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\subsection{Section~\ref{sec:3}: the loop-adapted grid}\label{S:B2}

\paragraph{Construction of $\mathfrak G(C)$ (Definition~\ref{def:3.4}).}\label{S:B2.grid} With $\Lbox=\prod_\mu[a_\mu,b_\mu]$ as in Definition~\ref{def:3.2}, we construct the objects in the following order.
\begin{enumerate}
\item \emph{Special planes.} In direction $\mu$ the special planes are the lattice planes $\{x_\mu=s\}$, $s\in\mathcal S_\mu$, with $\mathcal S_1=\{a_1,r_1,r_1+R,b_1\}$, $\mathcal S_2=\{a_2,t_1,t_1+T,b_2\}$, $\mathcal S_3=\{a_3,z_0,b_3\}$. A \emph{special interval} is the fine interval between two consecutive special planes of one direction.
\item \emph{Sites.} A special interval of fine length $D'$ is divided into $\lfloor D'/r_0\rfloor$ consecutive intervals: $\lfloor D'/r_0\rfloor-1$ of length $r_0$, followed, at the end with the larger coordinate, by one of length $r_0+(D'\bmod r_0)\in[r_0,\bar r_0]$. The sites (coarse cells) are the boxes $\prod_\mu I_\mu$ with $I_\mu$ such intervals. Their fine sides lie in $[r_0,\bar r_0]$, and equal $r_0$ outside at most one exceptional layer per special interval. The sites form, combinatorially, a box of $\mathbb Z^3$. We denote by $E$ its nearest-neighbour bonds, and the coarse face dual to a bond is the common face of its two sites.
\item \emph{Blocks.} In coarse index units, a special interval of coarse length $n$ is divided into $p:=\lfloor n/L_c\rfloor$ consecutive intervals, and with $r:=n-pL_c$ the first $(r\bmod p)$ of them have length $L_c+\lceil r/p\rceil$ and the others $L_c+\lfloor r/p\rfloor$. Their endpoints are the \emph{block planes}. Blocks are the boxes between consecutive block planes of all three directions, and a \emph{block face} is the common face of two adjacent blocks.
\item \emph{Removal cells.} In each direction the block intervals, in increasing order, are grouped greedily into consecutive groups of total coarse length $\ge\ell$, and a final remainder of length $<\ell$ is merged into the last group. Removal cells $\Delta$ are the boxes $\prod_\mu J_\mu$ with $J_\mu$ such groups, and $|\partial\Delta|$ is the number of bonds of $\mathbb Z^3$ (in coarse index units) with exactly one end in $\Delta$, bonds leaving $\Lbox$ included.
\item \emph{Windows.} We index the blocks by their position in each direction. The block distance is the $\ell^\infty$ distance of these indices. For each block $Q$, $\hat Q(Q)$ is the union of the blocks at block distance at most $75$ from $Q$, a block-aligned box, and $\Qc$ is the set of windows.
\item \emph{Tube zones and tubes.} Let $F'$ be a block face between blocks $Q$, $Q'$, in the plane $P'$. Its tube zone $Z(F')$ is the set of sites of $Q\cup Q'$ among the $L_c/2$ layers nearest to $P'$ on each side. For a bond $b$ across $F'$, its tube $\tau_b$ is the segment of the line of $b$ formed by the $L_c$ sites of that line in $Z(F')$. We write $|F'|$ for the number of coarse bonds across $F'$.
\end{enumerate}
Where the rules leave a choice, we put the exceptional coarse layer at the end with the larger coordinate and the longer block intervals first. These conventions depend only on $C$ and the constants, and any others with this property would do.

\paragraph{Proof of Lemma~\ref{lem:3.3}(a) to (d).}\label{S:B2.lem33}
(a) The quantity $M_\Lambda$ is determined by $R$, $T$, $r_0$ and $\ell$, and the centre is determined by $C$. The grid is built from $\Lbox$ and the planes of $C$ by rules that involve only the constants.

(b) By (e) and $m\ge\min(R,T)\ge R_1\ge L$, the special planes $a_\mu$, $b_\mu$ are at fine distance $\ge L$ from the planes of $C$, and $R,T\ge R_1\ge L$.

(c) Step~2 cuts a special interval of fine length $D'$ into $\lfloor D'/r_0\rfloor$ coarse intervals. Hence a side of $\Lbox$ made of at most three special intervals has at least $M'_\Lambda/r_0-3\ge\ell$ coarse cells. Step~4 tiles each side of coarse length $\ge\ell$ by groups of block intervals, so $\Lbox$ is a union of removal cells. Removal cells are unions of blocks by construction. The bound $\ell\ge k_P+4L_c$ is \eqref{eq:3.Hell}.

(d) Each $n_\mu$ lies in $[M'_\Lambda/r_0-3,\ M'_\Lambda/r_0]$, and $M'_\Lambda/r_0\ge\ell+3\ge6$. Hence $\max_\mu n_\mu\le M'_\Lambda/r_0\le2(M'_\Lambda/r_0-3)\le2\min_\mu n_\mu$.

\paragraph{Proof of (G1) to (G5).}\label{S:B2.G}
(G1) By Lemma~\ref{lem:3.3}(b), a special interval has fine length $D'\ge L=r_0L_c$, hence coarse length $n=\lfloor D'/r_0\rfloor\ge L_c$ and $p\ge1$. Since $r<L_c$, each block interval has length in $[L_c,2L_c-1]$. A block lies in one special interval in each direction, so it meets at most one exceptional layer per direction, and its fine side lies in $[r_0L_c,(2L_c-1)r_0+(r_0-1)]=[L,2L-1]$.

(G2) This holds because the construction is a tensor product and the endpoints of special intervals are block planes.

(G3) A greedy group closes as soon as its length reaches $\ell$, and a block interval has length at most $2L_c-1$. Hence a group has length in $[\ell,\ell+2L_c-2]$. Adding a final remainder of length at most $\ell-1$ keeps it at most $2\ell+2L_c-3<3\ell$, by \eqref{eq:3.Hell}. Every side of $\Lbox$ is $\ge\ell$ by Lemma~\ref{lem:3.3}(c), so there is at least one group per direction. Then $|\partial\Delta|=2(s_1s_2+s_2s_3+s_3s_1)$ with $s_\mu\in[\ell,3\ell)$.

(G4) A window has at most $151$ blocks per direction, each of coarse side at most $2L_c-1$ by (G1), so its sides are at most $151(2L_c-1)<k_P$. If $u\in Q$ and $|y-u|_\infty\le75L_c$, the segment from $u$ to $y$ in each direction crosses at most $1+\lfloor(75L_c-1)/L_c\rfloor=75$ block planes, since consecutive parallel block planes are at least $L_c$ apart. Thus the block of $y$ is at block distance at most $75$ from $Q$, and $Q_\infty(u,75L_c)\cap\Lbox\subseteq\hat Q(Q)$. An adjacent block is at block distance $1$ and therefore lies in the window. Finally, a block lies in $\hat Q(Q)$ only for the at most $151^3$ blocks $Q$ at block distance at most $75$ from it.

(G5) This holds because blocks have at least $L_c$ layers and parallel block planes are at least $L_c$ apart.