%
%
\providecommand{\bGM}{\beta_{\mathrm{GM}}}
\providecommand{\mt}{\tilde m}
\providecommand{\Floc}{F_1^{\mathrm{loc}}}
\providecommand{\cU}{\mathcal U}
\providecommand{\cK}{\mathcal K}
\providecommand{\cN}{\mathcal N}
\providecommand{\cP}{\mathcal P}
\providecommand{\sF}{\sigma_F}
\providecommand{\epsE}{\varepsilon_E}
\providecommand{\Funit}{F_{\mathrm{unit}}}
\providecommand{\eps}{\varepsilon}
\providecommand{\Ad}{\mathrm{A}}
\providecommand{\Dcell}{D_{\mathrm{cell}}}
\providecommand{\etaseed}{\eta_{\mathrm{seed}}}
\providecommand{\Kflat}{\mathfrak K^{\flat}}
\providecommand{\Glong}{\Gamma_{\mathrm{long}}}
\providecommand{\Nlong}{\mathcal N^{\mathrm{long}}}

\subsection{The Kirkwood\textendash Salsburg system in the class algebra}
\label{sec:7.7}\label{ssec:7.7}

In this subsection we solve the Kirkwood\textendash Salsburg system of Theorem~\ref{thm:7.29}\,(b) by the argument
of \cite[Appendix~4]{BF80} in a weighted $\ell^1$ algebra over $\Glong$, with the area weight of
the classes (Theorem~\ref{thm:7.52}). The area decay of $\mathcal Z_p(X)/Z$ will then follow from that of
$Z_p/Z$, established in Subsection~\ref{sec:7.8}. Activities and outer partition functions are built with the triple
$(C',G^\natural,\fc\cN)$ of Lemma~\ref{lem:7.31}\,(f).

\begin{lemma}[Positivity]\label{lem:7.44}
At $k=0$ and for real $\psi$, the functions $E$, $G$ and $G^\natural$ of
Lemma~\ref{lem:7.wall} are real, and $e^{E(Y)}Q^\natural_N(Y)>0$ for every wall field
$N$ and every union $Y$ of its elements. Hence $\mathcal Z_p(X)\ge0$ for every $X$
and $p$, $Z>0$, and every activity with $n=1$ is positive.
\end{lemma}

\begin{proof}
By (S3) the terms of $m$ and $-m$ in the charge sums of $E$, $G_1$, $G_{2,1}$, $G_{2,2}$ are
complex conjugate, and $G_{2,3}^{\Dep}$, $G_{2,3}^\natural$, $\Floc$, $F_2$ are real. Hence $e^{E(Y)}Q^\natural_N(Y)$ is the
exponential of a real number. The outer partition functions $\mathcal Z_p(X)$ and the activities with $n=1$ are integrals of
such factors, with $\fc\cN>0$, and $Z$ contains $N=0$.
\end{proof}

\begin{lemma}\label{lem:7.46}
For an additive function $p$ on the cycle space of
$\Gp$ let $\Ad(p)$ be the minimum of $\sum_{e\in E}|N(e)|$ over the $N\in\mathbb Z^E$ with
$[N](\gamma)=p(\gamma)$ for every cycle $\gamma$ of $\Gp$, and $\Ad(p):=\infty$ if no such $N$ exists; for
$p\in\Glong$, $\Ad(p)$ is that of the restriction of $p$ to the cycles of $\Gp$. Then
$\Ad(0)=0$, $\Ad(-p)=\Ad(p)$, $\Ad(p+q)\le\Ad(p)+\Ad(q)$, and $\Ad([N])\le\sum_E|N|$ for every
$N\in\mathbb Z^E$; in particular $\Ad<\infty$ on $\Glong$. Fix $\epsE\in(0,c_0'/8]$ and put
$\sigma':=\epsE\sF$ and $\omega(p):=e^{\sigma'\Ad(p)}$. Then $\omega(0)=1\le\omega$ and $\omega(p+q)\le\omega(p)\,\omega(q)$
on $\Glong$, so $\mathcal A_\omega:=\{a:\Glong\to\mathbb C:\|a\|_\omega:=\sum_p|a(p)|\,\omega(p)<\infty\}$, with
$(a*b)(p):=\sum_qa(q)\,b(p-q)$, is a commutative Banach algebra with unit $\delta_0$, and
$\|a*b\|_\omega\le\|a\|_\omega\|b\|_\omega$.
\end{lemma}

