%
%
%
\providecommand{\Lbox}{\Lambda^{\circ}}
\providecommand{\Hset}{\mathbb{X}}
\providecommand{\Mset}{\mathbb{M}}
\providecommand{\Uf}{\mathbb{U}}
\providecommand{\Kc}{\mathfrak{K}}
\providecommand{\Sbar}{\bar S}
\providecommand{\Gp}{G'}
\providecommand{\Ereg}{\mathcal{E}}
\providecommand{\good}{\operatorname{good}}
\providecommand{\Qc}{\mathcal{Q}}
\providecommand{\Uc}{\mathcal{U}}

\section{The regularization: proofs}\label{sec:4}

In this section we prove Theorems~\ref{thm:3.20} and~\ref{thm:3.21} and Proposition~\ref{prop:3.19}. Several steps below are abbreviated, for instance in the proofs of Proposition~\ref{prop:4.4}, of Theorem~\ref{thm:4.7}(iii), (iv) and of Lemma~\ref{lem:4.17}, and we give complete proofs of all of them in Section~\ref{S4-S:S4.1}. We first fix the class of histories to which the arguments apply.

\begin{definition}[Admissible history]\label{def:4.1}
An \emph{admissible history} is a finite sequence of states $\sigma_t=(\Hset_t,\Uf_t,S_t)$, $0\le t\le T$, such that (H1) $\sigma_0=(\Mset,\emptyset,\emptyset)$ and the first step is a hull step; (H2) every other step is a hull step or, at a hull-closed state, a \emph{fill}, which adds a union $\Uf'$ of blocks to $\Uf$ and to $\Hset$ and converts the frozen bonds touching $\Uf'$, or a \emph{freeze}, which adds to $S$ a nonempty set of $\Gp$-bonds crossing block faces, each with its kind; (H3) every fill and every freeze is followed by a hull step. The \emph{full history} is Phase~1 followed by Phase~2.
\end{definition}

Phase~1 is admissible: its first outer step is a hull step, the fill of $\Uf^*$ and a hull step, every later outer step that changes $\Uf$ is a fill and a hull step (Lemmas~\ref{lem:3.minE}(b), (c) and~\ref{lem:4.2}), and a seal step is a freeze. The full history is admissible as well, since its removal steps are fills and its seal steps and cut firings are freezes.

\subsection{The hull}\label{ssec:4.1}

We first show in Lemma~\ref{lem:4.2} that the hull steps of an admissible history only enlarge the hole set, and then derive the geometric consequences of the minimality of the hull.

\begin{lemma}\label{lem:4.2}
Let $(\sigma_t)$ satisfy (H1) to (H3) of Definition~\ref{def:4.1}. At every hull-closed time $t$, $\Hset_t=H^{\Sbar_t}_\lambda(B_t)=H^{S_t}_\lambda(\Hset_t)$ and $S_t\subseteq\Sbar_t\subseteq E(\Lbox\setminus B_t)$, and every bond of $\Sbar_t\setminus S_t$ touches $\Hset_t$. Every hull step $t\to t+1$ satisfies $\Hset_{t+1}=H^{S_t}_\lambda(\Hset_t)\supseteq\Hset_t$.
\end{lemma}

\begin{proof}
We use that $H$ is a closure operator, jointly monotone in the base and in the set of missing bonds, as stated in Lemma~\ref{lem:3.9}(d). We argue by induction over the steps. By (H1) to (H3), each time is $0$, or hull-closed, or follows a fill or a freeze at a hull-closed time $t$ and is followed by a hull step. A hull step at a hull-closed time changes nothing, and the first hull step produces $H^\emptyset_\lambda(B_0)\supseteq B_0=\Hset_0$. A freeze at $t$ of a set $A$ of $\Gp$-bonds, which do not touch $\Hset_t\supseteq B_t$, keeps $B$ and $\Hset$, while $\Sbar_{t+1}=\Sbar_t\cup A$. Thus $\Hset_t=H^{\Sbar_t}_\lambda(B_t)\subseteq H^{\Sbar_{t+1}}_\lambda(B_t)=\Hset_{t+2}$. A fill of $\Uf'$ gives $B_{t+1}=B_t\cup\Uf'$ and drops from $\Sbar$ only bonds touching $B_{t+1}$. It follows that $\mathcal R(B_t,\Sbar_t)\subseteq\mathcal R(B_{t+1},\Sbar_{t+1})$ and $\Hset_t\subseteq\Hset_{t+2}$. In this case, moreover, $\Uf'\subseteq B_{t+1}\subseteq\Hset_{t+2}$. In both cases $B_{t+1}\subseteq\Hset_{t+1}\subseteq\Hset_{t+2}$, and by monotonicity and idempotence $H^{\Sbar_{t+1}}_\lambda(\Hset_{t+1})=\Hset_{t+2}$.

We now pass from $\Sbar$ to $S$. Since a bond of $S_{t'}$ does not touch $\Hset_{t'}\supseteq B_{t'}$, we have $S_{t'}\subseteq\Sbar_{t'}$. A bond of $\Sbar_{t'}\setminus S_{t'}$ was converted, so it touches an earlier hole set and hence $\Hset_{t'}$, and by Lemma~\ref{lem:3.9}(b) it may be dropped: $H^{\Sbar_{t'}}_\lambda(\Hset_{t'})=H^{S_{t'}}_\lambda(\Hset_{t'})$. With $t'=t+1$ this is the claim on hull steps. At a hull-closed time, with $B_t\subseteq\Hset_t=H^{\Sbar_t}_\lambda(B_t)$ and idempotence, the same identity implies $\Hset_t=H^{S_t}_\lambda(\Hset_t)$. The inclusion $\Sbar_t\subseteq E(\Lbox\setminus B_t)$ holds by definition.
\qed
\end{proof}

\paragraph{Proof of (R1) and (N1) to (N5).} These properties follow from Propositions~\ref{prop:4.4}(b), (d) and~\ref{prop:3.19}.

