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\section{The min-cut lower tail}
\label{sec:5}\label{sec:mincut}

In this section we show that the frozen set of the construction of Section~\ref{sec:3} lowers the min-cut area of Definition~\ref{def:2.11} by at most $\etaT RT$, except with probability $e^{-RT}$. The proof has two ingredients. Section~\ref{ssec:5.2} shows that, as $\Lc$ is acyclic, a minimizer can be taken constant on each piece of the hole set. In Sections~\ref{ssec:5.3} to~\ref{ssec:5.7} we then prove that the frozen plaquettes which a part of the marks attaches are paid for by the marks of that part. Complete versions of the proofs of this section, including proofs of the facts that are only stated here, are in Section~\ref{S4-S:S4.2}.

The units, $\Xi$, $\Lc_2$, $\partial_2\Lc$, $F(Y)$, $F_S$, $\Ffr$, $\Gp(D)$, $\XX$ and $S=S_1\sqcup\Scut$ are those of Sections~\ref{ssec:2.5}, \ref{sec:3} and~\ref{ssec:3.6}. A coarse face carries at most $\rbar^{2}$ fine plaquettes, $\rbar=2r_0-1$, and $\Xi$ is a union of coarse faces, since the plane of $C$ is a special plane of $\mathfrak G(C)$. We write $\PD:=\mathrm{Ber}(\delta)^{\otimes(\Lc_2\setminus\partial_2\Lc)}$ for the law of $D$ and $\ED$ for its expectation.

\subsection{Statement}
\label{ssec:5.1}

\begin{theorem}[Min-cut lower tail]\label{thm:5.1}
Fix $K>K_*$ and $\beta\ge\beta_0(K)$. For every rectangle $C$ with $\min(R,T)\ge R_1(\beta)$ and every $\sigma\in(0,1]$,
\begin{gather}\label{eq:5.1}
\PD\bigl(\AGp(C)\le(1-\etaT)RT\bigr)\le 2^{4|C|+5}e^{-RT},\\
\ED\exp\bigl(\sigma(RT-\AGp(C))\bigr)\le\exp\bigl(\sigma\etaT RT+4|C|+4\bigr),\notag
\end{gather}
where
\begin{equation}\label{eq:5.2}
\etaT:=\frac{3c_T''\rbar^{2}C_3}{|\log\eps''|},
\end{equation}
with $\eps''$ of \eqref{eq:5.epspp} and the constants $c_T''$ and $C_3$ of Section~\ref{S:const5}. Moreover, there is $\tau_0(K)$ such that for $\tau\ge\tau_0(K)$ we have $\etaT\le6c_T''\rbar^{2}C_3/(\gamma_T(K)\tau)$, with $\gamma_T(K)$ of Lemma~\ref{lem:5.16}, and $\etaT=O\bigl(\rbar^{2}(C_\star'+\Cresp)(1+1/c_s)/(\tau\log^2K)\bigr)$ with an absolute implied constant, uniformly in $K>K_*$; so $\etaT\to0$ as $K\to\infty$, and as $\beta\to\infty$ at fixed $K>K_*$.
\end{theorem}

The conditions that define $\beta_0(K)$ in item~(a) of Section~\ref{ssec:D.5} are lower bounds on $\tau$, which persist for larger $\tau$, while the constants $c_T''$, $C_3$, $\Cfree$, $C_\star'$, $\Cresp$ are free of $\beta$, $\tau$, $K$, $C$, $D$. The condition $\tau\ge\tau_0(K)$ is used only for the order of $\etaT$, not for \eqref{eq:5.1}.

\subsection{The deterministic side}
\label{ssec:5.2}

In this subsection we let $(\XX,S)$ be any state with $S\subseteq E(\Lc\setminus\XX)$ and compute $\AGp(C)$ with $\Ffr=F(\XX)\cup F_S\cup\partial_2\Lc$. Then $F_S\cap F(\XX)=\emptyset$.

