%
%
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\providecommand{\MY}{M_{Y}}
\providecommand{\lPV}{\lambda_{\mathrm{PV}}}
\providecommand{\Gp}{G'}
\providecommand{\XX}{\mathbb{X}}
\providecommand{\KK}{\mathfrak{K}}
\providecommand{\Dep}{\mathfrak{D}}
\providecommand{\RHK}{R_{\mathrm{HK}}}
\providecommand{\rcov}{r_{\mathrm{cov}}}
\providecommand{\Rdep}{R_{\mathrm{dep}}}
\providecommand{\etaw}{\eta_{w}}
\providecommand{\Zthree}{\mathbb{Z}^{3}}
\providecommand{\RR}{\mathbb{R}}
\providecommand{\ZZ}{\mathbb{Z}}
\providecommand{\rhoref}{\rho^{\mathrm{ref}}}
\providecommand{\vCb}{v_{\mathrm{Cb}}}
\providecommand{\eps}{\varepsilon}
\providecommand{\Ord}{\mathcal{O}}

\section{Stage 1: the Mayer expansion to the Debye scale on slit domains}
\label{sec:stage1}

In this section we prove \cite[Prop.~3]{GM82}, with the iterated Mayer expansion of \cite{GM81}, for the gas whose kernel is the Neumann Green function of a slit graph (Theorem~\ref{thm:6.5}), and we derive heat-kernel, covariance and depletion bounds on outputs. Since the Neumann field of a distant source is enhanced near a free edge of a frozen sheet, the only pointwise bounds on a Neumann kernel or its gradient that Sections~\ref{sec:stage1} and~\ref{sec:stage2} use near such an edge are the diagonal lower bounds of Lemma~\ref{lem:6.1} and heat-kernel decay from the isoperimetry (R2), in the form of Theorem~\ref{thm:6.9} and Lemma~\ref{lem:6.15}. The remaining inputs are the exact $L^{1}$ sums of Lemma~\ref{lem:6.2} and, in Lemma~\ref{lem:7.34}, the free-space solid angle. Complete proofs of Propositions~\ref{prop:6.3} and~\ref{prop:6.4} are given in Section~\ref{S4-S:S4.3}, the proof of the latter including \cite[Prop.~8]{GM81} on $\Zthree$.

\paragraph{Conventions.} The lattice is the fine lattice (spacing $1$), $\bGM=4\pi^{2}\tau$, and distances are Euclidean unless $d_{\Gp}$ is written. Slit graphs $\Gp=(V,E)$, $V=\Lambda'\setminus\XX$, $E=E(V)\setminus S$, their components $\KK$, the bound $d_{\Gp}(x,y)\ge\|x-y\|_{1}$ and the outputs $\Gp=\Gp(D)$ are those of \S\ref{ssec:2.5}. We define $(-\Delta_{\Gp}f)(y):=\sum_{y'\sim y}(f(y)-f(y'))$ over the $\Gp$-neighbours of $y$. For $m>0$, $v^{(m)}_{\Gp}:=(-\Delta_{\Gp}+m^{2})^{-1}$, $v_{\Gp}:=v^{(\MY)}_{\Gp}$, $v:=v^{(\MY)}_{\Zthree}$, $v_{\Lambda'}$ is $v^{(\MY)}$ of the box $\Lambda'$ with all its bonds, and $\vCb$ is the Coulomb kernel of $\Zthree$. On $\RR^{V}_{0}:=\{\varphi:\sum_{x\in\KK}\varphi(x)=0$ for all $\KK\}$, $G_{N}:=(-\Delta_{\Gp})^{+}$ is the inverse of $-\Delta_{\Gp}$. The \emph{image energies} are $\Phi(x)=\Phi_{\Gp}(x):=\frac12\bGM(v_{\Gp}(x,x)-v(0))$ and $I_{\Gp}(x):=v_{\Gp}(x,x)-v_{\Lambda'}(x,x)$, and the \emph{depleted set} is $\Dep:=\{x\in V:\Phi_{\Gp}(x)>\theta_{0}\}$, $\theta_{0}=0.01$. For a matrix $K$ on $V$ and $r\ge0$, $\|K\|_{r}$ is the larger of $\sup_{x}\sum_{y}|K(x,y)|e^{r\,d_{\Gp}(x,y)}$ and the same expression with $x$ and $y$ exchanged. This norm is submultiplicative, and $\|K\|_{2\to2}\le\|K\|_{0}\le\|K\|_{r}$ for all such $K$ and $r$. The activity of a unit monopole is $z:=e^{-\bGM\vCb(0)/2}$, and $\mD^{2}:=2\bGM z$ \cite[(1.8a)]{GM82}. Hence $\mD=(8\pi^{2}\tau)^{1/2}e^{-\pi^{2}\vCb(0)\tau}$. Numerical values are given in \S\ref{S:E1.mass}. The mass $\mt\in[\mD/2,2\mD]$ and the Pauli\textendash Villars mass $\MY=\mt/\lPV$ are fixed in \eqref{eq:6.mass}. Until then $\MY$ is any mass in $I_{Y}:=[\mD/(2\lPV),2\mD/\lPV]$, and all statements hold uniformly in that range. Besides the loop, the letter $C$ denotes, depending on the context, the parameter $C\in(0,1)$ of Theorem~\ref{thm:6.5} (the notation of \cite{GM82}), a generic constant, such as those of (S5), \eqref{eq:6.lead} and Proposition~\ref{prop:6.6}, and, in Theorem~\ref{thm:6.12} and Appendix~\ref{app:C}, the covariance $(C_{0}^{-1}+\nu)^{-1}$.

\subsection{Kernel facts on every slit graph}
\label{ssec:6.1}

\begin{lemma}[Thomson monotonicity]\label{lem:6.1}
Let $H\subseteq H'$ be graphs with $V(H)\subseteq V(H')$, $E(H)\subseteq E(H')$ and $V(H)$ finite; $H'=\Zthree$ is allowed. Let $f$ be supported on $V(H)$. (i) For every $m>0$, $\bigl(f,v^{(m)}_{H}f\bigr)\ge\bigl(f,v^{(m)}_{H'}f\bigr)$. (ii) If $f$ has zero sum on every component of $H$, then it has zero sum on every component of $H'$, and $\bigl(f,(-\Delta_{H})^{+}f\bigr)\ge\bigl(f,(-\Delta_{H'})^{+}f\bigr)$; for $H'=\Zthree$ the right side is $(f,\vCb f)$. (iii) For every slit graph $\Gp\subseteq\Lambda'$ and $x\in V$: $v^{(m)}_{\Gp}(x,x)\ge v^{(m)}_{\Zthree}(0)$ and $v_{\Gp}(x,x)\ge v_{\Lambda'}(x,x)$. Hence $\Phi_{\Gp}\ge0$ and $I_{\Gp}\ge0$.
\end{lemma}

\begin{proof}
In the variational formula $(f,v^{(m)}_{H}f)=\sup_{\varphi}[2(f,\varphi)-\sum_{\{y,y'\}\in E(H)}(\varphi(y)-\varphi(y'))^{2}-m^{2}\sum_{y\in V(H)}\varphi(y)^{2}]$, restricting $\varphi$ from $V(H')$ to $V(H)$ keeps $(f,\varphi)$ and does not raise the energy, which gives (i). In (ii), $f$ is orthogonal to the kernels of both Laplacians, and the formula holds at $m=0$; for $H'=\Zthree$ let $m\downarrow0$ in (i): the left side tends to $(f,(-\Delta_H)^{+}f)$, by the spectral theorem on $f\perp\ker(-\Delta_H)$, and the right side increases to $(f,\vCb f)$, by monotone convergence in the Fourier formula. We obtain (iii) by taking $f=\delta_{x}$ in (i).
\qed
\end{proof}

\begin{lemma}\label{lem:6.2}
Let $\Gp$ be a slit graph and $0<m<m'$. Then: (i) $v^{(m)}_{\Gp}$ is symmetric and entrywise nonnegative, and $v^{(m)}_{\Gp}(x,y)=0$ when $x$ and $y$ lie in different components; (ii) $\sum_{y\in V}v^{(m)}_{\Gp}(x,y)=m^{-2}$ for every $x\in V$; (iii) $v^{(m)}_{\Gp}-v^{(m')}_{\Gp}=(m'^{2}-m^{2})\,v^{(m)}_{\Gp}v^{(m')}_{\Gp}$, and this matrix is entrywise nonnegative and positive semidefinite, with row sums $m^{-2}-m'^{-2}$; (iv) $\operatorname{spec}(-\Delta_{\Gp})\subseteq[0,12]$.
\end{lemma}

\begin{proof}
Since $\deg_{\Gp}\le6$, the matrix $P:=I-(-\Delta_{\Gp})/6$ is symmetric, nonnegative and stochastic, with no entry across components, and the series $v^{(m)}_{\Gp}=(6+m^{2})^{-1}\sum_{n\ge0}(6/(6+m^{2}))^{n}P^{n}$ gives (i) and (ii). Part (iii) is the resolvent identity, and (iv) is Gershgorin's theorem.
\qed
\end{proof}