\begin{proposition}\label{prop:4.4}
(a) Order states by $\sigma\le\sigma'$ iff $\Hset\subseteq\Hset'$, $\Uf\subseteq\Uf'$ and $\mathcal R(\sigma)\subseteq\mathcal R(\sigma')$. Under (H1) to (H3), every step that changes the state strictly increases it; in particular (H4) $\Hset_t\subseteq\Hset_{t+1}$ and $\mathcal R(\sigma_t)\subseteq\mathcal R(\sigma_{t+1})$ for every $t$, so that $\Gp$ only shrinks, and a bond is frozen at most once along an admissible history. (b) Phase~1 and Phase~2 terminate, and their outputs are deterministic functions of $D$. (c) The terminal state of Phase~2 satisfies (S-hull), (S-rem) and (S-NF), and no surviving block has auxiliary material there. (d) $\Mset\subseteq B_t\subseteq\Hset_t$ at every time; hence, for the output, $D\subseteq F(\Mset)\subseteq F(\Hset)\subseteq\Fset$.
\end{proposition}

\begin{proof}[Sketch]
Hull steps do not shrink $\Hset$ by Lemma~\ref{lem:4.2}, a fill adds blocks to $\Uf$, a freeze adds $\Gp$-bonds to $S$, and converted bonds stay missing. This proves (a). Phase~1 terminates by Theorem~\ref{thm:4.30}, and Phase~2 terminates because each effective step strictly increases the state in a finite box. Smallest minimizers are canonical and the cut rule uses a fixed order, so the outputs are deterministic. Since a terminal pass of Phase~2 changes nothing, Lemmas~\ref{lem:4.2} and~\ref{lem:4.aux}(a) imply (c). Part (d) holds because the base stays in $\Hset$.
\end{proof}

Below, we denote by $f(A,A')$ the number of bonds of $E$ between disjoint site sets $A$, $A'$, whereas $f_{\Gp}$ of Definition~\ref{def:3.13} counts $\Gp$-bonds. Similarly, $s(A,A')$ denotes the number of bonds of $S$ between $A$ and $A'$, and $s(E(A))$ the number of those with both ends in $A$. We put $N:=|\Mset|$. A state is \emph{closed} if $\Hset=H^S_\lambda(\Hset)$ for the hull weight. By Lemma~\ref{lem:4.2}, each hull-closed state of an admissible history is closed, and so is the output.

\paragraph{Proof of (R2).} We apply Lemma~\ref{lem:4.5} and Theorem~\ref{thm:4.7}(i), (ii) at the output, which is closed by (N1).

\begin{lemma}\label{lem:4.5}
Let $(\Hset,\Uf,S)$ be closed, $V:=\Lbox\setminus\Hset$ and $W\subseteq V$. Then
\begin{equation}\label{eq:4.Ra}
f(W,\Hset)+\sum_{b\in S,\ |b\cap W|=1}(w_b-1)+\sum_{b\in S,\ b\subseteq W}w_b\ \le\ |\partial_{\Gp}W|+\lambda|W| .
\end{equation}
Consequently, with $(a_\eta,b_\eta)=(1+\frac1{1-\eta_w},\frac1{1-\eta_w})=(\frac{19}9,\frac{10}9)$, $|\partial_{\Lbox}W|\le a_\eta|\partial_{\Gp}W|+b_\eta\lambda|W|$, and $N_2(W):=f(W,\Hset)+s(W,V\setminus W)+2s(E(W))\le(|\partial_{\Gp}W|+\lambda|W|)/(1-\eta_w)$.
\end{lemma}

\begin{proof}
Filling $W$ does not lower the functional: $F^S_\lambda(\Hset\cup W)\ge F^S_\lambda(\Hset)$. Since bonds of $S$ do not touch $\Hset$, passing from $\Hset$ to $\Hset\cup W$ changes a bond between $W$ and $\Hset$ from $1$ to $0$, a $\Gp$-bond between $W$ and $V\setminus W$ from $0$ to $1$, a bond of $S$ between $W$ and $V\setminus W$ from $w_b$ to $1$ and a bond of $S$ inside $W$ from $w_b$ to $0$, and adds $\lambda|W|$. This is \eqref{eq:4.Ra}. The hull weight has $w_b-1=1-\eta_w$ and $w_b\ge2(1-\eta_w)$, and the bound on $N_2(W)$ follows. For the remaining bound, note that $|\partial_{\Lbox}W|=f(W,\Hset)+s(W,V\setminus W)+|\partial_{\Gp}W|\le|\partial_{\Gp}W|+N_2(W)$.
\qed
\end{proof}

\begin{theorem}[Geometry of the slit domain]\label{thm:4.7}\label{lem:4.6}
Let $(\Hset,\Uf,S)$ be closed and $c_I:=2^{2/3}$. For $W\subseteq\Lbox\setminus\Hset$: (i) $|\partial_{\Gp}W|\ge(9c_I/38)|W|^{2/3}$ whenever $|W|\le V_*=(1-\eta_w)^3\rho_\lambda^3/2$; (ii) no component of $\Gp$ has at most $V_*$ sites; (iii) $|B_{\Gp}(x,r)|\ge\min\{(c_Ir/(48a_\eta))^3,V_*\}$ for all $x\in\Lbox\setminus\Hset$ and integers $r\ge0$; (iv) $\lambda_1^D(\Omega)\ge(c_I^2/(48a_\eta^2))|\Omega|^{-2/3}$ for $\Omega\subseteq\Lbox\setminus\Hset$ with $|\Omega|\le V_*$.
\end{theorem}

\begin{proof}
(i) In a box $\prod_{i=1}^3\{1,\dots,n_i\}$, a line parallel to $e_i$ that meets $W$ carries a bond of the box with exactly one end in $W$ unless its $n_i$ sites lie in $W$, which happens for at most $|W|/n_i$ lines, and the lines meeting $W$ number at least $3|W|^{2/3}$ by the discrete Loomis\textendash Whitney inequality \cite[Theorem~2]{LW49} and the arithmetic-geometric mean inequality. In this box, such bonds therefore number at least $3|W|^{2/3}-|W|\sum_in_i^{-1}\ge c_I|W|^{2/3}$ if $|W|^{1/3}\le0.47\min_in_i$, since $1-2^{2/3}/3=0.4708\ldots$. Lemma~\ref{lem:4.5} controls the boundary in the box up to $b_\eta\lambda|W|$, which is small below $V_*=(c_I\rho_\lambda/(2b_\eta))^3$, an identity that follows from $(c_I/2)^3=\frac12$. Since $|W|^{1/3}\le V_*^{1/3}<\rho_\lambda=L_c/12$ and each side of $\Lbox$ is at least $\ell\ge308L_c$ by Lemma~\ref{lem:3.3}(c), this and Lemma~\ref{lem:4.5} give $c_I|W|^{2/3}\le|\partial_{\Lbox}W|\le a_\eta|\partial_{\Gp}W|+b_\eta\lambda|W|^{1/3}|W|^{2/3}\le a_\eta|\partial_{\Gp}W|+\tfrac{c_I}2|W|^{2/3}$. We conclude that $|\partial_{\Gp}W|\ge(c_I/(2a_\eta))|W|^{2/3}=(9c_I/38)|W|^{2/3}$.

(ii) A component $\Kc$ of $\Gp$ has $\partial_{\Gp}\Kc=\emptyset$, so (i) excludes $0<|\Kc|\le V_*$.