\begin{proof}
Sums and negatives of admissible cochains are admissible, $N$ is admissible for
$[N]$, and $\sum_p|(a*b)(p)|\,\omega(p)\le\sum_{p,q}|a(q)|\,\omega(q)\,|b(p-q)|\,\omega(p-q)$.
\end{proof}

\begin{lemma}\label{lem:7.47}
Let $\Omega$ be a cell of the output. For all sites $x,y\in\Omega$,
$d_{\Gp}(x,y)\le\Dcell:=7r_0(2t_1+1)$, where $t_1:=1+\lceil(12k_P/\chi_{\rm N3})\ln k_P^3\rceil$ ($k_P$ in
coarse units). If $c_L\le1.99\cdot10^{-4}\chi_{\rm N3}$, then $\Dcell<L'$ for $\bGM\ge\bGMo$.
\end{lemma}

\begin{proof}[Sketch]
By (N3) of Theorem~\ref{thm:3.20}, there is no $\chi_{\rm N3}$-weak link. Hence, in the component $\mathfrak K$ of
$\Gp\cap\hat Q(Q)$ with $\Omega=Q\cap\mathfrak K$ as in Definition~\ref{def:3.12}, where $|\mathfrak K|\le k_P^3$, balls with $|B_t|\le|\mathfrak K|/2$ grow
by the factor $1+\chi_{\rm N3}/(6k_P)$ per step. Balls of radius $t_1$ about $x$, $y$ therefore meet. A coarse
path of length $\ell$ gives a $\Gp$-path of at most $7r_0(\ell+1)$ fine steps. With $L_c\ge35$
and $c_L\le1.99\cdot10^{-4}\chi_{\rm N3}$, we obtain $\Dcell<L'$ (Section~\ref{S:H1}).
\end{proof}

\begin{lemma}\label{lem:7.prim}
Let $N$ be a long wall field, $w:=2\pi\bGM^{-1/2}N$, and let $\Omega$ be a cell with the set
$\Kflat_\Omega$ of Proposition~\ref{prop:6.10}\,(iv). Then there is $h':\Kflat_\Omega\to P\mathbb Z$, constant on
$\Omega$, with $\nabla h'=w$ on $E(\Kflat_\Omega)$.
\end{lemma}

\begin{proof}
The graph $(\Kflat_\Omega,E(\Kflat_\Omega))$ is
connected, of diameter $d_\flat\le2+0.469\,\bGM/\mt=2+0.235L'$ (Proposition~\ref{prop:6.10}\,(iv), (v)). We define $h'(x)$ as the sum of $w$ along the path from $x_0\in\Omega$
to $x$ in a breadth-first spanning tree. Then $h'$ is $P\mathbb Z$-valued. Such a primitive can fail only on closed walks of about twice the
diameter of $\Kflat_\Omega$, which is less than $L'$. A bond of $E(\Kflat_\Omega)$ off the
tree closes a walk of length at most $2d_\flat+1<2L'$ with two tree paths, on which $N$ has
period $0$. Hence $\nabla h'=w$ on $E(\Kflat_\Omega)$. For $x\in\Omega$, the tree path from $x_0$ to $x$ and the
same-cell step back form a closed m3-walk of length at most
$d_\flat+\Dcell\le2+0.235L'+L+0.964L'<2L'$, by Lemma~\ref{lem:7.47} and Section~\ref{S:H1}, using $L'\ge16$ and
$L\le L'/(4\bGM)$. Therefore $h'(x)=h'(x_0)$ for each such $x$.
\end{proof}

