%
%

\providecommand{\bGM}{\beta_{\mathrm{GM}}}
\providecommand{\mt}{\tilde m}
\providecommand{\MY}{M_Y}
\providecommand{\lPV}{\lambda_{\mathrm{PV}}}
\providecommand{\Lo}{\Lambda^{\circ}}
\providecommand{\Kc}{\mathfrak{K}}
\providecommand{\Df}{\mathfrak{D}}
\providecommand{\Ff}{\mathfrak{F}}
\providecommand{\Xh}{\mathbb{X}}
\providecommand{\EG}{E_{G'}}
\providecommand{\AG}{A_{G'}}
\providecommand{\Floc}{F_1^{\mathrm{loc}}}
\providecommand{\UU}{\mathcal{U}}
\providecommand{\sF}{\sigma_F}
\providecommand{\sV}{\sigma_V}
\providecommand{\Funit}{F_{\mathrm{unit}}}
\providecommand{\Kcol}{K_{\mathrm{col}}}
\providecommand{\Rf}{R_f}
\providecommand{\good}{\operatorname{good}}
\providecommand{\sat}{\operatorname{sat}}
\providecommand{\supp}{\operatorname{supp}}
\providecommand{\lk}{\operatorname{lk}}
\providecommand{\dGp}{d_{G'}}
\providecommand{\one}{\mathbf{1}}
\providecommand{\Z}{\mathbb{Z}}
\providecommand{\R}{\mathbb{R}}
\providecommand{\Glong}{\Gamma_{\mathrm{long}}}
\providecommand{\Nlong}{\mathcal N^{\mathrm{long}}}

\section{Stage 2: the single-scale expansion and the deterministic area law}
\label{sec:stage2}\label{sec:7}

In this section we prove Theorem~\ref{thm:7.61}: for each output $G'=G'(D)$ of the
construction and each rectangle $C$ with $\min(R,T)\ge R_1(\beta)$,
$W_{G'}(C)\le\exp(-\sV[\AG(C)-\eta_SRT]+c_N(R+T))$. We carry out the
single-scale analysis of \cite[Section~6, Appendices~C to E]{GM82}, with the lemmas
of \cite[Section~9]{BF80} that it uses, on $G'$. The structural changes are the following. First, the outer
partition functions are graded by the class of their walls (their periods around
the handles of the insulating set), and the Kirkwood\textendash Salsburg system of
\cite[Appendix~4]{BF80} is solved for all classes at once in a weighted $\ell^1$ algebra.
Second, the handles enter the bounds in a single place: the area weight of a class
comes from one bound at fixed class (Proposition~\ref{prop:7.57}).
In Subsections~\ref{ssec:7.1} to \ref{ssec:7.5} we derive the class-graded polymer expansion
(Corollary~\ref{cor:7.30}), and in Subsections~\ref{ssec:7.6} to \ref{ssec:7.9} we bound it. Sections~\ref{S4-S:S4.4} to~\ref{S4-S:S4.8} of Part~S4 write out the proofs of this section and of Appendices~\ref{app:A} to~\ref{app:C} in full, including the steps taken from \cite{GM82} and \cite{BF80}. Section~\ref{S4-S:S4.5} contains the proofs of Propositions~\ref{prop:7.3}, \ref{prop:7.4}, \ref{prop:7.8} and~\ref{prop:7.25}, and Section~\ref{S4-S:S4.4} those of Propositions~\ref{prop:7.5} (with explicit constants), \ref{prop:7.9} and~\ref{prop:7.22}, Lemmas~\ref{lem:7.6}, \ref{lem:7.12}, \ref{lem:7.13}, \ref{lem:7.18}, \ref{lem:7.24}, \ref{lem:7.26} and~\ref{lem:7.wall}, Theorems~\ref{thm:7.19} and~\ref{thm:7.29} and Corollary~\ref{cor:7.21}.

\setcounter{subsection}{-1}
\subsection{Standing conventions}
\label{ssec:7.0}\label{sec:7.0}

We use the following conventions in Section~\ref{sec:stage2} and Appendices~\ref{app:A} to \ref{app:C}. The \emph{cells}
of Definition~\ref{def:3.12} may be disconnected. Interfaces, walls and tubes are defined on
cells, and polymers, seeds and the Kirkwood\textendash Salsburg system on the $\UU$-cells
of Theorem~\ref{thm:3.20}\,(R7). By $|X|$ we mean the number of $\UU$-cells of $X$, not a Debye
volume, and the constants of Appendix~\ref{app:B} are per $\UU$-cell. We work in fine lattice units,
$a=1$. Sums over sites and bonds are plain sums, each bond counted once.
The collar scale $L'$ is fixed by $\mt L'=\Kcol\bGM$, $\Kcol=2$, where $\bGM=4\pi^2\tau$. The min-cut
area $\AG(C)$ of Definition~\ref{def:2.11} (a support count) and the forced
wall content $A(p)$ of Lemma~\ref{lem:7.46} (an $\ell^1$ quantity) are related by
Theorem~\ref{thm:7.17} and Lemma~\ref{lem:7.59}. Constants $C$, $c$ are absolute or depend
only on the constants fixed before $K$ (Section~\ref{sec:8}), and change from line to
line. We write $\langle\cdot,\cdot\rangle$ and $\|\cdot\|$ for the $\ell^2$ inner product and norm on $V$ or $E$, and
$\|\varphi\|_Y^2:=\sum_{x\in Y}\varphi(x)^2$.

\subsection{Representation}
\label{ssec:7.1}\label{sec:7.1}

Throughout, $G'=(V,E)$ is the fine slit graph of Subsection~\ref{ssec:2.5} for the set
\eqref{eq:2.FF}, with set of components $\mathrm{Comp}(G')\ni\Kc$. From
Subsection~\ref{ssec:7.2} on, $G'$ will be an output of the construction.

\begin{definition}[Operators and the loop on $G'$]\label{def:7.1}
(a) For an oriented bond $b=(x,y)\in E$ and $\varphi:V\to\R$, $\nabla\varphi(b):=\varphi(y)-\varphi(x)$, and
$\Delta:=-\nabla^*\nabla$ is the Neumann Laplacian of $G'$. (b) On $\R^V$,
$u^{-1}:=\MY^{-2}(-\Delta)(-\Delta+\MY^2)$ and $C_0:=(u^{-1}+\mt^2)^{-1}=F(-\Delta)$, $F(t):=(\MY^{-2}t^2+t+\mt^2)^{-1}$.
(c) For an integer $2$-chain $\Xi$ in $\Lo$ with $\partial\Xi=C$ and $b=(x,y)\in E$, $\xi_b$ is the
coefficient in $\Xi$ of the plaquette $p_b$ dual to $b$, oriented as $\partial c_y$ ($c_y$ the
cube of $y$), and $j(b):=2\pi\xi_b$ (the opposite orientation of Subsection~\ref{ssec:2.5}
amounts to $\Xi\mapsto-\Xi$ and leaves \eqref{eq:2.ZG} unchanged, $m\mapsto-m$). With $n=2\pi m$,
\eqref{eq:2.ZG} reads $Z(k,\Xi):=Z_{G'}(k,\Xi)=\sum_{[n]}\exp(-\frac1{2\bGM}\sum_{b\in E}(\nabla n(b)-kj(b))^2)$, $[n]$ running over $(2\pi\Z)^V$ modulo functions constant on each component;
$W_{G'}(C)=Z(1,\Xi)/Z(0,\Xi)$ does not depend on $\Xi$ (Proposition~\ref{prop:2.10}). (d) The
Neumann solid angle $f=f_{G'}$ solves $-\Delta f=\nabla^*j$; $\Rf:=\nabla f-j$ and $\EG:=\|\Rf\|^2$.
\end{definition}

The operator $C_0$ is positive definite, since $u^{-1}\ge0$. As $C_0$ and $u^{-1}$ are both functions
of $-\Delta$ and $\Delta\one_\Kc=0$, we have
\begin{equation}\label{eq:7.F2}
C_0\one_\Kc=\mt^{-2}\one_\Kc\quad(\Kc\in\mathrm{Comp}(G')),\qquad
\mt^2C_0-1=-C_0u^{-1}=-u^{-1}C_0 .
\end{equation}
Since both ends of each bond lie in one component, $\nabla^*j\perp\ker\Delta$. Hence $f$ exists, unique
up to one constant per component, and $\Rf$, $\EG$ do not depend on the choice.

\begin{lemma}\label{lem:7.2}\label{prop:7.15}
Let $\Pi$ be the orthogonal projection of $\ell^2(E)$ onto $\operatorname{ran}\nabla$. Then
$\nabla f=\Pi j$ and $\Rf=-(1-\Pi)j$; hence $\nabla^*\Rf=0$, $\langle\nabla\varphi,\Rf\rangle=0$ for every
$\varphi:V\to\R$, $\langle j,\Rf\rangle=-\EG$, and $\EG=\min_\varphi\|\nabla\varphi-j\|^2$. For every $n:V\to\R$ and
$k\in\Z$, $\sum_{E}(\nabla n-kj)^2=\sum_{E}(\nabla(n-kf))^2+k^2\EG$, so
$Z(k,\Xi)=e^{-k^2\EG/2\bGM}\,\tilde Z(k,\Xi)$ with
$\tilde Z(k,\Xi):=\sum_{[n]}\exp\bigl(-\frac1{2\bGM}\sum_E(\nabla(n-kf))^2\bigr)$.
\end{lemma}

\begin{proof}
The solid angle is the gradient part of the loop current, since $f$ solves the
Euler\textendash Lagrange equation $\nabla^*(\nabla\varphi-j)=0$ of $\|\nabla\varphi-j\|^2$. Hence
$\nabla f=\Pi j$ and $\Rf\in\operatorname{ran}(1-\Pi)=\ker\nabla^*$, and the identities follow. For the
last one we write $\nabla n-kj=\nabla(n-kf)+k\Rf$ and use $\Rf\perp\operatorname{ran}\nabla$.
\end{proof}