(iii) (Sketch.) We put $v(r):=|B_{\Gp}(x,r)|$. Each $\Gp$-bond leaving $B_{\Gp}(x,r)$ ends at distance $r+1$. Thus $|\partial_{\Gp}B_{\Gp}(x,r)|\le6(v(r+1)-v(r))$, and by (i), $v(r+1)\ge v(r)+\alpha v(r)^{2/3}$ with $\alpha:=c_I/(12a_\eta)\le\frac14$ while $v(r)\le V_*$. Consequently, $v(r+1)^{1/3}>v(r)^{1/3}+\alpha/4$, and $v(r)^{1/3}\ge1+\alpha r/4\ge c_Ir/(48a_\eta)$ until $v$ exceeds $V_*$.

(iv) By (i), each nonempty $A\subseteq\Omega$ has $|\partial_{\Gp}A|\ge h|A|$ with $h:=(9c_I/38)|\Omega|^{-1/3}$, and the Cheeger argument in a graph of maximal degree $6$, based on the coarea formula for $f^2$ and the Cauchy\textendash Schwarz inequality, yields $\lambda_1^D(\Omega)\ge h^2/12\ge(c_I^2/(48a_\eta^2))|\Omega|^{-2/3}$.
\qed
\end{proof}

\paragraph{Segment accounting.} In Lemma~\ref{lem:4.8} and Corollary~\ref{cor:4.10} we assume that $B=\Mset'\cup U$ with $\Mset'$ a set of sites and $U$ a union of blocks and that $S\subseteq E(\Lbox\setminus B)$ carries weights $w_b\in[0,2]$, and we let $\Hset:=H^S_\lambda(B)$ and $W\subseteq\Hset\setminus B$. By Lemma~\ref{lem:4.2}, this setting holds at a hull-closed time of an admissible history, with $\Mset'=\Mset$, $U=\Uf_t$ and $S=\Sbar_t$ carrying the hull weight. A \emph{segment} $R$ of $W$ is a maximal set of consecutive sites of $W$ on an axis-parallel line, and $m(R):=|R|$. It has $m(R)-1$ \emph{interior bonds} and two \emph{ends}, the positions just beyond its extremities. An end is an \emph{$\Hset$-end} if it is a bond to $\Hset\setminus W$, a \emph{frozen end} if it is a bond of $S$ to $\Lbox\setminus\Hset$, an \emph{open end} if it is a bond of $E\setminus S$ to $\Lbox\setminus\Hset$, and a \emph{box end} if it is on $\partial\Lbox$. With $x(R)$ the number of $\Hset$-ends and $o(R)$ the number of open ends, we put $c_W(R):=x(R)+\sum_{\text{frozen ends }b}(w_b-1)+\sum_{\text{interior }b\in S}w_b-o(R)-\lambda m(R)/3$. A \emph{grid bond} has its ends in different blocks. Such a bond lies on a block plane. The \emph{grid-type terms} of $R$ are its $\Hset$-ends, frozen ends and interior frozen bonds that are grid bonds, and its box ends. The number $x_{\rm ng}(R)$ counts the $\Hset$-ends, frozen ends and interior frozen bonds that are not grid bonds. Similarly, $S^{\rm n}$ is the set of bonds of $S$ that are not grid bonds, and $f_{\rm ng}(A,A')$ counts the bonds between $A$ and $A'$ that are not grid bonds. Frozen bonds cross block faces and are therefore grid bonds, so $S^{\rm n}=\emptyset$ at each hull-closed time of an admissible history.

\begin{lemma}\label{lem:4.8}
Let $\lambda\ge12/L_c$. Every segment $R$ of $W$ satisfies $c_W(R)\le2x_{\rm ng}(R)-2m(R)/L_c$.
\end{lemma}

\begin{proof}
Insulation on block planes, which are at least $L_c$ apart, contributes at most $2$ per plane met, while the volume term is $\lambda m/3\ge4m/L_c$. Indeed, since $o=2-x-e_{\rm fr}-e_b$, where $e_{\rm fr}$ is the number of frozen ends and $e_b$ that of box ends, we have
\begin{equation}\label{eq:4.cWid}
c_W(R)=2x+\sum_{\text{frozen ends}}w_b+\sum_{\text{interior frozen}}w_b+e_b-2-\lambda m/3 .
\end{equation}
The terms that are not grid bonds contribute at most $2x_{\rm ng}(R)$, since $w_b\le2$. Each grid-type term has coefficient at most $2$ and sits at its own bond position on a block plane perpendicular to the line. The $m+1$ bond positions of $R$ lie in an interval of length $m$, which meets at most $m/L_c+1$ block planes by (G1). Therefore $c_W(R)\le2x_{\rm ng}(R)+2(m/L_c+1)-2-4m/L_c$.
\qed
\end{proof}

\begin{corollary}[Block-local hull]\label{cor:4.10}\label{lem:4.9}\label{lem:4.27}
For every union $N$ of blocks, $W:=(\Hset\setminus B)\cap N$ is empty or satisfies $|W|<\frac{L_c}3\bigl(f_{\rm ng}(W,B)+\sum_{v\in W}\deg_{S^{\rm n}}(v)\bigr)$, and every bond counted in $f_{\rm ng}(W,B)$ ends in $\Mset'\setminus U$. Consequently, for every block $q$, $|(\Hset\setminus B)\cap q|<(L_c/3)(6m_q+s_q)$ unless it vanishes, with $m_q:=|\Mset'\cap q|$ and $s_q$ the number of endpoints in $q$ of bonds of $S^{\rm n}$, and $\Hset\subseteq B\cup\bigcup\{q:\ q\cap(\Mset'\cup V(S^{\rm n}))\ne\emptyset\}$. In other words, grid bonds of $S$ of any weight in $[0,2]$, filled blocks and box faces support no hull material, alone or jointly. At a hull-closed time of an admissible history, $m_q$ is the number of marks in $q$ and $S^{\rm n}=\emptyset$.
\end{corollary}

