%
\subsection{Constants of Sections~\ref{sec:7.7} to~\ref{sec:7.9} and Appendix~\ref{app:A}}\label{S:H1}

\paragraph{Lemma~\ref{lem:7.47}.}
Since $t_1\le2+(36k_P/\chi_{\rm N3})\ln k_P$, we have $\Dcell\le35r_0+504\,r_0k_P\ln k_P/\chi_{\rm N3}$. In
fine units $r_0k_P=304L\le304c_L/\mt$ and $k_P<1/\mt$, and therefore
$\ln k_P\le\ln(1/\mt)<0.0632\,\bGM$, as in the proof of Lemma~\ref{lem:7.35}.
Hence
$\Dcell\le35r_0+504\cdot304\cdot0.0632\,(c_L/\chi_{\rm N3})\,\bGM/\mt\le35r_0+1.927\,\bGM/\mt$
for $c_L\le1.99\cdot10^{-4}\chi_{\rm N3}$, which is less than $L'=2\bGM/\mt$ for
$\bGM\ge\bGMo$, since $L_c\ge35$ gives $35r_0\le L$. The choice of $c_L$
through $C_P'c_L^2\le(1-4/\pi^2)/8$ places it far below the threshold
$1.99\cdot10^{-4}\chi_{\rm N3}$.

\paragraph{Constants of the seeded sums (before Lemma~\ref{lem:7.53}).}
The constant $C_9$ exists because blocks have side at least $L$, a block has
fewer than $8/c_1$ cells and a cell has at most $6/c_g$ interfaces. We
set\footnote{The summand $1+c_4'+\log2$ of $c_S$ is already contained in $\log\Kcell$: in the proof of
Proposition~\ref{prop:A.8} the factor $e^{(1+c_4'+\log2)|Z|}$ is bounded by $e^{a|Z|}$, which enters
$c_2^L=a+\alpha$ and hence $\Kcell$. Counting it twice only enlarges $c_S$, which the later arguments allow.}
\[
c_S:=\log\Kcell+\alpha+C_{\Dep}\cdep+1+c_4'+\log2,\qquad
\etaseed:=6D_Ie^{-(c'/4)\Funit}.
\]
Since the constants $\log\Kcell$, $D_{\cU}$ and $\alpha$ are fixed numbers and
$\cdep=O(\log\tau)$ by Proposition~\ref{prop:6.18}, we have $c_S=O(\log\tau)$ and
$c_2=c_S+2=O(\log\tau)$. Moreover $\log D_I=O(\log\tau)$, as $g/L=O(\tau)$, while
$\Funit$ is exponentially large in $\tau$, so that $\etaseed$ is doubly
exponentially small.

\paragraph{Proposition~\ref{prop:7.56} (the count), Step~4.}
By Definition~\ref{def:3.17}, $c_3^{\mathrm{NF}}\kappa_{\mathrm{NF}}=c_0'/16$
and $\thNF=c_0'c_g/(90(c_g+4))$, and hence
\[
\frac{\ln16}{\ln2}\,c_3^{\mathrm{NF}}\kappa_{\mathrm{NF}}=4c_3^{\mathrm{NF}}\kappa_{\mathrm{NF}}=\frac{c_0'}4,\qquad
8\thNF\ln16=\frac{8\ln16}{90}\,\frac{c_g}{c_g+4}\,c_0'\le\frac{c_0'}8 .
\]
With $n\le\nu$, $n_0\le(4/c_g')\nu$ and $j_g=O(\ln\bGM)$, the terms
$C(n+n_0+\nu)$ and $(1+j_g)\kappa_{\mathrm{NF}}\nu\ln16$ of Steps~2 and~3 are at
most $C'(1+\ln\bGM)\nu$ with $C'$ depending only on $C_9$, $c_g$, $c_g'$, $c_L$, $C_{\cU}$ and
$\kappa_{\mathrm{NF}}$ (constants of Section~\ref{ssec:D.1}; $c_g'$ through $M\le(1+4/c_g')\nu$, and $c_L$, $C_{\cU}$ through
$j_g$). Hence
$\ln\#\le\frac38c_0'\Funit\nu+C'(1+\ln\bGM)\nu\le\frac{c'}2\Funit\nu$ as soon as
$\Funit\ge(8C'/c_0')(1+\ln\bGM)$, which holds for $\bGM\ge\bGMo$ because
$\Funit=(\pi^2c_g'/18)(\mt L)^3/(\mt\bGM)\ge(\pi^2c_g'/144)c_L^3/(\mt\bGM)$
($\mt L\ge c_L/2$) is exponentially large. In addition, $c'\ge c_0'$.

