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\title[Osterwalder--Schrader axioms for scalar QED$_3$ in the Higgs phase]{Osterwalder--Schrader axioms for scalar QED$_3$ in the Higgs phase}
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\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}}
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\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} on October 10th, 2026.}
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\begin{document}
\begin{abstract}
We study scalar quantum electrodynamics in three dimensions (QED$_3$) in its Higgs phase: the three-dimensional lattice abelian Higgs model in unitary gauge, with the Villain action, on lattices whose spacing tends
to zero with canonically scaled couplings. At each spacing we construct infinite-volume states, as limit points of the periodic states
and as limits of states in finite boxes; they carry the lattice symmetries, reflection positivity and bounds uniform in the volume.
For the smeared curvature and bounded functions of the density, the generating functionals converge along a subsequence of
spacings to a translation-invariant limit functional with the continuity required for the Osterwalder--Schrader reconstruction. For
the limits built on the box states, and on the periodic states along dyadic sequences of sides, the limit curvature
field satisfies the axioms OS0--OS3, with invariance under the full Euclidean group. The
cluster axiom OS4 is proved in a weak form, on almost every ergodic component.
The tools are Gaussian domination of the Villain curvature, Ginibre inequalities and reflection positivity.
\end{abstract}
\maketitle
\setcounter{tocdepth}{1}
\tableofcontents
\section{Introduction}\label{st:sec}

\subsection{The model and the problem}

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{st:eq:action}
S(A,\phi)=\int_{\R^3}\Big[\tfrac14F_{\mu\nu}F_{\mu\nu}+\tfrac12\big|(\partial_\mu+ieA_\mu)\phi\big|^2
+\lambda\big(|\phi|^2-\rho_0^2\big)^2\Big]\,dx,\qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu ,
\end{equation}
with the charge $e>0$, the quartic coupling $\lambda>0$ and $\rho_0^2=m_H^2/(8\lambda)$, $m_H>0$. In three dimensions $e^2$ and
$\lambda$ have the dimension of a mass, and the only ultraviolet counterterm is a mass counterterm for the scalar field. The potential
is minimal on the circle $|\phi|=\rho_0$. Writing $\phi=\rho e^{i\chi}$ with $\rho=|\phi|$ and absorbing the phase $\chi$ into the
vector field by a gauge transformation (the unitary gauge), one obtains the integrand
\[
\tfrac14F_{\mu\nu}F_{\mu\nu}+\tfrac12(\partial_\mu\rho)^2+\tfrac12e^2\rho^2A_\mu A_\mu+\lambda\big(\rho^2-\rho_0^2\big)^2 ,
\]
in which, to second order about $\rho=\rho_0$, the modulus $\rho$ has the mass $m_H$ and the vector field the mass $e\rho_0$. It is
the simplest gauge theory in which the gauge
field becomes massive through its coupling to a charged scalar field with a non-zero vacuum value
\cite{And63,EB64,Hig64}. In three
Euclidean dimensions its action is the free-energy functional of Ginzburg and Landau for a superconductor \cite{GL50}: the scalar field
is the order parameter of the condensate, the gauge field is the vector potential of the magnetic field, and the functional integral
is the partition function of the two with their thermal fluctuations \cite{HLM74}. The model has two phases. In the Coulomb phase the
scalar field has no vacuum value and the gauge field is massless; this is the normal state, which a magnetic field penetrates. In the
Higgs phase the scalar field has a non-zero vacuum value and the gauge field is massive; this is the superconducting state, in which
the magnetic field is expelled (the Meissner effect). The two phases require different constructions, and we
treat them in two papers: the present paper is devoted to the Higgs phase, and the companion paper \cite{CP} to the Coulomb 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. The lattice gauge action is of Villain's form \cite{Vil75} with a \emph{closed} integer $2$-form,
that is, the compact Villain model with every monopole removed. On a finite lattice with free boundary conditions a closed
integer $2$-form is exact, and for observables of integer charge this is the model with a real-valued gauge field on the bonds
and a quadratic gauge action \cite{BIJB84}; on a torus the sum over the closed forms includes the flux classes. The
distinction matters in three dimensions: in the compact pure gauge theory monopoles produce a mass gap and confinement at every
coupling \cite{GM82}, while the non-compact pure gauge theory is the free field.

In three dimensions the model is superrenormalisable, and its
lattice approximations are among the basic examples of a gauge theory with matter \cite{BFS79,BFS80,BFS81,Bal82a,Bal82b,Bal83,Kin86a,Kin86b,Sei82}. The
problem treated here is due to King \cite{Kin86a,Kin86b}: with the renormalisation-group method of Ba{\l}aban
\cite{Bal82a,Bal82b,Bal83} he constructed the continuum limit of the three-dimensional model in finite volume and then its
infinite-volume limit, established its translation invariance and reflection positivity, and addressed its rotation invariance through a
comparison of two rotated lattices. The Osterwalder--Schrader axioms for his state, rotation invariance included, are established
in the companion paper \cite{CP} by a different route; the present paper treats the Higgs phase. The
framework of the present paper is that of Brydges, Fr\"ohlich and Seiler \cite{BFS79,BFS80,BFS81} for the lattice abelian Higgs model:
the gauge-invariant fields, and the axioms obtained through lattice approximations; these authors completed the two-dimensional
case. The method is reflection positivity and Gaussian domination \cite{FSS76,FILS78}, together with Ginibre's inequalities
\cite{Gin70}. We consider the model
on the cubic lattices $\eN\Z^3$ of spacing $\eN=\azero5^{-N}$, $N=N_0,N_0+1,\dots$ (the \emph{levels}; $\azero>0$ is a fixed length and $N_0\ge0$ a fixed integer), in unitary gauge and with the Villain form of the
gauge action \cite{Vil75}. The variables are the radial part $\rho_x=\rhof+\tr_x\ge0$ of the scalar field at the sites, written about
a vacuum value $\rhof>0$, a gauge field $B_b$ on the bonds, and a closed $2$-form $v_p$ on the plaquettes with values in $(2\pi/\gph)\Z$. The
gauge-invariant fields are the scalar density and the \emph{Villain curvature} $\ff_p=(dB)_p+v_p$. The Boltzmann weight is a product of
a site factor, which contains a quartic potential and a bare mass counterterm $\delta m_B$, of a hopping factor on the bonds, and of the
Gaussian factor $e^{-\ff_p^2/2}$ on the plaquettes; it is written out in Section~\ref{sec:setting}. The couplings are scaled with the
spacing according to their canonical dimensions: the charge is $\gph=\eN^{1/2}e$, the quartic coupling is $\lamf=\eN\lambda$, and
$\rhof^2=\eN m_H^2/(8\lambda)$, with fixed $e,\lambda,m_H>0$. Every result of this paper
holds for arbitrary positive values of these lattice couplings at each level $N$ and for every real $\delta m_B$ (for the coefficient of the hopping factor see Section~\ref{sec:setting} and Lemma~\ref{lem:rescale}); the scaling above is the
case of interest.

The problem is to construct limits of the gauge-invariant fields as $N\to\infty$ and to verify for them the axioms of Osterwalder and
Schrader \cite{OS73,OS75}: regularity (OS0), Euclidean invariance (OS1), reflection positivity (OS2), symmetry (OS3) and clustering
(OS4). The observables are the smeared fields
\[
\Phi_N(\sigma)=\eN^{3/2}\sum_pf_{\mu\nu(p)}(x_p)\,\ff_p+\eN^{3}\sum_zh(z)\,g_{\rm d}(\tr_z),\qquad\sigma=(f,h),
\]
where $f$ is a real $C^\infty_c$ $2$-form on $\R^3$, $h\in C^\infty_c(\R^3;\R)$ and $g_{\rm d}$ is a bounded Borel function
(Definition~\ref{def:pin}), and the objects of study are the generating functionals $Z_N(\sigma)=\om_N(e^{i\Phi_N(\sigma)})$ in an
infinite-volume state $\om_N$ of level $N$. Translation invariance and reflection positivity about the coordinate planes are present
on the lattice and pass to the limit. Invariance under rotations is not a symmetry of any of the lattice theories and has to be
recovered in the limit. The two main points of the paper are the construction of the infinite-volume states at fixed spacing without
an expansion, and the Gaussian domination of the Villain curvature, which holds for all values of the couplings and gives the bounds,
uniform in the volume and in the spacing, on which the continuum limit and its regularity rest.

\subsection{Main results}

Two families of infinite-volume states occur: limit points $\om_N$ of the states of the periodic lattices (tori), and the limits
$\om^{\mathrm{box}}_N$ of the states of finite boxes outside of which the matter field is set to zero (the \emph{killed-box} states,
Definition~\ref{def:d2:cone}). We write $S$ and $S^{\mathrm{box}}$ for limit functionals built on them.

\smallskip\noindent\textbf{(1) Infinite-volume states.}
For every level $N$, along a subsequence of any prescribed sequence of sides, the torus states converge to a state $\om_N$ given by a
Borel probability measure on the configurations of the gauge-invariant variables. The state $\om_N$ is invariant under the lattice
translations, the reflections in the lattice planes $\{x_j=c\}$ and the cubic group; it is reflection positive about these planes; and
it satisfies the bounds of the torus states that are uniform in the volume (Theorem~\ref{thm:d2:LP}). The killed-box states converge
along every exhausting sequence of boxes, without the passage to a subsequence, to a state $\om^{\mathrm{box}}_N$ with the same
properties, which lies below every $\om_N$ on a cone of observables (Theorem~\ref{thm:d2:box}, Lemma~\ref{flh:lem:I3}). We do not know
whether the two families coincide in general.

\smallskip\noindent\textbf{(2) Gaussian domination and correlation inequalities.}
On every torus the Villain curvature is dominated by the Gaussian field with the identity covariance,
$\avL{\exp\sum_pa_p\ff_p}\le\exp\big(\tfrac12\sum_pa_p^2\big)$, with the constant one for all values of the couplings
(Theorem~\ref{thm:GD}). The torus states and the killed-box states satisfy Ginibre's inequalities on a cone of observables, with the
monotonicity in the couplings and in the box that follows from them (Theorem~\ref{thm:gin:K}).

\smallskip\noindent\textbf{(3) The limit functional.}
There are a sequence $N_1<N_2<\cdots$ and a linear functional $S$ on the algebra $\cA$ generated by the symbols $e_\sigma$ such that
$Z_{N_m}(\sigma)\to S(e_\sigma)$ for every test pair $\sigma$. The functional $S$ is invariant under the translations of $\R^3$ and
under the cubic group, and $t\mapsto S(\Th a\cdot\tau_tb)$ is continuous on $[0,\infty)$ for $a,b$ supported in the upper half space $\{x_3>0\}$, where $\Th$ is the reflection in the plane $\{x_3=0\}$ and $\tau_t$ the translation by $te_3$
(Theorem~\ref{thm:Hlim}). The same holds for the killed-box family (Lemma~\ref{flh:lem:I3}).

\smallskip\noindent\textbf{(4) The axioms OS0--OS3 without the rotations.}
For every limit functional, the generating functional of the curvature extends continuously to the Schwartz $2$-forms and is the
characteristic functional of a unique Borel probability measure $\mu$ on a space of distributions (Theorem~\ref{thm:osrest:measure}).
Its moments, the Schwinger functions $S_n$, are tempered distributions with
$|S_n(f_1,\dots,f_n)|\le e^n(n!)^{1/2}\prod_iR(f_i)$ for real $f_i$, $R$ being the seminorm of Definition~\ref{def:norms}; this gives the growth
condition of \cite{OS75} (Theorem~\ref{thm:osrest:OS0}). The functional $S$ is invariant under all translations, under the reflections in
the coordinate planes and under the cubic group about every point (Proposition~\ref{prop:osrest:OS1}), it is reflection positive about
every coordinate plane (Proposition~\ref{prop:osrest:OS2}), and the Schwinger functions are symmetric
(Proposition~\ref{prop:osrest:OS3}).

\smallskip\noindent\textbf{(5) Rotation invariance.}
Let $S^{\mathrm{box}}$ be a limit functional built on the killed-box states. For the curvature, $S^{\mathrm{box}}$ is invariant under
the orthogonal group $\mathrm{O}(3)$, hence under the Euclidean group; its measure is invariant under the Euclidean group;
and the axioms OS0--OS3 hold in full (Theorem~\ref{flh:thm:FLH1}). For a bounded
density function $g_{\rm d}$, the joint functional of the curvature and the density is invariant under the Euclidean group, coincident
points included (Theorem~\ref{flh:thm:FLH1p}). The same statements hold for every limit functional $S$ built on torus limit points
$\om_N$ whose sides run along a dyadic sequence $L_n=2^nL_*$ (Definition~\ref{n1:def:sides}, Theorem~\ref{flh:thm:FLH2}). No rotation
statement is made for torus limit points along other sequences of sides.

\smallskip\noindent\textbf{(6) Clustering.}
The measure $\mu$ of the limit curvature has an ergodic decomposition in which almost every component is invariant and ergodic under
the translations of $\R^3$, is reflection positive about the coordinate planes, clusters in the mean, and has a unique vacuum in the
reconstruction along each coordinate direction: the weak form of OS4 (Theorem~\ref{thm:osrest:OS4weak}).

\subsection{Methods}

\emph{Uniform bounds and the limit.} The Gaussian domination of Theorem~\ref{thm:GD} bounds the smeared curvature uniformly in the
volume and in the level, in the normalisation $\eN^{3/2}$ in which the plaquette action is a Riemann sum of the Maxwell action; the
density term is bounded because $g_{\rm d}$ is. This gives an estimate of $Z_N(\sigma)-Z_N(\sigma')$ by a seminorm of $\sigma-\sigma'$
that does not depend on $N$ (Section~\ref{sec:E}), and a diagonal argument with the lattice symmetries gives the limit functional
(Sections~\ref{sec:R1}--\ref{sec:main}). The exponential moments make the generating functional of the curvature entire, which gives
the Schwinger functions and their bounds; the measure is obtained from the Bochner--Minlos theorem \cite{GJ87} and the measure theory of
Appendix~\ref{ms:measure}.

\emph{Infinite volume at fixed spacing.} The torus states are reflection positive about the coordinate planes
(Theorem~\ref{thm:osrest:RPtorus}). From this we derive bounds on the local moments of the scalar density that are uniform in the
volume (Lemma~\ref{lem:d2:mom}); with the Gaussian domination they give tightness, and every limit point inherits the symmetries and the
reflection positivity (Section~\ref{sec:d2}). For the killed boxes, Ginibre's inequalities (Section~\ref{sec:gin}) make the expectations
of a cone of observables monotone in the box, and the values on the cone determine the state; this gives the convergence of the box
states.

\emph{Rotations.} Reflection positivity about sufficiently many planes forces rotation covariance: this is the content of
Theorem~\ref{i7:thm:main}, whose proof (Section~\ref{sec:i7:proof}) proceeds through an analytic continuation in the angle of a
planar rotation, the positivity of the continued reflection pairing, a convexity argument and an arithmetic lemma. In proving the rotation invariance we implement the original strategy of \cite{RPM1}; in a subsequent manuscript we shall provide a different strategy, adapting the proof from the companion paper \cite{CP} (Remark~\ref{flh:rem:strategy}). On a cubic torus
only the coordinate reflections have all their walls pointwise fixed, and no torus carries the nine families with pointwise-fixed walls
(Lemma~\ref{flh:lem:I0}). A cubic box has nine symmetry planes, three coordinate and six diagonal ones, and its killed-box state is
reflection positive about each of them (Theorem~\ref{flh:thm:I1}); the box limit is then reflection positive about every lattice plane
of the nine families (Lemma~\ref{flh:lem:I2}), and the limit theory built on it satisfies the hypotheses of Theorem~\ref{i7:thm:main}. This gives the covariance away from coincident points; the coincident points are covered by
the closedness of the limit curvature and a theorem on bounded multilinear forms (Theorems~\ref{flh:thm:I6} and~\ref{flh:thm:I8};
Section~\ref{flh:sec:main}). For the torus limit points the reflection positivity about the diagonal planes is proved in
Section~\ref{n1:sec}: the proof runs through an adapted torus with a family of diagonal reflection walls, a perturbed law on it that
is reflection positive across every wall, a convexity inequality for the perturbed free energy, a surface bound that removes the
perturbation in the limit, and the rigidity theorem of Section~\ref{rv:sec}, applied to a comparison state. The dyadic sides enter here.

\emph{What is not addressed.} We do not address the non-degeneracy of the limit (that $S$ is not the state of a constant field) or its
non-triviality; the non-triviality will be the subject of a subsequent manuscript. We do not prove the uniqueness of the infinite-volume state at fixed spacing, that is, the existence of the limit
\eqref{eq:D2}; for continuous observables it is equivalent to the agreement of all torus limit points (Corollary~\ref{cor:d2:iff}).
We do not address the exponential clustering of the limit, nor unbounded density functions beyond the cases of Section~\ref{hx:sec}. No renormalisation-group analysis
is used in this paper.

\subsection{Related work}

The axioms are those of \cite{OS73,OS75}; the Bochner--Minlos theorem is used in the form of \cite{GJ87}. Reflection positivity on the
lattice, Gaussian domination and the estimates derived from them go back to \cite{FSS76,FILS78}; the correlation inequalities of
Section~\ref{sec:gin} are inequalities of Ginibre's type \cite{Gin70} for the present model. The lattice abelian Higgs model and its
continuum limit in two dimensions were studied in \cite{BFS79,BFS80,BFS81}; in three dimensions the ultraviolet problem was treated by
renormalisation-group methods in \cite{Bal82a,Bal82b,Bal83,Kin86a,Kin86b}. Gibbs measures and their symmetries are treated in \cite{Geo11}.

The companion paper \cite{CP} establishes the Osterwalder--Schrader axioms for the same model in the Coulomb phase. Its proof of
rotation invariance is of a different kind and is original to that paper: the continuum limit is built on two lattices in rotated
position and the two are compared on periodic tori carrying a wall in the scalar mass; the walled-torus families are shown to have
a common continuum limit, and the infinite-volume state of King's construction \cite{Kin86a,Kin86b} is identified as their lower
envelope by a closing estimate resting on heavy-shell bounds, the shell mass being carried inside the covariance. The invariance is
proved for the curvature sector on all of King's coupling range, and for Wick-square sources on a tube below a threshold.

\subsection*{Added note}

For further information on the efforts leading to the production of this manuscript we refer to the GitHub repository
\url{https://github.com/marikgoldstein/scalar-qed-existence}.

\subsection{Organisation}

Sections~\ref{sec:setting}--\ref{sec:main} construct the limit functional; Section~\ref{hx:sec} extends it to unbounded density functions and other normalisations, in the cases verified there.
Section~\ref{sec:osrest} proves the axioms without the rotations and the statements on clustering. Sections~\ref{sec:gin}
and~\ref{sec:d2} contain the correlation inequalities and the infinite-volume states, which the preceding sections use through
Definition~\ref{def:omN}. Sections~\ref{flh:sec:main}--\ref{sec:i7:proof} prove the rotation invariance. Appendix~\ref{ms:measure} contains the measure theory.

\section{Setting and the scheme}\label{sec:setting}
Throughout, $\azero>0$ is a fixed length, $N_0\ge0$ is a fixed integer, and the \emph{levels} are the integers $N\ge N_0$. The lattice
$\eN\Z^3$ of spacing $\eN=\azero5^{-N}$ is called the \emph{fine lattice} of level $N$; the model defined on it and on its tori is the
\emph{fine model}, and its states are the \emph{fine states}. The Villain fine model at level $N$ lives on periodic lattices $\tilde T_L=(\eN\Z/L\Z)^3$ of spacing
$\eN=\azero5^{-N}$, $L/2\in\eN\Z$, with the variables $\tr_x\in[-\rhof,\infty)$ (sites), $B_b\in(-\pi/\gph,\pi/\gph]$ (bonds,
$B_{-b}=-B_b$), $v_p\in(2\pi/\gph)\Z$ (plaquettes, $v_{-p}=-v_p$) and the weight
\[
W=\prod_xg_{\mathrm{ct}}(\tr_x)\cdot\prod_bh(\tr_{b_-},\tr_{b_+},B_b)\cdot\prod_pw\big((dB)_p+v_p\big),\qquad
w(f)=e^{-f^2/2},
\]
where $g_{\mathrm{ct}}(\tr):=g(\tr)\,e^{-\frac12\delta m_B(\rhof+\tr)^2}$ is the site factor (its second factor is the bare mass counterterm,
explained below), with $g(\tr)=(\rhof+\tr)e^{-V(\tr)}$, $V(\tr)=\lamf(\rho^2-\rhof^2)^2$ and $\rho=\rhof+\tr$; the hopping factor is
\[
h(r,r',\vartheta)=\exp\big(-\tfrac12\kappa_N|e^{i\gph\vartheta}(\rhof+r')-(\rhof+r)|^2\big),
\]
and the sum over the Villain
variables runs over all closed $v$ ($dv=0$ on every cube), every flux class included. Here $(dB)_p:=\sum_{b\in\partial p}B_b$ and $(dv)_c:=\sum_{p\in\partial c}v_p$ with the signs of the boundaries.

Here $\gph:=\eN^{1/2}e$ is the phase constant of the hopping factor, $\lamf:=\eN\lambda$ the quartic coupling and $\rhof>0$ the vacuum value,
$\rhof^2:=\eN\rho_0^2=\eN m_H^2/(8\lambda)$, with $\kappa_N=1$. The parameters $e,\lambda,m_H>0$ are fixed. In
angle units $\theta_b=\gph B_b$ the plaquette weight reads $e^{-((d\theta)_p+\gph v_p)^2/(2\gph^2)}$, i.e.\ the Villain weight with the inverse coupling $1/\gph^2$. The spacing
$\eN$ and the phase constant $\gph$ are different quantities ($\gph^2/\eN=e^2$); the Riemann normalisations below use $\eN$ only.

The factor $e^{-\frac12\delta m_B\rho_x^2}$ of $g_{\mathrm{ct}}$, $\rho_x:=\rhof+\tr_x$, is the bare mass counterterm, with $\delta m_B\in\R$ one real parameter of either sign, which may depend on $N$ (all constants below that involve the
site factor are at fixed $N$, Remark~\ref{rem:letters}); every statement below is proved for every real $\delta m_B$. The site integral and all its Gaussian
moments stay finite because the quartic dominates: $\lamf(\rho^2-\rhof^2)^2\ge\tfrac12\lamf\rho^4-\lamf\rhof^4$, so
\begin{equation}\label{eq:ct}
g(\tr)\,e^{-\frac12\delta m_B\rho^2}\,e^{q\rho^2}\le\rho\,e^{\lamf\rhof^4}\,e^{-\frac12\lamf\rho^4+(q+\frac12|\delta m_B|)\rho^2}\qquad(\rho=\rhof+\tr\ge0,\ q\ge0),
\end{equation}
and the right side is integrable over $\tr\in[-\rhof,\infty)$ (Remark~\ref{rem:ct}).

We write $\avL{\cdot}$ for the normalised expectation with respect to the weight $W$, the Lebesgue measure on the variables $\tr_x$ and $B_b$ and the sum over the closed $v$ (the fine
torus state), and $\ff_p=(dB)_p+v_p$ for the Villain curvature. One would like to define the infinite-volume state of level $N$ as the limit of the torus states,
\begin{equation}\label{eq:D2}
\om_N(F)=\lim_{L\to\infty}\avL F\qquad\text{for every bounded local gauge-invariant }F
\end{equation}
($N\ge N_0$). We do not assume that this limit exists. The fine infinite-volume state is defined instead as a weak limit point of the torus states (Definition~\ref{def:omN}),
which exists for all positive couplings by Theorem~\ref{thm:d2:LP} of Section~\ref{sec:d2}; when the limit \eqref{eq:D2} exists, $\om_N$ is that
limit. We have set $\kappa_N=1$; everything below holds for every $\kappa_N\ge0$, $\lamf>0$, $\rhof>0$, every real $\delta m_B$, every $\gph>0$ and every $\eN$: the
estimates use no property of the couplings, and the symmetry and compactness arguments use only the properties of the limit point listed in
Theorem~\ref{thm:d2:LP}.
The one exception concerns $\kappa_N$: Sections~\ref{rv:sec} and~\ref{n1:sec} are written for $\kappa_N=1$. By Lemma~\ref{lem:rescale} their main
results (Theorems~\ref{rv:thm:RV} and~\ref{n1:thm:N10} and Corollary~\ref{n1:cor:N1}) hold for every $\kappa_N>0$, and so do the statements
that rest on them, namely the rotation
statements for the torus limit points along dyadic sequences of sides (Theorem~\ref{flh:thm:FLH2}, and Corollary~\ref{flh:cor:DENS}(b) for the
limits of its first sentence); for all these statements the value $\kappa_N=0$ is excluded.

\begin{remark}[the coefficients]\label{rem:letters}
With the exception stated just before this remark (for which see Lemma~\ref{lem:rescale}), every statement of this paper holds for arbitrary positive values of the coefficients $\gph$, $\lamf$, $\rhof$, for every $\kappa_N\ge0$ and for every real $\delta m_B$ (Remark~\ref{rem:ct}): the
coefficients enter the proofs as parameters only, and a constant depends on them only where this is stated (the constants of
Lemma~\ref{lem:d2:mom} and the finiteness majorants inside the proofs of Theorem~\ref{thm:GD}, Lemma~\ref{lem:osrest:dualC} and
Theorem~\ref{thm:osrest:RPtorus}, all at fixed $N$). In particular the
constant of Theorem~\ref{thm:GD}, which is uniform in $N$, is exactly $1$ in the lattice normalisation used there (the curvature $\ff_p=(dB)_p+v_p$
with the plaquette weight $e^{-\ff_p^2/2}$), for every value of the coefficients.
\end{remark}

\begin{remark}[the bare mass counterterm]\label{rem:ct}
For every real $\delta m_B$ the site factor $g_{\mathrm{ct}}(\tr_x)=g(\tr_x)e^{-\frac12\delta m_B\rho_x^2}$ has three properties:
it is a positive function of the one site variable, the same at every site; its integral against $e^{q\rho^2}$ is finite for every $q\ge0$, by
\eqref{eq:ct}; and on $\tr\in[0,1]$ it is bounded below by $\rhof e^{-\lamf(2\rhof+1)^2}e^{-\frac12|\delta m_B|(\rhof+1)^2}>0$.
Every argument of this paper that involves the site factor uses of it only these three properties. This includes the majorant
$(\int g)^{|\mathcal X|}$ of Lemma~\ref{lem:osrest:dualC}.

In the proof of Theorem~\ref{thm:Hlim} the site factor enters as follows. The proof of Theorem~\ref{thm:GD} integrates it into $\Omega(m)$
and uses, in Step~4, only $\Omega(m)\ge0$ and $\Omega(m)<\infty$; Steps~1--3 concern the plaquette sum and the gauge
integral, in which no site factor appears. Lemma~\ref{lem:R1} uses that the site factor is the same at every site.
Proposition~\ref{prop:E} uses Theorem~\ref{thm:GD} and a pointwise bound, and not the site factor, and \eqref{eq:unimod} concerns $\iota_N$ alone.
The existence of the limit point and its properties (a)--(e) (Theorem~\ref{thm:d2:LP}) rest on Theorem~\ref{thm:GD}; on the lattice symmetries,
which use that the site factor is the same at every site; on the reflection positivity of the torus states (Theorem~\ref{thm:osrest:RPtorus}),
whose proof uses the positivity of the site factor, its invariance under the reflections and its integrability; and on the uniform
density moments of Lemma~\ref{lem:d2:mom}, whose proof uses the finiteness of the site integrals and a positive lower bound of the site factor on $\tr\in[0,1]$.
The site integrals $G^{\mathrm{ct}}_q$ and the positive lower bound $c^{\mathrm{ct}}_g$ of the full site factor $g_{\mathrm{ct}}$ on $\tr\in[0,1]$ are the constants
of Lemma~\ref{lem:d2:mom} (Section~\ref{sec:d2}; see also Remark~\ref{rem:d2:ct}), and Lemma~\ref{lem:d2:mom} and all its consequences
hold with these constants. The counterterm therefore changes no statement: every statement holds for every real
$\delta m_B$, and the counterterm enters only the constants that depend on the coefficients (Remark~\ref{rem:letters}).

The later sections use the site factor $g_{\mathrm{ct}}$ in the same sense. It enters the measure
$\mu^\circ:=g_{\mathrm{ct}}\,e^{-3\kappa_N\rho^2}\,d\rho$ and the weight $W_\Lambda$ of Sections~\ref{sec:gin} and~\ref{sec:d2} (Definition~\ref{def:gin:box});
the weight restated in Subsection~\ref{rot:sec:main}; the function $\widetilde g:=g_{\mathrm{ct}}\,e^{-3\kappa_N\rho^2}$ and the site integral $G^{\mathrm{ct}}_0:=\int g_{\mathrm{ct}}$
of Section~\ref{flh:sec:main}; and the proof of Theorem~\ref{thm:osrest:RPtorus}, which uses that $g_{\mathrm{ct}}$ is integrable. Each of those arguments uses only the positivity,
the position independence (which gives the reflection invariance) and the exponential moments of the site measure, which $g_{\mathrm{ct}}$ has by \eqref{eq:ct}.
\end{remark}

\begin{definition}[the fine infinite-volume state: a limit point of the torus states]\label{def:omN}
Let $\Omega_N:=\R^{\mathcal P_N}\times[-\rhof,\infty)^{\mathcal X_N}$ be the configuration space of the gauge-invariant variables
$(\ff_p)_{p\in\mathcal P_N}$, $(\tr_x)_{x\in\mathcal X_N}$ of the infinite lattice $\eN\Z^3$ ($\mathcal P_N$ its plaquettes, one orientation
chosen for each, $\mathcal X_N$ its sites; Subsection~\ref{sec:d2:setting}), with the product topology and its Borel $\sigma$-algebra. A function
on $\Omega_N$ is local if it depends on the variables of finitely many cells; the bounded Borel local functions are the bounded local
gauge-invariant functions of \eqref{eq:D2}.
For each $N\ge N_0$ fix a sequence of sides $L_n=L^{(N)}_n\to\infty$ with $L_n/2\in\eN\Z$ and a
Borel probability measure $\mu_N$ on $\Omega_N$ such that the finite-dimensional laws of the gauge-invariant variables under the torus states
$\langle\cdot\rangle^{\mathrm{fine}}_{L_n}$ converge weakly to those of $\mu_N$, and put
\begin{equation}\label{eq:omN}
\begin{aligned}
&\om_N(F):=\int F\,d\mu_N\quad\text{for every bounded Borel local }F;\\
&\text{then}\quad\om_N(F)=\lim_{n\to\infty}\langle F\rangle^{\mathrm{fine}}_{L_n}\quad\text{for every bounded continuous local }F .
\end{aligned}
\end{equation}
\end{definition}

Such a pair $((L_n),\mu_N)$ exists for every $N$, every choice of positive couplings and any real $\delta m_B$, along a subsequence of any prescribed sequence of sides
(Theorem~\ref{thm:d2:LP}, whose proof uses Theorem~\ref{thm:GD}, the reflection positivity of the torus states and a bound on the local moments
of the density). The sides $L_n$ are taken along a dyadic sequence (Definition~\ref{n1:def:sides}); by Theorem~\ref{thm:d2:LP} a limit point exists along a subsequence of it, and every statement below holds for each such limit point
(Remark~\ref{rem:omN:D}).
Throughout, $\om_N$ is the state \eqref{eq:omN} of such a limit point, and
``$L\to\infty$'' for expectations at level $N$ means $n\to\infty$ along $(L_n)$.

The properties of $\om_N$ used below are those of
Theorem~\ref{thm:d2:LP}: (a) the limit relation \eqref{eq:omN} for bounded continuous local $F$, for bounded Borel local $F$ with $\mu_N$-null
discontinuity set, and for continuous local $F$ dominated by $Ce^{\sum_xq_x\rho_x^2}$ ($\rho_x=\rhof+\tr_x$); (b) $\om_N$ is a state on
the bounded Borel local functions ($\om_N(1)=1$, positive, $|\om_N(F)|\le\|F\|_\infty$), with the Gaussian domination \eqref{eq:GD} and the
uniform density moment bounds of the torus states; (c) $\mu_N$ is invariant under the translations by $\eN\Z^3$, the reflections in the planes
$\{x_j=c\}$, $c\in\frac12\eN\Z$, the cubic group about every site and the charge conjugation $(\ff,\tr)\mapsto(-\ff,\tr)$; (d) $\om_N$ is
reflection positive about every such plane, on the bounded Borel functions of cells in the closed half space and on their $L^2(\mu_N)$ closure;
(e) inequalities between torus expectations of bounded continuous local functions pass to $\om_N$, and $\int F\,d\mu_N\le\liminf_n\langle
F\rangle^{\mathrm{fine}}_{L_n}$ for continuous local $F\ge0$.

If the limit \eqref{eq:D2} exists for every bounded continuous local $F$, over all sides or
over a cofinal class of sides,
then all limit points along sides of that class coincide and $\om_N$, with $(L_n)$ taken in the class, is that limit (Corollary~\ref{cor:d2:iff}, whose limits
may be taken over any cofinal class of sides). Limit points in a wider sense, subnet limits of the torus states on all bounded Borel local functions, are treated in Proposition~\ref{prop:subnet} and Remark~\ref{rem:subnet}.
The fine infinite-volume
state is used on nothing more than the functions of the curvature and the density.

\begin{remark}[the dyadic sides]\label{rem:omN:D}
The sides of Definition~\ref{def:omN} are taken along a dyadic sequence (Definition~\ref{n1:def:sides}) because the reflection positivity of
$\om_N$ at the six diagonal site planes through $0$ (Theorem~\ref{n1:thm:N10}), on which the rotation statements of
Section~\ref{flh:sec:main} rest, is proved for the limit points along such sequences. Theorem~\ref{thm:d2:LP} provides a limit point along
a subsequence of every prescribed sequence of sides, in particular of every dyadic sequence. The choice of dyadic sides is not a
hypothesis, but it is a restriction of the result: the rotation statements of Section~\ref{flh:sec:main} for the torus states are stated
for the limit points along such sequences only, and no rotation statement is made in this paper for torus limit points along other
sequences of sides.
\end{remark}

\emph{Test pairs and the algebra.} A \emph{test pair} is $\sigma=(f,h)$ with $f$ a real $C^\infty_c$ $2$-form on $\R^3$ and $h\in C^\infty_c(\R^3;\R)$;
$\mathcal T$ is their real vector space, $\tau_t$ the translation by $te_3$, $\theta$ the time reflection: $\theta\sigma$ is the push-forward of $\sigma=(f,h)$ under $x\mapsto(x_1,x_2,-x_3)$, the $2$-form transforming as a $2$-form (its components with one index $3$ change sign).
The algebra $\cA=\C[\mathcal T]$ is the group
algebra with basis $e_\sigma$, $e_\sigma e_{\sigma'}=e_{\sigma+\sigma'}$, $e_\sigma^*=e_{-\sigma}$, $\Th e_\sigma=e_{-\theta\sigma}$ ($\Th$ antilinear),
$\tau_te_\sigma=e_{\tau_t\sigma}$, and $\Ap$ the span of the $e_\sigma$ supported in $\{x_3>0\}$.

\begin{definition}[the scheme]\label{def:pin}
The \emph{scheme} realises $e_\sigma$ on the fine lattice as $\iota_N(e_\sigma)=e^{i\Phi_N(\sigma)}$, extended linearly to $\cA$, where
\[
\Phi_N(\sigma)=c_N\sum_pf_{\mu\nu(p)}(x_p)\,\ff_p+c_N'\sum_zh(z)\,\mathfrak w_z,\qquad\mathfrak w_z=g_{\rm d}(\tr_z),
\]
with real numbers $c_N$, $c_N'$ (the lattice normalisations) and a real Borel function $g_{\rm d}$ of the scalar density (the density
function). We fix them as follows, and every statement below refers to this choice, the \emph{pinned scheme}, except in
Section~\ref{hx:sec}, which treats general normalisations and density functions:
\[
c_N=\eN^{3/2},\qquad c_N'=\eN^{3},\qquad g_{\rm d}\ \text{a fixed bounded real Borel function on }\R,\ \ \|g_{\rm d}\|_\infty<\infty .
\]
Here $x_p$ is the centre of the plaquette $p$ and $f_{\mu\nu(p)}$ the component of $f$ along the orientation of $p$; the sums run over the
plaquettes and sites of the fine lattice (finitely many terms, since $\sigma$ has compact support), so $\Phi_N(\sigma)$ is a real local
function of finitely many gauge-invariant variables, additive in $\sigma$, unbounded in the curvature part and bounded in the density part.
We write $Z_N(\sigma):=\om_N(e^{i\Phi_N(\sigma)})$ and $M(\sum_jc_je_{\sigma_j}):=\sum_j|c_j|$; since the exponentials are unimodular,
\begin{equation}\label{eq:unimod}
|\iota_N(a)|\le M(a)\quad\text{pointwise, for every }a\in\cA\text{ and }N .
\end{equation}
\end{definition}

\begin{remark}\label{rem:pin}
(i) Definition~\ref{def:pin} makes sense for every real Borel $g_{\rm d}$ and all real $c_N$, $c_N'$; outside Section~\ref{hx:sec} only the pinned scheme is used. (ii) $c_N=\eN^{3/2}$ is the Riemann normalisation of the curvature: the plaquette weight $e^{-\ff_p^2/2}$ corresponds to the
continuum Maxwell term when $\ff_p=\eN^{3/2}F_{\mu\nu}(x_p)(1+O(\eN))$, so that $\Phi_N(f,0)\to\int f_{\mu\nu}F_{\mu\nu}$. (iii) The choice of a
bounded $g_{\rm d}$ is discussed in Remark~\ref{rem:density}: it makes the density part of the estimate trivial.
\end{remark}

\begin{lemma}[rescaling the hopping coefficient]\label{lem:rescale}
Fix a level $N$ and coefficients $\kappa_N>0$, $\lamf>0$, $\rhof>0$, $\gph>0$ and $\delta m_B\in\R$, and put
\begin{equation}\label{eq:rescale}
\lamf^{\star}:=\kappa_N^{-2}\lamf,\qquad\rhof^{\star}:=\kappa_N^{1/2}\rhof,\qquad\delta m_B^{\star}:=\kappa_N^{-1}\delta m_B .
\end{equation}
The \emph{given model} is the model of this section at level $N$ with the coefficients $(\kappa_N,\lamf,\rhof,\delta m_B,\gph)$; the
\emph{rescaled model} is the same model with the coefficients $(1,\lamf^{\star},\rhof^{\star},\delta m_B^{\star},\gph)$, so that its hopping
coefficient is $1$. We mark the objects of the rescaled model with a star: its site variables are $\tr^{\star}_x\in[-\rhof^{\star},\infty)$, with
$\rho^{\star}_x:=\rhof^{\star}+\tr^{\star}_x$, its site factor is $g^{\star}_{\mathrm{ct}}$ and its hopping factor is $h^{\star}$. Let $r$ be the map
that multiplies every site variable by $\kappa_N^{-1/2}$ and leaves the other variables unchanged,
\[
\begin{gathered}
r(\tr^{\star},B,v):=(\kappa_N^{-1/2}\tr^{\star},B,v)\quad\text{on the lattice variables},\\
r(\ff,\tr^{\star}):=(\ff,\kappa_N^{-1/2}\tr^{\star})\quad\text{on the gauge-invariant variables}.
\end{gathered}
\]
Since $\kappa_N^{-1/2}\rhof^{\star}=\rhof$, it is a homeomorphism of the configuration space of the rescaled model onto that of the given model, on
every torus, in every box and on $\eN\Z^3$; and $\tr_x\circ r=\kappa_N^{-1/2}\tr^{\star}_x$, $\rho_x\circ r=\kappa_N^{-1/2}\rho^{\star}_x$, while $B_b$,
$v_p$ and $\ff_p=(dB)_p+v_p$ are unchanged. Then:
\begin{itemize}
\item[(a)] (\emph{The weight.}) For all $s,s'\in[-\rhof^{\star},\infty)$ and every real $\vartheta$,
\[
g_{\mathrm{ct}}(\kappa_N^{-1/2}s)=\kappa_N^{-1/2}\,g^{\star}_{\mathrm{ct}}(s),\qquad
h(\kappa_N^{-1/2}s,\kappa_N^{-1/2}s',\vartheta)=h^{\star}(s,s',\vartheta),
\]
and $(\kappa_N\rho_x^2)\circ r=(\rho^{\star}_x)^2$, $(\kappa_N\rho_{b_-}\rho_{b_+})\circ r=\rho^{\star}_{b_-}\rho^{\star}_{b_+}$. The plaquette weight, the
ranges of $B_b$ and $v_p$ and the constraint $dv=0$ are the same in the two models.
\item[(b)] (\emph{Finite volume.}) Let $\langle\cdot\rangle^{\star}_L$ and $\langle\cdot\rangle^{\star}_\Lambda$ be the torus states and the killed-box
states (Definition~\ref{def:d2:cone}) of the rescaled model. For every torus $\tilde T_L$, every box $\Lambda$ and every bounded Borel function $F$
of the lattice variables,
\[
\langle F\rangle^{\mathrm{fine}}_L=\langle F\circ r\rangle^{\star}_L,\qquad\langle F\rangle_\Lambda=\langle F\circ r\rangle^{\star}_\Lambda .
\]
\item[(c)] (\emph{Limit points.}) A pair $((L_n),\mu^{\star}_N)$ is a limit point of the torus states of the rescaled model in the sense of
Definition~\ref{def:omN} if and only if $((L_n),r_*\mu^{\star}_N)$ is one for the given model. Thus the fine infinite-volume states of the given
model are the states
\[
\om_N(F)=\om^{\star}_N(F\circ r)\qquad(F\text{ bounded Borel local}),
\]
with $\om^{\star}_N$ a fine infinite-volume state of the rescaled model along the same sides; in particular $\om_N$ is a limit point along a dyadic
sequence (Definition~\ref{n1:def:sides}) if and only if $\om^{\star}_N$ is. In the same way the measures of the killed-box limits of
Theorem~\ref{thm:d2:box} satisfy $\mu^{\mathrm{box}}_N=r_*\mu^{\star,\mathrm{box}}_N$.
\item[(d)] (\emph{Symmetries and reflections.}) The map $r$ commutes with the action of every element of the group $\mathcal G_N$ of lattice
symmetries and with the charge conjugation (Subsection~\ref{sec:d2:setting}). For every set of cells, $F\mapsto F\circ r$ maps the bounded Borel
functions of the gauge-invariant variables of these cells in the given model onto those in the rescaled model, and for $\mu_N=r_*\mu^{\star}_N$ it
is an isometry of $L^2(\mu_N)$ onto $L^2(\mu^{\star}_N)$. Consequently, for $\om_N$ and $\om^{\star}_N$ as in (c), for every reflection
$\theta\in\mathcal G_N$ and each closed half space bounded by its plane, each of the following four statements holds for $\om=\om_N$ if and only if
it holds for $\om=\om^{\star}_N$: $\om(\overline{F\circ\theta}\cdot F)\ge0$ for every bounded Borel function $F$ of finitely many cells in the half
space; $(F,G)\mapsto\om(\overline{F\circ\theta}\cdot G)$ is a positive semidefinite Hermitian form on these functions;
$\om(\overline{F\circ\theta})=\overline{\om(F)}$ for these functions; the first three statements extend to the $L^2$ closure of these functions.
\item[(e)] (\emph{The scheme.}) Let $c_N$, $c_N'$ be real numbers and $g_{\rm d}$ a real Borel function (Definition~\ref{def:pin},
Remark~\ref{rem:pin}(i)), and put
\[
g^{\star}_{\rm d}(t):=g_{\rm d}(\kappa_N^{-1/2}t)\qquad(t\in\R),
\]
a real Borel function with $\|g^{\star}_{\rm d}\|_\infty=\|g_{\rm d}\|_\infty$. Let $\Phi^{\star}_N(\sigma)$ be the function of
Definition~\ref{def:pin} in the rescaled model, with the same $c_N$, $c_N'$ and with the density function $g^{\star}_{\rm d}$. Then
$\Phi_N(\sigma)\circ r=\Phi^{\star}_N(\sigma)$ for every test pair $\sigma$, the curvature part being unchanged, and for $\om_N$, $\om^{\star}_N$ as
in (c)
\[
Z_N(\sigma)=\om_N\big(e^{i\Phi_N(\sigma)}\big)=\om^{\star}_N\big(e^{i\Phi^{\star}_N(\sigma)}\big)=:Z^{\star}_N(\sigma)\qquad\text{for every test pair }\sigma .
\]
The function $g^{\star}_{\rm d}$ depends on the level through $\kappa_N$. If $\kappa_N$ is the same at every level, then $g^{\star}_{\rm d}$ is one
density function for all levels, and every limit functional of the given models (Theorem~\ref{thm:Hlim}) is a limit functional of the rescaled
models with the density function $g^{\star}_{\rm d}$, along the same levels and with the states $\om^{\star}_N$ of (c).
\item[(f)] (\emph{Gibbs measures.}) A probability measure $\nu^{\star}$ on the configurations $(\tr^{\star},B,v)$ of $\eN\Z^3$ is a Gibbs measure of
the rescaled model, for the specification of Subsection~\ref{rv:ssec:S0} (the weight \eqref{rv:eq:W} read with the hopping coefficient of the model), if and only if $r_*\nu^{\star}$ is a Gibbs measure of the given model. The
two measures are invariant under the same translations of $\eN\Z^3$, and $r_*\nu^{\star}$ satisfies the conditions (I1)--(I3) of
Theorem~\ref{rv:thm:RV} if and only if $\nu^{\star}$ does.
\item[(g)] (\emph{The results proved at $\kappa_N=1$.}) Theorem~\ref{rv:thm:RV}, Theorem~\ref{n1:thm:N10} and Corollary~\ref{n1:cor:N1}, which
Sections~\ref{rv:sec} and~\ref{n1:sec} prove for $\kappa_N=1$ and arbitrary $\lamf>0$, $\rhof>0$, $\gph>0$ and real $\delta m_B$, hold as stated for
every $\kappa_N>0$.
\end{itemize}
\end{lemma}
\begin{proof}
(a) Let $s\in[-\rhof^{\star},\infty)$ and $\rho^{\star}:=\rhof^{\star}+s$. By \eqref{eq:rescale}, $\rho:=\rhof+\kappa_N^{-1/2}s=\kappa_N^{-1/2}\rho^{\star}$ and
\[
\lamf(\rho^2-\rhof^2)^2=\lamf\kappa_N^{-2}\big((\rho^{\star})^2-(\rhof^{\star})^2\big)^2=\lamf^{\star}\big((\rho^{\star})^2-(\rhof^{\star})^2\big)^2,\qquad
\tfrac12\delta m_B\rho^2=\tfrac12\delta m_B^{\star}(\rho^{\star})^2 .
\]
Since $g_{\mathrm{ct}}(\tr)=\rho\,e^{-\lamf(\rho^2-\rhof^2)^2}e^{-\frac12\delta m_B\rho^2}$ with $\rho=\rhof+\tr$, and the prefactor is
$\rho=\kappa_N^{-1/2}\rho^{\star}$, the first identity follows. For the second,
\[
\kappa_N\big|e^{i\gph\vartheta}(\rhof+\kappa_N^{-1/2}s')-(\rhof+\kappa_N^{-1/2}s)\big|^2
=\big|e^{i\gph\vartheta}(\rhof^{\star}+s')-(\rhof^{\star}+s)\big|^2,
\]
and $-\frac12$ times the right side is the exponent of $h^{\star}$, whose hopping coefficient is $1$. The two remaining identities follow from
$\rho_x\circ r=\kappa_N^{-1/2}\rho^{\star}_x$. The plaquette weight $w$, the intervals $(-\pi/\gph,\pi/\gph]$, the lattices $(2\pi/\gph)\Z$ and the
constraint $dv=0$ involve no site variable and no coefficient other than $\gph$, which is the same in the two models.

(b) Let $n$ be the number of sites of $\tilde T_L$ and $W^{\star}$ the weight of the rescaled model. By (a), $W\circ r=\kappa_N^{-n/2}W^{\star}$: each
site factor contributes $\kappa_N^{-1/2}$, and the hopping and plaquette factors contribute no constant. The substitution
$\tr_x=\kappa_N^{-1/2}\tr^{\star}_x$ at every site maps $[-\rhof^{\star},\infty)$ onto $[-\rhof,\infty)$, with $d\tr_x=\kappa_N^{-1/2}d\tr^{\star}_x$; the
variables $B_b$, $v_p$ and their reference measures are untouched. Hence, for every bounded Borel $F$,
\[
\sum_v\int F\,W\prod_xd\tr_x\prod_bdB_b=\kappa_N^{-n}\sum_v\int(F\circ r)\,W^{\star}\prod_xd\tr^{\star}_x\prod_bdB_b ,
\]
the sums running over the closed $v$. Dividing by the same identity for $F=1$ gives the first equality of (b): the constant $\kappa_N^{-n}$ cancels.
For a box $\Lambda$ with $n$ sites the weight $W_\Lambda$ of Definition~\ref{def:d2:cone} satisfies $W_\Lambda\circ r=\kappa_N^{-n/2}W^{\star}_\Lambda$ by
(a), applied to the site factors $g_{\mathrm{ct}}$, to the factors $e^{-3\kappa_N\rho_x^2}$ and to the factors
$e^{\kappa_N\rho_{b_-}\rho_{b_+}\cos(\gph B_b)}$ in turn, and the same computation gives the second equality.

(c) Let $S$ be a finite set of plaquettes and sites, and let $r_S$ be the map that multiplies the site variables of $S$ by $\kappa_N^{-1/2}$ and leaves
its curvature variables unchanged; it is a homeomorphism of the space of the gauge-invariant variables of $S$ in the rescaled model onto that
space in the given model, and the variables of $S$ at the configuration $r(\cdot)$ are $r_S$ of the variables of $S$. By (b) the law of the
variables of $S$ under $\langle\cdot\rangle^{\mathrm{fine}}_L$ is the image under $r_S$ of their law under $\langle\cdot\rangle^{\star}_L$. Weak
convergence is preserved by $r_S$ and by its inverse, and the marginal of $r_*\mu^{\star}_N$ on the variables of $S$ is the image under $r_S$ of the
marginal of $\mu^{\star}_N$. This proves the equivalence. Given a limit point $((L_n),\mu_N)$ of the given model we take
$\mu^{\star}_N:=(r^{-1})_*\mu_N$, and then $\om_N(F)=\int F\,d(r_*\mu^{\star}_N)=\om^{\star}_N(F\circ r)$. Definition~\ref{n1:def:sides} is a condition
on the sides alone, and the two lengths that it fixes are taken the same for the two models. For the killed boxes the same argument applies to
the laws under $\langle\cdot\rangle_\Lambda$ along an exhausting sequence of boxes, with Theorem~\ref{thm:d2:box}(b).

(d) An element of $\mathcal G_N$ acts on a configuration by permuting the cells, with a change of sign of the curvature variables whose orientation
it reverses, and the charge conjugation changes the sign of the curvature variables; $r$ multiplies every site variable by one and the same
number and leaves the curvature variables unchanged. These actions commute. The maps $r$ and $r^{-1}$ are Borel and act on the variables of
each cell separately, which gives the statement on the functions of a set of cells, and $\int|F\circ r|^2d\mu^{\star}_N=\int|F|^2d\mu_N$. For a
reflection $\theta\in\mathcal G_N$ we have $\overline{F\circ\theta}\circ r=\overline{(F\circ r)\circ\theta}$, hence by (c)
\[
\om_N\big(\overline{F\circ\theta}\cdot G\big)=\om^{\star}_N\big(\overline{(F\circ r)\circ\theta}\cdot(G\circ r)\big),\qquad
\om_N\big(\overline{F\circ\theta}\big)=\om^{\star}_N\big(\overline{(F\circ r)\circ\theta}\big),
\]
and since $F\mapsto F\circ r$ is onto, is isometric in $L^2$ and preserves the set of cells on which a function depends, the four statements
for $\om_N$ are the four statements for $\om^{\star}_N$.

(e) We have $\ff_p\circ r=\ff_p$ and $g_{\rm d}(\tr_z)\circ r=g_{\rm d}(\kappa_N^{-1/2}\tr^{\star}_z)=g^{\star}_{\rm d}(\tr^{\star}_z)$, and the numbers
$c_N$, $c_N'$ and the test pair do not involve the coefficients; so $\Phi_N(\sigma)\circ r=\Phi^{\star}_N(\sigma)$. The function
$e^{i\Phi_N(\sigma)}$ is bounded Borel local, and (c) gives $Z_N(\sigma)=Z^{\star}_N(\sigma)$. The supremum norms agree because
$t\mapsto\kappa_N^{-1/2}t$ is a bijection of $\R$. If $\kappa_N$ does not depend on $N$ and $S$ is a limit functional of the given models along
$N_1<N_2<\cdots$, then $Z^{\star}_{N_m}(\sigma)=Z_{N_m}(\sigma)\to S(e_\sigma)$, which is (a) of Theorem~\ref{thm:Hlim} for the rescaled models with
the one density function $g^{\star}_{\rm d}$; (b) and (c) of that theorem are properties of $S$ alone.

(f) For a finite $\Lambda$ the kernel $\gamma_\Lambda(\cdot\mid\omega)$ is the normalised integral of the weight \eqref{rv:eq:W} over the variables
carried by $V(\Lambda)$ at a fixed exterior configuration $\omega$; only the factors that contain such a variable matter. We apply (a) to each of
these factors (in the hopping factor of a bond with one end outside $\Lambda$ the exterior site variable is rescaled as well, and the second
identity of (a) holds for all values of its arguments) and make the substitution of (b) at the sites of $\Lambda$. This gives
\[
\gamma_\Lambda(F\mid r\omega^{\star})=\gamma^{\star}_\Lambda(F\circ r\mid\omega^{\star})
\]
for every bounded Borel $F$ and every configuration $\omega^{\star}$ of the rescaled model, the constant $\kappa_N^{-|\Lambda|}$ cancelling in the
normalisation. With $\nu:=r_*\nu^{\star}$ we obtain $\nu(\gamma_\Lambda F)=\nu^{\star}(\gamma^{\star}_\Lambda(F\circ r))$, while
$\nu(F)=\nu^{\star}(F\circ r)$; as $F\circ r$ runs over all bounded Borel functions, $\nu\gamma_\Lambda=\nu$ for every finite $\Lambda$ if and only
if $\nu^{\star}\gamma^{\star}_\Lambda=\nu^{\star}$ for every finite $\Lambda$. The map $r$ commutes with the translations. Condition (I2) concerns
the curvature alone, and $\nu(\rho_x^2)=\kappa_N^{-1}\nu^{\star}((\rho^{\star}_x)^2)$,
$\nu(e^{q\rho_x^2})=\nu^{\star}(e^{(q/\kappa_N)(\rho^{\star}_x)^2})$, where $q/\kappa_N$ runs over $(0,\infty)$ with $q$; so (I1) and (I3) hold for
$\nu$ if and only if they hold for $\nu^{\star}$.

(g) Sections~\ref{rv:sec} and~\ref{n1:sec} prove the three statements for the hopping coefficient $1$ and arbitrary values of the other
coefficients, hence for the rescaled model. Let $\nu$ be a Gibbs measure of the given model as in Theorem~\ref{rv:thm:RV}. By (f),
$\nu^{\star}:=(r^{-1})_*\nu$ is a Gibbs measure of the rescaled model, invariant under the same sublattice and satisfying (I1)--(I3); by
Theorem~\ref{rv:thm:RV} at $\kappa_N=1$ it is invariant under the translations by $2\eN e_j$, $j=1,2,3$, and by (f) so is $\nu$. Let $\om$ be a
limit point of the given model along a dyadic sequence. By (c), $\om=\om^{\star}(\,\cdot\circ r)$ with $\om^{\star}$ a limit point of the rescaled
model along the same dyadic sequence, for which Theorem~\ref{n1:thm:N10} and Corollary~\ref{n1:cor:N1} hold; the reflections in the diagonal
site planes belong to $\mathcal G_N$, and (d) gives the statements of that theorem and of that corollary for $\om$.
\end{proof}

\section{Gaussian domination of the Villain curvature}\label{sec:GD}
\begin{theorem}[Gaussian domination, identity covariance]\label{thm:GD}
On every fine torus $\tilde T_L$, for every level $N$, every choice of positive couplings and any real $\delta m_B$, and every real plaquette function $a=(a_p)$,
\begin{equation}\label{eq:GD}
\Big\langle\exp\Big(\sum_pa_p\ff_p\Big)\Big\rangle^{\mathrm{fine}}_L\ \le\ \exp\Big(\tfrac12\sum_pa_p^2\Big).
\end{equation}
Consequently, with $X=\sum_pa_p\ff_p$: $\avL X=0$, $\avL{X^2}\le\sum_pa_p^2$, and $\avL{X^{2n}}\le(2n)!\,(e\sum_pa_p^2/2n)^n$ for $n\ge1$.
\end{theorem}

\begin{proof}
Write $\|a\|^2=\sum_pa_p^2$ and $Z(a)=\int d\tr\int dB\sum_{v\ \mathrm{closed}}W\,e^{\sum_pa_p\ff_p}$, so that the left side of
\eqref{eq:GD} is $Z(a)/Z(0)$; $Z(a)\in(0,\infty)$ for every real $a$ by the bounds in Step~4 below.

\emph{Step 1: the plaquette sum.} For $x,s,a\in\R$ put $c_u:=(\gph/\sqrt{2\pi})\,e^{-\gph^2u^2/2}>0$ ($u\in\R$). Completing the square,
$-\tfrac12(x+v)^2+a(x+v)=\tfrac12a^2-\tfrac12(x-a+v)^2$, and Poisson summation applied to $n\mapsto e^{-(y+2\pi n/\gph)^2/2}e^{2\pi isn}$
($y=x-a$, $v=2\pi n/\gph$) gives
\begin{equation}\label{eq:poisson}
\sum_{v\in(2\pi/\gph)\Z}e^{-(x+v)^2/2+a(x+v)}\,e^{i\gph sv}=e^{a^2/2}\sum_{m\in\Z}c_{m-s}\,e^{i\gph(m-s)(x-a)} .
\end{equation}
The Fourier coefficient is
\[
\int e^{-(y+2\pi u/\gph)^2/2}e^{2\pi i(s-m)u}du=(\gph/2\pi)\int e^{-w^2/2}e^{i\gph(s-m)(w-y)}dw=
c_{m-s}e^{i\gph(m-s)y}.
\]
All coefficients $c_{m-s}$ are positive, for every real $s$.

\emph{Step 2: the closed-$v$ constraint.} For a cube $c$, $(dv)_c=\sum_{p\in\partial c}\pm v_p\in(2\pi/\gph)\Z$, so
$\mathbf 1[(dv)_c=0]=\int_0^1e^{i\gph t_c(dv)_c}\,dt_c$. Hence, with $t=(t_c)\in[0,1]^{\mathcal C}$ and $s:=d^*t$ the real plaquette function
$(d^*t)_p=\sum_{c\ni p}\pm t_c$ (the signs of $\partial$), $\sum_{v\ \mathrm{closed}}G(v)=\int_{[0,1]^{\mathcal C}}dt\sum_{\mathrm{all}\ v}G(v)
\,e^{i\gph\sum_ps_pv_p}$ for every absolutely summable $G$. Applying \eqref{eq:poisson} plaquette by plaquette with $x=(dB)_p$,
\[
\sum_{v\ \mathrm{closed}}\prod_pw(\ff_p)e^{a_p\ff_p}=e^{\|a\|^2/2}\int dt\sum_{m\in\Z^{\mathcal P}}\prod_pc_{m_p-s_p}\;
e^{i\gph\sum_p(m_p-s_p)(dB)_p}\;e^{-i\gph\sum_p(m_p-s_p)a_p}.
\]
Now $\sum_p(m_p-s_p)(dB)_p=\sum_b\big(d^*(m-s)\big)_bB_b=\sum_b(d^*m)_bB_b$, because $d^*s=d^*d^*t=0$ ($dd=0$); and $(d^*m)_b\in\Z$.

\emph{Step 3: the gauge integral.} The hopping factor depends on $B_b$ only through $\cos(\gph B_b)$:
\[
h(r,r',\vartheta)=e^{-\frac12\kappa_N(\rho^2+\rho'^2)}e^{\kappa_N\rho\rho'\cos(\gph\vartheta)}
\]
with $\rho=\rhof+r\ge0$,
$\rho'=\rhof+r'\ge0$ (the variables are $\ge-\rhof$). By $e^{z\cos\varphi}=\sum_{q\in\Z}I_q(z)e^{iq\varphi}$ with modified Bessel
functions $I_q(z)=I_{-q}(z)\ge0$ for $z\ge0$, and $\sum_q I_q(z)=e^z<\infty$,
\[
\int\prod_bdB_b\;e^{i\gph\sum_b(d^*m)_bB_b}\prod_bh(\tr_{b_-},\tr_{b_+},B_b)=\prod_b\frac{2\pi}{\gph}\,
e^{-\frac12\kappa_N(\rho_{b_-}^2+\rho_{b_+}^2)}\,I_{(d^*m)_b}\big(\kappa_N\rho_{b_-}\rho_{b_+}\big)\ \ge0 ,
\]
the integral over $(-\pi/\gph,\pi/\gph]$ picking the Fourier mode $q=-(d^*m)_b$ (an integer).

\emph{Step 4: assembly.} Define
\[
\Omega(m):=\int d\tr\prod_xg(\tr_x)e^{-\frac12\delta m_B\rho_x^2}\prod_b\frac{2\pi}{\gph}e^{-\frac12\kappa_N(\rho_{b_-}^2+\rho_{b_+}^2)}
I_{(d^*m)_b}(\kappa_N\rho_{b_-}\rho_{b_+})\ge0,
\]
which does not depend on $a$ or $t$. Then
\begin{equation}\label{eq:dual}
Z(a)=e^{\|a\|^2/2}\int_{[0,1]^{\mathcal C}}dt\sum_{m\in\Z^{\mathcal P}}\Big(\prod_pc_{m_p-(d^*t)_p}\Big)\,\Omega(m)\,
e^{-i\gph\,(m-d^*t)\cdot a}.
\end{equation}
The interchanges of the sums and integrals are justified by absolute convergence: $\sum_m\prod_pc_{m_p-s_p}=\prod_p\sum_mc_{m-s_p}$, and
$\sum_mc_{m-s}=\sum_ne^{-(2\pi n/\gph)^2/2}e^{2\pi isn}$ (Poisson again) is bounded by $K_{\gph}:=\sum_ne^{-(2\pi n/\gph)^2/2}<\infty$
uniformly in $s$. Moreover,
\begin{multline*}
0\le\Omega(m)\le\int d\tr\prod_xg(\tr_x)e^{-\frac12\delta m_B\rho_x^2}\prod_b\frac{2\pi}{\gph}e^{-\frac12\kappa_N(\rho_{b_-}-\rho_{b_+})^2}\\
\le(2\pi/\gph)^{|\mathcal B|}\big(\int_{-\rhof}^\infty g(r)e^{-\frac12\delta m_B(\rhof+r)^2}dr\big)^{|\mathcal X|}<\infty
\end{multline*}
since $I_q(z)\le e^z$ and by \eqref{eq:ct} at $q=0$; the lattice is finite. Since $Z(a)$ is
the integral of a positive weight it is real and positive, so it equals the real part of \eqref{eq:dual}:
\begin{align*}
Z(a)&=e^{\|a\|^2/2}\int dt\sum_m\Big(\prod_pc_{m_p-(d^*t)_p}\Big)\Omega(m)\cos\big(\gph(m-d^*t)\cdot a\big)\\
&\le e^{\|a\|^2/2}\int dt\sum_m\Big(\prod_pc_{m_p-(d^*t)_p}\Big)\Omega(m)=e^{\|a\|^2/2}Z(0),
\end{align*}
the last equality being \eqref{eq:dual} at $a=0$. This is \eqref{eq:GD}.

\emph{Step 5: moments.} Apply \eqref{eq:GD} to $sa$, $s\in\R$: $\varphi(s):=\avL{e^{sX}}\le e^{s^2\|a\|^2/2}$. Then $\avL{e^{|sX|}}\le
2e^{s^2\|a\|^2/2}<\infty$, so all moments of $X$ are finite and $\varphi$ is smooth with $\varphi'(0)=\avL X$, $\varphi''(0)=\avL{X^2}$
(dominated convergence). The function $s\mapsto e^{s^2\|a\|^2/2}-\varphi(s)$ is $\ge0$ with a zero at $s=0$, so its first derivative vanishes
there and its second derivative is $\ge0$: $\avL X=0$ and $\avL{X^2}\le\|a\|^2$. For the higher moments, $\avL{X^{2n}}\le(2n)!\,s^{-2n}
\avL{\cosh(sX)}\le(2n)!\,s^{-2n}e^{s^2\|a\|^2/2}$ for every $s>0$; the choice $s^2=2n/\|a\|^2$ gives the stated bound.
\end{proof}

\begin{remark}
No property of the couplings enters: $\kappa_N\ge0$, $\lamf>0$, $\rhof>0$, $\delta m_B\in\R$, $\gph>0$, $\eN$, $L$ are arbitrary; the site factor enters only
through $\Omega(m)\ge0$ and $\Omega(m)<\infty$ in Step~4. The two structural facts used
are that the plaquette weight is Gaussian in the curvature (the Villain form) and that in unitary gauge the gauge field enters the matter
weight only through $\cos(\gph B_b)$ with a non-negative coefficient, so that all Fourier coefficients are non-negative (the positivity
structure used by Brydges, Fr\"ohlich and Seiler \cite{BFS79} for the non-compact field).
\end{remark}

\section{The estimate}\label{sec:E}
\begin{definition}[seminorms]\label{def:norms}
For a real $C^\infty_c$ $2$-form $f$ and $h\in C^\infty_c(\R^3;\R)$ put
\[
\begin{gathered}
R(f):=\sup_{N\ge N_0}\Big(\eN^3\sum_pf_{\mu\nu(p)}(x_p)^2\Big)^{1/2},\qquad
E(h):=\sup_{N\ge N_0}\ \eN^3\sum_z|h(z)|,\\
\|\sigma\|:=R(f)+\|g_{\rm d}\|_\infty E(h),
\end{gathered}
\]
the sums over the plaquettes and sites of the fine lattice at level $N$ (Riemann sums of $f^2$ and $|h|$ at spacing $\eN$). A seminorm
$\|\cdot\|'$ on $\mathcal T$ is \emph{admissible} if $\sigma_n\to\sigma$ in $C^\infty_c$ (supports in a fixed compact set, all derivatives
uniformly) implies $\|\sigma_n-\sigma\|'\to0$.
\end{definition}
For an admissible seminorm $\|\cdot\|'$ also $\|\tau_t\sigma-\sigma\|'\to0$ as $t\to0$.
If $K$ is a compact set containing the supports and $K_{+\azero}$ its $\azero$-neighbourhood, then $R(f)^2\le3\,|K_{+\azero}|\,\|f\|_\infty^2$
and $E(h)\le|K_{+\azero}|\,\|h\|_\infty$ (the open cubes of side $\eN$ centred at the sites of the lattice that lie in $K$ are pairwise disjoint and lie in $K_{+\sqrt3\eN/2}\subset K_{+\azero}$, as $\eN\le\azero$, so that at most $|K_{+\azero}|/\eN^3$ sites lie in $K$; the centres $x_p$ of the plaquettes of one orientation form a translate of the lattice, so that at most three times as many plaquettes have their centres in $K$). Hence both are
finite seminorms, and $\sigma_n\to\sigma$ in $C^\infty_c$ (supports in a fixed compact set, uniform convergence of all derivatives; uniform
convergence alone suffices) implies $\|\sigma_n-\sigma\|\to0$: the seminorm $\|\cdot\|$ is admissible. (The Riemann sums converge to
$\sum_{\mu<\nu}\|f_{\mu\nu}\|_{L^2}^2$ and $\|h\|_{L^1}$ as $N\to\infty$; only finiteness and continuity are used.)

\begin{proposition}[the estimate]\label{prop:E}
For every test pair $\sigma=(f,h)$, every level $N\ge N_0$ and every torus side $L$ large enough for the support of $\sigma$ to embed in
$\tilde T_L$,
\begin{equation}\label{eq:E}
\big(\avL{\Phi_N(\sigma)^2}\big)^{1/2}\le\|\sigma\|,\qquad\text{and hence}\qquad
\om_N\big(|\Phi_N(\sigma)|\wedge2\big)\le\|\sigma\| .
\end{equation}
\end{proposition}
\begin{proof}
Write $\Phi_N(\sigma)=X+Y$ (notation local to this proof) with $X=\sum_pa_p\ff_p$, $a_p=\eN^{3/2}f_{\mu\nu(p)}(x_p)$, and $Y=\eN^3\sum_zh(z)g_{\rm d}(\tr_z)$. By
Theorem~\ref{thm:GD}, $\avL{X^2}\le\sum_pa_p^2=\eN^3\sum_pf_{\mu\nu(p)}(x_p)^2\le R(f)^2$. Pointwise, $|Y|\le\|g_{\rm d}\|_\infty\eN^3\sum_z|h(z)|
\le\|g_{\rm d}\|_\infty E(h)$. By Minkowski's inequality in $L^2(\avL\cdot)$, $\big(\avL{(X+Y)^2}\big)^{1/2}\le R(f)+\|g_{\rm d}\|_\infty E(h)=\|\sigma\|$.
Finally, by the Schwarz inequality in $L^2(\mu_N)$ and Minkowski's inequality there,
\[
\om_N(|\Phi_N(\sigma)|\wedge2)\le\big(\int\Phi_N(\sigma)^2
d\mu_N\big)^{1/2}\le\big(\int X^2d\mu_N\big)^{1/2}+\|g_{\rm d}\|_\infty E(h),
\]
the pointwise bound on $Y$ holding on $\Omega_N$ as on the torus
($Y$ is a bounded Borel local function; $g_{\rm d}$ need not be continuous). $X^2$ is a continuous local function $\ge0$ with $\avL{X^2}\le R(f)^2$
for all large $L$, so $\int X^2d\mu_N\le\liminf_n\langle X^2\rangle^{\mathrm{fine}}_{L_n}\le R(f)^2$ by Theorem~\ref{thm:d2:LP}(e). Hence
$\om_N(|\Phi_N(\sigma)|\wedge2)\le R(f)+\|g_{\rm d}\|_\infty E(h)=\|\sigma\|$.
\end{proof}

\section{Symmetries of the fine state}\label{sec:R1}
\begin{lemma}[lattice symmetries of the fine state]\label{lem:R1}
$\om_N$ is invariant under the translations by $\eN\Z^3$, under the lattice reflections in the site planes and under the cubic group about
every site: $\om_N(F\circ g)=\om_N(F)$ for every bounded local gauge-invariant $F$ and every such $g$. In particular
$\om_N(\iota_N(\tau_ta))=\om_N(\iota_N(a))$ for all $a\in\cA$ and $t\in\eN\Z$.
\end{lemma}
\begin{proof}
The weight of the fine torus of side $L$ (Section~\ref{sec:setting}) is a product of position-independent factors (the site factor
$g(\tr_x)e^{-\frac12\delta m_B\rho_x^2}$ included) over the sites, bonds and
plaquettes of the periodic lattice $(\eN\Z/L\Z)^3$, and the sum over the Villain variables runs over all closed $v$. Every lattice
translation, reflection and cubic rotation $g$ of the torus permutes the sites, bonds and plaquettes, maps closed $v$ to closed $v$ and
preserves $dB$ up to the orientation signs absorbed in $B_{-b}=-B_b$, $v_{-p}=-v_p$; the plaquette weight $w(f)=e^{-f^2/2}$ is even and
the hopping factor is symmetric under bond reversal, $h(r,r',\vartheta)=h(r',r,-\vartheta)$, so the reversed orientations that $g$ produces
do not change the weight. (We write $F\circ g$ for the observable $F$ evaluated on the transformed configuration; the convention is
immaterial since the symmetries form groups.) Hence $\langle F\circ g\rangle^{\mathrm{fine}}_L=\langle F\rangle^{\mathrm{fine}}_L$ for every
$L$ with $L/2\in\eN\Z$ and every $g$ of the torus; for a fixed local $F$ and a fixed $g$ of $\eN\Z^3$ (or a fixed reflection in a site plane, or
a fixed cubic rotation about a site) this holds for all large $L$, in particular along the sides $(L_n)$ of Definition~\ref{def:omN}.

For bounded
continuous local $F$, \eqref{eq:omN} applied to $F$ and to $F\circ g$ (bounded continuous local as well) gives $\om_N(F\circ g)=\om_N(F)$; so
$g_*\mu_N$ and $\mu_N$ have the same integrals of bounded continuous local functions, hence the same finite-dimensional marginals (a Borel
probability measure on a Polish space is determined by the integrals of the bounded continuous functions), and $g_*\mu_N=\mu_N$ (the cylinder
sets form a $\pi$-system generating the Borel $\sigma$-algebra). Therefore $\om_N(F\circ g)=\om_N(F)$ for every bounded Borel local $F$; this is
Theorem~\ref{thm:d2:LP}(c), whose proof is the argument just given.

The last sentence is the case $g=$ translation by $te_3$, $t\in\eN\Z$: $\iota_N(\tau_ta)$ is $\iota_N(a)$
evaluated on the translated configuration, because $\Phi_N(\tau_t\sigma)=\Phi_N(\sigma)\circ g^{-1}$ for a lattice translation
(substitute $p\mapsto p-te_3$ in the plaquette sum and $z\mapsto z-te_3$ in the site sum; $\tau_t\sigma$ has its centres at $x_p+te_3$).
The same intertwining holds for the lattice reflections and for the cubic group about a site, with the action on test pairs defined as the
push-forward of $(f,h)$ (the $2$-form transforming as a $2$-form), by the same substitution.
\end{proof}

\section{The passage from the estimate to the limit functional}\label{sec:R3}
\begin{proposition}[the passage to the limit]\label{prop:R3}
Assume that each $\om_N$, $N\ge N_0$, is a state on the bounded Borel local functions (linear, positive, $\om_N(1)=1$, hence $|\om_N(F)|\le\|F\|_\infty$), and assume the unimodularity bound \eqref{eq:unimod}, the conclusions of Lemma~\ref{lem:R1} and the estimate \eqref{eq:E}
with an admissible seminorm (Definition~\ref{def:norms}). Then there are $N_0\le N_1<N_2<\cdots$ and
a linear $S:\cA\to\C$ such that
\begin{itemize}
\item[(a)] $\lim_m\om_{N_m}(\iota_{N_m}(a))=S(a)$ for every $a\in\cA$;
\item[(b)] $S(\tau_ta)=S(a)$ for all $t\in\R$ and $a\in\cA$;
\item[(c)] for $a,b\in\Ap$ the function $t\mapsto S(\Th a\cdot\tau_tb)$ is continuous on $[0,\infty)$.
\end{itemize}
Moreover $S$ is invariant under all space translations and under the cubic group about every point of $\bigcup_m\varepsilon_{N_m}\Z^3$, and
$|S(a)|\le M(a)$.
\end{proposition}
The state \eqref{eq:omN} of Definition~\ref{def:omN} is a state on the bounded Borel local functions by Theorem~\ref{thm:d2:LP}(b).
For unbounded density functions and other normalisations see Section~\ref{hx:sec} (Proposition~\ref{hx:prop:R3},
Theorem~\ref{hx:thm:Hlim}).

\begin{proof}
\emph{Step 1: equicontinuity.} For $\sigma,\sigma'\in\mathcal T$ and every $N$, by additivity of $\Phi_N$,
$|e^{ix}-e^{iy}|=|e^{i(x-y)}-1|\le|x-y|\wedge2$, the positivity of the state $\om_N$ and \eqref{eq:E},
\begin{equation}\label{eq:R31}
|Z_N(\sigma)-Z_N(\sigma')|\le\om_N\big(|\Phi_N(\sigma-\sigma')|\wedge2\big)\le C\|\sigma-\sigma'\|.
\end{equation}
Here $C$ denotes the constant with which the estimate \eqref{eq:E} holds for the seminorm; in the form stated in Proposition~\ref{prop:E}, $C=1$.

\emph{Step 2: a countable dense set.} Let $K_1\subset K_2\subset\cdots$ be compact cubes exhausting $\R^3$. For each $n$ the space
$\mathcal T_{K_n}$ of test pairs supported in $K_n$ is a separable Fr\'echet space (a closed subspace of a finite product of spaces
$C^\infty_{K_n}(\R^3)$, each separable), so it has a countable dense subset $\mathcal D_n$; let $\mathcal D=\bigcup_n\mathcal D_n$. Every
$\sigma$ lies in some $\mathcal T_{K_n}$ and is a $C^\infty_c$-limit, with supports in $K_n$, of elements of $\mathcal D_n$; by admissibility,
for every $\sigma$ and $\delta>0$ there is $\sigma'\in\mathcal D$ with $\|\sigma-\sigma'\|<\delta$.

\emph{Step 3: the subsequence.} $|Z_N(\sigma)|\le1$ for all $N,\sigma$ by \eqref{eq:unimod}. A diagonal extraction gives $N_1<N_2<\cdots$ such that
$Z_{N_m}(\sigma')$ converges for every $\sigma'\in\mathcal D$. For arbitrary $\sigma$ and $\delta>0$ pick $\sigma'\in\mathcal D$ with
$\|\sigma-\sigma'\|<\delta$; by \eqref{eq:R31}, $\limsup_{m,m'}|Z_{N_m}(\sigma)-Z_{N_{m'}}(\sigma)|\le2C\delta$, so $Z_{N_m}(\sigma)$ is Cauchy and
converges, to a limit $Z(\sigma)$. Define $S(e_\sigma):=Z(\sigma)$ and extend linearly to $\cA$; this is well defined because the
$e_\sigma$ form a basis of $\cA=\C[\mathcal T]$ (the group algebra of the additive group $\mathcal T$). For
$a=\sum_jc_je_{\sigma_j}$, $\om_{N_m}(\iota_{N_m}(a))=\sum_jc_jZ_{N_m}(\sigma_j)\to S(a)$: this is (a), and $|S(a)|\le M(a)$.

\emph{Step 4: continuity in the translation parameter.} For $\sigma\in\mathcal T$ and $t,t'\in\R$, by \eqref{eq:R31} in the limit,
\[
|Z(\tau_t\sigma)-Z(\tau_{t'}\sigma)|\le C\|\tau_t\sigma-\tau_{t'}\sigma\|,
\]
and $\tau_t\sigma-\tau_{t'}\sigma\to0$ in $C^\infty_c$ as $t'\to t$ (the supports stay in a fixed compact set and all derivatives
converge uniformly), so the right side tends to $0$ by admissibility. Hence $t\mapsto Z(\tau_t\sigma)$ is continuous on $\R$, and so is
$t\mapsto S(\tau_ta)$ for every $a\in\cA$ (finite sums).

\emph{Step 5: (b).} Fix $a\in\cA$ and $m$. For $t\in\varepsilon_{N_m}\Z$ Lemma~\ref{lem:R1} gives $\om_{N_{m'}}(\iota_{N_{m'}}(\tau_ta))=
\om_{N_{m'}}(\iota_{N_{m'}}(a))$ for every $m'\ge m$, because $\varepsilon_{N_{m'}}\Z\supset\varepsilon_{N_m}\Z$ for $m'\ge m$
($\varepsilon_{N}=\azero5^{-N}$ and $N_{m'}\ge N_m$). Letting $m'\to\infty$, $S(\tau_ta)=S(a)$ for $t\in\varepsilon_{N_m}\Z$, for every
$m$; the union $\bigcup_m\varepsilon_{N_m}\Z$ is dense in $\R$, and $t\mapsto S(\tau_ta)$ is continuous (Step 4): so $S(\tau_ta)=S(a)$
for all real $t$. The same argument with the space translations gives their invariance; for the cubic group about a site of
$\varepsilon_{N_m}\Z^3$ no density is needed: Lemma~\ref{lem:R1} and the intertwining of $\iota_{N_{m'}}$ with the push-forward
action (the same substitution as in Lemma~\ref{lem:R1}, a signed permutation of the plaquettes and sites)
give the invariance for every $m'\ge m$, hence in the limit.

\emph{Step 6: (c).} Let $a,b\in\Ap$ and $t,t'\ge0$. Since $\Th a\cdot\tau_tb$ is a finite sum of basis elements and $S$ is linear, it
suffices to treat $a=e_{\sigma_1}$, $b=e_{\sigma_2}$: then $\Th a\cdot\tau_tb=e_{-\theta\sigma_1+\tau_t\sigma_2}$ and
\[
|S(\Th a\cdot\tau_tb)-S(\Th a\cdot\tau_{t'}b)|=|Z(-\theta\sigma_1+\tau_t\sigma_2)-Z(-\theta\sigma_1+\tau_{t'}\sigma_2)|
\le C\|\tau_t\sigma_2-\tau_{t'}\sigma_2\|
\]
by \eqref{eq:R31} in the limit, which tends to $0$ as $t'\to t$ by Step 4 (two-sided in $t'$, so right-continuity at $t=0$ is included). For a general
$a=\sum_jc_je_{\sigma_j}$ the bound carries the factor $M(a)$: the exponentials $\iota_N(\Th e_{\sigma_j})$ are unimodular, so the estimate
\eqref{eq:R31} applies to the second factor alone.
\end{proof}

\begin{remark}
(i) Only the truncated first-moment form \eqref{eq:E} is used, through \eqref{eq:R31}. (ii) Nothing is assumed about the uniqueness of the limit; different
subsequences may give different $S$, each of which satisfies (a)--(c). (iii) The continuity in (c) is uniform on compact $t$-sets, with the
modulus $C\|\tau_t\sigma-\tau_{t'}\sigma\|$. (iv) The cubic invariance of $S$ comes from Lemma~\ref{lem:R1} and the intertwining of $\iota_N$
with the cubic action on test pairs; for a finite group no density argument is needed.
\end{remark}
\section{Existence and continuity of the limit functional}\label{sec:main}
\begin{theorem}[the limit functional]\label{thm:Hlim}
Consider the Villain fine model of Section~\ref{sec:setting} with any positive couplings and any real $\delta m_B$, let the fine
infinite-volume states $\om_N$, $N\ge N_0$, be limit points of the torus states in the sense of Definition~\ref{def:omN}, and let the
scheme be the one of Definition~\ref{def:pin}. Then
there are $N_0\le N_1<N_2<\cdots$ and a linear $S:\cA\to\C$ with
\begin{itemize}
\item[(a)] $\lim_{m\to\infty}\om_{N_m}(\iota_{N_m}(a))=S(a)$ for every $a\in\cA$, i.e.\ $Z_{N_m}(\sigma)=\om_{N_m}(e^{i\Phi_{N_m}(\sigma)})\to Z(\sigma)=S(e_\sigma)$
for every test pair $\sigma$;
\item[(b)] $S(\tau_ta)=S(a)$ for all $t\in\R$, $a\in\cA$;
\item[(c)] $t\mapsto S(\Th a\cdot\tau_tb)$ is continuous on $[0,\infty)$ for all $a,b\in\Ap$.
\end{itemize}
Moreover $S$ is invariant under all space translations and under the cubic group about every point of $\bigcup_m\varepsilon_{N_m}\Z^3$, and
$|S(a)|\le M(a)$.
\end{theorem}
\begin{proof}
The hypotheses of Proposition~\ref{prop:R3} hold: each $\om_N$ is a state on the bounded Borel local functions by
Theorem~\ref{thm:d2:LP}(b); the bound \eqref{eq:unimod} holds for $\iota_N$; Lemma~\ref{lem:R1} gives the lattice symmetries; and
Proposition~\ref{prop:E} is the estimate \eqref{eq:E} with the admissible seminorm of Definition~\ref{def:norms}. Proposition~\ref{prop:R3}
therefore gives (a)--(c), the invariances and the bound $|S(a)|\le M(a)$.
\end{proof}

The states $\om_N$ exist for every choice of positive couplings and any real $\delta m_B$ by Theorem~\ref{thm:d2:LP}; when the limit
\eqref{eq:D2} exists, $\om_N$ is that limit. A functional $S$ as in Theorem~\ref{thm:Hlim} is called a \emph{limit functional}.

\begin{proposition}[the limit functional for limit points along subnets]\label{prop:subnet}
For each $N\ge N_0$ let $(L_k)$ be a sequence of sides with $L_k\to\infty$ and $L_k/2\in\eN\Z$, and let $(L_\alpha)$ be a subnet of it along which
the limit
\[
\om_N(F):=\lim_\alpha\langle F\rangle^{\mathrm{fine}}_{L_\alpha}
\]
exists for every bounded Borel local $F$ (a Tychonoff cluster point of the torus states). Then $\om_N$ is a positive normalised linear functional
on the bounded Borel local functions, and Theorem~\ref{thm:Hlim}, its last clause included, holds with these $\om_N$ in place of the limit points of
Definition~\ref{def:omN}. The same is true for every pointwise limit point of the torus states on the bounded Borel local functions along any
subnet of sides $L\to\infty$, $L/2\in\eN\Z$, or along any cofinal class of them.
\end{proposition}
\begin{proof}
The proof of Theorem~\ref{thm:Hlim} uses of $\om_N$ only the hypotheses of Proposition~\ref{prop:R3}, that is, the following three properties.
\begin{itemize}
\item[(i)] The state property. It enters at \eqref{eq:R31}, in Step~3 and in Step~6 of the proof of Proposition~\ref{prop:R3}. In Step~3 it gives
$|Z_N(\sigma)|\le1$, together with \eqref{eq:unimod}, which is a pointwise bound on $\iota_N$ in which no state enters; in Step~6 it enters as the
positivity, with the unimodular prefactor.
\item[(ii)] The invariances of Lemma~\ref{lem:R1}, which enter in Step~5.
\item[(iii)] The bound $\om_N(|\Phi_N(\sigma)|\wedge2)\le\|\sigma\|$ of \eqref{eq:E}, which enters in Step~1.
\end{itemize}
Every torus state has the properties (i)--(iii) exactly, at every side $L$ large enough for the supports. We show that each of them passes to a
pointwise limit on the bounded Borel local functions.

\emph{Property (i).} Linearity, positivity and $\langle1\rangle^{\mathrm{fine}}_L=1$ are equalities and inequalities between the values of the state on
fixed functions, and such relations pass to the limit.

\emph{Property (ii).} Let $F$ be a bounded Borel local function and let $g$ be a translation, a site-plane reflection or a cubic rotation. By the
torus argument in the proof of Lemma~\ref{lem:R1}, $\langle F\circ g\rangle^{\mathrm{fine}}_L=\langle F\rangle^{\mathrm{fine}}_L$ for all large $L$.
Since $L_\alpha\to\infty$, it follows that $\om_N(F\circ g)=\om_N(F)$.

\emph{Property (iii).} The function $|\Phi_N(\sigma)|\wedge2$ is bounded Borel local, and for all large $L$ the Schwarz
inequality and the torus half of \eqref{eq:E} give
\[
\langle|\Phi_N(\sigma)|\wedge2\rangle^{\mathrm{fine}}_L\le(\avL{\Phi_N(\sigma)^2})^{1/2}\le\|\sigma\|,
\]
whence $\om_N(|\Phi_N(\sigma)|\wedge2)\le\|\sigma\|$. In the proof of Proposition~\ref{prop:E} the measure $\mu_N$ is used only in the passage through
$L^2(\mu_N)$ and in the step $\int X^2d\mu_N\le\liminf_n\langle X^2\rangle^{\mathrm{fine}}_{L_n}$ from Theorem~\ref{thm:d2:LP}(e); the bound at every $L$
replaces both.

With (i)--(iii), Steps~1--6 of the proof of Proposition~\ref{prop:R3} and the proof of Theorem~\ref{thm:Hlim} go through with (i) in place of
Theorem~\ref{thm:d2:LP}(b) and (iii) in place of Proposition~\ref{prop:E}. This gives (a)--(c), the invariance of $S$ and $|S(a)|\le M(a)$ for the
functionals $\om_N$ of the statement. The same proof applies to every pointwise limit point named in the last sentence of the statement.
\end{proof}

\begin{remark}[limit points along subnets]\label{rem:subnet}
Proposition~\ref{prop:subnet} extends Theorem~\ref{thm:Hlim} to limit points of the torus states in a wider sense than Definition~\ref{def:omN}.
The functional $\om_N$ of the proposition is finitely additive and not necessarily unique. It agrees with a state of Definition~\ref{def:omN} on the
continuous functions. Indeed, along $(L_\alpha)$ the finite-dimensional laws converge weakly (they are tight by Theorem~\ref{thm:GD} and
Lemma~\ref{lem:d2:mom}) to those of a Borel probability measure $\mu_N$. The limit laws form a cluster point of the laws along $(L_k)$ in the metrisable
countable product of the spaces of laws of the finitely many site and plaquette variables of a finite set $\Lambda$ of cells,
hence they are the limit of the laws along a subsequence $(L_n)$ of $(L_k)$. For this pair $((L_n),\mu_N)$ of Definition~\ref{def:omN}, $\om_N(F)=\int F\,d\mu_N$ for
every bounded continuous local $F$, so the two functionals give the same $Z_N(\sigma)$ when $g_{\rm d}$ is continuous. For a discontinuous bounded
Borel $g_{\rm d}$ the two values may differ, and the subnet functional is covered by the direct proof of Proposition~\ref{prop:subnet}.

The scheme for which Theorem~\ref{thm:Hlim} is proved is that of Definition~\ref{def:pin}: $c_N=\eN^{3/2}$, $c_N'=\eN^3$ and
$\mathfrak w_z=g_{\rm d}(\tr_z)$ with $g_{\rm d}$ bounded Borel. The Riemann scaling $c_N=\eN^{3/2}$ enters the proof
of Proposition~\ref{prop:E} essentially, and $g_{\rm d}=\mathrm{id}$ is not covered (Remark~\ref{rem:density}(ii),(iv)).
Section~\ref{hx:sec} treats unbounded density functions and other normalisations in the cases verified there
(Theorem~\ref{hx:thm:Hlim}, Proposition~\ref{hx:prop:pairs}).
\end{remark}







\begin{remark}[regularity of the limit curvature]\label{rem:OS0}
For real $f_1,\dots,f_k$ and $s\in\R^k$, Theorem~\ref{thm:GD} with $a_p=\eN^{3/2}\sum_js_jf_j(x_p)$ gives the uniform exponential moment bound
\[
\Big\langle\exp\Big(\sum_js_j\Phi_N(f_j,0)\Big)\Big\rangle^{\mathrm{fine}}_L\le\exp\Big(\tfrac12R\Big(\sum_js_jf_j\Big)^2\Big)\qquad\text{for all }N,L.
\]
This bound has the following standard consequences. The laws of $(\Phi_N(f_j,0))_{j\le k}$ under the torus states are
tight with uniformly Gaussian tails, so along the subsequence of Theorem~\ref{thm:Hlim} the characteristic function $s\mapsto Z(\sum_js_jf_j,0)$
is that of a probability law $\mu_f$ on $\R^k$ (L\'evy's continuity theorem), $\int e^{s\cdot x}d\mu_f(x)\le\exp(\tfrac12R(\sum_js_jf_j)^2)$, the
moments of the torus laws converge to those of $\mu_f$ (uniform integrability from the exponential moments), and the even moments of $\mu_f$ obey
the bounds of Theorem~\ref{thm:GD}. The odd moments vanish: the weight is invariant under $(B,v)\mapsto(-B,-v)$ (the hopping factor depends on
$B$ through $\cos(\gph B_b)$, $w$ is even, closed $v$ go to closed $v$), which maps $\ff\mapsto-\ff$. These are finite-family moment bounds for the
curvature sector; the full regularity axiom is Theorem~\ref{thm:osrest:OS0}.
\end{remark}

\begin{remark}[the density component]\label{rem:density}
We collect five observations on the density component of the scheme.
\begin{itemize}
\item[(i)] With a bounded $g_{\rm d}$ and $c_N'=\eN^3$ the density part of the estimate is pointwise and trivial.
\item[(ii)] A genuine scalar sector, the centred density $\eN^{5/2}\sum_zh(z)(\tr_z-\om_N(\tr_0))$, is not covered.
Section~\ref{hx:sec} treats unbounded density functions and other normalisations in the cases verified there (Theorem~\ref{hx:thm:Hlim}, Proposition~\ref{hx:prop:pairs}).
\item[(iii)] Any other bounded Borel function of finitely many fine gauge-invariant
variables can be added to the scheme in the same way.
\item[(iv)] Since \eqref{eq:E} is used in first-moment form only, the boundedness of
$g_{\rm d}$ can be weakened to $\sup_N\om_N(|g_{\rm d}(\tr_0)|)<\infty$, by translation invariance and the inequality $(\sum_zu_z)\wedge2\le\sum_z(u_z\wedge2)$ for
$u_z\ge0$. For $g_{\rm d}=\mathrm{id}$ at $c_N'=\eN^3$ this condition is the level-uniform first-moment bound $\sup_N\om_N(|\tr_0|)<\infty$.
The weakening is proved in Section~\ref{hx:sec} (Theorem~\ref{hx:thm:Hlim}, Remark~\ref{hx:rem:suff}).
\item[(v)] We do not prove that the limit is non-degenerate, that is, that $S$ is not the state of a constant field.
\end{itemize}
\end{remark}



\providecommand{\cK}{\mathcal K}
\section{Unbounded density functions and general normalisations}\label{hx:sec}

Theorem~\ref{thm:Hlim} is stated for the scheme of Definition~\ref{def:pin}, that is, for a \emph{bounded} real Borel density function $g_{\rm d}$ and the
\emph{pinned} normalisations $c_N=\eN^{3/2}$, $c_N'=\eN^{3}$. Definition~\ref{def:pin} makes sense for every real Borel density function $g_{\rm d}$ and all real normalisations $c_N$, $c_N'$. Boundedness and the pinned normalisations
therefore enter only through the proof of the properties (a)--(c) of Theorem~\ref{thm:Hlim}.
This section proves these three properties without the two restrictions for every scheme that satisfies the conditions \eqref{hx:hyp:N} and \eqref{hx:hyp:Y} of Definition~\ref{hx:def:ext} (Theorem~\ref{hx:thm:Hlim}). Condition \eqref{hx:hyp:Y} is verified in the cases \textup{(V1)--(V4)} of Proposition~\ref{hx:prop:pairs}, and \eqref{hx:hyp:N} is a condition on the curvature normalisation alone, which holds for $c_N=\eN^{3/2}$: in each of these cases, for every $c_N$ that satisfies \eqref{hx:hyp:N}, all conclusions of Theorem~\ref{hx:thm:Hlim} hold.

The proof of Theorem~\ref{thm:Hlim} uses the boundedness of $g_{\rm d}$ and the pinned normalisations at one place only, the estimate of
Proposition~\ref{prop:E} with the seminorm of Definition~\ref{def:norms}. They enter there four times: through the factor
$\|g_{\rm d}\|_\infty$ in the seminorm; through the pointwise bound of the density sum by $\|g_{\rm d}\|_\infty\,\eN^3\sum_z|h(z)|$ (on the
tori and under the fine state); through the factor $\eN^{3}$ of the density sum, which makes that bound a Riemann sum; and through the weights
$a_p=\eN^{3/2}f_{\mu\nu(p)}(x_p)$ of the curvature sum, which make the bound of Theorem~\ref{thm:GD} a Riemann sum. The symmetry lemma
(Lemma~\ref{lem:R1}) and the abstract passage to the limit (Proposition~\ref{prop:R3}) do not involve the density function or the normalisations.

We replace the estimate by one in first-moment form (Lemma~\ref{hx:lem:E}) under two conditions, stated in Definition~\ref{hx:def:ext}:
\eqref{hx:hyp:N}, a bound on the ratio of the curvature normalisation to its pinned value, and \eqref{hx:hyp:Y}, a level-independent bound for
the truncated first moment of the density term. A level-uniform first moment of the normalised density variable, \eqref{hx:hyp:Y1}, suffices
for \eqref{hx:hyp:Y}.
Theorem~\ref{hx:thm:Hlim} then gives clauses (a)--(c), the invariances and the bound $|S(a)|\le M(a)$ for every real Borel $g_{\rm d}$ and
all real normalisations satisfying \eqref{hx:hyp:N} and \eqref{hx:hyp:Y}. No convergence of the normalisation ratios is needed, and the pinned scheme
is the case of equality (Corollary~\ref{hx:cor:norm}).

Two restrictions remain. The first is the pair \eqref{hx:hyp:N}, \eqref{hx:hyp:Y} itself. For an unbounded $g_{\rm d}$ at the Riemann
normalisation $c_N'=\eN^3$, clause \eqref{hx:hyp:Y1} is a level-uniform first-moment bound for $g_{\rm d}(\tr_0)$; this paper proves no such bound that holds for every value of the counterterm $\delta m_B(N)$
(the moment bounds that it proves for every value of the counterterm are uniform in the volume at a fixed level only,
Subsection~\ref{hx:sub:bounds}), and none can be derived without information on the level dependence of the counterterm
(Remark~\ref{hx:rem:counter}). The cases in which \eqref{hx:hyp:Y1} is verified are listed in Proposition~\ref{hx:prop:pairs}.
The second concerns the other results of this paper: those listed in Remark~\ref{hx:rem:scope}(A) extend to the wider class, while the statements on
exponential or second moments of the joint field keep the bounded class (Remark~\ref{hx:rem:scope}(C)).

\subsection{The extended scheme and the conditions}\label{hx:sub:hyp}
Throughout this section $g_{\rm d}$ is an arbitrary real Borel function on $\R$ (finite at every point), and $c_N$, $c_N'$, $N\ge N_0$, are
arbitrary real numbers. The functions $\Phi_N(\sigma)$, the realisation $\iota_N$ and $Z_N(\sigma)=\om_N(e^{i\Phi_N(\sigma)})$ are defined by
the formulas of Definition~\ref{def:pin}, with these $g_{\rm d}$, $c_N$, $c_N'$ in place of those of the pinned scheme. We write
\[
\zeta_N:=c_N\,\eN^{-3/2},\qquad k_N:=c_N'\,\eN^{-3}\qquad(N\ge N_0)
\]
for the ratios of the two normalisations to their pinned values. They are signed real numbers, and the pinned scheme is the case $\zeta_N=k_N=1$.
We split
\begin{gather*}
\Phi_N(\sigma)=X_N(f)+Y_N(h),\\
X_N(f):=c_N\sum_pf_{\mu\nu(p)}(x_p)\,\ff_p,\qquad Y_N(h):=c_N'\sum_zh(z)\,g_{\rm d}(\tr_z),
\end{gather*}
and we use the Riemann sum of Definition~\ref{def:norms},
\[
E_N(h):=\eN^3\sum_z|h(z)|\le E(h).
\]
If $K$ is a compact set containing the support of $h$, then $E(h)\le|K_{+\azero}|\,\|h\|_\infty$,
for the reason given after Definition~\ref{def:norms}: every site of $\eN\Z^3$ owns a cell of
volume $\eN^3$, the cells of the sites in $K$ are disjoint and lie in $K_{+\eN}\subset K_{+\azero}$. The same bound holds for every translate
and every reflection of $h$, since the volume of the neighbourhood does not change. Hence $E$ is a finite seminorm on
$C^\infty_c(\R^3;\R)$, free of $N$.

\emph{The scheme is well posed for every real Borel $g_{\rm d}$.} The sums defining $\Phi_N(\sigma)$ are finite, so $\Phi_N(\sigma)$ is a
finite real Borel local function on $\Omega_N$, additive in $\sigma$, and $e^{i\Phi_N(\sigma)}$ is a bounded Borel local function of modulus
one. Therefore $Z_N(\sigma)$ is defined with no integrability condition on $g_{\rm d}$, and the unimodularity bound \eqref{eq:unimod} holds.
For a Borel local function $F\ge0$ we write $\om_N(F):=\int F\,d\mu_N\in[0,\infty]$. Since the measure $\mu_N$ is invariant under the
translations by $\eN\Z^3$ (Theorem~\ref{thm:d2:LP}(c)),
\begin{equation}\label{hx:eq:site}
\om_N\big(|g_{\rm d}(\tr_z)|\big)=\om_N\big(|g_{\rm d}(\tr_0)|\big)\in[0,\infty]\qquad\text{for every site }z\text{ of the fine lattice.}
\end{equation}

\begin{definition}[the extended scheme; conditions (N), (Y), (Y1)]\label{hx:def:ext}
The \emph{extended scheme} is the scheme of Definition~\ref{def:pin} with an arbitrary real Borel function $g_{\rm d}$ on $\R$ and arbitrary
real normalisations $c_N$, $c_N'$, $N\ge N_0$. For it we consider the following conditions. All suprema run over the levels $N\ge N_0$, the
same set of levels for which the conditions are imposed and the conclusions drawn (raising $N_0$ is allowed).
\begin{align}
&\zeta:=\sup_{N\ge N_0}|\zeta_N|=\sup_{N\ge N_0}|c_N|\,\eN^{-3/2}<\infty ;\tag{N}\label{hx:hyp:N}\\
&\text{there is a seminorm }\mathsf p\text{ on }C^\infty_c(\R^3;\R)\text{, free of }N\text{ and admissible, such that}\notag\\
&\qquad\om_N\big(|Y_N(h)|\wedge2\big)\le\mathsf p(h)\qquad\text{for all }h\in C^\infty_c(\R^3;\R)\text{ and all }N\ge N_0 ;\tag{Y}\label{hx:hyp:Y}\\
&A_1:=\sup_{N\ge N_0}|k_N|\,\om_N\big(|g_{\rm d}(\tr_0)|\big)<\infty .\tag{Y1}\label{hx:hyp:Y1}
\end{align}
In \eqref{hx:hyp:Y1} and below, $0\cdot\infty:=0$: at a level with $c_N'=0$ the density term vanishes.
In \eqref{hx:hyp:Y}, \emph{admissible} is the notion of Definition~\ref{def:norms}, read for $h$ alone: $h_n\to h$ in $C^\infty_c(\R^3;\R)$
(supports in a fixed compact set, all derivatives uniformly) implies $\mathsf p(h_n-h)\to0$; equivalently, the seminorm
$(f,h)\mapsto\mathsf p(h)$ on $\mathcal T$ is admissible.
\end{definition}

Condition \eqref{hx:hyp:N} is the \emph{bounded curvature ratio}; \eqref{hx:hyp:Y} is the abstract \emph{density clause}; and \eqref{hx:hyp:Y1}
is the \emph{first-moment clause}. For the pinned normalisations,
\eqref{hx:hyp:N} holds with $\zeta=1$, and for bounded $g_{\rm d}$ clause \eqref{hx:hyp:Y1} holds with $A_1\le\|g_{\rm d}\|_\infty$: the
scheme of Definition~\ref{def:pin} is covered.

\begin{lemma}[the density clauses]\label{hx:lem:Y}
\textup{(1)} \eqref{hx:hyp:Y1} implies \eqref{hx:hyp:Y} with $\mathsf p=A_1E$.
\textup{(2)} Clause \eqref{hx:hyp:Y} is equivalent to the $L^1$ bound $\om_N(|Y_N(h)|)\le\mathsf p(h)$ for all $h$ and all $N\ge N_0$, with
the same seminorm. In particular \eqref{hx:hyp:Y} forces $\om_N(|g_{\rm d}(\tr_0)|)<\infty$ at every level $N\ge N_0$ with $c_N'\ne0$.
\end{lemma}
\begin{proof}
(1) Pointwise, $|Y_N(h)|\le|c_N'|\sum_z|h(z)|\,|g_{\rm d}(\tr_z)|$, a finite sum of Borel functions $\ge0$. Integrating term by term and
using \eqref{hx:eq:site},
\[
\om_N\big(|Y_N(h)|\big)\le|c_N'|\sum_z|h(z)|\,\om_N\big(|g_{\rm d}(\tr_0)|\big)=|k_N|\,\om_N\big(|g_{\rm d}(\tr_0)|\big)\,E_N(h)\le A_1E(h).
\]
The seminorm $E$ is admissible, by the bound $E(h)\le|K_{+\azero}|\,\|h\|_\infty$.

(2) An $L^1$ bound implies \eqref{hx:hyp:Y} trivially. Conversely, apply \eqref{hx:hyp:Y} to $sh$, $s>0$: since $\mathsf p$ is a seminorm and
$Y_N$ is linear in $h$, $\om_N(|Y_N(h)|\wedge(2/s))\le\mathsf p(h)$; let $s\downarrow0$ and use monotone convergence. For the last
assertion fix $N$ with $c_N'\ne0$ and choose $h\in C^\infty_c$ with $h(0)=1$ and support in the open ball of radius $\eN$ about the
origin: the origin is the only site of $\eN\Z^3$ in the support, so $Y_N(h)=c_N'g_{\rm d}(\tr_0)$ and
$\om_N(|g_{\rm d}(\tr_0)|)\le\mathsf p(h)/|c_N'|<\infty$.
\end{proof}

\subsection{The estimate in first-moment form and the theorem}\label{hx:sub:thm}
\begin{lemma}[the estimate in first-moment form]\label{hx:lem:E}
Assume \eqref{hx:hyp:N} and \eqref{hx:hyp:Y}. Then for every test pair $\sigma=(f,h)$ and every level $N\ge N_0$,
\begin{equation}\label{hx:eq:E}
\om_N\big(|\Phi_N(\sigma)|\wedge2\big)\le\zeta\,R(f)+\mathsf p(h)=:\|\sigma\|_{\rm x},
\end{equation}
and $\|\cdot\|_{\rm x}$ is an admissible seminorm on $\mathcal T$, free of $N$. Consequently, for all $\sigma,\sigma'\in\mathcal T$ and $N\ge N_0$,
\begin{equation}\label{hx:eq:R31}
|Z_N(\sigma)-Z_N(\sigma')|\le\om_N\big(|\Phi_N(\sigma-\sigma')|\wedge2\big)\le\|\sigma-\sigma'\|_{\rm x}.
\end{equation}
Under \eqref{hx:hyp:N} and \eqref{hx:hyp:Y1} this holds with $\|\sigma\|_{\rm x}=\zeta R(f)+A_1E(h)$.
\end{lemma}
\begin{proof}
Write $X=X_N(f)$, $Y=Y_N(h)$. Pointwise on $\Omega_N$, $|X+Y|\wedge2\le|X|+(|Y|\wedge2)$: if $|Y|\ge2$ the right side is at least $2$, and
otherwise $|X+Y|\le|X|+|Y|$. \emph{Curvature part.} $X=\sum_pa_p\ff_p$ with the real plaquette function $a_p=c_Nf_{\mu\nu(p)}(x_p)$ of finite
support, and $X^2$ is a continuous local function $\ge0$. Theorem~\ref{thm:GD}, which holds for every real plaquette function, gives for
every large side $L$
\[
\avL{X^2}\le\sum_pa_p^2=\zeta_N^2\,\eN^3\sum_pf_{\mu\nu(p)}(x_p)^2\le\zeta^2R(f)^2 ,
\]
hence $\int X^2d\mu_N\le\liminf_n\langle X^2\rangle^{\mathrm{fine}}_{L_n}\le\zeta^2R(f)^2$ by Theorem~\ref{thm:d2:LP}(e), exactly as in the
proof of Proposition~\ref{prop:E}, and $\om_N(|X|)\le\zeta R(f)$ by the Schwarz inequality. \emph{Density part.}
$\om_N(|Y|\wedge2)\le\mathsf p(h)$ is \eqref{hx:hyp:Y}. Adding the two parts gives \eqref{hx:eq:E}. The seminorm $R$ is admissible (the paragraph after
Definition~\ref{def:norms}) and $\mathsf p$ is admissible by \eqref{hx:hyp:Y}, so $\|\cdot\|_{\rm x}$ is. For \eqref{hx:eq:R31} use the additivity
of $\Phi_N$, the inequality $|e^{ix}-e^{iy}|\le|x-y|\wedge2$ and the positivity of $\om_N$. The last sentence is Lemma~\ref{hx:lem:Y}(1).
\end{proof}

Proposition~\ref{prop:R3} has the estimate \eqref{eq:E} among its hypotheses. The first half of \eqref{eq:E} is an $L^2$ bound on the
tori; it fails as soon as $g_{\rm d}(\tr_0)$ is not square integrable under the torus states, which \eqref{hx:hyp:Y} does not exclude. The proof
of Proposition~\ref{prop:R3} does not use that half, and we restate the proposition with the inequality it uses as its hypothesis.

\begin{proposition}[the passage to the limit, with \eqref{eq:R31} as hypothesis]\label{hx:prop:R3}
Let $\Phi_N(\sigma)$, $N\ge N_0$, be real Borel local functions on $\Omega_N$, additive in $\sigma\in\mathcal T$, and
$\iota_N(e_\sigma)=e^{i\Phi_N(\sigma)}$, $Z_N(\sigma)=\om_N(e^{i\Phi_N(\sigma)})$. Assume:
\begin{itemize}
\item[(h1)] each $\om_N$, $N\ge N_0$, is a state on the bounded Borel local functions;
\item[(h2)] the unimodularity bound \eqref{eq:unimod};
\item[(h3)] the conclusions of Lemma~\ref{lem:R1} for these $\om_N$ and $\iota_N$, including the intertwining of $\iota_N$ with the lattice
translations and the cubic group stated in its proof;
\item[(h4)] there is an admissible seminorm $\|\cdot\|'$ on $\mathcal T$, free of $N$, with
$\om_N(|\Phi_N(\sigma)|\wedge2)\le\|\sigma\|'$ for all $\sigma\in\mathcal T$ and $N\ge N_0$.
\end{itemize}
Then all conclusions of Proposition~\ref{prop:R3} hold: there are $N_0\le N_1<N_2<\cdots$ and a linear $S:\cA\to\C$ with clauses
\textup{(a)}, \textup{(b)}, \textup{(c)} there; $S$ is invariant under all space translations and under the cubic group about every
point of $\bigcup_m\varepsilon_{N_m}\Z^3$; $|S(a)|\le M(a)$; and $|Z(\sigma)-Z(\sigma')|\le\|\sigma-\sigma'\|'$ for
$Z(\sigma)=S(e_\sigma)$.
\end{proposition}
\begin{proof}
We follow the proof of Proposition~\ref{prop:R3}. That proof uses the estimate \eqref{eq:E} in its Step~1 only, and there only the
second half, to obtain \eqref{eq:R31} (see also part (i) of the remark that follows that proof). Under (h4), the argument of Step~1
(the additivity of $\Phi_N$, the inequality $|e^{ix}-e^{iy}|\le|x-y|\wedge2$ and the positivity of $\om_N$) gives \eqref{eq:R31} with
$\|\cdot\|'$ in place of $C\|\cdot\|$. Steps~2--6 use only \eqref{eq:R31}, the admissibility of the seminorm, \eqref{eq:unimod} and
Lemma~\ref{lem:R1}. Indeed, Step~2 uses separability and admissibility, and the diagonal extraction of Step~3 uses $|Z_N|\le1$ and
\eqref{eq:R31}. Step~4 uses \eqref{eq:R31} in the limit, and Step~5 uses Lemma~\ref{lem:R1} on the nested lattices
$\varepsilon_{N_m}\Z$ together with Step~4. Step~6 uses \eqref{eq:R31} in the limit, applied to the second factor. None of these
steps involves a density function, a normalisation or the tori, so they apply without change. The last inequality of the statement is
\eqref{eq:R31} in the limit.
\end{proof}

\begin{theorem}[the limit functional for the extended scheme]\label{hx:thm:Hlim}
Let the fine model and the fine infinite-volume states $\om_N$, $N\ge N_0$, be as in Theorem~\ref{thm:Hlim} (any positive couplings,
any real $\delta m_B$, which may depend on $N$; limit points of the torus states in the sense of Definition~\ref{def:omN}), and let the
scheme be the extended scheme of Definition~\ref{hx:def:ext}, with a real Borel density function $g_{\rm d}$ and real
normalisations $c_N$, $c_N'$ that satisfy \eqref{hx:hyp:N} and \eqref{hx:hyp:Y}; in particular, that satisfy \eqref{hx:hyp:N} and
\eqref{hx:hyp:Y1}. Then there are
$N_0\le N_1<N_2<\cdots$ and a linear $S:\cA\to\C$ with
\begin{itemize}
\item[(a)] $\lim_{m\to\infty}\om_{N_m}(\iota_{N_m}(a))=S(a)$ for every $a\in\cA$, i.e.\ $Z_{N_m}(\sigma)\to Z(\sigma)=S(e_\sigma)$ for every
test pair $\sigma$;
\item[(b)] $S(\tau_ta)=S(a)$ for all $t\in\R$, $a\in\cA$;
\item[(c)] $t\mapsto S(\Th a\cdot\tau_tb)$ is continuous on $[0,\infty)$ for all $a,b\in\Ap$.
\end{itemize}
Moreover $S$ is invariant under all space translations and under the cubic group about every point of $\bigcup_m\varepsilon_{N_m}\Z^3$,
$|S(a)|\le M(a)$, and
\begin{equation}\label{hx:eq:modulus}
|Z(\sigma)-Z(\sigma')|\le\zeta\,R(f-f')+\mathsf p(h-h')\qquad\text{for all test pairs }\sigma=(f,h),\ \sigma'=(f',h') .
\end{equation}
\end{theorem}
The ratios $\zeta_N$, $k_N$ may have either sign, need not converge and need not be bounded away from zero. Condition \eqref{hx:hyp:Y} is verified in the cases \textup{(V1)--(V4)} of Proposition~\ref{hx:prop:pairs}.
\begin{proof}
We verify the hypotheses of Proposition~\ref{hx:prop:R3}. (h1) is Theorem~\ref{thm:d2:LP}(b); it does not involve the scheme. (h2) holds
because $\Phi_N(\sigma)$ is real (Subsection~\ref{hx:sub:hyp}). (h3): the first assertion of Lemma~\ref{lem:R1}, the invariance of
$\om_N$ on bounded Borel local functions under the lattice translations, the reflections and the cubic rotations, is a statement about
$\mu_N$ alone. Its second assertion, the intertwining of $\iota_N$ with these maps, is a relabelling of the plaquettes and sites in the
two sums that define $\Phi_N(\sigma)$; it uses only that the same function $g_{\rm d}$ sits at every site and that $c_N$, $c_N'$ do not
depend on the position, and $e^{i\Phi_N(\sigma)}$ is a bounded Borel local function for every real Borel $g_{\rm d}$. So
Lemma~\ref{lem:R1} holds verbatim for the extended scheme. (h4) is Lemma~\ref{hx:lem:E} with $\|\cdot\|'=\|\cdot\|_{\rm x}$.
Proposition~\ref{hx:prop:R3} gives the assertions, and \eqref{hx:eq:modulus} is its last inequality.
\end{proof}

\subsection{Uniform integrability, and what survives without integrability}\label{hx:sub:UI}
The proof of Theorem~\ref{hx:thm:Hlim} uses a first moment, not uniform integrability and not a second moment: the smallness needed
for (b) and (c) comes from the test function, through the admissibility of the seminorm, and not from the tails of $g_{\rm d}$. Uniform
integrability adds one statement, the approximation of the limit by limits of bounded-density schemes.
\begin{lemma}[uniform integrability and truncation]\label{hx:lem:UI}
Assume \eqref{hx:hyp:N} and
\begin{equation}\tag{UI}\label{hx:hyp:UI}
\eta(\ell):=\sup_{N\ge N_0}\om_N\Big(|k_Ng_{\rm d}(\tr_0)|\,\mathbf 1\{|k_Ng_{\rm d}(\tr_0)|>\ell\}\Big)\longrightarrow0\qquad(\ell\to\infty).
\end{equation}
For $\ell\in\{1,2,\dots\}$ let $g_{N,\ell}:=g_{\rm d}\mathbf 1\{|k_Ng_{\rm d}|\le\ell\}$,
a Borel function with $|k_Ng_{N,\ell}|\le\ell$ (bounded if $k_N\ne0$), which depends on the level
$N$ unless $k_N$ is constant, and let
$\Phi^{(\ell)}_N(\sigma):=X_N(f)+c_N'\sum_zh(z)g_{N,\ell}(\tr_z)$, $Z^{(\ell)}_N(\sigma):=\om_N(e^{i\Phi^{(\ell)}_N(\sigma)})$ be the
\emph{truncated scheme} at height $\ell$. Then:
\begin{itemize}
\item[(i)] \eqref{hx:hyp:Y1} holds, with $A_1\le\ell+\eta(\ell)$ for every $\ell$; so Theorem~\ref{hx:thm:Hlim} applies.
\item[(ii)] $|Z^{(\ell)}_N(\sigma)-Z_N(\sigma)|\le E(h)\,\eta(\ell)$ for all $N\ge N_0$, all $\sigma=(f,h)$ and all $\ell$.
\item[(iii)] There are levels $N_0\le N_1<N_2<\cdots$ and linear functionals $S$, $S_\ell$ ($\ell=1,2,\dots$) on $\cA$ such that, along
$(N_m)$, $Z_{N_m}(\sigma)\to S(e_\sigma)$ and $Z^{(\ell)}_{N_m}(\sigma)\to S_\ell(e_\sigma)$ for every $\sigma$ and $\ell$; $S$ and every
$S_\ell$ have all the properties stated in Theorem~\ref{hx:thm:Hlim}; and
\begin{gather*}
|S(e_\sigma)-S_\ell(e_\sigma)|\le E(h)\,\eta(\ell),\\
\sup_{t\ge0}\big|S(\Th a\cdot\tau_tb)-S_\ell(\Th a\cdot\tau_tb)\big|\le\big(|K_{1,+\azero}|\,\|h_1\|_\infty+|K_{2,+\azero}|\,\|h_2\|_\infty\big)\eta(\ell)
\end{gather*}
for $a=e_{\sigma_1}$, $b=e_{\sigma_2}$, $\sigma_i=(f_i,h_i)$ and compact sets $K_i\supset\operatorname{supp}h_i$.
\end{itemize}
Thus $S=\lim_{\ell\to\infty}S_\ell$ pointwise on $\cA$, with the explicit rate $E(h)\eta(\ell)$ on the basis, along one and the same sequence of levels.
For the pinned normalisations the truncated density function is the fixed function $g_{\rm d}\mathbf 1\{|g_{\rm d}|\le\ell\}$, and $S_\ell$ is a limit functional of
Theorem~\ref{thm:Hlim} as stated.
\end{lemma}
\begin{proof}
(i) $|k_Ng_{\rm d}|\le\ell+|k_Ng_{\rm d}|\mathbf 1\{|k_Ng_{\rm d}|>\ell\}$; integrate, and choose $\ell$ with $\eta(\ell)<\infty$.

(ii) By $|e^{ix}-e^{iy}|\le|x-y|\wedge2$, the positivity of $\om_N$, the translation invariance of $\mu_N$ and $|c_N'|=|k_N|\eN^3$,
\begin{align*}
|Z^{(\ell)}_N(\sigma)-Z_N(\sigma)|&\le|c_N'|\sum_z|h(z)|\,\om_N\big(|g_{\rm d}(\tr_z)|\mathbf 1\{|k_Ng_{\rm d}(\tr_z)|>\ell\}\big)\\
&=E_N(h)\,\om_N\big(|k_Ng_{\rm d}(\tr_0)|\mathbf 1\{|k_Ng_{\rm d}(\tr_0)|>\ell\}\big)\le E(h)\,\eta(\ell).
\end{align*}

(iii) The truncated scheme at height $\ell$ has a bounded density term: $|c_N'\sum_zh(z)g_{N,\ell}(\tr_z)|\le\ell E_N(h)\le\ell E(h)$
pointwise (if $k_N=0$ the term vanishes). As in Lemma~\ref{hx:lem:E},
$\om_N(|\Phi^{(\ell)}_N(\sigma)|\wedge2)\le\zeta R(f)+\ell E(h)$, an admissible seminorm free of $N$. Neither Lemma~\ref{lem:R1} nor the proof
of Proposition~\ref{hx:prop:R3} uses that the density function is the same at different levels; they use only that it is the same at all
sites of one level. Hence hypotheses (h1)--(h4) hold for each truncated scheme, and for the untruncated one by (i). In Step~3 of the proof
of Proposition~\ref{prop:R3} the diagonal extraction can be made for countably many schemes at once: there is one sequence $(N_m)$
along which $Z_{N_m}(\sigma')$ and all $Z^{(\ell)}_{N_m}(\sigma')$, $\ell=1,2,\dots$, converge for every $\sigma'$ in the countable dense
set $\mathcal D$. Steps~3--6 then apply to each scheme along this sequence and give $S$ and $S_\ell$ with the stated properties. The first
bound is (ii) in the limit. For the second, $\Th a\cdot\tau_tb=e_{-\theta\sigma_1+\tau_t\sigma_2}$ has density component
$-\theta h_1+\tau_th_2$, and
\[
E(-\theta h_1+\tau_th_2)\le E(\theta h_1)+E(\tau_th_2)\le|K_{1,+\azero}|\|h_1\|_\infty+|K_{2,+\azero}|\|h_2\|_\infty
\]
for every $t$, because reflections and translations do not change the volume of the neighbourhood.
\end{proof}

\begin{remark}[sufficient conditions]\label{hx:rem:suff}
\eqref{hx:hyp:UI} holds, with $\eta(\ell)\le C\ell^{-\delta}$, if $\sup_{N\ge N_0}\om_N(|k_Ng_{\rm d}(\tr_0)|^{1+\delta})\le C$ for some
$\delta>0$; \eqref{hx:hyp:UI} holds with $\eta(\ell)=0$ for $\ell\ge\sup_N|k_N|\,\|g_{\rm d}\|_\infty$ if $g_{\rm d}$ is bounded and
$\sup_N|k_N|<\infty$. Every property of the $S_\ell$ that is stable under pointwise limits on $\cA$ passes to $S$. For clauses
(a)--(c) Lemma~\ref{hx:lem:UI} is contained in Theorem~\ref{hx:thm:Hlim}; what \eqref{hx:hyp:UI} adds is the approximation statement (iii).
Remark~\ref{rem:density}(iv) states the first-moment weakening for the pinned normalisations; Lemmas~\ref{hx:lem:Y}, \ref{hx:lem:E} and
Theorem~\ref{hx:thm:Hlim} are its proof, in the general normalisation.
\end{remark}

\begin{remark}[no integrability at all]\label{hx:rem:nointeg}
Let $g_{\rm d}$ be any real Borel function and $c_N$, $c_N'$ any real numbers, with no condition.
(1) Since $|Z_N(\sigma)|\le1$, the family $(Z_N)_{N\ge N_0}$ lies in the product of closed unit discs indexed by $\mathcal T$, which is
compact. Hence a \emph{subnet} $(N_\alpha)$ has $Z_{N_\alpha}(\sigma)\to Z(\sigma)$ for every $\sigma$, and $S(e_\sigma):=Z(\sigma)$ extends
to a linear functional on $\cA$ with $\om_{N_\alpha}(\iota_{N_\alpha}(a))\to S(a)$ and $|S(a)|\le M(a)$. This is not clause (a), which asks
for a sequence of levels; $\mathcal T$ is uncountable, and this argument produces no subsequence.
(2) For every prescribed \emph{countable} set $\mathcal D$ of test pairs, a diagonal extraction gives a subsequence along which
$Z_N(\sigma)$ converges for all $\sigma\in\mathcal D$; Step~3 of the proof of Proposition~\ref{prop:R3} uses only $|Z_N|\le1$ for this.
(3) For the limits of (1) and of (2), Lemma~\ref{lem:R1} gives $Z(\tau_t\sigma)=Z(\sigma)$ for the times
$t\in\azero\Z[1/5]=\bigcup_N\eN\Z$ (in (2), for $\sigma\in\mathcal D$ and $\mathcal D$ invariant under these translations): for such $t$
the invariance is exact at every large level, and no estimate is used.
(4) The remaining steps are the passage from $\mathcal D$ to all test pairs, the continuity of $t\mapsto S(\tau_ta)$, hence clause (b) for
all real $t$, and clause (c). In the proof of Proposition~\ref{prop:R3} each of them uses \eqref{eq:R31} and nothing else, and what
gives \eqref{eq:R31} here is \eqref{hx:hyp:Y}. Without \eqref{hx:hyp:Y} this argument does not give these steps; we do not
know whether they can fail for some $g_{\rm d}$.
\end{remark}

\subsection{General normalisations: the pinned scheme as the case of equality}\label{hx:sub:norm}
Theorem~\ref{hx:thm:Hlim} is stated for all real $c_N$, $c_N'$ with \eqref{hx:hyp:N} and \eqref{hx:hyp:Y}. The Riemann scaling
of the curvature normalisation, which enters the proof of Proposition~\ref{prop:E} (Remark~\ref{rem:subnet}), can thus be weakened to the \emph{bound} \eqref{hx:hyp:N}; the equality
$\zeta_N=1$ is not required. We do not know whether \eqref{hx:hyp:N} is necessary. The next statement describes the limit. Its
degenerate cases (1) and (3) require $\zeta_{N_m}\to0$ for the ratio $\zeta_N=c_N\eN^{-3/2}$, a condition on the \emph{normalisation} of the curvature;
they do not occur for the pinned normalisations, where $\zeta_N=k_N=1$. Case (2) is a condition on the density term, that is, on the ratio $k_N=c_N'\eN^{-3}$ and the density function together.

\begin{corollary}[the limit for general normalisations]\label{hx:cor:norm}
Assume \eqref{hx:hyp:N} and \eqref{hx:hyp:Y}, and let $(N_m)$, $S$ be as in Theorem~\ref{hx:thm:Hlim}.
\begin{itemize}
\item[(1)] If $\zeta_{N_m}\to0$, then $S(e_{(f,h)})=S(e_{(0,h)})$ for all test pairs: the curvature drops out, and $S$ is the limit of the
density sector alone.
\item[(2)] If $\om_{N_m}(|Y_{N_m}(h)|\wedge2)\to0$ for every $h$, for instance if $k_{N_m}\om_{N_m}(|g_{\rm d}(\tr_0)|)\to0$, then
$S(e_{(f,h)})=S(e_{(f,0)})$ for all test pairs: the density drops out, and $S$ is the limit of the curvature sector at the normalisation $c_N$.
\item[(3)] If both hold, then $S(e_\sigma)=1$ for every $\sigma$, that is, $S(\sum_jc_je_{\sigma_j})=\sum_jc_j$. Clauses (a)--(c) hold,
$S(\Th a\cdot b)=S(\Th a)S(b)$, and the Hilbert space reconstructed from $S$ (Section~\ref{sec:osrest}) is one-dimensional.
\item[(4)] Assume in addition $\sup_N|k_N|<\infty$ and $A_1^{\rm pin}:=\sup_{N\ge N_0}\om_N(|g_{\rm d}(\tr_0)|)<\infty$, and let
$\Phi^{\rm pin}_N$, $Z^{\rm pin}_N$ belong to the scheme with the same $g_{\rm d}$ and the pinned normalisations. Then
$\Phi_N(f,h)=\Phi^{\rm pin}_N(\zeta_Nf,k_Nh)$ exactly. If $\zeta_{N_m}\to\zeta_\infty$ and $k_{N_m}\to k_\infty$, which holds along a
subsequence of every sequence of levels, then
\[
S(e_{(f,h)})=S^{\rm pin}\big(e_{(\zeta_\infty f,\,k_\infty h)}\big)\qquad\text{for all test pairs }(f,h),
\]
where $S^{\rm pin}$ is a limit of the pinned scheme, given by Theorem~\ref{hx:thm:Hlim}, along $(N_m)$ if $\zeta_\infty k_\infty\ne0$ and
along a subsequence of $(N_m)$ otherwise. The dilation $D(f,h)=(\zeta_\infty f,k_\infty h)$ of test pairs commutes with $\theta$ and
$\tau_t$ and preserves supports, so it induces an algebra homomorphism of $\cA$ which commutes with $\Th$ and $\tau_t$ and maps $\Ap$ into
$\Ap$, and $S=S^{\rm pin}\circ D$. The pinned normalisations are the case of equality $\zeta_N=k_N=1$, $D=\mathrm{id}$.
\end{itemize}
\end{corollary}
The properties (a)--(c) of Theorem~\ref{thm:Hlim} do not exclude case (3); the non-degeneracy of the limit is a separate question. The additional assumptions of part~(4) hold in case~\textup{(V1)} of Proposition~\ref{hx:prop:pairs}.
\begin{proof}
(1) By \eqref{hx:eq:R31} and the curvature part of the proof of Lemma~\ref{hx:lem:E},
$|Z_N(f,h)-Z_N(0,h)|\le\om_N(|X_N(f)|\wedge2)\le|\zeta_N|R(f)$; let $N=N_m\to\infty$.
(2) Likewise $|Z_N(f,h)-Z_N(f,0)|\le\om_N(|Y_N(h)|\wedge2)$, and $\om_N(|Y_N(h)|)\le|k_N|\om_N(|g_{\rm d}(\tr_0)|)E(h)$ by the proof of
Lemma~\ref{hx:lem:Y}(1).
(3) By (1) and (2), $S(e_{(f,h)})=S(e_{(0,h)})=S(e_{(0,0)})=1$. Then $S$ is the character $\sum_jc_je_{\sigma_j}\mapsto\sum_jc_j$ of the
group algebra, which is multiplicative; the sesquilinear form $(a,b)\mapsto S(\Th a\cdot b)=S(\Th a)S(b)$ has rank one.
(4) The identity holds because $\Phi_N$ is linear in the test pair and $c_N=\zeta_N\eN^{3/2}$, $c_N'=k_N\eN^3$. The pinned scheme
satisfies \eqref{hx:hyp:N} with $\zeta=1$ and \eqref{hx:hyp:Y1} with $A_1=A_1^{\rm pin}$, so by \eqref{hx:eq:R31} for it,
\[
\big|Z^{\rm pin}_N(\zeta_Nf,k_Nh)-Z^{\rm pin}_N(\zeta_\infty f,k_\infty h)\big|\le|\zeta_N-\zeta_\infty|\,R(f)+|k_N-k_\infty|\,A_1^{\rm pin}E(h)\longrightarrow0
\]
along $(N_m)$. Hence $Z^{\rm pin}_{N_m}(\zeta_\infty f,k_\infty h)\to S(e_{(f,h)})$ for all $(f,h)$. If $\zeta_\infty k_\infty\ne0$ the
dilation $D$ is onto $\mathcal T$, so $Z^{\rm pin}_{N_m}$ converges on all of $\mathcal T$, and its limit $S^{\rm pin}$ has the properties of
Theorem~\ref{hx:thm:Hlim} by Steps~3--6 of the proof of Proposition~\ref{prop:R3} along $(N_m)$. Otherwise pass to a subsequence of
$(N_m)$ along which the pinned scheme converges (Theorem~\ref{hx:thm:Hlim}, whose proof starts from any infinite set of levels). The
properties of $D$ are read off its definition; bounded ratios have convergent subsequences.
\end{proof}

\subsection{What the available bounds give}\label{hx:sub:bounds}
We write $\rho_x=\rhof+\tr_x\ge0$ for the site field, as in Section~\ref{sec:setting}, and $\rhof(N)$ when the level of the vacuum value
must be explicit: $\rhof(N)^2=\eN\rho_0^2$ decreases with $N$. (The constant $\rho_0$ is the continuum vacuum value; to avoid a clash, the
site field is always written at a generic site $x$. All states below are translation invariant, so one-site moments do not depend on $x$.)

This paper proves no moment bound for the site field that is uniform in the level and holds for every $\delta m_B$: the one constant uniform
in $N$ is that of Theorem~\ref{thm:GD} (Remark~\ref{rem:letters}). Lemma~\ref{lem:d2:mom} and Theorem~\ref{thm:d2:LP}(b) give
$\avL{e^{q\rho_x^2}}\le C_N(q)$ and $\int e^{q\rho_x^2}d\mu_N\le C_N(q)$, uniformly in the volume at a fixed level, and the constant
diverges with the level:
\begin{equation}\label{hx:eq:CNq}
C_N(q)\ \ge\ \pi^{24}\exp\big(192/(\eN e^2)\big)\qquad\text{for every }q\ge0\text{ and every }N .
\end{equation}
Indeed, in the notation of Lemma~\ref{lem:d2:mom}, $C_N(q)=(G^{\mathrm{ct}}_q/G^{\mathrm{ct}}_0)C_1^8$ with $G^{\mathrm{ct}}_q\ge G^{\mathrm{ct}}_0$ and
\[
C_1\ge G^{\mathrm{ct}}_0(2\pi/\gph)^3K_{\gph}'^3/c_0=(G^{\mathrm{ct}}_0/c^{\mathrm{ct}}_g)(\pi/c_h)^3K_{\gph}'^3e^{24/\gph^2}\ge\pi^3e^{24/\gph^2},
\]
because $G^{\mathrm{ct}}_0\ge\int_0^1g_{\mathrm{ct}}(r)\,dr\ge c^{\mathrm{ct}}_g$, $c_h\le1$ and $K'_{\gph}\ge1$; and $\gph^2=\eN e^2$.
The divergence is not only an effect of the factor $C_1^8$: at $\delta m_B=0$ the site ratio itself satisfies
$G^{\mathrm{ct}}_q/G^{\mathrm{ct}}_0\ge\pi^{-1/2}\exp\big(q^2/(4\eN\lambda)-1\big)$ for every $q>0$. Indeed, with $u=\rho^2$,
$G^{\mathrm{ct}}_q=\frac12\int_0^\infty e^{-\lamf(u-\rhof^2)^2+qu}du$; the exponent is at least $q^2/(4\lamf)-1$ on the interval of length
$\lamf^{-1/2}$ to the right of its maximum point $u=\rhof^2+q/(2\lamf)$, so $G^{\mathrm{ct}}_q\ge\frac12\lamf^{-1/2}e^{q^2/(4\lamf)-1}$, while
$G^{\mathrm{ct}}_0\le\frac12(\pi/\lamf)^{1/2}$.

\begin{proposition}[pairs $(g_{\rm d},c_N')$ that satisfy the density clause]\label{hx:prop:pairs}
Let $c_N$ satisfy \eqref{hx:hyp:N}. In each of the cases \textup{(V1)--(V3)} condition \eqref{hx:hyp:UI} holds; hence \eqref{hx:hyp:Y1},
Theorem~\ref{hx:thm:Hlim} and Lemma~\ref{hx:lem:UI} apply. In case \textup{(V4)}, \eqref{hx:hyp:Y1} holds and Theorem~\ref{hx:thm:Hlim} applies.
\begin{itemize}
\item[(V1)] \emph{Bounded on the relevant half line.} $g_{\rm d}$ is bounded on $[-\rho_{\sup},\infty)$, where $\rho_{\sup}:=\sup_{N\ge N_0}\rhof(N)$ (on $\R$ if $\rho_{\sup}=\infty$; $\rho_{\sup}=\rhof(N_0)$ for the scaling of Section~\ref{sec:setting}), in particular if it is bounded, and
$\sup_N|k_N|<\infty$; the Riemann normalisation $c_N'=\eN^3$ is included. Then $\eta(\ell)=0$ for
$\ell\ge\sup_N|k_N|\sup_{r\ge-\rho_{\sup}}|g_{\rm d}(r)|$. Examples, for $\rho_{\sup}<\infty$: $e^{-r}$, the negative part of $r$, $\min(r,1)$.
\item[(V2)] \emph{Polynomial growth, from an exponential bound.} $|g_{\rm d}(r)|\le C_g(1+|r|)^{2m}$ with an integer $m\ge1$. For each
$N\ge N_0$ let $t_N>0$ and $B_N:=C_N(t_N)$, the constant of Lemma~\ref{lem:d2:mom}, so that $\avL{e^{t_N\rho_x^2}}\le B_N$ for all sides $L\ge4\eN$. Put $M_N:=1+\rhof^{2m}+m!\,t_N^{-m}B_N$ and let $|c_N'|\le C\,\eN^3/M_N$. Then
\[
\om_N\big(|k_Ng_{\rm d}(\tr_0)|^2\big)\le C^2C_g^2\,2^{8m-2}\tbinom{2m}{m}\quad(N\ge N_0),\qquad\eta(\ell)\le C^2C_g^2\,2^{8m-2}\tbinom{2m}{m}\,\ell^{-1}.
\]
\item[(V3)] \emph{Polynomial growth, from Lemma~\ref{hx:lem:X8} below.} $|g_{\rm d}(r)|\le C_g(1+|r|)^{p}$ with a real $p\ge0$, and
$|c_N'|\le C\,\eN^3\,(1+\rhof+b_N^{1/2})^{-p}$, with the scale $b_N$ of Lemma~\ref{hx:lem:X8}. Then, with $p':=\max(1,\lceil p\rceil)$ and the
constants $c_j$ of that lemma, $\om_N(|k_Ng_{\rm d}(\tr_0)|^2)\le C^2C_g^2\,c_{p'}^{\,p/p'}$ for $N\ge N_0$, and
$\eta(\ell)\le C^2C_g^2\,c_{p'}^{\,p/p'}\ell^{-1}$.
\item[(V4)] \emph{Some normalisation always works at the first-moment level.} If $a_N:=\om_N(|g_{\rm d}(\tr_0)|)<\infty$ for every $N\ge N_0$,
which holds whenever $|g_{\rm d}(r)|\le C_ge^{qr^2}$, then \eqref{hx:hyp:Y1} holds for every $c_N'$ with $|c_N'|\le C\eN^3/\max(1,a_N)$.
Clause \eqref{hx:hyp:Y1} holds at the Riemann normalisation $c_N'=\eN^3$ if and only if $\sup_Na_N<\infty$.
\end{itemize}
\end{proposition}
\begin{proof}
(V1) $\tr_0\ge-\rhof(N)\ge-\rho_{\sup}$ on $\Omega_N$, by the definition of $\rho_{\sup}$, so $|k_Ng_{\rm d}(\tr_0)|\le\sup_N|k_N|\sup_{r\ge-\rho_{\sup}}|g_{\rm d}(r)|$ pointwise.

(V2) $B_N\ge1$ because $e^{t\rho^2}\ge1$. By convexity, $(1+|r|)^{4m}\le2^{4m-1}(1+|r|^{4m})$ and
$|\tr_0|^{4m}\le(\rho_x+\rhof)^{4m}\le2^{4m-1}(\rho_x^{4m}+\rhof^{4m})$ at the site $x=0$. From $\rho^{4m}\le(2m)!\,t^{-2m}e^{t\rho^2}$ and
Theorem~\ref{thm:d2:LP}(e), $\int\rho_x^{4m}d\mu_N\le\liminf_n\langle\rho_x^{4m}\rangle^{\mathrm{fine}}_{L_n}\le(2m)!\,t_N^{-2m}B_N$. Hence
\begin{gather*}
\om_N\big((1+|\tr_0|)^{4m}\big)\le2^{8m-2}\big(1+\rhof^{4m}+(2m)!\,t_N^{-2m}B_N\big),\\
M_N^2\ge1+\rhof^{4m}+(m!)^2t_N^{-2m}B_N^2\ge1+\rhof^{4m}+(m!)^2t_N^{-2m}B_N ,
\end{gather*}
and the quotient of the two right sides is at most $2^{8m-2}(2m)!/(m!)^2$. With $|k_N|\le C/M_N$ this is the $L^2$ bound; a family bounded
in $L^2$ by $Q$ has $\eta(\ell)\le Q^2/\ell$.

(V3) For $p=0$ the function is bounded and $|k_N|\le C$. Let $p>0$. On the probability space $(\Omega_N,\mu_N)$, first by Lyapunov's
inequality ($2p\le2p'$) and then by Minkowski's inequality in $L^{2p'}$ ($2p'\ge2$), with $|\tr_0|\le\rho_x+\rhof$ and Lemma~\ref{hx:lem:X8},
\begin{align*}
\big\|(1+|\tr_0|)^p\big\|_{L^2}&=\big\|1+|\tr_0|\big\|_{L^{2p}}^{\,p}\le\big\|1+|\tr_0|\big\|_{L^{2p'}}^{\,p}\\
&\le\big(1+\rhof+c_{p'}^{1/(2p')}b_N^{1/2}\big)^p\le c_{p'}^{\,p/(2p')}\big(1+\rhof+b_N^{1/2}\big)^p ,
\end{align*}
since $c_{p'}\ge1$. Multiply by $|k_N|C_g\le CC_g(1+\rhof+b_N^{1/2})^{-p}$.

(V4) $|k_N|a_N\le C$. If $|g_{\rm d}(r)|\le C_ge^{qr^2}$, then $|g_{\rm d}(\tr_0)|\le C_ge^{2q\rhof^2}e^{2q\rho_x^2}$ and $a_N\le
C_ge^{2q\rhof^2}C_N(2q)<\infty$ by Theorem~\ref{thm:d2:LP}(b). The last sentence is the definition of \eqref{hx:hyp:Y1} at $k_N=1$.
\end{proof}

\begin{remark}[the scope of the cases]\label{hx:rem:hyp}
Proposition~\ref{hx:prop:pairs} covers the following density functions and normalisations, and nothing is asserted outside them.
Case (V1): density functions bounded on $[-\rho_{\sup},\infty)$, at every normalisation with $\sup_N|k_N|<\infty$, the Riemann normalisation
$c_N'=\eN^3$ included. Cases (V2) and (V3): polynomial growth, at the normalisations stated there, which compare with the Riemann normalisation as follows.
In (V2), by \eqref{hx:eq:CNq}, the normalisation is smaller than the Riemann one by at least
the factor $(t_N^m/m!)\,\pi^{-24}\exp(-192/(\eN e^2))$. In (V3), $b_N\ge(2\eN\lambda)^{-1/2}$, so that it is smaller by at least the factor
$(2\eN\lambda)^{p/4}$; under the additional condition $\sup_N(\delta m_B)_-\lamf^{-1/2}<\infty$ on the counterterm, $b_N\le C'\eN^{-1/2}$
and $c_N'=\eN^{3+p/4}$ is allowed in (V3). Case (V4): density functions with $a_N<\infty$ at every level, in particular those of growth at most
$C_ge^{qr^2}$, at the normalisations $|c_N'|\le C\eN^3/\max(1,a_N)$. For a density function that is unbounded on $[-\rho_{\sup},\infty)$ the cases
do not give the Riemann normalisation for every value of the counterterm $\delta m_B(N)$ (Remark~\ref{hx:rem:counter}): for $g_{\rm d}=\mathrm{id}$
or a polynomial of degree $p$, clause \eqref{hx:hyp:Y1} at the Riemann normalisation is the level-uniform moment bound $\sup_N\om_N(|\tr_0|^p)<\infty$.
In the cases (V2) and (V3) Theorem~\ref{hx:thm:Hlim} holds, but
Proposition~\ref{hx:prop:pairs} does not assert that the density term survives in the limit; this question is treated in
Remark~\ref{hx:rem:dropout}.
\end{remark}

\begin{lemma}[all moments of the site field at the scale $b_N$]\label{hx:lem:X8}
For every level $N$, every torus side $L$, every real $\delta m_B$, every $\kappa_N\ge0$, every site $x$ and every integer $j\ge1$,
\begin{gather*}
\avL{\rho_x^{2j}}\le c_j\,b_N^{\,j},\qquad b_N:=\rhof^2+\frac{(\delta m_B)_-}{4\lamf}+\frac1{(2\lamf)^{1/2}},\\
c_1=c_2=1,\qquad c_j=c_{j-1}+(j-1)c_{j-2}\ \ (j\ge3),
\end{gather*}
where $(\delta m_B)_-:=\max(0,-\delta m_B)$; and $\int\rho_x^{2j}d\mu_N\le c_jb_N^{\,j}$. In particular $c_3=3$, $c_4=6$, and $c_j\ge1$.
\end{lemma}
\begin{proof}
Fix $N$, $L$ and $x$, write $\langle\cdot\rangle=\avL\cdot$ and $m_i:=\langle\rho_x^i\rangle$, which is finite for every $i\ge0$
(Definition~\ref{def:gin:weights}: every function dominated by $Ce^{\sum_yq_y\rho_y^2}$ is integrable) and does not depend on $x$.
As a function of $\rho_x\in[0,\infty)$, with all other variables fixed, the weight is
\begin{gather*}
W=\rho_x\,e^{-U(\rho_x)}\prod_{b\ni x}\exp\Big(-\tfrac12\kappa_N\big(\rho_x^2+\rho_{y(b)}^2-2\rho_x\rho_{y(b)}\cos\vartheta_b\big)\Big)\times(\text{a factor free of }\rho_x),\\
U(\rho):=\lamf(\rho^2-\rhof^2)^2+\tfrac12\delta m_B\rho^2,
\end{gather*}
the product over the six bonds $b$ at $x$, with $y(b)$ the other end of $b$ and $\vartheta_b=\gph B_b$; the Villain sum and the integrals
over the bond variables do not involve $\rho_x$. Let $k\ge1$. The function $\rho_x^kW$ vanishes at $\rho_x=0$ and decays at infinity like
$e^{-\lamf\rho_x^4}$ times a polynomial, so $\int_0^\infty\partial_{\rho_x}[\rho_x^kW]\,d\rho_x=0$, and
\[
\partial_{\rho_x}\big[\rho_x^kW\big]=\Big((k+1)\rho_x^{k-1}-\rho_x^kU'(\rho_x)-\kappa_N\sum_{b\ni x}\big(\rho_x^{k+1}-\rho_x^k\rho_{y(b)}\cos\vartheta_b\big)\Big)W .
\]
Each term on the right is a polynomial in the site fields times a bounded function times $W$, hence integrable over all variables
(Definition~\ref{def:gin:weights} and the inequality \eqref{eq:ct}); by Fubini's theorem we may integrate over the other
variables and divide by the mass of $W$:
\begin{equation}\label{hx:eq:SD}
\big\langle\rho_x^kU'(\rho_x)\big\rangle=(k+1)\,m_{k-1}-\kappa_N\sum_{b\ni x}\big\langle\rho_x^{k+1}-\rho_x^k\rho_{y(b)}\cos\vartheta_b\big\rangle .
\end{equation}

\emph{The hopping sum is non-negative.} Since $\rho\ge0$ and $|\cos|\le1$, H\"older's inequality with the exponents $(k+1)/k$ and $k+1$ and
the translation invariance of the torus state give
\[
\langle\rho_x^k\rho_y\cos\vartheta_b\rangle\le\langle\rho_x^{k+1}\rangle^{k/(k+1)}\langle\rho_y^{k+1}\rangle^{1/(k+1)}=\langle\rho_x^{k+1}\rangle .
\]

\emph{The recursion.} $U'(\rho)=4\lamf\rho(\rho^2-\rhof^2)+\delta m_B\rho$, so
\[
\langle\rho_x^kU'(\rho_x)\rangle=4\lamf m_{k+3}-(4\lamf\rhof^2-\delta m_B)m_{k+1}\ge4\lamf m_{k+3}-\beta m_{k+1}
\]
with
$\beta:=4\lamf\rhof^2+(\delta m_B)_-$. With \eqref{hx:eq:SD} and $\kappa_N\ge0$,
\begin{equation}\label{hx:eq:rec}
4\lamf\,m_{k+3}\le(k+1)\,m_{k-1}+\beta\,m_{k+1}\qquad(k\ge1).
\end{equation}

\emph{$j=1,2$.} For $k=1$, with $m_0=1$ and $m_2\le m_4^{1/2}=:s$, \eqref{hx:eq:rec} reads $4\lamf s^2-\beta s-2\le0$, hence
\[
s\le\frac{\beta+(\beta^2+32\lamf)^{1/2}}{8\lamf}\le\frac{\beta}{4\lamf}+\frac{(32\lamf)^{1/2}}{8\lamf}=\rhof^2+\frac{(\delta m_B)_-}{4\lamf}+\frac1{(2\lamf)^{1/2}}=b_N ,
\]
that is, $m_4\le b_N^2$ and $m_2\le b_N$.

\emph{$j\ge3$.} Take $k=2j-3$ in \eqref{hx:eq:rec} and use induction:
\begin{align*}
m_{2j}&\le\frac{j-1}{2\lamf}\,m_{2(j-2)}+\frac{\beta}{4\lamf}\,m_{2(j-1)}\le\frac{j-1}{2\lamf}\,c_{j-2}b_N^{\,j-2}+\frac{\beta}{4\lamf}\,c_{j-1}b_N^{\,j-1}\\
&\le\big((j-1)c_{j-2}+c_{j-1}\big)b_N^{\,j},
\end{align*}
because $1/(2\lamf)\le b_N^2$ and $\beta/(4\lamf)\le b_N$.

\emph{The limit state.} $\rho_x^{2j}$ is a continuous local function $\ge0$,
so $\int\rho_x^{2j}d\mu_N\le\liminf_n\langle\rho_x^{2j}\rangle^{\mathrm{fine}}_{L_n}\le c_jb_N^{\,j}$ by Theorem~\ref{thm:d2:LP}(e).
\end{proof}
The scale $b_N$ is polynomially bounded in $\eN^{-1}$ when $(\delta m_B)_-$ stays bounded, against the exponential of \eqref{hx:eq:CNq}, but it is not uniform in the level:
$b_N\ge(2\eN\lambda)^{-1/2}$. The bound uses the hopping term only through its sign.

\begin{remark}[a level-uniform moment bound requires information on the counterterm]\label{hx:rem:counter}
Theorems~\ref{thm:Hlim}, \ref{thm:d2:LP} and \ref{hx:thm:Hlim} hold for every real $\delta m_B$, with any dependence on $N$. For some such
choices the level-uniform moments fail; so they cannot follow from statements that hold for every $\delta m_B$. Let
$\delta m_B(N)=-c$ for all $N$, with a fixed $c>0$, and $\kappa_N=1$. Then, uniformly in the side $L$,
\begin{align}
\avL{\rho_x^2}&\ \ge\ \frac{3c}{16\,\lambda\,\eN}-\frac6c\log\frac1{\eN\lambda}-C',\label{hx:eq:c2}\\
\avL{\rho_x}&\ \ge\ \frac{\big(\avL{\rho_x^2}\big)^{3/2}}{b_N}\ \ge\ \frac{3\sqrt3}{16}\Big(\frac{c}{\lambda\,\eN}\Big)^{1/2}\big(1-o(1)\big)\qquad(N\to\infty),\label{hx:eq:c1}
\end{align}
with $C'$ depending on $c,\lambda,e,\rho_0,\azero$ only and $3\sqrt3/16>0.32$. By Theorem~\ref{thm:d2:LP}(a) (both functions are continuous,
local and dominated by $e^{\rho_x^2}$) the same bounds hold for $\om_N(\rho_x^2)$ and $\om_N(\rho_x)$. Hence the second \emph{and the
first} moment of the site variable diverge, $\om_N(|\tr_0|)\ge\om_N(\rho_x)-\rhof\to\infty$, and \eqref{hx:hyp:Y1} fails for
$g_{\rm d}=\mathrm{id}$ at the Riemann normalisation $c_N'=\eN^3$: there it is a genuine condition. Moreover, since
$\avL{\rho_x^2}$ is non-increasing in $\delta m_B$ (below), \eqref{hx:eq:c2} holds whenever $\delta m_B(N)\le-c$; so a level-uniform
second moment forces $(\delta m_B)_-(N)\to0$.
\end{remark}

\begin{proof}[Proof of \eqref{hx:eq:c2} and \eqref{hx:eq:c1}]
Fix $N$ and $L$; the torus has $n:=(L/\eN)^3$ sites, $3n$ bonds and $3n$ plaquettes. For $d\in\R$ let $Z_L(d)$ be the mass of the weight
$W$ with $\delta m_B=d$, $\langle\cdot\rangle_{L,d}$ its state, and $F(d):=n^{-1}\log Z_L(d)$.

\emph{Convexity.} By the quartic domination \eqref{eq:ct} one may differentiate under the integral: $F'(d)=-\frac12\langle\rho_x^2\rangle_{L,d}$
(translation invariance) and $F''(d)=(4n)^{-1}\,\mathrm{Var}_{L,d}\big(\sum_y\rho_y^2\big)\ge0$, the variance being taken under $\langle\cdot\rangle_{L,d}$. So
$d\mapsto\langle\rho_x^2\rangle_{L,d}$ is non-increasing, and
\[
F(-c)-F(-c/2)=\tfrac12\int_{-c}^{-c/2}\langle\rho_x^2\rangle_{L,s}\,ds\le\tfrac c4\,\langle\rho_x^2\rangle_{L,-c}.
\]
\emph{Upper bound for $F(-c/2)$.} As in Step~1 of the proof of Lemma~\ref{lem:d2:mom}: the hopping factors are at most $1$; the sum over the
closed $v$ is at most the sum over all $v$, which is at most $K_{\gph}'^{\,3n}$; each bond variable ranges over an interval of length
$2\pi/\gph$; and, with $u=\rho^2$ and a completed square, the site integral at $d=-c/2$ is
\begin{align*}
\int_0^\infty\rho\,e^{-\lamf(\rho^2-\rhof^2)^2+\frac c4\rho^2}d\rho
&=\tfrac12e^{\frac c4\rhof^2+\frac{c^2}{64\lamf}}\int_0^\infty e^{-\lamf(u-\rhof^2-\frac{c}{8\lamf})^2}du\\
&\le\tfrac12\Big(\frac\pi{\lamf}\Big)^{1/2}e^{\frac c4\rhof^2+\frac{c^2}{64\lamf}} .
\end{align*}
Hence
\[
F(-c/2)\le\frac{c^2}{64\lamf}+\frac c4\rhof^2+\frac12\log\frac{\pi}{4\lamf}+3\log\frac{2\pi}{\gph}+3\log K'_{\gph} .
\]

\emph{Lower bound for $F(-c)$.} All terms of the sum over $v$ are $\ge0$; keep $v=0$ and restrict the integral to $\rho_y\in[R,R+1]$ for all
sites, $R^2:=\rhof^2+c/(4\lamf)$, and $|B_b|\le1/(\gph(R+1))$ for all bonds (a subset of the domain). On this set the site factor at
$d=-c$ is
\[
\rho\,e^{-\lamf(\rho^2-\rhof^2)^2+\frac c2\rho^2}=\rho\,e^{-\lamf(\rho^2-R^2)^2+\frac{c^2}{16\lamf}+\frac c2\rhof^2}
\ge R\,e^{\frac{c^2}{16\lamf}-8\lamf\rhof^2-2c-2\lamf},
\]
because $0\le\rho^2-R^2\le2R+1$ and $\lamf(2R+1)^2\le\lamf(8R^2+2)$; the hopping factor is at least $e^{-\kappa_N}=e^{-1}$, because
$|e^{i\gph B}\rho'-\rho|^2=(\rho'-\rho)^2+2\rho\rho'(1-\cos\gph B)\le1+(R+1)^2\gph^2B^2\le2$; and the plaquette factor is at least
$e^{-8/(\gph(R+1))^2}$, because $|(dB)_p|\le4/(\gph(R+1))$. The set has volume $1$ per site and $2/(\gph(R+1))$ per bond. Hence
\[
F(-c)\ge\frac{c^2}{16\lamf}+\log R-8\lamf\rhof^2-2c-2\lamf-3+3\log\frac{2}{\gph(R+1)}-\frac{24}{\gph^2(R+1)^2}.
\]

\emph{The difference.} The powers of $\gph$ cancel, $3\log\frac{2}{\gph(R+1)}-3\log\frac{2\pi}{\gph}=-3\log(\pi(R+1))$, and
$\gph^2R^2\ge\gph^2c/(4\lamf)=e^2c/(4\lambda)$. Therefore
\begin{align*}
F(-c)-F(-c/2)\ge\frac{3c^2}{64\lamf}-\Big[&3\log(\pi(R+1))-\log R+\tfrac12\log\frac{\pi}{4\lamf}+\big(\tfrac c4+8\lamf\big)\rhof^2\\
&+2c+2\lamf+3+3\log K'_{\gph}+\frac{96\lambda}{e^2c}\Big].
\end{align*}
In the bracket,
\[
3\log(R+1)-\log R=\log(\rhof^2+\frac{c}{4\lamf})+3\log(1+\frac1R)=\log\frac1{\lamf}+\log(\frac c4+\lamf\rhof^2)+3\log(1+\frac1R),
\]
and $\lamf\le\azero\lambda$, $\rhof^2\le\azero\rho_0^2$, $1/R\le(4\azero\lambda/c)^{1/2}$, while $K'_{\gph}$ is bounded uniformly in $N$ by the
bound given in Lemma~\ref{lem:d2:mom}, which increases with $\gph\le\azero^{1/2}e$. So the bracket is at most
$\frac32\log\frac1{\lamf}+C''$ with $C''=C''(c,\lambda,e,\rho_0,\azero)$, and with the convexity step,
\[
\langle\rho_x^2\rangle_{L,-c}\ge\frac4c\Big(\frac{3c^2}{64\lamf}-\frac32\log\frac1{\lamf}-C''\Big)=\frac{3c}{16\lamf}-\frac6c\log\frac1{\lamf}-\frac{4C''}{c},
\]
which is \eqref{hx:eq:c2} ($\lamf=\eN\lambda$).

\emph{The first moment.} By H\"older's inequality with the exponents $3/2$ and $3$,
$\langle\rho_x^2\rangle\le\langle\rho_x\rangle^{2/3}\langle\rho_x^4\rangle^{1/3}$, so
$\langle\rho_x\rangle\ge\langle\rho_x^2\rangle^{3/2}\langle\rho_x^4\rangle^{-1/2}\ge\langle\rho_x^2\rangle^{3/2}/b_N$ by
Lemma~\ref{hx:lem:X8}. At $\delta m_B=-c$, $b_N=\frac{c}{4\lamf}(1+o(1))$ and $\langle\rho_x^2\rangle_{L,-c}\ge\frac{3c}{16\lamf}(1-o(1))$,
which gives $(\frac{3c}{16\lamf})^{3/2}\frac{4\lamf}{c}(1-o(1))=\frac{3\sqrt3}{16}(c/\lamf)^{1/2}(1-o(1))$, uniformly in $L$.
\end{proof}

\begin{remark}[in the reduced normalisations the density can drop out]\label{hx:rem:dropout}
Let $g_{\rm d}$, $m$, $t_N$, $B_N$, $M_N$ and $c_N'$ be as in Proposition~\ref{hx:prop:pairs}(V2). By the convexity inequalities of its
proof and Lemma~\ref{hx:lem:X8}, $\om_N(|g_{\rm d}(\tr_0)|)\le C_g2^{4m-2}(1+\rhof^{2m}+c_mb_N^{\,m})$. By \eqref{hx:eq:CNq},
$|k_N|\le Ct_N^m\pi^{-24}\exp(-192/(\eN e^2))/m!$. So, as soon as
$t_N^m(1+b_N^{\,m})\exp(-192/(\eN e^2))\to0$, Corollary~\ref{hx:cor:norm}(2) applies: $S(e_{(f,h)})=S(e_{(f,0)})$, and the density drops out of $S$.

In (V3) the situation is different: for $g_{\rm d}(r)=r^2$, $c_N'=\eN^3(1+\rhof+b_N^{1/2})^{-2}$
and $\delta m_B(N)=-c$, the normalised mean satisfies $\liminf_Nk_N\om_N(\tr_0^2)\ge\frac34$ by \eqref{hx:eq:c2} and
$b_N=\frac{c}{4\lamf}(1+o(1))$, so the sufficient condition $k_N\om_N(|g_{\rm d}(\tr_0)|)\to0$ of
Corollary~\ref{hx:cor:norm}(2) fails, and so does its hypothesis: for $h\ge0$, $h\ne0$, the variable $Y_N(h)\ge0$ has mean at least
$(\frac34-o(1))E_N(h)$ and $L^2(\mu_N)$ norm at most $E(h)$ (Minkowski's inequality and Lemma~\ref{hx:lem:X8}), so
$\om_N(Y_N(h)\wedge2)$ stays bounded below by the Paley--Zygmund inequality. We do not know whether the limit density is more than a constant
field in this case.
\end{remark}

\subsection{Results that hold for the extended class}\label{hx:sub:scope}
\begin{remark}[results that extend]\label{hx:rem:scope}
The \emph{extended class} is the extended scheme of Definition~\ref{hx:def:ext} with \eqref{hx:hyp:N} and \eqref{hx:hyp:Y}. The
following three lists sort results of this paper according to their relation to this class: list (A) contains results that hold for the
extended class, list (B) results in which the density function does not enter, and list (C) results that are stated for bounded density
functions only.

\textup{(A)} \emph{Results that extend.} The following results hold for the extended class, in the form indicated. They use of the scheme
only clauses (a)--(c) or the inequality \eqref{eq:R31}, the unimodularity \eqref{eq:unimod}, and the fact that the density function and
the normalisations do not depend on the position.
\begin{itemize}
\item[(A1)] Theorem~\ref{thm:Hlim} holds in the form of Theorem~\ref{hx:thm:Hlim}, Proposition~\ref{prop:R3} in the form of
Proposition~\ref{hx:prop:R3}, and Lemma~\ref{lem:R1} verbatim.
\item[(A2)] Corollary~\ref{cor:d2:consumers}(i), which restates the chain from the
estimate to Theorem~\ref{thm:Hlim} for the limit points, extends by way of Theorem~\ref{hx:thm:Hlim}.
\item[(A3)] Part (i) of Theorem~\ref{thm:osrest:OS0}, the continuity of $Z$, holds with the seminorm of \eqref{hx:eq:modulus}.
\item[(A4)] The first assertion of Proposition~\ref{prop:osrest:OS3}(b), the evenness $Z(-f,h)=Z(f,h)$ under charge conjugation,
extends: in its proof the invariance of $\mu_N$ is applied to the function $e^{i\Phi_N(\sigma)}$, which is bounded and Borel for every
real Borel $g_{\rm d}$.
\item[(A5)] The inheritance of reflection positivity by $S$ (Proposition~\ref{prop:osrest:OS2}) extends. It follows from clause (a) and
the reflection positivity of $\om_N$ on the bounded Borel functions of the closed half space (Theorem~\ref{thm:d2:LP}(d)), because for a
test pair supported in $\{x_3>0\}$ the sites and plaquettes that enter $\Phi_N(\sigma)$ lie in the closed half space, whatever
$g_{\rm d}$ is. For the other coordinate planes the proof of Proposition~\ref{prop:osrest:OS2} uses Proposition~\ref{prop:osrest:OS1}, which extends as well: its proof uses of the scheme only the invariances of $S$ stated in Theorem~\ref{hx:thm:Hlim}.
\end{itemize}

\textup{(B)} \emph{Results that do not involve the density function.} The following results concern the curvature sector $Z(f,0)$ or
the fine state alone: Theorem~\ref{thm:GD}; parts (ii), (iii) of Theorem~\ref{thm:osrest:OS0}; Theorem~\ref{thm:osrest:measure};
Theorem~\ref{thm:osrest:OS4weak}; Sections~\ref{sec:gin}, \ref{sec:d2}, \ref{rv:sec} and~\ref{n1:sec};
Theorem~\ref{flh:thm:FLH1}, which is stated for $g_{\rm d}=0$; Theorem~\ref{flh:thm:I6}; and Theorem~\ref{i7:thm:main}, which is an
abstract statement. Those that involve the scheme are stated at the pinned value of $c_N$. For a general $c_N$ with \eqref{hx:hyp:N} one
has
\[
\Phi_N(f,0)=\Phi^{\rm pin}_N(\zeta_Nf,0)\qquad\text{and}\qquad
|Z^{\rm pin}_N(\zeta_Nf,0)-Z^{\rm pin}_N(\zeta_\infty f,0)|\le|\zeta_N-\zeta_\infty|R(f),
\]
with no condition on the density. Hence, along a sequence of levels with $\zeta_{N_m}\to\zeta_\infty$, the curvature sector of the limit is that
of the pinned scheme composed with $f\mapsto\zeta_\infty f$, and these results apply to it in that form.

\textup{(C)} \emph{Results that keep the bounded class.} The following results use $\|g_{\rm d}\|_\infty$ through exponential or second
moments of the density component of the joint field.
\begin{itemize}
\item[(C1)] The bounds \eqref{rot:eq:GDall} of Lemma~\ref{rot:lem:GDall} (Subsection~\ref{rot:sec:main}) and
Remark~\ref{rot:rem:gd0} belong to this list.
\item[(C2)] In Section~\ref{flh:sec:main}, on the killed-box family, the list contains Lemma~\ref{flh:lem:I5}; hypothesis (i) for the
joint field in the proof of Proposition~\ref{flh:prop:DR}, whereas the proposition itself concerns the curvature Schwinger functions and
belongs to list (B); Theorem~\ref{flh:thm:FLH1p}, with the joint Schwinger functions and the measure constructed in its proof; and the
part of Theorem~\ref{flh:thm:FLH2} that rests on it.
\end{itemize}
For the results of list (C) the natural substitute for boundedness is a level-uniform exponential moment of $k_Ng_{\rm d}(\tr_0)$: by
Jensen's inequality with the weights $\eN^3|h(z)|/E_N(h)$ and the translation invariance of $\mu_N$,
\[
\om_N\big(e^{t|Y_N(h)|}\big)\le\om_N\big(e^{tE(h)\,|k_Ng_{\rm d}(\tr_0)|}\big)\qquad(t\ge0).
\]
We do not pursue this, and none of the results of list (C) is asserted for unbounded $g_{\rm d}$.
\end{remark}

\providecommand{\cS}{\mathcal S}\providecommand{\cHe}{\mathcal H}\providecommand{\cB}{\mathcal B}\providecommand{\E}{\mathbb E}
\providecommand{\supp}{\operatorname{supp}}\providecommand{\Oh}{O_h}\providecommand{\cZ}{\mathcal Z}\providecommand{\wnorm}[1]{\|#1\|_{w}}
\providecommand{\Hinf}{\mathcal H_\infty}\providecommand{\POm}{P_\Omega}\providecommand{\spec}{\operatorname{spec}}

\section{The remaining Osterwalder--Schrader axioms for the limit state}\label{sec:osrest}

Throughout this section $S$ is a limit functional of Theorem~\ref{thm:Hlim} along its sequence $(N_m)$, $Z(\sigma)=S(e_\sigma)$ for test
pairs $\sigma=(f,h)$, and $\cZ(f):=Z(f,0)$ is the curvature sector ($f$ a real $C^\infty_c$ $2$-form); $\Phi_N(f):=\Phi_N(f,0)=
\eN^{3/2}\sum_pf_{\mu\nu(p)}(x_p)\ff_p$, extended complex-linearly to complex $f$. The fine state $\om_N$ at level $N$ is the state \eqref{eq:omN}
of the limit point of Definition~\ref{def:omN}, with its sequence of sides $(L_n)=(L^{(N)}_n)$ and its measure $\mu_N$ on $\Omega_N$, and with
the properties (a)--(e) of Theorem~\ref{thm:d2:LP}; every limit ``$L\to\infty$'' of torus expectations at level $N$ below is the limit $n\to\infty$
along $(L_n)$, and every bound stated ``for all large $L$'' is used along $(L_n)$ only. The rotation group is not treated in this section. Invariance under $SO(3)$ is
proved in Sections~\ref{flh:sec:main}--\ref{sec:i7:proof}: for the limits built on the killed-box states by Theorems~\ref{flh:thm:FLH1}
and~\ref{flh:thm:FLH1p}, and for the limits built on the torus limit points along dyadic sequences (Definition~\ref{n1:def:sides}) by Theorem~\ref{flh:thm:FLH2}.
Once that invariance is in hand, the measure $\mu$ of Theorem~\ref{thm:osrest:measure} is invariant under the full orthogonal group and reflection positive about every plane,
and its reconstruction in any direction is unitarily equivalent to the one in a coordinate direction (Remark~\ref{rem:osrest:rot}).

We write $\wnorm f:=\sup_x(1+|x|^2)^2\max_{\mu<\nu}
|f_{\mu\nu}(x)|$ for a $2$-form and $\wnorm h:=\sup_x(1+|x|^2)^2|h(x)|$ for a function. These are continuous seminorms on the Schwartz spaces
$\cS(\R^3;\Lambda^2)$ and $\cS(\R^3)$, dominated with constant $1$ by the weighted $C^4$ norm $|q|_4=\sup_{x,\,|\gamma|\le4}(1+|x|^2)^2|D^\gamma q(x)|$ of
Osterwalder and Schrader \cite{OS75} (Definition~\ref{ms:conv0}(a)), which we use throughout: $\wnorm\cdot$ is its derivative-free term. For a $2$-form $u$ we write $|u|_4:=(\sum_{\mu<\nu}|u_{\mu\nu}|_4^2)^{1/2}$, as in that definition.