\paragraph{The input from Stage 1.}
We use (S1) to (S5) of Theorem~\ref{thm:6.5}, with $\xi=(m,x)$ and $\int d\xi$ as there, $u$
and $d\mu_u$ of Proposition~\ref{prop:6.3} (on $\R^V_0$, where $u$ inverts $u^{-1}$), and $P:=2\pi\bGM^{-1/2}$. By \eqref{eq:6.S1} with
$\theta_\Kc=\bGM^{1/2}c_\Kc$ there is $\mathcal N_0>0$, independent of $k$ and $\Xi$, such that
\begin{equation}\label{eq:7.S1}
\tilde Z(k,\Xi)=\mathcal N_0\int_{[0,P)^{\mathrm{Comp}(G')}}\prod_\Kc\frac{dc_\Kc}{P}
\int d\mu_u(\varphi)\,e^{-V_{\mathrm{eff}}(\varphi+c\mid k\Xi)},\qquad
c:=\sum_\Kc c_\Kc\one_\Kc,
\end{equation}
\begin{equation}\label{eq:7.Veff}
-V_{\mathrm{eff}}(\varphi\mid k\Xi)=\rho_0+\sum_{s\ge1}\frac1{s!}\int d\xi_1\cdots d\xi_s\,
\rho_s(\xi_1,\dots,\xi_s)\,\varepsilon_k(\xi_1)\cdots\varepsilon_k(\xi_s),
\end{equation}
with $\varepsilon_k(\xi):=e^{im(\bGM^{1/2}\varphi(x)+kf(x))}-1$ and the real coefficients $\rho_s=\rho_s^{G'}$ of
(S1), and the series converges absolutely, uniformly in real $\varphi$. By (S3),
$V_{\mathrm{eff}}(\varphi+PI\mid k\Xi)=V_{\mathrm{eff}}(\varphi\mid k\Xi)$ for $I\in\Z^V$. On outputs the dipole kernel $\nu$
of (S5) has $\|\nu\|_{r_{\mathrm{cov}}}\le\varepsilon_\nu\mt^2$, with $2K_3\varepsilon_\nu\le1$. The space $\R^V_0$, the norms $\|\cdot\|_r$, the
image energy $\Phi=\Phi_{G'}\ge0$ and the depleted set $\Df$ are those of the conventions of
Section~\ref{sec:stage1}. The quantities $\rho^{\mathrm{ref}}$, $\mt$ and $\MY=\mt/\lPV$ are as in \eqref{eq:6.mass}, and (S4$'$)
denotes Proposition~\ref{prop:6.6} together with \eqref{eq:6.S4c} for $\rho^{\mathrm{ref}}$ in place of
$\rho_1(\cdot,x)$ (proved in Section~\ref{S4-S:S4.3}).

\begin{proposition}[{\cite[(6.1) to (6.4)]{GM82} on $G'$}]\label{prop:7.3}
Assume (S1) to (S3). (a) $Z(k,\Xi)=e^{-k^2\EG/2\bGM}\tilde Z(k,\Xi)$, with $\tilde Z$ given by
\eqref{eq:7.S1}: one zero-mode variable $c_\Kc\in[0,P)$ per component of $G'$. (b) The
integrand of \eqref{eq:7.S1} is $P$-periodic in each $c_\Kc$; hence (a) does not change
when $f$ is changed by a constant on each component. (c) $\rho_0$ cancels in $W_{G'}(C)$.
\end{proposition}

Lemma~\ref{lem:7.2} and \eqref{eq:7.S1} give part (a) as in \cite[(6.1) to (6.4)]{GM82}, except that there is
one zero mode per component (GM's Gaussian measure is defined on $(a\Z)^3$ and has
none), the loop entering only through the phases $\varepsilon_k$, and that
$e^{-k^2\EG/2\bGM}$ replaces $e^{-k^2(J,v_{Cb}J)/2\beta}$. Part (b) is (S3) with $I=\one_\Kc$. Part (c) holds as in
\cite[(6.5)]{GM82}, because $\rho_0$ gives the same factor in $Z(1,\Xi)$ and $Z(0,\Xi)$. By (c), we set $\rho_0=0$ from now on.

\paragraph{Cells, cell averages and block spins.}
We write $-V_{\mathrm{eff}}(\varphi\mid k\Xi)=\int d\xi\,\rho_1(\xi)\varepsilon_k(\xi)+E'(\varphi)$, where $E'$
collects the terms $s\ge2$ of \eqref{eq:7.Veff}. The cells $\Omega=\Omega_{Q,\hat\Kc}$ of
Definition~\ref{def:3.12} partition $V$, each in one component. For a cell $\Omega$ and
$x\in\Omega$ let $A(x)=A_\Omega$ and $\bar f(x)=\bar f_\Omega$ be the averages of $\varphi$ and $f$ over $\Omega$,
$\delta:=\varphi-A$, $\delta f:=f-\bar f$ and $\tilde\delta:=\delta+k\bGM^{-1/2}\delta f$. Then $\sum_{x\in\Omega}\tilde\delta(x)=0$ for
each cell, and $\bGM^{1/2}\varphi+kf=(\bGM^{1/2}A+k\bar f)+\bGM^{1/2}\tilde\delta$. The block spins are the
elements of $\mathcal H:=\{h:V\to P\Z\text{ constant on every cell}\}$.

\begin{proposition}\label{prop:7.4}
Assume (S3) and $\sum_m m^2|\rho^{\mathrm{ref}}(m)|<\infty$. (a) For every real $\varphi$ on $V$,
$e^{\int d\xi\,\rho_1(\xi)\varepsilon_k(\xi)}=\Theta(\varphi)\,e^{G(\varphi)}$ with
$\Theta(\varphi):=\sum_{h\in\mathcal H}e^{-\frac12\mt^2\|\varphi+k\bGM^{-1/2}f-h\|^2}$, $G=G_1+G_2$, $e^{G_1}=\prod_\Omega r_\Omega(A_\Omega)$, $G_2=G_{2,1}+G_{2,2}+G_{2,3}$,
\begin{equation}\label{eq:7.rOmega}
r_\Omega(A):=\frac{\exp\bigl\{|\Omega|\sum_m\rho^{\mathrm{ref}}(m)\bigl[e^{im(\bGM^{1/2}A+k\bar f_\Omega)}-1\bigr]\bigr\}}
{\sum_{n\in\Z}\exp\bigl\{-\frac12\mt^2|\Omega|\,(A+k\bGM^{-1/2}\bar f_\Omega-nP)^2\bigr\}},
\end{equation}
\begin{align}
G_{2,1}&:=\int d\xi\,\rho^{\mathrm{ref}}(m)\bigl[e^{im\bGM^{1/2}\tilde\delta(x)}-1-im\bGM^{1/2}\tilde\delta(x)+\tfrac12m^2\bGM\tilde\delta(x)^2\bigr],\notag\\
G_{2,2}&:=\int d\xi\,\rho^{\mathrm{ref}}(m)\bigl[e^{im(\bGM^{1/2}A(x)+k\bar f(x))}-1\bigr]\bigl[e^{im\bGM^{1/2}\tilde\delta(x)}-1-im\bGM^{1/2}\tilde\delta(x)\bigr],\label{eq:7.G2}\\
G_{2,3}&:=\int d\xi\,\bigl(\rho_1(m,x)-\rho^{\mathrm{ref}}(m)\bigr)\bigl[e^{im(\bGM^{1/2}\varphi(x)+kf(x))}-1\bigr].\notag
\end{align}
$\Theta$, $r_\Omega$, $G_1$ and the $G_{2,i}$ are real, $\Theta>0$, $r_\Omega>0$, and $e^G$ does not
depend on $h$. The identity holds for every partition of $V$ in place of the cells.
(b) Under (S1) to (S3),
\begin{equation}\label{eq:7.GM612}
\tilde Z(k,\Xi)=\mathcal N_0\sum_{h\in\mathcal H}\int\prod_\Kc\frac{dc_\Kc}{P}\int
d\mu_u(\varphi)\,e^{-\frac12\mt^2\|\varphi+c+k\bGM^{-1/2}f-h\|^2}\,e^{G(\varphi+c)}\,e^{E'(\varphi+c)}.
\end{equation}
\end{proposition}

We obtain part (a) as in \cite[(6.6), (6.8) to (6.11)]{GM82} and part (b) as in \cite[(6.12)]{GM82}, with
cells in place of blocks, $|\Omega|$ in place of $L^3$ and $\rho^{\mathrm{ref}}(m)$ in place of
$\rho_1(m,0)$ in \cite[(6.7), (6.9), (6.10)]{GM82}, since on $G'$ the one-body coefficient
depends on the site. The difference is the term $G_{2,3}$. Also, (6.12) needs a sign
correction. In (b) we move the sum over $h$ out of the integrals by Tonelli's theorem, as the
moduli of the $h$-terms sum to $e^{\int\rho_1\varepsilon_k}|e^{E'}|$.

\paragraph{The deficit term.}
Since $e^{im\bGM^{1/2}h}=1$ for $h\in\mathcal H$, (S3) for $\rho_1$ and $\rho^{\mathrm{ref}}$ gives, for each
$h\in\mathcal H$ and with $u_h:=\bGM^{1/2}(\varphi+k\bGM^{-1/2}f-h)$,
$G_{2,3}=\sum_m\sum_{x\in V}(\rho_1(m,x)-\rho^{\mathrm{ref}}(m))(\cos(m\,u_h(x))-1)$, and $G_{2,3}(Y)$ is the same
sum over $x\in Y$. For a union $Y$ of $\UU$-cells, $N_\Df(Y)$ denotes the number of its $\UU$-cells that
meet the depleted set $\Df=\{\Phi>\theta_0\}$, $\theta_0=0.01$.

\begin{proposition}\label{prop:7.5}
Assume (S3) and (S4$'$), and put
\begin{equation}\label{eq:7.thetaprime}
\theta':=1-e^{-\theta_0}+C\bigl(e^{-\mu}+e^{-2\kappa_S}\bigr).
\end{equation}
Uniformly in real $\varphi$, in $h\in\mathcal H$ and in $G'$: (a)
$|G_{2,3}(V\setminus\Df)|\le\theta'\cdot\frac12\mt^2\|\varphi+k\bGM^{-1/2}f-h\|^2_{V\setminus\Df}$; (b) for every $Y\subseteq V$,
$-C(e^{-\mu}+e^{-2\kappa_S})\frac{\mt^2}{\bGM}\#(\Df\cap Y)\le G_{2,3}(\Df\cap Y)\le C\frac{\mt^2}{\bGM}\#(\Df\cap Y)$; (c) if $G'$ is an
output of the construction, then $e^{-C\cdep N_\Df(Y)}\le e^{G_{2,3}(\Df\cap Y)}\le e^{C\cdep N_\Df(Y)}$ for every
union $Y$ of $\UU$-cells, with $\cdep$ the constant of Proposition~\ref{prop:6.18}.
\end{proposition}

\begin{proof}[Sketch]
Off $\Df$, $\Phi\le\theta_0$ and (S4$'$) give
$\sum_m|\rho_1(m,x)-\rho^{\mathrm{ref}}(m)|m^2\le\theta'\mt^2/\bGM$ (Section~\ref{S:constF}), and
$|\cos(mu)-1|\le\frac12m^2u^2$ gives (a). On $\Df$ the deviation is at most the density,
and its negative part is at most $Ce^{-\mu}\rho^{\mathrm{ref}}(1)$ plus the tails, as $e^{-\Phi}\le1$. This proves (b).
Part (c) follows from (b) and Proposition~\ref{prop:6.18}, which gives $(\mt^2/\bGM)\#(\Df\cap\Delta_\UU)\le\cdep$ for each $\UU$-cell $\Delta_\UU$.
\end{proof}

In Lemma~\ref{lem:7.31} we will move the quadratic part of $G_{2,3}(V\setminus\Df)$ into the
covariance. Throughout Section~\ref{sec:stage2} we assume the \emph{standing
hypothesis}
\begin{equation}\label{eq:7.standing}
\varepsilon_\nu+\theta'\le\tfrac17 ,
\end{equation}
which, by Lemma~\ref{lem:6.19} and Section~\ref{sec:8}, holds for $\theta_0=0.01$, $\mu$ and $\kappa_S$ large, and $\bGM\ge\bGMo$.