\begin{proposition}\label{prop:6.3}
Let $\Gp$ be a slit graph, $\bGM>0$, $k\in\RR$ and $f:V\to\RR$. Put $\tilde Z(k,f):=\sum_{[n]}\exp\bigl(-\frac{1}{2\bGM}\sum_{b\in E}(\nabla_{b}(n-kf))^{2}\bigr)$, where $n$ runs over $(2\pi\ZZ)^{V}$ modulo the addition of a constant in $2\pi\ZZ$ on each component. Let $u:=G_{N}-v_{\Gp}$ on $\RR^{V}_{0}$ and let $d\mu_{u}$ be the centred Gaussian measure on $\RR^{V}_{0}$ with covariance $u$. Then: (a) $u=\MY^{2}(-\Delta_{\Gp})^{+}(-\Delta_{\Gp}+\MY^{2})^{-1}$ on $\RR^{V}_{0}$, and $u$ is positive definite there. (b) $\tilde Z(k,f)=\mathcal{N}_{0}\sum_{q}e^{ik(q,f)}e^{-\bGM(q,G_{N}q)/2}$, the sum over $q\in\ZZ^{V}$ with zero sum on every component, where $\mathcal{N}_{0}=\prod_{\KK}\mathcal{N}_{\KK}>0$ depends only on $\Gp$ and $\bGM$. An isolated vertex carries only $q=0$ and has $\mathcal{N}_{\KK}=1$. (c) With $\theta:=\sum_{\KK}\theta_{\KK}1_{\KK}$,
\begin{equation}\label{eq:6.SG}
\sum_{q\ \mathrm{neutral}}e^{ik(q,f)}e^{-\bGM(q,G_{N}q)/2}=\int_{[0,2\pi)^{\mathcal{K}}}\prod_{\KK}\frac{d\theta_{\KK}}{2\pi}\int d\mu_{u}(\varphi)\,Z^{Y}\bigl(\varphi+k\bGM^{-1/2}f+\bGM^{-1/2}\theta\bigr),
\end{equation}
where $Z^{Y}(\psi):=\sum_{q\in\ZZ^{V}}e^{i\bGM^{1/2}(q,\psi)}e^{-\bGM(q,v_{\Gp}q)/2}$ is the Yukawa gas on $\Gp$, with no neutrality constraint. (d) $Z^{Y}(\psi)>0$ for real $\psi$, and $Z^{Y}=\prod_{\KK}Z^{Y}_{\KK}(\psi|_{\KK})$. Consequently the coefficients $\rho_{s}$ of the expansion of $\ln Z^{Y}$ in Theorem~\ref{thm:6.5} vanish unless all their points lie in one component.
\end{proposition}

\begin{proof}
On each component $\KK$ we carry out the argument of \cite[(2.4) to (2.12)]{GM82}, which is written there for a measure on $\Zthree$ with charges confined to $\Lambda$ and no zero mode (change (c) of Table~\ref{tab:6.subst}). We root $\KK$ at a point $x_{0}\in\KK$. Poisson summation over $(2\pi\ZZ)^{|\KK|-1}$ then gives the neutral charges $\hat q=(q,-\sum q)$, with $\langle q,A^{-1}q\rangle=\langle\hat q,G_{N}\hat q\rangle$ for the restriction $A$ of $-\Delta_{\KK}$ to $\KK\setminus\{x_{0}\}$. One variable $\theta_{\KK}$ for each component imposes neutrality on every component, for any number of components of any sizes. Finally, $G_{N}=v_{\Gp}+u$, and (a) follows from $(-\Delta_{\Gp}+\MY^{2})^{-1}1_{\KK}=\MY^{-2}1_{\KK}$, which holds by Lemma~\ref{lem:6.2}(ii). The positivity $Z^{Y}>0$ is cited in \cite[p.~21]{GM82}. We obtain it by Poisson summation over $\ZZ^{V}$, which writes $Z^{Y}(\psi)$ as a sum of positive Gaussians in $n-\bGM^{1/2}\psi$, $n\in(2\pi\ZZ)^{V}$, with covariance $\bGM v_{\Gp}$. The factorization and (d) hold because $v_{\Gp}$ and the potentials $v^{\ell}_{\Gp}$, defined as in \cite[(2.5), p.~6, and (4.1), p.~17]{GM81}, vanish across components by Lemma~\ref{lem:6.2}(i), and \cite[\S3]{GM82} builds $\rho_{s}$ from these potentials through the tree formula \cite[(3.7)]{GM81}.
\qed
\end{proof}


\begin{proposition}\label{prop:6.4}
Let $M_{1}\ge\MY$ be the mass of \cite[(3.20)]{GM82}, and on a slit graph $\Gp$ let $v^{1}_{\Gp}:=v^{(M_{1})}_{\Gp}$, with the coincident-site rule \cite[(5.1a)]{GM81}, and $v^{2}_{\Gp}:=v_{\Gp}-v^{1}_{\Gp}$. Then the stability bounds \cite[(4.2)]{GM81} hold on $\Gp$ with $\gamma_{1}=\frac12v^{1}_{\Zthree}(0)$, with the same $\eps_{1}>0$ as on $\Zthree$ \cite[Prop.~8]{GM81}, and with $\eps_{2}=\delta_{2}=0$. Moreover $v^{\ell}_{\Gp}(x,y)\ge0$, and
\begin{equation}\label{eq:6.vtilde}
\tilde v^{1}:=\sup_{y}\sum_{x}v^{1}_{\Gp}(x,y)=M_{1}^{-2},\qquad\tilde v^{2}:=\sup_{y}\sum_{x}v^{2}_{\Gp}(x,y)=\MY^{-2}-M_{1}^{-2}
\end{equation}
exactly, as on $\Zthree$ \cite[(5.3)]{GM81}.
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:6.1} each Yukawa energy on $\Gp$ is at least its $\Zthree$ value for the same configuration. Hence the margins of \cite[Prop.~8]{GM81} transfer. At level one, with the rule (5.1a) and $m(x):=\sum_{i}m_{i}\delta_{x_{i}x}$, $E_{\Gp}=\frac12(m,v^{1}_{\Gp}m)+\frac12\sum_{x_{i}=x_{j},\ \mathrm{sign}\,m_{i}=-\mathrm{sign}\,m_{j}}|m_{i}m_{j}|\,v^{1}_{\Gp}(x_{i},x_{i})\ge E_{\Zthree}$ by Lemma~\ref{lem:6.1}(i), (iii), and $E_{\Zthree}\ge\gamma_{1}+\eps_{1}(-1+\sum_{i}m_{i}^{2})$, coincident positions included. At level two, $\frac12(m,v^{2}_{\Gp}m)\ge0$ by Lemma~\ref{lem:6.2}(iii), that is (4.2) with $\eps_{2}=\delta_{2}=0$. Positivity and \eqref{eq:6.vtilde} follow from Lemma~\ref{lem:6.2}(i) to (iii).
\qed
\end{proof}

