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\subsection{Section~\ref{sec:7.6} and Appendix~\ref{app:B}}\label{S:G}

\subsubsection{The H\"older exponents and (L8-1)}\label{S:G.L8}
Section~\ref{ssec:D.1} fixes $\eta_H=0.62$, $p_1=p_2=20$, $p_3=1.2$, $p_4=15$,
$p'=1.3$ and $q'=15.6$. These values satisfy $\sum_i1/p_i=1$ and $1/p'+1/q'=1/p_3$. With the bound
$\alpha_D\le7/6$ of Lemma~\ref{lem:7.31}(c) we get
$\hat\eta=\alpha_Dp'\eta_H\le\frac76\cdot1.3\cdot0.62=0.94033\ldots<0.941$, so
(L8-1) holds. Moreover, $\eta_*\le\bar\eta\le1-4/\pi^2+2C_{\hat\rho}e^{-2\kappa_S}<0.62$ for $\kappa_S$
so large that $C_{\hat\rho}e^{-2\kappa_S}<0.0126$, since $1-4/\pi^2=0.5947\ldots$.

\subsubsection{Lemma~\ref{lem:7.34}(c): the shell constant $C_{\mathrm{sh}}=92$}\label{S:G.34}
With $r_*=\frac12+\frac{\sqrt3}2$, $\int_{1/2}^D(r+r_*)^2r^{-3}dr\le\log(2D)+4r_*+2r_*^2$
and $(r+r_*)/r\le1+2r_*$ for $r\ge\frac12$, the shell count of the proof gives,
for $D\ge1$,
\[
\sum_{d(b,C)\le D}d(b,C)^{-2}
\le6\pi|C|\Bigl[\log(2D)+4r_*+2r_*^2+\tfrac12(1+r_*)^2\Bigr]
+16\pi\bigl[(1+r_*)^3+2(1+2r_*)^3\bigr]D .
\]
The first bracket is at most $\log(2D)+12$, since $4r_*+2r_*^2+\frac12(1+r_*)^2=11.9951\ldots$, and
$16\pi[(1+r_*)^3+2(1+2r_*)^3]=16\pi\cdot117.207\ldots=5891.4\ldots\le5892$. At
$D=D_{\cN}=\sqrt3\,\kbox+1\le2\kbox$ and $|C|\ge8\kbox\ge4D_{\cN}$ the sum is at most
$6\pi|C|\bigl(\log\kbox+\log4+12+1473/(6\pi)\bigr)\le6\pi|C|(\log\kbox+92)$,
since $\log4+12+1473/(6\pi)=91.53\ldots$. Multiplying by $(2\pi)^2$ gives
$S(D_{\cN})\le24\pi^3|C|(\log\kbox+92)$.

\subsubsection{Lemma~\ref{lem:7.35}: the constants $c_8$ and $\cnear$}\label{S:G.35}
We set
\[
c_8:=\frac{\hat\eta(1+4/\eta_H)K_1C_u}{2\lPV(1-\nu_0)},\qquad
\cnear:=\frac{2\theta_PC_{\mathrm{BS}}(\log\kbox+C_{\mathrm{sh}})}{\bGM}
+\frac{1024\pi^2\hat\eta}{\eta_Hc_1}\,\frac{(2\rbox+5)^2c_L^2}{\bGM}.
\]

\emph{Uniformity of $\cnear$.} Since $\kbox=\rbox L\le\rbox c_L/\mt$,
$\log\kbox\le\log(\rbox c_L)+\log(1/\mt)$ with $\log(\rbox c_L)<0$. With
$\mt\ge m_D/2$, $m_D=(8\pi^2\tau)^{1/2}e^{-\pi^2\vCb(0)\tau}$,
$\pi^2\vCb(0)<2.4943551$ and $\bGM=4\pi^2\tau$,
\[
\frac{\log(1/\mt)}{\bGM}\le\frac{2.4943551}{4\pi^2}
-\frac{\frac12\log(8\pi^2\tau)-\log2}{4\pi^2\tau}<0.0632\qquad(\tau\ge1).
\]
Hence $2\theta_PC_{\mathrm{BS}}\log\kbox/\bGM<0.1264\,\theta_PC_{\mathrm{BS}}$, which gives the bound
on $\cnear$ in Lemma~\ref{lem:7.35}. The coefficient $\theta_P$ in that bound satisfies
$\theta_P\le(4\hat\eta/\eta_H)C_P'c_L^2\le\frac12$ by (L8-2), uniformly in $\bGM$.