\paragraph{Walls.}
For $h\in\mathcal H$ we put $n:=(\bGM^{1/2}/2\pi)h\in\Z^V$, define the \emph{wall field}
$w:=\nabla h-k\bGM^{-1/2}j$ on $E$, and put $N:=(\bGM^{1/2}/2\pi)w$, that is,
$N(b)=n(y)-n(x)-k\xi_b\in\Z$ for $b=(x,y)$. The wall bonds are $\Sigma_E:=\{b\in E:N(b)\ne0\}$, the
wall sites $\Sigma_V$ are their endpoints, and we put $N^\vee:=\sum_{b\in E}N(b)p_b$, with $p_b$ oriented as in
Definition~\ref{def:7.1}\,(c). A wall is where the cell-constant field fails to absorb the jump of the loop
current, which is constant on interfaces because they are aligned with $\Xi$.

\begin{lemma}[Walls]\label{lem:7.6}\label{lem:7.7}
Let $h\in\mathcal H$ and $k\in\Z$. (a) $N(b)\ne0$ only on interface bonds, and $N$ is
constant on each interface $I$; we write $N_I$. (b) Frozen bonds, bonds into $\Xh$ and
bonds leaving $\Lo$ are not in $E$; they carry no wall. (c) $\partial N^\vee=-kC+\partial Y$ for an
integer $2$-chain $Y$ supported in $\Ff=F(\Xh)\cup F_S\cup\partial_2\Lo$.
(d) For every cycle $\gamma$ of $G'$, $\sum_{b\in\gamma}N(b)=-k\,\lk(\gamma,C)$. Moreover $N(h)=N(h')$ if and
only if $h-h'$ is constant on every component of $G'$; so $N$ determines the class $[h]$
of $h$ modulo $\mathcal C_P:=\{\sum_\Kc Pn_\Kc\one_\Kc:n_\Kc\in\Z\}$. (e) \emph{(Relatively closed walls.)} For an
integer field $M:E\to\Z$ with $M^\vee:=\sum_{b\in E}M(b)p_b$, the following are equivalent: (i) $M$
is relatively closed: $\partial M^\vee=\partial Y$ for an integer $2$-chain $Y$ supported in $\Ff$; (ii) $M$
has zero periods: $\sum_{b\in\gamma}M(b)=0$ for every cycle $\gamma$ of $G'$; (iii) $M=\nabla n'$ for some
$n':V\to\Z$.
\end{lemma}

\begin{proof}
(a) We have $\nabla h=0$ on bonds inside a cell, which do not cross $\Xi$ by Lemma~\ref{lem:3.5}\,(A1).
Other bonds join two blocks, and by Theorem~\ref{thm:3.20}\,(R7) an interface lies in one block face, which lies in
$\Xi$ or misses it. (b) is the definition of $E$. (c) Extend $n$
by $0$ to $\Xh$ and let $U:=\sum_{x\in V}n(x)c_x$. A plaquette of a cube of $V$ not dual to a bond of
$E$ is a face of a hole cube, a frozen face or in $\partial_2\Lo$, so $\partial U=\sum_{b\in E}\nabla n(b)p_b+Y_1$
with $\supp Y_1\subseteq\Ff$. Likewise, $\sum_{b\in E}\xi_bp_b=\Xi-Y_2$, with $Y_2$ the part of $\Xi$ in $\Ff$, since $\Xi$ is
interior to $\Lo$ by Lemma~\ref{lem:3.5} and $F(\Xh)$ contains all faces of hole cubes. So
$N^\vee=\partial U-Y_1-k\Xi+kY_2$ and $\partial N^\vee=-kC+\partial(kY_2-Y_1)$. Thus the wall spans $C$ relative to the insulating set and may end on faces of hole cubes, on frozen faces and on $\partial\Lo$. (d) $\sum_{b\in\gamma}\nabla n(b)=0$, and
$\sum_{b\in\gamma}\xi_b$ counts the crossings of $\gamma$ with $\Xi$, that is, $\lk(\gamma,C)$. If $N(h)=N(h')$,
then $\nabla(n-n')=0$. (e) Since the box is acyclic, the relative cycle $M^\vee-Y$ is a boundary
$\partial W$, which a cycle of $G'$ crosses as often inwards as outwards. As plaquettes dual to
bonds of $E$ are not in $\Ff$, this proves (i)$\Rightarrow$(ii). Integrating along paths, we obtain (ii)$\Leftrightarrow$(iii),
and the computation of (c) with $k=0$ gives (iii)$\Rightarrow$(i).
\end{proof}

\paragraph{Handles.} In $\Z^3$ the period condition of Lemma~\ref{lem:7.6}\,(d) is local, and
by Lemma~\ref{lem:7.6}\,(e) walls with zero periods are removed by $h\mapsto h-Pn'$. In $G'$ a cycle $\gamma$
around a handle of the insulating set with $\lk(\gamma,C)=0$ may be crossed once each, with
opposite signs, by two separated wall pieces. We therefore write the shift in the source form of
Proposition~\ref{prop:7.9} and, in Definition~\ref{def:7.27}, grade the outer partition functions
by the class of their walls.

\begin{proposition}\label{prop:7.8}
Assume (S1) to (S3) and (S5). For each $h\in\mathcal H$ let $g=g_h:V\to\R$ be
\emph{equivariant}: $g_{h+c}=g_h+c$ for $c\in\mathcal C_P$. Put $H:=h-k\bGM^{-1/2}f$,
$C^{-1}:=C_0^{-1}+\nu$, $E(\psi):=E'(\psi+g)+\tfrac12\langle\psi,\nu\psi\rangle$, $g_c:=\mt^2C_0H$,
$F_1:=\tfrac12\mt^2\|g-H\|^2+\tfrac12\langle g,u^{-1}g\rangle$ and $F_2:=\langle\psi,C_0^{-1}(g-g_c)\rangle$. Then $C^{-1}\ge\frac12C_0^{-1}>0$, and with $d\mu_C$ the centred Gaussian measure
on $\R^V$ with covariance $C$,
\begin{equation}\label{eq:7.translated}
\tilde Z(k,\Xi)=\sum_{[h]\in\mathcal H/\mathcal C_P}\mathcal N\int d\mu_C(\psi)\,
e^{E(\psi)}\,e^{G(\psi+g)}\,e^{-F_1}\,e^{-F_2},
\end{equation}
where $\mathcal N>0$ depends on $G'$, $\MY$, $\mt$ and $\nu$, and not on $k$, $\Xi$ or
$h$. Each term of \eqref{eq:7.translated} does not depend on the representative of $[h]$.
\end{proposition}

The proof follows \cite[(6.13) to (6.20)]{GM82}, with $\chi_\Lambda\equiv1$ on $V$ and $E$ the $E$ of
\cite[(6.14c)]{GM82}, except that the sum over $h$ is unfolded over
$\mathcal H/\mathcal C_P$ and the mass acts at every site. The integrand of \eqref{eq:7.GM612} is
invariant under $(\varphi+c,h)\mapsto(\varphi+c+c',h+c')$, $c'\in\mathcal C_P$, since each cell lies in one
component, $r_\Omega$ is $P$-periodic, and the phases of $E'$ shift by multiples of $2\pi$.
Hence the sum over $c'$ turns the zero-mode integrals into Lebesgue measure on
$\R^{\mathrm{Comp}(G')}$. As $u^{-1}\one_\Kc=0$, the measure becomes
$e^{-\frac12\langle\hat\varphi,u^{-1}\hat\varphi\rangle}d\hat\varphi$ on $\R^V$, and $C_0=(u^{-1}+\mt^2)^{-1}$ is one operator on
$\R^V$, whereas in \cite[(6.15)]{GM82} the mass term is $\mt^2\chi_\Lambda$. By (S5), $C^{-1}\ge\frac12C_0^{-1}$,
and equivariance with \eqref{eq:7.F2} gives the independence of the representative.

\subsection{The shift and the localized $F_1$}
\label{ssec:7.2}\label{sec:7.2}

From now on $G'$ is an output of the construction and $\bGM\ge\bGMo$, so that $L'\ge16$
and $L/2<L'$. We write $d=\dGp$ for the graph distance of $G'$ ($d(x,y)=\infty$ for sites in
different components) and $N_r(A):=\{x\in V:d(x,A)\le r\}$. By Theorem~\ref{thm:3.20}\,(R7), each cell lies in one
$\UU$-cell and each $\UU$-cell lies in one component. A good tube of
an interface bond $b$ (Subsection~\ref{ssec:7.3}) is a straight path of $G'$ of $L$ sites
with middle bond $b$.

\paragraph{The wall neighbourhood.} The set $\hat\Sigma$ is the union of the $\UU$-cells $\Delta$
with $d(\Delta,\Sigma_V)<L'$, as on \cite[p.~47]{GM82} with $\UU$-cells for unit cubes and $d$ for
GM's distance $3^{-1/2}\|x-y\|_{1}$ of \cite[(1.11a), (3.23)]{GM82}. The \emph{wall components} $S_\beta$ are the unions of the
$\UU$-cells of the connected components of the graph on the $\UU$-cells in $\hat\Sigma$,
two being adjacent when a bond of $G'$ joins them. We put $\Sigma_\beta:=\Sigma_V\cap S_\beta$ and $w_\beta:=w\one_{E(S_\beta)}$, where $E(S_\beta)$
is the set of bonds of $E$ with both ends in $S_\beta$. The
\emph{wall sources} and the \emph{standard cutoff} are $s_\beta:=\MY^{-2}(-\Delta+\MY^2)\nabla^*w_\beta$ and
$\chi_\beta:=\zeta(d(\cdot,\Sigma_\beta))$, with $\zeta=1$ on $[0,L'/2]$, $\zeta(t)=3-4t/L'$ on $[L'/2,3L'/4]$
and $\zeta=0$ on $[3L'/4,\infty)$. The transition set $A_\beta$ is the set of endpoints of bonds
$\{x,y\}\in E$ with $\chi_\beta(x)\ne\chi_\beta(y)$. The shift, which near each wall component subtracts the response $C_0s_\beta$ to the wall source of that component, is
\begin{equation}\label{eq:7.shift}
g:=H-\sum_\beta\chi_\beta C_0s_\beta,\qquad D:=g-H=\sum_\beta D_\beta,\quad D_\beta:=-\chi_\beta C_0s_\beta .
\end{equation}