%
\addtocounter{theorem}{-1}\renewcommand{\thetheorem}{\thesection.\arabic{theorem}a}%
\begin{definition}[Primitive-admissible wall fields]\label{def:7.primadm}
Let $k\in\{0,1\}$, $X$ a union of $\cU$-cells ($X\supseteq X_1$ if $k=1$), and $N$ a wall field on
$E(X)$ whose wall polymers have their closures in $X$. Then $N$ is \emph{primitive-admissible at
coupling $k$} if for every union $Y\subseteq X$ of elements of $N$ and every cell $\Omega\subseteq Y$
there is $h':\Kflat_\Omega\to P\mathbb Z$, constant on $\Omega$, with $\nabla h'=P(N\one_{E(Y)}+k\xi)$ on
$E(\Kflat_\Omega)$.
\end{definition}

\addtocounter{theorem}{-1}\renewcommand{\thetheorem}{\thesection.\arabic{theorem}b}%
\begin{lemma}[Primitives for long and inside fields]\label{lem:7.insideprim}
(a) Every long wall field on $E(X)$ whose wall polymers have their closures in $X$ is
primitive-admissible at $k=0$. (b) If $X\supseteq X_1$ and $N_X$ is an inside field at $k=1$ with
$p_1(N_X)\in\Glong$, then $N_X$ is primitive-admissible at $k=1$.
\end{lemma}

\begin{proof}
(a) For $Y$ a union of elements, $N\one_{E(Y)}$ is long (Lemma~\ref{lem:7.45}\,(a)); apply
Lemma~\ref{lem:7.prim}. (b) Let $Y\subseteq X$ be a union of elements, $N_Y:=N_X\one_{E(Y)}$ and
$\Omega\subseteq Y$ a cell. Each wall of $N_X$ on $E(\Kflat_\Omega)$ lies in $Y$ (first step of the proof
of Lemma~\ref{lem:7.34}\,(a)), and $\xi$ vanishes off $E(X)$. Hence $N_Y+\xi=(N_X+\xi)\one_{E(X)}$ on
$E(\Kflat_\Omega)$, and its period on a closed m3-walk $\gamma$ in $\Kflat_\Omega$ is $-p_1(N_X)(\gamma)$, which
is $0$ if $|\gamma|<2L'$. The breadth-first-tree argument of Lemma~\ref{lem:7.prim}, whose closed walks
(fundamental cycles of length at most $2d_\flat+1$, same-cell closures of length at most $d_\flat+\Dcell$) are
shorter than $2L'$, gives $h'$ with values in $P\mathbb Z$.
\end{proof}
\renewcommand{\thetheorem}{\thesection.\arabic{theorem}}%

\begin{proposition}[Long wall fields]\label{prop:7.long}
At $k=0$, the bounds of Subsection~\ref{ssec:7.6} and of Appendices~\ref{app:A}
and~\ref{app:B}, and Lemma~\ref{lem:7.53}, hold with $\cN(X)$ and $\cN_\sigma(X)$
replaced by $\Nlong(X)$ and
$\Nlong_\sigma(X):=\{N_X\in\Nlong(X):N_X|_{E(Z)}=N_Z,\ Z\text{ a union of elements of }N_X\}$,
for seed configurations $\sigma=(Z,N_Z)$ with $N_Z\in\Nlong(Z)$, with the
integrand of Lemma~\ref{lem:7.wall} and the same constants. In
Lemma~\ref{lem:7.34}\,(a) the label is replaced by the primitive $h'$ of
Lemma~\ref{lem:7.prim}. The same holds at $k=1$ for the inside fields $N_X$ with $p_1(N_X)\in\Glong$, with the
primitive of Lemma~\ref{lem:7.insideprim}\,(b) and $\cP'=\cP'_G$; in general, the bounds hold for every class of
primitive-admissible wall fields (Definition~\ref{def:7.primadm}).
\end{proposition}

\begin{proof}[Sketch]
For the sums over configurations, in Lemma~\ref{lem:7.32}\,(b) and in steps (2), (3) of the proof of
Proposition~\ref{prop:A.9}, the injection $N_X\mapsto(N_I)_{I\subseteq X}$ into integer
interface values is all that is needed, and it is valid for all wall fields. At a fixed configuration the label
occurs in four places: the phases (Lemma~\ref{lem:7.wall}), $w=0$ inside cells, the walls in
\eqref{eq:7.6.untr} and, in Lemma~\ref{lem:7.34}\,(a), a primitive of $w_Y$ on $E(\Kflat_\Omega)$ constant on
$\Omega$. In the last case $N\one_{E(Y)}$ is long by Lemma~\ref{lem:7.45}\,(a), and Lemma~\ref{lem:7.prim} gives such a primitive
(details in Section~\ref{S:H2}). Only the existence of the primitive is used, so the argument applies to every
primitive-admissible field.
\end{proof}