\begin{proof}
Let $W\subseteq\Hset\setminus B$ be nonempty. Then the set $\Hset\setminus W\supseteq B$ is strictly smaller than the smallest minimizer $\Hset$. Since bonds of $S$ do not touch $B$ and each site of $W$ lies in three segments, passing from $\Hset$ to $\Hset\setminus W$ changes the contribution of an $\Hset$-end by $+1$, that of a frozen end by $w_b-1$, that of an open end by $-1$ and that of an interior frozen bond by $+w_b$, and it changes the volume term by $-\lambda|W|$, so that
\begin{equation}\label{eq:4.locrem}
F^S_\lambda(\Hset\setminus W)-F^S_\lambda(\Hset)=\sum_Rc_W(R)>0,
\end{equation}
over the segments of $W$ in the three directions. For $W=(\Hset\setminus B)\cap N$, an $\Hset$-end that is not a grid bond has both ends in one block $q\subseteq N$, so it ends in $\Hset\cap N\setminus W\subseteq B$, but not in $U$, since otherwise $q\subseteq U$. It is not frozen either, and it is therefore counted in $f_{\rm ng}(W,B)$. A frozen end that is not a grid bond is counted once in the degree sum, and an interior bond of $S^{\rm n}$ is counted twice. Combining \eqref{eq:4.locrem}, Lemma~\ref{lem:4.8} and $\sum_Rm(R)=3|W|$, we obtain $0<\sum_Rc_W(R)\le2\sum_Rx_{\rm ng}(R)-6|W|/L_c\le2(f_{\rm ng}(W,B)+\sum_{v\in W}\deg_{S^{\rm n}}(v))-6|W|/L_c$. For $N=q$, $f_{\rm ng}(W,B)\le6m_q$ and the degree sum is at most $s_q$, so the bound for a single block holds. At a hull-closed time, such as the output, the last assertion of Corollary~\ref{cor:4.10} applies, and the bound for a single block then gives (R3).
\qed
\end{proof}

For a hull-closed time $t$ and a block face $F'$ between surviving blocks $Q,Q'$, in the plane $P'$, we put $W_t:=(\Hset_t\setminus B_t)\cap(Q\cup Q')$ and $\mathrm{supp}_t(F'):=f_{\rm ng}(W_t,B_t)$. For a line $\ell\perp F'$, $b_\ell$ denotes its bond across $F'$. As $B_t\cap(Q\cup Q')$ consists of marks, $\mathrm{supp}_t(F')\le6(m_Q+m_{Q'})$.

\begin{lemma}\label{lem:4.11}
At every hull-closed time $t$ of an admissible history and for every such face $F'$: (a) the lines $\ell\perp F'$ such that $b_\ell\notin\Sbar_t$ and $\ell\cap Z(F')$ contains a site of $W_t$ and no mark, number fewer than $2\,\mathrm{supp}_t(F')$ unless there is none; (b) without the condition on $b_\ell$, these lines number fewer than $2\,\mathrm{supp}_t(F')/\eta_w$ unless there is none. A frozen bond lies on the tube of a line $\ell\perp F'$ only if it is $b_\ell$, that is, frozen bonds block no tube of another line.
\end{lemma}

\begin{proof}
In outline, a line on which hull material meets the tube zone carries a segment that is either long, and pays through Lemma~\ref{lem:4.8}, or short, and then has no grid-type term except $b_\ell$ and has deficit at least $\eta_w$. We put $d(R):=2x_{\rm ng}(R)-c_W(R)$ for the segments of $W_t$. By Lemma~\ref{lem:4.8}, $d(R)\ge2m(R)/L_c\ge0$, and \eqref{eq:4.locrem} implies $\sum_Rd(R)<2\sum_Rx_{\rm ng}(R)\le2\,\mathrm{supp}_t(F')$, as in the proof of Corollary~\ref{cor:4.10}. For a counted line $\ell$ let $R_\ell$ be the segment of $W_t$ along $\ell$ through a site of $\ell\cap Z(F')\cap W_t$. Distinct lines give distinct segments. If $m(R_\ell)\ge L_c/2$, then $d(R_\ell)\ge1$. Otherwise all bond positions of $R_\ell$ are at distance $<L_c$ from $P'$, so by (G1) the only possible grid-type term is $b_\ell$, which is not an $\Hset$-end, since its other end would lie in $\Hset_t\cap(Q\cup Q')\setminus W_t\subseteq B_t$, a mark of $\ell\cap Z(F')$. Thus, in this case, $d(R_\ell)=2+\lambda m/3-\beta_\ell$, with $\beta_\ell=2-\eta_w$ if $b_\ell\in\Sbar_t$ and $b_\ell$ is a bond position of $R_\ell$, and $\beta_\ell=0$ otherwise. It follows that $d(R_\ell)\ge1$ for each line counted in (a) and $d(R_\ell)\ge\eta_w$ for each line counted in (b). Summing these bounds, we obtain (a) and (b). For the last claim, recall that a frozen bond is a grid bond, and use (G5): a tube consists of $b_\ell$ and of bonds interior to $Q$ or to $Q'$, which are not grid bonds.
\qed
\end{proof}

\subsection{Cells, giants and the seal}\label{ssec:4.3}

In this subsection we control the cells by sparseness. Since forced blocks are filled (Lemma~\ref{lem:4.forced}), a surviving block has fewer than $\vartheta L_c^2$ marks near it, its hole sites lie on few lines by Corollary~\ref{cor:4.10}, and one component of its free sites, the giant, contains all but a small volume. The seal excludes thin sets (Lemma~\ref{lem:4.thin} and Theorem~\ref{thm:4.rest}), at a cost of at most $726$ frozen bonds per mark (Proposition~\ref{prop:4.sealcount}).

\begin{lemma}\label{lem:4.forced}
Let $\Sbar$ be as in Definition~\ref{def:3.forced}. The smallest minimizer $(\Hset^*,\Uf^*)$ of $\Ereg^{\Sbar}$ has $\bigcup\mathfrak B_F\subseteq\Uf^*$. Hence at every hull-closed time of the full history other than the first every forced block lies in $\Uf$, and every surviving block $q$ has $m_q\le N(3q)<\vartheta L_c^2$.
\end{lemma}

\begin{proof}
Suppose that $Q\in\mathfrak B_F$ and $Q\not\subseteq\Uf^*$. Then $Q\cap\Uf^*=\emptyset$, and we compare with $(\Hset^*\cup Q,\Uf^*\cup Q)$. Only the at most $|\partial Q|$ bonds leaving $Q$ gain weight, at most $1$ each, the volume terms change by $\lambda_U|Q|-\lambda|Q\cap\Hset^*|\le\lambda_U|Q|$, and the penalty drops by $\mu_{\rm B}L_c^2>|\partial Q|+\lambda_U|Q|$ (see Section~\ref{S:B1.forced}). This is a contradiction. For the second assertion we use that $\Uf$ never decreases along the full history, by Proposition~\ref{prop:4.4}(a), and that the second hull-closed time follows the first outer step of Phase~1. If that step fills nothing, then $\mathfrak B_F=\emptyset$.
\qed
\end{proof}

\begin{lemma}\label{lem:4.runs}
At every hull-closed time of an admissible history, every surviving block $q$ satisfies: (a) $|\Hset\cap q|<3L_cm_q$ unless $\Hset\cap q=\emptyset$; (b) unless there is none, fewer than $9m_q$ maximal runs of $\Hset$ in the three directions lie inside $q$ and are joined by both end bonds to sites of $V_q:=q\setminus\Hset$.
\end{lemma}