\emph{Box topology.} Let $B$ be a box of $\Zthree$ with its cubical complex. (i) Over $\ZZ$ and over $\ZZ_2$ this complex is the tensor product of the complexes of three intervals, each chain-homotopy equivalent to the coefficients in degree $0$; hence every $k$-cycle with $k\ge1$ is a boundary. (ii) $\partial$ is injective on $3$-chains: in a nonzero $3$-chain, a cube that is extreme in the lexicographic order has a face shared with no other cube of the chain, and the boundary has a nonzero coefficient there. (iii) $H_1(\partial B;\ZZ)=0$: if $z$ is an integer $1$-cycle on the boundary bonds, then $z=\partial c$ for a $2$-chain $c$ of $B$ by (i); the part of $c$ on the interior plaquettes is a closed $1$-cochain on the dual box, hence the coboundary of a $0$-cochain $f$ on the cubes, and $c-\partial f$ is a $2$-chain of boundary plaquettes with boundary $z$. (iv) By (i) over $\ZZ_2$, the boundaries of the unit squares generate the $\ZZ_2$ cycle space of the box graph.

\begin{lemma}\label{lem:5.3}
Every integer $2$-chain $\Sigma$ in $\Lc$ with $\partial\Sigma=C$ satisfies $\#(\supp\Sigma\setminus\partial_2\Lc)\ge RT$. Moreover $\AGp(C)\le RT$.
\end{lemma}

\begin{proof}
There are $RT$ disjoint closed dual paths that link $C$, namely nested closed paths of fine cubes in the planes perpendicular to $\Xi$ through its plaquette rows. They lie inside $\Lc$ off $\partial_2\Lc$ by Lemma~\ref{lem:3.3}(e), and each of them crosses $\Xi$ once (Section~\ref{S:proofs5}). As $\Lc$ is acyclic (Box topology (i)), $\Sigma-\Xi$ is a boundary. Each path therefore meets $\supp\Sigma$, with intersection number $\pm1$, and different paths meet it in different plaquettes. The second claim holds because $\AGp(C)\le|\supp\Xi|=RT$. \qed
\end{proof}

A \emph{piece partition} is a partition $\mathcal Y$ of the sites of $\XX$ into pieces, which need not be connected. For a site set $Y$, let $\partial Y$ be the set of coarse bonds with exactly one end in $Y$, $\operatorname{int}F(Y)$ the set of fine plaquettes whose two fine cubes lie in sites of $Y$, and $\partial^fY$ the set of those with one fine cube in a site of $Y$ and the other in a site of $\Lc\setminus Y$. Then
\begin{equation}\label{eq:5.3}
F(Y)=\operatorname{int}F(Y)\sqcup\partial^fY\sqcup\bigl(F(Y)\cap\partial_2\Lc\bigr),\qquad|\partial^fY|\le\rbar^{2}|\partial Y|.
\end{equation}
Indeed, a plaquette of $F(Y)$ lies in $\partial_2\Lc$ iff it is a face of only one fine cube of $\Lc$, and otherwise its second cube lies in a site of $Y$ or of $\Lc\setminus Y$; a plaquette of $\partial^fY$ lies in the common coarse face of two adjacent sites $y\in Y$, $y'\notin Y$, which is dual to the bond $\{y,y'\}\in\partial Y$ and carries at most $\rbar^{2}$ fine plaquettes.

\begin{lemma}\label{lem:5.4}
There is a minimizer $\tilde\Sigma$ of Definition~\ref{def:2.11} ($\partial\tilde\Sigma=C$, $\#(\supp\tilde\Sigma\setminus\Ffr)=\AGp(C)$), determined by $(\XX,S)$, such that for every $Y$ of every piece partition
\begin{equation}\label{eq:5.4}
\supp\tilde\Sigma\cap F(Y)\subseteq\partial^fY\cup\bigl(\supp\Xi\cap\operatorname{int}F(Y)\bigr)\cup\partial_2\Lc .
\end{equation}
\end{lemma}

\begin{proof}
By acyclicity a minimizer is the flat rectangle plus a boundary, and we may change the boundary freely on hole cubes, where all plaquettes are free. Let $\Sigma$ be the first minimizer in a fixed order and $\Sigma-\Xi=\partial k$, $k$ an integer function on the fine cubes of $\Lc$ (as in Lemma~\ref{lem:5.3}; $k$ is unique by Box topology (ii)). We define $\tilde k:=k$ on the fine cubes of sites of $\Lc\setminus\XX$, $\tilde k:=0$ on those of sites of $\XX$, and $\tilde\Sigma:=\Xi+\partial\tilde k$. Then $\partial\tilde\Sigma=C$, and $\tilde\Sigma$, $\Sigma$ differ only on faces of fine cubes of sites of $\XX$, which lie in $F(\XX)\subseteq\Ffr$. It follows that $\#(\supp\tilde\Sigma\setminus\Ffr)=\AGp(C)$. For $p\in\operatorname{int}F(Y)$ both cubes of $p$ carry $\tilde k=0$, and therefore $\tilde\Sigma_p=\Xi_p$. Together with \eqref{eq:5.3}, this is \eqref{eq:5.4}, and $\tilde\Sigma$ does not involve $\mathcal Y$. \qed
\end{proof}

