part is $\le C\theta_j\Phi\min(1,r)^{-s_*}e^{-\delta_Pr}$. The zero-order and block terms of $V_x$ carry no difference on $G_A$ and are bounded by (PW1) and the convolution bound; the term $E_xO'_zG_A$ by (PW1) and $\Phi'$; and the leading term $O^{(B_x)}_xO'_zG_{B_x}$ by the constant-background bound, since $1+o+o'\le3+2\alpha\le s_*$. Altogether $\Phi'\le C\Phi+C_0+C\theta_j(128^{s_*}\Phi'+\Phi)$, hence $\Phi'\le2(C_0+C\Phi)$ once $C\theta_j128^{s_*}\le\frac12$; with $\Phi\le2C_0$ this is (PW2), and $\theta_{\mathrm{PW}}$ is the smaller of the two thresholds. \emph{The face clause (PW3).} Let $x_P\in\mathsf F$ be the foot of $x$ on the nearest Dirichlet plane, $|x-x_P|=t(x)$. Since $G_A(x_P,\cdot)=0$, telescoping along the $t(x)/\vartheta$ normal bonds from $x_P$ to $x$ (each a covariant difference times a unimodular transporter) gives $|O'_zG_A(x,z)|\le t(x)\max_b|D_{A,b}O'_zG_A(\cdot,z)|$ over those bonds, each at distance $\ge r-t(x)\ge r/2$ from $E(O'_z)$, so (PW2) (or (PW1) if $O'_z$ is the identity) bounds the maximum by $C\min(1,r/2)^{-s_*}e^{-\delta_Pr/2}$; for $t(x)\ge1$ the plain bounds already have the stated form. At $A=0$ or at a constant $A$ the constant-background bounds and this telescoping give the three clauses directly, with no condition. \end{proof} \space% % \paragraph{The range.} The constants $C_{\mathrm{box}}$, $C_\flat$ and the range \eqref{eq:k02:range} of $\eta$, with its threshold $\eta_2\le\eta_1$, are those of \cref{r13:def:range}; all three conditions hold for $\eta$ small because $p(\eta)$ grows slower than every power of $\eta^{-1}$ and faster than every logarithm. Since $C_{\mathrm{box}}p$ bounds the side of every cluster box $\square'_K$ and also the side $\le5p+4M+2$ of the one box $\square'$ of a localisation with $R\le p$ (\cref{lem:k02:Zh}(e)), the third condition contains the convergence condition $Ce\eta^{1/2}p\sqrt3\,(5p+4M+2)\le\frac12$ of the expansion in $\tilde A$ on that one box (the operator bound of \cref{r13:def:range} with $\Omega_K$ replaced by $\Omega$), as well as the one on every cluster box. $J_*$ is taken so large that $\eta=5^{-(k-j)}\le\eta_2$ and $C_{\mathrm{PW}}e\eta^{1/2}p(\eta)\le\theta_{\mathrm{PW}}$, the condition of \cref{cpi:b7:lem:PW} for the pointwise bounds (PW1)--(PW3) on the background propagator and its first and second operations, whenever $k-j\ge J_*$; the least such integer is the largest of the thresholds of the two geometric conditions of \eqref{eq:k02:range}, of the convergence conditions of the expansions in the cluster fields and on the one box, and of the smallness of the bounds at the background. \begin{lemma}[the source vertices and the clusters]\label{lem:k02:Zh} Let $\eta\le\eta_2$, let $H$ be a member of $\mathcal N_h$, $s$ an $h$-vertex of $H$ with centre $z_s$, and $\tau$ a tuple of $H$. Then: \textup{(a)} $t(z_s)\ge\eta^{-1}-3$. \textup{(b)} If $\tau$ is in zone~B, then $R(\tau)\ge\eta^{-1}/2$. \textup{(c)} If $\tau$ is in zone~B, then $\tau$ has at least two clusters. \textup{(d)} Every cluster $K$ containing an $h$-vertex has $\dist(\square_K,\mathsf F)\ge2p+M+1$, $\Omega_K=\square'_K$ and $\dist(\square_K,\partial\Omega_K)\ge2p$: it is a zone-I cluster. Hence every zone-B cluster is source-free. \textup{(e)} Every $\square'_K$ has side $\le c_{\rm cl}N_*+4p+6M$, and the one box $\square'$ of a localisation with $R\le p$ has side $\le R+4p+4M+2\le5p+4M+2$; both are $\le C_{\mathrm{box}}\,p$, where $C_{\mathrm{box}}=\max(c_{\rm cl}N_*+6M+4,\,4M+7)$ is the constant of \cref{r13:def:range}. The same holds with a curvature vertex in place of the $h$-vertex. \end{lemma} \begin{proof} (a) $h(x)\ne0$ forces $\ell^\infty$-$\dist(x,\R^3\setminus\CJ)>1$ in physical units, that is $>\eta^{-1}$ unit lengths; the planes $\mathsf F$ lie within one fine spacing $5^{-j}\le1$ of $\partial\CJ$, and the vertex lies in $\supp\chi_{z_s}$, within $\frac23\sqrt3<2$ of its centre. (b) A tree of total length $R$ joins the points of $P(\tau)$, so all of them lie within $R$ of $z_s$; the vertices lie within $\frac23$ of these points in each coordinate, so $\square$ lies in the sup-norm cube of half-side $R+M+1$ about $z_s$, and $\square'$, which adds at most $2p+M$ on each face (as in (e)), in that of half-side $R+2p+2M+1\le R+2p+3M$. If $\square'$ meets $\mathsf F$, then $t(z_s)\le R+2p+3M$, and with (a) and \eqref{eq:k02:range} $R\ge\eta^{-1}-2p-3M-3\ge\eta^{-1}/2$. (c) With one cluster $\operatorname{diam}P(\tau)\le c_{\rm cl}(n_\tau-1)$ (the diameter bound of the cut by clusters), so $R(\tau)\le(n_\tau-1)\operatorname{diam}P(\tau)\le c_{\rm cl}(N_*-1)^2<\eta^{-1}/2$ by \eqref{eq:k02:range}, against (b). (d) If $s\in K$, every point of $P_K$ lies within $c_{\rm cl}(n_\tau-1)$ of $z_s$, so $\square_K$ lies in the cube of half-side $c_{\rm cl}n_\tau+M$ about $z_s$ and $\square'_K$ in that of half-side $c_{\rm cl}n_\tau+2p+3M$. By (a) and \eqref{eq:k02:range}, $t(z_s)-(c_{\rm cl}n_\tau+2p+3M)\ge\eta^{-1}-3-c_{\rm cl}n_\tau-2p-3M>2p+M+1$. So $\square'_K$ misses $\mathsf F$, it is not truncated, it contains every large block within $2p$ of $\square_K$, and $\dist(\square_K,\mathsf F)\ge t(z_s)-c_{\rm cl}n_\tau-M\ge2p+M+1$. (e) $\square_K$ has side $\le\operatorname{diam}P_K+\frac43+2M\le c_{\rm cl}N_*+2M$ (the vertices lie within $\frac23$ of the points in each coordinate, $\operatorname{diam}P_K\le c_{\rm cl}(N_*-1)$ by the diameter bound of the cut by clusters, and $c_{\rm cl}>\frac43$) and $\square'_K$ adds at most $2p+M\le2p+2M$ per side, as for the one box below; truncation at $\partial T_\eta$ only shrinks it. For the one box: the points of $P(\tau)$ lie within $R$ of one another, and every vertex lies in the support of its localisation weight, within $\frac23$ of its centre in each coordinate; so $\square$ has side $\le R+\frac43+2M\le R+2M+2$, and $\square'$ adds on each face at most the large blocks within $2p$ of $\square$, that is at most $2p+M$; hence its side is $\le R+4p+4M+2$. Since $p\ge1$, $c_{\rm cl}N_*+4p+6M\le(c_{\rm cl}N_*+6M+4)p$ and $5p+4M+2\le(4M+7)p$. The curvature source $f$ is supported at the same distance from $\R^3\setminus\CJ$, so the same arithmetic applies to its vertex. \end{proof} \begin{lemma}[cut lines vanish on the Dirichlet planes]\label{lem:k02:face} Let $\ell$ be a scalar, dressed or external cut line of the cut by clusters with its end $u$ in a cluster $K$, in either model, at the background $A^{(j)}$ or at zero background. Put $m(u')\defeq\min(1,t(u'))$ for a scalar or an external line and $m(u')\defeq m_\alpha(u';1)=\min(1,t(u'))^\alpha$ for a dressed line, $m_\alpha(x;\rho)\defeq\min(1,t(x)/\rho)^\alpha$ being the H\"older gain of a slice propagator at a live site $x$ at distance $t(x)$ from a face (half the exponent with a pair difference); in every case $m(u')\le m_\alpha(u';1)$. \textup{(a)} For every evaluation point $u'$ of the $u$-end with $t(u')\le\frac14$ and every operation $O_v$ at the far end (a difference or H\"older quotient of order $\le1+\alpha$, or the identity) whose evaluation set $E_v$ lies at distance $\ge\frac12$ from $u'$, \[ |O_v\ell(u',\cdot)|\ \le\ C\,m(u')\,e^{-\delta\dist(u',E_v)} . \] \textup{(b)} For a scalar or dressed cut line and evaluation points $u',v'$ of its two ends, both undifferentiated, with $|u'-v'|\ge1$ and $t(u'),t(v')\le\frac14$, \[ |\ell(u',v')|\ \le\ C\,m(u')\min(1,t(v'))\,e^{-\delta|u'-v'|}\ \le\ C\,m_\alpha(u';1)\,m_\alpha(v';1)\,e^{-\delta|u'-v'|} . \] The constants depend on those of \cref{cpi:lem:a02:XL}(b) and, for dressed lines, also on the H\"older constants of the slice propagators up to the faces, on the propagator bounds (PW1)--(PW3) of \cref{cpi:b7:lem:PW} and on the convolution bound for two kernels of the form of (PW1). \end{lemma} \begin{proof} Each of the three kinds of line vanishes when its $u$-end lies on $\mathsf F$: a scalar line by the Dirichlet condition of the cube propagator, a dressed line and an external line through their first factor, which is such a propagator. \emph{(a) for scalar and external lines.} Let $u'_{\mathsf F}\in\mathsf F$ be the foot of $u'$ and telescope along the $t(u')/5^{-j}$ bonds of the normal segment from $u'_{\mathsf F}$ to $u'$; the parallel transporters along a bond are unimodular, so $|O_v\ell(u',\cdot)|\le t(u')\max_b|O_vD_b\ell(\cdot,\cdot)|$, the maximum over the bond differences $D_b$ on that segment. Every such bond lies within $\frac14$ of $u'$, hence at distance $\ge\frac14$ from $E_v$, and the mixed bound of \cref{cpi:lem:a02:XL}(b) ($C^{1,\alpha}\otimes C^{1,\alpha}$ at unit scale in both arguments, decay $e^{-\delta_1|u-v|}$, no distance condition) bounds each by $Ce^{-\delta\dist(u',E_v)}$. \emph{(a) for dressed lines.} Write the line as $\sum_x\eps_j^3G_A(u',x)\,c_{\ge j}\,G_A(x,\cdot)$, with $\eps_j=5^{-j}$ the fine spacing (chains iterated likewise). Its first factor is not cut off near $u'$, so the face factor is taken from the slices of the first factor at all scales. Each slice $G_{(i)}$ obeys the face clause (the H\"older gain of a slice propagator at a live site) $|G_{(i)}(u',x)|\le C\rho_i^{-1}m_\alpha(u';\rho_i)e^{-\delta|u'-x|/\rho_i}$: its proof uses only the factorisation $G_{(i)}=a_i^2\rho_i^{-4}(G_iQ_i^*)C^{(i)}(Q_iG_i)$ \cite[(2.17)]{K1}, the H\"older clause of the outer factor $G_iQ_i^*$ up to the faces and its vanishing on $\mathsf F$, and the exponential decay of the coarse factor $C^{(i)}$. At the background $A^{(j)}$ the outer-factor clauses hold at every level $i$ by \cite[(5.2)--(5.3)]{K2} (\cref{r13:def:boxlines}), the background being regular at level $i$. The coarse factor $C^{(i)}(A)=K_i(A)^{-1}$ of \cref{cpi:def:a02:setting} is the lower right block of the inverse of the block operator with diagonal entries $M(A)$ and $D_i\defeq a_i+a5^{-2}Q^*Q$ and off-diagonal entries $-a_iQ_i^*$ and $-a_iQ_i$, which is positive, its quadratic form being $\langle g_1,M_0(A)g_1\rangle+a_i\norm{g_2-Q_ig_1}^2+a5^{-2}\norm{Qg_2}^2$. The Schur complement of $D_i$, with $QQ^*=I$, $QQ_i=Q_{i+1}$ and $a_{i+1}^{-1}=a^{-1}+a_i^{-1}5^{-2}$, gives $C^{(i)}=D_i^{-1}+a_i^2D_i^{-1}Q_iG'_{i+1}Q_i^*D_i^{-1}$ with $G'_{i+1}\defeq(M_0(A)+a_{i+1}5^{-2}Q_{i+1}^*Q_{i+1})^{-1}$, the propagator of level $i+1$ in the units of level $i$. Since $D_i^{-1}=a_i^{-1}(I-Q^*Q)+(a_i+a5^{-2})^{-1}Q^*Q$ is block-local and bounded, the decay of $C^{(i)}$ is the outer-factor bound at level $i+1$, applied to the block-local bounded sources $Q_i^*D_i^{-1}\delta_y$. Since $m_\alpha(u';\rho_i)\le\rho_i^{-\alpha}m_\alpha(u';1)$ for $\rho_i\le1$, the sum over the slices gives \[ |G_A(u',x)|\ \le\ C\,m_\alpha(u';1)\,(|u'-x|+\eps_j)^{2-d-\alpha}\,e^{-\delta|u'-x|/2}, \] a kernel that is locally summable in $d=3$. The sum over $x$ against the constant $c_{\ge j}$ and the last factor $O_vG_A(x,\cdot)$, which (PW1) of \cref{cpi:b7:lem:PW} bounds, is the convolution of two kernels of the form of (PW1), and the convolution of two such kernels, each locally summable in $d=3$ and exponentially decaying, is bounded by $C\,m_\alpha(u';1)\,e^{-\delta\dist(u',E_v)}$, since $\dist(u',E_v)\ge\frac12$. \emph{(b).} Put $f(w)\defeq\ell^\flat(u',w)=\bar\psi(u'-w)\ell(u',w)$; then $f(v')=\ell(u',v')$, since $|u'-v'|\ge1$. Let $v'_{\mathsf F}\in\mathsf F$ be the foot of $v'$; $f(v'_{\mathsf F})=0$ by the Dirichlet condition in the second variable (for a dressed line through its last factor). Since $|u'-v'|\ge1$ (on the supports even $|u'-v'|\ge R_{\rm cl}-4\cdot5^{-j}>1$, \cref{cpi:lem:a02:XL}(a)) and $t(v')\le\frac14$, every point of the segment from $v'_{\mathsf F}$ to $v'$ lies at distance $\ge\frac34$ from $u'$. Telescope along its $t(v')/5^{-j}$ bonds, with unimodular transporters: $|f(v')|\le t(v')\max_b|D_bf|$. On each bond, $D_bf$ is $\bar\psi(u'-\cdot)D_b\ell(u',\cdot)$ plus the difference of the fixed smooth cutoff $\bar\psi$, of size $\le\norm{\nabla\bar\psi}_\infty$, times $\ell(u',\cdot)$ at an end of the bond; $\bar\psi$ is differentiated once only. Every point involved lies at distance $\ge\frac34-5^{-j}\ge\frac12$ from $u'$ and within $\frac14+5^{-j}$ of $v'$, so (a), with $O_v\defeq D_b$ and with $O_v$ the identity, bounds both terms by $C\,m(u')e^{-\delta|u'-v'|}$. Hence $|\ell(u',v')|\le C\,m(u')\min(1,t(v'))e^{-\delta|u'-v'|}$, and the second inequality of (b) is $m(u')\le m_\alpha(u';1)$ and $\min(1,t)\le\min(1,t)^\alpha$. Where $t>\frac14$ at an end, the factor of that end is $\ge\frac14$: if $t(v')>\frac14$, (a) with $O_v$ the identity gives the bound; if $t(u')>\frac14$, the telescoping at $v'$ above, with $D_bf=D_b\ell^\flat(u',\cdot)$ bounded by \cref{cpi:lem:a02:XL}(b), gives it; if both, \cref{cpi:lem:a02:XL}(b) alone. \end{proof} % % % \begin{proposition}[the branch $R\le p$: zone~I]\label{prop:k02:KhI} Let $H$ be a member of $\mathcal N_h$ and $\tau$ a localisation of $H$ with $R\le p$; by \cref{cpi:lem:a02:Z}(i) such a localisation lies in zone~I. Then, for each model, on $\supp\chi_j$, \begin{equation}\label{eq:k02:I1} |E^{(j)}(H;\tau)|\ \le\ C(1+k-j)^{p_0}\,\eta^{\rho(H)}\,p^{n_1}\,e^{-\delta R}, \end{equation} which is \cite[(3.38)]{K1} for the renormalised family at the tuple, the factor $(1+k-j)^{p_0}$ being King's power of $\ln(L^k\eps)^{-1}$ \cite[p.~666]{K1}, and across the pair $(k,j)$, $(k+1,j+1)$, on the common unit lattice, \begin{equation}\label{eq:k02:I2} |E'^{(j+1)}(H;\tau)-E^{(j)}(H;\tau)|\ \le\ C\,\eta^{\rho(H)-1/2}\,p^{n_1}\big[5^{-\gamma j}+\eta^{3+\sigma+1/2}\big]\,e^{-\delta R}. \end{equation} The constants depend on the point of King's range, $J$, $\bar n$, $\norm h_{C^1}$ and, for members with curvature vertices, $\max\norm{f_{\mu\nu}}_{C^{2,\alpha}}$; not on $j$, $k$, $\tau$. \end{proposition} \begin{proof} A zone-I localisation with $R\le p$ is reduced to one King box: King's steps \cite[(3.43)--(3.55)]{K1} on the box $\square'$ of side $\le5p+4M+2$, as in Step~1 of the proof of \cref{prop:k02:KhR} for a single cluster (the constant shift $B=A_j(x_0)$ and $\tilde A=A_j-B$; the gauge transformation \cite[(3.46)]{K1}, which puts the phases at the scalar ends of the external objects and acts trivially on the $h$-vertex, whose two scalar ends carry cancelling phases; the Taylor expansion of the propagator in $\tilde A$ to a fixed order with the explicit remainder \cite[(3.48)--(3.49)]{K1}, convergent by the third condition of \eqref{eq:k02:range}; the resolvent identity \cite[(3.52)]{K1} and the splitting \cite[(3.54)]{K1}, which produce the lines $\delta G_j(\square',A)$). The result is a finite sum over graphs $\tilde H$ with at most $C(\bar n)$ new vertices carrying the common block field $\tilde A$, the same graphs with the same fields for both members of the pair. Each $\tilde H$ equals its free $f_D$-flow version plus terms carrying an image line of any prescribed degree, of size $\le Ce^{-\delta_0p/2}\le C\eta^{3+\sigma+1/2}$ times its bound (the exponent of the logarithm in $p(\eta)$ being $6\ge2$), and the $\delta G_j(\square',A)$ lines are $O(e^{-\delta_0p})$ by \cite[(5.3)]{K2}, since $\dist(\square,\partial\square')\ge2p$. The free version obeys \eqref{eq:k02:I1} by \cref{cpi:thm:a02:S3}(a) with the root held at the $h$-vertex, which is \cite[(3.38)]{K1} with the depth factor $(1+k-j)^{p_0}$: every prefix component has positive degree, the stages containing the $h$-vertex by the margin \eqref{eq:k02:margin}, the others by the degree count of \cref{cpi:thm:a02:T}. For \eqref{eq:k02:I2} the two members consist of the same elements and are compared element by element as in \cite[pp.~663--665]{K1}: a line of the finer member of index $-1$ gives $5^{-\gamma j}$ through the first ultraviolet index sum ($\gamma<\alpha_0$); otherwise the difference is a telescoping sum over the factors of the graph, one factor replaced at a time, the external and internal lines by \cite[(3.71), (3.73)--(3.75)]{K1} for the free $f_D$ flow, the block-product point sums by \cref{cpi:thm:a02:S3}(b), the local coefficients of the rewritten units at the rate $\gamma_{\mathrm c}$, the $h$-vertex by sampling, $|h(x')-h(x)|\le\sqrt3\,\eps_k\norm{\nabla h}_\infty\le C5^{-\gamma j}\norm h_{C^1}$, the only factor that depends on $h$, and the phases by \cite[(3.72)]{K1}, which costs in addition the factor $\eta^{-1/2}$ of the field bound \cite[(3.2)]{K1}: this is the power $\rho(H)-\frac12$ of \cite[(3.56)--(3.57)]{K1}; the remainders, the $\delta G_j$ lines and the image terms are bounded separately in each model and give the term $\eta^{3+\sigma+1/2}$. (A zone-I localisation with $R>p$ has at least two clusters; the localisations with $R>p$ are the subject of Proposition~\ref{prop:k02:KhR}, the cut by clusters.) \end{proof} % \begin{lemma}[the zone-B bound with a cut line as the external field]\label{cpi:b9:Bcut} Let $K$ be a zone-B cluster of a tuple $\tau$ with at least two clusters. Then $K$ is source-free, every connected component of every cluster graph $H_K$ met in the nesting carries a cut-line leg, and the renormalised family of $H_K$, at $B_K=0$ and $A'=\tilde A_K$ on $\Omega_K$, rewritten by Definition~\ref{cpi:def:a02:RC} with the cut lines as external fields of index $j$ and with the factor $m_\beta(\cdot;1)$ of every cut-line end charged by the charging rule moved to the remainder vertex before any bound is taken, obeys Theorem~\ref{cpi:thm:a02:Bsc}(a) with no vertex held: for either model, on $\supp\chi_j$, \[\sum_{\text{positions of }K}\mathfrak E_K\ \le\ C\,\eta^{\rho_K}\,p^{n_K}\,(1+k-j)^{p_{0,K}} .\] \end{lemma} \begin{proof} $K$ is source-free because the source and curvature vertices lie in zone-I clusters, and every component of $H_K$ carries a cut leg by the degree census of the cut. The zone-B bound, Theorem~\ref{cpi:thm:a02:Bsc}, is stated for a renormalised family of King's expansion at zero background with at least one external scalar field; here the family is that of $H_K$ on the box $\Omega_K$ of side $O(p)$, its internal lines the zero-background Dirichlet lines, and the external field of every component is a cut leg. It remains to check that the proof of Theorem~\ref{cpi:thm:a02:Bsc} uses an external scalar leg only in ways a cut line satisfies, and the argument has one non-trivial point, the \emph{order of operations}: the rewriting acts on the cut-line legs as functions, entering through their sup and lattice $(1,\alpha)$-norms on $\Omega_K$ up to the face bonds in the gauge of $\tilde A_K$ (a covariant difference of a cut line is the phase times a difference covariant in $A^{(j)}$ plus a multiplier bounded by $e_j|\tilde A_K|\le C$); where the charging rule charges a cut-line leg at an end $q$, the vanishing factor $m_\beta(q;1)$, $\beta<\alpha/2$, demanded at a remainder vertex of degree $-1$ is provided by the face factor $m(q)\le m_\alpha(q;1)$ of the cut line with its decay $e^{-\delta\dist}$, and it is moved to the remainder vertex before any bound is taken, the leftover $\ell/m_\beta(q;1)\le Cm_{\alpha-\beta}(q;1)e^{-\delta\dist}\le Ce^{-\delta\dist}$ keeping the standard bound of the leg. A cut line joining two zone-B clusters may be charged at both ends, by the rules of the two clusters, and then both face factors are provided; equivalently the rule is run once on the multigraph whose nodes are the degree $-1$ remainder vertices of all zone-B clusters and whose edges are the member and cut lines joining two of them: every node has degree at most $2$, every component is a path or a cycle, and orienting each component and charging every node to its left leg end charges every line at most once and no cut line at both ends. A cycle of degree $-1$ remainder vertices of $H_K$ would use both scalar legs of each of its nodes and so be a whole component of $H_K$ without a cut leg, which does not occur; hence no vacuum component is dropped. In the degree count a cut leg, scalar or photon, is an undifferentiated external leg of index $j$ with no credit, so every source-free stage has an external field and the vacuum exception is absent; a photon cut line is an even image sum of free lines and has the $C^{1,\alpha}$ regularity up to the Neumann faces that the photon two-point sites need; the infrared lines lie inside free counterterm units within one cluster or on dressed cut lines, the latter costing $(1+k-j)^5$ per unlocalised vertex. No step uses a source vertex, a comparison across the pair or a localisation of diameter larger than $C(\bar n)$. The remaining estimates are those of the proof of Theorem~\ref{cpi:thm:a02:Bsc}: the bulk-type graphs by the degree bound with the Ward rows run inside $H_K$, the remainder graphs by the ordered induction with no vertex held, the last vertex summed over $\Omega_K$ with $|\Omega_K|\le Cp^3$, and one factor $C(1+k-j)$ per infrared sum. \end{proof} \space% % \begin{proposition}[the branch $R>p$: the cut by clusters]\label{prop:k02:KhR} Let $\eta\le\eta_2$, let $H$ be a member of $\mathcal N_h$ and $\tau$ a tuple of $H$ with at least two clusters. By \cref{lem:k02:Zh}(c) this covers every zone-B tuple, and by the diameter bound of the cut by clusters and \eqref{eq:k02:range} every tuple with $R>p$. Then, for each model separately, on $\supp\chi_j$, \begin{equation}\label{eq:k02:R1} |E^{(j)}_N(H;\tau)|\ \le\ C(1+k-j)^{p_0}\,\eta^{\rho(H)}\,p^{n_1}\,e^{-\delta\,\dist(P(\tau))}, \end{equation} $E_N$ the field-dependent part (the whole value for a field-independent member); this is \cite[(3.38)]{K1} at the tuple, with the same logarithm. Moreover, for every unit cube $y_*$ and $0<\delta'\le\delta/2$, \begin{equation}\label{eq:k02:R2} \sum_{\substack{\tau\ \text{in zone B}\\ \text{with an $h$-vertex in }y_*}}|E^{(j)}_N(H;\tau)|\,e^{\delta'R(\tau)}\ \le\ C(1+k-j)^{p_0}\,\eta^{\rho(H)}\,p^{n_1}\,e^{-\delta/(8\eta)} . \end{equation} The constants depend on the point of King's range, $J$, $\bar n$, $\norm h_\infty$ and, for members with curvature vertices, $\max\norm{f_{\mu\nu}}_{C^{2,\alpha}}$. \end{proposition} \begin{proof} We sketch the argument first. At a tuple with at least two clusters every connected component of every cluster graph carries a cut-line leg, so the vacuum members and the component-wise cancellation of \cite[p.~667]{K1} do not arise, and $E(H;A^{(j)};\tau)$ and $E(H;0;\tau)$ are bounded separately; at zero background the cut lines are the cube propagators at $A=0$ and $B_K=0$. Fix one of the two terms. \emph{Step 0: regularised cut lines.} Replace every cut line $\ell$ by $\ell^\flat$; by \cref{cpi:lem:a02:XL}(a) this changes nothing. \emph{Step 1: reduction per cluster.} Each $H_K$ is reduced by \cite[(3.43)--(3.55)]{K1} with its own box $\Omega_K$, centre $x_0^K$, gauge $B_K$ and field $\tilde A_K=A^{(j)}-B_K$ on $\Omega_K$ (the cut by clusters of \cref{r13:def:clusters}), its cut lines keeping the background and receiving the unitary factors \cite[(3.46)]{K1} at their ends. The expansion of the reduced propagator in $\tilde A_K$ converges on $\Omega_K$: $e_j\sup_{\Omega_K}|\tilde A_K|\le Ce_jp\operatorname{diam}\Omega_K\le C_\flat e\eta^{1/2}p^2\le\frac12$ by the difference bound \cite[(3.45)]{K1}, \cref{lem:k02:Zh}(e) and \eqref{eq:k02:range}: this is the reduction of the branch $R\le p$ on a box of side $O(p)$, run inside each cluster. By construction of the cut by clusters the partition, the boxes, the gauges and the set of cut lines are the same for every member of every family met in the nesting and for every position, so renormalisation families are formed inside each $H_K$. A zone-I cluster, in particular every cluster containing a source vertex (\cref{lem:k02:Zh}(d)), satisfies $\dist(\square_K,\partial\Omega_K)\ge2p$ and $\Omega_K\cap\mathsf F=\emptyset$, so $H_K$ is its free $f_D$-flow version plus terms carrying an image line of any prescribed degree (replacing the cube slices of a zone-I graph by the free $f_D$ slices leaves image-line terms of size $e^{-4\delta_0p/\rho_i}$, which is $O(\eta^N)$ for every $N$ because $p(\eta)$ is a power $\ge2$ of $\ln\eta^{-1}$), with the cut lines as external legs; the vertices of the expansion in $\tilde A_K$ that lie in $\square'_K\setminus\square_K$ are joined to a localised vertex of $K$ by a path of lines one of which is long, and that line provides a credit of any order, as in King's Ward-row reduction \cite[(3.39)--(3.55)]{K1} run inside $H_K$ with the cut lines as external legs, where each Ward-row component is zero, a local polynomial of degree $\le-D(S)$ with coefficients $\le C\rho^{D(S)+|a|}$, or a remainder with a line credit $\rho^\theta$ against a leg debit $\le\theta$, keeping $D'(S)>0$. A zone-B cluster is source-free and is rewritten by \cref{cpi:def:a02:RC} within $\Omega_K$. \emph{Step 2: the cut-line factors.} Every evaluation point of a cut-line end in an output term is joined to a localised anchor of its own cluster by at most $C(\bar n)$ lines of that cluster's rewritten units (the Ward-row reduction inside the cluster), and by \cref{cpi:lem:a02:XL}(b) the phase-dressed cut line satisfies $|O_uO_v\ell^{\flat,\mathrm{ph}}|\le Ce^{-\delta_2d(K,K')}\times$(transfer factors on the lines of those paths), for all operations $O_u,O_v$ of order $\le1+\alpha$ at its ends, and $\le C$ crudely. The quotient graph (clusters as nodes, cut lines as edges) is connected because $H$ is; bound the cut lines of one spanning tree $T$ of it by the first bound and the others by $C$. \emph{Step 3: factorisation.} The output term is bounded by $\prod_K\mathfrak E_K\times\prod_{\ell\in T}Ce^{-\delta_2d(K,K')}\times C^{\#\text{cut lines}}$, where $\mathfrak E_K$ depends on the positions of $K$ only; the transfer factors cost one extra decay draw per line of a cluster, within the decay budget of $\bar n+4$ draws per line of the Ward-row reduction. \emph{Step 4: the cluster factors.} \emph{Zone-I clusters.} The cluster graph is a free $f_D$-flow graph with its cut lines as undifferentiated external legs of unit scale (\cref{cpi:lem:a02:XL}(b)). For the cluster containing the $h$-vertex, the census is that of \cref{prop:k02:KhI}; since the cluster carries a cut leg, every component has an external leg and the excepted field-independent family does not occur. For the source-free zone-I clusters, the degree count holds at every tuple by \cref{cpi:thm:a02:T} and the Ward-row reduction inside the cluster, and block points by the split count of \cref{cpi:thm:a02:S3} (a block point with $s$ scalar and $p_\gamma$ photon fields weighs $3-s/2+p_\gamma/2\ge5/2$, which makes every block-point component convergent). The multiscale bound \cref{cpi:thm:a02:S3}(a) with the root held, summed over the root in $\Omega_K$ (volume $\le Cp^3$ by the cluster geometry), gives $\sum_{\text{positions of }K}\mathfrak E_K\le C\eta^{\rho_K}p^{n_K}(1+k-j)^{p_{0,K}}$, with $\rho_K$ the coupling power of the vertices of $K$. \emph{Zone-B clusters.} The cluster is source-free (\cref{lem:k02:Zh}(d)); apply \cref{cpi:thm:a02:Bsc}(a) and \cref{cpi:thm:a02:BVP} with no vertex held to the renormalised family of $H_K$, localised at $P_K$ with the box $\Omega_K$ of side $O(p)$; the hypothesis ``at least one external field'' is the cut leg. That the proof of \cref{cpi:thm:a02:Bsc} uses an external scalar leg only in ways a cut line satisfies, the borrowed factor at a $D'=-1$ remainder vertex included, is Lemma~\ref{cpi:b9:Bcut}; after the borrowing, the cut-line factor left in the output term is an undifferentiated evaluation bounded by $Ce^{-\delta\dist}$ (\cref{lem:k02:face}), which Steps~2--3 treat like every other cut-line factor. The resulting bound is again $\le C\eta^{\rho_K}p^{n_K}(1+k-j)^{p_{0,K}}$, the logarithm coming from the infrared sums inside $\Omega_K$. \emph{Step 5: the tree.} By the tree bound $\dist(P(\tau))\le\sum_{\ell\in T}|\ell|+c_{\rm cl}n_\tau$ of the cut by clusters (\cref{r13:def:clusters}), $\prod_{\ell\in T}e^{-\delta_2d(K,K')}\le e^{\delta_2c_{\rm cl}n_\tau}e^{-\delta_2\dist(P(\tau))}$. \emph{Step 6: exponents.} $\rho(H)=\sum_K\rho_K$, the count being vertex-local (an unlocalised mass-counterterm vertex on a cut dressed line keeps its own coupling power); $n_1\defeq\sum_Kn_K$ and $p_0\defeq\sum_Kp_{0,K}$ plus the dressed-line factor of \cref{cpi:lem:a02:XL}(b); the number of clusters is $\le N_*$ and $\delta\defeq\delta_2$. This is \eqref{eq:k02:R1}. \emph{The root sum.} On the tuples of \eqref{eq:k02:R2}, $R(\tau)\ge\eta^{-1}/2$ by \cref{lem:k02:Zh}(b). The number of tuples of $H$ with a point in $y_*$ and $R\in[n,n+1)$ is $\le C(n+1)^{3N_*}$, and $e^{\delta'R}e^{-\delta R}\le e^{-\delta R/2}$. So the sum is $\le C(1+k-j)^{p_0}\eta^{\rho(H)}p^{n_1}\sum_{n\ge\eta^{-1}/2-1}(n+1)^{3N_*}e^{-\delta n/2}\le C'(1+k-j)^{p_0}\eta^{\rho(H)}p^{n_1}e^{-\delta/(8\eta)}$, since $(n+1)^{3N_*}e^{-\delta n/4}\le C$ and $\eta^{-1}\ge8$. Members with curvature vertices: the curvature vertex lies in a zone-I cluster (\cref{lem:k02:Zh}(d)), is invariant under the reduction, being gauge invariant, and is bounded through a discrete summation by parts against an explicit weight $w$ with $d^*w=0$. \end{proof} \begin{proof}[Proof of \cref{hyp:k02:Kh}] For a member $H$ of $\mathcal N_h$ and $J_*\le j\le k-J_*$ (so that $\eta\le\eta_2$), the proof runs as follows, using the two propositions above (\cref{prop:k02:KhI,prop:k02:KhR}). \emph{First clause.} \eqref{eq:k02:I1} is \cite[(3.38)]{K1} with the factor $(1+k-j)^{p_0}$ for the localisations in zone~I, which are those at distance from the faces larger than the small-field scale $2p(\eta)$ of the localisation boxes (\cref{cpi:def:a02:zones}). \emph{Second clause.} A tuple with one cluster has $R\le c_{\rm cl}(N_*-1)^2\le p$, lies in zone~I by \cref{cpi:lem:a02:Z}(i), and obeys \eqref{eq:k02:I1}; a tuple with at least two clusters obeys \eqref{eq:k02:R1}. Summed over the localisations sharing the cube $y_*$ of an $h$-vertex: the zone-I tuples by $\sum_n(n+1)^{3N_*}e^{-(\delta-\delta')n}\le C$, the zone-B tuples by \eqref{eq:k02:R2}. This is \cite[(3.38)]{K1} with the factor $(1+k-j)^{p_0}$. \emph{Third clause.} Here $J_*\le j\le k-J_*$ and $\eta=5^{j-k}$. Split the localisations at $R=p$. Those with $R\le p$ are in zone~I and are reduced on one box of side $\le5p+4M+2$; there \eqref{eq:k02:I2} is the comparison of \cite[pp.~663--665]{K1} made element by element, in which the only element depending on $h$ is the coefficient of the $h$-vertex, replaced by sampling at the cost $C\eps_k\norm{\nabla h}_\infty$, the points of the block products being never source vertices. Sum \eqref{eq:k02:I2} over the localisations, the position sum costing $\eta^{-3}|\CJ|$ in the branch $R\le p$: the result is $\le C\eta^{\rho(H)-1/2}p^{n_1}(1+k-j)^{p_0}\eta^{-3}|\CJ|[5^{-\gamma j}+\eta^{3+\sigma+1/2}]$, whose first term is $\le C5^{-\gamma'j}\eta^{-\beta'}|\CJ|$ for every $\gamma'\le\gamma$ and $\beta'>\frac72-\rho(H)$, and whose second term is $\le C\eta^{\rho(H)+\sigma}p^{n_1}(1+k-j)^{p_0}|\CJ|\le C\eta^{\sigma'}|\CJ|$ for every $\sigma'<\sigma$, since $p^{n_1}(1+k-j)^{p_0}$ is bounded by a constant multiple of a power of $1+\ln\eta^{-1}$, hence $\le C_\epsilon\eta^{-\epsilon}$ for every $\epsilon>0$. Those with $R>p$ have at least two clusters; bound the two members of the pair separately by \eqref{eq:k02:R1}, with the roots at the $h$-vertex, $\le C\eta^{-3}|\CJ|$ of them. For the zone-I tuples among them the factor $e^{-(\delta-\delta')p/2}$ left by the position sum in the branch $R>p$ is $\le C_N\eta^N$ for every $N$, in particular $\le C\eta^{3+\sigma}$, since the exponent of the logarithm in $p(\eta)$ is $6\ge2$, and the sum is $\le C\eta^{-3}|\CJ|\eta^{\rho(H)}p^{n_1}(1+k-j)^{p_0}\eta^{3+\sigma}\le C\eta^{\sigma'}|\CJ|$; for the zone-B tuples \eqref{eq:k02:R2} gives $\le C\eta^{-3}|\CJ|\eta^{\rho(H)}p^{n_1}(1+k-j)^{p_0}e^{-\delta/(8\eta)}\le C_N\eta^N|\CJ|$ for every $N$. In both cases $\rho(H)\ge0$, and the external fields on $\supp\chi_j$ are accounted for in $\rho(H)$ by \eqref{eq:k02:margin}. Adding the three parts gives the third clause, with $C$ independent of $j$ and $k$. The constants depend on the point of King's range, so the lemma holds at every point of the range, in particular at the second mass $M_2$ of \cref{ssec:k02:setting}. \end{proof} \space% % \space% % \subsection{The estimates with scalar sources}\label{ssec:k02:inputs} For a source $s=(u,h)$ put \[ \mathsf S(s)\defeq\sum_i|x_i|\max_{\mu\nu}\norm{f_{i,\mu\nu}}_{C^{2,\alpha}}+\norm h_\infty, \] a size; a compact family of sources is contained in $\{\mathsf S(s)\le S_0\}$. \begin{proposition}[\Uplus: the small-field lines at a real scalar source]\label{prop:k02:Uplus} Fix $J\ge J_0$, $\alpha\in(0,\frac12)$ and $S_0>0$. There are $k_0$ and $C,\sigma>0$, depending on the couplings, $J$, $\alpha$, $S_0$ and on nothing else, such that for every real source $s=(u,h)$ with $\mathsf S(s)\le S_0$, every $k\ge k_0$ and both members $(k',j')\in\{(k,j(k)),(k+1,j(k)+1)\}$ of the diagonal pair, \begin{equation}\label{eq:k02:SF} Z_{k'}(s)\ \ge\ e^{-C\eta_k^{\sigma}}\,I^{(j')}_{k'}(s)\qquad\text{and}\qquad \int P\,d\mu_{k',s}\ \le\ e^{C\eta_k^{\sigma}}\,I^{(j')}_{k'}(s). \end{equation} \end{proposition} The proof runs Ba{\l}aban's inductive small-field lower bound and the no-large-field term of his upper bound on King's cube with the source inserted through King's single vertex \cite[(3.21)]{K1}, whose coupling carries two powers of the running scale where the other vertices carry at most one and a half; every step of the argument is either blind to the source or a bound on the interaction families which that margin leaves unchanged, and the remaining bookkeeping is routine. These are the two small-field lines of \eqref{eq:k01:H0rel} with the scalar source inserted into the small-field analysis through the vertex \cite[(3.21)]{K1}; the large-field remainder at $s\ne0$ is not part of the statement, and is provided instead by \cref{lem:k02:HLF}. The source enters the small-field steps only through its vertex, and the sign of $h$ plays no part in them; the statement is applied below at non-negative scalar sources, where the large-field remainder is provided by \cref{lem:k02:HLF}. \begin{proposition}[\Bh: one-step bound with the scheme $c$-number]\label{prop:k02:Bh} Fix $J\ge J_0$ and a compact family $\Sigma$ of real sources. For $\sigma\in\Sigma\cup\{0\}$ and $h_\sigma$ its scalar part, there is a sequence $\theta_k(\sigma)\le\theta_k$, with $\theta_k$ independent of $\sigma\in\Sigma$ and decaying geometrically, such that for $k\ge k_0$, on the support of $\chi_{j(k)}$, \begin{equation}\label{eq:k02:B} \big|S^{(j(k))}_{N,k}(\cdot;\sigma)+c_k(h_\sigma)-S^{(j(k)+1)}_{N,k+1}(\cdot;\sigma)-c_{k+1}(h_\sigma)\big|\ \le\ \theta_k(\sigma). \end{equation} \end{proposition} The proof is given in Section~\ref{cpi:ssec:Bh}. The field-dependent and the $h$-nonlinear parts are \cref{prop:k01:Kpos}, the graphs with curvature vertices at $s=(u,h)$, and \Kh. The field-independent $h$-linear part of the three lowest orders is a $c$-number $\bar\beta^{(j)}_k\int h$ up to a localisation error, and $c_k(h)$ is exactly the $c$-number that makes $\bar\beta^{(j)}_k+z_k$ converge along the diagonal pairs at the rate $5^{-\gamma'j}\eta^{-\beta'}+\eta^{\sigma'}$; this is the sense in which $z$ is the scheme adapted to King's flow. At $\sigma=0$, \eqref{eq:k02:B} is \eqref{eq:k01:Kpos} with $j=j(k)$. \begin{lemma}[$c$-numbers]\label{lem:k02:cnum} \textup{(a)} $|z_k|\le C(1+k)$. \textup{(b)} For every $M>0$, $\Delta_k(M)\defeq2(\sW_{\eps_k}(m)-\sW_{\eps_k}(M))+z_k(m)-z_k(M)$ is bounded in $k$, where $\sW_\eps(M)=(-\Delta_{\eps\Z^3}+M^2)^{-1}(0,0)$ and $z_k(M)$ is the reference scheme at mass $M$. \end{lemma} Both are statements about lattice Feynman integrals, obtained by direct computation: (a) is the logarithmic growth of the order-$e^2$ coefficient of $\langle|\phi|^2\rangle$ with the counterterm inserted; in (b) the Wick difference is a convergent lattice integral in three dimensions, and the scheme difference converges. \begin{proposition}[\NEGinf: the bare coefficient grows with the mass]\label{prop:k02:NEG} For every point of King's range ($e,\lambda>0$, $m,\lmu>0$) and every $M>m$, $u_\infty(M)\defeq\lim_ku_k(M)$ exists and \[ 2u_\infty(M)\ \ge\ (M-m)\Big(M+m+\frac{4\lambda}{\pi}-\frac{e^2}{4\pi}\Big)-\Big(\frac{e^4}{4\pi^2}+\frac{8\lambda^2}{\pi^2}\Big)\ln\frac Mm ; \] in particular $s_-(M)=u_\infty(M)\to\infty$ as $M\to\infty$, and $s_-(M_2)\ge\frac12$ for $M_2\defeq\max\{m,\ \frac{e^2}{4\pi}+(\frac{e^4}{4\pi^2}+\frac{8\lambda^2}{\pi^2})/m\}+1$. \end{proposition} Only $\sup_Ms_-(M)=\infty$ is used below; the proof rests on the exact continuum limit of King's free-boundary counterterm at each of its five orders (\cref{lem:s14:CT}). % % % % \begin{proof}[Proof of \cref{prop:k02:NEG}] The coefficient $\alpha_k(M)=M^2+\dm_{\eps_k}(M)$ of \cref{ssec:k02:setting} is King's free-boundary bare coefficient of \cref{conv:v065d_s02_setting:1}, the $\alpha^\infty_k(M)$ of \cref{lem:s14:CT}; so $u_k(M)=\frac12[\alpha^\infty_k(M)-\alpha^\infty_k(m)]$, and by \cref{lem:s14:CT}, whose proof is a computation of lattice Feynman integrals that uses nothing of Sections~\ref{sec:k01}--\ref{sec:k02}, the limit $u_\infty(M)=\frac12[\alpha_\infty(M)-\alpha_\infty(m)]$ exists for every $M>0$ and, by \eqref{eq:s14:CTclosed}, \begin{equation}\label{cpi:fix:NEG:closed} 2u_\infty(M)=(M^2-m^2)+\lambda D_{01}+e^2D_{20}+e^4D_{40}+\lambda e^2D_{21}+\lambda^2D_{02}, \end{equation} where, by \eqref{eq:s14:CTF1}--\eqref{eq:s14:CTF40} with $\mu\defeq\lmu$ as there, the five order-by-order limits are \begin{gather*} D_{01}=\frac{4}{\pi}(M-m),\qquad D_{02}=-\frac{8}{\pi^2}\ln\frac Mm,\\ D_{20}=-\frac{(M-m)(Mm+M\mu+m\mu)}{4\pi(M+\mu)(m+\mu)}=-\frac{M-m}{4\pi}\Big(1-\frac{\mu^2}{(M+\mu)(m+\mu)}\Big),\\ D_{21}=\frac{2}{\pi^2}\ln\frac{2M+\mu}{2m+\mu}+\frac{\mu}{2\pi^2}\Big(\frac1{M+\mu}-\frac1{m+\mu}\Big),\\ D_{40}=-\frac{1}{4\pi^2}\ln\frac Mm+\Phi_{40}\Big(\frac\mu M\Big)-\Phi_{40}\Big(\frac\mu m\Big). \end{gather*} Term by term, for $M\ge m$: $\lambda D_{01}=\frac{4\lambda}\pi(M-m)$; $e^2D_{20}\ge-\frac{e^2}{4\pi}(M-m)$ by the second form of $D_{20}$; $\lambda e^2D_{21}\ge0$, since the $M^2$-derivative $y^\infty_{21}$ of $F_{21}$ is positive, \eqref{eq:s14:CTy}; $\lambda^2D_{02}=-\frac{8\lambda^2}{\pi^2}\ln\frac Mm$; and $e^4D_{40}\ge-\frac{e^4}{4\pi^2}\ln\frac Mm$ by \cref{cpi:fix:PhiMono} below. Adding, with $M^2-m^2=(M-m)(M+m)$, gives the displayed inequality. With $\ln\frac Mm\le\frac{M-m}m$ it gives $2u_\infty(M)\ge(M-m)\big[M+m+\frac{4\lambda}\pi-\frac{e^2}{4\pi}-\big(\frac{e^4}{4\pi^2}+\frac{8\lambda^2}{\pi^2}\big)/m\big]\ge(M-m)(M-M_2+1)$ for $M\ge M_2$, which is $\ge M_2-m\ge1$ at $M=M_2$ and tends to infinity with $M$; and $s_-(M)=u_\infty(M)$ since the limit exists. \end{proof} \begin{lemma}[monotonicity of $\Phi_{40}$]\label{cpi:fix:PhiMono} $\Phi_{40}'(x)<0$ for every $x\ge0$. Consequently, for $M\ge m$, $D_{40}\ge-\frac1{4\pi^2}\ln\frac Mm$; and $y^\infty_{40}>-\frac1{8\pi^2M^2}$ for all $M,\mu>0$. \end{lemma} \begin{proof} Differentiating the closed form \eqref{eq:s14:CTF40} gives, for $x\ne1$, \[ \Phi_{40}'(x)=\frac{N(x)}{32\pi^2(1-x^2)^3(1+x)^2(2+x)(1+2x)},\qquad N(x)=(x^2-1)Q(x)+P(x)\ln\frac{1+2x}{2+x}, \] with $Q(x)=16x^7+56x^6+24x^5-100x^4-83x^3+82x^2+109x+40$ and $P(x)=8x(x-2)(x+1)^2(x+2)^2(2x+1)$; the denominator has the sign of $1-x$, so the claim is $\operatorname{sign}N(x)=\operatorname{sign}(x-1)$. The substitution $x=\frac{1+\xi}{1-\xi}$, $\xi\in[-1,1)$, gives $\ln\frac{1+2x}{2+x}=2\operatorname{artanh}\frac\xi3$ and $\tilde N(\xi)\defeq N(x(\xi))(1-\xi)^9=A(\xi)+2B(\xi)\operatorname{artanh}\frac\xi3$ with the polynomials \begin{gather*} A(\xi)=576\xi-2496\xi^2+13264\xi^3-3056\xi^4+224\xi^5-544\xi^6+272\xi^7-48\xi^8,\\ B(\xi)=32(\xi-3)^2(\xi-1)^2(\xi+1)(\xi+3)(3\xi-1), \end{gather*} and $\operatorname{sign}(x-1)=\operatorname{sign}\xi$, so the claim is $\tilde N(\xi)/\xi^3>0$ on $[-1,1]$. Expanding $2\operatorname{artanh}\frac\xi3=\sum_{j\ge0}\frac{2\xi^{2j+1}}{3^{2j+1}(2j+1)}$, the coefficients of $\xi^0$, $\xi^1$, $\xi^2$ of $\tilde N$ vanish exactly, and $\tilde N(\xi)/\xi^3=\sum_{k\ge0}c_k\xi^k$ on $|\xi|\le1$ with the exact rational coefficients $c_0=\frac{34160}3>11386$, $c_1=-\frac{46448}9$, $c_2=\frac{386528}{135}$, $c_3=-\frac{397664}{405}$, $c_4=\frac{256048}{2835}$, $c_5=-\frac{20144}{8505}$, so that $\sum_{k=1}^5|c_k|<9101$, while for $k\ge6$ only the series of the logarithm contributes ($A$ has degree $8$) and $|c_k|\le\frac{2\cdot1100736}{(k-4)\,3^{k+3}}$, so that $\sum_{k\ge6}|c_k|<84$. Hence $\tilde N(\xi)/\xi^3\ge c_0-\sum_{k\ge1}|c_k|>2201>0$ on $[-1,1]$, and for $\xi\ne0$ \[ \Phi_{40}'(x(\xi))=-\frac{\tilde N(\xi)/\xi^3}{2048\pi^2(1-\xi)^3(1+x)^2(2+x)(1+2x)}<0, \] with $\Phi_{40}'(1)=-\frac{c_0}{2048\pi^2\cdot36}<0$ at $\xi=0$ by continuity ($\Phi_{40}$ is analytic across $x=1$, \cref{lem:s14:CT}). For $M\ge m$, $\mu/M\le\mu/m$, so $\Phi_{40}(\mu/M)\ge\Phi_{40}(\mu/m)$, which is $D_{40}\ge-\frac1{4\pi^2}\ln\frac Mm$; and $y^\infty_{40}=-\frac1{8\pi^2M^2}-\frac{\mu}{2M^3}\Phi_{40}'(\frac{\mu}M)>-\frac1{8\pi^2M^2}$ by \eqref{eq:s14:CTy}. \end{proof} \space% % \begin{proposition}[{\Wsh$[s_W]$: the positive-source bound}]\label{prop:k02:Wsharp} For couplings in the coupling region of \cref{hyp:s03:T3} (the sign clause \Tthree), where \Itwo\ holds (\cref{prop:s10:U}), and with \Ithree\ (\cref{prop:s10:RP}): with $s_W\defeq s_1$ (the amplitude of \cref{prop:s11:W1prime}) and $\kappa=C_U/s_1$: for every $k$, $J$ and every lattice function $0\le q\le s_W$ on $\XJ$, \[ \ln n_k(0,q)\ \le\ \kappa\,\norm q_{1,k}. \] In particular $\sup_kn_k(0,q)<\infty$ for every $0\le q\in C^\infty_c$ with $\sup q\le s_W$ and $\supp q$ at distance $>1$ from $\R^3\setminus\CJ$. \end{proposition} \begin{proof} $e^{\sum_x\eps^3q(x)r_x^2}$ is a class-$\cI$ observable, so, since killing the matter on $\TJ$ does not increase a class-$\cI$ observable (Step~1 of the proof of \cref{lem:s04:G}; \cref{rem:s04:F}), $n_k(0,q)\le N^z_{k,\CJ}(0,q)$, the comparison-cube functional, which \cref{prop:s11:W1prime} at $X=\XJ$ bounds by $e^{\kappa\norm q_{1,k}}$; the scheme multiplier is common (\cref{lem:s02:alg}(a)). For $q\in C^\infty_c$, $\norm q_{1,k}\le C_0'\wnorm q$ by \cref{lem:s02:E}(E3). \end{proof} Any other threshold with the same property serves below in place of $s_1$; a second instance can be obtained from Ba{\l}aban's torus bounds through the comparison of the cube with King's tori. % \subsection{Proof of Proposition~\ref{prop:k02:Bh}: the scheme constant makes the $h$-linear vacuum constant local}\label{cpi:ssec:Bh} The notation is that of Sections~\ref{ssec:k01:setting} and \ref{ssec:k02:setting}: $k$ is the spacing index, $j$ counts renormalisation steps, $\eta=5^{j-k}$ is the block scale, the diagonal pair at $k$ consists of the $k$-model after $j(k)$ steps and the $(k+1)$-model after $j(k)+1$ steps, both with block scale $\eta_k=5^{j(k)-k}$, on the same unit lattice and with the same indicator $\chi_{j(k)}$; $S^{(j)}_{N,k}(\cdot;s)=S^{(j)}_k(\cdot;s)-v_{k,j}$ is the normalised action, $\sS_k(h)=\sum_{x\in\Lk\cap\CJ}\eps_k^3h(x)$ and $c_k(h)=z_k\sS_k(h)$. On the support of $\chi_j$ King's effective action after $j$ steps is \cite[(3.14)]{K1} (cube form \cite[(3.12)]{K2}), and its whole source dependence sits in $P^{(j),1}$, the sum of the connected renormalised graphs $H$ up to order $\bar n=13$: the sources enter only through the curvature vertex \cite[(3.20)]{K1}, linear in $u$, and the scalar-source vertex \cite[(3.21)]{K1}, linear in $h$. For a graph $H$ let $n_u(H)$, $n_h(H)$ be its numbers of curvature and scalar source vertices, $(\alpha(H),\beta(H))$ its order in $(e^2,\lambda)$, $E(H;A_j,\phi_j;s)$ its value and $E_N(H;\cdot;s)\defeq E(H;\cdot;s)-E(H;0,0;s)$ its field-dependent part; $E(H;\cdot;(u,h))$ is a homogeneous polynomial of degree $n_u(H)$ in $u$ and $n_h(H)$ in $h$, and the number of graphs depends on $\bar n$ only. \begin{definition}[the three families]\label{cpi:def:families} Put $P_0\defeq\{(0,0),(0,1),(2,0)\}$ and \[\mathcal T_h\defeq\{H:\ n_h(H)=1,\ n_u(H)=0,\ (\alpha(H),\beta(H))\in P_0\},\qquad \cV^{(j)}_k(h)\defeq\sum_{H\in\mathcal T_h}E(H;0,0;(0,h)),\] the \emph{$h$-linear vacuum constant of the three lowest orders}, linear in $h$ and independent of $u$. The family $\mathcal N_h$ consists of the field-dependent parts $E_N(H;\cdot;(u,h))$ of all graphs with $n_h(H)\ge1$ and of the field-independent values $E(H;0,0;(u,h))$ of all graphs with $n_h(H)\ge1$ and $H\notin\mathcal T_h$. The excepted graphs are exactly those of $\mathcal T_h$, which carry no curvature vertex; the $h$-linear graphs of the orders $P_0$ which do carry curvature vertices belong to $\mathcal N_h$. \end{definition} \begin{lemma}[the split]\label{cpi:lem:split} On the support of $\chi_j$, for every real source $(u,h)$, $S^{(j)}_{N,k}(\cdot;(u,h))=S^{(j)}_{N,k}(\cdot;(u,0))+\sum_{T\in\mathcal N_h}T+\cV^{(j)}_k(h)$. \end{lemma} \begin{proof} A graph with $n_h(H)=0$ takes the same value at $(u,h)$ and at $(u,0)$, a graph with $n_h(H)\ge1$ vanishes at $h=0$, and for $H\in\mathcal T_h$ one has $n_u(H)=0$, so $E(H;0,0;(u,h))=E(H;0,0;(0,h))$. \end{proof} \emph{(K0) and (SRC): the two bounds across the pair.} For a diagonal pair write $D[T]\defeq T|_{(k,j(k))}-T|_{(k+1,j(k)+1)}$. For a compact family $\Sigma=\{(u_x,h_y):(x,y)\in Y\}$, $h_y=\sum_jy_jh_j$, there are $k_0$ and constants such that for $k\ge k_0$, $j=j(k)$, on the support of $\chi_j$, \begin{gather*}\big|S^{(j)}_{N,k}(\cdot;(u_x,0))-S^{(j+1)}_{N,k+1}(\cdot;(u_x,0))\big|\le\theta^{\mathrm{K0}}_k\defeq C^{\mathrm{K0}}\big(5^{-\gamma j}\eta_k^{-\beta}+\eta_k^{\sigma}\big), \\ \Big|\sum_{T\in\mathcal N_{h_y}}D[T]\Big|\le\theta^{\mathrm{SRC}}_k\defeq C^{\mathrm{SRC}}\big(5^{-\gamma'j}\eta_k^{-\beta'}+\eta_k^{\sigma'}\big):\end{gather*} at a fixed source the first is \eqref{eq:k01:Kpos} and the second is the third clause of Lemma~\ref{hyp:k02:Kh} summed over the finitely many members of $\mathcal N_h$, and the uniformity on $\Sigma$ follows by polarisation, since a function of the source which is, at every field configuration, a homogeneous polynomial of degree $d\le\bar n$ is bounded on $\Sigma$ by $C(Y)$ times its maximum over the finite set of sources $\sum_{i\le d}\epsilon_iw_i$, $w_i\in\{(f_i,0),(0,h_j)\}$, $\epsilon_i=\pm1$. \emph{(LOW): the $h$-linear constant is a one-point function.} Let $\langle\cdot\rangle^{\mathrm{fl}}_{k,j}$ be King's fluctuation measure \cite[(3.17)]{K1} on $\CK$ at zero block fields $A_j=\phi_j=0$ and $[\,\cdot\,]_{P_0}$ the sum of the Taylor coefficients at the orders $P_0$ in $(e^2,\lambda)$. Differentiating $E'_j=-\ln\int\prod d\mu\,e^{V^{(0)}}$ in the source parameter under the finite-dimensional integral, the vertex \cite[(3.21)]{K1} being $\sum_x\eps_k^3h(x)(r_x^2-2\sW_{\eps_k})$ in unscaled units, \[\cV^{(j)}_k(h)=\sum_{x\in\Lk\cap\CJ}\eps_k^3h(x)\,v^{(j)}_k(x),\qquad v^{(j)}_k(x)=-\Big[\big\langle r_x^2-2\sW_{\eps_k}\big\rangle^{\mathrm{fl}}_{k,j}\Big]_{P_0}.\] \begin{definition}[the bulk density]\label{cpi:def:bulk} Let King's block-spin flow at zero field on the infinite lattice $\Lk=\eps_k\Z^3$ be that of \cref{def:k01:bulkflow}: the Dirichlet weight $f_D$ at the finest step and the uniform weight above it, the composite average $Q_j$ onto the blocks of side $\eta=5^j\eps_k$, the fluctuation covariance $G_j$ with the stiffness $\mathfrak a_j$, the split $C=G_j+\Gamma_j$ of the free propagator and, for the photon, $D=D_j+\Gamma^{\mathrm{ph}}_j$ with $D_j=\sum_{i1$ from $\R^3\setminus\CJ$ and $0\le j\le k-J_0$, \[\Big|\cV^{(j)}_k(h)+z_k\sS_k(h)-\sum_{x\in\Lk}\eps_k^3h(x)\,\beta^{(j)}_k(x)\Big|\le C\,\norm h_{L^1}\,\eps_k^{-\nu}\,e^{-c/\eta},\] with $C,\nu,c>0$ depending on the couplings and $J$ only. (b) For an $\eta\Z^3$-periodic $\beta$ on $\Lk$ with block mean $\bar\beta$, $h\in C^\infty_c(\R^3)$ and every integer $r\ge2$, \begin{gather*}\Big|\sum_{x\in\Lk}\eps_k^3h(x)\beta(x)-\bar\beta\,\sS_k(h)\Big|\le C_r\norm\beta_\infty\norm{\Delta^rh}_{L^1}\,\eta^{2r},\\ \big|\sS_k(h)-\textstyle\int h\big|\le C_r\norm{\Delta^rh}_{L^1}\,\eps_k^{2r}.\end{gather*} \end{lemma} \begin{proof} (a) The density $v^{(j)}_k(x)$ is a finite sum of graphs of King's fluctuation measure with one vertex \cite[(3.21)]{K1} at $x$ and lines $G_j(\CK)$, $G^{\mathrm{ph}}_{(i)}(\CK)$; the counterterm $\dm_{\eps_k}$ and $\sW_{\eps_k}$ are free-boundary numbers, the same on $\CK$ and on $\Lk$. The region operators are signed image sums of the free ones over a reflection group (\cite[(3.7)]{K2} for the photon, \cite[(3.9)]{K2} for the scalar with the weight $f_D$, which is why $f_D$ and not $f\equiv1$ is needed), every non-identity image at distance at least $\dist(x,\partial\CK)+\dist(y,\partial\CK)$ from the pair $(x,y)$ (\cite[Prop.~5.1]{K2}). Every free fine line decays at the block scale: for $H=-\Delta_\eps+m^2+b\eta^{-2}Q_f^*Q_f$ with a weight $f\ge0$ of block averages $1$, a discrete Poincar\'e inequality on each block gives $H\ge c_0\eta^{-2}$, and the Combes--Thomas argument with the conjugation $e^{\alpha w\cdot x/\eta}$ gives $|H^{-1}(x,y)|\le2c_0^{-1}\eta^2\eps^{-3}e^{-\alpha_0|x-y|_\infty/\eta}$. Expanding every region line as the free line plus its images and every vertex sum over $\CK$ as the sum over $\Lk$ minus its complement, every term other than the infinite-lattice graph carries a path of fine lines of total length at least $\dist(x,\partial\CK)\ge1$, hence a factor $e^{-\alpha_0/(2\eta)}$, while the singular prefactors are at most $C\eps_k^{-\nu}$; the infinite-lattice graphs give $\sum_x\eps_k^3h(x)v^{(j)}_{k,\infty}(x)$, and $z_k\sS_k(h)$ adds $z_k$ per site. (b) is Poisson summation on $\Lk$ with the finite Fourier expansion of the periodic $\beta$; the non-zero frequencies have modulus at least $2\pi/\eta$ and $|\hat h(p)|\le \norm{\Delta^rh}_{L^1}|p|^{-2r}$. \end{proof} \begin{lemma}[the bulk density at $A=0$]\label{cpi:lem:bulkform} With $c_\lambda>0$ the combinatorial tadpole constant of King's order-$\lambda$ counterterm (${\dm}^{(\lambda)}=-c_\lambda\lambda\sW_{\eps_k}$, \cite[(3.33)]{K1}), the three orders of $\beta^{(j)}_k=\beta_{(0,0)}+\beta_{(0,1)}+\beta_{(2,0)}$ are \begin{gather*}\beta_{(0,0)}(x)=2\,\Gamma_j(x,x),\qquad\beta_{(0,1)}(x)=-2c_\lambda\lambda\sum_y\eps_k^3G_j(x,y)^2\,\Gamma_j(y,y),\\ \beta_{(2,0)}(x)=e^2\big[\mathfrak g_x(C,C,C;D)-\mathfrak g_x(G_j,G_j,G_j;D_j)\big]-\mathfrak a_x,\end{gather*} where $\mathfrak g_x(G_1,G_2,G_3;D')$ is the coefficient of $e^2$ in $\langle r_x^2\rangle$ of the Gaussian matter model with outer lines $G_1,G_3$, internal scalar line $G_2$, photon covariance $D'$, King's bond coupling (exchange and seagull) and the free counterterm ${\dm}^{(e^2)}$, so that $e^2\mathfrak g_x(C,C,C;D)=z_k$, and $\mathfrak a_x$ is the order-$e^2$ contribution to $\langle r_x^2\rangle^{\mathrm{fl}}$ of the $A$-dependence of King's averaging vertices \cite[(3.24)--(3.26)]{K1} with the contour convention \cite[(2.19), (3.27)]{K1}. \end{lemma} \begin{proof} Order $1$: $\langle r_x^2\rangle^{\mathrm{fl}}=2G_j(x,x)$ and $G_j(x,x)-\sW_{\eps_k}=-\Gamma_j(x,x)$. Order $\lambda$: first-order perturbation theory with the vertices \cite[(3.18)--(3.19)]{K1} at $\phi^{(j)}=0$. Order $e^2$: the $\theta$-linear vacuum graphs of order $e^2$ built from the original vertices are the exchange graph, the seagull graph and the $\dm$ insertion with the lines $G_j$, $D_j$; the photon tadpole vanishes by charge conjugation; the graphs with averaging vertices form $\mathfrak a_x$; $z_k$ has no part of order $1$ or $\lambda$. \end{proof} \begin{lemma}[the bulk sequence]\label{cpi:lem:BULK} Fix $\gamma_b\in(0,\frac12)$ and put $\beta_b\defeq1$. There is $C$ such that for all $k$ and $1\le j\le k-J_0$, $|\bar\beta^{(j)}_k-\bar\beta^{(j+1)}_{k+1}|\le C\,5^{-\gamma_bj}\eta^{-\beta_b}$, and along the diagonal pairs, for $k\ge k_0$ and $j=j(k)$, $\norm{\beta^{(j)}_k}_\infty\le C\eta_k^{-2}$. \end{lemma} \begin{proof} \emph{Unit block scale.} Put $t\defeq\eps_k/\eta=5^{-j}$, $m'\defeq m\eta$, $b\defeq\mathfrak a_j$; the map $x\mapsto x/\eta$ carries $-\Delta_{\eps_k}+m^2+\mathfrak a_j\eta^{-2}Q_j^*Q_j$ to $\eta^{-2}(-\Delta_t+m'^2+bQ^*Q)$ on $t\Z^3$ with unit blocks, kernels scaling as $\Gamma_j(x,x')=\eta^{-1}\Gamma_t(x/\eta,x'/\eta)$, so $\bar\beta_{(0,0)}=\eta^{-1}F(t;m',b)$ with $F(t)\defeq2t^3\sum_{x\in t\Z^3\cap[0,1)^3}\Gamma_t(x,x)$, and a diagonal pair changes $(t,b)$ to $(t/5,\mathfrak a_{j+1})$ at fixed $m'$, with $|\mathfrak a_j-\mathfrak a_{j+1}|\le C5^{-2j}$. \emph{Comparison with the continuum block spin.} Woodbury's identity gives $\Gamma_t=C_tQ^*(b^{-1}+QC_tQ^*)^{-1}QC_t$, hence, in Fourier variables on the unit torus $\mathbb T=[-\pi,\pi)^3$ with $p=P+2\pi n$ running over the aliases $n\in N_t$ (the $t^{-3}$ values with $p$ in the Brillouin zone), the averaging symbol $s_t(p)=t^3\sum_{x\in t\Z^3\cap[0,1)^3}f(x)e^{ip\cdot x}$, $\hat C_t(p)=(\omega_t(p)+m'^2)^{-1}$, where $\omega_t(p)\defeq\sum_\mu4t^{-2}\sin^2(tp_\mu/2)$ is the symbol of $-\Delta_t$ (and $\omega_0(p)=|p|^2$), and $W_t(P)=\sum_{n\in N_t}|s_t(p)|^2\hat C_t(p)$, \[\Gamma_t(x,x')=\int_{\mathbb T}\frac{\overline{\hat v^t_x(P)}\,\hat v^t_{x'}(P)}{b^{-1}+W_t(P)}\,\frac{dP}{(2\pi)^3},\qquad \hat v^t_x(P)\defeq\sum_{n\in N_t}s_t(p)\hat C_t(p)e^{-ip\cdot x},\] \begingroup\emergencystretch=3em\relax and the continuum block-spin covariance $\Gamma_0$ is the same formula at $t=0$ ($N_0=\Z^3$, $s_0(p)=\int_{[0,1)^3}e^{iu\cdot p}du$). The elementary symbol bounds, $|s_t(p)|\le\prod_\mu\min(1,\pi|\sin(p_\mu/2)|/|p_\mu|)$, $\frac4{\pi^2}|p|^2\le\omega_t(p)\le|p|^2$, $|s_t-s_0|\le C\min(1,t|p|)\prod_\mu\min(1,\pi|\sin(p_\mu/2)|/|p_\mu|)$ and $|\hat C_t-\hat C_0|\le Ct(|p|^2+m'^2)^{-1/2}$, summed over the aliases, give $0\le\Gamma_t(x,x)\le C$, $|\Gamma_t(x,x')-\Gamma_0(x,x')|\le Ct$, $|\nabla_x\Gamma_0(x,x)|\le C$ and $|\partial_b\Gamma_t|\le C$ for the uniform weight. With the Dirichlet weight $f_D$ at the finest step the symbol factorises, $s^f_t(p)=s_{5t}(p)\varphi(tp)$ with $|\varphi|\le1$ and $|\varphi-1|\le C|tp|$, and the extra aliases $p=q+2\pi\ell/(5t)$, $\ell\ne0$, contribute $Ct^2(1+\ln t^{-1})^3$: the comparison becomes $|\Gamma_t-\Gamma_0|\le Ct(1+\ln t^{-1})^3\le C't^{1/2}$, which is why the rate is $5^{-j/2}$ and $\gamma_b<\frac12$. \emph{Order $1$.} $|F(t)-F(0)|\le2\sup|\Gamma_t-\Gamma_0|+Ct$ (the Riemann-sum error, by the gradient bound), so \[|\bar\beta^{(j)}_{(0,0),k}-\bar\beta^{(j+1)}_{(0,0),k+1}|\le C\eta^{-1}(t(1+\ln t^{-1})^3+5^{-2j})\le C'\eta^{-1}5^{-j/2}\]\endgroup and $\norm{\beta_{(0,0)}}_\infty\le C\eta^{-1}$. \emph{Order $\lambda$.} At unit block scale $\beta_{(0,1)}=-2c_\lambda\lambda\,t^3\sum_y G_t(x,y)^2\Gamma_t(y,y)$ is dimensionless; the fine line $G_t=C_t-\Gamma_t$ is compared with $G_0=C_0-\Gamma_0$ through the local limit theorem with a rate, $|C_t(r;m')-C_0(r;m')|\le C_*t^2|r|^{-3}$ uniformly in $m'\ge0$, and the previous bound, the Combes--Thomas decay of $G_t$ at the block scale controlling the part $|x-y|>R_t\defeq C\ln t^{-1}$; the logarithmic sum of $|r|^{-2}$ over $|r|\le R_t$ gives $\norm{\beta_{(0,1)}}_\infty\le C\lambda(1+j)^3$, and the increment is $\le C\lambda t^{1/2}(1+\ln t^{-1})^3\le C\lambda5^{-\gamma''j}$ for every $\gamma''<\frac12$. \emph{Order $e^2$.} At unit block scale, with $\tau\in\{t,t/5\}$ the spacing of the member of the pair, $C_\tau=(-\Delta_\tau+m'^2)^{-1}$, $\Gamma_\tau$ the coarse line and $G_\tau=C_\tau-\Gamma_\tau$ the fluctuation line, $D_\tau$, $\Gamma^{\mathrm{ph}}_\tau$, $G^{\mathrm{ph}}_\tau=D_\tau-\Gamma^{\mathrm{ph}}_\tau$ their photon analogues at the photon mass, and $m_0$ an upper bound for the two masses at unit block scale, the matter at $\lambda=0$ and zero block fields is Gaussian given $A$ with quadratic form $M_A=M_0+e'M_1+e'^2(M_2+\delta)+O(e'^3)$, $M_1$ and $M_2$ the first- and second-order terms of the covariant Laplacian in the gauge field and $\delta$ King's free order-$e^2$ counterterm \cite[(3.33)]{K1}, the same number in the full and in the fluctuation measure; with matter covariance $G$ and photon covariance $D'$ the coefficient of $e'^2$ in $\langle r_x^2\rangle$ is $-2(G\,\Pi[G,D']\,G)(x,x)$ with $\Pi[G,D']\defeq\E_{D'}[M_2]+\delta-\E_{D'}[M_1GM_1]$ (seagull plus counterterm minus exchange), so that $\beta_{(2,0)}(x)+\mathfrak a_x=-2e^2[(C_\tau\Pi_CC_\tau)(x,x)-(G_\tau\Pi_GG_\tau)(x,x)]$ with $\Pi_C\defeq\Pi[C_\tau,D_\tau]$, which is translation invariant, and $\Pi_G\defeq\Pi[G_\tau,G^{\mathrm{ph}}_\tau]$. The two sides are not compared graph by graph: with $\Gamma=\Gamma_\tau=C_\tau-G_\tau$ one writes the sandwich \begin{gather*}C\Pi_CC-G\Pi_GG=(\Gamma\Pi_CC+C\Pi_C\Gamma)-\Gamma\Pi_C\Gamma+G(\Pi_C-\Pi_G)G,\\ \Pi_C-\Pi_G=\Sigma^{\mathrm{sg}}[\Gamma^{\mathrm{ph}}]-\Sigma^{\mathrm{ex}}[\Gamma,D]-\Sigma^{\mathrm{ex}}[G,\Gamma^{\mathrm{ph}}],\end{gather*} where $\Sigma^{\mathrm{ex}}[G,D']\defeq\E_{D'}[M_1GM_1]$ and $\Sigma^{\mathrm{sg}}[D']\defeq\E_{D'}[M_2]$ (bilinearity in $(G,D')$); the counterterm $\delta$ cancels in the last term, and only the first two terms contain the renormalised translation-invariant self-energy $\Pi_C$, whose symbol $\hat\Pi_\tau$ vanishes at $p=0$ (King's counterterm makes the order-$e^2$ one-particle-irreducible two-point function vanish at zero momentum on the infinite lattice), is bounded by $C(|p|+m_0)$ on the Brillouin zone and differs from $-2$ times the continuum renormalised one-loop self-energy by at most $C\tau(1+|p|)(|p|+m_0)$, uniformly in $\tau\in(0,1]$ and in the masses in $(0,m_0]$. Every exchange term is a triangle of lines in which, after one summation by parts when both differences of the current vertices fall on the inner matter line, each line carries at most one difference; the fluctuation lines $G$, $\nabla G$, $D$, $\nabla D$ decay as $(|r|+\tau)^{-1}$, $(|r|+\tau)^{-2}$ and the coarse lines $\Gamma$, $\Gamma^{\mathrm{ph}}$ and their differences are bounded, which gives the power counting of each term. Each of the three terms is then compared across the diagonal pair at unit block scale, which keeps $\eta$, $m'$, the photon mass and the unit cell fixed and changes $(\tau,b,\text{weight})$ from $(t,\mathfrak a_j,f$ on $5t$-blocks$)$ to $(t/5,\mathfrak a_{j+1},f$ on $t$-blocks$)$, with the photon lines $D_j$ against $D$: the first two terms through the symbol bounds of $\hat\Pi_\tau$ and the alias sums of the coarse lines, the last through the line bounds above; this gives the increment bound $C_\gamma''' e^2 5^{-\gamma''' j}\eta^{-1}$ for every $\gamma'''\in(0,1)$ and the sup bound $Ce^2\eta^{-1}(1+j)^q$ for the original-vertex part; the averaging part $\mathfrak a_x$, whose vertices carry the contours of King's block averaging, obeys the same two bounds separately, with $\gamma'''<\frac12$. \emph{Conclusion.} Adding the three orders with $\gamma''=\gamma'''=\gamma_b$ gives the increment bound (as $\eta\le1$); for the sup bound along the diagonal, $\eta_k^{-1}=5^{k-j(k)}\ge5^{\alpha k-1}$ absorbs $(1+k)^3$ and $(1+k)^q$. The remaining estimates are routine given the symbol bounds and the local limit theorem. \end{proof} \begin{proof}[Proof of Proposition~\ref{prop:k02:Bh}] Take the diagonal pairs with $0<\alpha<\min\{\frac12,\gamma/(\gamma+\beta),\gamma'/(\gamma'+\beta'),\gamma_b/(\gamma_b+\beta_b)\}$; the Cauchy arguments of Theorems~\ref{thm:k01:CA} and \ref{thm:k02:CAh} hold for every $\alpha\in(0,\frac12)$ with $\alpha<\gamma/(\gamma+\beta)$, so the two further restrictions cost nothing. Fix $k\ge k_0$, $j=j(k)$, $\sigma=(u_x,h_y)\in\Sigma$ and $h=h_y$. By Lemma~\ref{cpi:lem:split} for both members of the pair, on the support of $\chi_j$, \begin{align*} S^{(j)}_{N,k}(\cdot;\sigma)+c_k(h)-S^{(j+1)}_{N,k+1}(\cdot;\sigma)-c_{k+1}(h) &=\big[S^{(j)}_{N,k}(\cdot;(u_x,0))-S^{(j+1)}_{N,k+1}(\cdot;(u_x,0))\big]+\sum_{T\in\mathcal N_h}D[T]\\ &\quad+\Big(\big[\cV^{(j)}_k(h)+c_k(h)\big]-\big[\cV^{(j+1)}_{k+1}(h)+c_{k+1}(h)\big]\Big). \end{align*} The first two terms are at most $\theta^{\mathrm{K0}}_k+\theta^{\mathrm{SRC}}_k$. The third is field-independent, linear in $h$ and vanishes at $h=0$; by Lemma~\ref{cpi:lem:loc} with $r=2$, $\cV^{(j)}_k(h)+c_k(h)=\bar\beta^{(j)}_k\int h+E_k(h)$ with $|E_k(h)|\le C\norm h_{L^1}\eps_k^{-\nu}e^{-c/\eta_k}+C\norm{\beta^{(j)}_k}_\infty\norm{\Delta^2h}_{L^1}(\eta_k^4+\eps_k^4)$, the same for the $(k+1)$-model with $\eps_{k+1}=\eps_k/5$ and the same $\eta_k$, so by Lemma~\ref{cpi:lem:BULK} the third term is at most \[\theta^{\mathrm{LOW}}_k(h)\defeq C\,5^{-\gamma_bj}\eta_k^{-\beta_b}\Big|\int h\Big|+C\norm h_{L^1}5^{\nu k}e^{-c5^{\alpha k-1}} +C\norm{\Delta^2h}_{L^1}\big(\eta_k^{2}+\eta_k^{-2}\eps_k^{4}\big),\] bounded on $\{h_y:y\in Y\}$ by a $\theta^{\mathrm{LOW}}_k$ independent of $\sigma$, with $\theta^{\mathrm{LOW}}_k(0)=0$. Along the diagonal pairs $\eta_k\le5^{-\alpha k}$, $j(k)\ge(1-\alpha)k-1$ and $k-j(k)\le\alpha k+1$, so $\eta_k^{\sigma}$, $\eta_k^{\sigma'}$, $\eta_k^2$ and $\eta_k^{-2}\eps_k^4$ decay geometrically, $5^{\nu k}e^{-c5^{\alpha k-1}}$ decays faster than any geometric sequence, and $5^{-\gamma_*j(k)}\eta_k^{-\beta_*}\le5^{\gamma_*+\beta_*}\,5^{-(\gamma_*(1-\alpha)-\beta_*\alpha)k}$ decays geometrically for each of the three pairs $(\gamma_*,\beta_*)$ by the choice of $\alpha$. Hence $\tilde\theta_k\defeq\theta^{\mathrm{K0}}_k+\theta^{\mathrm{SRC}}_k +\theta^{\mathrm{LOW}}_k$ decays geometrically and does not depend on $\sigma$, and the left side of \eqref{eq:k02:B} is bounded on the support of $\chi_{j(k)}$ by $\tilde\theta_k(\sigma)\defeq\theta^{\mathrm{K0}}_k+\theta^{\mathrm{SRC}}_k+\theta^{\mathrm{LOW}}_k(h_\sigma)\le\tilde\theta_k$ for $\sigma\in\Sigma$, and by $\theta^{\mathrm{K0}}_k$ at $\sigma=0$. \end{proof} \space% % \subsection{Three lemmas on positive and negative scalar sources}\label{ssec:k02:lemmas} For a mass $M$ let \[ Y^{(M)}_k(u,h)\defeq e^{-c_k(h)}\langle e^{F(u)+\Wick{h}}\rangle^{(M)} \] denote the functional of the mass-$M$ state \emph{in the conventions of the mass $m$} (free Wick constant $\sW_{\eps_k}(m)$ and scheme $z_k=z_k(m)$), so that $Y^{(m)}_k=n_k$ and $Y^{(M)}_k(u,h)=e^{-\Delta_k(M)\sS_k(h)}n^{(M)}_k(u,h)$, with $n^{(M)}_k$ the functional in the mass-$M$ conventions and $\Delta_k(M)$ as in \cref{lem:k02:cnum}(b). Write $r^2(q)\defeq\sum_{x}\eps_k^3q(x)r_x^2$ over the live sites. \begin{lemma}[mass shift and the correlation inequality]\label{lem:k02:GM} Let $M>0$, $k\ge0$, and let $0\le q$ be a lattice function on the live sites with $q(x)\le u_k(M)$ for all $x$. Then \[ \begin{gathered} n_k(0,-q)\ \le\ \frac1{Y^{(M)}_k(0,q)},\qquad\text{and}\qquad \frac1{n_k(0,q)}\ \le\ n_k(0,-q)\ \le\ \frac1{Y^{(M)}_k(0,q)},\\ \text{hence}\quad Y^{(M)}_k(0,q)\le n_k(0,q). \end{gathered} \] \end{lemma} \begin{proof} \emph{Mass shift.} In King's action the scalar mass enters only through $\sum_x\eps^3\frac12(M^2+\dm_\eps(M))r_x^2$: $\dm_\eps$ is one number per spacing and mass (\cref{def:s02:sources}), the Dirichlet terms $\frac\eps2r_x^2$ do not depend on $M$, and the photon does not see $M$. Hence, with $u\defeq u_k(M)$ and $r^2(1)\defeq\sum_x\eps_k^3r_x^2$, $e^{-S^{(m)}}=e^{-S^{(M)}}e^{u\,r^2(1)}$, and for every $\Phi$ integrable against both measures $\langle\Phi\rangle^{(m)}=\langle\Phi e^{ur^2(1)}\rangle^{(M)}/\langle e^{ur^2(1)}\rangle^{(M)}$; the quartic term makes $e^{\pm r^2(\cdot)}$ integrable. \emph{Correlation inequality.} Put $g_1\defeq u\mathbf 1-q\ge0$ and $g_2\defeq q\ge0$. The mass-$M$ state is a BFS measure, and so is the state with its single-site measures multiplied by $e^{t\,r^2(g_1)}$, $t\in[0,1]$ (\cref{def:s02:acube} and the paragraph on BFS measures following it: the exponential moments stay finite because $\lambda>0$). By \cref{lem:s02:BFS}(c), multiplying the single-site measures by $e^{-tr^2(g_1)}$ does not increase $\langle e^{r^2(g_2)}\rangle$; read from $t=1$ down to $t=0$, $\langle e^{r^2(g_2)}\rangle^{(M)}_{t=1}\ge\langle e^{r^2(g_2)}\rangle^{(M)}_{t=0}$, that is \[ \langle e^{r^2(g_1)}e^{r^2(g_2)}\rangle^{(M)}\ \ge\ \langle e^{r^2(g_1)}\rangle^{(M)}\langle e^{r^2(g_2)}\rangle^{(M)}. \] \emph{Conclusion.} With $\Phi=e^{-r^2(q)}$: $\langle e^{-r^2(q)}\rangle^{(m)}=\langle e^{r^2(g_1)}\rangle^{(M)}/\langle e^{r^2(g_1+g_2)}\rangle^{(M)}\le1/\langle e^{r^2(q)}\rangle^{(M)}$. Multiply both sides by $e^{(2\sW_{\eps_k}(m)+z_k)\sS_k(q)}$: the left side becomes $n_k(0,-q)$ and the right side $1/Y^{(M)}_k(0,q)$, both in the conventions of the mass $m$. The left inequality of the second display is \cref{lem:s02:alg}(c), and the last inequality follows. \end{proof} \begin{lemma}[the large-field remainder at a source, from the one at zero source]\label{lem:k02:HLF} Let $s$ be a real source and $p>1$, and assume \eqref{eq:k02:SF} at $s$ and \Rzero\ of \cref{prop:k01:H0rel} for the member $(k,j)$, with $a\defeq C\eta^\sigma$ and $\rho\defeq\rho(\eta)$. Then \[ e^{-a}\,\Ik{j}(s)\ \le\ \Zk(s)\ \le\ e^{a}\,\Ik{j}(s)+\rho^{1-1/p}\,\langle e^{X(ps)}\rangle_k^{1/p}\,\Zk(0). \] \end{lemma} \begin{proof} The lower line is the first inequality of \eqref{eq:k02:SF}. For the upper line, $\Zk(s)=\int P\,d\mu_{k,s}+\int(1-P)e^{X(s)}d\mu_{k,0}$. The first term is at most $e^{a}\Ik{j}(s)$ by \eqref{eq:k02:SF}. For the second, Hölder's inequality for the positive measure $(1-P)d\mu_{k,0}$ with the functions $1$ and $e^{X(s)}$, then $1-P\le1$ in the second factor and $pX(s)=X(ps)$ (linearity), give \[ \begin{split} \int(1-P)e^{X(s)}d\mu_{k,0}&\ \le\ \Big[\int(1-P)d\mu_{k,0}\Big]^{1-1/p}\Big[\int e^{X(ps)}d\mu_{k,0}\Big]^{1/p}\\ &\quad\le\ \big[\rho\Zk(0)\big]^{1-1/p}\big[\Zk(0)\langle e^{X(ps)}\rangle_k\big]^{1/p}, \end{split} \] using \Rzero\ for the first bracket. \end{proof} At a negative scalar source the same step would need a bound on $\langle e^{X(ps)}\rangle$ at a negative source of larger amplitude, which is what is to be proved; \cref{thm:k02:UB} avoids this through the mass shift. \begin{lemma}[positivity of the limit at a positive source]\label{lem:k02:POS} Fix a mass $M$, a function $0\le q\in C^\infty_c$ as in \cref{prop:k02:Wsharp}, and $\bar w\in(0,1]$. Assume that at mass $M$, in its own conventions, the hypotheses of \cref{thm:k02:CAh} below hold at $\sigma=(0,wq)$ for every $w\in[0,\bar w]$ and at $\sigma=0$, with sequences $b_k$, $\theta_k(wq)$, $r_k(0)$ and $e^{-c_k(wq)}r_k(wq)$ dominated by $w$-independent summable sequences, for all $k\ge k_0$ with $k_0$ independent of $w$. Then $\inf_{k\ge k_0}n^{(M)}_k(0,q)>0$. \end{lemma} \begin{proof} Steps~1--4 of the proof of \cref{thm:k02:CAh} use the hypotheses at $\{wq,0\}$ only and give the lower half of \eqref{eq:k02:rec} with $w$-independent summable $\bar\delta_k$, $\bar\epsilon_k$: $n_{k+1}(wq)\ge(1-\bar\delta_k)n_k(wq)-\bar\epsilon_k$. By induction, $n_k(wq)\ge\prod_{i=k_1}^{k-1}(1-\bar\delta_i)\,n_{k_1}(wq)-\sum_{i\ge k_1}\bar\epsilon_i$ for $k\ge k_1$. Choose $k_1\ge k_0$ with $\prod_{i\ge k_1}(1-\bar\delta_i)\ge\frac12$ and $\sum_{i\ge k_1}\bar\epsilon_i\le\frac18$. At this fixed $k_1$, $w\mapsto n_{k_1}(wq)$ is continuous (a finite-dimensional integral) and equals $1$ at $w=0$; choose $w_0\in(0,\bar w]$ with $n_{k_1}(w_0q)\ge\frac12$. Then $n_k(w_0q)\ge\frac18$ for all $k\ge k_1$. Since $\ln n_k(wq)$ is convex in $w$ and vanishes at $w=0$, $n_k(w_0q)\le n_k(q)^{w_0}$, so $n_k(q)\ge8^{-1/w_0}$ for $k\ge k_1$; the finitely many $k_0\le k0$. \end{proof} \space% % % % \subsection{The Cauchy argument with a $c$-number}\label{ssec:k02:CAh} \begin{theorem}[the Cauchy property with the scheme $c$-number]\label{thm:k02:CAh} Fix a real source $s=(u,h)$ (notation of \cref{ssec:k01:setting} and \cref{ssec:k02:inputs}) and let $\sigma$ range over $\{s,0\}$. Suppose there are $k_0$ and non-negative sequences $b_k$, $r_k(\sigma)$, $\theta_k(\sigma)$ such that for all $k\ge k_0$: \begin{itemize}[leftmargin=2em] \item[\textup{(U)}] for every $\sigma$ and both members $(k',j')$ of the diagonal pair at $k$, \[ e^{-b_k}I^{(j')}_{k'}(\sigma)\le Z_{k'}(\sigma)\le e^{b_k}I^{(j')}_{k'}(\sigma)+r_k(\sigma)Z_{k'}(0), \] with $\sum_kb_k<\infty$, $r_k(0)\le\frac12$, $\sum_kr_k(0)<\infty$ and $\sum_ke^{C_c(h)(k+2)}r_k(s)<\infty$, where $|c_k(h)|\le C_c(h)(1+k)$ (\cref{lem:k02:cnum}\textup{(a)}); \item[\textup{(B)}] on the support of $\chi_{j(k)}$, \eqref{eq:k02:B} holds at $\sigma$ with $\sum_k\theta_k(\sigma)<\infty$. \end{itemize} Put \begin{gather*} \Theta_k\defeq\theta_k(s)+\theta_k(0),\qquad \delta_k\defeq(4b_k+\Theta_k+2r_k(0))\,e^{4b_k+\Theta_k+2r_k(0)}\\ \text{and}\qquad \epsilon_k\defeq(e^{-c_k(h)}+e^{-c_{k+1}(h)})\,r_k(s). \end{gather*} Then for all $k\ge k_0$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $n_k(s)\le\mathsf M(s)\defeq\exp(\sum_{i\ge k_0}\delta_i)\big(n_{k_0}(s)+\sum_{i\ge k_0}\epsilon_i\big)<\infty$; \item $|n_{k+1}(s)-n_k(s)|\le\delta_k\mathsf M(s)+\epsilon_k$, so $n_k(s)$ converges to a limit $n(s)\in[0,\mathsf M(s)]$; \item if \textup{(U)} and \textup{(B)} hold at $-s$ as well, $n_k(s)\ge1/\mathsf M(-s)$ and $n(s)\in[1/\mathsf M(-s),\mathsf M(s)]\subset(0,\infty)$. \end{enumerate} If $b_k$, $\Theta_k$ decay geometrically and $r_k(\sigma)\le e^{-ck^2+Ck}$, then $|n_k(s)-n(s)|\le C'5^{-\kappa k}$ for some $\kappa>0$. \end{theorem} \begin{proof} Fix $k\ge k_0$, $j\defeq j(k)$, and let primes denote the $(k+1)$-model after $j+1$ steps. \emph{Step 1 (one model, \textup{(U)} only).} For one member write $Z(\sigma)$, $I(\sigma)$, $b=b_k$, $r(\sigma)=r_k(\sigma)$. From (U) at $\sigma=0$, $(1-r(0))Z(0)\le e^{b}I(0)$ and $Z(0)\ge e^{-b}I(0)$, so $I(0)>0$. With $\varrho(\sigma)\defeq I(\sigma)/I(0)$ and $N(\sigma)=Z(\sigma)/Z(0)$, \begin{equation}\label{eq:k02:onemodel} e^{-2b}(1-r(0))\,\varrho(\sigma)\ \le\ N(\sigma)\ \le\ e^{2b}\varrho(\sigma)+r(\sigma): \end{equation} the upper bound from $Z(\sigma)\le e^{b}I(\sigma)+r(\sigma)Z(0)$ and $1/Z(0)\le e^{b}/I(0)$, the lower one from $Z(\sigma)\ge e^{-b}I(\sigma)$ and $1/Z(0)\ge e^{-b}(1-r(0))/I(0)$. No lower bound on $Z(\sigma)$ for $\sigma\ne0$ is used: the remainder enters relative to $Z(0)$ only. \emph{Step 2 (the vacuum constant).} $\varrho(\sigma)=\hat I(\sigma)/\hat I(0)$, since $I=e^{-v}\hat I$ with one $v$ for all $\sigma$; likewise for the primed member. \emph{Step 3 (across the pair, \textup{(B)} only).} On the support of $\chi_j$, (B) at $\sigma=s$ reads $S'_N(\cdot;s)=S_N(\cdot;s)+c_k(h)-c_{k+1}(h)+E$ with $|E|\le\theta_k(s)$; the integrals $\hat I(s)$, $\hat I'(s)$ run over the same variables with the same $\chi_j$, so $\hat I'(s)\in e^{c_{k+1}(h)-c_k(h)}[e^{-\theta_k(s)},e^{\theta_k(s)}]\,\hat I(s)$. At $\sigma=0$ ($c_k(0)=0$), $\hat I'(0)\in[e^{-\theta_k(0)},e^{\theta_k(0)}]\,\hat I(0)$. With $\varrho\defeq\varrho(s)$ for the $k$-model and $\varrho'$ for the $(k+1)$-model, \begin{equation}\label{eq:k02:pair} e^{-c_{k+1}(h)}\varrho'\ \in\ [e^{-\Theta_k},e^{\Theta_k}]\;e^{-c_k(h)}\varrho . \end{equation} \emph{Step 4 (two-sided recursion).} Write $n\defeq n_k(s)$, $n'\defeq n_{k+1}(s)$, $r_0\defeq r_k(0)$, $r_s\defeq r_k(s)$, $\Theta\defeq\Theta_k$. Apply \eqref{eq:k02:onemodel} to the primed member, multiply by $e^{-c_{k+1}(h)}$, use \eqref{eq:k02:pair}, and then \eqref{eq:k02:onemodel} for the $k$-model in the forms $\varrho\le e^{2b}(1-r_0)^{-1}N(s)$ and $\varrho\ge e^{-2b}(N(s)-r_s)$: \begin{align*} n'&\le e^{2b}\,e^{-c_{k+1}}\varrho'+e^{-c_{k+1}}r_s\ \le\ e^{2b+\Theta}e^{-c_k}\varrho+e^{-c_{k+1}}r_s\ \le\ e^{4b+\Theta}(1-r_0)^{-1}\,n+e^{-c_{k+1}}r_s,\\ n'&\ge e^{-2b}(1-r_0)\,e^{-c_{k+1}}\varrho'\ \ge\ e^{-2b-\Theta}(1-r_0)\,e^{-c_k}\varrho\ \ge\ e^{-4b-\Theta}(1-r_0)\big(n-e^{-c_k}r_s\big). \end{align*} For $t\le\frac12$, $(1-t)^{-1}\le e^{2t}$ and $1-t\ge e^{-2t}$; with $e^{y}-1\le ye^{y}$ and $1-e^{-y}\le y$ this gives $e^{4b+\Theta}(1-r_0)^{-1}-1\le\delta_k$ and $1-e^{-4b-\Theta}(1-r_0)\le\delta_k$, hence \begin{equation}\label{eq:k02:rec} (1-\delta_k)\,n_k(s)-\epsilon_k\ \le\ n_{k+1}(s)\ \le\ (1+\delta_k)\,n_k(s)+\epsilon_k . \end{equation} \emph{Step 5 (boundedness and convergence).} By induction from the upper half of \eqref{eq:k02:rec}, $n_k\le\prod_{i=k_0}^{k-1}(1+\delta_i)\big(n_{k_0}+\sum_{i1$ from $\R^3\setminus\CJ$. Then $\sup_kn_k(0,-q)<\infty$. Moreover, for every real $2$-form $u$ with the same support condition, $n_k(u,-q)\le n_k(2u,0)^{1/2}n_k(0,-2q)^{1/2}$, so $\sup_kn_k(u,-q)<\infty$. \end{theorem} \begin{proof} By \cref{prop:k02:NEG} choose $M_2>m$ with $s_-(M_2)>\sup q$. \Cref{prop:k01:H0rel,prop:k02:Uplus,prop:k02:Bh} and \cref{lem:k02:cnum} hold at the point $(e,\lambda,M_2,\lmu)$ as well; $\Delta\defeq\sup_k|\Delta_k(M_2)|<\infty$. \emph{Step 1.} Choose $\bar w\in(0,\frac12]$ with $2\bar w\sup q0$, hence $Y^{(M_2)}_k(0,q)\ge e^{-\Delta\sS_k(q)}c$ for $k\ge k_*$. \emph{Step 5 (the mass shift).} By \cref{lem:k02:GM} at $q$, $n_k(0,-q)\le1/Y^{(M_2)}_k(0,q)\le e^{\Delta\sS_k(q)}/c\le e^{\Delta C_0'\wnorm q}/c$ for $k\ge k_*$; the finitely many $k1$ from $\R^3\setminus\CJ$, $J\ge J_0$. For $\zeta=(z,w)\in\C^{n+p}$ put $k_1(\zeta)\defeq\sum_j(\Re w_j)_+h_j$, $k_2(\zeta)\defeq\sum_j(\Re w_j)_-h_j$ (both $\ge0$, smooth, with $\sum_j(\Re w_j)h_j=k_1-k_2$), and \[ D_J(s_W)\ \defeq\ \big\{\zeta\in\C^{n+p}:\ \sup_x2k_1(\zeta)(x)0$ with $4\delta\sup_x\sum_jh_j(x)-\infty$ on $U$. If $g(t)=-\infty$ for some $t\in D_\R$, pick $t'\in U$ and, $D_\R$ being open, a point $t''\in D_\R$ beyond $t'$ on the ray from $t$ through $t'$, so that $t'=\lambda t+(1-\lambda)t''$ with $\lambda\in(0,1)$; then $g(t')\le\lambda g(t)+(1-\lambda)g(t'')=-\infty$, a contradiction. Hence $F_J>0$ on $D_\R$: (ii). \end{proof} \begin{corollary}[King's cube limits with scalar sources in the reference scheme, in the form used later]\label{cor:k02:W1} Under the hypotheses of \cref{thm:k02:W1}: \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item For every finite family of real $2$-forms $f_i$ and real functions $h_j$ (of either sign) in $C^\infty_c$ with the support condition, and every real coefficient vector $(x,y)$ with $2\norm{(\sum_jy_jh_j)_+}_\infty1$ from $\R^3\setminus\CJ$, and $\delta>0$ with $2(\norm{\mathsf k_+}_\infty+\delta)1$ from the cell's complement, \cref{cor:k02:W1}(b) and \cref{rem:k02:scheme} give \Bcell\ for every scheme with $\sup_k|a_k-z_k|<\infty$, and \cref{cor:k02:W1}(a) gives the convergence of the King-convention cell functional $Z^{z,K}_{k,C_{L_c}}(0,-2H'\chi_0)$ as well; for the comparison-convention cell functional only the boundedness is claimed (\cref{rem:k01:compfamily}). \end{remark} \begin{remark}[what is and is not claimed]\label{rem:k02:scope} Beyond the statements of \cref{ssec:k01:open} and \cref{hyp:k02:Kh}, the only result used in this section is the threshold $s_W$ of \cref{prop:k02:Wsharp}, provided by \Itwo\ and \Ithree. Nothing is claimed at a non-negative scalar source of amplitude $\ge s_W/2$: there the positive-source bound is the content of \Wtwo\ above the threshold (\cref{hyp:s03:W2}, which is open). The negative scalar sources are unrestricted, because the mass shift of \cref{lem:k02:GM} turns a negative source of the mass-$m$ model into a positive source of a heavier model, where the small-field analysis applies with no threshold on the amplitude, and \cref{prop:k02:NEG} provides a heavy enough mass for every amplitude. \end{remark} \space% % \part*{The axioms} % % \section{Regularity in the curvature sector}\label{sec:s05} This section proves the regularity axiom $\Ezz$ for the curvature field of King's state $\ZKinf$ (the infinite-volume state of King's construction \cite{K1,K2}), which lives on King's convention $\NDK$ (\cref{def:s02:Kcube}); on the comparison convention $\NDa$ (\cref{def:s02:acube}) it proves the finite-volume bound \cref{thm:s05:C} only. Gaussian domination (\cref{lem:s02:GD}) and a bound on the Neumann photon form of the cube (\cref{lem:s05:C1}) give a bound on every lattice generating functional, uniform in the spacing, the cube and the scalar sector, with absolute constants (\cref{rem:s05:absolute}); by the existence theorem \cref{hyp:s03:EC} it passes to the limits of King's state by Montel's theorem, and Cauchy's estimates turn it into $\Ezz$ with $s=4$. The section closes with the curvature-sector properties of the limits of King's state (locally uniform convergence, monotonicity in the cube, positivity, continuity and translations), proved with cone-direction correlation inequalities only (\cref{lem:s05:TWC}), and with a remark on the comparison family (\cref{rem:s05:comparison}). \refstepcounter{convention}\label{conv:v063_s05_os0_curv:1}\paragraph{Convention~\theconvention.}Throughout, $f_1,\dots,f_n$ are real $2$-forms in $C^\infty_c(\R^3;\Lambda^2)$ with supports at $\ell^\infty$-distance $>1$ from $\R^3\setminus C_{J_0}$, $J\ge J_0$, $z\in\C^n$, and $g(z)\defeq\sum_j(\Re z_j)f_j$, so that $\Re\sum_jz_jF(f_j)=F(g(z))$ (\cref{lem:s02:alg}(b)). For a box $X$, $\ip AB_k$ and $\dstar_kg$ are as in \eqref{eq:s02:riemann}: $\dstar_kg$ is the lattice $1$-form with $F_{\Lk}(g)=\ip{\dstar_kg}{A}_k$, defined whenever every plaquette at which $g\ne0$ has its four corners in $X$. The support condition implies this for $X=\XJ$: the plaquette centre is at $\ell^\infty$-distance $>1$ from $\R^3\setminus\CJ$ and the corners are within $\eps/2\le\frac12$ of it. \subsection{The Neumann cube photon form} \begin{lemma}[the Neumann cube photon form]\label{lem:s05:C1} Let $X=\CJ\cap\Lk$, let $C$ be the photon covariance of the cube states on $X$ (the same for $\NDa_{k,J}$ and $\NDK_{k,J}$) with respect to $\ip\cdot\cdot_k$, let $g$ be a real $2$-form on the plaquettes $p=(x;\alpha<\beta)$ all four of whose corners lie in $X$, and $v\defeq\dstar_kg$. Then \[ \ip v{Cv}_k\ \le\ \min\big\{\,2\norm{g}_{2,k}^2,\ \lmu^{-2}\norm{v}_{2,k}^2\,\big\}. \] \end{lemma} \begin{proof} The Gaussian density of the photon field is $e^{-Q(A)/2}$ with \[ \begin{split} Q(A)&=\ip A{C^{-1}A}_k=\sum_{\alpha,\beta}\ \sum_{x,\,x+\eps e_\beta\in X}\eps\,(\partial_\beta A_\alpha(x))^2+\lmu^2\norm A_{2,k}^2,\\ &\qquad \partial_\beta A_\alpha(x)\defeq A_\alpha(x+\eps e_\beta)-A_\alpha(x): \end{split} \] the Neumann rule keeps exactly the gradient terms of pairs inside $X$. By Legendre duality, $\ip v{Cv}=\sup_A\big[2\ip vA-Q(A)\big]$, attained at $A=Cv$. \emph{Second bound.} $Q(A)\ge\lmu^2\norm A^2$, so $C\le\lmu^{-2}$ as forms. \emph{First bound.} Let $P_g$ be the set of plaquettes on which $g\ne0$. For $p=(x;\alpha<\beta)\in P_g$ both gradient terms $(\partial_\alpha A_\beta(x))^2$ (pair $(x,x+\eps e_\alpha)$, component $\beta$) and $(\partial_\beta A_\alpha(x))^2$ (pair $(x,x+\eps e_\beta)$, component $\alpha$) are present in $Q$, because the pairs lie in $X$. With $F(p)=\eps^{-1}(\partial_\alpha A_\beta(x)-\partial_\beta A_\alpha(x))$, \[ \begin{split} \norm{dA}^2_{P_g}&\defeq\sum_{p\in P_g}\eps^3F(p)^2=\sum_{p\in P_g}\eps\,(\partial_\alpha A_\beta(x)-\partial_\beta A_\alpha(x))^2\\ &\quad\le2\sum_{p\in P_g}\eps\big[(\partial_\alpha A_\beta(x))^2+(\partial_\beta A_\alpha(x))^2\big]\le2Q(A), \end{split} \] where the last step is a counting statement: the ordered triple (site $x$, component $\beta$, direction $\alpha\ne\beta$) determines the plaquette $(x;\{\alpha,\beta\})$, so distinct plaquettes use distinct gradient terms of $Q$ and no term is counted twice. Also $\ip vA_k=F_{\Lk}(g)=\sum_{p\in P_g}\eps^3g_pF(p)=:\ip g{dA}_{P_g}$. Hence \[ \ip v{Cv}\le\sup_A\big[2\ip g{dA}_{P_g}-\tfrac12\norm{dA}^2_{P_g}\big]\le\sup_{B\in\R^{P_g}}\big[2\ip gB_{P_g}-\tfrac12\norm B^2_{P_g}\big]=2\norm g_{2,k}^2, \] the last supremum being attained at $B=2g$. The inequality $Q(A)-\frac12\norm{dA}^2_{P_g}\ge0$ used in the middle is in fact the identity \[ \begin{aligned} Q(A)-\tfrac12\norm{dA}^2_{P_g}&=\tfrac12\sum_{p\in P_g}\eps\,(\partial_\alpha A_\beta(x)+\partial_\beta A_\alpha(x))^2\\ &\quad+\big(\text{gradient terms of $Q$ used by no $p\in P_g$}\big)+\lmu^2\norm A^2_{2,k}.\qedhere \end{aligned} \] \end{proof} \begin{remark}[on the constant $2$]\label{rem:s05:sharp} On a King torus (\cref{def:s10:torus}) the Hodge identity of the periodic lattice gives the constant $1$ in place of $2$ (a direct computation: the source $\dstar g$ is co-exact and the massive photon form commutes with $\dstar$); on the Neumann cube no Hodge argument is available, and whether $2$ is the best constant for the exact quantity $\rho\defeq\sup_g\Var F(g)/\norm g^2_{2,k}$ (which depends on $\eps$ and $\lmu$ only through $\lmu\eps$) is not known; nothing below depends on it. The second bound is the cube form of the two-point bound $0\le S_2(f)\le\lmu^{-2}\norm{\dstar f}^2_{L^2}$; it is the worse of the two at small $\lmu$. \end{remark} \subsection{The comparison convention}\label{conv:v063_s05_os0_curv:2} \begin{lemma}[a local bound free of hypotheses]\label{lem:s05:C2} For either convention and every compact $K\subset\C^n$, \[ \sup_{k\ge0}\ \sup_{J\ge J_0}\ \sup_{z\in K}\ \big|\langle e^{\sum_jz_jF_{\Lk}(f_j)}\rangle_{k,J}\big|\ \le\ \exp\Big(3C_0\sup_{z\in K}\fournorm{g(z)}^2\Big)<\infty, \] with $C_0$ the constant of \cref{lem:s02:E}\textup{(E1)}. No hypothesis and no correlation inequality is used. \end{lemma} \begin{proof} $|\langle e^{\sum z_jF(f_j)}\rangle|\le\langle e^{F(g(z))}\rangle$. Both cube states are BFS measures whose Gaussian is the Neumann photon Gaussian of $\XJ$ with covariance $C$ (the matter on $\TJ$ does not touch the photon variables), so \cref{lem:s02:GD} with $s=1$ and $v=\dstar_kg(z)$ gives $\langle e^{F(g(z))}\rangle\le e^{\ip v{Cv}_k/2}\le e^{\norm{g(z)}^2_{2,k}}$ by \cref{lem:s05:C1}. The plaquette centres of each of the three orientations form a translate of $\eps_k\Z^3$, and $|g_{c(p)}(e_\alpha,e_\beta)|\le(1+|c(p)|^2)^{-2}\fournorm g$, so by (E1) each orientation contributes at most $C_0\fournorm g^2$ to $\norm g^2_{2,k}$ and $\norm{g(z)}^2_{2,k}\le3C_0\fournorm{g(z)}^2$ for every $k$ (this is the count behind (E3) of \cref{lem:s02:E}). Finally $\fournorm{g(z)}\le\sum_j|\Re z_j|\fournorm{f_j}$ is bounded on $K$. \end{proof} The existence theorem \cref{hyp:s03:EC} is posed on King's convention. The finite-volume bound below holds on both conventions and needs no hypothesis; the limit statements are proved for King's state in \cref{thm:s05:CK}. \begin{theorem}[finite-volume regularity, comparison convention]\label{thm:s05:C} Let $f_1,\dots,f_n$, $J_0$, $g(z)$ be as in \cref{conv:v063_s05_os0_curv:1}. For all $k\ge0$, $J\ge J_0$, $\lmu>0$, $z\in\C^n$, and in the presence of any real scalar source $h\in C^\infty_c$, \[ \Bigg|\frac{\ava{e^{\sum_jz_jF_{\Lk}(f_j)}e^{\Wick h}}}{\ava{e^{\Wick h}}}\Bigg| \ \le\ \frac{\ava{e^{F_{\Lk}(g(z))}e^{\Wick h}}}{\ava{e^{\Wick h}}} \ \le\ \exp\Big(\min\big\{\norm{g(z)}^2_{2,k},\ \tfrac12\lmu^{-2}\norm{\dstar_kg(z)}^2_{2,k}\big\}\Big). \] Nothing beyond the definition of the cube state enters. \end{theorem} \begin{proof} The first inequality is $\Re\sum_jz_jF(f_j)=F(g(z))$. The measure $e^{\Wick h}d\langle\cdot\rangle^{(a)}_{k,J}$, normalised, is a BFS measure whose single-site weights carry the extra factor $e^{\eps^3h(x)r_x^2}$ and whose photon covariance is the same $C$; \cref{lem:s02:GD} with $s=1$ and $v=\dstar_kg(z)$ bounds the middle expression by $e^{\ip v{Cv}_k/2}$, and \cref{lem:s05:C1} gives both terms of the minimum. \end{proof} \begin{remark}[the constants are absolute]\label{rem:s05:absolute} The bound $e^{\norm{g}_2^2}$ of \cref{thm:s05:C,thm:s05:CK} and the constants $(\sqrt2\,e)^n(n!)^{1/2}$, $(e\pi/2)^n(n!)^{1/2}$, $\sigma_n$ of $\Ezz$ depend on none of $e$, $\lambda$, $m$, $\lmu$ and $M$: the only ingredients are Gaussian domination, the Neumann form's constant $2$ and the elementary facts (E1)--(E4). The second bound of \cref{thm:s05:C} and of \cref{thm:s05:CK}(a),(b) is the one that carries $\lmu^{-2}$. The proof of \cref{thm:s05:CK}(b) uses, besides the existence clause \cref{hyp:s03:EC}, only the cube monotonicity of \cref{cor:s04:G1} (cone-direction correlation inequalities), Gaussian domination and \cref{lem:s05:C1}; it uses neither \cite[Thm.~6.2(3),(4)]{BFS1} (\cref{lem:s02:BFS}(d)) nor any local bound resting on that reversed inequality. \end{remark} \subsection{King's convention}\label{conv:v063_s05_os0_curv:3} For the K family write $a^K_{k,J}(z)\defeq\avK{e^{\sum_jz_jF(f_j)}}$ as in \cref{cor:s04:G2}, and $\ZKJ$, $\ZKinf$ for its limits. \begin{theorem}[regularity of King's state]\label{thm:s05:CK} Let $f=(f_1,\dots,f_n)$, $J_0$, $g(z)$ be as in \cref{conv:v063_s05_os0_curv:1}. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item \textup{(No hypothesis.)} For all $k\ge0$, $J\ge J_0$, $z\in\C^n$, \[ |a^K_{k,J}(z)|\ \le\ \avK{e^{F(g(z))}}\ \le\ \exp\Big(\min\big\{\norm{g(z)}^2_{2,k},\ \tfrac12\lmu^{-2}\norm{\dstar_kg(z)}^2_{2,k}\big\}\Big), \] and for every real $2$-form $u$ whose plaquettes have their corners in $\XJ$: $\avK{F(u)}=0$ and $\avK{F(u)^2}\le\ip v{Cv}_k\le2\norm u^2_{2,k}$, $v=\dstar_ku$. \item \textup{(Limits.)} $\ZKJ(z)\defeq\lim_ka^K_{k,J}(z)$ exists locally uniformly on $\C^n$ and is entire; $\ZKinf(z)\defeq\lim_J\ZKJ(z)$ exists locally uniformly on $\C^n$ and is entire; $|\ZKJ(z)|,|\ZKinf(z)|\le e^{\norm{g(z)}_2^2}$ and also $\le\exp(\frac12\lmu^{-2}\norm{\dstar g(z)}^2_{L^2})$; $J\mapsto\ZKJ(\mathrm it)$ is nondecreasing on $J\ge J_0$ with values in $[0,1]$ for $t\in\R^n$; and $\ZKinf(x)\ge1$ at every real $x\in\R^n$. \item \textup{(Schwinger functions.)} The Schwinger functions $S^K_n(f_1,\dots,f_n)\defeq\partial_{z_1}\cdots\partial_{z_n}\ZKinf(0)$ satisfy $|S^K_n|\le(\sqrt2\,e)^n(n!)^{1/2}\prod_i\norm{f_i}_2\le(e\pi/2)^n(n!)^{1/2}\prod_i\fournorm{f_i}$ for real $f_i$; the lattice Schwinger functions $S^{K,(k,J)}_n\defeq\partial_{z_1}\cdots\partial_{z_n}a^K_{k,J}(0)$ satisfy the same bound at every $(k,J)$ with $\norm{f_i}_{2,k}$ in place of $\norm{f_i}_2$, and $S^{K,(k,J)}_n\to S^K_n$ as $k\to\infty$ and then $J\to\infty$. Each $S^K_n$ extends to a continuous symmetric multilinear form on $\cS(\R^3;\Lambda^2_\C)^n$ with the same bound, carrying the factor $2^n$ for complex test functions: $\Ezz$ holds with $s=4$ and $\sigma_n=(e\pi)^n(n!)^{1/2}\le\alpha_\beta(n!)^\beta$ for every $\beta>\frac12$. \end{enumerate} \end{theorem} \begin{proof} (a) The two inequalities are the bound of \cref{lem:s05:C2} for the K family, with \cref{lem:s05:C1} kept in both forms: $\Re\sum z_jF(f_j)=F(g(z))$; $\langle\cdot\rangle^K_{k,J}$ is a BFS measure (the killed sites carry point masses, which \cref{lem:s02:GD} allows: at a killed site $I_q(0)=\delta_{q0}$ and the currents vanish on its bonds), and its Gaussian is the Neumann photon Gaussian of $\XJ$, the same as the comparison cube's, so \cref{lem:s02:GD} with $s=1$, $v=\dstar_kg(z)$ and \cref{lem:s05:C1} give $\avK{e^{F(g(z))}}\le e^{\ip v{Cv}_k/2}\le\exp(\frac12\min\{2\norm g^2_{2,k},\lmu^{-2}\norm v^2_{2,k}\})$. \emph{Second-moment clause.} $\avK{F(u)}=0$ because the state is invariant under $(A,\vartheta)\mapsto(-A,-\vartheta)$ and $F(u)$ is odd. For real $s$, $e^x+e^{-x}\ge2+x^2$ pointwise and $e^{\pm sF(u)}$ are integrable, so $2+s^2\avK{F(u)^2}\le\avK{e^{sF(u)}}+\avK{e^{-sF(u)}}\le2e^{s^2\ip v{Cv}_k/2}$ (\cref{lem:s02:GD} at $\pm s$; $\pm v$ give the same quadratic form). Hence $\avK{F(u)^2}\le2(e^{s^2\ip v{Cv}_k/2}-1)/s^2\to\ip v{Cv}_k$ as $s\downarrow0$, and \cref{lem:s05:C1} applies since the corners lie in $\XJ$. (b) \emph{Step 1: $k\to\infty$ at fixed $J\ge J_0$.} The functions $a^K_{k,J}$ are entire on $\C^n$: finite-dimensional integrals with Gaussian decay in $A$ ($\lmu>0$) and $e^{-\lambda\eps^3r^4}$ decay in $r$ at the alive sites, converging absolutely and locally uniformly in $z$. By (a) they are locally uniformly bounded, uniformly in $k$ and $J\ge J_0$ ($\norm g^2_{2,k}\le3C_0\fournorm g^2$ by (E3) of \cref{lem:s02:E}); by \cref{hyp:s03:EC} they converge at every real point. $\R^n$ is a set of uniqueness for entire functions on $\C^n$ (a holomorphic function vanishing on $\R^n$ has all its Taylor coefficients at $0$ equal to $0$). By Montel's theorem every subsequence has a locally uniformly convergent sub-subsequence; all subsequential limits are entire and agree on $\R^n$, hence everywhere \cite{Hor90}; so $a^K_{k,J}\to\ZKJ$ locally uniformly on $\C^n$ and $\ZKJ$ is entire (this is \cref{cor:s04:G2}(ii)($\alpha$)). The bounds of (a) pass to the limit: $|\ZKJ(z)|\le\lim_ke^{\norm{g(z)}^2_{2,k}}=e^{\norm{g(z)}_2^2}$ by (E3), and $|\ZKJ(z)|\le\lim_k\exp(\frac12\lmu^{-2}\norm{\dstar_kg(z)}^2_{2,k})=\exp(\frac12\lmu^{-2}\norm{\dstar g(z)}^2_{L^2})$ by \eqref{eq:s02:riemann}. Both bounds are free of $J$. \emph{Step 2: monotonicity in $J$ on the imaginary axes.} For $t\in\R^n$ and $f\defeq\sum_jt_jf_j$, $a^K_{k,J}(\mathrm it)=\avK{e^{\mathrm iF(f)}}$ is real (the state is invariant under $(A,\vartheta)\mapsto(-A,-\vartheta)$, which maps $F(f)$ to $-F(f)$) and of modulus $\le1$, and $J\mapsto a^K_{k,J}(\mathrm it)$ is nondecreasing on $J\ge J_0$ at fixed $k$: this is the cube monotonicity of \cref{cor:s04:G1} for the class-$\cI$ observable $\Theta(\mathrm if,0)$, whose plaquettes lie in $X_{J_0}$ by the support condition (\eqref{eq:s04:S} holds for $X_{J_0}$). It is $\ge0$ by the current representation of \cref{lem:s02:GD}: with $\Psi(A)=\sum_qw(q)\cos\ip{\kappa(q)}A$, $w(q)\ge0$, and the Gaussian identity at $a=\mathrm iv\pm\mathrm i\kappa$, \[ \avK{e^{\mathrm i\ip vA}}=e^{-\ip v{Cv}_k/2}\,\E^\sharp\cosh\ip{Cv}{\kappa}\ \ge\ 0,\qquad v=\dstar_k\textstyle\sum_jt_jf_j, \] the killed sites being covered as in (a). Letting $k\to\infty$ (Step 1), $J\mapsto\ZKJ(\mathrm it)$ is nondecreasing with values in $[0,1]$, hence convergent, for every $t\in\R^n$. \emph{Step 3: $J\to\infty$.} $\{\ZKJ\}_{J\ge J_0}$ is locally uniformly bounded on $\C^n$ by the $J$-free bounds of Step 1, and converges on $(\mathrm i\R)^n$ (Step 2), a totally real subspace of maximal dimension, hence a set of uniqueness for entire functions (at a point $\mathrm it_0$ the restriction to $\mathrm it_0+\mathrm i\R^n$ determines the Taylor series). Montel's theorem and the identity theorem as in Step 1 give $\ZKJ\to\ZKinf$ locally uniformly on $\C^n$, $\ZKinf$ entire, and both bounds of Step 1 pass to the limit (this is \cref{cor:s04:G2}(ii)($\beta$)). \emph{Step 4: $\ZKinf(x)\ge1$.} For real $x$, $a^K_{k,J}(x)=\avK{e^{F(g(x))}}\ge e^{\avK{F(g(x))}}=1$ by Jensen's inequality and (a); the inequality is closed under both limits. (c) Let $\norm{f_j}_{2,k}>0$ for all $j$ (otherwise $F_{\Lk}(f_j)=0$ and $S^{K,(k,J)}_n=0$). Cauchy's estimates on the polydisc $|z_j|\le\rho_j\defeq\rho/(n\norm{f_j}_{2,k})$ with the bound (a): there $\norm{g(z)}_{2,k}\le\sum_j|z_j|\norm{f_j}_{2,k}\le\rho$, so $|a^K_{k,J}|\le e^{\rho^2}$ and $|S^{K,(k,J)}_n|\le e^{\rho^2}\prod_j\rho_j^{-1}=e^{\rho^2}n^n\rho^{-n}\prod_j\norm{f_j}_{2,k}$; at $\rho^2=n/2$ the prefactor is $e^{n/2}2^{n/2}n^{n/2}\le e^{n/2}2^{n/2}e^{n/2}(n!)^{1/2}=(\sqrt2\,e)^n(n!)^{1/2}$ by (E4). The Taylor coefficients at $0$ converge under the locally uniform convergences of (b) (Weierstrass), and $\norm{f_j}_{2,k}\to\norm{f_j}_2$ by (E3), which gives the bound for $S^K_n$; (E2) gives the $\fournorm\cdot$ form. $S^K_n$ is symmetric and multilinear (derivatives of an entire function of a linear combination) and continuous for the $L^2$ norms, so it extends by density from $C^\infty_c$ to $\cS$; for complex $f_i=f_i'+\mathrm if_i''$ with real parts, multilinearity writes $S^K_n(f_1,\dots,f_n)$ as a sum of $2^n$ terms $\mathrm i^mS^K_n(f^\sharp_1,\dots,f^\sharp_n)$, each bounded by $(e\pi/2)^n(n!)^{1/2}\prod\fournorm{f^\sharp_i}\le(e\pi/2)^n(n!)^{1/2}\prod\fournorm{f_i}$, since $\fournorm{\Re f},\fournorm{\Im f}\le\fournorm f$; the kernel theorem gives the tempered distribution. The axiom $\Ezz$ of \cite[p.~287]{OS75} asks for $|S_n(f_1,\dots,f_n)|\le\sigma_n\prod_i|f_i|_s$ on $\cS$ with $\sigma_n\le\alpha(n!)^\beta$; here $s=4$: on scalar test functions $\fournorm\cdot$ is the norm $|\cdot|_4$ of \cite[(2.1), p.~284]{OS75} (\cref{def:s02:tube}), and a $2$-form $f$ whose only non-zero components are $f_{\alpha\beta}=-f_{\beta\alpha}=q$ ($\alpha<\beta$) has $\fournorm f=\fournorm q$ by the definition of $\fournorm\cdot$ on $2$-forms, so the bound, read on such test functions, is $\Ezz$ for the Schwinger functions of the three component fields $F_{\alpha\beta}(q)\defeq F(f)$, $\alpha<\beta$, mixed ones included (each slot may carry a different component); and $\sigma_n=(e\pi)^n(n!)^{1/2}\le\alpha_\beta(n!)^\beta$ exactly when $(e\pi)^n/(n!)^{\beta-1/2}$ is bounded, that is for every $\beta>\frac12$. \end{proof} \subsection{The limits of King's state, with cone-direction inequalities only} For King's state and an admissible scheme $a$, the limit functionals of the Wick sector on the tube have the following properties, established in \cref{thm:s13:TWK,prop:s13:WK,lem:s11:TWtube}: (i) locally uniform convergence of the fixed-cube functionals to a holomorphic $G^K_J$; (ii) on the cone, $J\mapsto G^K_J$ nondecreasing; (iii) the cube limit $G^K_\infty=\lim_JG^K_J$, locally uniform and holomorphic, positive at real sources, with $G^K_J\le G^K_\infty$ on the cone; (iv) $\ln G^K_\infty$ finite, convex and continuous on the real tube; (v) translation invariance; (vi) the change of scheme. At $h=0$ these conclusions hold by the existence clause \cref{hyp:s03:EC} alone; the natural local bound, $|\ZKJ(\zeta)|\le\ZKJ(\Re\zeta)\le Z^K_{J_0}(\Re\zeta)$, would use the class-$\cD$ form of the cube monotonicity, that is \cref{lem:s02:BFS}(d). The following lemma states the curvature sector with that use removed; the Wick sector on the tube is \cref{lem:s11:TWtube}. \begin{lemma}[the limits of King's state, curvature sector, with cone-direction inequalities only]\label{lem:s05:TWC} Let $f_1,\dots,f_N$ be a finite family of real $2$-forms in $C^\infty_c$ supported at $\ell^\infty$-distance $>1$ from $\R^3\setminus C_{J_0}$, put $Z^K_{k,J}(\sum_jz_jf_j,0)\defeq\avK{e^{\sum_jz_jF(f_j)}}$, let $\ZKJ$, $\ZKinf$ be its limits of \cref{thm:s05:CK}, and $g(z)=\sum_j(\Re z_j)f_j$. Then: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item $Z^K_{k,J}(\cdot,0)\to\ZKJ(\cdot,0)$ locally uniformly on $\C^N$ as $k\to\infty$, and $\ZKJ(\cdot,0)$ is entire, for every $J\ge J_0$; \item $J\mapsto\ZKJ(\mathrm if,0)$ is nondecreasing on $J\ge J_0$ for every real $f=\sum_jt_jf_j$; \item for all $k\ge0$, $J\ge J_0$ and $z\in\C^N$, \[ \begin{split} |Z^K_{k,J}(\textstyle\sum_jz_jf_j,0)|&\le\avK{e^{F(g(z))}}\le e^{\frac12\ip v{Cv}_k}\\ &\quad\le\exp\Big(\tfrac12\lmu^{-2}\norm{\dstar_kg(z)}^2_{2,k}\Big)\\ &\quad\le\exp\Big(\tfrac12\lmu^{-2}\sup_{k'}\norm{\dstar_{k'}g(z)}^2_{2,k'}\Big),\quad v=\dstar_kg(z), \end{split} \] a bound uniform in $k$ and $J$ and locally uniform in $z$; consequently $\ZKinf(\cdot,0)=\lim_J\ZKJ(\cdot,0)$ exists locally uniformly on $\C^N$, is entire, satisfies $\ZKinf(f,0)>0$ for real $f$, and $\ZKJ(\mathrm if,0)\le\ZKinf(\mathrm if,0)$ for real $f$; \item for every $\rho>0$, $\Phi^K(f)\defeq\ln\ZKinf(f,0)$ is finite, convex and continuous on the Fr\'echet space $C^\infty_c(\bar B_\rho;\Lambda^2)$; \item $\ZKinf(\tau_bf,0)=\ZKinf(f,0)$ for all $b\in\R^3$ and all complex coefficient vectors, $\tau_b$ acting on every $f_j$. \end{enumerate} Clauses \textup{(i)}, \textup{(iii)}--\textup{(v)} use, besides the existence clause \cref{hyp:s03:EC}, only the cube monotonicity of \cref{cor:s04:G1} and the translation sandwich of \cref{lem:s06:TK} (cone-direction correlation inequalities), \cref{lem:s02:GD}, \cref{lem:s02:alg}\textup{(b)}, \eqref{eq:s02:riemann}, \cref{lem:s02:baire}, Montel's theorem and the identity theorem. The nonincreasing clause at $h=0$, $J\mapsto\ZKJ(f,0)$ nonincreasing for real $f$, would follow from the class-$\cD$ form of \cref{cor:s04:G1}, that is \cref{lem:s02:BFS}\textup{(d)}, at $g=0$ with $k\to\infty$ by \textup{(i)}; it is not used by the curvature sector of this paper. The clause on the change of scheme is void at $h=0$, no scheme entering. \end{lemma} \begin{proof} (i) and (ii) are Steps 1 and 2 of the proof of \cref{thm:s05:CK}(b). (iii) The first inequality is \cref{lem:s02:alg}(b), the second is \cref{lem:s02:GD} with $s=1$ (King's cube state is a BFS measure, the killed sites carrying point masses, as in the proof of \cref{thm:s05:CK}(a)), the third is $C\le\lmu^{-2}$ as forms (the photon form is bounded below by $\lmu^2\norm A^2_{2,k}$, the second bound of \cref{lem:s05:C1}), and the supremum over $k'$ is finite by \eqref{eq:s02:riemann}. The bound passes to $\ZKJ$ by (i) and makes $\{\ZKJ(\cdot,0)\}_{J\ge J_0}$ locally bounded on $\C^N$; with the convergence on $(\mathrm i\R)^N$ from (ii) and the bound $|\ZKJ(\mathrm if,0)|\le1$, Montel's theorem and the identity theorem give the locally uniform convergence to the entire $\ZKinf(\cdot,0)$, as in Step 3 of the proof of \cref{thm:s05:CK}(b). Positivity: $Z^K_{k,J}(f,0)\ge e^{\avK{F(f)}}=1$ for real $f$ by Jensen's inequality and $\avK{F(f)}=0$ (the symmetry $(A,\vartheta)\mapsto(-A,-\vartheta)$), closed under both limits. The last clause is (ii) in the limit. (iv) $\Phi^K$ is finite by (iii). Convexity: for $f,f'\in C^\infty_c(\bar B_\rho;\Lambda^2)$ and $t\in[0,1]$ apply (i) and (iii) to the family $(f,f')$ at the real coefficients $(t,1-t)$; at fixed $k,J$, $\ln Z^K_{k,J}(\cdot,0)$ is convex in the coefficients by H\"older's inequality, and convexity survives the two pointwise limits. Borel measurability: fix $J$ with $MJ>\rho+1$, so that every $f\in C^\infty_c(\bar B_\rho;\Lambda^2)$ is supported at $\ell^\infty$-distance $>1$ from $\R^3\setminus\CJ$; at each $k$, $\ln Z^K_{k,J}(f,0)$ depends on $f$ only through the finitely many sampled values $f_{c(p)}(e_\alpha,e_\beta)$, continuous functionals of $f$ in the $C^\infty$ topology, and the finite-dimensional integral is continuous in them (dominated convergence); so $\Phi^K=\lim_J\lim_k$ of continuous functions is Borel (of Baire class $\le2$). $C^\infty_c(\bar B_\rho;\Lambda^2)$ is a closed subspace of $C^\infty(\bar B_{\rho+1};\Lambda^2)$, hence Fr\'echet. \cref{lem:s02:baire} gives the continuity. (v) is \cref{cor:s06:TK}, whose proof uses the lattice sandwich of \cref{lem:s06:TK} between King's cubes of indices $J-1$ and $J+1$, $k\to\infty$ by (i) (\cref{hyp:s03:EC} covers the translated family), $J\to\infty$ by (iii), the identity theorem with the uniqueness set $(\mathrm i\R)^N$, the Lipschitz bound from the second-moment clause of \cref{thm:s05:CK}(a) and the density of $\bigcup_k5^{-k}\Z^3$. \end{proof} \begin{remark}[the comparison family]\label{rem:s05:comparison} The proofs of \cref{thm:s05:CK}(b),(c) and \cref{lem:s05:TWC} use, of the K family, only the local bound of \cref{thm:s05:CK}(a) (which \cref{lem:s05:C2} gives for the comparison family with the same constant), the existence of the fixed-cube limits at real coefficients (\cref{hyp:s03:EC}), the cube monotonicity on the cone (\cref{cor:s04:G1}; for the comparison family the first inequality of \cref{lem:s02:cut}), the translation sandwich (\cref{lem:s06:TK}; for the comparison family \cref{prop:s06:translations}), Montel's theorem and the identity theorem. As noted in Section~\ref{sec:s03}, they apply to the comparison family $\NDa$ wherever its fixed-cube limits exist at real coefficients (Remark~\ref{rem:s04:notfollow}). In every clause above the inequalities used are the cone directions of \cref{lem:s02:BFS}(a),(b) and of \cref{lem:s04:G}; the reversed inequalities of \cref{lem:s02:BFS}(d) enter the curvature sector of this paper nowhere. \end{remark} \space% % % % \section{Translations, the cubic group, and rotations}\label{sec:s06} This section proves the clause \OSo\ of King's state (the infinite-volume state of King's construction \cite{K1,K2}) as far as the lattice allows: invariance under all translations and under the cubic group $\Oh$ about every point; full $O(3)$ invariance comes from the rotation theorem (\cref{hyp:s03:RC}, proved in the rotation part, \cref{rot:thm:main}) and is stated at the end. No cube is invariant under a non-trivial translation or a reflection; what replaces exactness is a sandwich between a smaller and a larger cube of the same family, in the direction of the cone inequalities of \cref{lem:s02:cut,lem:s04:G3}, followed by the limits $k\to\infty$, $J\to\infty$. Coordinate permutations alone are exact (\cref{lem:s06:PK}). \emph{Conventions.}\refstepcounter{convention}\label{conv:v065b_s06_os1:1} We use the notation of \cref{def:s02:sources,cor:s04:G2}: for a finite family $f=(f_1,\dots,f_N)$ of real $2$-forms in $C^\infty_c(\R^3;\Lambda^2)$ and $z\in\C^N$, $a^K_{k,J}(z)=\avK{e^{\sum_jz_jF(f_j)}}$ and $a^{(a)}_{k,J}(z)=\ava{e^{\sum_jz_jF(f_j)}}$ are the lattice functionals, $\ZKJ$, $\ZKinf$ the limits of the first (\cref{thm:s05:CK}(b)), and $g(z)=\sum_j(\Re z_j)f_j$. Since $F_{\Lk}$ is linear in the source (extended $\C$-linearly), a lattice functional depends on a family only through its total source, and we use the same symbol for a coefficient vector and for its total source: $a^K_{k,J}(z)=Z^K_{k,J}(\sum_jz_jf_j)$ with $Z^K_{k,J}(u)\defeq\avK{e^{F(u)}}$ for a complex-valued $2$-form $u$; in particular, for a real $2$-form $u$, $Z^K_{k,J}(\mathrm iu)$ is the one-member functional at the coefficient $\mathrm i$, and the same convention is carried to $\ZKJ$, $\ZKinf$ (so $\ZKinf(\mathrm iu_t)=Z^{K,f}_\infty(\mathrm it)$ for $u_t=\sum_jt_jf_j$). When several families are in play we write $a^{K,f}_{k,J}$, $Z^{K,f}_J$, $Z^{K,f}_\infty$. $J_0(f)$ is the least $J$ such that every $\supp f_j$ lies at $\ell^\infty$-distance $>1$ from $\R^3\setminus\CJ$; for $J\ge J_0(f)$ the support condition \eqref{eq:s04:S} holds for $\XJ$. The transport $(O\cdot f)_x(u,v)=f_{O^{-1}x}(O^{-1}u,O^{-1}v)$ and the translation $(\tau_bf)_x=f_{x-b}$ are those of \cref{def:s02:sources}; $O\cdot(O'\cdot f)=(OO')\cdot f$, and both are linear in $f$. Translations and the cubic group act on the Schwinger functions by this signed push-forward of $2$-forms: for an improper $O$ no extra sign appears. \subsection{The translation sandwich on the comparison convention}\label{conv:v065b_s06_os1:2} \begin{lemma}[translation sandwich, comparison convention]\label{prop:s06:translations} Let $f$ be a finite family of real $2$-forms, $k_0\ge0$, $b\in\Lambda_{k_0}$ with $|b|_\infty\le M$, and $k\ge k_0$. Then, for every $J\ge J_0(f)+1$, the lattice covariance \eqref{eq:s06:cov} holds for all coefficients, $X_{J-1}\subset\XJ-b\subset X_{J+1}$, and for every $t\in\R^N$ \[ a^{(a)}_{k,J-1}(\mathrm it)\ \le\ a^{(a),\tau_bf}_{k,J}(\mathrm it)\ \le\ a^{(a)}_{k,J+1}(\mathrm it). \] Only the first inequality of \cref{lem:s02:cut} enters: the sandwich is taken on the cone. \end{lemma} \begin{proof} \emph{Step 1 (lattice covariance).} Let $k_0\ge0$, $b\in\Lambda_{k_0}$ and $k\ge k_0$. Then $\tau_b$ maps $\Lk$ onto itself, and the substitution $A'_\alpha(x)\defeq A_\alpha(x+b)$, $\phi'(x)\defeq\phi(x+b)$ is a bijection of the configurations of the comparison state on $\XJ$ onto those of the comparison state on $\XJ-b$ (\cref{def:s02:acube} on the translated box) which carries the Lebesgue measure and the action to themselves: the action is a sum of local terms with site-independent coefficients ($\dm_\eps$, $\sW_\eps$ depend on $\eps$ only), and the boundary conditions are transported. The plaquette $p$ corresponds to $p-b$ with the same orientation and $F(p)(A)=F(p-b)(A')$, so $F_{\Lk}(\tau_bf)(A)=F_{\Lk}(f)(A')$. Hence, for all coefficients, \begin{equation}\label{eq:s06:cov} Z_{k,\XJ}\Big(\sum_jz_j\tau_bf_j\Big)=Z_{k,\XJ-b}\Big(\sum_jz_jf_j\Big)\qquad(b\in\Lambda_{k_0},\ k\ge k_0), \end{equation} the right side being the functional of the comparison state on $\XJ-b$. \emph{Step 2 (interlacing of the translated box).} For $|b|_\infty\le M$: $X_{J-1}\subset\XJ-b\subset X_{J+1}$, since $-MJ-b_\mu\le-M(J-1)$, $MJ-b_\mu\ge M(J-1)$, $-MJ-b_\mu\ge-M(J+1)$ and $MJ-b_\mu\le M(J+1)$ for every $\mu$. \emph{Step 3 (the sandwich on the imaginary axes).} Let $J\ge J_0(f)+1$, $t\in\R^N$ and $u_t=\sum_jt_jf_j$, so that $(u_t,0)\in\cK$ is supported at distance $>1$ from $\R^3\setminus C_{J-1}$. Apply the first inequality of \cref{lem:s02:cut} twice: with $Y=\XJ-b$ and $X=X_{J-1}$, and with $Y=X_{J+1}$ and $X=\XJ-b$. The support condition of the lemma holds because the plaquettes carrying $u_t$ have their corners in $X_{J-1}$; no site is killed, so each cut-out factor is the comparison state on the smaller box (the Dirichlet terms of the smaller state are the diagonal parts of the cut bonds, and where the boundaries of two of the boxes touch they coincide). With \eqref{eq:s06:cov}, \[ a^{(a)}_{k,J-1}(\mathrm it)\ \le\ a^{(a),\tau_bf}_{k,J}(\mathrm it)\ \le\ a^{(a)}_{k,J+1}(\mathrm it)\qquad(k\ge k_0,\ J\ge J_0(f)+1). \] \end{proof} \begin{remark}[the comparison family]\label{rem:s06:comparison} From a lattice sandwich such as \cref{prop:s06:translations}, the translation invariance of a limit follows by the argument of \cref{cor:s06:TK} below: the limits $k\to\infty$ and $J\to\infty$ at the two ends and in the middle, the identity theorem off the imaginary axes, composition over lattice translations and continuity in $b$. On King's convention, where \cref{hyp:s03:EC} provides the limits, it gives \cref{cor:s06:TK}; as noted in Section~\ref{sec:s03}, it applies to the comparison family wherever its fixed-cube limits exist, with \cref{prop:s06:translations} in place of \cref{lem:s06:TK}. The translation clause for the joint field on the tube, by the same sandwich at the cone points of the tube, is \cref{lem:s11:TWtube}(v). \end{remark} \subsection{The translation sandwich on King's convention}\label{conv:v065b_s06_os1:3} On King's convention the translated cube has its killed sites on $\TJ-b$, which neither contains nor is contained in $T_{J\pm1}$, so \cref{lem:s02:cut} does not compare it with $C^K_{J\pm1}$ directly; one kill step (\cref{lem:s02:BFS}(c) through \cref{lem:s02:Kprime}) and one cut-out with killed sites (\cref{lem:s04:G3}) at each end do. \emph{Lattice covariance on King's convention.} For $b\in\Lambda_{k_0}$ and $k\ge k_0$ let $\av{\cdot}^{K,-b}_{k,J}$ denote the comparison state on $\XJ-b$ with the matter killed on $\TJ-b$ (King's cube translated by $-b$). The substitution of Step~1 in the proof of \cref{prop:s06:translations} is a bijection of the configurations on $\XJ$ killed on $\TJ$ onto those on $\XJ-b$ killed on $\TJ-b$, carrying the point masses to the point masses and the action to the action (the kill set translates with the box). Hence, for every finite family and all coefficients, \begin{equation}\label{eq:s06:covK} Z^{a,K}_{k,J}\Big(\sum_jz_j\tau_bf_j,\sum_jw_j\tau_bh_j\Big)=Z^{a,K,-b}_{k,J}\Big(\sum_jz_jf_j,\sum_jw_jh_j\Big)\qquad(b\in\Lambda_{k_0},\ k\ge k_0), \end{equation} the right side being the scheme functional of $\av{\cdot}^{K,-b}_{k,J}$ with the same multiplier ($\sS_{\Lk}(\tau_bh)=\sS_{\Lk}(h)$). \begin{lemma}[translation sandwich, King's convention]\label{lem:s06:TK} Let $(f,g)\in\cK$ be supported at $\ell^\infty$-distance $>1$ from $\R^3\setminus C_{J_0}$, and let $b\in\Lambda_{k_0}$ with $|b|_\infty\le M$. Then for every $k\ge k_0$ and every $J\ge J_0+1$, \[ Z^{a,K}_{k,J-1}(\mathrm if,g)\ \le\ Z^{a,K}_{k,J}(\mathrm i\,\tau_bf,\tau_bg)\ \le\ Z^{a,K}_{k,J+1}(\mathrm if,g), \] and, for the class-$\cD$ observable $\Theta(f,-g)$ (real curvature coefficient), the reversed inequalities $Z^{a,K}_{k,J-1}(f,-g)\ge Z^{a,K}_{k,J}(\tau_bf,-\tau_bg)\ge Z^{a,K}_{k,J+1}(f,-g)$ hold. \end{lemma} \begin{proof} By \eqref{eq:s06:covK} the middle term is $Z^{a,K,-b}_{k,J}(\mathrm if,g)$; the multipliers of the three terms are the same positive constant (\cref{lem:s02:alg}(a); $\sS$ is translation invariant), so it suffices to prove the inequalities for $\av{\Theta(\mathrm if,g)}$, and the reversed ones for $\Theta(f,-g)$; all these expectations are real. \emph{Geometry.} Here $b_\mu\in\eps_{k_0}\Z$, $|b_\mu|\le M$, and all sets are subsets of $\Lk$. (G1) $\alive^K_J-b=\prod_\mu\{-MJ-b_\mu,\dots,MJ-2\eps-b_\mu\}\subset\alive^K_{J+1}=\{-MJ-M,\dots,MJ+M-2\eps\}^3$, since $-MJ-b_\mu\ge-MJ-M$ and $MJ-2\eps-b_\mu\le MJ+M-2\eps$. (G2) $\XJ-b\subset X_{J+1}$: the same arithmetic with $\eps$ in place of $2\eps$. (G3) $\alive^K_{J-1}=\{-MJ+M,\dots,MJ-M-2\eps\}^3\subset\alive^K_J-b$, since $-MJ+M\ge-MJ-b_\mu$ and $MJ-M-2\eps\le MJ-2\eps-b_\mu$. (G4) $X_{J-1}\subset\XJ-b$, likewise. All four inclusions hold with margin $M-|b_\mu|\ge0$ on every face, at every spacing, including $k=0$, $M=1$, $|b_\mu|=M$; non-strict inclusions are all that \cref{lem:s04:G3} needs. \emph{Sources.} By \eqref{eq:s04:S} for $X_{J-1}$, every plaquette carrying $f$ has its corners in $X_{J-1}$ and every site carrying $g$ lies in $\alive^K_{J-1}$; by (G1)--(G4) these lie in the site sets and alive sets of all three states. So the observable is one function of variables present and alive in all three states, and its scalar factor equals $1$ at every site killed below. \emph{Upper end.} Start from $\av{\cdot}^K_{k,J+1}$, the comparison state on $X_{J+1}$ killed on $T_{J+1}$, a BFS measure of King type (\cref{def:s02:Kcube}). (i) Kill $U_+\defeq\alive^K_{J+1}\setminus(\alive^K_J-b)$. These are alive sites of the state, $g=0$ on $U_+$ (by (G3), $U_+\cap\alive^K_{J-1}=\emptyset$), and $\Theta(\mathrm if,g)$ is continuous with $|\Theta|=e^{\sum_xg_xr_x^2}$; so by \cref{lem:s02:BFS}(c), through \cref{lem:s02:Kprime} (which allows the already killed sites $T_{J+1}$), killing $U_+$ does not increase $\av{\Theta(\mathrm if,g)}$, and by \cref{lem:s02:BFS}(d) with \cref{lem:s02:Kprime} it does not decrease $\av{\Theta(f,-g)}$. The result is the comparison state on $X_{J+1}$ killed on $T'\defeq T_{J+1}\cup U_+=X_{J+1}\setminus(\alive^K_J-b)$ (by (G1); nothing is revived). (ii) Apply \cref{lem:s04:G3} with $V=X_{J+1}$, $T=T'$ and $X=\XJ-b\subset V$ (by (G2)); the lemma allows any $T\subset V$ and any $X\subset V$. The cut-out factor is the comparison state on $\XJ-b$ killed on $(\XJ-b)\cap T'=(\XJ-b)\setminus(\alive^K_J-b)=\TJ-b$, that is $\av{\cdot}^{K,-b}_{k,J}$; the support condition (S$_X$) holds for $X=\XJ-b$ by \emph{Sources}. Hence $\av{\Theta(\mathrm if,g)}^{K,-b}_{k,J}\le\av{\Theta(\mathrm if,g)}^{(a),T'}_{X_{J+1}}\le\av{\Theta(\mathrm if,g)}^K_{k,J+1}$, and $\ge$ for class $\cD$. \emph{Lower end.