\begin{proof}
(a) The set $\Hset\cap q$ consists of the $m_q$ marks of $q$ and, by Corollary~\ref{cor:4.10}, fewer than $2L_cm_q$ sites of hull material. (b) At most $3m_q$ runs contain a mark. A run without marks is a segment $R$ of $W:=(\Hset\setminus B)\cap q$ with two open ends inside $q$, since in-block bonds are not frozen, and with no grid-type term. For such a segment, $d(R):=2x_{\rm ng}(R)-c_W(R)=2+\lambda m(R)/3$ by \eqref{eq:4.cWid}. As in Lemma~\ref{lem:4.11}, we have $d\ge0$ for each segment of $W$ and $\sum_Rd(R)<2f_{\rm ng}(W,B)\le12m_q$. Therefore fewer than $6m_q$ runs have no mark, which proves (b).
\qed
\end{proof}

\begin{lemma}\label{lem:4.LW}\label{lem:4.planar}
(a) Let $\mathsf Q$ be a box with sides $L_1,L_2,L_3$, $Z\subseteq\mathsf Q$, $\mathsf Q\setminus Z=A\sqcup B$, and $n_\mu$ the number of lines of $\mathsf Q$ in direction $\mu$ meeting both $A$ and $B$. Then $\min(|A|,|B|)\le4|Z|+6.4\max_\mu(n_\mu L_\mu)$. (b) Let $\mathsf R$ be a rectangle of $\mathbb Z^2$ with sides $s_1,s_2\ge s$, and $\sigma\subseteq\mathsf R$ with $|\sigma|\le s^2/4$. At least $2|\sigma|^{1/2}$ bonds of $\mathsf R$ have exactly one end in $\sigma$.
\end{lemma}

\begin{proof}
(a) Let $a_\mu$ be the number of lines in direction $\mu$ meeting $A$ but not $B$. Then $|A|\le L_\mu(a_\mu+n_\mu)$, and the lines meeting $B$ are among the other $|\mathsf Q|/L_\mu-a_\mu$. By the discrete Loomis\textendash Whitney inequality \cite[Theorem~2]{LW49}, it follows that $|B|^2\le\prod_\mu(|\mathsf Q|/L_\mu-a_\mu)\le\prod_\mu(|\mathsf Q|-|A|+n_\mu L_\mu)/L_\mu$. With $\alpha:=|A|/|\mathsf Q|$, $\beta:=1-\alpha$, $\zeta:=|Z|/|\mathsf Q|$ and $\nu:=\max_\mu n_\mu L_\mu/|\mathsf Q|$ this reads $(\beta-\zeta)^2\le(\beta+\nu)^3$. If $\nu>0.1$, then $\min(|A|,|B|)\le|\mathsf Q|/2<6.4\nu|\mathsf Q|$. If $\nu\le0.1$, then $(\beta+\nu)^{3/2}\le\beta^{3/2}+1.6\nu$, so $\beta(1-\sqrt\beta)\le\zeta+1.6\nu$. Since $1-\sqrt\beta\ge(1-\beta)/2$, in this case $\alpha(1-\alpha)\le2\zeta+3.2\nu$, and $\min(|A|,|B|)\le2|\mathsf Q|\alpha(1-\alpha)\le4|Z|+6.4\nu|\mathsf Q|$.

(b) Each row and each column that meets $\sigma$ without lying in it carries such a bond, a different one for each. If no row or column lies in $\sigma$, the $r$ rows and $c$ columns meeting $\sigma$ give $r+c\ge2(rc)^{1/2}\ge2|\sigma|^{1/2}$ bonds. If a row lies in $\sigma$ (the case of a column is symmetric), all columns meet $\sigma$, and if moreover a column lies in $\sigma$, at most $|\sigma|/s_2+|\sigma|/s_1\le s/2$ rows and columns lie in $\sigma$, so that either way there are at least $s\ge2|\sigma|^{1/2}$ bonds.
\qed
\end{proof}

\begin{proposition}[The giant]\label{prop:4.giant}
At every hull-closed time of the full history other than the first, every surviving block $q$ has $|V_q|>(1-3\vartheta)L_c^3$, and the graph $\Gp_q$ of the $\Gp$-bonds with both ends in $q$ has a unique component $C_q$, the \emph{giant}, with $|V_q\setminus C_q|<128L_cm_q$ unless $V_q=C_q$.
\end{proposition}

\begin{proof}
Since bonds inside a block are never frozen, on a line of $q$ that meets two disjoint unions $A$, $B$ of components of $\Gp_q$, two consecutive sites of $V_q$ in $A$ and in $B$ are separated by a run of Lemma~\ref{lem:4.runs}(b), a different run for each line. If $A\sqcup B=V_q$, Lemma~\ref{lem:4.LW}(a) with $Z=\Hset\cap q$, $|Z|<3L_cm_q$ and $L_\mu<2L_c$ yields $\min(|A|,|B|)<12L_cm_q+6.4\cdot9m_q\cdot2L_c<128L_cm_q$. By Lemma~\ref{lem:4.forced}, $m_q<\vartheta L_c^2$, so $|V_q|>(1-3\vartheta)L_c^3>3\cdot128L_cm_q$. If each component had fewer than $|V_q|/3$ sites, a union $A$ of components with $|V_q|/3\le|A|<2|V_q|/3$ would violate the bound. The largest component $C_q$ therefore has at least $|V_q|/3$ sites, and the bound with $A=C_q$ implies $|V_q\setminus C_q|<128L_cm_q$.
\qed
\end{proof}

A \emph{thin set} of a surviving block $j$ is a nonempty $A\subseteq V_j$ with $|A|<\varepsilon_{\rm th}L_c^3:=0.008L_c^3$ and $\mathrm{cut}_j(A)<|A|/L_c$, $\mathrm{cut}_j(A)$ being the number of $\Gp$-bonds inside $j$ between $A$ and $V_j\setminus A$.