\subsection{Regularity (OS0)}

\begin{lemma}[the seminorms on Schwartz space]\label{lem:osrest:riemann}
For every $f\in\cS(\R^3;\Lambda^2)$, $h\in\cS(\R^3)$ and $N\ge N_0$,
\[
\eN^3\sum_pf_{\mu\nu(p)}(x_p)^2\le3c_{\azero}\wnorm f^2,\qquad\eN^3\sum_z|h(z)|\le c_{\azero}\wnorm h,\qquad c_{\azero}:=\pi^2\big(2+\tfrac32\azero^2\big)^2 .
\]
Hence $R$ and $E$ of Definition~\ref{def:norms}, defined by the same formulas, are finite seminorms on the Schwartz spaces with
$R(f)\le(3c_{\azero})^{1/2}\wnorm f$, $E(h)\le c_{\azero}\wnorm h$, and $\|\cdot\|$ is continuous for the Schwartz topology.
\end{lemma}
\begin{proof}
The centres of the plaquettes of one orientation form a translate of $\eN\Z^3$; let $Q_p$ be the cube of side $\eN$ centred at $x_p$ (disjoint
for distinct $p$ of one orientation). For $y\in Q_p$, $|y|\le|x_p|+\tfrac{\sqrt3}2\eN$, so $1+|y|^2\le(1+|x_p|^2)(2+\tfrac32\azero^2)$ and
$\eN^3(1+|x_p|^2)^{-2}\le(2+\tfrac32\azero^2)^2\int_{Q_p}(1+|y|^2)^{-2}dy$. Summing over one orientation and using $\int_{\R^3}(1+|y|^2)^{-2}dy=\pi^2$
gives, with $f_{\mu\nu}(x)^2\le\wnorm f^2(1+|x|^2)^{-4}\le\wnorm f^2(1+|x|^2)^{-2}$,
$\eN^3\sum_pf_{\mu\nu(p)}(x_p)^2\le\wnorm f^2\sum_{\mu<\nu}\sum_p\eN^3(1+|x_p|^2)^{-2}\le3c_{\azero}\wnorm f^2$. The site sum is estimated in the same
way, with one lattice in place of the three orientations.
\end{proof}

\begin{lemma}[the dual representation at complex sources]\label{lem:osrest:dualC}
The identity \eqref{eq:dual} holds for every complex plaquette function $a$, with $\|a\|^2:=\sum_pa_p^2$ (no conjugation). In particular,
for every real plaquette function $b$,
\[
\Big\langle\exp\Big(i\sum_pb_p\ff_p\Big)\Big\rangle^{\mathrm{fine}}_L=\frac{e^{-\|b\|^2/2}}{Z(0)}\int_{[0,1]^{\mathcal C}}dt\sum_{m\in\Z^{\mathcal P}}
\Big(\prod_pc_{m_p-(d^*t)_p}\Big)\Omega(m)\,e^{\gph(m-d^*t)\cdot b}\ \in[0,1].
\]
\end{lemma}
\begin{proof}
Both sides of \eqref{eq:dual} are entire functions of $a\in\C^{\mathcal P}$: the left side because $|e^{\sum a_p\ff_p}|=e^{\sum(\Re a_p)\ff_p}$ is
integrable against $W$ locally uniformly in $a$ (the bounds of Step~4 of the proof of Theorem~\ref{thm:GD}), the right side because
$|e^{-i\gph(m-s)\cdot a}|=e^{\gph(m-s)\cdot\Im a}$ is controlled by the completed square $c_{m-s}e^{\gph(m-s)y}=\frac{\gph}{\sqrt{2\pi}}e^{y^2/2}e^{-(\gph(m-s)-y)^2/2}$
and $\sum_{m\in\Z}e^{-(\gph m-x)^2/2}\le1+\sqrt{2\pi}/\gph$ ($x\in\R$), which give $\sum_{m\in\Z}c_{m-s}e^{\gph|m-s|\,|y|}\le2e^{y^2/2}(1+\gph/\sqrt{2\pi})$ for every
real $s$ and locally uniformly in $y$ (here $s=d^*t$, $|s_p|\le2$); so the $m$-series converges absolutely and locally uniformly in $a$, while
$0\le\Omega(m)\le(2\pi/\gph)^{|\mathcal B|}(\int g_{\mathrm{ct}})^{|\mathcal X|}$ and the $t$-integral runs over a compact cube. No uniformity in $L$ is needed, since the lemma
is a statement at fixed $N$ and $L$. The two sides agree for real $a$ (Theorem~\ref{thm:GD}), hence
everywhere. At $a=ib$ every factor of the right side is non-negative, and the left side has modulus at most $Z(0)$.
\end{proof}

\begin{lemma}[analytic generating functions and their limits]\label{lem:osrest:entire}
Let $f=(f_1,\dots,f_k)$ be real $C^\infty_c$ $2$-forms, $N\ge N_0$, $L$ large enough for their supports to embed in $\tilde T_L$, and for
$z\in\C^k$ put $\Psi_{N,L}(z):=\avL{\exp(\sum_jz_j\Phi_N(f_j))}$ and $\rho(z):=R(\sum_j(\Re z_j)f_j)\le\sum_j|\Re z_j|R(f_j)$.
\begin{itemize}
\item[(a)] $\Psi_{N,L}$ is entire, $|\Psi_{N,L}(z)|\le\Psi_{N,L}(\Re z)\le e^{\rho(z)^2/2}$, $\Psi_{N,L}(-z)=\Psi_{N,L}(z)$,
$\Psi_{N,L}(x)\ge1$ for real $x$, and $0\le\Psi_{N,L}(it)\le1$ for real $t$.
\item[(b)] As $n\to\infty$, $\Psi_{N,L_n}\to\Psi_N$ locally uniformly on $\C^k$; $\Psi_N$ is entire, satisfies the bounds and symmetries of (a),
and $\Psi_N(it)=Z_N(\sum_jt_jf_j,0)$ for real $t$. The laws $\mu_{N,L_n}$ of $(\Phi_N(f_j))_{j\le k}$ under the torus states converge weakly to
the law $\mu^{(f)}_N$ of $(\Phi_N(f_j))_{j\le k}$ under $\mu_N$, a law on $\R^k$ with $\int e^{z\cdot x}d\mu^{(f)}_N(x)=\Psi_N(z)$ for all $z\in\C^k$.
\item[(c)] As $m\to\infty$, $\Psi_{N_m}\to\Psi_f$ locally uniformly on $\C^k$; $\Psi_f$ is entire, satisfies the bounds and symmetries of (a),
$\Psi_f(it)=\cZ(\sum_jt_jf_j)$, and $\mu^{(f)}_{N_m}\to\mu_f$ weakly, where $\mu_f$ is a symmetric law on $\R^k$ with $\int e^{z\cdot x}d\mu_f=\Psi_f(z)$.
\item[(d)] If $f'$ is a subfamily of $f$, $\Psi_{f'}$ is the restriction of $\Psi_f$ to the coordinates of $f'$ (the other coefficients set
to $0$); if $f_1=g+g'$, then $\Psi_{(f_1,f_2,\dots)}(z_1,z_2,\dots)=\Psi_{(g,g',f_2,\dots)}(z_1,z_1,z_2,\dots)$.
\end{itemize}
\end{lemma}
\begin{proof}
(a) For $s\in[0,\infty)^k$, 
\[
\avL{e^{\sum_js_j|\Phi_N(f_j)|}}\le\sum_{\epsilon\in\{\pm1\}^k}\avL{e^{\Phi_N(\sum_j\epsilon_js_jf_j)}}\le2^k \max_\epsilon e^{R(\sum_j\epsilon_js_jf_j)^2/2}<\infty
\]
by additivity of $\Phi_N$ and \eqref{eq:GD} with $a_p=\eN^{3/2}\sum_j\epsilon_js_jf_j(x_p)$;
so $\Psi_{N,L}$ is a finite-dimensional integral, together with a countable absolutely convergent $v$-sum, of an entire integrand with a locally
uniform integrable majorant, hence entire. For the bound,
$|e^{\sum z_j\Phi_N(f_j)}|=e^{\Phi_N(g)}$ with $g=\sum_j(\Re z_j)f_j$, and \eqref{eq:GD} gives $\avL{e^{\Phi_N(g)}}\le e^{\eN^3\sum_pg(x_p)^2/2}
\le e^{R(g)^2/2}$. For the evenness, the map $(B,v)\mapsto(-B,-v)$ preserves the reference measure (up to the null set on which some $B_b$ sits at the endpoint $\pi/\gph$ of
$(-\pi/\gph,\pi/\gph]$) and the weight (the hopping factor depends on $B_b$ only
through $\cos(\gph B_b)$, $w$ is even, closed $v$ go to closed $v$) and maps every $\ff_p$ to $-\ff_p$. Jensen's inequality and $\avL{\Phi_N(g)}=0$
(Theorem~\ref{thm:GD}) give $\Psi_{N,L}(x)=\avL{e^{\Phi_N(g)}}\ge1$. The last clause is Lemma~\ref{lem:osrest:dualC} with
$b_p=\eN^{3/2}\sum_jt_jf_j(x_p)$.

(b) The family $(\Psi_{N,L_n})_n$ is locally uniformly bounded on $\C^k$ by (a). By Montel's theorem every subsequence of $(L_n)$ has a further
subsequence along which $\Psi_{N,L_n}$ converges locally uniformly to an entire function; on $i\R^k$ every such limit equals $t\mapsto
\om_N(e^{i\sum_jt_j\Phi_N(f_j)})=\int e^{i\sum_jt_j\Phi_N(f_j)}d\mu_N$, by \eqref{eq:omN} (Theorem~\ref{thm:d2:LP}(a)) applied to the bounded
continuous local function $e^{i\sum_jt_j\Phi_N(f_j)}$ (a continuous function of the finitely many $\ff_p$ in the supports). An entire function $F$ on
$\C^k$ vanishing on $i\R^k$ vanishes identically (its Taylor coefficients at $0$ are, up to powers of $i$, the derivatives of the real-analytic
function $t\mapsto F(it)$ at $0$), so all subsequential limits agree, and $\Psi_{N,L_n}\to\Psi_N$ locally uniformly; the bounds and symmetries of
(a) pass to the limit. The characteristic function of $\mu^{(f)}_N$ is $t\mapsto\int e^{it\cdot\Phi_N(f)}d\mu_N=\Psi_N(it)$, and the
characteristic functions $t\mapsto\Psi_{N,L_n}(it)$ of $\mu_{N,L_n}$ converge to it, so $\mu_{N,L_n}\to\mu^{(f)}_N$ weakly by L\'evy's continuity
theorem (alternatively, $\mu_{N,L_n}$ is the image of the law of the variables in the supports under the continuous linear map $\Phi_N(f)$, and
Theorem~\ref{thm:d2:LP}(a) gives the weak convergence of that law). For $z\in\C^k$ the continuous functions $e^{z\cdot x}$ are uniformly
integrable for $(\mu_{N,L_n})_n$, because $\sup_n\int e^{2(\Re z)\cdot x}d\mu_{N,L_n}=\sup_n\Psi_{N,L_n}(2\Re z)\le e^{2\rho(z)^2}$; hence
$\int e^{z\cdot x}d\mu^{(f)}_N=\lim_n\Psi_{N,L_n}(z)=\Psi_N(z)$.

(c) The same argument applies: $(\Psi_{N_m})_m$ is locally uniformly bounded, and on $i\R^k$, $\Psi_{N_m}(it)=Z_{N_m}(\sum_jt_jf_j,0)\to Z(\sum_jt_jf_j,0)$
by Theorem~\ref{thm:Hlim}(a). The limit law $\mu_f$ has the even characteristic function $\Psi_f(i\,\cdot)$, so it is symmetric.

(d) On the torus both identities are the additivity of $\Phi_N$; they pass to the limits.
\end{proof}

\begin{theorem}[OS0: regularity of the limit curvature]\label{thm:osrest:OS0}
\begin{itemize}
\item[(i)] (Continuity.) $|Z(\sigma)-Z(\sigma')|\le\|\sigma-\sigma'\|$ for all test pairs; in particular $|\cZ(f)-\cZ(f')|\le R(f-f')\le
(3c_{\azero})^{1/2}\wnorm{f-f'}$, so $\cZ$ extends uniquely to a continuous function on $\cS(\R^3;\Lambda^2)$, with the same bound;
$0\le\cZ(f)\le1$ for every real $f$, and $\cZ(0)=1$.
\item[(ii)] (Entire extension.) For real $C^\infty_c$ $2$-forms $f_1,\dots,f_k$ the function $\Psi_f$ of Lemma~\ref{lem:osrest:entire} is entire
on $\C^k$ with $\Psi_f(it)=\cZ(\sum_jt_jf_j)$ and
\[
|\Psi_f(z)|\le\Psi_f(\Re z)\le\exp\Big(\tfrac12\Big(\sum_j|\Re z_j|R(f_j)\Big)^2\Big);
\]
for $k=1$ the generating functional $\zeta\mapsto\cZ(\zeta f):=\Psi_{(f)}(i\zeta)=\int e^{i\zeta x}d\mu_f(x)$ is entire with
$|\cZ(\zeta f)|\le e^{|\zeta|^2R(f)^2/2}$.
\item[(iii)] (Schwinger functions.) $S_n(f_1,\dots,f_n):=\partial_{z_1}\cdots\partial_{z_n}\Psi_f(0)=\int x_1\cdots x_n\,d\mu_f(x)$ is well
defined (independent of the family containing $f_1,\dots,f_n$), symmetric and multilinear, real for real $f_i$, and
\[
|S_n(f_1,\dots,f_n)|\le e^n(n!)^{1/2}\prod_{i=1}^nR(f_i)\le\big(e\sqrt{3c_{\azero}}\big)^n(n!)^{1/2}\prod_{i=1}^n\wnorm{f_i}\qquad(f_i\text{ real}).
\]
$S_n$ extends to a continuous symmetric multilinear form on $\cS(\R^3;\Lambda^2_\C)^n$, with the factor $2^n$ in the bound for complex test
functions, and is a tempered distribution. In the product form $|S_n(f_1,\dots,f_n)|\le\sigma_n\prod_i|f_i|_s$ on product test functions this is the
growth condition (E0$'$) of Osterwalder and Schrader with $s=4$ (for the three component fields, mixed entries included) and $\sigma_n=
(2e\sqrt{3c_{\azero}})^n(n!)^{1/2}\le\alpha_\beta(n!)^\beta$ for every $\beta>\frac12$, with the norm $|\cdot|_4$ of Definition~\ref{ms:conv0}(a) (the norm of \cite{OS75} with $p=4$; the growth condition is that of \cite{OS75} with $s=4$). $S_1=0$, and $S_2(f,f)\le R(f)^2$.
\item[(iv)] (The lattice moments converge.) The torus Schwinger functions $\avL{\prod_i\Phi_N(f_i)}=\partial_{z_1}\cdots\partial_{z_n}\Psi_{N,L}(0)$
obey the bound of (iii) with $(\eN^3\sum_pf_{i}(x_p)^2)^{1/2}$ in place of $R(f_i)$, and converge to $S_n(f_1,\dots,f_n)$ as $L\to\infty$ along
$(L_n)$ and then $m\to\infty$ along $N_m$.
\end{itemize}
\end{theorem}
\begin{proof}
(i) The continuity bound is \eqref{eq:R31} in the limit, with $C=1$ by Proposition~\ref{prop:E}. The extension to $\cS$ follows by uniform continuity and the density of
$C^\infty_c$, with Lemma~\ref{lem:osrest:riemann}, and $\cZ(f)=\Psi_{(f)}(i)\in[0,1]$ by Lemma~\ref{lem:osrest:entire}(a),(c). (ii) is
Lemma~\ref{lem:osrest:entire}(c), with $\Re(i\zeta)=-\Im\zeta$ for the last bound.

(iii) By Lemma~\ref{lem:osrest:entire}(d) the derivative does not
depend on the family, and multilinearity in each entry follows from the second identity of (d) (and from $\Psi_{(cf)}(z)=\Psi_{(f)}(cz)$); symmetry
is the symmetry of mixed partial derivatives; reality follows from $\Psi_f$ being real on $\R^k$ (a limit of real numbers), so that its Taylor coefficients at
$0$ are real. For the bound let first $R(f_i)>0$ for all $i$ and normalise $R(f_i)=1$ by multilinearity; on the polydisc $|z_i|\le r$,
$\sum_j|\Re z_j|R(f_j)\le nr$, so $|\Psi_f|\le e^{n^2r^2/2}$ there, and Cauchy's estimate for the mixed first derivative gives
$|S_n|\le r^{-n}e^{n^2r^2/2}=(en)^{n/2}$ at $r=n^{-1/2}$; $n^n\le e^nn!$ gives $e^n(n!)^{1/2}$. If $R(f_i)=0$ then $\Phi_N(f_i)=0$ for every $N$,
$\Psi_f$ does not depend on $z_i$ and $S_n=0$. The second bound is Lemma~\ref{lem:osrest:riemann}.

The form is continuous for the seminorm
$\wnorm\cdot$ in each entry, so it extends from $C^\infty_c$ to $\cS$; for complex $f_i=f_i'+if_i''$ multilinearity gives $2^n$ terms, each bounded
with $\wnorm{f_i'},\wnorm{f_i''}\le\wnorm{f_i}$; the nuclear theorem (\cite[Chapter~I]{GV64}, \cite[Chapter~V]{RS72}) turns a continuous multilinear form on $\cS^n$ into a tempered distribution on
$\R^{3n}$ (one for each choice of components). On scalar test functions $\wnorm\cdot\le|\cdot|_4$ with constant $1$, since $\wnorm\cdot$ is the derivative-free term of the weighted $C^4$ norm
recalled at the start of the section. (For another weighted-$C^4$ convention a constant $C$ enters, which multiplies $\sigma_n$ by $C^n$ and leaves the
class $\beta>\frac12$ unchanged.) This gives (E0$'$) in product form with $s=4$ and the stated $\sigma_n$; $(e')^n(n!)^{1/2}\le\alpha_\beta(n!)^\beta$ for a constant $e'$ exactly when $\beta>\frac12$. $S_1=0$ and
$S_2(f,f)\le R(f)^2$ are the first two moments of Theorem~\ref{thm:GD}, passed to $\mu_f$ by the convergence of moments (uniform integrability
from the exponential moments).

(iv) The torus bound is the Cauchy estimate above with the torus bound of Lemma~\ref{lem:osrest:entire}(a), and
the derivatives at $0$ converge under the locally uniform convergences of Lemma~\ref{lem:osrest:entire}(b),(c), by the Weierstrass convergence theorem.
\end{proof}

\begin{remark}
The constants are absolute or depend on $\azero$ alone: only Theorem~\ref{thm:GD} (identity covariance), the Riemann-sum count of Lemma~\ref{lem:osrest:riemann} and the
subsequence convergence of Theorem~\ref{thm:Hlim}(a) enter; no property of the couplings is used. The joint field with the density component and its mixed Schwinger functions are treated in Section~\ref{flh:sec:main}: for the killed-box family in Lemma~\ref{flh:lem:I5} and Theorem~\ref{flh:thm:FLH1p}, and for the torus limit points along dyadic sequences in Theorem~\ref{flh:thm:FLH2}.
\end{remark}

\subsection{Euclidean invariance without rotations (OS1)}

A Euclidean motion $\gamma x=Ox+y$ with $O\in\Oh$ (the cubic group, the $48$ signed permutation matrices) acts on test pairs by push-forward,
$(\gamma\sigma)=(\gamma_*f,h\circ\gamma^{-1})$ with the $2$-form transforming as a $2$-form, and on $\cA$ by $\gamma e_\sigma:=e_{\gamma\sigma}$;
the time reflection $\theta$ of Section~\ref{sec:setting} (the paragraph before Definition~\ref{def:pin}) is the case $O=\mathrm{diag}(1,1,-1)$,
and $\tau_{te_3}$ the case $O=1$, $y=te_3$.

\begin{proposition}[OS1, rotations excepted]\label{prop:osrest:OS1}
$S(\gamma a)=S(a)$ for every $a\in\cA$ and every Euclidean motion $\gamma$ whose linear part lies in $\Oh$: all translations of $\R^3$, the
reflections in all coordinate planes $\{x_j=c\}$ ($j=1,2,3$, $c\in\R$) and the cubic group about every point. Equivalently $Z(\gamma\sigma)=Z(\sigma)$
for every test pair.
\end{proposition}
\begin{proof}
\emph{Translations.} Theorem~\ref{thm:Hlim} states the invariance under all space and time translations; we write the argument for a direction
$e_j$. For $\sigma\in\mathcal T$ and $t,t'\in\R$, \eqref{eq:R31} in the limit gives $|Z(\tau_{te_j}\sigma)-Z(\tau_{t'e_j}\sigma)|\le\|\tau_{te_j}\sigma
-\tau_{t'e_j}\sigma\|$, and $\tau_{t'e_j}\sigma\to\tau_{te_j}\sigma$ in $C^\infty_c$ as $t'\to t$, so $t\mapsto Z(\tau_{te_j}\sigma)$ is continuous by
the admissibility of $\|\cdot\|$ (Definition~\ref{def:norms}). For $t\in\varepsilon_{N_m}\Z$ and $m'\ge m$, Lemma~\ref{lem:R1} (the translation by
$te_j$ is a lattice translation at level $N_{m'}$, and $\Phi_N(\tau_{te_j}\sigma)=\Phi_N(\sigma)\circ g^{-1}$ by the substitution of its proof)
gives $\om_{N_{m'}}(\iota_{N_{m'}}(\tau_{te_j}a))=\om_{N_{m'}}(\iota_{N_{m'}}(a))$; letting $m'\to\infty$, $S(\tau_{te_j}a)=S(a)$ on the dense
set $\bigcup_m\varepsilon_{N_m}\Z$, hence for all $t$ by continuity. Composing the three directions gives all translations of $\R^3$.

\emph{The cubic group.} Theorem~\ref{thm:Hlim}
gives the invariance under the cubic group about every point $x_0\in\bigcup_m\varepsilon_{N_m}\Z^3$ (Step~5 of the proof of
Proposition~\ref{prop:R3}, which rests on Lemma~\ref{lem:R1} and on the intertwining of $\iota_N$ with the push-forward, a signed permutation of the
plaquettes and sites). The cubic group about an arbitrary point $x$ is $\tau_{x-x_0}G_{x_0}\tau_{x_0-x}$, where $G_{x_0}$ is the cubic group about $x_0$, and $S$ is translation invariant. The
reflection in $\{x_j=c\}$ is the element $O=1-2e_je_j^{\mathsf T}$ of the cubic group about any point of that plane. Finally, every motion with
linear part in $\Oh$ is $\tau_y\circ O$ with $O$ in the cubic group about $0\in\varepsilon_{N_m}\Z^3$, and $\gamma e_\sigma=e_{\gamma\sigma}$ is a group
action (the push-forward is covariant), so the invariances proved give $S(\gamma a)=S(a)$ for the whole group.
\end{proof}

Invariance under $SO(3)$ is the subject of Sections~\ref{flh:sec:main}--\ref{sec:i7:proof} (Theorems~\ref{flh:thm:FLH1}, \ref{flh:thm:FLH1p} and~\ref{flh:thm:FLH2}).

\subsection{Reflection positivity (OS2)}

\emph{Reflections of the torus.} Fix $N$ and $L$ ($L/2\in\eN\Z$) and $c\in\frac12\eN\Z$. The map $R_c:x\mapsto(x_1,x_2,2c-x_3)$ is an involutive
symmetry of $\tilde T_L$ fixing pointwise the two planes $P_c=\{x_3=c\}$ and $P_{c+L/2}$; it permutes the sites, the oriented bonds, the oriented
plaquettes and the cubes, commuting with the boundary operator. It acts on configurations by $(R_c\xi)_\kappa:=\xi_{R_c\kappa}$ for every cell
$\kappa$, with the orientation conventions $B_{-b}=-B_b$, $v_{-p}=-v_p$; then $\tr_x\circ R_c=\tr_{R_cx}$, $\ff_p\circ R_c=\ff_{R_cp}$, closed $v$
go to closed $v$, and the weight and the reference measure are invariant (as in the proof of Lemma~\ref{lem:R1}, since $w$ is even and $h(r,r',\vartheta)=
h(r',r,-\vartheta)$), so $\avL{F\circ R_c}=\avL F$. Let $\Lambda^+_c:=\{x\in\tilde T_L:x_3\in[c,c+L/2]\}$ (coordinates taken in a fundamental
interval) and $\Lambda^-_c:=R_c\Lambda^+_c$; a cell lies in $\Lambda^\pm_c$ if all its vertices do. $\mathfrak A^+_L(c)$ is the algebra of bounded
measurable functions of the variables $(\tr_x,B_b,v_p)$ of the cells lying in $\Lambda^+_c$, and $\Th_cF:=\overline{F\circ R_c}$. When the support of $F$ is small compared with $L$, its cells lie in
$\Lambda^+_c$ exactly when they lie in $\{x_3\ge c\}$.

\begin{theorem}[reflection positivity of the fine torus state]\label{thm:osrest:RPtorus}
For every $N$, every $L\ge4\eN$ with $L/2\in\eN\Z$, every $c\in\frac12\eN\Z$ and all $F,G\in\mathfrak A^+_L(c)$,
\[
\avL{\Th_cF\cdot G}=\overline{\avL{\Th_cG\cdot F}},\qquad\avL{\Th_cF\cdot F}\ge0 .
\]
Both statements extend to all $\Lambda^+_c$-measurable $F,G$ with $\avL{|F|^2},\avL{|G|^2}<\infty$. By the cubic symmetry the same holds for the
reflections in the planes $\{x_j=c\}$, $j=1,2$, with the slabs taken in the $j$-th coordinate.
\end{theorem}
\begin{proof}
The Hermitian property is the $R_c$-invariance of the state: $\avL{\overline{F\circ R_c}\,G}=\avL{\overline F\,(G\circ R_c)}=
\overline{\avL{\overline{G\circ R_c}\,F}}$. For the positivity we treat $c=0$ and $c=\eN/2$; the other $c$ follow by the lattice translations,
which are symmetries of the torus state and conjugate $R_c$ to one of these. We write the sum over the Villain variables as an integral against
the counting measure on $((2\pi/\gph)\Z)^{\mathcal P}$ with the closedness constraint $\prod_{\mathrm{cubes}}\mathbf 1[(dv)_c=0]$ as a factor of
the weight; so the weight is a product of one factor per site, bond, plaquette and cube, and the constraint is local.

\emph{Site plane, $c=0$.} Call a cell a plane cell if all its vertices lie in $P_0\cup P_{L/2}$. Every other cell lies in exactly one of
$\Lambda^\pm_0$ (a cell has $x_3$-extent at most $\eN$ and $0,L/2\in\eN\Z$), and every cube, with its six faces, lies in one of them. $R_0$ fixes
every plane cell pointwise and maps the non-plane cells of $\Lambda^+_0$ bijectively onto those of $\Lambda^-_0$. Write $\xi=(\xi_0,\xi_+,\xi_-)$ for
the variables of the plane cells, of the other cells of $\Lambda^+_0$, of the other cells of $\Lambda^-_0$; the reference measure is the product
$d\xi_0\,d\xi_+\,d\xi_-$, and $R_0$ carries $d\xi_-$ to $d\xi_+$. Let $W_+(\xi_0,\xi_+)$ be the product of the factors of the non-plane cells and
the cubes of $\Lambda^+_0$, $W_-(\xi_0,\xi_-)$ the same for $\Lambda^-_0$, and $W_0(\xi_0)$ the product of the factors of the plane cells. Then
$W=W_0W_+W_-$ and $W_-(\xi)=W_+(R_0\xi)$, because $R_0$ maps the cells and cubes of $\Lambda^-_0$ onto those of $\Lambda^+_0$ and preserves
every factor. For $F\in\mathfrak A^+_L(0)$, $F(\xi)=F(\xi_0,\xi_+)$ and $(F\circ R_0)(\xi)=F(\xi_0,\eta)$ with $\eta:=(R_0\xi)_+$, a
measure-preserving bijective function of $\xi_-$. Hence, by Fubini's theorem (the integrand is dominated by $W\|F\|_\infty^2$),
\begin{align*}
\avL{\Th_0F\cdot F}&=\frac1Z\int d\xi_0\,W_0(\xi_0)\int d\xi_+W_+(\xi_0,\xi_+)F(\xi_0,\xi_+)\int d\xi_-W_-(\xi_0,\xi_-)\overline{F(\xi_0,(R_0\xi)_+)}\\
&=\frac1Z\int d\xi_0\,W_0(\xi_0)\,\big|\mathcal G_F(\xi_0)\big|^2\ge0,
\end{align*}
with $\mathcal G_F(\xi_0):=\int d\xi_+W_+(\xi_0,\xi_+)F(\xi_0,\xi_+)$: after the substitution $\eta=(R_0\xi)_+$ and $W_-(\xi_0,\xi_-)=W_+(\xi_0,\eta)$,
the last integral is $\overline{\mathcal G_F(\xi_0)}$.

\emph{Bond mid-plane, $c=\eN/2$.} Now no site lies on the fixed planes, and the sites are partitioned between $\Lambda^+:=\{x_3\in[\eN,L/2]\}$ and
$\Lambda^-:=R\Lambda^+$, $R:=R_{\eN/2}$ (these are the slabs $\Lambda^\pm_{\eN/2}$ of the general notation; we drop the subscript in this case). The crossing cells, those lying in neither slab, are the time-like bonds $(x,x+\eN e_3)$ with $x_3\in\{0,L/2\}$,
the time-like plaquettes with vertices at $x_3\in\{0,\eN\}$ or $\{L/2,L/2+\eN\}$, and the cubes between these slices; $R$ maps every crossing cell to
itself with reversed orientation, so on them $B_b\mapsto-B_b$, $v_p\mapsto-v_p$, and it maps the cells and cubes of $\Lambda^+$ bijectively onto those of
$\Lambda^-$. Write $\xi=(\xi_+,\xi_-,\xi_\times)$ with $\xi_\times=(B_b,v_p)_{\mathrm{crossing}}$, and $W=W_+(\xi_+)W_-(\xi_-)W_\times$, where $W_\pm$
collect the factors of the cells and cubes of $\Lambda^\pm$ (the faces of a cube of $\Lambda^\pm$ lie in $\Lambda^\pm$), $W_-=W_+\circ R$, and
\[
W_\times:=\prod_{b\times}h(\tr_{b_-},\tr_{b_+},B_b)\prod_{p\times}w(\ff_p)\prod_{c\times}\mathbf 1[(dv)_c=0],\qquad
K(\xi_-,\xi_+):=\int\prod_{b\times}dB_b\sum_{(v_p)_{p\times}}W_\times ,
\]
the products over the crossing bonds, plaquettes and cubes; $K$ depends on $\xi_\pm$ only through the two boundary slices.

As in the first case,
with $\eta:=(R\xi)_+$ and $\widetilde K(\eta,\xi_+):=K(R\eta,\xi_+)$ ($R\eta$ the $\Lambda^-$-configuration obtained from $\eta$ by reflection),
\[
\avL{\Th F\cdot F}=\frac1Z\int d\eta\,d\xi_+\,W_+(\eta)W_+(\xi_+)\,\widetilde K(\eta,\xi_+)\,\overline{F(\eta)}F(\xi_+),
\]
and it suffices to show that $\widetilde K$ is a kernel of positive type, $\widetilde K(\eta,\xi)=\int d\nu(k)\,\overline{A_k(\eta)}A_k(\xi)$ with
a positive measure $\nu$ and $\int d\nu(k)|A_k(\eta)||A_k(\xi)|$ bounded uniformly in $(\eta,\xi)$: then the double integral equals $\int d\nu(k)|\int d\xi\,W_+(\xi)A_k(\xi)F(\xi)|^2
\ge0$ by Fubini's theorem. Here $F$ is bounded and $W_+$ is integrable, since $g_{\mathrm{ct}}$ is integrable, $h\le1$, each $B_b$ ranges over a bounded interval, the
indicators are at most $1$, and each Villain sum $\sum_{v_p}w(\ff_p)$ is at most $K_{\gph}$ by \eqref{eq:poisson} at $a=0$ and Step~4 of the proof of
Theorem~\ref{thm:GD}.

Each crossing cell has one end in each of the two slabs; these ends are the two endpoints of a crossing bond, the two spatial edges of a crossing
plaquette, and the top and bottom faces of a crossing cube. We write $\uparrow$ for the end of a crossing cell in $\Lambda^+$ and $\downarrow$ for its end in $\Lambda^-$;
for a crossing bond $b$ the two ends are written $b_+$ and $b_-$ accordingly (at the slices $L/2$, $L/2+\eN$ this interchanges $b_\pm$). Thus the
spatial edges of a crossing plaquette $p$ with vertices at $x_3\in\{0,\eN\}$ are $p^\uparrow$ (at $x_3=\eN$) and $p^\downarrow$ (at $x_3=0$), and
the top and bottom faces of a crossing cube $c$ between these slices are $c^\uparrow$ and $c^\downarrow$; at the slices $L/2$, $L/2+\eN$ the
$\Lambda^+$ edge or face, which carries the label $\uparrow$, is the lower one (at $x_3=L/2$). The $\uparrow$ variables belong to $\xi_+$, the
$\downarrow$ variables to $\xi_-$.

We expand $K$ by the computation of Theorem~\ref{thm:GD} restricted to the crossing
cells. Write $\mathbf 1[(dv)_c=0]=\int_0^1dt_c\,e^{i\gph t_c(dv)_c}$ for the crossing cubes, put $t_c:=0$ for the others and $s_p:=(d^*t)_p$. The two
cubes containing a crossing plaquette are crossing, and \eqref{eq:poisson} at $a=0$ gives $\sum_{v_p}w(\ff_p)e^{i\gph s_pv_p}=\sum_{m_p\in\Z}
c_{m_p-s_p}e^{i\gph(m_p-s_p)(dB)_p}$ with $c_u>0$. The four plaquettes containing a crossing bond $b$ are crossing, and $\sum_{p\ni b}\pm(m_p-s_p)=
(d^*m)_b-(d^*d^*t)_b=(d^*m)_b=:q_b\in\Z$, so the integral over $B_b$ gives $(2\pi/\gph)e^{-\frac12\kappa_N(\rho_{b_-}^2+\rho_{b_+}^2)}
I_{q_b}(\kappa_N\rho_{b_-}\rho_{b_+})$ (Step~3 of the proof of Theorem~\ref{thm:GD}). What remains of the phases involves the spatial edges of the
crossing plaquettes and the top and bottom faces of the crossing cubes: for every crossing plaquette $p$ and every crossing cube $c$,
\[
(dB)_p=\pm(B_{p^\uparrow}-B_{p^\downarrow})+(\text{crossing bonds}),\qquad(dv)_c=\pm(v_{c^\uparrow}-v_{c^\downarrow})+(\text{crossing plaquettes}).
\]
Hence
\begin{multline*}
K(\xi_-,\xi_+)=\int dt\sum_{m}\Big(\prod_{p\times}c_{m_p-s_p}\Big)\prod_{b\times}\frac{2\pi}{\gph}e^{-\frac12\kappa_N(\rho_{b_-}^2+\rho_{b_+}^2)}
I_{q_b(m)}(\kappa_N\rho_{b_-}\rho_{b_+})\\
\prod_{p\times}e^{\pm i\gph(m_p-s_p)(B_{p^\uparrow}-B_{p^\downarrow})}\prod_{c\times}e^{\pm i\gph t_c(v_{c^\uparrow}-v_{c^\downarrow})}.
\end{multline*}

The reflection $R$ maps $p^\downarrow$ to $p^\uparrow$, $c^\downarrow$ to $c^\uparrow$ and $b_-$ to $b_+$, with the spatial orientations preserved; so in $\widetilde K(\eta,\xi)=K(R\eta,\xi)$ every
$\downarrow$ variable of $R\eta$ is the corresponding $\uparrow$ variable of $\eta$. Each factor is then of positive type in $(\eta,\xi)$: the phases
are $\overline{a(\eta)}a(\xi)$ with $a=e^{\pm i\gph(m_p-s_p)B_{p^\uparrow}}$ or $a=e^{\pm i\gph t_cv_{c^\uparrow}}$; the factor $e^{-\frac12\kappa_N
(\rho^2+\rho'^2)}$ splits; and 
\[
I_q(\kappa\rho\rho')=\sum_{k\ge0}\frac{(\kappa/2)^{2k+|q|}}{k!\,(k+|q|)!}\rho^{2k+|q|}\rho'^{2k+|q|}
\]
is a series with
positive coefficients of products $\varphi(\rho_{b_+}(\eta))\varphi(\rho_{b_+}(\xi))$ of one real function.

Products, non-negative combinations and
integrals over $t$ of kernels of this form are of this form (for products, by the Schur product theorem), which gives $\widetilde K=\int d\nu(k)\overline{A_k}\otimes A_k$ with
$k=(t,m,(k_b))$ and 
\[
d\nu=dt\prod_pc_{m_p-s_p}\prod_b\frac{2\pi}{\gph}\frac{(\kappa_N/2)^{2k_b+|q_b|}}{k_b!(k_b+|q_b|)!};
\]
$\nu$ is a finite measure, of mass at
most $(2\pi e^{\kappa_N}/\gph)^{|b\times|}K_{\gph}^{|p\times|}$, since $\sum_{k\ge0}\frac{(\kappa/2)^{2k+|q|}}{k!(k+|q|)!}=I_{|q|}(\kappa)\le e^{\kappa}$. The absolute series is the
same expansion with the phases replaced by $1$, i.e.
\[
\int dt\sum_m\prod_pc_{m_p-s_p}\prod_b\frac{2\pi}{\gph}e^{-\frac12\kappa_N(\rho^2+\rho'^2)} I_{q_b}(\kappa_N\rho\rho')\le(2\pi/\gph)^{|b\times|}K_{\gph}^{|p\times|},
\]
by $e^{-\frac12\kappa(\rho^2+\rho'^2)}I_q(\kappa\rho\rho')\le
e^{-\frac12\kappa(\rho-\rho')^2}\le1$ and $\sum_mc_{m-s}\le K_{\gph}$ (Step~4 of the proof of Theorem~\ref{thm:GD}). This is the required bound.

\emph{The $L^2$ extension.} Bounded $\Lambda^+_c$-measurable functions are dense in the $\Lambda^+_c$-measurable part of $L^2(\avL\cdot)$, and
$|\avL{\Th_cF\cdot F}-\avL{\Th_cG\cdot G}|\le\|F-G\|_2(\|F\|_2+\|G\|_2)$ by the Schwarz inequality and the $R_c$-invariance of the state; both sides of the Hermitian identity are $\|\cdot\|_2$-continuous in $(F,G)$ as well.
\end{proof}

\begin{corollary}[reflection positivity of the fine state at the lattice planes]\label{cor:osrest:D1}
For $N\ge N_0$ and $c\in\frac12\eN\Z$ let $\mathfrak A^+_N(c)$ be the bounded Borel local functions of the gauge-invariant variables $(\ff_p,\tr_x)$ of the
cells lying in $\{x_3\ge c\}$, and $\Th_cF:=\overline{F\circ R_c}$. Then $\om_N(\Th_cF\cdot G)=\overline{\om_N(\Th_cG\cdot F)}$ and $\om_N(\Th_cF\cdot F)\ge0$
for $F,G\in\mathfrak A^+_N(c)$; $\om_N\circ R_c=\om_N$; and $\om_N$ is invariant under $\tau_{2\azero}$, indeed under $\eN\Z^3$ (Lemma~\ref{lem:R1}).
In particular the fine state is reflection positive about the site planes $t=0$ and $t=\azero$ ($\azero=5^N\eN\in\eN\Z$) and invariant under
$R_0$, $R_{\azero}=\tau_{2\azero}R_0$.
\end{corollary}
\begin{proof}
For $n$ large the cells of $F$ and $G$ lie in $\Lambda^+_c$ of $\tilde T_{L_n}$, and their set $S$ together with $R_cS$ lies in a cube of side $<L_n$,
so that the torus expectations are those of the fixed functions (Subsection~\ref{sec:d2:setting}); on these cells the torus reflection $R_c$ agrees with the
reflection of the infinite lattice; $\Th_cF\cdot G$, $\Th_cF\cdot F$ and $F\circ R_c$ are bounded local functions of the gauge-invariant variables
($\ff_p\circ R_c=\ff_{R_cp}$, $\tr_x\circ R_c=\tr_{R_cx}$), the same functions of the same finitely many cells for all large $n$. If $F$ and $G$ are
continuous, so are these functions, and \eqref{eq:omN} (Theorem~\ref{thm:d2:LP}(a)), applied to their real and imaginary parts, passes the
identities of Theorem~\ref{thm:osrest:RPtorus} and the invariance $\langle F\circ R_c\rangle^{\mathrm{fine}}_{L_n}=\langle F\rangle^{\mathrm{fine}}_{L_n}$
to $\om_N$; positive semidefiniteness on finite families follows by applying the inequality to linear combinations.

Let now $F,G$ be bounded Borel functions of
the cells of a finite set $S$. The bounded continuous functions of the variables of $S$ are dense in $L^2$ of the $S$-marginal of $\mu_N$ (a
finite Borel measure on a Polish space is Radon), a bounded Borel $F$ is the $L^2(\mu_N)$-limit of continuous $F^{(i)}$ with $\|F^{(i)}\|_\infty
\le\|F\|_\infty$ (by truncation), and both sides of each identity are continuous in the $L^2(\mu_N)$-norms of $F$ and $G$, by the Schwarz inequality and
$\|F\circ R_c\|_{L^2(\mu_N)}=\|F\|_{L^2(\mu_N)}$ (the $R_c$-invariance of $\mu_N$, Theorem~\ref{thm:d2:LP}(c)). The invariance $\om_N\circ R_c=\om_N$
on the bounded Borel class and the invariance under $\eN\Z^3$ are Theorem~\ref{thm:d2:LP}(c) (Lemma~\ref{lem:R1}). The corollary is Theorem~\ref{thm:d2:LP}(c),(d)
for $j=3$, whose proof is the argument just given.
\end{proof}

\begin{proposition}[OS2 for $S$: inheritance]\label{prop:osrest:OS2}
For $j\in\{1,2,3\}$ and $c\in\R$ let $\Th_{j,c}$ be the reflection in $\{x_j=c\}$ on $\cA$ ($\Th_{j,c}e_\sigma:=e_{-\gamma\sigma}$, $\gamma$ the
reflection) and $\cA_+^{(j,c)}$ the span of the $e_\sigma$ with $\supp\sigma\subset\{x_j>c\}$; $\Th_{3,0}=\Th$, $\cA_+^{(3,0)}=\Ap$. Then
$\langle a,b\rangle:=S(\Th_{j,c}a\cdot b)$ is a positive semidefinite Hermitian sesquilinear form on $\cA_+^{(j,c)}$, and $S(\Th_{j,c}a)=\overline{S(a)}$
on $\cA$.
\end{proposition}
\begin{proof}
For $a=\sum_i\gamma_ie_{\sigma_i}$, $b=\sum_j\delta_je_{\sigma'_j}$ in $\Ap$ and $N=N_m$ with $\eN$
smaller than the distance of the supports from the plane, $\iota_N(a)$ and $\iota_N(b)$ are functions of the cells in $\{x_3\ge0\}$, and
$\Phi_N(\sigma)\circ R_0=\Phi_N(\theta\sigma)$: $R_0$ permutes the plaquettes with the orientation sign, $\ff_p\circ R_0=\ff_{R_0p}$, and the push-forward
of a $2$-form carries exactly this sign ($-1$ on the components with one index $3$, whose plaquettes $R_0$ reverses), while $\tr_z\circ R_0=
\tr_{R_0z}$. Hence $\iota_N(\Th a\cdot b)=\Th_0\iota_N(a)\cdot\iota_N(b)$, and by Theorem~\ref{thm:Hlim}(a), $S(\Th a\cdot b)=\lim_m\om_{N_m}
(\Th_0\iota_{N_m}(a)\,\iota_{N_m}(b))$. Each term is a positive semidefinite Hermitian form in $(a,b)$ by Corollary~\ref{cor:osrest:D1}, and these
properties of the Gram matrices $(\langle a_i,a_j\rangle)$ are closed under pointwise limits. Likewise $S(\Th a)=\lim\om_{N_m}(\Th_0\iota_{N_m}(a))
=\lim\overline{\om_{N_m}(\iota_{N_m}(a))}=\overline{S(a)}$ by the reflection invariance of $\om_N$. For $(j,c)$ arbitrary choose a Euclidean motion
$\gamma$ with linear part in $\Oh$ mapping $\{x_3>0\}$ onto $\{x_j>c\}$; then $\cA_+^{(j,c)}=\gamma\Ap$, $\Th_{j,c}=\gamma\Th\gamma^{-1}$, and
Proposition~\ref{prop:osrest:OS1} gives $S(\Th_{j,c}a\cdot b)=S(\Th(\gamma^{-1}a)\cdot\gamma^{-1}b)$.
\end{proof}

\begin{theorem}[OS2 in matrix form, complex coefficients]\label{thm:osrest:OS2matrix}
Let $j\in\{1,2,3\}$ and $c\in\R$, let $f_1,\dots,f_k$ be real $C^\infty_c$ $2$-forms supported in $\{x_j>c\}$, $\gamma$ the reflection in $\{x_j=c\}$, $F:=(\gamma_*f_1,\dots,
\gamma_*f_k,f_1,\dots,f_k)$ and $\Psi_F$ its entire function on $\C^{2k}$ (Lemma~\ref{lem:osrest:entire}). Then for all $z^{(1)},\dots,z^{(n)}\in\C^k$
and $\lambda\in\C^n$,
\[
\sum_{i,l=1}^n\bar\lambda_i\lambda_l\,\Psi_F\big(\overline{z^{(i)}},z^{(l)}\big)\ \ge\ 0 .
\]
At purely imaginary $z^{(i)}=it^{(i)}$ this is $\sum_{i,l}\bar\lambda_i\lambda_lZ(\sigma_l-\gamma\sigma_i)\ge0$, $\sigma_i:=(\sum_rt^{(i)}_rf_r,0)$,
the positivity of Proposition~\ref{prop:osrest:OS2} on the exponentials.
\end{theorem}
\begin{proof}
It suffices to take $(j,c)=(3,0)$. Indeed, let $\gamma_1$ be a Euclidean motion with linear part in $\Oh$ carrying $\{x_3>0\}$ onto $\{x_j>c\}$
(as in the proof of Proposition~\ref{prop:osrest:OS2}), so that $\gamma_1^{-1}\gamma\gamma_1=\theta$, put $g_r:=(\gamma_1^{-1})_*f_r$, supported in
$\{x_3>0\}$, and $F':=(\theta_*g_1,\dots,\theta_*g_k,g_1,\dots,g_k)=((\gamma_1^{-1})_*F_1,\dots,(\gamma_1^{-1})_*F_{2k})$. Then $\Psi_{F'}=\Psi_F$ on
$\C^{2k}$: both are entire (Lemma~\ref{lem:osrest:entire}(c)) and agree on $i\R^{2k}$, where $\Psi_{F'}(it)=\cZ((\gamma_1^{-1})_*\sum_rt_rF_r)=
\cZ(\sum_rt_rF_r)=\Psi_F(it)$ by Proposition~\ref{prop:osrest:OS1}, hence on $\C^{2k}$ (identity theorem, as in the proof of
Lemma~\ref{lem:osrest:entire}(b)); so the inequality for $F$ is the inequality for $F'$ at $(3,0)$. On the torus,
for $\eN$ smaller than the distance of the supports from the plane and $L$ large, $A_i:=\exp(\sum_rz^{(i)}_r\Phi_N(f_r))$ is $\Lambda^+_0$-measurable
with $\avL{|A_i|^2}<\infty$ (Lemma~\ref{lem:osrest:entire}(a)), and $\Th_0A_i=\exp(\sum_r\overline{z^{(i)}_r}\Phi_N(\gamma_*f_r))$ because
$\Phi_N(f)\circ R_0=\Phi_N(\gamma_*f)$ is real. So $\avL{\Th_0A_i\cdot A_l}=\Psi^F_{N,L}(\overline{z^{(i)}},z^{(l)})$, where $\Psi^F_{N,L}$ denotes the
function $\Psi_{N,L}$ of Lemma~\ref{lem:osrest:entire} for the family $F$, and the $L^2$ form of
Theorem~\ref{thm:osrest:RPtorus} applied to $A:=\sum_l\lambda_lA_l$ gives the inequality at level $(N,L)$. It passes to the limit $L\to\infty$ along $(L_n)$
and then to $m\to\infty$ by the locally uniform convergences of Lemma~\ref{lem:osrest:entire}(b),(c) at the finitely many points $(\overline{z^{(i)}},z^{(l)})$.
\end{proof}

\begin{proposition}[OS2 in polynomial form]\label{prop:osrest:poly}
Let $P=\sum_\alpha c_\alpha\prod_{i\le n_\alpha}\Phi(f_{\alpha i})$ be a formal polynomial in the smeared curvature with real $C^\infty_c$ sources
supported in $\{x_j>c\}$, $\gamma$ the reflection in $\{x_j=c\}$, $\theta P:=\sum_\alpha\bar c_\alpha\prod_i\Phi(\gamma_*f_{\alpha i})$, and let $S$
assign to a monomial its Schwinger function (Theorem~\ref{thm:osrest:OS0}(iii)). Then
\[
S(\theta(P)P)=\sum_{\alpha,\beta}\bar c_\alpha c_\beta\,S_{n_\alpha+n_\beta}(\gamma_*f_{\alpha1},\dots,\gamma_*f_{\alpha n_\alpha},f_{\beta1},\dots,f_{\beta n_\beta})\ \ge\ 0 .
\]
\end{proposition}
\begin{proof}
Again it suffices to take $(j,c)=(3,0)$: with $\gamma_1$ as in the previous proof, differentiating at $0$ the identity $\Psi_{F'}=\Psi_F$ (for the family
$F$ of all sources and their reflections named below) gives $S_n((\gamma_1^{-1})_*u_1,\dots,(\gamma_1^{-1})_*u_n)=S_n(u_1,\dots,u_n)$ for all
$u_1,\dots,u_n$ in $F$ (Theorem~\ref{thm:osrest:OS0}(iii)), which carries the sum of the statement to the one for the sources $(\gamma_1^{-1})_*f_{\alpha i}$,
supported in $\{x_3>0\}$, and the reflection $\theta$. On the torus, with $\eN$ and $L$ as in the previous proof,
$P_N:=\sum_\alpha c_\alpha\prod_i\Phi_N(f_{\alpha i})$ is $\Lambda^+_0$-measurable and square integrable (all moments
are finite by Theorem~\ref{thm:GD}), and $\Th_0P_N=\sum_\alpha\bar c_\alpha\prod_i\Phi_N(\gamma_*f_{\alpha i})$; Theorem~\ref{thm:osrest:RPtorus} gives
$\avL{\Th_0P_N\cdot P_N}\ge0$. This expectation is $\sum_{\alpha,\beta}\bar c_\alpha c_\beta$ times the mixed moment of the family $F$ of all sources
and their reflections, a mixed derivative at $0$ of $\Psi^F_{N,L}$ (differentiation under the integral, justified by the exponential moments), and
these derivatives converge to those of $\Psi_F$ at $0$, the Schwinger functions, by Theorem~\ref{thm:osrest:OS0}(iv).
\end{proof}

\subsection{Symmetry (OS3)}

\begin{proposition}[OS3: symmetry]\label{prop:osrest:OS3}
(a) Each $S_n$ is symmetric and multilinear, and $S_n(\bar f_1,\dots,\bar f_n)=\overline{S_n(f_1,\dots,f_n)}$; $S_n$ is real on real test
functions. (b) $Z(-f,h)=Z(f,h)$ for every test pair; $\Psi_f(-z)=\Psi_f(z)$; and $S_n=0$ for odd $n$.
\end{proposition}
\begin{proof}
(a) is Theorem~\ref{thm:osrest:OS0}(iii), which states the symmetry, the multilinearity and the reality on real test functions; the conjugation
identity $S_n(\bar f_1,\dots,\bar f_n)=\overline{S_n(f_1,\dots,f_n)}$ follows from the complex-multilinear extension (the $2^n$-term expansion in the
proof of (iii)) and that reality.

(b) The symmetry used is the charge conjugation $C\colon(\tr,B,v)\mapsto(\tr,-B,-v)$ of the fine torus
measure, a bijection of the torus configuration space with three properties: (1) it
preserves the reference measure (up to the null set on which some $B_b$ sits at the endpoint $\pi/\gph$ of $(-\pi/\gph,\pi/\gph]$); (2) it
preserves the weight, because the site factor $g_{\mathrm{ct}}$ does not see $B$, the hopping factor of the model of Section~\ref{sec:setting},
$h(r,r',\vartheta)=\exp(-\frac12\kappa_N(\rho^2+\rho'^2-2\rho\rho'\cos(\gph\vartheta)))$, is even in $\vartheta$ and so depends on $B_b$ only through
$\cos(\gph B_b)$, the plaquette weight $w$ is even, and $d$ is linear, so closed $v$ go to closed $v$ (the sum runs over all closed $v$); (3) it
maps $\ff_p$ to $-\ff_p$ and fixes $\tr$, hence $\Phi_N(f,h)\circ C=\Phi_N(-f,h)$. So $C$ preserves the fine torus measure, $\avL{F\circ C}=\avL F$
for every bounded Borel $F$, in particular $\avL{e^{i\Phi_N(f,h)}}=\avL{e^{i\Phi_N(-f,h)}}$ for
all $N$, $L$.

On $\Omega_N$ the charge conjugation is $C\colon(\ff,\tr)\mapsto(-\ff,\tr)$, $\Phi_N(f,h)\circ C=\Phi_N(-f,h)$, and $\mu_N$ is
$C$-invariant by Theorem~\ref{thm:d2:LP}(c). The proof of that theorem for $C$ passes the torus invariance just proved to the limit point through \eqref{eq:omN},
and Lemma~\ref{lem:d2:sym} takes the torus step from the present proof, so that the argument is not circular. Hence
\[
Z_N(-f,h)=\int e^{i\Phi_N(-f,h)}d\mu_N= \int e^{i\Phi_N(f,h)} d(C_*\mu_N)=Z_N(f,h)
\]
for every $N$ and every bounded Borel $g_{\rm d}$, and Theorem~\ref{thm:Hlim}(a) gives $Z(-f,h)=Z(f,h)$. The evenness of
$\Psi_f$ is Lemma~\ref{lem:osrest:entire}(a),(c), and the odd derivatives of an
even entire function vanish at $0$. (The density component carries no such sign: the argument gives no relation between $Z(f,-h)$ and $Z(f,h)$.)
\end{proof}

\subsection{The measure}

We use the field space $X:=H^{-r}_{\mathrm{loc}}(\R^3;\Lambda^2)$ (throughout this section $X$ denotes this space) and the Hermite--Sobolev space $\cHe^{-r}(\R^3;\Lambda^2)$ (the dual of the
completion of $\cS$ under $\|u\|_r^2=\sum_{\alpha,c}\lambda_\alpha^r\langle h_{\alpha,c},u\rangle^2$, $h_{\alpha,c}$ the Hermite basis,
$\lambda_\alpha=2|\alpha|+4$), with the push-forward actions of the Euclidean motions, $(\gamma\Phi)(u)=\Phi(\gamma^{-1}\cdot u)$, the local
$\sigma$-algebras $\Sigma(U)$ generated by $\Phi\mapsto\Phi(u)$, $u\in C^\infty_c(U;\Lambda^2)$ real, and the restriction map $\iota:\cHe^{-r}\to X$,
exactly as in Appendix~\ref{ms:measure} (probability measures on spaces of distributions; Definition~\ref{ms:fielddef}). Of its abstract results we use in particular
Lemma~\ref{ms:carrier} (the Hermite carrier), Proposition~\ref{ms:unique} (uniqueness), Proposition~\ref{ms:discharge} (the verification of (H1)--(H3)) and Lemma~\ref{ms:transfer} (transfer); they are proved there and do not involve the
lattice.

\begin{theorem}[the measure of the limit curvature]\label{thm:osrest:measure}
Let $r\ge14$ be even.
\begin{itemize}
\item[(a)] There is exactly one Borel probability measure $\mu^H$ on $\cHe^{-r}(\R^3;\Lambda^2)$ whose mixed moments on real $C^\infty_c$ $2$-forms
are the Schwinger functions $S_n$. Its characteristic functional on $C^\infty_c$, indeed on $\cS$, is $\cZ$: $\int e^{i\Psi(u)}d\mu^H(\Psi)=\cZ(u)$;
its moments on $\cS$ are the continuous extensions of the $S_n$ and obey the bounds of Theorem~\ref{thm:osrest:OS0}(iii); the law of
$(\Psi(f_1),\dots,\Psi(f_k))$ under $\mu^H$ is the law $\mu_f$ of Lemma~\ref{lem:osrest:entire}.
\item[(b)] $\mu:=\iota_*\mu^H$ is the unique Borel probability measure on $X$ with characteristic functional $\cZ$ on $C^\infty_c$, and also the
unique one whose mixed moments on $C^\infty_c$ are the $S_n$; $\mu^H$ is the unique measure on $\cHe^{-r}$ with image $\mu$.
\item[(c)] $\mu$ and $\mu^H$ are invariant under all translations, all coordinate reflections and the cubic group about every point.
\end{itemize}
\end{theorem}
\begin{proof}
The proof applies the results of Appendix~\ref{ms:measure}; we verify their hypotheses in turn. \emph{Positive definiteness.} For real $u_1,\dots,u_n\in C^\infty_c$ and
$c\in\C^n$, $\sum_{l,l'}\bar c_lc_{l'}\cZ(u_{l'}-u_l)=\lim_m\om_{N_m}(|\sum_lc_le^{i\Phi_{N_m}(u_l)}|^2)\ge0$ by additivity of $\Phi_N$ and
Theorem~\ref{thm:Hlim}(a); $\cZ(0)=1$.

\emph{Continuity.} $|\cZ(u)-\cZ(u')|\le(3c_{\azero})^{1/2}\wnorm{u-u'}$ (Theorem~\ref{thm:osrest:OS0}(i)),
and $\wnorm\cdot$ is a continuous seminorm on $\cS$; so $\cZ$ extends to a continuous positive definite function on $\cS(\R^3;\Lambda^2)$ (positive definiteness is
closed under pointwise limits). By the Bochner--Minlos theorem \cite{GJ87}, in the form stated after Definition~\ref{ms:fielddef} ($\cS$ is nuclear) there is exactly one probability measure $\mu'$ on
$\cS'(\R^3;\Lambda^2)$ with its cylinder $\sigma$-algebra and $\int e^{i\Phi(u)}d\mu'=\cZ(u)$ for real $u\in\cS$.

\emph{Moments.} For
$u_1,\dots,u_n\in C^\infty_c$ the law of $(\Phi(u_l))_l$ under $\mu'$ has the characteristic function $t\mapsto\cZ(\sum_lt_lu_l)=\Psi_u(it)$, so it is
the law $\mu_u$ of Lemma~\ref{lem:osrest:entire}(c), whose mixed moments are the $S_n(u_1,\dots,u_n)$ and whose exponential moments are finite
(Theorem~\ref{thm:osrest:OS0}(iii)). For $u_l\in\cS$ take cut-offs $\chi_R=\chi(\cdot/R)$, $\chi\in C^\infty_c$, $\chi=1$ on the unit ball,
$0\le\chi\le1$. Then $\Phi(\chi_Ru_l)\to\Phi(u_l)$ for every $\Phi\in\cS'$, and 
\[
\int\prod_l\Phi(\chi_Ru_l)^2d\mu'=S_{2n}(\chi_Ru_1,\chi_Ru_1,\dots) \le\sigma_{2n}\prod_l\wnorm{\chi_Ru_l}^2\le\sigma_{2n}\prod_l\wnorm{u_l}^2
\]
uniformly in $R\ge1$ (the seminorm $\wnorm\cdot$ carries no derivative),
so the products are uniformly integrable and $\int\prod_l\Phi(u_l)d\mu'=\lim_RS_n(\chi_Ru_1,\dots)=S_n(u_1,\dots,u_n)$ by the continuity of $S_n$
on $\cS^n$.

\emph{The two-point bound and the carrier.} $\int\Phi(u)^2d\mu'=S_2(u,u)\le R(u)^2\le3c_{\azero}\wnorm u^2\le3c_{\azero}C^2\|u\|_{10}^2$
on $\cS$, where $\wnorm u\le|u|_4$ (its derivative-free term) and $|u|_4\le C\|u\|_{10}$ (Lemma~\ref{ms:hermite}(b), with the norm $|u|_4$ of a $2$-form as in Definition~\ref{ms:conv0}(a)). The Hermite carrier, Lemma~\ref{ms:carrier}, whose hypothesis ($r\ge14$
even, $\mu'$ a probability measure on $\cS'$ with its cylinder $\sigma$-algebra, $\int\Phi(u)^2d\mu'\le C_2\|u\|_{10}^2$ for every real $u\in\cS$) holds with $C_2=3c_{\azero}C^2$, gives $\mu^H$, the
image of $\mu'$ under the map $\Pi$ of that lemma: a Borel probability measure on $\cHe^{-r}$ with the same characteristic functional and the same mixed moments on $\cS$ as $\mu'$.

\emph{The factorial
moment bound and uniqueness.} Proposition~\ref{ms:unique} gives uniqueness (a) by the characteristic functional on real $C^\infty_c$, on $X$ or on $\cHe^{-r}$, and
(b) by the mixed moments on $C^\infty_c$, when these exist and
$|\int\Phi(u)^nd\mu_1|\le c(n+1)K^nn!$ for all $n\ge0$ and every real $u\in C^\infty_c$, with $c,K\ge0$ depending on $u$. It applies with $c=1$ and $K=eR(u)$, since
\[
|\int\Phi(u)^nd\mu^H|=|S_n(u,\dots,u)|\le e^n(n!)^{1/2}R(u)^n\le(n+1)K^nn!.
\]
This gives the uniqueness in (a); for (b), $\mu:=\iota_*\mu^H$ has
characteristic functional $\cZ$ and moments $S_n$ on $C^\infty_c$ (the restriction map is the identity on $C^\infty_c$), both uniqueness clauses
hold on $X$ by the same proposition, and Lemma~\ref{ms:transfer}(a) ($\mu^H$ is the only Borel probability measure on $\cHe^{-r}$ whose image under
$\iota$ is $\iota_*\mu^H$) gives the last clause.

(c) The invariance of $\mu$ follows from Proposition~\ref{ms:discharge}(a) (if $\gamma$ is a Euclidean motion and
$\cZ(\gamma\cdot u)=\cZ(u)$ for all real $u$, then $\gamma_*\mu=\mu$) and Proposition~\ref{prop:osrest:OS1}, and that of
$\mu^H$ from Lemma~\ref{ms:transfer}(b). The law of $(\Psi(f_j))_j$ is $\mu_f$ as in the moment step.
\end{proof}

\subsection{Clustering in the mean and the vacuum: the weak form of OS4}

\emph{Reconstruction from $S$.} By Proposition~\ref{prop:osrest:OS2} the form $\langle a,b\rangle:=S(\Th a\cdot b)$ is positive
semidefinite and Hermitian on $\Ap$. Let $\Hinf$ be the completion of $\Ap$ modulo the null space of this form, $[a]\in\Hinf$ the class
of $a$, and $\Omega:=[1]$. For $t\ge0$ the translation $\tau_t$ maps $\Ap$ into itself,
and by Theorem~\ref{thm:Hlim}(b),(c), $|S(a)|\le M(a)$ and the argument of Remark~\ref{rem:osrest:recon} the map $[a]\mapsto[\tau_ta]$ is
a strongly continuous self-adjoint contraction semigroup on $\Hinf$; we write it $e^{-tH}$, with $H\ge0$ self-adjoint and
$e^{-tH}\Omega=\Omega$.

\emph{Reconstruction from $\mu$.} For $j\in\{1,2,3\}$ let $L^2_{j,+}:=L^2(\Sigma(\{x_j>0\}),\mu)$, $U(a)G:=G\circ\tau_{-a}$, $\Theta_jG:=G\circ\theta_j$
($\theta_j$ the coordinate reflection on $X$), and $\langle F,G\rangle_j:=\E_\mu[\overline{\Theta_jF}\,G]$ on $L^2_{j,+}$. By Theorem~\ref{thm:osrest:OS4weak}, $\mu$ satisfies (H1)--(H3) of Subsection~\ref{ms:H}, so the reconstruction in a coordinate direction applies (Proposition~\ref{ms:recon}; see Remark~\ref{rem:osrest:recon}): $\cH_j$ is the completion of
$L^2_{j,+}$ modulo the null space of $\langle\cdot,\cdot\rangle_j$, $\Omega_j=[1]$, $[U(te_j)G]=e^{-tH_j}[G]$ ($t\ge0$) with $H_j\ge0$ self-adjoint,
$[U(y)G]=V_j(y)[G]$ with $V_j(y)$ unitary for $y\perp e_j$, and $\E_\mu[FG]=\langle[\overline{\Theta_jF}],[G]\rangle_j$ for $F\in L^2(\Sigma(\{x_j<0\}),\mu)$,
$G\in L^2_{j,+}$. For $j=3$, $\cH_3$ is the curvature sector of $\Hinf$: the map $[e_{(f,0)}]\mapsto[e^{i\Phi(f)}]$, $\supp f\subset
\{x_3>0\}$, preserves the forms ($S(\Th e_{(f,0)}e_{(f',0)})=\cZ(f'-\theta f)=\E_\mu[\overline{e^{i\Phi(f)}\circ\theta_3}e^{i\Phi(f')}]$) and has dense
range (the exponential algebra is dense in $L^2_{3,+}$, by Lemma~\ref{ms:density}), so it extends to a unitary of the closed
$e^{-tH}$-invariant subspace $\Hinf^{\rm curv}\ni\Omega$ generated by the curvature exponentials onto $\cH_3$, carrying $\Omega$ to $\Omega_3$ and
$e^{-tH}$ to $e^{-tH_3}$.

\begin{remark}[the reconstruction]\label{rem:osrest:recon}
The two statements used in the preceding paragraph, the reconstruction proposition (Proposition~\ref{ms:recon}) and the density lemma (Lemma~\ref{ms:density}), are elementary consequences of (H1)--(H3) of Subsection~\ref{ms:H}. They are proved in Appendix~\ref{ms:measure}; we recall the reasons. The relation $\Theta_jU(a)=U(\rho_ja)\Theta_j$ and the invariance of $\mu$ give $\langle U(te_j)F,U(te_j)G\rangle_j=\langle F,
U(2te_j)G\rangle_j$ and $\langle F,U(te_j)G\rangle_j=\langle U(te_j)F,G\rangle_j$ on $L^2_{j,+}$ for $t\ge0$ ($U(te_j)L^2_{j,+}\subset L^2_{j,+}$ by (H2)).
With $\|[G]\|_j^2=\E_\mu[\overline{\Theta_jG}\,G]\le\|G\|_{L^2(\mu)}^2$ (the Schwarz inequality and the $\theta_j$-invariance of $\mu$) and the usual
iteration of $\|[U(te_j)G]\|_j^2=\langle G,U(2te_j)G\rangle_j$, the map $[G]\mapsto[U(te_j)G]$ is a self-adjoint positive contraction semigroup
on $\cH_j$, strongly continuous because $U$ is strongly continuous on $L^2(\mu)$ (bounded continuous functions are dense, $X$ being Polish, and
dominated convergence applies); this is $e^{-tH_j}$ with $H_j\ge0$. For $y\perp e_j$, $U(y)$ maps $L^2_{j,+}$ onto itself by (H2), since $\{x_j>0\}+y=\{x_j>0\}$. 

\emph{The transverse translations.} For $y\perp e_j$, $\Theta_j$ commutes with $U(y)$ by \eqref{ms:eqcommute}, so $\langle U(y)F,U(y)G\rangle_j=\E[\overline{U(y)\Theta_jF}\,U(y)G]=\langle F,G\rangle_j$ by invariance of $\mu$. Thus $V_j(y)$ preserves the null space of $\langle\cdot,\cdot\rangle_j$, is isometric on the quotient and has the isometric inverse $V_j(-y)$; it extends to a unitary operator, and $y\mapsto V_j(y)$ is a representation, strongly continuous by the same estimate as for $e^{-tH_j}$. The operators $e^{-tH_j}$ and $V_j(y)$ commute because the $U(a)$ do, and $e^{-tH_j}\Omega_j=[U(te_j)1]=[1]=\Omega_j$, $V_j(y)\Omega_j=\Omega_j$.

The identity $\E_\mu[FG]=\langle[\overline{\Theta_jF}],[G]\rangle_j$ is the
definition of the form, $\Theta_j$ being an involution commuting with conjugation and $\overline{\Theta_jF}\in L^2_{j,+}$ by (H2). The span of the
$e^{i\Phi(u)}$, $u\in C^\infty_c(\{x_j>0\};\Lambda^2)$ real, is a unital $*$-algebra of bounded functions generating $\Sigma(\{x_j>0\})$, so the
functional monotone class theorem puts every bounded $\Sigma(\{x_j>0\})$-measurable function in its $L^2(\mu)$-closure; this is the density lemma.
\end{remark}

\begin{theorem}[the weak form of OS4]\label{thm:osrest:OS4weak}
$\mu$ on $X$ (and $\mu^H$ on $\cHe^{-r}$) satisfies the hypotheses \textup{(H1)--(H3)} of Subsection~\ref{ms:H}:
\begin{itemize}
\item[(H1)] $X$ is Polish, the translations act jointly continuously, the coordinate reflections $\theta_j$ are involutive homeomorphisms with
$\theta_j\tau_a=\tau_{\rho_ja}\theta_j$, and $\mu$ is invariant under every $\tau_a$ and every $\theta_j$;
\item[(H2)] the local $\sigma$-algebras $\Sigma(U)$ have the monotonicity, exhaustion, countable generation and covariance properties stated in \textup{(H2)} of Subsection~\ref{ms:H};
\item[(H3)] $\E_\mu[\overline{F\circ\theta_{j,c}}\,F]\ge0$ for every $j$, $c\in\R$ and $F\in L^2(\Sigma(\{x_j>c\}),\mu)$, $\theta_{j,c}=\tau_{ce_j}\theta_j
\tau_{-ce_j}$;
\end{itemize}
and $\mu$ is invariant under the cubic group about every point. Consequently the results of Subsection~\ref{ms:quoted} apply to $\mu$: the ergodic decomposition $\mu=\int\mu_\xi\,d\nu(\xi)$ (Theorem~\ref{ms:erg:decomp}) with $\nu$-a.e.\ component translation invariant and ergodic for $\R^3$
(the weak form of OS4), invariant under $\theta_1,\theta_2,\theta_3$, reflection positive about every coordinate plane and satisfying \textup{(H2)}; clustering
in the mean and under diagonal separation on $\nu$-almost every component (Theorems~\ref{ms:mean} and~\ref{ms:diag}, Corollary~\ref{ms:components}(ii)); the reconstruction in each coordinate direction from $\nu$-almost every component, with a unique joint
vacuum (Proposition~\ref{ms:fixed}(iii), Corollary~\ref{ms:components}(iii)); the regrouping of the components into cubic-invariant ones (Theorem~\ref{ms:erg:Ohdecomp}); and the growth bound for the Schwartz moments of the components of $\mu^H$ (Theorem~\ref{ms:erg:growth}), whose
moment hypothesis \eqref{ms:erg:eq:hermitebound} ($\int\Psi(h_{\alpha,c})^{2n}d\mu^H\le c_S(2n)!(C_0\|h_{\alpha,c}\|_{10})^{2n}$) holds with $c_S=1$, $C_0=eC\sqrt{3c_{\azero}}$ ($C$ the constant of Lemma~\ref{ms:hermite}(b)) by
Theorem~\ref{thm:osrest:OS0}(iii).
\end{theorem}
\begin{proof}
In (H1) the properties of $X$ and of the actions are structural for the push-forward actions (Lemma~\ref{ms:borel}(a) and Lemma~\ref{ms:struct}(a)), and the invariance of $\mu$ is Theorem~\ref{thm:osrest:measure}(c). (H2) is structural (Lemma~\ref{ms:struct}(b), which rests on Lemma~\ref{ms:borel}(b),(c)). (H3) holds by Proposition~\ref{ms:discharge}(c), whose two hypotheses ($\mu$ is $\theta_{j,c}$-invariant; $\sum_{k,l\le n}\bar a_ka_l\cZ(u_l-\rho_{j,c}\cdot u_k)\ge0$
for all $n$, all real $u_1,\dots,u_n\in C^\infty_c(\{x_j>c\};\Lambda^2)$ and all $a\in\C^n$)
are Theorem~\ref{thm:osrest:measure}(c) and Theorem~\ref{thm:osrest:OS2matrix} at purely imaginary coefficients (i.e.\ Proposition~\ref{prop:osrest:OS2}
on the exponentials $e_{(u,0)}$). The invariance of $\mu$ under the cubic group about every point is Theorem~\ref{thm:osrest:measure}(c), and the structural hypotheses on the cubic action of the regrouping result (the standing assumption stated before Theorem~\ref{ms:erg:Ohdecomp}, namely that the cubic group acts by homeomorphisms with $\rho_j=\theta_j$, $g\tau_ag^{-1}=\tau_{ga}$ and $\Sigma(U)\mapsto\Sigma(gU)$) are identities for the push-forward action (Definition~\ref{ms:fielddef}, Lemma~\ref{ms:struct}).

The conclusions are the theorems of Subsection~\ref{ms:quoted}, which assume nothing but (H1)--(H3) (and the cubic invariance for
the regrouping, the moment bound for the growth clause). Between $X$ and $\cHe^{-r}$ the hypotheses are carried by Lemma~\ref{ms:transfer}(c), and the conclusions by Proposition~\ref{ms:erg:transfer} (the decomposition of an image law) and Corollary~\ref{ms:erg:growthX}, $\iota$ being a Borel isomorphism onto a Borel set of full measure that intertwines the actions and the local $\sigma$-algebras (Lemma~\ref{ms:borel}(d), Lemma~\ref{ms:transfer}(a), Remark~\ref{ms:met}).
\end{proof}

\begin{remark}[consequences of rotation invariance]\label{rem:osrest:rot}
Suppose that $S$ is invariant under $SO(3)$, hence, together with Proposition~\ref{prop:osrest:OS1}, under the whole Euclidean group. Then $\mu$ is $O(3)$-invariant
(Proposition~\ref{ms:discharge}(a) again) and reflection positive about every plane:
the half-space algebra and reflection of a tilted plane are the rotated ones. Moreover the reconstruction in any direction $n$ is unitarily equivalent to the one in
direction $e_3$, with the same Hamiltonian spectrum. Of the rotation invariance, both statements use only the invariance of $\mu$ under the
Euclidean group; we give the argument. Let $g$ be a Euclidean motion, acting on $X$ by $(g\Phi)(u)=\Phi(g^{-1}\cdot u)$. By the
invariance of $\mu$, $G\mapsto G\circ g^{-1}$ is a unitary of $L^2(\mu)$, and it maps $L^2(\Sigma(V),\mu)$ onto
$L^2(\Sigma(gV),\mu)$ for every open set $V$, because $\Sigma(V)$ is generated by the functions $\Phi\mapsto\Phi(u)$, $u\in C^\infty_c(V;\Lambda^2)$ real.

(i) \emph{Reflection positivity about a plane $\Pi$.} Choose $g$ such that $V_\Pi:=g\{x_3>0\}$ is one of the two open half spaces
of $\Pi$; then $\theta_\Pi:=g\theta_3g^{-1}$ is the reflection in $\Pi$. For $F\in L^2(\Sigma(V_\Pi),\mu)$ put $F':=F\circ g$, a
function in $L^2(\Sigma(\{x_3>0\}),\mu)$. By the invariance of $\mu$ and by (H3) of Theorem~\ref{thm:osrest:OS4weak},
$\E_\mu[\overline{F\circ\theta_\Pi}\,F]=\E_\mu[\overline{F'\circ\theta_3}\,F']\ge0$.

(ii) \emph{The reconstruction in a direction $n$.} For a unit vector $n$ let $g$ be a rotation with $ge_3=n$. The unitary
$G\mapsto G\circ g^{-1}$ carries $L^2_{3,+}$ onto $L^2(\Sigma(\{x\cdot n>0\}),\mu)$, the reflection $\Theta_3$ to the reflection in
the plane $\{x\cdot n=0\}$ and the translations $U(te_3)$ to $U(tn)$, by $g\tau_ag^{-1}=\tau_{ga}$. Hence it carries the
reconstruction of $\mu$ in the direction $e_3$ (the paragraph before Remark~\ref{rem:osrest:recon}) to the reconstruction of
$\mu$ in the direction $n$, which is defined in the same way from the half space $\{x\cdot n>0\}$, and the Hamiltonian $H_3$
to its Hamiltonian: the two reconstructions are unitarily equivalent, with the same spectrum.

Without rotations, what is missing from the full axiom OS1 is exactly the rotation
invariance, and from OS2 the reflection positivity about non-coordinate planes; OS0, OS3, the measure and the weak form of OS4 (Theorem~\ref{thm:osrest:OS4weak}) do not depend on it.
The rotation invariance is proved in Sections~\ref{flh:sec:main}--\ref{sec:i7:proof}:
for the killed-box states by Theorems~\ref{flh:thm:FLH1} and~\ref{flh:thm:FLH1p}, and for the torus limit points along dyadic sequences
(Definition~\ref{n1:def:sides}) by Theorem~\ref{flh:thm:FLH2}, whose proof uses the reflection positivity at the six diagonal site planes
through $0$ (Theorem~\ref{n1:thm:N10}).
\end{remark}



\providecommand{\cK}{\mathcal K}\providecommand{\cP}{\mathcal P}\providecommand{\cX}{\mathcal X}\providecommand{\cB}{\mathcal B}
\providecommand{\cC}{\mathcal C}\providecommand{\Lam}{\Lambda}\providecommand{\avB}[2]{\langle #1\rangle_{#2}}
\providecommand{\avLt}[1]{\langle #1\rangle_{L,t}}\providecommand{\eps}{\varepsilon}\providecommand{\avG}[1]{\langle #1\rangle}
\providecommand{\Kc}{\mathsf K}

\section{Correlation inequalities for the Villain fine model}\label{sec:gin}

This section proves the four correlation inequalities (K1)--(K4) that Section~\ref{sec:d2} uses: the first and second inequalities of
Ginibre \cite{Gin70} (for abelian Higgs models see also \cite{BFS79}, \cite{DDG77}) on a cone of observables, the monotonicity of the cone expectations in every coupling, and the monotonicity in the volume for the
killed boxes together with the comparison of a killed box with the torus. The proofs use the duplicate-variable argument, adapted to the
Villain form of the plaquette weight by a duplication identity of that weight (Lemma~\ref{lem:gin:dup}). Throughout, $N$ is fixed,
$\eps:=\eN$ is the spacing and $\gph$ the phase constant; the couplings $\kappa_N\ge0$, $\lamf>0$, $\rhof>0$ and the bare mass counterterm
$\delta m_B\in\R$ of Section~\ref{sec:setting} (of either sign) are arbitrary. We write
$\rho_x:=\rhof+\tr_x\in[0,\infty)$ for the modulus, $\theta_b:=\gph B_b\in(-\pi,\pi]$ for the angle of a bond and $n_p:=\gph v_p/2\pi\in\Z$
for the Villain integer of a plaquette, so that the curvature is
\[
\ff_p=(dB)_p+v_p=\gph^{-1}\big((d\theta)_p+2\pi n_p\big).
\]
The coboundaries are the standard ones: for a bond $b=(x,x+\eps e_i)$ and a plaquette $p=(x;ij)$, $i<j$, with the four bonds
$(x,i),(x+\eps e_i,j),(x+\eps e_j,i),(x,j)$, $(da)_{x;ij}=a_{x,i}+a_{x+\eps e_i,j}-a_{x+\eps e_j,i}-a_{x,j}$; for a cube $c=(x;123)$,
$(dn)_{x;123}=\sum_{(ijk)\ \mathrm{cyclic}}\big(n_{x+\eps e_i;jk}-n_{x;jk}\big)$ with $n_{x;kj}:=-n_{x;jk}$; $dd=0$. A cochain is closed if its
coboundary vanishes on every cube (or plaquette) of the lattice in question; $v$ closed means $n$ closed.

\subsection{The class of weights, the states, the cone}\label{sec:gin:setting}

\begin{definition}[finite lattices; the class of weights]\label{def:gin:weights}
A \emph{finite lattice} $G$ is a finite disjoint union $G=G_1\sqcup\dots\sqcup G_r$ ($r\ge1$) of \emph{components}, each of which is
either a fine torus $\tilde T_L$ or a \emph{box} $\Lam\subset\eps\Z^3$, a product of three finite intervals of sites; the components are
distinct copies, and the case $r=1$, a single torus or box, is the case of interest. The cells of a component $G_i$ are: the sites
$\cX(G_i)$; the bonds $\cB(G_i)$ (for a box, the bonds with both ends in $\Lam$); the plaquettes $\cP(G_i)$ (for a box, those with all four
bonds in $\cB(\Lam)$); the cubes $\cC(G_i)$ (for a box, those with all six faces in $\cP(\Lam)$). The cells of $G$ are the disjoint unions
$\cX(G):=\bigsqcup_i\cX(G_i)$, $\cB(G)$, $\cP(G)$, $\cC(G)$ of those of its components; every cell of $G$ lies in exactly one component,
and every face of a cube of $G$ lies in the component of that cube.

The variables are
$\rho_x\in[0,\infty)$ ($x\in\cX(G)$), $B_b\in(-\pi/\gph,\pi/\gph]$ ($b\in\cB(G)$) and $v\in(2\pi/\gph)\Z^{\cP(G)}$ closed on $\cC(G)$, i.e.\ with
$(dv)_c=0$ for every cube $c\in\cC(G)$; since every cube lies in one component, $v$ is closed on $\cC(G)$ exactly when each restriction
$v|_{\cP(G_i)}$ is closed on $\cC(G_i)$, and the closed cochains of $G$ are the products of the closed cochains of its components.

Given
finite positive Borel measures $\mu_x$ on $[0,\infty)$ with
\begin{equation}\label{eq:gin:moments}
\int_0^\infty e^{\alpha\rho^2}\mu_x(d\rho)<\infty\qquad\text{for every }\alpha\in\R ,
\end{equation}
coefficients $\kappa_b\ge0$ ($b\in\cB(G)$) and $\beta_p>0$ ($p\in\cP(G)$), the \emph{weight} $W=W_G(\mu,\kappa,\beta)$ is the measure
\[
\prod_{x\in\cX(G)}\mu_x(d\rho_x)\prod_{b\in\cB(G)}e^{\kappa_b\rho_{b_-}\rho_{b_+}\cos\theta_b}\,dB_b\sum_{v\ \mathrm{closed}}\ \prod_{p\in\cP(G)}
e^{-\beta_p\ff_p^2/2}
\]
and $\avG F:=\int F\,W/\int W$ is its \emph{state}. For $r\ge2$ every factor of $W$ is indexed by a cell of one component and the sum over
the closed $v$ factorises over the components, so $W=W_{G_1}\otimes\dots\otimes W_{G_r}$ is the product of the weights of the components (each
with the restriction of $\mu,\kappa,\beta$ to its cells), and the state of $G$ is the product state: functions of the variables of different
components are independent under $\avG\cdot$.
\end{definition}

The mass $Z:=\int W$ is finite and positive: for each $p$, $\sum_{n\in\Z}e^{-\beta_p(y+2\pi n)^2/
(2\gph^2)}\le K_p:=\sum_{m\in\Z}e^{-\beta_p(\pi m)^2/(2\gph^2)}$ for every real $y$ (the term with $y+2\pi n\in(-\pi,\pi]$ is at most $1$, and
the $m$-th term on either side of it has $|y+2\pi n|\ge2\pi|m|-\pi\ge\pi|m|$), so the $v$-sum is at most $\prod_pK_p$ uniformly in $B$; and
$e^{\kappa_b\rho\rho'}\le e^{\kappa_b(\rho^2+\rho'^2)/2}$ with \eqref{eq:gin:moments} bounds the $\rho$-integral. Every $F$ with $|F|\le
C\,e^{\sum_xq_x\rho_x^2}$ is $W$-integrable, by the same two bounds.

Let $\mu^\circ(d\rho):=g_{\mathrm{ct}}(\rho-\rhof)\,e^{-3\kappa_N\rho^2}\,d\rho$ on $[0,\infty)$, where $g_{\mathrm{ct}}(\tr):=g(\tr)\,e^{-\frac12\delta m_B(\rhof+\tr)^2}$,
$g(\tr)=(\rhof+\tr)e^{-\lamf(\rho^2-\rhof^2)^2}$, is the site factor of Section~\ref{sec:setting} with its bare mass counterterm, $\delta m_B\in\R$ of either sign.
The measure $\mu^\circ$ is finite and positive and satisfies \eqref{eq:gin:moments} for every $\alpha\in\R$ and every real $\delta m_B$, because the quartic dominates
(\eqref{eq:ct} at $q=|\alpha|$, times $e^{-3\kappa_N\rho^2}\le1$): for $\rho\ge0$,
\[
g_{\mathrm{ct}}(\rho-\rhof)\,e^{-3\kappa_N\rho^2}\,e^{\alpha\rho^2}\le\rho\,e^{\lamf\rhof^4}\,e^{-\frac12\lamf\rho^4+(|\alpha|+\frac12|\delta m_B|)\rho^2},
\]
by $\lamf(\rho^2-\rhof^2)^2\ge\frac12\lamf\rho^4-\lamf\rhof^4$ and $e^{\alpha\rho^2}e^{-\frac12\delta m_B\rho^2}\le e^{(|\alpha|+\frac12|\delta m_B|)\rho^2}$; the right side is
integrable over $[0,\infty)$ since $\lamf>0$.

\begin{definition}[torus states and killed-box states]\label{def:gin:box}
\begin{itemize}
\item[(a)] The \emph{torus state} $\avL\cdot$ of Section~\ref{sec:setting} is the state of $W_{\tilde T_L}(\mu^\circ,\kappa_N,1)$: every site of
the torus is an end of six bonds, and the hopping factor $h$ of each bond contributes the diagonal factor $e^{-\kappa_N\rho^2/2}$ to each of its
ends, so $\prod_xg_{\mathrm{ct}}\prod_bh=\prod_xg_{\mathrm{ct}}(\tr_x)e^{-3\kappa_N\rho_x^2}\prod_be^{\kappa_N\rho_{b_-}\rho_{b_+}\cos\theta_b}$.
\item[(b)] For a box $\Lam$ the \emph{killed-box state} $\avB\cdot\Lam$ is the state of $W_\Lam(\mu^\circ,\kappa_N,1)$: the weight
\[
W_\Lam=\prod_{x\in\Lam}g_{\mathrm{ct}}(\tr_x)e^{-3\kappa_N\rho_x^2}\prod_{b\in\cB(\Lam)}e^{\kappa_N\rho_{b_-}\rho_{b_+}\cos(\gph B_b)}\prod_{p\in\cP(\Lam)}
e^{-\ff_p^2/2}
\]
in the variables $\tr_x$ ($x\in\Lam$), $B_b$ ($b\in\cB(\Lam)$), $v_p$ ($p\in\cP(\Lam)$), the sum over the $v$ closed on the cubes of $\Lam$.
It is the weight of Section~\ref{sec:setting} with the matter killed outside $\Lam$ ($\rho_y=0$ for $y\notin\Lam$, which turns the hopping
factor of every bond leaving $\Lam$ into the diagonal factor $e^{-\kappa_N\rho_x^2/2}$ of its inner end, so that every site of $\Lam$ carries
the diagonal halves of all six of its bonds) and with no gauge or Villain variables outside $\cB(\Lam)$, $\cP(\Lam)$.
\end{itemize}
A function is \emph{supported in} $\Lam$ if it depends only on $\ff_p$ ($p\in\cP(\Lam)$), $B_b$ ($b\in\cB(\Lam)$) and $\rho_x$ ($x\in\Lam$).
\end{definition}

\begin{definition}[the cone $\cK_N$ and the enlarged cone $\cK_N^+$]\label{def:gin:cone}
For a real finitely supported plaquette function $a=(a_p)$, an integer finitely supported bond function $k=(k_b)$, a polynomial $P$ in finitely
many $\rho_x$ with non-negative coefficients and a finitely supported $q=(q_x)$ with $q_x\ge0$, put
\[
\Theta_{a,k,P,q}:=\cos\Big(\sum_pa_p\ff_p+\sum_bk_b\theta_b\Big)\,P(\rho)\,e^{\sum_xq_x\rho_x^2},\qquad\Theta_{a,P,q}:=\Theta_{a,0,P,q}.
\]
The \emph{cone} $\cK_N$ is the set of finite non-negative linear combinations of the $\Theta_{a,P,q}$, and the \emph{enlarged cone} $\cK_N^+$
that of the $\Theta_{a,k,P,q}$; $\cK_N\subset\cK_N^+$. The \emph{generators} of $\cK_N$ are the $\Theta_{a,1,q}$ (this is the sense of
Section~\ref{sec:d2}; the polynomial factors are in the cone but are not needed to generate the state). Both cones are closed under products
($\cos u\cos v=\frac12[\cos(u+v)+\cos(u-v)]$, and products of polynomials with non-negative coefficients and of the exponentials stay in the
class). In particular, for every site $x$, bond $b$ and plaquette $p$ and every $q\ge0$,
\[
\rho_x^2\in\cK_N,\qquad e^{q\rho_x^2}\in\cK_N,\qquad\rho_x^2e^{q\rho_x^2}\in\cK_N,\qquad\cos\Big(\sum_pa_p\ff_p\Big)\in\cK_N,\qquad
\rho_{b_-}\rho_{b_+}\cos\theta_b\in\cK_N^+
\]
($a=0$, $P=\rho_x^2$ or $P=1$; the last with $a=0$, $k=\delta_b$, $P=\rho_{b_-}\rho_{b_+}$).

Every element of $\cK_N^+$ is continuous, is
local, and satisfies $|\Theta|\le Ce^{\sum_xq'_x\rho_x^2}$ for some $C<\infty$ and finitely supported $q'\ge0$; hence it is integrable for
every weight of Definition~\ref{def:gin:weights}. On a finite lattice $G$ the cone $\cK_N$ and the enlarged cone $\cK_N^+$ \emph{of $G$} are defined by the same formulas with $a$, $k$, $P$, $q$ indexed by the plaquettes, bonds and sites of $G$ (cells of different components may occur in one element: the functions are evaluated on the cells of $G$, and only elements
whose plaquettes, bonds and sites are cells of $G$ are considered). The energy densities $\ff_p^2$ are not in $\cK_N^+$; their covariances
with the cone have a sign of their own, (K3) of Theorem~\ref{thm:gin:K}.
\end{definition}

\subsection{The duplication identity and three lemmas}\label{sec:gin:lemmas}

Fix a finite lattice $G$ and write $\cB=\cB(G)$, $\cP=\cP(G)$, $\cC=\cC(G)$ ($G$ may have several components; every statement of this
subsection is about the cells of $G$, with closedness imposed cube by cube, and holds component by component). Two independent copies $(\rho,\theta,n)$ and $(\rho',\theta',n')$
of the variables are called the \emph{duplicate variables}; $F'$ denotes the function $F$ evaluated in the primed copy. The new variables are
\begin{equation}\label{eq:gin:newvars}
u_x:=\tfrac12(\rho_x+\rho'_x),\quad v_x:=\tfrac12(\rho'_x-\rho_x),\qquad\alpha_b:=\tfrac12(\theta_b+\theta'_b),\quad\eta_b:=\tfrac12(\theta_b-\theta'_b),
\end{equation}
so that $\rho=u-v$, $\rho'=u+v$ with $u_x\ge|v_x|$, and $\theta=\alpha+\eta$, $\theta'=\alpha-\eta$.

\begin{lemma}[duplication identity of the Villain weight]\label{lem:gin:dup}
Let $\sigma$ range over $\{0,1\}^{\cP}$ and, for each $\sigma$, let $\Kc_\sigma:=\{k\in\Z^{\cP}:\ 2dk+d\sigma=0\ \text{on }\cC\}$ (possibly empty).
If $G$ has the components $G_i$, write $\Kc^{(i)}_\sigma$ for the set defined by the same condition for $\sigma|_{\cP(G_i)}$ on the cubes of $G_i$; then $k\in\Kc_\sigma$ exactly when each restriction $k|_{\cP(G_i)}$ lies in $\Kc^{(i)}_\sigma$,
because the condition is imposed cube by cube and every cube lies in one component: $\Kc_\sigma=\prod_i\Kc^{(i)}_\sigma$.
The map $(k,l,\sigma)\mapsto(n,n'):=(k+l+\sigma,\;k-l)$ is a bijection from $\{(k,l,\sigma):\ k,l\in\Kc_\sigma\}$ onto the pairs of closed integer
cochains $(n,n')$, with inverse $\sigma_p=(n_p+n'_p)\bmod2$, $k=\frac12(n+n'-\sigma)$, $l=\frac12(n-n'-\sigma)$. Under it, with
\begin{equation}\label{eq:gin:XY}
\begin{gathered}
X_p:=(d\alpha)_p+\pi\sigma_p+2\pi k_p,\qquad Y_p:=(d\eta)_p+\pi\sigma_p+2\pi l_p,\\
\xi_p:=X_p/\gph,\qquad\upsilon_p:=Y_p/\gph,
\end{gathered}
\end{equation}
one has $(d\theta)_p+2\pi n_p=X_p+Y_p$ and $(d\theta')_p+2\pi n'_p=X_p-Y_p$, i.e.\ $\ff_p=\xi_p+\upsilon_p$, $\ff'_p=\xi_p-\upsilon_p$, and
\begin{equation}\label{eq:gin:dup}
\begin{gathered}
\sum_{n,n'\ \mathrm{closed}}\ \prod_{p\in\cP}e^{-\beta_p(\ff_p^2+\ff_p'^2)/2}=\sum_{\sigma}\ \sum_{k\in\Kc_\sigma}\sum_{l\in\Kc_\sigma}
V_\sigma(\alpha,k)\,V_\sigma(\eta,l),\\
V_\sigma(\alpha,k):=\prod_{p\in\cP}e^{-\beta_pX_p^2/\gph^2}\ge0 ,
\end{gathered}
\end{equation}
where $V_\sigma(\eta,l)$ is the same function evaluated at $(\eta,l)$. Moreover $\sum_{k\in\Z^{\cP}}V_\sigma(\alpha,k)\le\prod_pK_p$ and
$\sum_{k\in\Z^{\cP}}|X_p|^jV_\sigma(\alpha,k)\le C_j<\infty$ uniformly in $\alpha$ and $\sigma$, for every $j\ge0$ and $p$.
\end{lemma}
\begin{proof}
Given closed $n,n'$, the integers $n_p+n'_p$ and $n_p-n'_p$ have the same parity, so $\sigma_p:=(n_p+n'_p)\bmod2\in\{0,1\}$ makes
$k:=\frac12(n+n'-\sigma)$ and $l:=\frac12(n-n'-\sigma)$ integer cochains; $2dk=dn+dn'-d\sigma=-d\sigma$ and $2dl=-d\sigma$ on $\cC$, so
$k,l\in\Kc_\sigma$, and $k+l+\sigma=n$, $k-l=n'$. Conversely, for $k,l\in\Kc_\sigma$ the cochains $n:=k+l+\sigma$ and $n':=k-l$ are integer
and closed: $dn=dk+dl+d\sigma=-\frac12d\sigma-\frac12d\sigma+d\sigma=0$ and $dn'=dk-dl=0$; and the two maps are inverse to each other
($(n+n')\bmod2=(2k+\sigma)\bmod2=\sigma$). Then $(d\theta)_p+2\pi n_p=(d\alpha)_p+(d\eta)_p+2\pi(k_p+l_p+\sigma_p)=X_p+Y_p$ and
$(d\theta')_p+2\pi n'_p=(d\alpha)_p-(d\eta)_p+2\pi(k_p-l_p)=X_p-Y_p$. Since $(X+Y)^2+(X-Y)^2=2X^2+2Y^2$,
$e^{-\beta_p[(X_p+Y_p)^2+(X_p-Y_p)^2]/(2\gph^2)}=e^{-\beta_pX_p^2/\gph^2}e^{-\beta_pY_p^2/\gph^2}$, which is \eqref{eq:gin:dup} after
the reindexing of the double sum.

For the uniform bounds, observe that for all real $y_p$
\[
\sum_{k\in\Z^{\cP}}\prod_pe^{-\beta_p(y_p+2\pi k_p)^2/\gph^2}=\prod_p\sum_{k\in\Z} e^{-\beta_p(y_p+2\pi k)^2/\gph^2}\le\prod_pK_p ,
\]
because each summand is at most the summand $e^{-\beta_p(y_p+2\pi k)^2/(2\gph^2)}$ of
Definition~\ref{def:gin:weights}, whose sum over $k$ is at most $K_p$ (here $K_p$ is the constant of the halved stiffness).
With a factor $|y_p+2\pi k_p|^j$ in one plaquette,
the one-dimensional sum $\sum_k|y+2\pi k|^je^{-\beta_p(y+2\pi k)^2/\gph^2}$ is bounded uniformly in $y$ by the same comparison with
$\pi|m|$, since the function $s\mapsto s^je^{-\beta_ps^2/\gph^2}$ is bounded on $[0,\infty)$ and summable along $s=\pi m$.
The identity \eqref{eq:gin:dup} is a product over the plaquettes of $G$ of per-plaquette factors and the bijection is defined plaquette by plaquette, with closedness imposed cube by cube;
so for a lattice with several components it is the product of the identities of the components, with $\Kc_\sigma$ the product set above.
\end{proof}

\begin{lemma}[the angle domain]\label{lem:gin:angle}
Let $F$ be an integrable function on $(-\pi,\pi]^{\cB}\times(-\pi,\pi]^{\cB}$ which is $2\pi$-periodic in each of its $2|\cB|$ arguments.
Then
\[
\iint_{(-\pi,\pi]^{2\cB}}F(\theta,\theta')\,d\theta\,d\theta'=\iint_{(-\pi,\pi]^{2\cB}}F(\alpha+\eta,\alpha-\eta)\,d\alpha\,d\eta .
\]
\end{lemma}
\begin{proof}
Regard $F$ as a function on the compact group $\mathbb T:=(\R/2\pi\Z)^{2\cB}$ (here $\cB$ may be any
finite set). The map $\Psi(\alpha,\eta):=(\alpha+\eta,\alpha-\eta)$ is a
continuous homomorphism $\mathbb T\to\mathbb T$, and it is onto: $(\theta,\theta')=\Psi(\frac12(\theta+\theta'),\frac12(\theta-\theta'))$.
The image under $\Psi$ of the Haar measure $m$ of $\mathbb T$ is translation invariant: for $y=\Psi(x)$ and Borel $A$, $\Psi_*m(A+y)=
m(\Psi^{-1}(A)+x)=m(\Psi^{-1}(A))=\Psi_*m(A)$, using $\Psi^{-1}(A+\Psi(x))=\Psi^{-1}(A)+x$. By the uniqueness of the Haar measure $\Psi_*m$ is
a multiple of $m$, and the multiple is $1$ because both have total mass $m(\mathbb T)$. Hence $\int F\circ\Psi\,dm=\int F\,dm$, which is the
claim in the coordinates $(-\pi,\pi]^{2\cB}$ of $\mathbb T$.
\end{proof}

\begin{lemma}[closed cochains on a box are exact; extension]\label{lem:gin:ext}
Let $\Lam$ be a box and $n\in\Z^{\cP(\Lam)}$ closed on $\cC(\Lam)$. Then there is $a\in\Z^{\cB(\Lam)}$ with $da=n$ on $\cP(\Lam)$.
Consequently, if $G$ is any finite lattice whose cells contain those of $\Lam$ (for instance a box $\Lam'\supset\Lam$, or a torus $\tilde T_L$ in which the
sites, bonds, plaquettes and cubes of $\Lam$ are cells, i.e.\ $\Lam$ a translate of a sub-box of the
fundamental box $\Lam_L$ of \textup{(K4)} of Theorem~\ref{thm:gin:K}), then every $n$ closed on $\cC(\Lam)$ is the restriction to $\cP(\Lam)$ of a cochain closed on $\cC(G)$, and two
such extensions differ by an element of the group $\Kc(G,\Lam):=\{k\in\Z^{\cP(G)}:\ k\ \text{closed on }\cC(G),\ k=0\ \text{on }\cP(\Lam)\}$.
\end{lemma}
\begin{proof}
Let $\Lam=\prod_{i=1}^3\{m_i,m_i+\eps,\dots,M_i\}$. Put $a_{x,1}:=0$ for every bond in direction $1$. On the face $x_1=m_1$ put $a_{x,3}:=0$
and $a_{x,2}:=-\sum_{m_3\le y_3<x_3}n_{(m_1,x_2,y_3);23}$, so that $(da)_{x;23}=a_{x,2}-a_{x+\eps e_3,2}=n_{x;23}$ there. For $x_1>m_1$ put
$a_{x,j}:=a_{(m_1,x_2,x_3),j}+\sum_{m_1\le y_1<x_1}n_{(y_1,x_2,x_3);1j}$ for $j=2,3$. Then $(da)_{x;1j}=a_{x+\eps e_1,j}-a_{x,j}=n_{x;1j}$
for $j=2,3$ on every plaquette of $\cP(\Lam)$. For the $23$-plaquettes let $D_x:=(da)_{x;23}-n_{x;23}$; then $D_x=0$ at $x_1=m_1$, and
\[
D_{x+\eps e_1}-D_x=n_{x;12}+n_{x+\eps e_2;13}-n_{x+\eps e_3;12}-n_{x;13}-\big(n_{x+\eps e_1;23}-n_{x;23}\big)=-(dn)_{x;123}=0
\]
by the formulas for $da$ and $dn$ (the four differences of $a$ are the $12$- and $13$-increments just computed); so $D\equiv0$. For the
extension, extend $a$ by zero to $\cB(G)$ and put $\tilde n:=da$ on $\cP(G)$: it is closed ($dd=0$), and on a plaquette of $\cP(\Lam)$,
whose four bonds lie in $\cB(\Lam)$, it equals $n$. If $\tilde n_1,\tilde n_2$ are two closed extensions, $\tilde n_1-\tilde n_2$ is closed
and vanishes on $\cP(\Lam)$.
\end{proof}

\begin{lemma}[the endpoint of the interpolation]\label{lem:gin:end}
Let $\Lam$ be a box and $G$ any finite lattice whose cells contain those of $\Lam$ (for instance a box $\Lam'\supset\Lam$, or a torus $\tilde T_L$ in
which the sites, bonds, plaquettes and cubes of $\Lam$ are cells, as in Lemma~\ref{lem:gin:ext}), and let $\mu_x$ ($x\in\cX(G)$), $\kappa_b\ge0$ ($b\in\cB(G)$), $\beta_p>0$ ($p\in\cP(G)$)
be as in Definition~\ref{def:gin:weights}. For $t\in(0,1]$ let $\avG\cdot_t$ be the state of the weight $W_G(\mu,\kappa^{(t)},\beta^{(t)})$
with $\kappa^{(t)}_b=\kappa_b$ for $b\in\cB(\Lam)$, $\kappa^{(t)}_b=t\kappa_b$ for $b\notin\cB(\Lam)$, $\beta^{(t)}_p=\beta_p$ for $p\in\cP(\Lam)$,
$\beta^{(t)}_p=t\beta_p$ for $p\notin\cP(\Lam)$, and let $\avB\cdot\Lam$ be the state of $W_\Lam(\mu,\kappa,\beta)$ (the restrictions of
$\mu,\kappa,\beta$ to the cells of $\Lam$). Then for every $\Theta$ supported in $\Lam$ with $|\Theta|\le Ce^{\sum_xq_x\rho_x^2}$,
\[
\lim_{t\downarrow0}\avG\Theta_t=\avB\Theta\Lam .
\]
\end{lemma}
\begin{proof}
Write $\cP_{\mathrm{new}}:=\cP(G)\setminus\cP(\Lam)$, $\cB_{\mathrm{new}}$, $\cX_{\mathrm{new}}$ likewise, and $\Kc:=\Kc(G,\Lam)$, a subgroup of
$\Z^{\cP_{\mathrm{new}}}$, free of some rank $r\ge0$. For each $n^\circ$ closed on $\cC(\Lam)$ fix one extension $E(n^\circ)$ closed on $\cC(G)$
(Lemma~\ref{lem:gin:ext}); by that lemma the cochains closed on $\cC(G)$ are exactly the $E(n^\circ)+k$, $k\in\Kc$, each once. On
$\R^{\cP_{\mathrm{new}}}$ use the inner product $\langle z,w\rangle_\beta:=\sum_{p\in\cP_{\mathrm{new}}}\beta_pz_pw_p/\gph^2$ and the norm
$Q(z):=\langle z,z\rangle_\beta$; let $V$ be the real span of $\Kc$ ($\dim V=r$) and $z=z_V+z_\perp$ the orthogonal decomposition. For
$y\in\R^{\cP_{\mathrm{new}}}$ put
\[
S_t(y):=\sum_{k\in\Kc}\prod_{p\in\cP_{\mathrm{new}}}e^{-t\beta_p(y_p+2\pi k_p)^2/(2\gph^2)}=\sum_{k\in\Kc}e^{-tQ(y+2\pi k)/2}
=e^{-tQ(y_\perp)/2}\sum_{k\in\Kc}e^{-tQ(y_V+2\pi k)/2},
\]
since $2\pi k\in V$.

Poisson summation on the lattice $2\pi\Kc$ of the Euclidean space $(V,\langle\cdot,\cdot\rangle_\beta)$ (the
$r$-dimensional form of the identity behind \eqref{eq:poisson}; for $r=0$ the sum is the single term $1$) gives
\[
\sum_{k\in\Kc}e^{-tQ(y_V+2\pi k)/2}=\frac{(2\pi/t)^{r/2}}{\mathrm{vol}(V/2\pi\Kc)}\sum_{\lambda\in(2\pi\Kc)^*}e^{-Q(\lambda)/(2t)}
e^{i\langle\lambda,y_V\rangle_\beta}=:c_\Kc\,t^{-r/2}\big(1+\varepsilon_t(y_V)\big),
\]
with $(2\pi\Kc)^*:=\{\lambda\in V:\ \langle\lambda,2\pi k\rangle_\beta\in2\pi\Z\ \text{for all }k\in\Kc\}$ the dual lattice (the convention
that goes with the characters $e^{i\langle\lambda,\cdot\rangle_\beta}$ in the identity) and $c_\Kc:=(2\pi)^{r/2}/\mathrm{vol}(V/2\pi\Kc)\in(0,\infty)$,
whose value cancels in the quotient at the end of the proof. With $\delta^2:=\min\{Q(\lambda):\lambda\in
(2\pi\Kc)^*\setminus\{0\}\}>0$ and $t\le1$, $|\varepsilon_t(y_V)|\le\sum_{\lambda\ne0}e^{-Q(\lambda)/2}e^{-\delta^2(1/t-1)/2}=:\bar\varepsilon_t$,
which is finite and tends to $0$ as $t\downarrow0$, uniformly in $y$.

Now fix the old variables $\rho_x$ ($x\in\Lam$), $B_b$ ($b\in\cB(\Lam)$), $n^\circ$ closed on $\cC(\Lam)$ and the new variables $\rho_x$
($x\in\cX_{\mathrm{new}}$), $B_b$ ($b\in\cB_{\mathrm{new}}$), and put $y_p:=(d\theta)_p+2\pi E(n^\circ)_p$ for $p\in\cP_{\mathrm{new}}$. Every
closed cochain of $G$ is $E(n^\circ)+k$ with $k=0$ on $\cP(\Lam)$, so on $\cP(\Lam)$ the curvature is $\ff_p=\gph^{-1}((d\theta)_p+2\pi
n^\circ_p)$, independent of $k$; hence $\Theta$ (supported in $\Lam$) and the plaquette factors of $\cP(\Lam)$ do not depend on $k$, and the
sum over $k$ of the new plaquette factors is $S_t(y)$. Therefore, with 
\[
W^\circ:=\prod_{x\in\Lam}\mu_x(d\rho_x)\prod_{b\in\cB(\Lam)} e^{\kappa_b\rho_{b_-}\rho_{b_+}\cos\theta_b}dB_b\prod_{p\in\cP(\Lam)}e^{-\beta_p\ff_p^2/2}
\]
(the weight of $\avB\cdot\Lam$ at the given
$n^\circ$),
\begin{multline*}
\frac{t^{r/2}}{c_\Kc}\int\Theta\,W_G(\mu,\kappa^{(t)},\beta^{(t)})=\int\prod_{x\in\cX_{\mathrm{new}}}\mu_x(d\rho_x)\prod_{b\in\cB_{\mathrm{new}}}dB_b\\
\sum_{n^\circ}\int\Theta\,W^\circ\ \prod_{b\in\cB_{\mathrm{new}}}e^{t\kappa_b\rho_{b_-}\rho_{b_+}\cos\theta_b}\,e^{-tQ(y_\perp)/2}\big(1+\varepsilon_t(y_V)\big).
\end{multline*}

The last three factors tend to $1$ as $t\downarrow0$, for every fixed value of the variables ($y$ does not depend on $t$), and their product
is bounded by $(1+\bar\varepsilon_1)\prod_{b\in\cB_{\mathrm{new}}}e^{\kappa_b(\rho_{b_-}^2+\rho_{b_+}^2)/2}$. The remaining integrand,
$|\Theta|\,W^\circ$ times this bound times the new site measures, is integrable: the new $B_b$ range over a bounded set, $\sum_{n^\circ}
\prod_{p\in\cP(\Lam)}e^{-\beta_p\ff_p^2/2}\le\prod_pK_p$, and the $\rho$-integral is finite by \eqref{eq:gin:moments} and $|\Theta|\le
Ce^{\sum q_x\rho_x^2}$. By dominated convergence the right side tends to $(2\pi/\gph)^{|\cB_{\mathrm{new}}|}\prod_{x\in\cX_{\mathrm{new}}}
\mu_x([0,\infty))\int\Theta\,W_\Lam(\mu,\kappa,\beta)$. The same holds with $\Theta$ replaced by $1$; the common factor is in $(0,\infty)$, and the
quotient of the two limits is $\avB\Theta\Lam$.
\end{proof}

\subsection{The inequalities}\label{sec:gin:thm}

\begin{theorem}[Ginibre's inequalities for the Villain fine model]\label{thm:gin:K}
Let $\avG\cdot$ be the state of a weight $W_G(\mu,\kappa,\beta)$ of Definition~\ref{def:gin:weights} on a finite lattice $G$, a finite
disjoint union of fine tori and boxes with the product weight (for instance the disjoint union of two copies of one torus carrying the same
weight, whose state is the product of two independent copies of one state of a torus weight); in particular a
torus state $\avL\cdot$ or a killed-box state $\avB\cdot\Lam$ of Definition~\ref{def:gin:box}, for every $\gph>0$, every $\kappa_N\ge0$,
$\lamf>0$, $\rhof>0$, every real $\delta m_B$ and every side $L$. For $\Theta,\Theta'\in\cK_N^+$ whose cells are cells of $G$ (in particular for $\Theta,\Theta'\in\cK_N$):
\begin{itemize}
\item[(K1)] $\avG\Theta\ge0$;
\item[(K2)] $\avG{\Theta;\Theta'}:=\avG{\Theta\Theta'}-\avG\Theta\avG{\Theta'}\ge0$;
\item[(K3)] $\avG\Theta$ is a continuously differentiable function of each $\kappa_b\in[0,\infty)$, of each $\beta_p\in(0,\infty)$ and of
$q\in[0,\infty)$ under $\mu_x\mapsto e^{q\rho^2}\mu_x$ (one site $x$), the other parameters held fixed, and its partial derivatives are
continuous functions of all the parameters $(\kappa_b)_b$, $(\beta_p)_p$, $q$ jointly (so that $\avG\Theta$ is continuously differentiable
along every continuously differentiable path of parameters), with
\begin{equation}\label{eq:gin:deriv}
\partial_{\kappa_b}\avG\Theta=\avG{\Theta;\rho_{b_-}\rho_{b_+}\cos\theta_b}\ge0,\qquad
\partial_{\beta_p}\avG\Theta=-\tfrac12\avG{\Theta;\ff_p^2}\ge0,\qquad
\partial_q\avG\Theta=\avG{\Theta;\rho_x^2}\ge0 ;
\end{equation}
hence $\avG\Theta$ is non-decreasing in each of them. For the torus states and the killed-box states this means that the expectation of $\Theta$ is non-decreasing in each hopping coefficient
$\kappa_b\ge0$ (the coefficient of $\rho_{b_-}\rho_{b_+}\cos(\gph B_b)$, bond by bond, the diagonal factors held fixed), in each plaquette
stiffness $\beta_p>0$ ($w_p(f)=e^{-\beta_pf^2/2}$) and in $q\ge0$ under the change of the site weight $g_{\mathrm{ct}}(\tr_x)\mapsto g_{\mathrm{ct}}(\tr_x)e^{q\rho_x^2}$.
\item[(K4)] For boxes $\Lam\subset\Lam'$ and $\Theta$ supported in $\Lam$: $\avB\Theta\Lam\le\avB\Theta{\Lam'}$. Identify the sites of
$\tilde T_L$ with the box $\Lam_L:=\{0,\eps,\dots,L-\eps\}^3$, its bonds and plaquettes with the pairs (site, direction) and triples (site,
two directions) based at those sites, so that $\cB(\Lam_L)\subset\cB(\tilde T_L)$ and $\cP(\Lam_L)\subset\cP(\tilde T_L)$. Then for every box
$\Lam\subset\Lam_L$ and $\Theta$ supported in $\Lam$: $\avB\Theta\Lam\le\avB\Theta{\Lam_L}\le\avL\Theta$. In particular, if $\Lam$ lies in a cube of side less
than $L$ (a fortiori if $\Lam$ together with its nearest-neighbour sites does), a
translation of the torus (which leaves $\avL\cdot$ invariant) brings $\Lam$ into $\Lam_L$, and $\avB\Theta\Lam\le\avL\Theta$.
\end{itemize}
\end{theorem}

The proof occupies the rest of this subsection. We derive (K1) from the positivity of the Fourier coefficients of the weight, as in the proof of
Theorem~\ref{thm:GD}, then (K2) and the three signs of (K3) from the duplication identity, and finally (K4) from (K3) and Lemma~\ref{lem:gin:end}. By
linearity it suffices to treat $\Theta=\Theta_{a,k,P,q}$ and $\Theta'=\Theta_{a',k',P',q'}$. Every step of the proofs of (K1)--(K3) is cell by
cell: the site, bond and plaquette factors are indexed by the cells of $G$, the closure constraint is imposed cube by cube, the duplication
identity is a product over the plaquettes and the angle domain a product over the bonds; so the proofs apply without change when $G$ has several components,
and (K1)--(K3) hold on every finite lattice of Definition~\ref{def:gin:weights}. (K4) is a statement about boxes and the torus, and the
lattice $G$ of Lemma~\ref{lem:gin:end} in its proof is a single box or a single torus.

\begin{proof}[Proof of (K1)]
Let $\Theta=\Theta_{a,k,P,q}$ and $\Phi:=\sum_pa_p\ff_p+\sum_bk_b\theta_b$. Put $Z(\Theta):=\int P(\rho)e^{\sum_xq_x\rho_x^2}e^{i\Phi}\,W$, so
that $\avG\Theta=\mathrm{Re}\,Z(\Theta)/Z$; we show $Z(\Theta)\ge0$.

\emph{The plaquette sum.} For $\beta>0$, real $x,s,a$ and $c^\beta_u:=(\gph/\sqrt{2\pi\beta})\,e^{-\gph^2u^2/(2\beta)}>0$,
\begin{equation}\label{eq:gin:poissonbeta}
\sum_{v\in(2\pi/\gph)\Z}e^{-\beta(x+v)^2/2}e^{ia(x+v)}e^{i\gph sv}=e^{-a^2/(2\beta)}\sum_{m\in\Z}c^\beta_{m-s}\,e^{i\gph(m-s)x}\,e^{\gph(m-s)a/\beta}.
\end{equation}
This is Poisson summation for $n\mapsto e^{-\beta(x+2\pi n/\gph)^2/2}e^{ia(x+2\pi n/\gph)}e^{2\pi isn}$ (a Schwartz function of $n\in\R$), whose
Fourier coefficient at $m$ is, after the substitution $w=x+2\pi u/\gph$,
\[
\frac{\gph}{2\pi}e^{i\gph(m-s)x}\int e^{-\beta w^2/2}e^{i(a+\gph(s-m))w}dw=\frac{\gph}{\sqrt{2\pi\beta}}e^{i\gph(m-s)x}e^{-(a+\gph(s-m))^2/(2\beta)},
\]
and $(a+\gph(s-m))^2=a^2-2a\gph(m-s)+\gph^2(m-s)^2$. (For $\beta=1$ this is \eqref{eq:poisson} with $a$ replaced by $ia$.) The closure
constraint is resolved as in the proof of Theorem~\ref{thm:GD}: $\mathbf 1[(dv)_c=0]=\int_0^1e^{i\gph t_c(dv)_c}dt_c$ for every cube $c\in\cC$,
so with $t\in[0,1]^{\cC}$ and $s:=d^*t$, $(d^*t)_p=\sum_{c\ni p}\pm t_c$ (at most two cubes, $|s_p|\le2$),
\[
\sum_{v\ \mathrm{closed}}\prod_pe^{-\beta_p\ff_p^2/2}e^{ia_p\ff_p}=e^{-\sum_pa_p^2/(2\beta_p)}\int_{[0,1]^{\cC}}dt\sum_{m\in\Z^{\cP}}
\prod_pc^{\beta_p}_{m_p-s_p}e^{\gph(m_p-s_p)a_p/\beta_p}\;e^{i\sum_b(d^*m)_b\theta_b},
\]
by \eqref{eq:gin:poissonbeta} plaquette by plaquette with $x=(dB)_p$ and $\gph\sum_p(m_p-s_p)(dB)_p=\sum_b(d^*(m-s))_b\theta_b=\sum_b(d^*m)_b
\theta_b$, because $d^*s=d^*d^*t=0$; here $(d^*m)_b\in\Z$.

\emph{The gauge integral.} By $e^{z\cos\varphi}=\sum_{j\in\Z}I_j(z)e^{ij\varphi}$ with $I_j(z)=I_{-j}(z)\ge0$ for $z\ge0$ and
$\sum_jI_j(z)=e^z$, for every bond $b$ and integer $j_b:=(d^*m)_b+k_b$,
\[
\int_{(-\pi/\gph,\pi/\gph]}dB_b\;e^{ij_b\theta_b}\,e^{\kappa_b\rho_{b_-}\rho_{b_+}\cos\theta_b}=\frac{2\pi}{\gph}\,I_{j_b}\big(\kappa_b\rho_{b_-}\rho_{b_+}\big)\ \ge0 .
\]

\emph{Assembly.} Hence, with 
\begin{gather}
\Omega_{k,P,q}(m):=\int\prod_x\mu_x(d\rho_x)\,P(\rho)e^{\sum_xq_x\rho_x^2}\prod_b\frac{2\pi}{\gph} I_{(d^*m)_b+k_b}(\kappa_b\rho_{b_-}\rho_{b_+})\ge0,\notag\\
\label{eq:gin:dual}
Z(\Theta)=e^{-\sum_pa_p^2/(2\beta_p)}\int_{[0,1]^{\cC}}dt\sum_{m\in\Z^{\cP}}\Big(\prod_pc^{\beta_p}_{m_p-s_p}e^{\gph(m_p-s_p)a_p/\beta_p}\Big)\,
\Omega_{k,P,q}(m)\ \ge0 .
\end{gather}
The interchanges are justified by absolute convergence: 
\[
\sum_m\prod_pc^{\beta_p}_{m_p-s_p}e^{\gph(m_p-s_p)|a_p|/\beta_p}=\prod_p\sum_{m\in\Z} c^{\beta_p}_{m-s_p}e^{\gph(m-s_p)|a_p|/\beta_p}
\]
is bounded uniformly in $|s_p|\le2$ (a Gaussian sum against an exponential), and
\[
0\le\Omega_{k,P,q}(m)\le\int\prod_x\mu_x(d\rho_x)P(\rho)e^{\sum q_x\rho_x^2}\prod_b\frac{2\pi}{\gph}e^{\kappa_b(\rho_{b_-}^2+\rho_{b_+}^2)/2}<\infty
\]
by $I_j(z)\le e^z$ and \eqref{eq:gin:moments}; the lattice is finite. So $Z(\Theta)$ is real and non-negative, and $\avG\Theta=Z(\Theta)/Z\ge0$.
\end{proof}

\begin{proof}[Proof of (K2) and of the signs in (K3)]
Write $\Theta_1:=\Theta=F(\rho)\cos\Phi_1$ and $\Theta_2:=\Theta'=G(\rho)\cos\Phi_2$ with $F(\rho):=P(\rho)e^{\sum_xq_x\rho_x^2}$,
$G(\rho):=P'(\rho)e^{\sum_xq'_x\rho_x^2}$, $\Phi_1:=\sum_pa_p\ff_p+\sum_bk_b\theta_b$, $\Phi_2$ likewise with $(a',k')$; from now on a prime
denotes the duplicate copy of Section~\ref{sec:gin:lemmas}. For integrable $F_1,F_2$,
\begin{equation}\label{eq:gin:dupcov}
Z^2\avG{F_1;F_2}=\tfrac12\iint(F_1-F_1')(F_2-F_2')\,W\,W' ,
\end{equation}
since the right side is $\frac12\big(2Z\int F_1F_2W-2\int F_1W\int F_2W\big)$.

\emph{The integrand in the new variables.} Expand $F(\rho)=\sum_mc_m\rho^m$ (multi-indices $m$, $c_m\ge0$, convergent everywhere; $P$ has
non-negative coefficients and $e^{q\rho^2}$ an entire series with non-negative coefficients). In the variables \eqref{eq:gin:newvars},
$F(u+v)=\sum_mc_m\prod_x\sum_{j_x\le m_x}\binom{m_x}{j_x}u_x^{m_x-j_x}v_x^{j_x}$; let $F_{\mathrm e}(u,v)$ be the sum of the terms with
$\sum_xj_x$ even and $F_{\mathrm o}(u,v)$ that of the terms with $\sum_xj_x$ odd. Both are power series in $(u,v)$ with non-negative
coefficients, absolutely convergent and dominated by $F(u+|v|)$, and
\[
F(\rho')=F(u+v)=F_{\mathrm e}+F_{\mathrm o},\qquad F(\rho)=F(u-v)=F_{\mathrm e}-F_{\mathrm o},
\]
because $v\mapsto-v$ changes the sign of exactly the odd terms. Likewise $G=G_{\mathrm e}\mp G_{\mathrm o}$.

In the parametrisation of
Lemma~\ref{lem:gin:dup}, $\Phi_1=\Xi_1+H_1$ and $\Phi_1'=\Xi_1-H_1$ with
\begin{gather*}
\Xi_1:=\sum_pa_p\xi_p+\sum_bk_b\alpha_b\quad\text{(a function of $(\alpha,k,\sigma)$)},\\
H_1:=\sum_pa_p\upsilon_p+\sum_bk_b\eta_b\quad
\text{(the same function of $(\eta,l,\sigma)$)},
\end{gather*}
by $\ff=\xi+\upsilon$, $\ff'=\xi-\upsilon$, $\theta=\alpha+\eta$, $\theta'=\alpha-\eta$; and $\Xi_2,H_2$ likewise. Hence, by the addition
formulas,
\begin{equation}\label{eq:gin:pm}
\begin{aligned}
\Theta_1+\Theta_1'&=2F_{\mathrm e}\cos\Xi_1\cos H_1+2F_{\mathrm o}\sin\Xi_1\sin H_1,\\
\Theta_1-\Theta_1'&=-2F_{\mathrm o}\cos\Xi_1\cos H_1-2F_{\mathrm e}\sin\Xi_1\sin H_1 ,
\end{aligned}
\end{equation}
and the same for $\Theta_2$.

For a bond $b=(x,y)$, 
\[
\rho_x\rho_y\cos\theta_b+\rho'_x\rho'_y\cos\theta'_b=2(u_xu_y+v_xv_y)\cos\alpha_b\cos\eta_b +2(u_xv_y+v_xu_y)\sin\alpha_b\sin\eta_b,
\]
so the pair of hopping factors is
\begin{multline*}
e^{\kappa_b(\rho_x\rho_y\cos\theta_b+\rho'_x\rho'_y\cos\theta'_b)}=\sum_{m,m'\ge0}\frac{(2\kappa_b)^{m+m'}}{m!\,m'!}(u_xu_y+v_xv_y)^m(u_xv_y+v_xu_y)^{m'}\\
\big(\cos^m\alpha_b\sin^{m'}\alpha_b\big)\big(\cos^m\eta_b\sin^{m'}\eta_b\big),
\end{multline*}
a series each term of which is the product of a non-negative coefficient, a polynomial in $u,v$ with non-negative coefficients and a factor $T(\alpha_b)T(\eta_b)$ with the
same trigonometric monomial $T$ in both copies; the absolute values of the terms sum to at most $e^{2\kappa_b(u_x+|v_x|)(u_y+|v_y|)}\le e^{\kappa_b
(\rho_x^2+\rho_x'^2+\rho_y^2+\rho_y'^2)}$. The pair of Villain weights is \eqref{eq:gin:dup}, and the site measures $\prod_x\mu_x(d\rho_x)
\mu_x(d\rho'_x)$ are invariant under $\rho_x\leftrightarrow\rho'_x$, i.e.\ under $v_x\mapsto-v_x$ with $u$ fixed, for each $x$ separately.

\emph{The positive structure.} By \eqref{eq:gin:pm},
\[
(\Theta_1-\Theta_1')(\Theta_2-\Theta_2')=4\big[F_{\mathrm o}\cos\Xi_1\cos H_1+F_{\mathrm e}\sin\Xi_1\sin H_1\big]
\big[G_{\mathrm o}\cos\Xi_2\cos H_2+G_{\mathrm e}\sin\Xi_2\sin H_2\big],
\]
which consists of four terms, each the product of a series in $u,v$ with non-negative coefficients and a factor $\tau(\alpha,k,\sigma)\,\tau(\eta,l,\sigma)$, where $\tau$ is a
product of two of the functions $\cos\Xi_i,\sin\Xi_i$, the same function in both copies. Multiplying out with the hopping series and
\eqref{eq:gin:dup}, the full integrand of \eqref{eq:gin:dupcov} becomes
\begin{equation}\label{eq:gin:terms}
\sum_\iota c_\iota\,M_\iota(u,v)\sum_\sigma\sum_{k,l\in\Kc_\sigma}\tau_\iota(\alpha,k,\sigma)V_\sigma(\alpha,k)\;\tau_\iota(\eta,l,\sigma)V_\sigma(\eta,l),
\end{equation}
with $c_\iota\ge0$, $M_\iota$ a monomial $\prod_xu_x^{a_x}v_x^{b_x}$ and $\tau_\iota$ a product of cosines and sines of the $\alpha_b$, $\Xi_1$, $\Xi_2$.
The series converges absolutely, uniformly on the angle domain, and is integrable, because $|\tau_\iota|\le1$, 
\[
\sum_\iota c_\iota|M_\iota|\le 4F(u+|v|)G(u+|v|)\prod_be^{\kappa_b(\rho_{b_-}^2+\rho_{b_-}'^2+\rho_{b_+}^2+\rho_{b_+}'^2)}
\]
with $F(u+|v|)=F(\max(\rho,\rho'))\le P(\rho+\rho')e^{\sum_xq_x(\rho_x^2
+\rho_x'^2)}$ (the maximum taken site by site; $P$ has non-negative coefficients), $\sum_\sigma\sum_{k,l\in\Z^{\cP}}V_\sigma(\alpha,k)V_\sigma(\eta,l)
\le2^{|\cP|}(\prod_pK_p)^2$ by Lemma~\ref{lem:gin:dup}, and the site measures satisfy
\eqref{eq:gin:moments}.

\emph{The integral.} The function of $(\theta,\theta')$ obtained from the integrand of \eqref{eq:gin:dupcov} at fixed $(\rho,\rho')$ by the
sums over $n,n'$ is $2\pi$-periodic in every $\theta_b$ and $\theta'_b$: the shift $\theta_b\mapsto\theta_b+2\pi$ together with $n\mapsto
n-d\delta_b$ (a bijection of the closed cochains, $dd=0$) leaves $\ff$, $\cos\theta_b$ and $e^{ik\cdot\theta}$, hence $\Theta_i$ and $W$,
unchanged. By Lemma~\ref{lem:gin:angle} (with $dB_b=d\theta_b/\gph$) and Fubini, $2Z^2\avG{\Theta_1;\Theta_2}$ equals the integral of
\eqref{eq:gin:terms} over $\prod_x\mu_x(d\rho_x)\mu_x(d\rho'_x)\,\gph^{-2|\cB|}d\alpha\,d\eta$ on $[0,\infty)^{2\cX}\times(-\pi,\pi]^{2\cB}$,
and term by term (absolute convergence) this is
\[
\begin{aligned}
2Z^2\avG{\Theta_1;\Theta_2}={}&\gph^{-2|\cB|}\sum_\iota c_\iota\Big(\int M_\iota(u,v)\prod_x\mu_x(d\rho_x)\mu_x(d\rho'_x)\Big)\\
&\quad\sum_\sigma\Big(\sum_{k\in\Kc_\sigma}\int_{(-\pi,\pi]^{\cB}}\tau_\iota(\alpha,k,\sigma)V_\sigma(\alpha,k)\,d\alpha\Big)^2 .
\end{aligned}
\]
The angular factor is a square. The radial factor is $\prod_x\iint u_x^{a_x}v_x^{b_x}\mu_x(d\rho_x)\mu_x(d\rho'_x)$: it vanishes if some $b_x$
is odd (the symmetry $v_x\mapsto-v_x$) and is $\ge0$ otherwise ($u_x\ge0$, $v_x^{b_x}\ge0$). Every term is $\ge0$; this is (K2).

\emph{The signs of (K3).} The functions $\rho_{b_-}\rho_{b_+}\cos\theta_b$ and $\rho_x^2$ are in $\cK_N^+$ (Definition~\ref{def:gin:cone}), so
$\avG{\Theta;\rho_{b_-}\rho_{b_+}\cos\theta_b}\ge0$ and $\avG{\Theta;\rho_x^2}\ge0$ are instances of (K2). For the plaquette energy,
$\ff_p^2-\ff_p'^2=(\xi_p+\upsilon_p)^2-(\xi_p-\upsilon_p)^2=4\xi_p\upsilon_p$, so by \eqref{eq:gin:pm}
\[
(\Theta-\Theta')(\ff_p^2-\ff_p'^2)=-8\big[F_{\mathrm o}\cos\Xi_1\cos H_1+F_{\mathrm e}\sin\Xi_1\sin H_1\big]\,\xi_p\,\upsilon_p ,
\]
which has the structure of \eqref{eq:gin:terms} with $\tau_\iota(\alpha,k,\sigma)$ replaced by $\tau_\iota(\alpha,k,\sigma)\,\xi_p$ (again the
same function of $(\alpha,k,\sigma)$ and of $(\eta,l,\sigma)$) and an overall sign $-$; the absolute convergence holds with the second bound of
Lemma~\ref{lem:gin:dup} ($j=1$). The same computation expresses $Z^2\avG{\Theta;\ff_p^2}$ as $-4\gph^{-2|\cB|}$ times a sum over $\iota$ of products of $c_\iota\ge0$, a non-negative radial factor
and a sum over $\sigma$ of squares; hence $\avG{\Theta;\ff_p^2}\le0$.
\end{proof}

\begin{proof}[Proof of the derivative formulas of (K3)]
Let $\lambda$ be one of the parameters $\kappa_b$, $\beta_p$, $q$ (the last for $\mu_x\mapsto e^{q\rho_x^2}\mu_x$ at one site $x$), and let
$W_\lambda$ be the weight; then $\partial_\lambda W_\lambda=E_\lambda W_\lambda$ with $E_{\kappa_b}=\rho_{b_-}\rho_{b_+}\cos\theta_b$,
$E_{\beta_p}=-\frac12\ff_p^2$, $E_q=\rho_x^2$. On a compact interval of parameters ($\kappa_b\in[0,\bar\kappa]$, $\beta_p\in[\beta_0,\bar\beta]$
with $\beta_0>0$, $q\in[0,\bar q]$) the functions $|\Theta|W_\lambda$ and $|\Theta E_\lambda|W_\lambda$ are dominated by one integrable
function: $e^{\kappa_b\rho\rho'\cos\theta_b}\le e^{\bar\kappa\rho\rho'}$ and $\rho\rho'e^{\bar\kappa\rho\rho'}$ is integrable by
\eqref{eq:gin:moments}; $\sum_{n_p\in\Z}(1+\ff_p^2)e^{-\beta_p\ff_p^2/2}\le\sum_{n_p}(1+\ff_p^2)e^{-\beta_0\ff_p^2/2}$ is bounded uniformly
in $(dB)_p$ (Lemma~\ref{lem:gin:dup}, second bound); $(1+\rho_x^2)e^{q\rho_x^2}\le(1+\rho_x^2)e^{\bar q\rho_x^2}$ is $\mu_x$-integrable.

So
$\lambda\mapsto\int\Theta W_\lambda$ and $\lambda\mapsto Z_\lambda$ are continuously differentiable with derivatives $\int\Theta E_\lambda
W_\lambda$ and $\int E_\lambda W_\lambda$ (differentiation under the integral, dominated convergence), the partial derivatives being continuous
in all parameters jointly (the same domination over a compact box of parameters), and the quotient rule gives $\partial_\lambda\avG\Theta=
\avG{\Theta E_\lambda}-\avG\Theta\avG{E_\lambda}=\avG{\Theta;E_\lambda}$. This is \eqref{eq:gin:deriv}; the signs were proved above, and a
continuously differentiable function with non-negative derivative on an interval is non-decreasing. Every intermediate weight belongs to the
class of Definition~\ref{def:gin:weights}, and for the torus states and the killed-box states the three parameters are exactly the hopping coefficient of a bond
with its diagonal factors fixed, the stiffness of a plaquette and the site factor $e^{q\rho_x^2}$.
\end{proof}

\begin{proof}[Proof of (K4)]
Let $\Lam\subset\Lam'$ be boxes and $\Theta\in\cK_N^+$ supported in $\Lam$. Take $G=\Lam'$ with the parameters of the model ($\mu_x=\mu^\circ$,
$\kappa_b=\kappa_N$, $\beta_p=1$) and let $\avG\cdot_t$, $t\in(0,1]$, be the interpolated states of Lemma~\ref{lem:gin:end}: the hopping
coefficients of the bonds of $\cB(\Lam')\setminus\cB(\Lam)$ at $t\kappa_N$ and the stiffnesses of the plaquettes of $\cP(\Lam')\setminus\cP(\Lam)$
at $t$, everything else fixed; $\avG\cdot_1=\avB\cdot{\Lam'}$. By (K3) the function $\varphi(t):=\avG\Theta_t$ is continuously differentiable
on $(0,1]$ (the parameters depend linearly on $t$, and the partial derivatives are jointly continuous by (K3)) with
\[
\varphi'(t)=\sum_{b\in\cB(\Lam')\setminus\cB(\Lam)}\kappa_N\avG{\Theta;\rho_{b_-}\rho_{b_+}\cos\theta_b}_t-\frac12\sum_{p\in\cP(\Lam')\setminus\cP(\Lam)}
\avG{\Theta;\ff_p^2}_t\ \ge0 .
\]
Hence $\varphi$ is non-decreasing on $(0,1]$, and $\avB\Theta{\Lam'}=\varphi(1)\ge\lim_{t\downarrow0}\varphi(t)=\avB\Theta\Lam$ by
Lemma~\ref{lem:gin:end} (whose conditions are met: $\Lam'\supset\Lam$ as in Lemma~\ref{lem:gin:ext}, $|\Theta|\le Ce^{\sum q'_x\rho_x^2}$, and the
parameters on the cells of $\Lam$ are those of $\avB\cdot\Lam$).

For the torus, the identification of the statement makes the sites, bonds and plaquettes of $\Lam_L$ cells of $\tilde T_L$ (the bonds $(x,i)$
with $x_i\le L-2\eps$ and the plaquettes $(x;ij)$ with $x_i,x_j\le L-2\eps$), the cubes of $\Lam_L$ cubes of the torus, and $\Lam_L$ is a box;
so Lemma~\ref{lem:gin:ext} and Lemma~\ref{lem:gin:end} apply to $G=\tilde T_L\supset\Lam_L$. The torus state is the state of
$W_{\tilde T_L}(\mu^\circ,\kappa_N,1)$ and the killed-box state of $\Lam_L$ that of $W_{\Lam_L}(\mu^\circ,\kappa_N,1)$
(Definition~\ref{def:gin:box}: the same site measure, since in both every site carries the diagonal halves of six bonds). The argument of the
first paragraph, with the bonds of $\cB(\tilde T_L)\setminus\cB(\Lam_L)$ and the plaquettes of $\cP(\tilde T_L)\setminus\cP(\Lam_L)$ interpolated,
gives $\avB\Theta{\Lam_L}\le\avL\Theta$ for every $\Theta$ supported in $\Lam_L$; combined with the first clause for $\Lam\subset\Lam_L$ this
is $\avB\Theta\Lam\le\avB\Theta{\Lam_L}\le\avL\Theta$. Finally, if $\Lam$ lies in a cube of side $s<L$ ($s\in\eps\Z$; the nearest-neighbour sites of $\Lam$ play no role), then
$s\le L-\eps$ and the cube has at most $L/\eps$ sites along each axis, so some translate $y+\Lam$ ($y\in\eps\Z^3$) lies in $\Lam_L$; the torus
weight is invariant under the translations of $\tilde T_L$ and the killed-box weight is carried along by them, so $\avL\Theta=\avL{\Theta_y}$
and $\avB\Theta\Lam=\avB{\Theta_y}{y+\Lam}$ for the translate $\Theta_y\in\cK_N^+$ of $\Theta$, supported in $y+\Lam$. The argument of the first paragraph uses only (K3) and Lemma~\ref{lem:gin:end}, which hold for every weight of Definition~\ref{def:gin:weights};
it therefore gives the same comparison for any such weight on a finite lattice $G$ whose cells contain those of $\Lam$ and whose parameters on the cells of $\Lam$ are those of the $\Lam$-weight.
\end{proof}

\begin{remark}[the hypotheses used by each clause]\label{rem:gin:hyp}
(i) (K1) uses only the positivity of the Fourier coefficients of the weight: the coefficients $c^\beta_u>0$ of the Villain plaquette weight
(its Gaussian form, through Poisson summation) and the Bessel coefficients $I_j\ge0$ of the hopping factor, which depends on the gauge field
only through $\cos(\gph B_b)$ with the non-negative coefficient $\kappa_b$; the site measures, the polynomial $P$ and the factor
$e^{\sum q_x\rho_x^2}$ enter as non-negative weights, and the closure constraint only through the phases $e^{i\gph t_c(dv)_c}$. It holds for
every $\gph>0$ and gives the first inequality alone: positivity of the Fourier coefficients does not imply (K2) (the weight $1+\cos t\cos2t$
on the circle has non-negative Fourier coefficients, and $\langle\cos t;\cos3t\rangle=0-\frac14\cdot\frac14=-\frac1{16}$).

(ii) (K2)--(K4) rest on the
duplication identity \eqref{eq:gin:dup}, which uses that the Villain integers are summed over \emph{all} closed cochains: the closed integer
cochains form a group, so that $n+n'$ and $n-n'$ are closed again and the parametrisation of Lemma~\ref{lem:gin:dup} exists; a sum restricted
to one nonzero flux class of the torus (a coset, not a group) does not have this property. The sum over all $v$ without the closure constraint has it
as well. The phases of the enlarged cone carry integer bond coefficients $k_b$ so that the integrand is a function on the angle torus
(Lemma~\ref{lem:gin:angle}), and the frequencies $a_p$ of the curvature are arbitrary real numbers because $\ff_p$ is a real variable.

(iii) The
argument of (K2) applied to $\Theta+\Theta'$ in \eqref{eq:gin:pm} gives (K1) a second time.

(iv) Nothing in (K1)--(K3) uses the form of the
site measure beyond \eqref{eq:gin:moments}; the killed-box convention (every site with all six diagonal halves) is what makes the site
measure the same in every volume, which (K4) needs.

(v) For the torus states and the killed-box states every statement holds for every $\gph>0$, $\kappa_N\ge0$,
$\lamf>0$, $\rhof>0$, $\delta m_B\in\R$ and every $L$. These are statements at a fixed level $N$, which suffices for Section~\ref{sec:d2}, where the limit $L\to\infty$ is taken at fixed $N$.

(vi) Definition~\ref{def:gin:weights} admits lattices with several components, and Theorem~\ref{thm:gin:K} covers them because the proofs of (K1)--(K3) are cell by cell.
\end{remark}

\begin{remark}[the bare mass counterterm]\label{rem:gin:ct}
The site measure $\mu^\circ=g_{\mathrm{ct}}(\rho-\rhof)e^{-3\kappa_N\rho^2}d\rho$, $g_{\mathrm{ct}}=g\,e^{-\frac12\delta m_B\rho^2}$ (Definition~\ref{def:gin:box}), carries the bare mass
counterterm of Section~\ref{sec:setting}, $\delta m_B\in\R$ of either sign. Every statement of this section holds for every real
$\delta m_B$: Definition~\ref{def:gin:weights}, Lemma~\ref{lem:gin:end} and the proofs of (K1)--(K3) use of the site measures only that they are finite positive measures on
$[0,\infty)$ with \eqref{eq:gin:moments} (item (iv) of Remark~\ref{rem:gin:hyp}), which $\mu^\circ$ has by \eqref{eq:ct}. The proof of (K4) uses in
addition only that $\mu^\circ$ is the same measure at every site of every box and of the torus (the counterterm is a per-site factor, and the killed-box convention keeps it at
every site); no monotonicity, no identity and no sign of a derivative of $g$ is used.
\end{remark}

\providecommand{\cS}{\mathcal S}\providecommand{\cK}{\mathcal K}\providecommand{\cP}{\mathcal P}\providecommand{\cX}{\mathcal X}
\providecommand{\cB}{\mathcal B}\providecommand{\cE}{\mathcal E}\providecommand{\cG}{\mathcal G}\providecommand{\E}{\mathbb E}
\providecommand{\supp}{\operatorname{supp}}\providecommand{\dist}{\operatorname{dist}}\providecommand{\Lam}{\Lambda}
\providecommand{\OmN}{\Omega_N}\providecommand{\avB}[2]{\langle #1\rangle_{#2}}\providecommand{\ombox}{\om^{\mathrm{box}}_N}
\providecommand{\mubox}{\mu^{\mathrm{box}}_N}\providecommand{\avLt}[1]{\langle #1\rangle_{L,t}}
\providecommand{\eps}{\varepsilon}\providecommand{\Lset}{\mathcal L}

\section{The existence of the fine infinite-volume state}\label{sec:d2}

The relation \eqref{eq:D2} would define the fine infinite-volume state $\om_N$ as the limit of the torus states; in this paper $\om_N$
is a limit point of the torus states (Definition~\ref{def:omN}), and the existence of the limit is not assumed. The results of this
section are the following. First, limit points of the torus states exist along subsequences of sides, and every limit point carries all the
lattice symmetries, the reflection positivity and the uniform bounds of the torus states (Theorem~\ref{thm:d2:LP}); the only new estimate
is a bound on the local moments of the scalar density, uniform in the volume, which we derive from the reflection positivity of the torus
states (Lemma~\ref{lem:d2:mom}). Second, no statement of this paper requires the existence of the limit
\eqref{eq:D2}: the chain from the estimate to the limit functional and the statements of Section~\ref{sec:osrest} are proved for limit points of this kind (Corollary~\ref{cor:d2:consumers}).

Third,
on the cone of observables for which Ginibre's inequalities hold, the killed-box states converge, every torus limit point dominates the box
limit, and the values on the cone determine the state (Theorem~\ref{thm:d2:box}, Lemma~\ref{lem:d2:cone}); hence \eqref{eq:D2}, in its continuous form (ii)
of Corollary~\ref{cor:d2:iff}, is equivalent to the agreement of the torus limit points on the cone generators (the form (i) implies it).

Throughout, $N\ge N_0$ is fixed, $\eps:=\eN$ is the spacing, $\gph$ the phase constant, and the couplings $\kappa_N\ge0$, $\lamf>0$, $\rhof>0$ and the bare mass counterterm $\delta m_B\in\R$ (Remark~\ref{rem:ct}) are arbitrary: nothing in this
section depends on their values. All constants may depend on $N$ through $(\eps,\gph,\kappa_N,\lamf,\rhof,\delta m_B)$; this is sufficient,
since the limit $L\to\infty$ is taken at fixed $N$.

\subsection{Configuration space, local functions, the symmetries of the torus states}\label{sec:d2:setting}

Let $\cP_N$ be the set of plaquettes of $\eps\Z^3$, one orientation chosen for each (the opposite orientation carries $\ff_{-p}=-\ff_p$), and
$\cX_N$ its set of sites. The configuration space of the gauge-invariant variables is
\[
\OmN:=\R^{\cP_N}\times[-\rhof,\infty)^{\cX_N},\qquad\xi=\big((f_p)_{p\in\cP_N},(\tr_x)_{x\in\cX_N}\big),\qquad\rho_x:=\rhof+\tr_x\ge0,
\]
with the product topology and its Borel $\sigma$-algebra; $\OmN$ is a countable product of Polish spaces, hence Polish, and its Borel
$\sigma$-algebra is the product $\sigma$-algebra, generated by the cylinder sets. For a finite set $S$ of plaquettes and sites put
$E_S:=\R^{\cP_S}\times[-\rhof,\infty)^{\cX_S}$ and let $\pi_S\colon\OmN\to E_S$ be the projection. A function $F$ on $\OmN$ is \emph{local
with support $S$} if $F=F_S\circ\pi_S$; it is bounded Borel local, bounded continuous local, etc., according to $F_S$. The bounded Borel
local functions are the ``bounded local gauge-invariant functions'' of \eqref{eq:D2}.

If $S$ lies in a cube of side less than $L$, the quotient map $\eps\Z^3\to\tilde T_L$ is injective on the cells of $S$, and the gauge-invariant
variables of $\tilde T_L$ at the image cells define a measurable map from the torus configurations to $E_S$; its law under $\avL\cdot$ is a
Borel probability measure $\nu^S_L$ on $E_S$, and for $F$ local with support $S$ we write $\avL F:=\int F_S\,d\nu^S_L$. This is the meaning
of $\avL F$ for local $F$ throughout; by the translation invariance of the torus state it does not depend on the translate of $S$ used. Let
$\cG_N$ be the group generated by the translations by $\eps\Z^3$, the reflections $R_{j,c}$ in the planes $\{x_j=c\}$, $j\in\{1,2,3\}$,
$c\in\frac12\eps\Z$, and the cubic group about every site, acting on cells and on $\OmN$ by $(g\xi)_\kappa:=\xi_{g\kappa}$ with the
orientation conventions ($f$ changes sign on the plaquettes whose orientation $g$ reverses: this is the time-like sign on the curvature);
and let $C\colon(f,\tr)\mapsto(-f,\tr)$ be the charge conjugation.

\begin{lemma}[symmetries of the torus states]\label{lem:d2:sym}
For every $g\in\cG_N\cup\{C\}$ and every bounded Borel local $F$, $\avL{F\circ g}=\avL F$ for all $L$ with $L/2\in\eps\Z$ so large that
$S$ and $gS$ lie in a cube of side less than $L$.
\end{lemma}
\begin{proof}
For the translations and the cubic group this is the proof of Lemma~\ref{lem:R1}; for the reflections $R_{j,c}$ it is the paragraph before
Theorem~\ref{thm:osrest:RPtorus} (every $R_{j,c}$ with $c\in\frac12\eps\Z$ is an involutive symmetry of $\tilde T_L$ for $L/2\in\eps\Z$, and the
weight and the reference measure are invariant); for $C$ it is the proof of Proposition~\ref{prop:osrest:OS3}(b).
\end{proof}

\subsection{Tightness: the uniform local moments}\label{sec:d2:tight}

For the curvature the uniform bound is Theorem~\ref{thm:GD}: $\avL{e^{s\ff_p}}\le e^{s^2/2}$ for all real $s$, hence
$\avL{e^{s|\ff_p|}}\le2e^{s^2/2}$. For the density we use the reflection positivity of the torus states about the bond mid-planes.

\begin{lemma}[uniform local moments of the density]\label{lem:d2:mom}
Let $g_{\mathrm{ct}}(\tr):=g(\tr)\,e^{-\frac12\delta m_B(\rhof+\tr)^2}$ be the full site factor of the model of Section~\ref{sec:setting}, the bare mass counterterm
included ($\delta m_B\in\R$ of either sign; Remark~\ref{rem:ct}). Put, for $q\ge0$,
\[
\begin{gathered}
G^{\mathrm{ct}}_q:=\int_{-\rhof}^\infty g_{\mathrm{ct}}(r)\,e^{q(\rhof+r)^2}\,dr<\infty,\\
K'_{\gph}:=\sup_{x\in\R}\sum_{v\in(2\pi/\gph)\Z}e^{-(x+v)^2/2}\le1+2\sum_{n\ge1}e^{-2\pi^2(n-\frac12)^2/\gph^2},\\
c_g:=\rhof{}e^{-\lamf(2\rhof+1)^2},\quad c^{\mathrm{ct}}_g:=c_g\,e^{-\frac12|\delta m_B|(\rhof+1)^2},\quad c_h:=e^{-\kappa_N(1+(\rhof+1)^2)/2},\\
c_0:=c^{\mathrm{ct}}_g\,(2c_h/\gph)^3e^{-24/\gph^2},\quad C_1:=\max\Big\{1,\ \frac{G^{\mathrm{ct}}_0(2\pi/\gph)^3K_{\gph}'^3}{c_0}\Big\}.
\end{gathered}
\]
($K'_{\gph}$ is a constant distinct from the $K_{\gph}$ of the proof of Theorem~\ref{thm:GD}.) Then for every $q\ge0$, every site $x$ and
every $L\ge4\eps$ with $L/2\in\eps\Z$,
\begin{equation}\label{eq:d2:mom}
\avL{e^{q\rho_x^2}}\ \le\ C_N(q):=\frac{G^{\mathrm{ct}}_q}{G^{\mathrm{ct}}_0}\,C_1^{\,8}.
\end{equation}
\end{lemma}
\begin{proof}
Write $\cX_L,\cB_L,\cP_L$ for the sites, bonds and plaquettes of $\tilde T_L$, so that $|\cB_L|=|\cP_L|=3|\cX_L|$, and $M:=L/\eps$ (an even
integer $\ge4$). The integral $G^{\mathrm{ct}}_q$ is finite for every $q\ge0$ and every real $\delta m_B$ by the inequality \eqref{eq:ct}: $\lamf(\rho^2-\rhof^2)^2\ge\frac12\lamf\rho^4-\lamf\rhof^4$ and
$e^{-\frac12\delta m_B\rho^2}\le e^{\frac12|\delta m_B|\rho^2}$ give $g_{\mathrm{ct}}(r)\,e^{q\rho^2}\le\rho\,e^{\lamf\rhof^4}\,e^{-\frac12\lamf\rho^4+(q+\frac12|\delta m_B|)\rho^2}$, $\rho=\rhof+r\ge0$, which is
integrable over $r\in[-\rhof,\infty)$ since $\lamf>0$.

\emph{Step 1: crude bounds on partition functions.} For a set $A\subset\cX_L$ and $q\ge0$ let $Z(A,q):=\int d\tr\int dB\sum_{v\ \mathrm{closed}}
W\prod_{y\in A}e^{q\rho_y^2}$, so that $Z=Z(\emptyset,0)$ and $\avL{\prod_{y\in A}e^{q\rho_y^2}}=Z(A,q)/Z$. All terms are non-negative ($g_{\mathrm{ct}}\ge0$). Since
$h\le1$, since the sum over closed $v$ is at most the sum over all $v$, and $\sum_{v_p}w((dB)_p+v_p)\le K'_{\gph}$ for every value of $(dB)_p$,
\[
Z(A,q)\le\Big(\int d\tr\prod_xg_{\mathrm{ct}}(\tr_x)\prod_{y\in A}e^{q\rho_y^2}\Big)\Big(\frac{2\pi}{\gph}\Big)^{|\cB_L|}K_{\gph}'^{\,|\cP_L|}
=(G^{\mathrm{ct}}_q)^{|A|}(G^{\mathrm{ct}}_0)^{|\cX_L|-|A|}\Big(\frac{2\pi}{\gph}\Big)^{3|\cX_L|}K_{\gph}'^{\,3|\cX_L|}.
\]
For a lower bound on $Z$ keep only the term $v\equiv0$ (which is closed) and restrict the integration to $\tr_x\in[0,1]$ for all $x$ and
$B_b\in[-1/\gph,1/\gph]\subset(-\pi/\gph,\pi/\gph]$ for all $b$. There $g_{\mathrm{ct}}(\tr_x)\ge c^{\mathrm{ct}}_g$ (the exponent $(\rhof+r)^2-\rhof^2$ lies in
$[0,2\rhof+1]$ for $r\in[0,1]$, so $g(r)\ge c_g$, and $\frac12\delta m_B(\rhof+r)^2\le\frac12|\delta m_B|(\rhof+1)^2$), $|(dB)_p|\le4/\gph$ so $w((dB)_p)\ge e^{-8/\gph^2}$, and, with $\rho,\rho'\in[\rhof,\rhof+1]$
and $|\gph B_b|\le1$,
\[
|e^{i\gph B_b}\rho'-\rho|^2=(\rho-\rho')^2+2\rho\rho'(1-\cos\gph B_b)\le1+\rho\rho'\gph^2B_b^2\le1+(\rhof+1)^2,
\]
so $h\ge c_h$. Hence
$Z\ge(c^{\mathrm{ct}}_g)^{|\cX_L|}(2c_h/\gph)^{|\cB_L|}e^{-8|\cP_L|/\gph^2}=c_0^{|\cX_L|}$.

\emph{Step 2: doubling by reflection.} Fix $x$ and $q$, put $F:=e^{q\rho_x^2}$, and for $n=(n_1,n_2,n_3)$ with $n_j=2^{k_j}\le M$ let $B(n):=
\{x+\eps(m_1,m_2,m_3):0\le m_j<n_j\}$ (a box of $|B(n)|=n_1n_2n_3$ distinct sites of $\tilde T_L$) and $\Pi_n:=\prod_{y\in B(n)}e^{q\rho_y^2}$. We
claim $\big(\avL{F}\big)^{|B(n)|}\le\avL{\Pi_n}$. For $n=(1,1,1)$ this is an identity.

Let it hold for $n$ with $n_j\le M/2$ for some $j$, and let $n'$ be $n$
with $n_j$ doubled. Take the bond mid-plane $\{x_j=c\}$, $c:=x_j-\eps/2$, and the slab $\Lam^+_c$ of Theorem~\ref{thm:osrest:RPtorus} in the
$j$-th coordinate, whose sites have $x'_j\in[x_j,\,x_j-\eps/2+L/2]$; $B(n)\subset\Lam^+_c$ because $(n_j-1)\eps\le L/2-\eps/2$. The reflection
$R_{j,c}$ maps the sites of $B(n)$ onto the adjacent box $\{x'_j\in\{x_j-\eps,\dots,x_j-n_j\eps\}\}$, disjoint from $B(n)$, and $B(n)\cup
R_{j,c}B(n)$ is a translate of $B(n')$. The function $\Pi_n$ is $\Lam^+_c$-measurable with $\avL{\Pi_n^2}=Z(B(n),2q)/Z<\infty$ by Step~1, so the
$L^2$ clause of Theorem~\ref{thm:osrest:RPtorus} applies: the form $(A,A')\mapsto\avL{\Th_cA\cdot A'}$ is positive semidefinite and Hermitian, and
the Schwarz inequality for it, with $A=\Pi_n$ and $A'=1$, gives
\[
\big(\avL{\Th_c\Pi_n}\big)^2\le\avL{\Th_c\Pi_n\cdot\Pi_n}\,\avL{\Th_c1\cdot1}=\avL{(\Pi_n\circ R_{j,c})\,\Pi_n}=\avL{\Pi_{n'}},
\]
by the reflection and translation invariance of the torus state (Lemma~\ref{lem:d2:sym}; $\Pi_n\circ R_{j,c}=\prod_{y\in R_{j,c}B(n)}e^{q\rho_y^2}$ is
real). Also $\avL{\Th_c\Pi_n}=\avL{\Pi_n\circ R_{j,c}}=\avL{\Pi_n}$. Hence $\big(\avL{F}\big)^{|B(n')|}=\big(\big(\avL{F}\big)^{|B(n)|}\big)^2\le\big(\avL{\Pi_n}\big)^2\le
\avL{\Pi_{n'}}$, which is the claim for $n'$.

Now take $k_1=k_2=k_3=k:=\lfloor\log_2M\rfloor$: every doubling on the way starts from a box with
$n_j\le2^{k-1}\le M/2$, the final box has $n_j=2^k\in(M/2,M]$ and $|B(n)|=2^{3k}>M^3/8=|\cX_L|/8$.

\emph{Step 3: assembly.} By Steps 1 and 2, with $n$ the final box,
\[
\avL{\Pi_n}=\frac{Z(B(n),q)}Z\le\Big(\frac{G^{\mathrm{ct}}_q}{G^{\mathrm{ct}}_0}\Big)^{|B(n)|}\Big(\frac{G^{\mathrm{ct}}_0(2\pi/\gph)^3K_{\gph}'^3}{c_0}\Big)^{|\cX_L|}
\le\Big(\frac{G^{\mathrm{ct}}_q}{G^{\mathrm{ct}}_0}\Big)^{|B(n)|}C_1^{|\cX_L|},
\]
so
\[
\avL F\le\big(\avL{\Pi_n}\big)^{1/|B(n)|}\le(G^{\mathrm{ct}}_q/G^{\mathrm{ct}}_0)\,C_1^{|\cX_L|/|B(n)|}\le(G^{\mathrm{ct}}_q/G^{\mathrm{ct}}_0)\,C_1^{\,8},
\]
since $C_1\ge1$.
\end{proof}

\begin{remark}
\label{rem:d2:mom}(i) The bound is a multiple-reflection bound of the chessboard kind \cite{FILS78}, in a weak form that suffices here: it has the exponent $8$ in place
of $1$ and concerns one observable only, so that no limiting argument on the number of reflections is needed; the ingredients are the reflection
positivity about the bond mid-planes in the three directions (Theorem~\ref{thm:osrest:RPtorus}) and the per-cell bounds of Step~1. (ii) The
same bound gives $\avL{\prod_{y\in A}e^{q\rho_y^2}}\le\prod_{y\in A}\big(\avL{e^{|A|q\rho_y^2}}\big)^{1/|A|}\le C_N(|A|q)$ for every finite set $A$ of
sites by H\"older's inequality, and the
bound $\avL{e^{\sum_xq_x\rho_x^2}}\le\max_x C_N(|X|q_x)$ for finitely many sites $X$ by the convexity of the exponential,
$e^{\sum_{x\in X}q_x\rho_x^2}\le|X|^{-1}\sum_{x\in X}e^{|X|q_x\rho_x^2}$. (iii) The constants depend on $N$; this is sufficient, because
the limit $L\to\infty$ is taken at fixed $N$.
\end{remark}

\begin{remark}[the bare mass counterterm in this section]\label{rem:d2:ct}
The site factor of the model is $g_{\mathrm{ct}}=g\,e^{-\frac12\delta m_B\rho^2}$, $\delta m_B\in\R$ of either sign (Remark~\ref{rem:ct}). Every statement of this section
holds for every real $\delta m_B$, with the constants $G^{\mathrm{ct}}_q$, $c^{\mathrm{ct}}_g$ (hence $c_0$, $C_1$, $C_N(q)$), at fixed $N$. Indeed, the
proofs use only three properties of the site factor. The first is that $g_{\mathrm{ct}}\ge0$, $G^{\mathrm{ct}}_q<\infty$ for every $q\ge0$ and $g_{\mathrm{ct}}\ge c^{\mathrm{ct}}_g$ on $\tr\in[0,1]$; it is used in Step~1 of the proof of Lemma~\ref{lem:d2:mom}.
The second is that $g_{\mathrm{ct}}$ is one positive function of the one site variable, the same at every site and independent of $f$. It is used in Lemma~\ref{lem:d2:sym}, hence in Theorem~\ref{thm:d2:LP}(c);
in Step~2 of the proof of Lemma~\ref{lem:d2:mom} and in Theorem~\ref{thm:d2:LP}(d), through Theorem~\ref{thm:osrest:RPtorus}, whose proof uses the positivity, the reflection invariance and the integrability of the site factor; and in Theorem~\ref{thm:GD},
which uses only that the site integral is non-negative and finite (Remark~\ref{rem:ct}). The third, for Definition~\ref{def:d2:cone} and Theorem~\ref{thm:d2:GIN}, is that
$g_{\mathrm{ct}}(\rho-\rhof)e^{-3\kappa_N\rho^2}d\rho$ is a finite positive measure on $[0,\infty)$ with $\int e^{\alpha\rho^2}<\infty$ for every $\alpha\in\R$ (by \eqref{eq:ct}),
which is the condition \eqref{eq:gin:moments} that Section~\ref{sec:gin} imposes on the site measure. No monotonicity of $g$, no moment identity and no sign of a derivative of $g$ is used.
\end{remark}

\subsection{Limit points of the torus states}\label{sec:d2:limitpoint}

\begin{theorem}[limit points, with every symmetry]\label{thm:d2:LP}
Fix $N$ and a sequence of sides $L'_n\to\infty$ with $L'_n/2\in\eps\Z$. There are a subsequence $(L_n)$ and a Borel probability measure $\mu_N$
on $\OmN$ with the following properties.
\begin{itemize}
\item[(a)] (\emph{Convergence.}) For every finite $S$, $\nu^S_{L_n}\to(\pi_S)_*\mu_N$ weakly. Equivalently, with
\[
\om_N(F):=\int F\,d\mu_N\qquad\text{for every bounded Borel local }F,
\]
one has $\om_N(F)=\lim_n\langle F\rangle^{\mathrm{fine}}_{L_n}$ for every bounded continuous local $F$. The same limit relation holds for every
bounded Borel local $F$ whose set of discontinuities is $\mu_N$-null, and for every continuous local $F$ with
$|F|\le Ce^{\sum_{x\in X}q_x\rho_x^2}$ for some finite $X$, $C<\infty$, $q\ge0$.
\item[(b)] (\emph{State; uniform bounds.}) $\om_N$ is linear and positive on the bounded Borel local functions, $\om_N(1)=1$,
$|\om_N(F)|\le\|F\|_\infty$; and for every real finitely supported $a$ and every $q\ge0$, $x$,
\[
\int e^{\sum_pa_p f_p}\,d\mu_N\le e^{\frac12\sum_pa_p^2},\qquad\int e^{q\rho_x^2}\,d\mu_N\le C_N(q).
\]
\item[(c)] (\emph{Symmetries.}) $g_*\mu_N=\mu_N$ for every $g\in\cG_N\cup\{C\}$; hence $\om_N(F\circ g)=\om_N(F)$ for every bounded Borel
local $F$. In particular the conclusion of Lemma~\ref{lem:R1} holds for $\om_N$.
\item[(d)] (\emph{Reflection positivity.}) For $j\in\{1,2,3\}$ and $c\in\frac12\eps\Z$ let $\mathfrak A^+_N(j,c)$ be the bounded Borel functions of the
variables of finitely many cells lying in $\{x_j\ge c\}$ (all vertices in the closed half space) and $\Th_{j,c}F:=\overline{F\circ R_{j,c}}$. Then
$(F,G)\mapsto\om_N(\Th_{j,c}F\cdot G)$ is a positive semidefinite Hermitian form on $\mathfrak A^+_N(j,c)$, and $\om_N(\Th_{j,c}F)=\overline{\om_N(F)}$.
Both statements extend to functions of those cells in $L^2(\mu_N)$.
\item[(e)] (\emph{Inequalities pass.}) If $F,G$ are bounded continuous local functions with $\langle F\rangle^{\mathrm{fine}}_L\le\langle G
\rangle^{\mathrm{fine}}_L$ for all large $L$, then $\om_N(F)\le\om_N(G)$; and $\int F\,d\mu_N\le\liminf_n\langle F\rangle^{\mathrm{fine}}_{L_n}$
for every continuous local $F\ge0$.
\end{itemize}
Moreover (a)--(e) hold for every pair $((L_n),\mu_N)$, $L_n\to\infty$ with $L_n/2\in\eps\Z$, for which the weak convergence of (a) holds.
\end{theorem}

\begin{proof}
\emph{Tightness.} Let $S$ be finite and $K_R:=[-R,R]^{\cP_S}\times[-\rhof,R]^{\cX_S}$, a compact subset of $E_S$. By Theorem~\ref{thm:GD}
and Markov's inequality, $\nu^S_L(|f_p|>R)\le e^{-R}\avL{e^{|\ff_p|}}\le2e^{1/2-R}$; by Lemma~\ref{lem:d2:mom}, $\nu^S_L(\tr_x>R)\le
e^{-(\rhof+R)^2}\avL{e^{\rho_x^2}}\le C_N(1)e^{-R^2}$. Hence $\sup_L\nu^S_L(E_S\setminus K_R)\le2|\cP_S|e^{1/2-R}+|\cX_S|C_N(1)e^{-R^2}\to0$
as $R\to\infty$: the family $(\nu^S_L)_L$ is tight on the Polish space $E_S$, hence relatively compact for the weak topology (Prokhorov).

\emph{Diagonal extraction and consistency.} Let $S_k$ be the set of cells of $\eps\Z^3$ lying in the cube $[-k\azero,k\azero]^3$, $k\ge1$; every
finite $S$ is contained in some $S_k$. A diagonal argument gives a subsequence $(L_n)$ of $(L'_n)$ along which $\nu^{S_k}_{L_n}\to\nu^{S_k}_\infty$
weakly for every $k$. For $k\le k'$ the $S_k$-marginal of $\nu^{S_{k'}}_{L_n}$ is $\nu^{S_k}_{L_n}$, and the image of a weakly convergent sequence
under the continuous projection $E_{S_{k'}}\to E_{S_k}$ converges weakly to the image of the limit; so the $S_k$-marginal of $\nu^{S_{k'}}_\infty$
is $\nu^{S_k}_\infty$: the family $(\nu^{S_k}_\infty)_k$ is consistent. By Kolmogorov's extension theorem (a countable product of Polish spaces)
there is a unique Borel probability measure $\mu_N$ on $\OmN$ with $(\pi_{S_k})_*\mu_N=\nu^{S_k}_\infty$ for all $k$, hence
$(\pi_S)_*\mu_N=\lim_n\nu^S_{L_n}$ weakly for every finite $S$.

\emph{(a).} For $F$ bounded continuous local with support $S$, $\langle F\rangle^{\mathrm{fine}}_{L_n}=\int F_S\,d\nu^S_{L_n}\to\int F_S\,
d(\pi_S)_*\mu_N=\om_N(F)$ by the definition of weak convergence. For bounded Borel $F_S$ with a $(\pi_S)_*\mu_N$-null discontinuity set this is the
portmanteau theorem. For continuous $F_S$ with $|F_S|\le Ce^{\sum_{x\in X}q_x\rho_x^2}$: by Remark~\ref{rem:d2:mom}(ii),
$\sup_L\int e^{2\sum_xq_x\rho_x^2}d\nu^S_L<\infty$, so $(F_S)$ is uniformly integrable for $(\nu^S_{L_n})_n$, and weak convergence together with
uniform integrability gives the convergence of the integrals.

\emph{(b).} Linearity, positivity, normalisation and the bound are those of the integral against a probability measure. The function
$e^{\sum_pa_pf_p}$ is continuous and non-negative, so by the portmanteau theorem for lower semicontinuous functions bounded below (apply weak
convergence to $\min(e^{\sum a_pf_p},m)$ and let $m\to\infty$) $\int e^{\sum a_pf_p}d\mu_N\le\liminf_n\langle e^{\sum_pa_p\ff_p}\rangle^{\mathrm{fine}}
_{L_n}\le e^{\frac12\sum a_p^2}$ by \eqref{eq:GD}; likewise $\int e^{q\rho_x^2}d\mu_N\le\liminf_n\langle e^{q\rho_x^2}\rangle^{\mathrm{fine}}_{L_n}
\le C_N(q)$ by \eqref{eq:d2:mom}. This also proves the second clause of (e); the first clause of (e) is (a) applied to $F$ and $G$.

\emph{(c).} Let $g\in\cG_N\cup\{C\}$ and $F$ bounded continuous local. Then $F\circ g$ is bounded continuous local, and by Lemma~\ref{lem:d2:sym}
$\langle F\circ g\rangle^{\mathrm{fine}}_{L_n}=\langle F\rangle^{\mathrm{fine}}_{L_n}$ for all large $n$; by (a), $\int F\circ g\,d\mu_N=\int F\,d\mu_N$,
i.e.\ $\int F\,d(g_*\mu_N)=\int F\,d\mu_N$ for every bounded continuous local $F$. A Borel probability measure on the Polish space $E_S$ is determined by
the integrals of the bounded continuous functions, so $g_*\mu_N$ and $\mu_N$ have the same finite-dimensional marginals; the cylinder sets form a
$\pi$-system generating the Borel $\sigma$-algebra of $\OmN$, so $g_*\mu_N=\mu_N$. The conclusion of Lemma~\ref{lem:R1} for $\om_N$ follows, since $\eps\Z^3$, the reflections in the
site planes and the cubic group about every site lie in $\cG_N$.

