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\subsection{Constants of Section~\ref{sec:3} and Subsections~\ref{ssec:4.1} and~\ref{ssec:4.3}}\label{S:B1}

We group the constants by the main-text statement in which they appear.

\paragraph{Definition~\ref{def:3.1}: the scales.}\label{S:B1.scales} The screening mass satisfies $m_D/2\le\tilde m\le2m_D$, so that $\log(1/\tilde m)=\pi^2v_{\rm Cb}(0)\,\tau+O(\log\tau)$ with $\pi^2v_{\rm Cb}(0)=2.49435508\ldots$, and $\tilde m^{-1}$ is the Debye length. The scales are fixed at item~11 of the ordered list in Section~\ref{ssec:D.1}. Since $c_L\le1/308$, \eqref{eq:3.Hell} follows from $\ell\ge\tilde m^{-1}/r_0\ge L_c/c_L$. Of the conditions $L_c\ge4$ and $c_L\tilde m^{-1}\ge4r_0$ of Section~\ref{ssec:D.5}(e), the first implies $c_g':=c_g(1+1/L_c)^{-2}\ge\frac9{16}c_g$, and by the second, $L>c_L\tilde m^{-1}-2r_0\ge c_L\tilde m^{-1}/2$, that is, $\tilde mL\ge c_L/2$.

\paragraph{Definition~\ref{def:3.4}: windows.} A window is the union of the blocks at block distance at most $75$ from a block. It therefore has at most $151$ blocks per direction, each of coarse side at most $2L_c-1$, and its sides are at most $151(2L_c-1)<302L_c<k_P$. A block lies in the windows of the at most $151^3=3.44\cdot10^6$ blocks at block distance at most $75$ from it, and $151^3<27\cdot304^3=7.6\cdot10^8=27(k_P/L_c)^3$, the overlap bound of (R6).

\paragraph{Definition~\ref{def:3.forced}, Lemma~\ref{lem:4.forced}, Theorem~\ref{thm:4.30}: the cost of a fill.}\label{S:B1.forced} A block $Q$ has $|\partial Q|\le6(2L_c-1)^2<24L_c^2$ and $|Q|<8L_c^3$. Hence, with $\lambda_U=2c_s/\ell$, $c_s\le10$ and $\ell\ge308L_c$, we have $\lambda_U|Q|<16c_sL_c^3/\ell\le0.52L_c^2$ and $|\partial Q|+\lambda_U|Q|<24.6L_c^2<\mu_{\rm B}L_c^2=25L_c^2$. Also $\lambda_U=2c_s/\ell\le20/(308L_c)<\lambda=12/L_c$. A removal cell $\Delta$ with sides $a,b,c\in[\ell,3\ell)$ has $|\Delta|/|\partial\Delta|=abc/(2(ab+bc+ca))\le(abc)^{1/3}/6<\ell/2$ by the arithmetic-geometric mean inequality, so $\lambda_U|\Delta|<c_s|\partial\Delta|$ and $|\partial\Delta|+\lambda_U|\Delta|<w_*$. The number of forced blocks is at most $27|\Mset|/(\vartheta L_c^2)$, since a mark lies in at most $27$ sets $3Q$. Hence filling them costs at most $27\cdot24.6/\vartheta=664.2/\vartheta$ per mark.

\paragraph{Lemma~\ref{lem:4.5}: constants that depend on the hull weight.}\label{S:B1.hull} Frozen bonds have the hull weight $2-\eta_w$ of Definition~\ref{def:3.8}, and the hull weight affects the constants in two ways. Through Lemma~\ref{lem:4.5} it determines $(a_\eta,b_\eta)=(1+1/(1-\eta_w),1/(1-\eta_w))$ and the factor $1/(1-\eta_w)$ in the bound on $N_2(W)$. Table~\ref{tab:S:B1.hull} lists the constants that depend on $\eta_w$ through Lemma~\ref{lem:4.5}, with their formula in $\eta_w$ and their value at $\eta_w=1/10$; the other constants, among them the constant $c_0$ of the density bound, the bound on $c_L$, $K_*$, $\sigma(\beta)$ and $R_1(\beta)$, do not depend on $\eta_w$ through Lemma~\ref{lem:4.5}. In the cell analysis the hull weight appears, by Lemma~\ref{lem:4.11}(b), as the deficit $\eta_w$ of a short segment through a frozen bond, which yields the factor $120=12/\eta_w$ of Proposition~\ref{prop:4.sealcount}. Through this factor, $C_{\rm seal}=726$, the number $134$ of non-good lines per unit of $m_Q+m_{Q'}$ and the $158$ in the expansion bound behind $h_\star$ depend on $\eta_w$ (see the arithmetic in Section~\ref{S:B1.cells}), and so do the constants built from them, such as $\Cst$, $C_{\mathrm{free}}$ and $C_{\mathrm{resp}}$, and the constraint on the weak-link constant $\chi$ (items~9 and~10 of Section~\ref{ssec:D.1}).

