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\subsection{Constants of Section~\ref{sec:stage1} and Appendix~\ref{app:F}}\label{S:E1}

\paragraph{Masses (Section~\ref{sec:stage1}).}\label{S:E1.mass}
The Coulomb kernel at the origin is
\[
\vCb(0)=\tfrac12W_{S}=(\sqrt6/(192\pi^{3}))\Gamma(\tfrac1{24})\Gamma(\tfrac5{24})\Gamma(\tfrac7{24})\Gamma(\tfrac{11}{24})=0.2527310098\ldots,
\]
where $W_{S}=\pi^{-3}\int_{[0,\pi]^{3}}(3-\sum_{i}\cos x_{i})^{-1}dx$ is Watson's integral \cite{Wat39}, in the Gamma-function form of \cite[(3.6)]{Zu11}. Numerically, $\pi^{2}\vCb(0)=2.4943550871\ldots$. Since
$\mD=(8\pi^{2}\tau)^{1/2}e^{-\pi^{2}\vCb(0)\tau}$ and $2(8\pi^{2})^{1/2}=17.77153\ldots$, it follows that
\begin{equation}\label{S:E1.mt}
\mt\le2\mD\le17.7716\,\tau^{1/2}e^{-2.494\tau}\qquad(\mD/2\le\mt\le2\mD).
\end{equation}
Here $2.494$ is $\pi^{2}\vCb(0)$ rounded down to a lower bound by the rule of Section~\ref{ssec:D.4}, and the hypothesis of Lemma~\ref{lem:6.19} has the same $2.494$ on its left.

\paragraph{Theorem~\ref{thm:6.5}: the Debye condition.}\label{S:E1.debye}
In \cite[(4.24)]{GM82}, at $A=(1-\delta)M$, \cite{GM82} use $\|v^{2}\|_{A}\le2(\delta M)^{-2}$ \cite[Lemma~5, (3.26)]{GM82}, and the Debye condition \cite[p.~34]{GM82} reads $1\le\frac14\delta^{2}C^{3}(1-C)^{8}e^{-2\kappa'}M^{2}/\mD^{2}$, where $\kappa'=\kappa_{S}+\frac12\mu$ by the choice made on \cite[p.~35]{GM82}. On $\Gp$ the bound $\|v^{2}_{\Gp}\|_{A}\le3\MY^{-2}$ holds by Proposition~\ref{prop:6.11} on outputs for $A\le\sqrt3\,\kappa_{Y}\MY$, and by $\|v^{2}_{\Gp}\|_{0}\le\MY^{-2}$ on all slit graphs. It replaces $2(\delta M)^{-2}$, that is, $\frac12\delta^{2}$ by $\frac13$, and hence $\frac14\delta^{2}$ becomes $\frac16$:
\[
1\le\tfrac16C^{3}(1-C)^{8}e^{-2\kappa_{S}-\mu}\MY^{2}/\mD^{2}.
\]
Since $\MY\ge\mD/(2\lPV)$ on $I_{Y}$, $\MY^{2}/\mD^{2}\ge1/(4\lPV^{2})$, and the condition holds for all $\MY\in I_{Y}$ once
\begin{equation}\label{S:E1.lPV}
\lPV^{2}\le C^{3}(1-C)^{8}e^{-2\kappa_{S}-\mu}/24 .
\end{equation}
In \cite[(4.23)]{GM82} the bound $\|v^{1}_{\Gp}\|_{A}\le3M_{1}^{-2}$, with $M_{1}=-4\pi\bGM^{-1}\ln(1-C)$, is a power of $\bGM$, which $e^{\bGM\eps_{1}}$ absorbs for $\bGM\ge\bGMo$. Neither condition depends on $\Gp$.