\begin{proposition}\label{prop:7.9}
For every $h\in\mathcal H$ and $k\in\{0,1\}$: (a) every wall bond lies in $E(S_\beta)$ for
exactly one $\beta$, so $w=\sum_\beta w_\beta$ with disjoint supports; $\{x:d(x,\Sigma_\beta)<L'\}\subseteq S_\beta$;
every interface carrying a wall lies in one $E(S_\beta)$, and the good tubes of its bonds
lie in $S_\beta$; (b) $\chi_\beta=0$ off $S_\beta$, $\chi_\beta=1$ on $\Sigma_\beta$, $d(A_\beta,\Sigma_\beta)\ge L'/2-1\ge7$,
$N_1(A_\beta)\subseteq S_\beta$, and $\chi_\beta$ is $(4/L')$-Lipschitz for $d$; (c) $u^{-1}H=\sum_\beta s_\beta$ and
$\supp s_\beta\subseteq N_1(\Sigma_\beta)$; (d) $g=H$ off $\bigcup_\beta\supp\chi_\beta$, in particular $g+k\bGM^{-1/2}f=h$
on $V\setminus\hat\Sigma$; on $S_\beta$, $g=H-\chi_\beta C_0s_\beta$; and $g$ is equivariant; (e) $F_2':=C_0^{-1}(g-g_c)$
satisfies
\begin{equation}\label{eq:7.F2prime}
F_2'=\sum_\beta F'_{2,\beta},\qquad F'_{2,\beta}:=-[C_0^{-1},\chi_\beta]\,C_0s_\beta,\qquad
\supp F'_{2,\beta}\subseteq N_1(A_\beta)\subseteq S_\beta,
\end{equation}
$C_0^{-1}D_\beta+s_\beta=F'_{2,\beta}$ (as $\chi_\beta=1$ on $\supp s_\beta$), and on $N_1(A_\beta)$ one has $s_\beta=0$ and $F_2'=-C_0^{-1}\chi_\beta C_0s_\beta$; (f) given $S_\beta$, the
functions $s_\beta$, $D|_{S_\beta}$ and $F'_{2,\beta}$ depend on $h$ only through $N|_{E(S_\beta)}$.
\end{proposition}

\begin{proof}
Parts (a), (b), (d) and (f) are support checks. A geodesic of length less than $L'$ to $\Sigma_\beta$
stays in $\hat\Sigma$, hence in one wall component, and good tubes lie within $L/2<L'$ of their
middle bond. The function $\zeta$ is $(4/L')$-Lipschitz, which gives the Lipschitz bound of (b). Since $A_\beta\subseteq\{L'/2-1<d(\cdot,\Sigma_\beta)\le3L'/4+1\}$
and $L'\ge16$, we have $N_1(A_\beta)\subseteq\{d(\cdot,\Sigma_\beta)<L'\}\subseteq S_\beta$. Finally, $\nabla c=0$ for $c\in\mathcal C_P$, and
$w_\beta=2\pi\bGM^{-1/2}N$ on $E(S_\beta)$, whose wall bonds have endpoint set $\Sigma_\beta$.

(c) Inserting $-\Delta f=\nabla^*j$, we find $-\Delta H=\nabla^*\nabla h-k\bGM^{-1/2}\nabla^*j=\nabla^*w$, so $u^{-1}H=\MY^{-2}(-\Delta+\MY^2)\nabla^*w=\sum_\beta s_\beta$ by (a).
Roughly speaking, on each bond with
$N=0$ the jumps of $h$ across $\Xi$ cancel $j$, and hence $u^{-1}H$ is a pure wall source. Moreover, $\nabla^*w_\beta$ is supported in $\Sigma_\beta$, and $-\Delta+\MY^2$ enlarges supports by at most one step.

(e) By \eqref{eq:7.F2}, $C_0^{-1}g_c=\mt^2H$ and $C_0^{-1}H-\mt^2H=u^{-1}H$, and by (c) we obtain
$F_2'=\sum_\beta(s_\beta-C_0^{-1}\chi_\beta C_0s_\beta)$. A site $x\in N_1(\Sigma_\beta)\setminus\Sigma_\beta$ is adjacent to some
$z\in\Sigma_\beta$, and $\chi_\beta(x)=1$, since otherwise $z\in A_\beta$, against (b). Thus $\chi_\beta=1$ on
$N_1(\Sigma_\beta)\supseteq\supp s_\beta$, and hence $s_\beta-C_0^{-1}\chi_\beta C_0s_\beta=-[C_0^{-1},\chi_\beta]C_0s_\beta$. The kernel
$C_0^{-1}(x,y)=(\MY^{-2}\Delta^2-\Delta+\mt^2)(x,y)$ vanishes unless $d(x,y)\le2$. Hence
$([C_0^{-1},\chi_\beta]v)(x)=\sum_yC_0^{-1}(x,y)(\chi_\beta(y)-\chi_\beta(x))v(y)$ vanishes unless $\chi_\beta$ changes
along a path of at most two bonds from $x$, which puts $x$ in $N_1(A_\beta)\subseteq S_\beta$. Since
$d(A_\beta,\Sigma_\beta)\ge7\ge3$, the sets $N_1(A_\beta)$ and $N_1(\Sigma_\beta)$ are disjoint, so $s_\beta=0$ on $N_1(A_\beta)$, and
there $F_2'=-C_0^{-1}\chi_\beta C_0s_\beta$.
\end{proof}

The construction above uses no extension of $h|_{S_\beta}$ to $V$. A construction through an extension $h^e_\beta$ of $h|_{S_\beta}$ that is
constant on the components of $V\setminus S_\beta$, as in \cite[(6.24)]{GM82}, is not available
on $G'$: such an extension exists only if $w_\beta$ has zero periods on all cycles of
$G'$, and a wall spanning the central hole of a solid torus of $\Xh$ has period $\pm1$
on a meridian of the torus. When an extension exists, \eqref{eq:7.shift} agrees with
\cite[(6.24), (6.25)]{GM82} up to the choice of cutoff and $C_0$ for $C_{00}$.

\begin{proposition}\label{prop:7.10}
For every $h\in\mathcal H$, $k\in\Z$ and $D:V\to\R$, with $g:=H+D$,
\begin{equation}\label{eq:7.exactF1}
2F_1=\mt^2\|D\|^2+\|w+\nabla D\|^2+\MY^{-2}\|\Delta D-\nabla^*w\|^2-k^2\bGM^{-1}\EG .
\end{equation}
\end{proposition}

\begin{proof}
As $u^{-1}=-\Delta+\MY^{-2}\Delta^2$ and $-\Delta=\nabla^*\nabla$, we have $\langle g,u^{-1}g\rangle=\|\nabla g\|^2+\MY^{-2}\|\Delta g\|^2$.
The idea is that the gradient of $g$ differs from the wall field plus $\nabla D$ by a multiple of the
Biot\textendash Savart field, which is orthogonal to gradients and has inner product
$-\EG$ with $j$. Now $\nabla g=\nabla h-k\bGM^{-1/2}(j+\Rf)+\nabla D=(w+\nabla D)-k\bGM^{-1/2}\Rf$ and, by
Lemma~\ref{lem:7.2}, $\langle w+\nabla D,\Rf\rangle=-k\bGM^{-1/2}\langle j,\Rf\rangle=k\bGM^{-1/2}\EG$, so
$\|\nabla g\|^2=\|w+\nabla D\|^2-k^2\bGM^{-1}\EG$, and the perimeter energy separates exactly. Next $\Delta g=\Delta h+k\bGM^{-1/2}\nabla^*j+\Delta D=\Delta D-\nabla^*w$.
Adding $\mt^2\|g-H\|^2=\mt^2\|D\|^2$ gives \eqref{eq:7.exactF1}.
\end{proof}

\begin{corollary}[Localization]\label{cor:7.11}
Let $D$ be the shift \eqref{eq:7.shift}. For each wall component put
\begin{equation}\label{eq:7.F1Sbeta}
2F_1(S_\beta):=\mt^2\|D_\beta\|^2+\|w_\beta+\nabla D_\beta\|^2+\MY^{-2}\|\Delta D_\beta-\nabla^*w_\beta\|^2,
\end{equation}
and for every union $Y$ of wall components and of $\UU$-cells outside $\hat\Sigma$,
$\Floc(Y):=\sum_{\beta:S_\beta\subseteq Y}F_1(S_\beta)$. Then: (a) $F_1=\Floc(V)-\frac12k^2\bGM^{-1}\EG$;
(b) $F_1(S_\beta)\ge0$, and given $S_\beta$ it depends on $h$ only through $N|_{E(S_\beta)}$; if
$E(S_\beta)$ contains no $\Xi$-bond, it does not depend on $(k,\Xi)$; (c) for every such $Y$,
\begin{equation}\label{eq:7.H6p}
\Floc(Y)\ge\tfrac12\mt^2\sum_{x\in Y}(g-H)^2+\tfrac12\sum_{b\in E(Y)}\bigl|\nabla g+k\bGM^{-1/2}\Rf\bigr|^2;
\end{equation}
(d) $Z(k,\Xi)$ is the right side of \eqref{eq:7.translated} with $e^{-F_1}$ replaced by
$e^{-\Floc(V)}$, and the prefactor $e^{-k^2\EG/2\bGM}$ of Lemma~\ref{lem:7.2} cancels.
\end{corollary}

\begin{proof}
Each term of \eqref{eq:7.exactF1} but the last is a sum of squares of functions
supported in the wall components: $\nabla D_\beta$ is supported on $E(S_\beta)$ and $\Delta D_\beta$ on $S_\beta$, since a bond
leaving $S_\beta$ from $\supp\chi_\beta$ would end in $A_\beta\subseteq S_\beta$. This gives (a). Part (b) is
Proposition~\ref{prop:7.9}\,(f), and off $\Xi$-bonds $N=\nabla n$ does not involve $k$. For (c) we drop
the terms with $\MY^{-2}$, as $\nabla g+k\bGM^{-1/2}\Rf=w+\nabla D$. Part (d) follows from Lemma~\ref{lem:7.2},
Proposition~\ref{prop:7.8} and (a), and since the prefactor cancels, $\EG$ does not enter the perimeter constant.
\end{proof}

\subsection{The wall tension and the spanning area}
\label{ssec:7.3}\label{sec:7.3}

\paragraph{Tubes on the fine lattice.} Let $b=(x,y)$ be a bond of an interface
$I=I(\Omega,\Omega')$, oriented from $\Omega$ to $\Omega'$, with $y=x+\sigma e_\mu$, $\sigma=\pm1$. We put
$\tau_b^-:=\{x-i\sigma e_\mu:0\le i<L/2\}$, $\tau_b^+:=\{y+i\sigma e_\mu:0\le i<L/2\}$ and
$\tau_b:=\tau_b^-\cup\tau_b^+$, a segment of $L$ sites ($L$ is even). The bond $b$ is
\emph{good} if $\tau_b\subseteq V$ and the $L-1$ consecutive bonds of $\tau_b$ lie in $E$.
We write $\good(I)$ for the number of good bonds of $I$. On the fine lattice the face rule gives, by
Theorem~\ref{thm:3.20}\,(R8) and Lemma~\ref{lem:4.17},
$\good(I)\ge c_g'\max(|I|,L^2)$ for each interface, with $c_g':=c_g(1+1/L_c)^{-2}\ge\frac9{16}c_g$.