\subsection{The Stage 1 theorem}
\label{ssec:6.2}

\begin{theorem}[Stage 1 on slit domains]\label{thm:6.5}
Fix $\kappa_{S},\mu<\infty$ and $C,\delta_{\mathrm{GM}}\in(0,1)$. There are $\lambda_{0}=\lambda_{0}(\kappa_{S},\mu,C,\delta_{\mathrm{GM}})>0$ and, for $\lPV\le\lambda_{0}$, $\bGMo=\bGMo(\lPV,\kappa_{S},\mu,C,\delta_{\mathrm{GM}})$ such that for $\bGM\ge\bGMo$ and every $\MY\in I_{Y}$ the following hold on every slit graph $\Gp$, with constants independent of $\Gp$ and $\MY$. Write $\xi=(m,x)$ with $m\in\ZZ\setminus\{0\}$ and $x\in V$, $-\xi:=(-m,x)$, and $\int d\xi:=\sum_{m}\sum_{x\in V}$.
\par(S1) \emph{Expansion.} For real $\psi$ on $V$, $Z^{Y}(\psi)=\exp(-V_{\mathrm{eff}}(\psi))$ with
\begin{equation}\label{eq:6.Veff}
-V_{\mathrm{eff}}(\psi)=\rho_{0}+\sum_{s\ge1}\frac{1}{s!}\int d\xi_{1}\cdots d\xi_{s}\,\rho_{s}(\xi_{1},\dots,\xi_{s})\,\eps(\xi_{1})\cdots\eps(\xi_{s}),
\end{equation}
where $\eps(\xi):=e^{im\bGM^{1/2}\psi(x)}-1$,
the coefficients $\rho_{s}=\rho_{s}^{\Gp}$ are real and symmetric, $\rho_{0}$ is a constant, and the series converges absolutely, uniformly in real $\psi$. With Proposition~\ref{prop:6.3}, for every $k$ and $f$,
\begin{equation}\label{eq:6.S1}
\tilde Z(k,f)=\mathcal{N}_{0}\int\prod_{\KK}\frac{d\theta_{\KK}}{2\pi}\int d\mu_{u}(\varphi)\,\exp\bigl[-V_{\mathrm{eff}}\bigl(\varphi+k\bGM^{-1/2}f+\bGM^{-1/2}\theta\bigr)\bigr].
\end{equation}
\par(S2) \emph{Coefficient bounds.} For $s\ge1$ and every $a_{1}\subseteq V$,
\begin{multline}\label{eq:6.S2}
\int_{x_{1}\in a_{1}}\!\!d\xi_{1}\int d\xi_{2}\cdots\int d\xi_{s}\,|\rho_{s}(\xi_{1},\dots,\xi_{s})|\,e^{2\kappa_{S}(-1+\sum_{i}|m_{i}|)}\,e^{(1-\delta_{\mathrm{GM}})A\,L_{\Gp}(x_{1},\dots,x_{s})}\\
\le s!\,|a_{1}|\,\frac{\mD^{2}}{\bGM}\,e^{-\mu(s-1)}(1-C)^{-s-10}
\end{multline}
with $A=0$, where $L_{\Gp}(a_{1},\dots,a_{n})$ is the tree length of \cite[(1.11a), (1.11b)]{GM82} computed with the distance $3^{-1/2}d_{\Gp}$ and auxiliary points in $V$, $L_{\Gp}(x_{1},\dots,x_{s}):=L_{\Gp}(\{x_{1}\},\dots,\{x_{s}\})$, and $0\cdot\infty:=0$. Moreover, for all $a_{1},\dots,a_{s}\subseteq V$
\begin{multline}\label{eq:6.S2sets}
\int_{x_{1}\in a_{1}}\!\!d\xi_{1}\cdots\int_{x_{s}\in a_{s}}\!\!d\xi_{s}\,|\rho_{s}(\xi_{1},\dots,\xi_{s})|\,e^{2\kappa_{S}(-1+\sum_{i}|m_{i}|)}\\
\le s!\,|a_{1}|\,\frac{\mD^{2}}{\bGM}\,e^{-(1-\delta_{\mathrm{GM}})A\,L_{\Gp}(a_{1},\dots,a_{s})-\mu(s-1)}(1-C)^{-s-10}.
\end{multline}
\par(S2$^+$) \emph{Charge moments.} If $\mu\ge\ln3$, then for every $a_{1}\subseteq V$
\[
\sum_{s\ge1}\frac{2^{s}}{s!}\int_{x_{1}\in a_{1}}\!\!d\xi_{1}\int d\xi_{2}\cdots\int d\xi_{s}\,|\rho_{s}(\xi_{1},\dots,\xi_{s})|\,e^{2\kappa_{S}\sum_{i}|m_{i}|}
\le|a_{1}|\,\frac{\mD^{2}}{\bGM}\,e^{2\kappa_{S}+\mu}(1-C)^{-9};
\]
in particular $\sum_{s\ge1}\frac{2^{s}}{s!}\int|\rho_{s}|(\sum_{i}|m_{i}|)^{n}\le n!(2\kappa_{S})^{-n}|V|\frac{\mD^{2}}{\bGM}e^{2\kappa_{S}+\mu}(1-C)^{-9}$ for every $n\ge0$.
\par(S3) \emph{Symmetries.} $\rho_{s}(-\xi_{1},\dots,-\xi_{s})=\rho_{s}(\xi_{1},\dots,\xi_{s})$; in particular $\sum_{m}m\,\rho_{1}(m,x)=0$ at every $x\in V$. Moreover $V_{\mathrm{eff}}(\psi+2\pi\bGM^{-1/2}I)=V_{\mathrm{eff}}(\psi)$ for every $I\in\ZZ^{V}$.
\par(S4) \emph{One-body term.} The bounds of Proposition~\ref{prop:6.6} hold.
\par(S5) \emph{Dipole kernel.} $\nu(x_{1},x_{2}):=\bGM\sum_{m_{1},m_{2}}m_{1}m_{2}\,\rho_{2}((m_{1},x_{1}),(m_{2},x_{2}))$ is real and symmetric, and $\|\nu\|_{2\to2}\le\|\nu\|_{0}\le Ce^{-\mu}\mD^{2}$. If $Ce^{-\mu}\le\frac18$, the right side is at most $\frac12\mt^{2}$.
\par With $\MY$ fixed in \eqref{eq:6.mass}, on every output (S2) holds also with $A=\sqrt3\,\kappa_{Y}\MY$ (Proposition~\ref{prop:6.11}), and, if $\delta_{\mathrm{GM}}\le\frac12$, (S5) in the form $\|\nu\|_{\rcov}\le Ce^{-\mu}\mD^{2}$ ($\rcov$ of Theorem~\ref{thm:6.12}).
\end{theorem}

\begin{proof}[Proof of Theorem~\ref{thm:6.5}]
By Lemma~\ref{lem:6.1}, removing bonds raises all quadratic energies, and by Lemma~\ref{lem:6.2} it conserves the mass of the walk. The stability margins of \cite{GM81} and the pinned $L^{1}$ norms of the links are therefore the same as on $\Zthree$, without image factors. Hence the proof of \cite[Prop.~3]{GM82}, together with the results of \cite{GM81} that it uses, runs on $\Gp$ with the changes (a) to (h) of Table~\ref{tab:6.subst}, and its other steps carry over without change, for the same reason. Numbers refer to the DESY preprints identified in \S\ref{ssec:1.4}.

\begin{center}
\footnotesize
\begin{tabular}{@{}p{0.1\textwidth}p{0.85\textwidth}@{}}
\hline\noalign{\smallskip}
& other steps\\
\noalign{\smallskip}\hline\noalign{\smallskip}
\cite{GM81} & \S2, (2.5) to (2.25), Prop.~2, (2.26) (its finite pieces are left arbitrary until the estimates are made, p.~6), Prop.~3, (2.30), App.~A (only (2.27) is used), Prop.~4, (3.7), App.~B, (B.4) (stated on p.~32 for any real symmetric pair array $w(ij)$, and used here for the $v^{\ell}_{\Gp}$, $\ell\ge1$), (3.8) to (3.12), (4.3) (with (4.2) for all subconfigurations, by Proposition~\ref{prop:6.4}), Lemma~5, (4.8), App.~C (combinatorial), Prop.~6, Cor.~7, Lemma~9 (through the pinned norms of change (a) only)\\
\noalign{\smallskip}
\cite{GM82} & (3.1) to (3.18) (for real $\psi$, (3.14) uses only $|1+\eps(\xi)|=1$, p.~25), (4.1) to (4.11), Lemma~6 (any metric), (4.14) to (4.22), Prop.~7, Cor.~8 (through $\|v^{\ell}\|_{A}$ only), (4.25) ($\gamma_{1}$ is the constant of $\Zthree$), Prop.~9, (4.26) to (4.30), (4.32) (through the preceding steps, with $L_{\Gp}$ and the rate $(1-\delta_{\mathrm{GM}})A$)\\
\noalign{\smallskip}\hline
\end{tabular}
\end{center}

We read the statement of \cite[Prop.~3]{GM82} with two corrections: $\eps(\xi)=-1+\exp[im\bGM^{1/2}\varphi(x)]$, as in its (3.4), and $2\bGM\mD^{-2}\rho_{1}(\pm1,x)\to1$ at its end. The constants depend only on $(\lPV,\kappa_{S},\mu,C,\delta_{\mathrm{GM}})$ and, on outputs, on those of Theorem~\ref{thm:6.9}.

A self-contained proof is given in Section~\ref{S4-S:S4.3}. There the tree formula \cite[Prop.~4, (3.7), App.~B]{GM81} and the tree estimate \cite[Lemma~5, App.~C]{GM81} are replaced by the forest formula of Brydges, Kennedy, Abdesselam and Rivasseau \cite{BK87,AR95}, its decoupling property and Cayley's formula; the recursive bound obtained there is slightly stronger than \cite[(4.16)]{GM82}, so that the analogues of \cite[Cor.~8, Prop.~9]{GM82} follow by their printed arguments.

\begin{table}[!htb]
\caption{Changes to \cite[Prop.~3]{GM82} and the results of \cite{GM81} it uses, on a slit graph $\Gp$.}\label{tab:6.subst}
\footnotesize
\begin{tabular}{@{}p{0.135\textwidth}p{0.31\textwidth}p{0.525\textwidth}@{}}
\hline\noalign{\smallskip}
change & steps & use of $\Zthree$ $\to$ on $\Gp$\\
\noalign{\smallskip}\hline\noalign{\smallskip}
(a) pinning & \cite[(4.5a), (4.6), pp.~18 to 20]{GM81}; \cite[(4.12), (4.13), p.~32]{GM82} & translation invariance $\to$ pinned norms in max form, re-rooted sums\\
(b) stability & \cite[Prop.~8, (5.1a) to (5.13), App.~D]{GM81}; \cite[Lemma~5, (3.25)]{GM82} & Fourier analysis $\to$ Proposition~\ref{prop:6.4}: same $\gamma_{1}$, $\eps_{1}$; $\eps_{2}=\delta_{2}=0$\\
(c) Gaussian & \cite[(2.4) to (2.12), pp.~20, 21]{GM82} & measure on $\Zthree$, no zero mode, $Z_{\Lambda}(\phi)>0$ cited $\to$ Proposition~\ref{prop:6.3}: one zero mode per component, $Z^{Y}>0$ proved\\
(d) resolvents & \cite[(3.19) to (3.22)]{GM82} & resolvents of $\Zthree$ $\to$ of $\Gp$; rule (3.19) ($=$ \cite[(5.1a)]{GM81}), same $M_{1}$ (3.20)\\
(e) metric & \cite[(3.23), (3.24)]{GM82} & $3^{-1/2}\|x-y\|_{1}$ $\to$ $\|x-y\|_{\Gp}:=3^{-1/2}d_{\Gp}(x,y)$, in $\|\cdot\|_{A}$ and $L_{\Gp}$\\
(f) link norms & \cite[Lemma~5, (3.26); (4.23), (4.24), p.~34]{GM82} & momentum space $\to$ \eqref{eq:6.vtilde}, Proposition~\ref{prop:6.11}; Debye condition $\lPV^{2}\le C^{3}(1-C)^{8}e^{-2\kappa_{S}-\mu}/24$\\
(g) $\sigma^{R}$ & \cite[(4.33), (4.34), p.~36]{GM82} & $v(0)$ $\to$ $v_{\Gp}(x,x)$ in $\sigma^{R}=\exp[-\bGM m^{2}v(0)/2]$\\
(h) tree length & \cite[(4.30) to (4.32), pp.~35, 36; (4.12), p.~32]{GM82} & $L(a_{1},\dots,a_{s})$ on the right of (4.31) $\to$ $L_{\Gp}(x_{1},\dots,x_{s})$ inside the sum over all positions\\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}