\paragraph{The scales of \S\ref{ssec:6.3}: the rounding of $L$.}\label{S:E1.scales}
By Definition~\ref{def:3.1}, $L=r_{0}L_{c}$ with $L_{c}$ the largest even integer $\le c_{L}\mt^{-1}/r_{0}$, and $c_{L}\mt^{-1}\ge4r_{0}$. Hence
\begin{equation}\label{S:E1.L}
c_{L}/(2\mt)\le L\le c_{L}/\mt .
\end{equation}
In fine units $r_{0}\rho_{\lambda}=L/12=c_{\rho}L$, where $c_{\rho}=1/12$, while $c_{I}=2^{2/3}$, $V_{*}=(1-\etaw)^{3}\rho_{\lambda}^{3}/2$ in coarse units and $V_{*}^{f}=r_{0}^{3}V_{*}/2$, so $\RHK=(1-\etaw)c_{I}r_{0}\rho_{\lambda}/3$ and $4\RHK^{3}/V_{*}^{f}=64/27$ exactly. The constant
\[
K_{0}:=(1-\etaw)c_{I}c_{\rho}c_{L}/6\qquad(=0.2381\,c_{\rho}c_{L}=0.01984\,c_{L}\ \text{at}\ \etaw=0.1)
\]
satisfies $K_{0}\le\mt\RHK\le2K_{0}$ by \eqref{S:E1.L}. We use $K_{0}$ only through the lower bound $\mt\RHK\ge K_{0}$, in the following places.
\begin{itemize}
\item[(a)] In Proposition~\ref{prop:6.11}, $(\eps_{1}^{\mathrm{HK}})^{1/2}\MY\RHK\ge(\eps_{1}^{\mathrm{HK}})^{1/2}K_{0}/\lPV$, so the rate of \eqref{eq:6.LW} at $M=\MY$ is at least $\kappa_{Y}\MY$.
\item[(b)] In (H2) of Theorem~\ref{thm:6.12}, at the mass $r_{-}^{1/2}\ge\mt$ the rate of \eqref{eq:6.LW} is at least $\kappa_{W}\mt\min(1,(\eps_{1}^{\mathrm{HK}})^{1/2}\mt\RHK)\ge\kappa_{0}\mt$, using $(\eps_{1}^{\mathrm{HK}})^{1/2}K_{0}<1$.
\item[(c)] In (H1) and (H1$'$) of Theorem~\ref{thm:6.12}, $\RHK^{-3}r_{-}^{-1}\le\mt^{-2}\RHK^{-3}\le\mt K_{0}^{-3}=\lPV K_{0}^{-3}\MY$.
\item[(d)] In Lemma~\ref{lem:6.15}, $\RHK^{-1}+\MY^{-2}\RHK^{-3}\le\mt K_{0}^{-1}+\lPV^{2}\mt K_{0}^{-3}=\eta_{G}$.
\item[(e)] In Appendix~\ref{app:F}, $\mt\le K_{0}(2C_{H})^{-1/3}$ gives $\RHK\ge(2C_{H})^{1/3}$.
\end{itemize}
By \eqref{S:E1.L}, $\MY r_{0}\rho_{\lambda}=c_{\rho}\MY L\in[c_{\rho}c_{L}/(2\lPV),\,c_{\rho}c_{L}/\lPV]$. We will use the lower bound in (f) to (h) below.

\paragraph{Theorem~\ref{thm:6.9}.}\label{S:E1.HK}
We set $c_{N}:=c_{F}^{2}/(24\cdot2^{4/3})=1.006\cdot10^{-7}$, $C_{H}:=(c_{N}/3)^{-3/2}=1.628\cdot10^{11}$ and $\eps_{1}^{\mathrm{HK}}:=(\log2)^{2}/512=9.38\cdot10^{-4}$. In the proof, given in Appendix~\ref{app:F}, the constant of Lemma~\ref{lem:F.3} is $A_{N}:=24\cdot2^{2/3}/c_{F}^{2}$, so that $c_{N}=1/(2^{2/3}A_{N})$, and (HK0) uses $6(\cosh\frac14-1)=0.1885<\frac14-\frac1{17}=0.1912$. The constants $C_{\mathrm{mo}}\ge1$ of the moment bound, $c_{\mathrm{ex}}:=\min((\log2)/(2A_{1}),\frac12)$ with $A_{1}:=512C_{\mathrm{mo}}^{2}/(\log2)^{2}$ (so $\eps_{1}^{\mathrm{HK}}=C_{\mathrm{mo}}^{2}/A_{1}$), $C_{\mathrm{gs}}:=8C_{H}e^{c_{\mathrm{ex}}/2}$, $c_{\mathrm{gs}}:=c_{\mathrm{ex}}/8$ and $C_{M}$, which contains the factor $e^{1/(2c_{\mathrm{ex}})}$, depend only on $C_{H}$. Hence $\kappa_{W}=\min(1,1/(2C_{M}))$ of Proposition~\ref{prop:6.11} is extremely small and fixed before $\bGM$; what matters is that it is a constant independent of $\tau$, $K$, $C$ and $D$.