} Start from $\av{\cdot}^{K,-b}_{k,J}$, the comparison state on $\XJ-b$ killed on $\TJ-b$ (of King type, a translate of a state of \cref{def:s02:Kcube}). (i) Kill $U_-\defeq(\alive^K_J-b)\setminus\alive^K_{J-1}$, alive sites at which $g=0$ by \emph{Sources}; as above, $\av{\Theta(\mathrm if,g)}$ does not increase and $\av{\Theta(f,-g)}$ does not decrease, and the result is the comparison state on $\XJ-b$ killed on $T''\defeq(\XJ-b)\setminus\alive^K_{J-1}$ (by (G3)). (ii) \cref{lem:s04:G3} with $V=\XJ-b$, $T=T''$ and $X=X_{J-1}\subset V$ (by (G4)): the factor is the comparison state on $X_{J-1}$ killed on $X_{J-1}\cap T''=X_{J-1}\setminus\alive^K_{J-1}=T_{J-1}$, that is $\av{\cdot}^K_{k,J-1}$; (S$_X$) for $X_{J-1}$ is the hypothesis. Hence $\av{\Theta(\mathrm if,g)}^K_{k,J-1}\le\av{\Theta(\mathrm if,g)}^{K,-b}_{k,J}$, and $\ge$ for class $\cD$. There is no edge case because a translation by $b\in\Lambda_{k_0}$, $|b|_\infty\le M$, moves King's cube rigidly, with the alive sets nested at margins $M-|b_\mu|\ge0$ at every spacing; the reflection of \cref{lem:s04:C} shifts the cube by one lattice layer instead, which is why its one-step form needs $M\ge2\eps_k$. \end{proof} The class-$\cI$ clause uses \cref{lem:s02:BFS}(a),(b),(c),(e) and \cref{lem:s02:Kprime} only; the class-$\cD$ clause uses \cref{lem:s02:BFS}(d) in addition. Only the class-$\cI$ clause is used in this section. \begin{corollary}[translations of King's state]\label{cor:s06:TK} For every finite family $f$ of real $2$-forms, every $b\in\R^3$ and every $z\in\C^N$, $\ZKinf(\sum_jz_j\tau_bf_j)=\ZKinf(\sum_jz_jf_j)$. Moreover, for all real $2$-forms $u,u'\in C^\infty_c$, \[ |\ZKinf(\mathrm iu)-\ZKinf(\mathrm iu')|\le\sqrt2\,\norm{u-u'}_2, \] the $L^2$ norm of a $2$-form being computed in a basis $(e_c)$ of $\Lambda^2$ orthonormal for the norm formula. \end{corollary} \begin{proof} Write $u_t=\sum_jt_jf_j$ for $t\in\R^N$. (1) \emph{$b\in\Lambda_{k_0}$, $|b|_\infty\le M$, on the imaginary axes.} For $J\ge J_0(f)+1$ and $k\ge k_0$, \cref{lem:s06:TK} at $(u_t,0)\in\cK$ gives $a^K_{k,J-1}(\mathrm it)\le a^{K,\tau_bf}_{k,J}(\mathrm it)\le a^K_{k,J+1}(\mathrm it)$. Let $k\to\infty$: the ends converge by \cref{thm:s05:CK}(b) for $f$ at $J\mp1\ge J_0(f)$; the middle term by \cref{thm:s05:CK}(b) for the family $\tau_bf$, which is admissible at $J$ since $\supp f_j+b$ lies at distance $>1$ from $\R^3\setminus\CJ$ ($C_{J-1}+b\subset\CJ$ for $|b|_\infty\le M$), so that \cref{hyp:s03:EC}, posed for every finite family, applies to it. Hence $Z^K_{J-1}(\mathrm it)\le Z^{K,\tau_bf}_J(\mathrm it)\le Z^K_{J+1}(\mathrm it)$, and $J\to\infty$ (\cref{cor:s04:G2}(ii)($\beta$) at both ends) gives $Z^{K,\tau_bf}_\infty(\mathrm it)=Z^{K,f}_\infty(\mathrm it)$ for all $t\in\R^N$. (2) \emph{All coefficients, same $b$.} $D(z)\defeq Z^{K,\tau_bf}_\infty(z)-Z^{K,f}_\infty(z)$ is entire on $\C^N$ (\cref{thm:s05:CK}(b) for both families) and vanishes on $(\mathrm i\R)^N$, a totally real subspace of maximal real dimension, so every partial derivative of $D$ at $0$ vanishes and $D\equiv0$. (3) \emph{All $b\in\bigcup_k\Lk$.} Such a $b$ lies in some $\Lambda_{k_0}$ and is a finite sum $b_1+\dots+b_n$ of vectors $b_i\in\Lambda_{k_0}$ with $|b_i|_\infty\le M$ (each coordinate splits into steps $\pm M$ and a remainder in $\eps_{k_0}\Z$ of modulus $1$ interior $(-MJ+1,MJ-1)^3$ of $\CJ$ is $P$-invariant, so $J_0(P\cdot f)=J_0(f)$, and \cref{thm:s05:CK}(b) for both families gives $Z^{K,P\cdot f}_\infty=Z^{K,f}_\infty$ on $\C^N$. \emph{Step 2 (the reflection $\sigma_3$ on the imaginary axes).} Let $\sigma_3=\diag(1,1,-1)$, $t\in\R^N$, $u_t=\sum_jt_jf_j$, $J_0=J_0(f)$. \cref{lem:s04:C}(b), for every $k\ge0$, every $M\ge1$ and every $J\ge J_0+2$, gives \[ a^{K,f}_{k,J-2}(\mathrm it)\ \le\ a^{K,\sigma_3\cdot f}_{k,J}(\mathrm it)\ \le\ a^{K,f}_{k,J+2}(\mathrm it), \] since $\sigma_3\cdot u_t=\sum_jt_j\sigma_3\cdot f_j$. The family $\sigma_3\cdot f$ is admissible at $J$, the interior being $\sigma_3$-symmetric, so $J_0(\sigma_3\cdot f)=J_0(f)$. Let $k\to\infty$ (\cref{thm:s05:CK}(b) for $f$ at $J\mp2$ and for $\sigma_3\cdot f$ at $J$), then $J\to\infty$ (\cref{cor:s04:G2}(ii)($\beta$) at both ends): $Z^{K,\sigma_3\cdot f}_\infty(\mathrm it)=Z^{K,f}_\infty(\mathrm it)$ for all $t\in\R^N$. The one-step sandwich of \cref{lem:s04:C}(a) would serve whenever $M\ge2\eps_k$, that is for all but finitely many $k$, which the $k$-limit ignores; the two-step form is used because it needs no case distinction, and the $J$-limit absorbs the extra step. This is the use King makes of the correlation inequalities in the proof of his Theorem~3.2 \cite{K2}, with the kill steps added; no reversed inequality enters. \emph{Step 3 ($\sigma_1,\sigma_2$).} $\sigma_\nu=P\sigma_3P^{-1}$ with $P$ the transposition of the axes $\nu$ and $3$. Since the transport is a left action, for every real $2$-form $u$, \[ \ZKinf(\mathrm i\sigma_\nu\!\cdot\!u)=\ZKinf(\mathrm iP\!\cdot\!(\sigma_3P^{-1}\!\cdot\!u))=\ZKinf(\mathrm i\sigma_3\!\cdot\!(P^{-1}\!\cdot\!u))=\ZKinf(\mathrm iP^{-1}\!\cdot\!u)=\ZKinf(\mathrm iu), \] by Step 1 for the family $\sigma_3P^{-1}\cdot u$, Step 2 for the family $P^{-1}\cdot u$, and Step 1 again; each step is a statement about every family. \emph{Step 4 (the group, on the imaginary axes).} Every $O\in\Oh$ is a product of a permutation matrix and some of $\sigma_1,\sigma_2,\sigma_3$, and if $\ZKinf(\mathrm iO\cdot u)=\ZKinf(\mathrm iu)$ for all real $u$ holds for $O_1$ and $O_2$, it holds for $O_1O_2$. By Steps 1--3, $Z^{K,O\cdot f}_\infty(\mathrm it)=Z^{K,f}_\infty(\mathrm it)$ for every $O\in\Oh$, every family $f$ and every $t\in\R^N$. \emph{Step 5 (all coefficients).} $D(z)\defeq Z^{K,O\cdot f}_\infty(z)-Z^{K,f}_\infty(z)$ is entire on $\C^N$ (\cref{thm:s05:CK}(b); $J_0(O\cdot f)=J_0(f)$) and vanishes on $(\mathrm i\R)^N$, a set of uniqueness; so $D\equiv0$. No splitting of sources is needed, there being no scalar source. \emph{Step 6 (about every point).} $\Oh$ about $b$ is $\tau_bO\tau_{-b}$, and translations are \cref{cor:s06:TK} (every family, every $b\in\R^3$, all coefficients): $\ZKinf(\sum_jz_j\tau_bO\cdot f_j)=\ZKinf(\sum_jz_jO\cdot f_j)=\ZKinf(\sum_jz_jf_j)$. \emph{Step 7 (Schwinger functions).} Differentiating the identity of entire functions at $z=0$ gives $S^K_n(\tau_bO\cdot f_1,\dots,\tau_bO\cdot f_n)=S^K_n(f_1,\dots,f_n)$ for all $n$ and $(b,O)\in\Cub$. \end{proof} \begin{remark}[what cubic symmetry needs]\label{rem:s06:K4} King's cube is exactly symmetric under the coordinate permutations (Step 1) but not under the reflections; there the sandwich of \cref{lem:s04:C} replaces exactness, as that of \cref{lem:s02:refl} does on the comparison convention. So $\Oh$-invariance of King's state follows from the uniqueness of the monotone cube limit (\cref{cor:s04:G1} with \cref{thm:s05:CK}(b)) together with the cut sandwich, and does not need the uniqueness of the infinite-volume state. Only class-$\cI$ clauses enter, and no scheme or Wick constant. \end{remark} \subsection{Rotations} \begin{theorem}[rotations]\label{thm:s06:O3} $\ZKinf$ is $\Euc$-invariant: for every finite family $f$, every $z\in\C^N$ and every $(b,R)\in\Euc$, $\ZKinf(\sum_jz_j\tau_bR\cdot f_j)=\ZKinf(\sum_jz_jf_j)$, and the Schwinger functions $S^K_n$ are Euclidean covariant, $F$ transforming as a $2$-form. \end{theorem} \begin{proof} By \cref{rot:thm:main}, whose object is King's state $\ZKinf(\cdot,0)$ of \cref{thm:s05:CK} (\cref{r13:def:objectM}), at every point of King's range, $\ZKinf(\cdot,0)$ is $O(3)$-invariant for all complex coefficients; translations are \cref{cor:s06:TK}; the Schwinger functions follow by differentiation at $z=0$. By \cref{cor:s04:G2}(iii), with the fixed-cube limits of \cref{hyp:s03:EC}, the limit state $Z^{\mathrm{LIM}}_\infty(\cdot,0)$ of the comparison family equals $\ZKinf(\cdot,0)$ and is therefore $\Euc$-invariant as well; this is \cref{hyp:s03:RC}. \end{proof} The rotation theorem is proved in the rotation part (\cref{sec:rot}, \cref{rot:thm:main}). Nothing in \cref{cor:s06:TK,thm:s06:S1K} depends on it. Its consequences for reflection positivity about every plane and for the components of the measure are drawn in \cref{thm:s09:S2K}(c). \space% % % % \section{Reflection positivity I: the modified box}\label{sec:s07} King's Neumann--Dirichlet cube is not reflection symmetric about any plane, and the natural symmetrisation of its photon form is not reflection positive on gauge-invariant functionals (\cref{rem:s07:R1}). Reflection positivity is therefore proved first for the modified box state of \cite[(3.38)]{K2} and then carried to the cube states by a two-sided comparison on the cone: the cube $C_J$ sits inside the box (\cref{lem:s07:L}), and the box sits inside a larger cube once a margin proportional to $\lmu^{-1}$ is allowed (\cref{lem:s07:U}). The passage to the limits is in \cref{thm:s08:OS2C}. Throughout, $\eps=\eps_k$, the lattice is $\Lk\subset\R^d$ with $d=3$ (the certificates of \cref{lem:s07:R} are also given for $d=2$), a \emph{lattice box} is a product $\Omega=\prod_\mu\{a_\mu,a_\mu+\eps,\dots,b_\mu\}\subset\Lk$, $\Omega^*$ is the set of its internal bonds (both ends in $\Omega$) and $\Omega^{**}$ the set of its plaquettes (all four corners in $\Omega$). \subsection{The lattice reflection and the modified box} \begin{definition}[mid-planes and the lattice reflection]\label{def:s07:reflection} The \emph{mid-planes} are the planes $\{x_\nu=c\}$ with $c\in D\defeq\bigcup_{k\ge0}\eps_k(\Z+\frac12)$. If $c\in\eps_{k_0}(\Z+\frac12)$ then $c\in\eps_k(\Z+\frac12)$ for every $k\ge k_0$, because $5^j(m+\frac12)=5^jm+\frac{5^j-1}2+\frac12$ and $5^j$ is odd; the same set $D$ is obtained from any tail $k\ge k_1$ of the spacings. $D$ is dense in $\R$ but not all of $\R$. For $c\in D$ let $k_0(c)$ be the least $k$ with $c\in\eps_k(\Z+\frac12)$, and for $\delta>0$ let $k_0(c,\delta)$ be the least $k\ge k_0(c)$ with $\eps_k<2\delta$. Fix $\nu$, a mid-plane $c\in\eps_k(\Z+\frac12)$ and a lattice box $\Omega\subset\Lk$ with $\rho\Omega=\Omega$, $\rho=\rho_{\nu,c}$ the reflection of \cref{def:s02:sources}. On the fields put $(\hat\rho\phi)(x)\defeq\phi(\rho x)$ and $(\hat\rho A)(b)\defeq A(\rho b)$, the image bond taken with its orientation: $(\hat\rho A)_\alpha(x)=A_\alpha(\rho x)$ for $\alpha\ne\nu$ and $(\hat\rho A)_\nu(x)=-A_\nu(\rho x-\eps e_\nu)$. Then $\hat\rho$ is a signed permutation of the variables attached to $\Omega$ and $\Omega^*$ (it preserves Lebesgue measure and maps $\Omega^*$, $\Omega^{**}$ to themselves); the \emph{crossing bonds} $B_0\defeq\{\langle x^-,x^-+\eps e_\nu\rangle:\ x^-_\nu=c-\eps/2\}$ are mapped to their reversals, $(\hat\rho A)(b)=-A(b)$ for $b\in B_0$. Let $\Omega_\pm\defeq\{x\in\Omega:\ \pm(x_\nu-c)>0\}$ (no site lies on the plane), $B_+$ the bonds with both ends in $\Omega_+$, $\mathfrak B_+$ the Borel functions of $(\phi|_{\Omega_+},A|_{B_+})$, and \[ \theta G\defeq\overline{G\circ\hat\rho}\qquad(G\in\mathfrak B_+). \] \end{definition} On the smeared fields, $F_{\Lk}(f)\circ\hat\rho=F_{\Lk}(\rho^*f)$ and $W^a_{\Lk}(h)\circ\hat\rho=W^a_{\Lk}(h\circ\rho)$. Indeed, $F(p)\circ\hat\rho=\sigma_pF(\rho p)$ with $\sigma_p=-1$ exactly when $\nu\in\{\alpha,\beta\}$ (the orientation of the image plaquette, whose base point is shifted back one layer), the centres satisfy $c(\rho p)=\rho c(p)$, and $(\rho^*f)_{c(\rho p)}(e_\alpha,e_\beta)=f_{c(p)}(\rho e_\alpha,\rho e_\beta)=\sigma_pf_{c(p)}(e_\alpha,e_\beta)$; reindexing $p\mapsto\rho p$ gives the first identity. The scalar case is a relabelling of sites, and the $c$-numbers $2\sW_\eps\sS(h)$ and $a_k\sS(h)$ are invariant because $\rho$ permutes $\Lk$ ($2(x_\nu-c)\in\eps\Z$). Consequently, for families $\zeta_i$ whose lattice observables $\Theta(\zeta_i)$ lie in $\mathfrak B_+$, \begin{equation}\label{eq:s07:thetaTheta} \theta(\Theta(\zeta_i))\,\Theta(\zeta_j)=\Theta(\theta\zeta_i^*\sqcup\zeta_j),\qquad m(\theta\zeta_i^*\sqcup\zeta_j)=\overline{m(\zeta_i)}\,m(\zeta_j) \end{equation} for the scheme multiplier $m(\zeta)\defeq e^{-\sum_jw_ja_k\sS(h_j)}$ ($a_k$ real), so that for any state $\av{\cdot}$ on the box \begin{equation}\label{eq:s07:matrix} \sum_{i,j}\bar\gamma_i\gamma_j\,m(\theta\zeta_i^*\sqcup\zeta_j)\av{\Theta(\theta\zeta_i^*\sqcup\zeta_j)}=\sum_{i,j}\overline{(\gamma_im(\zeta_i))}\,(\gamma_jm(\zeta_j))\,\big\langle\theta(\Theta(\zeta_i))\,\Theta(\zeta_j)\big\rangle . \end{equation} \emph{Support.} If the sources of $\zeta_i$ are supported in $\{x_\nu\ge c+\delta\}$ with $\delta>0$, then for $\eps_k<2\delta$ every site carrying a sampled $h$ and every corner of every plaquette carrying a sampled $f$ (centre in the support, corners at $x_\nu\ge c+\delta-\eps_k/2>c$) lies in $\Omega_+$, so $\Theta(\zeta_i)\in\mathfrak B_+$. The splitting $h=h^+-h^-$ of \cref{def:s08:enlarged} is taken with cutoffs supported in $\{x_\nu\ge c+\delta/2\}$, so that $h^\pm$ are supported in $\{x_\nu>c\}$ as well. \begin{definition}[the modified box state]\label{def:s07:box} Let $\Omega\subset\Lk$ be a lattice box, $F(p)=\eps^{-1}\sum_{b\in\partial p}A(b)$, $\Omega(y)$ the indicator of $\Omega$, and $x^*$ the nearest neighbours of $x$ in $\Lk$. King's modified action \cite[(3.38)]{K2} on $\Omega$ is \begin{align*} S^\eps_\Omega(A,\phi)&=\tfrac12\sum_{p\in\Omega^{**}}\eps^d|F(p)|^2+\tfrac12\lmu^2\sum_{b\in\Omega^*}\eps^d|A(b)|^2+\tfrac12\sum_{x\in\Omega}\eps^d\Big|\sum_{y\in x^*}\eps^{-1}\Omega(y)A(\langle x,y\rangle)\Big|^2\\ &\quad+\tfrac12\sum_{\langle x,y\rangle}\eps^{d-2}\big[(\Omega r)(x)^2+(\Omega r)(y)^2-2(\Omega r)(x)(\Omega r)(y)\cos(\vartheta_x-e\eps A(\langle x,y\rangle)-\vartheta_y)\big]\\ &\quad+\sum_{x\in\Omega}\eps^d\big[\tfrac12m^2r_x^2+\lambda r_x^4+\tfrac12\dm_\eps\,r_x^2\big], \end{align*} with the \emph{constant} free-boundary counterterm $\dm_\eps$ of \cref{def:s02:Kcube} at every site, and with the free Wick constant $\sW_\eps$ in the Wick observables. (In \cite{K2} the last term carries a site-dependent counterterm; it is not used here, and the statements below are made for the constant one.) The photon variables are the internal bonds $\Omega^*$ only. The matter part is exactly the Dirichlet matter of the Neumann--Dirichlet state on $X=\Omega$: cosines on the internal bonds, and the diagonal $\frac12\eps^{d-2}r_x^2$ for every bond leaving $\Omega$. The \emph{modified box state} $\langle\cdot\rangle^{(3.38)}_{k,\Omega}$ is the normalised measure $e^{-S^\eps_\Omega}\prod_{x\in\Omega}d^2\phi_x\prod_{b\in\Omega^*}dA_b$; it is a BFS measure (the Gaussian is non-degenerate since $\lmu>0$, the single-site measures have all exponential moments since $\lambda>0$). We write $Z^{a,(3.38)}_{k,\Omega}(\zeta)\defeq m(\zeta)\langle\Theta(\zeta)\rangle^{(3.38)}_{k,\Omega}$ with the same observables and multiplier as in \cref{def:s02:sources}. The third term is the graph divergence of $(\Omega,\Omega^*)$: with $d_0:\R^\Omega\to\R^{\Omega^*}$, $(d_0\chi)(\langle x,x+\eps e_\alpha\rangle)=\eps^{-1}(\chi(x+\eps e_\alpha)-\chi(x))$, one has $\sum_{y\in x^*}\Omega(y)A(\langle x,y\rangle)=-\eps(d_0^TA)(x)$ (only internal bonds contribute). With $d_1A\defeq(\eps F(p))_{p\in\Omega^{**}}$, the \emph{unitary form} on the box and the modified form are \[ Q^u_\Omega(A)\defeq\tfrac12\sum_{p\in\Omega^{**}}\eps^d|F(p)|^2+\tfrac12\lmu^2\sum_{b\in\Omega^*}\eps^d|A(b)|^2,\qquad Q^{(3.38)}_\Omega=Q^u_\Omega+\tfrac12\eps^d\|d_0^TA\|^2 , \] \ie the modified form is the unitary form plus a gauge-fixing term depending on $A$ only through $d_0^TA$. The modified action is invariant under $\hat\rho$ for every mid-plane $c$ with $\rho\Omega=\Omega$: every term is written through $A(\langle x,y\rangle)$ and nearest-neighbour pairs, which $\rho$ permutes, $(d_0^T\hat\rho A)(x)=(d_0^TA)(\rho x)$, and $\dm_\eps$ is site independent. \end{definition} \emph{The boxes.} Fix $c\in D$, $k\ge k_0(c)$, $J\ge J_0$ such that every source of the enlarged family of \cref{def:s08:enlarged} and its reflection is supported at $\ell^\infty$-distance $>1$ from $\R^3\setminus C_{J_0}$, an integer $n\ge1$, and put $a_k\defeq c+\eps_k/2\in\eps_k\Z$ and \begin{equation}\label{eq:s07:margin} \Omega_{J,k}\defeq(C_{J+n}+a_ke_\nu)\cap\Lk,\qquad Mn\ \ge\ |c|+2+R(\lmu),\qquad R(\lmu)\defeq\frac{\pi\sqrt3}{2\lmu}. \end{equation} $C_{J+n}\cap\Lk$ is symmetric in direction $\nu$ about $-\eps_k/2$ (sites $x_\nu\in\{-M(J+n),\dots,M(J+n)-\eps_k\}$, and $M(J+n)\in\eps_k\Z$), so $\Omega_{J,k}$ is a lattice box symmetric about $\{x_\nu=c\}$ with an even number $2M(J+n)5^k$ of layers. Under the margin condition of \eqref{eq:s07:margin}, for every $k$: (i) $C_J$ plus one layer is contained in $\Omega_{J,k}$ (this needs $Mn\ge|c|+\frac32\eps_k$); (ii) $C_{J+2n}$ contains every site at $\ell^\infty$-distance $\le R(\lmu)+\eps_k$ from $\Omega_{J,k}$ (this needs $Mn\ge|c|+R(\lmu)+\frac32\eps_k$). We fix such an $n=n(\lmu,c)$, independent of $k$ and $J$. The two margins are exactly the hypotheses of \cref{lem:s07:L,lem:s07:U} below; by \cref{lem:s04:D} they serve King's convention as they stand. \subsection{Reflection positivity of the modified box} \begin{lemma}[lattice reflection positivity of the modified box]\label{lem:s07:RPbox} Let $\lmu>0$, $c$ a mid-plane and $\Omega$ a lattice box with $\rho\Omega=\Omega$, $\rho=\rho_{\nu,c}$. \begin{enumerate}[label=\textup{(\alph*)},nosep] \item For every gauge-invariant $G\in\mathfrak B_+$ that is bounded, or of the form $\Theta(\zeta)$ with sources supported in $\Omega_+$, or more generally with $\langle|G|^2\rangle^{(3.38)}_{k,\Omega}<\infty$: $\langle\theta(G)G\rangle^{(3.38)}_{k,\Omega}\ge0$. \item $(G,G')\mapsto\langle\theta(G')G\rangle^{(3.38)}_{k,\Omega}$ is a Hermitian positive semidefinite form on such $G$; hence $\sum_{i,j}\bar\gamma_i\gamma_j\langle\theta(G_i)G_j\rangle^{(3.38)}_{k,\Omega}\ge0$, and by \eqref{eq:s07:matrix} and the support clause of \cref{def:s07:reflection}, $\sum_{i,j}\bar\gamma_i\gamma_jZ^{a,(3.38)}_{k,\Omega}(\theta\zeta_i^*\sqcup\zeta_j)\ge0$ for families $\zeta_i$ with sources supported in $\{x_\nu\ge c+\delta\}$, $\eps_k<2\delta$. \end{enumerate} \end{lemma} \begin{proof} \emph{Step 1: gauge-invariant expectations of the modified form equal those of the unitary form.} The graph $(\Omega,\Omega^*)$ is connected, $\ker d_0$ consists of the constants, $\R^{\Omega^*}=\operatorname{ran}d_0\oplus\ker d_0^T$ orthogonally, and $d_1d_0=0$. Write $A=A^T+d_0\chi$ with $d_0^TA^T=0$ and $\chi\perp1$; $(A^T,\chi)\mapsto A$ is a linear bijection with constant Jacobian. Then $Q^u_\Omega(A)=Q^u_\Omega(A^T)+\frac{\lmu^2\eps^d}2\|d_0\chi\|^2$ (orthogonality and $d_1d_0=0$) and $Q^{(3.38)}_\Omega(A)=Q^u_\Omega(A^T)+q(\chi)$ with $q(\chi)=\frac{\lmu^2\eps^d}2\|d_0\chi\|^2+\frac{\eps^d}2\|d_0^Td_0\chi\|^2$; both $\chi$-forms are positive definite on $1^\perp$ because $\lmu>0$. A gauge transformation is $A\mapsto A+d_0\chi$ together with $\vartheta\mapsto\vartheta-e\chi$: the hopping phase $\vartheta_x-e\eps A_b-e(\chi(y)-\chi(x))-\vartheta_y$ equals $(\vartheta_x+e\chi_x)-e\eps A_b-(\vartheta_y+e\chi_y)$ and the Dirichlet diagonals carry no $A$, so for gauge-invariant $G$ the matter integral $\Psi(A)\defeq\int\prod_xd^2\phi_x\,(Ge^{-S_m})(A,\phi)$ satisfies $\Psi(A^T+d_0\chi)=\Psi(A^T)$ (rotate $\phi$ site by site; $\prod d^2\phi$ is invariant). Hence $\int dA\,e^{-Q^{(3.38)}_\Omega}\Psi=\int dA^Te^{-Q^u_\Omega(A^T)}\Psi(A^T)\int d\chi\,e^{-q(\chi)}$, and the same with $Q^u_\Omega$ in place of $Q^{(3.38)}_\Omega$ and a different finite $\chi$-factor; the factors cancel in the normalised expectation. Fubini applies, since $|G|\le Ce^{a\sum_b|A(b)|}e^{C\sum_xr_x^2}$ for $G=\Theta(\zeta)$ (trivially for bounded $G$) is dominated by the Gaussian and by $e^{-\lambda\eps^d r^4}$. This is the mechanism of \cite[Thm.~5.1, p.~270]{BFS1}, carried to the box graph. \emph{Step 2: the unitary box measure is reflection positive on all of $\mathfrak B_+$.} Let $S^u_\Omega\defeq Q^u_\Omega+S_m$, write $\omega=(\omega_+,\omega_-,a)$ with $a=(a_b)_{b\in B_0}$ the crossing-bond variables, and put $\mathsf M\defeq\lmu^2\eps^d$. The terms of $S^u_\Omega$ are of four kinds. Single-site terms (local matter and Dirichlet diagonals), hopping cosines on bonds of $B_\pm$, masses of bonds of $B_\pm$, and plaquettes with all corners in $\Omega_+$ or all in $\Omega_-$ involve one side only; by $\hat\rho$-invariance the minus-side terms are the $\hat\rho$-transport of the plus-side terms. The crossing masses are $\frac{\mathsf M}2a_b^2$, $b\in B_0$. A crossing plaquette $p=(x^-;\alpha,\nu)$, $\alpha\ne\nu$, is present iff $x^-,x^-+\eps e_\alpha\in\Omega$ (then all four corners are in $\Omega$ by symmetry); with $x^+\defeq x^-+\eps e_\nu$, $b=\langle x^-,x^+\rangle$ and $b'=\langle x^-+\eps e_\alpha,x^++\eps e_\alpha\rangle$ it reads $\eps F(p)=\pm[A_\alpha(x^-)+a_{b'}-A_\alpha(x^+)-a_b]$, the overall sign immaterial after squaring. A crossing hopping term is $-\eps^{d-2}r_{x^-}r_{x^+}\cos(\vartheta_{x^-}-e\eps a_b-\vartheta_{x^+})$, $b\in B_0$. Split each crossing variable: $\int_\R g(a_b)e^{-\frac{\mathsf M}2a_b^2}da_b=\sqrt{2\pi/\mathsf M}\ \E\,g(\alpha_b-\alpha'_b)$ with $\alpha_b,\alpha'_b$ independent $N(0,\frac1{2\mathsf M})$, whose difference is $N(0,\frac1{\mathsf M})$. Put $\xi\defeq(\omega_+,\alpha)$ and $\xi'\defeq(\omega_-^\rho,\alpha')$, where $\omega_-^\rho$ is the $\hat\rho$-image of the minus data transported to the plus side ($(\omega^\rho_-)_\alpha(x^+)=A_\alpha(x^-)$ for the tangential bonds, $\phi^\rho(x^+)=\phi(x^-)$, the $\nu$-bonds of $B_-$ carried with the sign of \cref{def:s07:reflection}); by $\hat\rho$-invariance the minus-side one-sided terms are the plus-side terms evaluated on $\omega_-^\rho$, and the a priori measures of $\xi$ and $\xi'$ coincide. For a crossing plaquette put $X_p(\xi)\defeq A_\alpha(x^+)-\alpha_{b'}+\alpha_b$, a function of plus variables and $\alpha$ (note $A_\alpha(x^+)\in B_+$ since $x^+,x^++\eps e_\alpha\in\Omega_+$), and the same function on $\xi'$, $X_p(\xi')=A_\alpha(x^-)-\alpha'_{b'}+\alpha'_b$; then $\eps F(p)=\pm[X_p(\xi')-X_p(\xi)]$ and \[ \tfrac{\eps^d}2F(p)^2=\tfrac{\eps^{d-2}}2X_p(\xi)^2+\tfrac{\eps^{d-2}}2X_p(\xi')^2-\eps^{d-2}X_p(\xi)X_p(\xi'). \] For a crossing hopping term set $\psi_b(\xi)\defeq\phi(x^+)e^{\mathrm ie\eps\alpha_b}$, $\psi_b(\xi')=\phi(x^-)e^{\mathrm ie\eps\alpha'_b}$; then $r_{x^-}r_{x^+}\cos(\vartheta_{x^-}-e\eps(\alpha_b-\alpha'_b)-\vartheta_{x^+})=\Re[\psi_b(\xi')\overline{\psi_b(\xi)}]=\sum_{i=1,2}\psi^i_b(\xi)\psi^i_b(\xi')$. Hence, with $S'$ the action minus the crossing masses, \[ e^{-S'}=\cG(\xi)\,\cG(\xi')\,\exp\Big(\sum_\iota\gamma_\iota Y_\iota(\xi)Y_\iota(\xi')\Big),\qquad\cG(\xi)\defeq e^{-S_+(\xi)-\frac{\eps^{d-2}}2\sum_pX_p(\xi)^2}, \] where $S_+$ collects the one-sided plus terms and $\{(\gamma_\iota,Y_\iota)\}=\{(\eps^{d-2},X_p)\}_p\cup\{(\eps^{d-2},\psi^i_b)\}_{b,i}$: all $\gamma_\iota>0$, all $Y_\iota$ real. For $G\in\mathfrak B_+$ we have $G(\omega)=G(\xi)$ and $\theta G(\omega)=\overline{G(\xi')}$, the same function on the transported data ($G$ depends neither on $a$ nor on $\alpha,\alpha'$). Expanding the exponential, \[ Z\,\langle\theta(G)G\rangle^u_{k,\Omega}=K\sum_{n\in\N^I}\prod_\iota\frac{\gamma_\iota^{n_\iota}}{n_\iota!}\Big|\int d\xi\,G(\xi)\cG(\xi)\prod_\iota Y_\iota(\xi)^{n_\iota}\Big|^2\ \ge0,\qquad K>0, \] the two integrals of each term being complex conjugates because measures and functions coincide on the two copies. The interchange of sum and integral is justified by $\sum_n\prod_\iota\gamma_\iota^{n_\iota}|Y_\iota Y'_\iota|^{n_\iota}/n_\iota!\le\exp(\frac12\sum_\iota\gamma_\iota(Y_\iota^2+Y_\iota'^2))$: the plaquette parts cancel against $\cG$ exactly, $|\psi_b|=r_{x^+}$, and \[\int|G|e^{-S_+(\xi)+\frac{\eps^{d-2}}2\sum_br_{x^+}^2}d\xi<\infty\] by $\lambda>0$ (a quartic term at every plus site), $\lmu>0$ (a mass on every $B_+$ bond and the $\alpha$-Gaussians) and the growth bound on $G$. The argument is the positive cross-term criterion of \cite{GJ,FILS78}, with the crossing bond variables split into two independent Gaussians and the Dirichlet diagonals as single-site terms. The Hermitian property $\overline{\langle\theta(G')G\rangle}=\langle(G'\circ\hat\rho)\bar G\rangle=\langle G'\,\overline{G\circ\hat\rho}\rangle=\langle\theta(G)G'\rangle$ follows from the $\hat\rho$-invariance of the measure. \emph{Step 3: combination and $L^2$ extension.} For gauge-invariant $G\in\mathfrak B_+$, bounded or equal to $\Theta(\zeta)$, the product $\theta(G)G$ is gauge invariant ($\hat\rho$ maps gauge transformations to gauge transformations: $\hat\rho\,d_0\chi=d_0(\chi\circ\rho)$), so $\langle\theta(G)G\rangle^{(3.38)}_{k,\Omega}=\langle\theta(G)G\rangle^u_{k,\Omega}\ge0$ by Steps 1--2. For gauge-invariant $G\in\mathfrak B_+$ with $\langle|G|^2\rangle<\infty$ (for instance polynomials in the plus-side smeared fields, \cref{prop:s08:poly}), $G_n\defeq G\,\mathbf 1\{|G|\le n\}$ is bounded, gauge invariant and in $\mathfrak B_+$, so $\langle\theta(G_n)G_n\rangle\ge0$; moreover $|\theta(G_n)G_n|\le|\theta G||G|$ and $\langle|\theta G||G|\rangle\le\langle|G|^2\rangle^{1/2}\langle|\theta G|^2\rangle^{1/2}=\langle|G|^2\rangle$ by Cauchy--Schwarz and $\hat\rho$-invariance, so dominated convergence gives (a). The modified measure is itself $\hat\rho$-invariant (\cref{def:s07:box}), so the computation of the Hermitian property in Step 2 holds verbatim for it on $L^2$; the form is sesquilinear, Hermitian and nonnegative on the diagonal, hence positive semidefinite (apply (a) to $G=\sum_i\gamma_iG_i$). With \eqref{eq:s07:matrix} and the support clause this is (b). \end{proof} \begin{remark}[on King's Proposition 3.3] \Cref{lem:s07:RPbox} is the statement of \cite[Prop.~3.3]{K2}, which King derives from \cite[Cor.~5.4]{BFS1}. That corollary (with \cite[Lemma~5.2, Thm.~5.3]{BFS1}) concerns the lattice actions of \cite[\S5]{BFS1}, whose photon Gaussian is translation invariant with a reflection-positive boundary term; King's finite-box Gaussian with its masked divergence is not one of them, the obstruction being the box geometry of the gauge-fixing term. So his Proposition 3.3 is true, and it is proved here rather than derived from \cite[Cor.~5.4]{BFS1}. The gauge-fixing term $\|d_0^TA\|^2$ is not itself compatible with the mechanism of Step 2 (at a site next to the plane it produces cross terms $Y(\xi)Z(\xi')+Z(\xi)Y(\xi')$ of indefinite sign); on the torus it drops out for gauge-invariant observables, on King's cube it does not reduce to a divergence at the boundary (\cref{lem:s07:B}), and this is why the photon form has to be changed before reflecting (\cref{rem:s07:R1}). \end{remark} \subsection{The same box: the boundary identity and the reversed inequality} In this subsection we use lattice units, $\eps=1$ (every term of the three photon forms carries the same weight $\frac12\eps^{d-2}$), and write $\mu\defeq\lmu\eps$. For a lattice box $\Omega$ and $A\in\R^{\Omega^*}$ let $K^{\rm int}_\Omega(A)\defeq\sum_\alpha\sum_\nu\sum|A_\alpha(x+e_\nu)-A_\alpha(x)|^2$, the sum over pairs with both bonds $(x,\alpha),(x+e_\nu,\alpha)$ in $\Omega^*$ (the componentwise Neumann gradient form on the internal bonds), and $N_\partial(A)\defeq\sum|A_b|^2$, the sum over the normal crossing pairs: for $b=(x,\alpha)\in\Omega^*$ one term for each of $(x\pm e_\alpha,\alpha)$ that is not in $\Omega^*$. Split $N_\partial=N^{\rm up}_\partial+N^{\rm low}_\partial$ according to whether the missing partner is $(x+e_\alpha,\alpha)$ (\ie $x+2e_\alpha\notin\Omega$) or $(x-e_\alpha,\alpha)$ (\ie $x-e_\alpha\notin\Omega$). \begin{lemma}[the boundary identity]\label{lem:s07:B} For every lattice box $\Omega$ and every $A\in\R^{\Omega^*}$, \[ \|d_1A\|^2_{\Omega^{**}}+\|d_0^TA\|^2=K^{\rm int}_\Omega(A)+N_\partial(A). \] Consequently $Q^{(3.38)}_\Omega=\frac12\big[K^{\rm int}_\Omega+N_\partial+\mu^2\|\cdot\|^2_{\Omega^*}\big]$: the modified photon form is the internal Neumann--Feynman form plus the positive boundary term $N_\partial$. (In the notation of the proof of \cref{lem:s07:R}(a), the marginal of the Neumann--Dirichlet Gaussian of $\Omega$ on the internal bonds has the precision $M^{(3.38)}-N^{\rm low}_\partial-P_{12}P_{22}^{-1}P_{21}$: the modified form minus the squares of the near-normal bonds and minus the Schur term of the far-face variables.) \end{lemma} \begin{proof} Extend $A$ by $0$ to a $1$-form $a$ on $\Z^d$. For finitely supported $1$-forms, $\sum_p|da(p)|^2+\sum_x|d^*a(x)|^2=\sum_\alpha\sum_\nu\sum_x|a_\alpha(x+e_\nu)-a_\alpha(x)|^2$ (the Hodge Laplacian on $\Z^d$ is the componentwise Laplacian; summation by parts). On the left, $d^*a(x)$ is the masked divergence for $x\in\Omega$ and $0$ outside, since no internal bond has an end outside $\Omega$; a plaquette $p\notin\Omega^{**}$ meeting $\Omega^*$ has exactly one internal bond $b$ (two parallel internal edges put all four corners in $\Omega$; two adjacent ones put three corners in $\Omega$, and a product of intervals then contains the fourth), so $|da(p)|^2=|A_b|^2$. On the right, pairs with both bonds internal give $K^{\rm int}_\Omega$; a pair $(x,\alpha),(x+e_\nu,\alpha)$ with exactly one internal bond contributes $|A_b|^2$; for $\nu\ne\alpha$ the partner is non-internal iff $x+e_\nu\notin\Omega$ (as $x,x+e_\alpha\in\Omega$, and in a box $x+e_\nu\in\Omega\iff x+e_\alpha+e_\nu\in\Omega$), \ie iff the plaquette spanned by $b$ and $e_\nu$ lies outside $\Omega^{**}$; these pairs are in bijection with the plaquettes counted on the left and cancel; for $\nu=\alpha$ they are the terms of $N_\partial$. \end{proof} \begin{lemma}[the same box: the second inequality of King's (3.42) is reversed]\label{lem:s07:R} Let $\Omega$ be a lattice box, $\langle\cdot\rangle^{\ND}_\Omega$ the Neumann--Dirichlet state on $X=\Omega$ (variables $A_\alpha(x)$, $x\in\Omega$, all outgoing bonds; photon form $Q^{\ND}_\Omega=\frac12[K_\Omega+\mu^2\|\cdot\|^2]$ of \cref{def:s02:acube}) and $\langle\cdot\rangle^{(3.38)}_\Omega$ the modified box state of \cref{def:s07:box} (variables $\Omega^*$); both carry the same Dirichlet matter on $\Omega$ (constant $\dm_\eps$, free Wick constant), coupled to $A$ through the internal bonds only. Let $\Sigma^{\ND}_\Omega$ and $\Sigma^u_\Omega$ be the covariances of $F=(d_1A(p))_{p\in\Omega^{**}}$ under the two photon Gaussians. Then: \begin{enumerate}[label=\textup{(\alph*)},nosep] \item $\Sigma^u_\Omega=\Sigma^{(3.38)}_\Omega\le\Sigma^{\ND}_\Omega$ on $\R^{\Omega^{**}}$; \item for every gauge-invariant $Y\in\cI$ with sources in $\Omega$ and every $e\ge0$: $\langle Y\rangle^{\ND}_\Omega\le\langle Y\rangle^{(3.38)}_\Omega$, and $\langle X\rangle^{\ND}_\Omega\ge\langle X\rangle^{(3.38)}_\Omega$ for $X\in\cD$. This is the reverse of the second inequality of \cite[(3.42)]{K2}, whose middle term is the modified box state on $C_{J+1}$; it holds weakly, for the modified state with the constant $\dm_\eps$ (nothing is claimed for the site-dependent counterterm of \cite{K2}); \item King's inequality fails strictly in the Gaussian sector: at $e=0$ and $Y=e^{\mathrm it\langle v,F\rangle}$ it reads $v^T\Sigma^{\ND}_\Omega v\le v^T\Sigma^u_\Omega v$, while $\Delta_\Omega\defeq\Sigma^{\ND}_\Omega-\Sigma^u_\Omega\ge0$ is nonzero. In exact rational arithmetic the inertia (negative, zero, positive) of $\Delta_\Omega$ is $(0,1,8)$ on the $4\times4$ box ($d=2$) for $\mu^2\in\{\frac1{20},\frac14,1\}$ and $(0,3,13)$ on the $4\times2\times2$ box ($d=3$) for $\mu^2\in\{\frac1{20},\frac14\}$; single-plaquette certificates are $\Delta_{pp}=\Var^{\ND}F(p)-\Var^{(3.38)}F(p)=\frac{14285248}{44314767}$ at $p=((0,0);0,1)$ on $4\times4$, $\mu^2=\frac14$, and $\frac{121897601600}{395580222961}$ at $p=((0,0,0);1,2)$ on $4\times2\times2$, $\mu^2=\frac1{20}$. \end{enumerate} \end{lemma} \begin{proof} (a) Write the Neumann--Dirichlet precision $P\defeq K_\Omega+\mu^2$ in blocks, the internal variables $\Omega^*$ first and the outgoing boundary variables second. Its internal block is $P_{11}=K^{\rm int}_\Omega+N^{\rm up}_\partial+\mu^2$: pairs of $K_\Omega$ with both bonds internal give $K^{\rm int}_\Omega$; a pair $(x,\alpha),(x+e_\nu,\alpha)$ with exactly one internal bond is impossible for $\nu\ne\alpha$ (box convexity, as in \cref{lem:s07:B}) and for $\nu=\alpha$ is $(x,\alpha)$ internal with $(x+e_\alpha,\alpha)$ outgoing, \ie $x+2e_\alpha\notin\Omega$ (the other assignment would need $x+e_\alpha\notin\Omega$ and $x+e_\alpha\in\Omega$ at once); its diagonal contribution to the internal block is $|A_\alpha(x)|^2$, which is $N^{\rm up}_\partial$. Hence, by \cref{lem:s07:B}, $M^{(3.38)}\defeq K^{\rm int}_\Omega+N_\partial+\mu^2=P_{11}+N^{\rm low}_\partial\ge P_{11}$. The marginal covariance of the internal variables under the Neumann--Dirichlet Gaussian is the inverse of the Schur complement, $(P^{-1})_{11}=(P_{11}-P_{12}P_{22}^{-1}P_{21})^{-1}\ge P_{11}^{-1}\ge(M^{(3.38)})^{-1}$, by operator monotonicity of the inverse on positive definite matrices and $P_{12}P_{22}^{-1}P_{21}\ge0$. Since $F=d_1A$ on $\Omega^{**}$ uses internal bonds only, $\Sigma^{\ND}_\Omega=D_1(P^{-1})_{11}D_1^T\ge D_1(M^{(3.38)})^{-1}D_1^T=\Sigma^{(3.38)}_\Omega$, and $\Sigma^{(3.38)}_\Omega=\Sigma^u_\Omega$ by Step 0 in the proof of \cref{lem:s07:U} (the field strength depends only on the $\ker d_0^T$-component, whose law is the same under both Gaussians, Step 1 of \cref{lem:s07:RPbox}). (b) Both states carry the Dirichlet matter of $\Omega$ coupled to $\Omega^*$, and their field-strength laws on $\Omega^{**}$ are ordered by (a); so the argument of the proof of \cref{lem:s07:U}(b) after its removal step (gauge invariance, the realisation of the field-strength law, the $\delta$-regularisation and \cref{lem:s02:BFS}(a),(d)) applies verbatim, with $\Sigma^{(3.38)}_\Omega$ and $\Sigma^{\ND}_\Omega$ in the roles of $\Sigma^{\ND}_B$ and $\Sigma^u_\Omega$. (c) At $e=0$ the matter decouples from $A$ and is identical in the two states, so for $Y=e^{\mathrm it\langle v,F\rangle}$ both sides equal $\exp(-\frac{t^2}2v^T\Sigma v)$ with the respective $\Sigma$; King's $\le$ becomes $v^T\Sigma^{\ND}_\Omega v\le v^T\Sigma^u_\Omega v$, which is false for every $v$ with $v^T\Delta_\Omega v>0$, for instance $v=e_p$ at the plaquettes listed. The inertias and the certificates are obtained from the two covariance matrices in exact rational arithmetic (symmetric elimination with diagonal pivoting over $\mathbb Q$). \end{proof} \emph{Scope.} (i) King's passage from the box states to the cube limit on \cite[p.~336]{K2} needs the sandwich for the box functionals with purely imaginary curvature coefficients, \ie for the whole class $\cI$; (c) refutes the second inequality on that class exactly. (ii) For the scalar observable $\Theta(0,2h_+)$ of the display in \cite{K2}, (b) shows that the inequality can hold only with equality; at $e=0$ it does (identical decoupled matter), and at $e>0$ it is not decided here whether the equality fails. Both clauses are for the modified state with the constant $\dm_\eps$: if King's site-dependent counterterm differs from the free-boundary number, the two matter measures differ, the mechanism of (b) does not apply, and neither clause is proved for the objects of \cite{K2}; (c) alone is independent of the counterterm, the matter decoupling from $A$ at $e=0$. \begin{remark}[what is false in King's route]\label{rem:s07:R1} (i) King's Neumann--Dirichlet cube state is not reflection symmetric about any plane: the site set $C_J\cap\Lk$ is symmetric in direction $\nu$ about $x_\nu=-\eps_k/2$, the set of $\nu$-bonds about $x_\nu=0$, and no reflection maps the variable set to itself (\cref{lem:s02:refl} accordingly obtains the reflection symmetry of the limit from a cut sandwich, not from a lattice symmetry). (ii) The natural symmetrisation, the Feynman gradient form on the internal bonds $K^{\rm int}_\Omega+\mu^2$ (the modified form without $N_\partial$; reflection symmetric), is \emph{not} reflection positive on gauge-invariant functionals. This is exact at $e=0$: on the $4\times4$ box at $\mu^2=\frac1{20}$ the reflection kernel $R_{pq}=\sigma_p\Sigma[\rho p,q]$ of the plus-side plaquettes is negative definite (inertia $(3,0,0)$), and on $4\times2\times2$ at $\mu^2=\frac1{20}$ it has inertia $(2,2,2)$ with $R_{pp}=\Cov(\theta F(p),F(p))<0$ for $p=((2,0,0);0,1)$. Indeed, at $e=0$ the matter decouples, and the gauge-invariant plus-side functionals $G_1=e^{\mathrm it\langle v,F_+\rangle}$, $G_2=1$ give the matrix $(\langle\theta(G_i)G_j\rangle)$ with determinant $e^{-t^2v^T\Sigma_{++}v}(e^{t^2v^TRv}-1)<0$ for every $t>0$ when $v^TRv<0$, since $\langle\theta(G_1)G_1\rangle=\exp(-t^2v^T\Sigma_{++}v+t^2v^TRv)$ and $\langle\theta(G_1)\rangle=\langle G_1\rangle=\exp(-\frac12t^2v^T\Sigma_{++}v)$ by the reflection symmetry $\Sigma_{--}=\Sigma_{++}$. By continuity at fixed spacing the symmetrised cube state is not reflection positive for small $e>0$ either. A proof must work at every coupling, so reflection positivity of King's own cube form, or of this symmetrisation, at fixed spacing is not available as a route: any route through cube states at fixed spacing has to change the photon form, as the modified box does (a reflection-symmetric, reflection-positive modification whose values converge to the same limit, \cref{thm:s08:OS2C}). Nothing is claimed about the symmetrised form at a given larger $e$ or about other proofs; the route taken here needs the comparison of \cref{lem:s07:L,lem:s07:U}, and its reflecting plane must be a lattice mid-plane for the half-bond mechanism of \cref{lem:s07:RPbox} (King's cubes have an even number of layers). (iii) Accordingly, the second inequality of \cite[(3.42)]{K2} is false as stated there, in the following sense: false for the class of observables that the argument on \cite[p.~336]{K2} needs (exactly, \cref{lem:s07:R}(c)), and never strictly true for the modified state with the constant counterterm (\cref{lem:s07:R}(b)). Its valid first half is \cref{lem:s07:L}; the invalid half is replaced by \cref{lem:s07:U} with the margin $R(\lmu)$ of \eqref{eq:s07:margin}, $n(\lmu,c)$ cube steps instead of one. \end{remark} \begin{remark}[the box form is the larger one]\label{rem:s07:K8} King describes the modified box action as obtained from the cube action by omitting some terms. By \cref{lem:s07:B} the relation is the reverse: on one box the modified photon form is the internal Neumann--Dirichlet form $P_{11}$ plus the positive term $N^{\rm low}_\partial$ and without the outgoing boundary variables, so the modified state is the one with the \emph{larger} form, the smaller field-strength covariance, and the larger $\cI$-expectations: $\langle Y\rangle^{\ND}_{C}\le\langle Y\rangle^{(3.38)}_{C}$ on the cone for every box $C$ (\cref{lem:s07:R}(b)), the direction that \cref{lem:s02:cut} requires for consistency with \cref{lem:s07:L,lem:s07:U}. For King's convention the same comparison holds with Step~1 of \cref{lem:s04:G} prepended (\cref{rem:s04:F}): $\avK{Y}\le\ava{Y}\le\langle Y\rangle^{(3.38)}_{k,\XJ}$ on $\cI$. \end{remark} \subsection{The sandwich} \begin{lemma}[the lower half]\label{lem:s07:L} Let $C\subset\Omega$ be lattice boxes with $C$ plus one layer contained in $\Omega$, so that the variables $A_\mu(x)$, $x\in C$, of the Neumann--Dirichlet state on $C$ (the outgoing bonds) are internal bonds of $\Omega$. Then \begin{enumerate}[label=\textup{(\alph*)},nosep] \item $Q^{(3.38)}_\Omega\ge Q^{\ND}_C\oplus\frac{\mu^2}2I_{\rm rest}$ as quadratic forms on $\R^{\Omega^*}=\R^{C\text{-variables}}\oplus\R^{\rm rest}$; \item for sources such that every plaquette and every site at which they do not vanish lies in $C$, \[ \langle Y\rangle^{\ND}_C\le\langle Y\rangle^{(3.38)}_\Omega\ \ (Y\in\cI),\qquad\langle X\rangle^{\ND}_C\ge\langle X\rangle^{(3.38)}_\Omega\ \ (X\in\cD), \] and the same for $Z^a$ (equal positive multipliers, \cref{lem:s02:alg}(a)). The $\cD$ half uses \cref{lem:s02:BFS}(d); the $\cI$ half only (a), (b), (e). \end{enumerate} \end{lemma} \begin{proof} (a) $K^{\rm int}_\Omega$ contains every gradient pair of $K_C$ (both base points in $C$, both bonds internal to $\Omega$); the remaining pairs and $N_\partial$ are nonnegative; the mass term splits. (b) Start from $\langle\cdot\rangle^{(3.38)}_\Omega$ and remove the hopping cosines across $\partial C$ and all cosines not internal to $C$; by \cref{lem:s02:BFS}(b),(d) $\langle Y\rangle$ does not increase and $\langle X\rangle$ does not decrease, every intermediate measure being a BFS measure. The matter is now the Dirichlet matter of $C$: the diagonal parts $\frac12r^2$ of the cut bonds stay and are exactly the Dirichlet terms, as in the proof of \cref{lem:s02:cut}; it is coupled to $A$ only through bonds internal to $C$, which are $C$-variables, and the matter of $\Omega\setminus C$ is decoupled and cancels (\cref{lem:s02:BFS}(e)). Then pass from the form $Q^{(3.38)}_\Omega$ to the smaller form $Q^{\ND}_C\oplus\frac{\mu^2}2I_{\rm rest}$: by (a) the difference is positive semidefinite, a finite sum $\sum_is_i|\ell_i(A)|^2$ with $s_i\ge0$, and adding each term does not decrease $\langle Y\rangle$ and does not increase $\langle X\rangle$ (\cref{lem:s02:BFS}(a),(d)). So the state with the larger form, the modified one, has the larger $\langle Y\rangle$ and the smaller $\langle X\rangle$. The state with the smaller form factorises as $\langle\cdot\rangle^{\ND}_C$ times an independent Gaussian on the rest, which cancels. \end{proof} For King's convention the lower end is used in the form $\avK{Y}\le\ava{Y}\le\langle Y\rangle^{(3.38)}_{\Omega}$ on $\cI$ (Step~1 of \cref{lem:s04:G}, then \cref{lem:s07:L}; \cref{rem:s04:F}), and the upper end needs no change (\cref{lem:s04:D}). \begin{lemma}[the upper half, with a margin]\label{lem:s07:U} Let $\Omega\subset B$ be lattice boxes at spacing $\eps$ such that $B$ contains every site within $\ell^\infty$-distance $N$ lattice steps of $\Omega$, and let $\mu=\lmu\eps$ satisfy $d(\pi/2N)^2\le\mu^2$, \ie \[ N\eps\ \ge\ \frac{\pi\sqrt d}{2\lmu}\qquad\big(=R(\lmu)=\tfrac{\pi\sqrt3}{2\lmu}\text{ in }d=3\big). \] Let $\Sigma_X$ denote the covariance of $(\eps F(p))_{p\in\Omega^{**}}$ under the photon Gaussian of the state on $X$. Then \begin{enumerate}[label=\textup{(\alph*)},nosep] \item $\Sigma^{\ND}_B\le\Sigma^{(3.38)}_\Omega=\Sigma^u_\Omega$ on $\R^{\Omega^{**}}$; \item for sources supported in $\Omega$ as in \cref{lem:s07:L}, \[ \langle Y\rangle^{(3.38)}_\Omega\le\langle Y\rangle^{\ND}_B\ \ (Y\in\cI),\qquad\langle X\rangle^{(3.38)}_\Omega\ge\langle X\rangle^{\ND}_B\ \ (X\in\cD), \] and the same for $Z^a$. Only the $\cI$ half is used in the sequel. \end{enumerate} \end{lemma} \begin{proof} \emph{(a), in lattice units.} Step 0: $\Sigma^{(3.38)}_\Omega=\Sigma^u_\Omega$, because $d_1A=d_1A^T$ depends only on the $\ker d_0^T$-component, whose law $\propto e^{-Q^u_\Omega(A^T)}$ is the same under both Gaussians (Step 1 of \cref{lem:s07:RPbox}). Step 1 (variational identity): for a Gaussian with density $\propto e^{-\frac12A^TM_XA}$, $\langle g,\Sigma_Xg\rangle=\Var\langle g,d_1A\rangle=\sup_A[2\langle g,d_1A\rangle-A^TM_XA]$, the supremum attained at $A=M_X^{-1}d_1^Tg$. So it suffices to show that for every $A$ on the $B$-variables there is $A'\in\R^{\Omega^*}$ with $d_1A'=d_1A$ on $\Omega^{**}$ and $\|d_1A'\|^2_{\Omega^{**}}+\mu^2\|A'\|^2_{\Omega^*}\le K_B(A)+\mu^2\|A\|^2_B$; take $A'\defeq A|_{\Omega^*}$ (the plaquettes of $\Omega^{**}$ use only $\Omega^*$-bonds). Step 2 (cutoff): let $\psi(x)=\prod_\nu\phi_\nu(x_\nu)$ with $\phi_\nu=1$ on the $\nu$-range of $\Omega$ and on the first layer outside, $\phi_\nu(t)=\cos(\pi(t-1)/2N)$ at the $t$-th layer outside for $1\le t\le N+1$ (so $\phi_\nu(N+1)=0$), and $0$ beyond; then $0\le\psi\le1$, $\psi=1$ on $\Omega$ and its neighbours, $\supp\psi\subset B$. Put $a\defeq\psi A$ on $\Z^d$ (the bond $\langle x,x+e_\mu\rangle$ weighted by $\psi(x)$; $A\defeq0$ off the $B$-variables). Step 3 (Hodge on $\Z^d$): $\|d_1A\|^2_{\Omega^{**}}=\|d_1A'\|^2_{\Omega^{**}}\le\sum_p|da(p)|^2\le\sum_p|da|^2+\sum_x|d^*a|^2=\sum_\mu\sum_\nu\sum_x|a_\mu(x+e_\nu)-a_\mu(x)|^2$, since the bonds of plaquettes in $\Omega^{**}$ have base points in $\Omega$, where $\psi=1$. Step 4 (discrete IMS identity): for finitely supported $\psi f$ and a direction $e$, \[ \begin{split} \sum_x|\psi(x+e)f(x+e)-\psi(x)f(x)|^2&=\sum_x\psi(x)\psi(x+e)|f(x+e)-f(x)|^2\\ &\quad+\sum_x\psi(x)\big(2\psi(x)-\psi(x+e)-\psi(x-e)\big)f(x)^2 , \end{split} \] exactly (per bond the difference of the two sides is $\psi'(\psi'-\psi)f'^2+\psi(\psi-\psi')f^2$; collect by site). With $f=A_\mu$ and $e=e_\nu$: $\psi\psi'\le1$, and $\psi\psi'>0$ only for pairs with both base points in $B$, which are gradient pairs of $K_B$; so the first sums, over $\mu,\nu$, total at most $K_B(A)$. The potential is $V(x)=\sum_\nu\big(\prod_{\kappa\ne\nu}\phi_\kappa^2\big)\phi_\nu(-\Delta\phi_\nu)(x_\nu)$. Step 5 (the profile): with $\varpi\defeq\pi/2N$, $\phi(-\Delta\phi)=0$ on the range of $\Omega$ (all neighbours equal $1$), $=1-\cos\varpi\le\varpi^2/2$ at $t=1$, $=2\cos^2(\varpi(t-1))(1-\cos\varpi)\le\varpi^2$ for $2\le t\le N$ (using $\phi(N+1)=\cos(\varpi N)=0$ at $t=N$), and $=0$ for $t\ge N+1$; hence $V=0$ on $\Omega$ and $V\le d\varpi^2$ everywhere. Step 6: $\|d_1A'\|^2_{\Omega^{**}}\le K_B(A)+d\varpi^2\sum_{x\in B\setminus\Omega}|A(x)|^2$; the variables with base point in $B\setminus\Omega$ and those of $\Omega^*$ (base point in $\Omega$) are disjoint subsets of the $B$-variables, so $\|d_1A'\|^2_{\Omega^{**}}+\mu^2\|A'\|^2_{\Omega^*}\le K_B(A)+\mu^2\|A\|^2_B$ as soon as $d\varpi^2\le\mu^2$. \emph{(b).} Start from $\langle\cdot\rangle^{\ND}_B$ and remove all hopping cosines touching $B\setminus\Omega$ (\cref{lem:s02:BFS}(b): $\cI$ does not increase; (d): $\cD$ does not decrease). The matter is now the Dirichlet matter of $\Omega$ (the diagonals of the bonds leaving $\Omega$ kept; every such bond ends in $B$ since $N\ge1$), coupled to $A$ on $\Omega^*$, and the sites of $B\setminus\Omega$ decouple. This matter is identical to that of the modified box state: the same cosines on $\Omega^*$, the same diagonals $\frac12r_x^2$ for every bond leaving $\Omega$, the same single-site weights with the constant $\dm_\eps$ and the same Wick constant. For a gauge-invariant observable with sources in $\Omega$ the matter integral at fixed $A$, $\Psi_Y(A)=\int\prod_{x\in\Omega}d^2\phi_x\,Y\,e^{-S_m}$, is invariant under $A\mapsto A+d_0\chi$, $\chi\in\R^\Omega$ (as in Step 1 of \cref{lem:s07:RPbox}), hence a function of the class of $A|_{\Omega^*}$ modulo $\operatorname{ran}d_0=\ker d_1$ (a box has $H^1=0$: the cycle space of a grid graph is generated by its $4$-cycles), hence a function $\tilde\Psi_Y(d_1A|_{\Omega^{**}})$ of the field strengths on $\Omega^{**}$. So $\langle Y\rangle=\E[\tilde\Psi_Y(F)]/\E[\tilde\Psi_1(F)]$ depends on the photon law only through the centred Gaussian law of $F=(d_1A(p))_{p\in\Omega^{**}}$, with covariance $\Sigma^{\ND}_B$ on the Neumann--Dirichlet side after the removal and $\Sigma^u_\Omega$ on the modified side (Step 0). Both laws live on $\operatorname{ran}d_1$. Let $T:\R^{\Omega^{**}}\to\R^{\Omega^*}$ be a linear map with $d_1T=\mathrm{id}$ on $\operatorname{ran}d_1$, extended linearly to all of $\R^{\Omega^{**}}$ (any extension serves, both laws being supported on $\operatorname{ran}d_1$), and realise each law as the photon Gaussian of $A''\defeq TF\in\R^{\Omega^*}$, covariance $T\Sigma T^T$: then $d_1A''=F$ almost surely, the gauge-invariant expectations are reproduced, and $T\Sigma^{\ND}_BT^T\le T\Sigma^u_\Omega T^T$ by (a). These covariances are degenerate; regularise to $T\Sigma T^T+\delta I$, $\delta>0$, which are non-degenerate, so the regularised states are BFS measures, the ordering of the covariances is preserved, and the law of $F''=d_1A''$ is $N(0,\Sigma+\delta d_1d_1^T)$, still ordered. \Cref{lem:s02:BFS}(a) by interpolation (the difference of the precision matrices is positive semidefinite by operator monotonicity of the inverse, hence a finite sum of terms $s|\ell(A)|^2$, $s\ge0$) gives $\langle Y\rangle_{\Sigma^u+\delta}\le\langle Y\rangle_{\Sigma^{\ND}+\delta}$ for $Y\in\cI$, and the reverse for $\cD$ by \cref{lem:s02:BFS}(d). \emph{The limit $\delta\downarrow0$.} For the $\cI$ half, $\tilde\Psi_Y$ is bounded and continuous: $|\Psi_Y(A)|\le\int e^{r^2(g)}e^{\sum\eps^{d-2}r_xr_y}\prod_xd\rho_x<\infty$ independently of $A$ ($|e^{\mathrm iF}|=1$; the hopping bound $e^{\eps^{d-2}r_xr_y}\le e^{\frac12\eps^{d-2}(r_x^2+r_y^2)}$ cancels the diagonals and $\lambda>0$ does the rest), and so is $\tilde\Psi_1$, with $\tilde\Psi_1>0$; the Gaussians $N(0,\Sigma+\delta d_1d_1^T)$ converge weakly to $N(0,\Sigma)$, so numerator and denominator converge and the inequality passes to the limit. For the $\cD$ half (not used below), $X=\Theta(f,-g)$ gives $\tilde\Psi_X(F)=e^{\langle v,F\rangle}\tilde\Psi^0_X(F)$ with $v$ the plaquette vector of $f$ and $0<\tilde\Psi^0_X\le e^{2\sW_\eps\sS(g)}\tilde\Psi_1\le C$ bounded and continuous, so $\tilde\Psi_X$ is continuous but unbounded; under $N(0,\Sigma+\delta d_1d_1^T)$ one has $\E[e^{2\langle v,F''\rangle}]=e^{2v^T(\Sigma+\delta d_1d_1^T)v}\le e^{2v^T\Sigma v+2\|d_1^Tv\|^2}$ for $\delta\le1$, so $\{\tilde\Psi_X(F'')\}_{0<\delta\le1}$ is bounded in $L^2$, hence uniformly integrable, and weak convergence with uniform integrability and continuity gives $\E[\tilde\Psi_X(F'')]\to\E[\tilde\Psi_X(F)]$. Altogether $\langle Y\rangle^{(3.38)}_\Omega\le\langle Y\rangle^{\text{after removal}}\le\langle Y\rangle^{\ND}_B$, and the reverse for $\cD$. \end{proof} \begin{remark}[why a margin, and why a fixed photon mass] Part (a) of \cref{lem:s07:U} fails without a margin: for $B=\Omega$ the comparison is reversed (\cref{lem:s07:R}). No fixed margin works at $\lmu=0$: there $\Sigma^u_\Omega$ is the projector onto exact $2$-forms, while $\Var_B\langle g,dA\rangle>\|g\|^2$ for a Coulomb potential whose gradient leaks out of $B$. So \cref{lem:s07:U} is a statement at fixed $\lmu>0$ with $R(\lmu)\to\infty$ as $\lmu\to0$; a limit of vanishing photon mass inherits reflection positivity, if it exists, as a pointwise limit of positive semidefinite matrices. On one box the three states are ordered on $\cI$ as $\langle Y\rangle^{\ND}_\Omega\le\langle Y\rangle^{(3.38)}_\Omega\le\langle Y\rangle^{\ND}_B$ (\cref{lem:s07:R,lem:s07:U}), as the monotonicity of \cref{lem:s02:cut} requires. The sandwich with the boxes \eqref{eq:s07:margin} is assembled in the proof of \cref{thm:s08:OS2C}; for King's convention its lower end carries the kill step of \cref{rem:s04:F} and its upper end is unchanged (\cref{lem:s04:D}). \end{remark} \space% % % % \section{Reflection positivity II: the limit, \texorpdfstring{$\OSth$}{OS3} and the polynomial form}\label{sec:s08} The lattice statements of the previous section (\cref{sec:s07}) are now carried to the limits $k\to\infty$ and $J\to\infty$. The scheme of proof is the same for the curvature sector and for the Wick tube: uniform local bounds for the modified-box functionals at complex coefficients (\cref{lem:s08:claim4}), Montel's theorem in $k$ at fixed $J$, the two-sided sandwich of \cref{lem:s07:L,lem:s07:U} on the cone as $J\to\infty$ with the identity theorem off it, and a translation for the planes that are not mid-planes. The statement specific to each sector is the convergence of the cube functionals themselves: the existence theorem \cref{hyp:s03:EC} in the curvature sector, and on the Wick tube the positive-leg bound of \cref{cor:s11:W2prime} together with the existence theorem with scalar sources \cref{hyp:s03:W1} (\cref{prop:s08:T}). All statements are for King's state, the infinite-volume state of King's construction \cite{K1,K2}; the two places where the cube convention is seen are the ends of the sandwich \eqref{eq:s08:sandwich} (\cref{lem:s04:D}(c), \cref{rem:s04:F}), and \cref{rem:s08:comparison} states what the arguments give for the comparison family. What is proved of the Wick sector is its tube form; reflection positivity at arbitrary complex Wick coefficients is not treated, the fixed-cube limits and a uniform positive-source bound above the amplitude threshold of \cref{hyp:s03:W1} being open (\cref{sec:s23}). \subsection{The enlarged family} \begin{definition}[the enlarged family, its functionals and its matrix points]\label{def:s08:enlarged} Fix $\nu\in\{1,2,3\}$, a mid-plane $c\in D$ (\cref{def:s07:reflection}), and finite families $\zeta_1,\dots,\zeta_N$ of real sources $f_{il}\in C^\infty_c(\R^3;\Lambda^2)$, $h_{il}\in C^\infty_c(\R^3)$ supported in $\{x_\nu\ge c+\delta\}$ for some $\delta>0$, with complex coefficients $z_{il},w_{il}$. Split each scalar source as $h_{il}=h^+_{il}-h^-_{il}$ with $h^-_{il}\defeq\norm{h_{il}}_\infty\chi_{il}$, where $\chi_{il}\in C^\infty_c$, $0\le\chi_{il}\le1$, $\chi_{il}=1$ on $\supp h_{il}$ and $\supp\chi_{il}\subset\{x_\nu\ge c+\delta/2\}$, and $h^+_{il}\defeq h_{il}+h^-_{il}\ge0$. The \emph{enlarged family} is the family of real sources \[ \cF\defeq\{\rho^*f_{il}\}\cup\{f_{il}\}\cup\{h^\pm_{il}\circ\rho\}\cup\{h^\pm_{il}\},\qquad\text{coefficient space }\C^m,\ m=m_F+m_h , \] whose scalar members $\tilde h_j\ge0$ are all non-negative. For a real coefficient vector $\xi\in\R^m$ the total source is $(f_\xi,k_1(\xi)-k_2(\xi))$ with $k_1(\xi)\defeq\sum_j(\xi_j)_+\tilde h_j$, $k_2(\xi)\defeq\sum_j(\xi_j)_-\tilde h_j$, both smooth and non-negative ($x_\pm\defeq\max(\pm x,0)$). Let $J_0$ be the least $J$ such that every member of $\cF$ is supported at $\ell^\infty$-distance $>1$ from $\R^3\setminus C_J$; for $J\ge J_0$, $n=n(\lmu,c)$ and the boxes $\Omega_{J,k}$ of \eqref{eq:s07:margin} put \[ \Phi_k(\zeta)\defeq Z^{a,(3.38)}_{k,\Omega_{J,k}}(\zeta),\qquad\zeta\in\C^m , \] an entire function (a finite-dimensional integral with Gaussian decay in $A$ and $e^{-\lambda\eps^3r^4}$ decay in $r$). The \emph{matrix points} $\zeta^*_{ij}\in\C^m$ carry $\bar z_{il}$ on $\rho^*f_{il}$, $z_{jl}$ on $f_{jl}$, $\pm\bar w_{il}$ on $h^\pm_{il}\circ\rho$, $\pm w_{jl}$ on $h^\pm_{jl}$ and $0$ elsewhere; their total source is that of $\theta\zeta_i^*\sqcup\zeta_j$, so $\Phi_k(\zeta^*_{ij})$ is the $(i,j)$ entry of \cref{lem:s07:RPbox}(b). Since reflected and unreflected members have disjoint supports, $\norm{h^+}_\infty\le2\norm h_\infty$ and $\norm{h^-}_\infty=\norm h_\infty$, \begin{equation}\label{eq:s08:ampbound} \norm{k_1(\Re\zeta^*_{ij})}_\infty\ \le\ 2\max\big(\amp(\zeta_i),\amp(\zeta_j)\big). \end{equation} Every lattice functional, and hence every limit below, depends on a family only through its total sources; the splitting is therefore immaterial at the matrix points and at the Taylor coefficients at $0$. \end{definition} \begin{definition}[the Wick tube, the bounded domain and the uniqueness set]\label{r13:def:s08:domains} Let $\zeta_1,\dots,\zeta_N$, $\cF$, $\Phi_k$ and the matrix points $\zeta^*_{ij}$ be as in \cref{def:s08:enlarged}. \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*,nosep] \item The \emph{Wick tube} of $\cF$ is $T_\cF\defeq\{\zeta\in\C^m:\ 2\norm{k_1(\Re\zeta)}_\infty<\sigma_+/2\}$, with $\sigma_+=s_1$ the amplitude of \Itwo\ (\cref{def:s10:I2}, \eqref{eq:s11:constants}). It is open, convex ($\xi\mapsto\norm{k_1(\xi)}_\infty$ is a supremum of convex functions), connected, invariant under $z\mapsto-z$ on the curvature coordinates, contains $0$, and by \eqref{eq:s08:ampbound} contains every $\zeta^*_{ij}$ as soon as $\amp(\zeta_i)<\sigma_+/8=:\sigma_W$ for every $i$. \item The \emph{bounded domain}: let $B_1\subset\R^m$ be the open convex hull of the unit $\ell^\infty$-balls about $0$ and about the real parts $\Re\zeta^*_{ij}$; let $\beta_j\defeq2\sup_{\xi\in B_1}|\xi_j|$ for the scalar coordinates (so $\beta_j=2(|\Re w_{il}|+1)$ for the coordinate of $h^\pm_{il}$ or of $h^\pm_{il}\circ\rho$), $B''\defeq\prod_j[0,\beta_j]$ over the scalar coordinates, and $q_v\defeq\sum_jv_j\tilde h_j$ for $v\in\vertex(B'')$, the \emph{vertex sources}. All $q_v$ are pointwise $\le q_\zeta\defeq\sum_{i,l}2(|\Re w_{il}|+1)(h^+_{il}+h^-_{il}+h^+_{il}\circ\rho+h^-_{il}\circ\rho)$. Put $\cT_1\defeq\{\zeta\in\C^m:\Re\zeta\in B_1\}$: open, convex, connected, containing every $\zeta^*_{ij}$ and the unit polydisc about $0$, so that $\cT_1\cap T_\cF$ is a non-empty open set. For $\xi\in B_1$ both $2\xi_+$ and $\xi_+$ lie in $B''$. \item The \emph{uniqueness set} $U\subset\C^m$ is the set of coefficient vectors whose curvature coefficients are purely imaginary and whose scalar coefficients are positive reals. It is a non-empty, relatively open subset of the totally real subspace $(\mathrm i\R)^{m_F}\times\R^{m_h}$ of maximal real dimension $m$ (it is not open in $\C^m$), hence a set of uniqueness for holomorphic functions on any connected open set meeting it: a holomorphic function vanishing on a relatively open piece of a maximal totally real subspace has all derivatives zero there \cite{Hor90}. $U\cap\cT_1\ne\emptyset$ and $U\cap T_\cF\ne\emptyset$ (small positive scalar coefficients). On $U$ the total source is $(\mathrm if_\zeta,g_\zeta)\in\cK$, so $\Theta(\zeta)\in\cI$; the scheme multipliers are positive reals, equal for the Neumann--Dirichlet and the modified states; and every lattice value is real (charge conjugation, \cref{def:s02:sources}: $\overline{Z^a(\mathrm if,g)}=Z^a(-\mathrm if,g)=Z^a(\mathrm if,g)$). \end{enumerate} \end{definition} \subsection{Uniform local bounds} The uniform local bounds below are stated for King's state: in the Wick sector the first factor of the Cauchy--Schwarz splitting is controlled through the fixed-cube limits of \cref{hyp:s03:W1} at one cube, the second through the positive-leg bound \cref{cor:s11:W2prime}. \begin{lemma}[uniform local bounds for the modified-box functionals]\label{lem:s08:claim4} Let $\cF$, $\Phi_k$, $\cT_1$, $T_\cF$ be as in \cref{def:s08:enlarged,r13:def:s08:domains}. \begin{itemize}[leftmargin=*,nosep] \item[(T)] \textup{(The tube.)} Using, of the K-family functionals $Z^{a,K}_{k,J}$ (admissible scheme $a$, couplings in the coupling region of \cref{hyp:s03:T3}), only their fixed-cube limits of \cref{hyp:s03:W1}, together with \Itwo, \Ithree\ and the correlation inequalities of \cref{lem:s02:BFS}(a)--(e): for every compact $Q\subset T_\cF$ there is $C_Q<\infty$, independent of $k$ and of $J\ge J_0$, with $\limsup_k\sup_Q|\Phi_k|\le C_Q$; and at fixed $J$, $\sup_k\sup_Q|\Phi_k|<\infty$. No further threshold on $J$ enters. \item[(C)] \textup{(Curvature sector, $h\equiv0$; no hypothesis.)} $|\Phi_k(z)|\le\langle e^{F(f_{\Re z})}\rangle^{(3.38)}_{k,\Omega_{J,k}}\le\exp(\tfrac12\Var^{(3.38)}F(f_{\Re z}))$ and $\Var^{(3.38)}F(f)\le\sum_p\eps^3f_{c(p)}^2\to\norm f_{L^2}^2$, uniformly in $k$ and $J$. \end{itemize} \end{lemma} \begin{proof} $|\Phi_k(\zeta)|\le\Phi_k(\Re\zeta)$ by the proof of \cref{lem:s02:alg}(b) ($|\Theta(\zeta)|=\Theta(\Re\zeta)$ and the modulus of the multiplier), which is valid for any probability measure with the same real multiplier, in particular for the modified box state, which is not one of the states of \cref{def:s02:acube,def:s02:Kcube}. At a real $\xi$ the proof of \cref{lem:s02:alg}(d) (Cauchy--Schwarz with the multiplier identity, again valid for any such measure) gives \begin{equation}\label{eq:s08:split} \Phi_k(\xi)\ \le\ Z^{a,(3.38)}_{k,\Omega_{J,k}}(2f_\xi,-2k_2(\xi))^{1/2}\;Z^{a,(3.38)}_{k,\Omega_{J,k}}(0,2k_1(\xi))^{1/2}. \end{equation} \emph{First factor} (class $\cD$). By the $\cD$ half of \cref{lem:s07:L} with $C=C_J\cap\Lk$ (its hypothesis is \eqref{eq:s07:margin}(i)), read for King's cube with Step~3 of \cref{lem:s04:G} prepended (\cref{rem:s04:F}), it is $\le Z^{a,K}_{k,J}(2f_\xi,-2k_2)\le Z^{a,K}_{k,J_0}(2f_\xi,-2k_2)$, the second step being the class-$\cD$ cube monotonicity of \cref{cor:s04:G1} at fixed $k$. The map $\xi\mapsto\eta(\xi)\defeq$ (coefficient vector of $(2f_\xi,-2k_2(\xi))$) is continuous, so $\eta(\Re Q)$ lies in a real box $B'$; $\eta\mapsto\ln Z^{a,K}_{k,J_0}(\eta)$ is convex (\cref{lem:s02:alg}(b)), so $\sup_\xi Z^{a,K}_{k,J_0}(\eta(\xi))\le\max_{v\in\vertex(B')}Z^{a,K}_{k,J_0}(v)$, finitely many values, each convergent in $k$ by \cref{hyp:s03:W1} (the sources here have negative Wick coefficients, on which that theorem has no amplitude restriction; for the admissible scheme $a$ through the convergent factor of \cref{def:s03:adm}), and convergence in $k$ implies boundedness in $k$. So the first factor is bounded uniformly in $k$, $\xi$ and $J\ge J_0$. \emph{Second factor} (class $\cI$). By the $\cI$ half of \cref{lem:s07:U} with $B=C_{J+2n}\cap\Lk$ and $N\defeq\lceil R(\lmu)/\eps_k\rceil$ (hypothesis \eqref{eq:s07:margin}(ii)), with King's state on $B$ (\cref{lem:s04:D}(b): the box $\Omega_{J,k}$ does not meet the killed layers of $C_{J+2n}$), it is $\le Z^{a,K}_{k,J+2n}(0,2k_1(\xi))$. For $\xi\in\Re Q$, $Q\subset T_\cF$, one has $0\le2k_1(\xi)\le\sigma_+/2\le s_1$ on the lattice (the sampled supremum is at most the continuum one), so \cref{cor:s11:W2prime} on King's convention (\cref{rem:s04:F}, \cref{rem:s11:Kconvention}), $\ln N^{a,K}_{k,C_J}(0,q)\le\kappa\norm q_{1,k}$ for $0\le q\le s_1$ on $C_J\cap\Lk$, uniformly in $k$ and $J$, applied with $N^{a,K}_{k,X}(0,q)=Z^{a,K}_{k,X}(0,q)$ (the same functional with the same scheme multiplier, so no $J$-dependent factor intervenes), gives $Z^{a,K}_{k,J+2n}(0,2k_1(\xi))\le\exp(\kappa\norm{2k_1(\xi)}_{1,k})\le\exp\big(2\kappa\sum_j|\xi_j|\sup_{k'}\norm{\tilde h_j}_{1,k'}\big)$, bounded uniformly in $k$, $J$ and $\xi\in\Re Q$ because $\norm{\tilde h_j}_{1,k'}\to\norm{\tilde h_j}_{L^1}$ (Riemann sums, \eqref{eq:s02:riemann}). No lower bound on $J$ beyond $J_0$ enters: \cref{cor:s11:W2prime} holds for every $k$ and $J$. \emph{(C).} At $h\equiv0$ the first inequality is \cref{lem:s02:alg}(b) as above and the second is \cref{lem:s02:GD} with $s=1$, the modified box state being a BFS measure (\cref{def:s07:box}). Its photon Gaussian has the unitary form for gauge-invariant observables (Step 1 of \cref{lem:s07:RPbox}), and $\Var^u\langle v,d_1A\rangle=\eps^{-1}\langle d_1^Tv,(d_1^Td_1+\lmu^2\eps^2)^{-1}d_1^Tv\rangle\le\eps^{-1}\norm v^2$; with $v_p=\eps^2f_{c(p)}$ this is $\Var F(f)\le\sum_p\eps^3f_{c(p)}^2$. \end{proof} \subsection{King's state on the Wick tube} \begin{proposition}[the Wick-tube object of King's state: existence and translations]\label{prop:s08:T} Let the couplings lie in the coupling region of \cref{hyp:s03:T3} and let $a$ be an admissible scheme (\cref{def:s03:adm}), so that \Itwo\ and \Ithree\ hold (\cref{prop:s10:U,prop:s10:RP}) and the fixed-cube limits $Z^{a,K}_J\defeq\lim_kZ^{a,K}_{k,J}$ of the K-family functionals exist, locally uniformly and holomorphic, on every Wick tube below (\cref{hyp:s03:W1} in the form \cref{thm:k02:W1}(i), whose domain $\{\sup_x2k_10$ for real $\xi\in T_{\cF'}$ with $-\xi\in T_{\cF'}$; and $Z^{a,K}_{\infty,T}(-z,w)=Z^{a,K}_{\infty,T}(z,w)$. \item \textup{(Translations.)} $Z^{a,K}_{\infty,T}(\tau_b\cdot\zeta)=Z^{a,K}_{\infty,T}(\zeta)$ for all $b\in\R^3$ and $\zeta\in T_{\cF'}$, where $\tau_b$ acts on all sources; the translated family has the same tube. \end{enumerate} The proof uses, besides the fixed-cube limits of \cref{hyp:s03:W1}, only \cref{cor:s11:W2prime} on King's convention (\cref{rem:s04:F}; hence \Itwo, \Ithree\ and \cite[Thm.~6.2(1),(2)]{BFS1}), the cube monotonicity of \cref{cor:s04:G1} in both classes (its class-$\cD$ half resting on \cref{lem:s02:BFS}(d)), the translation sandwich of \cref{lem:s06:TK}, \cref{lem:s02:alg,lem:s02:baire}, Montel's theorem and the identity theorem. \end{proposition} \begin{proof} (a) \emph{Local boundedness} of $\{Z^{a,K}_J\}_{J\ge J_0}$ on $T_{\cF'}$: \begin{multline*} |Z^{a,K}_J(\zeta)|\le Z^{a,K}_J(\Re\zeta)\le Z^{a,K}_J(2f,-2k_2)^{1/2}Z^{a,K}_J(0,2k_1)^{1/2}\\ \le Z^{a,K}_{J_0}(2f,-2k_2)^{1/2}\exp(\kappa\norm{k_1}_{L^1}), \end{multline*} by \cref{lem:s02:alg}(b),(d), the nonincreasing $\cD$-factor (the class-$\cD$ cube monotonicity of \cref{cor:s04:G1} at fixed $k$, then $k\to\infty$), and \cref{cor:s11:W2prime} on King's convention (\cref{rem:s04:F}) in the limit $k\to\infty$ for the $\cI$-factor, valid since the source $2k_1$ has $2\sup(2k_1)=4\norm{k_1}_\infty<\sigma_+=s_1$; the first factor is bounded on compacts by the vertex argument of \cref{lem:s08:claim4} with the fixed-cube limits of \cref{hyp:s03:W1} at $C_{J_0}$ (negative Wick coefficients only). \emph{Convergence on $U\cap T_{\cF'}$}: there the values are real, nondecreasing in $J$ (\cref{cor:s04:G1}, then $k\to\infty$) and bounded, since $Z^{a,K}_J(\mathrm if_\zeta,g_\zeta)\le Z^{a,K}_J(0,g_\zeta)$ by \cref{lem:s02:alg}(b) and $Z^{a,K}_J(0,g_\zeta)\le\exp(\kappa\norm{g_\zeta}_{L^1})$ by \cref{cor:s11:W2prime} on King's convention in the limit $k\to\infty$ ($0\le g_\zeta\le\sigma_+/4$ there, so $2\sup g_\zeta0$ let $E_\rho\defeq C^\infty_c(\bar B_\rho;\Lambda^2)\times C^\infty_c(\bar B_\rho)$ (a Fr\'echet space) and $O_\rho\defeq\{(f,h)\in E_\rho:\ 4\norm h_\infty<\sigma_+/2\}$, open, convex and symmetric under $s\mapsto-s$. For $s=(f,h)\in O_\rho$ the one-member family with its splitting has the coefficient vector $(1,1,-1)$ in its tube ($2\norm{k_1}_\infty=2\norm{h^+}_\infty\le4\norm h_\infty<\sigma_+/2$), so $\Phi_T(s)\defeq\ln Z^{a,K}_{\infty,T}(f,h)$ is defined and finite on $O_\rho$ by (a). $\Phi_T$ is convex (a pointwise limit of the functions $\ln Z^{a,K}_{k,J}$, convex by H\"older's inequality) and Borel (a pointwise limit of continuous functions of finitely many sampled values). By \cref{lem:s02:baire}, $\Phi_T$ is continuous on $O_\rho$. For $s\in O_\rho$ and $b\in\R^3$ pick $b_n\to b$ in $\bigcup_k5^{-k}\Z^3$; then $\tau_{b_n}s,\tau_bs\in O_{\rho+|b|+1}$, $\tau_{b_n}s\to\tau_bs$ in $E_{\rho+|b|+1}$ (translations act continuously on $C^\infty_c$), and $\Phi_T(\tau_{b_n}s)=\Phi_T(s)$; so $\Phi_T(\tau_bs)=\Phi_T(s)$. \emph{Complex coefficients.} For a family $\cF'$ the two holomorphic functions $\zeta\mapsto Z^{a,K}_{\infty,T}(\tau_b\cdot\zeta)$ and $\zeta\mapsto Z^{a,K}_{\infty,T}(\zeta)$ on $T_{\cF'}$ agree at the real points of $\{\xi\in\R^m:4\norm{h_\xi}_\infty<\sigma_+/2\}\cap T_{\cF'}$, where both limits exist and both equal the one-member value $\exp\Phi_T(f_\xi,h_\xi)$ (the same $J$-limit of the same $Z^{a,K}_J$). This intersection is open in $\R^m$ and contains $0$ (the first set alone need not lie in the tube: with $\tilde h_1=\tilde h_2$ and $\xi=(1,-1)$, $h_\xi=0$ while $k_1=\tilde h_1$), hence is a set of uniqueness in the connected $T_{\cF'}$; so the two functions agree on $T_{\cF'}$. \end{proof} \subsection{The two limits and the passage to every plane} \label{conv:v063_s08_os2_limit:1}Throughout this subsection $\cF$, $\Phi_k$, $\zeta^*_{ij}$, $\cT_1$, $T_\cF$, $U$ are those of \cref{def:s08:enlarged,r13:def:s08:domains} for a mid-plane $c\in D$, and ``version'' refers to \cref{lem:s08:claim4}: (C) is the curvature sector of King's state (the uniform local bound being \cref{thm:s05:CK}(a),(b)), (T) the tube of King's state under \cref{hyp:s03:W1}, \Itwo, \Ithree. The cube functionals are those of the K family, with the fixed-cube limits $\ZKJ$ (version (C)) and $Z^{a,K}_J$ (version (T)); $Z^{a,K}_{\infty(,T)}$ denotes $\ZKinf(\cdot,0)$ in version (C) and $Z^{a,K}_{\infty,T}$ in version (T). \emph{Step 1 ($k\to\infty$ at fixed $J$).} By the fixed-$J$ bound of \cref{lem:s08:claim4} and Montel's theorem there is a subsequence $k_m\to\infty$ along which $\Phi_{k_m}\to\Phi^J$ locally uniformly on $\C^m$, with $\Phi^J$ entire. No convergence of the full sequence is claimed or needed. By \cref{lem:s07:RPbox}(b), $\sum_{ij}\bar\gamma_i\gamma_j\Phi_k(\zeta^*_{ij})\ge0$ for every $k\ge k_0(c,\delta)$ and $\gamma\in\C^N$; positive semidefiniteness of a Hermitian matrix is closed under entrywise limits, so $\sum_{ij}\bar\gamma_i\gamma_j\Phi^J(\zeta^*_{ij})\ge0$. The uniform bounds of \cref{lem:s08:claim4} pass to $\Phi^J$: $\sup_Q|\Phi^J|\le C_Q$ in (T) and (C), uniformly in $J\ge J_0$, whatever subsequence was taken. (The box $\Omega_{J,k}$ depends on $J$, so $\Phi^J$ does; the subsequence may depend on $J$.) \emph{Step 2 ($J\to\infty$: sandwich on the cone, identity theorem off it).} By the $\cI$ halves of \cref{lem:s07:L,lem:s07:U}, whose hypotheses are \eqref{eq:s07:margin}(i),(ii), read for King's cubes (\cref{lem:s04:D}(c): the lower end with Step~1 of \cref{lem:s04:G} prepended, \cref{rem:s04:F}; the upper end with King's state on $X_{J+2n}$, \cref{lem:s04:D}(b)), for every $k$ and every $\zeta\in U$, \begin{equation}\label{eq:s08:sandwich} Z^{a,K}_{k,J}(\zeta)\ \le\ \Phi_k(\zeta)\ \le\ Z^{a,K}_{k,J+2n}(\zeta). \end{equation} Let $k\to\infty$ along the subsequence of Step 1. The outer terms converge along the full sequence (version (C): \cref{thm:s05:CK}(b); (T): the fixed-cube limits of \cref{hyp:s03:W1}, the points of $U$ carrying non-negative scalar sources below the threshold; each with Montel's theorem), so $Z^{a,K}_J\le\Phi^J\le Z^{a,K}_{J+2n}$ on $U$ for every subsequential $\Phi^J$. As $J\to\infty$ the outer terms converge on the respective uniqueness set: in (C) on $U=(\mathrm i\R)^{m_F}$ to $\ZKinf(\cdot,0)$ (\cref{thm:s05:CK}(b)); in (T) on $U\cap T_\cF$ to $Z^{a,K}_{\infty,T}$ (\cref{prop:s08:T}(a); there $J\mapsto Z^{a,K}_J(\zeta)$ is real, nondecreasing and bounded by $Z^{a,K}_J(0,g_\zeta)\le\exp(\kappa\norm{g_\zeta}_{L^1})$, since $0\le g_\zeta\le\sigma_+/4$, so $2\sup g_\zetac\}$. If $c\in D$, \eqref{eq:s08:midplane} is the result. Otherwise pick $c'\in D$ and $b\defeq(c'-c)e_\nu$. The translated sources $\tau_b\zeta_i$ are supported in $\{x_\nu>c'\}$, and $\theta_{c'}(\tau_b\zeta)^*=\tau_b(\theta_c\zeta^*)$, because $\rho_{c'}(x+b)=\rho_c(x)+b$ for this $b$ and the linear parts of $\rho_c,\rho_{c'}$ agree. By \eqref{eq:s08:midplane} for the plane $c'$ and the translation invariance of the limit functional, \[ \sum\bar\gamma_i\gamma_jZ^{a,K}_{\infty(,T)}(\theta_c\zeta_i^*\sqcup\zeta_j)=\sum\bar\gamma_i\gamma_jZ^{a,K}_{\infty(,T)}\big(\tau_b(\theta_c\zeta_i^*\sqcup\zeta_j)\big)=\sum\bar\gamma_i\gamma_jZ^{a,K}_{\infty(,T)}\big(\theta_{c'}(\tau_b\zeta_i)^*\sqcup\tau_b\zeta_j\big)\ \ge\ 0 . \] The translation invariance is provided in (C) by \cref{cor:s06:TK}, and in (T) by \cref{prop:s08:T}(b) on $T_\cF$ (the amplitude $\amp$ is translation invariant, so the matrix points stay in the tube). \subsection{The theorems} \begin{theorem}[reflection positivity of King's state, curvature sector]\label{thm:s08:OS2C}\label{lem:s08:OS2K} Let $\lmu>0$, $\nu\in\{1,2,3\}$, $c\in\R$, and let $\zeta_1,\dots,\zeta_N$ be finite families of real $2$-form sources $f_{il}\in C^\infty_c(\R^3;\Lambda^2)$ supported in the open half-space $\{x_\nu>c\}$, with arbitrary complex coefficients $z_{il}$ and no scalar sources. Then for all $\gamma\in\C^N$ \[ \sum_{i,j=1}^N\bar\gamma_i\gamma_j\,\ZKinf\big(\theta\zeta_i^*\sqcup\zeta_j,\,0\big)\ \ge\ 0 ; \] moreover \cref{prop:s08:OS3,prop:s08:poly} below hold for $\ZKinf(\cdot,0)$. The results used are \cref{hyp:s03:EC}, \cite[Thm.~6.2(1),(2)]{BFS1} (through the cube monotonicity \cref{cor:s04:G1} and the $\cI$ halves of \cref{lem:s07:L,lem:s07:U}), \cref{thm:s05:CK}(a),(b) with \cref{lem:s05:C2}, \cref{lem:s04:D}(c) and \cref{rem:s04:F} for the two places where the cube convention is seen, \cref{cor:s06:TK} for the translations, and \cref{lem:s02:GD}, \cref{lem:s02:alg}(b),(c), \cref{lem:s02:BFS}(e), \cref{lem:s02:baire}; no inequality of class $\cD$ enters. \end{theorem} \begin{proof} At $h\equiv0$ there is no Wick constant and no scheme, and $\cF$ consists of the $f_{il}$ and $\rho^*f_{il}$ with $U=(\mathrm i\R)^{m}$. What the argument of Steps 1--3 needs of the cube functionals is: the existence of $\ZKJ(\cdot,0)=\lim_ka^K_{k,J}$ locally uniformly on $\C^m$, entire; the monotonicity of $J\mapsto\ZKJ(\mathrm if,0)$ on the cone; the existence of $\ZKinf(\cdot,0)=\lim_J\ZKJ(\cdot,0)$, entire, locally uniformly on $\C^m$; and translation invariance. The first three are \cref{thm:s05:CK}(b), whose local bound is the $\lmu$-free bound of \cref{thm:s05:CK}(a) and \cref{lem:s05:C2} rather than a local bound resting on the class-$\cD$ half of the cube monotonicity and so on \cite[Thm.~6.2(3),(4)]{BFS1}; the fourth is \cref{cor:s06:TK}. \Cref{lem:s07:RPbox,lem:s07:B} concern the modified box state alone and the lattice reflection, and \cref{lem:s08:claim4}(C) bounds $\Phi_k$ on the modified photon form: these are convention-free, as is Step 1 (Montel in $k$; positivity passes). The two places where the Neumann--Dirichlet cube states enter are the ends of the sandwich \eqref{eq:s08:sandwich}, \ie the $\cI$ halves of \cref{lem:s07:L,lem:s07:U}; on King's convention they read $Z^K_{k,J}(\zeta)\le\Phi_k(\zeta)\le Z^K_{k,J+2n}(\zeta)$ for every $k$ and $\zeta\in U=(\mathrm i\R)^m$: the lower end is \cref{lem:s07:L} with Step 1 of \cref{lem:s04:G} prepended (\cref{rem:s04:F}: $\avK{\cdot}\le\ava{\cdot}\le\langle\cdot\rangle^{(3.38)}_{k,\Omega_{J,k}}$ on $\cI$), the upper end is \cref{lem:s07:U} with $\avK{\cdot}$ on $X_{J+2n}$ in place of the comparison state (\cref{lem:s04:D}(b),(c): the box $\Omega_{J,k}$ does not meet the killed layers $T_{J+2n}$, and part (a) of \cref{lem:s07:U} is photon-only), under the hypotheses of \eqref{eq:s07:margin}. Here $J_0$ is the $J_0$ of the enlarged family (the sources and their reflections), which exceeds that of the sources alone when $c\ne0$; \cref{thm:s05:CK}(b) is applied to the enlarged family at $J$ and $J+2n$, $J\ge J_0$. Along Step 1's subsequence the ends converge along the full sequence by \cref{thm:s05:CK}(b), so $Z^K_J\le\Phi^J\le Z^K_{J+2n}$ on $U$ for every subsequential $\Phi^J$; as $J\to\infty$ both ends tend to $\ZKinf$ on $U$, $\{\Phi^J\}_J$ is locally bounded on $\C^m$ by \cref{lem:s08:claim4}(C), and $U$ is a set of uniqueness, so $\Phi^J\to\ZKinf$ locally uniformly on $\C^m$ as in Step 2, in particular at the matrix points; positivity passes to the limit. Step 3 uses \cref{cor:s06:TK}. \Cref{prop:s08:OS3,prop:s08:poly} use only the existence, holomorphy near $0$ and lattice symmetries of the limit, and the locally uniform convergences just established; their proofs below apply with $\ZKinf$, the charge-conjugation invariance being exact for $\avK{\cdot}$. \end{proof} \begin{theorem}[reflection positivity on the Wick tube, King's state]\label{thm:s08:OS2WT} Let the couplings lie in the coupling region of \cref{hyp:s03:T3} and let $a$ be an admissible scheme, so that \Itwo\ and \Ithree\ hold (the hypotheses of \cref{cor:s11:W2prime}) and the fixed-cube limits of \cref{hyp:s03:W1} are available as in \cref{prop:s08:T}; the $\cD$-class steps use in addition \cref{lem:s02:BFS}(d). Let $\sigma_+=s_1$ be the amplitude of \Itwo\ and $\sigma_W\defeq\sigma_+/8$. Let $\lmu>0$, $\nu$, $c\in\R$, and let $\zeta_1,\dots,\zeta_N$ be finite families of real sources $f_{il}\in C^\infty_c(\R^3;\Lambda^2)$, $h_{il}\in C^\infty_c(\R^3)$ supported in $\{x_\nu>c\}$, with complex coefficients $z_{il},w_{il}$ such that $\amp(\zeta_i)<\sigma_W$ for every $i$; in particular, all purely imaginary $w_{il}$ with arbitrary $z_{il}$ are admitted. Then the values $Z^{a,K}_{\infty,T}(\theta\zeta_i^*\sqcup\zeta_j)=\lim_JZ^{a,K}_J(\theta\zeta_i^*\sqcup\zeta_j)$ exist (\cref{prop:s08:T}) and \[ \sum_{i,j=1}^N\bar\gamma_i\gamma_j\,Z^{a,K}_{\infty,T}\big(\theta\zeta_i^*\sqcup\zeta_j\big)\ \ge\ 0\qquad\text{for all }\gamma\in\C^N . \] On $T_\cF\cap\Dtube$, $Z^{a,K}_{\infty,T}=G^K_\infty$. No lower bound on $J$ beyond $J_0$ enters, and no bound at positive scalar sources beyond the cone $\sup q\le s_1$ of \cref{cor:s11:W2prime} is used. \end{theorem} \begin{proof} With the splitting of \cref{def:s08:enlarged} the matrix points lie in $T_\cF$ by \eqref{eq:s08:ampbound}. \Cref{lem:s08:claim4}(T) gives the bounds; Steps 1 and 2 in version (T), with the sandwich \eqref{eq:s08:sandwich} in its King form (\cref{lem:s04:D}(c)) and \cref{prop:s08:T}(a) for the limit on $U\cap T_\cF$ and its holomorphy on $T_\cF$, give \eqref{eq:s08:midplane} for $c\in D$; Step 3 with \cref{prop:s08:T}(b) gives every plane. The $\cD$-class steps are the first factor of \eqref{eq:s08:split} (the $\cD$ half of \cref{lem:s07:L} with Step~3 of \cref{lem:s04:G} prepended, and the class-$\cD$ cube monotonicity of \cref{cor:s04:G1}) and the nonincreasing factor in \cref{prop:s08:T}(a). \end{proof} \subsection{\texorpdfstring{$\OSth$}{OS3} and the polynomial form} \begin{proposition}[symmetry, realness and charge conjugation]\label{prop:s08:OS3} Let $Z$ be one of $\ZKinf(\cdot,0)$ (\cref{thm:s05:CK}), $Z^{a,K}_{\infty,T}$ (\cref{prop:s08:T}) and $G^K_\infty$ (\cref{thm:s13:TWK}), or a limit state of the comparison family along a continuum subsequence, in either sector (\cref{thm:s12:LIM}). The Schwinger functions \[ S_{n,p}(f_1,\dots,f_n;h_1,\dots,h_p)\defeq\partial_{z_1}\cdots\partial_{z_n}\partial_{w_1}\cdots\partial_{w_p}\,Z\big(\textstyle\sum_iz_if_i,\sum_lw_lh_l\big)\big|_{0} \] (with $p=0$ and $S_n$ in the curvature sector) are (a) symmetric multilinear in the $f$'s and in the $h$'s; (b) real for real test functions; (c) zero for odd $n$. The proof uses only that $Z$ exists and is holomorphic near $0$, with the stated symmetries of the lattice functionals; the existence is established by the result cited for each. \end{proposition} \begin{proof} (a) $Z$ depends on a family only through its total sources (so does every lattice functional, hence both limits), so its Taylor coefficients at $0$ are symmetric multilinear forms in the $f$'s and in the $h$'s; $Z^{a,K}_{\infty,T}$ is holomorphic on a tube containing $0$. The definition of $S_{n,p}$ does not depend on the family used to compute it (restrict a larger family to the given sources). (b) For real coefficients every lattice functional is real (real integrand and multiplier, \cref{lem:s02:alg}(a)), so $Z$ is real at the real points near $0$ (for $Z^{a,K}_{\infty,T}$ on $\R^m\cap T$, an open piece of $\R^m$ containing $0$); a holomorphic function real on the real points near $0$ has real Taylor coefficients at $0$. (c) Every cube state and every modified box state is invariant under $(A,\vartheta)\mapsto(-A,-\vartheta)$ (the photon terms are even in $A$, the masked divergence included; $\cos(\vartheta_x-e\eps A-\vartheta_y)$ is invariant; $W^a$ depends on $r$ only; Lebesgue measure is preserved), which maps $F_{\Lk}(f)\mapsto-F_{\Lk}(f)$ and fixes $W^a_{\Lk}(h)$. Hence $Z_{k,X}(-z,w)=Z_{k,X}(z,w)$ for all $k$, $X$, and in both limits $Z(-z,w)=Z(z,w)$ (on the $z\mapsto-z$ invariant tube for $Z^{a,K}_{\infty,T}$). So $\partial_z^{(n)}\partial_w^{(p)}Z(0)=0$ for odd $n$. \end{proof} \begin{proposition}[the polynomial form of reflection positivity]\label{prop:s08:poly} Let $P=\sum_\alpha c_\alpha\prod_iF(f_{\alpha i})\allowbreak\prod_lW^a(h_{\alpha l})$ be a polynomial in the smeared fields with real sources supported in $\{x_\nu>c\}$, and $\theta P\defeq\sum_\alpha\bar c_\alpha\prod_iF(\rho^*f_{\alpha i})\allowbreak\prod_lW^a(h_{\alpha l}\circ\rho)$. Then $S(\theta(P)P)\ge0$, where $S$ assigns to a monomial the corresponding Schwinger function of \cref{prop:s08:OS3}: the Taylor coefficient at $0$ of $Z^{a,K}_{\infty,T}$ (\cref{thm:s08:OS2WT}), resp.\ of $\ZKinf(\cdot,0)$ (\cref{thm:s08:OS2C}), where at $h\equiv0$ the $W^a$ factors are absent. \end{proposition} \begin{proof} Let all sources of $P$ be supported in $\{x_\nu\ge c+\delta\}$ and let $\cF'$ be the family of all sources of $P$ and their reflections. Its scalar members have no sign, so Steps 1--2 are applied to the split family $\cF'_{\rm split}$ (members $h^\pm_{\alpha l}$, $h^\pm_{\alpha l}\circ\rho$, all $\ge0$); its tube contains $0$ ($k_1(0)=0$), and the unsplit functional is the composition of the split one with the linear map $w_{\alpha l}\mapsto(w^+_{\alpha l},w^-_{\alpha l})=(w_{\alpha l},-w_{\alpha l})$, so by the chain rule the derivatives at $0$ of the unsplit $\Phi_k$, $\Phi^J$, $Z^{a,K}_{\infty(,T)}$ are finite linear combinations of those of the split ones, and locally uniform convergence near $0$ passes through the linear map. (i) \emph{Lattice.} For $k\ge k_0$ the sampled $P$ is a gauge-invariant polynomial in the plus-side smeared fields, lies in $L^2$ of the modified box state (Gaussian moments; $\lambda>0$) and in $\mathfrak B_+$ (\cref{def:s07:reflection}), and $\theta(P)=\overline{P\circ\hat\rho}$ by \eqref{eq:s07:thetaTheta}. By \cref{lem:s07:RPbox}(a) (its $L^2$ extension), $\langle\theta(P)P\rangle^{(3.38)}_{k,\Omega_{J,k}}\ge0$. (ii) \emph{Moments as derivatives.} $\langle\theta(P)P\rangle^{(3.38)}=\sum_{\alpha,\beta}\bar c_\alpha c_\beta\langle\prod_iF(\rho^*f_{\alpha i})\prod_lW^a(h_{\alpha l}\circ\rho)\prod_iF(f_{\beta i})\prod_lW^a(h_{\beta l})\rangle^{(3.38)}$, and each mixed moment is a mixed partial derivative at $0$ of $\Phi_k$ for the family $\cF'$ (differentiate under the integral; the scheme multiplier $e^{-wa_k\sS(h)}$ is the $-a_k$ inside $W^a$). (iii) \emph{Limits.} By Steps 1--2 for $\cF'_{\rm split}$, $\Phi_{k_m}\to\Phi^J$ and $\Phi^J\to Z^{a,K}_{\infty(,T)}$ locally uniformly on a neighbourhood of $0$, so by Weierstrass' theorem all derivatives at $0$ converge, and $\sum\bar c_\alpha c_\beta\,\partial^{(\alpha,\beta)}Z^{a,K}_{\infty(,T)}(0)\ge0$; the left side is $S(\theta(P)P)$ by the family-independence of the Schwinger functions (\cref{prop:s08:OS3}(a)). Every plane: Step 3 (translation of the sources; $S$ is translation invariant by \cref{prop:s08:T}(b), resp.\ \cref{cor:s06:TK}). The curvature sector is the case $h\equiv0$ with no splitting. This is the analogue, for the curvature and the renormalised Wick square with reflection in the planes $\{x_\nu=c\}$, of the positivity condition \cite[Axiom~E2, p.~88]{OS73} in its form for finite sums of tensor products of test functions supported in a half-space. \end{proof} \begin{remark}[which form is used, and the amplitude constants]\label{rem:s08:amplitudes} The main theorem (\cref{thm:s13:main}) states reflection positivity of the Wick sector in the tube form of \cref{thm:s08:OS2WT}, with \cref{prop:s08:OS3,prop:s08:poly} for the Schwinger functions; the results it uses for this clause are \cref{hyp:s03:W1}, \Itwo, \Ithree and \cite[Thm.~6.2(1)--(4)]{BFS1}. This suffices for its use in the proof of \cref{prop:s13:WK}(d): the measure and the weak form of $\OSf$ for the joint field use reflection positivity for exponentials $e^{\mathrm i\Phi(f)}$ of the smeared fields, whose Wick coefficients are purely imaginary, so amplitude $0$ only. Three positive, $\eps$-free amplitude constants occur in the Wick sector: $\sigma_W=\sigma_+/8$ here; $H/2=\sigma_+/32$, the negative-part radius of the tube $\Dtube$ of \cref{thm:s11:W,thm:s13:TWK} ($H=s_1/16$); and $\sigma_0=H/4$ in the bound $\Ezz$ of \cref{thm:s13:TWK}. On $T_\cF\cap\Dtube$ the objects $Z^{a,K}_{\infty,T}$ and $G^K_\infty$ coincide (\cref{prop:s08:T}(a)). The tube statements of this section are stated on King's convention; their convention-sensitive steps are those listed in \cref{rem:s11:Kconvention} and \cref{prop:s13:WK}(c), and the bound \eqref{eq:s11:W} enters on King's convention as \eqref{eq:s13:WK} (\cref{thm:s13:TWK}). The margin $R(\lmu)$ of \cref{lem:s07:U} enters the proof, not the conclusion: reflection positivity is proved at every fixed $\lmu>0$; the limit of vanishing photon mass is treated in a subsequent paper. \end{remark} \begin{remark}[the comparison family]\label{rem:s08:comparison} The proofs of this section use of the cube functionals only their fixed-cube limits (at real coefficients in the curvature sector, at the real points of the tube in the Wick sector), the cube monotonicity on the cone, the positive-leg bound \cref{cor:s11:W2prime} and the bounds of \cref{lem:s08:claim4}; all of these are available for the comparison family as well, with the first inequality of \cref{lem:s02:cut} in place of \cref{cor:s04:G1} and \cref{prop:s06:translations} in place of \cref{lem:s06:TK}. As noted in Section~\ref{sec:s03}, they apply to the comparison family $\NDa$ wherever its fixed-cube limits exist. The limit states of the comparison family along continuum subsequences, which exist without that clause, are treated in \cref{sec:s12}, and they coincide with the objects of King's state (\cref{cor:s12:KK}, \cref{prop:s13:LIMK}). \end{remark} \space% % % % % \section{The measure and weak clustering}\label{sec:s09}\label{sec:s09:measure} This section constructs the measure of King's state (the infinite-volume state of King's construction \cite{K1,K2}) in the curvature sector and derives from it the consequences of reflection positivity: the ergodic decomposition, clustering, the unique joint vacuum on a component, the growth of the components' moments, and weak \OSf\ on almost every component. The general results used are those of \HPapp\ (the measure on the Hermite--Sobolev space and its uniqueness; the ergodic decomposition, obtained by conditioning on the translation-invariant $\sigma$-algebra, almost every component of which inherits the reflections and reflection positivity, the components regrouping into $\Oh$-invariant ones when the measure is $\Oh$-invariant and the moment bounds passing to almost every component; the Osterwalder--Schrader reconstruction in each coordinate direction and the clustering statements), cited where they are used, and the rotation corollary, Corollary~\ref{cpi:rot:cor}. The joint field with the Wick square is the subject of the tube sections. The measure is constructed from the characteristic functional, not from the lattice laws, which converge only locally (\cref{rem:s09:S2K}). \emph{Conventions.}\refstepcounter{convention}\label{conv:s09_measure:1} $\cV\defeq\Lambda^2\R^3$ with its Euclidean fibre norm. For a real $u\in C^\infty_c(\R^3;\Lambda^2)$ put $\cZ^K(u)\defeq\ZKinf(\mathrm iu)$, the one-member functional at the coefficient $\mathrm i$ (\cref{thm:s05:CK}(b), with the total-source convention of \cref{thm:s06:S1K}). Fix an even $r\ge14$. $\cH^{-r}(\R^3;\Lambda^2)$ is the Hermite--Sobolev space, the dual of the completion of $\cS(\R^3;\Lambda^2)$ under $\norm u_r^2=\sum_{\alpha,c}\lambda_\alpha^{r}\ip{h_{\alpha,c}}u^2$, where $(h_{\alpha,c})_{\alpha\in\N^3,\,c\le3}$ is the Hermite basis of $L^2(\R^3;\Lambda^2)$ and $\lambda_\alpha=2|\alpha|+4$ are the eigenvalues of $-\Delta+|x|^2+1$; $\fournorm\cdot$ is the weighted $C^4$ norm of \cref{lem:s02:E}, and $\fournorm u\le C\norm u_{10}$ on $\cS$ (\cite[\HPrefraw{ms:hermite}(b)]{HP}). $X\defeq H^{-r}_{\mathrm{loc}}(\R^3;\Lambda^2)$, a Fr\'echet space with the seminorms $\|\chi_n\Phi\|_{H^{-r}}$ for a fixed sequence of cut-offs $\chi_n\in C^\infty_c(\R^3)$ with $\chi_n=1$ on the ball of radius $n$, carries the actions $(\tau_a\Phi)(u)=\Phi(\tau_{-a}u)$, $(g\Phi)(u)=\Phi(g^{-1}\cdot u)$ for $g\in\Oh$ (or $O(3)$; with the translations, the push-forward action of the Euclidean group), $(\theta_j\Phi)(u)=\Phi(\sigma_j\cdot u)$ for the coordinate reflections $\sigma_j$ of $\R^3$ (written $\rho_j$ in Definition~\ref{cpi:imp:H} and in Section~\ref{cpi:rot:sec}), with the signed transport of $2$-forms of \cref{def:s02:sources}, and the local $\sigma$-algebras $\Sigma(U)$ generated by $\Phi\mapsto\Phi(u)$, $u\in C^\infty_c(U;\Lambda^2)$ real; $\theta_{j,c}\defeq\tau_{ce_j}\theta_j\tau_{-ce_j}$ is the reflection in the plane $x_j=c$ and $\rho_{j,c*}u$ the reflected source (\cref{def:s07:reflection}). The \emph{lattice curvature field} of King's cube is the random distribution $\Phi^K_{k,J}:u\mapsto F_{\Lk}(u)$ under $\avK\cdot$. $J_0(u)$ is the least $J$ with $\supp u$ at $\ell^\infty$-distance $>1$ from $\R^3\setminus\CJ$. % Throughout, $\mu$ is a translation-invariant Borel probability measure on $X$, and $U(a)G:=G\circ\tau_{-a}$ is the induced unitary representation of the translation group $\R^3$ on $L^2(\mu)$; $e_1,e_2,e_3$ is the standard basis of $\R^3$, $H^+_{j,0}:=\{x\in\R^3:x_j>0\}$, and the expectation $\E_\mu[G'G]$ is taken without conjugation. \begin{definition}[the cluster property]\label{cpi:imp:OS4} For $j\in\{1,2,3\}$ consider the two conditions: \begin{enumerate} \item[(OS4$_j$)] (ergodicity of the translations in the direction $e_j$) every $G\in L^2(\mu)$ with $U(te_j)G=G$ for all $t\in\R$ is $\mu$-almost surely constant; \item[(OS4$'_j$)] (clustering along the axis $e_j$) $\E_\mu[G'\,U(te_j)G]\to\E_\mu[G']\,\E_\mu[G]$ as $\abs t\to\infty$, for all $G,G'\in L^2(\mu)$. \end{enumerate} The measure $\mu$ satisfies \emph{\OSf} if (OS4$_j$) holds for $j=1,2,3$. A family of Schwinger functions satisfies \OSf\ if it is the family of moments of exactly one translation-invariant probability measure $\mu$ on $X$, and this $\mu$ satisfies \OSf. \end{definition} \begin{definition}[forms of the cluster axiom; the weak form]\label{cpi:imp:OS4forms} Let $X$, $\mu$ and $U$ be as in Definition~\ref{cpi:imp:OS4}, and let $j\in\{1,2,3\}$ be the direction chosen as Euclidean time. In (i) assume moreover that $X$ is a space of distributions with values in a finite-dimensional real vector space $\cV$ (for the curvature field $\cV=\Lambda^2\R^3$, for the joint field with the Wick square $\cV=\Lambda^2\R^3\oplus\R$), that $\mu$ is $\theta_j$-invariant, and let $\mathcal P_j^+$ be the set of polynomials in the fields $\Phi(f)$, $f\in C_c^\infty(H^+_{j,0};\cV)$ real, that belong to $L^2(\mu)$. \begin{enumerate} \item[(i)] (cluster property, \cite[E4, p.~88]{OS73}) For all $F,G\in\mathcal P_j^+$ and every $a\in\R^3\setminus\{0\}$ with $a\perp e_j$, \[ \lim_{s\to\infty}\E_\mu\big[\overline{F\circ\theta_j}\;U(sa)G\big]=\E_\mu\big[\overline F\,\big]\,\E_\mu[G]. \] \item[(ii)] (ergodicity of the time translations, \cite[(6.1.10), p.~92]{GJ}) Condition (OS4$_j$) of Definition~\ref{cpi:imp:OS4} holds. \item[(iii)] (weak form) $\mu$ is ergodic for the action of the translation group $\R^3$: every $G\in L^2(\mu)$ with $U(a)G=G$ for all $a\in\R^3$ is $\mu$-almost surely constant. \end{enumerate} We call (i) and (ii) the single-axis forms of \OSf, and say that $\mu$ satisfies \emph{\OSf\ in the weak form} if (iii) holds. \end{definition} \begin{remark}[relations between the forms]\label{cpi:imp:OS4rem} (a) Condition (OS4$_j$) for one $j$ implies the weak form (iii), since an $\R^3$-invariant function is invariant under the translations along $e_j$; the converse does not hold in general, the weak form controlling the joint fixed space of the reconstruction and not the kernel of a single Hamiltonian. (b) If the coordinate permutations act measurably on $X$, as in the setting of the field spaces used here, and $\mu$ is invariant under them (under the cubic group, as the measure of King's state is), then (OS4$_1$), (OS4$_2$) and (OS4$_3$) are equivalent: for such a permutation $g$, $G\mapsto G\circ g^{-1}$ is a unitary of $L^2(\mu)$ that preserves the constants and carries the functions invariant along $e_j$ onto those invariant along $ge_j$, since $g\,\tau_a\,g^{-1}=\tau_{ga}$. In general they are three statements. (c) Under the hypotheses of the reconstruction in the coordinate directions (translation invariance, local $\sigma$-algebras generated by the bounded regions, reflection positivity about the coordinate planes) the conditions (OS4$_j$) and (OS4$'_j$) are equivalent for each $j$. Writing $P_j$ for the orthogonal projection of $L^2(\mu)$ onto the $e_j$-invariant functions, (OS4$_j$) says that $P_jG=\E_\mu G$ for every $G\in L^2(\mu)$ (an $e_j$-invariant $G$ equals $P_jG$, and $P_jG$ is $e_j$-invariant), while $\E_\mu[G'\,U(te_j)G]\to\E_\mu[G'\,P_jG]$ as $\abs t\to\infty$ by the spectral theorem for the self-adjoint contraction semigroup of the reconstruction in the direction $e_j$. If $P_jG=\E_\mu G$ for all $G$, the limit is $\E_\mu[G']\,\E_\mu[G]$. Conversely, if (OS4$'_j$) holds, then $\E_\mu[G'(P_jG-\E_\mu G)]=0$ for all $G'$, and the choice $G'=\overline{P_jG-\E_\mu G}$ gives $P_jG=\E_\mu G$. \end{remark} \space% % % For $j\in\{1,2,3\}$ and $c\in\R$ let $H^\pm_{j,c}\defeq\{x\in\R^3:\pm(x_j-c)>0\}$ be the open half spaces, $\rho_jx\defeq x-2x_je_j$ the reflection in the plane $\{x_j=0\}$ (the reflection written $\sigma_j$ in the Conventions of Section~\ref{sec:s09:measure}) and $\rho_{j,c}$ the reflection in $\{x_j=c\}$; $K$ denotes a bounded open subset of $\R^3$ and $K\Subset U$ means $\overline K\subset U$. The ergodic decomposition, the clustering theorems and the reconstruction of the vacuum are proved in \HPapp\ for an abstract \emph{reflection-positive stationary system}, which we now define. \begin{definition}[the abstract setting]\label{cpi:imp:H} A \emph{reflection-positive stationary system} $(X,\tau,\theta,\Sigma,\mu)$ consists of the following data, subject to (H1)--(H3). \begin{enumerate} \item[(H1)] (\emph{Space, actions, measure.}) $X$ is a Polish space with a jointly continuous action $(a,x)\mapsto\tau_ax$ of $\R^3$ and three involutive homeomorphisms $\theta_1,\theta_2,\theta_3$ of $X$ with $\theta_j\tau_a=\tau_{\rho_ja}\theta_j$; $\mu$ is a Borel probability measure on $X$ invariant under every $\tau_a$ and every $\theta_j$. \item[(H2)] (\emph{Local structure.}) To every open $U\subset\R^3$ is assigned a sub-$\sigma$-algebra $\Sigma(U)$ of the Borel $\sigma$-algebra $\cB$ of $X$ such that $\Sigma(U)\subset\Sigma(U')$ if $U\subset U'$; $\Sigma(U)=\bigvee_{K\Subset U}\Sigma(K)$; $\bigvee_K\Sigma(K)=\cB$, the join over all bounded open $K$; each $\Sigma(U)$ is generated by a countable family of bounded continuous functions on $X$; and, with \begin{equation}\label{cpi:imp:eq:cov} U(a)G\defeq G\circ\tau_{-a},\qquad\Theta_jG\defeq G\circ\theta_j, \end{equation} $U(a)$ maps $\Sigma(U)$-measurable functions to $\Sigma(U+a)$-measurable ones and $\Theta_j$ maps $\Sigma(U)$-measurable functions to $\Sigma(\rho_jU)$-measurable ones. \item[(H3)] (\emph{Coordinate reflection positivity.}) For every $j$, every $c\in\R$ and every $F\in L^2(\Sigma(H^+_{j,c}),\mu)$, \begin{equation}\label{cpi:imp:eq:RP} \E_\mu\big[\,\overline{F\circ\theta_{j,c}}\;F\,\big]\ \ge\ 0,\qquad\theta_{j,c}\defeq\tau_{ce_j}\theta_j\tau_{-ce_j}. \end{equation} \end{enumerate} \end{definition} By (H1), $a\mapsto U(a)$ is a unitary representation of $\R^3$ on $L^2(\mu)$, each $\Theta_j$ is a unitary involution, and $\Theta_jU(a)=U(\rho_ja)\Theta_j$; in particular $\Theta_jU(te_j)=U(-te_j)\Theta_j$ and $\Theta_jU(y)=U(y)\Theta_j$ for $y\perp e_j$. By translation invariance, \eqref{cpi:imp:eq:RP} for $c=0$ implies it for every $c$: for $F\in L^2(\Sigma(H^+_{j,c}),\mu)$ the function $F_0\defeq U(-ce_j)F$ is $\Sigma(H^+_{j,0})$-measurable by (H2), $\overline{F\circ\theta_{j,c}}\,F=(\overline{F_0\circ\theta_j}\,F_0)\circ\tau_{-ce_j}$, and $\tau_{-ce_j}$ preserves $\mu$. On a space of distributions we use $(\tau_a\Phi)(u)\defeq\Phi(\tau_{-a}u)$ with $\tau_bu=u(\cdot-b)$, the push-forward of $\Phi$ by $x\mapsto x+a$; then $U(a)[\Phi(u)]=\Phi(\tau_au)$, whose test function is supported in $\supp u+a$, as (H2) requires. A Borel probability measure $m$ on $X$ invariant under every $\tau_a$ is \emph{ergodic} if $m(B)\in\{0,1\}$ for every Borel set $B$ with $\tau_aB=B$ for all $a\in\R^3$; for a sub-$\sigma$-algebra $\mathcal F\subset\cB$, $L^2(\mathcal F,m)$ denotes the closed subspace of classes with an $\mathcal F$-measurable representative. In the application $X=H^{-r}_{\mathrm{loc}}(\R^3;\cV)$ with $r\ge14$ even, $\Sigma(U)$ is generated by the evaluations $\Phi\mapsto\Phi(u)$, $\supp u\subset U$, and the actions are the push-forwards of the Euclidean motions; $\mu$ is the measure of the curvature field, for which (H1)--(H3) are proved in Section~\ref{sec:s09:measure}. \space% % \begin{theorem}[the measure, the local lattice laws, weak clustering]\label{thm:s09:S2K} The limits below exist by \cref{hyp:s03:EC}. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item \textup{(The measure.)} For every even $r\ge14$ there is exactly one Borel probability measure $\mu^{K,H}$ on $\cH^{-r}(\R^3;\Lambda^2)$ whose mixed moments on $C^\infty_c$ are the Schwinger functions $S^K_n$ of King's state. Its characteristic functional on $C^\infty_c$, indeed on $\cS$, is $\cZ^K$ (extended by continuity); its moments on $\cS$ are the continuous extensions of the $S^K_n$ and satisfy the bounds of \cref{thm:s05:CK}(c). Its image $\mu^K_S$ under the restriction map $\iota:\cH^{-r}\to X$ is the unique Borel probability measure on $X$ with characteristic functional $\cZ^K$ on $C^\infty_c$, and also the unique one whose mixed moments on $C^\infty_c$ are the $S^K_n$. \emph{Local convergence of the lattice laws:} for every real $\chi\in C^\infty_c(\R^3)$ and every box $Q$ containing $\supp\chi$ in its interior, the laws $\nu^K_{k,J,\chi}$ of the localised lattice fields $\chi\Phi^K_{k,J}$ on $H^{-r}_{\mathrm{per}}(Q;\Lambda^2)$ converge weakly, along the full sequence $k\to\infty$ at every fixed $J\ge J_0(\chi)$, to a law $\nu^K_{J,\chi}$, and $\nu^K_{J,\chi}\Rightarrow\nu^K_\chi$ as $J\to\infty$, where $\nu^K_\chi$ is the law of $\chi\Phi$ under $\mu^K_S$. Nothing is claimed about the laws of the unlocalised lattice fields at fixed $J$ (\cref{rem:s09:S2K}). \item \textup{(Invariance and weak clustering.)} $\mu^K_S$ satisfies the hypotheses (H1)--(H3) of Definition~\ref{cpi:imp:H}, is invariant under $\Cub$ about every point, and the general results of \HPapp\ on the ergodic decomposition and on the Osterwalder--Schrader reconstruction in each coordinate direction from reflection positivity about the coordinate planes apply to it (Steps~3 and~4 of the proof): $\mu^K_S=\int\mu_\xi\,d\nu(\xi)$ with components that are translation-invariant and ergodic, $\theta_j$-invariant, reflection positive about every coordinate plane and satisfy (H2); clustering in the mean and under diagonal separation; the Osterwalder--Schrader reconstruction in each coordinate direction with a unique joint vacuum on every component; the growth bound $|S^\xi_N(u_1,\dots,u_N)|\le\sigma^\xi_N\prod_i\norm{u_i}_{r+2}$, $\sigma^\xi_N\le\alpha_\beta(\xi)(N!)^\beta$ for every $\beta>1$, for the components' Schwartz moments; and the cubic-invariant decomposition into mixtures of $k(\zeta)\in\{1,2,3,6\}$ ergodic measures. The same holds on $\cH^{-r}$ for $\mu^{K,H}$. $\nu$-almost every component satisfies \OSf\ in the weak form, ergodicity under the translation group $\R^3$ (Definition~\ref{cpi:imp:OS4forms}(iii)). \item \textup{(The rotation corollary.)} $\cZ^K$ is $O(3)$-invariant, $\cZ^K(R\cdot u)=\cZ^K(u)$ for all $R\in O(3)$ and real $u$, by \cref{thm:s06:O3} (the rotation theorem \cref{hyp:s03:RC}); the clauses below use only this invariance. Then for $\nu$-a.e.\ $\xi$: $\mu_\xi$ is invariant under $\Euc$; $\E_{\mu_\xi}[\overline{F\circ\theta_{n,c}}\,F]\ge0$ for every unit vector $n$, every $c\in\R$ and every $F\in L^2(\Sigma(H^+_{n,c}),\mu_\xi)$, $H^+_{n,c}=\{x\cdot n>c\}$; and $\E_{\mu_\xi}[G'\,U(a)G]\to\E_{\mu_\xi}[G']\,\E_{\mu_\xi}[G]$ as $|a|\to\infty$ in any manner, for all $G,G'\in L^2(\mu_\xi)$. Consequently $\mu_\xi$ satisfies \OSf\ in each of the forms of Definition~\ref{cpi:imp:OS4}, Definition~\ref{cpi:imp:OS4forms} and Definition~\ref{cpi:rot:forms} below (the cluster property on the positive-time algebra, ergodicity and clustering of the translations along the time direction, ergodicity under $\R^3$) for every choice of the time direction, coordinate or not, and in the reconstruction from $\mu_\xi$ in any direction $n$ the joint fixed space is $\ker H_n=\C\Omega_n$ (Step~5 of the proof). The moments of $\mu_\xi$ are real and symmetric; the vanishing of its odd curvature moments is \emph{not} claimed for a component. \end{enumerate} \end{theorem} \noindent\emph{Outline of the proof.} Clause (a) uses \cref{thm:s05:CK,cor:s06:TK} only (Steps~1, 2 and~$2'$). Clause (b) uses also \cref{thm:s06:S1K,lem:s08:OS2K}, which give (H1)--(H3) in Step~3; the general results of \HPapp\ applied in Step~4 are \HPref{ms:erg:decomp} (the decomposition), \HPref{ms:mean} and \HPref{ms:diag} (clustering in the mean and under diagonal separation), \HPref{ms:recon} with \cite[\HPrefraw{ms:components}(iii)]{HP} (the reconstruction and the unique joint vacuum), \HPref{ms:erg:growth} and \HPref{ms:erg:growthX} (the growth bound), \HPref{ms:erg:Ohdecomp} (the cubic decomposition), \HPref{ms:transfer} (the passage to $\cH^{-r}$) and Definition~\ref{cpi:imp:OS4forms}(iii) with \cite[\HPrefraw{ms:components}(i)]{HP} (weak \OSf). Clause (c) uses only the $O(3)$-invariance of $\cZ^K$ and is Corollary~\ref{cpi:rot:cor}, applied in Step~5. \subsection{Proof of (a)} \emph{Step 1 (moment bounds, and where they hold).} For real $2$-forms $f_1,\dots,f_n$ at distance $>1$ from $\R^3\setminus C_{J_0}$, \cref{thm:s05:CK}(c) gives, at every $k$ and $J\ge J_0$, $|S^{K,(k,J)}_n|\le(\sqrt2\,e)^n(n!)^{1/2}\prod_j\norm{f_j}_{2,k}$ with $\norm f_{2,k}^2\le3C_0\fournorm f^2$ uniformly in $k$ (\cref{lem:s02:E}), and in the limit the bound of \cref{thm:s05:CK}(c); by \cref{thm:s05:CK}(a), $\avK{F(u)^2}\le2\norm u^2_{2,k}$ and hence $S^K_2(u,u)\le2\norm u_2^2$. The lattice Schwinger functions are the Taylor coefficients at $0$ of $a^K_{k,J}$, that is the moments of $\Phi^K_{k,J}$, and converge to the $S^K_n$. These bounds are uniform in $k$ and $J\ge J_0$ for sources at distance $>1$ inside $C_{J_0}$; at a fixed $J$ they are not available for sources reaching $\partial\CJ$, and neither is the convergence in $k$: \cref{hyp:s03:EC} and \cref{thm:s05:CK}(a) are stated for sources whose plaquettes have their corners in $\XJ$. \emph{Step 2 (the measure).} (i) \emph{The functional on $\cS$.} For real $u_1,\dots,u_N\in C^\infty_c$ and $c\in\C^N$, at every $(k,J)$ with $J\ge J_0(u)$, $\sum_{l,m}\bar c_lc_m\,Z^K_{k,J}(\mathrm i(u_m-u_l))=\avK{|\sum_mc_me^{\mathrm iF(u_m)}|^2}\ge0$ by the linearity of $F_{\Lk}$ in the source. The limits $k\to\infty$, then $J\to\infty$, exist at these points by \cref{thm:s05:CK}(b) for the family $u$, so $\cZ^K$ is positive definite on $C^\infty_c$ with $\cZ^K(0)=1$. By \cref{cor:s06:TK}'s Lipschitz bound and (E2), $|\cZ^K(u)-\cZ^K(u')|\le\sqrt2\norm{u-u'}_2\le\frac\pi2\fournorm{u-u'}$. Since $\fournorm\cdot$ is a continuous seminorm on $\cS(\R^3;\Lambda^2)$ and $C^\infty_c$ is dense in $\cS$, $\cZ^K$ extends uniquely to a continuous function on $\cS$, still positive definite with $\cZ^K(0)=1$. By the Bochner--Minlos theorem \cite{GJ,Jan97} ($\cS$ is nuclear) there is exactly one Borel probability measure $\mu'$ on $\cS'(\R^3;\Lambda^2)$, with its cylinder $\sigma$-algebra, such that $\int e^{\mathrm i\Phi(u)}\,d\mu'=\cZ^K(u)$ for all real $u\in\cS$. (ii) \emph{Moments on $C^\infty_c$, and on $\cS$ by uniform integrability.} For real $u_1,\dots,u_N\in C^\infty_c$ the characteristic function of $(\Phi(u_l))_l$ under $\mu'$ is $\varphi(t)=\cZ^K(\sum_lt_lu_l)=Z^{K,u}_\infty(\mathrm it)$, the restriction to $\mathrm i\R^N$ of the entire function $Z^{K,u}_\infty$ (\cref{thm:s05:CK}(b)). A characteristic function real-analytic near $0$ has moments of all orders, with $\int\prod_l\Phi(u_l)\,d\mu'=(-\mathrm i)^N\partial_{t_1}\cdots\partial_{t_N}\varphi(0)$ \cite{Luk70}; the chain rule gives $\partial_{t_1}\cdots\partial_{t_N}\varphi(0)=\mathrm i^N\partial_{z_1}\cdots\partial_{z_N}Z^{K,u}_\infty(0)$, so the factors cancel and \begin{equation}\label{eq:s09:moments} \int\prod_{l=1}^N\Phi(u_l)\,d\mu'=S^K_N(u_1,\dots,u_N);\qquad\int\Phi(u)^2\,d\mu'=S^K_2(u,u)\le2\norm u_2^2\le\tfrac{\pi^2}4\fournorm u^2\le C_2''\norm u_{10}^2 \end{equation} by Step~1, (E2) and $\fournorm u\le C\norm u_{10}$. \emph{Extension to $\cS$.} By \cref{thm:s05:CK}(c) each $S^K_N$ extends to a continuous symmetric multilinear form on $\cS^N$ with $|S^K_N|\le\sigma_N\prod_l\fournorm{u_l}$. For $u_l\in\cS$ take cut-offs $\chi_R=\chi(\cdot/R)$, $\chi\in C^\infty_c$, $\chi=1$ on the unit ball, $0\le\chi\le1$, $R\ge1$: $\chi_Ru_l\to u_l$ in $\cS$, so $\Phi(\chi_Ru_l)\to\Phi(u_l)$ for every $\Phi\in\cS'$; and $\fournorm{\chi_Ru}\le C_\chi\fournorm u$ uniformly in $R\ge1$ with $C_\chi=2^4\norm\chi_{C^4}$, by Leibniz with $|R^{-|\beta|}(D^\beta\chi)(\cdot/R)|\le\norm\chi_{C^4}$ (the norm $\fournorm\cdot$ carries derivatives up to order $4$, so the cut-off does not pass through it with constant $1$). Hence $\int\prod_l\Phi(\chi_Ru_l)^2\,d\mu'=S^K_{2N}(\chi_Ru_1,\chi_Ru_1,\dots)\le\sigma_{2N}C_\chi^{2N}\prod_l\fournorm{u_l}^2$ uniformly in $R$: the products $\prod_l\Phi(\chi_Ru_l)$ are bounded in $L^2(\mu')$, hence uniformly integrable, and converge pointwise, so $\int\prod_l\Phi(u_l)\,d\mu'=\lim_RS^K_N(\chi_Ru_1,\dots)=S^K_N(u_1,\dots,u_N)$ by the continuity of $S^K_N$ on $\cS^N$. (iii) \emph{The Hermite carrier.} Put $M(\Phi)\defeq\sum_{\alpha,c}\lambda_\alpha^{-r}\Phi(h_{\alpha,c})^2$. By \eqref{eq:s09:moments} with $\norm{h_{\alpha,c}}_{10}^2=\lambda_\alpha^{10}$, $\int M\,d\mu'\le3C_2''\sum_\alpha\lambda_\alpha^{10-r}<\infty$ as soon as $r-10>3$ (there are $(n+1)(n+2)/2$ multi-indices with $|\alpha|=n$), that is for every even $r\ge14$; so $\mu'(M<\infty)=1$. For $\Phi\in\cS'$ and $u\in\cS$ the Hermite expansion $u=\sum_{\alpha,c}\ip{h_{\alpha,c}}u\,h_{\alpha,c}$ converges in $\cS$ (\cite[\HPrefraw{ms:hermite}(c)]{HP}), so $\Phi(u)=\sum_{\alpha,c}\ip{h_{\alpha,c}}u\,\Phi(h_{\alpha,c})$ and, on $\{M<\infty\}$, $|\Phi(u)|\le M(\Phi)^{1/2}\norm u_r$ by Cauchy--Schwarz: $\Phi$ extends to $\Psi(\Phi)\in\cH^{-r}$ with $\norm{\Psi(\Phi)}_{-r}^2=M(\Phi)$ and $\Psi(\Phi)=\Phi$ on $\cS$. The map $\Phi\mapsto\Psi(\Phi)$ ($\defeq0$ off $\{M<\infty\}$) is measurable from the cylinder $\sigma$-algebra to the Borel $\sigma$-algebra of the separable Hilbert space $\cH^{-r}$, its coordinates being evaluations. Put $\mu^{K,H}\defeq\Psi_*\mu'$. Then $\int e^{\mathrm i\Psi(u)}\,d\mu^{K,H}=\cZ^K(u)$ and $\int\prod_l\Psi(u_l)\,d\mu^{K,H}=S^K_N(u_1,\dots,u_N)$ for all $u,u_l\in\cS$, by (ii). This is \HPref{ms:carrier}, applied with the two-point bound \eqref{eq:s09:moments} on $\cS$. (iv) \emph{Uniqueness on $\cH^{-r}$.} By \cite[\HPrefraw{ms:unique}(b)]{HP} on $\cH^{-r}$, $\mu^{K,H}$ is the only Borel probability measure on $\cH^{-r}$ whose mixed moments on $C^\infty_c$ exist and equal the $S^K_n$, provided the moments $m_n$ of $Z=\sum_lt_l\Psi(u_l)$ satisfy $|m_n|\le c_S(n+1)K^nn!$ for constants $c_S$, $K$ depending on the $u_l$ and $t$. Our moments give this with $c_S=1$: $m_n=S^K_n(g,\dots,g)$ for $g=\sum_lt_lu_l$, and $|m_n|\le(\sqrt2\,e)^n(n!)^{1/2}\norm g_2^n\le(n+1)K^nn!$ with $K=\sqrt2\,e\norm g_2$. (v) \emph{The law on $X$ and its two uniqueness statements.} The map $\iota:\cH^{-r}\to X$, $\iota\Psi\defeq\Psi|_{C^\infty_c}$, is continuous, linear and injective (\cite[\HPrefraw{ms:borel}(d)]{HP}): $|\Psi(\chi f)|\le\norm\Psi_{-r}\norm{\chi f}_r\le C_\chi\norm\Psi_{-r}\norm f_{H^r}$ for a cut-off $\chi$, and $C^\infty_c$ is dense in $\cS$, which is dense in $\cH^r$. Put $\mu^K_S\defeq\iota_*\mu^{K,H}$; its characteristic functional on real $C^\infty_c$ is $\cZ^K$. By \cite[\HPrefraw{ms:unique}(a)]{HP} on $X$ (for the countable $(f_n)\subset C^\infty_c$ of \cite[\HPrefraw{ms:borel}(c)]{HP} the sets $\{\Phi:(\Phi(f_1),\dots,\Phi(f_N))\in B\}$ form a $\pi$-system generating the Borel $\sigma$-algebra, and the law of each such vector is determined by $t\mapsto\cZ^K(\sum_kt_kf_k)$), $\mu^K_S$ is the only Borel probability measure on $X$ with this characteristic functional. \emph{Moment uniqueness on $X$:} if $\mu''$ is any Borel probability measure on $X$ whose mixed moments on $C^\infty_c$ are the $S^K_n$, \cite[\HPrefraw{ms:unique}(b)]{HP} on $X$ applies with the moment bound of (iv), so $\mu''=\mu^K_S$. And $\mu^{K,H}$ is the only measure on $\cH^{-r}$ with $\iota_*\mu^{K,H}=\mu^K_S$, since $\iota$ is a Borel isomorphism onto the Borel set $\iota\cH^{-r}$. Nothing in Step~2 sees the lattice: the measure clauses of (a) depend on King's state only through $\cZ^K$ and the $S^K_n$. \emph{Step $2'$ (local convergence of the lattice laws).} Fix a real $\chi\in C^\infty_c(\R^3)$ and a box $Q$ with $\supp\chi$ compact in the interior of $Q$, and let $(e_{\alpha,c})_{\alpha\in\Z^3,\,c\le3}$ be the Fourier basis of $L^2(Q;\Lambda^2)$, $|e_{\alpha,c}(x)|=|Q|^{-1/2}$. For a real $Q$-periodic smooth $2$-form $\psi$ put $\ip{\chi\Phi^K_{k,J}}\psi\defeq\Phi^K_{k,J}(\chi\psi)=F_{\Lk}(\chi\psi)$, where $\chi\psi\in C^\infty_c$ is supported in $\supp\chi$ (for the complex basis vectors, used only in (1), take real and imaginary parts); this identifies $\chi\Phi^K_{k,J}$ with an element of the real Hilbert space $H_\chi\defeq H^{-r}_{\mathrm{per}}(Q;\Lambda^2)$, and $\nu^K_{k,J,\chi}$ denotes its law. (1) \emph{Second moments, uniformly.} For $J\ge J_0(\chi)$ and every $k$, every plaquette carrying $\chi e_{\alpha,c}$ has its centre in $\supp\chi$, hence its corners in $\XJ$, so the second-moment clause of \cref{thm:s05:CK}(a) applies to the real and imaginary parts: $\E^K_{k,J}|F_{\Lk}(\chi e_{\alpha,c})|^2\le2\norm{\chi\Re e_{\alpha,c}}^2_{2,k}+2\norm{\chi\Im e_{\alpha,c}}^2_{2,k}\le4\norm{\chi e_{\alpha,c}}^2_{2,k}$. \emph{Riemann-sum bound:} the plaquettes of one orientation have their centres on a translate of $\eps_k\Z^3$; those carrying $\chi e_{\alpha,c}$ have centres in $\supp\chi$; the cubes of side $\eps_k\le1$ about these centres are disjoint and lie in the $\frac12$-neighbourhood $N_{1/2}(\supp\chi)$; so $\norm{\chi e_{\alpha,c}}^2_{2,k}\le3\,\mathrm{vol}(N_{1/2}(\supp\chi))\,\norm\chi_\infty^2/|Q|=:K_\chi$, uniformly in $\alpha$, $c$, $k$ and $J\ge J_0(\chi)$ (no derivative of $\chi$ enters). Hence, for $s_0>\frac32$, \begin{equation}\label{eq:s09:secondmoment} \E^K_{k,J}\norm{\chi\Phi^K_{k,J}}^2_{H^{-s_0}_{\mathrm{per}}(Q)}=\sum_{\alpha,c}(1+|\alpha|^2)^{-s_0}\,\E^K_{k,J}|F_{\Lk}(\chi e_{\alpha,c})|^2\ \le\ 12K_\chi\sum_{\alpha\in\Z^3}(1+|\alpha|^2)^{-s_0}=:C'_\chi<\infty, \end{equation} uniformly in $k$ and $J\ge J_0(\chi)$ (the constants $4$ and $12K_\chi$ are valid but not sharp). (2) \emph{Tightness.} The embedding $H^{-s_0}_{\mathrm{per}}(Q)\hookrightarrow H^{-r}_{\mathrm{per}}(Q)$ is compact for $r>s_0$ (Rellich on the torus); by Chebyshev, $\nu^K_{k,J,\chi}(\norm\cdot_{H^{-s_0}}>\rho)\le C'_\chi/\rho^2$, and the $H^{-s_0}$-ball of radius $\rho$ is compact in $H_\chi$; so $(\nu^K_{k,J,\chi})_{k\ge0,\,J\ge J_0(\chi)}$ is tight on the separable Hilbert space $H_\chi$, and Prokhorov's theorem \cite{Bil99} gives weak limit points along $k$ at fixed $J$. (3) \emph{Identification at fixed $J$: the full sequence, and where \cref{hyp:s03:EC} enters.