(a) \emph{Pinned norms.} Where \cite[p.~18]{GM81} and \cite[p.~32]{GM82} use translation invariance, the former after a preparatory majorization by functions of differences of the arguments, we take instead, on $\Gp$, the norms in the max form \cite[(4.5b)]{GM81}, $\|\sigma_{\alpha}\|(m_{\alpha}):=\max_{i\in\alpha}\sup_{x\in V}\sum_{x_{\alpha}\in V^{n}:\,x_{i}=x}|\sigma_{\alpha}(\xi_{\alpha})|$, with the weights of \cite[(4.12)]{GM82} inserted in the same way, and $\tilde v^{\ell}:=\sup_{y}\sum_{x}|v^{\ell}_{\Gp}(x,y)|$. We sum the positions as in the trimming of the tree in \cite[p.~18]{GM81}, after re-rooting the tree at the vertex containing the pinned constituent, in order of decreasing tree distance from it. Each summed vertex is then a leaf attached by one link $(j,k)$, and $\sum_{x_{j}}|v^{\ell}_{\Gp}(x_{j},x_{k})|\le\tilde v^{\ell}$ uniformly in $x_{k}$. The $s$-integrals and tree sums are bounded by \cite[Lemma~5]{GM81} independently of the positions. Hence \cite[(4.7), (4.10), (4.13), (4.16), (4.17)]{GM81} hold without change, and $\int d\xi_{\alpha}|\sigma_{\alpha}|\le|V|\,\|\sigma_{\alpha}\|$.

(f) \emph{Link norms.} For $A\ge0$ we put $\|v^{\ell}_{\Gp}\|_{A}:=\sup_{y}\sum_{x}|v^{\ell}_{\Gp}(x,y)|\,e^{A\|x-y\|_{\Gp}}$. At $A=0$ the values are \eqref{eq:6.vtilde}, on all slit graphs. On an output, Proposition~\ref{prop:6.11} gives $\|v_{\Gp}\|_{A},\|v^{2}_{\Gp}\|_{A}\le3\MY^{-2}$ and $\|v^{1}_{\Gp}\|_{A}\le3M_{1}^{-2}$ for $A\le\sqrt3\,\kappa_{Y}\MY$, since $\|x-y\|_{\Gp}=3^{-1/2}d_{\Gp}(x,y)$, the rate of \eqref{eq:6.LW} is nondecreasing in the mass, and $\MY\le M_{1}\le m_{W}$. These bounds replace \cite[(3.26)]{GM82} in \cite[(4.23)]{GM82} and in \cite[(4.24)]{GM82}. In \cite[(4.23)]{GM82}, $3M_{1}^{-2}$ is a power of $\bGM$, absorbed by $e^{\bGM\eps_{1}}$. In \cite[(4.24)]{GM82}, $3\MY^{-2}$ replaces $2(\delta M)^{-2}$ at $A=(1-\delta)M$, so the factor $\frac14\delta^{2}$ of the Debye condition \cite[p.~34]{GM82} becomes $\frac16$. The condition then holds on $I_{Y}$ once $\lPV^{2}\le C^{3}(1-C)^{8}e^{-2\kappa_{S}-\mu}/24$ (\S\ref{S:E1.debye}). This threshold does not depend on $\Gp$, and all other thresholds of the proof are those of $\Zthree$.

(g) \emph{The leading one-body term.} For a single constituent the recursion \cite[(3.12)]{GM82} gives $\sigma^{\ell+1}=\sigma^{\ell}\exp[-\bGM m^{2}v^{\ell}_{\Gp}(x,x)/2]$, hence $\sigma^{R}(m,x)=\exp[-\bGM m^{2}v_{\Gp}(x,x)/2]$ in place of \cite[(4.33)]{GM82}. The argument on \cite[p.~36]{GM82}, through the pinned norm of \cite[Prop.~9]{GM82}, bounds the remainder in \cite[(4.34)]{GM82} by $e^{-\mu}(1-e^{-\mu})^{-2}\mD^{2}\bGM^{-1}(1-C)^{-9}$, uniformly in $x$. With the weights $e^{2\kappa_{S}(|m|-1)}$ the same argument gives this bound up to a constant factor, since we use the pinned norm with weight $e^{(2\kappa_{S}+\mu)\sum_{j}|m_{j}|}$ and $\sum_{j\ge2}|m_{j}|\ge t-1$. Thus, with $A=0$ and uniformly in $x$ and $\Gp$,
\begin{equation}\label{eq:6.lead}
\rho_{1}(m,x)=e^{-\bGM m^{2}v_{\Gp}(x,x)/2}+R(m,x),\qquad\sum_{m}|R(m,x)|\,e^{2\kappa_{S}(|m|-1)}\le Ce^{-\mu}\mD^{2}/\bGM .
\end{equation}
It replaces the final assertion of \cite[Prop.~3]{GM82}, whose error at fixed $\mu$ is of order $e^{-\mu}$.

(h) \emph{The tree length inside the sum.} We state the bound (S2) with the tree length of the points inside the integral, which is what the chain \cite[(4.30) to (4.32)]{GM82} gives. By \cite[(4.30)]{GM82}, $|\rho_{s}(\xi_{1},\dots,\xi_{s})|\le s!\sum_{t\ge s}\binom ts\sum_{[\alpha]:\,C(\alpha)=\{1,\dots,t\}}\int d\xi_{s+1}\cdots d\xi_{t}\,|\sigma^{R}_{\alpha}(\xi_{\alpha})|$. Applied to the sets $\{x_{i}\}$, \cite[(4.9)]{GM82} gives $L_{\Gp}(x_{1},\dots,x_{s})\le\tilde L(x_{1},\dots,x_{t})$ for all $t\ge s$ and $x_{s+1},\dots,x_{t}\in V$, where $\tilde L$ is the least length of a tree on exactly the vertices $x_{1},\dots,x_{t}$ \cite[(4.8)]{GM82}, computed with $3^{-1/2}d_{\Gp}$. In \cite[(4.31)]{GM82} the restriction $x_{i}\in a_{i}$ ($i\ge2$) enters only through $L(a_{1},\dots,a_{s})\le\tilde L(x_{1},\dots,x_{t})$, since the pinned norm \cite[(4.12)]{GM82}, taken in the max form of change (a), sums all positions with the weight $e^{A\tilde L(x_{\alpha})}$. Hence (4.31) holds with $x_{2},\dots,x_{s}$ summed over $V$ and the factor $e^{(1-\delta_{\mathrm{GM}})AL_{\Gp}(x_{1},\dots,x_{s})}$ on the left in place of $e^{-(1-\delta_{\mathrm{GM}})AL(a_{1},\dots,a_{s})}$ on the right. Combining this with (4.32), \cite[Prop.~9]{GM82} and the sum over $x_{1}\in a_{1}$, we obtain \eqref{eq:6.S2}. The bound \eqref{eq:6.S2sets}, which has the form of \cite[(1.18)]{GM82}, follows from it, since $L_{\Gp}(a_{1},\dots,a_{s})\le L_{\Gp}(x_{1},\dots,x_{s})$ whenever $x_{i}\in a_{i}$.

With these changes the proof of \cite[Prop.~3]{GM82} gives (S1) and (S2). Since the Debye condition is the only threshold that involves $\MY$, they hold uniformly in $\MY\in I_{Y}$. By Proposition~\ref{prop:6.3}(d), $\rho_{s}=0$ in (S2) unless $x_{1},\dots,x_{s}$ lie in one component. Together with Proposition~\ref{prop:6.3}, (S1) gives \eqref{eq:6.S1}. For (S3), the activities are equal, the potentials depend on the charges through $m_{i}m_{j}$ and the rule (3.19), both invariant under $m\mapsto-m$, and $\eps(\xi)$ is $2\pi\bGM^{-1/2}$-periodic in $\psi(x)$. (S4) is Proposition~\ref{prop:6.6}. For (S5), $\nu$ is real and symmetric because $\rho_{2}$ is, and it vanishes across components by Proposition~\ref{prop:6.3}(d). Since $|m_{1}m_{2}|\le Ce^{2\kappa_{S}(|m_{1}|+|m_{2}|-1)}$, \eqref{eq:6.S2sets} at $s=2$ with $a_{1}=\{x_{1}\}$ and any $a_{2}\subseteq V$ gives $\sum_{x_{2}\in a_{2}}|\nu(x_{1},x_{2})|\le Ce^{-\mu}\mD^{2}e^{-(1-\delta_{\mathrm{GM}})A\,L_{\Gp}(\{x_{1}\},a_{2})}$. With $A=0$ and $a_{2}=V$ this is $\|\nu\|_{0}\le Ce^{-\mu}\mD^{2}$, at most $\frac18\mD^{2}\le\frac12\mt^{2}$ once $Ce^{-\mu}\le\frac18$. On outputs we take $A=\sqrt3\,\kappa_{Y}\MY$ and, for an integer $k\ge0$, $a_{2}=\{x_{2}:d_{\Gp}(x_{1},x_{2})\ge k\}$, whose tree length with $\{x_{1}\}$ is at least $3^{-1/2}k$. With these choices and $\gamma_{\nu}:=(1-\delta_{\mathrm{GM}})\kappa_{Y}\MY$, the tail sums $T_{k}:=\sum_{x_{2}:\,d_{\Gp}(x_{1},x_{2})\ge k}|\nu(x_{1},x_{2})|$ satisfy $T_{k}\le Ce^{-\mu}\mD^{2}e^{-\gamma_{\nu}k}$. Summation by parts, $e^{rd}=1+\sum_{1\le k\le d}(e^{rk}-e^{r(k-1)})$, gives for $0\le r<\gamma_{\nu}$
\begin{align*}
\sum_{x_{2}}|\nu(x_{1},x_{2})|\,e^{r\,d_{\Gp}(x_{1},x_{2})}&=T_{0}+\sum_{k\ge1}(e^{rk}-e^{r(k-1)})T_{k}\\
&\le\Bigl(1+\frac{1-e^{-r}}{e^{\gamma_{\nu}-r}-1}\Bigr)Ce^{-\mu}\mD^{2}\le\Bigl(1+\frac{r}{\gamma_{\nu}-r}\Bigr)Ce^{-\mu}\mD^{2},
\end{align*}
which is at most $2Ce^{-\mu}\mD^{2}$ for $r\le\gamma_{\nu}/2$. For $\delta_{\mathrm{GM}}\le\frac12$ we have $\gamma_{\nu}\ge\frac12\kappa_{Y}\MY\ge2\kappa_{0}\mt\ge2\rcov$, where the middle inequality follows from the last assertion of Proposition~\ref{prop:6.11}, and hence $\|\nu\|_{\rcov}\le2Ce^{-\mu}\mD^{2}$. This is (S5) on outputs, with the constant $C$ doubled. Finally, (S2$^+$) follows from the pinned-norm bound of \cite[Prop.~9]{GM82} at $\kappa_{S}+\frac12\ln3$, since $\sum_{s=1}^{t}\binom ts2^{s}\le3^{t}\le\prod_{i\le t}3^{|m_{i}|}$, and the moments from $(\sum_{i}|m_{i}|)^{n}\le n!(2\kappa_{S})^{-n}e^{2\kappa_{S}\sum_{i}|m_{i}|}$ (Section~\ref{S4-S:S4.3}).
\qed
\end{proof}


