%
%
\providecommand{\bGM}{\beta_{\mathrm{GM}}}
\providecommand{\Floc}{F_1^{\mathrm{loc}}}
\providecommand{\cU}{\mathcal U}
\providecommand{\sF}{\sigma_F}
\providecommand{\epsE}{\varepsilon_E}
\providecommand{\Funit}{F_{\mathrm{unit}}}
\providecommand{\Ad}{\mathrm{A}}
\providecommand{\RNF}{R_{\mathrm{NF}}}
\providecommand{\thNF}{\theta_{\mathrm{NF}}}
\providecommand{\cNF}{c_3^{\mathrm{NF}}}
\providecommand{\fS}{\mathfrak S}
\providecommand{\Glong}{\Gamma_{\mathrm{long}}}
\providecommand{\Nlong}{\mathcal N^{\mathrm{long}}}

\subsection{Frozen flux costs area: group charge and the count}
\label{sec:7.8}\label{ssec:7.8}

In this subsection we prove that the partition function at fixed class $r$ carries
the area weight of $r$ (Proposition~\ref{prop:7.57}). The bounds of this subsection are the only ones that use the handles of the
insulating set and (NF$_*$), which holds by (R10) of Theorem~\ref{thm:3.20}. In a configuration of class $r$, a group of wall polymers
realizing $r$ is re-expanded as a seed at $X=\emptyset$, and its entropy is controlled by
Proposition~\ref{prop:7.56}.

\paragraph{Polymers, realizations and the reference realization.}
Fix $r\in\Glong\setminus\{0\}$ with $Z_r\neq0$. For a configuration $N$ of $Z_r$
(Definition~\ref{def:7.27}), $\mathcal P(N)$ is the set of its wall polymers, and we put
$N_{\mathcal S}:=N\one_{E(\mathcal S)}$ for $\mathcal S\in\mathcal P(N)$ and $N_R:=\sum_{\mathcal S\in R}N_{\mathcal S}$, $q_R:=[N_R]$, $|R|:=\|N_R\|_1$ for
$R\subseteq\mathcal P(N)$. By Proposition~\ref{prop:7.9}\,(a), each wall bond lies in exactly one wall polymer.
Hence $\sum_{\mathcal S\in\mathcal P(N)}N_{\mathcal S}=N$ and $q_{\mathcal P(N)}=r$, and $q_R\in\Glong$
by Lemma~\ref{lem:7.45}\,(a). A family $R$ \emph{realizes} $r$ if $q_R=r$, and is
\emph{irreducible} if no nonempty $T\subseteq R$ has $q_T=0$. An inclusion-minimal realizing
subfamily of $\mathcal P(N)$ is irreducible: if $\emptyset\neq T\subseteq R$ had $q_T=0$, then $T\neq R$ as $r\neq0$,
and $R\setminus T$ would realize $r$. Let $\fS(r)$ be the set of families $R$, with their wall
configurations, that occur as irreducible realizations of $r$ in some configuration
of $Z_r$. We fix $R_0\in\fS(r)$ of least area by a deterministic choice, so that
$|R|\ge|R_0|\ge\Ad(r)$ for all $R\in\fS(r)$, the last inequality by Lemma~\ref{lem:7.46}. A polymer is determined by
its wall interfaces and their values, since closures and links are functions of the
walls. Thus $R$ is determined by $N_R$: the wall components of $N_R$ are those of
the members of $R$, closures from different members are not linked (they are distinct wall
polymers of one configuration), so the wall polymers of $N_R$ are exactly the members of $R$. We also set $\nu(R):=\sum_{I\subseteq\supp N_R}N_I^2$, $c_w:=c'/16$,
$\lambda_E:=2c_w\sF=\frac{c'}8\sF$ and
$\Theta_4(r):=\sum_{R\in\fS(r)}e^{-\lambda_E|R|}\prod_{\mathcal S\in R}e^{-(c'/2)\Floc(\mathcal S)}$. The notion of c-connectedness is that of
Definition~\ref{def:3.17}, and distances between bond sets are Euclidean.