\begin{lemma}[$C$-rooted minimizer]\label{lem:5.5}
There is a minimizer $\Sigma^\circ$, determined by $(\XX,S)$ alone, with the following properties, in which $\supp\Sigma^\circ$ denotes its support in $\Lc_2\setminus\partial_2\Lc$ and $a_\Sigma:=|\supp\Sigma^\circ|$. We have $\supp\Sigma^\circ\subseteq\supp\tilde\Sigma$, and every component of $\supp\Sigma^\circ$ for edge adjacency contains a plaquette incident to an edge of $C$. Moreover, \eqref{eq:5.4} holds for $\Sigma^\circ$ for every piece partition, and $a_\Sigma\ge RT$.
\end{lemma}

\begin{proof}
In a fixed order, we replace each component of $\supp\tilde\Sigma\setminus\partial_2\Lc$ with no plaquette at an edge of $C$ by a free $2$-chain on $\partial_2\Lc$ with the same boundary, which exists because that boundary lies on $\partial\Lc$ (Box topology (iii)) (the plaquettes of $\supp\tilde\Sigma$ at an edge off $C\cup\partial\Lc$ are pairwise adjacent). Then \eqref{eq:5.4} passes to subsets, and $a_\Sigma\ge RT$ follows from Lemma~\ref{lem:5.3}. Since $(\XX,S)$ is a function of $D$ by Theorem~\ref{thm:3.20}(R1), so is $\Sigma^\circ$. \qed
\end{proof}

\begin{proposition}\label{prop:5.7}\label{cor:5.8}\label{lem:5.6}
For a piece partition $\mathcal Y$, call $Y\in\mathcal Y$ \emph{touched} if $\supp\Sigma^\circ\cap F(Y)\ne\emptyset$, and $b\in S$ \emph{touched} if $\supp\Sigma^\circ$ meets the coarse face dual to $b$. Then
\begin{equation}\label{eq:5.5}
a_\Sigma-\AGp(C)=\#(\supp\Sigma^\circ\cap\Ffr)\le\rbar^{2}\Bigl[\sum_{Y\ \mathrm{touched}}\bigl(2|\partial Y|+|Y|/\ell\bigr)+\#\{b\in S\ \mathrm{touched}\}\Bigr].
\end{equation}
In particular, if $\eta\in(0,1]$ and $\AGp(C)\le(1-\eta)RT$, then the right side of \eqref{eq:5.5} is at least $a_\Sigma-(1-\eta)RT\ge\eta RT$.
\end{proposition}

\begin{proof}
The identity holds as $\partial_2\Lc\subseteq\Ffr$ and $\Sigma^\circ$ is a minimizer, and $a_\Sigma\ge RT$ by Lemma~\ref{lem:5.5}. Off $\partial_2\Lc$ the free set is $F(\XX)\sqcup F_S$, and we bound the contributions of the two sets in turn. Each touched bond of $S$ contributes at most $\rbar^{2}$ plaquettes. A face of $\Xi$ carrying a plaquette of $\operatorname{int}F(Y)$ has both its sites in $Y$. Its coarse column perpendicular to the plane of $C$, which is disjoint from those of the other faces, therefore either lies in $Y$ or contains a bond of $\partial Y$, and in the first case it has $n_3\ge\ell$ sites, in view of Lemma~\ref{lem:3.3}(c). Consequently, $|\supp\Xi\cap\operatorname{int}F(Y)|\le\rbar^{2}(|\partial Y|+|Y|/\ell)$, and \eqref{eq:5.4} and \eqref{eq:5.3} show that a touched $Y$ contributes at most $\rbar^{2}(2|\partial Y|+|Y|/\ell)$. \qed
\end{proof}