\emph{(d).} Let first $F,G$ be bounded \emph{continuous} functions of the variables of finitely many cells lying in $\{x_j\ge c\}$. For $n$ large
these cells lie in the slab $\Lam^+_c$ of $\tilde T_{L_n}$ (in the $j$-th coordinate), so Theorem~\ref{thm:osrest:RPtorus} (with the cubic symmetry for
$j\in\{1,2\}$) gives $\langle\Th_{j,c}F\cdot G\rangle^{\mathrm{fine}}_{L_n}=\overline{\langle\Th_{j,c}G\cdot F\rangle^{\mathrm{fine}}_{L_n}}$ and
$\langle\Th_{j,c}F\cdot F\rangle^{\mathrm{fine}}_{L_n}\ge0$; the functions $\Th_{j,c}F\cdot G$, $\Th_{j,c}F\cdot F$ and $F\circ R_{j,c}$ are bounded
continuous local, so by (a) the identities hold for $\om_N$, and $\om_N(\Th_{j,c}F)=\overline{\om_N(F)}$ by (c). Positive semidefiniteness of the
form on finite families follows by applying the inequality to linear combinations. For the extension let $S$ be a finite set of cells lying in
$\{x_j\ge c\}$. The bounded continuous functions on $E_S$ are dense in $L^2(E_S,(\pi_S)_*\mu_N)$ (a finite Borel measure on a Polish space is
Radon), and a bounded Borel $F_S$ can be approximated by continuous $F_S^{(i)}$ with $\|F_S^{(i)}\|_\infty\le\|F_S\|_\infty$ (truncate). Both
sides of the identities are continuous in the $L^2(\mu_N)$-norms of $F$ and $G$, by the Schwarz inequality and $\|F\circ R_{j,c}\|_{L^2(\mu_N)}=
\|F\|_{L^2(\mu_N)}$ (the $R_{j,c}$-invariance of $\mu_N$, (c)). This gives (d) for bounded Borel $F,G$ and for $F,G$ in $L^2(\mu_N)$ measurable
with respect to the cells of $S$.

\emph{The last clause.} The proofs of (b)--(e) above use only (a) and the torus statements along $(L_n)$.
\end{proof}

\begin{remark}[uniqueness and the choice of sides]\label{rem:d2:LP}
(i) Different sequences of sides may give different limit points; the theorem claims no uniqueness, and \eqref{eq:D2} in its continuous form (ii) of
Corollary~\ref{cor:d2:iff} is exactly the statement that all limit points coincide, the form (i) implying it. (ii) Any cofinal class of sides may be
prescribed. (iii) The proof of Theorem~\ref{thm:d2:LP} uses only Theorem~\ref{thm:GD}, Theorem~\ref{thm:osrest:RPtorus},
Lemma~\ref{lem:d2:sym} and Lemma~\ref{lem:d2:mom}, which hold for arbitrary positive couplings and every real $\delta m_B$.
\end{remark}

\subsection{Limit points along arbitrary sequences of sides}\label{sec:d2:consumers}

\begin{corollary}[the results hold for every limit point]\label{cor:d2:consumers}
For each $N\ge N_0$ let $\om_N$ be the state of a limit point as in Theorem~\ref{thm:d2:LP}, along any sequence of sides $(L^{(N)}_n)_n$, and
write $Z_N(\sigma):=\om_N(e^{i\Phi_N(\sigma)})=\int e^{i\Phi_N(\sigma)}d\mu_N$.
\begin{itemize}
\item[(i)] Proposition~\ref{prop:E}, Lemma~\ref{lem:R1}, Proposition~\ref{prop:R3} and Theorem~\ref{thm:Hlim} hold for these $\om_N$; the density function $g_{\rm d}$ of Definition~\ref{def:pin} may be any bounded Borel function.
\item[(ii)] Every statement of Section~\ref{sec:osrest} holds for these $\om_N$, with ``$L\to\infty$'' read as ``$n\to\infty$ along $L^{(N)}_n$'': Lemma~\ref{lem:osrest:entire}, Theorem~\ref{thm:osrest:OS0}, Proposition~\ref{prop:osrest:OS1}, Corollary~\ref{cor:osrest:D1},
Propositions~\ref{prop:osrest:OS2}, \ref{prop:osrest:poly} and~\ref{prop:osrest:OS3}, Theorems~\ref{thm:osrest:OS2matrix}, \ref{thm:osrest:measure}
and~\ref{thm:osrest:OS4weak}.
\end{itemize}
\end{corollary}
\begin{proof}
(i) \emph{Proposition~\ref{prop:E}.} The only steps of its proof that use $\om_N$ are the final ones (from ``Finally'' on), and they rest on
Theorem~\ref{thm:d2:LP}(e). Indeed, write $\Phi_N(\sigma)=X+Y$ as in that proof, where $X$ is the curvature
part, a continuous function of finitely many $f_p$, and $Y$ is the density part, a bounded Borel function with $|Y|\le\|g_{\rm d}\|_\infty E(h)$
pointwise. The Schwarz inequality in $L^2(\mu_N)$ and Minkowski's inequality give
\[
\om_N(|\Phi_N(\sigma)|\wedge2)\le\Big(\int\Phi_N(\sigma)^2d\mu_N\Big)^{1/2}\le\Big(\int X^2d\mu_N\Big)^{1/2}+\|g_{\rm d}\|_\infty E(h),
\]
and, since $X^2$ is continuous and non-negative, Theorem~\ref{thm:d2:LP}(e) and Theorem~\ref{thm:GD} give
\[
\int X^2d\mu_N\le\liminf_n\langle X^2\rangle^{\mathrm{fine}}_{L_n}\le R(f)^2.
\]
So \eqref{eq:E} holds. The continuity of $g_{\rm d}$ is not needed, since $\om_N$ is the integral against $\mu_N$.

\emph{Lemma~\ref{lem:R1}} is Theorem~\ref{thm:d2:LP}(c). \emph{Proposition~\ref{prop:R3}} uses of $\om_N$ only
that it is a state, which gives $|Z_N(\sigma)|\le1$ and the positivity in Step~1, Lemma~\ref{lem:R1} and \eqref{eq:E}.
\emph{Theorem~\ref{thm:Hlim}} follows from these statements.

(ii) \emph{Lemma~\ref{lem:osrest:entire}(b).} The family $(\Psi_{N,L_n})_n$ is locally uniformly bounded by part (a) of that lemma. By Montel's
theorem it has locally uniformly convergent subsequences, and every subsequential limit agrees on $i\R^k$ with
\[
t\mapsto\int e^{i\sum_jt_j\Phi_N(f_j)}d\mu_N=\lim_n\langle e^{i\sum_jt_j\Phi_N(f_j)}\rangle^{\mathrm{fine}}_{L_n},
\]
by Theorem~\ref{thm:d2:LP}(a) applied to a bounded continuous function of finitely many $f_p$. Two entire functions that agree on $i\R^k$ coincide, so
$\Psi_{N,L_n}\to\Psi_N$ locally uniformly with $\Psi_N(it):=\int e^{it\cdot\Phi_N(f)}d\mu_N$, and the law $\mu_N^{(f)}$ of $(\Phi_N(f_j))_j$ under $\mu_N$ is
the weak limit of the torus laws, with $\int e^{z\cdot x}d\mu_N^{(f)}=\Psi_N(z)$ by the same uniform integrability as in the proof of that lemma. Every later
use of the lemma (Theorem~\ref{thm:osrest:OS0}(ii)--(iv), Theorem~\ref{thm:osrest:OS2matrix}, Proposition~\ref{prop:osrest:poly}) takes the limit along $(L_n)$.

\emph{Proposition~\ref{prop:osrest:OS1}} uses of $\om_N$ only Lemma~\ref{lem:R1}, Proposition~\ref{prop:R3} and Theorem~\ref{thm:Hlim}, which hold for these $\om_N$ by (i).
In \emph{Corollary~\ref{cor:osrest:D1}}, the identities for bounded Borel $F,G$ of cells in $\{x_3\ge c\}$ and the invariances are Theorem~\ref{thm:d2:LP}(d),(c),
and the extension to the bounded Borel class comes from the $L^2$ clause of
(d). \emph{Proposition~\ref{prop:osrest:OS2}} uses Theorem~\ref{thm:Hlim}(a) and Corollary~\ref{cor:osrest:D1}. For \emph{Proposition~\ref{prop:osrest:OS3}(b)},
$\Phi_N(f,h)\circ C=\Phi_N(-f,h)$, so $Z_N(-f,h)=Z_N(f,h)$ by the $C$-invariance of $\mu_N$, for every bounded Borel $g_{\rm d}$; the rest of its proof is unchanged.
\emph{Theorem~\ref{thm:osrest:measure}} and \emph{Theorem~\ref{thm:osrest:OS4weak}} use only the statements already covered.
\end{proof}

\begin{remark}[the existence statement that is used]\label{rem:d2:weakest}
The statements of this paper use the following existence statement and no more: \emph{for each $N$ there are a sequence of sides $L_n\to\infty$, $L_n/2\in\eps\Z$, and a probability measure $\mu_N$
on $\OmN$ such that the finite-dimensional laws of the gauge-invariant variables under the torus states of sides $L_n$ converge weakly to those of
$\mu_N$}; $\om_N$ is then the integral against $\mu_N$. The symmetries, the reflection positivity and the uniform bounds come with it
(Theorem~\ref{thm:d2:LP}). No averaging over the symmetry group is needed, because every lattice symmetry is a symmetry of every torus state.
The existence of the limit \eqref{eq:D2}, and a fortiori its form for bounded Borel $F$ (a setwise convergence), is not used in any
statement of this paper; it is needed
only to speak of \emph{the} fine state.
\end{remark}



\subsection{The cone class: the box limit, the floor, and the reduction of the limit \eqref{eq:D2} to the cone}\label{sec:d2:cone}

\begin{definition}[the cone; the killed boxes]\label{def:d2:cone}
For a real finitely supported $a=(a_p)_{p\in\cP_N}$, a polynomial $P$ in finitely many $\rho_x$ with non-negative coefficients and a finitely
supported $q=(q_x)_{x\in\cX_N}$ with $q_x\ge0$ put
\[
\Theta_{a,P,q}:=\cos\Big(\sum_pa_pf_p\Big)\,P(\rho)\,e^{\sum_xq_x\rho_x^2},
\]
and let $\cK_N$ be the set of finite non-negative linear combinations of such functions. The \emph{generators} are the $\Theta_{a,1,q}$.

For a finite box $\Lam\subset\eps\Z^3$ (a product of three intervals of sites) let $\cB(\Lam)$ be the bonds with both ends in $\Lam$ and
$\cP(\Lam)$ the plaquettes with all four bonds in $\cB(\Lam)$. The \emph{killed-box state} $\avB\cdot\Lam$ is the normalised expectation of the
weight
\[
W_\Lam:=\prod_{x\in\Lam}g_{\mathrm{ct}}(\tr_x)e^{-3\kappa_N\rho_x^2}\prod_{b\in\cB(\Lam)}e^{\kappa_N\rho_{b_-}\rho_{b_+}\cos(\gph B_b)}\prod_{p\in\cP(\Lam)}w\big((dB)_p+v_p\big)
\]
in the variables $\tr_x$ ($x\in\Lam$), $B_b$ ($b\in\cB(\Lam)$), $v_p$ ($p\in\cP(\Lam)$), the sum over the $v$ with $dv=0$ on every cube of $\Lam$. It is
the weight of the model of Section~\ref{sec:setting} with the matter killed outside $\Lam$ ($\rho_y=0$ for $y\notin\Lam$, which turns the hopping factor of
every bond leaving $\Lam$ into the diagonal factor $e^{-\kappa_N\rho_x^2/2}$ of its inner end, so that every site of $\Lam$ carries the diagonal
halves of all six of its bonds) and with no gauge or Villain variables outside $\cB(\Lam)$, $\cP(\Lam)$. A function is supported in $\Lam$ if it
depends only on $f_p$, $p\in\cP(\Lam)$, and $\rho_x$, $x\in\Lam$. A sequence of boxes $(\Lam_n)_n$ is \emph{exhausting} if every finite subset of $\eps\Z^3$ lies in $\Lam_n$ for all large $n$.
\end{definition}

The cone $\cK_N$ is closed under products ($\cos u\cos v=
\frac12[\cos(u+v)+\cos(u-v)]$), consists of continuous local functions, and each element satisfies $|\Theta|\le Ce^{\sum_xq'_x\rho_x^2}$ for
some $C<\infty$ and finitely supported $q'\ge0$; so Theorem~\ref{thm:d2:LP}(a) gives $\om_N(\Theta)=\lim_n\langle\Theta\rangle^{\mathrm{fine}}_{L_n}$
for every limit point and every $\Theta\in\cK_N$.
Gaussian domination holds for the box states by Lemma~\ref{rot:lem:GDall}.

\begin{theorem}[Ginibre's inequalities for the Villain fine model]\label{thm:d2:GIN}
For the torus states and for the killed-box states, and for $\Theta,\Theta'\in\cK_N$:
\begin{itemize}
\item[(K1)] $\langle\Theta\rangle\ge0$;\quad \textup{(K2)} $\langle\Theta;\Theta'\rangle:=\langle\Theta\Theta'\rangle-\langle\Theta\rangle\langle\Theta'\rangle\ge0$;
\item[(K3)] $\langle\Theta\rangle$ is non-decreasing in each hopping coefficient $\kappa_b\ge0$ (the coefficient of $\rho_{b_-}\rho_{b_+}\cos(\gph B_b)$,
bond by bond, the diagonal factors held fixed), in each plaquette stiffness $\beta_p>0$ ($w_p(f)=e^{-\beta_pf^2/2}$) and in $q\ge0$ under the change
of the site weight $g_{\mathrm{ct}}(\tr_x)\mapsto g_{\mathrm{ct}}(\tr_x)e^{q\rho_x^2}$; explicitly $\partial_{\kappa_b}\langle\Theta\rangle=\langle\Theta;\rho_{b_-}\rho_{b_+}
\cos(\gph B_b)\rangle\ge0$ and $\partial_{\beta_p}\langle\Theta\rangle=-\frac12\langle\Theta;f_p^2\rangle\ge0$;
\item[(K4)] for boxes $\Lam\subset\Lam'$ and $\Theta$ supported in $\Lam$: $\avB\Theta\Lam\le\avB\Theta{\Lam'}$; and if $\Lam$ together with its
nearest-neighbour sites lies in a cube of side less than $L$: $\avB\Theta\Lam\le\avL\Theta$.
\end{itemize}
\end{theorem}

These statements are proved, by the duplicate-variable argument of Ginibre \cite{Gin70} adapted to the Villain form of the plaquette weight, in Section~\ref{sec:gin} (Theorem~\ref{thm:gin:K}); we restate them here and do not repeat the proofs. For (K4) the proof interpolates the couplings of the cells of
$\Lam'$ (or of $\tilde T_L$) outside $\Lam$ from their values to $0$ and uses (K3), the endpoint being the box state. The classical input is positivity of the Fourier coefficients of the
weight, which gives (K1) alone; (K2)--(K4) need the duplication identity of the Villain weight.

\begin{theorem}[the box limit on the cone]\label{thm:d2:box}
\begin{itemize}
\item[(a)] For $\Theta\in\cK_N$ the net $\Lam\mapsto\avB\Theta\Lam$, over the boxes containing the support of $\Theta$ ordered by inclusion, is
non-decreasing and bounded; hence
\[
\ombox(\Theta):=\lim_{\Lam\uparrow\eps\Z^3}\avB\Theta\Lam=\sup_\Lam\avB\Theta\Lam
\]
exists, along every exhausting sequence of boxes, and $\ombox(\Theta)\le\om_N(\Theta)$ for every limit point $\om_N$ of Theorem~\ref{thm:d2:LP}
(along any sequence of sides).
\item[(b)] The box states converge: for every finite $S$ the laws $\nu^S_\Lam$ of the variables of $S$ under $\avB\cdot\Lam$ converge weakly, along
every exhausting sequence of boxes, to the marginal of one Borel probability measure $\mubox$ on $\OmN$; $\ombox(F):=\int F\,d\mubox$ extends
$\ombox$ from $\cK_N$ to all bounded Borel local $F$ and all $F$ as in Theorem~\ref{thm:d2:LP}(a) (with $\mubox$-null in place of $\mu_N$-null discontinuity sets); $\mubox$ is invariant under $\cG_N$ and $C$, and
$\int e^{\sum a_pf_p}d\mubox\le e^{\frac12\sum a_p^2}$, $\int e^{q\rho_x^2}d\mubox\le C_N(q)$.
\end{itemize}
\end{theorem}
\begin{proof}
(a) Monotonicity is (K4). Boundedness: for $\Theta=\Theta_{a,P,q}$ and a box $\Lam$ containing its support, $|\Theta|\le Ce^{\sum_xq'_x\rho_x^2}$ with
$q'>q$ on the sites of $P$ and $q'=q$ elsewhere, and $e^{\sum_xq'_x\rho_x^2}\in\cK_N$ is supported in $\Lam$, so by (K4) with a torus of side larger
than $\Lam$ and its neighbours, and Remark~\ref{rem:d2:mom}(ii), $\avB{|\Theta|}\Lam\le C\avL{e^{\sum_xq'_x\rho_x^2}}\le C\max_xC_N(|X|q'_x)
=:C(\Theta)$, uniformly in $\Lam$; the same bound holds for non-negative combinations. A bounded non-decreasing net of reals converges to its supremum,
and the values along any exhausting sequence of boxes converge to the same supremum (every box lies in all late members of the sequence, so by monotonicity the limit inferior along the sequence is at least the supremum over all boxes). For the comparison
with a limit point: given a box $\Lam$ containing the support of $\Theta$, (K4) gives $\avB\Theta\Lam\le\langle\Theta\rangle^{\mathrm{fine}}_{L_n}$ for
all large $n$, and the right side converges to $\om_N(\Theta)$ (Theorem~\ref{thm:d2:LP}(a)); so $\avB\Theta\Lam\le\om_N(\Theta)$ for every $\Lam$, and
$\ombox(\Theta)\le\om_N(\Theta)$.

(b) Tightness of $(\nu^S_\Lam)_\Lam$: by (K4), $\avB{e^{\rho_x^2}}\Lam\le\sup_L\avL{e^{\rho_x^2}}\le C_N(1)$, and $\avB{e^{s\ff_p}}\Lam\le e^{s^2/2}$ by
Theorem~\ref{thm:GD} for the box states (Lemma~\ref{rot:lem:GDall}). The argument of the proof of Theorem~\ref{thm:d2:LP} applies and gives,
along every exhausting sequence, a subsequence and a limit measure $\mu$ with the properties (a), (b) of that theorem (with ``box'' for ``torus''),
and $\mu$ is $C$-invariant because every box state is. By (a) of the present theorem and the uniform integrability of Theorem~\ref{thm:d2:LP}(a),
$\int\Theta\,d\mu=\lim\avB\Theta\Lam=\ombox(\Theta)$ along the subsequence, for every $\Theta\in\cK_N$. By Lemma~\ref{lem:d2:cone} below, any two such
limit measures coincide; hence the whole net converges, to one measure $\mubox$, and the extension statement follows.

For the invariance under a translation
$\tau$, note that $\tau$ maps an exhausting sequence of boxes to an exhausting sequence of boxes and $\avB{F\circ\tau}\Lam=\avB F{\tau^{-1}\Lam}$, so the limits agree.
Under a reflection or a cubic rotation $g$ about a site the same holds: $g$ maps boxes to boxes and exhausting sequences of boxes to exhausting sequences of boxes, and the box weights are covariant, the weight of $g\Lam$ being the image of $W_\Lam$ under $g$. Indeed every factor of $W_\Lam$ is indexed by a cell of $\Lam$, is the same function at every cell, and is even in $B_b$ and in $f_p$; a cell map commutes with the coboundaries up to the orientation signs, so the closedness of $v$ is preserved; the diagonal factor $e^{-3\kappa_N\rho_x^2}$ is the same at every site. In each case $\int F\circ g\,d\mubox=\int F\,d\mubox$ for every bounded continuous local $F$, and $g_*\mubox=\mubox$ follows as in the proof of Theorem~\ref{thm:d2:LP}(c). The bounds are Theorem~\ref{thm:d2:LP}(e) for the
box sequence.
\end{proof}

\begin{lemma}[the cone determines the state]\label{lem:d2:cone}
Let $\mu,\mu'$ be Borel probability measures on $\OmN$, both $C$-invariant, with $\int e^{\sum_xq_x\rho_x^2}d\mu<\infty$ and
$\int e^{\sum_xq_x\rho_x^2}d\mu'<\infty$ for every finitely supported $q\ge0$, and suppose $\int\Theta_{a,1,q}\,d\mu=\int\Theta_{a,1,q}\,d\mu'$ for all
generators. Then $\mu=\mu'$.
\end{lemma}
\begin{proof}
Fix finite sets $P\subset\cP_N$, $X\subset\cX_N$, and for $a\in\R^P$, $z\in\C^X$ put $G(a,z):=\int e^{i\sum_{p\in P}a_pf_p}e^{\sum_{x\in X}z_x\rho_x^2}d\mu$ and
$G'$ likewise for $\mu'$. For $z$ in a compact subset of $\C^X$ the integrand is dominated by $e^{\sum_xq_x\rho_x^2}$ with $q_x:=\max\{0,\max\Re z_x\}$, the
maximum over the compact set, which is integrable; so $G(a,\cdot)$ is entire on $\C^X$ (differentiation under the integral), and so is $G'(a,\cdot)$. By $C$-invariance the imaginary
part $\int\sin(a\cdot f)e^{z\cdot\rho^2}d\mu$ vanishes for real $z$, so for $z=q\in[0,\infty)^X$ the hypothesis gives $G(a,q)=\int\Theta_{a,1,q}d\mu=
\int\Theta_{a,1,q}d\mu'=G'(a,q)$. Two entire functions on $\C^X$ that agree on the set $[0,\infty)^X$, which has non-empty interior in $\R^X$,
coincide (identity theorem, one variable at a time). At $z=it$, $t\in\R^X$, $G(a,it)$ is the characteristic function of the random vector
$\big((f_p)_{p\in P},(\rho_x^2)_{x\in X}\big)$ under $\mu$; so the joint law of this vector is the same under $\mu$ and $\mu'$. Since $\rho_x\ge0$,
$\tr_x=\sqrt{\rho_x^2}-\rhof$ is a measurable function of $\rho_x^2$, and the joint law of $\big((f_p)_{p\in P},(\tr_x)_{x\in X}\big)$ is the same
under $\mu$ and $\mu'$. This holds for all finite $P,X$, so all finite-dimensional marginals agree, and $\mu=\mu'$ ($\pi$-$\lambda$ theorem on the
cylinder sets).
\end{proof}

\begin{corollary}[equivalent forms of the existence of the limit]\label{cor:d2:iff}
Fix $N$. The following are equivalent, the limits being over all $L$ with $L/2\in\eps\Z$ (or over any cofinal class of such sides):
\begin{itemize}
\item[(ii)] $\lim_{L\to\infty}\avL F$ exists for every bounded continuous local $F$;
\item[(iii)] $\lim_{L\to\infty}\avL{\Theta_{a,1,q}}$ exists for every generator;
\item[(iv)] all limit points of Theorem~\ref{thm:d2:LP}, along all sequences of sides, are one measure $\mu_N$.
\end{itemize}
Each is implied by \eqref{eq:D2} as stated (the limit for every bounded Borel local $F$), which we call (i). When they hold, $\om_N(F)=\lim_L\avL F$
for every bounded continuous local $F$, for every $F$ as in Theorem~\ref{thm:d2:LP}(a), and for every bounded Borel local $F$ whose discontinuity
set is $\mu_N$-null; and $\ombox\le\om_N$ on $\cK_N$.
\end{corollary}
\begin{proof}
(i)$\Rightarrow$(ii) is trivial. (ii)$\Rightarrow$(iii): under (ii) the laws $\nu^S_L$ converge weakly as $L\to\infty$ (a tight family whose integrals of
bounded continuous functions converge), and the generators are uniformly integrable by Lemma~\ref{lem:d2:mom}, so their expectations converge.
(iii)$\Rightarrow$(iv): two limit points $\mu_N,\mu_N'$ satisfy $\int\Theta_{a,1,q}d\mu_N=\lim_L\avL{\Theta_{a,1,q}}=\int\Theta_{a,1,q}d\mu'_N$ (Theorem~\ref{thm:d2:LP}(a)), are $C$-invariant and have all exponential moments (Theorem~\ref{thm:d2:LP}(b),(c)); Lemma~\ref{lem:d2:cone} gives $\mu_N=\mu'_N$.
(iv)$\Rightarrow$(ii): if the bounded continuous local $F$ had $\avL F\not\to\int Fd\mu_N$, a subsequence of sides would keep $|\avL F-\int Fd\mu_N|\ge\delta$;
Theorem~\ref{thm:d2:LP} applied to it gives a limit point, equal to $\mu_N$ by (iv), along a further subsequence of which $\avL F\to\int Fd\mu_N$, a
contradiction. The last sentence is Theorem~\ref{thm:d2:LP}(a) and Theorem~\ref{thm:d2:box}(a).
\end{proof}

\begin{remark}[a floor]\label{rem:d2:max}
On the cone the monotonicity (K3), (K4) runs one way: enlarging the volume, raising a hopping coefficient, a plaquette stiffness or a site factor
$e^{q\rho^2}$ raises every cone expectation. The killed box is therefore \emph{minimal} among the finite-volume states compared in (K4), and its limit is a floor under
every torus limit point (Theorem~\ref{thm:d2:box}(a)).
\end{remark}



\begin{remark}[summary of the section]\label{rem:d2:summary}
Limit points of the torus states exist, with every lattice symmetry, reflection positivity about all coordinate planes
through sites and bond midpoints, the Gaussian domination of the curvature and uniform local moments of the density (Theorem~\ref{thm:d2:LP}).
They suffice for Theorem~\ref{thm:Hlim} and for the statements of Section~\ref{sec:osrest} listed in
Corollary~\ref{cor:d2:consumers}. The killed-box states converge to a state $\ombox$ with the same symmetries, which lies below every torus limit
point on the cone $\cK_N$ (Theorem~\ref{thm:d2:box}); and \eqref{eq:D2}, in its continuous form (ii) of Corollary~\ref{cor:d2:iff}, is equivalent to the agreement
of the torus limit points on the generators of the cone, the form (i) implying it.
\end{remark}

\providecommand{\Rgen}{\mathcal R}
\providecommand{\Ggen}{G_0}
\providecommand{\SO}{\mathrm{SO}(3)}
\providecommand{\Oth}{\mathrm{O}(3)}
\providecommand{\Lset}{\mathcal L}
\providecommand{\TL}{T_L}
\providecommand{\supp}{\operatorname{supp}}
\providecommand{\avT}[2]{\langle #2\rangle_{#1}}

\providecommand{\cS}{\mathcal S}\providecommand{\cP}{\mathcal P}\providecommand{\cB}{\mathcal B}\providecommand{\cK}{\mathcal K}
\providecommand{\cG}{\mathcal G}\providecommand{\cC}{\mathcal C}\providecommand{\cD}{\mathcal D}\providecommand{\cH}{\mathcal H}
\providecommand{\cX}{\mathcal X}\providecommand{\cZ}{\mathcal Z}\providecommand{\E}{\mathbb E}\providecommand{\eps}{\varepsilon}
\providecommand{\Lam}{\Lambda}\providecommand{\OmN}{\Omega_N}\providecommand{\avB}[2]{\langle #1\rangle_{#2}}
\providecommand{\ombox}{\om^{\mathrm{box}}_N}\providecommand{\mubox}{\mu^{\mathrm{box}}_N}\providecommand{\supp}{\operatorname{supp}}
\providecommand{\dist}{\operatorname{dist}}\providecommand{\wnorm}[1]{\|#1\|_{w}}\providecommand{\Oh}{O_h}
\providecommand{\Oth}{\mathrm{O}(3)}\providecommand{\SO}{\mathrm{SO}(3)}\providecommand{\Sbox}{S^{\mathrm{box}}}
\providecommand{\Zbox}{Z^{\mathrm{box}}}\providecommand{\mub}{\mu^{\mathrm{box}}}\providecommand{\Lt}{\Lambda^2}
\providecommand{\Rp}{\operatorname{Re}}\providecommand{\Ip}{\operatorname{Im}}\providecommand{\bbone}{\mathbf 1}
\providecommand{\fA}{\mathfrak A}\providecommand{\wt}{\widetilde}
\providecommand{\gph}{\gamma_N}\providecommand{\lamf}{\lambda_0}\providecommand{\rhof}{\rho_{0,0}}

\section{Rotation invariance}\label{flh:sec:main}

Section~\ref{sec:osrest} left one piece of the Euclidean invariance open: the invariance of the limit theory under the rotations of
$\R^3$. For the limit theory built on the killed-box states of Definition~\ref{def:d2:cone} we prove it here. The corresponding statement for the limits built on the torus limit points of Definition~\ref{def:omN} is Theorem~\ref{flh:thm:FLH2}.
Section~\ref{flh:ssec:thms} states the theorems and gives a guide to the proof; Section~\ref{rot:sec:main} collects the preliminaries;
Sections~\ref{flh:ssec:I0}--\ref{flh:ssec:I8} prove the ingredients and Section~\ref{flh:ssec:proof} assembles them. The rigidity theorem, the diagonal
reflection positivity of the torus limit points and the general theorem on random fields are the subjects of Sections~\ref{rv:sec}, \ref{n1:sec} and~\ref{sec:i7:proof}.

\begin{remark}\label{flh:rem:strategy}
In proving the rotation invariance, in this section and in Sections~\ref{rv:sec}, \ref{n1:sec} and~\ref{sec:i7:proof}, we implement the original strategy of \cite{RPM1}. In a subsequent manuscript we shall provide a different strategy, adapting the proof from the companion paper \cite{CP}.
\end{remark}

\subsection{Setting and the theorems}\label{flh:ssec:thms}

Test pairs are $\sigma=(f,h)$ as in Section~\ref{sec:setting}; the transport $O\cdot\sigma$ of a test pair by $O\in\Oth$, the automorphism
$\gamma_O$ of $\cA$, the Euclidean group $E(3)$ and its action $\gamma\cdot\sigma$ are those of Definition~\ref{rot:def:sample} and
Proposition~\ref{rot:cor:E3} below (the $2$-form transported as a $2$-form, $h$ as a scalar). $\ombox$ is the killed-box limit of Theorem~\ref{thm:d2:box}
(it exists along every exhausting sequence of boxes), $\mubox$ its measure on $\OmN$. By Lemma~\ref{flh:lem:I3} below, Theorem~\ref{thm:Hlim} and
the statements of Section~\ref{sec:osrest} hold with $\ombox$ in place of $\om_N$, in the form given in that lemma; we call $\Sbox$ any limit of Theorem~\ref{thm:Hlim} so
obtained along a sequence $N_1<N_2<\cdots$, write $\Zbox(\sigma):=\Sbox(e_\sigma)$, $\Zbox(f):=\Zbox(f,0)$, $S^{\mathrm{box}}_n$ for its
Schwinger functions (Theorem~\ref{thm:osrest:OS0}(iii)) and $\mub$ for its measure (Theorem~\ref{thm:osrest:measure}, Lemma~\ref{rot:rem:minlos}).

\begin{theorem}[rotation invariance for the killed-box states]\label{flh:thm:FLH1}
Let the couplings be any $\kappa_N\ge0$, $\lamf>0$, $\rhof>0$, $\gph>0$ and any real $\delta m_B$ (notation of Remark~\ref{rem:letters}), and take $g_{\rm d}\equiv0$ in
Definition~\ref{def:pin} (the curvature scheme, Remark~\ref{rot:rem:gd0}). Let $\Sbox$ be a limit along $N_1<N_2<\cdots$. Then
\[
\begin{gathered}
\Zbox(R\cdot f)=\Zbox(f)\quad\text{for every }R\in\Oth\text{ and every real }C^\infty_c\ 2\text{-form }f,\\
\text{i.e.}\qquad\Sbox\circ\gamma_R=\Sbox ;
\end{gathered}
\]
hence (with Theorem~\ref{thm:Hlim}) $\Sbox(e_{\gamma\cdot\sigma})=\Sbox(e_\sigma)$ for all $\gamma\in E(3)$; $\gamma_*\mub=\mub$ for all
$\gamma\in E(3)$ (Proposition~\ref{rot:cor:E3}); $\mub$ is reflection positive across every affine plane of $\R^3$, on $L^2$ of either open side,
and the reconstruction in any direction is unitarily the one in direction $e_3$ (Remark~\ref{rem:osrest:rot}). With OS0
(Theorem~\ref{thm:osrest:OS0}), OS3 (Proposition~\ref{prop:osrest:OS3}) and the measure (Theorem~\ref{thm:osrest:measure}), $\Sbox$ satisfies
OS0--OS3 in full.
\end{theorem}

\begin{theorem}[bounded functions of the density]\label{flh:thm:FLH1p}
Let $g_{\rm d}$ be any bounded Borel function (the joint scheme $(f,h)$ of Definition~\ref{def:pin}). Then the curvature marginal of $\Sbox$ is as in
Theorem~\ref{flh:thm:FLH1}, and every mixed Schwinger function $S_n(\sigma_1,\dots,\sigma_n)$ of the joint field (the moments $\int\prod_i\Phi(\sigma_i)d\mu$ of the measure $\mu$ of Lemma~\ref{flh:lem:I5}) satisfies
\[
S_n(R\cdot\sigma_1,\dots,R\cdot\sigma_n)=S_n(\sigma_1,\dots,\sigma_n)\qquad\text{for all }R\in\Oth
\]
whenever the supports of $\sigma_1,\dots,\sigma_n$ are pairwise disjoint (the Osterwalder--Schrader form of Euclidean covariance, away from
coincident points). The coincident points are covered as well (Lemma~\ref{flh:lem:DENS}, Corollary~\ref{flh:cor:DENS}(a)): the mixed Schwinger functions extend to bounded multilinear forms,
on $L^2(\R^3;\Lt_\C)$ in the curvature entries and on $L^1(\R^3;\C)$ in the density entries, the displayed identity holds for all
such entries with no condition on the supports, and
\[
\begin{gathered}
\Zbox(R\cdot\sigma)=\Zbox(\sigma)\quad\text{for every }R\in\Oth\text{ and every test pair }\sigma=(f,h),\\
\text{i.e.}\qquad\Sbox\circ\gamma_R=\Sbox\ \text{ on }\cA ;
\end{gathered}
\]
hence (Theorem~\ref{thm:Hlim}, Proposition~\ref{rot:cor:E3}) $\Sbox(e_{\gamma\cdot\sigma})=\Sbox(e_\sigma)$ for all $\gamma\in E(3)$ and all test
pairs, and the joint measure of Lemma~\ref{flh:lem:I5} is invariant under $E(3)$.
\end{theorem}

\begin{theorem}[rotation invariance for the torus limit points]\label{flh:thm:FLH2}
Let $\om_N$ be a cubic-torus limit point (Definition~\ref{def:omN}, whose sides run along a dyadic sequence,
Definition~\ref{n1:def:sides}) and $S$ a limit functional of Theorem~\ref{thm:Hlim}. Then Theorems~\ref{flh:thm:FLH1}
and~\ref{flh:thm:FLH1p} hold for $S$ for $\kappa_N>0$ at every level (by Lemma~\ref{lem:rescale}; the value $\kappa_N=0$ is excluded), with the other couplings and the conclusions as stated there.
\end{theorem}

\begin{remark}
The reflection positivity of $\om_N$ at the six diagonal site planes $\{x_i\pm x_j=0\}$ through $0$, in the sense of
Theorem~\ref{thm:d2:LP}(d), is Corollary~\ref{n1:cor:N1} at $c=0$ (Theorem~\ref{n1:thm:N10} with the Hermitian form, the conjugation
identity and the $L^2$ extension) for the limit points along dyadic sequences; it is proved (for $\kappa_N\ne1$ with Lemma~\ref{lem:rescale}) through Theorem~\ref{rv:thm:RV} and
Lemma~\ref{n1:lem:SV}. Every other step of the proof below holds for the torus limit points as it stands.
\end{remark}

\emph{Guide to the proof.} The mechanism is the one of
Remark~\ref{rem:osrest:rot} read backwards: reflection positivity in enough planes forces rotation covariance. On the cubic torus
the reflection argument of Theorem~\ref{thm:osrest:RPtorus} applies at the coordinate planes only (the second wall of the torus at a diagonal plane is a glide, not a mirror; no period lattice
carries both a coordinate and a diagonal family of pointwise-fixed walls with the same coordinate $j$, that is $\{x_j=c\}$ and $\{x_j\pm x_k=c\}$, Lemma~\ref{flh:lem:I0}), whereas the killed box is reflection positive
at a symmetry plane of each of the nine families of lattice planes $\{x_j=c\}$, $\{x_j\pm x_k=c\}$ (Theorem~\ref{flh:thm:I1}). The box limit $\ombox$ of Theorem~\ref{thm:d2:box}
inherits the reflection positivity, at every lattice plane of the nine families (Lemma~\ref{flh:lem:I2}), the continuum passage of Sections~\ref{sec:setting}--\ref{sec:main} and~\ref{sec:osrest} runs on the box family verbatim,
and a general theorem on random fields with the Gaussian bound, translation invariance and reflection positivity in the nine planes
(Theorem~\ref{flh:thm:I7}, proved for all $n$ in Section~\ref{sec:i7:proof}, its case $n\le2$ also in
Section~\ref{flh:ssec:I7}) gives the rotation covariance of the
Schwinger functions away from coincident points. The closedness of the limit curvature (Theorem~\ref{flh:thm:I6}) and an elementary theorem
on bounded multilinear forms (Theorem~\ref{flh:thm:I8}) remove the coincident part.

For a bounded density function the coincident points of the density entries are treated in Lemma~\ref{flh:lem:DENS} and Corollary~\ref{flh:cor:DENS}.
For the limits built on the torus limit points the assembly is the same (Theorem~\ref{flh:thm:FLH2}): the reflection positivity at the six diagonal planes
through the origin, which the reflection argument on the cubic torus does not supply, is the subject of Section~\ref{n1:sec} and rests on the rigidity theorem of Section~\ref{rv:sec}.

\emph{Structure of the proof.} Theorem~\ref{flh:thm:I7} applied to the measure $\mu$ of Lemma~\ref{flh:lem:I5} (of which $\mub$, read on $\cS'$, is the curvature marginal), whose hypotheses are supplied by
Lemma~\ref{flh:lem:I5} (the Gaussian bound), Proposition~\ref{prop:osrest:OS1} for the box family together with Lemma~\ref{rot:rem:minlos} (translations) and
Lemma~\ref{flh:lem:I4} (reflection positivity in the nine plane families), gives the covariance away from coincidences (Theorem~\ref{flh:thm:FLH1p}). Theorem~\ref{flh:thm:I8}
with Lemma~\ref{flh:lem:I5} (moments bounded on $L^2$) and Theorem~\ref{flh:thm:I6} (closed field) removes the coincident-point part
(Proposition~\ref{flh:prop:DR}). The uniqueness from moments of Theorem~\ref{thm:osrest:measure} gives the invariance of $\Zbox$;
Proposition~\ref{rot:cor:E3} and Remark~\ref{rem:osrest:rot} give the rest.

\emph{Notation for the planes.} We write $\eps=\eN$. The nine families of lattice planes are $\{\phi=c\}$ with $\phi(x)=x_j$ ($j\in\{1,2,3\}$,
$c\in\frac12\eps\Z$: site planes for $c\in\eps\Z$, bond mid-planes otherwise) and $\phi(x)=x_i+\theta x_j$ ($i<j$, $\theta\in\{\pm1\}$,
$c\in\eps\Z$: the diagonal site planes). For such a plane $P$, $R_P$ is the orthogonal reflection ($R_Px=x-2(x_j-c)e_j$, resp.\
$R_Px=x-(x_i+\theta x_j-c)(e_i+\theta e_j)$); it maps $\eps\Z^3$ onto itself and lies in the group $\cG_N$ of Section~\ref{sec:d2:setting}
(a signed permutation of the coordinates about a site composed with a lattice translation). The closed side is $P^+:=\{\phi\ge c\}$; a cell
(site, bond, plaquette, cube) lies in $P^+$ if all its vertices do. Through the origin the nine planes are $\{x_j=0\}$ and $\{x_j\pm x_k=0\}$,
and the corresponding reflections are the signed permutation matrices $1-2e_je_j^{\mathsf T}$ and the matrices exchanging $e_j$ and $\mp e_k$.
A box is a product of three intervals of sites of $\eps\Z^3$ (Definition~\ref{def:d2:cone}); a \emph{cube of a box $\Lam$} is a unit cube of
$\eps\Z^3$ with all eight vertices in $\Lam$ (then its twelve bonds lie in $\cB(\Lam)$ and its six faces in $\cP(\Lam)$), and the closedness
constraint of Definition~\ref{def:d2:cone} is imposed on these cubes. A plane $P$ is a \emph{symmetry plane} of $\Lam$ if $R_P\Lam=\Lam$;
for the cube of sites $\Lam=\alpha+\eps\{0,\dots,M\}^3$, $\alpha\in\eps\Z^3$, the symmetry planes among the nine families are the three
coordinate planes $\{x_j=\alpha_j+M\eps/2\}$ and the six diagonal planes $\{x_i-x_j=\alpha_i-\alpha_j\}$, $\{x_i+x_j=\alpha_i+\alpha_j+M\eps\}$, $i<j$.

\subsection{Preliminaries: lattices on tori, the transport of test pairs, Gaussian domination, Euclidean invariance}\label{rot:sec:main}
This section collects what the proofs below take from the lattice models and from the limit functional:
the Villain model of a lattice on a torus and the transport of test pairs (Definitions~\ref{rot:def:model}, \ref{rot:def:sample}, Lemma~\ref{rot:lem:pass});
the choice $g_{\rm d}\equiv0$ (Remark~\ref{rot:rem:gd0}); Gaussian domination on every lattice, torus and killed box (Lemma~\ref{rot:lem:GDall});
Euclidean invariance from invariance under $\Oth$, for the functional and for its measure (Proposition~\ref{rot:cor:E3}, Lemma~\ref{rot:rem:minlos}).
They are proved from the statements of Sections~\ref{sec:setting}--\ref{sec:d2} alone;
Proposition~\ref{rot:cor:E3} carries the invariance of $S$ under $\Oth$ as its hypothesis.
Notation: for every level $N$, $\Lambda_N:=\eN\Z^3$ is the fine lattice, and $\sigma_\nu$ is the reflection $x_\nu\mapsto-x_\nu$ ($\nu=1,2,3$).

\begin{definition}[lattices, tori and the model]\label{rot:def:model}
For $O\in\Oth$ the \emph{lattice} $\Lambda:=O\Lambda_N$ carries the \emph{frame} $f_\alpha:=Oe_\alpha$, $\alpha=1,2,3$. Its cells are the
sites $x\in\Lambda$; the bonds $(x;\alpha)$ from $x$ to $x+\eN f_\alpha$, with the reversed bond $-(x;\alpha)$ from $x+\eN f_\alpha$ to
$x$; the plaquettes $p=(x;\alpha,\beta)$, $\alpha<\beta$, the squares with corners $x$, $x+\eN f_\alpha$, $x+\eN f_\beta$,
$x+\eN(f_\alpha+f_\beta)$, with centre $c(p):=x+\frac\eN2(f_\alpha+f_\beta)$, oriented boundary
$\partial p=(x;\alpha)+(x+\eN f_\alpha;\beta)-(x+\eN f_\beta;\alpha)-(x;\beta)$ and reversed plaquette $-p$; and the cubes
$x+\eN[0,1]^3$ in the frame, with the usual oriented boundary (six plaquettes with signs), so that $\partial\partial=0$. A \emph{period
lattice} for $\Lambda$ is a lattice $P\subset\Lambda$ of rank $3$ whose non-zero vectors have length at least $5\eN$; $P\subset\Lambda$ is
the same as $\Lambda+p=\Lambda$ for $p\in P$, and translation by $P$ permutes the cells of $\Lambda$, preserving the frame indices, the
orientations and the boundary operators. The quotient $\Lambda/P$ is a finite cubical complex (two cells of one cube are never
identified: a translate of a cell by $q\in P\setminus\{0\}$ lies at distance $|q|\ge5\eN>\sqrt3\eN$ from it), and $T:=\R^3/P$ is its torus.

The \emph{Villain model of $\Lambda$ on $T$} is the model of Section~\ref{sec:setting} (notation of
Remark~\ref{rem:letters}) on the cell complex $\Lambda/P$: the variables $\tr_x\in[-\rhof,\infty)$ at the sites,
$B_b\in(-\pi/\gph,\pi/\gph]$ on the bonds with $B_{-b}=-B_b$, and $v_p\in(2\pi/\gph)\Z$ on the plaquettes with $v_{-p}=-v_p$, of $\Lambda/P$;
$(dB)_p:=\sum_{b\in\partial p}B_b$ and $(dv)_c:=\sum_{p\in\partial c}v_p$ with the signs of the boundaries; $v$ is \emph{closed} if
$(dv)_c=0$ for every cube. The weight is
\[
W=\prod_xg_{\mathrm{ct}}(\tr_x)\cdot\prod_bh(\tr_{b_-},\tr_{b_+},B_b)\cdot\prod_pw\big((dB)_p+v_p\big)
\]
with the same factors $g_{\mathrm{ct}},h,w$ and coefficients $\gph,\lamf,\rhof,\kappa_N,\delta m_B$ as in Section~\ref{sec:setting}, the products over the
sites, the unoriented bonds and the unoriented plaquettes of $\Lambda/P$, and the sum over the Villain variables runs over all closed $v$.
Only the cell complex differs from Section~\ref{sec:setting}. $\avT{\Lambda,T}{\cdot}$ denotes the normalised
expectation and $\ff_p:=(dB)_p+v_p$ the curvature. For $\Lambda=\Lambda_N$ and $P=L\Z^3$, $L\in\eN\Z$, this is the fine torus state
$\avL\cdot$ of the model of Section~\ref{sec:setting} (on $\tilde T_L=\Lambda_N/L\Z^3$).
\end{definition}

The weight is well defined on unoriented cells: $h(r,r',\vartheta)=h(r',r,-\vartheta)$, so the factor of a bond does not depend on its
orientation; $w$ is even, so neither does the factor of a plaquette; and the closedness of $v$ does not depend on the orientations of
the plaquettes and cubes. Consequently the model depends only on the point set $\Lambda$ and on $P$, not on the frame: replacing
$(f_\alpha)$ by $(\pm f_{\pi(\alpha)})$, $\pi$ a permutation, relabels the cells and reverses some orientations, which changes the signs
of the variables on the reversed bonds and plaquettes and nothing else.

\begin{definition}[sampling of a test pair on a lattice; transport]\label{rot:def:sample}
For a test pair $\sigma=(f,h)$ let $\rho(\sigma)$ be the smallest $\rho\ge0$ with $\supp f\cup\supp h\subset\bar B_\rho$, the closed
Euclidean ball of radius $\rho$ about $0$. Let $\Lambda=O\Lambda_N$ with frame $(f_\alpha)$ and let $P$ be a period lattice for $\Lambda$
whose non-zero vectors have length $>2\rho(\sigma)$. The \emph{sampling of $\sigma$ on $\Lambda$}, read on $T=\R^3/P$, is
\begin{equation}\label{rot:eq:sample}
\Phi^{\Lambda,T}_N(\sigma):=\eN^{3/2}\sum_{p=(x;\alpha,\beta)}f_{c(p)}(f_\alpha,f_\beta)\,\ff_{\bar p}
+\eN^3\sum_{z\in\Lambda}h(z)\,g_{\rm d}(\tr_{\bar z}),
\end{equation}
where $f_c(u,v)$ is the $2$-form $f$ at the point $c$ evaluated on the vectors $u,v$, the first sum runs over the plaquettes of $\Lambda$
in $\R^3$ with $\alpha<\beta$, the second over the sites of $\Lambda$, and $\bar p,\bar z$ are the images in $\Lambda/P$; finitely many
terms are non-zero. For the fine lattice $\Lambda_N$ itself, $f_{c(p)}(e_\mu,e_\nu)=f_{\mu\nu}(c(p))$ and \eqref{rot:eq:sample} is
$\Phi_N(\sigma)$ of Definition~\ref{def:pin} with $x_p=c(p)$. For a rotated lattice $R^{-1}\Lambda_N$, $R\in\Oth$, the $2$-form is sampled as
\[
f_{c(p)}\big(R^{-1}e_\alpha,R^{-1}e_\beta\big)\qquad\text{at the centres }c(p)=R^{-1}\big(y+\tfrac\eN2(e_\alpha+e_\beta)\big),\ y\in\Lambda_N,
\]
of its own plaquettes, and the density at its own sites $R^{-1}y$.

The \emph{transport} of a test pair by $O\in\Oth$ is the push-forward
\[
(O\cdot f)_x(u,v):=f_{O^{-1}x}(O^{-1}u,O^{-1}v),\qquad(O\cdot h)(x):=h(O^{-1}x),\qquad O\cdot\sigma:=(O\cdot f,O\cdot h);
\]
$(O,\sigma)\mapsto O\cdot\sigma$ is an action of $\Oth$ by linear maps of $\mathcal T$, with $\supp(O\cdot\sigma)=O\supp\sigma$ and
$\rho(O\cdot\sigma)=\rho(\sigma)$; for $O=\sigma_3$ it is the time reflection $\theta$ of Section~\ref{sec:setting}, and for $O$ in the
cubic group about $0$ it is the push-forward action of Lemma~\ref{lem:R1}. For $O\in\Oth$ let $\gamma_O$ be the $*$-automorphism of
$\cA$ with $\gamma_Oe_\sigma:=e_{O\cdot\sigma}$ (well defined because $O\cdot$ is additive on $\mathcal T$).
\end{definition}

The sampling \eqref{rot:eq:sample} has three properties. (i) It does not depend on the frame: for the square with corner $x$ spanned by $u=f_\alpha$,
$v=f_\beta$, the term $f_{c}(u,v)\,\ff_{(x;u,v)}$ is unchanged if the same square is described as $(x;v,u)$ (both factors change sign,
$f$ being alternating and $\ff$ odd under reversal) or as $(x+\eN u;-u,v)$ (again both change sign, the boundary being reversed); so
\eqref{rot:eq:sample} is a function of the point set $\Lambda$, of the $2$-form and of the plaquette variables alone. (ii) Since the
non-zero vectors of $P$ are longer than $2\rho(\sigma)$ and the sampled cells have their centres in $\bar B_{\rho(\sigma)}$, distinct
sampled cells have distinct images in $\Lambda/P$ (two cells differing by $q\in P\setminus\{0\}$ have centres at distance $|q|>2\rho(\sigma)$):
$e^{\mathrm i\Phi^{\Lambda,T}_N(\sigma)}$ is the bounded local gauge-invariant observable $e^{\mathrm i\Phi^\Lambda_N(\sigma)}$ of the
infinite lattice $\Lambda$, read on the torus. (iii) It is real and additive in $\sigma$, so \eqref{eq:unimod} holds for every lattice
and torus.

\begin{lemma}[passive covariance]\label{rot:lem:pass}
Let $O\in\Oth$, let $\Lambda=O'\Lambda_N$ be a lattice with frame $(f_\alpha)$, $P$ a period lattice for $\Lambda$ and $T=\R^3/P$. Then
$O\Lambda$ is the lattice with frame $(Of_\alpha)$, $OP$ is a period lattice for it, and $O$ induces an isomorphism of cubical complexes
$\Lambda/P\to O\Lambda/OP$. The induced bijection of configurations,
\[
\tr'_{Ox}:=\tr_x,\qquad B'_{Ob}:=B_b,\qquad v'_{Op}:=v_p ,
\]
maps closed $v$ onto closed $v'$, preserves the weight $W$, and carries the Villain model of $\Lambda$ on $T$ onto the Villain model of
$O\Lambda$ on $OT:=\R^3/OP$: for every bounded Borel function $F$ of finitely many cell variables of $\Lambda/P$, writing $F^O$ for the same
function of the corresponding variables of $O\Lambda/OP$ (that is, $F^O(\tr',B',v')=F(\tr,B,v)$),
\begin{equation}\label{rot:eq:pass}
\avT{O\Lambda,OT}{F^O}=\avT{\Lambda,T}{F}.
\end{equation}
For test pairs $\sigma_1,\dots,\sigma_k$ with $2\rho(\sigma_j)$ smaller than the non-zero vectors of $P$, the sampled fields correspond,
$\Phi^{O\Lambda,OT}_N(O\cdot\sigma_j)(\tr',B',v')=\Phi^{\Lambda,T}_N(\sigma_j)(\tr,B,v)$, so that for every bounded Borel $G$ on $\R^k$
\begin{equation}\label{rot:eq:passsrc}
\Big\langle G\big(\Phi^{O\Lambda,OT}_N(O\cdot\sigma_1),\dots,\Phi^{O\Lambda,OT}_N(O\cdot\sigma_k)\big)\Big\rangle_{O\Lambda,OT}
=\Big\langle G\big(\Phi^{\Lambda,T}_N(\sigma_1),\dots,\Phi^{\Lambda,T}_N(\sigma_k)\big)\Big\rangle_{\Lambda,T}.
\end{equation}
\end{lemma}

\begin{proof}
$O$ is an isometry of $\R^3$ with $O\Lambda=OO'\Lambda_N$, $O(x+\eN f_\alpha)=Ox+\eN Of_\alpha$ and
$O\big(x+\frac\eN2(f_\alpha+f_\beta)\big)=Ox+\frac\eN2(Of_\alpha+Of_\beta)$: it maps the site $x$, the bond $(x;\alpha)$, the plaquette
$(x;\alpha,\beta)$ and the cube at $x$ of $\Lambda$ to the cells with the same labels of $O\Lambda$ in the frame $(Of_\alpha)$, and
$c(Op)=Oc(p)$. The boundary operators are defined by the labels, so $O$ commutes with $\partial$, signs included. $OP\subset O\Lambda$ is
a lattice of rank $3$ whose non-zero vectors have the same lengths as those of $P$, and $O(x+p)=Ox+Op$, so $O$ descends to a bijection
of the cells of $\Lambda/P$ onto those of $O\Lambda/OP$ commuting with $\partial$. Hence the map of configurations is a bijection which
preserves the product Lebesgue measure in $(\tr,B)$ and the counting measure in $v$, satisfies $(dB')_{Op}=(dB)_p$ and
$(dv')_{Oc}=(dv)_c$, so maps closed $v$ onto closed $v'$, and satisfies $W(\tr',B',v')=W(\tr,B,v)$, because $W$ is the product of the
same position-independent factors over corresponding cells. Therefore the normalisations agree and \eqref{rot:eq:pass} holds. For the
sources, $\ff'_{Op}=\ff_p$, $(O\cdot f)_{c(Op)}(Of_\alpha,Of_\beta)=f_{O^{-1}Oc(p)}(f_\alpha,f_\beta)=f_{c(p)}(f_\alpha,f_\beta)$ and
$(O\cdot h)(Oz)=h(z)$; summing over the plaquettes and sites of $O\Lambda$, which are the images of those of $\Lambda$, gives the
correspondence of the sampled fields, and \eqref{rot:eq:passsrc} is \eqref{rot:eq:pass} for $F=G(\Phi^{\Lambda,T}_N(\sigma_1),\dots)$.
The support condition is the same on both sides since $\rho(O\cdot\sigma_j)=\rho(\sigma_j)$.
\end{proof}

\begin{remark}[the density source and the choice $g_{\rm d}\equiv0$]\label{rot:rem:gd0}
Definition~\ref{def:pin} allows every bounded Borel $g_{\rm d}$, in particular $g_{\rm d}\equiv0$. With that choice
$\Phi_N(\sigma)=\eN^{3/2}\sum_pf_{\mu\nu(p)}(x_p)\ff_p$ depends on $f$ alone: the density terms disappear from \eqref{rot:eq:sample}
and from the seminorms
($\|\sigma\|=R(f)$); the limit theory is that of the smeared curvature, and Theorem~\ref{thm:Hlim}
applies to this scheme as to any other.
\end{remark}

\begin{lemma}[Gaussian domination on every lattice and torus]\label{rot:lem:GDall}
Let $\Lambda=O\Lambda_N$ be a lattice, $P$ a period lattice for it and $T=\R^3/P$. For every real function $a=(a_p)$ on the plaquettes of
$\Lambda/P$,
\[
\Big\langle\exp\Big(\sum_pa_p\ff_p\Big)\Big\rangle_{\Lambda,T}\le\exp\Big(\tfrac12\sum_pa_p^2\Big).
\]
The same inequality holds for the killed-box states $\langle\cdot\rangle_{\Lambda'}$ of Definition~\ref{def:d2:cone}, $\Lambda'$ a finite
box of $\Lambda_N$, for every real $a$ on the plaquettes of the box, and more generally for the normalised expectation of every weight
that is a product of the factors $g_{\mathrm{ct}}$, $h$ (or, for a bond with one end outside, the diagonal half $e^{-\kappa_N\rho^2/2}$ at its inner
end) and $w$ over the sites, bonds and plaquettes of a finite cubical complex with $\partial\partial=0$, the sum over the $v$ closed on
its cubes. Let $\sigma$ be a test pair with $2\rho(\sigma)$ smaller than the non-zero vectors of $P$, and define the seminorms of $\Lambda$ by
$R_\Lambda(f)^2:=\sup_{N'\ge N_0}\varepsilon_{N'}^3\sum_pf_{c(p)}(f_\alpha,f_\beta)^2$ (over the plaquettes of the lattice $O\Lambda_{N'}$),
$E_\Lambda(h):=\sup_{N'\ge N_0}\varepsilon_{N'}^3\sum_{z\in O\Lambda_{N'}}|h(z)|$, $\|\sigma\|_\Lambda:=R_\Lambda(f)+\|g_{\rm d}\|_\infty E_\Lambda(h)$.
Consequently, for all real $s$ and all $N$,
\begin{equation}\label{rot:eq:GDall}
\big\langle e^{s\Phi^{\Lambda,T}_N(\sigma)}\big\rangle_{\Lambda,T}\le e^{|s|\,\|g_{\rm d}\|_\infty E_\Lambda(h)}\,e^{s^2R_\Lambda(f)^2/2},\qquad
\big\langle\Phi^{\Lambda,T}_N(\sigma)^2\big\rangle_{\Lambda,T}^{1/2}\le\|\sigma\|_\Lambda ,
\end{equation}
and $|\langle e^{\mathrm i\Phi^{\Lambda,T}_N(\sigma)}\rangle_{\Lambda,T}-\langle e^{\mathrm i\Phi^{\Lambda,T}_N(\sigma')}\rangle_{\Lambda,T}|
\le\|\sigma-\sigma'\|_\Lambda$. The seminorms obey $R_\Lambda(f)\le\sqrt{3|K_{+\azero}|}\,\|f\|_\infty$ and
$E_\Lambda(h)\le|K_{+\azero}|\,\|h\|_\infty$ for a compact $K$ containing the supports; here $\|f\|_\infty$ is the supremum over $\R^3$ of the Euclidean norm $(\sum_{\mu<\nu}f_{\mu\nu}^2)^{1/2}$, so that $|f_c(u,v)|\le\|f\|_\infty$ for orthonormal $u,v$.
\end{lemma}

\begin{proof}
Steps 1--4 of the proof of Theorem~\ref{thm:GD} use only that the weight is a product over the sites, bonds and plaquettes of a finite
cubical complex with $\partial\partial=0$ (whence $d^*d^*=0$ in Step 2), the Fourier expansion of the hopping factor bond by bond
(Step 3) and the sum over closed $v$; nothing refers to the frame or to the period lattice. They apply word for word to the complex
$\Lambda/P$ and give the first inequality. They apply equally to the killed boxes of Definition~\ref{def:d2:cone} and to the general
weights described in the statement: the diagonal halves $e^{-\kappa_N\rho_x^2/2}$ attached to the sites do not depend on the gauge
variables, so the Fourier expansion of Step~3 is applied bond by bond to the remaining hopping factors, and Steps~1, 2 and~4 use only the
complex and the plaquette weight $w$. The cells of a box $\Lambda'$ form such a complex: every cube of $\cC(\Lambda')$ has its six faces in $\cP(\Lambda')$ and every plaquette of $\cP(\Lambda')$ its four bonds in $\cB(\Lambda')$, Definition~\ref{def:gin:weights}; and every bond of $\cB(\Lambda')$ carries the hopping factor with both matter variables.

The exponential moment bound follows as in the proof of Proposition~\ref{prop:E}:
$\Phi^{\Lambda,T}_N(\sigma)=X+Y$ with $X=\sum_pa_p\ff_p$, $a_p=\eN^{3/2}f_{c(p)}(f_\alpha,f_\beta)$, $|Y|\le\|g_{\rm d}\|_\infty E_\Lambda(h)$ and
$\sum_pa_p^2\le R_\Lambda(f)^2$; the second-moment bound is Step~5 of the proof of Theorem~\ref{thm:GD} with Minkowski's inequality, and the Lipschitz bound
is \eqref{eq:R31} with the Schwarz inequality, $|e^{\mathrm ix}-e^{\mathrm iy}|\le|x-y|$. The bounds on the seminorms are those
following Definition~\ref{def:norms}: the open cubes of side $\varepsilon_{N'}$ centred at the sites of the lattice that lie in $K$ are pairwise disjoint and lie in
$K_{+\sqrt3\varepsilon_{N'}/2}\subset K_{+\azero}$ (as $\varepsilon_{N'}\le\azero$), so there are at most $|K_{+\azero}|/\varepsilon_{N'}^3$ such sites; the centres of the plaquettes of one
orientation form a translate of the lattice, so at most three times as many plaquettes have their centres in $K$.
\end{proof}

\begin{proposition}[Euclidean invariance from invariance under $\Oth$; invariance of the measure]\label{rot:cor:E3}
Let $S$ be a limit of Theorem~\ref{thm:Hlim} (for the torus family, or for any family of fine states satisfying its hypotheses), with
$Z(\sigma)=S(e_\sigma)$, and assume that $S$ is invariant under the orthogonal group, $S\circ\gamma_R=S$ for all $R\in\Oth$.
\begin{enumerate}
\item[(a)] Let $E(3)=\R^3\rtimes\Oth$ be the Euclidean group, $\gamma=(b,R)$ acting on $\R^3$ by $\gamma x=Rx+b$ and on test pairs by
the push-forward $\gamma\cdot\sigma:=\tau_b(R\cdot\sigma)$, $\tau_b$ the translation by $b$. Then $S(e_{\gamma\cdot\sigma})=S(e_\sigma)$
for every $\gamma\in E(3)$ and every test pair $\sigma$.
\item[(b)] Let $X$ be a set of real linear functionals $\Phi$ on $\mathcal T$, stable under $(\gamma\Phi)(\sigma):=\Phi(\gamma^{-1}\cdot\sigma)$
for $\gamma\in E(3)$, with the $\sigma$-algebra generated by the evaluations $\Phi\mapsto\Phi(\sigma)$, and let $\mu$ be a probability
measure on $X$ with $S(e_\sigma)=\int_Xe^{\mathrm i\Phi(\sigma)}\,d\mu(\Phi)$ for every $\sigma$. Then $\gamma_*\mu=\mu$ for every
$\gamma\in E(3)$; in particular $\mu$ is $\Oth$-invariant.
\end{enumerate}
\end{proposition}

\begin{proof}
(a) $R\cdot(\tau_{b'}\sigma)=\tau_{Rb'}(R\cdot\sigma)$ (substitute $R^{-1}x-b'=R^{-1}(x-Rb')$), so
$\gamma\cdot(\gamma'\cdot\sigma)=\tau_b\tau_{Rb'}((RR')\cdot\sigma)=(\gamma\gamma')\cdot\sigma$ with $\gamma\gamma'=(b+Rb',RR')$: this
is an action of $E(3)$, and $E(3)$ is generated by the translations and $\Oth$. $S$ is invariant under all translations by
Theorem~\ref{thm:Hlim} (the time translations by its clause (b), the space translations by its last clause), and under $\Oth$ by
hypothesis; so it is invariant under $E(3)$. (b) $\Phi\mapsto\gamma\Phi$ is measurable, since $(\gamma\Phi)(\sigma)$ is
the evaluation at $\gamma^{-1}\cdot\sigma$. For $\sigma_1,\dots,\sigma_k\in\mathcal T$ and $s\in\R^k$, by the linearity of $\Phi$ and (a),
\[
\begin{aligned}
\int_Xe^{\mathrm i\sum_js_j(\gamma\Phi)(\sigma_j)}\,d\mu(\Phi)&=\int_Xe^{\mathrm i\Phi(\gamma^{-1}\cdot\sum_js_j\sigma_j)}\,d\mu(\Phi)\\
&=S\big(e_{\gamma^{-1}\cdot\sum_js_j\sigma_j}\big)=S\big(e_{\sum_js_j\sigma_j}\big)=\int_Xe^{\mathrm i\sum_js_j\Phi(\sigma_j)}\,d\mu(\Phi).
\end{aligned}
\]
So the laws of $(\Phi(\sigma_j))_{j\le k}$ under $\gamma_*\mu$ and under $\mu$ have the same Fourier transform on $\R^k$ and coincide;
the cylinder sets $\{\Phi:(\Phi(\sigma_j))_{j\le k}\in B\}$ form a $\pi$-system generating the $\sigma$-algebra, on which the two
probability measures agree; hence $\gamma_*\mu=\mu$.
\end{proof}

Every limit $S^{\mathrm{box}}$ of the killed-box family is invariant under the orthogonal group, by Theorems~\ref{flh:thm:FLH1} and~\ref{flh:thm:FLH1p}, and so is every limit $S$ built on the torus limit points of Definition~\ref{def:omN} along dyadic sequences of sides, by Theorem~\ref{flh:thm:FLH2}.

Nothing above needs $\mu$ to exist; it does, by the following lemma.

\begin{lemma}[existence of the measure]\label{rot:rem:minlos}
Let $S$ be a limit of Theorem~\ref{thm:Hlim} (for the torus family, or for any family of fine states satisfying its hypotheses), with $Z(\sigma)=S(e_\sigma)$. There is a unique probability measure on the dual $\mathcal D'(\R^3;\Lambda^2\R^3\oplus\R)$,
with the cylinder $\sigma$-algebra, whose characteristic functional is $Z$; $E(3)$ acts on that dual by the transposed action, and Proposition~\ref{rot:cor:E3}(b)
applies to this measure when $S$ is invariant under $\Oth$. Moreover, a probability measure on any such space of linear functionals, with the cylinder
$\sigma$-algebra, is determined by its characteristic functional.
\end{lemma}
\begin{proof}
$Z$ is the pointwise limit of the characteristic functionals
$\sigma\mapsto Z_{N_m}(\sigma)=\om_{N_m}(e^{\mathrm i\Phi_{N_m}(\sigma)})$ of the real random variables $\Phi_{N_m}(\sigma)$, which are
additive in $\sigma$, so $Z$ is positive definite on the additive group $\mathcal T$ with $Z(0)=1$; and $Z$ is continuous on each
Fr\'echet space $\mathcal T_K$ of test pairs supported in a compact $K$, by \eqref{eq:R31} and the admissibility of the seminorm. The
Bochner--Minlos theorem, in Minlos' form on $\mathcal D'$ (\cite{Min59}, \cite[Chapter~IV]{GV64}), then gives the measure and its uniqueness. The argument of Proposition~\ref{rot:cor:E3}(b) shows that a probability measure on any such space of linear functionals, with the cylinder
$\sigma$-algebra, is determined by its characteristic functional.
\end{proof}

\subsection{No torus carries the nine families}\label{flh:ssec:I0}

\begin{lemma}[reflection walls of a torus]\label{flh:lem:I0}
Let $\Gamma\subset\R^3$ be a lattice of rank $3$, $n$ a unit vector, $\theta_nx:=x-2(n\cdot x)n$ and $\pi_nx:=(n\cdot x)n$.
\begin{itemize}
\item[(a)] If $\theta_n\Gamma=\Gamma$, the $\theta_n$-invariant walls of $\R^3/\Gamma$ are the images of the planes $\{n\cdot x=c\}$ with
$2c\in n\cdot\Gamma$; such a wall is pointwise fixed iff $2cn\in\Gamma$; all of them are pointwise fixed iff $\pi_n(\Gamma)\subset\Gamma$,
i.e.\ iff $\pi_n(\Gamma)=\Gamma\cap\R n$ (and this inclusion implies $\theta_n\Gamma=\Gamma$).
\item[(b)] No lattice satisfies $\pi_{e_j}(\Gamma)\subset\Gamma$ and $\pi_n(\Gamma)\subset\Gamma$ with $n=(e_j\pm e_k)/\sqrt2$, $k\ne j$.
\end{itemize}
Hence no torus $\R^3/\Gamma$ carries a coordinate family and a diagonal family with the same coordinate $j$ (the families $\{x_j=c\}$ and $\{x_j\pm x_k=c\}$, $k\ne j$) with pointwise-fixed walls, a
fortiori none carries the nine families. (The cubic lattice $\Z^3$ passes the coordinate test only, the lattice $\Z(e_1-e_2)+\Z(e_1+e_2)+\Z e_3$
the test for $(e_1-e_2)/\sqrt2$ but not the one for $e_1$.)
\end{lemma}
\begin{proof}
(a) The image of $\{n\cdot x=c\}$ in $\R^3/\Gamma$ is the set $\{x:n\cdot x\in c+n\cdot\Gamma\}$; $\theta_n$ maps it to
$\{n\cdot x\in-c+n\cdot\Gamma\}$, which is the same set iff $2c\in n\cdot\Gamma$. A point $x$ of it is fixed iff $\theta_nx-x=-2(n\cdot x)n\in
\Gamma$. Write $n\cdot x=c+n\cdot\gamma$ with $\gamma\in\Gamma$; since $2(n\cdot\gamma)n=\gamma-\theta_n\gamma\in\Gamma$ (here
$\theta_n\Gamma=\Gamma$ enters), $x$ is fixed iff $2cn\in\Gamma$, independently of $x$ on the wall. If $\pi_n(\Gamma)\subset\Gamma$ and
$2c=n\cdot\gamma$, then $2cn=\pi_n\gamma\in\Gamma$; conversely every $\gamma\in\Gamma$ gives the admissible $c=\frac12n\cdot\gamma$ with
$\pi_n\gamma=2cn$, so if all walls are fixed then $\pi_n(\Gamma)\subset\Gamma$. Always $\Gamma\cap\R n\subset\pi_n(\Gamma)$, so
$\pi_n(\Gamma)\subset\Gamma$ is the same as $\pi_n(\Gamma)=\Gamma\cap\R n$; and then $\theta_n\gamma=\gamma-2\pi_n\gamma\in\Gamma$.

(b) Take $j=1$ and $n=(e_1-e_2)/\sqrt2$; the other cases follow by a signed permutation of the coordinates. Since $\Gamma$ has rank $3$,
$\pi_{e_1}(\Gamma)\ne0$ and $\pi_n(\Gamma)\ne0$, so under the two hypotheses $\Gamma\cap\R e_1=p\Z e_1$ and $\Gamma\cap\R n=s\Z(e_1-e_2)$ with
$p,s>0$, and the hypotheses read: $\gamma_1\in p\Z$ and $\frac12(\gamma_1-\gamma_2)\in s\Z$ for every $\gamma\in\Gamma$. For
$\gamma=s(e_1-e_2)$ the first gives $s\in p\Z$, $s=kp$; for $\gamma=pe_1$ the second gives $p\in2s\Z$, $p=2ls$. Then $p=2klp$, i.e.\ $2kl=1$ with
integers $k,l\ge1$: impossible. The families $\{x_1=c\}$ and $\{x_1-x_2=c\}$ are among the nine, which proves the last sentence; the two
examples are checked by the same computation ($\Z^3$: $\pi_{e_1}\Z^3=\Z e_1\subset\Z^3$, but $\pi_n(e_1)=\frac12(e_1-e_2)\notin\Z^3$; the
second lattice: $\pi_n$ of its generators lies in $\Z(e_1-e_2)$, but $\pi_{e_1}(e_1-e_2)=e_1$ is not in it).
\end{proof}

\subsection{Reflection positivity of the killed-box state at its nine symmetry planes}\label{flh:ssec:I1}

\emph{Setting.} Fix $N$, $\eps:=\eN$ (the spacing), $\gph$ the phase constant, $\kappa_N$ the hopping coefficient (the unindexed symbol $\kappa$ denotes a cell below), and a cube of sites $\Lam=\alpha+\eps\{0,\dots,M\}^3$, $\alpha\in\eps\Z^3$, $M\ge1$;
$\cB(\Lam)$, $\cP(\Lam)$ are as in Definition~\ref{def:d2:cone} and $\cC(\Lam)$ is the set of cubes of $\Lam$. Cells are oriented, with
incidence numbers $[\kappa:\sigma]\in\{0,\pm1\}$, $\partial\kappa=\sum_\sigma[\kappa:\sigma]\sigma$, $\partial\partial=0$; the variables
$\xi=(\tr_x,B_b,v_p)$ obey $u_{-\sigma}=-u_\sigma$, $(dB)_p=\sum_b[p:b]B_b$, $(dv)_c=\sum_p[c:p]v_p$, $\ff_p=(dB)_p+v_p$. With
$\wt g(r):=g_{\mathrm{ct}}(r)e^{-3\kappa_N\rho^2}$ ($\rho=\rhof+r$) and $\wt h(\rho,\rho',B):=e^{\kappa_N\rho\rho'\cos(\gph B)}$, the weight of
Definition~\ref{def:d2:cone} is
\[
W_\Lam=\prod_{x\in\Lam}\wt g(\tr_x)\prod_{b\in\cB(\Lam)}\wt h(\rho_{b_-},\rho_{b_+},B_b)\prod_{p\in\cP(\Lam)}w(\ff_p)\prod_{c\in\cC(\Lam)}\bbone[(dv)_c=0],
\]
against the reference measure $m_\Lam=\prod_xd\tr_x\prod_bdB_b\prod_p(\text{counting measure on }(2\pi/\gph)\Z)$, $B_b\in(-\pi/\gph,\pi/\gph]$;
$Z_\Lam:=\int W_\Lam\,dm_\Lam\in(0,\infty)$ (Step~4 below) and $\avB F\Lam:=Z_\Lam^{-1}\int FW_\Lam\,dm_\Lam$.

Let $\Pi$ be a symmetry plane of
$\Lam$, $\varphi$ the affine function $x_j-\alpha_j-M\eps/2$, resp.\ $x_i\mp x_j-c^\mp_{ij}$ with $c^-_{ij}=\alpha_i-\alpha_j$,
$c^+_{ij}=\alpha_i+\alpha_j+M\eps$ (we write $\Pi^-_{ij}$, $\Pi^+_{ij}$ for these two diagonal planes), so that $\Pi=\{\varphi=0\}$; $n:=\nabla\varphi$, $R=R_\Pi\colon x\mapsto x-2\varphi(x)n/|n|^2$ the reflection,
$H_\pm:=\{\pm\varphi\ge0\}$. $R$ maps $\eps\Z^3$ onto itself ($2\varphi/|n|^2\in\eps\Z$ on sites, as $c^\pm_{ij}\in\eps\Z$), maps $\Lam$ onto
$\Lam$ ($\alpha_j+\eps m\mapsto\alpha_j+\eps(M-m)$; $(\alpha_i+\eps m_i,\alpha_j+\eps m_j)\mapsto(\alpha_i+\eps m_j,\alpha_j+\eps m_i)$, resp.\
$(\alpha_i+\eps(M-m_j),\alpha_j+\eps(M-m_i))$), hence maps $\cB(\Lam)$, $\cP(\Lam)$, $\cC(\Lam)$ onto themselves, and commutes with $\partial$:
$[R\kappa:R\sigma]=[\kappa:\sigma]$. It acts on configurations by $(R\xi)_\kappa:=\xi_{R\kappa}$, so that $(dB)_p\circ R=(dB)_{Rp}$,
$\ff_p\circ R=\ff_{Rp}$, $(dv)_c\circ R=(dv)_{Rc}$; it maps the configuration space $(-\pi/\gph,\pi/\gph]^{\cB(\Lam)}$ onto itself up to the
$m_\Lam$-null set where some $B_b=\pi/\gph$, so $F\circ R$ is defined almost everywhere, which suffices.

A cell $\kappa$ with vertex set $V(\kappa)$ is a \emph{plane cell} if $V(\kappa)\subset\Pi$, a \emph{$(+)$-cell} if $V(\kappa)\subset H_+$ and
it is not plane, a \emph{$(-)$-cell} likewise, and \emph{crossing} otherwise. Write $\xi_0,\xi_+,\xi_-,\xi_\times$ for the variables of the
four classes and $\cB_\times,\cP_\times,\cC_\times$ for the crossing bonds, plaquettes, cubes (no site crosses). $\fA^+_\Lam(\Pi)$ is the
algebra of bounded Borel functions of $(\xi_0,\xi_+)$, i.e.\ of the variables of the cells lying in $H_+$, and $\Th_\Pi F:=\overline{F\circ R_\Pi}$.

\begin{lemma}[cells at a symmetry plane]\label{flh:lem:I1cells}
(a) $R$ fixes every plane cell (orientation included), maps the $(+)$-cells bijectively onto the $(-)$-cells, and maps every crossing cell
$\kappa$ onto itself with reversed orientation, $R\kappa=-\kappa$. (b) No face of a crossing cell is a plane cell; every cell having a crossing
face is crossing; the faces of a plane or $(+)$-cell are plane or $(+)$-cells. (c) At a coordinate site plane ($M$ even) there are no crossing cells.
At a coordinate bond mid-plane ($M$ odd) there are no plane cells; the crossing cells are the $e_j$-bonds, the $e_je_k$-plaquettes and the cubes
with vertices on both slices $\varphi=\pm\eps/2$. At a diagonal plane $\Pi$ the plane cells are the sites of $\Pi$ and the $e_k$-bonds
($k\ne i,j$) in $\Pi$; \emph{no bond crosses}; the crossing cells are the $e_ie_j$-plaquettes whose vertices carry the $\varphi$-values
$\{-\eps,0,0,\eps\}$ (bisected by $\Pi$ along a diagonal) and the cubes having such a face; a crossing plaquette has two bonds in $H_+$ and two in
$H_-$; a crossing cube has two crossing faces and two side faces in each of $H_\pm$; no plaquette is a plane cell.
\end{lemma}
\begin{proof}
(c) On the sites $\varphi$ takes values in $\eps\Z$ (site and diagonal planes) or in $\eps\Z+\eps/2$ (mid-plane), and
$|\varphi(x+\eps e_l)-\varphi(x)|\in\{0,\eps\}$, so the $\varphi$-range of a cell is $\eps$ times the number of its directions $e_l$ with
$n_l\ne0$. A cell crosses iff $\varphi$ takes both signs on $V(\kappa)$. At a coordinate plane the range is at most $\eps$; with values in $\eps\Z$
no cell crosses; with values in $\eps\Z+\eps/2$ a cell crosses iff it has vertices on both slices, and no cell is plane. At a diagonal plane
crossing needs the values $\eps$ and $-\eps$, hence range $2\eps$, hence both directions $e_i,e_j$; for the $e_ie_j$-plaquette with corner $a$ the
values are $\varphi(a),\varphi(a)+\eps,\varphi(a)-\eps,\varphi(a)$ ($\Pi^-$), resp.\ $\varphi(a),\varphi(a)+\eps,\varphi(a)+\eps,\varphi(a)+2\eps$
($\Pi^+$), and it crosses iff they are $\{-\eps,0,0,\eps\}$; a cube crosses iff its $e_ie_j$-faces do. The bonds of a crossing plaquette and the
$e_ie_k$-, $e_je_k$-faces of a crossing cube take the values $\{0,\eps\}$ or $\{0,-\eps\}$, two of each kind; a plane cell lies in $\Pi$, which
contains no square and no bond of direction $e_i$ or $e_j$.

(a) $R$ fixes $\Pi$ pointwise, hence every vertex of a plane cell and the oriented cell. A $(\pm)$-cell has a vertex in the open side
$\{\pm\varphi>0\}$, which $R$ carries to the other open side, and $R(H_\pm)=H_\mp$; with $R^2=1$ the bijection follows. Let $\kappa$ cross.
At a mid-plane, $V(\kappa)=V'\times\{c\mp\eps/2\}$ and $R$ swaps the two slices, fixing the other coordinates. At a diagonal plane the two vertices off
$\Pi$ are $a+\eps e_i$, $a+\eps e_j$ ($\Pi^-$, $\varphi(a)=0$), resp.\ $a$, $a+\eps e_i+\eps e_j$ ($\Pi^+$, $\varphi(a)=-\eps$), and
$R(x)=x-\varphi(x)(e_i\mp e_j)$ exchanges them; for cubes add $\{0,\eps\}e_k$, fixed by $R$. So $V(\kappa)$ is $R$-invariant and $R\kappa=\pm\kappa$.
$R$ restricted to the affine span $E$ of $\kappa$ fixes $E\cap\Pi$ pointwise, a hyperplane of $E$ ($E\not\subset\Pi$, and $E\cap\Pi\ne\emptyset$
since $\kappa$ meets both sides), and moves a vertex: it is the reflection of $E$ in that hyperplane and reverses the orientation, $R\kappa=-\kappa$.

(b) If a face $\sigma$ of a crossing $\kappa$ were a plane cell, then $R\sigma=\sigma$ and $[\kappa:\sigma]=[R\kappa:R\sigma]=[-\kappa:\sigma]
=-[\kappa:\sigma]$, so $[\kappa:\sigma]=0$. A cell with a face having vertices on both open sides has such vertices itself. The vertices of a face
are vertices of the cell, which gives the last clause.
\end{proof}

\begin{lemma}[splitting of the coboundaries at a symmetry plane]\label{flh:lem:I1split}
Let $\kappa\in\cP_\times\cup\cC_\times$ and let $u$ be the variables of its faces ($B$ for a plaquette, $v$ for a cube). Put
$U_\times(\kappa):=\sum_{\sigma\ \mathrm{crossing}}[\kappa:\sigma]u_\sigma$, $U_+(\kappa):=\sum_{\sigma\ (+)}[\kappa:\sigma]u_\sigma$,
$U_-(\kappa):=-\sum_{\sigma\ (-)}[\kappa:\sigma]u_\sigma$. Then $(du)_\kappa=U_\times(\kappa)+U_+(\kappa)-U_-(\kappa)$, and
$U_+(\kappa)\circ R=U_-(\kappa)$, $U_-(\kappa)\circ R=U_+(\kappa)$. For $t\in\R^{\cC_\times}$, $s_p:=\sum_{c\in\cC_\times}[c:p]t_c$
($p\in\cP_\times$) and every crossing bond $b$: $\sum_{p\in\cP_\times}[p:b]s_p=0$. We write $B_\times(p)$, $B_\pm(p)$ for $U_\times(p)$, $U_\pm(p)$ when $u=B$, and $V_\pm(c)$ for $U_\pm(c)$ when $u=v$.
\end{lemma}
\begin{proof}
The first identity is Lemma~\ref{flh:lem:I1cells}(b): no face is plane, so the faces are crossing, $(+)$ or $(-)$. $R$ maps the $(+)$-faces of
$\kappa$ bijectively onto its $(-)$-faces (Lemma~\ref{flh:lem:I1cells}(a); $-\kappa$ has the faces of $\kappa$), and
$[\kappa:R\sigma]=-[R\kappa:R\sigma]=-[\kappa:\sigma]$; so
\[
U_+(\kappa)(R\xi)=\sum_{\sigma(+)}[\kappa:\sigma]\xi_{R\sigma}=-\sum_{\sigma(+)}[\kappa:R\sigma]u_{R\sigma}(\xi)=-\sum_{\sigma'(-)}
[\kappa:\sigma']u_{\sigma'}(\xi)=U_-(\kappa)(\xi);
\]
apply $R$ once more for the other identity. Finally
$\sum_{p\in\cP_\times}[p:b]s_p=\sum_ct_c\sum_p[c:p][p:b]=0$: every plaquette of $\Lam$ containing the crossing bond $b$ is crossing
(Lemma~\ref{flh:lem:I1cells}(b)), every face of a cube of $\Lam$ is a plaquette of $\Lam$, and $\partial\partial=0$.
\end{proof}

\begin{theorem}[reflection positivity of the killed-box state at the nine planes]\label{flh:thm:I1}
Let $\Pi$ be a symmetry plane of the cube $\Lam$ (a coordinate plane or a diagonal plane). For all $F,G\in\fA^+_\Lam(\Pi)$,
\[
\avB{F\circ R_\Pi}\Lam=\avB F\Lam,\qquad\avB{\Th_\Pi F\cdot G}\Lam=\overline{\avB{\Th_\Pi G\cdot F}\Lam},\qquad\avB{\Th_\Pi F\cdot F}\Lam\ge0,
\]
and the Hermitian identity and the positivity extend to all $(\xi_0,\xi_+)$-measurable $F,G$ with $\avB{|F|^2}\Lam,\avB{|G|^2}\Lam<\infty$.
\end{theorem}
\begin{proof}
\emph{Step 0 (invariance).} We have $W_\Lam\circ R=W_\Lam$, because $R$ permutes $\Lam$, $\cB(\Lam)$, $\cP(\Lam)$, $\cC(\Lam)$; $\wt g$ is one function of the
site variable; $\wt h$ is symmetric in $(\rho,\rho')$ and even in $B$; $w$ is even and $\ff_p\circ R=\ff_{Rp}$; $\bbone[(dv)_c\circ R=0]=
\bbone[(dv)_{Rc}=0]$. The reference measure is invariant (a permutation of the coordinates with sign changes of some $B_b$, $v_p$; $dB$ on
$(-\pi/\gph,\pi/\gph]$ is invariant under $B\mapsto-B$ up to a null set, the counting measure exactly). Hence $\avB{F\circ R}\Lam=\avB F\Lam$
and $\avB{\overline{F\circ R}\,G}\Lam=\avB{\overline F\,(G\circ R)}\Lam=\overline{\avB{\overline{G\circ R}\,F}\Lam}$.

\emph{Step 1 (factorisation).} Grouping the factors of $W_\Lam$ by the class of their cell,
$W_\Lam=W_0(\xi_0)W_+(\xi_0,\xi_+)W_-(\xi_0,\xi_-)W_\times(\xi_+,\xi_-,\xi_\times)$: the faces of a plane or $(+)$-cell are plane or $(+)$, and,
by Lemma~\ref{flh:lem:I1cells}(b), no variable of $W_\times:=\prod_{b\in\cB_\times}\wt h\prod_{p\in\cP_\times}w(\ff_p)\prod_{c\in\cC_\times}
\bbone[(dv)_c=0]$ is a plane variable. By Step~0 and Lemma~\ref{flh:lem:I1cells}(a), $W_-(\xi_0,\xi_-)=W_+(\xi_0,(R\xi)_+)$, where
$\eta:=(R\xi)_+$, $\eta_\kappa=\xi_{R\kappa}$ ($\kappa$ a $(+)$-cell), is a measure-preserving bijection of the $(-)$-variables onto the
$(+)$-variables, and $(R\xi)_0=\xi_0$. For $F\in\fA^+_\Lam(\Pi)$: $F(R\xi)=F(\xi_0,\eta)$.

\emph{Step 2 (the kernel).} Let $K(\xi_-,\xi_+):=\int\prod_{b\in\cB_\times}dB_b\sum_{(v_p)_{p\in\cP_\times}}W_\times$. Write
$\bbone[(dv)_c=0]=\int_0^1dt_c\,e^{i\gph t_c(dv)_c}$ for $c\in\cC_\times$ (as $(dv)_c\in(2\pi/\gph)\Z$; Step~2 of the proof of
Theorem~\ref{thm:GD}); by Lemma~\ref{flh:lem:I1split} for $v$, $\sum_ct_c(dv)_c=\sum_{p\in\cP_\times}s_pv_p+\sum_ct_c(V_+(c)-V_-(c))$.
Fubini's theorem ($\sum_v\prod_pw\le K_{\gph}'^{|\cP_\times|}$, $K'_{\gph}$ of Lemma~\ref{lem:d2:mom}; $t$ ranges over a probability space) and
\eqref{eq:poisson} at $a=0$, $x=(dB)_p$, $s=s_p$ (absolutely convergent, $\sum_mc_{m-s}\le K_{\gph}$ uniformly in $s$, Step~4 of the proof of
Theorem~\ref{thm:GD}) give
\[
\sum_{(v_p)_{p\times}}\prod_{p\times}w(\ff_p)\prod_{c\times}\bbone[(dv)_c=0]=\int dt\sum_{m}\prod_{p\times}c_{m_p-s_p}e^{i\gph(m_p-s_p)(dB)_p}
\prod_{c\times}e^{i\gph t_c(V_+(c)-V_-(c))}.
\]
By Lemma~\ref{flh:lem:I1split} for $B$, $(dB)_p=B_\times(p)+B_+(p)-B_-(p)$ and $\sum_{p\times}(m_p-s_p)B_\times(p)=\sum_{b\in\cB_\times}q_bB_b$ with
$q_b:=\sum_p[p:b]m_p\in\Z$ (the $s$-part drops by the last clause of that lemma). With $e^{z\cos\phi}=\sum_{q\in\Z}I_q(z)e^{iq\phi}$ ($I_q\ge0$,
$\sum_qI_q(z)=e^z$; Step~3 of the proof of Theorem~\ref{thm:GD}),
\[
\int_{-\pi/\gph}^{\pi/\gph}dB_b\,\wt h(\rho_{b_-},\rho_{b_+},B_b)e^{i\gph q_bB_b}
=(2\pi/\gph)I_{q_b}(\kappa_N\rho_{b_-}\rho_{b_+}).
\]
Hence
\[
\begin{aligned}
K=\int dt\sum_m&\prod_{p\times}c_{m_p-s_p}\prod_{b\times}\tfrac{2\pi}{\gph}I_{q_b(m)}(\kappa_N\rho_{b_-}\rho_{b_+})\\
&\prod_{p\times}e^{i\gph(m_p-s_p)(B_+(p)-B_-(p))}\prod_{c\times}e^{i\gph t_c(V_+(c)-V_-(c))}.
\end{aligned}
\]
At a diagonal plane $\cB_\times=\emptyset$ (Lemma~\ref{flh:lem:I1cells}(c)) and no Bessel function enters; at a coordinate site plane $K\equiv1$.

\emph{Step 3 (positive type).} With $I_q(z)=\sum_{k\ge0}a_{k,q}z^{2k+|q|}$, $a_{k,q}=2^{-2k-|q|}/(k!\,(k+|q|)!)\ge0$, put, for
$k=(t,m,(k_b)_{b\times})$,
\[
\begin{gathered}
A_k(\xi_+):=\prod_{p\times}e^{i\gph(m_p-s_p)B_+(p)}\prod_{c\times}e^{i\gph t_cV_+(c)}\prod_{b\times}\rho_{b_+}^{\,2k_b+|q_b|},\\
d\nu(k):=dt\prod_{p\times}c_{m_p-s_p}\prod_{b\times}\tfrac{2\pi}{\gph}\,a_{k_b,q_b(m)}\,\kappa_N^{2k_b+|q_b|},
\end{gathered}
\]
a positive measure. By Lemma~\ref{flh:lem:I1split} and Lemma~\ref{flh:lem:I1cells}(a) ($Rb=-b$ swaps the ends $b_\pm$ of a crossing bond):
$B_-(p)=B_+(p)\circ R$, $V_-(c)=V_+(c)\circ R$, $\rho_{b_-}=\rho_{b_+}\circ R$, functions of $\eta=(R\xi)_+$ alone. Therefore
$K(\xi_-,\xi_+)=\wt K(\eta,\xi_+):=\int d\nu(k)\,\overline{A_k(\eta)}A_k(\xi_+)$, and
\[
\begin{aligned}
\int d\nu(k)\,|A_k(\eta)||A_k(\xi)|&=\int dt\sum_m\prod_{p\times}c_{m_p-s_p}\prod_{b\times}\tfrac{2\pi}{\gph}I_{q_b}\big(\kappa_N\rho_{b_+}(\eta)\rho_{b_+}(\xi)\big)\\
&\le K_{\gph}^{|\cP_\times|}\big(\tfrac{2\pi}{\gph}\big)^{|\cB_\times|}\prod_{b\times}e^{\frac{\kappa_N}2(\rho_{b_+}(\eta)^2+\rho_{b_+}(\xi)^2)}
\end{aligned}
\]
by $I_q(z)\le e^z$ and $2\rho\rho'\le\rho^2+\rho'^2$.

\emph{Step 4 (positivity).} Summing $\xi_\times$ first (Tonelli on $W_\Lam\ge0$, then $|F|\le\|F\|_\infty$) and substituting $\eta=(R\xi)_+$ (Step~1),
\[
\begin{aligned}
Z_\Lam\avB{\Th_\Pi F\cdot F}\Lam={}&\int d\xi_0W_0\iint d\eta\,d\xi_+\,W_+(\xi_0,\eta)W_+(\xi_0,\xi_+)\\
&\quad\int d\nu(k)\,\overline{A_k(\eta)F(\xi_0,\eta)}\,A_k(\xi_+)F(\xi_0,\xi_+).
\end{aligned}
\]
By Step~3 the absolute integrand is at most $\|F\|_\infty^2K_{\gph}^{|\cP_\times|}(2\pi/\gph)^{|\cB_\times|}$ times
$W_0W_+(\xi_0,\eta)W_+(\xi_0,\xi_+)e^{\frac{\kappa_N}2\sum_{b\times}(\rho_{b_+}(\eta)^2+\rho_{b_+}(\xi_+)^2)}$, whose integral is, undoing the
substitution, $\int W_0W_+W_-e^{\frac{\kappa_N}2\sum_{b\times}(\rho_{b_-}^2+\rho_{b_+}^2)}$ over the non-crossing variables. Bounding
$\wt h\le e^{\frac{\kappa_N}2(\rho^2+\rho'^2)}$ on every non-crossing bond, all bond factors together are at most
$\prod_{x\in\Lam}e^{\frac{\kappa_N}2\deg(x)\rho_x^2}\le\prod_xe^{3\kappa_N\rho_x^2}$ ($\deg(x)\le6$ bonds of $\cB(\Lam)$ at $x$), which cancels the site
factors $e^{-3\kappa_N\rho_x^2}$; the indicators are $\le1$, each $dB_b$ integrates to $2\pi/\gph$, and $\sum_{v_p}w(\ff_p)\le K'_{\gph}$ per plaquette.
So the integral is at most $(G^{\mathrm{ct}}_0)^{|\Lam|}(2\pi/\gph)^{|\cB(\Lam)\setminus\cB_\times|}K_{\gph}'^{\,|\cP(\Lam)\setminus\cP_\times|}<\infty$,
$G^{\mathrm{ct}}_0:=\int g_{\mathrm{ct}}$ (Lemma~\ref{lem:d2:mom}, Remark~\ref{rem:d2:ct}); the same bound with all cells gives $Z_\Lam<\infty$, and $Z_\Lam>0$ since $W_\Lam>0$ on $\{v\equiv0\}$.
Fubini's theorem now yields
\[
Z_\Lam\avB{\Th_\Pi F\cdot F}\Lam=\int d\xi_0\,W_0(\xi_0)\int d\nu(k)\Big|\int d\xi_+\,W_+(\xi_0,\xi_+)A_k(\xi_+)F(\xi_0,\xi_+)\Big|^2\ \ge0 .
\]

\emph{Step 5 ($L^2$).} Bounded $(\xi_0,\xi_+)$-measurable functions are dense in the $(\xi_0,\xi_+)$-measurable part of $L^2(\avB\cdot\Lam)$,
and $|\avB{\Th F\cdot F}\Lam-\avB{\Th G\cdot G}\Lam|\le\|F-G\|_2(\|F\|_2+\|G\|_2)$ by the Schwarz inequality and $\|F\circ R\|_2=\|F\|_2$ (Step~0);
the Hermitian identity is $\|\cdot\|_2$-continuous likewise.
\end{proof}

\begin{remark}[the crossing kernel]\label{flh:rem:I1kernel}
At a diagonal plane the crossing kernel $\wt K$ of Step~3 of the proof of Theorem~\ref{flh:thm:I1} is of positive type by the Poisson step alone:
\[
\begin{gathered}
\wt K(\eta,\xi)=\int_{[0,1]^{\cC_\times}}dt\sum_{m\in\Z^{\cP_\times}}\prod_{p\in\cP_\times}c_{m_p-s_p}\ \overline{A_{t,m}(\eta)}\,A_{t,m}(\xi),\\
A_{t,m}:=\prod_{p\in\cP_\times}e^{i\gph(m_p-s_p)B_+(p)}\prod_{c\in\cC_\times}e^{i\gph t_cV_+(c)},
\end{gathered}
\]
with $c_u=(\gph/\sqrt{2\pi})e^{-\gph^2u^2/2}>0$ (equation~\eqref{eq:poisson}) and $|A_{t,m}|=1$. At a coordinate mid-plane the factors
$(2\pi/\gph)I_{q_b}(\kappa_N\rho_{b_-}\rho_{b_+})$ of the crossing bonds are added (the Bessel step of Theorem~\ref{thm:GD}).
\end{remark}

\begin{corollary}[matrix and exponential forms, complex coefficients]\label{flh:cor:I1exp}
For $(\xi_0,\xi_+)$-measurable $A_1,\dots,A_n$ with $\avB{|A_i|^2}\Lam<\infty$ and $\lambda\in\C^n$:
$\sum_{i,l}\bar\lambda_i\lambda_l\avB{\Th_\Pi A_i\cdot A_l}\Lam=\avB{\Th_\Pi A\cdot A}\Lam\ge0$ with $A:=\sum_l\lambda_lA_l$ ($\Th_\Pi$ is
antilinear); the Gram matrix $(\avB{\Th_\Pi A_i\cdot A_l}\Lam)_{i,l}$ is Hermitian positive semidefinite. In particular, for real plaquette
functions $a^{(1)},\dots,a^{(k)}$ supported on the plaquettes of $\cP(\Lam)$ lying in $H_+$, $X_r:=\sum_pa^{(r)}_p\ff_p$,
$X_r^R:=X_r\circ R_\Pi=\sum_pa^{(r)}_p\ff_{R_\Pi p}$ (the reflected source, with the $2$-form sign), and $z^{(1)},\dots,z^{(n)}\in\C^k$,
\[
\sum_{i,l=1}^n\bar\lambda_i\lambda_l\Big\langle\exp\Big(\sum_r\overline{z^{(i)}_r}X^R_r+\sum_rz^{(l)}_rX_r\Big)\Big\rangle_\Lam\ \ge\ 0 .
\]
\end{corollary}
\begin{proof}
$A_i:=\exp(\sum_rz^{(i)}_rX_r)$ is $(\xi_0,\xi_+)$-measurable (the bonds of a plaquette in $H_+$ lie in $H_+$), $\Th_\Pi A_i=
\exp(\sum_r\overline{z^{(i)}_r}X^R_r)$ because $X_r$ is real, and $\avB{|A_i|^2}\Lam=\avB{e^{2\sum_r(\Rp z^{(i)}_r)X_r}}\Lam\le
\exp(2\|\sum_r(\Rp z^{(i)}_r)a^{(r)}\|^2)<\infty$ by Theorem~\ref{thm:GD} for the box states (Lemma~\ref{rot:lem:GDall}:
its proof uses only $\partial\partial=0$ on the complex and the Fourier positivity of the bond and plaquette factors). Apply the theorem to
$A=\sum_l\lambda_lA_l$. With $a^{(r)}_p=\eps^{3/2}f^{(r)}_{\mu\nu(p)}(x_p)$ for real $C^\infty_c$ $2$-forms supported in the open side at distance
$>\eps$ from $\Pi$ and inside the cube, this is the box version of Theorem~\ref{thm:osrest:OS2matrix} at level $(N,\Lam)$, which
Lemma~\ref{flh:lem:I4} uses; the plaquettes carrying such $a^{(r)}$ lie in $H_+$ because the crossing plaquettes have their centres on $\Pi$.
\end{proof}

\subsection{Reflection positivity of the box limit at every lattice plane of the nine families}\label{flh:ssec:I2}

\begin{lemma}[reflection positivity of the box limit at the lattice planes of the nine families]\label{flh:lem:I2}
Fix $N$ and a lattice plane $P=\{\phi=c\}$ of one of the nine families (notation of Section~\ref{flh:ssec:thms}: $c\in\frac12\eps\Z$ for
$\phi=x_j$, $c\in\eps\Z$ for $\phi=x_i\pm x_j$); write $R:=R_P$. Let $\fA^+_N(P)$ be the bounded Borel functions of the gauge-invariant
variables $(f_p,\tr_x)$ of finitely many cells of $\eps\Z^3$ lying in $P^+=\{\phi\ge c\}$, $\Sigma_N(P^+)$ the $\sigma$-algebra generated by all
such variables, and $\Th_PF:=\overline{F\circ R}$. Then:
\begin{itemize}
\item[(i)] $\avB{F\circ R}\Lam=\avB F\Lam$ for every cube $\Lam$ with $R\Lam=\Lam$ and every bounded Borel $F$ supported in $\Lam$;
\item[(ii)] for $F,G\in\fA^+_N(P)$: $\ombox(\Th_PF\cdot G)=\overline{\ombox(\Th_PG\cdot F)}$, $\ombox(\Th_PF\cdot F)\ge0$ and
$\ombox(\Th_PF)=\overline{\ombox(F)}$; the form $(F,G)\mapsto\ombox(\Th_PF\cdot G)$ is positive semidefinite Hermitian;
\item[(iii)] (ii) holds for all $F,G\in L^2(\Sigma_N(P^+),\mubox)$.
\end{itemize}
The statements do not depend on any exhaustion.
\end{lemma}
\begin{proof}
(i) is Step~0 of the proof of Theorem~\ref{flh:thm:I1} ($R$ is a symmetry of $\Lam$ in the sense there; for a lattice plane of the nine
families with $R\Lam=\Lam$ it is one of the symmetry planes listed in Section~\ref{flh:ssec:thms}).

(ii) \emph{Symmetric exhaustion.} Let $I_k:=\{-k\eps,\dots,k\eps\}$ and $J_k:=\{-(k-1)\eps,\dots,k\eps\}$. For $P=\{x_j=c\}$ with $c\in\eps\Z$
let $\Lam_k$ be the cube with side $I_k$ in the directions $\ne j$ and $c+I_k$ in the direction $j$; for $c\in\eps\Z+\frac\eps2$ take $J_k$ and
$c-\frac\eps2+J_k$; for $\{x_i-x_j=c\}$ take $I_k$ in the directions $j,l$ and $c+I_k$ in the direction $i$; for $\{x_i+x_j=c\}$ take $I_k$ in
the directions $j,l$ and $c-I_k=c+I_k$ in the direction $i$. In each case $R\Lam_k=\Lam_k$ (for $\{x_i-x_j=c\}$: $(Rx)_i=x_j+c\in c+I_k$,
$(Rx)_j=x_i-c\in I_k$; for $\{x_i+x_j=c\}$: $(Rx)_i=c-x_j\in c+I_k$, $(Rx)_j=c-x_i\in I_k$; for the mid-planes $c-\frac\eps2+J_k$ is symmetric
about $c$), $\Lam_k\subset\Lam_{k+1}$ and $\bigcup_k\Lam_k=\eps\Z^3$: an exhausting sequence of cubes, each having $P$ as a symmetry plane.

\emph{Continuous $F$.} Let $F$ be a bounded continuous function of the gauge-invariant variables of a finite set $S$ of cells in $P^+$. For $k$
large, $S\cup RS\subset\Lam_k$ (all cells of $S\cup RS$ are cells of $\Lam_k$), so $F$, $F\circ R$ and $\Th_PF\cdot F$ are functions of the box
variables, $F\in\fA^+_{\Lam_k}(P)$ (a function of $(f_p,\tr_x)=((dB)_p+v_p,\tr_x)$ for cells in $P^+$ is a function of the variables of cells in
$P^+$), and Theorem~\ref{flh:thm:I1} gives $\avB{\Th_PF\cdot F}{\Lam_k}\ge0$. By Theorem~\ref{thm:d2:box}(b) the laws of the variables of
$S\cup RS$ under $\avB\cdot{\Lam_k}$ converge weakly to the marginal of $\mubox$, and $\Th_PF\cdot F$ is bounded continuous, so
$\ombox(\Th_PF\cdot F)=\lim_k\avB{\Th_PF\cdot F}{\Lam_k}\ge0$. The Hermitian identity and $\ombox(\Th_PF)=\overline{\ombox(F)}$ hold for all
bounded Borel $F,G$ of finitely many cells: $\mubox$ is $R$-invariant ($R\in\cG_N$, Theorem~\ref{thm:d2:box}(b)), and
$(\overline F\cdot(G\circ R))\circ R=\overline{F\circ R}\cdot G$. Positive semidefiniteness on finite families follows by applying the inequality
to $\sum_ic_iF_i$.

\emph{Bounded Borel functions and $L^2$.} As in the proof of Theorem~\ref{thm:d2:LP}(d): for a finite set $S$ of cells in $P^+$ the bounded
continuous functions on $E_S$ are dense in $L^2(E_S,(\pi_S)_*\mubox)$ (a finite Borel measure on a Polish space is Radon), a bounded Borel $F$
of the cells of $S$ is the $L^2(\mubox)$-limit of continuous $F^{(i)}$ with $\|F^{(i)}\|_\infty\le\|F\|_\infty$ (truncate), and both sides of each
identity are continuous in $\|F\|_{L^2}$, $\|G\|_{L^2}$ by the Schwarz inequality and $\|F\circ R\|_{L^2(\mubox)}=\|F\|_{L^2(\mubox)}$. This gives
(ii), and (iii) for functions of finitely many cells. For (iii) in general, the bounded $\Sigma_N(P^+)$-measurable functions that are
$L^2(\mubox)$-limits of bounded functions of finitely many cells of $P^+$ form a vector space containing the constants, closed under bounded
monotone limits (dominated convergence), and containing the multiplicative class of the bounded functions of finitely many cells, which generates
$\Sigma_N(P^+)$; by the functional monotone class theorem it contains every bounded $\Sigma_N(P^+)$-measurable function, and these are dense in
$L^2(\Sigma_N(P^+),\mubox)$ (truncation); the continuity of both sides concludes the proof of (iii). The statements do not depend on the exhaustion, because $\mubox$ is one measure
(Theorem~\ref{thm:d2:box}(b)).
\end{proof}

\subsection{The continuum passage for the box family}\label{flh:ssec:I3}

\begin{lemma}[the continuum passage for the box family]\label{flh:lem:I3}
Let $(\Lam_k)$ be any exhausting sequence of boxes. The pair $((\Lam_k),\mubox)$ has the properties (a), (b), (c), (d), (e) of
Theorem~\ref{thm:d2:LP}, with ``box'' for ``torus'' and ``$k\to\infty$'' for ``$L\to\infty$'', and in addition the reflection positivity (d) at
every lattice plane of the nine families (Lemma~\ref{flh:lem:I2}). Consequently Proposition~\ref{prop:E}, Lemma~\ref{lem:R1},
Proposition~\ref{prop:R3} and Theorem~\ref{thm:Hlim} hold for $\ombox$ with any bounded Borel $g_{\rm d}$; and every statement of
Section~\ref{sec:osrest} holds for the box family, with Theorem~\ref{thm:osrest:RPtorus} replaced by Theorem~\ref{flh:thm:I1} and
Corollary~\ref{cor:osrest:D1} by Lemma~\ref{flh:lem:I2}. $\Sbox$, $\Zbox$, $\Psi^{\mathrm{box}}_f$, $S^{\mathrm{box}}_n$, $\mub$ denote the objects so obtained along $(N_m)$.
\end{lemma}
\begin{proof}
\emph{The properties (a)--(e).} (a) The weak convergence of all finite-dimensional laws is Theorem~\ref{thm:d2:box}(b), and the Borel clause
follows from the portmanteau theorem. For a continuous $F$ with
$|F|\le Ce^{\sum_{x\in X}q_x\rho_x^2}$ the uniform integrability needed is
\[
\sup_\Lam\avB{e^{2\sum_xq_x\rho_x^2}}\Lam\le\max_xC_N(2|X|q_x).
\]
This is (K4) of Theorem~\ref{thm:d2:GIN}, where $e^{\sum_xq_x\rho_x^2}\in\cK_N$ and the comparison is with a torus of side larger than $\Lam$ and its
neighbours, together with Remark~\ref{rem:d2:mom}(ii), exactly as in the proof of Theorem~\ref{thm:d2:box}(a).

(b) The functional $\ombox$ is the integral against a probability measure, and the Gaussian and density bounds are in Theorem~\ref{thm:d2:box}(b).

(c) The invariance under $\cG_N$ and $C$ is Theorem~\ref{thm:d2:box}(b), and the conclusion of Lemma~\ref{lem:R1} follows as in the proof of
Theorem~\ref{thm:d2:LP}(c).

(d) For the planes $\{x_j=c\}$, $c\in\frac12\eps\Z$, this is Lemma~\ref{flh:lem:I2}, and the additional clause is Lemma~\ref{flh:lem:I2} for the
diagonal planes.

(e) The first clause follows from (a); the second is the portmanteau theorem for continuous $F\ge0$, as in the proof of Theorem~\ref{thm:d2:LP}(b).

\emph{The statements of Sections~\ref{sec:E}--\ref{sec:main}.} For Proposition~\ref{prop:E}, Theorem~\ref{thm:GD} for the box states
(Lemma~\ref{rot:lem:GDall}) and Minkowski's inequality give $\avB{\Phi_N(\sigma)^2}\Lam^{1/2}\le\|\sigma\|$. The bound
$\ombox(|\Phi_N(\sigma)|\wedge2)\le\|\sigma\|$ follows from the Schwarz and Minkowski inequalities, from $\int X^2d\mubox\le R(f)^2$, which holds by
(e) or by Step~5 of the proof of Theorem~\ref{thm:GD} applied to the bound of (b), and from the pointwise bound $|Y|\le\|g_{\rm d}\|_\infty E(h)$.
Lemma~\ref{lem:R1} is (c). Proposition~\ref{prop:R3} uses of the state only the
state property, \eqref{eq:unimod}, Lemma~\ref{lem:R1} and \eqref{eq:E}. Theorem~\ref{thm:Hlim} follows from these statements, including the
invariance under the space translations and under the cubic group about $\bigcup_m\varepsilon_{N_m}\Z^3$ and the bound $|S(a)|\le M(a)$.

\emph{The statements of Section~\ref{sec:osrest}.} Lemma~\ref{lem:osrest:riemann} is a statement of lattice counting. For
Lemma~\ref{lem:osrest:dualC}, the identity \eqref{eq:dual} holds on every
finite cubical complex, the killed box included, by the proof of Lemma~\ref{rot:lem:GDall}, with the same entire continuation.

Part (a) of Lemma~\ref{lem:osrest:entire} holds for the box states for the following reasons. The bound follows from
Lemma~\ref{rot:lem:GDall} on the box. The evenness follows from the substitution $(B,v)\mapsto(-B,-v)$, since $W_\Lam$ depends on $B$ through
$\cos(\gph B_b)$, $w$ is even and closed $v$ go to closed $v$. The inequality $\Psi\ge1$ on $\R^k$ follows from Jensen's inequality and
$\avB{\Phi_N}\Lam=0$ (Step~5 of the proof of Theorem~\ref{thm:GD}), and $0\le\Psi(it)\le1$ follows from the dual representation on the box.
Part (b) of Lemma~\ref{lem:osrest:entire} follows from Montel's theorem and its part (a) for the box states; the limit is identified on $i\R^k$
by the property (a) of the present lemma, and the uniform integrability is obtained as in the torus case. Parts (c) and (d) of
Lemma~\ref{lem:osrest:entire} hold without change.

Theorem~\ref{thm:osrest:OS0} follows from \eqref{eq:R31} with
Proposition~\ref{prop:E} and Lemma~\ref{lem:osrest:entire}. Proposition~\ref{prop:osrest:OS1} follows from Theorem~\ref{thm:Hlim} and (c).
Proposition~\ref{prop:osrest:OS2} follows from Theorem~\ref{thm:Hlim}(a) and Lemma~\ref{flh:lem:I2}; the argument is written out in
Lemma~\ref{flh:lem:I4}(i) below.

For Theorem~\ref{thm:osrest:OS2matrix} and Proposition~\ref{prop:osrest:poly}, the functions $A_i=\exp(\sum_rz^{(i)}_r
\Phi_N(f_r))$ and the polynomial $P_N$ on $\OmN$ lie in $L^2(\Sigma_N(P^+),\mubox)$ by (b). The intertwining
$\Phi_N(f)\circ R_P=\Phi_N(\gamma_*f)$ of the proof of Proposition~\ref{prop:osrest:OS2} gives
\[
\Th_PA_i=\exp(\sum_r\overline{z^{(i)}_r}\Phi_N(\gamma_*f_r)),\qquad
\Th_PP_N=\sum_\alpha\bar c_\alpha\prod_i\Phi_N(\gamma_*f_{\alpha i}).
\]
Lemma~\ref{flh:lem:I2}(iii) gives the positivity at level $N$, Lemma~\ref{lem:osrest:entire}(b) identifies
$\int\Th_PA_i\cdot A_l\,d\mubox=\Psi^F_N(\overline{z^{(i)}},z^{(l)})$ and the moments as derivatives of $\Psi^F_N$, and Lemma~\ref{lem:osrest:entire}(c),
with Weierstrass' theorem for the derivatives, gives the passage $m\to\infty$.

Proposition~\ref{prop:osrest:OS3} follows from (c), applied to $C$. Theorem~\ref{thm:osrest:measure}
uses only Theorem~\ref{thm:osrest:OS0}, Theorem~\ref{thm:Hlim}(a) and Proposition~\ref{prop:osrest:OS1}, together with the abstract measure-theoretic
statements of Appendix~\ref{ms:measure} and the Bochner--Minlos theorem, which do not depend on the family. Theorem~\ref{thm:osrest:OS4weak} uses only
Theorem~\ref{thm:osrest:measure}(c), Theorem~\ref{thm:osrest:OS2matrix} and, for the growth bound of its last clause, Theorem~\ref{thm:osrest:OS0}(iii).
\end{proof}

\subsection{Reflection positivity of the limit theory at the nine plane families}\label{flh:ssec:I4}

\begin{lemma}[reflection positivity of the limit theory at the nine plane families]\label{flh:lem:I4}
Let $\Sbox$ be a limit along $(N_m)$, let $P_0$ be one of the nine planes through $0$, $n$ a unit normal, $c\in\R$, $P_c:=P_0+cn$, $\gamma$ the
reflection $R_{P_c}$ acting on test pairs by push-forward ($\gamma\sigma=(\gamma_*f,h\circ\gamma)$, the $2$-form transported as a $2$-form),
$\Th$ the antilinear map of $\cA$ with $\Th e_\sigma=e_{-\gamma\sigma}$, and $\cA_+$ the span of the $e_\sigma$ with $\supp\sigma$ in the open
side $\{n\cdot x>c\}$.
\begin{itemize}
\item[(i)] $\langle a,b\rangle:=\Sbox(\Th a\cdot b)$ is a positive semidefinite Hermitian sesquilinear form on $\cA_+$, and
$\Sbox(\Th a)=\overline{\Sbox(a)}$ on $\cA$.
\item[(ii)] For real $C^\infty_c$ $2$-forms $f_1,\dots,f_k$ supported in the open side, $F:=(\gamma_*f_1,\dots,\gamma_*f_k,f_1,\dots,f_k)$,
$z^{(1)},\dots,z^{(n)}\in\C^k$ and $\lambda\in\C^n$: $\sum_{i,l}\bar\lambda_i\lambda_l\Psi^{\mathrm{box}}_F(\overline{z^{(i)}},z^{(l)})\ge0$.
\item[(iii)] For real Schwartz pairs $\sigma_1,\dots,\sigma_n$ supported in the closed side $\{n\cdot x\ge c\}$ and $\lambda\in\C^n$:
$\sum_{i,l}\bar\lambda_i\lambda_l\Zbox(\sigma_l-\gamma\sigma_i)\ge0$, $\Zbox(-\gamma\sigma)=\overline{\Zbox(\sigma)}$ and $\Zbox(\gamma\sigma)=\Zbox(\sigma)$,
$\Zbox$ being the continuous extension of Theorem~\ref{thm:osrest:OS0}(i) to Schwartz pairs.
\end{itemize}
\end{lemma}

Here (ii) is the complex-coefficient form for $C^\infty_c$ sources in the open side; (iii) is the characteristic-function form, with complex
coefficients in front of the exponentials, for Schwartz pairs in the closed side; the hypothesis (iii) of Theorem~\ref{flh:thm:I7} is (iii)
restricted to the open side.

\begin{proof}
\emph{Reduction to $c=0$.} $P_c=\tau_yP_0$ with $y=cn$, $\gamma_c=\tau_y\gamma_0\tau_{-y}$, $\cA_+(P_c)=\tau_y\cA_+(P_0)$, $\Th_c=\tau_y\Th_0\tau_y^{-1}$,
and $\Sbox\circ\tau_y=\Sbox$ for every $y\in\R^3$ (Proposition~\ref{prop:osrest:OS1} for the box family, Lemma~\ref{flh:lem:I3}); so (i) for $P_c$
is (i) for $P_0$ at $\tau_{-y}a,\tau_{-y}b$. For (ii), $\Psi^{\mathrm{box}}_{\tau_yF}=\Psi^{\mathrm{box}}_F$: both are entire on $\C^{2k}$ and agree
on $i\R^{2k}$ (Lemma~\ref{lem:osrest:entire}(c) and the identity theorem). For (iii), $\Zbox\circ\tau_y=\Zbox$ on Schwartz pairs by continuity.

\emph{(i), $c=0$.} $R:=R_{P_0}$ is a signed permutation matrix, so $R\varepsilon_N\Z^3=\varepsilon_N\Z^3$ and $P_0$ is a lattice plane of its family
for every $N$. For $\sigma$ supported in $\{n\cdot x\ge\delta\}$ and $\varepsilon_N<\delta$, every cell carrying a term of $\Phi_N(\sigma)$ lies in
$\{n\cdot x>0\}$ (the vertices of a plaquette are within $\varepsilon_N/\sqrt2$ of its centre, so $n\cdot x\ge\delta-\varepsilon_N/\sqrt2>0$ on
them), and $\Phi_N(\sigma)\circ R=\Phi_N(\gamma\sigma)$: $R$ permutes the oriented plaquettes and the sites, the push-forward of the $2$-form carries
exactly the orientation sign, and $\tr_z\circ R=\tr_{Rz}$ (Definition~\ref{rot:def:sample} and the properties following it, Lemma~\ref{rot:lem:pass}
with $O\Lam=\Lam$; this is the intertwining of the proof of Proposition~\ref{prop:osrest:OS2} for the
diagonal reflections as well).

Hence $\iota_N(\Th a\cdot b)=\Th_P\iota_N(a)\cdot\iota_N(b)$ for $a,b\in\cA_+$, with $\iota_N(a)$ a bounded Borel
function of the gauge-invariant variables of cells in $P_0^+$ (bounded continuous when $g_{\rm d}$ is continuous; for a bounded Borel $g_{\rm d}$
only bounded Borel, which Lemma~\ref{flh:lem:I2}(ii) covers). (Lemma~\ref{flh:lem:I2} is stated for the side $\{\phi\ge c\}$; the other side follows from the $R$-invariance of $\mubox$: $\ombox(\Th_PF\cdot G)=\ombox(\Th_P(F\circ R)\cdot(G\circ R))$.) By Lemma~\ref{flh:lem:I2}(ii) the Gram matrices
$(\om^{\mathrm{box}}_{N_m}(\Th_P\iota_{N_m}(a_i)\,\iota_{N_m}(a_j)))_{ij}$ are Hermitian positive semidefinite; they converge entrywise to
$(\Sbox(\Th a_i\cdot a_j))_{ij}$ by Theorem~\ref{thm:Hlim}(a) for the box family (Lemma~\ref{flh:lem:I3}), and both properties are closed under
limits. $\Sbox(\Th a)=\lim_m\overline{\om^{\mathrm{box}}_{N_m}(\iota_{N_m}(a))}=\overline{\Sbox(a)}$ by Lemma~\ref{flh:lem:I2}(ii).

\emph{(ii), $c=0$.} This is the step for Theorem~\ref{thm:osrest:OS2matrix} in the proof of Lemma~\ref{flh:lem:I3}, with $\gamma$ the diagonal or
coordinate reflection: the Gram matrices at level $N$ are positive semidefinite by Lemma~\ref{flh:lem:I2}(iii) (Corollary~\ref{flh:cor:I1exp} at
the level of the boxes), equal to $(\Psi^F_N(\overline{z^{(i)}},z^{(l)}))_{il}$ by Lemma~\ref{lem:osrest:entire}(b), and converge to
$(\Psi^{\mathrm{box}}_F(\overline{z^{(i)}},z^{(l)}))_{il}$ by Lemma~\ref{lem:osrest:entire}(c).

\emph{(iii), $c=0$.} For a Schwartz pair $\sigma=(f,h)$ supported in $P_0^+$ put $\sigma^{(\nu)}:=\chi(\cdot/\nu)\,\tau_{n/\nu}\sigma$ with
$\chi\in C^\infty_c$, $\chi=1$ on the unit ball and $0\le\chi\le1$: $\sigma^{(\nu)}\in C^\infty_c$ is supported in $\{n\cdot x\ge1/\nu\}$, and
\[
\wnorm{\sigma^{(\nu)}-\sigma}\le\sup_{|x|\ge\nu}(1+|x|^2)^2|\sigma(x)|+\nu^{-1}\,9\sup_y(1+|y|^2)^2|\nabla\sigma(y)|\to0
\]
(mean value theorem; $1+|x|^2\le3(1+|y|^2)$ for $|x-y|\le1$; $0\le\chi\le1$ for the first term). Here $\wnorm\cdot$ is the derivative-free seminorm
of Section~\ref{sec:osrest} applied to each component of the pair. $\gamma$ is a linear isometry acting on the components by a signed permutation,
so $\wnorm{\gamma\sigma^{(\nu)}-\gamma\sigma}=\wnorm{\sigma^{(\nu)}-\sigma}$. By Theorem~\ref{thm:osrest:OS0}(i) and Lemma~\ref{lem:osrest:riemann}
(its pair version: $|Z(\sigma)-Z(\sigma')|\le\|\sigma-\sigma'\|\le(3c_{\azero})^{1/2}\wnorm{f-f'}+\|g_{\rm d}\|_\infty c_{\azero}\wnorm{h-h'}$, which
extends $\Zbox$ continuously to real Schwartz pairs), $\Zbox(\sigma^{(\nu)}_l-\gamma\sigma^{(\nu)}_i)\to\Zbox(\sigma_l-\gamma\sigma_i)$, while each
$\sum_{i,l}\bar\lambda_i\lambda_l\Zbox(\sigma^{(\nu)}_l-\gamma\sigma^{(\nu)}_i)=\Sbox(\Th a_\nu\cdot a_\nu)\ge0$ with
$a_\nu:=\sum_l\lambda_le_{\sigma^{(\nu)}_l}\in\cA_+$, by (i). The two identities pass from $C^\infty_c$ pairs (by (i), and by
Proposition~\ref{prop:osrest:OS1} together with the cubic invariance of Theorem~\ref{thm:Hlim}, $\gamma$ lying in the cubic group $\Oh$ about a
point of $P_0$) to Schwartz pairs by the same continuity.
\end{proof}

\subsection{The Gaussian bound in continuum norms}\label{flh:ssec:I5}

\begin{lemma}[the Gaussian bound in continuum norms]\label{flh:lem:I5}
$\Zbox$, extended to real Schwartz pairs by Theorem~\ref{thm:osrest:OS0}(i) and Lemma~\ref{lem:osrest:riemann} (pair version, as in the proof of
Lemma~\ref{flh:lem:I4}(iii)), is positive definite and continuous on $\cS(\R^3;\Lt\oplus\R)$; let $\mu$ be the probability measure on
$\cS'(\R^3;\Lt\oplus\R)$, with its cylinder $\sigma$-algebra, such that $\int e^{i\Phi(\sigma)}d\mu=\Zbox(\sigma)$ for all real Schwartz pairs
(Bochner--Minlos, as in the proof of Theorem~\ref{thm:osrest:measure} and in Lemma~\ref{rot:rem:minlos}; its curvature marginal is the measure
of Theorem~\ref{thm:osrest:measure} read on $\cS'$). Then for every real Schwartz pair $\sigma=(f,h)$,
\[
\int e^{\Phi(f,h)}\,d\mu\ \le\ \exp\Big(\tfrac12\|f\|_{L^2}^2+\|g_{\rm d}\|_\infty\|h\|_{L^1}\Big),\qquad\|f\|_{L^2}^2:=\sum_{\mu<\nu}\|f_{\mu\nu}\|_{L^2}^2 .
\]
Consequently the curvature Schwinger functions $S_n(f_1,\dots,f_n)=\int\prod_i\Phi(f_i,0)\,d\mu$ (Theorem~\ref{thm:osrest:OS0}(iii),
Theorem~\ref{thm:osrest:measure}(a)) satisfy $|S_n(f_1,\dots,f_n)|\le e^n(n!)^{1/2}\prod_i\|f_i\|_{L^2}$ for real $f_i\in\cS(\R^3;\Lt)$, and
$S_n$ extends uniquely to a bounded $n$-linear form on $L^2(\R^3;\Lt_\C)$, with the same bound on real entries and $2^{n/2}$ times it on complex ones.
\end{lemma}
\begin{proof}
\emph{Positive definiteness.} We have
\[
\sum_{l,l'}\bar c_lc_{l'}\Zbox(\sigma_{l'}-\sigma_l)=\lim_m\om^{\mathrm{box}}_{N_m}(|\sum_lc_le^{i\Phi_{N_m}(\sigma_l)}|^2)
\ge0
\]
by the additivity of $\Phi_N$ and Theorem~\ref{thm:Hlim}(a); positive definiteness is preserved under the continuous extension.
Continuity on $\cS$ is the pair version of Theorem~\ref{thm:osrest:OS0}(i) just mentioned, and $\cS(\R^3;\Lt\oplus\R)$ is nuclear, so the
Bochner--Minlos theorem gives $\mu$.

\emph{Level $N$.} $\Phi_N(\sigma)=X+Y$ with $X=\sum_pa_p\ff_p$, $a_p=\varepsilon_N^{3/2}f_{\mu\nu(p)}(x_p)$, and $|Y|\le\|g_{\rm d}\|_\infty E_N(h)$ on
$\OmN$, $E_N(h):=\varepsilon_N^3\sum_z|h(z)|$. By Theorem~\ref{thm:d2:box}(b), $\int e^Xd\mubox\le e^{\frac12\sum_pa_p^2}$, so
$\int e^{\Phi_N(\sigma)}d\mubox\le\exp(\frac12R_N(f)^2+\|g_{\rm d}\|_\infty E_N(h))$ with $R_N(f)^2:=\varepsilon_N^3\sum_pf_{\mu\nu(p)}(x_p)^2$.

\emph{The limit, $\sigma\in C^\infty_c$.} The law $\nu_m$ of $\Phi_{N_m}(\sigma)$ under $\mu^{\mathrm{box}}_{N_m}$ has the characteristic function
$t\mapsto Z_{N_m}(t\sigma)\to\Zbox(t\sigma)=\int e^{it\Phi(\sigma)}d\mu$ (Theorem~\ref{thm:Hlim}(a)), which is continuous at $0$ by
$|\Zbox(t\sigma)-1|\le|t|\,\|\sigma\|$; by L\'evy's continuity theorem $\nu_m\to\nu$ weakly, $\nu$ the law of $\Phi(\sigma)$ under $\mu$. For the
continuous function $e^x\ge0$, $\int e^xd\nu=\lim_M\int(e^x\wedge M)d\nu\le\liminf_m\int e^xd\nu_m$. The centres of the plaquettes of one
orientation form a translate of $\varepsilon_N\Z^3$, so $R_N(f)^2\to\|f\|_{L^2}^2$ and $E_N(h)\to\|h\|_{L^1}$ (Riemann sums of continuous compactly
supported functions). The bound follows.

\emph{Schwartz $\sigma$.} Take $\sigma_\nu:=\chi(\cdot/\nu)\sigma\in C^\infty_c$ ($\chi$ as in the proof of Lemma~\ref{flh:lem:I4}); $\sigma_\nu\to\sigma$
in $\cS$, so $\Phi(\sigma_\nu)\to\Phi(\sigma)$ for every $\Phi\in\cS'$, and $\|f_\nu\|_{L^2}\to\|f\|_{L^2}$, $\|h_\nu\|_{L^1}\to\|h\|_{L^1}$; Fatou's
lemma gives
\[
\int e^{\Phi(\sigma)}d\mu\le\liminf_\nu\int e^{\Phi(\sigma_\nu)}d\mu\le\exp(\frac12\|f\|^2_{L^2}+\|g_{\rm d}\|_\infty\|h\|_{L^1}).
\]

\emph{Moments.} For real $f_1,\dots,f_n\in\cS$ and $z\in\C^n$ put $\Psi(z):=\int e^{\sum_jz_j\Phi(f_j,0)}d\mu$. Since
$\int e^{s|\Phi(f,0)|}d\mu\le2e^{s^2\|f\|_{L^2}^2/2}$ for all real $s$, the integrand has a locally uniform integrable majorant, so $\Psi$ is entire
with $\partial_{z_1}\cdots\partial_{z_n}\Psi(0)=\int\prod_j\Phi(f_j,0)\,d\mu=S_n(f_1,\dots,f_n)$ (the law of $(\Phi(f_j,0))_j$ under $\mu$ is $\mu_f$,
as in the moment step of the proof of Theorem~\ref{thm:osrest:measure}), and
\[
|\Psi(z)|\le\Psi(\Rp z)\le\exp(\frac12\|\sum_j(\Rp z_j)f_j\|_{L^2}^2)\le\exp(\frac12(\sum_j|\Rp z_j|\,\|f_j\|_{L^2})^2).
\]
If some $\|f_j\|_{L^2}=0$
then $f_j=0$ and $S_n=0$; otherwise normalise $\|f_j\|_{L^2}=1$: on the polydisc $|z_j|\le r$, $|\Psi|\le e^{n^2r^2/2}$, and Cauchy's estimate gives
$|S_n|\le r^{-n}e^{n^2r^2/2}=(en)^{n/2}\le e^n(n!)^{1/2}$ at $r=n^{-1/2}$, by $n^n\le e^nn!$. Real $\cS$ is dense in real $L^2$, so the $n$-linear
$S_n$ extends uniquely by continuity with the same bound; the complex $n$-linear extension splits each entry into real and imaginary parts, and
$\|u\|+\|v\|\le\sqrt2\|u+iv\|$.
\end{proof}

\subsection{Closedness of the limit curvature}\label{flh:ssec:I6}

\emph{Conventions.} We write $\eps=\eN$. The plaquettes are $p=(x;\mu,\nu)$, $\mu<\nu$, with centre $x_p$ and
$\partial p=(x;\mu)+(x+\eps e_\mu;\nu)-(x+\eps e_\nu;\mu)-(x;\nu)$ (Definition~\ref{rot:def:model}); $f_{\mu\nu(p)}=f_{\mu\nu}$. The cubes are
$c(y)=y+\eps[0,1]^3$ with centre $x_c=y+\frac\eps2(1,1,1)$ and oriented boundary
\[
\partial c(y)=\sum_{\lambda=1}^3\epsilon_{\lambda\mu\nu}\big[(y+\eps e_\lambda;\mu,\nu)-(y;\mu,\nu)\big],\qquad\{\mu<\nu\}=\{1,2,3\}\setminus\{\lambda\},
\]
the usual oriented boundary, with $\partial\partial=0$. $[\kappa:\kappa']\in\{0,\pm1\}$ is the coefficient of the cell $\kappa$ in $\partial\kappa'$ (the smaller cell is written first here; Section~\ref{flh:ssec:I1} writes the larger cell first: there $[\kappa:\sigma]$ is the coefficient of $\sigma$ in $\partial\kappa$).
A finite cubical complex $\cK$ is a torus $(\eps\Z/L\Z)^3$ or a killed box $\Lam$ of Definition~\ref{def:d2:cone}, whose cubes are the cubes of
$\Lam$ in the sense of Section~\ref{flh:ssec:thms} (all eight vertices in $\Lam$; all six faces then lie in $\cP(\Lam)$). For functions $B$ on
bonds, $v,f$ on plaquettes and $a$ on cubes we put $(dB)_p=\sum_b[b:p]B_b$, $(df)_c=\sum_p[p:c]f_p$, $(d^*a)_p:=\sum_{c\in\cK}[p:c]a_c$ (the operator
of Step~2 of the proof of Theorem~\ref{thm:GD}). The Hodge star is $(*du)_{\mu\nu}=\epsilon_{\lambda\mu\nu}\partial_\lambda u$ for $\mu<\nu$, $\lambda$
the third index, so that $\int\sum_{\mu<\nu}F_{\mu\nu}(*du)_{\mu\nu}=-\int u\,(dF)_{123}$ for smooth $F$. $K_{+t}$ is the closed $t$-neighbourhood
of a compact set $K$, and $\|u\|_{C^3}:=\max_{|\gamma|\le3}\|D^\gamma u\|_\infty$.

\begin{lemma}[lattice Bianchi identity]\label{flh:lem:bianchi}
Let $\cK$ be a finite cubical complex, $B$ real on its bonds, $v$ on its plaquettes with $(dv)_c=0$ for every cube $c$ of $\cK$, and
$f:=dB+v$. Then for every real $a$ on the cubes of $\cK$,
\[
\sum_{p\in\cP(\cK)}(d^*a)_p\,f_p=\sum_{c\in\cC(\cK)}a_c\,(df)_c=0 .
\]
\end{lemma}
\begin{proof}
All sums are finite and every face of a cube of $\cK$ is a plaquette of $\cK$, so $\sum_p\sum_c[p:c]a_cf_p=\sum_ca_c\sum_p[p:c]f_p=
\sum_ca_c(df)_c$. Further $(df)_c=(ddB)_c+(dv)_c$, $(dv)_c=0$ by hypothesis, and $(ddB)_c=\sum_p[p:c]\sum_b[b:p]B_b=\sum_bB_b\sum_p[p:c][b:p]=0$,
since $\sum_p[p:c][b:p]$ is the coefficient of $b$ in $\partial\partial c=0$.
\end{proof}

\begin{lemma}[sampling of $*du$]\label{flh:lem:sample}
Let $u\in C^\infty_c(\R^3)$, $K:=\supp u$, and $a_c:=\eps^{1/2}u(x_c)$ on all cubes of $\eps\Z^3$. For $p=(x;\mu,\nu)$ with third index $\lambda$,
\[
(d^*a)_p=\epsilon_{\lambda\mu\nu}\eps^{1/2}\big[u(x_p-\tfrac\eps2e_\lambda)-u(x_p+\tfrac\eps2e_\lambda)\big]
=-\eps^{3/2}(*du)_{\mu\nu}(x_p)+r_p,\qquad|r_p|\le\tfrac1{24}\eps^{7/2}\|\partial_\lambda^3u\|_\infty ,
\]
$r_p=0$ unless $x_p\in K_{+\eps/2}$, and $\sum_pr_p^2\le\frac1{192}|K_{+2\azero}|\,\eps^4\|u\|_{C^3}^2$.
\end{lemma}
\begin{proof}
$p$ is the face $(x;\mu,\nu)$ of $c(x)$, with $[p:c(x)]=-\epsilon_{\lambda\mu\nu}$, and the face $(x;\mu,\nu)=(y+\eps e_\lambda;\mu,\nu)$ of $c(y)$,
$y=x-\eps e_\lambda$, with $[p:c(y)]=+\epsilon_{\lambda\mu\nu}$; no other cube contains $p$. The centres are $x_{c(x)}=x_p+\frac\eps2e_\lambda$ and
$x_{c(y)}=x_p-\frac\eps2e_\lambda$. This gives the first equality. With $g(t):=u(x_p+te_\lambda)$ and $h=\eps/2$, Taylor's formula gives
$g(h)-g(-h)-2hg'(0)=\frac{h^3}6(g'''(\xi_+)+g'''(\xi_-))$, hence
\[
|u(x_p+\frac\eps2e_\lambda)-u(x_p-\frac\eps2e_\lambda)-\eps\partial_\lambda u(x_p)|\le\frac{\eps^3}{24}\|\partial_\lambda^3u\|_\infty;
\]
multiplying by
$\eps^{1/2}$ and using $(*du)_{\mu\nu}=\epsilon_{\lambda\mu\nu}\partial_\lambda u$ gives the second equality and the bound on $r_p$. If
$x_p\notin K_{+\eps/2}$, the three points $x_p$, $x_p\pm\frac\eps2e_\lambda$ lie outside $K$, so $r_p=0$. The centres of the plaquettes of one
orientation form a translate of $\eps\Z^3$; the open cubes $Q_p$ of side $\eps$ centred at $x_p$ are pairwise disjoint, and $x_p\in K_{+\eps/2}$
implies $Q_p\subset K_{+\eps/2+\sqrt3\eps/2}\subset K_{+2\eps}\subset K_{+2\azero}$ ($\eN\le\azero$). So at most $|K_{+2\azero}|/\eps^3$ plaquettes
of each of the three orientations have $r_p\ne0$, and $\sum_pr_p^2\le3\,\frac{|K_{+2\azero}|}{\eps^3}\,\frac{\eps^7}{576}\|u\|_{C^3}^2$.
\end{proof}

\begin{proposition}[closedness at every level]\label{flh:prop:I6level}
Fix $N$, $u\in C^\infty_c(\R^3)$, $s\in\R$, and let $\mu$ be either $\mubox$ (Theorem~\ref{thm:d2:box}(b)) or the measure $\mu_N$ of a torus limit
point (Definition~\ref{def:omN}, Theorem~\ref{thm:d2:LP}); $Z_N(f):=\int e^{i\Phi_N(f,0)}d\mu$ with $\Phi_N(f,0)=\eps^{3/2}\sum_pf_{\mu\nu(p)}(x_p)\ff_p$
(Definition~\ref{def:pin}). With $a$, $r$ as in Lemma~\ref{flh:lem:sample}, $X_0:=\sum_p(d^*a)_p\ff_p$, $X_r:=\sum_pr_p\ff_p$ and $\sigma^2:=\sum_pr_p^2$:
\[
\begin{gathered}
X_0=0\ \ \mu\text{-a.s.},\qquad\int\Phi_N(*du,0)^2\,d\mu\le\sigma^2,\\
|Z_N(s\,{*du})-1|\le|s|\,\sigma\le|s|\Big(\frac{|K_{+2\azero}|}{192}\Big)^{1/2}\eN^2\,\|u\|_{C^3}.
\end{gathered}
\]
The same bounds hold for the killed-box states $\avB\cdot\Lam$ with $\Lam$ containing every site within distance $\eps$ of $K$, and for the torus
states $\avL\cdot$ with $L$ large.
\end{proposition}
\begin{proof}
\emph{Finite volume.} If $a_c\ne0$ then $x_c\in K$ and the vertices of $c$ lie within $\frac{\sqrt3}2\eps<\eps$ of $x_c$, so $c$ is a cube of $\Lam$;
every $p$ with $(d^*a)_p\ne0$ is a face of such a cube, hence lies in $\cP(\Lam)$, and $X_0$ is supported in $\Lam$ in the sense of
Definition~\ref{def:d2:cone}; so are $\Phi_N(*du,0)$ (its plaquettes have centres in $K$, so their vertices are within $\eps/\sqrt2$ of $K$) and $X_r=X_0+\Phi_N(*du,0)$. The box sum runs
over $v$ with $dv=0$ on every cube of $\Lam$ (Definition~\ref{def:d2:cone}), so Lemma~\ref{flh:lem:bianchi} with $\cK=\Lam$ gives $X_0=0$ on every
configuration, and $\avB{e^{isX_0}}\Lam=1$. On a torus whose side exceeds the diameter of the finite set of cells $\{c:a_c\ne0\}$ and their faces,
these cells embed injectively; Lemma~\ref{flh:lem:bianchi} with $\cK$ the torus and $a$ transported (zero elsewhere) gives $X_0=0$ pointwise, so
$\avL{e^{isX_0}}=1$ for all large $L$.

\emph{Passage.} $e^{isX_0}$ is a bounded continuous function of finitely many $\ff_p$; by Theorem~\ref{thm:d2:box}(b) (weak convergence of the
box laws along every exhausting sequence) and Theorem~\ref{thm:d2:LP}(a), $\int e^{isX_0}d\mu=1$ for every real $s$. The law of $X_0$ under $\mu$
has characteristic function $1$, so it is $\delta_0$: $X_0=0$ $\mu$-a.s.

\emph{Remainder.} By Theorem~\ref{thm:d2:box}(b), resp.\ Theorem~\ref{thm:d2:LP}(b), $\varphi(t):=\int e^{tX_r}d\mu\le e^{t^2\sigma^2/2}$ for all real
$t$ ($tr$ is a real finitely supported plaquette function). Since $|X_r|^ke^{tX_r}\le k!\,(e^{2X_r}+e^{-2X_r})$ for $|t|\le1$, dominated convergence
makes $\varphi\in C^2(-1,1)$ with $\varphi'(0)=\int X_rd\mu$ and $\varphi''(0)=\int X_r^2d\mu$. The function $\psi(t):=e^{t^2\sigma^2/2}-\varphi(t)$ is
$\ge0$ with $\psi(0)=0$, so $\psi'(0)=0$ and $\psi''(0)\ge0$: $\int X_r^2d\mu\le\sigma^2$ (the argument of Step~5 of the proof of
Theorem~\ref{thm:GD}). By Lemma~\ref{flh:lem:sample} and Definition~\ref{def:pin}, $\Phi_N(*du,0)=\sum_p\eps^{3/2}(*du)_{\mu\nu(p)}(x_p)\ff_p=-X_0+X_r$,
whence $\int\Phi_N(*du,0)^2d\mu=\int X_r^2d\mu\le\sigma^2$. Finally $|e^{is(-X_0+X_r)}-e^{-isX_0}|\le|s||X_r|$ and $\int e^{-isX_0}d\mu=1$, so by the
Schwarz inequality $|Z_N(s\,{*du})-1|\le|s|\int|X_r|d\mu\le|s|\sigma$; Lemma~\ref{flh:lem:sample} bounds $\sigma$. In finite volume the same
steps run with Theorem~\ref{thm:GD} (valid for the box states, Lemma~\ref{rot:lem:GDall}) in place of (b).
\end{proof}

\begin{theorem}[closedness of the limit curvature]\label{flh:thm:I6}
Let $\Sbox$ be a limit of Theorem~\ref{thm:Hlim} along $N_1<N_2<\cdots$ with $\ombox$ in place of $\om_N$ (Lemma~\ref{flh:lem:I3}), and let $S$ be
a limit of Theorem~\ref{thm:Hlim} for the torus limit points, $\cZ(f)=Z(f,0)$. Then for every $u\in C^\infty_c(\R^3)$ and $s\in\R$,
\[
\begin{gathered}
\Zbox(s\,{*du})=1,\qquad\cZ(s\,{*du})=1,\\
\lim_{m\to\infty}\int\Phi_{N_m}(*du,0)^2\,d\mu^{\mathrm{box}}_{N_m}=\lim_{m\to\infty}\int\Phi_{N_m}(*du,0)^2\,d\mu_{N_m}=0 .
\end{gathered}
\]
\end{theorem}
\begin{proof}
$s\,{*du}$ is a real $C^\infty_c$ $2$-form, so $(s\,{*du},0)$ is a test pair. By Theorem~\ref{thm:Hlim}(a) (for the box family, the relation
defining $\Sbox$), $\Zbox(s\,{*du})=\lim_mZ^{\mathrm{box}}_{N_m}(s\,{*du})$, and Proposition~\ref{flh:prop:I6level} gives
$|Z^{\mathrm{box}}_{N_m}(s\,{*du})-1|\le|s|(|K_{+2\azero}|/192)^{1/2}\varepsilon_{N_m}^2\|u\|_{C^3}\to0$, since $\eN=\azero5^{-N}$. The same argument applies to $\cZ$
and to the second moments. The density component of $(s\,{*du},0)$ is zero, so $g_{\rm d}$ plays no role: the statement holds for the joint scheme with
any bounded Borel $g_{\rm d}$.
\end{proof}

\begin{corollary}\label{flh:cor:I6}
(a) Let $\mu$ be a probability measure on a set $X$ of linear functionals $\Phi$ on the real $C^\infty_c$ $2$-forms, with the cylinder
$\sigma$-algebra and $\int e^{i\Phi(f)}d\mu=\Zbox(f)$ for all $f$ (Theorem~\ref{thm:osrest:measure}(a),(b), Lemma~\ref{rot:rem:minlos},
Proposition~\ref{rot:cor:E3}(b)). Then $\Phi(*du)=0$ $\mu$-a.s.\ for every $u\in C^\infty_c(\R^3)$; if $X\subset\cD'(\R^3;\Lt)$, then $\mu$-a.e.\ $\Phi$
satisfies $\Phi(*du)=0$ for all $u$ simultaneously, i.e.\ $d\Phi=0$ in the sense of distributions ($*\Phi$ is divergence-free). (b) The
Schwinger functions of $\Sbox$ (Theorem~\ref{thm:osrest:OS0}(iii)) satisfy $S_n(f_1,\dots,f_n)=0$ whenever some $f_i={*du}$, $u\in C^\infty_c(\R^3)$;
likewise for $S$. (c) (a), (b) and $\Zbox(s\,{*du})=1$ hold for $u\in\cS(\R^3)$.
\end{corollary}
\begin{proof}
(a) The characteristic function of the real random variable $\Phi(*du)$ is $s\mapsto\Zbox(s\,{*du})=1$, that of $\delta_0$. Let $K_1\subset K_2
\subset\cdots$ be compact cubes exhausting $\R^3$; each $C^\infty_{K_n}$ is a separable Fr\'echet space, so there is a countable $D\subset C^\infty_c$
such that every $u$ is a limit in some $C^\infty_{K_n}$ of elements of $D$. The union $\mathcal N$ of the null sets $\{\Phi(*du)\ne0\}$, $u\in D$, is
null; for $\Phi\notin\mathcal N$ and $u_k\to u$ in $C^\infty_{K_n}$, $*du_k\to*du$ in $C^\infty_{K_n}(\Lt)$ and $\Phi(*du_k)\to\Phi(*du)$ by the
continuity of $\Phi$ on test forms, so $\Phi(*du)=0$. For distributions $\langle\Phi,*du\rangle=-\langle(d\Phi)_{123},u\rangle$ by definition, so
$d\Phi=0$. (b) By Lemma~\ref{lem:osrest:entire}(c) and Theorem~\ref{thm:osrest:OS0}(iii), $S_n(f_1,\dots,f_n)=\int x_1\cdots x_n\,d\mu_f(x)$ with
$\mu_f$ the law on $\R^n$ with characteristic function $t\mapsto\Zbox(\sum_jt_jf_j)$ (all exponential moments finite). If $f_1=*du$, the first
marginal has characteristic function $t_1\mapsto\Zbox(t_1{*du})=1$, so $x_1=0$ $\mu_f$-a.s.\ and $S_n=0$. (c) By Theorem~\ref{thm:osrest:OS0}(i),
$\Zbox$ extends continuously to $\cS(\R^3;\Lt)$; for cut-offs $\chi_R=\chi(\cdot/R)$, $*d(\chi_Ru)\to*du$ in $\cS$, so
$\Zbox(s\,{*du})=\lim_R\Zbox(s\,{*d}(\chi_Ru))=1$; (a) and (b) follow as before, $S_n$ being continuous on $\cS^n$ (Theorem~\ref{thm:osrest:OS0}(iii)).
\end{proof}

\begin{remark}
(i) With the opposite orientation of the cubes, $(d^*a)_p=+\eps^{3/2}(*du)_{\mu\nu}(x_p)-r_p$; nothing else changes. (ii) No property of the
couplings, of $L$ or of $\Lam$ enters: Lemma~\ref{flh:lem:bianchi} is exact, Lemma~\ref{flh:lem:sample} is calculus, and the only
probabilistic input is the Gaussian domination of Theorem~\ref{thm:GD} as carried by Theorems~\ref{thm:d2:box}(b) and~\ref{thm:d2:LP}(b).
\end{remark}

\subsection{The two-point function: rotation covariance from reflection positivity in nine planes}\label{flh:ssec:I7}

\emph{Setting.} Let $T=\Lt\R^3\oplus\R$ or $T=\Lt\R^3$, with its orthogonal $\Oth$-action $\pi$ ($\pi(R)=\Lt R\,(\oplus1)$); we write
$R^*\chi:=\pi(R)^{-1}\,\chi\circ R$ for the pull-back of $\chi\colon\R^3\to T_\C$, $\tau_a\chi:=\chi(\cdot-a)$, and $\|\chi\|:=\|\chi\|_{L^1}+\|\chi\|_{L^2}$.
For a unit vector $u$ put $\theta_ux:=x-2(u\cdot x)u$, $\Theta_u:=\theta_u^*$ (so $\Theta_u\chi=\pi(\theta_u)\,\chi\circ\theta_u$), $\Omega_u^+:=\{u\cdot x>0\}$.
$\mathcal N:=\{e_j,\ (e_j\pm e_k)/\sqrt2:\ j\ne k\}$ is the set of the nine unit normals (up to sign). Let $\mu$ be a probability measure on
$\cS'(\R^3;T)$ with the $\sigma$-algebra generated by the evaluations $\Phi\mapsto\Phi(\chi)$, $\chi\in\cS(\R^3;T)$ real; $\Phi(\chi)$ for complex
$\chi$ is defined by linearity. We consider the following hypotheses:
\begin{itemize}
\item[(i)] $\int e^{\Phi(\chi)}d\mu\le e^{C(\|\chi\|_1+\|\chi\|_2^2)}$ for all real $\chi\in\cS(\R^3;T)$;
\item[(ii)] $\tau_{a*}\mu=\mu$ for all $a\in\R^3$, where $(\tau_a\Phi)(\chi):=\Phi(\tau_{-a}\chi)$;
\item[(iii)] $\sum_{a,b}\bar z_az_b\int e^{i\Phi(\chi_a-\Theta_u\chi_b)}d\mu\ge0$ for every $u\in\mathcal N$, every finite family of real
$\chi_a\in C^\infty_c(\Omega^+_u;T)$ and all $z_a\in\C$.
\end{itemize}
We write $S_n(\chi_1,\dots,\chi_n):=\int\prod_i\Phi(\chi_i)\,d\mu$ (complex-multilinear) and $Z(\chi):=\int e^{i\Phi(\chi)}d\mu$. Note that $(R_1R_2)^*=R_2^*R_1^*$.

\begin{theorem}[rotation covariance of the two-point function]\label{flh:thm:I7n2}
Under (i)--(iii):
\begin{itemize}
\item[(a)] $\int|\Phi(\chi)|^2d\mu\le c_0\|\chi\|^2$ for real $\chi$, $c_0:=3C^2+15C$; all $S_n$ exist and are continuous for $\|\cdot\|$ in each entry.
\item[(b)] $\mu$ is invariant under all translations and under the hyperoctahedral group $B_3$ of signed permutation matrices, acting by
$\Phi\mapsto\Phi\circ g^*$; in particular $S_n(g^*\chi_1,\dots,g^*\chi_n)=S_n(\chi_1,\dots,\chi_n)$ for $g\in B_3$ and all $\chi_i$.
\item[(c)] For $u\in\mathcal N$ the reconstruction $(\cH_u,H_u\ge0,V_u,E_u)$ from $L^2(\Sigma(\Omega^+_u),\mu)$ exists, and
$E_u(\{(\omega,p):\omega<|p\cdot m|\})=0$ whenever $m\perp u$ is a unit vector with $(u+m)/\sqrt2\in\mathcal N$; for $u=e_j$ this is $\omega\ge|p|_\infty$.
\item[(d)] $S_1(R^*\chi)=S_1(\chi)$ and $S_2(R^*\chi_1,R^*\chi_2)=S_2(\chi_1,\chi_2)$ for every $R\in\Oth$ and all $\chi_1,\chi_2\in\cS(\R^3;T_\C)$
with disjoint supports.
\end{itemize}
\end{theorem}
\begin{proof}
\emph{Step 1 (moments).} Put $X=\Phi(\chi)$ for real $\chi$, $a=C\|\chi\|_1$, $b=C\|\chi\|_2^2$. By (i) at $\pm\lambda\chi$ and Markov's inequality,
$\mu(|X|>t)\le2e^{-\lambda t+a\lambda+b\lambda^2}\le2e^{-(t-a)^2/4b}$ for $t>a$ ($\lambda=(t-a)/2b$), so
$\int X^2d\mu\le a^2+4\int_0^\infty(s+a)e^{-s^2/4b}ds=a^2+8b+4a\sqrt{\pi b}\le3a^2+15b$. Hence $\|\Phi(\chi)-\Phi(\chi')\|_{L^2(\mu)}\le
\sqrt{c_0}\|\chi-\chi'\|$ for real $\chi,\chi'$, and $\|\Phi(\chi)\|_{L^2(\mu)}\le2\sqrt{c_0}\|\chi\|$ for complex $\chi$; all exponential moments
are finite by (i), so all $S_n$ exist (H\"older), and they, as well as $Z$, are $\|\cdot\|$-continuous; this proves (a). Consequently $\Phi$ extends by
continuity to $(L^1\cap L^2)(\R^3;T_\C)$ with values in $L^2(\mu)$, and $S_n$ to $(L^1\cap L^2)^n$.

\emph{Step 2 (reflections).} For $\psi_1,\psi_2\in C_c^\infty(\Omega^+_u;T)$ real the $2\times2$ matrix $(Z(\psi_a-\Theta_u\psi_b))_{ab}$ is positive
semidefinite by (iii), hence Hermitian:
\[
Z(\psi_2-\Theta_u\psi_1)=\overline{Z(\psi_1-\Theta_u\psi_2)}=Z(\Theta_u\psi_2-\psi_1)=Z(\Theta_u(\psi_2-\Theta_u\psi_1)),
\]
using $\overline{Z(\chi)}=Z(-\chi)$ and $\Theta_u^2=1$. Every real $\chi\in C^\infty_c$ vanishing in a neighbourhood of the plane $\{u\cdot x=0\}$
is of the form $\psi_2-\Theta_u\psi_1$, so $Z\circ\Theta_u=Z$ on such $\chi$; for real $\chi\in\cS$, the cut-offs $\chi\,\vartheta(u\cdot x/\delta)$
($\vartheta\in C^\infty(\R)$, $\vartheta=0$ on $[-1,1]$, $\vartheta=1$ off $[-2,2]$, $0\le\vartheta\le1$) converge to $\chi$ in $\|\cdot\|$ as
$\delta\downarrow0$ (dominated convergence), and $\Theta_u$ is an isometry for $\|\cdot\|$, so $Z\circ\Theta_u=Z$ on $\cS$ by (a). Two probability
measures on $\cS'$ with equal characteristic functionals coincide (the laws of $(\Phi(\chi_j))_{j\le k}$ have the same Fourier transforms, and the
cylinder sets form a $\pi$-system generating the $\sigma$-algebra; the argument of Proposition~\ref{rot:cor:E3}(b)), so $\theta_{u*}\mu=\mu$, where
$(\theta_u\Phi)(\chi):=\Phi(\Theta_u\chi)$. The nine reflections generate $B_3$, and (ii) gives the translations; this proves the invariance in (b), and the moment identity follows
by the change of variables.

\emph{Step 3 (reconstruction).} Fix $u\in\mathcal N$. Let $\Sigma(\Omega_u^+)$ be the $\sigma$-algebra generated by the $\Phi(\chi)$,
$\chi\in C^\infty_c(\Omega^+_u;T)$ real, $L^2_{u,+}:=L^2(\Sigma(\Omega^+_u),\mu)$, $\Theta_uA:=A\circ\theta_u$ (unitary on $L^2(\mu)$ by Step~2) and
$\langle A,B\rangle_u:=\int\overline{\Theta_uA}\,B\,d\mu$ on $L^2_{u,+}$. On $A=\sum_az_ae^{i\Phi(\chi_a)}$ the form is
$\langle A,A\rangle_u=\sum_{a,b}\bar z_az_bZ(\chi_b-\Theta_u\chi_a)\ge0$ by (iii); such $A$ form a unital $*$-algebra of bounded functions generating
$\Sigma(\Omega_u^+)$, hence dense in $L^2_{u,+}$ by the functional monotone class theorem (Remark~\ref{rem:osrest:recon}), and the form is
$L^2$-continuous ($|\langle A,B\rangle_u|\le\|A\|_{L^2(\mu)}\|B\|_{L^2(\mu)}$), so it is positive semidefinite on $L^2_{u,+}$. Let $\cH_u$ be the
completion of $L^2_{u,+}$ modulo the null space, $[A]$ the class of $A$; the class map is a contraction.

With $U(a)A:=A\circ\tau_{-a}$ (so that
$U(a)\Phi(\chi)=\Phi(\tau_a\chi)$), $U$ is a strongly continuous unitary group on $L^2(\mu)$ by (ii) and (a) (strong continuity on the dense
exponentials by $|e^{ix}-e^{iy}|\le|x-y|$), $\Theta_uU(a)=U(\theta_ua)\Theta_u$, and $U(tu)L^2_{u,+}\subset L^2_{u,+}$ for $t\ge0$. Hence
$\langle U(tu)A,U(tu)B\rangle_u=\langle A,U(2tu)B\rangle_u=\langle U(2tu)A,B\rangle_u$ for $t\ge0$ and $\langle U(a)A,U(a)B\rangle_u=\langle A,B\rangle_u$ for
$a\perp u$. As in Remark~\ref{rem:osrest:recon} ($\|[U(tu)A]\|^2=\langle A,U(2tu)A\rangle_u\le\|[A]\|\,\|[U(2tu)A]\|$, iterated), $[A]\mapsto[U(tu)A]$
defines a strongly continuous self-adjoint contraction semigroup $e^{-tH_u}$ on $\cH_u$ with $H_u\ge0$, and $[A]\mapsto[U(a)A]$, $a\perp u$, a
strongly continuous unitary group $V_u(a)$ commuting with it; let $E_u$ be their joint spectral measure on $[0,\infty)\times u^\perp$, so that
$e^{-tH_u}V_u(a)=\int e^{-t\omega+ip\cdot a}dE_u(\omega,p)$. For $f,f'\in C_c^\infty(\Omega^+_u;T_\C)$, $t\ge0$, $a\perp u$, since
$\overline{\Theta_u\Phi(f)}=\Phi(\overline{\Theta_uf})$,
\begin{equation}\label{flh:eq:spec}
S_2\big(\overline{\Theta_uf},\tau_{tu+a}f'\big)=\langle[\Phi(f)],e^{-tH_u}V_u(a)[\Phi(f')]\rangle_u=\int e^{-t\omega+ip\cdot a}\,d\langle[\Phi(f)],E_u[\Phi(f')]\rangle_u .
\end{equation}

\emph{Step 4 (the cone).} Let $u\in\mathcal N$ and $m\perp u$ a unit vector with $u':=(u+m)/\sqrt2\in\mathcal N$ (for $u=e_j$: $m=\pm e_i$,
$i\ne j$; for $u=(e_j+e_i)/\sqrt2$: $m=\pm(e_j-e_i)/\sqrt2$, $u'=e_j$ or $e_i$); put $u'':=(u-m)/\sqrt2\perp u'$. Fix $d>0$ and let $A$ be a
bounded $\Sigma(Q_d)$-measurable function, $Q_d:=\{x:0<u\cdot x<d,\ |x-(u\cdot x)u|<d\}$; such classes $[A]$ are dense in $\cH_u$: the
$\sigma$-algebras $\Sigma(Q_d)$ increase to $\Sigma(\Omega^+_u)$ as $d\to\infty$, since every $\chi\in C^\infty_c(\Omega^+_u;T)$ is supported in some
$Q_d$; by the martingale convergence theorem the bounded $\Sigma(Q_d)$-measurable functions are dense in the bounded $\Sigma(\Omega^+_u)$-measurable
ones for $L^2(\mu)$, which are dense in $L^2_{u,+}$, and the class map is a contraction. For $a,c\ge0$ put
$b(a,c):=2du+\sqrt2\,a\,u'+\sqrt2\,c\,u''=(2d+a+c)u+(a-c)m$ and $G(a,c):=\langle[A],U(b(a,c))[A]\rangle_u=\int\overline{\Theta_uA}\;U(b(a,c))A\,d\mu$.

\emph{First representation:} $U(b(a,c))=U((2d+a+c)u)U((a-c)m)$ on $L^2_{u,+}$, so $G(a,c)=\int e^{-(2d+a+c)\omega+i(a-c)p\cdot m}d\mu_A(\omega,p)$ with
$\mu_A:=\langle[A],E_u(\cdot)[A]\rangle_u$, a finite positive measure. With $z:=\omega-ip\cdot m$ this is $\int e^{-az-c\bar z}e^{-2d\omega}d\mu_A$.

\emph{Second representation:} put $y_0:=\frac d2(u+m)$, $A^{(1)}:=\Theta_{u'}U(-y_0)\Theta_uA$ and $A^{(2)}:=U(2du-y_0)A$. A point $x\in Q_d$ has
$0<u\cdot x<d$ and $|m\cdot x|<d$; the point $\theta_ux-y_0$ has $u$-component in $(-\frac32d,-\frac12d)$ and $m$-component in $(-\frac32d,\frac12d)$,
hence lies in $\{u'\cdot x<0\}$, which $\theta_{u'}$ maps onto $\{u'\cdot x>0\}$; the point $x+2du-y_0$ has $u$-component in $(\frac32d,\frac52d)$
and $m$-component in $(-\frac32d,\frac12d)$, hence lies in $\{u'\cdot x>0\}$; and $\sqrt2au'+\sqrt2cu''$ has non-negative $u'$-component. So
$A^{(1)},A^{(2)}\in L^2_{u',+}$, and $U(\sqrt2au'+\sqrt2cu'')A^{(2)}\in L^2_{u',+}$. By (ii) (translating both factors by $-y_0$) and
$\Theta_{u'}^2=1$,
\[
\begin{aligned}
G(a,c)&=\int\overline{U(-y_0)\Theta_uA}\;U(b(a,c)-y_0)A\,d\mu=\int\overline{\Theta_{u'}A^{(1)}}\;U(\sqrt2au'+\sqrt2cu'')A^{(2)}\,d\mu\\
&=\langle[A^{(1)}],e^{-\sqrt2aH_{u'}}V_{u'}(\sqrt2cu'')[A^{(2)}]\rangle_{u'},
\end{aligned}
\]
which is $\int e^{-\sqrt2a\omega'+i\sqrt2c\,p'\cdot u''}d\varrho_{12}(\omega',p')$ with $\varrho_{12}:=\langle[A^{(1)}],E_{u'}(\cdot)[A^{(2)}]\rangle_{u'}$,
a complex measure of total variation at most $\|[A^{(1)}]\|_{u'}\|[A^{(2)}]\|_{u'}\le\|A\|_{L^2(\mu)}^2$. Let $\varrho$ be the image of
$e^{-2d\omega}\mu_A$ under $(\omega,p)\mapsto(\omega,p\cdot m)$ and $\varrho'$ the image of $\varrho_{12}$ under $(\omega',p')\mapsto(\sqrt2\omega',
\sqrt2\,p'\cdot u'')$. Then $\int e^{-az-c\bar z}d\varrho=\int e^{-a\omega'+icp'}d\varrho'$ for all $a,c\ge0$, and Lemma~\ref{flh:lem:cone} below gives
$\varrho(\{\omega<|q|\})=0$, i.e.\ $\mu_A(\{\omega<|p\cdot m|\})=0$ ($e^{-2d\omega}>0$). By density and the boundedness of $E_u(\{\omega<|p\cdot m|\})$,
$E_u(\{\omega<|p\cdot m|\})=0$, which is (c). For $u=e_j$ the admissible $m$ are $\pm e_i$, $i\ne j$, which gives $\omega\ge|p|_\infty$.

\emph{Step 5 (planar kernels).} Fix a permutation $(j,i,l)$ of $(1,2,3)$ and write $x=(y,z)$ with $y=(x_j,x_i)\in\R^2$ (coordinates in the
basis $\hat e_j,\hat e_i$ of the plane) and $z=x_l$. Let $\hat u$ range over the eight planar unit vectors $\pm\hat e_j$, $\pm\hat e_i$,
$\pm\hat n$, $\pm\hat m$, $\hat n:=(\hat e_j+\hat e_i)/\sqrt2$, $\hat m:=(\hat e_j-\hat e_i)/\sqrt2$, let $u\in\R^3$ be the corresponding vector
(so $\pm u\in\mathcal N$, $\Theta_{-u}=\Theta_u$) and $\hat u^\perp$, $u^\perp$ the rotation of $\hat u$, $u$ by $+\pi/2$ in the plane. In each case
$(u+u^\perp)/\sqrt2\in\pm\mathcal N$, so Step~4 (with $m=u^\perp$; hypothesis (iii) for $-u$ follows from (iii) for $u$ by the invariance of
$Z$ under $\Theta_u$, Step~2) gives: $E_u$ is carried by $\{\omega\ge|p\cdot u^\perp|\}$. For $\phi\in C^\infty_c(\R^2;\C)$ and
$w\in\mathcal W:=C^\infty_c(\R;T_\C)$ put $(\phi\otimes w)(x):=\phi(y)w(z)$, and $J_uw:=\pi(\theta_u)\bar w$, an antilinear involution of $\mathcal W$
($\pi(\theta_u)$ is a real orthogonal involution). Since $\theta_u$ fixes $z$, $\Theta_u(\phi\otimes w)=(\phi\circ\hat\theta_u)\otimes\pi(\theta_u)w$
and $\overline{\Theta_u(\phi\otimes w)}=\overline{\phi\circ\hat\theta_u}\otimes J_uw$, $\hat\theta_u$ the planar reflection.

\emph{(5a) The spectral measures and a Laplace bound.} Fix $w\in\mathcal W$. Let $\psi_k\in C^\infty_c(\R^2)$, $\psi_k\ge0$, $\int\psi_k=1$,
$\supp\psi_k\subset B(s_k\hat u,s_k/2)$, $s_k\downarrow0$, and $f_k:=\psi_k\otimes w\in C^\infty_c(\Omega^+_u;T_\C)$. Let $\nu_k$ be the image of the
finite positive measure $\langle[\Phi(f_k)],E_u(\cdot)[\Phi(f_k)]\rangle_u$ under $(\omega,p)\mapsto(\omega,q:=p\cdot u^\perp)$; it is carried by
the cone $\mathsf C:=\{\omega\ge|q|\}$. For $\phi\in C^\infty_c(\{\hat u\cdot y>0\})$ put $\mathcal L\phi(\omega,q):=\int\phi(y)e^{-(\hat u\cdot y)\omega
+i(\hat u^\perp\cdot y)q}dy$. Identify $y\in\R^2$ with the vector $(\hat u\cdot y)u+(\hat u^\perp\cdot y)u^\perp\in\R^3$ and write $(\mathcal L\phi)(H_u,P\cdot u^\perp):=\int\mathcal L\phi(\omega,p\cdot u^\perp)\,dE_u(\omega,p)$. For $\hat u\cdot y\ge0$,
\[
[\Phi(\tau_yf_k)]=[U(y)\Phi(f_k)]=e^{-(\hat u\cdot y)H_u}V_u((\hat u^\perp\cdot y)u^\perp)[\Phi(f_k)],
\]
and $\int\phi(y)\tau_yf_k\,dy=(\phi*\psi_k)\otimes w$ as a Bochner integral in $L^1\cap L^2$, so by Step~1, the continuity of the class map and the
spectral calculus,
\[
[\Phi((\phi*\psi_k)\otimes w)]=\int\phi(y)[\Phi(\tau_yf_k)]dy=(\mathcal L\phi)(H_u,P\cdot u^\perp)[\Phi(f_k)]
\]
and
\[
\begin{gathered}
\int|\mathcal L\phi|^2\,d\nu_k=\|[\Phi((\phi*\psi_k)\otimes w)]\|_u^2\le4c_0\|(\phi*\psi_k)\otimes w\|^2\le K_w\big(\|\phi\|_1^2+\|\phi\|_2^2\big),\\
K_w:=8c_0\max(\|w\|_1^2,\|w\|_2^2),
\end{gathered}
\]
by Young's inequality ($\|\psi_k\|_1=1$), uniformly in $k$.

