%
%
%
\providecommand{\bGM}{\beta_{\mathrm{GM}}}
\providecommand{\mt}{\tilde m}
\providecommand{\Gp}{G'}
\providecommand{\cU}{\mathcal U}
\providecommand{\sF}{\sigma_F}
\providecommand{\sV}{\sigma_V}
\providecommand{\epsE}{\varepsilon_E}
\providecommand{\Ad}{\mathrm{A}}
\providecommand{\cnear}{c_{\mathrm{near}}}
\providecommand{\rbox}{r_{\square}}

\subsection{The deterministic area law}
\label{sec:7.9}\label{ssec:7.9}

\providecommand{\Floc}{F_1^{\mathrm{loc}}}
\providecommand{\cP}{\mathcal P}
\providecommand{\fc}{\mathfrak c}
\providecommand{\cN}{\mathcal N}
\providecommand{\cK}{\mathcal K}
\providecommand{\kbox}{k_{\square}}

We now prove the deterministic area law, Theorem~\ref{thm:7.61}.

\begin{lemma}\label{lem:7.59}
Let $X\supseteq X_1$ be a union of $\cU$-cells, $N_X$ an inside field at $k=1$ on $X$, and
$p:=p_1(N_X)$ its outer class (Definition~\ref{def:7.27}). Then
$\sum_{e\in E(X)}|N_X(e)|+\Ad(p)\ge A_{\Gp}(C)$.
\end{lemma}

\begin{proof}
Since linking numbers are additive, the inside walls and a cheapest cochain with the
forced periods form a wall field at $k=1$, and by Theorem~\ref{thm:7.17} this field spans $C$ modulo the frozen complex. Let $N'\in\mathbb Z^E$ realize $\Ad(p)$, which is finite since $-\xi-N_X1_{E(X)}$ has periods $p$. By definition of $p_1$, $N_X1_{E(X)}$ has periods
$-\mathrm{lk}(\cdot,C)-p$, and hence $M:=N_X1_{E(X)}+N'$ has periods $-\mathrm{lk}(\cdot,C)$. Since $\xi$ has periods
$\mathrm{lk}(\cdot,C)$ by Lemma~\ref{lem:7.6}(d), we get $M+\xi=\nabla m$ for some $m:V\to\mathbb Z$, and $M=\nabla m-\xi$ is
the $k=1$ wall field of $m$. By Theorem~\ref{thm:7.17}, $\sum_E|M|\ge\#\{M\neq0\}\ge A_{\Gp}(C)$, and
$\sum_E|M|\le\sum_{E(X)}|N_X|+\sum_E|N'|=\sum_{E(X)}|N_X|+\Ad(p)$.
\end{proof}

\begin{lemma}\label{lem:7.60}
With $C_{\cU}$ and $c_u$ as in (R7) and $C_X:=3C_{\cU}/c_u$,
$|X_1|\le C_X(\mt^2RT+3C_{\cU}\mt(R+T)+9C_{\cU}^2)$.
\end{lemma}

\begin{proof}
The $\cU$-cells of $X_1$ meet bonds crossing $\Xi$, by Definition~\ref{def:7.23}. Thus they lie
within Euclidean distance $D_E+2$ of $\Xi$, where $D_E=C_{\cU}\mt^{-1}\ge4$, in a set of volume at most
$3D_E(R+3D_E)(T+3D_E)$. Each of these cells has at least $c_u\mt^{-3}$ sites by (R7).
\end{proof}

For a rectangle $C$ of sides $R,T$ we put $R_1(\beta):=(2\rbox+5)L$, where $L$ is
the block scale of Definition~\ref{def:3.1} at $\tau=K\beta$.

\begin{theorem}[Deterministic area law]\label{thm:7.61}
Fix the constants of Section~\ref{ssec:D.1} of the supplement. There is $\bGMo$ such that for
$\bGM\ge\bGMo$, for every output $\Gp=\Gp(D)$ of the construction, for every
$D$, and for every rectangle $C$ with $\min(R,T)\ge R_1(\beta)=(2\rbox+5)L$,
\[
W_{\Gp}(C)\le\exp\bigl(-\sV[A_{\Gp}(C)-\eta_SRT]+c_N(R+T)\bigr),
\]
with
\[
\sV=\epsE\sF=\frac{c_g'\,\mt L}{9216}\,\frac{\mt}{\tau}\quad(\epsE=\tfrac1{128}),\qquad
\eta_S=\frac{18c_2C_X}{\epsE\pi^2c_g'\,\mt L}\,\mt\bGM=O(\mt\bGM\log\tau),
\]
and $c_N$ as in Remark~\ref{rem:7.62}. The
degenerate output $\mathbb X=\Lambda^\circ$ ($\Gp=\emptyset$, $W_{\Gp}(C)=1$,
$A_{\Gp}(C)=0$) is included.
\end{theorem}

Section~\ref{S:H1} gives the values of $\sV$ and $\eta_S$. The geometric input of the proof consists of (R2),
(R5) to (R10), Theorems~\ref{thm:6.5} and~\ref{thm:6.9} to \ref{thm:6.12}, Proposition~\ref{prop:6.18}, the frame,
the coarse alignment of the holes and frozen bonds, Lemma~\ref{lem:4.5}, in Lemma~\ref{lem:6.14}, and (N3) of
Theorem~\ref{thm:3.20}, in Lemma~\ref{lem:7.47}. The frame, which does not depend on $D$, is made up of the box
(Lemma~\ref{lem:3.3}), the grid with its windows and removal cells (Definition~\ref{def:3.4}) and the alignment
(A1) to (A3) (Lemma~\ref{lem:3.5}). The holes and frozen bonds are unions of coarse sites and of whole coarse
faces (Section~\ref{ssec:2.5}).