\begin{lemma}\label{lem:7.12}
Let $z_1,\dots,z_L$ be a path, let $\tilde H=c_-$ on $\{z_1,\dots,z_{L/2}\}$ and
$\tilde H=c_+$ on $\{z_{L/2+1},\dots,z_L\}$, $J:=c_+-c_-$, and let
$m\,(L/2)\le1/\sqrt2$. Then for every $g$,
$\frac12m^2\sum_{i\le L}(g(z_i)-\tilde H(z_i))^2+\frac12\sum_{i<L}(g(z_{i+1})-g(z_i))^2\ge\frac16\,m^2\,(L/2)\,J^2$.
\end{lemma}

\begin{proof}[Sketch]
With $\bar g_\pm$ the averages of $g$ on the two halves, we write
$J=(c_+-\bar g_+)+(\bar g_+-\bar g_-)+(\bar g_--c_-)$ and apply Cauchy\textendash Schwarz to each
term. Roughly speaking, the mass term supplies the lower bound if $g$ does not follow the jump, and the gradient term
supplies it if $g$ does.
\end{proof}

\begin{lemma}[Tube geometry]\label{lem:7.13}
(a) If $b\in I(\Omega,\Omega')$ is good, then $\tau_b^-\subseteq\Omega$ and $\tau_b^+\subseteq\Omega'$. (b) Good tubes on
a common lattice line are site-disjoint. Hence every site lies in at most $3$ good
tubes and every bond in at most one. (c) No bond of $\tau_b$ other than $b$ crosses $\Xi$,
and $\xi$, $\nabla h$, $N$ are constant on every interface.
\end{lemma}

\begin{proof}[Sketch]
A good tube is a straight path of $G'$ crossing one block face at its middle bond,
and by (G1) of Definition~\ref{def:3.4} blocks are at least $L$ long. The chain $\Xi$ lies in grid planes
by Lemma~\ref{lem:3.5}\,(A1), and for (c) we use Lemma~\ref{lem:7.6}\,(a).
\end{proof}

\begin{theorem}[Wall tension]\label{thm:7.14}
Let $h\in\mathcal H$, $k\in\{0,1\}$, and let $Y$ be a union of wall components and
of $\UU$-cells outside $\hat\Sigma$. With $J_I:=2\pi\bGM^{-1/2}N_I$,
\begin{equation}\label{eq:7.sigmaF}
\Floc(Y)\ge\frac1{18}\,\mt^2\,\frac L2\sum_{I\subseteq Y}\good(I)\,J_I^2\ge2\sF\sum_{b\in E(Y)}N(b)^2,
\qquad\sF:=\frac{\pi^2c_g'}{18}\,\frac{\mt^2L}{\bGM},
\end{equation}
where $I\subseteq Y$ means that both cells of $I$ lie in $Y$. We put $\Funit:=\sF L^2$.
\end{theorem}

\begin{proof}
For a good bond $b$ of an interface $I\subseteq Y$, Lemma~\ref{lem:7.13}\,(a), (c) shows that $\tau_b\subseteq Y$ and, with
$\vartheta_b:=D+J_b\one_{\tau_b^+}$, that $w+\nabla D=\nabla\vartheta_b$ on the bonds of $\tau_b$ and
$g-H=\vartheta_b-J_b\one_{\tau_b^+}$ on $\tau_b$. Hence, as $\mt L/2\le c_L/2\le1/\sqrt2$,
Lemma~\ref{lem:7.12} with $m=\mt$, $c_-=0$ and $c_+=J_b$ bounds the
part of the right side of \eqref{eq:7.H6p} on $\tau_b$ below by $\frac16\mt^2\frac L2J_b^2$.
Summing over good bonds, we count each site at most three times and each bond at
most once, by Lemma~\ref{lem:7.13}\,(b). The second inequality follows from
$\sum_{E(Y)}N^2=\sum_{I\subseteq Y}|I|N_I^2$ and $\good(I)\ge c_g'|I|$.
The constant $\frac1{18}$ is the $\frac16$ of Lemma~\ref{lem:7.12} divided by the overlap
$3$ of good tubes, and the remaining factor $2$ in front of $\sF$ is used in
Subsection~\ref{ssec:7.8}. By comparison, \cite[Lemma~12, (C.9)]{GM82} gives $\frac1{60}$ together with a perimeter
term, which the bound of the theorem does not contain.
\end{proof}

\begin{theorem}[Walls span the loop]\label{thm:7.17}
Let $k=1$, let $n:V\to\Z$ be arbitrary, and $N(b):=\nabla n(b)-\xi_b$ on $E$. Then
$\sum_{b\in E}N(b)^2\ge\#\{b\in E:N(b)\ne0\}\ge\AG(C)$.
\end{theorem}

\begin{proof}
With $U$ as in the proof of Lemma~\ref{lem:7.6}\,(c), we correct $\Xi$ by the boundary of the integer field
and obtain the integer $2$-chain $\Sigma:=\Xi-\partial U$
in $\Lo$, with $\partial\Sigma=C$. The coefficient of $p_b$ in $\Sigma$ is $\xi_b-\nabla n(b)=-N(b)$ for
$b\in E$, and all other plaquettes of $\Lo$ are faces of hole cubes, frozen faces or
faces in $\partial_2\Lo$, hence lie in $\Ff$. So the support of this chain outside the free set $\Ff$ is exactly the
set of wall bonds, $\supp\Sigma\setminus\Ff=\{p_b:b\in E,\ N(b)\ne0\}$, and
Definition~\ref{def:2.11}, applied to this $2$-chain spanning $C$, gives $\AG(C)\le\#\{N\ne0\}\le\sum_EN^2$, since $N$ is integer valued.
\end{proof}

\subsection{The fluctuation term $F_2$ is small}
\label{ssec:7.4}\label{sec:7.4}

Throughout this subsection we assume $\lPV\le\frac14$ and $\mt\le\MY\le1$, which hold for $\bGM\ge\bGMo$,
and put $R_s:=(-\Delta+s)^{-1}$ for $s>0$ and
\begin{equation}\label{eq:7.partialfrac}
r_\pm:=\tfrac12\MY^2\bigl(1\pm(1-4\lPV^2)^{1/2}\bigr),\qquad
\mathcal G:=C_0\MY^{-2}(-\Delta+\MY^2).
\end{equation}
As $r_++r_-=\MY^2$ and $r_+r_-=\mt^2\MY^2$, partial fractions in $t=-\Delta$ give
$C_0=\varpi(R_{r_-}-R_{r_+})$ and $\mathcal G=c_+R_{r_-}-c_-R_{r_+}$, with $\varpi:=\MY^2/(r_+-r_-)$ and
$c_\pm:=r_\pm/(r_+-r_-)$. Bounds on these quantities, the derivation of the numerical constants of
Lemma~\ref{lem:7.18} and the arithmetic in the proofs of Theorem~\ref{thm:7.19} and
Corollary~\ref{cor:7.21} are given in Section~\ref{S:constF}. For $X$ a union of wall
components and of $\UU$-cells outside $\hat\Sigma$ we put
$F_2(X):=\langle\one_X\psi,F_2'\rangle=\sum_{\beta:S_\beta\subseteq X}\langle\psi,F'_{2,\beta}\rangle$, where the last equality holds by Proposition~\ref{prop:7.9}\,(e).

\begin{lemma}\label{lem:7.18}
(a) \emph{(Combes\textendash Thomas.)} Let $\rho:V\to\R$ satisfy $|\rho(x)-\rho(y)|\le\alpha$ on every bond
of $G'$, and let $\kappa_{\mathrm{CT}}:=s-6(\cosh\alpha-1)>0$. Then $\|e^\rho R_se^{-\rho}\|\le\kappa_{\mathrm{CT}}^{-1}$.
(b) Let $B\subseteq E$ have endpoint set $V_B$, and let $A\subseteq V$ with $d(A,V_B)\ge r\ge0$. Then
$\|\one_A\mathcal G\nabla^*\one_B\|\le8.71\,\mt^{-2}e^{-\mt r/3}$. (c) If $\chi:V\to[0,1]$ is
$\Lambda_\chi$-Lipschitz for $d$, then $\|[C_0^{-1},\chi]\|\le\Lambda_\chi(144\MY^{-2}+6)\le150\,\Lambda_\chi\MY^{-2}$.
\end{lemma}

\begin{proof}
All three are operator norms from the Schur test and the degree bound $6$, so missing
bonds enter only through the degree. (a) The bound follows by the method of \cite[Theorem~1, p.~257]{CT73},
in a lattice form not stated there. With $e^\rho(-\Delta+s)e^{-\rho}=-\Delta+s+\mathfrak B$, pairing the two
orientations of each bond gives $\mathrm{Re}\langle f,\mathfrak Bf\rangle=-\sum_{\text{bonds}}2(\cosh t-1)\mathrm{Re}(\bar f(x)f(y))$,
$t:=\rho(x)-\rho(y)$, which is at least $-6(\cosh\alpha-1)\|f\|^2$. As $-\Delta\ge0$, it follows that
$\mathrm{Re}\langle f,(-\Delta+s+\mathfrak B)f\rangle\ge\kappa_{\mathrm{CT}}\|f\|^2$. The antisymmetric part of
$\mathfrak B$ drops out, so $\rho$ may have kinks, such as those of $d$ at slit edges and hole corners, and
the second-order form matters: in (b), at $s=r_-\le1.08\,\mt^2$ and $\alpha=\mt/3$, the first-order
quantity $s-6(e^\alpha-1)$ is negative. (b) We apply (a) with $\rho:=\frac{\mt}3\min\{d(\cdot,V_B),r\}$ to the two
resolvents of $\mathcal G$, and the Schur test to $e^\rho\nabla^*e^{-\rho}$. (c) We use
$[C_0^{-1},\chi]=\MY^{-2}(\Delta[\Delta,\chi]+[\Delta,\chi]\Delta)-[\Delta,\chi]$.
\end{proof}

\begin{theorem}[{$F_2$ is controlled by $F_1$; \cite[Lemma~14]{GM82} on $G'$}]\label{thm:7.19}
For every $h\in\mathcal H$, $k\in\{0,1\}$ and every union $X$ of wall components and of
$\UU$-cells outside $\hat\Sigma$,
$\sum_{x\in X}F_2'(x)^2=\sum_{\beta:S_\beta\subseteq X}\|F'_{2,\beta}\|^2\le\mt^2\,\varepsilon(L')\,\Floc(X)$, where
\begin{equation}\label{eq:7.epsL}
\varepsilon(L'):=C_\varepsilon\,c_g^{-1}\,\lPV^4\,\mt^{-9}\,(\mt L/2)^{-1}(\mt L')^{-2}e^{-\mt L'/3},
\end{equation}
with the absolute constant $C_\varepsilon$ of Section~\ref{S:constF}.
\end{theorem}

\begin{proof}
By Proposition~\ref{prop:7.9}\,(e),
$F'_{2,\beta}=-[C_0^{-1},\chi_\beta]\one_{A'}\mathcal G\nabla^*\one_Bw_\beta$ with $A':=N_1(A_\beta)$, $B:=\supp w_\beta$,
$d(A',V_B)\ge L'/2-2$ and $\Lambda_\chi=4/L'$. The idea is that the commutator is localized where $\chi_\beta$ changes, at distance at least
$L'/2-1$ from the walls, and that at that distance the Combes\textendash Thomas decay outweighs the polynomial loss, because $\mt L'\propto\bGM$.
We apply Lemma~\ref{lem:7.18}\,(b), (c), and bound
$\sum_\beta\|w_\beta\|^2$ by Theorem~\ref{thm:7.14} with $\good(I)\ge c_g'|I|$.
\end{proof}

