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\section{Proof of the main theorem}\label{sec:8}

\subsection{The chain}\label{ssec:8.1}

The deterministic area law holds for each frozen set $D$, and the min-cut
lower tail is exactly the exponential moment of the area deficit
$RT-\AG(C)$. Multiplying the two, we obtain $e^{-\sigma RT}\cdot\EE_De^{\sigma(RT-A)}$,
and the two deficits $\etaS$ and $\etaT$ add. We fix $K>\Kst$ and $\beta\ge\beta_0(K)$ as in Section~\ref{app:D}. With $\tau=K\beta$, the
conditions defining $\beta_0(K)$ include $\beta\ge\pi^{-2}$, $\delta<1$ and the condition \eqref{eq:2.threshold} of Lemma~\ref{lem:2.6}, that is
$\beta\ge\beta_1(K)$, as well as the hypotheses of
Theorems~\ref{thm:5.1} and~\ref{thm:7.61} for $\beta$ and $\bGM=4\pi^2\tau$. They also include
$\etaT+\etaS\le\frac12$ and $\sigV\le1$, which are true for large $\tau$, since $\etaT\to0$
by Theorem~\ref{thm:5.1}, $\etaS=O(\mt\bGM\log\tau)\to0$ by Theorem~\ref{thm:7.61},
and $\sigV=(\cgp\mt L/9216)\mt/\tau\to0$. Let $C$ be a rectangle with
$\min(R,T)\ge\Rone(\beta)=(2\rbox+5)L$ (Definition~\ref{def:3.1}), $\Lamo=\Lamo(C)$
the box of Definition~\ref{def:3.2}, and $\Lam\supseteq C$ any finite box.

\begin{proof}[Proof of Theorem~\ref{thm:1.2}]
We prove the chain \eqref{eq:8.1} to \eqref{eq:8.5}, where $\EE_D$ is the
expectation over $D\sim\Ber(\delta)^{\otimes(\Lamo_2\setminus\bdtwo\Lamo)}$.

\emph{Step \eqref{eq:8.1}.} Lemma~\ref{lem:3.3}(e) gives
$\dist(C,\partial\Lamo)\ge\max(R,T)\ge1$. Hence Proposition~\ref{prop:2.9} applies
with $\Lam'=\Lamo$ and $\delta'=\delta$. By Lemma~\ref{lem:3.3}(a), the box $\Lamo$ is determined by $C$ and the
constants, so it is the same for all $\Lam$ and $D$ and is fixed before
$D$ is drawn.

\emph{Step \eqref{eq:8.2}.} Theorem~\ref{thm:3.20} shows that for each $D\subseteq\Lamo_2\setminus\bdtwo\Lamo$ the
output $(\XX(D),S(D))$ is a deterministic function of $D$ with $D\subseteq\FF(D)$. Hence Proposition~\ref{prop:2.10} applies to each $D$.

\emph{Step \eqref{eq:8.3}.} Theorem~\ref{thm:7.61} applies to each output
$\Gp(D)$ and each rectangle with $\min(R,T)\ge\Rone(\beta)$, pointwise inside
$\EE_D$. We need it for every $D$, because $\Ber(\delta)$
charges each subset of $\Lamo_2\setminus\bdtwo\Lamo$ and the construction is
deterministic. Its functional $A_{\Gp(D)}(C)$ is the min-cut area of
Definition~\ref{def:2.11} for the same output on the same box $\Lamo$.

\emph{Step \eqref{eq:8.4}} is the identity $-\sigV A=-\sigV RT+\sigV(RT-A)$.

\emph{Step \eqref{eq:8.5}.} Theorem~\ref{thm:5.1} at $\sigma=\sigV\in(0,1]$, applied to the same random set $D$, the box $\Lamo$, the output $\Gp(D)$ and the functional $A_{\Gp(D)}(C)$
of Steps \eqref{eq:8.2} and \eqref{eq:8.3}, bounds
the expectation in \eqref{eq:8.4} by $\exp(\sigV\etaT RT+4\abs{C}+4)$.