\begin{lemma}\label{lem:7.54}
Let $R\in\fS(r)$. Then $N_R-N_{R_0}=\sum_k\eps_k\nabla1_{P_k}$ with $\eps_k\in\{-1,1\}$, where (i) each $P_k$ is
a nonempty c-connected cell set with c-connected complement in its component $\mathfrak K$
of $\Gp$; (ii) $\partial_{\Gp}P_k\subseteq\supp N_R\cup\supp N_{R_0}$; (iii)
$\sum_k|\partial_{\Gp}P_k|=\|N_R-N_{R_0}\|_1\le|R|+|R_0|\le2|R|$; and (iv) every $P_k$ is joined to $\supp N_{R_0}$
by a chain $P_k=P_{k_0},\dots,P_{k_j}$ with $\dist(\partial P_{k_{i-1}},\partial P_{k_i})\le g$ and
$\dist(\partial P_{k_j},\supp N_{R_0})\le g$.
\end{lemma}

\begin{proof}
(1) Since $[N_R-N_{R_0}]=q_R-q_{R_0}=0$, Definition~\ref{def:7.27} gives $N_R-N_{R_0}=\nabla f$ with $f:V\to\mathbb Z$ constant on each cell.

(2) By the coarea formula for the level sets of $f$, $\nabla f=\sum_t\pm\nabla1_{U_t}$ with
$\sum_t|\partial U_t|=\|\nabla f\|_1$. We split each $U_t$ into its c-components $W$ and write
$\nabla1_W=-\sum_i\nabla1_{C_i}$ over the c-components $C_i$ of $\mathfrak K\setminus W$, where $\mathfrak K$ is the component of $W$.
Since each $C_i$ is c-connected with c-connected complement and the cuts partition, (i)
to (iii) hold.

(3) For (iv) we use irreducibility, which forbids groups of pieces
far from $R_0$, since these would be sub-families of $R$ of class zero. We group the pieces into clusters by the relation ``cuts
within distance $g$''. Let $\mathcal V$ be a cluster no cut of which is within $g$ of $\supp N_{R_0}$,
and let $B:=\bigcup_{k\in\mathcal V}\partial P_k$. Within distance $g$ of $B$, $N_{R_0}=0$ and the other pieces
contribute nothing. It follows that $N_R=N_{\mathcal V}:=\sum_{k\in\mathcal V}\eps_k\nabla1_{P_k}$ in this region, and each wall bond of
$N_R$ within distance $g$ of $B$ lies in $B$. Two wall bonds of one polymer are joined by
a chain of wall bonds of that polymer with consecutive gaps at most $g$, since walls of
one $S_\beta$ have gaps less than $2L'+2D_E$ and linked closures add at most $2D_E+2$.
Hence the polymers of $R$ meeting $B$ form a sub-family $T$ with $N_T=N_R1_B=N_{\mathcal V}$, the
gradient of a cell-constant function, and therefore $q_T=0$. By irreducibility $T=\emptyset$, so that
$N_{\mathcal V}=0$. By (iii) the terms of $N_R-N_{R_0}$ do not cancel, and hence neither do those of the
sub-sum: $\|N_{\mathcal V}\|_1=\sum_{k\in\mathcal V}|\partial P_k|$. So $N_{\mathcal V}=0$ forces $\partial P_k=\emptyset$ for every
$k\in\mathcal V$. This is impossible, since by (i) each $P_k$ is a nonempty proper cell set of its
component $\mathfrak K$. Therefore no such cluster exists: every cluster has a cut within $g$ of
$\supp N_{R_0}$, and chaining inside the cluster gives (iv).
\end{proof}

For a finite set $B$ of interfaces and $s>0$, the $s$-clusters of $B$ and their number
$\varphi_s(B)$ are those of Definition~\ref{def:3.17} (for a cut, of its set of interfaces),
and $\mathrm{MST}(B)$ denotes a minimal spanning tree of the complete graph on $B$ with edge
lengths $\ell_e$ the distances of interfaces. Let $j(\ell):=\#\{j\ge0:2^jL<\ell\}$.

\begin{lemma}[Kruskal's identity]\label{lem:7.55}
For every $s>0$, $\#\{e\in\mathrm{MST}(B):\ell_e>s\}=\varphi_s(B)-1$, and
$\sum_{e\in\mathrm{MST}(B)}j(\ell_e)=\sum_{j\ge0}(\varphi_{2^jL}(B)-1)$.
\end{lemma}

\begin{proof}
The edges of $\mathrm{MST}(B)$ of length $\le s$ form a spanning forest of the threshold
graph $\{\ell\le s\}$, since Kruskal's algorithm processes them first. Thus they number
$|B|-\varphi_s(B)$. For the second identity we sum $1[\ell_e>2^jL]$ over $j\ge0$ and $e$.
\end{proof}