This proof follows \cite[Appendix~D, (D.3) to (D.16)]{GM82}, with the following changes.
\begin{itemize}
\item[(1)] The operator
norms of Lemma~\ref{lem:7.18} replace GM's pointwise bounds \cite[(D.10)]{GM82}, which come from
\cite[Lemma~B.3]{GM82} and fail within distance $2$ of a missing bond. This loses a
factor $\MY^{-4}\mt^{-4}$.
\item[(2)] The source form of Proposition~\ref{prop:7.9}\,(e) replaces
\cite[(D.3) to (D.5)]{GM82}, since $h^e_\beta$ need not exist on $G'$.
\item[(3)]
Theorem~\ref{thm:7.14} replaces \cite[Lemma~12, (C.21)]{GM82}, so the bound has no
perimeter term $P_F$ and is uniform in $\Lo$, $C$, $\Xi$, the hole set $\Xh$, $S$, $h$, $k$
and $X$.
\end{itemize}

\begin{corollary}[Gaussian $F_2$ factors]\label{cor:7.20}
Let $C'$ be positive semidefinite on $\R^V$ with $\|C'\|\le t_C\mt^{-2}$, and let
$d\mu_{C'}$ be the centred Gaussian measure with covariance $C'$. For every
$p\in\R$ and every $X$ as in Theorem~\ref{thm:7.19},
\[
\int d\mu_{C'}(\psi)\,e^{-pF_2(X)}=\exp(\tfrac12p^2\langle\one_XF_2',C'\one_XF_2'\rangle)\le\exp(\tfrac12p^2t_C\,\varepsilon(L')\,\Floc(X)).
\]
\end{corollary}

\begin{proof}
Since $F_2(X)=\langle\psi,\one_XF_2'\rangle$ is linear in $\psi$, its Laplace transform is Gaussian.
The bound then follows from $\langle F,C'F\rangle\le\|C'\|\|F\|^2$ and Theorem~\ref{thm:7.19}. We will apply the corollary with the $C'$ of
Lemma~\ref{lem:7.31}, for which $C'\le\alpha_DC_0\le\frac76\mt^{-2}$, so that $t_C=\frac76<2$, and with block-diagonal
restrictions $\sum_j\one_{Y_j}C_0\one_{Y_j}$, for which $t_C=1$.
\end{proof}

\begin{corollary}[Collar]\label{cor:7.21}
Let $p_2$, $p_3$, $p'$, $q'$ and $c_\lambda$ be the Hölder exponents and the constant of
Subsection~\ref{ssec:7.6}, and $\varepsilon_0:=\min\{(1-1/p_3-c_\lambda)/(2p_2),\,2(p'/q')^2\}$
(values in Section~\ref{S:constF}). There is $\bGMo$ such that $\varepsilon(L')\le\varepsilon_0$ for
$\bGM\ge\bGMo$.
\end{corollary}

\begin{proof}
We have $\mt L'=2\bGM$, while for $\varepsilon(L')\le\varepsilon_0$ we need only
$\mt L'\ge27\log(1/\mt)+O(1)$ (Section~\ref{S:constF}), with $\log(1/\mt)\le2.4944\tau$ and
$\Kcol=2>27\cdot2.4944/(4\pi^2)=1.706$. A collar with $\mt L'$
fixed, as in \cite[p.~47]{GM82}, does not suffice on $G'$.
\end{proof}

\begin{proposition}\label{prop:7.22}
Let $D_E:=C_\UU\mt^{-1}$ bound the Euclidean diameter of $\UU$-cells
(Theorem~\ref{thm:3.20}\,(R7)) and $\eta_{\mathrm{col}}:=(2\sqrt2L+D_E+3)/L'$. For every wall
component $S_\beta$, with $n_I(\beta)$ the number of interfaces of $S_\beta$ carrying a wall:
(a) $F_1(S_\beta)\ge\frac{\pi^2c_g'}9\frac{\mt^2L^3}{\bGM}n_I(\beta)=2\Funit\,n_I(\beta)$;
(b) $S_\beta$ has at most $n_I(\beta)\frac{4\pi}{3c_u}(\mt L')^3(1+\eta_{\mathrm{col}})^3$ $\UU$-cells;
(c) for $\bGM\ge\bGMo$, every $\UU$-cell of $\hat\Sigma$ receives a share at least $\Phi_\UU$ of
$\Floc$, that is, $F_1(S_\beta)\ge\Phi_\UU\cdot\#\{\UU\text{-cells of }S_\beta\}$, where the constant
$\Phi_\UU$ (Section~\ref{S:constF}) satisfies $\Phi_\UU\to\infty$ as $\bGM\to\infty$.
\end{proposition}

\begin{proof}[Sketch]
(a) is Theorem~\ref{thm:7.14} with $Y=S_\beta$, $J_I^2\ge4\pi^2/\bGM$ and $\good(I)\ge c_g'L^2$. (b) Each
$\UU$-cell of $S_\beta$ lies within $d$-distance $L'$ of a wall interface, and by (R7) this interface lies in one
block face and $\UU$-cells are disjoint with at least $c_u\mt^{-3}$ sites. (c) We divide
(a) by (b). Then $\Phi_\UU\to\infty$, because the energy is exponential in $\bGM$, whereas the collar
is polynomial in $\mt L'\propto\bGM$.
\end{proof}

\subsection{The class-graded expansion}
\label{ssec:7.5}\label{sec:7.5}

By Corollary~\ref{cor:7.11}\,(d), $Z(k,\Xi)=\sum_{[h]}\mathcal N\int d\mu_C\,e^{E}Q_h$ with
$Q_h:=e^{G(\psi+g)}e^{-\Floc(V)}e^{-F_2}$. We expand the Gaussian integral at fixed $h$
and then resum the walls, graded by their class.

\begin{definition}[Closures, wall polymers, seeds]\label{def:7.23}
Fix $h\in\mathcal H$. (a) The closure $\operatorname{cl}(S)$ of a union $S$ of $\UU$-cells is $S$
together with the $\UU$-cells joined to $S$ by a bond of $G'$. Closures of wall
components are \emph{linked} if they share a $\UU$-cell; a \emph{wall polymer} $\mathcal S_\gamma(h)$
is the union of a maximal linked family of closures. (b) The wall polymers and the
$\UU$-cells contained in no wall polymer form a partition $\tilde{\mathcal Y}(h)$ of $V$, whose
members are the \emph{elements}. A set is \emph{admissible} (for $h$) if it is a union of
elements; $\sat_h(S)$ is the union of the elements that meet $S$. (c) $X_1$ is the
union of the $\UU$-cells that contain an endpoint of a bond of $G'$ crossing $\Xi$, and
$\Delta_1$ is a fixed $\UU$-cell. The \emph{seeds} are $Y_1(h):=\sat_h(X_1)$ in the expansion of
$Z(1,\Xi)$ and $Y_1(h):=\sat_h(\Delta_1)$ in that of $Z(0,\Xi)$. (d)
$\mathfrak M(h):=\{X\supseteq Y_1(h):X\text{ admissible}\}$. (e) For admissible $Y$,
$G(Y):=\sum_{\Omega\subseteq Y}\ln r_\Omega(A_\Omega)+\sum_iG_{2,i}(Y)$ (the site sums of \eqref{eq:7.G2}
restricted to $Y$) and $Q_h(Y):=e^{G(Y)}e^{-\Floc(Y)}e^{-F_2(Y)}$ ($F_2(Y)$ as in Subsection~\ref{ssec:7.4});
$E(Y)$ and $E(Y,s)$ are defined in Appendix~\ref{app:C}.
\end{definition}

An admissible set is a union of wall components and of $\UU$-cells outside $\hat\Sigma$,
as a $\UU$-cell of $\hat\Sigma$ joined by a bond to $S_\beta$ belongs to $S_\beta$. Hence $\Floc(Y)$ and
$F_2(Y)$ are defined. An element that meets $X_1$ lies in each admissible $X\supseteq X_1$, since the elements partition $V$ and admissible sets are unions of elements.
Therefore $\mathfrak M(h)=\{X\supseteq X_1:X\text{ admissible}\}$, as in the rule of
\cite[p.~50]{GM82}, and with $\Delta_1$ for $k=0$, as on \cite[p.~51]{GM82}.

\begin{lemma}\label{lem:7.24}
Let $Y,Y'$ be disjoint admissible sets. Then
$Q_h(Y\cup Y')=Q_h(Y)Q_h(Y')$; in particular $Q_h=Q_h(X)Q_h(V\setminus X)$ for
admissible $X$. Moreover $Q_h(Y)$, and every term of $E(Y,s)$, depend on
$(\psi,h)$ only through $\psi|_Y$ and $N|_{E(Y)}$. The same holds with $G^\natural$
of Lemma~\ref{lem:7.31} in place of $G$.
\end{lemma}

\begin{proof}[Sketch]
We follow \cite[(E.4)]{GM82}, with admissible sets in place of unions of unit cubes. The action is
a sum of cell terms ($\ln r_\Omega$), site terms ($G_{2,i}$, the terms of $G^\natural$ and of $E$) and
wall-component terms ($F_1(S_\beta)$, $F'_{2,\beta}$), and an admissible set contains each cell and
wall component entirely or not at all. As $g+k\bGM^{-1/2}f=h+D$ with $h$ cell-constant and $P\Z$-valued, each phase is $e^{im\bGM^{1/2}(\psi+D)}$, and $r_\Omega$ is $P$-periodic. By Proposition~\ref{prop:7.9}\,(f) and Corollary~\ref{cor:7.11}\,(b), $D|_Y$, $\Floc(Y)$ and
$F_2(Y)$ depend on $h$ only through $N|_{E(Y)}$.
\end{proof}

Since $\Floc$ does not split inside a wall polymer, this factorization holds for admissible sets
only, so we saturate the seed, as $X_1$ itself need not be admissible.