\paragraph{Lemma~\ref{lem:7.60} and Theorem~\ref{thm:7.61}.}
Under the weight $2-\eta_w$ of the hull functional the constant $c_u$ of (R7)
is the value fixed in Section~\ref{ssec:D.1}, and only the value of $C_X=3C_{\cU}/c_u$
depends on it. Since small components of $\Gp$ are $\cU$-cells of Debye volume at
least $c_u$, they are counted in Lemma~\ref{lem:7.60}. In
Theorem~\ref{thm:7.61}, with $\sF=(\pi^2c_g'/18)(\mt L)\mt/\bGM$
from Definition~\ref{def:3.17} and with $\bGM=4\pi^2\tau$,
\[
\sV=\epsE\,\frac{\pi^2c_g'\mt L}{18}\,\frac{\mt}{\bGM}=\epsE\,\frac{c_g'\mt L}{72}\,\frac{\mt}{\tau}
=\frac{c_g'\mt L}{9216}\,\frac{\mt}{\tau}\quad\text{at }\epsE=\tfrac1{128}=\tfrac{c_0'}8,
\qquad\eta_S=\frac{c_2C_X\mt^2}{\sV},
\]
which is the displayed expression, and it is $O(\mt\bGM\log\tau)$ because
$c_2=O(\log\tau)$.

\paragraph{Remark~\ref{rem:7.62} (the perimeter constant).}
By Lemma~\ref{lem:7.35}, $\cnear\le0.1264\,\theta_PC_{\mathrm{BS}}+O(1/\bGM)$ with
$C_{\mathrm{BS}}=24\pi^3$, uniformly for $\bGM\ge\bGMo$. The coefficient
$0.1264\,\theta_PC_{\mathrm{BS}}$ decreases as $c_L$ decreases, since $\theta_P\propto c_L^2$. Since
$c_2=O(\log\tau)$, $\mt=e^{-\pi^2\vCb(0)\tau(1+o(1))}$ and
$R_1(\beta)=(2\rbox+5)L\ge(2\rbox+5)c_L/(2\mt)$, the terms $3c_2C_XC_{\cU}\mt$ and
$9c_2C_XC_{\cU}^2/R_1(\beta)$ of $c_N$ tend to $0$.

\paragraph{Lemma~\ref{lem:A.7} and the order of $c_{10}$, $c''_{10}$, $\mu$.}
We fix $c_{10}$ first, independently of
$\bar\delta$, with $c_{10}\le\frac79r_C$. Then we fix $c''_{10}$, which depends only on $c_{10}$ and on the growth constants
$C_{\cU}$, $c_u$, with
$e^{c''_{10}}\ge16e\max(1,c_{c_{10}/2})$, and finally we fix the Mayer parameter $\mu$ with
$\mu\ge1+2c''_{10}$.

\paragraph{Proposition~\ref{prop:A.8} (the constant $c_2^L$).}
The constant $c_2^L=a+\alpha$, where $a=\log(2eD_{\cU})+4+c_4'+a''$ with
$e^{a''}\ge4C_\varphi$, is built from $D_{\cU}$, $c_4'$, $C_\varphi$ and
$\alpha$. Of these, $c_4'$ depends on $\tau$, but it is doubly exponentially small and decreasing in
$\tau$, hence bounded uniformly for $\tau\ge\tau_0$, and in this sense $c_2^L$ is uniform in $L'$,
$\bGM$ and $\tau$. Lemma~\ref{lem:A.11} provides a finite threshold $\tau_0$, which grows with $c_2^L$.

\paragraph{Proposition~\ref{prop:A.9}: the conditions on $\bGMo$.}
The conditions on $\bGMo$ used in Proposition~\ref{prop:A.9} are
\begin{equation}\label{eq:A.beta}
\begin{split}
&(\beta1)\ \ \tfrac{c'}{32}\Phi_{\cU}\ge(1+D_{\cU})\bigl(c_S+c_B(\bD)\bigr),\\
&(\beta3)\ \ 3eD_Ie^{-(c'/4)\Funit}\le\tfrac12,\qquad(\beta4)\ \ c_{16}^D\le1 .
\end{split}
\end{equation}