\paragraph{Proposition~\ref{prop:6.10}.}\label{S:E1.ctf}
The box inequalities of the sketch in \S\ref{ssec:F.1} are as follows. For a box graph $B$ with sides in $[n_{\min},n_{\max}]$, $\varrho:=n_{\max}/n_{\min}$, and $U\subseteq B$, canonical paths give $|\partial_{B}U|\ge4|U||B\setminus U|/(n_{\max}|B|)\ge2\min(|U|,|B\setminus U|)/n_{\max}$, and the Loomis\textendash Whitney inequality gives $|\partial_{B}U|\ge3|U|^{2/3}-3|U|/n_{\min}$. Hence $|\partial_{B}U|\ge c_{\mathrm{box}}(\varrho)|U|^{2/3}$ for $|U|\le|B|/2$, with $c_{\mathrm{box}}(\varrho):=6/(3\varrho+2)$, as one sees by minimizing over $|U|^{1/3}=t\,n_{\min}$. In the face count, for a heavy cell $c$ and a light neighbour $c'$, the set $D:=B_{c}\cup B_{c'}$ has $n_{\max}(D)<4r_{0}$ and $|U||D\setminus U|/|D|\ge r_{0}^{3}/4$ for $U:=A\cap D$, so $D$ carries at least $\gamma_{D}r_{0}^{2}$ bonds of $\partial A$, $\gamma_{D}=\frac14$. Summing over $\partial_{\Gamma}H$, where $\Gamma=(V_{c},E_{c})$ is the coarse slit graph of \S\ref{ssec:6.3}, counts the bonds inside a coarse cell at most six times. Hence $|\partial A|\ge(\gamma_{D}r_{0}^{2}/6)|\partial_{\Gamma}H|$. With the constant $a:=9c_{I}/38$ of (R2), and using $c_{\mathrm{box}}(\varrho)>\frac34$ for $\varrho<2$ and $\gamma_{D}=\frac14$, the two cases of (i) give $|\partial A|\ge2^{-2/3}c_{\mathrm{box}}|A|^{2/3}\ge0.47|A|^{2/3}$ and $|\partial A|\ge(\gamma_{D}a/6)16^{-2/3}|A|^{2/3}=(a/152.39\ldots)|A|^{2/3}$, so $c_{F}:=a/152.4=2.467\cdot10^{-3}$. The constant $C_{\mathrm{bl}}:=384$ comes from $\gamma_{D}r_{0}^{3}/(12v_{\max})\ge1/384$ with $v_{\max}<8r_{0}^{3}$. Further, $C_{W}^{f}:=108C_{\mathrm{bl}}^{2}/\chi^{2}$, $k_{\square}:=r_{0}(k_{P}+3)$, $h^{f}:=\chi/(3C_{\mathrm{bl}}k_{\square})$, $12/(h^{f})^{2}=C_{W}^{f}k_{\square}^{2}$ and $N_{\mathrm{ov}}\le27(k_{P}/L_{c})^{3}$. For (v), (L8-2) reads $C_{W}^{f}r_{\square}^{2}N_{\mathrm{ov}}(\mt L)^{2}\le\eta_{H}/8$ and $C_{W}^{f}r_{\square}^{2}(\mt L)^{2}=12\mt^{2}/(h^{f})^{2}$, so $1/h^{f}\le96^{-1/2}\mt^{-1}\le0.103\,\mt^{-1}$, since $\eta_{H}<1\le N_{\mathrm{ov}}$. Moreover $\log k_{\square}\le\log(1/\mt)\le2.4944\,\tau=0.0632\,\bGM$. Also $|\KK^{\flat}|\le|V\cap\hat Q(Q)|\le k_{\square}^{3}$: a window has fewer than $k_P$ coarse sites per direction (G4), all of fine side $r_0$ except at most one exceptional layer per special interval (Section~\ref{S:B2.grid}), of fine side at most $2r_0-1$, and it meets at most three special intervals per direction, so its fine sides are at most $r_0k_P+3(r_0-1)<k_{\square}$. Hence the diameter bound $2+(24/h^{f})\cdot3\log k_{\square}\le2+24\cdot0.103\cdot3\cdot0.0632\,\bGM/\mt<2\bGM/\mt=L'$ holds.