\begin{proposition}[{Cluster expansion at fixed $h$; \cite[Lemma~E.1, Proposition~E.2]{GM82} on $G'$}]\label{prop:7.25}\label{prop:C.8}\label{lem:C.1}\label{lem:C.2}\label{lem:C.6}\label{lem:C.7}
Assume (S1) to (S5) and the hypotheses of Lemma~\ref{lem:C.4}, and let $\mathbf C\in\{C,C'\}$
($C'$ of Lemma~\ref{lem:7.31}). Let $U=V$, or $U=V\setminus X_0$ for a union $X_0$ of $\UU$-cells;
let $N$ be a wall field on $E(U)$ (see the paragraph on wall fields before Lemma~\ref{lem:7.wall}, where $E(Y,s)$ and $Q_N$
are defined for wall fields) whose wall polymers have their closures in $U$ (for $U=V$,
$N=N(h)$ with any $h\in\mathcal H$), with $D$, $S_\beta$, $F_1(S_\beta)$, $F'_{2,\beta}$ and the elements defined from $N$
(Proposition~\ref{prop:7.9}\,(f), Corollary~\ref{cor:7.11}\,(b)); and let $Y_1\subseteq U$ be
nonempty and admissible for $N$ on $U$ (it may depend on $N$ and need not be
connected). With $\mathbf C_U:=\one_U\mathbf C\one_U$ on $\R^U$,
\begin{equation}\label{eq:7.clusterexp}
\int d\mu_{\mathbf C_U}(\psi)\,e^{E(U)}Q_N(U)=\sum_{X}\mathcal K_N(X)\int d\mu_{\mathbf C_U}(\psi)\,
e^{E(U\setminus X)}Q_N(U\setminus X),
\end{equation}
where $X$ runs over the admissible subsets of $U$ containing $Y_1$,
$\mathcal K_N(X):=\sum_{\bar y:\,X(\bar y)=X}\mathcal K_{\bar y}$ over the sequences $\bar y=(Y_1,\dots,Y_n)$ of pairwise disjoint
nonempty admissible sets with union $X(\bar y)=X$, and
\begin{equation}\label{eq:7.activity}
\mathcal K_{\bar y}:=\int_{[0,1]^{n-1}}ds\int d\mu^{\bar y}_s(\psi)\,
e^{E^{\bar y}(X(\bar y),s)}\,\kappa(\bar y,s)\,Q_N(X(\bar y)),
\end{equation}
with $d\mu^{\bar y}_s$ (covariance \eqref{eq:C.E3} at $s_n=0$), $E^{\bar y}$ and the operator $\kappa(\bar y,s)$ of
Appendix~\ref{app:C}, built from $\mathbf C_U$.
Every sequence has length at most $e_N+1$, $e_N$ the number of elements of $U$ not
contained in $Y_1$; all sums are finite, all integrals converge absolutely, and
$\int d\mu_{\mathbf C_U}e^{E(Y)}Q_N(Y)\in(0,\infty)$ for every admissible $Y\subseteq U$. $\mathcal K_{\bar y}$ depends only
on $\bar y$, the $X(\bar y)\times X(\bar y)$ block of $\mathbf C$ and $N|_{E(X(\bar y))}$, not on $U$.
\end{proposition}

We give the proof in Appendix~\ref{app:C}.

\paragraph{Admissibility in terms of walls.} For a set $W$ of wall bonds with endpoint
set $V(W)$, $\hat\Sigma(W)$ is the union of the $\UU$-cells $\Delta$ with $d(\Delta,V(W))<L'$. For a
union $X$ of $\UU$-cells, $\partial X$ is the set of bonds of $G'$ joining $X$ to $X^c:=V\setminus X$,
and we put $W_X:=\Sigma_E\cap E(X)$ and $W_{X^c}:=\Sigma_E\cap E(X^c)$. We consider the conditions (in)
$\operatorname{cl}(\hat\Sigma(W_X))\subseteq X$, (out) $\operatorname{cl}(\hat\Sigma(W_{X^c}))\subseteq X^c$, and ($\partial$) $N=0$ on $\partial X$.

\begin{lemma}\label{lem:7.26}
Let $k\in\{0,1\}$, $h\in\mathcal H$, and let $X$ be a union of $\UU$-cells with
$X\supseteq X_1$ if $k=1$ and $X\supseteq\Delta_1$ if $k=0$. Then
$X\in\mathfrak M(h)$ if and only if (in), (out) and ($\partial$) hold. In that case
the elements of $\tilde{\mathcal Y}(h)$ inside $X$ are determined by $X$ and $N|_{E(X)}$,
and those inside $X^c$ by $X$ and $N|_{E(X^c)}$.
\end{lemma}

\begin{proof}[Sketch]
By Proposition~\ref{prop:7.9}\,(a), each wall component, with its closure, belongs to the side that contains its walls.
No element straddles $\partial X$ when $\partial X$ carries no wall, since
under ($\partial$) we have $\hat\Sigma=\hat\Sigma(W_X)\cup\hat\Sigma(W_{X^c})$, no $\UU$-cell of one being adjacent to one of the
other by (in) and (out), and the wall components inside $X$ are the components of
$\hat\Sigma(W_X)$, with $W_X=\supp N|_{E(X)}$.
\end{proof}

At $k=1$, since both ends of each $\Xi$-bond lie in $X_1\subseteq X$, the set $\partial X$ contains no $\Xi$-bond,
and ($\partial$) is equivalent to $h(x)=h(y)$ for all $\{x,y\}\in\partial X$, as at $k=0$. By Lemmas~\ref{lem:7.24}
and \ref{lem:7.26}, for admissible $X\supseteq X_1$ the activity $\mathcal K_h(X)$ with first set
$\sat_h(X_1)$ depends on $h$ only through $N|_{E(X)}$.

\paragraph{Wall fields.} A \emph{wall field} on a union $U$ of $\UU$-cells is a function
$N:E(U)\to\Z$, extended by $0$ to $E$, that vanishes on the bonds inside cells and is
constant on each interface. By Lemma~\ref{lem:7.6}\,(a), the restriction $N(h)|_{E(U)}$ is a
wall field for $h\in\mathcal H$ and $k\in\{0,1\}$. The objects $S_\beta$, $\chi_\beta$, $s_\beta$, $D$,
$F_1(S_\beta)$, $\Floc$, $F'_{2,\beta}$ and $F_2(Y)$ of Subsection~\ref{ssec:7.2} are functions of $w:=2\pi\bGM^{-1/2}N$ alone,
as Proposition~\ref{prop:7.9}\,(f) and Corollary~\ref{cor:7.11}\,(b) show. For a wall field $N$ we define $G(Y)$,
$G^\natural(Y)$ (Lemma~\ref{lem:7.31}), $E(Y)$ and $E(Y,s)$ by their formulas with
$\psi+g+k\bGM^{-1/2}f$ replaced by $\psi+D$ in all phases and all cell averages, and put $Q_N(Y):=e^{G(Y)}e^{-\Floc(Y)}e^{-F_2(Y)}$. Under this replacement,
$A_\Omega+k\bGM^{-1/2}\bar f_\Omega$ becomes $\operatorname{avg}_\Omega(\psi+D)$, and $\tilde\delta$ becomes
$(\psi+D)-\operatorname{avg}_\Omega(\psi+D)$.

\begin{lemma}[Wall form]\label{lem:7.wall}
(a) If $U$ is admissible for $h\in\mathcal H$ ($U=V$, or $U=X^c$ with $X\in\mathfrak M(h)$), $k\in\{0,1\}$ and $N=N(h)|_{E(U)}$, then the elements of $N$ are the elements of $h$ inside $U$, $Q_N(Y)=Q_h(Y)$, and
$E(Y,s)$ is the same in both definitions, for every admissible $Y\subseteq U$.
(b) Definition~\ref{def:7.23} applies, and Lemma~\ref{lem:7.24}, Proposition~\ref{prop:7.25} and
Lemma~\ref{lem:7.26} hold, for every wall field $N$ on $E(U)$ whose wall polymers have their
closures in $U$, with $V$ replaced by $U$ and without the condition on the seed in
Lemma~\ref{lem:7.26}; Theorem~\ref{thm:7.14} holds for every wall field; and
Proposition~\ref{prop:7.5} holds with $\|\varphi+k\bGM^{-1/2}f-h\|$ replaced by $\|\psi+D\|$.
\end{lemma}

\begin{proof}[Sketch]
(a) As in the proof of Lemma~\ref{lem:7.24}, $h$ drops out. (b) Since the action consists of cell, site and wall-component
terms, these proofs use $h$ only
through (a) and the walls. In Theorem~\ref{thm:7.14}, \eqref{eq:7.H6p} becomes
\eqref{eq:7.F1Sbeta} without its last term, and $w=0$ inside cells. The proof of
Proposition~\ref{prop:7.5} is pointwise in $\bGM^{1/2}(\psi+D)(x)$, and Appendix~\ref{app:C} uses Lemma~\ref{lem:C.3}
and the Poincaré inequality on cells, pointwise in $(\psi,D)$ (details in Section~\ref{S:F2}).
\end{proof}

\begin{definition}[Classes and outer partition functions]\label{def:7.27}
(a) The \emph{m3-graph} has the sites of $V$ as vertices, joined by the bonds of $E$ and
by the pairs of sites in a common cell; a closed walk $\gamma$ in it has the length $|\gamma|$,
the sum of the $d_{G'}$-distances of its steps. The \emph{class} of a function $N$ on $E$
(extended by $0$ to the same-cell pairs) is $[N]:\gamma\mapsto\sum_{e\in\gamma}N(e)$ (oriented sum) on
closed m3-walks. (b) A wall field $N$ is \emph{long} if $[N](\gamma)=0$ for every closed
m3-walk $\gamma$ with $|\gamma|<2L'$. $\Nlong(U)$ is the set of long wall fields on $E(U)$, and
$\Glong:=\{[N]:N\in\Nlong(V)\}$; it is a group. (c) For a union $X$ of $\UU$-cells and
$p\in\Glong$,
\begin{equation}\label{eq:7.Zprime}
\mathcal Z_p(X):=\sum_N\mathcal N\int d\mu_C(\psi)\,e^{E(X^c)}\,Q_N(X^c),
\end{equation}
the sum over the $N\in\Nlong(X^c)$ that satisfy (out) and have $[N]=p$, with the
integrand of Lemma~\ref{lem:7.wall}; $\mathcal Z(X):=(\mathcal Z_p(X))_{p\in\Glong}$, $\mathcal Z_p(X):=0$
for $p\notin\Glong$, $Z_p:=\mathcal Z_p(\emptyset)$ and $Z:=Z_0$. (d) For $k\in\{0,1\}$ and a union $X$ of
$\UU$-cells, with $X\supseteq X_1$ if $k=1$, an \emph{inside field} at coupling $k$ is a wall field
$N_X$ on $E(X)$ that satisfies (in) and for which $[(N_X+k\xi)\one_{E(X)}]$ vanishes on the
closed m3-walks in $X$; its \emph{outer class} is $p_k(N_X):=-[(N_X+k\xi)\one_{E(X)}]$.
\end{definition}

Integrating along m3-paths, we see that, for $N$ vanishing on the bonds inside cells (for instance a wall field), $[N]=0$ if and only if $N=\nabla n$ with $n:V\to\Z$ constant
on each cell (on the cycles of $G'$ alone $n$ need not be constant on cells, which
may be disconnected). With Lemmas~\ref{lem:7.6}\,(d) and~\ref{lem:7.wall}\,(a), this characterization gives
$Z=Z(0,\Xi)$: the configurations of $Z_0$ are the walls of the $[h]\in\mathcal H/\mathcal C_P$ at $k=0$.
A long wall field has period $0$ around each plaquette of $G'$, which is a closed walk of
length $4$. For $X\supseteq X_1$, $\xi$ vanishes off $E(X)$, so by Lemma~\ref{lem:7.6}\,(d) an inside
field at $k=1$ has the periods $-\lk(\cdot,C)$ on the cycles of $G'$ in $X$, and $p_1(N_X)(\gamma)=-\lk(\gamma,C)-\sum_{e\in\gamma\cap E(X)}N_X(e)$ on the cycles of $G'$.
In general, such a field is not long.