\begin{lemma}[Thin sets]\label{lem:4.thin}
At every hull-closed time of the full history other than the first, every thin set $A$ of a surviving block $j$ satisfies $F^{\rm aux}_j(\Hset\cup A)<F^{\rm aux}_j(\Hset)$. Hence a surviving block without auxiliary material has no thin set, and $V_j=C_j$ is connected in $\Gp_j$.
\end{lemma}

\begin{proof}
A thin set is enclosed by in-block insulation in each planar slice, and planar isoperimetry makes this insulation exceed the volume penalty. We fix a direction $\nu$. A line of $j$ in direction $\nu$ that meets $A$ either meets $V_j\setminus A$, and then carries a bond counted in $\mathrm{cut}_j(A)$ or a run of Lemma~\ref{lem:4.runs}(b), or lies in $A\cup(\Hset\cap j)$, which happens for at most $(|A|+3L_cm_j)/L_c$ lines. A slice $\sigma$ of $A$ perpendicular to $\nu$ meets each line once. Therefore, with $m_j<\vartheta L_c^2$ from Lemma~\ref{lem:4.forced}, $|\sigma|\le a:=\mathrm{cut}_j(A)+9m_j+(|A|+3L_cm_j)/L_c<(2\varepsilon_{\rm th}+12\vartheta)L_c^2\le0.0164\,L_c^2\le L_c^2/4$. The bonds of the cross-section of $j$ (sides at least $L_c$) with exactly one end in $\sigma$ end in $\Hset$ or in $V_j\setminus A$, and there are at least $2|\sigma|^{1/2}\ge2|\sigma|/a^{1/2}$ of them, where the first bound is Lemma~\ref{lem:4.LW}(b). Summing over the slices, we obtain $f_j(A,\Hset)+\mathrm{cut}_j(A)\ge2|A|/a^{1/2}\ge15.6|A|/L_c$, with $f_j(A,\Hset)$ the number of bonds inside $j$ from $A$ to $\Hset$.

Passing from $\Hset$ to $\Hset\cup A$ changes $F^{\rm aux}_j$ by $-f_j(A,\Hset)$ on these hole faces, by at most $0$ on the bonds of $\partial j$ at $A$ (from $w_b$ to $1$, or from $1$ to $0$), by $+\mathrm{cut}_j(A)$ on the in-block bonds to $V_j\setminus A$, and by $+\lambda|A|$. In total the change is at most $-15.6|A|/L_c+2\,\mathrm{cut}_j(A)+12|A|/L_c<0$. If $j$ has no auxiliary material, $\Hset$ minimizes $F^{\rm aux}_j$, so $j$ has no thin set. Moreover, a component of $\Gp_j$ other than $C_j$ has $\mathrm{cut}_j=0$ and, by Proposition~\ref{prop:4.giant}, fewer than $128L_cm_j<\varepsilon_{\rm th}L_c^3$ sites, so it would be thin.
\qed
\end{proof}

\begin{lemma}[The seal]\label{lem:4.aux}
Let $t$ be a hull-closed time of an admissible history and $j$ a surviving block with $A_j\neq\emptyset$. (a) Some $\Gp$-bond that crosses a face between $j$ and a surviving block has an end in $A_j$. (b) If a seal step at $t$ is followed by a hull step with the same $\Uf$, then $A_j\subseteq\Hset_{t+2}$.
\end{lemma}

\begin{proof}
On the sets between $\Hset:=\Hset_t$ and $\Hset\cup A_j$, the functionals $F:=F^{\Sbar_t}_\lambda$ and $F^{\rm aux}_j$ differ by a constant unless a $\Gp$-bond leaves $A_j$ across a face of $j$. Indeed, since a block adjacent to $j$ that is not surviving lies in $\Hset$, the other bonds of $\partial j$ at $A_j$ touch $\Hset$ or are frozen. Those at $V_j\setminus A_j$ keep their values. (a) Otherwise $F(\Hset\cup A_j)-F(\Hset)=F^{\rm aux}_j(M_j)-F^{\rm aux}_j(\Hset)<0$, since $M_j\ne\Hset$ is the smallest minimizer. This contradicts $\Hset=H^{\Sbar_t}_\lambda(B_t)$, as $\Hset\cup A_j\supseteq B_t$.

(b) Let $X':=\Hset_{t+2}\supseteq\Hset$ (Lemma~\ref{lem:4.2}), $F':=F^{\Sbar_{t+2}}_\lambda$, $A':=A_j\setminus X'$ and $C:=\Hset\cup(A_j\cap X')\subseteq X'$. On the sets between $C$ and $M_j=C\cup A'$ only the sites of $A'$ vary. The bonds inside $j$ at these sites count alike in $F'$ and $F^{\rm aux}_j$, and after the seal each bond of $\partial j$ at them is frozen or touches $\Hset$, in both. Consequently, $F'-F^{\rm aux}_j$ is constant on these sets. If $A'\neq\emptyset$, then $C\subsetneq M_j$, and submodularity of $F'$ implies $F'(X'\cup A')-F'(X')\le F'(C\cup A')-F'(C)=F^{\rm aux}_j(M_j)-F^{\rm aux}_j(C)<0$, contradicting the minimality of $X'=H^{\Sbar_{t+2}}_\lambda(B_t)$.
\qed
\end{proof}

\begin{proposition}\label{prop:4.sealcount}
Along the full history, a block face $F'$ between blocks $Q$, $Q'$ carries at most $121(m_Q+m_{Q'})$ bonds frozen by seal steps, converted ones included. Hence seal steps freeze at most $C_{\rm seal}N$ bonds, all on faces $F'$ with $m_Q+m_{Q'}>0$, where $C_{\rm seal}:=6\cdot121=726$.
\end{proposition}

\begin{proof}
Each seal bond on $F'$ has an end in auxiliary material, which the next hull step fills while $Q$ and $Q'$ still survive. This follows from Lemma~\ref{lem:4.aux}(b), since in Phase~1 $\Uf$ is fixed during the inner loop and in Phase~2 the hull step follows the seal before any removal. Consequently, at the last hull-closed time $T$ at which $Q$ and $Q'$ both survive, each seal bond on $F'$ has an end in $W_T$ in the layer next to $F'$, hence in $Z(F')$, and distinct bonds lie on distinct lines $\ell\perp F'$. At most $m_Q+m_{Q'}$ of these lines carry a mark in $Z(F')$, and fewer than $2\,\mathrm{supp}_T(F')/\eta_w\le120(m_Q+m_{Q'})$ of the others meet $W_T$ in $Z(F')$, the first inequality being Lemma~\ref{lem:4.11}(b). Since a block has six faces, summing over all block faces yields the bound on the total number of seal bonds.
\qed
\end{proof}

\begin{theorem}[Cells at rest]\label{thm:4.rest}
Let $t$ be a hull-closed time of the full history other than the first at which no surviving block has auxiliary material. Then: (a) every surviving block $q$ carries exactly one cell, $\Omega_q=V_q$, connected in $q$, with $(1-3\vartheta)L_c^3<|\Omega_q|<8L_c^3$, and every other block lies in $\Hset$; (b) every block face $F'$ between surviving blocks contains no $\Gp$-bond, or carries exactly one interface $I$; in the latter case $F'$ contains no cut bond and $\good(I)\ge(1-134\vartheta)|F'|$; in particular (S-face) holds; (c) every component $\Kc$ of $\Gp\cap\hat Q$, $\hat Q\in\Qc$, satisfies $f_{\Gp}(A,\Kc\setminus A)\ge h_\star|A|$ for $0<|A|\le|\Kc|/2$, with $h_\star:=2.2\cdot10^{-11}/L_c$, and (S-plug) holds.
\end{theorem}

\begin{proof}
(a) By Lemma~\ref{lem:4.thin}, $V_q=C_q$ is connected in $q$, and therefore lies in one component of $\Gp\cap\hat Q(q)$ and is the only cell of $q$. The volume bounds follow from Proposition~\ref{prop:4.giant} and (G1). Finally, a block that is not surviving lies in $\Uf\subseteq\Hset$, which is the last assertion of (a).