\emph{Constants.} Since $\abs{C}=2(R+T)$ and $R+T\ge2$, we have $4\abs{C}+4\le10(R+T)$,
and \eqref{eq:8.5} gives \eqref{eq:1.main} with
$\sigma(\beta):=(1-\etaT-\etaS)\sigV$, $\Rone(\beta)=(2\rbox+5)L$, both depending on
$K$ through $\tau=K\beta$, and $c:=c_N+10$. This constant is absolute in the sense of
Theorem~\ref{thm:1.2}, since, by Remark~\ref{rem:7.62}, $c_N$ depends only on constants fixed before $K$, namely those of
items~(0) to~(14) of the ordered list in the next subsection. Using $\etaT+\etaS\le\frac12$ and the bound
$\cgp\mt L\ge\frac{9}{16}\cg\cL$ of Section~\ref{ssec:D.4}, we obtain
\begin{equation}\label{eq:8.sigma}
\sigma(\beta)\ge\frac{\cg\cL\,\mt}{2^{15}\tau}
\ge\frac{(8\pi^2)^{1/2}}{2^{16}}\,\cg\cL\,\tau^{-1/2}e^{-\pi^2\vCb(0)\tau}
\ge1.35\cdot10^{-4}\,\cg\cL\,\tau^{-1/2}e^{-2.49436\,\tau},
\end{equation}
by $\mt\ge\mD/2$ and the formula $\mD=(8\pi^2\tau)^{1/2}e^{-\pi^2\vCb(0)\tau}$
of Section~\ref{sec:stage1}, with the Debye exponent
$\pi^2\vCb(0)=2.4943550871\ldots$ rounded up and the prefactor rounded down
(see Section~\ref{ssec:D.4}). This proves the second assertion of
Theorem~\ref{thm:1.2} for $\beta\ge\beta_0(K)$. Remark~\ref{rem:8.1} gives the case
$\beta<\beta_0(K)$ at the same $K$, and the first assertion follows with one
fixed $K>\Kst$, for instance $K=378$ (any sufficiently large $K$ also works, by the remark after
Lemma~\ref{lem:2.6}).
\qed
\end{proof}

\begin{remark}[Small $\beta$ and the infinite-volume limit]\label{rem:8.1}\label{rem:8.2}
For fixed $K>\Kst$ and $\beta<\beta_0(K)$, Theorem~\ref{thm:2.1}(b), applied pointwise in $\Lam$ and $C$,
gives Theorem~\ref{thm:1.2} at $\beta$ with the three constants
$\sigma(\beta_0(K))$, $c$, $\Rone(\beta_0(K))$. Only the Wilson side is monotone
in $\beta$: raising $\beta$ raises $\tau$ but lowers $\delta$, so the right
side of Theorem~\ref{thm:2.5} is not monotone. The bound depends on $\Lam$ only
through Proposition~\ref{prop:2.9}, on $D$ only inside $\EE_D$ and on $C$ only
through $R,T$, and by Theorem~\ref{thm:2.1}(c) it holds for the infinite-volume
limit $\ex{\cos\theta_C}^{W}_{\beta}$, which is nondecreasing in $\beta$. Hence the string
tension
$\sigma_\infty(\beta):=\liminf_{\min(R,T)\to\infty}-\frac{1}{RT}\log\ex{\cos\theta_{C_{R,T}}}^{W}_{\beta}\in[0,\infty]$,
where $C_{R,T}$ is a coordinate rectangle of sides $R,T$, is nonincreasing in $\beta$
and at least $\sigma(\beta)$.
\end{remark}

\begin{remark}[Loops below the non-fragmentation scale]\label{rem:8.3}
If $\diam\Lamo\le\RNF$, for instance if both $3\sqrt3\max(R,T)+\sqrt3$ and
$\sqrt3(r_0(\ell+3)+1)$ are at most $\RNF$, then Phase~2 makes no firing by
Proposition~\ref{prop:3.19}, and Proposition~\ref{prop:5.15}, with the deficit
$\etaT^{(1)}$, replaces Theorem~\ref{thm:5.1} in Step~\eqref{eq:8.5}. The string
tension, by contrast, uses Theorem~\ref{thm:5.1} in full.
\end{remark}