\begin{table}[ht]
\caption{Constants depending on the hull weight; values at $\eta_w=1/10$, $c_I=2^{2/3}$, $\rho_\lambda=L_c/12$.}\label{tab:S:B1.hull}
\centering\footnotesize
\begin{tabular}{p{2.9cm}p{3.5cm}p{3.3cm}p{2.3cm}}
\hline
quantity & formula & value at $\eta_w=1/10$ & used in\\
\hline
$(a_\eta,b_\eta)$ & $(1+1/(1-\eta_w),\,1/(1-\eta_w))$ & $(19/9,10/9)$ & Lemma~\ref{lem:4.5}, (R2)\\
isoperimetry for $|W|\le V_*$ & $(c_I/(2a_\eta))|W|^{2/3}$ & $(9c_I/38)|W|^{2/3}=0.376|W|^{2/3}$ & Theorem~\ref{thm:4.7}(i)\\
$V_*$ & $(1-\eta_w)^3\rho_\lambda^3/2$ & $0.3645\rho_\lambda^3$ & Definition~\ref{def:3.1}, (R2)\\
volume growth constant & $(c_I/(48a_\eta))^3$ & $3.844\cdot10^{-6}$ & Theorem~\ref{thm:4.7}(iii)\\
saturation radius of (iii) & $24a_\eta(1-\eta_w)\rho_\lambda$ & $45.6\rho_\lambda$ & Theorem~\ref{thm:4.7}(iii)\\
Faber\textendash Krahn constant & $c_I^2/(48a_\eta^2)$ & $c_I^2/213.93$ & Theorem~\ref{thm:4.7}(iv)\\
$N_2(W)$ factor & $1/(1-\eta_w)$ & $10/9$ & Lemmas~\ref{lem:4.5}, \ref{lem:6.14}\\
$c_1$ (second entry) & $((1-\eta_w)c_I/24)^3$ & $2.109\cdot10^{-4}$ & (R5)\\
$c_u$ & $((1-\eta_w)c_I\tilde mL/24)^3$ & $(0.0595\,\tilde mL)^3$ & (R7)\\
$R_{\mathrm{HK}}$, $K_0$ & $(1-\eta_w)c_I\rho_\lambda/3$, $K_0=(1-\eta_w)c_Ic_\rho c_L/6$ & $\tilde mR_{\mathrm{HK}}=0.0397\tilde mL\in[K_0,2K_0]$, $K_0=0.01984c_L$ & Theorem~\ref{thm:6.9}\\
$c_F$ (fine isoperimetry) & $(c_I/(2a_\eta))/152.4$ & $2.467\cdot10^{-3}$ & Proposition~\ref{prop:6.10}\\
$C_{\mathcal U}$, $N_{\mathrm{near}}$, $A_2$ & through $1/c_1$ and $1/c_u$, at most the factor $(1-\eta_w)^{-3}$ & factor at most $1.372$ & (R7), Section~\ref{sec:stage2}\\
$c_{\mathrm{sh}}$ & $c_1/216$ & $9.77\cdot10^{-7}$ & Lemma~\ref{lem:5.22}\\
\hline
\end{tabular}
\end{table}

\noindent At $\eta_w=1/10$, $c_1$ equals its second entry, with the factor $(1-\eta_w)^3=0.729$, when $c_g\ge4.22\cdot10^{-4}$, and $c_1=c_g/2$ otherwise. Hence the factor $(1-\eta_w)^{-3}=1.372$ in the table is an upper bound. The Nash constant $A_N=24\cdot2^{2/3}/c_F^2$ of Appendix~\ref{app:F} depends on $\eta_w$ only through $c_F$.

\paragraph{Theorem~\ref{thm:4.7}.}\label{S:B1.geom} We have $V_*=(c_I\rho_\lambda/(2b_\eta))^3$, since $(c_I/2)^3=\frac12$. Hence $b_\eta\lambda V_*^{1/3}=c_I/2$. In (iii), $\alpha:=c_I/(12a_\eta)=0.0626\ldots\le\frac14$, and the two expressions in the minimum are equal at $r=24(a_\eta/b_\eta)\rho_\lambda=45.6\rho_\lambda$.

\paragraph{Lemma~\ref{lem:4.19}.}\label{S:B1.U} We set
\[
C_{\Uc}:=8\sqrt3\,(2+6/c_g)/(c_1c_L^2),
\]
which bounds $2\sqrt3(2+6/c_g)/(c_1(\tilde mL)^2)$ when $\tilde mL\ge c_L/2$.