\begin{proposition}[Additive one-body bound]\label{prop:6.6}
Let $\rhoref(m):=\lim_{\Lambda\uparrow\Zthree}\rho^{\Lambda}_{1}(m,0)$, the one-point coefficient of the Yukawa gas on $\Zthree$: $\rho^{\Lambda}_{1}(m,x)$ is the series of \cite[Prop.~3]{GM82} over the configurations in $\Lambda$, whose terms do not depend on $\Lambda$, so the limit exists by dominated convergence, does not depend on the base point, and satisfies \eqref{eq:6.lead} and \eqref{eq:6.S4c} with the same constants (Section~\ref{S4-S:S4.3}). Let $\Phi=\Phi_{\Gp}$. Uniformly in $x\in V$ and $\Gp$,
\begin{align}
\bigl|\rho_{1}(\pm1,x)-\rhoref(1)e^{-\Phi(x)}\bigr|&\le Ce^{-\mu}\rhoref(1),\label{eq:6.S4a}\\
\rho_{1}(\pm1,x)&\le\rhoref(1)\bigl(1+Ce^{-\mu}\bigr),\label{eq:6.S4b}\\
\sum_{|m|\ge2}|\rho_{1}(m,x)|\,m^{2}e^{2\kappa_{S}(|m|-1)}&\le C\mD^{2}/\bGM .\label{eq:6.S4c}
\end{align}
Consequently $|\rho_{1}(\pm1,x)-\rhoref(1)|\le\rhoref(1)\bigl(1-e^{-\Phi(x)}+Ce^{-\mu}\bigr)$.
\end{proposition}

\begin{proof}
Subtracting from \eqref{eq:6.lead} its form on $\Zthree$, $\rhoref(1)=e^{-\bGM v(0)/2}+R^{\infty}$, $|R^{\infty}|\le Ce^{-\mu}\mD^{2}/\bGM$, gives \eqref{eq:6.S4a} and \eqref{eq:6.S4b}. Here we use $e^{-\bGM v_{\Gp}(x,x)/2}=e^{-\bGM v(0)/2}e^{-\Phi(x)}$: the coefficient $\rhoref(1)$ carries the same self-energy factor as $\rho_{1}$ up to the image term $e^{-\Phi}$, so the subtraction involves no error of order $\bGM\MY$. We also use the bound $\Phi\ge0$ of Lemma~\ref{lem:6.1} and $\mD^{2}/\bGM\le2\rhoref(1)(1-2Ce^{-\mu})^{-1}$, which holds because $e^{-\bGM v(0)/2}\ge z$. In \eqref{eq:6.S4c} the leading terms are at most $e^{-2\bGM v(0)}\le e^{-\mu}\mD^{2}/\bGM$, and in the remainders $m^{2}$ is absorbed by the $\mu$-part of the weight $e^{(2\kappa_{S}+\mu)\sum_{j}|m_{j}|}$ of change (g), since $m^{2}e^{-\mu|m|}\le4e^{-2\mu}$ for $|m|\ge2$ and $\mu\ge1$, so that the Debye condition of change (f) suffices at the same $\kappa_{S}$.
\qed
\end{proof}

\begin{remark}\label{rem:6.7}
We state Proposition~\ref{prop:6.6} in additive form only, because the multiplicative form $\rho_{1}(\pm1,x)=\rhoref(1)e^{-\Phi(x)}(1+\Ord(e^{-\mu}))$ fails at a first-layer corner site of a box. There a tight neutral pair, which feels only its dipole image, makes the one-site density exponentially larger in $\bGM$ (see the computation for corner sites in \S\ref{ssec:E.5}).
\end{remark}

\paragraph{The masses $\mt$ and $\MY$.} For $M\in I_{Y}$ let $\rho^{\mathrm{ref}}_{M}$ be the one-point coefficient of the Yukawa gas on $\Zthree$ with kernel $(-\Delta_{\Zthree}+M^{2})^{-1}$, and $\mt(M)^{2}:=\bGM\sum_{m}m^{2}\rho^{\mathrm{ref}}_{M}(m)$. By \eqref{eq:6.lead} and \eqref{eq:6.S4c} on $\Zthree$, in which the terms with $|m|\ge2$ carry the weight $e^{2\kappa_{S}(|m|-1)}\ge e^{2\kappa_{S}}$, we have $\mt(M)^{2}=2\bGM e^{-\bGM v^{(M)}_{\Zthree}(0)/2}+\Ord(e^{-\mu}+e^{-2\kappa_{S}})\mD^{2}=\mD^{2}(1+\Ord(\bGM M))+\Ord(e^{-\mu}+e^{-2\kappa_{S}})\mD^{2}$, as $v^{(M)}_{\Zthree}(0)=\vCb(0)-\Ord(M)$. Since $\bGM\mD\to0$, $\mt(M)\in[\mD/2,2\mD]$ once $\mu$ and $\kappa_{S}$ exceed a threshold that depends on $C$ through the factor $(1-C)^{-9}$ in the remainder of \eqref{eq:6.lead} \cite[p.~36]{GM82}, and $\bGM\ge\bGMo$. Since $M_{1}$ and the rule (3.19) do not depend on $M$, the mass $M$ appears only in $v^{2}_{M}=v^{(M)}-v^{(M_{1})}$, so each term of the expansion of $\rho^{\mathrm{ref}}_{M}$ is continuous in $M$. The $t$-th term is $\Ord(te^{-\mu(t-1)})$ uniformly in $M\in I_{Y}$ \cite[p.~36]{GM82}. With $M_{-}:=\mD/(2\lPV)$, we have $0\le v^{2}_{M}\le v^{2}_{M_{-}}$ entrywise by the resolvent identity, so at fixed $t$ the terms are dominated by those with $v^{2}_{M_{-}}$ in place of $v^{2}_{M}$, whose bound in \cite{GM82} needs exactly the Debye condition of change (f) at $M=M_{-}$. By dominated convergence, $M\mapsto\mt(M)/\lPV$ is a continuous map of $I_{Y}$ into itself. Its fixed points form a nonempty closed set. We let $\MY$ be the smallest fixed point and put
\begin{equation}\label{eq:6.mass}
\mt:=\mt(\MY)=\lPV\MY\in[\mD/2,2\mD],\qquad\rhoref:=\rho^{\mathrm{ref}}_{\MY},\qquad\mt^{2}=\bGM\sum_{m}m^{2}\rhoref(m).
\end{equation}
The inverse mass $\mt^{-1}$, which like $\rhoref$ is determined by the gas on $\Zthree$ alone, before any construction, is the unit of length of Sections~\ref{sec:3} to~\ref{sec:mincut}, and $\rhoref$ is the reference density of Proposition~\ref{prop:6.6} and of Section~\ref{sec:stage2}.