\subsection{The order of the constants}\label{app:D}

We fix the constants once, in the following order. (0) Absolute
constants, among them $\pi^2\vCb(0)$, the threshold $14.9662$ of
Lemma~\ref{lem:2.6} and the absolute constants quoted from
\cite{GM82,GM81,BF80} (those that depend on the parameters $C$,
$\delta_{\mathrm{GM}}$, $\mu$, $\kappa_S$, $\lPV$ of Theorem~\ref{thm:6.5} are fixed
with them, in (12) and (13));
(1) $r_0$; (2) $\cg=\frac14$ (all $\cg\in(0,\frac14]$ work); (3) $\eta_w=\frac1{10}$;
(4) $c_\rho=\frac1{12}$, that is $\lambda=12/L_c$; (5) $k_P/L_c=304$;
(6) $c_1$ and $c_{\mathrm{sh}}$; (7) $c_s\in[3.3\cdot10^{-7},10]$; (8) the forcing threshold
$\vartheta$, the penalty $\mu_{\rm B}$ and the thin-set constant
$\varepsilon_{\rm th}$; (9) the weak-link constants $\chi=2.0\cdot10^{-8}$ of
(R6) and $\chi_{\rm N3}=\chi/3$ of (N3); (10) the constants derived from these,
among them $C_\star'$, $\Cfree$, $C_a$, $C_b$, $C_c$ and $\Cresp$;
(11) $\cL=1.6\cdot10^{-19}$, and then the scales $L_c$, $L$, $k_P$, $\ell$,
which depend on $\beta$ only through $\mt$; (12) the Stage~1 and Stage~2
constants: $C\in(0,\frac14]$ and $\delta_{\mathrm{GM}}\le\frac12$ of
Theorem~\ref{thm:6.5}, $\eta_H=0.62$, the rate $c_{10}\le\frac79r_C$ with $r_C$
taken at $\mt L=\cL/2$, then $c''_{10}$, then $\mu\ge1+2c''_{10}$ and $\kappa_S$,
and $c'_0=\frac1{16}$; (13) $\lPV$, and with it $\MY$, $\mt=\lPV\MY$ and the constants of
\cite{GM82,GM81,BF80} and of Stage~1 that depend on $\lPV$;
(14) the cut-rule constants $\theta_{\mathrm{NF}}$ and $c_3^{\mathrm{NF}}$;
(15) $K>\Kst$; (16) $\beta_0(K)$, defined by the conditions listed in Section~\ref{ssec:D.5}. The values of these constants, and the statements that use each of them, are given in Section~\ref{ssec:D.1}.

Four placements are forced. First, $k_P/L_c$,
$\vartheta$ and $\varepsilon_{\rm th}$ precede $\chi$, because the expansion
constant of Theorem~\ref{thm:4.rest}(c), which bounds $\chi$, is computed from
them. Second, $\chi$ precedes $\cL$, because the condition
$C_P'\cL^2\le(1-4/\pi^2)/8$ that fixes $\cL$ has
$C_P'\le2916\rbox^5C_{\mathrm{bl}}^2/\chi^2$. Since $1-4/\pi^2<\eta_H$ and
$\mt L\le\cL$, the condition that fixes $\cL$ also gives the condition $C_P'(\mt L)^2\le\eta_H/8$ of
Stage~2. Third, $C$, $\delta_{\mathrm{GM}}$, $c_{10}$, $c''_{10}$, $\mu$ and $\kappa_S$
precede $\lPV$, because the Debye condition of Theorem~\ref{thm:6.5} gives
$\lambda_0=\lambda_0(\kappa_S,\mu,C,\delta_{\mathrm{GM}})$ and each earlier
constant uses $\lPV$ only through $\lPV\le\frac14$. Fourth, $\theta_{\mathrm{NF}}$ and
$c_3^{\mathrm{NF}}$ are built from the pure number $c'_0$, and no earlier
constant depends on them.

Each condition defining $\beta_0(K)$ is, at fixed $K$, a lower bound on $\tau=K\beta$ that
persists for larger $\tau$. Those of Section~\ref{sec:2} are $\beta\ge\pi^{-2}$,
$\beta\ge\beta_1(K)$ and $\delta<1$, which \eqref{eq:2.cdelta} gives for all
large $\beta$. In all the others the left side is a fixed function of $\tau$ (built from $\mt$, $\tau$,
$\log\tau$ and the doubly exponential scale $\RNF$) tending monotonically to its
limit. The value $\beta_0(K)$ is the largest of these thresholds divided by $K$. It is very large ($\abs{\log\epspp}\gtrsim10^{14}$ in
Theorem~\ref{thm:5.1}), and the constants are far from sharp, but the proof uses
only that those fixed before $K$ do not depend on $\tau$.