Let $\kappa_{\mathrm{NF}}:=4(c_g+4)/c_g^2$, and let $R_{\mathrm{NF}}$, $\thNF$, $\cNF$, $J$ be as in Definition~\ref{def:3.17}.
Applying (R10) in the form of Proposition~\ref{prop:3.19} to the pieces $P_k$ of Lemma~\ref{lem:7.54}, we obtain
\begin{equation}\label{eq:7.8.NF}
J(P_k)\le\thNF\,\Funit\,|\partial_{\Gp}P_k|/L^2 ,
\end{equation}
with the cut measured in fine bonds, of which a coarse face carries at least $r_0^2$. Here $J$ depends
only on the cut.

\begin{proposition}[The count]\label{prop:7.56}
For $\bGM\ge\bGMo$, every $r\in\Glong\setminus\{0\}$ and every $\nu\ge1$,
$\#\{R\in\fS(r):\nu(R)=\nu\}\le\exp((c'/2)\Funit\,\nu)$. Consequently $\Theta_4(r)\le\frac14$.
\end{proposition}

\begin{proof}
The idea is that, by Kruskal's identity, specifying the positions of the cut patches costs, per dyadic scale, the number of
clusters minus one. Below $R_{\mathrm{NF}}$ this is at most $\log_2(R_{\mathrm{NF}}/L)$ per interface, within the
factor $e^{(c'/2)\Funit}$, and above $R_{\mathrm{NF}}$ it is $J$, bounded by the cut \eqref{eq:7.8.NF}.
We first bound the count and then prove the second assertion, $\Theta_4\le\frac14$.

Fix $R$ with $\nu(R)=\nu$. Let $S$, $S_0$ be the interface sets of $\supp N_R$ and
$\supp N_{R_0}$, put $n:=|S|\le\nu$, $n_0:=|S_0|$ and $T:=S\cup S_0$, and let $I_k$ be the interface set of $\partial P_k$.
An interface has between $c_g'L^2$ and $4L^2$ bonds by (R8) and since block sides are less than
$2L$, and $|R_0|\le|R|\le4L^2\nu$. Hence $n_0\le(4/c_g')\nu$, and by Lemma~\ref{lem:7.54}(iii)
$\sum_k|\partial P_k|\le8L^2\nu$, $\sum_k|I_k|\le(8/c_g')\nu\le\kappa_{\mathrm{NF}}\nu$. The last inequality follows from $8/c_g'\le128/(9c_g)\le4(c_g+4)/c_g^2$, which holds since $c_g'\ge\frac9{16}c_g$ by Section~\ref{ssec:D.5}(e) and $c_g\le\frac14$.
Since a bond where $N_R\neq N_{R_0}$ lies on some $\partial P_k$, we have $S\subseteq S_0\cup\bigcup_kI_k\subseteq T$.

\emph{Step 1 (a spanning forest).} For each $k$ let $t_k$ be the least $j$ for which a chain
$P_k=P_{k_0},\dots,P_{k_j}$ as in Lemma~\ref{lem:7.54}(iv) exists. Let $H$ be the graph on $T$ whose edges are the
edges of the $\mathrm{MST}(I_k)$ and, for each $k$, one parent edge at distance at most $2g$: from an
interface of $I_k$ to one of $S_0$ if $t_k=0$, and otherwise to one of $I_{k'}$, where $P_{k'}$ is the
second piece of a chain of length $t_k$, so that $t_{k'}=t_k-1$. Here we use that a bond and its dual coarse face are within distance
$\sqrt3\,r_0\le g/2$. Each node of $T\setminus S_0$ lies in some $I_k$, and $t$ decreases along parent
edges. Hence the parent edges lead to $S_0$. Consequently $H$ contains a spanning forest $F$ of $T$ with roots in $S_0$,
as one sees by adding a virtual node joined to all of $S_0$ and removing it from a spanning
tree. Let
$B:=\sum_{e\in F}j(\ell_e)\le\sum_{e\in H}j(\ell_e)$.