\emph{Claim:} every positive measure $\nu$ on $\mathsf C$ with
$\int|\mathcal L\phi|^2d\nu\le K(\|\phi\|_1^2+\|\phi\|_2^2)$ for all such $\phi$ satisfies $\int e^{-\tau\omega}d\nu\le C_*K(1+\tau^{-2})$ for all $\tau>0$,
with $C_*$ an absolute constant. Indeed, fix $\phi_0\in C^\infty_c(\R^2)$, $\phi_0\ge0$, $\int\phi_0=1$, $\supp\phi_0\subset B(\hat u,\frac18)$, put
$A:=8\pi/3$, $\kappa_0:=\frac94$ and $\phi_\sigma:=\sigma^{-2}\phi_0(\cdot/\sigma)$ ($\sigma>0$), so that $\|\phi_\sigma\|_1=1$,
$\|\phi_\sigma\|_2^2=\sigma^{-2}\|\phi_0\|_2^2$, and $\hat u\cdot y\in[\frac78\sigma,\frac98\sigma]$, $|\hat u^\perp\cdot y|\le\frac\sigma8$ on
$\supp\phi_\sigma$. On $\{\sigma\omega\le A\}\cap\mathsf C$ the phase satisfies $|(\hat u^\perp\cdot y)q|\le\frac\sigma8\omega\le\frac A8=\frac\pi3$, so
$\Rp\mathcal L\phi_\sigma\ge\frac12e^{-\kappa_0\sigma\omega/2}$ there and $\int_{\{\sigma\omega\le A\}}e^{-\kappa_0\sigma\omega}d\nu\le4K(1+\sigma^{-2}
\|\phi_0\|_2^2)$. For $\tau>0$ put $\sigma_m:=\tau/(\kappa_02^m)$, $m\ge0$, and split $\mathsf C$ into $S_0:=\{\tau\omega\le\kappa_0A\}$ and
$S_m:=\{2^{m-1}\kappa_0A<\tau\omega\le2^m\kappa_0A\}$, $m\ge1$. On $S_m$, $\sigma_m\omega\le A$ and $e^{-\tau\omega}=e^{-\kappa_0\sigma_m\omega}
e^{-(2^m-1)\kappa_0\sigma_m\omega}\le e^{-\kappa_0\sigma_m\omega}e^{-2^{m-2}\kappa_0A}$ for $m\ge1$. Summing,
\[
\int e^{-\tau\omega}d\nu\le4K\sum_{m\ge0}e^{-2^{m-2}\kappa_0A\,\bbone_{m\ge1}}(1+\kappa_0^24^m\tau^{-2}\|\phi_0\|_2^2)\le C_*K(1+\tau^{-2}),
\]
the
series converging since $\kappa_0A=6\pi$. Hence $\int e^{-\tau\omega}d\nu_k\le C_*K_w(1+\tau^{-2})$ for all $k$ and $\tau>0$.

\emph{(5b) The limit measure.} Fix $l\ge1$. The measures $e^{-\omega/l}\nu_k$ have total mass at most $C_*K_w(1+l^2)$, satisfy
\[
(e^{-\omega/l}\nu_k)(\{\omega>R\})\le e^{-R/2l}\int e^{-\omega/2l}d\nu_k\le e^{-R/2l}C_*K_w(1+4l^2),
\]
and on $\{\omega\le R\}$ they are carried by the
compact set $\mathsf C\cap\{\omega\le R\}$: they are tight. By Prokhorov's theorem and a diagonal argument there is a subsequence along which
$e^{-\omega/l}\nu_k\to\mu_l$ weakly for every $l$, $\mu_l$ a finite measure on $\mathsf C$. For $l'<l$ the function $e^{-\omega(1/l'-1/l)}$ is bounded
and continuous on $\mathsf C$, so $\mu_{l'}=e^{-\omega(1/l'-1/l)}\mu_l$ and $\nu_w:=e^{\omega/l}\mu_l$ does not depend on $l$: a positive measure on
$\mathsf C$ with $\int e^{-\tau\omega}d\nu_w\le C_*K_w(1+\tau^{-2})$ for every $\tau>0$ (apply the weak convergence to the bounded continuous function
$e^{-(\tau-1/l)\omega}$, $l>1/\tau$).

For $v\in\R^2$ with $\hat u\cdot v>0$, along the subsequence,
\[
\begin{aligned}
g_k(v):=S_2\big(\overline{\Theta_uf_k},\tau_vf_k\big)&=\int e^{-(\hat u\cdot v)\omega+i(\hat u^\perp\cdot v)q}\,d\nu_k(\omega,q)\\
&\ \longrightarrow\ \hat W_w(v):=\int e^{-(\hat u\cdot v)\omega+i(\hat u^\perp\cdot v)q}\,d\nu_w(\omega,q),
\end{aligned}
\]
the first equality by \eqref{flh:eq:spec}, the limit by the weak convergence against the bounded continuous function
$e^{\omega/l}e^{-(\hat u\cdot v)\omega+i(\hat u^\perp\cdot v)q}$ on $\mathsf C$ for $l>1/(\hat u\cdot v)$. $\hat W_w$ is continuous on $\{\hat u\cdot v>0\}$
(indeed holomorphic in $\zeta\in\C^2$ on $\{\Rp(\hat u\cdot\zeta)>|\Ip(\hat u^\perp\cdot\zeta)|\}$, where the integral converges absolutely and locally
uniformly, since $|e^{-(\hat u\cdot\zeta)\omega+i(\hat u^\perp\cdot\zeta)q}|\le e^{-(\Rp(\hat u\cdot\zeta)-|\Ip(\hat u^\perp\cdot\zeta)|)\omega}$ on $\mathsf C$),
and $|g_k(v)|,|\hat W_w(v)|\le C_*K_w(1+\delta^{-2})$ on $\{\hat u\cdot v\ge\delta\}$.

\emph{(5c) Identification.} Let $\phi,\phi'\in C^\infty_c(\R^2;\C)$ with $\supp\phi'-\supp\phi\subset\{\hat u\cdot v>0\}$; by compactness
$\supp\phi'-\supp\phi\subset\{\hat u\cdot v\ge\delta\}$ for some $\delta>0$. By (ii), $g_k(y'-y)=S_2(\tau_y\overline{\Theta_uf_k},\tau_{y'}f_k)$, and
$\int\phi(y)\tau_y(\psi_k\circ\hat\theta_u)\,dy=\big((\phi\circ\hat\theta_u)*\psi_k\big)\circ\hat\theta_u$, so, the integrals being Bochner integrals
in $L^1\cap L^2$ and $S_2$ continuous bilinear,
\[
\begin{aligned}
\iint\phi(y)\phi'(y')g_k(y'-y)\,dy\,dy'&=S_2\Big(\big[((\phi\circ\hat\theta_u)*\psi_k)\circ\hat\theta_u\big]\otimes J_uw,\ (\phi'*\psi_k)\otimes w\Big)\\
&\ \longrightarrow\ S_2\big(\phi\otimes J_uw,\ \phi'\otimes w\big),
\end{aligned}
\]
while the left side tends to $\iint\phi(y)\phi'(y')\hat W_w(y'-y)\,dy\,dy'$ by dominated convergence (the bound of (5b) on $\{\hat u\cdot v\ge\delta\}$).
Hence
\begin{equation}\label{flh:eq:kerdiag}
\iint\phi(y)\phi'(y')\,\hat W_w(y'-y)\,dy\,dy'=S_2\big(\phi\otimes J_uw,\ \phi'\otimes w\big)\qquad(\supp\phi'-\supp\phi\subset\{\hat u\cdot v>0\}).
\end{equation}
A continuous function on $\{\hat u\cdot v>0\}$ is determined by these integrals (approximate identities concentrating $\phi$ at $y_0$ and $\phi'$ at
$y_0+v_0$), so $\hat W_w$ does not depend on the subsequence.

\emph{(5d) Polarisation.} The map $(w_1,w_2)\mapsto S_2(\phi\otimes J_uw_1,\phi'\otimes w_2)$ is sesquilinear on $\mathcal W$ (antilinear in $w_1$), and
\eqref{flh:eq:kerdiag} gives its diagonal. By the polarisation identity $s(w_1,w_2)=\frac14\sum_{k=0}^3i^ks(w_2+i^kw_1,w_2+i^kw_1)$, the complex
measure $\nu_{w_1,w_2}:=\frac14\sum_ki^k\nu_{w_2+i^kw_1}$ on $\mathsf C$ and its transform $\hat W_{w_1,w_2}(v):=\int e^{-(\hat u\cdot v)\omega+i(\hat u^\perp
\cdot v)q}d\nu_{w_1,w_2}$ on $\{\hat u\cdot v>0\}$ satisfy $\int e^{-\tau\omega}d|\nu_{w_1,w_2}|\le C(1+\tau^{-2})$ and
$\iint\phi\phi'\hat W_{w_1,w_2}(y'-y)=S_2(\phi\otimes J_uw_1,\phi'\otimes w_2)$ for the pairs of \eqref{flh:eq:kerdiag}. For $\alpha,\beta\in T_\C$ and
$\eta,\eta'\in C^\infty_c(\R;\C)$ put $W^{u}_{\eta\alpha,\eta'\beta}:=\hat W_{J_u(\eta\alpha),\eta'\beta}$ (so $J_uw_1=\eta\alpha$). Then, for
$\supp\phi'-\supp\phi\subset\{\hat u\cdot v>0\}$,
\begin{equation}\label{flh:eq:ker}
\iint\phi(y)\phi'(y')\,W^{u}_{\eta\alpha,\eta'\beta}(y'-y)\,dy\,dy'=S_2\big((\phi\otimes\eta)\alpha,\ (\phi'\otimes\eta')\beta\big),
\end{equation}
$W^u_{\eta\alpha,\eta'\beta}$ is complex-bilinear in $(\eta\alpha,\eta'\beta)\in\mathcal W\times\mathcal W$ (the right side is, and the kernel is unique),
it is the transform of a complex measure $\nu^u_{\eta\alpha,\eta'\beta}$ on $\mathsf C$ with $\int e^{-\tau\omega}d|\nu^u|\le C(1+\tau^{-2})$, and
$\nu^u_{\eta\alpha,\eta'\beta}\ge0$ whenever $\eta\alpha=J_u(\eta'\beta)$, in particular whenever $\eta'=\eta$ is real and $\alpha=\pi(\theta_u)\bar\beta$.

\emph{(5e) One kernel on $\R^2\setminus\{0\}$.} The four open half-planes $\{\hat u\cdot v>0\}$, $\hat u\in\{\pm\hat e_j,\pm\hat e_i\}$, cover
$\R^2\setminus\{0\}$; on an overlap the two kernels agree, since both satisfy \eqref{flh:eq:ker} for the pairs with $\supp\phi'-\supp\phi$ in the
overlap, which determine a continuous function there. So there is one continuous $W_{\eta\alpha,\eta'\beta}$ on $\R^2\setminus\{0\}$, bilinear in
$(\eta\alpha,\eta'\beta)$, with \eqref{flh:eq:ker} in every frame, including the diagonal frames $\pm\hat n,\pm\hat m$ (same argument). It satisfies
\eqref{flh:eq:ker} for \emph{all} $\phi,\phi'$ with disjoint supports: with $D:=\dist(\supp\phi,\supp\phi')>0$, $\delta<D/4$ and a smooth partition of
unity $(\varrho_a)$ of $\R^2$ with $\operatorname{diam}\supp\varrho_a<\delta$, write $\phi=\sum_a\varrho_a\phi$, $\phi'=\sum_b\varrho_b\phi'$. For a pair
$(a,b)$ of non-zero pieces pick $y_a\in\supp\varrho_a\phi$, $y'_b\in\supp\varrho_b\phi'$; $v_0:=y'_b-y_a$ has $|v_0|\ge D$, so $|\hat u\cdot v_0|\ge D/\sqrt2>2\delta$
for $\hat u=\hat e_j$ or $\hat e_i$, and $\supp\varrho_b\phi'-\supp\varrho_a\phi\subset B(v_0,2\delta)\subset\{\pm\hat u\cdot v>0\}$. Apply
\eqref{flh:eq:ker} in that frame to each pair and sum (both sides are bilinear).

Three consequences, each by the uniqueness of the kernel on open
sets: (1) \emph{symmetry}: $S_2(f,f')=S_2(f',f)$ gives $W_{\eta\alpha,\eta'\beta}(v)=W_{\eta'\beta,\eta\alpha}(-v)$; (2) \emph{conjugation}:
$S_2(\bar f,\bar f')=\overline{S_2(f,f')}$ ($\mu$ is a measure on real fields) gives $W_{\bar\eta\bar\alpha,\bar\eta'\bar\beta}=\overline{W_{\eta\alpha,\eta'\beta}}$;
(3) \emph{dihedral covariance}: for $g\in B_3$ fixing $e_l$, acting on the plane by $\hat g$, (b) and $g^*((\phi\otimes\eta)\alpha)=((\phi\circ\hat g)\otimes\eta)
\pi(g)^{-1}\alpha$ give $W_{\eta\alpha,\eta'\beta}(\hat gv)=W_{\eta\pi(g)^{-1}\alpha,\,\eta'\pi(g)^{-1}\beta}(v)$ for all $v\ne0$.

\emph{Step 6 (the angle).} Let $R_t$ be the rotation by $t$ in the $(j,i)$-plane ($R_te_j=\cos t\,e_j+\sin t\,e_i$, $R_te_i=-\sin t\,e_j+\cos t\,e_i$,
$R_te_l=e_l$), $\hat R_t$ its planar action, and $n_t:=(\cos t,\sin t)$. $T_\C$ splits into weight vectors $\pi(R_t)\epsilon=e^{iwt}\epsilon$:
$e_j\wedge e_i$ and the scalar component with $w=0$, and $\epsilon_\pm:=(e_j\wedge e_l\mp ie_i\wedge e_l)/\sqrt2$ with $w=\pm1$ (indeed
$\Lt R_t$ rotates the pair $(e_j\wedge e_l,e_i\wedge e_l)$). One computes $\bar\epsilon_\pm=\epsilon_\mp$, $\pi(\theta_j)\epsilon_\pm=-\epsilon_\mp$,
and, for $n:=(e_j+e_i)/\sqrt2$ ($\theta_ne_j=-e_i$, $\theta_ne_i=-e_j$), $\pi(\theta_n)\epsilon_+=i\epsilon_-$, $\pi(\theta_n)\epsilon_-=-i\epsilon_+$.

Fix $\eta,\eta'$ and weight vectors $a,b$ with weights $w_a,w_b$, $w:=w_a+w_b\in\{0,\pm1,\pm2\}$, and write $W_{ab}:=W_{\eta a,\eta'b}$. By
(5e)(3) with $g=R_{\pi/2}\in B_3$: $W_{ab}(\hat R_{\pi/2}v)=e^{-iw\pi/2}W_{ab}(v)$. Fix $r>0$. By \eqref{flh:eq:ker} in the frame $\hat e_j$,
$F_0(\zeta):=\int e^{-r\cos\zeta\,\omega+ir\sin\zeta\,q}\,d\nu^{e_j}_{\eta a,\eta'b}(\omega,q)$ is holomorphic on the strip $|\Rp\zeta|<\pi/2$ and
$F_0(t)=W_{ab}(rn_t)$ for real $t$ there: on $\mathsf C$, with $\zeta=t+i\beta$, $\Rp(-r\cos\zeta\,\omega+ir\sin\zeta\,q)=-r\cos t\,(\omega\cosh\beta+
q\sinh\beta)\le-r\cos t\,e^{-|\beta|}\omega$, so the integral converges absolutely and locally uniformly, and
$|F_0(t+i\beta)|\le C(1+(r\cos t)^{-2}e^{2|\beta|})$ by the Laplace bound of (5d).

The functions $\zeta\mapsto e^{-iwk\pi/2}F_0(\zeta-k\pi/2)$,
$k\in\Z$, defined on the strips $|\Rp\zeta-k\pi/2|<\pi/2$, agree with $W_{ab}(rn_t)$ on the real points of their strips by the quarter-turn
covariance, hence agree on the overlaps of consecutive strips by the identity theorem, and define an entire function $h$ with $h(t)=W_{ab}(rn_t)$
for real $t$, $h(\zeta+\pi/2)=e^{-iw\pi/2}h(\zeta)$ and $M(\beta):=\sup_t|h(t+i\beta)|\le C(1+2r^{-2}e^{2|\beta|})$ (the bound on $|t|\le\pi/4$ and
the periodicity). Thus $e^{iw\zeta}h(\zeta)$ is entire with period $\pi/2$, so $h(\zeta)=\sum_{k\in\Z}c_ke^{i(4k-w)\zeta}$ with
$|c_k|\le M(\beta)e^{(4k-w)\beta}$ for every real $\beta$; letting $\beta\to\mp\infty$ shows $c_k=0$ unless $|4k-w|\le2$. For $w\in\{0,\pm1\}$ only
$k=0$ survives: $W_{ab}(\hat R_tv)=e^{-iwt}W_{ab}(v)$ for all $t$ and $v\ne0$. For $w=2$ ($a=b=\epsilon_+$) the survivors are $k\in\{0,1\}$:
$h(\zeta)=c_0e^{-2i\zeta}+c_1e^{2i\zeta}$, and for $w=-2$ symmetrically $h=c_0e^{2i\zeta}+c_{-1}e^{-2i\zeta}$.

\emph{The sign step, $w=2$, first for $\eta'=\eta$ real.} By (5d), $\nu^{e_j}_{\eta\alpha,\eta\epsilon_+}\ge0$ for $\alpha=\pi(\theta_j)\bar\epsilon_+=
\pi(\theta_j)\epsilon_-=-\epsilon_+$, i.e.\ $-\nu^{e_j}_{++}\ge0$, so $-h(i\beta)=-F_0(i\beta)=\int e^{-r(\omega\cosh\beta+q\sinh\beta)}d(-\nu^{e_j}_{++})\ge0$ for all
real $\beta$; as $-h(i\beta)=-c_0e^{2\beta}-c_1e^{-2\beta}$, this forces $c_0,c_1$ real and $\le0$. In the frame $\hat n$, (5d) gives
$\nu^n_{\eta\alpha,\eta\epsilon_+}\ge0$ for $\alpha=\pi(\theta_n)\epsilon_-=-i\epsilon_+$, i.e.\ $-i\nu^n_{++}\ge0$. Since $n_t=\cos(t-\frac\pi4)\hat n+
\sin(t-\frac\pi4)\hat n^\perp$, \eqref{flh:eq:ker} in the frame $\hat n$ gives $-ih(t)=F^n(t)$ for $t\in(-\frac\pi4,\frac{3\pi}4)$, where
$F^n(\zeta):=\int e^{-r\cos(\zeta-\pi/4)\omega+ir\sin(\zeta-\pi/4)q}d(-i\nu^n_{++})$ is holomorphic on $|\Rp\zeta-\frac\pi4|<\frac\pi2$; by the identity
theorem $-ih=F^n$ on that strip, and at $\zeta=\frac\pi4+i\beta$, $F^n(\frac\pi4+i\beta)=\int e^{-r(\omega\cosh\beta+q\sinh\beta)}d(-i\nu^n_{++})\ge0$,
while $-ih(\frac\pi4+i\beta)=-c_0e^{2\beta}+c_1e^{-2\beta}$. Letting $\beta\to-\infty$ gives $c_1\ge0$, hence $c_1=0$: $h(\zeta)=c_0e^{-2i\zeta}$, which
is the covariance $W_{++}(\hat R_tv)=e^{-2it}W_{++}(v)$ for $\eta'=\eta$ real.

\emph{General $\eta,\eta'$.} The coefficient $c_1=c_1^{\eta\eta'}$ is
bilinear in $(\eta,\eta')$ (the kernel is, and the Fourier coefficient is a linear functional of $h$). By (5e)(1), $h^{\eta\eta'}(t)=W_{\eta\epsilon_+,
\eta'\epsilon_+}(rn_t)=W_{\eta'\epsilon_+,\eta\epsilon_+}(-rn_t)=h^{\eta'\eta}(t+\pi)$, and $e^{i(4k-2)\pi}=1$, so $c_1^{\eta\eta'}=c_1^{\eta'\eta}$. For real
$\eta,\eta'$: $0=c_1^{\eta+\eta',\eta+\eta'}=c_1^{\eta\eta}+2c_1^{\eta\eta'}+c_1^{\eta'\eta'}=2c_1^{\eta\eta'}$; complex $\eta,\eta'$ follow by bilinearity.
The channel $w=-2$ follows from (5e)(2) ($W_{\bar\eta\epsilon_-,\bar\eta'\epsilon_-}=\overline{W_{\eta\epsilon_+,\eta'\epsilon_+}}$), or by the same argument.
Hence, for all $\alpha,\beta\in T_\C$, all $\eta,\eta'$, all $t\in\R$ and $v\ne0$ (expanding $\alpha,\beta$ in weight vectors),
\begin{equation}\label{flh:eq:cov}
W_{\eta\alpha,\eta'\beta}(\hat R_tv)=W_{\eta\pi(R_t)^{-1}\alpha,\ \eta'\pi(R_t)^{-1}\beta}(v).
\end{equation}

\emph{Step 7 (assembly).} \emph{(7a) Planar-disjoint tests.} For $f=(\phi\otimes\eta)\alpha$, $f'=(\phi'\otimes\eta')\beta$ with
$\supp\phi\cap\supp\phi'=\emptyset$: $R_t^*f=((\phi\circ\hat R_t)\otimes\eta)\pi(R_t)^{-1}\alpha$ ($R_t$ fixes $z$), the planar supports of
$\phi\circ\hat R_t$, $\phi'\circ\hat R_t$ are disjoint, and \eqref{flh:eq:ker} (for all disjoint planar supports, (5e)), then the substitution $y\mapsto\hat R_t^{-1}y$, $y'\mapsto\hat R_t^{-1}y'$
and \eqref{flh:eq:cov} for $-t$ at $v=y'-y$, give
\[
\begin{aligned}
S_2(R_t^*f,R_t^*f')&=\iint\phi(\hat R_ty)\phi'(\hat R_ty')W_{\eta\pi(R_t)^{-1}\alpha,\eta'\pi(R_t)^{-1}\beta}(y'-y)\,dy\,dy'\\
&=\iint\phi(y)\phi'(y')W_{\eta\alpha,\eta'\beta}(y'-y)\,dy\,dy'=S_2(f,f').
\end{aligned}
\]
For disjoint open $U,U'\subset\R^2$ the finite sums of such products with $\phi\in C^\infty_c(U)$, resp.\ $\phi'\in C^\infty_c(U')$, are dense in
$C^\infty_c(U\times\R;T_\C)$, resp.\ $C^\infty_c(U'\times\R;T_\C)$, for the $C^\infty_c$ topology, hence for $\|\cdot\|$; $R_t^*$ is a $\|\cdot\|$-isometry
and $S_2$ is $\|\cdot\|$-continuous (Step~1). Hence, for each of the three coordinate planes: \emph{(P)} $S_2(R_t^*\chi_1,R_t^*\chi_2)=S_2(\chi_1,\chi_2)$
for all $t\in\R$ and all $\chi_1,\chi_2\in C^\infty_c(\R^3;T_\C)$ whose supports have disjoint orthogonal projections onto that plane.

\emph{(7b) Small supports.} Let $\chi_1,\chi_2\in C^\infty_c(\R^3;T_\C)$ be supported in the closed balls $\bar B(x_1,\delta)$, $\bar B(x_2,\delta)$ with
$|x_2-x_1|>3\delta$, and put $F(R):=S_2(R^*\chi_1,R^*\chi_2)-S_2(\chi_1,\chi_2)$ for $R\in\SO$. $F$ is $C^\infty$ on $\SO$: in a local chart
$\vartheta\mapsto R(\vartheta)$ the function $(\vartheta,x)\mapsto(R(\vartheta)^*\chi)(x)=\pi(R(\vartheta))^{-1}\chi(R(\vartheta)x)$ is $C^\infty$ with
$x$-support in a fixed compact set for $\vartheta$ in a compact neighbourhood, so $\vartheta\mapsto R(\vartheta)^*\chi$ is $C^\infty$ into $L^1\cap L^2$
(its difference quotients converge uniformly, with uniformly bounded supports), and $S_2$ is a continuous bilinear form on $(L^1\cap L^2)^2$
(Step~1).

For $a\in\{1,2,3\}$ let $R^{(a)}_s=\exp(sL_a)$ be the rotation by $s$ about $e_a$, $(L_a)_{bc}=-\epsilon_{abc}$, and $X_a$ the
left-invariant vector field $(X_aG)(R):=\frac d{ds}G(RR^{(a)}_s)|_{s=0}$; then $[X_a,X_b]=X_{[L_a,L_b]}=\pm X_c$ for $\{a,b,c\}=\{1,2,3\}$, and
$X_1,X_2,X_3$ span the tangent space at every point. Let $P^\perp_a$ be the orthogonal projection onto $e_a^\perp$ and
$U_a:=\{R\in\SO:|P^\perp_aR^{-1}(x_2-x_1)|>2\delta\}$, an open set. For $R\in U_a$ the pair $(R^*\chi_1,R^*\chi_2)$ is supported in
$\bar B(R^{-1}x_1,\delta)$, $\bar B(R^{-1}x_2,\delta)$, whose projections along $e_a$ are disjoint, so (P) for the plane $e_a^\perp$ gives
$S_2(R^{(a)*}_sR^*\chi_1,R^{(a)*}_sR^*\chi_2)=S_2(R^*\chi_1,R^*\chi_2)$ for all $s$, i.e.\ $F(RR^{(a)}_s)=F(R)$ since $(RR^{(a)}_s)^*=R^{(a)*}_sR^*$; hence
$X_aF=0$ on $U_a$. Every $R\in\SO$ lies in at least two of $U_1,U_2,U_3$: with $w:=R^{-1}(x_2-x_1)$, $\sum_a|P^\perp_aw|^2=2|w|^2>18\delta^2$, and
if two of the three terms were $\le4\delta^2$ we would get $2|w|^2\le8\delta^2+|w|^2$, i.e.\ $|w|^2\le8\delta^2<9\delta^2$. On the open set $U_a\cap U_b$,
$X_aF=X_bF=0$, hence $X_cF=\pm(X_a(X_bF)-X_b(X_aF))=0$ ($F\in C^2$); so $dF=0$ on $U_a\cap U_b$, and every point of $\SO$ lies in such a set. Thus
$dF\equiv0$ on the connected manifold $\SO$, and $F\equiv F(1)=0$.

\emph{(7c) Conclusion.} Let $\chi_1,\chi_2\in C^\infty_c(\R^3;T_\C)$ have disjoint supports, $D:=\dist(\supp\chi_1,\supp\chi_2)>0$, $\delta:=D/6$, and
let $(\varrho_a)$ be a smooth partition of unity of $\R^3$ with $\supp\varrho_a\subset\bar B(y_a,\delta)$. Then $\chi_1=\sum_a\varrho_a\chi_1$,
$\chi_2=\sum_b\varrho_b\chi_2$ are finite sums, every pair $(\varrho_a\chi_1,\varrho_b\chi_2)$ of non-zero pieces has $|y_a-y_b|\ge D-2\delta=4\delta>3\delta$, and
(7b) with the bilinearity of $S_2$ and of $R^*$ gives $S_2(R^*\chi_1,R^*\chi_2)=S_2(\chi_1,\chi_2)$ for $R\in\SO$. For $R\in\Oth\setminus\SO$ write
$R=\theta_3R'$ with $R'\in\SO$; then $R^*=R'^*\Theta_3$, the supports of $\Theta_3\chi_1,\Theta_3\chi_2$ are disjoint, and (b) gives
$S_2(R^*\chi_1,R^*\chi_2)=S_2(\Theta_3\chi_1,\Theta_3\chi_2)=S_2(\chi_1,\chi_2)$. For Schwartz $\chi_1,\chi_2$ with disjoint supports the cut-offs
$\vartheta(\cdot/\nu)\chi_i$ ($\vartheta\in C^\infty_c$, $\vartheta=1$ on the unit ball) converge to $\chi_i$ in $\|\cdot\|$ with supports inside the
original ones, and $R^*$ is a $\|\cdot\|$-isometry; Step~1 concludes.

Finally $S_1$: for real $\chi\in C^\infty_c$ and $\rho\in C^\infty_c$,
$\rho\ge0$, $\int\rho=1$, $\rho_R:=R^{-3}\rho(\cdot/R)$, (ii) and the Bochner integral give $S_1(\chi*\rho_R)=S_1(\chi)$, while
\[
\|\chi*\rho_R-(\textstyle\int\chi)\rho_R\|\le\int|\chi(y)|\,\|\tau_y\rho_R-\rho_R\|\,dy\le R^{-1}(\|\nabla\rho\|_1+\|\nabla\rho\|_2)\int|y||\chi(y)|dy\to0;
\]
applied to $\chi=\rho_{R_1}e_r$ ($e_r$ a basis vector of $T$) this shows that $m_r:=S_1(\rho_Re_r)$ does not depend on $R$, and then
$S_1(\chi)=m(\int\chi)$ with $m:=\sum_rm_re_r^*\in T^*$. By (b), $m\circ\pi(g)^{-1}=m$ for $g\in B_3$; the sign flip $g=1-2e_je_j^{\mathsf T}$ acts by $-1$
on $e_j\wedge e_k$, so $m$ vanishes on $\Lt\R^3$ and lives on the scalar component, on which $\pi$ is trivial; hence $S_1(R^*\chi)=m(\pi(R)^{-1}\int\chi)
=S_1(\chi)$ for all $R\in\Oth$ (and $S_1=0$ for $T=\Lt\R^3$). This proves (d).
\end{proof}

The lemma used in Step~4 is the following statement of complex analysis; its proof is independent of everything above. For $h\in L^1(0,\infty)$
put $\tilde h(z):=\int_0^\infty h(a)e^{-az}da$ for $\Rp z\ge0$ (continuous, bounded by $\|h\|_{L^1}$, holomorphic for $\Rp z>0$); powers are
principal, $z^\nu:=\exp(\nu\operatorname{Log}z)$ for $z\notin(-\infty,0]$ with $\arg z\in(-\pi,\pi)$.

\begin{lemma}[two representations of one correlation]\label{flh:lem:cone}
Let $\varrho$ be a finite positive Borel measure on $[0,\infty)\times\R$ and $\varrho'$ a complex Borel measure on $[0,\infty)\times\R$ of finite total
variation $\|\varrho'\|$. Suppose that, with $z:=\omega-iq$,
\[
\int e^{-az-c\bar z}\,d\varrho(\omega,q)=\int e^{-a\omega'+icq'}\,d\varrho'(\omega',q')\qquad\text{for all }a,c\ge0 .
\]
Then $\varrho(\{(\omega,q):\omega<|q|\})=0$.
\end{lemma}
\begin{proof}
\emph{An inequality.} Let $h\in L^1(0,\infty)$. Multiply the hypothesis by $h(a)\overline{h(c)}$ and integrate over $a,c\ge0$; on the left
$|e^{-az-c\bar z}|=e^{-(a+c)\omega}\le1$ and $\varrho$ is finite, on the right the integrand has modulus at most $1$ and $\|\varrho'\|<\infty$, so
Fubini's theorem applies on both sides. Since $\int_0^\infty\overline{h(c)}e^{-c\bar z}dc=\overline{\tilde h(z)}$ and
$\int_0^\infty\overline{h(c)}e^{icq'}dc=\overline{\tilde h(iq')}$,
\begin{equation}\label{flh:eq:kernelineq}
\int|\tilde h(z)|^2\,d\varrho=\int\tilde h(\omega')\,\overline{\tilde h(iq')}\,d\varrho'\le\|\varrho'\|\,\sup_{t\ge0}|\tilde h(t)|\,\sup_{s\in\R}|\tilde h(is)| .
\end{equation}

\emph{The test functions.} Fix $\epsilon\in\{1,-1\}$ and an integer $N\ge1$, and put $\nu:=1-4i\epsilon N/\pi$ ($\Rp\nu=1$, $\nu\notin\{0,1,2,\dots\}$).
We construct $h_N\in L^1(0,\infty)$ with $\tilde h_N(z)=e^{-N}z^\nu(1+z)^{-3}$ for $\Rp z\ge0$, $z\ne0$, and $\tilde h_N(0)=0$. The binomial
coefficients $b_k:=\binom\nu k$ satisfy $b_k=(-1)^k\Gamma(k-\nu)/(\Gamma(-\nu)\Gamma(k+1))$ and $|\Gamma(k-\nu)/\Gamma(k+1)|\sim k^{-\Rp\nu-1}=k^{-2}$
(Stirling), so $\sum_k|b_k|<\infty$ and $\sum_kb_k(-u)^k$ converges absolutely and uniformly on the closed unit disc. Its sum is $(1-u)^\nu$ for
$|u|<1$ (Taylor series of the holomorphic function $u\mapsto(1-u)^\nu$, $\Rp(1-u)>0$ there), hence also for $|u|\le1$, $u\ne1$, by continuity
of $w\mapsto w^\nu$ on the closed right half-plane minus $0$; and as $u\to1$ in the disc, $|(1-u)^\nu|=|1-u|e^{-\Ip\nu\arg(1-u)}\le|1-u|e^{2N}\to0$, so
the sum is $0$ at $u=1$.

For $k\ge1$ let $\gamma_k(a):=a^{k-1}e^{-a}/(k-1)!$, so $\gamma_k\ge0$, $\int_0^\infty\gamma_k=1$,
$\int_0^\infty\gamma_k(a)e^{-az}da=(1+z)^{-k}$ for $\Rp z\ge0$, and let $m:=\delta_0+\sum_{k\ge1}(-1)^kb_k\gamma_k(a)\,da$, a complex measure on
$[0,\infty)$ of total variation at most $1+\sum_k|b_k|$. For $\Rp z\ge0$ put $u:=(1+z)^{-1}$; then $|u|\le1$ (as $|1+z|^2=1+2\Rp z+|z|^2\ge1$) and
$u=1$ only for $z=0$, so $\int e^{-az}dm(a)=\sum_kb_k(-1)^k(1+z)^{-k}=(z/(1+z))^\nu$ for $z\ne0$ and $\int dm=0$.

Next let
$q(a):=a^{2-\nu}e^{-a}/\Gamma(3-\nu)$, $a>0$; since $\Rp(2-\nu)=1$, $|q(a)|=ae^{-a}/|\Gamma(3-\nu)|$, so $q\in L^1(0,\infty)$, and
$\int_0^\infty q(a)e^{-az}da=(1+z)^{\nu-3}$ (for real $z>-1$ by the Gamma integral, then for $\Rp z>-1$ by analytic continuation). Put
$h_N:=e^{-N}m*q$, i.e.\ $h_N(a)=e^{-N}(q(a)+\sum_{k\ge1}(-1)^kb_k\int_0^a\gamma_k(a-b)q(b)db)$; then $h_N\in L^1(0,\infty)$ and, by Fubini's theorem,
$\tilde h_N(z)=e^{-N}(z/(1+z))^\nu(1+z)^{\nu-3}$ for $\Rp z\ge0$, $z\ne0$, and $\tilde h_N(0)=0$. For $\Rp z\ge0$, $z\ne0$, the arguments of $z$ and
$1+z$ lie in $[-\pi/2,\pi/2]$ and have the same sign ($\Ip z=\Ip(1+z)$), so $\operatorname{Log}z-\operatorname{Log}(1+z)$ is the principal logarithm
of $z/(1+z)$, and $(z/(1+z))^\nu(1+z)^{\nu-3}=\exp(\nu\operatorname{Log}z-3\operatorname{Log}(1+z))=z^\nu(1+z)^{-3}$.

\emph{Bounds.} For $\Rp z\ge0$, $z\ne0$: $|z^\nu|=|z|\exp(-\Ip\nu\arg z)$ with $-\Ip\nu=4\epsilon N/\pi$, so
$|\tilde h_N(z)|=e^{-N}|z|\,|1+z|^{-3}\exp(4\epsilon N\arg z/\pi)$. Since $|1+z|\ge\max(1,|z|)$ on the closed right half-plane, $|z||1+z|^{-3}\le1$
there. On $(0,\infty)$, $\arg z=0$, so $\sup_{t\ge0}|\tilde h_N(t)|\le e^{-N}$; on $i\R\setminus\{0\}$, $\arg z=\pm\pi/2$, so
$\sup_s|\tilde h_N(is)|\le e^{-N}e^{2N}=e^N$; the product of the two suprema is at most $1$. On the sector
$\Sigma_{\epsilon,\delta}:=\{z\ne0:\Rp z\ge0,\ \epsilon\arg z\ge\pi/4+\delta\}$, $\delta>0$: $|\tilde h_N(z)|^2\ge e^{-2N}e^{8N(\pi/4+\delta)/\pi}|z|^2|1+z|^{-6}
=e^{8\delta N/\pi}|z|^2|1+z|^{-6}$.

\emph{Conclusion.} By \eqref{flh:eq:kernelineq} with $h=h_N$ and the bounds, $\int_{\Sigma_{\epsilon,\delta}}|z|^2|1+z|^{-6}d\varrho\le e^{-8\delta N/\pi}
\|\varrho'\|$ for all $N\ge1$, $\Sigma_{\epsilon,\delta}$ read as a set of points $(\omega,q)$ through $z=\omega-iq$. Letting $N\to\infty$, and since the
weight is strictly positive on $\Sigma_{\epsilon,\delta}$, $\varrho(\Sigma_{\epsilon,\delta})=0$. The union of the sets $\Sigma_{\pm1,1/k}$, $k\ge1$, is
$\{z\ne0:\Rp z\ge0,\ |\arg z|>\pi/4\}$, and for $z=\omega-iq$ with $\omega\ge0$ the condition $|\arg z|>\pi/4$ is $|q|>\omega$ (for $\omega>0$,
$|\arg z|=\arctan(|q|/\omega)$; for $\omega=0$, $z\ne0$ means $q\ne0$).
\end{proof}

\begin{theorem}[rotation covariance of all moments away from coincident points]\label{flh:thm:I7}
Let $T=\Lt\R^3$ or $\Lt\R^3\oplus\R$ with its orthogonal $\Oth$-action and let $\mu$ be a probability measure on $\cS'(\R^3;T)$, with the $\sigma$-algebra generated by the evaluations, with the
properties (i), (ii), (iii) of Section~\ref{flh:ssec:I7}. Then
\[
\begin{gathered}
S_n(R^*\chi_1,\dots,R^*\chi_n)=S_n(\chi_1,\dots,\chi_n)\\
\text{for all }n\ge1,\ R\in\Oth\text{ and }\chi_1,\dots,\chi_n\in\cS(\R^3;T_\C)\text{ with pairwise disjoint supports.}
\end{gathered}
\]
\end{theorem}

\begin{proof}
Theorem~\ref{flh:thm:I7} is Theorem~\ref{i7:thm:main} of Section~\ref{sec:i7:proof}, proved for all $n$ from (i)--(iii) alone; for
$n\le2$ it is also Theorem~\ref{flh:thm:I7n2}(d). Two conventions of Section~\ref{sec:i7:proof} differ from those of
Section~\ref{flh:ssec:I7}. Hypothesis (iii) is asked there of finite families of real Schwartz tests
supported in one open side of the plane; (iii) above, with $C^\infty_c$ tests, implies it: for real Schwartz $\chi_a$ supported in
$\Omega^+_u$ the cut-offs $\chi_a\vartheta(\cdot/q)$ ($\vartheta\in C^\infty_c(\R^3)$, $\vartheta=1$ on $B(0,1)$) lie in
$C^\infty_c(\Omega^+_u;T)$ and converge to $\chi_a$ in $\|\cdot\|$ as $q\to\infty$, and
$|Z(\chi)-Z(\chi')|\le\int|\Phi(\chi-\chi')|\,d\mu\le c_0^{1/2}\|\chi-\chi'\|$ for real $\chi,\chi'$ (Theorem~\ref{flh:thm:I7n2}(a)), so each
finite Gram sum of (iii) passes to the limit. The measure there is a Borel probability measure on $\cS'$; the proofs of
Section~\ref{sec:i7:proof} use $\mu$ only through the random variables $\Phi(\chi)$, $\chi\in\cS$ real, and the $\sigma$-algebras they
generate (Remark~\ref{i7:rem:sigma}), so they apply to every probability measure on the cylinder $\sigma$-algebra, in particular to the measure of Lemma~\ref{flh:lem:I5}, with the reading given in that remark.
The conclusion there is stated, as above, for Schwartz tests with pairwise disjoint supports (complex ones by multilinearity, the real and
imaginary parts keeping the supports).
\end{proof}



\subsection{Bounded multilinear forms vanishing off the diagonal and on gradients}\label{flh:ssec:I8}

\emph{Notation.} Let $U\subset\R^3$ be open and $H(U):=L^2(U;\C^3)$ (elements extended by $0$ to $\R^3$); $|E|$ denotes the Lebesgue measure and $1_E$ the multiplication
by the indicator; ``$b$ vanishes on $E$'' means almost everywhere; an $n$-linear form $D$ on $H(U)$ ($\C$-linear in each entry) is bounded if
$|D(b_1,\dots,b_n)|\le C\prod_i\|b_i\|_{L^2}$. We consider two hypotheses on $D$:
\begin{itemize}
\item[(H1$_U$)] $D(b_1,\dots,b_n)=0$ whenever $b_1,\dots,b_n\in C^\infty_c(U;\C^3)$ have pairwise disjoint supports (for $n=1$: for all $b_1$);
\item[(H2$_U$)] $D(b_1,\dots,b_n)=0$ whenever one entry is $\nabla u$, $u\in C^\infty_c(U;\C)$, the others arbitrary in $H(U)$.
\end{itemize}
For $U=\R^3$ we write (H1) and (H2) for (H1$_U$) and (H2$_U$).

\begin{lemma}[locality gives a multiplier]\label{flh:lem:I8mult}
Let $\beta$ be a bounded bilinear form on $H(U)$ with constant $C$ and $\beta(b_1,b_2)=0$ whenever $b_1,b_2\in C^\infty_c(U;\C^3)$ have disjoint
supports. Then there is a unique $M\in L^\infty(U;M_3(\C))$ with $\beta(b_1,b_2)=\int_Ub_1^{\mathsf T}Mb_2\,dx$ for all $b_1,b_2\in H(U)$, and
$|M(x)v|\le\sqrt3\,C|v|$ almost everywhere.
\end{lemma}
\begin{proof}
(i) $(b_1,b_2)\mapsto\beta(\bar b_1,b_2)$ is bounded sesquilinear, so $\beta(\bar b_1,b_2)=\langle b_1,Tb_2\rangle$ for a unique bounded operator $T$
on $H(U)$, $\|T\|\le C$; $\langle b_1,Tb_2\rangle=0$ for $b_1,b_2\in C^\infty_c(U)$ with disjoint supports ($\bar b_1$ has the support of $b_1$), and
likewise for $T^*$.

(ii) \emph{Claim: if $V\subset U$ is open and $b\in H(U)$ vanishes on $V$, then $Tb$ vanishes on $V$.} Let $K\subset V$ be compact. If $U\setminus V=
\emptyset$ then $b=0$. Otherwise $\delta:=\dist(K,U\setminus V)>0$ (a sequence of $U\setminus V$ approaching $K$ would accumulate at a point of
$K\subset V$, while $U\setminus V$ is closed in $U$). Put $W:=\{x\in U:\dist(x,K)<\delta/3\}$ and $V':=\{x\in U:\dist(x,U\setminus V)<\delta/3\}$: open,
disjoint, $K\subset W$, $b\in H(V')$. Take $b^{(k)}\in C^\infty_c(V';\C^3)$ with $b^{(k)}\to b$ in $L^2$. For $b_1\in C^\infty_c(W;\C^3)$,
$\langle b_1,Tb^{(k)}\rangle=0$, hence $\langle b_1,Tb\rangle=0$; $C^\infty_c(W)$ is dense in $H(W)$, so $Tb=0$ on $W\supset K$; $V$ is a countable union
of compact sets.

(iii) By (ii), $1_VT(1-1_V)=0$; by (ii) for $T^*$, $(1-1_V)T1_V=0$; so $T1_V=1_VT$ for every open $V\subset U$. The class of Borel $E\subset U$ with
$T1_E=1_ET$ contains the open sets, is closed under complements ($1_{U\setminus E}=1-1_E$) and under countable disjoint unions ($\sum_{k\le m}1_{E_k}b
\to1_Eb$ in $L^2$, $T$ bounded); by Dynkin's theorem it is the Borel $\sigma$-algebra. By linearity and $\|M_\phi\|\le\|\phi\|_\infty$, $TM_\phi=M_\phi T$
for every bounded Borel $\phi$.

(iv) Let $U_k:=U\cap\{|x|<k\}$ and $m_j^{(k)}:=T(1_{U_k}e_j)$. For $k\le k'$, $1_{U_k}m_j^{(k')}=T(1_{U_k}e_j)=m_j^{(k)}$ and $(1-1_{U_k})m_j^{(k)}=0$,
so $m_j^{(k)}=1_{U_k}m_j$ for one measurable $m_j$, and $T(1_Ee_j)=1_ET(1_{U_k}e_j)=1_Em_j$ for Borel $E\subset U_k$. Hence $\int_E|m_j|^2=
\|T(1_Ee_j)\|^2\le\|T\|^2|E|$ for every bounded Borel $E\subset U$, so $|m_j|\le\|T\|$ a.e.\ (otherwise integrate over a bounded subset of positive
measure of $\{|m_j|>\|T\|\}$). With $M:=(m_1|m_2|m_3)$, $|Mv|\le\sum_j|v_j||m_j|\le\sqrt3\|T\||v|$, and $Tb=Mb$ on the dense set of finite sums
$\sum c_{ij}1_{E_{ij}}e_j$ (bounded Borel $E_{ij}$); $T$ and the multiplication by $M$ are bounded, so $T=M$, i.e.\
$\beta(b_1,b_2)=\langle\bar b_1,Mb_2\rangle=\int_Ub_1^{\mathsf T}Mb_2$. Uniqueness: $\int b_1^{\mathsf T}Mb_2=0$ for all $b_1,b_2$ gives $M1_{U_k}e_j=0$
a.e.\ for all $j,k$.
\end{proof}

\begin{lemma}[gradients kill the multiplier]\label{flh:lem:I8grad}
If in addition $\beta(b_1,\nabla u)=0$ for all $b_1\in H(U)$ and $u\in C^\infty_c(U)$, then $M=0$ and $\beta=0$.
\end{lemma}
\begin{proof}
$b_1=\overline{M\nabla u}\in H(U)$ gives $\int_U|M\nabla u|^2=0$, so $M\nabla u=0$ a.e. For a ball $B$ with $\bar B\subset U$ take $\chi\in C^\infty_c(U)$,
$\chi=1$ near $\bar B$, and $u=\chi x_k$: $Me_k=0$ a.e.\ on $B$, $k=1,2,3$. $U$ is a countable union of such balls.
\end{proof}

\begin{theorem}[bounded multilinear forms vanishing off the diagonal and on gradients]\label{flh:thm:I8}
For every $n\ge1$ and every open $U\subset\R^3$, a bounded $n$-linear form on $H(U)$ satisfying (H1$_U$) and (H2$_U$) is zero.
\end{theorem}
\begin{proof}
We argue by strong induction on $n$, for all open sets at once. For $n=1$, $D=0$ on the dense set $C^\infty_c(U;\C^3)$. For $n=2$ apply Lemma~\ref{flh:lem:I8mult} with
(H1$_U$), then Lemma~\ref{flh:lem:I8grad} with (H2$_U$). Let $n\ge3$, assume the theorem for all $m<n$ and all open sets, and let $D$ be bounded
$n$-linear on $H(U)$, with constant $C$, satisfying (H1$_U$), (H2$_U$).

\emph{Step 1 (one confined group).} Let $\emptyset\ne J\subsetneq\{1,\dots,n\}$, $F$ its complement, $K\subset U$ compact, $b_i\in H(U)$ vanishing
outside $K$ for $i\in J$ and on $K$ for $i\in F$. Then $D(b_1,\dots,b_n)=0$. Indeed, let $U':=U\setminus K$ and let $E$ be the $|F|$-linear form on
$H(U')$ obtained from $D$ by fixing the entries $J$ at the $b_i$; $E$ is bounded, $1\le|F|\le n-1$, and (H2$_{U'}$) is inherited. For (H1$_{U'}$), let
$c_j\in C^\infty_c(U';\C^3)$, $j\in F$, have pairwise disjoint supports, fix $j_0\in F$ with $c_{j_0}\ne0$, $T:=\supp c_{j_0}\subset U'$, and let $G$ be
the $(n-1)$-linear form on $H(U\setminus T)$ obtained from $D$ by fixing entry $j_0$ at $c_{j_0}$. $G$ is bounded, (H2$_{U\setminus T}$) is inherited,
and (H1$_{U\setminus T}$) holds because its $n-1$ arguments together with $c_{j_0}$ are $n$ elements of $C^\infty_c(U;\C^3)$ with pairwise disjoint
supports (H1$_U$). By the theorem for $n-1$, $G=0$. The $c_j$, $j\in F\setminus\{j_0\}$, and the $b_i$, $i\in J$, vanish on $T$ ($T$ is disjoint from
$K$ and from the other supports), so $E((c_j)_{j\in F})=G(\dots)=0$. By the theorem for $|F|$ on $U'$, $E=0$ on $H(U')^{|F|}$, and
$(b_j)_{j\in F}\in H(U')^{|F|}$.

\emph{Step 2 (one confined entry).} For $i\ne j$: $D(b_1,\dots,b_n)=0$ whenever $b_i$ vanishes outside a compact $S\subset U$ and $b_j$ vanishes on
$S$, the other entries arbitrary. Expand $b_l=1_Sb_l+1_{U\setminus S}b_l$ for $l\notin\{i,j\}$: each of the $2^{n-2}$ terms is of the form of Step~1
with $K=S$, $J=\{i\}\cup\{l:1_Sb_l\text{ chosen}\}$, $j\notin J$. The statement extends to Borel $S\subset U$: choose compacts $K_m\uparrow$ in $S$ with
$|S\setminus\bigcup_mK_m|=0$; then $1_{K_m}b_i\to b_i$ in $L^2$ and $D$ is bounded.

\emph{Step 3 (the diagonal).} Fix $b_3,\dots,b_n\in H(U)$. $\beta(b_1,b_2):=D(b_1,\dots,b_n)$ is bounded with constant $C\prod_{i\ge3}\|b_i\|$ and
vanishes for $b_1,b_2\in C^\infty_c(U)$ with disjoint supports (Step~2 with $(i,j)=(1,2)$, $S=\supp b_1$). By Lemma~\ref{flh:lem:I8mult},
$\beta(b_1,b_2)=\int_Ub_1^{\mathsf T}M[b_3,\dots,b_n]\,b_2$ with $M[\cdot]$ multilinear (uniqueness) and $\|M[b_3,\dots,b_n]\|_{L^\infty}\le
\sqrt3C\prod_{i\ge3}\|b_i\|$. If $b_3$ vanishes outside a Borel $E\subset U$, Step~2 with $(i,j)=(3,1)$ gives $\beta(b_1,b_2)=0$ for all
$b_1\in H(U\setminus E)$, $b_2\in H(U)$, hence $M[b_3,\dots]=0$ on $U\setminus E$. Given $\eta>0$ choose $\delta>0$ with $\int_E|b_3|^2\le\eta^2$
whenever $|E|\le\delta^3$ (absolute continuity of the integral) and let $(E_\alpha)$ be the countable partition of $U$ by half-open cubes of side
$\delta$. Then $b_3=\sum_\alpha1_{E_\alpha}b_3$ in $L^2$, so $M[b_3,\dots]=\sum_\alpha M[1_{E_\alpha}b_3,\dots]$ in $L^\infty$; the summands vanish outside
the disjoint sets $E_\alpha$ and each has $L^\infty$ norm at most $\sqrt3C\eta\prod_{i\ge4}\|b_i\|$, so $\|M[b_3,\dots,b_n]\|_{L^\infty}\le\sqrt3C\eta
\prod_{i\ge4}\|b_i\|$. Letting $\eta\downarrow0$, $M[b_3,\dots,b_n]=0$, i.e.\ $D(b_1,\dots,b_n)=0$.
\end{proof}

\begin{remark}
For $n\ge3$ the hypothesis (H1) alone does not confine $D$ to the full diagonal: $D(b_1,b_2,b_3)=\langle b_1,b_2\rangle\ell(b_3)$, $\ell$ a bounded linear functional on $H(\R^3)$,
is bounded, vanishes on pairwise disjoint supports and is carried by $\{x_1=x_2\}$; (H2) excludes it, and is needed in every entry ($n=1$ is the
case $D(b)=\int b\cdot w$, excluded by the $n=1$ case of (H1)). (H2) enters the induction only through the bilinear case, inside Step~1.
\end{remark}

\emph{The Hodge map and the transport.} For $f\in L^2(\R^3;\Lt_\C)$ put $\flat f:=(f_{23},f_{31},f_{12})$, so that $f_x(u,v)=\langle\flat f(x),u\times v\rangle$
(bilinear pairing), $\|\flat f\|_{L^2}^2=\sum_{\mu<\nu}\|f_{\mu\nu}\|_{L^2}^2$ (the norm of Lemma~\ref{flh:lem:I5}) and $\flat(*du)=\nabla u$. With the
transport $(R\cdot f)_x(u,v)=f_{R^{-1}x}(R^{-1}u,R^{-1}v)$ of Definition~\ref{rot:def:sample} and $(R^{-1}u)\times(R^{-1}v)=\det R\;R^{-1}(u\times v)$,
\[
\flat(R\cdot f)(x)=\det R\;R\,\flat f(R^{-1}x)=:(\rho(R)\flat f)(x),\qquad\rho(R)\nabla u=\det R\,\nabla(u\circ R^{-1}),
\]
$\rho(R)$ an isometry of $H(\R^3)$ preserving $C^\infty_c$; thus $R\cdot(*du)=\det R\,{*d}(u\circ R^{-1})$, and $R\cdot f=(R^{-1})^*f$ is the pull-back by
$R^{-1}$ with the tensor action (notation of Section~\ref{flh:ssec:I7}).

\begin{proposition}[the coincident part vanishes]\label{flh:prop:DR}
Let $S_n$ be the curvature Schwinger functions of a limit $\Sbox$ of Theorem~\ref{thm:Hlim} for the box family (Lemma~\ref{flh:lem:I3}), with any
bounded Borel $g_{\rm d}$, extended to $L^2(\R^3;\Lt_\C)$ (Lemma~\ref{flh:lem:I5}). For all $n\ge1$, $R\in\Oth$ and $f_1,\dots,f_n\in L^2(\R^3;\Lt_\C)$:
$S_n(R\cdot f_1,\dots,R\cdot f_n)=S_n(f_1,\dots,f_n)$. Consequently $\Zbox(R\cdot f)=\Zbox(f)$ for every real $C^\infty_c$ $2$-form $f$.
\end{proposition}
\begin{proof}
Put
\[
D_R(b_1,\dots,b_n):=S_n(\flat^{-1}\rho(R)b_1,\dots,\flat^{-1}\rho(R)b_n)-S_n(\flat^{-1}b_1,\dots,\flat^{-1}b_n)
\]
on $H(\R^3)$; since
$\flat^{-1}\rho(R)b=R\cdot\flat^{-1}b$, $D_R=S_n(R\cdot\,\cdot)-S_n(\cdot)$ read on vector fields. $D_R$ is bounded by Lemma~\ref{flh:lem:I5}
($|S_n|\le e^n(n!)^{1/2}\prod\|f_i\|_{L^2}$, with the factor $2^{n/2}$ for complex entries) and the isometries.

\emph{Verification of \textup{(H1)}.} If $b_i\in C^\infty_c(\R^3;\C^3)$
have pairwise disjoint supports, the $f_i=\flat^{-1}b_i$ are Schwartz $2$-forms with pairwise disjoint supports and $R\cdot f=(R^{-1})^*f$. The
measure $\mu$ of Lemma~\ref{flh:lem:I5} satisfies (i) (with $C=\max(\frac12,\|g_{\rm d}\|_\infty)$, as $\|h\|_{L^1}\le\|\chi\|_{L^1}$ and
$\|f\|_{L^2}\le\|\chi\|_{L^2}$ for $\chi=(f,h)$), (ii) (Proposition~\ref{prop:osrest:OS1} for the box family on Schwartz pairs by continuity, and the
uniqueness of a measure with given characteristic functional, Lemma~\ref{rot:rem:minlos}, the hypothesis-free form of the argument of
Proposition~\ref{rot:cor:E3}(b)) and (iii) (Lemma~\ref{flh:lem:I4}(iii) on the open
side; the push-forward by a reflection is its pull-back), and its moments on $(f,0)$ are the $S_n$ (Lemma~\ref{flh:lem:I5}). So
Theorem~\ref{flh:thm:I7} with $R^{-1}$ gives $D_R=0$ on such $b_i$ (real $f_i$ first, complex by multilinearity, the real and imaginary parts
keeping the supports).

\emph{Verification of \textup{(H2)}.} For $b_i=\nabla u$, $\flat^{-1}b_i={*du}$ and $\flat^{-1}\rho(R)b_i=\det R\,{*d}(u\circ R^{-1})$. By
Theorem~\ref{flh:thm:I6}, $\Zbox(s\,{*dw})=1$ for all real $s$ and real $w\in C^\infty_c$; for the measure $\mu$ of
Theorem~\ref{thm:osrest:measure} (box family by Lemma~\ref{flh:lem:I3}), $\int e^{is\Phi({*dw})}d\mu=\Zbox(s\,{*dw})=1$, so $\Phi({*dw})=0$ $\mu$-a.s.;
the moments of $\mu$ on real $C^\infty_c$ $2$-forms are the $S_n$ (Theorem~\ref{thm:osrest:measure}(a)), hence $S_n(g_1,\dots,{*dw},\dots,g_n)=
\int\Phi(g_1)\cdots\Phi({*dw})\cdots d\mu=0$ for real $C^\infty_c$ $g_j$, then for complex $g_j\in L^2$ (multilinearity, boundedness, density) and
complex $w$ (linearity). Both terms of $D_R$ vanish. Theorem~\ref{flh:thm:I8} with $U=\R^3$ gives $D_R=0$.

For the last claim,
$\zeta\mapsto\Zbox(\zeta f)=\Psi_{(f)}(i\zeta)$ is entire (Theorem~\ref{thm:osrest:OS0}(ii)); $\Psi_{(f,\dots,f)}(z_1,\dots,z_n)=\Psi_{(f)}(z_1+\dots+z_n)$
(at level $N$ the exponent is $(\sum_jz_j)\Phi_N(f)$, and the limits are those of Lemma~\ref{lem:osrest:entire}(b),(c); compare (d)), so $\Psi^{(n)}_{(f)}(0)=S_n(f,\dots,f)$ (Theorem~\ref{thm:osrest:OS0}(iii)) and
$\Zbox(f)=\sum_{n\ge0}\frac{i^n}{n!}S_n(f,\dots,f)$, the same for $R\cdot f$, with equal coefficients. (Equivalently: the image of $\mu$ under
$\Phi\mapsto\Phi(R^{-1}\cdot\,\cdot)$ has the same moments on $C^\infty_c$ as $\mu$, with the factorial bound of Theorem~\ref{thm:osrest:OS0}(iii), so it
equals $\mu$ by the uniqueness from moments in Theorem~\ref{thm:osrest:measure}(b), and the characteristic functionals agree.)
\end{proof}

\emph{The density component at coincident points.} In Lemma~\ref{flh:lem:DENS}, Corollary~\ref{flh:cor:DENS} and their proofs, $X:=\cS'(\R^3;\Lt\oplus\R)$ with its cylinder
$\sigma$-algebra, $\Phi(\sigma)$ is the evaluation at a real Schwartz pair $\sigma=(f,h)$ ($f$ a $2$-form, $h$ a function), and $A(f):=\Phi(f,0)$, $D(h):=\Phi(0,h)$, so that
$\Phi(f,h)=A(f)+D(h)$. The symbols $A$, $D$, $M$, $M_R$, $\Delta_R$, $E$, $B$, $B_0$, $C_m$ are local to these statements and proofs (the generic form called $D$ in the notation of
this section does not occur below). The transport $R\cdot f$, $R\cdot h$, $R\cdot\sigma$ of Definition~\ref{rot:def:sample} is given by the same formulas on $L^2$ $2$-forms and on $L^1$
functions: $\|R\cdot f\|_{L^2}=\|f\|_{L^2}$ ($\rho(R)$ is an isometry), $\|R\cdot h\|_{L^1}=\|h\|_{L^1}$ ($|\det R|=1$); it preserves the real Schwartz pairs and $C^\infty_c$, and
$\supp(R\cdot\sigma)=R\supp\sigma$. Here $\|f\|_{L^2}^2:=\sum_{\mu<\nu}\|f_{\mu\nu}\|_{L^2}^2$, as in Lemma~\ref{flh:lem:I5}.

\begin{lemma}[an $L^1$-bounded entry has no coincident-point part]\label{flh:lem:DENS}
Let $c\ge0$ and let $\mu$ be a probability measure on $X$ with:
\begin{itemize}
\item[(P1)] $\int e^{\Phi(f,h)}\,d\mu\le\exp\big(\tfrac12\|f\|_{L^2}^2+c\,\|h\|_{L^1}\big)$ for every real Schwartz pair $(f,h)$;
\item[(P2)] for all $m,k\ge0$ with $m+k\ge1$, every $R\in\Oth$, and all real $C^\infty_c$ $2$-forms $f_1,\dots,f_m$ and real $C^\infty_c$ functions $h_1,\dots,h_k$ whose
$m+k$ supports are pairwise disjoint: $\int\prod_iA(R\cdot f_i)\prod_jD(R\cdot h_j)\,d\mu=\int\prod_iA(f_i)\prod_jD(h_j)\,d\mu$;
\item[(P3)] $\int e^{isA(*du)}\,d\mu=1$ for every real $s$ and every real $u\in C^\infty_c(\R^3)$.
\end{itemize}
Then $\int e^{i\Phi(R\cdot\sigma)}\,d\mu=\int e^{i\Phi(\sigma)}\,d\mu$ for every $R\in\Oth$ and every real Schwartz pair $\sigma$. Moreover every mixed moment
$M(f_1,\dots,f_m;h_1,\dots,h_k):=\int\prod_iA(f_i)\prod_jD(h_j)\,d\mu$ extends to a bounded multilinear form on $L^2(\R^3;\Lt_\C)^m\times L^1(\R^3;\C)^k$, and
$M(R\cdot f_1,\dots,R\cdot f_m;R\cdot h_1,\dots,R\cdot h_k)=M(f_1,\dots,f_m;h_1,\dots,h_k)$ for all such entries, with no condition on the supports.
\end{lemma}

\begin{corollary}[bounded functions of the density, coincident points included]\label{flh:cor:DENS}
Let $g_{\rm d}$ be any bounded Borel function (the joint scheme of Definition~\ref{def:pin}), with the couplings as in Theorem~\ref{flh:thm:FLH1}.
\begin{itemize}
\item[(a)] (The box family.) Every limit $\Sbox$ of Theorem~\ref{thm:Hlim} for the box family satisfies $\Sbox(e_{R\cdot\sigma})=\Sbox(e_\sigma)$ for every $R\in\Oth$ and every test
pair $\sigma=(f,h)$, that is $\Sbox\circ\gamma_R=\Sbox$ on $\cA$; the mixed Schwinger functions of the joint field extend to bounded multilinear forms, on $L^2(\R^3;\Lt_\C)$ in the
curvature entries and on $L^1(\R^3;\C)$ in the density entries, and the identity of Theorem~\ref{flh:thm:FLH1p} holds for all such entries, with no condition on the supports.
\item[(b)] (The torus family.) Let $S$ be a limit of Theorem~\ref{thm:Hlim} built on cubic-torus limit points $\om_N$ whose sides run along dyadic sequences
(Definition~\ref{n1:def:sides}), and $Z(\sigma)=S(e_\sigma)$. Then $Z$, extended to the real Schwartz pairs as in Lemma~\ref{flh:lem:I5}, is the characteristic functional of a
probability measure $\mu$ on $\cS'(\R^3;\Lt\oplus\R)$ with its cylinder $\sigma$-algebra, and for every real Schwartz pair $(f,h)$
\[
\int e^{\Phi(f,h)}\,d\mu\ \le\ \exp\Big(\tfrac12\|f\|_{L^2}^2+\|g_{\rm d}\|_\infty\|h\|_{L^1}\Big)
\]
(the display of Lemma~\ref{flh:lem:I5}, for the torus family). The statements of (a), with $S$ in place of $\Sbox$, hold for every limit $S$ on torus limit points that satisfies the covariance identity of Theorem~\ref{flh:thm:FLH1p} on pairwise disjoint supports; for the limits $S$ of the first sentence this identity is proved, for $\kappa_N>0$, in the proof of Theorem~\ref{flh:thm:FLH2} (Section~\ref{flh:ssec:proof}), before the present statement is used there.
\end{itemize}
In both cases Proposition~\ref{rot:cor:E3} then gives the invariance of the limit functional under $E(3)$ on all test pairs, and the invariance under $E(3)$ of the joint measure
(the measure of Lemma~\ref{flh:lem:I5} in (a), the measure $\mu$ in (b)).
\end{corollary}
\begin{proof}[Proof of Corollary~\ref{flh:cor:DENS}, given Lemma~\ref{flh:lem:DENS}]
(a) Let $\mu$ be the measure of Lemma~\ref{flh:lem:I5}: $\int e^{i\Phi(\sigma)}d\mu=\Zbox(\sigma)$ for all real Schwartz pairs, and $\Zbox(\sigma)=\Sbox(e_\sigma)$ for test pairs
(Theorem~\ref{thm:Hlim}(a)). (P1) with $c=\|g_{\rm d}\|_\infty$ is the display of Lemma~\ref{flh:lem:I5}.
(P2) is the identity of Theorem~\ref{flh:thm:FLH1p} on pairwise disjoint supports (proved there from Theorem~\ref{flh:thm:I7}, independently of its sentence on the coincident points), for the pairs $\sigma_i=(f_i,0)$,
$\sigma_{m+j}=(0,h_j)$: these are real $C^\infty_c$ pairs with pairwise disjoint supports, $R\cdot\sigma_i=(R\cdot f_i,0)$, $R\cdot\sigma_{m+j}=(0,R\cdot h_j)$, and the mixed Schwinger
functions of the joint field are the moments $\int\prod_i\Phi(\sigma_i)\,d\mu$ of this measure (proof of Theorem~\ref{flh:thm:FLH1p}). (P3) is Theorem~\ref{flh:thm:I6}:
$\int e^{isA(*du)}d\mu=\Zbox(s\,{*du},0)=1$. By Lemma~\ref{flh:lem:DENS}, for a test pair $\sigma$ (then $R\cdot\sigma$ is a test pair),
$\Sbox(e_{R\cdot\sigma})=\int e^{i\Phi(R\cdot\sigma)}d\mu=\int e^{i\Phi(\sigma)}d\mu=\Sbox(e_\sigma)$, and by linearity on the basis $(e_\sigma)$ of $\cA$, $\Sbox\circ\gamma_R=\Sbox$.
The statement on the mixed Schwinger functions is the last sentence of the lemma. Proposition~\ref{rot:cor:E3} applies, its hypothesis being $\Sbox\circ\gamma_R=\Sbox$ for all $R\in\Oth$.

(b) The proof of Lemma~\ref{flh:lem:I5} applies to the torus family with Theorem~\ref{thm:d2:LP}(b) in place of Theorem~\ref{thm:d2:box}(b): the two theorems state the same
Gaussian bound, and the continuity on pairs (Theorem~\ref{thm:osrest:OS0}(i)) and the convergence (Theorem~\ref{thm:Hlim}(a)) are stated for the torus family. This gives the
measure $\mu$, with $\int e^{i\Phi(\sigma)}d\mu=Z(\sigma)$ on the real Schwartz pairs (Bochner--Minlos), and the display; so (P1) holds with $c=\|g_{\rm d}\|_\infty$. (P3) is
Theorem~\ref{flh:thm:I6} for $\cZ(f)=Z(f,0)$: $\int e^{isA(*du)}d\mu=Z(s\,{*du},0)=1$. Now let $S$ be a limit on a torus limit point that satisfies the covariance identity of Theorem~\ref{flh:thm:FLH1p} on pairwise disjoint supports. The mixed Schwinger
functions in the conclusion of Theorem~\ref{flh:thm:FLH1p} for $S$ are the moments of this same measure $\mu$ (their definition in the proof of Theorem~\ref{flh:thm:FLH1p}, carried
over by the proof of Theorem~\ref{flh:thm:FLH2}); so (P2) holds, and (P1), (P2), (P3) are properties of one measure. Lemma~\ref{flh:lem:DENS} applies, and the rest is as in (a).
\end{proof}
\begin{proof}[Proof of Lemma~\ref{flh:lem:DENS}]
Throughout, $\mu$ satisfies (P1)--(P3), ``a.s.''\ means $\mu$-almost surely, and $C_m:=2e^{1/2}m!$ for $m\ge1$, $C_0:=1$.

\emph{Step 1: the joint $L^2\times L^1$ bound.} (1a) \emph{For every real Schwartz $h$, $|D(h)|\le c\|h\|_{L^1}$ a.s.} By (P1) for the pair $(0,sh)$, $s$ real,
$\int e^{sD(h)}d\mu\le e^{c|s|\|h\|_{L^1}}$. For $\eta>0$ and $s>0$, by the Chebyshev inequality,
$\mu\big(D(h)\ge c\|h\|_{L^1}+\eta\big)\le e^{-s(c\|h\|_{L^1}+\eta)}\int e^{sD(h)}d\mu\le e^{-s\eta}$, which tends to $0$ as $s\to\infty$. So $D(h)<c\|h\|_{L^1}+\eta$ a.s.\ for
every $\eta=1/n$, hence $D(h)\le c\|h\|_{L^1}$ a.s.; the same for $-h$, $D$ being linear.

(1b) \emph{For every real Schwartz $2$-form $f$ and every integer $p\ge1$, $\int|A(f)|^pd\mu\le C_p\|f\|_{L^2}^p$.} If $f=0$ then $A(f)=0$. Otherwise, for $t>0$ and real $x$,
$|x|^p\le p!\,t^{-p}e^{t|x|}\le p!\,t^{-p}(e^{tx}+e^{-tx})$, and (P1) for the pairs $(tf,0)$ and $(-tf,0)$ gives $\int e^{\pm tA(f)}d\mu\le e^{t^2\|f\|_{L^2}^2/2}$; so
$\int|A(f)|^pd\mu\le2\,p!\,t^{-p}e^{t^2\|f\|_{L^2}^2/2}$, and $t:=1/\|f\|_{L^2}$ gives $2e^{1/2}p!\,\|f\|_{L^2}^p$.

(1c) \emph{For real Schwartz $f_1,\dots,f_m$, $h_1,\dots,h_k$ the product $\prod_iA(f_i)\prod_jD(h_j)$ is $\mu$-integrable, and
$|M(f_1,\dots,f_m;h_1,\dots,h_k)|\le C_m\prod_i\|f_i\|_{L^2}\prod_j\big(c\|h_j\|_{L^1}\big)$.} By (1a), outside a null set (a finite union),
$|\prod_jD(h_j)|\le\prod_jc\|h_j\|_{L^1}$. By the H\"older inequality with $m$ exponents equal to $m$ and (1b) with $p=m$,
$\int\prod_i|A(f_i)|\,d\mu\le\prod_i\big(\int|A(f_i)|^md\mu\big)^{1/m}\le C_m\prod_i\|f_i\|_{L^2}$ (for $m=0$ there is no curvature factor, and the constant is $C_0=1$).

(1d) \emph{Extension.} $M$ is real-multilinear on real Schwartz entries, every $\Phi\in X$ being linear. Extend it $\C$-multilinearly to complex Schwartz entries
($A(f):=A(\Rp f)+iA(\Ip f)$, likewise $D$): expanding into $2^{m+k}$ real terms, $|M|\le B_0\prod_i\|f_i\|_{L^2}\prod_j\|h_j\|_{L^1}$ with $B_0:=2^{m+k}C_mc^k$. The complex
Schwartz functions are dense in $L^2(\R^3;\Lt_\C)$ and in $L^1(\R^3;\C)$, so $M$ extends uniquely to a $\C$-multilinear form on $L^2(\R^3;\Lt_\C)^m\times L^1(\R^3;\C)^k$ with the same
bound. This is the first half of the last sentence of the lemma.

(1e) $M_R(f;h):=M(R\cdot f_1,\dots,R\cdot f_m;R\cdot h_1,\dots,R\cdot h_k)$ is $\C$-multilinear with the same bound ($R\cdot$ is linear, isometric on $L^2$ and on $L^1$, and
commutes with real and imaginary parts), and on real Schwartz entries it is $\int\prod_iA(R\cdot f_i)\prod_jD(R\cdot h_j)\,d\mu$. Put $\Delta_R:=M_R-M$; then
$|\Delta_R(f;h)|\le B\prod_i\|f_i\|_{L^2}\prod_j\|h_j\|_{L^1}$ with $B:=2B_0=2^{m+k+1}C_mc^k$.

\emph{Step 2: closedness in every curvature entry.} \emph{Let $m\ge1$, $1\le i\le m$ and $w\in C^\infty_c(\R^3;\C)$. If $f_i={*dw}$, then $M$ and $M_R$, hence $\Delta_R$, vanish at
$(f_1,\dots,f_m;h_1,\dots,h_k)$, for all $f_l\in L^2(\R^3;\Lt_\C)$ ($l\ne i$), all $h_j\in L^1(\R^3;\C)$ and all $R\in\Oth$.} Real $w$ first. By (P3) the law of the real random
variable $A(*dw)$ has the characteristic function $s\mapsto\int e^{isA(*dw)}d\mu=1$, so it is the unit mass at $0$: $A(*dw)=0$ a.s. For real Schwartz $f_l$ and $h_j$ the integrand
of $M$ contains the factor $A(*dw)$, is integrable by (1c) and vanishes a.s.; so $M=0$ on real Schwartz other entries, then on complex Schwartz ones (multilinearity), then on all of
$L^2$ and $L^1$ (density and the bound of (1d)). For complex $w=w_1+iw_2$: ${*dw}={*dw_1}+i\,{*dw_2}$ and linearity in the entry $i$. For $M_R$:
$R\cdot({*dw})=\det R\,{*d}(w\circ R^{-1})$ and $w\circ R^{-1}\in C^\infty_c(\R^3;\C)$, so
\[
M_R(\dots,{*dw},\dots;\dots)=\det R\cdot M(R\cdot f_1,\dots,{*d}(w\circ R^{-1}),\dots,R\cdot f_m;R\cdot h_1,\dots,R\cdot h_k)=0
\]
by the first part.

\emph{Step 3: entries carried by pairwise disjoint Borel sets.} \emph{Let $E_1,\dots,E_m,G_1,\dots,G_k$ be pairwise disjoint Borel subsets of $\R^3$, $f_i\in L^2(\R^3;\Lt_\C)$ with
$f_i=0$ a.e.\ outside $E_i$, and $h_j\in L^1(\R^3;\C)$ with $h_j=0$ a.e.\ outside $G_j$. Then $\Delta_R(f_1,\dots,f_m;h_1,\dots,h_k)=0$ for every $R\in\Oth$.}
(1) \emph{Smooth entries.} For real $C^\infty_c$ entries with pairwise disjoint supports, $\Delta_R=0$ is (P2). For complex $C^\infty_c$ entries with pairwise disjoint supports,
expand in real and imaginary parts: each of them has its support inside the support of the entry, so every term vanishes.

(2) \emph{Compact carriers.} Suppose in addition that the $E_i$ and $G_j$ are compact. Pairwise disjoint compact sets have pairwise positive distances; let $d>0$ be the smallest
(if fewer than two of the sets are non-empty there is no such distance: then any $r>0$ serves below). Let $\eta_r$, $0<r<d/3$, be a standard mollifier supported in the ball of
radius $r$. Then $f_i*\eta_r$ (componentwise) and $h_j*\eta_r$ are $C^\infty_c$, supported in the closed $r$-neighbourhoods of $E_i$ and $G_j$, which are pairwise disjoint (distances
$\ge d-2r>0$), and $f_i*\eta_r\to f_i$ in $L^2$, $h_j*\eta_r\to h_j$ in $L^1$ as $r\to0$. By (1), $\Delta_R$ vanishes on the mollified entries, and $\Delta_R$ is continuous for
these norms (1e); so $\Delta_R(f;h)=0$.

(3) \emph{Borel carriers.} The finite Borel measures $|f_i|^2dx$ and $|h_j|\,dx$ on $\R^3$ are inner regular: for every $\epsilon>0$ there are compact $K_i\subset E_i$ and
$K'_j\subset G_j$ with $\|f_i-f_i1_{K_i}\|_{L^2}<\epsilon$ and $\|h_j-h_j1_{K'_j}\|_{L^1}<\epsilon$. The entries $f_i1_{K_i}$, $h_j1_{K'_j}$ are carried by the pairwise disjoint
compact sets $K_i$, $K'_j$, so $\Delta_R$ vanishes on them by (2); letting $\epsilon\to0$ and using (1e) again, $\Delta_R(f;h)=0$. This step requires neither a positive distance between the carriers
nor a smooth partition of unity; it rests on the joint bound alone.

\emph{Step 4: the curvature entries, the density entries being spectators.} \emph{Let $K\subset\R^3$ be compact, $G_1,\dots,G_k$ pairwise disjoint Borel subsets of $K$, and
$h_j\in L^1(\R^3;\C)$ with $h_j=0$ a.e.\ outside $G_j$ ($k\ge0$; for $k=0$ any compact $K$, the empty set included). Then $\Delta_R(f_1,\dots,f_m;h_1,\dots,h_k)=0$ for every
$R\in\Oth$ and all $f_1,\dots,f_m\in L^2(\R^3;\Lt_\C)$ that vanish a.e.\ on $K$.} The case $m=0$ is Step~3. Let $m\ge1$, fix $R$ and the $h_j$, and put $U:=\R^3\setminus K$, an
open set. With the Hodge map $\flat$ (an isometry of $L^2(\R^3;\Lt_\C)$ onto $L^2(\R^3;\C^3)$, $\flat({*du})=\nabla u$) define on $H(U)$, its elements extended by $0$ to $\R^3$,
\[
E(b_1,\dots,b_m):=\Delta_R\big(\flat^{-1}b_1,\dots,\flat^{-1}b_m;h_1,\dots,h_k\big),
\]
the difference of $M\big(R\cdot\flat^{-1}b_1,\dots,R\cdot\flat^{-1}b_m;R\cdot h_1,\dots,R\cdot h_k\big)$ and $M\big(\flat^{-1}b_1,\dots,\flat^{-1}b_m;h_1,\dots,h_k\big)$.
$E$ is an $m$-linear form on $H(U)$, $\C$-linear in each entry, and bounded: $|E(b_1,\dots,b_m)|\le\big(B\prod_j\|h_j\|_{L^1}\big)\prod_i\|b_i\|_{L^2}$ by (1e).

\emph{Verification of \textup{(H1$_U$)} for $E$.} Let $b_1,\dots,b_m\in C^\infty_c(U;\C^3)$ have pairwise disjoint supports. The sets $\supp b_i$ are compact subsets of $U$, so they are disjoint from $K$, hence
from every $G_j$; the $m+k$ sets $\supp b_1,\dots,\supp b_m,G_1,\dots,G_k$ are pairwise disjoint Borel sets carrying the entries $\flat^{-1}b_i$ and $h_j$, and Step~3 gives
$E(b_1,\dots,b_m)=0$ (for $m=1$: $E(b_1)=0$ for all $b_1\in C^\infty_c(U;\C^3)$).

\emph{Verification of \textup{(H2$_U$)} for $E$.} Let one entry be $b_i=\nabla u$, $u\in C^\infty_c(U;\C)$, the others arbitrary in $H(U)$. Then $\flat^{-1}b_i={*du}$ with $u\in C^\infty_c(\R^3;\C)$, and Step~2
gives $\Delta_R(\dots,{*du},\dots;h_1,\dots,h_k)=0$ (both terms vanish separately).

By Theorem~\ref{flh:thm:I8} with $n=m$ and this $U$, $E=0$ on $H(U)^m$. If $f_i$ vanishes a.e.\ on $K$, then $b_i:=(\flat f_i)|_U$ is in $H(U)$, its extension by $0$ is
$\flat f_i$, and $\Delta_R(f;h)=E(b_1,\dots,b_m)=0$. (Theorem~\ref{flh:thm:I8} is applied once, on $U=\R^3\setminus K$, to the difference $E$ of two forms of the unrotated
arguments; (H2$_U$) holds for each of the two terms, (H1$_U$) for the difference only. The density entries are fixed inside the definition of $E$.)

\emph{Step 5: no mass on coincidences of an $L^1$-bounded entry.} \emph{$\Delta_R(f_1,\dots,f_m;h_1,\dots,h_k)=0$ for every $R\in\Oth$, all $m,k\ge0$ with $m+k\ge1$, all
$f_i\in L^2(\R^3;\Lt_\C)$ and all $h_j\in L^1(\R^3;\C)$} (the second half of the last sentence of the lemma). For $h\in L^1$ and $f\in L^2$ put
$\omega^1_h(t):=\sup\{\|h1_E\|_{L^1}:\ E\text{ Borel},\ |E|\le t\}$ and $\omega^2_f(t):=\sup\{\|f1_E\|_{L^2}:\ E\text{ Borel},\ |E|\le t\}$; both tend to $0$ as $t\to0$ (absolute
continuity of the integrals of $|h|$ and of $|f|^2$). If $k=0$, the claim is Step~4 with $K=\emptyset$. Let $k\ge1$. Fix $\delta>0$ and let $\mathcal Q_\delta$ be the partition
of $\R^3$ into the half-open cubes $\delta(z+[0,1)^3)$, $z\in\Z^3$. For $h\in L^1$, $h=\sum_Qh1_Q$ with $\sum_Q\|h1_Q\|_{L^1}=\|h\|_{L^1}$, so the series converges in $L^1$.
$\Delta_R$ is linear and continuous in each density entry (1e); expanding the entries one after the other,
\begin{equation}\label{flh:eq:DENScubes}
\Delta_R(f;h_1,\dots,h_k)=\sum_{(Q_1,\dots,Q_k)\in\mathcal Q_\delta^k}\Delta_R\big(f;h_11_{Q_1},\dots,h_k1_{Q_k}\big),
\end{equation}
and the series converges absolutely (so in any order): the sum of the moduli of its terms is at most
\[
B\prod_i\|f_i\|_{L^2}\sum_{(Q_1,\dots,Q_k)}\prod_j\|h_j1_{Q_j}\|_{L^1}=B\prod_i\|f_i\|_{L^2}\prod_j\|h_j\|_{L^1}.
\]

(a) \emph{Tuples with a repeated cube} (only if $k\ge2$). For a pair $i<j$ the tuples with $Q_i=Q_j$ contribute at most
\[
B\prod_l\|f_l\|_{L^2}\Big(\sum_Q\|h_i1_Q\|_{L^1}\|h_j1_Q\|_{L^1}\Big)\prod_{l\notin\{i,j\}}\|h_l\|_{L^1}\ \le\ B\prod_l\|f_l\|_{L^2}\;\omega^1_{h_i}(\delta^3)\prod_{l\ne i}\|h_l\|_{L^1},
\]
since $\|h_i1_Q\|_{L^1}\le\omega^1_{h_i}(\delta^3)$ for every cube and $\sum_Q\|h_j1_Q\|_{L^1}=\|h_j\|_{L^1}$. Hence the sum over the tuples with a repeated cube is at most
$a(\delta):=B\prod_l\|f_l\|_{L^2}\sum_{i<j}\omega^1_{h_i}(\delta^3)\prod_{l\ne i}\|h_l\|_{L^1}$ in modulus.

(b) \emph{Tuples of pairwise distinct cubes.} Fix such a tuple and let $K$ be the union of the closures of $Q_1,\dots,Q_k$: $K$ is compact, $|K|=k\delta^3$, and the $Q_j$ are
pairwise disjoint Borel subsets of $K$. Write $f_i=f_i1_K+f_i1_{\R^3\setminus K}$ in every curvature entry and expand the term of \eqref{flh:eq:DENScubes} into $2^m$ terms. The
term in which every entry carries $f_i1_{\R^3\setminus K}$ vanishes by Step~4 (with $G_j:=Q_j$). Each of the other $2^m-1$ terms carries $f_i1_K$ in at least one entry $i$ and
entries of norm $\le\|f_l\|_{L^2}$ in the others, so by (1e) it is at most $B\,\omega^2_{f_i}(k\delta^3)\prod_{l\ne i}\|f_l\|_{L^2}\prod_j\|h_j1_{Q_j}\|_{L^1}$ in modulus. Hence the
term of \eqref{flh:eq:DENScubes} is at most $(2^m-1)\,B\,W(\delta)\prod_j\|h_j1_{Q_j}\|_{L^1}$ in modulus, where
$W(\delta):=\max_i\omega^2_{f_i}(k\delta^3)\prod_{l\ne i}\|f_l\|_{L^2}$ does not depend on the tuple. Summing over the tuples of distinct cubes, the sum is at most
$b(\delta):=(2^m-1)\,B\,W(\delta)\prod_j\|h_j\|_{L^1}$ in modulus. (For $m=0$ every such term vanishes by Step~3, and $b=0$.)

So $|\Delta_R(f;h)|\le a(\delta)+b(\delta)$ for every $\delta>0$; the left side does not depend on $\delta$, and $a(\delta)+b(\delta)\to0$ as $\delta\to0$. Hence $\Delta_R(f;h)=0$.

\emph{Step 6: from the moments to the characteristic functional.} Let $\sigma=(f,h)$ be a real Schwartz pair and $R\in\Oth$; $R\cdot\sigma=(R\cdot f,R\cdot h)$ is a real Schwartz
pair. By (P1) for $\sigma$ and for $-\sigma$,
\[
\int e^{|\Phi(\sigma)|}d\mu\le\int\big(e^{\Phi(\sigma)}+e^{-\Phi(\sigma)}\big)d\mu\le2\exp\big(\tfrac12\|f\|_{L^2}^2+c\|h\|_{L^1}\big)<\infty,
\]
and the same holds for $R\cdot\sigma$, with the same right side. For real $x$, $\big|\sum_{n\le N}(ix)^n/n!\big|\le e^{|x|}$; so by dominated convergence
$\int e^{i\Phi(\sigma)}d\mu=\sum_{n\ge0}\frac{i^n}{n!}\int\Phi(\sigma)^nd\mu$, an absolutely convergent series ($\sum_n\frac1{n!}\int|\Phi(\sigma)|^nd\mu=\int e^{|\Phi(\sigma)|}d\mu$),
and the same for $R\cdot\sigma$. Since $\Phi(\sigma)=A(f)+D(h)$,
$\int\Phi(\sigma)^nd\mu=\sum_{j=0}^n\binom njM(f,\dots,f;h,\dots,h)$ with $j$ entries $f$ and $n-j$ entries $h$, every term being finite by (1c); likewise
$\int\Phi(R\cdot\sigma)^nd\mu=\sum_{j=0}^n\binom njM_R(f,\dots,f;h,\dots,h)$. By Step~5 ($\Delta_R=0$, with the entries $f$ and $h$ repeated: coincident supports in every entry) the
two sums are equal for every $n\ge1$; for $n=0$ both are $1$. Hence $\int e^{i\Phi(R\cdot\sigma)}d\mu=\int e^{i\Phi(\sigma)}d\mu$. This argument uses no uniqueness theorem for a moment problem:
the two characteristic functionals are compared coefficient by coefficient under one measure, the series converging by the exponential bound (P1).
\end{proof}

\subsection{Proofs of the theorems}\label{flh:ssec:proof}

\begin{proof}[Proof of Theorem~\ref{flh:thm:FLH1}]
By Lemma~\ref{flh:lem:I3} the box family satisfies the hypotheses of Theorem~\ref{thm:Hlim} and of every statement of Section~\ref{sec:osrest} for
the couplings of the theorem and any bounded Borel $g_{\rm d}$, so $\Sbox$, its Schwinger functions $S_n$ and its measure $\mub$ exist, and
Lemmas~\ref{flh:lem:I4}, \ref{flh:lem:I5} and Theorem~\ref{flh:thm:I6} hold for them. Proposition~\ref{flh:prop:DR}, whose inputs are these three
together with Theorem~\ref{flh:thm:I7} and Theorem~\ref{flh:thm:I8}, gives $\Zbox(R\cdot f)=\Zbox(f)$ for every $R\in\Oth$ and every real $C^\infty_c$
$2$-form $f$. With the choice $g_{\rm d}\equiv0$, $\Phi_N(\sigma)$ depends on $f$ alone (Remark~\ref{rot:rem:gd0}), so $\Zbox(f,h)=\Zbox(f,0)$ for every
test pair, and $Z^{\mathrm{box}}(R\cdot\sigma)=Z^{\mathrm{box}}(\sigma)$ for every test pair $\sigma$; by linearity on the basis $(e_\sigma)$ of $\cA$,
$\Sbox\circ\gamma_R=\Sbox$ for all $R\in\Oth$.

Proposition~\ref{rot:cor:E3}, whose hypothesis is the invariance of $S$ under all translations
(Theorem~\ref{thm:Hlim}, here for the box family) and under $\Oth$ (just proved), applies to $\Sbox$. Its part (a) gives $\Sbox(e_{\gamma\cdot\sigma})=\Sbox(e_\sigma)$ for all $\gamma\in E(3)$,
and its part (b) gives $\gamma_*\mub=\mub$ for every $\gamma\in E(3)$ and every measure representing $\Zbox$ on a set of linear functionals with the cylinder
$\sigma$-algebra. This holds in particular for the measure of Theorem~\ref{thm:osrest:measure}, either through Proposition~\ref{ms:discharge}(a) or directly
through (b) applied to $\mu$ on $X$, whose Borel $\sigma$-algebra is generated by the evaluations (Lemma~\ref{ms:borel}(c)), and Lemma~\ref{ms:transfer}(b); it holds also for the Minlos measure of Lemma~\ref{rot:rem:minlos}.

Remark~\ref{rem:osrest:rot}
now applies to $\Sbox$: $\mub$ is reflection positive across every affine plane, on $L^2$ of either open side (the half-space algebra and the
reflection of a tilted plane are the rotated ones, and $\mub$ is $E(3)$-invariant), and the reconstruction in any direction is unitarily the one in
direction $e_3$. OS0 is Theorem~\ref{thm:osrest:OS0}, OS1 is Proposition~\ref{prop:osrest:OS1} together with the $\Oth$-invariance just proved, OS2 is
Proposition~\ref{prop:osrest:OS2} and Theorem~\ref{thm:osrest:OS2matrix} together with the reflection positivity across every plane, OS3 is
Proposition~\ref{prop:osrest:OS3}, and the measure is Theorem~\ref{thm:osrest:measure}, all for the box family by Lemma~\ref{flh:lem:I3}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{flh:thm:FLH1p}]
Let $g_{\rm d}$ be bounded Borel. \emph{The curvature marginal.} Lemmas~\ref{flh:lem:I3}, \ref{flh:lem:I4}, \ref{flh:lem:I5}, Theorem~\ref{flh:thm:I6}
and Proposition~\ref{flh:prop:DR} hold for any bounded Borel $g_{\rm d}$ (the density component of $(f,0)$ and of $(s\,{*du},0)$ is zero), so
$\Zbox(R\cdot f)=\Zbox(f)$ for all $R\in\Oth$ and all real $C^\infty_c$ $2$-forms $f$, i.e.\ the curvature functional $f\mapsto\Zbox(f,0)$ and its measure
(the curvature marginal of $\mu$ of Lemma~\ref{flh:lem:I5}, which is the measure of Theorem~\ref{thm:osrest:measure}) have all the properties of
Theorem~\ref{flh:thm:FLH1}, by the proof just given.

\emph{The joint field.} Let $\mu$ be the measure of Lemma~\ref{flh:lem:I5} on
$\cS'(\R^3;\Lt\oplus\R)$, with $\int e^{i\Phi(\sigma)}d\mu=\Zbox(\sigma)$ for real Schwartz pairs. It satisfies (i) of Section~\ref{flh:ssec:I7} with
$C=\max(\frac12,\|g_{\rm d}\|_\infty)$ (Lemma~\ref{flh:lem:I5}: $\|h\|_{L^1}\le\|\chi\|_{L^1}$ and $\|f\|_{L^2}\le\|\chi\|_{L^2}$ for $\chi=(f,h)$); (ii)
by Proposition~\ref{prop:osrest:OS1} for the box family (translation invariance of $\Zbox$ on test pairs, extended to Schwartz pairs by the
continuity of Theorem~\ref{thm:osrest:OS0}(i) in its pair version) and the uniqueness of a probability measure on $\cS'$ with a given
characteristic functional (Lemma~\ref{rot:rem:minlos}, the hypothesis-free form of the argument of Proposition~\ref{rot:cor:E3}(b)); (iii) by
Lemma~\ref{flh:lem:I4}(iii) for the planes through $0$ and tests in the open side
(for a reflection the push-forward of a test pair is its pull-back with the tensor action, $\Theta_u=\theta_u^*$).

Its moments
$\int\prod_i\Phi(\sigma_i)d\mu$ are the mixed Schwinger functions $S_n(\sigma_1,\dots,\sigma_n)$ of the joint field: for real Schwartz pairs the law of
$(\Phi(\sigma_i))_i$ under $\mu$ has the characteristic function $t\mapsto\Zbox(\sum_it_i\sigma_i)$, which is the limit of the characteristic functions
of $(\Phi_{N_m}(\sigma_i))_i$ under $\mu^{\mathrm{box}}_{N_m}$ (Theorem~\ref{thm:Hlim}(a)), and the moments pass to the limit by the uniform exponential
bounds of Lemma~\ref{flh:lem:I5} at level $N$ (the density part is bounded by $\|g_{\rm d}\|_\infty E_N(h)$, the curvature part by
Theorem~\ref{thm:d2:box}(b)). The Schwinger functions of the joint field are defined as these moments, which for $h_i\equiv0$ are the $S_n$ of
Theorem~\ref{thm:osrest:OS0}(iii). Theorem~\ref{flh:thm:I7} with $T=\Lt\R^3\oplus\R$ and $R^{-1}$ ($R\cdot\sigma=(R^{-1})^*\sigma$) gives
$S_n(R\cdot\sigma_1,\dots,R\cdot\sigma_n)=S_n(\sigma_1,\dots,\sigma_n)$ for all $R\in\Oth$ whenever the supports of the $\sigma_i$ are pairwise disjoint.
The density entries carry no closedness identity, so Theorem~\ref{flh:thm:I8} does not apply to them; their coincident-point parts are removed by the $L^1$ bound of Lemma~\ref{flh:lem:I5} instead
(Lemma~\ref{flh:lem:DENS}, Corollary~\ref{flh:cor:DENS}(a)).
\end{proof}

\begin{proof}[Proof of Theorem~\ref{flh:thm:FLH2}]
We run the proofs of Theorems~\ref{flh:thm:FLH1} and~\ref{flh:thm:FLH1p} for the torus limit points $\om_N$ themselves, and go through
their ingredients in turn. Theorem~\ref{thm:Hlim} and Section~\ref{sec:osrest} hold for the torus limit points as stated.

Lemma~\ref{flh:lem:I4} needs of the level-$N$ states only the reflection positivity of
Theorem~\ref{thm:d2:LP}(d) at the lattice planes of the nine families through $0$: its part (i) uses it through Lemma~\ref{flh:lem:I2}(ii), and
its part (ii) uses it, in the $L^2$ form, through Lemma~\ref{flh:lem:I2}(iii). At the coordinate planes this reflection positivity is
Theorem~\ref{thm:d2:LP}(d) itself. At the six diagonal site planes through $0$ it is Corollary~\ref{n1:cor:N1} at $c=0$, the Hermitian form and the
$L^2$ closure included, and Theorem~\ref{n1:thm:N10} for the bounded Borel functions. Here the sides of
Definition~\ref{def:omN} run along a dyadic sequence (Definition~\ref{n1:def:sides}), as that corollary requires. Section~\ref{n1:sec} proves these statements for $\kappa_N=1$; for $\kappa_N>0$ they are Lemma~\ref{lem:rescale}(g). Explicitly, at each level $N$
the state $\om_N$ is $F\mapsto\om^{\star}_N(F\circ r)$, where $r$ multiplies the site variables by $\kappa_N^{-1/2}$ and $\om^{\star}_N$ is a limit point,
along the same dyadic sequence, of the torus states of the model with hopping coefficient $1$ and the coefficients \eqref{eq:rescale}
(Lemma~\ref{lem:rescale}(c)); the statements hold for $\om^{\star}_N$, and they pass to $\om_N$ because $r$ commutes with the lattice reflections and
preserves the functions of the cells of a half space and the $L^2$ norms (Lemma~\ref{lem:rescale}(d)). This is the only step of the proof in which the
value of $\kappa_N$ enters, and it is taken level by level: the limit functional $S$, the scheme and the density function $g_{\rm d}$ are those of the
given models, and every other step holds for every $\kappa_N\ge0$. For a diagonal reflection $R$ the
intertwining $\Phi_N(\sigma)\circ R=\Phi_N(\gamma\sigma)$ and the $R$-invariance of $\mu_N$ hold by Theorem~\ref{thm:d2:LP}(c), since $R\in\cG_N$.

Lemma~\ref{flh:lem:I5} uses only Theorem~\ref{thm:d2:LP}(b), in place of Theorem~\ref{thm:d2:box}(b). Theorem~\ref{flh:thm:I6} is proved for both
families. Theorems~\ref{flh:thm:I7} and~\ref{flh:thm:I8} are statements about the limit law and about multilinear forms, and
Proposition~\ref{flh:prop:DR} and the assembly of these ingredients in the two proofs use nothing else.

Finally, Corollary~\ref{flh:cor:DENS}(b) uses, besides Lemma~\ref{flh:lem:DENS}, only the display of Lemma~\ref{flh:lem:I5} in its torus form,
which is stated in that corollary, Theorem~\ref{flh:thm:I6}, Theorem~\ref{flh:thm:I8} and the covariance on pairwise disjoint supports, which the
paragraph on the joint field in the proof of Theorem~\ref{flh:thm:FLH1p} has just given for the torus limit points. Hence the conclusions of
Theorems~\ref{flh:thm:FLH1} and~\ref{flh:thm:FLH1p} hold for $S$.
\end{proof}



\providecommand{\C}{\mathbb C}\providecommand{\FD}{\mathrm{FD}}\providecommand{\cper}{c}\providecommand{\Tper}{\mathbb T}\providecommand{\Om}{\Omega}

\section{Rigidity of the Villain--Higgs Gibbs measures on $\eN\Z^3$}\label{rv:sec}

This section proves a rigidity theorem (Theorem~\ref{rv:thm:RV}): a Gibbs measure of the fine Villain--Higgs specification on $\eN\Z^3$
(the model of Section~\ref{sec:setting}, in unitary gauge) which is invariant under some finite-index sublattice of $\eN\Z^3$ and has
the three regularity properties (I1)--(I3) below is invariant under the translations by $2\eN e_j$, $j=1,2,3$. The main ingredient is
the statement that two consecutive layers determine the Gibbs measure (Theorem~\ref{rv:thm:DV}, Subsection~\ref{rv:ssec:S8}), proved on
$\eN\Z^3$. Two elementary steps, a conjugation of reflections by coordinate transpositions and a composition of two parallel
reflections, are proved in Subsection~\ref{rv:ssec:S9} (Proposition~\ref{rv:prop:YM4}); the relative-entropy estimate on which the argument
rests is the tangent inequality of Lemma~\ref{rv:lem:TV}.
Throughout, $N$ is fixed, $\eps:=\eN$, and all constants may depend on $N$, $\eps$, the couplings and the finite sets involved.

\subsection{Statement}\label{rv:ssec:S0}

\paragraph{The model on $\eN\Z^3$.} Sites $x\in\eps\Z^3$ carry the radial field $\tr_x\in[-\rhof,\infty)$, $\rho_x:=\rhof+\tr_x$; links $b$ carry
$B_b\in\FD:=(-\pi/\gph,\pi/\gph]$, $B_{-b}=-B_b$; plaquettes $p$ carry the Villain $2$-form $v_p\in(2\pi/\gph)\Z$, $v_{-p}=-v_p$, closed ($dv=0$ on every cube).
Here $\gph>0$ is the phase constant of Section~\ref{sec:setting}, a general coefficient (Remark~\ref{rem:letters}), and $\kappa_N=1$ is the value of the hopping coefficient for which this section and the next are written (Lemma~\ref{lem:rescale}(g) extends Theorem~\ref{rv:thm:RV} to every $\kappa_N>0$); the local weight is that of Section~\ref{sec:setting}:
\begin{equation}\label{rv:eq:W}
\begin{gathered}
W=\prod_x g_{\mathrm{ct}}(\tr_x)\prod_b h(\tr_{b_-},\tr_{b_+},B_b)\prod_p w\big((dB)_p+v_p\big),\\
w(f)=e^{-f^2/2},\qquad
h(r,r',\vartheta)=\exp\big(-\tfrac12\kappa_N|e^{i\gph\vartheta}(\rhof+r')-(\rhof+r)|^2\big),
\end{gathered}
\end{equation}
$h(r,r',\cdot)$ being $2\pi/\gph$-periodic and $g_{\mathrm{ct}}e^{q\rho^2}$ integrable on $[-\rhof,\infty)$ for every $q\ge0$ by~\eqref{eq:ct}. The curvature is
$f_p:=(dB)_p+v_p$; a plaquette is temporal if it contains a time link ($x_3$ is the time coordinate, $e_3$ the time direction), spatial otherwise. We write $\cper:=2\pi/\gph$ for the Villain
period and $\Tper:=\R/\cper\Z$.

For a finite $\Lambda\subset\eps\Z^3$ let $V(\Lambda)$ be the set of sites in $\Lambda$ together with the bonds and plaquettes meeting
$\Lambda$. The specification $\gamma_\Lambda(\cdot\mid\omega)$ is the normalised integral of~\eqref{rv:eq:W} over the variables carried by $V(\Lambda)$ (the $v_p$ summed over
the closed integer $2$-forms with the exterior $v$ fixed), the exterior configuration $\omega$ held fixed. The specification $\{\gamma_\Lambda\}$ is consistent on the cofinal family
$\{V(\Lambda)\}$, and a probability measure $\nu$ on the configurations $(\tr,B,v)$ of $\eps\Z^3$ is a \emph{Gibbs measure} if $\nu\gamma_\Lambda=\nu$ for every finite
$\Lambda$ (equivalently for the full specification, by consistency); this is the Dobrushin--Lanford--Ruelle (DLR) equation, see \cite{Dob68,LR69} and \cite[Chapters~1 and~2]{Geo11}. For $a\in\eps\Z^3$, $\tau_a$ is the translation of configurations; $\nu$ is $\Gamma$-invariant,
$\Gamma\subset\eps\Z^3$ a sublattice, if $\nu\circ\tau_g=\nu$ for all $g\in\Gamma$. For $\mu\in\{1,2,3\}$ and $c\in\eps(\Z+\tfrac12)$, $\theta_{\mu,c}$ is the reflection in
the mid-plane $\{x_\mu=c\}$ (half-way between two adjacent layers of sites), acting on configurations by transport of the forms. In this section, for a lattice isometry $g\in\eps\Z^3\rtimes B_3$ or a mid-plane reflection $g=\theta_{\mu,c}$, we also write $\hat g$ for the map of configurations induced by $g$ by transport of the forms, and $\nu\circ\hat g$ for the measure $f\mapsto\nu(f\circ\hat g)$; thus $\hat\theta_{\mu,c}$ and $\theta_{\mu,c}$ denote the same map of configurations.

\begin{theorem}[rigidity]\label{rv:thm:RV}
Let $N$ be fixed and let $\nu$ be a Gibbs measure of the fine Villain--Higgs specification on $\eps\Z^3$, $\eps=\eN$, which is invariant under a sublattice
$\Gamma\subset\eps\Z^3$ of finite index containing a time translation, and which satisfies
\begin{itemize}
\item[(I1)] $\nu(f_p^2)<\infty$ and $\nu(\rho_x^2)<\infty$ for every plaquette $p$ and site $x$ (finite local second moments);
\item[(I2)] joint Gaussian domination of the curvature over all plaquette orientations: there is $C<\infty$ with
$\nu\big(\exp\sum_p s_pf_p\big)\le C\exp(|s|^2/2)$ for every finitely supported real $s$ on the plaquettes of $\eps\Z^3$, temporal and spatial alike;
\item[(I3)] $\sup_x\nu\big(\exp(q\rho_x^2)\big)<\infty$ for every $q>0$.
\end{itemize}
Then $\nu$ is invariant under the translations $\tau_{2\eps e_j}$, $j=1,2,3$.
\end{theorem}

In Theorem~\ref{rv:thm:RV} the clause ``containing a time translation'' is redundant: $\Gamma\supset k\eps\Z^3$ with $k=[\eps\Z^3:\Gamma]$ (Subsection~\ref{rv:ssec:S9}, Step~2 of the proof of the theorem).

\begin{remark}[where (I1)--(I3) hold]\label{rv:rem:where}
(I1)--(I3) hold for every cubic-torus limit point $\om_N$ of Theorem~\ref{thm:d2:LP}: (I2) is Theorem~\ref{thm:GD} (constant $1$, all plaquettes jointly),
(I1) and (I3) are Lemma~\ref{lem:d2:mom}. They also hold for the comparison states of Section~\ref{n1:sec} (Lemma~\ref{n1:lem:JGD}, Proposition~\ref{n1:prop:I123}).
\end{remark}

\subsection{Phase restoration and the bijection of Gibbs measures}\label{rv:ssec:S1}

Let $\Om^u$ be the unitary-gauge configuration space $\{(\tr,B,v)\}$ of Subsection~\ref{rv:ssec:S0} and $\wt\Om:=\Om^u\times\Tper^{\mathrm{sites}}$, with Higgs phases $\psi_x\in\Tper$ and
normalised Haar measure on each $\psi_x$ as reference measure. The \emph{phase-restored} weight keeps $g_{\mathrm{ct}}$, $w((dB)_p+v_p)$ and the closure constraint, and
replaces the hopping factor by $h(\tr_{b_-},\tr_{b_+},B_b+\psi_{b_+}-\psi_{b_-})$; it is well defined on $\Tper$ because $h(r,r',\cdot)$ is $\cper$-periodic.

\begin{lemma}[phase-restored specification]\label{rv:lem:phase}
Let $G:=\Tper^{\mathrm{sites}}$. For $\lambda\in G$ with real lift $\wt\lambda\in\FD^{\mathrm{sites}}$ put
\begin{equation}\label{rv:eq:gauge}
\begin{gathered}
\lambda.(\tr,\psi,B,v):=\big(\tr,\ \psi-\lambda,\ [B+d\wt\lambda]_{\FD},\ v+\cper\,dn\big),\\
n_b:=\big(B_b+(d\wt\lambda)_b-[B_b+(d\wt\lambda)_b]_{\FD}\big)/\cper\in\Z,
\end{gathered}
\end{equation}
where $[\cdot]_{\FD}$ is the representative in $\FD$ modulo $\cper$. Then:
\begin{itemize}
\item[(a)] \eqref{rv:eq:gauge} is independent of the lift, is a group action of $G$ on $\wt\Om$, keeps $v$ closed ($d\,dn=0$), leaves the phase-restored weight invariant
factor by factor and preserves the reference measure.
\item[(b)] The gauge fixing $\Phi(\wt\om):=\psi.\wt\om$, read in $\Om^u$, and the Haar randomisation $\Psi(\om,\psi):=(-\psi).(\tr,0,B,v)$ satisfy $\Phi(\lambda.\wt\om)=\Phi(\wt\om)$,
$\wt\om=(-\psi).\Phi(\wt\om)$, $\Phi\circ\Psi=\mathrm{id}$ and $\lambda.\Psi(\om,\psi)=\Psi(\om,\psi-\lambda)$; the $G$-orbits are the fibres of $\Phi$, and the $G$-invariant
functions on $\wt\Om$ are exactly the functions $f\circ\Phi$.
\item[(c)] Let $\gamma_\Lambda$ be the unitary-gauge specification of Subsection~\ref{rv:ssec:S0} and $\wt\gamma_\Lambda$ the phase-restored one (Haar integral over $\psi_\Lambda$, phase-restored
hopping), both on the cofinal family $\{V(\Lambda)\}$. $\Phi$ is local outside $V(\Lambda)$, and for every bounded $F$ on $\wt\Om$
\begin{equation}\label{rv:eq:intertw}
\wt\gamma_\Lambda(F\mid\wt\om)=\int\mathrm{Haar}(d\psi_\Lambda)\int\gamma_\Lambda\big(d\om_{\mathrm{int}}\mid\Phi(\wt\om)\big)\,
F\big(\Psi(\om_{\mathrm{int}}\cup\Phi(\wt\om)_{\mathrm{ext}},\ \psi_\Lambda\cup\psi_{\mathrm{ext}})\big);
\end{equation}
in particular $\wt\gamma_\Lambda(f\circ\Phi\mid\wt\om)=\gamma_\Lambda(f\mid\Phi(\wt\om))$.
\item[(d)] If $\wt\mu$ is a Gibbs measure for $\wt\gamma$ then $\Phi_*\wt\mu$ is a Gibbs measure for $\gamma$; if $\mu$ is a Gibbs measure for $\gamma$ then
$\wt\mu:=\Psi_*(\mu\times\mathrm{Haar}^{\mathrm{sites}})$ is a $G$-invariant Gibbs measure for $\wt\gamma$ with $\Phi_*\wt\mu=\mu$; and $\Phi_*$, $\Psi_*(\cdot\times\mathrm{Haar})$ are mutually
inverse bijections between the $G$-invariant Gibbs measures of $\wt\gamma$ and the Gibbs measures of $\gamma$.
\item[(e)] $\Phi\circ\tau_a=\tau_a\circ\Phi$ and $\Psi\circ(\tau_a\times\tau_a)=\tau_a\circ\Psi$ for every $a\in\eps\Z^3$; hence $\mu$ is $\tau_a$-invariant if and only if $\wt\mu$ is. The same
holds for the lattice reflections $\theta_{\mu,c}$, up to the Lebesgue-null endpoint of $\FD$, and, without exception, for the transpositions $g$ of two coordinates; for such a $g$, moreover, $\Psi_*\big((\mu\circ\hat g)\times\mathrm{Haar}^{\mathrm{sites}}\big)=\wt\mu\circ\hat g$.
\end{itemize}
\end{lemma}

\begin{proof}
(a) Two lifts differ by $\cper m$, $m$ integer, which changes $d\wt\lambda$ by $\cper\,dm$ and $n$ by $+dm$, leaving $[B+d\wt\lambda]_{\FD}$ and $v+\cper\,dn$ unchanged; the composition
rule follows from the same bookkeeping, and $d\,dn=0$ keeps $v$ closed. The hopping factor is invariant since $B+d\psi$ is unchanged modulo $\cper$; the plaquette factor since
$d[B+d\wt\lambda]_{\FD}+v+\cper\,dn=dB+v$; $g_{\mathrm{ct}}$ does not see $\lambda$. The reference measure is preserved: $B_b\mapsto[B_b+(d\wt\lambda)_b]_{\FD}$ is a rotation of $\FD$,
$\psi\mapsto\psi-\lambda$ a Haar shift, and at fixed $B$, $v\mapsto v+\cper\,dn$ is a bijection of the closed integer $2$-forms.

(b) is read off from~\eqref{rv:eq:gauge}: $\Phi(\wt\om)$ has $\psi=0$ and is constant on orbits, $(-\psi).\Phi(\wt\om)$ restores $\psi$.

(c) Locality: $\Phi$ changes only variables carried by the bonds and plaquettes at the sites where $\psi\ne0$ enters, and $\psi_\Lambda$ is integrated out. For~\eqref{rv:eq:intertw}
make the interior change of variables $(B,v)_{\mathrm{int}}\mapsto(\psi.(\cdot))_{\mathrm{int}}$: it is bijective and measure preserving by (a), maps the closure constraint onto itself,
turns the phase-restored weight into the unitary-gauge weight $W^u_\Lambda$ at boundary condition $\Phi(\wt\om)$, and leaves the normalisation free of $\psi$. The special case is
$F=f\circ\Phi$ with $\Phi\circ\Psi=\mathrm{id}$.

(d) The first claim is the special case of (c) integrated against $\wt\mu$; the second is~\eqref{rv:eq:intertw} with Fubini and $\mu\gamma_\Lambda=\mu$, the $G$-invariance from
$\lambda.\Psi(\om,\psi)=\Psi(\om,\psi-\lambda)$ and the shift invariance of Haar; the third from $\wt\mu=\Psi_*(\Phi_*\wt\mu\times\mathrm{Haar})$ for $G$-invariant $\wt\mu$.

(e) $\Phi$ and $\Psi$ are given by the same local formula at every site, and $\mathrm{Haar}^{\mathrm{sites}}$ is shift invariant. For a reflection, $[-x]_{\FD}=-[x]_{\FD}$ except at
$x=\pi/\gph$, and $B_b$ has a Lebesgue density under every Gibbs measure because every $\gamma_\Lambda(\cdot\mid\om)$ has one. A transposition $g$ of two coordinates maps every bond from $x$ to $x+\eps e_\alpha$ to a bond of the same form, so that no $B_b$ changes sign; the formula~\eqref{rv:eq:gauge}, which is built from $d$ and from the bond-wise representative $[\cdot]_{\FD}$, therefore commutes with $\hat g$ without exception; a plaquette whose orientation $g$ reverses (every plaquette of the plane of the two swapped coordinates is one) carries $-v$ and $-dn$ after the transport, because the Villain form is transported as a $2$-cochain and $d$ commutes with the transport, and no representative $[\cdot]_{\FD}$ is taken on the plaquettes, whose variables range over the symmetric set $\cper\Z$. The last identity follows from $\Psi\circ(\hat g\times\hat g)=\hat g\circ\Psi$ and from the invariance of $\mathrm{Haar}^{\mathrm{sites}}$ under the relabelling of the sites.
\end{proof}

By Lemma~\ref{rv:lem:phase}(d)--(e), Theorem~\ref{rv:thm:RV} for the unitary-gauge specification is equivalent to the same statement for the $G$-invariant Gibbs measures of
the phase-restored specification. The hypotheses (I1)--(I3) concern $\tr$ and $f=dB+v$ only, which $\Phi$ and $\Psi$ leave unchanged ($\tr$ untouched, $f$ gauge invariant
by (a)), so $\nu$ satisfies (I1)--(I3) if and only if $\wt\nu:=\Psi_*(\nu\times\mathrm{Haar})$ does. \emph{From here on $\nu$ denotes $\wt\nu$.}

\subsection{Slab, temporal gauge, compact twist and the slab kernel}\label{rv:ssec:S2}

Fix two adjacent time layers $n$ and $n+1$ and the slab between them. The lower layer carries $c=(\tr,\psi,B,v)$ (its sites, spatial bonds and spatial plaquettes),
the upper layer $c'=(\tr',\psi',B',v')$; the temporal links carry $B_t(x)\in\FD$ (Lebesgue measure on $\FD$, which is Haar) and the temporal plaquettes carry the Villain
integers $v^t_b\in\cper\Z$ ($b$ a spatial bond) with the closure relation $d_sv^t=v-v'$ ($d_s$ the spatial exterior derivative). The slab weight is
\begin{equation}\label{rv:eq:slabW}
\prod_x h\big(\rho_x,\rho'_x,B_t(x)+\psi'_x-\psi_x\big)\prod_b w\big(B_b-B'_b+(d_sB_t)_b+v^t_b\big),
\end{equation}
where here and below $h(\rho,\rho',\vartheta)$ abbreviates $h(\tr,\tr',\vartheta)$ with $\rho=\rhof+\tr$.

\begin{lemma}[temporal gauge, slab kernel, compact twist]\label{rv:lem:slab}
\begin{itemize}
\item[(a)] (Temporal gauge.) The gauge transformation~\eqref{rv:eq:gauge} with $\lambda=0$ on layer $n$ and $\lambda=-B_t$ on layer $n+1$ sends $B_t\mapsto0$, $\psi'\mapsto\wt\psi':=\psi'+B_t$,
$B'\mapsto u:=[B'-d_sB_t]_{\FD}$, $v'\mapsto\wt v':=v'+\cper\,d_sn'$, $v^t\mapsto\wt v^t:=v^t-\cper\,n'$, where $n'$ are the wrap integers of $u$; one has $B-u+\wt v^t=B-B'+d_sB_t+v^t$
and $d_s\wt v^t=v-\wt v'$. Write $\wt c':=(-B_t).c'$ and $\Theta_\sigma(\om):=(c,\wt c',\wt v^t)$: the two temporal-gauge layers \emph{together with} the temporal Villain integers.
\item[(b)] (Slab kernel.) Under every finite-volume slab integral the slab data $(c,\wt c',\wt v^t)$ have the law
\[
W(c)W(\wt c')\prod_xh(\rho_x,\wt\rho'_x,\wt\psi'_x-\psi_x)\prod_bw(B_b-u_b+\wt v^t_b)\,\bbone[d_s\wt v^t=v-\wt v']
\]
with $\psi'$ an independent uniform spectator, $W(c)$ being the one-layer weight, that is, the product of the factors of the phase-restored weight that belong to the sites, spatial bonds and spatial plaquettes of the layer; summing out
$\wt v^t$ gives the two-layer law $W(c)W(\wt c')K(c,\wt c')$ with
\begin{equation}\label{rv:eq:K}
K(c,\wt c'):=\prod_xh\big(\rho_x,\wt\rho'_x,\wt\psi'_x-\psi_x\big)\sum_{\wt v^t:\ d_s\wt v^t=v-\wt v'}\ \prod_bw\big(B_b-u_b+\wt v^t_b\big).
\end{equation}
$K$ is symmetric, $K(c,\wt c')=K(\wt c',c)$, and invariant under the diagonal layer gauge action, $K(\lambda.c,\lambda.\wt c')=K(c,\wt c')$. The gauge-invariant slab functions are the
functions of $(c,\wt c',\wt v^t)$ invariant under the diagonal action.
\item[(c)] (Compact twist.) Let $\theta$ be the reflection in the mid-plane of the slab and let $\mathrm{sw}$ swap the two layers and reverse the sign of $\wt v^t$. Then
$\Theta_\sigma(\theta\om)=B_t.\big(\mathrm{sw}\,\Theta_\sigma(\om)\big)$, the twist being the diagonal action of the compact element $g=B_t\in\Tper^{\mathrm{sites}}$. Consequently, for every
diagonal-gauge-invariant $F$, $F(\Theta_\sigma\theta\om)=F(\mathrm{sw}\,\Theta_\sigma\om)$.
\end{itemize}
\end{lemma}

\begin{proof}
(a) is the computation of~\eqref{rv:eq:gauge} at the two layers; $d\wt\lambda$ on a temporal link is $-B_t$, which kills $B_t$, and on a spatial bond of the upper layer it is $-(d_sB_t)_b$.
(b) The map $(\psi',B_t)\mapsto(\psi',\wt\psi')$ preserves Haar on $\Tper\times\Tper$, and at fixed phases $(B',v',v^t)\mapsto(u,\wt v',\wt v^t)$ is the measure-preserving bijection of
Lemma~\ref{rv:lem:phase}(a), which preserves the closure set; the weight~\eqref{rv:eq:slabW} is then $\psi'$-free and reads as stated. Symmetry of $K$: $h(r,r',\vartheta)=h(r',r,-\vartheta)$,
$w$ is even, and $\wt v^t\mapsto-\wt v^t$ maps the coset $\{d_s\wt v^t=v-\wt v'\}$ onto $\{d_s\wt v^t=\wt v'-v\}$. Diagonal gauge invariance: the diagonal action moves $\wt v^t$ by the
wrap difference of the two layers and maps the coset onto the new one, while each factor is invariant by Lemma~\ref{rv:lem:phase}(a).
(c) The mid-plane reflection swaps the layers and reverses $B_t$ and $v^t$; its temporal gauge needs $\lambda=+B_t$ on the new upper layer, which gives
$\Theta_\sigma(\theta\om)=(c',B_t.c,-\wt v^t)=B_t.(\mathrm{sw}\,\Theta_\sigma(\om))$. The last claim follows since $F$ is invariant under the diagonal action of $B_t$.
\end{proof}

\begin{remark}[the Villain coset sum]\label{rv:rem:coset}
The sum in~\eqref{rv:eq:K} runs over the coset of closed integer $1$-forms fixed by $v-\wt v'$; it is bounded above by the free periodised Gaussian $\prod_bw_{\mathrm{per}}(B_b-u_b)$ but is
not bounded below uniformly in $(v,\wt v')$: it decays like $\exp(-\dist(\text{coset},\FD^S-\FD^S)^2/2)$. This is why the insertion calculus of Subsections~\ref{rv:ssec:S3}--\ref{rv:ssec:S4} works with
$\Z$-periodic (frame-blind) insertions and why the Villain sector is treated separately in Subsection~\ref{rv:ssec:S6}.
\end{remark}

\subsection{Averaging identity, integer residual frame, Zak feature map}\label{rv:ssec:S3}
\providecommand{\Tr}{\operatorname{Tr}}\providecommand{\sw}{\mathrm{sw}}\providecommand{\Haar}{\mathrm{Haar}}

Let $S$ be a finite spatial bond set and $K$ the slab kernel~\eqref{rv:eq:K}; in Lemma~\ref{rv:lem:zak}(c) and its proof one layer of the periodic measure is a finite spatial torus, and $S$ is there the set of all spatial bonds of the layer. 
The pair coordinates of the slab over $S$ are $B_b\in\FD$ and $u_b:=\wt B'_b-\wt v^t_b\in\R$, $b\in S$; $(\wt B',\wt v^t)\leftrightarrow u$ is a bijection
($\wt B'=[u]_{\FD}$, $\wt v^t=[u]_{\FD}-u$) and $f:=B-u=B-\wt B'+\wt v^t$ is the temporal plaquette curvature of Lemma~\ref{rv:lem:slab}(a). Put $h^x(\xi):=(2/\pi)^{|S|/4}e^{-|\xi-x|^2}$
on $\R^S$, so that $\prod_{b\in S}w(B_b-\wt B'_b+\wt v^t_b)=e^{-|B-u|^2/2}=\langle h^B,h^u\rangle_{L^2(\R^S)}$: the slab bond factor factorises through the Gaussian dressing, the closure
constraint restricting only the integer parts $[u]_{\FD}-u$. For a bounded operator $X$ on $L^2(\R^S)$ the \emph{dressed insertion} and its ratio are
\begin{equation}\label{rv:eq:kX}
k_X(B,u):=\langle h^B,Xh^u\rangle,\qquad r_X:=k_X/k_1,\qquad k_1(B,u)=e^{-|B-u|^2/2}.
\end{equation}
$X$ is \emph{$\Z$-periodic} if $X\tau_a=\tau_aX$ for all $a\in\cper\Z^S$ ($\tau_a$ the translation of $L^2(\R^S)$, $(\tau_ag)(\xi)=g(\xi+a)$), and $X^a:=\tau_aX\tau_a^*$.

\begin{lemma}[frame, periodic insertions, gauge covariance]\label{rv:lem:frame}
\begin{itemize}
\item[(a)] (Residual frame.) After the compact twist of Lemma~\ref{rv:lem:slab}(c) the reflection acts on $(B,\wt B',\wt v^t)$ as $(\wt B',B,-\wt v^t)$, i.e.\ on the pair as
$R(B,u)=T_a\,\sw(B,u)$ with the diagonal shift $T_a$ by $a:=[u]_{\FD}-u=\wt v^t\in\cper\Z^S$. $R$ is an involution of $\FD^S\times\R^S$, $k_1\circ R=k_1$ and
$k_X\circ R=\overline{k_{(X^a)^*}}$. The shift $a=\wt v^t$ (the temporal Villain flux) is called the \emph{frame}; it is part of the configuration, and no averaging over $a$ is available.
\item[(b)] For $\Z$-periodic $X$, $X^a=X$ and $k_X\circ R=\overline{k_{X^*}}$: the reflection acts on dressed $\Z$-periodic insertions with no frame.
\item[(c)] Under the diagonal layer gauge action with $\lambda\in\Tper^{\mathrm{sites}}$ the pair moves by the diagonal translation $(B,u)\mapsto(B+d\wt\lambda-\cper n,u+d\wt\lambda-\cper n)$;
for $\Z$-periodic $X$ the wrap $n$ drops, $\lambda\mapsto X^{d\wt\lambda}$ is an action of the compact group $\Tper^{\mathrm{sites}}$, and the gauge average
$\bar X:=\int_{\Tper^{\mathrm{sites}}}d\lambda\,X^{d\wt\lambda}$ satisfies $\|\bar X\|\le\|X\|$.
\end{itemize}
\end{lemma}

\begin{proof}
(a) With $(\wt B',B,-\wt v^t)$ the new pair is $([u]_{\FD},\,B+[u]_{\FD}-u)=T_a(u,B)$. Applying $R$ twice: the new frame is $[B+a]_{\FD}-(B+a)=-a$ (as $B\in\FD$), so $R^2=\mathrm{id}$.
$k_1\circ R=k_1$ since $|u+a-(B+a)|=|B-u|$. For $k_X$:
\[
k_X(R(B,u))=\langle h^{u+a},Xh^{B+a}\rangle=\langle\tau_a^*h^u,X\tau_a^*h^B\rangle=\langle h^u,X^ah^B\rangle=\overline{\langle
h^B,(X^a)^*h^u\rangle}.
\]
(b) is (a) with $X^a=X$. (c) The diagonal action of Lemma~\ref{rv:lem:phase}(a) shifts both $B$ and $\wt B'$ by $d\wt\lambda$ modulo the wrap, and $\wt v^t$ by the wrap difference,
which is the stated diagonal translation of $(B,u)$; $k_X$ transforms into $k_{X^{d\wt\lambda-\cper n}}=k_{X^{d\wt\lambda}}$ for $\Z$-periodic $X$; the average of a strongly continuous
unitary conjugation over a compact group has norm at most $\|X\|$.
\end{proof}

\begin{lemma}[Zak feature map; averaging identity]\label{rv:lem:zak}
Let $(Zh)(\theta,\xi):=\sum_{k}e^{-2\pi i\theta\cdot k}h(\xi+\cper k)$, with $\xi\in\FD^{\mathrm{bonds}}$ and $\theta\in[0,1)^{\mathrm{bonds}}$, be the Zak transform, extended to all $\theta\in\R^{\mathrm{bonds}}$ by periodicity and defined at every $\theta$ when $h$ is regular in the sense of (b), and let $\wt d$ be
the adjoint of $d_s$.
\begin{itemize}
\item[(a)] The closure constraint has the integral representation
$\bbone[d_s\wt v^t=v-\wt v']=\int_{[0,1)^{\mathrm{plaq}}}d\alpha\,e^{-2\pi i\alpha\cdot(d_s\wt v^t-v+\wt v')/\cper}$. Put $\theta:=\wt d\alpha$. The Zak transform turns the integer frame
into a phase,
$Zh^{x-\cper n}(\theta)=e^{2\pi i\theta\cdot n}Zh^x(\theta)$, and the Villain part of~\eqref{rv:eq:K} is a positive kernel in the data $(B,v)$ of the two layers:
\begin{equation}\label{rv:eq:zak}
\begin{gathered}
K^{\mathrm{Vill}}(c,c')=\langle\Phi(c),\Phi(c')\rangle_{\cH},\qquad \cH:=\int^\oplus_{[0,1)^{\mathrm{plaq}}}L^2(\FD^{\mathrm{bonds}})\,d\alpha,\\
\Phi(c)(\alpha):=e^{-2\pi i\alpha\cdot v/\cper}\,(Zh^B)(\wt d\alpha).
\end{gathered}
\end{equation}
The hopping factor enters the kernel as a further tensor factor, through the Bessel factorisation
$h(r,r',\vartheta)=e^{-\kappa_N(\rho^2+\rho'^2)/2}\sum_{n\in\Z}I_n(\kappa_N\rho\rho')e^{in\gph\vartheta}$.
\item[(b)] (Fibred operators.) For $y\in\R^S$ let $k(y):=(y-[y]_{\FD})/\cper\in\Z^S$ be the cell index of $y$, and for $a,\theta\in\R^S$ let $\tau_a(\theta)$ be the unitary operator on $L^2(\FD^S)$
\[
(\tau_a(\theta)F)(\xi):=e^{2\pi i\theta\cdot k(\xi+a)}F([\xi+a]_{\FD}).
\]
A function $g\in L^2(\R^S)$ is \emph{regular} if $\sum_{k\in\Z^S}\|g(\cdot+\cper k)\|_{L^2(\FD^S)}<\infty$; then the series that defines $(Zg)(\theta,\cdot)$ converges in $L^2(\FD^S)$ uniformly in $\theta$, and $\theta\mapsto(Zg)(\theta,\cdot)$ is norm-continuous. A bounded operator $X$ on $L^2(\R^S)$ is \emph{fibred} if there are bounded operators $X(\theta)$ on $L^2(\FD^S)$, $\theta\in\R^S$, its \emph{fibres}, with $\sup_\theta\|X(\theta)\|<\infty$ and $X(\theta+j)=X(\theta)$ for $j\in\Z^S$, such that $\theta\mapsto\langle F,X(\theta)G\rangle$ is continuous for all $F,G\in L^2(\FD^S)$ and
\begin{equation}\label{rv:eq:fib}
\langle g_1,Xg_2\rangle=\int_{[0,1)^S}\big\langle(Zg_1)(\theta,\cdot),X(\theta)(Zg_2)(\theta,\cdot)\big\rangle_{L^2(\FD^S)}\,d\theta\qquad\text{for all }g_1,g_2\in L^2(\R^S).
\end{equation}
A fibred $X$ satisfies the \emph{summability condition} if $\sum_{a\in\cper\Z^S}|k_X(B,u-a)|<\infty$ for all $B,u\in\R^S$ (Remark~\ref{rv:rem:abs}). The following hold.
\begin{itemize}
\item[(b1)] The fibres of a fibred $X$ are determined by $X$, $\|X(\theta)\|\le\|X\|$ for every $\theta$, and $X$ is $\Z$-periodic. The fibred operators form a linear space, closed under adjoints, with $X^*(\theta)=X(\theta)^*$, and under limits in the operator norm.
\item[(b2)] Every translation $\tau_a$, $a\in\R^S$, is fibred with the fibres $\tau_a(\theta)$; these are norm-continuous in $\theta$, they satisfy $\tau_a(\theta)\tau_b(\theta)=\tau_{a+b}(\theta)$, and $(a,\theta)\mapsto\tau_a(\theta)F$ is continuous for every $F$. Every operator $\sum_{a\in\cper\Z^S}c_a\tau_a$ with $\sum_a|c_a|<\infty$ is fibred, with the scalar fibres $\sum_ac_ae^{2\pi i\theta\cdot a/\cper}$. Every operator $\sum_{a\in\cper\Z^S}\tau_a^*X_0\tau_a$ with $X_0$ bounded and $X_0=P_KX_0P_K$ for some $K<\infty$, where $P_K$ is the multiplication by $\bbone_{[-K,K]^S}$, is bounded and fibred. If $X$ is fibred, then so are: the operator $X^a=\tau_aX\tau_a^*$ for $a\in\R^S$, with the fibres $\tau_a(\theta)X(\theta)\tau_a(\theta)^*$; the gauge average $\bar X$ of Lemma~\ref{rv:lem:frame}(c), with the fibres $\int d\lambda\,\tau_{d\wt\lambda}(\theta)X(\theta)\tau_{d\wt\lambda}(\theta)^*$ (weak integrals; here and below, in $X^{d\wt\lambda}$ and in these fibres, $d\wt\lambda$ stands for its restriction $(d\wt\lambda)_S$ to the bonds of $S$); the operator $M_\sigma^{-1}XM_\sigma$ for $\sigma\in\R^S$, where $M_\sigma$ is the multiplication by $e^{2\pi i\sigma\cdot k(y)}$, with the fibres $X(\theta+\sigma)$; and the complex conjugate $X^\sharp:=\mathsf{C}X\mathsf{C}$, where $\mathsf{C}$ is the complex conjugation, with the fibres $\mathsf{C}X(-\theta)\mathsf{C}$. If $S'$ is a finite bond set disjoint from $S$ and $X'$ is fibred on $L^2(\R^{S'})$, then $X\otimes X'$ is fibred on $L^2(\R^{S\cup S'})$, with the fibres $X(\theta_S)\otimes X'(\theta_{S'})$.
\item[(b3)] For a fibred $X$ there is exactly one bounded operator $\hat X$ on $\cH$ with
\[
\langle G,\hat XF\rangle=\int_{[0,1)^{\mathrm{plaq}}}\big\langle G(\alpha),\big(X((\wt d\alpha)_S)\otimes1\big)F(\alpha)\big\rangle\,d\alpha\qquad\text{for all }F,G\in\cH,
\]
where the factor $1$ acts on the bonds outside $S$ and on the Higgs
factor of Lemma~\ref{rv:lem:fullmap}; we say that $\hat X$ has the fibres $X((\wt d\alpha)_S)\otimes1$. The map $X\mapsto\hat X$ is linear, $(X^*)^\wedge=(\hat X)^*$, $\|\hat X\|\le\|X\|$, and $\langle G,\widehat{\bar X}F\rangle=\int d\lambda\,\langle G,(X^{d\wt\lambda})^\wedge F\rangle$ for all $F,G\in\cH$. Moreover $(\hat XF)(\alpha)=\big(X((\wt d\alpha)_S)\otimes1\big)F(\alpha)$ for almost every $\alpha$, for every $F\in\cH$; consequently, for fibred operators $X_i$ on pairwise disjoint bond sets $S_i$, $\prod_i\hat X_i=(\bigotimes_iX_i)^\wedge$, in any order.
\item[(b4)] Let $X$ be fibred and let $g_1,g_2$ be regular, with $\sum_{m\in\Z^S}|\langle g_1,X\tau_{\cper m}g_2\rangle|<\infty$. Then, for every $\theta\in\R^S$,
\[
\big\langle(Zg_1)(\theta,\cdot),X(\theta)(Zg_2)(\theta,\cdot)\big\rangle=\sum_{m\in\Z^S}e^{-2\pi i\theta\cdot m}\langle g_1,X\tau_{\cper m}g_2\rangle.
\]
For a fibred $X$ put $K^X(c,c'):=\langle\Phi(c),\hat X\Phi(c')\rangle$; if $X$ satisfies the summability condition, $K^X$ is the kernel with the insertion~\eqref{rv:eq:kX}.
\item[(b5)] The gauge action on the feature map is $\Phi(\lambda.c)=R(\lambda)\Phi(c)$, where $(R(\lambda)F)(\alpha):=\tau_{-d\wt\lambda}(\wt d\alpha)F(\alpha)$, with the operators $\tau_a(\theta)$ on $L^2(\FD^{\mathrm{bonds}})$ of all the bonds of the layer; the operator $R(\lambda)$ is unitary and independent of the configuration, and $R(\lambda)^*\hat
XR(\lambda)=(X^{d\wt\lambda})^\wedge$ for every fibred $X$.
\end{itemize}
\item[(c)] (Averaging identity.) Let $L:=L^2(dc)$ be the layer space, $Q$ the multiplication by the one-layer weight $W(c)$, $T:=Q^{1/2}KQ^{1/2}$ the transfer operator on $L$, and
$\mathsf{B}:L\to\cH$ the dressing $(\mathsf{B}\psi):=\int W(c)^{1/2}\Phi(c)\psi(c)\,dc$, so that $\mathsf{B}^*\mathsf{B}=T$. Here, and in the proof, the feature map $\Phi$ and the space $\cH$ include the Higgs tensor factor that reproduces the hopping factor of~\eqref{rv:eq:K} (see (a); it is written out in Lemma~\ref{rv:lem:fullmap}), so that $\langle\Phi(c),\Phi(c')\rangle=K(c,c')$; every $\hat X$ acts as the identity on that factor, and $R(\lambda)$ includes the rotation $U(\lambda)$ of that lemma. Let $R_L(\lambda)\psi:=\psi((-\lambda).c)$ be the gauge
representation on $L$ and $P_G:=\int dG\,R_L(G)$ the
projection onto the gauge-invariant layer subspace $L_G$. Let $X_1,\dots,X_k$ be fibred insertions at distinct heights that satisfy the summability condition of (b), and let $Z_M(X_1,\dots,X_k)$ be the partition
function of the periodic measure with $M$ layers and with these insertions, that is, the integral of the weight of this measure in which, for each $s$, the slab bond factor $k_1(B,u)$ of the slab at the height of $X_s$, written in the pair coordinates $(B,u)$ of that slab, is replaced by $k_{X_s}(B,u)$; in this integral the sum over the temporal Villain variables $\wt v^t$ of each slab is taken first, where it converges absolutely by the summability condition and gives the kernel $K^{X_s}$ of (b4), and the remaining integral over the layers converges absolutely by the bound $|K^{X_s}(c,c')|\le\|X_s\|\sqrt{K(c,c)K(c',c')}$ of the proof. Then
\begin{equation}\label{rv:eq:avg}
Z_M(X_1,\dots,X_k)=\Tr_{L_G}\Big(\prod_s\mathsf{B}^*\widehat{\bar X_s}\mathsf{B}\Big).
\end{equation}
\end{itemize}
\end{lemma}

\begin{proof}
(a) The form $d_s\wt v^t-v+\wt v'$ takes its values in $\cper\Z^{\mathrm{plaq}}$, so the indicator of the closure constraint is equal to
$\int_{[0,1)^{\mathrm{plaq}}}e^{-2\pi i\alpha\cdot(d_s\wt v^t-v+\wt v')/\cper}d\alpha$. The indicator is real, so the sign of the exponent is a convention; the sign chosen is the one
consistent with~\eqref{rv:eq:zak}. Write
$\wt v^t=\cper m$ with $m\in\Z^S$, and note that $\alpha\cdot d_sm=(\wt d\alpha)\cdot m=\theta\cdot m$. The sum over $m$ is then free of constraints and factorises over the bonds. For
one bond, with $x,y\in\FD$,
\[
\sum_{m\in\Z}e^{-2\pi i\theta m}\,w(x-y+\cper m)=\sum_{m\in\Z}e^{-2\pi i\theta m}\langle h^x,h^{y-\cper m}\rangle_{L^2(\R)}=\big\langle(Zh^x)(\theta,\cdot),(Zh^y)(\theta,\cdot)\big\rangle_{L^2(\FD)},
\]
by the definition of $Z$, the rearrangement $\sum_{k,k'}=\sum_k\sum_{m=k'-k}$ and the decomposition $\bigcup_k(\FD+\cper k)=\R$. The remaining phase splits as
$e^{2\pi i\alpha\cdot(v-\wt v')/\cper}=\overline{e^{-2\pi i\alpha\cdot
v/\cper}}\,e^{-2\pi i\alpha\cdot\wt v'/\cper}$,
and~\eqref{rv:eq:zak} follows. The frame identity is the relation $h^{x-\cper n}(\xi)=h^x(\xi+\cper n)$, after a shift of the summation index in the sum that defines $Z$. The Bessel
factorisation is the generating function
$e^{z\cos\varphi}=\sum_nI_n(z)e^{in\varphi}$ applied to $|e^{i\gph\vartheta}\rho'-\rho|^2=\rho^2+\rho'^2-2\rho\rho'\cos(\gph\vartheta)$; it exhibits the hopping factor as a
positive-definite kernel in the data $(\rho,\psi)$ of the two layers, which enters the kernel as a tensor factor.

(b) \emph{Preliminaries.} For almost every $\xi$ the function $\theta\mapsto(Zg)(\theta,\xi)$ is the Fourier series of the square-summable sequence $(g(\xi+\cper k))_k$. By Parseval's identity, which makes the Fourier series an isometric isomorphism of $\ell^2(\Z^S)$ onto $L^2([0,1)^S)$, and by $\bigcup_k(\FD^S+\cper k)=\R^S$, the map $Z$ is unitary from $L^2(\R^S)$ onto $L^2([0,1)^S\times\FD^S)$; moreover $Z\tau_{\cper k}g=e^{2\pi i\theta\cdot k}Zg$, and $(Zg)(\theta,\cdot)$ is periodic in $\theta$ with the period lattice $\Z^S$. For a regular $g$ the tail of the series is bounded in $L^2(\FD^S)$ by $\sum_{|k|>n}\|g(\cdot+\cper k)\|_{L^2(\FD^S)}$, uniformly in $\theta$, which gives the uniform convergence and the norm-continuity. \emph{Measurability.} Let $A(\theta)$ be operators on a separable Hilbert space $\mathcal K$, bounded uniformly in $\theta$, with $\theta\mapsto\langle F,A(\theta)G\rangle$ continuous for all $F,G\in\mathcal K$, and let $G_1,G_2$ be square-integrable $\mathcal K$-valued functions. The finite sums $\sum_j\varphi_j(\theta)F_j$ are dense in the $L^2$ space of such functions, and a sequence that converges there has a subsequence that converges in $\mathcal K$ for almost every $\theta$. For finite sums the function $\theta\mapsto\langle G_1(\theta),A(\theta)G_2(\theta)\rangle$ is a finite sum of products of measurable functions and continuous functions, and the uniform bound on $A$ carries the convergence almost everywhere over to these functions. So $\theta\mapsto\langle G_1(\theta),A(\theta)G_2(\theta)\rangle$ is measurable, of modulus at most $\sup_\theta\|A(\theta)\|\,\|G_1(\theta)\|\,\|G_2(\theta)\|$. In particular the right side of~\eqref{rv:eq:fib} is a sesquilinear form in $(g_1,g_2)$ bounded by $\sup_\theta\|X(\theta)\|\,\|g_1\|\,\|g_2\|$.

(b1) Let $F,G\in L^2(\FD^S)$ and $\varphi\in L^2([0,1)^S)$, and insert $g_1:=Z^*(\varphi\otimes F)$ and $g_2:=Z^*(\varphi\otimes G)$ in~\eqref{rv:eq:fib}: $\int|\varphi(\theta)|^2\langle F,X(\theta)G\rangle\,d\theta=\langle g_1,Xg_2\rangle$, which has modulus at most $\|X\|\,\|\varphi\|^2\|F\|\,\|G\|$. Let $|\varphi|^2$ run through the normalised indicator functions of the intersections of $[0,1)^S$ with balls that shrink to a point $\theta_0$ of the closed cube: the continuity of $\theta\mapsto\langle F,X(\theta)G\rangle$ gives $|\langle F,X(\theta_0)G\rangle|\le\|X\|\,\|F\|\,\|G\|$, and by periodicity this holds for every $\theta_0\in\R^S$. If two families of fibres belong to the same $X$, their difference $D(\theta)$ satisfies $\int|\varphi|^2\langle F,D(\theta)G\rangle\,d\theta=0$ for all $\varphi$, so the continuous periodic function $\langle F,D(\theta)G\rangle$ vanishes. By~\eqref{rv:eq:fib} and $Z\tau_{\cper k}g=e^{2\pi i\theta\cdot k}Zg$ one has $\langle g_1,X\tau_{\cper k}g_2\rangle=\langle\tau_{\cper k}^*g_1,Xg_2\rangle$, so $X$ is $\Z$-periodic. Linear combinations and adjoints satisfy~\eqref{rv:eq:fib} with the same combinations and the adjoints of the fibres. If fibred operators $X_n$ converge to $X$ in norm, the bound, applied to $X_n-X_m$, gives $\|X_n(\theta)-X_m(\theta)\|\le\|X_n-X_m\|$ for every $\theta$; the fibres converge in norm, uniformly in $\theta$, their limits are bounded, periodic and weakly continuous, and~\eqref{rv:eq:fib} passes to the limit.

(b2) \emph{Translations.} For $g$ of compact support the sums are finite, and with $\xi+a=[\xi+a]_{\FD}+\cper\,k(\xi+a)$ a shift of the summation index gives
\[
(Z\tau_ag)(\theta,\xi)=e^{2\pi i\theta\cdot k(\xi+a)}(Zg)(\theta,[\xi+a]_{\FD})=\big(\tau_a(\theta)(Zg)(\theta,\cdot)\big)(\xi).
\]
The map $\xi\mapsto[\xi+a]_{\FD}$ is a measure-preserving bijection of $\FD^S$, so $\tau_a(\theta)$ is unitary and both sides are isometric in $g$; the identity therefore holds for all $g\in L^2(\R^S)$ as an identity in $L^2([0,1)^S\times\FD^S)$, and~\eqref{rv:eq:fib} for $\tau_a$ follows from the unitarity of $Z$. The fibres are periodic because $k$ is integer valued. On $\FD^S$ one has $|k(\xi+a)|\le|a|/\cper+\sqrt{|S|}$, hence $\|\tau_a(\theta)-\tau_a(\theta_0)\|\le2\pi|\theta-\theta_0|\,(|a|/\cper+\sqrt{|S|})$. For fixed $\theta$ the function $\tau_a(\theta)F$ is the restriction to $\FD^S$ of the translate by $a$ of the locally square-integrable function $E_\theta F(y):=e^{2\pi i\theta\cdot k(y)}F([y]_{\FD})$, $y\in\R^S$, and translations act continuously on $L^2$ of bounded sets; the two facts give the continuity of $(a,\theta)\mapsto\tau_a(\theta)F$. Since $\tau_a^*=\tau_{-a}$, the uniqueness in (b1) gives $\tau_a(\theta)^*=\tau_{-a}(\theta)$. The group law follows from $k(y-\cper j)=k(y)-j$ and $[y-\cper j]_{\FD}=[y]_{\FD}$ for $j\in\Z^S$, applied to $y=\xi+a+b$ and $j=k(\xi+a)$. For $a=\cper j$, $j\in\Z^S$, one has $k(\xi+a)=j$ and $[\xi+a]_{\FD}=\xi$ on $\FD^S$, so $\tau_a(\theta)=e^{2\pi i\theta\cdot a/\cper}$; the series $\sum_ac_a\tau_a$ converges in norm, and (b1) gives the claim on these series.

\emph{Localised sums.} Let $X=\sum_{a\in\cper\Z^S}\tau_a^*X_0\tau_a$ with $X_0=P_KX_0P_K$, and let $N$ be a number, depending only on $K$, $\cper$ and $|S|$, such that every translate of $[-K,K]^S$ or of $\FD^S$ meets at most $N$ of the sets $[-K,K]^S+a$ and at most $N$ of the cells $\FD^S+a$, $a\in\cper\Z^S$. The function $\tau_a^*X_0\tau_ag$ is supported in $[-K,K]^S+a$ and equals $\tau_a^*X_0P_K\tau_ag$, so $\|Xg\|^2\le N\sum_a\|X_0P_K\tau_ag\|^2\le N^2\|X_0\|^2\|g\|^2$: the operator $X$ is bounded, and a shift of the summation index shows that it commutes with the $\tau_{\cper k}$. Put
\[
X(\theta)F:=\sum_{a\in\cper\Z^S}\big(\tau_a^*X_0P_K\tau_aE_\theta F\big)\big|_{\FD^S},\qquad F\in L^2(\FD^S),
\]
a sum with at most $N$ non-zero terms. The map $F\mapsto P_K\tau_aE_\theta F$ has norm at most $N^{1/2}$ and is norm-continuous in $\theta$, as for the translations, because $k$ is bounded on $[-K,K]^S+a$. So the operators $X(\theta)$ are bounded uniformly in $\theta$ and norm-continuous in $\theta$, and they are periodic because $E_{\theta+j}=E_\theta$. For $g$ of compact support one has $E_\theta\big((Zg)(\theta,\cdot)\big)=\sum_ke^{-2\pi i\theta\cdot k}\tau_{\cper k}g$ on $\R^S$, a locally finite sum, and $Xg$ has compact support; therefore, all sums being finite,
\[
X(\theta)(Zg)(\theta,\cdot)=\sum_ke^{-2\pi i\theta\cdot k}\big(X\tau_{\cper k}g\big)\big|_{\FD^S}=\sum_ke^{-2\pi i\theta\cdot k}\big(\tau_{\cper k}Xg\big)\big|_{\FD^S}=(ZXg)(\theta,\cdot)
\]
for every $\theta$. With the unitarity of $Z$ this is~\eqref{rv:eq:fib} for $g_2$ of compact support and all $g_1$; both sides of~\eqref{rv:eq:fib} are bounded sesquilinear forms, so~\eqref{rv:eq:fib} holds for all $g_2$.

\emph{Stability.} For $X^a$: the identity for $Z\tau_a$, applied to $\tau_a^*=\tau_{-a}$, and $\tau_{-a}(\theta)=\tau_a(\theta)^*$ turn~\eqref{rv:eq:fib} into $\langle g_1,X^ag_2\rangle=\langle\tau_a^*g_1,X\tau_a^*g_2\rangle=\int\langle\tau_a(\theta)^*(Zg_1)(\theta,\cdot),X(\theta)\tau_a(\theta)^*(Zg_2)(\theta,\cdot)\rangle\,d\theta$. The family $\tau_a(\theta)X(\theta)\tau_a(\theta)^*$ is bounded and periodic, and $\langle F,\tau_a(\theta)X(\theta)\tau_a(\theta)^*G\rangle=\langle\tau_a(\theta)^*F,X(\theta)\tau_a(\theta)^*G\rangle$ is continuous in $(a,\theta)$, because the vectors $\tau_a(\theta)^*F=\tau_{-a}(\theta)F$ and $\tau_a(\theta)^*G$ are norm-continuous in $(a,\theta)$ and $X(\theta)$ is bounded and weakly continuous. For $\bar X=\int d\lambda\,X^{d\wt\lambda}$: the lift $\lambda\mapsto\wt\lambda$ is Borel, so $(\lambda,\theta)\mapsto\langle F,X^{d\wt\lambda}(\theta)G\rangle$ is Borel and bounded; the weak integrals over $\lambda$ define bounded periodic operators, weakly continuous in $\theta$ by dominated convergence; and~\eqref{rv:eq:fib} for $\bar X$ follows from~\eqref{rv:eq:fib} for the operators $X^{d\wt\lambda}$ by Fubini's theorem, the integrand being measurable in $(\lambda,\theta)$ by the argument of the preliminaries. For $M_\sigma$: for $g$ of compact support, $k(\xi+\cper k)=k$ on $\FD^S$ gives $(ZM_\sigma g)(\theta,\cdot)=(Zg)(\theta-\sigma,\cdot)$, both sides are isometric in $g$, and $M_\sigma$ is unitary; inserting this in~\eqref{rv:eq:fib} for $\langle M_\sigma g_1,XM_\sigma g_2\rangle$ and shifting the integration variable, which the periodicity allows, gives~\eqref{rv:eq:fib} for $M_\sigma^{-1}XM_\sigma$ with the fibres $X(\theta+\sigma)$. For $X^\sharp$: $(Z\mathsf{C}g)(\theta,\cdot)=\mathsf{C}(Zg)(-\theta,\cdot)$ and $\langle g_1,\mathsf{C}X\mathsf{C}g_2\rangle=\overline{\langle\mathsf{C}g_1,X\mathsf{C}g_2\rangle}$ give~\eqref{rv:eq:fib} with the fibres $\mathsf{C}X(-\theta)\mathsf{C}$, after the substitution $\theta\mapsto-\theta$. For $X\otimes X'$: the family $X(\theta_S)\otimes X'(\theta_{S'})$ is bounded and periodic; the function $\langle\Psi_1,(X(\theta_S)\otimes X'(\theta_{S'}))\Psi_2\rangle$ is continuous for finite sums $\Psi_1,\Psi_2$ of product vectors and hence, by density and the uniform bound, for all vectors; and~\eqref{rv:eq:fib} holds for products $g_i=g_{i,S}\otimes g_{i,S'}$, the Zak transform of a product being the product of the Zak transforms, hence for finite sums of products, and for all $g_i$ because both sides are bounded sesquilinear forms.

(b3) The family $A(\alpha):=X((\wt d\alpha)_S)\otimes1$ is bounded by $\sup_\theta\|X(\theta)\|\le\|X\|$ and weakly continuous in $\alpha$, by the argument just given for tensor products, $\alpha\mapsto(\wt d\alpha)_S$ being continuous. By the preliminaries the right side of the formula in (b3) is a sesquilinear form on $\cH$ bounded by $\|X\|\,\|G\|\,\|F\|$, and a bounded sesquilinear form on a Hilbert space is $\langle G,\hat XF\rangle$ for exactly one bounded operator $\hat X$, whose norm is at most the bound of the form (Riesz). Linearity and the formula for the adjoint are read off from the form. Testing the form with $G(\alpha)=\varphi(\alpha)G_j$, where $\varphi$ is square-integrable and $G_j$ runs through a countable dense set of vectors, gives $\langle G_j,(\hat XF)(\alpha)-A(\alpha)F(\alpha)\rangle=0$ for almost every $\alpha$ and every $j$, which is the pointwise formula. Applied to two operators on disjoint bond sets it gives $(\hat X_1\hat X_2F)(\alpha)=A_1(\alpha)A_2(\alpha)F(\alpha)$ almost everywhere, and $A_1(\alpha)A_2(\alpha)$ is the fibre of $(X_1\otimes X_2)^\wedge$ by (b2); the product formula follows by induction on the number of factors. The identity for $\widehat{\bar X}$ is Fubini's theorem for the function $(\lambda,\alpha)\mapsto\langle G(\alpha),(X^{d\wt\lambda}((\wt d\alpha)_S)\otimes1)F(\alpha)\rangle$, which is measurable by the continuity in $(a,\theta)$ shown in (b2), the Borel lift and the argument of the preliminaries, and is bounded by $\|X\|\,\|G(\alpha)\|\,\|F(\alpha)\|$.

(b4) The function $u(\theta):=\langle(Zg_1)(\theta,\cdot),X(\theta)(Zg_2)(\theta,\cdot)\rangle$ is periodic, and it is continuous because the vectors $(Zg_i)(\theta,\cdot)$ are norm-continuous and $X(\theta)$ is bounded and weakly continuous. By $Z\tau_{\cper m}g_2=e^{2\pi i\theta\cdot m}Zg_2$ and~\eqref{rv:eq:fib}, $\int_{[0,1)^S}e^{2\pi i\theta\cdot m}u(\theta)\,d\theta=\langle g_1,X\tau_{\cper m}g_2\rangle$. These Fourier coefficients are absolutely summable by assumption, so the series on the right of the claim converges uniformly to a continuous periodic function with the same Fourier coefficients as $u$. A continuous periodic function is determined by its Fourier coefficients (Fej\'er's theorem); this proves the claim for every $\theta$. For the kernel, the functions $h^x$ are regular, and $\langle h^B,X\tau_{\cper m}h^u\rangle=k_X(B,u-\cper m)$ because $\tau_ah^u=h^{u-a}$. By the definition of $\hat X$,
\[
\langle\Phi(c),\hat X\Phi(c')\rangle=\int d\alpha\,e^{2\pi i\alpha\cdot(v-\wt v')/\cper}\big\langle(Zh^B)(\theta,\cdot),\big(X(\theta_S)\otimes1\big)(Zh^{\wt B'})(\theta,\cdot)\big\rangle,\qquad\theta=\wt d\alpha .
\]
The scalar product is the product of its factor over the bonds of $S$, to which the claim applies, of the factors of the other bonds, which are given by (a), and of the factor of the Higgs variables, which does not depend on $\alpha$. Under the summability condition the resulting series over $m\in\Z^{\mathrm{bonds}}$ converges absolutely, uniformly in $\alpha$, and is integrated term by term. Since $\theta\cdot m=\alpha\cdot d_sm$, the integral over $\alpha$ selects the terms with $\cper\,d_sm=v-\wt v'$, that is, $\wt v^t:=\cper m$ with $d_s\wt v^t=v-\wt v'$:
\[
\langle\Phi(c),\hat X\Phi(c')\rangle=\sum_{\wt v^t:\,d_s\wt v^t=v-\wt v'}k_X(B,\wt B'-\wt v^t),
\]
which is the kernel with the insertion~\eqref{rv:eq:kX}.

(b5) For the gauge action, the configuration $\lambda.c$ has $B\mapsto B+d\wt\lambda-\cper n$ and
$v\mapsto v+\cper\,dn$. By the frame identity of (a),
\[
\Phi(\lambda.c)(\alpha)=e^{-2\pi i\alpha\cdot v/\cper}e^{-2\pi i\alpha\cdot dn}e^{2\pi i\theta\cdot n}(Zh^{B+d\wt\lambda})(\theta),
\]
and $\alpha\cdot dn=\theta\cdot n$, so the two wrap phases cancel; this is where $\Z$-periodicity enters. Since $h^{B+a}=\tau_{-a}h^B$, and since the identity for $Z\tau_a$ of (b2) holds pointwise for the functions $h^B$, whose series converge absolutely, $(Zh^{B+d\wt\lambda})(\theta,\cdot)=\tau_{-d\wt\lambda}(\theta)(Zh^B)(\theta,\cdot)$ for every $\theta$. Hence $\Phi(\lambda.c)=R(\lambda)\Phi(c)$. On functions of $(\alpha,\xi)$ the operator $R(\lambda)$ is $(R(\lambda)F)(\alpha,\xi)=e^{2\pi i(\wt d\alpha)\cdot k(\xi-d\wt\lambda)}F(\alpha,[\xi-d\wt\lambda]_{\FD})$; it maps measurable functions to measurable functions, is unitary and does not depend on the
configuration. The operator $\tau_{-a}(\theta)$ of all the bonds is the tensor product of the operator $\tau_{-a_S}(\theta_S)$ of the bonds of $S$ with that of the other bonds, and $\tau_{-a_S}(\theta_S)^*=\tau_{a_S}(\theta_S)$. So, with $a=d\wt\lambda$ and $\theta=\wt d\alpha$, the form of (b3) gives $\langle G,R(\lambda)^*\hat XR(\lambda)F\rangle=\int\langle G(\alpha),(\tau_{a_S}(\theta_S)X(\theta_S)\tau_{a_S}(\theta_S)^*\otimes1)F(\alpha)\rangle\,d\alpha=\langle G,(X^{d\wt\lambda})^\wedge F\rangle$, by the fibres of $X^a$ in (b2).

(c) Here the sans-serif letter $\mathsf{B}$ denotes the dressing operator, while $B$ remains the link field. The proof has three steps.

\emph{The trace structure.} The kernel $K=\langle\Phi,\Phi\rangle$ is positive, so $T=Q^{1/2}KQ^{1/2}=\mathsf{B}^*\mathsf{B}\ge0$ with the dressing $\mathsf{B}$ of the statement. The
operator $\mathsf{B}$
is Hilbert--Schmidt, with
$\|\mathsf{B}\|_2^2=\int W(c)K(c,c)\,dc=Z_1<\infty$, because
\[
K(c,c)=\prod_xh(\rho_x,\rho_x,0)\sum_{\text{closed }\wt v^t}\prod_bw(\wt v^t_b)\le(\sum_{m\in\Z}e^{-\cper^2m^2/2})^{|S|}
\]
and $\int W\,dc<\infty$; the latter follows from~\eqref{eq:ct} for the radial integrals, from the convergence of the Gaussian Villain sums and from the compactness of the ranges of
$B$ and $\psi$. Hence $T$ and every operator $\mathsf{B}^*Y\mathsf{B}$, with $Y$ bounded on $\cH$, are trace class. The bound
$|K^Y(c,c')|\le\|Y\|\sqrt{K(c,c)K(c',c')}$ makes the layer integrals absolutely convergent, and a cyclic integral of $M\ge2$ such kernels is the trace of the product of the
corresponding operators.

\emph{The intertwining relations.} One has $\mathsf{B}R_L(\lambda)=R(\lambda)\mathsf{B}$, by the change of variables $c=\lambda.c'$ and the invariance of $W$. Both $R_L$ and $R$ are
strongly continuous unitary
representations of the
compact abelian group $\Tper^{\mathrm{sites}}$; the operator $R(\lambda)$ does not depend on the lift, because two lifts of $d\wt\lambda$ differ by $\cper d_sk$, whose Zak phase is
$e^{-2\pi i\alpha\cdot d_sd_sk}=1$. Consequently $\int dG\,R_L(G)=P_G$
and $P_G\mathsf{B}^*=\mathsf{B}^*P^{\cH}_G$, where $P^{\cH}_G:=\int dG\,R(G)$ is the projection onto the $R$-invariant vectors of $\cH$.

\emph{The change of variables.} Let $\mu_M$ be the periodic measure with $M$ layers $c^{(1)},\dots,c^{(M)}$ and temporal links $B_t^{(s)}$ between the layers $s$ and $s+1$, where
$M+1\equiv1$. Put
$G_1:=0$, $G_{s+1}:=G_s-B_t^{(s)}$ and $W^{(s)}:=G_s.c^{(s)}$. The map $(B_t^{(s)})_s\mapsto(G_{s+1})_s$ is a triangular bijection of $(\Tper^{\mathrm{sites}})^M$ that preserves the Haar
measure, and the map $c^{(s)}\mapsto
W^{(s)}$ preserves
$W(c)\,dc$ (Lemma~\ref{rv:lem:phase}(a)). The three-dimensional closure $dv=0$ holds if and only if every slab satisfies $d_s\wt v^{t,(s)}=v^{(s)}-\wt v^{(s+1)}$, so the Villain sum
factorises into the sum over the
layer $2$-forms times the independent slab coset sums carried by the kernels. The operator $X_s^{-d\wt G_s}$ is fibred by (b2) and satisfies the summability condition together with $X_s$, because $k_{X^a}(B,u)=k_X(B+a,u+a)$ for every $a\in\R^S$. By (b), the slab $s<M$ with the insertion $X_s$ contributes
\[
K^{X_s^{-d\wt G_s}}(W^{(s)},W^{(s+1)})=\langle\Phi(W^{(s)}),(X_s^{-d\wt G_s})^\wedge\Phi(W^{(s+1)})\rangle,
\]
and the closing slab carries the holonomy $G:=G_{M+1}$: its kernel is that
of $\mathsf{B}^*\hat X_M\mathsf{B}\,R_L(G)$. Integrating out the layers gives
\[
Z_M(X_1,\dots,X_k)=\int dG_2\cdots dG_{M+1}\,\Tr\big(\prod_s\mathsf{B}^*(X_s^{-d\wt G_s})^\wedge
\mathsf{B}\cdot R_L(G)\big),
\]
with the insertion $1$ at the
heights that carry no $X_s$.

Conjugating the integrand by $R_L(g)$ shifts every $G_s$ by $g$ and leaves the trace unchanged; after a renaming of the variables, by the invariance of
the Haar measure, there is one independent Haar variable for each
insertion. For each of them $\int dg\,(X^{-d\wt g})^\wedge=\widehat{\bar X}$, by (b) and Lemma~\ref{rv:lem:frame}(c); the integral over $g$ is exchanged with the trace by dominated
convergence for trace-class operators. Finally
$\int dG\,R_L(G)=P_G$, and $P_G$ commutes with every $\mathsf{B}^*\widehat{\bar X}\mathsf{B}$, because $\bar X$ is gauge invariant. This proves~\eqref{rv:eq:avg}. For $X_s=1$ the
identity reads
$Z_M=\Tr_{L_G}(T^M)=\Tr(P_GT^M)\le\Tr(T^M)$.
\end{proof}

\begin{remark}[absolute convergence]\label{rv:rem:abs}
The Zak form of Lemma~\ref{rv:lem:zak}(b) needs $\sum_{\wt v^t}|k_X(B,\wt B'-\wt v^t)|<\infty$; it holds for the bounded class $\cC$ of Subsection~\ref{rv:ssec:S5} (by Lemma~\ref{rv:lem:growth}, $|k_X(B,u)|\le C_Xe^{c_X|f|-|f|^2/2}$ with $f=B-u$, and the sum over $\wt v^t$ moves $f$ through the lattice $\cper\Z^S$) and for the corrector $Z$ of Lemma~\ref{rv:lem:corr} ($\sum_a\alpha_a<\infty$).
\end{remark}

\subsection{The Schatten--H\"older bound, the corrector and the bond pair}\label{rv:ssec:S4}

Translations of $L^2(\R^S)$ are written $(\tau_ag)(\xi):=g(\xi+a)$, so that $\tau_ah^u=h^{u-a}$, $k_{\tau_a}(B,u)=e^{-|f+a|^2/2}$ and $r_{\tau_a}=e^{-a\cdot f-|a|^2/2}$, $f=B-u$.
We first prove the relative-entropy estimate of this subsection, the tangent inequality of Lemma~\ref{rv:lem:TV}, which is used in Lemmas~\ref{rv:lem:step2}
and~\ref{rv:lem:step2full}: it compares a Gibbs measure that is invariant under a sublattice of finite index with the periodic measures of cubic tori whose
three sides grow together. Because the closure constraint ties the Villain variables across the boundary of a box (on the torus it holds also on the cubes that
straddle the seam), the law of a whole fundamental box under the Gibbs measure need not be absolutely continuous with respect to its law under the torus
measure; the comparison is therefore made on the closed cell complex of an inner box, on which both laws are carried by the same configurations, and the shell
around that complex is summed out on both sides.

\begin{lemma}[tangent inequality]\label{rv:lem:TV}
Let $\nu$ be a Gibbs measure of the phase-restored specification, invariant under a sublattice $\Gamma'\subset\eps\Z^3$ of finite index, which satisfies \textup{(I1)}
and \textup{(I2)} of Theorem~\ref{rv:thm:RV}. Let $v$ be a local Borel function (a Borel function of the variables of finitely many cells), bounded below, and let
$\ell_v$ be the sup-norm diameter, in units of $\eps$, of its cell support. For an integer $\ell\ge\max\{8,\ell_v+2\}$ with $\ell\eps\Z^3\subset\Gamma'$ let
$\eps\Z^3_\ell:=\eps\Z^3/\ell\eps\Z^3$ be the cubic torus of side $\ell$ ($\ell$ time layers, each a layer torus of side $\ell$) and let $\mu_\ell$ be its
periodic measure in the phase-restored variables: the probability measure on the configurations of $\eps\Z^3_\ell$ ($\tr_x\in[-\rhof,\infty)$ and
$\psi_x\in\Tper$ at the sites, $B_b\in\FD$ on the bonds, $v_p\in\cper\Z$ on the plaquettes) given by the normalised phase-restored weight of
Subsection~\ref{rv:ssec:S1} on the cells of $\eps\Z^3_\ell$, with Lebesgue measure on each $\tr_x$ and each $B_b$ and normalised Haar measure on each $\psi_x$, the sum
over the Villain variables running over all closed $(v_p)$ (closed on every cube of $\eps\Z^3_\ell$), every flux class included. Let
$V_\ell:=\sum_gv\circ\tau_g$ be the sum over the $n_\ell=\ell^3/[\eps\Z^3:\Gamma']$ distinct $\Gamma'$-translates of $v$ on $\eps\Z^3_\ell$ ($g$ running
over $\Gamma'/\ell\eps\Z^3$). Then
\[
\nu(v)\ \le\ \liminf_{\ell\to\infty}\ n_\ell^{-1}\log\mu_\ell\big(e^{V_\ell}\big),
\]
the lower limit being taken along the integers $\ell$ with $\ell\eps\Z^3\subset\Gamma'$, and both sides having values in $(-\infty,+\infty]$.
More precisely, let $v\ge-c_v$ with a constant $c_v\ge0$. There are a constant $C_v$, depending only on $\ell_v$ and on the index $[\eps\Z^3:\Gamma']$,
integers $n^-_\ell\le n_\ell$ with $n_\ell-n^-_\ell\le C_v\ell^2$, and a constant $C_{\mathrm{TV}}$, depending only on $N$, the couplings, the constant $C$ of
\textup{(I2)} and $m_\nu:=\sup_x\nu(\rho_x^2)$, such that for every $\ell$ as above
\begin{equation}\label{tv:A:TV}
n^-_\ell\,\nu(v)\ \le\ \log\mu_\ell\big(e^{V_\ell}\big)+c_vC_v\,\ell^2+C_{\mathrm{TV}}\,\ell^2(1+\log_2\ell)^2 .
\end{equation}
\end{lemma}

Here $m_\nu<\infty$ by (I1) and the $\Gamma'$-invariance of $\nu$ (the sites fall into finitely many classes modulo $\Gamma'$). Hypothesis (I2) enters in its
joint form, with one constant $C$ for all finitely supported $s$: it bounds the second moments of the fluxes through the boundary spheres of boxes whose size
grows with $\ell$ (Claim~3 in the proof).

\begin{proof}
\emph{Conventions.} Lengths and coordinates are in units of $\eps$, so that $\Gamma'\subset\Z^3$ in coordinates. The symbol $v$ without an index is the local
function of the statement; the Villain variable of a plaquette $p$ is always written with its index, $v_p$, or $v_p(\zeta)$ for a configuration $\zeta$.
A \emph{cell} is a site, a bond, a plaquette or a cube. Every bond $b$ and every plaquette $p$ has a reference orientation, in which $B_b$, $v_p$ and
$f_p=(dB)_p+v_p$ are read, and $[p:b]\in\{0,\pm1\}$ is the incidence number of $b$ in the boundary of $p$: $(dB)_p=\sum_b[p:b]B_b$, and likewise
$(d\mathfrak a)_p:=\sum_b[p:b]\mathfrak a_b$ for a function $\mathfrak a$ on the bonds. Fix $\ell$ as in the statement and put $\ell_-:=\ell-3\ge5$ and
$\ell_+:=\ell+1$.