(b) By (a), the $\Gp$-bonds across $F'$, if any, join the cell of $Q$ to the cell of $Q'$ and form one interface. Cuts fire only at such times, where by (a) a cut, which is a union of whole interfaces by Lemma~\ref{lem:3.18}(a), freezes all $\Gp$-bonds of each face its bonds cross. As $\Gp$ only shrinks, a face with a $\Gp$-bond carries no cut bond. A line $\ell\perp F'$ is good unless $\ell\cap Z(F')$ contains a mark, or $b_\ell$ is a seal bond, or else $b_\ell\notin\Sbar_t$ and $\ell\cap Z(F')$ contains a site of $W_t$. Indeed, on each remaining line the tube has no hole site, and its bonds are $\Gp$-bonds, since bonds inside a block are never frozen. The first case holds for at most $m_Q+m_{Q'}$ lines and, by Proposition~\ref{prop:4.sealcount}, the second holds for at most $121(m_Q+m_{Q'})$ lines. The third holds for fewer than $2\,\mathrm{supp}_t(F')\le12(m_Q+m_{Q'})$ lines, where the first inequality comes from Lemma~\ref{lem:4.11}(a). With $m_Q+m_{Q'}\le N(3Q)<\vartheta L_c^2\le\vartheta|F'|$, we obtain $\good(I)\ge(1-134\vartheta)|F'|\ge c_g|F'|$.

(c) Expansion can fail only through a thin set, and by Lemma~\ref{lem:4.thin} there is none. Let $A\subseteq\Kc$, $0<|A|\le|\Kc|/2$, and $\mathrm{cut}:=f_{\Gp}(A,\Kc\setminus A)$. Since windows are unions of blocks, by (a) $\Kc$ is a union of sets $V_j$ of surviving blocks $j\subseteq\hat Q$, connected through faces that contain $\Gp$-bonds. We put $A_j:=A\cap V_j$ and $B_j:=V_j\setminus A_j$. A line of $j$ meeting $A_j$ and $B_j$ carries a bond counted in $\mathrm{cut}$ or a run of Lemma~\ref{lem:4.runs}(b). If $\min(|A_j|,|B_j|)\ge\varepsilon_{\rm th}L_c^3$ for some $j$, then $\varepsilon_{\rm th}L_c^3\le12L_cm_j+12.8L_c(\mathrm{cut}+9m_j)$ by Lemma~\ref{lem:4.LW}(a), so $\mathrm{cut}\ge(\varepsilon_{\rm th}-127.2\vartheta)L_c^2/12.8\ge\varepsilon_{\rm th}L_c^2/25.6$. Otherwise each block of $\Kc$ is either \emph{in} ($|B_j|<\varepsilon_{\rm th}L_c^3$) or \emph{out} ($|A_j|<\varepsilon_{\rm th}L_c^3$), but not both, since $|V_j|>2\varepsilon_{\rm th}L_c^3$. If an in-block and an out-block share a face $F'$ that contains $\Gp$-bonds, we follow each line $\perp F'$ from the face into either block until it reaches the majority side of that block through a bond counted in $\mathrm{cut}$ or a run, or stays in the minority side and $\Hset$ for the whole block. In this case the minority side of each block meets the face layer in at most $\mathrm{cut}+9\vartheta L_c^2+(\varepsilon_{\rm th}+3\vartheta)L_c^2$ sites, all other bonds among the $\Gp$-bonds across $F'$, of which there are at least $(1-134\vartheta)L_c^2$ by (b), join $A$ to $\Kc\setminus A$, and $3\,\mathrm{cut}\ge(1-158\vartheta-2\varepsilon_{\rm th})L_c^2$. Otherwise all blocks are in, which is impossible since then $|A|\ge\sum_j(|V_j|-\varepsilon_{\rm th}L_c^3)>|\Kc|/2$, or all are out. In the latter case each nonempty $A_j$ has fewer than $\varepsilon_{\rm th}L_c^3$ sites and is not thin, so $\mathrm{cut}\ge\sum_j\mathrm{cut}_j(A_j)\ge|A|/L_c$. In all cases $\mathrm{cut}\ge\min(3.1\cdot10^{-4}L_c^2,|A|/L_c)\ge h_\star|A|$, where we use that $|A|<k_P^3/2$ by (G4), together with the arithmetic of Section~\ref{S:B1.cells}. As $3k_Ph_\star\ge\chi$, we have $\chi_{\rm N3}/k_P=\chi/(3k_P)\le h_\star$, and (S-plug) holds.
\qed
\end{proof}

\begin{theorem}[The Phase-1 output]\label{thm:4.30}\label{prop:3.16}
Phase~1 terminates. Its output $(\Hset^1,\Uf^1,S^1)$ is the smallest minimizer of $\Ereg^{\Sbar^1}$, no surviving block has auxiliary material there, and: (i) every bond of $\Sbar^1$ touches $\Hset^1$, so $S^1=\emptyset$, and crosses a face between two blocks that together contain a mark; (ii) the four terminal properties (S-hull), (S-rem), (S-face), (S-plug) hold.
\end{theorem}

\begin{proof}
By Lemma~\ref{lem:3.minE}(c), each outer step returns $\Uf^*\supseteq\Uf$, since $\Sbar$ only grows, and by Lemma~\ref{lem:4.aux}(a) each inner round freezes at least one bond. Each outer step except the last enlarges $\Uf$ or $\Sbar$, which only grow in a finite box. Phase~1 thus terminates. In the last outer step the inner loop performs no seal step, so $\Sbar=\Sbar^1$ throughout that step. Thus step (1) of the last outer step returns the smallest minimizer of $\Ereg^{\Sbar^1}$, with hole set $H^{\Sbar^1}_\lambda(\Mset\cup\Uf^1)$ by Lemma~\ref{lem:3.minE}(b). This pair is the output, and step (2) finds no auxiliary material at it.

(i) Each bond of $\Sbar^1$ has an end in auxiliary material, which the next hull step fills by Lemma~\ref{lem:4.aux}(b). For the faces crossed by these bonds we use Proposition~\ref{prop:4.sealcount}.