} For a real trigonometric polynomial $\psi$, $\E^K_{k,J}e^{\mathrm i\ip{\chi\Phi^K_{k,J}}\psi}=Z^K_{k,J}(\mathrm i\chi\psi)\to\ZKJ(\mathrm i\chi\psi)$ as $k\to\infty$, along the full sequence, by \cref{thm:s05:CK}(b) (that is \cref{cor:s04:G2}(ii)) for the one-member family $(\chi\psi)$ at $J\ge J_0(\chi)\ge J_0(\chi\psi)$: this is \cref{hyp:s03:EC} for that family on King's cube, together with the local bound of \cref{thm:s05:CK}(a) and Montel's theorem. A Borel probability measure on $H_\chi$ is determined by its characteristic functional on the dual $H^r_{\mathrm{per}}(Q;\Lambda^2)$, which is continuous there (dominated convergence), hence by its values on the dense set of real trigonometric polynomials. So every weak limit point of $(\nu^K_{k,J,\chi})_k$ is the one probability measure $\nu^K_{J,\chi}$ on $H_\chi$ with characteristic functional $\psi\mapsto\ZKJ(\mathrm i\chi\psi)$, and $\nu^K_{k,J,\chi}\Rightarrow\nu^K_{J,\chi}$ along the whole sequence $k\to\infty$ by tightness and the uniqueness of the limit point. Without \cref{hyp:s03:EC} the same argument gives the convergence along every continuum subsequence (\cref{thm:s12:LIM,cor:s12:KK}), not along the full sequence; the full-sequence clause is what \cref{hyp:s03:EC} adds for King's state. (4) \emph{$J\to\infty$.} $\norm\cdot^2_{H^{-s_0}_{\mathrm{per}}}=\sum_{\alpha,c}(1+|\alpha|^2)^{-s_0}|\ip\cdot{e_{\alpha,c}}|^2$ is lower semicontinuous on $H_\chi$ (a supremum of finite sums of continuous functions), so by the portmanteau theorem \cite{Bil99} the bound \eqref{eq:s09:secondmoment} passes to $\nu^K_{J,\chi}$: $\int\norm\cdot^2_{H^{-s_0}}\,d\nu^K_{J,\chi}\le C'_\chi$ for every $J\ge J_0(\chi)$; hence $(\nu^K_{J,\chi})_J$ is tight on $H_\chi$ by the argument of (2), and $\ZKJ(\mathrm i\chi\psi)\to\ZKinf(\mathrm i\chi\psi)=\cZ^K(\chi\psi)$ (\cref{thm:s05:CK}(b)) identifies every limit point: $\nu^K_{J,\chi}\Rightarrow\nu^K_\chi$, the measure with characteristic functional $\psi\mapsto\cZ^K(\chi\psi)$. (5) \emph{$\nu^K_\chi$ is the law of $\chi\Phi$ under $\mu^K_S$.} For $\Phi\in X$, $\chi\Phi\in H^{-r}(\R^3;\Lambda^2)$ is supported in $\supp\chi$, compact in the interior of $Q$, so its periodisation is an element of $H_\chi$ with $\ip{\chi\Phi}\psi=\Phi(\chi\psi)$ for $Q$-periodic smooth $\psi$; the map $\Phi\mapsto\chi\Phi$ is continuous from $X$ to $H_\chi$, since $|\Phi(\chi\psi)|\le\norm{\tilde\chi\Phi}_{H^{-r}}\norm{\chi\psi}_{H^r}\le c_\chi\norm{\tilde\chi\Phi}_{H^{-r}}\norm\psi_{H^r_{\mathrm{per}}(Q)}$ for a cut-off $\tilde\chi=1$ on $Q$. And $\int e^{\mathrm i\ip{\chi\Phi}\psi}\,d\mu^K_S=\int e^{\mathrm i\Phi(\chi\psi)}\,d\mu^K_S=\cZ^K(\chi\psi)$ by Step~2(v). (6) \emph{Moments.} For every trigonometric polynomial $\psi$ (indeed every $\psi\in H^r_{\mathrm{per}}(Q;\Lambda^2)$, by the uniform-integrability argument of Step~2(ii) with $\chi\psi_n\to\chi\psi$), all moments of $\ip{\chi\Phi^K_{k,J}}\psi=F_{\Lk}(\chi\psi)$ are bounded uniformly in $k$ and $J\ge J_0(\chi)$ (Step~1; $\fournorm{\chi\psi}\le d_\chi\norm\psi_{C^4}$ by Leibniz), so by uniform integrability the moments of $\nu^K_{J,\chi}$ are the Taylor coefficients of $\ZKJ$ at $\chi\psi$, and those of $\nu^K_\chi$ are the $S^K_n(\chi\psi,\dots,\chi\psi)$, consistently with \eqref{eq:s09:moments}. This proves (a). \qed \begin{remark}[why the convergence clause is local]\label{rem:s09:S2K} The convergence $Z^K_{k,J}(\mathrm iu)\to\ZKJ(\mathrm iu)$ is available only for sources $u$ whose plaquettes have their corners in $\XJ$, that is for $u$ at distance $>1$ inside $\CJ$, where \cref{hyp:s03:EC} speaks. For a $u$ reaching $\partial\CJ$ the lattice functional is defined only after a convention for the plaquettes with a corner outside $\XJ$ (for instance, drop them), and for it no bound uniform in $k$ and no convergence is claimed: the second-moment bound of \cref{thm:s05:CK}(a) carries the corner hypothesis, and the crude bound $\avK{e^{F(\tilde u)}}\le e^{\frac12\lmu^{-2}\norm{\dstar_k\tilde u}^2_{2,k}}$ (\cref{lem:s02:GD} with $C\le\lmu^{-2}$ as forms) is not uniform in $k$ for a source $\tilde u$ truncated at $\partial\CJ$: $\dstar_k\tilde u$ is of order $\norm u_\infty/\eps_k$ on the $O(\eps_k^{-2}|\partial\CJ|)$ bonds of the truncation layer, so $\norm{\dstar_k\tilde u}^2_{2,k}=O(\eps_k^{-1})$. In particular the fixed-$J$ laws of the unlocalised lattice field on $X$ are not claimed to be tight; nothing below depends on them. \end{remark} \subsection{Proof of (b) and (c)} \emph{Step 3 ((H1)--(H3)).} The hypotheses (H1)--(H3) are those of Definition~\ref{cpi:imp:H}. On $X$ with the actions of the Conventions, the structural parts of (H1) (a Polish space with a jointly continuous $\R^3$-action and involutive homeomorphisms $\theta_j$ with $\theta_j\tau_a=\tau_{\rho_ja}\theta_j$) and all of (H2) hold by \HPref{ms:struct}, which uses only the space and the push-forward actions. \emph{Invariance of $\mu^K_S$:} by the uniqueness in Step~2(v) (\cite[\HPrefraw{ms:discharge}(a)]{HP}) it suffices that $\cZ^K(\tau_au)=\cZ^K(u)$ and $\cZ^K(\sigma_j\cdot u)=\cZ^K(u)$ for real $u$; the first is \cref{cor:s06:TK} at the coefficient $\mathrm i$, the second \cref{thm:s06:S1K} at $z=\mathrm i$ with $O=\sigma_j$, and the same theorem with $O=\tau_bg\tau_{-b}$ gives the $\Oh$-invariance about every point. \emph{(H3):} for $E_a=e^{\mathrm i\Phi(u_a)}$ with $u_a$ real in $C^\infty_c(\{x_j>c\};\Lambda^2)$, $\overline{E_a\circ\theta_{j,c}}\,E_b=e^{\mathrm i\Phi(u_b-\rho_{j,c*}u_a)}$ and \[ \E_{\mu^K_S}\big[\overline{E_a\circ\theta_{j,c}}\,E_b\big]=\cZ^K(u_b-\rho_{j,c*}u_a)=\ZKinf(\theta\zeta_a^*\sqcup\zeta_b),\qquad\zeta_a=(\mathrm i,u_a), \] the conjugated coefficient $-\mathrm i$ sitting on the reflected source as in \cref{def:s07:reflection}. By \cref{lem:s08:OS2K}, for the plane $x_j=c$ and every $c\in\R$, this Gram matrix is positive semidefinite. The exponential algebra is dense in $L^2(\Sigma(H^+_{j,c}),\mu^K_S)$ (\HPref{ms:density}: an algebra of bounded complex functions, closed under conjugation and containing the constants, is dense in $L^2$ of the $\sigma$-algebra it generates), and $\theta_{j,c}$ preserves $\mu^K_S$, so (H3) follows by \cite[\HPrefraw{ms:discharge}(c)]{HP}. Finally \HPref{ms:transfer} carries (H1)--(H3) and the $\Oh$-invariance about every point from $(X,\mu^K_S)$ to $(\cH^{-r},\mu^{K,H})$. \emph{Step 4 (weak clustering).} The general results of \HPapp\ used here, the ergodic decomposition and the Osterwalder--Schrader reconstruction \cite{OS73,OS75}, \cite[Thm.~6.1.3]{GJ} in each coordinate direction from reflection positivity about the coordinate planes ($\R^3$-ergodicity makes the joint fixed space $\C\Omega$, and the self-adjoint contraction semigroups then give clustering in the mean and under diagonal separation), assume only that $\mu$ satisfies (H1)--(H3) of Definition~\ref{cpi:imp:H}. So \HPref{ms:erg:decomp} gives the ergodic decomposition $\mu^K_S=\int\mu_\xi\,d\nu$ with components translation-invariant and ergodic, $\theta_j$-invariant, satisfying (H3) and (H2): the decomposition is the regular conditional probability given the translation-invariant $\sigma$-algebra, and every invariant set lies, modulo null sets, in $\Sigma(H^+_j)\cap\Sigma(H^-_j)$ by the local structure, so that reflection positivity applied to $\mathbf 1_B-\mathbf 1_B\circ\theta_j$ forces the $\theta_j$-invariance of the invariant functions, and almost every component is reflection invariant and reflection positive. \HPref{ms:mean} and \HPref{ms:diag} give clustering in the mean and under diagonal separation (von Neumann's mean ergodic theorem for the strongly continuous unitary $\R^3$-action on $L^2(\mu)$: the cube averages of $U(a)$ converge strongly to the projection onto the invariant functions, the conditional expectation given the invariant $\sigma$-algebra; by the coordinate-direction reconstruction, translation by $a_j$ acts as the contraction $e^{-|a_j|H_j}$ and the transverse components as unitaries, so that as $\min_j|a_j|\to\infty$ the correlation tends to the projection onto the joint fixed space, the conditional expectation given the invariant $\sigma$-algebra). \HPref{ms:recon} is the reconstruction in direction $j$, and on an ergodic component the joint fixed space is $\C\Omega_j$ (\cite[\HPrefraw{ms:fixed}(iii)]{HP}, \cite[\HPrefraw{ms:components}(iii)]{HP}: the mean ergodic theorem on $\cH_j$ and on $L^2(\mu)$ takes the Ces\`aro means of $T_j(t)$ and $V_j(y)$ to the class of the conditional expectation on the invariant $\sigma$-algebra, which lies in $L^2_{j,+}$, so the joint fixed space is $\C\Omega_j$ if and only if $\mu$ is ergodic). The hypothesis of \HPref{ms:erg:growth}, a moment bound on the Hermite basis, is provided for $\mu^{K,H}$ by \cref{thm:s05:CK}(c) through \eqref{eq:s09:moments}: $\int\Psi(h_{\alpha,c})^{2n}\,d\mu^{K,H}=S^K_{2n}(h_{\alpha,c},\dots,h_{\alpha,c})\le(e\pi/2)^{2n}((2n)!)^{1/2}\fournorm{h_{\alpha,c}}^{2n}\le(e\pi/2)^{2n}(2n)!\,\fournorm{h_{\alpha,c}}^{2n}$, and by \HPref{ms:hermite}, with the inequality $\fournorm u\le C\norm u_{10}$ of the Conventions, that bound holds with $c_S=1$ and $C_0=Ce\pi/2$. So \HPref{ms:erg:growth} gives the growth bound of (b) for the components of $\mu^{K,H}$ (the factorial growth passes to almost every component by Chebyshev's inequality and Borel--Cantelli in $\nu$: from the Hermite bound $\E_\mu\|\Psi\|_{-r}^{2N}\le c_S(4C_1)^N(N!)^2$, $C_1=w_rC_0^2$ with the Hermite weight sum $w_r\defeq3\sum_{\alpha\in\N^3}\lambda_\alpha^{10-r}$, finite for $r\ge14$, obtained by Minkowski's inequality in $L^N$ over the Hermite expansion of the $H^{-r}$ norm, $\int(\sigma^\xi_N)^2\,d\nu\le\E_\mu\|\Psi\|_{-r}^{2N}\le c_S(4C_1)^N(N!)^2$ (the constant $\sigma^\xi_N\defeq\E_{\mu_\xi}\|\Psi\|_{-r}^N$ serves, as $|\Psi(u)|\le\|\Psi\|_{-r}\|u\|_r$, and $(\E_{\mu_\xi}\|\Psi\|_{-r}^N)^2\le\E_{\mu_\xi}\|\Psi\|_{-r}^{2N}$ by Jensen's inequality), so $\nu\{\sigma^\xi_N>Nc_S^{1/2}(4C_1)^{N/2}N!\}\le N^{-2}$, whence the bound for $N\ge n_0(\xi)$, $\nu$-almost everywhere), and \HPref{ms:erg:growthX}, with the restriction map $\iota$ of Step~2(v), for those of $\mu^K_S$; the translation invariance of $\mu^{K,H}$ and the joint continuity of the translations on $\cH^{-r}$ assumed there are (H1) for $(\cH^{-r},\mu^{K,H})$, from Step~3. The norm $\norm{u_i}_{r+2}$ of (b) dominates $\norm{u_i}_r$. \HPref{ms:erg:Ohdecomp}, the ergodic decomposition for $\R^3\rtimes\Oh$, in which a component is the $\Oh$-orbit average of an $\R^3$-ergodic measure invariant under the coordinate reflections, so that the orbit is an $S_3$-orbit of size $1$, $2$, $3$ or $6$, assumes in addition that $\Oh$ acts on $X$ by homeomorphisms with $\rho_j=\theta_j$, $g\tau_ag^{-1}=\tau_{ga}$, $\mu$ invariant under every $g$, and $G\mapsto G\circ g^{-1}$ mapping $\Sigma(U)$-measurable functions to $\Sigma(gU)$-measurable ones: structural for the push-forward action ($(g\Phi)(u)=\Phi(g^{-1}\cdot u)$ is a linear homeomorphism of $X$, the relations are identities on test functions, $\supp g\cdot u=g\supp u$) together with the invariance of Step~3; it gives the cubic-invariant decomposition with $k(\zeta)\in\{1,2,3,6\}$. Weak \OSf\ is Definition~\ref{cpi:imp:OS4forms}(iii), ergodicity for the action of $\R^3$, the ergodicity clause of the decomposition for $\nu$-a.e.\ component. \emph{Step 5 (the rotation corollary).} Corollary~\ref{cpi:rot:cor} assumes (H1)--(H3) and an admissible action of $O(3)$ on $X$ (Definition~\ref{cpi:rot:adm}: homeomorphisms $x\mapsto Rx$, continuous in $R$ for each $x$, with $\rho_jx=\theta_jx$ for the coordinate reflections, $R\tau_aR^{-1}=\tau_{Ra}$, and $G\mapsto G\circ R^{-1}$ mapping $\Sigma(U)$-measurable functions to $\Sigma(RU)$-measurable ones), under which $\mu$ is invariant. (H1)--(H3) hold by Step~3. The push-forward $(R\Phi)(u)=\Phi(R^{-1}\cdot u)$ is an admissible action on $X=H^{-r}_{\mathrm{loc}}(\R^3;\Lambda^2)$ by Lemma~\ref{cpi:rot:action}(i), with $\cV=\Lambda^2$. By Lemma~\ref{cpi:rot:action}(ii), the invariance of $\mu^K_S$ under every $R\in O(3)$ is the identity $\cZ^K(R\cdot u)=\cZ^K(u)$ for real $u$, the hypothesis of (c), that is the $O(3)$-invariance of $\ZKinf$ at imaginary coefficients, provided by \cref{thm:s06:O3}; this section does not derive it. So Corollary~\ref{cpi:rot:cor} applies to $\mu^K_S$ and its ergodic decomposition of Step~4, and gives the conclusions quoted in (c): the Euclidean invariance and the reflection positivity about every plane of almost every component (Corollary~\ref{cpi:rot:cor}(a),(b)), clustering as $|a|\to\infty$ in any manner (Corollary~\ref{cpi:rot:cor}(c)), every form of \OSf\ for every time direction, coordinate or not, and $\ker H_n=\C\Omega_n$ in every direction $n$ (the final clause of Corollary~\ref{cpi:rot:cor}). They contain nothing on \OSth: the vanishing of the odd curvature moments of a component is not among them, and is not known for a single component. \emph{Step 6 (curvature alone).} Everything above runs on $X=H^{-r}_{\mathrm{loc}}(\R^3;\Lambda^2)$ for the curvature field alone: the regularity statement is \cref{thm:s05:CK}, the reflection positivity statement \cref{lem:s08:OS2K}, the invariance statement \cref{thm:s06:S1K}, the $O(3)$ statement that of (c). The result is an ergodic decomposition of the law of the curvature field; nothing is claimed about the components of a joint law with the Wick square. \qed \begin{remark}[the hypotheses of the general results, matched]\label{rem:s09:v1match} For each general result of \HPapp\ and each statement from the literature used above we list the step that establishes its hypotheses. (H1)--(H3) of Definition~\ref{cpi:imp:H}: Step~3 (structure from \HPref{ms:struct}; invariance from \cref{cor:s06:TK,thm:s06:S1K} through \cite[\HPrefraw{ms:discharge}(a)]{HP}; positivity from \cref{lem:s08:OS2K}, \HPref{ms:density} and \cite[\HPrefraw{ms:discharge}(c)]{HP}). \HPref{ms:hermite}: no hypothesis; Step~2(iii). The two-point bound $\E\Phi(f)^2\le C\norm f_{10}^2$ of \HPref{ms:carrier}: Step~2(ii). The moment bound of \cite[\HPrefraw{ms:unique}(b)]{HP}, $r\ge14$ even: Step~2(iv), with $K=\sqrt2\,e\norm g_2$ and $c_S=1$. The uniqueness of a measure on $X$ with a given characteristic functional (\cite[\HPrefraw{ms:unique}(a)]{HP}, with the Borel structure of \HPref{ms:borel}): Step~2(v). The transfer between $X$ and $\cH^{-r}$ (\HPref{ms:transfer}): Steps~2(v) and~3. The standing hypothesis ``$\mu$ satisfies (H1)--(H3)'' of \HPref{ms:erg:decomp}, \HPref{ms:mean}, \HPref{ms:diag}, \HPref{ms:recon} and \HPref{ms:fixed}: Step~3; the ergodicity needed in \cite[\HPrefraw{ms:fixed}(iii)]{HP} holds on a component by \HPref{ms:erg:decomp}. \HPref{ms:erg:Ohdecomp} (the $\Oh$ action and invariance): Step~4 and \cref{thm:s06:S1K}. The moment bound on the Hermite basis assumed in \HPref{ms:erg:growth}: \cref{thm:s05:CK}(c) through Step~2(ii), with $c_S=1$ and $C_0=Ce\pi/2$ (Step~4). Corollary~\ref{cpi:rot:cor} (the admissible $O(3)$ action and the invariance): Step~5, the structure from Lemma~\ref{cpi:rot:action}(i), the invariance from the hypothesis of (c) through Lemma~\ref{cpi:rot:action}(ii). The forms of \OSf\ (Definition~\ref{cpi:imp:OS4}, Definition~\ref{cpi:imp:OS4forms}): definitions, Steps~4 and~5; Definition~\ref{cpi:imp:OS4forms}(iii) is the ergodicity clause of \HPref{ms:erg:decomp}. The same matching is made for the joint field with the Wick square in the proof of \cref{prop:s13:WK}(d) and for the limit states of the comparison family in the proof of \cref{thm:s12:LIM}(v),(vi); for Corollary~\ref{cpi:rot:cor} it is made in the proof of \cref{thm:s12:LIM}(vi), where the admissible push-forward action (Lemma~\ref{cpi:rot:action}(i)) and the $O(3)$-invariance of the characteristic functional are established. The vanishing of the odd moments is not transferred to the components (clause (c)). \end{remark} \begin{remark}[the comparison family]\label{rem:s09:components} The same conclusions hold for the limit states of the comparison family $\NDa$: for every continuum subsequence $\mathbf{k}$ (\cref{def:s12:LIMstate}) the limit state $\ZLIM{\mathbf{k}}$ carries a measure with the properties of \cref{thm:s09:S2K}(a) and (b), the local lattice laws converging along $\mathbf{k}$ instead of the full sequence, and (c) (\cref{thm:s12:LIM}, clauses (v)--(vii)). By \cref{hyp:s03:EC} every such limit state equals $\ZKinf$ (\cref{cor:s04:G2}(iii)), so for King's state the two routes to the measure agree. Whether the comparison family's own fixed-cube limits exist is not decided here (Remark~\ref{rem:s04:notfollow}, \cref{prop:s12:ID}(b)); nothing beyond \cref{cor:s04:G2}(iii) is claimed about its iterated limit. \end{remark} \space% % % % \subsection{Rotation invariance: reflection positivity about every plane and clustering in every direction}\label{cpi:rot:sec} A law with the properties (H1)--(H3) which is moreover invariant under $O(3)$ has almost all of its ergodic components invariant under the Euclidean group, reflection positive about every plane, and clustering as the separation tends to infinity in any manner; hence every form of \OSf\ holds on almost every component for every choice of the time direction, coordinate or not. Reflection positivity is essential: without it an $O(3)$-average of rotated copies of a translation-ergodic law is $O(3)$-invariant with components that are not. The two ideas are that a translation-invariant function is fixed by every reflection (Lemma~\ref{cpi:rot:inv}), which passes the $O(3)$-invariance of $\mu$ to its components, and that an $O(3)$-invariant ergodic law clusters along every sequence of separations, because the sequence can be rotated onto the diagonal, where clustering is known (Lemma~\ref{cpi:rot:diag}). \emph{Setting.} $\mu$ is a Borel probability measure on a Polish space $X$ satisfying (H1)--(H3) (as in Theorem~\ref{thm:s09:S2K}(b)), with the translations $\tau_a$, the coordinate reflections $\theta_j$, the local $\sigma$-algebras $\Sigma(U)$, and $U(a)G\defeq G\circ\tau_{-a}$, $\Theta_jG\defeq G\circ\theta_j$ on $L^2(\mu)$; the functions in $L^2(\Sigma(K),\mu)$, $K$ bounded, are dense in $L^2(\mu)$. $\mu=\int\mu_\xi\,d\nu(\xi)$ is its ergodic decomposition and $\mathcal J$ the $\sigma$-algebra of translation-invariant Borel sets completed by the $\mu$-null sets; we use that $\xi\mapsto\E_{\mu_\xi}[G]$ is a version of $\E_\mu[G\mid\mathcal J]$, that is $\E_\mu[WG]=\int W(\xi)\,\E_{\mu_\xi}[G]\,d\nu(\xi)$ for bounded $\mathcal J$-measurable $W$, and that for $\nu$-almost every $\xi$ the component $\mu_\xi$ is translation invariant, ergodic and satisfies (H1)--(H3). For a coordinate direction $j$, $L^2_{j,+}\defeq L^2(\Sigma(H^+_{j,0}),\mu)$, and $\cH_j$, $[\,\cdot\,]$, $\Omega_j$, $H_j$ and the transverse translations $V_j(y)$ are the objects of the Osterwalder--Schrader reconstruction in direction $j$. For a unit vector $n\in\R^3$ and $c\in\R$ put $H^+_{n,c}\defeq\{x:x\cdot n>c\}$, $H^-_{n,c}\defeq\{x:x\cdot n0$ let $\varphi_{k,q,m}(x)\defeq\min\big(1,m\,d(x,X\setminus B(x_k,q))\big)$, and let $\cD_0$ be the countable set of finite products of these bounded continuous functions. As $m\to\infty$, $\varphi_{k,q,m}\uparrow\mathbf 1_{B(x_k,q)}$, so two Borel probability measures that agree on $\cD_0$ agree on the finite intersections of the balls $B(x_k,q)$, a $\pi$-system generating the Borel sets, and are equal. Let $\mathcal R_{\mathrm d}\subset O(3)$ be a countable dense subgroup; for instance the group generated by the coordinate reflection $\rho_1$ together with the countable group $\Gz$ of Section~\ref{sec:rot}, which is generated by the finite set $\Rgen$ of the rotations $R_{\mu\nu}(\pm\thz)$ introduced there and is dense in $SO(3)$. Fix $R\in\mathcal R_{\mathrm d}$, $G\in\cD_0$ and a bounded $\mathcal J$-measurable $W$. Then $W$ agrees $\mu$-a.e.\ with a translation-invariant Borel function, so $U(a)W=W$ in $L^2(\mu)$ for every $a$, and $\Gamma_RW=W$ by Lemma~\ref{cpi:rot:inv}(b). By $R$-invariance of $\mu$, \[ \E_\mu[W\,\Gamma_RG]=\E_\mu[(\Gamma_RW)(\Gamma_RG)]=\E_\mu[\Gamma_R(WG)]=\E_\mu[WG]. \] Disintegrating both sides, $\int W(\xi)\,D(\xi)\,d\nu(\xi)=0$ with $D(\xi)\defeq\E_{\mu_\xi}[\Gamma_RG]-\E_{\mu_\xi}[G]$, which is bounded, real and $\mathcal J$-measurable; $W=\mathbf 1_{\{D>0\}}$ and $W=\mathbf 1_{\{D<0\}}$ give $D=0$ $\nu$-a.e. The pairs $(R,G)$ are countably many, so off one $\nu$-null set the image of $\mu_\xi$ under $R^{-1}$ agrees with $\mu_\xi$ on $\cD_0$ for all $R\in\mathcal R_{\mathrm d}$, and $\mu_\xi$ is invariant under $\mathcal R_{\mathrm d}$. For such $\xi$ and bounded continuous $G$, $R\mapsto\E_{\mu_\xi}[G\circ R^{-1}]$ is continuous on $O(3)$ (continuity of $R\mapsto R^{-1}x$ and dominated convergence) and equals $\E_{\mu_\xi}[G]$ on the dense set $\mathcal R_{\mathrm d}$, hence everywhere: $\mu_\xi$ is $O(3)$-invariant, and with its translation invariance this is (a). For $\nu$-a.e.\ $\xi$ the component satisfies (H1)--(H3), so Lemma~\ref{cpi:rot:frame}(a) applied to $\mu_\xi$ gives (b). \emph{Step 2: (c).} Fix such a $\xi$ and write $m\defeq\mu_\xi$: ergodic under $\R^3$, $O(3)$-invariant, with (H1)--(H3). Each $\Gamma_R$ is unitary on $L^2(m)$, and $R\mapsto\Gamma_R$ is strongly continuous: for bounded continuous $G$ by dominated convergence, in general by density and $\norm{\Gamma_R}=1$. Put $n_0\defeq(1,1,1)/\sqrt3$, let $\abs{a^{(l)}}\to\infty$, and choose $R_l\in O(3)$ with $R_la^{(l)}=b_l\defeq\abs{a^{(l)}}\,n_0$. By $R_l$-invariance and \eqref{eq:cpi:rot:cov}, \[ \E_m\big[G'\,U(a^{(l)})G\big]=\E_m\big[\Gamma_{R_l}\big(G'\,U(a^{(l)})G\big)\big]=\E_m\big[(\Gamma_{R_l}G')\,U(b_l)(\Gamma_{R_l}G)\big]. \] Every subsequence has a further subsequence along which $R_l\to R\in O(3)$, and along it, since $\abs{\E_m[A\,U(b)B]}\le\norm A_2\norm B_2$, \begin{multline*} \big|\E_m\big[(\Gamma_{R_l}G')\,U(b_l)(\Gamma_{R_l}G)\big]-\E_m\big[(\Gamma_RG')\,U(b_l)(\Gamma_RG)\big]\big|\\ \le\norm{\Gamma_{R_l}G'-\Gamma_RG'}_2\norm G_2+\norm{G'}_2\norm{\Gamma_{R_l}G-\Gamma_RG}_2\to0 . \end{multline*} Every coordinate of $b_l$ equals $\abs{a^{(l)}}/\sqrt3\to\infty$, so Lemma~\ref{cpi:rot:diag} for the ergodic law $m$ gives $\E_m[(\Gamma_RG')\,U(b_l)(\Gamma_RG)]\to\E_m[\Gamma_RG']\,\E_m[\Gamma_RG]=\E_m[G']\,\E_m[G]$. The limit does not depend on the subsequence, so (c) follows. \emph{Step 3: the forms of the cluster property.} Fix $m$ as in Step~2 and a unit vector $n$. (C$_n$) is (c) with $a=tn$. If $U(tn)G=G$ for all $t$, then (c) with $G'=\overline G$ gives $\E_m\abs G^2=\E_m[\overline G\,U(tn)G]\to\abs{\E_mG}^2$, so $G$ is a.s.\ constant: (E$_n$) holds, and $\ker H_n=\C\Omega_n$ by Proposition~\ref{cpi:rot:kerH}; the joint fixed space contains $\Omega_n$ and lies in $\ker H_n$. For (OS$_n$), let $F,G\in\mathcal P^+_n$ and $a\perp n$, $a\ne0$; then $\overline{F\circ\rho_n}\in L^2(m)$, and (c) along $\lambda a$ gives the limit $\E_m[\overline{F\circ\rho_n}]\,\E_m[G]=\E_m[\overline F]\,\E_m[G]$ by the $\rho_n$-invariance of $m$. \end{proof} \space% % \space% % % % \section{The Wick tube I: torus theorems and the extensive bound}\label{sec:s10} The Wick leg controls the joint functional of the curvature and the Wick square on the tube $\Dtube$ of \cref{def:s02:tube}. Its positive leg (sources $q\ge0$) and negative leg (sources $-q$) are proved in the next section from three statements on \emph{King's tori}: lattice reflection positivity (\cref{prop:s10:RP}), the chessboard estimate it yields (\cref{lem:s10:CB}), and an \emph{extensive bound} \Itwo\ for constant positive scalar sources in the fixed-spacing volume limit (\cref{def:s10:I2}). This section proves the first two and derives the third from the torus theorems \cref{hyp:s03:T1,hyp:s03:T2} and the sign clause \Tthree\ (\cref{prop:s10:U}); a variant with the auxiliary mass on either side of $m$ (\cref{prop:s10:Upm}) serves the discussion of the coupling range; no continuum limit is taken here. \subsection{King's tori, the two conventions and mid-plane pairs}\label{conv:s10_tube_inputs:1} \begin{definition}[King's tori]\label{def:s10:torus} A \emph{King torus} is $T=\R^3/(\ell\Z)^3$ with $\ell\in2M\N$; $|T|\defeq\ell^3$ is its volume, and $\mathbb T_J\defeq\R^3/(2MJ\Z)^3$, $J\in\N$, is the torus with fundamental cell $\CJ$. Since $\ell\Z^3\subset\Z^3\subset\Lk$, the lattice $\Lk$ descends to $T$, with $\ell\,5^k$ sites per side, an even number, and $|T|/\eps_k^3$ sites in all. The \emph{periodic state} $\langle\cdot\rangle_{k,T}$ is King's action of Section~\ref{sec:s02} summed over the finite graph $\Lk/(\ell\Z)^3$, with periodic conditions for both fields \cite[(4.1)]{K2}, \cite[(2.21)]{K1}: Feynman-form gradient terms for every pair of parallel neighbouring bonds, the photon mass $\lmu^2$, hopping terms on every bond, the single-site weights $e^{-\eps^3[\frac12(m^2+\dm)r^2+\lambda r^4]}$. Two normalisations of the counterterm are used: \begin{enumerate}[label=\textup{(\roman*)},nosep] \item the \emph{free convention} of Section~\ref{sec:s02} ($\dm=\dm_\eps$, Wick constant $\sW_\eps$; the convention of every cube state of this paper), with $N^a_{k,T}(0,q)\defeq e^{-(a_k+2\sW_\eps)\sS(q)}\langle e^{\sum_x\eps^3q(x)r_x^2}\rangle_{k,T}$ for a real function $q$ on the sites of $T$, exactly as on boxes (\cref{def:s02:sources}), and \[ \alpha^a_{k,T}(u)\defeq|T|^{-1}\ln N^a_{k,T}(0,u\mathbf 1_T)\qquad(u\in\R), \] the \emph{free energy density} of a constant source; \item \emph{King's torus convention} \cite[(2.4), (3.33), (3.35)]{K1}: the counterterm $\dm_{k,T}(\mathsf M)$ and the Wick constant $c_{k,T}(\mathsf M)$ computed with the periodic propagators of $T$ at scalar mass $\mathsf M$, together with the field-independent constants $E(\mathsf M)=E_0(\mathsf M)+E_1(\mathsf M)$ of King's torus action. $Z^{(\mathsf M)}_{k,T}\defeq\int e^{-S}$ is the source-free partition function of this model at scalar mass $\mathsf M$ (bare coefficient $\mathsf M^2+\dm_{k,T}(\mathsf M)$; \cref{hyp:s03:T1} is a statement about these numbers), and $N^{z,\mathrm{tor}}_{k,T}(0,q)$ the normalised source functional of the torus model at mass $m$ in the reference scheme $z$, with $c_{k,T}=c_{k,T}(m)$ in place of $\sW_\eps$. \end{enumerate} In either convention the coefficients are the same at every site of $T$. A \emph{mid-plane pair} is $\pi=\{x_\nu=c\}\cup\{x_\nu=c+\ell/2\}$ with $\nu\in\{1,2,3\}$ and $c\in\eps_k(\Z+\frac12)$ (both planes lie between lattice planes, since $\ell5^k$ is even); $\theta$ denotes the reflection of $T$ in $\pi$, which exchanges the two slabs of $T\setminus\pi$ (``the two sides of $\pi$''), acting on the fields as in \cref{def:s07:reflection}. \end{definition} \subsection{Reflection positivity of King's tori} \begin{proposition}[lattice reflection positivity of King's tori]\label{prop:s10:RP} Let $T$ be a King torus, $k\ge0$, $\pi$ a mid-plane pair and $\theta$ the reflection in $\pi$. For every finite family $G_1,\dots,G_N$ of gauge-invariant Borel functions of the fields on one side of $\pi$ with $\langle|G_i|^2\rangle_{k,T}<\infty$ and every $\gamma\in\C^N$, \[ \sum_{i,l=1}^N\bar\gamma_i\gamma_l\,\langle\theta(G_i)\,G_l\rangle_{k,T}\ \ge\ 0 . \] The periodic state is invariant under $\theta$ and under all lattice translations. This holds in both conventions of \cref{def:s10:torus}: reflection positivity does not see the value of the (site-independent) counterterm. \end{proposition} \begin{proof} The proof is the mechanism of \cref{lem:s07:RPbox} with a pair of planes in place of one and without Dirichlet diagonals, in three steps. \emph{Step 1: from the Feynman form to the Proca form.} On the torus write $A=Z+d_0\chi$ with $d_0^TZ=0$, $\chi\perp1$ (Hodge decomposition on the finite periodic graph). The Feynman form equals $\frac12\langle Z,(d_1^Td_1+\lmu^2)Z\rangle$ plus a quadratic form in $\chi$, and the Proca form $\frac12\langle A,(d_1^Td_1+\lmu^2)A\rangle$ equals the same $Z$-form plus another $\chi$-form; both $\chi$-forms are positive definite on $1^\perp$ because $\lmu>0$. For gauge-invariant $G$ the matter integral at fixed $A$ is invariant under $A\mapsto A+d_0\chi$ (rotate $\phi$ site by site), so the $\chi$-integrals factor and cancel in the normalised expectation: gauge-invariant expectations in the Feynman form equal those in the Proca form. Fubini's theorem applies by $\lmu>0$, $\lambda>0$ and a growth bound $|G|\le Ce^{a\sum_b|A(b)|}e^{C\sum_xr_x^2}$, satisfied by bounded $G$ and by the observables $\Theta(\zeta)$; so this step uses $\lmu>0$, gauge invariance, $\lambda>0$ and the growth bound, and nothing about the value of the counterterm. \emph{Step 2: the Proca torus measure is reflection positive about $\pi$.} In the Proca form the action couples the two sides of $\pi$ only through the plaquettes and hopping terms crossing one of the two planes of $\pi$ (no gradient terms between parallel bonds across a plane; the mass term is local). Write each crossing bond variable as $a_b=\alpha_b-\alpha'_b$ with $\alpha_b,\alpha'_b$ independent centred Gaussians of variance $(2\lmu^2\eps^3)^{-1}$, attached to the two sides. Then $\theta$ becomes the swap of the two data sets $\xi=(\omega_+,\alpha)$, $\xi'=(\omega_-^\theta,\alpha')$, whose a priori measures coincide by the $\theta$-invariance of the one-sided terms, and exactly as in Step~2 of \cref{lem:s07:RPbox} every crossing plaquette term is $\frac\eps2X_p(\xi)^2+\frac\eps2X_p(\xi')^2-\eps X_p(\xi)X_p(\xi')$ and every crossing hopping term is $-\eps\sum_{i=1,2}\psi^i_b(\xi)\psi^i_b(\xi')$, with the same real functions on both copies. So the Boltzmann factor is $\cG(\xi)\cG(\xi')\exp(\sum_\iota\gamma_\iota Y_\iota(\xi)Y_\iota(\xi'))$ with all $\gamma_\iota=\eps>0$ and all $Y_\iota$ real, and for $G$ a function of one side $\theta(G)G=\overline{G(\xi')}G(\xi)$. Expanding the exponential, $Z\langle\theta(G)G\rangle$ is a sum with nonnegative coefficients of squared moduli $|\int d\xi\,G(\xi)\cG(\xi)\prod_\iota Y_\iota(\xi)^{n_\iota}|^2$, the interchange of sum and integral being justified as in \cref{lem:s07:RPbox} ($\lambda>0$, $\lmu>0$ and the growth bound). The Hermitian property $\overline{\langle\theta(G')G\rangle}=\langle\theta(G)G'\rangle$ follows from the $\theta$-invariance of the measure, so $(G,G')\mapsto\langle\theta(G')G\rangle$ is a positive semidefinite Hermitian form on bounded gauge-invariant one-sided $G$. \emph{Step 3: $L^2$ observables.} For gauge-invariant one-sided $G$ with $\langle|G|^2\rangle<\infty$ apply Step~2 to $G_n\defeq G\mathbf 1\{|G|\le n\}$; since $|\theta(G_n)G_n|\le|\theta G||G|$ and $\langle|\theta G||G|\rangle\le\langle|G|^2\rangle$ by Cauchy--Schwarz and $\theta$-invariance, dominated convergence gives the inequality for $G=\sum_i\gamma_iG_i$, hence the form statement. \emph{Invariance.} Under $\theta$ and under every lattice translation each term of the periodic action, in either convention, goes to a term of the same kind with the same coefficient, and the a priori measure is invariant; the value of the site-independent bare coefficient enters nowhere above. \end{proof} \begin{remark}[reflection positivity in every gauge, and the chessboard estimate] (i) \cite[Cor.~5.4]{BFS1} (reflection positivity ``in every gauge'', for covariances $C_F=C_u+dFd^*\ge0$ with $F$ of either sign) is the mechanism of the proof above: on the torus the Feynman covariance has this form with $F=-\lmu^{-2}(\Delta_0+\lmu^2)^{-1}\le0$ on functions orthogonal to the constants. It is not a literal instance (the region of \cite[\S5]{BFS1} is a bounded subset of the lattice with a boundary condition, and reflection positivity of periodic conditions is claimed there in a remark), and \cref{prop:s10:RP} does not rest on it. (ii) The chessboard estimate uses \cref{prop:s10:RP} for functions of the $r_x$ only; nothing in this section uses reflection positivity of a cube or of a limit state. \end{remark} \subsection{The chessboard estimate on King's tori} \begin{lemma}[chessboard estimate]\label{lem:s10:CB} Let $T$ be a King torus at spacing $\eps=\eps_k$, $a$ a scheme, and $q$ a real function on the sites of $T$. In the free convention, \begin{equation}\label{eq:s10:CB} \ln N^a_{k,T}(0,q)\ \le\ \sum_{x}\eps^3\,\alpha^a_{k,T}(q(x)), \end{equation} and $u\mapsto\alpha^a_{k,T}(u)$ is convex on $\R$ with $\alpha^a_{k,T}(0)=0$. In King's torus convention \[ u\mapsto|T|^{-1}\ln N^{z,\mathrm{tor}}_{k,T}(0,u\mathbf 1_T) \] is likewise convex with value $0$ at $u=0$. \end{lemma} \begin{proof} We apply the chessboard estimate \cite[Thm.~4.2]{FILS78} (lattice-gauge form: \cite[Thm.~2.10]{BFS1}), whose hypotheses we verify: the site set of $T$ is a rectangular lattice box with sides of even length $\ell5^k$; the periodic state is invariant under translations modulo the sides and reflection positive with respect to the reflections in all planes perpendicular to the coordinate axes running midway between neighbouring sites (\cref{prop:s10:RP}; on the torus such a reflection is the reflection in a mid-plane pair, the antipodal plane $c+\ell/2$ being again a mid-plane). Take as the ``spin at $x$'' the gauge-invariant variable $r_x$ and \[ G_x\defeq\exp\big(\eps^3q(x)(r_x^2-2\sW_\eps-a_k)\big),\qquad\text{so that}\qquad\Big\langle\prod_xG_x\Big\rangle_{k,T}=N^a_{k,T}(0,q). \] The $G_x$ are positive with all moments ($\lambda>0$), and the Schwarz inequality for such unbounded gauge-invariant functions, all that \cite[Thm.~4.1]{FILS78} uses beyond cyclic invariance, follows from the bounded case by truncation and monotone convergence. The conclusion is $|\langle\prod_xG_x\rangle|\le\prod_x\langle\prod_yG_x^{(y)}\rangle^{1/|\Lambda|}$, $G_x^{(y)}$ the copy of $G_x$ carried to $y$ by reflections, $|\Lambda|=|T|/\eps^3$ the number of sites. Because $\sW_\eps$, $a_k$ and $\dm_\eps$ are the same at every site, $\prod_yG_x^{(y)}=\exp(q(x)\sum_y\eps^3(r_y^2-2\sW_\eps-a_k))$ has expectation $N^a_{k,T}(0,q(x)\mathbf 1_T)=e^{|T|\alpha^a_{k,T}(q(x))}$ in the same state; taking logarithms gives \eqref{eq:s10:CB}. Convexity: $|T|\alpha^a_{k,T}(u)$ is the logarithm of a Laplace transform in the linear parameter $u$ plus a linear term, and vanishes at $u=0$; the same computation gives the convexity statement in the torus convention. \end{proof} \subsection{Constant sources on large tori: the extensive bound} \begin{definition}[the extensive bound \Itwo]\label{def:s10:I2} We say that \Itwo\ (one-sided, in the volume limit at fixed spacing) holds for the scheme $a$ in the free convention if there are $s_1>0$ and $C_U\in[0,\infty)$ such that \begin{equation}\label{eq:s10:I2} \sup_{k\ge0}\ \limsup_{J\to\infty}\ \alpha^a_{k,\mathbb T_J}(s_1)\ \le\ C_U . \end{equation} \end{definition} The positive leg of the next section uses exactly this, with $\kappa\defeq C_U/s_1$ (by convexity and $\alpha(0)=0$, $\limsup_J\alpha^a_{k,\mathbb T_J}(u)\le\kappa u$ for $0\le u\le s_1$). Throughout, $M_1\in(0,m)$ is an auxiliary scalar mass, fixed at the end of this paragraph. We recall from Definition~\ref{hyp:s03:T3} King's free-boundary bare coefficient $\alpha^\infty_k(\mathsf M)=\mathsf M^2+\dm_k(\mathsf M)$ on $\Lk$ at scalar mass $\mathsf M$ (it is the fixed-$k$ volume limit of the torus coefficient $\mathsf M^2+\dm_{k,\mathbb T_J}(\mathsf M)$, \cref{lem:s10:V}), the difference $s^\infty_k(M_1)=\tfrac12\big(\alpha^\infty_k(m)-\alpha^\infty_k(M_1)\big)$ and $\sigma_1=\sigma_1(M_1)=\liminf_{k\to\infty}s^\infty_k(M_1)$, and we put, in the reference scheme $z$, \[ B^z_{k,T}(\mathsf M)\defeq E(\mathsf M)-E(m)-\tfrac12\big(2c_{k,T}+z_k\big)\big(\alpha^{\mathrm{tor}}_{k,T}(m)-\alpha^{\mathrm{tor}}_{k,T}(\mathsf M)\big)|T|, \] \[ b^\infty_k(M_1)\defeq\lim_{J\to\infty}|\mathbb T_J|^{-1}B^z_{k,\mathbb T_J}(M_1),\qquad \beta_1=\beta_1(M_1)\defeq\limsup_{k\to\infty}b^\infty_k(M_1), \] where $\alpha^{\mathrm{tor}}_{k,T}(\mathsf M)\defeq \mathsf M^2+\dm_{k,T}(\mathsf M)$ is the bare coefficient of King's torus convention and the limit defining $b^\infty_k$ exists by \cref{lem:s10:V}. That $\sigma_1(M_1)$ and $\beta_1(M_1)$ are finite is Lemma~\ref{hyp:s03:T2} below; \Tthree\ says that $\sigma_1(M_1)>0$ for some $M_1\in(0,m)$ (\cref{hyp:s03:T3}), and we fix $M_1$ as such a mass, so that $\sigma_1(M_1)>0$ and the interval $(0,\sigma_1)$ of \cref{prop:s10:U} is not empty. \begin{proposition}[torus stability at two scalar masses]\label{hyp:s03:T1} For each scalar mass $\mathsf M\in\{m,M_1\}$, $M_1$ the auxiliary mass fixed after Definition~\ref{def:s10:I2} (the sans-serif letter $\mathsf M$ denotes a scalar mass throughout, $M$ being King's large-block integer), there are constants $E_\pm(\mathsf M)<\infty$, a tail $K_1$ and, for each $k\ge K_1$, a $J_*(k)$ such that the source-free partition functions of King's tori $T=T_J=\R^3/(2MJ\Z)^3$ (Definition~\ref{def:s10:torus}; $J\in\N$ is the torus parameter) at spacing $\eps_k$ and at scalar mass $\mathsf M$ satisfy, in King's torus convention, \[ e^{-E_-(\mathsf M)\abs{T}}\le Z^{(\mathsf M)}_{k,T}\le e^{E_+(\mathsf M)\abs{T}}\qquad(k\ge K_1,\ J\ge J_*(k)). \] \end{proposition} These are Ba{\l}aban's lower and upper stability bounds, \cite[(1.14)]{Ba1} and \cite[(1.3)]{Ba2}, at a fixed scalar mass, with block factor $5$ and constants uniform in the spacing and in the torus, used as stated there and read at the two masses $m$ and $M_1$ and in King's torus normalisation of the counterterm. Proposition~3.1 of \cite{Ba2}, behind the upper bound, is stated under a cap on the dimensionless scalar mass at the scales at which it is applied; it is cited under that cap, one more one-sided cap on the stopping scale and no condition on the bare mass (Remark~\ref{rem:s14:BALm}). \begin{lemma}[the $c$-number statement]\label{hyp:s03:T2} For the auxiliary mass $M_1$ fixed after Definition~\ref{def:s10:I2}, the quantities $\sigma_1(M_1)$ (Definition~\ref{hyp:s03:T3}) and $\beta_1(M_1)$ defined above are finite. \end{lemma} The proof is a computation with lattice Feynman integrals: the quantity is the difference of two mass counterterms at the same photon mass, whose divergent parts ($1/\eps$ at one loop, $\ln(1/\eps)$ at two loops) do not depend on the scalar mass and cancel in the difference, and the remaining lattice sums converge as Riemann sums of massive lattice Feynman integrals, by Reisz's power-counting theorems (Remark~\ref{rot:rem:Reisz}). Only the relative limit $\alpha_\infty(\mathsf M)-\alpha_\infty(m)=\lim_k[\alpha^\infty_k(\mathsf M)-\alpha^\infty_k(m)]$ exists, each coefficient alone carrying the $1/\eps$ and $\ln(1/\eps)$ divergences of the counterterm; this relative limit, its description as a function of the couplings and the coupling region it determines are given in Lemmas~\ref{lem:s14:CT} and~\ref{lem:s14:CW}. The finiteness of $\sigma_1(M_1)$ also follows from Lemma~\ref{lem:s14:CT} (Section~\ref{sec:s14}), which gives the limit in closed form. \begin{proposition}[\Itwo\ from the torus theorems]\label{prop:s10:U} Let the couplings lie in the coupling region of \cref{hyp:s03:T3} (the sign clause \Tthree). Then \Itwo\ holds in the reference scheme $z$ for every $s_1\in(0,\sigma_1)$, with a constant $C_U=C_U(s_1)<\infty$ depending on $(e,\lambda,m,\lmu,M)$, on $M_1$ and on $s_1$. For a scheme $a\ne z$ with $\sup_k|z_k-a_k|<\infty$ (an admissible scheme, \cref{def:s03:adm}) it holds with the constant $C_U+s_1\sup_k|z_k-a_k|$. \end{proposition} The proof uses the torus theorems \cref{hyp:s03:T1}, at the two masses $m$ and $M_1$, and \cref{hyp:s03:T2}, and assembles four lemmas on King's torus model at fixed spacing, stated in the form used and with their short proofs where elementary. Notation at fixed $k$ (spacing $\eps$) on $T=\mathbb T_J$: $f^{\mathrm{tor}}(u)\defeq\ln N^{z,\mathrm{tor}}_{k,T}(0,u\mathbf 1)$ (King's torus model at mass $m$, Wick constant $c_{k,T}$, scheme $z$) and $f^F(u)\defeq\ln N^{z}_{k,T}(0,u\mathbf 1)$ (free convention); both are convex with $f(0)=0$ (\cref{lem:s10:CB}). Write $\alpha^{\mathrm{tor}}\defeq m^2+\dm_{k,T}(m)$, $\alpha^F\defeq m^2+\dm_\eps$, $\sigma_{k,T}\defeq\frac12(\alpha^{\mathrm{tor}}-\alpha^F)$ and $\Delta c_T\defeq\sW_\eps-c_{k,T}$ (the two \emph{convention offsets}), and $s_{k,T}(M_1)\defeq\frac12(\alpha^{\mathrm{tor}}_{k,T}(m)-\alpha^{\mathrm{tor}}_{k,T}(M_1))$. \begin{lemma}[mass shift]\label{lem:s10:M} If $M''>0$ satisfies $\alpha^{\mathrm{tor}}_{k,T}(M'')=\alpha^{\mathrm{tor}}_{k,T}(m)-2u$, then \[ N^{z,\mathrm{tor}}_{k,T}(0,u\mathbf 1)=e^{B}\,Z^{(M'')}_{k,T}\big/Z^{(m)}_{k,T},\qquad B=E(M'')-E(m)-(2c_{k,T}+z_k)\,u\,|T| . \] \end{lemma} \begin{proof} The source $u\sum_x\eps^3r_x^2$ shifts the bare coefficient of the torus model at mass $m$ by $-2u$, that is to the bare coefficient of the model at mass $M''$; beyond the fixed terms, King's torus action \cite[(2.4)]{K1} contains only the counterterm and the field-independent constants $E_0,E_1$, so the two Boltzmann factors differ by the constant $e^{E(M'')-E(m)}$, and the Wick and scheme multipliers $e^{-(2c_{k,T}+z_k)u|T|}$ of $N^{z,\mathrm{tor}}$ are bookkeeping. \end{proof} \begin{lemma}[chord]\label{lem:s10:C} If $f$ is convex on $\R$ with $f(0)=0$, then $f(u)\le(u/s)f(s)$ for $0\le u\le s$, $s>0$. \end{lemma} \begin{proof} $u=(1-u/s)\cdot0+(u/s)\cdot s$ and $f(0)=0$. \end{proof} \begin{lemma}[the volume limit at fixed spacing]\label{lem:s10:V} Fix $k$. As $J\to\infty$: $\alpha^{\mathrm{tor}}_{k,\mathbb T_J}(\mathsf M)\to\alpha^\infty_k(\mathsf M)$ uniformly for $\mathsf M\in[M_1,m]$; $s_{k,\mathbb T_J}(M_1)\to s^\infty_k(M_1)$; $|\mathbb T_J|^{-1}B^z_{k,\mathbb T_J}(M_1)\to b^\infty_k(M_1)$, a finite limit; and the offsets vanish, $\sigma_{k,\mathbb T_J}\to0$ and $\Delta c_{\mathbb T_J}\to0$. Moreover (the \emph{boundedness clause}) $\sup_{J\ge1}\sup_{\mathsf M\in[M_1,m]}\big(|\dm_{k,\mathbb T_J}(\mathsf M)|+|c_{k,\mathbb T_J}(\mathsf M)|\big)<\infty$; in particular $\dm_{k,\mathbb T_J}(m)$ and $c_{k,\mathbb T_J}$ are bounded in $J\ge1$. \end{lemma} The proof is the convergence of Riemann sums: at fixed $k$ the torus quantities are sums over the dual lattice of $\mathbb T_J$ of smooth periodic functions of the momentum (the propagators are massive lattice covariances at fixed spacing), bounded uniformly in $J$ and convergent as $J\to\infty$ to the corresponding Brillouin-zone integrals; this gives the limits, the boundedness clause and the vanishing of the offsets. No rate and no uniformity in the source are claimed or used. Of the clauses listed, Step~2 of the proof of \cref{prop:s10:U} uses the convergence of $s_{k,\mathbb T_J}(M_1)$ and the $\limsup_J$ of $|\mathbb T_J|^{-1}B^z_{k,\mathbb T_J}(M_1)$, and Step~3 the two offset clauses. \begin{lemma}[the fixed-spacing extensive bound]\label{lem:s10:F} Fix $k$ and $v>0$. There is $C_k(v)<\infty$ such that for every King torus $T=\mathbb T_J$, $J\ge1$, in either convention, $f(\pm v)\le C_k(v)|T|$; hence, by convexity, $|f(w)|\le C_k(v)\,|w|\,|T|/v$ for $|w|\le v$. \end{lemma} \begin{proof} Let $I(b)\defeq\int_\C e^{-\eps^3[\frac12b|\phi|^2+\lambda|\phi|^4]}d^2\phi<\infty$ for real $b$ ($\lambda>0$), $\alpha$ the bare coefficient and $c$ the Wick constant of the convention in use, and $Z_\gamma$ the photon Gaussian integral, finite because $\lmu>0$ (at $\lmu=0$ the constant mode is in the kernel of the Feynman form). The hopping term is $\ge0$, so dropping it bounds the numerator: $\int e^{-S+w\sum\eps^3r^2}\le Z_\gamma I(\alpha-2w)^{|T|/\eps^3}$. The hopping term of a bond is $\le\eps(r_x^2+r_y^2)$, and each site lies on six bonds, so the hopping part of the action is $\le6\eps^{-2}\sum_x\eps^3r_x^2=\sum_x\eps^3\cdot\frac12(12\eps^{-2})r_x^2$ and $\int e^{-S}\ge Z_\gamma I(\alpha+12\eps^{-2})^{|T|/\eps^3}$. Hence $\ln\langle e^{w\sum\eps^3r^2}\rangle\le\eps^{-3}|T|[\ln I(\alpha-2w)-\ln I(\alpha+12\eps^{-2})]$ and \[ f(w)\le\big\{\eps^{-3}[\ln I(\alpha-2w)-\ln I(\alpha+12\eps^{-2})]+(2|c|+|z_k|)|w|\big\}|T| . \] In the free convention $\alpha$ and $c$ do not depend on $T$; in the torus convention they converge as $J\to\infty$ and are bounded in $J\ge1$ (the boundedness clause of \cref{lem:s10:V}). So $C_k(v)$, the supremum over the sign and over $J\ge1$ of the brace at $w=\pm v$, is finite. The last claim is convexity with $f(0)=0$: $f(w)\le(|w|/v)f(\pm v)$ and $f(w)\ge-f(-w)\ge-(|w|/v)f(\mp v)$. \end{proof} \begin{proof}[Proof of \cref{prop:s10:U}] \emph{Step 1: the exact convention bridge.} On the same torus the free-convention action differs from the torus-convention action by \[ \tfrac12(\alpha^F-\alpha^{\mathrm{tor}})\sum_x\eps^3r_x^2=-\sigma_{k,T}\sum_x\eps^3r_x^2 \] plus field-independent constants. Hence $\langle e^{u\sum\eps^3r^2}\rangle^F=\langle e^{(u+\sigma)\sum\eps^3r^2}\rangle^{\mathrm{tor}}/\langle e^{\sigma\sum\eps^3r^2}\rangle^{\mathrm{tor}}$ with $\sigma=\sigma_{k,T}$, and inserting the Wick and scheme multipliers, \begin{equation}\label{eq:s10:bridge} f^F(u)=f^{\mathrm{tor}}(u+\sigma)-f^{\mathrm{tor}}(\sigma)-2\,\Delta c_T\,u\,|T| . \end{equation} \emph{Step 2: the bound in the torus convention.} Let $0\le u<\sigma_1$ and $\eta>0$. By the definitions of $\sigma_1$ and $\beta_1$ there is $K_1=K_1(u,\eta)$, which we take at least as large as the tail of \cref{hyp:s03:T1}, such that $s^\infty_k(M_1)>u$ and $b^\infty_k(M_1)<\beta_1+\eta$ for all $k\ge K_1$. Fix such a $k$. By \cref{lem:s10:V} there is $J_*\ge J_*(k)$ with $s\defeq s_{k,\mathbb T_J}(M_1)>u$ for $J\ge J_*$. The chord (\cref{lem:s10:C}) at this $s$ gives $f^{\mathrm{tor}}(u)\le(u/s)f^{\mathrm{tor}}(s)\le[f^{\mathrm{tor}}(s)]_+$. \Cref{lem:s10:M} with $M''=M_1$ applies exactly, since $\alpha^{\mathrm{tor}}_{k,\mathbb T_J}(M_1)=\alpha^{\mathrm{tor}}_{k,\mathbb T_J}(m)-2s$ by the definition of $s$: \[ f^{\mathrm{tor}}(s)=B^z_{k,\mathbb T_J}(M_1)+\ln Z^{(M_1)}_{k,\mathbb T_J}-\ln Z^{(m)}_{k,\mathbb T_J}\le B^z_{k,\mathbb T_J}(M_1)+\big(E_+(M_1)+E_-(m)\big)|\mathbb T_J| \] by \cref{hyp:s03:T1}, used as an upper bound at $M_1$ and as a lower bound at $m$. Divide by $|\mathbb T_J|$ and let $J\to\infty$ (\cref{lem:s10:V} for the first term): \[ \limsup_{J\to\infty}|\mathbb T_J|^{-1}f^{\mathrm{tor}}_{k,\mathbb T_J}(u)\le\big[\beta_1+\eta+E_+(M_1)+E_-(m)\big]_+\qquad(k\ge K_1). \] For each of the finitely many $k0$ serves); then \[ C^{\mathrm{tor}}(u)\defeq\sup_{k\ge0}\ \limsup_{J\to\infty}|\mathbb T_J|^{-1}f^{\mathrm{tor}}_{k,\mathbb T_J}(u)<\infty\qquad\text{for every }u\in[0,\sigma_1). \] \emph{Step 3: transfer to the free convention.} Fix $u\in[0,\sigma_1)$ and $p>1$ with $pu<\sigma_1$, and let $p'=p/(p-1)$. Since $u+\sigma=\frac1p(pu)+\frac1{p'}(p'\sigma)$, convexity gives $f^{\mathrm{tor}}(u+\sigma)\le\frac1pf^{\mathrm{tor}}(pu)+\frac1{p'}f^{\mathrm{tor}}(p'\sigma)$, whatever the sign of $\sigma=\sigma_{k,\mathbb T_J}$. Once $p'|\sigma_{k,\mathbb T_J}|\le v$, which holds for all large $J$ by \cref{lem:s10:V}, \cref{lem:s10:F} gives $|f^{\mathrm{tor}}(p'\sigma)|,|f^{\mathrm{tor}}(\sigma)|\le C_k(v)p'|\sigma_{k,\mathbb T_J}||\mathbb T_J|/v$. Inserting this into \eqref{eq:s10:bridge}, \[ |\mathbb T_J|^{-1}f^F_{k,\mathbb T_J}(u)\ \le\ \tfrac1p|\mathbb T_J|^{-1}f^{\mathrm{tor}}_{k,\mathbb T_J}(pu)+\tfrac{2C_k(v)p'}{v}|\sigma_{k,\mathbb T_J}|+2|\Delta c_{\mathbb T_J}|\,u \ \xrightarrow[J\to\infty]{\ \limsup\ }\ \tfrac1pC^{\mathrm{tor}}(pu), \] because $\sigma_{k,\mathbb T_J}\to0$ and $\Delta c_{\mathbb T_J}\to0$ at fixed $k$ (\cref{lem:s10:V}; the factor $\frac1{p'}\le1$ in the middle term has been dropped). Hence $\sup_k\limsup_J\alpha^z_{k,\mathbb T_J}(u)\le\frac1pC^{\mathrm{tor}}(pu)<\infty$ for every $u\in[0,\sigma_1)$. Take $u=s_1\in(0,\sigma_1)$ and $C_U\defeq[\inf_{p>1:\,ps_1<\sigma_1}\frac1pC^{\mathrm{tor}}(ps_1)]_+$; this is \eqref{eq:s10:I2} in the scheme $z$. For a scheme $a\ne z$, $N^a_{k,T}(0,q)=e^{-(a_k-z_k)\sS(q)}N^z_{k,T}(0,q)$ gives $\alpha^a_{k,T}(u)=\alpha^z_{k,T}(u)+(z_k-a_k)u\le\alpha^z_{k,T}(u)+|z_k-a_k|\,u$, so \eqref{eq:s10:I2} holds with $C_U+s_1\sup_k|z_k-a_k|$. \end{proof} \begin{proposition}[the auxiliary mass on either side of $m$]\label{prop:s10:Upm} Let $M_1>0$, $M_1\ne m$, be a scalar mass for which $s^\infty_k(M_1)=\frac12(\alpha^\infty_k(m)-\alpha^\infty_k(M_1))$, of either sign, has $\sigma_1(M_1)=\liminf_ks^\infty_k(M_1)>0$ and $\beta_1(M_1)<\infty$, the two quantities being defined by the formulas above with this $M_1$ (the computation of \cref{hyp:s03:T2} is at a fixed scalar mass). Then, with the stability bounds of \cref{hyp:s03:T1} at $m$ and at this $M_1$ (Ba{\l}aban's bounds hold at every fixed scalar mass; \cref{hyp:s03:T1} states them at $m$ and at the mass $M_1\in(0,m)$ fixed after \cref{def:s10:I2}), \Itwo\ holds in the scheme $z$ for every $s_1\in(0,\sigma_1(M_1))$, and for $a\ne z$ with the shifted constant, exactly as in \cref{prop:s10:U}. \end{proposition} \begin{proof} The proof of \cref{prop:s10:U} is repeated with \cref{lem:s10:V} stated on the interval $I\defeq[\min(M_1,m),\max(M_1,m)]$, and we verify that no step uses $M_10$, so it holds on $I$ word for word; Step~2 uses it at the two masses $m$ and $M_1$ only. \Cref{lem:s10:F} carries no mass order, and \eqref{eq:s10:bridge} is an identity. In Step~2 the sign clause is $s^\infty_k(M_1)>u\ge0$, the chord is applied at $s=s_{k,\mathbb T_J}(M_1)>u\ge0$, \cref{lem:s10:M} at $M''=M_1$ is exact by the definition of $s$, and \cref{hyp:s03:T1} is used as an upper bound at $M_1$ and a lower bound at $m$; none of this uses $M_1m$ has $\sigma_1(M_1)>0$. The main results use \cref{prop:s10:U} with $M_1\in(0,m)$, as \Tthree\ is posed. \begin{remark}[the convention of \cref{hyp:s03:T1} and how much of it is used]\label{rem:s10:T1convention} (i) \Cref{hyp:s03:T1} is stated in King's torus convention (counterterm and Wick constant computed with the periodic propagators of the torus itself). For the source-free partition function this is also the convention of Bałaban's torus model \cite[(1.1)--(1.13)]{Ba1}, which is the same action: his $E_0$ is the Gaussian normalisation and his $E_1$ \cite[(1.13)]{Ba1} is King's field-independent constant on the torus, the same connected vacuum graphs with the same counterterm vertices \cite[(1.24)]{Ba3}; so the partition function of \cref{hyp:s03:T1} is exactly the quantity bounded in \cite[(1.14)]{Ba1} and \cite[(1.3)]{Ba2}, and no change of normalisation is involved in reading those bounds as \cref{hyp:s03:T1}. This is what the proof needs: the offsets $\sigma_{k,\mathbb T_J}$, $\Delta c_{\mathbb T_J}$ between King's torus convention and the free convention vanish in the volume limit at fixed spacing (\cref{lem:s10:V}), and Step~3 of the proof of \cref{prop:s10:U} transfers the bound through them; between other normalisations they need not vanish. (ii) Only a tail of \cref{hyp:s03:T1} is used: Step~2 of that proof invokes it for $k\ge K_1$ and $J\ge J_*(k)$, the finitely many $k1$ from $\R^3\setminus\CJ$, $\limsup_kZ^a_{k,J}(0,q)\le e^{\kappa\norm{q}_{L^1}}$, uniformly in $J$. On King's convention the same bounds hold for $N^{a,K}_{k,\CJ}(0,q)$ and $\limsup_kZ^{a,K}_{k,J}(0,q)$ (\cref{rem:s04:F}: Step~1 of \cref{lem:s04:G} prepended to \cref{prop:s11:W1prime}), and for $2\sup q2\ \forall\mu\}$ are open and cover $\R^3$ (the $2$-neighbourhoods of $W_\mu(0)$, $W_\mu(1)$ are disjoint as $ML_c\ge5>4$). Fix a smooth partition of unity $0\le\chi_s\le1$, $\sum_s\chi_s=1$, $\supp\chi_s\subset G_s$; so $\chi_s$ vanishes within distance $2$ of every wall of $\cT_s$. Fix $\chi_0\in C^\infty_c$, $0\le\chi_0\le1$, $\chi_0=1$ on $\{\dist_\infty(x,\R^3\setminus C_{L_c})\ge2\}$, $\supp\chi_0\subset\{\dist_\infty(x,\R^3\setminus C_{L_c})>1\}$, and put $\chi_c\defeq\tau_b\chi_0$ for the cell $c=C_{L_c}+b$. For sources supported in $Q_R$ let $J_1(R)\defeq\min\{J\in\N:MJ\ge R+\ell_c\}$; a cell meeting $Q_R$ has $w_\mu\in[-R-\ell_c,R]$, hence lies in $[-R-\ell_c,R+\ell_c)^3\subset\CJ$ for $J\ge J_1(R)$, so every cell meeting $Q_R$ is a full cell inside $\CJ$ and the cells clipped by the faces of $\CJ$ carry no source. \end{definition} The constants of the Wick leg are \begin{equation}\label{eq:s11:constants} \begin{gathered} \sigma_+\defeq s_1,\quad H'\defeq\frac{s_1}2,\quad H\defeq\frac{H'}8=\frac{s_1}{16},\quad \kappa\defeq\frac{C_U}{s_1},\quad B_{\rm cell}\defeq\sup_k\ln N^a_{k,C_{L_c}}(0,-2H'\chi_0),\\ C_{\rm cell}\defeq\tfrac12B_{\rm cell}+\kappa H'\ell_c^3,\qquad C_\Sigma\defeq\frac{(2+6\ell_c^2)^2\pi^2}{\ell_c^3},\qquad K_-\defeq\frac{8\,[C_{\rm cell}]_+\,C_\Sigma}{H'} . \end{gathered} \end{equation} $B_{\rm cell}$ is finite by \cref{hyp:s12:Bcell}, that is by \cref{cor:k02:W1}(b): at the single cube $C_{L_c}$ and the single real source $-2H'\chi_0$ the comparison-convention functional is bounded by the King-convention one (the $\cD$ direction of \cref{lem:s04:G}), which converges by \cref{hyp:s03:W1} ($\supp\chi_0$ is at $\ell^\infty$-distance $>1$ from $\R^3\setminus C_{L_c}$ as \cref{hyp:s03:W1} requires, and the theorem puts no amplitude restriction on a negative Wick source, a position-dependent increase of the scalar mass squared). This boundedness statement is \Bcell\ (\cref{hyp:s12:Bcell}) and the only place where the existence part (Sections~\ref{sec:k01} and~\ref{sec:k02}) enters the bound \eqref{eq:s11:W}; no fixed-cube limit of the comparison family is used. \subsection{The negative leg on cubes} \begin{lemma}[wall insertion]\label{lem:s11:W3} Let $X_1,\dots,X_N$ be the site sets $c\cap\XJ$ of the cells $c$ of one tiling $\cT_s$ that meet $\CJ$, and $q\ge0$ a lattice function on $\XJ$. Then $N^a_{k,\CJ}(0,-q)\le\prod_{i=1}^NN^a_{k,X_i}(0,-q|_{X_i})$, with $N^a_{k,X_i}$ the functional of the Neumann--Dirichlet state of \cref{def:s02:acube} on $X_i$. \end{lemma} \begin{proof} $e^{-\sum\eps^3qr^2}$ is the class-$\cD$ observable $\Theta^-_{F,g}$, $F=0$, $g=\eps^3q\ge0$. Delete, for every nearest-neighbour pair with ends in different cells, the three photon gradient terms and the cosine part of the hopping term, one at a time: by \cref{lem:s02:BFS}(d) (the inequalities \cite[Thm.~6.2(3),(4)]{BFS1}, proved in \cref{sec:s02}) neither deletion decreases $\langle\Theta^-\rangle$, and each intermediate measure is a BFS measure. The result is a product over the cells (\cref{lem:s02:BFS}(e)) whose factors are the Neumann--Dirichlet states on the $X_i$: the kept diagonal parts of the deleted bonds are their Dirichlet terms, the remaining gradient terms their Neumann terms, and $\dm_\eps$, $\sW_\eps$, $a_k$ are site-independent. The observable and the multiplier $e^{a_k\sS(q)}=\prod_ie^{a_k\sS(q|_{X_i})}$ factorise. \end{proof} \begin{lemma}[one cell]\label{lem:s11:W4} Let \Itwo\ hold (\cref{def:s10:I2}), and let $c=C_{L_c}+b$ be a cell ($b\in ML_c\Z^3$) and $0\le\tilde q\le H'$ a lattice function supported in $\{x:\dist_\infty(x,\R^3\setminus c)\ge2\}$. Then $\ln N^a_{k,c}(0,-\tilde q)\le C_{\rm cell}$ for every $k$. \end{lemma} \begin{proof} $\chi_c=1$ on $\supp\tilde q$, so $-\tilde q=k_1-k_2$ with $k_1\defeq H'\chi_c-\tilde q\ge0$, $k_2\defeq H'\chi_c$, and \cref{lem:s02:alg}(d) at $f=0$ gives $N^a_{k,c}(0,-\tilde q)\le N^a_{k,c}(0,-2H'\chi_c)^{1/2}N^a_{k,c}(0,2k_1)^{1/2}$. \emph{First factor:} exact lattice translation covariance ($b\in\Lk$; the change of variables $x\mapsto x-b$ transports the boundary conditions, and the action, $\dm_\eps$, $\sW_\eps$, $a_k$ are site-independent) gives $N^a_{k,c}(0,-2H'\chi_c)=N^a_{k,C_{L_c}}(0,-2H'\chi_0)\le e^{B_{\rm cell}}$. \emph{Second factor:} \cref{prop:s11:W1prime} on the box $c\cap\Lk$ with $q'\defeq2k_1$, $0\le q'\le2H'=s_1$, gives at most $\exp(\kappa\norm{q'}_{1,k})\le\exp(2\kappa H'\ell_c^3)$, the cell having volume $\ell_c^3$. Hence $\ln N^a_{k,c}(0,-\tilde q)\le\frac12B_{\rm cell}+\kappa H'\ell_c^3$. \end{proof} \begin{lemma}[chord on the cell]\label{lem:s11:W5} Under the hypotheses of \cref{lem:s11:W4}: $\ln N^a_{k,c}(0,-\tilde q)\le([C_{\rm cell}]_+/H')\sup\tilde q$. \end{lemma} \begin{proof} Let $t\defeq\sup\tilde q/H'\in(0,1]$, $\hat q\defeq\tilde q/t$ ($0\le\hat q\le H'$, same support). $\varphi(u)\defeq\ln N^a_{k,c}(0,-u\hat q)$ is convex (\cref{lem:s02:alg}(b)), $\varphi(0)=0$, $\varphi(1)\le C_{\rm cell}$ (\cref{lem:s11:W4}); so $\varphi(t)\le t\varphi(1)\le t[C_{\rm cell}]_+$. \end{proof} \begin{lemma}[eight families and the sum over cells]\label{lem:s11:W67} \textup{(a)} For a lattice function $q\ge0$ on $\XJ$: $N^a_{k,\CJ}(0,-q)\le\prod_{s\in\{0,1\}^3}N^a_{k,\CJ}(0,-8\chi_sq)^{1/8}$. If $q$ is supported in $Q_R$ and $J\ge J_1(R)$, then for each $s$ every cell of $\cT_s$ meeting $\supp(\chi_sq)$ is a full cell inside $\CJ$, on which $\chi_sq$ is supported at $\ell^\infty$-distance $>2$ from the complement of the cell. \textup{(b)} For any tiling of $\R^3$ by cubes of side $\ell_c$: $\sum_c\sup_{x\in c}(1+|x|^2)^{-2}\le C_\Sigma$. \end{lemma} \begin{proof} (a) H\"older's inequality with eight exponents $8$ in the one state $\langle\cdot\rangle^{\ND}_{k,\CJ}$ for $e^{-\sum\eps^3qr^2}=\prod_se^{-\sum\eps^3\chi_sqr^2}$; the multipliers match since $\sS(q)=\sum_s\sS(\chi_sq)$ at lattice points. Support: every cell meeting $Q_R$ is a full cell inside $\CJ$ for $J\ge J_1(R)$ (\cref{def:s11:cells}), and a point of $G_s$ in the cell $\prod_\mu[w_\mu,w_\mu+\ell_c)$ has $\dist(x_\mu,\{w_\mu,w_\mu+\ell_c\})>2$ for every $\mu$. (b) For $x,y$ in one cell $|y|\le|x|+\sqrt3\ell_c$, so $1+|y|^2\le(2+6\ell_c^2)(1+|x|^2)$ and $\sup_{x\in c}(1+|x|^2)^{-2}\le(2+6\ell_c^2)^2\ell_c^{-3}\int_c(1+|y|^2)^{-2}dy$; sum over the disjoint cells, with $\int_{\R^3}(1+|y|^2)^{-2}dy=\pi^2$. \end{proof} \begin{proposition}[the negative leg]\label{prop:s11:W8} Let \Itwo\ hold (\cref{def:s10:I2}). For every $k$, every $J\ge J_1(R)$ and every lattice function $q\ge0$ supported in $Q_R$ with $\sup q\le H$: \[ \ln N^a_{k,\CJ}(0,-q)\ \le\ K_-\,\wnorm{q}. \] \end{proposition} \begin{proof} By \cref{lem:s11:W67}(a), then \cref{lem:s11:W3} for each tiling $\cT_s$ (the clipped cells carry no source and contribute $\ln N^a_{k,X_i}(0,0)=0$), then \cref{lem:s11:W5}, applicable since $8\chi_sq\le8\sup q\le H'$ and by the support clause of \cref{lem:s11:W67}(a): \[ \begin{split} \ln N^a_{k,\CJ}(0,-q)&\le\frac18\sum_s\sum_{c\in\cT_s}\ln N^a_{k,c}(0,-8\chi_sq|_c)\\ &\quad\le\frac18\sum_s\sum_c\frac{[C_{\rm cell}]_+}{H'}\,8\sup_cq\le\frac{[C_{\rm cell}]_+}{H'}\,\wnorm{q}\sum_s\sum_c\sup_{x\in c}(1+|x|^2)^{-2}\\ &\quad\le\frac{8[C_{\rm cell}]_+C_\Sigma}{H'}\,\wnorm{q}, \end{split} \] using $\sup_cq\le\wnorm{q}\sup_{x\in c}(1+|x|^2)^{-2}$ (a sampled supremum is at most the continuum one, \cref{lem:s02:E}(E3)) and \cref{lem:s11:W67}(b) for each of the eight tilings. \end{proof} The output norm is $\wnorm q$: the chord bound on a cell sees only $\sup_cq$, so no $L^1$ information inside a cell survives, and the tube bound carries $K_-\wnorm{k_-}$ rather than an $L^1$ norm of $k_-$. \subsection{The uniform bound on the tube and its limits} For a family with supports in $Q_R$ and $\zeta=(z,w)\in\C^{n+p}$ put $G^{(k,\CJ)}(\zeta)\defeq Z^a_{k,J}(\sum_iz_if_i,\sum_jw_jh_j)$, and let $\Dtube$ be the tube of \cref{def:s02:tube} with $\sigma_+=s_1$, $H=s_1/16$. Two facts from several complex variables \cite{Hor90} are used: (C2) (the identity theorem) a holomorphic function on a connected open $D\subset\C^m$ vanishing on a non-empty relatively open subset of $D\cap\Sigma$, $\Sigma$ an affine totally real subspace of real dimension $m$ (such as $\R^m$ or $(\mathrm i\R)^{m_1}\times\R^{m_2}$), vanishes on $D$ (all its derivatives vanish at such a point); (C3) (Montel--Vitali) a locally bounded family of holomorphic functions on $D$ is normal, and if it converges on a set as in (C2) it converges locally uniformly on $D$ to a holomorphic limit. \begin{theorem}[the tube bound]\label{thm:s11:W} Let the couplings lie in the coupling region of \cref{hyp:s03:T3}, so that \Itwo\ holds (\cref{prop:s10:U}, from the sign clause and the torus theorems \cref{hyp:s03:T1,hyp:s03:T2}), for the reference scheme $z$ or for an admissible scheme $a$ (\cref{def:s03:adm}); here and below \Ithree\ and \Ifour\ denote the proved statements \cref{prop:s10:RP} and \cref{lem:s02:BFS}. The constants $\sigma_+,H,\kappa,K_-$ of \eqref{eq:s11:constants} depend on $(e,\lambda,m,\lmu,M,a)$ and on the fixed auxiliary choices $M_1$, $\chi_0$, $s_1\in(0,\sigma_1)$, and on nothing else; no uniformity in $\lmu$ is claimed. For every family with supports in $Q_R$, every $k\ge0$, $J\ge J_1(R)$ and $\zeta\in\Dtube$: \begin{equation}\label{eq:s11:W} |G^{(k,\CJ)}(\zeta)|\ \le\ G^{(k,\CJ)}(\Re\zeta)\ \le\ \exp\Big(\norm{g(\zeta)}_{2,k}^2+\kappa\norm{k_+(\zeta)}_{1,k}+K_-\wnorm{k_-(\zeta)}\Big). \end{equation} On King's convention the bound holds in the form \eqref{eq:s13:WK} for the K-family functionals $Z^{a,K}_{k,J}$ (\cref{thm:s13:TWK}), and their fixed-cube limits exist on the tube (\cref{hyp:s03:W1}); consequently, for King's state (the infinite-volume state of King's construction \cite{K1,K2}): \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item \textup{(Limits on the whole tube.)} $G^K_J\defeq\lim_kZ^{a,K}_{k,J}$ exists locally uniformly on $\Dtube$ for every $J\ge J_1(R)$, $G^K_\infty\defeq\lim_JG^K_J$ exists locally uniformly on $\Dtube$ (the limit along $J\ge J_1(R+\frac12)$, hence along any tail), both are holomorphic there, and \[ |G^K_\infty(\zeta)|\le\exp\big(\norm{g(\zeta)}_2^2+\kappa\norm{k_+(\zeta)}_{L^1}+K_-\wnorm{k_-(\zeta)}\big)\qquad(\zeta\in\Dtube). \] \item \textup{($\Ezz$.)} With $\sigma_0\defeq H/4$, $K'\defeq\kappa\pi^2+K_-$, the mixed Schwinger functions $S_{n,p}(f;h)\defeq\partial^{(n)}_z\partial^{(p)}_wG^K_\infty(0)$ satisfy, for real test functions, \[ \begin{split} |S_{n,p}(f_1,\dots,f_n;h_1,\dots,h_p)|&\le e^{K'\sigma_0}(\sqrt2e)^n(n!)^{1/2}(e/\sigma_0)^p\,p!\prod_i\norm{f_i}_2\prod_j\wnorm{h_j}\\ &\quad\le\sigma_{n+p}\prod_i\fournorm{f_i}\prod_j\fournorm{h_j}, \end{split} \] $\sigma_N\defeq e^{K'\sigma_0}C_1^NN!$, $C_1\defeq2\max(e\pi/2,e/\sigma_0)$; for complex test functions the first bound carries the factor $2^{n+p}\le2^N$, $N=n+p$, absorbed in $C_1$. This is $\Ezz$ with $s=4$ and every $\beta>1$; the $S_{n,p}$ are tempered distributions. \end{enumerate} \end{theorem} $\sigma_1$, $C_U$ depend on $(e,\lambda,m,\lmu,M_1)$ through the torus stability constants of \cref{hyp:s03:T1} at $m$, $M_1$, the $c$-number $\beta_1$ of \cref{hyp:s03:T2} and finitely many fixed-spacing constants; $B_{\rm cell}$ depends on everything, including $\lmu$, $M$, $a$. The curvature term of \eqref{eq:s11:W} rests on the constant $2$ of \cref{lem:s05:C1}, which is not known to be sharp (\cref{rem:s05:sharp}). \begin{proof} \emph{The bound \eqref{eq:s11:W}.} The first inequality is \cref{lem:s02:alg}(b). Write $g=g(\zeta)$, $k=k(\zeta)$ (real). \emph{Step 1} (curvature off): \cref{thm:s05:C} in the presence of the real scalar source $k$ gives $N^a(g,k)\le e^{\norm g_{2,k}^2}N^a(0,k)$; the supports of the $f_i$ are at distance $>1$ from $\R^3\setminus\CJ$ since $MJ\ge R+\ell_c>R+1$. \emph{Step 2}: $N^a(0,k_+-k_-)\le N^a(0,2k_+)^{1/2}N^a(0,-2k_-)^{1/2}$ (\cref{lem:s02:alg}(d) at $f=0$). \emph{Step 3} (positive leg): $\zeta\in\Dtube$ and (E3) give $0\le2k_+\le s_1$ on the lattice, so $\frac12\ln N^a(0,2k_+)\le\kappa\norm{k_+}_{1,k}$ by \cref{cor:s11:W2prime}. \emph{Step 4} (negative leg): $0\le2k_-\le H$, supported in $Q_R$, $J\ge J_1(R)$, so $\frac12\ln N^a(0,-2k_-)\le\frac12K_-\wnorm{2k_-}=K_-\wnorm{k_-}$ by \cref{prop:s11:W8}. \emph{(i), (ii).} The bound \eqref{eq:s13:WK} on King's convention is Steps 1--6 of the proof of \cref{thm:s13:TWK}, the sequence of estimates above carried out on $\NDK$ with the kill step of \cref{rem:s04:F} in the positive leg and the cell statement of \cref{lem:s04:E} in the negative leg. From it, from the fixed-cube limits of \cref{hyp:s03:W1} at the real points of the tube (stated on King's convention) and from the cube monotonicity of \cref{cor:s04:G1} on the cone, Steps 7--11 of that proof give (i) and (ii): Montel's theorem in $k$ on the tube with the real points as the uniqueness set (C3), the monotone cube limit on a sub-tube of the split family and the identity theorem (C2) on the whole tube, and Cauchy's estimates on a polydisc inside the tube, with the extension to $\cS$, to complex test functions and to tempered distributions as in \cref{thm:s05:CK}(c). \end{proof} \begin{lemma}[the tube object of King's state: limit, continuity, translations]\label{lem:s11:TWtube} Let the couplings lie in the coupling region of \cref{hyp:s03:T3} and let $a$ be an admissible scheme, so that the bound \eqref{eq:s13:WK} and the fixed-cube limits $G^K_J$ of \cref{thm:s11:W}(i) are available. Write $\cD=\Dtube$ for the tube of a family with supports in $Q_R$; the items are numbered in continuation of \cref{thm:s11:W}(i),(ii). \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*,start=3] \item \textup{(The $J$-limit on the tube.)} For $J\ge J_1(R+\frac12)$, which exceeds the least $J_0$ with the supports at $\ell^\infty$-distance $>1$ from $\R^3\setminus\CJ$ (as $MJ_1(R')-R'\ge\ell_c\ge10>1$), $G^K_\infty=\lim_JG^K_J$ exists locally uniformly on $\cD$, is holomorphic on $\cD$, satisfies the bound of \cref{thm:s11:W}\textup{(i)}, is $>0$ at every real point of $\cD$, and $G^K_J\le G^K_\infty$ at the cone points of $\cD$ ($z\in(\mathrm i\R)^n$, $w\in\R^p$, $k(\zeta)\ge0$: the real scalar source $k(\zeta)\ge0$ sits in the slot of the cone). \item \textup{(Continuity on the real tube.)} Let $E_\rho\defeq C^\infty_c(\bar B_\rho;\Lambda^2)\times C^\infty_c(\bar B_\rho)$ with its $C^\infty$ topology, a Fr\'echet space, and $U_\rho\defeq\{(f,h)\in E_\rho:\norm{h_+}_\infty<\sigma_+/2,\ \norm{h_-}_\infty0$ with $-\epsilon\zeta\in\cD$ ($\cD$ is an open neighbourhood of $0$). $\varphi_J(t)\defeq\ln G^K_J(t\zeta)$ is convex on $[-\epsilon,1]$ (\cref{lem:s02:alg}(b) in the limit) with $\varphi_J(0)=0$, so $0\le\frac{\epsilon}{1+\epsilon}\varphi_J(1)+\frac1{1+\epsilon}\varphi_J(-\epsilon)$, i.e.\ $\varphi_J(1)\ge-\varphi_J(-\epsilon)/\epsilon\ge-C_\epsilon/\epsilon$, $C_\epsilon$ the limit bound of \eqref{eq:s13:WK} at $-\epsilon\zeta$; hence $G^K_J(\zeta)\ge e^{-C_\epsilon/\epsilon}$ uniformly in $J$ and $G^K_\infty(\zeta)>0$. \emph{Cone points.} There $G^K_J\le G^K_{J+1}$ (\cref{cor:s04:G1} with \cref{lem:s02:alg}(a) at fixed $k$, then $k\to\infty$; it needs $J\ge J_0$, which the range of (iii) ensures), so $G^K_J\le G^K_\infty$. (iv) \emph{Finite:} (iii) for the two-source family $(f;h)$ at $\zeta=(1,1)$, in $\cD$ by the definition of $U_\rho$; the value is the pointwise iterated limit and does not depend on the family used. \emph{Convex:} for $s,s'\in U_\rho$, $t\in[0,1]$, use the family $(f,f';h,h')$ at the real point $(t,1-t,t,1-t)$, in $\cD$ because $(th+(1-t)h')_\pm\le th_\pm+(1-t)h'_\pm$ pointwise; the convexity of $\ln Z^{a,K}_{k,J}$ in the real coefficients (\cref{lem:s02:alg}(b)) passes to the limits. \emph{Borel:} a pointwise limit of functions each depending continuously on finitely many values of the sources. \emph{Continuous:} \cref{lem:s02:baire} on the open convex $U_\rho$. (v) \emph{Lattice translations at cone points.} Let $b\in5^{-k_0}\Z^3$, $|b|_\infty\le M$, and $\zeta$ a cone point of $\cD$ ($k_-=0$, $q\defeq k(\zeta)\ge0$, $\norm q_\infty<\sigma_+/2$). The translation sandwich of \cref{lem:s06:TK} at the cone point $(\mathrm if_\zeta,q)$, with the real scalar source $q$ in the slot of the cone (lattice covariance \eqref{eq:s06:covK}: $\tau_b$ preserves $\Lk$ and $\sS_{\Lk}$ for $k\ge k_0$, and King's cube translated by $-b$ is interlaced between King's cubes of indices $J\mp1$ by the kill steps and cut-outs of its proof), gives $Z^{a,K}_{k,C_{J-1}}(\zeta)\le Z^{a,K}_{k,\CJ}(\tau_b\cdot\zeta)\le Z^{a,K}_{k,C_{J+1}}(\zeta)$ for $J\ge J_0+1$. Let $k\to\infty$ by \cref{thm:s11:W}(i) for the three families (the translated family is supported at distance $>1$ from $\R^3\setminus C_{J_0+1}$), then $J\to\infty$ by (iii), the three terms being cone points of their tubes: $G^K_\infty(\tau_b\cdot\zeta)=G^K_\infty(\zeta)$ at the cone points of $\cD$. \emph{All $\zeta\in\cD$.} For the split family of Step~8 of the proof of \cref{thm:s13:TWK} and its translate (with the sets of that step: $\cD''$ the convex tube of the split family, $\Omega''_+\subset\cD''$ its points with imaginary curvature coefficients and positive scalar coefficients, and $\cD'\subset\Dtube$ the sub-tube on which the diagonal $w^+=w$, $w^-=-w$ recovers the original family), $F^{K,\tau_b}_\infty-F^K_\infty$ is holomorphic on the connected $\cD''$ and vanishes on $\Omega''_+$, whose points are cone points of the split family's tube and which is an open subset of the maximal totally real subspace $(\mathrm i\R)^n\times\R^{2p}$; by (C2) it vanishes on $\cD''$, and the diagonal gives $G^{K,\tau_b}_\infty=G^K_\infty$ on $\cD'$. Then $G^{K,\tau_b}_\infty-G^K_\infty$ is holomorphic on the connected $\cD$ ((iii) for both families) and vanishes on the open set $\cD'$, hence on $\cD$. \emph{All $b\in\bigcup_k5^{-k}\Z^3$:} iterate; the tube is the same at every step. \emph{All $b\in\R^3$ at real points.} For real $\zeta\in\cD$ the source pair $s\defeq(\sum_iz_if_i,\sum_jw_jh_j)$ lies in some $U_\rho$, $\tau_bs\in U_{\rho'}$ for $|b|\le\rho'-\rho$, and $b\mapsto\tau_bs$ is continuous into $E_{\rho'}$; by (iv) and the density of $\bigcup_k5^{-k}\Z^3$, $\Phi^{a,K}(\tau_bs)=\Phi^{a,K}(s)$. \emph{Complex $\zeta\in\cD$.} $G^{K,\tau_b}_\infty-G^K_\infty$ is holomorphic on $\cD$ and vanishes on the real points of $\cD$, a non-empty open subset of $\R^{n+p}$; (C2) gives the claim. \end{proof} \begin{remark}[King's convention]\label{rem:s11:Kconvention} On King's RG cube $\NDK_{k,J}$ the negative leg holds as stated: by \cref{lem:s04:E}, the wall insertion of \cref{lem:s11:W3} runs on $\NDK_{k,J}$ with factors that are comparison box states killed on $X_i\cap\TJ$, and every cell on which a cutoff source $8\chi_sq$ does not vanish is a full cell inside $\CJ$ meeting no killed site, because $\chi_s$ vanishes within distance $2$ of every cell wall; so \cref{lem:s11:W4,lem:s11:W5,lem:s11:W67,prop:s11:W8} apply with the constant $K_-$ unchanged. The direction matters: killing sites of a cell increases a class-$\cD$ functional (\cref{lem:s04:E}(e)), so $B_{\rm cell}$ does not control a source-carrying cell with killed sites. For the positive leg, \cref{rem:s04:F} prepends the interlacing inequality $\avK{\cdot}\le\ava{\cdot}$ on class $\cI$ (Step~1 of \cref{lem:s04:G}) to \cref{prop:s11:W2,prop:s11:W1prime}, so \cref{cor:s11:W2prime} holds for $N^K_{k,\CJ}(0,q)$ at every $(k,J)$ and, for the fixed-cube limits, in the form $\limsup_kZ^{a,K}_{k,J}(0,q)\le e^{\kappa\norm q_{L^1}}$. The assembly of \eqref{eq:s11:W} on $\NDK$ from these pieces, with the check that every step of its proof runs in the right direction on King's convention, is Steps 1--6 of the proof of \cref{thm:s13:TWK} (the bound \eqref{eq:s13:WK}); with the fixed-cube limits of \cref{hyp:s03:W1} it gives the limit statements \cref{thm:s11:W}(i),(ii) and \cref{lem:s11:TWtube} for King's state, on which the Wick sector of the main theorem is stated (\cref{prop:s13:WK}). \end{remark} \begin{remark}[the comparison family]\label{rem:s11:comparison} The bound \eqref{eq:s11:W} is proved for the comparison convention, and the arguments behind \cref{thm:s11:W}(i),(ii) and \cref{lem:s11:TWtube} use of the cube functionals only such a bound, the fixed-cube limits at the real points of the tube and the cube monotonicity on the cone (for the comparison family the first inequality of \cref{lem:s02:cut} and the sandwich of \cref{prop:s06:translations}). As noted in Section~\ref{sec:s03}, they apply to the comparison family wherever its fixed-cube limits exist at the real points of the tube. Its limit states along continuum subsequences, which exist without that clause, are treated in \cref{sec:s12} and coincide with $G^K_\infty$ (\cref{prop:s13:LIMK}). \end{remark} \space% % % % % \section{Continuum subsequences and LIM states}\label{sec:s12} The axioms of the preceding sections rest on the existence of the fixed-cube limits $\lim_k$: on \cref{hyp:s03:EC} in the curvature sector, on \cref{hyp:s03:W1} on the Wick tube, both proved in the existence part (Sections~\ref{sec:k01} and~\ref{sec:k02}) on King's convention. This section shows that the axioms hold without the existence theorems, for the limit state along any continuum subsequence of spacings, and identifies every such state with the iterated limit. The $k$-uniform bounds of \cref{lem:s05:C2} and \eqref{eq:s11:W} make the fixed-cube functionals normal families, so one subsequence of spacings serves every cube and every family at once; along it the cube limit $J\to\infty$ exists because the cone functionals are monotone in the cube at fixed $k$ (\cref{lem:s02:cut}). The limit state of such a subsequence (a \emph{LIM state}) satisfies every axiom proved above, with no fixed-cube limit used in the curvature sector and, on the Wick tube, for every admissible scheme at couplings in the coupling region of \cref{hyp:s03:T3} (through two properties of the scheme, the cell bound and the scheme bound, proved below); \cref{hyp:s03:EC} (resp.\ \cref{hyp:s03:W1}) adds exactly the identification with the iterated limit and the independence of the subsequence. The section is written on the comparison convention $\NDa$ (\cref{def:s02:acube}), on which the tube bound \eqref{eq:s11:W} and the rotation part are written; King's convention is \cref{cor:s12:KK}, and the Wick-sector LIM states are identified with King's tube object in \cref{prop:s13:LIMK}. Notation is that of \cref{def:s02:sources,def:s02:tube,def:s11:cells} and \eqref{eq:s11:constants}: $a^{(a)}_{k,J}(z)=\ava{e^{\sum_jz_jF(f_j)}}$ for a family $f=(f_1,\dots,f_N)$ of real $2$-forms; $Z^a_{k,J}(\zeta)$ for $\cF=(f_1,\dots,f_n;h_1,\dots,h_p)$ and $\zeta=(z,w)\in\C^{n+p}$ in the scheme $a$; $g(\zeta)$, $k(\zeta)=k_+-k_-$, $\amp(\zeta)$; $U_J\defeq\{x:\dist_\infty(x,\R^3\setminus\CJ)>1\}$, $J_0(f)$ the least $J$ with all supports in $U_J$, and $R(\cF)$ the least $R$ with all supports in $Q_R=[-R,R]^3$. Two standard facts \cite{Hor90} are used by name, the \emph{identity theorem} and \emph{Montel--Vitali}: they are the facts (C2) and (C3) stated before \cref{thm:s11:W}. \subsection{Definitions, the cell bound and the scheme bound} \begin{definition}[continuum subsequences]\label{def:s12:contsub} An increasing sequence $\mathbf{k}=(k_m)_{m\ge1}\subset\N$ of spacings is a \emph{continuum subsequence} (of the comparison family) if for every finite family $f$ of real $2$-forms and every $J\ge J_0(f)$ the sequence $(a^{(a)}_{k_m,J})_m$ converges locally uniformly on $\C^N$; the limit is written $Z^{\mathbf{k}}_J(\sum_jz_jf_j,0)$ and depends on the family only through the total complex source, as each $a^{(a)}_{k,J}$ does. Fix a scheme $a$. $\mathbf{k}$ is a \emph{continuum subsequence for the Wick sector} (Form~(B)) if it is a continuum subsequence and, for every finite family $\cF$ of real $2$-forms and scalars and every $J$ with $MJ>R(\cF)+\ell_c$, the sequence $(Z^a_{k_m,J})_m$ converges locally uniformly on the tube $\Dtube$ of $\cF$; the limit is $Z^{a,\mathbf{k}}_J$. (The strict inequality replaces the threshold $J_1(R)$ of \cref{thm:s11:W} by at most one cube step; only tails in $J$ are used.) Convergence on all of $\C^{n+p}$ for the same $J$ is not used here. \end{definition} \begin{definition}[LIM states]\label{def:s12:LIMstate} The \emph{LIM state} of a continuum subsequence $\mathbf{k}$ is $\ZLIM{\mathbf{k}}\defeq\lim_JZ^{\mathbf{k}}_J$ on $\C^N$ (curvature sector), resp.\ $Z^{a,\mathbf{k}}_\infty\defeq\lim_JZ^{a,\mathbf{k}}_J$ on the tube (Wick sector); \cref{lem:s12:LIM1} gives existence. A subsequence of a continuum subsequence is a continuum subsequence with the same fixed-cube limits, hence the same LIM state. \end{definition} \begin{definition}[the limit state of the rotation theorem]\label{def:s12:o3LIM} For a family $f$ and $J\ge J_0(f)$ let $L^f_J$ be the set of locally uniform limits on $\C^N$ of subsequences of $(a^{(a)}_{k,J})_k$. The rotation theorem (\cref{hyp:s03:RC}) gives an entire $Z^{\mathrm{LIM}}_\infty(\cdot,0)$ to which \emph{every} choice $\zeta_J\in L^f_J$ converges locally uniformly as $J\to\infty$, the subsequence being allowed to depend on $J$. With $J$-dependent subsequences one only has $\liminf_ka^{(a)}_{k,J}(\mathrm iu)\le\zeta_J(\mathrm iu)\le\limsup_ka^{(a)}_{k,J}(\mathrm iu)$, and the $J$-limit is pinned only if the $J$-limits of the two bounds agree, which is the content of that theorem. A LIM state in the sense of \cref{def:s12:LIMstate} uses one $\mathbf{k}$ common to all $J$ and all families, along which the fixed-cube limits are monotone in $J$ (\cref{lem:s12:LIM1}), and uses no fixed-cube limit; \cref{prop:s12:ID}(a) relates the two. \end{definition} The curvature sector uses no fixed-cube limit. On the Wick tube the bound \eqref{eq:s11:W} of \cref{thm:s11:W} is used; it rests on the sign clause \Tthree\ and the torus theorems (\cref{hyp:s03:T1,hyp:s03:T2,hyp:s03:T3}, which give \Itwo\ with amplitude $s_1\in(0,\sigma_1)$ and constant $C_U$ through \cref{prop:s10:U}; \Ithree\ is \cref{prop:s10:RP} and \Ifour\ is \cref{lem:s02:BFS}) and on one instance of \cref{hyp:s03:W1}, isolated as follows. We say that the scheme $a$ has the \emph{cell bound} \Bcell\ if $B_{\mathrm{cell}}\defeq\sup_k\ln N^a_{k,C_{L_c}}(0,-2H'\chi_0)<\infty$, for the one cell $C_{L_c}$ (side $\ell_c\ge10$) and the one negative Wick source $-2H'\chi_0$ of \cref{def:s11:cells}, and that it has the \emph{scheme bound} \SCH\ if either $a=z$, the reference scheme in which \cref{hyp:s03:T2} is computed, or $\sup_k|z_k-a_k|<\infty$. The cell bound is the boundedness form of \cref{hyp:s03:W1} at one cube and one source, and the only place where that theorem enters the proof of \eqref{eq:s11:W} (through \cref{lem:s11:W4,lem:s11:W5,prop:s11:W8}; $K_-$ is built from $B_{\mathrm{cell}}$). Both properties hold for the schemes of the main theorem: \begin{lemma}[\Bcell, the cell bound]\label{hyp:s12:Bcell} Let the couplings lie in the coupling region of \cref{hyp:s03:T3} and let $a$ be an admissible scheme (\cref{def:s03:adm}). Then $a$ has the cell bound: $B_{\mathrm{cell}}=\sup_k\ln N^a_{k,C_{L_c}}(0,-2H'\chi_0)<\infty$. \end{lemma} \begin{proof} By \cref{def:s11:cells} the cell $C_{L_c}=[-ML_c,ML_c)^3$ is a King cube and $\supp\chi_0$ lies at $\ell^\infty$-distance $>1$ from $\R^3\setminus C_{L_c}$; the source $q\defeq2H'\chi_0$ is a non-negative function of $C^\infty_c(\R^3)$ with this support condition. At couplings in the coupling region the hypotheses \Itwo\ and \Ithree\ of the existence theorem with scalar sources hold (\cref{prop:s10:U}, \cref{prop:s10:RP}), so \cref{cor:k02:W1}(b) applies and gives $\sup_kN^z_{k,C_{L_c}}(0,-q)<\infty$ for the comparison-convention functional in the reference scheme $z$. For a scheme $a$ with $\sup_k|a_k-z_k|<\infty$ the two functionals differ by the factor $e^{-(a_k-z_k)\sS_k(-q)}$ (\cref{rem:s02:scheme}), bounded uniformly in $k$ since $\sS_k(q)\to\int q$; this is \cref{rem:k02:scheme}, and an admissible scheme has $\sup_k|z_k-a_k|<\infty$ by \cref{def:s03:adm}. Hence $\sup_kN^a_{k,C_{L_c}}(0,-2H'\chi_0)<\infty$, and $B_{\mathrm{cell}}<\infty$ because each term is positive (\cref{rem:k02:cell}). \end{proof} \begin{lemma}[\SCH, the scheme bound]\label{hyp:s12:SCH} An admissible scheme has the scheme bound. If $a\ne z$ has the scheme bound, the constants of \eqref{eq:s11:W} hold with $C_U$ replaced by $C_U+s_1\sup_k|z_k-a_k|$. \end{lemma} \begin{proof} For an admissible scheme either $a=z$ or $\lim_k(z_k-a_k)$ exists, so $\sup_k|z_k-a_k|<\infty$ (\cref{def:s03:adm}). The replacement of the constant is the last clause of \cref{prop:s10:U}, carried through \cref{thm:s11:W}. \end{proof} \begin{proposition}[\UNIQ, independence of the subsequence]\label{hyp:s12:UNIQ} Any two continuum subsequences $\mathbf{k},\mathbf{k}'$ have the same LIM state: $\ZLIM{\mathbf{k}}(\cdot,0)=\ZLIM{\mathbf{k}'}(\cdot,0)$ for every family. \end{proposition} \begin{proof} This is \cref{prop:s12:ID}(a) below, from the rotation theorem \cref{hyp:s03:RC}: every continuum subsequence's LIM state is the limit state $Z^{\mathrm{LIM}}_\infty(\cdot,0)$ of that theorem. It also follows from \cref{hyp:s03:EC} through \cref{cor:s04:G2}(iii) (\cref{cor:s12:KK}(b)): every LIM state of the comparison family is King's state $\ZKinf$, the infinite-volume state of King's construction \cite{K1,K2}. \end{proof} The proofs of the axioms below hold for each LIM state separately and do not use \UNIQ; it enters only in the Euclidean clause (vii) of \cref{thm:s12:LIM}, through \cref{prop:s12:ID}(a). Without the rotation theorem and the existence theorem one would only know that every LIM state lies on $(\mathrm i\R)^N$ between $\lim_J\liminf_ka^{(a)}_{k,J}$ and $\lim_J\limsup_ka^{(a)}_{k,J}$ (both exist, by monotonicity in $J$), with equality of the LIM states if these coincide at every real argument, an entire function being determined on $(\mathrm i\R)^N$. \subsection{Existence of continuum subsequences and the cube limit} \begin{lemma}[continuum subsequences exist]\label{lem:s12:LIM0} \leavevmode \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*,nosep] \item \textup{(Curvature sector; no fixed-cube limit used.)} Every increasing sequence of spacings has a continuum subsequence. For every continuum subsequence $\mathbf{k}$, family $f$ and $J\ge J_0(f)$: $Z^{\mathbf{k}}_J(\cdot,0)$ is entire, $|Z^{\mathbf{k}}_J(\sum_jz_jf_j,0)|\le e^{\norm{g(z)}_2^2}$, $Z^{\mathbf{k}}_J(x,0)$ is real and $\ge1$ at real $x$, and $0\le Z^{\mathbf{k}}_J(\mathrm it,0)\le1$ at imaginary $\mathrm it$. \emph{Several models at once:} for every countable set $\cM$ of lattice models of King's type (Neumann--Dirichlet cube states with a site-independent real bare scalar coefficient, photon mass $\mu>0$, $\lambda>0$; for instance the models at the photon masses of a countable set, with the counterterm fixed) one $\mathbf{k}$ is a continuum subsequence for every member of $\cM$. \item \textup{(Wick sector, Form~(B).)} Let the couplings lie in the coupling region of \cref{hyp:s03:T3} and let the scheme $a$ have the cell bound \Bcell\ and the scheme bound \SCH\ (both hold for every admissible scheme, \cref{hyp:s12:Bcell,hyp:s12:SCH}). Then every increasing sequence of spacings has a continuum subsequence for the Wick sector, and for every such $\mathbf{k}$, family $\cF$ and $J$ with $MJ>R(\cF)+\ell_c$, $Z^{a,\mathbf{k}}_J$ is holomorphic on $\Dtube$ and \begin{equation}\label{eq:s12:Winf} |Z^{a,\mathbf{k}}_J(\zeta)|\le\exp\big(\norm{g(\zeta)}_2^2+\kappa\norm{k_+(\zeta)}_{L^1}+K_-\wnorm{k_-(\zeta)}\big)\qquad(\zeta\in\Dtube), \end{equation} with the constants of \eqref{eq:s11:constants}. \emph{Several models at once:} for a countable set of models all satisfying \eqref{eq:s11:W} with one set of constants, one $\mathbf{k}$ serves all. \end{enumerate} \end{lemma} The subsequence is chosen in Step~2 below and nowhere else (\cref{rem:s12:K9}). \begin{proof} \emph{(a).} Statements used: \cref{lem:s05:C2} ($\sup_{k\ge0,\,J\ge J_0(f)}\sup_K|a^{(a)}_{k,J}|<\infty$ for every family and compact $K\subset\C^N$); \cref{lem:s05:C1} at second order ($\ava{F(u)^2}\le2\norm u^2_{2,k}$ for real $u$ with plaquettes in $\XJ$); charge conjugation $(A,\vartheta)\mapsto(-A,-\vartheta)$, a symmetry of every Neumann--Dirichlet state, so $\ava{F(u)}=0$ and $\ava{e^{\mathrm iF(u)}}$ is real. \emph{Step 1 (a countable core).} Let $\cB$ be the countable set of finite unions of closed boxes with rational corners, for $B\in\cB$ let $S_0(B)$ be a countable $\norm\cdot_\infty$-dense subset of $C^\infty_c(\operatorname{int}B;\Lambda^2)$, and let $\cP$ be the countable set of pairs $(J,u')$, $u'$ a finite family from $\bigcup_BS_0(B)$, $J\ge J_0(u')$. \emph{Step 2 (Montel and the diagonal).} For $(J,u')\in\cP$, $(a^{u'}_{k,J})_k$ is locally bounded on $\C^N$ uniformly in $k$ (\cref{lem:s05:C2}), hence normal; enumerate $\cP$, extract successively, and let $\mathbf{k}=(k_m)$ be the diagonal sequence: $(a^{u'}_{k_m,J})_m$ converges locally uniformly for every $(J,u')\in\cP$. \emph{Step 3 (every family, imaginary coefficients).} Let $f$ be a family, $J\ge J_0(f)$, and $B\in\cB$ with $\bigcup_j\supp f_j\subset\operatorname{int}B\subset B\subset U_{J_0(f)}$ (so every $u'\in S_0(B)$ is admissible at every $J\ge J_0(f)$). For real $2$-forms $u,u',g$ supported in $B$ and every $k$ with $\eps_k\le1$, \[ \begin{gathered} |\ava{e^{\mathrm iF(u)}}-\ava{e^{\mathrm iF(u')}}|\le{\ava{F(u-u')^2}}^{1/2}\le\sqrt2\norm{u-u'}_{2,k},\\ \norm g^2_{2,k}\le V_B\norm g_\infty^2,\quad V_B\defeq3\operatorname{vol}(N_{1/2}(B)), \end{gathered} \] the first by $|e^{\mathrm ix}-1|\le|x|$ and the second-order bound, the second because the cubes of side $\eps_k$ about the centres of the plaquettes of one orientation on which $g\ne0$ are disjoint and lie in the closed $\ell^\infty$-neighbourhood $N_{1/2}(B)$. Given $t\in\R^N$ and $\eta>0$ pick $u'\in S_0(B)$ with $\norm{\sum_jt_jf_j-u'}_\infty\le\eta$; then $|a^f_{k_m,J}(\mathrm it)-a^f_{k_{m'},J}(\mathrm it)|\le2\sqrt{2V_B}\,\eta+|{\ava{e^{\mathrm iF(u')}}}_{k_m}-{\ava{e^{\mathrm iF(u')}}}_{k_{m'}}|$, and the last term tends to $0$ by Step~2. So $(a^f_{k_m,J}(\mathrm it))_m$ converges for every $t$. \emph{Step 4 (all complex coefficients).} $(a^f_{k_m,J})_m$ is locally bounded on $\C^N$ and converges on the maximal totally real $(\mathrm i\R)^N$; Montel--Vitali gives locally uniform convergence to an entire $Z^{\mathbf{k}}_J(\cdot,0)$. The bound $|a^{(a)}_{k,J}(z)|\le e^{\norm{g(z)}^2_{2,k}}$ (\cref{thm:s05:C}) passes to the limit with $\norm g_{2,k}\to\norm g_2$ \eqref{eq:s02:riemann}; realness at real points passes; $a^{(a)}_{k,J}(x)=\ava{e^{F(u_x)}}\ge e^{\ava{F(u_x)}}=1$ by Jensen, and $[1,\infty)$ is closed; $0\le a^{(a)}_{k,J}(\mathrm it)\le1$ by the current representation of \cref{lem:s02:GD}(b), $\ava{e^{\mathrm i\ip vA}}=e^{-\ip v{Cv}/2}\,\E^\sharp\cosh\ip{Cv}{\kappa}\ge0$, and $|e^{\mathrm iF}|=1$. Dependence only on the total source holds at every $(k,J)$, hence in the limit. \emph{Step 5 (several models).} The constant of \cref{lem:s05:C2}, the bound $e^{\norm g^2_{2,k}}$ and the Lipschitz constant $\sqrt2$ are model-free: \cref{thm:s05:C} is \cref{lem:s02:GD}(a) for a measure of the class of \cref{lem:s02:BFS} plus \cref{lem:s05:C1} for the Neumann--Dirichlet photon covariance, valid for every real site-independent bare coefficient (the matter weight does not enter) and every $\mu>0$. For countable $\cM$ run Step~2 over $\cM\times\cP$ and Steps~3--4 member by member. \emph{(b).} Statements used: under \Itwo--\Ifour\ and $B_{\mathrm{cell}}<\infty$, for every family with supports in $Q_R$, every $k$, $J\ge J_1(R)$ and $\zeta\in\Dtube$, \eqref{eq:s11:W} gives $|Z^a_{k,J}(\zeta)|\le Z^a_{k,J}(\Re\zeta)\le\exp(\norm g^2_{2,k}+\kappa\norm{k_+}_{1,k}+K_-\wnorm{k_-})$, with \Itwo\ from \Tthree\ by \cref{prop:s10:U} and \SCH\ for the scheme; on compacts of the tube the right side is bounded uniformly in $k$ (\cref{lem:s02:E}). \emph{Steps 1--2.} As in (a), with $S^{\sW}_0(B)$ a countable $\norm\cdot_\infty$-dense subset of $C^\infty_c(\operatorname{int}B;\Lambda^2)\times C^\infty_c(\operatorname{int}B)$ and $\cP^{\sW}$ the countable set of pairs $(J,\cF')$, $\cF'$ a finite family from $\bigcup_BS^{\sW}_0(B)$, $MJ>R(\cF')+\ell_c$. For such a pair $J\ge J_1(R)$ for every $R\in(R(\cF'),MJ-\ell_c]$, so \eqref{eq:s11:W} holds for $\cF'$ at every $k$ and $(Z^a_{k,J})_k$ is normal on the tube of $\cF'$; the diagonal over $\cP^{\sW}$, together with that of (a), gives $\mathbf{k}$. \emph{Step 3 (every family, imaginary coefficients).} Let $\cF=(f_i;h_j)$, $MJ>R(\cF)+\ell_c$, $R_J\defeq MJ-\ell_c>R(\cF)$, so that all supports lie in the open cube $Q^\circ_{R_J}$ and $J\ge J_1(R_J)$; choose $B\in\cB$ with the supports in $\operatorname{int}B\subset B\subset Q^\circ_{R_J}$, so that $(J,\cF')\in\cP^{\sW}$ for every $\cF'$ from $S^{\sW}_0(B)$. For $(s,s')\in\R^{n+p}$ put $\sigma\defeq(\sum_is_if_i,\sum_js'_jh_j)$, so that $Z^a_{k,J}(\mathrm is,\mathrm is')=\ava{e^{\mathrm i\Phi^a_k(\sigma)}}$ with $\Phi^a_k(g,h)\defeq F_{\Lk}(g)+W^a_{\Lk}(h)$. \emph{Lipschitz bound.} For a real pair $q=(g_q,h_q)$ supported in $Q_{R_J}$, $(t_1,t_2)\mapsto Z^a_{k,J}(t_1g_q,t_2h_q)$ is holomorphic, and on the polydisc $|t_1|\le\norm{g_q}_{2,k}^{-1}$, $|t_2|\le\sigma_0/\wnorm{h_q}$ one has $\norm{k_\pm}_\infty\le|\Re t_2|\,\norm{h_q}_\infty\le\sigma_00$ pick $\sigma'\in S^{\sW}_0(B)$ with $\norm{g_\sigma-g_{\sigma'}}_\infty,\norm{h_\sigma-h_{\sigma'}}_\infty\le\eta$; with $V_B$ and $\wnorm h\le(1+3R_J^2)^2\norm h_\infty$ for $h$ supported in $Q_{R_J}$, $|Z^a_{k,J}(\mathrm i\sigma)-Z^a_{k,J}(\mathrm i\sigma')|\le C(\sqrt{V_B}+(1+3R_J^2)^2)\eta$ for every $k$, while $Z^a_{k_m,J}(\mathrm i\sigma')$, the core functional at the point $(\mathrm i,\mathrm i)$ of its tube, converges by Step~2. So $(Z^a_{k_m,J})_m$ converges on $(\mathrm i\R)^{n+p}$. \emph{Step 4 (the tube).} $(Z^a_{k_m,J})_m$ is locally bounded on the connected open $\Dtube$ and converges on the maximal totally real $(\mathrm i\R)^{n+p}\subset\Dtube$; Montel--Vitali gives locally uniform convergence to a holomorphic $Z^{a,\mathbf{k}}_J$, and \eqref{eq:s11:W} passes to the limit with $\norm g_{2,k}\to\norm g_2$, $\norm{k_+}_{1,k}\to\norm{k_+}_{L^1}$ (Riemann sums; $k_+$ is continuous with compact support) and sampled $\wnorm{k_-}\le$ continuum $\wnorm{k_-}$: this is \eqref{eq:s12:Winf}. \emph{Step 5 (several models).} With one set of constants in \eqref{eq:s11:W} for all members, $C$ is model-free and Step~2 runs over the product of the model set with $\cP^{\sW}$. \end{proof} \begin{lemma}[the cube limit along $\mathbf{k}$ is a full limit]\label{lem:s12:LIM1} Let $\mathbf{k}$ be a continuum subsequence. \textup{(Curvature sector.)} For every family, $Z^{\mathbf{k}}_J(\cdot,0)\to\ZLIM{\mathbf{k}}(\cdot,0)$ locally uniformly on $\C^N$ as $J\to\infty$; the limit is entire, $|\ZLIM{\mathbf{k}}(\sum_jz_jf_j,0)|\le e^{\norm{g(z)}_2^2}$, real and $\ge1$ at real points, in $[0,1]$ at imaginary points; $J\mapsto Z^{\mathbf{k}}_J(\mathrm it,0)$ is nondecreasing, so $Z^{\mathbf{k}}_J\le\ZLIM{\mathbf{k}}$ on $(\mathrm i\R)^N$. \textup{(Wick sector, Form~(B).)} Under the hypotheses of \cref{lem:s12:LIM0}(b), $Z^{a,\mathbf{k}}_J\to Z^{a,\mathbf{k}}_\infty$ locally uniformly on $\Dtube$; the limit is holomorphic, obeys \eqref{eq:s12:Winf}, is $>0$ at the real points of the tube, and $Z^{a,\mathbf{k}}_J\le Z^{a,\mathbf{k}}_\infty$ at the cone points ($z\in(\mathrm i\R)^n$, $w$ real, $k(\zeta)\ge0$). \end{lemma} \begin{proof} \emph{Curvature sector.} For $t\in\R^N$, $J\ge J_0(f)$ and every $k$: $a^{(a)}_{k,J}(\mathrm it)\le a^{(a)}_{k,J+1}(\mathrm it)\le1$ (first inequality of \cref{lem:s02:cut} with $Y=C_{J+1}\cap\Lk$, $X=\XJ$; the cut-out factor is the Neumann--Dirichlet state on $\CJ$ because $\dm_\eps$ is site-independent; realness by charge conjugation). Both sides converge along $\mathbf{k}$, so $J\mapsto Z^{\mathbf{k}}_J(\mathrm it,0)$ is nondecreasing, bounded by $1$, convergent. $\{Z^{\mathbf{k}}_J\}_J$ is locally bounded on $\C^N$ by the $J$-free bound $e^{\norm{g(z)}_2^2}$ and converges on $(\mathrm i\R)^N$; Montel--Vitali gives the entire limit with the same bound; realness, $\ge1$ at real points and $[0,1]$ at imaginary points pass to the limit. \emph{Wick sector.} (1) \emph{Split family.} Write $h_j=h_j^+-h_j^-$, $h_j^-\defeq\norm{h_j}_\infty\chi_j$, $h^+_j\defeq h_j+h^-_j\ge0$, with $\chi_j\in C^\infty_c$, $0\le\chi_j\le1$, $\chi_j=1$ on $\supp h_j$, $\supp\chi_j\subset N_{1/2}(\supp h_j)$; put $F_{k,J}(z,w^+,w^-)\defeq Z^a_{k,J}(\sum_iz_if_i,\sum_j(w^+_jh^+_j+w^-_jh^-_j))$ and $\cD''\defeq\{|\Re w^\pm_j|<\delta_j\}$ with $\sum_j\delta_j(\norm{h^+_j}_\infty+\norm{h^-_j}_\infty)R(\cF)+\frac12+\ell_c$, with \eqref{eq:s12:Winf}. (2) On $\Omega_+\defeq\cD''\cap((\mathrm i\R)^n\times(0,\infty)^{2p})$ the observable has imaginary curvature coefficients and nonnegative total scalar source, so $J\mapsto F_{k,J}$ is nondecreasing at every $k$ (first inequality of \cref{lem:s02:cut} with \cref{lem:s02:alg}(a), the scheme multiplier being common); along $\mathbf{k}$, $J\mapsto F^{\mathbf{k}}_J$ is nondecreasing and bounded on $\Omega_+$, hence convergent. (3) $\Omega_+$ is a non-empty open subset of the maximal totally real $(\mathrm i\R)^n\times\R^{2p}$ and $\cD''$ is connected: $F^{\mathbf{k}}_J\to F^{\mathbf{k}}_\infty$ locally uniformly on $\cD''$. (4) At $w^+=w$, $w^-=-w$ the lattice functionals equal $Z^a_{k,J}(\zeta)$ exactly (equal total sources), so $Z^{a,\mathbf{k}}_J$ converges locally uniformly on $\cD'\defeq\{|\Re w_j|<\delta_j\}\times\C^n\subset\Dtube$; $\{Z^{a,\mathbf{k}}_J\}_J$ is locally bounded on $\Dtube$ by the $J$-free \eqref{eq:s12:Winf} and converges on the open $\cD'$; Montel--Vitali on the connected tube gives $Z^{a,\mathbf{k}}_J\to Z^{a,\mathbf{k}}_\infty$, holomorphic, with \eqref{eq:s12:Winf}. (5) \emph{Positivity at real tube points.} For real $\zeta\in\Dtube$ pick $\epsilon>0$ with $-\epsilon\zeta\in\Dtube$ and let $C_\epsilon\defeq\sup_k(\norm g^2_{2,k}+\kappa\norm{k_+}_{1,k}+K_-\wnorm{k_-})|_{-\epsilon\zeta}<\infty$, the exponent of \eqref{eq:s11:W} at $-\epsilon\zeta$ (bounded in $k$ by \cref{lem:s02:E}, free of $J$). At every $k$, $t\mapsto\ln Z^a_{k,J}(t\zeta)$ is convex on $\R$ (H\"older, \cref{lem:s02:alg}(b)) and vanishes at $t=0$, so $1\le Z^a_{k,J}(\zeta)^{\epsilon/(1+\epsilon)}Z^a_{k,J}(-\epsilon\zeta)^{1/(1+\epsilon)}\le Z^a_{k,J}(\zeta)^{\epsilon/(1+\epsilon)}e^{C_\epsilon/(1+\epsilon)}$, i.e.\ $Z^a_{k,J}(\zeta)\ge e^{-C_\epsilon/\epsilon}$ for every $k$ and $J$; the bound passes along $\mathbf{k}$ and then in $J$: $Z^{a,\mathbf{k}}_\infty(\zeta)\ge e^{-C_\epsilon/\epsilon}>0$. (The product inequality of \cref{lem:s02:alg}(c) is not used: the tube is asymmetric, $H=\sigma_+/16$, and for real $\xi\in\Dtube$ with $\norm{k_+(\xi)}_\infty\ge H/2$ the point $-\xi$ lies outside it, where \cref{def:s12:contsub} gives no limit.) (6) At a cone point of the original family $J\mapsto Z^a_{k,J}(\zeta)$ is nondecreasing at every $k$ (first inequality of \cref{lem:s02:cut} with \cref{lem:s02:alg}(a)); pass along $\mathbf{k}$. \end{proof} \subsection{The axioms for LIM states} Throughout this subsection, let $e>0$, $\lambda,m,\lmu>0$, $\eps_k=5^{-k}$, $\CJ=[-MJ,MJ)^3$, and let $\NDa_{k,J}$ be the states of \cref{def:s02:acube}. For a continuum subsequence $\mathbf{k}$ (\cref{def:s12:contsub}) and a finite family of real $2$-forms the LIM state (\cref{def:s12:LIMstate}) is \[ \ZLIM{\mathbf{k}}\Big(\sum_jz_jf_j,0\Big)=\lim_{J\to\infty}\lim_{m\to\infty}\Big\langle e^{\sum_jz_jF_{\Lambda_{k_m}}(f_j)}\Big\rangle^{(a)}_{k_m,J} \] with the Schwinger functions $S^{\mathbf{k}}_n(f_1,\dots,f_n)\defeq\partial_{z_1}\cdots\partial_{z_n}\ZLIM{\mathbf{k}}(\sum_jz_jf_j,0)|_{z=0}$; in the Wick sector, $Z^{a,\mathbf{k}}_\infty$ is the LIM state of $\mathbf{k}$ for a family $\cF=(f_1,\dots,f_n;h_1,\dots,h_p)$ of real $2$-forms and real scalars on its tube $\Dtube$, with the joint Schwinger functions $S^{\mathbf{k}}_{n,p}(f;h)=\partial^{(n)}_z\partial^{(p)}_wZ^{a,\mathbf{k}}_\infty(0)$, and $\sigma_0=H/4$, $K'=\kappa\pi^2+K_-$ with the constants of \eqref{eq:s11:constants}. \begin{theorem}[LIM states of the comparison family at fixed photon mass]\label{thm:s12:LIM} \textbf{Curvature sector (no fixed-cube limit used).} Every increasing sequence of spacings has a continuum subsequence. For every continuum subsequence $\mathbf{k}$ and finite family of real $2$-forms, $\ZLIM{\mathbf{k}}(\sum_jz_jf_j,0)$ exists locally uniformly in $z\in\C^N$, is entire, and depends on the family only through the total source, and its Schwinger functions $S^{\mathbf{k}}_n$ satisfy: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*,nosep] \item \textup{(\OSz.)} $|\ZLIM{\mathbf{k}}(\sum_jz_jf_j,0)|\le e^{\norm{g(z)}_2^2}$ and $\le\exp(\half\lmu^{-2}\norm{\sum_j(\Re z_j)\dstar f_j}_2^2)$ (the first bound independent of $\lmu,e,\lambda,m$); $|S^{\mathbf{k}}_n|\le(\sqrt2e)^n(n!)^{1/2}\prod_j\norm{f_j}_2\le(e\pi/2)^n(n!)^{1/2}\prod_j\fournorm{f_j}$; the $S^{\mathbf{k}}_n$ are tempered distributions, and \Ezz\ holds with $s=4$ and every $\beta>\half$. \item \textup{(\OSo.)} $S^{\mathbf{k}}_n$ is invariant under all translations and under $\Oh$ about every point, $F$ transforming as a $2$-form. \item \textup{(\OSt.)} Reflection positivity about every coordinate plane $\{x_\nu=c\}$: for exponentials $E_a=e^{\sum_lz_{al}F(f_{al})}$ with arbitrary complex $z_{al}$ and $f_{al}$ supported in $\{x_\nu>c\}$, $(S(\overline{\theta_cE_a}E_b))_{ab}$ is positive semidefinite, and $S(\theta_c(P)P)\ge0$ for polynomials $P$ in such fields. \item \textup{(\OSth.)} $S^{\mathbf{k}}_n$ is real, symmetric, and zero for odd $n$. \item \textup{(The measure.)} For every even $r\ge14$ there is exactly one Borel probability measure $\mu^{\mathbf{k}}_F$ on $\cH^{-r}(\R^3;\Lambda^2)$ whose moments on $C^\infty_c$ are the $S^{\mathbf{k}}_n$; its characteristic functional is $\ZLIM{\mathbf{k}}(\mathrm i\,\cdot,0)$ extended by continuity to $\cS$, and its image on $H^{-r}_{\mathrm{loc}}$ is the unique measure there with this characteristic functional and the unique one with these moments. For every $\chi\in C^\infty_c$ and box $Q$ containing $\supp\chi$ in its interior, the laws of $\chi F_{\Lambda_{k_m}}$ under $\NDa_{k_m,J}$ converge weakly on $H^{-r}_{\mathrm{per}}(Q;\Lambda^2)$, \emph{first as $m\to\infty$ along $\mathbf{k}$} at every fixed $J\ge J_0(\chi)$ and then as $J\to\infty$, to the law of $\chi\Phi$ under $\mu^{\mathbf{k}}_F$. \item \textup{(Weak \OSf.)} $\mu^{\mathbf{k}}_F$ is $\Cub$-invariant and satisfies the clauses of \cref{thm:s09:S2K}(b) with $\mu^{\mathbf{k}}_F$ in place of $\mu^K_S$. \item \textup{(Euclidean clause.)} $\ZLIM{\mathbf{k}}$ does not depend on $\mathbf{k}$, equals the limit state $Z^{\mathrm{LIM}}_\infty$ of \cref{def:s12:o3LIM}, is $\Euc$-invariant for all complex coefficients (\cref{hyp:s03:RC}), and the Euclidean forms of (ii) (full $\Euc$), (iii) (reflection positivity about every plane) and (vi) (almost every component $\Euc$-invariant, reflection positive about every plane, clustering in any manner, with a unique vacuum in every direction; the vanishing of a component's odd moments is not claimed) hold. \end{enumerate} \textbf{Wick sector, Form~(B).} Let the couplings lie in the coupling region of \cref{hyp:s03:T3} and let the scheme $a$ have the cell bound \Bcell\ and the scheme bound \SCH\ (\cref{hyp:s12:Bcell,hyp:s12:SCH}; every admissible scheme does). Every increasing sequence of spacings has a continuum subsequence for the Wick sector, and for every such $\mathbf{k}$ and family the LIM state $Z^{a,\mathbf{k}}_\infty$ exists locally uniformly on $\Dtube$, is holomorphic there, obeys \eqref{eq:s12:Winf}, and (i)--(vi) hold for the joint Schwinger functions $S^{\mathbf{k}}_{n,p}$ with: in (i) the mixed bounds $|S^{\mathbf{k}}_{n,p}|\le e^{K'\sigma_0}(\sqrt2e)^n(n!)^{1/2}(e/\sigma_0)^p\,p!\prod_i\norm{f_i}_2\prod_j\wnorm{h_j}$, and \Ezz\ with $s=4$ and every $\beta>1$; in (ii) $W^a$ a scalar, the identity holding on the tube; in (iii) exponentials with arbitrary complex curvature coefficients and Wick coefficient vectors $\zeta_1,\dots,\zeta_N$ of \emph{amplitude $\amp(\zeta_i)c\}$ with $\amp(\zeta_i)c\}$; (vi) is Corollary~\ref{cpi:rot:cor}, with the admissible push-forward action of Lemma~\ref{cpi:rot:action}(i) and the $O(3)$-invariance of the measure, which by Lemma~\ref{cpi:rot:action}(ii) is that of $\cZ^{\mathbf{k}}$, as in \cref{thm:s09:S2K}(c), with its proviso on the odd moments of a component. \end{proof} \begin{proposition}[LIM states of two schemes]\label{r13:prop:s12schemes} Let the couplings lie in the coupling region of \cref{hyp:s03:T3} and let the schemes $a,a'$ both have the cell bound \Bcell\ and the scheme bound \SCH. Then every continuum subsequence for the Wick sector in the scheme $a$ has a further subsequence $\mathbf{k}'$ along which $a'_k-a_k\to\kappa^{\mathbf{k}'}$, and $Z^{a',\mathbf{k}'}_\infty(\zeta)=e^{-\kappa^{\mathbf{k}'}\sum_jw_j\int h_j}Z^{a,\mathbf{k}'}_\infty(\zeta)$: the two LIM states differ by the shift $W\mapsto W-\kappa^{\mathbf{k}'}\int(\cdot)$ of the Wick component, and joint cumulants of order $\ge2$ are scheme-free. \end{proposition} \begin{proof} $\sup_k|a'_k-a_k|\le\sup_k|a'_k-z_k|+\sup_k|z_k-a_k|<\infty$ by \SCH\ for both. For a continuum subsequence $\mathbf{k}$ for $a$, Bolzano--Weierstrass gives $\mathbf{k}'\subset\mathbf{k}$ with $a'_k-a_k\to\kappa^{\mathbf{k}'}\in\R$. At every $(k,J)$, $Z^{a'}_{k,J}(\zeta)=e^{-\sum_jw_j(a'_k-a_k)\sS_{\Lk}(h_j)}Z^a_{k,J}(\zeta)$ (\cref{rem:s02:scheme}) with $\sS_{\Lk}(h_j)\to\int h_j$; so along $\mathbf{k}'$ the $a'$-functionals converge wherever the $a$-functionals do, $\mathbf{k}'$ is a continuum subsequence for $a'$ with $Z^{a',\mathbf{k}'}_J=e^{-\kappa^{\mathbf{k}'}\sum_jw_j\int h_j}Z^{a,\mathbf{k}'}_J$, and then $J\to\infty$. (For admissible $a,a'$, $a'_k-a_k$ converges along the full sequence and $\kappa^{\mathbf{k}'}=\kappa$.) The cumulant statement is that of \cref{thm:s13:main}. \Bcell\ transfers between schemes with bounded difference: $B^{a'}_{\mathrm{cell}}\le B^a_{\mathrm{cell}}+\sup_k\big(|a'_k-a_k|\,\norm{2H'\chi_0}_{1,k}\big)$. \end{proof} \begin{lemma}[a $k$- and $J$-uniform bound for the modified-box functionals on the tube of the enlarged family]\label{lem:s12:claim4prime} For couplings in the coupling region of \cref{hyp:s03:T3}, under \Bcell\ and \SCH, for every $k$, every $J\ge\max(J_0,J_1(R_\cF))$ and every $\zeta\in\cD^{\cF}_{\sigma_+,H}$ (the tube of $\cF$ in the sense of \cref{def:s02:tube}), \[ |\Phi_k(\zeta)|\le\exp\big(2\norm{g(\zeta)}^2_{2,k}+\kappa\norm{k_+(\zeta)}_{1,k}+K_-\wnorm{k_-(\zeta)}\big). \] \end{lemma} The curvature constant is $2$, not the $1$ of \eqref{eq:s11:W}, because the Cauchy--Schwarz split below doubles the curvature source and \cref{thm:s05:C} is quadratic in it; nothing depends on its value. \begin{proof} $|\Phi_k(\zeta)|\le\Phi_k(\xi)$, $\xi\defeq\Re\zeta$ (\cref{lem:s02:alg}(b)). Let $k(\xi)$ be the total real scalar source at $\xi$ and $k_1\defeq k(\xi)_+$, $k_2\defeq k(\xi)_-$ its \emph{pointwise} parts, sampled on the lattice. Cauchy--Schwarz at the lattice level (\cref{lem:s02:alg}(d); the multiplier $e^{-a_k\sS(k_1-k_2)}$ splits by linearity of $\sS$): $\Phi_k(\xi)\le Z^{a,(3.38)}(2f_\xi,-2k_2)^{1/2}Z^{a,(3.38)}(0,2k_1)^{1/2}$. \emph{First factor} (class $\cD$): the $\cD$-half of \cref{lem:s07:L}, which needs only that the sources vanish off $\XJ$ ($J\ge J_0$), gives $\le Z^a_{k,\CJ}(2f_\xi,-2k_2)$; \cref{thm:s05:C} with the real scalar source $-2k_2$ gives $\le e^{\norm{2f_\xi}^2_{2,k}}N^a_{k,\CJ}(0,-2k_2)$ (the same scheme multiplier on both sides); \cref{prop:s11:W8} with $q\defeq2k_2\ge0$, supported in $Q_{R_\cF}$, $J\ge J_1(R_\cF)$ and $\sup q=2\norm{k(\xi)_-}_\infty0$~\cite[(2.5)]{K2}, using the ultraviolet stability bounds of Ba{\l}aban~\cite{Ba1,Ba2,Ba3,Ba4}. Rotation invariance is not automatic, since all lattices in the construction have the same orientation. Theorem~2.4 of~\cite{K2} states that on the tori $D_J$ the lattices $\eps_k\Z^3$ and $R(\eps_k\Z^3)$, with $R$ the rotation by $\theta_0=\arctan(3/4)$ in a coordinate plane, have the same continuum limit; since $\theta_0/\pi$ is irrational these rotations generate a dense subgroup of $SO(3)$. The statement as given in~\cite[Thm.~2.4]{K2} is false in two ways (Remark~\ref{cpi:rot:errata}): the source it pairs with the plaquettes is one scalar function on both lattices, which couples the field to two different $2$-forms, and the Wick constant it uses misses an order-$e^2$ logarithm. We therefore call the statement of~\cite[Thm.~2.4]{K2} \emph{King's torus conjecture}, and we use instead the corrected statement, for $2$-form sources sampled in each lattice's own frame and for the two lattices at the same spacing, which is proved in Theorem~\ref{rot:thm:CT} with \cref{cpi:ct:Vvac}. The ultraviolet analysis of~\cite{K1,K2} needs boundaries that are lattice hyperplanes~\cite[p.~326]{K2}, and no bounded region has that property for both lattices~\cite[p.~326]{K2}. Where to place the two-lattice comparison is therefore the whole question. This part places it on periodic tori carrying a smooth mass wall, and proves $O(3)$ invariance of King's state in the curvature sector, on all of King's range; the results used in the proof that are not theorems of the cited literature are stated, each with its proof, where first used; the corrections to the cited sources are Remark~\ref{cpi:rot:errata}. \subsection{The torus route} On each lattice separately King's cubes are compared with tori carrying a \emph{smooth mass wall}. Fix $R$ in the generating set $\Rgen=\{R_{\mu\nu}(\pm\theta_0)\}$ and a real $2$-form $f$, smooth with compact support. For a walled torus $\fc=(J',w)$, the periodic model on $D_{J'}$ in which the scalar mass squared is raised by $2w$ with $w\ge0$ smooth and vanishing near the support of $f$, the chain is \begin{equation}\label{rot:eq:chain} Z_{C_L}(\mathrm i\,R\cdot f,0)\;=\;Z^{R^{-1}\Lambda}_{R^{-1}C_L}(\mathrm if,0)\;\le\;T^{R^{-1}\Lambda}_{\fc}(f) \;\xrightarrow[\eps\to0]{}\;\cW_\fc(f,0)\;\xleftarrow[\eps\to0]{}\;T^{\Lambda}_{\fc}(f). \end{equation} Here $Z_{C_L}$ is King's cube functional on the axis lattice $\Lambda$, the first equality is a passive rotation, the inequality is a correlation inequality on the rotated lattice $R^{-1}\Lambda$, for which $R^{-1}C_L$ is a cube with faces on lattice hyperplanes, and the two continuum limits agree by the torus comparison. What is missing is an upper bound for walled tori on the axis lattice by King's state: \begin{equation}\label{rot:eq:missing} \text{for every }\eta>0\text{ there is an admissible }\fc\text{ with }\cW_\fc(f,0)\le Z_\infty(\mathrm if,0)+\eta . \end{equation} This is the statement (TIGHT$_0$) of Section~\ref{rot:sec:env}. It involves one lattice and one orientation, and the reverse inequality holds in the limit (Lemma~\ref{rot:lem:free}), so (TIGHT$_0$) says that King's state is the lower envelope of the walled tori. Granting it, letting $L\to\infty$ in~\eqref{rot:eq:chain} gives $Z_\infty(\mathrm i\,R\cdot f,0)\le Z_\infty(\mathrm if,0)$, and applying this to $R^{-1}\in\Rgen$ gives equality. The passage from the dense subgroup to $O(3)$ uses a continuity lemma for convex functionals (Lemma~\ref{lem:s02:baire}), so no regularity of the Schwinger functions is needed. We isolate the local estimate that implies (TIGHT$_0$). In the walled torus let the source sit in a box $H$ and let $U$ be a thick shell around $H$ inside the heavy plateau, where the wall is at least $1$. Raise the scalar mass squared on the shell further by $2K$, through a second smooth profile, and let $\Psi_k(n,K)$ be the resulting expectation at lattice spacing $\eps_k$, with $\Psi_k(n,\infty)$ the expectation with the matter removed on the shell. The statement (RES$_0$) of Section~\ref{rot:sec:shell} asks that for one such configuration and one finite $K$ \[ \liminf_{k\to\infty}\big[\Psi_k(n,K)-\Psi_k(n,\infty)\big]\le\eta , \] with the width of the shell chosen so that a photon boundary term is below $\eta$ as well. At every fixed spacing the bracket tends to $0$ as $K\to\infty$ (Lemma~\ref{rot:lem:tailprop}); the content of (RES$_0$) is that the heavy-shell limit and the continuum limit can be interchanged, at one geometry and with no rate. A two-line sandwich (Lemma~\ref{rot:lem:TL}) shows that the heavy shell is bounded below by a higher wall, and once the matter on the shell is dead the photon on the far side can be pinned and then replaced by King's Neumann photon on a larger cube at a cost exponentially small in the width of the shell, uniformly in the spacing (Lemma~\ref{rot:lem:GB}); so (RES$_0$) implies (TIGHT$_0$) (Theorem~\ref{rot:thm:E}). In the curvature sector (RES$_0$) at one configuration is itself proved (Theorem~\ref{rot:thm:shell}): the shell mass $\kappa$ is put inside the matter covariance of King's block-spin flow, and four estimates on that flow, two-sided bounds on the stopping-scale integral uniformly in $\kappa$, a disc bound for the heavy shell, a $\kappa$-Cauchy property of the reduced small-field action, and a threshold clause, give the tail bound at one configuration. Each of the four is proved in Theorems~\ref{cpi:thm:main} and~\ref{cpi:swc:main}, with the shell-model statements they rest on (Proposition~\ref{cpi:prop:TD}, Definition~\ref{cpi:swc:capset}); the statements used inside those proofs are stated in Sections~\ref{rot:ssec:shellest} and~\ref{rot:sec:shellestB}. The torus comparison is reduced to one-sided bounds at a single lattice spacing (Section~\ref{rot:sec:K}). The passive map $S=R^{-1}\sigma_\nu$ is an involution that exchanges the two lattices and maps the torus $D_J$ to the torus $R^{-1}D_J$. A forward bound of King's type on each of the two tori therefore gives both directions of the comparison on $D_J$. The forward bounds and the existence of the axis-lattice limits are the content of the corrected torus theorem (Theorem~\ref{rot:thm:CT}; \cref{cpi:ct:Vvac}), whose proof establishes them with the large-field clause of King's upper bound in place of King's torus bound~\cite[(4.2)]{K2}, which is vacuous at a non-zero wall; we state the theorem and its hypotheses. An alternative placement of the comparison, on a Neumann--Dirichlet box aligned with one lattice and a staircase for the other, is possible in principle and is not used here. \subsection{Guide to the proof} The route is rows (R0)--(R4) of the roadmap table (Table~\ref{tab:intro:roadmap}), and the sections of this part prove them in that order. Section~\ref{rot:sec:model} fixes the lattices, the states, the sources and the standing lemmas; Section~\ref{rot:sec:wick} defines the object, King's state on curvature sources (Theorem~\ref{rot:thm:W}, Definition~\ref{rot:def:object}), from King's cube limit \cite[Thm.