\subsection{Heat kernel and covariance on outputs of the construction}
\label{ssec:6.3}

We prove the statements of this subsection in Appendix~\ref{app:F}, from (R2) and, for Proposition~\ref{prop:6.10}, from (R5) and (R6), without any Poincar\'e inequality or volume-growth hypothesis. Full proofs of Proposition~\ref{prop:6.11} and Theorem~\ref{thm:6.12} are given in Section~\ref{S:E2}. The \emph{coarse slit graph} of an output is $\Gamma=(V_{c},E_{c})$, $V_{c}:=\Lambda^{\circ}\setminus\XX$, $E_{c}:=E(V_{c})\setminus S$, the graph $\Gp$ of Sections~\ref{sec:3} to~\ref{sec:mincut} (\S\ref{ssec:2.5}). For $W\subseteq V_{c}$, $\partial_{\Gamma}W$ is the set of bonds of $E_{c}$ with exactly one end in $W$. A \emph{coarse cell} is a site of the grid of Definition~\ref{def:3.4}, a box of fine sides in $[r_{0},\bar r_{0}]$, not a cell of Definition~\ref{def:3.12}. In fine units we put
\begin{equation}\label{eq:6.scales}
V_{*}^{f}:=r_{0}^{3}V_{*}/2,\qquad\RHK:=(1-\etaw)c_{I}c_{\rho}L/3,\qquad K_{0}:=(1-\etaw)c_{I}c_{\rho}c_{L}/6\le\mt\RHK\le2K_{0},
\end{equation}
with $c_{I}=2^{2/3}$, $c_{\rho}=1/12$, $V_{*}=(1-\etaw)^{3}\rho_{\lambda}^{3}/2$ (coarse units). The bounds on $\mt\RHK$ follow from Definition~\ref{def:3.1} (\S\ref{S:E1.scales}). Let $p_{t}(x,y)$ be the kernel of $e^{t\Delta_{\Gp}}$, the transition probability of the walk $X_{t}$ that jumps across each bond of $\Gp$ at rate $1$, and let $\tau(x,\rho):=\inf\{t:d_{\Gp}(x,X_{t})>\rho\}$.

\begin{theorem}[Heat kernel at scale $\RHK$]\label{thm:6.9}
Let $\Gp$ be an output, and let $C_{H}$ (a function of $c_{F}$ of Proposition~\ref{prop:6.10}) and $\eps_{1}^{\mathrm{HK}}$ be the absolute constants of \S\ref{S:E1.HK}. There are $C_{\mathrm{mo}},c_{\mathrm{ex}},C_{\mathrm{gs}},c_{\mathrm{gs}},C_{M}>0$, depending only on $C_{H}$, such that for $\bGM\ge\bGMo$, $x,y\in V$ and $d:=d_{\Gp}(x,y)$: (HK0) $p_{t}(x,y)\le\exp(-\alpha d+6(\cosh\alpha-1)t)$ for all $\alpha,t\ge0$; in particular $p_{t}(x,y)\le e^{-d/17}$ for $t\le d$; (HK1) $p_{t}(x,y)\le\max\bigl(C_{H}t^{-3/2},4/V_{*}^{f}\bigr)\le C_{H}\max\bigl(t^{-3/2},\RHK^{-3}\bigr)$ for all $t>0$; (HK1$'$) $\mathbb{E}_{x}d_{\Gp}(x,X_{t})\le C_{\mathrm{mo}}t^{1/2}$ for $0\le t\le\RHK^{2}$; (HK2) $\mathbb{P}_{x}(\tau(x,\rho)<t)\le\sqrt2\exp(-c_{\mathrm{ex}}\min(\rho^{2}/t,\rho))$ for $\rho\ge1$ and $0<t\le\eps_{1}^{\mathrm{HK}}\RHK^{2}$; (HK3) $p_{t}(x,y)\le C_{\mathrm{gs}}t^{-3/2}\exp(-c_{\mathrm{gs}}\min(d^{2}/t,d))$ for $1\le t\le2\eps_{1}^{\mathrm{HK}}\RHK^{2}$; (HK4) $\mathbb{E}_{x}e^{\theta d_{\Gp}(x,X_{s})}\le1+C_{M}\theta s^{1/2}$ for $0<s\le\eps_{1}^{\mathrm{HK}}\RHK^{2}$ and $0\le\theta\le\min(s^{-1/2},c_{\mathrm{ex}}/2)$.
\end{theorem}

\begin{proposition}[Coarse-to-fine transfer]\label{prop:6.10}
Let $\Gp$ be an output, read on the fine lattice. (i) \emph{Isoperimetry.} $|\partial_{\Gp}A|\ge c_{F}|A|^{2/3}$ for every $A\subseteq V$ with $|A|\le V_{*}^{f}$, where the absolute constant $c_{F}$ (\S\ref{S:E1.ctf}) does not depend on $r_{0}$. (ii) \emph{Faber\textendash Krahn, components.} For $\Omega\subseteq V$ with $|\Omega|\le V_{*}^{f}$, $\lambda_{1}^{D}(\Omega)\ge(c_{F}^{2}/12)|\Omega|^{-2/3}$, where $\lambda^{D}_{1}(\Omega)$ is the smallest eigenvalue of $-\Delta_{\Gp}$ on functions vanishing off $\Omega$; and every component of $\Gp$ has more than $2V_{*}^{f}$ vertices. (iii) \emph{Cheeger transfer.} If a set $K$ of coarse cells satisfies $|\partial_{K}W|\ge h_{c}\min(|W|,|K\setminus W|)$ for every $W\subseteq K$, then its blow-up $K^{f}$ satisfies $|\partial_{K^{f}}A|\ge r_{0}^{-1}\min\{\frac12,h_{c}/C_{\mathrm{bl}}\}\min(|A|,|K^{f}\setminus A|)$ for every $A\subseteq K^{f}$, with the absolute constant $C_{\mathrm{bl}}$ of \S\ref{S:E1.ctf}. (iv) \emph{Cell Poincar\'e inequality.} For every fine cell $\Omega$ of a block $Q$ there is a connected set $\KK^{\flat}$ with $\Omega\subseteq\KK^{\flat}\subseteq V\cap\hat Q(Q)$ such that $\sum_{x\in\Omega}|\varphi(x)-\bar\varphi_{\Omega}|^{2}\le C_{W}^{f}k_{\square}^{2}\sum_{b\in E(\KK^{\flat})}(\nabla_{b}\varphi)^{2}$ for every $\varphi:V\to\RR$, with $C_{W}^{f}:=108C_{\mathrm{bl}}^{2}/\chi^{2}$ and $k_{\square}:=r_{0}(k_{P}+3)$; every fine site and every fine bond lies in at most $N_{\mathrm{ov}}\le27(k_{P}/L_{c})^{3}$ of the sets $\KK^{\flat}$, resp.\ $E(\KK^{\flat})$; and $|\Omega|\ge c_{1}L^{3}$. (v) \emph{Diameter.} Under (L8-2) of \S\ref{sec:7.6}, every graph $(\KK^{\flat},E(\KK^{\flat}))$ is connected, of diameter $d_\flat\le2+0.469\,\bGM/\mt=2+0.235L'$ in its own graph distance; in particular every $\KK^{\flat}$ has $d_{\Gp}$-diameter less than $L'$.
\end{proposition}

\begin{proposition}\label{prop:6.11}
Let $\Gp$ be an output, $\kappa_{W}:=\min(1,1/(2C_{M}))$ and $m_{W}:=\min(1,c_{\mathrm{ex}}/2)$. For every mass $0<M\le m_{W}$,
\begin{equation}\label{eq:6.LW}
\sup_{x}\sum_{y}v^{(M)}_{\Gp}(x,y)e^{\theta_{M}d_{\Gp}(x,y)}\le3M^{-2},\qquad\theta_{M}:=\kappa_{W}M\min\bigl(1,(\eps_{1}^{\mathrm{HK}})^{1/2}M\RHK\bigr).
\end{equation}
In particular, if $\MY\le m_{W}$,
\begin{equation}\label{eq:6.CY}
\sup_{x}\sum_{y}v_{\Gp}(x,y)e^{\kappa_{Y}\MY d_{\Gp}(x,y)}\le3\MY^{-2},\qquad\kappa_{Y}:=\kappa_{W}\min\bigl(1,(\eps_{1}^{\mathrm{HK}})^{1/2}K_{0}/\lPV\bigr),
\end{equation}
and the same bound holds for $v^{2}_{\Gp}$ (since $0\le v^{2}_{\Gp}\le v_{\Gp}$ entrywise) and for $v^{1}_{\Gp}$ at mass $M_{1}\le m_{W}$. Moreover $\kappa_{Y}\MY\ge4\kappa_{0}\mt$, where $\kappa_{0}:=\kappa_{W}(\eps_{1}^{\mathrm{HK}})^{1/2}K_{0}$.
\end{proposition}

We use only one property of the constant $\kappa_{W}$: since $C_{M}$ depends only on the absolute constant $C_{H}$ (Theorem~\ref{thm:6.9}), $\kappa_{W}$ does not depend on $\tau$, $K$, $C$ or $D$.

\begin{theorem}[Covariance bounds]\label{thm:6.12}
Let $\Gp$ be an output and $C_{0}:=(\MY^{-2}\Delta_{\Gp}^{2}-\Delta_{\Gp}+\mt^{2})^{-1}$ on $\RR^{V}$. Assume $\lPV\le\frac14$, $\MY\le m_{W}$ and $\mt\le\min(1/68,m_{W}/2)$ (all true for $\bGM\ge\bGMo$). Put $\kappa_{0}':=\min(\kappa_{0}/2,(2c_{\mathrm{gs}})^{1/2})$ and $\rcov:=\min(\kappa_{0},\kappa_{0}')\mt$, and let $K_{1},K_{1}',K_{1}''$ and $K_{3}\le3.5$ be the constants of \S\ref{S:E1.cov}, which depend only on $(c_{F},c_{\rho},c_{L},\lPV)$. Then for all $x,y\in V$: (H1) $C_{0}(x,x)\le K_{1}\MY$ (the zero modes included); (H1$'$) $0\le C_{0}(x,y)\le K_{1}'\MY e^{-\kappa_{0}'\mt d_{\Gp}(x,y)}$; (H2) $\|C_{0}\|_{\kappa_{0}\mt}\le K_{3}\mt^{-2}$; (H$_{C}$) if $\nu$ is symmetric with $\|\nu\|_{\rcov}\le\eps_{\nu}\mt^{2}$ and $2K_{3}\eps_{\nu}\le1$, then $C:=(C_{0}^{-1}+\nu)^{-1}$ exists and is positive definite, $C(x,x)\le2K_{1}\MY$, $|C(x,y)|\le2K_{1}''\MY e^{-\rcov d_{\Gp}(x,y)}$ and $\|C\|_{\rcov}\le2K_{3}\mt^{-2}$; the same bounds hold for every matrix $C_{s}$ with $|C_{s}(x,y)|\le|C(x,y)|$ entrywise and $C_{s}(x,x)\le C(x,x)$, such as the interpolated covariances of Appendix~\ref{app:C}.
\end{theorem}