\emph{Step 0 (geometry).} Let
\[
\Lambda^-:=\{0,\dots,\ell_-\}^3\ \subset\ \Lambda:=\{-1,\dots,\ell-2\}^3\ \subset\ \Lambda^+:=\{-2,\dots,\ell-1\}^3 .
\]
$\Lambda$ has $\ell^3$ sites and is a fundamental domain of sites of $\eps\Z^3_\ell$. The \emph{closed complex} of a box consists of its sites and of the
bonds, plaquettes and cubes all of whose vertices lie in the box; $\mathfrak C^-$ and $\mathfrak C^+$ are the closed complexes of $\Lambda^-$ and of
$\Lambda^+$. The \emph{boundary sphere} $\Sigma$ of $\mathfrak C^-$ consists of the cells of $\mathfrak C^-$ contained in the boundary of the solid cube
$[0,\ell_-]^3$; it has $6\ell_-^2$ plaquettes, $12\ell_-^2$ bonds and $6\ell_-^2+2$ sites. $\Sigma^+$ is the boundary sphere of $\mathfrak C^+$, of side
$\ell_+$. With $V(\Lambda)$ as in Subsection~\ref{rv:ssec:S0}, the \emph{shells} are $\fA:=V(\Lambda)\setminus\mathfrak C^-$ and, on the torus, the set $\fA_\ell$
of the sites, bonds and plaquettes of $\eps\Z^3_\ell$ that are not in $\mathfrak C^-$ (see (G3)).
\begin{itemize}
\item[(G1)] The cubes having a face in $V(\Lambda)$ are the cubes of $\mathfrak C^+$: a cube with a vertex in $\Lambda$ lies in $\Lambda^+$, and a cube
$\{y_1,y_1+1\}\times\{y_2,y_2+1\}\times\{y_3,y_3+1\}$ in $\Lambda^+$ has a vertex in $\Lambda$ (in each coordinate take $y_i+1$ if $y_i=-2$ and $y_i$
otherwise). The faces of these cubes that are not in $V(\Lambda)$ are the plaquettes of $\Sigma^+$. Indeed, a plaquette of $\mathfrak C^+$ in a plane
$\{x_i=-2\}$ or $\{x_i=\ell-1\}$ has no vertex in $\Lambda$; and a plaquette of $\mathfrak C^+$ with normal direction $i$ whose coordinate $x_i$ lies in
$[-1,\ell-2]$ has a vertex in $\Lambda$: if it had none, and $(y_j,y_k)$, $(y_j+1,y_k+1)$ are its extreme corners in the two tangential coordinates, then
$y_j=-2$ and $y_k=\ell-2$, or the same with $j$ and $k$ exchanged, and the corner with tangential coordinates $(y_j+1,y_k)=(-1,\ell-2)$ would lie in
$\Lambda$.
\item[(G2)] Every cell of $\fA$ has a vertex in $\Lambda\setminus\Lambda^-$: a bond or plaquette of $\fA$ meets $\Lambda$ and not all its vertices lie in
$\Lambda^-$, so if it had no vertex in $\Lambda\setminus\Lambda^-$ it would have a vertex in $\Lambda^-$ and a vertex outside $\Lambda$, and these are at
sup-distance at least $2$. $\Lambda\setminus\Lambda^-$ has $\ell^3-(\ell-2)^3\le6\ell^2$ sites, each of which lies in $6$ bonds and $12$ plaquettes, so
$|\fA|\le114\,\ell^2$. The same count gives $|\fA_\ell|\le114\,\ell^2$.
\item[(G3)] In $\eps\Z^3_\ell$ two layers of sites separate opposite faces of $\Lambda^-$ (the layers $x_i=\ell-2$ and $x_i=\ell-1\equiv-1$), so no bond,
plaquette or cube of the torus joins $\Lambda^-$ to itself around the torus: the cells of $\eps\Z^3_\ell$ with all vertices in $\Lambda^-$ are the cells of
$\mathfrak C^-$, and $\mathfrak C^-$ is embedded in $\eps\Z^3_\ell$.
\item[(G4)] A face of a cube not in $\mathfrak C^-$ which belongs to $\mathfrak C^-$ lies in $\Sigma$ (the cube has a vertex outside $\Lambda^-$ in the
direction normal to that face). This holds in $\eps\Z^3$ and in $\eps\Z^3_\ell$.
\item[(G5)] A plaquette not in $\mathfrak C^-$ contains at most one bond of $\mathfrak C^-$, and such a bond lies in $\Sigma$ (two bonds of one plaquette
with all ends in the product set $\Lambda^-$ force its four corners into $\Lambda^-$; and the plaquette leaves $\Lambda^-$ in a direction normal to the
bond). A bond of $\Sigma$ lies in exactly one plaquette not in $\mathfrak C^-$ if it is interior to a face of the cube (of the four plaquettes around it two
lie in the face and one in the interior of $\mathfrak C^-$), and in exactly two if it lies on an edge of the cube. Likewise a bond of $\Sigma^+$ lies in
exactly one plaquette of $\mathfrak C^+$ that is not in $\Sigma^+$ if it is interior to a face, and in none if it lies on an edge. No plaquette contains a
bond of $\Sigma$ and a bond of $\Sigma^+$ (their sites are at sup-distance at least $2$).
\end{itemize}

\emph{The surface coboundary.} For a plaquette $p$ of $\Sigma$ let $o_p=\pm1$ be its orientation relative to the outward normal of the cube. Every bond $b$
of $\Sigma$ lies in exactly two plaquettes $p$, $p'$ of $\Sigma$, and, $\Sigma$ being a closed oriented surface,
\begin{equation}\label{tv:A:orient}
o_p[p:b]=-o_{p'}[p':b].
\end{equation}
For a real function $\mathfrak r$ on the bonds of $\Sigma$ put $(d_\Sigma\mathfrak r)_p:=o_p\sum_b[p:b]\mathfrak r_b$, a function on the plaquettes of
$\Sigma$. By~\eqref{tv:A:orient} every $d_\Sigma\mathfrak r$ has zero sum, and $\mathcal L_\Sigma:=d_\Sigma d_\Sigma^*$ is the graph Laplacian of the dual
graph of $\Sigma$, whose vertices are the $6\ell_-^2$ plaquettes of $\Sigma$, two of them adjacent when they share a bond (the graph is $4$-regular and
connected). Hence the range of $d_\Sigma$ is the space of the functions of zero sum, and $\mathcal L_\Sigma^{-1}$ denotes the inverse of $\mathcal L_\Sigma$
on that space, extended by $0$ to the constants. $|\cdot|$ and $\langle\cdot,\cdot\rangle$ are the Euclidean norm and scalar product of real functions on
the bonds, or on the plaquettes, of $\Sigma$. The same notation is used on $\Sigma^+$, with $\ell_+$ in place of $\ell_-$.

\emph{Step 1 (reduction to an entropy bound on the interior complex).} Let $x_v$ be a vertex of the cell support of $v$ and $k':=[\eps\Z^3:\Gamma']$, so
that $k'\eps\Z^3\subset\Gamma'$ and $n_\ell=\ell^3/k'$. The $\Gamma'$-translates of $v$ on $\eps\Z^3$ are indexed by the position $z\in x_v+\Gamma'$ of the
translate of $x_v$. If $z\in\{\ell_v,\dots,\ell_--\ell_v\}^3$, the cell support of the translate lies in $\mathfrak C^-$. These translates are pairwise
incongruent modulo $\ell\Z^3$ (the differences of their positions have coordinates of modulus $<\ell$), so by (G3) they are distinct terms of $V_\ell$; let
$n^-_\ell$ be their number and $V^-$ their sum, a function of the variables of $\mathfrak C^-$. Every cube of $k'^3$ sites contains $k'^2$ points of each
coset of $\Gamma'$, so $n^-_\ell\ge(\ell-2\ell_v-2-k')^3/k'$ whenever $\ell\ge2\ell_v+2+k'$, and in every case
\[
n_\ell-n^-_\ell\le C_v\,\ell^2,\qquad C_v:=3(2\ell_v+2+k')/k'.
\]
On $\eps\Z^3_\ell$, $V_\ell-V^-$ is the sum of the other $n_\ell-n^-_\ell$ translates, so $V_\ell\ge V^--c_v(n_\ell-n^-_\ell)\ge V^--c_vC_v\ell^2$. By the
$\Gamma'$-invariance of $\nu$, $\nu(V^-)=n^-_\ell\,\nu(v)$, both sides in $(-\infty,+\infty]$.

Let $\Om^-$ be the space of the configurations $\zeta$ of the variables of $\mathfrak C^-$: $\tr_y$ and $\psi_y$ at its sites, $B_b$ on its bonds and
$v_p$ on its plaquettes, the Villain variables being closed on the cubes of $\mathfrak C^-$ and subject to no other constraint. Let $\nu^-$ and $\mu^-_\ell$
be the laws of these variables under $\nu$ and under $\mu_\ell$ (through the embedding (G3)), and write $\mathrm{Ent}(\cdot\mid\cdot)$ for the relative
entropy of probability measures on $\Om^-$. If $\nu^-\ll\mu^-_\ell$, the entropy inequality
\[
\nu^-(F)\le\log\mu^-_\ell\big(e^F\big)+\mathrm{Ent}(\nu^-\mid\mu^-_\ell)
\]
holds for every bounded measurable $F$ on $\Om^-$, and it extends to $F$ bounded below by monotone convergence on $\min(F,j)$, $j\to\infty$. For $F=V^-$,
since $\mu^-_\ell(e^{V^-})=\mu_\ell(e^{V^-})\le e^{c_vC_v\ell^2}\mu_\ell(e^{V_\ell})$,
\begin{equation}\label{tv:A:red}
n^-_\ell\,\nu(v)=\nu^-(V^-)\le\log\mu^-_\ell\big(e^{V^-}\big)+\mathrm{Ent}(\nu^-\mid\mu^-_\ell)
\le\log\mu_\ell\big(e^{V_\ell}\big)+c_vC_v\,\ell^2+\mathrm{Ent}(\nu^-\mid\mu^-_\ell).
\end{equation}
So~\eqref{tv:A:TV} follows from the \emph{entropy bound}
\begin{equation}\label{tv:A:ent}
\nu^-\ll\mu^-_\ell\qquad\text{and}\qquad\mathrm{Ent}(\nu^-\mid\mu^-_\ell)\le C_{\mathrm{TV}}\,\ell^2(1+\log_2\ell)^2,
\end{equation}
which is proved in Steps 2--8. The statement on the lower limit follows from~\eqref{tv:A:TV}. Indeed $n_\ell^{-1}\log\mu_\ell(e^{V_\ell})\ge-c_v$,
because $V_\ell\ge-c_vn_\ell$. If the lower limit is $+\infty$ there is nothing to prove. Otherwise choose a sequence of sides along which
$n_\ell^{-1}\log\mu_\ell(e^{V_\ell})$ converges to the lower limit, divide~\eqref{tv:A:TV} by $n^-_\ell$, and use that $n_\ell/n^-_\ell\to1$ and
$\ell^2(1+\log_2\ell)^2/n^-_\ell\to0$.

No closure is asked of a configuration across the seam of the torus: $\nu^-$ and $\mu^-_\ell$ live on the same space $\Om^-$, whose Villain variables are
closed on the cubes of $\mathfrak C^-$ only; the cubes at the seam and the translates of $v$ that wrap around the torus belong to the shell, where they have
been summed out (in $\mu^-_\ell$) or bounded below by $-c_v$ (in $V_\ell-V^-$).

\emph{Step 2 (densities on $\Om^-$; decomposition of the entropy).} Let $d\zeta$ be the reference measure on $\Om^-$: Lebesgue measure on $[-\rhof,\infty)$
and normalised Haar measure on $\Tper$ at every site, Lebesgue measure on $\FD$ on every bond, and counting measure on the $\cper\Z$-valued Villain variables
that are closed on the cubes of $\mathfrak C^-$. Each factor of the phase-restored weight belongs to one cell and depends on the variables of that cell and
of its faces: $g_{\mathrm{ct}}(\tr_y)$ to the site $y$, $h(\tr_{b_-},\tr_{b_+},B_b+\psi_{b_+}-\psi_{b_-})$ to the bond $b$, $w((dB)_p+v_p)$ to the plaquette
$p$; and each closure constraint belongs to one cube. Let $W^-(\zeta)$ be the product of the factors of the sites, bonds and plaquettes of $\mathfrak C^-$.

\emph{Shell factors.} For $\zeta\in\Om^-$ and a configuration $\wt\om$ of $\eps\Z^3$, of which only the variables outside $V(\Lambda)$ enter, let
$\cZ^{\mathrm{sh}}(\zeta,\wt\om)$ be the integral over the variables of the shell $\fA$ ($\tr$ and $\psi$ at the sites of $\Lambda\setminus\Lambda^-$, $B$ on
the bonds of $\fA$, and the Villain variables on the plaquettes of $\fA$, summed over the values for which every cube of $\mathfrak C^+$ not in
$\mathfrak C^-$ is closed, its faces in $\Sigma$ carrying the values of $\zeta$ and its faces in $\Sigma^+$ those of $\wt\om$) of the product of the factors
of the cells of $\fA$, the variables on $\mathfrak C^-$ being those of $\zeta$ and the variables outside $V(\Lambda)$ those of $\wt\om$. Let
$\cZ^{\mathrm{sh}}_\ell(\zeta)$ be the same on the torus: the integral over the variables of $\fA_\ell$, the Villain variables being summed over the values
for which every cube of $\eps\Z^3_\ell$ not in $\mathfrak C^-$ is closed, of the product of the factors of the cells of $\fA_\ell$.

By (G1), (G3) and (G4), the weight of the kernel $\wt\gamma_\Lambda(\cdot\mid\wt\om)$ of the phase-restored specification (Lemma~\ref{rv:lem:phase}(c)) and
the weight of $\mu_\ell$ are $W^-(\zeta)$ times the factors of the shell cells, and the closure constraints split into those of the cubes of $\mathfrak C^-$,
which are carried by $d\zeta$, and those of the other cubes, which are in the shell factors. Hence the laws of the variables of $\mathfrak C^-$ under
$\wt\gamma_\Lambda(\cdot\mid\wt\om)$ and under $\mu_\ell$ are
\begin{equation}\label{tv:A:dens}
\wt\gamma^-_\Lambda(d\zeta\mid\wt\om)=\frac{W^-(\zeta)\,\cZ^{\mathrm{sh}}(\zeta,\wt\om)}{\cZ_\Lambda(\wt\om)}\,d\zeta,\qquad
\mu^-_\ell(d\zeta)=\frac{W^-(\zeta)\,\cZ^{\mathrm{sh}}_\ell(\zeta)}{\cZ_\ell}\,d\zeta,
\end{equation}
with $\cZ_\Lambda(\wt\om):=\int W^-(\zeta)\,\cZ^{\mathrm{sh}}(\zeta,\wt\om)\,d\zeta$ and $\cZ_\ell:=\int W^-(\zeta)\,\cZ^{\mathrm{sh}}_\ell(\zeta)\,d\zeta$.
By the Gibbs property of $\nu$ for the box $\Lambda$, that is, the equation $\nu\wt\gamma_\Lambda=\nu$ of the definition of a Gibbs measure in
Subsection~\ref{rv:ssec:S0}, read for the phase-restored specification,
\[
\nu^-=\int\nu(d\wt\om)\,\wt\gamma^-_\Lambda(\cdot\mid\wt\om).
\]

Both laws in~\eqref{tv:A:dens} have densities with respect to $W^-(\zeta)\,d\zeta$, and by Step 4 the shell factors are strictly positive:
$\cZ^{\mathrm{sh}}_\ell(\zeta)>0$ for every $\zeta\in\Om^-$, and $\cZ^{\mathrm{sh}}(\zeta,\wt\om)>0$ for every $\zeta$ and $\nu$-almost every $\wt\om$. So
$\wt\gamma^-_\Lambda(\cdot\mid\wt\om)$, $\nu^-$ and $\mu^-_\ell$ are mutually absolutely continuous; what is used is $\nu^-\ll\mu^-_\ell$. The relative
entropy being convex,
\[
\mathrm{Ent}(\nu^-\mid\mu^-_\ell)\le\int\nu(d\wt\om)\,\mathrm{Ent}\big(\wt\gamma^-_\Lambda(\cdot\mid\wt\om)\mid\mu^-_\ell\big),
\]
and by~\eqref{tv:A:dens} the logarithm of the density of $\wt\gamma^-_\Lambda(\cdot\mid\wt\om)$ with respect to $\mu^-_\ell$ is
$\log\cZ^{\mathrm{sh}}(\zeta,\wt\om)-\log\cZ^{\mathrm{sh}}_\ell(\zeta)+\log\cZ_\ell-\log\cZ_\Lambda(\wt\om)$ (the factor $W^-$ cancels). Therefore
\begin{equation}\label{tv:A:E1}
\begin{aligned}
\mathrm{Ent}(\nu^-\mid\mu^-_\ell)&\le\int\nu(d\wt\om)\int\wt\gamma^-_\Lambda(d\zeta\mid\wt\om)\,
\big[\log\cZ^{\mathrm{sh}}(\zeta,\wt\om)-\log\cZ^{\mathrm{sh}}_\ell(\zeta)\big]\\
&\quad+\int\nu(d\wt\om)\,\big[\log\cZ_\ell-\log\cZ_\Lambda(\wt\om)\big].
\end{aligned}
\end{equation}
Since $\cZ_\Lambda(\wt\om)/\cZ_\ell=\int\mu^-_\ell(d\zeta')\,\cZ^{\mathrm{sh}}(\zeta',\wt\om)/\cZ^{\mathrm{sh}}_\ell(\zeta')$ by~\eqref{tv:A:dens}, Jensen's
inequality for the logarithm gives
\begin{equation}\label{tv:A:E2}
\log\cZ_\ell-\log\cZ_\Lambda(\wt\om)\le\int\mu^-_\ell(d\zeta')\,\big[\log\cZ^{\mathrm{sh}}_\ell(\zeta')-\log\cZ^{\mathrm{sh}}(\zeta',\wt\om)\big].
\end{equation}

It remains to bound the shell factors from above (Step 3) and from below (Step 4) and to bound the means of the lower bounds under $\nu$ and under
$\mu_\ell$ (Steps 5--7); the splitting of the integrals in~\eqref{tv:A:E1} and~\eqref{tv:A:E2} is justified in Step 8.

\emph{Step 3 (upper bounds for the shell factors).} Dropping the closure constraints enlarges the sum over the Villain variables of the shell to the free
sum. Then, the Villain sums being done first, at fixed $B$, the integrals over the $B_b$ next and the site integrals last, each shell plaquette contributes
$\sum_{v_p\in\cper\Z}w((dB)_p+v_p)\le K'_{\gph}$, whatever the value of $(dB)_p$; each shell bond contributes $\int_{\FD}h\,dB_b\le2\pi/\gph$, as $h\le1$; and
each shell site contributes $\int g_{\mathrm{ct}}=G^{\mathrm{ct}}_0$, the Haar measure having mass $1$. Here $K'_{\gph}$ and $G^{\mathrm{ct}}_0$ are the
constants of Lemma~\ref{lem:d2:mom}. With $C_{\mathrm{sh}}:=\max\{1,G^{\mathrm{ct}}_0,2\pi/\gph,K'_{\gph}\}$,
\begin{equation}\label{tv:A:U}
\cZ^{\mathrm{sh}}(\zeta,\wt\om)\le C_{\mathrm{sh}}^{\,|\fA|},\qquad\cZ^{\mathrm{sh}}_\ell(\zeta)\le C_{\mathrm{sh}}^{\,|\fA_\ell|}\qquad\text{for all }\zeta\text{ and }\wt\om .
\end{equation}

\emph{Step 4 (one closed extension; lower bounds for the shell factors).} For $\zeta\in\Om^-$ put
\[
\mathfrak m(\zeta)_p:=o_p\,v_p(\zeta)/\cper\qquad\text{for the plaquettes }p\text{ of }\Sigma,
\]
an integer function on the plaquettes of $\Sigma$. It has zero sum: $\sum_{p\in\Sigma}o_pv_p(\zeta)$ is the sum, over the cubes of $\mathfrak C^-$, of the
coboundary of the Villain variables of $\zeta$ (the interior faces cancel), and every term of that sum vanishes. Likewise, for a configuration $\wt\om$ of
$\eps\Z^3$ with closed Villain variables (as they are $\nu$-almost surely) put $\mathfrak m^+(\wt\om)_p:=o_p\,v_p(\wt\om)/\cper$ for the plaquettes $p$ of
$\Sigma^+$; it has zero sum, by the closure on the cubes of $\mathfrak C^+$.

\emph{Claim~1 (closed extension).} Let $\mathfrak a$ be an integer-valued function on the bonds which vanishes off the bonds of $\Sigma$ and satisfies
$d_\Sigma\mathfrak a=\mathfrak m(\zeta)$. Then the Villain variables equal to $v_p(\zeta)$ on the plaquettes $p$ of $\mathfrak C^-$ and to
$\cper\,(d\mathfrak a)_p$ on the plaquettes $p$ not in $\mathfrak C^-$ are closed on every cube, in $\eps\Z^3$ and in $\eps\Z^3_\ell$. Outside
$\mathfrak C^-$ they equal $\pm\cper\,\mathfrak a_b$ on the one or two plaquettes that contain the bond $b$ of $\Sigma$, see (G5), and $0$ elsewhere.
Two-sided form: if moreover $\mathfrak a^+$ is an integer-valued function on the bonds which vanishes off the bonds of $\Sigma^+$ and satisfies
$d_{\Sigma^+}\mathfrak a^+=\mathfrak m^+(\wt\om)$, then the Villain variables equal to $v_p(\zeta)$ on the plaquettes of $\mathfrak C^-$, to $v_p(\wt\om)$
on the plaquettes outside $V(\Lambda)$ and to $\cper\,(d(\mathfrak a+\mathfrak a^+))_p$ on the plaquettes of $\fA$ are closed on every cube of $\eps\Z^3$.
On $\fA$ they equal $\pm\cper\,\mathfrak a_b$ on the plaquettes at the bonds $b$ of $\Sigma$, $\pm\cper\,\mathfrak a^+_b$ on the plaquette at a bond $b$
interior to a face of $\Sigma^+$, and $0$ elsewhere, with one exception, the plaquettes of $\fA$ that contain two bonds of $\Sigma^+$. A plaquette $p$ of $\fA$ has a vertex $y\in\Lambda$, and its two bonds at $y$ are not in $\Sigma^+$, whose sites lie outside $\Lambda$; so $p$ contains at most two bonds of $\Sigma^+$, its bonds $b$, $b'$ at the vertex $z$ opposite to $y$. Both are in $\Sigma^+$ if and only if $z$ lies on an edge of the cube $[-2,\ell-1]^3$: two bonds of $p$ in a common face would force $y$ into that face, and if $z$ lies on an edge, $y$ differs from $z$ in the two coordinates that are extremal at $z$, so that each of $b$, $b'$ keeps one of them and lies in one of the two faces that meet at the edge. There are $12\ell$ such plaquettes, $\ell$ at each edge, and each of them carries $\cper\,([p:b]\,\mathfrak a^+_b+[p:b']\,\mathfrak a^+_{b'})$.

\emph{Proof of Claim~1.} For a plaquette $p$ of $\Sigma$, all of whose bonds are bonds of $\Sigma$,
$\cper\,(d\mathfrak a)_p=\cper\sum_b[p:b]\mathfrak a_b=\cper\,o_p(d_\Sigma\mathfrak a)_p=\cper\,o_p\,\mathfrak m(\zeta)_p=v_p(\zeta)$. A cube not in
$\mathfrak C^-$ has its faces either outside $\mathfrak C^-$ or in $\Sigma$, by (G4); so the variables equal $\cper\,d\mathfrak a$ on all its faces, and
their coboundary on the cube is $\cper\,d\,d\mathfrak a=0$. The cubes of $\mathfrak C^-$ are closed because $\zeta\in\Om^-$. The values outside
$\mathfrak C^-$ are read off from (G5). Two-sided form: $d\mathfrak a^+$ vanishes on the plaquettes of $\Sigma$ and $d\mathfrak a$ on those of $\Sigma^+$,
by (G5), and $\cper\,(d\mathfrak a^+)_p=v_p(\wt\om)$ on the plaquettes of $\Sigma^+$ as before. So on every face of a cube of $\mathfrak C^+$ not in
$\mathfrak C^-$ the variables equal $\cper\,d(\mathfrak a+\mathfrak a^+)$, and such a cube is closed; by (G1) these are all the cubes that have a face in
$\fA$, and every other cube outside $\mathfrak C^-$ has all its faces outside $V(\Lambda)$ and is closed because the Villain variables of $\wt\om$ are.
This proves Claim~1.

\emph{Flux classes.} An $\mathfrak a$ as in Claim~1 exists for every $\zeta\in\Om^-$: by Step 5 the only condition is that $\mathfrak m(\zeta)$ has zero
sum, that is, that the total flux of the Villain variables of $\zeta$ through the sphere $\Sigma$ vanishes, and this holds by the closure on the cubes of
$\mathfrak C^-$. On the torus the extension of Claim~1 vanishes on every plaquette lying in one of the three planes $\{x_i=\ell-2\}$, $i=1,2,3$, which
contain no bond of $\Sigma$ and no cell of $\mathfrak C^-$; so its fluxes through these three planes vanish, and it lies in the class of zero flux. In any
case the Villain sum of $\mu_\ell$ runs over all closed Villain variables, every flux class included: the extension is one of its terms, and no condition
on the flux class arises.

\emph{Lower bounds.} Let
\[
\cE_\Sigma(\mathfrak m):=\min\{|\mathfrak a|^2:\ \mathfrak a\text{ integer valued on the bonds of }\Sigma,\ d_\Sigma\mathfrak a=\mathfrak m\}
\]
for an integer function $\mathfrak m$ of zero sum on the plaquettes of $\Sigma$ (it is finite by Step 5), and $\cE_{\Sigma^+}$ likewise on $\Sigma^+$. Restrict
the integral defining a shell factor to the one value of the Villain variables of the shell given by Claim~1, with minimising $\mathfrak a$ (and
$\mathfrak a^+$), and to $\tr\in[0,1]$ at the shell sites; $\psi$ and $B$ are not restricted.

A shell site then contributes $\int_0^1g_{\mathrm{ct}}\ge
c^{\mathrm{ct}}_g$, because $g_{\mathrm{ct}}\ge c^{\mathrm{ct}}_g$ on $[0,1]$ (proof of Lemma~\ref{lem:d2:mom}, Step 1). A shell bond contributes at least
$(2\pi/\gph)\,e^{-\kappa_N(\rho^2+\rho'^2)}$, where $\rho$, $\rho'$ are the values of $\rho$ at its two ends, because
$h\ge e^{-\kappa_N(\rho^2+\rho'^2)}$ for every value of its third argument (as $|e^{i\gph\vartheta}\rho'-\rho|^2\le2\rho^2+2\rho'^2$); here $\rho\le\rhof+1$
at an end which is a shell site, a site $y$ of $\Sigma$ lies on at most $3$ shell bonds, and a site $y$ of $\Sigma^+$ lies on at most one: only the
$6\ell^2$ sites interior to the faces of $\Sigma^+$ lie on a shell bond, and the sums over $y\in\Sigma^+$ below run over these. A shell plaquette $p$
contributes $w\big((dB)_p+\cper(d(\mathfrak a+\mathfrak a^+))_p\big)\ge\exp\big(-\cper^2(d(\mathfrak a+\mathfrak a^+))_p^2-4\cper^2\big)$, because
$|(dB)_p|\le4\pi/\gph=2\cper$; and the sum over the shell plaquettes of $\cper^2(d(\mathfrak a+\mathfrak a^+))_p^2$ is at most
$2\cper^2|\mathfrak a|^2+2\cper^2|\mathfrak a^+|^2$, by the values in Claim~1: a bond of $\Sigma$ lies in at most two shell plaquettes and a bond of $\Sigma^+$ in at most one, by (G5), and $(\alpha+\beta)^2\le2\alpha^2+2\beta^2$ on the plaquettes that contain two bonds of $\Sigma^+$ (on the torus $\mathfrak a^+$ is absent). With
$c_{\mathrm{sh}}:=|\log c^{\mathrm{ct}}_g|+|\log(2\pi/\gph)|+2\kappa_N(\rhof+1)^2+4\cper^2$ this gives, for every $\zeta\in\Om^-$ and every $\wt\om$ with
closed Villain variables,
\begin{align}
-\log\cZ^{\mathrm{sh}}_\ell(\zeta)&\le c_{\mathrm{sh}}|\fA_\ell|+3\kappa_N\sum_{y\in\Sigma}\rho_y(\zeta)^2+2\cper^2\,\cE_\Sigma(\mathfrak m(\zeta)),\label{tv:A:LT}\\
-\log\cZ^{\mathrm{sh}}(\zeta,\wt\om)&\le c_{\mathrm{sh}}|\fA|+3\kappa_N\sum_{y\in\Sigma}\rho_y(\zeta)^2+\kappa_N\sum_{y\in\Sigma^+}\rho_y(\wt\om)^2
+2\cper^2\,\cE_\Sigma(\mathfrak m(\zeta))+2\cper^2\,\cE_{\Sigma^+}(\mathfrak m^+(\wt\om)).\label{tv:A:LK}
\end{align}
In particular $\cZ^{\mathrm{sh}}_\ell(\zeta)>0$ for every $\zeta$ and $\cZ^{\mathrm{sh}}(\zeta,\wt\om)>0$ for every $\zeta$ and $\nu$-almost every $\wt\om$,
as used in Step 2.

\emph{Step 5 (integer functions on the bonds of the sphere with prescribed coboundary).}

\emph{Claim~2 (rounding).} Let $\mathfrak r$ be a real function on the bonds of $\Sigma$ with $d_\Sigma\mathfrak r$ integer valued. Then there is an
integer-valued function $\mathfrak a$ on the bonds of $\Sigma$ with $d_\Sigma\mathfrak a=d_\Sigma\mathfrak r$ and
$\lfloor\mathfrak r_b\rfloor\le\mathfrak a_b\le\lceil\mathfrak r_b\rceil$ for every bond $b$; hence $|\mathfrak a|^2\le2|\mathfrak r|^2+24\,\ell_-^2$.

\emph{Proof of Claim~2.} Let $\mathrm{Fr}$ be the set of the bonds whose value is not an integer. At every plaquette the signed sum of its four values is an
integer, so no plaquette has exactly one bond in $\mathrm{Fr}$: in the dual graph the edge set $\mathrm{Fr}$ has no vertex of degree $1$ and, if it is not
empty, it contains a cycle $p_0,b_1,p_1,\dots,b_m,p_m=p_0$, where $b_i$ is the bond shared by $p_{i-1}$ and $p_i$ and the $b_i$ are distinct. Add
$t\,o_{p_i}[p_i:b_i]$ to the value on $b_i$, $i=1,\dots,m$. This does not change $d_\Sigma$: at $p_i$ the bonds $b_i$ and $b_{i+1}$ contribute
\[
t\,(o_{p_i}[p_i:b_i])^2+t\,o_{p_i}[p_i:b_{i+1}]\,o_{p_{i+1}}[p_{i+1}:b_{i+1}]=t-t=0,
\]
by~\eqref{tv:A:orient} (indices modulo $m$). Increase $t$ from $0$
until the first bond of the cycle reaches an integer: no value has crossed an integer, and $\mathrm{Fr}$ has lost a bond. Bonds outside $\mathrm{Fr}$ are
never changed again. After at most $12\,\ell_-^2$ rounds $\mathrm{Fr}$ is empty. Each final value lies between the floor and the ceiling of the original
one, so $|\mathfrak a_b|\le|\mathfrak r_b|+1$ on each of the $12\,\ell_-^2$ bonds. This proves Claim~2.

\emph{Minimal real energy.} For an integer function $\mathfrak m$ of zero sum on the plaquettes of $\Sigma$, the function
$\mathfrak r:=d_\Sigma^*\mathcal L_\Sigma^{-1}\mathfrak m$ satisfies $d_\Sigma\mathfrak r=\mathcal L_\Sigma\mathcal L_\Sigma^{-1}\mathfrak m=\mathfrak m$ and
$|\mathfrak r|^2=\langle\mathfrak m,\mathcal L_\Sigma^{-1}\mathfrak m\rangle$. By Claim~2,
\begin{equation}\label{tv:A:R1}
\cE_\Sigma(\mathfrak m)\le2\,\langle\mathfrak m,\mathcal L_\Sigma^{-1}\mathfrak m\rangle+24\,\ell_-^2<\infty .
\end{equation}

For $\mathfrak m=\mathfrak m(\zeta)$ let $f^\Sigma_p:=o_pf_p$ for the plaquettes $p$ of $\Sigma$, with $f_p=(dB)_p+v_p$ the curvature of $\zeta$, and let
$B^\Sigma$ be the restriction of $B$ to the bonds of $\Sigma$. Then $o_p(dB)_p=(d_\Sigma B^\Sigma)_p$, so that
$\cper\,\mathfrak m(\zeta)=f^\Sigma-d_\Sigma B^\Sigma$; in particular $f^\Sigma$ has zero sum. Since $d_\Sigma^*\mathcal L_\Sigma^{-1}d_\Sigma$ is the
orthogonal projection of the functions on the bonds of $\Sigma$ onto the range of $d_\Sigma^*$, the identity
$\cper\,d_\Sigma^*\mathcal L_\Sigma^{-1}\mathfrak m(\zeta)=d_\Sigma^*\mathcal L_\Sigma^{-1}f^\Sigma-d_\Sigma^*\mathcal L_\Sigma^{-1}d_\Sigma B^\Sigma$ and
$|B_b|\le\pi/\gph=\cper/2$ on the $12\,\ell_-^2$ bonds of $\Sigma$ give
\begin{equation}\label{tv:A:R2}
\cper^2\,\langle\mathfrak m(\zeta),\mathcal L_\Sigma^{-1}\mathfrak m(\zeta)\rangle\le2\,\langle f^\Sigma,\mathcal L_\Sigma^{-1}f^\Sigma\rangle+2\,|B^\Sigma|^2
\le2\,\langle f^\Sigma,\mathcal L_\Sigma^{-1}f^\Sigma\rangle+6\,\cper^2\ell_-^2 .
\end{equation}
Together,
\begin{equation}\label{tv:A:R3}
\cE_\Sigma(\mathfrak m(\zeta))\le\frac4{\cper^2}\,\langle f^\Sigma,\mathcal L_\Sigma^{-1}f^\Sigma\rangle+36\,\ell_-^2 .
\end{equation}
The same holds on $\Sigma^+$ for $\mathfrak m^+(\wt\om)$, with $\ell_+$ in place of $\ell_-$ and with the curvature $f^{\Sigma^+}_p:=o_pf_p$ and the link
variables of $\wt\om$ on $\Sigma^+$.

\emph{Step 6 (the mean of the surface energy, where the joint form of (I2) enters; the trace bound).}

\emph{Claim~3 (mean of the surface energy).} Let $\mathbb P$ be a probability measure on the configurations of $\eps\Z^3$, or of $\eps\Z^3_\ell$, with
closed Villain variables, and suppose that with a constant $C_{\mathbb P}$
\begin{equation}\label{tv:A:hypM}
\mathbb P\Big(\Big(\sum_ps_pf_p\Big)^2\Big)\le C_{\mathbb P}\,|s|^2\qquad\text{for every real function }s\text{ on the plaquettes of }\Sigma .
\end{equation}
Then $\mathbb P\big(\langle f^\Sigma,\mathcal L_\Sigma^{-1}f^\Sigma\rangle\big)\le C_{\mathbb P}\Tr\mathcal L_\Sigma^{-1}$.

\emph{Proof of Claim~3.} Let $(\phi_i)$ be an orthonormal basis of eigenfunctions of $\mathcal L_\Sigma$ on the functions of zero sum, with eigenvalues
$\varkappa_i>0$. Then $\langle f^\Sigma,\mathcal L_\Sigma^{-1}f^\Sigma\rangle=\sum_i\varkappa_i^{-1}\langle\phi_i,f^\Sigma\rangle^2$, and
$\langle\phi_i,f^\Sigma\rangle=\sum_p(\phi_i)_po_pf_p$ is a sum as in~\eqref{tv:A:hypM} with a coefficient function of norm $1$. Hence the mean is at most
$C_{\mathbb P}\sum_i\varkappa_i^{-1}=C_{\mathbb P}\Tr\mathcal L_\Sigma^{-1}$. This proves Claim~3.

The hypothesis~\eqref{tv:A:hypM} holds in the two cases that are needed.

(a) For $\mathbb P=\nu$, with $C_{\mathbb P}=2Ce^{1/2}$, $C$ being the constant of (I2). Indeed, for $|s|=1$ the inequality $\cosh y\ge1+y^2/2$ and (I2),
applied to $s$ and to $-s$ (extended by $0$ to the other plaquettes of $\eps\Z^3$), give
\[
1+\tfrac12\,\nu\Big(\Big(\sum_ps_pf_p\Big)^2\Big)\le\tfrac12\,\nu\big(e^{\sum_ps_pf_p}\big)+\tfrac12\,\nu\big(e^{-\sum_ps_pf_p}\big)\le Ce^{1/2},
\]
so that $\nu((\sum_ps_pf_p)^2)\le2(Ce^{1/2}-1)$; the general case follows by scaling $s$. The same holds on $\Sigma^+$. Hypothesis (I2) is used here in its
joint form: the spheres $\Sigma$ and $\Sigma^+$ grow with $\ell$, so that one constant $C$ valid for all finitely supported $s$ is required.

(b) For $\mathbb P=\mu_\ell$, with $C_{\mathbb P}=1$. Let $\mu^u_\ell$ be the unitary-gauge periodic measure of $\eps\Z^3_\ell$: the normalised
weight~\eqref{rv:eq:W} on the cells of the torus, the sum over the Villain variables running over all closed $(v_p)$, every flux class included. As
$\ell\ge8$, the lattice $\ell\eps\Z^3$ is a period lattice for $\eps\Z^3$ in the sense of Definition~\ref{rot:def:model} (its non-zero vectors have length
at least $8\eps$), and $\mu^u_\ell$ is the Villain model of $\eps\Z^3$ on the torus $\R^3/\ell\eps\Z^3$ of that definition. By Lemma~\ref{rot:lem:GDall},
whose proof (Steps 1--4 of the proof of Theorem~\ref{thm:GD}) is made for exactly this sum over all closed Villain variables, every flux class included,
\[
\mu^u_\ell\big(e^{\sum_pa_pf_p}\big)\le e^{|a|^2/2}\qquad\text{for every real function }a\text{ on the plaquettes of }\eps\Z^3_\ell .
\]
Applying this to $a=\pm ts$, $t>0$, and using $\cosh y\ge1+y^2/2$ gives $\tfrac{t^2}2\,\mu^u_\ell((\sum_ps_pf_p)^2)\le e^{t^2|s|^2/2}-1$, whence
$\mu^u_\ell((\sum_ps_pf_p)^2)\le|s|^2$ as $t\downarrow0$.

Finally, $(\tr,f)$ has the same law under $\mu_\ell$ and under $\mu^u_\ell$, by the change of
variables of Lemma~\ref{rv:lem:phase}(a), which is given by the cell-local formula~\eqref{rv:eq:gauge} and applies on the torus. In detail: at fixed
phases $\psi$ with lift $\wt\psi\in\FD^{\mathrm{sites}}$ put
\[
B'_b:=[B_b+(d\wt\psi)_b]_{\FD},\qquad n_b:=\big(B_b+(d\wt\psi)_b-B'_b\big)/\cper\in\Z,\qquad v'_p:=v_p+\cper\,(dn)_p .
\]
Then $(dB')_p+v'_p=\big((dB)_p-\cper(dn)_p\big)+\big(v_p+\cper(dn)_p\big)=f_p$; $(v'_p)$ is closed if and only if $(v_p)$ is, and in the same flux class,
$\cper\,dn$ being exact; $h(\cdot,\cdot,B_b+\psi_{b_+}-\psi_{b_-})=h(\cdot,\cdot,B'_b)$ by the $\cper$-periodicity of $h$; the map $B\mapsto B'$ is a
rotation of each $\FD$ and preserves Lebesgue measure; and, at fixed $B$, $(v_p)\mapsto(v'_p)$ is a bijection of the closed Villain variables. So,
conditionally on $\psi$, the variables $\tr$, $B'$ and $(v'_p)$ have the law $\mu^u_\ell$. Hence $(\tr,f)$ has the same law under $\mu_\ell$ and under
$\mu^u_\ell$, and the two partition functions coincide.

\emph{Claim~4 (trace bound).} $\Tr\mathcal L_\Sigma^{-1}\le144\,\ell_-^2(1+\log_2\ell_-)^2$.

\emph{Proof of Claim~4.} $\Tr\mathcal L_\Sigma^{-1}=\sum_p\mathcal L_\Sigma^{-1}(p,p)$, the sum over the plaquettes of $\Sigma$. For a set $A$ of
plaquettes of $\Sigma$ write $\bar\bbone_A:=|A|^{-1}\bbone_A$, and $\bbone_p$ for the indicator of the plaquette $p$. Since $\mathcal L_\Sigma^{-1}$
annihilates the constants,
\[
\mathcal L_\Sigma^{-1}(p,p)=\big\langle\bbone_p-\bar\bbone_\Sigma,\mathcal L_\Sigma^{-1}(\bbone_p-\bar\bbone_\Sigma)\big\rangle
=\min\big\{|\mathfrak r|^2:\ \mathfrak r\text{ real on the bonds of }\Sigma,\ d_\Sigma\mathfrak r=\bbone_p-\bar\bbone_\Sigma\big\},
\]
the minimiser being $d_\Sigma^*\mathcal L_\Sigma^{-1}(\bbone_p-\bar\bbone_\Sigma)$. So it suffices to exhibit, for every plaquette $p$ of $\Sigma$, a
real $\mathfrak r$ with $d_\Sigma\mathfrak r=\bbone_p-\bar\bbone_\Sigma$ and $|\mathfrak r|\le(2\sqrt2+2)(1+\log_2\ell_-)$.

(SF) \emph{Strip flows.} Let $\Pi$ be a set of plaquettes of $\Sigma$ arranged in $q$ rows, each row a chain of $m$ plaquettes in which consecutive ones
share a bond (inside one face of the cube, or continuing across edges of the cube in an unfolded strip of faces). Let $\mathfrak t$ be a real function on
$\Pi$ whose sum over every row is $0$ and whose partial sums along every row have modulus at most $\mathfrak t^*$. Then the function $\mathfrak r$ which,
on the bond between two consecutive plaquettes of a row, equals the partial sum of $\mathfrak t$ up to the first of them, with the sign
$o_p[p:b]$ of that bond $b$ in that plaquette $p$, and which vanishes on all other bonds, satisfies $d_\Sigma\mathfrak r=\mathfrak t$ (by~\eqref{tv:A:orient})
and $|\mathfrak r|^2\le q(m-1)(\mathfrak t^*)^2$. In particular, if $\Pi'$ and $\Pi''$ are subsets of $\Pi$ each consisting of full columns of $\Pi$ (the
$i$-th plaquettes of the rows forming the $i$-th column), then $\mathfrak t=\bar\bbone_{\Pi'}-\bar\bbone_{\Pi''}$ has row sums $0$ and partial sums in
$[-1/q,1/q]$, so that $|\mathfrak r|^2\le m/q$.

\emph{Inside the face.} Let $\Sigma_1$ be the face of the cube that contains $p$, an $\ell_-\times\ell_-$ array of plaquettes. Put
$\ell_j:=\min(2^j,\ell_-)$ and $j_*:=\lceil\log_2\ell_-\rceil$, and choose rectangles $\Pi_0=\{p\}\subset\Pi_1\subset\dots\subset\Pi_{2j_*}=\Sigma_1$, where
$\Pi_{2j-1}$ has $\ell_j$ columns and $\ell_{j-1}$ rows and $\Pi_{2j}$ has $\ell_j$ columns and $\ell_j$ rows (intervals of columns and of rows containing
the previous ones). For the step from $\Pi_{2j-2}$ to $\Pi_{2j-1}$, (SF) on $\Pi_{2j-1}$ with its $q=\ell_{j-1}$ rows of length $m=\ell_j$ gives
$\mathfrak r$ with $d_\Sigma\mathfrak r=\bar\bbone_{\Pi_{2j-2}}-\bar\bbone_{\Pi_{2j-1}}$ and $|\mathfrak r|^2\le\ell_j/\ell_{j-1}\le2$. For the step from
$\Pi_{2j-1}$ to $\Pi_{2j}$, (SF) with the $q=\ell_j$ columns of $\Pi_{2j}$ as chains of length $m=\ell_j$ gives
$d_\Sigma\mathfrak r=\bar\bbone_{\Pi_{2j-1}}-\bar\bbone_{\Pi_{2j}}$ and $|\mathfrak r|^2\le1$. These are $2j_*$ steps, each of norm at most $\sqrt2$.

\emph{From the face to the sphere.} Let $\Sigma_2,\dots,\Sigma_5$ be the faces adjacent to $\Sigma_1$ and $\Sigma_6$ the opposite face. For each
$i=2,\dots,6$ we give $\mathfrak r$ with $d_\Sigma\mathfrak r=(\bar\bbone_{\Sigma_1}-\bar\bbone_{\Sigma_i})/6$. For $i\le5$: (SF) on the unfolded
$\ell_-\times2\ell_-$ strip $\Sigma_1\cup\Sigma_i$ with its $\ell_-$ rows across the common edge; the partial sums are at most $\ell_-/(6\ell_-^2)$, so
$|\mathfrak r|^2\le\ell_-(2\ell_-)(6\ell_-)^{-2}=1/18$. For $i=6$: the $\ell_-\times3\ell_-$ strip $\Sigma_1\cup\Sigma_2\cup\Sigma_6$ through the adjacent
face $\Sigma_2$; $|\mathfrak r|^2\le\ell_-(3\ell_-)(6\ell_-)^{-2}=1/12$. The sum of the five has
$d_\Sigma=\tfrac56\bar\bbone_{\Sigma_1}-\tfrac16\sum_{i=2}^6\bar\bbone_{\Sigma_i}=\bar\bbone_{\Sigma_1}-\bar\bbone_\Sigma$ and norm at most
$4/\sqrt{18}+1/\sqrt{12}<2$.

Adding all these functions gives $\mathfrak r$ with
\[
d_\Sigma\mathfrak r=(\bbone_p-\bar\bbone_{\Pi_1})+\dots+(\bar\bbone_{\Pi_{2j_*-1}}-\bar\bbone_{\Sigma_1})
+(\bar\bbone_{\Sigma_1}-\bar\bbone_\Sigma)=\bbone_p-\bar\bbone_\Sigma
\]
and $|\mathfrak r|\le2\sqrt2\,j_*+2\le(2\sqrt2+2)(1+\log_2\ell_-)$. So
$\mathcal L_\Sigma^{-1}(p,p)\le24\,(1+\log_2\ell_-)^2$, and $\Sigma$ has $6\ell_-^2$ plaquettes. This proves Claim~4. The same bound holds on $\Sigma^+$
with $\ell_+$ in place of $\ell_-$.

\emph{Step 7 (the means of the lower bounds).} By~\eqref{tv:A:R3}, Claim~3 and Claim~4, for $\mathbb P=\nu$ (with $C_{\mathbb P}=2Ce^{1/2}$) and for
$\mathbb P=\mu_\ell$ (with $C_{\mathbb P}=1$), $\mathfrak m$ being the function $\mathfrak m(\zeta)$ of the variables of $\mathfrak C^-$,
\begin{equation}\label{tv:A:M1}
\mathbb P\big(\cE_\Sigma(\mathfrak m)\big)\le\frac4{\cper^2}\,C_{\mathbb P}\Tr\mathcal L_\Sigma^{-1}+36\,\ell_-^2
\le576\,C_{\mathbb P}\,\cper^{-2}\,\ell_-^2(1+\log_2\ell_-)^2+36\,\ell_-^2 ,
\end{equation}
and for $\mathbb P=\nu$ the same bound holds for $\cE_{\Sigma^+}(\mathfrak m^+)$ with $\ell_+$ in place of $\ell_-$.

For the density,
\begin{equation}\label{tv:A:M2}
\nu(\rho_y^2)\le m_\nu,\qquad\mu_\ell(\rho_y^2)\le C_\rho:=\log\max\Big\{1,\frac{G^{\mathrm{ct}}_1(2\pi/\gph)^3K_{\gph}'^{\,3}}{c_0}\Big\}
\end{equation}
for every site $y$, with the constants of Lemma~\ref{lem:d2:mom}. The first bound is the definition of $m_\nu$. For the second, $\rho$ has the same law
under $\mu_\ell$ and under $\mu^u_\ell$ (Step 6(b)), and $\mu^u_\ell$ is invariant under the translations of the torus, which act by automorphisms of the
cell complex preserving the cell-local weight and the reference measure. So, by Jensen's inequality,
\[
\mu^u_\ell(\rho_y^2)=\ell^{-3}\,\mu^u_\ell\Big(\sum_x\rho_x^2\Big)\le\ell^{-3}\log\mu^u_\ell\big(e^{\sum_x\rho_x^2}\big)=\ell^{-3}\log\frac{\cZ^{(1)}_\ell}{\cZ^{(0)}_\ell},
\]
where $\cZ^{(q)}_\ell$, $q=0,1$, is the integral of the unitary-gauge weight of the torus with $g_{\mathrm{ct}}(\tr_x)$ replaced by
$g_{\mathrm{ct}}(\tr_x)e^{q\rho_x^2}$ at every site, the sum running over all closed Villain variables. The crude bounds of Step 1 of the proof of
Lemma~\ref{lem:d2:mom} hold on $\eps\Z^3_\ell$, which has three bonds and three plaquettes per site:
$\cZ^{(1)}_\ell\le\big(G^{\mathrm{ct}}_1(2\pi/\gph)^3K_{\gph}'^{\,3}\big)^{\ell^3}$, because $h\le1$, the sum over the closed Villain variables is at most the
free sum and $\sum_{v_p}w((dB)_p+v_p)\le K'_{\gph}$; and $\cZ^{(0)}_\ell\ge c_0^{\,\ell^3}$, from the term with all Villain variables equal to $0$, which is
closed, and the integration restricted to $\tr_x\in[0,1]$ and $|\gph B_b|\le1$. Reflection positivity is not used in these bounds.

\emph{Step 8 (assembly).} In the first term of~\eqref{tv:A:E1}, $\zeta$ and the variables of $\wt\om$ outside $V(\Lambda)$, the only ones on which the
integrand depends, have by the Gibbs property their joint law under $\nu$; in particular $\zeta$ has the law $\nu^-$. Insert~\eqref{tv:A:U} for $\cZ^{\mathrm{sh}}$
and~\eqref{tv:A:LT}. In the second term insert~\eqref{tv:A:E2}, then~\eqref{tv:A:U} for $\cZ^{\mathrm{sh}}_\ell$ and~\eqref{tv:A:LK}, $\zeta'$ having the
law $\mu^-_\ell$. This gives
\begin{multline}\label{tv:A:asm}
\mathrm{Ent}(\nu^-\mid\mu^-_\ell)\le(|\fA|+|\fA_\ell|)(\log C_{\mathrm{sh}}+c_{\mathrm{sh}})
+3\kappa_N\sum_{y\in\Sigma}\big[\nu(\rho_y^2)+\mu_\ell(\rho_y^2)\big]+\kappa_N\sum_{y\in\Sigma^+}\nu(\rho_y^2)\\
+2\cper^2\big[\nu\big(\cE_\Sigma(\mathfrak m)\big)+\mu_\ell\big(\cE_\Sigma(\mathfrak m)\big)\big]+2\cper^2\,\nu\big(\cE_{\Sigma^+}(\mathfrak m^+)\big).
\end{multline}
By (G2), \eqref{tv:A:M1} and~\eqref{tv:A:M2}, with at most $7\ell^2$ sites of $\Sigma$ and $6\ell^2$ sites of $\Sigma^+$ on shell bonds, $\ell_-\le\ell$
and $\ell_+\le2\ell$,
\[
\mathrm{Ent}(\nu^-\mid\mu^-_\ell)\le C_{\mathrm{TV}}\,\ell^2(1+\log_2\ell)^2,\qquad
C_{\mathrm{TV}}:=228\,(\log C_{\mathrm{sh}}+c_{\mathrm{sh}})+7\kappa_N(4m_\nu+3C_\rho)+C_{\mathrm{fl}},
\]
\[
C_{\mathrm{fl}}:=39168\,Ce^{1/2}+1152+432\,\cper^2 .
\]

Here $C_{\mathrm{fl}}$ collects the terms with $\cE$: by~\eqref{tv:A:M1},
\[
2\cper^2[\nu(\cE_\Sigma(\mathfrak m))+\mu_\ell(\cE_\Sigma(\mathfrak m))]
\le(2304\,Ce^{1/2}+1152)\,\ell_-^2(1+\log_2\ell_-)^2+144\,\cper^2\ell_-^2
\]
and
$2\cper^2\nu(\cE_{\Sigma^+}(\mathfrak m^+))\le2304\,Ce^{1/2}\ell_+^2(1+\log_2\ell_+)^2+72\,\cper^2\ell_+^2$, with $\ell_+^2\le4\ell^2$ and
$(1+\log_2\ell_+)^2\le4(1+\log_2\ell)^2$. All the terms being finite, the splittings of the integrals in~\eqref{tv:A:E1} and~\eqref{tv:A:E2} are
legitimate: $\log\cZ^{\mathrm{sh}}$ and $\log\cZ^{\mathrm{sh}}_\ell$ are bounded above by~\eqref{tv:A:U} and below by integrable functions, and
$\cZ_\Lambda(\wt\om)\le C_{\mathrm{sh}}^{\,|\fA|}\int W^-(\zeta)\,d\zeta<\infty$. This is the entropy bound~\eqref{tv:A:ent}; by Step 1 it
gives~\eqref{tv:A:TV} and the lemma.
\end{proof}

\begin{lemma}[Schatten--H\"older bound]\label{rv:lem:UV}
For fibred insertions $X_1,\dots,X_k$ at distinct heights that satisfy the summability condition of Lemma~\ref{rv:lem:zak}(b) (insertions at the same height on disjoint bond sets tensor, $\|X\otimes X'\|=\|X\|\|X'\|$, the dressing being
a product over bonds), the periodic partition function with $M$ layers satisfies
\[
|Z_M(X_1,\dots,X_k)|=|\Tr_{L_G}(\prod_s\mathsf{B}^*\widehat{\bar X_s}\mathsf{B})|\le\prod_s\|\bar X_s\|\cdot\Tr_{L_G}(T^M)\le\prod_s\|X_s\|\,Z_M,
\]
uniformly in the number of insertions.
\end{lemma}

The only input of Lemma~\ref{rv:lem:UV} is $Z_M=\Tr(P_GT^M)\le\Tr(T^M)<\infty$, the finite torus partition function at fixed $N$ (\eqref{eq:ct} and the Gaussian Villain sums).

\begin{proof}
The equality is~\eqref{rv:eq:avg}. With $D:=\mathsf{B}\mathsf{B}^*\ge0$ on $\cH$, $\Tr D^M=\Tr T^M$, $D$ commutes with $P^{\cH}_G$, cyclicity moves the trace to $\Tr(\prod_s\widehat{\bar
X_s}D\,P^{\cH}_G)$, and H\"older's
inequality for Schatten norms \cite{Sim05} with exponents $p_s=M$ gives the bound $\prod_s\|\bar X_s\|\,\Tr(P^{\cH}_GD^M)=\prod_s\|\bar X_s\|\,Z_M$; finally $\|\bar X_s\|\le\|X_s\|$ (Lemma~\ref{rv:lem:frame}(c)).
\end{proof}

\begin{lemma}[growth of $r_Y$]\label{rv:lem:growth}
Let $Y=\sum_{a\in\cper\Z^S}\tau_a^*Y_0\tau_a$ with $Y_0$ bounded and $Y_0=P_KY_0P_K$, $P_K:=\bbone_{[-K,K]^S}(\xi)$. Then $Y$ is bounded, fibred (Lemma~\ref{rv:lem:zak}(b2)), $\Z$-periodic, and
$|r_Y(B,u)|\le C_Ye^{c_Y|f|}$ with $c_Y=2K\sqrt{|S|}$, $f=B-u$.
\end{lemma}

\begin{proof}
$|k_{\tau_a^*Y_0\tau_a}(B,u)|=|\langle P_Kh^{B-a},Y_0P_Kh^{u-a}\rangle|\le\|Y_0\|\,\|P_Kh^{B-a}\|\,\|P_Kh^{u-a}\|\le C\|Y_0\|e^{-d_1^2-d_2^2}$ with $d_i$ the distances of $B-a$, $u-a$ to
$Q_K:=[-K,K]^S$. Since $|f|\le d_1+d_2+2K\sqrt{|S|}$, $d_1^2+d_2^2\ge\frac12(d_1+d_2)^2\ge\frac12|f|^2-2K\sqrt{|S|}\,|f|$, and the sum over $a$ of the remaining Gaussian factors is
bounded uniformly; dividing by $k_1=e^{-|f|^2/2}$ gives the claim. Boundedness of $Y$ and $\Z$-periodicity are clear from the form of $Y$ (the translates $\tau_a^*P_K$ are
pairwise orthogonal for $|a|$ large, and finitely many overlap).
\end{proof}

\begin{lemma}[corrector]\label{rv:lem:corr}
Put $D:=\cper\sqrt{|S|}/2$ (half-diameter of a cell of $\cper\Z^S$), $\beta:=2c_Y+D+1$ and $Z:=\delta+\sum_{a\in\cper\Z^S}\alpha_a\tau_a$ with $\alpha_a:=C_Ze^{\beta|a|-|a|^2/2}$, $\delta>0$.
Then $Z$ is bounded ($\sum_a\alpha_a<\infty$), fibred (Lemma~\ref{rv:lem:zak}(b2)), $\Z$-periodic and real self-adjoint ($\alpha_a=\alpha_{-a}$), $r_Z(B,u)=\delta+\sum_a\alpha_ae^{-a\cdot f-|a|^2/2}$, and for $C_Z\ge C_Y^2e^{\beta
D+D^2}/4$
\begin{equation}\label{rv:eq:corr}
\delta+\tfrac14r_Y^2\le r_Z\le\delta+Ce^{|f|^2/4+(\beta/2)|f|}.
\end{equation}
Hence $X_t:=1+tY+t^2Z$ has $r_{X_t}=1+tr_Y+t^2r_Z\ge(1+tr_Y/2)^2+\delta t^2>0$ for every real $t$, and $\|X_t\|\le1+t^2(\|Z\|+\|Y\|^2/2)+O(t^3)$ for real skew-adjoint $Y$.
\end{lemma}

\begin{proof}
Lower bound: let $a^*$ be the lattice point nearest to $-f$, $|a^*+f|\le D$. With $a^*=-f+e$, $|e|\le D$: $\beta|a^*|\ge\beta|f|-\beta D$ and $-|a^*|^2-a^*\!\cdot f=-a^*\!\cdot e\ge-D|f|-D^2$, so the
$a^*$-term alone gives
\[
r_Z\ge\delta+C_Ze^{(\beta-D)|f|-\beta D-D^2}=\delta+C_Ze^{(2c_Y+1)|f|-\beta D-D^2}\ge\delta+\tfrac14C_Y^2e^{2c_Y|f|}\ge\delta+\tfrac14r_Y^2
\]
(Lemma~\ref{rv:lem:growth}).
Upper bound: $-|a|^2-a\cdot f=\tfrac14|f|^2-|a+f/2|^2$ and $\beta|a|\le\beta|a+f/2|+\tfrac12\beta|f|$, and $\sum_ae^{\beta|a+f/2|-|a+f/2|^2}$ is bounded uniformly in $f$. The bound on $\|X_t\|$ follows from
$\|1+tY\|^2=\|1-t^2Y^2\|\le1+t^2\|Y\|^2$ for real skew-adjoint $Y$.
\end{proof}

\begin{lemma}[moments from (I2)]\label{rv:lem:moments}
Let $\nu$ satisfy \textup{(I2)} and let $f=(f_p)_{p\in P}$ be the curvature on a finite plaquette set $P$. Then $\nu(e^{s|f_p|})\le2Ce^{s^2/2}$ for all $s\ge0$, and
$\nu(e^{a|f|^2/2})\le C(1-a)^{-|P|/2}$ for every $a\in(0,1)$: Gaussian moments at every rate $a/2<1/2$, jointly over $P$.
\end{lemma}

\begin{proof}
The first claim follows from (I2) and $e^{s|f_p|}\le e^{sf_p}+e^{-sf_p}$. For the second claim (Hubbard--Stratonovich) $e^{a|f|^2/2}=(2\pi)^{-|P|/2}\int e^{\sqrt a\,s\cdot f-|s|^2/2}ds$, and (I2) under the $s$-integral gives
$\nu(e^{a|f|^2/2})\le C(2\pi)^{-|P|/2}\int e^{(a-1)|s|^2/2}ds=C(1-a)^{-|P|/2}$.
\end{proof}

For a fibred $X$ that satisfies the summability condition of Lemma~\ref{rv:lem:zak}(b), the insertion of $X$ at one slab multiplies the slab weight by $r_X\circ\Theta_\sigma$ (the slab data $(B,u)$ have density $k_1(B,u)$ per bond set,
Lemma~\ref{rv:lem:slab}(b), and the inserted kernel of Lemma~\ref{rv:lem:zak}(b) has $k_X$ in its place), so that for insertions $X_s$ at distinct heights, the sum over the variables $\wt v^t$ being taken first in the expectation on the left (for non-negative ratios $r_{X_s}$, as for the operators $X_t$ of Lemmas~\ref{rv:lem:step2} and~\ref{rv:lem:step2full}, the expectation is then an ordinary integral, by Tonelli's theorem),
\begin{equation}\label{rv:eq:insert}
\mu_M\Big(\prod_sr_{X_s}\circ\Theta_\sigma^{(s)}\Big)=Z_M(X_1,\dots,X_k)/Z_M.
\end{equation}

\begin{lemma}[$\nu(r_Y)=0$ for the bond pair]\label{rv:lem:step2}
Let $\nu$ be as in Theorem~\ref{rv:thm:RV} (after Lemma~\ref{rv:lem:phase}), $Y$ real skew-adjoint of the class of Lemma~\ref{rv:lem:growth}, $Z$ the corrector of
Lemma~\ref{rv:lem:corr} and $v_t:=\log(1+tr_Y+t^2r_Z)\circ\Theta_\sigma$, a local slab function. Then $v_t\ge\log(\delta t^2)$,
$v_t\le|t|\,|r_Y|+t^2r_Z\le C(1+|t|e^{c_Y|f|})+t^2Ce^{|f|^2/4+(\beta/2)|f|}$, $\nu(v_t)=O(t^2)$, and $\nu(r_Y)=0$, i.e.\ $\nu\big(k_Y\circ\Theta_\sigma/k_1\circ\Theta_\sigma\big)=0$.
\end{lemma}

\begin{proof}
The two bounds on $v_t$ are~\eqref{rv:eq:corr} and $\log(1+x)\le x$.

\emph{Upper bound.} Choose a sublattice $\Gamma'\subset\Gamma$ (still of finite index, containing a time translation,
$\nu$ being $\Gamma'$-invariant) such that the translates of the window $S$ at equal height are pairwise disjoint; then same-height insertions tensor and Lemma~\ref{rv:lem:UV} applies: the insertion $X_t=1+tY+t^2Z$ is fibred by Lemma~\ref{rv:lem:zak}(b1)--(b2), $Y$ being a localised sum and $Z$ a summable series of lattice translations, and it satisfies the summability condition by Remark~\ref{rv:rem:abs} (the condition holds for $1$ and is stable under sums); so do its tensor products over disjoint bond sets, which are fibred by (b2) and for which the sum factorises.
Lemma~\ref{rv:lem:TV} applies to $v_t$ (bounded below; (I1), (I2) and the $\Gamma'$-invariance of $\nu$ give its hypotheses).
The lemma is applied on its cubic tori $\eps\Z^3_\ell$, $\ell\eps\Z^3\subset\Gamma'$: the measure $\mu_\ell$ is the periodic measure $\mu_M$ with $M=\ell$ layers over the layer torus of side
$\ell$ (so that $\mu_M=\mu_\ell$, $V_M=V_\ell$ and $n_M=n_\ell$ below), and on $\eps\Z^3_\ell$ the translates of the window $S$ at equal height stay pairwise disjoint, because they are the
images of distinct $\Gamma'$-translates of $S$ in $\eps\Z^3$ and $\ell\eps\Z^3\subset\Gamma'$.
By~\eqref{rv:eq:insert} and Lemma~\ref{rv:lem:UV} on these tori, $\mu_M(e^{V_M})=Z_M(X_t,\dots,X_t)/Z_M\le\|X_t\|^{n_M}$,
so $\nu(v_t)\le\log\|X_t\|\le t^2(\|Z\|+\|Y\|^2/2)+O(t^3)$ by Lemma~\ref{rv:lem:corr}.

\emph{Lower bound.} On $A_t:=\{|t|\,|r_Y|\le\tfrac12\}$, $r_Z\ge0$ gives $v_t\ge\log(1+tr_Y)\ge tr_Y-t^2r_Y^2$;
off $A_t$, $v_t\ge\log(\delta t^2)$. Hence, for $|t|$ small,
\[
t\,\nu(r_Y)\le\nu(v_t)+t^2\nu(r_Y^2)+|t|\,\nu(|r_Y|\bbone_{A_t^c})+\log\big(1/(\delta t^2)\big)\,\nu(A_t^c).
\]
Here $\nu(r_Y^2)<\infty$ by Lemma~\ref{rv:lem:growth} and the exponential moments of Lemma~\ref{rv:lem:moments}; $A_t^c$ forces $|f|\ge c_Y^{-1}\log(1/(2C_Y|t|))$, so by the Gaussian
moments of Lemma~\ref{rv:lem:moments} $\nu(A_t^c)=O(e^{-a'\log^2(1/|t|)})$ is $o(t^k)$ for every $k$, and $|t|\nu(|r_Y|\bbone_{A_t^c})\le|t|\nu(r_Y^2)^{1/2}\nu(A_t^c)^{1/2}=o(t^k)$. The right side is
$O(t^2)$ for both signs of $t$, so $\nu(r_Y)=0$. The argument uses no moment of $r_Z$.
\end{proof}

\begin{remark}[what Subsection~\ref{rv:ssec:S4} uses]\label{rv:rem:S4inputs}
Hypotheses (I1) and (I2) are used in Lemma~\ref{rv:lem:TV}, the latter in its joint form, with one constant for all finitely supported $s$.
Hypothesis (I2) is used once more in the proof of Lemma~\ref{rv:lem:step2}, for $\nu(r_Y^2)<\infty$ and for the tail of $A_t$. The growth of $r_Z$ at the rate $1/4$ in~\eqref{rv:eq:corr} is used only in
Lemma~\ref{rv:lem:step2full} (for the set $A_t$ of its proof), together with the Gaussian moments at rates $>3/8$ of Lemma~\ref{rv:lem:moments}.
\end{remark}

\subsection{Bond-sector separation}\label{rv:ssec:S5}

\begin{lemma}[bond pair law]\label{rv:lem:bondlaw}
Let
\[
\cC:=\{\sum_{a\in\cper\Z^S}\tau_a^*X_0\tau_a:\ X_0\text{ bounded on }L^2(\R^S),\ X_0=P_KX_0P_K\text{ for some }K\}.
\]
$\cC$ is closed under adjoints, complex conjugation and
real-linear combinations; every $X\in\cC$ is fibred (Lemma~\ref{rv:lem:zak}(b2)), hence $\Z$-periodic and bounded, with $r_X$ exponential-linear (Lemma~\ref{rv:lem:growth}). Let $\lambda$ be the law of the pair
$(B_S,u_S)=(B_S,\wt B'_S-\wt v^t_S)$ under $\nu$ (after $\Theta_\sigma$) and $m:=k_1^{-1}\lambda$, a $\sigma$-finite measure on $\FD^S\times\R^S$. Then $m(|k_X|)=\lambda(|r_X|)<\infty$ for
$X\in\cC$, and $(R_*m)(k_X)=m(k_X)$ for every $X\in\cC$, $R$ the twisted reflection of Lemma~\ref{rv:lem:frame}(a).
\end{lemma}

\begin{proof}
Decompose $X=A+Y+iA'+iY'$ with $A,A'$ real self-adjoint and $Y,Y'$ real skew-adjoint, all in $\cC$. By Lemma~\ref{rv:lem:frame}(b), $k_A\circ R=k_A$, $k_{iA'}\circ R=k_{iA'}$,
$k_Y\circ R=-k_Y$, $k_{iY'}\circ R=-k_{iY'}$. Lemma~\ref{rv:lem:step2} gives $m(k_Y)=\nu(r_Y)=0$ for the real skew-adjoint elements, hence $m(k_X\circ R)=m(k_X)$. Finiteness is
Lemma~\ref{rv:lem:growth} with the exponential moments of Lemma~\ref{rv:lem:moments}.
\end{proof}

\begin{lemma}[separation on the bond pair]\label{rv:lem:bondsep}
Let $X_0=|g_1\rangle\langle g_2|$ with $g_i\in L^2(\R^S)$ supported in $[-K,K]^S$, let $X=\sum_a\tau_a^*X_0\tau_a\in\cC$, and put $G_i:=e^{-|\cdot|^2}*g_i$. Let
$\hat m:=\sum_{a\in\cper\Z^S}(T_a)_*m$ be the unfolding of $m$ to
$\R^S\times\R^S$, where $T_a$ is the diagonal shift; the set $\FD^S\times\R^S$ is a fundamental domain, so $m\leftrightarrow\hat m$ is a bijection. Then the following hold.
\begin{itemize}
\item[(i)] Up to the normalisation of $h$,
$k_X(B,u)=\sum_{a\in\cper\Z^S}G_1(B-a)\overline{G_2(u-a)}$, so that $m(k_X)=\hat m(G_1\otimes\overline{G_2})$.
\item[(ii)] $\hat m(e^{-\delta(|B|^2+|u|^2)})\le C\lambda(e^{(1-\delta)|f|^2/2})<\infty$ for every $\delta\in(0,1)$.
\item[(iii)] The measure $\hat m$ is determined by the numbers $\hat m(G_1\otimes\overline{G_2})$.
\item[(iv)] $R_*m=m$ and $\lambda\circ R=\lambda$.
\end{itemize}
\end{lemma}

\begin{proof}
(i) The formula for $k_X$ is the sum over $a$ of the terms $\langle h^{B-a},g_1\rangle\langle g_2,h^{u-a}\rangle$, and the identity $m(k_X)=\hat m(G_1\otimes\overline{G_2})$ follows from
it by the definition of $\hat m$.

(ii) The identity $|B+a|^2+|u+a|^2=2|a+(B+u)/2|^2+\tfrac12|f|^2$ gives $\sum_ae^{-\delta(|B+a|^2+|u+a|^2)}\le
C_\delta e^{-\delta|f|^2/2}$ uniformly in $(B,u)$. Since $m=k_1^{-1}\lambda$ and $k_1=e^{-|f|^2/2}$, it follows that
\[
\hat m(e^{-\delta(|B|^2+|u|^2)})=\int\lambda(dB\,du)\,e^{|f|^2/2}\sum_ae^{-\delta(|B+a|^2+|u+a|^2)}\le C_\delta\lambda(e^{(1-\delta)|f|^2/2}),
\]
and the right-hand side is finite by Lemma~\ref{rv:lem:moments} at the rate
$(1-\delta)/2<1/2$.

(iii) Let $(\hat m*\mathrm{Gauss})(\xi,\eta):=\int e^{-|\xi-B|^2-|\eta-u|^2}\hat m(dB\,du)$ be the Gaussian smoothing of $\hat m$; by (ii) it is finite, continuous and bounded on compact sets. By Fubini's theorem,
$\hat m(G_1\otimes\overline{G_2})=\int g_1(\xi)\overline{g_2(\eta)}\,(\hat m*\mathrm{Gauss})(\xi,\eta)\,d\xi\,d\eta$. The linear span of the products $g_1\otimes\overline{g_2}$ is dense in $L^2$ of every compact subset of $\R^S\times\R^S$,
so the numbers $\hat m(G_1\otimes\overline{G_2})$ determine the Gaussian smoothing. The smoothing in turn determines the measure. Indeed, $\hat m*\mathrm{Gauss}$
is $e^{-|\xi|^2-|\eta|^2}$ times the two-sided Laplace transform of the finite measure $e^{-|B|^2-|u|^2}\hat m(dB\,du)$; here the weight with $\delta=1$ is reached from the weight with
$\delta=\frac12$ of (ii) by a Gaussian factor. This Laplace transform
is finite everywhere and entire, and its restriction to imaginary arguments is the Fourier transform of the measure. Hence two measures with the same Gaussian smoothing coincide.

(iv) By
Lemma~\ref{rv:lem:bondlaw}, the measures $(R_*m)^\wedge$ and $\hat m$ give the same numbers. The measure $R_*m$ satisfies the same growth bound as $m$, because $k_1\circ R=k_1$ and $R$
maps $f$ to $-f$. Hence $(R_*m)^\wedge=\hat m$ by (iii), and therefore $R_*m=m$ and
$\lambda\circ R=\lambda$.
\end{proof}

The conclusion of this subsection is the law of the \emph{bond} pair only: for every finite $S$ the law under $\nu$ of the frame-reduced pair $(B_S,u_S)$ is invariant under the
twisted reflection $R$. The insertions of $\cC$ act as $X\otimes1$ on the Higgs tensor factor of Lemma~\ref{rv:lem:zak}(a), so the joint law of $(\tr,\tr',\wt\psi'-\psi,B,u)$ is not
separated by Lemma~\ref{rv:lem:bondsep}; the full statement is Theorem~\ref{rv:thm:EV}.

\subsection{Higgs and Villain sectors}\label{rv:ssec:S6}

\paragraph{(A) Higgs sector.} Let $V:=V(S)$ be the finite site set of the slab window, that is, the set of the end points of the bonds of $S$; $\mathrm{plaq}(S)$ denotes the set of the spatial plaquettes of the layer all four bonds of which lie in $S$, and $S$ is \emph{closed under the bonds of its plaquettes} if every spatial plaquette of the layer whose four corners lie in $V(S)$ belongs to $\mathrm{plaq}(S)$ (as for the set of the links of a rectangle).

\begin{lemma}[Gaussian hopping kernel, no frame]\label{rv:lem:higgskernel}
With $\Phi_x:=\rho_xe^{i\gph\psi_x}$ and $\wt\Phi'_x:=\wt\rho'_xe^{i\gph\wt\psi'_x}$ one has $|e^{i\gph(\wt\psi'_x-\psi_x)}\wt\rho'_x-\rho_x|=|\wt\Phi'_x-\Phi_x|$, so the slab Higgs factor is
$\prod_xh=\prod_xe^{-|\wt\Phi'_x-\Phi_x|^2/2}=c_V\langle h^\Phi,h^{\wt\Phi'}\rangle_{L^2(\C^V)}$ with $h^\Phi(\Xi):=e^{-|\Xi-\Phi|^2}$, and the layer reference measure
$\prod_xg_{\mathrm{ct}}(\tr_x)\,d\tr_x\,d\psi_x$ is $\prod_x\gph^{-1}e^{-V(\tr_x)-\delta m_B\rho_x^2/2}\,d^2\Phi_x$ (polar Lebesgue measure; the factor $\rho$ of $g(\tr)=\rho e^{-V}$ is the Jacobian).
Here $\Phi$ ranges over all of $\C^V$.
\end{lemma}

\begin{proof}
$|e^{i\gph\vartheta}\rho'-\rho|$ is invariant under the common rotation by $e^{i\gph\psi}$, and $d^2\Phi=\gph\rho\,d\rho\,d\psi$ on $[0,\infty)\times\Tper$. The Gaussian identity is that of
Subsection~\ref{rv:ssec:S3}
with $c_V$ the normalisation constant.
\end{proof}

\begin{lemma}[Higgs class and ratio]\label{rv:lem:higgsclass}
Let $\cC_H:=\{X_H\text{ bounded on }L^2(\C^V)=L^2(\R^{2V}):\ X_H=P_KX_HP_K\}$ with $P_K:=\bbone_{\{|\Xi_x|\le K\ \forall x\}}$, $k_{X_H}(\Phi,\wt\Phi'):=\langle h^\Phi,X_Hh^{\wt\Phi'}\rangle$,
$k^H_1=c_Ve^{-|f_H|^2/2}$ with $f_H:=\wt\Phi'-\Phi$, and $r_{X_H}:=k_{X_H}/k^H_1$. Then $|r_{X_H}|\le C_K\|X_H\|e^{c_Y|f_H|}$ with $c_Y=4\sqrt V\,K$: exponential-linear in $f_H$.
\end{lemma}

\begin{proof}
As in Lemma~\ref{rv:lem:growth} with the ball $\{|\Xi_x|\le K\ \forall x\}$ in place of $Q_K$: its Euclidean half-diameter is $\sqrt VK$ (attained at $\Phi_1=\dots=\Phi_V=K$), so the same computation gives the bound with $2\sqrt VK$ in place of $c_Y$, and a fortiori with the constant $c_Y=4\sqrt VK$ of the statement.
\end{proof}

\begin{lemma}[Higgs corrector]\label{rv:lem:higgscorr}
For $a\in\Z^{2V}$, $r_{\tau_a}(f_H)=e^{a\cdot f_H-|a|^2/2}$. Let $Z_H:=\delta+\sum_{a\in\Z^{2V}}\alpha_a\tau_a$, $\alpha_a:=C_Ze^{\beta|a|-|a|^2/2}$, $D_H:=\sqrt{2V}/2$, $\beta\ge2c_Y+3D_H$ and
$C_Z\ge\tfrac14C_Y^2e^{\beta D_H+D_H^2}$. Then $Z_H$ is bounded and real self-adjoint, $r_{Z_H}\ge\delta+C_Ze^{(\beta-3D_H)|f_H|-\beta D_H-D_H^2}\ge\delta+\tfrac14r_Y^2$ whenever $|r_Y|\le
C_Ye^{c_Y|f_H|}$,
so $1+tr_Y+t^2r_{Z_H}\ge(1+tr_Y/2)^2+\delta t^2>0$, and $r_{Z_H}\le Ce^{|f_H|^2/4+\beta|f_H|/2}$ (rate $1/4$).
\end{lemma}

\begin{proof}
The proof of Lemma~\ref{rv:lem:corr} applies word for word on the lattice $\Z^{2V}$ (cell half-diameter $D_H$; the nearest lattice point $a^*$ to $-f_H$ has $|a^*+f_H|\le D_H$).
\end{proof}

\begin{lemma}[full feature map and averaging identity]\label{rv:lem:fullmap}
Let $\cH_{\mathrm{full}}:=\cH\otimes L^2(\C^V)$ and $\Phi_{\mathrm{full}}(c):=\Phi(c)\otimes h^{\Phi(c)}$. Then $K(c,\wt c')=\langle\Phi_{\mathrm{full}}(c),\Phi_{\mathrm{full}}(\wt c')\rangle$
is the slab
kernel~\eqref{rv:eq:K} (the Gaussian factorisation of Lemma~\ref{rv:lem:higgskernel} replacing the Bessel one; the same positive kernel, so $T=\mathsf{B}^*\mathsf{B}$ and $Z_M=\Tr(P_GT^M)$ are
unchanged). The diagonal gauge action rotates $\Phi_x\mapsto e^{i\gph\lambda_x}\Phi_x$; with $(U(\lambda)g)(\Xi):=g(e^{-i\gph\lambda}\Xi)$, unitary and configuration independent,
$\Phi_{\mathrm{full}}(\lambda.c)=(R(\lambda)\otimes U(\lambda))\Phi_{\mathrm{full}}(c)$, and Lemma~\ref{rv:lem:zak}(c) holds for every bounded $X$ on $\cH_{\mathrm{full}}$ with
$\bar X:=\int d\lambda\,(R\otimes U)(\lambda)^*X(R\otimes U)(\lambda)$, where for such insertions $Z_M(X_1,\dots,X_k)$ denotes the partition function of the periodic measure in which the slab kernel $K(c,\wt c')$ at the height of $X_s$ is replaced by $\langle\Phi_{\mathrm{full}}(c),X_s\Phi_{\mathrm{full}}(\wt c')\rangle$; no summability condition enters, and the layer integrals converge absolutely because this kernel is bounded by $\|X_s\|\sqrt{K(c,c)K(\wt c',\wt c')}$. So does Lemma~\ref{rv:lem:UV}. For $X\otimes X_H$: $k=k_Xk_{X_H}$ and $r=r_Xr_{X_H}$.
\end{lemma}

\begin{proof}
The kernel identity is Lemma~\ref{rv:lem:zak}(a) with the hopping factor written by Lemma~\ref{rv:lem:higgskernel}; the covariance of $h^\Phi$ under rotations is
$h^{e^{i\gph\lambda}\Phi}=U(\lambda)h^\Phi$;
the proof of Lemma~\ref{rv:lem:zak}(c) used only the covariance $\Phi(\lambda.c)=R(\lambda)\Phi(c)$ and the measure preservation of Lemma~\ref{rv:lem:phase}(a).
\end{proof}

\emph{What the Higgs sector uses.} (I3) gives, by $|f_{H,x}|^2\le2\rho_x^2+2\wt\rho_x'^2$ and H\"older over the $2V$ factors, $\nu(e^{a\sum_{x\in V}|f_{H,x}|^2})<\infty$ for every $a$
(every rate, in particular the rate $1/4$ of Lemma~\ref{rv:lem:higgscorr}), and $\nu(\rho_x^2)<\infty$ for Lemma~\ref{rv:lem:TV}.

\paragraph{(B) Villain sector: the layer integers $v$ individually.}

In the Zak form~\eqref{rv:eq:zak} the kernel of every $\alpha$-decomposable insertion, that is, of every $\Z$-periodic $X\in\cC$ and of every multiplication by a function $m(\alpha)$,
depends on the Villain integers of the two layers only through $\wt v'-v$. The translations in $\alpha$ of the next lemma carry, in addition, a character of $v$ itself.

\begin{lemma}[$\alpha$-translations carry the character of the lower layer's $v$]\label{rv:lem:char}
For
$s\in[0,1)^{\mathrm{plaq}(S)}$ let $(T_sF)(\alpha):=F(\alpha-s)$, a unitary operator on $\cH$, and put $\sigma:=\wt ds$. Let $k(y):=(y-[y]_{\FD})/\cper\in\Z^S$ be the cell index of $y$,
and let $u=\wt B'-\wt v^t$ be the pair coordinate. Then
\begin{equation}\label{rv:eq:char}
\begin{gathered}
k_{T_s}(c,\wt c')=e_s(v)\,\langle h^B,M_sh^u\rangle_{L^2(\R^S)},\\
e_s(v):=e^{+2\pi is\cdot v/\cper},\qquad M_s:=\text{multiplication by }\chi_s(y):=e^{+2\pi i\sigma\cdot k(y)},
\end{gathered}
\end{equation}
and, for
fibred $X$ (Lemma~\ref{rv:lem:zak}(b)) such that $\sum_{a\in\cper\Z^S}|\langle h^B,M_sXh^{u-a}\rangle|<\infty$ for all $B,u\in\R^S$, in particular for $X\in\cC$, $k_{T_sX}(c,\wt c')=e_s(v)\langle h^B,M_sXh^u\rangle$. Moreover $|r_{T_s}|\le1$ and, for $X\in\cC$, $|r_{T_sX}|\le Ce^{c_Y|f|}$ as in
Lemma~\ref{rv:lem:growth}.
Thus the kernel of $T_sX$ is the character $e_s(v)$ of the closed $\cper\Z$-valued Villain $2$-form of the lower layer times a pair-space insertion with the same growth bound. The
operator $T_s$ is bounded, so
Lemmas~\ref{rv:lem:UV} and~\ref{rv:lem:fullmap} apply to it. In pair coordinates, $\chi_s(u)=e^{-2\pi i\sigma\cdot\wt v^t/\cper}$.
\end{lemma}

\begin{proof}
By the definitions of $\Phi$ and of $T_s$,
\[
\langle\Phi(c),T_s\Phi(\wt c')\rangle=\int d\alpha\,e^{2\pi i\alpha\cdot v/\cper}e^{-2\pi i(\alpha-s)\cdot\wt v'/\cper}\langle Zh^B(\theta),Zh^{\wt B'}(\theta-\sigma)\rangle_{L^2(\FD^S)}.
\]
The computation
of Lemma~\ref{rv:lem:zak}(a), with the extra phase, gives
\[
\langle Zh^B(\theta),Zh^{\wt B'}(\theta-\sigma)\rangle=\sum_me^{-2\pi i(\theta-\sigma)\cdot m}\langle h^B,M_sh^{\wt B'-\cper m}\rangle,
\]
because a point
$\eta\in\FD^S+\cper k$ has the cell index $k(\eta)=k$ and $e^{2\pi i\sigma\cdot(k+m)}=e^{2\pi i\sigma\cdot m}\chi_s(\eta)$. Since $\theta\cdot m=\alpha\cdot d_sm$ and
$\sigma\cdot m=s\cdot d_sm$, the integral over $\alpha$
selects $\cper m=\wt v^t$ with $d_s\wt v^t=v-\wt v'$, and the remaining phase is $e^{2\pi is\cdot(\wt v'+d_s\wt v^t)/\cper}=e_s(v)$. This is~\eqref{rv:eq:char}. For fibred $X$ as in the statement the same computation applies. Since $T_s^*=T_{-s}$, the form that defines $\hat X$ gives $\langle\Phi(c),T_s\hat X\Phi(\wt c')\rangle=\int d\alpha\,\langle\Phi(c)(\alpha),(X(\theta-\sigma)\otimes1)\Phi(\wt c')(\alpha-s)\rangle$ with $\theta=\wt d\alpha$. Lemma~\ref{rv:lem:zak}(b4), applied at the point $\theta-\sigma$ to $g_1:=M_s^{-1}h^B$, which is regular with $(Zg_1)(\theta-\sigma,\cdot)=(Zh^B)(\theta,\cdot)$, and to $g_2:=h^{\wt B'}$, gives $\langle(Zh^B)(\theta,\cdot),X(\theta-\sigma)(Zh^{\wt B'})(\theta-\sigma,\cdot)\rangle=\sum_me^{-2\pi i(\theta-\sigma)\cdot m}\langle h^B,M_sXh^{\wt B'-\cper m}\rangle$ for every $\theta$; the assumption makes this series absolutely convergent, so that it may be exchanged with the integral over $\alpha$. For $X\in\cC$ the assumption follows from the growth bound proved next, because $\sum_{a\in\cper\Z^S}e^{c_Y|f+a|-|f+a|^2/2}<\infty$.

Since $|\chi_s|=1$ and $h\ge0$, one has $|r_{T_s}|\le1$. For the growth bound of $r_{T_sX}$, each localised piece
$\langle\tau_ah^B,(\tau_aM_s\tau_a^*)X_0\tau_ah^u\rangle$ has $|\tau_aM_s\tau_a^*|=1$, and $M_s$ commutes with $P_K$, both being multiplications; the proof of
Lemma~\ref{rv:lem:growth} therefore applies. The character $e_s$ and the
multiplier $\chi_s$ carry the same sign in the exponent; both signs may be reversed together with the sign convention of~\eqref{rv:eq:zak}.
\end{proof}

\begin{lemma}[quasi-periodicity]\label{rv:lem:quasi}
For $a\in\cper\Z^S$ one has $\chi_s(y+a)=e^{2\pi i\sigma\cdot a/\cper}\chi_s(y)$. Consequently, for $X_0=|g_1\rangle\langle g_2|$ and $X=\sum_a\tau_a^*X_0\tau_a$,
\begin{equation}\label{rv:eq:quasi}
\begin{gathered}
k_{T_sX}(c,\wt c')=e_s(v)\sum_{a\in\cper\Z^S}e^{2\pi is\cdot(d_sa)/\cper}\,G^s_1(B-a)\,\overline{G_2(u-a)},\\
G^s_1:=e^{-|\cdot|^2}*(\chi_sg_1),\quad G_2:=e^{-|\cdot|^2}*g_2.
\end{gathered}
\end{equation}
The map $g_1\mapsto\chi_sg_1$ is a bijection of the compactly supported $L^2$ functions, so that, for fixed $s$, the functions $G^s_1$ range over the same set as the functions $G_1$
of Lemma~\ref{rv:lem:bondsep}.
\end{lemma}

\begin{proof}
The cell index satisfies $k(y+a)=k(y)+a/\cper$, which gives the first claim. Hence $\tau_aM_s\tau_a^*$ is the multiplication by $\chi_s(\cdot+a)=e^{2\pi i\sigma\cdot a/\cper}\chi_s$, and
\[
\langle h^B,M_s\tau_a^*X_0\tau_ah^u\rangle=e^{2\pi i\sigma\cdot a/\cper}\langle h^{B-a},\chi_sg_1\rangle\overline{\langle h^{u-a},g_2\rangle}.
\]
Summation over $a$ and Lemma~\ref{rv:lem:char} give~\eqref{rv:eq:quasi}, since $\sigma\cdot a=s\cdot d_sa$ by adjointness.
\end{proof}

\subsection{Joint separation and the exchange theorem}\label{rv:ssec:S7}

\begin{lemma}[full class; Hermitian relation]\label{rv:lem:fullclass}
Let $\cC_{\mathrm{full}}$ be the complex span of $\{T_s(X\otimes X_H):\ s\in[0,1)^{\mathrm{plaq}(S)},\ X\in\cC,\ X_H\in\cC_H\}$ on $\cH_{\mathrm{full}}$. The class
$\cC_{\mathrm{full}}$ is closed under adjoints; indeed
$T_s^*=T_{-s}$ and
\[
(T_s(X\otimes X_H))^*=(X^*\otimes X_H^*)T_{-s}=T_{-s}\big(T_s\hat X^*T_{-s}\otimes X_H^*\big),
\]
with $T_{-s}\hat XT_s=(M_s^{-1}XM_s)^\wedge\in\cC$ and $T_s\hat
XT_{-s}=(M_sXM_s^{-1})^\wedge\in\cC$. It is also closed under
$J:=J_\alpha\otimes J_H$, where $(J_\alpha F)(\alpha):=\overline{F(-\alpha)}$ and $J_H$ is the complex conjugation on $L^2(\C^V)$. One has
$J\Phi_{\mathrm{full}}(c)=\Phi_{\mathrm{full}}(c)$, and therefore
$k_{JXJ}=\overline{k_X}$ and
$k_X\circ\sw=\overline{k_{X^*}}$ for every bounded $X$ on $\cH_{\mathrm{full}}$. The ratios satisfy $|r_X|\le Ce^{c_Y(|f|+|f_H|)}$ for
$X\in\cC_{\mathrm{full}}$. Finally, $\cC_{\mathrm{full}}$ contains, with each of its elements $X$, the operators $(X+JXJ)/2$ and $(X-JXJ)/(2i)$, and, with each of its elements $W$, the
operators $(W+W^*)/2$ and $(W-W^*)/2$; these are the $J$-real and the $J$-imaginary part of $X$ and the self-adjoint and the skew-adjoint part of $W$.
\end{lemma}

\begin{proof}
By Lemma~\ref{rv:lem:char}, in the form $Z(M_sXg)(\theta)=X(\theta-\sigma)Zg(\theta-\sigma)$, the operator
$T_{-s}\hat XT_s$ corresponds under the Zak transform to $M_s^{-1}XM_s$: by Lemma~\ref{rv:lem:zak}(b2) the operator $M_s^{-1}XM_s$ is fibred with the fibres $X(\theta+\sigma)$, and these are the fibres of $T_{-s}\hat XT_s$, because $T_{-s}^*=T_s$ shifts $\alpha$ by $s$ in the form that defines $\hat X$. The function $\chi_s$ commutes with $P_K$, both being
multiplications, so $M_s^{-1}X_0M_s$ is again bounded and $P_K$-localised, and $M_s^{-1}XM_s\in\cC$; the same holds with $-s$ in place of $s$. This gives the closure under adjoints.
The class is also closed under $J$. Indeed, $J_\alpha T_sJ_\alpha=T_{-s}$ and $J_\alpha\hat XJ_\alpha=\widehat{X^\sharp}$, where $X^\sharp\in\cC$ is the complex conjugate of $X$ (Lemma~\ref{rv:lem:bondlaw}; the Zak transform turns the complex conjugation of $L^2(\R^S)$ into the map $F(\theta)\mapsto\overline{F(-\theta)}$, and the fibres of $X^\sharp$ are the complex conjugates of the operators $X(-\theta)$, by Lemma~\ref{rv:lem:zak}(b2)), and $J_HX_HJ_H\in\cC_H$ because $P_K$ is a real multiplication. Hence $J\,T_s(X\otimes X_H)\,J=T_{-s}(X^\sharp\otimes J_HX_HJ_H)$, and the complex span is preserved because $J$ is antilinear.
The identity $J\Phi_{\mathrm{full}}=\Phi_{\mathrm{full}}$ holds because $h$ is real and
$\overline{Zh^B(-\theta)}=Zh^B(\theta)$. It gives
\[
k_{JXJ}(c,\wt c')=\overline{\langle J\Phi_{\mathrm{full}}(c),XJ\Phi_{\mathrm{full}}(\wt c')\rangle}=\overline{\langle\Phi_{\mathrm{full}}(c),X\Phi_{\mathrm{full}}(\wt c')\rangle}
\]
and
\[
k_X(\sw(c,\wt c'))=\langle\Phi_{\mathrm{full}}(\wt c'),X\Phi_{\mathrm{full}}(c)\rangle=\overline{k_{X^*}(c,\wt c')}.
\]
No frame appears in the second identity, because the feature map of a layer depends on the data of that layer only. The bound on the ratios follows from
Lemmas~\ref{rv:lem:growth}, \ref{rv:lem:higgsclass} and~\ref{rv:lem:char}. The last assertion follows from the closure of $\cC_{\mathrm{full}}$ under adjoints, under $J$ and under complex linear combinations.
\end{proof}

\begin{lemma}[$\nu(r_Y)=0$ for $\cC_{\mathrm{full}}$]\label{rv:lem:step2full}
Let $\nu$ be as in Theorem~\ref{rv:thm:RV} (after Lemma~\ref{rv:lem:phase}), let the bond set $S$ be closed under the bonds of its plaquettes, and let $Y\in\cC_{\mathrm{full}}$ be real skew-adjoint, with $|r_Y|\le C_Ye^{c_Y(|f|+|f_H|)}$. Then
$\lambda(r_Y)=\nu(r_Y\circ\Theta_\sigma)=0$.
\end{lemma}

\begin{proof}
Let $Z:=\delta+Z_b\otimes Z_H$, where $Z_b$ is the corrector of Lemma~\ref{rv:lem:corr} and $Z_H$ that of Lemma~\ref{rv:lem:higgscorr}. The operator $Z$ is bounded and real
self-adjoint, and
$r_Z=\delta+r_{Z_b}(f)\,r_{Z_H}(f_H)\ge\delta+\tfrac14r_Y^2$; the product of the two correctors is taken here because it dominates $e^{2c_Y(|f|+|f_H|)}$, which a sum would not do.
Consequently $1+tr_Y+t^2r_Z\ge(1+tr_Y/2)^2+\delta t^2>0$. Moreover
$r_Z\le Ce^{(|f|^2+|f_H|^2)/4+c(|f|+|f_H|)}\le Ce^{3Q/8}$ with $Q:=|f|^2+|f_H|^2$, because $e^{c(|f|+|f_H|)}\le e^{Q/8+4c^2}$. Put $X_t:=1+tY+t^2Z$, so that
$\|X_t\|\le1+O(t^2)$, and
$v_t:=\log(1+tr_Y+t^2r_Z)\circ\Theta_\sigma$, so that $|v_t|\le|\log(\delta t^2)|+C(1+Q)$.

\emph{Upper bound.} We apply Lemma~\ref{rv:lem:TV}, whose hypotheses follow from (I1) and (I2), with the periodised sum over a coarser sublattice $\Gamma'\subset\Gamma$, which exists because $S$ and its site set $V(S)$ are finite, for which the
translates of $S$ and of $V(S)$ at equal height are pairwise
disjoint, and then~\eqref{rv:eq:insert} with Lemmas~\ref{rv:lem:fullmap} and~\ref{rv:lem:UV}; the sums over $\wt v^t$ in the kernels of $Y$ and of $Z$ converge absolutely, by Lemma~\ref{rv:lem:char} for the terms $T_s(X\otimes X_H)$ of $Y$ and by Remark~\ref{rv:rem:abs} for $Z_b$; for the products over the pairwise disjoint translates of the window at one height the kernel is the product of the kernels of the windows and the sum factorises (Lemma~\ref{tv:lem:win}(b),(c), whose proof does not use the present lemma; its hypotheses hold because the translates of $S$ are closed under the bonds of their plaquettes, as $S$ is, and are pairwise disjoint in bonds and in sites). This gives $\nu(v_t)\le\log\|X_t\|=O(t^2)$.
As in the proof of Lemma~\ref{rv:lem:step2}, Lemma~\ref{rv:lem:TV} is applied on its cubic tori $\eps\Z^3_\ell$ with $\ell\eps\Z^3\subset\Gamma'$, where $\mu_\ell$ is the periodic
measure with
$M=\ell$ layers over the layer torus of side $\ell$; on these tori the translates of $S$ and of $V(S)$ at equal height stay pairwise disjoint, because $\ell\eps\Z^3\subset\Gamma'$.

\emph{Lower bound.} The lower bound uses no moment of $r_Z$. On the set $A_t:=\{|t|\,|r_Y|\le\tfrac12,\ t^2r_Z\le\tfrac12\}$ the inequality $r_Z\ge0$ gives
$v_t\ge\log(1+tr_Y)\ge tr_Y-t^2r_Y^2$, and off $A_t$ one has $v_t\ge\log(\delta t^2)$. The complement of $A_t$ satisfies
\[
A_t^c\subset\{|f|+|f_H|\ge c_Y^{-1}\log(1/(2C_Y|t|))\}\cup\{Q\ge\tfrac{16}3\log(1/|t|)-C\}.
\]
The first set has a super-polynomially small $\nu$-measure, by the Gaussian moments. For the second set we use that
$\nu(e^{aQ})<\infty$ for every $a<\tfrac12$: the bond factor is controlled by Lemma~\ref{rv:lem:moments} at a rate $a'<\tfrac12$, the Higgs factor by (I3) at every rate, and the two
are combined by H\"older's inequality with an exponent
$p\downarrow1$ on the bond factor. Hence $\nu(A_t^c)\le C|t|^{16a/3}$. For $a\in(\tfrac38,\tfrac12)$ this gives $\nu(A_t^c)\log(1/(\delta t^2))=o(t^2)$ and
$|t|\nu(|r_Y|\bbone_{A_t^c})\le|t|\nu(r_Y^2)^{1/2}\nu(A_t^c)^{1/2}=o(t^2)$. As in the proof of Lemma~\ref{rv:lem:step2}, it follows that $t\nu(r_Y)\le O(t^2)$ for both signs of $t$,
so $\lambda(r_Y)=0$.

The second set calls for a rate $>\tfrac38$, which the constant
$\tfrac12$ in the exponent of (I2) gives. The condition $t^2r_Z\le\tfrac12$ in $A_t$ is not needed for the lower bound (compare the proof of Lemma~\ref{rv:lem:step2}), so this
requirement is a feature of the present proof.
\end{proof}

\begin{lemma}[Hermitian property]\label{rv:lem:hermit}
Let $S$ be closed under the bonds of its plaquettes, and let $m:=K^{-1}\lambda$ on the $S$-local slab data, where
$K^{-1}$ denotes the division by
the single-term unfolded kernel $w(B-u)e^{-|f_H|^2/2}>0$. The $S$-local slab data are $B\in\FD^S$, $u\in\R^S$, the closed form $v\in(\cper\Z)^{\mathrm{plaq}(S)}$ and
$\Phi,\wt\Phi'\in\C^V$; the form $\wt v'=v-d_s([u]_{\FD}-u)$ is determined by them. Then $m(|k_X|)=\lambda(|r_X|)<\infty$ and $m(k_X\circ\sw)=m(k_X)$ for every
$X\in\cC_{\mathrm{full}}$.
\end{lemma}

\begin{proof}
The finiteness follows from Lemma~\ref{rv:lem:fullclass}, Lemma~\ref{rv:lem:moments} and (I3). For the identity, write $X=X_r+iX_i$ with $X_r:=(X+JXJ)/2$ and
$X_i:=(X-JXJ)/(2i)$, which both commute with $J$, and write a $J$-real operator $W$ as
$W=A+Y$, where $A:=(W+W^*)/2$ is real self-adjoint and $Y:=(W-W^*)/2$ is real skew-adjoint. All these operators belong to $\cC_{\mathrm{full}}$
(Lemma~\ref{rv:lem:fullclass}). For a $J$-real $W$ the kernel $k_W$ is real, since
$k_{JWJ}=\overline{k_W}$, and
\[
k_W\circ\sw=\overline{k_{W^*}}=k_{W^*}=k_A-k_Y,
\]
so that $m(k_W\circ\sw)=m(k_W)-2\lambda(r_Y)=m(k_W)$ by Lemma~\ref{rv:lem:step2full}. Applying this to $X_r$ and to $X_i$ gives the claim.
\end{proof}

\begin{lemma}[separation, the layer integers $v$ included]\label{rv:lem:fullsep}
The measure $m$ of Lemma~\ref{rv:lem:hermit} satisfies $\sw_*m=m$ on the full $S$-local slab data.
\end{lemma}

\begin{proof}
Fix $s\in[0,1)^{\mathrm{plaq}(S)}$. For $X_0=|g_1\rangle\langle g_2|$, $X_H=|g^H_1\rangle\langle g^H_2|$ and $X=T_s(\sum_a\tau_a^*X_0\tau_a\otimes X_H)$, Lemmas~\ref{rv:lem:quasi}
and~\ref{rv:lem:higgsclass} give
\[
k_X=e_s(v)\Big[\sum_{a\in\cper\Z^S}e^{2\pi is\cdot(d_sa)/\cper}G^s_1(B-a)\overline{G_2(u-a)}\Big]G^H_1(\Phi)\overline{G^H_2(\wt\Phi')}.
\]
The swapped kernel is $k_X\circ\sw=e_s(\wt v')e^{2\pi i\sigma\cdot a_0/\cper}\langle h^u,M_sXh^B\rangle\cdots$, where $a_0:=[u]_{\FD}-u$ is the frame and the dots stand for the
Higgs factors. Since $\wt v'=v-d_sa_0$, the quasi-periodicity phase of $M_s$
(Lemma~\ref{rv:lem:quasi}) converts $e_s(\wt v')$ into $e_s(v)$. Thus both sides of the Hermitian identity carry the same character.

Define the twisted unfolding
$\hat m_s:=\sum_{a\in\cper\Z^S}e^{2\pi is\cdot(d_sa)/\cper}(T_a)_*\big(e_s(v)\,m\big)$, where $T_a$ is the diagonal shift of $(B,u)$, which fixes $v,\Phi,\wt\Phi'$. It is a complex
measure on
$\R^S\times\R^S\times\C^V\times\C^V$ with $|\hat m_s|\le\hat m:=\sum_a(T_a)_*m$. The measure $\hat m$ satisfies the following growth bound for every $\delta\in[\tfrac34,1)$; such a
weight dominates the Gaussian smoothing at rate $1$:
\[
\hat m\big(e^{-\delta(|B|^2+|u|^2+|\Phi|^2+|\wt\Phi'|^2)}\big)\le C_\delta\lambda\big(e^{|f|^2/4}\big)^{1/2}\lambda\big(e^{|f_H|^2/4}\big)^{1/2}<\infty,
\]
by Lemma~\ref{rv:lem:moments} at the
joint rate $1/4$, used in the Cauchy--Schwarz form, and by (I3). A joint rate $1/8$
would suffice with this weight.

The identities of Lemma~\ref{rv:lem:hermit}, written for the unfolded pair kernels, state that the integral
$\int G^s_1\otimes\overline{G_2}\otimes G^H_1\otimes\overline{G^H_2}\,d\hat m_s$ is equal to the
same integral with respect to
$(\sw_*m)^\wedge_s$, for all compactly supported $g_1$, $g_2$, $g^H_1$, $g^H_2$. The linear span of the products is dense in $L^2$ of every compact set of $(B,u,\Phi,\wt\Phi')$, so, as in the proof of Lemma~\ref{rv:lem:bondsep}(iii), the Gaussian smoothings
of the weighted measures agree.
The argument of Lemma~\ref{rv:lem:bondsep} applies: the measures are finite complex measures after multiplication by the Gaussian weight, their two-sided Laplace transforms are
entire, and the restriction to imaginary arguments is the Fourier transform. It gives $\hat m_s=(\sw_*m)^\wedge_s$
as complex measures, for every $s$.

Now restrict this identity to $B\in\FD^S$. The sets $\FD^S+a$ are disjoint, and $\sw_*m$ is supported on $B\in\FD^S$ as $m$ is, so the term $a=0$ is selected on both sides:
\[
\int e_s(v)\,m(\cdot,dv)=\int e_s(v)\,(\sw_*m)(\cdot,dv)
\]
on $\FD^S\times\R^S\times\C^V\times\C^V$, for every $s$. The characters $e_s$ separate $(\cper\Z)^{\mathrm{plaq}(S)}$, and Fourier
inversion in $s$ on bounded sets of the slab data, where both measures are finite, recovers the $v$-marginal. Hence $\sw_*m=m$.
\end{proof}

\begin{theorem}[exchange theorem, $S$-local form]\label{rv:thm:EV}
Let $\nu$ be as in Theorem~\ref{rv:thm:RV}, passed to the phase-restored specification by Lemma~\ref{rv:lem:phase}, and let $S$ be a finite spatial bond set closed under the bonds of
its plaquettes. Then $\nu$ and $\nu\circ\hat\theta$, $\theta$ the mid-plane reflection of the slab, agree on all bounded gauge-invariant $S$-local slab functions (Higgs, Villain and bond
sectors jointly, temporal plaquettes included).
\end{theorem}

\begin{proof}
$K\circ\sw=K$ and $\sw_*m=m$ (Lemma~\ref{rv:lem:fullsep}) give $\sw_*\lambda=\lambda$ on the $S$-local temporal-gauge slab data $\Theta_\sigma|_S=(c|_S,\wt c'|_S,\wt v^t|_S)$. For a
diagonal-gauge-invariant $S$-local $F$, Lemma~\ref{rv:lem:slab}(c) gives $F(\Theta_\sigma\theta\om)=F(\sw\,\Theta_\sigma\om)$, hence
$\nu(F\circ\Theta_\sigma\circ\theta)=\lambda(F\circ\sw)=\lambda(F)=\nu(F\circ\Theta_\sigma)$.
The gauge-invariant $S$-local slab functions are exactly the $F\circ\Theta_\sigma$ with $F$ diagonal-invariant and $S$-local (Lemma~\ref{rv:lem:slab}(b)).
\end{proof}

\begin{remark}\label{rv:rem:EVscope}
Theorem~\ref{rv:thm:EV} is stated for $S$-local functions, for every finite $S$ closed under the bonds of its plaquettes; the passage to the laws of the full two-layer data over a box happens inside the proof of Theorem~\ref{rv:thm:DV}
(Lemma~\ref{rv:lem:S71}), not here. It applies to Gibbs measures of the unperturbed specification (the comparison states, by Proposition~\ref{n1:prop:I123}), and not to the
limit points $\nu_t$, $t\ne0$, of the perturbed laws of Section~\ref{n1:sec} (Lemma~\ref{n1:lem:gibbs}).
\end{remark}

\subsection{Two consecutive layers determine the Gibbs measure}\label{rv:ssec:S8}

Throughout this subsection $\wt\gamma$ is the phase-restored specification of Subsection~\ref{rv:ssec:S1} on $\eps\Z^3=(\text{time }\eps\Z)\times(\text{space }\eps\Z^2)$, layer $k$ is the set of sites at
time $k\eps$, and $\Z_g:=\cper\Z$. \emph{Convention.} Every Gibbs measure of $\wt\gamma$ is carried by configurations satisfying all cube constraints ($dv=0$ on every cube), so the flux
identities of Lemma~\ref{rv:lem:F} below hold almost surely; on the infinite layer there is no total flux class, and the proofs below use none. The DLR kernels of finite sub-sets $\Delta$ of
$V(\Lambda)$ are the Fubini conditionals of $\wt\gamma_\Lambda$ (Lemma~\ref{rv:lem:phase}(c)). The statement holds for all Gibbs measures of $\wt\gamma$: it requires neither temperedness nor moment bounds nor
$G$-invariance, and it does not depend on (I1)--(I3).

\begin{theorem}[two consecutive layers determine the Gibbs measure]\label{rv:thm:DV}
Let $\nu,\nu'$ be Gibbs measures of $\wt\gamma$ on $\eps\Z^3$ and $n_0\in\Z$. Suppose that for every finite spatial set $S$ (equivalently every rectangle $B$) $\nu$ and $\nu'$ agree on
every bounded function of the variables of the slab $[n_0,n_0+1]$ over $S$, namely the Higgs fields $(\tr,\psi)$ of layers $n_0$, $n_0+1$ at the sites of $S$, the spatial links of
both layers inside $S$ and the spatial Villain integers of both layers over the set $P(S)$ of the plaquettes with all four corners in $S$, the temporal links at the sites of $S$, and the temporal plaquettes with their Villain integers over
the links of $S$, that is invariant under the gauge transformations supported in $S\times\{n_0,n_0+1\}$ (equivalently, under all gauge transformations). Then $\nu=\nu'$.
\end{theorem}

\begin{lemma}[Gaussian convolution is injective on positive measures]\label{rv:lem:G}
Let $d\ge1$, $\Omega\subset\R^d$ nonempty open, and $M,M'$ positive Borel measures on $\R^d$ with $(WM)(x):=\int e^{-|x-y|^2/2}M(dy)=(WM')(x)<\infty$ for Lebesgue-a.e.\ $x\in\Omega$. Then $M=M'$.
\end{lemma}

\begin{proof}
Put $g:=e^{-|y|^2/2}M$, $g':=e^{-|y|^2/2}M'$, so that $WM(x)=e^{-|x|^2/2}L_g(x)$ with $L_g(x):=\int e^{x\cdot y}g(dy)$. (i) By H\"older, $C:=\{L_g<\infty\}$ is convex and contains a.e.\ point of
$\Omega$; a point $x_0\in\Omega\setminus C$ would give, by separation, a nonempty open subset of $\Omega$ disjoint from $C$, so $\Omega\subset C$, likewise $\Omega\subset C'$, and
$O:=\operatorname{int}(C\cap C')$ is open, convex and contains $\Omega$. (ii) $F(z):=\int e^{z\cdot y}(g-g')(dy)$ is holomorphic on the tube $\{\Rp z\in O\}$ (domination of $e^{\Rp z\cdot y}$ by the
vertex exponentials of a small cube, Morera, Hartogs). (iii) $L_g=L_{g'}$ a.e.\ on $\Omega$ and both are continuous on $O$, hence equal on $\Omega$; all real partial derivatives of $F$
vanish at a point of $\Omega$, so $F=0$ on the connected tube. Fix $x_0\in O$: $\xi\mapsto F(x_0+i\xi)$ is the Fourier transform of the finite signed measure $e^{x_0\cdot y}(g-g')(dy)$ on
$\R^d$, which therefore vanishes; so $g=g'$ and $M=M'$. (The restriction to $x_0+i\R^d$ with $x_0\in O$ is the general form; in every use below $0\in O$.)
\end{proof}

Lemma~\ref{rv:lem:G} assumes no moment bound beyond the finiteness almost everywhere; the case of any other Gaussian rate follows by scaling, and a positive factor $f(x)f(y)$ in the kernel can be absorbed.

\paragraph{Rectangles and straddling plaquettes.} A \emph{rectangle} is a set of sites $\Lambda=\{a_1,\dots,a_1+L_1\}\times\{a_2,\dots,a_2+L_2\}$ in $\eps\Z^2$ (in units of $\eps$; $L_i\ge1$).
We write $\mathrm{Bo}(\Lambda)$ for the set of links with both ends in $\Lambda$, $P(\Lambda)$ for the set of plaquettes with all four corners in $\Lambda$,
$\partial\mathrm{Bo}(\Lambda)$ for the set of the $2(L_1+L_2)$ boundary links of the rectangle, and $\mathrm{Bo}^\circ(\Lambda):=\mathrm{Bo}(\Lambda)\setminus\partial\mathrm{Bo}(\Lambda)$.
A plaquette that is not in $P(\Lambda)$ and has a link in $\mathrm{Bo}(\Lambda)$ is called \emph{straddling}. A straddling plaquette has exactly one link in $\mathrm{Bo}(\Lambda)$, and this link is a
boundary link; conversely, every boundary link lies in exactly one straddling plaquette (the corner plaquettes have no link in $\mathrm{Bo}(\Lambda)$). The straddling plaquettes are thus in
bijection $p'\leftrightarrow b(p')$ with $\partial\mathrm{Bo}(\Lambda)$; we write $p'(b)$ for the straddling plaquette that contains the boundary link $b$, and we put
$P^+(\Lambda):=P(\Lambda)\cup\{\text{straddling plaquettes}\}$. The three other links of a straddling plaquette are its \emph{collar} links. They belong to $\mathrm{Bo}(\Lambda^+)$, where
$\Lambda^+:=\{\text{sites at sup-distance}\le\eps\text{ from }\Lambda\}$ is again a rectangle.

For $w\in\Z_g^{\mathrm{Bo}(\Lambda)}$ the Stokes identity reads
\[
\sum_{p\in P(\Lambda)}(d_sw)_p=\mathrm{circ}(w):=\sum_{b\in\partial\mathrm{Bo}(\Lambda)}\epsilon_bw_b,
\]
where $\epsilon_b=\pm1$ is the boundary orientation. A boundary link has a fixed orientation in its straddling plaquette, so the straddling cube constraints and the flux invariant
defined below carry fixed signs $\pm1$, one for each boundary link. In the lemmas below these signs only relabel $\Z_g$-valued coordinates, and we suppress them from here on.

\paragraph{Transported slab data.} Let $\nu$ and $\nu'$ be Gibbs measures of $\wt\gamma$. For the slab $[m,n]$ put $\lambda^{(k)}_x:=-(B_t^{(m)}+\dots+B_t^{(k-1)})_x$ for $m\le k\le n$, let
$\wt c_k:=\lambda^{(k)}.c_k$ be the transported configuration of layer $k$, and let $\wt v^{t,(k)}$ be the correspondingly shifted temporal integers; the temporal links of the slab then
become $0$. Here $c_k$ is the configuration of layer $k$ in the notation of Subsection~\ref{rv:ssec:S2}, and we write $\wt\Phi^{(k)}$, $\wt B^{(k)}$ and $\wt v^{(k)}$ for the Higgs field
of $\wt c_k$ (in the complex form of Lemma~\ref{rv:lem:higgskernel}), its spatial links and its spatial Villain integers. Layer $m$, on which $\lambda^{(m)}=0$, is called the \emph{base
layer} of these data; the data with another base layer $k\in[m,n]$ are defined in the same way, with the gauge transformation that vanishes on layer $k$ and
removes the temporal links of the slab. For a rectangle $B$ let
\[
\Theta_{m,n,B}:=\big(\wt\Phi^{(k)}|_B,\ \wt B^{(k)}|_{\mathrm{Bo}(B)},\ \wt v^{(k)}|_{P(B)},\ m\le k\le n;\ \ \wt v^{t,(k)}|_{\mathrm{Bo}(B)},\ m\le k\le n-1\big),
\]
and let $H(m,n)$ be the statement that the laws of $\Theta_{m,n,B}$ under $\nu$ and $\nu'$ coincide for every rectangle $B$. Every finite spatial set lies in a rectangle, so by
Lemma~\ref{rv:lem:S71} below the hypothesis of Theorem~\ref{rv:thm:DV} is $H(n_0,n_0+1)$.

\paragraph{The DLR equation of a rectangle and the patch kernel.} Fix $m\le n-1$ and a rectangle $\Lambda$. Let $\Delta$ be the following set of variables: on layer $n$, the Higgs variables
$\{\tr_x,\psi_x:x\in\Lambda\}$, the links $\{B_b:b\in\mathrm{Bo}(\Lambda)\}$ and the Villain integers $\{v_p:p\in P^+(\Lambda)\}$; and the temporal integers
$\{v^{t,(n-1)}_b,v^{t,(n)}_b:b\in\mathrm{Bo}(\Lambda)\}$ of both adjacent slabs over $\mathrm{Bo}(\Lambda)$, boundary links included. The temporal links do not belong to $\Delta$; they stay exterior.

We collect the transported coordinates in three groups, which are denoted by $v$, $Y$ and $u$ until the end of this subsection. The first group consists of the layer-$n$ coordinates over $\Lambda$,
\[
v:=(\wt\Phi^{(n)}|_\Lambda,\wt B^{(n)}|_{\mathrm{Bo}(\Lambda)},\wt v^{(n)}|_{P^+(\Lambda)}).
\]
The second group, $Y$, consists of all coordinates of $\Theta_{m,n,\Lambda^+}$ other than those of $v$ and other than $\wt v^{t,(n-1)}|_{\mathrm{Bo}(\Lambda)}$. The third group is
\[
u:=(\wt\Phi^{(n+1)}|_\Lambda,\wt B^{(n+1)}|_{\mathrm{Bo}(\Lambda)},\wt v^{(n+1)}|_{P^+(\Lambda)},\wt v^{t,(n)}|_{\text{collar links}}).
\]
Every coordinate of $Y$ and of $u$ is a function of exterior variables, that is, of variables that do not belong to $\Delta$. Indeed, the coordinates of the layers $m,\dots,n-1$ and of the slabs $m,\dots,n-2$ over $\Lambda^+$ involve only $B_t$
and layer-$k$ data with $k<n$. The layer-$n$ coordinates outside $\Lambda$ are $\wt\Phi$ at the sites of $\Lambda^+\setminus\Lambda$, $\wt B_b$ for $b\notin\mathrm{Bo}(\Lambda)$ and $\wt v_p$ for
$p\notin P^+(\Lambda)$, whose $\FD$-shifts involve only links not in $\mathrm{Bo}(\Lambda)$. The slab-$(n-1)$ integers in $Y$ are those on links not in $\mathrm{Bo}(\Lambda)$. Finally, $u$
involves layer $n+1$ and the collar links only. This is the reason for including the straddling plaquettes in $\Delta$. Since $(Y,v)$ is a sub-family of $\Theta_{m,n,\Lambda^+}$, the
statement $H(m,n)$, applied to the rectangle $\Lambda^+$, implies that the law of $(Y,v)$ is the same under $\nu$ and $\nu'$.

The factors of the weight $W$ that contain a variable of $\Delta$ are the following: the factors $g_{\mathrm{ct}}(\tr_x)$, $x\in\Lambda$; the spatial factors $h$ on $\mathrm{Bo}(\Lambda)$ and
on the links with one end in $\Lambda$; the spatial factors $w$ on $P(\Lambda)$ and on the straddling plaquettes; the temporal factors $h$ at the sites $x\in\Lambda$ and the temporal factors
$w$ on $\mathrm{Bo}(\Lambda)$, for both slabs; and the cube constraints on $P^+(\Lambda)$, for both slabs. The integrand of $\wt\gamma_\Delta$ is gauge invariant, and for a fixed exterior
configuration the gauge action is a measure-preserving bijection of the space of the variables of $\Delta$ (Lemma~\ref{rv:lem:phase}(a)). We apply the gauge transformation
$\lambda=(\lambda^{(k)})_k$ and transport the variables $(c_n|_\Lambda,v^t|_{\mathrm{Bo}(\Lambda)})$. For bounded functions $A$, $a$, $f$ this gives
\begin{equation}\label{rv:eq:patchDLR}
\begin{gathered}
\wt\gamma_\Delta\big(A(Y)a(v)f(u)\big)(\om)=A(Y)f(u)\,Z(Y,u)^{-1}\!\int a(v')\,c(v';Y)\,K_\Lambda(v',u)\,\lambda_\Lambda(dv'),\\
\lambda_\Lambda:=\mathrm{Leb}(\C^\Lambda)\times\Haar(\Tper^{\mathrm{Bo}(\Lambda)})\times\mathrm{counting}(\Z_g^{P^+(\Lambda)}),
\end{gathered}
\end{equation}
with $G(z):=e^{-|z|^2/2}$, $c(v';Y):=W_\Lambda(v';Y)K^{\mathrm{low}}_\Lambda(Y,v')$ and
\[
\begin{gathered}
K^{\mathrm{low}}_\Lambda(Y,v'):=\prod_{x\in\Lambda}G(\Phi'_x-\wt\Phi^{(n-1)}_x)\sum_{v^t:\ \text{lower cube constraints on }P^+(\Lambda)}\ \prod_{b\in\mathrm{Bo}(\Lambda)}w(B'_b-\wt
B^{(n-1)}_b+v^t_b),\\
K_\Lambda(v',u):=\prod_{x\in\Lambda}G(\wt\Phi^{(n+1)}_x-\Phi'_x)\sum_{v^t:\ \text{upper cube constraints on }P^+(\Lambda)}\ \prod_{b\in\mathrm{Bo}(\Lambda)}w(\wt B^{(n+1)}_b-B'_b+v^t_b),
\end{gathered}
\]
where the integration variable is written $v'=(\Phi',B',(v'_p)_{p\in P^+(\Lambda)})$ and $Z(Y,u)$ denotes the integral on the right side of~\eqref{rv:eq:patchDLR} with $a=1$. The function
$W_\Lambda(v';Y)$ is the product of the factors $g_{\mathrm{ct}}$ on $\Lambda$ and of the spatial factors $h$ and $w$ that contain a variable of $\Delta$, written in transported variables. The
exterior layer-$n$ data that enter the factors at the boundary links and plaquettes are coordinates of $Y$, and the site Jacobian $\prod_x(\gph\rho'_x)^{-1}$ of
Lemma~\ref{rv:lem:higgskernel} is absorbed into $W_\Lambda$. We call $K_\Lambda$ the \emph{patch kernel}.

The upper cube constraints, over which the sum in $K_\Lambda$ runs, read as follows. On $p\in P(\Lambda)$ the constraint is $(d_sv^t)_p=\wt v^{(n+1)}_p-v'_p$. On a straddling plaquette $p'$
with boundary link $b=b(p')$ it is $v^t_b=\gamma_{p'}(u)-v'_{p'}$, where
\[
\gamma_{p'}(u):=\wt v^{(n+1)}_{p'}-(\text{signed sum of the three collar
integers }\wt v^{t,(n)}\text{ of }u\text{ at }p')
\]
is defined by this exterior expression in $u$. Hence $K_\Lambda$ depends on $u$ only through
\[
\bar u:=(\wt\Phi^{(n+1)}|_\Lambda,\wt B^{(n+1)}|_{\mathrm{Bo}(\Lambda)},\wt v^{(n+1)}|_{P(\Lambda)},\gamma)\in
U_\Lambda:=\C^\Lambda\times\Tper^{\mathrm{Bo}(\Lambda)}\times\Z_g^{P(\Lambda)}\times\Z_g^{\partial\mathrm{Bo}(\Lambda)},
\]
and so does $Z$. We write $K_\Lambda(v',\bar u)$ and $Z(Y,\bar u)$ accordingly.

Since $\nu$ and $\nu'$ are Gibbs measures, integrating~\eqref{rv:eq:patchDLR} against $\rho\in\{\nu,\nu'\}$ gives
\begin{equation}\label{rv:eq:patchint}
\begin{gathered}
\rho\big(A(Y)a(v)f(\bar u)\big)=\int A(Y)f(\bar u)\Psi_a(Y,\bar u)\,Q_\rho(dY\,d\bar u),\\
\Psi_a:=Z^{-1}\!\int a(v')c(v';Y)K_\Lambda(v',\bar u)\lambda_\Lambda(dv'),
\end{gathered}
\end{equation}
where $Q_\rho$ is the law of $(Y,\bar u)$ under $\rho$; under $H(m,n)$ we write $Q^Y$ for the common law of $Y$ under $\nu$ and $\nu'$. The function $Z$ is finite everywhere: one has
$h,w,G\le1$, each coset sum is bounded by a product of periodised Gaussians, the sum over $v'_{P^+}$ of the spatial $w$-factors converges, and $\int g_{\mathrm{ct}}G\,d^2\Phi<\infty$
by~\eqref{eq:ct}. Moreover $Z$ lies in $(0,\infty)$ $\rho$-almost surely. The condition $m\le n-1$ ensures that $\wt c_{n-1}|_\Lambda$ belongs to $Y$.

\begin{lemma}[relative flux on a rectangle]\label{rv:lem:F}
Let $\beta\in\Z_g^{\partial\mathrm{Bo}(\Lambda)}$ and $f\in\Z_g^{P(\Lambda)}$. The set $\Gamma_\Lambda(f;\beta):=\{w\in\Z_g^{\mathrm{Bo}(\Lambda)}:\ w|_{\partial\mathrm{Bo}}=\beta,\ d_sw=f\text{
on }P(\Lambda)\}$ is nonempty if and only if
$\sum_{p\in P(\Lambda)}f_p=\mathrm{circ}(\beta)$, and then it is a coset of the countable subgroup $\Gamma_\Lambda(0;0)$.
\end{lemma}

\begin{proof}
Necessity follows from the Stokes identity $\sum_{p\in P(\Lambda)}(d_sw)_p=\mathrm{circ}(w)$. Sufficiency: let $w_0:=\beta$ on $\partial\mathrm{Bo}$, $0$ on $\mathrm{Bo}^\circ$, and $g:=f-d_sw_0$, so $\sum_pg_p=0$. List $P(\Lambda)$ along a boustrophedon path
$p_1,\dots,p_M$;
consecutive plaquettes share an interior link $b_j$ lying in no other plaquette of $P(\Lambda)$ and entering $(d_sw)_{p_j}$, $(d_sw)_{p_{j+1}}$ with opposite signs. Choose $w$ on $b_1$ so
that $(d_sw)_{p_1}=g_{p_1}$, which changes the requirement at $p_2$ to $g_{p_2}+g_{p_1}$; continue; at $p_M$ the requirement is $\sum_pg_p=0$, met with the remaining interior links $0$.
Then $w_0+w\in\Gamma_\Lambda(f;\beta)$. The coset statement follows from the linearity of the two conditions.
\end{proof}

\paragraph{Flux invariant.} Let $V^+:=\Z_g^{P^+(\Lambda)}$ be the space of the integer components $(v'_p)_{p\in P^+(\Lambda)}$ of the layer-$n$ variable $v'$; we write $v'$ also for these
components. For $v'\in V^+$ put
\[
F(v'):=\sum_{p\in P(\Lambda)}v'_p-\mathrm{circ}\big((v'_{p'(b)})_{b\in\partial\mathrm{Bo}}\big),
\]
where the integers of the straddling plaquettes are read as boundary data, and for $\bar u$ put
\[
F(\bar u):=\sum_{p\in P(\Lambda)}\wt v^{(n+1)}_p-\mathrm{circ}(\gamma).
\]
We call $F$ the \emph{flux invariant}. The sum over the upper cube constraints in $K_\Lambda(v',\bar u)$ runs over a set of the form considered in Lemma~\ref{rv:lem:F}, whose boundary values
are $\gamma_{p'}-v'_{p'}$ and whose $f$ is $\wt v^{(n+1)}-v'$ on $P(\Lambda)$; we call it the upper coset. By Lemma~\ref{rv:lem:F} the upper coset is nonempty if and only if $F(\bar u)=F(v')$. In the same
way the lower coset, over which the sum in $K^{\mathrm{low}}_\Lambda(Y,v')$ runs, is nonempty if and only if $F(v')=F(Y)$, where $F(Y)$ is the analogous expression in the layer-$(n-1)$ and
slab-$(n-1)$ integers; it is a function of $Y$. Every Gibbs measure of $\wt\gamma$ is carried by the configurations that satisfy all cube constraints (the convention stated at the beginning
of this subsection), so under a Gibbs measure both constraints hold almost surely, and $F(v)=F(\bar u)=F(Y)$ almost surely. Finally, $c(v';Y)>0$ if and only if $F(v')=F(Y)$ and
$\rho'_x>0$ for all $x$, where the second condition holds $\lambda_\Lambda$-almost everywhere; indeed all factors $h,w,g_{\mathrm{ct}}$ are $>0$ there.

For the bond variables let
\[
\mathfrak B_\Lambda:=\Tper^{\mathrm{Bo}(\Lambda)}\times\Z_g^{P(\Lambda)}\times\Z_g^{\partial\mathrm{Bo}(\Lambda)},
\]
with elements $(\wt B,\wt v,\gamma)$ and with the measure $\lambda_{\mathfrak B}:=\Haar\times\mathrm{counting}$, and for $(B',v')\in\Tper^{\mathrm{Bo}(\Lambda)}\times V^+$ put
\[
K_{\mathrm{bond},\Lambda}\big((B',v'),(\wt B,\wt v,\gamma)\big):=\sum_{w\in\Gamma_\Lambda(\wt v-v'|_{P(\Lambda)};\,\gamma-v'|_{\partial\mathrm{Bo}})}\ \prod_{b\in\mathrm{Bo}(\Lambda)}e^{-(\wt
B_b-B'_b+w_b)^2/2}
\]
where the link variables are taken as their $\FD$ representatives. Then $K_\Lambda(v',\bar u)=G(\wt\Phi-\Phi')K_{\mathrm{bond},\Lambda}$.

\begin{lemma}[the bond sector of the patch kernel]\label{rv:lem:BL}
Fix $F_0\in\Z_g$ and let $\mu,\mu'$ be positive Borel measures on $\mathfrak B_\Lambda$ carried by the class $C(F_0):=\{F(\bar u)=F_0\}$. Suppose that for one $v'\in V^+$ with $F(v')=F_0$
and for Haar-almost every $B'\in\Tper^{\mathrm{Bo}(\Lambda)}$,
\[
(K_{\mathrm{bond},\Lambda}\mu)(B',v')=(K_{\mathrm{bond},\Lambda}\mu')(B',v')<\infty.
\]
Then $\mu=\mu'$.
\end{lemma}

\begin{proof}
Let $v'$ be as in the statement and put
\[
D:=\{(\wt B,\wt v,\gamma,w):\ w\in\Gamma_\Lambda(\wt v-v';\gamma-v'_{\partial\mathrm{Bo}})\}.
\]
We first show that the map $T(\wt B,\wt v,\gamma,w):=\wt B+w\in\R^{\mathrm{Bo}(\Lambda)}$ is injective on $D$. Indeed, if $U=T(\wt B,\wt v,\gamma,w)$, then $\wt B=[U]_{\FD}$ and
$w=U-[U]_{\FD}$ on all of $\mathrm{Bo}(\Lambda)$; the boundary values of $w$ then give $\gamma_{p'(b)}=w_b+v'_{p'(b)}$, and $\wt v=v'+d_sw$ on $P(\Lambda)$. So $U$ determines the point of $D$.

Next let $M:=T_*(\mu\otimes\mathrm{counting}_D)$, where $\mu\otimes\mathrm{counting}_D$ is the measure on $D$ obtained by integrating against $\mu$ the counting measures of the cosets, and
define $M'$ from $\mu'$ in the same way. By Tonelli's theorem,
\[
(K_{\mathrm{bond},\Lambda}\mu)(B',v')=\int\prod_be^{-(U_b-B'_b)^2/2}M(dU)=(WM)(B')
\]
on the open box $\operatorname{int}(\FD)^{\mathrm{Bo}(\Lambda)}$, in the notation of Lemma~\ref{rv:lem:G}, and the same holds for $\mu'$ and $M'$. By the hypothesis, $WM=WM'<\infty$ almost
everywhere on this open box, so Lemma~\ref{rv:lem:G} gives $M=M'$.

It remains to recover $\mu$ from $M$. The pairs $(\wt v,\gamma)$ that occur in $C(F_0)$ form a countable set, and for each of them the coset
$\Gamma_\Lambda(\wt v-v';\gamma-v'_{\partial\mathrm{Bo}})$ is nonempty by Lemma~\ref{rv:lem:F}, because $F(v')=F_0$; we choose an element $\mathrm{gam}(\wt v,\gamma)$ of it. For a Borel set
$A\subset C(F_0)$ put
\[
A_{\mathrm{gam}}:=\{(\wt B,\wt v,\gamma,\mathrm{gam}(\wt v,\gamma)):(\wt B,\wt v,\gamma)\in A\}.
\]
Then $(\mu\otimes\mathrm{counting})(A_{\mathrm{gam}})=\mu(A)$. The set $T(A_{\mathrm{gam}})$ is Borel, because $T|_D$ is a Borel injection between Polish spaces (the Lusin--Souslin
theorem \cite{Kec95}), and $T^{-1}(T(A_{\mathrm{gam}}))=A_{\mathrm{gam}}$. Therefore
\[
\mu(A)=M(T(A_{\mathrm{gam}}))=M'(T(A_{\mathrm{gam}}))=\mu'(A).
\]
Since $\mu$ and $\mu'$ are carried by $C(F_0)$, this gives $\mu=\mu'$. One $v'$ suffices in the hypothesis because the carrier of $\mu$ and $\mu'$ is a single class.
\end{proof}

\begin{lemma}[local gauge invariance]\label{rv:lem:Eloc}
For a gauge transformation $\Lambda_G$ supported on a finite set of sites, $\nu(F\circ\Lambda_G)=\nu(F)$ for every Gibbs measure $\nu$ of $\wt\gamma$ and bounded $F$.
\end{lemma}

\begin{proof}
$\Lambda_G$ moves only the variables of the finite set $V(\mathrm{supp})$ (Higgs phases at the sites, adjacent links, Villain integers of adjacent plaquettes), where it acts as a
measure-preserving bijection at fixed exterior with invariant weight and preserved closure (Lemma~\ref{rv:lem:phase}(a)); so
$\wt\gamma_{V(\mathrm{supp})}(F\circ\Lambda_G)=\wt\gamma_{V(\mathrm{supp})}(F)$ pointwise,
and one integrates against $\nu$.
\end{proof}

\begin{lemma}[injectivity of the patch kernel on positive measures]\label{rv:lem:IVL}
Let $F_0\in\Z_g$ and let $m,m'$ be positive Borel measures on $U_\Lambda=\C^\Lambda\times\mathfrak B_\Lambda$ carried by $\{F(\bar u)=F_0\}$ with $(K_\Lambda m)(v')=(K_\Lambda m')(v')<\infty$ for
$\lambda_\Lambda$-a.e.\ $v'\in\{F(v')=F_0\}$. Then $m=m'$.
\end{lemma}

\begin{proof}
(i) Higgs sector. For $b'=(B',v')$ put $m_{b'}(d\wt\Phi):=\int K_{\mathrm{bond},\Lambda}(b',\wt b)\,m(d\wt\Phi\,d\wt b)$, so that $(K_\Lambda m)(\Phi',b')=(G*m_{b'})(\Phi')$ (Tonelli). By Fubini
on $\lambda_\Lambda$
restricted to $\{F(v')=F_0\}$, for $\lambda_{\mathfrak B}$-a.e.\ such $b'$ (outside a null set $N_1$) the hypothesis holds for Lebesgue-a.e.\ $\Phi'$; Lemma~\ref{rv:lem:G} with
$\Omega=\R^{2|\Lambda|}$ gives $m_{b'}=m'_{b'}$, and $G*m_{b'}$ is finite everywhere because $\Omega$ is the whole space (for a general open $\Omega$ Lemma~\ref{rv:lem:G} gives finiteness
only on the convex hull $O$), so $m_{b'}$ is finite on bounded sets.

(ii) Bond sector. For bounded Borel $E\subset\C^\Lambda$ let $\mu_E(d\wt b):=m(E\times d\wt b)$, carried by $C(F_0)$, with $(K_{\mathrm{bond},\Lambda}\mu_E)(b')=m_{b'}(E)=m'_{b'}(E)<\infty$ for
$b'\notin N_1$.
Pick $v'_0\in V^+$ with $F(v'_0)=F_0$ (e.g.\ $F_0$ on one plaquette of $P(\Lambda)$, $0$ elsewhere); $N_1$ is $\lambda_{\mathfrak B}$-null, so for Haar-a.e.\ $B'$ the pair $(B',v'_0)$ is outside
$N_1$, and Lemma~\ref{rv:lem:BL} gives $\mu_E=\mu'_E$.

(iii) $\sigma$-finiteness and $\pi$-system. The pairs $(\wt v,\gamma)$ that occur in $C(F_0)$ form a countable set; let $(\wt v_j,\gamma_j)_j$ be an enumeration of it. For each $j$ the coset $\Gamma_\Lambda(\wt v_j-v'_0;\gamma_j-v'_{0,\partial\mathrm{Bo}})$ is nonempty
(Lemma~\ref{rv:lem:F}), so $K_{\mathrm{bond},\Lambda}((B',v'_0),(\wt B,\wt v_j,\gamma_j))$ is a positive continuous function of $(B',\wt B)$ on the compact
$\overline{\FD}^{\mathrm{Bo}(\Lambda)}\times\overline{\FD}^{\mathrm{Bo}(\Lambda)}$
(continuity on the closed box of $\FD$-representatives, where the Gaussian sum converges uniformly; not on the torus, where the representative jumps), bounded below by $c_j>0$. Hence
$\mu_E(\Tper^{\mathrm{Bo}(\Lambda)}\times\{(\wt v_j,\gamma_j)\})\le(K_{\mathrm{bond},\Lambda}\mu_E)(B',v'_0)/c_j<\infty$ for a $B'$ outside $N_1$. The sets
$\mathrm{ball}(0,k)\times\Tper^{\mathrm{Bo}(\Lambda)}\times\{(\wt v_j,\gamma_j):j\le k\}$
exhaust the carrier with finite $m$-mass, and $m,m'$ agree on the $\pi$-system of products $E\times F$; the $\pi$-$\lambda$ theorem gives $m=m'$.
\end{proof}

\begin{lemma}[upward step]\label{rv:lem:upward}
$H(m,n)$ implies $H(m,n+1)$ for every $m\le n-1$.
\end{lemma}

\begin{proof}
Assume $H(m,n)$. Fix a rectangle $\Lambda$ and use the notation of~\eqref{rv:eq:patchDLR}--\eqref{rv:eq:patchint}. We show that the laws of $\Theta_{m,n+1,\Lambda}$ under $\nu$ and $\nu'$
coincide.

\emph{Step 1: disintegration.} The space $U_\Lambda$ and the space of $Y$ are Polish. Let $q_\rho(Y,d\bar u)$ be a regular conditional law of $\bar u$ given $Y$ under $Q_\rho$, and put
$m^Y_\rho:=Z(Y,\cdot)^{-1}q_\rho(Y,\cdot)$. For $Q^Y$-almost every $Y$ this is a positive $\sigma$-finite measure carried by $\{F(\bar u)=F(Y)\}$. Indeed, $Z<\infty$ everywhere, and by
Lemma~\ref{rv:lem:F} one has $Z>0$ if and only if $F(\bar u)=F(Y)$, an equality that holds almost surely.

\emph{Step 2: the conditional laws of $\bar u$ given $Y$ coincide.} Taking $f=1$ in~\eqref{rv:eq:patchint} and using Tonelli's theorem, we obtain
\[
\rho(A(Y)a(v))=\int Q^Y(dY)A(Y)\int a(v')c(v';Y)(K_\Lambda m^Y_\rho)(v')\lambda_\Lambda(dv'),
\]
and $\int c\,(K_\Lambda m^Y_\rho)\,d\lambda_\Lambda=\int ZZ^{-1}q_\rho(Y,d\bar u)=1$. So $\pi^Y_\rho:=c(\cdot;Y)(K_\Lambda m^Y_\rho)\lambda_\Lambda$ is a probability measure on the space of
$v'$, which is locally compact and second countable. The left side of the display depends only on the law of $(Y,v)$. This law is common to $\nu$ and $\nu'$ by $H(m,n)$ on $\Lambda^+$,
the coordinates of $Y$ and $u$ being functions of exterior variables. Applying the display to a countable dense set of functions $a$ in $C_c$ and using the Riesz representation
theorem, we get $\pi^Y_\nu=\pi^Y_{\nu'}$ for $Y$ outside a $Q^Y$-null set. Fix such a $Y$. Then
\[
c(v';Y)(K_\Lambda m^Y_\nu)(v')=c(v';Y)(K_\Lambda m^Y_{\nu'})(v')<\infty
\]
for $\lambda_\Lambda$-almost every $v'$. On the set where $c>0$, that is, where $F(v')=F(Y)$ and $\rho'>0$, this is the hypothesis of Lemma~\ref{rv:lem:IVL} with $F_0=F(Y)$; both measures
$m^Y_\nu$ and $m^Y_{\nu'}$ are carried by $\{F=F(Y)\}$. On the set where $F(v')\ne F(Y)$ the coset is empty, so that $K_\Lambda(v',\bar u)=0$ for every $\bar u$ in the carrier, and there is
nothing to verify. Lemma~\ref{rv:lem:IVL} gives $m^Y_\nu=m^Y_{\nu'}$. Hence $q_\nu(Y,\cdot)=q_{\nu'}(Y,\cdot)$ for $Q^Y$-almost every $Y$, and therefore $Q_\nu=Q_{\nu'}$.

\emph{Step 3: the joint law of $(Y,v,\bar u)$.} By~\eqref{rv:eq:patchint} and $Q_\nu=Q_{\nu'}$, the number $\rho(A(Y)a(v)f(\bar u))$ is the same for $\rho=\nu$ and $\rho=\nu'$, for all
bounded Borel $A,a,f$. Since the product sets form a $\pi$-system, the laws of $(Y,v,\bar u)$ under $\nu$ and $\nu'$ coincide.

\emph{Step 4: the temporal integers over $\mathrm{Bo}(\Lambda)$.} For $b\in\partial\mathrm{Bo}(\Lambda)$ with straddling plaquette $p'(b)$ the upper cube constraint reads
$\wt v^{t,(n)}_b=\gamma_{p'(b)}-\wt v^{(n)}_{p'(b)}$, which is a function of $(v,\bar u)$.

Let $\Delta_2:=\{v^{t,(n-1)}_b,v^{t,(n)}_b:b\in\mathrm{Bo}(\Lambda)\}$. The factors of the weight that contain a variable of $\Delta_2$ are the temporal factors $w$ on $\mathrm{Bo}(\Lambda)$
of the two slabs and the cube constraints on $P^+(\Lambda)$ of the two slabs. Every other variable is exterior to $\Delta_2$, and in particular so is all of $(Y,v,\bar u)$: the layer-$n$
links are now exterior, so that the transported straddling integers of layer $n$ are functions of exterior variables, and $\gamma$ is the exterior expression fixed above. After the gauge
transport, which is a bijection of the integer sets, we have for bounded $g$
\begin{multline*}
\wt\gamma_{\Delta_2}(g)(\om)=\mathcal N^{-1}\!\!\sum_{w_1\in\Gamma_\Lambda(\wt v^{(n)}-\wt v^{(n-1)};\,\wt v^{(n)}_{p'}-\wt v^{(n-1)}_{p'}-\mathrm{collar}^{(n-1)})}\ \
\sum_{w_2\in\Gamma_\Lambda(\wt v^{(n+1)}-\wt v^{(n)};\,\gamma-\wt v^{(n)}_{p'})}\\
g(w_1,w_2)\prod_bw(\wt B^{(n)}_b-\wt B^{(n-1)}_b+w_{1,b})\,w(\wt B^{(n+1)}_b-\wt B^{(n)}_b+w_{2,b})=:G^\sharp[g](Y,v,\bar u),
\end{multline*}
with $\mathcal N\in(0,\infty)$ almost surely, because the cosets are nonempty by Lemma~\ref{rv:lem:F} and the convention stated at the beginning of this subsection, and the sums are bounded
by periodised Gaussians. Here $\mathrm{collar}^{(n-1)}$ stands for the signed sum of the three collar integers $\wt v^{t,(n-1)}$ at $p'$. Writing
$\wt\Delta_2:=(\wt v^{t,(n-1)}|_{\mathrm{Bo}(\Lambda)},\wt v^{t,(n)}|_{\mathrm{Bo}(\Lambda)})$ for the transported variables of $\Delta_2$, the DLR equation gives
\[
\rho(A(Y,v,\bar u)g(\wt\Delta_2))=\rho(A(Y,v,\bar u)G^\sharp[g](Y,v,\bar u)),
\]
and by Step~3 the right side is the same for $\nu$ and $\nu'$. Hence the laws of $(Y,v,\bar u,\wt v^{t,(n-1)}|_{\mathrm{Bo}(\Lambda)},\wt v^{t,(n)}|_{\mathrm{Bo}(\Lambda)})$ under $\nu$ and
$\nu'$ coincide.

\emph{Step 5: conclusion.} The family $\Theta_{m,n+1,\Lambda}$ is a sub-family of these coordinates. The layers $m,\dots,n$ on $\Lambda$ come from $(Y,v)$, because
$P(\Lambda)\subset P^+(\Lambda)$ and $\Lambda\subset\Lambda^+$. The slabs $m,\dots,n-2$ on $\mathrm{Bo}(\Lambda)$ come from $Y$, and the slab $n-1$ on $\mathrm{Bo}(\Lambda)$ comes from Step~4.
The layer $n+1$, that is, $\wt\Phi$ on $\Lambda$, $\wt B$ on $\mathrm{Bo}(\Lambda)$ and $\wt v$ on $P(\Lambda)$, comes from $\bar u$, and the slab $n$ on $\mathrm{Bo}(\Lambda)$ comes from
Step~4. So the laws of $\Theta_{m,n+1,\Lambda}$ under $\nu$ and $\nu'$ coincide. Since $\Lambda$ was arbitrary, this proves $H(m,n+1)$.
\end{proof}

\begin{lemma}[transported slab data and gauge-invariant slab functions]\label{rv:lem:S71}
Let $\nu,\nu'$ be Gibbs measures of $\wt\gamma$, let $B$ be a rectangle and let $m<n$. The following are equivalent:
\begin{itemize}
\item[(i)] the laws of $\Theta_{m,n,B}$ (base layer $m$) under $\nu$ and $\nu'$ coincide;
\item[(ii)] $\nu=\nu'$ on every bounded function of the slab-$[m,n]$ variables over $B$ (sites of $B$, links of $\mathrm{Bo}(B)$, plaquettes of $P(B)$, temporal links at the sites of $B$ and
temporal plaquettes over $\mathrm{Bo}(B)$ between layers $m$ and $n$) that is invariant under all gauge transformations supported in $B\times[m,n]$;
\item[(iii)] the statement (i) with any other base layer $k\in[m,n]$.
\end{itemize}
Moreover, if $\nu=\nu'$ on all bounded gauge-invariant local functions, then $\nu=\nu'$.
\end{lemma}

\begin{proof}
We first determine how the slab data transform. Under any gauge transformation $G$, the rule $B_t^{(k)}\mapsto B_t^{(k)}+G^{(k+1)}-G^{(k)}$ gives
$\lambda^{(k)}\mapsto\lambda^{(k)}-G^{(k)}+G^{(m)}$, so that $\wt c_k\mapsto G^{(m)}.\wt c_k$ for every $k$. Thus $\Theta=\Theta_{m,n,B}$ transforms by the diagonal action of $G^{(m)}|_B$;
here we use that $\Tper$ is abelian, and the integers shift consistently.

\emph{(ii) implies (i).} For a bounded Borel function $H$ let
\[
\bar H(\om):=\int_{\Tper^B}H(g.\Theta(\om))\,\Haar(dg).
\]
By the invariance of the Haar measure, $\bar H$ is a function of the slab variables over $B$ that is invariant under every gauge transformation. For $g\in\Tper^B$ let $G_g$ be the gauge
transformation that equals $g$ on layer $m$ over $B$ and $0$ elsewhere. It is finitely supported and acts on $\Theta$ as the diagonal action of $g$, so Lemma~\ref{rv:lem:Eloc} gives
$\nu(H(\Theta))=\nu(H(g.\Theta))$ for every $g$, and the same for $\nu'$. Integrating over $g$ and using (ii) for $\bar H$, we obtain
\[
\nu(H(\Theta))=\nu(\bar H)=\nu'(\bar H)=\nu'(H(\Theta)).
\]

\emph{(i) implies (ii).} Let $F$ be as in (ii). For $\lambda=(\lambda^{(k)}_x)_{x\in B,\,m\le k\le n}$ let $G_\lambda$ be the gauge transformation that equals $\lambda$ on the slab over $B$ and
$0$ elsewhere. For each fixed value of $\lambda$ it is finitely supported and $F$ is invariant under it; therefore $F(\om)=F(G_{\lambda(\om)}.\om)$ pointwise. The configuration
$G_{\lambda(\om)}.\om$ has temporal links $0$ on the slab over $B$ and slab data $\Theta(\om)$. So $F=F^\sharp(\Theta_{m,n,B})$ with
\[
F^\sharp(\theta):=F(\text{the configuration with slab data }\theta\text{ and temporal links }0).
\]
Hence (i) gives $\nu(F)=\nu'(F)$.

\emph{(i) and (iii) are equivalent.} Condition (ii) does not mention the base layer, and the two implications just proved hold, with the same proof, for every base layer $k\in[m,n]$.

\emph{The last assertion.} Let $F$ be a bounded local function, and let $\bar F$ be its Haar average over the finitely many sites whose links and plaquettes $F$ depends on. Then $\bar F$
is local and gauge invariant, and $\nu(F)=\nu(\bar F)$ by Lemma~\ref{rv:lem:Eloc}. So if $\nu$ and $\nu'$ agree on the bounded gauge-invariant local functions, then they agree on all
bounded local functions, and hence $\nu=\nu'$.
\end{proof}

\begin{lemma}[downward step]\label{rv:lem:downward}
$H(m,n)$ implies $H(m-1,n)$ for every $m\le n-1$.
\end{lemma}

\begin{proof}
Let $\theta:(x,k)\mapsto(x,-k-1)$ be the time reflection in a mid-plane; it is a lattice automorphism of $\eps\Z^3$. The weight is invariant under $\theta$ factor by factor: the factors
$g_{\mathrm{ct}}$ and the spatial factors $h$, $w$ are relabelled; the temporal factor $h$ satisfies $h(r,r',\vartheta)=h(r',r,-\vartheta)$; the temporal factor $w$ is even; and the maps
$B_t\mapsto-B_t$, $v^t\mapsto-v^t$ preserve $\mathrm{Leb}\times\Haar\times\mathrm{counting}$ and the closure constraint. Consequently
$\wt\gamma_\Delta(F\circ\theta)=(\wt\gamma_{\theta\Delta}F)\circ\theta$ for every finite $\Delta$, the measure $\theta_*\nu$ is a Gibbs measure, and $\theta$ conjugates finitely supported
gauge transformations to finitely supported gauge transformations.

The reflection $\theta$ maps the slab $[m,n]$ over $B$ onto the slab $[-n-1,-m-1]$ over $B$ and preserves the class of functions in Lemma~\ref{rv:lem:S71}(ii). Since condition (ii) of
that lemma does not involve the base layer, the equivalence of (i) and (ii) shows that $H(m,n)$ for $(\nu,\nu')$ is equivalent to $H(-n-1,-m-1)$ for $(\theta_*\nu,\theta_*\nu')$. Since
$-n-1\le(-m-1)-1$, Lemma~\ref{rv:lem:upward} for the reflected pair gives $H(-n-1,-m)$, and reflecting back gives $H(m-1,n)$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{rv:thm:DV}]
A function of the slab-$[m,n]$ variables over a rectangle $B$ sees a gauge transformation $G$ only through $G$ at the sites of $B\times[m,n]$, so for such functions invariance under
the gauge transformations supported in $B\times[m,n]$ is invariance under all gauge transformations; Lemma~\ref{rv:lem:S71} (ii)$\Rightarrow$(i) turns the hypothesis into $H(n_0,n_0+1)$.
Lemma~\ref{rv:lem:upward} gives $H(n_0,n)$ for all $n>n_0$ and Lemma~\ref{rv:lem:downward} gives $H(m,n)$ for all $m\le n_0<n$. Every bounded local $F$ depends on finitely many variables, all
contained in the slab $[m,n]$ over some rectangle $B$; if $F$ is gauge invariant, Lemma~\ref{rv:lem:S71} (i)$\Rightarrow$(ii) with $H(m,n)$ gives $\nu(F)=\nu'(F)$; by the last claim of
Lemma~\ref{rv:lem:S71}, $\nu=\nu'$.
\end{proof}

\subsection{Conclusion: proof of Theorem~\ref{rv:thm:RV}}\label{rv:ssec:S9}

The proof uses the following elementary statement.

\begin{proposition}[covariance; conjugation and composition of reflections]\label{rv:prop:YM4}
\begin{itemize}
\item[(i)] (Covariance.) The fine Villain--Higgs specification, in unitary gauge and phase-restored alike, is covariant under the lattice isometries $g\in\eps\Z^3\rtimes B_3$ and the
mid-plane reflections $\theta_{\mu,c}$, $c\in\eps(\Z+\tfrac12)$: $\gamma_{g^{-1}\Delta}(f\circ\hat g)=(\gamma_\Delta f)\circ\hat g$. Hence $\nu\circ\hat g$ is Gibbs whenever $\nu$ is, and
$\Gamma$-invariance of $\nu$
becomes $g\Gamma$-invariance of $\nu\circ\hat g$.
\item[(ii)] (Conjugation.) Let $\nu$ be a Gibbs measure such that $\nu$ and $\nu\circ\hat g$, for $g\in B_3$ the transpositions of the coordinates $\mu$ and $3$ ($\mu=1,2$), are each invariant under
every $\theta_{3,c}$, $c\in\eps(\Z+\tfrac12)$. Then $\nu$ is invariant under every $\theta_{\mu,c}$, $\mu=1,2,3$.
\item[(iii)] (Composition.) $\theta_{\mu,c}\circ\theta_{\mu,c'}$ maps $x_\mu$ to $x_\mu+2(c-c')$ and fixes the other coordinates; with $c-c'=\eps$ it is $\tau_{2\eps e_\mu}$. Hence a measure
invariant under all
$\theta_{\mu,c}$ is invariant under $\tau_{\pm2\eps e_\mu}$, $\mu=1,2,3$, i.e.\ under $2\eps\Z^3$.
\end{itemize}
\end{proposition}

\begin{proof}
(i) $h(r,r',\vartheta)$ is symmetric in $(r,r')$ and even in $\vartheta$, so with $B_{-b}=-B_b$ the hopping factor is a function of the unoriented bond; $w$ is even and $v_{-p}=-v_p$, so the
plaquette factor is a function of the unoriented plaquette; $g_{\mathrm{ct}}$ is a site function; the constraint $dv=0$ is mapped to itself by any lattice isometry (cubes to cubes,
orientation signs cancel); the reference measures are invariant under the relabelling-with-sign bijection. Hence $Z_{g^{-1}\Delta}(\hat g\xi)=Z_\Delta(\xi)$ and the covariance. The
mid-plane reflections map sites to sites (no site on the plane) and bonds to bonds. For the phase-restored specification the phase field is a site field and $h$ is even in $\vartheta$.
(ii) $g\theta_{\mu,c}g=\theta_{3,c}$. The $\theta_{3,c}$-invariance of $\nu\circ\hat g$ reads $\nu(f\circ\hat\theta_{3,c}\circ\hat g)=\nu(f\circ\hat g)$ for all bounded $f$; replacing $f$ by
$f\circ\hat g$ and using
$\hat g\hat\theta_{3,c}\hat g=\hat\theta_{\mu,c}$ gives $\nu(f\circ\hat\theta_{\mu,c})=\nu(f)$. (iii) is the computation $x_\mu\mapsto2c-(2c'-x_\mu)$ and $\widehat{g_1g_2}=\hat g_1\hat g_2$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{rv:thm:RV}]
\emph{Step 0.} Let $\nu$ be as in the statement. By Lemma~\ref{rv:lem:phase}(d), $\wt\nu:=\Psi_*(\nu\times\Haar^{\mathrm{sites}})$ is a $G$-invariant Gibbs measure of $\wt\gamma$ with the same law
of $(\tr,f)$ as $\nu$, so (I1)--(I3) hold for $\wt\nu$; by Lemma~\ref{rv:lem:phase}(e) $\wt\nu$ is $\Gamma$-invariant, and $\nu$ is $\tau_a$-invariant if and only if $\wt\nu$ is. We work with $\wt\nu$,
called $\nu$.

\emph{Step 1 (time reflections).} Let $c\in\eps(\Z+\tfrac12)$, $\theta:=\theta_{3,c}$ and $\nu':=\nu\circ\hat\theta$, a Gibbs measure of $\wt\gamma$ by Proposition~\ref{rv:prop:YM4}(i). Let
$n_0$ be the layer
below the plane, so that $\theta$ is the mid-plane reflection of the slab $[n_0,n_0+1]$. Let $B$ be a rectangle and $S:=\mathrm{Bo}(B)$ the set of its links (a bond set closed under the bonds of its
plaquettes $P(B)$), and let $f$ be a bounded function of the slab-$[n_0,n_0+1]$ variables over $B$ invariant under the gauge transformations supported in $B\times\{n_0,n_0+1\}$,
equivalently under all gauge transformations.
Such an $f$ is a bounded gauge-invariant $S$-local slab function in the sense of Lemma~\ref{rv:lem:slab}(b)
($f=F\circ\Theta_\sigma|_S$ with $F$ diagonal-invariant,
$\wt v^t$ included), so Theorem~\ref{rv:thm:EV} (which uses (I1)--(I3) and the $\Gamma$-invariance of $\nu$, including a time translation, through Lemma~\ref{rv:lem:TV}) gives
$\nu(f\circ\hat\theta)=\nu(f)$, i.e.\ $\nu'(f)=\nu(f)$, for every rectangle $B$. This is the hypothesis of Theorem~\ref{rv:thm:DV}, in its form for rectangles (the passage
from $S$-local agreement to the laws of
$\Theta_{n_0,n_0+1,B}$ is made in the proof of that theorem, by Lemma~\ref{rv:lem:S71}). Theorem~\ref{rv:thm:DV} gives $\nu'=\nu$: $\nu$ is $\theta_{3,c}$-invariant for every $c$.

\emph{Step 2 (the other directions).} Let $\mu\in\{1,2\}$ and $g\in B_3$ the transposition of the coordinates $\mu$ and $3$. By Proposition~\ref{rv:prop:YM4}(i), $\nu\circ\hat g$ is a Gibbs measure
invariant under $g\Gamma$, which contains a time translation because a finite-index sublattice $\Gamma$ of $\eps\Z^3$ contains $k\eps\Z^3$, $k=[\eps\Z^3:\Gamma]$, hence translations along
every axis; (I1) and (I3) are invariant under relabelling, and (I2) is stated for all plaquettes, so it holds for $\nu\circ\hat g$ (whose temporal
plaquettes are $\nu$'s plaquettes that contain a link in the direction $e_\mu$). By Lemma~\ref{rv:lem:phase}(e), $\nu\circ\hat g$ is the phase restoration of the unitary-gauge measure $(\Phi_*\nu)\circ\hat g$, which is a Gibbs measure by Proposition~\ref{rv:prop:YM4}(i), is invariant under $g\Gamma$ and satisfies (I1)--(I3) for the same reasons. So Step 1 applies to $\nu\circ\hat g$ and gives its $\theta_{3,c}$-invariance, and Proposition~\ref{rv:prop:YM4}(ii) gives the invariance of $\nu$
under every $\theta_{\mu,c}$, $\mu=1,2,3$.

\emph{Step 3 (translations).} By Proposition~\ref{rv:prop:YM4}(iii), $\nu$ is invariant under $\tau_{2\eps e_\mu}$, $\mu=1,2,3$. (The argument uses only reflections in mid-planes of links, so it does not give the invariance under the
translations by $\eps$.)

\emph{Step 4.} By Step 0 and Lemma~\ref{rv:lem:phase}(e) the unitary-gauge measure $\nu$ is $\tau_{2\eps e_j}$-invariant, $j=1,2,3$.
\end{proof}

\begin{remark}[where each hypothesis enters]\label{rv:rem:hyp}
The hypotheses of Theorem~\ref{rv:thm:RV} enter its proof as follows. The Gibbs property is used in Theorem~\ref{rv:thm:EV}, by way
of Lemma~\ref{rv:lem:TV} and the averaging identity through the torus states, and in Theorem~\ref{rv:thm:DV}, by way of the DLR equation~\eqref{rv:eq:patchDLR}
(Lemma~\ref{rv:lem:upward}) and Lemma~\ref{rv:lem:Eloc}; in both it is used for $\nu$ and for $\nu\circ\hat g$. The invariance under a sublattice
of finite index is used only in Lemma~\ref{rv:lem:TV}, which is applied in Lemmas~\ref{rv:lem:step2} and~\ref{rv:lem:step2full} and
compares the measure with the $\Gamma'$-periodic torus states; it is used for $\nu$ and for $\nu\circ\hat g$.

Hypothesis (I1), through $\sup_x\nu(\rho_x^2)$, and hypothesis (I2) in its joint form enter the entropy bound of
Lemma~\ref{rv:lem:TV}; the joint form provides one constant for all finitely supported $s$, and this bounds the second moments of the
fluxes through spheres of growing size. Hypothesis (I2), jointly over all plaquettes, also gives the exponential moments of $f$, which
are used in Lemma~\ref{rv:lem:bondlaw} and for $\nu(r_Y^2)$ in Lemmas~\ref{rv:lem:step2} and~\ref{rv:lem:step2full}, and its Gaussian
moments at every rate $<\tfrac12$. The latter are used at the rate $\tfrac14$ for the growth of $\hat m$ in
Lemma~\ref{rv:lem:bondsep} and of $\hat m_s$ in Lemma~\ref{rv:lem:fullsep}, and for the tail bound $\nu(A_t^c)\le C|t|^{16a/3}$,
$a\in(\tfrac38,\tfrac12)$, in Lemma~\ref{rv:lem:step2full}. The proof of Lemma~\ref{rv:lem:step2full} requires a rate $>\tfrac38$, which
the constant $\tfrac12$ in the exponent of (I2) provides; this requirement is a feature of that proof, as explained at its end.
Hypothesis (I3) gives the Gaussian moments of $f_H$ at every rate, which are used in Lemma~\ref{rv:lem:higgscorr}, for the tails in
Lemma~\ref{rv:lem:step2full} and for the growth in Lemma~\ref{rv:lem:fullsep}; it also gives $\nu(\rho_x^2)$ for
Lemma~\ref{rv:lem:TV}.

Theorem~\ref{rv:thm:DV} and Subsection~\ref{rv:ssec:S1} use none of (I1)--(I3). The proof of Theorem~\ref{rv:thm:RV} uses nothing
beyond (I1)--(I3), the Gibbs property and the $\Gamma$-invariance; in particular, it uses no temperedness, uniqueness or clustering.
The theorem applies to Gibbs measures of the unperturbed specification: in this paper it is applied to the comparison states of Section~\ref{n1:sec}
(Lemma~\ref{n1:lem:JGD}, Proposition~\ref{n1:prop:I123}; compare Remark~\ref{rv:rem:where}). It does not apply to the limit points
$\nu_t$, $t\ne0$, of the perturbed laws (Lemma~\ref{n1:lem:gibbs}).
\end{remark}

\providecommand{\cA}{\mathcal A}\providecommand{\cW}{\mathcal W}\providecommand{\cH}{\mathcal H}\providecommand{\cC}{\mathcal C}
\providecommand{\cG}{\mathcal G}\providecommand{\cX}{\mathcal X}\providecommand{\cB}{\mathcal B}\providecommand{\cP}{\mathcal P}

\section{Diagonal reflection positivity of the torus limit points}
\label{n1:sec}

This section proves the reflection positivity at the diagonal planes for the limit states $\omega$ of
Definition~\ref{def:omN}, taken along the dyadic sequences of sides introduced in Definition~\ref{n1:def:D}.
The proof runs through an adapted torus with a family of reflection walls, a perturbed twisted law on it which is
reflection positive across every wall, a fold identity and a convexity inequality for the perturbed free energy, a
surface bound which removes the perturbation in the limit, and the rigidity theorem of the preceding section,
Theorem~\ref{rv:thm:RV}, applied to a sourceless comparison state; as in that section, $\kappa_N=1$ (Lemma~\ref{lem:rescale}(g) extends Theorem~\ref{n1:thm:N10} and Corollary~\ref{n1:cor:N1} to every $\kappa_N>0$).