Heuristically, a fixed fraction of the $\Floc$ of each configuration accounts for its walls,
which carry at least the area of their class.

\begin{proposition}[Absolute convergence]\label{prop:7.51}
There is $c<\infty$, depending on $\tau$ and the fixed constants only, such that
$\|\mathcal Z(X)\|_\omega\le\fc\cN\,e^{(c+c_4')|X^c|}$ for every union $X$ of $\cU$-cells, with $c_4'$ from
Lemma~\ref{lem:7.32} at $c_4=c'/2$. Moreover
$\sum_Y\sum_{N_Y}|\cK^{(Z)}_{N_Y}(Y)|\,\omega([N_Y])\,e^{\alpha|Y|}<\infty$ for the seeds $Z$ of
Theorem~\ref{thm:7.29}\,(b), (c), and $\sum_{N_X}|\cK_{N_X}(X)|<\infty$ in Theorem~\ref{thm:7.29}\,(a),
the sum over the inside fields $N_X$ at $k=1$ with $p_1(N_X)\in\Glong$.
Hence the sums of Theorem~\ref{thm:7.29} converge absolutely.
\end{proposition}

\begin{proof}
By H\"older's inequality as in
Proposition~\ref{prop:7.37}\,(b), (d) at $k=0$, with Lemma~\ref{lem:7.33} at $N=0$ and
Proposition~\ref{prop:7.long}, the term of $N$ in $\mathcal Z_p(X)$ is at most $\fc\cN e^{c|X^c|}e^{-c'\Floc(X^c,N)}$.
Moreover, $\omega([N])\le e^{\sigma'\sum|N|}\le e^{(\epsE/2)\Floc(X^c,N)}$, the first inequality by Lemma~\ref{lem:7.46} and the second by Theorem~\ref{thm:7.14}
and $|N|\le N^2$. We also have $\epsE\le c'$, and by Lemma~\ref{lem:7.32}\,(b) the sum of $e^{-(c'/2)\Floc(X^c,N)}$ over the wall fields on $E(X^c)$
is at most $e^{c_4'|X^c|}$. We bound the seeded sums by
Lemma~\ref{lem:7.53}\,(a), (c$'$), with $\eps=\epsE/2$ at $k=0$ and $\eps=0$ at $k=1$. At $k=1$ the
inside fields with $p_1(N_X)\notin\Glong$ contribute $0$ to Theorem~\ref{thm:7.29}\,(a), as
$\mathcal Z_p(X)=0$ for $p\notin\Glong$ by Definition~\ref{def:7.27}(c). For the inside fields at $k=1$ with
$p_1(N_X)\in\Glong$, Lemma~\ref{lem:7.53}\,(a) applies by Lemma~\ref{lem:7.insideprim}\,(b) and
Proposition~\ref{prop:7.long}.
\end{proof}

\paragraph{Constants of the seeded sums.}
The constant $C_9$ bounds the number of interfaces within Euclidean distance $r$ of a given interface
or site by $C_9(r/L+1)^3$. The Euclidean link range is $g:=2L'+4D_E+2$ ($D_E$ of
Proposition~\ref{prop:7.22}). We put $c_2:=c_S+2=O(\log\tau)$ and $c_{16}:=c_{16}^D+\etaseed\to0$ as $\tau\to\infty$,
where $c_{16}^D$ is defined in \eqref{eq:7.6.c16D} and $c_S$, $\etaseed$ in Section~\ref{S:H1}. Let $\cP':=\cP'_G$ if $k=1$ and $\cP':=1$ if $k=0$. For a seed configuration $\sigma=(Z,N_Z)$, $Z^\Sigma$ is the union of its
wall polymers and $Z^o:=Z\setminus Z^\Sigma$. For an anchor $S_0\subseteq U$, that is, a nonempty union of
$\cU$-cells with $S_0\supseteq X_1$ if $k=1$, and for $N_X\in\cN(X)$, $\sat_{N_X}(S_0)$ is the union of the
elements of $N_X$ meeting $S_0$.