\paragraph{Theorem~\ref{thm:6.12}.}\label{S:E1.cov}
Let $\varpi:=(1-4\lPV^{2})^{-1/2}\in[1,1.16]$ and $r_{\pm}:=\frac12\MY^{2}(1\pm(1-4\lPV^{2})^{1/2})$, so that $r_{-}=2\mt^{2}/(1+(1-4\lPV^{2})^{1/2})\in[\mt^{2},1.08\,\mt^{2}]$ for $\lPV\le\frac14$. The constants, which depend only on $(c_{F},c_{\rho},c_{L},\lPV)$, are
\begin{align*}
K_{1}&:=\tfrac12+2C_{H}(1+\varpi)+\varpi C_{H}\lPV K_{0}^{-3},\qquad K_{3}:=3\varpi\mt^{2}/r_{-}\le3.5,\\
K_{1}'&:=\max\bigl\{eK_{1},\ \varpi\bigl(313+C_{\mathrm{gs}}(2\pi/c_{\mathrm{gs}})^{1/2}+4C_{H}(\eps_{1}^{\mathrm{HK}})^{-3/2}\lPV K_{0}^{-3}\bigr)\bigr\},\\
K_{1}''&:=\max(eK_{1},K_{1}').
\end{align*}
The bound $de^{-3d/68}\le\MY d^{2}e^{-3d/68}\le(136/3)^{2}e^{-2}\MY<279\,\MY$, valid for $d\ge\MY^{-1}$, together with $279\le313$, accounts for the number $313$.

\paragraph{Section~\ref{ssec:6.4}: the depletion constants.}\label{S:E1.dep}
Both $A_{0}^{\mathrm{AD}}$ and $C_{F}:=C_{\mathrm{Gr}}C_{L}'$ are upper-bound constants, taken $\ge1$; we also take $C_{\mathrm{Gr}}\ge1$ and $C_{L}'\ge C_{\rm im}$, the constant of the far images in step (iv) of the proof of Theorem~\ref{thm:6.17}, so that $C_{F}\ge C_{\rm im}$. Let $\eps_{0}:=\theta_{0}/(2\pi^{2}\tau)$, $\eps_{F}:=\eps_{0}/(4C_{F})$,
\[
L_{g}:=3+\log_{2}^{+}\bigl(8\lPV/(c_{\rho}c_{L})\bigr),\qquad C_{R}:=256\pi^{2}A_{0}^{\mathrm{AD}}C_{F}L_{g},\qquad C_{S}:=CA_{0}^{\mathrm{AD}}\bigl(1+\lPV/(c_{\rho}c_{L})\bigr),
\]
so that $32A_{0}^{\mathrm{AD}}L_{g}/\eps_{F}=C_{R}\tau/\theta_{0}$ and $\Rdep=\lceil C_{R}\tau/\theta_{0}\rceil$. Moreover, $2/\eps_{F}=16\pi^{2}C_{F}\tau/\theta_{0}$ and $\Rdep\le2C_{R}\tau/\theta_{0}$. Since $A_{0}^{\mathrm{AD}}\ge1$ and $L_{g}\ge3$, $C_{R}\ge768\pi^{2}C_{F}\ge24\pi^{2}C_{\rm im}$.
\begin{itemize}
\item[(f)] In Theorem~\ref{thm:6.16}, the shells $2^{k}\le r(y)<2^{k+1}$ with $2^{k+1}\le r_{0}\rho_{\lambda}$ contribute at most $8A_{0}^{\mathrm{AD}}$ each to $S_{2}(x)$ by Lemma~\ref{lem:6.14}, and hence at most $CA_{0}^{\mathrm{AD}}\log(2/\MY)$ up to $r\le\MY^{-1}$. The shells beyond $\MY^{-1}$ are summed with $e^{-\MY r/2}$, and the growth term gives $\sum_{k}(16A_{0}^{\mathrm{AD}}/(r_{0}\rho_{\lambda}))2^{k}e^{-\MY2^{k}/2}\le CA_{0}^{\mathrm{AD}}/(\MY r_{0}\rho_{\lambda})\le2CA_{0}^{\mathrm{AD}}\lPV/(c_{\rho}c_{L})$. Since $\log(2/\MY)\ge1$, $S_{2}(x)\le C_{S}\log(2/\MY)$.
\item[(g)] In step (ii) of the proof of Theorem~\ref{thm:6.17}, with $R\le r_{0}\rho_{\lambda}$, the far pairs contribute at most $16A_{0}^{\mathrm{AD}}/R+(16A_{0}^{\mathrm{AD}}/(r_{0}\rho_{\lambda}))(2+\log_{2}^{+}(4/(\MY r_{0}\rho_{\lambda})))\le(16A_{0}^{\mathrm{AD}}/R)(3+\log_{2}^{+}(8\lPV/(c_{\rho}c_{L})))=16A_{0}^{\mathrm{AD}}L_{g}/R$, which is at most $\eps_{F}/2$ for $R=\Rdep$. In the second sum, the shells with $2^{k+1}R>r_{0}\rho_{\lambda}$ give $\sum_{j\ge0}e^{-a2^{j}}$ with $a>\MY r_{0}\rho_{\lambda}/4$, which is at most $2+\log_{2}^{+}(1/a)$.
\item[(h)] For Lemma~\ref{lem:6.19}, see the proof below.
\end{itemize}
As a function of $\Rdep$, the covering number $N_{\mathrm{cov}}$ of Proposition~\ref{prop:6.18} takes the same value for all $\Rdep\le\ell_{D}$, so it does not depend on $\theta_{0}$. It is a very large constant, and it raises $\bGMo$ through a single condition, that of Theorem~\ref{thm:7.42}.