\subsection{Dyadic sequences of sides}
\label{n1:sub:D}

\begin{definition}[dyadic sequences of sides]\label{n1:def:D}\label{n1:def:sides}
For each level $N$ fix two lengths $L_{\mathrm{IV}}>0$ and $L_{\min}>0$ with $L_{\mathrm{IV}}/a_0$ rational, and call a side $L$ \emph{admissible} if
$L\in4L_{\mathrm{IV}}\mathbb N$, $L/2\in a_0\mathbb Z$ and $L\ge L_{\min}$. Fix an admissible side $L_*$.
The \emph{dyadic sequence} generated by $L_*$ is the sequence of sides
\[
L_n := 2^n L_*,\qquad n\ge 0,
\]
and the associated \emph{wall spacings} are
\[
h_j := 2^{j}\,\frac{L_*}{2},\qquad j\ge 0,\qquad\text{so that } h_j \text{ divides } \tfrac{L_n}{2}\ \text{ for every } n\ge j .
\]
A \emph{dyadic limit point} is a weak-$*$ limit point of the torus states $(\langle\cdot\rangle^{\mathrm{fine}}_{L_n})_{n\ge0}$ of Definition~\ref{def:omN}
along a dyadic sequence.
\end{definition}

\begin{remark}\label{n1:rem:D}
Admissible sides exist: writing $L_{\mathrm{IV}}/a_0=p/q$ with integers $p,q\ge1$, the side $L=4qkL_{\mathrm{IV}}=4pka_0$ is admissible for every integer $k\ge L_{\min}/(4pa_0)$.
Every $L_n$ is admissible ($4L_{\mathrm{IV}}\mathbb N$ is closed under multiplication by $2$, $L_n/2=2^{n-1}L_*\in a_0\mathbb Z$ for
$n\ge1$, while $L_0/2=L_*/2\in a_0\mathbb Z$ by assumption, and $L_n\ge L_*\ge L_{\min}$). Definition~\ref{def:omN} fixes a sequence of
sides with $L_n/2\in\eN\Z$ and a limit law along a subsequence of it, any cofinal prescription of sides being allowed
(Theorem~\ref{thm:d2:LP}, Remark~\ref{rem:d2:LP}(ii)); here that prescribed sequence is \emph{chosen} to be a dyadic sequence, which
is a choice and not a hypothesis. The divisibility $h_j\mid L_n/2$ ($n\ge j$) survives the passage to any subsequence, so every dyadic
limit point keeps it; and (a)--(e) of Theorem~\ref{thm:d2:LP} hold for it along that subsequence (its ``Moreover'' clause).
\end{remark}

\begin{theorem}[diagonal reflection positivity through the origin]\label{n1:thm:N1zero}\label{n1:thm:N10}
Let $\omega$ be a dyadic limit point. Let $P$ be one of the six diagonal site planes through $0$,
\[
P = P_{ij}^{\sigma} := \{x\in\mathbb R^3 : x_i = \sigma x_j\},\qquad 1\le i<j\le 3,\quad \sigma\in\{+1,-1\},
\]
let $\theta_P$ be the reflection in $P$ and $\overline{H}_P$ one of the two closed half spaces bounded by $P$.
Then for every bounded Borel function $F$ of the gauge-invariant variables $(\ff_p,\tr_x)$ of finitely many cells (plaquettes, sites) all of whose
vertices lie in $\overline{H}_P$,
\[
\omega\big(\overline{F\circ\theta_P}\cdot F\big)\ \ge\ 0 ,
\]
$\theta_P$ acting on cells and fields as $R_{j,c}$ does in Theorem~\ref{thm:d2:LP}(d) (so $\overline{F\circ\theta_P}$ is the
antilinear reflection $\Th_P F$ of that theorem, now at a diagonal plane).
\end{theorem}

Theorem~\ref{n1:thm:N1zero} is proved in Subsection~\ref{n1:sub:N1}.

\begin{remark}[the planes and the observables]\label{n1:rem:scope}
The proof of Theorem~\ref{flh:thm:FLH2} uses, at $c=0$ (Remark~\ref{n1:rem:FLH}), the reflection positivity of $\om_N$ at the six diagonal site-plane
families $\{x_i\pm x_j=c\}$, $c\in\eN\Z$, in the sense of Theorem~\ref{thm:d2:LP}(d). Theorem~\ref{n1:thm:N1zero} is the case $c=0$
with bounded Borel observables, real or complex, of finitely many cells; Corollary~\ref{n1:cor:N1} in Subsection~\ref{n1:sub:N1} extends it to all
$c\in\eN\Z$ (by translation invariance of the limit points), to the Hermitian form and to the $L^2$ closure, which is the form
Theorem~\ref{thm:d2:LP}(d) uses. Remark~\ref{n1:rem:FLH} states the consequence for Theorems~\ref{flh:thm:FLH1}, \ref{flh:thm:FLH1p} and~\ref{flh:thm:FLH2}.
Throughout, $\theta_P$ acts on a configuration by pulling back the fields along the reflection in $P$ (a site plane, so that the
cells of $P$ are mapped to themselves), and on a function $F$ of the configuration by composition, $\theta_P F:=F\circ\theta_P$.
\end{remark}

\subsection{The adapted torus and its reflection walls}
\label{n1:sub:torus}

\emph{Notation.} Throughout $N$ is fixed, $\eN$ is the spacing of the fine lattice $\eN\Z^3$, the cells are its sites $x$, bonds $b$ and
plaquettes $p$, the fields are $(\tr_x,B_b,v_p)$ of Section~\ref{sec:setting} with $\rho_x=\rhof+\tr_x$ and $\ff_p=(dB)_p+v_p$, and the
weight is the local weight $W$ of Section~\ref{sec:setting} ($v$ closed on every cube). We use the coordinates $u:=x_1-x_2$, $v:=x_1+x_2$
(the symbol $v$ for this coordinate is always written with its argument, $v(x)$, where confusion with the Villain variable is possible) and $x_3$.
For a plane $\Pi$, $\theta_\Pi$ is the reflection of cells and fields in $\Pi$ (the action of Theorem~\ref{thm:d2:LP}(d)); a function ``of
cells in a closed half space'' depends on the variables of finitely many cells all of whose vertices lie in it; $\cA_+(\Pi)$ denotes the
bounded continuous such functions for the closed half space on the $+$ side of $\Pi$. The cubic torus $T^{\mathrm{cub}}_L:=\eN\Z^3/L\Z^3$ is
the fine torus $\tilde T_L$ of Section~\ref{sec:setting}; we write $\langle\cdot\rangle^{\mathrm{cub}}_L$ for its state
$\langle\cdot\rangle^{\mathrm{fine}}_L$ and $Z^{\mathrm{cub}}_L$ for its partition function.

\begin{definition}[the adapted torus]\label{n1:def:tw}
For $L$ with $L/2\in\eN\Z$ let $\Gamma(L):=\tfrac L2(e_1-e_2)\Z+L(e_1+e_2)\Z+Le_3\Z\subset\eN\Z^3$ and $T^{\mathrm{tw}}_L:=\eN\Z^3/\Gamma(L)$.
Since $|\det\Gamma(L)|=L^3$, $T^{\mathrm{tw}}_L$ has $(L/\eN)^3$ sites and three times as many bonds and plaquettes. Its state
$\langle\cdot\rangle^{\mathrm{tw}}_L$ is the probability law with weight $W$ on the cells of $T^{\mathrm{tw}}_L$ (sum over closed $v$), with
partition function $Z^{\mathrm{tw}}_L$; $\mu^{\mathrm{tw}}_L$ denotes this law.
\end{definition}

\begin{lemma}[symmetries and walls of the adapted torus]\label{n1:lem:walls}
For $c\in\eN\Z$ put $r_c(x):=(x_2+c,\,x_1-c,\,x_3)$ and $s_c(x):=(c-x_2,\,c-x_1,\,x_3)$, and for $c\in\tfrac12\eN\Z$ put
$q_c(x):=(x_1,\,x_2,\,2c-x_3)$. Each maps $\eN\Z^3$ onto itself, its linear part fixes $\Gamma(L)$, and it descends to an involutive
automorphism of the cell complex $T^{\mathrm{tw}}_L$ with $W\circ R=W$ cell by cell and an invariant reference measure. The pointwise fixed
walls are: $\{u=c\}$ and $\{u=c+L/2\}$ for $r_c$; $\{v(x)=c\}$ and $\{v(x)=c+L\}$ for $s_c$; $\{x_3=c\}$ and $\{x_3=c+L/2\}$ for $q_c$.
On $T^{\mathrm{tw}}_L$ the coordinates $u$, $v(x)$, $x_3$ are defined modulo $L$, $2L$, $L$, and $r_{c+L/2}=r_c$. The translations by
$\eN\Z^3$ act on $T^{\mathrm{tw}}_L$ and preserve $W$. Moreover $r_cr_{c'}$, $s_cs_{c'}$, $q_cq_{c'}$ are the translations by
$(c-c')(e_1-e_2)$, $(c-c')(e_1+e_2)$, $2(c-c')e_3$.
\end{lemma}
\begin{proof}
The linear part of $r_c$ is the coordinate swap $(1\,2)$, which sends $\tfrac L2(e_1-e_2)$ to its negative and fixes $e_1+e_2$, $e_3$; that of
$s_c$ is $-(1\,2)$, which fixes $e_1-e_2$ and negates $e_1+e_2$; that of $q_c$ is $\mathrm{diag}(1,1,-1)$. So each fixes $\Gamma(L)$ and
descends to the quotient complex. For the fixed points, $r_cx-x=(c-u)(e_1-e_2)$ vanishes for $u=c$ and equals $-\tfrac L2(e_1-e_2)\in\Gamma(L)$ for
$u=c+L/2$; $s_cx-x=(c-v(x))(e_1+e_2)$ is $0$ for $v(x)=c$ and $-L(e_1+e_2)\in\Gamma(L)$ for $v(x)=c+L$; $q_cx-x=2(c-x_3)e_3$ is $0$ for
$x_3=c$ and $-Le_3\in\Gamma(L)$ for $x_3=c+L/2$. The products are $r_cr_{c'}x=x+(c-c')(e_1-e_2)$, $s_cs_{c'}x=x+(c-c')(e_1+e_2)$,
$q_cq_{c'}x=x+2(c-c')e_3$; with $c'=c+L/2$ the first is the translation by $-\tfrac L2(e_1-e_2)\in\Gamma(L)$, the identity of
$T^{\mathrm{tw}}_L$, so $r_{c+L/2}=r_c$. Invariance of $W$ is Step~0 of the proof of Theorem~\ref{flh:thm:I1}, which uses only that
$g_{\mathrm{ct}}$ is one function of the site variable, $h$ is symmetric in $(\rho,\rho')$ and even in $B$, $w$ is even, $\ff_p\circ R=\ff_{Rp}$
and the closedness constraint is $R$-invariant; the reference measure is invariant, because $R$ acts on the variables by a permutation of the coordinates with sign changes of some $B_b$, $v_p$.
That both walls of each reflection are pointwise fixed is Lemma~\ref{flh:lem:I0}(a) for the lattice $\Gamma(L)$.
\end{proof}

\begin{lemma}[reflection positivity of the adapted torus at its walls]\label{n1:lem:RPtw}
Let $L\ge6\eN$, $L/2\in\eN\Z$, and let $R$ be one of $r_c$ ($c\in\eN\Z$), $s_c$ ($c\in\eN\Z$), $q_c$ ($c\in\tfrac12\eN\Z$), with its two walls
$\Pi$, $\Pi'$ of Lemma~\ref{n1:lem:walls} and the two closed halves of $T^{\mathrm{tw}}_L$ they bound ($\{c\le u\le c+L/2\}$ and
$\{c+L/2\le u\le c+L\}$ for $r_c$; likewise in $v(x)$ with $L$ in place of $L/2$, and in $x_3$). Then for all bounded Borel $F,G$ of
cells in one closed half,
\[
\langle F\circ R\rangle^{\mathrm{tw}}_L=\langle F\rangle^{\mathrm{tw}}_L,\qquad
\langle\overline{F\circ R}\cdot G\rangle^{\mathrm{tw}}_L=\overline{\langle\overline{G\circ R}\cdot F\rangle^{\mathrm{tw}}_L},\qquad
\langle\overline{F\circ R}\cdot F\rangle^{\mathrm{tw}}_L\ge0 .
\]
\end{lemma}
\begin{proof}
For $L\ge6\eN$ no cell meets both walls of $R$: the $u$-, $v(x)$- and $x_3$-ranges of a cell are at most $2\eN$, $2\eN$, $\eN$, while the
walls are $L/2$, $L$, $L/2$ apart. The proof of Theorem~\ref{flh:thm:I1} uses of its box only: (i) that $R$ is an automorphism of the
complex fixing its wall(s) pointwise, exchanging the two open halves, with $W\circ R=W$ cell by cell and an invariant reference measure
(Step~0; here Lemma~\ref{n1:lem:walls}); (ii) the local classification of Lemma~\ref{flh:lem:I1cells} of the cells by the sign pattern of
$\varphi=u-c$, $v(x)-c$ or $x_3-c$ on their vertices ($|\varphi|\le2\eN$ on a cell), which is local and holds at every wall of
Lemma~\ref{n1:lem:walls}; (iii) the splitting Lemma~\ref{flh:lem:I1split} and the Poisson and Bessel identities of Steps~1--4 of the proof of
Theorem~\ref{thm:GD}, again local. With ``the two closed halves'' as in the statement, Steps~0--4 of the proof of Theorem~\ref{flh:thm:I1} run on $T^{\mathrm{tw}}_L$
unchanged and give the three relations. At the diagonal walls $\{u=c\}$, $\{v(x)=c\}$ no bond crosses and the crossing kernel is of
positive type by the Poisson step alone; at $\{x_3=c\}$ with $c\in\eN\Z+\eN/2$ the Bessel factors of the crossing bonds are added, as in
Remark~\ref{flh:rem:I1kernel}.
\end{proof}

\begin{remark}[two walls, two crossing kernels]\label{n1:rem:twokernels}
On the torus each reflection $R$ has two walls, $\Pi$ and $\Pi'$, hence two families of crossing cells. Steps~2--3 of the proof of
Theorem~\ref{flh:thm:I1} give one kernel of positive type per family, $K_0(\eta,\xi)=\int d\nu(k)\,\overline{A_k(\eta)}A_k(\xi)$ at $\Pi$ and
$K'(\eta',\xi')=\int d\nu'(k')\,\overline{A'_{k'}(\eta')}A'_{k'}(\xi')$ at $\Pi'$ ($|A_k|=|A'_{k'}|=1$ at the diagonal walls), and the full crossing weight is their
\emph{product}, of positive type as the explicit integral $\int d\nu\,d\nu'\,\overline{(A_kA'_{k'})}(A_kA'_{k'})$ (the Schur product of two
kernels of positive type). This is the form in which the fold of Proposition~\ref{n1:prop:fold} writes the weight.
\end{remark}

\begin{lemma}[uniform moments and limit points of the adapted torus states]\label{n1:lem:twmom}
(a) Theorem~\ref{thm:GD} holds on $T^{\mathrm{tw}}_L$ for every $L\ge8\eN$ with $L/2\in\eN\Z$: $\langle e^{\sum_pa_p\ff_p}\rangle^{\mathrm{tw}}_L\le e^{\frac12\sum_pa_p^2}$ for every real
plaquette function $a$. (b) With the constants $G^{\mathrm{ct}}_q$, $C_1$
of Lemma~\ref{lem:d2:mom}, for every $q\ge0$, every site $x$ and every $L\ge6\eN$ with $L/2\in\eN\Z$,
\[
\langle e^{q\rho_x^2}\rangle^{\mathrm{tw}}_L\le C'_N(q):=(G^{\mathrm{ct}}_q/G^{\mathrm{ct}}_0)\,C_1^{\,16}.
\]
(c) The family $(\langle\cdot\rangle^{\mathrm{tw}}_L)_L$ has weak limit points $\om^{\mathrm{tw}}$ on $\OmN$ and on the full configuration
space $E:=[-\rhof,\infty)^{\cX_N}\times(-\pi/\gph,\pi/\gph]^{\cB_N}\times((2\pi/\gph)\Z)^{\cP_N}$ of $\eN\Z^3$; every limit point is
invariant under the reflections of Lemma~\ref{n1:lem:walls} and reflection positive at every plane $\{u=c\}$, $\{v(x)=c\}$ ($c\in\eN\Z$)
and $\{x_3=c\}$ ($c\in\tfrac12\eN\Z$), in the sense of Theorem~\ref{thm:d2:LP}(d).
\end{lemma}
\begin{proof}
(a) is Lemma~\ref{rot:lem:GDall}, since $T^{\mathrm{tw}}_L$ is a finite cubical complex with $\partial\partial=0$ carrying the weight $W$, the product of the
local factors of Section~\ref{sec:setting} over its cells with $v$ closed on its cubes.

(b) \emph{Step~1 of the proof of Lemma~\ref{lem:d2:mom} does not depend on the cell complex.} With $Z(A,q):=\int d\tr\int dB\sum_{v\ \mathrm{closed}}W\prod_{y\in A}e^{q\rho_y^2}$
on $T^{\mathrm{tw}}_L$ and $|\cB|=|\cP|=3|\cX|$ (Definition~\ref{n1:def:tw}), the same crude bounds give
\[
Z(A,q)\le(G^{\mathrm{ct}}_q)^{|A|}(G^{\mathrm{ct}}_0)^{|\cX|-|A|}(2\pi/\gph)^{3|\cX|}K_{\gph}'^{\,3|\cX|}\qquad\text{and}\qquad Z\ge c_0^{|\cX|},
\]
hence
$Z(A,q)/Z\le(G^{\mathrm{ct}}_q/G^{\mathrm{ct}}_0)^{|A|}C_1^{\,|\cX|}$.

\emph{Doubling.} Fix $x$ and $q$, $F:=e^{q\rho_x^2}$. For $n=(n_1,n_2,n_3)$, $n_i=2^{k_i}$, let
$B(n):=\{x+\eN m_1(e_1-e_2)+\eN m_2(e_1+e_2)+\eN m_3e_3:0\le m_i<n_i\}$ (sites with $u$, $v(x)$, $x_3$ in arithmetic progressions of
spacings $2\eN$, $2\eN$, $\eN$) and $\Pi_n:=\prod_{y\in B(n)}e^{q\rho_y^2}$. We claim $(\langle F\rangle^{\mathrm{tw}}_L)^{|B(n)|}\le\langle\Pi_n\rangle^{\mathrm{tw}}_L$
whenever the extents $2\eN(n_1-1)$, $2\eN(n_2-1)$, $\eN(n_3-1)$ are less than $L$, $2L$, $L$ (so that $B(n)$ consists of distinct sites of
$T^{\mathrm{tw}}_L$). For $n=(1,1,1)$ it is an identity. Suppose it holds for $n$ and double $n_1$: the wall $\{u=u(x)-\eN\}$ of
$r_{u(x)-\eN}$ passes through no site of $B(n)$, and $B(n)$ lies in the closed half $\{u(x)-\eN\le u\le u(x)-\eN+L/2\}$ as soon as
$2\eN n_1\le L/2$ (the condition under which we double $n_1$); its reflection is the adjacent set with $u\in u(x)-2\eN\{1,\dots,n_1\}$, disjoint from $B(n)$, and the union is a
translate (by $-\eN n_1(e_1-e_2)\in\eN\Z^3$) of $B(n')$, $n'$ the doubled triple. Lemma~\ref{n1:lem:RPtw} applied to the bounded
$\Pi_n\wedge M$ gives
\[
(\langle\Pi_n\wedge M\rangle^{\mathrm{tw}}_L)^2\le\langle(\Pi_n\wedge M)\circ R\cdot(\Pi_n\wedge M)\rangle^{\mathrm{tw}}_L\le
\langle(\Pi_n\circ R)\,\Pi_n\rangle^{\mathrm{tw}}_L=\langle\Pi_{n'}\rangle^{\mathrm{tw}}_L
\]
by the Schwarz inequality for the form of Lemma~\ref{n1:lem:RPtw} with
$G=1$, the reflection invariance and then the translation invariance of Lemma~\ref{n1:lem:walls}; then let $M\to\infty$, by monotone convergence.

Doubling $n_2$
across the wall of $s_{v(x)-\eN}$ (the plane where the coordinate $v$ equals $v(x)-\eN$; closed half of $v$-length $L$, condition
$2\eN n_2\le L$) and $n_3$ across the wall of $q_{x_3(x)-\eN/2}$ (closed half of length $L/2$, condition $\eN n_3\le L/2$) is the same. Doubling
each $n_i$ as long as its condition holds ends with $n_1>L/(4\eN)$, $n_2>L/(2\eN)$, $n_3>L/(2\eN)$, and the extents are still below the
periods ($n_1\le L/(2\eN)$, $n_2\le L/\eN$, $n_3\le L/\eN$), so the final $A:=B(n)$ has $|A|>L^3/(16\eN^3)=|\cX|/16$ distinct sites. Then
\[
\langle F\rangle^{\mathrm{tw}}_L\le(Z(A,q)/Z)^{1/|A|}\le(G^{\mathrm{ct}}_q/G^{\mathrm{ct}}_0)\,C_1^{\,|\cX|/|A|}\le C'_N(q).
\]

(c) With (a) and (b) in place of Theorem~\ref{thm:GD} and Lemma~\ref{lem:d2:mom}, the tightness, diagonal extraction and Kolmogorov steps
of the proof of Theorem~\ref{thm:d2:LP} apply verbatim to the laws of the variables of finite sets of cells under
$\langle\cdot\rangle^{\mathrm{tw}}_L$, on $\OmN$ and on $E$. For the laws on $E$, the variable $B_b$ ranges over a bounded interval and $v_p=\ff_p-(dB)_p$ is tight because $\ff_p$ is; the limits are taken with $B_b$ in the closed interval $[-\pi/\gph,\pi/\gph]$. Under every limit point the end point $-\pi/\gph$ has measure zero, by the identity of Lemma~\ref{n1:lem:gibbs}(i) for a $\Delta$ containing $b$, whose kernel integrates $B_b$ against Lebesgue measure. It suffices to have that identity for bounded continuous local functions on the closed space, and for these the first step of the proof of that lemma, which at $t=0$ involves no datum and applies along every sequence of sides, gives it without using that the limit points are carried by $E$, because the factors of the kernel are continuous functions of the bond variables on the closed interval. Hence the limit points are carried by $E$, and the convergence holds for the bounded functions that are continuous off this null set.

Each reflection $R$ of Lemma~\ref{n1:lem:walls} satisfies $\langle F\circ R\rangle^{\mathrm{tw}}_L=\langle F\rangle^{\mathrm{tw}}_L$
for bounded continuous local $F$ and all large $L$ (Lemma~\ref{n1:lem:RPtw}); both sides pass to the limit, so $\om^{\mathrm{tw}}$ is
$R$-invariant. For bounded continuous $F$ of cells in the closed half space, $\overline{F\circ R}\cdot F$ is bounded continuous local, so
$\om^{\mathrm{tw}}(\overline{F\circ R}\cdot F)=\lim\langle\overline{F\circ R}\cdot F\rangle^{\mathrm{tw}}_L\ge0$; such $F$ are dense in
$L^2$ of the bounded Borel functions of those cells (Lusin, the marginal being a finite Borel measure on a Polish product), and
$F\mapsto\om^{\mathrm{tw}}(\overline{F\circ R}\cdot F)$ is $L^2$-continuous because $F\mapsto F\circ R$ is an isometry of $L^2(\om^{\mathrm{tw}})$
by $R$-invariance. This is reflection positivity in the sense of Theorem~\ref{thm:d2:LP}(d).
\end{proof}

\subsection{The perturbed twisted law: invariance, reflection positivity, fold and convexity}
\label{n1:sub:pert}

\begin{definition}[datum, wall group, perturbed law]\label{n1:def:datum}
Let $L/2\in\eN\Z$, $L\ge6\eN$. A \emph{datum} is $D=(c_0,c_+,c_3;h;O)$ with $c_0,c_+\in\eN\Z$, $c_3\in\tfrac12\eN\Z$, $h\in\eN\Z$ with
$h\mid L/2$, and $O$ a real bounded continuous function of the variables of finitely many cells (sites, bonds, plaquettes) all of whose
vertices lie in the open cell $C_0:=\{c_0<u<c_0+h,\ c_+<v(x)<c_++h,\ c_3<x_3<c_3+h\}$. The \emph{walls} of $D$ are the planes
$\{u=c_0+jh\}$, $\{v(x)=c_++jh\}$, $\{x_3=c_3+jh\}$, $j\in\Z$; since $h\mid L$ and $h\mid2L$ they close up on $T^{\mathrm{tw}}_L$. $\cW$ is the
group generated by the reflections $r_{c_0+jh}$, $s_{c_++jh}$, $q_{c_3+jh}$ of Lemma~\ref{n1:lem:walls}, a finite group of automorphisms of
$T^{\mathrm{tw}}_L$. The \emph{$\cW$-cells} are the components of the complement of the walls, open boxes of side $h$ in $(u,v(x),x_3)$;
there are $n_c:=(L/h)(2L/h)(L/h)=2L^3/h^3$ of them, and $C_0$ is one. Put $O_c:=O\circ w_c^{-1}$ for a $\cW$-cell $c$, where $w_c\in\cW$
is the element with $w_cC_0=c$, and $O_d:=\sum_cO_c$. For real $t$ let
\[
Z_t:=\langle e^{tO_d}\rangle^{\mathrm{tw}}_L\in\big[e^{-|t|n_c\|O\|_\infty},e^{|t|n_c\|O\|_\infty}\big],\qquad
\mu^t_L:=Z_t^{-1}e^{tO_d}\mu^{\mathrm{tw}}_L,\qquad \om^t_L(F):=\mu^t_L(F).
\]
\end{definition}

\begin{lemma}[the wall group acts simply transitively on the cells]\label{n1:lem:Wcells}
$\cW$ permutes the walls of $D$ and the $\cW$-cells, and acts simply transitively on the $\cW$-cells; so $w_c$ is well defined,
$|O_d|\le n_c\|O\|_\infty$, and $O_d\circ w=O_d$ for every $w\in\cW$.
\end{lemma}
\begin{proof}
By Lemma~\ref{n1:lem:walls} the products $r_cr_{c'}$, $s_cs_{c'}$, $q_cq_{c'}$ of two generators of one family are the translations by
$(c-c')(e_1-e_2)$, $(c-c')(e_1+e_2)$, $2(c-c')e_3$, which shift $u$, $v(x)$, $x_3$ by $2(c-c')\in2h\Z$ and fix the other two coordinates;
each generator maps the wall family it belongs to onto itself and fixes the walls of the other two families setwise. So $\cW$ permutes
the walls, hence the cells. In each coordinate the action is that of the dihedral group generated by the reflections in the points of
$h\Z$ on the circle of length $L$ (resp.\ $2L$, $L$), with $L/h$ (resp.\ $2L/h$, $L/h$) even because $h\mid L/2$: its translations are the
shifts by $2h\Z$ and its reflections have their fixed points in $h\Z$, so no element other than the identity maps an open cell of length
$h$ onto itself, and the action on the cells of that circle is simply transitive. The product action on the $\cW$-cells is then simply
transitive too. Finally $O_{wc}=O\circ w_{wc}^{-1}=O\circ(ww_c)^{-1}=O_c\circ w^{-1}$ gives $O_d\circ w^{-1}=\sum_cO_{wc}=O_d$.
\end{proof}

\begin{lemma}[$\cW$-invariance and reflection positivity of the perturbed law, every real $t$]\label{n1:lem:Winv}
Let $D$ be a datum and $t\in\R$. (i) $\mu^t_L(F\circ w)=\mu^t_L(F)$ for every $w\in\cW$ and bounded Borel $F$; in particular
$\om^t_L(O_c)=\om^t_L(O)$ for every $\cW$-cell $c$. (ii) Let $\Pi$ be a wall of $D$, $\theta_\Pi\in\cW$ the reflection in it, and $\Pi'$ its
second wall (Lemma~\ref{n1:lem:walls}), also a wall of $D$ since $h\mid L/2$. For every bounded Borel $G$ of cells in one of the two closed
halves of $T^{\mathrm{tw}}_L$ bounded by $\Pi$ and $\Pi'$,
\[
\mu^t_L\big(\overline{G\circ\theta_\Pi}\cdot G\big)\ge0,\qquad \mu^t_L(G\circ\theta_\Pi)=\mu^t_L(G).
\]
\end{lemma}
\begin{proof}
(i) We have
\[
Z_t\,\mu^t_L(F\circ w)=\mu^{\mathrm{tw}}_L\big((F\circ w)e^{tO_d}\big)=\mu^{\mathrm{tw}}_L\big((Fe^{tO_d})\circ w\big)=\mu^{\mathrm{tw}}_L(Fe^{tO_d})=Z_t\,\mu^t_L(F),
\]
by $O_d\circ w=O_d$ (Lemma~\ref{n1:lem:Wcells}) and the $\cW$-invariance of $\mu^{\mathrm{tw}}_L$ (Lemmas~\ref{n1:lem:walls}
and~\ref{n1:lem:RPtw}; $\cW$ is generated by the reflections). Then $\om^t_L(O_c)=\mu^t_L(O\circ w_c^{-1})=\om^t_L(O)$.
(ii) $\theta_\Pi$ exchanges the two open halves bounded by $\Pi$ and $\Pi'$ and maps the $\cW$-cells of one onto those of the other (a
$\cW$-cell meets no wall, so it lies in one open half). Hence $O_d=O^\Pi_++O^\Pi_+\circ\theta_\Pi$ with $O^\Pi_+:=\sum_{c\subset H_+}O_c$, the
sum over the $\cW$-cells of the open half $H_+$ containing the cells of $G$: a real bounded function of cells inside $H_+$. Therefore
$e^{tO_d}=e^{tO^\Pi_+}\cdot\big(e^{tO^\Pi_+}\circ\theta_\Pi\big)$ and, with $G':=Ge^{tO^\Pi_+}$, a bounded Borel function of cells in the closed half,
\[
\mu^t_L\big(\overline{G\circ\theta_\Pi}\cdot G\big)=Z_t^{-1}\mu^{\mathrm{tw}}_L\big(\overline{G'\circ\theta_\Pi}\cdot G'\big)\ge0
\]
by Lemma~\ref{n1:lem:RPtw}. The invariance is (i).
\end{proof}

\begin{proposition}[the fold: comparison of the two tori of the same side]\label{n1:prop:fold}
Let $D$ be a datum with $c_0=0$ and $L\ge6\eN$, $h\mid L/2$. Define $O_d$ on the cubic torus $T^{\mathrm{cub}}_L$ by the same formula
$O_d:=O_++O_+\circ r_0$, $O_+:=\sum_{c\subset\{0<u<L/2\}}O_c$ (both tori contain the strip $0\le u\le L/2$ with its
$\cW$-cells and $r_0$ as a symmetry). Then for every real $t$
\[
\langle e^{tO_d}\rangle^{\mathrm{cub}}_L\ \le\ e^{\phi_L}\,Z_t,\qquad \phi_L:=\log\frac{Z^{\mathrm{tw}}_L}{Z^{\mathrm{cub}}_L}\ \ge\ 0 .
\]
\end{proposition}
\begin{proof}
\emph{Two quotients of one cylinder.} Let $\Gamma_\parallel:=L(e_1+e_2)\Z+Le_3\Z=\Gamma(L)\cap\{u=0\}=L\Z^3\cap\{u=0\}$ (a vector $(a,a,b)$
lies in $L\Z^3$ if and only if $a,b\in L\Z$) and $\cC:=\eN\Z^3/\Gamma_\parallel$, a cylinder on which $u\in\eN\Z$ is a global coordinate. Then
$T^{\mathrm{tw}}_L=\cC/\langle\tau_2\rangle$ with $\tau_2$ the translation by $\tfrac L2(e_1-e_2)$, and $T^{\mathrm{cub}}_L=\cC/\langle\tau_1\rangle$
with $\tau_1$ the translation by $Le_1$; both shift $u$ by $L$. On $\cC$ one checks $r_{L/2}\tau_2=r_0$ and $r_{L/2}\tau_1=Yr_0$, where $Y$
is the translation by $\tfrac L2(e_1+e_2)$: indeed $r_{L/2}(x_1+L,x_2,x_3)=(x_2+L/2,x_1+L/2,x_3)=Y(x_2,x_1,x_3)$; $Y^2$ is the
translation by $L(e_1+e_2)\in\Gamma_\parallel$, so $Y$ is an involution of $\cC$ preserving $u$, the strip $S_+:=\{0\le u\le L/2\}$, both walls
$\Pi_0=\{u=0\}$, $\Pi_{L/2}=\{u=L/2\}$ and the wall structure of $D$ (it shifts $v(x)$ by $L\in h\Z$ and fixes $x_3$).

\emph{Fold variables.} On $\cC$ let $\xi_0$, $\xi'$ be the variables of the cells lying in the walls $\Pi_0$, $\Pi_{L/2}$ (sites and
$e_3$-bonds in the wall, Lemma~\ref{flh:lem:I1cells}), $\xi_+$ those of the cells with all vertices in $S_+$ and not all in a wall, $\xi_-$
the same for $S_-:=\{-L/2\le u\le0\}$; the remaining cells are the crossing cells at $\Pi_0$ and at the seam $\Pi_{L/2}\equiv\Pi_{-L/2}$:
the $e_1e_2$-plaquettes bisected by the wall and the cubes having them as faces; no bond is a crossing cell, and the factors of the crossing cells contain no wall variable.
Torus $i$ ($i=2$: $T^{\mathrm{tw}}_L$; $i=1$: $T^{\mathrm{cub}}_L$) is the strip $\{-L/2\le u\le L/2\}$ of $\cC$ with $\Pi_{-L/2}$ glued to
$\Pi_{L/2}$ by $\tau_i$; on $\Pi_{-L/2}$, $\tau_2=r_0$ and $\tau_1=Yr_0$ (by the identities $r_{L/2}\tau_2=r_0$, $r_{L/2}\tau_1=Yr_0$ and $r_{L/2}=\mathrm{id}$ on
$\Pi_{L/2}$). Its weight is
\[
W_0(\xi_0)W_0'(\xi')\,W_+(\xi_0,\xi',\xi_+)\,W_-(\xi_0,\tau_i^*\xi',\xi_-)\,K_0\,K'_i,
\]
where $W_0,W_0'$ collect
the factors of the wall cells, $W_\pm$ those of the $(\pm)$-cells (which see the wall variables of the walls they touch, the $(-)$-cells
seeing $\Pi_{L/2}$ through $\tau_i$), and $K_0$, $K'_i$ are the products of the factors of the crossing cells at $\Pi_0$ and at the seam.

\emph{Reflection and kernels.} Put $\eta:=r_0^*\xi_-$, the $(+)$-configuration obtained by reflecting $\xi_-$ in $\Pi_0$. By $W\circ R=W$ cell
by cell (Lemma~\ref{n1:lem:walls}), $W_-(\xi_0,\tau_2^*\xi',\xi_-)=W_+(\xi_0,\xi',\eta)$ and $W_-(\xi_0,\tau_1^*\xi',\xi_-)=W_+(\xi_0,Y^*\xi',\eta)$.
Steps~2--3 of the proof of Theorem~\ref{flh:thm:I1} (Poisson resummation at a diagonal wall, Remark~\ref{n1:rem:twokernels}) write
$K_0=\int d\nu(k)\,\overline{A_k(\eta)}A_k(\xi_+)$ and $K'_2=\int d\nu'(k')\,\overline{A'_{k'}(\eta)}A'_{k'}(\xi_+)$, with positive measures
$\nu,\nu'$ on auxiliary variables and $|A_k|=|A'_{k'}|=1$, $A_k$ depending on the $(+)$-cells at $\Pi_0$ and $A'_{k'}$ on those at $\Pi_{L/2}$;
for the cubic torus the seam is glued through $\tau_1^{-1}r_{L/2}=r_0Y^{-1}$ instead of $r_0$, so $K'_1=\int d\nu'(k')\,\overline{A'_{k'}(Y^*\eta)}A'_{k'}(\xi_+)$.

\emph{The two scalar products.} Let $\cH:=L^2\big(d\xi_0\,W_0\;d\xi'\,W_0'\;d\nu(k)\,d\nu'(k')\big)$ and, for a bounded $F=F(\xi_+)$,
\[
\begin{gathered}
\Xi_F(\xi_0,\xi',k,k'):=\int d\xi_+\,W_+(\xi_0,\xi',\xi_+)\,A_k(\xi_+)A'_{k'}(\xi_+)\,F(\xi_+),\\
(Y\Xi)(\xi_0,\xi',k,k'):=\Xi(\xi_0,Y^*\xi',k,Yk'),
\end{gathered}
\]
where $Y$ permutes the wall cells of $\Pi_{L/2}$ and the seam's crossing cells, $W_0'$, the reference measure and $\nu'$ are $Y$-invariant
and $A'_{Yk'}(\eta)=A'_{k'}(Y^*\eta)$; so $Y$ is a unitary involution of $\cH$. Take $F:=e^{tO_+}$, a real bounded function of $\xi_+$. On
both tori $O_d=O_++O_+\circ r_0$ (the $\cW$-cells of $\{-L/2<u<0\}$ are the $r_0$-images of those of the strip and $O_{r_0c}=O_c\circ r_0$,
Lemma~\ref{n1:lem:Wcells}; on $T^{\mathrm{cub}}_L$ this is the definition), and $O_+\circ r_0$, as a function of $\xi_-$, is $O_+(\eta)$. Since
$O_+\circ Y=O_+$ ($Y=s_{c_++L/2}\,s_{c_+}\in\cW$), the substitution $\xi_-\mapsto\eta$ (measure preserving) and the integration over
$\xi_+$ and $\eta$ give
\[
Z^{\mathrm{tw}}_L\langle e^{tO_d}\rangle^{\mathrm{tw}}_L=\langle\Xi_F,\Xi_F\rangle_{\cH},\qquad
Z^{\mathrm{cub}}_L\langle e^{tO_d}\rangle^{\mathrm{cub}}_L=\langle Y\Xi_F,\Xi_F\rangle_{\cH}.
\]
By the Cauchy--Schwarz inequality and $\|Y\|=1$, $Z^{\mathrm{cub}}_L\langle e^{tO_d}\rangle^{\mathrm{cub}}_L\le Z^{\mathrm{tw}}_LZ_t$, which is the claim;
$t=0$ gives $Z^{\mathrm{cub}}_L\le Z^{\mathrm{tw}}_L$, i.e.\ $\phi_L\ge0$.
\end{proof}

\begin{lemma}[surface bound for the adapted torus]\label{n1:lem:SV}
Fix the level $N$ and the couplings of the fine Villain--Higgs model. Let $L>0$ with $L/2\in\eN\mathbb Z$ and $L\ge 8\eN$, and write $M:=L/\eN\in 2\mathbb Z$, $M\ge 8$.
Let $Z^{\mathrm{cub}}_L$ be the partition function of the model on the cubic torus $T^{\mathrm{cub}}_L=\eN\mathbb Z^3/L\mathbb Z^3$, the fine torus $\tilde T_L$ of Section~\ref{sec:setting},
and let $Z^{\mathrm{tw}}_L$ be the partition
function of the same model (same $N$, same couplings, same local weights) on the adapted torus $T^{\mathrm{tw}}_L=\eN\mathbb Z^3/\Gamma(L)$ of the same side $L$. Then
\[
\phi_L:=\log\frac{Z^{\mathrm{tw}}_L}{Z^{\mathrm{cub}}_L}
\qquad\text{satisfies}\qquad
0\;\le\;\phi_L\;\le\;\Big(\frac{L}{\eN}\Big)^{2}\Big(5\log\frac{L}{\eN}+A_N\Big),
\]
where $A_N<\infty$ depends only on $N$ and the couplings (not on $L$). In particular $\phi_L=o(L^3)$ as $L\to\infty$ at fixed $N$.
\end{lemma}

\begin{proof}
The lower bound $\phi_L\ge0$ is Proposition~\ref{n1:prop:fold} at $t=0$; here we prove the upper bound.
Throughout, $N$ and the couplings are fixed, $\gph$ is the phase constant, $\kappa_N$ the hopping coefficient, $g_{\mathrm{ct}}$ the site factor and $\rho_x:=\rhof+\tr_x$,
so that the weight on either torus is
\[
W=\prod_xg_{\mathrm{ct}}(\tr_x)\prod_bh(\tr_{b_-},\tr_{b_+},B_b)\prod_pw((dB)_p+v_p).
\]
Here $w(f)=e^{-f^2/2}$, $B_b\in(-\pi/\gph,\pi/\gph]$,
$v_p\in(2\pi/\gph)\mathbb Z$, the sum runs over the closed $v$ ($dv=0$ on every cube), and
\[
\begin{gathered}
e^{-\kappa_N(\rho^2+\rho'^2)}\ \le\ h(\tr,\tr',\vartheta)=\exp\big(-\tfrac12\kappa_N|e^{i\gph\vartheta}\rho'-\rho|^2\big)\ \le\ 1\\
(|e^{i\gph\vartheta}\rho'-\rho|^2\le(\rho+\rho')^2\le2(\rho^2+\rho'^2)).
\end{gathered}
\]
For $q\in\mathbb R$ put $G_q:=\int_{-\rhof}^{\infty}g_{\mathrm{ct}}(r)\,e^{q(\rhof+r)^2}\,dr$, finite for every real $q$ because the quartic dominates (\eqref{eq:ct}; for $q\ge0$ this is
the $G^{\mathrm{ct}}_q$ of Lemma~\ref{lem:d2:mom}); let $K'_{\gph}$ and $c_0$ be the constants of Lemma~\ref{lem:d2:mom} ($\sum_{v\in(2\pi/\gph)\mathbb Z}w(f+v)\le K'_{\gph}$ for every
real $f$, and $Z\ge c_0^{|X|}$ on any torus with site set $X$). The proof has nine steps.

\smallskip\noindent\emph{Step 0: seam geometry.} Write $u:=x_1-x_2$, $v:=x_1+x_2$ and let $\Gamma'_\parallel:=L(e_1-e_2)\mathbb Z+L(e_1+e_2)\mathbb Z+Le_3\mathbb Z$ (index $2$ in $L\mathbb Z^3$
and in $\Gamma(L)=\frac L2(e_1-e_2)\mathbb Z+L(e_1+e_2)\mathbb Z+Le_3\mathbb Z$). On the finite cylinder $C':=\eN\mathbb Z^3/\Gamma'_\parallel$
(the quotient of the cylinder $\cC$ of Proposition~\ref{n1:prop:fold} by the translation $L(e_1-e_2)$: $\Gamma'_\parallel=\Gamma_\parallel+L(e_1-e_2)\mathbb Z$) the coordinate $u$ is
defined mod $2L$ and $v$ mod $2L$.
Both tori are quotients of $C'$ by a single translation: $T^{\mathrm{tw}}:=\eN\mathbb Z^3/\Gamma(L)=C'/\tau_{\mathrm{tw}}$ with $\tau_{\mathrm{tw}}=\frac L2(e_1-e_2)\colon(u,v,x_3)\mapsto(u+L,v,x_3)$,
and $T^{\mathrm{cub}}:=\eN\mathbb Z^3/L\mathbb Z^3=C'/\tau_{\mathrm{cub}}$ with $\tau_{\mathrm{cub}}=Le_1\colon(u,v,x_3)\mapsto(u+L,v+L,x_3)$. On both tori $u$ is defined mod $L$, and the
\emph{layer} $\{u\equiv c\}$ has exactly $M^2$ sites, labelled by $(v,x_3)$ with $v\in\eN\mathbb Z/2L\mathbb Z$, $v\equiv c\pmod{2\eN}$, and $x_3\in\eN\mathbb Z/L\mathbb Z$.

An $e_1$-bond raises $u$ by $\eN$, an $e_2$-bond lowers it by $\eN$, an $e_3$-bond keeps it; so the vertices of any cell (bond, plaquette, cube) occupy at most three consecutive
layers. Call a cell \emph{seam} if it has a vertex in the layer $u\equiv0$ and \emph{interior} if all its vertices lie in the layers $u\in\{\eN,\dots,L-\eN\}$; every cell is one or
the other. The quotient maps $C'\to T^{\mathrm{tw}}$, $C'\to T^{\mathrm{cub}}$ are injective on the strip $\eN\le u\le L-\eN$ (its $\tau$-image lies in $L+\eN\le u\le2L-\eN$), so the
interior cells, their incidences and their labels are the same for both tori; the two tori differ only in how the cells touching $u\equiv0$ are glued to the strip.

\emph{Counts.} The plane $u\equiv0$ has $M^2$ sites. The seam bonds are $4M^2$ bonds of type $e_1,e_2$ (four at each plane site, each with exactly one plane vertex) and $M^2$ plane
$e_3$-bonds, $5M^2$ in all. The seam plaquettes are, in terms of the $u$-coordinate $u_x$ of the base corner $x$, the $e_1e_2$-plaquettes with $u_x\in\{-\eN,0,\eN\}$, the $e_1e_3$-plaquettes with $u_x\in\{-\eN,0\}$
and the $e_2e_3$-plaquettes with $u_x\in\{0,\eN\}$, $7M^2$ in all. The seam cubes are the cubes with $u_x\in\{-\eN,0,\eN\}$, called $\Sigma_-,\Sigma_0,\Sigma_+$ respectively (middle layer $u\equiv-\eN,0,\eN$), $3M^2$ in all.
The six faces of the cube with base corner $x$ meet its middle layer: the two $e_1e_2$-faces (base $u_x$), the $e_1e_3$-face at $x$ ($u\in\{u_x,u_x+\eN\}$), the $e_1e_3$-face at
$x+\eN e_2$ ($u\in\{u_x-\eN,u_x\}$), the $e_2e_3$-face at $x$ ($u\in\{u_x-\eN,u_x\}$) and the $e_2e_3$-face at $x+\eN e_1$ ($u\in\{u_x,u_x+\eN\}$). Hence a $\Sigma_+$-cube
has exactly two interior faces ($e_1e_3$ at $x$, $e_2e_3$ at $x+\eN e_1$, both with $u\in\{\eN,2\eN\}$), a $\Sigma_-$-cube exactly two ($e_1e_3$ at $x+\eN e_2$, $e_2e_3$ at $x$,
$u\in\{L-2\eN,L-\eN\}$) and a $\Sigma_0$-cube none. We call these $4M^2$ interior faces of seam cubes the \emph{data faces}; each data face lies in exactly one seam cube (its
other cube has base $u\in\{2\eN,L-2\eN\}$ and is interior since $M\ge4$), and we write $c_p$ for that seam cube and $c'_p$ for the interior one. Every seam plaquette lies in
exactly two seam cubes (for an $e_1e_2$-plaquette the cubes at $x$ and $x-\eN e_3$, for an $e_1e_3$-plaquette those at $x$ and $x-\eN e_2$, for an $e_2e_3$-plaquette those at $x$ and $x-\eN e_1$), with opposite incidence signs
$[c\!:\!p]=-[c'\!:\!p]$ (two cubes induce opposite orientations on a common face). Every interior plaquette lies in at least one interior cube (again $M\ge4$).

\emph{The seam-cube graph.} Let $G^i$ ($i\in\{\mathrm{cub},\mathrm{tw}\}$) be the multigraph with vertex set the $3M^2$ seam cubes of $T^i$ and edge set the $7M^2$ seam
plaquettes, each joining the two seam cubes containing it. Label the $\Sigma_s$-cubes by the $(v,x_3)$ of their base corner, lifting $\Sigma_-$ to base $u=L-\eN$ in $C'$. The
$e_1e_2$-faces give the column edges $(v,x_3)\sim(v,x_3+\eN)$ in each of the three layers; the $e_1e_3$- and $e_2e_3$-faces give $\Sigma_+(v,x_3)\sim\Sigma_0(v\pm\eN,x_3)$ in both tori,
and $\Sigma_-(v,x_3)\sim\Sigma_0(v\pm\eN-\delta_i,x_3)$ with $\delta_{\mathrm{tw}}=0$, $\delta_{\mathrm{cub}}=L$, because the far faces of the lifted $\Sigma_-$-cubes have vertices at
$u=L$ and are glued to the plane $u=0$ by $\tau_i^{-1}$. Since $L\in2\eN\mathbb Z$ preserves the parity class of $v$, the map $\Sigma_-(v,x_3)\mapsto\Sigma_-(v-L,x_3)$, identity on
$\Sigma_0\cup\Sigma_+$, is a graph isomorphism $G^{\mathrm{cub}}\cong G^{\mathrm{tw}}$; both graphs are connected: the column edges join all values of $x_3$, and the paths $(v,0)\sim(v+\eN,+)\sim(v+2\eN,0)$ join all values of $v$.

\smallskip\noindent\emph{Step 1: seam decomposition.} Write $\xi_{\mathrm{int}}$ for the interior variables ($\tr_x$ at the sites of the layers $u\in\{\eN,\dots,L-\eN\}$, $B_b$ on the
interior bonds, $v_p$ on the interior plaquettes) and
\[
W_{\mathrm{int}}(\xi_{\mathrm{int}}):=\prod_{x\ \mathrm{int}}g_{\mathrm{ct}}(\tr_x)\prod_{b\ \mathrm{int}}h\prod_{p\ \mathrm{int}}w\big((dB)_p+v_p\big)\prod_{c\ \mathrm{int}}\mathbf 1[(dv)_c=0],
\]
the same function of the same variables for both tori (Step 0). The seam variables are $\tr_x$ at the $M^2$ plane sites, $B_b$ on the $5M^2$ seam bonds and $v_p$ on the $7M^2$
seam plaquettes. For a seam cube $c$ of $T^i$ every face is a seam plaquette or a data face, so
\[
(dv)_c=\operatorname{div}_{G^i}v_{\mathrm{seam}}(c)-q(c),\qquad \operatorname{div}_{G}v(c):=\sum_{\mathrm{seam}\ p\subset c}[c\!:\!p]\,v_p,\qquad q(c):=-\sum_{\mathrm{data}\ p\subset
c}[c\!:\!p]\,v_p ,
\]
where $q$ depends on $\xi_{\mathrm{int}}$ only (data faces are interior), $q=0$ on $\Sigma_0$, and $\operatorname{div}_G$ is the signed incidence (divergence) operator of the graph $G$
with the edge $p$ oriented from the cube with $[c\!:\!p]=-1$ to the cube with $[c\!:\!p]=+1$. Thus $v$ is closed on $T^i$ if and only if it is closed on the interior cubes and
$\operatorname{div}_{G^i}v_{\mathrm{seam}}=q$.

For $i\in\{\mathrm{cub},\mathrm{tw}\}$ define the \emph{seam factor}
\[
\Sigma^i(\xi_{\mathrm{int}}):=\int\prod_{u(x)=0}g_{\mathrm{ct}}(\tr_x)\,d\tr_x\int\prod_{\mathrm{seam}\ b}dB_b\prod_{\mathrm{seam}\ b}h\sum_{v_{\mathrm{seam}}:\
\operatorname{div}_{G^i}v_{\mathrm{seam}}=q}\ \prod_{\mathrm{seam}\ p}w\big((dB)_p+v_p\big).
\]
Every cell weight of $T^i$ appears exactly once in $W_{\mathrm{int}}\Sigma^i$ and all terms are non-negative, so by Tonelli $Z^i=\int
d\xi_{\mathrm{int}}\,W_{\mathrm{int}}\,\Sigma^i(\xi_{\mathrm{int}})$.
(The seam weights depend on $\xi_{\mathrm{int}}$: a seam plaquette may contain an interior bond, and a seam bond of type $e_1,e_2$ has one end in the layer $u=\pm\eN$; this
dependence is the only place, besides the constraint graph $G^i$, where the two tori differ, through the gluing $\delta_i$ of Step 0.)

\emph{The data charges sum to zero.} If $v$ is closed on the interior cubes, then
\[
0=\sum_{c'\ \mathrm{int}}(dv)_{c'}=\sum_{c'\ \mathrm{int}}\sum_{p\subset c'}[c'\!:\!p]v_p.
\]
An interior
plaquette lying in two interior cubes contributes $0$ (opposite signs); one lying in exactly one interior cube is a data face $p$ with interior cube $c'_p$ (its other cube is the
seam cube $c_p$); seam plaquettes lie in no interior cube. Hence
\[
0=\sum_{\mathrm{data}\ p}[c'_p\!:\!p]v_p=-\sum_{\mathrm{data}\ p}[c_p\!:\!p]v_p=\sum_cq(c).
\]
The same computation
with $f:=dB+v$ (closed on interior cubes, as $d\,dB=0$) and with $dB$ alone gives $\sum_cQ(c)=0$ for $Q(c):=-\sum_{\mathrm{data}\ p\subset c}[c\!:\!p]f_p$ and
$\sum_c\sum_{\mathrm{data}\ p\subset c}[c\!:\!p](dB)_p=0$.

\smallskip\noindent\emph{Step 2: Jensen.} Let $\mu(d\xi_{\mathrm{int}}):=W_{\mathrm{int}}\Sigma^{\mathrm{tw}}d\xi_{\mathrm{int}}/Z^{\mathrm{tw}}_L$, the law of the interior variables under
$\langle\cdot\rangle^{\mathrm{tw}}_L$, and $R:=\Sigma^{\mathrm{cub}}/\Sigma^{\mathrm{tw}}$. By Step 1, $Z^{\mathrm{cub}}_L/Z^{\mathrm{tw}}_L=\int R\,d\mu$. For every
$\xi_{\mathrm{int}}$ with $W_{\mathrm{int}}>0$ (the
others form a $\mu$-null set) both seam factors lie in $(0,\infty)$: finiteness is Step 3 and positivity is Step 5 (each applies to either torus, since $G^{\mathrm{cub}}\cong
G^{\mathrm{tw}}$ and the cell counts agree).
By the concavity of $\log$ (Jensen),
\[
\phi_L=-\log\int R\,d\mu\ \le\ -\int\log R\,d\mu\ =\ \big\langle\log\Sigma^{\mathrm{tw}}-\log\Sigma^{\mathrm{cub}}\big\rangle^{\mathrm{tw}}_L ,
\]
where $\log R$ is $\mu$-integrable: by Steps 3 and 5,
\[
|\log\Sigma^i|\le\mathrm{const}\cdot M^2+\langle Q,\Delta^+_{G^i}Q\rangle+2\kappa_N\sum_{u(x)=\pm\eN}\rho_x^2\qquad\text{for both }i,
\]
and the right side has finite $\mu$-mean by Step 6 (with $\operatorname{Tr}\Delta^+_{G^{\mathrm{cub}}}=\operatorname{Tr}\Delta^+_{G^{\mathrm{tw}}}$ by the isomorphism of Step 0).

\smallskip\noindent\emph{Step 3: upper bound on the seam factor.} In $\Sigma^i$ use $h\le1$, drop the constraint $\operatorname{div}_{G^i}v_{\mathrm{seam}}=q$ and sum each $v_p$ separately
($\sum_{v_p}w((dB)_p+v_p)\le K'_{\gph}$); each of the $5M^2$ seam bonds contributes $\int dB_b=2\pi/\gph$ and each of the $M^2$ plane sites $\int g_{\mathrm{ct}}=G_0$. Hence
\[
\log\Sigma^i\ \le\ 5M^2\log\frac{2\pi}{\gph}+M^2\log G_0+7M^2\log K'_{\gph}\qquad(i\in\{\mathrm{cub},\mathrm{tw}\}),
\]
uniformly in $\xi_{\mathrm{int}}$.

\smallskip\noindent\emph{Step 4: a rounding lemma.} \emph{Let $(V,E)$ be a finite multigraph without loops (parallel edges allowed), each edge oriented, $D\colon\mathbb R^E\to\mathbb R^V$ its
signed incidence (divergence) operator, and $j\in\mathbb R^E$ with $Dj\in\mathbb Z^V$. Then there is $n\in\mathbb Z^E$ with $Dn=Dj$ and $|n_e-j_e|\le1$ for every $e\in E$.}
Indeed, $\phi:=j-\lfloor j\rfloor\in[0,1)^E$ has $D\phi=Dj-D\lfloor j\rfloor\in\mathbb Z^V$. The set $\{\phi'\in[0,1]^E:D\phi'=D\phi\}$ contains $\phi$; choose in it a $\phi'$ with the fewest
non-integer coordinates and let $F:=\{e:\phi'_e\notin\{0,1\}\}$. Suppose $F\ne\emptyset$. At a vertex $x$, $(D\phi')_x$ is an integer and the integer-valued edges contribute an integer, so
the signed sum of the values $\phi'_e\in(0,1)$ over the $F$-edges at $x$ is an integer; a vertex with exactly one $F$-edge is therefore impossible. Thus every vertex has $F$-degree $0$
or $\ge2$, and $F$ contains a cycle (a closed trail without repeated edges; two parallel edges form one). Let $\gamma\in\{0,\pm1\}^E$ be its circulation (the sign of each cycle edge
according to its orientation along a traversal, $0$ off the cycle), so $D\gamma=0$. For real $t$ the vector $\phi'+t\gamma$ has the same divergence; take $|t|$ minimal such that some cycle
edge reaches $0$ or $1$ (every cycle edge starts in the open interval $(0,1)$). Then $\phi'+t\gamma\in[0,1]^E$ and it has fewer non-integer coordinates than $\phi'$: a contradiction.
Hence $\phi'\in\{0,1\}^E$, and $n:=\lfloor j\rfloor+\phi'\in\mathbb Z^E$ satisfies $Dn=D\lfloor j\rfloor+D\phi=Dj$ and $n_e-j_e=\phi'_e-\phi_e\in[-1,1]$.

\smallskip\noindent\emph{Step 5: lower bound on the seam factor by one closed extension.} Fix $\xi_{\mathrm{int}}$ with $W_{\mathrm{int}}>0$ (so $v$ is closed on the interior cubes) and fix
$B_{\mathrm{seam}}$. Put $f:=dB+v$ on the interior plaquettes and $Q(c):=-\sum_{\mathrm{data}\ p\subset c}[c\!:\!p]f_p$ for the seam cubes $c$; $Q$ depends on $\xi_{\mathrm{int}}$ only
(a data face is interior, and so are its four bonds), and $\sum_cQ(c)=0$ by Step 1. Let $G:=G^{\mathrm{cub}}$ with the edge orientation of Step 1, $D$ its divergence,
$\nabla=D^{\mathsf T}$ the gradient, $\Delta:=DD^{\mathsf T}$ its graph Laplacian and $\Delta^+$ the pseudo-inverse. $G$ is connected, so $\ker\Delta$ is the constants and
$\Delta\Delta^+Q=Q$. Put $\psi:=\Delta^+Q$ and $j^*:=\nabla\psi$; then $\operatorname{div}j^*=\Delta\psi=Q$ and $|j^*|^2=\langle\psi,\Delta\psi\rangle=\langle Q,\Delta^+Q\rangle$.
Since $(d\,dB)_c=0$ at every seam cube, $\operatorname{div}\big((dB)|_{\mathrm{seam}}\big)(c)=-\sum_{\mathrm{data}\ p\subset c}[c\!:\!p](dB)_p$, hence $j':=j^*-(dB)|_{\mathrm{seam}}$ has
\[
\operatorname{div}j'(c)=Q(c)+\sum_{\mathrm{data}\ p\subset c}[c\!:\!p](dB)_p=-\sum_{\mathrm{data}\ p\subset c}[c\!:\!p]\,v_p=q(c)\in\frac{2\pi}{\gph}\mathbb Z .
\]

Step 4, applied to $(\gph/2\pi)j'$, gives $v^\dagger\in((2\pi/\gph)\mathbb Z)^{\mathrm{seam}}$ with $\operatorname{div}_{G^{\mathrm{cub}}}v^\dagger=q$ and $|v^\dagger_p-j'_p|\le2\pi/\gph$
for every seam plaquette $p$. By Step 1, $(v_{\mathrm{int}},v^\dagger)$ is closed on $T^{\mathrm{cub}}$, and $f^\dagger:=dB+v^\dagger$ on the seam plaquettes obeys
$f^\dagger-j^*=v^\dagger-j'$, so $|f^\dagger-j^*|_\infty\le2\pi/\gph$ and, on the $7M^2$ seam plaquettes,
\[
|f^\dagger|^2\le2|j^*|^2+2\cdot7M^2(2\pi/\gph)^2=2\langle Q,\Delta^+Q\rangle+56\pi^2M^2/\gph^2,
\]
uniformly in $B_{\mathrm{seam}}$. Keep in $\Sigma^{\mathrm{cub}}$ only the term
$v_{\mathrm{seam}}=v^\dagger$:
\[
\prod_{\mathrm{seam}\ p}w(f^\dagger_p)=e^{-|f^\dagger|^2/2}\ge\exp\big(-\langle Q,\Delta^+Q\rangle-28\pi^2M^2/\gph^2\big).
\]
Bound each seam-bond factor by
$h\ge e^{-\kappa_N(\rho^2+\rho'^2)}$: a plane site lies in six seam bonds ($\pm e_1,\pm e_2,\pm e_3$) and a site of the layers $u=\pm\eN$ in exactly two ($-e_1,+e_2$ at $u=\eN$;
$+e_1,-e_2$ at $u=-\eN$), so
\[
\prod_{\mathrm{seam}\ b}h\ge\prod_{u(x)=0}e^{-6\kappa_N\rho_x^2}\prod_{u(x)=\pm\eN}e^{-2\kappa_N\rho_x^2}.
\]
Integrating over the plane sites
($\int g_{\mathrm{ct}}e^{-6\kappa_N\rho^2}=G_{-6\kappa_N}>0$) and the seam bonds ($\int dB_b=2\pi/\gph$), we obtain
\[
\log\Sigma^{\mathrm{cub}}\ \ge\ -\langle Q,\Delta^+_{G^{\mathrm{cub}}}Q\rangle-\frac{28\pi^2M^2}{\gph^2}+5M^2\log\frac{2\pi}{\gph}+M^2\log G_{-6\kappa_N}-2\kappa_N\!\!\sum_{u(x)=\pm\eN}\!\!\rho_x^2 .
\]
The same argument with $G^{\mathrm{tw}}$ bounds $\Sigma^{\mathrm{tw}}$ from below, which gives $\Sigma^{\mathrm{tw}}>0$ in Step 2. In particular both constraint sets are non-empty for every
interior datum: the only condition for the existence of a closed extension is $\sum_cq(c)=0$.

\smallskip\noindent\emph{Step 6: moments under the twisted state.} Write $\langle\cdot\rangle^{\mathrm{tw}}_L$ for the normalised state of weight $W$ on $T^{\mathrm{tw}}$; the interior
variables have law $\mu$ under it (Step 2).

(a) \emph{Gaussian domination.} By the first clause of Lemma~\ref{rot:lem:GDall}, for every real function $b$ on the plaquettes of $\eN\mathbb Z^3/P$, $P$ a period lattice for $\eN\mathbb Z^3$ (a rank-$3$ sublattice of
$\eN\mathbb Z^3$ whose non-zero vectors have length $\ge5\eN$), $\langle e^{\sum_pb_pf_p}\rangle\le e^{|b|^2/2}$, with $f_p=(dB)_p+v_p$ (the $\ff_p$ of Section~\ref{sec:setting}). The lattice
$\Gamma(L)\subset\eN\mathbb Z^3$ (as $L/2\in\eN\mathbb Z$) has rank $3$ and is a period lattice for $L\ge8\eN$: its
non-zero vectors $a\frac L2(e_1-e_2)+bL(e_1+e_2)+cLe_3$ have squared length $L^2(a^2/2+2b^2+c^2)\ge L^2/2$, and $L/\sqrt2\ge5\eN$ holds for $L\ge8\eN$ (as $L\in2\eN\mathbb Z$,
$L\ge5\sqrt2\,\eN$ means $L\ge8\eN$). Applying the bound to $\pm sb$ and using $\cosh y\ge1+y^2/2$ gives $\tfrac{s^2}2\langle(\sum_pb_pf_p)^2\rangle^{\mathrm{tw}}_L\le e^{s^2|b|^2/2}-1$ for
every $s>0$, whence $\langle(\sum_pb_pf_p)^2\rangle^{\mathrm{tw}}_L\le|b|^2$ ($s\downarrow0$). Let $S_{cc'}:=\langle Q(c)Q(c')\rangle^{\mathrm{tw}}_L$ be the second-moment matrix of the data
charges. For $a\in\mathbb R^{V}$ ($V$ the seam cubes), $\sum_ca_cQ(c)=\sum_{\mathrm{data}\ p}b_pf_p$ with $b_p:=-a_{c_p}[c_p\!:\!p]$ ($c_p$ the unique seam cube of the data face $p$), so
$\langle a,Sa\rangle\le|b|^2=\sum_{\mathrm{data}\ p}a_{c_p}^2\le2|a|^2$, since a seam cube has at most two data faces. Thus $S\le2\,\mathrm{Id}$ as quadratic forms, and, $\Delta^+$
being positive semi-definite,
\[
\big\langle\langle Q,\Delta^+Q\rangle\big\rangle^{\mathrm{tw}}_L=\operatorname{Tr}(\Delta^+S)=\operatorname{Tr}\big((\Delta^+)^{1/2}S(\Delta^+)^{1/2}\big)\le2\operatorname{Tr}\Delta^+ .
\]

(b) \emph{Density moments.} The translations by $\eN\mathbb Z^3$ act transitively on the sites of $T^{\mathrm{tw}}$ by automorphisms of the cell complex which preserve the cell-local
weights and the product reference measure, so $\langle\cdot\rangle^{\mathrm{tw}}_L$ is translation invariant and $\langle\rho_x^2\rangle^{\mathrm{tw}}_L$ does not depend on $x$. Let $X$ be the
site set ($|X|=M^3$). By Jensen's inequality ($\langle Y\rangle\le\log\langle e^{Y}\rangle$), in the notation $Z(A,q):=\int d\tr\int dB\sum_{v\ \mathrm{closed}}W\prod_{y\in A}e^{q\rho_y^2}$ of Step~1 of the
proof of Lemma~\ref{lem:d2:mom},
\begin{align*}
\langle\rho_x^2\rangle^{\mathrm{tw}}_L&=M^{-3}\Big\langle\sum_{y\in X}\rho_y^2\Big\rangle^{\mathrm{tw}}_L\le M^{-3}\log\frac{Z(X,1)}{Z^{\mathrm{tw}}_L}\\
&\le\log\frac{G_1(2\pi/\gph)^3K_{\gph}'^{\,3}}{c_0}\le C^{(2)}_N:=\log\max\Big\{1,\frac{G_1(2\pi/\gph)^3K_{\gph}'^{\,3}}{c_0}\Big\}.
\end{align*}
Here the crude bounds $Z(A,q)\le G_q^{|A|}G_0^{|X|-|A|}(2\pi/\gph)^{3|X|}K_{\gph}'^{\,3|X|}$ and $Z\ge c_0^{|X|}$ of that Step~1 are used on $T^{\mathrm{tw}}$. Their proof uses only
$g_{\mathrm{ct}}\ge0$, $h\le1$, the sum over closed $v$ being at most the free sum, $\sum_{v_p}w((dB)_p+v_p)\le K'_{\gph}$, the term $v\equiv0$ (closed on every complex) with the
integration restricted to $\tr_x\in[0,1]$, $|\gph B_b|\le1$, and the counts $|\mathcal B|=|\mathcal P|=3|X|$, which hold on $T^{\mathrm{tw}}$ (three bonds and three plaquettes per site on any
quotient of $\eN\mathbb Z^3$ by a rank-$3$ lattice). Consequently
\[
2\kappa_N\sum_{u(x)=\pm\eN}\langle\rho_x^2\rangle^{\mathrm{tw}}_L\le2\kappa_N\cdot2M^2\cdot C^{(2)}_N=4\kappa_NM^2C^{(2)}_N.
\]

\smallskip\noindent\emph{Step 7: the trace bound.} By the isomorphism of Step 0, $\operatorname{Tr}\Delta^+_{G^{\mathrm{cub}}}=\operatorname{Tr}\Delta^+_{G^{\mathrm{tw}}}$, so we bound the
latter. Relabel the $\Sigma_\pm$-cubes of $G:=G^{\mathrm{tw}}$ by $w:=v-\eN$ (so that $w\equiv0\pmod{2\eN}$, like the $v$ of the $\Sigma_0$-cubes). With $\delta_{\mathrm{tw}}=0$ the graph of
Step 0 has the vertex set $V=\mathbb Z_M^2\times\{-,0,+\}$ ($w\in2\eN\mathbb Z/2L\mathbb Z\cong\mathbb Z_M$, $x_3\in\eN\mathbb Z/L\mathbb Z\cong\mathbb Z_M$), the column edges $(w,x_3,s)\sim(w,x_3+\eN,s)$
in each layer $s$, and the edges $(w,x_3,\pm)\sim(w,x_3,0)$, $(w,x_3,\pm)\sim(w+2\eN,x_3,0)$; it has $3M^2+4M^2=7M^2$ edges. Let $H:=T_M\,\square\,P_3$ be the Cartesian product of the $M\times M$ torus
grid $T_M$ (vertex set $\mathbb Z_M^2$, nearest-neighbour edges in both directions) with the path $P_3$ on $\{-,0,+\}$ (edges $-\!\sim\!0$, $0\!\sim\!+$), a graph on the same vertex set $V$.
Its column edges and its vertical edges $(w,x_3,\pm)\sim(w,x_3,0)$ are edges of $G$. Its in-layer $w$-edges are not, but the two ends of each are joined by a path of two $G$-edges
($x_3$ suppressed): $(w,0)\!-\!(w,+)\!-\!(w+2\eN,0)$ in the layer $0$, and $(w,\pm)\!-\!(w+2\eN,0)\!-\!(w+2\eN,\pm)$ in the layers $\pm$; every non-column $G$-edge is used by at most
two of these paths. With the Dirichlet forms $\mathcal E_G(\psi):=\sum_{\{a,b\}\in E(G)}(\psi_a-\psi_b)^2=\langle\psi,\Delta_G\psi\rangle$ and the inequality
$(\psi_a-\psi_b)^2\le2(\psi_a-\psi_c)^2+2(\psi_c-\psi_b)^2$ on each $w$-edge,
\[
\begin{gathered}
\mathcal E_H(\psi)\ \le\ \mathcal E_G(\psi)+2\cdot2\sum_{\text{non-column }\{a,b\}\in E(G)}(\psi_a-\psi_b)^2\ \le\ 5\,\mathcal E_G(\psi)\qquad(\psi\in\mathbb R^V),\\
\text{i.e.}\quad\Delta_H\le5\Delta_G .
\end{gathered}
\]

Both graphs are connected, so both Laplacians have kernel the constants; on its orthogonal complement $0<\Delta_H\le5\Delta_G$ gives $\Delta_G^+\le5\Delta_H^+$ (monotonicity of the
inverse), hence $\operatorname{Tr}\Delta_G^+\le5\operatorname{Tr}\Delta_H^+$. The spectrum of $\Delta_H$ is $\{\lambda_T(k)+\mu_j:k\in\mathbb Z_M^2,\ j=1,2,3\}$ with
$\lambda_T(k)=4\sin^2(\pi k_1/M)+4\sin^2(\pi k_2/M)$ and $\mu_j\in\{0,1,3\}$ the eigenvalues of the Laplacian of $P_3$. Writing $\tilde k_i\in(-M/2,M/2]$ for the representative of
$k_i$ and using $\sin\pi t\ge2t$ on $[0,\frac12]$, $\lambda_T(k)\ge16|\tilde k|^2/M^2$. At most $8r$ points $k$ have $|\tilde k|_\infty=r$ ($1\le r\le M/2$), and $|\tilde k|^2\ge r^2$
there; the terms with $\mu_j\in\{1,3\}$ are at most $M^2(1+\frac13)$. Hence
\[
\operatorname{Tr}\Delta_H^+=\sum_{(k,j):\ \lambda_T(k)+\mu_j>0}\frac1{\lambda_T(k)+\mu_j}\ \le\ \frac{M^2}{16}\sum_{r=1}^{M/2}\frac{8r}{r^2}+\frac43M^2\ \le\
\frac{M^2}2\Big(1+\log\frac M2\Big)+\frac43M^2 ,
\]
and therefore, since $5-5\log2+\frac{40}3=14.87\ldots\le15$,
\[
2\operatorname{Tr}\Delta^+_{G^{\mathrm{cub}}}=2\operatorname{Tr}\Delta^+_{G}\ \le\ 10\operatorname{Tr}\Delta_H^+\ \le\ 5M^2\log M+\Big(5-5\log2+\frac{40}3\Big)M^2\ \le\ 5M^2\log M+15M^2 .
\]

\smallskip\noindent\emph{Step 8: assembly.} Insert in Step 2 the bound of Step 3 for $\log\Sigma^{\mathrm{tw}}$ and the bound of Step 5 for $\log\Sigma^{\mathrm{cub}}$, and take the
$\langle\cdot\rangle^{\mathrm{tw}}_L$-mean; the terms $5M^2\log(2\pi/\gph)$ cancel, and Steps 6 and 7 bound the two non-constant terms:
\begin{align*}
\phi_L&\le\big\langle\langle Q,\Delta^+_{G^{\mathrm{cub}}}Q\rangle\big\rangle^{\mathrm{tw}}_L+2\kappa_N\!\!\sum_{u(x)=\pm\eN}\!\!\langle\rho_x^2\rangle^{\mathrm{tw}}_L
+M^2\Big[\frac{28\pi^2}{\gph^2}+7\log K'_{\gph}+\log\frac{G_0}{G_{-6\kappa_N}}\Big]\\
&\le5M^2\log M+M^2\Big[15+4\kappa_NC^{(2)}_N+\frac{28\pi^2}{\gph^2}+7\log K'_{\gph}+\log\frac{G_0}{G_{-6\kappa_N}}\Big]=M^2\big(5\log M+A_N\big),
\end{align*}
with
\[
A_N:=15+4\kappa_NC^{(2)}_N+\frac{28\pi^2}{\gph^2}+7\log K'_{\gph}+\log\frac{G_0}{G_{-6\kappa_N}},
\]
which depends on $N$ and the couplings only (through $\gph$, $\kappa_N$, $K'_{\gph}$, $c_0$ and the site moments $G_q$), not on $L$. Since $M=L/\eN$ and $N$ is fixed,
$M^2(5\log M+A_N)=O(L^2\log L)=o(L^3)$. Together with $\phi_L\ge0$ (Proposition~\ref{n1:prop:fold} at $t=0$) this proves the lemma.
\end{proof}

\begin{remark}[the logarithm in the surface bound]\label{n1:rem:SV}
The bound of Lemma~\ref{n1:lem:SV} is $O(L^2\log L)$. The logarithm comes from the energy $\langle Q,\Delta^+Q\rangle$ of the single
extension of Step 5: its mean under $\langle\cdot\rangle^{\mathrm{tw}}_L$ is at most $2\operatorname{Tr}\Delta^+$ by the covariance bound
$S\le2\,\mathrm{Id}$ of Step 6(a), and $\operatorname{Tr}\Delta^+$ is at most of order $M^2\log M$ (Step 7). Of this bound the argument of
this section uses only $\phi_L=o(L^3)$.
\end{remark}

\begin{proposition}[Jensen and convexity: the comparison inequality at finite volume]\label{n1:prop:conv}
Let $D=(c_0,c_+,c_3;h;O)$ be a datum on $T^{\mathrm{tw}}_L$, $L\ge6\eN$, $h\mid L/2$. Then for every real $t$
\[
t\,n_c\,\langle O\rangle^{\mathrm{cub}}_L\ \le\ \phi_L+t\,n_c\,\om^t_L(O),\qquad\text{hence for }t>0:\qquad
\langle O\rangle^{\mathrm{cub}}_L\ \le\ \om^t_L(O)+\frac{h^3\,\phi_L}{2\,t\,L^3}.
\]
\end{proposition}
\begin{proof}
\emph{Reduction to $c_0=0$.} The translation $\tau$ by $-c_0e_1$ lies in $\eN\Z^3$, acts on both tori and preserves both states
(Lemma~\ref{n1:lem:walls}; Lemma~\ref{lem:d2:sym}); it maps $D$ to the datum $(0,c_+-c_0,c_3;h;O\circ\tau^{-1})$, carries $\mu^t_L$ to the
perturbed law of that datum, and leaves $\langle O\rangle^{\mathrm{cub}}_L$, $Z_t$, $\phi_L$ and $\om^t_L(O)$ unchanged. So let $c_0=0$ and define
$O_d$ on $T^{\mathrm{cub}}_L$ as in Proposition~\ref{n1:prop:fold}.

\emph{Jensen on the cubic torus.} Each of the $n_c$ summands of $O_d$ on $T^{\mathrm{cub}}_L$ is $O\circ w^{-1}$ with $w$ a composition of
the maps $r_c,s_c,q_c$ and $r_0$, each an automorphism of $T^{\mathrm{cub}}_L$ preserving $W$ and the reference measure (a signed coordinate
permutation about a site composed with a translation of $\eN\Z^3$, Lemma~\ref{lem:d2:sym}); hence $\langle O\circ w^{-1}\rangle^{\mathrm{cub}}_L=
\langle O\rangle^{\mathrm{cub}}_L$ and $\langle O_d\rangle^{\mathrm{cub}}_L=n_c\langle O\rangle^{\mathrm{cub}}_L$. Jensen's inequality and
Proposition~\ref{n1:prop:fold} give
\[
t\,n_c\langle O\rangle^{\mathrm{cub}}_L=t\langle O_d\rangle^{\mathrm{cub}}_L\le\log\langle e^{tO_d}\rangle^{\mathrm{cub}}_L\le\phi_L+\log Z_t.
\]

\emph{Convexity on the adapted torus.} $p(s):=\log Z_s$ is finite ($|O_d|\le n_c\|O\|_\infty$) and convex in $s$ (H\"older), $p(0)=0$, and
$p'(s)=Z_s^{-1}\mu^{\mathrm{tw}}_L(O_de^{sO_d})=\mu^s_L(O_d)=n_c\,\om^s_L(O)$ by Lemma~\ref{n1:lem:Winv}(i) (differentiation under the integral
sign, $O_d$ bounded). Convexity gives $p(t)\le p(0)+t\,p'(t)=t\,n_c\,\om^t_L(O)$. Combining, $t\,n_c\langle O\rangle^{\mathrm{cub}}_L\le\phi_L+t\,n_c\om^t_L(O)$;
for $t>0$ divide by $t\,n_c=2tL^3/h^3$.
\end{proof}

\begin{remark}[use along the sides]\label{n1:rem:sides}
To pass Proposition~\ref{n1:prop:conv} to $\om_N$ the fold must run on the adapted torus of the \emph{same} side $L_n$ as the cubic torus of
Definition~\ref{def:omN}, with $h\mid L_n/2$, and the error $h^3\phi_{L_n}/(2tL_n^3)$ must vanish. Under the choice of dyadic sides of
Definition~\ref{n1:def:D}, for a datum with $h=h_j$ one has $h_j\mid L_n/2$ for all $n\ge j$, so the fold runs for all large $n$ along
$(L_n)$ and along every subsequence of it, in particular along the one defining a dyadic limit point; and Lemma~\ref{n1:lem:SV} gives
$h^3\phi_{L_n}/(2tL_n^3)\to0$ at fixed $h$ and $t>0$. No other consequence of Lemma~\ref{n1:lem:SV} is used below.
\end{remark}

\subsection{Limits: the comparison states and the hypotheses of the rigidity theorem}
\label{n1:sub:lim}

Fix a dyadic sequence $(L_n)$, a datum $D=(c_0,c_+,c_3;h;O)$ with $h=h_j$ one of its wall spacings, and $n\ge j$, so that $h\mid L_n/2$ and
$D$ is a datum on every $T^{\mathrm{tw}}_{L_n}$. For a finite set of cells of $\eN\Z^3$ and $L_n$ larger than its diameter, the cells embed in
$T^{\mathrm{tw}}_{L_n}$ and $\mu^t_{L_n}$ induces a law of their variables; in this sense $(\mu^t_{L_n})_n$ is a sequence of laws on the full
configuration space $E$ of Lemma~\ref{n1:lem:twmom}(c). We write $\langle\cdot\rangle_t$ for $\mu^t_{L_n}$, $L=L_n$.

\begin{lemma}[block doubling under the perturbed law]\label{n1:lem:block}
Let $Q$ be a block of $2^{k_1}\times2^{k_2}\times2^{k_3}$ consecutive $\cW$-cells (the walls between them included), with
$2^{k_1}h\le L/2$, $2^{k_2}h\le L$, $2^{k_3}h\le L/2$, and let $G'\ge0$ be a bounded Borel function of cells inside the open block $Q$. Then there
is a finite $A\subset\cW$ with the blocks $w(Q)$, $w\in A$, pairwise disjoint and $|A|\,|Q|>n_c/8$ ($|Q|=2^{k_1+k_2+k_3}$ the number of cells of $Q$),
such that
\[
\langle G'\rangle_t\ \le\ \Big\langle\prod_{w\in A}G'\circ w^{-1}\Big\rangle_t^{1/|A|}
\ \le\ \Big(e^{2|t|n_c\|O\|_\infty}\Big\langle\prod_{w\in A}G'\circ w^{-1}\Big\rangle^{\mathrm{tw}}_L\Big)^{1/|A|}.
\]
The same holds for unbounded Borel $G'\ge0$.
\end{lemma}
\begin{proof}
Let $\Pi$ be one of the two bounding walls of $Q$ in the $u$-direction, a wall of $D$; by the hypothesis on the $u$-extent, $Q$ lies in one
of the two closed halves bounded by $\Pi$ and its second wall. By Lemma~\ref{n1:lem:Winv}(ii) the form $(G,\tilde G)\mapsto\langle\overline{G\circ\theta_\Pi}\,\tilde G\rangle_t$
is positive semidefinite and Hermitian on the bounded Borel functions of cells in that closed half (Hermitian by the $\theta_\Pi$-invariance),
so the Schwarz inequality with $\tilde G=1$ gives $\langle G'\rangle_t^2\le\langle(G'\circ\theta_\Pi)\,G'\rangle_t$, and $(G'\circ\theta_\Pi)G'$
is a function of cells inside the doubled open block $Q\cup\Pi\cup\theta_\Pi(Q)$, a wall-bounded block of twice the cells, whose cells are
distinct modulo the torus since its $u$-extent is at most $L$. We iterate this doubling, first in $u$ while the current $u$-extent is at most $L/2$, then in $v(x)$
(period $2L$, halves of length $L$) while the $v$-extent is at most $L$, then in $x_3$ while the $x_3$-extent is at most $L/2$. The final
block has extents greater than $L/2$, $L$, $L/2$, so it covers more than $(L/2)\cdot L\cdot(L/2)=L^3/4$ of the volume $2L^3$ of the torus in
the coordinates $(u,v(x),x_3)$: with $A$ the set of the $2^m$ elements of $\cW$ (products of the reflections used) carrying $Q$ onto the
$2^m$ constituent blocks, $|A|\,|Q|>n_c/8$ and $\langle G'\rangle_t^{2^m}\le\langle\prod_{w\in A}G'\circ w^{-1}\rangle_t$. The second
inequality is the removal of the source: for $H\ge0$,
\[
\langle H\rangle_t=Z_t^{-1}\mu^{\mathrm{tw}}_L(e^{tO_d}H)\le e^{2|t|n_c\|O\|_\infty}\langle H\rangle^{\mathrm{tw}}_L
\]
by the bounds on $Z_t$ and $e^{tO_d}$ in Definition~\ref{n1:def:datum}. For unbounded $G'\ge0$ apply the two inequalities to $G'\wedge M$ and let $M\to\infty$ (monotone convergence).
\end{proof}

\begin{lemma}[uniform bounds and tightness of the perturbed laws]\label{n1:lem:tight}
For a site $x$ let $Q_x$ be the smallest wall-bounded block of $\cW$-cells containing $x$ in its interior ($|Q_x|\in\{1,2,4,8\}$, one cell
if $x$ lies on no wall of $D$, doubled in each coordinate direction in which $x$ lies on a wall). Then for every $q\ge0$, every real $t$
and all large $n$,
\[
\langle e^{q\rho_x^2}\rangle_t\ \le\ e^{16|t|\,|Q_x|\,\|O\|_\infty}\,\frac{G^{\mathrm{ct}}_q}{G^{\mathrm{ct}}_0}\,C_1^{\,4|Q_x|(h/\eN)^3}
\ \le\ e^{128|t|\,\|O\|_\infty}\,\frac{G^{\mathrm{ct}}_q}{G^{\mathrm{ct}}_0}\,C_1^{\,32(h/\eN)^3}=:C''_N(q,t,h),
\]
uniformly in $x$ and $n$. For a finite set $P$ of plaquettes and $s\in\R^P$, with $Q_P$ a block as in Lemma~\ref{n1:lem:block} whose six
bounding walls have every vertex of every $p\in P$ strictly between them (it exists for all large $n$),
\[
\langle e^{\sum_{p\in P}s_p\ff_p}\rangle_t\le e^{16|t|\,|Q_P|\,\|O\|_\infty}e^{|s|^2/2}.
\]
Consequently $(\mu^t_{L_n})_n$ is tight on $E$ and has weak
limit points $\nu_t$ (along subsequences of $(L_n)$).
\end{lemma}
\begin{proof}
\emph{Sites.} Apply Lemma~\ref{n1:lem:block} to $Q=Q_x$ and $G'=e^{q\rho_x^2}$. The sites $wx$, $w\in A$, are pairwise distinct ($x$ is interior
to $Q_x$ and the blocks $w(Q_x)$ are disjoint), so $\prod_{w\in A}G'\circ w^{-1}=\prod_{y\in A\cdot x}e^{q\rho_y^2}$ with $|A\cdot x|=|A|$, and
Step~1 of the proof of Lemma~\ref{lem:d2:mom} on $T^{\mathrm{tw}}_L$ (proof of Lemma~\ref{n1:lem:twmom}(b)) gives
\[
\langle\prod_{y\in A\cdot x}e^{q\rho_y^2}\rangle^{\mathrm{tw}}_L\le(G^{\mathrm{ct}}_q/G^{\mathrm{ct}}_0)^{|A|}C_1^{\,|\cX|}\qquad\text{with }|\cX|=(L/\eN)^3.
\]
Since
$|A|>n_c/(8|Q_x|)$ and $n_c=2L^3/h^3$, we have $|\cX|/|A|<8|Q_x|(L/\eN)^3h^3/(2L^3)=4|Q_x|(h/\eN)^3$ and $2|t|n_c\|O\|_\infty/|A|<16|t|\,|Q_x|\,\|O\|_\infty$.

\emph{Plaquettes.} Apply Lemma~\ref{n1:lem:block} to $Q=Q_P$ and $G'=e^{\sum_{p\in P}s_p\ff_p}$, a continuous function $\ge0$ of cells inside
$Q_P$ ($\ff_p$ depends on $p$ and its bonds). $G'\circ w^{-1}=e^{\sum_{p'}s'_{p'}\ff_{p'}}$, where the sum runs over the plaquettes $p'=wp$ of $w(Q_P)$ and
$s'=\pm s$ (the reflections may reverse orientations), so that $|s'|=|s|$; the $p'$ are distinct over $w\in A$ (disjoint blocks). By
Lemma~\ref{n1:lem:twmom}(a), $\langle\prod_{w\in A}G'\circ w^{-1}\rangle^{\mathrm{tw}}_L\le e^{|A|\,|s|^2/2}$, so
\[
\langle G'\rangle_t\le\big(e^{2|t|n_c\|O\|_\infty}e^{|A||s|^2/2}\big)^{1/|A|}\le e^{16|t|\,|Q_P|\,\|O\|_\infty}e^{|s|^2/2}.
\]

\emph{Tightness.} The bounds for $e^{\rho_x^2}$ and $e^{\pm\ff_p}$ give, by Markov's inequality, uniform tail bounds for $\tr_x$ and $\ff_p$. The variable $B_b$ ranges over a bounded interval and $v_p=\ff_p-(dB)_p$ is tight because $\ff_p$ is; the limits are taken with $B_b$ in the closed interval $[-\pi/\gph,\pi/\gph]$. Under every limit point $\nu_t$ each of the two end points $\pm\pi/\gph$ has measure zero. To prove this, let $\Delta=\{b\}$ and let $\gamma_\Delta$, $\gamma^t_\Delta$, $I_\Delta$ and $S_\Delta$ be as in the paragraph on the fine specification that precedes Lemma~\ref{n1:lem:gibbs}. For a bounded continuous local function $f\ge0$ on the closed space (the space $E$ with the closed intervals $[-\pi/\gph,\pi/\gph]$ for the bond variables) and all large $n$, integrating the variable $B_b$ first gives $\langle f\rangle_t=\langle\gamma^t_\Delta f\rangle_t\le e^{2|t|S_\Delta}\langle\gamma_\Delta f\rangle_t$, where the inequality holds because $e^{-|t|S_\Delta}\le e^{t\sum_{c\in I_\Delta}O_c}\le e^{|t|S_\Delta}$. The function $\gamma_\Delta f$ is bounded and continuous on the closed space, because the factors of $W$ are continuous functions of the bond variables on the closed interval, and therefore $\nu_t(f)\le e^{2|t|S_\Delta}\nu_t(\gamma_\Delta f)$. For $f_k:=\max\{0,1-k(\pi/\gph-|B_b|)\}$ the functions $\gamma_\Delta f_k\le1$ tend to $0$ pointwise as $k\to\infty$, since $\gamma_\Delta$ integrates $B_b$ against a density with respect to Lebesgue measure, so that $\nu_t(|B_b|=\pi/\gph)\le\nu_t(f_k)\le e^{2|t|S_\Delta}\nu_t(\gamma_\Delta f_k)\to0$ by dominated convergence. Hence the limit points are carried by $E$, and the convergence holds for the bounded functions that are continuous off this null set. The finite-dimensional laws are tight on the Polish factors of $E$,
and Prokhorov's theorem with the diagonal argument of the proof of Theorem~\ref{thm:d2:LP} gives weak limit points on $E$.
\end{proof}

\begin{lemma}[joint Gaussian domination of the comparison states]\label{n1:lem:JGD}
Let $\nu_t$ be a weak limit point of $(\mu^t_{L_n})_n$. For every finite set $P$ of plaquettes of $\eN\Z^3$ and every $s\in\R^P$,
\[
\nu_t\big(e^{\sum_{p\in P}s_p\ff_p}\big)\le e^{16|t|\,|Q_P|\,\|O\|_\infty}e^{|s|^2/2}.
\]
If $\nu_0$ is a weak limit point of $(\nu_{1/k})_k$, then
$\nu_0\big(e^{\sum_{p\in P}s_p\ff_p}\big)\le e^{|s|^2/2}$ for every finite $P$ and $s\in\R^P$; this is hypothesis \textup{(I2)} of Theorem~\ref{rv:thm:RV} with $C=1$.
\end{lemma}
\begin{proof}
$G:=e^{\sum_ps_p\ff_p}$ is continuous and $\ge0$ on $E$, so along the subsequence with $\mu^t_{L_n}\to\nu_t$ the Portmanteau theorem (lower
semicontinuity for non-negative continuous functions) gives $\nu_t(G)\le\liminf_n\langle G\rangle_t\le e^{16|t||Q_P|\|O\|_\infty}e^{|s|^2/2}$ by
Lemma~\ref{n1:lem:tight}, and then $\nu_0(G)\le\liminf_k\nu_{1/k}(G)\le e^{|s|^2/2}$.
\end{proof}

\emph{The fine specification.} For a finite set $\Delta$ of cells of $\eN\Z^3$ the kernel $\gamma_\Delta$ integrates the variables of
$\Delta$ against the product of those local factors of $W$ (site, bond and plaquette factors and the closedness indicators of the cubes)
that involve a cell of $\Delta$, normalised by $Z_\Delta(\xi_{\mathrm{out}})$, the outside variables held fixed; this is the specification of
Theorem~\ref{rv:thm:RV} (unitary gauge). The walls, $\cW$-cells and functions $O_c$ are defined on $\eN\Z^3$ as in Definition~\ref{n1:def:datum}
(the reflections of Lemma~\ref{n1:lem:walls} act on $\eN\Z^3$; the group they generate is again simply transitive on the $\cW$-cells, by the
dihedral argument of Lemma~\ref{n1:lem:Wcells} on the line), and $\gamma^t_\Delta$ is the kernel with the extra factor $e^{t\sum_{c\in I_\Delta}O_c}$,
$I_\Delta$ the finite set of $\cW$-cells $c$ for which $O_c$ depends on a variable of $\Delta$; $S_\Delta:=|I_\Delta|\,\|O\|_\infty$.

\begin{lemma}[the comparison states: Gibbs, $\cW$-invariant, reflection positive]\label{n1:lem:gibbs}
Let $\nu_t$ be a weak limit point of $(\mu^t_{L_n})_n$ and $\nu_0$ a weak limit point of $(\nu_{1/k})_k$ (which exists: the bounds of
Lemma~\ref{n1:lem:tight} are uniform in $|t|\le1$). (i) $\nu_t$ is carried by closed $v$, and $\nu_t(f)=\nu_t(\gamma^t_\Delta f)$ for every finite
$\Delta$ and bounded Borel $f$ on $E$. (ii) $\nu_t$ is invariant under the reflections generating $\cW$ and reflection positive at every wall
of $D$ in the sense of Theorem~\ref{thm:d2:LP}(d) (positive semidefinite Hermitian form, conjugation identity, $L^2(\nu_t)$ extension).
(iii) $\nu_0$ is a Gibbs measure of the fine specification ($\nu_0(f)=\nu_0(\gamma_\Delta f)$ for all finite $\Delta$, bounded Borel $f$), carried by
closed $v$, invariant under the reflections generating $\cW$, reflection positive at every wall of $D$, and satisfies
$\nu_0(e^{q\rho_x^2})\le(G^{\mathrm{ct}}_q/G^{\mathrm{ct}}_0)\,C_1^{\,32(h/\eN)^3}$ for every site $x$ and $q\ge0$, and the Gaussian bound of Lemma~\ref{n1:lem:JGD}.
\end{lemma}
\begin{proof}
(i) For $L=L_n$ larger than the diameter of $\Delta$ and of the cells of the $O_c$, $c\in I_\Delta$, integrating the $\Delta$-variables first
under $\mu^t_L$ gives $\mu^t_L(f)=\mu^t_L(\gamma^t_\Delta f)$ for bounded continuous local $f$: the torus weight times $e^{tO_d}$ is the product of
the factors involving $\Delta$ (those of $\gamma^t_\Delta$, including $e^{t\sum_{c\in I_\Delta}O_c}$) and factors independent of the $\Delta$-variables.
The kernel is Feller: the factors are continuous in the outside variables, $Z_\Delta(\xi_{\mathrm{out}})\in(0,\infty)$ on the outside
configurations that are restrictions of closed configurations (the closed configuration itself provides an admissible term; finiteness by
\eqref{eq:ct}), and dominated convergence applies; so $\gamma^t_\Delta f$ is bounded continuous, both sides pass to the limit, and
$\nu_t(f)=\nu_t(\gamma^t_\Delta f)$ for bounded continuous local $f$. For each cube $c$ the set $\{(dv)_c=0\}$ is clopen in $E$ (finitely many
discrete variables), so $\nu_t((dv)_c=0)=\lim_n\mu^t_{L_n}((dv)_c=0)=1$: $\nu_t$ is carried by closed $v$, where the kernel is defined. Now
$f\mapsto\nu_t(\gamma^t_\Delta f)$ is a Borel probability measure on $E$ ($\gamma^t_\Delta$ is a Markov kernel) agreeing with $\nu_t$ on the bounded
continuous local functions; these determine every finite-dimensional marginal (bounded continuous functions determine a finite Borel
measure on a metric space), and the cylinder sets form a $\pi$-system generating the Borel $\sigma$-algebra of the Polish product $E$, so
the two measures coincide: the identity holds for every bounded Borel $f$.

(ii) For a generating reflection $R$ and bounded continuous local $F$, $\langle F\circ R\rangle_t=\langle F\rangle_t$ for all large $n$
(Lemma~\ref{n1:lem:Winv}(i)); both sides converge, so $\nu_t$ is $R$-invariant. For $F$ bounded continuous of cells in a closed half space
bounded by a wall $\Pi$ of $D$, the cells lie in one closed half of $T^{\mathrm{tw}}_{L_n}$ for all large $n$, $\overline{F\circ\theta_\Pi}\cdot F$ is
bounded continuous local, and $\nu_t(\overline{F\circ\theta_\Pi}F)=\lim_n\langle\overline{F\circ\theta_\Pi}F\rangle_t\ge0$ by Lemma~\ref{n1:lem:Winv}(ii).
The Hermitian form, the identity $\nu_t(\overline{F\circ\theta_\Pi})=\overline{\nu_t(F)}$ and the $L^2$ extension follow as in the proof of
Lemma~\ref{n1:lem:twmom}(c) (polarisation; $\theta_\Pi$-invariance; Lusin density and $L^2$-continuity).

(iii) The source bound $e^{-|t|S_\Delta}\le e^{t\sum_{c\in I_\Delta}O_c}\le e^{|t|S_\Delta}$ gives $\|\gamma^t_\Delta f-\gamma_\Delta f\|_\infty\le2\|f\|_\infty(e^{2|t|S_\Delta}-1)$
(write $\gamma^t_\Delta f-\gamma_\Delta f=\gamma^t_\Delta(f-\gamma_\Delta f)$ and compare the two normalised weights). Hence for bounded continuous local
$f$:
\[
\nu_0(f)=\lim_k\nu_{1/k}(f)=\lim_k\nu_{1/k}(\gamma^{1/k}_\Delta f)=\lim_k\nu_{1/k}(\gamma_\Delta f)=\nu_0(\gamma_\Delta f)
\]
($\gamma_\Delta f$ bounded
continuous by the Feller property), and the extension to bounded Borel $f$ is the determining-class argument of (i). Closedness of $v$,
$R$-invariance and reflection positivity are closed conditions on bounded continuous local functions and pass from $\nu_{1/k}$ to $\nu_0$ as
in (ii). For the per-site bound, $\nu_{1/k}(e^{q\rho_x^2})\le\liminf_n\langle e^{q\rho_x^2}\rangle_{1/k}\le C''_N(q,1/k,h)$ (Portmanteau, Lemma~\ref{n1:lem:tight}),
and $\nu_0(e^{q\rho_x^2})\le\liminf_k\nu_{1/k}(e^{q\rho_x^2})\le\lim_kC''_N(q,1/k,h)=(G^{\mathrm{ct}}_q/G^{\mathrm{ct}}_0)C_1^{32(h/\eN)^3}$.
\end{proof}

\begin{proposition}[the comparison states satisfy the hypotheses of Theorem~\ref{rv:thm:RV}]\label{n1:prop:I123}
Let $\nu_0=\nu_0(D)$ be as in Lemma~\ref{n1:lem:gibbs}(iii). Then $\nu_0$ is a Gibbs measure of the fine specification on $\eN\Z^3$, invariant
under the sublattice $\Lambda':=h(e_1-e_2)\Z+h(e_1+e_2)\Z+2he_3\Z$ of $\eN\Z^3$, which has finite index (it contains $2h\Z^3$) and contains
the time translation $2he_3$, and satisfies \textup{(I1)}, \textup{(I2)} and \textup{(I3)} of Theorem~\ref{rv:thm:RV}. Consequently $\nu_0$ is invariant under the
translations $\tau_{2\eN e_j}$, $j=1,2,3$.
\end{proposition}
\begin{proof}
The Gibbs property is Lemma~\ref{n1:lem:gibbs}(iii). For the periodicity, by Lemma~\ref{n1:lem:walls} the products $r_cr_{c'}$, $s_cs_{c'}$, $q_cq_{c'}$ of
generating reflections ($c-c'\in h\Z$) are the translations by $h\Z(e_1-e_2)$, $h\Z(e_1+e_2)$, $2h\Z e_3$, under which $\nu_0$ is invariant by
Lemma~\ref{n1:lem:gibbs}(iii); $2he_1=h(e_1-e_2)+h(e_1+e_2)$ and $2he_2=h(e_1+e_2)-h(e_1-e_2)$ show $2h\Z^3\subset\Lambda'$. Hypothesis (I2) is Lemma~\ref{n1:lem:JGD}
with $C=1$. Hypothesis (I3) is the per-site bound of Lemma~\ref{n1:lem:gibbs}(iii), uniform in $x$, for every $q>0$. For (I1), with $s=\pm1$ on the single
plaquette $p$, (I2) gives $\nu_0(\ff_p^2)\le2\nu_0(\cosh\ff_p)\le2e^{1/2}$ (as $f^2\le2\cosh f$), and $\nu_0(\rho_x^2)\le\nu_0(e^{\rho_x^2})<\infty$ by (I3).
Theorem~\ref{rv:thm:RV} then gives the invariance under $\tau_{2\eN e_j}$, $j=1,2,3$.
\end{proof}

\begin{proposition}[the comparison inequality]\label{n1:prop:comp}
Let $\om$ be a dyadic limit point, formed along the subsequence $(L_{n_k})_k$ of the dyadic sequence $(L_n)$, let $D$ be a datum with $h\in\{h_j\}$
whose observable $O$ is a function of the gauge-invariant variables $(\ff_p,\tr_x)$ of its cells (evaluated under a law on $E$ through
$\ff_p=(dB)_p+v_p$), and let $\nu_t$, $\nu_0$ be weak limit points as in Lemma~\ref{n1:lem:gibbs} formed along subsequences of $(L_{n_k})_k$. Then
\[
\om(O)\ \le\ \nu_0(O).
\]
\end{proposition}
\begin{proof}
Fix $t>0$. For all large $k$, $h\mid L_{n_k}/2$ (Remark~\ref{n1:rem:sides}) and Proposition~\ref{n1:prop:conv} gives
\[
\langle O\rangle^{\mathrm{cub}}_{L_{n_k}}\le\om^t_{L_{n_k}}(O)+h^3\phi_{L_{n_k}}/(2tL_{n_k}^3).
\]
Along the subsequence with $\mu^t_{L_{n_k}}\to\nu_t$ the
left side tends to $\om(O)$ (Definition~\ref{def:omN}, $O$ bounded continuous local), $\om^t_{L_{n_k}}(O)\to\nu_t(O)$, and the error tends to $0$
by Lemma~\ref{n1:lem:SV}. So $\om(O)\le\nu_t(O)$ for every $t>0$, and $\nu_{1/k}(O)\to\nu_0(O)$ along the subsequence defining $\nu_0$.
\end{proof}

\begin{lemma}[reflection positivity from one-sided domination]\label{n1:lem:RPcone}
Let $\Pi$ be a plane, $\theta:=\theta_\Pi$, and let $\nu$ be a law on $\OmN$ or on $E$ such that for every real bounded continuous local $O$ there
is a law $\nu_O$, $\theta$-invariant and reflection positive at $\Pi$ on $\cA_+(\Pi)$, with $\nu(O)\le\nu_O(O)$. Then $\nu$ is $\theta$-invariant
on the bounded continuous local functions and $\nu(\overline{F\circ\theta}\cdot F)\ge0$ for every real $F\in\cA_+(\Pi)$.
\end{lemma}
\begin{proof}
For real $F\in\cA_+(\Pi)$ take $O:=-(F\circ\theta)F$, which is real, bounded, continuous and local; then $\nu(-(F\circ\theta)F)\le\nu_O(-(F\circ\theta)F)\le0$. For a real
bounded continuous local $G$ take $O:=\pm(G\circ\theta-G)$; then $\pm\nu(G\circ\theta-G)\le\pm\nu_O(G\circ\theta-G)=0$, so $\nu(G\circ\theta)=\nu(G)$.
\end{proof}

\subsection{A second proof of the tangent inequality for the comparison states}
\label{tv:sub:T}

This subsection gives a direct proof, for the comparison states, of the inequality which the proofs of Lemmas~\ref{rv:lem:step2} and~\ref{rv:lem:step2full} take
from the tangent inequality, Lemma~\ref{rv:lem:TV}; the result is Proposition~\ref{tv:prop:T}. The proof does not use Lemma~\ref{rv:lem:TV} and uses no relative
entropy: the comparison states are limits of laws on tori, and on a torus the inequality follows from Jensen's inequality and the transfer calculus of
Section~\ref{rv:sec}. Its main ingredients are Lemmas~\ref{rv:lem:phase}, \ref{rv:lem:zak}, \ref{rv:lem:UV} and~\ref{rv:lem:fullmap} (with Lemmas~\ref{rv:lem:char}, \ref{rv:lem:fullclass} and Remark~\ref{rv:rem:abs} for the full class), the insertion identity~\eqref{rv:eq:insert}, the covariance of the weight under the permutations of the axes (the first part of the proof of Proposition~\ref{rv:prop:YM4}(i)), the
fold at zero source (Proposition~\ref{n1:prop:fold}), the lower bound of the partition function in the proof of Lemma~\ref{lem:d2:mom}, and
Lemmas~\ref{n1:lem:walls}, \ref{n1:lem:Winv} and~\ref{n1:lem:gibbs}. Neither Theorem~\ref{rv:thm:RV} nor the upper bound of Lemma~\ref{n1:lem:SV} enters the proof, and no other part of this paper depends on this subsection (Remark~\ref{tv:rem:second}).

\emph{Notation.} In this subsection $t$ is the parameter of the operators $X_t$ of Lemmas~\ref{rv:lem:step2} and~\ref{rv:lem:step2full}, as in
Section~\ref{rv:sec}; the source strength of Subsection~\ref{n1:sub:lim} is always written $1/k$. The symbol $T$ without an index is the transfer operator of
Lemma~\ref{rv:lem:zak}(c), and tori carry an index. The symbol $L$ is a side length; the layer space of Lemma~\ref{rv:lem:zak}(c) is not given a symbol,
and $R_L$, $P_G$ denote, as there, its gauge representation and its gauge projection.

\emph{Tori and partition functions.} For a sublattice $\Gamma\subset\eN\Z^3$ of finite index all of whose non-zero elements have length at least
$4\eN$ let $T_\Gamma:=\eN\Z^3/\Gamma$, let $\mu_\Gamma$ be the probability law with weight $W$ on the cells of $T_\Gamma$ (sum over closed $v$) and
\[
Z(\Gamma):=\int d\tr\int dB\sum_{v\ \mathrm{closed}}W
\]
its partition function (Lebesgue measure on the $\tr_x$ and on the $B_b\in(-\pi/\gph,\pi/\gph]$, counting measure on the closed $v$: the convention of
Step~1 of the proof of Lemma~\ref{lem:d2:mom}). Thus $Z^{\mathrm{cub}}_L=Z(L\Z^3)$, $Z^{\mathrm{tw}}_L=Z(\Gamma(L))$ and $\mu^{\mathrm{tw}}_L=\mu_{\Gamma(L)}$.
For a permutation $g$ of the coordinate axes the relabelling $\hat g$ maps the cell complex, the weight, the closure constraint and the reference measure
of $T_\Gamma$ onto those of $T_{g\Gamma}$ (first part of the proof of Proposition~\ref{rv:prop:YM4}(i)), so that
\begin{equation}\label{tv:eq:cov}
Z(g\Gamma)=Z(\Gamma),\qquad \mu_{g\Gamma}=\mu_\Gamma\circ\hat g .
\end{equation}
For $a_1,a_2,a_3\in\eN\mathbb N$, each at least $4\eN$, we write $Z(a_1,a_2,a_3):=Z(a_1e_1\Z+a_2e_2\Z+a_3e_3\Z)$ for the partition function of the product
torus with sides $a_1,a_2,a_3$, which has $a_1a_2a_3/\eN^3$ sites; $Z^{\mathrm{cub}}_L=Z(L,L,L)$.

\emph{Tori with a twist.} With $e_3$ as time direction, as in Section~\ref{rv:sec}, let $\Gamma_s\subset\eN(\Z e_1+\Z e_2)$ be a sublattice of finite index,
$\eN(\Z e_1+\Z e_2)/\Gamma_s$ the \emph{layer torus}, $n_{\mathrm{lay}}:=[\eN(\Z e_1+\Z e_2):\Gamma_s]$ its number of sites, $M\ge2$ an integer and
$\zeta\in\eN(\Z e_1+\Z e_2)$ (the \emph{twist}), and put
\[
\Gamma(\Gamma_s,M,\zeta):=\Gamma_s+\Z\,(M\eN e_3+\zeta).
\]
We assume, as above, that all non-zero elements of $\Gamma(\Gamma_s,M,\zeta)$ have length at least $4\eN$.
The torus $T_{\Gamma(\Gamma_s,M,\zeta)}$ consists of $M$ layers, each a copy of the layer torus, and its layer $M+1$ is the layer $1$ translated by
$\zeta$: in terms of the layer configurations $c^{(s)}$ of Subsection~\ref{rv:ssec:S3}, $c^{(M+1)}=\tau_\zeta c^{(1)}$ with $(\tau_\zeta c)(y):=c(y-\zeta)$. The
\emph{$\zeta$-twisted periodic measure} $\mu^\zeta_M$ is the probability law with the phase-restored weight of Subsection~\ref{rv:ssec:S1} on the phase-restored
configurations of $T_{\Gamma(\Gamma_s,M,\zeta)}$. For $\zeta=0$ it is the periodic measure $\mu_M$ with $M$ layers of the proof of
Lemma~\ref{rv:lem:zak}(c), for the layer torus at hand: the statements and proofs of Lemmas~\ref{rv:lem:zak}(c), \ref{rv:lem:UV}, \ref{rv:lem:fullmap}
and of~\eqref{rv:eq:insert} use the layer only as a finite complex (its sites, spatial bonds and spatial plaquettes, the gauge group
$\Tper^{\mathrm{sites}}$ and the one-layer weight, of finite integral) and hold for every layer torus. For general $\zeta$ the only change is that the
closing slab joins $c^{(M)}$ to $\tau_\zeta c^{(1)}$ instead of $c^{(1)}$.

For insertions $X_s$ at the heights $s\in\{1,\dots,M\}$ ($X_s:=1$ where there
is none; fibred in the sense of Lemma~\ref{rv:lem:zak}(b), or bounded on $\cH_{\mathrm{full}}$ as in Lemma~\ref{rv:lem:fullmap}), $Z^\zeta_M(X_1,\dots,X_M)$ denotes the partition
function of $\mu^\zeta_M$ with the slab kernel $K$ replaced by $K^{X_s}$ (Lemma~\ref{rv:lem:zak}(b)), or by $\langle\Phi_{\mathrm{full}}(c),X_s\Phi_{\mathrm{full}}(\wt c')\rangle$ for an insertion on $\cH_{\mathrm{full}}$ (Lemma~\ref{rv:lem:fullmap}), at the slab $s$, in the normalisation of
Lemma~\ref{rv:lem:zak}(c), and $Z^\zeta_M:=Z^\zeta_M(1,\dots,1)$; for $\zeta=0$, and for insertions that satisfy the summability condition of Lemma~\ref{rv:lem:zak}(b) or act on $\cH_{\mathrm{full}}$, these are the quantities $Z_M(\cdot)$ and $Z_M$ of~\eqref{rv:eq:avg}.

\begin{lemma}[phase restoration on a finite torus]\label{tv:lem:pr}
Let $T_\Gamma$ be a torus as above and let $\wt\mu_\Gamma$ be the probability law with the phase-restored weight of Subsection~\ref{rv:ssec:S1} on the
phase-restored configurations of $T_\Gamma$ (normalised Haar measure on each phase), with partition function $\wt Z(\Gamma)$. The formulas of
Lemma~\ref{rv:lem:phase} are local and define the gauge action and the Haar randomisation $\Psi$ on the configurations of $T_\Gamma$. Then
\begin{itemize}
\item[(a)] $\wt Z(\Gamma)=Z(\Gamma)$ and $\wt\mu_\Gamma=\Psi_*(\mu_\Gamma\times\Haar^{\mathrm{sites}})$;
\item[(b)] $\Psi\circ(\tau_a\times\tau_a)=\tau_a\circ\Psi$ for $a\in\eN\Z^3$; so for every law $\mu$ on the configurations of $T_\Gamma$ the law
$\Psi_*(\mu\times\Haar^{\mathrm{sites}})$ is invariant under every translation which leaves $\mu$ invariant;
\item[(c)] if $\Gamma=\Gamma(\Gamma_s,M,\zeta)$, then $\wt\mu_\Gamma=\mu^\zeta_M$ and $Z(\Gamma)=\varkappa^{\,n_{\mathrm{lay}}M}Z^\zeta_M$, where the constant
$\varkappa\in(0,\infty)$ depends only on the normalisation of the reference measure of the temporal links (in $Z(\Gamma)$ each of the
$n_{\mathrm{lay}}M$ temporal links carries the Lebesgue measure of $\FD$, of mass $\cper$; in~\eqref{rv:eq:avg} it carries the normalised Haar measure)
and not on $\Gamma_s$, $M$, $\zeta$. ($\varkappa$ is not the hopping constant $\kappa_N$.)
\end{itemize}
\end{lemma}
\begin{proof}
(a) Let $F$ be a bounded Borel function of the phase-restored configurations $(\om,\psi)$ of $T_\Gamma$ and fix $\psi$. By~\eqref{rv:eq:gauge},
$\Psi(\om,\psi)=(-\psi).(\om,0)$ has the phases $\psi$, and by Lemma~\ref{rv:lem:phase}(a) the map $\om\mapsto\om^\psi$, the unitary-gauge part of
$\Psi(\om,\psi)$, is a bijection of the configurations with closed $v$ which preserves the reference measure (its inverse is the unitary-gauge part of
$\psi.(\,\cdot\,,\psi)$), and the phase-restored weight is invariant: its value at $\Psi(\om,\psi)$ is its value at $(\om,0)$, which is $W(\om)$ (at
zero phases the phase-restored hopping factor is the unitary-gauge one). Hence, with $d\om$ the reference measure of the unitary-gauge variables and
$\wt W$ the phase-restored weight,
\[
\int d\om\,W(\om)\,F\big(\Psi(\om,\psi)\big)=\int d\om'\,\wt W(\om',\psi)\,F(\om',\psi)\qquad\text{for every }\psi .
\]
Integrating over $\psi$ against the normalised Haar measure gives $Z(\Gamma)\,\Psi_*(\mu_\Gamma\times\Haar^{\mathrm{sites}})(F)=\wt Z(\Gamma)\,\wt\mu_\Gamma(F)$;
$F=1$ gives $\wt Z(\Gamma)=Z(\Gamma)$, and then the two laws agree.

(b) The identity is Lemma~\ref{rv:lem:phase}(e), a statement about the local
formulas; it gives $\Psi_*(\mu\times\Haar^{\mathrm{sites}})\circ\tau_a=\Psi_*((\mu\circ\tau_a)\times\Haar^{\mathrm{sites}})$, the Haar measure being
translation invariant.

(c) $\wt\mu_\Gamma$ and $\mu^\zeta_M$ are both the normalised phase-restored weight on the same torus, written in the layers
$c^{(1)},\dots,c^{(M)}$, the temporal links and the temporal Villain integers; the two partition functions differ only by the normalisation of the
reference measure of the temporal links, one factor $\varkappa$ for each of the $n_{\mathrm{lay}}M$ temporal links, and $\wt Z(\Gamma)=Z(\Gamma)$ by (a).
\end{proof}

\begin{lemma}[twisted averaging identity and twisted Schatten--H\"older bound]\label{tv:lem:tw}
Let $\Gamma_s$, $M\ge2$, $\zeta$ be as above, let $T=\mathsf{B}^*\mathsf{B}$, $R_L$, $P_G$ be the transfer operator, the gauge representation and the gauge
projection of Lemma~\ref{rv:lem:zak}(c) for the layer torus $\eN(\Z e_1+\Z e_2)/\Gamma_s$, and let $(\mathsf{S}_\zeta\psi)(c):=\psi(\tau_\zeta^{-1}c)$ on the
layer space. Let $X_s$, $1\le s\le M$, be insertions at the $M$ heights, each fibred (Lemma~\ref{rv:lem:zak}(b)) or bounded on $\cH_{\mathrm{full}}$ ($X_s=1$
where there is none); for an insertion on $\cH_{\mathrm{full}}$ the symbol $\widehat{\bar X_s}$ stands for its gauge average $\bar X_s$ of
Lemma~\ref{rv:lem:fullmap}, which satisfies $\|\bar X_s\|\le\|X_s\|$. Then
\begin{itemize}
\item[(i)] (trace identity) $\mathsf{S}_\zeta$ is unitary and commutes with $T$ and with $P_G$, and
\begin{equation}\label{tv:eq:tw}
Z^\zeta_M(X_1,\dots,X_M)=\Tr\Big(\prod_{s=1}^{M}\mathsf{B}^*\widehat{\bar X_s}\mathsf{B}\cdot P_G\mathsf{S}_\zeta\Big),\qquad\text{in particular}\quad
Z^\zeta_M=\Tr\big(P_GT^M\mathsf{S}_\zeta\big);
\end{equation}
\item[(ii)] (H\"older bound) $|Z^\zeta_M(X_1,\dots,X_M)|\le\prod_s\|X_s\|\cdot\Tr(P_GT^M)=\prod_s\|X_s\|\cdot Z^0_M$, where $Z^0_M=Z_M$ is the partition
function of the untwisted torus with the same layer torus and the same number of layers; in particular $Z^\zeta_M\le Z^0_M$;
\item[(iii)] (insertions) if every fibred $X_s$ satisfies the summability condition of Lemma~\ref{rv:lem:zak}(b), and every $X_s$ on $\cH_{\mathrm{full}}$ is one of the full operators of Lemma~\ref{tv:lem:win}(b) to which Lemma~\ref{tv:lem:win}(c) applies, then, the sum over the variables $\wt v^t$ being taken first as in~\eqref{rv:eq:insert}, $\mu^\zeta_M\big(\prod_sr_{X_s}\circ\Theta^{(s)}_\sigma\big)=Z^\zeta_M(X_1,\dots,X_M)/Z^\zeta_M$, hence
$\big|\mu^\zeta_M\big(\prod_sr_{X_s}\circ\Theta^{(s)}_\sigma\big)\big|\le\prod_s\|X_s\|\cdot Z^0_M/Z^\zeta_M$.
\end{itemize}
\end{lemma}
\begin{proof}
\emph{The shift.} The one-layer weight $W(c)$, the layer reference measure, the slab kernel $K$ and the gauge action are given by the same local
formulas at every cell of the layer torus, so $W(\tau_\zeta c)=W(c)$, the reference measure is $\tau_\zeta$-invariant, $K(\tau_\zeta c,\tau_\zeta c')=K(c,c')$
and $\tau_\zeta(\lambda.c)=(\tau_\zeta\lambda).(\tau_\zeta c)$, where $\tau_\zeta\lambda$ is the translated gauge function. Hence $\mathsf{S}_\zeta$ is unitary on
the layer space $L^2(W(c)\,dc)$, commutes with $Q$ and with $T=Q^{1/2}KQ^{1/2}$, and $\mathsf{S}_\zeta R_L(\lambda)\mathsf{S}_\zeta^{-1}=R_L(\tau_\zeta\lambda)$;
since $\lambda\mapsto\tau_\zeta\lambda$ is an automorphism of the compact group $\Tper^{\mathrm{sites}}$, it preserves the Haar measure, and
$\mathsf{S}_\zeta P_G\mathsf{S}_\zeta^{-1}=\int dG\,R_L(\tau_\zeta G)=P_G$.

\emph{Change of variables.} We repeat the proof of Lemma~\ref{rv:lem:zak}(c) up to the closing slab. Put $G_1:=0$, $G_{s+1}:=G_s-B_t^{(s)}$ and
$W^{(s)}:=G_s.c^{(s)}$ for $1\le s\le M$. In temporal gauge the slab $s$ has the layers $c^{(s)}=(-G_s).W^{(s)}$ and $(-B_t^{(s)}).c^{(s+1)}$. For $s<M$,
$(-B_t^{(s)}).c^{(s+1)}=(-B_t^{(s)}-G_{s+1}).W^{(s+1)}=(-G_s).W^{(s+1)}$, and by Lemma~\ref{rv:lem:zak}(b) the slab contributes, as in that proof,
$\langle\Phi(W^{(s)}),(X_s^{-d\wt G_s})^\wedge\Phi(W^{(s+1)})\rangle$. For the closing slab, $c^{(M+1)}=\tau_\zeta c^{(1)}=\tau_\zeta W^{(1)}$ (as $G_1=0$), so
its upper layer is $(-B_t^{(M)}).\tau_\zeta W^{(1)}=(-G_M).\big(G.\tau_\zeta W^{(1)}\big)$ with the holonomy $G:=G_{M+1}=G_M-B_t^{(M)}$, and it contributes
\[
\big\langle\Phi(W^{(M)}),(X_M^{-d\wt G_M})^\wedge\Phi\big(G.\tau_\zeta W^{(1)}\big)\big\rangle .
\]
For a bounded operator $\hat X$ on $\cH$, the kernel $(c,c'')\mapsto W(c)^{1/2}\langle\Phi(c),\hat X\Phi(G.\tau_\zeta c'')\rangle W(c'')^{1/2}$ is the integral
kernel of $\mathsf{B}^*\hat X\mathsf{B}\,R_L(G)\mathsf{S}_\zeta$: the substitution $c'=G.\tau_\zeta c''$ preserves $W(c)\,dc$ and turns $\psi(c'')$ into
$\psi(\tau_\zeta^{-1}((-G).c'))=(R_L(G)\mathsf{S}_\zeta\psi)(c')$. Integrating out the layers (the trace-class structure and the absolute convergence are
those of that proof; $R_L(G)\mathsf{S}_\zeta$ is unitary) gives
\[
Z^\zeta_M(X_1,\dots,X_M)=\int dG_2\cdots dG_{M+1}\,\Tr\Big(\prod_{s=1}^{M}\mathsf{B}^*(X_s^{-d\wt G_s})^\wedge\mathsf{B}\cdot R_L(G_{M+1})\mathsf{S}_\zeta\Big).
\]

\emph{Averaging.} Conjugate the operator under the trace by $R_L(\lambda)$, $\lambda\in\Tper^{\mathrm{sites}}$; the trace is unchanged. By
$\mathsf{B}R_L(\lambda)=R(\lambda)\mathsf{B}$ and $R(\lambda)^*\hat XR(\lambda)=(X^{d\wt\lambda})^\wedge$ the factor $s$ becomes
$\mathsf{B}^*(X_s^{-d(\wt G_s-\wt\lambda)})^\wedge\mathsf{B}$ (for $s=1$: $\mathsf{B}^*(X_1^{d\wt\lambda})^\wedge\mathsf{B}$), and
\[
R_L(\lambda)^{-1}R_L(G_{M+1})\mathsf{S}_\zeta R_L(\lambda)=R_L(G_{M+1}-\lambda+\tau_\zeta\lambda)\mathsf{S}_\zeta.
\]
Integrate over $\lambda$ against the Haar
measure and, at fixed $\lambda$, substitute $G_s':=G_s-\lambda$ ($2\le s\le M$) and $G':=G_{M+1}-\lambda+\tau_\zeta\lambda$, a Haar-preserving bijection:
$\lambda,G_2',\dots,G_M',G'$ are then independent Haar variables, one for each height and one for the holonomy. The integrand is multilinear in the
insertions and linear in $R_L(G')$; exchanging the integrals with the trace (dominated convergence for trace-class operators, as in the proof of Lemma~\ref{rv:lem:zak}(c))
and using $\int d\lambda\,(X^{\pm d\wt\lambda})^\wedge=\widehat{\bar X}$ (Lemma~\ref{rv:lem:zak}(b), Lemma~\ref{rv:lem:frame}(c)) and
$\int dG'\,R_L(G')=P_G$ gives~\eqref{tv:eq:tw}. With $X_s=1$ for all $s$ it reads $Z^\zeta_M=\Tr(T^MP_G\mathsf{S}_\zeta)=\Tr(P_GT^M\mathsf{S}_\zeta)$. For
insertions on $\cH_{\mathrm{full}}$ the same computation and the H\"older step below apply with $R\otimes U$ in place of $R$ and with $P^{\cH}_G=\int dG\,(R\otimes U)(G)$, since they use of Lemma~\ref{rv:lem:fullmap} only the covariance of
the feature map and the measure preservation.

\emph{H\"older.} $\|\widehat{\bar X_s}\|\le\|\bar X_s\|\le\|X_s\|$ (Lemma~\ref{rv:lem:zak}(b), Lemma~\ref{rv:lem:frame}(c)). $\bar X_s$ is gauge
invariant, so $\widehat{\bar X_s}$ commutes with every $R(\lambda)$ and with the orthogonal projection $P^{\cH}_G=\int dG\,R(G)$; $P_G$ commutes with
every $\mathsf{B}^*\widehat{\bar X_s}\mathsf{B}$, and $P_G\mathsf{B}^*=\mathsf{B}^*P^{\cH}_G$ (proof of Lemma~\ref{rv:lem:zak}(c)). Hence, with $P_G=P_G^M$,
\[
\prod_{s=1}^M\mathsf{B}^*\widehat{\bar X_s}\mathsf{B}\cdot P_G=\prod_{s=1}^MA_s,\qquad
A_s:=P_G\mathsf{B}^*\widehat{\bar X_s}\mathsf{B}=(P^{\cH}_G\mathsf{B})^*\,\widehat{\bar X_s}\,(P^{\cH}_G\mathsf{B}).
\]
By H\"older's inequality for Schatten norms \cite{Sim05} (exponent $M$ on each $A_s$, the operator norm on the unitary $\mathsf{S}_\zeta$),
$|\Tr(A_1\cdots A_M\mathsf{S}_\zeta)|\le\prod_s\|A_s\|_M$, and $\|A_s\|_M\le\|\widehat{\bar X_s}\|\,\|P^{\cH}_G\mathsf{B}\|_{2M}^2$ with
\[
\|P^{\cH}_G\mathsf{B}\|_{2M}^{2M}=\Tr\big((\mathsf{B}^*P^{\cH}_G\mathsf{B})^M\big)=\Tr\big((P_GT)^M\big)=\Tr(P_GT^M).
\]
So
$|Z^\zeta_M(X_1,\dots,X_M)|\le\prod_s\|X_s\|\cdot\Tr(P_GT^M)$, and $\Tr(P_GT^M)=Z_M=Z^0_M$ by~\eqref{rv:eq:avg} with all insertions equal to $1$. This is
(ii); without insertions it gives $Z^\zeta_M=|Z^\zeta_M|\le Z^0_M$.

\emph{Insertions.} The identity~\eqref{rv:eq:insert} rests on the statement preceding it, that the insertion of $X$ at one slab multiplies the weight of
that slab by $r_X\circ\Theta_\sigma$; this concerns one slab at a time and holds for every slab of $\mu^\zeta_M$, the closing one included (its slab data
are those of the pair $c^{(M)}$, $\tau_\zeta c^{(1)}$). For a full operator $X$ of Lemma~\ref{tv:lem:win}(b),(c) the same statement holds: by~\eqref{rv:eq:char} and Lemma~\ref{rv:lem:fullmap} its kernel $\langle\Phi_{\mathrm{full}}(c),X\Phi_{\mathrm{full}}(\wt c')\rangle$ is the sum over $\wt v^t$ of the pair kernel $k_X$, absolutely convergent by Lemma~\ref{tv:lem:win}(c), whose proof does not use the present lemma. This gives the identity in (iii), and the bound follows from (ii).
\end{proof}

\begin{lemma}[several windows at one height; the full class]\label{tv:lem:win}
Let $S_1,\dots,S_\ell$ be pairwise disjoint finite spatial bond sets at one slab, and $S:=S_1\cup\dots\cup S_\ell$.
\begin{itemize}
\item[(a)] (Bond class.) For each $i$ let $X^{(i)}$ be a fibred operator on $L^2(\R^{S_i})$ (Lemma~\ref{rv:lem:zak}(b)). Then $X:=\bigotimes_iX^{(i)}$ on
$L^2(\R^S)=\bigotimes_iL^2(\R^{S_i})$ is fibred (Lemma~\ref{rv:lem:zak}(b2)), hence bounded and $\Z$-periodic, $\|X\|=\prod_i\|X^{(i)}\|$, and
\[
k_X(B,u)=\prod_ik_{X^{(i)}}(B_{S_i},u_{S_i}),\qquad r_X=\prod_ir_{X^{(i)}} ;
\]
if every $X^{(i)}$ has the absolute convergence required in Remark~\ref{rv:rem:abs}, so has $X$.
\item[(b)] (Full class.) Assume moreover that every $S_i$ is closed under the bonds of its plaquettes and that the site sets $V(S_i)$ are pairwise
disjoint. Let $V$ be a finite set of sites of the layer containing $\bigcup_iV(S_i)$ (on a layer torus: all its sites), and let $\cH_{\mathrm{full}}$,
$\Phi_{\mathrm{full}}$ be defined as in Lemma~\ref{rv:lem:fullmap} with this $V$; that lemma is stated for the site set of one window, and its statement
and proof hold for every finite $V$. For each $i$ let $X^{(i)}$ be a finite linear combination of operators $T_s(X'\otimes X'_H)$ with
$s\in[0,1)^{\mathrm{plaq}(S_i)}$, $X'$ fibred on $L^2(\R^{S_i})$ (Lemma~\ref{rv:lem:zak}(b)) with $\sum_{a\in\cper\Z^{S_i}}|\langle h^B,M_sX'h^{u-a}\rangle|<\infty$ for all $B,u\in\R^{S_i}$, as in Lemma~\ref{rv:lem:char}, and $X'_H$ bounded on $L^2(\C^{V(S_i)})$, acting on
$\cH_{\mathrm{full}}$ as the identity on the bonds outside $S_i$ and on the sites outside $V(S_i)$. Then the $X^{(i)}$ commute, the \emph{full operator}
$X:=\prod_iX^{(i)}$ satisfies $\|X\|\le\prod_i\|X^{(i)}\|$, and
\[
k_X=\prod_ik_{X^{(i)}},\qquad r_X=\prod_ir_{X^{(i)}} ,
\]
where $k_X$, $r_X$ are taken over the union of the windows (the pair coordinates over $S$, the Villain integers over the plaquettes of the $S_i$ and the
Higgs coordinates over $\bigcup_iV(S_i)$) and $k_{X^{(i)}}$, $r_{X^{(i)}}$ over the window $S_i$.
\item[(c)] (Absolute convergence in the full class.) The operators $X_t=1+tY+t^2Z$ of Lemma~\ref{rv:lem:step2full}, with $Y\in\cC_{\mathrm{full}}$ and
$Z=\delta+Z_b\otimes Z_H$, are of the form in (b), and the sum over $\wt v^t$ of Remark~\ref{rv:rem:abs} converges absolutely for them and for the full
operators built from them in (b).
\end{itemize}
\end{lemma}
\begin{proof}
(a) The Gaussian $h^x(\xi)=(2/\pi)^{|S|/4}e^{-|\xi-x|^2}$ is a product over the bonds, $h^x=\bigotimes_ih^{x_{S_i}}$, so
$\langle h^B,Xh^u\rangle=\prod_i\langle h^{B_{S_i}},X^{(i)}h^{u_{S_i}}\rangle$; for $X^{(i)}=1$ this is the factorisation of $k_1$, and the one of
$r_X=k_X/k_1$ follows. The translations by $\cper\Z^S$ are tensor products of translations, which gives the $\Z$-periodicity; the norm of a tensor product
is the product of the norms; and the sum over $\wt v^t\in\cper\Z^S$ of $|k_X(B,\wt B'-\wt v^t)|$ is the product of the sums over the windows. (This is
the tensoring used in Lemma~\ref{rv:lem:UV}.)

(b) Consider first one term $\Xi_i=T_{s_i}(X_i\otimes X_{H,i})$ for each window. One has $T_sT_{s'}=T_{s+s'}$. For $i'\ne i$, $T_{s_{i'}}$ commutes with the
decomposable operator $\hat X_i$: its fibres are $X_i((\wt d\alpha)_{S_i})$ (Lemma~\ref{rv:lem:zak}(b)), and $T_{s_{i'}}$ shifts $\alpha$ by $s_{i'}$,
hence $\wt d\alpha$ by $\wt ds_{i'}$, which vanishes on $S_i$, because $s_{i'}$ is supported on $\mathrm{plaq}(S_{i'})$, the plaquettes of
$\mathrm{plaq}(S_{i'})$ have all their bonds in $S_{i'}$, and $S_{i'}\cap S_i=\emptyset$. The Higgs factors $X_{H,i}$ act on different tensor factors of
$L^2(\C^V)$ and commute with all the $T_s$ and $\hat X_{i'}$. Hence $\prod_i\Xi_i=T_s\big((\bigotimes_iX_i)\otimes(\bigotimes_iX_{H,i})\big)$ with
$s:=\sum_is_i$, in any order; in particular the $X^{(i)}$ commute. By~\eqref{rv:eq:char} and Lemma~\ref{rv:lem:fullmap},
\[
k_{T_s(X'\otimes X'_H)}(c,\wt c')=e_s(v)\,\langle h^B,M_sX'h^u\rangle\,k_{X'_H}(\Phi,\wt\Phi').
\]
For $s=\sum_is_i$: $e_s(v)=\prod_ie_{s_i}(v)$, the $i$-th factor depending on $v$ over $\mathrm{plaq}(S_i)$ only; $\chi_s=\prod_i\chi_{s_i}$, where
$\chi_{s_i}$ depends on the coordinates of the bonds of $S_i$ only ($\wt ds_i$ is supported on $S_i$), so $M_s=\bigotimes_iM_{s_i}$; $h^B$ and $h^u$
factorise over the disjoint bond sets as in (a), and $h^\Phi$, $h^{\wt\Phi'}$ over the disjoint site sets. Therefore $k_{\prod_i\Xi_i}=\prod_ik_{\Xi_i}$. Both
sides are multilinear in the $X^{(i)}$, so $k_X=\prod_ik_{X^{(i)}}$; with $X^{(i)}=1$ this is the factorisation of $k_1$, and $r_X=\prod_ir_{X^{(i)}}$
follows. The norm bound is the submultiplicativity of the operator norm.

(c) $Y\in\cC_{\mathrm{full}}$ is a finite linear combination of operators $T_s(X'\otimes X'_H)$ with $X'\in\cC$, $X'_H\in\cC_H$ (Lemma~\ref{rv:lem:fullclass}),
and $Z_b\otimes Z_H=T_0(Z_b\otimes Z_H)$, $1=T_0(1\otimes1)$; so $X_t$ is of the form in (b). For one term,
$|k_{T_s(X'\otimes X'_H)}|=|\langle h^B,M_sX'h^u\rangle|\,|k_{X'_H}(\Phi,\wt\Phi')|$: the Higgs factor does not depend on $\wt v^t$, the character $e_s(v)$
is unimodular, and $M_s$ is the multiplication by a unimodular function, which commutes with the localisations $\tau_a^*P_K\tau_a$ in the proof of
Lemma~\ref{rv:lem:growth} and preserves norms, so the Gaussian bound of that proof, which gives the absolute convergence for the class $\cC$
(Remark~\ref{rv:rem:abs}), holds for $\langle h^B,M_sX'h^u\rangle$ as for $\langle h^B,X'h^u\rangle$. For the term $Z_b\otimes Z_H$ the sum over $\wt v^t$ is
that of the bond corrector $Z_b$, which converges by Remark~\ref{rv:rem:abs}, and for the term $1$ it is a Gaussian sum. For the full operators of (b) the sum over $\wt v^t\in\cper\Z^S$ factorises
over the windows.
\end{proof}

\begin{lemma}[free energy of the product tori: halving a side]\label{tv:lem:fe}
Let $a_1,a_2,a_3\in\eN\mathbb N$, each at least $4\eN$.
\begin{itemize}
\item[(i)] (Symmetry.) $Z(a_1,a_2,a_3)$ is symmetric under the permutations of $(a_1,a_2,a_3)$.
\item[(ii)] (Halving.) $Z(a_1,a_2,2a_3)\le Z(a_1,a_2,a_3)^2$, and likewise in the first and in the second argument.
\item[(iii)] (Cubic tori.) Let $\mathsf{p}(L):=(\eN/L)^3\log Z^{\mathrm{cub}}_L$ be the free energy per site of the cubic torus of side $L\in\eN\mathbb N$,
$L\ge4\eN$. Then $\mathsf{p}(2L)\le\mathsf{p}(L)$, and $\mathsf{p}(L)\ge\log c_0$ with the constant $c_0>0$ of Lemma~\ref{lem:d2:mom} if moreover
$L/2\in\eN\Z$. Consequently, along a dyadic sequence $(L_n)$, the sequence $\mathsf{p}(L_n)$, $n\ge2$, is non-increasing and bounded below, and
\[
\eta_n:=\mathsf{p}(L_{n-1})-\mathsf{p}(L_n)\ \ge\ 0\qquad(n\ge3)\qquad\text{tends to }0\text{ as }n\to\infty .
\]
\item[(iv)] (With the volumes.) For $L$ with $L/2\in\eN\mathbb N$, $L/2\ge4\eN$,
\[
\begin{gathered}
\log Z\big(\tfrac L2,2L,L\big)\ \le\ 8\log Z^{\mathrm{cub}}_{L/2},\\
\text{so that}\qquad\log Z\big(\tfrac L2,2L,L\big)-\log Z^{\mathrm{cub}}_L\ \le\ \Big(\frac{L}{\eN}\Big)^3\big(\mathsf{p}(L/2)-\mathsf{p}(L)\big).
\end{gathered}
\]
\end{itemize}
\end{lemma}
\begin{proof}
(i) is~\eqref{tv:eq:cov} for the permutations of the coordinate axes.

(ii) By (i) it suffices to treat the third argument. Let $\Gamma_s:=a_1e_1\Z+a_2e_2\Z$ and $M:=a_3/\eN\ge4$. The positivity used is that of the transfer
operator, as it is given in Lemma~\ref{rv:lem:zak}(c) and its proof, for the layer torus $\eN(\Z e_1+\Z e_2)/\Gamma_s$: $T=\mathsf{B}^*\mathsf{B}\ge0$
is trace class; $P_G$ is an orthogonal projection (the average of a unitary representation of a compact group) which commutes with $T$ (it commutes
with every $\mathsf{B}^*\widehat{\bar X}\mathsf{B}$, in particular for $X=1$); and $Z_{M'}=\Tr(P_GT^{M'})$ for every number $M'\ge2$ of layers. Hence
$T_G:=P_GT=P_GTP_G$ is self-adjoint, non-negative and trace class, $\Tr(P_GT^{M'})=\Tr(T_G^{M'})=\sum_i\lambda_i^{M'}$ with $\lambda_i\ge0$ the eigenvalues
of $T_G$, and
\[
\Tr(T_G^{2M})=\sum_i\lambda_i^{2M}\ \le\ \Big(\sum_i\lambda_i^{M}\Big)^2=\big(\Tr(T_G^{M})\big)^2 .
\]
By Lemma~\ref{tv:lem:pr}(c) with $\zeta=0$, $Z(a_1,a_2,M'\eN)=\varkappa^{\,n_{\mathrm{lay}}M'}\Tr(T_G^{M'})$ for $M'\ge4$; the two sides of the claim carry
the same power of $\varkappa$, so
\[
Z(a_1,a_2,2a_3)=\varkappa^{\,2n_{\mathrm{lay}}M}\Tr(T_G^{2M})\le\varkappa^{\,2n_{\mathrm{lay}}M}\Tr(T_G^M)^2=Z(a_1,a_2,a_3)^2.
\]

(iii) Three applications of (ii) give $Z(2L,2L,2L)\le Z(2L,2L,L)^2\le Z(2L,L,L)^4\le Z(L,L,L)^8$. The torus of side $2L$ has $8(L/\eN)^3$ sites, so
\[
\mathsf{p}(2L)=\tfrac18(\eN/L)^3\log Z^{\mathrm{cub}}_{2L}\le(\eN/L)^3\log Z^{\mathrm{cub}}_L=\mathsf{p}(L).
\]
The lower bound is the last line of Step~1 of
the proof of Lemma~\ref{lem:d2:mom}: $Z^{\mathrm{cub}}_L\ge c_0^{\,(L/\eN)^3}$ for $L\ge4\eN$ with $L/2\in\eN\Z$ (keep the term $v\equiv0$ and restrict
the integrations). Along a dyadic sequence $L_n=2L_{n-1}$, and $L_n\ge4\eN$, $L_n/2\in\eN\Z$ for $n\ge2$ (as $L_*\in\eN\mathbb N$); so $\mathsf{p}(L_n)$,
$n\ge2$, is non-increasing and bounded below by $\log c_0$, hence convergent, and $\eta_n\to0$ (indeed $\sum_{n\ge3}\eta_n\le\mathsf{p}(L_2)-\log c_0$).

(iv) By (ii) in the second argument, twice, and then in the third:
\[
Z(\tfrac L2,2L,L)\le Z(\tfrac L2,L,L)^2\le Z(\tfrac L2,\tfrac L2,L)^4\le
Z(\tfrac L2,\tfrac L2,\tfrac L2)^8=(Z^{\mathrm{cub}}_{L/2})^8.
\]
The cubic torus of side $L/2$ has $\tfrac18(L/\eN)^3$ sites, so
$8\log Z^{\mathrm{cub}}_{L/2}=(L/\eN)^3\mathsf{p}(L/2)$, while $\log Z^{\mathrm{cub}}_L=(L/\eN)^3\mathsf{p}(L)$.
\end{proof}

\begin{lemma}[the adapted torus in the three coordinate directions; the twist costs $o(L^3)$]\label{tv:lem:tws}
Let $g_3:=\mathrm{id}$ and, for $j=1,2$, let $g_j\in B_3$ be the transposition of the coordinates $j$ and $3$. Let $L$ with $L/2\in\eN\mathbb N$,
$L/2\ge4\eN$.
\begin{itemize}
\item[(i)] (Geometry.) $g_j\Gamma(L)=\Gamma(\Gamma_{s,j},M_j,\zeta_j)$ for $j=1,2,3$, with
\[
\begin{array}{llll}
j=3:\quad&\Gamma_{s,3}=\tfrac L2(e_1-e_2)\Z+L(e_1+e_2)\Z,&M_3=L/\eN,&\zeta_3=0;\\[2pt]
j=1:\quad&\Gamma_{s,1}=Le_1\Z+2Le_2\Z,&M_1=L/(2\eN),&\zeta_1=-\tfrac L2e_2;\\[2pt]
j=2:\quad&\Gamma_{s,2}=2Le_1\Z+Le_2\Z,&M_2=L/(2\eN),&\zeta_2=-\tfrac L2e_1.
\end{array}
\]
For $j=1,2$ the untwisted torus with the same layer torus and the same number of layers, $T_{\Gamma(\Gamma_{s,j},M_j,0)}$, is the product torus with sides
$(L,2L,\tfrac L2)$, resp.\ $(2L,L,\tfrac L2)$; we write $Z^{\mathrm{rect}}_L:=Z(\tfrac L2,2L,L)$ for its partition function (Lemma~\ref{tv:lem:fe}(i)).
\item[(ii)] (Twist comparison.) For $j=1,2$, in the notation of Lemma~\ref{tv:lem:tw} for $(\Gamma_{s,j},M_j,\zeta_j)$,
\[
1\ \le\ \frac{Z^0_{M_j}}{Z^{\zeta_j}_{M_j}}=\frac{Z^{\mathrm{rect}}_L}{Z^{\mathrm{tw}}_L}\ \le\ \exp\Big[\Big(\frac{L}{\eN}\Big)^3\big(\mathsf{p}(L/2)-\mathsf{p}(L)\big)\Big].
\]
\item[(iii)] (The surface term of the fold.) With $\phi_L=\log(Z^{\mathrm{tw}}_L/Z^{\mathrm{cub}}_L)$ as in Proposition~\ref{n1:prop:fold},
\[
0\ \le\ \phi_L\ \le\ \Big(\frac{L}{\eN}\Big)^3\big(\mathsf{p}(L/2)-\mathsf{p}(L)\big).
\]
\end{itemize}
Along a dyadic sequence, for all large $n$: $0\le\log Z^{\mathrm{rect}}_{L_n}-\log Z^{\mathrm{tw}}_{L_n}\le(L_n/\eN)^3\eta_n$ and
$0\le\phi_{L_n}\le(L_n/\eN)^3\eta_n$, with $\eta_n\to0$ (Lemma~\ref{tv:lem:fe}(iii)).
\end{lemma}
\begin{proof}
(i) $\Gamma(L)=\tfrac L2(e_1-e_2)\Z+L(e_1+e_2)\Z+Le_3\Z$ (Definition~\ref{n1:def:tw}), which is the case $j=3$. The transposition $g_1$ gives
$g_1\Gamma(L)=\tfrac L2(e_3-e_2)\Z+L(e_3+e_2)\Z+Le_1\Z$, and $L(e_3+e_2)=2\cdot\tfrac L2(e_3-e_2)+2Le_2$, so
$g_1\Gamma(L)=Le_1\Z+2Le_2\Z+\Z\big(\tfrac L2e_3-\tfrac L2e_2\big)$. The transposition $g_2$ gives
$g_2\Gamma(L)=\tfrac L2(e_1-e_3)\Z+L(e_1+e_3)\Z+Le_2\Z$, and $L(e_1+e_3)=2Le_1-2\cdot\tfrac L2(e_1-e_3)$, so
$g_2\Gamma(L)=2Le_1\Z+Le_2\Z+\Z\big(\tfrac L2e_3-\tfrac L2e_1\big)$. The untwisted tori are $\eN\Z^3/(Le_1\Z+2Le_2\Z+\tfrac L2e_3\Z)$ and
$\eN\Z^3/(2Le_1\Z+Le_2\Z+\tfrac L2e_3\Z)$; all these tori have $(L/\eN)^3$ sites.

(ii) By Lemma~\ref{tv:lem:pr}(c), \eqref{tv:eq:cov} and (i), $\varkappa^{(L/\eN)^3}Z^{\zeta_j}_{M_j}=Z(g_j\Gamma(L))=Z(\Gamma(L))=Z^{\mathrm{tw}}_L$ and
$\varkappa^{(L/\eN)^3}Z^0_{M_j}=Z^{\mathrm{rect}}_L$, which is the equality. The lower bound is Lemma~\ref{tv:lem:tw}(ii) without insertions:
$Z^{\mathrm{tw}}_L\le Z^{\mathrm{rect}}_L$. For the upper bound, Proposition~\ref{n1:prop:fold} gives $Z^{\mathrm{cub}}_L\le Z^{\mathrm{tw}}_L$ (its statement
contains $\phi_L\ge0$; it applies, as $L\ge8\eN$, to the datum $(0,0,0;\tfrac L2;0)$ with zero source), and Lemma~\ref{tv:lem:fe}(iv) gives
$\log Z^{\mathrm{rect}}_L-\log Z^{\mathrm{cub}}_L\le(L/\eN)^3(\mathsf{p}(L/2)-\mathsf{p}(L))$.

(iii) $\phi_L\ge0$ is part of Proposition~\ref{n1:prop:fold}, and by (ii) and Lemma~\ref{tv:lem:fe}(iv),
\[
\phi_L=\log Z^{\mathrm{tw}}_L-\log Z^{\mathrm{cub}}_L\le\log Z^{\mathrm{rect}}_L-\log Z^{\mathrm{cub}}_L\le(L/\eN)^3(\mathsf{p}(L/2)-\mathsf{p}(L)).
\]

Along a dyadic sequence $L_n/2=L_{n-1}$, so $\mathsf{p}(L_n/2)-\mathsf{p}(L_n)=\eta_n$, and the hypotheses on $L=L_n$ hold for all large $n$.
\end{proof}

\begin{lemma}[bounded Borel local functions converge along the limits]\label{tv:lem:conv}
In the setting of Subsection~\ref{n1:sub:lim}, for $k\in\mathbb N$ let $\nu_{1/k}$ be the weak limit of $(\mu^{1/k}_{L_n})_n$ along a subsequence of $(L_n)$ (which
may depend on $k$), and let $\nu_0$ be the weak limit of $(\nu_{1/k})_k$ along a subsequence (Lemma~\ref{n1:lem:gibbs}). Let $f$ be a bounded Borel
function on $E$ of the variables of finitely many cells. Then $\mu^{1/k}_{L_n}(f)\to\nu_{1/k}(f)$ along the subsequence defining $\nu_{1/k}$, for every
$k$, and $\nu_{1/k}(f)\to\nu_0(f)$ along the subsequence defining $\nu_0$.
\end{lemma}
\begin{proof}
Let $\Delta$ be a finite set of cells containing the cells of $f$, so that $f$ is a function of the $\Delta$-variables only, and write
$\gamma^{1/k}_\Delta$ for the kernel of Subsection~\ref{n1:sub:lim} with the source strength $1/k$. For $L=L_n$ larger than the diameter of $\Delta$ and of the cells
of the $O_c$, $c\in I_\Delta$, integrating the $\Delta$-variables first under $\mu^{1/k}_L$ gives $\mu^{1/k}_L(f)=\mu^{1/k}_L(\gamma^{1/k}_\Delta f)$: this is the
first step of the proof of Lemma~\ref{n1:lem:gibbs}(i), an application of Fubini's theorem which uses of $f$ only that it is bounded and Borel. The
function $\gamma^{1/k}_\Delta f$ depends on the outside variables of finitely many cells only, and it is bounded and continuous: this is the Feller argument
of the same proof (the factors of the kernel are continuous in the outside variables, $Z_\Delta(\xi_{\mathrm{out}})\in(0,\infty)$, dominated convergence),
in which $f$ enters only as a bounded Borel function of the integration variables. Hence, by the weak convergence and by Lemma~\ref{n1:lem:gibbs}(i)
for the bounded Borel $f$, along the subsequence defining $\nu_{1/k}$,
\[
\mu^{1/k}_{L_n}(f)=\mu^{1/k}_{L_n}(\gamma^{1/k}_\Delta f)\ \longrightarrow\ \nu_{1/k}(\gamma^{1/k}_\Delta f)=\nu_{1/k}(f).
\]

For the second limit, $\nu_{1/k}(f)=\nu_{1/k}(\gamma^{1/k}_\Delta f)$ by Lemma~\ref{n1:lem:gibbs}(i),
$\|\gamma^{1/k}_\Delta f-\gamma_\Delta f\|_\infty\le2\|f\|_\infty(e^{2S_\Delta/k}-1)$ by the comparison of the two normalised weights in the proof of
Lemma~\ref{n1:lem:gibbs}(iii) (valid for bounded Borel $f$), and $\gamma_\Delta f$ is bounded and continuous (the Feller argument without source). So,
along the subsequence defining $\nu_0$, $\nu_{1/k}(f)-\nu_{1/k}(\gamma_\Delta f)\to0$ and $\nu_{1/k}(\gamma_\Delta f)\to\nu_0(\gamma_\Delta f)=\nu_0(f)$, the
last equality by Lemma~\ref{n1:lem:gibbs}(iii).
\end{proof}

\begin{proposition}[the tangent inequality of the comparison states, without relative entropy]\label{tv:prop:T}
Let $\nu_0=\nu_0(D)$ be a comparison state as in Lemma~\ref{n1:lem:gibbs}(iii), formed along a dyadic sequence $(L_n)$ with a datum $D=(c_0,c_+,c_3;h;O)$.
For $j\in\{1,2,3\}$ let $g_j$ be as in Lemma~\ref{tv:lem:tws} and let
\[
\nu^{(j)}:=\Psi_*\big((\nu_0\circ\hat g_j)\times\Haar^{\mathrm{sites}}\big)
\]
be the phase restoration (Lemma~\ref{rv:lem:phase}(d)) of the Gibbs measure $\nu_0\circ\hat g_j$ (Proposition~\ref{rv:prop:YM4}(i)): for $j=3$ the phase
restoration of $\nu_0$ itself; for $j=1,2$ its time slabs are the slabs of $\nu_0$ between adjacent layers orthogonal to $e_j$. Let $\nu=\nu^{(j)}$ for
some $j\in\{1,2,3\}$, let $[n_0,n_0+1]$ be a slab between two adjacent time layers and $t\ne0$ real. Then:
\begin{itemize}
\item[(a)] for every finite spatial bond set $S$, every real skew-adjoint $Y$ of the class of Lemma~\ref{rv:lem:growth} and the corrector $Z$ of
Lemma~\ref{rv:lem:corr}, the slab function $v_t=\log(1+tr_Y+t^2r_Z)\circ\Theta_\sigma$ of Lemma~\ref{rv:lem:step2} satisfies
\begin{equation}\label{tv:eq:T}
\nu(v_t)\ \le\ \log\|X_t\|,\qquad X_t=1+tY+t^2Z;
\end{equation}
\item[(b)] for every finite spatial bond set $S$ closed under the bonds of its plaquettes, every real skew-adjoint $Y\in\cC_{\mathrm{full}}$ and the
corrector $Z=\delta+Z_b\otimes Z_H$ of the proof of Lemma~\ref{rv:lem:step2full}, the slab function $v_t$ of that proof satisfies~\eqref{tv:eq:T}.
\end{itemize}
\end{proposition}

The inequality~\eqref{tv:eq:T} is the upper bound which the proof of Lemma~\ref{rv:lem:step2} obtains from Lemma~\ref{rv:lem:TV},
\eqref{rv:eq:insert} and Lemma~\ref{rv:lem:UV}, and which the proof of Lemma~\ref{rv:lem:step2full} obtains from these and Lemma~\ref{rv:lem:fullmap}; here it is obtained in each direction $e_j$ from the construction of $\nu_0$.

\begin{proof}
Fix $j$ and put $g:=g_j$; fix the case (a) or (b), $S$, the slab, $Y$, $Z$ and $t\ne0$, and write $X:=X_t$ and $v:=v_t=\log r_X\circ\Theta_\sigma$. By
Lemma~\ref{rv:lem:corr}, resp.\ the proof of Lemma~\ref{rv:lem:step2full}, $r_X\ge\delta t^2>0$, so $v\ge\log(\delta t^2)$. The function $v$ depends on
the phase-restored variables of a finite set $\Delta_S$ of cells of the slab (the sites, bonds and plaquettes of the window in the two layers, the
temporal links and the temporal plaquettes between them), and it is Borel ($r_X$ is continuous in the pair and Higgs coordinates of the slab for each
value of the integer variables, and $\Theta_\sigma$ is a Borel map). Let $h=h_{j_0}$ be the wall spacing of $D$.

\emph{Step 1 (reduction to bounded Borel functions on $E$).} For $m\in\mathbb N$ let $v\wedge m:=\min(v,m)$ and
\[
f_m(\om):=\int\Haar(d\psi)\,(v\wedge m)\big(\Psi(\hat g\om,\psi)\big),
\]
the integral being over the finitely many phases on which the integrand depends. Since $\Psi$ and $\hat g$ are local, $f_m$ is a bounded Borel function
on $E$ of the variables of finitely many cells, and by the definition of $\nu=\nu^{(j)}$ and Fubini's theorem $\nu(v\wedge m)=\nu_0(f_m)$. As
$v\wedge m\uparrow v$ and $v\wedge m\ge\min(\log(\delta t^2),1)$, monotone convergence gives
\begin{equation}\label{tv:eq:P1}
\nu(v)=\sup_m\nu_0(f_m)\in(-\infty,+\infty].
\end{equation}
It therefore suffices to prove $\nu_0(f_m)\le\log\|X\|$ for every $m$.

\emph{Step 2 (the torus laws).} Let $k\in\mathbb N$, $n\ge j_0$ and $L=L_n$, so large that the cells of $f_m$ embed in $T^{\mathrm{tw}}_L$ and
Lemma~\ref{tv:lem:tws} applies. Let $\mu':=\mu^{1/k}_L\circ\hat g$, a law on the configurations of the torus $T_{g\Gamma(L)}$, and
$\wt\mu':=\Psi_*(\mu'\times\Haar^{\mathrm{sites}})$. The computation of Step~1, on the torus, gives
\begin{equation}\label{tv:eq:P2}
\mu^{1/k}_L(f_m)=\wt\mu'(v\wedge m)\ \le\ \wt\mu'(v).
\end{equation}
We bound $\wt\mu'(v)$.

\emph{Step 3 (a sublattice of periods; the periodised sum).} Let $\Lambda':=h(e_1-e_2)\Z+h(e_1+e_2)\Z+2he_3\Z$. By Lemma~\ref{n1:lem:walls} the products
of the reflections in two adjacent parallel walls of $D$ are the translations by $h(e_1-e_2)$, $h(e_1+e_2)$ and $2he_3$; they belong to $\cW$, so
$\mu^{1/k}_L$ is invariant under the translations by $\Lambda'$ (Lemma~\ref{n1:lem:Winv}(i)). Every non-zero element of $\Lambda'$ has length at least
$h$. Let $d_S$ be the diameter of $\Delta_S$, choose an integer $q\ge0$ with $2^qh>d_S+2\eN$, and put
\[
\Gamma':=2^q\,g\Lambda',\qquad I':=[\eN\Z^3:\Gamma']=4\cdot8^q\,(h/\eN)^3 .
\]
$\Gamma(L)\subset2^q\Lambda'$ holds if and only if $2^qh$ divides $L/2$, that is, for $L=L_n$ with $n\ge j_0+q$ ($L_n/2=2^{n-j_0}h$); we assume this
from now on. Then $g\Gamma(L)\subset\Gamma'$, $n_L:=[\Gamma':g\Gamma(L)]=(L/\eN)^3/I'$, and $\mu'$ is invariant under the translations by
$g\Lambda'\supset\Gamma'$ ($\tau_{ga}\circ\hat g=\hat g\circ\tau_a$); so is $\wt\mu'$ (Lemma~\ref{tv:lem:pr}(b)).

For $\gamma\in\Gamma'$ the function $v\circ\tau_\gamma$ on the phase-restored configurations of $T_{g\Gamma(L)}$ depends only on the class of $\gamma$ modulo
$g\Gamma(L)$. It is the slab function $\log r_{X^\gamma}\circ\Theta^{(s)}_\sigma$ of the translated slab, of height $s=s(\gamma)\in\{1,\dots,M_j\}$, and of the
translated window $S^\gamma$, where $X^\gamma$ is the copy of $X$ on $S^\gamma$, an operator of the same class and the same norm. Let
\[
V_L:=\sum_{\gamma\in\Gamma'/g\Gamma(L)}v\circ\tau_\gamma ,
\]
a sum of $n_L$ terms, bounded below. The cell sets $\Delta_S+\gamma$ of distinct classes are at distance at least $2\eN$ from each other on
$T_{g\Gamma(L)}$: if $\Delta_S+\gamma$ and $\Delta_S+\gamma'+\gamma''$, $\gamma''\in g\Gamma(L)$, are at distance less than $2\eN$ in $\eN\Z^3$, then the element
$\gamma'+\gamma''-\gamma$ of $\Gamma'$ has length less than $d_S+2\eN<2^qh$, hence is zero. In particular the windows $S^\gamma$ at one height are pairwise
disjoint, and so are their site sets. By the invariance of $\wt\mu'$ (applied to the truncations of $v$, then monotone convergence),
\begin{equation}\label{tv:eq:P3}
\wt\mu'(v)=n_L^{-1}\,\wt\mu'(V_L),
\end{equation}
both sides in $(-\infty,+\infty]$.

\emph{Step 4 (Jensen's inequality; removal of the source).} For a probability measure, the integral of a function bounded below is at most the logarithm of the integral of its exponential, both
sides in $(-\infty,+\infty]$ (Jensen's inequality for the truncations of the function from above, then monotone convergence on the left). Thus
$\wt\mu'(V_L)\le\log\wt\mu'(e^{V_L})$. The density of $\mu^{1/k}_L$ with respect to $\mu^{\mathrm{tw}}_L$ is $Z_{1/k}^{-1}e^{O_d/k}\le e^{2n_c\|O\|_\infty/k}$
(Definition~\ref{n1:def:datum}; this is the removal of the source in the proof of Lemma~\ref{n1:lem:block}), and $\mu^{\mathrm{tw}}_L\circ\hat g=\mu_{g\Gamma(L)}$
by~\eqref{tv:eq:cov}. Hence, for every Borel $F\ge0$ on the phase-restored configurations of $T_{g\Gamma(L)}$,
\[
\wt\mu'(F)=\int\mu^{1/k}_L(d\om)\int\Haar(d\psi)\,F\big(\Psi(\hat g\om,\psi)\big)\ \le\ e^{2n_c\|O\|_\infty/k}\,\wt\mu_{g\Gamma(L)}(F)
=e^{2n_c\|O\|_\infty/k}\,\mu^{\zeta_j}_{M_j}(F),
\]
by Lemma~\ref{tv:lem:pr}(a),(c) and Lemma~\ref{tv:lem:tws}(i). With~\eqref{tv:eq:P3},
\begin{equation}\label{tv:eq:P4}
\wt\mu'(v)\ \le\ n_L^{-1}\Big[\frac{2n_c\|O\|_\infty}{k}+\log\mu^{\zeta_j}_{M_j}\big(e^{V_L}\big)\Big],\qquad
\frac{n_c}{n_L}=\frac{2L^3}{h^3}\cdot\frac{I'\eN^3}{L^3}=8^{q+1}.
\end{equation}

\emph{Step 5 (insertions on the twisted torus).} For each height $s\in\{1,\dots,M_j\}$ let $X^{[s]}$ be the full operator of the copies $X^\gamma$ with
$s(\gamma)=s$: their tensor product in the case (a) (Lemma~\ref{tv:lem:win}(a)), their product in the case (b) (Lemma~\ref{tv:lem:win}(b); the windows
$S^\gamma$ are closed under the bonds of their plaquettes, as $S$ is, and by Step~3 they are pairwise disjoint in bonds and in sites); $X^{[s]}:=1$ if
there is no such $\gamma$. In the case (b) the feature map is that of Lemma~\ref{tv:lem:win}(b) with $V$ the set of all sites of the layer torus, the
same at all heights. By Lemma~\ref{tv:lem:win}, $r_{X^{[s]}}$ is the product of the $r_{X^\gamma}$ with $s(\gamma)=s$, $\|X^{[s]}\|$ is at most the
product of their norms, and $X^{[s]}$ has the absolute convergence on which~\eqref{rv:eq:insert} rests (Remark~\ref{rv:rem:abs} in the case (a),
Lemma~\ref{tv:lem:win}(c) in the case (b)). Hence
\[
e^{V_L}=\prod_{\gamma\in\Gamma'/g\Gamma(L)}r_{X^\gamma}\circ\Theta^{(s(\gamma))}_\sigma=\prod_{s=1}^{M_j}r_{X^{[s]}}\circ\Theta^{(s)}_\sigma,\qquad
\prod_{s=1}^{M_j}\|X^{[s]}\|\le\|X\|^{n_L},
\]
and Lemma~\ref{tv:lem:tw}(iii) gives
\[
\mu^{\zeta_j}_{M_j}\big(e^{V_L}\big)=\frac{Z^{\zeta_j}_{M_j}(X^{[1]},\dots,X^{[M_j]})}{Z^{\zeta_j}_{M_j}}\ \le\ \|X\|^{n_L}\,\frac{Z^0_{M_j}}{Z^{\zeta_j}_{M_j}} .
\]

For $j=3$ the twist $\zeta_3$ vanishes and the last ratio is $1$ (this is Lemma~\ref{rv:lem:UV} with~\eqref{rv:eq:insert}). For $j=1,2$ it is at most
$\exp[(L/\eN)^3\eta_n]$ by Lemma~\ref{tv:lem:tws}(ii), $L=L_n$. Since $(L/\eN)^3/n_L=I'$, \eqref{tv:eq:P4} becomes
\begin{equation}\label{tv:eq:P5}
\wt\mu'(v)\ \le\ \log\|X\|+\frac{2\cdot8^{q+1}\|O\|_\infty}{k}+I'\,\eta_n ,
\end{equation}
where $q$ and $I'$ depend on the window, the slab function and $h$, but not on $n$, $k$ or $m$ (for $j=3$ the last term is absent).

\emph{Step 6 (the limits: first $n\to\infty$ at fixed $k$, then $k\to\infty$).} By~\eqref{tv:eq:P2} and~\eqref{tv:eq:P5},
$\mu^{1/k}_{L_n}(f_m)\le\log\|X\|+2\cdot8^{q+1}\|O\|_\infty/k+I'\eta_n$ for all large $n$. Let $n\to\infty$ along the subsequence defining $\nu_{1/k}$: by
Lemma~\ref{tv:lem:conv} the left side tends to $\nu_{1/k}(f_m)$, and $\eta_n\to0$ along the whole sequence (Lemma~\ref{tv:lem:fe}(iii)), so
$\nu_{1/k}(f_m)\le\log\|X\|+2\cdot8^{q+1}\|O\|_\infty/k$ for every $k$. Let $k\to\infty$ along the subsequence defining $\nu_0$: by
Lemma~\ref{tv:lem:conv}, $\nu_0(f_m)\le\log\|X\|$. This holds for every $m$, and~\eqref{tv:eq:P1} gives $\nu(v)\le\log\|X\|$, which is~\eqref{tv:eq:T}.
\end{proof}

\begin{remark}[a second proof]\label{tv:rem:second}
(i) \emph{A second proof of the invariance used in Proposition~\ref{n1:prop:straddle}.} For the comparison states, Proposition~\ref{tv:prop:T}
gives a proof of Lemmas~\ref{rv:lem:step2} and~\ref{rv:lem:step2full}, of Theorem~\ref{rv:thm:EV} and of the conclusion of
Theorem~\ref{rv:thm:RV} which does not use the tangent inequality, Lemma~\ref{rv:lem:TV}. Indeed, let $\nu_0$ be a comparison state. In the
proof of Theorem~\ref{rv:thm:RV} for $\nu_0$ (Subsection~\ref{rv:ssec:S9}), Lemma~\ref{rv:lem:TV} is applied only to the three measures
$\nu^{(j)}$ of Proposition~\ref{tv:prop:T} (in Step~1 to the phase restoration of $\nu_0$, in Step~2 to those of $\nu_0\circ\hat g_1$ and
$\nu_0\circ\hat g_2$), and only through the upper bounds~\eqref{tv:eq:T} in the proofs of Lemmas~\ref{rv:lem:step2}
and~\ref{rv:lem:step2full} (Remark~\ref{rv:rem:hyp}); and Proposition~\ref{tv:prop:T} gives these bounds directly. This applies in particular
to the invariance of $\nu_{O'}$ under $\tau_{2\eN e_1}$ in step~(3) of the proof of Proposition~\ref{n1:prop:straddle}. For that invariance
the case $j=1$ of Proposition~\ref{tv:prop:T} suffices: Step~1 of the proof of Theorem~\ref{rv:thm:RV} for $\nu_0\circ\hat g_1$ gives its
invariance under the time reflections, the computation in the proof of Proposition~\ref{rv:prop:YM4}(ii) turns it into the invariance of
$\nu_0$ under the reflections $\theta_{1,c}$, and Proposition~\ref{rv:prop:YM4}(iii) gives the invariance under $\tau_{2\eN e_1}$.

(ii) \emph{The surface term.} The convergence which the argument of this section takes from the surface bound, Lemma~\ref{n1:lem:SV},
also follows from Lemma~\ref{tv:lem:tws}(iii). By that lemma, $0\le\phi_{L_n}\le(L_n/\eN)^3\eta_n$ with $\eta_n\to0$ along a dyadic sequence,
hence $\phi_{L_n}/L_n^3\to0$. In the notation of Remark~\ref{n1:rem:sides}, where $t$ is the source strength, this is the statement
$h^3\phi_{L_n}/(2tL_n^3)\to0$ at fixed $h$ and $t>0$ of that remark, which is the only consequence of Lemma~\ref{n1:lem:SV} that the argument
of this section uses (Remark~\ref{n1:rem:sides}, the proof of Proposition~\ref{n1:prop:comp}, Remark~\ref{n1:rem:SV}). Here it follows
from $\phi_L\ge0$ (Proposition~\ref{n1:prop:fold}), the H\"older bound of Lemma~\ref{tv:lem:tw} without insertions and the halvings of
Lemma~\ref{tv:lem:fe}, with no analysis of the seam. The bound of Lemma~\ref{n1:lem:SV} itself, of the order $L^2\log L$, is not obtained in
this way, since Lemma~\ref{tv:lem:tws}(iii) gives no rate for $\eta_n$.

(iii) \emph{Restriction to the comparison states.} Proposition~\ref{tv:prop:T} is a statement about the comparison states only: its proof
uses their construction as limits of the torus laws $\mu^{1/k}_{L_n}$ and gives nothing for other Gibbs measures. Section~\ref{rv:sec} does
not depend on it; there the tangent inequality is Lemma~\ref{rv:lem:TV}.
\end{remark}

\subsection{Straddling: the rigidity step}
\label{n1:sub:straddle}

\begin{proposition}[reflection positivity of the dyadic limit points at the planes $\{u=c_0\}$]\label{n1:prop:straddle}
Let $\om$ be a dyadic limit point, $c_0\in\eN\Z$, $\Pi:=\{u=c_0\}$ and
$\theta:=\theta_\Pi=r_{c_0}$. Then $\om\big((F\circ\theta)\,F\big)\ge0$ for every real bounded continuous function $F$ of the variables of
finitely many cells with all vertices in $\{u\ge c_0\}$ (a function on $\OmN$, in the sense of Theorem~\ref{thm:d2:LP}(d)), and $\om(G\circ\theta)=\om(G)$
for every bounded continuous local $G$.
\end{proposition}

For Theorem~\ref{n1:thm:N1zero} only $c_0=0$ is needed.

\begin{proof}
The invariance is Theorem~\ref{thm:d2:LP}(c), since $r_{c_0}$ is a signed coordinate permutation about a site composed with a translation of $\eN\Z^3$,
an element of $\cG_N$. Let $F$ be as stated and $O:=-(F\circ\theta)F$. This function is real, bounded, continuous and local, a function of the gauge-invariant
variables $(\ff_p,\tr_x)$, but of cells on \emph{both} sides of $\Pi$ (those of $F$ and their reflections), so no datum with the wall $\Pi$
admits it.

\emph{Translate.} Let $u_{\min}$ be the least value of $u$ over the vertices of the cells of $O$ and pick $a\in\eN\Z$ with
$u_{\min}+2a\in\{c_0+\eN,c_0+2\eN\}$ (one of two consecutive multiples of $\eN$ lies in $2\eN\Z$); let $\tau$ be the translation by $2ae_1\in\eN\Z^3$
(it raises $u$ and $v(x)$ by $2a$ and fixes $x_3$) and $O':=O\circ\tau^{-1}$, a real bounded continuous function of the gauge-invariant variables
of cells with all vertices in $\{u\ge c_0+\eN\}$. Choose $h\in\{h_j\}$ (unbounded, Definition~\ref{n1:def:D}) larger than each of the $u$-,
$v(x)$- and $x_3$-ranges of the vertices of the cells of $O'$ plus $2\eN$, and put $c_+:=\min v(x)-\eN\in\eN\Z$, $c_3:=\min x_3-\eN/2\in\tfrac12\eN\Z$
over those vertices. Then the cells of $O'$ lie inside the open cell $C_0=\{c_0<u<c_0+h,\ c_+<v(x)<c_++h,\ c_3<x_3<c_3+h\}$ of the datum
$D_{O'}:=(c_0,c_+,c_3;h;O')$, whose wall set contains $\Pi$. Let $\nu_{O'}:=\nu_0(D_{O'})$ be a comparison state of Lemma~\ref{n1:lem:gibbs}(iii),
formed along subsequences of the subsequence defining $\om$.