\begin{lemma}[Long fields]\label{lem:7.45}
(a) A closed m3-walk $\gamma$ with $|\gamma|<2L'$ that contains a wall bond of a wall
component $S_\beta$ has all its sites in $S_\beta$. Hence, if $N\in\Nlong(U)$ has its wall
polymers' closures in $U$ and $Y\subseteq U$ is a union of elements of $N$, then $N\one_{E(Y)}$
and $N\one_{E(U\setminus Y)}$ are long, and their classes add up to $[N]$. (b) Let $h\in\mathcal H$,
$k=1$ and $X\in\mathfrak M(h)$. Then $N(h)|_{E(X)}$ is an inside field at $k=1$, $N(h)|_{E(X^c)}$ is
long and satisfies (out), and $[N(h)|_{E(X^c)}]=p_1(N(h)|_{E(X)})\in\Glong$.
\end{lemma}

\begin{proof}
A closed walk of length less than $2L'$ stays at $d_{G'}$-distance less than $L'$
from each of its sites. (a) So all sites of $\gamma$ lie in $S_\beta$
(Proposition~\ref{prop:7.9}\,(a)), all wall bonds of $\gamma$ lie in $E(S_\beta)$, and $S_\beta$ lies in
one element. Thus the period of $N\one_{E(Y)}$ on $\gamma$ is $0$ or that of $N$, and $N$
vanishes on the bonds between $Y$ and $U\setminus Y$ by Lemma~\ref{lem:7.26}. (b) We put
$N:=N(h)=\nabla n-\xi$, $N_X:=N|_{E(X)}$ and $N':=N|_{E(X^c)}$. By Lemma~\ref{lem:7.26}, $N_X$
satisfies (in), $N'$ satisfies (out), and $N=0$ on $\partial X$. Since $\xi$ vanishes off $E(X)$, we have
$N_X+\xi\one_{E(X)}=\nabla n$ on $E(X)$ and $[N']=-[N_X+\xi\one_{E(X)}]=p_1(N_X)$, as $[N+\xi]=0$. A short
closed walk through a site of $X$ carries no wall of $N'$, by (out), and on one inside
$X^c$ the period of $N'=\nabla n$ is $0$. Hence $N'$ is long.
\end{proof}

For unions $Z\subseteq Y$ of $\UU$-cells and a wall field $N_Y$ on $E(Y)$ of which $Z$ is a union
of elements, we write $\mathcal K^{(Z)}_{N_Y}(Y)$ for the activity of Proposition~\ref{prop:7.25} at $N_Y$ with
first set $Z$, and $\tilde{\mathcal K}(\Delta)$ for that of a wall-free $\UU$-cell $\Delta$ ($n=1$). The finiteness
of $\sum_{N_Y}|\mathcal K^{(Z)}_{N_Y}(Y)|$ and of $\sup_p\mathcal Z_p(X)$ used below is Proposition~\ref{prop:7.51},
proved without Theorem~\ref{thm:7.29} for the activities and outer partition functions built with $(C',G^\natural,\mathfrak c\mathcal N)$
(Lemma~\ref{lem:7.31}\,(f)), the only ones used from Lemma~\ref{lem:7.31} on.

\begin{theorem}[Resummation]\label{thm:7.29}\label{lem:7.49}
Assume the hypotheses of Proposition~\ref{prop:7.25}, and that the sums below converge absolutely; for the objects built with
$(C',G^\natural,\mathfrak c\mathcal N)$ of Lemma~\ref{lem:7.31}\,(f) they do, by Proposition~\ref{prop:7.51}. (a) For $X\supseteq X_1$ and an inside field $N_X$ at $k=1$, let
$\mathcal K_{N_X}(X):=\mathcal K^{(\sat(X_1))}_{N_X}(X)$. Then
\[
Z(1,\Xi)=\sum_{X\supseteq X_1}\sum_{N_X}\mathcal K_{N_X}(X)\,\mathcal Z_{p_1(N_X)}(X),
\]
the inner sum over the inside
fields at $k=1$. (b) \emph{(Kirkwood\textendash Salsburg
identity.)} Let $X_0\subsetneq V$ be a union of
$\UU$-cells and $\Delta\subseteq X_0^c$ a $\UU$-cell. For every $p\in\Glong$,
\begin{equation}\label{eq:7.7.KS}
\mathcal Z_p(X_0)=\sum_{Y\ni\Delta}\ \sum_{N_Y}\mathcal K^{(\sat(\Delta))}_{N_Y}(Y)\,
\mathcal Z_{p-[N_Y]}(X_0\cup Y),
\end{equation}
the sums over the unions $Y\subseteq X_0^c$ of $\UU$-cells containing $\Delta$ and the
$N_Y\in\Nlong(Y)$ satisfying (in), with first set $\sat_{N_Y}(\Delta)$. The term $Y=\Delta$, $N_Y=0$ is $\tilde{\mathcal K}(\Delta)\mathcal Z_p(X_0\cup\Delta)$. (c) \emph{(Prescribed
walls.)} Let $R$ be a family of disjoint unions $\mathcal S$ of
$\UU$-cells, each with a long wall field $N_{\mathcal S}$ on $E(\mathcal S)$ of which it is a wall
polymer, $\sigma:=(\bigcup R,\sum_{\mathcal S}N_{\mathcal S})$, and for $r\in\Glong$ let $Z_r(R)$ be the part of
$Z_r$ from the configurations $N$ in which every $\mathcal S\in R$, with $N|_{E(\mathcal S)}=N_{\mathcal S}$,
is an element. Then
$Z_r(R)=\sum_{Y\supseteq\bigcup R}\sum_{N_Y}\mathcal K^{(\bigcup R)}_{N_Y}(Y)\,\mathcal Z_{r-[N_Y]}(Y)$, the sum over
the $N_Y\in\Nlong(Y)$ satisfying (in) with $N_Y|_{E(\bigcup R)}=\sum_{\mathcal S}N_{\mathcal S}$ of which $\bigcup R$
is a union of elements.
\end{theorem}

\begin{proof}
(a) By Corollary~\ref{cor:7.11}\,(d) and Proposition~\ref{prop:7.25} with $U=V$ and first set
$\sat_h(X_1)$, $Z(1,\Xi)=\sum_{[h]}\mathcal N\sum_{X\in\mathfrak M(h)}\mathcal K_h(X)\int d\mu_C\,e^{E(X^c)}Q_h(X^c)$.
Fix $X\supseteq X_1$. The map $[h]\mapsto(N(h)|_{E(X)},N(h)|_{E(X^c)})$ is a bijection from
$\{[h]:X\in\mathfrak M(h)\}$ onto the pairs $(N_X,N')$ of an inside field at $k=1$ and a
configuration $N'$ of $\mathcal Z_{p_1(N_X)}(X)$. It maps into this set by Lemma~\ref{lem:7.45}\,(b), and it is
injective by Lemma~\ref{lem:7.6}\,(d). It is onto, since $N:=N_X+N'$ has $[N+\xi]=[(N_X+\xi)\one_{E(X)}]+[N']=0$,
hence $N=N(Pn)$ with $n$ cell-constant, and (in), (out), ($\partial$) give $X\in\mathfrak M(Pn)$
(Lemma~\ref{lem:7.26}). The activity depends only on $N_X$ (Lemmas~\ref{lem:7.24},
\ref{lem:7.26}). Since both ends of each $\Xi$-bond lie in $X_1\subseteq X$, the outer integral
depends only on $N'$ and is the summand of \eqref{eq:7.Zprime} (Lemma~\ref{lem:7.wall}\,(a)).
The outer integrals are $\ge0$ (Lemma~\ref{lem:7.44}), and the absolute convergence allows
the sums to be exchanged. Only inside fields with $p_1(N_X)\in\Glong$ occur, by Lemma~\ref{lem:7.45}\,(b),
and for them $\sum_{N_X}|\mathcal K_{N_X}(X)|<\infty$ (Proposition~\ref{prop:7.51}, for the objects built with $(C',G^\natural,\mathfrak c\mathcal N)$). (b), (c) In (b) we fix a configuration $N$ of $\mathcal Z_p(X_0)$
and expand on $U=X_0^c$ with first set $\sat_N(\Delta)$ by Proposition~\ref{prop:7.25} in wall form (Lemma~\ref{lem:7.wall}\,(b)).
In (c) we fix a configuration $N$ of $Z_r(R)$ and expand in the same way on $U=V$ with first set $\bigcup R$, possibly
disconnected. The index sets agree: if each $\mathcal S\in R$ is an element of $N$, then $\bigcup R$ is admissible;
conversely, every site within distance $L'$ of a wall of $N_{\mathcal S}$ lies in a wall component of
$N_{\mathcal S}$, hence in $\mathcal S$ (Proposition~\ref{prop:7.9}\,(a)), so the wall components of
$\sum_{\mathcal S}N_{\mathcal S}$ are those of the $N_{\mathcal S}$, closures from different $\mathcal S$ are
disjoint, and the elements of $N_Y$ in $\bigcup R$ are the $\mathcal S$ (Lemma~\ref{lem:7.26}). By
Lemma~\ref{lem:7.26} in wall form and Lemma~\ref{lem:7.45}\,(a), the map
$N\mapsto(Y,N\one_{E(Y)},N\one_{E(U\setminus Y)})$ is a bijection onto the triples of $Y$, $N_Y$ as stated
and a configuration of $\mathcal Z_{p-[N_Y]}(X_0\cup Y)$ in (b), and of $\mathcal Z_{r-[N_Y]}(Y)$ in (c). The summand
factorizes (Lemma~\ref{lem:7.24}), and the absolute convergence allows the sums to be
exchanged. For $Y=\Delta$ and $N_Y=0$, the $\UU$-cell $\Delta$ is a wall-free element.
\end{proof}

At fixed walls the expansion factorizes over the elements, admissibility splits
into conditions on the two sides (Lemma~\ref{lem:7.26}), and the class of the walls
is additive. This class is therefore all that the inside constrains on the outer side.
In (b) the removal of $Y$ moves the class by $[N_Y]$, and each pair $(X,p)$, $X\ne\emptyset$,
enters the identity for $(X\setminus\Delta,p)$ through the wall-free term $Y=\Delta$, the form of
\cite[(A4.3), (A4.5)]{BF80}. The class cannot be dropped: removing the polymer of one
of the two walls in the paragraph headed \emph{Handles} (Section~\ref{sec:7.1}) leaves outer walls of nonzero class, which
are not the walls of a height on the complement.

\begin{corollary}[Loop expansion]\label{cor:7.30}
Under the hypotheses of Theorem~\ref{thm:7.29}, in the finite box,
\begin{equation}\label{eq:7.loopexp}
W_{G'}(C)=\sum_{X\supseteq X_1}\ \sum_{N_X}\mathcal K_{N_X}(X)\,\frac{\mathcal Z_{p_1(N_X)}(X)}Z,
\end{equation}
the inner sum over the inside fields at $k=1$, and $\mathcal Z_{p_1(N_X)}(X)\ge0$.
\end{corollary}

\begin{proof}
We divide Theorem~\ref{thm:7.29}\,(a) by $Z=Z(0,\Xi)>0$ and use Lemma~\ref{lem:7.44}. Here
$\mathcal Z_p(X)/Z$ is a ratio of two absolutely convergent
sums in the finite box, which Theorem~\ref{thm:7.52} only bounds.
\end{proof}