\begin{lemma}[Seeded sums at $k=0$ and $k=1$]\label{lem:7.53}
Let $(U,k)$ be admissible, $\bGM\ge\bGMo$, and $\eps\in[0,c'/2]$. (a) \emph{(Variable first set.)} For every anchor $S_0\subseteq U$,
\[
\sum_{S_0\subseteq X\subseteq U}\ \sum_{N_X\in\cN(X)}\bigl|\cK^{(\sat_{N_X}(S_0))}_{N_X}(X)\bigr|\,e^{\alpha|X|+\eps\Floc(X,N_X)}\le\cP'\,e^{c_2|S_0|}.
\]
(b) \emph{(Strict containment.)} The sum in (a) without the term $X=S_0$, $N_X=0$ is at
most $c_{16}\,\cP'\,e^{c_2|S_0|}$. (c$'$) \emph{(Prescribed seed walls, $k=0$.)} For every seed
configuration $\sigma=(Z,N_Z)$ in $(U,0)$ with $Z^o=\emptyset$, every
$\eps\in[0,c'/8]$ and $c_w:=c'/16$,
\[
\sum_{Z\subseteq Y\subseteq U}\ \sum_{N_Y\in\cN_\sigma(Y)}|\cK^{(Z)}_{N_Y}(Y)|\,e^{\alpha|Y|+\eps\Floc(Y,N_Y)}\,e^{c_w\Floc(Z,N_Z)}
\le\prod_{\mathcal S}e^{-(c'/2)\Floc(\mathcal S)},
\]
the product over the wall polymers $\mathcal S$ of $N_Z$.
\end{lemma}

We prove Lemma~\ref{lem:7.53} in Appendix~\ref{app:A}. Proposition~\ref{prop:A.9} treats a fixed seed
configuration, and for (a), (b) the entropy of the first set is controlled by its walls.

\begin{theorem}[{The Kirkwood\textendash Salsburg system in $\mathcal A_\omega$; \cite[Lemma~A4.1]{BF80} on $\Gp$}]\label{thm:7.52}\label{prop:7.52}
Fix a linear order on the $\cU$-cells; for $X\ne\emptyset$ let $\ell(X)$ be its last
$\cU$-cell and $X^*:=X\setminus\ell(X)$. For a $\cU$-cell $\Delta$ and a union $Y\ni\Delta$
of $\cU$-cells let $\cK'_\Delta(Y)\in\mathcal A_\omega$ be the sequence
$q\mapsto\sum\cK^{(\sat(\Delta))}_{N_Y}(Y)$, the sum over the $N_Y$ of
\eqref{eq:7.7.KS} with $[N_Y]=q$ and $(Y,N_Y)\ne(\Delta,0)$. There is exactly one
family $(r(X))_X$ in $\mathcal A_\omega$ with $r(\emptyset)=\delta_0$ and
$\max_{X\ne\emptyset}e^{-\alpha|X|}\|r(X)\|_\omega<\infty$ such that, for every $X\ne\emptyset$,
\begin{equation}\label{eq:7.7.r}
r(X^*)=\tilde\cK(\ell(X))\,r(X)+\sum_{Y\ni\ell(X),\ Y\cap X^*=\emptyset}\cK'_{\ell(X)}(Y)*r(X^*\cup Y),
\end{equation}
and it satisfies, with $\alpha=\ln8$: (a) $\|r(X)\|_\omega\le\frac8{31}e^{\alpha|X|}$ for every $X\ne\emptyset$;
(b) $\mathcal Z(X)=r(X)*\mathcal Z(\emptyset)$ for every $X$, that is, $\mathcal Z_p(X)=\sum_qr(X)(q)\,Z_{p-q}$.
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:7.29}\,(b) the system has the form of \cite[(A4.5)]{BF80} with values in
$\mathcal A_\omega$, and we run the contraction of \cite[(A4.6) to (A4.13)]{BF80} with $\|\cdot\|_\omega$ in place of $|\cdot|$. The
normalization $r(\emptyset)=\delta_0$ replaces the division by $Z$, which is not available for the sequence
$\mathcal Z(\emptyset)$, and (b) follows from uniqueness in the finite box.