\subsubsection{Lemma~\ref{lem:7.36}: the condition \eqref{eq:7.6.cl}}\label{S:G.36}
At $p_4=15$ and $p_3=1.2$, so that $c_\lambda=\frac14(1-1/p_3)=\frac1{24}$, and with $\alpha_D\le\frac76$
and $c_E\le\frac1{70}$, we have $2\alpha_Dp_4c_E\le2\cdot\frac76\cdot\frac{15}{70}=\frac12$ and
$p_4\alpha_Dc_E^2/(1-2\alpha_Dp_4c_E)+c_E\le\frac{15\cdot(7/6)}{4900\cdot(1/2)}+\frac1{70}
=0.00714\ldots+0.01428\ldots=0.02142\ldots\le\frac1{24}$.
In Proposition~\ref{prop:7.37}(d), $c_0'=\frac12(1-1/p_3-c_\lambda)=\frac12(\frac16-\frac1{24})=\frac1{16}$.

\subsubsection{Theorem~\ref{thm:7.42}: the threshold $\tau_0$}\label{S:G.42}
With $\pi^2\vCb(0)=2.49435508\ldots$ and $p_0=\frac16$, the lone-cell condition
in the proof of Theorem~\ref{thm:7.42} reads
\[
0.4157\,\tau\ge C_{\Dep}A_1^D\log(C'\tau/\theta_0)+\alpha+O(\log\tau),
\]
where $\cdep=A_0^D+A_1^D\log(C'\tau/\theta_0)$ is the depletion constant of Proposition~\ref{prop:6.18}.
The threshold $\tau_0$ is explicit, and it is large because $A_1^D$ is large, as $\theta_0$ is
small and the covering number of Proposition~\ref{prop:6.18} is large. We need only that
this threshold exists, and the same holds for the threshold of Lemma~\ref{lem:A.11}. For
the wall-polymer cells, Proposition~\ref{prop:7.22} gives
$\Phi_{\cU}\ge(\pi c_gc_uc_L^3/24)/(64\bGM^3(11/10)^3\bD)$.

\subsubsection{Lemma~\ref{lem:7.40} and Proposition~\ref{prop:7.41}: constants and fractions}\label{S:G.B11}
\emph{Constants on class-A cells.} They, and the constants $C_{\mathcal{E}}$, $C_{G}$ of Proposition~\ref{prop:B.10}, are independent of $L'$ and depend only on the following quantities:
\begin{itemize}
\item $c_{7},c_{7}',c_{7}''$ of Lemma~\ref{lem:7.33} (through $v_{\pm}$, $\eta_{H}$, $\kappa_{S}$, that is, on the grid scale $L$);
\item $C_{k}$, $\theta'$, $\cdep$ of Lemma~\ref{lem:B.1}(d);
\item $\mu,\kappa_{S},\delta_{\mathrm{GM}},C$ and the Mayer decay;
\item $K_{1}'',K_{3},\alpha_{D}K_{1},\lPV,r_{C}$ of the lines;
\item $C_{r}(c_{p})$ and $c(\qpin,c_{p},K_{g},C_{\cU})$ of the count and the pinning;
\item $c_{T}$ of the Wick bound;
\item $c_{13}$ and $C_{\Dep}\cdep$ of the H\"older split;
\item the geometry constants $c_{u},C_{u},C_{\cU},D_{\cU},v_{\pm},K_{g}$.
\end{itemize}

\emph{Fractions of $F_{1}^{\mathrm{loc}}$.} They are those of Table~\ref{tab:A.1} in Section~\ref{S:H1}. Proposition~\ref{prop:7.41} uses the entries for the numerical factors, which are $c'/16$ on $X\setminus Z$ (Lemma~\ref{lem:B.8}) and $c'/32$ on $Z$, and the entry $c'/8$ on $X\setminus Z$ for the wall polymers.

\emph{Cells.} Put $K_{B}:=K_{0}^{B}\max(1,K_\ell)$, with the line constant $K_\ell$ of Proposition~\ref{prop:B.7}. Each cell carries the per-cell constants $e^{c_{2}^{L}+c_{13}}$ and at most the factor $\max(1,K_\ell)$ of the line constants, as the links inject into the cells. Since $C_{u}\ge1$, $c_{2}^{L}+c_{13}\ge0$ and $\alpha>0$, it follows that $e^{c_{2}^{L}+c_{13}}\le e^{\alpha}e^{(c_{2}^{L}+c_{13})C_{u}}$. With $\Kcell=K_{B}e^{(c_{2}^{L}+c_{13})C_{u}}\max(1,\lPV^{1/6})$ and $\bD^{1/6}=\lPV^{1/6}(\bD/\lPV)^{1/6}$, Proposition~\ref{prop:B.7} shows that a non-seed lone cell carries at most $\Kcell e^{\alpha+C_{\Dep}\cdep}(\bD/\lPV)^{1/6}$, that a non-seed cell of a wall polymer carries at most $\Kcell e^{\alpha+C_{\Dep}\cdep+c_{B}(\bD)}e^{-(c'/8)\Phi_{\cU}/(1+D_{\cU})}$, and that a seed cell carries at most $\Kcell e^{\alpha+C_{\Dep}\cdep}$, times $e^{c_{B}(\bD)}$ if it is of class B.