~2.1]{K2} (Theorem~\ref{thm:k01:EC}); Section~\ref{rot:sec:K} proves the torus comparison (K$_0$) (Theorem~\ref{rot:hyp:K}) by the corrected torus theorem (Theorem~\ref{rot:thm:CT}), whose hypotheses (N1)--(N6) are stated, each with its proof, in Section~\ref{rot:ssec:inputsN}, and whose positivity condition is Proposition~\ref{rot:thm:POS} (Section~\ref{rot:ssec:openin}); Section~\ref{rot:sec:env} introduces the walled tori and the closing statement (TIGHT$_0$) (Theorem~\ref{rot:hyp:TIGHT}); Section~\ref{rot:sec:shell} proves it: the heavy-shell estimate (RES$_0$) at one configuration (Theorem~\ref{rot:thm:shell}) rests on the four shell estimates on King's block-spin flow with the shell mass inside the covariance (Theorems~\ref{cpi:thm:main} and~\ref{cpi:swc:main}, Lemma~\ref{rot:lem:SW2}, Corollary~\ref{rot:cor:SWT}; Sections~\ref{rot:ssec:shellest} and~\ref{rot:sec:shellestB}), and Theorem~\ref{rot:thm:E} derives (TIGHT$_0$) from (K$_0$), (RES$_0$) and the dead-shell bound (Theorem~\ref{rot:thm:SHT}). The invariance under the generators is obtained by letting $L\to\infty$ in the chain of inequalities~\eqref{rot:eq:chain} (Theorem~\ref{rot:thm:B}), the passage from the dense subgroup to $O(3)$ is Theorem~\ref{rot:thm:C}, and Theorem~\ref{rot:thm:curv} states the result for the curvature sector (Section~\ref{rot:sec:rotinv}). Every statement used is proved or cited where it is first used. \subsection{The main theorem} We fix $e>0$, $\lambda,m,\mu_0>0$ in King's range and King's block factor $5$ (King's $L$; below, $L$ denotes only the index of a King cube $C_L$). The proof of the theorem of this part runs through two statements about King's walled tori, (K$_0$) and (TIGHT$_0$) (Theorems~\ref{rot:hyp:K} and~\ref{rot:hyp:TIGHT}): \begin{itemize} \item[(K$_0$)] \emph{The torus comparison, curvature sector}: for every $R\in\Rgen$, every walled configuration $\fc$ of the class $\mathcal C$ (Definition~\ref{rot:def:class}) and every real smooth periodic $2$-form $G$, $\lim_k\langle e^{F(G)}\rangle_{\fc,\Lambda_k}$ and $\lim_k\langle e^{F(G)}\rangle_{\fc,R^{-1}\Lambda_k}$ exist and are equal. \item[(TIGHT$_0$)] \emph{The closing statement, curvature sector}: $\cW^{\env}(f,0)\le Z_\infty(\mathrm if,0)$ for every real $2$-form $f\in C^\infty_c$; that is, King's state is the lower envelope of the walled tori. \end{itemize} Both are proved: (K$_0$) is proved from the hypotheses (N1)--(N6) of Section~\ref{rot:ssec:inputsN}, on the class $\mathcal C$, by the corrected torus theorem (Theorem~\ref{rot:thm:CT}); (TIGHT$_0$) is proved from (K$_0$), the heavy-shell estimate (RES$_0$) at one configuration and the dead-shell bound (Theorem~\ref{rot:thm:SHT}) at that configuration (Theorem~\ref{rot:thm:E}), and (RES$_0$) is proved from the four shell estimates (Theorems~\ref{cpi:thm:main} and~\ref{cpi:swc:main}, Lemma~\ref{rot:lem:SW2}, Corollary~\ref{rot:cor:SWT}) by Theorem~\ref{rot:thm:shell}. Let $Z_\infty(\cdot,0)$ denote King's state on curvature sources (Definition~\ref{rot:def:object}, Theorem~\ref{rot:thm:W}): the entire function $z\mapsto Z_\infty(\sum_jz_jf_j,0)$ of the coefficients of real $2$-forms $f_j\in C^\infty_c$, whose derivatives at $z=0$ are the curvature Schwinger functions $\langle F(f_1)\cdots F(f_N)\rangle$. For $R\in O(3)$ and $b\in\R^3$, sources are transported as $2$-forms, $R\cdot f$ and $\tau_bf$ (Definition~\ref{def:s02:sources}). \begin{theorem}[$O(3)$ invariance of the curvature sector]\label{rot:thm:main} Let $e>0$ and $\lambda,m,\mu_0>0$ be any point of King's range, at block factor $5$. Then \textup{(K$_0$)} and \textup{(TIGHT$_0$)} hold outside the set $E_W$ of Proposition~\ref{rot:thm:POS}, and King's state on curvature sources \textup{(}Definition~\ref{rot:def:object}\textup{)} is $O(3)$-invariant at every point of King's range, at the points of $E_W$ by right-continuity in the scalar mass: for every $R\in O(3)$, all real $2$-forms $f_1,\dots,f_N\in C^\infty_c$ and all $z\in\C^N$, \[ Z_\infty\Big(\sum_jz_j\,R\cdot f_j,\,0\Big)=Z_\infty\Big(\sum_jz_jf_j,\,0\Big), \] so all curvature Schwinger functions are $O(3)$-covariant; with the translations (Theorem~\ref{rot:thm:W}(v)) the same holds for every Euclidean motion $x\mapsto Rx+b$, with $\tau_bR\cdot f_j$ in place of $R\cdot f_j$, at the points of $E_W$ again by right-continuity in the scalar mass (Theorem~\ref{cpi:ew:EWC}): the curvature sector is $E(3)$-invariant. For the curvature two-point function the weaker closing condition \textup{(TIGHT$_2$)} suffices (Corollary~\ref{rot:cor:twopoint}). \end{theorem} Theorem~\ref{rot:thm:main} has no hypothesis beyond King's range and the block factor $5$. Its proof is the following assembly of the results of Sections~\ref{rot:sec:K}--\ref{rot:sec:shell}: the implication (N1)--(N6) $\Rightarrow$ (K$_0$), on the class $\mathcal C$, is Theorem~\ref{rot:thm:CT}, its hypotheses (N1)--(N6) being the statements of Section~\ref{rot:ssec:inputsN}; the implication from the four shell estimates (Theorems~\ref{cpi:thm:main} and~\ref{cpi:swc:main}, Lemma~\ref{rot:lem:SW2}, Corollary~\ref{rot:cor:SWT}) to (RES$_0$) is Theorem~\ref{rot:thm:shell}; (RES$_0$), (K$_0$) and the dead-shell bound (Theorem~\ref{rot:thm:SHT}) give (TIGHT$_0$) by Theorem~\ref{rot:thm:E}; and (K$_0$) and (TIGHT$_0$) give the invariance by Theorem~\ref{rot:thm:curv}. King's cube limit \cite[Thm.~2.1]{K2} (Theorem~\ref{thm:k01:EC}) defines the object, King's state on curvature sources on King's cubes (Definition~\ref{rot:def:object}, Theorem~\ref{rot:thm:W}). At the points of $E_W$, where the positivity condition of Proposition~\ref{rot:thm:POS} fails, the argument for (K$_0$) does not apply and the invariance follows from the nearby scalar masses by right-continuity of King's state in the scalar mass (Theorem~\ref{cpi:ew:EWC}). In one sentence: (K$_0$) is a one-sided comparison of effective actions at equal spacing of the kind sketched in~\cite[\S4]{K2} for King's torus conjecture, carried out in Theorem~\ref{rot:thm:CT} and \cref{cpi:ct:Vvac} from the hypotheses (N1)--(N6) of Section~\ref{rot:ssec:inputsN}, on the class $\mathcal C$, outside the set $E_W$; (RES$_0$) is the uniform-in-the-spacing decoupling of a heavy weakly coupled shell from a light core across a heavy plateau, proved from the four shell estimates (Theorems~\ref{cpi:thm:main} and~\ref{cpi:swc:main}, Lemma~\ref{rot:lem:SW2}, Corollary~\ref{rot:cor:SWT}) on King's block-spin flow with the shell mass inside the covariance. Every statement used is proved in this part or in the existence part (Theorem~\ref{thm:k01:EC}), or is cited from the literature at its point of use. \space% % % \section{The model}\label{rot:sec:model} This section fixes the objects of the rotation part: the generating rotations and the large-block integer (Definition~\ref{rot:def:geom}), with the geometry that lets the rotated lattice descend to the torus (Lemma~\ref{rot:lem:geom}); the lattices with their frames, the tori and the walled tori (Definitions~\ref{rot:def:lattice} and~\ref{rot:def:states}) and the configuration class on which the torus comparison is proved (Definition~\ref{rot:def:class}); the two conventions for the counterterm and the mismatch vertex of the shell model (Remark~\ref{rot:rem:CT}, Definition~\ref{cpi:swab:M}); two facts about King's sources that fix the form of the statements, the logarithm in the Wick square and the failure of the source rotated as a scalar (Propositions~\ref{cpi:rot:log} and~\ref{cpi:rot:scalar}); and the standing lemmas of the setting in the walled forms used below (Lemmas~\ref{rot:lem:pass}, \ref{rot:lem:GD}, \ref{rot:lem:cut} and Corollary~\ref{rot:cor:Kmono}). Apart from Definition~\ref{cpi:swab:M}, whose objects are those of the shell model of Section~\ref{rot:sec:shell}, and the two propositions, which are computations in perturbation theory and for the free field, nothing here goes beyond the setting of Section~\ref{sec:s02} and the correlation inequalities of Lemma~\ref{lem:s02:BFS}. \subsection{Geometry} The rotations of the generating set and the block integer are chosen so that the rotated lattice descends to the torus $D_J$; this subsection fixes them and proves the geometric facts used later (Lemma~\ref{rot:lem:geom}). \begin{definition}\label{rot:def:geom} Let $\theta_0\in(0,\pi/2)$ with $\cos\theta_0=\frac45$, $\sin\theta_0=\frac35$. For $1\le\mu<\nu\le3$ let $R_{\mu\nu}(\theta)$ be the rotation with $R e_\mu=\cos\theta\,e_\mu+\sin\theta\,e_\nu$, $Re_\nu=-\sin\theta\,e_\mu+\cos\theta\,e_\nu$, $Re_\rho=e_\rho$ for $\rho\ne\mu,\nu$. Put $\Rgen:=\{R_{\mu\nu}(\pm\theta_0):1\le\mu<\nu\le3\}$ and let $G_0$ be the group generated by $\Rgen$. Let $\eps_k:=5^{-k}$ and $\Lambda_k:=\eps_k\Z^3$, $k\ge0$. Let $M$ be the large-block integer of~\cite[\S2.3]{K1}, fixed throughout, chosen as in the setting: $M\in50\N$ and $M\ge M_{\min}$ (Section~\ref{sec:s02}). Then $M=5M'$ with $M'$ divisible by the prime $\mathfrak p:=2$ and $M'\ge2\mathfrak p$, that is, $M'$ even and $M'\ge4$ (indeed $M'=M/5\in10\N$), for instance $M=20\cdot5^b$ with $b$ large (the prime $\mathfrak p$ of the grid offset of the class $\mathcal C$ (Definition~\ref{rot:def:class}) is $2$: the box $H'$ of Definition~\ref{rot:def:shell} is centred, so the offset $c\equiv M'/\mathfrak p\pmod{M'}$ at both corners $\pm c$ forces $\mathfrak p=2$, and for an odd prime the class would be empty). This is admissible: every condition that the constructions of~\cite{K1,K2,Ba1,Ba2,Ba3,Ba4} place on $M$ is a lower bound, apart from divisibility conditions on the sides of King's cubes and tori, such as that the side $2MJ$ of $C_J$ be a multiple of every block side~\cite[(3.4)]{K2}, which hold for every $M$ by the definitions of $C_J$ and $D_J$ below; it does not change King's state (Definition~\ref{r13:def:objectM}), and it is used, with the grid offset of the box $H'$ in the class $\mathcal C$, by the sub-domain form of the perturbative bound (Proposition~\ref{cpi:prop:TD}). The \emph{King cubes} are $C_J:=\{x\in\R^3:-MJ\le x_\mu0$. \item[(d)] $\theta_0/\pi$ is irrational, and $G_0$ is dense in $SO(3)$. \end{enumerate} \end{lemma} \begin{proof} (a) The entries of $R$ are $0,1,\pm\frac35,\pm\frac45$, and $R_{\mu\nu}(\theta)^{-1}=R_{\mu\nu}(-\theta)$; conjugation by $\sigma_\nu$ reverses the orientation of the $(x_\mu,x_\nu)$-plane and fixes the third axis. (b) For $p\in50MJ\,\Z^3$ we have $R^{\pm1}p=\frac15(5R^{\pm1})p\in10MJ\,\Z^3\subset\Lambda_k$. Hence $R^{-1}\Lambda_k+p=R^{-1}(\Lambda_k+Rp)=R^{-1}\Lambda_k$, and likewise for $R\Lambda_k$. (c) The sums of the absolute values of the entries in each row of $R$ are at most $\frac75$. If $Rx\in Q_b$ then $|x|_\infty=|R^{-1}(Rx)|_\infty\le\frac75b$, since $R^{-1}\in\Rgen$; if $|x|_\infty\le\frac57b$ then $|Rx|_\infty\le b$. (d) If $\theta_0\in\pi\mathbb Q$, then $e^{\mathrm i\theta_0}=(4+3\mathrm i)/5$ is a root of unity, hence an algebraic integer. Its minimal polynomial over $\mathbb Q$ is $x^2-\frac85x+1$, which does not have integer coefficients; this is a contradiction (compare~\cite[pp.~337--338]{K2}). Hence $\{n\theta_0\}_{n\in\Z}$ is dense modulo $2\pi$, and the closure of $\{R_{\mu\nu}(\theta_0)^n\}_{n\in\Z}$ is the group of all rotations in the $(x_\mu,x_\nu)$-plane. Every element of $SO(3)$ is a product $R_{12}(\alpha)R_{23}(\beta)R_{12}(\gamma)$ (Euler angles), so $\overline{G_0}=SO(3)$. \end{proof} \subsection{Lattices, states, sources and the standing lemmas}\label{rot:ssec:standing} This subsection fixes the lattices with their frames, the states on tori and walled tori and the configuration class of the torus comparison, describes the two counterterm conventions and the mismatch vertex of the shell model, proves two facts about King's sources (the logarithm in the Wick square and the failure of the source rotated as a scalar), and restates the standing lemmas of the setting (the passive covariance, the current representation with Gaussian domination, the cutting inequality and the comparison of King's cubes with the boxes) in the walled forms that the torus comparison and the heavy shell use. \begin{definition}[lattices and fields]\label{rot:def:lattice} A \emph{lattice of spacing $\eps_k$} is $\Lambda=O\Lambda_k$ with $O\in O(3)$, equipped with the frame $f_\alpha:=Oe_\alpha$, $\alpha=1,2,3$, and with the nearest-neighbour bonds $\langle x,x+\eps_kf_\alpha\rangle$. A field configuration on a set $X$ of sites consists of $\phi(x)=r_xe^{\mathrm i\vartheta_x}\in\C$ and $A_\alpha(x)\in\R$, $x\in X$; the bond $\langle x,x+\eps f_\alpha\rangle$ carries $A_{\langle x,x+\eps f_\alpha\rangle}:=A_\alpha(x)$ and the reversed bond carries $-A_\alpha(x)$ (Feynman form of the vector field,~\cite[(2.5)]{K2}). A plaquette is $p=(x;\alpha,\beta)$ with $\alpha<\beta$; its centre is $c(p):=x+\frac\eps2(f_\alpha+f_\beta)$ and its field strength is \[ F(p):=\eps^{-1}\big(A_\alpha(x)+A_\beta(x+\eps f_\alpha)-A_\alpha(x+\eps f_\beta)-A_\beta(x)\big) \qquad\text{\cite[(2.11)]{K2}}. \] The action on $\Lambda$ is the action of the setting (Section~\ref{sec:s02}) written in the frame $(f_\alpha)$, the gradient and hopping terms running along the bonds $\langle x,x+\eps f_\beta\rangle$, with the free-boundary mass counterterm $\delta m^2_\eps$ (King's~\cite[(3.32)--(3.34)]{K1}, a polynomial in $e^2$ and $\lambda$ whose coefficients depend only on $\eps$, $m$ and $\mu_0$) and the free Wick constant $\sW_\eps$ of the setting (\cref{sec:s02}); King's field-independent constants $E_0$, $E_1$~\cite[(2.4), (3.35)]{K1}, \cite[p.~325]{K2} cancel in every normalised expectation and are omitted. The sources $F_\Lambda(f)$, $\sS_\Lambda(h)$, $\Wick{h}$ and the observables $\Theta$ of \cref{def:s02:sources} are formed on $\Lambda$ with the plaquettes and the frame of $\Lambda$: $F_\Lambda(f)=\sum_p\eps^3f_{c(p)}(f_\alpha,f_\beta)F(p)$ (Remark~\ref{rot:rem:frame}); on a torus smooth periodic sources are used and the sums run over one fundamental cell. The cone $\cK$ of \cref{def:s02:sources} is invariant under $O(3)$, under translations, under $(f,g)\mapsto(tf,tg)$, $t\ge0$, and under sums. \end{definition} \begin{definition}[boxes, tori, walled tori and killed sites]\label{rot:def:states} Let $\Lambda$ be a lattice of spacing $\eps$ and $d\nu:=\prod_x\prod_\alpha dA_\alpha(x)\,d^2\phi(x)$ over the sites in question. \begin{enumerate} \item[(a)] \emph{Boxes and King's cubes.} For a finite set $X$ of sites of $\Lambda$ the Neumann--Dirichlet box $\langle\cdot\rangle^\ND_{\Lambda,X}$ is the state of \cref{def:s02:acube} on $\Lambda$, with the frame $(f_\beta)$ of $\Lambda$ in place of $(e_\beta)$ (for $\Lambda=\Lambda_k$ and $X=C_J\cap\Lambda_k$ written $\langle\cdot\rangle^\ND_{k,J}$), and King's cube $C^K_J$ with its state $\langle\cdot\rangle^K_{k,J}$ is \cref{def:s02:Kcube}. \item[(b)] \emph{Tori.} If $\Lambda$ descends to $D_J$, $\langle\cdot\rangle_{\Lambda,D_J}$ is the normalised measure with the action $S$ summed over the sites and bonds of the finite graph $\Lambda/(50MJ\,\Z)^3$ (periodic conditions for both fields). \item[(c)] \emph{Walled tori.} A \emph{walled configuration} is $\fc=(J',w)$ with $J'\in\N$ and $w\in C^\infty(D_{J'};[0,\infty))$, the \emph{wall}. If $\Lambda$ descends to $D_{J'}$, $\langle\cdot\rangle_{\fc,\Lambda}$ is the torus state of (b) with the density multiplied by $\prod_x\exp(-\eps^3w(x)|\phi(x)|^2)$. The wall raises the scalar mass squared by $2w$ and acts in the single-site measures. We write $(J',\psi,n):=(J',n\psi)$ for $n\ge0$. \item[(d)] \emph{Killed sites.} For any of the states above and a set $U$ of sites, the state \emph{with the matter killed on $U$} is obtained by replacing the single-site measure $d^2\phi(x)$ times its weight, for $x\in U$, by the point mass at $\phi(x)=0$. All hopping terms with an endpoint in $U$ then reduce to their diagonal parts at the other endpoint; the photon variables are unchanged. \end{enumerate} \end{definition} \begin{definition}[the unnormalised functional and the walled expectation]\label{r13:def:unnormZ} For a lattice $\Lambda$ descending to a torus $T$ we also use the \emph{unnormalised} generating functional $\cZ_{\Lambda,T}(G,h):=\int e^{-S(A,\phi)+F_\Lambda(G)+{:}\phi^2{:}_\Lambda(h)}\,d\nu$ (${:}\phi^2{:}_\Lambda(h)=\Wick{h}$ formed on $\Lambda$, \cref{def:s02:sources}), with King's constants $E_0,E_1$ omitted. Two lattices of the same spacing descending to $T$ have the same number of sites in a fundamental cell, so any field-independent constant per site multiplies both by the same factor. All the states of Definition~\ref{rot:def:states} are BFS measures of King type in the sense of the setting (\cref{conv:v065d_s02_setting:2}; the wall multiplies the single-site measure at $x$ by $e^{-\eps^3w(x)r^2}$ with $w\ge0$), and the charge conjugation $(A,\vartheta)\mapsto(-A,-\vartheta)$ leaves each of them invariant; for a walled configuration $\fc=(J',w)$, a lattice $\Lambda$ descending to $D_{J'}$ and a real $2$-form $f$ supported in one fundamental cell we write $T_{\fc,\Lambda}(f):=\langle\Theta(\mathrm if,0)\rangle_{\fc,\Lambda}=\langle e^{\mathrm iF_\Lambda(f)}\rangle_{\fc,\Lambda} =\cZ_{\Lambda,D_{J'}}(\mathrm if,-w)/\cZ_{\Lambda,D_{J'}}(0,-w)$ for the \emph{walled expectation} of $f$ (the Wick constants cancel in the ratio); it is real. \end{definition} \begin{definition}[the configuration class]\label{rot:def:class} Let $R\in\Rgen$, let $C_L$, $L\in\N$, be a King cube, and let $X_K:=\Lambda_K$ be the axis lattice of spacing $\eps_K$. The \emph{configuration class} $\mathcal C$, on which the torus comparison (Theorem~\ref{rot:thm:CT}) is proved and over which the infimum of the closing statement \textup{(TIGHT$_0$)} (Theorem~\ref{rot:hyp:TIGHT}) is taken, consists of the walled configurations $\fc=(J,\psi,n)$ on the King tori $T:=D_J$ with the following properties: $J\in5^{k_*}\N$ with $J\ge\max(J_0(\eps_0),J_{\rm pos})$ ($k_*$ the number of large-field scales of the rotated flow, $J_0(\eps_0)$ the non-wrapping threshold of the skew torus at the coarsest scale $\eps_0$, $J_{\rm pos}$ the threshold of the positivity condition of Proposition~\ref{rot:thm:POS}, above which its margin holds); $\psi\ge0$ smooth and $D_J$-periodic, $\psi=0$ on the $2$-neighbourhood of $R^{-1}C_L$, and $n>0$ (a mass-increasing wall); the wall condition $n\langle\psi\rangle_{X_K}>\Delta_*|T|/2$ for every $K$, with $\Delta_*$ the bound of the convention bridge (CT)(b) (Remark~\ref{rot:rem:CT}, Theorem~\ref{cpi:ct:CTB}); block alignment: $T$ and the box $H'$ (Definition~\ref{rot:def:shell}) are unions of blocks of the final scale, whose side we denote $\ell_f$; and the grid offset: with $M=5M'$ and the prime $\mathfrak p=2$ dividing $M'$ of Definition~\ref{rot:def:geom}, every corner coordinate of $H'$ is $c\,\ell_f$ with $c\equiv M'/\mathfrak p\pmod{M'}$ (the sides $50MJ$ of $D_J$ are multiples of $5M\ell_f$). The class carries the cap of Section~\ref{rot:ssec:openin} on the coarsest scale, $\eps_0\le\eps_{\rm sys}(M,R,\rho_{\rm B},\delta_0)$ ($\rho_{\rm B}$ Ba{\l}aban's radius and $\delta_0$ the finest block size, as there), which does not depend on $J$ and is closed downward, and under it the non-wrapping threshold at that spacing equals $1$ for every $J$ of the residue class, so that no further restriction of $J$ is needed. \end{definition} \begin{remark}[the thresholds of the class]\label{r11:rem:classthr} The threshold $J_{\rm pos}$ of Proposition~\ref{rot:thm:POS} belongs to the class because the margin of the positivity condition is proved for $J\ge J_{\rm pos}$ only. The large-field clause of Theorem~\ref{rot:thm:BAL5} at the wall on the skew torus takes one step at the final spacing of Ba{\l}aban's fixed-mass bounds; the cap on the coarsest scale that the class carries is a proviso of the torus comparison (Theorem~\ref{rot:thm:CT}) on its parameters (Section~\ref{rot:ssec:openin}). Theorem~\ref{rot:thm:E} produces configurations in this class (Remark~\ref{rot:rem:class}). \end{remark} \begin{remark}[the counterterm]\label{rot:rem:CT} King's $\delta m^2_\eps$ is the free-boundary counterterm of the setting (\cref{sec:s02}, \eqref{eq:s14:CTdef}), one number per spacing; it is the convention under which King's cut of a cube out of a larger cube is ``exactly'' the smaller Neumann--Dirichlet state~\cite[p.~334]{K2}. The theorems of~\cite{K1} are proved for the torus conventions of~\cite[(3.33), (2.8), (3.21), (3.35)]{K1}; the bridge between the two conventions is the statement (CT) (the torus counterterms of~\cite{K1} and the free-boundary ones of~\cite{K2} differ by $\Delta_T=\delta m^2-\delta m^2_T$, $\Delta E_T$ and $\Delta c_T$, so that the torus model is the free-boundary model with the mass term shifted by $\frac12\Delta_T\sum s^3|\phi|^2$ ($s$ the spacing) and multiplied by $e^{-\Delta E_T|T|-N\Delta c_T\langle h\rangle}$ ($N$ the number of sites of $T$), exactly; the bound $\Delta_*$ on $|\Delta_T(s)|$, $s\le s_0$, and the decay of the difference terms are Theorem~\ref{cpi:ct:CTB}) on the King side and (CT)$_{\rm sh}$ (Definition~\ref{r13:def:Moffset} and Lemma~\ref{cpi:swab:M123}: the offset $\Delta_T$ is a constant added to the counterterm-mismatch vertex; no cross-convention equality of states is used) on the shell side (Lemma~\ref{cpi:swab:M123} and Theorem~\ref{cpi:swc:main} of Sections~\ref{rot:ssec:shellest} and~\ref{rot:sec:shellestB}, respectively). \end{remark} \begin{definition}[the mismatch vertex]\label{cpi:swab:M} The shell model keeps the bulk mass counterterm $\delta m^2_{\mathrm{bulk}}$, King's counterterm \cite[(3.32)--(3.33)]{K1} at zero shell mass and zero wall, which is $\kappa$-free. Relative to the $\kappa$-adapted local subtraction $\delta m^2_\kappa(x)$, the counterterm number solving $-\delta m^2(x)+\sum_{x'}\eta^d\Gamma(x,x')=0$ order by order with the two-leg graphs $\Gamma$ of order at most four built from the full fine-lattice lines $C^{\eta,\kappa}=(-\Delta_A+V)^{-1}$ at zero background, the action carries the \emph{mismatch vertex} \[ \mathcal M=-\tfrac12\ell_k^2\sum_x\eta^d\,\Delta\delta(x)\,|\phi'(x)+\phi^{(k)}_\kappa(x)|^2,\qquad \Delta\delta(x)=\delta m^2_{\mathrm{bulk}}-\delta m^2_\kappa(x), \] where $\phi'$ is the fluctuation field and $\phi^{(k)}_\kappa$ the background of the block data. Only the matter lines differ between the two counterterm numbers ($\Delta\delta$ is a finite sum, over King's two-point graphs of orders at most four in $(e,\lambda)$ \cite[(3.32)]{K1}, of differences of bulk and shell-model graphs, in which every member without a matter line, the $e^2$ tadpole, the tadpole part of the order-two seagull and the $e^4$ double tadpole, cancels exactly), so $\Delta\delta=[\delta m^2_{\mathrm{bulk}}-\delta m^2_0(x)]+[\delta m^2_0(x)-\delta m^2_\kappa(x)]$, the first bracket the wall term at zero shell mass and the second minus the zeroth moment of $\Gamma^\kappa-\Gamma^0$ summed over the counterterm graphs; at one loop, with $c_\lambda=4\lambda(N+2)$, $\Delta\delta_\lambda=-c_\lambda[C^{\eta,\mathrm{bulk}}-C^{\eta,\kappa}](x,x)\le0$, but no step uses a sign of $\Delta\delta$, only $|\Delta\delta|$. \end{definition} \begin{definition}[the size convention and the bounds on the mismatch]\label{r13:def:Msize} Let $\Delta\delta$ be as in Definition~\ref{cpi:swab:M}. The size convention is, with $t(x)$ the distance of $x$ to $\partial U$ and the wall floor $C_n\le C\sqrt n$ coming from the $\kappa$-independent part $n\psi$ of the profile, \[ W(x):=\min\{\eps^{-1},\sqrt\kappa,t(x)^{-1}\}+C_n\quad(x\in H'),\qquad W(x):=\min\{\eps^{-1},\sqrt\kappa\}+C_n\quad(x\in U) \] (no decay deep in $U$, and none is needed), the one-loop part obeying $|\Delta\delta_\lambda(x)|\le C\lambda\,W(x)$ and the full counterterm difference \begin{gather*} |\Delta\delta(x)|\le C(\lambda+e^2)\,W(x)+C(\lambda+e^2)^2\,\mathcal L(x)^{q_F},\\ \mathcal L=1+\log_+(1/t)\ \text{on }H',\quad \mathcal L=1+\log\big(1+\min(\kappa,\eps^{-2})/m^2\big)\ \text{on }U, \end{gather*} with $q_F\in\{1,2\}$ the logarithm exponent of the two-loop family ($q_F=1$ for the $\lambda^2$, $e^2\lambda$ and nested $e^4$ families, $q_F=2$ for the crossed $e^4$ family). \end{definition} \begin{definition}[the counterterm offset and the bridge (CT)$_{\mathrm{sh}}$]\label{r13:def:Moffset} The \emph{counterterm offset}: if the counterterm of Definition~\ref{cpi:swab:M} is $\delta m^2+c(\eps)$ with $c=\sum_{1\le\alpha+\beta\le4}c(\alpha,\beta)$ graded in $(e,\lambda)$ as $\delta m^2$ is, with $\Delta_c<\infty$ as in Theorem~\ref{cpi:thm:main}, then the $\kappa$-adapted subtraction is unchanged ($c$ is bounded uniformly in $\eps$, so a $c$-vertex lowers the superficial degree by two and occurs in no divergent two-leg graph), $\Delta\delta\to\Delta\delta+c$ exactly, every statement about the shell model holds with constants depending also on $\Delta_c$, and $\partial_\kappa c=0$. This is the bridge \emph{(CT)$_{\mathrm{sh}}$} of Remark~\ref{rot:rem:CT} between the torus convention of the counterterm and the shell model's: the offset used is the free-boundary offset $c=\Delta_T(\eps)$ of that remark, bounded uniformly in $\eps$ by $\Delta_*$; a constant added to the counterterm-mismatch vertex, with no sign of $\Delta_T$ and no equality of states across the two conventions used. \end{definition} \begin{remark}[the frame convention]\label{rot:rem:frame} In~\cite{K1,K2} the curvature source is written \[ F(g)=\sum_p\eps^3g(p)F(p) \] with one function $g$ on all plaquettes \cite[(2.12)]{K2}. On one lattice a $2$-form sampled as above is a function on plaquettes, and King's proofs treat the source terms componentwise (``the sources $g_{\mu\nu}(x'),h(x')$ \dots can be replaced by $g_{\mu\nu}(x),h(x)$'', \cite[p.~665]{K1}; the sentence is claimed there without an argument). We use King's single-lattice statements at $h=0$ for $2$-form sources in this sense: Theorem~2.1 of~\cite{K2} holds, on King's cubes $C^K_J$ (Definition~\ref{def:s02:Kcube}), for real smooth $2$-form sources sampled at plaquette centres in the lattice's own frame, supported at distance $>1$ from $\R^3\setminus C_J$ (\cite[Thm.~2.1]{K2}, stated there without a complete argument; proved as \cref{thm:k01:EC}); the statements of~\cite[Thms.~2.2, 2.3, (3.40)--(3.44)]{K2} that we use, all at $h=0$, are proved here from it and from Lemmas~\ref{rot:lem:cut} and \ref{lem:s04:G} (Section~\ref{rot:sec:wick}; clauses (i)--(iii) of Theorem~\ref{rot:thm:W} are \cref{thm:s05:CK}(b) with \cref{cor:s04:G1}), and~\cite[(3.42)--(3.43)]{K2} are not used, and the torus statement~\cite[Thm.~2.1]{K1} at a wall is Theorem~\ref{cpi:thm:hrs:H6W}. The two-lattice statement is different: Proposition~\ref{cpi:rot:scalar} shows that it fails for a single scalar function on all plaquettes, and Theorem~\ref{rot:hyp:K} is formulated for $2$-forms sampled in each lattice's own frame. \end{remark} \begin{proposition}[the logarithm]\label{cpi:rot:log} In formal perturbation theory, King's model on $\eps\Z^3$ with the free Wick constant $\sW_\eps$ alone satisfies, at order $e^2\lambda^0$, \[ \big\langle{:}\phi^2{:}_{\eps\Z^3}(h)\big\rangle=\frac{e^2}{4\pi^2}\,\ln(1/\eps)\int h+O(1). \] The coefficient does not depend on $m$ or $\lmu$. \end{proposition} \begin{proof} In the normalisation of the action used throughout, each real component of $\phi$ has the propagator $(p^2+m^2)^{-1}$ at $e=0$. Write the one-loop correction to the inverse propagator as $\Sigma(p)$; the mass counterterm removes $\Sigma(0)$. The photon-exchange part is $-e^2\int\frac{d^3k}{(2\pi)^3}\frac{(2p-k)^2}{((p-k)^2+m^2)(k^2+\lmu^2)}$; writing $(2p-k)^2=2p^2+2(p-k)^2-k^2$ and using $\int\frac{d^3k}{(2\pi)^3}\frac1{k^2(p-k)^2}=\frac1{8\abs p}$, one finds $\Sigma(p)-\Sigma(0)=-\frac{e^2}4\abs p+O(m+\lmu)$ for $1\ll\abs p\ll\eps^{-1}$. Inserting, \[ \langle\abs{\phi(0)}^2\rangle-2\sW_\eps=-2\int\frac{d^3p}{(2\pi)^3}\frac{\Sigma(p)-\Sigma(0)}{(p^2+m^2)^2}+O(e^4)=\frac{e^2}{2}\cdot\frac1{2\pi^2}\ln(1/\eps)+O(1) \] at order $e^2$. The lattice regularisation changes only momentum-independent terms and terms of order $\eps p^2$; carrying it through the two integrals is routine. \end{proof} Two consequences. (a) For $h$ with $\int h\ne0$ King's generating functionals with scalar sources diverge as $\eps\to0$ in perturbation theory, beyond the curvature sector: for $h\ge0$, $h\not\equiv0$, Jensen's inequality gives $\langle e^{\Wick{h}}\rangle\ge e^{\langle\Wick{h}\rangle}\to\infty$. This affects \cite[Thm.~2.1]{K1} and \cite[Thms.~2.1--2.3, (3.43), (4.2)]{K2} for scalar sources with $\int h\ne0$, and the statement of \cite[p.~677]{K1} that the sources are proportional to $(5^k\eps)^\alpha$, $\alpha>0$, in the small-field region does not hold for the marginal graph that produces the logarithm. Walled ratios and the curvature sector are unaffected, since the $h$-linear $c$-numbers cancel in them; for scalar sources a further vacuum counterterm, linear in $h$, is needed (the scheme $\Theta^\sharp$ of Section~\ref{rot:sec:gensrc}). (b) King's proof of \cite[Thm.~2.4]{K2} compares the effective action generated from spacing $\eps$ by a first block-spin step with rotated blocks with the one generated from the rotated lattice of spacing $5\eps$ \cite[(4.3)--(4.9)]{K2}. If the logarithm is removed by a vacuum counterterm $a_\eps\sS(h)$, linear in $h$, the two generating functionals carry the multipliers $e^{-a_\eps\sS(h)}$ and $e^{-a_{5\eps}\sS(h)}$ with $a_\eps-a_{5\eps}\to\frac{e^2}{4\pi^2}\ln5$, so the comparison \cite[(4.6)]{K2} must carry the $c$-number $\frac{e^2}{4\pi^2}\ln5\int h$, and as stated in \cite{K2} it cannot hold for $\int h\ne0$; since a walled ratio contains the wall source $-w$ with $\int w>0$, this enters the proof of walled comparisons even though their statement is free of the counterterm. In the equal-spacing formulation of Section~\ref{rot:sec:K} the two lattices carry the same counterterm, and the issue does not arise. Rotating a source given by one function on all plaquettes, as in Theorem~2.4 of \cite{K2}, fails already for the free field. \begin{proposition}[the source rotated as a scalar]\label{cpi:rot:scalar} Let $e=0$ and $F^{sc}_\Lambda(g)\defeq\sum_p\eps^3g(c(p))F(p)$, with $c(p)$ the centre of the plaquette $p$ and a single real function $g$ on all plaquettes. Let $R=R_{12}(\thz)$ be the rotation about the $3$-axis by the angle $\thz=\arctan(3/4)$, let $\ell=50MJ$ be the side of the torus $\DJ$, and let $g(x)=\cos(2\pi x_1/\ell)$. Then \[ \lim_k\av{e^{F^{sc}(g)}}_{\Lk,\DJ}\ne\lim_k\av{e^{F^{sc}(g)}}_{R\Lk,\DJ}. \] \end{proposition} \begin{proof} At $e=0$ the photon is the centred Gaussian field with covariance $\delta_{\alpha\beta}(-\Delta_\eps+\lmu^2)^{-1}$ on the torus, so $\av{e^{F^{sc}(g)}}=e^{\frac12\Var F^{sc}(g)}$. On a lattice with orthonormal frame $(f_\alpha)$ the sum of the three plaquette components is $\sum_{\alpha<\beta}F(f_\alpha,f_\beta)=\ip uB$ for a continuum field strength $F=dA$ with $B=\star F$, where $u\defeq\sum_{\alpha<\beta}f_\alpha\times f_\beta$. For the plane wave $g$ only the momenta $\pm k_0$, $k_0=(2\pi/\ell)e_1$, contribute, and the lattice symbols converge, so \[ \lim_k\Var F^{sc}(g)=\frac{\ell^3}{2}\;\frac{\abs{k_0}^2\big(\abs u^2-u_1^2\big)}{\abs{k_0}^2+\lmu^2}. \] For $\Lk$ one has $u=(1,-1,1)$ and $\abs u^2-u_1^2=2$; for $R\Lk$, $u=R(1,-1,1)=(\tfrac75,-\tfrac15,1)$ and $\abs u^2-u_1^2=\tfrac{26}{25}$. The limiting variances differ by the factor $\tfrac{25}{13}$. \end{proof} With $2$-form sources sampled in each lattice's own frame (Definition~\ref{def:s02:sources}) the limit on either lattice is the variance of $\int\ip GF$, which does not depend on the frame, and the obstruction disappears; this form is consistent with King's proof, which treats the source components separately, and it is the form of Theorem~\ref{rot:hyp:K} and of Theorem~\ref{rot:thm:CT}. The second error in King's statement concerns the Wick constant (Proposition~\ref{cpi:rot:log}). Since the statement is false on both counts, we refer to it in that form as \emph{King's original torus comparison} (Remark~\ref{cpi:rot:errata}(ii)). Theorem~2.4 of \cite{K2} is not, by itself, a statement about King's infinite-volume state: the passage from tori to the state \cite[(2.24)--(2.25)]{K2} is conjectural there, and it is not used here. \begin{lemma}[passive covariance]\label{rot:lem:pass} Let $O\in O(3)$, let $\Lambda$ be a lattice of spacing $\eps$ with frame $(f_\alpha)$, and let $X$ be a box, a torus or a walled torus for $\Lambda$. Equip $O\Lambda$ with the frame $(Of_\alpha)$ and $OX$ with the transported boundary conditions and wall $O\cdot w$. Then, for all coefficients $z$, $u$, \[ \big\langle\Theta(z\,O\cdot f,u\,O\cdot h)\big\rangle_{O\Lambda,OX}=\big\langle\Theta(zf,uh)\big\rangle_{\Lambda,X},\qquad \sS_{O\Lambda}(O\cdot h)=\sS_\Lambda(h), \] and likewise for finite families. Moreover, on every torus the periodic model on $\Lambda$ does not change if the frame $(f_\alpha)$ is replaced by $(\pm f_{\pi(\alpha)})$, for a permutation $\pi$ and any choice of signs. \end{lemma} \begin{proof} The map $\phi'(Ox)=\phi(x)$, $A'_\alpha(Ox)=A_\alpha(x)$, with components taken in the frames $(f_\alpha)$ and $(Of_\alpha)$, is a bijection of the configuration spaces preserving $d\nu$. The action is written in the frame, and $\delta m^2_\eps$ and $\sW_\eps$ depend on $\eps$ only. Plaquettes, their centres and field strengths are transported, and $(O\cdot f)_{Oc}(Of_\alpha,Of_\beta)=f_c(f_\alpha,f_\beta)$. For the second statement, a permutation of the frame relabels the components; reversing $f_\alpha$ replaces $A_\alpha(x)$ by $-A_\alpha(x-\eps f_\alpha)$, a bijection of the torus variables that preserves the action, and changes the sign of both $F(p)$ and $f_{c(p)}(f_\alpha,f_\beta)$ while preserving the set of plaquette centres. For a box or a torus without wall this is \cref{lem:s02:pass}. \end{proof} \begin{lemma}[current representation and Gaussian domination]\label{rot:lem:GD} Let $\langle\cdot\rangle$ be a BFS measure whose Gaussian has covariance $C$, in the notation of \cref{lem:s02:GD} ($B$ its set of bonds, $\ell_{xy}(A)$ its bond functionals of the vector field and $\R^m$ the space of the vector-field configurations). Integrating out the matter at fixed $A$, the matter integral is $\Psi(A)=\sum_qw(q)\cos\ip{\kappa(q)}{A}$, where $q$ runs over integer-valued divergence-free currents on $B$, $w(q)\ge0$, $\sum_qw(q)<\infty$, and $\kappa(q)\in\R^m$ is defined by $\ip{\kappa(q)}A:=\sum_{xy}q_{xy}\ell_{xy}(A)$. Let $\E^\sharp$ be the probability measure on currents with weights proportional to $w(q)e^{-\ip{\kappa(q)}{C\kappa(q)}/2}$. Then for $v\in\R^m$: \begin{enumerate} \item[(a)] $\langle e^{\mathrm i\ip vA}\rangle=e^{-\ip v{Cv}/2}\,\E^\sharp\cosh\ip{Cv}{\kappa}$ and \[ \langle e^{s\ip vA}\rangle=e^{s^2\ip v{Cv}/2}\,\E^\sharp\cos(s\ip{Cv}{\kappa})\le e^{s^2\ip v{Cv}/2} \] for $s\in\R$; \item[(b)] $\langle\ip vA^2\rangle=\ip v{Cv}-\E^\sharp\ip{Cv}{\kappa}^2\in[0,\ip v{Cv}]$. \end{enumerate} \end{lemma} \begin{proof} The representation and the two formulas of (a) are \cref{lem:s02:GD}(b) and (a). Replacing $v$ by $tv$ in the first formula of (a) and comparing the terms of order $t^2$ (all sums converge absolutely at fixed lattice spacing) gives (b). \end{proof} \begin{lemma}[cutting]\label{rot:lem:cut} Let $\langle\cdot\rangle_Y$ be a box, torus or walled-torus state on a lattice $\Lambda$, possibly with killed sites, and let $X$ be a set of sites of $\Lambda$ such that $X$ and its nearest neighbours lie in one fundamental cell. Let $\langle\cdot\rangle^\ND_{X}$ be the Neumann--Dirichlet state on $X$ whose single-site measures (wall, killed sites and diagonal parts of the bonds leaving $X$ included) are those of $\langle\cdot\rangle_Y$. Let $f,g$ be sources with $g\ge0$ such that every plaquette and site at which they do not vanish lies in $X$. Then, at every lattice spacing, \[ \langle\Theta(\mathrm if,g)\rangle^\ND_{X}\le\langle\Theta(\mathrm if,g)\rangle_Y,\qquad \langle\Theta(f,-g)\rangle^\ND_{X}\ge\langle\Theta(f,-g)\rangle_Y . \] If the wall of $Y$ vanishes on $X$ and no site of $X$ is killed, $\langle\cdot\rangle^\ND_X=\langle\cdot\rangle^\ND_{\Lambda,X}$ is the state of Definition~\ref{def:s02:acube}. For a box of $\Lambda_k$ or a King torus without wall this is \cref{lem:s02:cut}. \end{lemma} \begin{proof} Delete from $Y$ every photon gradient term coupling a site of $X$ to a site outside $X$, and the cosine parts of all hopping terms between $X$ and its complement, keeping their diagonal parts $\frac\eps2(r_x^2+r_y^2)$. By Lemma~\ref{lem:s02:BFS}(a),(b),(d) this does not increase the first expectation and does not decrease the second. The resulting measure is a product (Lemma~\ref{lem:s02:BFS}(e)); its $X$-factor is $\langle\cdot\rangle^\ND_{X}$, since the diagonal parts of the cut bonds are exactly the Dirichlet terms and the remaining photon terms are the Neumann ones. The free Wick constants and $\delta m^2_\eps$ are the same in both states. \end{proof} \begin{remark}[King's cube in King's coordinates]\label{rot:rem:Kcoord} Let $X_J:=C_J\cap\Lambda_k$ and let $T_J$ be its three top face layers (\cref{def:s02:Kcube}). The translation by $(MJ+\eps_k)(1,1,1)$ carries $X_J$ onto the box $\{01$ from $\R^3\setminus C_{J_0}$. \begin{enumerate} \item[(a)] For all $k\ge0$ and $J\ge J_0$, \[ \langle\Theta(\mathrm if,g)\rangle^K_{k,J}\le\langle\Theta(\mathrm if,g)\rangle^K_{k,J+1}\quad\text{and}\quad \langle\Theta(f,-g)\rangle^K_{k,J}\ge\langle\Theta(f,-g)\rangle^K_{k,J+1}: \] King's monotonicity in the volume~\cite[Thm.~3.2]{K2}, in the form his proof proves, with one $J$-independent Wick constant, holds on King's cubes. \item[(b)] At $g=0$ and every fixed $k$, the families $J\mapsto\langle e^{\mathrm iF(f)}\rangle^K_{k,J}$ and $J\mapsto\langle e^{\mathrm iF(f)}\rangle^\ND_{k,J}$ are nondecreasing on $J\ge J_0$, bounded by $1$, and have the same limit as $J\to\infty$. \item[(c)] If $Z^K_J(\mathrm if,0):=\lim_k\langle e^{\mathrm iF(f)}\rangle^K_{k,J}$ exists for every $J\ge J_0$, then $\lim_J\liminf_k\langle e^{\mathrm iF(f)}\rangle^\ND_{k,J}=\lim_J\limsup_k\langle e^{\mathrm iF(f)}\rangle^\ND_{k,J}=\lim_JZ^K_J(\mathrm if,0)$; if instead $\lim_k\langle e^{\mathrm iF(f)}\rangle^\ND_{k,J}$ exists for every $J\ge J_0$, the same holds with the two families exchanged. \end{enumerate} \end{corollary} \begin{proof} (a) The hypothesis of Lemma~\ref{lem:s04:G} holds for every $J\ge J_0$; combine its two inequalities (this is \cref{cor:s04:G1}). (b) Both families are nondecreasing, by (a) and by Lemma~\ref{rot:lem:cut} with $Y=C_{J+1}\cap\Lambda_k$, $X=C_J\cap\Lambda_k$; they are real and bounded by $1$; and two nondecreasing sequences with $a_J\le b_J\le a_{J+1}$ (Lemma~\ref{lem:s04:G}) have the same limit. (c) Lemma~\ref{lem:s04:G} at every $k$ gives $Z^K_J\le\liminf_k\langle e^{\mathrm iF(f)}\rangle^\ND_{k,J}\le\limsup_k\langle e^{\mathrm iF(f)}\rangle^\ND_{k,J}\le Z^K_{J+1}$, and $J\mapsto Z^K_J(\mathrm if,0)$ is nondecreasing and bounded by (a); for the exchanged statement use $\langle\cdot\rangle^\ND_{k,J-1}\le\langle\cdot\rangle^K_{k,J}\le\langle\cdot\rangle^\ND_{k,J}$, $J\ge J_0+1$ ((b) is \cref{cor:s04:G2}(i); under \cref{hyp:s03:EC} the first statement of (c) is \cref{cor:s04:G2}(ii)($\gamma$)). \end{proof} \begin{remark}[what the interlacing gives]\label{rot:rem:Kcube} Lemma~\ref{lem:s04:G} relates the two families at every spacing and every $J$, so that they have the same infinite-volume state (Corollary~\ref{rot:cor:Kmono}), but it gives no information on the fixed-$J$ continuum limit of one family from that of the other: the two bounds $\langle\cdot\rangle^K_{k,J}$ and $\langle\cdot\rangle^K_{k,J+1}$ differ macroscopically. King's cube limit (Theorem~\ref{thm:k01:EC}) is therefore posed on the cubes on which King's renormalisation group is run, and every other comparison of cubes in this part is transferred to them through Lemma~\ref{lem:s04:G}, at the cost of one index at each end: the comparisons of a cube with walled tori (Theorems~\ref{rot:thm:curv} and~\ref{rot:thm:SHT}), the monotonicity and the translations (Theorem~\ref{rot:thm:W}), and the reflection (Lemma~\ref{rot:lem:refl}). \end{remark} \space% % % \section{King's state on curvature sources}\label{rot:sec:wick} This section defines the object of Theorem~\ref{rot:thm:main}. From King's cube limit \cite[Thm.~2.1]{K2} (Theorem~\ref{thm:k01:EC}), Theorem~\ref{rot:thm:W} gives the infinite-volume limit of the cube functionals along King's cubes as an entire function of the coefficients, with its monotonicity in $J$, its positivity at real sources, the convexity and continuity of its logarithm and its translation invariance; Definition~\ref{rot:def:object} fixes King's state on curvature sources. The torus comparison of Section~\ref{rot:sec:K} and the closing statement of Section~\ref{rot:sec:env} (the inequality (TIGHT$_0$), which bounds this state from below by the lower envelope of the walled tori) are statements about this object. For $\rho>0$ let $E_\rho:=C^\infty_c(\bar B_\rho;\Lambda^2)$ be the space of smooth real $2$-forms supported in the closed ball $\bar B_\rho$, with the $C^\infty$ topology; it is a Fr\'echet space and it is invariant under $O(3)$. For a finite family of real $2$-forms $f_1,\dots,f_N\in C^\infty_c$ and $z\in\C^N$ write \[ Z^K_{k,J}\Big(\sum_jz_jf_j,0\Big):=\Big\langle\exp\Big(\sum_jz_jF_{\Lambda_k}(f_j)\Big)\Big\rangle^K_{k,J} \] for the functional of King's cube $C^K_J$ (Definition~\ref{def:s02:Kcube}) at spacing $\eps_k$, and $Z^\ND_{k,J}(\cdot,0)$ for that of the box $C_J\cap\Lambda_k$ of Definition~\ref{def:s02:acube}. If the sources are supported at $\ell^\infty$-distance $>1$ from $\R^3\setminus C_J$, King's cube limit (Theorem~\ref{thm:k01:EC}) gives $Z^K_J(\sum_jz_jf_j,0):=\lim_kZ^K_{k,J}(\sum_jz_jf_j,0)$ in $(0,\infty)$ for real $z$. \begin{theorem}[King's state on curvature sources]\label{rot:thm:W} Let $f_1,\dots,f_N\in C^\infty_c$ be real $2$-forms supported at $\ell^\infty$-distance $>1$ from $\R^3\setminus C_{J_0}$. \begin{enumerate} \item[(i)] For $J\ge J_0$, $Z^K_{k,J}(\sum_jz_jf_j,0)\to Z^K_J(\sum_jz_jf_j,0)$ locally uniformly on $\C^N$ as $k\to\infty$, and $z\mapsto Z^K_J(\sum_jz_jf_j,0)$ is entire. \item[(ii)] For a real $2$-form $f$ supported at distance $>1$ from $\R^3\setminus C_{J_0}$, $J\mapsto Z^K_J(\mathrm if,0)$ is nondecreasing and $J\mapsto Z^K_J(f,0)$ is nonincreasing for $J\ge J_0$. \item[(iii)] The limit $Z_\infty(\sum_jz_jf_j,0):=\lim_{J\to\infty}Z^K_J(\sum_jz_jf_j,0)$ exists locally uniformly on $\C^N$ and is entire; $Z_\infty(f,0)>0$ for every real $2$-form $f$; and $Z^K_J(\mathrm if,0)\le Z_\infty(\mathrm if,0)$ for $J\ge J_0$. \item[(iv)] For every $\rho>0$, $\Phi(f):=\ln Z_\infty(f,0)$ is finite, convex and continuous on $E_\rho$. \item[(v)] $Z_\infty(\sum_jz_j\,\tau_bf_j,0)=Z_\infty(\sum_jz_jf_j,0)$ for all $b\in\R^3$ and $z\in\C^N$. \end{enumerate} \end{theorem} \begin{proof} (i)--(iii) are \cref{thm:s05:CK}(b): the two limits exist locally uniformly on $\C^N$ and are entire, $J\mapsto Z^K_J(\mathrm if,0)$ is nondecreasing, and $Z_\infty(f,0)\ge1>0$ at real $f$; the nonincreasing clause of (ii) is \cref{cor:s04:G1} at $g=0$ and every $k$, passed to the limit $k\to\infty$ by (i); and $Z^K_J(\mathrm if,0)\le Z_\infty(\mathrm if,0)$ follows from (ii). (iv) Finiteness and positivity by (iii); convexity as a pointwise limit of convex functions (Lemma~\ref{lem:s02:alg}(b)). For $J$ so large that $N_1(\bar B_\rho)\subset C_J$, and each $k$, the function $f\mapsto\ln Z^K_{k,J}(f,0)$ on $E_\rho$ depends continuously on finitely many sampled values of $f$, hence is continuous; so $\Phi=\lim_J\lim_k$ of continuous functions is Borel. Apply Lemma~\ref{lem:s02:baire}. (v) Let $b\in5^{-k_0}\Z^3$ with $|b|_\infty\le M$ and $k\ge k_0$, so that $\tau_b$ preserves $\Lambda_k$. Write $\langle\cdot\rangle^{\ND,b}_{k,J}$ and $\langle\cdot\rangle^{K,b}_{k,J}$ for the states of the box and of King's cube translated by $-b$, which are again states on $\Lambda_k$, so that $\langle\Theta(\tau_b\cdot s)\rangle^K_{k,J}=\langle\Theta(s)\rangle^{K,b}_{k,J}$, and let $s=\mathrm i\sum_jt_jf_j$ with $t\in\R^N$, the sources being supported at distance $>1$ from $\R^3\setminus C_{J_0}$. As sets of sites, $C_{J-1}\cap\Lambda_k\subset(C_J-b)\cap\Lambda_k\subset C_{J+1}\cap\Lambda_k$, so Lemma~\ref{rot:lem:cut} (with $X=C_{J-1}\cap\Lambda_k$ in the state on $(C_J-b)\cap\Lambda_k$, then $X=(C_J-b)\cap\Lambda_k$ in the state on $C_{J+1}\cap\Lambda_k$) gives, for $J\ge J_0+1$ (so that, the corners of a plaquette lying within $\ell^\infty$-distance $\eps/2\le\frac12$ of its centre, every plaquette at which $s$ does not vanish lies in $C_{J-1}$), $\langle\Theta(s)\rangle^\ND_{k,J-1}\le\langle\Theta(s)\rangle^{\ND,b}_{k,J}\le\langle\Theta(s)\rangle^\ND_{k,J+1}$. Lemma~\ref{lem:s04:G}, applied to the cubes and to their translates, and this sandwich at $J$ and at $J-1$ give for $J\ge J_0+3$ \begin{align*} \langle\Theta(s)\rangle^K_{k,J-2}\le\langle\Theta(s)\rangle^\ND_{k,J-2}\le\langle\Theta(s)\rangle^{\ND,b}_{k,J-1}&\le \langle\Theta(s)\rangle^{K,b}_{k,J}\\ &\le\langle\Theta(s)\rangle^{\ND,b}_{k,J}\le\langle\Theta(s)\rangle^\ND_{k,J+1}\le \langle\Theta(s)\rangle^K_{k,J+2}. \end{align*} Let $k\to\infty$ by Theorem~\ref{thm:k01:EC} and then $J\to\infty$ by (iii): $Z_\infty(\tau_b\cdot s,0)=Z_\infty(s,0)$ on $(\mathrm i\R)^N$. The identity extends to all $z\in\C^N$, since $(\mathrm i\R)^N$ is a set of uniqueness for entire functions (\cref{cor:s04:G2}(ii)($\beta$)), to all $b\in\bigcup_k5^{-k}\Z^3$ by iteration, and to all $b\in\R^3$ at real coefficients by (iv), since $b\mapsto\tau_bf\in E_{\rho'}$ is continuous; then to complex coefficients by the identity theorem. (King states translation invariance as~\cite[Thm.~2.3(i)]{K2}; his argument is not used.) \end{proof} \begin{definition}[the object of Theorem~\ref{rot:thm:main}: the infinite-volume state of King's construction~\cite{K1,K2}]\label{rot:def:object} Fix $e>0$, $\lambda,m,\mu_0>0$ in King's range, the block factor $5$ and the large-block integer $M$ of Definition~\ref{rot:def:geom} ($M\in50\N$, $M\ge M_{\min}$), of the form $M=5M'$ with $M'$ divisible by the prime $\mathfrak p=2$ and $M'\ge2\mathfrak p$, and let $\Lambda_k=\eps_k\Z^3$, $\eps_k=5^{-k}$. For $J\in\N$ let $\langle\cdot\rangle^K_{k,J}$ be the state of King's cube $C^K_J$ at spacing $\eps_k$ (Definition~\ref{def:s02:Kcube}): the Neumann photon on $C_J\cap\Lambda_k$, and the scalar field with Dirichlet conditions on the planes $x_\mu=-MJ-\eps_k$ and $x_\mu=MJ-\eps_k$, that is, the cube of~\cite[(3.3), (3.7), (3.9), p.~329]{K2} with $L_\mu=2MJ$, where (3.9) is interpreted, after the translation of Remark~\ref{rot:rem:Kcoord}, as setting the scalar field to zero on the planes $x_\mu=0$ and $x_\mu=L_\mu$, for the action in the free convention (Section~\ref{sec:s02}), whose mass counterterm $\delta m^2_{\eps_k}$ is the free-boundary one, computed on the infinite lattice $\eps_k\Z^3$ and the same at every site (Remark~\ref{rot:rem:CT}). For finite families of real $2$-forms $f_j\in C^\infty_c$, sampled at plaquette centres in the lattice's own frame (Definition~\ref{def:s02:sources}), and $z\in\C^N$, \emph{King's state on curvature sources} is \[ Z_\infty\Big(\sum_jz_jf_j,0\Big):=\lim_{J\to\infty}\lim_{k\to\infty}\Big\langle\exp\Big(\sum_jz_jF_{\Lambda_k}(f_j)\Big)\Big\rangle^K_{k,J}. \] The inner limit is King's cube limit~\cite[Thm.~2.1]{K2} at $h=0$ along the spacings $5^{-k}$, at fixed $\mu_0$ and fixed $J$, taken for the $J$ at which the sources have $\ell^\infty$-distance $>1$ from $\R^3\setminus C_J$; King's proof runs his block-spin flow on $C^K_J$, whose block averages of the scalar field carry the Dirichlet weight $f_D$, and the limit is \cref{thm:k01:EC}. The outer limit runs along the family $(C^K_J)_J$. Both limits exist, locally uniformly in $z$, given Theorem~\ref{thm:k01:EC} (Theorem~\ref{rot:thm:W}). \end{definition} \begin{definition}[the box limits, the choice of $M$ and the identification of the object]\label{r13:def:objectM} Let $Z_\infty(\cdot,0)$ be King's state on curvature sources (Definition~\ref{rot:def:object}). On imaginary coefficients the lower and upper iterated limits of the functionals $\langle e^{\mathrm iF(f)}\rangle^\ND_{k,J}$ of the boxes $C_J\cap\Lambda_k$ of Definition~\ref{def:s02:acube}, $\lim_J\liminf_k$ and $\lim_J\limsup_k$, both equal $Z_\infty(\mathrm if,0)$ (Corollary~\ref{rot:cor:Kmono}(c)); whether the fixed-cube limits $\lim_k$ of the box functionals exist is not decided (Remark~\ref{rem:s04:notfollow}, \cref{prop:s12:ID}(b)). The state does not depend on the admissible choice of $M$: the states $\langle\cdot\rangle^K_{k,J}$ depend on $M$ only through $MJ$, so if $M_1$ divides $M_2$ the cubes of $M_2$ are the cubes $C^K_{JM_2/M_1}$ of $M_1$, and the limits along the two families agree, on imaginary coefficients by Corollary~\ref{rot:cor:Kmono}(a) and then everywhere by Theorem~\ref{rot:thm:W}(iii); two admissible values divide their least common multiple, which is again admissible. This is King's state~\cite[(2.15)]{K2} at $h=0$, the state $\ZKinf(\cdot,0)$ of \cref{thm:s05:CK} (the same cubes, action and iterated limit, for the same $M$), and it is the object of Theorem~\ref{rot:thm:main}; the admissible values of $M$ are those of the setting, and the independence of $M$ just shown is \cref{rem:s02:Mindep}. \end{definition} We do not address the regularity of the Schwinger functions (they are distributions), the limit $\mu_0\to0$, or the uniqueness of the vacuum. \space% % % \section{The torus comparison}\label{rot:sec:K} This section proves (K$_0$), the equality of the continuum limits of a walled torus on the axis lattice and on its rotated copy (Theorem~\ref{rot:hyp:K}). The involution of Lemma~\ref{rot:lem:inv} reduces the comparison to one-sided bounds at a single spacing (Theorem~\ref{rot:thm:sandwich}), and the bounds are provided by the corrected torus theorem, Theorem~\ref{rot:thm:CT}, whose hypotheses (N1)--(N6) are provided by the statements of Section~\ref{rot:ssec:inputsN} on the class $\mathcal C$. (K$_0$) is used by the closing statement (Theorem~\ref{rot:hyp:TIGHT}: the inequality (TIGHT$_0$), which bounds King's state from below by the lower envelope of the walled tori) and by the invariance theorem (Theorem~\ref{rot:thm:B}). \subsection{The statement} \begin{theorem}[\textup{(K$_0$)}: the torus comparison, curvature sector]\label{rot:hyp:K} Let the couplings lie outside the set $E_W$ of Proposition~\ref{rot:thm:POS}. For every $R\in\Rgen$, every walled configuration $\fc$ of the class $\mathcal C$ (the class on which (K$_0$) enters Theorem~\ref{rot:thm:main}; Definition~\ref{rot:def:class}, Remark~\ref{rot:rem:class}) and every real smooth periodic $2$-form $G$, $\lim_k\langle e^{F(G)}\rangle_{\fc,\Lambda_k}$ and $\lim_k\langle e^{F(G)}\rangle_{\fc,R^{-1}\Lambda_k}$ exist and are equal. \end{theorem} \begin{proof} Theorem~\ref{rot:thm:CT}(ii) and (iii), from the statements of Section~\ref{rot:ssec:inputsN} on the class $\mathcal C$, the large-field clause of Theorem~\ref{rot:thm:BAL5} at the wall and Theorem~\ref{cpi:thm:hrs:H6W} holding outside $E_W$ for $J\ge J_{\rm pos}$ (Proposition~\ref{rot:thm:POS}, Remark~\ref{rot:rem:Kstatus}). \end{proof} Below, ``under (K$_0$)'' means: for the configurations for which the two limits exist and agree. Here $R^{-1}\Lambda_k$ carries the frame $(R^{-1}e_\alpha)$ and descends to $D_J$ by Lemma~\ref{rot:lem:geom}(b); both lattices have spacing $\eps_k$ and carry the same $\delta m^2_{\eps_k}$. Since $\Rgen=\Rgen^{-1}$, (K$_0$) for all $R\in\Rgen$ is the same as the statement for the lattices $R\Lambda_k$ used in~\cite{K2}. Formulated with a single scalar function on all plaquettes, as in~\cite[(2.12), Thm.~2.4]{K2}, the comparison is false already at $e=0$ (Proposition~\ref{cpi:rot:scalar}); the $2$-form formulation is consistent with King's proof, which treats the source components separately (Remark~\ref{rot:rem:frame}). \begin{lemma}\label{rot:lem:K} Under (K$_0$) the convergence holds locally uniformly in $z\in\C$: the two limits $\lim_k\langle e^{zF(G)}\rangle_{\fc,\Lambda_k}$ and $\lim_k\langle e^{zF(G)}\rangle_{\fc,R^{-1}\Lambda_k}$ exist and are the same entire function of $z$. In particular, for every walled configuration $\fc$ for which (K$_0$) holds (on the class $\mathcal C$, by Theorem~\ref{rot:hyp:K}) and every real $2$-form $f\in C^\infty_c$ supported in one cell, $\lim_kT_{\fc,\Lambda_k}(f)$ and $\lim_kT_{\fc,R^{-1}\Lambda_k}(f)$ exist and are equal. \end{lemma} \begin{proof} As in Theorem~\ref{rot:thm:W}(i): local boundedness from $|\langle e^{zF}\rangle|\le\langle e^{\Re z\,F}\rangle$ and convexity, Montel's theorem, and $\R$ as a set of uniqueness. For the second claim take $G$ the periodised $f$ and $z=\mathrm i$. \end{proof} \subsection{Reduction to one-sided bounds at one lattice spacing} King's proof of his Theorem~2.4 compares the lattice of spacing $\eps$ with the rotated lattice of spacing $5\eps$, and obtains the reverse inequality by ``switching around the argument''~\cite[p.~337]{K2}. We compare the two lattices at the \emph{same} spacing, and obtain the reverse inequality from an exact symmetry. \begin{lemma}[the involution]\label{rot:lem:inv} Let $R=R_{\mu\nu}(\pm\theta_0)\in\Rgen$, let $\sigma:=\sigma_\nu$ be the reflection $x_\nu\mapsto-x_\nu$, and put $S:=R^{-1}\sigma$. Then $S^2=1$, $S\Lambda_k=R^{-1}\Lambda_k$ and $S(R^{-1}\Lambda_k)=\Lambda_k$, and $S$ maps $D_J=\R^3/P_J$, $P_J=(50MJ\Z)^3$, onto the torus $R^{-1}D_J:=\R^3/R^{-1}P_J$. Both lattices descend to both tori, and for all real smooth $D_J$-periodic sources $s=(G,h)$, \begin{gather*} \cZ_{R^{-1}\Lambda_k,D_J}(s)=\cZ_{\Lambda_k,R^{-1}D_J}(S\cdot s),\qquad\cZ_{\Lambda_k,D_J}(s)=\cZ_{R^{-1}\Lambda_k,R^{-1}D_J}(S\cdot s),\\ \sS_{R^{-1}\Lambda_k,D_J}(h)=\sS_{\Lambda_k,R^{-1}D_J}(S\cdot h). \end{gather*} \end{lemma} \begin{proof} By Lemma~\ref{rot:lem:geom}(a), $\sigma R\sigma=R^{-1}$, so $S^2=R^{-1}(\sigma R^{-1}\sigma)=R^{-1}R=1$; $\sigma\Lambda_k=\Lambda_k$ gives $S\Lambda_k=R^{-1}\Lambda_k$, and then $S(R^{-1}\Lambda_k)=S^2\Lambda_k=\Lambda_k$. Also $\sigma P_J=P_J$, so $SP_J=R^{-1}P_J$. The lattice $R^{-1}\Lambda_k$ is invariant under $R^{-1}P_J$ trivially, and $\Lambda_k$ is invariant under $R^{-1}P_J$ because $R^{-1}p\in10MJ\,\Z^3$ for $p\in P_J$ (Lemma~\ref{rot:lem:geom}(b)). The map $S$ carries the frame $(R^{-1}e_\alpha)$ to $(SR^{-1}e_\alpha)=(\sigma e_\alpha)$ and the frame $(e_\alpha)$ to $(R^{-1}\sigma e_\alpha)$; these differ from the frames of $\Lambda_k$ and $R^{-1}\Lambda_k$ only by signs, which do not change the periodic models (Lemma~\ref{rot:lem:pass}). Lemma~\ref{rot:lem:pass} with $O=S$, applied to the unnormalised integrals (the bijection of configurations preserves $d\nu$), gives the identities. \end{proof} \begin{definition}[\textup{(F)} and \textup{(F$_0$)}: one-sided bounds at one spacing]\label{rot:hyp:F} Let $R\in\Rgen$ and let $J\in\N$. The following two named bounds are the abstract form to which the torus comparison is reduced below; (F$_0$) is established, on the class $\mathcal C$ and outside $E_W$, by the corrected torus theorem (Theorem~\ref{rot:thm:CT}, Remark~\ref{rot:rem:closing}), and (F) at general sources is not used. Write $T_1:=D_J$, $T_2:=R^{-1}D_J$. \textbf{(F)}: for $i=1,2$ there are real numbers $v_k(T_i)$ and $c_k(T_i)$, $k\ge k_0(J)$, independent of the fields and the sources, with $|c_k(T_i)|\le C\eps_k^{-q}$ for some $q<\infty$ and $v_k(T_i)$ bounded, such that for every real smooth $T_i$-periodic source $s=(G,h)$, \[ \cZ_{R^{-1}\Lambda_k,T_i}(s)\le\big(1+\delta_k(T_i,s)\big)\exp\!\big(-v_k(T_i)-c_k(T_i)\,\sS_{\Lambda_k,T_i}(h)\big)\,\cZ_{\Lambda_k,T_i}(s), \qquad\lim_{k\to\infty}\delta_k(T_i,s)=0 . \] \textbf{(F$_0$)} is (F) restricted to sources with $h=0$ and to the sources $(G,h_0)$, where $h_0=-w\le0$ is a wall, in total form ($v_k\equiv0$), on $T_1$ for these sources and on $T_2$ for their images under the involution $S$ of Lemma~\ref{rot:lem:inv}, together with \textbf{(C$_0$)}: $\limsup_kc_k(T_1)\le0$ and $c_k(T_2)-c_k(T_1)\to0$. The sources $(G,-h_0)$, with a mass-decreasing wall, are not needed. \end{definition} On each torus, (F) bounds the functional of the rotated lattice from above by that of the axis lattice. The rotated lattice is the one whose flow is King's standard block-spin flow in its own frame, and the axis lattice is the one run through King's hybrid flow; this is the direction of~\cite[(4.7)--(4.9)]{K2}. Since smooth periodic functions have rapidly convergent Riemann sums, $|\sS_{\Lambda,T}(h)-\int_{\Gamma}h|\le C_N(h)\eps_k^N$ for every $N$ and every lattice $\Lambda$ of spacing $\eps_k$ descending to $T$, $\Gamma$ a fundamental cell; with $N>q$ the error $c_k(\sS_{\Lambda,T}(h)-\int h)$ tends to $0$. \begin{theorem}[closing at one spacing]\label{rot:thm:sandwich} Let $R\in\Rgen$, $J\in\N$, and assume \textup{(F)}. Write $v=v_k(T_1)$, $v'=v_k(T_2)$, $c=c_k(T_1)$, $c'=c_k(T_2)$, and $\rho_k(s):=\cZ_{R^{-1}\Lambda_k,D_J}(s)/\cZ_{\Lambda_k,D_J}(s)$. \begin{enumerate} \item[(a)] For every real smooth $D_J$-periodic source $s=(G,h)$, $e^{v'+c'\int h}(1+o(1))\le\rho_k(s)\le e^{-v-c\int h}(1+o(1))$ as $k\to\infty$. \item[(b)] If (F) holds in total form ($v=v'=0$) on $T_1$ for the sources $s_t=(tG,h_0)$, $t\in\R$, with $\int h_0<0$, and on $T_2$ for their $S$-images, and $\limsup_k(c+c')\le0$, then $c+c'\to0$ and $\rho_k(s_t)e^{c\int h_0}\to1$ for every real $t$; in particular, for $s_0=(0,h_0)$ and $s_1=(tG,h_0)$, \[ \frac{\cZ_{R^{-1}\Lambda_k,D_J}(s_1)}{\cZ_{R^{-1}\Lambda_k,D_J}(s_0)}=\frac{\cZ_{\Lambda_k,D_J}(s_1)}{\cZ_{\Lambda_k,D_J}(s_0)}\,(1+o(1)). \] \end{enumerate} No convergence of either family is assumed. \end{theorem} \begin{proof} (a) The upper bound is (F) on $T_1$, with $c\,\sS_{\Lambda_k,D_J}(h)=c\int h+o(1)$. Apply (F) on $T_2$ to the source $S\cdot s$ and use Lemma~\ref{rot:lem:inv}: $\cZ_{R^{-1}\Lambda_k,T_2}(S\cdot s)=\cZ_{\Lambda_k,D_J}(s)$, $\cZ_{\Lambda_k,T_2}(S\cdot s)=\cZ_{R^{-1}\Lambda_k,D_J}(s)$ and $\sS_{\Lambda_k,T_2}(S\cdot h)=\sS_{R^{-1}\Lambda_k,D_J}(h)=\int h+O(\eps_k^N)$. This is the lower bound. (b) By (a) at $s_t$ the interval is non-empty, so $(c+c')\int h_0\le o(1)$; as $\int h_0<0$, $c+c'\ge-o(1)$, and with $\limsup_k(c+c')\le0$ this gives $c+c'\to0$. The two ends of the interval in (a) then differ by a factor $e^{o(1)}$, so $\rho_k(s_t)e^{c\int h_0}\to1$; dividing the statement for $s_1$ by the one for $s_0$ cancels the prefactor. \end{proof} The proof uses (F) only at the sources named. In particular, under (F$_0$) the conclusions (a) and (b) hold for the sources $(tG,h_0)$ (where $\int h_0<0$ unless $w=0$) and, with no condition on $c,c'$, (a) holds in total form for the sources with $h=0$. \begin{corollary}[walled curvature ratios]\label{rot:cor:walled} Assume \textup{(F$_0$)} for $R$ and $J'$. Then for every walled configuration $\fc=(J',w)$ whose wall $w$ is one for which \textup{(F$_0$)} holds at $h_0=-w$, and every real smooth periodic $2$-form $G$, $\langle e^{zF(G)}\rangle_{\fc,R^{-1}\Lambda_k}-\langle e^{zF(G)}\rangle_{\fc,\Lambda_k}\to0$ locally uniformly in $z\in\C$. If moreover $\lim_k\langle e^{tF(G)}\rangle_{\fc,\Lambda_k}$ exists for real $t$, then \textup{(K$_0$)} holds for $\fc$ and $G$. \end{corollary} \begin{proof} With $h_0=-w$, $\langle e^{tF(G)}\rangle_{\fc,\Lambda}=\cZ_\Lambda(tG,h_0)/\cZ_\Lambda(0,h_0)$ (the Wick constants cancel), a balanced ratio. By Theorem~\ref{rot:thm:sandwich}(b) (at $w=0$ directly by (a) in total form), \[ u_k(t):=\langle e^{tF(G)}\rangle_{\fc,R^{-1}\Lambda_k}/\langle e^{tF(G)}\rangle_{\fc,\Lambda_k}\to1 \] for real $t$. By Lemma~\ref{rot:lem:GD}(a) and the symmetry $(A,\vartheta)\mapsto(-A,-\vartheta)$, $1\le\langle e^{tF(G)}\rangle\le e^{t^2\sigma_\Lambda/2}$ for real $t$, where $\sigma_\Lambda=\ip{v_G}{Cv_G}$ is the variance of $F(G)$ under the free periodic photon on the lattice, $v_G$ the vector with $\ip{v_G}A=F(G)$ and $C$ the photon covariance; $\sigma_\Lambda\le\mu_0^{-2}\|\delta_\eps G\|^2$ is bounded uniformly in $k$, $\delta_\eps$ the lattice codifferential. Hence the difference of the two functions tends to $0$ on $\R$, and it is bounded on every strip $|\Re z|\le r$ uniformly in $k$, because $|\langle e^{zF}\rangle|\le\langle e^{\Re z\,F}\rangle$. Vitali's theorem gives locally uniform convergence to $0$ on $\C$. The last statement follows by the same argument. \end{proof} \begin{remark}[what this subsection proves]\label{rot:rem:closing} Lemma~\ref{rot:lem:inv}, Theorem~\ref{rot:thm:sandwich} and its corollary are a reduction of the torus comparison to the abstract one-sided bound (F): they are proved, and they provide none of the bounds. In the curvature sector the bound (F$_0$) that Corollary~\ref{rot:cor:walled} uses, the one-sided forward bound in total form on the mass-increasing family with its source-linear constant on $D_J$ of upper limit at most zero and the constant on the rotated torus $R^{-1}D_J$ equal to it up to $o(1)$, comes from the proof of the corrected torus theorem itself (Proposition~\ref{cpi:ct:F}, Section~\ref{rot:ssec:CTproof}), in which King's torus bound~\cite[(4.2)]{K2} is not used, because at a non-zero wall it is vacuous; its place is taken by the large-field clause of King's upper bound at blocking factor $5$, Theorem~\ref{rot:thm:BAL5}, and at a mass-increasing wall the polynomial lower bound (N5) is the wall instance Proposition~\ref{cpi:ct:H5}. \end{remark} \subsection{The corrected torus theorem}\label{rot:ssec:CT} The bound (F$_0$) and the existence of the axis-lattice limits for walled curvature ratios are the content of the corrected torus theorem, which we now state in the form used. Write $X_K:=\Lambda_K$ and $Y_K:=R\Lambda_K$, $R=R_{12}(\theta_0)$, so that (K$_0$) for $R^{-1}$ is the statement for the pair $(X_K,Y_K)$; in this subsection and in Proposition~\ref{cpi:ct:F} the spacing index is written $K$, $\eps_K=5^{-K}$ (it is the index $k$ of the other sections, not the shell mass of Sections~\ref{rot:sec:intro}, \ref{rot:sec:env} and~\ref{rot:sec:shell}). Configurations are taken in the class $\mathcal C$ of Definition~\ref{rot:def:class}: $J\in5^{k_*}\N$ with $J\ge\max(J_0(\eps_0),J_{\rm pos})$, $\fc=(J,\psi,n)$ with $n>0$, $\psi$ vanishing on the $2$-neighbourhood of $R^{-1}C_L$, the wall condition $n\langle\psi\rangle_{X_K}>\Delta_*|T|/2$, block alignment and the grid offset of the corners of $H'$. \begin{theorem}[the corrected torus theorem, curvature sector]\label{rot:thm:CT} Let the couplings lie outside the set $E_W$ of Proposition~\ref{rot:thm:POS}. Let $e^2,\lambda,m^2,\mu_0^2>0$ be in King's range, the block factor $5$, and $\fc=(J,\psi,n)$ a walled configuration of the class $\mathcal C$ with wall source $h_0=-n\psi$. Then for every real smooth $D_J$-periodic $2$-form $G$: \begin{enumerate} \item[(i)] $\langle e^{zF(G)}\rangle_{\fc,Y_K}-\langle e^{zF(G)}\rangle_{\fc,X_K}\to0$ as $K\to\infty$, locally uniformly in $z\in\C$; in particular $\langle F(G)^p\rangle_{\fc,Y_K}-\langle F(G)^p\rangle_{\fc,X_K}\to0$ for every moment order $p\in\N$; \item[(ii)] both families converge, locally uniformly on $\C$, to the same entire function $W_\fc(zG)$, and all moments converge to the same limits (this uses (N6), the existence of $\lim_K\langle e^{tF(G)}\rangle_{\fc,X_K}$ for real $t$, Theorem~\ref{cpi:thm:hrs:H6W}); \item[(iii)] the same holds with $R$ replaced by any $R'\in\Rgen$. \end{enumerate} At $e=0$ the statement holds without (N1)--(N6) and without the condition on $E_W$. \end{theorem} The hypotheses (N1)--(N5) of the forward bound (Proposition~\ref{cpi:ct:F}, Section~\ref{rot:ssec:CTproof}) are the statements of Section~\ref{rot:ssec:inputsN}, for the sources $(tG,h_0)$, $t\in\R$, on $D_J$, for their $S^*$-images on $RD_J$, and for their images under the coordinate reflections and permutations on $D_J$ together with the $S^*$-images of those on $RD_J$, where $S^*:=R\sigma$ is the involution of Lemma~\ref{rot:lem:inv} for $R^{-1}$ in place of $R$ (it exchanges $X_K$ and $Y_K$ and maps $D_J$ onto $RD_J$); the proof of Theorem~\ref{rot:thm:CT} is Section~\ref{rot:ssec:CTproof}. Its steps are: the involution of Lemma~\ref{rot:lem:inv} and exact passive-rotation reductions; an equal-spacing comparison, in total form (every field-independent constant included), of the effective actions of the two block-spin flows on King's small-field domain, the standard flow of $Y_K$ and King's hybrid flow of $X_K$, whose first step averages over rotated blocks; a one-sided forward bound of the form (F) on each torus, from King's upper bound~\cite[Thm.~3.1]{K1} for $Y_K$ and the hybrid lower bound~\cite[Prop.~4.1]{K2} for $X_K$, the additive large-field term, in the form of the large-field clause of Theorem~\ref{rot:thm:BAL5}, being absorbed on the left by the polynomial lower bound (N5), which at a mass-increasing wall is the wall instance Proposition~\ref{cpi:ct:H5}; and the closing by Theorem~\ref{rot:thm:sandwich}(b) on the mass-increasing family, where the vacuum constants are absent (the comparison is in total form) and the source-linear constants of the pair differ by $o(1)$, the axis-lattice one tending to zero. King's torus bound~\cite[(4.2)]{K2} is not used anywhere in this argument. The sector in which the sources are coupled through a single scalar function on all plaquettes is false (Proposition~\ref{cpi:rot:scalar}) and is not what the theorem states. The scalar (Wick-source) sector of the torus theorem, which needs the convergence on the torus of the Wick-source functionals with a renormalised Wick square (Section~\ref{rot:sec:open}), is outside Theorem~\ref{rot:thm:main}; its walled form on the cone, at $g\ne0$ and outside $E_W$, is proved in Section~\ref{rot:ssec:Ksupply} (Theorem~\ref{rot:thm:Kwsupply}). % % \space% % % \subsection{Proof of the corrected torus theorem}\label{rot:ssec:CTproof} The proof compares, in total form, the effective actions produced by the two block-spin flows at one common spacing. We first bound the field-independent part of that comparison, the vacuum constants of the two flows. For the vacuum families, the connected source-free graphs of the effective action together with the vacuum-energy counterterm $E_1$ of \cite[(3.35)]{K1}, the two flows are compared in the \emph{$C/\Gamma$ form}: each line $G^F_k=C-\Gamma^F$ is split into the common free propagator $C$ and the flow-dependent part $\Gamma^F=H^FB^FH^{F*}$ ($H^F$ the external-line kernels of the flow, $B^F$ its block kernel; $\sup|\Gamma^F|\le C_1$), and the renormalised family of each flow is written, exactly, as the sum over connected vacuum graphs $G$ and non-empty line subsets $S\subset G$ of $(-1)^{|S|}G[S\mapsto\Gamma^F,\ G\setminus S\mapsto C]$, plus the graphs with contour vertices: the all-$C$ monomials ($S=\emptyset$) are cancelled exactly by $E_1$ within each flow, and the $\Gamma$-lines play the roles of legs and recipients in Theorem~\ref{cpi:ct:H1S}. The vacuum constant of the effective-action comparison on $T$ splits as $v_K(T)=v^{\rm G}_K(T)+v^{\rm I}_K(T)$, the Gaussian part and the interaction part, with $v^{\rm I}_K(T)=|T|\Delta p_K+r_K(T)$, $\Delta p_K$ the difference of the two flows' vacuum pressures per coarse site, and $v^{\rm G}_K(T)=\ln N^Y_k-\ln N^X_k+\ln N^Y_k(0)-\ln N^X_k(0)$ the difference of the Gaussian normalisations of the two flows' effective actions at zero fields (\cite[(3.89)]{K1} at $A_k=\phi_k=0$), $\ln N_k=-\frac32\sum_p\ln(\Delta^{(k)}_{\mu_0}(p)/2\pi)$ and $\ln N_k(0)=-\frac N2\sum_p\ln(\Delta^{(k)}_m(p)/2\pi)$ being the sums over the dual of the common coarse torus of the logarithms of the flow's coarse symbol at the photon and the scalar mass. Here $X$ is the standard and $Y$ the hybrid flow, $k=k(K)$ the level of the construction; in the unit scale of the level $k$ (the physical spacing $\delta_k=5^k\eps_K$ of the common coarse lattice is the unit, so $\delta_k=5^{k-K}$ in the unit scale of the whole construction) the fine spacing is $\eta'\defeq5^{-k}$ and $\eta\defeq5\eta'=5^{1-k}$ is the spacing of the first-step coarse lattice, the letters of the hybrid kernels (Definition~\ref{cpi:hyb:cont} and the lines before it); $\tau$ is the bijection between the first-step cells of the two lattices (so that $\tau$-matched points are corresponding cell points), $\theta_1$ the exponent of Theorem~\ref{cpi:ct:H1S}, $m_0\defeq\min(m,\mu_0)$, and $\vartheta_K$ the rate of the forward bound, Proposition~\ref{cpi:ct:F}, which tends to $0$. \begin{theorem}[the vacuum-family two-flow comparison]\label{cpi:ct:Vvac} There are $p=p(\bar n)$, $p'=p'(\bar n)$, $\beta_G=\beta_G(\bar n)\le3\cdot(\text{number of lines fixed at the top scale})$ and $C=C(a,m,\mu_0,N,\bar n)$ such that: \begin{enumerate} \item[(i)] (\emph{per coarse site}) for every order at most $\bar n$, every renormalised vacuum family $R$ in the $C/\Gamma$ form, every $\theta\le\theta_1$, at $\tau$-matched points of the two lattices, \[|E^X(R)-E^Y(R)|\le C\,p_k^2\delta_k^{-1/2}\,\eta'^{\theta}\,(1+K)^p\,(m_0\delta_k)^{-\beta_G}\cdot\,\mathcal B_R,\] where $\mathcal B_R\defeq e_k^{d_v}\lambda_k^{d_s}$ times the per-flow constants of \cite[(1.33)]{Ba3} is the coupling factor of the family, $d_v$ and $d_s$ the numbers of its charge and quartic vertices, and the prefactor $p_k^2\delta_k^{-1/2}$ is present only when a contour element is switched ($e>0$); \item[(ii)] (\emph{per torus}) $|\Delta p_K|\le C\,5^{-\theta k}(m_0\delta_k)^{-\beta_G}(1+K)^{p'}$ per coarse site, and $|v^{\rm I}_K(T)|\le\vartheta_K|T|$ for every $T\in\{D_J,RD_J\}$; \item[(iii)] (\emph{the form used}) $|v^{\rm G}_K(T)|\le12.5\,a\,5^{3K-5k}|T|=12.5\,a\,5^{-2K}\delta_k^{-5}|T|$, which tends to $0$ at the level $k=k(K)$, where $\delta_k^{-1}$ grows at most polynomially in $K$; hence $\Delta p_K\to0$ and $v_K(T)\to0$ on each torus $T\in\{D_J,RD_J\}$ as $K\to\infty$; in particular $\liminf_K[v_K(D_J)+v_K(RD_J)]\ge0$ for every admissible $J$. \end{enumerate} The constants are not uniform in the masses: they depend on $m,\mu_0$ through $(m_0\delta_k)^{-\beta_G}$, the price of the lines fixed at the top scale, whose decay is at the rate $m_0\delta_k$ only. \end{theorem} Only (iii) is used: the forward bound of Proposition~\ref{cpi:ct:F} uses $\vartheta_K$ through $\delta_K\to0$, and Theorem~\ref{rot:thm:sandwich} needs only $v_K(D_J)+v_K(RD_J)\ge-o(1)$, since in the comparison in total form the vacuum constants are absent from the ratio. \begin{proof} \emph{The interaction part, (i)--(ii).} The bound is Theorem~\ref{cpi:ct:H1S} in its vacuum instance: the bound $|\Gamma^X-\Gamma^Y\circ\tau|\le C\eta^{1/2}$ at matched pairs, pointwise, is combined with each $\Gamma^F$'s decay at the rate $m_0\delta_k$ (the kernel bounds (N2): \cite[Thm.~3.3]{K1} for the standard flow and Theorem~\ref{cpi:hyb:BD} for the hybrid flow) into the decaying form $|\Delta\Gamma(u)|\le C\eta^{1/4}e^{-m_0\delta_k|u|/2}$ (the geometric mean), the all-$C$ components of the vacuum monomials carry at least two $\Gamma$-ends, and the zero-degree all-$C$ subgraphs are enumerated by the census of Theorem~\ref{cpi:ct:H1S}; the orders $\lambda$ and $\lambda^2e^0$ are treated directly. The per-torus bound $|v^{\rm I}_K(T)|\le\vartheta_K|T|$ follows by summing the per-site bound over the $|T|\delta_k^{-3}$ coarse sites, the finitely many families and ordering classes per order; the torus remainder $r_K(T)$ consists of the winding images of the infinite-lattice kernels, each torus kernel being the periodisation of its infinite-lattice kernel, and is at most $C\eps^{-q}e^{-c\delta_k^{-1}}$ because the shortest period of the torus is at least $50\delta_k^{-1}$ in unit scale; $\Delta p_K$ is a difference of infinite-lattice vacuum densities, the same for both tori. \emph{The Gaussian part.} By \cite[(3.89)]{K1} at $A_k=\phi_k=0$ the field-independent part of each flow's effective action is $\ln N_k+\ln N_k(0)+P^{(k)}(0)$, and the counterterm $E_1$ is the same $c$-number for both lattices. The coarse symbols $A^X(p)$, $A^Y(p)$ of the two flows obey $|A^X(p)-A^Y(p)|\le0.20\,a\,\eta^2A^X(p)A^Y(p)$ at every momentum $p$ and every mass (Woodbury's formula for the composite block average, whose symbol is the product of the two averaging symbols; the terms $\eta^2/a$ cancel in the difference, and the sum over the aliases of the squared averaging symbol is one), so $|\ln(A^X(p)/A^Y(p))|\le0.20\,a\eta^2$; summing over the $|T|\delta_k^{-3}$ momenta of the dual coarse torus and the $3+N=5$ components with the factor $\frac12$ gives $|v^{\rm G}_K(T)|\le\tfrac12\,a\,\eta^2\delta_k^{-3}|T|$ (the factor is $0.20\cdot5\cdot\tfrac12=\tfrac12$); with $\eta=5^{1-k}$ and $\delta_k=5^{k-K}$ one has $\eta^2\delta_k^{-3}=25\cdot5^{-2k}\cdot5^{3(K-k)}=25\cdot5^{3K-5k}$, which gives the bound $12.5\,a\,5^{3K-5k}|T|$ of (iii). \emph{The background determinants.} Besides the perturbative families, the effective actions of the two flows differ in the normalisations of their fluctuation integrals at the background including the Cameron--Martin shift of the curvature source; given the kernel bounds of both flows (\cite[Thm.~3.3]{K1}, Theorem~\ref{cpi:hyb:BD}), the logarithms of the two determinant ratios differ on $\supp\chi_k$, for $T\in\{D_J,RD_J\}$ and at the level $k=k(K)$, by $o(|T|)$ times a polynomial in $K$, uniformly in $J$ and the torus (each ratio is split over the depths and sites of the flow and written as a second-order Duhamel integral in the background, the first order vanishing by charge-conjugation parity, and the difference of the two flows is telescoped through the resolvents, the kernel difference bounds supplying the gain), a difference which the forward bound of Proposition~\ref{cpi:ct:F} absorbs. \emph{Clause (iii).} With $v^{\rm G}_K\to0$, $r_K\to0$, $\Delta p_K$ the same for both tori and $|D_J|=|RD_J|$, $v_K(D_J)+v_K(RD_J)=2|D_J|\Delta p_K+o(1)$, so $\liminf_K[v_K(D_J)+v_K(RD_J)]\ge0$ is equivalent to $\liminf_K\Delta p_K\ge0$, hence, with the upper half $v+v'\le o(1)$ of Theorem~\ref{rot:thm:sandwich}(a), to $\Delta p_K\to0$ and to $v_K(T)\to0$ on each torus. \end{proof} Theorem~\ref{rot:thm:CT} is proved below from Theorems~\ref{cpi:ct:H1S}--\ref{cpi:thm:hrs:H6W}, with Theorems~\ref{cpi:ct:Vvac} and~\ref{cpi:ct:CTB} and the cumulant bound (Theorem~\ref{cpi:thm:cumulant}) applied at each depth of the flow. King's statements enter as citations (\cite[Thm.~3.1, Thm.~3.3, Prop.~3.7 within its stated scope, Prop.~3.10, Lemma~4.5]{K1}), except the two proved in Section~\ref{rot:ssec:inputsN} in the forms used (Theorems~\ref{cpi:thm:hrs:H6W} and~\ref{cpi:thm:78}); Ba{\l}aban's steps left without argument are Theorems~\ref{rot:thm:BAL5} and~\ref{r13:thm:BAL5compl}. The two-lattice content of the proof is the following one-sided bound, in which the large-field term is absorbed on the left. \begin{proposition}[the forward bound]\label{cpi:ct:F} Let $T\in\{D_J,RD_J\}$ and let $s=(zG,h)$ be a real source at which the hypotheses (N3), (N4), (N5) of Section~\ref{rot:ssec:inputsN} and the comparison of the two effective actions on $\supp\chi_k$ hold, the latter in the form $e^{-S^{(k)}}\le e^{-\hat s^{(k)}-c_K\langle h\rangle_{X_K}+\vartheta_K|T|}$ with $\vartheta_K\to0$ and a constant $c_K$ independent of $T$ and of the source. Then \[Z_{Y_K}(T;s)\ \le\ (1+\delta_K)\,e^{-c_K\langle h\rangle_{X_K}}\,Z_{X_K}(T;s),\qquad\delta_K=\delta_K(T,s)\to0 .\] No bound on $c_K$ is used. \end{proposition} \begin{proof} Write $I_Y\defeq\int\chi_ke^{-S^{(k)}}$ and $I_X\defeq\int\chi_ke^{-\hat s^{(k)}}$. By (N3), King's upper bound with the large-field clause of Theorem~\ref{rot:thm:BAL5}, $Z_Y\le e^{C\delta_k^\sigma|T|}I_Y+B$ with $B\defeq e^{-c_*p(\delta_k)^2+C|T|}\eps_K^{-q}$, $c_*>0$. By (N5), $B\le b\,Z_Y$ with $b\defeq e^{-c_*p(\delta_k)^2+(C+C_1)|T|}\eps_K^{-2q}$; since $p(\delta_k)^2$ grows at least quadratically in $K$ while $\ln\eps_K^{-2q}=2qK\ln5$ grows linearly, $b\to0$, so $Z_Y\le(1-b)^{-1}e^{C\delta_k^\sigma|T|}I_Y$. The large-field term is thus absorbed on the \emph{left}, which is why no bound on $c_K$ and no lower bound for the axis lattice is needed. On $\supp\chi_k$ the comparison of the effective actions gives $I_Y\le e^{-c_K\langle h\rangle+\vartheta_K|T|}I_X$, and by (N4), Proposition~\ref{cpi:ct:H4}, $I_X\le e^{C(1+K)^{p'}\delta_k^\sigma|T|}Z_X$. So $\delta_K=(1-b)^{-1}e^{(2C(1+K)^{p'}\delta_k^\sigma+\vartheta_K)|T|}-1\to0$, since $(1+K)^{p'}\delta_k^\sigma\to0$. \end{proof} \begin{proof}[Proof of Theorem~\textup{\ref{rot:thm:CT}}]\label{cpi:ct:closing} Fix $\fc=(J,\psi,n)$ and $G$, and let $\cS^-\defeq\{(tG,h_0):t\in\R\}$, $h_0=-n\psi$, the \emph{mass-increasing family}; the sources $(tG,+n\psi)$ do not occur. \emph{Step 1 (comparison of the effective actions).} The two flows, the standard flow of $Y_K$ and the hybrid flow of $X_K$, are run to one common spacing, and their effective actions on $\supp\chi_k$ are compared term by term in total form, every field-independent constant included: by the two-flow comparison bound (N1), Theorem~\ref{cpi:ct:H1S}, applied to every renormalised family of \cite[\S3.5]{K1} in per-flow normal form, by the kernel bounds (N2) of both flows and by the difference bounds of Theorem~\ref{cpi:hyb:BD}, $e^{-S^{(k)}}\le e^{-\hat s^{(k)}-c_K\langle h\rangle_{X_K}+\vartheta_K|T|}$ on $D_J$ for $s\in\cS^-$ and on $RD_J$ for $S^*s$ (Lemma~\ref{rot:lem:pass} transports the own-frame sources, and the $S^*$-image of a wall is a wall on $RD_J$), with one constant $c_K$ for both tori; the $h$-linear field-independent families, excepted from Theorem~\ref{cpi:ct:H1S}, are exactly the terms collected in $c_K\langle h\rangle$, and the vacuum constants are bounded by Theorem~\ref{cpi:ct:Vvac}(iii). \emph{Step 2 (forward bounds).} Proposition~\ref{cpi:ct:F} on $D_J$ and on $RD_J$ for these sources. Its hypotheses (N3), (N4), (N5) hold on both tori: on $D_J$ every $Y$-frame step is a skew step, so (N3) is the instance [SKEW-U] and (N4), (N5) are instances of [SKEW-L]; on $RD_J$ only the first hybrid step is skew, with image terms $O(e^{-c\,50MJ/\eps})$. The wall instance of (N5) at $h_0=-n\psi$ is Proposition~\ref{cpi:ct:H5}. \emph{Step 3 (sandwich).} Pull the forward bound on $RD_J$ back to $D_J$ by the involution $S$ of Lemma~\ref{rot:lem:inv} ($S^{*2}=1$): $Z_{X_K}(D_J;s)\le(1+o(1))e^{-c'_K\langle h\rangle_{Y_K}}Z_{Y_K}(D_J;s)$ with $c'_K=c_K+o(1)$. Together with the forward bound on $D_J$ this gives $e^{c'_K\int h}(1+o(1))\le Z_{Y_K}(D_J;s)/Z_{X_K}(D_J;s)\le e^{-c_K\int h}(1+o(1))$ for $s\in\cS^-$, the lattice sums $\langle h\rangle_\Lambda$ differing from $\int h$ by $O(\eps_K^N)$. The axis-lattice constant has upper limit at most zero, $\limsup_Kc_K\le0$ (Section~\ref{rot:ssec:CT}), so $\limsup_K(c_K+c'_K)\le0$, which is the hypothesis of Theorem~\ref{rot:thm:sandwich}(b): since $\int h_0<0$ the interval is non-empty only if $c_K+c'_K\to0$, and then the ratio identity of the same clause, applied to the balanced pair $s_1=(tG,h_0)$, $s_0=(0,h_0)$, gives \[u_K(t)\defeq\langle e^{tF(G)}\rangle_{\fc,Y_K}-\langle e^{tF(G)}\rangle_{\fc,X_K}\to0\qquad\text{for every real }t ;\] no closure of $\cS^-$ under $h\mapsto-h$ is needed, and for $n=0$ there is no $h$ and no $c$. \emph{Step 4 (complex coefficient, moments).} By Gaussian domination, $1\le\langle e^{tF}\rangle\le e^{t^2q_\Lambda/2}$ with $q_\Lambda\to q(G)$ on both lattices, and $|\langle e^{zF}\rangle|\le\langle e^{\Re z\,F}\rangle$; so $u_K$ is bounded on strips uniformly in $K$ and tends to $0$ on $\R$, Vitali's theorem gives (i), and Cauchy's estimates the moments. \emph{Step 5 (existence).} Under (N6), Theorem~\ref{cpi:thm:hrs:H6W}, the axis family converges for real $t$; by the same domination it converges locally uniformly on $\C$ to an entire function $W_\fc(zG)$, and by (i) so does the rotated family, to the same limit. This is (ii). \emph{Step 6 (other generators).} $\sigma X_K=X_K$, $\sigma D_J=D_J$ and $\sigma R^{-1}(\eps\Z^3)=R(\eps\Z^3)$, and $R_{\mu\nu}(\pm\theta_0)=\Pi R^{\pm1}\Pi^{-1}$ for a permutation matrix $\Pi$ preserving $X_K$ and $D_J$; by Lemma~\ref{rot:lem:pass} with $O=\sigma$, resp.\ $O=\Pi$, the statement for $(R,D_J)$ and all sources is equivalent to the statement for $(R',D_J)$ and all sources, the hypotheses for the images being those the theorem assumes. This is (iii). At $e=0$ the hopping term does not contain $A$, so $\langle e^{zF(G)}\rangle_{\fc,\Lambda}=e^{z^2q_\Lambda(G)/2}$ exactly for both lattices, with $q_\Lambda(G)\to q(G)\defeq\int_{D_J}\langle d^*G,(-\Delta+\mu_0^2)^{-1}d^*G\rangle\,dx$, a frame-free quantity. \end{proof} The closing uses no reverse step, no comparison of two spacings, no Wick scheme, no convergence of the rotated family on its own and no infinite-volume limit; its two-lattice content is Steps~1--4, and existence is a one-lattice question, Step~5. The group generated by $\Rgen$ and the cube group is dense in $O(3)$, because $\theta_0/2\pi$ is irrational. \begin{remark}[the content of \textup{(K$_0$)}]\label{rot:rem:Kstatus} Theorem~\ref{rot:hyp:K} is the curvature sector of~\cite[Thm.~2.4]{K2} with the corrections of Remark~\ref{cpi:rot:errata}, for walled tori and with free-boundary counterterms; by Corollary~\ref{rot:cor:walled} its two-lattice content is the one-sided bound (F$_0$), and its one-lattice content is the existence of the walled limits on the axis lattice. By Theorem~\ref{rot:thm:CT}, (K$_0$) holds for every $R\in\Rgen$ and every configuration of the class $\mathcal C$, by the statements of Section~\ref{rot:ssec:inputsN} with Propositions~\ref{rot:prop:CR} and~\ref{rot:thm:POS} of Section~\ref{rot:ssec:openin}, outside the set $E_W$ there; the interpretations (B1)--(B6) enter as clauses of Theorem~\ref{rot:thm:BAL5}. Thus (K$_0$) is proved in this section (Section~\ref{rot:sec:K}) alone, through Theorem~\ref{rot:thm:CT}, from the statements of Section~\ref{rot:ssec:inputsN} and the proof in Section~\ref{rot:ssec:CTproof}; nothing from Sections~\ref{rot:sec:env}--\ref{rot:sec:shell} enters it. The rotation enters the whole argument of this part only here. \end{remark} \begin{remark}[what the torus comparison alone does not give]\label{rot:rem:shape} Given (K$_0$), the walled values of a configuration $\fc$ on $D_J$ agree on the two lattices, and by passive covariance the walled value of $R\cdot\fc$ on the rotated torus $R\,D_J$ for the axis lattice equals the walled value of $\fc$ on $D_J$ for the rotated lattice. Rotation invariance of a limit along tori would therefore follow from (K$_0$) if the limit did not depend on the shape and orientation of the tori, which is the content of King's expected identity~\cite[(2.24)]{K2} between torus and cube limits. That identity is a uniqueness statement for the infinite-volume limit, and we do not use it. The walls of Section~\ref{rot:sec:env} replace it by a one-sided statement about one lattice, (TIGHT$_0$), whose local form (RES$_0$) is the hypothesis of Theorem~\ref{rot:thm:E}. \end{remark} \space% % % \section{The hypotheses of the torus theorem}\label{rot:ssec:inputsN} Theorem~\ref{rot:thm:CT} is proved in Section~\ref{rot:ssec:CTproof} from the exact reductions, Theorem~\ref{cpi:ct:Vvac} and Proposition~\ref{cpi:ct:F}, and the statements of this section: the hypotheses (N1)--(N6), the skew-torus instances (SK) of King's and Ba{\l}aban's bounds, Ba{\l}aban's estimates at blocking factor five, (BAL)$_5$, and the convention bridge (CT), stated and proved in the order in which each rests on the ones before it; the corrections to the cited sources are collected in Remark~\ref{cpi:rot:errata}. Throughout, $\fc=(J,\psi,n)$ is a configuration of the class $\mathcal C$ (Definition~\ref{rot:def:class}), $k$ the depth of a flow, $K$ the number of scales, and [SKEW-U], [SKEW-L] the skew-torus instances of (N3)--(N5); the shell-side estimates are in Sections~\ref{rot:ssec:shellest} and~\ref{rot:sec:shellestB}. \begin{convention}[the exponent of the small-field function]\label{cpi:rot:convP} The small-field function of King's block-spin bounds \cite[Thm.~3.1]{K1} is $p(\delta)=b_0(1+\log\delta^{-1})^{p}$ with a fixed integer exponent, $p=6$ throughout; any fixed integer $p\ge4$ would do, the lower bounds in the literature being $p>1$ (\cite[(3.41)]{K1}, King's standing inequality), $p\ge2$ (\cite[(3.54)]{Ba1}, kept in \cite[p.~676]{K1}) and $p\ge4$ (\cite[pp.~557, 594]{Ba2}: ``$p>2$'' and ``$2p\ge(d+1)r$'' with $r\ge2$, $d=3$). Every large-field bound of this part uses $p$ only through $p(\delta_{r(k)})^{2}$, which grows at least quadratically in $k$, and \cite[(3.41)]{K1} uses only $p>1$. \end{convention} \subsection{Ba{\l}aban's estimates at blocking factor five} The estimates of~\cite{Ba1,Ba2,Ba3,Ba4} enter at block factor $L=5$ ($d=3$, $a=1$, $r=2$, $p=6$), on the aligned tori and on the skew tori $D_J$, $RD_J$ of Section~\ref{rot:sec:model}; since the proofs of~\cite{Ba2,Ba3,Ba4} leave steps to the reader and several of Ba{\l}aban's statements are used in an interpretation that his text does not fix, the following definition fixes the estimates meant, the stopping depth and the large-field constant, and the two theorems after it collect what is used. \begin{definition}[the estimates used, the stopping depth and the large-field constant]\label{r13:def:BAL5} The \emph{estimates used} are the estimates of~\cite{Ba1,Ba2,Ba3,Ba4} that are claimed there without a complete argument and that the proofs of this part use: the four steps of~\cite{Ba2,Ba3} left without argument, namely the cube step~\cite[p.~433]{Ba3}, the cut-off $\zeta^{(k)}$ of~\cite[(2.44)]{Ba2}, the boxes of~\cite[Lemma~2.4]{Ba2} and the cube partition~\cite[(3.43)--(3.45)]{Ba2}; the steps behind~\cite[Thm.~3.3]{K1}, \cite[(1.14)]{Ba1}, \cite[Prop.~3.6]{K1} and~\cite[Thm.~3.1]{K1} named in Theorem~\ref{r13:thm:BAL5compl}(l), in Proposition~\ref{cpi:ct:H5}, in Theorem~\ref{cpi:thm:78} and in Theorem~\ref{cpi:ct:CTB}(c); the completions of~\cite[pp.~433--445]{Ba3} and~\cite[\S3C]{Ba2} named in the proof of Theorem~\ref{rot:thm:BAL5}; and the interpretations (a)--(f) of Ba{\l}aban's statements in Theorem~\ref{rot:thm:BAL5}. The \emph{stopping depth} $r(k)$ of a flow of depth $k\ge k_0$ is such that $k-r(k)$, the number of steps left to the flow, is a fixed fraction, less than one ninth, of $k$, rounded down. Throughout, $p(\delta)=b_0(1+\log\delta^{-1})^p$ at the fixed integer $p$ of Convention~\ref{cpi:rot:convP} (the function $p(\cdot)$ and the integer $p$ are told apart by the argument). The \emph{large-field constant} is a constant $c_*>0$: $c_*=c_0'/2$ at zero source, $c_0'$ the constant of Ba{\l}aban's large-field bound~\cite[(3.41)]{Ba2} at block factor $5$ (at most $\min\{a/8,\gamma_*/4,1\}$), any $c_*0$: $p(\delta_{r(k)})^2$ grows at least quadratically in $k$). \end{definition} \begin{theorem}[(BAL)$_5$: Ba{\l}aban's estimates at blocking factor five]\label{rot:thm:BAL5} Every estimate used (Definition~\ref{r13:def:BAL5}) holds as stated at blocking factor $5$, with per-step constants that depend on the step only through the running scale (scale-local), hence uniformly in the initial spacing, including the large-field clause of King's upper bound~\cite[(3.4)]{K1} on the skew torus: the large-field contribution to the partition function at the stopping depth $r(k)$ of the flow, $k\ge k_0$ (and at every admissible depth for the standard flow on $D_J$), is at most $\exp(-c_*p(\delta_{r(k)})^2+C|T|+o(K^2))$ (Remark~\ref{cpi:rot:errata}(i)), with the large-field constant $c_*>0$ of Definition~\ref{r13:def:BAL5}. In particular the following hold. \begin{enumerate} \item[(a)] (\emph{Vector-field propagator bounds on the periodic torus}, \cite[(2.10)--(2.12), p.~426]{Ba3}.) The torus kernels of Ba{\l}aban's vector propagators obey the pointwise bound $Cl_j^{-1}e^{-\delta_1|x-x'|_T/l_j}$, the bounds with one and two differences ($l_j^{-1}$ per difference) and the H\"older clause in the quotient metric; a leg integrated along a contour is bounded by the sum of the pointwise bounds along it. \item[(b)] (\cite[p.~437]{Ba3}.) The massive nearest-neighbour Green's function and its first differences decay exponentially, at a rate $c_{\rm G}>0$ depending only on the mass and the dimension (its value is not used). \item[(c)] (\emph{The local term and the first-moment step}, \cite[(3.33)--(3.38), (3.26)--(3.28)]{Ba3}.) The local term of the primitively divergent classes with two scalar legs and one vector leg is bounded, window by window, and its cube sum converges at every window; and with the wall in the covariance, at $A=0$, on any rectangular torus and at zero shell mass, the first moment $\Pi[G](y)$ of the vacuum-polarisation bubble satisfies $\sup_y|\Pi[G](y)|\le C\operatorname{Tr}q^2$ uniformly in the spacing, the torus and the wall. \item[(d)] (\emph{The integral over the vector fields in the completion of}~\cite[\S3C, p.~591]{Ba2}.) Given the scalar half of~\cite[(3.39)]{Ba2}, the vector half of the passage to~\cite[(3.40)]{Ba2} closes without vector characteristic functions, with $\gamma_*=\min(\gamma_0^s,\gamma_v)$ in~\cite[(3.41)]{Ba2}, at the cost $(d/2)\ln2$ per site, the Gaussian renormalisation cost, and with no remainder; the constants of the completion of~\cite[\S3C]{Ba2} at block factor $5$ are explicit, and the bounds that~\cite{Ba2} writes as $O(1)$ without qualification are uniform in the scale index. The count~\cite[(3.45)]{Ba2} and the first inequality of~\cite[(3.46)]{Ba2} are false as stated in~\cite{Ba2}, at every block factor, and are not used: a corrected count of the cubes replaces them. \item[(e)] (\emph{Regularity of the gauge background}, \cite[Prop.~2.1]{Ba1}, \cite[(1.7)]{Ba4}, \cite[Lemma~2.3]{Ba2}.) On the propagator domain the gauge background is regular at block factor $5$ (Proposition~\ref{cpi:asa:B5Omega}), and the uncut background is regular within one block of it: one block beyond the propagator domain $|\partial^\eta_\mu A^{(k)}_\nu|\le C_Zp_k$ with $C_Z=C_1c_1(1+\log5)^p+1$, so that King's regularity~\cite[(3.5)]{K1} holds at scale $k$ on the margin with a constant free of $k$, $\eps$, $\Omega$ and $\kappa$. \item[(f)] (\emph{Carrying the scalar block-field bound to the next step}, \cite{Ba2}.) The cap is inherited, and the gradient line $|D_{\bar A^{(k)}}\phi_k(b)|\le c_1^{\rm sc}p_{k-1}$ holds on the unit-$k$ bonds $b$ of $\Lambda_2^{(k-1)\prime}$ at $k\ge2$, with $c_1^{\rm sc}=3d\cdot5^{3/2}+5C'_\Delta e(1+\lambda^{-1/4})\kappa_*$, $\kappa_*=\sup_{\ell\le1}\ell^{1/4}p(\ell)$, scale-free. \end{enumerate} \end{theorem} \begin{proof} The proofs of~\cite{Ba1,Ba2,Ba3,Ba4} are repeated at block factor $5$, each scale treated as the first in its running spacing (whence scale-local constants), with the four steps written out: the cube step of~\cite[p.~433]{Ba3} by the end-functional bound of the shell-model repetition at zero shell mass (Theorem~\ref{cpi:swab:B5K}, resting on clauses (c), (e), (f) of this theorem; this is the completion of~\cite[pp.~433--445]{Ba3}), after which the bounds hold on the torus and on every propagator domain with domain-independent constants; the cut-off $\zeta^{(k)}$ at $\theta=2/5$, where Proposition~\ref{cpi:asa:B5Omega} gives the regularity of the background; the boxes of~\cite[Lemma~2.4]{Ba2} and the cube partition with the corrected cube count of (d) (the completion of~\cite[\S3C]{Ba2}); and the identification step, in which the per-step determinants compose exactly, by~\cite[(2.105)]{Ba2}, into King's matter determinant, bounded through the first-order clause of Theorem~\ref{cpi:swab:DOM}, King's action at Ba{\l}aban's cut and at the uncut background differing on the small-field set by at most $Ct_k^s|T^{(k)}|$ (Proposition~\ref{cpi:prop:TD}(b)). Clause by clause: (a) is periodisation of the infinite-lattice kernels of~\cite{Ba1,Ba2}; (b) is Combes--Thomas; (c) follows from the $\ell^1$ row sum of the window resolvent and the $\ell^2$ bound of the photon line, from Theorem~\ref{cpi:thm:Kpt} with the Ward identities (Lemma~\ref{cpi:lem:ward}(b), (c)) for the cube sum, and from the reflection $\Pi[G]=-\Pi[G^I]$ up to the reflected kernel and the two wrap-around planes; (d) follows from (a), Proposition~\ref{cpi:asa:B5Omega} and the completions named in (d)--(f); (e) is Proposition~\ref{cpi:asa:B5Omega} on the propagator domain and, one block beyond, the McShane extension of the restricted field with~\cite[Lemma~2.3]{Ba2} at $\Lambda_0=T$ through the three-layer containment~\cite[(2.7)--(2.8)]{Ba2}; (f) compares the transport $U(e_ka)$ along the bond with Ba{\l}aban's line $\bar A^{(k)}_b$. In the large-field clause the bound on the free massive lattice propagator and its first three differences, uniform in the spacing and the mass, is standard, H\"older's inequality separates the wall term (whence $c_*=(c_0'/2)(1-1/p_H)$), the propagators of~\cite[Prop.~2.6]{Ba2} are replaced on the skew tori through Theorem~\ref{r13:thm:BAL5compl}(k), the forms for the contour families of the flows and at the walls follow from Propositions~\ref{rot:prop:CR} and~\ref{rot:thm:POS} outside $E_W$ under the final-scale cap of the class $\mathcal C$ (Proposition~\ref{rot:prop:CR} also through (N4) and the comparison of the effective actions), and Ba{\l}aban's bulk estimate~\cite[(2.113)]{Ba2} on King's cube with the Dirichlet weight acquires a face correction and a first-scale lift form by the proof of~\cite[Lemma~2.7]{Ba2} run from the exact identity $aL^{-2}(C_\Lambda f)-f=-C_\Lambda\Lambda\Delta^{(k)}\Lambda f$. \end{proof} \begin{theorem}[(BAL)$_5$, continued: the completions on the skew tori, at the wall and across domains, and the propagator theorem of~\cite{Ba4}]\label{r13:thm:BAL5compl} The following hold at blocking factor $5$, with per-step constants that are scale-local as in Theorem~\ref{rot:thm:BAL5}, whose lettering continues. \begin{enumerate} \item[(g)] (\cite[p.~412]{Ba3}.) The coarse-covariance decay bound holds in product form. \item[(h)] Ba{\l}aban's steps behind the estimates of Theorem~\ref{rot:thm:BAL5} and of this theorem and behind the kernel bounds of Theorem~\ref{cpi:hyb:BD} hold on the skew tori and in the boundary boxes. \item[(i)] The H\"older clause of the shell estimates (Theorem~\ref{cpi:thm:main}) and of the renormalisation bound for the hybrid flow (Proposition~\ref{cpi:ct:H4}) holds. \item[(j)] (\emph{The renormalisation bound with the scalar wall}, \cite[Prop.~1, (1.33)]{Ba3}.) Ba{\l}aban's renormalisation bound holds for the operator with the scalar wall $0\le v\le\bar v$, with constants polynomial in $\bar v$. \item[(k)] (\emph{The change of domain}, \cite[Prop.~2.6]{Ba2}.) On the skew tori the change of domain of Ba{\l}aban's propagators between nested domains decays with the depth of its two points, exponentially in both depths. \item[(l)]\label{r13:thm:Ba4prop} (\emph{The propagator theorem of~\cite{Ba4}}.) The Theorem of~\cite{Ba4}, (1.9)--(1.12) there (equivalently \cite[Prop.~2.1]{Ba1} and \cite[Thm.~3.3]{K1}), holds at $L=5$, for every $\tilde a\in[a(1-L^{-2}),a]$ and every constant mass $m^2\ge0$, in its restricted scope: at points at distance at least $R_0$ from the complement of the domain, with no restriction on a torus, and with $\delta_0$, $c_0^{\rm Ba}$, $R_0$ independent of $A$, $k$ and the domain (the unrestricted clause of~\cite{Ba4} for rectangular parallelepipeds at a general regular $A$ is neither claimed nor used). The three steps that~\cite{Ba4} states without argument hold: (i) the renormalisation-group identity~\cite[(2.34)]{Ba4}, which is~\cite[(2.43)]{Ba1}, \[G_k(\Box)=C^{(0),\eta}(\Box)+\sum_{j=1}^{k-1}a_j^2(L^j\eta)^{-4}\,G^\eta_jQ_j^*C^{(j),L^j\eta}Q_jG^\eta_j,\] the iterate of the one-step identity $G_{j+1}=G_j+a_j^2G_jQ_j^*C^{(j)}Q_jG_j$ with $C^{(j)}=(\Delta^{(j)}+aL^{-2}P)^{-1}$ and $\Delta^{(j)}=a_j-a_j^2Q_jG_jQ_j^*$, holds exactly on every domain $\Omega$ (Neumann conditions preserved at all scales), for every regular $A$ and every constant $m^2\ge0$, in any normalisation in which $P_k=Q_k^*Q_k$ and $P=Q^*Q$ are projections and $Q_{k+1}=QQ_k$; (ii) the analyticity neighbourhood in~\cite[Lemma~2.4]{Ba4} (p.~586) is uniform in $j$: the symbol of the block-averaged propagator is analytic and at least $\frac12a_*(2/\pi)^{2d}$ in modulus on a strip of width $\kappa_*$ independent of $j$ and $L$, which gives the decay $e^{-\kappa_*|x-y|}$ and the H\"older clause of~\cite[(2.35)--(2.36)]{Ba4} with $\delta_0=\kappa_*$; (iii) the gap ``$\pi^2$'' of~\cite[(2.27)]{Ba4} is false for a finite block, the lattice Neumann gap being $4N^2\sin^2(\pi/2N)\in[8,\pi^2)$ for a block of $N$ points per side, and the lower bound $4$ suffices for every use. With these three steps the proof of~\cite[\S\S2--3]{Ba4} runs as written, its constants depending on $d$, $a$, $M$, $L$ ($c_0^{\rm Ba}$ also on the H\"older exponent $\alpha$) and not on $A$, $k$ or $\Omega$; for $\tilde a$ off Ba{\l}aban's orbit of coefficients, the bare value $a'=\tilde a(1-L^{-2k})/(1-L^{-2})$ has $a'_k=\tilde a$ with all $a'_j\in[0.9216\,a,1.0417\,a]$. \end{enumerate} \end{theorem} \begin{proof} (g)--(i) are the completions of Ba{\l}aban's steps on the skew tori and in the boundary boxes, with the H\"older constant kept explicit; (j) holds since the wall enters~\cite[(1.33)]{Ba3} only through the scalar propagators, whose bounds improve under a non-negative potential, and the walled counterterms; (k) is the resolvent identity through the boundary of the inner domain with Ba{\l}aban's Green-function and block-operator decay. In (l) the renormalisation-group identity is the Woodbury identity for $H_{k+1}=H_k-Q_k^*WQ_k$, $W=a_k-a_{k+1}L^{-2}P$, the $j$-uniformity is Paley--Wiener and the gap is the explicit lattice Neumann eigenvalue; the bounds of~\cite[\S\S2--4]{Ba4} pass to the admissible lattice tori and to unions of large blocks by the random-walk transfer, only the restricted clause holding within $R_0$ of the complement. \end{proof} \begin{proposition}[the regular background on the propagator domain at block factor five]\label{cpi:asa:B5Omega} Let $L=5$, $r=2$, let the localisation radius satisfy $R\ge\max\{5(2M+1),C_Z(M+s)\}$ with $C_Z$ an absolute constant and $R\ge10(M+1)$, and let Ba{\l}aban's smoothing cut-off $\zeta_\theta$ of \cite[(2.44)]{Ba2} be taken at $\theta=2/5$ (the cut-off condition (Z$_\theta$)). At every step $k$ of the upper bound, on the propagator domain $\Omega=B^k(\Lambda_2^{(k-1)\prime})$ of that scale, the gauge background obeys \cite[Lemma~2.3, (2.59)--(2.60)]{Ba2} with $O(p(L^k\eps))$ replaced by $O_L(p(L^k\eps))$: for $x\in B^k(y)$, $y\in\Lambda_2^{(k-1)\prime}$, \[A^{(k)}(x)=A(y)+O_L(p(L^k\eps))=(Q_k^*A)(x)+O_L(p(L^k\eps)),\quad(\partial^\eta_\mu A^{(k)})(x)=O_L(p(L^k\eps))\quad(x\in\Omega),\] where the constant contains the factor $(1+\ln L)^{p}=315.7$ ($p=6$) and otherwise only the constants of \cite[(2.55)]{Ba2} and of \cite[Prop.~2.2]{Ba2}. The cut field $\theta_kA^{(k)}$ obeys the same bounds on $\Omega$, since $\theta_k=1$ there, together with their H\"older clause. Consequently the configuration $A^{(k)}$ on $\Omega$ satisfies the regularity assumption of \cite[Prop.~2.1]{Ba1}, which is what the propagator bounds of the step require. On the tori of the construction the same holds in the quotient metric. \end{proposition} \begin{proof} For $y\in\Lambda_2^{(k-1)\prime}$ and $x\in B^k(y)$ the $\zeta_\theta$-ball lies in the rescaled $\Lambda_{-1}^{(k-1)\prime}$, where the restrictions~\cite[(2.55)]{Ba2} hold (this needs $\theta<3/L$), and there the argument of~\cite[Lemma~2.3]{Ba2} runs with $\zeta_\theta$, the previous scale's restrictions carrying $p(t_{k-1})\le(1+\ln L)^pp(t_k)$, the origin of the constant. The split~\cite[(2.89)]{Ba2} is exact with $\zeta^{(k+1)}_\theta$ only if $L\theta\le2$, so $\theta=2/5$ is the largest admissible value at $L=5$, and the lower bounds on $R$ give the transition width $\frac12\theta r(t_k)-2M\ge2$; with the cut-off as written in~\cite{Ba2} the lemma fails at $L=5$ (the restrictions reach only depth $3r(t_{k-1})/L-2$, which exceeds $r(t_k)-2M$ at every scale only if $L\le3$), and the repair costs $(1+\ln L)^p$, absorbed into the class constants fixed before $M$. \end{proof} \subsection{The convention bridge (CT)} (CT) (Remark~\ref{rot:rem:CT}) bridges the torus counterterm conventions of~\cite{K1} and the free-boundary conventions of~\cite[p.~330]{K2}: (a) exactly, the torus model is the free-boundary model with the mass term shifted by $\frac12\Delta_T\sum s^3|\phi|^2$ and multiplied by $e^{-\Delta E_T|T|-N\Delta c_T\langle h\rangle}$; (b) the bound clause, Theorem~\ref{cpi:ct:CTB}, the decay of the difference terms, carries King's upper bound and the hybrid lower bound from the torus-convention model to the free-boundary model. The bridge constants enter through a single bound $\Delta_*$ on $|\Delta_T(s)|$ for $s\le s_0$ (also the constant of the wall condition of the class $\mathcal C$, and all that (CT)$_{\rm sh}$, Lemma~\ref{cpi:swab:M123}, uses on the shell side) and a bound on the shift $\Delta c_T(s)$ of the Wick constant for $s\le s_0$; clause (c) of the theorem is used by (N3) and (N4), the rate clause (d) only through (N6). In the effective action at level $k$ the mass shift generates the \emph{difference terms} $D^F_k$: the renormalised families of King's expansion (King order at most $13$) with $j\ge1$ vertices $u^{(\alpha,\beta)}$ carrying the order-$(\alpha,\beta)$ part of $\Delta_T$, $|u^{(\alpha,\beta)}|\le u_0\defeq\sup_{s\le1}|\Delta_T(s)|/2$; counted with the $u$-vertices, their free-boundary order $\sum_it_i+n_h+n_\lambda+m_\Sigma$ is at least $14$, where $m_\Sigma\defeq\sum_vm_v$ is the sum of the photon orders $m_v$ of the charge vertices. \begin{theorem}[the bound clause of the convention bridge]\label{cpi:ct:CTB} For $K\ge K_0$, $k=k(K)$, both flows $F\in\{X,Y\}$, both tori $T\in\{D_J,RD_J\}$, every configuration in $\supp\chi_k$ and every $|u^{(\alpha,\beta)}|\le u_0$, \[\sup_{\supp\chi_k}|D^F_k|\ \le\ C_\sigma\,\delta_k^{\sigma}\,|T|\qquad\text{for every }\sigma<1,\] two-sided, the field-free ($A_k$-dependent, $\phi_k$-free) part included; $C_\sigma$ is independent of $K$, $T$ and the configuration, and uniform in the source through $u_0$ and $S_0$. In particular the upper bound (N3) and the hybrid lower bound (N4) below, proved for the torus-convention model, hold for the free-boundary model, the forward bound of Section~\ref{rot:ssec:CTproof} tolerating any fixed $\sigma>0$ and any polynomial prefactor in $K$. In addition: \begin{enumerate} \item[(c)] (\emph{Uniformity under a constant source shift.}) The constant of King's upper bound~\cite[Thm.~3.1]{K1} is uniform under a shift of the scalar source by a bounded constant (King's statement that the constant depends on $g$, $h$ and $|T|$), and King's statement that the small-field analysis includes the sources~\cite[p.~677]{K1} holds in the form used: the small-field half of~\cite[Thm.~3.1]{K1}, in King's vertex form~\cite[(3.14)--(3.21)]{K1}, holds at every fixed source of bounded size, on King's aligned tori and on the rotated torus, for King's flow and for the centred flow, and for the hybrid flow in its lower half, with a cap depending on the number of scales. \item[(d)] (\emph{The rate.}) $|\Delta_T(s)-\Delta_T(0)|\le Cs$ for spacings $s\le s_0$, where $\Delta_T(s)$ is the constant of the convention bridge at spacing $s$ and $\Delta_T(0)$ its limit; the rate is read off the counterterm differences $\Delta_T$, $\Delta E_T$, $\Delta c_T$ of the two conventions. \end{enumerate} \end{theorem} \begin{proof} In the unit scale of level $k$ a connected graph of the difference terms with $j\ge1$ vertices $u$, $n_h$ source vertices, $n_\lambda$ quartic vertices, charge vertices of total photon order $m_\Sigma$ and $s$ external scalar legs carries $\delta_k^{P}$, $P=2j+2n_h+n_\lambda+m_\Sigma/2$ (\cite[p.~657]{K1}, \cite[(1.33)]{Ba3}), the position sums against the external legs cost $\delta_k^{-1/4}$ each and the volume $\delta_k^{-3}|T|$, and connectivity with the order bound $\sum_it_i+n_h+n_\lambda+m_\Sigma\ge14$ gives $Q=P-s/4-3\ge j\ge1$; the logarithms go into $\delta_k^\sigma$, the uniformity in $K$, $T$ and the source is that of King's small-field bounds given $u_0$ and $S_0$, the position sum of the families with external legs is the renormalisation bound~\cite[(1.33)]{Ba3} at depth $k$ (slice lines through Theorem~\ref{cpi:thm:78}), and the field-free families of the hybrid flow need a vacuum localisation with one $u$-vertex held fixed, which gives $C(1+K)^p\delta_k^4|T|$, the form that (N4) needs. Clause (c) is the block-factor-five repetition of the proof of~\cite[Thm.~3.1]{K1} with Theorems~\ref{rot:thm:BAL5} and~\ref{r13:thm:BAL5compl} and King's statements~\cite[Thm.~3.3, Props.~3.7--3.10, Lemma~4.5]{K1}, the small-field half in King's vertex form absorbing the source at every scale. Clause (d) is the Lipschitz dependence on the spacing $s$ of the constant of the convention bridge, read off the counterterm differences of the two conventions by Reisz's power counting for the Wick-ordered torus one-point graphs through order $13$, uniformly in the scalar mass on a compact interval of King's range, only the coefficient of order $e^2\lambda^0$ being logarithmic. \end{proof} \subsection{(N1) The two-flow comparison bound} (N1) is the comparison of the values on the standard and on the hybrid flow of the renormalised families of~\cite[\S3.5]{K1} with lines taken from the two flows, in per-flow normal form, Theorem~\ref{cpi:ct:H1S}; its one-flow instance is King's, in the proof of~\cite[Prop.~3.6]{K1}, p.~665 (Theorem~\ref{cpi:thm:78}). \begin{theorem}[the two-flow comparison bound]\label{cpi:ct:H1S} Let $\alpha\in(0,1/\bar n)$ be a H\"older exponent for which the kernel bounds of Theorem~\ref{cpi:hyb:BD}(ii) hold (any exponent smaller than King's serves, the H\"older clause at a given exponent implying it at every smaller one on pairs at distance at most $1$), and put \[\theta_1\defeq\frac{\alpha^2}{4(1+\alpha)^2}.\] There are $q_1=q_1(\bar n)$, $\beta'=\beta'(\bar n)\ge0$ and $C$ such that for every renormalised family $\mathcal R$ of order at most $\bar n$ in per-flow normal form that the comparison meets, that is, for the three kinds of family that occur in it (the $h$-linear families other than the field-independent ones, the families of King's constant-background reduction at $e>0$, and the vacuum families in their $C/\Gamma$ form with the $\Gamma$-lines as legs and recipients), for $0<\theta\le\theta_1$, uniformly in $k$, $K$ and the torus, \[|E^X(\mathcal R)-E^Y(\mathcal R)|\ \le\ C\,n_{\mathcal R}\,p_k^2\,\delta_k^{-1/2}\,\eta'^{\theta}\,(1+K)^{q_1}\,\delta_k^{-\beta'}\cdot\big[\text{the right side of \cite[(1.33)]{Ba3} for }\mathcal R\big],\] with $\beta'=0$ for the $h$-families and the families of the constant-background reduction, and $\beta'\le\beta_G\le3\cdot(\text{number of lines fixed at the top scale})$ for the vacuum families, whose constants depend on the masses through $m_0^{-\beta_G}$, $m_0\defeq\min(m,\mu_0)$; the other constants are uniform in the masses. $\theta_1$ does not depend on $\bar n$, and any $\theta>0$ with any polynomial factor suffices for the applications below. \end{theorem} \begin{proof} Fix an ordering of the scale-summed lines and telescope the difference of the two flows element by element, each insertion bundled with the earliest of its legs; a restricted subtraction rule removes from every connected component of degree $\le0$ formed in an initial segment its common zero- and first-order moments, which agree for the two flows by isotropy and the zero-momentum counterterm conditions~\cite[(3.32)--(3.33)]{K1} (for the vacuum polarisation by the Ward identities and~\cite[(2.18)--(2.28), Prop.~2.1]{Ba3}), and a census by exact integer arithmetic over the graphs of order at most $\bar n$ finds no component of non-positive degree that the rule does not handle. The difference elements carry the positive power of the scale ratio of Theorem~\ref{cpi:hyb:BD}(iii), the scale sums give the gain $\eta'^{\theta_1}$ with the margin $\min(\alpha,1)-\theta_1>0$, a switched contour element costs $p_k^2\delta_k^{-1/2}$ (\cite[(3.46)]{K1}), the sums over slices and localisations the polynomial factor, and the top-scale lines of the vacuum families $(m_0\delta_k)^{-\beta_G}$. The remaining estimates use Theorem~\ref{cpi:hyb:BD}(ii), the census of the zero-degree all-$C$ subgraphs and the local limit law for the simple random walk~\cite[Chap.~2]{LL10}. \end{proof} \subsection{(N2) Decay and regularity of the block-spin operators} (N2) is the decay and regularity of the block-spin operators, \cite[Thm.~3.3]{K1}. For the standard flow it is proved in~\cite[Lemmas~2.1, 2.2, 2.4, Cor.~2.3]{Ba4} up to the three steps claimed there without argument, the coefficients $a_k\in[a(1-5^{-2}),a]$, the uniformity in a constant $m^2\ge0$ and the transfer to the admissible lattice tori (Theorem~\ref{r13:thm:BAL5compl}(l), with the random-walk transfer named in its proof), and up to the photon clause of~\cite[Thm.~3.3]{K1} (p.~656, ``of course with $A=0$''); King's steps of the torus construction stated without a complete argument enter by citation of~\cite[Thm.~3.1, Thm.~3.3, Prop.~3.7 (within its stated scope, the $K$-slice treated separately), Prop.~3.10, Lemma~4.5]{K1}, King's conventions for the sources (pp.