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\subsection{Constants of Sections~\ref{ssec:4.9} and~\ref{ssec:4.4}}\label{S:C1}

We first list the sizes on the loop-adapted grid and then the constants, grouped by main-text statement.

\paragraph{Sizes on the loop-adapted grid.} Table~\ref{tab:4.1} lists the sizes on $\Grid(C)$ and, for orientation only, their values on the uniform grid, in which all coarse cells have side $r_0$ and all block spacings are $L_c$; the constant $c_1$ in the table is that of item~6 of Section~\ref{ssec:D.1}. The opacity property (S-face) of Definition~\ref{def:3.14} is normalized per face, with the threshold $c_g|F(I)|$, and hence the good fraction and the cell degree are independent of the face area.

\begin{table}[ht]
\caption{Sizes on the loop-adapted grid $\Grid(C)$, in coarse units unless stated, with the values on the uniform grid for orientation and the statements in which each size is used.}\label{tab:4.1}
\centering\footnotesize
\begin{tabular}{p{3.0cm}p{2.9cm}p{2.2cm}p{2.7cm}}
\hline
quantity & on $\Grid(C)$ & uniform grid & used in\\
\hline
block side & $[L_c,2L_c-1]$ & $L_c$ & (G1)\\
block and cell volume & $<8L_c^3$ & $L_c^3$ & (R5); Lemma~\ref{lem:4.19} ($\tilde mL\le\frac12$)\\
face area $|F'|$ & $[L_c^2,4L_c^2)$ & $L_c^2$ & (S-face), Lemma~\ref{lem:4.forced}\\
block diameter & $<2\sqrt3L_c$ & $\sqrt3L_c$ & Cor.~\ref{cor:4.10}\\
window side; windows per site & $<k_P$; $\le151^3$ & $151L_c$; $151^3$ & (G4), (R6)\\
removal cell side & $[\ell,3\ell)$ & $\ell$ & Lemma~\ref{lem:5.32}\\
$|\partial\Delta|$ & $[6\ell^2,54\ell^2)$ & $6\ell^2$ & (R9), $c_0$ of Section~\ref{S:B1.c0}\\
reach $R_*$ & Theorem~\ref{thm:4.42} & same & Theorem~\ref{thm:3.21}\\
$k_P$ & $304L_c$ & same & (G4), (R6)\\
$c_1$; good fraction; cell degree & $2.109\cdot10^{-4}$; $c_g$; $6/c_g$ & same & (R5), (R8)\\
fine good fraction & $c_g(1+1/L_c)^{-2}$ & $c_g$ & Lemma~\ref{lem:4.17}\\
$q_m$ & $3(\bar r_0+1)^3\delta^{1/2}\le24r_0^3\delta^{1/2}$ & $3(r_0+1)^3\delta^{1/2}$ & Lemma~\ref{lem:5.11}\\
fine area of a coarse face & $\le\bar r_0^2<4r_0^2$ & $r_0^2$ & Section~\ref{sec:5} ($\bar r_0^2$ in $\eta_T$)\\
$c_I$ for $|W|\le V_*$ & $2^{2/3}$ & $2^{2/3}$ & Theorem~\ref{thm:4.7}\\
\hline
\end{tabular}
\end{table}

\paragraph{Lemma~\ref{lem:4.density}(c) and Theorem~\ref{thm:4.42}: the reach.} In Lemma~\ref{lem:4.density}(c), $\lambda^{-1}+\lambda_U^{-1}=L_c/12+\ell/(2c_s)\le\ell/c_s$ because $c_s\le10<6\ell/L_c$, and $w\ge120\ell/c_s$ gives the factor $30\ell/(c_sw)\le\frac14$. In Theorem~\ref{thm:4.42}, $w'\le121\ell/c_s+2L_c\le122\ell/c_s$, since $2L_c\le\ell/c_s$, and for $\rho\ge4L_c$ we have $g(\rho)\le54\rho^2+54c_s\rho^3/\ell$. Therefore $g((J+1)w')\le10^8(J+1)^3(\ell/c_s)^2$, and $J$ is at most the least integer with $5^J>10^8(J+1)^3(\ell/c_s)^2$, that is, $J=2\log_5(\ell/c_s)+O(\log\log(\ell/c_s))$. Hence
\[
R_*=(J+1)w'+6L_c\le122(J+1)\ell/c_s+6L_c=O\bigl((\ell/c_s)\log(\ell/c_s)\bigr);
\]
with $\log\ell=\pi^2\vCb(0)\,\tau+O(\log\tau)$ this is $R_*\approx380\,\tau\,\ell/c_s$, and $\log R_*=\log(1/\tilde m)+O(\log\tau)$, the only property of $R_*$ that the threshold $K_*$ uses, in Lemma~\ref{lem:5.16}. In the clearing-out step, each site of $\XX^1$ weighs at least $\lambda_U$ in $f$, because $\lambda\ge\lambda_U$. Further, $|B_{w'}(x)|\ge\ell^3$ since $w'\ge\ell$ and each side of $\Lamo$ is at least $\ell$, and $\lambda_U\ell^3=2c_s\ell^2>1$.

\paragraph{Proposition~\ref{prop:4.39}: the area per part.} The competitor costs at most $(6+\lambda)N(P)$ for the marks, $24.6L_c^2$ for each of at most $27N(P)/(\vartheta L_c^2)$ forced blocks, and $2$ for each of at most $C_{\rm seal}N(P)=726N(P)$ seal bonds. We therefore set
\[
C_1:=7+27\cdot24.6/\vartheta=7+664.2/\vartheta=2.214\cdot10^7,\qquad\Cst:=C_1+2C_{\rm seal}=C_1+1452=2.2141\cdot10^7
\]
at $\vartheta=3\cdot10^{-5}$, and $\Cfree:=\Cst(2+1/(2c_s)+1/3696)=5.536\cdot10^7$ at $c_s=1$, where $1/(\lambda\ell)=L_c/(12\ell)\le1/3696$ by \eqref{eq:3.Hell}. The marks of a set $3Q$ are within distance $3(2L_c-1)<6L_c$ of each other, and a bond of $\bar S^1$ has both ends within $4L_c-3$ of a mark of its two blocks, so $R_*+4L_c<r'=2R_*+2$.

\paragraph{Lemma~\ref{lem:4.21}.} The comparison base is $\XX^1\cup R_t$ rather than the base $B_t=\MM\cup\Uf^1\cup R_t$. A bond from a block of $\Uf^1$ to hull material of $\XX^1$ lies in $\partial\Uf^1$ but not in $\mathrm{ins}^1=\partial\XX^1$, and the marks do not bound such bonds, so only the comparison with $\XX^1\cup R_t$ gives a bound by $\mathrm{ins}^1$.