\paragraph{Proof of Lemma~\ref{lem:6.19}.}\label{S:E1.619}
The hypothesis says $C_{*}^{\mathrm{dep}}\tau^{5/2}e^{-2.494\tau}\le\theta_{0}$, and we use the bound \eqref{S:E1.mt} throughout.

We first prove $\Rdep\le r_{0}\rho_{\lambda}$. Since $A_{0}^{\mathrm{AD}},C_{F}\ge1$ and $L_{g}\ge3$, $C_{R}\ge768\pi^{2}>7.5\cdot10^{3}$. Hence $y:=C_{R}\tau/\theta_{0}\ge1.5\cdot10^{6}$ and $\Rdep=\lceil y\rceil\le(1+10^{-6})y$. By \eqref{S:E1.L}, $r_{0}\rho_{\lambda}=c_{\rho}L\ge c_{\rho}c_{L}/(2\mt)$, so it suffices that $2(1+10^{-6})C_{R}\tau\mt\le c_{\rho}c_{L}\theta_{0}$. Using $\tau\ge2$,
\begin{align*}
2(1+10^{-6})C_{R}\tau\mt&\le2(1+10^{-6})\,17.7716\,C_{R}\tau^{3/2}e^{-2.494\tau}\le17.8\,C_{R}\tau^{5/2}e^{-2.494\tau}\\
&\le c_{\rho}c_{L}C_{*}^{\mathrm{dep}}\tau^{5/2}e^{-2.494\tau}\le c_{\rho}c_{L}\theta_{0}.
\end{align*}

Next, $\Rdep\le\ell_{D}$ because $r_{0}\rho_{\lambda}=c_{\rho}L<L\le\ell_{D}$.

Finally, for (F), $\log(2/\MY)=\ln(2\lPV/\mt)\le\ln(1/\mD)=\pi^{2}\vCb(0)\tau-\frac12\ln(8\pi^{2}\tau)\le2.4944\,\tau$, using $\lPV\le\frac14$ and $\mt\ge\mD/2$. With $\eta_{G}=\mt(K_{0}^{-1}+\lPV^{2}K_{0}^{-3})$,
\[
C_{F}\eta_{G}C_{S}\log(2/\MY)\le C_{F}C_{S}(K_{0}^{-1}+\lPV^{2}K_{0}^{-3})\cdot17.7716\,\tau^{1/2}e^{-2.494\tau}\cdot2.4944\,\tau,
\]
and this is at most $\eps_{0}/4=\theta_{0}/(8\pi^{2}\tau)$ because $8\pi^{2}\cdot17.7716\cdot2.4944<3500.2\le3.6\cdot10^{3}$ and $C_{*}^{\mathrm{dep}}\ge3.6\cdot10^{3}C_{F}C_{S}(K_{0}^{-1}+\lPV^{2}K_{0}^{-3})$.
\qed