\emph{Step 2 (encoding).} We record the roots (at most $2^{n_0}$ choices), the shape of the
plane forest on $|T|\le n+n_0$ nodes (at most $4^{n+n_0}$), and for each child the number
$j_e$ and the child among the at most $C_9(2^{j_e}+1)^3\le8C_9\,8^{j_e}$ interfaces within
distance $2^{j_e}L$ of its parent. Sequences of $|F|$ nonnegative integers with sum at
most $B_{\max}$ (the bound of Step~3, a function of $\nu$ only) number at most
$2^{B_{\max}+|F|}$. We then record $S\subseteq T$ (at most $2^{n+n_0}$) and the values $N_I$ with
$\sum N_I^2=\nu$ (at most $4^\nu$, counting signs and compositions of $\sum|N_I|\le\nu$). Since $R$ is
determined by $N_R$, we obtain $\ln\#\{R:\nu(R)=\nu\}\le C(n+n_0+\nu)+B_{\max}\ln16$ with $C:=\ln(64C_9)+2$.

\emph{Step 3 (the bound on $B$).} By Lemma~\ref{lem:7.55},
$\sum_{e\in\mathrm{MST}(I_k)}j(\ell_e)=\sum_{j\ge0}(\varphi_{2^jL}(I_k)-1)$. The scales with $2^jL<R_{\mathrm{NF}}$
number at most $j_*:=\cNF\Funit/\ln2+1$, each contributing at most $|I_k|-1$, while the scales
with $2^jL\ge R_{\mathrm{NF}}$ contribute exactly $J(P_k)$, by Definition~\ref{def:3.17}. A parent edge
has $j\le j_g:=\log_2(2g/L)+1=O(\ln\bGM)$, and there are at most $\sum_k|I_k|\le\kappa_{\mathrm{NF}}\nu$
pieces. With \eqref{eq:7.8.NF},
\[
B\le(j_*+j_g)\kappa_{\mathrm{NF}}\nu+\thNF\Funit\sum_k|\partial P_k|/L^2
\le\Bigl(\frac{\cNF\kappa_{\mathrm{NF}}}{\ln2}+8\thNF\Bigr)\Funit\,\nu+(1+j_g)\kappa_{\mathrm{NF}}\nu=:B_{\max}.
\]

\emph{Step 4.} By Definition~\ref{def:3.17}, $4\cNF\kappa_{\mathrm{NF}}=c_0'/4$ and $8\thNF\ln16\le c_0'/8$.
Hence $\ln\#\le\frac38c_0'\Funit\nu+O((1+\ln\bGM)\nu)\le\frac{c'}2\Funit\nu$ for
$\bGM\ge\bGMo$, as $\Funit$ is exponentially large by Section~\ref{S:H1}.

It remains to prove $\Theta_4\le\frac14$, which rests on the factor $2$ of Lemma~\ref{lem:7.32}(a), that is, on the constant $1/18$ of
Theorem~\ref{thm:7.14}, set against the bound of the count. By Lemma~\ref{lem:7.32}(a), $\Floc(\mathcal S)\ge2\Funit\,\nu(\mathcal S)$ for each
wall polymer $\mathcal S$. Hence the summand of $\Theta_4$ is at most $e^{-c'\Funit\nu(R)}$, and
$\Theta_4(r)\le\sum_{\nu\ge1}e^{(c'/2)\Funit\nu}e^{-c'\Funit\nu}\le2e^{-(c'/2)\Funit}\le\frac14$.
\end{proof}

\begin{proposition}[Frozen flux costs area]\label{prop:7.57}
For every $r\in\Glong$, $Z_r\le e^{-\sigma'\Ad(r)}Z$.
\end{proposition}

\begin{proof}
We extract an irreducible realization from each configuration and
re-expand it as a seed, which pays for its own cells and shape from its own
$\Floc$ and leaves the fraction $c'/2$ of it. The outer factors are bounded by
Theorem~\ref{thm:7.52} and by the quantity being bounded, which closes the
argument. We put $C_{\mathrm{FF}}:=\sup_{s\in\Glong}\omega(s)Z_s/Z$. Then $1\le C_{\mathrm{FF}}<\infty$, by taking
$s=0$ and by Proposition~\ref{prop:7.51}. Let $r\neq0$ with $Z_r\neq0$.

(1) \emph{Union bound.} For each configuration we choose one irreducible realization
$R(N)\subseteq\mathcal P(N)$ by a fixed rule. All terms are nonnegative by Lemma~\ref{lem:7.44}. Hence
$Z_r\le\sum_{R\in\fS(r)}Z_r(R)$, with $Z_r(R)$ as in Theorem~\ref{thm:7.29}\,(c). Dropping the
condition $R=R(N)$ only enlarges the right side.