\paragraph{Proposition~\ref{prop:A.9}: the booking of $\Floc$.}
In the proof of the core estimate \eqref{eq:A.core}, we book the factor
$e^{-(3c'/4-\eps)\Floc(X)}$ left by Proposition~\ref{prop:7.37} and the
extraction according to Table~\ref{tab:A.1}, as follows.
\begin{itemize}
\item \emph{Type-iii factors, per region.} The factors $|D|$ and $|F_2'|$ at derivatives in a region involve $D$ and $F_2'$ on that region only, because $\supp D_\beta$ lies in the element of $S_\beta$. For a wall component with $N$ factors $|D|$, $n_\Delta$ of them in the cell $\Delta$, let $a_\Delta:=\|D\|_{L^2(\Delta)}$, so that $\sum_\Delta a_\Delta^2\le2\Floc/\mt^2$. The weighted arithmetic-geometric mean inequality and the identity $\sup_{F>0}(2F/(\mt^2N))^{N/2}e^{-\delta F}=(e\delta\mt^2)^{-N/2}$ give, for every $\delta>0$,
\[
\prod_\Delta a_\Delta^{n_\Delta}\,e^{-\delta\Floc}\le(\delta\mt^2)^{-N/2}\prod_\Delta(n_\Delta!)^{1/2},
\]
and $F_2'$ is handled in the same way. These constants enter $\Kcell$ and the pinning factor of Proposition~\ref{prop:B.7} and Lemma~\ref{lem:B.8}. We take $\delta=c'/16$ on $X\setminus Z$ and $\delta=c'/32$ on $Z$.
\item \emph{(M-$\Sigma$) on $X\setminus Z$.} We use $c'/8$ of $\Floc(X\setminus Z)$. A wall polymer $\mathcal S$ is the union of the closures of its components, $|\operatorname{cl}(S_\beta)|\le(1+D_{\cU})|S_\beta|$, and $F_1(S_\beta)\ge\Phi_{\cU}|S_\beta|$ by Proposition~\ref{prop:7.22}(c). Hence $\Floc(\mathcal S)\ge\Phi_{\cU}|\mathcal S|/(1+D_{\cU})$, and each $\cU$-cell of a wall polymer of $X\setminus Z$ receives the factor $e^{-(c'/8)\Phi_{\cU}/(1+D_{\cU})}$ of $\eps_\Delta$ in Proposition~\ref{prop:7.41}. Lone cells of $X\setminus Z$ carry $\eps_\Delta=\Kcell e^{\alpha+C_{\Dep}\cdep}(\bD/\lPV)^{1/6}$. The remaining fraction $9c'/16-\eps\ge c'/16$ of $\Floc(X\setminus Z)$ contributes a factor bounded by $1$.
\item \emph{Seed.} We use $c'/32$ of $\Floc(Z,N_Z)$ for the $\cU$-cells of $Z^\Sigma$. Each carries at most $\Kcell e^{\alpha+C_{\Dep}\cdep+c_B(\bD)}\le e^{c_S+c_B(\bD)}$ by Proposition~\ref{prop:7.41}, and $\Floc(Z^\Sigma)\ge\Phi_{\cU}|Z^\Sigma|/(1+D_{\cU})$ pays it under ($\beta1$). Lone seed cells carry at most $\Kcell e^{\alpha+C_{\Dep}\cdep}\le e^{c_S}$. What remains is
\[
e^{-(c'-c'/4-\eps-c'/32-c'/32)\Floc(Z,N_Z)}=e^{-(11c'/16-\eps)\Floc(Z,N_Z)}.
\]
\end{itemize}
Each non-seed cell carries at most $\eps_\Delta\le\frac12c_{16}^D\le\frac12$ by ($\beta4$).
In (a) and (b) of Lemma~\ref{lem:7.53}, $c'/8$ of the fraction $11c'/16-\eps$ left on the seed is used for the entropy of the first set, and at least $c'/16$ is left unused. In (c$'$) of that lemma this fraction pays $\eps\le c'/8$ and $c_w=c'/16$ and leaves at least $c'/2$.

\begin{remark}[The $F_1$ ledger]\label{rem:A.10}
Table~\ref{tab:A.1} lists, for the proofs of Lemma~\ref{lem:7.53} and Proposition~\ref{prop:7.41}, the fractions of $\Floc$ assigned to the non-seed elements ($X\setminus Z$) and to the seed elements ($Z$). In each column the entries add up to $1$, and no fraction is used twice. Since the walls on $Z$ are fixed on $\cN_\sigma(X)$ and no wall lies on an interface between $Z$ and $X\setminus Z$, the weight $e^{c_1'\Floc(X\setminus Z)}$ suffices in the sum over $N_X$ in the proof of Proposition~\ref{prop:A.8}, and the entry for the $h$-entropy on $Z$ is optional. We keep it, at the cost of $c'/4$ on the seed that would otherwise be left unused, so that Lemma~\ref{lem:7.39} can be stated with $e^{c_1'\Floc(X,N_X)}$.