~664--665, 677) are Theorem~\ref{cpi:thm:78} and Theorem~\ref{cpi:ct:CTB}(c), and the coercivity bound (4.33) behind~\cite[Lemma~4.5]{K1} is proved in Section~\ref{rot:ssec:errata}, item (iv). For the hybrid flow (N2) is Theorem~\ref{cpi:hyb:BD} (periodic form, two-torus clause in the smaller quotient metric; the Neumann variant on digitised rotated domains is neither claimed nor used), whose first step averages over rotated blocks, a block-spin step by Lemma~\ref{cpi:ct:tile}. \subsection{The hybrid kernels at a regular background} Fix $K$ and $2\le k\le K$ and work in the unit scale of level $k$: the physical spacing $\delta_k=5^k\eps_K$ is the unit, the fine spacing is $\eta':=5^{-k}$, $\eta:=5\eta'$ and $\ell_j:=5^j\eta'$ (so $\ell_k=1$). With $R$ the rotation of the frame and $f_\alpha=Re_\alpha$, the coarse lattices are $\Gamma_j:=R(\ell_j\Z^3)$, $0\le j\le k$; $\Lambda_Y:=\Gamma_0=R\eta'\Z^3$ is the fine lattice of the standard flow $Y$ and $\Lambda_X:=\eta'\Z^3$ that of the hybrid flow $X$; since $5R\in\Z^{3\times3}$, $\Gamma_1\subset\Lambda_X\cap\Lambda_Y$. For $z\in\Gamma_j$ the \emph{level-$j$ block} of the flow $F\in\{X,Y\}$ is $B^F_j(z):=\Lambda_F\cap(z+R(-\frac{\ell_j}2,\frac{\ell_j}2)^3)$, a cube of $5^{3j}$ points for $F=Y$ and the digitised rotated cube for $F=X$ (no lattice point lies on its boundary; $B^X_1(z)$ is the tile); blocks are nested and \emph{centred}, the root of every block being its centre, and for $x\in B^F_k(z)$ we write $x_j$ for the centre of the level-$j$ block containing $x$ ($x_k=z$). The matching $\tau\colon\Lambda_X\to\Lambda_Y$ maps the tile $B^X_1(x)$ onto the cube $B^Y_1(x)$ for every $x\in\Gamma_1$, with $|\tau x'-x'|\le c_\tau\eta'$, $c_\tau=\sqrt{13}+2\sqrt3<7.1$. \begin{definition}[the two contour systems]\label{cpi:hyb:cont} (Y) For $x\in B^Y_k(z)$ the \emph{composite contour} is King's \cite[(2.12)]{K1}, $\Gamma^{Y,(k)}_{z,x}=\Gamma^Y_{z,x_{k-1}}\cup\Gamma^Y_{x_{k-1},x_{k-2}}\cup\dots\cup\Gamma^Y_{x_1,x}$, where $\Gamma^Y_{u,v}$ is the L-path from $u$ to $v$ that changes the $f_3$-coordinate first, then $f_2$, then $f_1$, each leg a straight run of fine $\Lambda_Y$-bonds. (X) For $x'\in B^X_k(z)$ the \emph{composite contour} is $\Gamma^{X,(k)}_{z,x'}=\Gamma^X_{z,x_{k-1}}\cup\dots\cup\Gamma^X_{x_2,x_1}\cup\Gamma^X_{x_1,x'}$, where for $i\ge1$ the piece $\Gamma^X_{x_{i+1},x_i}$ is the L-path $\Gamma^Y_{x_{i+1},x_i}$, whose legs are concatenations of $\eta$-bonds $(u,u\pm\eta f_\alpha)$ with $u\in\Gamma_1$, in which every $\eta$-bond $(u,u+\eta f_\alpha)$ is replaced by the \emph{fixed} fine axis staircase $u+\eta'\gamma_\alpha$ (and $(u,u-\eta f_\alpha)$ by its reverse), with \begin{gather*} \gamma_1:\ e_1,e_2,e_1,e_2,e_1,e_2,e_1\ (\text{from }0\text{ to }(4,3,0)),\\ \gamma_2:\ e_2,-e_1,e_2,-e_1,e_2,-e_1,e_2\ (\text{to }(-3,4,0)),\qquad \gamma_3:\ e_3^{\,5}\ (\text{to }(0,0,5)), \end{gather*} and $\Gamma^X_{x_1,x'}$ is a fine axis path inside the tile $B^X_1(x_1)$ of at most $5$ bonds, chosen once as a function of $x'-x_1$ only, so that the block and contour system is invariant under the translations of $\Gamma_1$, in particular under the periods of the tori. The gauge field along a composite contour is read in King's convention \cite[(2.19)]{K1}: the stage-$j$ fluctuation is integrated along $\Gamma^{(j+1)}_{x_{j+1},x}$ only, the lifted coarse background along the whole contour. \end{definition} The first step of the hybrid flow averages over the \emph{tiles} $B(y)\defeq y+\eps B_0$, $y\in Y_{K-1}$, where $B_0\defeq\Z^3\cap R(-\tfrac52,\tfrac52)^3$ is the set of integer points of the open rotated cube of side $5$; the frame of $Y_K$ is $f_\alpha\defeq Re_\alpha$. The block averages of the step are $(\hat QA)_\alpha(y)\defeq\frac1{125}\sum_{x\in B(y)}f_\alpha\cdot A(x)$ on bond fields and $(Q(A)\phi)(y)\defeq\frac1{125}\sum_{x\in B(y)}U(A(\Gamma_{yx}))\phi(x)$ on scalar fields, $\Gamma_{yx}$ the first-step contour from $y$ to $x$ and $U$ the parallel transport along it, $|U|=1$; the pairings are $\langle u,v\rangle_\Lambda\defeq\sum_x s^3u(x)\cdot v(x)$ on a lattice of spacing $s$. \begin{lemma}[the tile average is a block-spin average]\label{cpi:ct:tile} \begin{enumerate} \item[(i)] $B_0$ is a complete residue system of $\Z^3/5R\Z^3$ with $|B_0|=125$; the tiles $B(y)$, $y\in Y_{K-1}$, partition $X_K$, and for each of the period lattices $P\in\{P_J,RP_J\}$, which satisfy $P\subset Y_{K-1}$, the tiles $B(y)$, $y\in Y_{K-1}/P$, partition $X_K/P$. \item[(ii)] $B_0=-B_0$, $\max_{x\in B_0}|x|^2=13$, $|x|_\infty\le3$ on $B_0$, and every $x\in B_0$ is joined to $0$ by a nearest-neighbour path inside $B_0$, monotone in each coordinate, of length $|x|_1\le5$. So a contour inside the tile, of at most $5$ fine bonds, exists for every point of the tile; it is the first-step contour $\Gamma_{yx}$. \item[(iii)] $\hat Q\hat Q^*=\mathbf 1$ on coarse bond fields and $Q(A)Q(A)^*=\mathbf 1$ on coarse scalar fields, for every bond field $A$. Hence $\hat Q^*\hat Q$ and $Q(A)^*Q(A)$ are orthogonal projections, the Gaussian kernels $\exp[-\frac{a_{\mathrm{bs}}}2\|A_1-\hat QA\|^2]$ and $\exp[-\frac{a_{\mathrm{bs}}}2\|\phi_1-Q(A)\phi\|^2]$, $a_{\mathrm{bs}}>0$ the constant of King's block-spin kernels (the subscript keeps it apart from the scheme $a=(a_k)$ of Definition~\ref{def:s02:sources}), have the normalisations of \cite[(2.10)--(2.14)]{K1}, and the first hybrid step is King's renormalisation transformation with $B(y)$ in place of the cube. \item[(iv)] The composite of the first step with the $j-1$ standard steps that follow it is the equal-weight average over the composite blocks $B^{(j)}(y)$, $y\in Y_{K-j}$, unions of $125^{j-1}$ tiles, which partition $X_K$ and have diameter at most $C5^j\eps$; the composite Gaussian kernel has King's coefficient $a_{\mathrm{bs},j}$ (\cite[p.~653]{K1}), as for the standard flow. \end{enumerate} \end{lemma} \begin{proof} (i): the open rotated cube is a fundamental domain of $5R\Z^3$ whose boundary planes carry no integer point, so $B_0$ is a complete residue system with $|B_0|=|\det5R|=125$, and the partition descends to $X_K/P$ since $P\subset Y_{K-1}$; (ii) is a finite check over the $125$ points. (iii): with respect to the pairings $(\hat Q^*\psi)(x)=\sum_\alpha\psi_\alpha(y(x))f_\alpha$ and $(Q(A)^*\chi)(x)=\overline{U(A(\Gamma_{y(x)x}))}\chi(y(x))$, since $(5\eps)^3/125=\eps^3$, so $f_\alpha\cdot f_\beta=\delta_{\alpha\beta}$ and $U\bar U=1$ give the identities, and the Gaussian integrals over $A_1$, $\phi_1$ are then field-independent constants, all that~\cite[(2.10)--(2.14)]{K1} use of the block. (iv): equal-weight averages over successive partitions compose, and the coefficient is King's computation~\cite[p.~653]{K1}, which uses only the form of the kernels. \end{proof} A \emph{background} is a real bond field $A$ on $\Lambda_F$, $U(A(\Gamma))=\exp[ie_kA(\Gamma)]$ with $A(\Gamma)=\sum_{b\in\Gamma}\eta'A_b$ and the running charge $e_k=e\,\delta_k^{1/2}$; it is \emph{regular} if $|\partial_\mu A_\nu|\le Ce_k^{\beta-1}$ for some $\beta>0$ \cite[Def.~3.2]{K1}. The pinned operators of the flow $F$ are \begin{gather*} Q_F(A)\phi(z):=\sum_{x\in B^F_k(z)}\eta'^3\,U(A(\Gamma^{F,(k)}_{z,x}))^{-1}\phi(x),\\ G^F_k(A):=\big(-\Delta_A+m_k^2+a_{\mathrm{bs},k}Q_F(A)^*Q_F(A)\big)^{-1}, \end{gather*} $m_k=m\delta_k$, $a_{\mathrm{bs},k}=a_{\mathrm{bs}}(1-5^{-2})(1-5^{-2k})^{-1}$ ($a_{\mathrm{bs}}$ the constant of King's block-spin kernels, as in Lemma~\ref{cpi:ct:tile}), and with $G_A:=(-\Delta_A+m_k^2)^{-1}$ we put $A^F(A):=(a_{\mathrm{bs},k}^{-1}+Q_FC_AQ_F^*)^{-1}$ (King's effective Laplacian $\Delta^{(k)}(A)$), $B^F:=(A^F)^{-1}$, $H^F:=a_{\mathrm{bs},k}G^F_kQ_F^*$ (the lift operator), $\Gamma^F:=G_A-G^F_k=H^FB^FH^{F*}$, $P(A):=Q^{(1)}(A)^*Q^{(1)}(A)$ the pin of the next step and $C^{(k),F}(A)=(A^F(A)+a_{\mathrm{bs}}L^{-2}P(A))^{-1}$ the next fluctuation covariance; these identities are exact for every $A$ by the Woodbury identity. The photon kernels are three copies of the scalar kernels at $A=0$ and mass $\mu_0\delta_k$: no photon kernel sees a background. A \emph{comparable pair} consists of two tori, or of a torus and the infinite lattice, both fitted by both lattices, of side at least $10L$ in unit scale, compared on a common fundamental region $F$ that is a union of level-$(k+1)$ blocks, with boundary $\partial$ and the same background on it; $d_\wedge$ denotes the smaller of the two quotient metrics. For a subset $\Lambda$ of the coarse lattice we put $\hat C^{(k)}_\Lambda:=((A^X(A)+a_{\mathrm{bs}}L^{-2}P(A))|_\Lambda)^{-1}$. \begin{theorem}[the hybrid block-spin kernels at a regular background]\label{cpi:hyb:BD} Let $k\ge2$. \begin{enumerate} \item[(i)] \emph{(Pairing; block-spin algebra.)} The hybrid composite contours lie inside their level-$k$ blocks, every level-$i$ piece inside its level-$(i+1)$ block, and have length at most $2.1$; for every background $A$, $Q_X(A)Q_X(A)^*=\mathbf 1$, $Q_X(A)^*Q_X(A)$ is an orthogonal projection, gauge covariance holds, and $Q_X(A)=Q^{\mathrm{rot}}_{k-1}(A)\circ\hat Q(A)$. The contours are \emph{rotation-paired}: for $z\in\Gamma_k$, $x'\in B^X_k(z)$, $x''=\tau x'$, the Hausdorff distance between $\Gamma^X_{x_{i+1},x_i}$ and $\Gamma^Y_{x_{i+1},x_i}$ is at most $\eta'$ for $i\ge1$ and at most $2\sqrt{13}\,\eta'$ for $i=0$, and the closed polygon $\Gamma^{X,(k)}_{z,x'}\cup[x',x'']\cup(\Gamma^{Y,(k)}_{z,x''})^{-1}$ bounds a piecewise planar surface $\Sigma(z,x')$ with $\operatorname{area}(\Sigma)\le1.5\,\eta'+33\,\eta'^2$. The roots of the two flows are matched. \item[(ii)] \emph{(Decay and regularity; periodic form.)} Let $A$ be regular on $\Lambda_X$ or on $\Lambda_X/P$ for a torus $P$ fitted by both lattices, and let $e_k^\beta\le c(a_{\mathrm{bs}})$. Then, with constants depending on $d=3$, $L=5$, $a_{\mathrm{bs}}$, $M$ and $\alpha$ only, except the H\"older constant, which is $C_\alpha\log^{3\alpha}(1/\eta')$, \begin{gather*} |A^X(A)(z,w)|,\ |C^{(k),X}(A)(z,w)|\le Ce^{-\delta_0|z-w|},\\ |(G^X_k(A)f)(x)|,\ |(D_{A,\mu}G^X_k(A)f)(x)|\le Ce^{-\delta_0\dist(x,\supp f)}\|f\|_\infty,\\ \begin{split} |x-y|^{-\alpha}\big|U(A(\Gamma_{x,y}))(D_{A,\mu}G^X_k(A)f)(y)-(D_{A,\mu}G^X_k(A)f)(x)\big|&\\ \le Ce^{-\delta_0\dist(\{x,y\},\supp f)}\|f\|_\infty&\quad(0<\alpha<1), \end{split} \end{gather*} that is, \cite[(3.6)--(3.8)]{K1} for the hybrid flow. The differences between the two members of a comparable pair obey the same bounds with $\dist(\cdot,\supp f)$ taken in $d_\wedge$ and the additional factor $e^{-\delta(\dist(\{x,y\},\partial)+\dist(\supp f,\partial))}$, $\delta=(2\sqrt3M)^{-1}$; for $A^X$ and $C^{(k),X}$ the decay of the difference is in $d_\wedge$, with the factor $e^{-\delta(\dist(z,\partial)+\dist(w,\partial))}$. For every subset $\Lambda$ of the coarse lattice, in particular every sum of big blocks, $\hat C^{(k)}_\Lambda$ obeys the bound \cite[(5.54)]{K2}. In addition $|H^X(A)(x,z)|,|D_AH^X(A)(x,z)|\le Ce^{-\delta_0|x-z|}$ with the H\"older clause (constant $C_\alpha\log^{3\alpha}(1/\eta')$), $|B^X(A)(z,w)|\le C(1+|z-w|)^{-1}$ and $|\Gamma^X(A)(x,y)|\le C(1+|x-y|)^{-1}$, with one covariant derivative and the H\"older clause in the fine variables; $B^X$ and $\Gamma^X$ decay only at the coarse mass. The same statements hold for the standard flow \cite[Thm.~3.3]{K1}, \cite{Ba4}, there with King's uniform constants. \item[(iii)] \emph{(Gain at $A\ne0$.)} Let $A_k$ be in the small-field class and $A^F$ its lifts. After King's constant-background reduction \cite[(3.43)--(3.55)]{K1}, applied to both flows on their infinite lattices with the torus lines as exact periodisations, every $X$--$Y$ difference element at matched points carries a positive power of the scale ratio: the $A=0$ kernel differences gain $\eta'^{1/2}$ (slice-weighted), the contour lines gain $(\eta'/\ell_j)^{1/2}$ on slice $j$ (the shape part of it $\eta'/\ell_j$), the double contour lines the same against their size $C\ell_j$, the contour vertices and the lifted external fields gain $\eta^{1/2}$ (the former times $e\,p_k^2\delta_k^{1/2}$), and the constant-background phases gain $\eta'$ (times $e\,p_k\mu_0^{-1}\delta_k^{-1/2}$); the remainders are bounded per flow by (ii). Hence the comparison of the two flows holds at $e>0$ as at $e=0$, with the gain exponent $\theta=\frac12$ and a prefactor polynomial in $p_k$ and $\delta_k^{-1/2}$. \end{enumerate} Part (ii), in the form used by the hybrid lower bound at its first step, holds also at the levels $k\in\{0,1\}$. \end{theorem} \begin{proof} (i) is the geometry of Definition~\ref{cpi:hyb:cont}: each staircase lies within $\eta'$ of its segment and bounds with it an area at most $5\eta'^2$, the pieces stay in their blocks since the $\eta$-bonds keep distance at least $\frac52\eta'$ from the faces, $Q_X(A)Q_X(A)^*=\mathbf 1$ follows from $|U|=1$ and the unit block volume, and $\Sigma(z,x')$ is a union of at most $0.3/\eta'$ slivers and one polygon in the ball of radius $\sqrt{13}\eta'$ about $x_1$. (ii) is proved by a coercivity lemma that localises on axis cubes of side $M$ and gauges the constant part of $A$ away cube by cube, by Combes--Thomas decay and by Ba{\l}aban's random-walk assembly~\cite[(2.6)]{Ba4} with periodic cells fitted by both lattices in the form of~\cite[\S5.3]{K2}, whose $\log(1/\eta')$ scales give the logarithm in the H\"older constant; the two-tori clause compares the random walks of the two cells on their common fundamental region, the bounds of~\cite[Prop.~5.6, (5.54)]{K2} apply verbatim to the hybrid operator, Theorem~\ref{r13:thm:BAL5compl}(h) gives the standard-flow bounds in the same form (a lift-distance clause for the domain difference is false and not used), and the same proofs give the levels $k\in\{0,1\}$, since they use $k\ge2$ nowhere. (iii): after King's constant-background reduction every compared line is an $A=0$ line and the new difference elements are the contour vertices and the one-ended and double contour lines~\cite[(3.26)]{K1}, whose mismatch is the flux through the sliver between the paired contours, of area $O(\eta')$ by (i); with the cubic-Hermite cardinal interpolation of the lattice kernels as the shared continuum field, both flows have, after the isometry, one fine lattice, propagator, vertices and counterterms, the zero-degree all-$C$ subgraphs of the $C/\Gamma$ normal form are enumerated and bounded by a multiscale bound, and the gains carry the small-field bound~\cite[(3.2)]{K1}. \end{proof} \subsection{(N3) King's upper bound} (N3) is King's upper bound~\cite[Thm.~3.1, (3.4)]{K1} for $Y_K$ with the additive large-field term in the form of the large-field clause of Theorem~\ref{rot:thm:BAL5}: on $RD_J$ it follows from Theorems~\ref{rot:thm:BAL5} and~\ref{r13:thm:BAL5compl} and the statements of King cited with (N2), given (CT), carried from the axis lattice on $D_J$ by the rotation (Lemma~\ref{rot:lem:pass} with $O=R$); on $D_J$ it is the instance [SKEW-U], proved relative to the aligned bounds with the propagators of~\cite[Prop.~2.6]{Ba2} replaced through Theorem~\ref{r13:thm:BAL5compl}(k); for the contours of the flow of $Y_K$ it uses Proposition~\ref{rot:prop:CR}, and at the wall Proposition~\ref{rot:thm:POS} for $J\ge J_{\rm pos}$, on $D_J$ under the final-scale cap of the class $\mathcal C$, through the large-field clause of Theorem~\ref{rot:thm:BAL5} (Section~\ref{rot:ssec:openin}); Theorem~\ref{cpi:ct:CTB} carries it to the free-boundary model, clause (c) giving the uniformity of its constant at the shifted constant source. \subsection{(N4) The hybrid lower bound} (N4) is the hybrid lower bound, the analogue of~\cite[Prop.~4.1, (4.5)]{K2} for the hybrid flow, Proposition~\ref{cpi:ct:H4}, proved from Theorems~\ref{rot:thm:BAL5} and~\ref{r13:thm:BAL5compl}, (CT), Theorem~\ref{cpi:hyb:BD} and, on $D_J$, [SKEW-L], with Proposition~\ref{rot:prop:CR} for the composite contours; its argument needs a cumulant bound for the fluctuation integral of one step, for a Gaussian field with exponentially decaying precision on $\Z^3$, on cubic tori and on the skew tori, with constants free of the spacing and of the shell mass, which the cluster-expansion lemma of~\cite{BCGNOPS} (the nearest-neighbour Markov field, thresholds growing with the volume) does not give, and which is used again in Theorem~\ref{cpi:thm:main}. \begin{definition}[the setting of the cumulant bound]\label{r13:def:cumulant} Let $d\ge1$, let $\mathbb L=\Z^d$ or $\mathbb L=(\Z/N\Z)^d$ with its Euclidean, resp.\ torus Euclidean, distance $|x-y|$, let $\Lambda\subset\mathbb L$ be finite and let $z=(z_x)_{x\in\Lambda}$, $z_x\in\R^{n_c}$, be the field; operators on $(\R^{n_c})^\Lambda$ are written through their $n_c\times n_c$ block kernels $P(x,y)$, with the block operator norm $\|P(x,y)\|$ (a complex-Hermitian operator on $\C^\Lambda$ is read in its real form, which has the same spectrum and the same block norms). The conditions on the Gaussian measure $\mu$ and on the interaction $V$ are \begin{itemize} \item[(Pr)] $\mu=N(0,C)$ on $(\R^{n_c})^\Lambda$, $C=P^{-1}$, $P$ real symmetric with $\gamma_P\le P\le\gamma_1$; \item[(Dec)] $\|P(x,y)\|\le c_Pe^{-\delta_P|x-y|}$ for all $x,y\in\Lambda$; \item[(V)] $V=\sum_{Y\subset J}V_Y$ with $J\subset\Lambda$, every $V_Y$ a real polynomial in $(z_y)_{y\in Y}$ of degree $\le D$ without constant term, and $\mathsf A\defeq\|V\|_\nu\defeq\sup_{x\in\mathbb L}\sum_{Y\ni x}e^{\nu\ell(Y)}|V_Y|_1$, where $\ell(Y)$ is the length of a shortest connected graph containing $Y$ and $|V_Y|_1$ the sum of the moduli of the coefficients of $V_Y$; \end{itemize} and the cut-off is $\chi_b\defeq\prod_{x\in\Lambda}\mathbf 1\{|z_x|\le b\}$, or $\chi^K_b\defeq\prod_{x\in\Lambda}\mathbf 1\{z_x\in bK\}$ with $K\subset\R^{n_c}$ symmetric, convex, $B(1)\subset K\subset B(c_K)$ (Ba{\l}aban's cut-off~\cite[(3.43), (3.50)]{Ba1}, which restricts the photon and the matter components separately, is of this form with $c_K=\sqrt2$; this $K$ is a convex body, not the scale index). $\mathcal E^T_n(V)$ are the cumulants of $V$ under $\mu$, with no cut-off, and with $\tilde c(\theta)\defeq\sum_{z\in\Z^d}e^{-\theta|z|}$ we put \begin{gather*} \mu_*\defeq\min\Big(\frac{\delta_P}2,\ \frac{\gamma_P}{2c_P\sum_{z\in\Z^d}|z|e^{-\delta_P|z|/2}}\Big),\qquad M_1\defeq\frac{2c_P}{\gamma_P}\,\tilde c(\mu_*/2)^2,\\ \gamma\defeq\min\{\tfrac12,(2M_1)^{-1}\},\qquad \gamma_K\defeq\min\{\tfrac12,(2c_KM_1)^{-1}\}, \end{gather*} and $\bar\mu\defeq\min(\mu_*,\nu)$; $\bar n\ge1$ is the order of the expansion. \end{definition} \begin{theorem}[the cumulant bound]\label{cpi:thm:cumulant} In the setting of Definition~\ref{r13:def:cumulant}, assume (Pr), (Dec) and (V). There are $b_*<\infty$ and $S<\infty$, depending only on $(d,n_c,D,\bar n,\nu,\gamma_P,c_P,\delta_P)$ (and on $c_K$ for the cut-off $\chi^K_b$, for which $\gamma_K$ replaces $\gamma$), such that for every finite $\Lambda\subset\mathbb L$, every $J\subset\Lambda$, every $b\ge b_*$ and every $V$ with $\mathsf A\,b^{D+2d}\le1$, \begin{align} \int d\mu\,\chi_b\,e^{V}&\le\exp\Big\{\sum_{n=1}^{\bar n}\frac{\mathcal E^T_n(V)}{n!}+S|J|\big[(\mathsf A\,b^{D+2d})^{\bar n+1}+e^{-(\bar\mu/16)b^{3/2}}\big]\Big\},\tag{CU}\\ \int d\mu\,\chi_b\,e^{V}&\ge\exp\Big\{\sum_{n=1}^{\bar n}\frac{\mathcal E^T_n(V)}{n!}-S|J|\big[(\mathsf A\,b^{D+2d})^{\bar n+1}+e^{-(\bar\mu/16)b^{3/2}}\big]-4n_c|\Lambda|e^{-\gamma_P\gamma^{2(d+1)}b^2/(2n_c)}\Big\}.\tag{CL} \end{align} The constants do not depend on $\Lambda$, $J$, the torus side, the lattice spacing, the step or the background fields of the application; translation invariance is not used, and the upper bound $\gamma_1$ of (Pr) is not used. The same bounds hold, with the same constants, on every finite $\Lambda$ of a skew torus $T=\Z^3/G$ ($d=3$), $G=mG_+$ or $G=mG_-$ with $m\in\N$ and $G_\pm\defeq\langle(40,\mp30,0),(\pm30,40,0),(0,0,50)\rangle$ (the tori $R^{-1}D_J$ and $RD_J$ in the units of the step in which they are skew), with the quotient Euclidean distance $|x-y|_T\defeq\min_{g\in G}|\tilde x-\tilde y-g|$ ($\tilde x,\tilde y$ lifts to $\Z^3$) in place of $|x-y|$ in (Dec) and in $\ell(Y)$. \end{theorem} \begin{proof} The proof of the cluster-expansion lemma of Benfatto et al.~\cite[p.~152, (4.6)--(4.7)]{BCGNOPS} is re-run for a Gaussian measure whose precision is only bounded below and exponentially decaying: the Markov factorisation across corridors is replaced by a decoupling estimate across corridors of width $w=\lceil b^{3/2}\rceil$ ((Dec) and the Schur test), the nearest-neighbour formula for the conditional mean by Combes--Thomas decay of the conditioned Gaussian, and the superstability bound by the Khatri--\v Sid\'ak bound $\mu(\chi_{\gamma^{d+1}b})\ge\exp(-4n_c|\Lambda|e^{-\gamma_P\gamma^{2(d+1)}b^2/(2n_c)})$, the last term of (CL). On each cell the cumulant expansion of the bounded interaction to order $\bar n$, the replacement of the conditioned law by $\mu$ at the cost $O(e^{-\mu_*w/8})$ and the connected Wick expansion (anchored tree sums at the rate $\bar\mu$, where $\|V\|_\nu$ enters) telescope over the $d+1$ shifted pavements to $\sum_{n\le\bar n}\mathcal E^T_n(V)/n!$. On the skew tori the pavement is built coordinatewise on the cycles of the period lattice, and of the quotient metric only the $1$-Lipschitz distance functions, the domination of its sums by the $\Z^3$ sums, $\ell(Y)\ge|x-y|_T$ and the existence of the pavement are used; translation invariance is used nowhere. \end{proof} For a source $s=(G,h)$ we write $S(s)\defeq\max_{\mu\nu}(\|G_{\mu\nu}\|_\infty+\|\nabla G_{\mu\nu}\|_\infty)+\|h\|_\infty$ for its \emph{size}, and for $S_0>0$ we let $k_X(K,S_0)$ be the largest $k$ such that $(1+K)^{p_{\rm pb}}\ell_j^{\beta_0}p_1(\ell_j)^{D+2d}\le c_{\rm B}$ for all $j\le k$, where $\ell_j=5^j\eps_K$, $p_1(t)=b_1(1+\ln t^{-1})^{\varpi_1}$, with a fixed exponent $\varpi_1\ge1$, is Ba{\l}aban's fluctuation cut-off (\cite[(3.43), (3.50)]{Ba1}), $\beta_0=\frac14-\beta$, and $D$, $d$ are the degree and the dimension of the cumulant bound, Theorem~\ref{cpi:thm:cumulant}, as it is used in (L3) below, and $c_{\rm B}$ the constant for which the cap implies the norm condition $\mathsf A\,b^{D+2d}\le1$ of that theorem for the step interaction ($D$ is not a degree of a graph). The cap contains the level $k=k(K)$ of the construction: for every fixed $\varrho>0$ it holds at $k=k(K)$ once $K\ge K_0(\varrho)$. \begin{proposition}[the hybrid lower bound]\label{cpi:ct:H4} Let $T\in\{D_J,RD_J\}$ and let the hybrid flow of $X_K$ on $T$ be run to the level $k\le k_X(K,S_0)$. There are $p'=p'(p_{\rm pb},\bar n)\le14p_{\rm pb}+c_{\bar n}$ ($c_{\bar n}$ depending only on $\bar n$), $\sigma>0$ and, for every $S_0>0$, a constant $C(S_0)$ such that for every source $s=(G,h)$ with $S(s)\le S_0$ \[Z_{X_K}(T;s)\ \ge\ e^{-C(S_0)(1+K)^{p'}\delta_k^{\sigma}|T|}\int\chi_k\,e^{-\hat s^{(k)}_T},\] uniformly in $K$, $T$ and the source; for curvature sources in the Cameron--Martin form the integral is posed, like King's upper bound (N3), on the common set $\chi_k(\cdot+a^Y_k)\subset\chi_k^{(1+\eps_k)}$ and carries the factor $e^{z^2q_F/2}$. Here $p_{\rm pb}$ is the exponent of the whole-torus perturbative bound for the hybrid flow (the hybrid form of \cite[Prop.~1]{Ba3}; it is not the exponent $p$ of Convention~\ref{cpi:rot:convP}), $\chi_k$ is King's small-field indicator and $\hat s^{(k)}_T$ the hybrid effective action; $(1+K)^{p'}\delta_k^\sigma\to0$ along $k=k(K)$. \end{proposition} \begin{proof} Ba{\l}aban's lower-bound argument~\cite[\S3]{Ba1} is run along the hybrid flow on four facts: (L1) the first step is a block-spin step and the later composite contours are rotation-paired (Lemma~\ref{cpi:ct:tile}, Theorem~\ref{cpi:hyb:BD}(i)); (L2) the hybrid analogues of~\cite[Props.~2.1--2.3]{Ba1} are Theorem~\ref{cpi:hyb:BD}(ii) at every level, $0$ and $1$ included; (L3) the cumulant lower bound for the step Gaussian is Theorem~\ref{cpi:thm:cumulant} for Ba{\l}aban's two-component cut-off ($c_K=\sqrt2$, $\gamma_K$), with (Pr), (Dec) from (L2) and (V) from the coefficient bounds of the step interaction~\cite[\S5.3]{K2} under the cap on $k_X(K,S_0)$, its additional term at the step $j$ (threshold $b_j=p_1(t_j)$) being at most $C_\kappa t_j^\kappa|T^{(j)}_1|$ for every $\kappa$, with sum at most $C'_\sigma\delta_k^\sigma|T|$, and its pavement on the skew tori built coordinatewise on the cycles of the period lattice; (L4) Ba{\l}aban's renormalisation bound~\cite[Prop.~1]{Ba3} is proved again for the hybrid kernels on the whole torus, with the unweakened degree count and the H\"older clause of Theorem~\ref{r13:thm:BAL5compl}(i), whose H\"older constant $C_\alpha\log^{3\alpha}(1/\eta')$ is the origin of $C(1+K)^{p_{\rm pb}}$. On $D_J$ the steps $2,\dots,k$ are the skew-torus instance [SKEW-L], on $RD_J$ only step $1$ is skew, on the torus with period lattice $RP_J=R(50MJ\,\Z^3)$, whose blocks tile $RD_J$ (Lemma~\ref{cpi:ct:tile}(i)), whose kernels have torus images $O(e^{-c\,50MJ/\eps})$ and which lands on $Y_{K-1}\subset RD_J$, and with curvature sources the bound is used on the common set named in the statement, through Theorem~\ref{cpi:ct:CTB}(c) and the Cameron--Martin form of the effective action. Given (L1)--(L4) the remaining estimates are those of~\cite[\S3]{Ba1}, with $C(S_0)$ uniform over $S(s)\le S_0$ because the source enters the step interaction only through its sup norms. \end{proof} \subsection{(N5) The polynomial lower bound} (N5) is a polynomial lower bound $\cZ_{Y_K}(T;s)\ge e^{-C_1|T|}\eps_K^{q(s)}$: at $h=0$ Ba{\l}aban's lower bound~\cite[(1.14)]{Ba1}, the steps of whose proof~\cite[\S\S3--4]{Ba1} are completed by Theorems~\ref{rot:thm:BAL5} and~\ref{r13:thm:BAL5compl}, given (CT), on $RD_J$ directly and on $D_J$ relative to it; at the mass-increasing wall source $h_0=-n\psi$ Proposition~\ref{cpi:ct:H5}, from the source-free bounds at two other fixed masses, the torus one-point Wick bound (Remark~\ref{rot:rem:Reisz}), the skew instances under the final-scale cap and exact identities. The Wick bound rests on Reisz's theorems~\cite[Thm.~1]{Rei88}, \cite[Renormalization Theorem, p.~92]{Rei88b} in the corrected three-dimensional form of Theorem~\ref{cpi:thm:R3} below, in which a naturalness condition proved for the graphs used replaces Reisz's naturalness clause, false as stated, and the forest identity~\cite[Lemma~5.1]{Rei88b} is Zimmermann's forest formula ($R_\Gamma$ below) with its hypothesis~\cite[(2-24)]{Rei88b} checked. \emph{Reisz's power counting in three dimensions.} In the notation of~\cite{Rei88,Rei88b}, with lattice spacing $a>0$, $\mathcal C^c$ is the class of finite sums of functions $V(u,w;a)=a^{-m}F(au,aw)$, $F$ smooth, possessing a continuum limit as $a\to0$, and the integrand class $\mathcal F$ consists of the quotients $V/C$ with $V\in\mathcal C^c$ (coefficients polynomial in the masses) and $C=\prod_i[e_i(l_i;a)+\mu_i^2]$, $\mu_i>0$, where $e_i(l;a)=a^{-2}\eta_i(al)$ with $\eta_i$ smooth, $2\pi$-periodic, positive on the Brillouin zone off its single zero at $0$, and $l_i\ne0$ linear in the loop and external momenta. A Zimmermann subspace $H$ of the loop momenta $k\in\R^{3m}$ is given by a basis $(u,v)$ of a set of line momenta, $u$ its variable part, and $\operatorname{degr}_u$ is Reisz's lattice ultraviolet degree. For a subgraph $\gamma$ with $m(\gamma)$ loops, $\omega_{\mathrm{loc}}(\gamma)$ is the sum of the degrees of its propagators and vertex functions and $\omega_3(\gamma)\defeq\omega_{\mathrm{loc}}(\gamma)+3m(\gamma)$; a \emph{forest} $U$ is a set of one-particle-irreducible subgraphs, pairwise nested or disjoint, and $\bar\gamma(U)$ is $\gamma$ with the maximal elements $\gamma_i$ of $U$ below $\gamma$ contracted. The subtraction operators $t^\delta_\gamma$, $\delta\in\N_0$, are Reisz's periodic lattice Taylor operators in the external momenta of $\gamma$, $t^\delta\defeq0$ for $\delta<0$, and $R_\Gamma F\defeq\sum_U\prod_{\gamma\in U}(-t^{\delta(\gamma)}_\gamma)F$ is the forest formula, the internal momenta $k^\gamma$ of a subgraph being fixed by Reisz's chord rule for a line ordering ($L_j$ is a chord of $\gamma$ if the lines $L_j,L_{j+1},\dots$ of $\gamma$ contain a cycle through $L_j$). A set of line momenta is \emph{natural} if every maximal linearly independent subset expresses the others integrally, and $L_{\mathrm{eff}}(\Gamma)\defeq\{l_L:L\in\Gamma\}\cup\bigcup_{\gamma:\,\delta(\gamma)\ge0}\{k^\gamma_L:L\in\gamma\}$. The degree conditions on the integer labels $\delta(\gamma)$ are \begin{equation}\label{cpi:eq:R3deg} \delta(\gamma)\ge\omega_3(\gamma)\qquad\text{and}\qquad \delta(\gamma)\ge\omega_3(\bar\gamma(U))+\sum_i\delta(\gamma_i)\ \text{ for every forest }U, \end{equation} the three-dimensional form of~\cite[(4-3), (4-4)]{Rei88b}; they hold with equality for $\delta\equiv\omega_3$ and, by additivity and the Euler identity, for every label scheme $w\ge\omega$ on lines and vertices with $\delta(\gamma)=\sum_{L\in\gamma}w(L)+\sum_{B\in\gamma}w(B)+3m(\gamma)$, in particular for the mass derivatives with the labels of the undifferentiated graph. The family $\mathsf{Fam}$ to which the torus one-point Wick bound (Remark~\ref{rot:rem:Reisz}) applies consists of the rings through the Wick vertex at zero external momentum, the insertion graphs of that bound, the two-point graphs at zero momentum with one scalar line doubled (the counterterm derivatives), and the photon two-point graphs at the fixed momenta of a basis of the dual lattice $P^*$ of the torus, up to order $13$, each with its derivatives $\partial^b$, $|b|\le3m+2$, $m$ the number of loops of the graph, with the labels $\delta(\gamma)\defeq\omega_3(\gamma)$ on every one-particle-irreducible $\gamma$, $\Gamma$ included (for a ring the top label is $1-n\le-1$, $n$ its order, so $t^\delta=0$ at the top), which satisfy \eqref{cpi:eq:R3deg} with equality, so that the subtracted set $\{\gamma:\delta(\gamma)\ge0\}$ consists of one-loop subgraphs and of two-loop two-point subgraphs only, all with $\omega_3\le1$; the family is restricted to the graphs with no one-particle-irreducible subgraph having no external scalar leg and an odd number of photon legs (insertion vertices contracted), the omitted graphs contributing $0$ to every torus amplitude and counterterm at fixed $\eps$, by reversing the charge arrows of one closed scalar component. \begin{theorem}[the renormalisation theorem in three dimensions]\label{cpi:thm:R3} Let $\Gamma$ be a one-particle-irreducible lattice graph with $m$ loops in three dimensions: propagators that are products of regular denominators $e_{Lj}(la)+\mu_{Lj}^2$, $\mu_{Lj}>0$, with numerators in $\mathcal C^c$; periodic vertex functions in $\mathcal C^c$; internal momenta of subgraphs by the chord rule; lines and vertices may carry indices, the degrees being the maxima over the index values. Let integers $\delta(\gamma)$, for every one-particle-irreducible $\gamma$ including $\Gamma$, satisfy \eqref{cpi:eq:R3deg}, and let $R_\Gamma$ be the forest formula with the subtraction operators $t^{\delta}$ (components of the $q^\gamma$ in $\R^3$). Then: \begin{enumerate}\emergencystretch=3em\relax \item[(a)] $R_\Gamma=V/(B_1B_2)$ as in \cite[(5-1)--(5-3)]{Rei88b}: $R_\Gamma\in\mathcal F$, $(2\pi/a)$-periodic in $k_1,\dots,k_m$, all masses positive; \item[(b)] for every Zimmermann subspace $H$ of the set of all line and internal momenta, pa\-ram\-e\-trised by $t=(t_1,\dots,t_h)$, $t_j\in\R^3$: $\operatorname{degr}_tR_\Gamma\le-3h-1$, i.e.\ $3h+\operatorname{degr}_tR_\Gamma<0$; \item[(c)] if $L_{\mathrm{eff}}(\Gamma)$ is natural for the ordering used, then $\lim_{a\to0}\int_{B_a^m}R_\Gamma$ exists and equals $\int_{\R^{3m}}\lim_{a\to0}R_\Gamma$, $B_a=[-\pi/a,\pi/a)^3$, and $\sup_{02H_0$ exists at every coupling and wall height, the coefficients of $\partial_s\delta m^2$ being non-negative at order $\lambda$, $O(s^{-1/2})$ at order $e^2$ and $O(s^{-1})$ at two loops, uniformly in $\eps\le\eps_0$ (\cite[Thm.~1]{Rei88} and, for the subtracted two-loop integrands, Theorem~\ref{cpi:thm:R3}, as in the proof of Lemma~\ref{lem:s14:CT}), and since $-\partial_\gamma\ln\hat Z(\gamma)=\frac12\langle\sum s^3|\phi|^2\rangle_\gamma$ is non-increasing, $\ln\hat Z(\gamma')\ge\ln\hat Z(a_M)+u|T|(2c_M+w_K(M))$ with $u\defeq\frac12(a_M-\gamma')\in[0,C]$ and $\ln\hat Z(a_M)\ge-C|T|-qK\ln5$ by Ba{\l}aban's lower line~\cite[(1.14)]{Ba1} at $M^2$; the \emph{$c$-numbers} collapse, by the integrated counterterm identity $P_M-P_m=-|T|\int_{m^2}^{M^2}c_s\,\partial_sa_s\,ds+\mathcal R_{m,M}$, to $2u(c_M-c_m)|T|=O(|T|)$ plus a term $\ge-C|T|$ and the perturbative remainder $\mathcal R_{m,M}\ge-C(1+K)|T|$ (King's vacuum-energy counterterm~\cite[(3.32)--(3.35)]{K1}, of which only the order-$e^2$ coefficient is logarithmic, by the torus one-point Wick bound of Remark~\ref{rot:rem:Reisz}); and the \emph{mirror bound} $w_K(M)\ge-C(1+K)$ follows by the same monotonicity argument run upwards to a King point $M_+^2$, from the lower line at $M^2$ and the upper line~\cite[(1.3)]{Ba2} at $M_+^2$. Altogether $\ln Z^{m^2}_\Lambda(T;0,h)\ge-C|T|-C(1+K)|T|+q\ln\eps_K$, the source $tG$ is added as in the statement, and on $D_J$ the skew instances [SKEW-L] replace the aligned lines. \end{proof} \subsection{(SK) The skew-torus instances} The skew-torus instances [SKEW-U], [SKEW-L] of King's and Ba{\l}aban's bounds for the rotated lattice on $D_J$, not aligned with its frame, are proved relative to the aligned bounds at blocking factor $5$ (Theorem~\ref{rot:thm:BAL5}), given the skew-torus clause of Theorem~\ref{cpi:thm:cumulant} and, for [SKEW-U], the change of domain, Theorem~\ref{r13:thm:BAL5compl}(k); they are the instances named in Step~2 of the proof of Theorem~\ref{rot:thm:CT} and in (N3)--(N5) above. The last hypothesis, (N6), the existence of the axis-lattice limit $\lim_K\langle e^{tF(G)}\rangle_{\fc,X_K}$, is Theorem~\ref{cpi:thm:hrs:H6W}, which rests on the statements that follow. \space% % % \subsection{(N6) The walled axis-lattice limit} (N6) is the existence of $\lim_K\langle e^{tF(G)}\rangle_{\fc,X_K}$ along the axis lattices $X_K$ of spacing $\eps_K$ (Section~\ref{rot:sec:model}), King's Theorem~2.1 of~\cite{K1} at a wall for the normalised ratio; its proof uses the large-field clause of the walled envelope at the wall (Theorem~\ref{cpi:thm:gs:corv2}), the wall instance of (N5) (Proposition~\ref{cpi:ct:H5}), the rate clause of the convention bridge (Theorem~\ref{cpi:ct:CTB}(d)) and the restricted form of King's renormalisation theorem, Section~\ref{r11:ssec:restricted}. \begin{theorem}[the walled axis-lattice limit]\label{cpi:thm:hrs:H6W} Let $\fc=(J,\psi,n)$ be a configuration of the class $\mathcal C$ (Definition~\ref{rot:def:class}) with $n>0$ (the renormalisation bound for the operator with the scalar wall and the mismatch-vertex bound, Theorem~\ref{r13:thm:BAL5compl}(j) and Proposition~\ref{r13:prop:B5Kfurther}(b), hold for every such configuration), let the scalar mass be a point of King's range at which the positivity condition of \cref{rot:thm:POS} holds, with $J\ge J_{\mathrm{pos}}$, and let $G$ be a real smooth $P_J$-periodic $2$-form. Put $r_K(z)\defeq Z_{X_K}(T;zG,h_0)/Z_{X_K}(T;0,h_0)$, the normalised ratio at the wall $h_0=-n\psi$ in the free-boundary conventions of this paper, so that $r_K(t)=\langle e^{tF(G)}\rangle_{\fc,X_K}$ for real $t$, and let $C_G\defeq\sup_Kq_{X_K}(G)/2$, $q_{X_K}(G)$ the variance of $F(G)$ under the free photon Gaussian of $X_K$ on $T$, finite by the convergence of the lattice variances. Then: \begin{itemize}[nosep] \item[(i)] for every real $t$ the limit $W_\fc(tG)\defeq\lim_{K\to\infty}r_K(t)$ exists along $\eps_K=5^{-K}$, and $1\le W_\fc(tG)\le e^{C_Gt^2}$; \item[(ii)] the limit is the same number in King's torus conventions at the shifted wall: $W_\fc(tG)=\lim_KZ^{\mathrm{tor}}_{X_K}(T;tG,h_\infty)/Z^{\mathrm{tor}}_{X_K}(T;0,h_\infty)$ with $h_\infty=h_0-\frac12\Delta_T(0)$, a constant shift of the wall of size $O(e^{-c\,50MJ})$; \item[(iii)] $r_K$ converges locally uniformly on $\C$ to an entire function $W_\fc(zG)$, and \[ \sup_K\sup_{|z|\le R}|r_K(z)|\le e^{C_GR^2} \] for every $R>0$; in particular $r_K(\mathrm i)\to W_\fc(\mathrm iG)$. \end{itemize} The statement is at fixed $\fc$, $G$ and $t$, with $K\to\infty$ first, and does not depend on the later infimum over configurations or on the wall-height index of the closing estimate. The free-boundary limit and the torus-convention limit at the same wall $h_0$ are not claimed equal, nor is the case $n=0$ claimed; the positivity condition and $J\ge J_{\mathrm{pos}}$ enter only through the large-field clause of the walled envelope at the wall, and at the points of King's range where the condition fails Theorem~\ref{rot:thm:main} is obtained by right-continuity in the scalar mass without the present theorem. \end{theorem} \begin{proof}\emergencystretch=3em\relax In either convention $1\le Z_{X_K}(T;tG,h)/Z_{X_K}(T;0,h)\le\exp[t^2q_{X_K}(G)/2]\le e^{C_Gt^2}$ (Jensen's inequality and the free photon variance) and $|r_K(z)|\le r_K(\Re z)$, so (iii) follows from (i) by Vitali's theorem. By the convention bridge, $r_K(t)=Z_K(t;h_K)/Z_K(0;h_K)$ with $Z_K(t;h)\defeq Z^{\mathrm{tor}}_{X_K}(T;tG,h)$ and the moving wall $h_K\defeq h_0-\frac12\Delta_T(\eps_K)$, the $h$-linear $c$-numbers cancelling in the ratio. With a depth $k=k(K)\to\infty$ so slow that $e^{-c\,p(\delta_k)^2}$ beats every fixed power of $\eps_K$, and $\rho_K(t;h)\defeq I_K(t;h)/I_K(0;h)$ the ratio of the small-field integrals $I_K(t;h)\defeq\int(dA_k)(d\phi_k)\,\chi_k\,e^{-S^{(k),1}_K(tG,h)}$ of King's effective action at depth $k$, for $h\in\{h_K,h_\infty\}$: (S1) $e^{-C\delta_k^\sigma|T|}I_K(t;h)\le Z_K(t;h)\le e^{C\delta_k^\sigma|T|}I_K(t;h)+e^{-c_*p(\delta_k)^2+C|T|}\eps_K^{-q}$ with constants independent of $K$ and of $h$ in the family (King's upper bound \cite[Thm.~3.1]{K1} in its walled envelope form, \cref{cpi:thm:gs:corv2}, and the lower bound by the small-field integral); (S2) $Z_K(0;h)\ge e^{-C_1|T|}\eps_K^{q_1}$ (Proposition~\ref{cpi:ct:H5}), whence $Z_K(t;h)/Z_K(0;h)-\rho_K(t;h)\to0$; (S3) $|S^{(k),1}_K(tG,h_K)-S^{(k),1}_K(tG,h_\infty)|\le C(1+k)|h_K-h_\infty|p(\delta_k)^{n_1}\delta_k^{-3}|T|+2C\delta_k^\sigma|T|\to0$ on $\supp\chi_k$, since $h_K-h_\infty=\frac12(\Delta_T(0)-\Delta_T(\eps_K))=O(\eps_K)$ by the rate clause of the convention bridge (Theorem~\ref{cpi:ct:CTB}(d), used here and nowhere else), whence $\rho_K(t;h_K)-\rho_K(t;h_\infty)\to0$; (S4) King's two-cutoff estimate at the wall $h_\infty$, $|S^{(k),1}_K(tG,h_\infty)-S^{(k+1),1}_{K+1}(tG,h_\infty)-d_K|\le C(5^{-\gamma k}\delta_k^{-\beta}+\delta_k^\sigma)|T|$ on $\supp\chi_k$ with a field- and $t$-independent constant $d_K$, the step $n=1$ of \cite[Thm.~3.4]{K1} from $X_K$ at depth $k$ to $X_{K+1}$ at depth $k+1$, whose hypotheses at a wall are the restricted estimates (Theorem~\ref{cpi:thm:Ri}), the routing of the sources through the renormalised classes with Theorem~\ref{cpi:hrs:G27}, the torus one-point Wick bound beyond one loop, and Lemma~\ref{cpi:lem:hrs:NP1} with Theorem~\ref{cpi:thm:hrs:NP3} for the non-polynomial terms; (S5) by (S4) $s_K(t)\defeq Z_K(t;h_\infty)/Z_K(0;h_\infty)$ is Cauchy, $d_K$ cancelling in the ratio, and by (S2), (S3) $r_K(t)-s_K(t)\to0$, so $\lim_Kr_K(t)=\lim_Ks_K(t)=:W_\fc(tG)$, which is (i) and (ii). \end{proof} \section{The restricted renormalisation theorem}\label{r11:ssec:restricted}\label{cpi:ssec:RS}\label{cpi:ssec:Rthm} King's renormalisation theorem \cite[Thm.~3.5]{K1} is stated for every graph of his expansion, whatever its degree, scalar and curvature sources included. With the notation of \cite[(3.37)]{K1}, the contribution of a graph $H$ with $s$ external scalar legs to the effective action after $k$ steps is $E^{(k)}(H)=\sum_{\{y_i\},\{z_q\}\subset T^{(k)}}\phi_k(y_1)\cdots\phi_k(y_s)\,E^{(k)}(H,A^{(k)};\{y_i\},\{z_q\})$; $\dist(\{v_i\})$ is the length of the shortest tree connecting the points, $\delta_k\defeq L^k\eps$ with $L=5$, and $p$ abbreviates $p(\delta_k)$, the small-field function of Convention~\ref{cpi:rot:convP} at $\delta_k$. The theorem asserts that for some $n_1,\delta,\gamma,\sigma>0$ depending only on the order cap $\bar n$ and on $d$, and for every $\rho\ge s/4$, \begin{align} &|E^{(\theta)}(H,A^{(\theta)};\{y_i\},\{z_q\})|\le C\delta_k^{\rho}\,p^{n_1}\exp[-\delta\dist(\{y_i\},\{z_q\})],\qquad\theta\in\{k,k+n\},\label{cpi:eq:K338}\\ &|E^{(k+n)}(H,A^{(k+n)};\cdot)-E^{(k)}(H,A^{(k)};\cdot)|\le C\delta_k^{\rho-1/2}p^{n_1}\big[L^{-\gamma k}+\delta_k^{d+\sigma+1/2}\big]e^{-\delta\dist}\quad\text{when }\dist\le p,\label{cpi:eq:K339} \end{align} King's (3.38) and (3.39), which \cite[(3.40)--(3.42)]{K1} sum over the positions to $|E^{(k+n)}(H)-E^{(k)}(H)|\le C[L^{-\gamma k}\delta_k^{-\beta}+\delta_k^{\sigma'}]|T|$. In that generality the theorem is false, failing on the order-$e^2$ $h$-linear vacuum family with its counterterm partner (Proposition~\ref{cpi:prop:F}); all that is used is its restricted form, Theorem~\ref{cpi:thm:Ri}, which excludes that family and rests on Theorems~\ref{cpi:ir:IRG}, \ref{cpi:ir:S3}, \ref{cpi:thm:78} and~\ref{cpi:thm:TBA3} below and on the census, in its proof, of the subgraphs of non-positive degree that contain a source vertex; the proof of (N6) uses besides Theorem~\ref{cpi:hrs:G27}, Lemma~\ref{cpi:lem:hrs:NP1} and Theorem~\ref{cpi:thm:hrs:NP3}. Throughout, the lines of a graph carry slice indices \cite[(3.58)]{K1}, an ordering lists them, $H_i$ is the prefix of the first $i$ lines, the degree of a connected subgraph is Ba{\l}aban's \cite[(2.1)--(2.2)]{Ba3}, equal to King's \cite[(3.30)--(3.31)]{K1} on subgraphs without block vertices, the \emph{strict ordered degree condition} (DEG) is $D(H_i)>0$ for every prefix of every admitted ordering (\cite[(3.77)]{K1}, \cite[(3.5)]{Ba3}), and $\varrho_j\defeq L^j\eta$ is the range of a line of index $j$ in the units of level $k$ ($\eta$ the spacing). \emph{The growth form for the mass-counterterm blobs.} The mass counterterm vertex \cite[(3.19)]{K1} is $-\frac12(L^k\eps)^2\sum_x\eta^d\,\delta m^2(x;e'e,\lambda'\lambda)|\phi'+\phi^{(k)}|^2$ with $\delta m^2=\sum_{2\le\alpha+2\beta\le4}e^\alpha\lambda^\beta\,\delta m^2(x;\alpha,\beta)$ \cite[(3.32)--(3.33)]{K1}, built recursively from the one-particle-irreducible two-point graphs with two undifferentiated external scalar lines, whose lines $C^\eta$ are decomposed as $\sum_{j=0}^KG^\eta_{(j)}$; a term with some index $j\ge k$ is a \emph{coarse} term of the blob, for which no partition of unity is introduced \cite[p.~667]{K1}. With $\delta m^2_{\ge k}$ the sum of the coarse terms, their coefficient in the effective interaction $P^{(k),1}$ in the units of level $k$ is $c^{(k)}_{\ge k}(x)\defeq\tfrac12\ell^2\sum_{\alpha,\beta}e^\alpha\lambda^\beta\,\delta m^2_{\ge k}(x;\alpha,\beta)$, $\ell\defeq L^k\eps$, and by scaling $\frac12\ell^2e^\alpha\lambda^\beta\,\delta m^2(x;\alpha,\beta)=e_k^\alpha\lambda_k^\beta\,\Gamma^{(k)}_{\alpha,\beta}(x)$ with the running couplings $e_k=e\ell^{1/2}$, $\lambda_k=\lambda\ell$ and $\Gamma^{(k)}_{\alpha,\beta}$ the same graph in the units of level $k$ with the couplings stripped. \begin{theorem}[the growth form for every mass-counterterm blob]\label{cpi:ir:IRG} For every order $(\alpha,\beta)$ and every $k\le K$, uniformly in $x$, \begin{gather*} |\Gamma^{(k)}_{\alpha,\beta,\ge k}(x)|\le C\,(K-k+1)^5\,\ell^{-\max(0,D_{\max}(\alpha,\beta))},\\ D_{\max}=-1\text{ for }\alpha+2\beta=2,\quad D_{\max}=+1\text{ for }\alpha+2\beta=4; \end{gather*} hence \begin{equation}\label{cpi:eq:Clog} |c^{(k)}_{\ge k}(x)|\le C\,\ell\,(K-k+1)^5\le C'\,\ell\,(1+\ln\ell^{-1})^5, \end{equation} Ba{\l}aban's bound \cite[(1.29)]{Ba3} up to a power of the logarithm. Moreover: (a) in a graph of $P^{(k),1}$ the counterterm vertex splits as $c_{0$ the least positive ordered degree over the finitely many rewritten shapes of the blobs and $\gamma_{3.9}$ the exponent of \cite[Prop.~3.9]{K1}, $|c^{(k+n)}_{\ge k}-c^{(k)}_{\ge k}|\le C\,L^{-\gamma k}\,\ell\,(1+\ln\ell^{-1})^5$ with $C$ independent of $n$, $k$, $K$ and $\eps$; (d) the bounds hold on the skew torus and on the aligned torus alike. \end{theorem} \begin{proof} A finite census of the blobs (at most five lines each) gives the maximal quotient degree over the fine sets $F$ and the coarse prefix sets $S$ as $-1$ at order two and $+1$ at order four, so every positive quotient needs a fine or earlier-coarse divergent sub-piece; with the lines ordered as in \cite[(3.76)]{K1}, each divergent fine component is rewritten by the renormalisation theorem in its free-sum form (the zero-momentum gauge covariance identity inside the order-$e^2$ vacuum polarisation being applied with the fine indices summed, never to a fixed fine ordering), and the coarse sum is at most the raw sum over the coarse indices, $(K-k+1)^m\ell^{-\max(0,D_{\max})}$ with $m\le5$, by \cite[Prop.~3.7]{K1}, which gives \eqref{cpi:eq:Clog} since $e_k^\alpha\lambda_k^\beta\ell^{-D^+_{\max}}=O(\ell)$. For (c) the difference of the two models telescopes one factor at a time, each factor carrying $L^{-\gamma k}$ (the shrinking induction, the difference form of the renormalisation theorem, \cite[Prop.~3.9]{K1}), the weakening $\gamma<\alpha_1$ being absorbed by the geometric sums; the census is torus-free, which gives (d). \end{proof} \emph{The split degree count.} For the classes with block vertices the count that governs the bounds is the split count of Definition~\ref{cpi:ir:split} (a block product represented by its two summation points, a vector leg of a block vertex credited only on an internal line of the subgraph), in which every component with a block point has degree at least one (Lemma~\ref{cpi:ir:BLKnat}); the following theorem says that this count, and not King's per-vertex credit, is what the multiscale bound sees. \begin{definition}[the condition $(\mathrm{DEG}^{\mathrm{nat}})$]\label{r13:def:DEGnat} Let $\mathcal R$ be a renormalised family of King's expansion of order at most $\bar n$, with the source vertices \cite[(3.20), (3.21)]{K1} admitted, written as a finite sum of graphs $G$. The family satisfies the condition $(\mathrm{DEG}^{\mathrm{nat}})$ if, for each admitted ordering of the scale-summed lines of the split graph $G^{\mathrm{sp}}$, every component of every ultraviolet prefix (indices $j0$ after the rewriting of Theorem~\ref{cpi:ir:S3}(iii) and the infrared counterterm lines (indices $k\le j\le K$) obey the growth form of Theorem~\ref{cpi:ir:IRG}; $\alpha_0>0$ is then the minimum of these degrees over the finitely many graphs and orderings. \end{definition} \begin{theorem}[the split count governs the multiscale bound]\label{cpi:ir:S3} Let $\mathcal R$ be a renormalised family as in Definition~\ref{r13:def:DEGnat} satisfying $(\mathrm{DEG}^{\mathrm{nat}})$, with the minimal degree $\alpha_0>0$ of that definition, and let the lines obey King's bounds \cite[(3.63)--(3.65)]{K1}, the infrared slices \cite[p.~666]{K1} included, the contour and contour-H\"older lines with the contour factor $\varrho_{j_l}$ booked on the line, for the finer model's slices as well. Then: \begin{enumerate} \item[(i)] the sum over orderings, compatible integer assignments and positions of $|E^{(k)}(G)|$ is at most $C\,(L^k\eps)^{\rho}\,p(L^k\eps)^{n_1}\,(1+K)^{p_0}\exp[-\delta_0\dist(\text{external vertices})]$, the bound \cite[(3.38)]{K1} with King's exponents, decay rate and prefactor, and the same holds with one element weakened by $\theta<\alpha_0$ (a line $l$ with $c_l\mapsto c_l\varrho_{j_l}^{-\theta}$, a vertex with an explicit $\eta$-power $a_v\ge\theta$ reduced to $a_v-\theta$, or an external field replaced by one of the same $(1,\alpha)$-norm); \item[(ii)] for $0<\gamma<\alpha_0$, $|E^{(k+n)}(G)-E^{(k)}(G)|\le C\,(L^k\eps)^{\rho'}L^{-\gamma k}\exp[-\delta_0\dist]$, uniformly in $n$; \item[(iii)] every rewritten unit is source-free, point-free and a component of a prefix of $G^{\mathrm{sp}}$, and $D'(\gamma)\ge D^{\mathrm{sp}}(\gamma)$ for every component of every prefix; \item[(iv)] the exponent of every stage of the ordered bound before the $\kappa$-lines is $D'^{\mathrm{nat}}$ of the stage, raised by $d$ per merging $\kappa$-line from a $\kappa$-line on (immaterial, since $\varrho_k=1$); no per-vertex credit $+m$ enters. \end{enumerate} The constants depend on $\bar n$, $d$, $\alpha$, $\theta$ (resp.\ $\gamma$), $\delta_0$ and King's constants only. Parts (i), (iii), (iv) hold verbatim on a cube with the slices of the Dirichlet flow, and part (ii) holds there for the diagonal pair $(K,k)$ against $(K+1,k+1)$ on the common unit lattice, at $h=0$ and on the small-field set, in the form that the sum over positions of $|E^{(k+1)}_{K+1}(G)-E^{(k)}_K(G)|$ is at most $C(L^{-\gamma k}\eta^{-\beta}+\eta^\sigma)$ times the volume of the cube, for some $\gamma,\sigma>0$ and $\beta\ge0$ (King's two-term bracket \cite[(3.56)]{K1}). \end{theorem} \begin{proof} (i) is the shrinking induction \cite[(3.66)--(3.70)]{K1}, read per ordering, on $G^{\mathrm{sp}}$ with the dimensions and weights of Definition~\ref{cpi:ir:split}: a connected all-scale-summed subgraph containing a $\kappa$-line has $D^{\mathrm{MS}}=\sum_iD^{\mathrm{sp}}(c_i)+d(r-1)\ge1$ by Lemma~\ref{cpi:ir:BLKnat}, so the hypothesis of the induction is exactly $(\mathrm{DEG}^{\mathrm{nat}})$, and shrinking the kernel $\kappa$ leaves the infrared differences \cite[(3.80)]{K1} and the growth form of Theorem~\ref{cpi:ir:IRG} unchanged. (ii) is King's replacement scheme \cite[pp.~663--665, (3.70)--(3.75), (3.88)]{K1} on $G^{\mathrm{sp}}$, each error term being the same graph with one element weakened by $\gamma$ and the prefactor $L^{-\gamma k}$, bounded by the weakened clause of (i); (iii) holds because a rewritten unit is divergent, hence point-free by Lemma~\ref{cpi:ir:BLKnat} and a graph of Ba{\l}aban's divergent classes \cite[pp.~428--429]{Ba3}; (iv) is the telescoping identity of the induction. On the cube the same holds through the split-graph forms of the line bounds, (ii) as the diagonal-pair comparison with King's two-term bracket, a contour line being a weakened scale-summed line with gain $(\eta'/\varrho_j)^{1/2}$ whose degree requirement holds with margin $\frac12$ by Lemma~\ref{cpi:ir:BLKnat}. \end{proof} \emph{Ba{\l}aban's renormalisation rewriting in King's scheme.} The plain degree of a connected component $\gamma$ is $D(\gamma)=\sum_{v\in\gamma}u_v-3+\frac12E_\gamma$, with $u_v\defeq3+a_v-\frac12n_v-\delta_v$ for a vertex with $n_v$ legs, $a_v$ extra powers of $\eta$ beyond $\eta^d$ and $\delta_v$ differentiated internal ends, and $E_\gamma$ the number of legs at vertices of $\gamma$ that are not ends of lines of $\gamma$ ($u=2$ for the $h$-vertex, $u=\frac52$ for the summed curvature vertex). A \emph{unit} is a connected prefix component $S$ of an admitted ordering with ordered degree $D'(S)\le0$, $D'$ being the plain degree with the credits and debits of the earlier rewritings; a unit contains no $h$-, $g$- or source vertex. Units are rewritten finest first, on their admitted set $A_S$ (all internal indices at most $j_0$, the lowest index among the legs of $S$) and with their inside family $F_S$ (all connected members of the same order and profile with the same legs at the same outside points, $\dm$- and Ward--Takahashi partners included); $\varrho_j^\theta$, $\theta>0$, is a \emph{credit} on a line of index $j$, where $\varrho_j\defeq L^j\eta$ is the range of a line of index $j$ in unit-$k$ units ($\eta$ the spacing, so $\varrho_j=\ell_j/\ell_k$ with $\ell_j=L^j\eps$ the physical block scales). The rewriting of a unit, its sum over $F_S$ and $A_S$, is an exact identity whose output terms are of three shapes: (U0) zero; (U1) collapsed local terms $\sum_x\eta^dc_a(x)\prod\partial^a\Phi(x)$, $|a|\le n\defeq-D(S)$, times fixed localisation weights ($\Phi$ the external fields of $S$); (U2) remainders with one credit $\varrho_{j_c}^\theta$ on a line of $S$ and one debit of total order at most $\theta$ on one leg of $S$ or on a smooth weight, $\theta=n+\alpha$, with $D'(S)>0$. The long-line units (zero debit, $\theta=3$) and the face units ($\theta=1$) are units. \begin{theorem}[the strict degree property, componentwise]\label{cpi:thm:78} Run the construction of \cite[\S\S2--3]{Ba3} in King's scheme, with $h$ and $g$ counted as vertices. Then: \begin{enumerate} \item[(i)] (\emph{unit shapes}) the rewriting of each unit $S$ is an exact identity with output terms of the shapes (U0)--(U2); in (U1), $|c_a(x)|\le C\varrho_{j_0}^{D(S)+|a|}$ uniformly in $x$, $j_0$, $k$, $\eta$, with $c_0\equiv0$ for the one-particle-irreducible two-scalar-leg units and for the tree \cite[(2.4)]{Ba3}; in (U2), $D'(S)\ge\alpha$ and $D'(\gamma)\ge D(\gamma)$ for every prefix component $\gamma\subsetneq S$. The constants depend only on $\bar n=13$, $d=3$, $L=5$, $\alpha\in(0,1)$ and King's constants \cite[Thm.~3.3, Props.~3.7--3.9]{K1}; \item[(ii)] (\emph{strictness, componentwise}) for every output term and every stage $i$, every source-free connected component $\gamma$ of $H_i$ has $D'(\gamma)>0$, and the minimum $\alpha_0$ of these values, the \emph{minimal ordered degree}, is positive and depends only on $\bar n$, $d$, $\alpha$ ($\alpha_0\ge\alpha$ for $\alpha<1/N(\bar n)$, $N(\bar n)$ the maximal leg count of a component). Every component meeting $h$ or $g$ has $D'(\gamma)\ge1$ off the excluded shapes of the census (the whole vacuum graph and the two shapes that vanish identically, in the proof of Theorem~\ref{cpi:thm:Ri}). Hence $D'(H_i)=\sum_\gamma D'(\gamma)\ge\min(\alpha_0,1)>0$ for every prefix: the strict ordered degree condition (DEG), with this $\alpha_0$; \item[(iii)] (\emph{one-particle-reducible components}) (i)--(ii) include the one-particle-reducible source-free components; for the scalar chains with one-particle-irreducible pieces the zeroth moment vanishes exactly for every scale assignment; \item[(iv)] (\emph{embedded form; far ends}) the legs of a unit may be any lines of the graph ($h$-, $g$-, counterterm-, block- and $\delta G$-lines included) or external fields; the rewriting uses a leg only through its index, its value and its Taylor or H\"older quotient in its $S$-end variable (uniformly in the far end) and the cube of its far end, and never acts on the far-end vertex; $h$ and the summed $g$ put no derivative on their lines; \item[(v)] (\emph{King's scheme}) the counterterm partners are King's $\dm$ \cite[(3.32)--(3.34)]{K1} and $E_1$ \cite[(3.35)]{K1}, which on the ultraviolet slices coincide graph by graph with Ba{\l}aban's $\dm_k$, $E_k$ \cite[(1.26)--(1.29)]{Ba3}; \item[(vi)] (\emph{generic legs}) (i)--(ii) hold with the product of the external fields of $S$ replaced by any function $\Psi$ of the leg positions, the bounds depending on $\Psi$ through sup norms and the norm \cite[(1.32)]{Ba3} in each variable only; in the form used, $h$ and $w=\eta^{-1}d^*g$ enter only as far-end vertex functions, through $\|h\|_\infty$, $\|w\|_\infty$; \item[(vii)] (\emph{difference form}) for the two lattices of \eqref{cpi:eq:K339}, $|c_a^{(k+n)}-c_a^{(k)}|\le CL^{-\gamma k}\varrho_{j_0}^{D(S)+|a|-\gamma}$; \item[(viii)] (\emph{cube}) on King's cube $C^K_J$, in the zone of the free flow and under the range condition $L^k\eps\le\eps_0$, (i)--(vi) hold with the cube form of the renormalised-degree structure (Theorem~\ref{cpi:thm:a02:T}) in place of its torus form. \end{enumerate} \end{theorem} \begin{proof} (i) is the renormalisation rewriting of a unit (the resummation of its inside family $F_S$, the transport and Taylor identities in the debited variable, the moment computation of the collapsed coefficients), whose hypotheses hold for every unit by the prefix criterion and the single-debit clause of Theorem~\ref{cpi:thm:TBA3} and by the split count (Lemma~\ref{cpi:ir:BLKnat}, Theorem~\ref{cpi:ir:S3}), units being rewritten finest first. (ii): by the stage bookkeeping of Theorem~\ref{cpi:thm:TBA3} a rewriting never lowers $D'$ of a prefix component, a source-free component of current degree $\le0$ would itself be a unit, already rewritten, $\alpha_0=\min D'$ is a minimum over a finite subset of $\mathbb Z+\alpha\mathbb Z_{\ge0}$, and the components meeting $h$ or $g$ have $D'\ge D_{\mathrm{plain}}\ge\frac12$ by the census, hence $\ge1$ by integrality, off the whole vacuum graph and the shapes that vanish by $\Tr q=0$ (neither the bookkeeping nor the census uses a strict degree property, so there is no circularity). (iii)--(vi) are read off the rewriting, which sees a leg only through its index, its value, its quotient at the $S$-end and the cube of its far end, (vii), (viii) are its difference form for the two lattices of \eqref{cpi:eq:K339} and its cube form through Theorem~\ref{cpi:thm:a02:T}, and the componentwise form is Ba{\l}aban's own \cite[pp.~425, 428, 433]{Ba3}, every step of \cite[\S3]{Ba3} stated there without argument being replaced by a lemma of the rewriting, the Ward identity being applied only to the $C^\xi$ replacement of the resummed two-photon family, and the vector-propagator clause of \cite[p.~426]{Ba3} being the vector-propagator bound of Theorem~\ref{rot:thm:BAL5}. \end{proof} \emph{Ba{\l}aban's construction with the source vertices counted.} The restricted estimates rest on four statements. (I1) King's two-cutoff estimate for renormalised graphs, \cite[Prop.~3.6]{K1} in its one-flow instance: for the graphs $\tilde H$ of \cite[(3.55)]{K1} satisfying (DEG), read on the split graph (source vertices included, $g$ at $\frac52$), with the growth form of the coarse counterterm attachments (Theorem~\ref{cpi:ir:IRG}), \begin{multline}\label{cpi:eq:K356} |E^{(k+n)}(\tilde H;\{y\},\{z\},\{w\})-E^{(k)}(\tilde H;\cdot)|\\ \le C\delta_k^{\rho-1/2}p^{n_2}\big[L^{-\gamma k}+\delta_k^{d+\sigma+1/2}\big]\exp[-\delta\dist(\{y\},\{z\},\{w\})], \end{multline} King's (3.56), with $\gamma$ smaller than the minimal degree (for hosts with host-level divergent units the collapsed coefficients differ between the two models by Theorem~\ref{cpi:thm:78}(vii), an admissible weakened element). (I2) The multiscale bound of type \eqref{cpi:eq:K338} for a single model on the split graph (points as vertices, contour lines with $d_l=-1+\#\{\text{contour ends}\}$, the block kernel a line of index $k$), read per ordering, with the infrared logarithm $C(1+K-k)$ per divergent subgraph. (I3) The strict degree property of the source-free renormalised components under Ba{\l}aban's rewriting, Theorem~\ref{cpi:thm:78}. (I4) King's propagator and remainder bounds \cite[Thm.~3.3, Props.~3.7--3.9]{K1} on the torus slices $j\le k-1$ and the infrared slices $k\le j\le K$ \cite[p.~666]{K1}; the clauses of \cite[(3.71), (3.73)--(3.75), Lemma~4.5]{K1} that King states without proof hold with a smaller exponent $\gamma'>0$, and every use needs only $0<\gamma'<\alpha_0$. The rewriting of (I3) acts only on source-free divergent subgraphs ($x$ and $g$ carry no counterterm off $P_0$, and the Wick constant of \cite[(3.21)]{K1} is a $c$-number entering only the $(0,0)$ tadpole) and does not lower the degree of an ordered subgraph containing $x$ or $g$; this \emph{rewriting property} (RW) is not automatic and is a consequence of the following theorem, in which the construction is run on the whole graph with $x$ and $g$ counted as vertices. \begin{theorem}[Ba{\l}aban's construction with source vertices]\label{cpi:thm:TBA3} Let $\tilde H$ be a connected graph of King's family \cite[(3.37), (3.55)]{K1} at $B=0$, with an external leg or a $g$-vertex; $x$ the $h$-vertex ($u_x=2$), $g$ the curvature vertex with the curl summed onto $g$ ($u_g=\frac52$), the counterterm graphs \cite[(3.34)]{K1} with their merged attachment vertex $x_S$; all lines ($x$-, $g$- and counterterm lines included) carry slice indices, and the degrees are the plain degrees by legs. Run Ba{\l}aban's construction \cite[Ch.~3]{Ba3} on $\tilde H$: ``divergent'' means a connected prefix component with current degree $\le0$; the first divergent component is rewritten by Ba{\l}aban's rule for its type; degree-zero single lines \cite[(2.4)]{Ba3} are postponed into a containing divergent component. Then (i) every rewritten $S$ is source-free and is a prefix component, i.e.\ it is rewritten only in configurations where every $S$-index is at most every external-leg index ($x$- and $g$-lines included); (ii) for every output term, stage $i$ and component $\gamma$ of $H_i$: $D'(\gamma)\ge D_{\mathrm{plain}}(\gamma)\ge\frac12$ if $\gamma$ meets $x$ or $g$, and $D'(\gamma)>0$ if $\gamma$ is source-free; (iii) hence $D'(H_i)\ge\alpha_0>0$ at every stage with a strict index increase, $\alpha_0=\min(\frac12,\alpha,\text{Ba{\l}aban's source-free margins})$; (iv) the H\"older exponent $\alpha\in(0,1)$ is free. The same holds, with the same four statements, for a connected vacuum graph $\tilde H$ at $B=0$ with $m_h\ge1$ $h$-vertices, no $g$-vertex and either $m_h\ge2$ or total order $\alpha_e+2\beta_\lambda\ge4$ in the order $(\alpha_e,\beta_\lambda)$ in $(e,\lambda)$ (the orders of the counterterm units included), i.e.\ for the $E=0$, $G$-free members outside $P_0\defeq\{(0,0),(0,1),(2,0)\}$. \end{theorem} \begin{proof} A connected $S$ is a component of some prefix if and only if every adjacent outside line is later than every line of $S$ (the prefix criterion), so a component containing a debited leg contains all of $S$ and $D'(\gamma)\ge D_{\mathrm{plain}}(\gamma)$ for every prefix component, wherever the credit sits (the stage bookkeeping); a leg receives at most one Taylor debit $1+\alpha$ plus one $\alpha$ per unit having it as an external leg, and a source line at most one debit (the single-debit lemma); scalar-closed subgraphs with an odd number of photon legs vanish by $\Tr q^{\mathrm{odd}}=0$ (the parity lemma). Inducting over units, finest first, every $x$- or $g$-component has $D_{\mathrm{plain}}\ge\frac12$ by the census in the proof of Theorem~\ref{cpi:thm:Ri} (its case \textup{(T)} needs $E=0$, impossible with an external leg or a $g$, and in the vacuum case the order is off $P_0$ or $m_h\ge2$; its two degree-zero exceptions kill the graph), so the first divergent component is source-free and Ba{\l}aban's type list applies, whence (i); for a source-free component the prefix criterion makes the construction Ba{\l}aban's embedded one, so $D'>0$ is Theorem~\ref{cpi:thm:78}(ii) componentwise, whence (ii); (iii) and (iv) follow by summing components. \end{proof} \begin{proposition}[the failing class]\label{cpi:prop:F} The bounds \eqref{cpi:eq:K338}--\eqref{cpi:eq:K339} fail on the order-$e^2$ $h$-linear vacuum family $T_h^{(2,0)}$: the photon exchange on the scalar loop with one $h$-insertion, the seagull tadpole and the two-photon current loop, with their $\dm(2,0)$ partner, which have degree $0$ as whole graphs and no counterterm in King's scheme. Precisely, with $c_Z\defeq e^2/(4\pi^2)$ the coefficient of the logarithmic growth of the order-$e^2$ Wick constant: (a) at fixed $\delta=\delta_k$ the position sum of the family grows like $c_Zk\ln5\int h$, while \eqref{cpi:eq:K338} summed over positions is at most $C(\delta)|T|$ uniformly in $k$; (b) \eqref{cpi:eq:K338} and \eqref{cpi:eq:K339} cannot both hold there: with King's position sum \cite[(3.40)--(3.42)]{K1} the per-step difference would be at most $C(1+K-k)^{p_0}[5^{-\gamma k}\delta^{-\beta}+\delta^{\sigma'}]|T|\to C'\delta^{\sigma'}|T|$ as $k\to\infty$ at fixed $\delta$, while it tends to $c_Z\ln5\int h\ne0$. \end{proposition} \begin{proof} The order-$e^2$ $h$-linear field-free part of the effective action changes per step by an amount tending to $c_Z\ln5\int h$, the logarithmic growth $z_\eps-z_{5\eps}\to c_Z\ln5$ of the order-$e^2$ Wick constant (Proposition~\ref{cpi:rot:log}; the schemes of Definition~\ref{rot:def:scheme} absorb it), so its value after $k$ steps is $c_Zk\ln5\int h+O(1)$: the whole graph is the one stage of degree $0$ of the ordered bound, whose index sum $\sum_{jd+\frac12$, $\sigma'<\sigma$, with $C$ independent of $k$, $K$, $n$. The classes covered are the field-dependent $h$-families, the families with an $h$- and a curvature vertex, the $h$-independent families (source-free and with $g$), and the field-dependent members of every other family. Not covered are the field-independent, $G$-independent constants (the $h$-linear vacuum values at zero field, the $h$-free vacuum energy, the $h$-nonlinear $g$-free vacuum graphs); they are $t$- and $G$-independent. \item[(ii)]\label{cpi:thm:Rii} (\emph{Fixed scalar source, curvature source zero.}) At $G=0$ and a fixed smooth source $h'$, let $N_{\mathrm{ii}}$ consist of every $E_N(H)$ of (i) and of every field-independent graph $H$ depending on $h'$ except the $h$-linear ones of order in $P_0$; it contains the $h$-linear vacuum classes of order $\alpha_e+2\beta_\lambda\ge4$ (every order outside $P_0$, $\alpha_e$ being even for vacuum classes at $G=0$) and the $h$-nonlinear vacuum graphs (two or more $h$-vertices). For every member, \eqref{cpi:eq:K339} holds, \eqref{cpi:eq:K338} holds with $(1+K-k)^{p_0}$, and the position sum is at most $C[5^{-\gamma'k}\delta_k^{-\beta'}+\delta_k^{\sigma'}]|T|$. \item[(iii)]\label{cpi:thm:Riii} (\emph{The one-cutoff bound for the constant-source shift.}) (a) For every graph $H$ with at least one $h$-vertex other than the $D=0$ members of the order-$(2,0)$ $h$-linear family $T_h^{(2,0)}$, and for the $(0,0)$, $(0,1)$ members, \[|E^{(k)}(H;\{y\},\{z\})|\le C(1+K-k)^{p_0}\delta_k^{\rho}p^{n_1}e^{-\delta\dist}\] with $C$ independent of $k$, $K$. The order-$(2,0)$ $h$-linear field-free members with a block-spin vertex belong to \textup{(a)}: every prefix component has split degree at least $2$, and the plain multiscale bound bounds them by $O(1)$. (b) For $T_h^{(2,0)}$ the position-summed bound $\sum_{\mathrm{positions,\,indices}}|E'|\le C(1+k)$ of Theorem~\ref{cpi:hrs:G27} holds; the factor $1+k$ is necessary (Proposition~\ref{cpi:prop:F}), and the per-position exponential-decay clause of \textup{(a)} is not claimed for this family and is not used. For a graph $H$ at non-zero background, \eqref{cpi:eq:K338} is read through \cite[(3.55)]{K1}: the core $\tilde H_0$ is the $A_k=0$ value, \textup{(b)} if $H\in T_h^{(2,0)}$ and \textup{(a)} otherwise (the torus value and King's box-localised core differ by graphs with a $\partial G_k$ line, which \textup{(a)} covers); every other $\tilde H$ has an external leg, so \textup{(a)} applies, with the $w$-sum of Lemma~\ref{cpi:lem:R0}. The factor $(1+K-k)^{p_0}$ is a power of $\log\delta_k^{-1}$, absorbed by enlarging $n_1$. \end{enumerate} \end{theorem} \begin{proof} \emph{The census.} With the degree by legs $D(\gamma)=\sum_{v\in\gamma}u_v-3+\frac12E_\gamma$ and the vertex values $u=2$ ($h$-vertex), $1$ (quartic vertex), $\frac12$ or $\frac32$ (one-photon current vertex), $\ge1$ (current and seagull vertices with more photon legs), $\frac52$ (curvature vertex, after the exact summation by parts onto $g$) and $\ge\frac52$ (block point carrying a line), a connected field-independent $h$-linear vacuum graph without curvature vertex has $D=\frac12(\alpha_e+2\beta_\lambda)-1$, so $D\le0$ exactly for the orders in $P_0$ (the subtracted tadpole, $D=-1$; the figure-eight and the family $T_h^{(2,0)}$, $D=0$), of which $(0,0)$ and $(0,1)$ have exact counterterms and $(2,0)$ none; a connected $\gamma$ with exactly one $h$-vertex $x$ has $D(\gamma)=-1+\sum_{v\ne x}u_v+\frac12E_\gamma\le0$ only in the case \textup{(T)}, $E_\gamma=0$ and $\gamma$ a whole $h$-linear vacuum graph of order in $P_0$ or one of its counterterm partners, and in the case \textup{(F)}, $\gamma=\gamma_F$ the $h$-vertex joined by both lines to a one-photon current vertex whose photon leg is external ($D=0$); a connected $\gamma$ with a curvature vertex and no $h$-vertex has $D\ge\frac12$ except $\gamma_G$, the curvature vertex joined to a one-photon current vertex with a scalar loop ($D=0$); two or more $h$-vertices, or an $h$- and a curvature vertex, give $D\ge\frac12$; every graph of \cite[(3.55)]{K1} in which $\gamma_F$ or $\gamma_G$ occurs with internal lines vanishes identically, its closed scalar loop carrying $\Tr q=0$ (Furry's theorem at graph level); and a counterterm insertion is a unit, treated by the census as a point vertex. \emph{The bounds.} Expanding $E(H)$ by \cite[(3.43)--(3.55)]{K1} at $A_k$ (and at $A_k=0$ when $s=0$ and $H$ has no $g$-vertex), the core graph cancels in $E_N(H)$, so every $\tilde H$ entering $E_N(H)$ is connected with an external leg or a curvature vertex, and every connected component of an ordered subgraph of $\tilde H$ has positive degree: with a block point by Lemma~\ref{cpi:ir:BLKnat}; with an $h$- or a curvature vertex by the census, case \textup{(T)} being excluded by the external leg or $g$-vertex (in (ii) by the order off $P_0$ or $m_h\ge2$) and the cases \textup{(F)}, $\gamma_G$ vanishing; and source-free by (I3), whose rewriting does not lower the degrees used in the other cases (the rewriting property (RW), by the stage bookkeeping of Theorem~\ref{cpi:thm:TBA3}, on the $E=0$, $G$-free members through its vacuum form); so every non-vanishing $\tilde H$ satisfies (DEG) with a positive minimum $\alpha_0$ over the finitely many graphs and orderings, the infrared indices occurring only on counterterm lines, and (I1) gives \eqref{cpi:eq:K356} with $\gamma<\alpha_0$, Lemma~\ref{cpi:lem:R0} gives \eqref{cpi:eq:K339}, (I2) gives \eqref{cpi:eq:K338} with $(1+K-k)^{p_0}$, $p_0\le\bar n$, and King's position sum \cite[(3.40)--(3.42)]{K1} gives the volume bound, with $\beta=d+\frac12$ and $\sigma'=\sigma$ worsened by an arbitrarily small amount to absorb $p^{n_1}$ and the logarithms. In (iii)(a) the $(0,0)$ member is the subtracted tadpole at zero coarse field and the $(0,1)$ member the figure-eight minus its $\dm_1$ partner, both bounded directly; in (iii)(b) only the whole graph of a member of $T_h^{(2,0)}$ has $D=0$, so the last geometric sum of the multiscale bound, $\sum_{j\le k-1}(L^j\eta)^0=k$, is the factor $1+k$, and the Taylor step with the $\dm(2,0)$ partner and the lifting of the degree-zero line are Theorem~\ref{cpi:hrs:G27}; the $h$-free, field-free vacuum constants cancel in the ratio of partition functions, and (ii) makes no claim about them. \end{proof} \emph{The position-summed bound on the failing family.} For the family $T_h^{(2,0)}$, on which the per-position bound \eqref{cpi:eq:K338} fails, the proof of (N6) uses a position-summed bound. The members of the family at $A_k=0$, with the source vertex $x$ carrying the two scalar legs $l_e,l_{e'}$, are the self-energy insertion \textup{(E)} (a scalar line $s$ and a photon line $p$ between the two legs, the \emph{unit} $S=\{s,p\}$, in the four derivative placements $dd$, $du$, $ud$, $uu$ of the two current derivatives), the two-photon-current tadpole \textup{(T2)}, the seagull tadpole \textup{(T3)} and the counterterm member \textup{(P)}, in which the mass counterterm $\delta m^2$ of \cite[(3.33)]{K1} sits between the legs and which splits into the self-energy unit $P_E$ and the seagull-tadpole unit $P_{T3}$; every line carries a slice index $j0$ (at least $1$ for the components containing $x$) and whole-graph degree $0$. \end{theorem} \begin{proof} \emph{Slice matching.} The unit of \textup{(E)} with slices $(j_s,j_p)$, $j_s,j_p0$. \begin{lemma}[the bound on $B_1$]\label{cpi:lem:hrs:NP1} On $\supp\chi_k$, $|B_1-B_1'|\le\frac12\|\Delta^{(k)}_T-\Delta'^{(k+1)}_T\|\,\|A_k\|^2\le C\lmu^{-2}p^2\,5^{-2k}\delta^{-5}|T|$. \end{lemma} \begin{proof} Both operators are King's zero-field photon operators on the unit torus $T^{(k)}$, diagonal in Fourier space, whose free symbols differ by at most $Ca5^{-2k}$ uniformly in the unit mass $\mu_k^2=\lmu^2\delta^2$; $\|A_k\|^2\le C\lmu^{-2}p^2\delta^{-5}|T|$ is the field bound \cite[(3.2)]{K1} in unit-lattice counting. \end{proof} \begin{theorem}[the bounds on $B_2$ and $B_3$]\label{cpi:thm:hrs:NP3} On $\supp\chi_k$, for every $\gamma_L<\frac12$ and $s>0$: \begin{enumerate} \item[(a)] $|B_2-B_2'|\le\frac12\|\Delta^{(k)}_T(A^{(k)})-\Delta'^{(k+1)}_T(A'^{(k+1)})\|\,\|\phi_k\|^2\le C_s\lambda^{-1/2}p^{c+2}\delta^{-7/2}(5^{-\gamma_Lk}+\delta^s)|T|$; \item[(b)] $|B_3-B_3'|\le C_sNe^2(1+\lmu^{-1})p^{c+2}\delta^{-3}\big(5^{-\gamma_Lk}+5^{-k}+\delta^s\big)|T|$. \end{enumerate} \end{theorem} \begin{proof}\emergencystretch=3em\relax On the universal cover $\Lambda\defeq\eps_K\Z^3$ of the fine torus $T_\eps=\Lambda/P_J$, periodisation commutes with products and inverses and is bounded on $\ell^2$ by the $\ell^1$ kernel norm (Schur's test), every operator of \cref{cpi:def:a02:setting} on $T$ is the periodisation of its cover version (block-local coercivity and the Combes--Thomas bound on $\supp\chi_k$), and the per-site traces $g^\Lambda_{j,x}\defeq\operatorname{Tr}_N[\ln K_{j,\Lambda}](x,x)$ are invariant under the gauge transformations $B\cdot(z-x_0)$ of the cover; the two models' lifts $A^{(k)}=HA_k$ and $A'^{(k+1)}=H'A_k$ of the same unit-lattice field differ by at most $Cp(5^{-\gamma_Lk}+\lmu\delta\,5^{-2k})$ at every common fine site (the constant mode exactly, the rest by the kernel comparison of \cite[(3.71)]{K1} at zero field against the first-difference bound of \cite[(3.2)]{K1}). \emph{(a)} is the field bound on $\|\phi_k\|^2$ \cite[(3.2)]{K1} times the operator bound $\|\Delta^{(k)}_T(A^{(k)})-\Delta'^{(k+1)}_T(A'^{(k+1)})\|\le C_sp^c(5^{-\gamma_Lk}+\delta^s)$, proved on the cover by gauging both models at a common coarse site and expanding the gauged kernels in the coupling to an order $n_T\ge2s+2$, compared chain by chain through the zero-field infinite-lattice lines at the rate $5^{-\gamma_Lk}$, with Taylor remainders at most $C\delta^s$. \emph{(b)} $B_3=\frac12\sum_{j0$: the factor $\abs{p'}^2$ must read $\abs{p'}^2+m^2(L^k\eps)^2$, since at $p'=0$ the right side as stated vanishes while the left side is positive for King's massive scalar; with this correction Propositions~3.8 and 3.9 of \cite{K1} go through, only the truncation of (4.21) to the alias-free term being used, and (4.28) is not used. The step (4.36)--(4.37) in the proof of \cite[Lemma~4.5]{K1} is replaced by the explicit bound of Remark~\ref{cpi:rot:errata}(iv). \item[(iv)] The statement \cite[p.~677]{K1} that the sources are proportional to $(5^k\eps)^\alpha$ in the small-field region does not hold for the marginal graph of Proposition~\ref{cpi:rot:log}, and the sentence there that the inclusion of the sources in Theorem~3.1 is straightforward carries no argument; the inclusion of the sources is used only as the small-field half of \cite[Thm.~3.1]{K1} at fixed sources, in the form of Theorem~\ref{cpi:ct:CTB}(c), and at a source with a positive part the walled envelope takes its place on the cone $2\sup h_+0$, $\psi=0$ on the $2$-neighbourhood of $R^{-1}C_1$ for every $R\in\Rgen$, the wall condition $n\langle\psi\rangle_{X_K}>\Delta_*|T|/2$ for every $K$, the block alignment and the grid offset of the corners of $H'$; the conditions other than the vanishing of $\psi$ near $R^{-1}C_1$ do not depend on $R$, $J_{\rm pos}$ being the maximum over the finite set $\Rgen$ and the three torus pairs, Proposition~\ref{rot:thm:POS}) with $N_1(X)\subset(-25MJ',25MJ')^3$ and $w=0$ on $N_1(X)$; the cell $(-25MJ',25MJ')^3$ is a fundamental domain of $D_{J'}$ for both lattices, $R^{-1}\Lambda_k$ descending to $D_{J'}$ by Lemma~\ref{rot:lem:geom}(b), and sources supported in $X$ are periodised from it. Put $b_\circ:=\frac75M+2$; for $X\supset Q_{b_\circ}$ the vanishing of $\psi$ near $R^{-1}C_1$ follows from $w=0$ on $N_1(X)$, since $R^{-1}C_1\subset Q_{7M/5}$ (Lemma~\ref{rot:lem:geom}(c)), so that $\cA(X)$ then consists of all walled configurations of the class $\mathcal C$ vanishing on $N_1(X)$. If $X\subset Y$ then $\cA(Y)\subset\cA(X)$. Assume (K$_0$), which holds for every $\fc\in\cA(X)$ when the couplings lie outside the set $E_W$ of Proposition~\ref{rot:thm:POS} (Theorem~\ref{rot:hyp:K}). For a real $2$-form $f\in C^\infty_c$ supported in $X$ and $\fc\in\cA(X)$, the source restricted to the cell and periodised is smooth, and by Lemma~\ref{rot:lem:K} the \emph{walled value} \[ \cW_\fc(f,0):=\lim_{k\to\infty}T_{\fc,\Lambda_k}(f)=\lim_{k\to\infty}T_{\fc,R^{-1}\Lambda_k}(f)\qquad(R\in\Rgen) \] exists. For $b>0$ and $f$ supported in $Q_b$ let $I_b(f):=\inf_{\fc\in\cA(Q_b)}\cW_\fc(f,0)$, which is nondecreasing in $b$, and \[ \cW^{\env}(f,0):=\lim_{b\to\infty}I_b(f). \] \end{definition} \begin{lemma}[the free direction]\label{rot:lem:free} Assume (K$_0$). \begin{enumerate} \item[(a)] If $f$ is a real $2$-form supported at distance $>1$ from $\R^3\setminus C_L$ and $\fc\in\cA(\bar C_L)$, then $\cW_\fc(f,0)\ge Z^K_L(\mathrm if,0):=\lim_k\langle e^{\mathrm iF(f)}\rangle^K_{k,L}$. \item[(b)] $\cW^{\env}(f,0)\ge Z_\infty(\mathrm if,0)>0$ for every real $2$-form $f\in C^\infty_c$. \end{enumerate} \end{lemma} \begin{proof} (a) At each $k$, Lemma~\ref{rot:lem:cut} with $X=C_L\cap\Lambda_k$ and Lemma~\ref{lem:s04:G} give $T_{\fc,\Lambda_k}(f)\ge\langle e^{\mathrm iF(f)}\rangle^\ND_{k,L}\ge\langle e^{\mathrm iF(f)}\rangle^K_{k,L}$; let $k\to\infty$, the limit of the right side existing by King's cube limit \cite[Thm.~2.1]{K2} (Theorem~\ref{thm:k01:EC}) with Theorem~\ref{rot:thm:W}(i). (b) If $\bar C_L\subset Q_b$ then $\cA(Q_b)\subset\cA(\bar C_L)$, so $I_b\ge Z^K_L(\mathrm if,0)$; let $b\to\infty$ and then $L\to\infty$ (Theorem~\ref{rot:thm:W}(iii)). Positivity: by Lemma~\ref{rot:lem:GD}(a), $\langle e^{\mathrm iF(f)}\rangle^K_{k,L}\ge e^{-\ip v{Cv}/2}$ with $v$ the vector with $\ip vA=F(f)$ and $C$ the Neumann photon covariance of the cube, and $\ip v{Cv}\le\mu_0^{-2}\|d^*_\eps f\|^2$, with $\|d^*_\eps f\|$ the lattice $L^2(\eps_k)$ norm of the lattice coderivative $d^*_\eps f$ of $f$, is bounded uniformly in $k$ and $L$. \end{proof} \begin{lemma}[independence of the orientation]\label{rot:lem:Rind} Assume (K$_0$). For $R\in\Rgen$ and every real $2$-form $f\in C^\infty_c$, $\lim_{L\to\infty}\inf_{\fc\in\cA(R^{-1}Q_{ML})}\cW_\fc(f,0)=\cW^{\env}(f,0)$, the infimum being defined as soon as $\supp f\subset Q_{5ML/7}$. \end{lemma} \begin{proof} By Lemma~\ref{rot:lem:geom}(c), $Q_{5ML/7}\subset R^{-1}Q_{ML}\subset Q_{7ML/5}$, hence $\cA(Q_{7ML/5})\subset\cA(R^{-1}Q_{ML})\subset\cA(Q_{5ML/7})$, and \[ I_{5ML/7}(f)\le\inf_{\cA(R^{-1}Q_{ML})}\cW_\fc(f,0)\le I_{7ML/5}(f). \] Both bounds tend to $\cW^{\env}(f,0)$. \end{proof} \begin{theorem}[\textup{(TIGHT$_0$)} and \textup{(TIGHT$_2$)}: the closing statement]\label{rot:hyp:TIGHT} Let the couplings lie outside the set $E_W$ of Proposition~\ref{rot:thm:POS}, so that (K$_0$) holds for every configuration of the class $\mathcal C$, hence for every member of every $\cA(X)$ (Theorem~\ref{rot:hyp:K}, Definition~\ref{rot:def:walls}). Then: \textbf{(TIGHT$_0$)}: $\cW^{\env}(f,0)\le Z_\infty(\mathrm if,0)$ for every real $2$-form $f\in C^\infty_c$. Equivalently: for every $b>0$, every real $2$-form $f$ supported in $Q_b$ and every $\eta>0$ there is $\fc\in\cA(Q_b)$ with $\cW_\fc(f,0)\le Z_\infty(\mathrm if,0)+\eta$; it involves only (K$_0$) and King's state (Definition~\ref{rot:def:object}). For $b\ge\frac75ML+2$ such a configuration has $\psi=0$ on the $2$-neighbourhood of $R^{-1}C_L$ for every $R\in\Rgen$ (Lemma~\ref{rot:lem:geom}(c)), so it lies in the class $\mathcal C$ for $R$ and $L$ as well. \textbf{(TIGHT$_2$)}: for every real $2$-form $f\in C^\infty_c$, $\lim_{b\to\infty}\sup_{\fc\in\cA(Q_b)}\sigma^2_\fc(f)\ge S_2(f)$, where $\sigma^2_\fc(f):=\lim_k\langle F_{\Lambda_k}(f)^2\rangle_{\fc,\Lambda_k}$ and $S_2(f):=\lim_J\lim_k\langle F_{\Lambda_k}(f)^2\rangle^K_{k,J}$, on King's cubes (Definition~\ref{def:s02:Kcube}). \end{theorem} \begin{proof} (TIGHT$_0$) is Theorem~\ref{rot:thm:E}, from (K$_0$) (Theorem~\ref{rot:hyp:K}), the heavy-shell estimate (RES$_0$) at one configuration (Theorem~\ref{rot:thm:shell}) and the dead-shell chain (Theorem~\ref{rot:thm:SHT}) at that configuration; (TIGHT$_2$) follows from (TIGHT$_0$) by Lemma~\ref{rot:lem:rel}. \end{proof} Below, ``assume (TIGHT$_0$)'' refers to this statement. The equivalence in (TIGHT$_0$) holds because $I_b$ is nondecreasing in $b$. By Lemma~\ref{rot:lem:free}, (TIGHT$_0$) is equivalent to $\cW^{\env}(f,0)=Z_\infty(\mathrm if,0)$ for every real $2$-form $f$: \emph{King's state is the limit of the lower envelopes of walled tori, the continuum limit being taken first}. The limits defining $\sigma^2_\fc$ and $S_2$ exist, by Lemma~\ref{rot:lem:K} under (K$_0$) and by Theorem~\ref{rot:thm:W} respectively, since second derivatives of locally uniformly convergent entire functions converge. \begin{remark}[the configuration class]\label{rot:rem:class} (K$_0$) is proved by Theorem~\ref{rot:thm:CT} for the configurations of the class $\mathcal C$ of Definition~\ref{rot:def:class}, of which the sets $\cA(X)$ of Definition~\ref{rot:def:walls} consist. The configuration that Theorem~\ref{rot:thm:E} produces lies in it: its wall is $(n+K)\psi$ with $n+K\ge n_*>0$, mass-increasing; the wall condition is linear in the height and holds for $K$ large at fixed $T$, which Remark~\ref{rot:rem:Eproof}(iii) allows; $J$ (the $J'$ of Theorem~\ref{rot:thm:shell}), the block alignment of $H'$, $T$ and the grid offset of the corners of $H'$ are choices of the torus and of the box that Theorem~\ref{rot:thm:shell} makes ($J'\ge\max(J_0,J_{\rm pos})$ among them), the last moving $H'$ by less than $M'\ell_f$ ($\ell_f$ the side of the blocks of the final scale, Definition~\ref{rot:def:class}). \end{remark} \begin{lemma}\label{rot:lem:rel} Under (K$_0$), (TIGHT$_0$) implies (TIGHT$_2$), and $\lim_b\sup_{\cA(Q_b)}\sigma^2_\fc(f)\le S_2(f)$ always; so (TIGHT$_2$) is an equality. \end{lemma} \begin{proof} Fix $f$, $b$ and $\eta'>0$. All functions of $t$ below are real and even. Since $\cos x\ge1-\frac{x^2}2$ and $\cos x\le1-\frac{x^2}2+\frac{x^4}{24}$ at every lattice level, and derivatives converge in the limits, \[ \cW_\fc(tf,0)\ge1-\tfrac{t^2}2\sigma^2_\fc(f),\qquad Z_\infty(\mathrm itf,0)\le1-\tfrac{t^2}2S_2(f)+\tfrac{t^4}{24}M_4(f), \] with $M_4(f):=\partial_t^4Z_\infty(tf,0)|_{t=0}<\infty$. By (TIGHT$_0$) applied to $tf$, for every $t>0$ and $\eta>0$ there is $\fc\in\cA(Q_b)$ with $\cW_\fc(tf,0)\le Z_\infty(\mathrm itf,0)+\eta$, hence $\sigma^2_\fc(f)\ge S_2(f)-\frac{t^2}{12}M_4(f)-\frac{2\eta}{t^2}$. Choosing $t$ with $t^2M_4(f)/12<\eta'/2$ and then $\eta=t^2\eta'/4$ gives $\sup_{\cA(Q_b)}\sigma^2_\fc(f)\ge S_2(f)-\eta'$. For the second claim, let $\fc\in\cA(\bar C_L)$ with $f$ supported well inside $C_L$. At every $k$, Lemma~\ref{lem:s04:G} and Lemma~\ref{rot:lem:cut}, as in the proof of Lemma~\ref{rot:lem:free}(a), give $T_{\fc,\Lambda_k}(tf)\ge\langle e^{\mathrm itF(f)}\rangle^K_{k,L}$; so $\cW_\fc(tf,0)\ge Z^K_L(\mathrm itf,0)=\lim_k\langle e^{\mathrm itF(f)}\rangle^K_{k,L}$ for all real $t$, with equality at $t=0$, and both sides are even and smooth, so $\sigma^2_\fc(f)\le\langle F(f)^2\rangle_{C^K_L}:=\lim_k\langle F(f)^2\rangle^K_{k,L}$. Hence $\sup_{\cA(Q_b)}\sigma^2_\fc(f)\le\langle F(f)^2\rangle_{C^K_L}$ whenever $\bar C_L\subset Q_b$. By Theorem~\ref{rot:thm:W}(ii), comparing second derivatives at $t=0$, the cube variances are nonincreasing in $L$, and they converge to $S_2(f)$. \end{proof} % \section{Rotation invariance}\label{rot:sec:rotinv} \begingroup\emergencystretch=3em\relax This section derives the invariance of King's state from (K$_0$) and (TIGHT$_0$): Theorem~\ref{rot:thm:B} gives the invariance under the generating rotations by letting $L\to\infty$ in the chain~\eqref{rot:eq:chain}, Lemma~\ref{rot:lem:refl} adds a reflection, Theorem~\ref{rot:thm:C} passes from the dense subgroup to $O(3)$ by the continuity lemma for convex functionals, and Theorem~\ref{rot:thm:curv} states the result for the curvature sector, with the translations of Theorem~\ref{rot:thm:W}(v). Theorem~\ref{rot:thm:main} is this theorem once (K$_0$) and (TIGHT$_0$) are proved (Sections~\ref{rot:sec:K} and \ref{rot:sec:shell}).\par\endgroup \begin{theorem}[invariance under the generators]\label{rot:thm:B} Assume (K$_0$) and (TIGHT$_0$) (Theorems~\ref{rot:hyp:K} and~\ref{rot:hyp:TIGHT}). Then for every $R\in\Rgen$ and every real $2$-form $f\in C^\infty_c$, \[ Z_\infty(\mathrm i\,R\cdot f,0)=Z_\infty(\mathrm if,0), \] and consequently the same holds for every $R\in G_0$. \end{theorem} \begin{proof} Fix $R\in\Rgen$ and a real $2$-form $f$ supported in $Q_\rho$. \emph{Step 0 (geometry).} Take $L$ with $ML\ge\frac75\rho+2$. By Lemma~\ref{rot:lem:geom}(c), $R\cdot f$ is supported in $Q_{7\rho/5}$, at distance $>1$ from $\R^3\setminus C_L$, and $f$ is supported in $Q_{5ML/7}\subset R^{-1}Q_{ML}$. Let $\fc\in\cA(R^{-1}Q_{ML})$. Then $R^{-1}\Lambda_k$ descends to $D_{J'}$ (Lemma~\ref{rot:lem:geom}(b)), and $B'_k:=R^{-1}(C_L\cap\Lambda_k)$ is a set of sites of $R^{-1}\Lambda_k$ which, with its nearest neighbours, lies in the open cell, and on which $w=0$. Since $R\cdot f$ has Euclidean distance $>1$ from $\R^3\setminus C_L$, every plaquette of $R^{-1}\Lambda_k$ at which $f$ does not vanish lies in $B'_k$. \emph{Step 1 (cutting on the rotated lattice).} By Lemma~\ref{rot:lem:cut}, at every $k$, \[ \langle e^{\mathrm iF(f)}\rangle^\ND_{R^{-1}\Lambda_k,B'_k}\le T_{\fc,R^{-1}\Lambda_k}(f). \] \emph{Step 2 (passive rotation, and King's cube).} By Lemma~\ref{rot:lem:pass} with $O=R$, $\langle e^{\mathrm iF(f)}\rangle^\ND_{R^{-1}\Lambda_k,B'_k}=\langle e^{\mathrm iF(R\cdot f)}\rangle^\ND_{k,L}$, and by Lemma~\ref{lem:s04:G} the functional of King's cube $C^K_L$ for the source $R\cdot f$ is bounded by that of the box: $\langle e^{\mathrm iF(R\cdot f)}\rangle^K_{k,L}\le\langle e^{\mathrm iF(R\cdot f)}\rangle^\ND_{k,L}$. \emph{Step 3 (continuum limits).} As $k\to\infty$ the left side converges by King's cube limit \cite[Thm.~2.1]{K2} (Theorem~\ref{thm:k01:EC}) with Theorem~\ref{rot:thm:W}(i) and the right side by (K$_0$) (Definition~\ref{rot:def:walls}): \[ Z^K_L(\mathrm i\,R\cdot f,0)\le\cW_\fc(f,0)\qquad\text{for every }\fc\in\cA(R^{-1}Q_{ML}). \] \emph{Step 4 (infimum, volume limit, closing estimate).} Take the infimum over $\fc\in\cA(R^{-1}Q_{ML})$ and let $L\to\infty$. By Theorem~\ref{rot:thm:W}(iii), Lemma~\ref{rot:lem:Rind} and (TIGHT$_0$), \[ Z_\infty(\mathrm i\,R\cdot f,0)=\lim_{L\to\infty}Z^K_L(\mathrm i\,R\cdot f,0)\le\cW^{\env}(f,0)\le Z_\infty(\mathrm if,0). \] \emph{Step 5 (the inverse).} $R^{-1}\in\Rgen$ and $R\cdot f$ is a real $2$-form, so Steps 0--4 for $(R^{-1},R\cdot f)$ give $Z_\infty(\mathrm if,0)\le Z_\infty(\mathrm i\,R\cdot f,0)$. Since the real $2$-forms are $G_0$-invariant, the claim extends to $G_0$. \end{proof} The order of limits is: at fixed $R$, $f$, $L$ and $\fc$, first $k\to\infty$; then the infimum over $\fc$; then $L\to\infty$. The continuum limit is always taken first, at fixed volume and fixed wall, as in King's construction~\cite[p.~336]{K2}, and (K$_0$) is used only as an equality of continuum limits, not at fixed lattice spacing. \begin{lemma}[reflection]\label{rot:lem:refl} Let $\sigma_3=\mathrm{diag}(1,1,-1)$. Then $Z_\infty(\mathrm i\,\sigma_3\cdot f,0)=Z_\infty(\mathrm if,0)$ for every real $2$-form $f\in C^\infty_c$. \end{lemma} \begin{proof} $\sigma_3$ preserves $\Lambda_k$, and maps $C_J\cap\Lambda_k$ onto $X_+:=(C_J+\eps_ke_3)\cap\Lambda_k$. By Lemma~\ref{rot:lem:pass}, $\langle e^{\mathrm iF(\sigma_3\cdot f)}\rangle^\ND_{k,J}$ equals the corresponding expectation in the Neumann--Dirichlet state on $X_+$ defined with the frame $(e_1,e_2,-e_3)$. Written in the frame $(e_\alpha)$ (with $A'_3(x)=-A_3(x-\eps e_3)$), this is the BFS measure in which $\phi$, $A_1$, $A_2$ live on $X_+$ and $A_3$ on $X_+-\eps e_3=C_J\cap\Lambda_k$, with Dirichlet conditions for $\phi$ on $X_+$ and photon gradient terms within these sets. It is obtained from the Neumann--Dirichlet state on $C_{J+1}\cap\Lambda_k$ by deleting photon gradient terms and cosine parts of hopping terms, since $X_+\cup(X_+-\eps e_3)\subset C_{J+1}$; and the state on $C_{J-1}\cap\Lambda_k$ is obtained from it in the same way, since $C_{J-1}\subset C_J\cap(C_J+\eps e_3)$. As in Lemma~\ref{rot:lem:cut}, for $J$ large and every $k$, $\langle e^{\mathrm iF(f)}\rangle^\ND_{k,J-1}\le\langle e^{\mathrm iF(\sigma_3\cdot f)}\rangle^\ND_{k,J}\le\langle e^{\mathrm iF(f)}\rangle^\ND_{k,J+1}$ (this is \cref{lem:s02:refl} at zero scalar source). On King's cubes: this sandwich at $J$ and at $J-1$, and Lemma~\ref{lem:s04:G}, applied at its two ends and twice to the source $\sigma_3\cdot f$ in the middle, give for $J$ large and every $k\ge0$ \begin{multline*} \langle e^{\mathrm iF(f)}\rangle^K_{k,J-2}\le\langle e^{\mathrm iF(f)}\rangle^\ND_{k,J-2}\le\langle e^{\mathrm iF(\sigma_3\cdot f)}\rangle^\ND_{k,J-1} \le\langle e^{\mathrm iF(\sigma_3\cdot f)}\rangle^K_{k,J}\\ \le\langle e^{\mathrm iF(\sigma_3\cdot f)}\rangle^\ND_{k,J} \le\langle e^{\mathrm iF(f)}\rangle^\ND_{k,J+1}\le\langle e^{\mathrm iF(f)}\rangle^K_{k,J+2}; \end{multline*} let $k\to\infty$ by Theorem~\ref{thm:k01:EC} and then $J\to\infty$ (Theorem~\ref{rot:thm:W}(iii)). The comparison of $C^K_J$ with $C^K_{J\mp1}$ is also obtained by deletions and kills when $M\ge2\eps_k$, which holds at every $k$ for the $M$ of Definition~\ref{rot:def:geom} (\cref{lem:s04:C}(a)); the two-step form used here needs no such condition. \end{proof} \begin{theorem}[closure: from the generators to $O(3)$]\label{rot:thm:C} Suppose $Z_\infty(\mathrm i\,R\cdot f,0)=Z_\infty(\mathrm if,0)$ for all real $2$-forms $f\in C^\infty_c$ and $R\in\Rgen$. Then for every $R\in O(3)$, every $N$, all real $2$-forms $f_j\in C^\infty_c$ and all $z\in\C^{N}$, \[ Z_\infty\Big(\sum_jz_j\,R\cdot f_j,0\Big)=Z_\infty\Big(\sum_jz_jf_j,0\Big). \] \end{theorem} \begin{proof} \emph{Complex coefficients.} The hypothesis passes to $G_0$, since the real $2$-forms are invariant under $O(3)$ and $\Rgen=\Rgen^{-1}$. For $R\in G_0$ and a finite family, the difference of the two sides, as a function of $z\in\C^{N}$, is entire (Theorem~\ref{rot:thm:W}(iii)) and vanishes on $(\mathrm i\R)^N$, where $\sum_jz_jf_j=\mathrm i\sum_jt_jf_j$ with $t\in\R^N$ and the hypothesis applies to the real $2$-form $\sum_jt_jf_j$; hence it vanishes identically. \emph{Closure in $SO(3)$.} At real coefficients, $\Phi(R\cdot f)=\Phi(f)$ for $f\in E_\rho$ and $R\in G_0$. The map $R\mapsto R\cdot f\in E_\rho$ is continuous on $O(3)$, and $\Phi$ is continuous on $E_\rho$ by Theorem~\ref{rot:thm:W}(iv). Hence $G:=\{R\in O(3):\Phi(R\cdot f)=\Phi(f)\text{ for all }\rho\text{ and }f\in E_\rho\}$ is a closed subgroup of $O(3)$ containing $G_0$, so $G\supset SO(3)$ by Lemma~\ref{rot:lem:geom}(d). \emph{Reflection.} Lemma~\ref{rot:lem:refl} and the first step (with $\sigma_3$ in place of $R$) give $\sigma_3\in G$, so $G=O(3)=SO(3)\cup\sigma_3SO(3)$. \emph{Conclusion.} For $R\in O(3)$ both sides of the claim are entire in $z\in\C^{N}$ and agree on $\R^{N}$. \end{proof} \begin{theorem}[the curvature sector]\label{rot:thm:curv} Assume (K$_0$) and (TIGHT$_0$) (Theorems~\ref{rot:hyp:K} and~\ref{rot:hyp:TIGHT}). Then King's state on curvature sources is $O(3)$-invariant: for every $R\in O(3)$, all real $2$-forms $f_j\in C^\infty_c$ and all $z\in\C^N$, $Z_\infty(\sum_jz_j\,R\cdot f_j,0)=Z_\infty(\sum_jz_jf_j,0)$, and all curvature Schwinger functions are $O(3)$-covariant. Moreover (Theorem~\ref{rot:thm:W}(v)), for every Euclidean motion $x\mapsto Rx+b$, $Z_\infty(\sum_jz_j\,\tau_bR\cdot f_j,0)=Z_\infty(\sum_jz_jf_j,0)$. \end{theorem} \begin{proof} Theorem~\ref{rot:thm:B} gives invariance under $\Rgen$ on imaginary coefficients; Theorem~\ref{rot:thm:C} gives $O(3)$ and the entire extension. Translations are Theorem~\ref{rot:thm:W}(v), proved there from Theorem~\ref{thm:k01:EC}, Lemma~\ref{lem:s04:G} and the cut sandwich; King's argument~\cite[Thm.~2.3(i)]{K2} is not used. \end{proof} \begin{corollary}[the curvature two-point function]\label{rot:cor:twopoint} Assume (K$_0$) and (TIGHT$_2$). Then \[ S_2(f_1,f_2):=\lim_J\lim_k\langle F_{\Lambda_k}(f_1)F_{\Lambda_k}(f_2)\rangle^K_{k,J} \] satisfies $S_2(R\cdot f_1,R\cdot f_2)=S_2(f_1,f_2)$ for all $R\in O(3)$ and real $2$-forms $f_1,f_2\in C^\infty_c$. \end{corollary} \begin{proof} Let $R\in\Rgen$. Steps 0--3 of the proof of Theorem~\ref{rot:thm:B}, with source $tf$, give $Z^K_L(\mathrm it\,R\cdot f,0)\le\cW_\fc(tf,0)$ for all real $t$ and every $\fc\in\cA(R^{-1}Q_{ML})$. Both sides are even and smooth in $t$ and equal $1$ at $t=0$, so $\langle F(R\cdot f)^2\rangle_{C^K_L}\ge\sigma^2_\fc(f)$. Since $\cA(Q_{7ML/5+2})\subset\cA(R^{-1}Q_{ML})$ (Lemma~\ref{rot:lem:geom}(c)) and $b\mapsto\sup_{\cA(Q_b)}\sigma^2_\fc(f)$ is nonincreasing, \[ \langle F(R\cdot f)^2\rangle_{C^K_L}\ge\sup_{\cA(Q_{7ML/5+2})}\sigma^2_\fc(f)\ge\lim_{b\to\infty}\sup_{\cA(Q_b)}\sigma^2_\fc(f)\ge S_2(f) \] by (TIGHT$_2$). The cube variances converge to $S_2$ (proof of Lemma~\ref{rot:lem:rel}), so $S_2(R\cdot f)\ge S_2(f)$, and the inverse generator gives equality; hence $S_2$ is $G_0$-invariant. By Lemma~\ref{rot:lem:GD}, $\langle F(f)^2\rangle^K_{k,J}\le\ip v{Cv}_\eps$ with $v=d^*_\eps f$, where $C$ is the photon covariance of the cube, the Neumann photon of $C_J\cap\Lambda_k$; the photon form is bounded below by $\mu_0^2$, so $0\le S_2(f)\le\mu_0^{-2}\|d^*f\|^2_{L^2}$. Since $S_2$ is a positive semidefinite quadratic form, $|S_2(f)-S_2(f')|\le S_2(f-f')^{1/2}S_2(f+f')^{1/2}$, and $R\mapsto S_2(R\cdot f)$ is continuous on $O(3)$. Density of $G_0$ gives $SO(3)$, Lemma~\ref{rot:lem:refl} (expanded to second order) gives $\sigma_3$, and polarisation gives the bilinear form. \end{proof} % \space% % % \section{The heavy-shell decoupling}\label{rot:sec:shell} This section proves the closing statement (TIGHT$_0$), the lower bound on King's state by the lower envelope of the walled tori. Lemma~\ref{rot:lem:GB} switches the photon boundary condition across a matter-free collar at a cost exponentially small in the width of the collar, uniformly in the spacing; Theorem~\ref{rot:thm:SHT} bounds the expectation with the matter dead on a shell by King's state plus the collar term; Theorem~\ref{rot:thm:E} derives (TIGHT$_0$) from the heavy-shell decoupling (RES$_0$) at one configuration; and Theorem~\ref{rot:thm:shell} proves (RES$_0$) from the four shell estimates (SW1$^\circ$)$_{\rm int}$, (SW2$^{\min}$), (SW3$^\circ$)$_{\rm int}$ and (SWT) on King's block-spin flow with the shell mass inside the matter covariance, which are stated and proved in Sections~\ref{rot:ssec:shellest} and~\ref{rot:sec:shellestB}. Throughout this section $\Lambda=\Lambda_k$ is the axis lattice, $\eps=\eps_k\le1/\mu_0$, and $m_\eps:=\frac2\eps\operatorname{arcsinh}(\eps\mu_0/2)$; since $t\mapsto\operatorname{arcsinh}(t)/t$ is decreasing, $m_\eps\ge0.96\,\mu_0$. Let $G_{\Z^3}$ be the Green's function of simple random walk on $\Z^3$ and \[ \mathsf a:=\sup_{z\in\Z^3\setminus\{0\}}|z|_\infty\,G_{\Z^3}(0,z), \] which is finite by the Green's function asymptotics of simple random walk in three dimensions~\cite[Chap.~4]{LL10}. For a set $X\subset\R^3$ put $\mathrm{vol}_\eps(X):=\eps^3\#(X\cap\Lambda)$. \subsection{Photon boundary conditions across a matter-free collar} The photon in Feynman form is Gaussian, so the comparison of two boundary conditions across a region without matter is a statement about Gaussian measures and lattice Green's functions; this subsection makes it quantitative, uniformly in the spacing (Lemma~\ref{rot:lem:GB}), which is what allows a walled torus to be compared with King's Neumann cube. The photon in Feynman form consists of three independent scalar Gaussian fields $A_\alpha$, $\alpha=1,2,3$, and on a finite graph $\mathcal G$ of sites of $\Lambda$ each has the precision $K_{\mathcal G}=\eps L_{\mathcal G}+\mu_0^2\eps^3$, with $L_{\mathcal G}$ the graph Laplacian of $\mathcal G$ with unit conductances and $\ip uv=\sum_xu(x)v(x)$; King's Neumann photon on a cube is the case of the induced subgraph~\cite[(2.5), p.~333]{K2}, and the periodic photon the case of the torus graph. \begin{lemma}[photon boundary switch]\label{rot:lem:GB} Let $H=Q_{b'}$ be a closed cube and $C_L$ a King cube with $\dist_\infty(H,\R^3\setminus C_L)=D+3$, where $D\ge\max(1,3/\mu_0)$ and $\eps\le\min(1/\mu_0,1/4)$. Let $\langle\cdot\rangle_\natural$ and $\langle\cdot\rangle_N$ be BFS measures with the same matter part: the sites of $H\cap\Lambda$, BFS single-site measures (arbitrary; wall and killed sites included), and hopping terms only between neighbouring sites of $H$, coupled to the photon variables of the bonds of $H$ as in King's action. Their photon parts are: \begin{itemize} \item[($\natural$)] the variables $A_\alpha(x)$ with $x,x+\eps e_\alpha\in C_L$, with the law of the periodic photon of a torus whose cell contains $N_1(C_L)$, conditioned on $A_\beta(y)=0$ for all other $(\beta,y)$; \item[(N)] King's Neumann photon on $C_L\cap\Lambda$. \end{itemize} Let $\Theta=e^{\mathrm iF(f)}e^{\sum_x\eps^3g(x)r_x^2-c}$ with $f$ supported on plaquettes of $H$, $g\ge0$ supported in $H$, and $c\in\R$; both expectations of $\Theta$ below are real, since the charge conjugation $(A,\vartheta)\mapsto(-A,-\vartheta)$ leaves each measure invariant and carries $\Theta$ to its complex conjugate. Then \[ 0\le\langle\Theta\rangle_\natural-\langle\Theta\rangle_N\le\langle|\Theta|^2\rangle_N^{1/2}\big(e^{\tau}-1\big)^{1/2},\qquad \tau\le C_{\rm GB}\,\mathsf a\,\mathrm{vol}_\eps(N_2(H))\,D^{-1}e^{-\frac32m_\eps D}, \] with an absolute constant $C_{\rm GB}$. \end{lemma} \begin{proof} \emph{Step 1 (marginals and order).} Fix a component $\alpha$. Let $X_\alpha:=\{x\in C_L\cap\Lambda:x+\eps e_\alpha\in C_L\}$; conditioning a Gaussian on the vanishing of some coordinates leaves the principal submatrix of the precision on the others, so in ($\natural$) the variables $A_\alpha(x)$, $x\in X_\alpha$, have precision $K_\natural=\eps(6-\mathrm{Adj}_{X_\alpha})+\mu_0^2\eps^3$, where $\mathrm{Adj}_{X_\alpha}$ is the adjacency matrix of $X_\alpha$. In (N) the variables are $A_\alpha(x)$, $x\in C_L\cap\Lambda$, with precision $K_N=\eps L_{C_L}+\mu_0^2\eps^3$. The principal submatrix of $K_N$ on $X_\alpha$ is $\eps(\deg_{C_L}-\mathrm{Adj}_{X_\alpha})+\mu_0^2\eps^3\le K_\natural$, and the Schur complement of $K_N$ on $X_\alpha$ is at most its principal submatrix; hence $C_\natural:=K_\natural^{-1}\le C_N|_{X_\alpha}$, $C_N:=K_N^{-1}$. Let $\hat X:=\{(\alpha,x):x,x+\eps e_\alpha\in H\}$. The integrands depend on $A$ only through $A_{\hat X}$, whose laws are $N(0,C_\natural|_{\hat X})$ and $N(0,C_N|_{\hat X})$, with $C_\natural|_{\hat X}\le C_N|_{\hat X}$. Put $M:=(C_\natural|_{\hat X})^{-1}-(C_N|_{\hat X})^{-1}\ge0$. The $\natural$-measure is obtained from the $N$-measure, after integrating out the photon variables off $\hat X$, by adding $\frac12\ip AMA$ to the action; by Lemma~\ref{lem:s02:BFS}(a) along $s\mapsto\frac s2\ip AMA$, $\langle\Theta\rangle_\natural\ge\langle\Theta\rangle_N$. \emph{Step 2 (Cauchy--Schwarz).} With $Q:=\ip AMA$ and $\varsigma:=e^{-Q/2}$, $\langle\Theta\rangle_\natural=\langle\Theta\varsigma\rangle_N/\langle\varsigma\rangle_N$, so \[ \langle\Theta\rangle_\natural-\langle\Theta\rangle_N=\frac{\langle(\Theta-\langle\Theta\rangle_N)(\varsigma-\langle\varsigma\rangle_N)\rangle_N}{\langle\varsigma\rangle_N} \le\langle|\Theta|^2\rangle_N^{1/2}\Big(\frac{\langle e^{-Q}\rangle_N}{\langle e^{-Q/2}\rangle_N^2}-1\Big)^{1/2} \le\langle|\Theta|^2\rangle_N^{1/2}\big(e^{\langle Q\rangle_N}-1\big)^{1/2}, \] by $\langle e^{-Q}\rangle_N\le1$ and Jensen's inequality. Writing $M=\sum_j\lambda_jv_jv_j^T$, Lemma~\ref{rot:lem:GD}(b) in the $N$-measure gives $\langle Q\rangle_N\le\sum_j\lambda_j\ip{v_j}{C_Nv_j}=\tr(MC_N|_{\hat X})=:\tau$. \emph{Step 3 (the trace).} Let $\Delta:=C_N|_{X_\alpha}-C_\natural\ge0$, for each $\alpha$. Then $MC_N|_{\hat X}=(C_\natural|_{\hat X})^{-1}\Delta_{\hat X}$, so $\tau=\E\ip{Y}{(C_\natural|_{\hat X})^{-1}Y}$ with $Y\sim N(0,\Delta_{\hat X})$. Since the inverse of a principal block of the covariance is the minimum of the precision form over extensions, $\ip a{(C_\natural|_{\hat X})^{-1}a}=\min\{\ip u{K_\natural u}:u|_{\hat X}=a\}$. Let $\tilde Y\sim N(0,\Delta)$ on $X_\alpha$ and $\eta(x):=(1-\dist_\infty(x,H))_+$; then $\eta\tilde Y$ is an extension of $\tilde Y|_{\hat X}$, so $\tau\le\sum_\alpha\E\ip{\eta\tilde Y}{K_\natural\eta\tilde Y}$. Let $\mathrm{int}$ be the set of $x\in X_\alpha$ whose six neighbours lie in $X_\alpha$; it contains $N_2(H)\cap\Lambda$. At $x\in\mathrm{int}$ the rows of $K_\natural$, $K_N$ and of the infinite-lattice precision $K_\infty$ coincide, so $(K_\infty\Delta)(x,w)=(K_NC_N)(x,w)-(K_\natural C_\natural)(x,w)=0$ for all $w\in X_\alpha$, hence $\mathrm{Var}((K_\infty\tilde Y)(x))=0$ and $K_\infty\tilde Y=0$ on $\mathrm{int}$ almost surely. For $u$ with $K_\infty u=0$ on $\mathrm{int}\supset\supp\eta$, summation by parts and the identity $(\eta_xu_x-\eta_yu_y)^2-(u_x-u_y)(\eta_x^2u_x-\eta_y^2u_y)=(\eta_x-\eta_y)^2u_xu_y$ give $\ip{\eta u}{K_\natural\eta u}=\ip{\eta u}{K_\infty\eta u}=\eps\sum_{\langle xy\rangle}(\eta_x-\eta_y)^2u_xu_y$. Since $|\eta_x-\eta_y|\le\eps$ and $|\Delta(x,y)|\le\frac12(\Delta(x,x)+\Delta(y,y))$, \[ \tau\le\sum_\alpha\eps^3\sum_{\langle xy\rangle\subset N_2(H)}|\Delta(x,y)|\le9\,\mathrm{vol}_\eps(N_2(H))\sup_{\alpha,\ x\in N_2(H)}\Delta(x,x). \] \emph{Step 4 (decay of $\Delta$).} For $K=\mathcal D-\eps\mathrm{Adj}_{\mathcal G}$ with $\mathcal D$ diagonal, $K^{-1}(x,y)=\sum_\omega\prod_{(u\to v)\in\omega} (\eps/\mathcal D(u))\cdot\mathcal D(y)^{-1}$, the sum over walks $\omega$ from $x$ to $y$ in $\mathcal G$. For $K_\natural$ and $K_N$ the weights of steps from sites of $\mathrm{int}$ coincide. Decomposing walks from $x\in N_2(H)$ at their first visit to $\Sigma:=X_\alpha\setminus\mathrm{int}$, $K^{-1}(x,y)=G^{\mathrm{int}}(x,y)+\sum_{z\in\Sigma}\mathsf H(x,z)K^{-1}(z,y)$ with $G^{\mathrm{int}}$ and $\mathsf H$ the same for both precisions. Hence \[ \Delta(x,x)=\sum_{z\in\Sigma}\mathsf H(x,z)\big(K_N^{-1}(z,x)-K_\natural^{-1}(z,x)\big)\le\Big(\sum_{z\in\Sigma}\mathsf H(x,z)\Big)\sup_{z\in\Sigma}K_N^{-1}(z,x). \] The first factor is the probability that simple random walk from $x$, killed with probability $\mu_0^2\eps^2/(6+\mu_0^2\eps^2)$ per step, reaches $\Sigma$, which lies at $\ell^\infty$-distance at least $D$ from $x$. With $\cosh\lambda=1+\mu_0^2\eps^2/2$, that is $\lambda=\eps m_\eps$, $e^{\pm\lambda S^\beta_n}\mathbf 1_{\{\text{alive at }n\}}$ is a martingale for each coordinate $S^\beta$; optional stopping and a union over the six half-spaces give $\sum_z\mathsf H(x,z)\le6e^{-m_\eps D}$. The same argument, with the decomposition at the first exit from the $\ell^\infty$-ball of radius $|y|_\infty/2$ and the bound $K^{-1}_{\eps\Z^3,\mu_0}\le K^{-1}_{\eps\Z^3,0}=(6\eps)^{-1}G_{\Z^3}(\cdot/\eps)$ entrywise, gives the free bound \[ G_\eps(y):=K^{-1}_{\eps\Z^3}(0,y)\le4\mathsf a\,|y|_\infty^{-1}e^{-m_\eps|y|_\infty/2}\qquad(|y|_\infty\ge4\eps). \] King's Neumann precision on the box $C_L\cap\Lambda$ has the Green's function $K_N^{-1}(z,x)=\sum_{x'}G_\eps(z-x')$, the sum over the images $x'$ of $x$ under the reflections of $\eps\Z^3$ in the planes half a lattice spacing outside the faces of the box (method of images; the mass term is uniform). Every image other than $x$ lies outside $C_L$ at $\ell^\infty$-distance at least $D$ from every site of $C_L$, and so does $x$ from $z\in\Sigma$. If $s\ge2D$ is the side of $C_L$, at most $8(j+3)^3$ images satisfy $|z-x'|_\infty0$. A \emph{shell configuration} for $Q_b$ is $\fs=(\fc,b',L')$, where $\fc=(J',\psi,n)$ with $\psi\in C^\infty(D_{J'};[0,\infty))$ and $n\ge0$ is a walled configuration (Definition~\ref{rot:def:states}(c); $\fc=(J',n\psi)$ in the notation of Definition~\ref{rot:def:walls}) with $N_1(Q_b)\subset(-25MJ',25MJ')^3$ and $\psi=0$ on $N_1(Q_b)$, $H:=Q_{b'}$ with $b'\ge b+3$, and $C_{L'}$ is a King cube with $\dist_\infty(H,\R^3\setminus C_{L'})=D(\fs)+3$, $D(\fs)\ge\max(1,3/\mu_0)$, and $N_1(\bar C_{L'})\subset(-25MJ',25MJ')^3$. Its \emph{shell} is a set $U$ with $C_{L'}\setminus H\subset U\subset T\setminus H$, where $T=D_{J'}$ is the torus of the configuration and $C_{L'}$ is understood in its cell (the two cases used are the cubic shell $U=C_{L'}\setminus H$ and the whole exterior $U=T\setminus H$, Section~\ref{rot:ssec:W8}), and we require the \emph{plateau condition} $\psi\ge1$ on $N_1(U)$ (compatible with $\psi=0$ on $N_1(Q_b)$, since $\dist_\infty(U,Q_b)\ge3$). Put \[ \tau(\fs):=C_{\rm GB}\,\mathsf a\,\mathrm{vol}(N_3(H))\,D(\fs)^{-1}e^{-1.44\,\mu_0D(\fs)} . \] \end{definition} \begin{definition}[shell profiles and two-level configurations]\label{r13:def:shellprofile} Let $\fs$ be a shell configuration for $Q_b$ with the shell $U$ (Definition~\ref{rot:def:shell}). A \emph{shell profile} for $\fs$ is $\chi\in C^\infty(D_{J'};[0,1])$ with $\chi=1$ on $U$ and $\supp\chi\subset N_1(U)$, or the \emph{sharp profile} $\chi=\mathbf 1_U$ (Remark~\ref{rot:rem:Eproof}(ii): the lattice statements of this section hold for it verbatim); then $\chi\le\psi$ automatically, and $\chi=0$ on $N_1(Q_b)$. Its \emph{dead set} is $D_\chi:=\{\chi>0\}\subset N_1(U)$; every site of $C_{L'}$ within $\eps$ of $\R^3\setminus C_{L'}$, and every site of $U$, lies in $D_\chi$. We call $(\fs,\chi)$ a \emph{two-level configuration} for $Q_b$. \end{definition} \begin{definition}[conventions for shell configurations]\label{r13:def:shellconv} In Definition~\ref{rot:def:shell}, the conditions on $\fc$ are the cell and support conditions of Definition~\ref{rot:def:walls}; the conditions of the class $\mathcal C$ (Definition~\ref{rot:def:class}) are imposed, in Theorem~\ref{rot:hyp:RES}, on the single-level configuration $(J',\psi,n+K)$, not on $\fc$. On the lattice, $H$ is the site set of the half-open box $[-b',b')^3$, a union of blocks as in Theorem~\ref{cpi:thm:main}: the sites of the faces $x_\mu=+b'$ belong to the shell $U$ and those of the faces $x_\mu=-b'$ to $H$, $\partial U$ being the midplane between the last $U$-layer and the first $H$-layer of sites, as there; no clause of Definition~\ref{rot:def:shell} depends on this convention, which fixes the site set of the sharp shell profile $\chi=\mathbf 1_U$ of Remark~\ref{rot:rem:Eproof}(ii) and of the shell mass in the shell estimates (Sections~\ref{rot:ssec:shellest} and~\ref{rot:sec:shellestB}) as the one of Theorem~\ref{cpi:thm:main}. The King cube $C_{L'}$ is the shell's cube, with its own index $L'$, written $C_{L'}$ to keep it apart from the source cube $C_L$ of the class $\mathcal C$ and of Theorems~\ref{rot:thm:B} and~\ref{rot:thm:curv}; Lemma~\ref{rot:lem:GB} is applied with its cube equal to $C_{L'}$. \end{definition} \begin{definition}[two-level states and the tail]\label{rot:def:twolevel} Let $(\fs,\chi)$ be a two-level configuration for $Q_b$, $f$ a real $2$-form in $C^\infty_c$ supported in $Q_b$, and $k$ with $\eps_k\le\min(1/\mu_0,1/4)$. For $K\in[0,\infty)$ let \[ \Psik(n,K):=T_{(J',\,n\psi+K\chi),\Lambda_k}(f) \] be the walled expectation of the configuration with the two-level wall $n\psi+K\chi$, which vanishes on $N_1(Q_b)$ (for the sharp profile the wall is a lattice function and the state is defined by the same single-site weights, Definition~\ref{rot:def:states}(c)); and let $\Psik(n,\infty):=\langle e^{\mathrm iF(f)}\rangle$ in the walled torus state of $\fc=(J',n\psi)$ with the matter killed on $D_\chi\cap\Lambda_k$ (Definition~\ref{rot:def:states}(d)). Put $\tail_k(K):=\Psik(n,K)-\Psik(n,\infty)$. \end{definition} \begin{lemma}[elementary properties]\label{rot:lem:tailprop}\emergencystretch=3em\relax \begin{enumerate} \item[(a)] $K\mapsto\Psik(n,K)$ is nonincreasing on $[0,\infty)$, $\Psik(n,K)\ge\Psik(n,\infty)$, and $\Psik(n,K)\to\Psik(n,\infty)$ as $K\to\infty$. Hence $\tail_k(K)\ge0$ is nonincreasing in $K$ and tends to $0$ as $K\to\infty$, at every fixed $k$. \item[(b)] (two-level sandwich) For $n,K\ge0$: $\Psik(n+K,0)\le\Psik(n,K)\le\Psik(n,0)$. \end{enumerate} \end{lemma} \begin{proof}\emergencystretch=4em\relax (a) Increasing $K$ multiplies the single-site measures on $D_\chi$ by $e^{-(K'-K)\eps^3\chi(x)r_x^2}$; Lemma~\ref{lem:s02:BFS}(c). Killing $D_\chi$ is the limit (Lemma~\ref{lem:s02:Kprime} with the rates $\chi>0$ on $D_\chi$), and it does not increase the expectation. (b) The wall of $\Psik(n+K,0)$ exceeds that of $\Psik(n,K)$ by $K(\psi-\chi)\ge0$, since $\chi\le\psi$; the wall of $\Psik(n,K)$ exceeds that of $\Psik(n,0)$ by $K\chi\ge0$. Apply Lemma~\ref{lem:s02:BFS}(c). \end{proof} Part (b) is the two-level sandwich (Lemma~\ref{rot:lem:TL} below in the form used); it is the only place where $\chi\le\psi$ is needed. In words: raising the matter mass on the shell alone, by any amount, gives a state above the one in which the whole plateau is raised by the same amount, and below the one with the shell untouched. \begin{theorem}[the dead-shell bound]\label{rot:thm:SHT} For every two-level configuration $(\fs,\chi)$ for $Q_b$, every real $2$-form $f\in C^\infty_c$ supported in $Q_b$ and every $k$ with $\eps_k\le\min(1/\mu_0,1/4)$ and $\mathrm{vol}_{\eps_k}(N_2(H))\le\mathrm{vol}(N_3(H))$, \begin{equation}\label{rot:eq:deadbound} \Psik(n,\infty)\le\langle e^{\mathrm iF(f)}\rangle^\ND_{k,L'}+\big(e^{\tau(\fs)}-1\big)^{1/2} \le\langle e^{\mathrm iF(f)}\rangle^K_{k,L'+1}+\big(e^{\tau(\fs)}-1\big)^{1/2}, \end{equation} $C_{L'}$ the shell's cube, and hence $\limsup_k\Psik(n,\infty)\le Z_\infty(\mathrm if,0)+(e^{\tau(\fs)}-1)^{1/2}$. \end{theorem} \begin{proof} Fix such a $k$, and write $\eps=\eps_k$ and $\Theta=e^{\mathrm iF(f)}$. Let $E$ be the set of sites of the torus outside $C_{L'}$. The dead set enters only through $C_{L'}\setminus H\subset U\subset D_\chi$: the matter being dead on all of $D_\chi$ rather than on the cubic shell $C_{L'}\setminus H$ alone does not increase $\Psik(n,\infty)$, since killing is the limit of raising the mass (Lemma~\ref{lem:s02:Kprime}) and raising the mass does not increase the expectation (Lemma~\ref{lem:s02:BFS}(c)); the steps below are run for $D_\chi$ directly and use of it only that it contains $U$ and every site of $C_{L'}$ adjacent to $E$. \emph{Step 1 (pinning).} In the state with the matter killed on $D_\chi\cap\Lambda_k$, add $s|A_\beta(y)|^2$ for every photon variable on a bond with an endpoint in $E$, and let $s\to\infty$. By Lemma~\ref{lem:s02:BFS}(a) the expectation of $\Theta$ does not decrease, and in the limit the photon is the conditioned law ($\natural$) of Lemma~\ref{rot:lem:GB}. In this state the hopping terms within $E$ do not involve $A$, and there are no hopping terms between $E$ and live sites: every site of $C_{L'}$ adjacent to $E$ lies in $D_\chi$ and is dead. So the matter on $E$ is a factor independent of $A$ and cancels (Lemma~\ref{lem:s02:BFS}(e)), or is dead altogether when $E\subset U$. What remains is the $\natural$-measure of Lemma~\ref{rot:lem:GB}, with the matter on $H$ (dead on $D_\chi\cap H$), the wall, and Dirichlet conditions at $\partial H$, since $U\subset D_\chi$ is dead. Hence $\Psik(n,\infty)\le\langle\Theta\rangle_\natural$. \emph{Step 2 (photon switch).} By Lemma~\ref{rot:lem:GB}, $\langle\Theta\rangle_\natural\le\langle\Theta\rangle_N+\langle|\Theta|^2\rangle_N^{1/2}(e^{\tau(\fs)}-1)^{1/2}$, and $|\Theta|^2=1$; here $m_\eps\ge0.96\mu_0$ and the assumption $\mathrm{vol}_\eps(N_2(H))\le\mathrm{vol}(N_3(H))$ give $\tau\le\tau(\fs)$. \emph{Step 3 (to the box $C_{L'}$, and to King's cube).} The $N$-measure is the limit $s\to\infty$ of the walled Neumann--Dirichlet state on $C_{L'}\cap\Lambda_k$ with the single-site measures on $D_\chi$ multiplied by $e^{-s\eps^3\chi(x)r^2}$ (Lemma~\ref{lem:s02:Kprime}). By Lemma~\ref{lem:s02:BFS}(c) the expectation of $\Theta$ does not decrease as $s$ decreases to $0$, nor when the wall is then removed. The result is the box state $\langle\cdot\rangle^\ND_{k,L'}$ of Definition~\ref{def:s02:acube}: this is the first inequality of~\eqref{rot:eq:deadbound}. The second is Lemma~\ref{lem:s04:G}, since $f$ is supported in $H$, at distance $>1$ from $\R^3\setminus C_{L'}$. \emph{Step 4 (limits).