(2) \emph{Re-expansion.} By Theorems~\ref{thm:7.29}\,(c) and~\ref{thm:7.52}\,(b), with
$\sigma:=(\bigcup R,N_R)$,
$Z_r(R)\le\sum_{Y\supseteq\bigcup R}\sum_{N_Y\in\Nlong_\sigma(Y)}|\cK^{(\bigcup R)}_{N_Y}(Y)|\sum_q|r(Y)(q)|\,Z_{r-[N_Y]-q}$.

(3) \emph{Weights.} By Lemma~\ref{lem:7.46}, $\omega(r)\le\omega([N_Y])\,\omega(q)\,\omega(r-[N_Y]-q)$, and
$\omega([N_Y])\le e^{(\epsE/2)\Floc(Y)}$ by Theorem~\ref{thm:7.14}. The definition of $C_{\mathrm{FF}}$ and
Theorem~\ref{thm:7.52}\,(a), which applies as $Y\supseteq\bigcup R\neq\emptyset$, give
$\sum_q|r(Y)(q)|\,\omega(q)\,\omega(r-[N_Y]-q)Z_{r-[N_Y]-q}\le\frac8{31}e^{\alpha|Y|}C_{\mathrm{FF}}Z$.

(4) \emph{Activity.} So that the sum takes the form $\Theta_4(r)$ of Proposition~\ref{prop:7.56}, we insert
$1\le e^{c_w\Floc(\bigcup R)}e^{-\lambda_E|R|}$, which is valid since
$\Floc(\bigcup R)\ge2\sF|R|$ and $2c_w\sF=\lambda_E$. The final estimate of this step holds without this insertion as well, since the proof of $\Theta_4\le\frac14$ bounds $e^{-\lambda_E|R|}$ by $1$ and Lemma~\ref{lem:7.53}\,(c$'$) without the factor $e^{c_w\Floc(Z,N_Z)}\ge1$ follows from (c$'$). Lemma~\ref{lem:7.53}\,(c$'$) over long fields,
which holds by Proposition~\ref{prop:7.long}, applied with the seed configuration $\sigma$, all of whose $\cU$-cells lie
in wall polymers, and with $\eps=\epsE/2\le c'/8$, bounds the sum over $Y$ and $N_Y$, including
$e^{\alpha|Y|}e^{(\epsE/2)\Floc(Y)}e^{c_w\Floc(\bigcup R)}$, by $\prod_{\mathcal S\in R}e^{-(c'/2)\Floc(\mathcal S)}$. Collecting (1)
to (4), we obtain $\omega(r)Z_r/Z\le\frac8{31}C_{\mathrm{FF}}\Theta_4(r)\le\frac2{31}C_{\mathrm{FF}}$ by Proposition~\ref{prop:7.56}.

(5) \emph{The supremum.} With $s=0$, we obtain $C_{\mathrm{FF}}\le\max(1,\frac2{31}C_{\mathrm{FF}})$. Since $C_{\mathrm{FF}}<\infty$, this gives
$C_{\mathrm{FF}}=1$.
\end{proof}

\medskip\noindent\emph{Remark.} The position entropy is part of the count of $\fS(r)$, and so the bound just
proved carries no position factor, such as a sum over a cycle detecting $r$ or $|\gamma|$, even for a long handle with a tapered thin neck, since its cuts of one
class are compared with $R_0$ by Lemma~\ref{lem:7.54}.
\medskip

\begin{lemma}\label{lem:7.58}
Let $\alpha=\ln8$, $\epsE\in(0,c_0'/8]$, $\sigma'=\epsE\sF$ and $\bGM\ge\bGMo$. For every nonempty union
$X$ of $\cU$-cells and every $p\in\Glong$, $0\le\mathcal Z_p(X)/Z\le\frac8{31}e^{\alpha|X|}e^{-\sigma'\Ad(p)}$. The
bound is uniform in $\Gp$, $\Lambda^\circ$ and $C$.
\end{lemma}

\begin{proof}
By Theorem~\ref{thm:7.52}\,(b) and
Lemma~\ref{lem:7.46}, we have $\omega(p)\mathcal Z_p(X)/Z\le\sum_q|r(X)(q)|\,\omega(q)\cdot\omega(p-q)Z_{p-q}/Z$, and this is at most $\|r(X)\|_\omega\le\frac8{31}e^{\alpha|X|}$ by Proposition~\ref{prop:7.57}
and Theorem~\ref{thm:7.52}\,(a). Nonnegativity follows from Lemma~\ref{lem:7.44}.
\end{proof}