\emph{The chain of relations.} The following four relations hold.
\begin{itemize}
\item[(1)] $\om(O)=\om(O')$, because the law $\mu_N$ of $\om$ (Definition~\ref{def:omN}) is invariant under the translations by $\eN\Z^3$ (Theorem~\ref{thm:d2:LP}(c)) and $2ae_1\in\eN\Z^3$.
\item[(2)] $\om(O')\le\nu_{O'}(O')$, by Proposition~\ref{n1:prop:comp} ($O'$ is a bounded continuous function of the gauge-invariant variables
of cells inside $C_0$; under $\nu_{O'}$ it is evaluated through $\ff_p=(dB)_p+v_p$).
\item[(3)] $\nu_{O'}(O')=\nu_{O'}(O)$. Indeed, by Proposition~\ref{n1:prop:I123} (the sourceless $\nu_{O'}$ is Gibbs, $\Lambda'$-periodic with the time
translation $2he_3$, and satisfies (I1)--(I3)), Theorem~\ref{rv:thm:RV} makes $\nu_{O'}$ invariant under $\tau_{2\eN e_1}$, hence under
$\tau=\tau_{2ae_1}$ ($a\in\eN\Z$). A second proof of this invariance, which does not use Lemma~\ref{rv:lem:TV}, is obtained with Proposition~\ref{tv:prop:T} in the place of Lemma~\ref{rv:lem:TV}; see Remark~\ref{tv:rem:second}(i).
\item[(4)] $\nu_{O'}(O)=-\nu_{O'}\big((F\circ\theta)F\big)\le0$, by the reflection positivity of $\nu_{O'}$ at its wall $\Pi$ (Lemma~\ref{n1:lem:gibbs}(iii)),
$F$ being a real bounded continuous function of cells in the closed half space $\{u\ge c_0\}$ (as a function on $E$ through $\ff_p=(dB)_p+v_p$;
the reflection acts on $(\tr,B,v)$ and induces on $\ff$ the action used on $\OmN$).
\end{itemize}
Hence $\om\big((F\circ\theta)F\big)=-\om(O)\ge0$. (Equivalently: $\nu_{O'}$ is the dominating state of Lemma~\ref{n1:lem:RPcone} for this $O$,
after the translation $\tau$ has been undone by the rigidity theorem.)
\end{proof}

\subsection{Conclusion: the six planes through $0$ and all their translates}
\label{n1:sub:N1}

\begin{lemma}[complex form, Hermitian form, Borel functions and the $L^2$ closure]\label{n1:lem:herm}
Let $\om$ be a dyadic limit point, $c_0\in\eN\Z$, $R:=r_{c_0}$ and $\Th F:=\overline{F\circ R}$ for functions $F$ of cells in $\{u\ge c_0\}$.
(a) For complex bounded continuous such $F,G$: $\om(\Th F\cdot F)\ge0$, $\om(\Th F)=\overline{\om(F)}$, and $(F,G)\mapsto\om(\Th F\cdot G)$ is a
positive semidefinite Hermitian form. (b) The same holds for bounded Borel $F,G$ of finitely many cells in $\{u\ge c_0\}$ and for their
$L^2(\mu_N)$-closure.
\end{lemma}
\begin{proof}
(a) Write $F=F_1+iF_2$ with $F_1,F_2$ real. Then
\[
\om(\Th F\cdot F)=\om((F_1\circ R)F_1)+\om((F_2\circ R)F_2)+i\big[\om((F_1\circ R)F_2)-\om((F_2\circ R)F_1)\big],
\]
and the bracket vanishes by the
$R$-invariance of $\mu_N$ (Theorem~\ref{thm:d2:LP}(c), $R\in\cG_N$): $\om((F_2\circ R)F_1)=\om\big(((F_2\circ R)F_1)\circ R\big)=\om(F_2\,(F_1\circ R))$.
So $\om(\Th F\cdot F)\ge0$ by Proposition~\ref{n1:prop:straddle}, and $\om(\Th F)=\overline{\om(F\circ R)}=\overline{\om(F)}$. The Hermitian symmetry is the identity
\[
\om(\Th F\cdot G)=\om(\overline{F\circ R}\,G)=\om(\overline F\,(G\circ R))=\overline{\om(\overline{G\circ R}\,F)}=\overline{\om(\Th G\cdot F)},
\]
again by
$R$-invariance; a sesquilinear form with Hermitian symmetry and non-negative diagonal is positive semidefinite (polarisation).
(b) Bounded Borel functions of finitely many cells are $L^2(\mu_N)$-limits of bounded continuous functions of the same cells (the marginal is
a finite Borel measure on a Polish product; Lusin's theorem), and $(F,G)\mapsto\om(\Th F\cdot G)$ is $L^2$-continuous because
$F\mapsto F\circ R$ is an isometry of $L^2(\mu_N)$ by $R$-invariance, so $\Th$ is an antiunitary involution of $L^2(\mu_N)$. The three
statements pass to the limits, and then to the $L^2(\mu_N)$-closure.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{n1:thm:N1zero}]
Let $P=P^\sigma_{ij}=\{x_i=\sigma x_j\}$, $1\le i<j\le3$, $\sigma\in\{\pm1\}$, and $k$ the third index. Let $R'$ be the signed coordinate
permutation with $R'e_1=e_i$, $R'e_2=\sigma e_j$, $R'e_3=e_k$: for $y=R'x$ one has $y_i-\sigma y_j=x_1-\sigma^2x_2=u(x)$, so $R'$ maps
$\{u=0\}$ onto $P$ and $\{u\ge0\}$ onto the closed half space $\{x_i-\sigma x_j\ge0\}$. For the other closed half space use $R'\circ(1\,2)$,
which maps $\{u=0\}$ onto $P$ and $\{u\ge0\}$ onto $\{x_i-\sigma x_j\le0\}$. Call the chosen map $g$; it fixes $0$, so $g\in\cG_N$
(Theorem~\ref{thm:d2:LP}(c)), and $\theta_P=g\,\theta_0\,g^{-1}$ with $\theta_0:=r_0$ the reflection in $\{u=0\}$ (both sides are the involutive
isometry of $\eN\Z^3$ fixing $P$ pointwise). For $F$ bounded Borel of cells in $\overline{H}_P=g\{u\ge0\}$, $F\circ g$ is bounded Borel of
cells in $\{u\ge0\}$, and by $g$-invariance of $\mu_N$ and Lemma~\ref{n1:lem:herm}(b) at $c_0=0$,
\[
\om\big(\overline{F\circ\theta_P}\cdot F\big)=\om\big((\overline{F\circ\theta_P}\cdot F)\circ g\big)=\om\big(\overline{(F\circ g)\circ\theta_0}\cdot(F\circ g)\big)\ \ge\ 0 ,
\]
using $\theta_P\circ g=g\circ\theta_0$. As $(i<j,\sigma)$ vary, all six diagonal site planes through $0$ are reached.
\end{proof}

\begin{corollary}[all diagonal site planes; Hermitian form; $L^2$ closure]\label{n1:cor:N1}
Let $\om$ be a dyadic limit point. For every $c\in\eN\Z$ and every plane $\Pi=\{x_i-\sigma x_j=c\}$ ($1\le i<j\le3$, $\sigma\in\{\pm1\}$), with
$\theta_\Pi$ the reflection in $\Pi$ and $\Th_\Pi F:=\overline{F\circ\theta_\Pi}$: $\om(\Th_\Pi F\cdot F)\ge0$ for every bounded Borel $F$ of finitely
many cells in either closed half space bounded by $\Pi$; $(F,G)\mapsto\om(\Th_\Pi F\cdot G)$ is a positive semidefinite Hermitian form on these
functions; $\om(\Th_\Pi F)=\overline{\om(F)}$; and all three statements extend to the $L^2(\mu_N)$-closure.
\end{corollary}
\begin{proof}
With $g$ as in the proof of Theorem~\ref{n1:thm:N1zero} put $g_c(x):=gx+ce_i$; since $ce_i\in\eN\Z^3$, $g_c\in\cG_N$ (Theorem~\ref{thm:d2:LP}(c)),
$g_c\{u=0\}=\Pi$ ($y=gx+ce_i$ has $y_i-\sigma y_j=u(x)+c$), $g_c\{u\ge0\}$ is the chosen closed half space and $\theta_\Pi=g_c\theta_0g_c^{-1}$. The
computation of that proof, with $g_c$ in place of $g$, transports the three statements of Lemma~\ref{n1:lem:herm}(b) at $c_0=0$ to $\Pi$
(for the Hermitian form and the conjugation identity: $\om(\Th_\Pi F\cdot G)=\om(\Th_0(F\circ g_c)\cdot(G\circ g_c))$ and $\om(\Th_\Pi F)=\om(\Th_0(F\circ g_c))$
by $g_c$-invariance), and $F\mapsto F\circ g_c$ is an isometry of $L^2(\mu_N)$, so the $L^2$-closure statement transports too. Beyond
Theorem~\ref{n1:thm:N1zero}, the proof uses only the $\eN\Z^3$-translation invariance of $\mu_N$.
\end{proof}

\subsection{Consequence for the rotation theorems}
\label{n1:sub:FLH}

\begin{remark}[consequence for the rotation theorems]\label{n1:rem:FLH}
Let $\om_N$ be a dyadic limit point and $S$ a limit functional of Theorem~\ref{thm:Hlim} built on it. Then Theorems~\ref{flh:thm:FLH1}
and~\ref{flh:thm:FLH1p} hold for $S$; this is Theorem~\ref{flh:thm:FLH2}. Indeed, apart from the reflection positivity of $\om_N$ at the
six diagonal site-plane families, every step of the proofs of Theorems~\ref{flh:thm:FLH1} and~\ref{flh:thm:FLH1p} holds for the torus
limit points as it stands, with Theorem~\ref{thm:d2:LP}(a)--(d) in place of the corresponding statements for the box states: the Gaussian
bound of Lemma~\ref{flh:lem:I5} follows from Theorem~\ref{thm:d2:LP}(b), the translation invariance and the coordinate reflections are
given by Theorem~\ref{thm:d2:LP}(c),(d), and the continuum passage is given by Theorem~\ref{thm:Hlim} and Section~\ref{sec:osrest}, which
are stated for $\om_N$. Lemma~\ref{flh:lem:I4} requires reflection positivity only at the nine planes through $0$, and the Hermitian
identities which it uses follow from Theorem~\ref{thm:d2:LP}(c). Theorem~\ref{n1:thm:N1zero} therefore suffices for the proof, while
Corollary~\ref{n1:cor:N1} gives the reflection positivity at all diagonal site planes. Theorem~\ref{n1:thm:N1zero} rests in turn on the
rigidity theorem (Theorem~\ref{rv:thm:RV}) and on the surface bound (Lemma~\ref{n1:lem:SV}). One and the same dyadic limit point $\om_N$
must be used throughout, in Definition~\ref{def:omN} and in the sequence $(N_m)$ of Theorem~\ref{thm:Hlim}.
\end{remark}


\providecommand{\cS}{\mathcal S}\providecommand{\cH}{\mathcal H}\providecommand{\cW}{\mathcal W}\providecommand{\cC}{\mathcal C}
\providecommand{\cB}{\mathcal B}\providecommand{\cX}{\mathcal X}\providecommand{\cD}{\mathcal D}\providecommand{\cN}{\mathcal N}
\providecommand{\Hc}{\cH_\circ}\providecommand{\Thc}{\Theta_\circ}\providecommand{\re}{\operatorname{Re}}\providecommand{\im}{\operatorname{Im}}
\providecommand{\supp}{\operatorname{supp}}\providecommand{\Arg}{\operatorname{Arg}}\providecommand{\E}{\mathbb E}
\providecommand{\ip}[2]{\langle #1,#2\rangle}\providecommand{\norm}[1]{\lVert #1\rVert}\providecommand{\Lc}{L^2_{\circ,+}}
\providecommand{\dist}{\operatorname{dist}}\providecommand{\id}{\mathrm{id}}

\section{Rotation covariance of the moments from reflection positivity}\label{i7:sec}\label{sec:i7:proof}

Throughout this section $T$ is $\Lambda^2\R^3$ or $\Lambda^2\R^3\oplus\R$ with its orthogonal $O(3)$-action $\pi$ (on the $2$-form
summand $\pi(g)=\det(g)\,g$ under $\Lambda^2\R^3\cong\R^3$, on the scalar summand the identity), $T_\C$ its complexification with the
Hermitian extension of the inner product, and $\cS=\cS(\R^3;T)$ the real Schwartz space; $\cS'(\R^3;T)$ carries its Borel $\sigma$-algebra.
For a Borel probability measure $\mu$ on $\cS'(\R^3;T)$ put $F(\chi)(\phi):=\phi(\chi)$ and
$S_n(\chi_1,\dots,\chi_n):=\int\prod_{j=1}^nF(\chi_j)\,d\mu$ whenever the integral converges absolutely. For $g\in O(3)$ the pull-back is
$(g^*\chi)(x):=\pi(g)^{-1}\chi(gx)$, and for $a\in\R^3$, $(\tau_a\chi)(x):=\chi(x-a)$. For a reflection $\Theta$ in an affine plane,
$\Theta^*\chi$ is the pull-back $(\Theta^*\chi)(x)=\pi(\Theta_0)\chi(\Theta x)$, $\Theta_0$ the linear part. The constants of this section depend only on
the constant $C$ of hypothesis (i) below (and on the number of fields where this is stated), never on $\mu$ otherwise. The symbols
$\cW$, $\cN$, $N$ and $\mu$ of this section are local to it (elsewhere in this paper $N$ is the level and $\cN$ the set of the nine
normals).

\begin{theorem}[rotation covariance of the moments away from coincident points]\label{i7:thm:main}
Let $\mu$ be a Borel probability measure on $\cS'(\R^3;T)$ such that
\begin{itemize}
\item[(i)] $\int e^{F(\chi)}\,d\mu\le e^{C(\|\chi\|_{L^1}+\|\chi\|_{L^2}^2)}$ for all real $\chi\in\cS(\R^3;T)$, with one constant $C<\infty$;
\item[(ii)] $\mu$ is invariant under all translations of $\R^3$;
\item[(iii)] $\sum_{a,b}\bar z_az_b\int e^{iF(\chi_a-\Theta^*\chi_b)}\,d\mu\ge0$ for every reflection $\Theta$ in one of the nine planes
$\{x_j=0\}$, $\{x_j\pm x_k=0\}$ ($1\le j<k\le3$), all finite families $\chi_a\in\cS$ supported in one open side of the plane and all $z_a\in\C$.
\end{itemize}
Then $S_n(R^*\chi_1,\dots,R^*\chi_n)=S_n(\chi_1,\dots,\chi_n)$ for all $n\ge1$, all $R\in O(3)$ and all $\chi_1,\dots,\chi_n\in\cS(\R^3;T)$ with
pairwise disjoint supports.
\end{theorem}

For the two representations of the theorem the planar rotation weights are $0,\pm1$ (the $2$-forms) and $0$ (the scalar), and the
proof below is written for weights in $\{-2,\dots,2\}$ wherever the weights enter (Proposition~\ref{i7:prop:W10}).
The proof occupies the
rest of the section: Subsection~\ref{i7:sub:setting} fixes the notation, Subsections~\ref{i7:sub:W1}--\ref{i7:sub:W17} prove the intermediate
statements in order, and Subsection~\ref{i7:sub:W18} assembles them.

\begin{remark}[the tests in (iii)]\label{i7:rem:tests}
Hypothesis (iii) with real $C^\infty_c$ tests supported in one open side implies (iii) as stated, with Schwartz tests supported in that
open side: for such $\chi_a$ the cut-offs $\chi_a\vartheta(\cdot/q)$ ($\vartheta\in C^\infty_c(\R^3)$, $\vartheta=1$ on $B(0,1)$) are
compactly supported in the open side and converge to $\chi_a$ in the norm $N$ of Subsection~\ref{i7:sub:setting} as $q\to\infty$, and the characteristic functional $Z$ is $N$-continuous (Lemma~\ref{i7:lem:W1}(b)),
so each finite Gram sum of (iii) passes to the limit. Conversely the stated form contains the compactly supported one. Only the
compactly supported form is ever used below.
\end{remark}

\begin{remark}[the $\sigma$-algebra]\label{i7:rem:sigma}
The proofs of this section use $\mu$ only through the random variables $F(\chi)$, $\chi\in\cS$ real, and the $\sigma$-algebras they generate
(the $\Sigma(K)$ and $\Sigma\{u\cdot v>0\}$ of Subsection~\ref{i7:sub:setting}, and in Lemma~\ref{i7:lem:W1}(e) the uniqueness of a
probability measure with a given characteristic functional on the $\sigma$-algebra generated by the evaluations). They apply therefore
to a probability measure on $\cS'(\R^3;T)$ carrying the cylinder $\sigma$-algebra generated by the evaluations, which is
the form in which the measures of Sections~\ref{sec:osrest} and~\ref{flh:sec:main} are given: ``Borel'' is then read as ``on that $\sigma$-algebra'', and the image measures used in the proofs (under translations, reflections and the rotation by $\pi/4$ of a coordinate plane) are again of this form, because every evaluation of the image of $\phi$ is an evaluation of $\phi$.
\end{remark}

\subsection{Setting and notation}\label{i7:sub:setting}

\emph{Fields.} For $\chi\in(L^1\cap L^2)(\R^3;T_\C)$ we put $N(\chi):=\|\chi\|_1+\|\chi\|_2$, the norms being taken with the Hermitian norm on $T_\C$; we note that $\|\re\chi\|_p\le\|\chi\|_p$.
By Lemma~\ref{i7:lem:W1} below, $F$ extends to $(L^1\cap L^2)(\R^3;T_\C)$, complex-linearly, with $F(\chi)\in\bigcap_{p<\infty}L^p(\mu)$,
$\overline{F(\chi)}=F(\bar\chi)$, and $S_n$ is defined on $(L^1\cap L^2)_\C^{\,n}$. $Z(\chi):=\int e^{iF(\chi)}\,d\mu$ is the characteristic functional.

\emph{Coordinates.} A coordinate pair is an ordered pair $(t,x)=(x_j,x_k)$ of distinct coordinates, $y=x_l$ the third; $e_t,e_x,e_y$ are the
unit vectors, and a point of $\R^3$ is written $(t,x,y)$ or $(z,y)$ with $z=(t,x)\in\R^2$ the planar part. $\theta_\circ(t,x,y):=(-t,x,y)$ is the
reflection in $\{t=0\}$, $\theta_c(t,x,y):=(2c-t,x,y)$ the reflection in $\{t=c\}$, $n:=(e_t+e_x)/\sqrt2$, $n':=(e_t-e_x)/\sqrt2$, and $\theta_n$ the
reflection in $\{t+x=0\}$ (so $\theta_nn=-n$, $\theta_nn'=n'$, $\theta_\circ n=-n'$). On tests we put $(\theta_c^*\chi)(v):=\pi(\theta_\circ)\chi(\theta_cv)$ and
$(\theta_n^*\chi)(v):=\pi(\theta_n)\chi(\theta_nv)$. Since $\pi(\theta_\circ)$ and $\pi(\theta_n)$ are real orthogonal involutions, each $\theta^*$ is an
involutive isometry of $L^1$ and of $L^2$, $\re\theta_c^*k=\theta_c^*\re k$, and $\theta_c^*\tau_a\theta_c^*=\tau_{\theta_\circ a}$ for $a\in\R^3$.
We put $J_c\chi:=\overline{\theta_c^*\chi}$ and $J:=J_0$, and for a transverse profile $f\in(L^1\cap L^2)(\R;T_\C)$ we put $Jf:=\pi(\theta_\circ)\bar f$, so that
$J(\alpha f)=\tilde\alpha\,Jf$, where $(\alpha f)(t,x,y):=\alpha(t,x)f(y)$ and $\tilde\alpha(t,x):=\overline{\alpha(-t,x)}$.

\emph{Observables.} The translations and the reflections act on $\cS'$ by $(\tau_a\phi)(\chi):=\phi(\tau_{-a}\chi)$ and $(\theta\phi)(\chi):=\phi(\theta^*\chi)$ for $\theta\in\{\theta_c,\theta_n\}$,
and on functions $G$ of $\phi$ by $U(a)G:=G\circ\tau_{-a}$, $\Theta_cG:=G\circ\theta_c$, $\Thc:=\Theta_0$ and $\Theta_nG:=G\circ\theta_n$. Hence
$U(a)F(\chi)=F(\tau_a\chi)$, $\Theta_cF(\chi)=F(\theta_c^*\chi)$, $\Theta_nF(\chi)=F(\theta_n^*\chi)$, $U(a)(GG')=U(a)G\,U(a)G'$, $U(a)U(b)=U(a+b)$,
and, from the affine identities $\theta_c\theta_0=\tau_{2ce_t}$, $\theta_c\tau_{ce_t}=\tau_{ce_t}\theta_0$, $\theta_0\tau_{-ce_t}=\tau_{-ce_t}\theta_c$,
$\theta\tau_a=\tau_{\theta_0a}\theta$ ($\theta_0$ the linear part),
\begin{equation}\label{i7:eq:intertw}
\begin{gathered}
\Theta_c\Thc=U(2ce_t),\quad\Theta_cU(ce_t)=U(ce_t)\Thc,\quad\Thc U(-ce_t)=U(-ce_t)\Theta_c,\\
\Theta_cU(a)=U(\theta_\circ a)\Theta_c,\quad\Theta_nU(a)=U(\theta_na)\Theta_n .
\end{gathered}
\end{equation}
These identities hold on the generators $F(\chi)$ by the point-map identities, and the $F(\chi)$ separate the points of $\cS'$.
$\Sigma(O)$ denotes the $\sigma$-algebra generated by the $F(\chi)$, $\chi\in\cS$ real with $\supp\chi\subset O$ ($O$ open); it coincides with the one
generated by the $F(\chi)$, $\chi\in C^\infty_c(O;T)$, since $\chi\,\varphi(\cdot/j)\to\chi$ in $\cS$ for a cut-off $\varphi$. We put $L^2_c:=L^2(\Sigma\{t>c\},\mu)$,
$\Lc:=L^2_0$ and $L^2_{n,+}:=L^2(\Sigma\{n\cdot v>0\},\mu)$.

\emph{The OS space.} We define the forms $\ip{G}{G'}_c:=\E_\mu[\overline{\Theta_cG}\,G']$ on $L^2_c$, $\ip{\cdot}{\cdot}_\circ:=\ip{\cdot}{\cdot}_0$, and $\ip{G}{G'}_n:=
\E_\mu[\overline{\Theta_nG}\,G']$ on $L^2_{n,+}$. The space $\Hc$ is the completion of $\Lc$ modulo the null space $\cN$ of $\ip{\cdot}{\cdot}_\circ$ (Lemma~\ref{i7:lem:recon});
we write $[G]$ or $[G]_\circ$ for the class of $G$, and $\Omega:=[1]$, so that $\ip{\Omega}{[G]}_\circ=\E_\mu[G]$. The space $\cH_n$ and the classes $[G]_n$ are defined likewise for the plane $\{t+x=0\}$.
The operators $H\ge0$ and $P$ are the self-adjoint generators with $[U(te_t+xe_x)G]_\circ=e^{-tH}e^{ixP}[G]_\circ$ for $t\ge0$, $x\in\R$, and $\Pi$ is their joint spectral measure on
$[0,\infty)\times\R$ (Lemma~\ref{i7:lem:recon}); $H_n,P_n,\Pi_n$ are the same objects for $\{t+x=0\}$, with $n$ as time and $n'$ as space.

\emph{Complex translations.} For $z=(t,x)\in\C^2$ we put $q(z):=\re t-|\im x|$, $\cW:=\{q>0\}$, $\cC_+:=\{(t,x)\in\R^2:t>|x|\}$, $\Thc z:=(-t,x)$ and
$\bar z:=(\bar t,\bar x)$. The function $q$ is superadditive, $q(z+z')\ge q(z)+q(z')$, and $q(-\Thc\bar z)=q(z)$. For $q(z)\ge0$ we put $V(z):=\int e^{-t\omega+ixp}\,d\Pi(\omega,p)$
(Lemma~\ref{i7:lem:W4}). We write $R(\theta)$ for the rotation of the $(t,x)$-plane by $\theta$, acting on $\R^3$ with $y$ fixed, and $\varrho(\theta):=\pi(R(\theta))$ for its tensor action,
so that $(R(\theta)^*\chi)(z,y)=\varrho(-\theta)\chi(R(\theta)z,y)$; both extend to complex $\theta$ as entire functions (polynomials in $e^{\pm i\theta}$).

\emph{Narrow and admissible configurations.} A finite planar configuration $(z_1,\dots,z_N)\in(\R^2)^N$ is narrow if for some ordering $\sigma$ and
some unit vector $u\in\R^2$ all differences $z_{\sigma(j+1)}-z_{\sigma(j)}$ lie in the open cone of half-angle $\pi/4$ about $u$. An ordered tuple of
tests $\chi_j=\alpha_jf_j$ with compact planar supports $K_j=\supp\alpha_j$ is admissible if $K_1\subset\cC_+$ and $K_{j+1}-K_j\subset\cC_+$ for all $j$. The symbols $K_1$, $K_6$, $K_7$ and $K_8$ also denote the constants of this section listed in Remark~\ref{i7:rem:constants}: as a numerical factor in an estimate $K_n$ is that constant, and in an inclusion or under a map it is the support $\supp\alpha_n$.

\begin{remark}[the four planes of a coordinate pair]\label{i7:rem:fourplanes}
Every statement of Subsections~\ref{i7:sub:W1}--\ref{i7:sub:W15} is proved for one coordinate pair $(t,x)$ from (i), (ii) and hypothesis (iii) restricted to
the four planes $\{t=0\}$, $\{x=0\}$, $\{t+x=0\}$, $\{t-x=0\}$ of that pair; where a proof uses fewer planes this is said. Hypothesis (iii) of
Theorem~\ref{i7:thm:main} supplies this four-plane hypothesis for every coordinate pair, since for $(t,x)=(x_j,x_k)$ the four planes are
$\{x_j=0\}$, $\{x_k=0\}$, $\{x_j+x_k=0\}$, $\{x_j-x_k=0\}$, all among the nine. The four-plane form is what Proposition~\ref{i7:prop:W15} needs: it
applies Subsections~\ref{i7:sub:W1}--\ref{i7:sub:W13} to the law $\tilde\mu$ obtained from $\mu$ by the rotation by $\pi/4$ in the $(t,x)$-plane, which
has (iii) at the four planes of the pair but not at the other five. Proposition~\ref{i7:prop:W17} uses Theorem~\ref{i7:thm:W3} for the
six ordered coordinate pairs, that is (iii) at the three coordinate planes and at the three planes $\{x_j+x_k=0\}$, $j<k$ (Theorem~\ref{i7:thm:W3}
for the ordered pair $(t,x)$ uses the planes $\{t=0\}$ and $\{t+x=0\}$ only, and the two orderings of a pair share the second); the three planes
$\{x_j-x_k=0\}$ enter through Proposition~\ref{i7:prop:W15} (the plane $P_3$ of each pair) and through the $B_3$-invariance of
Lemma~\ref{i7:lem:W2} used in step (4) of the assembly. The hypothesis of Theorem~\ref{i7:thm:main} supplies all of them.
\end{remark}

\begin{lemma}[reconstruction]\label{i7:lem:recon}
Let $\theta$ be the reflection in a plane $\{u\cdot v=0\}$ through $0$, with $\theta_*\mu=\mu$, $\mu$ translation invariant, and (iii) for $\theta$. Put
$L^2_{u,+}:=L^2(\Sigma\{u\cdot v>0\},\mu)$, $\Theta_uG:=G\circ\theta$ and $\ip{G}{G'}_u:=\E_\mu[\overline{\Theta_uG}\,G']$. Then $\ip{\cdot}{\cdot}_u$ is a positive
semidefinite Hermitian form on $L^2_{u,+}$; on the Hilbert space $\cH_u$ (completion modulo the null space) $\|[G]_u\|\le\|G\|_{L^2(\mu)}$; the
classes of bounded functions in $L^2(\Sigma(K),\mu)$, $K\subset\{u\cdot v>0\}$ bounded open, are dense in $\cH_u$; and there are a self-adjoint $H_u\ge0$ and,
for each unit $m\perp u$, a self-adjoint $P_{u,m}$, all strongly commuting, with joint spectral measure $\Pi_u$ on $[0,\infty)\times\R$ for each
$(H_u,P_{u,m})$, such that $[U(su+s'm)G]_u=\int e^{-s\omega+is'p}\,d\Pi_u(\omega,p)\,[G]_u$ for $s\ge0$, $s'\in\R$, $G\in L^2_{u,+}$.
\end{lemma}
\begin{proof}
Positivity: for $G=\sum_bz_be^{iF(\chi_b)}$ with real $\chi_b\in\cS$ supported in $\{u\cdot v>0\}$, $\Theta_ue^{iF(\chi)}=e^{iF(\theta^*\chi)}$, so
$\E_\mu[\overline{\Theta_uG}\,G]=\sum_{a,b}\bar z_az_bZ(\chi_b-\theta^*\chi_a)$, which is $\sum_{a,b}\bar z_az_bM_{ba}$ with the matrix $M_{ab}=Z(\chi_a-\theta^*\chi_b)$
of (iii); $M$ is positive semidefinite, hence so is $M^{\mathsf T}=\bar M$, and the expectation is $\ge0$. These $G$ form a unital $*$-algebra $\cA_u$ of bounded
functions generating $\Sigma\{u\cdot v>0\}$; by the functional monotone class theorem the $L^2(\mu)$-closure of $\cA_u$ (closed under bounded pointwise
sequential limits by dominated convergence) contains every bounded $\Sigma\{u\cdot v>0\}$-measurable function, and bounded functions are dense in
$L^2_{u,+}$. The form is $L^2\times L^2$-continuous ($|\E[\overline{\Theta_uG}G']|\le\|G\|_2\|G'\|_2$, $\Theta_u$ being an isometry of $L^2(\mu)$ by $\theta_*\mu=\mu$),
so it is positive semidefinite on $L^2_{u,+}$; a positive semidefinite sesquilinear form is Hermitian and obeys the Schwarz inequality, and
$|\ip{G}{G}_u|\le\|\Theta_uG\|_2\|G\|_2=\|G\|_2^2$.

For $s\ge0$, $U(su)L^2_{u,+}\subset L^2_{u,+}$ and, by $\Theta_uU(su)=U(-su)\Theta_u$ and (ii),
$\ip{U(su)G}{G'}_u=\E[\overline{U(-su)\Theta_uG}\,G']=\E[\overline{\Theta_uG}\,U(su)G']=\ip{G}{U(su)G'}_u$; hence $\|[U(su)G]\|^2=\ip{G}{U(2su)G}_u\le
\|[G]\|\,\|[U(2su)G]\|$ and, iterating, $\|[U(su)G]\|\le\|[G]\|^{1-2^{-m}}\|G\|_{L^2}^{2^{-m}}\to\|[G]\|$. So $[G]\mapsto[U(su)G]$ is a well-defined
symmetric contraction semigroup on $\cH_u$; it is strongly continuous because $U$ is strongly continuous on $L^2(\mu)$ (for $G=e^{iF(\chi)}$,
$U(a)G=e^{iF(\tau_a\chi)}\to G$ pointwise and boundedly as $a\to0$, $\phi$ being continuous on $\cS$; such $G$ span a dense set; the $U(a)$ are unitary);
thus it is $e^{-sH_u}$ with $H_u\ge0$ self-adjoint (the spectral theorem \cite{RS72}). For $m\perp u$, $\Theta_uU(s'm)=U(s'm)\Theta_u$ gives $\ip{U(s'm)G}{U(s'm)G'}_u=\ip{G}{G'}_u$; the induced
operators form a strongly continuous unitary group commuting with $e^{-sH_u}$ (both compositions equal $[U(su+s'm)G]_u$), and $e^{is'P_{u,m}}$ is that
group (Stone's theorem \cite{RS72}). A unitary commuting with the bounded self-adjoint $e^{-H_u}$ commutes with its spectral projections, which are those of $H_u$; so $H_u$ and
$P_{u,m}$ strongly commute, and the joint spectral measure exists, carried by $[0,\infty)\times\R$.

Density of the localised bounded classes: the
$\Sigma(K)$ are countably generated and $\Sigma\{u\cdot v>0\}=\bigvee_K\Sigma(K)$ over bounded open $K\subset\{u\cdot v>0\}$, so by martingale convergence
and truncation such functions are dense in $L^2_{u,+}$, and the class map is a contraction with dense range.
\end{proof}
For $u=e_t$ the objects of Lemma~\ref{i7:lem:recon} are $\Hc$, $H$, $P:=P_{e_t,e_x}$ and $\Pi$; for $u=n$ they are $\cH_n$, $H_n$, $P_n:=P_{n,n'}$ and $\Pi_n$. The invariances $\theta_{\circ*}\mu=\mu$,
$\theta_{n*}\mu=\mu$ required are supplied by Lemma~\ref{i7:lem:W2} from (i), (ii) and (iii) at the respective plane. The identification
$[U(te_t+xe_x)G]_\circ=e^{-tH}e^{ixP}[G]_\circ$ ($t\ge0$) is the last formula of Lemma~\ref{i7:lem:recon} with $u=e_t$, $m=e_x$.

\subsection{Fields and moments}\label{i7:sub:W1}

\begin{lemma}[moments]\label{i7:lem:W1}
Assume (i), and put $K_1:=17(1+C)$. Then:
\begin{itemize}
\item[(a)] $\|F(\chi)\|_{L^p(\mu)}\le K_1(\|\chi\|_1+\sqrt p\,\|\chi\|_2)$ for all real $\chi\in\cS$ and $1\le p<\infty$;
\item[(b)] $F$ extends uniquely to a map $(L^1\cap L^2)(\R^3;T_\C)\to\bigcap_{p<\infty}L^p(\mu)$, complex-linear, continuous for $N$ in each $L^p$, with (a)
for real arguments, $\|F(\chi)\|_p\le2K_1(\|\chi\|_1+\sqrt p\|\chi\|_2)$ for complex ones, $\overline{F(\chi)}=F(\bar\chi)$, and (i) holds for all real
$\chi\in L^1\cap L^2$;
\item[(c)] $|S_n(\chi_1,\dots,\chi_n)|\le(2K_1\sqrt n)^n\prod_{j=1}^nN(\chi_j)$ for $\chi_j\in(L^1\cap L^2)_\C$; $S_n$ is a symmetric $n$-linear form, continuous in each
entry for $N$, and its restriction to $\cS^n$ is a tempered distribution on $\R^{3n}$;
\item[(d)] $\E_\mu e^{|F(\chi)|}<\infty$ for real $\chi\in L^1\cap L^2$, and $Z(\chi)=\sum_{n\ge0}i^nS_n(\chi,\dots,\chi)/n!$, the series converging absolutely;
\item[(e)] for an affine isometry $g$ of $\R^3$ with linear part in $O(3)$ (acting on tests by the pull-back with the tensor action), $g_*\mu=\mu$ on the $\sigma$-algebra generated by the evaluations $\phi\mapsto\phi(\chi)$, $\chi\in\cS$ (the sense in which the invariance of a measure under $g$ is used in this section; Remark~\ref{i7:rem:sigma}), if and
only if $Z\circ g^*=Z$ on $\cS$, if and only if $S_n(g^*\chi_1,\dots,g^*\chi_n)=S_n(\chi_1,\dots,\chi_n)$ for all $n$ and all $\chi_j\in\cS$.
\end{itemize}
\end{lemma}
\begin{proof}
(a) Let $\chi\ne0$, $X:=F(\chi)$, $a:=C\|\chi\|_1$, $b:=C\|\chi\|_2^2>0$. Hypothesis (i) at $\pm\lambda\chi$ ($\lambda>0$) and the Chebyshev inequality give
$\mu(|X|>s)\le2e^{-\lambda s+a\lambda+b\lambda^2}$; for $s>a$ the choice $\lambda=(s-a)/(2b)$ yields $\mu(|X|>s)\le2e^{-(s-a)^2/(4b)}$. Hence
\begin{align*}
\E|X|^p&=p\int_0^\infty s^{p-1}\mu(|X|>s)\,ds\le a^p+2p\int_0^\infty(a+r)^{p-1}e^{-r^2/(4b)}\,dr\\
&\le a^p+2^pp\,a^{p-1}\sqrt{\pi b}+2^{p-1}p\,(4b)^{p/2}\Gamma(p/2),
\end{align*}
using $(a+r)^{p-1}\le2^{p-1}(a^{p-1}+r^{p-1})$, $\int_0^\infty e^{-r^2/4b}dr=\sqrt{\pi b}$ and $\int_0^\infty r^{p-1}e^{-r^2/4b}dr=\tfrac12(4b)^{p/2}\Gamma(p/2)$.
Taking $p$-th roots, with $(u+v+w)^{1/p}\le u^{1/p}+v^{1/p}+w^{1/p}$, $p^{1/p}\le e^{1/e}<1.45$, $\pi^{1/(2p)}\le\sqrt\pi<1.78$, $a^{(p-1)/p}b^{1/(2p)}\le a+\sqrt b$
(Young's inequality) and $\Gamma(p/2)^{1/p}\le2\sqrt p$ (for $p\ge2$ from $\Gamma(x)\le x^{x-1}$, $x\ge1$; for $1\le p\le2$ from $\Gamma(p/2)\le\sqrt\pi$):
\[
\|X\|_p\le a+5.2\,(a+\sqrt b)+2\cdot1.45\cdot2\sqrt p\cdot2\sqrt b\le6.2\,a+16.8\sqrt p\,\sqrt b\le7C\|\chi\|_1+17\sqrt C\sqrt p\,\|\chi\|_2 ,
\]
which is (a) since $7C\le K_1$ and $17\sqrt C\le K_1$.

(b) $F$ is linear on $\cS$ ($\phi$ is linear), so (a) gives $\|F(\chi)-F(\chi')\|_p\le K_1(1+\sqrt p)N(\chi-\chi')$; $\cS$ is dense in $L^1\cap L^2$ for $N$, and the
extension by uniform continuity is unique; put $F(\chi_1+i\chi_2):=F(\chi_1)+iF(\chi_2)$ for real $\chi_1,\chi_2$. The conjugation identity is clear.
For real $\chi\in L^1\cap L^2$ and Schwartz $\chi_j\to\chi$ in $N$, a subsequence of $F(\chi_j)$ converges a.e., and Fatou's lemma gives
$\E e^{F(\chi)}\le\liminf_je^{C(\|\chi_j\|_1+\|\chi_j\|_2^2)}=e^{C(\|\chi\|_1+\|\chi\|_2^2)}$.

(c) The bound follows from Hölder's inequality with exponents $n,\dots,n$ and (b) at $p=n$. Symmetry and multilinearity are clear; continuity in each entry follows from (b); on $\cS$ the
norm $N$ is dominated by finitely many Schwartz seminorms, so $S_n$ is a separately continuous $n$-linear form on the Fréchet space $\cS$, hence jointly
continuous, hence a tempered distribution on $\R^{3n}$ by the kernel theorem \cite[Chapter~I]{GV64}, \cite[Chapter~V]{RS72}.

(d) The first assertion follows from $e^{|X|}\le e^X+e^{-X}$ and (b). By (c), $\sum_n|S_n(\chi,\dots,\chi)|/n!\le\sum_n(2K_1\sqrt nN(\chi))^n/n!<\infty$ ($n^{n/2}/n!\le e^{n/2}/\sqrt{n!}$), and
dominated convergence (the partial sums of $e^{iX}$ are bounded by $e^{|X|}$) gives the expansion.

(e) The cylinder sets $\{\phi:(\phi(\chi_1),\dots,\phi(\chi_N))\in B\}$, $\chi_j\in\cS$, $B\subset\R^N$ Borel, form a $\pi$-system generating the $\sigma$-algebra generated by the evaluations $\phi\mapsto\phi(\chi)$, $\chi\in\cS$, and by the linearity of $\phi$ the characteristic functional determines the characteristic function $s\mapsto Z(\sum_js_j\chi_j)$ of each vector $(\phi(\chi_1),\dots,\phi(\chi_N))$, hence its law; so two
probability measures with the same characteristic functional on $\cS$ coincide on that $\sigma$-algebra (Dynkin's $\pi$--$\lambda$ theorem, as in the proof of Proposition~\ref{ms:unique}(a)), which is the one through which $\mu$ is used (Remark~\ref{i7:rem:sigma}); $\int e^{iF(\chi)}d(g_*\mu)=Z(g^*\chi)$ (with the pull-back convention,
$(g\phi)(\chi)=\phi(g^*\chi)$ up to the orientation of the action, which does not affect the equivalence since $g^{-1}$ is again such an isometry). This gives the
first equivalence. If $Z\circ g^*=Z$ on $\cS$ then, by (d) applied to $\sum_js_j\chi_j$ and the identification of the coefficient of $s_1\cdots s_n$ (polarisation),
$S_n(g^*\chi_1,\dots,g^*\chi_n)=S_n(\chi_1,\dots,\chi_n)$; conversely the invariance of all $S_n$ gives that of $Z$ through (d).
\end{proof}

\subsection{The discrete symmetries}\label{i7:sub:W2}

\begin{lemma}[reflections and the one-point function]\label{i7:lem:W2}
Assume (i), (ii), and (iii) for one reflection $\theta$ in a plane through $0$. Then $\theta_*\mu=\mu$. Consequently, under (iii) for the nine planes, $\mu$ is
invariant under the group $B_3$ of signed permutations of the coordinates, and $S_1(\chi)=\langle m,\int\chi\rangle$ with a vector $m\in T$ fixed by
$\pi(B_3)$; $m$ has no $2$-form component, so $\pi(R)m=m$ and $S_1(R^*\chi)=S_1(\chi)$ for all $R\in O(3)$. Under (iii) for the four planes of a coordinate
pair only, $\mu$ is invariant under the four reflections and the dihedral group $D_4$ they generate (the rotations by multiples of $\pi/2$ of that plane
among them), $S_1(\chi)=\langle m,\int\chi\rangle$ with $m$ fixed by $\pi(D_4)$, and $S_1(R(\theta)^*\chi)=S_1(\chi)$ for every rotation $R(\theta)$ of that plane.
\end{lemma}
\begin{proof}
Let $\psi_1,\psi_2\in\cS$ be real, supported in the open positive side of the plane. The matrix $(Z(\psi_a-\theta^*\psi_b))_{a,b=1,2}$ is positive
semidefinite by (iii), hence Hermitian: $Z(\psi_2-\theta^*\psi_1)=\overline{Z(\psi_1-\theta^*\psi_2)}=Z(\theta^*\psi_2-\psi_1)=Z(\theta^*(\psi_2-\theta^*\psi_1))$,
using $\overline{Z(\chi)}=Z(-\chi)$ and $(\theta^*)^2=1$. Every real $\chi\in\cS$ vanishing on a neighbourhood of the plane is of the form $\psi_2-\theta^*\psi_1$
(its restrictions to the two open sides), so $Z\circ\theta^*=Z$ on such tests. For general real $\chi\in\cS$ let $\chi_\delta:=\chi\cdot(1-\varphi(u\cdot v/\delta))$,
$\varphi\in C^\infty_c(\R)$ equal to $1$ near $0$, $u$ the unit normal; $\chi_\delta\in\cS$ vanishes near the plane and $N(\chi-\chi_\delta)\to0$ as
$\delta\to0$ by dominated convergence; $Z$ is $N$-continuous (Lemma~\ref{i7:lem:W1}(b)) and $\theta^*$ is an $N$-isometry, so $Z\circ\theta^*=Z$ on $\cS$, and
Lemma~\ref{i7:lem:W1}(e) gives $\theta_*\mu=\mu$. The nine reflections generate $B_3$, the four reflections of a pair the dihedral group of order $8$ in
that plane, and invariance passes to products.

$S_1$ is a translation-invariant tempered distribution (Lemma~\ref{i7:lem:W1}(c) and (ii)), so its first derivatives vanish and $S_1(\chi)=\int\langle m,\chi(v)\rangle dv$
for a vector $m\in T$. For each of the nine reflections, $S_1(\theta^*\chi)=S_1(\chi)$ gives $\langle m,\pi(\theta)\int\chi\rangle=\langle m,\int\chi\rangle$ for all
$\chi$, i.e. $\pi(\theta)m=m$ ($\pi(\theta)$ symmetric), hence $\pi(g)m=m$ for $g\in B_3$. On the $2$-form summand $\pi(g)=\det(g)g$; for
$g=\operatorname{diag}(-1,1,1)$, $\pi(g)=\operatorname{diag}(1,-1,-1)$, which fixes only the first axis, and the three coordinate sign changes together fix only $0$: the
$2$-form component of $m$ vanishes. The scalar summand carries the trivial action, so $\pi(R)m=m$ for all $R\in O(3)$, and
$S_1(R^*\chi)=\langle m,\pi(R)^{-1}\int\chi(Rv)\,dv\rangle=\langle\pi(R)m,\int\chi\rangle=S_1(\chi)$. Under the four planes of one pair only,
the same argument gives $\pi(g)m=m$ for $g\in D_4$, in particular for the rotation by $\pi/2$ of the plane, whose tensor action is
$\varrho(\pi/2)=\sum_ai^{w_a}P_a$ in the weight decomposition $\varrho(\theta)=\sum_ae^{iw_a\theta}P_a$ of the conventions of Subsection~\ref{i7:sub:W10} ($w_a\in\{-2,\dots,2\}$); since $i^w=1$ with $|w|\le2$ only for $w=0$, $m$
lies in the weight-$0$ subspace, so $\varrho(\theta)m=m$ for all $\theta$ and $S_1(R(\theta)^*\chi)=\langle\varrho(\theta)m,\int\chi\rangle=S_1(\chi)$.
\end{proof}

\subsection{The spectral cone}\label{i7:sub:W3}

\begin{theorem}[spectral cone]\label{i7:thm:W3}
Assume (i), (ii) and (iii) for the two planes $\{t=0\}$ and $\{t+x=0\}$ of a coordinate pair. Then the joint spectral measure $\Pi$ of $(H,P)$ on $\Hc$ satisfies
\[
\Pi\big(\{(\omega,p):\omega<|p|\}\big)=0,\qquad\text{hence}\qquad D(H)\subset D(P),\quad\ip{\psi}{H\psi}\ge\ip{\psi}{|P|\psi}\ \ (\psi\in D(H)).
\]
\end{theorem}
Only the invariances $\theta_{\circ*}\mu=\theta_{n*}\mu=\mu$ (Lemma~\ref{i7:lem:W2} for these two planes), (ii), (iii) at the two planes, and
Lemma~\ref{i7:lem:recon} enter; (i) enters only through Lemma~\ref{i7:lem:W2}, that is through the $N$-continuity of $Z$ (Lemma~\ref{i7:lem:W1}(b)).
Throughout the proof $z(\omega,p):=(\omega-ip)/\sqrt2$, so $\re z=\omega/\sqrt2\ge0$ and $\im z=-p/\sqrt2$ on the
carrier of $\Pi$.

\begin{lemma}[one pairing, two quantisations]\label{i7:lem:W3two}
Let $K\subset\{t>0\}$ be bounded open, $A\in L^2(\Sigma(K),\mu)$, $G(b):=\E_\mu[\overline{\Thc A}\,U(b)A]$ for $b\in\R^3$, $\alpha:=\inf_Kn\cdot v$,
$\alpha':=\inf_Kn'\cdot v$, $s_0:=-\alpha-\alpha'$, and $\nu_A:=\ip{[A]_\circ}{\Pi(\cdot)[A]_\circ}_\circ$ (a positive measure of mass $\le\|A\|_{L^2(\mu)}^2$).
\begin{itemize}
\item[(a)] For $s,s'\ge0$: $\ip{[U(s'n)A]_\circ}{[U(sn)A]_\circ}_\circ=G(sn+s'n')$ and $[U(sn)A]_\circ=\int e^{-s\,z(\omega,p)}\,d\Pi(\omega,p)\,[A]_\circ$.
\item[(b)] There is a complex Borel measure $\rho$ on $[0,\infty)\times\R$ with $|\rho|([0,\infty)\times\R)\le\|A\|_{L^2(\mu)}^2$ such that
$G(sn+s'n')=\int e^{-(s-s_0)\omega'+is'p'}\,d\rho(\omega',p')$ for all $s\ge s_0$, $s'\in\R$.
\end{itemize}
\end{lemma}
\begin{proof}
(a) $U(sn)A\in L^2(\Sigma(K+sn))\subset\Lc$ for $s\ge0$, and $[U(sn)A]_\circ=e^{-(s/\sqrt2)H}e^{i(s/\sqrt2)P}[A]_\circ=\int e^{-s(\omega-ip)/\sqrt2}d\Pi\,[A]_\circ$ by
Lemma~\ref{i7:lem:recon}. Since $\Thc U(s'n)=U(-s'n')\Thc$ by \eqref{i7:eq:intertw} ($\theta_\circ n=-n'$), (ii) gives
$\ip{U(s'n)A}{U(sn)A}_\circ=\E[\overline{U(-s'n')\Thc A}\,U(sn)A]=\E[\overline{\Thc A}\,U(sn+s'n')A]$.

(b) For $v\in K$, $n\cdot(\theta_\circ v+\alpha'n)=-n'\cdot v+\alpha'\le0$, so the open set $\theta_\circ K+\alpha'n$ lies in the open half-space $\{n\cdot v<0\}$;
likewise $K-\alpha n\subset\{n\cdot v>0\}$. Put $A_1:=\Theta_nU(\alpha'n)\Thc A\in L^2(\Sigma(\theta_n(\theta_\circ K+\alpha'n)))\subset L^2_{n,+}$ and
$A_2:=U(-\alpha n)A\in L^2_{n,+}$; $\|A_i\|_{L^2(\mu)}=\|A\|_{L^2(\mu)}$. For $b=sn+s'n'$ and $c:=\alpha'n$, $b+c=(s-s_0)n+s'n'-\alpha n$, so by (ii) and
$\Theta_nA_1=U(c)\Thc A$,
\[
G(b)=\E[\overline{U(c)\Thc A}\,U(b+c)A]=\E[\overline{\Theta_nA_1}\,U((s-s_0)n+s'n')A_2]=\ip{[A_1]_n}{e^{-(s-s_0)H_n}e^{is'P_n}[A_2]_n}_n
\]
for $s\ge s_0$ (Lemma~\ref{i7:lem:recon} with $u=n$, $m=n'$). Set $\rho(\Delta):=\ip{[A_1]_n}{\Pi_n(\Delta)[A_2]_n}_n$; the Schwarz inequality over finite partitions
gives $|\rho|\le\|[A_1]_n\|\,\|[A_2]_n\|\le\|A\|_{L^2(\mu)}^2$.
\end{proof}

\begin{lemma}[test inequality]\label{i7:lem:W3test}
In the setting of Lemma~\ref{i7:lem:W3two} let $s_1:=\max(s_0,0)$, $h\in L^1(\R)$ vanishing on $(-\infty,s_1)$, and $\tilde h(w):=\int h(s)e^{-sw}ds$ for
$\re w\ge0$. Then
\[
\int|\tilde h(z(\omega,p))|^2\,d\nu_A(\omega,p)\le\|A\|_{L^2(\mu)}^2\ \sup_{\omega'\ge0}e^{s_0\omega'}|\tilde h(\omega')|\ \sup_{p'\in\R}|\tilde h(ip')| .
\]
\end{lemma}
\begin{proof}
Let $\psi_h:=\int h(s)[U(sn)A]_\circ\,ds$ (Bochner integral; the integrand is continuous and bounded by $\|[A]_\circ\|$). By Lemma~\ref{i7:lem:W3two}(a) and Fubini
against $\ip{\varphi}{\Pi(\cdot)[A]_\circ}$ ($|h(s)e^{-sz}|\le|h(s)|$), $\psi_h=\tilde h(z(H,P))[A]_\circ$, so $\|\psi_h\|^2=\int|\tilde h(z)|^2d\nu_A$. Also
$\|\psi_h\|^2=\iint\overline{h(s')}h(s)\,G(sn+s'n')\,ds'ds$, and for $s,s'\ge s_1\ge s_0$ Lemma~\ref{i7:lem:W3two}(b) applies with $|e^{-(s-s_0)\omega'}|\le1$, so the
integrals interchange: $\|\psi_h\|^2=\int e^{s_0\omega'}\tilde h(\omega')\,\overline{\tilde h(ip')}\,d\rho(\omega',p')$. Bound by $|\rho|$; the first supremum is finite
since $|e^{s_0\omega'}\tilde h(\omega')|\le e^{(s_0-s_1)\omega'}\|h\|_1\le\|h\|_1$.
\end{proof}

\begin{lemma}[the Laplace test]\label{i7:lem:W3lap}
For $\kappa\in\R\setminus\{0\}$, $\varepsilon>0$ let $h(s):=(s-s_1)^{-i\kappa}e^{-\varepsilon(s-s_1)}$ for $s>s_1$ and $h(s):=0$ otherwise. Then $\|h\|_1=1/\varepsilon$ and,
for $\re w\ge0$ (principal branch),
\[
\tilde h(w)=e^{-s_1w}\,\Gamma(1-i\kappa)\,(w+\varepsilon)^{-1+i\kappa},\qquad
|\tilde h(w)|=e^{-s_1\re w}\,|\Gamma(1-i\kappa)|\,|w+\varepsilon|^{-1}e^{-\kappa\Arg(w+\varepsilon)} .
\]
\end{lemma}
\begin{proof}
Substitute $\sigma=s-s_1$ and use $\int_0^\infty\sigma^{a-1}e^{-\upsilon\sigma}d\sigma=\Gamma(a)\upsilon^{-a}$ for $\re a>0$, $\re\upsilon>0$ (analytic continuation from
real $\upsilon$), with $a=1-i\kappa$, $\upsilon=w+\varepsilon$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{i7:thm:W3}]
Fix $K$, $A$ as in Lemma~\ref{i7:lem:W3two}. Insert Lemma~\ref{i7:lem:W3lap} into Lemma~\ref{i7:lem:W3test}: $e^{s_0\omega'}|\tilde h(\omega')|\le|\Gamma(1-i\kappa)|/\varepsilon$
(as $s_1\ge s_0$ and $\Arg(\omega'+\varepsilon)=0$) and $|\tilde h(ip')|\le|\Gamma(1-i\kappa)|\varepsilon^{-1}e^{|\kappa|\pi/2}$ (as $|\Arg(\varepsilon+ip')|<\pi/2$); dividing by
$|\Gamma(1-i\kappa)|^2=\pi\kappa/\sinh(\pi\kappa)>0$,
\begin{equation}\label{i7:eq:W3star}
\int e^{-\sqrt2s_1\omega}\,|z+\varepsilon|^{-2}\,e^{-2\kappa\Arg(z+\varepsilon)}\,d\nu_A(\omega,p)\le\|A\|_{L^2(\mu)}^2\,\varepsilon^{-2}e^{|\kappa|\pi/2},\qquad z=z(\omega,p).
\end{equation}

For $\delta\in(0,\pi/4)$ and $r>0$ put $S^\pm_{\delta,r}:=\{(\omega,p):|z|\ge r,\ \pm\Arg z\ge\pi/4+\delta\}$. For $\re z\ge0$,
$|\partial_\varepsilon\Arg(z+\varepsilon)|=|\im z|/|z+\varepsilon|^2\le1/|z|$, so $|\Arg(z+\varepsilon)-\Arg z|\le\varepsilon/r$ on $\{|z|\ge r\}$. Choose
$\varepsilon:=r\delta/2$ (independent of $\kappa$). On $S^-_{\delta,r}$, $\Arg(z+\varepsilon)\le-\pi/4-\delta/2$, so for $\kappa>0$ the integrand of
\eqref{i7:eq:W3star} is at least $e^{-\sqrt2s_1\omega}|z+\varepsilon|^{-2}e^{\kappa(\pi/2+\delta)}$ there, and
\[
\int_{S^-_{\delta,r}}e^{-\sqrt2s_1\omega}\,|z+\varepsilon|^{-2}\,d\nu_A\le4\,\|A\|_{L^2(\mu)}^2\,(r\delta)^{-2}e^{-\kappa\delta}\xrightarrow[\kappa\to\infty]{}0 .
\]
The weight is strictly positive, hence $\nu_A(S^-_{\delta,r})=0$; letting $\kappa\to-\infty$ gives $\nu_A(S^+_{\delta,r})=0$. Since $\re z=\omega/\sqrt2$ and
$\im z=-p/\sqrt2$, $\{\omega<p\}=\{\Arg z<-\pi/4\}=\bigcup_{m\ge2}S^-_{1/m,1/m}$ and $\{\omega<-p\}=\bigcup_{m\ge2}S^+_{1/m,1/m}$, so $\nu_A(\Delta)=0$ for
$\Delta:=\{\omega<|p|\}$, i.e. $\Pi(\Delta)[A]_\circ=0$. The classes of such $A$ are dense in $\Hc$ (Lemma~\ref{i7:lem:recon}) and $\Pi(\Delta)$ is bounded, so
$\Pi(\Delta)=0$. For $\psi\in D(H)$, $\int p^2\,d\Pi_\psi\le\int\omega^2\,d\Pi_\psi<\infty$ and $\int|p|\,d\Pi_\psi\le\int\omega\,d\Pi_\psi$.
\end{proof}

\subsection{Complex translations}\label{i7:sub:W4}

\begin{lemma}[complex translations]\label{i7:lem:W4}
Under the hypotheses of Theorem~\ref{i7:thm:W3}, $V(z):=\int e^{-t\omega+ixp}\,d\Pi(\omega,p)$, $z=(t,x)\in\C^2$ with $q(z)\ge0$, is a contraction on $\Hc$, strongly continuous
on $\{q\ge0\}$, operator-norm holomorphic on $\cW$, with $V(z)V(z')=V(z+z')$, $V(0)=1$, $V(z)^*=V(-\Thc\bar z)=V(\bar t,-\bar x)$, $V(z)\Omega=V(z)^*\Omega=\Omega$,
$V(t,x)[G]_\circ=[U(te_t+xe_x)G]_\circ$ for real $t\ge0$, $x\in\R$, and
\begin{equation}\label{i7:eq:Vderiv}
\big\|\partial_t^a\partial_x^bV(z)\big\|\le\sup_{\omega\ge0}\omega^{a+b}e^{-q(z)\omega}=\Big(\frac{a+b}{e\,q(z)}\Big)^{a+b}\qquad(z\in\cW,\ a+b\ge1).
\end{equation}
\end{lemma}
\begin{proof}
On the carrier $\{\omega\ge|p|\}$ of $\Pi$, $|e^{-t\omega+ixp}|=e^{-(\re t)\omega-(\im x)p}\le e^{-q(z)\omega}\le1$, so $V(z)$ is a contraction; the semigroup law and
the adjoint formula are those of the symbols; the real case is Lemma~\ref{i7:lem:recon}; $V(z)\Omega=\Omega$ since $\Pi$ has the atom $\{(0,0)\}$ at $\Omega$
($e^{-tH}\Omega=\Omega$ because $U(te_t)1=1$, and $e^{ixP}\Omega=\Omega$ likewise), and $V(z)^*\Omega=\Omega$ by the adjoint formula. Strong continuity on $\{q\ge0\}$:
the symbols are bounded by $1$ and converge pointwise, so dominated convergence in the spectral integral applies. Holomorphy: for $q(z)>0$ and $|h|<q(z)/2$,
$|e^{-h\omega}-1+h\omega|\le|h|^2\omega^2e^{|h|\omega}/2$ and $\sup_\omega\omega^2e^{-(q(z)-|h|)\omega}<\infty$, so the difference quotients of the symbol in $t$
converge uniformly on the carrier to $-\omega e^{-t\omega+ixp}$; the same for $x$ with $|p|\le\omega$; thus $V$ is norm-holomorphic in each variable separately,
hence jointly (Hartogs' theorem for Banach-space-valued functions, or directly from the uniform convergence of the two-variable Taylor remainder of the symbol),
with $\partial_t^a\partial_x^bV(z)=\int(-\omega)^a(ip)^be^{-t\omega+ixp}d\Pi$; \eqref{i7:eq:Vderiv} follows from $|p|\le\omega$ and the elementary maximum.
\end{proof}

\subsection{The slab-source bound and the bounded slab insertion}\label{i7:sub:W5}

By Lemma~\ref{i7:lem:W2}, hypotheses (i), (ii) and (iii) at $\{t=0\}$ give $\theta_{\circ*}\mu=\mu$; hence, with (ii), every
$\Theta_c$ is an isometry of $L^2(\mu)$ exchanging $L^2(\Sigma\{t<c\})$ and $L^2_c$. The following two facts, which hold under (i), (ii) and (iii) at $\{t=0\}$ alone, are used throughout this subsection:
\begin{itemize}
\item[(F1)] $\ip{G}{G}_0\ge0$ on $\Lc$ (Lemma~\ref{i7:lem:recon}); for $G\in L^2_c$, $G':=U(-ce_t)G\in\Lc$ and $\overline{\Theta_cG}\,G=U(ce_t)(\overline{\Thc G'}\,G')$ by
\eqref{i7:eq:intertw}, so $\ip{G}{G}_c\ge0$ by (ii). Hence $\ip{\cdot}{\cdot}_c$ is Hermitian and obeys the Schwarz inequality. For $F_-\in L^2(\Sigma\{t<c\})$ and
$G_+\in L^2_c$, $\E[F_-G_+]=\ip{\overline{\Theta_cF_-}}{G_+}_c$, whence
\begin{equation}\label{i7:eq:RPS}
|\E[F_-G_+]|^2\le\E[F_-\,\overline{\Theta_cF_-}]\ \E[\overline{\Theta_cG_+}\,G_+],\qquad\text{both factors }\ge0 .
\end{equation}
\item[(F2)] For real $a\ge0$, $b\in\R$ and $G\in\Lc$: $V(a,b)[G]_\circ=[U(ae_t+be_x)G]_\circ$ (Lemma~\ref{i7:lem:recon}).
\end{itemize}

\begin{lemma}[localisation]\label{i7:lem:W5loc}
Let $k\in(L^1\cap L^2)(\R^3;T_\C)$ vanish a.e.\ outside $\{t\le c\}$ (resp.\ $\{t\ge c\}$). Then $F(k)\in L^2(\Sigma\{t<c\})$ (resp.\ $L^2_c$), $|e^{F(k)}|=e^{F(\re k)}$,
and $\E\,e^{pF(\re k)}\le e^{C(p\|\re k\|_1+p^2\|\re k\|_2^2)}$ for $p>0$.
\end{lemma}
\begin{proof}
The function $k_j:=\psi_{1/2j}*\big(1_{\{|v|\le j\}}\tau_{-e_t/j}k\big)$, with $\psi_\varepsilon$ a mollifier supported in the $\varepsilon$-ball, lies in $C^\infty_c(\{t<c\};T_\C)$ and $k_j\to k$
in $L^1\cap L^2$; by Lemma~\ref{i7:lem:W1}(b), $F(k_j)\to F(k)$ in $L^2(\mu)$, and $L^2(\Sigma\{t<c\})$ is closed. $F(k)=F(\re k)+iF(\im k)$ with real summands gives the
modulus; the bound is (i) for $p\re k$ (Lemma~\ref{i7:lem:W1}(b)).
\end{proof}

\begin{theorem}[slab-source bound]\label{i7:thm:W5}
Assume (i), (ii) and (iii) at $\{t=0\}$. Let $l>0$, $h\in(L^1\cap L^2)(\R^3;T_\C)$ vanish a.e.\ outside $\{0\le t\le l\}$, and $A\in L^\infty(\Sigma\{t>0\},\mu)$. Then
$e^{F(h)}U(le_t)A\in\Lc$ and
\[
\big\|[e^{F(h)}U(le_t)A]_\circ\big\|\le\exp\big(C(\|\re h\|_1+\|\re h\|_2^2)\big)\,\|[A]_\circ\| ,
\]
with the constant $C$ of (i).
\end{theorem}
\begin{proof}
Fix a version with $|A|\le\|A\|_\infty$ everywhere; $A':=U(le_t)A\in L^\infty(\Sigma\{t>l\})$, $\Thc A'\in L^\infty(\Sigma\{t<-l\})$, $|\Thc A'|\le\|A\|_\infty$;
$G:=e^{F(h)}A'\in\Lc$ by Lemma~\ref{i7:lem:W5loc}. Put $\kappa:=\|\re h\|_1+\|\re h\|_2^2$.

\emph{Step 1 (dyadic sources).} Put $c_n:=(2^{n+1}-1)l$, $k_0:=h+Jh$, $k_{n+1}:=k_n+J_{c_n}k_n$. By induction $k_n$ vanishes a.e.\ outside $\{-l\le t\le c_n\}$:
$\theta_{c_n}$ maps $[-l,c_n]$ onto $[c_n,2c_n+l]=[c_n,c_{n+1}]$. The two summands of $k_{n+1}$ (and of $k_0$: slabs $[0,l]$ and $[-l,0]$) have a.e.\ disjoint supports,
and $\theta_c^*$ preserves $\|\re\cdot\|_1$ and $\|\re\cdot\|_2$; hence $\|\re k_n\|_1=2^{n+1}\|\re h\|_1$ and $\|\re k_n\|_2^2=2^{n+1}\|\re h\|_2^2$.

\emph{Step 2.} Put $E_n:=\E\big[e^{F(k_n)}\,\overline{\Thc A'}\,U(c_ne_t)A\big]$, which is finite by Lemma~\ref{i7:lem:W5loc}. Since $\overline{\Thc e^{F(h)}}=e^{\overline{F(\theta_0^*h)}}=e^{F(Jh)}$,
$\overline{\Thc G}\,G=e^{F(h+Jh)}\,\overline{\Thc A'}\,A'$, i.e.\ $E_0=\|[G]_\circ\|^2$.

\emph{Step 3 (one reflection).} Claim: $E_{n+1}\ge0$ and $|E_n|\le\|[A]_\circ\|\,E_{n+1}^{1/2}$. Let $c=c_n$, $F_-:=e^{F(k_n)}\overline{\Thc A'}\in L^2(\Sigma\{t<c\})$
(Lemma~\ref{i7:lem:W5loc}; $-l<c$), $G_+:=U(ce_t)A\in L^\infty(\Sigma\{t>c\})$, so $E_n=\E[F_-G_+]$ and \eqref{i7:eq:RPS} applies. Second factor:
$\Theta_cU(ce_t)=U(ce_t)\Thc$ and (ii) give $\E[\overline{\Theta_cG_+}G_+]=\E[U(ce_t)(\overline{\Thc A}A)]=\|[A]_\circ\|^2$. First factor:
$\overline{\Theta_cF_-}=e^{F(J_ck_n)}\,\Theta_c\Thc A'=e^{F(J_ck_n)}\,U((2c+l)e_t)A$ with $2c_n+l=c_{n+1}$, so $\E[F_-\overline{\Theta_cF_-}]=E_{n+1}$, and this is
$\ip{\overline{\Theta_cF_-}}{\overline{\Theta_cF_-}}_c\ge0$.

\emph{Step 4 (terminal bound).} $|E_n|\le\|A\|_\infty^2\,\E\,e^{F(\re k_n)}\le\|A\|_\infty^2\exp(2^{n+1}C\kappa)$ by Lemma~\ref{i7:lem:W5loc} and Step 1.

\emph{Step 5.} Iterating Step 3, $E_0\le\|[A]_\circ\|^{2-2^{1-n}}E_n^{2^{-n}}$ (induction: $2-2^{1-n}+2^{-n}=2-2^{-n}$), hence
\[
\|[G]_\circ\|^2=E_0\le\|[A]_\circ\|^{2-2^{1-n}}\,\|A\|_\infty^{2^{1-n}}\,e^{2C\kappa}\qquad(n\ge1).
\]
If $\|[A]_\circ\|=0$ the right side is $0$; if $A=0$ a.e.\ everything vanishes; otherwise let $n\to\infty$. Thus $\|[G]_\circ\|\le e^{C\kappa}\|[A]_\circ\|$.
\end{proof}

\begin{theorem}[bounded slab insertion]\label{i7:thm:W6}
Assume (i), (ii) and (iii) at $\{t=0\}$. Let $\delta>0$, $\chi\in(L^1\cap L^2)(\R^3;T_\C)$ vanish a.e.\ outside $\{|t|\le\delta\}$, $h:=\tau_{\delta e_t}\chi$, and $K_6:=e^{2C}$.
\begin{itemize}
\item[(a)] For $A\in L^\infty(\Sigma\{t>0\})$: $F(h)U(2\delta e_t)A\in\Lc$ and $\|[F(h)U(2\delta e_t)A]_\circ\|\le K_6\,N(\chi)\,\|[A]_\circ\|$. The map
$[A]_\circ\mapsto[F(h)U(2\delta e_t)A]_\circ$ is well defined on the dense subspace $\cD:=\{[A]_\circ:A\in L^\infty(\Sigma\{t>0\})\}$ and extends uniquely to
$B_\delta(\chi)\in\cB(\Hc)$ with $\|B_\delta(\chi)\|\le K_6(\|\chi\|_1+\|\chi\|_2)$, complex-linear in $\chi$.
\item[(b)] $B_\delta(\chi)^*=B_\delta(J\chi)$.
\item[(c)] $V(r+t,x)\,B_\delta(\chi)\,V(r-t,-x)=B_{\delta+r}(\tau_{(t,x,0)}\chi)$ for real $r\ge|t|$, $x\in\R$.
\item[(d)] For $A\in L^4(\Sigma\{t>0\},\mu)$: $F(h)U(2\delta e_t)A\in\Lc$ and $B_\delta(\chi)[A]_\circ=[F(h)U(2\delta e_t)A]_\circ$. Consequently, for $\chi_j$ vanishing
outside $\{|t|\le\delta_j\}$, $j=1,\dots,m$, with $s_j:=\delta_j+2\sum_{i<j}\delta_i$,
\[
\begin{gathered}
B_{\delta_1}(\chi_1)\cdots B_{\delta_m}(\chi_m)\Omega=\Big[\prod_{j=1}^mF(\tau_{s_je_t}\chi_j)\Big]_\circ,\\
\ip{\Omega}{B_{\delta_1}(\chi_1)\cdots B_{\delta_m}(\chi_m)\Omega}_\circ=S_m(\tau_{s_1e_t}\chi_1,\dots,\tau_{s_me_t}\chi_m).
\end{gathered}
\]
\end{itemize}
\end{theorem}
\begin{proof}
(a) Let $A':=U(2\delta e_t)A$, $G_m:=F(h)^mA'/m!$. By Lemma~\ref{i7:lem:W1}(b), $\|G_m\|_{L^2}\le\|A\|_\infty\|F(h)\|_{2m}^m/m!\le\|A\|_\infty(2\sqrt2K_1N(h))^mm^{m/2}/m!$,
and $m^{m/2}/m!\le(e/\sqrt m)^m$, so $g(s):=\sum_ms^mG_m$ converges absolutely in $\Lc$ for every $s\in\C$ and equals $e^{sF(h)}A'=e^{F(sh)}A'$ a.e. By (F1),
$f(s):=[g(s)]_\circ=\sum_ms^m[G_m]_\circ$ is an absolutely convergent power series in $\Hc$; termwise integration gives $[G_1]_\circ=\frac1{2\pi i}\oint_{|s|=r}f(s)s^{-2}ds$,
hence $\|[F(h)A']_\circ\|\le r^{-1}\max_{|s|=r}\|f(s)\|$. The source $sh$ vanishes outside $\{0\le t\le2\delta\}$ and $|\re(sh)|\le|s||h|$ pointwise, so
Theorem~\ref{i7:thm:W5} ($l=2\delta$) gives $\|f(s)\|\le\exp(C(r\|h\|_1+r^2\|h\|_2^2))\|[A]_\circ\|$ for $|s|=r$. If $N(h)=0$ there is nothing to prove; otherwise
$r:=1/N(h)$ yields $\|[F(h)A']_\circ\|\le e^{2C}N(h)\|[A]_\circ\|$, and $N(h)=N(\chi)$. If $[A]_\circ=[A_2]_\circ$ then
$\|[F(h)U(2\delta e_t)(A-A_2)]_\circ\|\le K_6N(\chi)\|[A-A_2]_\circ\|=0$. Density: for $G\in\Lc$, $G_M:=G1_{\{|G|\le M\}}$ satisfies $\|[G-G_M]_\circ\|\le\|G-G_M\|_{L^2}\to0$,
and $\Lc/\cN$ is dense in $\Hc$. Extend by uniform continuity; linearity in $\chi$ is that of $F$.

(b) For $A,A'\in L^\infty(\Sigma\{t>0\})$: $\ip{[A']_\circ}{B_\delta(\chi)[A]_\circ}_\circ=\E[\overline{\Thc A'}\,F(\tau_{\delta e_t}\chi)\,U(2\delta e_t)A]$, while
\[
\ip{B_\delta(J\chi)[A']_\circ}{[A]_\circ}_\circ=\E\big[\overline{\Thc\big(F(\tau_{\delta e_t}J\chi)U(2\delta e_t)A'\big)}\,A\big]
=\E\big[F(J\tau_{\delta e_t}J\chi)\,\overline{U(-2\delta e_t)\Thc A'}\,A\big],
\]
using $\overline{\Thc F(k)}=F(Jk)$ and $\Thc U(2\delta e_t)=U(-2\delta e_t)\Thc$. Here $J\tau_{\delta e_t}J\chi=\theta_0^*\tau_{\delta e_t}\theta_0^*\chi=\tau_{-\delta e_t}\chi$, and
(ii) applied with $U(2\delta e_t)$ turns the last expectation into the first. Both operators are bounded and $\cD$ is dense.

(c) For $A\in L^\infty(\Sigma\{t>0\})$, (F2) ($r\mp t\ge0$) gives $V(r-t,-x)[A]_\circ=[A_1]_\circ$, $A_1:=U((r-t)e_t-xe_x)A\in L^\infty(\Sigma\{t>0\})$, so
$B_\delta(\chi)[A_1]_\circ=[F(\tau_{\delta e_t}\chi)U(2\delta e_t)A_1]_\circ$ and, by (F2) applied to this element of $\Lc$, $U(a)(GG')=U(a)G\,U(a)G'$ and $U(a)F(k)=F(\tau_ak)$,
\[
V(r+t,x)B_\delta(\chi)V(r-t,-x)[A]_\circ=\big[F(\tau_{(\delta+r+t,x,0)}\chi)\,U(2(\delta+r)e_t)A\big]_\circ=B_{\delta+r}(\tau_{(t,x,0)}\chi)[A]_\circ ,
\]
because $\tau_{(t,x,0)}\chi$ vanishes outside $\{|t'|\le\delta+|t|\}\subset\{|t'|\le\delta+r\}$ and $\tau_{(\delta+r)e_t}\tau_{(t,x,0)}=\tau_{(\delta+r+t,x,0)}$. Conclude by density.

(d) $\|F(h)U(2\delta e_t)A\|_{L^2}\le\|F(h)\|_4\|A\|_4$ (Hölder and (ii)), and measurability follows from Lemma~\ref{i7:lem:W5loc}. With $A_M:=A1_{\{|A|\le M\}}$: $A_M\to A$ in $L^4$ and
$L^2$, so $B_\delta(\chi)[A_M]_\circ\to B_\delta(\chi)[A]_\circ$, while $\|[F(h)U(2\delta e_t)(A-A_M)]_\circ\|\le\|F(h)\|_4\|A-A_M\|_4\to0$; since
$B_\delta(\chi)[A_M]_\circ=[F(h)U(2\delta e_t)A_M]_\circ$, the identity follows. The iterated formula follows by induction from the right: the vector at each stage is the
class of a product of fields whose tests vanish outside $\{t\ge0\}$ (the shifts $s'_j-\delta_j=2\sum_{2\le i<j}\delta_i\ge0$ at the inner stage, and $s'_j+2\delta_1=s_j$ after the
outer insertion), which lies in $L^4(\Sigma\{t>0\})$ by Lemma~\ref{i7:lem:W1}(b), Hölder and Lemma~\ref{i7:lem:W5loc}; finally $\ip{\Omega}{[G]_\circ}_\circ=\E[G]$.
\end{proof}

\subsection{The planar point insertion}\label{i7:sub:W7}

In this subsection $e_t=(1,0)$ also denotes the planar unit vector; $\tau_z$ translates planar functions by $z\in\R^2$ and tests by $(z,0)$.

\begin{theorem}[planar point insertion]\label{i7:thm:W7}
Assume the hypotheses of Theorem~\ref{i7:thm:W3}, and let $K_6$ be the constant of Theorem~\ref{i7:thm:W6}. Let $f\in(L^1\cap L^2)(\R;T_\C)$, $(\xi,\eta)\in\cW^2$,
$q_*:=\min(q(\xi),q(\eta))$. Then there is a unique $A_f(\xi,\eta)\in\cB(\Hc)$ with
\[
V(\xi-\delta_ne_t)\,B_{\delta_n}(\alpha_nf)\,V(\eta-\delta_ne_t)\longrightarrow A_f(\xi,\eta)\quad\text{in operator norm}
\]
for every sequence $\alpha_n\in L^2(\R^2)$ with $\int\alpha_n=1$, $\sup_n\|\alpha_n\|_1<\infty$, $\supp\alpha_n\subset\{|z|\le s_n\}$, $s_n\to0$, and every choice
$\delta_n\in[s_n,q_*]$ (for the $n$ with $s_n\le q_*$). Moreover, with $K_7:=14K_6$:
\begin{itemize}
\item[(a)] $f\mapsto A_f(\xi,\eta)$ is linear and $\|A_f(\xi,\eta)\|\le K_7(\|f\|_1+\|f\|_2/q_*)$;
\item[(b)] $(\xi,\eta)\mapsto A_f(\xi,\eta)$ is operator-norm holomorphic on $\cW^2$;
\item[(c)] $V(\kappa)A_f(\xi,\eta)V(\kappa')=A_f(\kappa+\xi,\eta+\kappa')$ for $q(\kappa),q(\kappa')\ge0$;
\item[(d)] $A_f(\xi,\eta)^*=A_{Jf}(-\Thc\bar\eta,-\Thc\bar\xi)$;
\item[(e)] $A_f(\xi,a)A_g(b,\eta)=A_f(\xi,a')A_g(b',\eta)$ whenever $a,b,a',b'\in\cW$ and $a+b=a'+b'$;
\item[(f)] for $\alpha\in L^2(\R^2)$ with compact support in $\{|t|<q_*\}$ and every $\delta\in[\sup_{\supp\alpha}|t|,q_*]$,
$V(\xi-\delta e_t)B_\delta(\alpha f)V(\eta-\delta e_t)=\int\alpha(z)A_f(\xi+z,\eta-z)\,dz$ (norm-convergent Bochner integral); in particular
$B_\delta(\alpha f)=\int\alpha(z)A_f(\delta e_t+z,\delta e_t-z)\,dz$ for $\alpha$ compactly supported in $\{|t|<\delta\}$.
\end{itemize}
\end{theorem}
\begin{proof}
\emph{Step 0 (calculus of $V$).} By Lemma~\ref{i7:lem:W4}, $\|V(w)\|\le1$ for $q(w)\ge0$, $V(w)V(w')=V(w+w')$, $V(w)^*=V(-\Thc\bar w)$, and $V$ is norm-holomorphic on $\cW$ with
\eqref{i7:eq:Vderiv}.

\emph{Step 1 (smeared insertions).} For $\beta\in L^2(\R^2)$ with compact support let $t_\beta:=\sup\{|t|:(t,x)\in\supp\beta\}$; for $q_*>t_\beta$ and $\delta\in(0,q_*]$ with
$\delta\ge t_\beta$ (``admissible'') put $T_\beta(\xi,\eta):=V(\xi-\delta e_t)B_\delta(\beta f)V(\eta-\delta e_t)$.
(1.i) Independence of $\delta$: for admissible $\delta<\delta'$, Theorem~\ref{i7:thm:W6}(c) with $(t,x)=0$ gives $B_{\delta'}(\beta f)=V((\delta'-\delta)e_t)B_\delta(\beta f)V((\delta'-\delta)e_t)$;
the semigroup law finishes.
(1.ii) $\|T_\beta(\xi,\eta)\|\le K_6(\|\beta\|_1\|f\|_1+\|\beta\|_2\|f\|_2)$ since $\|\beta f\|_p=\|\beta\|_p\|f\|_p$; $T_\beta$ is linear in $\beta$ and in $f$.
(1.iii) $T_\beta$ is norm-holomorphic on $\{(\xi,\eta):q_*>t_\beta\}$: near a point with $q_*>\delta$ it is a product of norm-holomorphic factors in independent variables, and
near a point with $t_\beta<q_*\le\delta$ one first passes by (1.i) to an admissible $\delta''<q_*$. Also $V(\kappa)T_\beta(\xi,\eta)V(\kappa')=T_\beta(\kappa+\xi,\eta+\kappa')$ for
$q(\kappa),q(\kappa')\ge0$ (semigroup law; $q$ is superadditive).
(1.iv) By Theorem~\ref{i7:thm:W6}(b), Step 0 and $-\Thc\overline{(\eta-\delta e_t)}=-\Thc\bar\eta-\delta e_t$: $T^f_\beta(\xi,\eta)^*=T^{Jf}_{\tilde\beta}(-\Thc\bar\eta,-\Thc\bar\xi)$,
where $q(-\Thc\bar\eta)=q(\eta)$ and $t_{\tilde\beta}=t_\beta$.

(1.v) Convolution identity. Let $\gamma\in L^1(\R^2)$ with compact support and $\sup_{\supp\gamma}|t|\le s<r$, $\delta\ge t_\beta$, $\delta+r\le q_*$, and
$\Lambda(z):=V(r+t,x)B_\delta(\beta f)V(r-t,-x)$ for real $z=(t,x)$ with $|t|\le r$. By Theorem~\ref{i7:thm:W6}(c), $\Lambda(z)=B_{\delta+r}(\tau_z(\beta f))$, norm-continuous in $z$
($\tau_z$ is strongly continuous on $L^1\cap L^2$ and $B_{\delta+r}$ is bounded linear). The Bochner integral $\int\gamma(z)\tau_z(\beta f)dz$ equals $(\gamma*\beta)f$ in $L^1\cap L^2$
(pair with bounded compactly supported functions and use Fubini), and $B_{\delta+r}$ commutes with Bochner integrals, so $\int\gamma\Lambda=B_{\delta+r}((\gamma*\beta)f)$; as
$t_{\gamma*\beta}\le t_\beta+s<\delta+r\le q_*$, $\delta+r$ is admissible for $\gamma*\beta$, and
\begin{equation}\label{i7:eq:W7conv1}
T_{\gamma*\beta}(\xi,\eta)=V(\xi-(\delta+r)e_t)\Big[\int\gamma(z)\Lambda(z)\,dz\Big]V(\eta-(\delta+r)e_t).
\end{equation}
Since $V(\xi-(\delta+r)e_t)V(r+t,x)=V(\xi+z-\delta e_t)$ (both factors have $q\ge0$) and likewise on the right,
$V(\xi-(\delta+r)e_t)\Lambda(z)V(\eta-(\delta+r)e_t)=T_\beta(\xi+z,\eta-z)$ for real $z$ with $|t|\le s$ (here $q(\xi+z)=q(\xi)+t\ge\delta+r-s>t_\beta$), whence
\begin{equation}\label{i7:eq:W7conv2}
T_{\gamma*\beta}(\xi,\eta)=\int\gamma(z)\,T_\beta(\xi+z,\eta-z)\,dz .
\end{equation}

(1.vi) Taylor bounds. Let now $\supp\gamma\subset\{|z|\le s\}$. $\Lambda$ is norm-$C^2$ on $\{|t|<r\}$; by the Leibniz rule and \eqref{i7:eq:Vderiv} (the arguments of the two
$V$'s have $q=r\pm t\ge r-|z|$) every first derivative is bounded by $2\|B\|/(e(r-|z|))$ and every second derivative by $(4+2+4)\|B\|/(e^2(r-|z|)^2)$, $B:=B_\delta(\beta f)$.
The mean value inequality and Taylor's formula with integral remainder give, for $|z|\le s<r$,
\begin{equation}\label{i7:eq:W7taylor}
\|\Lambda(z)-\Lambda(0)\|\le\frac{2\sqrt2\,|z|\,\|B\|}{e(r-s)},\qquad\|\Lambda(z)-\Lambda(0)-D\Lambda(0)z\|\le\frac{10\,|z|^2\|B\|}{e^2(r-s)^2}.
\end{equation}
Hence, if $\int\gamma=0$ and $\int\gamma(z)z_i\,dz=0$ ($i=1,2$), then $\|\int\gamma\Lambda\|\le10s^2\|\gamma\|_1\|B\|/(e^2(r-s)^2)$.

\emph{Step 2 (existence, bound, holomorphy, (c)--(e)).} Let $\rho_\varepsilon:=(\pi\varepsilon^2)^{-1}\mathbf 1_{\{|z|\le\varepsilon\}}$, $\beta_\varepsilon:=\rho_\varepsilon*\rho_\varepsilon$; then $\beta_\varepsilon$ is even,
$\int\beta_\varepsilon=1$, its support lies in $\{|z|\le2\varepsilon\}$, and $\|\beta_\varepsilon\|_2\le\|\rho_\varepsilon\|_2=(\pi\varepsilon^2)^{-1/2}$. Fix $(\xi,\eta)$, $0<q\le q_*$,
$\varepsilon\le q/4$. Then $\beta_\varepsilon-\beta_{\varepsilon/2}=\gamma*\beta'$ with $\gamma:=\rho_\varepsilon-\rho_{\varepsilon/2}$ (even, $\int\gamma=0$, $\|\gamma\|_1\le2$) and
$\beta':=\rho_\varepsilon+\rho_{\varepsilon/2}$ ($\|\beta'\|_1=2$, $\|\beta'\|_2\le3\pi^{-1/2}\varepsilon^{-1}$), both supported in $\{|z|\le\varepsilon\}$. Apply
\eqref{i7:eq:W7conv1} and (1.vi) with $\delta=\varepsilon$, $r=q-\varepsilon$, $s=\varepsilon$ ($r-s\ge q/2$), $\|V\|\le1$ and Theorem~\ref{i7:thm:W6}(a):
\begin{equation}\label{i7:eq:W7dyad}
\|T_{\beta_\varepsilon}-T_{\beta_{\varepsilon/2}}\|\le\frac{80\varepsilon^2}{e^2q^2}K_6\Big(2\|f\|_1+\frac{3\|f\|_2}{\sqrt\pi\,\varepsilon}\Big)
=\frac{160K_6}{e^2}\frac{\varepsilon^2}{q^2}\|f\|_1+\frac{240K_6}{e^2\sqrt\pi}\frac{\varepsilon}{q^2}\|f\|_2 .
\end{equation}

Let $\varepsilon_k:=2^{-k}$ and $k_0$ be least with $2^{-k_0}\le q_*/4$, so $\varepsilon_{k_0}>q_*/8$; $\sum_{k\ge k_0}\varepsilon_k^2\le q_*^2/12$ and $\sum_{k\ge k_0}\varepsilon_k\le q_*/2$.
By \eqref{i7:eq:W7dyad} ($q=q_*$), $(T_{\beta_{\varepsilon_k}}(\xi,\eta))_{k\ge k_0}$ is norm-Cauchy; set $A_f(\xi,\eta):=\lim_kT_{\beta_{\varepsilon_k}}(\xi,\eta)$. With (1.ii),
$\|T_{\beta_{\varepsilon_{k_0}}}\|\le K_6(\|f\|_1+8\pi^{-1/2}\|f\|_2/q_*)$, so
\[
\|A_f(\xi,\eta)\|\le K_6\Big[\Big(1+\tfrac{40}{3e^2}\Big)\|f\|_1+\tfrac{8+120/e^2}{\sqrt\pi}\,\tfrac{\|f\|_2}{q_*}\Big]\le K_6\big(2.81\,\|f\|_1+13.68\,\|f\|_2/q_*\big),
\]
which is (a); linearity in $f$ from (1.ii).

On $\{(\xi,\eta):q_*\ge q\}$ the tail of \eqref{i7:eq:W7dyad} is uniform, and each $T_{\beta_{\varepsilon_k}}$ is norm-holomorphic
there for $2\varepsilon_k<q$ (1.iii); a locally uniform limit of norm-holomorphic functions is norm-holomorphic: (b). (c) follows from (1.iii) in the limit. (d):
$\tilde\beta_\varepsilon=\beta_\varepsilon$ ($\rho_\varepsilon$ real and even), $\|Jf\|_p=\|f\|_p$, so (1.iv) passes to the limit, the adjoint being norm-continuous. (e): for
$2\varepsilon$ below $q(\xi),q(\eta),q(a),q(b),q(a'),q(b')$, the semigroup law gives
$T^f_{\beta_\varepsilon}(\xi,a)T^g_{\beta_\varepsilon}(b,\eta)=V(\xi-2\varepsilon e_t)B_{2\varepsilon}(\beta_\varepsilon f)V(a+b-4\varepsilon e_t)B_{2\varepsilon}(\beta_\varepsilon g)V(\eta-2\varepsilon e_t)$,
the same expression for $(a',b')$; products of norm-convergent sequences converge.

\emph{Step 3 ((f) and uniqueness).} Let $\alpha$ be as in (f), $t_\alpha:=\sup_{\supp\alpha}|t|<q_*$ ($t_\alpha>0$ unless $\alpha=0$, in which case both sides vanish), $\delta=t_\alpha$,
$r=q_*-t_\alpha$, $0<\varepsilon<r/4$. (i) By \eqref{i7:eq:W7conv1} with $(\gamma,\beta)=(\beta_\varepsilon,\alpha)$ (here $\supp\gamma\subset\{|z|\le2\varepsilon\}$, $s=2\varepsilon<r$) and
$\int\beta_\varepsilon=1$, $T_\alpha(\xi,\eta)=V(\xi-q_*e_t)\Lambda(0)V(\eta-q_*e_t)$ and \eqref{i7:eq:W7taylor} with $s=2\varepsilon$:
$\|T_{\beta_\varepsilon*\alpha}(\xi,\eta)-T_\alpha(\xi,\eta)\|\le4\sqrt2\,\varepsilon\|B_\delta(\alpha f)\|/(e(r-2\varepsilon))\to0$.
(ii) By \eqref{i7:eq:W7conv2} with $(\gamma,\beta)=(\alpha,\beta_\varepsilon)$, $\delta'=2\varepsilon$, $r'=q_*-2\varepsilon$ (so that $\sup_{\supp\alpha}|t|=t_\alpha<r'$ and
$\delta'+r'=q_*$): $T_{\alpha*\beta_\varepsilon}(\xi,\eta)=\int\alpha(z)T_{\beta_\varepsilon}(\xi+z,\eta-z)dz$; here $\min(q(\xi+z),q(\eta-z))\ge q_*-t_\alpha=r$ on $\supp\alpha$, so by
Step 2 with $q=r$ the integrand converges to $A_f(\xi+z,\eta-z)$ uniformly on $\supp\alpha$ along $\varepsilon=\varepsilon_k$; the limit is norm-continuous in $z$ by (b), and
$\|\alpha\|_1<\infty$. Since $\alpha*\beta_\varepsilon=\beta_\varepsilon*\alpha$, (i) and (ii) give $T_\alpha(\xi,\eta)=\int\alpha(z)A_f(\xi+z,\eta-z)dz$, i.e.\ (f); for $\xi=\eta=\delta e_t$,
$q_*=\delta$ and $T_\alpha(\delta e_t,\delta e_t)=B_\delta(\alpha f)$ by (1.i).

Uniqueness: for $(\alpha_n)$ as in the statement and $n$ so large that $s_n<q_*$, (f) and $\int\alpha_n=1$ give
$T_{\alpha_n}(\xi,\eta)-A_f(\xi,\eta)=\int\alpha_n(z)[A_f(\xi+z,\eta-z)-A_f(\xi,\eta)]dz$, of norm at most
$\sup_n\|\alpha_n\|_1\sup_{|z|\le s_n}\|A_f(\xi+z,\eta-z)-A_f(\xi,\eta)\|\to0$ by (b); with (1.i) the limit depends neither on the approximate identity nor on $\delta_n$.
\end{proof}

\subsection{The ordered holomorphic kernel and the one kernel}\label{i7:sub:W8}

Here Theorems~\ref{i7:thm:W6} and~\ref{i7:thm:W7} are used for the coordinate pair $(t,x)$ and for the pair $(x,t)$ (same transverse coordinate $y$); the second
requires (iii) at $\{x=0\}$ and $\{x+t=0\}$, so this subsection uses the four planes of the pair. $K_7$ is the constant of Theorem~\ref{i7:thm:W7}, the same for both
pairs, and $K_8:=2K_7$.

\begin{lemma}[end entries]\label{i7:lem:W8end}
For a profile $f$, $a,b\in\cW$ and $\psi\in\Hc$: $\xi\mapsto\ip{\Omega}{A_f(\xi,a)\psi}$ and $\eta\mapsto A_f(b,\eta)\Omega$ are constant on $\cW$.
\end{lemma}
\begin{proof}
Theorem~\ref{i7:thm:W7}(c) with $\kappa'=0$ gives $A_f(\xi+\kappa,a)=V(\kappa)A_f(\xi,a)$ for $q(\kappa)\ge0$, so
$g(\xi+\kappa):=\ip{\Omega}{A_f(\xi+\kappa,a)\psi}=\ip{V(\kappa)^*\Omega}{A_f(\xi,a)\psi}=g(\xi)$. For $\xi,\xi'\in\cW$, $g(\xi)=g(\xi+\xi')=g(\xi')$ ($\cW$ is a convex cone).
Likewise $A_f(b,\eta+\kappa')\Omega=A_f(b,\eta)V(\kappa')\Omega=A_f(b,\eta)\Omega$.
\end{proof}

\begin{lemma}[unbounded vectors]\label{i7:lem:W8unb}
If $\chi\in(L^1\cap L^2)_\C$ vanishes outside $\{t\ge0\}$ then $F(\chi)\in\Lc\cap\bigcap_{p<\infty}L^p(\mu)$. If $\chi$ vanishes outside $\{|t|\le\delta\}$ and $A\in\Lc\cap L^4(\mu)$,
then $F(\tau_{\delta e_t}\chi)U(2\delta e_t)A\in\Lc$ and $B_\delta(\chi)[A]_\circ=[F(\tau_{\delta e_t}\chi)U(2\delta e_t)A]_\circ$.
\end{lemma}
\begin{proof}
This follows from Lemma~\ref{i7:lem:W5loc} and Theorem~\ref{i7:thm:W6}(d) (with $U(2\delta e_t)\Lc\subset\Lc$).
\end{proof}

\begin{proposition}[ordered holomorphic kernel]\label{i7:prop:W8}
Let $N\ge2$, $f_1,\dots,f_N\in(L^1\cap L^2)(\R;T_\C)$, $\mathcal Z_N:=\{Z=(z_1,\dots,z_N)\in(\C^2)^N:\Delta_j:=z_{j+1}-z_j\in\cW,\ 1\le j<N\}$. For $Z\in\mathcal Z_N$ and $\kappa\in\cW$
put
\[
S_N(Z;f):=\big\langle\Omega,\,A_{f_1}(\kappa,\tfrac{\Delta_1}2)\,A_{f_2}(\tfrac{\Delta_1}2,\tfrac{\Delta_2}2)\cdots A_{f_{N-1}}(\tfrac{\Delta_{N-2}}2,\tfrac{\Delta_{N-1}}2)\,
A_{f_N}(\tfrac{\Delta_{N-1}}2,\kappa)\,\Omega\big\rangle_\circ
\]
(ordered product). Then: (a) $S_N(Z;f)$ does not depend on $\kappa$; (b) it is holomorphic on the open set $\mathcal Z_N$ and $S_N(Z+(c,\dots,c);f)=S_N(Z;f)$ for $c\in\C^2$;
(c) it is $N$-linear in $(f_1,\dots,f_N)$; (d) $|S_N(Z;f)|\le K_8^N\prod_j(\|f_j\|_1+q_*^{-1}\|f_j\|_2)$ with $q_*:=\min_jq(\Delta_j)$; (e) if $\chi_j=\alpha_j(t,x)f_j(y)$ with
$\alpha_j\in L^2(\R^2)$ of compact support $K_j$ and $\max_{K_j}t<\min_{K_{j+1}}t$ ($1\le j<N$), then
\[
S_N(\chi_1,\dots,\chi_N)=\int_{(\R^2)^N}\prod_j\alpha_j(z_j)\,S_N(Z;f)\,dZ\qquad\text{(absolutely convergent).}
\]
\end{proposition}
\begin{proof}
(a) follows from Lemma~\ref{i7:lem:W8end}, applied first to the last entry and then to the first entry with $\psi=A_{f_2}(\cdot)\cdots A_{f_N}(\cdot,\kappa)\Omega$. (b) $Z\mapsto\Delta_j/2$ is
linear, each factor is operator-norm holomorphic by Theorem~\ref{i7:thm:W7}(b), products in $\cB(\Hc)$ and $\ip{\Omega}{\cdot\,\Omega}$ preserve holomorphy; only differences
enter. (c) follows from Theorem~\ref{i7:thm:W7}(a). (d) Take $\kappa=\Delta_1/2$; $q(\Delta_j/2)=q(\Delta_j)/2\ge q_*/2$, so Theorem~\ref{i7:thm:W7}(a) gives
$\|A_{f_j}(\cdot,\cdot)\|\le K_7(\|f_j\|_1+2q_*^{-1}\|f_j\|_2)\le2K_7(\|f_j\|_1+q_*^{-1}\|f_j\|_2)$ for every factor, and $\|\Omega\|=1$.

(e) \emph{Slabs.} Choose $a_0<a_1<\dots<a_N$ with $K_j\subset\{a_{j-1}<t<a_j\}$, put $\delta_j:=(a_j-a_{j-1})/2$ and $s_j:=(a_{j-1}+a_j)/2-a_0=2\sum_{i<j}\delta_i+\delta_j$. Both
sides are invariant under the common shift $\chi_j\mapsto\tau_{-a_0e_t}\chi_j$ (left: (ii) and Lemma~\ref{i7:lem:W1}; right: substitution and (b)), so let $a_0=0$. Put
$\alpha^0_j:=\tau_{-s_je_t}\alpha_j$ and $\chi^0_j:=\alpha^0_jf_j$: compact support in $\{|t|<\delta_j\}$, $\chi_j=\tau_{s_je_t}\chi^0_j$, $\|\chi^0_j\|_1=\|\alpha_j\|_1\|f_j\|_1$,
$\|\chi^0_j\|_2=\|\alpha_j\|_2\|f_j\|_2$.

\emph{Iteration.} Put $G_N:=F(\tau_{\delta_Ne_t}\chi^0_N)$, $G_j:=F(\tau_{\delta_je_t}\chi^0_j)\,U(2\delta_je_t)G_{j+1}$. Each $G_j$ is a product of fields with tests vanishing outside
$\{t\ge0\}$, hence in $\Lc\cap\bigcap_pL^p$ (Lemma~\ref{i7:lem:W8unb}, Hölder), so $B_{\delta_N}(\chi^0_N)\Omega=[G_N]_\circ$ and $B_{\delta_j}(\chi^0_j)[G_{j+1}]_\circ=[G_j]_\circ$.
Multiplicativity of $U$ and $U(a)F(\chi)=F(\tau_a\chi)$ give $G_1=\prod_jF(\tau_{s_je_t}\chi^0_j)=\prod_jF(\chi_j)$; thus
$\ip{\Omega}{B_{\delta_1}(\chi^0_1)\cdots B_{\delta_N}(\chi^0_N)\Omega}_\circ=\int G_1\,d\mu=S_N(\chi_1,\dots,\chi_N)$.

\emph{Insertion.} Let $A_j(z):=A_{f_j}(\delta_je_t+z,\delta_je_t-z)$, norm-continuous for $|t|<\delta_j$ and bounded by $K_7(\|f_j\|_1+\|f_j\|_2/(\delta_j-|t|))$, hence bounded
on $\supp\alpha^0_j$. By Theorem~\ref{i7:thm:W7}(f), $B_{\delta_j}(\chi^0_j)=\int\alpha^0_j(z)A_j(z)\,dz$. For fixed bounded $Y$ and $\psi\in\Hc$, $X\mapsto\ip{Y^*\Omega}{X\psi}$ is a
bounded functional on $\cB(\Hc)$ and passes through Bochner integrals; applied $N$ times from the left,
\[
\ip{\Omega}{\textstyle\prod_jB_{\delta_j}(\chi^0_j)\Omega}_\circ=\int\alpha^0_1(z_1)\cdots\int\alpha^0_N(z_N)\,\ip{\Omega}{A_1(z_1)\cdots A_N(z_N)\Omega}_\circ\,dz_N\cdots dz_1 .
\]
The inner function is continuous in $Z$ on $\prod_j\{|t_j|<\delta_j\}$ and bounded on the compact set $\prod_j\supp\alpha^0_j$, and $\alpha^0_j\in L^1$; Fubini turns the
iterated integral into one over $(\R^2)^N$.

\emph{Identification.} Fix real $Z$ with $|t_j|<\delta_j$, $w_j:=z_j+s_je_t$, $W:=(w_j)_j$. Then $a_j:=\delta_je_t-z_j$ and $b_j:=\delta_je_t+z_j$ lie in $\cW$, and
$a_j+b_{j+1}=(\delta_j+\delta_{j+1})e_t+z_{j+1}-z_j=w_{j+1}-w_j=:\Delta_j\in\cW$ (as $s_{j+1}-s_j=\delta_j+\delta_{j+1}$), so $W\in\mathcal Z_N$. In
$A_1(z_1)\cdots A_N(z_N)=A_{f_1}(b_1,a_1)A_{f_2}(b_2,a_2)\cdots A_{f_N}(b_N,a_N)$ apply Theorem~\ref{i7:thm:W7}(e) to the adjacent pairs $(a_j,b_{j+1})\mapsto(\Delta_j/2,\Delta_j/2)$,
$j=1,\dots,N-1$ in turn (each step touches only those two entries), then Lemma~\ref{i7:lem:W8end} to $b_1$ and $a_N$: $\ip{\Omega}{A_1(z_1)\cdots A_N(z_N)\Omega}_\circ=S_N(W;f)$.
Substituting $w_j=z_j+s_je_t$, $\int\prod_j\alpha^0_j(z_j)S_N(Z+(s_je_t)_j;f)\,dZ=\int\prod_j\alpha_j(w_j)S_N(W;f)\,dW$.
\end{proof}

\emph{Charts.} For $\sigma\in\mathfrak S_N$ we put $Z_\sigma:=(z_{\sigma(1)},\dots,z_{\sigma(N)})$, and $f_\sigma$ likewise. We put $\mathrm{sw}(t,x):=(x,t)$,
$D^t_\sigma:=\{Z:Z_\sigma\in\mathcal Z_N\}=\{\re(t_{\sigma(j+1)}-t_{\sigma(j)})>|\im(x_{\sigma(j+1)}-x_{\sigma(j)})|\ \forall j\}$, $D^x_\sigma:=\{Z:(\mathrm{sw}Z)_\sigma\in\mathcal Z_N\}$ and
$\cD_N:=\bigcup_\sigma(D^t_\sigma\cup D^x_\sigma)$. We write $S'_N$ for the kernel of Proposition~\ref{i7:prop:W8} for the coordinate pair $(x,t)$, and we put
$\cD^\sharp_\R:=\{Z\in(\R^2)^N:\text{the }t_j\text{ pairwise distinct and the }x_j\text{ pairwise distinct}\}$. Note, with $\mathrm{rev}\in\mathfrak S_N$ the order reversal $\mathrm{rev}(j):=N+1-j$, that $D^{t}_{\sigma\circ\mathrm{rev}}=\{Z:-Z_\sigma\in\mathcal Z_N\}$
(the chart with time $-t$) and likewise for $x$, so the charts with the four time directions $\pm e_t,\pm e_x$ are all among the $D^t_\sigma$, $D^x_\sigma$.

\begin{proposition}[one kernel]\label{i7:prop:W9}
(a) $D^t_\sigma\cap D^t_{\sigma'}=\emptyset=D^x_\sigma\cap D^x_{\sigma'}$ for $\sigma\ne\sigma'$; all $D^t_\sigma$, $D^x_\sigma$, $D^t_\sigma\cap D^x_{\sigma'}$ are convex, open and
nonempty; $\cD_N$ is connected and $\cD_N\cap(\R^2)^N=\{\text{all }t_j\text{ distinct}\}\cup\{\text{all }x_j\text{ distinct}\}$.
(b) $\mathcal S_N(Z;f):=S_N(Z_\sigma;f_\sigma)$ on $D^t_\sigma$ and $:=S'_N((\mathrm{sw}Z)_\sigma;f_\sigma)$ on $D^x_\sigma$ is consistent and defines a holomorphic function on
$\cD_N$, $N$-linear in $f$, with $|\mathcal S_N(Z;f)|\le K_8^N\prod_j(\|f_j\|_1+q_*^{-1}\|f_j\|_2)$ on each chart ($q_*$ the minimum of $q$ over the successive differences of
that chart, computed in its coordinates), such that for product tests $\chi_j=\alpha_jf_j$, $\alpha_j\in L^2(\R^2)$ with compact planar supports pairwise strictly ordered in $t$ or
pairwise strictly ordered in $x$ (in any order), $S_N(\chi_1,\dots,\chi_N)=\int\prod_j\alpha_j(z_j)\,\mathcal S_N(Z;f)\,dZ$.
(c) $\mathcal S_N(Z_\pi;f_\pi)=\mathcal S_N(Z;f)$ for all $\pi\in\mathfrak S_N$, $Z\in\cD_N$.
(d) $R(\theta)\cD_N=\cD_N$ and $\mathcal S_N(R(\theta)Z;\varrho(\theta)f)=\mathcal S_N(Z;f)$ for $\theta\in\frac\pi2\Z$ (with $\varrho(\theta)=\pi(R(\theta))$ acting pointwise on the
profiles); in particular $\mathcal S_N(R(-\pi/2)Z;\varrho(-\pi/2)f)=\mathcal S_N(Z;f)$.
\end{proposition}
\begin{proof}
(a) On $D^t_\sigma$, $\re t_{\sigma(1)}<\dots<\re t_{\sigma(N)}$, which determines $\sigma$; the same for $x$. Convexity: $\cW$ is convex and the defining maps are linear.
$O_{\sigma\sigma'}:=\{Z\text{ real}:t\text{ ordered by }\sigma,\ x\text{ ordered by }\sigma'\}$ is nonempty, open in $\R^{2N}$ and lies in $D^t_\sigma\cap D^x_{\sigma'}$ (for real
$\Delta$, $q(\Delta)$ is the $t$-component). Every $t$-chart meets every $x$-chart, so $\cD_N$ is connected; the real points are as stated.

(b) By (a) consistency concerns only $D^t_\sigma\cap D^x_{\sigma'}$. Product tests: if the supports are ordered in $t$ along $\sigma$, Proposition~\ref{i7:prop:W8}(e) for
$(\chi_{\sigma(1)},\dots,\chi_{\sigma(N)})$, the symmetry of $S_N$ and the substitution $W=Z_\sigma$ give $S_N(\chi_1,\dots,\chi_N)=\int\prod_j\alpha_j(z_j)S_N(Z_\sigma;f_\sigma)dZ$ (the
support lies in $D^t_\sigma$); for supports ordered in $x$, the same functions read $(\alpha_j\circ\mathrm{sw})(x,t)f_j(y)$ in the chart $(x,t)$, and Proposition~\ref{i7:prop:W8}(e)
there, with the substitution $Z\mapsto\mathrm{sw}Z$, gives the formula with $S'_N((\mathrm{sw}Z)_{\sigma'};f_{\sigma'})$. Let $Z_0\in O_{\sigma\sigma'}$, $\phi\in C^\infty_c(\R^2)$, $\phi\ge0$,
$\int\phi=1$, support in the unit disc, $\alpha_j:=\epsilon^{-2}\phi((\cdot-z_{0,j})/\epsilon)$. For small $\epsilon$ both orderings hold, so
$\int\prod_j\alpha_j(z_j)\,[S_N(Z_\sigma;f_\sigma)-S'_N((\mathrm{sw}Z)_{\sigma'};f_{\sigma'})]\,dZ=0$; both functions are continuous at $Z_0$, so $\epsilon\to0$ gives equality at $Z_0$.
Two holomorphic functions agreeing on the open subset $O_{\sigma\sigma'}$ of $\R^{2N}$ have equal complex Taylor coefficients at its points (real derivatives of the difference
vanish), hence agree on the connected open set $D^t_\sigma\cap D^x_{\sigma'}$. Holomorphy, multilinearity and the bound are chartwise (Proposition~\ref{i7:prop:W8}(b)--(d)).

(c) $(Z_\pi)_\rho=Z_{\pi\rho}$, so $Z\in D^t_\sigma\iff Z_\pi\in D^t_{\pi^{-1}\sigma}$ and $\mathcal S_N(Z_\pi;f_\pi)=S_N((Z_\pi)_{\pi^{-1}\sigma};(f_\pi)_{\pi^{-1}\sigma})=S_N(Z_\sigma;f_\sigma)$;
the same on $x$-charts.

(d) $R(\pi/2)(t,x)=(-x,t)$ gives $R(\pi/2)D^t_\sigma=D^x_\sigma$ and $R(\pi/2)D^x_\sigma=D^t_{\sigma\circ\mathrm{rev}}$, so $R(\theta)\cD_N=\cD_N$ and $R(\theta)\cD^\sharp_\R=\cD^\sharp_\R$.
$G_\theta(Z):=\mathcal S_N(R(\theta)Z;\varrho(\theta)f)$ is holomorphic on $\cD_N$. Let $Z_0\in\cD^\sharp_\R$, $\alpha_j$ as in (b), $\chi_j:=\alpha_jf_j$. Substituting $Z'=R(\theta)Z$ and
using (b) (the supports of $\alpha_j\circ R(-\theta)$ are the $\epsilon$-discs about the points of $R(\theta)Z_0\in\cD^\sharp_\R$, strictly $t$-ordered for small $\epsilon$):
\[
\begin{aligned}
\int\prod_j\alpha_j(z_j)G_\theta(Z)\,dZ&=\int\prod_j(\alpha_j\circ R(-\theta))(z'_j)\,\mathcal S_N(Z';\varrho(\theta)f)\,dZ'\\
&=S_N(R(-\theta)^*\chi_1,\dots,R(-\theta)^*\chi_N),
\end{aligned}
\]
since $(R(-\theta)^*\chi_j)(z,y)=\pi(R(\theta))\alpha_j(R(-\theta)z)f_j(y)$. The map $(z,y)\mapsto(R(-\theta)z,y)$, $\theta\in\frac\pi2\Z$, is a product of two of the four
reflections $\{t=0\}$, $\{x=0\}$, $\{t\pm x=0\}$, so Lemma~\ref{i7:lem:W2} (four-plane form) with Lemma~\ref{i7:lem:W1}(e) gives $S_N(R(-\theta)^*\chi)=S_N(\chi)$ for Schwartz tests,
and then for $L^1\cap L^2$ tests by the continuity of $S_N$ and the isometry of the pull-back; so the left side equals $\int\prod_j\alpha_j(z_j)\mathcal S_N(Z;f)dZ$. Letting
$\epsilon\to0$, $G_\theta=\mathcal S_N$ on $\cD^\sharp_\R$, which meets every convex chart in a nonempty open subset of $\R^{2N}$; the identity theorem chart by chart gives
$G_\theta=\mathcal S_N$ on $\cD_N$.
\end{proof}

\subsection{The entire angular continuation}\label{i7:sub:W10}

\emph{Conventions.} The map $\varrho(\theta)=\pi(R(\theta))$ is a continuous unitary one-parameter group on $T_\C$ with $\varrho(2\pi)=1$, so $\varrho(\theta)=\sum_ae^{iw_a\theta}P_a$ with
integers $w_a$ and mutually orthogonal projections $P_a$, $\sum_aP_a=1$. We put $w_{\max}:=\max_a|w_a|$, which is $1$ for the representations of Theorem~\ref{i7:thm:main}; the proof is written for
$w_{\max}\le2$. For $\zeta\in\C$ we put $\varrho(\zeta):=\sum_ae^{iw_a\zeta}P_a$ and $R(\zeta):=\cos\zeta\,I+\sin\zeta\,J_\circ$ on $\C^2$, where $J_\circ:=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, so that
$R(\zeta)R(\zeta')=R(\zeta+\zeta')$. We write $z\cdot w$ for the bilinear pairing on $\C^2$ and put $E:=\{\pm e_t,\pm e_x\}$; for $e\in E$ we put $q_e(z):=\re(z\cdot e)-|\im(z\cdot J_\circ e)|$ and
$\cW_e:=\{q_e>0\}$ (so $\cW_{e_t}=\cW$, $\cW_{-e}=-\cW_e$, $\cW_{e_x}=J_\circ\cW$). We put $\mathcal P:=(L^1\cap L^2)(\R;T_\C)$, and $\mathcal A$ is the class of bounded measurable planar factors
with compact support. For $N\ge2$, $e\in E$ and $\sigma\in\mathfrak S_N$ we put
\[
\cD^{e,\sigma}_N:=\{Z\in(\C^2)^N:\ z_{\sigma(j+1)}-z_{\sigma(j)}\in\cW_e,\ 1\le j<N\},\qquad q^{e,\sigma}_*(Z):=\min_{j<N}q_e(z_{\sigma(j+1)}-z_{\sigma(j)}).
\]
These are the charts of Proposition~\ref{i7:prop:W9}: $\cD^{e_t,\sigma}_N=D^t_\sigma$, $\cD^{-e_t,\sigma}_N=D^t_{\sigma\circ\mathrm{rev}}$, $\cD^{e_x,\sigma}_N=D^x_\sigma$,
$\cD^{-e_x,\sigma}_N=D^x_{\sigma\circ\mathrm{rev}}$. The facts used from Subsections~\ref{i7:sub:W1}--\ref{i7:sub:W8} are:
\begin{itemize}
\item[(K1)] the kernel $\mathcal S_N$ of Proposition~\ref{i7:prop:W9} on $\cD_N=\bigcup_{e,\sigma}\cD^{e,\sigma}_N$, holomorphic in $Z$, $N$-linear in $f\in\mathcal P^N$, invariant under
simultaneous permutations of $(Z,f)$;
\item[(K2)] $|\mathcal S_N(Z;f)|\le K_8^N\prod_j(\|f_j\|_1+q^{e,\sigma}_*(Z)^{-1}\|f_j\|_2)$ for $Z\in\cD^{e,\sigma}_N$;
\item[(K3)] if $\chi_j=\alpha_jf_j$ with $\alpha_j\in\mathcal A$, $f_j\in\mathcal P$, and $\sup_{\supp\alpha_{\sigma(j)}}z\cdot e<\inf_{\supp\alpha_{\sigma(j+1)}}z\cdot e$ for $j<N$, then
$S_N(\chi_1,\dots,\chi_N)=\int_{(\R^2)^N}\prod_j\alpha_j(z_j)\,\mathcal S_N(Z;f)\,dZ$ (Proposition~\ref{i7:prop:W9}(b), the orderings along $-e_t$, $-e_x$ being the reversed
orderings along $e_t$, $e_x$);
\item[(K4)] $S_N(R(\pi/2)^*\chi)=S_N(\chi)$ and $|S_N(\chi)|\le(2K_1\sqrt N)^N\prod_jN(\chi_j)$ for all $\chi\in((L^1\cap L^2)(\R^3;T_\C))^N$ (Lemma~\ref{i7:lem:W2} for the
four planes of the pair, since $R(\pi/2)$ is the product of two of the four reflections, with Lemma~\ref{i7:lem:W1}(c),(e)).
\end{itemize}

\emph{Narrow configurations with data.} A configuration $X=(x_1,\dots,x_N)\in(\R^2)^N$ is narrow with data $(u,\gamma)$ if, after relabelling, $d_j:=x_{j+1}-x_j$ satisfies $\angle(d_j,u)\le\gamma<\pi/4$
for all $j$ ($u$ a unit vector); this is the notion of Subsection~\ref{i7:sub:setting}. Put $\delta:=\min_j|d_j|>0$, $c:=\cos(\gamma+\pi/4)>0$, $r_0:=\delta c/2$.

\begin{proposition}[entire angular continuation]\label{i7:prop:W10}
Let $N\ge2$, $X$ narrow with data $(u,\gamma)$, $0<r<r_0$, $m:=\delta c-2r>0$, $K_j:=\bar B(x_j,r)$, $K:=\prod_jK_j$. Let $\chi_j=\alpha_jf_j$ with $\alpha_j\in\mathcal A$,
$\supp\alpha_j\subset K_j$, $f_j\in\mathcal P$, and $D_N[\chi](\theta):=S_N(R(\theta)^*\chi)-S_N(\chi)$ for $\theta\in\R$.
\begin{itemize}
\item[(a)] $D_N[\chi]$ is the restriction to $\R$ of a unique entire function, again written $D_N[\chi]$, which is $\pi/2$-periodic (hence $2\pi$-periodic) and satisfies
\[
|D_N[\chi](\theta+i\beta)|\le C_\chi\,e^{(1+w_{\max})N|\beta|},\qquad C_\chi:=2K_8^N\prod_j\|\alpha_j\|_1\prod_j\big(\|f_j\|_1+m^{-1}\|f_j\|_2\big),
\]
with $C_\chi$ independent of $\theta$ and $\beta$.
\item[(b)] $c_\ell[\chi]:=\frac1{2\pi}\int_0^{2\pi}D_N[\chi](\theta)e^{-i\ell\theta}d\theta$ vanishes unless $\ell\in4\Z$ and $|\ell|\le(1+w_{\max})N$, and
\[
D_N[\chi](\zeta)=\sum_{\ell\in4\Z,\ 0<|\ell|\le(1+w_{\max})N}c_\ell[\chi]\,(e^{i\ell\zeta}-1)\qquad(\zeta\in\C).
\]
\item[(c)] (b) holds, with the same index set and $N$-linear $c_\ell$, for all $\chi_j\in(L^1\cap L^2)(\R^3;T_\C)$ vanishing outside $K_j\times\R$.
\end{itemize}
For $w_{\max}\le2$ the exponent is $3N$; for the weights $0,\pm1$ of Theorem~\ref{i7:thm:main} it is $2N$.
\end{proposition}

\begin{proof}
\emph{Claim A.} For $d\in\R^2$, a unit vector $e\in\R^2$ and $\zeta=\theta+i\beta$: $\re(R(\zeta)d\cdot e)=\cosh\beta\,(R(\theta)d\cdot e)$ and
$\im(R(\zeta)d\cdot J_\circ e)=\sinh\beta\,(R(\theta)d\cdot e)$; hence $q_e(R(\zeta)d)=e^{-|\beta|}\,R(\theta)d\cdot e$ whenever $R(\theta)d\cdot e\ge0$. Indeed $\cos(i\beta)=\cosh\beta$,
$\sin(i\beta)=i\sinh\beta$, so $R(i\beta)=\cosh\beta\,I+i\sinh\beta\,J_\circ$ and $R(\zeta)=R(i\beta)R(\theta)$; with $d':=R(\theta)d\in\R^2$, $R(\zeta)d=\cosh\beta\,d'+i\sinh\beta\,J_\circ d'$,
and $J_\circ d'\cdot J_\circ e=d'\cdot e$. So $q_e(R(\zeta)d)=\cosh\beta\,d'\cdot e-|\sinh\beta|\,|d'\cdot e|=(\cosh\beta-|\sinh\beta|)\,d'\cdot e=e^{-|\beta|}d'\cdot e$.

\emph{Claim B.} For every $\phi\in\R$ there is $e(\phi)\in E$ with $\angle(R(\phi)u,e(\phi))\le\pi/4$, and for all $Z\in K$, $j<N$: $R(\phi)(z_{j+1}-z_j)\cdot e(\phi)\ge m$.
Consequently, for $\zeta=\phi+i\beta$, $R(\zeta)Z\in\cD^{e(\phi),\id}_N\subset\cD_N$ with $q^{e(\phi),\id}_*(R(\zeta)Z)\ge e^{-|\beta|}m$, and the sets $R(\phi)K_1,\dots,R(\phi)K_N$ are
strictly increasing in the $e(\phi)$-coordinate with gaps $\ge m$. Indeed, the four directions of $E$ are $\pi/2$ apart, so one of them is within $\pi/4$ of $R(\phi)u$; by the triangle
inequality for angles, $\angle(R(\phi)d_j,e(\phi))\le\angle(d_j,u)+\pi/4\le\gamma+\pi/4<\pi/2$, so $R(\phi)d_j\cdot e(\phi)\ge|d_j|\,c\ge\delta c$. For $Z\in K$ write $z_j=x_j+\epsilon_j$,
$|\epsilon_j|\le r$: $R(\phi)(z_{j+1}-z_j)\cdot e(\phi)\ge\delta c-2r=m$. Claim A with $d=z_{j+1}-z_j$, $\theta=\phi$, $e=e(\phi)$ gives
$q_{e(\phi)}(R(\zeta)(z_{j+1}-z_j))=e^{-|\beta|}R(\phi)(z_{j+1}-z_j)\cdot e(\phi)\ge e^{-|\beta|}m>0$. The ordering statement is the case $\beta=0$ with $z_j\in K_j$, $z_{j+1}\in K_{j+1}$
arbitrary.

\emph{Step 1 (the entire function along the orbit).} For $Z\in K$ let $g_Z(\zeta):=\mathcal S_N(R(\zeta)Z;\varrho(\zeta)f)$, where $\varrho(\zeta)f:=(\varrho(\zeta)f_1,\dots,\varrho(\zeta)f_N)$,
$\varrho(\zeta)$ acting pointwise in $y$. By Claim B, $R(\zeta)Z\in\cD_N$ for every $\zeta\in\C$. By $N$-linearity,
\[
g_Z(\zeta)=\sum_{a_1,\dots,a_N}e^{i(w_{a_1}+\dots+w_{a_N})\zeta}\,\mathcal S_N\big(R(\zeta)Z;P_{a_1}f_1,\dots,P_{a_N}f_N\big),
\]
a finite sum; each term is an exponential times the composition of the entire map $\zeta\mapsto R(\zeta)Z\in\cD_N$ with a holomorphic function (K1). Hence $g_Z$ is entire,
$(\zeta,Z)\mapsto g_Z(\zeta)$ is continuous on $\C\times K$, and $g_Z(\zeta+2\pi)=g_Z(\zeta)$ (the weights are integers, $R(\zeta+2\pi)=R(\zeta)$). Bound:
$\|\varrho(\zeta)v\|^2=\sum_ae^{-2w_a\beta}\|P_av\|^2\le e^{2w_{\max}|\beta|}\|v\|^2$, so $\|\varrho(\zeta)f_j\|_p\le e^{w_{\max}|\beta|}\|f_j\|_p$; (K2) on $\cD^{e(\phi),\id}_N$ with
$q_*\ge e^{-|\beta|}m$ gives
\[
\begin{gathered}
|g_Z(\zeta)|\le K_8^N\prod_j\big(e^{w_{\max}|\beta|}\|f_j\|_1+e^{(1+w_{\max})|\beta|}m^{-1}\|f_j\|_2\big)\le G\,e^{(1+w_{\max})N|\beta|},\\
G:=K_8^N\prod_j\big(\|f_j\|_1+m^{-1}\|f_j\|_2\big).
\end{gathered}
\]

\emph{Step 2 (smearing at every real angle).} For real $\theta$, $(R(\theta)^*\chi_j)(z,y)=\varrho(\theta)^{-1}\alpha_j(R(\theta)z)f_j(y)$, i.e.\ $R(\theta)^*\chi_j=(\alpha_j\circ R(\theta))\,\varrho(-\theta)f_j$,
a product test with $\alpha_j\circ R(\theta)\in\mathcal A$ supported in $R(-\theta)K_j$. By Claim B with $\phi=-\theta$ these supports are strictly increasing in the $e(-\theta)$-coordinate, so
(K3) with $e=e(-\theta)$, $\sigma=\id$ gives $S_N(R(\theta)^*\chi)=\int\prod_j\alpha_j(R(\theta)z_j)\,\mathcal S_N(Z;\varrho(-\theta)f)\,dZ$. The substitution $Z=R(-\theta)Z'$ (orthogonal on
$(\R^2)^N$) yields
\[
S_N(R(\theta)^*\chi)=\int_K\prod_j\alpha_j(z'_j)\,g_{Z'}(-\theta)\,dZ',\qquad S_N(\chi)=\int_K\prod_j\alpha_j(z_j)\,g_Z(0)\,dZ\ \ (\theta=0).
\]
Hence $D_N[\chi](\theta)=\mathcal E(\theta)$ for all real $\theta$, where $\mathcal E(\zeta):=\int_K\prod_j\alpha_j(z_j)\,[g_Z(-\zeta)-g_Z(0)]\,dZ$.

\emph{Step 3 ($\mathcal E$ is entire, bounded, periodic).} The integrand is jointly continuous and bounded by $|\prod_j\alpha_j(z_j)|\,2Ge^{(1+w_{\max})N|\beta|}$, so $\mathcal E$ is
continuous (dominated convergence) and $\oint_{\partial\Delta}\mathcal E\,d\zeta=0$ for every triangle $\Delta$ (Fubini and Cauchy); by Morera's theorem $\mathcal E$ is entire, and
$|\mathcal E(\theta+i\beta)|\le\prod_j\|\alpha_j\|_1\,(Ge^{(1+w_{\max})N|\beta|}+G)\le C_\chi e^{(1+w_{\max})N|\beta|}$. $\mathcal E(\zeta+2\pi)=\mathcal E(\zeta)$ by Step 1. Since
$R(\theta+\pi/2)=R(\theta)R(\pi/2)$ and $(R_1R_2)^*=R_2^*R_1^*$, (K4) gives $S_N(R(\theta+\pi/2)^*\chi)=S_N(R(\pi/2)^*R(\theta)^*\chi)=S_N(R(\theta)^*\chi)$, so
$\mathcal E(\theta+\pi/2)=\mathcal E(\theta)$ on $\R$, hence on $\C$ by the identity theorem, which also gives the uniqueness of the entire extension. This is (a).

\emph{Step 4 (Fourier coefficients).} As $\mathcal E$ is entire and $2\pi$-periodic, $\mathcal F(w):=\mathcal E(\zeta)$ for $w=e^{i\zeta}$ is well defined and holomorphic on $\C\setminus\{0\}$
(local inverses of $\zeta\mapsto e^{i\zeta}$ differ by multiples of $2\pi$), with Laurent expansion $\mathcal F(w)=\sum_{\ell\in\Z}c_\ell w^\ell$, where for every $\beta\in\R$
\[
c_\ell=\frac1{2\pi i}\oint_{|w|=e^{-\beta}}\mathcal F(w)w^{-\ell-1}dw=\frac1{2\pi}\int_0^{2\pi}\mathcal E(\theta+i\beta)e^{-i\ell(\theta+i\beta)}d\theta ;
\]
at $\beta=0$ this is $c_\ell[\chi]$, and $\mathcal E(\zeta)=\sum_\ell c_\ell e^{i\ell\zeta}$ absolutely. Thus $|c_\ell|\le C_\chi e^{(1+w_{\max})N|\beta|+\ell\beta}$ for all $\beta$:
$\beta\to-\infty$ gives $c_\ell=0$ for $\ell>(1+w_{\max})N$, $\beta\to+\infty$ gives $c_\ell=0$ for $\ell<-(1+w_{\max})N$. The $\pi/2$-periodicity gives $c_\ell(e^{i\ell\pi/2}-1)=0$, so
$c_\ell=0$ for $\ell\notin4\Z$. $\mathcal E(0)=0$ gives $c_0=-\sum_{\ell\ne0}c_\ell$, whence (b).

\emph{Step 5 (all tests with the same planar supports).} Let $\cX:=\prod_j(L^1\cap L^2)(K_j\times\R;T_\C)$ with the norm $N$ in each factor. $R(\theta)^*$ preserves $\|\cdot\|_1$ and
$\|\cdot\|_2$ (orthogonal change of variables, unitary $\varrho(-\theta)$), so (K4) and telescoping give, uniformly in $\theta\in\R$,
\[
|D_N[\chi](\theta)-D_N[\chi'](\theta)|\le2(2K_1\sqrt N)^N\sum_j N(\chi_j-\chi'_j)\prod_{i\ne j}\max(N(\chi_i),N(\chi'_i)).
\]
Finite sums of product tests $\alpha f$, $\alpha\in\mathcal A$ supported in $K_j$, are dense in $(L^1\cap L^2)(K_j\times\R;T_\C)$: truncate to $|y|\le M$ (convergence in $N$), and on the
bounded set $K_j\times[-M,M]$ the $L^2$ norm dominates the $L^1$ norm while simple tensors with bounded factors are $L^2$-dense. By (b) and $N$-linearity, $D_N[\chi^{(n)}]$ is, for
approximants $\chi^{(n)}$ in these spans, a trigonometric polynomial with frequencies in $\Lambda:=\{\ell\in4\Z:0<|\ell|\le(1+w_{\max})N\}\cup\{0\}$ vanishing at $0$; $D_N[\chi]$ is its
uniform limit, hence of the same form (the space of such polynomials is finite-dimensional), and $c_\ell[\chi]=\lim_nc_\ell[\chi^{(n)}]$, which is $N$-linear in $\chi$. This is (c).
\end{proof}

For $N=1$, $D_1\equiv0$ by Lemma~\ref{i7:lem:W2}. The threshold is explicit: $r<\frac12\min_j|d_j|\cos(\gamma+\pi/4)$, and $C_\chi$ depends on the neighbourhood only through
$m=\delta c-2r$. The extension of (a)--(c) from small product neighbourhoods to arbitrary compact supports with successive differences in one open $\pi/4$-cone (``narrow tuples'') is
Lemma~\ref{i7:lem:W12D} below.

\subsection{Continuation of positivity}\label{i7:sub:W11}

\begin{lemma}[derivative positivity]\label{i7:lem:W11A}
Let $I\subset\R$ be open, $k\in C^\infty(I\times I)$ with $\sum_{j,l}\bar\lambda_j\lambda_lk(x_j,x_l)\ge0$ for all finite families $x_j\in I$, $\lambda_j\in\C$. Then
$\sum_{a,b\le M}\bar\lambda_a\lambda_b\,\partial_1^a\partial_2^bk(x_0,x_0)\ge0$ for $x_0\in I$, $M\in\mathbb N$, $\lambda\in\C^{M+1}$.
\end{lemma}
\begin{proof}
For $\phi\in C_c(I)$, $Q(\phi):=\iint\overline{\phi(x)}\phi(x')k(x,x')\,dx\,dx'\ge0$: the Riemann sums $h^2\sum_{j,l}\overline{\phi(x_j)}\phi(x_l)k(x_j,x_l)$ over grids of mesh $h$ are of the
assumed form and converge to $Q(\phi)$ (continuous integrand on a compact square). Take $\eta\in C^\infty_c(\R;\R)$, $\int\eta=1$, $\eta_\varepsilon:=\varepsilon^{-1}\eta((\cdot-x_0)/\varepsilon)$,
$\phi_\varepsilon:=\sum_{a\le M}\lambda_a(-1)^a\eta_\varepsilon^{(a)}$. Integrating by parts,
$Q(\phi_\varepsilon)=\sum_{a,b}\bar\lambda_a\lambda_b\iint\eta_\varepsilon(x)\eta_\varepsilon(x')\,\partial_1^a\partial_2^bk(x,x')\,dx\,dx'\to\sum_{a,b}\bar\lambda_a\lambda_b\,\partial_1^a\partial_2^bk(x_0,x_0)$ as
$\varepsilon\to0$, and $Q(\phi_\varepsilon)\ge0$.
\end{proof}

\begin{lemma}[disc propagation of kernel positivity]\label{i7:lem:W11B}
Let $O\subset\C$ be open with $O=\{\bar z:z\in O\}$, $\Phi$ holomorphic on $O\times O$, $z_1\in O$, $B(z_1,r)\subset O$, and suppose that $K(z,\bar w):=\Phi(z,\bar w)$ is positive semidefinite on
a segment $z_1+(-\varepsilon,\varepsilon)$, meaning $\sum_{j,l}\bar\lambda_j\lambda_l\Phi(z_j,\bar z_l)\ge0$ for all finite families $z_j$ in the segment. Then $K$ is positive semidefinite on $B(z_1,r)$.
\end{lemma}
\begin{proof}
$k(x,x'):=\Phi(z_1+x,\bar z_1+x')=K(z_1+x,\overline{z_1+x'})$ is a smooth positive semidefinite kernel on $(-\varepsilon,\varepsilon)$. Real-direction derivatives of a holomorphic function are
its complex derivatives, so Lemma~\ref{i7:lem:W11A} gives that $c_{ab}:=\frac1{a!b!}\partial_\zeta^a\partial_\xi^b\Phi(z_1,\bar z_1)$ satisfies $\sum_{a,b\le M}\bar\mu_a\mu_bc_{ab}\ge0$ for all
$M,\mu$. Since $\{\bar w:w\in B(z_1,r)\}=B(\bar z_1,r)\subset O$, $\Phi$ is holomorphic on the polydisc $B(z_1,r)\times B(\bar z_1,r)$, hence
$\Phi(\zeta,\xi)=\sum_{a,b}c_{ab}(\zeta-z_1)^a(\xi-\bar z_1)^b$, absolutely convergent there. For $z^{(1)},\dots,z^{(m)}\in B(z_1,r)$, $\lambda\in\C^m$, with
$v_b:=\sum_l\lambda_l\overline{(z^{(l)}-z_1)}^{\,b}$,
\[
\sum_{j,l}\bar\lambda_j\lambda_l\,\Phi\big(z^{(j)},\overline{z^{(l)}}\big)=\sum_{a,b}c_{ab}\,\overline{v_a}\,v_b=\lim_{M\to\infty}\sum_{a,b\le M}c_{ab}\overline{v_a}v_b\ \ge0 ,
\]
the double series being absolutely convergent and each square partial sum nonnegative.
\end{proof}

\begin{proposition}[continuation of positivity]\label{i7:prop:W11}
Let $\cX$ be a complex vector space, $O\subset\C$ open and connected with $O=\{\bar z:z\in O\}$ and $0\in O$, and $(\zeta,\xi)\mapsto\ip{A}{B}_{\zeta,\xi}$ holomorphic on $O\times O$ for all
$A,B\in\cX$, sesquilinear in $(A,B)$ (antilinear in $A$). Assume that every $A\in\cX$ has a map $[0,\delta_A)\ni s\mapsto A(s)\in\Lc$ with $\ip{A}{B}_{r,s}=\ip{A(r)}{B(s)}_\circ$ for
$0<r<\delta_A$, $0<s<\delta_B$. Then $\ip{A}{A}_{z,\bar z}\ge0$ for all $z\in O$, $A\in\cX$.
\end{proposition}
\begin{proof}
Fix $A$; $\Phi(\zeta,\xi):=\ip{A}{A}_{\zeta,\xi}$ is holomorphic on $O\times O$. As $0\in O$ is interior, $O\supset(0,\delta')$ with $0<\delta'\le\delta_A$, and for $r,s\in(0,\delta')$,
$\Phi(r,\bar s)=\Phi(r,s)=\ip{A(r)}{A(s)}_\circ$; thus $\sum_{j,l}\bar\lambda_j\lambda_l\Phi(r_j,\bar r_l)=\big\langle\sum_j\lambda_jA(r_j),\sum_l\lambda_lA(r_l)\big\rangle_\circ\ge0$ by the
positivity of the form $\ip{\cdot}{\cdot}_\circ$ on $\Lc$ (Lemma~\ref{i7:lem:recon}), i.e.\ $K$ is positive semidefinite on $(0,\delta')$. Let
$G:=\{z\in O:\ K\text{ is positive semidefinite on some open }N_z\ni z,\ N_z\subset O\}$. $G$ is open; $G\ne\emptyset$ by Lemma~\ref{i7:lem:W11B} at $r_1\in(0,\delta')$ with
$r=\dist(r_1,\C\setminus O)$; $G$ is closed in $O$: if $z^*\in O$, $B(z^*,2\rho)\subset O$ and $z_1\in G\cap B(z^*,\rho)$, Lemma~\ref{i7:lem:W11B} (a segment inside $N_{z_1}$, the disc
$B(z_1,\rho)\subset O$) makes $K$ positive semidefinite on $B(z_1,\rho)\ni z^*$. Connectedness gives $G=O$, and the one-point case at $z\in N_z$ gives $\Phi(z,\bar z)=\ip{A}{A}_{z,\bar z}\ge0$.
\end{proof}

\subsection{Positivity of the continued reflection pairing}\label{i7:sub:W12}

\emph{Conventions.} We use $R(\zeta)=\begin{pmatrix}\cos\zeta&-\sin\zeta\\ \sin\zeta&\cos\zeta\end{pmatrix}$ on $\C^2$ and $\varrho(\zeta)=\sum_we^{iw\zeta}P_w$ as in Subsection~\ref{i7:sub:W10}, together with
$(R(\theta)^*\chi)(t,x,y)=\varrho(-\theta)\chi(R(\theta)(t,x),y)$ and $J\chi=\pi(\theta_\circ)(\bar\chi\circ\theta_\circ)$, so that $\overline{\Thc F(\chi)}=F(J\chi)$. In this subsection and the
next two a product test is $\chi=\alpha f$ with $\alpha\in C^\infty_c(\R^2;\R)$ and $f\in C^\infty_c(\R;T)$ real, with planar support $K=\supp\alpha$; then $R(\theta)^*(\alpha f)=
(\alpha\circ R(\theta))\,\varrho(-\theta)f$ has planar support $R(-\theta)K$ and $J(\alpha f)=(\alpha\circ\theta_\circ)\,\pi(\theta_\circ)f$ has planar support $\theta_\circ K$.
We put $\Sigma_{\pi/4}:=\{\zeta\in\C:|\re\zeta|<\pi/4\}$. For $\Phi$ holomorphic on $M\times\bar M$, ``$K(z,\bar w):=\Phi(z,\bar w)$ is positive semidefinite on $M$'' means
$\sum_{j,l}\bar\lambda_j\lambda_l\Phi(z_j,\bar z_l)\ge0$ for all finite families $z_j\in M$, $\lambda_j\in\C$. For a tuple $\chi=(\chi_1,\dots,\chi_a)$ we put $J\chi:=(J\chi_a,\dots,J\chi_1)$
(reflected, conjugated, reversed), and $D_N[\chi](\zeta)$ is the entire function of Proposition~\ref{i7:prop:W10} where it is defined.

\begin{proposition}[the continued reflection pairing is positive]\label{i7:prop:W12}
Let $\mathfrak A$ be the set of admissible tuples $\chi=(\chi_1,\dots,\chi_a)$, $a\ge0$, of real product tests (planar supports $K_j\subset\cC_+$, $K_{j+1}-K_j\subset\cC_+$), and $\cX$
the free complex vector space on $\mathfrak A$ (the empty tuple is the basis vector $\Omega$). For $\chi,\eta\in\mathfrak A$ of lengths $a,b$ and $N:=a+b$:
\begin{itemize}
\item[(a)] there is $\Psi_{\chi,\eta}$ holomorphic on $\Sigma_{\pi/4}\times\Sigma_{\pi/4}$ with $\Psi_{\chi,\eta}(\theta_1,\theta_2)=S_N(R(\theta_1)^*J\chi,R(\theta_2)^*\eta)$ for real $|\theta_i|<\pi/4$,
and $\Psi_{\chi,\eta}(\zeta,\zeta)=S_N(J\chi,\eta)+D_N[J\chi,\eta](\zeta)$ on $\Sigma_{\pi/4}$;
\item[(b)] the sesquilinear extension of $\ip{\chi}{\eta}_{\zeta,\xi}:=\Psi_{\chi,\eta}(-\zeta,\xi)$ is jointly holomorphic on $\Sigma_{\pi/4}^2$ and equals $\ip{A(r)}{B(s)}_\circ$ at
$(r,s)\in(0,\pi/4)^2$, where $A(s):=\sum_ic_i\prod_jF(R(s)^*\chi^{(i)}_j)$ for $A=\sum_ic_i\chi^{(i)}$;
\item[(c)] for every $\beta\in\R$ the sesquilinear form $\ip{\chi}{\eta}_\beta:=S_N(J\chi,\eta)+D_N[J\chi,\eta](i\beta)$ is positive semidefinite on $\cX$; more generally
$\ip{A}{A}_{z,\bar z}\ge0$ for all $z\in\Sigma_{\pi/4}$.
\end{itemize}
\end{proposition}
The proof uses, besides (i), (ii) and Lemma~\ref{i7:lem:W2} for $S_1$, only the plane $\{t=0\}$ (through Lemma~\ref{i7:lem:recon}) and Propositions~\ref{i7:prop:W8} and~\ref{i7:prop:W10}.
Write $\chi^{\rm tot}:=(J\chi_a,\dots,J\chi_1,\eta_1,\dots,\eta_b)=(\alpha_mf_m)_{m\le N}$ with planar supports $K^{\rm tot}_m$, block index $\iota(m)=1$ for $m\le a$ and $2$ for $m>a$, and
for $Z'\in\prod_mK^{\rm tot}_m$ put $Z(\zeta_1,\zeta_2;Z'):=(R(-\zeta_{\iota(m)})z'_m)_m$.

\begin{lemma}[geometry of complex rotations]\label{i7:lem:W12G}
(i) $K^{\rm tot}_{m+1}-K^{\rm tot}_m\subset\cC_+$ for all $m$ (the tuple $(J\chi,\eta)$ is narrow with unit vector $e_t$). (ii) For real $v\in\cC_+$ and $\zeta=\theta+i\beta$ with $|\theta|<\pi/4$:
$(R(-\theta)v)_t>0$ and $q(R(-\zeta)v)=e^{-|\beta|}(R(-\theta)v)_t>0$. (iii) For $(\zeta_1,\zeta_2)\in\Sigma_{\pi/4}^2$ and $Z'\in\prod K^{\rm tot}_m$ all successive differences of
$Z(\zeta_1,\zeta_2;Z')$ lie in $\cW$; for real $|\theta_i|<\pi/4$ the sets $R(-\theta_{\iota(m)})K^{\rm tot}_m$ are compact, strictly ordered in $t$, in $\{t<0\}$ for $m\le a$ and in $\{t>0\}$ for $m>a$.
\end{lemma}
\begin{proof}
(i) $\theta_\circ\cC_+=-\cC_+$; for $m<a$, $K^{\rm tot}_{m+1}-K^{\rm tot}_m=-\theta_\circ(K_{a-m+1}-K_{a-m})\subset\cC_+$; for $m=a$, $K'_1-\theta_\circ K_1\subset\cC_++\cC_+=\cC_+$ ($K'_1$ the first
support of $\eta$); for $m>a$ by admissibility. (ii) $v=|v|(\cos\phi,\sin\phi)$ with $|\phi|<\pi/4$, so $|\phi-\theta|<\pi/2$. With $w:=R(-\theta)v$ real,
$R(-i\beta)w=(\cosh\beta\,w_t+i\sinh\beta\,w_x,\,-i\sinh\beta\,w_t+\cosh\beta\,w_x)$, hence $q=w_t(\cosh\beta-|\sinh\beta|)=e^{-|\beta|}w_t$. (iii) Within a block,
$\Delta_m=R(-\zeta_\iota)(z'_{m+1}-z'_m)$ with $z'_{m+1}-z'_m\in\cC_+$, so (ii) applies. At the junction, $\Delta_a=R(-\zeta_2)z'_{a+1}+R(-\zeta_1)(-z'_a)$ with $z'_{a+1}\in K'_1\subset\cC_+$ and
$-z'_a\in-\theta_\circ K_1\subset\cC_+$, so the claim follows from (ii) and the superadditivity of $q$. Real angles: the $t$-components of all rotated differences are positive and continuous on compact sets, so their
minimum is positive; $R(-\theta)\cC_+\subset\{t>0\}$ by (ii).
\end{proof}

\begin{lemma}[the two-angle pairing]\label{i7:lem:W12H}
For $N\ge2$ put $\Psi_{\chi,\eta}(\zeta_1,\zeta_2):=\int_{\prod K^{\rm tot}_m}\prod_m\alpha_m(z'_m)\,S_N\big(Z(\zeta_1,\zeta_2;Z');(\varrho(-\zeta_{\iota(m)})f_m)_m\big)\,dZ'$ with the kernel of
Proposition~\ref{i7:prop:W8}; for $N=1$, $\Psi:=S_1(\chi^{\rm tot}_1)$; for $N=0$, $\Psi:=1$. Then (a) of Proposition~\ref{i7:prop:W12} holds, except the last identity.
\end{lemma}
\begin{proof}
By multilinearity, $S_N(Z;(\varrho(-\zeta_{\iota(m)})f_m)_m)=\sum_{w_1,\dots,w_N}e^{-i\sum_mw_m\zeta_{\iota(m)}}S_N(Z;(P_{w_m}f_m)_m)$, a finite sum; $Z(\cdot;Z')$ is entire with values in
$\mathcal Z_N$ on $\Sigma_{\pi/4}^2$ (Lemma~\ref{i7:lem:W12G}(iii)), where $S_N(\cdot;f)$ is holomorphic. So the integrand is holomorphic in $(\zeta_1,\zeta_2)\in\Sigma_{\pi/4}^2$ for each $Z'$
and continuous on $\Sigma_{\pi/4}^2\times\prod K^{\rm tot}_m$. Hence $\Psi$ is continuous, $\oint_\gamma\Psi\,d\zeta_i=\int\oint_\gamma(\dots)d\zeta_i\,dZ'=0$ for closed triangles
$\gamma\subset\Sigma_{\pi/4}$ (Fubini, Cauchy), so $\Psi$ is separately holomorphic (Morera) and jointly holomorphic (Osgood). For real $|\theta_i|<\pi/4$,
$\tilde\chi_m:=R(\theta_{\iota(m)})^*\chi^{\rm tot}_m=(\alpha_m\circ R(\theta_{\iota(m)}))\,\varrho(-\theta_{\iota(m)})f_m$ are product tests with compact planar supports strictly ordered in
$t$ (Lemma~\ref{i7:lem:W12G}(iii)); Proposition~\ref{i7:prop:W8}(e) gives $S_N(\tilde\chi)=\int\prod_m\alpha_m(R(\theta_{\iota(m)})z_m)\,S_N(Z;(\varrho(-\theta_{\iota(m)})f_m)_m)\,dZ$, and the
substitution $z_m=R(-\theta_{\iota(m)})z'_m$ (Jacobian $1$) yields $\Psi(\theta_1,\theta_2)$; $S_N(\tilde\chi)=S_N(R(\theta_1)^*J\chi,R(\theta_2)^*\eta)$ by the symmetry of $S_N$. For $N=1$,
$S_1(R(\theta)^*\psi)=S_1(\psi)$ by Lemma~\ref{i7:lem:W2}.
\end{proof}

\begin{lemma}[narrow tuples]\label{i7:lem:W12D}
Let $\psi=(\psi_1,\dots,\psi_N)$, $N\ge2$, be a tuple of tests $\psi_m\in(L^1\cap L^2)(\R^3;T_\C)$ vanishing outside $L_m\times\R$, with compact planar sets $L_m$ such that, for some unit
vector $u$ and some ordering, all differences $L_{m+1}-L_m$ lie in the open cone of half-angle $\pi/4$ about $u$ (a narrow tuple). Then $D_N[\psi]$ extends to an entire $\pi/2$-periodic
function with $|D_N[\psi](\theta+i\beta)|\le C_\psi e^{(1+w_{\max})N|\beta|}$ and the finite Fourier form of Proposition~\ref{i7:prop:W10}(b), with $N$-linear coefficients $c_\ell[\psi]$. In
particular this holds for $\chi^{\rm tot}=(J\chi,\eta)$, and $\Psi_{\chi,\eta}(\zeta,\zeta)=S_N(\chi^{\rm tot})+D_N[\chi^{\rm tot}](\zeta)$ for $\zeta\in\Sigma_{\pi/4}$.
\end{lemma}
\begin{proof}
On the compact sets $L_{m+1}-L_m$ inside the open cone, $\angle(d,u)\le\gamma_0<\pi/4$ and $|d|\ge\delta_0>0$ uniformly, so every configuration $X\in\prod_mL_m$ is narrow with data
$(u,\gamma_0)$ and $r_0(X)\ge r_*:=\frac12\delta_0\cos(\gamma_0+\pi/4)$. Take a smooth partition of unity $\{\phi_\nu\}$ of $\R^2$ with $\operatorname{diam}\supp\phi_\nu<r_*$ and expand
$D_N[\psi]=\sum_{\nu_1,\dots,\nu_N}D_N[(\phi_{\nu_m}\psi_m)_m]$ (a finite sum, by $N$-linearity, the $L_m$ being compact); each term has planar supports in balls of radius $<r_*$ about a
configuration of $\prod L_m$, so Proposition~\ref{i7:prop:W10}(a)--(c) applies to it (with (c) for non-product tests); sums of finitely many entire $\pi/2$-periodic functions with the growth
bound and the finite Fourier form are of the same kind, and the coefficients add. For $\chi^{\rm tot}$ the hypothesis is Lemma~\ref{i7:lem:W12G}(i) with $u=e_t$. Both sides of the last
identity are holomorphic on the connected set $\Sigma_{\pi/4}$ and agree for real $\zeta\in(-\pi/4,\pi/4)$ by Lemma~\ref{i7:lem:W12H} with $\theta_1=\theta_2$ (for $N\le1$, $D_N\equiv0$).
\end{proof}

\begin{lemma}[curves in the OS space]\label{i7:lem:W12C}
For $\chi\in\mathfrak A$ and $0\le s<\pi/4$: $A_\chi(s):=\prod_jF(R(s)^*\chi_j)\in\Lc$, and $\ip{A_\chi(r)}{A_\eta(s)}_\circ=\Psi_{\chi,\eta}(-r,s)$ for $r,s\in[0,\pi/4)$.
\end{lemma}
\begin{proof}
$R(s)^*\chi_j$ is a real test supported in $R(-s)K_j\times\R\subset\{t>0\}$ (Lemma~\ref{i7:lem:W12G}(ii)), so $A_\chi(s)$ is $\Sigma(\{t>0\})$-measurable and in $L^2$ by
Lemma~\ref{i7:lem:W1}(b) and Hölder. Since $\theta_\circ R(r)=R(-r)\theta_\circ$ and $\pi(\theta_\circ)\varrho(-r)=\varrho(r)\pi(\theta_\circ)$, $J(R(r)^*\psi)=R(-r)^*(J\psi)$ for real tests and
real $r$. $\Thc$ is multiplicative, so $\overline{\Thc A_\chi(r)}=\prod_jF(R(-r)^*J\chi_j)$ and
$\ip{A_\chi(r)}{A_\eta(s)}_\circ=\E_\mu\big[\prod_jF(R(-r)^*J\chi_j)\prod_kF(R(s)^*\eta_k)\big]=S_N(R(-r)^*J\chi,R(s)^*\eta)=\Psi_{\chi,\eta}(-r,s)$ by Lemma~\ref{i7:lem:W12H}.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{i7:prop:W12}]
(a) is Lemmas~\ref{i7:lem:W12H} and~\ref{i7:lem:W12D}. (b): $\ip{\chi}{\eta}_{\zeta,\xi}=\Psi_{\chi,\eta}(-\zeta,\xi)$ is holomorphic on $\Sigma_{\pi/4}^2$ ($-\Sigma_{\pi/4}=\Sigma_{\pi/4}$);
finite sums preserve this; both sides of the identity in (b) are sesquilinear in $(A,B)$ and agree on basis pairs by Lemma~\ref{i7:lem:W12C}. (c): $\Sigma_{\pi/4}$ is open, connected,
conjugation-symmetric, $0\in\Sigma_{\pi/4}$, so Proposition~\ref{i7:prop:W11} with $O=\Sigma_{\pi/4}$, $\delta_A=\pi/4$ gives $\ip{A}{A}_{z,\bar z}\ge0$ for $z\in\Sigma_{\pi/4}$; at $z=-i\beta$,
$\ip{\chi}{\eta}_{-i\beta,i\beta}=\Psi_{\chi,\eta}(i\beta,i\beta)=S_N(J\chi,\eta)+D_N[J\chi,\eta](i\beta)=\ip{\chi}{\eta}_\beta$ by Lemma~\ref{i7:lem:W12D}, for every $\beta\in\R$.
\end{proof}

\subsection{Testing and convexity}\label{i7:sub:W13}

\emph{Conventions.} For a unit vector $u\in\R^2$ let $\Gamma(u):=\{p\in\R^2:\ p\cdot u>|p|/\sqrt2\}$, the open cone of half-angle $\pi/4$ about $u$; $\Gamma(e_t)=\cC_+$,
$\cC_-:=-\cC_+=\theta_\circ\cC_+$, $\cC_\pm+\cC_\pm\subset\cC_\pm$, $R\,\Gamma(u)=\Gamma(Ru)$ for planar rotations $R$. A tuple is $\chi=(\chi_1,\dots,\chi_N)$, $N\ge0$, of real product tests
$\chi_j=\alpha_jf_j$ with compact planar supports $K_j:=\supp\alpha_j$; $S_0:=1$. $R^*$, $\tau_v\chi=\chi(\cdot-v)$ and $\chi_\sigma:=(\chi_{\sigma(1)},\dots,\chi_{\sigma(N)})$ act entrywise.
$J\chi_j=\pi(\theta_\circ)(\bar\chi_j\circ\theta_\circ)$ with $\pi(\theta_\circ)$ a real orthogonal involution, so $J$ maps real product tests to real product tests, $J^2=1$, the planar
support of $J\chi_j$ is $\theta_\circ K_j$, and $J\chi=(J\chi_a,\dots,J\chi_1)$ for a tuple of length $a$. A tuple $\chi$ is narrow if there are a permutation $\sigma$ and a unit vector $u$
with $K_{\sigma(j+1)}-K_{\sigma(j)}\subset\Gamma(u)$ for $1\le j<N$ (every tuple of length $\le1$ is narrow); this is the hypothesis of Lemma~\ref{i7:lem:W12D}. Every configuration
$(X_j)$ with $X_j\in K_j$ is then narrow; conversely, for a narrow configuration $X$ with data $(\sigma,u)$ and $0<\delta<\frac12\min_j\dist(X_{\sigma(j+1)}-X_{\sigma(j)},\partial\Gamma(u))$,
every tuple with $K_j\subset B(X_j,\delta)$ is narrow with the same data. For admissible $\chi$ (length $a$) and $\eta$ (length $b$), $(J\chi,\eta)$ is narrow with $u=e_t$, $\sigma=\id$
(Lemma~\ref{i7:lem:W12G}(i)).

\emph{Regular tuples and their coefficients.} A tuple $\chi$ of length $N$ is regular if $D_N[\chi](\theta)=S_N(R(\theta)^*\chi)-S_N(\chi)$, $\theta\in\R$, is the restriction of an entire function
$D_N[\chi](\zeta)$ with $\sup_{\theta\in\R}|D_N[\chi](\theta+i\beta)|\le C_0e^{3N|\beta|}$ for all $\beta\in\R$ and some $C_0<\infty$. The extension is unique (identity theorem) and
$2\pi$-periodic, since $D_N[\chi](\zeta+2\pi)-D_N[\chi](\zeta)$ is entire and vanishes on $\R$ ($R(2\pi)=1=\varrho(2\pi)$). Hence $D_N[\chi](\zeta)=\sum_{\ell\in\Z}d_\ell e^{i\ell\zeta}$ (Laurent
series in $e^{i\zeta}$, absolutely convergent) with $d_\ell=\frac1{2\pi}\int_0^{2\pi}D_N[\chi](\theta+i\beta)e^{-i\ell(\theta+i\beta)}\,d\theta$ for every $\beta$; so
$|d_\ell|\le C_0e^{3N|\beta|+\ell\beta}$ and ($\beta\to\mp\infty$) $d_\ell=0$ for $|\ell|>3N$. Put $c_\ell[\chi]:=d_\ell$ ($\ell\ne0$); $D_N[\chi](0)=0$ gives
\begin{equation}\label{i7:eq:fourierform}
D_N[\chi](\zeta)=\sum_{0<|\ell|\le3N}c_\ell[\chi]\,(e^{i\ell\zeta}-1).
\end{equation}
For $\ell\ne0$, $c_\ell[\chi]=\frac1{2\pi}\int_0^{2\pi}D_N[\chi](\theta)e^{-i\ell\theta}d\theta$ is intrinsic. The facts used below are:
\begin{itemize}
\item[(A)] every narrow tuple is regular, and $c_\ell[\chi]=0$ for $\ell\notin4\Z$ (Lemma~\ref{i7:lem:W13A} below);
\item[(B)] for admissible $\chi,\eta$ of lengths $a,b$ and $\beta\in\R$, $\ip{\chi}{\eta}_\beta:=S_{a+b}(J\chi,\eta)+D_{a+b}[J\chi,\eta](i\beta)$ (the empty tuple $\emptyset$ allowed,
$\ip{\emptyset}{\emptyset}_\beta=S_0=1$) satisfies $\sum_{i,k}\bar z_iz_k\ip{\chi^{(i)}}{\chi^{(k)}}_\beta\ge0$ for all finite families of admissible tuples and all $z\in\C^n$
(Proposition~\ref{i7:prop:W12}(c));
\item[(C)] $S_N(\tau_v\chi)=S_N(\chi)$ for all $v\in\R^3$ (Lemma~\ref{i7:lem:W1}(e) and (ii)); $S_N$ is symmetric.
\end{itemize}

\begin{lemma}[testing]\label{i7:lem:W13}
Assume \textup{(C)}.
\begin{itemize}
\item[(a)] If $\chi$ is regular, so are $\chi_\sigma$, $\tau_v\chi$ ($v\in\R^3$) and $R(\phi)^*\chi$ ($\phi\in\R$), and
$c_\ell[\chi_\sigma]=c_\ell[\chi]$, $c_\ell[\tau_v\chi]=c_\ell[\chi]$, $c_\ell[R(\phi)^*\chi]=e^{i\ell\phi}c_\ell[\chi]$. These operations map narrow tuples to narrow tuples.
\item[(b)] Let $\chi$ be narrow of length $N$ with data $(\sigma,u)$, $u=R(\phi)e_t$, and $a+b=N$. Then there are admissible $\chi',\eta'$ of lengths $a,b$ with
$(J\chi',\eta')=\tau_vR(\phi)^*\chi_\sigma$ for some $v\in\R^3$; if $\chi$ is regular, so is $(J\chi',\eta')$, with $c_\ell[J\chi',\eta']=e^{i\ell\phi}c_\ell[\chi]$. In particular, if
$\ell\ne0$ and $c_\ell[\chi]\ne0$ for a narrow regular $\chi$ of length $N$, then for every $a+b=N$ there are admissible $\chi',\eta'$ of lengths $a,b$ with $c_\ell[J\chi',\eta']\ne0$.
\end{itemize}
\end{lemma}
\begin{proof}
(a) Permutations: $\prod_jF(\chi_j)$ is a commutative product and $R(\theta)^*$ acts entrywise, so $D_N[\chi_\sigma]=D_N[\chi]$ on $\R$; the entire extensions and their coefficients coincide.
Translations: $(R^*\tau_v\chi)(x)=\pi(R)^{-1}\chi(Rx-v)=(R^*\chi)(x-R^{-1}v)$, i.e.\ $R^*\tau_v=\tau_{R^{-1}v}R^*$; by (C), $S_N(R(\theta)^*\tau_v\chi)=S_N(R(\theta)^*\chi)$ and
$S_N(\tau_v\chi)=S_N(\chi)$, so $D_N[\tau_v\chi]=D_N[\chi]$ on $\R$. Rotations: $\pi$ is a representation and planar rotations commute, so $R(\theta)^*R(\phi)^*=(R(\phi)R(\theta))^*=R(\theta+\phi)^*$ and
\begin{align*}
D_N[R(\phi)^*\chi](\theta)&=\big(S_N(R(\theta+\phi)^*\chi)-S_N(\chi)\big)-\big(S_N(R(\phi)^*\chi)-S_N(\chi)\big)\\
&=D_N[\chi](\theta+\phi)-D_N[\chi](\phi).
\end{align*}
The right side is the restriction of the entire function $\zeta\mapsto D_N[\chi](\zeta+\phi)-D_N[\chi](\phi)$, bounded by $(C_0+|D_N[\chi](\phi)|)e^{3N|\beta|}$; so $R(\phi)^*\chi$ is regular with
$D_N[R(\phi)^*\chi](\zeta)=\sum_\ell c_\ell[\chi](e^{i\ell(\zeta+\phi)}-e^{i\ell\phi})=\sum_\ell c_\ell[\chi]e^{i\ell\phi}(e^{i\ell\zeta}-1)$, and uniqueness of Laurent coefficients gives
$c_\ell[R(\phi)^*\chi]=e^{i\ell\phi}c_\ell[\chi]$. Finally $R(\phi)^*(\alpha f)=(\alpha\circ R(\phi))\,\varrho(\phi)^{-1}f$ is a real product test ($\varrho(\phi)$ real orthogonal) with planar
support $R(\phi)^{-1}K$, so the cone condition is transported to $R(\phi)^{-1}\Gamma(u)=\Gamma(R(\phi)^{-1}u)$; $\tau_v$ shifts all $K_j$ by the planar part of $v$, leaving differences
unchanged; $\chi_\sigma$ permutes the $K_j$, and the ordering composes.

(b) Step 1. $\psi:=R(\phi)^*(\chi_\sigma)$ has supports $M_j:=R(\phi)^{-1}K_{\sigma(j)}$ with $M_{j+1}-M_j\subset R(\phi)^{-1}\Gamma(u)=\Gamma(e_t)=\cC_+$.

Step 2 (splitting). With $\lambda_\pm(t,x):=t\pm x$, $\cC_+=\{\lambda_+>0,\ \lambda_->0\}$. If $1\le a<N$: $\lambda_\pm(p')>\lambda_\pm(p)$ for $p\in M_a$, $p'\in M_{a+1}$, so by compactness
$m^\pm:=\max_{M_a}\lambda_\pm<\min_{M_{a+1}}\lambda_\pm=:m'^\pm$. Let $w\in\R^2$ have $\lambda_\pm(w)=\frac12(m^\pm+m'^\pm)$ (if $a=0$: $\lambda_\pm(w)=m'^\pm-1$; if $a=N$:
$\lambda_\pm(w)=m^\pm+1$). Then $M_a-w\subset\cC_-$ and $M_{a+1}-w\subset\cC_+$; since $M_j-M_{j+1}\subset\cC_-$ and $\cC_\pm+\cC_\pm\subset\cC_\pm$, induction downward and upward gives
$M_j-w\subset\cC_-$ ($j\le a$) and $M_j-w\subset\cC_+$ ($j>a$).

Step 3. Let $v:=(-w,0)\in\R^3$ and $\psi':=\tau_v\psi$, so that $\psi'_j$ has planar support $M_j-w$. Define $\chi'_i:=J\psi'_{a+1-i}$ ($1\le i\le a$), $\eta'_k:=\psi'_{a+k}$ ($1\le k\le b$):
tuples of real product tests with $J\chi'=(J^2\psi'_1,\dots,J^2\psi'_a)=(\psi'_1,\dots,\psi'_a)$, hence $(J\chi',\eta')=\psi'=\tau_vR(\phi)^*\chi_\sigma$. Admissibility:
$\supp\chi'_i=\theta_\circ(M_{a+1-i}-w)\subset\theta_\circ\cC_-=\cC_+$; $\supp\chi'_{i+1}-\supp\chi'_i=\theta_\circ(-(M_{a+1-i}-M_{a-i}))\subset\theta_\circ\cC_-=\cC_+$;
$\supp\eta'_k=M_{a+k}-w\subset\cC_+$ and $\supp\eta'_{k+1}-\supp\eta'_k=M_{a+k+1}-M_{a+k}\subset\cC_+$. If $\chi$ is regular, (a) gives the regularity of $\psi'$ and
$c_\ell[J\chi',\eta']=c_\ell[\psi']=e^{i\ell\phi}c_\ell[\chi]$.
\end{proof}

\begin{lemma}[narrow tuples are regular]\label{i7:lem:W13A}
Every narrow tuple $\chi$ is regular, with $c_\ell[\chi]=0$ for $\ell\notin4\Z$, and $|D_N[\chi](\theta+i\beta)|\le C_0e^{(1+w_{\max})N|\beta|}$. Moreover, if $c_\ell[\chi]\ne0$ for a narrow
tuple $\chi$ of length $N$, then $c_\ell[\chi^\flat]\ne0$ for some tuple $\chi^\flat$ of length $N$ with planar supports in balls $\bar B(x_j,r)$, $r<r_0$, about a narrow configuration
$X$ as in Proposition~\ref{i7:prop:W10}; and conversely every such tuple is narrow.
\end{lemma}
\begin{proof}
Let $\chi$ be narrow with data $(\sigma,u)$, $u=R(\phi)e_t$. Lemma~\ref{i7:lem:W13}(b), Steps 1--3, uses only the support geometry and produces admissible $\chi',\eta'$ with
$(J\chi',\eta')=\psi'=\tau_vR(\phi)^*\chi_\sigma$. By Lemma~\ref{i7:lem:W12D}, $\psi'$ is regular, with the growth bound and $c_\ell[\psi']=0$ for $\ell\notin4\Z$. The proof of
Lemma~\ref{i7:lem:W13}(a) uses only regularity, so $\chi_\sigma=R(-\phi)^*\tau_{-v}\psi'$ and hence $\chi$ are regular with $c_\ell[\chi]=e^{-i\ell\phi}c_\ell[\psi']$, which vanishes off $4\Z$
(and the growth bound transports with a larger constant). Alternatively, and for the second claim: by compactness of $\prod_j(K_{\sigma(j+1)}-K_{\sigma(j)})\subset\Gamma(u)^{N-1}$ there are
$\delta_*>0$ and $\gamma_*<\pi/4$ bounding all drawn configurations, so $r_0\ge r_*:=\frac12\delta_*\cos(\gamma_*+\pi/4)$ uniformly; a smooth planar partition of unity with pieces in balls of
radius $<r_*$ centred in the $K_j$ and the $N$-linearity of $D_N$ express $D_N[\chi]$ as a finite sum of $D_N[\chi^\flat]$ with $\chi^\flat$ as in Proposition~\ref{i7:prop:W10}, the
coefficients adding, so some piece has $c_\ell[\chi^\flat]\ne0$. Conversely, for $d\in\Gamma(u)$ at angle $\alpha\le\gamma$ from $u$, $\dist(d,\partial\Gamma(u))=|d|\sin(\pi/4-\alpha)=
|d|\cos(\pi/4+\alpha)\ge2r_0>2r$, so $\bar B(d_j,2r)\subset\Gamma(u)$ and tuples supported in the balls $\bar B(x_j,r)$ are narrow.
\end{proof}
Consequently, for each $\ell\ne0$ and each $N$, $c_\ell[\chi]\ne0$ for some narrow tuple $\chi$ of length $N$ if and only if $c_\ell[\chi]\ne0$ for some tuple $\chi$ of length $N$ supported in the balls of Proposition~\ref{i7:prop:W10};
in particular the maximum $M_N$ of Proposition~\ref{i7:prop:W14} is the same for both classes of tuples.

\begin{proposition}[convexity]\label{i7:prop:W14}
Let
\[
M_N:=\max\{|\ell|:\ c_\ell[\chi]\ne0\ \text{for some narrow tuple }\chi\text{ of length }N\},
\]
and $M_N:=0$ if there is none. Then $M_0=0$, $M_N\in4\Z$, $0\le M_N\le3N$
(indeed $\le(1+w_{\max})N$), and $2M_{a+b}\le M_{2a}+M_{2b}$ for all $a,b\ge0$.
\end{proposition}
\begin{proof}
Bounds. By \eqref{i7:eq:fourierform} and Lemma~\ref{i7:lem:W13A}, $c_\ell[\chi]=0$ for $|\ell|>(1+w_{\max})N$, so $M_N\le3N$ and $M_0=0$; by (A) only $\ell\in4\Z$ occur, so $M_N\in4\Z$ ($0$ included).

Convexity. Let $N=a+b$, $m:=M_N$; if $m=0$ there is nothing to prove. Otherwise pick $\ell_0$ with $|\ell_0|=m$ and $c_{\ell_0}[\chi]\ne0$ for a narrow tuple $\chi$ of length $N$;
Lemma~\ref{i7:lem:W13}(b) gives admissible $\chi',\eta'$ of lengths $a,b$ with $\gamma:=c_{\ell_0}[J\chi',\eta']\ne0$. The tuples $(J\chi',\eta')$, $(J\chi',\chi')$, $(J\eta',\eta')$ are narrow,
of lengths $N,2a,2b$; hence their coefficients $c_\ell$, $c^A_\ell$, $c^B_\ell$ vanish for $|\ell|>m$, $|\ell|>M_{2a}$, $|\ell|>M_{2b}$ respectively, by the definition of $M$. At $\zeta=i\beta$,
\begin{align*}
\ip{\chi'}{\eta'}_\beta&=S_N(J\chi',\eta')+\sum_{0<|\ell|\le m}c_\ell\,(e^{-\ell\beta}-1),\\
\ip{\chi'}{\chi'}_\beta&=S_{2a}(J\chi',\chi')+\sum_{0<|\ell|\le M_{2a}}c^A_\ell\,(e^{-\ell\beta}-1),
\end{align*}
and likewise for $\eta'$ (for $a=0$: $\ip{\emptyset}{\emptyset}_\beta=1$). With $C_A:=|S_{2a}(J\chi',\chi')|+2\sum_{0<|\ell|\le M_{2a}}|c^A_\ell|$ and $C_B$ likewise, $|e^{-\ell\beta}-1|\le2e^{|\ell||\beta|}$ gives
$|\ip{\chi'}{\chi'}_\beta|\le C_Ae^{M_{2a}|\beta|}$ and $|\ip{\eta'}{\eta'}_\beta|\le C_Be^{M_{2b}|\beta|}$ for $\beta\in\R$.

Lower bound: let $s:=-\operatorname{sgn}\ell_0$, $\beta=s\tau$, $\tau>0$. Then $e^{-\ell_0\beta}=e^{m\tau}$, $e^{\ell_0\beta}=e^{-m\tau}\le1$, and every other $\ell$ with $0<|\ell|\le m$ has
$|\ell|\le m-4$ (all frequencies lie in $4\Z$, and $m\ge4$). With $C':=|S_N(J\chi',\eta')|+2\sum_{0<|\ell|\le m}|c_\ell|$,
\[
|\ip{\chi'}{\eta'}_{s\tau}|\ \ge\ |\gamma|e^{m\tau}-C'e^{(m-4)\tau}\ \ge\ \tfrac12|\gamma|e^{m\tau}\qquad\text{for }\tau\ge\tau_0:=\max\{0,\tfrac14\log(2C'/|\gamma|)\}.
\]

Positivity: by (B) for the family $\{\chi',\eta'\}$, the $2\times2$ matrix $G_\beta:=(\ip{\cdot}{\cdot}_\beta)$ satisfies $z^*G_\beta z\ge0$ for all $z\in\C^2$; thus $z^*(G_\beta-G_\beta^*)z=0$ for
all $z$, so by polarisation $G_\beta$ is Hermitian, positive semidefinite, and $\det G_\beta\ge0$: $|\ip{\chi'}{\eta'}_\beta|^2\le\ip{\chi'}{\chi'}_\beta\,\ip{\eta'}{\eta'}_\beta$. Hence
$\frac14|\gamma|^2e^{2m\tau}\le C_AC_B\,e^{(M_{2a}+M_{2b})\tau}$ for all $\tau\ge\tau_0$; letting $\tau\to\infty$ forces $2m\le M_{2a}+M_{2b}$.
\end{proof}
Only two-element families (and the empty tuple when $a=0$ or $b=0$, giving $2M_b\le M_{2b}$) of (B) are used. This subsection uses, beyond (i), (ii) and the four planes of the pair through
Propositions~\ref{i7:prop:W10} and~\ref{i7:prop:W12}, nothing else.