} The limit of $\langle e^{\mathrm iF(f)}\rangle^K_{k,L'+1}$ as $k\to\infty$ exists by Theorem~\ref{rot:thm:W}(i) (Theorem~\ref{thm:k01:EC}, Lemma~\ref{lem:s02:alg}(b), Montel) and is at most $Z_\infty(\mathrm if,0)$ (Theorem~\ref{rot:thm:W}(iii)). This gives the $\limsup$ bound. \end{proof} \subsection{The main statement} This subsection states the heavy-shell decoupling (RES$_0$) (Theorem~\ref{rot:hyp:RES}) and proves that, together with (K$_0$), it implies the closing statement (TIGHT$_0$) (Theorem~\ref{rot:thm:E}), by the two-level sandwich of Lemma~\ref{rot:lem:TL} and the dead-shell bound. \begin{lemma}[two-level sandwich]\label{rot:lem:TL} For every two-level configuration, every real $2$-form $f\in C^\infty_c$ supported in $Q_b$, every $k$ and $n,K\ge0$, \[ \Psik(n+K,0)\;\le\;\Psik(n,K)\;=\;\Psik(n,\infty)+\tail_k(K)\;\le\;\Psik(n,0),\qquad\tail_k(K)\ge0 . \] \end{lemma} \begin{proof} Lemma~\ref{rot:lem:tailprop}. \end{proof} \begin{theorem}[\textup{(RES$_0$)}: heavy-shell decoupling at one configuration]\label{rot:hyp:RES} For every $b>0$, every real $2$-form $f\in C^\infty_c$ supported in $Q_b$ and every $\eta>0$ there are a two-level configuration $(\fs,\chi)$ for $Q_b$ with the shell $U=T\setminus H$, the whole exterior of the box $H=Q_{b'}$ in the torus $T=D_{J'}$ of the configuration, the sharp profile $\chi=\mathbf 1_U$ and $\tau(\fs)\le\eta$, and a shell mass $K\in[0,\infty)$ such that the single-level configuration $(J',\psi,n+K)$ lies in $\cA(Q_b)$ and \[ \liminf_{k\to\infty}\tail_k(K)=\liminf_{k\to\infty}\big[\Psik(n,K)-\Psik(n,\infty)\big]\le\eta . \] \end{theorem} \begin{proof} Theorem~\ref{rot:thm:shell}, from the shell estimates (Remark~\ref{rot:rem:shellbounds}), at one two-level configuration with the shell $U=T\setminus H'$, $H'=H$; the quantifiers are met in the order stated there, and $(J',\psi,n+K)$ is placed in $\cA(Q_b)$ by the choice of $K$ (Remark~\ref{rot:rem:class}). For $b0$ there are a stopping scale $\eta_0$ and a shell mass $K<\infty$ with $\limsup_{k}\tail_k(K)\le\eta$, in the quantifier order of \textup{(RES$_0$)}: given $b$, a source $f$ supported in $Q_b$ and $\eta>0$, the configuration $(\fs,\chi)$ is chosen first, with $D(\fs)$ so large that $\tau(\fs)\le\eta$, and then $\eta_0$ and $K$; so \textup{(RES$_0$)} holds at $(\fs,\chi)$ with this $K$, the single-level configuration $(J',\psi,n+K)$ lying in $\cA(Q_b)$ for $b\ge b_\circ$ once $K$ is taken large enough for the wall condition of the class $\mathcal C$ (Remark~\ref{rot:rem:Eproof}(iii); the vanishing of $\psi$ near $R^{-1}C_1$ follows from $b\ge b_\circ$, Definition~\ref{rot:def:walls}), and under \textup{(K$_0$)}, \textup{(TIGHT$_0$)} holds (Theorem~\ref{rot:thm:E}). \end{theorem} \begin{proof} Fix $(\fs,\chi)$, $f$ and $g=0$, put $\Psi_\kappa(z):=Z_\kappa(zf,0)/Z_\kappa(0,0)$ for $\kappa\in[0,\infty]$ and $z\in\C$ ($Z_\infty$ the partition function of the state with the matter killed on $D_\chi$), so that, the shell mass $\kappa\mathbf 1_U$ being the second level $\kappa\chi$ of the wall, Definition~\ref{rot:def:twolevel} gives $\Psi_K(\mathrm i)=\Psik(n,K)$, $\Psi_\infty(\mathrm i)=\Psik(n,\infty)$ and $\tail_k(K)=\Psi_K(\mathrm i)-\Psi_\infty(\mathrm i)$, put $J_\kappa(z):=I^{\rm cap}_\kappa(zf,0)/I^{\rm cap}_\kappa(0,0)$ for $\kappa\ge\kappa_0$ and real $z$, and let $a=a(\eta_0)$, $b=b(\eta_0)$, $\eta_1$, $\eta_2$, $\eps_0(\eta_0)$, $c_0$ and $r(\kappa;\eta_0)$ be the constants of Theorems~\ref{cpi:thm:main} and~\ref{cpi:swc:main}, every bound below being uniform in $\eps\le\eps_0(\eta_0)$. By (SW2$^{\min}$), $|\Psi_\kappa(z)|\le B_\Psi$ on $|z|\le2$ uniformly in $\kappa\in[0,\infty]$ and in $\eps$, and $\Psi_\kappa(z)\to\Psi_\infty(z)$ as $\kappa\to\infty$ at fixed $\eps$ for real $z$ (the substitution and dominated convergence of the proof of Lemma~\ref{lem:s02:Kprime}, $e^{zF(f)}$ being integrable against the photon Gaussian by the same bound); for $\eta_0\le\min(\eta_1,\eta_2)$ with $b(\eta_0)\le\frac12$, $\kappa\ge\kappa_0$ and $z\in[-1,1]$, (SW1$^\circ$)$_{\rm int}$ at $(0,0)$ and at $(zf,0)$ gives $e^{-2a}(1-b)J_\kappa(z)\le\Psi_\kappa(z)\le e^{2a}J_\kappa(z)+b$, hence $J_\kappa(z)\le2e^{2a}B_\Psi=:\bar J$ and $|\Psi_\kappa(z)-J_\kappa(z)|\le\bar J(e^{2a}-e^{-2a}+b)+b=:m_1(\eta_0)$, which tends to $0$ with $\eta_0$ because $a,b\to0$; for $\kappa_0\le\kappa\le\kappa'<\infty$, $J_\kappa(z)/J_{\kappa'}(z)$ is the quotient of the integrals of $e^{-\tilde S_\kappa(b;z)-\tilde S_{\kappa'}(b';0)}$ and of $e^{-\tilde S_{\kappa'}(b;z)-\tilde S_\kappa(b';0)}$ over $\tilde\chi\times\tilde\chi$, $\tilde S_\kappa(\cdot;z)$ the reduced small-field action at shell mass $\kappa$ and $\tilde\chi$ the capped test set, the same for all $\kappa\ge\kappa_0$ (this is where (SWT) enters: no threshold term arises between $\kappa$ and $\kappa'$), whose integrands differ pointwise by the factor $e^{\Delta_z(b)-\Delta_0(b')}\in[e^{-2r},e^{2r}]$, $r:=r(\kappa;\eta_0)$, by (SW3$^\circ$)$_{\rm int}$, so that $|J_\kappa(z)-J_{\kappa'}(z)|\le\bar J(e^{2r}-1)$ and, with $\bar r(K;\eta_0):=\sup_{\kappa\ge K}r(\kappa;\eta_0)$, which tends to $0$ as $K\to\infty$ at fixed $\eta_0$, \[ |\Psi_\kappa(z)-\Psi_{\kappa'}(z)|\le m(K,\eta_0):=2m_1(\eta_0)+\bar J\big(e^{2\bar r(K;\eta_0)}-1\big)\qquad(\kappa'\ge\kappa\ge K\ge\kappa_0,\ z\in[-1,1]), \] and letting $\kappa'\to\infty$ at fixed $\eps$, $|\Psi_K(z)-\Psi_\infty(z)|\le m(K,\eta_0)$ for $z\in[-1,1]$. Thus $D:=\Psi_K-\Psi_\infty$ is entire with $|D|\le2B_\Psi$ on $\{|z|\le2\}$ and $|D|\le m(K,\eta_0)$ on $[-1,1]$, so that the two-constants theorem on $\{|z|<2\}\setminus[-1,1]$ ($\log|D|$ subharmonic, dominated by the harmonic function with boundary values $\log(2B_\Psi)$ on $|z|=2$ and $\log m(K,\eta_0)$ on the slit) gives $|D(\mathrm i)|\le(2B_\Psi)^{1-\omega}m(K,\eta_0)^{\omega}$, $\omega\in(0,1)$ the harmonic measure of the slit $[-1,1]$ at $\mathrm i$, an absolute constant; the map $(A,\phi)\mapsto(-A,\bar\phi)$ preserves the action, the wall, the shell mass, the gauge fixing and the measure and reverses the sign of $F(f)$, so $\Psi_\kappa(-z)=\Psi_\kappa(z)=\overline{\Psi_\kappa(\bar z)}$ and $\Psi_\kappa(\mathrm i)$ is real, whence $0\le\tail_k(K)=D(\mathrm i)\le(2B_\Psi)^{1-\omega}\,m(K,\eta_0)^{\omega}$ for every $\eps_k\le\eps_0(\eta_0)$; given $\eta>0$, take $\delta>0$ with $(2B_\Psi)^{1-\omega}\delta^{\omega}\le\eta$, then $\eta_0\le\min(\eta_1,\eta_2)$ with $b(\eta_0)\le\frac12$ and $2m_1(\eta_0)\le\delta/2$, then $K\ge\kappa_0$ with $\bar J(e^{2\bar r(K;\eta_0)}-1)\le\delta/2$, and only then $\eps\le\eps_0(\eta_0)$: then $m(K,\eta_0)\le\delta$ and $\tail_k(K)\le\eta$ for all $\eps_k\le\eps_0(\eta_0)$, so $\limsup_k\tail_k(K)\le\eta$, which is (RES$_0$) at $(\fs,\chi)$ with this $K$ in its quantifier order (one configuration, one $K$, no rate, the limit along the spacings last), and since $\tail_k$ is nonincreasing in $K$ (Lemma~\ref{rot:lem:tailprop}(a)) $K$ may be enlarged at will (Remark~\ref{rot:rem:Eproof}(iii)), which places $(J',\psi,n+K)$ in the class $\mathcal C$, hence in $\cA(Q_b)$ for $b\ge b_\circ$ (Remark~\ref{rot:rem:class}, Definition~\ref{rot:def:walls}), and Theorem~\ref{rot:thm:E} then gives (TIGHT$_0$) under (K$_0$). \end{proof} \begin{remark}[the shell bounds]\label{rot:rem:shellbounds} (SW1$^\circ$)$_{\rm int}$, (SW2$^{\min}$), (SW3$^\circ$)$_{\rm int}$ and (SWT) are Theorem~\ref{cpi:thm:main}, Lemma~\ref{rot:lem:SW2}, Theorem~\ref{cpi:swc:main} and Corollary~\ref{rot:cor:SWT} of Sections~\ref{rot:ssec:shellest} and~\ref{rot:sec:shellestB}. Theorem~\ref{rot:thm:shell} uses Theorem~\ref{cpi:swc:LDT} only through the decay $r(\kappa;\eta_0)\to0$ of the bound in (SW3$^\circ$)$_{\rm int}$ as $\kappa\to\infty$, so a change of the rate of that decay, its exponent or a logarithmic factor, leaves the theorem unchanged; the dead-shell bound (Theorem~\ref{rot:thm:SHT}) enters only through Theorem~\ref{rot:thm:E} at the configuration of Theorem~\ref{rot:thm:shell}, and none of the auxiliary statements of Sections~\ref{rot:ssec:shellest} and~\ref{rot:sec:shellestB} enters the statement of (RES$_0$) that Theorem~\ref{rot:thm:E} uses. \end{remark} % \space% % % \section{The shell model with the shell mass inside the covariance: the two-sided bound}\label{rot:ssec:shellest} The shell estimates (SW1$^\circ$)$_{\rm int}$, (SW2$^{\min}$), (SW3$^\circ$)$_{\rm int}$ and (SWT) of Section~\ref{rot:ssec:W8} are King's renormalisation bounds \cite[Thm.~3.1]{K1} and Ba{\l}aban's lower and upper bounds \cite{Ba1,Ba2,Ba3} run for the \emph{shell model}, in which the shell mass $\kappa\mathbf 1_U$ sits inside every matter covariance. This section sets the model up and proves the first of the four. We fix the model, its two stopping-scale tests and the notation; we then bound the mismatch vertex, which carries the difference between the counterterms of the shell model and those of the bulk, introduce the capped stopping-scale set, one set of block fields for all heavy shell masses, and state the exact mass shift on a finite lattice. With these we prove the two-sided bound (Theorem~\ref{cpi:thm:main}) and the perturbative bound for the shell model on which it rests (Proposition~\ref{cpi:prop:TD}). Ba{\l}aban's estimates for the shell model and the $\kappa$-Cauchy property (Theorem~\ref{cpi:swc:main}), with the threshold clause (Corollary~\ref{rot:cor:SWT}) and the disc bound (Lemma~\ref{rot:lem:SW2}), follow in Section~\ref{rot:sec:shellestB}. \paragraph{The shell model and the stopping-scale tests.} Throughout, $T$, $\eps=L^{-K}$ ($L=5$) and $\ell_i:=L^i\eps$ are the torus, the spacing and the block scales of Definitions~\ref{rot:def:lattice} and~\ref{rot:def:states}, $a_i:=a(1-L^{-2})(1-L^{-2i})^{-1}\in[a(1-L^{-2}),a]$ the averaging constants of King's block-spin transformation \cite[(2.10)--(2.17)]{K1}, and \emph{unit-$i$ units} means the lattice rescaled so that $i$-blocks have side one. Ba{\l}aban's \emph{final scale} is $K_f:=\max\{i:\ell_i\le\eps_0^{\rm Ba}\}$, $\eps_0^{\rm Ba}$ his fixed final spacing \cite[(3.66)]{Ba1}, \cite[(2.116)]{Ba2}; the \emph{final block scale} $\ell_f:=\ell_{K_f}\in(\eps_0^{\rm Ba}/L,\eps_0^{\rm Ba}]$ is a fixed length independent of $\eps$, $\eta_0$ and $\kappa$, and all scales below are $\le K_f\le K$. The box $H'\subset T$, the \emph{shell} $U:=T\setminus H'$, the wall profile $\psi\colon T\to[0,1]$ ($0$ on the source region $V\subset H'$, $1$ off a neighbourhood of $V$, hence on $U$) and the sources $f,g$ supported in $V$ at distance $\ge\eta_0$ from $U$ are those of Section~\ref{rot:sec:shell}. The shell model with shell mass $\kappa\in[0,\infty)$ is King's model with the profile \begin{equation}\label{cpi:eq:profile} V(x):=m^2+n\psi(x)+\kappa\mathbf 1_U(x)\ \ge m^2>0,\qquad n\ge1\ \text{fixed}, \end{equation} inside every matter covariance: each block-spin step is run with $-\Delta_A+V$, so that $\kappa$ sits in no vertex and enters the effective actions only through the lines, the legs $\phi^{(i)}_\kappa=a_i\ell_i^{-2}G^\kappa_iQ_i^*\phi_i$ of the block fields and the determinants. In unit-$i$ units the shell mass is $w_i:=\kappa\ell_i^2$; the scale $i$ is \emph{heavy} when $w_i\ge c_0$ and \emph{light} otherwise, where $c_0\in(0,c_0^*]$ is a fixed threshold, $c_0^*:=1/(2C_{\rm far})$ with $C_{\rm far}$ the constant of the modulus form $|\phi^{(i)}_\kappa(x)|\ge(1-C_{\rm far}c_0)|\phi_i(y)|-C_1p_i$ of \cite[(2.65)]{Ba2}, so that $1-C_{\rm far}c_0\ge\frac12$. The stopping scale $k$, $\ell_k=\eta_0$, is heavy for $\kappa\ge c_0/\eta_0^2$; as in Section~\ref{rot:sec:shell}, $\kappa_0:=\max(c_0,1)/\eta_0^2$, so that on $[\kappa_0,\infty)$ the stopping scale is heavy with $w_k\ge\max(c_0,1)$ (every statement below on $[\kappa_0,\infty)$ uses only $w_k\ge c_0$ and holds on $[c_0/\eta_0^2,\infty)$ as well). We write $p(t)=b_0(1+\log t^{-1})^p$ \cite[(3.1)]{K1} ($b_0$, $p$ King's constants), $p_i:=p(\ell_i)$, $M_i:=p_i(\lambda\ell_i)^{-1/4}$, $U^{(i)}$ for the $i$-blocks in $U$ and $L_1^{(i)}$ for the \emph{first light layer}, the $i$-blocks of $H'$ adjacent to $U^{(i)}$. \emph{Block alignment and torus size.} The \emph{block-alignment condition} (BA): $T$ and $H'$ are unions of $\ell_f$-blocks with the grid offset of the class $\mathcal C$ (Definition~\ref{rot:def:class}: the large-block size is $M=LM'$ with $M'$ having a prime factor $q_0\nmid L$ and $M'\ge2q_0$, and every corner coordinate of $H'$ is $c\,\ell_f$ with $c\equiv M'/q_0\pmod{M'}$, counted from the physical origin, the anchor of Ba{\l}aban's block grids), each side of $H'$ at least $2\ell_f$ and the shell at least $3\ell_f$ thick. Under (BA), $H'$ and $U$ are unions of $i$-blocks at every scale $i\le K_f$, and every face of every union of large blocks of scale $i-1$ or $i$ lies at least two unit-$i$ blocks from every parallel face of $\partial U$. The \emph{torus-size condition} (T.nw): each side of $T$ exceeds the fixed length $2\big(8r(\eps_0^{\rm Ba})+s_b\big)\eps_0^{\rm Ba}$, where $r(t)=R(1+\log t^{-1})^{\varrho}$ is Ba{\l}aban's localisation radius (in blocks), $R$ his radius constant, $\varrho=2$, and $s_b=LM$ the side of his large blocks in block units; like (BA) it is chosen after $\eps_0^{\rm Ba}$, and it gives torus sides $N_\mu>16\,r(\eps_0^{\rm Ba})+2s_b$ in unit-$i$ units at every scale $i\le K_f$, so that no localisation box wraps the torus. Besides King's small-field indicator, the shell model carries two tests on the block fields, one on the shell blocks at each scale and one on the first light layer at the stopping scale, both imposed on the small-field integral at the stopping scale $k$; their thresholds depend on the shell mass through the coefficients $\gamma$, $\gamma_2$ defined now. Fix shares $\vartheta_1\in(0,\tfrac12)$ and $\vartheta_2\in(0,\vartheta_1/4]$ with $\vartheta_1+\vartheta_2<1$, and put $\gamma(w):=a_iw/(a_i+w)$, $\gamma_2(w):=(a_k^{-1}+c_P^{-1}+w^{-1})^{-1}$, $c_P$ the Poincar\'e constant of a unit block. The \emph{near-$U$ test} at scale $i$ and the \emph{face-layer test} at the stopping scale $k$ are \begin{equation}\label{cpi:eq:tests} \begin{gathered} \tau_i:=\prod_{y\in U^{(i)}}\chi\big(|\phi_i(y)|\le\bar\tau_i\big),\quad \bar\tau_i:=\Big(\frac{2}{\vartheta_1\gamma(w_i)}\Big)^{1/2}p_i;\\ \lambda_k:=\prod_{y'\in L_1^{(k)}}\chi\big(|\phi_k(y')|\le\sigma_k\big),\quad \sigma_k:=\Big(\frac{2}{\vartheta_2\gamma_2(w_k)}\Big)^{1/2}p_k , \end{gathered} \end{equation} Both thresholds decrease in $\kappa$; at a heavy scale $\bar\tau_i\le C_7p_i$ and $\sigma_k\le C_8p_k$ with the constants $C_7$, $C_8$ of Definition~\ref{cpi:swc:capset}. Let $\chi_k$ be the indicator of King's small-field set \cite[(3.2)]{K1} (the block-sup clause, the covariant gradient clause $|D_{A_k}\phi_k(b)|\le p_{k-1}$ on every block bond $b=(y,y')$ of $T^{(k)}$, where $D_{A_k}\phi_k(b):=\phi_k(y')-U(A_k(b))\,\phi_k(y)$ with $U(A_k(b))$ the parallel transporter of $A_k$ along $b$, in unit-$k$ units, and the photon clauses), $\chi_k^\flat$ the same with the gradient clause imposed only on bonds with both ends in $H'^{(k)}$, $S^{(k),\kappa}(A_k,\phi_k,zf,g')$ the effective action after $k$ steps with the profile inside, and \begin{equation}\label{cpi:eq:Ikappa} \begin{gathered} I_\kappa(z,g'):=\int(dA_k)(d\phi_k)\,\chi_k^\flat\tau_k\lambda_k\,e^{-S^{(k),\kappa}},\\ I^\sharp_\kappa(z,g'):=\int(dA_k)(d\phi_k)\,\chi_k\tau_k\lambda_k\,e^{-S^{(k),\kappa}}\ \le\ I_\kappa(z,g'). \end{gathered} \end{equation} At $\kappa=0$ both thresholds are $+\infty$, so $I^\sharp_0$ is King's small-field integral. \emph{Unit-$k$ notation.} Write $\eta:=\eps/\eta_0$ for the spacing in unit-$k$ units, $\ell_j=L^j\eps$ for the block scales (written $\delta_j$ in the rest of this part), $\hat\kappa:=w_k=\kappa\eta_0^2$, $V=V_0+\kappa\mathbf 1_U$ with $V_0=m^2+n\psi$ its $\kappa$-free part and $m_V^2:=\inf_{H'}V_0\ge m^2$ the plateau gap; $Z_\kappa(zf,g')$ is the partition function with the sources $zf$ (real $z$) and $g'\in\{0,g\}$, supported in $H'$ at distance $\ge\eta_0$ from $U$, $A^{(k)}$ the lifted background, $P^{(k),\kappa}$ the renormalised perturbative part of $S^{(k),\kappa}$, $M_k:=p_{k-1}(\lambda\ell_{k-1})^{-1/4}$ King's block sup threshold \cite[(3.2)]{K1}, and $|U|_k$, $|\partial H'|_k$ the numbers of unit-$k$ cubes in $U$ and meeting $\partial U$. All constants $C$ depend on $(L,a,N,\lambda,e,m,\mu_0,\alpha_0,\bar n,\vartheta_1,\vartheta_2,c_0)$ and King's constants unless a further dependence is displayed; none depends on $\kappa$, $\eps$ or $k$. \subsection{The mismatch vertex, the capped set and the mass shift} The shell model subtracts its counterterms $\kappa$-adaptedly (Proposition~\ref{cpi:prop:TD}); the difference to the bulk counterterm is the mismatch vertex $\mathcal M$ of Definitions~\ref{cpi:swab:M} and~\ref{r13:def:Moffset}, with the offset $\Delta_T$ of the shell-side convention bridge (CT)$_{\rm sh}$, $|\Delta_T(s)|\le\Delta_*$ for $s\le s_0$, treated as an additional two-leg vertex. In unit-$k$ units, $C_{\mathcal M}:=C(\lambda+e^2)\ell_k$ is the coupling of one mismatch vertex, $r_j:=L^{j-k}$ the range of a line of scale $j$ ($r_j=\varrho_j$ in the notation of Theorem~\ref{cpi:thm:78}) and $r_*:=r_{j_*}$ that of the finer line $j_*$ of a vertex. The mismatch vertex enters the perturbative bound as an additional vertex of the graphs (Proposition~\ref{cpi:prop:TD}). The following lemma collects the three bounds on it that are used: a bound on its tree term (M1), its one-line form (M2) and a lossless form of its two-line density (M3), in which each line keeps the square root of its own amplitude. The lossless form is needed because the lossy bound $\rho_{j_*}^{1/2}\rho_{j_1}^{1/2}\le C\rho_{j_*}$ would leave a deficit $L^{k-j_*}\min(\kappa_{j_*}^{1/2},\kappa_{j_*}^{-1/2})$ deep in $U$, not summable over the scales, whereas the deep part $\hat\kappa^{1/2}m(j_*)m(j_1)$ of the lossless density is absorbed in Theorem~\ref{cpi:swab:TDc}. \begin{lemma}[the three clauses of the mismatch vertex]\label{cpi:swab:M123}\label{cpi:lem:mismatch} For King's legs of block data $b$ on the small-field set, $|b|_\infty\le p_{k-1}(\lambda\ell_{k-1})^{-1/4}$, and for internal lines with their half-amplitude bounds, uniformly in $\kappa\in[0,\infty)$ and $\eps$ where no scale is named, in unit-$k$ units: \begin{enumerate} \item[(M1)] at a heavy scale ($\hat\kappa\ge c_0$) the tree term $\frac12\ell_k^2\sum_x\eta^d|\Delta\delta(x)||\phi^{(k)}_\kappa(x)|^2$ is at most \begin{multline*} C(\lambda+e^2)\lambda^{-1/2}\ell_k^{1/2}p_{k-1}^2\big[|\partial H'|_k(1+\hat\kappa^{-1}\log(2+\hat\kappa))+|U|_k\hat\kappa^{-1}\big]\\ +C(\lambda+e^2)^2\lambda^{-1/2}\ell_k^{3/2}(1+\log(1/\ell_k))^{q_F}p_{k-1}^2|T^{(k)}| \end{multline*} plus the non-decaying wall-floor term $CC_n(\lambda+e^2)\lambda^{-1/2}L^{1/2}\ell_k^{3/2}p_{k-1}^2|T^{(k)}|$; at a light scale ($\hat\kappa0$ of the free-convention limit $s_{\rm inf}(\mathsf M)$ of this coefficient (Proposition~\ref{cpi:prop:S}). On a finite lattice the mass shift is exact: in the action \cite[(2.5)]{K2} the scalar mass enters only through $\frac12\sum_xs^3(\mathsf M^2+\delta m^2(\mathsf M))|\phi(x)|^2$ and the field-independent constants. The proof of Theorem~\ref{cpi:thm:T3F} uses the lemma to write the moment of the scalar source as a ratio of torus-convention functionals, and the threshold proposition of \cref{rot:ssec:openin} (Proposition~\ref{cpi:prop:S}) and the proof of (W$^\sharp$2) in \cref{rot:ssec:Wcone} use it again. \begin{lemma}[the exact mass shift on a finite lattice]\label{cpi:lem:massshift} Let $\langle\cdot\rangle^{(\mathsf M)}$ be the state of the same lattice, boundary condition and convention at the scalar mass $\mathsf M$ in place of $m$, and $u\defeq\frac12(\alpha(\mathsf M)-\alpha(m))$. Then, for every $F$ integrable against both measures, $\langle F\rangle^{(m)}=\langle F\,e^{u\,r^2(1)}\rangle^{(\mathsf M)}/\langle e^{u\,r^2(1)}\rangle^{(\mathsf M)}$, $r^2(1)\defeq\sum_xs^3|\phi(x)|^2$, and $f(s_{K,T}(\mathsf M))=B^z_{K,T}(\mathsf M)+\ln Z^{(\mathsf M)}-\ln Z^{(m)}$ with $B^z_{K,T}(\mathsf M)\defeq E(\mathsf M)-E(m)-(2c_{K,T}+z_K)\,s_{K,T}(\mathsf M)\,|T|$, where $E(\mathsf M)=E_0(\mathsf M)+E_1(\mathsf M)$ are the field-free constants of King's action at the mass $\mathsf M$ \cite[(2.6), (3.35)]{K1}. \end{lemma} \begin{proof} By the form of the action, the two Gibbs densities differ by the factor $e^{u\,r^2(1)}$ and a constant, $e^{\pm r^2(\cdot)}$ being integrable by the quartic term; the second identity is the first at $F=e^{r^2(s_{K,T}(\mathsf M)\cdot1)}$ with the constants written out. \end{proof} \subsection{The two-sided bound and the perturbative bound} This subsection proves the two-sided bound (SW1$^\circ$)$_{\rm int}$ of Theorem~\ref{cpi:thm:main}, King's bound \cite[Theorem~3.1]{K1} for the shell model, whose shell mass $\kappa\mathbf 1_U$ sits inside every matter covariance, with constants uniform in $\kappa\in[0,\infty)$, in the spacing and in the source coefficient. The mechanism is one: Ba{\l}aban's lower and upper bounds are run with the profile \eqref{cpi:eq:profile} in every covariance, and at each step the shell mass is handled by an exact Gaussian identity, by a positivity or by one of the bounds on the shell model proved below, Ba{\l}aban's four replaced objects being those of the shell model (the notation is collected in the closing paragraph of this subsection). The bounds behind the replacements are three, each stated here in the form in which the proof uses it, and proved here, except the mismatch factor, which is proved in Section~\ref{rot:sec:shellestB}: the first-order determinant term (Theorem~\ref{cpi:swab:DOM}), the perturbative bound with the mismatch factor and the vacuum sector (Proposition~\ref{cpi:prop:TD}), and the class of small fields carried through the lower bound (Lemma~\ref{cpi:lem:appclass}); the first, and the two displayed bounds of the vacuum sector, hold up to the endpoint $\kappa=\infty$. We begin with the fixed-torus moment of the scalar source (Theorem~\ref{cpi:thm:T3F}), which the upper bound needs at $g'=g$, then fix the standing data and state the two-sided bound, prove the three bounds, and close with the notation for Ba{\l}aban's steps, which Section~\ref{rot:sec:shellestB} runs for the shell model (Theorems~\ref{cpi:swab:B5K} and~\ref{cpi:thm:Kpt}). The upper bound of Theorem~\ref{cpi:thm:main} at $g'=g$ carries, through H\"older's inequality, a normalised moment of the Wick-ordered scalar source on the torus, and its constant $b_g$ in \eqref{cpi:eq:bg} is finite exactly when that moment is. The following theorem provides the finiteness. It is a by-product of the proof of Theorem~\ref{cpi:ct:mainW}(ii) run on the bare torus: by the mass-shift lemma the normalised functional of the source is a ratio of two torus-convention functionals, times convergent prefactors, and both functionals converge in $(0,\infty)$ along the spacings, so the ratio is bounded. Only the bound is used; no limit value, rate or uniformity in $T$ enters. \begin{theorem}[the fixed-torus moment of the scalar source]\label{cpi:thm:T3F} Let $T=D_{J'}$ with $J'\in5^{k_*}\N$, and let $0\le q\in C^\infty_c$ with $2\sup q0$, $e$, $m^2>0$, $\mu_0^2>0$ and $L=5$; $d=3$, $a=1$ (King's averaging constant), $\varrho=2$ and $p=6$ are the values at which (BAL)$_5$ is stated and its four steps are repeated (Theorem~\ref{rot:thm:BAL5}). Fix the shares $\vartheta_1,\vartheta_2$, a geometry $(H',T)$ satisfying (BA) with its grid offset and (T.nw), the wall height $n\ge1$, the wall profile $\psi$, the sources $f,g$ and the Wick-ordered source scheme of Section~\ref{rot:sec:gensrc}, and let the counterterm be $\delta m^2_\eps+c(\eps)$ with $c=\sum_{1\le\alpha+\beta\le4}c_{\alpha\beta}$, $c_{\alpha\beta}$ of order $e^\alpha\lambda^\beta$, and $\Delta_c:=\sum_{\alpha,\beta}\sup_{\eps\le\eps_0}|c_{\alpha\beta}(\eps)|<\infty$. Throughout, $\eps_0\in(0,1]$ is the fixed upper bound on the spacings of the class $\mathcal C$ (Definition~\ref{rot:def:class}), and every threshold $\eps_0(\eta_0)$ below is at most $\eps_0$. Three constants recur: $M_k'=L^{1/4}(p_{k-1}/p_k)M_k$ is the block-sup threshold of \cite[(3.2)]{K1}; $B_\eps(f)$, half the variance of the plaquette source $F(f)$ under the free photon measure at spacing $\eps$, the constant of the Gaussian domination $\langle e^{tF(f)}\rangle\le e^{t^2B_\eps(f)}$ of Lemma~\ref{rot:lem:GD}(a), with $B:=\sup_{\eps\le\eps_0}B_\eps(f)\le C(f)/(2\mu_0^2)$; and $M_T(q)$, the fixed-torus moment of Theorem~\ref{cpi:thm:T3F} at the amplitude $q$ ($s_*(e)$, $\bar\Delta$, $D_{J'}$ and $k_*$ as in Section~\ref{rot:sec:gensrc}). The provisions on the choice of the constants are those of Definition~\ref{r13:def:B5Kset}. The two-sided bound compares the partition function $Z_\kappa(zf,g')$ of the shell model with the stopping-scale integral $I_\kappa(z,g')$ and its variant $I^\sharp_\kappa(z,g')$: a lower bound by $e^{-\mathsf a}I^\sharp_\kappa$ at every $\kappa$, a lower bound by $e^{-\mathsf a}I_\kappa$ once the shell mass is heavy at the stopping scale, and an upper bound by $e^{\mathsf a}I_\kappa$ up to a large-field remainder $b_{g'}Z_\kappa(0,0)$. What is used later is that $\mathsf a$, $b$ and $b_g$ depend neither on $\kappa$ nor on the spacing nor on $z$, and vanish as the stopping scale $\eta_0$ tends to zero: in this form the bound is the shell estimate (SW1$^\circ$)$_{\rm int}$, one of the four shell estimates on which the heavy-shell decoupling (Theorem~\ref{rot:thm:shell}) rests. The statement also gives the bounds with the capped stopping-scale integral of Definition~\ref{cpi:swc:capset} in place of $I_\kappa$, and the order in which the constants are chosen. \begin{theorem}[two-sided bound with the shell mass inside: \textup{(SW1$^\circ$)$_{\rm int}$}]\label{cpi:thm:main}\label{r13:def:standing} Under the standing data there are $\eta_1>0$ (depending on $n$ as well as on the constants listed below, through the condition $(m^2+n)\ell_k^2M_k'\le p_k$ ($M_k'=L^{1/4}(p_{k-1}/p_k)M_k$ the block-sup threshold of \cite[(3.2)]{K1}) at the stopping scale and through the threshold of the walled model at the fixed wall height), $\sigma>0$, $c_1>0$, for every $\eta_0\in L^{-\N}$ with $\eta_0\le\eta_1$ a threshold $\eps_0(\eta_0)>0$, and a constant $C<\infty$ depending on $(L,a,N,\lambda,e,m,\mu_0,|T|,n,f,g,H',\vartheta_1,\vartheta_2,\Delta_c,c_0)$ but not on $\eta_0$, $\kappa$ or $\eps$, such that for all $\kappa\in[0,\infty)$, $\eps=L^{-K}\le\eps_0(\eta_0)$, $z\in[-1,1]$ and $g'\in\{0,g\}$, \begin{align} Z_\kappa(zf,g')&\ \ge\ e^{-\mathsf a}\,I^\sharp_\kappa(z,g')\qquad\ (\kappa\in[0,\infty)),\label{cpi:eq:lowsharp}\\ Z_\kappa(zf,g')&\ \ge\ e^{-\mathsf a}\,I_\kappa(z,g')\qquad\ \ \,(\kappa\in[\kappa_0,\infty)),\label{cpi:eq:low}\\ Z_\kappa(zf,g')&\ \le\ e^{\mathsf a}\,I_\kappa(z,g')+b_{g'}\,Z_\kappa(0,0)\qquad(\kappa\in[0,\infty)),\label{cpi:eq:up} \end{align} with $\mathsf a=C\eta_0^\sigma|T|$ and $b=e^{-c_1\,p(\eta_0)^2+C|T|}$; in \eqref{cpi:eq:up}, $b_{g'}$ denotes $b$ at $g'=0$ and $b_g$ at $g'=g$, where \begin{equation}\label{cpi:eq:bg} b_g:=b^{1/q}\,e^{q'B}\,M_T(q'g)^{1/q'},\qquad q'=q'(g)\in(1,2],\ 2q'\sup g0$, independent of $\kappa\in[0,\infty]$, of the spacing, of $i$, of $\Omega$ and of $B$ (depending on the parameters of the model, among them $m$, the wall height $n$ and $\lmu$, and $C_\sigma$ on $\sigma$), such that: \begin{enumerate} \item[(a)] $DF^\kappa_\Omega(B)[a]=\sum_{x\in\Omega}d^\kappa_x$ with $d^\kappa_x\defeq-\int_0^\infty Dg^{\kappa,s}_x(B)[a]\,ds$, where \[ g^{\kappa,s}_x(B)\defeq(\eps/\ell_i)^3\Tr(M^\kappa_\Omega(B)+s)^{-1}(x,x) \] is invariant under every gauge transformation of $B$ on the sites of $\Omega$; and for every block $Q$ of scale $i$ in $\Omega$ (the blocks at the boundary of $\Omega$, next to its holes and at its re-entrant edges included), every $r\ge1$ and every $\sigma>0$, \[ \sum_{x\in Q}\abs{d^\kappa_x}\le Ce_i^2\Big[\sup_{Q_r}\abs{\partial B}\sup_{Q_r}\abs{\partial a}+e^{-\delta r/2}\sup_\Omega\abs{\partial B}\sup_\Omega\abs{\partial a}\Big]+C_\sigma e_i^2p(\ell_i)^2\ell_i^\sigma, \] $Q_r$ the set of points of $\Omega$ joined to $Q$ by a path in $\Omega$ of length $\le r$; in particular the left side is $\le Ce_i^2p(\ell_i)^2$, and the same holds for the shell-mass residual $d^\kappa_x-d^0_x$; no H\"older quotient of $B$ or $a$ enters; \item[(b)] on the two domains of the kept terms write $DF^\kappa_\Omega(B)[b]=\sum_{y\in\Omega}b(y)\cdot j^\kappa_B(y)$; at every block $Q$ carrying a vertex of the kept terms (which lies deep inside the domain, at distance $>r(\ell_i)-1$, respectively $\ge5r(\ell_i)-5$, from its complement, $r(\cdot)$ Ba{\l}aban's localisation function \cite[(2.7)]{Ba2}), if the H\"older quotients of the first differences of $B$ are $\le Cp(\ell_i)$ at the points at distance $0$, and the zero-shell-mass current $j^0_B$ obeys Ba{\l}aban's per-step bound \cite[Prop.~3.2, (3.58)]{Ba1}, \cite[pp.~578--579]{Ba2}; \item[(c)] on the torus $T$, besides (a), the bound of (b) holds at every block of $T$ at King's backgrounds, whose size is bounded by \cite[(3.2), (A.1)]{K1}. \end{enumerate} \end{theorem} \begin{proof} The Gaussian normalisations of the earlier steps telescope, and the one-shot form $\Delta^{(i)}=a_i-a_i^2Q_iG_iQ_i^*$ gives $\prod_{j0$, \begin{equation} \rho_j(x):=\begin{cases}\min\{1,\ \dist(x,\partial U)+w_j^{-1/2}\},& x\in H',\\[2pt] \dfrac{w_j^{-1/2}+\eta_j}{1+w_j\eta_j^2}\,C_d\,e^{-\delta^\flat_j(\dist(x,\partial U)-3\eta_j/2)_+}+w_j^{-1},& x\in U,\end{cases} \end{equation} and $\rho_j\equiv1$ at a light scale: the \emph{amplitude}, so that every leg $u$ of block data $b$ obeys $|u(x)|\le C|b|_\infty\rho_j(x)$ (the linear vanishing of the minimiser at the heavy shell); $\rho_j\le C_\rho$ everywhere, and on $U$ in the continuum regime $w_j\eta_j^2\le1$, $\rho_j(x)\le Cw_j^{-1/2}e^{-c\sqrt{w_j}\dist(x,\partial U)}+w_j^{-1}$. With the H\"older exponent $\alpha\in(0,\tfrac12]$ of the norm, fixed once for all (we take $\alpha=\tfrac13$; $\alpha_0$ denotes the minimal ordered degree of Theorem~\ref{cpi:thm:78}), and a fixed rate $\delta_4>0$, the \emph{half-amplitude norm} of a function $\Phi$ at scale $j$ relative to a source set $S$ ($S=T^{(j)}$ for true external legs, where the decay weight is $1$) is \begin{equation}\label{cpi:eq:kanorm} \begin{split} \|\Phi\|_{\kappa,j;S}:=\sup_x\frac{e^{\delta_4\dist(x,S)}|\Phi(x)|}{\rho_j(x)^{1/2}}&+\sup_be^{\delta_4\dist(b,S)}|D_A\Phi(b)|\\ &+\sup_{0<|x-y|\le1}e^{\delta_4\dist(x,S)}\rho_j(x)^{1/2}\,\frac{|U(A(\Gamma_{x,y}))D_A\Phi(y)-D_A\Phi(x)|}{|x-y|^{\alpha}} . \end{split} \end{equation} At light scales this is Ba{\l}aban's norm \cite[(1.32)]{Ba3} with decay. Its two properties are the \emph{leg bounds}, $\|a_j\ell_j^{-2}G^\kappa_jQ_j^*b\|_{\kappa,j;\supp b}\le C|b|_\infty$ and $\|G^\kappa_{(j)}(\cdot,z)\|_{\kappa,j;\{z\}}\le C\rho_j(z)^{1/2}$ uniformly in $\kappa\in[0,\infty)$, and the \emph{pairing property}: in a product of two legs the H\"older quotient of one, carrying the weight $\rho_j^{1/2}$, is paired with the half-amplitude $\rho_j^{1/2}$ of the other, so that the pair is bounded by the product of the two norms times the full amplitude $\rho_j$, in place of every heavy-side H\"older use of \cite{Ba3}. \emph{Notation of the bound.} In Ba{\l}aban's notation $h,h'$ are the localisations, $d_v,d_s$ the orders in $e$ and $\lambda$ and $d(\cdot)$ the length of the shortest tree through the cubes; the source term $zF(f)$ enters as an external photon field, with $f$-dependent constants. On the photon side nothing is converted: $A^{(k)}$ enters every family of $P^{(k),\kappa}$ only through lines, transporters, covariant differences, block averages, legs and the plaquette curl of the source vertex, so it is the regular line field $B$ of the norm form below, and no sup bound $|A_k|\le Cp$ uniform in the scale is used. $P^{(k),\kappa}_0$ denotes the field-independent part of $P^{(k),\kappa}$, its value at zero block fields, every term of which carries a line of scale $\ge k$, and ${\rm gen}_j(\square)$ the constants generated at step $j$ on a unit-$j$ cube $\square$. The perturbative bound comes in two forms, King's block-data form (a), for legs of block data in King's small-field set, and Ba{\l}aban's norm form (b), in which the scalar external fields are measured in the half-amplitude norm \eqref{cpi:eq:kanorm}, with the leg bounds and the pairing property stated after it; clause (c) is the gain of a factor $C(\lambda+e^2)\ell_k$ at every mismatch vertex, and clause (d) controls the field-independent part per step and through its dependence on the background, no absolute bound per block being claimed. In the proof of Theorem~\ref{cpi:thm:main} the proposition, with Theorem~\ref{cpi:swab:DOM}, stands in place of \cite[Proposition~1]{Ba3}; its leg estimate enters the class of Lemma~\ref{cpi:lem:appclass}(c), and the estimates of Section~\ref{rot:sec:shellestB} draw on it. The proof repeats Ba{\l}aban's \cite[Proposition~1]{Ba3} for the shell model; the bound differs from his \cite[(1.33)]{Ba3} only in the norm on the scalar external fields, and the wall profile enters only through the scalar propagators, whose bounds improve under a non-negative potential. \begin{proposition}[the perturbative bound for the shell model]\label{cpi:prop:TD} There are $\delta_0>0$ and constants $O(1)$ depending on $\alpha$, $\bar n$, $d$, $L$, $a$, $N$, on $(m,n,\mu_0)$ through King's constants at the one fixed wall height, and, for the families containing a source vertex \cite[(3.20)--(3.21)]{K1}, on the sources through $\|f\|_{C^1}$ and $\|g\|_\infty$ (a factor $(1+\|g\|_\infty)^{n_h}$ for $n_h$ vertices \cite[(3.21)]{K1}), but \emph{not} on $\eps$, $k$, $\kappa\in[0,\infty)$, the domains $\Omega,\Omega_1,\Omega_2$ of the construction, or the regular background field $B$, such that: \begin{enumerate} \item[(a)] (block-data form) For every renormalised family $H_{\rm ren}$ of $P^{(k),\kappa}$ of order $\le13$ with at least one external field, with the shell-model propagators and the legs $\phi^{(k)}_\kappa$ of block data in King's small-field set \cite[(3.2)]{K1}, of whose photon clauses only the derivative clause $|\partial^\eta A^{(k)}|+[\partial^\eta A^{(k)}]_{\alpha}\le C_kp_{k-1}$ is used, through the regularity of the line field, \begin{equation} |E^\kappa(H_{\rm ren},A^{(k)};\{y_i\},\{z_q\})|\le O(1)\,(L^k\eps)^{\rho}\,p(L^k\eps)^{n_1}\,e^{-\delta_0\dist(\{y_i\},\{z_q\})},\qquad\rho\ge s/4, \end{equation} with King's exponents $\rho$, $n_1$, $s$ \cite[(3.38)]{K1}, in King's $\ell^1$-in-localisations form; the vacuum family is excluded, the field-independent part being recentred by $E_1^\kappa$ (clause (d)). \item[(b)] (norm form) On the torus, for generalised graphs of any finite order, with constants depending on their size, with external photon fields satisfying Ba{\l}aban's conditions, their H\"older clause included, in Ba{\l}aban's form \cite[(1.33), Proposition~2.2]{Ba3} with the half-amplitude norm on the scalar external fields and the norm $\|h'A_{\rm ext}\|_{1,\alpha}$ on the photon ones, \begin{equation}\label{cpi:eq:TDb} \begin{split} |E^\kappa(G_{\rm ren},\{\square(v)\},\Phi_{\rm ext},A_{\rm ext})|&\le O(1)\,(e\ell_k^{1/2})^{d_v}(\lambda\ell_k)^{d_s}\,e^{-\delta_0d(\{\square(v)\})}\\ &\qquad\times\prod\|h\Phi_{\rm ext}\|_{\kappa,k;S_{\rm ext}}\prod\|h'A_{\rm ext}\|_{1,\alpha}. \end{split} \end{equation} For King's legs $\|h\phi^{(k)}_\kappa\|_{\kappa,k}\le C|\phi_k|_\infty$, so (b) implies (a) for true legs. \item[(c)] (mismatch factor) On the torus, at King's backgrounds $B=uA^{(k)}$, $u\in[0,1]$, $A_k$ in the small-field set, every graph containing $r\ge1$ mismatch vertices $\mathcal M$ (Definition~\ref{cpi:swab:M}) and at least one external field obeys the bound \eqref{cpi:eq:TDb} with the additional factor $(C(\lambda+e^2)\ell_k)^{r}$, $C$ free of $\kappa\in[0,\infty)$, $\eps$, $k$ and $A_k$ (with a counterterm offset $c$, $C(\lambda+e^2)$ is enlarged by a constant multiple of $\Delta_c$, as in Lemma~\ref{cpi:swab:M123}), for the family total over the scale assignments of the graph, not per member. \item[(d)] (vacuum sector) $P^{(k),\kappa}_0$ is controlled per step: $|{\rm gen}_j(\square)|\le C\ell_j^{1/2}$ on every unit-$j$ cube, uniformly in $\kappa$ and $\eps$, so that $|P^{(k),\kappa}_0(X)-P^{(K_f),\kappa}_0(X)|\le C'\ell_k^{1/2}|X|_k$ for every union $X$ of unit-$k$ cubes, and the common constant $P^{(K_f),\kappa}_0(T)$ cancels between the two lines of Theorem~\ref{cpi:thm:main}; no absolute bound on $P^{(k),\kappa}_0$ per block, free of the final spacing, is claimed or used. At a regular block configuration $A_k$ (only derivative bounds of $A^{(k)}$ enter), every renormalised vacuum family of order $\le13$ with $r\ge0$ mismatch vertices obeys, per unit-$k$ root cube $\square$, \begin{equation} \begin{split} |E^\kappa(H_{\rm ren},A^{(k)};\square)-E^\kappa(H_{\rm ren},0;\square)|&\le O(1)(e\ell_k^{1/2})^{d_v}(\lambda\ell_k)^{d_s}\big(C(\lambda+e^2)\ell_k\big)^r\big(1+C^\flat p(\ell_k)\big)^{n_1}\\ &\le O(1)\,\ell_k^{\beta_0}, \end{split} \end{equation} uniformly in $\kappa\in[0,\infty]$, $\eps$ and $k$, $C^\flat$ fixed by the cube side of the cube step. For a source-free renormalised vacuum family $F$ of order $\le13$ with $r\ge1$ mismatch vertices, a unit-$k$ cube $Q$ and two backgrounds $B_0,B_1$ of King's class $R_k(C_R)$, with $b:=B_1-B_0$ and $N_k(b):=\|\partial b\|_\infty+[\partial b]_{\alpha}+e^{-\delta_0N_T/16}\|b\|_\infty$, $N_T$ the smallest side of $T^{(k)}$, \[ |F(B_1;Q)-F(B_0;Q)|\le C_{\mathrm{vac}}\,\big(C(\lambda+e^2)\ell_k\big)^r\,e_k\,N_k(b), \] with $C_{\mathrm{vac}}$ depending on $C_R$, King's constants ($\mu_0$ included), $\bar n$, $L$ and $\alpha$, and not on $\kappa\in[0,\infty]$, $\eps$, $k$, $B_0$, $B_1$, $Q$ or the torus. \end{enumerate} \end{proposition} \begin{proof} For King's legs $\|h\phi^{(k)}_\kappa\|_{\kappa,k}\le C|\phi_k|_\infty\le Cp(\ell_{k-1})(\lambda\ell_{k-1})^{-1/4}$ on the small-field set, Ba{\l}aban's bound on his class \cite[p.~421]{Ba3}. Since the bound \eqref{cpi:eq:TDb} differs from Ba{\l}aban's \cite[(1.33)]{Ba3} only in the norm on the scalar external fields, every consequence of (1.33) holds for the shell model with the same constants: the coefficients of \cite[Propositions~3.1--3.2]{Ba1} are $O(\ell_k^{\beta_0})e^{-\delta_0d}$ with $\beta_0=\tfrac14-\beta$, $\beta\in(0,\tfrac1{28})$ the exponent of King's regularity condition \cite[(3.5)]{K1}, fixed once for all, and the remainders $O(1)\ell_k^{\kappa_*}|T^{(k)}|$ with $\kappa_*=\min(\tfrac{13}4-\beta,14\beta_0)>3$. The proposition is thus \cite[Proposition~1]{Ba3} run for the shell model with the leg bounds and the pairing property of \eqref{cpi:eq:kanorm} wherever the shell mass enters a line or leg bound, the Taylor step at a divergent two-scalar-leg subgraph (its linear term is the $\kappa$-adapted counterterm), Lemma~\ref{cpi:lem:ward} for the photon and mixed classes, the mismatch vertex as an additional vertex (Lemma~\ref{cpi:swab:M123}, Theorem~\ref{cpi:swab:TDc}) and the step determinants per step \cite[(3.46)--(3.47)]{Ba1}, \cite[(2.99)--(2.107)]{Ba2}, their first-order term by Theorem~\ref{cpi:swab:DOM}, their second-order term as a photon two-leg class, their third-order term by the gauge identity of the full determinant in a slot of the fluctuation field, and the absolute ratio only at the final scale \cite[(3.66)--(3.69)]{Ba1}. Clause (c) is proved as Theorem~\ref{cpi:swab:TDc}: the closed matter blocks with an odd number of photon-type legs are first rewritten by charge conjugation on their full step-$k$ lines, no Ward identity being used for a class containing $\mathcal M$. Finally, on the propagator sub-domains the torus form passes to the vertices at depth $\ge R_{\rm Ba}$ \cite[pp.~411--412]{Ba3}, \cite[pp.~570, 579--580]{Ba2}, with the boundary-difference clause \cite[(1.20)]{Ba4} through the layer-source identity $\delta G^\kappa_k(\Omega,\Omega_2)F=G^\kappa_k(\Omega_2)\,\sigma$. \end{proof} The third bound is the class of small fields carried through the lower bound. Ba{\l}aban's lower bound \cite[\S3]{Ba1} is an induction over the scales on such a class, King's class (A.1) of \cite[p.~676]{K1}; with the shell mass inside the covariance its clause on the covariant Laplacian of the background fails near $\partial U$ at heavy scales, and we replace it by the two consequences that the power counting uses, a bound on the covariant gradient of the background and a bound on the mismatch between the block field and the block average of its background. The lemma defines the class and shows that it is what the induction needs: the backgrounds compose exactly from one scale to the next; the class at scale $i+1$ together with a small fluctuation gives the class at scale $i$, under lower bounds on its constants and caps on Ba{\l}aban's final spacing that give the remainder of the expansion of the background in the fluctuation gauge field its share of the room, as his compatibility \cite[(3.53)--(3.54)]{Ba1} requires; Ba{\l}aban's coefficient bounds hold on it; and the supports of the stopping-scale cutoffs of Theorem~\ref{cpi:thm:main}, the capped set included, lie in it, which is what fixes the ranges of \eqref{cpi:eq:low} and \eqref{cpi:eq:lowsharp} and gives the cap form. The room in the three clauses comes from the ratio of their thresholds at consecutive scales, and the clause on the stopping scale is King's near-face analysis at the shell face. \begin{lemma}[the class for the shell model, with the gradient and mismatch clauses]\label{cpi:lem:appclass}\label{r13:def:appclass} Let $G^\kappa_i$ be King's operator \cite[(2.13)--(2.17)]{K1} at scale $i$ with the profile $V$ of \eqref{cpi:eq:profile} in place of $m^2$, $\phi^{(i)}_\kappa:=a_i\ell_i^{-2}G^\kappa_iQ_i^*\phi_i$ the \emph{background} (the minimiser of the one-step functional) of the block field $\phi_i$, $Q_i=Q_i(A^{(i)})$ the covariant block average, $p_1(t):=b_1(1+\log t^{-1})^{p_{\mathrm{fl}}}$, $b_1\ge b_0$, $p_{\mathrm{fl}}\le p-1$, Ba{\l}aban's bound on the fluctuation field \cite[(3.50), (3.54)]{Ba1}, $\varkappa_i:=aL^{-2}/(a_i+aL^{-2})\le L^{-2}$, and $C_{\rm sup}$, $C_D$, $C_{Q4}$ the constants, independent of $\kappa$, $\eps$ and $i$, in $|\phi^{(i)}_\kappa|_\infty\le C_{\rm sup}|\phi_i|_\infty$, $\sup_b|D_Au(b)|\le C_D|b|_\infty$ and $|Q_i(A)u|_\infty\le C_{Q4}|b|_\infty$ for the leg $u=a_i\ell_i^{-2}G^\kappa_iQ_i^*b$ of block data $b$. For $i\le K_f$ let $(\mathrm{A.1})^{(i)}_\kappa$ be the class of pairs $(A^{(i)},\phi_i)$ such that, in unit-$i$ units, \begin{gather*} \sup_{x}|\phi^{(i)}_\kappa(x)|\le C_1M_i,\qquad \sup_{y\in T^{(i)}}|\phi_i(y)|\le C_1M_i,\\ (\mathrm G)_i:\ \sup_{b}\big|D_{A^{(i)}}\phi^{(i)}_\kappa(b)\big|\le C_1'p_i,\qquad (\mathrm E)_i:\ \sup_{y\in T^{(i)}}\big|\phi_i(y)-Q_i(A^{(i)})\phi^{(i)}_\kappa(y)\big|\le C_1''p_i,\\ (\Gamma)_i:\ |\partial A^{(i)}|_\infty\le C_\Gamma'p_i, \end{gather*} the supremum in $(\mathrm G)_i$ over all fine bonds $b$ on both sides of $\partial U$, with no clause on $\Delta_A\phi^{(i)}_\kappa$ and no near-$U$ clause; $(\mathrm G)_i$ is the \emph{background-gradient clause} and $(\mathrm E)_i$ the \emph{block-mismatch clause}, both linear in $\phi_i$, and $(\mathrm E)_i$ is the block average of the Laplacian clause, since $a_i(\phi_i-Q_i\phi^{(i)}_\kappa)=Q_i[(-\Delta_A+V)\phi^{(i)}_\kappa]$ exactly. Then, in unit-$i$ units: \begin{enumerate} \item[(a)] (exact composition) Under the translation $\phi_i=aL^{-2}C^{(i),\kappa}Q^*\psi+\phi_i'$ of the $(i+1)$-st fluctuation step, $\psi=\phi_{i+1}$, the backgrounds compose exactly, $\phi^{(i)}_\kappa(\phi_i)=\phi^{(i+1)}_\kappa(\psi)+a_i\ell_i^{-2}G^\kappa_iQ_i^*\phi_i'$, and $|a_i\ell_i^{-2}G^\kappa_iQ_i^*\phi_i'|_\infty\le C_{\rm sup}|\phi_i'|_\infty$. \item[(b)] (compatibility with coefficient one) If $\psi\in(\mathrm{A.1})^{(i+1)}_\kappa$ and $|\phi_i'|_\infty\le p_1(\ell_i)$, then $\phi_i\in(\mathrm{A.1})^{(i)}_\kappa$, provided $L^{3/4}\ge3$ (true for every integer $L\ge5$), \begin{gather*} C_1\ge3\max(C_{\rm sup},1)\max(1,b_1/b_0),\qquad C_1'\ge3C_D\max(1,b_1/b_0),\\ C_1''\ge\max\{2(1+C_{Q4})\max(1,b_1/b_0),\,C_1'\}, \end{gather*} and the three caps on Ba{\l}aban's final spacing $\eps_0^{\rm Ba}$ (chosen after $C_1$, $C_1'$, $b_1$, $p_{\mathrm{fl}}$) for the remainder of the expansion of the background in the fluctuation gauge field, with $C_{\rm rem}$ the remainder constant of \eqref{cpi:eq:remainder}, \begin{gather*} C_{\rm rem}\,e\,C_1\lambda^{-1/4}(\eps_0^{\rm Ba})^{1/4}p_1(\eps_0^{\rm Ba})\le\tfrac16C_1',\qquad C_{\rm rem}\,e\,(\eps_0^{\rm Ba})^{1/2}p_1(\eps_0^{\rm Ba})\le\tfrac16,\\ C_{\rm rem}\,e\,p_1(\eps_0^{\rm Ba})\big(\lambda^{-1/4}(\eps_0^{\rm Ba})^{1/4}+(C_1'/C_1)(\eps_0^{\rm Ba})^{1/2}\big)\le\tfrac13, \end{gather*} together with $\eps_0^{\rm Ba}\le e^{1-4p_{\mathrm{fl}}}$, so that by monotonicity they hold at every scale $\ell_i\le\eps_0^{\rm Ba}$. \item[(c)] (the fluctuation expansion closes) On $(\mathrm{A.1})^{(i)}_\kappa$ the external legs satisfy $\|\phi^{(i)}_\kappa\|_{\kappa,i}\le CC_1M_i$ in the half-amplitude norm of Proposition~\ref{cpi:prop:TD}, $\lambda\ell_i|\phi^{(i)}_\kappa|^4\le C_1^4p_i^4$ pointwise, and at every block $y$, every $x\in B(y)$ and every fine bond $b$, (P-grad) $|D_{A^{(i)}}\phi^{(i)}_\kappa(b)|\le C_1'p_i$ and (P-block) $|\phi^{(i)}_\kappa(x)-U(A^{(i)}(\Gamma_{x,y}))\phi_i(y)|\le C_Pp_i$, $C_P:=C_1''+2dC_1'$, $U(A^{(i)}(\Gamma_{x,y}))$ the transport along a path from $y$ to $x$ inside the block. These are the conditions under which Ba{\l}aban's coefficient bounds \cite[(3.58)]{Ba1}, \cite[p.~421]{Ba3} hold for the vertex families with external scalar legs (the current vertex through (P-grad), the kernel vertices through (P-block), every other family through the sup clause alone), hence under which \cite[Propositions~3.1--3.2]{Ba1}, taken to order $\bar n=13$, hold for the shell model with remainder terms $O(1)\ell_i^{\kappa_*}|T^{(i)}|$, $\kappa_*=\min(\tfrac{13}4-\beta,14\beta_0)>3$, and coefficients $O(1)\ell_i^{\beta_0}$, $\beta_0=\tfrac14-\beta$, with $\beta<\tfrac1{28}$ the exponent of King's regularity condition \cite[(3.5)]{K1}, as in Proposition~\ref{cpi:prop:TD}. \item[(d)] (final restriction) At the stopping scale $k$, with $M_k'=L^{1/4}(p_{k-1}/p_k)M_k$ the block-sup threshold of \cite[(3.2)]{K1}, as in Theorem~\ref{cpi:thm:main}: (i) for $w_k\ge c_0$, $\supp(\chi_k^\flat\tau_k\lambda_k)\subset(\mathrm{A.1})^{(k)}_\kappa$, and the same for the larger cap set $\supp(\chi_k^\flat\tau_k^{\rm cap}\lambda_k^{\rm cap})$ of Definition~\ref{cpi:swc:capset}, once $C_1\ge\max(C_{\rm sup},1)L^{1/4}(1+\log L)^p$, $C_1'\ge C_{\rm G}(1+\log L)^p$ and $C_1''\ge C_E(1+\log L)^p$, with constants $C_{\rm G}$, $C_E$ free of $\kappa$, $\eps$ and $k$, provided $(m^2+n)\ell_k^2M_k'\le p_k$ and $e_kM_k\le1$ at $\ell_k=\eta_0$ (both are conditions on $\eta_1$); (ii) for every $\kappa\in[0,\infty)$, $\supp(\chi_k\tau_k\lambda_k)\subset(\mathrm{A.1})^{(k)}_\kappa$ under the conditions of (i), those of the $w$-linear bound on the $\kappa$-difference of the background (the derivative clause for small-field data, $\ell_k\le\eps_0^{\rm Ba}$, Ba{\l}aban's radius constant $R\ge R_*$, the largest of the lower bounds on $R$ of the cutoff listed in Definition~\ref{r13:def:B5Kset}) and with $C_1'$ at least the constant of that bound; (iii) at the final scale, $\{|\phi_{K_f}(y)|\le1\ \forall y\}\subset(\mathrm{A.1})^{(K_f)}_\kappa$. \end{enumerate} \end{lemma} \begin{proof} Clause (a) is the composition law of the kernels $\exp(-\tfrac a2\ell^{d-2}|\psi-Q\phi|^2)$, $V$ entering only through $-\Delta_A+V$, with the $\kappa$-uniform sup bound of the legs. For (b), in unit-$i$ units the thresholds of $(\mathrm{A.1})^{(i+1)}_\kappa$ are at most $L^{-3/4}$, $L^{-3/2}$ and $\varkappa_iL^{-1/2}\le L^{-5/2}$ times those of $(\mathrm{A.1})^{(i)}_\kappa$ for the sup clauses, $(\mathrm G)$ and $(\mathrm E)$ (the last by the push-down identity $aL^{-2}C^{(i),\kappa}Q^*\psi=Q_i\Phi_\psi+\varkappa_iQ^*(\psi-Q_{i+1}\Phi_\psi)$, $\Phi_\psi=\phi^{(i+1)}_\kappa(\psi)$), and each room is split into three shares, for the scale-$(i+1)$ clause, the fluctuation leg and the remainder $\delta$ of the expansion of the background in the fluctuation gauge field $A'=A'^{(i)}$, \begin{equation}\label{cpi:eq:remainder} \begin{gathered} G^\kappa_i(A'+B)-G^\kappa_i(B)=G^\kappa_i(A'+B)\,D_B^*\big(U(A')-1\big)D_B\,G^\kappa_i(B),\\ |U(A')-1|\le e\eta_i|A'|,\qquad |\delta|_\infty\le C_{\rm rem}\,e\,p_1(\ell_i)\big(C_1\lambda^{-1/4}\ell_i^{1/4}+C_1'\ell_i^{1/2}\big)p_i, \end{gathered} \end{equation} with $|A'|_\infty,|\partial A'|_\infty\le C_\gamma p_1(\ell_i)$ on Ba{\l}aban's fluctuation set. In (c) the norm bound is the leg estimate of Proposition~\ref{cpi:prop:TD}, (P-grad) is $(\mathrm G)_i$ and (P-block) follows from $(\mathrm E)_i$ and $(\mathrm G)_i$ along a path inside the block. Clause (d) is King's near-face analysis \cite[(5.41)--(5.49)]{K2}, \cite[\S5.2]{K2} at the shell face, with $\phi^{(k)}_\kappa=v+\Delta\phi$, $v$ the background at $\kappa=0$ controlled by \cite[Lemma~2.4, (2.65)]{Ba2} and an interior estimate, $|\Delta\phi|+|D_A\Delta\phi|\le C_\Delta p_{k-1}$ on every block, whence $C_{\rm G}:=C_{G0}+C_\Delta$ and $C_E:=C+C_\Delta$; (ii) follows by the $w$-linear route $\Delta\phi=-(M^0_k)^{-1}(w_k\mathbf 1_U\phi^{(k)}_\kappa)$ for $w_k0$ depending only on $d$, $L$, $a$, $c_0$, $c_V$ and King's constants, and not on $\kappa$, $\eta$, $j$ or $A$, such that for every $\kappa\in[0,\infty]$, every scale $j\le K_f$, every $A$ regular at scale $j$, all fine sites $x,y$ of $T$ (on either side of $\partial U$, in any position relative to the faces, edges and corners of $H'$) and all bonds $b$ at $x$, $b'$ at $y$, \begin{align} |G(x,y)|&\le C|x-y|_\eta^{-1}e^{-c|x-y|},\label{cpi:eq:K0}\\ |D^x_AG(b,y)|+|D^y_AG(x,b')|&\le C|x-y|_\eta^{-2}e^{-c|x-y|},\label{cpi:eq:K1}\\ |D^x_AD^y_AG(b,b')|&\le C|x-y|_\eta^{-3}e^{-c|x-y|}.\label{cpi:eq:K2} \end{align} The constants depend on $c_0$ only through $W_*\defeq\max(64,c_0)$, and no lower bound on $c_0$ is used. The same bounds hold for every partial sum of the slices of the scale decomposition \eqref{cpi:eq:slices}, with the same constants, and for every single slice $G^\kappa_{(i)}$ with constants of the same form (any rate $c<\delta_6$, the rate of the single-scale bounds). No H\"older clause and no second difference in the same variable is claimed: $D^xD^xG$ is not uniform in $\kappa$ near $\partial U$, its normal part jumping by about $w_jG$ across the face. \end{theorem} \begin{proof} We first bound one slice, in unit-$i$ units. Every slice is a pair of legs, \[ G'_{(i)}(x,y)=\sum_{u,v\in T^{(i)}}\mathsf l(x,u)\,C'(u,v)\,\overline{\mathsf l(y,v)},\qquad \mathsf l(\cdot,u)\defeq a_i(M^\kappa_i)^{-1}Q_i(A)^*e_u, \] so that each covariant difference acts on one leg. The legs obey decaying sup and first-difference bounds on both sides of $\partial U$, uniform in $i$, $\kappa$, $\eta_i$ and $A$: for $w_i>W_*$ from the first-difference bound of the shell model, for $w_i\le W_*$ from the interior estimate on $B_{1/4}(x)$; and $|C'(u,v)|\le C_2e^{-\delta_1|u-v|}$. Hence \[ |D_x^{n_x}D_y^{n_y}G'_{(i)}(x,y)|\le C_3e^{-\delta_6|x-y|}\quad(n_x,n_y\in\{0,1\}),\qquad \delta_6\defeq\tfrac14\min(\delta_0,\delta_1,\delta_2,\delta_3). \] The finest term $G^\kappa_1$ obeys the same bound by the Combes--Thomas estimate. Next we sum over the slices. Rescaled by the side $s_i\defeq L^{i-j}$ of an $i$-block, the slice bound reads $C_4s_i^{-1-n_x-n_y}e^{-\delta_6|x-y|/s_i}$, and $\sum_{s\in\Lambda}s^{-\nu}e^{-\delta r/s}\le C\max(r,\eta)^{-\nu}e^{-\delta r/2}$ over $\Lambda=\{L^m\eta\le1/L\}$ sums the slices to \eqref{cpi:eq:K0}--\eqref{cpi:eq:K2} with $c=\delta_6/2$; the same sum over any set of slices gives the bounds for the partial sums, and the slice bound gives them for a single slice with any rate $c<\delta_6$. At $\kappa=\infty$ the bounds pass to the limit, $(M^\kappa_j)^{-1}$ being decreasing in the Loewner order. It remains to pass to sub-domains. On a union $\Omega$ of $j$-blocks whose faces lie at least two blocks from the parallel faces of $\partial U$ the same holds in sub-domain form: the heavy-scale amplitude comes from the barrier of the shell model periodised by an image sum, the light scales from the interior estimate, and the gauge field reflected at Neumann faces is regular after a gauge transformation. On Ba{\l}aban's scale-dependent propagator domains the per-scale bound is applied domain by domain. \end{proof} The step-line bounds do not by themselves control the classes with photon external legs. These are bounded only after the Ward--Takahashi combination of the seagull and exchange graphs, step by step, never for a fixed ordering or scale assignment of the lines: the face seagull term $e^2\sum_x\eta^d[C^{\eta,\kappa}-C^{\rm bulk}](x,x)|L(x)|^2$ of a class with photon legs $L$ is bounded only after the combination, cancelled against the face part of the exchange graph. The next lemma collects what the perturbative bound of Proposition~\ref{cpi:prop:TD} needs from these classes, uniformly in $\kappa$: the Ward--Takahashi identities of the shell model (a), the pointwise bound and the moment form behind the Taylor step at a divergent photon two-leg subgraph (b), and the first-moment bounds of the two-photon and mixed kernels with the transport bound (c). The identities follow from gauge invariance alone; the bounds rest on the step-line bounds of Theorem~\ref{cpi:thm:Kpt} on both sides of $\partial U$, on Ba{\l}aban's bulk statements read at $\kappa=0$ with the wall and on heavy suppression on the shell side. \begin{lemma}[the photon two-leg and mixed classes of the shell model]\label{cpi:lem:ward} \begin{enumerate} \item[(a)] (Ward identities) The matter action of the shell model, $\tfrac12\langle\phi,(-\Delta_A+V)\phi\rangle+\sum\eps^3(\tfrac12\delta m^2|\phi|^2+\lambda|\phi|^4)$, and the block kernel $a_kQ_k(A)^*Q_k(A)$ are invariant under $\phi\to e^{eq\theta}\phi$, $A\to A+\partial\theta$, and the lattice Ward--Takahashi identities of \cite[\S2]{Ba3} hold for the vacuum-polarisation families of the shell model verbatim: at each step $j$ the step-$j$ effective action, in particular the logarithm of the determinant of the step-$j$ operator with its block vertices, is exactly gauge invariant, the seagull and exchange graphs combine into transverse families, and the \emph{zeroth moment} of the combined two-photon kernel $\Pi^\kappa_j$ of the step-$j$ family, built with the full step-$j$ lines, vanishes up to the torus zero mode, $\sum_y\Pi^\kappa_j(x,y)=0$ up to a holonomy response of size $O(e^{-\delta|T|^{1/3}/\eta_0})$. \item[(b)] (Taylor step) The combined two-photon kernel obeys $|\Pi^\kappa_j(x,y)|\le C|x-y|^{-4}$ and the moment form $\sup_x\sum_{y\ne x}\eta_j^3|x-y|^{1+\alpha}\big[|D^x_AG|\,|D^y_AG|+|G|\,|D^x_AD^y_AG|\big](x,y)\le C$, uniformly in $\kappa$, $\alpha$ the H\"older exponent of \eqref{cpi:eq:kanorm}; hence the Taylor step at a divergent photon two-leg subgraph, which acts on the \emph{photon} legs, whose norms $\|\cdot\|_{1,\alpha}$ do not see $\kappa$, has a remainder of degree $\alpha$ with $\kappa$-uniform constants. \item[(c)] (first moment) Split $\Pi^\kappa_j=\Pi^0_j+(\Pi^\kappa_j-\Pi^0_j)$, $\Pi^0_j$ the kernel at $\kappa=0$ with the wall, and likewise the mixed kernel with two scalar and one photon leg. For the first moment of the two-photon kernel \cite[(3.26)--(3.30)]{Ba3} and of the mixed kernel \cite[(3.36)--(3.38)]{Ba3}, the first moment of $\Pi^0_j$ vanishes, and the residual, of degree $0$, obeys two bounds, uniform in $\kappa$, $\eps$ and $k$, with no translation invariance on $U$ and no constant-mass kernel. On the light side, for $x\in H'$ at unit-$j$ distance $d\le1$ from $\partial U$, \[ \Big|\sum_y(\Pi^\kappa_j-\Pi^0_j)(x,y)(y-x)_\mu\Big|\le C\min(1,w_j)\big(1+\log_+\tfrac{1}{\max(d,w_j^{-1/2},\eta_j)}\big)^{1/2}, \] exponentially small beyond, so that, summed over the layer of unit-$j$ volume $\ell_j|\partial H'|$ against $\sup_Q|L_1|\sup_Q|\partial L_2|\le C_{\mathcal L}^2p^2(1+d_Q)^2$ for the photon legs on each unit-$k$ cube $Q$ ($d_Q$ the distance of $Q$ from the gauge base point) and over the scales, $\sum_j\ell_j\min(1,w_j)\le C$. On the shell side, at a heavy scale, the full kernel is bounded by heavy suppression: $|\Pi^\kappa_j(x,y)|\le Ce_j^2w_j^{-1}e^{-\delta'|x-y|}$ at depth $\ge1$ and $|c_\mu(x)|:=|\sum_y\eta^d\Pi^\kappa_j(x,y)(y-x)_\mu|\le Ce_j^2w_j^{-1}$ on every deep unit-$j$ cube, $|c_\mu(x)|\le Ce_j^2$ on the at most $C|\partial H'|_j$ cubes of the skin, so that $\sum_{x\in U}\eta^d|c_\mu(x)|\le Ce_j^2[\,|\partial H'|_j+|U|_jw_j^{-1}]$ at every heavy scale and the sum over the heavy scales is $\le Ce^2p^2(|\partial H'|+|U|)$ in unit-$k$ units, while at light scales the relative size is $w_j$ and $\sum_{w_jr_k:=r(\ell_k)$ and $\le2r_k+3s_b$, of volume $\le(3r_k)^d$ ($r_k\ge R\ge10s_b+10$) and with corridor $22r_k+3s_b$; it is defined from $(L,M,R,\ell_k,\bar n)$ and the regions of the run alone, so that it contains no $\kappa$. The \emph{lower bounds on $R$ of the cutoff} are $R\ge5(2M+1)$, $R\ge10$, $R\ge M_\theta+2$, $R\ge(M_\theta+6)/11.6$, together with $R\ge\max\{5(2M+1),C_Z(M+s_b)\}$ and $R\ge10(M+1)$ above. The \emph{cutoff background} is $A^{(k)}=a_k\ell_k^{-2}\zeta^{(k)}_\theta G^{\rm vec}_kQ_k^*A$ (\cite[(2.44)]{Ba2} with the cutoff $\zeta_\theta$), and $\tilde B^{(k)}$, $\theta_{k+1}B^{(k+1)}$ are the fields of \cite[(2.88)--(2.90)]{Ba2}; these three are photon objects, formed from the photon block field, the vector operator $G^{\rm vec}_k$ (Ba{\l}aban's $G_k$ for the vector operator, with the photon mass $\mu_0^2$ in place of the scalar mass $m^2$), the vector covariance $C^{(k)}$, the averages at zero background \cite[p.~608]{Ba1}, the cutoffs and the sets $\Lambda_0^{(k)\prime}$, none of which contains $\kappa$. The \emph{regularity cap} is $\sup_{t\le\eps_0^{\rm Ba}}C_Zb_0(1+\ln(1/t))^6(et^{1/2})^{1-\beta}\le C$, a downward-closed cap on the final block size (satisfiable, the function tending to $0$ as $t\to0$; $\beta$ the exponent of King's regularity condition, as in Proposition~\ref{cpi:prop:TD}), with the constant $C_Z$ of Theorem~\ref{cpi:swab:B5K}(ii). For \cite[Lemma~2.4]{Ba2} let $i\le K_f$, in unit-$i$ units, $y\in\Lambda_7^{(i-1)\prime}$, $x\in B^i(y)$, $\Omega=B^i(\Lambda_2^{(i-1)\prime})$ (the propagator domain of \cite[(2.56)]{Ba2}) or $T$, $\phi_i$ the block field with the restrictions \cite[(2.55)]{Ba2} on $\Lambda_{-1}^{(i-1)\prime}$ ($|D_{\tilde A}\phi_i|\le c_1p_{i-1}$ and $|\phi_i|\le\Upsilon_i:=c_1p_{i-1}(\lambda\ell_{i-1})^{-1/4}$, $c_1$ Ba{\l}aban's constant), $\phi^{(i)}_\kappa:=a_iG^\kappa_i(\Omega,A^{(i)})Q_i^*\Lambda_6\phi_i$ its background and $\phi^{(i)}_0$ the walled background at $\kappa=0$; the \emph{edge conditions} are a lower bound on $R$ and a cap on the final block size. In the cube step the cube $\square$ is replaced by the \emph{face-adapted box} $\square^F\supset\square$ of large blocks; the lines $G^\kappa_k(\Omega,\tilde B)$ of a renormalised class are split as $G^\kappa_k(\square^F,\tilde B)+\delta G^\kappa_k(\Omega,\square^F,\tilde B)$ on $\square^F$, and $B_0=\tilde B(c)$ is the constant field at the centre $c$. \end{definition} \begin{proposition}[further bounds of Ba{\l}aban's argument for the shell model]\label{r13:prop:B5Kfurther} In the setting of Definition~\ref{r13:def:B5Kset}, under the hypotheses of Theorem~\ref{cpi:thm:main} with (BA) and (T.nw), the following bounds of Ba{\l}aban's argument hold for the shell model, uniformly in $\kappa\in[0,\infty)$, $\eps$ and $k$: \begin{enumerate} \item[(a)] (the scale-free sub-domain bound) on sub-domains whose faces lie two blocks from $\partial U$ the heavy-scale leg bound holds scale-free, with decay from $\partial U$ at the rate $\delta^\flat_j=\Lambda_*(\kappa\eps^2)/(4\eta_j)$, $\Lambda_*(\xi)=\max\{\lambda_*(\xi),\log(1+\xi/d)\}$, $\lambda_*(\xi)$ determined by $\lambda\max(\lambda,4\eta_{\max})+\tfrac12\lambda^2(e^\lambda+1)\le\xi/d$, and $C_d=(2e^{5/2})^d$ in the amplitude of Proposition~\ref{cpi:prop:TD}; \item[(b)] (vacuum families with the mismatch vertex) at King's backgrounds, on the torus and on the propagator domains, the renormalised vacuum families with $r\ge0$ mismatch vertices obey the bounds of Proposition~\ref{cpi:prop:TD}(d): a vacuum family differs from its zero-background value by $O(1)\ell_k^{\beta_0}$, each mismatch vertex costing $C(\lambda+e^2)\ell_k$, and between two regular backgrounds it changes by at most $Ce_k(C(\lambda+e^2)\ell_k)^r$ times $\|\partial b\|_\infty+[\partial b]_{\alpha_0}$, plus the constant part of $b$ weighted by $e^{-\delta_0N_T/16}$; \item[(c)] (the odd sector at higher orders) Ba{\l}aban's enumeration runs block by block at the orders up to $2\bar n$ for the graphs of the odd sector of Theorem~\ref{cpi:swab:TDc}, and his inductive bound \cite[pp.~424--428]{Ba3}, in the form in which the proof of Proposition~\ref{cpi:prop:TD}(b) runs it, applies to their subgraphs free of mismatch vertices; \item[(d)] (the sub-domain form at order fourteen) the sub-domain form of the perturbative bound holds at order fourteen for the removal of the tail of the background \cite[(2.89)--(2.90)]{Ba2}; \item[(e)] (the photon external fields) the cut-off photon slice lines and King's photon legs are Ba{\l}aban's external fields in his localised $C^{1,\alpha_0}$ norm \cite[(1.32)]{Ba3}, with exponential decay in the block distance, componentwise and uniformly in the spacing, the scale and the scale mass; no moment bound of \cite[(A.2)--(A.6)]{K1} is used; \item[(f)] (uniformity in the scale) the strip bounds of \cite[\S3.3]{BFS1} hold uniformly in the scale; \item[(g)] (the completed steps of \cite[\S3.3]{BFS1}) the $L^1$ block-data clause, the photon two-leg kernel bounds (Theorem~\ref{cpi:thm:Kpt}), the zeroth-moment identity in resummed form (Lemma~\ref{cpi:lem:ward}(a)), the light-side localisation near edges (clause (j)), the external field of the class of Lemma~\ref{cpi:lem:appclass}, and the site-dependent diamagnetic inequality $\log\det M_U\ge\log\det M_1$ for the Gaussian determinant at a unitary connection $U$ hold for the shell model; \item[(h)] (the random-walk steps) the ballot bound next to the half-line, the lifted-walk identification and the martingale moment bound behind the strip bounds hold as stated there; \item[(j)] (light-side localisation, \textup{(LOC-R)}) the light-side estimates hold on sub-domains at deep points, edges and corners of $H'$ included: Ba{\l}aban's constant-mass bounds \cite[(3.7)]{K1}, \cite[(2.65), (2.66), (2.113), (2.114)]{Ba2} hold for $(M^{(m)}+N)^{-1}$ with the wall profile $N=n\psi\ell_j^2\in[0,n\ell_j^2]$ added to the mass, and the difference $u_\Omega-u_T$ between the sub-domain and the torus legs is bounded by $C_RBe^{-\delta_R\rho_{\rm B}}$ uniformly in $\kappa$, $\Omega$ and $A$, $\rho_{\rm B}$ Ba{\l}aban's radius. \end{enumerate} \end{proposition} \begin{proof} Clause (a) is a Combes--Thomas argument: conjugating the heavy-scale operator by $e^{\lambda\dist(\cdot,\partial U)}$ perturbs it by at most $\lambda\max(\lambda,4\eta_{\max})+\tfrac12\lambda^2(e^\lambda+1)$ per unit of mass, which the shell mass $\xi=\kappa\eps^2$ dominates for $\lambda\le\lambda_*(\xi)$. For (b), at a regular background only the derivative bounds of $A^{(k)}$ enter King's per-graph estimate; the second bound is the difference of two backgrounds read through Proposition~\ref{cpi:prop:TD}(d). For (c) and (d), each block is set at its own background by interpolating the backgrounds over $u\in[0,1]^{n_o}$, each parameter raising the order by one (order $\le2\bar n$). Clause (e) follows from the vector-field propagator bounds on the periodic torus (Theorem~\ref{rot:thm:BAL5}) by the pointwise and H\"older bounds of the propagators, the bilinear sums closing by the slice-line decay; (f) is the Montel--Vitali step of \cite[\S3.3]{BFS1}. For (g), the diamagnetic inequality is the Dyson expansion of $\Tr e^{-tM_U}$, a sum over closed paths whose matrix traces are maximal at $U\equiv1$; the photon two-leg kernel bounds, the zeroth-moment identity, the light-side localisation and the external field of the class are Theorem~\ref{cpi:thm:Kpt}, Lemma~\ref{cpi:lem:ward}(a), clause (j) and Lemma~\ref{cpi:lem:appclass}, each proved where it stands; the $L^1$ block-data clause is the completed step of \cite[\S3.3]{BFS1}. (h) consists of the three random-walk steps of \cite[\S3.3]{BFS1}, which hold as stated there. Finally, for (j), the walled bounds follow from $\phi_0=\phi_m-(M^{(m)})^{-1}N\phi_0$ with $0\le N\le n\ell_j^2$. For the localisation, $u_\Omega-u_T$ solves the homogeneous equation on the cube of radius $4\rho_{\rm B}$ about $x_0$, so a lattice Caccioppoli inequality iterated over shells, with the decaying torus leg bounds, gives $C_RBe^{-\delta_R\rho_{\rm B}}$. \end{proof} The theorem below states Ba{\l}aban's estimates in the form in which the proof of Theorem~\ref{cpi:thm:main} uses them. Its proof rests on the mechanism of the subsection, which we now make precise. Every step at which the shell model differs from King's is reduced to resolvent kernels, their $B$-derivatives, block averages and the composed background of the walled operator $M^w_\Omega(B)=D_B^*D_B+V^0+V_D+w\mathbf 1_{U\cap\Omega}+a_kP_\Omega(B)$ on a sub-domain $\Omega$ ($V_D$ the Dirichlet face term, $w=\kappa\ell_k^2$, $P_\Omega(B)$ the block projection). For this operator the shell mass is a non-negative scalar potential, so coercivity, decay and derivative bounds at $w=0$ hold at every $w\ge0$ with the same constants. Ba{\l}aban's statements at the replaced objects, by contrast, are not uniform in $w$: deep in $U$ \cite[(2.65)]{Ba2} fails for $|\phi_0|$ of the size $\Upsilon_k\gg p(\ell_k)$; under the cap $|\phi_0|\le\bar\tau_k$ of the near-$U$ test the defect is $\frac w{a+w}\bar\tau_k\le(2/(\vartheta_1a_k))^{1/2}p_k$, and the uncapped blocks are paid by the test. The proof therefore treats the four replaced objects one by one. \begin{theorem}[\textup{(BAL)}$_5^{\kappa}$ in the form in which it is used]\label{cpi:swab:B5K} In the setting of Definition~\ref{r13:def:B5Kset}, under the hypotheses of Theorem~\ref{cpi:thm:main} with (BA) and (T.nw), the shell model runs the steps D5--D9 and D11--D15 of Ba{\l}aban's lower and upper bounds (every step except D10, the step of the upper bound that uses the tail removal, treated separately) and the cube step of the perturbative bound on the four replaced objects, with constants independent of $\kappa\in[0,\infty)$, $\eps$, $k$ and $T$; the case $\kappa=\infty$ follows by the endpoint of the heavy-shell estimates. In detail, for every $\kappa\in[0,\infty)$ and every scale $k\le K_f$: \begin{enumerate} \item[(i)] (the partition) D14 holds in the shell model with the product partition and $\kappa$-uniform constants; the shell mass enters D14 only through the paid families and their coefficients, fixed before the count by D7, D12 and the face-adapted placement, with constants uniform in $\kappa$ there; \item[(ii)] (the cutoff) under the lower bounds on $R$ of the cutoff, the cutoff background $A^{(k)}$ and the fields $\tilde B^{(k)}$ and $\theta_{k+1}B^{(k+1)}$ obey the bounds (Z1)--(Z7) in the shell model verbatim, with Ba{\l}aban's constants and for every admissible sequence of large-field regions; on $\Omega=B^k(\Lambda_2^{(k-1)\prime})$ one has $\abs{\partial^\eta A^{(k)}}\le C_Zp(\ell_k)$ with $C_Z=O((1+\ln5)^6)$, so that under the regularity cap $A^{(k)}$ is regular at scale $k$ in the sense of \cite[(3.5)]{K1} for every $\ell_k\le\eps_0^{\rm Ba}$, and every operator statement of the shell model applies with the background $A^{(k)}$ on $\Omega$ with its $\kappa$-uniform constants, which do not depend on $\kappa$, $\eta_k$, $k$, $\Omega$ or $A$; \item[(iii)] (Ba{\l}aban's Lemma 2.4) D7 and D8 hold with \cite[Lemma~2.4]{Ba2}, with its box, applied to the walled background at $\kappa=0$ only, in the following form, with constants free of $\kappa$, $\eps$ and $i$. (a) Under the edge conditions the lemma holds for $\phi^{(i)}_0$: $|\phi^{(i)}_0(x)-U(A(\Gamma_{x,y}))\phi_i(y)|\le C_0p_i$ and $|D_A\phi^{(i)}_0(b)|\le C_0p_i$ for bonds $b\subset B^i(\Lambda_7^{(i-1)\prime})$, independently of the position of $y$ relative to $\partial\Lambda_7$. (b) At light scales ($w_iM_i$, $|\phi^{(i)}_\kappa(x)|^4\ge\frac12(1-C_\Delta c_0)^4|\phi_i(y)|^4$, the light-scale clause of D7. (c) At heavy scales, for $x$ in the collar with $\dist(x,U)>R_0\ge3$ and at depth $\ge R_{\rm Ba}$ in $\Omega$, $|\phi^{(i)}_\kappa(x)-\phi^{(i)}_0(x)|\le Ce^{-\delta_6R_0}\Upsilon_i$ (D7), and for every fine bond $b$ with $\dist(b,\Omega^c)\ge\max(3,R_{\rm Ba})$ and every block $y$ with unit datum $\delta_y$, $|a_iD_A[G^w_i-G^0_i]Q_i^*\delta_y(b)|\le Ce^{-\delta_6(\dist(b,U)+\dist(y,U)+d(b,y))}$ (D8); \item[(iv)] (the cube step) the cube step inside the perturbative bound and its sub-domain form hold on $T$ and on every propagator domain of the sub-domain form of Proposition~\ref{cpi:prop:TD} (the torus, $B^k(\Lambda_2^{(k-1)\prime})$, $B^k(\Lambda_2^{(k)})$ and the splitting sub-domains of D5), faces at distance $\le1$ from $\partial U$ included, with the face-adapted box $\square^F$ in place of the cube $\square$ and the torus kernel at zero background $G^\kappa_k(T,0)$, carrying the wall, in place of Ba{\l}aban's $G_k(\eta\Z^d,0)$; on $\square^F$ one has the kernel bounds $|D^a_xD^b_{x'}G^w_k(\square^F,0)(x,x')|\le C_K(\eta+\rho)^{-p(a,b)}e^{-\delta_K\rho}$ at every pair of points of $\square^F$, $\rho=|x-x'|$, $a,b\in\{0,1\}$, $p(a,b)=1+a+b$, with $C_K,\delta_K>0$ depending only on $d$, $L$, $a$, the H\"older exponent of the norm \eqref{cpi:eq:kanorm}, $c_0$, an upper bound $c_V$ for $(m^2+n)\ell_k^2$ and King's constants, free of $\kappa$, $\eta$, $k$, the box and the positions; the difference lines contribute $\kappa$-free factors in the norm \eqref{cpi:eq:kanorm}, at base points and sources at depth $\ge r_\flat\ge R_{\rm Ba}+12$ in $\square^F$; \item[(v)] (no new containment across scales) every geometric condition that the steps of the shell model add to Ba{\l}aban's is a condition between quantities of one scale, a constant measured against the amplitude profile $\rho_k$, or a cap on the final block size, never a containment of a block of one scale in a region of another scale: the shell model adds only constants (depth $3$, two large blocks per face, $R_{\rm Ba}\ge3$), so none of its conditions can fail as $\eps\to0$ at $L=5$, and the conditions on $L$ are those of Ba{\l}aban's sequence of steps. \end{enumerate} \end{theorem} \begin{proof} For (i), D14 uses the partition only through the covering, the minimal arc $\ge r_k+s_b$, the volume bound and the corridor step \cite[p.~593]{Ba2}, and the product partition contains no $\kappa$ (Definition~\ref{r13:def:B5Kset}). For (ii), the three photon objects are bounded by \cite[Prop.~2.2]{Ba2}, the photon restrictions \cite[(2.55)]{Ba2} and arithmetic, none of them containing $\kappa$; the factor $(1+\ln5)^6$ is absorbed by the cap $e\ell_f^{1/2}\le e_0$ on the final block size. Regularity is the only hypothesis on the gauge field in the operator bounds of the shell model, so every operator statement applies with the background $A^{(k)}$ on $\Omega$. For (iii), the defect of the reproduction of constants is $O(p_i)$ by the cap $(m^2+n)\ell_i^2M_i'\le p_i$ of Theorem~\ref{cpi:thm:main}. At heavy scales D7 and D8 use the split $\phi^{(i)}_\kappa=\phi^{(i)}_0+\Delta\phi$, $\Delta\phi=-G^0_iw_i\mathbf 1_U\phi^{(i)}_\kappa$, with (a) for $\phi^{(i)}_0$ and (c) for $\Delta\phi$. For (iv), the constant field $B_0$ is gauged out exactly ($\square^F$ does not wrap); the lines are expanded in $\tilde B-B_0$ at Ba{\l}aban's order with King's cutoff, the line-created vertices lying at depth $\ge R_{\rm Ba}+2$ in $\square^F$; the chains are bounded by \cite[Proposition~2.1]{Ba3} on $\square^F$, from the slice bounds of the lines or from the kernel bounds of (iv). Finally, the shell model adds to Ba{\l}aban's conditions only constants (depth $3$, two large blocks per face, $R_{\rm Ba}\ge3$), which gives (v). With (i)--(v) and the bounds of Proposition~\ref{r13:prop:B5Kfurther} the steps D5--D9 and D11--D15 and the cube step run for the shell model. The case $\kappa=\infty$ is the endpoint of the heavy-shell estimates. \end{proof} Beyond (i)--(v) and the bounds of Proposition~\ref{r13:prop:B5Kfurther}, the proof uses only (BAL)$_5$ for the walled operator at $\kappa=0$ (Theorems~\ref{rot:thm:BAL5} and~\ref{r13:thm:BAL5compl}(j)), the operator lemmas of the shell model on the torus and on sub-domains, and the perturbative bound with its sub-domain form; no bound on the face-localised difference under a flat-face hypothesis on $U$, no H\"older bound on the heavy side and no Ward identity for a single ordering enters. The last statement of the subsection is the mismatch clause of the perturbative bound, Proposition~\ref{cpi:prop:TD}(c): the perturbative bound for the graphs with at least one external field and $r\ge1$ mismatch vertices, with the additional factor $C_{\mathcal M}^r$ of the couplings of the mismatch vertices, uniformly in $\kappa$. Its proof is a degree count in Ba{\l}aban's induction: a mismatch vertex counts as a plain vertex of degree $2$, so that every stage of the even sector containing one has degree exceeding the deficit of its deep part, except for the stages of effective degree zero, whose gap sums, taken first, are uniform in $\kappa$ and $k$. The frame is Ba{\l}aban's induction \cite[pp.~426--428]{Ba3}: his degree \cite[(2.1)--(2.3)]{Ba3} of a connected graph $S$ is \[D(S)=\sum_{v\in S}[p_v-\frac12\#(\text{legs of }v\text{ on lines of }S)-\mathbf 1(\text{the derivative of }v\text{ acts on a line of }S)]-3,\] $p_v$ the number of $\eta$-factors of $v$, a mass counterterm of a graph $G_0$ adding $D(G_0)$, and $\mathcal M$ counts as a plain vertex of the quartic type ($p=3$, no derivative; $D(\mathcal M)=2$). For an ordering $l(1),\dots,l(m)$ of the lines with scales $j(1)\le\dots\le j(m)\le k-1$, the stage $i$ is $G_i=l(1)\cup\dots\cup l(i)$ with its components, $D_i$ the sum of the degrees of the components of $G_i$ and $d_i:=D_i-D_{i-1}$; after the localisation sums the summand is $O(1)\prod_ir_{j(i)}^{d_i}$ \cite[(2.15)--(2.16)]{Ba3}, summable iff every $D_i>0$, and in the gap variables $n_i:=j(i+1)-j(i)$, $\prod_ir_{j(i)}^{d_i}=L^{-\sum_in_iD_i}$. For the $\mathcal M$-free divergent subgraphs the frame's transformations (integration by parts for the photon class, the Taylor step for the two-scalar-leg class, the Ward identity for the photon and mixed classes) leave every stage with an effective degree at least $c_{\mathrm{frame}}>0$, depending on $\alpha_0$ and the order only. \begin{theorem}[the mismatch clause of the perturbative bound]\label{cpi:swab:TDc} On the torus, for $\kappa\in[0,\infty)$, every graph of $P^{(k),\kappa}$ with at least one external field and $r\ge1$ mismatch vertices obeys the perturbative bound with the additional factor $(C(\lambda+e^2)\ell_k)^r$, $C$ free of $\kappa$, $\eps$ and $k$ (with the counterterm offset, $C(\lambda+e^2)$ replaced by $C(\lambda+e^2+\Delta_c)$), for the total over those scale assignments in which no photon-closed one-slot stage contains a mismatch vertex (the \emph{even sector}). The statement holds for classes of any finite order with constants depending on the class only through its size, and its field-independent part is a constant absorbed in the normalisation; the bound is Proposition~\ref{cpi:prop:TD}(c). The members of the \emph{odd sector}, those with a photon-closed one-slot stage containing a mismatch vertex, are bounded by rewriting the closed odd matter blocks by charge conjugation on full lines before the scale decomposition, which yields families of order at most $14+n_o$, $n_o$ the number of rewritten blocks, to which the even-sector bound applies. \end{theorem} \begin{proof} We first determine the degree gained at $\mathcal M$ and its deficit. Inserting $\mathcal M$ raises the plain degree of every subgraph containing it, by $2$ if both its lines lie in the subgraph and by $2-n_{\mathrm{der}}\in\{1,2\}$ if one does ($n_{\mathrm{der}}=1$ iff the derivative of the far vertex acts on the line of $\mathcal M$). Against this, the deep part $\hat\kappa^{1/2}m(j_*)m(j_1)$ of the lossless density of Lemma~\ref{cpi:swab:M123}(M3) is a power of $L$ brought into the stages between the entry of the finer line and the closing of the host, and a lone $\mathcal M$ closing a one-loop core of degree $-1$ gives a stage of effective degree $0$. Next we bound the degrees of the stages of the even sector. By the census of the $\mathcal M$-free graphs \cite[(2.17), p.~429]{Ba3}, a stage $S$ of the even sector with $n_S$ mismatch vertices has $D(S)\ge1$ if $n_S=1$ (with equality for a two-line $\mathcal M$ iff the core is a degree-$(-1)$ graph with an internal matter line), and $D(S)$ exceeds its deficit (the number of two-line vertices carrying their deep part plus the number of face weights integrated in $S$) by at least $1$ if $n_S\ge2$; here the hypothesis on $S$ excludes the one-photon current tadpole with $\mathcal M$ on its loop ($D=0$), under which a one-photon-leg core occurs only with a source vertex \cite[(3.20)]{K1}, of degree $\ge0$. We now sum over the stages. With the composite vertex containing an $\mathcal M$ as the root of each stage at which it is joined to a composite without one \cite[p.~427]{Ba3}, the density of (M3) is summed exactly over the subcubes and split into its floor, face and deep parts. This bounds the localised summand by $O(1)^mC_{\mathcal M}^r\,L^{-\sum_ic_in_i}$ times the deep factors and the face polynomials, with $c_i\ge\min(1,c_{\mathrm{frame}})$ outside the zero stages (one two-line deep $\mathcal M$ with a core of degree $-1$ per component), whose gap sums, taken first, have a deep factor equal to $1$ on a window of values and geometric outside, hence are uniform in $\kappa$ and $k$. Finally, the (M1)- and (M2)-vertices are plain vertices with stages $D\ge1$ and no deficit, so Ba{\l}aban's \cite[(2.13)]{Ba3} gives the perturbative bound times $C_{\mathcal M}^r$, and $\kappa=\infty$ follows by continuity on the finite lattice. \end{proof} \space% % % % \subsection{The determinant tail, King's mechanism and the $\kappa$-Cauchy property} This subsection proves the $\kappa$-Cauchy property (SW3$^\circ$)$_{\rm int}$ of the capped reduced action, Theorem~\ref{cpi:swc:main}: at fixed spacing, the capped reduced actions at two shell masses $\kappa\le\kappa'$ differ, up to a constant free of the fields, by at most a quantity $R(\kappa,\kappa')$ which tends to zero as $\kappa\to\infty$, uniformly in the spacing. The proof bounds the $\kappa$-derivative of the capped reduced action, one insertion of the two-valent vertex $\mathbf 1_U$, whose three terms are a quadratic form of the background, the derivative of the determinant and the renormalised graphs carrying the inserted vertex. The estimates behind these terms come first: the tail in $\kappa$ of the block-free determinant (Theorem~\ref{cpi:swc:LDT}), Ba{\l}aban's all-orders enumeration with one inserted vertex (Theorem~\ref{cpi:thm:10f}) and King's mechanism for the graphs carrying the insertion (Theorem~\ref{cpi:swcw:Rka0}). The threshold clause (SWT) (Corollary~\ref{rot:cor:SWT}) and the disc bound (SW2$^{\min}$) (Lemma~\ref{rot:lem:SW2}) close the subsection. \emph{Standing assumptions.} We work in the shell model of Section~\ref{rot:sec:shell} (Definition~\ref{rot:def:shell}): $T$ is the torus, $H'\subset T$ the box, $U\defeq T\setminus H'$ the shell and $V_0=m^2+n\psi\ge m^2$ the scalar potential with the wall; $\eta_0=\ell_k=L^k\eps$ is the stopping scale. Fixed throughout are $L=5$, $a$, $N$, $\lambda>0$, $e$, $m^2>0$, $\mu_0^2>0$, $\alpha_0\in(0,\frac12]$ (a H\"older exponent; not the minimal ordered degree $\alpha_0$ of Theorem~\ref{cpi:thm:78}), the shares $\vartheta_1,\vartheta_2$, the heavy-scale threshold $c_0>0$ (no lower bound on $c_0$ is assumed) with $\hat c_0=\max(c_0,1)$, the torus $T$ and the box $H'$, block-aligned and with the grid offset of the corners of $H'$ and the torus-side condition of the stopping-scale bounds, $n\ge1$, the wall profile $\psi$, the sources $f,g$ supported in $H'$ at distance at least $\eta_0$ from $U$, and the scheme of the Wick constant. Here $\eta=\eps/\ell_k$ is the lattice spacing in unit-$k$ units and $\rho_j$ the length of scale $j$; $\kappa_0=\hat c_0/\eta_0^2$ and $w_k=\kappa\eta_0^2$ are as in Definition~\ref{cpi:swc:capset}, and the threshold $\kappa_0$, the Compton scales $\kappa_j$ and intervals $\bar I_j$, the stopping-scale bounds and King's constants are those of Definition~\ref{cpi:swc:capset} and Section~\ref{rot:ssec:W8}. With $\eta_1$ the threshold of the stopping-scale bounds and $m_V^2=\inf_{H'}V_0$ the plateau gap, let \[ \eta_2:=\min\big(\eta_1,\ L/(\lambda C_7^4),\ (\hat c_0/(4m_V^2))^{1/2}\big), \] $c_L:=1+(L^2-1)(1+2\log L)^2/(1-\alpha_0)$ and $c'_L:=L^{1-\alpha_0}\sum_{i\ge0}L^{-(1-\alpha_0)i}(1+2i\log L)^2$. The \emph{inserted vertex} is the two-valent vertex $\mathbf 1_U$ produced by the derivative in the shell mass $\kappa$, placed on one internal line or on one leg. The first estimate concerns the block-free factor of the determinant, $\det H_\kappa(A)$ with $H_\kappa(A)=-\Delta_A+V_0+\kappa\mathbf 1_U$. The theorem bounds the tail $T(\kappa;A)$ of its dependence on the gauge field, the part that the shell mass has not yet killed, by an $\eps$-free function of $\kappa$ decaying like $\kappa^{-1/2}$ up to a logarithm. In the proof of Theorem~\ref{cpi:swc:main} the higher orders in $A$ of the $\kappa$-derivative of this factor are bounded through the tail, read at $\kappa/2$: this is the monotone term of the derivative, and the origin of the term $\tau_{\mathrm{bf}}(\kappa)$ in \eqref{cpi:eq:swcmain}, which does not vanish as $\kappa'\downarrow\kappa$. The proof of the theorem expands the inverse of the block-free operator in random walks, so that the gauge field enters only through the holonomies around closed walks rooted in the shell, and the three terms of \eqref{cpi:eq:LDT} bound three classes of walks: those confined to the shell, those touching the box without traversing the shell, and those traversing a shell arc. The hypothesis $\eps_0\le\ell_H/2$ enters only once, for the third class, through the distance $\min(w_U,\ell_H/2)$ in the return Green's function. \begin{theorem}[the tail of the block-free determinant on the torus]\label{cpi:swc:LDT} Let $T$ be the torus, $H'\subset T$ the box, $U=T\setminus H'$ the shell, of width $w_U$ in each coordinate direction, $\ell_H$ the smallest side of $H'$, $V_0\ge m^2$ on $T$ with $V_0\ge m_V^2$ on $H'$, $H_\kappa(A)=-\Delta_A+V_0+\kappa\mathbf 1_U$ the block-free operator at lattice spacing $\eps$ (physical units), and \[ \Phi^{\mathrm{bf}}(\kappa):=\ln\det H_\kappa(A)-\ln\det H_\kappa(0),\qquad T(\kappa;A):=\Phi^{\mathrm{bf}}(\kappa)-\Phi^{\mathrm{bf}}(\infty). \] Let $A$ be any lattice gauge field on $T$ with $|\partial_\mu A_\nu|\le C_A$ (no bound on $A$ itself and no condition on its torus holonomy), and let $\eps_0\le\min(w_U,\ell_H/2)$, as at the value $\eps_0=\min(\eps_0(\eta_0),\eta_0)$ at which Theorem~\ref{cpi:swc:main} uses the theorem, $T$ and $H'$ being block-aligned with each side of $H'$ at least two blocks of the final scale $\ge\eta_0$. Then for $\kappa\ge2m_V^2$ and all $\eps\le\eps_0$ \begin{equation}\label{cpi:eq:LDT} \begin{split} 0\le T(\kappa;A)&\le C\,e^2C_A^2\,|U|\,\kappa^{-1/2}+C(T,H')\,e^2C_A^2\,m_V^{-1}\kappa^{-1/2}\big(1+\log(\kappa/m_V^2)\big)^{c_\star}\\ &\quad+C(T,H',m_V)\Big[e^{-c\,w_U\sqrt\kappa}+\Big(1+\frac{\eps_0^2\kappa}{8d}\Big)^{-w_U/(2\eps_0)}\Big], \end{split} \end{equation} with an absolute exponent $c_\star\le5$, $C(T,H')$ proportional to the cross-sections of the torus, and every constant free of $\eps$, $\kappa$ and $A$. Consequently \begin{equation}\label{cpi:eq:LDTmaj} 0\le T(\kappa;A)\le C(T,H',m_V,\eps_0)\,(1+e^2C_A^2)\,\kappa^{-1/2}\big(1+\log(\kappa/m_V^2)\big)^{c_\star}\qquad(\kappa\ge2m_V^2), \end{equation} an $\eps$-free function of $\kappa$ dominated by $C_\delta\kappa^{-1/2+\delta}$ for every $\delta>0$. The same bound, with the third term replaced by $e^2(C_a^2K_a+C_A^2K_A)e^{-\Lambda/2}$ for fields with $|A_\nu|\le C_a$ in the given gauge, $\Lambda:=(2w_U/\eps_0)\operatorname{arcsinh}(\eps_0\sqrt\kappa/2)$ and $K_a,K_A$ depending on $(T,H',m_V,\eps_0)$ only, holds for the second-order tail $T_2(\kappa;A):=\lim_{s\to0}s^{-2}T(\kappa;sA)$. \end{theorem} \begin{proof} The random-walk expansion of $G_\kappa=H_\kappa^{-1}$ in $\partial_\kappa\Phi^{\mathrm{bf}}=\Tr\mathbf 1_U[G_\kappa(A)-G_\kappa(0)]$ writes $T(\kappa;A)$ as a sum over closed walks $\gamma$ rooted in $U$ of $1-\cos\Phi(\gamma)\ge0$ times a positive weight, $\Phi(\gamma)$ the holonomy of $eA$ around $\gamma$. In particular $T\ge0$ (Kato's inequality). For the walks confined to $U$, the L\'evy-area bound $E_x[\Phi^2;\gamma\subset U,\,Y_n=x]\le Ce^2C_A^2\eps^4n^2$, against the return probability of the walk killed at rate $\kappa$, gives the first term of \eqref{cpi:eq:LDT}. For the walks touching $H'$ without traversing the shell, H\"older's inequality with $q\asymp\log(\kappa/m_V^2)$ against a free-bridge moment bound of order $(Cq)^{2q}(e^2C_A^2\eps^4n^2)^q$ gives the second term, whence the logarithm. For the walks traversing a shell arc, winding walks included, $1-\cos\Phi\le2$ removes $A$; the discounted probability of crossing the slab before returning is $\le C(\eps/w_U)e^{-\Lambda/2}$, and the strong Markov property at the exit sends the remainder into the return Green's function at sup-distance $\ge\min(w_U,\ell_H/2)$. This gives the third term and proves \eqref{cpi:eq:LDT}. Since $w_U\ge\eps_0$, the bracket in \eqref{cpi:eq:LDT} is at most $C\eps_0^{-1}\kappa^{-1/2}$, which gives \eqref{cpi:eq:LDTmaj}. The second-order tail follows with $\Phi^2/2$ in place of $1-\cos\Phi$. \end{proof} We turn to the graphs. The derivative of the graph part $P^{(k),\kappa}$ of the explicit action consists of renormalised graphs of King's kind with one extra two-valent vertex $\mathbf 1_U$, the inserted vertex, and these have to be bounded uniformly in $\kappa$ and $\eps$ at the single value $\kappa$ of the shell mass. Two theorems do this. The first says that Ba{\l}aban's all-orders enumeration \cite[\S\S2--3]{Ba3} survives both the inserted vertex and our setting of block factor $5$, tori and the seagull vertex. The graph pass of the proof of Theorem~\ref{cpi:swc:main} uses (i)--(iv), and (iv) in turn rests on the census, engine and pinning bounds of the second theorem, King's mechanism (Theorem~\ref{cpi:swcw:Rka0}). The idea of the proof is that the inserted vertex, being two-valent and of King's degree two, can take the place of the source vertex in the degree count of Theorem~\ref{cpi:thm:Ri}, so that the census without the insertion carries over; the renormalisation is done at zero shell mass, so that no scale is both renormalised and heavy, and on the heavy side the graphs with the insertion are bounded at the single value $\kappa$. \begin{theorem}[Ba{\l}aban's all-orders enumeration with one inserted vertex]\label{cpi:thm:10f} Ba{\l}aban's all-orders enumeration \cite[\S\S2--3]{Ba3} to order $13$ with one inserted vertex, as used in the proofs of \cite[Thms.~3.4, 3.5]{K1}, holds at block factor $5$ on tori, with King's vertex list including the seagull vertex of order two: \begin{enumerate} \item[(i)] (\emph{order cap}) $\bar n=13$ suffices for the cumulant expansion with King's field class; \item[(ii)] (\emph{census}) the classification of the divergent classes, with and without the inserted vertex, which does not involve the shell mass; \item[(iii)] (\emph{renormalisation}) the renormalisation of every class in every ordering at zero shell mass; \item[(iv)] (\emph{bound}) in the shell model at $\hat\kappa=w_k\ge\hat c_0$, the bound of the renormalised graphs with the insertion in the form used by the $\kappa$-derivative estimate of the shell model, the census, engine and pinning bounds of Theorem~\ref{cpi:swcw:Rka0}; \item[(v)]\label{r13:thm:10fcube} (\emph{cubes}) on King's cubes $C^K_J$, at zero shell mass and without the inserted vertex, (i)--(iii) hold at every localisation and (iv) is replaced by the cube bound (iv)$_C$: for the classes (i), (ii), (iii)(a) of Theorem~\ref{cpi:thm:Ri}, the two-field classes 2F-R (the classes with a curvature source $g$ and the mixed classes with $g$ and $h$) and the source-free field-dependent classes, at every localisation tuple $\tau$ with $R=\dist(P(\tau))>p(\delta_k)$, on the scales $0\le k\le K-J_*$ and for each of the two models $\theta=(K,k)$ and $\theta=(K+n,k+n)$ of Theorem~\ref{cpi:thm:Ri}, \begin{equation}\label{cpi:eq:10f:ivC} |E^{(\theta)}_N(H_{\rm ren};\tau)|\le C\,(1+K-k)^{p_0}\,\delta_k^{\rho(H)}\,p(\delta_k)^{n_1}\,e^{-\delta\dist(P(\tau))}. \end{equation} Here the localisation tuple $\tau$, its point set $P(\tau)$ and $\dist(P(\tau))$ are as in Definition~\ref{cpi:def:a02:zones}, $\delta_k=5^{k-K}$, $p(\cdot)$ is the small-field function of Convention~\ref{cpi:rot:convP} (as in Proposition~\ref{rot:prop:BV}) and $J_*$ the step offset of Section~\ref{ssec:k01:setting}; $H_{\rm ren}$ is the renormalised graph, $E_N$ is as in Theorem~\ref{cpi:thm:Ri} ($E_N\defeq E$ for the field-independent members of the class (ii), for the class (iii)(a) and for the $g$- and source-free classes), $\rho(H)$, $n_1$ and $p_0$ are the exponents of King's bound \cite[(3.38)]{K1} in the restricted form of Theorem~\ref{cpi:thm:Ri} (with its factor $(1+K-k)^{p_0}$, now on King's cube at $R>p(\delta_k)$), and the constants depend on the point of King's range, on $J$, on the bound on the sources, on $\bar n$, $d$, $L$ and the H\"older exponent $\alpha$ only, uniformly in $K$. The same bound holds for the finite family $\mathcal F_{14}$ of the families of order at most $14$ with one inserted external photon field (a vertex of King's list with one more external photon leg), on the scales $0\le k\le K-J_*(\mathcal F_{14})$ and with constants depending on $\mathcal F_{14}$ in place of $\bar n$, where $J_*(\mathcal F_{14})\ge J_*$ is the finite threshold, not quantified here, at which the geometric ranges of the cube chain hold for $\mathcal F_{14}$: the range row (ZB), read as the convergence condition (ZB$^\Sigma$) of (iii) (the renormalisation paragraph of the proof below) with $C_1=C_1(\mathcal F_{14})$, and the zone-B constant $C_1^B$ in zone~B of Definition~\ref{cpi:def:a02:zones}. At $R\le p(\delta_k)$ the bound of the renormalised classes on the cube is King's one-box route (Theorems~\ref{cpi:thm:a02:B4} and~\ref{cpi:thm:a02:Bsc}, Proposition~\ref{prop:k02:KhI}) under its own condition (ZB$_\le$), the convergence condition of King's expansion in the block field on the one box of a localisation with $R\le p(\delta_k)$ (the third condition of the range \eqref{eq:k02:range} of Definition~\ref{r13:def:range}), and the scales $k>K-J_*$ lie outside the statement; so $J_0\ge\max(J_*,J({\rm ZB}_\le))$ is required of the cube index wherever the bound is used, $J({\rm ZB}_\le)$ being the least offset such that (ZB$_\le$) holds on every scale $k\le K-J({\rm ZB}_\le)$. \end{enumerate} \end{theorem} \begin{proof} For the order cap (i), the coefficients $O_\beta(\ell_j^{1/4-\beta})$ of the effective interaction at scale $j$ make the remainder of the cumulant expansion truncated at $\bar n=13$ of order $\ell_j^{14(1/4-\beta)}|T|$, and $14(\tfrac14-\beta)>3$ (Lemma~\ref{cpi:lem:appclass}; Proposition~\ref{prop:k01:H0rel} on the cubes). For the census (ii), the degree-by-legs count of Theorem~\ref{cpi:thm:Ri} with the inserted vertex in the place of the source vertex gives degree $\ge\tfrac12$ to every connected subgraph containing it, except the field-free vacuum components and the parity shape, which vanishes at zero shell mass. The renormalisation (iii) is Ba{\l}aban's Taylor and Ward steps and King's reduction \cite[(3.37)]{K1}, which rewrite each divergent class in every ordering into zero, collapsed local terms and remainders of positive degree. King's expansion \cite[(3.52)--(3.53)]{K1} on the box $\Omega$ of a localisation converges absolutely class by class under the condition (ZB$^\Sigma$): $C_1\theta_\Omega\le\tfrac12$, where $\theta_\Omega\defeq C_{ZB}\,e_k\,p\operatorname{diam}(\Omega)$ is the norm of one term \cite[(3.49)]{K1} and $C_1=C_1(\mathcal F)\ge1$ a constant of the finite family. For the bound (iv), on the heavy side the graphs with the insertion are a finite sum of renormalised graphs of King's kind at the single value $\kappa$, and they are bounded by Theorem~\ref{cpi:swcw:Rka0}(a)--(c) with the counterterms of zero shell mass. Finally, the cube bound (v) is the cut by clusters of Proposition~\ref{prop:k02:KhR} with the restricted Theorem~\ref{cpi:thm:Ri} as the per-cluster bound and the tree bound of Definition~\ref{r13:def:clusters}. The inserted leg of $\mathcal F_{14}$ is one more external field admitted by every leg clause, the three numerical uses of $\bar n=13$ hold at order $14$, and the row (ZB) at $C_1(\mathcal F_{14})$ fixes $J_*(\mathcal F_{14})$. \end{proof} King's mechanism, Theorem~\ref{cpi:swcw:Rka0}, is the engine of the graph pass. For each member of the $\kappa$-derivative and each ordering of its lines it gives the census (a), the multiscale bound (b) with its pinned form (c), the gains (d), the position sum (e) and the window moments (f), the coefficient bounds of the local vertices in (a); clauses (a)--(c) are what Theorem~\ref{cpi:thm:10f}(iv) and the graph pass of the proof of Theorem~\ref{cpi:swc:main} use. The statement needs the vocabulary of the definition that precedes it: the members and their degrees; the prefix components of an ordering; the odd-block step, the fundamental theorem of calculus in the background along the full lines of a closed matter block with an odd number of photon ends (which vanishes at zero field, by Furry's theorem), inserting one background leg; the extended list of small divergent subgraphs, which enter not as factors of positive degree but as local vertices at their window scale; and the window kernel, whose moments are their coefficients. The last sentence of the theorem marks a limitation: the bounds are uniform in the shell mass but carry no smallness in $w_k$; in the graph pass the decay in $\kappa$ comes from the inserted line, through its heavy-kernel bounds per unit $w$. The proof runs the degree count of Theorem~\ref{cpi:thm:Ri} with the inserted vertex as the source vertex, proves the multiscale bound in ordered form, one prefix component at a time, and reads the window moments off the gauge invariance and the Ward identity of the window class; it uses no Ward identity per ordering, no H\"older clause on the heavy side and no lower bound on $c_0$, the split degree count (Lemma~\ref{cpi:ir:BLKnat}, Theorem~\ref{cpi:ir:S3}) replacing King's per-vertex credit \cite[(3.68)--(3.70)]{K1}. \begin{definition}[members, degrees, the odd-block step, the extended list and the window kernel]\label{r13:def:window} A \emph{member} is a graph of the expansion of the $\kappa$-derivative of the shell functional after the regroupings of the $\kappa$-Cauchy estimate, with the inserted vertex $u$ (the $\kappa$-derivative of $\kappa\mathbf 1_U$) on the \emph{inserted line} $l_0$; the members form the \emph{graph pass} (III), the term $\partial_{\hat\kappa}P^{(k),\kappa}$ of the derivative formula in the proof of Theorem~\ref{cpi:swc:main}, and the \emph{source-dependent dressings} are the members with a curvature source $g'$ attached. $M_u$ is the matter block containing $u$, \emph{open} if a vertex of it carries an external scalar end and \emph{closed} otherwise. The \emph{degree} of a connected subgraph $\gamma$ is the integer $D(\gamma)=3(\abs{V(\gamma)}-1)+\sum_va_v-\#\mathrm{lines}(\gamma)-\sum_v\delta_v$ ($a_v$ the vertex weights, $\delta_v$ the differenced ends), the \emph{prefix component} $H_i$ of an ordering $l(1),\dots,l(n)$ of the lines compatible with the scale assignment is the connected component of $\{l(1),\dots,l(i)\}$ containing $l(i)$, and the \emph{window} of a divergent subgraph is the smallest scale $J$ of the fluctuation photon lines with exactly one end on it. $c_l$ is the clause constant of a line $l$ (on the inserted line $l_0$: the propagator bounds per unit $w$, the bound of the inserted line differenced at both ends and the bounds of the inserted leg per unit $w$, as re-posed in the $\kappa$-Cauchy estimate); in unit-$k$ units, $t(y)\defeq\dist(y,U)$, $\tau_0\defeq\max(w_k^{-1/2},\eta)$, $\tau_U(y)\defeq\tau_0+t(y)$ on $H'$ and $\tau_U\defeq\tau_0$ on $U$. The \emph{odd-block step}: a closed matter block with an odd number of photon-type ends vanishes at zero field, so $V_\beta(A_k)=\int_0^1V^{(1)}_\beta(tA_k)[A_k]\,dt$ with one background leg inserted; it is performed once per closed odd block, before the ordering is chosen. The \emph{extended list} consists of the divergent two-scalar-leg subgraphs of orders $\le4$, the two-photon-leg subgraphs of orders $\le4$ with two line slots or with one line slot and one background leg (nested chains included), and the mixed subgraphs of order $3$ with a fluctuation photon end or with the background leg. For a renormalised family of the extended list with one background leg, attached at the vertex $x$ of the family: its \emph{slot line} $s$ is its fluctuation photon line, of index $J$; its \emph{background leg} $B''$ is the background $A_k$ re-gauged to vanish at the cube of $x$, with base point $x_0$; its \emph{window kernel} $\Pi_{[J]}(x,y)$ is the kernel of the \emph{window class}, the sum of the family's members over the internal scale assignments with all indices below $J$, the matter lines at the background $tA_k$ of the odd-block step, read as the bilinear form $\sum_{x,y}\eta^3\eta^3\,s(x)\Pi_{[J]}(x,y)B''(y)$. With $e_J^2\defeq e^2\rho_J$ and $R_J$ a quarter of the side of the torus, both in the units of step $J$, $N$ the side of the torus in unit-$k$ units and $p$ the small-field bound of the block fields, the Taylor expansion of $B''$ about $x$ splits the family into the \emph{zeroth moment} $\zeta_J(x)\defeq\sum_y\eta^3\Pi_{[J]}(x,y)$ of the window kernel in the background slot, the \emph{first moment} $c_\nu(x)\defeq\sum_y\eta^3\Pi_{[J]}(x,y)(y-x)_\nu$ and the \emph{second-order Taylor remainder} $\sum_y\eta^3\Pi_{[J]}(x,y)[B''(y)-B''(x)-(y-x)\cdot\partial A_k(x)]$. \end{definition} \begin{theorem}[King's mechanism for the $\kappa$-model]\label{cpi:swcw:Rka0} Let $w_k\ge\hat c_0$, $\eps\le\eps_0(\eta_0)$, $(A_k,\phi_k)\in\mathcal X^{\mathrm{cap}}$, the capped test set of Definition~\ref{cpi:swc:capset}, $\abs z\le2$ and $g'\in\{0,g,2g\}$. For every member of (III), or of the source-dependent dressings, after the regroupings, and every ordering compatible with its scale assignment, the following hold. \begin{enumerate} \item[(a)] (census) Every prefix component containing the insertion $u$ has $D\ge1$ if $M_u$ is open. If $M_u$ is closed, the only exceptions are (T) the whole graph and (F) the subgraph $\gamma_F(u)$ formed by $u$, by the vertex $y$ of King's type \cite[(3.22)]{K1} at which both $u$-lines end, and by these two lines, with its photon slot external; both are bounded separately within the class of the members whose insertion lies on a closed matter component, which the proof of Theorem~\ref{cpi:swc:main} bounds as a class. After the odd-block step (Definition~\ref{r13:def:window}), which raises the degree of every containing subgraph by at least one, and Ba{\l}aban's summation by parts \cite[(2.8)]{Ba3} per ordering, every prefix component not containing $u$ is a factor of integer degree $\ge1$ or lies inside a renormalised family of the extended list. The latter are not factors of positive degree: their linear and first-moment parts, King's symmetry-cancelled degree-$0$ classes \cite[(3.84)]{K1}, are not cancelled per ordering in the $\kappa$-model and enter only as \emph{local vertices at their window scale} with the $\kappa$-uniform coefficient bounds of (f), pointwise or averaged over the cubes of that scale, plus remainder members whose stages have positive degree, the containing prefix components keeping their census degree or gaining. The two-scalar-leg classes are paired at step $k$ with the part of King's $\kappa$-free $\dm$ whose indices are all $\le k-1$. \item[(b)] (engine) The member is bounded by the multiscale bound: the sum over the positions but one held vertex and over the scales is $\le C\prod_lc_l\prod_v(\text{vertex factors})\,e^{-\delta\dist(\text{held})}(1+\log_L\eta_0^{-1})^{p_0}$, with $C=C(\text{graph},\delta)$ free of $w_k$, $\eps$ and $k$. The clause constants are $\kappa$-uniform for undifferenced lines and legs, for one difference and for a difference at each end, and the local vertices of (a) enter with their coefficient bounds. The bound is linear in each clause constant and may be evaluated on configuration sets of one vertex and with any fixed fraction of the decay rates. \item[(c)] (fixed insertion scale, pinning) With $l_0$ at a fixed scale $i$, the sum over the other scales is $\le C_{d'}\rho_i^{d'}\prod c_l$ for every $d'p(\delta_k)$ in units of the blocks of scale $k$ ($p(\delta)$ the length threshold of that passage), in the zone of the faces of the cube and away from it, and (R2) its summed form (the weights $e^{-\delta\,\mathrm{dist}(P(\tau))}$ of the zone-B tuples summed with $e^{\delta'R}$, $\delta'\le\delta/2$); and King's long-tree bound~\cite[(3.41)]{K1}, as (R3) the bound $C(5^{-\gamma k}\delta_k^{-\beta}+\delta_k^{\sigma})|C_J|$ on the long trees in the difference clause of the split degree count on King's cubes at the two cutoffs of a diagonal pair. The localisations $\tau$, the zone of the faces and the diagonal pair of cutoffs below are those of (R1)--(R3). \begin{proposition}[block vertices at long trees in the families with one inserted field]\label{rot:prop:BV} On King's cube $C^K_J$ with the flow of weight $f_D$, let $\eps=5^{-K}$, $\delta_k:=5^{k-K}$, $0\le k\le K-J_*$ and $p(\delta)$ be as in (R1), and let the scalar source be zero. Consider the families of the perturbative part at a background $B=B'+\sigma X$ on $C^K_J$, $\sigma\in[0,1]$, whose first differences and their H\"older quotients of exponent $\alpha_0$ are bounded by $C_1p(\delta_k)$ (the class of~\cite[pp.~412--413]{Ba3}), among them the backgrounds $A_\theta+\sigma X_0$, with $A_\theta$ Ba{\l}aban's cut-off photon background and $X_0$ its difference from King's background, that carry one inserted vertex whose additional leg is the external field $X$ (the field $X_0$ in that case), and whose graphs, with every block vertex split into its two summation points as in the split count (Theorem~\ref{cpi:ir:S3}), contain a block vertex. Then at every localisation $\tau$ of such a family whose tree has length $d_\tau>p(\delta_k)$ in units of the blocks of scale $k$, in the zone of the faces of the cube and away from it, the sum $E(\tau)$ of the renormalised family at $\tau$ obeys at each of the two cutoffs of a diagonal pair the bound of (R1) multiplied by $\|X\|_{1,\alpha_0}$, the supremum of $X$, of its first differences and of their H\"older quotients of exponent $\alpha_0$, with constants independent of $k$, $K$, $B'$ and $X$ (within the class, at fixed $C_1$); so that, by~\cite[(3.41)]{K1}, these localisations contribute at most $C_N\delta_k^N\|X\|_{1,\alpha_0}|C_J|$ for every $N$. \end{proposition} \begin{proof} The bound is (R3) for these families, with the three further rows that the inserted field requires (one cluster for each product of block vertices, the cut lines paired at the ends of the products, the cut photon lines with an end on a contour, and the ordering bound with the split count), each at the per-tuple bound of (R1) times $\|X\|_{1,\alpha_0}$ and summed by~\cite[(3.41)]{K1}. \end{proof} \begin{proposition}[the fluctuation estimates for centre-rooted and hybrid in-block contours]\label{rot:prop:CR} Ba{\l}aban's construction transports the fields inside each block along a contour $\Gamma_{r,x}$ from a point $x$ of the block to its reference point $r$~\cite[(2.1)--(2.3)]{Ba1}, and King's flow uses axial contours rooted at a corner of the block~\cite[(2.12)]{K1}; the flows of the torus theorem use in-block contours of two other families, the axial contours rooted at the centre of each block, of length at most $\frac32$ times the side of the block, the coarse variable being attached to the centre, and the composite contours of the hybrid flow, of length at most $2.1$ times the side of the block of each level. Every estimate of the fluctuation integrals of~\cite{Ba1,Ba2,Ba3}, among them~\cite[Prop.~1]{Ba3}, and King's upper bound~\cite[Thm.~3.1]{K1}, in the forms that the torus theorem uses (Section~\ref{rot:ssec:CT}), holds for these two families in place of King's, with constants that depend on the family through the bound $c_\Gamma\in\{\frac32,2.1\}$ on the length of its contours in units of the block side and, for the hybrid family, also through the hybrid kernel bounds (Theorem~\ref{cpi:hyb:BD}) and the geometry of the tiles. \end{proposition} \begin{proof} Ba{\l}aban's and King's proofs use the contour family only through properties shared by the three families ($QQ^*=\mathbf 1$ for the block average, unitarity of the contour phases, gauge covariance with the phase at the root, path independence at a constant field, length bounds, containment in the block, King's tail convention~\cite[(2.19)]{K1}, a segment property used once at~\cite[p.~560, (2.16)]{Ba2}, and the Green's-function bounds of~\cite{Ba4}), with constants at most King's. The Green's-function bounds pass to the centre-rooted family by a change of root, a diagonal unitary on the coarse variables leaving $P_k=Q_k^*Q_k$, the Green's functions and the moduli of all kernels invariant, and for the hybrid family (Definition~\ref{cpi:hyb:cont}, Lemma~\ref{cpi:ct:tile}) they are the hybrid kernel bounds (Theorem~\ref{cpi:hyb:BD}(ii)); the segment property is not used there, and Ba{\l}aban's bound for the fluctuation field~\cite[Lemma~2.4, (2.65)--(2.66)]{Ba2} follows by the steps of his proof with the hybrid lengths. \end{proof} \emph{The threshold proposition.} In the notation introduced before Lemma~\ref{cpi:lem:massshift} (a King torus $T$ at spacing $s=\eps_K$ in the torus convention; $N(0,h)=N^{{\rm tor},z}_{K,T}(0,h)$ with $z_K=a^\sharp_K$ the scheme constant of Definition~\ref{rot:def:scheme}, the reference scheme $z=a^\sharp$; the convex $h\mapsto\ln N(0,h)$ and $f(u)=\ln N(0,u\cdot1)$; the full torus coefficient $\alpha^T_s(\mathsf M)$ of a scalar mass $\mathsf M$ and the relative coefficient $s_{K,T}(\mathsf M)$), let $\alpha_\eps(\mathsf M)=\mathsf M^2+\delta m^2_\eps(\mathsf M)$ be the full coefficient of $\frac12\sum_xs^3|\phi(x)|^2$ on $\eps\Z^3$ in the free convention and $s^{\rm f}_\eps(\mathsf M)\defeq\frac12(\alpha_\eps(m)-\alpha_\eps(\mathsf M))$. Its limit $s_{\rm inf}(\mathsf M)\defeq\lim_{\eps\to0}s^{\rm f}_\eps(\mathsf M)$, which exists by part (i) below, is the relative coefficient of Proposition~\ref{rot:thm:POS}, with $\alpha_{\rm inf}(\mathsf M)-\alpha_{\rm inf}(m)=-2s_{\rm inf}(\mathsf M)$ and $s_{\rm inf}=s_\infty$ of Section~\ref{sec:s14}; the \emph{threshold} $s_*(e)=\sup_{\mathsf M>0}s_{\rm inf}(\mathsf M)\in[0,\infty]$ is that of Definition~\ref{rot:def:cone} (equal to $s_*$ of \eqref{eq:s14:EW}) and depends on $(\lambda,m,\mu_0)$ as well. \begin{proposition}[the threshold proposition]\label{cpi:prop:S} \begin{enumerate} \item[(i)] For every scalar mass $\mathsf M>0$ the limit $\lim_{\eps\to0}\big(\delta m^2_\eps(m)-\delta m^2_\eps(\mathsf M)\big)$ of the free-boundary counterterm differences on $\eps\Z^3$ exists; hence $s_{\rm inf}(\mathsf M)=\lim_{\eps\to0}s^{\rm f}_\eps(\mathsf M)$ exists, is a polynomial of degree at most two in $(e^2,\lambda)$ with finite coefficients, and is Lipschitz in $\mathsf M$ on compact subsets of $(0,\infty)$, with $s_{\rm inf}(m)=0$. This is a statement about the free-boundary coefficient on $\eps\Z^3$ alone. \item[(ii)] Let $0\le q\in C^\infty_c(\R^3)$ with $\bar q\defeq\sup q\bar q$, there are $l_0\in2M\N$ ($\supp q$ in one cell), $K_0$ and two masses whose torus coefficients satisfy $\bar q0$, let $\Delta_\tau(s;M)$ be the constant of the convention bridge of Proposition~\ref{rot:thm:POS}, computed at the scalar mass $M$ in place of $m$, on the pair $\tau$ of a lattice and a torus of that proposition (so that $\Delta_\tau(s;m)$ is its constant $\Delta_T(s)$ on the torus $T$ of the pair), and put \[ \kappa_M\defeq\tfrac12\min(M,\mu_0,2),\qquad \Phi_\kappa(l)\defeq\big(1+(\kappa l)^{-1}\big)^6l^{-2}e^{-\kappa l/\sqrt3},\qquad l_0(M)\defeq2\sqrt3\,\pi/\kappa_M . \] There is $K_\Delta(M)<\infty$, depending on $(e,\lambda,\mu_0,M)$ and on nothing else (not on $k$, on $J$ or on the frame), such that for each of the three pairs $\tau$, with torus $D_J$ or $RD_J$ of side $l=l_J\defeq50M_{\rm b}J\ge l_0(M)$ (here $M_{\rm b}$ denotes the block integer, written $M$ elsewhere in this paper, since in this lemma $M$ is the scalar mass), and every admissible spacing $s=\eps_k\le1$, \begin{equation}\label{cpi:eq:ew:BR} |\Delta_\tau(s;M)|\le K_\Delta(M)\,\Phi_{\kappa_M}(l); \end{equation} the limit $\Delta_\tau(0;M)\defeq\lim_{s\to0}\Delta_\tau(s;M)$ exists and obeys the same bound. The function $\Phi_\kappa(l)$ is strictly decreasing in $l$ and non-increasing in $\kappa$, and tends to $0$ as $l\to\infty$. \end{lemma} \begin{proof} The convention bridge ((CT)(b), with the rate clause of Theorem~\ref{cpi:ct:CTB}(d)) uses the scalar mass only through the scalar propagators, the strip width of the strip coercivity and the condition $s\kappa\le1$, so it holds at the mass $M$ with the strip width $\kappa_M$ for every $s\le1$, frame by frame: $\sup_{s\le1}|\Delta_\tau(s;M)|<\infty$, $\Delta_\tau(0;M)$ exists and $|\Delta_\tau(s;M)-\Delta_\tau(0;M)|\le Cs$. Up to King's overall sign, $\Delta_\tau(s;M)$ is a finite sum over the one- and two-loop graphs of King's counterterm of differences (torus value) $-$ (free value), each an image sum over the period lattice $P$ of an integrand analytic in a strip of width $\kappa_M$, a Riemann sum over the dual lattice of an outer factor times such an image sum (bounded for $l\ge l_0(M)$, where the dual Voronoi cells have circumradius $\sqrt3\,\pi/l\le\kappa_M/2$), or an exact cancellation. Poisson summation bounds each image sum by a constant free of $l$, $k$, $J$ and the frame times $\sum_{\gamma\in P\setminus0}e^{-\kappa_M|\gamma|_\infty}|\gamma|_\infty^{-2}\le81\,\Phi_{\kappa_M}(l)$, a bound that uses only the shortest vector $l$ of $P$; it passes to $s\to0$, and the monotonicity of $\Phi_\kappa$ is read off its factors. \end{proof} \emph{Right-continuity in the scalar mass at $E_W$.