\begin{table}[ht]
\centering\small
\begin{tabular}{p{0.35\textwidth}p{0.2\textwidth}p{0.2\textwidth}p{0.14\textwidth}}
\hline
item & $X\setminus Z$ (non-seed) & $Z$ (seed) & where\\
\hline
H\"older split and \cite[Lemma~13]{GM82} ($T_2,T_3,T_4$) & $1-c'$ & $1-c'$ & Prop.~\ref{prop:7.37}\\
$h$-entropy, $c_1'$ & $c'/4$ & $c'/4$ (optional) & Prop.~\ref{prop:A.8}\\
extraction $\eps$ ($\eps\le c'/2$; the chain uses $\eps\le\epsE$) & $\eps$ & $\eps$ & Lemma~\ref{lem:7.53}\\
type-iii factors $|D|$, $|F_2'|$ & $c'/16$ & $c'/32$ & Prop.~\ref{prop:A.9}\\
(M-$\Sigma$): cells of wall polymers & $c'/8$ & none & Prop.~\ref{prop:7.41}\\
seed cells of $Z^\Sigma$ & none & $c'/32$ & ($\beta1$)\\
left over & $9c'/16-\eps\ge c'/16$ (discarded) & $11c'/16-\eps$ (retained) & \\
\hline
\end{tabular}
\caption{The $F_1$ ledger: fractions of $\Floc$ per region.}
\label{tab:A.1}
\end{table}
\end{remark}

\begin{lemma}[Constant check]\label{lem:A.11}
There is $\tau_0$ such that for $\tau\ge\tau_0$ the conditions \eqref{eq:A.beta} hold and $c_{16}e^{c_2}\le\frac18$.
\end{lemma}

\begin{proof}
Each condition compares an energy that is exponentially large in $\tau$ with a cost that is logarithmic in $\tau$ or of smaller exponential order. Since $\mt\le2\mD$ and $\mD=(8\pi^2\tau)^{1/2}e^{-\pi^2\vCb(0)\tau}$, $\log(1/\bD)\ge\pi^2\vCb(0)\,\tau-\frac32\log\tau-C$, with $\pi^2\vCb(0)=2.49435508\ldots$. For the constants of the seeded sums we showed (Section~\ref{S:H1}) that $\log D_I=O(\log\tau)$ and, by Proposition~\ref{prop:6.18}, that $c_S=O(\log\tau)$ and $c_2=c_S+2=O(\log\tau)$. By Proposition~\ref{prop:7.22}, $\Phi_{\cU}\ge c\,\bD^{-1}\bGM^{-3}$ with $c>0$ fixed, while $c_B(\bD)\le C\bD^{-1/2}$ by Lemma~\ref{lem:7.40}. Moreover, since $\mt L\ge c_L/2$, we have $\Funit=(\pi^2c_g'/18)(\mt L)^3(\mt\bGM)^{-1}\ge(\pi^2c_g'/144)c_L^3(\mt\bGM)^{-1}$. Hence ($\beta1$) and ($\beta3$) hold for $\tau\ge\tau_0$, and ($\beta4$) is the content of Theorem~\ref{thm:7.42}. For the product, the first branch of $c_{16}^D$ gives
\[
\log\bigl(2\Kcell e^{\alpha+C_{\Dep}\cdep}(\bD/\lPV)^{1/6}e^{c_2}\bigr)=O(\log\tau)-\tfrac16\pi^2\vCb(0)\,\tau ,
\]
and the second gives $O(\log\tau)+c_B(\bD)-(c'/8)\Phi_{\cU}/(1+D_{\cU})$, which tends to $-\infty$ faster. Consequently, $c_{16}^De^{c_2}=\exp(O(\log\tau)-\pi^2\vCb(0)p_0\tau(1+o(1)))$ with $p_0=1/6$. Also $\etaseed e^{c_2}=6D_Ie^{c_S+2}e^{-(c'/4)\Funit}$ is doubly exponentially small. Both are at most $\frac1{16}$ for $\tau\ge\tau_0$. \qed
\end{proof}