\paragraph{Proposition~\ref{prop:4.22}: the constant $c_0$.}\label{S:B1.c0} The fine plaquettes dual to the insulating bonds with an end in a removal cell $\Delta$, those of frozen bonds counted twice, number at most $\bar r_0^{\,2}(3+c_s)|\partial\Delta|$, since a coarse bond is dual to at most $\bar r_0^{\,2}$ fine plaquettes. Hence they number at most $c_0/(z\tau)$ with $c_0:=z\tau\,\bar r_0^{\,2}(3+c_s)\max_\Delta|\partial\Delta|$, where $z$ is the monopole activity of Section~\ref{sec:stage1}, $m_D^2=2\beta_{\rm GM}z$ and $z\tau=m_D^2/(8\pi^2)$. Using $\bar r_0<2r_0$, $r_0\ell=\ell_D<\tilde m^{-1}+L\le(1+c_L)\tilde m^{-1}$, $m_D\le2\tilde m$, the bound $|\partial\Delta|<54\ell^2$ of (G3) and $c_L\le\frac12$, we obtain
\[
c_0<\frac{4\tilde m^2}{8\pi^2}\cdot4\cdot54\,(3+c_s)(1+c_L)^2\tilde m^{-2}\le\frac{243}{\pi^2}(3+c_s)<25(3+c_s),
\]
independently of $\beta$. On a grid without exceptional layers and with cubic removal cells, $c_0=6(3+c_s)(m_D\ell_D)^2/(8\pi^2)$.

\paragraph{Lemma~\ref{lem:4.runs}, Proposition~\ref{prop:4.giant}, Lemmas~\ref{lem:4.thin}, \ref{lem:4.aux}, Proposition~\ref{prop:4.sealcount}, Theorem~\ref{thm:4.rest}: the cell analysis.}\label{S:B1.cells} With $\vartheta=3\cdot10^{-5}$ and $\varepsilon_{\rm th}=0.008$:
\begin{itemize}
\item \emph{Giant.} $12+6.4\cdot9\cdot2=127.2<128$, and $(1-3\vartheta)L_c^3>384\vartheta L_c^3=3\cdot128\vartheta L_c^3$ since $384\vartheta=0.01152$.
\item \emph{Thin sets.} $2\varepsilon_{\rm th}+12\vartheta=0.01636\le0.0164\le\frac14$, and $2/\sqrt{0.0164}=15.62\ge15.6>14$. Non-giant components are thin since $128\vartheta=0.00384<\varepsilon_{\rm th}$.
\item \emph{Seal.} Lemma~\ref{lem:4.11}(b) with $\mathrm{supp}\le6(m_Q+m_{Q'})$ and $\eta_w=1/10$ gives $120(m_Q+m_{Q'})$ lines, plus $m_Q+m_{Q'}$ lines with a mark, in total $121$ per unit of $m_Q+m_{Q'}$, and $C_{\rm seal}=6\cdot121=726$.
\item \emph{Interfaces.} $1+121+12=134$ non-good lines per unit of $m_Q+m_{Q'}<\vartheta L_c^2$, so $\good(I)\ge(1-134\vartheta)|F'|=0.99598|F'|\ge c_g|F'|$.
\item \emph{Expansion.} For two large sides, $\varepsilon_{\rm th}-127.2\vartheta=0.004184\ge\varepsilon_{\rm th}/2$, so $\mathrm{cut}\ge\varepsilon_{\rm th}L_c^2/25.6=3.125\cdot10^{-4}L_c^2$. For adjacent in/out blocks, $\mathrm{cut}\ge(1-158\vartheta-2\varepsilon_{\rm th})L_c^2/3=0.32642L_c^2$. Both are at least $3.1\cdot10^{-4}L_c^2$, and $|A|<k_P^3/2=1.4047\cdot10^7L_c^3$. Hence $h_\star\ge3.1\cdot10^{-4}/(1.4047\cdot10^7L_c)=2.207\cdot10^{-11}/L_c\ge2.2\cdot10^{-11}/L_c$. Then $3k_Ph_\star=912\cdot2.2\cdot10^{-11}=2.006\cdot10^{-8}\ge\chi=2.0\cdot10^{-8}$, and $\chi_{\rm N3}=\chi/3=6.67\cdot10^{-9}\le k_Ph_\star=6.69\cdot10^{-9}$.
\end{itemize}
Theorem~\ref{thm:4.rest} shows that the construction provides $c_1$-type volume $(1-3\vartheta)L_c^3$ per cell, one interface per block face and six neighbouring cells per cell. These facts imply (R5) and (R8) with the constants $c_1$, $1/c_g$ and $6/c_g$, which we keep.