\subsection{Depletion}
\label{ssec:6.4}

In this subsection we bound the number of depleted sites per cell. The face kernel of Theorem~\ref{thm:6.16} contains all collective effects of the walls. Chebyshev's inequality turns (R9) into a site count, and sparse insulation costs one logarithm, which Section~\ref{sec:stage2} absorbs. Throughout, $\Lambda'=\Lambda^{\circ}$, $M:=\MY$, and $u_{x}:=v_{\Lambda'}(\cdot,x)$ is the kernel of the box without holes. The \emph{missing pairs} are the $(y,y')$ with $y\in V$, $|y-y'|=1$, and $y'\in\XX$ (the set $\mathfrak{M}_{\XX}$) or $\{y,y'\}\in S$ (the set $\mathfrak{M}_{S}$). Their union is $\mathfrak{M}:=\mathfrak{M}_{\XX}\cup\mathfrak{M}_{S}$. A frozen bond gives one pair at each end (frozen bonds are \emph{two-sided}), and $N_{2}(U):=\#\{(y,y')\in\mathfrak{M}:y\in U\}$. For $x,y\in V$ we put $g_{x}(y):=\sum_{y':\,(y,y')\in\mathfrak{M}}(u_{x}(y)-u_{x}(y'))$.

\begin{lemma}\label{lem:6.13}
For every slit graph $\Gp\subseteq\Lambda'$ and $x\in V$: (a) for $y\in V$,
\[
\bigl((-\Delta_{\Gp}+M^{2})u_{x}|_{V}\bigr)(y)=\delta_{x}(y)-g_{x}(y);
\]
(b) $I_{\Gp}(x)=\sum_{(y,y')\in\mathfrak{M}}v_{\Gp}(x,y)\bigl[u_{x}(y)-u_{x}(y')\bigr]$.
\end{lemma}

\begin{proof}
For (a), at $y\in V$ the operator $-\Delta_{\Lambda'}+M^{2}$ is $-\Delta_{\Gp}+M^{2}$ plus the terms of the pairs of $\mathfrak{M}$ at $y$, and $(-\Delta_{\Lambda'}+M^{2})u_{x}=\delta_{x}$. For (b) we apply $v_{\Gp}$ to (a) at $x$.
\qed
\end{proof}

\begin{lemma}\label{lem:6.14}
There is an absolute constant $A_{0}^{\mathrm{AD}}$ such that on every output, for every $z$ and every $r\ge1$ (fine units), $N_{2}(V\cap B(z,r))\le A_{0}^{\mathrm{AD}}r^{2}+A_{0}^{\mathrm{AD}}r^{3}/(r_{0}\rho_{\lambda})$.
\end{lemma}

\begin{proof}
At the final state Lemma~\ref{lem:4.5} applies to all sets $W\subseteq V_{c}$ of coarse cells, with $\partial_{\Gamma}W$ as the coarse cut $\partial_{\Gp}W$. In \eqref{eq:4.Ra} a frozen bond with one end in $W$ carries $w_{b}-1=1-\etaw$ and one inside $W$ carries $w_{b}\ge2(1-\etaw)$, where $w_{b}=2-\etaw$ is the hull weight. Hence the coarse count $N_{2}^{c}(W)$ of missing pairs with first entry in $W$ is at most $(|\partial_{\Gamma}W|+\lambda|W|)/(1-\etaw)$. Informally, the two-sided count is forced, and it reduces the density bound to the minimality of the hull on a ball. We first consider $r\ge r_{0}$, and let $W$ be the set of coarse cells of $V_{c}$ that meet $B(z,r)$. Then $|W|\le C(r/r_{0})^{3}$, and each bond of $\partial_{\Gamma}W$ joins a coarse cell meeting $B(z,r)$ to a coarse cell of $V_{c}$ that does not, so $|\partial_{\Gamma}W|\le C(r/r_{0})^{2}$. Since a coarse missing pair carries fewer than $4r_{0}^{2}$ fine ones, $N_{2}(V\cap B(z,r))\le4r_{0}^{2}\cdot\frac{10}{9}\bigl(C(r/r_{0})^{2}+C(r/r_{0})^{3}/\rho_{\lambda}\bigr)$. For $1\le r<r_{0}$, the fine missing pairs with first entry in $B(z,r)$ cross at most six coarse grid planes, each in at most $Cr^{2}$ bonds, counted with both orientations.
\qed
\end{proof}

\begin{lemma}\label{lem:6.15}
On every output, for all $x,y\in V$, $v_{\Gp}(x,y)\le C_{\mathrm{Gr}}\bigl[(1+|x-y|)^{-1}+\eta_{G}\bigr]$ with $\eta_{G}:=\mt(K_{0}^{-1}+\lPV^{2}K_{0}^{-3})$ and $C_{\mathrm{Gr}}$ depending only on the constants of Theorem~\ref{thm:6.9}; and $v_{\Gp}(x,y)=0$ when $x$ and $y$ lie in different components.
\end{lemma}

\begin{proof}
We split the time integral in $v_{\Gp}=\int_{0}^{\infty}e^{-M^{2}t}p_{t}\,dt$, with $R:=\RHK$. On $[0,1]$ we use $p_{t}\le1$ and (HK0). On $[1,2\eps_{1}^{\mathrm{HK}}R^{2}]$, (HK3), split at $t=d_{\Gp}(x,y)$, gives $C/(1+|x-y|)$. Beyond this interval, (HK1) gives $C(R^{-1}+M^{-2}R^{-3})\le C\eta_{G}$, since $R\ge K_{0}/\mt$ and $M=\mt/\lPV$. Across components $v_{\Gp}=0$ by Lemma~\ref{lem:6.2}(i). The zero mode of a component $\KK$ contributes $M^{-2}/|\KK|\le1/(2M^{2}V_{*}^{f})$, by Proposition~\ref{prop:6.10}(ii), and this lies inside $M^{-2}R^{-3}$.
\qed
\end{proof}

\begin{theorem}[Face-kernel bound]\label{thm:6.16}
On every output, for every $x\in V$, with $r(y):=|x-y|$, $I_{\Gp}(x)\le C_{F}S_{3}(x)+C_{F}\eta_{G}S_{2}(x)$, where $S_{3}(x):=\sum_{(y,y')\in\mathfrak{M}}(1+r(y))^{-3}e^{-\MY r(y)/2}$, $S_{2}(x):=\sum_{(y,y')\in\mathfrak{M}}(1+r(y))^{-2}e^{-\MY r(y)/2}$, $C_{F}:=C_{\mathrm{Gr}}C_{L}'$ with $C_{L}'$ the constant of Lemma~\ref{lem:F.7}, and $S_{2}(x)\le C_{S}\log(2/\MY)$ with $C_{S}:=CA_{0}^{\mathrm{AD}}(1+\lPV/(c_{\rho}c_{L}))$ (\S\ref{S:E1.dep}).
\end{theorem}

\begin{proof}
We take absolute values in Lemma~\ref{lem:6.13}(b), so no cancellation across a sheet is used and the Neumann kernel appears only through its values. For each pair, $0\le v_{\Gp}(x,y)\le C_{\mathrm{Gr}}[(1+r(y))^{-1}+\eta_{G}]$ by Lemma~\ref{lem:6.15}, and $|u_{x}(y)-u_{x}(y')|\le C_{L}'(1+r(y))^{-2}e^{-\MY r(y)/2}$ by (L2) of Lemma~\ref{lem:F.7}. That lemma applies with $M=\MY$, because $|y-y'|=1$, the sides of $\Lambda^{\circ}$ are, by Lemma~\ref{lem:3.3}(c), at least $\ell_{D}\ge\frac12\mt^{-1}\ge\MY^{-1}$, and $\MY\le\frac15$ for $\bGM\ge\bGMo$. A frozen bond appears twice, each time with its own first entry. For the bound on $S_{2}$ we sum Lemma~\ref{lem:6.14} over dyadic shells $2^{k}\le r<2^{k+1}$. Each shell inside $r_{0}\rho_{\lambda}$ gives at most $8A_{0}^{\mathrm{AD}}$, and up to $r\le\MY^{-1}$ these contributions yield the logarithm. For the shells beyond we use $e^{-\MY r/2}$, and the growth term gives at most $CA_{0}^{\mathrm{AD}}/(\MY r_{0}\rho_{\lambda})\le2CA_{0}^{\mathrm{AD}}\lPV/(c_{\rho}c_{L})$ (\S\ref{S:E1.dep}(f)).
\qed
\end{proof}