\begin{proof}
We bound the outer factors of the loop expansion by Lemma~\ref{lem:7.58}. If no
bond of $\Gp$ crosses $\Xi$ (for instance if $\Gp=\emptyset$), then $X_1=\emptyset$ and the bound holds
trivially. Indeed, in this case $j=0$ on $E$, which gives $Z(1,\Xi)=Z(0,\Xi)$ and $W_{\Gp}(C)=1$. Moreover, by the proof of Lemma~\ref{lem:7.6}(c), each plaquette of $\Xi$ is then a
face of a hole cube or a frozen face, hence lies in $\FF$, and Definition~\ref{def:2.11} with $\Sigma=\Xi$ gives
$A_{\Gp}(C)=0$. Finally, $\sV\eta_SRT+c_N(R+T)\ge0$.
Otherwise $X_1\neq\emptyset$, and
\begin{align}
W_{\Gp}(C)&=\sum_{X\supseteq X_1}\sum_{N_X}\cK_{N_X}(X)\,\frac{\mathcal Z_{p_1(N_X)}(X)}Z
\le\frac8{31}\sum_{X\supseteq X_1}\sum_{N_X}|\cK_{N_X}(X)|\,e^{\alpha|X|}e^{-\sigma'\Ad(p_1(N_X))}\label{eq:7.9.82}\\
&\le\frac8{31}\,e^{-\epsE\sF A_{\Gp}(C)}\sum_{X\supseteq X_1}\sum_{N_X}|\cK_{N_X}(X)|\,e^{\alpha|X|+\epsE\Floc(X,N_X)}
\le\frac8{31}\,\cP'_G\,e^{c_2|X_1|}e^{-\sV A_{\Gp}(C)}\nonumber\\
&\le\exp\Bigl(-\sV A_{\Gp}(C)+\sV\eta_SRT+\Bigl[\frac{2\cnear}{p'}+3c_2C_XC_{\cU}\mt
+\frac{9c_2C_XC_{\cU}^2}{R_1(\beta)}\Bigr](R+T)\Bigr),\nonumber
\end{align}
and the bracket, $b(\beta)$, is at most $c_N$. The equality follows from Corollary~\ref{cor:7.30} together with
Lemma~\ref{lem:7.31}(f). Its terms with $p_1(N_X)\notin\Glong$ vanish by Definition~\ref{def:7.27}(c).
Consequently, in \eqref{eq:7.9.82}, each sum over $N_X$ runs over the
inside fields at $k=1$ with $p_1(N_X)\in\Glong$. Lemma~\ref{lem:7.58} gives the first inequality. The second uses Lemma~\ref{lem:7.59}, $\sigma'=\epsE\sF$ and $\Floc(X,N_X)\ge2\sF\sum_{E(X)}|N_X|$, which holds by Theorem~\ref{thm:7.14} since $X$ is
a union of elements. The third
follows from Lemma~\ref{lem:7.53}(a) with $k=1$, $U=V$, $S_0=X_1$, $\eps=\epsE\le c'/2$, applied to these fields.
These fields are primitive-admissible at $k=1$ (Lemma~\ref{lem:7.insideprim}\,(b)), so Lemma~\ref{lem:7.53}(a)
applies to them by Proposition~\ref{prop:7.long}. The last inequality uses $\log(8/31)<0$, Lemma~\ref{lem:7.60} with $c_2C_X\mt^2=\sV\eta_S$
and $R+T\ge2R_1(\beta)$, and $\cP'_G=e^{\fn'(V)/p'}\le e^{2\cnear(R+T)/p'}$ (Lemma~\ref{lem:7.35}).
\end{proof}

\begin{remark}[The perimeter constant]\label{rem:7.62}
Schematically, the proof above obtains $\epsE\sF A_{\Gp}(C)$ from a
fraction $\epsE$ of the wall energy of the seeded polymer, with the area weight, by Lemma~\ref{lem:7.59}; the seed contributes $e^{c_2|X_1|}$, whose bulk part gives $\eta_S$ and whose
remainder is a perimeter term, and the near cells give the perimeter factor $\cP'_G$. Let $c_N:=\sup_{\bGM\ge\bGMo}b(\beta)$, where $b(\beta)$ is the bracket in \eqref{eq:7.9.82} (see also the remark after the proof of Proposition~\ref{prop:7.37}) and $p'=1.3$. It is
finite and depends only on the constants fixed before $K$, since $\cnear$ is bounded uniformly
for $\bGM\ge\bGMo$ by Lemma~\ref{lem:7.35} together with $\log(1/\mt)/\bGM<0.0632$, and the last two terms tend to $0$
(Section~\ref{S:H1}). Since
$\beta$ and $K$ appear in Sections~\ref{sec:3}, \ref{sec:stage1} and~\ref{sec:stage2} only in the combination $\tau=K\beta$,
that is, in $\bGM=4\pi^2\tau$, and every condition on $\bGM$ used there is a lower bound
in terms of items~0 to~14 of Section~\ref{ssec:D.1}, it follows that Theorem~\ref{thm:7.61}, its threshold
$\bGMo$ and $c_N$ use items~0 to~14, not $K$ and $\beta_0(K)$ (items~15 and~16). Section~\ref{sec:8} needs this, since there $c=c_N+10$ must be absolute.
\end{remark}