} Fix $(e,\lambda,\lmu)$; for a scalar mass $M>0$ the \emph{King point} $\tau_M\defeq(e,\lambda,M,\lmu)$ is King's model with scalar mass $M$ and everything else equal, and $\tau_*\defeq\tau_m$. On King's cube $C^K_J$ (Definition~\ref{def:s02:Kcube}) at spacing $\eps_K$ the lattice measures of $\tau_M$ and $\tau_m$ differ, up to field-independent constants, only by the single-site factors $\exp(-\frac{\eps_K^3}2t_K(M)r_x^2)$ at the live sites, $t_K(M)\defeq\alpha_K(M)-\alpha_K(m)$ with the bare coefficient $\alpha_K(M)=M^2+\dm_{\eps_K}(M)$; as $K\to\infty$, $t_K(M)\to t_\infty(M)\defeq\alpha_\infty(M)-\alpha_\infty(m)$ for every $M>0$, which defines $\alpha_\infty$ up to an additive constant as a function real-analytic on $(0,\infty)$ (the closed form of Section~\ref{sec:s14}; Remark~\ref{cpi:ew:rightcont}), and $E_W$ of \eqref{eq:s14:EW} is the set of couplings at which $m$ is a global minimiser of $\alpha_\infty$. A \emph{source} $u$ is a real smooth compactly supported $2$-form, $F(u)$ the curvature source and $J_0(u)$ as in Theorem~\ref{thm:k01:EC}; for $J\ge J_0(u)$ put $G_{K,J}(M)\defeq\langle e^{\mathrm iF(u)}\rangle^K_{K,J;\tau_M}=\langle\cos F(u)\rangle^K_{K,J;\tau_M}\in[-1,1]$ (real by charge conjugation), $\Gamma_J(M)\defeq\lim_KG_{K,J}(M)$, which exists by Theorem~\ref{thm:k01:EC} at $\tau_M$, and $\Gamma(M)\defeq\lim_J\Gamma_J(M)$, which exists by Lemma~\ref{cpi:ew:mono}(b); $\Gamma(M;u)$ is King's state at $\tau_M$ on $e^{\mathrm iF(u)}$. For a finite family $f=(f_1,\dots,f_N)$ of sources and $z\in\C^N$ put $a_{K,J}(z;M)\defeq\langle\exp\sum_jz_jF(f_j)\rangle^K_{K,J;\tau_M}$, $Z^f_J(z;M)\defeq\lim_Ka_{K,J}(z;M)$ and $Z^f_\infty(z;M)\defeq\lim_JZ^f_J(z;M)$, King's curvature functional at $\tau_M$ (the object $\ZKinf$ of Theorem~\ref{thm:s05:CK}); by Lemma~\ref{lem:k01:free}, $\abs{a_{K,J}(z;M)}\le a_{K,J}(\Re z;M)\le e^{\beta_f\abs{\Re z}^2}$ with $\beta_f=C_0\sum_j\fournorm{f_j}^2$ free of $K$, $J$ and the couplings, and the bound passes to the limits. \begin{lemma}[the two monotonicities]\label{cpi:ew:mono} \begin{enumerate} \item[(a)] (in the bare coefficient) For every $K$, every $J\ge J_0(u)$ and every $M$ with $t_K(M)\ge0$: $G_{K,J}(M)\le G_{K,J}(m)$. \item[(b)] (in the cube) For every $K$ and $J\ge J_0(u)$: $G_{K,J}(M)\le G_{K,J+1}(M)$. Hence at every King point $J\mapsto\Gamma_J(M)$ is non-decreasing, $\Gamma(M)=\sup_J\Gamma_J(M)$ exists, and $Z^f_\infty(\,\cdot\,;M)$ exists, is entire and obeys $\abs{Z^f_\infty(z;M)}\le e^{\beta_f\abs{\Re z}^2}$. \end{enumerate} \end{lemma} \begin{proof} (a) The measures with single-site factors $\exp(-\frac{\eps_K^3}2\theta r_x^2)$ at the live sites, $0\le\theta\le t_K(M)$, interpolate between $\tau_m$ and $\tau_M$ within the class of Lemma~\ref{lem:s02:BFS}, whose part (c) says that such a factor does not increase $\langle\cos F(u)\rangle$. (b) Passing from $C^K_{J+1}$ to $C^K_J$ removes photon-gradient and hopping terms, none of which increases $\langle\cos F(u)\rangle$ (Lemma~\ref{lem:s02:BFS}(a),(b); \cite[Thm.~3.2, (3.32)]{K2} at $h=0$), the single-site measures being the same on both cubes. Hence $Z^f_J(\mathrm it;M)=\Gamma_J(M;\sum_jt_jf_j)$ converges for $t\in\R^N$, and the family $Z^f_J(\,\cdot\,;M)$, normal by the bound $e^{\beta_f\abs{\Re z}^2}$, converges locally uniformly on $\C^N$ to an entire limit with the same bound, $(\mathrm i\R)^N$ being a set of uniqueness. \end{proof} \begin{theorem}[continuity in the offset, uniformly in the spacing]\label{cpi:ew:alpha} In the geometry of the proof of Theorem~\ref{cpi:ew:EWC}(a) below (torus $T$, box $H'$ with the dead set $U\defeq T\setminus H'$, wall $n\psi$, source $u$ supported in $V\subset H'$), let the \emph{dead-shell model with offset} $c\in[0,c_{\max}]$ be the endpoint $\kappa=\infty$ of the heavy-shell model whose counterterm is King's free-boundary $\dm_{\eps_K}(m)$ plus the constant $c$, declared of order $\lambda$ (in King's complexified counterterm vertex \cite[(3.19)]{K1} it reads $\dm(x;e'e,\lambda'\lambda)+\lambda'c$): periodic photon on $T$, matter alive on $H'$ with Dirichlet condition at $\partial H'$, single-site coefficient $\alpha_K(m)+c+n\psi(x)$; and let $X^{[c]}_K$ be its expectation of $\cos F(u)$ at the spacing $\eps_K$. Then \[ \lim_{c\downarrow0}\ \limsup_{K\to\infty}\ \big|X^{[c]}_K-X^{[0]}_K\big|=0 . \] \end{theorem} \begin{proof} Let $a^{[c]}_K(x)\defeq\langle e^{xF(u)}\rangle$ in the offset model, entire in $x$ with $X^{[c]}_K=a^{[c]}_K(\mathrm i)$ by charge conjugation, and $I^{[c]}_D(x)$ the small-field integral of the dead-shell effective action at a stopping scale $\eta_0\le\eta_1$ over its $c$-independent test set $\mathsf T_k$; the two-sided heavy-shell bound at the endpoint (Theorem~\ref{cpi:thm:main} at $\kappa=\infty$), whose offset clause covers the constant $c$ declared of order $\lambda$ as the single graded summand $c_{01}=c$, holds with one set of constants for all $c\in[0,c_{\max}]$: for $\eps_K\le\eps_0(\eta_0)$, \begin{equation}\label{eq:cpi:ew:sandwich} e^{-a'}\,\frac{I^{[c]}_D(x)}{Z^{[c]}_\infty(0)}\ \le\ \frac{Z^{[c]}_\infty(x)}{Z^{[c]}_\infty(0)}\ \le\ e^{a}\,\frac{I^{[c]}_D(x)}{Z^{[c]}_\infty(0)}+b,\qquad a=C\eta_0^\sigma\abs T,\quad a'\le2a,\quad b=e^{-c_{\rm lf}\,p(\eta_0)^2+C\abs T}, \end{equation} with $c_{\rm lf}>0$ the large-field constant of the heavy-shell estimates. The offset enters only as the factor $\lambda'c$ of one vertex, so the two effective actions differ on $\mathsf T_k$ by a polynomial of degree at most $13$ in $c$ with coefficients bounded on $\mathsf T_k$ uniformly in $K$, which Markov's inequality bounds by $C_W(\eta_0)\,c$ up to a constant cancelling in the ratios $I^{[c]}_D(x)/I^{[c]}_D(0)$, which Gaussian domination on the dead-shell torus (Lemma~\ref{lem:s02:GD}) bounds uniformly in $c$ and $K$. The sandwich at $x$ and at $0$ gives $\sup_{\abs x\le1}\abs{a^{[c]}_K(x)-a^{[0]}_K(x)}\le\Phi(c,\eta_0)$ with $\lim_{c\downarrow0}\Phi(c,\eta_0)\to0$ as $\eta_0\to0$, the left side being free of $\eta_0$, and a normal-family argument carries the convergence to $z=\mathrm i$. \end{proof} \begin{theorem}[right-continuity of King's state in the scalar mass at $E_W$]\label{cpi:ew:EWC}\label{cpi:ew:JUMP}\emergencystretch=4em\relax Let $\tau_*=(e,\lambda,m,\lmu)\in E_W$. \begin{enumerate} \item[(a)] (the lower half) For every source $u$, $\liminf_{M\downarrow m}\Gamma(M;u)\ge\Gamma(m;u)$. \item[(b)] For every finite family $f=(f_1,\dots,f_N)$ of sources, \[ Z^f_\infty(\,\cdot\,;M)\longrightarrow Z^f_\infty(\,\cdot\,;m)\qquad\text{locally uniformly on }\C^N\text{ as }M\downarrow m; \] in particular each smeared curvature Schwinger function converges, $S_n(u_1,\dots,u_n;\tau_M)\to S_n(u_1,\dots,u_n;\tau_*)$. \item[(c)] The nearby King points $\tau_M$, $0<\abs{M-m}<\delta$, lie off $E_W$. \end{enumerate} \end{theorem} \begin{proof} (c) At $\tau_*\in E_W$, $t_\infty=\alpha_\infty-\alpha_\infty(m)$ is real-analytic, non-negative, vanishes at $m$ and is not constant ($t_\infty(M)=M^2+O(M)$ as $M\to\infty$ by the closed form of Section~\ref{sec:s14}), so $t_\infty>0$ on $0<\abs{M-m}<\delta$ and $t_K(M)>0$ for $K\ge K_1(M)$. (a) Fix $u$, $J\ge J_0(u)$ and an admissible geometry of the heavy-shell estimates: a torus $T$, a box $H'\subset T$ containing a source region $V\supset C_J$ with $\supp u\subset V$ at distance $\ge1$ from the dead set $U\defeq T\setminus H'$, a wall $n\psi$ ($n\ge1$, $\psi\ge0$ vanishing on $V$) and a King cube $C_{J''}\supset H'$ with collar width $w_c\defeq\dist(H',\partial C_{J''})\ge1$ whose translates by the periods of $T$ have disjoint interiors; let $X^{[c]}_K$ be $\langle\cos F(u)\rangle$ in the \emph{dead-shell model} at spacing $\eps_K$ (periodic photon on $T$, matter alive on $H'$ with Dirichlet condition at $\partial H'$, single-site coefficient $\alpha_K(m)+c+n\psi(x)$, $c\ge0$ declared of order $\lambda$ as in Theorem~\ref{cpi:ew:alpha}). The photon switch across the matter-free collar (Neumann images) changes expectations of $e^{\mathrm iF(u)}$ by at most $\delta_g(w_c)\to0$ as $w_c\to\infty$, uniformly in the single-site terms and in $T$, and the monotonicities of Lemma~\ref{lem:s02:BFS} give $G_{K,J}(m)\le X^{[0]}_K+\delta_g(w_c)$ and $G_{K,J''}(M)\ge X^{[c]}_K-\delta_g(w_c)$ whenever $0\le t_K(M)\le c$; with $c=2t_\infty(M)$, $M\in(m,m+\delta)$, and $K\to\infty$, $\Gamma(M)\ge\Gamma_{J''}(M)\ge\Gamma_J(m)-2\delta_g(w_c)-\limsup_K\abs{X^{[c]}_K-X^{[0]}_K}$ (Lemma~\ref{cpi:ew:mono}(b)), the last term tending to $0$ as $M\downarrow m$ by Theorem~\ref{cpi:ew:alpha} at the fixed geometry; then $w_c\to\infty$ at fixed $J$, then $J\to\infty$. (b) Lemma~\ref{cpi:ew:mono}(a) and (a) give $\Gamma(M;u)\to\Gamma(m;u)$, that is, convergence of $Z^f_\infty(\mathrm it;M)=\Gamma(M;\sum_jt_jf_j)$ on the set of uniqueness $(\mathrm i\R)^N$; the bound $e^{\beta_f\abs{\Re z}^2}$ of Lemma~\ref{cpi:ew:mono}(b), uniform in $M$, makes the family normal, so the convergence is locally uniform on $\C^N$ and the Taylor coefficients $S_n$ converge. \end{proof} Any identity between smeared curvature Schwinger functions holding at the nearby King points transfers to $\tau_*$: if $\Gamma(M;g\cdot u)=\Gamma(M;u)$ for a Euclidean motion $g$, every source $u$ and all $M$ with $00$ let $\alpha_k(m'):=m'^2+\delta m^2_{T,k}(m')$, where $\delta m^2_{T,k}(m')$ is King's mass counterterm~\cite[(3.32)--(3.33)]{K1} at the mass $m'$, computed with the propagators of the torus $T$ at spacing $\eps_k$, and let $\Delta_T(s)$, $s\le s_0$, be the constants of the convention bridge (CT)(b) on $T$, with $s_0=1$ (so that every spacing $\eps_k\le1$ is covered). \begin{proposition}[the positivity condition on the constant of the convention bridge]\label{rot:thm:POS} For each $R\in\Rgen$ and each of the three pairs of a lattice and a torus used, the axis lattice on $D_J$ and the lattice rotated by $R$ on $D_J$ and on $RD_J$, there are $p_H>1$, a mass $m_1>0$ with $(e,\lambda,m_1,\mu_0)$ in King's range and a threshold, depending on the couplings, the block integer $M$, $p_H$, $m_1$ and $R$ (equal to $1$ if $\Delta_T(s)\ge0$ for all $s\le s_0$ and every $J$), such that for every $J$ above it either $\Delta_T(s)\ge0$ for all $s\le s_0$, or \[ \liminf_{k\to\infty}\big(\alpha_k(m)-\alpha_k(m_1)\big)>p_H\sup_{s\le s_0}\max\{0,-\Delta_T(s)\}; \] $J_{\rm pos}$ denotes the maximum of these thresholds over the finite set $\Rgen$ and the three pairs. This holds at every point of King's range, $e>0$, outside the set $E_W$ of the points at which the scalar mass $m$ minimises, at fixed $(e,\lambda,\mu_0)$, the function $\alpha_\infty$ of the mass defined, up to an additive constant, by the limits $\alpha_\infty(m')-\alpha_\infty(m)=\lim_k\big(\alpha^{\rm f}_k(m')-\alpha^{\rm f}_k(m)\big)$, where $\alpha^{\rm f}_k$ is $\alpha_k$ with King's counterterm computed in the free-boundary conventions; $E_W$ is the set of the masses minimising that function: for fixed $(e,\lambda,\mu_0)$ these masses form a non-empty finite set, and for fixed $(\lambda,m,\mu_0)$ the set $E_W$ contains at most two values of $e^2$ (both read off the closed form of $\alpha_\infty$: the four definitions of the free-boundary coefficient coincide, and its mass differences converge to the closed form $\alpha_{\inf}(m')-\alpha_{\inf}(m)=-2s_{\inf}(m')$, where $s_{\inf}(m'):=\lim_k\frac12\big(\alpha^{\rm f}_k(m)-\alpha^{\rm f}_k(m')\big)$ is a polynomial of degree at most two in $(e^2,\lambda)$ (Proposition~\ref{cpi:prop:S}), real-analytic in the mass (Remark~\ref{cpi:ew:rightcont}), so that $\alpha_\infty$ and $\alpha_{\inf}$ differ by a mass-independent constant and $E_W$ is the set of global minimisers of $\alpha_{\inf}$). \end{proposition} \begin{proof} Fix $R\in\Rgen$ and a pair $\tau$ with torus $T$, and write $\Delta_\tau(s;m')$ for the bridge constant at the scalar mass $m'$ (Lemma~\ref{cpi:lem:ew:bridge}), $\Delta_T(s)=\Delta_\tau(s;m)$; if $\Delta_T(s)\ge0$ for all $s\le s_0$ and every $J$ the threshold is $1$, and otherwise the exact bridge $\alpha_k(m')=\alpha^{\rm f}_k(m')-\Delta_\tau(\eps_k;m')$ (Remark~\ref{rot:rem:CT}) gives \[ \alpha_k(m)-\alpha_k(m_1)=\big(\alpha^{\rm f}_k(m)-\alpha^{\rm f}_k(m_1)\big)-\Delta_\tau(\eps_k;m)+\Delta_\tau(\eps_k;m_1), \] where the bracket tends to $2s_{\inf}(m_1)$ (Proposition~\ref{cpi:prop:S}, a result on $\eps_k\Z^3$ alone) and $|\Delta_\tau(s;m')|\le K_\Delta(m')\,\Phi_{\kappa_{m'}}(50MJ)$ for $s\le1$ once $50MJ\ge l_0(m')$ (Lemma~\ref{cpi:lem:ew:bridge}). Outside $E_W$ there is $m_1>0$ with $\sigma_1:=s_{\inf}(m_1)>0$; with $\kappa_1:=\frac12\min(m,m_1,\mu_0,2)$, $K_1:=\max\{K_\Delta(m),K_\Delta(m_1)\}$ and $J_1$ the least $J$ with $\ell=50MJ\ge2\sqrt3\,\pi/\kappa_1$ and $(2+p_H)K_1\Phi_{\kappa_1}(\ell)\le\sigma_1$, for $J\ge J_1$ the $\liminf_k$ of the left side exceeds $p_H\sup_{s\le s_0}\max\{0,-\Delta_T(s)\}\le p_HK_1\Phi_{\kappa_1}(\ell)$ by at least $\sigma_1$, where $\kappa_1$, $K_1$, $\sigma_1$ are free of $\tau$, $R$ and $J$, so one $m_1$ serves all pairs, and no sign of $\Delta_T$ is used. The statements on $E_W$ follow from the closed form: $\alpha_{\inf}$ is real-analytic and not constant (Remark~\ref{cpi:ew:rightcont}), with $\alpha_{\inf}(m')=m'^2+O(m')$ as $m'\to\infty$ (Section~\ref{sec:s14}), so its global minimisers are isolated and finitely many, and for fixed $(\lambda,m,\mu_0)$ the condition $\partial_{m'}\alpha_{\inf}(m)=0$ is a polynomial equation of degree at most two in $e^2$. \end{proof} The scalar-mass variable is written $m'$ above and $M$ or $\mathsf M$, $M_1$ in Remark~\ref{cpi:rem:gs:torusalpha}, Proposition~\ref{cpi:prop:S}, Theorem~\ref{cpi:ew:EWC} and Lemma~\ref{cpi:lem:ew:bridge}. The constant $c_*$ of the large-field clause of Theorem~\ref{rot:thm:BAL5} at a non-zero wall is $(c_0'/2)(1-1/p_H)$ ($c_*=c_0'/4$ at $p_H=2$ if $\Delta_T\ge0$), with $c_0'=\min\{c_0'^{\rm AS},c_0'^{\rm D}\}\in(0,1]$, where $c_0'^{\rm AS}=\min\{a/8,\gamma_*/4,1\}$ ($a$ the block-spin coefficient of~\cite{K1}, $\gamma_*=\min(\gamma_0^s,\gamma_v)$ the exponent of~\cite[(3.41)]{Ba2}) is the constant of Ba{\l}aban's kept convergence factors~\cite[(3.41)]{Ba2} at block factor $5$ and $c_0'^{\rm D}$ the constant of the upper-bound estimates of the shell model. \begin{remark}[the exceptional set and the scope of the general-source results]\label{cpi:rem:gs:torusalpha} The torus coefficients $\alpha^\tau_k(M)$ enter \cref{rot:thm:POS} only through its $\liminf$ clause, where the exact bridge $\alpha^\tau_s(M)=\alpha^{\mathrm f}_s(M)-\Delta_T(s;M)$ between the torus and the free-boundary coefficient at the spacing $s$ is used with the decay bound on the bridge constants $\Delta_T$, as a bound and not as an identity; the free-boundary coefficient is one number however it is computed (as $\alpha^{\mathrm f}_k(M)$ of \cref{rot:thm:POS}, at the spacing $\eps_k$ in the torus conventions, through the closed form of $\alpha_\infty$, or on the infinite lattice $\eps_k\Z^3$), its mass differences converge to $\alpha_\infty(M)-\alpha_\infty(m)=-2s_{\inf}(M)$ in the notation of \cref{rot:def:cone}, so that $\alpha_\infty-\alpha_{\inf}$ is independent of the mass, $E_W=\{m\in\operatorname{argmin}_M\alpha_\infty\}=\{s_*(e)=0\}$ exactly, and $s_*(e)>0$ at every other point of King's range. On $E_W$ the cone, the tube and the split domain of \cref{rot:def:cone,r13:def:srcsplit} are empty, their defining inequalities being strict, and nothing of the general-source results is claimed there, the rotation theorem being obtained by right-continuity of King's state in the scalar mass; $E_W$ does meet King's range (a numerical evaluation of the closed form of $\alpha_\infty$ finds points of $E_W$ there, for instance at $\lambda=0.05$, $m=\mu_0=1$, $e^2\approx12.086$), for fixed $(\lambda,m,\mu_0)$ it contains at most two values of $e^2$, since $E_W\subset\{\alpha'_\infty(m)=0\}$ and $e^2\mapsto\alpha'_\infty(m)$ is a polynomial of degree at most two (\cref{r13:lem:s14:failset}(iii)), and what happens at its points beyond the right-continuity argument is problem~1 of \cref{rot:sec:open}. \end{remark} \begin{proposition}[the positivity margin]\label{cpi:prop:gs:I6bg} Fix $(e,\lambda,m,\mu_0)$ in King's range, $p\ge1$ and $\Gamma\ge0$ with $p\Gamma0}s_{\inf}(M)$, $s_{\inf}$ as in \Cref{rot:def:cone} (so outside $E_W$). Pick a King point $M_1$ with $\sigma_1\defeq s_{\inf}(M_1)>p\Gamma$ and put $\kappa_1\defeq\min(m,M_1,\mu_0,2)/2$ and $K_1\defeq\max\{K_\Delta(m),K_\Delta(M_1)\}$, where $K_\Delta(M)$ is the constant of the decay bound \[|\Delta_\tau(s;M)|\le K_\Delta(M)\Phi_{\kappa(M)}(l)\] of Lemma~\ref{cpi:lem:ew:bridge} on the bridge constants of \Cref{rot:thm:POS} at the scalar mass $M$ in the torus side $l\ge l_0(M)$, with $\Phi_\kappa$ as there and the rate $\kappa(M)=\frac12\min(M,\mu_0,2)\ge\kappa_1$ for $M\in\{m,M_1\}$. Let the \emph{margin threshold} $J_{\mathrm{mar}}(\Gamma)$ be the least $J$ with $l_J\ge2\sqrt3\pi/\kappa_1$ ($l_J$ the side of $D_J$, \Cref{rot:def:geom}; then $l_J\ge l_0(m)$ and $l_J\ge l_0(M_1)$) and $(2+p)K_1\Phi_{\kappa_1}(l_J)\le\sigma_1-p\Gamma$. Write $-2u_k(M)\defeq\alpha_k(m)-\alpha_k(M)$ for the torus coefficients of \Cref{rot:thm:POS} and \[ 2V^{(p)}_+(\tau)\defeq p\sup_{s\in S}\max\{0,-\Delta_\tau(s;m)\}, \] $S$ the set of spacings $s\le s_0$ of that proposition and $\tau$ one of its three pairs of a lattice and a torus. Then for $J\ge J_{\mathrm{mar}}(\Gamma)$, on each of the three pairs, \[ \liminf_{k\to\infty}\big(-2u_k(M_1)\big)\ \ge\ 2V^{(p)}_+(\tau)+2p\Gamma+(\sigma_1-p\Gamma), \] that is, the positivity condition of \Cref{rot:thm:POS} holds at the one point $M_1$ with the margin $2p\Gamma$ to spare; with $\bar V\defeq\frac12\sup_{s\in S}|\Delta_\tau(s;m)|$ in place of $V^{(1)}_+(\tau)$, both signs of the bridge constant being admitted, the same bound holds with $2p\bar V$ in place of $2V^{(p)}_+(\tau)$. \end{proposition} \begin{proof} $\Phi_\kappa(l)$ decreases strictly to $0$, so $J_{\mathrm{mar}}(\Gamma)$ exists. The bridge $\alpha_k(M)=\alpha^{\mathrm f}_k(M)-\Delta_\tau(\eps_k;M)$ at $m$ and $M_1$ with the decay bound of Lemma~\ref{cpi:lem:ew:bridge} ($\kappa_1\le\kappa(m),\kappa(M_1)$) gives $-2u_k(M_1)\ge2s^{\mathrm f}_{\eps_k}(M_1)-2K_1\Phi_{\kappa_1}(l_J)$ with $s^{\mathrm f}_{\eps_k}(M_1)\defeq\frac12(\alpha^{\mathrm f}_k(m)-\alpha^{\mathrm f}_k(M_1))\to\sigma_1$ (\Cref{cpi:rem:gs:torusalpha}), and $2V^{(p)}_+(\tau)$, $2p\bar V\le pK_1\Phi_{\kappa_1}(l_J)$. Subtracting, $\liminf_k(-2u_k(M_1))-2V^{(p)}_+(\tau)\ge2\sigma_1-(2+p)K_1\Phi_{\kappa_1}(l_J)\ge\sigma_1+p\Gamma$ for $J\ge J_{\mathrm{mar}}(\Gamma)$, with $M_1$, $K_1$, $\kappa_1$ free of $\tau$ and $J$. \end{proof} Here and below $\Delta_T(s;M)$ and $\Delta_\tau(s;M)$ are the bridge constants of Lemma~\ref{cpi:lem:ew:bridge} on the torus $T$ and on the pair $\tau$, $\Delta_T(s):=\Delta_T(s;m)$, $\Delta_T(0;M):=\lim_{s\to0}\Delta_T(s;M)$, and $S$, $V^{(p)}_+(\tau)$ and $-2u^\tau_k(M)$ (the coefficient $-2u_k(M)$ on the pair $\tau$) are as in Proposition~\ref{cpi:prop:gs:I6bg}. The source families of the walled torus comparison at a configuration $\fc=(J,\psi,n)$ and a source $g$ (Definition~\ref{rot:def:EXW}) are the moving families $h_K=g-n\psi-\frac12\Delta_T(\eps_K)$ and $h_{0,K}=-n\psi-\frac12\Delta_T(\eps_K)$ with their limits at $s=0$, the sources $(tG,g-n\psi+u\cdot1)$ and $(0,-n\psi+u\cdot1)$ with $u=u(s)\defeq-\frac12\Delta_\tau(s;m)$, $s\in S$, their images under the reflections and the rotation of the pair (which preserve sup norms and positive parts), and the mirrors $-\beta h_K$, $-\beta h_{0,K}$ with their limits, $\beta\in(0,1]$; $\Gamma_g\defeq\sup g^+$, $\Delta_*\ge\sup_{s\in S}|\Delta_T(s;m)|$ is the two-sided bound on the bridge constants, and $\bar H\defeq\norm g_\infty+n\sup\psi+\frac12\Delta_*$. \begin{theorem}[the choice rule and the envelope on the cone]\label{cpi:thm:gs:corv2}\label{r13:def:Jpos} Fix a point of King's range outside $E_W$, $p_H>1$, and $g$ with $p_H\Gamma_gp_H\Gamma_g$ ($M_1(0)$ the King point of \Cref{rot:thm:POS}), let $\kappa_1$, $K_1$ be as in \Cref{cpi:prop:gs:I6bg}, and set \begin{gather*} J_{\mathrm{pos}}(g)\defeq\max\big\{J_{\mathrm{pos}},\ J_{\mathrm{mar}}(\Gamma_g)\big\},\\ J_{\mathrm{mar}}(\Gamma_g)=\min\{J:\ l_J\ge2\sqrt3\pi/\kappa_1,\ (2+p_H)K_1\Phi_{\kappa_1}(l_J)\le\sigma_1-p_H\Gamma_g\}, \end{gather*} $J_{\mathrm{pos}}$ the threshold of \Cref{rot:thm:POS} and $J_{\mathrm{mar}}(\Gamma_g)$ the margin threshold of \Cref{cpi:prop:gs:I6bg} at $p=p_H$, $\Gamma=\Gamma_g$; this is the definition of $J_{\mathrm{pos}}(g)$ used throughout Section~\ref{rot:sec:gensrc}. Then $J_{\mathrm{pos}}(g)$ depends on $g$ through $\Gamma_g$ only, not on $n$, $\psi$, $\beta$, $\tau$ or $R$; it is finite, so that the admissible $J$ are cofinal; and $J_{\mathrm{pos}}(0)=J_{\mathrm{pos}}$. Let $\fc=(J,\psi,n)$ with $J\ge J_{\mathrm{pos}}(g)$, $J\in5^{k_*}\N$ and $J\ge J_0(\eps_0)$. Then on each of the three pairs: \begin{itemize}[nosep] \item[(A)] (margin) $\liminf_k(-2u^\tau_k(M_1))-2V^{(p_H)}_+(\tau)\ge2\sigma_1-(2+p_H)K_1\Phi_{\kappa_1}(l_J)\ge\sigma_1+p_H\Gamma_g>2p_H\Gamma_g$. Hence the positivity condition of \Cref{rot:thm:POS} holds at $M_1$ with the margin $p_H(\Gamma_g+V^{(1)}_+(\tau))$, and the walled large-field envelope, King's upper bound \cite[Thm.~3.1, (3.3)--(3.4)]{K1} in the envelope form of (BAL)$_5$ (Theorem~\ref{rot:thm:BAL5}), holds at every source of the direct families and their images and at $\{h_K\}\cup\{h_\infty\}$, with the large-field constant $c_*=(c_0'/2)(1-1/p_H)$ and an exponent $q$ free of $K$; no $\beta$ enters. \item[(B)] (mirrors) For the same $M_1$ and $J$, and every $\beta\in(0,1]$ with \[ (\beta)\qquad 2p_H\,\beta \bar H<\sigma_1+p_H\Gamma_g\qquad(p_H=2:\ 4\beta \bar H<\sigma_1+2\sup g^+), \] the positivity condition holds at $M_1$ with the margin $p_H\beta \bar H$, hence the envelope holds at $-\beta h_K$, $-\beta h_{0,K}$ and their limits, which is the upper line of clause (a) of (EX$^\sharp_W{}'(\beta)$) (Definition~\ref{r13:def:EXWcond}), also at $(K+1,k+1)$. The admissible $\beta$ form $(0,1]\cap(0,\beta_0)$ with $\beta_0\defeq(\sigma_1+p_H\Gamma_g)/(2p_H\bar H)>\Gamma_g/\bar H$, a non-empty set also at $\Gamma_g=0$. \end{itemize} \end{theorem} \begin{proof} (A) is \Cref{cpi:prop:gs:I6bg} at $p=p_H$, $\Gamma=\Gamma_g$ for $J\ge J_{\mathrm{pos}}(g)\ge J_{\mathrm{mar}}(\Gamma_g)$, with the same $K_1$, $\kappa_1$ on every pair, and $\sigma_1>p_H\Gamma_g$. By $(a+b)^+\le a^++b^+$ the direct families and their images have positive part at most $\Gamma_g+V^{(1)}_+(\tau)$ and every member is $\ge-\bar H$ with oscillation $\le2\bar H$, so each direct source at the amplitude $p_H$ is $v\cdot1-q'$ with $-p_H\bar H\le v\le p_H(\Gamma_g+V^{(1)}_+(\tau))$ and $0\le q'\le2p_H\bar H$, its positive part is paid from the margin (A), and King's envelope at a non-positive source applies with an exponent $q=q(G,\Gamma_h,V_-,|T|,p_H)$ independent of the shape of the source. (B) The mirrors have positive part at most $\beta \bar H$ and (A) gives $\liminf_k(-2u^\tau_k(M_1))\ge\sigma_1+p_H\Gamma_g>2p_H\beta \bar H$ under $(\beta)$; as $J_{\mathrm{pos}}(g)$ is free of $n$, $\psi$ and $\beta$, the configuration $(n+K)\psi$ stays in the class for every height $K$ with any $\beta<\min(1,\beta_0(K))$, and the admissible $J$ are cofinal. \end{proof} \emph{The final-scale cap.} Ba{\l}aban's lower and upper bounds~\cite[(1.14)]{Ba1}, \cite[(1.3)]{Ba2} for the partition functions at the fixed masses end at a final spacing $t_K\in(\eps_0/5,\eps_0]$, at which the large-field clause of Theorem~\ref{rot:thm:BAL5} at the wall on the skew torus $D_J$ takes one step; so that the skew tori do not wrap at that scale, the coarsest scale $\eps_0$ of the flow is at most a threshold $\eps_{\rm sys}(M,R,\rho_{\rm B},\delta_{\mathrm{fb}})$ fixed by the large-block integer $M$, the rotation $R$, Ba{\l}aban's radius $\rho_{\rm B}$ and the finest block size $\delta_{\mathrm{fb}}$, independent of $J$ and of the scalar mass and closed downward, under which the non-wrapping threshold at the final scale equals $1$ for every $J$ in the residue class of Definition~\ref{rot:def:class}, so that every configuration of the class $\mathcal C$ meets it and the wall condition holds on every skew torus $R^{-1}D_J$. The cap is a proviso of the torus theorem (Theorem~\ref{rot:thm:CT}) on its parameters, carried by the class $\mathcal C$ (Definition~\ref{rot:def:class}, Remark~\ref{rot:rem:class}), not an assumption of Theorem~\ref{rot:thm:main}. \emph{The odd-block clause of the degree census.}\label{rot:par:oddblock} The degree census of the $\kappa$-Cauchy estimate (Theorem~\ref{cpi:swc:main}) has a clause for the closed matter blocks of degree $-2$ or $-1$ with one photon end: after the rewriting they appear as classes with two photon-type ends, a background leg and a line slot, of degree exactly $-1$ or $0$, entering the enumeration as black boxes, and after the summation by parts on the trees with one background slot every proper prefix sub-component of a renormalised member has integer degree at least one or lies inside a black box, which enters the enumeration as a local vertex at the window scale. This is Theorem~\ref{cpi:swcw:Rka0}(a), proved from Furry's theorem with the fundamental theorem of calculus in the background and the Ward identity for the window class, which controls the zeroth and first moments of the two-photon kernel in the background slot (Theorem~\ref{cpi:swcw:Rka0}(f)); Theorem~\ref{cpi:swcw:Rka0}(e) bounds the tail of the coarse counterterm at orders two to four that its one-loop form leaves. \space% % % \section{King's state with scalar sources, and its Euclidean invariance}\label{rot:sec:gensrc} This section extends Theorem~\ref{rot:thm:main} to neutral scalar sources: outside the exceptional set $E_W$, King's state with one further vacuum counterterm, the scheme $\Theta^\sharp$, exists on an explicit tube of complex coefficients around the origin and is invariant under every Euclidean motion there (Theorem~\ref{rot:thm:mainGS}). It uses the argument of Sections~\ref{rot:sec:K}--\ref{rot:sec:shell} carried out on a cone of scalar sources, together with the cube limits with scalar sources of the existence part; nothing in it is used by Theorem~\ref{rot:thm:main}, and at $h=0$ every statement below reduces to the corresponding statement of Sections~\ref{rot:sec:wick}--\ref{rot:sec:shell}. For scalar sources with $\int h\ne0$ King's Wick ordering has no finite continuum limit beyond the curvature sector: with the free Wick constant alone the expectation of the Wick square of Definition~\ref{def:s02:sources} is $(e^2/4\pi^2)\ln(1/\eps)\int h+O(1)$ at order $e^2\lambda^0$ in formal perturbation theory, so it diverges at order $e^2$ (\Cref{cpi:rot:log}), and the Wick constant must absorb this logarithm, which grows by $(e^2/4\pi^2)\ln5$ per step of the block factor $5$. The object with scalar sources is therefore King's state with one further vacuum counterterm, linear in $h$ and depending on the lattice spacing only, which we call a \emph{scheme}; the state is constructed, in the scheme $\Theta^\sharp$ below, on a tube of complex coefficients around the origin whose real points are the sources whose mass-decreasing part is below a threshold fixed by the couplings (Definition~\ref{rot:def:cone}). The torus comparison in its walled form is provided by the torus theorem on a mixed source family (\Cref{cpi:ct:mainW}; Theorem~\ref{rot:thm:Kwsupply}), and the closing statement on the cone (the lower bound of the state by the envelope of the walled values, Section~\ref{rot:ssec:shellg}) is obtained from the heavy-shell estimate at one configuration, which the shell estimates at the scalar source provide. \subsection{Schemes, the threshold, the cone and the tube}\label{rot:ssec:scheme}\label{rot:ssec:cone} \begin{definition}[schemes]\label{rot:def:scheme} A \emph{scheme} is a real sequence $a=(a_k)_{k\ge0}$ such that $a_k$ depends only on $\eps_k$ and on $(e,\lambda,m,\mu_0)$. For a state $\langle\cdot\rangle_{\Lambda,X}$ of Definition~\ref{rot:def:states} on a lattice of spacing $\eps_k$ put \[ Z^a_{\Lambda,X}\Big(\sum_jz_jf_j,\sum_jw_jh_j\Big):=e^{-\sum_jw_ja_k\sS_\Lambda(h_j)}\Big\langle\Theta\Big(\sum_jz_jf_j,\sum_jw_jh_j\Big)\Big\rangle_{\Lambda,X}, \] that is, the expectation of $\exp(\sum_jz_jF(f_j)+\sum_jw_j\sum_x\eps^3h_j(x)[|\phi(x)|^2-2\sW_\eps-a_k])$. We write $\Theta^a$ for the corresponding observable. For walled tori $\fc=(J',w)$ and $(f,g)$ with $w=0$ on the support of $g$ put \[ T^a_{\fc,\Lambda}(f,g):=\langle\Theta^a(\mathrm if,g)\rangle_{\fc,\Lambda} =\frac{Z^a_{\Lambda,D_{J'}}(\mathrm if,g-w)}{Z^a_{\Lambda,D_{J'}}(0,-w)} . \] Examples. (i) King's scheme $a\equiv0$. (ii) The scheme $\Theta^\sharp$: $a^\sharp_k:=e^2X_{\eps_k}$, where $X_\eps$ is the coefficient of $e^2$ in the perturbative expansion of $\langle|\phi(0)|^2\rangle$ for the model on $\eps\Z^3$ at $\lambda=0$, with the order-$e^2$ part of $\delta m^2_\eps$ inserted; we write $\Theta^\sharp:=\Theta^{a^\sharp}$. The scheme and the reference scheme $z=a^\sharp$ are those of \Cref{def:s02:sources} (the functional $Z^a_{k,X}$ there is $Z^a_{\Lambda_k,X}$ here), so $\Theta^\sharp$ is the reference scheme of \Cref{r13:def:s13wick}. \end{definition} The threshold is defined from the function $\alpha_\infty$ of Proposition~\ref{rot:thm:POS}; the curvature sector carries no threshold. \begin{definition}[the threshold, the cone and the tube]\label{rot:def:cone} Fix $(e,\lambda,m,\mu_0)$ in King's range and let $\alpha_\infty$ be the function of the scalar mass of Proposition~\ref{rot:thm:POS} (defined up to an additive constant). Put \[ s_*(e):=\tfrac12\sup_{M>0}\big(\alpha_\infty(m)-\alpha_\infty(M)\big)\in[0,\infty]; \] $s_*(e)=0$ exactly when $m$ is a global minimiser of $\alpha_\infty$, that is, exactly at the points of the set $E_W$ of Proposition~\ref{rot:thm:POS}, and $s_*(e)>0$ at every other point of King's range ($s_*(e)$ is the threshold $s_*$ of \eqref{eq:s14:EW}, and $E_W$ is the set defined there). The \emph{cone} is \[ \Kcs:=\{(f,g)\in\cK:\ \sup\nolimits_x2g(x)0$ and, in Section~\ref{rot:ssec:shellg}, through the exponent $q'(g)$ of \eqref{cpi:eq:bg} and the threshold $J_1(g)$ defined there, both taken in $[0,\infty]$ ($q'(g)=2$ and $J_1(g)=1$ at $s_*(e)=+\infty$), so nothing below depends on its finiteness. The function $\alpha_\infty$ is King's counterterm~\cite[(3.32)--(3.33)]{K1} with the propagators of $\eps\Z^3$ and King's free boundary condition~\cite[pp.~329--330]{K2}; that $E_W=\{s_*(e)=0\}$ exactly follows from the closed form of $\alpha_\infty$ (the four definitions of the free-boundary coefficient coincide, and the mass differences of $\alpha_\infty$ converge, as the spacing tends to zero, to the closed form $\alpha_{\inf}(M)-\alpha_{\inf}(m)=-2s_{\inf}(M)$, where $s_{\inf}(M)$ is the limit of the free-boundary counterterm difference $\tfrac12(m^2-M^2)+\tfrac12[\delta m^2_\eps(m)-\delta m^2_\eps(M)]$; so $\alpha_\infty$ and $\alpha_{\inf}$ differ by a mass-independent constant, $s_*(e)=\sup_Ms_{\inf}(M)$, and $E_W$, the set of global minimisers of $\alpha_{\inf}$, is exactly $\{s_*(e)=0\}$, with $s_*(e)>0$ off $E_W$; the torus form of $\alpha_\infty$ and the scope of the statements with scalar sources are the subject of Remark~\ref{cpi:rem:gs:torusalpha}). \subsection{The condition (W$^\sharp$) and King's state on the cone}\label{rot:ssec:Wcone} \begin{definition}[\textup{(W$^\sharp$)}: the scheme condition on the cone]\label{rot:hyp:Wsharp} Let $a$ be a scheme and write $Z^{a,K}_{k,J}$ for the functionals $Z^a$ of Definition~\ref{rot:def:scheme} in the state $\langle\cdot\rangle^K_{k,J}$ of King's cube $C^K_J$ (Definition~\ref{def:s02:Kcube}). \begin{enumerate} \item[(W$^\sharp$1)] For every King cube $C^K_J$, every finite family of real $2$-forms $f_j\in C^\infty_c$, $j\le N$, sampled at plaquette centres in the lattice's own frame, and of non-negative scalar sources $0\le g_i\in C^\infty_c$, $i\le N'$, all supported at $\ell^\infty$-distance $>1$ from $\R^3\setminus C_J$, the limit $\lim_kZ^{a,K}_{k,J}(\sum_jz_jf_j,\sum_iv_ig_i)$ exists locally uniformly on the open convex set $\{(z,v)\in\C^N\times\C^{N'}:\ \sup_x2\sum_i(\Re v_i)_+g_i(x)1$ from $\R^3\setminus C_J$. \end{enumerate} A scheme $a$ with (W$^\sharp$1) and (W$^\sharp$2) is called \emph{admissible$_{0$ at order $e^2$. \begin{theorem}[the scheme $\Theta^\sharp$ is admissible$_{1$ absorbing the passage to the free convention; and the sign of the threshold is a lattice computation: $\delta m^2_\eps(m)-\delta m^2_\eps(M_1)$ is a finite sum of massive lattice integrals with at most two loops, Reisz's absolute bound~\cite{Rei88} gives its $\eps\to0$ limit, so that $s_{\inf}(M_1)$ exists and is Lipschitz in $M_1$, with a torus correction $O(l^{-4})$, and an $M_1$ with $s_{\inf}(M_1)>0$ exists exactly when $m$ is not a global minimiser of $\alpha_{\inf}$. \begin{theorem}[King's state in an admissible$_{1$ from $\R^3\setminus C_{J_0}$, and write $Z^{a,K}_J(\zeta):=\lim_kZ^{a,K}_{k,J}(\zeta)$ for $\zeta=(z,w)$. \begin{enumerate} \item[(i)] For $J\ge J_0+1$, $Z^{a,K}_{k,J}\to Z^{a,K}_J$ locally uniformly on $\Om(h)$; $Z^{a,K}_J$ is holomorphic on $\Om(h)$ and lies in $(0,\infty)$ at its real points. For a one-member family $(f;g)$ with $g\ge0$ the same holds already for $J\ge J_0$. \item[(ii)] For $(f,g)\in\Kcs$ supported at $\ell^\infty$-distance $>1$ from $\R^3\setminus C_{J_0}$, $J\mapsto Z^{a,K}_J(\mathrm if,g)$ is nondecreasing for $J\ge J_0$; if $s_*(e)>0$, then for every $(f,g)\in\cK$ so supported, $J\mapsto Z^{a,K}_J(f,-g)$ is nonincreasing for $J\ge J_0$. \item[(iii)] $Z^a_\infty(\zeta):=\lim_{J\to\infty}Z^{a,K}_J(\zeta)$ exists locally uniformly on $\Om(h)$ and is holomorphic there; $Z^a_\infty(f,h)\in(0,\infty)$ at every real point of $\Om(h)$; and $Z^{a,K}_J\le Z^a_\infty$ on $\Kcs$. \item[(iv)] For every $\rho>0$, $\Phi^a(s):=\ln Z^a_\infty(s)$ is finite, convex and continuous on the open convex set $U_\rho:=\{(f,h)\in E_\rho\times C^\infty_c(\bar B_\rho):\sup2h_+0$ and $a'$ is admissible$_{0$ (the wall condition of the class $\mathcal C$ of Definition~\ref{rot:def:class} is not assumed), $G$ a real smooth $D_J$-periodic $2$-form and $g$ a real smooth $D_J$-periodic function, with $J\ge J_{\rm pos}(g)$, the threshold of Theorem~\ref{cpi:thm:gs:corv2} defined before Theorem~\ref{rot:thm:Kwsupply} (a function of $\sup g^+$ alone), and, when $g$ has a positive part, $2\sup g0$ a \emph{fixed reference photon mass}, independent of $K$, at which the scheme constant is computed; it need not equal the photon mass $\mu_0$ of the model, and $z_K$ differs from $a^\sharp_K=e^2X_{\eps_K}(m;\mu_0)$ by $e^2[X_{\eps_K}(m;\mu_0)-X_{\eps_K}(m;\mu_*)]$, so that $a^\sharp_K-z_K$ converges, by statement (4) of Section~\ref{rot:ssec:Kw}). \end{definition} \begin{theorem}[the torus comparison on the mixed source family]\label{cpi:ct:mainW} In the setting of Definition~\ref{r13:def:mixedW}, assume the hypotheses (N1)--(N5) of Section~\ref{rot:ssec:inputsN} at the sources of $\cS_W$ on $D_J$, at their $S^*$-images on $RD_J$, and at their coordinate-reflected and permuted images on $D_J$ with the $S^*$-images of these on $RD_J$. Then, along $\eps_K=5^{-K}$ and in the scheme $z_K$: \begin{enumerate} \item[(i)] (\emph{two-lattice comparison, scheme-free}) $Z_{Y_K}(D_J;s)/Z_{X_K}(D_J;s)\to1$ for every $s\in\cS_W$. Consequently $\langle e^{zF(G)}\rangle_{\fc,g,Y_K}-\langle e^{zF(G)}\rangle_{\fc,g,X_K}\to0$ locally uniformly in $z\in\C$, all moments $\langle F(G)^m\rangle_{\fc,g}$ agree in the limit, $T^{Y_K}_\fc(tG,g)/T^{X_K}_\fc(tG,g)\to1$ for every real $t$, and for complex $z$, $T^{Y_K}_\fc(zG,g)-T^{X_K}_\fc(zG,g)=\rho_K\cdot o(1)$ locally uniformly, with $\rho_K\defeq T^{X_K}_\fc(0,g)$. \item[(ii)] (\emph{existence on the axis lattice}) If $\lim_K\langle e^{tF(G)}\rangle_{\fc,g,X_K}$ exists for every real $t$ (the walled existence (N6$_W$) at the mixed source), both families converge, locally uniformly on $\C$, to the same entire function $W_{\fc,g}(zG)$, with all moments. If moreover the single-torus hypotheses (EX$^\sharp_W{}'(\beta)$) of Definition~\ref{r13:def:EXWcond} hold for one amplitude $\beta\in(0,1]$, then \[ \rho_K=e^{-z_K\langle g\rangle_{X_K}}Z_{X_K}(D_J;0,h)/Z_{X_K}(D_J;0,h_0)\to\rho_\infty\in(0,\infty), \] hence $T^{X_K}_\fc(zG,g)$ and $T^{Y_K}_\fc(zG,g)$ converge, locally uniformly in $z\in\C$, to $T_\fc(zG,g)\defeq\rho_\infty W_{\fc,g}(zG)$; at $zG=\mathrm if$ this is the axis-lattice limit of the walled ratio of Definition~\ref{rot:def:scheme}. \item[(iii)] (\emph{generators}) (i) and (ii) hold with $R$ replaced by any $R'\in\Rgen$. \end{enumerate} The order of limits is $\eps\to0$ along $\eps_K$ at fixed $\fc$, $g$, $G$, and $\beta$ is chosen at fixed $\fc$ and $g$. \end{theorem} In the application $g\ge0$ and $\psi=0$ on the $1$-neighbourhood of $\supp g$; neither the sign of $g$ nor any support property of $g$ is used in the theorem. The proof is that of Theorem~\ref{rot:thm:CT} (the forward bound and the closing of Section~\ref{rot:ssec:CT}) run on the family $\cS_W$ in place of $\{(tG,h_0)\}$: the instance of (N5) at the mixed source is Proposition~\ref{cpi:ct:H5}, whose wall bound holds for every bounded real wall, $\|h\|_\infty\le\sup|g|+n\sup\psi$, with no sign of $h$, no condition on $\min h$ and no condition on $\langle h\rangle$; the source-linear constant $c_K$ tends to zero, so no closure of $\cS_W$ under $(G,h)\mapsto(G,-h)$ is needed; and (ii) adds the identification of the limit at the limiting source $h_\infty=h-\frac12\Delta_T(0)$ of the convention bridge ($\Delta_T(s)=\Delta_T(s;m)$ the bridge constant of Proposition~\ref{rot:thm:POS} on $T=D_J$, Definition~\ref{rot:def:EXW}) with the Vitali upgrade to complex coefficients. In (i) the scheme factors differ by $e^{-z_K(\langle g\rangle_{Y_K}-\langle g\rangle_{X_K})}=1+O((1+K)\eps_K^N)$; no Wick scheme enters (i) or (iii) beyond the growth bound $|z_K|\le C(1+K)$. For $0\le g\in C^\infty_c$ let $J_{\rm pos}(g)\defeq\max\{J_{\rm pos},J_{\rm mar}(\sup g)\}$ be the threshold of Theorem~\ref{cpi:thm:gs:corv2}: the larger of the threshold $J_{\rm pos}$ of Proposition~\ref{rot:thm:POS} and the least $J$ at which the positivity margin of Proposition~\ref{cpi:prop:gs:I6bg} exceeds $2p_H\sup g$, $p_H=2$ (the right side of the display of Proposition~\ref{rot:thm:POS} enlarged by $2p_H\sup g$), at a King point $M_1=M_1(\sup g)$ (a scalar mass with $(e,\lambda,M_1,\mu_0)$ in King's range, as in Proposition~\ref{rot:thm:POS}) with $\tfrac12(\alpha_\infty(m)-\alpha_\infty(M_1))>p_H\sup g$, which exists exactly when $p_H\sup g0$, $J'\in5^{k_*}\N$ and $J'\ge\max(J_0(\eps_0),J_{\rm pos}(g))$, under the cap of Section~\ref{rot:ssec:openin} imposed for every $R\in\Rgen$ at once; the class $\mathcal C$ of Definition~\ref{rot:def:class} for $R$ and $L$ is contained in $\cAW(X;0)$ whenever $X\subset R^{-1}C_L$. The \emph{walled value} $\cW^a_\fc(f,g)$ is the common limit of the walled ratios $T^a_{\fc,\Lambda_k}(f,g)$ and $T^a_{\fc,R^{-1}\Lambda_k}(f,g)$ of Definition~\ref{rot:def:scheme}, when it exists, and the \emph{envelope} $\cW^{a,\env}(f,g)$ is the limit in $b$ of $\inf_{\fc\in\cAW(Q_b;g)}\cW^a_\fc(f,g)$ for $(f,g)\in\Kcs$ supported in $Q_b$. \begin{theorem}[the walled torus comparison holds on the cone]\label{rot:thm:Kwsupply} At every point of King's range outside $E_W$, for every $R\in\Rgen$, every bounded $X\subset\R^3$, every $(f,g)\in\Kcs$ with $g\ne0$ supported in $X$ and every $\fc\in\cAW(X;g)$, the limits $\lim_kT^{a^\sharp}_{\fc,\Lambda_k}(f,g)$ and $\lim_kT^{a^\sharp}_{\fc,R^{-1}\Lambda_k}(f,g)$ exist and are equal; the same holds for every scheme $a$ with $a_k-a^\sharp_k$ convergent. At $g=0$ the same holds for every $\fc$ of the class $\mathcal C$ (Theorem~\ref{rot:hyp:K}; the walled ratio does not depend on the scheme there). \end{theorem} \begin{definition}[the moving families, their limits and their mirrors]\label{rot:def:EXW} Let $\fc=(J,\psi,n)$, $n\ge0$, $g$, $h=g-n\psi$ and $h_0=-n\psi$ be as in Definition~\ref{r13:def:mixedW} (there $n>0$; $n=0$, the bare torus, is admitted here), let $\Delta_T(s)$ be the constant of the convention bridge of Proposition~\ref{rot:thm:POS} on the torus $T=D_J$ at spacing $s$, with $\Delta_T(0):=\lim_{s\to0}\Delta_T(s)$ (Lemma~\ref{cpi:lem:ew:bridge}), and put $h_K:=h-\tfrac12\Delta_T(\eps_K)$, $h_{0,K}:=h_0-\tfrac12\Delta_T(\eps_K)$ (the \emph{moving families}) and $h_\infty:=h-\tfrac12\Delta_T(0)$, $h_{0,\infty}:=h_0-\tfrac12\Delta_T(0)$ (their limits), so that $h_K-h_{0,K}=h_\infty-h_{0,\infty}=g$; fix $\beta\in(0,1]$, write $a^\sharp_K=e^2X_{\eps_K}(m;\mu_0)$ for the scheme constant of Definition~\ref{rot:def:scheme} at spacing $\eps_K$ ($X_\eps(m;\mu)$ the coefficient $X_\eps$ of that definition at the scalar mass $m$ and the photon mass $\mu$, as in statement (4) below), and let $H':=\|g\|_\infty+n\sup\psi+\tfrac12\Delta_*$ be the sup-norm bound of the families with their bridge shift, $\Delta_*$ the bound of the convention bridge of the class $\mathcal C$ of Definition~\ref{rot:def:class}. The four moving families $h_K$, $h_{0,K}$, $-\beta h_K$, $-\beta h_{0,K}$ and their limits lie in one bounded set of King's class, of sup norm at most $H'$, the \emph{mirrors} $-\beta h_K$, $-\beta h_{0,K}$ having sup norm at most $\beta H'$. \end{definition} \begin{definition}[\textup{(EX$^\sharp_W{}'(\beta)$)}: the single-torus statements with a mirror of amplitude $\beta$]\label{r13:def:EXWcond} Let the moving families $h_K$, $h_{0,K}$, their limits $h_\infty$, $h_{0,\infty}$, the mirrors of amplitude $\beta\in(0,1]$, the scheme constant $a^\sharp_K$ and the bound $H'$ be as in Definition~\ref{rot:def:EXW}. The condition \emph{(EX$^\sharp_W{}'(\beta)$)} is the conjunction of the following four statements for King's torus-convention model on the axis lattice $X_K$ and the torus $D_J$, along $\eps_K=5^{-K}$: \begin{enumerate} \item[(a)] the two-sided single-torus bound at the four families $h_K$, $h_{0,K}$, $-\beta h_K$, $-\beta h_{0,K}$, at the depth $k(K)$ and at $(K+1,k+1)$: its upper line is the walled large-field envelope, King's upper bound~\cite[Thm.~3.1]{K1} in the envelope form of Theorem~\ref{rot:thm:BAL5}, with the large-field coefficient $c_*=(c_0'/2)(1-1/p_H)$; its lower line is the lower bound by the small-field integral, the small-field half of King's bound with sources~\cite[Thm.~3.1]{K1} at the fixed source size $S_0=H'$, for both signs of the source and at any amplitude (Theorem~\ref{cpi:ct:CTB}(c)); \item[(b)] King's low-field positivity~\cite[Thm.~3.4]{K1} with $n=1$ at the four limiting sources $h_\infty$, $h_{0,\infty}$, $-\beta h_\infty$, $-\beta h_{0,\infty}$, for the effective action minus only its field-independent $h$-linear constant at the orders $1$, $\lambda$, $e^2$, in the restricted form of Remark~\ref{cpi:rot:errfurther}(i); \item[(c)] King's vacuum-energy bound~\cite[Thm.~2.1]{K1} at zero source, in torus conventions; \item[(d)] the bulk bound at order one: the source-linear field-independent constant of the effective action at the orders $1$, $\lambda$, $e^2$, in the free conventions and for the constant weight, taken together with the scheme term $a^\sharp_K$, is bounded by $C\delta_k^{-2}$, and its increments between the depth $k$ at spacing $\eps_K$ and the depth $k+1$ at spacing $\eps_{K+1}$ are summable along $k(K)$ (Lemma~\ref{cpi:lem:BULK}). \end{enumerate} \end{definition} The mirrors enter the proof of Theorem~\ref{cpi:ct:mainW}(ii) only through its telescoping step. On the cone, for $J\ge J_{\rm pos}(g)$ and outside $E_W$, the upper line of (a) is provided by Theorem~\ref{cpi:thm:gs:corv2}, (A) at the direct families and (B) at the mirrors for every $\beta$ with the clause ($\beta$) there; (b), (c) and (d) involve $\beta$ only through the choice of the sources. The statement at $g\ne0$ is the torus theorem on the \emph{mixed source family} $\{(tG,g-n\psi):t\in\R\}\cup\{(0,-n\psi)\}$ (\Cref{cpi:ct:mainW}), for the scheme $z^{\mu_*}_K=e^2X_{\eps_K}(m;\mu_*)$ of that theorem (the sequence written $z_K$ in its statement; the superscript distinguishes it from the reference scheme $z=a^\sharp$ of Definition~\ref{rot:def:scheme}, which is $e^2X_{\eps_K}(m;\mu_0)$; $\mu_*>0$ is the fixed reference photon mass of that scheme, Definition~\ref{r13:def:mixedW}), evaluated at the point $(\mathrm i;1)$: its two-lattice comparison and the existence of the axis-lattice limit give the two limits and their equality. Its hypotheses are provided at every such configuration by four statements, each given in the form used, with its proof: (1) Theorem~\ref{cpi:ct:mainW} itself on the mixed family, with its composition step with the two-flow comparison (N1$^*$) in the form of a bound in the sup norms of $h$ and its derivatives; (2) (N1)--(N5) at the family and its images (notation of Section~\ref{rot:ssec:CT}), (N6$_W$), that is, the convergence of the mixed-source walled ratio along $\eps_K=5^{-K}$, which the walled torus theorem gives at the shifted source $h_\infty=h-\tfrac12\Delta_T(0)$, uniform bounds on compact $t$-sets upgrading it, by Vitali's theorem, to an entire limit with all moments, and the clauses of the single-torus statement (EX$^\sharp_W{}'(\beta)$) of Definition~\ref{r13:def:EXWcond}, proved by the method of the torus theorem (the passive-rotation reductions, the equal-spacing comparison of the hybrid and the standard flow, the one-sided forward bound and the final step, the involution at $\theta_0=\arctan(3/4)$; \Cref{cpi:ct:Vvac}) from the single-torus statements off $E_W$: for $2\sup g0$, $(f,g)\in\Kcs$ supported in $Q_b$ and $\eta>0$ there is $\fc\in\cAW(Q_b;g)$ with $\cW^a_\fc(f,g)\le Z^a_\infty(\mathrm if,g)+\eta$. \begin{proposition}[the closing statement on the cone]\label{rot:prop:closingcone} The closing statement on the cone holds for the scheme $\Theta^\sharp$ at every $(f,g)\in\Kcs$, in the pointwise form at the box $Q_b$ of the source: for every $\eta>0$ there is a single-level configuration $\fc=(J',\psi,n+K)\in\cAW(Q_b;g)$ with $\cW^{a^\sharp}_\fc(f,g)\le Z^{a^\sharp}_\infty(\mathrm if,g)+\eta$. \end{proposition} \begin{proof} The statement is obtained, as in Section~\ref{rot:sec:shell}, from the heavy-shell decoupling at one configuration: for every such $b$, $(f,g)$ and $\eta$ there are a two-level configuration $(\fs,\chi)$ for $Q_b$ with $\tau(\fs)\le\eta$ and a shell mass $K<\infty$ with $\liminf_k[\Psi_k(n,K)-\Psi_k(n,\infty)]\le\eta$, where $\Psi_k(n,K):=T^{a}_{(J',n\psi+K\chi),\Lambda_k}(f,g)$ is the walled ratio of the two-level configuration (wall height $n$, shell mass $K$) at spacing $\eps_k$ and $\Psi_k(n,\infty)$ its value with the matter removed on the shell, as in Definition~\ref{rot:def:twolevel}, the states now carrying the source $(zf,g)$ with the multiplier $e^{-a^\sharp_k\sS(g)}$ of the scheme. The two-level sandwich and the dead-shell bound on the cone, whose constant $\MW(2g):=\sup_{J}\limsup_kZ^a_{\Lambda_k,C_J\cap\Lambda_k}(0,2g)$ is finite by (W$^\sharp$2), give the closing statement from the decoupling, under the walled comparison of Theorem~\ref{rot:thm:Kwsupply}. The decoupling is proved from the shell estimates at the scalar source $g'=g$: the two-sided bounds (SW1$^\circ$)$_{\rm int}$ at the source $g$ on the capped stopping-scale integral, with the exponent $q'(g)$ of \eqref{cpi:eq:bg} in place of the constants of (SW1$^\circ$)$_{\rm int}$; the disc bound (SW2$^{\min}$) at $g$, with the constant $2e^{4B}M_T(g)$, where $M_T(q)$ is the supremum over the spacings of the Wick-source expectation of the bare torus, finite at $2\sup q0$, a stopping scale and a shell mass $K$ with $\limsup_k\tail_k(K)\le\eta$, in the quantifier order of Theorem~\ref{rot:thm:shell}. Consequently the closing statement holds for $\Theta^\sharp$ at every $(f,g)\in\Kcs$, in the pointwise form at the box $Q_b$ of the source, with the single-level configuration $(J',\psi,n+K)\in\cAW(Q_b;g)$ as the witness. \end{proof} \subsection{The theorem}\label{rot:ssec:Aprime} \begin{theorem}[$E(3)$ invariance of King's state with general uncharged sources]\label{rot:thm:mainGS} Let $e>0$ and $\lambda,m,\mu_0>0$ be a point of King's range outside the set $E_W$ of Proposition~\ref{rot:thm:POS}, at block factor $5$, so that $s_*(e)>0$. Let $f_1,\dots,f_N$ be real $2$-forms and $h_1,\dots,h_N$ real functions in $C^\infty_c$, with $\int h_j$ arbitrary, and let $\Om(h)$ be the tube of Definition~\ref{rot:def:cone}. Then: \begin{enumerate} \item[(i)] King's state in the scheme $\Theta^\sharp$, \[ Z^\sharp_\infty\Big(\sum_jz_jf_j,\sum_jw_jh_j\Big)=\lim_J\lim_kZ^{a^\sharp,K}_{k,J}\Big(\sum_jz_jf_j,\sum_jw_jh_j\Big), \] exists locally uniformly on $\Om(h)\ni0$, is holomorphic there, and lies in $(0,\infty)$ at the real points of $\Om(h)$. \item[(ii)] For every Euclidean motion $x\mapsto Rx+b$, $R\in O(3)$, $b\in\R^3$, and all $(z,w)\in\Om(h)=\Om(\tau_bR\cdot h)$, \[ Z^\sharp_\infty\Big(\sum_jz_j\,\tau_bR\cdot f_j,\sum_jw_j\,\tau_bR\cdot h_j\Big)=Z^\sharp_\infty\Big(\sum_jz_jf_j,\sum_jw_jh_j\Big). \] \item[(iii)] Consequently every Schwinger function with curvature and Wick-square insertions, \[\partial_{z_1}\cdots\partial_{z_p}\partial_{w_1}\cdots\partial_{w_q}Z^\sharp_\infty|_{(z,w)=0},\] is $E(3)$-covariant, and the state at real sources $(f,h)$ with $\sup2h_+0$ and all positive couplings $e,\lambda,m$ (\Cref{lem:s14:CW}). Both sectors are proved on King's convention $\NDK$, on which the existence theorems \Cref{hyp:s03:EC,hyp:s03:W1} are posed. In the Wick sector (Leg~W), proved in its tube form, Form~(B), the tube bound \eqref{eq:s11:W} is proved on the comparison convention $\NDa$ (\Cref{def:s02:acube}). Its transfer to King's convention, the bound \eqref{eq:s13:WK}, and the fixed-cube limits on the tube for the K-family functionals $Z^{a,K}_{k,J}$ are \Cref{thm:s13:TWK}, on which the limit statements \Cref{thm:s11:W}(i),(ii), \Cref{lem:s11:TWtube}, \Cref{prop:s08:T} and \Cref{thm:s08:OS2WT} for King's state rest. After that the Wick sector is assembled on $\NDK$ (\Cref{prop:s13:WK}), and the comparison family enters through its LIM states, which coincide with King's tube object (\Cref{prop:s13:LIMK}). The torus theorems are \Cref{hyp:s03:T1,hyp:s03:T2}, and the couplings of the Wick leg lie in the coupling region of \Cref{hyp:s03:T3} (the set on which the sign clause \Tthree\ holds), which contains the explicit regions of \Cref{prop:s14:T3}; $\cK=\{(\mathrm if,g):g\ge0\}$ is the cone. The theorem announced in the introduction is \Cref{thm:s13:main} below. Throughout this section, let $e>0$, $\lambda,m,\lmu>0$, $\eps_k=5^{-k}$, $\CJ=[-MJ,MJ)^3$ with King's large-block integer $M\in50\N$, $M\ge M_{\min}$ (the objects below do not depend on such $M$, the tube limits of part~C coinciding where both are defined, \Cref{rem:s02:Mindep}), and let $\NDK_{k,J}$ be King's cube state. For a finite family $f=(f_1,\dots,f_N)$ of real $2$-forms in $C^\infty_c(\R^3;\Lambda^2)$ and $z\in\C^N$ we write $g(z)=\sum_j(\Re z_j)f_j$, and $S^K_n(f_1,\dots,f_n)=\partial_{z_1}\cdots\partial_{z_n}\ZKinf\Big(\sum_jz_jf_j,0\Big)\Big|_{z=0}$ for the Schwinger functions of King's state. \begin{definition}[the data of the Wick leg]\label{r13:def:s13wick} Let the couplings lie in the coupling region of \Cref{hyp:s03:T3}, the set on which \Tthree\ holds, which contains the explicit regions of \Cref{prop:s14:T3}, and fix an admissible scheme $a$ (\Cref{def:s03:adm}; $z=a^\sharp$ the reference scheme). Let $\sigma_+=s_1$, $H=s_1/16$, $\kappa=C_U/s_1$, $K_-$ be the constants of \eqref{eq:s11:constants} ($K_-$ built from $B_{\rm cell}$; for $a\ne z$ with $C_U$ replaced by $C_U+s_1\sup_k|z_k-a_k|$), $\sigma_0=H/4$ and $K'=\kappa\pi^2+K_-$. For a finite family $\cF=(f_1,\dots,f_n;h_1,\dots,h_p)$ of real $2$-forms and real scalars and $\zeta=(z,w)\in\C^{n+p}$ (the notation of \Cref{def:s02:tube}) the tube is \[ \Dtube=\{\norm{k_+(\zeta)}_\infty<\sigma_+/2,\ \norm{k_-(\zeta)}_\infty\frac12$. \item \emph{\OSo.} The $S^K_n$ are invariant under all translations of $\R^3$ and under the cubic group $\Oh$ about every point, $F$ transforming as a $2$-form. Moreover $\ZKinf$ is invariant under the Euclidean group $\Euc$ and the $S^K_n$ are Euclidean covariant. \item \emph{\OSt.} Reflection positivity about every coordinate plane $\{x_\nu=c\}$, $\nu\in\{1,2,3\}$, $c\in\R$: for finite families $\zeta_1,\dots,\zeta_N$ of real $2$-form sources supported in $\{x_\nu>c\}$ with arbitrary complex coefficients, $\sum_{i,j}\bar\gamma_i\gamma_j\ZKinf(\theta\zeta_i^*\sqcup\zeta_j,0)\ge0$ for all $\gamma\in\C^N$ (the matrix form for exponentials), and $S(\theta(P)P)\ge0$ for every polynomial $P$ in the smeared curvature with sources supported in $\{x_\nu>c\}$ (the polynomial form). The same holds about every plane $\{x\cdot n=c\}$, $n$ a unit vector. \item \emph{\OSth.} $S^K_n$ is real for real test functions, symmetric, and zero for odd $n$. \item \emph{The measure.} For every even $r\ge14$ there is exactly one Borel probability measure $\mu^{K,H}$ on $\cH^{-r}(\R^3;\Lambda^2)$ whose moments on $C^\infty_c$ are the $S^K_n$; its characteristic functional is $\ZKinf(\mathrm i\,\cdot,0)$, extended by continuity to $\cS$; its image $\mu^K_S$ on $H^{-r}_{\rm loc}(\R^3;\Lambda^2)$ is the unique Borel probability measure there with this characteristic functional, and the unique one with these moments. For every real $\chi\in C^\infty_c(\R^3)$ and every box $Q$ containing $\supp\chi$ in its interior, the laws of the localised lattice curvature fields $\chi\Phi^K_{k,J}$ under $\NDK_{k,J}$ on $H^{-r}_{\rm per}(Q;\Lambda^2)$ converge weakly, along the full sequence $k\to\infty$ at every fixed $J\ge J_0(\chi)$ and then as $J\to\infty$, to the law of $\chi\Phi$ under $\mu^K_S$. Nothing is claimed about the laws of the unlocalised lattice fields at fixed $J$. \item \emph{Weak \OSf.} $\mu^K_S$ is $\Cub$-invariant and is an integral $\int\mu_\xi\,d\nu(\xi)$ of translation-ergodic, reflection-positive, $\theta_j$-invariant components which cluster in the mean and under diagonal separation, have a unique joint vacuum in each coordinate direction, and satisfy the factorial growth bound $|S^\xi_N(u_1,\dots,u_N)|\le\sigma^\xi_N\prod_i\norm{u_i}_{r+2}$, $\sigma^\xi_N\le\alpha_\beta(\xi)(N!)^\beta$ for every $\beta>1$, with component constants (no tube bound); there is an $\Oh$-invariant decomposition into mixtures of $k\in\{1,2,3,6\}$ ergodic measures; and $\nu$-almost every component satisfies \OSf\ in the weak form (ergodicity under $\R^3$). Moreover $\nu$-almost every component is $\Euc$-invariant, reflection positive about every plane, and satisfies \OSf\ in each of its forms for every time direction, with $\ker H_n=\C\Omega_n$ in every direction $n$; its moments are real and symmetric. The vanishing of a component's odd moments is not claimed. \end{enumerate} \smallskip\noindent\textbf{C. Leg~W, Form~(B), on King's convention.} Let the couplings lie in the coupling region of \Cref{hyp:s03:T3} and let $a$ be an admissible scheme, with the constants $\sigma_+,H,\kappa,K_-,\sigma_0,K'$ and the tube $\Dtube$ of \Cref{r13:def:s13wick}. Then for every finite family $\cF=(f_1,\dots,f_n;h_1,\dots,h_p)$ of real $2$-forms and real scalars the tube object $G^K_\infty$ exists locally uniformly on the tube and is holomorphic there; on the Wick tube of \Cref{r13:def:s08:domains} the limit $Z^{a,K}_{\infty,T}=\lim_J\lim_kZ^{a,K}_{k,J}$ exists and equals $G^K_\infty$ where both are defined. The joint Schwinger functions $S_{n,p}$ of the pair $(F,W^a)$ satisfy: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item \emph{\OSz.} $|G^K_\infty(\zeta)|\le\exp(\norm{g(\zeta)}_2^2+\kappa\norm{k_+(\zeta)}_{L^1}+K_-\wnorm{k_-(\zeta)})$ on $\Dtube$, and the same bound with lattice norms holds for $Z^{a,K}_{k,J}$ uniformly in $k\ge0$ and $J\ge J_1(R)$ when the supports lie in $Q_R$; for real test functions \[ |S_{n,p}(f_1,\dots,f_n;h_1,\dots,h_p)|\le e^{K'\sigma_0}(\sqrt2\,e)^n(n!)^{1/2}(e/\sigma_0)^p\,p!\prod_{i=1}^n\norm{f_i}_2\prod_{j=1}^p\wnorm{h_j}, \] with the factor $2^{n+p}$ for complex test functions; the $S_{n,p}$ are tempered distributions and \Ezz\ holds with $s=4$ and every $\beta>1$. The constants $\sigma_+,H,\kappa,K_-$ depend on $(e,\lambda,m,\lmu,M,a)$, on the fixed auxiliary choices $M_1$, $\chi_0$ and the amplitude $s_1\in(0,\sigma_1)$ of \Itwo, and on nothing else; they are not uniform in $\lmu$. \item \emph{\OSo.} $G^K_\infty(\tau_bR\cdot\zeta)=G^K_\infty(\zeta)$ on $\Dtube$ for every $b\in\R^3$ and $R\in\Oh$, $F$ transforming as a $2$-form and $W^a$ as a scalar (the transformed family has the same tube); hence the $S_{n,p}$ are invariant under translations and under $\Oh$ about every point. No Euclidean clause is claimed. \item \emph{\OSt.} For finite families $\zeta_1,\dots,\zeta_N$ of real sources supported in $\{x_\nu>c\}$ with complex coefficients of amplitude $\amp(\zeta_i)<\sigma_W\defeq\sigma_+/8$ (in particular all purely imaginary Wick coefficients with arbitrary complex curvature coefficients), $\sum_{i,j}\bar\gamma_i\gamma_jZ^{a,K}_{\infty,T}(\theta\zeta_i^*\sqcup\zeta_j)\ge0$ about every coordinate plane; and the polynomial form $S(\theta(P)P)\ge0$ for polynomials in the smeared $F$ and $W^a$ with sources in $\{x_\nu>c\}$. \item \emph{\OSth.} The $S_{n,p}$ are symmetric multilinear in the $f$'s and in the $h$'s, real for real test functions, and zero for odd $n$. \item[\textup{(v),(vi)}] \emph{The measure and weak \OSf.} Clauses (v) and (vi) of Leg~C, without the Euclidean sentence of (vi) (the sentence beginning ``Moreover''; that is, \Cref{thm:s09:S2K}(a),(b) for the joint field), hold for the joint field on $\cH^{-r}(\R^3;\Lambda^2\oplus\R)$ and $H^{-r}_{\rm loc}(\R^3;\Lambda^2\oplus\R)$, every even $r\ge14$, with moments $S_{n,p}$, characteristic functional $(u,v)\mapsto G^K_\infty(\mathrm iu,\mathrm iv)$ on real test functions (purely imaginary coefficients have $k(\zeta)=0$ and lie in the tube), the component growth bound with the constants of (i), and the local convergence of the laws of the scheme-shifted lattice fields $\chi(F_{\Lk},W^a_{\Lk})$ under $\NDK_{k,J}$ along the full sequence $k\to\infty$ at every fixed $J\ge\max(J_0(\chi),J_1(R_\chi))$, then $J\to\infty$. No Euclidean clause is claimed. Two admissible schemes $a,a'$ give objects differing by the shift $W\mapsto W-\kappa_{a'a}\int(\cdot)$ of the Wick component, $\kappa_{a'a}=\lim_k(a'_k-a_k)$; joint cumulants of order $\ge2$ are scheme-free. \end{enumerate} \smallskip\noindent\textbf{D. The comparison family.} Under the conditions of~C, every increasing sequence of spacings has a continuum subsequence $\mathbf{k}$ for the Wick sector of the comparison family $\NDa$ (\Cref{def:s12:contsub}), and for every such $\mathbf{k}$ and every family the LIM state $Z^{a,\mathbf{k}}_\infty=\lim_J\lim_{k\in\mathbf{k}}Z^a_{k,J}$ exists on $\Dtube$ and equals $G^K_\infty$ (\Cref{prop:s13:LIMK}). Hence clauses (i)--(vi) of~C hold for the Wick-sector LIM states of the comparison family, which coincide with King's tube object and do not depend on $\mathbf{k}$; in (v) the laws of the localised lattice fields $\chi(F_{\Lk},W^a_{\Lk})$ under $\NDa_{k,J}$ converge along $\mathbf{k}$ at every fixed $J$ with $MJ>R(\chi)+\ell_c$, then as $J\to\infty$, to the law of the joint field under the measure of~C(v). \end{theorem} \begin{proof} \emph{A.} \Cref{thm:s05:CK}(b): by \Cref{hyp:s03:EC} the two limits exist locally uniformly on $\C^N$ and are entire, $|\ZKJ|,|\ZKinf|\le e^{\norm{g(z)}_2^2}$, $J\mapsto\ZKJ(\mathrm it)$ is nondecreasing with values in $[0,1]$, and $\ZKinf(x)\ge1$ at real $x$; realness at real $z$ holds because the integrand $e^{\sum_jz_jF_{\Lk}(f_j)}$ is then real under the probability measure $\NDK_{k,J}$, and realness passes to the limits; the symmetry $(A,\vartheta)\mapsto(-A,-\vartheta)$ of every $\NDK_{k,J}$ gives $\ZKinf(-x)=\ZKinf(x)$ at real $x$. Each lattice functional depends on the family only through its total source, hence so do the limits. The bound is absolute (\Cref{rem:s05:absolute}). \emph{B.} (i) is \Cref{thm:s05:CK}: the finite-volume bound (a) with $\min\{\norm{g}^2_{2,k},\frac12\lmu^{-2}\norm{\dstar_kg}^2_{2,k}\}$ at every $(k,J)$; the two limit bounds are (b), the second carried from the lattice by $\norm{\dstar_kg}_{2,k}\to\norm{\dstar g}_2$, \eqref{eq:s02:riemann}; the Schwinger bounds, the extension to $\cS$, the factor $2^n$ and \Ezz\ (c). (ii): translations are \Cref{cor:s06:TK} (the lattice sandwich \Cref{lem:s06:TK}, the two limits, the identity theorem, density); the cubic group about every point is \Cref{thm:s06:S1K}; the Euclidean clause is \Cref{thm:s06:O3}, proved from \Cref{rot:thm:main} and \Cref{cor:s04:G2} (the rotation theorem \Cref{hyp:s03:RC} at every point of King's range: \Cref{rot:thm:curv} off the exceptional set $E_W$ of \Cref{rot:thm:POS,rot:def:cone}, Theorem~\ref{cpi:ew:EWC} and Remark~\ref{cpi:ew:rightcont} at the points of $E_W$) with \Cref{cor:s06:TK}. (iii): the matrix form about every coordinate plane and the polynomial form are \Cref{thm:s08:OS2C} with \Cref{prop:s08:poly}; a plane $\{x\cdot n=c\}$ is the image of a coordinate plane under a Euclidean motion $\gamma$, and by (ii) $\ZKinf(\theta_\gamma\zeta_i^*\sqcup\zeta_j,0)=\ZKinf(\gamma^{-1}\cdot(\theta_\gamma\zeta_i^*\sqcup\zeta_j),0)$ is a Gram entry about the coordinate plane for the sources $\gamma^{-1}\cdot\zeta_i$, supported in the corresponding coordinate half-space. (iv) is \Cref{prop:s08:OS3} for $\ZKinf(\cdot,0)$ (through \Cref{thm:s08:OS2C}; charge conjugation gives the vanishing for odd $n$). (v) is \Cref{thm:s09:S2K}(a), including the local convergence of the lattice laws along the full sequence, with \Cref{rem:s09:S2K} for the unlocalised fields; (vi) is \Cref{thm:s09:S2K}(b) and (c), whose $O(3)$ hypothesis is (ii)'s Euclidean clause; the disclaimer on odd moments is that of \Cref{thm:s09:S2K}(c). \emph{C.} Under \Tthree, \Cref{prop:s10:U} gives \Itwo\ with amplitude $s_1\in(0,\sigma_1)$ and constant $C_U$ from the torus theorems \Cref{hyp:s03:T1,hyp:s03:T2}; for $a\ne z$ the difference $z_k-a_k$ converges by admissibility, so $\sup_k|z_k-a_k|<\infty$ and $C_U$ is replaced as stated in \Cref{r13:def:s13wick} (\Cref{hyp:s12:SCH}); \Ithree\ is \Cref{prop:s10:RP}, the correlation inequalities are \Cref{lem:s02:BFS}, and the cell constant $B^{(a)}_{\rm cell}$ is finite (\Cref{hyp:s12:Bcell}). \emph{(i)}: \Cref{thm:s13:TWK} gives the bound \eqref{eq:s13:WK} for every $k\ge0$ and $J\ge J_1(R)$, the fixed-cube limits $G^K_J$ and the cube limit $G^K_\infty$ locally uniformly on $\Dtube$, their holomorphy, the bound in the limit, the Schwinger bounds, the factor $2^{n+p}$ and \Ezz, with the stated dependence of the constants and no uniformity in $\lmu$. On the Wick tube $T_{\cF'}$ of the enlarged family of \Cref{def:s08:enlarged,r13:def:s08:domains}, whose scalar members are nonnegative and whose points have $2\norm{k_1}_\infty<\sigma_+/2$, the fixed-cube limits of $Z^{a,K}_{k,J}$ exist locally uniformly by \Cref{thm:k02:W1}(i), whose domain $\{\sup_x2k_10 \] for $0<\abs{M-m}<\delta$, because $\alpha_\infty$ is real-analytic and not constant, so that its other global minimisers, if any, are isolated. Both monotonicities of Lemma~\ref{cpi:ew:mono} raise $\langle\cos F(u)\rangle$ when terms are added, cube growth adding hopping and photon gradient terms and a smaller bare coefficient adding weight to $r^2$; so along King points near $m$ King's state is a non-increasing function of the bare coefficient. What is used in finite volume is the one-sided statement of Theorem~\ref{cpi:ew:EWC}(a), proved through the dead-shell torus. On $E_W$ right-continuity is exactly what is needed, and nothing to the left of $m$ is used; only the limit $M\downarrow m$ is claimed (on $E_W$, $t_\infty>0$ on both sides of $m$, so both one-sided limits approach $m$ from the side of the larger bare coefficient). \end{remark} \space% % \begin{remark}[the Euclidean clause]\label{rem:s13:rotation} The Euclidean clauses of B(ii), (iii), (vi) rest on the rotation theorem \Cref{hyp:s03:RC}, proved in the rotation part (\Cref{rot:thm:main}): the walled tori on the two lattices have the same continuum limits (\Cref{rot:hyp:K}), King's state is the lower envelope of the walled tori (\Cref{rot:hyp:TIGHT}), and that closing estimate (the lower-envelope inequality) comes from the heavy-shell estimates on King's block-spin flow (\Cref{rot:hyp:RES,rot:rem:shellbounds}), both the comparison and the closing estimate holding off the exceptional set $E_W$ of \Cref{rot:thm:POS,rot:def:cone}, the points of $E_W$ being reached by right-continuity in the scalar mass (Theorem~\ref{cpi:ew:EWC}, Remark~\ref{cpi:ew:rightcont}); the Euclidean clause of the Wick sector below the threshold is Theorem~\ref{rot:thm:mainGS}; above it, \Cref{hyp:s03:RW} is open. \end{remark} \begin{theorem}[the Wick tube on King's convention]\label{thm:s13:TWK} Let the couplings lie in the coupling region of \Cref{hyp:s03:T3} and let $a$ be an admissible scheme. Let $\sigma_+=s_1$, $H$, $\kappa$ be the constants of \eqref{eq:s11:constants} and let $K_-$ be the constant there, computed from the cell constant $B^{(a)}_{\rm cell}$ of the comparison convention (\Cref{hyp:s12:Bcell}), which is finite. Then for every finite family of real sources with supports in $Q_R$, every $k\ge0$, every $J\ge J_1(R)$ and every $\zeta\in\Dtube$, \begin{equation}\label{eq:s13:WK} |Z^{a,K}_{k,J}(\zeta)|\ \le\ Z^{a,K}_{k,J}(\Re\zeta)\ \le\ \exp\big(\norm{g(\zeta)}_{2,k}^2+\kappa\norm{k_+(\zeta)}_{1,k}+K_-\wnorm{k_-(\zeta)}\big), \end{equation} the bound \eqref{eq:s11:W} for the K-family functionals with the constants of the comparison convention; and the fixed-cube limits $G^K_J(\zeta)=\lim_{k\to\infty}Z^{a,K}_{k,J}(\zeta)$ exist for every $\zeta\in\Dtube$, locally uniformly, which gives \Cref{thm:s11:W}(i),(ii): $G^K_J$ and $G^K_\infty=\lim_JG^K_J$ exist locally uniformly on $\Dtube$ and are holomorphic, the bound \eqref{eq:s13:WK} passes to the limits, and the Schwinger bounds and \Ezz\ of \Cref{thm:s13:main}C(i) hold for $G^K_\infty$; moreover $G^K_\infty>0$ at the real points of $\Dtube$ and $G^K_J\le G^K_\infty$ at its cone points. \end{theorem} % % % \begin{proof} Throughout, $\cF=(f_1,\dots,f_n;h_1,\dots,h_p)$ is a family of real sources with supports in $Q_R$, $J\ge J_1(R)$, so that $MJ\ge R+\ell_c\ge R+10$: every source lies at $\ell^\infty$-distance $>1$ from $\R^3\setminus\CJ$, the support condition \eqref{eq:s04:S} holds for $\XJ$, and $J\ge J_0(\cF)$. For $\zeta\in\Dtube$ write $g=g(\zeta)$ and $k=k(\zeta)=k_+-k_-$ (\Cref{def:s02:tube}); by \Cref{lem:s02:E}(E3) the sampled sources satisfy $0\le2k_+\le s_1$ and $0\le2k_-\le H$ on $\Lk$. The statements used are: \Itwo\ with amplitude $s_1\in(0,\sigma_1)$ and constant $C_U$, from the sign clause \Tthree\ of the coupling region and the torus theorems \Cref{hyp:s03:T1,hyp:s03:T2} through \Cref{prop:s10:U}, for $a\ne z$ with the constant $C_U+s_1\sup_k|z_k-a_k|$, finite by \Cref{def:s03:adm} (\Cref{hyp:s12:SCH}); \Ithree, which is \Cref{prop:s10:RP}; the correlation inequalities \Ifour, which are \Cref{lem:s02:BFS}; and the cell constant $B^{(a)}_{\rm cell}<\infty$ of the comparison convention (\Cref{hyp:s12:Bcell}). Under these, \Cref{cor:s11:W2prime} and \Cref{prop:s11:W8} hold on the comparison convention and \Cref{lem:s04:E}(d) on King's. \emph{Step 1 (modulus).} \Cref{lem:s02:alg}(b) holds for every state of \Cref{def:s02:acube}, killed sites included: $|Z^{a,K}_{k,J}(\zeta)|\le Z^{a,K}_{k,J}(\Re\zeta)$. It therefore suffices to bound $Z^{a,K}_{k,J}$ at the real point $\Re\zeta$, where $Z^{a,K}_{k,J}(\Re\zeta)=N^{a,K}(g,k)\defeq e^{-a_k\sS(k)}\avK{e^{F(g)+\Wick k}}$. \emph{Step 2 (curvature off).} Let $\langle\cdot\rangle_{K,k}$ be King's cube with each single-site measure at an alive site multiplied by $e^{\eps^3k(x)r_x^2}$; it is a BFS measure (the factor keeps $\int e^{\alpha r^2}d\rho_x<\infty$ because $\lambda>0$, and the killed sites are unchanged), and $N^{a,K}(g,k)=N^{a,K}(0,k)\,\langle e^{F(g)}\rangle_{K,k}$. Its Gaussian is the Neumann photon Gaussian of $\XJ$, the same for both conventions, with covariance $C$ as in \Cref{lem:s05:C1}. By \eqref{eq:s04:S} every plaquette with $g\ne0$ has its corners in $\XJ$, so $F(g)=\ip vA_k$ with $v=\dstar_kg$, and \Cref{lem:s02:GD} (with $s=1$) and \Cref{lem:s05:C1} give $\langle e^{F(g)}\rangle_{K,k}\le e^{\ip v{Cv}_k/2}\le e^{\norm g_{2,k}^2}$. This is the argument of \Cref{thm:s05:CK}(a) in the presence of the real scalar source $k$. Hence $N^{a,K}(g,k)\le e^{\norm g_{2,k}^2}N^{a,K}(0,k)$. \emph{Step 3 (the split).} \Cref{lem:s02:alg}(d) at $f=0$ with $k_1=k_+$, $k_2=k_-$: $N^{a,K}(0,k_+-k_-)\le N^{a,K}(0,2k_+)^{1/2}N^{a,K}(0,-2k_-)^{1/2}$. \emph{Step 4 (the positive leg on King's convention).} Put $q\defeq2k_+$, a lattice function with $0\le q\le s_1$ supported in $Q_R\cap\Lk\subset\alive^K_J$. The observable $e^{\sum_x\eps^3q(x)r_x^2}$ is $\Theta_{0,q'}$ with $q'=\eps^3q\ge0$, of class $\cI$, and \eqref{eq:s04:S} holds for $\XJ$; Step~1 of the proof of \Cref{lem:s04:G} (the kill step on $\cI$) gives $\avK{\Theta_{0,q'}}\le\ava{\Theta_{0,q'}}$, and the multiplier $e^{-(a_k+2\sW_\eps)\sS(q)}$ is one positive constant for both states (\Cref{lem:s02:alg}(a)). Hence \[ N^{a,K}_{k,\CJ}(0,q)\ \le\ N^a_{k,\CJ}(0,q)\ \le\ e^{\kappa\norm q_{1,k}} \] by \Cref{cor:s11:W2prime}, whose hypotheses \Itwo, \Ithree\ hold (this is the cone instance of \Cref{rem:s04:F}, as in \Cref{rem:s11:Kconvention}). So $\frac12\ln N^{a,K}(0,2k_+)\le\kappa\norm{k_+}_{1,k}$. \emph{Step 5 (the negative leg on King's convention).} Put $q\defeq2k_-$, a lattice function with $0\le q\le H$ supported in $Q_R$, and $J\ge J_1(R)$. \Cref{lem:s04:E}(d) gives $\ln N^{a,K}_{k,\CJ}(0,-q)\le K_-\wnorm q$ with the constant $K_-$ of \eqref{eq:s11:constants}, built from the cell constant $B^{(a)}_{\rm cell}$ of the comparison cell, which is finite by \Cref{hyp:s12:Bcell}: by \Cref{lem:s04:E}(c) every cell on which one of the cutoff sources $8\chi_sq$ does not vanish is a full cell inside $\CJ$ containing no killed site, so each factor of the wall insertion of \Cref{lem:s04:E}(a) that carries a source is a comparison-cell functional, to which \Cref{lem:s11:W4,lem:s11:W5} apply as stated, and the sum over cells is \Cref{lem:s11:W67}; the scheme multiplier factorises over the cells (\Cref{lem:s02:alg}(a)). The direction of the kill matters here and is the content of \Cref{lem:s04:E}(e). Hence $\frac12\ln N^{a,K}(0,-2k_-)\le\frac12K_-\wnorm{2k_-}=K_-\wnorm{k_-}$. \emph{Step 6 (assembly).} Steps 1--5 give, for every $k\ge0$, $J\ge J_1(R)$ and $\zeta\in\Dtube$, \[ |Z^{a,K}_{k,J}(\zeta)|\le Z^{a,K}_{k,J}(\Re\zeta)\le\exp\big(\norm{g(\zeta)}_{2,k}^2+\kappa\norm{k_+(\zeta)}_{1,k}+K_-\wnorm{k_-(\zeta)}\big), \] which is \eqref{eq:s13:WK}. The constants are those of \eqref{eq:s11:constants}, with $B^{(a)}_{\rm cell}$ in $K_-$ and, for $a\ne z$, $C_U+s_1\sup_k|z_k-a_k|$ in $\kappa$. \emph{Step 7 (the fixed-cube limits).} Each $Z^{a,K}_{k,J}$ is entire in $\zeta\in\C^{n+p}$: it is a finite-dimensional integral with Gaussian decay in $A$ ($\lmu>0$) and $e^{-\lambda\eps^3r^4}$ decay in $r$ at every alive site, the killed sites carrying no variables, and differentiation under the integral sign is justified. On a compact $Q\subset\Dtube$ the family $\{Z^{a,K}_{k,J}\}_{k\ge0}$ is uniformly bounded, by \eqref{eq:s13:WK} and \Cref{lem:s02:E}(E1)--(E3): $\norm g_{2,k}^2\le C_0\fournorm g^2$, $\norm{k_+}_{1,k}\le C_0'\wnorm{k_+}$ and $\wnorm{k_\pm}\le\sum_j|\Re w_j|\wnorm{h_j}$, all continuous in $\zeta$. At every real point of $\Dtube$ the sequence converges: there $2\norm{k_+}_\infty<\sigma_+=s_1=s_W$ and the negative part is unrestricted, so \Cref{hyp:s03:W1}, which is posed on King's convention (\Cref{cor:k02:W1}(a)), gives the limit in $(0,\infty)$ for the reference scheme; for the admissible scheme $a$, $Z^{a,K}_{k,J}=e^{-(a_k-z_k)\sum_jw_j\sS_k(h_j)}Z^{z,K}_{k,J}$ with $a_k-z_k$ convergent and $\sS_k(h_j)\to\int h_j$ (\Cref{def:s03:adm}, \Cref{rem:k02:scheme}). The real points of $\Dtube$ form a non-empty open subset of $\R^{n+p}$ (\Cref{def:s02:tube}), a set of uniqueness in the connected open $\Dtube$; by Montel's and Vitali's theorems (the fact (C3) stated before \Cref{thm:s11:W}), $Z^{a,K}_{k,J}\to G^K_J$ locally uniformly on $\Dtube$, $G^K_J$ is holomorphic there, and \eqref{eq:s13:WK} passes to the limit with $\norm g_{2,k}\to\norm g_2$, $\norm{k_+}_{1,k}\to\norm{k_+}_{L^1}$ ($k_+$ is continuous with compact support) and the sampled $\wnorm{k_-}$ at most the continuum one (E3): \begin{equation}\label{eq:s13:WKlim} |G^K_J(\zeta)|\le\exp\big(\norm{g(\zeta)}_2^2+\kappa\norm{k_+(\zeta)}_{L^1}+K_-\wnorm{k_-(\zeta)}\big)\qquad(\zeta\in\Dtube,\ J\ge J_1(R)), \end{equation} a bound free of $J$. At real points $G^K_J>0$ (\Cref{hyp:s03:W1}). \emph{Step 8 (the cube limit on a sub-tube).} Split $h_j=h_j^+-h_j^-$ with $h_j^-\defeq\norm{h_j}_\infty\chi_j$, $\chi_j\in C^\infty_c$, $0\le\chi_j\le1$, $\chi_j=1$ on $\supp h_j$, $\supp\chi_j\subset Q_{R+1/2}$, and $h_j^+\defeq h_j+h_j^-\ge0$. For the split family $\cF''=(f;h^+,h^-)$ put $F^K_{k,J}(z,w^+,w^-)\defeq Z^{a,K}_{k,J}(\sum_iz_if_i,\sum_j(w_j^+h_j^++w_j^-h_j^-))$ and $\cD''\defeq\{(z,w^+,w^-)\in\C^{n+2p}:|\Re w_j^\pm|<\delta_j\}$ with $\sum_j\delta_j(\norm{h_j^+}_\infty+\norm{h_j^-}_\infty)0$ with $-\epsilon\zeta\in\Dtube$. The function $\varphi_J(t)\defeq\ln G^K_J(t\zeta)$ is finite and convex on $[-\epsilon,1]$ (a pointwise limit of the convex functions $\ln Z^{a,K}_{k,J}(t\zeta)$, \Cref{lem:s02:alg}(b), with positive limits at real points) and $\varphi_J(0)=0$, so $\varphi_J(1)\ge-\varphi_J(-\epsilon)/\epsilon\ge-C_\epsilon/\epsilon$, $C_\epsilon$ the exponent of \eqref{eq:s13:WKlim} at $-\epsilon\zeta$; hence $G^K_\infty(\zeta)\ge e^{-C_\epsilon/\epsilon}>0$. At the cone points of $\Dtube$ ($z\in(\mathrm i\R)^n$, $w\in\R^p$, $k(\zeta)\ge0$), \Cref{cor:s04:G1} at every $k$ and $k\to\infty$ give $G^K_J\le G^K_{J+1}\le G^K_\infty$. \emph{Step 11 (the Schwinger bounds and \Ezz).} Put $S^K_{n,p}(f;h)\defeq\partial^{(n)}_z\partial^{(p)}_wG^K_\infty(0)$, $\sigma_0=H/4$, $K'=\kappa\pi^2+K_-$. Normalise $\norm{f_i}_2=1$ and let $\rho_j\defeq\sigma_0/(p\wnorm{h_j})$. On the polydisc $\{|w_j|\le\rho_j\}\times\C^n$ we have $\norm k_\infty\le\sum_j|\Re w_j|\norm{h_j}_\infty\le\sum_j\rho_j\wnorm{h_j}=\sigma_01$: this is \Ezz\ with $s=4$ and every $\beta>1$, with the constants of \Cref{thm:s13:main}C(i). \end{proof} \space% % \begin{proposition}[the Wick sector on King's convention]\label{prop:s13:WK} Let the couplings lie in the coupling region of \Cref{hyp:s03:T3} and let $a$ be an admissible scheme. Then, with the constants of \Cref{r13:def:s13wick}: \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*,nosep] \item $G^K_J\defeq\lim_kZ^{a,K}_{k,J}$ exists locally uniformly on $\Dtube$ for every $J\ge J_1(R)$, $G^K_\infty\defeq\lim_JG^K_J$ exists locally uniformly on $\Dtube$, both are holomorphic, $|G^K_\infty(\zeta)|\le\exp(\norm{g(\zeta)}_2^2+\kappa\norm{k_+(\zeta)}_{L^1}+K_-\wnorm{k_-(\zeta)})$, and the Schwinger bounds and \Ezz\ of \Cref{thm:s13:main}C(i) follow. \item $G^K_\infty(\tau_bR\cdot\zeta)=G^K_\infty(\zeta)$ on $\Dtube$ for all $b\in\R^3$, $R\in\Oh$, $W^a$ a scalar. \item For every family $\cF'$ with nonnegative scalar members the Wick-tube object \[ Z^{a,K}_{\infty,T}\defeq\lim_J\lim_kZ^{a,K}_{k,J} \] exists locally uniformly on $T_{\cF'}$ and equals $G^K_\infty$ on $T_{\cF'}\cap\Dtube$; reflection positivity about every coordinate plane holds in the matrix form at amplitude $<\sigma_W$ and in the polynomial form, and \OSth, for $Z^{a,K}_{\infty,T}$, $G^K_\infty$ and their Schwinger functions. \item \emph{The measure and weak \OSf\ for the joint field.} For every even $r\ge14$ there is exactly one Borel probability measure on $\cH^{-r}(\R^3;\Lambda^2\oplus\R)$ whose moments on $C^\infty_c$ are the $S_{n,p}$; its characteristic functional is $(u,v)\mapsto G^K_\infty(\mathrm iu,\mathrm iv)$ on real test functions, and its image on $H^{-r}_{\rm loc}(\R^3;\Lambda^2\oplus\R)$ is the unique Borel probability measure there with this characteristic functional, and the unique one with these moments. For every real $\chi\in C^\infty_c(\R^3)$ the laws of the localised scheme-shifted lattice fields $\chi(F_{\Lk},W^a_{\Lk})$ under $\NDK_{k,J}$ converge weakly along the full sequence $k\to\infty$ at every fixed $J\ge\max(J_0(\chi),J_1(R_\chi))$, then as $J\to\infty$, to the law of the localised joint field under this measure. The measure is $\Cub$-invariant and is an integral of translation-ergodic, reflection-positive, $\theta_j$-invariant components which cluster in the mean and under diagonal separation, have a unique joint vacuum in each coordinate direction and satisfy the factorial growth bound of \Cref{thm:s13:main}B(vi) with the constants of (a); there is an $\Oh$-invariant decomposition into mixtures of $k\in\{1,2,3,6\}$ ergodic measures, and almost every component is ergodic under $\R^3$ (weak \OSf). No Euclidean clause is claimed. These are clauses (v) and (vi) of \Cref{thm:s13:main}C. \end{enumerate} \end{proposition} \begin{proof} Parts (a) and (c) are proved below. Parts (b) and (d) are proved by running the cited proofs of the curvature sector on the joint field; the paragraphs (b) and (d) below give the steps of these proofs and the result each step uses. (a) is \Cref{thm:s13:TWK}, whose proof contains the fixed-cube limits (Step~7), the cube limit on the connected tube with the cube monotonicity of \Cref{cor:s04:G1} in place of the first inequality of \Cref{lem:s02:cut} (Steps 8--9), the positivity at real points and the cone monotonicity (Step~10), and the Schwinger bounds by Cauchy's estimates on a polydisc in the tube (Step~11). (b) The steps of \Cref{thm:s06:S1K} with scalar sources. Coordinate permutations are exact on $\NDK$ for $(f,h)$ by \Cref{lem:s06:PK}. The reflection sandwich is \Cref{lem:s04:C}(b) at the cone points of the tube: for $(f,g)\in\cK$ with $\norm g_\infty<\sigma_+/2$ the point $(\mathrm i,1)$ lies in $\Dtube$, so (a) provides the three $k$-limits and the two $J$-limits of the two-step sandwich $C^K_{J\mp2}$. Conjugation and group closure on the cone are unchanged. Off the cone, the identity theorem on the split family's tube of \Cref{lem:s11:TWtube}, where (a) gives the limits, extends the identity from the cone points to the connected tube; so \Cref{lem:s04:C} is needed on $\cK$ only, and no class-$\cD$ form of it enters (the proof splits $h=h^+-h^-$ and never applies the sandwich to a negative scalar source). Translations are \Cref{lem:s06:TK}, stated on $\cK$ and applied at the cone points of the tube with (a) for the limits, then the continuity route of \Cref{cor:s06:TK} with the Wick-sector second-moment bound, which Cauchy's estimates give from \eqref{eq:s11:W} on $\NDK$; the translated family needs $J\ge\max(J_0+1,J_1(R+M+\frac12))$. Invariance about every point and differentiation concern the limit object. (c) is \Cref{thm:s08:OS2WT} with \Cref{prop:s08:T}, both stated for King's state. The convention-sensitive steps used there are: in the sandwich of the $J$-limit the upper box is \Cref{lem:s04:D} (the boxes $\CJ$, $C_{J+2n}$ as stated there) and the lower end carries the kill step of \Cref{rem:s04:F}; the positive-leg bound \Cref{cor:s11:W2prime} holds on $\NDK$ by \Cref{rem:s04:F} prepended to \Cref{prop:s11:W1prime} (\Cref{rem:s11:Kconvention}); the cube monotonicity in both classes is \Cref{cor:s04:G1}; the fixed-cube limits on $T_{\cF'}$ are \Cref{thm:k02:W1}(i), whose domain $\{\sup_x2k_1R+\frac12+\ell_c$. At a point of $\Omega''_+$ the curvature coefficients are purely imaginary and the total scalar source is nonnegative, so the observable is of class $\cI$, and the support condition \eqref{eq:s04:S} holds for $\XJ$; \Cref{lem:s04:G} (with its clause for the scheme functionals) gives, at every $k$, \[ F^K_{k,J}\ \le\ F^{(a)}_{k,J}\ \le\ F^K_{k,J+1}\qquad\text{on }\Omega''_+ . \] Along $\mathbf{k}$ the two ends converge, by \Cref{thm:s13:TWK} for the family $\cF''$ (whose fixed-cube limits exist along the full sequence, hence along $\mathbf{k}$), to the fixed-cube limits $F^K_J$, $F^K_{J+1}$ of Step~8 of that proof, and the middle term converges to the fixed-cube limit $F^{(a),\mathbf{k}}_J$ of the comparison family along $\mathbf{k}$ (\Cref{def:s12:contsub} for $\cF''$, whose tube contains $\cD''$ and whose threshold is $MJ>R(\cF'')+\ell_c$, $R(\cF'')\le R+\frac12$). Letting $J\to\infty$, the two ends tend to $F^K_\infty$ (Steps 8--9 of the proof of \Cref{thm:s13:TWK}) and the middle term to the LIM state $F^{(a),\mathbf{k}}_\infty$ of $\cF''$ (\Cref{lem:s12:LIM1}). Hence $F^{(a),\mathbf{k}}_\infty=F^K_\infty$ on $\Omega''_+$, a non-empty open subset of the maximal totally real subspace $(\mathrm i\R)^n\times\R^{2p}$, and both functions are holomorphic on the connected open set $\cD''$; by the identity theorem (the fact (C2) stated before \Cref{thm:s11:W}) they agree on $\cD''$. At $w^+_j=w_j$, $w^-_j=-w_j$ the lattice functionals of $\cF''$ coincide exactly with those of $\cF$ at $\zeta$, on both conventions and at every $k$ (equal total sources); so $Z^{a,\mathbf{k}}_\infty=G^K_\infty$ on $\cD'=\{\zeta:|\Re w_j|<\delta_j\}\subset\Dtube$. Both are holomorphic on the connected $\Dtube$ and agree on the open set $\cD'$, hence on $\Dtube$. The last clause is \Cref{def:s12:LIMstate}: every subsequential fixed-cube limit on the tube is the fixed-cube limit along some continuum subsequence. \end{proof} \space% % \begin{remark}[the three amplitude constants]\label{rem:s13:amplitudes} Three amplitudes occur in the Wick sector, all built from $s_1$: the matrix form of \OSt\ is proved at amplitude $\amp(\zeta_i)<\sigma_W=\sigma_+/8$ on the Wick tube (\Cref{thm:s08:OS2WT}); the tube $\Dtube$ has half-widths $\sigma_+/2$ for the positive and $H/2=\sigma_+/32$ for the negative part of the real Wick source, and the form of \OSt\ proved for LIM states (\Cref{thm:s12:LIM}) takes $\amp(\zeta_i)0$ is fixed throughout \Cref{thm:s13:main}. No relation $\lmu\le m$ is used or needed: the lattice reflection positivity of King's tori holds for every $\lmu>0$, its proof using no bound on $\lmu$ (\Cref{prop:s10:RP}, \Cref{lem:s14:CW}(d)). Reflection positivity of the cube limit needs the margin $R(\lmu)=\pi\sqrt3/(2\lmu)$ of \eqref{eq:s07:margin} inside the upper half of the sandwich (\Cref{lem:s07:U}) and nowhere in the conclusion. The constants of Leg~C are absolute; those of Leg~W depend on $\lmu$ and are not uniform in it. The couplings range over all of $e>0$, $\lambda,m,\lmu>0$: their quantifier is the sources' (\Cref{lem:s14:CW}(e), \Cref{rem:s14:quantifier}), the stopping scale of the lattice constructions behind \Cref{hyp:s03:EC,hyp:s03:W1,hyp:s03:T1} being chosen after the point and the datum, with the mass cap of \cite[Prop.~3.1]{Ba2} one more one-sided cap on that scale (\Cref{rem:s14:BALm}); so the theorem contains no sentence ``$e,\lambda$ sufficiently small'', and the only coupling condition of the Wick sector is the one-sided sign clause \Tthree\ (\Cref{lem:s14:CW}(b)), proved on the explicit regions of \Cref{prop:s14:T3}. \end{remark} \space% % % % \section{The coupling range}\label{sec:s14} \providecommand{\sxivNE}{\ensuremath{(\mathrm{NE})}} \providecommand{\sxivNEl}{\ensuremath{(\mathrm{NE}^{<})}} \providecommand{\sxivCT}{\ensuremath{(\mathrm{CT}_{\infty})}} This section collects the conditions on the couplings in the literature behind \Cref{thm:s13:main} and shows that the only coupling condition anywhere in this paper is the one-sided sign clause \Tthree\ of the Wick leg, which \Cref{prop:s14:T3} proves on explicit regions. It uses the $c$-number statement (Lemma~\ref{hyp:s03:T2}), the closed form of the limiting bare coefficient (Lemma~\ref{lem:s14:CT}) and the two-loop majorants of the lattice mass counterterm, proved below; its results enter the Wick leg through Proposition~\ref{prop:s10:U} and the statement of Theorem~\ref{thm:s13:main}, and nothing in the curvature sector depends on them. \medskip\noindent\textbf{King's range, units, two letters $J$.} Let $e>0$ and $\lambda,m,\lmu>0$. Every constant of the cited constructions, including the number $J_{\rm K}$ of renormalisation steps after which the running couplings are small and the stopping scale $5^{-J_{\rm K}}$ (a member of the family $\{5^{-k}\}$), may depend on $(e,\lambda,m,\lmu)$, on the large-block integer $M$ and on the cube or torus; no numerical restriction is placed on the couplings themselves. That a stopping scale exists at every positive point is part of the cited constructions (\cite[Thm.~3.1]{K1}, \cite{Ba1,Ba2,Ba3,Ba4}, and the existence and rotation parts of this paper), under the mass cap of \cite[Prop.~3.1]{Ba2}, a one-sided cap on the stopping scale (\Cref{rem:s14:BALm}). This is what ``King's range'' means here: all positive $(e,\lambda,m,\lmu)$, with a stopping scale chosen after them. The cube index $J$ of $\CJ$ is ours; King's stopping index, the ``fixed integer $J$'' of \cite[Thm.~3.1]{K1}, is written $J_{\rm K}$. Couplings are measured in the units in which the coarsest spacing is $1$; in $d=3$ all four of $e^2,\lambda,m,\lmu$ have the dimension of a mass, rescaling to the unit lattice at spacing $\eps$ replacing $m^2,\lambda,e$ by $m^2\eps^2,\lambda\eps,e\eps^{1/2}$ \cite[Sect.~0]{K1} and $\lmu^2$ like $m^2$. \medskip\noindent\textbf{The relative bare coefficient.} Let $\alpha^\infty_k(\mathsf M)=\mathsf M^2+\dm_k(\mathsf M)$, $s^\infty_k(M_1)=\frac12(\alpha^\infty_k(m)-\alpha^\infty_k(M_1))$ and $\sigma_1(M_1)=\liminf_ks^\infty_k(M_1)$ be as in \Cref{hyp:s03:T3}; the theorem is stated through $\sigma_1$ alone. Describing \emph{where} \Tthree\ fails uses the continuum limit of this coefficient, which the following lemma, of the same kind as \Cref{hyp:s03:T2}, gives in closed form. We write \sxivCT\ for its first assertion: for every $\mathsf M>0$ the limit $\alpha_\infty(\mathsf M)-\alpha_\infty(m)\defeq\lim_{k\to\infty}[\alpha^\infty_k(\mathsf M)-\alpha^\infty_k(m)]$ exists and is a polynomial of degree $\le2$ in $(e^2,\lambda)$ with finite coefficients, differentiable in $\mathsf M^2$. % % % % % \begin{lemma}[the limiting bare coefficient]\label{lem:s14:CT} Let $N=2$ (King's model), $e>0$, $\lambda,m,\lmu>0$, and write $\mu\defeq\lmu$. For every $M>0$ the limit \[ \alpha_\infty(M)-\alpha_\infty(m)\defeq\lim_{k\to\infty}\big[\alpha^\infty_k(M)-\alpha^\infty_k(m)\big] \] exists, and \begin{equation}\label{eq:s14:CTclosed} \begin{split} \alpha_\infty(M)-\alpha_\infty(m)&=(M^2-m^2)+\lambda D_{01}+e^2D_{20}+e^4D_{40}+\lambda e^2D_{21}+\lambda^2D_{02},\\ D_{\alpha\beta}&\defeq F_{\alpha\beta}(M)-F_{\alpha\beta}(m), \end{split} \end{equation} with \begin{gather} F_{01}(M)=\frac{4M}{\pi},\qquad F_{20}(M)=-\frac{M^2}{4\pi(M+\mu)},\qquad F_{02}(M)=-\frac{8}{\pi^2}\ln M,\label{eq:s14:CTF1}\\ F_{21}(M)=\frac{2}{\pi^2}\ln(2M+\mu)+\frac{\mu}{2\pi^2(M+\mu)},\qquad F_{40}(M)=-\frac{1}{4\pi^2}\ln M+\Phi_{40}(\mu/M),\notag\\ \begin{split} \Phi_{40}(x)&=\frac{-7+4x+8x^2-3x^3-3x^4+x^5}{32\pi^2(1+x)(1-x^2)^2}\\ &\quad+\frac{(5-2x^2)\ln(2+x)-(9-10x^2+4x^4)\ln(1+2x)}{16\pi^2(1-x^2)^2}, \end{split}\label{eq:s14:CTF40} \end{gather} where $\Phi_{40}$ extends analytically across $x=1$. Consequently $\alpha_\infty(M)=M^2+\lambda F_{01}(M)+e^2F_{20}(M)+e^4F_{40}(M)+\lambda e^2F_{21}(M)+\lambda^2F_{02}(M)$ up to an additive constant, a function of $(e,\lambda,\lmu)$ and $M$ alone, real-analytic on $(0,\infty)$; its derivatives in $s=M^2$ are \begin{equation}\label{eq:s14:CTy} \begin{gathered} y^\infty_{01}=\frac{2}{\pi M},\quad y^\infty_{20}=-\frac{M+2\mu}{8\pi(M+\mu)^2},\quad y^\infty_{02}=-\frac{4}{\pi^2M^2},\\ y^\infty_{21}=\frac{8M^2+14M\mu+7\mu^2}{4\pi^2M(M+\mu)^2(2M+\mu)}>0,\quad y^\infty_{40}=-\frac{1}{8\pi^2M^2}-\frac{\mu}{2M^3}\Phi_{40}'(\mu/M), \end{gathered} \end{equation} and $\alpha_\infty(M)-\alpha_\infty(m)=M^2+(4\lambda/\pi-e^2/4\pi)M-(e^4/4\pi^2-2\lambda e^2/\pi^2+8\lambda^2/\pi^2)\ln M+O(1)$ as $M\to\infty$. In particular the limit is a polynomial of degree $\le2$ in $(e^2,\lambda)$ with finite coefficients, differentiable in $M^2$ at every $M>0$, and, for $00$. \end{lemma} \begin{proof} \emph{Existence.} Each coefficient of the counterterm difference is \[ \dm_{(\alpha\beta)}(\eps_k;M)-\dm_{(\alpha\beta)}(\eps_k;m)=\int_{m^2}^{M^2}y_{\alpha\beta}(s;\eps_k)\,ds, \] where $y_{\alpha\beta}(s;\eps)=\frac{d}{ds}\dm_{(\alpha\beta)}(\eps;\sqrt s)$ is, by \eqref{eq:s14:CTdef}, the $s$-derivative of the zero-momentum one-particle-irreducible two-point function at that order without its own counterterm vertex, that is a sum over the scalar lines $l$ of its graphs of the integrand with the line $l$ doubled (a factor $-(\hat p_l^2+s)^{-1}$), the lower-order counterterm vertices and their $s$-derivatives inserted. Doubling a line lowers the lattice degree of divergence of every subgraph containing it by $2$, so every subtracted derivative integrand has negative overall degree and negative degrees on all Zimmermann subspaces (the subtraction terms that thereby become over-subtractions integrate to zero by parity in the loop momentum of the subgraph, or are cancelled by the derivative of the counterterm vertex that the forest formula carries). Reisz's power counting theorem \cite[Thm.~1]{Rei88}, in its three-dimensional form (the dimension enters his proof only through the measure and the degrees) and with the absolute bound $\sup_{\eps\le\eps_0}\int|F|\le K$ that its proof gives at fixed masses, bounds $y_{\alpha\beta}(s;\eps)$ uniformly in $\eps\le\eps_0$ and in $s$ on compact intervals $[s_1,s_2]\subset(0,\infty)$: on such an interval the integrand is dominated by $\sum_is_2^{\,i}|V_i|/C(s_1)$, with finitely many $s$-free trigonometric numerators $V_i$ over the common denominator $C(s)=\prod_L(e_L+\mu_L^2)^{n_L}$, $C(s)\ge C(s_1)$, a finite family of fixed integrands of negative degrees. Reisz's renormalisation theorem \cite{Rei88b}, applied to the subtracted integrands of King's graph classes (his forest formula with periodic subtraction operators produces King's recursive counterterms, since the order-$1$ Taylor term of the scalar two-point subtraction vanishes by parity and the photon two-point function at zero momentum by the lattice Ward identity; the one-loop vertex subgraphs of degree $0$ of the two crossed classes of order $e^4$, the crossed graph and the end seagull with both photons on the currents, subgraphs with two scalar legs and one photon leg, are subtracted at order $0$ as his condition on the subtraction degrees requires, and their subtraction term, the vertex at zero external momenta, is odd in the loop momentum of the subgraph, a free integration variable at fixed outer momenta, so it integrates to zero over the Brillouin zone and adds no counterterm beyond King's), gives the convergence $y_{\alpha\beta}(s;\eps)\to y^\infty_{\alpha\beta}(s)$ as $\eps\to0$ for every $s>0$, the limit being the absolutely convergent continuum integral of the pointwise limit of King's subtracted integrand (the lattice-only structures, vertices with $n\ge3$ photon legs carrying $\eps^{n-2}$, the cosine factors of the seagull and the phases, have pointwise limit $1$ or $0$); dominated convergence in $s$ then gives $\lim_kD_{\alpha\beta,\eps_k}=\int_{m^2}^{M^2}y^\infty_{\alpha\beta}(s)\,ds$. The limit is a polynomial of degree $\le2$ in $(e^2,\lambda)$ because the counterterm has the orders $\alpha+2\beta\le4$ with the odd powers of $e$ vanishing (\cref{cpi:tl:odd}). The divergent parts of the counterterm, $1/\eps$ at one loop and $\ln(1/\eps)$ at two loops, do not depend on the scalar mass and cancel in the differences; this is what the negative degrees of the derivative integrands express. \emph{Value.} Continuum conventions: $\int_k\defeq\int d^3k/(2\pi)^3$, lines $G(q)=(q^2+M^2)^{-1}$, $D(k)=(k^2+\mu^2)^{-1}\delta_{\mu\nu}$; for the complex field $\phi_c=(\phi_1+\mathrm i\phi_2)/\sqrt2$ the vertices are the current $-e(p+p')_\mu$, the seagull $2e^2\delta_{\mu\nu}$, the quartic $16\lambda$ and the counterterm, with King's prescription $\dm_{(\alpha\beta)}=\Sigma^{(\alpha\beta)}_{\rm int}(0)$ \cite[(3.32)--(3.33)]{K1}. \emph{One loop:} $D_{01}$ and $D_{20}$ are the limits of the Brillouin-zone integrals $16(M^2-m^2)\int_\BZ G_MG_m$ and $-(M^2-m^2)\int_\BZ\hat q^2\hat D\,G_MG_m$, the differences between the masses $M$ and $m$ of \cref{lem:s14:low}(i), (ii), with $G_M(p)=(\hat p^2+M^2)^{-1}$ and $\hat D$ the momentum-space propagators of that lemma, dominated by $(\pi^2/4)^2[(q^2+M^2)(q^2+m^2)]^{-1}$ since $\hat q^2\ge\frac4{\pi^2}q^2$ and $\hat q^2\hat D\le1$; the continuum integrals $\int_kG_MG_m=\frac1{4\pi(M+m)}$ and $\int_kk^2DG_MG_m$, by partial fractions, give $F_{01}$ and $F_{20}$. \emph{Order $\lambda^2$:} $y^\infty_{02}(s)=-384\iint G^2(q)G(k)G(q+k)=-384/(96\pi^2M^2)=-4/(\pi^2s)$, from the two-loop vacuum integral $\iint G_a(q)^2G_b(k)G_c(q+k)=\frac1{32\pi^2a(a+b+c)}$ (in position space $(G_a^2)^\sim(x)=e^{-a|x|}/8\pi a$ and $\tilde G_a(x)=e^{-a|x|}/4\pi|x|$). \emph{Order $e^2\lambda$:} the limit is $-16\int_kG^2\sigma$ up to an $M$-independent constant, where $\sigma(q)=2q^2B(q)+(\mu^2-2M^2)(B(q)-B(0))$ is the renormalised one-loop self-energy subtracted at zero momentum and $B(q)=\arctan(|q|/(M+\mu))/(4\pi|q|)$ the bubble with lines of masses $M,\mu$, the continuum limit of the exact lattice decomposition of the order-$e^2$ self-energy (\cref{cpi:tl:Pi}(a)); partial fractions in $q^2$ with $q^2G^2=G-M^2G^2$ reduce it to differences $V(a,b,c)-V(a',b',c')=-\frac1{16\pi^2}\ln\frac{a+b+c}{a'+b'+c'}$ of the vacuum integral with lines $q,k,q+k$ of masses $a,b,c$ (with $\mathsf s\defeq a+b+c$ and $\mathsf s'\defeq a'+b'+c'$ the mass sums of the two triples, the Frullani integral $\frac1{16\pi^2}\int_0^\infty(e^{-\mathsf sr}-e^{-\mathsf s'r})\frac{dr}r$ in position space) and to the integrals above, which gives $F_{21}$. \emph{Order $e^4$:} the twelve topologies of order $e^4$, with the order-$e^2$ counterterm inserted on the exchange line, reduce to five continuum classes, the rainbow with $\sigma$ on its inner line, $\int Dq^2G^2\sigma$; the crossed graph, $V(M,\mu,\mu)-M^2I^2$ with $I=\int_kDG=\frac1{4\pi(M+\mu)}$; the seagull at an end with both photons to the currents; the two-photon seagull sunset $6V(M,\mu,\mu)$; and the seagull tadpole with the one-loop vacuum polarisation $\Pi_{\mu\mu}(k)=-e^2[k^2B_{MM}(k)+4M^2(B_{MM}(k)-B_{MM}(0))]$, while the exchange with the vacuum-polarisation photon vanishes by the Ward identity $k_\mu\Pi_{\mu\nu}=0$ and the middle seagull by parity. Each class is reduced to $V(M,\mu,\mu)$, $V(M,M,\mu)$ and elementary integrals by the exact identities $k_1\cdot(k_1+2k_2)=G(k_1+k_2)^{-1}-G(k_2)^{-1}$, $k_2\cdot(2k_1+k_2)=G(k_1+k_2)^{-1}-G(k_1)^{-1}$ and partial fractions; the two classes with a logarithmically divergent vertex subgraph are evaluated with the product cutoff $|k_1|,|k_2|<\Lambda$, under which the zeroth Taylor term of the subgraph is odd in its loop momentum and the one shift needed has a surface term whose $s$-derivative tends to $0$. The coefficients of $V$ add to $1+6-6+1+2=4$, which is the universal $\frac1{4\pi^2}\ln(1/\eps)$ of the order $e^4$, cancelling in differences; the five class values add to $F_{40}$ as displayed. \emph{Analyticity and the derivatives.} $F_{01},F_{20},F_{02},F_{21}$ are analytic on $(0,\infty)$; $\Phi_{40}$ is built from rational functions and $\ln(2+x)$, $\ln(1+2x)$, and at $x=1$, where its denominators vanish, the coefficients of $(x-1)^{-4},\dots,(x-1)^{-1}$ of its Laurent expansion vanish identically, so the singularity is removable. The formulas \eqref{eq:s14:CTy} are the $s$-derivatives of \eqref{eq:s14:CTF1}--\eqref{eq:s14:CTF40}, and the large-$M$ form collects their terms. The identification $\sigma_1(M_1)=\lim_ks^\infty_k(M_1)$ for $00$. So $a_0$ and $a_1$ never vanish together, the polynomial $e^2\mapsto\alpha_\infty'(m)$ is never identically zero, and it has at most two zeros. \end{proof} \space% % \sxivCT\ enters \Cref{thm:s13:main} through the exceptional set $E_W$ of the rotation part (\Cref{rot:thm:POS,rot:def:cone}), defined in \eqref{eq:s14:EW} below, off which the rotation theorem is proved first and whose points are reached by right-continuity in the scalar mass, and through the description of the failure set of \Tthree. By \sxivCT\ put \[ s_\infty(M_1)\defeq\tfrac12\big(\alpha_\infty(m)-\alpha_\infty(M_1)\big),\qquad s_*\defeq\sup_{\mathsf M>0}s_\infty(\mathsf M)\in[0,\infty], \] \begin{equation}\label{eq:s14:EW} \begin{split} E_W&\defeq\{s_*=0\}=\{m\text{ is a global minimiser of }\mathsf M\mapsto\alpha_\infty(\mathsf M)\text{ on }(0,\infty)\},\\ E^<_W&\defeq\{m\text{ minimises }\alpha_\infty\text{ on }(0,m]\}\supseteq E_W, \end{split} \end{equation} subsets of the coupling space ($s_*\ge0$ because $\mathsf M=m$ gives $0$; the value $+\infty$ is harmless below). The two sign clauses are \begin{itemize}[nosep,leftmargin=*] \item[\sxivNE] $m$ is not a global minimiser of $\alpha_\infty$ on $(0,\infty)$, i.e.\ $s_*>0$, i.e.\ $(e,\lambda,m,\lmu)\notin E_W$; \item[\sxivNEl] some $M_1\in(0,m)$ has $\alpha_\infty(M_1)<\alpha_\infty(m)$, i.e.\ $(e,\lambda,m,\lmu)\notin E^<_W$ (the one-sided form; it implies \sxivNE). \end{itemize} \refstepcounter{convention}\label{conv:v065_s14:1}Throughout, $'$ denotes $d/dM^2$, $N$ is the number of real components of the scalar field, and ``one-loop order'' means the orders $e^2$ and $\lambda$. \smallskip\noindent Throughout \cref{lem:s14:CW} and its proof, \Cref{hyp:s03:EC} is the existence clause of \cite[Thm.~2.1]{K2} at zero scalar source, \Cref{lem:s02:BFS} is Theorem~6.2(1),(2) of \cite{BFS1}, and \emph{the results used in Form~(B)} are, beyond \Cref{hyp:s03:W1,hyp:s03:T1,hyp:s03:T2} and \Tthree, the statements \Cref{prop:s10:RP,lem:s10:CB,lem:s02:BFS,lem:s10:F,lem:s10:V} and the tube statements \Cref{thm:s11:W,lem:s11:TWtube,thm:s08:OS2WT}, whose conditions on the Wick amplitudes are not conditions on the couplings. \begin{lemma}[the coupling range]\label{lem:s14:CW} Let $e>0$ and $\lambda,m,\lmu>0$. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item \emph{The curvature sector.} The existence theorem of the curvature sector, \Cref{hyp:s03:EC}, carries no condition on $e,\lambda,m,\lmu$. The correlation inequalities it uses, \Cref{lem:s02:BFS}, have the standing hypotheses of a finite single-site measure, positive hopping coefficients and a positive photon covariance, which for King's action are $\lambda>0$ and $\lmu>0$; the margin $R(\lmu)=\pi\sqrt3/(2\lmu)$ of \Cref{lem:s07:U} needs $\lmu>0$ only; and no lemma proved in this paper for the curvature sector carries a condition on the couplings beyond those of the results it uses. Hence the curvature sector of \Cref{thm:s13:main} holds on all of King's range, with no exceptional set. The one proof-internal proviso in the Ba{\l}aban antecedents of \Cref{hyp:s03:EC}, the mass clause of \cite[Prop.~3.1]{Ba2}, is a cap on the dimensionless scalar mass at the stopping scales and no condition on the bare couplings (\Cref{rem:s14:BALm}). Of the bounds of the paper only the absolute bounds of \Cref{thm:s05:CK} are uniform in $\lmu$. \item \emph{The Wick sector, Form~(B).} The results used in Form~(B) state no condition on the bare couplings beyond positivity ($\lambda>0$, $\lmu>0$); \Cref{hyp:s03:T1} is used at the two masses $m$ and $M_1$. The only coupling condition of Form~(B) is therefore membership of the coupling region of \Cref{hyp:s03:T3}, the sign clause \Tthree, which \Cref{prop:s14:T3} proves on explicit regions. Its failure set is described in \cref{r13:lem:s14:failset}. \item \emph{The existence and rotation parts.} The existence theorems \Cref{hyp:s03:EC,hyp:s03:W1} and the rotation theorem \Cref{hyp:s03:RC} are proved for $e>0$, $\lambda,m,\lmu>0$ in King's range, and none carries a condition on the bare couplings beyond positivity: every smallness condition of \cite{Ba1,Ba2,Ba3,Ba4} that they use (of the three kinds classified in the proof of (b)) is a one-sided cap on a stopping scale $\beta\le1$, imposed last in the schedule of constants, and an admissible $\beta$ exists at every positive point.\item \emph{The photon mass.} No relation $\lmu\le m$ is needed: \Cref{prop:s10:RP} holds for every $\lmu>0$. The explicit regions of \Tthree\ at $\lmu=m$ and at every $\lmu\in(0,m]$ are those of \Cref{prop:s14:T3}(iii),(iv). \item \emph{The quantifier.} For all $e>0$, $\lambda,m,\lmu>0$ and each datum there are an integer $J_{\rm K}$ (\cite[Thm.~3.1]{K1}) and a stopping scale $\beta\le1$ such that the running couplings $(e\beta^{1/2},\lambda\beta,m\beta,\lmu\beta)$ satisfy finitely many one-sided caps on $\beta$ and on three of the four of them ($\lmu\beta$ is uncapped in the sources); no condition is placed on the bare couplings, and the theorems used from \cite{K1,K2} carry no coupling hypothesis. A sentence ``$e,\lambda$ sufficiently small, $m,\lmu>0$ fixed'', read as a hypothesis on the bare couplings, is strictly stronger and is stated in none of the sources (\Cref{rem:s14:quantifier}). \end{enumerate} \end{lemma} \begin{proof} \emph{(a).} Theorem~2.1 of \cite{K2} states, for $g,h$ supported in $C_J$ and all $z,w\in\C$, the existence of $\lim_K\langle\Theta(zg,wh)\rangle^{\eps_K}_{N,D,C_J}$; no hypothesis on the couplings precedes it. Its proof rests on \cite[Thm.~3.1]{K1} (the convergence of the effective action, referred there to \cite{Ba1,Ba2,Ba3,Ba4}), stated ``for some fixed integer $J$'', our $J_{\rm K}$; neither it nor \cite[Thms.~2.1, 3.4]{K1} carries a coupling hypothesis, the smallness in \cite{K1} being of the fields and of the stopping spacing. So the one proviso of \cite[Prop.~3.1]{Ba2} is upstream of \Cref{hyp:s03:EC}. For King's action the single-site measure $e^{-\eps^3[\frac12\alpha r^2+\lambda r^4]}r\,dr$ is finite iff $\lambda>0$ (or $\lambda=0$, $\alpha>0$), the hopping weights are positive, and the Feynman-form photon covariance is positive iff $\lmu>0$; the ``$g>0$'' of Theorem~6.2 of \cite{BFS1} is the sign of the scalar source, and \Cref{lem:s02:BFS} covers $g=0$. The margin of \Cref{lem:s07:U} concerns the photon Gaussians of the two boxes alone, so $\lmu$ is its only parameter (no fixed margin works at $\lmu=0$). The lemmas of this paper see only the form of the action. \emph{(b): the torus theorems.} \Cref{hyp:s03:T1} is Ba{\l}aban's stability theorem \cite[(1.14)]{Ba1}, \cite[(1.3)]{Ba2}, at fixed scalar mass, read at $m$ and at $M_1$ with block factor $5$ and King's normalisation, with constants independent of $\eps$ and of the torus, under $\lambda>0$, $\lmu^2>0$, $m^2>0$. Every smallness condition stated in \cite{Ba1,Ba2,Ba3,Ba4} is of one of three kinds. \emph{Running charge:} the bounds hold ``for $e(L^k\eps)$ sufficiently small'', $e(L^k\eps)=e\,(L^k\eps)^{(4-d)/2}$ \cite[(2.23)]{Ba1}; \cite{Ba4} proves them for a unit-lattice model whose charge is this running charge (its theorem is stated there as Proposition~2.1 of \cite{Ba1}), so ``$e$ sufficiently small'' there is a cap on $e\beta^{1/2}$ at the stopping scale $\beta$. \emph{Scale:} the stopping scale $\eps_0$ with $L^K\eps\le\eps_00$) and a finite photon Gaussian on the torus ($\lmu>0$; at $\lmu=0$ the constant mode lies in the kernel); \Cref{lem:s10:V} needs massive propagators; \Cref{prop:s10:RP} needs finite single-site moments and a positive Proca covariance and does not see the counterterm. The tube statements restrict Wick amplitudes only. So the only coupling condition of Form~(B) is the sign clause of \Cref{prop:s10:U}, i.e.\ \Tthree. \emph{(c).} The smallness conditions are of the three kinds listed in the proof of (b) (running charge, scale, mass); the order in which the constants of the existence and rotation parts are fixed, and the proof that an admissible stopping scale exists at every positive point, are as follows: the running couplings $(e\beta^{1/2},\lambda\beta,m\beta,\mu\beta)$ vanish as $\beta\to0$, so every smallness condition of the existence and rotation parts holds below a stopping scale $\beta_0(e,\lambda,m,\mu)>0$, the constants being fixed first; every smallness condition used is a one-sided cap, monotone in the stopping scale $\beta$, on $\beta$ or on the running couplings at $\beta$, so with the other constants fixed first a $\beta=5^{-J}$ below the finitely many positive caps exists at every positive point; and the constants are fixed in an acyclic order ($L=5$; the decay and coercivity constants before $M$; $M\in50\N$; the operator and class constants; the localisation radius $R$; $b_0$; the stopping scale $\beta$ last; $J\in5^{k_*}\N$, $\eta_0\in5^{-\N}$), each step imposing finitely many one-sided or divisibility conditions on one entry, with right sides depending on the couplings and on the earlier entries only, so an admissible choice exists at every positive point. \emph{(d).} The proof of \Cref{prop:s10:RP} uses site-independent coefficients, $\lambda>0$, $\lmu>0$ and even sides, and no bound on $\lmu$. The statements at $\lmu=m$ and for $\lmu\in(0,m]$ are \Cref{prop:s14:T3}(iii),(iv), whose proof uses \cref{r13:lem:s14:failset}(i) and neither \cref{r13:lem:s14:failset}(iii) nor (d). \emph{(e).} The quantifier is read off \cite[Thm.~3.1]{K1} (``for some fixed integer $J$, for all $K\ge J$, and all $k\le K-J$'') and the stopping-scale caps listed above; the last sentence is \Cref{rem:s14:quantifier}. \end{proof} \begin{lemma}[the failure set of the sign clause]\label{r13:lem:s14:failset} Let $e>0$ and $\lambda,m,\lmu>0$. By \sxivCT\ (Lemma~\ref{lem:s14:CT}): \begin{enumerate}[label=\textup{(\roman*)},nosep,leftmargin=*] \item $\sigma_1(M_1)=\lim_ks^\infty_k(M_1)=s_\infty(M_1)$ for every $M_1>0$. Hence \Tthree\ holds if and only if \sxivNEl: it fails exactly on $E^<_W$. Since \Cref{hyp:s03:T1,hyp:s03:T2} hold also at auxiliary masses $M_1>m$ (Ba{\l}aban's theorem at a fixed mass, and the $c$-number bound on a compact mass interval on either side of $m$; see the proof of \cref{lem:s14:CW}), by \Cref{prop:s10:Upm} the two-sided clause \sxivNE\ replaces \Tthree, with failure set exactly $E_W$; the branch $M_1>m$ is needed exactly on $E^<_W\setminus E_W$, for instance when $\alpha'_\infty(m)<0$ and $m$ minimises $\alpha_\infty$ on $(0,m]$. \item To one-loop order, for every $M_1>0$, \[ s_\infty(M_1)=\tfrac12(m^2-M_1^2)+\frac{\lambda(N+2)(m-M_1)}{2\pi}-e^2\,\frac{(m-M_1)\,(m\lmu+mM_1+M_1\lmu)}{8\pi(\lmu+M_1)(m+\lmu)}, \] so that \begin{equation}\label{eq:s14:alphaprime} \alpha'_\infty(m)=1+\frac{\lambda(N+2)}{2\pi m}-e^2\,\frac{m+2\lmu}{8\pi(m+\lmu)^2},\qquad \frac{m+2\lmu}{8\pi(m+\lmu)^2}\le\frac1{4\pi(m+\lmu)}, \end{equation} and to one-loop order \Tthree\ holds if and only if \begin{equation}\label{eq:s14:ec2} e^20$ iff \sxivNEl, and some $M_1>0$ has $\sigma_1(M_1)>0$ iff \sxivNE. \Cref{prop:s10:Upm} gives \Itwo\ from \Cref{hyp:s03:T1,hyp:s03:T2} at an auxiliary mass on either side of $m$, so with those statements \sxivNE\ suffices; on $E^<_W\setminus E_W$ every $M_1m$ works. \emph{(ii).} The closed form is the one-loop evaluation of King's counterterm \cite[(3.32)--(3.33)]{K1} (tadpole and zero-momentum photon-exchange graphs; the $1/\eps$ divergence cancels in the difference). It equals $\frac12\int_{M_1^2}^{m^2}\alpha'_\infty(\mu)\,d\mu^2$ with $\alpha'_\infty(\mu)=1+\frac{\lambda(N+2)}{2\pi\mu}-e^2\frac{\mu+2\lmu}{8\pi(\mu+\lmu)^2}$, an orientation-free definite integral, valid on either side of $m$; at $\mu=m$ this is \eqref{eq:s14:alphaprime}, whose inequality is $m\ge0$. For $M_1\in(0,m)$ write $s_\infty(M_1)=\frac{m-M_1}2\beta(M_1)$ with \[ \beta(M_1)=(m+M_1)+c-e^2\,\frac{m\lmu+(m+\lmu)M_1}{4\pi(m+\lmu)(\lmu+M_1)},\qquad c\defeq\frac{\lambda(N+2)}{\pi}; \] $s_\infty(M_1)>0$ iff $\beta(M_1)>0$ iff $e^20$, negative then positive with one zero. So $e_c^2(\cdot)$ is increasing, or decreasing then increasing, on $(0,m)$, and its supremum over the open interval is the larger of the two end values, not attained: $M_1\to m^-$ gives $\frac{8\pi(m+\lmu)^2}{m+2\lmu}(1+\frac c{2m})$, the zero $e_*^2$ of \eqref{eq:s14:alphaprime}, and $M_1\to0^+$ gives $4\pi(m+\lmu)(1+\frac cm)$. Hence to one-loop order \Tthree\ holds iff $e^2100$), which meets the explicit regions of \Cref{prop:s14:T3}, and would be a hypothesis of the theorem (1.3) of \cite{Ba2} that is not stated there, the only stated condition on $m$ being $m^2>0$; the proposition is not proved under that interpretation, and it is used nowhere. The clause sits upstream of \Cref{hyp:s03:T1} at both masses, of \Cref{hyp:s03:EC,hyp:s03:W1} (through \cite[Thm.~3.1]{K1}) and of the steps of the rotation part that rest on \cite{Ba1,Ba2}, as every cap on the stopping scale does; it is not a condition of this paper. \end{remark} \medskip\noindent\textbf{The explicit coupling region.