(ii) Property (S-hull) follows from Lemma~\ref{lem:4.2}, and (S-face), (S-plug) follow from Theorem~\ref{thm:4.rest}. For (S-rem), let $\Delta\not\subseteq B^1$ be a removal cell with $\Delta\not\subseteq\Hset^1$ (otherwise $w(\Delta)=0$). We compare with $(\Hset^1\cup\Delta,\Uf^1\cup\Delta)$, admissible since $\Delta$ is a union of blocks. The bonds of $E(\Delta)$ with one end in $\Hset^1$ lose their weight $1$, since by (i) no bond is frozen. At most $|\partial\Delta|$ bonds gain at most $1$, the volume terms grow by at most $\lambda_U|\Delta|$, and the penalty does not grow. Therefore $w(\Delta)\le|\partial\Delta|+\lambda_U|\Delta|\le(1+c_s)|\partial\Delta|=w_*$, since $|\Delta|/|\partial\Delta|<\ell/2$ for a box with sides in $[\ell,3\ell)$ (see Section~\ref{S:B1.forced}).
\qed
\end{proof}

\begin{lemma}\label{lem:4.16}
Let the state satisfy (S-plug), $\hat Q\in\Qc$, and $\Kc$ a component of $\Gp\cap\hat Q$. Then $f_{\Gp}(A,\Kc\setminus A)\ge\chi|A|/(3k_P)$ for $A\subseteq\Kc$ with $0<|A|\le|\Kc|/2$, and $\sum_{x\in\Kc}|f(x)-\bar f_{\Kc}|^2\le(108/\chi^2)k_P^2\sum_{\{x,y\}\in E_{\Gp}(\Kc)}(f(x)-f(y))^2$ for every $f$.
\end{lemma}

\begin{proof}
The first claim is (S-plug), since $\chi_{\rm N3}/k_P=\chi/(3k_P)$. The Poincar\'e inequality follows from the Cheeger argument and a median. Let $\mu$ be a median of $f$ on $\Kc$. Since each of $(f-\mu)_\pm$ is supported on at most $|\Kc|/2$ sites, the argument of Theorem~\ref{thm:4.7}(iv) in the graph $(\Kc,E_{\Gp}(\Kc))$, with $h:=\chi/(3k_P)$, bounds $\sum(f-\mu)_\pm^2$ by $12h^{-2}=108k_P^2/\chi^2$ times the Dirichlet form of $(f-\mu)_\pm$. We add the two bounds, using $(a-b)^2\ge(a_+-b_+)^2+(a_--b_-)^2$, and replace $\mu$ by the mean.
\qed
\end{proof}

\begin{lemma}[Good bonds]\label{lem:4.17}
At every state satisfying (S-face), (R8) holds, a fine bond across $F(I)$ being good if the segment of $L$ fine sites of its line centred at it is a path of the fine slit graph.
\end{lemma}

\begin{proof}[Sketch]
The first claim follows from (S-face) and (G1), and the second holds because bonds across block faces lie on block planes, at least $L_c$ apart. A good tube is a $\Gp$-path in $Z(F')\subseteq Q\cup Q'\subseteq\hat Q(Q)\cap\hat Q(Q')$, by (G5) and (G4). Thus its half in $Q$ lies in the component of $\Gp\cap\hat Q(Q)$ of its end, the cell of $I$ in $Q$. The interfaces of one face are disjoint with at least $c_g|F'|$ bonds each, and a block has six faces. Together, these facts bound the number of interfaces of a face and the number of neighbouring cells of a cell. In fine units, a block face meets at most one exceptional layer on each side, so $|F'|_{\rm fine}<r_0^2|F'|(1+1/L_c)^2$, while each coarse good bond carries at least $r_0^2$ fine good bonds.
\qed
\end{proof}

\begin{theorem}[Cells]\label{thm:4.18}
At every closed state with (S-face) and (S-plug), such as the output, (R5), (R6) and the cell part of (R7) hold, with $\Kc^\flat(\Omega):=\hat\Kc$ for the cell $\Omega=Q\cap\hat\Kc$ of a block $Q$.
\end{theorem}

\begin{proof}
For (R5), $|\Omega|\le|Q|<8L_c^3$ by (G1). An interface on a face $F'$ of $Q$ provides, by Lemma~\ref{lem:4.17}, at least $c_gL_c^2$ good half-tubes of $L_c/2$ sites in $\Omega$, on distinct lines. Without an interface no $\Gp$-bond leaves $\Omega$, since inside $Q$ a $\Gp$-bond joins two sites of one cell and across a face it belongs to an interface. In that case $\Omega$ is therefore a union of components of $\Gp$, each with more than $V_*=((1-\eta_w)c_I/24)^3L_c^3$ sites by Theorem~\ref{thm:4.7}(ii). For (R6), we apply Lemma~\ref{lem:4.16} in $\hat\Kc\supseteq\Omega$ and use that the mean minimizes the sum of squares. The overlap bound holds because a bond lies in at most $151^3\le27(k_P/L_c)^3$ windows (G4) and the components $\hat\Kc$ of distinct cells of one block are distinct. For (R7), we use that $\hat\Kc$ is connected in $\Gp$, together with Lemma~\ref{lem:3.5}(A2).
\qed
\end{proof}

\begin{lemma}[$\Uc$-cells]\label{lem:4.19}
Let $\mathcal T$ be a connected graph of maximal degree at most $D$, with vertex weights $w(v)\in(0,V)$ and total weight at least $V$. Then the vertex set of $\mathcal T$ splits into parts, each connected in $\mathcal T$, of weight in $[V,(2+D)V)$. Applied to the cell graphs of the output with $D=6/c_g$, cell weights $|\Omega|_{\rm fine}$ and $V=V_{\Uc}=\tilde m^{-3}$, it gives the $\Uc$-cell part of (R7), with $C_{\Uc}$ of Section~\ref{S:B1.U}, provided $\tilde mL\ge c_L/2$.
\end{lemma}

The proviso $\tilde mL\ge c_L/2$ is a lower bound on $\tau$.

\begin{proof}
We cut a rooted spanning tree from the leaves upward, cutting off a vertex with its uncut descendants once their weight reaches $V$. A cut part weighs less than $w(v)+DV<(1+D)V$, and the light remainder at the root joins an adjacent cut part. In the application, the cell graph is connected, since $\Gp$-bonds between cells are interface bonds, and Lemma~\ref{lem:4.17} bounds its degree by $6/c_g$. A cell weighs less than $8L^3\le\tilde m^{-3}$ by \eqref{eq:3.Hell}. By Theorem~\ref{thm:4.7}(ii), a light component has more than $V_*$ sites, and thus weight more than $r_0^3V_*=c_uV_{\Uc}$. Finally, a $\Uc$-cell holds fewer than $(2+6/c_g)\tilde m^{-3}/(c_1L^3)$ connected cells by (R5), in blocks of Euclidean diameter less than $2\sqrt3L$.
\qed
\end{proof}