We need two bounds. (K1) $\tilde\cK(\Delta)\ge\frac12$, by Lemma~\ref{lem:7.43}, and of the two bounds this is the one that rests on positivity, at $k=0$ in the proof of
Lemma~\ref{lem:7.43}. (K2)
$\sum_{Y\ni\Delta}\|\cK'_\Delta(Y)\|_\omega e^{\alpha|Y|}\le\frac18$. By Lemma~\ref{lem:7.46} and Theorem~\ref{thm:7.14},
$\omega([N_Y])\le e^{\sigma'\sum|N_Y|}\le e^{(\epsE/2)\Floc(Y,N_Y)}$. Hence (K2) follows from Lemma~\ref{lem:7.53}\,(b) with
$S_0=\Delta$, $k=0$ and $\eps=\epsE/2$, over long fields by Proposition~\ref{prop:7.long}, and from the bound
$c_{16}e^{c_2}\le\frac18$ of Lemma~\ref{lem:A.11}.

\emph{Step 1 (the map).} Let $\mathfrak B$ be the Banach space of families
$s=(s(X))_{X\ne\emptyset}$ in $\mathcal A_\omega$ with $|||s|||:=\max_{X\ne\emptyset}e^{-\alpha|X|}\|s(X)\|_\omega$, put $s(\emptyset):=\delta_0$,
and let $(Ts)(X)$ be $\tilde\cK(\ell(X))^{-1}$ times $s(X^*)$ minus the sum in \eqref{eq:7.7.r}.
A family satisfies \eqref{eq:7.7.r} if and only if $Ts=s$.

\emph{Step 2 (contraction).} By $\|a*b\|_\omega\le\|a\|_\omega\|b\|_\omega$, $|X^*\cup Y|=|X|-1+|Y|$,
$\|s(\emptyset)\|_\omega=1$, (K1) and (K2), we obtain
$e^{-\alpha|X|}\|(Ts)(X)\|_\omega\le2e^{-\alpha}[\max(1,|||s|||)+|||s|||/8]=\frac14\max(1,|||s|||)+|||s|||/32$, and in the same way $|||Ts-Ts'|||\le2e^{-\alpha}(1+\frac18)|||s-s'|||=\frac9{32}|||s-s'|||$, since
$(s-s')(\emptyset)=0$. Let $r$ be the fixed point of $T$.

\emph{Step 3 ((a)).} If $|||r|||\ge1$, Step 2 gives $|||r|||\le\frac9{32}|||r|||$, which is
impossible. Therefore $|||r|||\le\frac14+|||r|||/32$, that is, $|||r|||\le\frac8{31}$.

\emph{Step 4 ((b)).} We set $s(X):=r(X)*\mathcal Z(\emptyset)-\mathcal Z(X)$, which has $s(\emptyset)=0$. By associativity and
commutativity, $X\mapsto r(X)*\mathcal Z(\emptyset)$ satisfies \eqref{eq:7.7.r}, and so does $X\mapsto\mathcal Z(X)$, by
\eqref{eq:7.7.KS} with $X_0=X^*$ and $\Delta=\ell(X)$. Consequently $s$ satisfies the linear part of
\eqref{eq:7.7.r} with $s(\emptyset)=0$. Moreover, $|||s|||<\infty$, since there are finitely many $X$ and
$\|\mathcal Z(X)\|_\omega<\infty$ by Proposition~\ref{prop:7.51}. As in Step 2, $|||s|||\le\frac9{32}|||s|||$, and hence $s=0$.
\end{proof}

The bounds in the proof above are uniform in $\Gp$, $\Lambda^\circ$ and $C$. We obtain the invertibility of $\mathcal Z(\emptyset)$
in $\mathcal A_\omega$ as a consequence of Theorem~\ref{thm:7.52}\,(b): $r(V)*\mathcal Z(\emptyset)=\mathcal Z(V)=\fc\cN\,\delta_0$.