} The sign clause \Tthree\ is proved below on an explicit region of the $(\lambda,e^2)$-plane at every photon mass $\lmu\in(0,m]$, with $N=2$ and the auxiliary mass $M_1=0.7m$. The argument has two parts: the three low orders of King's counterterm, whose contributions to $s^\infty_k(M_1)$ are lattice integrals with exact limits (\Cref{lem:s14:struct,lem:s14:low,lem:s14:A,lem:s14:B,lem:s14:C}), and the two orders $e^2\lambda$ and $e^4$, whose contributions are bounded from below, uniformly in $k$, through the two-loop majorants of King's counterterm on periodic tori (the facts (G1)--(G5) stated in the subsection on the two-loop orders, before \Cref{lem:s14:E}, and proved there), read at the photon mass $\lmu$, with the torus floor removed (\Cref{lem:s14:E,lem:s14:TF}), made uniform on intervals of photon masses (\Cref{lem:s14:MON}) and, in a weaker closed form, uniform in the photon mass (\Cref{lem:s14:CF}). The resulting lower bound $\sigma_{\rm low}$ on $\sigma_1(0.7m)$, \eqref{eq:s14:sigmalow}, is explicit through \Cref{tab:s14:T3}. % % \providecommand{\BZ}{\ensuremath{\mathsf B}} \subsection*{King's counterterm order by order} \refstepcounter{convention}\label{conv:s14_t3_low:1}Throughout this subsection and the next, $\eps=\eps_k$, $\Lam=\Lk$, $\BZ=\BZ_\eps\defeq[-\pi/\eps,\pi/\eps)^3$ is the Brillouin zone with $\int_\BZ\defeq\int_\BZ\frac{d^3p}{(2\pi)^3}$, $\hat p_\mu\defeq\frac2\eps\sin\frac{\eps p_\mu}2$ and $\hat p^2\defeq\sum_\mu\hat p_\mu^2$ (periodic in $p$), and for a mass $M>0$ \[ \begin{gathered} G_M(p)\defeq\frac1{\hat p^2+M^2},\qquad \hat D(p)\defeq\frac1{\hat p^2+\lmu^2},\\ C_M(x)\defeq\int_\BZ e^{\mathrm ip\cdot x}G_M(p),\qquad D(x)\defeq\int_\BZ e^{\mathrm ip\cdot x}\hat D(p)\quad(x\in\Lam) \end{gathered} \] are the free scalar and photon propagators of King's action on $\Lam$ (per real scalar component and per photon component; $C_M(0)$ is the free Wick constant at mass $M$, $C_m(0)=\sW_\eps$). King's free-boundary counterterm \cite[(3.32)--(3.33)]{K1}, \cite[p.~330]{K2} at scalar mass $M$ is the polynomial \begin{equation}\label{eq:s14:CTdef} \dm_\eps(M)=\sum_{(\alpha,\beta)}e^\alpha\lambda^\beta\,\dm_{(\alpha\beta)}(\eps;M),\qquad(\alpha,\beta)\in\{(0,1),(2,0),(0,2),(2,1),(4,0)\}, \end{equation} over the superficially divergent orders $2\le\alpha+2\beta\le4$ with the odd powers of $e$ dropped (they vanish: an odd power of $e$ comes with an odd number of photon fields, whose centred Gaussian expectation is zero); the coefficient $\dm_{(\alpha\beta)}(\eps;M)$ is fixed recursively by the condition that the order-$e^\alpha\lambda^\beta$ part of the amputated one-particle-irreducible two-point function of $\phi$ at zero external momentum, computed on $\Lam$ with the propagators above and with all counterterm vertices of lower order inserted, be cancelled by the counterterm vertex of that order. In the sign convention in which a term of the expansion of $e^{-V}$ that reduces to $+\frac12\sum_{x,y}\eps^6\phi_a(x)\mathsf K(x-y)\phi_a(y)$ contributes $+\hat{\mathsf K}(p)$ to the self-energy $\Sigma(p)$, so that the inverse propagator per component is $\hat p^2+M^2-\Sigma(p)$ and the counterterm vertex $-\frac12\dm\sum\eps^3|\phi|^2$ contributes $-\dm$, the condition reads $\dm_{(\alpha\beta)}(\eps;M)=\Sigma^{(\alpha\beta)}(0;M)$, $\Sigma^{(\alpha\beta)}$ the order-$e^\alpha\lambda^\beta$ part of the sum of the one-particle-irreducible two-point graphs built from the vertices of King's action and from the counterterm vertices of lower order. Every such graph is a finite sum of absolutely convergent integrals over $\BZ^L$, $L\le2$ (all masses are $\ge\min(M,\lmu)>0$ and $\BZ$ is compact), so each coefficient is a finite real number, smooth in $M$ (a rational function of $M^2$ without poles on $(0,\infty)$ under each integral). With $\alpha^\infty_k(M)=M^2+\dm_{\eps_k}(M)$ as in \cref{hyp:s03:T3} and $M_1\in(0,m)$: \begin{lemma}[structure]\label{lem:s14:struct} Put $S_{\alpha\beta,k}\defeq\frac12\big[\dm_{(\alpha\beta)}(\eps_k;m)-\dm_{(\alpha\beta)}(\eps_k;M_1)\big]$, numbers free of the couplings. Then \begin{equation}\label{eq:s14:struct} s^\infty_k(M_1)=\tfrac12(m^2-M_1^2)+\lambda S_{01,k}+e^2S_{20,k}+\lambda^2S_{02,k}+e^2\lambda S_{21,k}+e^4S_{40,k}. \end{equation} In $d=3$, $S_{01,k}$ and $S_{20,k}$ have the dimension of a mass and $S_{02,k}$, $S_{21,k}$, $S_{40,k}$ are dimensionless. \end{lemma} \begin{proof} $s^\infty_k(M_1)=\frac12(\alpha^\infty_k(m)-\alpha^\infty_k(M_1))$ and \eqref{eq:s14:CTdef}. \end{proof} \begin{lemma}[the three low orders, exact]\label{lem:s14:low} For every $\eps$ and $M>0$: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*,nosep] \item $\dm_{(01)}(\eps;M)=-4(N+2)\,C_M(0)$; \item $\dm_{(20)}(\eps;M)=\int_\BZ\hat k^2\hat D(k)G_M(k)-3D(0)=-2D(0)-M^2\int_\BZ\hat DG_M$; \item $\dm_{(02)}(\eps;M)=32(N+2)\sum_{x\in\Lam}\eps^3C_M(x)^3$. \end{enumerate} Consequently, with \[ \begin{gathered} J_\eps\defeq\int_\BZ G_mG_{M_1},\qquad I_\eps\defeq\int_\BZ\hat k^2\hat D(k)G_m(k)G_{M_1}(k)\\ \text{and}\qquad F_\eps\defeq\sum_{x\in\Lam}\eps^3[C_{M_1}(x)^3-C_m(x)^3], \end{gathered} \] we have \begin{equation}\label{eq:s14:lowcoef} \begin{split} S_{01,k}&=2(N+2)\big(C_{M_1}(0)-C_m(0)\big)=2(N+2)(m^2-M_1^2)J_{\eps_k},\\ S_{20,k}&=-\tfrac12(m^2-M_1^2)I_{\eps_k},\qquad S_{02,k}=-16(N+2)F_{\eps_k}. \end{split} \end{equation} \end{lemma} \begin{proof} Write $\phi=(\phi_1,\dots,\phi_N)$ with $\phi\cdot\psi=\sum_a\phi_a\psi_a$ and, for $N=2$, $\phi\wedge\psi=\phi_1\psi_2-\phi_2\psi_1=J_{ab}\phi_a\psi_b$ with $J=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$, $J^2=-1$. With $y=x+\eps e_\mu$ and $\theta=e\eps A_\mu(x)$ the hopping term of King's action is $\frac12\eps|e^{\mathrm i\theta}\phi(y)-\phi(x)|^2=\frac12\eps(|\phi(x)|^2+|\phi(y)|^2)-\eps\cos\theta\,\phi(x)\cdot\phi(y)+\eps\sin\theta\,\phi(x)\wedge\phi(y)$; expanding $\cos\theta$ and $\sin\theta$ gives the free action $S_0=\frac12\sum_x\eps^3[\sum_\mu|\nabla_\mu\phi|^2+M^2|\phi|^2]+\frac12\sum_x\eps^3[\sum_{\mu,\nu}(\nabla_\nu A_\mu)^2+\lmu^2|A|^2]$, $\nabla_\mu f(x)=(f(x+\eps e_\mu)-f(x))/\eps$, whose covariances are $\langle\phi_a(x)\phi_b(y)\rangle_0=\delta_{ab}C_M(x-y)$ and $\langle A_\mu(x)A_\nu(y)\rangle_0=\delta_{\mu\nu}D(x-y)$, and the vertices \[ \begin{gathered} V_1=e\sum_{x,\mu}\eps^2A_\mu(x)\,\phi(x)\wedge\phi(y),\qquad V_2=\tfrac{e^2}2\sum_{x,\mu}\eps^3A_\mu(x)^2\,\phi(x)\cdot\phi(y),\\ V_\lambda=\lambda\sum_x\eps^3(\phi\cdot\phi)^2,\qquad V_{\rm ct}=\tfrac12\dm\sum_x\eps^3\phi\cdot\phi, \end{gathered} \] together with vertices of order $e^3$, $e^4$ and higher, which do not enter the three orders below. The graphs are those of $-V+\frac12V^2-\cdots$ with the normalisation fixed above; by $O(N)$ symmetry they are proportional to $\delta_{ab}$. (i) At order $\lambda$ the only one-particle-irreducible two-point graph is the tadpole of $-V_\lambda$. Contracting one pair of fields in $(\phi\cdot\phi)^2=\phi_a\phi_a\phi_b\phi_b$ gives $2(N+2)C_M(0)\,\phi\cdot\phi$. So $-V_\lambda$ contributes $-2(N+2)\lambda C_M(0)\sum_x\eps^3\phi\cdot\phi=\frac12\sum\phi_a\mathsf K\phi_a$ with $\hat{\mathsf K}=-4(N+2)\lambda C_M(0)$, and $\dm_{(01)}=\Sigma^{(01)}(0)/\lambda=-4(N+2)C_M(0)$. (ii) At order $e^2$ there are two graphs. \emph{Seagull:} contracting $A_\mu(x)^2$ to $D(0)$ in $-V_2$ gives $-\frac{e^2}2D(0)\sum_{x,\mu}\eps^3\phi(x)\cdot\phi(x+\eps e_\mu)$, i.e.\ $\Sigma_{\rm seag}(p)=-e^2D(0)\sum_\mu\cos(\eps p_\mu)$ after symmetrisation, and $\Sigma_{\rm seag}(0)=-3e^2D(0)$. \emph{Exchange:} in momentum space the vertex $V_1$ carries the symmetrised factor $W^\mu_{ij}(p,p')=\mathrm ie\,J_{ij}\,e^{-\mathrm i\eps k_\mu/2}\frac1\eps\sin\frac{\eps(p'-p)_\mu}2$ at momenta $k+p+p'=0$; in $\frac12V_1^2$ contract the photon pair and one scalar pair, so that with external pair $\hat\phi_i(p)\hat\phi_l(-p)$ the internal line carries $p+k$ and the two vertex factors multiply to $e^2\delta_{il}\frac1{\eps^2}\sin^2\frac{\eps(2p+k)_\mu}2$ ($J^2=-1$; the phases cancel). Hence $\Sigma_{\rm exch}(p)=e^2\int_\BZ\hat D(k)G_M(p+k)\sum_\mu\big(\frac2\eps\sin\frac{\eps(2p+k)_\mu}2\big)^2$ and $\Sigma_{\rm exch}(0)=e^2\int_\BZ\hat D(k)\hat k^2G_M(k)$. So $\dm_{(20)}=\int_\BZ\hat k^2\hat DG_M-3D(0)$; the second form follows from $\hat k^2\hat DG_M=\hat D(1-M^2G_M)$ and $\int_\BZ\hat D=D(0)$. (iii) At order $\lambda^2$ (and $e=0$) the one-particle-irreducible two-point graphs with two quartic vertices $u,v$ are the sunset (external legs at $u$ and at $v$, three lines $u$--$v$) and the tadpole with a tadpole inserted on its loop line (external legs at $u$, two lines $u$--$v$, one loop at $v$); the configurations with a single bridge are not one-particle irreducible and the rest are disconnected. The insertion graph is cancelled exactly by the graph with the counterterm vertex $-V^{(01)}_{\rm ct}$ on the loop line, because the order-$\lambda$ self-energy with its counterterm, $\Sigma^{(01)}(q)-\lambda\dm_{(01)}$, vanishes for every $q$ (the tadpole does not depend on the momentum). For the sunset, the contractions of $\frac12V_\lambda^2=\frac12\lambda^2\sum_{u,v}\eps^6(\phi\cdot\phi)^2_u(\phi\cdot\phi)^2_v$ with one external field at $u$ and one at $v$ carry the factor $16\cdot2(N+2)\delta_{ab}$; so $\frac12V_\lambda^2$ contributes $\frac12\sum_{u,v}\eps^6\phi_a(u)\mathsf K(u-v)\phi_a(v)$ with $\mathsf K(x)=32(N+2)\lambda^2C_M(x)^3$, and $\dm_{(02)}=\hat{\mathsf K}(0)/\lambda^2=32(N+2)\sum_x\eps^3C_M(x)^3$. The formulas \eqref{eq:s14:lowcoef} follow from (i)--(iii) and the resolvent identity \[ G_{M_1}-G_m=(m^2-M_1^2)G_{M_1}G_m; \] the $M$-independent tadpole $3D(0)$ and the term $-2D(0)$ cancel in $S_{20,k}$. \end{proof} At $N=2$ these coefficients are the terms $-16\lambda C_T(M)$, $-2e^2D_T-e^2J_T(M)$ and $128\lambda^2S_T(M)$ of the exact two-loop form of the torus coefficient, namely (G1) of the list before \cref{lem:s14:E} and Lemma~\ref{cpi:tl:odd}, read on $\Lam$ in place of the torus; the orders $e^2\lambda$ and $e^4$ are the terms $16e^2\lambda F_T(M)$ and $e^4Q_T(M)$ there. Every graph of these two orders is an absolutely convergent integral over $\BZ^2$ at fixed $\eps$ (massive propagators, bounded trigonometric vertex factors), so $S_{21,k}$ and $S_{40,k}$ are finite real numbers for every $k$; their bounds, uniform in $k$, are the subject of the next subsection. \space% % % % \begin{lemma}[four lattice facts]\label{lem:s14:L} \textup{(L1)} For $p\in\BZ$: $c\,p^2\le\hat p^2\le p^2$ with $c=4/\pi^2$. Hence, for a mass $a>0$, $\bar G_a(p)\defeq(p^2+a^2)^{-1}\le G_a(p)\le\bar G^c_a(p)\defeq(cp^2+a^2)^{-1}$ on $\BZ$, and for $a,b>0$: $\int_\BZ\bar G^c_a\bar G^c_b\le\int_{\R^3}\bar G^c_a\bar G^c_b\,\frac{d^3p}{(2\pi)^3}=\frac{\pi^2}{32(a+b)}$, $\int_\BZ(\bar G^c_a)^2\le\frac{\pi^2}{64a}$. \textup{(L2)} $C_M(x)=\eps^{-3}\int_0^\infty e^{-tM^2}(e^{t\Delta_\eps})_{x,0}\,dt$ with $(e^{t\Delta_\eps})_{x,y}=e^{-6t/\eps^2}\sum_{n\ge0}\frac{(t/\eps^2)^n}{n!}\#\{n\text{-step nearest-neighbour paths }y\to x\}>0$. Hence $C_M(x)>0$ for every $x\in\Lam$, $M\mapsto C_M(x)$ is strictly decreasing, $\sum_x\eps^3C_M(x)=M^{-2}$, and the resolvent identity $C_{M_1}-C_m=(m^2-M_1^2)\,C_{M_1}*C_m$ holds pointwise with a positive right side, $(f*g)(x)\defeq\sum_y\eps^3f(x-y)g(y)$. \textup{(L3)} With $\int\defeq\int_{\R^3}\frac{d^3p}{(2\pi)^3}$ and $a,b,c>0$: $\int\frac1{(p^2+a^2)(p^2+b^2)}=\frac1{4\pi(a+b)}$, $\int\frac1{(p^2+a^2)^2}=\frac1{8\pi a}$, $\int\frac1{(p^2+a^2)(p^2+b^2)(p^2+c^2)}=\frac1{4\pi(a+b)(b+c)(c+a)}$. \textup{(L4)} If $h_\eps$ on $\BZ_\eps$ satisfies $|h_\eps|\le\bar h\in L^1(\R^3)$ and $h_\eps(p)\to h(p)$ for every $p$ as $\eps\to0$, then $\int_{\BZ_\eps}h_\eps\to\int_{\R^3}h$. As $\eps\to0$, $\hat p^2\to p^2$ at every fixed $p$, so $G_M(p)\to\bar G_M(p)$ and $\hat D(p)\to\bar G_{\lmu}(p)$. \end{lemma} \begin{proof} (L1) $|\sin x|\le|x|$ and $\sin x\ge2x/\pi$ on $[0,\pi/2]$, at $x=\eps|p_\mu|/2\le\pi/2$; the integrals follow from (L3) by the scaling $p\mapsto c^{-1/2}p$, $c^{-3/2}=\pi^3/8$. (L2) $(-\Delta_\eps+M^2)^{-1}=\int_0^\infty e^{-tM^2}e^{t\Delta_\eps}dt$ as matrices on $\Lam$, with kernel $\eps^3C_M(x-y)$; the path expansion is $e^{t\Delta_\eps}=e^{-6t/\eps^2}e^{(t/\eps^2)\mathcal A}$ with $\mathcal A$ the adjacency matrix; $\sum_x(e^{t\Delta_\eps})_{x,0}=1$; the identity is $G_{M_1}-G_m=(m^2-M_1^2)G_{M_1}G_m$ in Fourier space. (L3) Radially, $\frac1{2\pi^2}\int_0^\infty\frac{p^2dp}{(p^2+a^2)(p^2+b^2)}=\frac1{2\pi^2}\cdot\frac{\pi}{2(a+b)}$, the second integral is the limit $b\to a$, and the third follows by partial fractions from the first. (L4) Dominated convergence after extension by $0$. \end{proof} \begin{lemma}[order $\lambda$]\label{lem:s14:A} For every $k$: \textup{(i)} $S_{01,k}>0$; \textup{(ii)} $J_\eps\le\frac{\pi^2}{32(m+M_1)}$, hence $S_{01,k}\le\frac{(N+2)\pi^2(m-M_1)}{16}$; \textup{(iii)} $J_{\eps_k}\to\frac1{4\pi(m+M_1)}$, hence $S_{01,k}\to\frac{(N+2)(m-M_1)}{2\pi}$ as $k\to\infty$. \end{lemma} \begin{proof} (i) \eqref{eq:s14:lowcoef} and $G_mG_{M_1}>0$. (ii) (L1). (iii) (L4) with $\bar h=\bar G^c_m\bar G^c_{M_1}$, and (L3). \end{proof} \begin{lemma}[order $e^2$]\label{lem:s14:B} For every $k$: \textup{(i)} $S_{20,k}<0$; \textup{(ii)} $00$, so $S_{02,k}<0$; \textup{(ii)} $F_\eps\le3(m^2-M_1^2)\big(\int_\BZ G_{M_1}^2\big)^2$ and so $F_\eps\le\frac{3\pi^4(m^2-M_1^2)}{4096M_1^2}$, hence $|S_{02,k}|\le\bar L\defeq\frac{3(N+2)\pi^4(m^2-M_1^2)}{256M_1^2}$; \textup{(iii)} $F_{\eps_k}\to\frac{\ln(m/M_1)}{16\pi^2}$, hence $S_{02,k}\to-L(M_1)\defeq-\frac{(N+2)\ln(m/M_1)}{\pi^2}$. The same holds with any $M>0$ in place of $M_1$: \[\lim_k[\dm_{(02)}(\eps_k;m)-\dm_{(02)}(\eps_k;M)]=-\frac{N+2}{\pi^2}\ln\frac{m^2}{M^2}\] (each $\dm_{(02)}(\eps_k;M)$ alone diverges like $\ln(1/\eps_k)$). \end{lemma} \begin{proof} (i) By (L2), $C_{M_1}(x)>C_m(x)>0$ pointwise, and the sum converges absolutely ($C_M\in\ell^1\cap\ell^\infty$). (ii) Pointwise $C_{M_1}^3-C_m^3=(C_{M_1}-C_m)(C_{M_1}^2+C_{M_1}C_m+C_m^2)\le3(C_{M_1}-C_m)C_{M_1}^2$, and by (L2) $C_{M_1}-C_m=(m^2-M_1^2)C_{M_1}*C_m\le(m^2-M_1^2)C_{M_1}*C_{M_1}$. For $f,g,h\in\ell^1(\Lam)$, $\sum_x\eps^3f(x)g(x)h(x)=\int_\BZ\!\int_\BZ\hat f(p)\hat g(q)\hat h(-p-q)$ (Fourier inversion and Fubini, $\hat h$ periodic); here $\hat f=G_{M_1}^2$, $\hat g=\hat h=G_{M_1}$, all even. By $G(q)G(p+q)\le\frac12[G(q)^2+G(p+q)^2]$ and the measure-preserving shift $q\mapsto p+q$ of $\BZ$, \[ \int_\BZ\!\int_\BZ G_{M_1}(p)^2G_{M_1}(q)G_{M_1}(p+q)\le\Big(\int_\BZ G_{M_1}^2\Big)^2\le\Big(\frac{\pi^2}{64M_1}\Big)^2 \] by (L1). Multiply by $3(m^2-M_1^2)$, and by $16(N+2)$ for $\bar L$. (iii) Write $G_1\defeq G_{M_1}$, $G_2\defeq G_m$ (at spacing $\eps$) and $\Delta\defeq G_1-G_2=(m^2-M_1^2)G_1G_2\in[0,(m^2-M_1^2)G_1^2]$, so that, by the formula of (ii), $F_\eps=\int_\BZ\!\int_\BZ[G_1G_1G_1-G_2G_2G_2](p,q,p+q)=T_1+T_2+T_3$ with $T_1=\iint\Delta(p)G_1(q)G_1(p+q)$, $T_2=\iint G_2(p)\Delta(q)G_1(p+q)$, $T_3=\iint G_2(p)G_2(q)\Delta(p+q)$, all $\ge0$ (telescoping). Consider $T_3$; the other two are the same argument after the measure-preserving change of variables $(p,q)\mapsto(q,-p-q\bmod\BZ)$ on $\BZ\times\BZ$. Split $\BZ\times\BZ=N_\eps\cup U_\eps$ with $U_\eps\defeq\{(p,q):|p_\mu+q_\mu|>\pi/\eps\text{ for some }\mu\}$, the set where $p+q$ is folded back into $\BZ$. On $U_\eps$, $\max(|p|,|q|)\ge\pi/(2\eps)$, so $\min(G_2(p),G_2(q))\le\eps^2$ by (L1), and with $\Delta\le(m^2-M_1^2)G_1G_2$ the contribution of $U_\eps$ is at most $2(m^2-M_1^2)\,\eps^2\int_\BZ G_2\int_\BZ G_1G_2=O(\eps)$ by (L1). On $N_\eps$ the integrand, extended by $0$, converges pointwise by (L4) to $(m^2-M_1^2)\bar G_m(p)\bar G_m(q)\bar G_{M_1}(p+q)\bar G_m(p+q)$ and is dominated by $(m^2-M_1^2)\bar G^c_{M_1}(p)\bar G^c_{M_1}(q)\bar G^c_{M_1}(p+q)^2\in L^1(\R^6)$ (substitute $r=p+q$ and use $\bar G^c(q)\bar G^c(r-q)\le\frac12[\bar G^c(q)^2+\bar G^c(r-q)^2]$). So $T_3\to T_3^\infty\defeq(m^2-M_1^2)\iint_{\R^6}\bar G_m(p)\bar G_m(q)\bar G_{M_1}(p+q)\bar G_m(p+q)$, and likewise $T_1$, $T_2$. \emph{Back to position space:} with the continuum Green's function $\bar C_M(x)=\frac{e^{-M|x|}}{4\pi|x|}$ (Fourier transform $\bar G_M$) and $\hat h\defeq\bar G_{M_1}\bar G_m\in L^1\cap L^2$, $h=\bar C_{M_1}*\bar C_m=(\bar C_{M_1}-\bar C_m)/(m^2-M_1^2)$, a bounded continuous function in $L^2$: for fixed $p$, $\int\bar G_m(q)\hat h(p+q)\,dq=\widehat{(\bar C_mh)}(p)$ as a Plancherel identity for the $L^2$ functions $\bar G_m$ and $\hat h$ ($\bar C_mh\in L^1\cap L^2$, being $O(|x|^{-1})$ near $0$ and exponentially small at infinity), and $\int\bar G_m(p)\widehat{(\bar C_mh)}(p)\,dp=\int d^3x\,\bar C_m(x)(\bar C_mh)(x)$ by Plancherel again ($\bar C_m\in L^2(\R^3)$). Hence $T_3^\infty=\int d^3x\,\bar C_m^2(\bar C_{M_1}-\bar C_m)$, $T_1^\infty=\int\bar C_{M_1}^2(\bar C_{M_1}-\bar C_m)$, $T_2^\infty=\int\bar C_m\bar C_{M_1}(\bar C_{M_1}-\bar C_m)$, each absolutely convergent (the integrands are $O(|x|^{-2})$ at $0$), and \[ \lim_{\eps\to0}F_\eps=\int_{\R^3}d^3x\,\big[\bar C_{M_1}(x)^3-\bar C_m(x)^3\big]=\frac{4\pi}{(4\pi)^3}\int_0^\infty\frac{e^{-3M_1x}-e^{-3mx}}{x}\,dx=\frac{\ln(m/M_1)}{16\pi^2} \] by Frullani's integral $\int_0^\infty(e^{-ax}-e^{-bx})\frac{dx}x=\ln\frac ba$. Multiplying by $-16(N+2)$ gives the limit of $S_{02,k}$. For an arbitrary $M>0$ in place of $M_1$ the same proof applies (for $M>m$ with the roles of the two masses exchanged); the divergence of a single $\dm_{(02)}(\eps;M)$ is that of $\sum_x\eps^3C_M(x)^3$, whose terms near $x=0$ are of size $\eps^{-3}$ on $O(1)$ sites, with the logarithm from the region $\eps\ll|x|\ll1/M$, where $C_M(x)\approx(4\pi|x|)^{-1}$. \end{proof} \begin{remark}[the one-loop form and the $e=0$ limit]\label{rem:s14:lowlimits} Inserted into \eqref{eq:s14:struct}, the limits of \cref{lem:s14:A,lem:s14:B,lem:s14:C} are the one-loop closed form of \cref{r13:lem:s14:failset}(ii), and at $e=0$ the limit $\lim_ks^\infty_k(M_1)$ of \cref{prop:s14:T3}(vi) is exact; of the three low orders only $I_\infty$ depends on $\lmu$, and $I_\infty(\lmu)\le I^*$ for every $\lmu>0$ (\cref{lem:s14:B}(iv)). \end{remark} \space% % % % \subsection*{The two-loop orders at every photon mass} From here to the end of the proof of \Cref{prop:s14:T3} we use the units $m=1$ ($\lambda$ and $e^2$ in units of $m$, $\sigma$ in units of $m^2$; Remark~\ref{cpi:tl:units}), $N=2$ and $M_1=0.7$; ``$\eps_k\le1$'' means $\eps_k\le1/m$. The two-loop coefficients $S_{21,k}$, $S_{40,k}$ are bounded from below through the facts (G1)--(G5) below, proved after the list ((G1) in Lemma~\ref{cpi:tl:odd} with \Cref{lem:s14:low} and Lemmas~\ref{cpi:tl:F}, \ref{cpi:tl:e4}; (G2) in Lemma~\ref{cpi:tl:oneloop}; (G3) in Lemma~\ref{cpi:tl:dens}; (G4) and (G5) where they are stated), which are written on periodic tori $T$ of side $\ell_T\ge\ell_0$ with $\ell_T/\eps\in2\N$, in units $m=1$, and carry the photon mass $\mu$ of the photon propagator $\hat D=(\hat k^2+\mu^2)^{-1}$ as a parameter; below we write it $\mu_\gamma$ (a parameter of (G1)--(G5) below; it is not the fixed reference photon mass $\mu_*$ of the scheme of Definition~\ref{r13:def:mixedW}) and read it at $\mu_\gamma=\lmu$. Its objects are the following. \begin{enumerate}[label=\textup{(G\arabic*)},leftmargin=*] \item \emph{The torus coefficient} $a_{K,T}(M)$: King's recursion \eqref{eq:s14:CTdef} computed on $T$ with the periodic propagators of $T$ at scalar mass $M$ and photon mass $\mu_\gamma$. Read at $\mu_\gamma=\lmu$, this is the bare coefficient $\alpha^{\mathrm{tor}}_{k,T}(M)=M^2+\dm_{k,T}(M)$ of King's torus convention (\Cref{def:s10:torus}(ii)): both are the same recursion with the same propagators. Its exact two-loop form (Lemma~\ref{cpi:tl:odd}: the odd orders vanish, and the orders $e^2\lambda$ and $e^4$ are the two-loop graphs) is, at $N=2$, \[ a_{K,T}(M)=M^2-16\lambda C_T(M)-2e^2D_T-e^2J_T(M)+128\lambda^2S_T(M)+16e^2\lambda F_T(M)+e^4Q_T(M), \] with $C_T(M)=\int_T\hat G_M$, $D_T=\int_T\hat D$, $J_T(M)=M^2\int_T\hat D\hat G_M$, $S_T(M)=\sum_{x\in T}\eps^3G_M(x)^3$, $F_T$ the order-$e^2\lambda$ and $Q_T$ the order-$e^4$ coefficient, the term $\frac34\eps^2D_T^2$ of $Q_T$ being independent of $M$; $M\mapsto a_{K,T}(M)$ is smooth. (The three low orders are \Cref{lem:s14:low} on the torus.) \item \emph{The one-loop majorants} (Lemma~\ref{cpi:tl:oneloop}): for $\alpha,\beta\ge0$ with $g\defeq\alpha+\beta>\frac32$, $a,b>0$, $\nu\defeq\min(a,b)$ and $\ell\in(0,\infty]$, \begin{equation}\label{eq:s14:Iab} \mathcal I_{\alpha,\beta}(a,b;\ell)\defeq c_NC_3+\frac{3\pi}{16\ell}C_2+\frac3{4\ell^2}C_1+\frac{a^{-2\alpha}b^{-2\beta}}{\ell^3},\qquad c_N=\frac{\pi^3}8, \end{equation} each $C_d=C_d(\alpha,\beta;a,b)$ the minimum of its admissible closed forms: for $C_3$ the $\nu$-form $\Gamma(g-\frac32)\nu^{3-2g}/((4\pi)^{3/2}\Gamma(g))$, the Beta form in $a$, $\frac1{4\pi^2}B(c,\alpha-c)a^{2c-2\alpha}$ with $c=\frac{3-2\beta}2$ if $\beta<\frac32$, the same with $(a,\alpha)\leftrightarrow(b,\beta)$ if $\alpha<\frac32$, the exact values $\frac1{4\pi(a+b)}$ at $(\alpha,\beta)=(1,1)$ and $\frac1{8\pi a(a+b)^2}$ at $(2,1)$, and the H\"older form $\gamma_3(g)a^{(3-2g)\alpha/g}b^{(3-2g)\beta/g}$, $\gamma_3(g)a^{3-2g}$ the closed form of $\int(u+a^2)^{-g}$; for $C_2$ the $\nu$-form $\nu^{2-2g}/(g-1)$, the Beta form $a^{2(1-g)}B(1-\beta,g-1)$ if $0<\beta<1$ and its symmetric form, and the H\"older form; for $C_1$ the $\nu$-form $\sqrt\pi\Gamma(g-\frac12)\nu^{1-2g}/\Gamma(g)$, the Beta form $a^{1-2g}B(\frac12-\beta,g-\frac12)$ if $0<\beta<\frac12$ and its symmetric form, and the H\"older form. For every torus of side $\ell$ and every $\eps$, $\int_T(\hat k^2+a^2)^{-\alpha}(\hat k^2+b^2)^{-\beta}\le\mathcal I_{\alpha,\beta}(a,b;\ell)$, on $\eps\Z^3$ with $\ell=\infty$; $\mathcal I_{\alpha,\beta}$ is decreasing in $\ell$, and at $(a,b)=(M,\mu)$ every admissible form is non-increasing in $M$. The majorants, all at $(M,\mu;\ell_0)$ with $\mu=\mu_\gamma$, are $\bar\tau=\mathcal I_{2,0}$, $\bar\psi=\mathcal I_{1,1}$, $\bar\delta=\mathcal I_{0,2}$, $\bar\Theta=\mathcal I_{1,3/4}+\mathcal I_{3/4,1}$, $\bar J_s=\mathcal I_{3-s/2,0}$ and $\bar J'_s=\mathcal I_{2-s/2,1}$ for $s\in\{0,1,\frac32\}$, $\bar\Lambda_0=\mathcal I_{3/4,1}$, $\bar\Lambda_1=\mathcal I_{5/4,1}+\mathcal I_{2,1/4}$, $\bar\Lambda_2=\mathcal I_{1/4,2}+\mathcal I_{1,5/4}$: fourteen distinct pairs $(\alpha,\beta)$, all with $g\ge\frac74>\frac32$; and $\bar\omega\defeq w+\hat E(\mu,\ell_0)$ with $w=0.252731$ (used as $w\le0.25273101$), $\hat E\defeq\min\{\bar E,\tilde E\}$, where $\bar E(\mu,\ell)=\frac{7.37}\ell+\frac{3\pi}{4\ell}E_1(\frac{\mu^2\ell^2}8)+\frac{12\pi\,\mathrm{erfc}(\mu\ell/(2\sqrt2))}{\mu\ell^2}+\frac{64e^{-\mu^2\ell^2/8}}{\mu^2\ell^3}$ (Lemma~\ref{cpi:tl:img}(a)) and $\tilde E\ge0$ the image bound of Lemma~\ref{cpi:tl:img}(b), defined for $\ell>\frac12$ (for $\ell\le\frac12$, $\hat E\defeq\bar E$); for $\eps\le1$ and $\ell_T\ge\ell_0$, $\eps D_T\le\bar\omega$ (Lemma~\ref{cpi:tl:img}(c)), and on $\eps\Z^3$ ($\ell_0=\infty$) one takes $\bar\omega=w$, since $\eps D_f\le w$ there (Lemma~\ref{cpi:tl:img}(c)). \item \emph{The slope bounds} (Lemma~\ref{cpi:tl:dens}): for $\eps\le1$, $\ell_T\ge\ell_0$, $M>0$, with the majorants at $(M,\mu_\gamma;\ell_0)$, $\bar\beta\defeq2M[2\bar\tau+8\bar\psi]$ and \begin{gather*} j\defeq2M\min\{\bar\tau,\bar\psi\},\qquad \varphi_3\defeq4M\big[2\sqrt2\,\bar\Theta\bar J_{3/2}+|\mu_\gamma^2-2M^2|\bar\psi\bar J_0+3\sqrt3\,\bar\omega\bar J_1\big]+\bar\beta\bar\tau,\\ b_4\defeq\varphi_A+28M\bar\delta\bar\tau+\varphi_{D2},\quad \varphi_A\defeq6M\big[2\sqrt2\,\bar\Theta\bar J'_{3/2}+|\mu_\gamma^2-2M^2|\bar\psi\bar J'_0+3\sqrt3\,\bar\omega\bar J'_1\big]+\bar\beta\bar\psi,\\ \varphi_{D2}\defeq2M\bar\psi(2\bar\psi+4\bar\delta)+\min\{8M\bar\psi(6\bar\psi+2\bar\tau+4\bar\delta),\ 2^{5/2}M(2\bar\Lambda_1+4\bar\Lambda_2)\bar\Lambda_0\}, \end{gather*} one has $0\le\partial_MJ_T\le j(M)$, $0\le-128\,\partial_MS_T\le768M\bar\tau(M)^2$, $|\partial_MF_T|\le\varphi_3(M)$ and $\partial_MQ_T\ge-b_4(M)$. \item \emph{The two-loop gap}: for $00$, every $\eps\le1$ and every torus of side $\ell_T\ge\ell_0$, $a_{K,T}(M_2)-a_{K,T}(M_1)\ge(M_2^2-M_1^2)+16\lambda h_T-e^2\bar j-\lambda^2\bar s-e^2\lambda\bar B_3-e^4\bar B_4^-$ with $h_T\defeq C_T(M_1)-C_T(M_2)\ge0$, $\bar j=\int_{M_1}^{M_2}j$, $\bar s=\int768M\bar\tau^2$, $\bar B_3=16\int\varphi_3$, $\bar B_4^-=\int b_4$; its proof writes each order as $\int_{M_1}^{M_2}\partial_M(\cdot)$ and applies (G3) \emph{order by order}: $-[J_T(M_2)-J_T(M_1)]\ge-\bar j$, $128[S_T(M_2)-S_T(M_1)]\ge-\bar s$, $16[F_T(M_2)-F_T(M_1)]\ge-\bar B_3$, $Q_T(M_2)-Q_T(M_1)\ge-\bar B_4^-$. \item \emph{The cell rule}: for a density $\phi\in\{j,768M\bar\tau^2,16\varphi_3,b_4\}$ and a cell $[a,b]\subset(0,\infty)$ let $\phi^{\rm cell}[a,b](\mu,\ell_0)$ be the number obtained from the formula for $\phi$ by evaluating every majorant at the left endpoint $a$, the prefactor $M$ at the right endpoint $b$, the prefactor $|\mu^2-2M^2|$ at its maximum over $\{a,b\}$ and $\bar\omega$ as it is; since the majorants are non-negative and non-increasing in $M$ and the formulas are sums, products and minima with non-negative coefficients, $\phi\le\phi^{\rm cell}[a,b]$ on $[a,b]$, and for a partition $M_1=M^{(0)}<\dots0$: with $\phi'(x')=s^{1/2}\phi(sx')$ and $A'(x')=s^{1/2}A(sx')$, King's action \cite[(2.4), p.~652]{K1} at spacing $\eps/s$, on the torus of side $\ell/s$, with masses $sM$, $s\mu$, couplings $se^2$, $s\lambda$ and bare coefficient $s^2a$, evaluated at $(\phi',A')$, equals the action at the original parameters evaluated at $(\phi,A)$, since each sum $\sum_{x'}(\eps/s)^3$ contributes $s^{-3}$, each lattice derivative $s$, each field $s^{1/2}$, and the phase $e^{\mathrm ie'\eps'A'}=e^{\mathrm ie\eps A}$. The renormalisation condition defining the counterterm, the vanishing of a zero-momentum two-point function, is covariant under this change, so $\dm_{(\alpha\beta)}$ is multiplied by $s^{2-\alpha/2-\beta}$ and $a_{K,T}(M)$ by $s^2$. Hence it suffices to prove every statement in units in which a fixed reference mass $m$ equals $1$; in these units ``$\eps\le1$'' means $\eps\le1/m$, a torus floor $\ell_0$ means $\ell_0/m$, and $\lambda,e^2$ stand for $\lambda/m,e^2/m$. \end{remark} \begin{lemma}[odd orders vanish]\label{cpi:tl:odd} Let $\dm_{(\alpha\beta),T}(\eps;M)$, $2\le\alpha+2\beta\le4$, be the coefficients of King's counterterm \cite[(3.32)--(3.33)]{K1} computed on a torus $T$ with the periodic propagators (the recursion \eqref{eq:s14:CTdef}: each coefficient is fixed, in the order in which the recursion determines it, by the vanishing of the order-$e^\alpha\lambda^\beta$ part of the amputated one-particle-irreducible two-point function of $\phi$ at zero momentum, all counterterm vertices of lower order inserted). Then $\dm_{(\alpha\beta),T}=0$ for odd $\alpha$; hence \[ \dm_T=e^2\dm_{(20),T}+\lambda\dm_{(01),T}+e^4\dm_{(40),T}+e^2\lambda\dm_{(21),T}+\lambda^2\dm_{(02),T}, \] and $a_{K,T}(M)=M^2+\dm_T(\eps;M)$ is a polynomial in $(e^2,\lambda)$, which is the form (G1). \end{lemma} \begin{proof} \emph{Vertices.} Expanding the hopping term of King's action in $e$, the part of order $e^n$ is homogeneous of degree $n$ in $A$ (a sum of terms $A_{x,\mu}^n$ times a quadratic form in $\phi$); the quartic and counterterm vertices contain no $A$. So a hopping vertex of order $e^n$ has exactly $n$ photon legs, every other vertex has none, and the photon covariance does not depend on $e$. \emph{Contractions.} Under the free measure $A$ and $\phi$ are independent and the centred Gaussian law of $A$ is invariant under $A\mapsto-A$; so every polynomial of odd degree in $A$ has expectation $0$, and a graph with an odd number of photon legs vanishes (the external legs are scalar). \emph{Induction.} Order the pairs $(\alpha,\beta)$ as the recursion determines them and assume $\dm_{(\alpha'\beta')}=0$ for the earlier pairs with $\alpha'$ odd. A graph of order $e^\alpha\lambda^\beta$ other than the single counterterm vertex, with hopping vertices of orders $n_1,\dots,n_r$ and counterterm vertices of orders $(\alpha_i,\beta_i)$, has $\alpha=\sum n_j+\sum\alpha_i$; it vanishes unless every $\alpha_i$ is even (the earlier pairs), and then unless $\sum n_j$, the number of photon legs, is even. So the order-$e^\alpha\lambda^\beta$ part of the two-point function vanishes for odd $\alpha$, and the recursion gives $\dm_{(\alpha\beta)}=0$. Of the pairs $(2,0),(0,1),(3,0),(1,1),(4,0),(2,1),(0,2)$ the five with even $\alpha$ remain. \end{proof} The two-loop form (G1) collects the five orders: the three low orders are Lemma~\ref{lem:s14:low} on the torus (its proof is a computation graph by graph with the free propagators and holds with $\int_T$ in place of the Brillouin-zone integral and the sites of $T$ in place of $\Lam$), and the coefficients $F_T$ and $Q_T$ of the orders $e^2\lambda$ and $e^4$ are the finite sums of two-loop lattice graphs with the periodic propagators of Lemmas~\ref{cpi:tl:F} and \ref{cpi:tl:e4} below; at fixed $\eps$ and $T$ each term is a finite sum of products of factors $\hat G_M(k)$ and of $M$-independent factors, a rational function of $M^2$ without poles on $(0,\infty)$, which is the smoothness of (G1). The one-loop majorants (G2) are Lemma~\ref{cpi:tl:oneloop} and the slope bounds (G3) are Lemma~\ref{cpi:tl:dens}, both proved below from the lattice tools (Lemmas~\ref{cpi:tl:dos}--\ref{cpi:tl:split}), the structure of the subtracted self-energy (Lemma~\ref{cpi:tl:Pi}) and the $e^4$ decomposition (Lemma~\ref{cpi:tl:e4}); (G4) and (G5) follow as stated there. \space% % % For a torus $T$ of side $\ell$ and $M>0$ let $G_{M,f}$ be the massive propagator $(-\Delta_\eps+M^2)^{-1}$ on $\eps\Z^3$ and $G_{M,T}$ the periodic one on $T$, and put $E_T(M)\defeq G_{M,T}(0)-G_{M,f}(0)=\sum_{z\in\ell\Z^3\setminus\{0\}}G_{M,f}(z)$, the image excess at coincident points (every term is positive by the random-walk representation of $G_{M,f}$); $c_N=\pi^3/8$ and $w=\eps\int_f\hat k^{-2}=0.2527310098\ldots$ are the constants of (G2) ($\int_f\defeq\int_{\mathsf B_\eps}\frac{d^3k}{(2\pi)^3}$ is the Brillouin-zone integral, as in the next subsection). \begin{lemma}[torus images; the constant $\bar\omega$]\label{cpi:tl:img} Let $T$ be a torus of side $\ell=\ell_T$ and $M>0$. \begin{enumerate} \item[(a)] $E_T(M)\le\bar E(M,\ell)\defeq\dfrac{7.37}\ell+\dfrac{3\pi}{4\ell}E_1\Big(\dfrac{M^2\ell^2}8\Big)+\dfrac{12\pi\,\mathrm{erfc}(M\ell/(2\sqrt2))}{M\ell^2}+\dfrac{64e^{-M^2\ell^2/8}}{M^2\ell^3}$, $E_1$ the exponential integral; $\bar E$ is decreasing in $\ell$ and does not depend on $\eps$. \item[(b)] For $z\in\eps\Z^3\setminus\{0\}$ and $r\defeq\abs z_\infty$: $G_{M,f}(z)\le\dfrac{c_N}{\pi(2r-\eps)}\Big(1+\dfrac{\eps M}{2\sqrt2}\Big)^{1-2r/\eps}$. Hence, for $\eps\le1$ and $\ell>\frac12$, with $\kappa(x)\defeq\ln\big(1+\frac x{2\sqrt2}\big)\big/\frac x{2\sqrt2}$, \[ E_T(M)\le\tilde E(M,\ell)\defeq\sum_{j\ge1}(24j^2+2)\,\frac{c_N(1+M/(2\sqrt2))}{\pi(2j\ell-1)}\,e^{-\kappa(M)Mj\ell/\sqrt2}, \] which is decreasing in $\ell>\frac12$. \item[(c)] Put $\hat E\defeq\min\{\bar E,\tilde E\}$ for $\ell>\frac12$ and $\hat E\defeq\bar E$ for $\ell\le\frac12$; $\hat E$ is non-increasing in $\ell$. For $\eps\le1$ and $\ell_T\ge\ell_0$: $\eps D_T\le\bar\omega\defeq w+\hat E(\mu,\ell_0)$, where $D_T=\int_T\hat D$ is the periodic photon propagator at coincident points. On $\eps\Z^3$, $\eps D_f\le w$ for every $\eps$. \end{enumerate} \end{lemma} \begin{proof} (a) By the random-walk representation, $G_{M,f}(x)=\int_0^\infty e^{-M^2t}\prod_\mu p_t(x_\mu)\,dt$, where $p_t(x)=\eps^{-1}\mathbb P(X_t=x)$ is the kernel of the continuous-time random walk $X_t$ on $\eps\Z$ with jumps $\pm\eps$ at rate $\eps^{-2}$ each, started at $0$, so that $\Var X_t=2t$; $p_t$ is non-increasing in $\abs x$ (for $P_n\defeq\mathbb P(X_t=n\eps)$ and $q_n\defeq P_n-P_{n+1}$, $n\ge0$, the forward equation gives $\dot q=Lq$ with $L+3\eps^{-2}$ entrywise non-negative, so $q(t)\ge0$ follows from $q(0)=(1,0,0,\dots)$), and $p_t(0)\le a_*t^{-1/2}$ with $a_*\defeq\sqrt\pi/4$ (from $\mathbb P(X_t=0)=\int_{-\pi}^\pi\frac{d\theta}{2\pi}e^{-v(1-\cos\theta)}$, $v=2t/\eps^2$, and $1-\cos\theta\ge2\theta^2/\pi^2$ on $[0,\pi]$). Let $p^0\defeq p_t(0)$ and $\delta(t)\defeq\sum_{n\ne0}p_t(n\ell)$. For $n\ge1$ the interval $(n\ell-\ell/2,n\ell]$ contains at least $\ell/(4\eps)$ sites, at each of which $p_t\ge p_t(n\ell)$, so $p_t(n\ell)\le\frac4\ell\mathbb P(X_t\in(n\ell-\frac\ell2,n\ell])$, and symmetrically for $n\le-1$; these intervals are disjoint, so by Chebyshev's inequality $\delta(t)\le\frac4\ell\mathbb P(\abs{X_t}\ge\frac\ell2)\le\frac4\ell\min(1,8t/\ell^2)$. By the image formula (Lemma~\ref{cpi:tl:pos}(b) below) and Tonelli, \[ E_T(M)=\int_0^\infty e^{-M^2t}\big[(p^0+\delta)^3-(p^0)^3\big]dt=\int_0^\infty e^{-M^2t}\big[3(p^0)^2\delta+3p^0\delta^2+\delta^3\big]dt. \] Split at $t_0\defeq\ell^2/8$, use $\min(1,8t/\ell^2)=t/t_0$ and $e^{-M^2t}\le1$ on $t\le t_0$, and $\min=1$ on $t\ge t_0$. The first term gives $\frac{3\pi}{4\ell t_0}\int_0^{t_0}dt+\frac{3\pi}{4\ell}\int_{t_0}^\infty\frac{e^{-M^2t}}tdt=\frac{3\pi}{4\ell}+\frac{3\pi}{4\ell}E_1(\frac{M^2\ell^2}8)$; the second gives $48a_*\ell^{-2}t_0^{-2}\int_0^{t_0}t^{3/2}dt+48a_*\ell^{-2}\int_{t_0}^\infty t^{-1/2}e^{-M^2t}dt=\frac{24\sqrt\pi}{5\sqrt8\,\ell}+\frac{12\pi\,\mathrm{erfc}(M\ell/(2\sqrt2))}{M\ell^2}$; the third gives $\frac{64}{\ell^3}\big[\frac{t_0}4+\frac{e^{-M^2t_0}}{M^2}\big]=\frac2\ell+\frac{64e^{-M^2\ell^2/8}}{M^2\ell^3}$. The constants add to $\frac{3\pi}4+\frac{24\sqrt\pi}{5\sqrt8}+2=7.3641\ldots\le7.37$. Each term of $\bar E$ is a product of positive decreasing functions of $\ell$. (b) By Lemma~\ref{cpi:tl:pos}(a) below, $G_M=\int_{M^2}^\infty(G_{\sqrt\sigma}*G_{\sqrt\sigma})\,d\sigma$. By the lattice symmetries we may take $z_1=r>0$. In $(G_{\sqrt\sigma}*G_{\sqrt\sigma})(z)=\int_fe^{\mathrm ik\cdot z}\hat G_{\sqrt\sigma}(k)^2$ shift $k_1\to k_1+\mathrm i\eta$; the integrand is periodic, so the vertical sides cancel, provided the denominator does not vanish. Since $\mathrm{Re}\,\frac4{\eps^2}\sin^2\frac{\eps(k_1+\mathrm i\eta)}2\ge\hat k_1^2-\frac4{\eps^2}\sinh^2\frac{\eps\eta}2$, the choice $\frac4{\eps^2}\sinh^2\frac{\eps\eta_\sigma}2=\frac\sigma2$ gives $|\hat G_{\sqrt\sigma}|\le(\hat k^2+\sigma/2)^{-1}$ on the shifted line, and the density-of-states bound (Lemma~\ref{cpi:tl:dos} below) gives $|(G_{\sqrt\sigma}*G_{\sqrt\sigma})(z)|\le e^{-\eta_\sigma r}c_N/(8\pi(\sigma/2)^{1/2})$, with $\eta_\sigma=\frac2\eps\operatorname{arsinh}\frac{\eps\sqrt\sigma}{2\sqrt2}\ge\frac2\eps\ln(1+\frac{\eps\sqrt\sigma}{2\sqrt2})$. With $\sigma=u^2$, $\int_M^\infty\frac{c_N\sqrt2}{4\pi}(1+\frac{\eps u}{2\sqrt2})^{-2r/\eps}du$ is the first bound. For $\tilde E$ use the image formula, $\#\{n\in\Z^3:\abs n_\infty=j\}=24j^2+2$ and $r=j\ell$; each factor of the first bound is non-decreasing in $\eps$ while $2j\ell-\eps>0$ ($\ln(1+y)/y$ is decreasing), and for $\ell>\frac12$ this holds for every $\eps\le1$, so $\eps\le1$ gives the value at $\eps=1$. (c) $D_T=D_f+E_T(\mu)$ by the image formula, $\eps D_f\le\eps\int_f\hat k^{-2}=w$, and $\eps E_T(\mu)\le E_T(\mu)\le\hat E(\mu,\ell_T)\le\hat E(\mu,\ell_0)$ for $\eps\le1$, by (a), (b) and the monotonicity of $\hat E$ in $\ell$. \end{proof} \space% % % % % % % \subsection*{The one-loop majorants and the slope bounds} In this subsection $\mathsf X\in\{T,f\}$ is a torus $T$ of side $\ell_T$ with $\ell_T/\eps\in2\N$ or the lattice $\eps\Z^3$ (written $f$); $\int_T\defeq|T|^{-1}\sum_{k\in T^*}$ over the dual lattice of $T$, $\int_f\defeq\int_{\mathsf B_\eps}\frac{d^3k}{(2\pi)^3}$ over the Brillouin zone $\mathsf B_\eps=[-\pi/\eps,\pi/\eps)^3$, $\hat k_\mu=\frac2\eps\sin\frac{\eps k_\mu}2$, $\hat k^2=\sum_\mu\hat k_\mu^2$, $\hat G=\hat G_M(k)=(\hat k^2+M^2)^{-1}$, $\hat D(k)=(\hat k^2+\mu^2)^{-1}$ with the photon mass $\mu=\mu_\gamma$, and $G_{M,\mathsf X}$, $D_{\mathsf X}$ are the position kernels (periodic on $T$), so that $C_{\mathsf X}=\int_{\mathsf X}\hat G_M=G_{M,\mathsf X}(0)$ and $D_{\mathsf X}=\int_{\mathsf X}\hat D$. The one-loop sums are $\tau_{\mathsf X}\defeq\int\hat G^2$, $\Xi_{\mathsf X}(p)\defeq\int_{\mathsf X,p'}\hat D(p-p')\hat G_M(p')$ (so $\Xi_{\mathsf X}(0)=\int\hat D\hat G$), $\delta_{\mathsf X}\defeq\int\hat D^2$ and $\Theta_{\mathsf X}\defeq\int\hat D^{3/4}\hat G+\int\hat D\hat G^{3/4}$. With $\psi\defeq(\phi_1+\mathrm i\phi_2)/\sqrt2$, so that $|\phi|^2=2|\psi|^2$, the order-$e^2$ part of the shift $\Sigma$ of the coefficient of $|\psi|^2$ (the inverse of the full propagator being $\hat p^2+M^2+\dm+\Sigma(p)$, so that $\Sigma$ is the negative of the self-energy in the convention of \eqref{eq:s14:CTdef} and the counterterm condition reads $\dm_{(\alpha\beta)}e^\alpha\lambda^\beta=-\Sigma^{(\alpha\beta)}(0)$) is $e^2\Sigma_{2,\mathsf X}$ with \[ \Sigma_{2,\mathsf X}(p;M)\defeq D_{\mathsf X}\sum_\mu\cos(\eps p_\mu)-\int_{\mathsf X,p'}\hat D(p-p')\,\widehat{(p+p')}^2\,\hat G_M(p'), \] the seagull and the exchange graph, so that \[ \dm_{(20),T}=-\Sigma_{2,T}(0;M)=-2D_T-J_T(M) \] (\cref{lem:s14:low}(ii) on $T$); the \emph{subtracted self-energy} is \[ \pi_{\mathsf X}(p;M)\defeq\Sigma_{2,\mathsf X}(p;M)-\Sigma_{2,\mathsf X}(0;M), \] explicitly \[ \pi_T(p;M)=-D_T\sum_\mu(1-\cos\eps p_\mu)-\int_{T,p'}\big[\hat D(p-p')\widehat{(p+p')}^2-\hat D(p')\hat p'^2\big]\hat G_M(p'). \] \begin{lemma}[density of states]\label{cpi:tl:dos} For every $\eps>0$ and every non-increasing $F:[0,\infty)\to[0,\infty)$, $\int_{\mathsf B_\eps}F(\hat k^2)\,d^3k\le c_N\int_{\R^3}F(|k|^2)\,d^3k$, $c_N=\pi^3/8$. \end{lemma} \begin{proof} On $[-\pi,\pi]$, $|\sin(y/2)|\ge|y|/\pi$, so $\hat k^2\ge(4/\pi^2)|k|^2$ on $\mathsf B_\eps$ and $\{k\in\mathsf B_\eps:\hat k^2\le E\}\subset\{|k|\le\frac\pi2E^{1/2}\}$, a set of volume $c_N\frac{4\pi}3E^{3/2}$. By the layer-cake formula $\int_{\mathsf B_\eps}F(\hat k^2)=\int_0^\infty|\{k\in\mathsf B_\eps:F(\hat k^2)>t\}|\,dt$; since $F$ is non-increasing, $\{u:F(u)>t\}$ is an interval $[0,E_t)$ or $[0,E_t]$, so the set in the integrand has measure at most $c_N\frac{4\pi}3E_t^{3/2}\le c_N|\{k\in\R^3:F(|k|^2)>t\}|$. Integrate in $t$. \end{proof} \begin{lemma}[torus Riemann sums]\label{cpi:tl:riem} Let $F$ be as in \cref{cpi:tl:dos} and $T$ a torus of side $\ell$. Then \[ \int_TF(\hat k^2)\le c_N\int_{\R^3}F(|k|^2)\tfrac{d^3k}{(2\pi)^3}+\tfrac{3\pi}{16\ell}\int_0^\infty F(u)\,du+\tfrac3{4\ell^2}\int_\R F(q^2)\,dq+\tfrac{F(0)}{\ell^3}. \] \end{lemma} \begin{proof} \emph{One coordinate.} Let $g$ be even and non-increasing in $|k|$ on $[-\pi/\eps,\pi/\eps]$, $h=2\pi/\ell$, $N'=\ell/\eps$. The points of $T^*$ in one coordinate are $jh$, $-N'/2\frac32$, let $a,b>0$, $\nu\defeq\min(a,b)$, $F(u)\defeq(u+a^2)^{-\alpha}(u+b^2)^{-\beta}$, and let $\mathcal I_{\alpha,\beta}(a,b;\ell)$, $\ell\in(0,\infty]$, be as in \eqref{eq:s14:Iab}, with $\gamma_d(g)x^{d-2g}$ the closed form of $\int_{\R^3}(|k|^2+x^2)^{-g}\frac{d^3k}{(2\pi)^3}$, $\int_0^\infty(u+x^2)^{-g}du$, $\int_\R(q^2+x^2)^{-g}dq$ for $d=3,2,1$ in the H\"older forms. Then: \textup{(a)} for every $\eps$ and every torus of side $\ell$, $\int_TF(\hat k^2)\le\mathcal I_{\alpha,\beta}(a,b;\ell)$, and $\int_fF(\hat k^2)\le\mathcal I_{\alpha,\beta}(a,b;\infty)=c_NC_3$; \textup{(b)} $\mathcal I_{\alpha,\beta}$ is decreasing in $\ell$; \textup{(c)} at $(a,b)=(M,\mu)$ every admissible form is non-increasing in $M$, hence so is $\mathcal I_{\alpha,\beta}(M,\mu;\ell)$. A factor $\hat p^s$ with $0\le s<2\alpha$ is absorbed by $\hat p^s\le(\hat p^2+a^2)^{s/2}$, i.e.\ $\alpha\to\alpha-\frac s2$. The one-loop majorants of \textup{(G2)}, all at $(M,\mu;\ell_0)$, bound the following sums ($s\in\{0,1,\frac32\}$): $\bar\tau=\mathcal I_{2,0}$ bounds $\tau_{\mathsf X}$; $\bar\psi=\mathcal I_{1,1}$ bounds $\Xi_{\mathsf X}(0)$; $\bar\delta=\mathcal I_{0,2}$ bounds $\delta_{\mathsf X}$; $\bar\Theta=\mathcal I_{1,3/4}+\mathcal I_{3/4,1}$ bounds $\Theta_{\mathsf X}$; $\bar J_s=\mathcal I_{3-s/2,0}$ bounds $\int\hat G^3\hat p^s$; $\bar J'_s=\mathcal I_{2-s/2,1}$ bounds $\int\hat D\hat G^2\hat p^s$; $\bar\Lambda_0=\mathcal I_{3/4,1}$ bounds $\int\hat k^{1/2}\hat G\hat D$; $\bar\Lambda_1=\mathcal I_{5/4,1}+\mathcal I_{2,1/4}$ bounds $\int\hat G^{5/4}\hat D+\int\hat G^2\hat D^{1/4}$; $\bar\Lambda_2=\mathcal I_{1/4,2}+\mathcal I_{1,5/4}$ bounds $\int\hat G^{1/4}\hat D^2+\int\hat G\hat D^{5/4}$. Every instance has $g\ge\frac74$, and by \textup{(a)}, \textup{(b)} each one-loop sum on a torus of side $\ell_T\ge\ell_0$, and on $\eps\Z^3$ with $\ell_0=\infty$, is at most its majorant. \end{lemma} \begin{proof} (a) $F$ is non-increasing, so \cref{cpi:tl:riem} applies (on $\eps\Z^3$, \cref{cpi:tl:dos}). Each closed form is the integral of a pointwise larger function: $F\le(u+\nu^2)^{-g}$ (the $\nu$-forms); or $(u+b^2)^{-\beta}\le u^{-\beta}$ (the Beta forms, and symmetrically); or H\"older's inequality with the exponents $g/\alpha$, $g/\beta$ (the H\"older forms). They are evaluated by $\int_0^\infty t^{c-1}(t+a^2)^{-x}dt=a^{2(c-x)}B(c,x-c)$ ($00$ and $\mathsf X\in\{T,f\}$. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*,nosep] \item $G_{M,\mathsf X}(x)>0$ for every $x$, and $-\partial_{M^2}G_{M,\mathsf X}=G_{M,\mathsf X}*G_{M,\mathsf X}\ge0$ pointwise. \item $G_{M,T}(x)=\sum_{n\in\Z^3}G_{M,f}(x+n\ell_T)$, the series converging absolutely, and likewise for $D$ and for $G*G$. In particular $E_T(M)=C_T(M)-C_f(M)=\sum_{n\ne0}G_{M,f}(n\ell_T)\ge0$ and $D_T-D_f=E_T(\mu)$. \item For $s,t>0$ the position kernels of $\hat G_M^s$ and $\hat D^t$ are pointwise positive. Hence, for every $q$ ($q\in T^*$ on a torus), \begin{gather*} \int_{\mathsf X,P}\hat G(P)^s\hat D(P\pm q)^t\le\int_{\mathsf X}\hat G^s\hat D^t,\qquad\int_{\mathsf X,P}\hat G(P)\hat G(P+q)\le\int_{\mathsf X}\hat G^2,\\ \int_{\mathsf X,P}\hat D(P-q)\hat D(P+q)\le\int_{\mathsf X}\hat D^2, \end{gather*} the first also with $s=0$. Also $|(K*K')(x)|\le(K*K')(0)$ whenever $\hat K,\hat K'\ge0$. \end{enumerate} \end{lemma} \begin{proof} (a) Let $P$ be the nearest-neighbour adjacency matrix of $\mathsf X$ (entries $\ge0$, row sums $6$). Then $-\Delta_\eps+M^2=(6\eps^{-2}+M^2)(1-P/(6+\eps^2M^2))$ and $\|P\|\le6<6+\eps^2M^2$, so the Neumann series of the inverse has non-negative terms, its zeroth term is positive on the diagonal and its $n$-th term reaches graph distance $n$; as $\mathsf X$ is connected, the kernel is positive everywhere. Moreover $-\partial_{M^2}(-\Delta_\eps+M^2)^{-1}=(-\Delta_\eps+M^2)^{-2}$, whose kernel is $G*G$. (b) $\hat G_M$, $\hat D$ and $\hat G_M^2$ are $C^\infty$ and $(2\pi/\eps)$-periodic in each coordinate, so their position kernels on $\eps\Z^3$ decay faster than every power and the image series converge absolutely; for $k\in T^*$, $\sum_{x\in T}\eps^3e^{-\mathrm ik\cdot x}\sum_nG_{M,f}(x+n\ell_T)=\sum_{y\in\eps\Z^3}\eps^3e^{-\mathrm ik\cdot y}G_{M,f}(y)=\hat G_M(k)$, so the image sum has the Fourier coefficients of $G_{M,T}$ on $T$. Positivity of $E_T$ follows from (a). (c) By subordination, $(-\Delta_\eps+M^2)^{-s}=\Gamma(s)^{-1}\int_0^\infty\vartheta^{s-1}e^{-\vartheta M^2}e^{\vartheta\Delta_\eps}d\vartheta$, and $e^{\vartheta\Delta_\eps}=e^{-6\vartheta/\eps^2}e^{\vartheta P/\eps^2}$ has a positive kernel; the same for $\hat D^t$. For even kernels $K,K'\ge0$, Parseval gives $\int_P\hat K(P)\hat K'(P-q)=\sum_x\eps^3K(x)K'(x)e^{\mathrm iq\cdot x}$, whose modulus is at most its value at $q=0$; for $s=0$ the left side does not depend on $q$. The last claim: $(K*K')(x)=\int e^{\mathrm ik\cdot x}\hat K\hat K'$. \end{proof} \begin{lemma}[splitting and lattice identities]\label{cpi:tl:split} \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*,nosep] \item For $q\ne0$ and $0<\theta\le\min(s,t)$ (with $\hat G^0\defeq1$, $\hat D^0\defeq1$): $\int_{\mathsf X,P}\hat G(P)^s\hat D(P-q)^t\le(2/\hat q)^{2\theta}\big[\int_{\mathsf X}\hat G^{s-\theta}\hat D^t+\int_{\mathsf X}\hat G^s\hat D^{t-\theta}\big]$. \item $|\widehat{(a+b)}_\nu|\le|\hat a_\nu|+|\hat b_\nu|$, hence $\widehat{(a+b)}^2\le2\hat a^2+2\hat b^2$; and $\widehat{(p+p')}^2\le2\widehat{(p-p')}^2+8\hat p'^2$. \item $|\hat D(P-q)-\hat D(P+q)|\le4\hat P\hat q\,\hat D(P-q)\hat D(P+q)$. \item Componentwise, $\widehat{(a+b)}_\mu^2+\widehat{(a-b)}_\mu^2=2\hat a_\mu^2+2\hat b_\mu^2-\eps^2\hat a_\mu^2\hat b_\mu^2$. \end{enumerate} \end{lemma} \begin{proof} (a) By $|\sin(x+y)|\le|\sin x|+|\sin y|$ componentwise and Minkowski, $\hat q\le\hat P+\widehat{(q-P)}$, so either $\hat P\ge\hat q/2$, and then $\hat G(P)^\theta\le(2/\hat q)^{2\theta}$, or $\widehat{(P-q)}\ge\hat q/2$, and then $\hat D(P-q)^\theta\le(2/\hat q)^{2\theta}$; bound the integrand by the sum of the two cases and apply \cref{cpi:tl:pos}(c). (b) The first claim is the same subadditivity; the second uses $|\sin2x|\le2|\sin x|$ with $p+p'=(p-p')+2p'$ and $(x+y)^2\le2x^2+2y^2$. (c) $\hat D(P-q)-\hat D(P+q)=\hat D(P-q)\hat D(P+q)[\widehat{(P+q)}^2-\widehat{(P-q)}^2]$ and $\sin^2A-\sin^2B=\sin(A+B)\sin(A-B)$ give $|\widehat{(P+q)}^2-\widehat{(P-q)}^2|=\sum_\mu\frac4{\eps^2}|\sin(\eps P_\mu)\sin(\eps q_\mu)|\le4\sum_\mu|\hat P_\mu||\hat q_\mu|\le4\hat P\hat q$. (d) $4-4\cos a\cos b=2(2-2\cos a)+2(2-2\cos b)-(2-2\cos a)(2-2\cos b)$, divided by $\eps^2$. \end{proof} \begin{lemma}[the order $e^2\lambda$]\label{cpi:tl:F} On a torus $T$, $\dm_{(21),T}=16F_T(M)$ with \[ F_T(M)\defeq\int_T\hat G_M^2\,\pi_T(\cdot;M). \] \end{lemma} \begin{proof} Write the hopping term of King's action as $\sum_x\eps^3\bar\psi(-\Delta_A\psi)$ and let $V_n$ be the part of $-\Delta_A+\Delta_\eps$ of degree $n$ in $A$: $(V_n\psi)(x)=c_n\sum_\mu[A_\mu(x)^n\psi(x+\eps e_\mu)+(-1)^nA_\mu(x-\eps e_\mu)^n\psi(x-\eps e_\mu)]$, $c_n=-\eps^{-2}(\mathrm i\eps e)^n/n!$; a vertex $\lambda$ carries $4\lambda|\psi|^4$ and a counterterm vertex $c_{\alpha,\beta}$ the factor $e^\alpha\lambda^\beta\dm_{(\alpha\beta)}$. The coefficient $\dm_{(21),T}e^2\lambda$ is minus the zero-momentum value of the sum of the amputated one-particle-irreducible two-point graphs of order $e^2\lambda$ other than the single vertex $c_{2,1}$ (the chain of scalar-irreducible parts sums to a geometric series, and a scalar-irreducible graph that is not one-particle irreducible contains a one-photon part, whose value vanishes by the inversion $x\mapsto-x$ of $T$, which multiplies each vertex of type $V_n$ by $(-1)^n$). Apart from $c_{2,1}$, the vertex sets of order $e^2\lambda$ with non-vanishing factors are $\{\lambda,V_2\}$, $\{\lambda,V_1,V_1'\}$, $\{\lambda,c_{2,0}\}$, $\{V_2,c_{0,1}\}$, $\{V_1,V_1',c_{0,1}\}$, $\{c_{2,0},c_{0,1}\}$ (the odd counterterms vanish, \cref{cpi:tl:odd}, and $V_n$ carries $e^n$), and the one-particle-irreducible graphs on them are exactly: (a$_1$)--(a$_3$) the order-$\lambda$ tadpole $16\lambda\int_T\hat G_M$ with the seagull, the exchange graph or the counterterm $c_{2,0}$ inserted on its loop line; (b$_1$) the exchange graph with the tadpole $16\lambda C_T$ inserted on its internal scalar line and (b$_2$) the same graph with the insertion $\lambda\dm_{(01),T}=-16\lambda C_T$; (c) the graphs with external legs at $\lambda$ and at $V_1$, one scalar line $\lambda$--$V_1$, two scalar lines $\lambda$--$V_1'$ and the photon line $V_1$--$V_1'$ (and the same with $V_1,V_1'$ exchanged); all other scalar pairings are disconnected or have a bridge. The sum of (a$_1$)--(a$_3$) is the tadpole with $\hat G$ replaced by $-\hat G(e^2\Sigma_{2,T}+e^2\dm_{(20),T})\hat G=-e^2\hat G^2\pi_T$, that is $-16e^2\lambda\int_T\hat G_M^2\pi_T$; (b$_1$) and (b$_2$) cancel; each graph (c) vanishes before the sum over the position $w$ of its $\lambda$-vertex, since for $V_1'$ on the bond $(z,z+\eps e_\mu)$ both its scalar lines end at $w$ and the sum over its two orientations is $c_1[G_{M,T}(z+\eps e_\mu-w)G_{M,T}(z-w)-G_{M,T}(z-w)G_{M,T}(z+\eps e_\mu-w)]=0$. Hence $\dm_{(21),T}=16\int_T\hat G_M^2\pi_T$. \end{proof} \begin{lemma}[structure of the subtracted self-energy]\label{cpi:tl:Pi} Put $Y_\mu(p)\defeq\int_{\mathsf X,p'}\hat D(p-p')\hat p'^2_\mu\hat G_M(p')$. Then $\Xi_{\mathsf X}(p)\in[0,\Xi_{\mathsf X}(0)]$, $Y_\mu(p)\in[0,D_{\mathsf X}]$, and: \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*,nosep] \item $\pi_{\mathsf X}(p;M)=-\big\{2\hat p^2\Xi_{\mathsf X}(p)+(\mu^2-2M^2)[\Xi_{\mathsf X}(p)-\Xi_{\mathsf X}(0)]-\eps^2\sum_\mu\hat p_\mu^2Y_\mu(p)+D_{\mathsf X}\tfrac{\eps^2}2\hat p^2\big\}$; \item $|\pi_{\mathsf X}(p;M)|\le2\sqrt2\,\Theta_{\mathsf X}(M)\hat p^{3/2}+|\mu^2-2M^2|\,\Xi_{\mathsf X}(0)+3\sqrt3\,(\eps D_{\mathsf X})\hat p$; \item $0\le\partial_M\Sigma_{2,\mathsf X}(p;M)\le\beta_{\mathsf X}(M)\defeq2M[2\tau_{\mathsf X}+8\Xi_{\mathsf X}(0)]$, and $|\partial_M\pi_{\mathsf X}(p;M)|\le\beta_{\mathsf X}(M)$ for all $p$. \end{enumerate} \end{lemma} \begin{proof} The ranges of $\Xi_{\mathsf X}$, $Y_\mu$ follow from \cref{cpi:tl:pos}(c) and $\hat p'^2_\mu\hat G\le1$. (a) By \cref{cpi:tl:split}(d) with $a=p$, $b=p'$, summed over $\mu$: $\widehat{(p+p')}^2=2\hat p^2+2\hat p'^2-\eps^2\sum_\mu\hat p_\mu^2\hat p'^2_\mu-\hat k^2$ with $k=p-p'$. With $\hat p'^2\hat G=1-M^2\hat G$ and $\hat k^2\hat D=1-\mu^2\hat D$, the exchange integral is $2\hat p^2\Xi(p)+(\mu^2-2M^2)\Xi(p)-\eps^2\sum_\mu\hat p_\mu^2Y_\mu(p)+2D_{\mathsf X}-C_{\mathsf X}$; the constant $2D_{\mathsf X}-C_{\mathsf X}$ cancels in $\pi=\Sigma_2(p)-\Sigma_2(0)$, and the seagull part of $\pi$ is $-D_{\mathsf X}\sum_\mu(1-\cos\eps p_\mu)=-D_{\mathsf X}\frac{\eps^2}2\hat p^2$. (b) \Cref{cpi:tl:split}(a) with $s=t=1$, $\theta=\frac14$ gives $\Xi(p)\le\sqrt2\,\hat p^{-1/2}\Theta$, which bounds $2\hat p^2\Xi(p)$; $|\Xi(p)-\Xi(0)|\le\Xi(0)$; the last two terms of (a) are at most $\frac32\eps^2D_{\mathsf X}\hat p^2$, and $\eps\hat p\le2\sqrt3$. (c) The seagull does not depend on $M$, and the derivative of the exchange part of $\Sigma_2(p)$ is $2M\int\hat D(p-p')\widehat{(p+p')}^2\hat G(p')^2\ge0$; by \cref{cpi:tl:split}(b), $\widehat{(p+p')}^2\le2\hat k^2+8\hat p'^2$, and $\hat D\hat k^2\le1$, $\hat p'^2\hat G\le1$, \cref{cpi:tl:pos}(c) bound it by $2M[2\tau+8\Xi(0)]$. So $\partial_M\Sigma_2(p)$ and $\partial_M\Sigma_2(0)$ lie in $[0,\beta_{\mathsf X}]$, and their difference $\partial_M\pi(p)$ is at most $\beta_{\mathsf X}$ in modulus. \end{proof} \begin{lemma}[the $e^4$ coefficient]\label{cpi:tl:e4} $\dm_{(40),T}=Q_T(M)=\frac34\eps^2D_T^2+(A)+(C)+(D1)+(D2)+(D3)$, where, with $s_\mu(k)\defeq\hat k_\mu$ and $\hat G=\hat G_M$, \begin{align*} (A)&=M^2\int_T\hat D\hat G^2\pi_T,\qquad(C)=-\tfrac{\eps^2D_T}2\big(D_T-J_T(M)\big),\\ (D1)&=\tfrac12\sum_\mu\sum_z\eps^3\big[D_T(z)^2+D_T(z+\eps e_\mu)^2\big]\big[G_{M,T}(z)+G_{M,T}(z+\eps e_\mu)\big],\\ (D2)&=-\sum_\mu\int_{T,k}\int_{T,P}s_\mu(k)\big[s_\mu(k)+s_\mu(2P+k)\big]\hat G(k)\hat G(P)\hat D(k)\hat D(P-k),\\ (D3)&=\int_T\hat D(k)^2\sum_\mu\Pi_{\mu\mu}(k;M),\qquad\Pi_{\mu\mu}(k;M)\defeq2G_{M,T}(\eps e_\mu)-\int_{T,q}\widehat{(2q+k)}_\mu^{\,2}\hat G(q)\hat G(q+k). \end{align*} \end{lemma} \begin{proof} The mechanism is a Ward identity. Let $\lambda=0$, $a=M^2+\dm$, $\mathcal G_{A,a}\defeq(-\Delta_A+a)^{-1}$, and let $\langle\cdot\rangle$ be the photon expectation of the model after the scalar field is integrated out (the Gaussian of covariance $\delta_{\mu\nu}D_T$ times $\det(-\Delta_A+a)^{-1}$, normalised), so that the full two-point function is $G_{\rm full}(x-y)=\langle\mathcal G_{A,a}(x,y)\rangle$; all identities are identities of formal power series in $e$. King's condition is equivalent to $\hat G_{\rm full}(0)=M^{-2}$ through order $e^4$. With $W_A\defeq-\Delta_A\mathbf1$ one has $(-\Delta_A+a)\mathbf1=a+W_A$, hence $a\,\hat G_{\rm full}(0)=1-\langle(\mathcal G_{A,a}W_A)(x)\rangle$ and $\dm=-M^2\langle(\mathcal G_{A,a}W_A)(x)\rangle$ through order $e^4$. Gauge covariance, $-\Delta_{A+\nabla^+\chi}=e^{-\mathrm ie\chi}(-\Delta_A)e^{\mathrm ie\chi}$, and Gaussian integration by parts in the direction $\nabla^+\chi_y$, $\chi_y=-D_T(y-\cdot)$, give $\langle\mathcal G_{A,a}(x,y)\,\mathrm{div}A(y)\rangle=-\mathrm ie(D_T(0)-D_T(x-y))G_{\rm full}(x-y)$, whence the \emph{Ward form} $\dm=e^2D_T-e^2M^2\int_T\hat D\hat G_{\rm full}-M^2\langle(\mathcal G_{A,a}R_A)(x)\rangle$, $R_A\defeq W_A+\mathrm ie\,\mathrm{div}A=R^{(2)}+R^{(3)}+R^{(4)}+O(e^5)$ with $R^{(2)}=\frac{e^2}2\sum_\mu[A_\mu(x)^2+A_\mu(x-\eps e_\mu)^2]$, $R^{(3)}=\frac{\mathrm i\eps^2e^3}6\sum_\mu\nabla^-_\mu(A_\mu^3)(x)$, $R^{(4)}=-\frac{\eps^2e^4}{24}\sum_\mu[A_\mu(x)^4+A_\mu(x-\eps e_\mu)^4]$ (the expansion of $1-\cos\vartheta$ and $\sin\vartheta-\vartheta$, $\vartheta=\eps eA$). Expanding $\mathcal G_{A,a}=G-GV_1G-G(V_2+e^2\dm_{(20)})G+GV_1GV_1G+O(e^3)$ and the determinant to order $e^2$ ($U_2=\frac12\langle A,\Pi A\rangle$, whose diagonal symbol is $e^2\Pi_{\mu\mu}(k;M)$), and using $\sum_y\eps^3G(x-y)=M^{-2}$, the order-$e^4$ part of the Ward form is the sum of five terms: the order-$e^4$ part of $-e^2M^2\int\hat D\hat G_{\rm full}$, which is $(A)$ since the order-$e^2$ part of $\hat G_{\rm full}$ is $-e^2\hat G^2\pi_T$; $-\E_0R^{(4)}(0)=\frac34\eps^2e^4D_T^2$ ($\E_0A_\mu^4=3D_T^2$); the term with $R^{(3)}$, which by $\E_0[L\,A_\mu^3]=3D_T\E_0[L\,A_\mu]$ for linear $L$ and the order-$e$ Ward identity is $(C)$; the terms with $R^{(2)}$ contracted with $V_2$ and with $V_1V_1$, which are $(D1)$ and $(D2)$ (the vertex factors multiply to $e^4s_\mu(k)[s_\mu(k)+s_\mu(2P+k)]$, from $e^{\mathrm i\eps k_\mu}-1=\mathrm i\eps e^{\mathrm i\eps k_\mu/2}s_\mu(k)$), the pairing of $R^{(2)}$ with itself contributing $-3e^4D_T\hat G(0)^2\pi_T(0)=0$; and the covariance of $R^{(2)}(0)$ with $U_2$, which is $(D3)$. The identity holds for King's counterterm through order $e^4$, and the coefficient of $e^4\lambda^0$ comes from graphs without a $\lambda$-vertex, so it is the same at $\lambda\ne0$. \end{proof} \begin{lemma}[slope bounds for each coupling order: the bounds \textup{(G3)}]\label{cpi:tl:dens} Let $\eps\le1$, let $T$ have side $\ell_T\ge\ell_0>0$, and let $M>0$. With the one-loop majorants of \cref{cpi:tl:oneloop}, all at $(M,\mu;\ell_0)$, the constant $\bar\omega=w+\hat E(\mu,\ell_0)$ of \cref{cpi:tl:img}(c), and $\bar\beta$, $j$, $\varphi_3$, $b_4$, $\varphi_A$, $\varphi_{D2}$ as defined in \textup{(G3)}: \textup{(o-$\lambda$)} $\partial_MC_T\le0$; \textup{(o-$e^2$)} $0\le\partial_MJ_T\le j(M)$; \textup{(o-$\lambda^2$)} $0\le-128\,\partial_MS_T\le768M\bar\tau(M)^2$; \textup{(o-$e^2\lambda$)} $|\partial_MF_T|\le\varphi_3(M)$; \textup{(o-$e^4$)} $\partial_MQ_T\ge-b_4(M)$. All the majorants and $\bar\omega$ are non-negative and independent of the couplings and of $\eps$; the one-loop majorants of \cref{cpi:tl:oneloop} and $\bar\omega$ are non-increasing in $M$ and in $\ell_0$, and $\bar\omega$ does not depend on $M$. \end{lemma} \begin{proof} Every one-loop sum on $T$ ($\ell_T\ge\ell_0$) is at most its majorant (\cref{cpi:tl:oneloop}(a),(b)); write $\tau=\tau_T$, $\Xi=\Xi_T$, $\delta=\delta_T$, $\Theta=\Theta_T$. The bound $\eps D_T\le\bar\omega$ (\cref{cpi:tl:img}(c)) is the only place where $\eps\le1$ enters. \emph{(o-$\lambda$).} $\partial_M\hat G_M=-2M\hat G_M^2$, so $\partial_MC_T=-2M\tau\le0$. \emph{(o-$e^2$).} $\partial_MJ_T=2M\int\hat D\hat G(1-M^2\hat G)=2M\int\hat D\hat k^2\hat G^2\ge0$. With $\hat D\hat k^2\le1$ it is at most $2M\tau\le2M\bar\tau$; with $\hat k^2\hat G\le1$ at most $2M\Xi(0)\le2M\bar\psi$. \emph{(o-$\lambda^2$).} By \cref{cpi:tl:pos}(a), $-\partial_MS_T=3\sum_x\eps^3G_M^2(-\partial_MG_M)=6M\sum_x\eps^3G_M(x)^2(G_M*G_M)(x)\ge0$. Next $(G*G)(x)\le(G*G)(0)=\tau$ (\cref{cpi:tl:pos}(c)) and $\sum_x\eps^3G_M^2=\tau$ (Parseval), so $-128\,\partial_MS_T\le768M\tau^2\le768M\bar\tau^2$. \emph{(o-$e^2\lambda$).} By \cref{cpi:tl:F}, $\partial_MF_T=-4M\int\hat G^3\pi_T+\int\hat G^2\partial_M\pi_T$. \Cref{cpi:tl:Pi}(b) bounds the first term by $4M[2\sqrt2\Theta J_{3/2}+|\mu^2-2M^2|\Xi(0)J_0+3\sqrt3(\eps D_T)J_1]$, $J_s=\int\hat G^3\hat p^s$, and \cref{cpi:tl:Pi}(c) bounds the second by $\beta_T\tau$. Replacing each factor by its majorant and $\eps D_T$ by $\bar\omega$ gives $\varphi_3$. \emph{(o-$e^4$).} By \cref{cpi:tl:e4}, $\partial_MQ_T$ is the sum of the following five derivatives. \begin{itemize}[leftmargin=*,nosep] \item[(C)] $\partial_M(C)=\frac{\eps^2D_T}2\partial_MJ_T\ge0$ by (o-$e^2$); it is dropped. \item[(A)] $(A)=M^2H$ with $H\defeq\int\hat D\hat G^2\pi_T$, so $\partial_M(A)=2MH-4M^3\int\hat D\hat G^3\pi_T+M^2\int\hat D\hat G^2\partial_M\pi_T$. By $M^2\hat G\le1$ and \cref{cpi:tl:Pi}(b),(c), $|\partial_M(A)|\le6M\int\hat D\hat G^2|\pi_T|+\beta_T\Xi(0)\le\varphi_A$, with $\int\hat D\hat G^2\hat p^s\le\bar J'_s$. \item[(D1)] $\partial_MG_{M,T}(z)=-2M(G*G)(z)\in[-2M\tau,0]$ (\cref{cpi:tl:pos}(a),(c)) and \[ \sum_z\eps^3D_T(z)^2=\sum_z\eps^3D_T(z+\eps e_\mu)^2=\delta; \] over three directions, $-12M\bar\delta\bar\tau\le\partial_M(D1)\le0$. \item[(D3)] $\sum_\mu\partial_M[2G_{M,T}(\eps e_\mu)]\in[-12M\tau,0]$. With $\hat G'\defeq\hat G(q+k)$, the derivative of the bubble part of $\sum_\mu\Pi_{\mu\mu}$ is $2M\int\widehat{(2q+k)}^2(\hat G^2\hat G'+\hat G\hat G'^2)\ge0$; by \cref{cpi:tl:split}(b) with $a=q$, $b=q+k$, $\hat q^2\hat G\le1$ and \cref{cpi:tl:pos}(c) it is at most $2M[4\int\hat G\hat G'+4\tau]\le16M\tau$. Since $\int\hat D^2=\delta$, $|\partial_M(D3)|\le16M\bar\delta\bar\tau$. \item[(D2)] Write $(D2)=-\sum_\mu\int_ks_\mu(k)\hat G(k)\hat D(k)S_\mu(k)$ with $S_\mu(k)\defeq\int_P[s_\mu(k)+s_\mu(2P+k)]\hat G(P)\hat D(P-k)$, and decompose $s_\mu(2P+k)=\frac2\eps[\sin(\eps P_\mu)\cos\frac{\eps k_\mu}2+\cos(\eps P_\mu)\sin\frac{\eps k_\mu}2]$. The first piece is odd in $P$ except through $\hat D(P-k)$; symmetrising $P\to-P$, $|\frac2\eps\sin\eps P_\mu|\le2|\hat P_\mu|$ and \cref{cpi:tl:split}(c) bound its contribution to $S_\mu$ by $\int4|\hat P_\mu|\hat P\hat k\,\hat G\hat D(P-k)\hat D(P+k)$. The other two pieces are each at most $|s_\mu(k)|\int\hat G\hat D(\cdot-k)\le|s_\mu(k)|\Xi(0)$. With $\sum_\mu|\hat k_\mu||\hat P_\mu|\le\hat k\hat P$, $\hat P^2\hat G\le1$ and \cref{cpi:tl:pos}(c), $\sum_\mu|s_\mu||S_\mu|\le\hat k^2(2\Xi(0)+4\delta)$. Likewise, with $\hat D\hat D'\le\frac12(\hat D^2+\hat D'^2)$ and \cref{cpi:tl:split}(a) for $\theta\in(\frac12,1]$ (at $k=0$ the factor $s_\mu(k)$ vanishes), $\sum_\mu|s_\mu||\partial_MS_\mu|\le2M\hat k^2(2/\hat k)^{2\theta}\big[2(\int\hat G^{2-\theta}\hat D+\int\hat G^2\hat D^{1-\theta})+4(\int\hat G^{1-\theta}\hat D^2+\int\hat G\hat D^{2-\theta})\big]$. When $\partial_M$ hits $\hat G(k)$, $\hat k^2\hat G^2\hat D\le\hat G\hat D$ bounds the contribution by $2M\Xi(0)(2\Xi(0)+4\delta)$. When it hits $\hat G(P)$, the contribution is at most $8M\Xi(0)(6\Xi(0)+2\tau+4\delta)$ at $\theta=1$ and at most $2^{5/2}M(2\Lambda_1+4\Lambda_2)\Lambda_0$ at $\theta=\frac34$, with $\Lambda_i$ the sums bounded by $\bar\Lambda_i$. So $|\partial_M(D2)|\le\varphi_{D2}$. \end{itemize} Summing, $\partial_MQ_T\ge-[\varphi_A+(12+16)M\bar\delta\bar\tau+\varphi_{D2}]=-b_4$. \emph{Properties of the majorants.} They are non-negative and coupling-free; the $\mathcal I$'s and $\bar\omega$ do not depend on $\eps$, and they are non-increasing in $M$ and in $\ell_0$ (\cref{cpi:tl:oneloop}(b),(c), \cref{cpi:tl:img}); $\bar\omega$ does not depend on $M$. \end{proof} \space% % \begin{lemma}[the four orders on $\Lk$ at every photon mass]\label{lem:s14:E} Let $\lmu\in(0,1]$, $\ell_0>0$, $n\ge1$. For every $k$ with $\eps_k\le1$: \[ 2S_{20,k}\ge-\bar j_n(\lmu,\ell_0),\qquad 2S_{02,k}\ge-\bar s_n,\qquad 2S_{21,k}\ge-\bar B_{3,n}(\lmu,\ell_0),\qquad 2S_{40,k}\ge-\bar B^-_{4,n}(\lmu,\ell_0). \] \end{lemma} \begin{proof} Fix such a $k$, and let $T=T_J$ be King's tori (\Cref{def:s10:torus}; side $2MJ$, $2MJ\cdot5^k$ sites per side, an even number), $J$ so large that $2MJ\ge\ell_0$. By (G1) at $\mu_\gamma=\lmu$, the coefficients of the torus-convention bare coefficient $\alpha^{\mathrm{tor}}_{k,T}(M)$ in the five orders are the terms of the exact two-loop form, so that for $S^T_{\alpha\beta,k}\defeq\frac12[\dm_{(\alpha\beta),T}(\eps_k;1)-\dm_{(\alpha\beta),T}(\eps_k;0.7)]$, \[ \begin{gathered} 2S^T_{20,k}=-[J_T(1)-J_T(0.7)],\quad 2S^T_{02,k}=128[S_T(1)-S_T(0.7)],\\ 2S^T_{21,k}=16[F_T(1)-F_T(0.7)],\quad 2S^T_{40,k}=Q_T(1)-Q_T(0.7), \end{gathered} \] the $M$-independent terms $-2D_T$ and $\frac34\eps^2D_T^2$ cancelling in the differences. The conditions of (G3) and (G4) are met: $\eps_k\le1$, $\ell_T\ge\ell_0$, $\mu_\gamma=\lmu\in(0,1]$, $M\in[0.7,1]$, and Lemmas~\ref{cpi:tl:oneloop}, \ref{cpi:tl:img} and \ref{cpi:tl:dens}, which prove (G2) and (G3), carry $\mu_\gamma$ as a parameter in their statements (Lemma~\ref{cpi:tl:oneloop}: $a,b>0$ and $g>\frac32$ for every instance; Lemma~\ref{cpi:tl:img}: $\eps\le1$ and $\ell_T\ge\ell_0$ for $\bar\omega$, and $\ell_T/\eps\in\N$, $\ell_T/\eps\ge2$ for $\bar E$). Hence, by the order-by-order bounds of (G4) with the majorants at $(\lmu,\ell_0)$ and by the cell rule \eqref{eq:s14:cell}, $2S^T_{20,k}\ge-\bar j_n(\lmu,\ell_0)$, $2S^T_{02,k}\ge-\bar s_n$, $2S^T_{21,k}\ge-\bar B_{3,n}(\lmu,\ell_0)$, $2S^T_{40,k}\ge-\bar B^-_{4,n}(\lmu,\ell_0)$, uniformly in $J$ ($\bar s$ does not depend on $\mu$: $\bar\tau=\mathcal I_{2,0}$ has $\beta=0$). Now let $J\to\infty$ at fixed $k$. By \Cref{lem:s10:V}, $\alpha^{\mathrm{tor}}_{k,T_J}(M)\to\alpha^\infty_k(M)$ for $M\in\{0.7,1\}$, and the same argument (the volume limit at fixed $\eps$ is the convergence of torus Riemann sums of smooth periodic functions, faster than any power, by Poisson summation) gives the convergence of the recursion \eqref{eq:s14:CTdef} order by order: $\dm_{(\alpha\beta),T_J}(\eps_k;M)\to\dm_{(\alpha\beta)}(\eps_k;M)$ for each of the five orders (alternatively: a polynomial of bounded degree in $(e^2,\lambda)$ that converges at every point of an open set converges coefficient by coefficient, and \Cref{lem:s10:V} holds at every point of King's range). So $S^{T_J}_{\alpha\beta,k}\to S_{\alpha\beta,k}$, and closed inequalities pass to the limit. \end{proof} \begin{lemma}[the torus floor can be removed]\label{lem:s14:TF} For every $\lmu\in(0,1]$, $n\ge1$ and every $k$ with $\eps_k\le1$, the four inequalities of \Cref{lem:s14:E} hold with $\bar X_n(\lmu,\infty)$, the upper sums formed with the $\eps\Z^3$ majorants $\mathcal I_{\alpha,\beta}(M,\lmu;\infty)=c_NC_3(M,\lmu)$ and with $\bar\omega=w$. \end{lemma} \begin{proof} Fix $k$, $\lmu$, $n$. By \Cref{lem:s14:E}, $2S_{\alpha\beta,k}\ge-\bar X_n(\lmu,\ell_0)$ for \emph{every} $\ell_0>0$: the torus floor is a free parameter of (G4), and the tori $T_J$ of the volume limit eventually exceed any $\ell_0$. It therefore suffices that $\bar X_n(\lmu,\ell_0)\to\bar X_n(\lmu,\infty)$ as $\ell_0\to\infty$. Each $\bar X_n$ is a finite sum of cell values, and each cell value is a fixed finite expression (sums, products and minima, with the fixed prefactors) in the majorants $\mathcal I_{\alpha,\beta}(a,\lmu;\ell_0)$ at the left endpoint $a$ of the cell and in $\bar\omega(\lmu,\ell_0)$, hence continuous in these. As $\ell_0\to\infty$, $\mathcal I_{\alpha,\beta}(a,\lmu;\ell_0)\to c_NC_3(a,\lmu)$ by \eqref{eq:s14:Iab}, since $C_2$ and $C_1$ are finite numbers that do not depend on $\ell_0$; and $0\le\hat E(\lmu,\ell_0)\le\bar E(\lmu,\ell_0)\to0$, each of the four terms of $\bar E$ being a positive constant times $\ell_0^{-1}$, $\ell_0^{-1}E_1(\lmu^2\ell_0^2/8)$, $\ell_0^{-2}\mathrm{erfc}(\lmu\ell_0/(2\sqrt2))$ or $\ell_0^{-3}e^{-\lmu^2\ell_0^2/8}$, so $\bar\omega\to w$. \end{proof} \begin{lemma}[monotonicity in the photon mass; interval bounds]\label{lem:s14:MON} Let $0<\mu_{\rm lo}\le\mu_{\rm hi}\le1$. Define the \emph{interval upper sum} $\bar X^{\rm int}_n([\mu_{\rm lo},\mu_{\rm hi}])$ as $\bar X_n(\cdot,\infty)$ with every majorant evaluated at $\mu=\mu_{\rm lo}$ and, on the cell $[a,b]$, the prefactor $|\mu^2-2M^2|$ replaced by $\max\{|\mu^2-2M^2|:\mu\in\{\mu_{\rm lo},\mu_{\rm hi}\},\,M\in\{a,b\}\}$. Then $\bar X_n(\mu,\infty)\le\bar X^{\rm int}_n([\mu_{\rm lo},\mu_{\rm hi}])$ for every $\mu\in[\mu_{\rm lo},\mu_{\rm hi}]$, so the inequalities of \Cref{lem:s14:TF} hold with $\bar X^{\rm int}_n([\mu_{\rm lo},\mu_{\rm hi}])$ for all $\lmu\in[\mu_{\rm lo},\mu_{\rm hi}]$ at once. Moreover, if $\mu_{\rm hi}\le0.989949$, the interval upper sum equals the point upper sum $\bar X_n(\mu_{\rm lo},\infty)$. \end{lemma} \begin{proof} A cell value is non-decreasing in each majorant and in the prefactor (sums, products and minima of non-negative quantities with non-negative coefficients). (i) Each admissible closed form of $C_3$ is non-increasing in $b=\mu$: the $\nu$-form, since $\nu=\min(a,b)$ is non-decreasing in $b$ and $3-2g<0$; the Beta form in $a$ is $b$-free; the Beta form in $b$ carries $b^{2c'-2\beta}$ with $c'=\frac{3-2\alpha}2<\beta$, as $g>\frac32$; the H\"older form carries $b^{(3-2g)\beta/g}$; the exact forms $\frac1{4\pi(a+b)}$ and $\frac1{8\pi a(a+b)^2}$ decrease in $b$. A minimum of non-increasing functions is non-increasing, so every $\mathcal I_{\alpha,\beta}(a,\mu;\infty)$ is non-increasing in $\mu$, and $\bar\omega=w$ does not depend on $\mu$. (ii) $\mu\mapsto|\mu^2-2M^2|$ is convex in $\mu^2$ and $M\mapsto|\mu^2-2M^2|$ convex in $M^2$, so the maximum over the rectangle $[\mu_{\rm lo},\mu_{\rm hi}]\times[a,b]$ is attained at a corner. Hence every ingredient of the cell value at $\mu$ is at most the corresponding ingredient of the interval cell value. For the last claim: on $[0.7,1]$ one has $2M^2\ge0.98>0.989949^2$, so $|\mu^2-2M^2|=2M^2-\mu^2$ on the whole rectangle, maximal at $(\mu_{\rm lo},b)$, which is also the point prefactor at $\mu_{\rm lo}$. (For the interval $[0.98,0.99]$ the corner $(0.99,0.7)$ of the first cell gives $|0.9801-0.98|=10^{-4}$, below the value $2\cdot0.70075^2-0.98^2=0.0217$ at $(0.98,0.70075)$, so the equality holds there too; only the interval $[0.99,1]$ differs, by $8.5786210$ against the point value $8.5785880$ of $\bar B_3$.) \end{proof} \begin{lemma}[a closed form, uniform in the photon mass]\label{lem:s14:CF} For $M\in[0.7,1]$ put $\bar\tau^*(M)\defeq\mathcal I_{2,0}(M;\infty)=\frac{c_N}{8\pi M}$, $\bar\psi^*(M)\defeq\frac{c_N}{4\pi M}$, and for $X\in\{\Theta,J_s,J'_s\}$ let $X^*(M)$ be $c_N$ times the minimum over the $\mu$-free admissible forms of $C_3$ only (the Beta forms in $a=M$ and, for $\beta=0$, the H\"older form $\gamma_3(g)M^{3-2g}$; the $\nu$-forms, the Beta forms in $b$, the exact form at $(2,1)$ and the H\"older forms with $\beta>0$ are dropped, and the exact form at $(1,1)$ is replaced by its bound $\frac1{4\pi a}$, which is $\bar\psi^*$). Put $\bar\beta^*\defeq2M[2\bar\tau^*+8\bar\psi^*]$, \begin{gather*} \varphi_3^*(M)\defeq4M\big[2\sqrt2\,\bar\Theta^*\bar J^*_{3/2}+2M^2\bar\psi^*\bar J^*_0+3\sqrt3\,w\,\bar J^*_1\big]+\bar\beta^*\bar\tau^*,\\ \varphi_A^*(M)\defeq6M\big[2\sqrt2\,\bar\Theta^*\bar J'^*_{3/2}+2M^2\bar\psi^*\bar J'^*_0+3\sqrt3\,w\,\bar J'^*_1\big]+\bar\beta^*\bar\psi^*,\\ P(M)\defeq\varphi_A^*+52M\bar\psi^{*2}+16M\bar\psi^*\bar\tau^*,\qquad Q(M)\defeq\frac{c_N}{8\pi}M\big(28\bar\tau^*+40\bar\psi^*\big). \end{gather*} Then $16\varphi_3(M;\mu,\infty)\le16\varphi_3^*(M)$ and $b_4(M;\mu,\infty)\le P(M)+Q(M)/\mu$ for all $M\in[0.7,1]$ and $\mu\in(0,1]$. Consequently, with the $400$-cell upper sums $B_3^*\defeq\sum h\,(16\varphi_3^*)^{\rm cell}$, $P^*\defeq\sum hP^{\rm cell}$, $Q^*\defeq\sum hQ^{\rm cell}$ (majorants at the left endpoint, the prefactors $M$ and $2M^2$ at the right endpoint), for every $\lmu\in(0,1]$ and every $k$ with $\eps_k\le1$: \[ \begin{gathered} 2S_{21,k}\ge-B_3^*,\qquad 2S_{40,k}\ge-\big(P^*+Q^*/\lmu\big),\\ B_3^*=13.187549,\quad P^*=4.7629000,\quad Q^*=0.77120851 . \end{gathered} \] \end{lemma} \begin{proof} Each starred majorant is an upper bound of the corresponding majorant at $(M,\mu;\infty)$ for every $\mu>0$: a minimum over a subset of the admissible forms is at least the minimum over all of them, and $\mathcal I_{1,1}(M,\mu;\infty)\le c_N\frac1{4\pi(M+\mu)}\le\frac{c_N}{4\pi M}$. Further $\bar\delta(\mu)=\mathcal I_{0,2}(M,\mu;\infty)\le c_N\frac1{4\pi^2}B(\frac32,\frac12)\mu^{-1}=\frac{c_N}{8\pi\mu}$ (the Beta form in $b$, admissible since $\alpha=0<\frac32$ and $\beta=2>\frac32$); $|\mu^2-2M^2|\le2M^2$, since $\mu^2\le1\le4M^2$; $\bar\omega=w$ (\Cref{lem:s14:TF}); and in $\varphi_{D2}$ the minimum is at most its first branch. Substituting these bounds into the formulas (G3), which are non-decreasing in every majorant, gives $16\varphi_3\le16\varphi_3^*$ and \[ b_4\le\varphi_A^*+28M\bar\tau^*\frac{c_N}{8\pi\mu}+2M\bar\psi^*\Big(2\bar\psi^*+\frac{4c_N}{8\pi\mu}\Big)+8M\bar\psi^*\Big(6\bar\psi^*+2\bar\tau^*+\frac{4c_N}{8\pi\mu}\Big)=P(M)+\frac{Q(M)}\mu . \] The starred majorants are non-increasing in $M$ (powers $M^{2c-2\alpha}$ with $c<\alpha$, and $1/M$), so the cell rule \eqref{eq:s14:cell} applies to $16\varphi_3^*$, $P$ and $Q$ with the prefactors $M$, $2M^2$ at the right endpoint. The two inequalities follow \emph{cellwise} from \Cref{lem:s14:TF}: for a cell $[a,b]$ the cell value of $16\varphi_3$ (resp.\ $b_4$) at $(\lmu,\infty)$ is a non-decreasing expression in the majorants at $(a,\lmu;\infty)$, in $\bar\omega$ and in the prefactor, each of which is at most its starred counterpart at $a$ (with $\bar\delta\le c_N/(8\pi\lmu)$, $\max_{M\in\{a,b\}}|\lmu^2-2M^2|\le2b^2$, $\bar\omega=w$); so the cell value is at most the cell value of $16\varphi_3^*$ (resp.\ of $P+Q/\lmu$). Summing over the $400$ cells, $\bar B_{3,400}(\lmu,\infty)\le B_3^*$ and $\bar B^-_{4,400}(\lmu,\infty)\le P^*+Q^*/\lmu$, and \Cref{lem:s14:TF} gives the two inequalities. The three numbers are the $400$-cell sums, computed as described after \Cref{tab:s14:T3}. (The slope bounds (G3) are not invoked with the starred majorants in place of the one-loop majorants; the domination above is all that is used.) \end{proof} \space% % % % \subsection*{The numbers} Let $\hat\mu=\lmu/m\in(0,1]$. The grid is $\{1,\allowbreak 0.99,\allowbreak 0.98,\allowbreak 0.95,\allowbreak 0.9,\allowbreak 0.85,\allowbreak \dots,\allowbreak 0.15,\allowbreak 0.1,\allowbreak 0.08,\allowbreak 0.06,\allowbreak 0.05,\allowbreak 0.04,\allowbreak 0.03,\allowbreak 0.02,\allowbreak 0.015,\allowbreak 0.01,\allowbreak 0.005,\allowbreak 0.002,\allowbreak 0.001,\allowbreak 0.0005,\allowbreak 0.0002,\allowbreak 0.0001\}$ ($35$ points), and the \emph{rows} of \Cref{tab:s14:T3} are the $34$ intervals $[\mu_{\rm lo},\mu_{\rm hi}]$ between consecutive grid points. For $\hat\mu\in[10^{-4},1]$ the \emph{row of $\hat\mu$} is the row containing $\hat\mu$ (at a grid point, where two rows contain $\hat\mu$, the row with $\mu_{\rm lo}=\hat\mu$), and $\bar B_3(\hat\mu)$, $\bar B_4^-(\hat\mu)$, $\bar j(\hat\mu)$ denote the interval upper sums $\bar B^{\rm int}_{3,400}$, $\bar B^{-,{\rm int}}_{4,400}$, $\bar j^{\rm int}_{400}$ of that row (\Cref{lem:s14:MON}), and $\bar s\defeq\bar s_{400}=6.5231964$; for $\hat\mu<10^{-4}$, below the table, we read $\bar B_3(\hat\mu)\defeq B_3^*$, $\bar B_4^-(\hat\mu)\defeq P^*+Q^*/\hat\mu$ and $\bar j(\hat\mu)\defeq0.092610085$ (\Cref{lem:s14:CF} for the first two; the last is the $\mu$-free branch $\int2M\bar\tau$ of $\bar j$, see the proof of \Cref{prop:s14:T3}(i)). With $A\defeq\frac12(1-M_1^2)=0.255$, $b\defeq\frac{4(1-M_1)}{2\pi}=0.1909859$, $L\defeq\frac{4\ln(1/M_1)}{\pi^2}=0.1445549$ (\Cref{lem:s14:A,lem:s14:B,lem:s14:C} at $N=2$, $M_1=0.7$), $I_\infty(\hat\mu)=\frac{0.7+1.7\hat\mu}{4\pi\cdot1.7\,(\hat\mu+0.7)(1+\hat\mu)}$ (so $I_\infty(1)=0.0330425$) and $I^*=\frac1{4\pi\cdot1.7}=0.0468103$, put \begin{equation}\label{eq:s14:sigmalow} \sigma_{\rm low}(\hat\lambda,\hat e^2;\hat\mu)\defeq A\big(1-\hat e^2I_\infty(\mu_{\rm lo})\big)+b\hat\lambda-L\hat\lambda^2-\tfrac12\bar B_3(\hat\mu)\,\hat e^2\hat\lambda-\tfrac12\bar B_4^-(\hat\mu)\,\hat e^4\qquad(\hat\mu\in[10^{-4},1]), \end{equation} \begin{equation}\label{eq:s14:sigmastar} \sigma^*(\hat\lambda,\hat e^2;\hat\mu)\defeq A\big(1-\hat e^2I^*\big)+b\hat\lambda-L\hat\lambda^2-\tfrac12B_3^*\,\hat e^2\hat\lambda-\tfrac12\big(P^*+Q^*/\hat\mu\big)\hat e^4\qquad(\hat\mu\in(0,1]), \end{equation} and $\sigma_{\rm low}(\cdot;\hat\mu)\defeq\sigma^*(\cdot;\hat\mu)$ for $\hat\mu<10^{-4}$. The derived columns of the table are: $e^2_{\max}$, the positive root of $\sigma_{\rm low}(0,\cdot;\hat\mu)$, $e^2_{\max}=\big(-AI_\infty+\sqrt{A^2I_\infty^2+2A\bar B_4^-}\big)/\bar B_4^-$; the side $c$ of the largest box $[0,c]^2$ on which $\sigma_{\rm low}\ge0$; and the value $\sigma_{\rm low}(0.1,0.1;\hat\mu)$, where a negative entry means only that the bound does not place $[0,0.1]^2$ in the region. \begin{table}[htbp] \centering\footnotesize \setlength{\tabcolsep}{4pt} \begin{tabular}{l|lll|l|lll} $[\mu_{\rm lo},\mu_{\rm hi}]$ & $\bar j$ & $\bar B_3$ & $\bar B_4^-$ & $AI_\infty(\mu_{\rm lo})$ & $e^2_{\max}(\lambda=0)$ & $c$ & $\sigma_{\rm low}(0.1,0.1)$\\\hline $[0.99,1]$ & 0.085313936 & 8.5786210 & 1.5854190 & 0.008458 & 0.5618 & 0.2390 & 0.2209\\ $[0.98,0.99]$ & 0.085757165 & 8.6036471 & 1.6023646 & 0.008491 & 0.5588 & 0.2385 & 0.2207\\ $[0.95,0.98]$ & 0.087020288 & 8.6804035 & 1.6554211 & 0.008589 & 0.5498 & 0.2369 & 0.2201\\ $[0.9,0.95]$ & 0.088887935 & 8.8139269 & 1.7519811 & 0.008757 & 0.5345 & 0.2341 & 0.2189\\ $[0.85,0.9]$ & 0.090427040 & 8.9551127 & 1.8603008 & 0.008930 & 0.5188 & 0.2313 & 0.2176\\ $[0.8,0.85]$ & 0.091595352 & 9.1048479 & 1.9826438 & 0.009108 & 0.5026 & 0.2283 & 0.2163\\ $[0.75,0.8]$ & 0.092342924 & 9.2641791 & 2.1218953 & 0.009291 & 0.4858 & 0.2251 & 0.2147\\ $[0.7,0.75]$ & 0.092610085 & 9.4343534 & 2.2812536 & 0.009480 & 0.4686 & 0.2217 & 0.2131\\ $[0.65,0.7]$ & 0.092610085 & 9.6168755 & 2.4625429 & 0.009673 & 0.4511 & 0.2182 & 0.2112\\ $[0.6,0.65]$ & 0.092610085 & 9.8135862 & 2.6698136 & 0.009871 & 0.4333 & 0.2145 & 0.2092\\ $[0.55,0.6]$ & 0.092610085 & 10.026776 & 2.9084862 & 0.010073 & 0.4152 & 0.2105 & 0.2069\\ $[0.5,0.55]$ & 0.092610085 & 10.259350 & 3.1856310 & 0.010279 & 0.3969 & 0.2063 & 0.2044\\ $[0.45,0.5]$ & 0.092610085 & 10.515087 & 3.5155645 & 0.010488 & 0.3779 & 0.2017 & 0.2014\\ $[0.4,0.45]$ & 0.092610085 & 10.799045 & 3.9183359 & 0.010697 & 0.3580 & 0.1967 & 0.1979\\ $[0.35,0.4]$ & 0.092610085 & 11.118258 & 4.4224644 & 0.010906 & 0.3371 & 0.1912 & 0.1938\\ $[0.3,0.35]$ & 0.092610085 & 11.477171 & 5.0733664 & 0.011111 & 0.3148 & 0.1850 & 0.1887\\ $[0.25,0.3]$ & 0.092610085 & 11.851059 & 5.9437544 & 0.011309 & 0.2910 & 0.1781 & 0.1825\\ $[0.2,0.25]$ & 0.092610085 & 12.193509 & 7.0632067 & 0.011495 & 0.2670 & 0.1709 & 0.1752\\ $[0.15,0.2]$ & 0.092610085 & 12.433452 & 8.5440078 & 0.011662 & 0.2429 & 0.1635 & 0.1665\\ $[0.1,0.15]$ & 0.092610085 & 12.659419 & 11.303445 & 0.011801 & 0.2113 & 0.1525 & 0.1516\\ $[0.08,0.1]$ & 0.092610085 & 12.756284 & 13.316343 & 0.011846 & 0.1948 & 0.1460 & 0.1411\\ $[0.06,0.08]$ & 0.092610085 & 12.857230 & 16.620716 & 0.011884 & 0.1744 & 0.1370 & 0.1240\\ $[0.05,0.06]$ & 0.092610085 & 12.909321 & 19.239396 & 0.011899 & 0.1621 & 0.1310 & 0.1107\\ $[0.04,0.05]$ & 0.092610085 & 12.962542 & 23.145186 & 0.011912 & 0.1479 & 0.1233 & 0.0909\\ $[0.03,0.04]$ & 0.092610085 & 13.016932 & 29.623519 & 0.011923 & 0.1308 & 0.1132 & 0.0582\\ $[0.02,0.03]$ & 0.092610085 & 13.072534 & 42.530531 & 0.011931 & 0.1092 & 0.0987 & $-0.0066$\\ $[0.015,0.02]$ & 0.092610085 & 13.100802 & 55.411533 & 0.011933 & 0.0957 & 0.0887 & $-0.0712$\\ $[0.01,0.015]$ & 0.092610085 & 13.129390 & 81.146532 & 0.011935 & 0.0791 & 0.0753 & $-0.2000$\\ $[0.005,0.01]$ & 0.092610085 & 13.158304 & 158.29597 & 0.011937 & 0.0566 & 0.0555 & $-0.5859$\\ $[0.002,0.005]$ & 0.092610085 & 13.175811 & 389.67594 & 0.011937 & 0.0361 & 0.0360 & $-1.7428$\\ $[0.001,0.002]$ & 0.092610085 & 13.181673 & 775.28604 & 0.011937 & 0.0256 & 0.0256 & $-3.6709$\\ $[0.0005,0.001]$ & 0.092610085 & 13.184609 & 1546.4975 & 0.011937 & 0.0181 & 0.0181 & $-7.5270$\\ $[0.0002,0.0005]$ & 0.092610085 & 13.186373 & 3860.1248 & 0.011937 & 0.0114 & 0.0114 & $-19.096$\\ $[0.0001,0.0002]$ & 0.092610085 & 13.186961 & 7716.1679 & 0.011937 & 0.0081 & 0.0081 & $-38.376$ \end{tabular} \caption{The interval upper sums $\bar j$, $\bar B_3$, $\bar B_4^-$ of \Cref{lem:s14:MON} on the $400$-cell partition of $[0.7,1]$ at $\ell_0=\infty$ (\Cref{lem:s14:TF}), valid for every $\hat\mu$ in the row; $\bar s=6.5231964$ on every row. The three upper sums are rounded upward to eight significant digits; the column $AI_\infty(\mu_{\rm lo})$ is rounded up, and the derived columns $e^2_{\max}$, $c$ and $\sigma_{\rm low}(0.1,0.1)$ are rounded down (towards $-\infty$). Units $m=1$, $N=2$, $M_1=0.7$.}\label{tab:s14:T3} \end{table} \paragraph{How the entries were computed, and what is exact.} The upper sums are the cell sums \eqref{eq:s14:cell} with the interval rule of \Cref{lem:s14:MON}. At $\ell_0=\infty$ only the $c_NC_3$ term of \eqref{eq:s14:Iab} and $\bar\omega=w\le0.25273101$ enter, so no series and no special function beyond $\Gamma$ and $B$ is involved. The evaluation was carried out in floating-point arithmetic with $50$ significant digits; every final upper sum was multiplied by $1+10^{-30}$ and rounded upward to eight significant digits, and the derived region numbers were computed from the rounded upper sums with coefficients rounded up and sizes rounded down. The rounding of the elementary and Gamma functions at $50$ digits is many orders below the safety factor and the final rounding; the entries are $50$-digit floating-point evaluations with the safety factor $1+10^{-30}$, not a certified (interval-arithmetic) computation; their validity as upper bounds rests on \Cref{lem:s14:MON} and on these finitely many evaluations. Internal consistency of the table: $\bar s_{400}=6.5231964$ at every $\hat\mu$ ($\bar s$ is $\mu$-free); $\bar j=0.092610085$ for all $\hat\mu\le0.7$, the $\mu$-free branch $\int2M\bar\tau$ of $\bar j$ (analytically $\frac{c_N}{4\pi}(1-M_1)=0.09253$ before the cell rounding); the interval values equal the point values at $\mu_{\rm lo}$ except on $[0.99,1]$ (\Cref{lem:s14:MON}); $\bar B^-_{4,400}(\hat\mu,\infty)\le P^*+Q^*/\hat\mu$ at every grid point ($7716.17\le7716.85$ at $\hat\mu=10^{-4}$, $1.5854\le5.534$ at $\hat\mu=1$); and $\hat\mu\,\bar B_4^-(\hat\mu)\to0.7716$ at $\hat\mu=10^{-4}$ against $Q^*=0.7712$. The closed form of \Cref{lem:s14:CF} is weaker than the table near $\hat\mu=1$ (it uses $\bar\psi^*$, $|\mu^2-2M^2|\le2M^2$ and the first branch of the minimum) and is within one per cent of it for $\hat\mu\le10^{-2}$. \space% % \smallskip\noindent Throughout \Cref{prop:s14:T3} and its proof, let $N=2$, $M_1=0.7m$ and $\lmu\in(0,m]$; write $\hat\lambda=\lambda/m$, $\hat e^2=e^2/m$, $\hat\mu=\lmu/m$. Let $\dm_k(\mathsf M)$ be King's counterterm \eqref{eq:s14:CTdef} on $\Lk$ at the photon mass $\lmu$, $S_{\alpha\beta,k}$ the coefficients of \Cref{lem:s14:struct}, $\sigma_1(M_1)=\liminf_ks^\infty_k(M_1)$ as in \Cref{hyp:s03:T3}, and let $\sigma_{\rm low}$, $\sigma^*$, $\bar j(\hat\mu)$, $\bar s$, $\bar B_3(\hat\mu)$, $\bar B_4^-(\hat\mu)$, $A$, $b$, $L$, $I_\infty$, $I^*$ be as defined before \Cref{tab:s14:T3}, $B_3^*$, $P^*$, $Q^*$ as in \Cref{lem:s14:CF}, and $\lambda_c\defeq\frac{b+\sqrt{b^2+4AL}}{2L}=2.14398\ldots$. \begin{proposition}[the explicit coupling region of \Tthree\ at every photon mass]\label{prop:s14:T3} \leavevmode \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item \emph{(The five orders.)} For every $k$ with $\eps_k\le1/m$, in units $m=1$: $S_{01,k}>0$; $2S_{20,k}\ge-\bar j(\hat\mu)\ge-0.092611$; $2S_{02,k}\ge-\bar s=-6.5231964$; $2S_{21,k}\ge-\bar B_3(\hat\mu)$ and $2S_{21,k}\ge-B_3^*$; $2S_{40,k}\ge-\bar B_4^-(\hat\mu)$ and $2S_{40,k}\ge-(P^*+Q^*/\hat\mu)$. \item \emph{(The lower bound.)} $\sigma_1(0.7m)\ge m^2\max\big\{\sigma_{\rm low}(\hat\lambda,\hat e^2;\hat\mu),\ \sigma^*(\hat\lambda,\hat e^2;\hat\mu)\big\}$. \item \emph{(The region.)} \Tthree\ holds on $R(\lmu)\defeq\{(\lambda,e^2):\sigma_{\rm low}(\hat\lambda,\hat e^2;\hat\mu)>0\}$, with $\sigma_1(0.7m)\ge m^2\sigma_{\rm low}>0$ there. $R(\lmu)$ contains the box $[0,c(\hat\mu)m]^2$ and the segment $\{\lambda=0,\ e^20$, with $\sigma_1(0.7m)\ge m^2\sigma^*$; in particular, for every $\lmu\in(0,m]$, on each of the boxes $\{\lambda\le\lambda_1m,\ e^2\le c_1\sqrt{\lmu m}\}$ with $(\lambda_1,c_1)\in\{(0,0.3014),\allowbreak(0.05,0.2533),\allowbreak(0.1,0.2152),\allowbreak(0.2,0.1616),\allowbreak(0.5,0.0884),\allowbreak(1,0.0447)\}$. \item \emph{(Every $k$.)} For every $k$ with $\eps_k\le1/m$: $s^\infty_k(0.7m)\ge m^2\big[A-\frac12\bar j(\hat\mu)\hat e^2-\frac12\bar s\hat\lambda^2-\frac12\bar B_3(\hat\mu)\hat e^2\hat\lambda-\frac12\bar B_4^-(\hat\mu)\hat e^4\big]$. \item \emph{($e=0$; exact.)} At $e=0$, for every $M_1\in(0,m)$, every $\lmu>0$ and every $N\ge1$: $\lim_ks^\infty_k(M_1)=\frac12(m^2-M_1^2)+\frac{(N+2)\lambda(m-M_1)}{2\pi}-\frac{(N+2)\lambda^2}{\pi^2}\ln\frac m{M_1}$ exactly, and \Tthree\ holds if and only if $\lambda<\frac\pi4\big(1+\sqrt{1+\frac{16}{N+2}}\big)m$, which is $\frac\pi4(1+\sqrt5)\,m=2.5416\ldots\,m$ for $N=2$; at $M_1=0.7m$, $N=2$, the exact window is $\lambda<\lambda_cm$, $\lambda_c=2.14398\ldots$, in particular $\lambda\le2.1439\,m$. \item \emph{(Two-sided.)} On each region of (iii), (iv) and (vi) the two-sided clause \sxivNE\ holds as well, with no use of \Cref{prop:s10:Upm}. \end{enumerate} \end{proposition} Outside these regions the sign clause \Tthree\ is not decided here (it fails on $E^<_W$, \Cref{r13:lem:s14:failset}(i), so the regions are not the whole coupling space). The region is explicit at every photon mass but not uniform in it: by \eqref{eq:s14:sigmastar}, $e^2_{\max}(\hat\mu)\sim\sqrt{2A/Q^*}\sqrt{\hat\mu}=0.81\sqrt{\hat\mu}$ as $\hat\mu\to0$ (\Cref{tab:s14:T3}: $\hat\mu\,\bar B_4^-(\hat\mu)\to0.77$), and $\bigcap_{\lmu\in(0,m]}R(\lmu)=\{e=0,\ \lambda<\lambda_cm\}$, since for $e>0$ the term $-\frac12(Q^*/\hat\mu)\hat e^4$ of $\sigma^*$ dominates as $\hat\mu\to0$. The $1/\hat\mu$ sits in the order-$e^4$ majorant alone: the low orders and $B_3^*$ are uniform in the photon mass. % % \begin{proof}[Proof of \Cref{prop:s14:T3}] Units $m=1$, $N=2$, $M_1=0.7$; $\hat\mu=\lmu\in(0,1]$. \emph{(i).} $S_{01,k}>0$ is \Cref{lem:s14:A}(i). The four bounds with interval upper sums are \Cref{lem:s14:TF} with \Cref{lem:s14:MON}, applied to the row of $\hat\mu$ (for $\hat\mu\ge10^{-4}$); the bounds with $B_3^*$ and $P^*+Q^*/\hat\mu$ are \Cref{lem:s14:CF}; for $\hat\mu<10^{-4}$ the first bound is the $\mu$-free branch of $\bar j$: $j\le2M\bar\tau$ gives $\bar j_{400}(\hat\mu,\infty)\le\sum h\,(2M\bar\tau^*)^{\rm cell}=0.092610085$ for every $\hat\mu$, by the cell rule \eqref{eq:s14:cell}. The numbers $\bar j\le0.092610085<0.092611$ and $\bar s=6.5231964$ are read off \Cref{tab:s14:T3}. The interval value $\bar B_4^-(\hat\mu)$, taken at $\mu_{\rm lo}$, and the closed form $P^*+Q^*/\hat\mu$, taken at $\hat\mu$, are two separate bounds, neither of which dominates the other on a whole row. \emph{(ii).} By \eqref{eq:s14:struct} and \Cref{lem:s14:A}(iii), \Cref{lem:s14:B}(iii), \Cref{lem:s14:C}(iii), the first four coefficients converge, so \[ \sigma_1(0.7)=\liminf_ks^\infty_k(0.7)=A+b\hat\lambda-\hat e^2AI_\infty(\hat\mu)-L\hat\lambda^2+\liminf_k\big(\hat e^2\hat\lambda S_{21,k}+\hat e^4S_{40,k}\big), \] and by (i), for all but finitely many $k$ (those with $\eps_k\le1$), $\hat e^2\hat\lambda S_{21,k}+\hat e^4S_{40,k}\ge-\frac12\bar B_3(\hat\mu)\hat e^2\hat\lambda-\frac12\bar B_4^-(\hat\mu)\hat e^4$; with $I_\infty(\hat\mu)\le I_\infty(\mu_{\rm lo})$ (\Cref{lem:s14:B}(iv)) this is $\sigma_1(0.7)\ge\sigma_{\rm low}$. The same argument with the closed-form bounds of (i) and $I_\infty(\hat\mu)\le I^*$ gives $\sigma_1(0.7)\ge\sigma^*$. (Cellwise, the interval value of $16\varphi_3$ at $\mu_{\rm lo}$ is at most the starred cell value, so $\bar B_3(\hat\mu)\le B_3^*$ always holds; for $b_4$ the interval value carries $\bar\delta(\mu_{\rm lo})\approx c_N/(8\pi\mu_{\rm lo})$ while the closed form carries $Q^*/\hat\mu$, so neither bound dominates the other on a whole row; hence the maximum in (ii).) \emph{(iii).} The first sentence is (ii) and the definition of \Tthree\ (\Cref{hyp:s03:T3}). $\sigma_{\rm low}$ is concave in $\hat\lambda$ at fixed $\hat e^2$ (the term $-L\hat\lambda^2$) and concave in $\hat e^2$ at fixed $\hat\lambda$ (the term $-\frac12\bar B_4^-\hat e^4$), so its minimum over a box $[0,c]^2$ is attained at a corner: the minimum over $\hat\lambda\in[0,c]$ of a concave function is attained at an endpoint, the two resulting functions of $\hat e^2$ are concave and so is their minimum, whose minimum over $[0,c]$ is again at an endpoint. The side $c(\hat\mu)$ of \Cref{tab:s14:T3} is the largest $c$ for which the four corner values are $\ge0$, rounded down; $e^2_{\max}$ is the positive root of $\sigma_{\rm low}(0,\cdot)=A(1-\hat e^2I_\infty)-\frac12\bar B_4^-\hat e^4$, rounded down; and at $e=0$, $\sigma_{\rm low}=A+b\hat\lambda-L\hat\lambda^2>0$ if and only if $\hat\lambda<\lambda_c=\frac{b+\sqrt{b^2+4AL}}{2L}=2.14398\ldots$ (as in (vi)). \emph{Monotonicity in $\hat\mu$:} on $[10^{-4},1]$, $\sigma_{\rm low}(\cdot;\hat\mu)$ is a step function of $\hat\mu$, built row by row from $I_\infty(\mu_{\rm lo})$ and the interval values $\bar B_3$, $\bar B_4^-$ of the row, which enter with negative coefficients; so it suffices that these three row values do not decrease from each row to the row below it. For every row but the top one the interval values are the point values at $\mu_{\rm lo}$ (\Cref{lem:s14:MON}, last clause, and the remark at the end of its proof for the row $[0.98,0.99]$), the point values are non-increasing in $\mu$ (\Cref{lem:s14:MON}(i), with the point prefactor $\max_{M\in\{a,b\}}|\mu^2-2M^2|=2b^2-\mu^2$ non-increasing in $\mu$ for $\mu\le0.989949$) and so is $I_\infty$ (\Cref{lem:s14:B}(iv)); hence the values of a row, taken at its $\mu_{\rm lo}$, are at most those of the row below, whose left end is smaller. For the top row $[0.99,1]$ this is read off the table ($8.5786210\le8.6036471$, $1.5854190\le1.6023646$, $0.008458\le0.008491$); by inspection each of the three columns is non-decreasing down the table through all $34$ rows. Across $\hat\mu=10^{-4}$: $\sigma^*$ is non-decreasing in $\hat\mu$, and $\sigma^*(\cdot;\hat\mu)\le\sigma_{\rm low}(\cdot;10^{-4})$ for $\hat\mu<10^{-4}$ because $I^*\ge I_\infty(10^{-4})$, $B_3^*\ge\bar B_3(10^{-4})=13.186961$ and $P^*+Q^*/\hat\mu>7716.84>\bar B_4^-(10^{-4})=7716.1679$. At a grid point both rows containing it give valid bounds, and the convention selects the row with $\mu_{\rm lo}=\hat\mu$. Hence $\hat\mu\mapsto\sigma_{\rm low}(\hat\lambda,\hat e^2;\hat\mu)$ is non-decreasing and $R(\lmu)\supseteq R(\lmu')$ for $\lmu\ge\lmu'$. \emph{The listed boxes:} $c\ge0.2063$ on the row $[0.5,0.55]$, $c\ge0.1525$ on $[0.1,0.15]$, $c\ge0.1132$ and $\sigma_{\rm low}(0.1,0.1)\ge0.0582$ on $[0.03,0.04]$, $c\ge0.0753$ on $[0.01,0.015]$ (\Cref{tab:s14:T3}), together with the monotonicity; at $\hat\mu=1$ the row $[0.99,1]$ gives $c\ge0.2390$ and $e^2_{\max}\ge0.5618$. \emph{(iv).} $\sigma^*$ is concave in each variable separately, so again the corners decide. Let $(\hat\lambda,\hat e^2)$ lie in the box $\{\hat\lambda\le\lambda_1,\ \hat e^2\le c_1\sqrt{\hat\mu}\}$ with $\hat\mu\le1$. At a corner $(\hat\lambda,c_1\sqrt{\hat\mu})$, $\hat\lambda\in\{0,\lambda_1\}$, \[ \begin{split} \sigma^*&=A-AI^*c_1\sqrt{\hat\mu}+b\hat\lambda-L\hat\lambda^2-\tfrac12B_3^*c_1\sqrt{\hat\mu}\,\hat\lambda-\tfrac12\big(P^*\hat\mu+Q^*\big)c_1^2\\ &\ge\ A-AI^*c_1+b\hat\lambda-L\hat\lambda^2-\tfrac12B_3^*c_1\hat\lambda-\tfrac12(P^*+Q^*)c_1^2=\sigma^*(\hat\lambda,c_1;1), \end{split} \] using $\sqrt{\hat\mu}\le1$ and $\hat\mu\le1$ in the negative terms; the corners with $e=0$ give $A>0$ and $A+b\lambda_1-L\lambda_1^2>0$ ($\lambda_1\le1<\lambda_c$). So it suffices that $\sigma^*(0,c_1;1)\ge0$ and $\sigma^*(\lambda_1,c_1;1)\ge0$ for each listed pair, which is how the $c_1$ were determined (bisection on $c_1$, rounded down); explicitly, $\sigma^*(0,c_1;1)=A(1-I^*c_1)-\frac12(P^*+Q^*)c_1^2=3.8\cdot10^{-5}$ at $c_1=0.3014$, and $\sigma^*(\lambda_1,c_1;1)$ equals $1.2\cdot10^{-4}$, $4.1\cdot10^{-5}$, $1.1\cdot10^{-4}$, $2.3\cdot10^{-4}$, $6.3\cdot10^{-4}$ at the five pairs with $\lambda_1>0$, each positive. \emph{(v).} \eqref{eq:s14:struct}, $\hat\lambda S_{01,k}\ge0$ (\Cref{lem:s14:A}(i)) and (i). \emph{(vi).} At $e=0$ only the orders $(0,1)$ and $(0,2)$ of \eqref{eq:s14:CTdef} are present, so by \Cref{lem:s14:A}(iii), \Cref{lem:s14:C}(iii) (for an arbitrary $M_1\in(0,m)$, and now in general units) $\lim_ks^\infty_k(M_1)=g_\lambda(M_1)\defeq(m-M_1)\big[\frac{m+M_1}2+\frac{(N+2)\lambda}{2\pi}\big]-\frac{(N+2)\lambda^2}{\pi^2}\ln\frac m{M_1}$ exactly, and $\sigma_1(M_1)=g_\lambda(M_1)$. As a function of $M_1\in(0,m]$: $g_\lambda(m)=0$, $g_\lambda'(M_1)=-M_1-\frac{(N+2)\lambda}{2\pi}+\frac{(N+2)\lambda^2}{\pi^2M_1}$, $g_\lambda''(M_1)=-1-\frac{(N+2)\lambda^2}{\pi^2M_1^2}<0$, so $g_\lambda$ is strictly concave. If $g_\lambda'(m)\ge0$, concavity gives $g_\lambda(M_1)0$ on a left neighbourhood of $m$, and \Tthree\ holds. Now $g'_\lambda(m)<0$ if and only if $x=\lambda/m$ lies below the positive root of $\frac{N+2}{\pi^2}x^2-\frac{N+2}{2\pi}x-1$, which is $x_*=\frac\pi4\big(1+\sqrt{1+\frac{16}{N+2}}\big)$, equal to $\frac\pi4(1+\sqrt5)=2.5416\ldots$ for $N=2$. At a fixed $M_1$, $g_\lambda(M_1)>0$ if and only if $L\lambda^2-b\lambda-A<0$ with $A,b,L$ at $M_1$, i.e.\ $\lambda<\lambda_c(M_1)m\defeq\frac{b+\sqrt{b^2+4AL}}{2L}m$; at $M_1=0.7m$, $\lambda_c=2.14398\ldots$. \emph{(vii).} By \sxivCT\ (Lemma~\ref{lem:s14:CT}) the relative limit $\alpha_\infty(M)-\alpha_\infty(m)$ exists for every $M>0$ and $\sigma_1(M_1)=\lim_ks^\infty_k(M_1)=\frac12(\alpha_\infty(m)-\alpha_\infty(M_1))$. On each region of (iii), (iv), (vi), $\sigma_1(0.7m)>0$ (resp.\ $\sigma_1(M_1)>0$), so $\alpha_\infty(0.7m)<\alpha_\infty(m)$: $m$ is not a minimiser of $\alpha_\infty$ on $(0,m]$, which is \sxivNEl, and \sxivNEl\ implies \sxivNE\ (\Cref{r13:lem:s14:failset}(i)). \Cref{prop:s10:Upm} is needed only on $E^<_W\setminus E_W$, which these regions avoid. \end{proof} \space% % \begin{remark}[the quantifier over the couplings]\label{rem:s14:quantifier} The theorems of this paper carry the sources' quantifier of \Cref{lem:s14:CW}(e), in the units of the coarsest spacing, with the cube family $\CJ$ not rescaled and $\beta=5^{-J_{\rm K}}$. A sentence ``$e,\lambda$ sufficiently small, $m,\lmu>0$ fixed'' describes \emph{where} the smallness lives (the running couplings, through Ba{\l}aban's $\eps_0$), but as a hypothesis on the bare couplings it is strictly stronger than what the sources need. (i) All four couplings have mass dimension one, so its dimensionless content is a bound on $e^2/m$ and $\lambda/m$, while the sources' hypothesis on the bare couplings is empty beyond positivity: the point $(e^2,\lambda,m,\lmu)=(100,100,1,1)$ is admitted by every source (with a large $J_{\rm K}$) and excluded by the phrase in every system of units. (ii) In the units in which the stopping scale is $1$, $m$ and $\lmu$ are not fixed but rescaled to $m\beta$ and $\lmu\beta$, of which $m\beta$ is capped (by the mass cap of \cite[Prop.~3.1]{Ba2} and by a further cap of the schedule) and $\lmu\beta$ is not: in those units the theorem reads ``three couplings small, the fourth rescaled''. (iii) Even at a fixed stopping scale $\beta_0$ the admissible set is not a product neighbourhood of $(0,0)$ in $(e,\lambda)$: one cap of the schedule of constants of the rotation part has the form $e\le\lambda^{1/4}c(\beta_0)$, a cusp that fails as $\lambda\to0$ at fixed $e$. So the phrase names no set exactly, and the sources' form survives because $\beta$ can be taken to $0$. \end{remark} \space% % % % % \section{Open problems}\label{sec:s23} \ifcsname openproblem\endcsname\else \theoremstyle{definition} \newtheorem{openproblem}[theorem]{Open Problem} \crefname{openproblem}{Open Problem}{Open Problems} \Crefname{openproblem}{Open Problem}{Open Problems} \fi This section collects what is not proved in this paper: the relation of King's state to the periodic-torus states, its uniqueness and ergodicity, non-triviality, the limit of vanishing photon mass, the Wick sector at arbitrary amplitudes with its Euclidean clause and the stability of the shifted torus model behind it, the rotation invariance with scalar sources at the points of $E_W$ and above the threshold, and the coupling condition. Each is posed as a question about the objects of Theorem~\ref{thm:s13:main} and of the rotation part; none is used anywhere in the paper. Throughout, $K$ denotes King's state of \cref{thm:s13:main}, the infinite-volume state of King's construction \cite{K1,K2}: the Neumann--Dirichlet cube limit on King's RG cube at block factor $5$, fixed photon mass $\lmu>0$, with an admissible scheme $a$ in the Wick-square sector; $S$ denotes a periodic-torus limit in the sense of Definition~\ref{cpi:per:def}(iii) below (dyadic spacings, periodic tori, the mass counterterm computed with the photon propagator at the mass $m$), at couplings at which one exists. No identification between the two is proved, and the existence of $S$ is not proved here. ``Symmetric phase'' is, for both, the expectation that the states lie above the surface predicted by one-loop perturbation theory; it is neither a hypothesis nor a conclusion of any theorem here, and the curvature leg of \cref{thm:s13:main} holds on all of King's coupling range (\cref{lem:s14:CW}(a)), the surface included, the Wick leg on the coupling region of \cref{hyp:s03:T3}, where the sign clause \Tthree\ holds (\cref{lem:s14:CW}(b), Section~\ref{ssec:s23:coupling}). The problems below are extensions of the main theorem, not conditions on it. \subsection{Relation to the periodic-torus states} For which couplings do the periodic-torus states of Definition~\ref{cpi:per:def} have continuum limits $S$, and does such a limit coincide with $K$, possibly at a matched bare mass? Three statements bear on this, none of them proved; we call them $(\mathrm{FAM}_L)$, King's cube construction exists with block factor $2$ along the dyadic spacings $\eps'_K$ of Definition~\ref{cpi:per:def}, with the counterterm defined there; $(\mathrm{KI}_L)$, the fixed-torus continuum limit exists in that family; $(\mathrm{CE})$, a convergent unit-scale cluster expansion with sources on the cubic tori, their tilted copies and their cuts along King's cubes, uniform in the lattice spacing. Whether $(\mathrm{CE})$ holds anywhere at $e>0$ is open. Two obstacles are structural. The counterterm of Definition~\ref{cpi:per:def} is computed with the photon propagator at the mass $m$, King's with the model's own $\lmu$; already at order $e^2$ the two relative bare coefficients, $s^{\rm per}_{j,K}(M_1)$ and King's $s^\infty_k(M_1)$, have different limits unless $\lmu=m$ (the first as $j\to\infty$ and then $K\to\infty$), the difference being at most of order $e^2|m-\lmu|$, of exactly that order for $\lmu$ in compact subsets of $(0,\infty)$, and not small when $|m-\lmu|$ is comparable to $m$ (the one-loop comparison stated after Definition~\ref{cpi:per:def}); and King's construction needs block factor $5$ for the rotation by $\arctan(3/4)$, while no lattice family admits both a dyadic structure and a rotation of infinite order. On the one-loop surface $\lambda m^2=e^6/(72\pi^2)(1+O(e^2/m))$ the two constructions are expected to select different phases (periodic tori the broken one, Dirichlet faces the symmetric one); this is a heuristic, not a theorem, and no statement of this paper uses the phase. % \begin{definition}[periodic-torus states with the counterterm at photon mass $m$]\label{cpi:per:def} Let $e>0$ and $\lambda,m,\lmu>0$. Fix an integer $n_0\ge0$ and an even integer $\ell_0\ge2$; for $K\ge0$ and $j\ge1$ put \[ \eps'_K\defeq2^{-K-n_0},\qquad\ell_j\defeq j!\,\ell_0,\qquad T_j\defeq\R^3/(\ell_j\Z)^3,\qquad T_{j,K}\defeq T_j\cap\eps'_K\Z^3 . \] Since $\ell_j/\eps'_K=j!\,\ell_0\,2^{K+n_0}$ is an even integer, $\eps'_K\Z^3$ descends to $T_j$ with an even number of sites per side; $\ell_j\mid\ell_{j+1}$ and $\ell_j\to\infty$. \begin{enumerate} \item[(i)] \emph{The counterterm.} For $M>0$ let $\dm_{{\rm per};j,K}(M)$ be the polynomial \eqref{eq:s14:CTdef} over the orders $(\alpha,\beta)\in\{(0,1),(2,0),(0,2),(2,1),(4,0)\}$, with every coefficient fixed by the recursive zero-momentum condition stated there, computed on the finite periodic graph $T_{j,K}$ with the periodic scalar propagator $(-\Delta_{\eps'_K}+M^2)^{-1}$ and the periodic photon propagator $(-\Delta_{\eps'_K}+m^2)^{-1}$, both without block averaging. The photon propagator of these graphs has the mass $m$, whatever the value of $\lmu$. Both propagators are massive, so every graph is a finite sum on the dual group of $T_{j,K}$ and each coefficient is a finite real number. \item[(ii)] \emph{The state.} $\langle\cdot\rangle^{\rm per}_{K,j}$ is King's action on $T_{j,K}$ with periodic conditions for both fields, exactly as the periodic state of Definition~\ref{def:s10:torus} with the spacing $\eps'_K$ in place of $\eps_k$: Feynman-form gradient terms for every pair of parallel neighbouring bonds, the photon mass $\lmu^2$, hopping terms on every bond, and the single-site weights $e^{-\eps_K'^3[\frac12b\,r^2+\lambda r^4]}$ with the site-independent bare coefficient \[ b=b^{\rm per}_{j,K}\defeq m^2+\dm_{{\rm per};j,K}(m). \] The normalisation is finite and positive because $\lambda>0$ and $\lmu>0$. The observables are the gauge-invariant functions of the fields, generated by the curvature $F(f)$ and the Wick square $\sum_x\eps_K'^3h(x)\big(r_x^2-2\sW_{\eps'_K}\big)$, with the free Wick constant $\sW_\eps=(-\Delta_{\eps\Z^3}+m^2)^{-1}(0,0)$ of the free convention. \item[(iii)] \emph{Continuum limits.} A \emph{periodic-torus limit} is a family $S$ of Schwinger functions of the curvature and the Wick square such that, along some sequence $(K_l,j_l)$ with $K_l\to\infty$ and $j_l\to\infty$, the lattice Schwinger functions of $\langle\cdot\rangle^{\rm per}_{K_l,j_l}$, at test functions in $C^\infty_c$ sampled on the lattice, converge to those of $S$ for all orders. \end{enumerate} \end{definition} The bare coefficient of King's state is computed on the infinite lattice with the photon propagator at $\lmu$ (\eqref{eq:s14:CTdef}); that of Definition~\ref{cpi:per:def} on the tori with the photon propagator at $m$, for every $\lmu$. The two differ already at order $e^2$: for $M_1\in(0,m)$ the relative bare coefficient $s^{\rm per}_{j,K}(M_1)\defeq\frac12[b^{\rm per}_{j,K}-(M_1^2+\dm_{{\rm per};j,K}(M_1))]$, the analogue of King's $s^\infty_k(M_1)$ of Lemma~\ref{lem:s14:struct}, has at order $e^2$ the limit $-e^2(m-M_1)R_m/8\pi$ as $j\to\infty$ and then $K\to\infty$, while King's has $-e^2(m-M_1)R_{\lmu}/8\pi$, where $R_\nu\defeq\frac{m\nu+mM_1+M_1\nu}{(\nu+M_1)(m+\nu)}=1-\frac{\nu^2}{(\nu+M_1)(\nu+m)}$ is strictly decreasing in $\nu$, tends to $1$ as $\nu\downarrow0$ and is Lipschitz with constant $2/m$ (a one-loop computation: the orders $\lambda$, $\lambda^2$ contain no photon line and agree, the order $e^2$ is $-\frac12(m^2-M_1^2)\int\hat k^2\hat D_\nu G_mG_{M_1}$ on the lattice, whose continuum limit is $R_\nu/(4\pi(m+M_1))$, and the torus sums are Riemann sums of this integral); so the two limits differ unless $\lmu=m$, by at most $e^2(m-M_1)\abs{m-\lmu}/(4\pi m)$ and by a positive multiple of $e^2(m-M_1)\abs{m-\lmu}$ for $\lmu$ in compact subsets of $(0,\infty)$. \space% % \subsection{Uniqueness and ergodicity of \texorpdfstring{$K$}{K}} Uniqueness of King's cube limit is part of the construction (monotonicity in the cube), but uniqueness of the infinite-volume state, in the sense of King's expected identity between cube limits and periodic-box limits, is not proved, and neither is ergodicity of $K$ under translations: \cref{thm:s13:main} gives clustering only on the ergodic components. The usual route, a cluster expansion with a mass gap, is $(\mathrm{CE})$. This is distinct from the independence of a LIM state (Section~\ref{sec:s12}) from its continuum subsequence, \UNIQ, which holds (\cref{hyp:s12:UNIQ}, from \cref{hyp:s03:RC} and from \cref{hyp:s03:EC}; \cref{prop:s12:ID}, \cref{cor:s04:G2}). Cubic symmetry of $K$ is a theorem (\cref{thm:s06:S1K}): it needs only the uniqueness of the monotone cube limit and the reflection sandwich (\cref{rem:s06:K4}). \subsection{Non-triviality} For a state with Schwinger functions of the curvature and the Wick square let $M(f;h)\defeq S^T_{2,1}(f,f;h)$ be the truncated function with two curvature arguments and one Wick-square argument. If the curvature and the Wick square are independent under the law of a state, all their mixed truncated functions vanish; if their joint law is Gaussian, all truncated functions of order three vanish. In either case $M\equiv0$; so $M(f;h)\ne0$ for one real $2$-form $f$ and one $h\ge0$ would show that under $K$ the curvature and the Wick square are neither independent (``decoupled'') nor jointly Gaussian. On every BFS measure, in particular on every lattice approximant of $K$, $\big\langle\sum_x\eps^3h(x)r_x^2;F(f)^2\big\rangle\le0$ for $h\ge0$: this is \cref{lem:s02:BFS}(a) for $\Theta=e^{t\sum_x\eps^3h(x)r_x^2}$ and $\ell=F(f)$, divided by $t>0$ and taken as $t\downarrow0$ (dominated convergence, $\lambda>0$). The sign is not strict. A route to $M(f;h)<0$ at small couplings has two steps, both open: (C) at $e=\lambda=0$ the coefficient of $e^2$ in that lattice covariance has a strictly negative continuum limit $c^{\rm free}(f,h)$ for $d^*f\not\equiv0$ and $h\ge0$, $h\not\equiv0$ (a one-loop statement); (R) $e^{-2}$ times the covariance differs from that coefficient by at most $\rho((e^2+\lambda)/m)\,\kappa(f,h)$, with $\rho(t)\to0$ as $t\downarrow0$, uniformly along the approximants of $K$. (C) and (R) together would give $M(f;h)<0$ at small couplings; both are open. The bounds of the existence theory are not uniform in the couplings, and the cited constructions state no asymptotics in $(e,\lambda)$, so neither yields (R). A non-vanishing curvature two-point function is also left open. For $K$ nothing is proved here beyond the non-strict lattice sign above. Whether $K$ is non-Gaussian, or even not decoupled, is open. \subsection{The limit of vanishing photon mass} Every constant of the Wick leg of \cref{thm:s13:main} depends on $\lmu$ and is not uniform in it (\cref{rem:s13:mu0}); the absolute bounds of the curvature leg are. The limit of vanishing photon mass, $\lmu\downarrow0$, taken along a family with fixed counterterm and scheme (both held at a reference mass, so that the members are monotone in $\mu$ on the cone by the correlation inequalities, \cref{lem:s02:BFS}(a)), is treated in a subsequent paper and is not taken here: in the curvature sector it would need the existence theorem for every member and nothing else; in the Wick sector it would need a cell bound uniform in the photon mass (the analogue, along the family, of the cell bound \Bcell\ of \cref{hyp:s12:Bcell}, which is proved at a fixed mass; the clause ``several models at once'' of \cref{lem:s12:LIM0} is formulated for this purpose), together with the torus theorems at the reference mass and the transfer of \cref{thm:s13:TWK} for the members on King's convention; the identification of the members with King's models at shifted scalar masses, masslessness proper (the vanishing of the photon mass in the limit theory) and the two-point function are further questions. None of these enters the present paper. Of the three statements proved for an admissible scheme at a fixed photon mass, the cell bound \Bcell, the scheme bound \SCH\ and the independence \UNIQ\ of the continuum-subsequence statements from the subsequence (\cref{hyp:s12:Bcell,hyp:s12:SCH,hyp:s12:UNIQ}, \cref{prop:s12:ID}), the first two would have to be re-established for a LIM state along a family of photon masses, as for a LIM state of a scheme that is not admissible. \subsection{The Wick sector above the amplitude threshold, and its Euclidean clause} Form~(B) of \cref{thm:s13:main} lives on the tube $\Dtube$: the only positive-source bound it uses is the one proved on the cone $\sup q\le s_1<\sigma_1(M_1)$ (\cref{cor:s11:W2prime}), and the only existence statement is \cref{hyp:s03:W1}, whose coefficient vectors have positive scalar part below the same threshold. Above the threshold the existence of the fixed-cube limits and a uniform positive-source bound are open (\cref{hyp:s03:W2}), and so are a Euclidean clause for the Wick sector (\cref{hyp:s03:RW}) and the corresponding statements on King's convention (\cref{hyp:s03:FAK}). % % % \begin{openproblem}[$\mathrm{W2}$: the Wick sector at arbitrary amplitudes]\label{hyp:s03:W2} Let $a$ be an admissible scheme, and let $Z^a_{k,J}$ denote the scheme functional of \cref{def:s02:sources} on King's convention or on the comparison convention. It is not known whether (a) for every finite family of real $2$-forms and real functions, every $J\ge J_0(f,h)$ and \emph{every} real coefficient vector $(z,w)$ the limit $\lim_kZ^a_{k,J}(z,w)$ exists and lies in $(0,\infty)$, nor whether (b) for every $q\in C^\infty_c(\R^3)$ with $q\ge0$ \[ \sup_{J\ge J_q}\ \limsup_{k\to\infty}Z^{a}_{k,J}(0,q)<\infty , \] where $J_q$ is the least $J$ with $\supp q$ at $\ell^\infty$-distance greater than $1$ from $\R^3\setminus\CJ$. \end{openproblem} What is proved is the part below the amplitude threshold, which is all that Form~(B) uses: (a) on King's convention for every real coefficient vector whose positive scalar part satisfies $2\norm{k_+}_\infty0$. It is not known whether, in the free convention and the scheme $a$, \[ \sup_{k\ge0}\ \limsup_{J\to\infty}\ \alpha^a_{k,T_J}(u)<\infty,\qquad\alpha^a_{k,T}(u)=|T|^{-1}\ln N^a_{k,T}(0,u\mathbf 1), \] the free-energy density of the torus functional at the constant scalar source $u$; that is, whether \Itwo\ of \cref{def:s10:I2} holds at the amplitude $s_1=u$. \end{openproblem} For $u$ below the threshold $s_*$ defined before \eqref{eq:s14:EW}, $\Bminus{u}$ is proved: it is \cref{prop:s10:U} (or \cref{prop:s10:Upm}) at an auxiliary mass $M_1$ with $\sigma_1(M_1)>u$, so Form~(B), whose constants are built from \Itwo\ at one such amplitude, already contains every amplitude at which the positive-source bound is known. For $u\ge s_*$ the statement is open, and it is the exact obstruction. The scheme enters only through the exact relation $\alpha^a_{k,T}(u)=\alpha^z_{k,T}(u)+(z_k-a_k)u$ (proof of \cref{prop:s10:U}); the discussion below is in the reference scheme $z$. By \cref{lem:s10:M}, exact at fixed $(k,T)$, $\alpha^z_{k,T}(u)$ is, up to explicit $c$-numbers, $|T|^{-1}$ times the difference of the free energies of King's torus model with bare scalar coefficient $\alpha^{\rm tor}_{k,T}(m)-2u$ and with $\alpha^{\rm tor}_{k,T}(m)$, both with the counterterms of the mass $m$: the second is \cref{hyp:s03:T1} at $m$, the first the ultraviolet stability of the shifted model. For $u0}\alpha^\infty_k(\mathsf M)$, and no positive scalar mass realises it. The proofs of \cite{Ba1,Ba2} expand about the Gaussian at the positive scalar mass $m$ and use its mass term in the small-field bounds; no published theorem covers a bare coefficient below every positive-mass value. The two open statements are related by \cref{prop:s11:W1prime}, whose steps do not depend on the amplitude: run at $s_1=u$ it turns $\Bminus{u}$ into the bound $\ln N^a_{k,X}(0,q)\le(C_U/u)\norm q_{1,k}$ for $0\le q\le u$ on every Neumann--Dirichlet box, that is into clause (b) of \cref{hyp:s03:W2} in $L^1$ form on $\sup q\le u$. The converse direction is not known: the inequalities available bound a constant-source density from below by local sources, never from above. Also open: a form of the tube bound \eqref{eq:s11:W} in which the negative part $k_-$ of the Wick source enters through its $L^1$ norm in place of the weighted norm $\wnorm{k_-}$. \subsection{Rotation invariance with scalar sources: the points of $E_W$ and the sources above the threshold}\label{rot:sec:open} Theorem~\ref{rot:thm:main} concerns curvature sources, and Theorem~\ref{rot:thm:mainGS} (Section~\ref{rot:sec:gensrc}, whose notation we use) curvature sources together with neutral scalar sources on the tube $\Om(h)$, outside the set $E_W$. The following two problems are not solved in this paper; neither is a hypothesis of Theorem~\ref{rot:thm:main} or of Theorem~\ref{rot:thm:mainGS}. \emph{1. The points of $E_W$.} At a point of $E_W$ (Proposition~\ref{rot:thm:POS}), $s_*(e)=0$ and the cone, the tube and the split domain of Definitions~\ref{rot:def:cone} and~\ref{r13:def:srcsplit} are empty; Theorem~\ref{rot:thm:main} covers $h=0$ there by right-continuity in the scalar mass (Theorem~\ref{cpi:ew:EWC}), and nothing with a scalar source is proved. The statement that would extend Theorem~\ref{rot:thm:mainGS} to such a point is: (W$^\sharp$2) at that point with some positive threshold $s'$ in place of $s_*(e)$ and (W$^\sharp$1) on the split domain of $s'$ there (Theorem~\ref{rot:thm:Wcone}(i) uses it), together with the two walled-torus statements of the Wick sector in the scheme $a^\sharp$ on the cone of $s'$ at that point: the walled torus comparison on the cone (Theorem~\ref{rot:thm:Kwsupply}, Section~\ref{rot:ssec:Kw}) and the closing statement on the cone (Section~\ref{rot:ssec:shellg}); both rest on the walled large-field envelope, statement (3) of Section~\ref{rot:ssec:Ksupply}, which is proved where $s_*(e)>0$. With these, every statement of Section~\ref{rot:sec:gensrc} holds at that point with $s'$ in place of $s_*(e)$, by the same proofs. Right-continuity in the scalar mass does not provide them: the tubes $\Om(h)$ at the nearby masses shrink to the empty set as the mass decreases to $m$. \emph{2. The generating functional above the threshold.} (W$^\sharp$1) at the points outside $D(h)$, where the positive part of the split source exceeds half the threshold, and (W$^\sharp$2) at every amplitude are not proved (the former would need the large-field half of the cube bound above the threshold, the latter the ultraviolet stability bound at a negative mass); they would extend the domain of $Z^\sharp_\infty$ beyond $\Om(h)$. The invariance there would need, in addition, the walled torus comparison and the closing statement of the scheme $a^\sharp$ above the cone (Sections~\ref{rot:ssec:Kw} and~\ref{rot:ssec:shellg}); the plain-torus comparison of King's type with scalar sources, the equality of the two lattice limits of $Z^a_{\Lambda_k,D_J}$ for every smooth periodic family at every real coefficient, which would give the walled form on all of $\cK$, is likewise not proved. None of this restricts a Schwinger function: every Schwinger function with curvature and Wick-square insertions is a derivative at the interior point $0$ of $\Om(h)$ (Theorem~\ref{rot:thm:mainGS}(iii)). \subsection{The coupling region}\label{ssec:s23:coupling} The sign clause \Tthree\ is proved on the explicit regions of \cref{prop:s14:T3} (at $\lmu=m$ on a region containing the box $0<\lambda,e^2\le0.239\,m$; at every $\lmu\in(0,m]$ on explicit boxes; exactly for $\lambda<\frac\pi4(1+\sqrt5)\,m$ at $e=0$), and the Wick leg is stated on the coupling region of \cref{hyp:s03:T3}, which contains them. 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