\subsection{Divisibility by eight and the arithmetic lemma}\label{i7:sub:W15}

\emph{Notation.} Recall that $(g^*\chi)(x)=\pi(g)^{-1}\chi(gx)$, so that $(gh)^*=h^*g^*$, $g^*$ preserves reality, $\|\cdot\|_1$ and $\|\cdot\|_2$, and $\supp g^*\chi=g^{-1}\supp\chi$. We put $(\tau_a^*\chi)(x):=\chi(x+a)=
(\tau_{-a}\chi)(x)$, hence $g^*\tau_a^*=\tau^*_{g^{-1}a}g^*$. We put $R_\diamond:=R(\pi/4)$ and $P_0:=\{t=0\}$, $P_1:=\{t+x=0\}$, $P_2:=\{x=0\}$, $P_3:=\{t-x=0\}$, and we write $\Theta_{P_k}$ for the reflection in $P_k$.
Then $R_\diamond P_k=P_{k+1}$ (indices mod $4$), $R_\diamond\{y=0\}=\{y=0\}$, and $\Theta_{gP}=g\Theta_Pg^{-1}$ for every plane $P$ through $0$. We put $\tilde\mu:=T_*\mu$ with $T\colon\cS'\to\cS'$,
$(T\omega)(\chi):=\omega(R_\diamond^*\chi)$ ($T$ is the transpose of the continuous linear bijection $R_\diamond^*$ of $\cS$, hence continuous, so $\tilde\mu$ is a Borel probability measure);
thus $\tilde Z(\chi)=Z(R_\diamond^*\chi)$ and $\tilde S_N(\chi_1,\dots,\chi_N)=S_N(R_\diamond^*\chi_1,\dots,R_\diamond^*\chi_N)$. Objects attached to $\tilde\mu$ carry a tilde. For a regular tuple
$\psi$ put $c_0[\psi]:=S_N(\psi)-\sum_{\ell\ne0}c_\ell[\psi]$, so that $S_N(R(\theta)^*\psi)=\sum_\ell c_\ell[\psi]e^{i\ell\theta}$ for real $\theta$. $\mathfrak A_a$ denotes the admissible tuples
of length $a$; by Lemma~\ref{i7:lem:W13A} every pair $(J\chi,\eta)$, $\chi\in\mathfrak A_a$, $\eta\in\mathfrak A_b$, is regular with frequencies in $4\Z$ and $|\ell|\le M_{a+b}$.

\begin{proposition}[divisibility by eight]\label{i7:prop:W15}
Let $\mu$ satisfy (i), (ii) and (iii) for the four planes $P_0,\dots,P_3$ of the coordinate pair $(t,x)$, and let $\tilde\mu$ be as above.
\begin{itemize}
\item[(1)] $\tilde\mu$ satisfies (i) with the same constant $C$, (ii), and (iii) for the reflections in $P_0,P_1,P_2,P_3$ (and in $\{y=0\}$ if $\mu$ has (iii) there); the four reflections,
hence the dihedral group they generate, which contains $R(\pi/2)$, preserve $\tilde\mu$.
\item[(2)] For every regular tuple $\psi$ (for $\mu$) and all $\zeta\in\C$: $\psi$ is regular for $\tilde\mu$, $\tilde D_N[\psi](\zeta)=D_N[\psi](\zeta+\pi/4)-D_N[\psi](\pi/4)$, entire and
$2\pi$-periodic, $|\tilde D_N[\psi](\theta+i\beta)|\le2C_0e^{3N|\beta|}$, and $\tilde c_\ell[\psi]=e^{i\ell\pi/4}c_\ell[\psi]=(-1)^{\ell/4}c_\ell[\psi]$ for $\ell\in4\Z\setminus\{0\}$.
\item[(3)] $c_{\pm M_{2a}}[J\chi,\chi]\ge0$ for all $\chi\in\mathfrak A_a$, and if $M:=M_{2a}>0$ there is $\chi\in\mathfrak A_a$ with $c_{-M}[J\chi,\chi]=c_M[J\chi,\chi]>0$.
\item[(4)] $M_{2a}\in8\Z$ for all $a\ge0$.
\end{itemize}
\end{proposition}
\begin{proof}
(1) (i): for real Schwartz $\chi$, $\int e^{F(\chi)}d\tilde\mu=\int e^{F(R_\diamond^*\chi)}d\mu\le e^{C(\|R_\diamond^*\chi\|_1+\|R_\diamond^*\chi\|_2^2)}=e^{C(\|\chi\|_1+\|\chi\|_2^2)}$, because
$R_\diamond^*\chi$ is real Schwartz and $|(R_\diamond^*\chi)(x)|_T=|\chi(R_\diamond x)|_T$ with $R_\diamond$ measure preserving.
(ii): $\tilde S_N(\tau_a^*\chi_1,\dots)=S_N(\tau^*_{R_\diamond^{-1}a}R_\diamond^*\chi_1,\dots)=S_N(R_\diamond^*\chi_1,\dots)=\tilde S_N(\chi_1,\dots)$ by the translation invariance of the $S_N$
(Lemma~\ref{i7:lem:W1}(e) for $\mu$); since $\tilde\mu$ satisfies (i), Lemma~\ref{i7:lem:W1}(e) applied to $\tilde\mu$ gives $\tau_{a*}\tilde\mu=\tilde\mu$.
(iii): let $P\in\{P_0,\dots,P_3\}$, $P':=R_\diamond^{-1}P\in\{P_0,\dots,P_3\}$, $\chi_a$ real Schwartz supported in one open side $P^+$ of $P$, $z_a\in\C$. From $\Theta_P=R_\diamond\Theta_{P'}R_\diamond^{-1}$,
$R_\diamond^*\Theta_P^*=(\Theta_PR_\diamond)^*=(R_\diamond\Theta_{P'})^*=\Theta_{P'}^*R_\diamond^*$, so
$\sum_{a,b}\bar z_az_b\int e^{iF(\chi_a-\Theta_P^*\chi_b)}d\tilde\mu=\sum_{a,b}\bar z_az_b\int e^{iF(\chi'_a-\Theta_{P'}^*\chi'_b)}d\mu\ge0$, with $\chi'_a:=R_\diamond^*\chi_a$ supported in
$R_\diamond^{-1}P^+$, one open side of $P'$, by (iii) for $\mu$ at $P'$. The same with $P=P'=\{y=0\}$. Invariance under the four reflections follows from Lemma~\ref{i7:lem:W2} applied to
$\tilde\mu$ (or directly: $\tilde S_N(\Theta_{P_k}^*\chi_1,\dots)=S_N(\Theta_{P_{k-1}}^*R_\diamond^*\chi_1,\dots)=S_N(R_\diamond^*\chi_1,\dots)$). $\Theta_{P_1}\Theta_{P_0}=R(\pi/2)$.

(2) $R_\diamond^*R(\theta)^*=(R(\theta)R_\diamond)^*=R(\theta+\pi/4)^*$, hence $\tilde S_N(R(\theta)^*\psi)=S_N(R(\theta+\pi/4)^*\psi)$ and, for real $\theta$,
$\tilde D_N[\psi](\theta)=[S_N(\psi)+D_N[\psi](\theta+\pi/4)]-[S_N(\psi)+D_N[\psi](\pi/4)]$. The right side is entire in $\theta$, so it is the continuation, with the stated period and bound.
Inserting \eqref{i7:eq:fourierform}, $\tilde D_N[\psi](\zeta)=\sum_{\ell\ne0}c_\ell[\psi](e^{i\ell(\zeta+\pi/4)}-e^{i\ell\pi/4})=\sum_{\ell\ne0}e^{i\ell\pi/4}c_\ell[\psi](e^{i\ell\zeta}-1)$;
uniqueness of Fourier coefficients gives $\tilde c_\ell=e^{i\ell\pi/4}c_\ell$, and $e^{i\ell\pi/4}=(e^{i\pi})^{\ell/4}=(-1)^{\ell/4}$ for $\ell\in4\Z$.

(3) Step 1 (reality). For a real tuple $\psi$ and real $\theta$ the tuple $R(\theta)^*\psi$ is real ($\pi(R(\theta))$ is a real matrix), so $S_N(R(\theta)^*\psi)=\int\prod_jF(R(\theta)^*\psi_j)\,d\mu\in\R$
($F$ is real on real tests). A real trigonometric polynomial has $c_{-\ell}=\overline{c_\ell}$. For real $\chi$, $J\chi$ is real ($\pi(\theta_\circ)$ is real), so this applies to $(J\chi,\eta)$.
Step 2 (leading Gram matrix). Let $N=2a$, $M=M_N$, $\chi^{(1)},\dots,\chi^{(m)}\in\mathfrak A_a$ and $G(\beta)_{jk}:=S_N(J\chi^{(j)},\chi^{(k)})+D_N[J\chi^{(j)},\chi^{(k)}](i\beta)$. By
Proposition~\ref{i7:prop:W12}(c), $G(\beta)\ge0$ for every real $\beta$. By \eqref{i7:eq:fourierform} at $\zeta=i\beta$ and the frequency bound,
$G(\beta)_{jk}=\sum_{\ell\in4\Z,\,|\ell|\le M}c_\ell[J\chi^{(j)},\chi^{(k)}]e^{-\ell\beta}$. Hence $e^{-M\beta}G(\beta)\to G_{-M}:=(c_{-M}[J\chi^{(j)},\chi^{(k)}])_{jk}$ as $\beta\to+\infty$ and
$e^{M\beta}G(\beta)\to G_M$ as $\beta\to-\infty$; the cone of positive semidefinite matrices is closed, so $G_{\pm M}\ge0$. With $m=1$: $c_{\pm M}[J\chi,\chi]\ge0$.
Step 3. Let $M>0$. By the definition of $M_N$ some $\ell\in\{M,-M\}$ has $c_\ell[\psi]\ne0$ for a narrow tuple $\psi$ of length $N$; by Step 1 then $\ell=-M$ does.
Lemma~\ref{i7:lem:W13}(b) with $(a,b)=(a,a)$ gives $\chi,\eta\in\mathfrak A_a$ with $c_{-M}[J\chi,\eta]\ne0$. Step 2 for $(\chi^{(1)},\chi^{(2)})=(\chi,\eta)$: the $2\times2$ matrix $G_{-M}\ge0$
has a nonzero off-diagonal entry, so $0<|c_{-M}[J\chi,\eta]|^2\le c_{-M}[J\chi,\chi]\,c_{-M}[J\eta,\eta]$, whence $c_{-M}[J\chi,\chi]>0$, and $c_M[J\chi,\chi]=\overline{c_{-M}[J\chi,\chi]}=c_{-M}[J\chi,\chi]$
by Step 1.

(4) Let $M=M_{2a}>0$ (otherwise there is nothing to prove) and $\chi$ as in (3). By (1), $\tilde\mu$ satisfies the hypotheses of Subsections~\ref{i7:sub:W1}--\ref{i7:sub:W12} for the
coordinate pair $(t,x)$ (Remark~\ref{i7:rem:fourplanes} and Lemma~\ref{i7:lem:W15b} below), so Proposition~\ref{i7:prop:W12}(c) holds for $\tilde\mu$:
$\tilde S_N(J\chi,\chi)+\tilde D_N[J\chi,\chi](i\beta)\ge0$ for all $\beta\in\R$ ($\chi$ is admissible for $\tilde\mu$ as well, admissibility being a property of the supports). By (2) this
reads $S_N(J\chi,\chi)+D_N[J\chi,\chi](\pi/4+i\beta)=c_0[J\chi,\chi]+\sum_{\ell\ne0}(-1)^{\ell/4}c_\ell[J\chi,\chi]e^{-\ell\beta}\ge0$ for all real $\beta$. Multiplying by $e^{-M\beta}$ and
letting $\beta\to+\infty$ gives $(-1)^{M/4}c_{-M}[J\chi,\chi]\ge0$. As $c_{-M}[J\chi,\chi]>0$, $(-1)^{M/4}=1$, i.e.\ $M/4$ is even: $M\in8\Z$.
\end{proof}

\begin{lemma}[the four planes of a pair suffice]\label{i7:lem:W15b}
Let $\nu$ be a Borel probability measure on $\cS'(\R^3;T)$ satisfying (i), (ii) and (iii) for the four planes $P_0,\dots,P_3$ of the coordinate pair $(t,x)$ only. Then every statement of
Subsections~\ref{i7:sub:W1}--\ref{i7:sub:W13} holds for $\nu$ and this pair, with the same proofs; in particular Proposition~\ref{i7:prop:W12}(c) holds for $\nu$.
\end{lemma}
\begin{proof}
We go through the statements in question and determine the planes and symmetries that their proofs use. Lemma~\ref{i7:lem:W1} uses (i)
only. In Lemma~\ref{i7:lem:W2} the first assertion needs (i), (ii) and (iii) for the one plane in question, and it gives for $\nu$ the
invariance under $\Theta_{P_0},\dots,\Theta_{P_3}$ and under the dihedral group $D_4$ they generate. The four-plane clause of that lemma
gives $S_1(R(\theta)^*\chi)=S_1(\chi)$ for all $\theta$, which is the only property of $S_1$ used later, namely in
Lemmas~\ref{i7:lem:W12H} and~\ref{i7:lem:W12D} and in the paragraph after the proof of Proposition~\ref{i7:prop:W10}.
Lemma~\ref{i7:lem:recon} uses (ii), (iii) and the invariance at the one plane $u$ in question, where $u\in\{e_t,n\}$ or, for the pair
$(x,t)$, $u\in\{e_x,(e_x+e_t)/\sqrt2\}$; these planes are all among $P_0,\dots,P_3$.

Theorem~\ref{i7:thm:W3} uses $P_0$ and $P_1$ for the pair $(t,x)$, and $P_2$ and $P_1$ for the pair $(x,t)$.
Lemma~\ref{i7:lem:W4} uses Theorem~\ref{i7:thm:W3} only. Theorems~\ref{i7:thm:W5} and~\ref{i7:thm:W6} use $P_0$ and its translates
$\{t=c\}$ for the pair $(t,x)$, and $P_2$ and its translates for the pair $(x,t)$, together with (i) and (ii).
Theorem~\ref{i7:thm:W7} uses Theorem~\ref{i7:thm:W3}, Lemma~\ref{i7:lem:W4} and Theorem~\ref{i7:thm:W6}.
Propositions~\ref{i7:prop:W8} and~\ref{i7:prop:W9} use Theorems~\ref{i7:thm:W6} and~\ref{i7:thm:W7} for the pairs $(t,x)$ and
$(x,t)$, and they use Lemma~\ref{i7:lem:W2} only through $R(\theta)$ with $\theta\in\frac\pi2\Z$, which is an element of $D_4$.
Proposition~\ref{i7:prop:W10} uses (K1)--(K3), which come from Proposition~\ref{i7:prop:W9}, and (K4), which is the invariance under
$R(\pi/2)\in D_4$. Proposition~\ref{i7:prop:W11} is abstract, apart from the positivity of the form at $P_0$.
Proposition~\ref{i7:prop:W12} and its lemmas use the form at $P_0$, Propositions~\ref{i7:prop:W8} and~\ref{i7:prop:W10}, and the
identity $S_1(R(\theta)^*\chi)=S_1(\chi)$. Lemmas~\ref{i7:lem:W13} and~\ref{i7:lem:W13A} and Proposition~\ref{i7:prop:W14} use (ii) and
Propositions~\ref{i7:prop:W10} and~\ref{i7:prop:W12}. Thus no proof uses a plane outside $\{P_0,P_1,P_2,P_3\}$ or a symmetry outside
$D_4$ and the translations.
\end{proof}

\begin{lemma}[arithmetic]\label{i7:lem:W16}
Let $(M_n)_{n\ge0}$ be non-negative integers with $M_0=0$, $M_n\le3n$, $2M_{a+b}\le M_{2a}+M_{2b}$ for all $a,b\ge0$, and $M_{2n}\in8\Z$ for all $n$. Then $M_n=0$ for all $n$.
\end{lemma}
\begin{proof}
Put $g(n):=M_{2n}$; then $g(0)=0$, $0\le g(n)\le6n$, $g(n)\in8\Z$. For $c\ge1$ the hypothesis with $(a,b)=(c-1,c+1)$ gives $2g(c)\le g(c-1)+g(c+1)$, so the increments $d_c:=g(c)-g(c-1)\in8\Z$
satisfy $d_{c+1}\ge d_c$. Suppose $g(k)>0$ for some $k$. Then $\sum_{c=1}^kd_c>0$, so some $d_j>0$, hence $d_j\ge8$, hence $d_c\ge8$ for all $c\ge j$ and
$g(n)=g(j-1)+\sum_{c=j}^nd_c\ge8(n-j+1)$ for $n\ge j$. At $n=4j$: $g(4j)\ge24j+8>24j\ge g(4j)$, a contradiction. Thus $M_{2n}=0$ for all $n$, and the hypothesis with $b=0$ gives
$2M_a\le M_{2a}+M_0=0$, so $M_a=0$.
\end{proof}

\begin{corollary}\label{i7:cor:Mzero}
Under (i), (ii) and (iii) for the four planes of a coordinate pair, $M_N=0$ for all $N$; hence $c_\ell[\chi]=0$ for all $\ell\ne0$ and every narrow tuple $\chi$, and $D_N[\chi]\equiv0$
on $\C$ for every narrow tuple and for every tuple of tests in $(L^1\cap L^2)(\R^3;T_\C)$ vanishing outside $K_j\times\R$ with $K_j$ the balls of Proposition~\ref{i7:prop:W10}.
\end{corollary}
\begin{proof}
Proposition~\ref{i7:prop:W14}, Proposition~\ref{i7:prop:W15}(4) and Lemma~\ref{i7:lem:W16} give $M_N=0$. Then \eqref{i7:eq:fourierform} gives $D_N[\chi]\equiv0$ for narrow tuples
(Lemma~\ref{i7:lem:W13A}), and Proposition~\ref{i7:prop:W10}(c) (with Lemma~\ref{i7:lem:W13A}: the tuples supported in the balls are narrow, so their coefficients vanish) for the non-product tests:
by Step~5 of the proof of Proposition~\ref{i7:prop:W10} the coefficients $c_\ell[\chi]$ are $N$-linear in $\chi$ and are limits of the coefficients of finite complex-linear combinations of real smooth
product tests supported in the same balls, which vanish, so $c_\ell[\chi]=0$ for $\ell\ne0$ and $D_N[\chi]\equiv0$ on $\cX$.
\end{proof}

\subsection{From narrow configurations to all distinct points}\label{i7:sub:W17}

Throughout this subsection $(t,x)=(x_j,x_k)$ is a coordinate pair and $y=x_l$ is the third coordinate. We write $P(x):=(x_j,x_k)$ for the planar projection and $R(\theta)$ for the rotation about $e_l$;
for $R=R(\theta)$ we have $\supp R^*\chi=R^{-1}\supp\chi$ and $R^*\tau_a=\tau_{R^{-1}a}R^*$. We put $\cS_N:=\cS(\R^3;T_\C)^N$ and $B_r(X):=\prod_mB(x_m,r)$ for $X=(x_m)_m\in(\R^3)^N$, and we say that a tuple $\chi$ is
supported in $\prod O_m$ if $\supp\chi_m\subset O_m$ for every $m$.

A \emph{move} from $X$ is a pair $(H,a)$ consisting of an affine coordinate hyperplane $H=\{x\cdot n=c\}$, $n\in\{\pm e_1,\pm e_2,\pm e_3\}$, with \emph{margin} $d_H(X):=\min_m|x_m\cdot n-c|>0$,
and of a vector $a=s_1n+v_1$ ($v_1\perp n$). Its \emph{forward group} is $F:=\{m:x_m\cdot n>c\}$, and the move takes $X$ to the configuration $X^a:=(x_m+a1_F(m))_m$. For a tuple $\chi$, $s\in\R$ and
$v\perp n$ we put $\chi_{(s,v)}:=T^*_{-sn-v}\chi:=(\tau_{sn+v}^{1_F(m)}\chi_m)_m$; this operation translates only the tests of the forward group. Finally, $V(X;r,\delta)$ denotes the statement
that $D_N[\chi](\theta)=0$ for all $|\theta|<\delta$ and all $\chi\in\cS_N$ supported in $B_r(X)$.

The proofs of this subsection use Theorem~\ref{i7:thm:W3} for the pairs $(x_j,x_k)$ and $(x_j,x_l)$, for every $j$, and hence hypothesis (iii) at the three coordinate planes and at the
three planes $\{x_j+x_k=0\}$, $j<k$ (Remark~\ref{i7:rem:fourplanes}).

\begin{proposition}[from narrow configurations to all distinct points]\label{i7:prop:W17}
Assume (i), (ii) and (iii) for the nine planes, $N\ge1$.
\begin{itemize}
\item[(a)] Let $X^0\in(\R^3)^N$, $H=\{x\cdot n=c\}$, $d:=d_H(X^0)>0$, $O:=B_{d/2}(X^0)$, $\varepsilon:=d/4$. For $\chi\in\cS_N$ supported in $O$, $(s,v)\mapsto S_N(T^*_{-sn-v}\chi)$ is
real-analytic on $(-\varepsilon,\infty)\times n^\perp$, the restriction of a bounded holomorphic function on $\{(s,v)\in\C\times n^\perp_\C:\re s+\varepsilon>|\im v|_1\}$.
\item[(b)] Let $X$ have pairwise distinct planar projections $P(x_m)$, $N\ge2$. There are $k\le N(N+1)/2$ moves $X=X^0\to X^1\to\dots\to X^k$ with $P(X^k)$ narrow, radii $r_i>0$ and a
$\delta>0$ such that for $|\theta|<\delta$ and $\chi$ supported in $B_{r_i}(X^{i-1})$ both $(s,v)\mapsto S_N(\chi_{(s,v)})$ and $(s,v)\mapsto S_N(R(\theta)^*\chi_{(s,v)})$ are real-analytic
on a connected open set $\Omega_{i,\theta}$ containing $(0,0)$ and $(s_i,v_i)$ (the $i$-th move).
\item[(c)] If for every $\varphi\in\R$ there are $r_\varphi,\delta_\varphi>0$ with $V(R(\varphi)X;r_\varphi,\delta_\varphi)$, then $V(X;r_*,\infty)$ for some $r_*>0$.
\item[(d)] If $V(X;r_X,\infty)$ holds at every $X$ with pairwise distinct planar projections, it holds at every $X\notin\Delta:=\{X:\exists m\ne m',\ x_m=x_{m'}\}$.
\end{itemize}
\end{proposition}

\begin{proof}[Proof of (a)]
By (ii) and Lemma~\ref{i7:lem:W1}(e), $S_N$ is translation invariant; by Lemma~\ref{i7:lem:W2} it is $\theta_j^*$-invariant for the coordinate reflections, and $\theta_j^*\tau_a=\tau_{\theta_ja}\theta_j^*$.
Applying $\tau_{-cn}$ and, if $n=-e_j$, $\theta_j$ (which maps the forward group to the forward group), assume $H=\{x_j=0\}$, $n=e_j$. Let $B:=\{m:x^0_m\cdot e_j<0\}$ and $F:=\{m:x^0_m\cdot e_j>0\}$;
for $\chi$ supported in $O$, $\supp\chi_m\subset\{x_j<-d/2\}$ for $m\in B$ and $\subset\{x_j>d/2\}$ for $m\in F$. Put $A:=\prod_{m\in B}F(\chi_m)$ and $G:=\prod_{m\in F}F(\tau_{-\varepsilon e_j}\chi_m)$;
by Lemma~\ref{i7:lem:W1}(b), Hölder and Lemma~\ref{i7:lem:W5loc}, $A\in L^2(\Sigma(\{x_j<0\}),\mu)$ and $G\in L^2_{j,+}:=L^2(\Sigma\{x_j>0\},\mu)$. Lemma~\ref{i7:lem:recon} at the plane
$\{x_j=0\}$ ($u=e_j$) gives $\cH_j$, the contraction $G'\mapsto[G']_j$, $e^{-sH_j}$ with $H_j\ge0$, the strongly continuous two-parameter unitary group $V_j(v)$, $v\perp e_j$, with
$[U(se_j+v)G']_j=e^{-sH_j}V_j(v)[G']_j$ ($s\ge0$) and $\E_\mu[A'G']=\ip{[\overline{\Theta_jA'}]_j}{[G']_j}_j$ for $A'\in L^2(\Sigma\{x_j<0\})$, $G'\in L^2_{j,+}$ (the definition of the form,
$\overline{\Theta_jA'}\in L^2_{j,+}$). For $s>-\varepsilon$, $U((s+\varepsilon)e_j+v)G=\prod_{m\in F}F(\tau_{se_j+v}\chi_m)\in L^2_{j,+}$, hence
\[
S_N(\chi_{(s,v)})=\E_\mu[A\,U((s+\varepsilon)e_j+v)G]=\ip{\psi}{e^{-(s+\varepsilon)H_j}V_j(v)\phi}_j,\qquad\psi:=[\overline{\Theta_jA}]_j,\ \phi:=[G]_j .
\]

$e^{-sH_j}$ and $V_j(v)$ commute (translations commute on $L^2_{j,+}$), so $H_j$ and the Stone generators $P_k,P_l$ of $V_j$ have a joint spectral measure $\Pi_j$ on $\R^3$ (the spectral theorem and Stone's theorem \cite{RS72}) whose
$(\omega,p_k)$- and $(\omega,p_l)$-marginals are the joint spectral measures of $(H_j,P_k)$ and $(H_j,P_l)$. By Theorem~\ref{i7:thm:W3} for the coordinate pairs $(x_j,x_k)$ and $(x_j,x_l)$,
$\Pi_j$ is carried by $\Gamma:=\{\omega\ge|p_k|\}\cap\{\omega\ge|p_l|\}=\{\omega\ge|p|_\infty\}$. With $\rho:=\ip{\psi}{\Pi_j(\cdot)\phi}_j$, $|\rho|(\R^3)\le\|A\|_{L^2(\mu)}\|G\|_{L^2(\mu)}$ and
$S_N(\chi_{(s,v)})=\int_\Gamma e^{-(s+\varepsilon)\omega+ip\cdot v}d\rho$. For complex $(s,v)$ with $\re s+\varepsilon>|\im v|_1$ and $(\omega,p)\in\Gamma$:
$|e^{-(s+\varepsilon)\omega+ip\cdot v}|\le e^{-(\re s+\varepsilon-|\im v|_1)\omega}\le1$; the integral is continuous (dominated convergence), holomorphic in each variable (Morera and Fubini),
hence holomorphic (Osgood), bounded by $\|A\|_2\|G\|_2$; its real restriction is real-analytic.
\end{proof}

\begin{lemma}[propagation]\label{i7:lem:W17P}
Let $(H,a)$ be a move from $X$, $r_0>0$, $\delta\in(0,\infty]$, and for each $|\theta|<\delta$ let $\Omega_\theta\subset\R\times n^\perp$ be connected and open, containing $(0,0)$ and $(s_1,v_1)$,
such that $(s,v)\mapsto S_N(\chi_{(s,v)})$ and $(s,v)\mapsto S_N(R(\theta)^*\chi_{(s,v)})$ are real-analytic on $\Omega_\theta$ for all $\chi$ supported in $B_{r_0}(X)$. If $V(X^a;r',\delta')$
then $V(X;\min(r_0,r'/2),\min(\delta,\delta'))$.
\end{lemma}
\begin{proof}
Let $\chi$ be supported in $B_r(X)$, $r:=\min(r_0,r'/2)$, and $|\theta|<\min(\delta,\delta')$. $\chi_{(s,v)}$ is supported in $B_r(X^{(s,v)})$, and $\max_m|x^{(s,v)}_m-x^a_m|=|(s,v)-(s_1,v_1)|<r'/2$
gives $B_r(X^{(s,v)})\subset B_{r'}(X^a)$, so $D_N[\chi_{(s,v)}](\theta)=0$ on a nonempty open subset of $\Omega_\theta$; the real-analytic function $(s,v)\mapsto D_N[\chi_{(s,v)}](\theta)$
vanishes on $\Omega_\theta$ (identity theorem), in particular at $(0,0)$. (If $F=\emptyset$ then $X^a=X$ and there is nothing to prove.)
\end{proof}

\begin{lemma}[two classes of moves]\label{i7:lem:W17M}
Under the hypotheses of Proposition~\ref{i7:prop:W17}, let $(H,a)$ be a move from $X$, and put $d=d_H(X)$.
\begin{itemize}
\item[(M1)] If $n=\pm e_l$ and $a=v_1$ is planar, then the hypotheses of Lemma~\ref{i7:lem:W17P} hold with $r_0=d/2$, $\delta=\infty$ and $\Omega_\theta=(-d/4,\infty)\times n^\perp$.
\item[(M2)] If $n\in\{\pm e_j,\pm e_k\}$, $a=we_l$ with $w\in\R$, and $\varrho_X:=\max_m|P(x_m)|>0$, then the hypotheses of Lemma~\ref{i7:lem:W17P} hold with $r_0=d/4$ and $\delta:=d/(2\varrho_X)$.
\end{itemize}
\end{lemma}
\begin{proof}
Let $F$ be the forward group of the move and $B$ its complement. Since $R^*\tau_a=\tau_{R^{-1}a}R^*$, we have $R(\theta)^*\chi_{(s,v)}=(R(\theta)^*\chi)_{(s',v')}$ with $R(-\theta)(sn+v)=s'n+v'$,
provided $H$ separates $R(-\theta)X$ into the same groups $F,B$; and $R(\theta)^*\chi$ is supported in $B_r(R(-\theta)X)$ if $\chi$ is supported in $B_r(X)$.

(M1) The rotation $R(\theta)$ fixes $x_l$, so $R(-\theta)X$ has margin $d$ and forward group $F$, and $s'=s$, $v'=R(-\theta)v$. By Proposition~\ref{i7:prop:W17}(a), applied to $X$ and to
$R(-\theta)X$, the hypotheses of Lemma~\ref{i7:lem:W17P} hold with the stated $r_0$, $\delta$ and $\Omega_\theta$.

(M2) For $|\theta|<\delta$ we have $|R(-\theta)x_m-x_m|\le|\theta|\varrho_X<d/2$, so $H$ misses $R(-\theta)X$ with margin $d'>d/2$ and forward group $F$. With $n'$ a planar unit vector
perpendicular to $n$,
\[
R(-\theta)(sn+v_xn'+v_ye_l)=(s\cos\theta\mp v_x\sin\theta)n+(\pm s\sin\theta+v_x\cos\theta)n'+v_ye_l .
\]
By Proposition~\ref{i7:prop:W17}(a) for $R(-\theta)X$ (with $O'\supset B_{d/4}(R(-\theta)X)$ and $\varepsilon'>d/8$) and the linear substitution, $(s,v)\mapsto S_N(R(\theta)^*\chi_{(s,v)})$ is
real-analytic on the half-space $\{s\cos\theta\mp v_x\sin\theta>-d/8\}$ for $\chi$ supported in $B_{d/4}(X)$. By Proposition~\ref{i7:prop:W17}(a) for $X$, the map
$(s,v)\mapsto S_N(\chi_{(s,v)})$ is real-analytic on $\{s>-d/4\}$. The intersection $\Omega_\theta$ of these two half-spaces is convex and contains $(0,0)$ and $(0,we_l)$, so the hypotheses
of Lemma~\ref{i7:lem:W17P} hold with $r_0=d/4$ and this $\delta$.
\end{proof}

\begin{proof}[Proof of (b)]
\emph{Step 1.} For each pair $m<m'$ choose $n_{mm'}\in\{e_j,e_k\}$ with $x_m\cdot n_{mm'}\ne x_{m'}\cdot n_{mm'}$, which is possible as $P(x_m)\ne P(x_{m'})$, and choose $c_{mm'}$ strictly
between these two values and outside $\{x_i\cdot n_{mm'}\}_i$. The hyperplane $H_{mm'}:=\{x\cdot n_{mm'}=c_{mm'}\}$ misses $X$; let $d_{mm'}$ be its margin and $F_{mm'}$ its forward group, which
contains exactly one of $m,m'$. Perform, in any order, the moves $(H_{mm'},w_{mm'}e_l)$ of type (M2), with coefficients $w_{mm'}$ to be chosen. These moves do not change the planar projections,
so every $H_{mm'}$ keeps its margin and its forward group, and by Lemma~\ref{i7:lem:W17M} each of these moves satisfies the hypotheses of Lemma~\ref{i7:lem:W17P} for
$|\theta|<\delta:=\min_{m<m'}d_{mm'}/(2\varrho_X)$. The third coordinates after these moves are $y'_i=y_i+\sum_{m<m'}w_{mm'}1_{F_{mm'}}(i)$. For $i\ne i'$ the set $\{w:y'_i=y'_{i'}\}$ is a proper
affine hyperplane of $\R^{N(N-1)/2}$, because the coefficient of $w_{ii'}$ is $\pm1$. We choose $w$ outside the finite union of these hyperplanes; the resulting configuration $X^{(1)}$ then has
pairwise distinct third coordinates.

\emph{Step 2.} Order the third coordinates of $X^{(1)}$ as $y'_{\sigma(1)}<\dots<y'_{\sigma(N)}$, and put $a_m:=(y'_{\sigma(m-1)}+y'_{\sigma(m)})/2$, $H'_m:=\{x_l=a_m\}$ and
$z_m:=P(x_{\sigma(m)})$; the hyperplane $H'_m$ has margin $>0$ and forward group $\{\sigma(m),\dots,\sigma(N)\}$. Perform for $m=2,\dots,N$ the moves $(H'_m,v_m)$ of type (M1), where
$v_m:=e_j-(z_m-z_{m-1})$. These moves do not change the third coordinates, so each $H'_m$ keeps its margin and its forward group. The final projection of the point with index $\sigma(m)$ is
$z_m+\sum_{i=2}^mv_i=z_1+(m-1)e_j$. Hence all successive differences equal $e_j$, and $P(X^k)$ is narrow (with $u=e_j$ and angle $0<\pi/4$). By Lemma~\ref{i7:lem:W17M} all the moves satisfy
the hypotheses of Lemma~\ref{i7:lem:W17P} for $|\theta|<\delta$, and their number is $k\le N(N-1)/2+N-1$.
\end{proof}

\begin{proof}[Proof of (c)]
If $V(R(\varphi_0)X;r_0,\delta_0)$ and $\max_m|R(\varphi)x_m-R(\varphi_0)x_m|<r_0/2$, then $B_{r_0/2}(R(\varphi)X)\subset B_{r_0}(R(\varphi_0)X)$, so $V(R(\varphi)X;r_0/2,\delta_0)$ on an open interval
around $\varphi_0$. Compactness of $[0,2\pi]$ and $2\pi$-periodicity give $r_*,\delta_*>0$ with $V(R(\varphi)X;r_*,\delta_*)$ for all $\varphi$. Let $\chi$ be supported in $B_{r_*}(X)$,
$\Phi(\varphi):=D_N[\chi](\varphi)$. Planar rotations commute, so $R(\varphi+\theta)^*=R(\theta)^*R(\varphi)^*$ and $\Phi(\varphi+\theta)=D_N[R(\varphi)^*\chi](\theta)+\Phi(\varphi)$. $R(\varphi)^*\chi$ is
supported in $B_{r_*}(R(-\varphi)X)$, so the first term vanishes for $|\theta|<\delta_*$: $\Phi(\varphi+\theta)=\Phi(\varphi)$ for all $\varphi$, $|\theta|<\delta_*$; $\Phi$ is constant, $\Phi\equiv\Phi(0)=0$.
\end{proof}

\begin{proof}[Proof of (d)]
Let $X\notin\Delta$, and let $y^{(1)}<\dots<y^{(q)}$ be the distinct values of the third coordinates $y_m$ of its points. If $q=1$ the planar projections are distinct. Otherwise let
$H_i:=\{x_l=(y^{(i)}+y^{(i+1)})/2\}$, with forward group $F_i:=\{m:y_m>y^{(i)}\}$, and perform the moves $(H_i,w_ie_j)$, $i<q$, of type (M1); they leave the third coordinates unchanged. The
final planar projections are $z'_m=P(x_m)+(\sum_iw_i1_{F_i}(m))e_j$. If $y_m=y_{m'}$, then $z'_m-z'_{m'}=P(x_m)-P(x_{m'})\ne0$. If $y_m=y^{(\alpha)}<y_{m'}$, then
\[
\{w:z'_m=z'_{m'}\}\subset\{(P(x_m)-P(x_{m'}))\cdot e_j+\sum_iw_i(1_{F_i}(m)-1_{F_i}(m'))=0\},
\]
and the set on the right is a proper affine hyperplane, because the coefficient of $w_\alpha$ is $-1$. Choose $w$ outside the finite union of these hyperplanes. The resulting configuration
$X'$ has distinct planar projections, so $V(X';r_{X'},\infty)$ holds. Applying Lemma~\ref{i7:lem:W17P} to the $q-1$ moves in reverse order, for all $\theta$ since the moves of type (M1) have
$\delta=\infty$ (Lemma~\ref{i7:lem:W17M}), we obtain $V(X;r,\infty)$.
\end{proof}

\subsection{Assembly}\label{i7:sub:W18}

\begin{proof}[Proof of Theorem~\ref{i7:thm:main}]
The case $N=0$ is trivial and $N=1$ is Lemma~\ref{i7:lem:W2}. Let $N\ge2$; fix the coordinate pair $(x_1,x_2)$, $l=3$, and write $R(\theta)$ for the rotations about $e_3$.

(1) By Corollary~\ref{i7:cor:Mzero}, $D_N[\chi](\theta)=0$ for all real $\theta$ and all $\chi\in\cS_N$ with $\supp\chi_m\subset U_m\times\R$, $U_m$ the planar balls of
Proposition~\ref{i7:prop:W10} about any narrow configuration $Z$. Hence $V(X;r,\infty)$ whenever $P(X)$ is narrow and $r$ is small (take $r$ below the radius of the balls).

(2) Let $X$ have distinct planar projections. By Proposition~\ref{i7:prop:W17}(b) and (1), $V(X^k;r_k,\infty)$; Lemma~\ref{i7:lem:W17P} applied $k$ times gives $V(X;r,\delta)$ for some
$r,\delta>0$. Every $R(\varphi)X$ has distinct planar projections, so Proposition~\ref{i7:prop:W17}(c) gives $V(X;r_*,\infty)$; Proposition~\ref{i7:prop:W17}(d) gives $V(X;r_X,\infty)$ for
all $X\notin\Delta$.

(3) Let $\chi_m\in C^\infty_c(\R^3;T_\C)$ have pairwise disjoint supports $K_m$. $K:=\prod_mK_m\subset(\R^3)^N\setminus\Delta$ is compact, covered by finitely many
$B_{r_{X^{(i)}}/2}(X^{(i)})$; with $r:=\min_ir_{X^{(i)}}/2$, each $Y\in K$ has $B_r(Y)\subset B_{r_{X^{(i)}}}(X^{(i)})$ for some $i$. Smooth partitions of unity on $K_m$ with supports in balls
$B(y_m^{(\alpha)},r)$, $y_m^{(\alpha)}\in K_m$, give $\chi_m=\sum_\alpha\chi_m^{(\alpha)}$; by multilinearity $D_N[\chi](\theta)=\sum D_N[(\chi_1^{(\alpha_1)},\dots,\chi_N^{(\alpha_N)})](\theta)$, each
tuple supported in some $B_r(Y)$, $Y\in K$, hence each term vanishes: $S_N(R(\theta)^*\chi)=S_N(\chi)$ for all rotations about $e_3$ and all $\theta$.

(4) Other axes: for a cyclic permutation matrix $Q\in B_3$, $QR_{e_3}(\theta)Q^{-1}=R_{Qe_3}(\theta)$, $(QRQ^{-1})^*=(Q^{-1})^*R^*Q^*$, $Q^*$ preserves pairwise disjointness of supports, and
$S_N(Q^*\psi)=S_N(\psi)$ (Lemmas~\ref{i7:lem:W2}, \ref{i7:lem:W1}(e)): $S_N((QRQ^{-1})^*\chi)=S_N(R^*Q^*\chi)=S_N(Q^*\chi)=S_N(\chi)$. So (3) holds for the rotations about $e_1$ and $e_2$ as well.

(5) Euler angles: every $R\in SO(3)$ is $R_{e_3}(\alpha)R_{e_1}(\beta)R_{e_3}(\gamma)$, so $R^*=R_{e_3}(\gamma)^*R_{e_1}(\beta)^*R_{e_3}(\alpha)^*$, each factor preserving pairwise disjointness:
$S_N(R^*\chi)=S_N(\chi)$ on $SO(3)$. Since $O(3)=SO(3)\cup\theta_1SO(3)$ and $(\theta_1R)^*=R^*\theta_1^*$, Lemma~\ref{i7:lem:W2} extends the identity to $O(3)$.

(6) Schwartz tests: for $\chi_m\in\cS$ with pairwise disjoint supports, $\chi_m^{(q)}:=\vartheta(\cdot/q)\chi_m$ ($\vartheta\in C^\infty_c$, $\vartheta=1$ on $B(0,1)$) have compact pairwise
disjoint supports, and $\chi^{(q)}_m\to\chi_m$, $R^*\chi^{(q)}_m\to R^*\chi_m$ in $L^1\cap L^2$; the continuity of Lemma~\ref{i7:lem:W1}(c) gives
$S_N(R^*\chi)=\lim_qS_N(R^*\chi^{(q)})=\lim_qS_N(\chi^{(q)})=S_N(\chi)$. This is the theorem; the extension from real to complex Schwartz tests is by multilinearity.
\end{proof}

\begin{remark}[constants and the planes used]\label{i7:rem:constants}
The constants of the proof are explicit in terms of the constant $C$ of (i): $K_1=17(1+C)$ (Lemma~\ref{i7:lem:W1}), $K_6=e^{2C}$ (Theorem~\ref{i7:thm:W6}), $K_7=14K_6$
(Theorem~\ref{i7:thm:W7}), $K_8=2K_7$ (Proposition~\ref{i7:prop:W8}), and the geometric thresholds $r_0=\frac12\min_j|d_j|\cos(\gamma+\pi/4)$ (Proposition~\ref{i7:prop:W10}),
$\varepsilon=d/4$, $\delta=d/(2\varrho_X)$ (Proposition~\ref{i7:prop:W17}). Subsections~\ref{i7:sub:W1}--\ref{i7:sub:W15} use, for one coordinate pair, only the four planes of that pair
(Lemma~\ref{i7:lem:W15b}); Subsection~\ref{i7:sub:W17} uses Theorem~\ref{i7:thm:W3} for the six ordered coordinate pairs, i.e.\ the three coordinate planes and the three diagonal
planes $\{x_j+x_k=0\}$; the three planes $\{x_j-x_k=0\}$ enter through Proposition~\ref{i7:prop:W15} (the plane $P_3$ of each pair) and through the $B_3$-invariance of
Lemma~\ref{i7:lem:W2} used in step (4) of the assembly (a $3$-cycle is a product of two transposition reflections); step (5) of the assembly uses in addition the reflection
$\theta_1$. No property of $\mu$ beyond (i)--(iii) enters.
\end{remark}

\begin{remark}\label{i7:rem:credit}
The statement of Theorem~\ref{i7:thm:main} is Theorem~A of \cite{RPM1}; in proving it we implement the original strategy of that work (Remark~\ref{flh:rem:strategy}).
\end{remark}












\appendix
\providecommand{\N}{\ensuremath{\mathbb N}}\providecommand{\cV}{\ensuremath{\mathcal V}}\providecommand{\cI}{\ensuremath{\mathcal I}}
\providecommand{\Euc}{\ensuremath{\R^3\rtimes O(3)}}\providecommand{\Cub}{\ensuremath{\R^3\rtimes O_h}}
\providecommand{\defeq}{:=}\providecommand{\OSf}{\ensuremath{\mathrm{OS4}}}
\providecommand{\fournorm}[1]{\ensuremath{\lvert #1\rvert_{4}}}\providecommand{\abs}[1]{\ensuremath{\lvert #1\rvert}}
\providecommand{\cS}{\mathcal S}\providecommand{\cB}{\mathcal B}\providecommand{\cH}{\mathcal H}\providecommand{\cZ}{\mathcal Z}
\providecommand{\cG}{\mathcal G}\providecommand{\cN}{\mathcal N}\providecommand{\E}{\mathbb E}
\providecommand{\supp}{\operatorname{supp}}\providecommand{\Oh}{O_h}
\providecommand{\ip}[2]{\langle #1,#2\rangle}\providecommand{\norm}[1]{\lVert #1\rVert}

\section{Probability measures on spaces of distributions}\label{ms:measure}

This appendix contains the measure theory used in Theorems~\ref{thm:osrest:measure} and~\ref{thm:osrest:OS4weak}: the setting (H1)--(H3) in which reflection positivity of the law of a field is formulated; the field spaces and the action of the Euclidean group on them; Hermite--Sobolev norms; the Borel structure of the field spaces; the density of generating algebras; the passage from a measure on $\cS'$ to a measure on a Hermite--Sobolev space; the uniqueness of a measure with a given characteristic functional or with given moments; and the verification of (H1)--(H3) from properties of the characteristic functional. Nothing in it uses the lattice. Remark~\ref{ms:met} states where its hypotheses are met in the main text.

The Bochner--Minlos theorem is used in the form stated after Definition~\ref{ms:fielddef}. Definition~\ref{ms:conv0} fixes two
conventions. Subsection~\ref{ms:quoted} contains the results on the ergodic decomposition and on clustering that are used in
Theorem~\ref{thm:osrest:OS4weak}.

\begin{definition}[conventions: the weighted $C^4$ norm and the transport of test functions]\label{ms:conv0}
\textup{(a)} \emph{The weighted $C^4$ norm.} For a function $q$ and a $2$-form $g$ on $\R^3$:
$\fournorm{q}\defeq\sup_{x,|\gamma|\le4}(1+|x|^2)^2|D^\gamma q(x)|$ and $\fournorm{g}\defeq(\sum_{\alpha<\beta}\fournorm{g_{\alpha\beta}}^2)^{1/2}$; on scalar functions this is the norm $|q|_p$ of \cite{OS75} with $p=4$.
For a pair $u=(g,q)$ of a $2$-form and a function, $\fournorm u\defeq(\fournorm g^2+\fournorm q^2)^{1/2}$.
\textup{(b)} \emph{The transport of test functions.} For a function $h$ and a $2$-form $f$ on $\R^3$ and for $O\in O(3)$ put $(O\cdot h)(x)\defeq h(O^{-1}x)$ and $(O\cdot f)_x(u,v)\defeq f_{O^{-1}x}(O^{-1}u,O^{-1}v)$; for $b\in\R^3$ put $\tau_bh\defeq h(\cdot-b)$ and $(\tau_bf)_x\defeq f_{x-b}$. For $\nu\in\{1,2,3\}$ and $c\in\R$ let $\rho=\rho_{\nu,c}$ be the map $x\mapsto x-2(x_\nu-c)e_\nu$, and put $\theta h\defeq h\circ\rho$ and $\theta f\defeq\rho^*f$, where $(\rho^*f)_x(u,v)\defeq f_{\rho x}(\rho u,\rho v)$ (the components $f_{\mu\nu}$, $\mu\ne\nu$, change sign).
\end{definition}

\subsection{The abstract setting}\label{ms:H}

\emph{Notation.} The standard basis of $\R^3$ is $e_1,e_2,e_3$. For $j\in\{1,2,3\}$ and $c\in\R$ put $H^\pm_{j,c}\defeq\{x\in\R^3:\pm(x_j-c)>0\}$ and $\rho_jx\defeq x-2x_je_j$, and let $\rho_{j,c}$ be the reflection in the plane $\{x_j=c\}$. A bounded open subset of $\R^3$ is denoted $K$; $K\Subset U$ means that $\overline K\subset U$.

\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 $U\subset U'$; $\Sigma(U)=\bigvee_{K\Subset U}\Sigma(K)$; $\bigvee_K\Sigma(K)=\cB$; each $\Sigma(U)$ is generated by a countable family of bounded continuous functions on $X$; and, with
\begin{equation}\label{ms:eqcov}
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{ms:eqRP}
\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}

By (H1), $a\mapsto U(a)$ is a unitary representation of $\R^3$ on $L^2(\mu)$, $\Theta_j$ is a unitary involution, and
\begin{equation}\label{ms:eqcommute}
\Theta_jU(a)=U(\rho_ja)\Theta_j;\quad\text{in particular } \Theta_jU(te_j)=U(-te_j)\Theta_j,\ \ \Theta_jU(y)=U(y)\Theta_j\ (y\perp e_j).
\end{equation}
By translation invariance, \eqref{ms:eqRP} for $c=0$ implies it for every $c$: if $F\in L^2(\Sigma(H^+_{j,c}),\mu)$, then $F_0\defeq U(-ce_j)F=F\circ\tau_{ce_j}$ is $\Sigma(H^+_{j,0})$-measurable, $\overline{F\circ\theta_{j,c}}\,F=(\overline{F_0\circ\theta_j}\,F_0)\circ\tau_{-ce_j}$, and $\tau_{-ce_j}$ preserves $\mu$, so $\E_\mu[\overline{F\circ\theta_{j,c}}\,F]=\E_\mu[\overline{F_0\circ\theta_j}\,F_0]$.

\begin{remark}[conventions]\label{ms:conv}
On a space of distributions we use $(\tau_a\Phi)(u)\defeq\Phi(\tau_{-a}u)$ with $\tau_bu=u(\cdot-b)$ as in Definition~\ref{ms:conv0}(b): this is 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.
\end{remark}

\subsection{Field spaces, the Euclidean action and Hermite norms}\label{ms:spaces}

\emph{Fibre and transport.} $\cV$ is $\Lambda^2\defeq\Lambda^2\R^3$ with its Euclidean fibre norm, or $\Lambda^2\oplus\R$; $(e_c)_{c\le\dim\cV}$ is an orthonormal basis of $\cV$, and $\ip uv\defeq\int_{\R^3}u(x)\cdot v(x)\,dx$ is the $L^2$ pairing of $\cV$-valued functions. All test functions are real. A Euclidean motion $\gamma\in\Euc$, $\gamma x=Ox+b$, acts on $\cV$-valued functions by $\gamma\cdot u\defeq\tau_b(O\cdot u)$, with the transport $O\cdot$ of Definition~\ref{ms:conv0}(b) on each summand. Thus $(\gamma\cdot u)(x)=\pi(O)u(\gamma^{-1}x)$, where $\pi(O)\omega=\omega(O^{-1}\cdot,O^{-1}\cdot)$ on $\Lambda^2$ and $\pi(O)=1$ on $\R$; $\pi$ is an orthogonal representation of $O(3)$ on $\cV$, because the fibre norm of a $2$-form is the Frobenius norm of an antisymmetric matrix, up to a factor. Hence $\gamma\mapsto\gamma\cdot$ is an action of $\Euc$, $\supp(\gamma\cdot u)=\gamma\supp u$, and $\gamma\cdot$ preserves $C^\infty_c$, $\cS$, the $L^2$ pairing and the Sobolev norms $\norm u_{H^s}^2\defeq\int(1+|\xi|^2)^s|\hat u(\xi)|^2\,d\xi$, $s\in\N_0$; and $\gamma\mapsto\gamma\cdot u$ is continuous into $L^2$ for $u\in C^\infty_c$ (uniform continuity of $u$). The coordinate reflection $\sigma_j\cdot u$ is the case $\gamma=\rho_j$, and the reflected source $\rho_{j,c*}u$ is $\rho_{j,c}\cdot u$, which is $\theta u$ of Definition~\ref{ms:conv0}(b) for $\rho=\rho_{j,c}$.

\emph{Hermite norms.} $\cA\defeq-\Delta+|x|^2+1$ acts componentwise. The Hermite functions $h_\alpha$, $\alpha\in\N_0^3$, form an orthonormal basis of $L^2(\R^3)$, and $\cA h_\alpha=\lambda_\alpha h_\alpha$ with $\lambda_\alpha=2|\alpha|+4$; put $h_{\alpha,c}\defeq h_\alpha e_c$. For $k\in2\N_0$ and $u\in\cS(\R^3;\cV)$,
\[
\norm u_k\defeq\norm{\cA^{k/2}u}_{L^2}=\Big(\sum_{\alpha,c}\lambda_\alpha^k\ip{h_{\alpha,c}}u^2\Big)^{1/2},
\]
since $\cA$ is symmetric on $\cS$. $\cH^k$ is the completion of $\cS(\R^3;\cV)$ under $\norm\cdot_k$, and $\cH^{-k}$ is its dual, a separable Hilbert space with $\norm\Psi_{-k}^2=\sum_{\alpha,c}\lambda_\alpha^{-k}\Psi(h_{\alpha,c})^2$, because $(\lambda_\alpha^{-k/2}h_{\alpha,c})$ is an orthonormal basis of $\cH^k$ (Lemma~\ref{ms:hermite}(c)). These are the spaces used in Theorem~\ref{thm:osrest:measure}.

\begin{definition}[the field spaces]\label{ms:fielddef}
Fix an even integer $r\ge2$ (in the applications $r\ge14$). $X\defeq H^{-r}_{\mathrm{loc}}(\R^3;\cV)$ is the space of real linear functionals $\Phi$ on $C^\infty_c(\R^3;\cV)$ such that, for every $\chi\in C^\infty_c(\R^3)$, $u\mapsto\Phi(\chi u)$ is bounded for $\norm\cdot_{H^r}$, that is, $\chi\Phi\in H^{-r}(\R^3;\cV)$, the dual of $H^r(\R^3;\cV)$. Fix $\chi_n\in C^\infty_c(\R^3)$ with $0\le\chi_n\le1$, $\chi_n=1$ on the ball $B_n$ and $\supp\chi_n\subset B_{n+1}$, and give $X$ the seminorms $p_n(\Phi)\defeq\norm{\chi_n\Phi}_{H^{-r}}$. Since $\chi_nu=u$ when $\supp u\subset B_n$,
\begin{equation}\label{ms:star}
|\Phi(u)|\le p_n(\Phi)\norm u_{H^r}\qquad(u\in C^\infty_c(B_n;\cV)).
\end{equation}
For $\gamma\in\Euc$ put $(\gamma\Phi)(u)\defeq\Phi(\gamma^{-1}\cdot u)$ on $X$ and $(\gamma\Psi)(u)\defeq\Psi(\gamma^{-1}\cdot u)$, $u\in\cH^r$, on $\cH^{-r}$ (with $\gamma^{-1}\cdot$ extended to $\cH^r$ by Lemma~\ref{ms:hermite}(e)). In particular $\tau_a$ is $\gamma=(x\mapsto x+a)$, so $(\tau_a\Phi)(u)=\Phi(\tau_{-a}u)$; $\theta_j\defeq\rho_j$, so $(\theta_j\Phi)(u)=\Phi(\sigma_j\cdot u)$; $\theta_{j,c}=\tau_{ce_j}\theta_j\tau_{-ce_j}$ is $\rho_{j,c}$; and $O_h\subset O(3)$ acts as a subgroup. Since $\gamma\mapsto\gamma\cdot$ is an action, so is $\gamma\mapsto\gamma$ on $X$ and on $\cH^{-r}$; the relations $\theta_j^2=1$, $\theta_j\tau_a=\tau_{\rho_ja}\theta_j$ and $O\tau_aO^{-1}=\tau_{Oa}$ ($O\in O(3)$) are relations in $\Euc$.

The local $\sigma$-algebras are $\Sigma(U)\defeq\sigma(\Phi\mapsto\Phi(u):u\in C^\infty_c(U;\cV))$ on $X$ and $\Sigma^H(U)\defeq\sigma(\Psi\mapsto\Psi(u):u\in C^\infty_c(U;\cV))$ on $\cH^{-r}$, for open $U\subset\R^3$. The restriction map is $\iota:\cH^{-r}\to X$, $\iota\Psi\defeq\Psi|_{C^\infty_c}$. On $\cS'(\R^3;\cV)$ the cylinder $\sigma$-algebra is generated by $\Phi\mapsto\Phi(u)$, $u\in\cS(\R^3;\cV)$.
\end{definition}

We recall the Bochner--Minlos theorem \cite{Min59}, \cite[Chapter~IV]{GV64}, \cite{GJ87} in the form used in the proof of Theorem~\ref{thm:osrest:measure}. Let $\cZ:\cS(\R^3;\cV)\to\C$ be continuous for the Schwartz topology, positive definite ($\sum_{k,l\le N}\bar a_ka_l\cZ(u_l-u_k)\ge0$ for all $N$, all $u_1,\dots,u_N\in\cS(\R^3;\cV)$ and all $a\in\C^N$) and normalised ($\cZ(0)=1$). Then there is exactly one probability measure $\mu'$ on $\cS'(\R^3;\cV)$ with its cylinder $\sigma$-algebra such that $\int e^{\mathrm i\Phi(u)}\,d\mu'(\Phi)=\cZ(u)$ for all $u\in\cS(\R^3;\cV)$.

\begin{lemma}[Hermite norms]\label{ms:hermite}
Let $k\in2\N$.
\begin{enumerate}
\item[(a)] For $|\delta|+|\gamma|\le k$ and $u\in\cS(\R^3;\cV)$, $\norm{x^\delta\partial^\gamma u}_{L^2}\le k^{k/2}\norm u_k$. Conversely $\norm u_k\le C_k\sum_{|\delta|+|\gamma|\le k}\norm{x^\delta\partial^\gamma u}_{L^2}$.
\item[(b)] $\fournorm u\le C\norm u_{10}$ for $u\in\cS$, $\fournorm\cdot$ the weighted $C^4$ norm of Definition~\ref{ms:conv0}(a), taken componentwise.
\item[(c)] For $u\in\cS$ the Hermite expansion $\sum_{\alpha,c}\ip{h_{\alpha,c}}u\,h_{\alpha,c}$ converges to $u$ in $\cS$. $C^\infty_c$ is dense in $\cH^k$, and $(\lambda_\alpha^{-k/2}h_{\alpha,c})$ is an orthonormal basis of $\cH^k$.
\item[(d)] $\norm u_k\le C_{k,n}\norm u_{H^k}$ for $u\in C^\infty_c(B_n;\cV)$, and $\norm{\chi u}_k\le C_{k,\chi}\norm u_{H^k}$ for $\chi\in C^\infty_c(\R^3)$, $u\in\cS$.
\item[(e)] For $\gamma x=Ox+b$ and $u\in\cS$, $\norm{\gamma\cdot u}_k\le C_k(1+|b|)^k\norm u_k$, with equality and $C_k=1$ when $b=0$. So $\gamma\cdot$ extends to a bounded operator on $\cH^k$, and $\gamma$ acts on $\cH^{-k}$ with norm $\le C_k(1+|b|)^k$, unitarily if $b=0$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) In each coordinate, $x_j=(a_j+a_j^*)/\sqrt2$ and $\partial_j=(a_j-a_j^*)/\sqrt2$ on $\cS$, where $a_jh_\alpha=\sqrt{\alpha_j}\,h_{\alpha-e_j}$, $a_j^*h_\alpha=\sqrt{\alpha_j+1}\,h_{\alpha+e_j}$, and $a_j$, $a_j^*$ are adjoint to each other on $\cS$. So $x^\delta\partial^\gamma$ is a sum of $2^\ell$ words of length $\ell\defeq|\delta|+|\gamma|$ in the $a_j^\#$, each with coefficient $\pm2^{-\ell/2}$. A word $w$ maps $h_\alpha$ to $c_w(\alpha)h_{\alpha+v_w}$ (or to $0$) for a fixed shift $v_w$, with $|c_w(\alpha)|\le(|\alpha|+\ell)^{\ell/2}\le(\ell\lambda_\alpha/2)^{\ell/2}$, since each factor multiplies by at most the square root of $1$ plus the current $|\alpha|$, and $|\alpha|+\ell\le\ell(|\alpha|+2)$. Computing the Hermite coefficients of $wu$ by duality on $\cS$, $wu=\sum_{\alpha,c}c_w(\alpha)\ip{h_{\alpha,c}}u\,h_{\alpha+v_w,c}$, distinct $\alpha$ going to distinct basis vectors. Hence $\norm{wu}^2\le(\ell/2)^\ell\sum\lambda_\alpha^\ell\ip{h_{\alpha,c}}u^2\le(\ell/2)^\ell\norm u_k^2$ for $\ell\le k$, since $\lambda_\alpha\ge1$, and
\[
\norm{x^\delta\partial^\gamma u}\le2^\ell2^{-\ell/2}(\ell/2)^{\ell/2}\norm u_k=\ell^{\ell/2}\norm u_k\le k^{k/2}\norm u_k.
\]
Conversely, $\cA^{k/2}$ is a differential operator with polynomial coefficients; moving derivatives to the right with $\partial_jx_j=x_j\partial_j+1$ writes it as a finite sum of terms $c\,x^\delta\partial^\gamma$ with $|\delta|+|\gamma|\le k$.

(b) Let $|\gamma|\le4$ and $w\defeq(1+|x|^2)^2D^\gamma u_c$ for a component $u_c$. The Sobolev embedding $H^2(\R^3)\subset L^\infty$ gives $\norm w_\infty\le C\sum_{|\beta|\le2}\norm{\partial^\beta w}_{L^2}$, and each $\partial^\beta w$ is a finite sum of terms $x^\delta\partial^{\gamma'}u_c$ with $|\delta|\le4$, $|\gamma'|\le6$. Apply (a) with $k=10$.

(c) For $u\in\cS$ and $m\in\N$, $\cA^mu\in\cS$ and $\ip{h_{\alpha,c}}{\cA^mu}=\lambda_\alpha^m\ip{h_{\alpha,c}}u$, so $\sum\lambda_\alpha^{2m}\ip{h_{\alpha,c}}u^2=\norm{\cA^mu}^2<\infty$. Let $P_Nu$ be the sum over $|\alpha|\le N$. Then $u-P_Nu\in\cS$ and $\norm{u-P_Nu}_k^2=\sum_{|\alpha|>N}\lambda_\alpha^k\ip{h_{\alpha,c}}u^2\to0$ for every $k$; by (a), $x^\delta\partial^\gamma(u-P_Nu)\to0$ in $L^2$ for all $\delta,\gamma$, and by the Sobolev embedding as in (b), $\sup|x^\delta\partial^\gamma(u-P_Nu)|\to0$: this is convergence in $\cS$. $C^\infty_c$ is dense in $\cS$, whose topology is finer than that of $\norm\cdot_k$ by the converse in (a), and $\cS$ is dense in $\cH^k$. The family $(\lambda_\alpha^{-k/2}h_{\alpha,c})$ is orthonormal for $\norm\cdot_k$, and its span is dense in $\cS$, hence in $\cH^k$.

(d) By the converse in (a),
\[
\norm u_k\le C_k\sum\norm{x^\delta\partial^\gamma u}_{L^2}\le C_k\sum n^{|\delta|}\norm{\partial^\gamma u}_{L^2}\le C_{k,n}\norm u_{H^k}
\]
if $\supp u\subset B_n$. For $\chi u$, Leibniz writes $x^\delta\partial^\gamma(\chi u)$ as a sum of terms $(x^\delta\partial^{\gamma-\gamma'}\chi)\,\partial^{\gamma'}u$ with bounded first factors.

(e) If $b=0$, $\gamma\cdot$ is unitary on $L^2$ and commutes with $\cA$ ($\Delta$ and $|x|^2$ are $O(3)$-invariant and $\pi(O)$ acts on the fibre), so $\norm{O\cdot u}_k=\norm u_k$. For a translation, $x^\delta\partial^\gamma(\tau_bu)=\tau_b\big((y+b)^\delta\partial^\gamma u\big)$, and expanding $(y+b)^\delta$ gives
\[
\norm{x^\delta\partial^\gamma\tau_bu}_{L^2}\le2^{|\delta|}(1+|b|)^{|\delta|}\max_{\delta'\le\delta}\norm{y^{\delta'}\partial^\gamma u}_{L^2};
\]
the two halves of (a) give the bound. The last statements follow by density and duality.
\end{proof}

\subsection{Borel structure and the Euclidean action}\label{ms:borelsec}

\begin{lemma}[Borel structure of the field spaces]\label{ms:borel}
Let $r\ge2$ be even, and $X$, $\cH^{-r}$, $\Sigma$, $\Sigma^H$, $\iota$ as in Definition~\ref{ms:fielddef}.
\begin{enumerate}
\item[(a)] $X$ is a separable Fr\'echet space, hence Polish; $\cH^{-r}$ is a separable Hilbert space.
\item[(b)] For every open $U\subset\R^3$ there is a countable set $D_U\subset C^\infty_c(U;\cV)$ such that every $u\in C^\infty_c(U;\cV)$ is the limit in $H^r$ of a sequence $(u_k)$ in $D_U$ with supports in a fixed compact subset of $U$. For any such sequence, $\Phi(u_k)\to\Phi(u)$ for every $\Phi\in X$ and $\Psi(u_k)\to\Psi(u)$ for every $\Psi\in\cH^{-r}$. Hence $\Sigma(U)$, respectively $\Sigma^H(U)$, is generated by the countably many bounded continuous functions $\arctan\Phi(u)$, respectively $\arctan\Psi(u)$, $u\in D_U$.
\item[(c)] There are countable sets $D_X,D_H\subset C^\infty_c(\R^3;\cV)$ such that the Borel $\sigma$-algebra of $X$ is generated by $\Phi\mapsto\Phi(u)$, $u\in D_X$, and that of $\cH^{-r}$ by $\Psi\mapsto\Psi(u)$, $u\in D_H$. In particular both Borel $\sigma$-algebras equal $\Sigma(\R^3)=\bigvee_K\Sigma(K)$, respectively $\Sigma^H(\R^3)=\bigvee_K\Sigma^H(K)$.
\item[(d)] $\iota$ is continuous, linear and injective; $\iota\cH^{-r}$ is a Borel subset of $X$, and $\iota$ is a Borel isomorphism of $\cH^{-r}$ onto it. Moreover $\iota\gamma=\gamma\iota$ for $\gamma\in\Euc$, and $\Sigma^H(U)=\iota^{-1}\Sigma(U)$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) $X$ is a locally convex space with the countable family of seminorms $p_n$, which separates points by \eqref{ms:star}, so it is metrizable, with the metric $d(\Phi,\Phi')\defeq\sum_n2^{-n}\min(1,p_n(\Phi-\Phi'))$. It is complete: if $(\Phi_k)$ is Cauchy, then $\chi_n\Phi_k\to\Xi_n$ in $H^{-r}$ for every $n$; put $\Phi(u)\defeq\Xi_n(u)$ for $\supp u\subset B_n$, which is consistent because $(\chi_n\Phi_k)(u)=\Phi_k(u)$ for such $u$; then $|\Phi(\chi u)|\le\norm{\Xi_n}_{H^{-r}}\norm{\chi u}_{H^r}$ for $\supp\chi\subset B_n$, so $\Phi\in X$, and $p_n(\Phi_k-\Phi)=\sup_{\norm g_{H^r}\le1}|(\chi_{n+1}\Phi_k-\Xi_{n+1})(\chi_ng)|\to0$. It is separable, because $\Phi\mapsto(\chi_n\Phi)_n$ is a homeomorphism of $X$ onto a subspace of the separable metrizable space $\prod_nH^{-r}(\R^3;\cV)$. $\cH^{-r}$ has the countable orthonormal basis dual to that of Lemma~\ref{ms:hermite}(c).

(b) Choose compact sets $K_1\subset K_2\subset\dots$ with $K_m\subset\operatorname{int}K_{m+1}$ and $\bigcup_m\operatorname{int}K_m=U$, and a countable subset $D_m$ of $C^\infty_c(\operatorname{int}K_m;\cV)$ dense in it for $\norm\cdot_{H^r}$ (a subset of a separable metric space is separable). Put $D_U\defeq\bigcup_mD_m$. A $u\in C^\infty_c(U;\cV)$ has support in some $\operatorname{int}K_m$, and is the $H^r$-limit of a sequence in $D_m$, with supports in $K_m$. If $K_m\subset B_n$, then $|\Phi(u_k)-\Phi(u)|\le p_n(\Phi)\norm{u_k-u}_{H^r}$ by \eqref{ms:star}, and $|\Psi(u_k)-\Psi(u)|\le\norm\Psi_{-r}C_{r,n}\norm{u_k-u}_{H^r}$ by Lemma~\ref{ms:hermite}(d). So every generator $\Phi\mapsto\Phi(u)$ of $\Sigma(U)$ is a pointwise limit of the functions $\Phi\mapsto\Phi(u_k)$, $u_k\in D_U$, and $\Sigma(U)=\sigma(\Phi(u):u\in D_U)=\sigma(\arctan\Phi(u):u\in D_U)$; the same holds for $\Sigma^H(U)$. The functions are continuous by \eqref{ms:star}, respectively by $|\Psi(u)|\le\norm\Psi_{-r}\norm u_r$.

(c) Let $D\subset C^\infty_c(\R^3;\cV)$ be countable and dense in the unit ball of $H^r(\R^3;\cV)$ ($C^\infty_c$ is dense in $H^r$, which is separable). For $\Xi\in X$, $\chi_n\Xi\in H^{-r}$ is continuous on $H^r$, so $p_n(\Xi)=\sup_{g\in D}|\Xi(\chi_ng)|$. Put $D_X\defeq\{\chi_ng:n\in\N,\ g\in D\}$. For fixed $\Phi_0$, $\Phi\mapsto d(\Phi,\Phi_0)$ is then measurable for $\sigma(\Phi(u):u\in D_X)$. As $X$ is separable, every open set is a countable union of $d$-balls with centres in a countable dense set, so the Borel $\sigma$-algebra is contained in $\sigma(\Phi(u):u\in D_X)\subset\Sigma(\R^3)$, and the evaluations are continuous. For $\cH^{-r}$, let $D_H\subset C^\infty_c$ be countable and dense in the unit ball of $\cH^r$ (Lemma~\ref{ms:hermite}(c)); then $\norm{\Psi-\Psi_0}_{-r}=\sup_{g\in D_H}|\Psi(g)-\Psi_0(g)|$, and the same argument applies. Finally, every $u\in C^\infty_c$ is supported in a bounded open $K$, so $\Sigma(\R^3)=\bigvee_K\Sigma(K)$.

(d) For $\chi\in C^\infty_c(\R^3)$ and $u\in C^\infty_c$, $|\Psi(\chi u)|\le\norm\Psi_{-r}\norm{\chi u}_r\le C_{r,\chi}\norm\Psi_{-r}\norm u_{H^r}$ by Lemma~\ref{ms:hermite}(d); so $\iota\Psi\in X$ and $p_n(\iota\Psi)\le C_{r,\chi_n}\norm\Psi_{-r}$: $\iota$ is continuous and linear. It is injective because $C^\infty_c$ is dense in $\cH^r$. Let $D_*\defeq\bigcup_nD_{B_n}\setminus\{0\}$, with $D_{B_n}$ from (b), and $N(\Phi)\defeq\sup_{g\in D_*}|\Phi(g)|/\norm g_r$, a measurable function on $X$. If $\Phi=\iota\Psi$, then $N(\Phi)\le\norm\Psi_{-r}$. If $N(\Phi)<\infty$ and $u\in C^\infty_c(B_n;\cV)$, choose $u_k\in D_{B_n}$ as in (b); then $\Phi(u_k)\to\Phi(u)$ and $\norm{u_k}_r\to\norm u_r$ by Lemma~\ref{ms:hermite}(d), so $|\Phi(u)|\le N(\Phi)\norm u_r$; as $C^\infty_c$ is dense in $\cH^r$, $\Phi$ extends to some $\Psi\in\cH^{-r}$ with $\iota\Psi=\Phi$. So $\iota\cH^{-r}=\{N<\infty\}$ is Borel. The class of Borel sets $B\subset\cH^{-r}$ with $\iota B$ Borel is a $\sigma$-algebra ($\iota$ is injective and $\iota\cH^{-r}$ is Borel), and it contains the generators $\{\Psi:\Psi(g)\in A\}$, $g\in D_H$, of (c), whose images are $\{\Phi\in\iota\cH^{-r}:\Phi(g)\in A\}$. So $\iota^{-1}$ is Borel on $\iota\cH^{-r}$. For $u\in C^\infty_c$, $(\iota\gamma\Psi)(u)=\Psi(\gamma^{-1}\cdot u)=(\gamma\iota\Psi)(u)$, because $\gamma^{-1}\cdot u\in C^\infty_c$. Since $(\iota\Psi)(u)=\Psi(u)$, the preimages under $\iota$ of the generators of $\Sigma(U)$ are the generators of $\Sigma^H(U)$, so $\Sigma^H(U)=\iota^{-1}\Sigma(U)$.
\end{proof}

\begin{lemma}[the Euclidean action and (H2)]\label{ms:struct}
Let $r\ge2$ be even.
\begin{enumerate}
\item[(a)] For every $\gamma\in\Euc$, $\Phi\mapsto\gamma\Phi$ is a linear homeomorphism of $X$, and $\Euc\times X\to X$, $(\gamma,\Phi)\mapsto\gamma\Phi$, is jointly continuous. The same holds on $\cH^{-r}$. In particular $X$ and $\cH^{-r}$, with $\tau$ and the involutions $\theta_j$, have the structural properties required in \textup{(H1)}, and every $\theta_{j,c}$ is an involutive homeomorphism.
\item[(b)] $(X,\Sigma)$ and $(\cH^{-r},\Sigma^H)$ satisfy \textup{(H2)}. Moreover, for every $\gamma\in\Euc$ and open $U$, $G\mapsto G\circ\gamma^{-1}$ maps $\Sigma(U)$-measurable functions to $\Sigma(\gamma U)$-measurable ones, and likewise for $\Sigma^H$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) \emph{Equicontinuity on $X$.} Let $\gamma x=Ox+b$ with $|b|\le\beta$, and let $c_n$ be the norm of multiplication by $\chi_n$ on $H^r$. For $g\in C^\infty_c$ with $\norm g_{H^r}\le1$, $\gamma^{-1}\cdot(\chi_ng)$ is supported in $\gamma^{-1}B_{n+1}\subset B_m$, $m\defeq n+1+\lceil\beta\rceil$, and has $H^r$-norm $\norm{\chi_ng}_{H^r}\le c_n$; by \eqref{ms:star}, $p_n(\gamma\Phi)\le c_np_m(\Phi)$. The same estimate with any $\chi\in C^\infty_c$ in place of $\chi_n$ shows $\gamma\Phi\in X$. So each $\gamma$ is a continuous linear map of $X$ with continuous inverse $\gamma^{-1}$, and $\{\gamma:|b|\le\beta\}$ is equicontinuous.

\emph{Strong continuity on $X$.} Fix $\Phi$, $n$ and $\epsilon>0$, and let $|b|\le1$. For $g$ as above, $w\defeq\gamma^{-1}\cdot(\chi_ng)-\chi_ng$ is supported in $B_{n+2}$ with $\norm w_{H^r}\le2c_n$, and $(\gamma\Phi-\Phi)(\chi_ng)=\Phi(w)=\Xi(w)$, $\Xi\defeq\chi_{n+2}\Phi\in H^{-r}$. Choose $\psi\in C^\infty_c(\R^3;\cV)$ with $\norm{\Xi-\ip\psi\cdot}_{H^{-r}}\le\epsilon$ ($C^\infty_c$ is dense in $H^{-r}$, through the $L^2$ pairing). Since $\gamma\cdot$ preserves the $L^2$ pairing, $\ip\psi w=\ip{\gamma\cdot\psi-\psi}{\chi_ng}$, so $|\Phi(w)|\le2c_n\epsilon+c_n\norm{\gamma\cdot\psi-\psi}_{L^2}$ uniformly in $g$, and $\limsup_{\gamma\to\mathrm{id}}p_n(\gamma\Phi-\Phi)\le2c_n\epsilon$.
\emph{Joint continuity.} $\gamma'\Phi'-\gamma\Phi=\gamma'(\Phi'-\Phi)+\gamma\big((\gamma^{-1}\gamma')\Phi-\Phi\big)$; the first term is small by equicontinuity, the second by strong continuity and the continuity of $\gamma$.

\emph{On $\cH^{-r}$.} By Lemma~\ref{ms:hermite}(e) each $\gamma$ is bounded with norm $\le C_r(1+|b|)^r$, with inverse $\gamma^{-1}$. The functionals $\ip\psi\cdot$, $\psi\in C^\infty_c$, are dense in $\cH^{-r}$: for $\Psi\in\cH^{-r}$ the pairing with $\psi_N\defeq\sum_{|\alpha|\le N}\Psi(h_{\alpha,c})h_{\alpha,c}\in\cS$ differs from $\Psi$ by $\big(\sum_{|\alpha|>N}\lambda_\alpha^{-r}\Psi(h_{\alpha,c})^2\big)^{1/2}$, and $\norm{\ip\phi\cdot}_{-r}\le\norm\phi_{L^2}$ because $\norm u_{L^2}\le\norm u_r$. On these, $\gamma\ip\psi\cdot=\ip{\gamma\cdot\psi}\cdot$ and $\norm{\gamma\ip\psi\cdot-\ip\psi\cdot}_{-r}\le\norm{\gamma\cdot\psi-\psi}_{L^2}\to0$. Local boundedness and an $\epsilon/3$ argument give strong continuity, and joint continuity follows as on $X$.

(b) Monotonicity is clear. $\Sigma(U)=\bigvee_{K\Subset U}\Sigma(K)$ because every $u\in C^\infty_c(U)$ is supported in some bounded open $K\Subset U$. $\bigvee_K\Sigma(K)$ is the Borel $\sigma$-algebra by Lemma~\ref{ms:borel}(c), and the countable generation by bounded continuous functions is Lemma~\ref{ms:borel}(b). For a generator $G(\Phi)=\Phi(u)$ of $\Sigma(U)$, $(G\circ\gamma^{-1})(\Phi)=\Phi(\gamma\cdot u)$ with $\gamma\cdot u\in C^\infty_c(\gamma U)$; so $\gamma^{-1}$ is measurable from $(X,\Sigma(\gamma U))$ to $(X,\Sigma(U))$, and $G\circ\gamma^{-1}$ is $\Sigma(\gamma U)$-measurable for every $\Sigma(U)$-measurable $G$. With $\gamma=\tau_a$ this gives $U(a)G=G\circ\tau_a^{-1}$, $\Sigma(U+a)$-measurable; with $\gamma=\theta_j=\theta_j^{-1}$ it gives $\Theta_jG$, $\Sigma(\rho_jU)$-measurable. The same argument applies to $\Sigma^H$.
\end{proof}

\subsection{Density, the Hermite carrier and uniqueness}\label{ms:uniqsec}

\begin{lemma}[density of generating algebras]\label{ms:density}
Let $m$ be a probability measure on a measurable space $(\Omega,\mathcal F)$, and $\mathfrak A$ an algebra of bounded complex $\mathcal F$-measurable functions, closed under complex conjugation and containing the constants. Then $\mathfrak A$ is dense in $L^2(\Omega,\sigma(\mathfrak A),m)$. The same holds for a countable algebra $\mathfrak A_{\mathbb Q}$ over $\mathbb Q[\mathrm i]$, closed under conjugation and containing $1$, whose complex span is such an algebra.
\end{lemma}

\begin{proof}
Let $\mathcal L$ be the closure of $\mathfrak A$ in $L^2(\Omega,\sigma(\mathfrak A),m)$, a closed complex subspace, and $\mathfrak A_\R$ the set of real elements of $\mathfrak A$. Since $\mathfrak A$ is closed under conjugation, $\operatorname{Re} a,\operatorname{Im} a\in\mathfrak A_\R$ for $a\in\mathfrak A$; $\mathfrak A_\R$ is a real algebra containing the constants, and $\sigma(\mathfrak A_\R)=\sigma(\mathfrak A)$.
(i) Let $a_1,\dots,a_N\in\mathfrak A_\R$, $|a_i|\le M$, and $\varphi_1,\dots,\varphi_N$ continuous on $[-M,M]$. By Weierstrass' theorem there are polynomials $p_{i,k}\to\varphi_i$ uniformly on $[-M,M]$; then $\prod_ip_{i,k}(a_i)\in\mathfrak A$ converges uniformly to $\prod_i\varphi_i(a_i)$, which therefore lies in $\mathcal L$.
(ii) For $t_i\in\R$ the continuous functions $\varphi_{i,k}(s)\defeq\min(1,k(s-t_i)^+)$ increase to $\mathbf 1_{\{s>t_i\}}$; by dominated convergence $\prod_i\varphi_{i,k}(a_i)\to\prod_i\mathbf 1_{\{a_i>t_i\}}$ in $L^2$, so $\mathbf 1_P\in\mathcal L$ for every $P$ in $\mathcal P\defeq\{\bigcap_{i\le N}\{a_i>t_i\}:N\ge0,\ a_i\in\mathfrak A_\R,\ t_i\in\R\}$.

(iii) $\mathcal P$ is a $\pi$-system generating $\sigma(\mathfrak A_\R)$. The class of $B\in\sigma(\mathfrak A)$ with $\mathbf 1_B\in\mathcal L$ contains $\Omega$, is closed under complements ($\mathbf 1_{B^c}=1-\mathbf 1_B$), under proper differences ($\mathbf 1_{B\setminus A}=\mathbf 1_B-\mathbf 1_A$ for $A\subset B$, as $\mathcal L$ is linear) and under increasing countable unions (dominated convergence; $\mathcal L$ is closed). By Dynkin's $\pi$--$\lambda$ theorem it is all of $\sigma(\mathfrak A)$. So $\mathcal L$ contains the simple $\sigma(\mathfrak A)$-measurable functions, which are dense.
For $\mathfrak A_{\mathbb Q}$: if $a=\sum_kc_ka_k$ with $c_k\in\C$, $a_k\in\mathfrak A_{\mathbb Q}$, and $q_k\in\mathbb Q[\mathrm i]$, then $\norm{a-\sum_kq_ka_k}_\infty\le\sum_k|c_k-q_k|\norm{a_k}_\infty$; so $\mathfrak A_{\mathbb Q}$ is dense in its complex span, to which the first part applies.
\end{proof}

\begin{lemma}[the Hermite carrier]\label{ms:carrier}
Let $r\ge14$ be even, and let $\mu'$ be a probability measure on $\cS'(\R^3;\cV)$ with its cylinder $\sigma$-algebra such that $\int\Phi(u)^2\,d\mu'\le C_2\norm u_{10}^2$ for every real $u\in\cS(\R^3;\cV)$. Put $M(\Phi)\defeq\sum_{\alpha,c}\lambda_\alpha^{-r}\Phi(h_{\alpha,c})^2$. Then $\int M\,d\mu'\le C_2\dim\cV\sum_\alpha\lambda_\alpha^{10-r}<\infty$. On $\{M<\infty\}$, $\Phi|_\cS$ extends uniquely to an element $\Pi(\Phi)$ of $\cH^{-r}$, with $\norm{\Pi(\Phi)}_{-r}^2=M(\Phi)$. The map $\Pi$ (put $\Pi\defeq0$ on $\{M=\infty\}$) is measurable into the Borel $\sigma$-algebra of $\cH^{-r}$, and $\mu^H\defeq\Pi_*\mu'$ is a Borel probability measure on $\cH^{-r}$ under which, for all real $u_1,\dots,u_N\in\cS$, the vector $(\Psi(u_l))_{l\le N}$ has the law of $(\Phi(u_l))_{l\le N}$ under $\mu'$. In particular $\mu^H$ and $\mu'$ have the same characteristic functional and the same mixed moments on $\cS$.
\end{lemma}

\begin{proof}
$\norm{h_{\alpha,c}}_{10}^2=\lambda_\alpha^{10}$, so $\int M\,d\mu'\le C_2\sum_{\alpha,c}\lambda_\alpha^{10-r}$, and $\sum_\alpha\lambda_\alpha^{10-r}=\sum_{n\ge0}\frac{(n+1)(n+2)}2(2n+4)^{10-r}<\infty$ because $r-10>3$. Hence $\mu'(M<\infty)=1$. For $\Phi\in\cS'$ and $u\in\cS$ the Hermite expansion of $u$ converges in $\cS$ (Lemma~\ref{ms:hermite}(c)), so $\Phi(u)=\sum_{\alpha,c}\ip{h_{\alpha,c}}u\,\Phi(h_{\alpha,c})$, and by the Cauchy--Schwarz inequality $|\Phi(u)|\le M(\Phi)^{1/2}\norm u_r$. As $\cS$ is dense in $\cH^r$, $\Phi|_\cS$ extends uniquely to $\Pi(\Phi)\in\cH^{-r}$ when $M(\Phi)<\infty$, and $\norm{\Pi(\Phi)}^2_{-r}=\sum\lambda_\alpha^{-r}\Pi(\Phi)(h_{\alpha,c})^2=M(\Phi)$. The Borel $\sigma$-algebra of the separable Hilbert space $\cH^{-r}$ is generated by the coordinates $\Psi\mapsto\Psi(h_{\alpha,c})$, since $\norm{\Psi-\Psi_0}_{-r}$ is a measurable function of them; and $\Pi(\Phi)(h_{\alpha,c})=\mathbf 1_{\{M<\infty\}}\Phi(h_{\alpha,c})$ is measurable. Finally $\Pi(\Phi)(u)=\Phi(u)$ for $u\in\cS$ on the set $\{M<\infty\}$ of full measure.
\end{proof}

\begin{proposition}[uniqueness by the characteristic functional and by the moments]\label{ms:unique}
Let $r\ge2$ be even, let $Y$ be $X$ or $\cH^{-r}$, write $\Phi$ for its elements, and let $\mu_1,\mu_2$ be Borel probability measures on $Y$.
\begin{enumerate}
\item[(a)] If $\int e^{\mathrm i\Phi(u)}\,d\mu_1=\int e^{\mathrm i\Phi(u)}\,d\mu_2$ for every real $u\in C^\infty_c(\R^3;\cV)$, then $\mu_1=\mu_2$.
\item[(b)] Suppose that for all real $u_1,\dots,u_N\in C^\infty_c(\R^3;\cV)$ the mixed moments $\int\prod_{l\le N}\Phi(u_l)\,d\mu_i$ exist and are equal for $i=1,2$, and that for every real $u\in C^\infty_c$ there are constants $c,K\ge0$, depending on $u$, with $|\int\Phi(u)^n\,d\mu_1|\le c\,(n+1)K^nn!$ for all $n\ge0$. Then $\mu_1=\mu_2$.
\end{enumerate}
\end{proposition}

\begin{proof}
(a) Let $D_Y=\{u_1,u_2,\dots\}$ be the countable set of Lemma~\ref{ms:borel}(c). The sets $\{\Phi:(\Phi(u_1),\dots,\Phi(u_N))\in B\}$, $N\in\N$, $B\subset\R^N$ Borel, form a $\pi$-system generating the Borel $\sigma$-algebra of $Y$. Under $\mu_i$ the characteristic function of $(\Phi(u_1),\dots,\Phi(u_N))$ is $t\mapsto\int e^{\mathrm i\Phi(\sum_kt_ku_k)}\,d\mu_i$, by linearity of $\Phi$; so the two laws on $\R^N$ agree, and $\mu_1=\mu_2$ by Dynkin's $\pi$--$\lambda$ theorem.

(b) Fix $u$ and let $Z\defeq\Phi(u)$, $m_n\defeq\int Z^n\,d\mu_1=\int Z^n\,d\mu_2$; we may take $K>0$. Even moments are absolute moments, and the Cauchy--Schwarz inequality gives $\E|Z|^{2n+1}\le(m_{2n}m_{2n+2})^{1/2}\le c\,K^{2n+1}(2n+1)!\,\big((2n+3)(2n+2)\big)^{1/2}$, using $(2n)!\,(2n+2)!=((2n+1)!)^2\frac{2n+2}{2n+1}$. Hence $\E|Z|^n\le c\,(n+2)K^nn!$ for all $n$, under both measures. Let $\varphi_i(s)\defeq\int e^{\mathrm isZ}\,d\mu_i$. Then $\varphi_i^{(k)}(s)=\int(\mathrm iZ)^ke^{\mathrm isZ}\,d\mu_i$, and from $|e^{\mathrm iy}-\sum_{k<n}(\mathrm iy)^k/k!|\le|y|^n/n!$,
\[
\Big|\varphi_i(s+v)-\sum_{k<n}\varphi_i^{(k)}(s)\frac{v^k}{k!}\Big|\le\frac{|v|^n\,\E|Z|^n}{n!}\le c\,(n+2)(K|v|)^n\to0\qquad(|v|<1/K).
\]
So each $\varphi_i$ is the sum of its Taylor series on $(s-1/K,s+1/K)$, for every $s$. The set of $s$ near which $\varphi_1=\varphi_2$ is open; it contains $0$, because $\varphi_i^{(k)}(0)=\mathrm i^km_k$; and it is closed, because at a point $s'$ of it within $1/(2K)$ of a limit point $s$ all derivatives agree and the two Taylor series at $s'$ agree on a neighbourhood of $s$. So $\varphi_1=\varphi_2$ on $\R$; at $s=1$ this is $\int e^{\mathrm i\Phi(u)}\,d\mu_1=\int e^{\mathrm i\Phi(u)}\,d\mu_2$, and (a) applies.
\end{proof}

\subsection{(H1)--(H3) from the characteristic functional}\label{ms:dischargesec}

\begin{proposition}[(H1)--(H3) on the field space]\label{ms:discharge}
Let $r\ge2$ be even, $\mu$ a Borel probability measure on $X=H^{-r}_{\mathrm{loc}}(\R^3;\cV)$, and $\cZ(u)\defeq\int e^{\mathrm i\Phi(u)}\,d\mu(\Phi)$ for real $u\in C^\infty_c(\R^3;\cV)$.
\begin{enumerate}
\item[(a)] If $\gamma\in\Euc$ and $\cZ(\gamma\cdot u)=\cZ(u)$ for all real $u$, then $\gamma_*\mu=\mu$.
\item[(b)] For real $u_k,u_l\in C^\infty_c$ and $E_k\defeq e^{\mathrm i\Phi(u_k)}$, $\overline{E_k\circ\theta_{j,c}}\,E_l=e^{\mathrm i\Phi(u_l-\rho_{j,c}\cdot u_k)}$, so $\E_\mu[\overline{E_k\circ\theta_{j,c}}\,E_l]=\cZ(u_l-\rho_{j,c}\cdot u_k)$.
\item[(c)] Let $j\in\{1,2,3\}$ and $c\in\R$. If $\mu$ is $\theta_{j,c}$-invariant and $\sum_{k,l\le N}\bar a_ka_l\,\cZ(u_l-\rho_{j,c}\cdot u_k)\ge0$ for all $N$, all real $u_1,\dots,u_N\in C^\infty_c(H^+_{j,c};\cV)$ and all $a\in\C^N$, then $\E_\mu[\overline{F\circ\theta_{j,c}}\,F]\ge0$ for every $F\in L^2(\Sigma(H^+_{j,c}),\mu)$.
\item[(d)] Consequently, if $\cZ$ is invariant under the translations and the coordinate reflections $\rho_j$, and the condition of (c) holds for every $j$ and every $c$ (or for every $j$ and $c=0$), then $(X,\tau,\theta,\mu,\Sigma)$ satisfies \textup{(H1)--(H3)}. If moreover $\cZ$ is invariant under $\tau_bg\tau_{-b}$ for $g\in O_h$ and $b\in\R^3$, then $\mu$ is invariant under $O_h$ about every point.
\end{enumerate}
\end{proposition}

\begin{proof}
(a) The characteristic functional of $\gamma_*\mu$ is $u\mapsto\int e^{\mathrm i(\gamma\Phi)(u)}\,d\mu=\cZ(\gamma^{-1}\cdot u)=\cZ(u)$; apply Proposition~\ref{ms:unique}(a). (b) $\theta_{j,c}$ is the action of the involution $\rho_{j,c}$, so $E_k(\theta_{j,c}\Phi)=e^{\mathrm i\Phi(\rho_{j,c}\cdot u_k)}$, and $\Phi$ is linear. (c) For $F=\sum_la_lE_l$ with $u_l\in C^\infty_c(H^+_{j,c};\cV)$, (b) gives $\E_\mu[\overline{F\circ\theta_{j,c}}\,F]=\sum_{k,l}\bar a_ka_l\cZ(u_l-\rho_{j,c}\cdot u_k)\ge0$. These $F$ form an algebra $\mathfrak A$ of bounded continuous functions (products of exponentials are exponentials of sums), closed under conjugation ($\overline{E_l}=e^{\mathrm i\Phi(-u_l)}$) and containing $1$; it generates $\Sigma(H^+_{j,c})$, since $\Phi(u)=\lim_{t\to0}t^{-1}\operatorname{Im} e^{\mathrm it\Phi(u)}$. By Lemma~\ref{ms:density}, $\mathfrak A$ is dense in $L^2(\Sigma(H^+_{j,c}),\mu)$. If $F_n\to F$ in $L^2$ with $F_n\in\mathfrak A$, then $F_n\circ\theta_{j,c}\to F\circ\theta_{j,c}$ in $L^2$, as $\theta_{j,c}$ preserves $\mu$, so $\overline{F_n\circ\theta_{j,c}}F_n\to\overline{F\circ\theta_{j,c}}F$ in $L^1(\mu)$, and the inequality passes to the limit. (d) The structural parts of (H1) and all of (H2) are Lemma~\ref{ms:struct}; the invariances follow from (a), every $\theta_{j,c}=\tau_{ce_j}\theta_j\tau_{-ce_j}$ then preserving $\mu$; (H3) is (c), and the case $c=0$ suffices by the sentence after \eqref{ms:eqcommute}.
\end{proof}

\begin{lemma}[transfer to the Hermite--Sobolev space]\label{ms:transfer}
Let $r\ge2$ be even, $\mu^H$ a Borel probability measure on $\cH^{-r}$, and $\mu\defeq\iota_*\mu^H$.
\begin{enumerate}
\item[(a)] $\mu(\iota\cH^{-r})=1$; $\mu^H$ is the only Borel probability measure on $\cH^{-r}$ whose image under $\iota$ is $\mu$; and $\mu$, $\mu^H$ have the same characteristic functional and the same mixed moments on $C^\infty_c$.
\item[(b)] For $\gamma\in\Euc$: $\gamma_*\mu=\mu$ if and only if $\gamma_*\mu^H=\mu^H$.
\item[(c)] $(X,\tau,\theta,\mu,\Sigma)$ satisfies \textup{(H1)--(H3)} if and only if $(\cH^{-r},\tau,\theta,\mu^H,\Sigma^H)$ does.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) By Lemma~\ref{ms:borel}(d), $\iota B$ is Borel for every Borel $B$, so any $\nu$ with $\iota_*\nu=\mu$ satisfies $\nu(B)=\nu(\iota^{-1}\iota B)=\mu(\iota B)$; with $B=\cH^{-r}$, $\mu(\iota\cH^{-r})=1$. The functionals agree because $(\iota\Psi)(u)=\Psi(u)$ for $u\in C^\infty_c$. (b) $\iota\gamma=\gamma\iota$ gives $\gamma_*\mu=\iota_*(\gamma_*\mu^H)$; if $\gamma_*\mu^H=\mu^H$ this is $\mu$, and if $\gamma_*\mu=\mu$ then $\gamma_*\mu^H=\mu^H$ by the uniqueness in (a). (c) The structural parts of (H1) and (H2) hold on both spaces (Lemma~\ref{ms:struct}), and the invariances correspond by (b). For (H3): $\Sigma^H(U)=\iota^{-1}\Sigma(U)$ (Lemma~\ref{ms:borel}(d)), so the $\Sigma^H(H^+_{j,c})$-measurable functions are exactly the $F'\circ\iota$ with $F'$ $\Sigma(H^+_{j,c})$-measurable (factorisation through $\iota$). For such $F=F'\circ\iota$, $\int|F|^2d\mu^H=\int|F'|^2d\mu$, $F\circ\theta_{j,c}=F'\circ\theta_{j,c}\circ\iota$, and $\E_{\mu^H}[\overline{F\circ\theta_{j,c}}F]=\E_\mu[\overline{F'\circ\theta_{j,c}}F']$.
\end{proof}

\begin{remark}[where the hypotheses are met]\label{ms:met}
For the curvature field, $\cV=\Lambda^2$, the hypotheses of Lemma~\ref{ms:hermite}(b), Lemma~\ref{ms:carrier}, Proposition~\ref{ms:unique}, Proposition~\ref{ms:discharge}(a) and Lemma~\ref{ms:transfer} are verified in the proof of Theorem~\ref{thm:osrest:measure}, and those of Proposition~\ref{ms:discharge}(c),(d) are verified in the proof of Theorem~\ref{thm:osrest:OS4weak}.

Lemma~\ref{ms:transfer} carries (H1)--(H3) and the invariances between $X$ and $\cH^{-r}$. Every statement proved for a measure satisfying (H1)--(H3) therefore applies to $\mu$ on $X$ and to $\mu^H$ on $\cH^{-r}$ alike, and the conclusions correspond under $\iota$, which is a Borel isomorphism of $\cH^{-r}$ onto a Borel set of full $\mu$-measure that intertwines the actions of $\Euc$ and the local $\sigma$-algebras.
\end{remark}


\subsection{Ergodic decomposition and clustering}\label{ms:quoted}

This subsection proves the results on the ergodic decomposition, on the reconstruction in a coordinate direction and on clustering that are used in Section~\ref{sec:osrest}: in the reconstruction before Remark~\ref{rem:osrest:recon} and in Theorem~\ref{thm:osrest:OS4weak}. The hypotheses (H1)--(H3) are those of Subsection~\ref{ms:H}, and $U(a)$, $\Theta_j$ are the operators of \eqref{ms:eqcov}.

\emph{Notation.}
A Borel probability measure $m$ on $X$ invariant under every $\tau_a$ is \emph{ergodic} (for the action of $\R^3$) 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 $L^2(m)$ of classes with an $\mathcal F$-measurable representative.
For $J\subset\{1,2,3\}$ put $V_J\defeq\operatorname{span}\{e_j:j\in J\}$, with $V_\emptyset=\{0\}$.
For $a\in\R^3$ and $j$ we write $a=a_je_j+a_{\perp}$ with $a_\perp\perp e_j$. For a measure $m$ and an $m$-integrable function $G$ we write $m(G)\defeq\int G\,dm$.

\begin{remark}[countable determining algebras]\label{ms:x:detalg}
Let $d\le1$ be a complete metric on a Polish space $X$, $(x_n)$ a dense sequence, and $\cD$ the countable $\mathbb Q[\mathrm i]$-algebra generated by $1$ and the functions $x\mapsto d(x,x_n)$. Its elements are bounded and continuous, and $\sigma(\cD)=\cB$, because the balls $\{d(\cdot,x_n)<q\}$, $q\in\mathbb Q$, generate the open sets of the separable space $X$. By Lemma~\ref{ms:density}, $\cD$ is dense in $L^2(m)$ for every Borel probability measure $m$ on $X$; consequently two Borel probability measures $m,m'$ with $m(G)=m'(G)$ for all $G\in\cD$ are equal (for bounded Borel $F$ and $G\in\cD$, $\abs{m(F)-m'(F)}\le2\norm{F-G}_{L^2((m+m')/2)}$). We call $\cD$ a \emph{countable determining algebra} of $X$.
\end{remark}

\subsubsection{Translations and the mean-ergodic projections}\label{ms:x:projsec}

Throughout Subsections~\ref{ms:x:projsec} and~\ref{ms:x:decsec}, $\mu$ is a measure satisfying (H1)--(H3).

\begin{lemma}\label{ms:x:strongcont}
$a\mapsto U(a)$ is strongly continuous on $L^2(\mu)$.
\end{lemma}
\begin{proof}
For a bounded continuous $G$, $G(\tau_{-a}x)\to G(x)$ for every $x$ as $a\to0$, by the continuity of the action, and dominated convergence gives $U(a)G\to G$ in $L^2(\mu)$. Bounded continuous functions are dense in $L^2(\mu)$ (the countable determining algebra of Remark~\ref{ms:x:detalg}), and $\norm{U(a)}=1$; an $\epsilon/3$ argument gives $U(a)G\to G$ for every $G\in L^2(\mu)$, and the group law gives continuity at every point.
\end{proof}

For $J\subset\{1,2,3\}$ let $P_J$ be the orthogonal projection of $L^2(\mu)$ onto $\{G:U(a)G=G\ \forall a\in V_J\}$, with $P_\emptyset=1$ and $P_j\defeq P_{\{j\}}$; let $\cI_J$ be the $\sigma$-algebra of $V_J$-invariant Borel sets completed by $\mu$-null sets, and $\cI\defeq\cI_{\{1,2,3\}}$. Put $\Lambda^J_T\defeq[-T,T]^3\cap V_J$ and $\Lambda_T\defeq[-T,T]^3$.

\begin{lemma}\label{ms:x:proj}
\textup{(i)} For $\Lambda^J_T\defeq[-T,T]^3\cap V_J$ and every $G\in L^2(\mu)$, $\abs{\Lambda^J_T}^{-1}\int_{\Lambda^J_T}U(a)G\,da\to P_JG$ in $L^2(\mu)$ as $T\to\infty$, with $\abs{\Lambda^J_T}$ the Lebesgue measure of $\Lambda^J_T$ in $V_J$. The same holds for the one-sided means $T^{-1}\int_0^TU(te_j)G\,dt$, $J=\{j\}$.

\textup{(ii)} $P_J$ commutes with every $U(a)$ and every $\Theta_j$; $P_JP_{J'}=P_{J\cup J'}$; and $P_J=\E[\,\cdot\mid\cI_J]$. In particular $P_{\{1,2,3\}}=\E[\,\cdot\mid\cI]$.
\end{lemma}
\begin{proof}
(i) For one coordinate $j$, $t\mapsto U(te_j)$ is strongly continuous by Lemma~\ref{ms:x:strongcont}, and Stone's theorem \cite{RS72} gives $U(te_j)=e^{\mathrm itA_j}$ with $A_j$ self-adjoint, and $(2T)^{-1}\int_{-T}^Te^{\mathrm itA_j}\,dt=\phi_T(A_j)$ with $\phi_T(\lambda)=\sin(T\lambda)/(T\lambda)$, $\phi_T(0)=1$. Since $\abs{\phi_T}\le1$ and $\phi_T\to\mathbf 1_{\{0\}}$ pointwise, the functional calculus of the spectral theorem \cite{RS72} and dominated convergence give strong convergence to the spectral projection of $\{0\}$, which is the projection onto $\operatorname{Ker}A_j=\{G:U(te_j)G=G\ \forall t\}$. For the one-sided means use $\phi_T(\lambda)=(e^{\mathrm iT\lambda}-1)/(\mathrm iT\lambda)$. The mean over $\Lambda^J_T$ is the product of the commuting one-dimensional means over the coordinates in $J$. Products of uniformly bounded, strongly convergent families converge strongly to the product of the limits, and a product of commuting orthogonal projections is the orthogonal projection onto the intersection of their ranges, which here is the $V_J$-fixed space.

(ii) The means commute with every $U(a)$, and with $\Theta_j$ by \eqref{ms:eqcommute} and $\rho_jV_J=V_J$; strong limits of operators commuting with a fixed operator commute with it. $P_JP_{J'}=P_{J\cup J'}$ by the intersection statement in (i). It remains to show that the range of $P_J$ is $L^2(\cI_J,\mu)$; the orthogonal projection onto $L^2$ of a sub-$\sigma$-algebra is the conditional expectation.

Let $G\in L^2(\mu)$ satisfy $U(a)G=G$ for all $a\in V_J$, and fix a Borel version of $G$. The map $(a,x)\mapsto G(\tau_{-a}x)$ is Borel on $V_J\times X$. For each $a\in V_J$ the set $\{x:G(\tau_{-a}x)\neq G(x)\}$ is $\mu$-null, so by Fubini's theorem the set $X_0\defeq\{x:G(\tau_{-a}x)=G(x)\text{ for a.e.\ }a\in V_J\}$ is Borel and of full measure. Let $X_1$ be the set of $x$ for which $a\mapsto G(\tau_{-a}x)$ is a.e.\ constant on $V_J$, that is,
\[
\int_{\Lambda^J_n}\int_{\Lambda^J_n}\min\big(1,\abs{G(\tau_{-a}x)-G(\tau_{-b}x)}\big)\,da\,db=0\qquad\text{for all }n\in\N.
\]
By Tonelli's theorem $X_1$ is Borel; $X_0\subset X_1$; and $X_1$ is $V_J$-invariant, because $G(\tau_{-a}\tau_cx)=G(\tau_{-(a-c)}x)$ and a.e.\ constancy on $V_J$ is preserved under shifts of the variable. For $x\in X_1$ let $\tilde G(x)$ be the a.e.\ value of $a\mapsto G(\tau_{-a}x)$, that is $\tilde G(x)=\lim_{N\to\infty}\abs{\Lambda^J_1}^{-1}\int_{\Lambda^J_1}t_N\big(G(\tau_{-a}x)\big)\,da$, where $t_N$ truncates real and imaginary parts at level $N$ (for $J=\emptyset$ put $\tilde G\defeq G$); put $\tilde G\defeq0$ off $X_1$. Then $\tilde G$ is Borel, $\tilde G\circ\tau_c=\tilde G$ everywhere for $c\in V_J$, and $\tilde G=G$ on $X_0$, hence $\mu$-a.e. Thus $G$ is $\cI_J$-measurable. Conversely, an $\cI_J$-measurable function is a.e.\ equal to a function measurable for the $\sigma$-algebra of $V_J$-invariant Borel sets; such a function is $V_J$-invariant, hence fixed by every $U(a)$, $a\in V_J$.
\end{proof}

The next lemma is the only place where reflection positivity enters the decomposition: functions invariant under the translations in a direction live at infinity in that direction and are fixed by the corresponding reflection.

\begin{lemma}[localisation of invariant functions]\label{ms:x:invloc}
Let $j\in J\subset\{1,2,3\}$ and $W\in L^2(\mu)$ with $U(a)W=W$ for all $a\in V_J$. Then:
\textup{(i)} $W\in L^2(\Sigma(H^+_{j,c}),\mu)\cap L^2(\Sigma(H^-_{j,c}),\mu)$ for every $c\in\R$;
\textup{(ii)} $\Theta_jW=W$;
\textup{(iii)} if $G\in L^2(\Sigma(K),\mu)$, then $P_JG\in L^2(\Sigma(K+V_J),\mu)$.
\end{lemma}
\begin{proof}
(i) Let $B_n$ be the open ball of radius $n$ about $0$. Every bounded open $K$ lies in some $B_n$, so by (H2) the $\sigma$-algebras $\Sigma(B_n)$ increase and their join is $\cB$. The family of Borel sets $B$ for which $\mathbf 1_B$ lies in the closure of $\bigcup_nL^2(\Sigma(B_n),\mu)$ is closed under complements and countable disjoint unions and contains the algebra $\bigcup_n\Sigma(B_n)$; by Dynkin's $\pi$--$\lambda$ theorem it is $\cB$, and so $\bigcup_nL^2(\Sigma(B_n),\mu)$ is dense in $L^2(\mu)$. Given $\epsilon>0$, there are therefore a bounded $K$ and $G\in L^2(\Sigma(K))$ with $\norm{W-G}_2<\epsilon$. For every $t$, $\norm{W-U(te_j)G}_2=\norm{U(te_j)(W-G)}_2<\epsilon$, and $U(te_j)G\in L^2(\Sigma(K+te_j))\subset L^2(\Sigma(H^+_{j,c}))$ for $t$ large, by (H2). That space is closed, so it contains $W$. Letting $t\to-\infty$ gives the other half space.

(ii) $\Theta_jW$ is $V_J$-invariant by \eqref{ms:eqcommute} and $\rho_jV_J=V_J$, so by (i) both $W$ and $\Theta_jW$ lie in $L^2(\Sigma(H^+_{j,0}))$. Put $F\defeq W-\Theta_jW$. Then $\Theta_jF=-F$, and \eqref{ms:eqRP} gives $0\le\E[\overline{\Theta_jF}\,F]=-\E\abs F^2$, so $F=0$.

(iii) The means of Lemma~\ref{ms:x:proj}(i) applied to $G$ lie in the closed subspace $L^2(\Sigma(K+V_J))$, since $U(a)G\in L^2(\Sigma(K+a))$ and $K+a\subset K+V_J$ for $a\in V_J$; so does their limit $P_JG$.
\end{proof}

\subsubsection{The ergodic decomposition and inheritance}\label{ms:x:decsec}

\begin{lemma}[conditional probabilities]\label{ms:erg:rcp}
Let $X$ be Polish, $m$ a Borel probability measure on $X$ and $\mathcal F_0\subset\cB$ a $\sigma$-algebra. \textup{(i)} There are Borel probability measures $m_\xi$, $\xi\in X$, such that $\xi\mapsto m_\xi(B)$ is $\mathcal F_0$-measurable and $\int_Cm_\xi(B)\,dm(\xi)=m(B\cap C)$ for all $B\in\cB$ and $C\in\mathcal F_0$. \textup{(ii)} If $(m'_\xi)$ is a second family with these properties, then $m'_\xi=m_\xi$ for $m$-a.e.\ $\xi$.
\end{lemma}
\begin{proof}
(i) Let $d\le1$ be a complete metric on $X$ and $(x_n)$ a dense sequence. Then $\jmath(x)\defeq(d(x,x_n))_n$ is a continuous injection of $X$ into the compact metrisable space $Q\defeq[0,1]^\N$; by the Lusin--Souslin theorem \cite{Kec95} the image under $\jmath$ of every Borel subset of $X$ is Borel, so $\jmath$ is a Borel isomorphism onto the Borel set $\jmath X$. Let $\mathcal C_0$ be a countable $\mathbb Q$-linear subspace of the real space $C(Q)$, dense for the supremum norm and containing $1$, and choose $\mathcal F_0$-measurable real versions $L_\xi(G)$ of $\E_m[G\circ\jmath\mid\mathcal F_0]$, $G\in\mathcal C_0$. Off a set $N_0\in\mathcal F_0$ with $m(N_0)=0$ the countably many relations of $\mathbb Q$-linearity, positivity ($L_\xi(G)\ge0$ for $G\ge0$) and $L_\xi(1)=1$ hold. For $\xi\notin N_0$ and $G\in\mathcal C_0$, applying positivity to $q\pm G$, $q\in\mathbb Q$, $q\ge\norm G_\infty$, gives $\abs{L_\xi(G)}\le\norm G_\infty$; so $L_\xi$ extends to a positive linear functional on $C(Q)$ with $L_\xi(1)=1$, given by a Borel probability measure $\kappa_\xi$ on $Q$ (Riesz representation theorem). On $N_0$ put $\kappa_\xi\defeq\jmath_*m$. For $G\in C(Q)$, $\kappa_\xi(G)$ is an $\mathcal F_0$-measurable version of $\E_m[G\circ\jmath\mid\mathcal F_0]$, by uniform approximation from $\mathcal C_0$.

Let $\mathcal L$ be the family of Borel sets $A\subset Q$ for which $\xi\mapsto\kappa_\xi(A)$ is an $\mathcal F_0$-measurable version of $\E_m[\mathbf 1_A\circ\jmath\mid\mathcal F_0]$. It is closed under complements and countable disjoint unions (monotone convergence, for the measures and for conditional expectations), and it contains every closed set $A$, since $\mathbf 1_A$ is the decreasing limit of the continuous functions $\max(0,1-n\operatorname{dist}(\cdot,A))$. Closed sets form a $\pi$-system generating the Borel sets, so by Dynkin's $\pi$--$\lambda$ theorem $\mathcal L$ contains every Borel set. In particular $\kappa_\xi(\jmath X)=1$ off a set $N_1\in\mathcal F_0$ with $m(N_1)=0$. Put $m_\xi\defeq(\jmath^{-1})_*\kappa_\xi$ off $N_0\cup N_1$ and $m_\xi\defeq m$ on it. For Borel $B\subset X$, $m_\xi(B)=\kappa_\xi(\jmath B)$ off $N_0\cup N_1$, an $\mathcal F_0$-measurable version of $\E_m[\mathbf 1_B\mid\mathcal F_0]$; this is the claim.

(ii) Let $\cD$ be a countable determining algebra of $X$ (Remark~\ref{ms:x:detalg}). For $G\in\cD$, $m_\xi(G)$ and $m'_\xi(G)$ are $\mathcal F_0$-measurable versions of $\E_m[G\mid\mathcal F_0]$, so they agree off an $m$-null set; off the union of these countably many null sets $m_\xi$ and $m'_\xi$ agree on $\cD$, hence are equal.
\end{proof}

\begin{theorem}[ergodic decomposition with inheritance]\label{ms:erg:decomp}
Let $\mu$ satisfy \textup{(H1)--(H3)}. Let $\cI_0\subset\cB$ be the $\sigma$-algebra of Borel sets $B$ with $\tau_aB=B$ for all $a\in\R^3$ (its $\mu$-completion is $\cI$), let $\mu_\xi$, $\xi\in X$, be measures as in Lemma~\ref{ms:erg:rcp} for $m=\mu$ and $\mathcal F_0=\cI_0$, a regular conditional probability of $\mu$ given $\cI$, and put $(\Xi,\nu)\defeq(X,\mu)$. Then $\xi\mapsto\mu_\xi(B)$ is $\cI_0$-measurable for every Borel set $B$, and:
\begin{enumerate}
\item[\textup{(a)}] $\mu=\int_X\mu_\xi\,d\mu(\xi)$; for $G\in L^1(\mu)$ one has $G\in L^1(\mu_\xi)$ and $\E[G\mid\cI](\xi)=\E_{\mu_\xi}[G]$ for $\mu$-a.e.\ $\xi$; in particular $\E_\mu[WG]=\int W(\xi)\,\E_{\mu_\xi}[G]\,d\mu(\xi)$ for bounded $\cI$-measurable $W$;
\item[\textup{(b)}] for $\nu$-a.e.\ $\xi$, $\mu_\xi$ is invariant under every $\tau_a$ and ergodic for the action of $\R^3$;
\item[\textup{(c)}] for $\nu$-a.e.\ $\xi$, $\mu_\xi$ is invariant under $\theta_1,\theta_2,\theta_3$;
\item[\textup{(d)}] for $\nu$-a.e.\ $\xi$, $\mu_\xi$ satisfies \textup{(H3)};
\item[\textup{(e)}] every $\mu_\xi$ satisfies \textup{(H2)} with the same $\Sigma(U)$, and every $G\in L^2(\mu)$ lies in $L^2(\mu_\xi)$ for $\nu$-a.e.\ $\xi$.
\end{enumerate}
In particular $\nu$-almost every $\mu_\xi$ satisfies \textup{(H1)--(H3)} and is ergodic. By Lemma~\ref{ms:erg:rcp}(ii) the family $(\mu_\xi)$ is determined up to a $\nu$-null set of $\xi$.
\end{theorem}
\begin{proof}
(a) By Lemma~\ref{ms:erg:rcp}, $\E_{\mu_\xi}[G]$ is an $\cI_0$-measurable version of $\E[G\mid\cI]$ for bounded Borel $G$ (uniform approximation by simple functions), and for $G\ge0$ in $L^1(\mu)$ by monotone convergence applied to $\min(G,n)$, which also gives $\int\E_{\mu_\xi}[G]\,d\mu=\E_\mu G<\infty$; complex $G$ are treated by real and imaginary parts. Two Borel versions of $G$ differ on a Borel null set $N$, and $\int\mu_\xi(N)\,d\mu=\mu(N)=0$, so the statement does not depend on the version. Let $\cD$ be a countable determining algebra of $X$.

(b) \emph{Invariance.} For $B\in\cI_0$ and bounded Borel $G$, $\E_\mu[\mathbf 1_B\,G\circ\tau_a]=\E_\mu[(\mathbf 1_BG)\circ\tau_a]=\E_\mu[\mathbf 1_BG]$. Hence $\mu_\xi(G\circ\tau_a)$ and $\mu_\xi(G)$ are both versions of $\E[G\mid\cI]$, and there is a null set off which $\mu_\xi(G\circ\tau_a)=\mu_\xi(G)$ for all $G\in\cD$ and $a\in\mathbb Q^3$. Since $a\mapsto\mu_\xi(G\circ\tau_a)$ is continuous (joint continuity of the action and dominated convergence), this holds for all $a$, so $\mu_\xi$ is $\tau$-invariant.

\emph{Ergodicity.} For $G\in\cD$ let $q_G(\xi)\defeq\mu_\xi(G)$, a bounded $\cI_0$-measurable version of $\E_\mu[G\mid\cI]$. For a bounded $\cI_0$-measurable $h$, $\mu_\xi(h)$ is an $\cI_0$-measurable version of $\E[h\mid\cI]=h$, so $\mu_\xi(h)=h(\xi)$ for a.e.\ $\xi$. Applied to $h=q_G$ and $h=\abs{q_G}^2$ this gives $\mu_\xi(\abs{q_G-q_G(\xi)}^2)=0$: for a.e.\ $\xi$, $q_G$ is $\mu_\xi$-a.s.\ the constant $\mu_\xi(G)$. Let $\Lambda_T\defeq[-T,T]^3$ and $A_TG(x)\defeq\abs{\Lambda_T}^{-1}\int_{\Lambda_T}G(\tau_{-a}x)\,da$. By Fubini's theorem, $A_TG$ is a version of the mean $\abs{\Lambda_T}^{-1}\int_{\Lambda_T}U(a)G\,da$ of Lemma~\ref{ms:x:proj}, in $L^2$ of $\mu$ and of every $\tau$-invariant Borel probability measure (both have the same integrals against bounded Borel functions). So, by (a) and Lemma~\ref{ms:x:proj}, $\int\mu_\xi(\abs{A_TG-q_G}^2)\,d\mu(\xi)=\E_\mu\abs{A_TG-q_G}^2\to0$. Choose $T_k\to\infty$ such that this is at most $2^{-k}$ for the first $k$ elements of an enumeration of $\cD$. Then for a.e.\ $\xi$, $A_{T_k}G\to\mu_\xi(G)$ in $L^2(\mu_\xi)$ for every $G\in\cD$. The proof of Lemmas~\ref{ms:x:strongcont} and~\ref{ms:x:proj}(i) uses only the invariance of the measure and the continuity of the action, so for $\tau$-invariant $\mu_\xi$ the means $A_{T_k}G$ also tend in $L^2(\mu_\xi)$ to the projection of $G$ onto the $\tau$-fixed vectors of $L^2(\mu_\xi)$. Both this projection and $G\mapsto\mu_\xi(G)\mathbf 1$ are continuous on $L^2(\mu_\xi)$ and agree on the dense set $\cD$, so the projection is $G\mapsto\mu_\xi(G)\mathbf 1$. If $B\in\cI_0$, then $\mathbf 1_B$ is a fixed vector, hence a.s.\ constant, so $\mu_\xi(B)\in\{0,1\}$.

(e) (H2) is a condition on $X$ and $\Sigma$ alone and does not involve the measure, so it holds for every $\mu_\xi$. Also $\int\E_{\mu_\xi}\abs G^2\,d\nu=\E_\mu\abs G^2<\infty$ by (a).

(c) Let $W\ge0$ be bounded and $\cI$-measurable and $G$ bounded Borel. By Lemma~\ref{ms:x:invloc}(ii) with $J=\{1,2,3\}$, $\Theta_jW=W$, so $\E_\mu[W\,\Theta_jG]=\E_\mu[\Theta_j(WG)]=\E_\mu[WG]$ by the $\theta_j$-invariance of $\mu$. By (a), $\int W(\xi)\big(\E_{\mu_\xi}[\Theta_jG]-\E_{\mu_\xi}[G]\big)\,d\nu=0$ for all such $W$, hence $\E_{\mu_\xi}[\Theta_jG]=\E_{\mu_\xi}[G]$ for $\nu$-a.e.\ $\xi$. Taking $G\in\cD$ makes the exceptional set independent of $G$ and gives the $\theta_j$-invariance of $\mu_\xi$ for a.e.\ $\xi$.

(d) By the translation invariance of $\mu_\xi$ it suffices to take $c=0$. Let $W\ge0$ be bounded and $\cI$-measurable and $F\in L^2(\Sigma(H^+_{j,0}),\mu)$. By Lemma~\ref{ms:x:invloc}(i)--(ii), $\sqrt W$ has a $\Sigma(H^+_{j,0})$-measurable version and $\Theta_j\sqrt W=\sqrt W$. Hence $\sqrt WF\in L^2(\Sigma(H^+_{j,0}),\mu)$ and
\[
\E_\mu\big[W\,\overline{F\circ\theta_j}\,F\big]=\E_\mu\big[\overline{(\sqrt WF)\circ\theta_j}\,(\sqrt WF)\big]\ge0.
\]
With (a), $\E_{\mu_\xi}[\overline{F\circ\theta_j}\,F]\ge0$ for $\nu$-a.e.\ $\xi$. Let $\mathcal L$ be the countable $\mathbb Q[\mathrm i]$-algebra, closed under conjugation, generated by $1$ and a countable family of bounded continuous functions generating $\Sigma(H^+_{j,0})$ (H2). The exceptional set can be taken uniform over $F\in\mathcal L$. By Lemma~\ref{ms:density}, $\mathcal L$ is dense in $L^2(\Sigma(H^+_{j,0}),m)$ for every Borel probability measure $m$, and \eqref{ms:eqRP} extends from $\mathcal L$ to $L^2(\Sigma(H^+_{j,0}),\mu_\xi)$ by the bound $\abs{\E[\overline{F\circ\theta_j}\,G]}\le\norm F_2\norm G_2$, valid because $\mu_\xi$ is $\theta_j$-invariant by (c).

As for the final assertions, (H1) for $\mu_\xi$ is (b) and (c), the space and the actions being those of $\mu$.
\end{proof}

\begin{proposition}[the decomposition of an image law]\label{ms:erg:transfer}
Let $X$ and $X'$ be Polish spaces with jointly continuous actions $\tau$, $\tau'$ of $\R^3$, and $\iota:X'\to X$ a continuous injection with $\iota\circ\tau'_a=\tau_a\circ\iota$ for all $a$. Let $\mu'$ be a $\tau'$-invariant Borel probability measure on $X'$ and $\mu\defeq\iota_*\mu'$. Let $(\mu'_y)_{y\in X'}$ and $(\mu_x)_{x\in X}$ be families as in Lemma~\ref{ms:erg:rcp} for $(\mu',\cI_0(X'))$ and $(\mu,\cI_0(X))$, where $\cI_0(\cdot)$ is the $\sigma$-algebra of translation-invariant Borel sets. Then $\iota X'$ is a translation-invariant Borel set with $\mu(\iota X')=1$, and $\mu_{\iota y}=\iota_*\mu'_y$ for $\mu'$-a.e.\ $y$. In particular, if $\mu'$ and $\mu$ satisfy \textup{(H1)--(H3)}, the components of $\mu$ in Theorem~\ref{ms:erg:decomp} are, for $\mu$-a.e.\ $x$, the images under $\iota$ of the components of $\mu'$, and they are carried by $\iota X'$.
\end{proposition}
\begin{proof}
By the Lusin--Souslin theorem \cite{Kec95}, $\iota$ maps Borel sets to Borel sets, so $\iota X'$ is Borel, and it is invariant because $\tau_a\iota X'=\iota\tau'_aX'=\iota X'$; $\mu(\iota X')=\mu'(X')=1$. If $B'\in\cI_0(X')$, then $\iota B'$ is Borel and $\tau_a\iota B'=\iota\tau'_aB'=\iota B'$, so $\iota B'\in\cI_0(X)$; and $\iota^{-1}C\in\cI_0(X')$ for $C\in\cI_0(X)$. Define $m_x\defeq\iota_*\mu'_{\iota^{-1}x}$ for $x\in\iota X'$ and $m_x\defeq\mu_x$ otherwise. For bounded Borel $G$ on $X$, $\varphi(y)\defeq\mu'_y(G\circ\iota)$ is $\cI_0(X')$-measurable, so for a Borel set $A\subset\C$ the set $\{x:m_x(G)\in A\}=\iota\big(\varphi^{-1}(A)\big)\cup\big((X\setminus\iota X')\cap\{x:\mu_x(G)\in A\}\big)$ lies in $\cI_0(X)$. For $C\in\cI_0(X)$,
\[
\int_Cm_x(G)\,d\mu(x)=\int_{\iota^{-1}C}\mu'_y(G\circ\iota)\,d\mu'(y)=\E_{\mu'}\big[(G\circ\iota)\,\mathbf 1_{\iota^{-1}C}\big]=\E_\mu[G\,\mathbf 1_C].
\]
So $(m_x)$ is a family as in Lemma~\ref{ms:erg:rcp}(i) for $(\mu,\cI_0(X))$, and by Lemma~\ref{ms:erg:rcp}(ii) $m_x=\mu_x$ for $\mu$-a.e.\ $x$; since $\mu=\iota_*\mu'$, this is $\mu_{\iota y}=\iota_*\mu'_y$ for $\mu'$-a.e.\ $y$.
\end{proof}

\subsubsection{The decomposition with cubic symmetry}

Assume, in addition to (H1)--(H3), that the cubic group $\Oh$ acts on $X$ by homeomorphisms $x\mapsto gx$, with $\rho_jx=\theta_jx$ for the coordinate reflections $\rho_j\in\Oh$, $g\,\tau_a\,g^{-1}=\tau_{ga}$, $\mu$ invariant under every $g$, and $G\mapsto G\circ g^{-1}$ mapping $\Sigma(U)$-measurable functions to $\Sigma(gU)$-measurable ones. Let $\cG\defeq\Cub$, let $N\cong(\Z/2)^3$ be the subgroup of $\Oh$ generated by $\rho_1,\rho_2,\rho_3$, so that $\Oh/N\cong S_3$, and let $\cI_\cG\subset\cI$ be the $\sigma$-algebra of $\cG$-invariant Borel sets, completed by $\mu$-null sets.

\begin{theorem}[decomposition with cubic symmetry]\label{ms:erg:Ohdecomp}
There are Borel probability measures $\mu'_x$ on $X$, $x\in X$, with $x\mapsto\mu'_x(B)$ $\cI_\cG$-measurable for every Borel $B$ and $\mu=\int\mu'_x\,d\mu(x)$, such that for $\mu$-almost every $x$:
\begin{enumerate}
\item[\textup{(a)}] $\mu'_x$ is invariant under $\cG$ and ergodic for its action (every $\cG$-invariant Borel set has $\mu'_x$-measure $0$ or $1$), and satisfies \textup{(H1)--(H3)};
\item[\textup{(b)}] there are an integer $k=k(x)\in\{1,2,3,6\}$ and pairwise distinct measures $m_1,\dots,m_k$, each invariant under the translations and the coordinate reflections, ergodic for $\R^3$ and satisfying \textup{(H1)--(H3)}, such that $\Oh$ permutes $\{m_1,\dots,m_k\}$ transitively and $\mu'_x=k^{-1}\sum_{i=1}^km_i$; the $m_i$ are components $\mu_{gx}$, $g\in\Oh$, of the decomposition of Theorem~\ref{ms:erg:decomp}.
\end{enumerate}
Moreover $k(x)=1$ for $\mu$-almost every $x$ if and only if almost every component of Theorem~\ref{ms:erg:decomp} is $\Oh$-invariant.
\end{theorem}
\begin{proof}
\emph{Step 1: $N$ acts trivially on $\cI$, and $\Oh$ preserves $\cI$.} A bounded $\cI$-measurable $W$ is invariant along every coordinate axis, so $W\circ\theta_j=W$ a.s.\ for $j=1,2,3$ by Lemma~\ref{ms:x:invloc}(ii); hence $W\circ n=W$ a.s.\ for $n\in N$. For $g\in\Oh$ and a translation-invariant Borel set $B$, $\tau_a(gB)=g\,\tau_{g^{-1}a}B=gB$; and $g$ maps $\mu$-null sets to $\mu$-null sets. So $g$ maps $\cI$ onto itself, and $h\circ g$ is $\cI$-measurable whenever $h$ is.

\emph{Step 2: conditional expectations.} Let $\mu_x$ be the components of Theorem~\ref{ms:erg:decomp}, and let $\mu'_x$ be a regular conditional probability of $\mu$ given $\cI_\cG$: the measures of Lemma~\ref{ms:erg:rcp} for the $\sigma$-algebra of $\cG$-invariant Borel sets, whose $\mu$-completion is $\cI_\cG$. For bounded Borel $F$, $g\in\Oh$ and bounded $\cI$-measurable $W$, Step~1 and the $g$-invariance of $\mu$ give $\E\big[W\cdot(\E[F\mid\cI]\circ g)\big]=\E\big[(W\circ g^{-1})\,\E[F\mid\cI]\big]=\E\big[(W\circ g^{-1})\,F\big]=\E[W\cdot(F\circ g)]$, so $\E[F\circ g\mid\cI]=\E[F\mid\cI]\circ g$ a.s. Applied to $F$ in a countable determining algebra of $X$, and then as measures,
\begin{equation}\label{ms:x:eq:equivariant}
\mu_{gx}=g_*\mu_x\qquad\text{for $\mu$-a.e.\ $x$ and every }g\in\Oh.
\end{equation}
By Step~1 and \eqref{ms:x:eq:equivariant} with $g\in N$, $n_*\mu_x=\mu_{nx}=\mu_x$ for $n\in N$ and a.e.\ $x$, so $S_3=\Oh/N$ acts on the components by $\sigma\cdot\mu_x\defeq g_*\mu_x$, $g\in\sigma$.

For bounded $\cI$-measurable $h$ put $\bar h\defeq\abs{\Oh}^{-1}\sum_{g\in\Oh}h\circ g$. It is a.e.\ equal to a translation-invariant and $\Oh$-invariant Borel function, hence $\cI_\cG$-measurable, and for bounded $\cI_\cG$-measurable $W$, which satisfies $W\circ g=W$ a.s., $\E[W\bar h]=\abs{\Oh}^{-1}\sum_g\E[(W\circ g)(h\circ g)]=\E[Wh]$. So $\E[h\mid\cI_\cG]=\bar h$. Applied to $h=\E[F\mid\cI]$, with $\E[F\mid\cI_\cG]=\E\big[\E[F\mid\cI]\mid\cI_\cG\big]$ and \eqref{ms:x:eq:equivariant},
\[
\mu'_x=\frac1{48}\sum_{g\in\Oh}\mu_{gx}=\frac1{48}\sum_{g\in\Oh}g_*\mu_x=\frac16\sum_{\sigma\in S_3}\sigma\cdot\mu_x\qquad\text{for $\mu$-a.e.\ $x$},
\]
first evaluated on a countable determining algebra and then as measures. Grouping equal terms, $\mu'_x=k^{-1}\sum_{i=1}^km_i$, where $\{m_1,\dots,m_k\}$ is the $S_3$-orbit of $\mu_x$, so $k=6/\abs{\mathrm{Stab}(\mu_x)}\in\{1,2,3,6\}$, the orders of the subgroups of $S_3$ being $1,2,3,6$. Let $E$ be the set of full measure of those $x$ for which $\mu_x$ has the properties (b)--(e) of Theorem~\ref{ms:erg:decomp}; since every $g\in\Oh$ preserves $\mu$, $\bigcap_{g\in\Oh}g^{-1}E$ also has full measure, and for $x$ in it every $\mu_{gx}$ has these properties. This gives (b): each $m_i$ is translation invariant, ergodic, $\theta_j$-invariant and satisfies (H2) and (H3), hence (H1)--(H3). Since $k=1$ exactly when $\mu_x$ is fixed by $S_3$, that is $\Oh$-invariant, the last assertion follows.

\emph{Step 3: (a).} $\mu'_x$ is invariant under the translations and under $\Oh$, as the average over an $\Oh$-orbit of translation-invariant measures ($\tau_ag=g\tau_{g^{-1}a}$). If $B$ is a $\cG$-invariant Borel set, then $m_i(B)\in\{0,1\}$ by ergodicity, and $(g_*m_i)(B)=m_i(g^{-1}B)=m_i(B)$, so all $m_i(B)$ are equal by transitivity and $\mu'_x(B)\in\{0,1\}$. Each $m_i$ satisfies (H3), and (H3) is preserved under convex combinations: an $F\in L^2(\Sigma(H^+_{j,c}),\mu'_x)$ lies in every $L^2(m_i)$, since $m_i\le k\mu'_x$. (H2) holds as in Theorem~\ref{ms:erg:decomp}(e), and the $\theta_j=\rho_j$-invariance is part of the $\Oh$-invariance.
\end{proof}

\subsubsection{The growth of the moments of the components}

Let $\cV$ be a real Euclidean space of dimension $d$ with orthonormal basis $(e_c)_{c\le d}$. Let $h_\alpha$, $\alpha\in(\N\cup\{0\})^3$, be the Hermite functions on $\R^3$, $h_{\alpha,c}\defeq h_\alpha e_c$, and $\lambda_\alpha\defeq2\abs\alpha+4$ the eigenvalues of $-\Delta+\abs x^2+1$. For even $s\ge0$ put $\norm f_s^2\defeq\sum_{\alpha,c}\lambda_\alpha^s\abs{\ip{h_{\alpha,c}}f}^2$, let $\cH^s$ be the completion of $\cS(\R^3;\cV)$ in this norm and $\cH^{-s}$ its dual, a separable Hilbert space with $\norm\Psi_{-s}^2=\sum_{\alpha,c}\lambda_\alpha^{-s}\Psi(h_{\alpha,c})^2$ and $\abs{\Psi(f)}\le\norm\Psi_{-s}\norm f_s$. Note that $\norm{h_{\alpha,c}}_{10}=\lambda_\alpha^5$, and $\norm f_s\le\norm f_{s'}$ for $s\le s'$ since $\lambda_\alpha\ge1$. There are $(n+1)(n+2)/2$ multi-indices with $\abs\alpha=n$, so
\[
w_r\defeq d\sum_\alpha\lambda_\alpha^{10-r}<\infty\qquad\text{for every even }r\ge14 .
\]

\begin{theorem}[growth of the moments of the components]\label{ms:erg:growth}
Let $r\ge14$ be even, $\mu^H$ a Borel probability measure on $\cH^{-r}=\cH^{-r}(\R^3;\cV)$, and suppose that there are constants $c_S\ge1$ and $C_0>0$ with
\begin{equation}\label{ms:erg:eq:hermitebound}
\int\Psi(h_{\alpha,c})^{2n}\,d\mu^H(\Psi)\le c_S\,C_0^{2n}\,(2n)!\,\norm{h_{\alpha,c}}_{10}^{2n}\qquad\text{for all }\alpha,\ c\le d,\ n\in\N .
\end{equation}
Let $(\Xi,\nu)$ be a probability space and $\mu_\xi$, $\xi\in\Xi$, Borel probability measures on $\cH^{-r}$ with $\xi\mapsto\mu_\xi(B)$ measurable for every Borel $B$ and $\mu^H=\int\mu_\xi\,d\nu(\xi)$; for instance the components of Theorem~\ref{ms:erg:decomp} or of Theorem~\ref{ms:erg:Ohdecomp}, when $\mu^H$ satisfies the hypotheses there. Put $C_1\defeq w_rC_0^2$. Then for $\nu$-a.e.\ $\xi$, for every $N\in\N$ and all $f_1,\dots,f_N\in\cS(\R^3;\cV)$, the product $\prod_i\Psi(f_i)$ is $\mu_\xi$-integrable, and its integral $S^\xi_N(f_1,\dots,f_N)\defeq\E_{\mu_\xi}\prod_i\Psi(f_i)$ satisfies
\[
\abs{S^\xi_N(f_1,\dots,f_N)}\le\sigma^\xi_N\prod_{i=1}^N\norm{f_i}_r,\qquad\sigma^\xi_N\defeq\big(\E_{\mu_\xi}\norm\Psi_{-r}^{2N}\big)^{1/2}<\infty .
\]
There is $n_0(\xi)$ with $\sigma^\xi_N\le N\,c_S^{1/2}(4C_1)^{N/2}N!$ for $N\ge n_0(\xi)$, and consequently $\sigma^\xi_N\le\alpha_\beta(\xi)(N!)^\beta$ for all $N$, for every $\beta>1$, with $\alpha_\beta(\xi)\defeq\sup_N\sigma^\xi_N(N!)^{-\beta}<\infty$.
\end{theorem}
\begin{proof}
Put $M(\Psi)\defeq\norm\Psi_{-r}^2=\sum_{\alpha,c}\lambda_\alpha^{-r}\Psi(h_{\alpha,c})^2$, a continuous function on $\cH^{-r}$. With the probability weights $p_{\alpha,c}\defeq\lambda_\alpha^{10-r}/w_r$,
\[
M=\sum_{\alpha,c}p_{\alpha,c}\,w_r\lambda_\alpha^{-10}\Psi(h_{\alpha,c})^2,\qquad\text{so}\qquad M^n\le w_r^n\sum_{\alpha,c}p_{\alpha,c}\,\lambda_\alpha^{-10n}\Psi(h_{\alpha,c})^{2n}
\]
by Jensen's inequality for the convex function $t\mapsto t^n$ on $[0,\infty)$ (applied to the finite partial sums, renormalised, and passed to the limit). By \eqref{ms:erg:eq:hermitebound} and $\norm{h_{\alpha,c}}_{10}^{2n}=\lambda_\alpha^{10n}$,
\[
\E_{\mu^H}M^n\le w_r^n\sup_{\alpha,c}\lambda_\alpha^{-10n}\,\E_{\mu^H}\Psi(h_{\alpha,c})^{2n}\le c_S\,C_1^n\,(2n)! .
\]
Put $M_n(\xi)\defeq\E_{\mu_\xi}M^n\in[0,\infty]$, a measurable function of $\xi$ (monotone limit of $\mu_\xi(\min(M^n,k))$). Since $\int M_n\,d\nu=\E_{\mu^H}M^n<\infty$, $M_n(\xi)<\infty$ for all $n$ and $\nu$-a.e.\ $\xi$. By Markov's inequality, $\nu\{M_n>n^2c_SC_1^n(2n)!\}\le n^{-2}$, which is summable, so by the Borel--Cantelli lemma there is, for $\nu$-a.e.\ $\xi$, an $n_0(\xi)$ with $M_n(\xi)\le n^2c_SC_1^n(2n)!$ for $n\ge n_0(\xi)$. Fix such a $\xi$. For $f_i\in\cS\subset\cH^r$, $\abs{\Psi(f_i)}\le M^{1/2}\norm{f_i}_r$, so
\[
\E_{\mu_\xi}\prod_{i=1}^N\abs{\Psi(f_i)}\le\E_{\mu_\xi}\big[M^{N/2}\big]\prod_i\norm{f_i}_r\le M_N(\xi)^{1/2}\prod_i\norm{f_i}_r
\]
by the Cauchy--Schwarz inequality, and $\sigma^\xi_N=M_N(\xi)^{1/2}<\infty$. For $N\ge n_0(\xi)$, $\sigma^\xi_N\le N\,c_S^{1/2}C_1^{N/2}((2N)!)^{1/2}\le N\,c_S^{1/2}(4C_1)^{N/2}N!$, because $(2N)!\le4^N(N!)^2$. For $\beta>1$, $N(4C_1)^{N/2}(N!)^{1-\beta}\to0$, so $\alpha_\beta(\xi)<\infty$.
\end{proof}

\begin{corollary}[the components of a law on a space of local distributions]\label{ms:erg:growthX}
In the setting of Theorem~\ref{ms:erg:growth}, let $X$ be a Polish space with a jointly continuous action of $\R^3$, let $\R^3$ act on $\cH^{-r}$ jointly continuously and leave $\mu^H$ invariant, and let $\iota:\cH^{-r}\to X$ be a continuous injection with $\iota\circ\tau^H_a=\tau_a\circ\iota$. Put $\mu\defeq\iota_*\mu^H$, and let $\mu_x$ and $\mu^H_y$ be the components of $\mu$ and of $\mu^H$ given by Lemma~\ref{ms:erg:rcp} for the $\sigma$-algebras of translation-invariant Borel sets. Then for $\mu$-a.e.\ $x$ the component $\mu_x$ is carried by $\iota\cH^{-r}$, and with $\Phi(f)\defeq\Psi(f)$ for $\Phi=\iota\Psi$ and $f\in\cS(\R^3;\cV)$ its moments $S^x_N(f_1,\dots,f_N)\defeq\E_{\mu_x}\prod_i\Phi(f_i)$ satisfy the conclusions of Theorem~\ref{ms:erg:growth}.
\end{corollary}
\begin{proof}
By Proposition~\ref{ms:erg:transfer}, $\mu_{\iota y}=\iota_*\mu^H_y$ for $\mu^H$-a.e.\ $y$, and Theorem~\ref{ms:erg:growth} applies to $(\mu^H_y)$ with $(\Xi,\nu)=(\cH^{-r},\mu^H)$. Let $Y$ be the Borel set of full $\mu^H$-measure of the $y$ for which both statements hold. Then $\iota Y$ is Borel (Lusin--Souslin \cite{Kec95}) with $\mu(\iota Y)=1$, and for $x=\iota y\in\iota Y$ the moments of $\mu_x$ in the variables $\Phi(f)$ are those of $\mu^H_y$ in the variables $\Psi(f)$.
\end{proof}

\subsubsection{Reconstruction in a coordinate direction}\label{ms:x:reconsec}

\emph{Conventions.} In Subsections~\ref{ms:x:reconsec} and~\ref{ms:x:clustersec}, $X$, the translations $\tau_a$, the coordinate reflections $\theta_j$, the local $\sigma$-algebras $\Sigma(U)$ and the measure $\mu$ satisfy (H1)--(H3) of Subsection~\ref{ms:H}, with $U(a)G\defeq G\circ\tau_{-a}$, $\Theta_jG\defeq G\circ\theta_j$ and $\theta_{j,c}\defeq\tau_{ce_j}\theta_j\tau_{-ce_j}$. So $a\mapsto U(a)$ is a unitary representation of $\R^3$ on $L^2(\mu)$, $\Theta_j$ is a unitary involution, and $\Theta_jU(a)=U(\rho_ja)\Theta_j$ \eqref{ms:eqcommute}; in particular $\Theta_jU(te_j)=U(-te_j)\Theta_j$ and $\Theta_jU(y)=U(y)\Theta_j$ for $y\perp e_j$. Both $U(a)$ and $\Theta_j$ are multiplicative and commute with complex conjugation. The expectation $\E_\mu[G'G]$ is taken without conjugation throughout.
The union over $n\in\N$ of the spaces $L^2(\Sigma(B_n),\mu)$, $B_n$ the open ball of radius $n$, is dense in $L^2(\mu)$: every bounded $K$ lies in some $B_n$, so $\bigvee_n\Sigma(B_n)=\cB$ by (H2), and $\E[G\mid\Sigma(B_n)]\to G$ in $L^2(\mu)$ by the martingale convergence theorem. We call $G\in L^2(\Sigma(K),\mu)$ with $K$ bounded \emph{localised}.

Fix $\mu$ satisfying (H1)--(H3) and a direction $j$. Write $L^2_{j,+}\defeq L^2(\Sigma(H^+_{j,0}),\mu)$, a closed subspace of $L^2(\mu)$, and define
\begin{equation}\label{ms:eq:form}
\ip FG_j\defeq\E_\mu\big[\overline{\Theta_jF}\,G\big],\qquad F,G\in L^2_{j,+}.
\end{equation}
The form is positive semidefinite by (H3) with $c=0$. It is Hermitian: $\overline{\ip FG_j}=\E_\mu[\Theta_jF\,\overline G]=\E_\mu[\Theta_j(F\,\overline{\Theta_jG})]=\E_\mu[\overline{\Theta_jG}\,F]=\ip GF_j$, since $\Theta_j$ is a multiplicative involution and $\mu$ is $\theta_j$-invariant. It is bounded, $\abs{\ip FG_j}\le\norm F_2\norm G_2$, since $\Theta_j$ is unitary.

\begin{proposition}[reconstruction in direction $j$]\label{ms:recon}
Let $\cN_j\defeq\{F\in L^2_{j,+}:\ip FF_j=0\}$, $\cH_j$ the completion of $L^2_{j,+}/\cN_j$ for the form \eqref{ms:eq:form}, $[\,\cdot\,]:L^2_{j,+}\to\cH_j$ the quotient map, and $\Omega_j\defeq[1]$.
\begin{enumerate}
\item[(i)] For $t\ge0$ and $y\perp e_j$, $T_j(t)[G]\defeq[U(te_j)G]$ and $V_j(y)[G]\defeq[U(y)G]$ are well defined. $T_j$ is a strongly continuous semigroup of self-adjoint contractions, so $T_j(t)=e^{-tH_j}$ with $H_j\ge0$ self-adjoint. $V_j$ is a strongly continuous unitary representation of $e_j^\perp\cong\R^2$. All these operators commute, and $T_j(t)\Omega_j=V_j(y)\Omega_j=\Omega_j$; in particular $H_j\Omega_j=0$, and $\Omega_j$ is a unit vector.
\item[(ii)] For $G\in L^2_{j,+}$ and $a=a_je_j+a_\perp$ with $a_j\ge0$: $U(a)G\in L^2_{j,+}$ and $[U(a)G]=T_j(a_j)V_j(a_\perp)[G]$.
\item[(iii)] For $F\in L^2(\Sigma(H^-_{j,0}),\mu)$ and $G\in L^2_{j,+}$: $\overline{\Theta_jF}\in L^2_{j,+}$ and $\E_\mu[FG]=\ip{[\overline{\Theta_jF}]}{[G]}_{\cH_j}$.
\end{enumerate}
\end{proposition}
\begin{proof}
This is the Osterwalder--Schrader construction \cite{OS73}, \cite[Chapter~6]{GJ87}; we give the argument in the present setting. Two inequalities are used repeatedly: the Schwarz inequality for the positive semidefinite form \eqref{ms:eq:form}, and $\norm{[F]}_{\cH_j}^2=\ip FF_j\le\norm F_2^2$.

\emph{Invariance of $L^2_{j,+}$.} For $G\in L^2_{j,+}$ and $a$ with $a_j\ge0$, $U(a)G$ is $\Sigma(H^+_{j,0}+a)$-measurable by (H2), and $H^+_{j,0}+a=H^+_{j,a_j}\subset H^+_{j,0}$; so $U(a)G\in L^2_{j,+}$. This gives the first claim of (ii).

\emph{The semigroup $T_j$.} For $F,G\in L^2_{j,+}$ and $t\ge0$, by \eqref{ms:eqcommute} and the invariance of $\mu$,
\[
\ip{U(te_j)F}{G}_j=\E\big[\overline{U(-te_j)\Theta_jF}\;G\big]=\E\big[\overline{\Theta_jF}\;U(te_j)G\big]=\ip F{U(te_j)G}_j ;
\]
in particular $\ip{U(te_j)G}{U(te_j)G}_j=\ip G{U(2te_j)G}_j$. So $U(te_j)$ is symmetric for the form, and by the Schwarz inequality for the form, $\ip{U(te_j)G}{U(te_j)G}_j\le\ip GG_j^{1/2}\,\ip{U(2te_j)G}{U(2te_j)G}_j^{1/2}$. Applying this successively at $t,2t,\dots,2^{n-1}t$ and using $\ip{U(se_j)G}{U(se_j)G}_j\le\norm{U(se_j)G}_2^2=\norm G_2^2$ at the last step gives
\[
\ip{U(te_j)G}{U(te_j)G}_j\le\ip GG_j^{1-2^{-n}}\,\norm G_2^{2^{1-n}},
\]
and $n\to\infty$ gives $\ip{U(te_j)G}{U(te_j)G}_j\le\ip GG_j$. Hence $U(te_j)$ maps $\cN_j$ into $\cN_j$, $T_j(t)$ is well defined on the quotient and is a contraction there, and it is symmetric by the identity above; it extends to a self-adjoint contraction on $\cH_j$. The semigroup law $T_j(t)T_j(s)=T_j(t+s)$ is inherited from $U$, and $T_j(t)=T_j(t/2)^2\ge0$. For the strong continuity, $\norm{T_j(t)[G]-[G]}_{\cH_j}\le\norm{U(te_j)G-G}_2\to0$ as $t\downarrow0$ by Lemma~\ref{ms:x:strongcont}, on the dense set $[L^2_{j,+}]$, and the $T_j(t)$ are uniformly bounded. A strongly continuous semigroup of positive self-adjoint contractions with $T_j(0)=1$ is $e^{-tH_j}$ with $H_j\ge0$ self-adjoint (by the spectral theorem, as in \cite[Chapter~6]{GJ87}).

\emph{The transverse translations.} For $y\perp e_j$, $\Theta_j$ commutes with $U(y)$ by \eqref{ms:eqcommute}, so $\ip{U(y)F}{U(y)G}_j=\E[\overline{U(y)\Theta_jF}\,U(y)G]=\ip FG_j$ by invariance of $\mu$. Thus $V_j(y)$ preserves $\cN_j$, is isometric on the quotient and has the isometric inverse $V_j(-y)$; it extends to a unitary operator, and $y\mapsto V_j(y)$ is a representation, strongly continuous by the same estimate as for $T_j$. The operators $T_j(t)$ and $V_j(y)$ commute because the $U(a)$ do, and $T_j(t)\Omega_j=[U(te_j)1]=[1]=\Omega_j$, $V_j(y)\Omega_j=\Omega_j$. Hence $H_j\Omega_j=0$, and $\norm{\Omega_j}^2=\E[\overline{\Theta_j1}\,1]=1$. This proves (i); the identity in (ii) is the definition, $[U(a)G]=[U(a_je_j)U(a_\perp)G]=T_j(a_j)V_j(a_\perp)[G]$.

(iii) $F$ is $\Sigma(H^-_{j,0})$-measurable, so $\Theta_jF$ and $\overline{\Theta_jF}$ are $\Sigma(\rho_jH^-_{j,0})=\Sigma(H^+_{j,0})$-measurable by (H2). Since $\Theta_j$ is an involution commuting with conjugation, $\overline{\Theta_j\overline{\Theta_jF}}=F$, and $\ip{\overline{\Theta_jF}}G_j=\E[FG]$.
\end{proof}

\begin{proposition}[zero-energy vectors, invariant functions and the joint vacuum]\label{ms:fixed}
In the setting of Proposition~\ref{ms:recon}, let $Q_j$ be the orthogonal projection of $\cH_j$ onto the joint fixed space of $\{T_j(t)\}_{t\ge0}\cup\{V_j(y)\}_{y\perp e_j}$, and $P_{\ker H_j}$ that onto $\ker H_j$. For every $G\in L^2_{j,+}$:
\begin{enumerate}
\item[(i)] $P_jG\in L^2_{j,+}$, $P_{\ker H_j}[G]=[P_jG]$, and $\norm{[P_jG]}^2_{\cH_j}=\E_\mu\abs{P_jG}^2$;
\item[(ii)] $\E[G\mid\cI]\in L^2_{j,+}$ and $Q_j[G]=[\E[G\mid\cI]]$;
\item[(iii)] if $\mu$ is ergodic for the $\R^3$-action, the joint fixed space is $\C\Omega_j$;
\item[(iv)] conversely, if the joint fixed space is $\C\Omega_j$, then $\mu$ is ergodic for the $\R^3$-action.
\end{enumerate}
So the joint vacuum is unique, that is, the joint fixed space is $\C\Omega_j$, if and only if $\mu$ is ergodic for the $\R^3$-action.
\end{proposition}
\begin{proof}
(i) $P_jG$ is $e_j$-invariant, so $P_jG\in L^2_{j,+}$ by Lemma~\ref{ms:x:invloc}(i) with $J=\{j\}$. The quotient map is bounded and linear on $L^2_{j,+}$, and $t\mapsto U(te_j)G$ is continuous into $L^2_{j,+}$ by Lemma~\ref{ms:x:strongcont}; so the quotient map commutes with the Bochner integrals,
\[
\frac1T\int_0^TT_j(t)[G]\,dt=\Big[\frac1T\int_0^TU(te_j)G\,dt\Big].
\]
As $T\to\infty$ the left side tends to $P_{\ker H_j}[G]$ by the spectral theorem, since $T^{-1}\int_0^Te^{-t\lambda}dt\to\mathbf 1_{\{\lambda=0\}}$ boundedly on $[0,\infty)$; the right side tends to $[P_jG]$ by the one-sided means of Lemma~\ref{ms:x:proj}(i) and the continuity of the quotient map. Finally $\Theta_jP_jG=P_jG$ by Lemma~\ref{ms:x:invloc}(ii), so $\norm{[P_jG]}^2=\E[\overline{\Theta_jP_jG}\,P_jG]=\E\abs{P_jG}^2$.

(ii) $\E[G\mid\cI]=P_{\{1,2,3\}}G$ by Lemma~\ref{ms:x:proj}(ii); it is invariant under all translations, so it lies in $L^2_{j,+}$ by Lemma~\ref{ms:x:invloc}(i). Let $k,k'$ be the two indices other than $j$, and put $W(a)\defeq T_j(a_j)V_j(a_ke_k+a_{k'}e_{k'})$ for $a_j\ge0$. The means of $W(a)$ over $a\in[0,T]\times[-T,T]^2$ (coordinates ordered as $(a_j,a_k,a_{k'})$) are the product of three commuting means: the one-sided mean of $T_j$, which tends strongly to $P_{\ker H_j}$ as in (i), and the two-sided means of the unitary groups $s\mapsto V_j(se_k)$ and $s\mapsto V_j(se_{k'})$, which tend strongly to the projections onto their fixed spaces (by the spectral theorem, as in Lemma~\ref{ms:x:proj}(i)). These three projections commute, and a product of commuting, uniformly bounded, strongly convergent families converges strongly to the product of the limits, which is the projection onto the intersection of the three fixed spaces, that is $Q_j$. On the other hand $W(a)[G]=[U(a)G]$ by Proposition~\ref{ms:recon}(ii), and the means of $U(a)G$ over the same set are the product of the commuting means $T^{-1}\int_0^TU(te_j)\,dt$, $(2T)^{-1}\int_{-T}^TU(se_k)\,ds$ and $(2T)^{-1}\int_{-T}^TU(se_{k'})\,ds$, applied to $G$; by Lemma~\ref{ms:x:proj}(i) these converge strongly to $P_j$, $P_k$ and $P_{k'}$, so their product converges strongly to $P_jP_kP_{k'}=P_{\{1,2,3\}}$ (Lemma~\ref{ms:x:proj}(ii)). The means of $U(a)G$ lie in $L^2_{j,+}$ and are mapped by the quotient map to the means of $W(a)[G]$ (the quotient map commutes with Bochner integrals, as in (i)); by continuity of the quotient map the limit is $[P_{\{1,2,3\}}G]=[\E[G\mid\cI]]$.

(iii) If $\mu$ is ergodic, $\cI$ is trivial and $\E[G\mid\cI]=\E_\mu G$ a.s., so by (ii) $Q_j[G]=\E_\mu[G]\,\Omega_j=\ip{\Omega_j}{[G]}_{\cH_j}\Omega_j$, because $\ip{[1]}{[G]}_{\cH_j}=\E[\overline{\Theta_j1}\,G]=\E_\mu G$. Since $[L^2_{j,+}]$ is dense in $\cH_j$, $Q_j$ is the projection onto $\C\Omega_j$.

(iv) Suppose $\mu$ is not ergodic: there is a translation-invariant Borel set $B$ with $0<\mu(B)<1$. Put $W\defeq\mathbf 1_B-\mu(B)$. It is invariant under all translations, so $W\in L^2_{j,+}$ and $\Theta_jW=W$ by Lemma~\ref{ms:x:invloc}(i)--(ii). The vector $[W]$ is fixed by every $T_j(t)$ and $V_j(y)$, since $[U(a)W]=[W]$; it is orthogonal to $\Omega_j$, since $\ip{\Omega_j}{[W]}_{\cH_j}=\E_\mu W=0$; and $\norm{[W]}^2_{\cH_j}=\E[\overline{\Theta_jW}\,W]=\E\abs W^2=\mu(B)(1-\mu(B))>0$. So the joint fixed space is larger than $\C\Omega_j$.
\end{proof}

\subsubsection{Clustering}\label{ms:x:clustersec}

\begin{definition}[forms of clustering]\label{ms:ladder}
Let $\mu$ satisfy (H1). We say that $\mu$ satisfies
(CL1), \emph{clustering in the mean}, if the Ces\`aro averages of $\E_\mu[G'\,U(a)G]$ over the cubes $\Lambda_T$ converge to $\E_\mu[G']\,\E_\mu[G]$ as $T\to\infty$, and
(CL2), \emph{clustering under diagonal separation}, if $\E_\mu[G'\,U(a)G]\to\E_\mu[G']\,\E_\mu[G]$ as $\min_j\abs{a_j}\to\infty$,
in each case for all $G,G'\in L^2(\mu)$.
\end{definition}

A measure satisfies OS4 in the weak form if it is ergodic for the action of the translation group $\R^3$.

Throughout Subsection~\ref{ms:x:clustersec}, $\mu$ satisfies (H1)--(H3), and $\mu=\int\mu_\xi\,d\nu(\xi)$ is the decomposition of Theorem~\ref{ms:erg:decomp}. We say that a statement holds \emph{on almost every ergodic component} if it holds for $\mu_\xi$ for $\nu$-a.e.\ $\xi$.

\begin{theorem}[clustering in the mean]\label{ms:mean}
For all $G,G'\in L^2(\mu)$,
\[
\lim_{T\to\infty}\frac1{\abs{\Lambda_T}}\int_{\Lambda_T}\E_\mu\big[G'\,U(a)G\big]\,da=\E_\mu\big[\E[G'\mid\cI]\,\E[G\mid\cI]\big]=\int_\Xi\E_{\mu_\xi}[G']\,\E_{\mu_\xi}[G]\,d\nu(\xi).
\]
If $\mu$ is ergodic for the $\R^3$-action, the right side is $\E_\mu[G']\,\E_\mu[G]$; in particular \textup{(CL1)} holds on almost every ergodic component. For $\emptyset\ne J\subset\{1,2,3\}$ the first equality holds with $\Lambda_T$ replaced by $\Lambda^J_T$ and $\cI$ by $\cI_J$.
\end{theorem}
\begin{proof}
The map $H\mapsto\E_\mu[G'H]$ is a continuous linear functional on $L^2(\mu)$, and $a\mapsto U(a)G$ is continuous by Lemma~\ref{ms:x:strongcont}; so the mean of $\E_\mu[G'\,U(a)G]$ over $\Lambda^J_T$ equals $\E_\mu[G'\,A^J_TG]$, where $A^J_TG\defeq\abs{\Lambda^J_T}^{-1}\int_{\Lambda^J_T}U(a)G\,da$. By Lemma~\ref{ms:x:proj}(i)--(ii), $A^J_TG\to P_JG=\E[G\mid\cI_J]$ in $L^2(\mu)$, so the mean tends to $\E_\mu[G'\,\E[G\mid\cI_J]]=\E_\mu[\E[G'\mid\cI_J]\,\E[G\mid\cI_J]]$, the last because $\E[G\mid\cI_J]$ is square-integrable and $\cI_J$-measurable. For $J=\{1,2,3\}$, the function $H\defeq\E[G'\mid\cI]\,\E[G\mid\cI]$ is integrable and, by Theorem~\ref{ms:erg:decomp}(a), $H(\xi)=\E_{\mu_\xi}[G']\,\E_{\mu_\xi}[G]$ for $\nu$-a.e.\ $\xi$; since $\nu=\mu$, $\E_\mu H=\int\E_{\mu_\xi}[G']\,\E_{\mu_\xi}[G]\,d\nu(\xi)$. If $\mu$ is ergodic, $\cI$ is trivial and $\E[G\mid\cI]=\E_\mu G$. Almost every ergodic component satisfies (H1)--(H3) (Theorem~\ref{ms:erg:decomp}), so the statement applies to it with $G,G'\in L^2(\mu_\xi)$.
\end{proof}

The next lemma is the step in which reflection positivity enters the clustering: through the reconstruction, the part of a localised function orthogonal to the $e_\ell$-invariant functions decays under the semigroup $e^{-tH_\ell}$.

\begin{lemma}[decay of the non-invariant part along one axis]\label{ms:x:oneterm}
Let $\ell\in\{1,2,3\}$, $R>0$, $G'\in L^2(\Sigma(H^-_{\ell,0}),\mu)$, and $\tilde G\in L^2(\Sigma(D),\mu)$ for an open set $D\subset H^+_{\ell,-R}$. Then
\[
\E_\mu\big[G'\,U(a)(1-P_\ell)\tilde G\big]\to0\qquad\text{as }a_\ell\to+\infty,\ \text{uniformly in }a_\perp .
\]
\end{lemma}
\begin{proof}
Put $s\defeq2R$ and $Y\defeq U(se_\ell)\tilde G$. By (H2), $Y$ is $\Sigma(D+se_\ell)$-measurable, and $D+se_\ell\subset H^+_{\ell,R}\subset H^+_{\ell,0}$; so $Y\in L^2_{\ell,+}$. Also $P_\ell Y\in L^2_{\ell,+}$ by Lemma~\ref{ms:x:invloc}(i). Since $P_\ell$ commutes with the translations (Lemma~\ref{ms:x:proj}(ii)), for $a_\ell\ge s$,
\[
U(a)(1-P_\ell)\tilde G=U\big((a_\ell-s)e_\ell+a_\perp\big)(1-P_\ell)Y ,
\]
and the right side lies in $L^2_{\ell,+}$ by Proposition~\ref{ms:recon}(ii). By Proposition~\ref{ms:recon}(ii)--(iii), with $F\defeq\overline{\Theta_\ell G'}\in L^2_{\ell,+}$, and by Proposition~\ref{ms:fixed}(i), which gives $[(1-P_\ell)Y]=(1-P_{\ker H_\ell})[Y]$,
\[
\E_\mu\big[G'\,U(a)(1-P_\ell)\tilde G\big]=\Big\langle[F],\ V_\ell(a_\perp)\,e^{-(a_\ell-s)H_\ell}\,(1-P_{\ker H_\ell})[Y]\Big\rangle_{\cH_\ell}.
\]
Since $V_\ell(a_\perp)$ is unitary, the modulus is at most $\norm{[F]}\,\norm{e^{-(a_\ell-s)H_\ell}(1-P_{\ker H_\ell})[Y]}$, which does not depend on $a_\perp$ and tends to $0$ as $a_\ell\to\infty$ by the spectral theorem and dominated convergence ($e^{-t\lambda}\mathbf 1_{\{\lambda>0\}}\to0$ pointwise and boundedly on $[0,\infty)$).
\end{proof}

\begin{theorem}[clustering under diagonal separation]\label{ms:diag}
For all $G,G'\in L^2(\mu)$,
\[
\lim_{\min_j\abs{a_j}\to\infty}\E_\mu\big[G'\,U(a)G\big]=\E_\mu\big[\E[G'\mid\cI]\,\E[G\mid\cI]\big].
\]
If $\mu$ is ergodic for the $\R^3$-action, the limit is $\E_\mu[G']\,\E_\mu[G]$; in particular \textup{(CL2)} holds on almost every ergodic component.
\end{theorem}
\begin{proof}
\emph{Step 1 (reductions).} Since
\[
\abs{\E[G'U(a)G]-\E[G'_nU(a)G_n]}\le\norm{G'-G'_n}_2\norm G_2+\norm{G'_n}_2\norm{G-G_n}_2
\]
uniformly in $a$, and the right side of the claim is continuous in $(G,G')$ in the same way, it suffices to take $G\in L^2(\Sigma(K))$ and $G'\in L^2(\Sigma(K'))$ with $K,K'$ bounded (Subsection~\ref{ms:x:reconsec}). A sequence $a^{(n)}$ with $\min_j\abs{a^{(n)}_j}\to\infty$ splits into finitely many subsequences on each of which the signs $\sigma_j=\operatorname{sgn}a^{(n)}_j$ are fixed, and it suffices to prove the claim along each. On such a subsequence, replace $(G',G,a)$ by $(\Theta_jG',\Theta_jG,\rho_ja)$ for each $j$ with $\sigma_j=-1$. This leaves the left side unchanged, since $\E[\Theta_jG'\,U(\rho_ja)\Theta_jG]=\E[\Theta_j(G'\,U(a)G)]=\E[G'\,U(a)G]$ by \eqref{ms:eqcommute} and the $\theta_j$-invariance of $\mu$; it leaves the right side unchanged, since $\E[\Theta_jG\mid\cI]=\Theta_j\E[G\mid\cI]$ by Lemma~\ref{ms:x:proj}(ii); and it replaces $K,K'$ by the bounded sets $\rho_jK,\rho_jK'$ (H2). So we may assume $a_j\to+\infty$ for every $j$. By translation invariance and multiplicativity, $\E[G'\,U(a)G]=\E[(U(b)G')\,U(a)(U(b)G)]$ for every $b\in\R^3$. If $K\cup K'\subset(-r,r)^3$, the choice $b=-(r+1)(e_1+e_2+e_3)$ makes $U(b)G'$ measurable for $\Sigma(K'+b)$ with $K'+b$ in the open negative octant $\bigcap_jH^-_{j,0}$, and $U(b)G$ measurable for $\Sigma(K+b)$ with $K+b\subset\{x:\abs{x_j}<R\ \forall j\}$, $R\defeq2r+1$. Renaming, we may assume $K'\subset\bigcap_jH^-_{j,0}$ and $K\subset\{\abs{x_j}<R\ \forall j\}$.

\emph{Step 2 (decomposition).} By Lemma~\ref{ms:x:proj}(ii), $P_1P_2P_3=P_{\{1,2,3\}}$, so
\[
1=P_{\{1,2,3\}}+(1-P_1)+P_1(1-P_2)+P_1P_2(1-P_3).
\]
The first term contributes $\E[G'\,U(a)\E[G\mid\cI]]=\E[G'\,\E[G\mid\cI]]=\E[\E[G'\mid\cI]\,\E[G\mid\cI]]$ for every $a$, since $\E[G\mid\cI]$ is translation invariant. It remains to show that each of the other three terms tends to $0$ as $a_\ell\to+\infty$, uniformly in the other coordinates of $a$, where $\ell$ is the index of the factor $(1-P_\ell)$.

\emph{Step 3 (one term).} Fix $\ell$, let $J\defeq\{1,\dots,\ell-1\}$ and $\tilde G\defeq P_JG$ ($\tilde G=G$ for $\ell=1$). Since $P_J$ and $P_\ell$ commute, the term is $\E[G'\,U(a)(1-P_\ell)\tilde G]$. By Lemma~\ref{ms:x:invloc}(iii), $\tilde G\in L^2(\Sigma(K+V_J))$, and $K+V_J\subset H^+_{\ell,-R}$ because $e_\ell\notin V_J$ and $K\subset\{\abs{x_\ell}<R\}$. Since $G'\in L^2(\Sigma(H^-_{\ell,0}))$, Lemma~\ref{ms:x:oneterm} with $D=K+V_J$ gives the claim.

Finally, if $\mu$ is ergodic, $\E[G\mid\cI]=\E_\mu G$; and almost every ergodic component satisfies (H1)--(H3) (Theorem~\ref{ms:erg:decomp}), so the statement applies to it with $G,G'\in L^2(\mu_\xi)$.
\end{proof}

\begin{corollary}[the ergodic components]\label{ms:components}
For $\nu$-a.e.\ $\xi$:
\begin{enumerate}
\item[(i)] $\mu_\xi$ is translation invariant and ergodic for the $\R^3$-action, invariant under $\theta_1,\theta_2,\theta_3$, reflection positive about every coordinate plane in the sense of \textup{(H3)}, and satisfies \textup{(H2)}; so it satisfies \textup{(H1)--(H3)}, and \OSf\ in the weak form;
\item[(ii)] \textup{(CL1)} and \textup{(CL2)} hold for $\mu_\xi$: the Ces\`aro means over $\Lambda_T$ of $\E_{\mu_\xi}[G'\,U(a)G]$, and $\E_{\mu_\xi}[G'\,U(a)G]$ itself as $\min_j\abs{a_j}\to\infty$, converge to $\E_{\mu_\xi}[G']\,\E_{\mu_\xi}[G]$, for all $G,G'\in L^2(\mu_\xi)$;
\item[(iii)] in each coordinate direction $j$, the reconstruction of Proposition~\ref{ms:recon} from $\mu_\xi$ has a unique joint vacuum: the joint fixed space of the $e^{-tH_j}$, $t\ge0$, and of the transverse translations $V_j(y)$, $y\perp e_j$, is $\C\Omega_j$.
\end{enumerate}
\end{corollary}
\begin{proof}
(i) is Theorem~\ref{ms:erg:decomp}(b)--(e). (ii) is Theorems~\ref{ms:mean} and~\ref{ms:diag} for the ergodic measure $\mu_\xi$, and (iii) is Proposition~\ref{ms:fixed}(iii) for it.
\end{proof}


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\end{document}