We put $\eps_{0}:=\theta_{0}/(2\pi^{2}\tau)$, so that $\Phi(x)>\theta_{0}$ if and only if $v_{\Gp}(x,x)-v(0)>\eps_{0}$, and $\eps_{F}:=\eps_{0}/(4C_{F})$. With the constants $L_{g}$, $C_{R}$ of \S\ref{S:E1.dep}, let $\Rdep:=\lceil32A_{0}^{\mathrm{AD}}L_{g}/\eps_{F}\rceil=\lceil C_{R}\tau/\theta_{0}\rceil$, and let (F) be the condition $C_{F}\eta_{G}C_{S}\log(2/\MY)\le\eps_{0}/4$.
The constant $c_{0}:=z\tau\,\bar r_{0}^{\,2}(w_{*}+2|\partial\Delta|)$, maximized over removal cells $\Delta$, comes from the two-sided bound (R9), proved in Proposition~\ref{prop:4.22}. It satisfies $c_{0}<25(3+c_{s})$ (\S\ref{S:B1.c0}).

\begin{theorem}[Depletion per removal cell]\label{thm:6.17}
Let $\Gp$ be an output and assume $\Rdep\le r_{0}\rho_{\lambda}$, $\Rdep\le\ell_{D}$ and (F). Then for every removal cell $Q$, $\#(\Dep\cap Q)\le C(z\theta_{0})^{-1}[c_{0}+C_{\mathrm{box}}+c_{0}^{\mathrm{sp}}\log(C'\tau/\theta_{0})]$, where $c_{0}^{\mathrm{sp}}\le c_{0}$ is the part of the insulation lying in sparse blocks (defined in the proof), $C_{\mathrm{box}}$ is the box-face term, and $C,C'$ are absolute up to $C_{R}$ and $C_{F}$.
\end{theorem}

\begin{proof}
(i) \emph{Reduction to $S_{3}$.} Since $v_{\Gp}(x,x)-v(0)=I_{\Gp}(x)+[v_{\Lambda'}(x,x)-v(0)]$, we have $\Dep\subseteq\{I_{\Gp}>\eps_{0}/2\}\cup\{\Phi_{\Lambda'}>\theta_{0}/2\}$, where $\Phi_{\Lambda'}(x):=\frac12\bGM(v_{\Lambda'}(x,x)-v(0))$. On the first set Theorem~\ref{thm:6.16} and (F) give $S_{3}(x)>\eps_{F}$.

(ii) \emph{Far insulation.} The pairs at distance more than $R:=\Rdep$ from $x$ contribute to $S_{3}(x)$ at most $\sum_{k\ge0}(2^{k}R)^{-3}N_{2}(V\cap B(x,2^{k+1}R))e^{-\MY2^{k}R/2}$. By Lemma~\ref{lem:6.14}, with $R\le r_{0}\rho_{\lambda}$ and $\MY r_{0}\rho_{\lambda}\ge c_{\rho}c_{L}/(2\lPV)$, this is at most $16A_{0}^{\mathrm{AD}}L_{g}/R\le\eps_{F}/2$ (\S\ref{S:E1.dep}(g)). There the shells with $2^{k+1}R\le r_{0}\rho_{\lambda}$ give $16A_{0}^{\mathrm{AD}}/R$, and the others give a logarithm. Hence each $x\in\Dep$ outside the box set has $S_{3}^{\mathrm{near}}(x)>\eps_{F}/2$, where $S_{3}^{\mathrm{near}}$ counts the pairs with first entry within $R$ of $x$.

(iii) \emph{Blocks.} We cover $Q$ by grid cubes $q$ of side $R$. Let $3q$ be the block of $3\times3\times3$ such cubes centred at $q$, put $N_{3q}:=N_{2}(V\cap3q)$, and call $q$ dense if $N_{3q}\ge R^{2}$, sparse otherwise. A dense $q$ has $\#(\Dep\cap q)\le R^{3}\le RN_{3q}$. For a sparse $q$, Chebyshev's inequality gives $\#(\Dep\cap q)\le(2/\eps_{F})\sum_{x\in q}S_{3}^{\mathrm{near}}(x)\le(2/\eps_{F})N_{3q}(4\pi\log(4R)+C_{3})$ plus the box set. Each pair lies in at most $27$ blocks $3q$, and the pairs counted for $q\cap Q\neq\emptyset$ lie within $2R\le2r_{0}\rho_{\lambda}<\ell_{D}$ of $Q$, hence in the $27$ removal cells $Q'$ around $Q$, as the removal cells form a product grid with sides at least $\ell_{D}$. Each of these cells has $N_{2}(Q')\le4r_{0}^{2}(w_{*}+2|\partial\Delta|)\le4c_{0}/(z\tau)$ by (R9), since a coarse pair carries fewer than $4r_{0}^{2}$ fine ones and removed cells contain no site of $V$. Hence $A_{\mathrm{dense}}+A_{\mathrm{sparse}}\le4\cdot27^{2}c_{0}/(z\tau)$, where $A_{\bullet}:=\sum_{q\ \bullet}N_{3q}$. We put $c_{0}^{\mathrm{sp}}:=z\tau A_{\mathrm{sparse}}/(4\cdot27^{2})\le c_{0}$ and substitute $R\le2C_{R}\tau/\theta_{0}$ and $2/\eps_{F}=16\pi^{2}C_{F}\tau/\theta_{0}$.

(iv) \emph{Box faces.} By the image representation of the box kernel (proof of Lemma~\ref{lem:F.7}) and (L1), $v_{\Lambda'}(x,x)-v(0)\le C\sum_{F}d_{F}(x)^{-1}+3C_{\rm im}/\ell_{D}$, the sum over the faces $F$ of $\Lambda'$, $d_{F}(x)$ the layer distance of $x$ from $F$, and the last term from the images other than the $26$ nearest ones, which have a coordinate at distance at least $2n_\mu\ge2\ell_D$ from $x$ ($n_\mu$ the sides of $\Lambda'$). As $\Rdep\le\ell_{D}$ and $C_{R}\ge24\pi^{2}C_{\rm im}$ (\S\ref{S:E1.dep}), $3C_{\rm im}/\ell_{D}\le\theta_{0}/(2\bGM)$. Hence the set $\{\Phi_{\Lambda'}>\theta_{0}/2\}$ lies within distance $C\tau/\theta_{0}$ of $\partial\Lambda'$ and meets a removal cell, of side at most $3\ell_{D}$, in at most $C\ell_{D}^{2}\tau/\theta_{0}\le C_{\mathrm{box}}/(z\theta_{0})$ sites, since $\ell_{D}^{2}\tau\le C\tau/\mt^{2}\le C/z$.
\qed
\end{proof}

\begin{proposition}\label{prop:6.18}
Under the hypotheses of Theorem~\ref{thm:6.17}, for every $\mathcal{U}$-cell $\Delta_{\mathcal{U}}$ of the output, $(\mt^{2}/\bGM)\,\#(\Dep\cap\Delta_{\mathcal{U}})\le \cdep:=A_{0}^{D}+A_{1}^{D}\log(C'\tau/\theta_{0})$, with $A_{0}^{D}:=CN_{\mathrm{cov}}(c_{0}+C_{\mathrm{box}})/\theta_{0}$, $A_{1}^{D}:=CN_{\mathrm{cov}}c_{0}^{\mathrm{sp}}/\theta_{0}$ and $N_{\mathrm{cov}}\le(C_{\mathcal{U}}\mt^{-1}/\ell_{D}+6)^{3}$.
\end{proposition}

\begin{proof}
Steps (i) to (iv) of the proof of Theorem~\ref{thm:6.17} apply to any region $U$ in place of $Q$, the pairs being counted in the union of the blocks $3q$ over the cubes $q$ meeting $U$. By (R7) the cell $\Delta_{\mathcal{U}}$ has diameter at most $C_{\mathcal{U}}\mt^{-1}$. Hence that union lies in a cube of side $C_{\mathcal{U}}\mt^{-1}+4R$, which meets at most $N_{\mathrm{cov}}$ removal cells, each with sides $\ge\ell_{D}\ge R$ and with $N_{2}\le4c_{0}/(z\tau)$. Finally $\mt^{2}/\bGM\le4\mD^{2}/\bGM=8z$. Since a $\mathcal{U}$-cell lies in one component and depletion is counted site by site, the bound does not depend on the number of components.
\qed
\end{proof}

\begin{lemma}\label{lem:6.19}
Let $\theta_{0}=0.01$ and
\[
C_{*}^{\mathrm{dep}}:=\max\{17.8\,C_{R}/(c_{\rho}c_{L}),\,3.6\cdot10^{3}C_{F}C_{S}(K_{0}^{-1}+\lPV^{2}K_{0}^{-3})\}.
\]
If $\tau\ge2$, $2.494\,\tau\ge\ln(C_{*}^{\mathrm{dep}}/\theta_{0})+2.5\ln\tau$ and $\mD/2\le\mt\le2\mD$, then $\Rdep\le r_{0}\rho_{\lambda}$, $\Rdep\le\ell_{D}$ and (F) hold.
\end{lemma}

\begin{proof}
By hypothesis, $C_{*}^{\mathrm{dep}}\tau^{5/2}e^{-2.494\tau}\le\theta_{0}$. With $\mt\le2\mD$, $r_{0}\rho_{\lambda}=c_{\rho}L\ge c_{\rho}c_{L}/(2\mt)$ and $\log(2/\MY)\le\ln(1/\mD)\le2.4944\,\tau$, the three conclusions follow by arithmetic (\S\ref{S:E1.619}).
\qed
\end{proof}

The value $\theta_{0}=0.01$ is dictated by Section~\ref{sec:stage2}, which needs $\eps_{\nu}+\theta'\le1/7$ with $\theta'\ge1-e^{-\theta_{0}}$. The condition of Lemma~\ref{lem:6.19} is a lower bound on $\tau$ that persists.