\subsection{The Phase-1 Peierls estimate}
\label{ssec:5.3}

In this subsection we work with the Phase-1 output $(\XX^1,\Uf^1)$, which has no frozen bond, take $\cP$, $r'=2R_*+2$ and the pieces $Y_P$, $P\in\cP$, from Theorem~\ref{thm:3.21}, and compute $\Ffr^1$, $A^1:=\AGp(C)$ and $\Sigma^\circ$ for $(\XX^1,\emptyset)$. For a plaquette set $\Xi'$, let $Z(\Xi')$ be the set of sites with a fine cube having a face in $\Xi'$. Then $|Z(\Xi')|\le2|\Xi'|$, as a plaquette is a face of at most two fine cubes. We also write $Z(\Sigma^\circ):=Z(\supp\Sigma^\circ)$. A part $P$ is \emph{touched} if $Y_P$ is.

\begin{lemma}\label{lem:5.10}\label{lem:5.9}\label{eq:5.6}
$\#(\supp\Sigma^\circ\cap\Ffr^1)\le\rbar^{2}\sum_{P\ \mathrm{touched}}(2|\partial Y_P|+|Y_P|/\ell)\le\rbar^{2}\Cfree\sum_{P\ \mathrm{touched}}N(P)$, and every touched part satisfies $\dist(P,Z(\Sigma^\circ))\le R_*$.
\end{lemma}

\begin{proof}
The two inequalities follow from Proposition~\ref{prop:5.7}, applied to $\mathcal Y^1:=\{Y_P\}$, and from Theorem~\ref{thm:3.21}(a). For the last assertion, note that a touched plaquette is a face of a fine cube of a site of $Y_P$. Thus $Z(\Sigma^\circ)$ meets $Y_P\subseteq P^{+R_*}$ (Theorem~\ref{thm:3.21}(b)). \qed
\end{proof}

\begin{lemma}[Marks]\label{lem:5.11}
Put $q_m:=3(\rbar+1)^3\delta^{1/2}\le24r_0^3\delta^{1/2}$. For every finite set $W$ of sites, $\PD(W\subseteq\Mm)\le q_m^{|W|}$.
\end{lemma}

\begin{proof}
A plaquette marks at most two sites, and a site has at most $3(\rbar+1)\rbar^2\le3(\rbar+1)^3$ plaquettes on its fine cubes. If $W\subseteq\Mm$, a choice for each $x\in W$ of such a plaquette in $D$ takes at least $|W|/2$ distinct values, and there are at most $(3(\rbar+1)^3)^{|W|}$ choices. \qed
\end{proof}

\begin{lemma}[Animals]\label{lem:5.12}
In a graph of maximal degree $D_{\max}$, the number of connected vertex sets of size $n$ containing a given vertex is at most $(eD_{\max})^{n-1}$. The number of plaquette sets $\Xi'\subseteq\Lc_2$ with $|\Xi'|=a$ each of whose components (for edge adjacency) contains a plaquette incident to an edge of $C$ is at most $2^{4|C|}(2\kpl)^a$, with $\kpl:=12e$.
\end{lemma}

\begin{proof}
Section~\ref{S:proofs5} gives both counts in more detail. By \cite[Lemma~2.1 and the sentence after it, p.~141]{GK04} the first number is at most $\binom{Dn}{n-1}/n\le(eD)^{n-1}$, where $D:=D_{\max}$ denotes the maximal degree of the graph. For the second, we choose the first plaquette at an edge of $C$ of each component, the sizes and the components. There are $2^{4|C|}$ choices for the first plaquettes and $2^a$ for the sizes, and the components are counted by the first count with $D_{\max}=12$. \qed
\end{proof}

We put $\epsc:=8e\,(2R_*+1)^3(2r'+1)^3q_m$.

\begin{proposition}\label{prop:5.13}\label{eq:5.7}\label{eq:5.8}
Assume $\epsc\le\frac12$. For every finite set $Z$ of sites and every $u_0>0$, $\PD\bigl(\sum_{P\in\cP:\ \dist(P,Z)\le R_*}N(P)\ge u_0\bigr)\le2^{|Z|+1}\epsc^{u_0}$.
\end{proposition}

\begin{proof}
Since the partition is random, we take the union bound over all candidate families of mark sets, not over the partition, and we use only that parts are pairwise disjoint subsets of $\Mm$, $r'$-connected by definition (the graph on $P$ joining sites at distance at most $r'$ is connected, see Theorem~\ref{thm:3.21}). Thus the event lies in the union, over $u\ge u_0$ and all families $(P_1,\dots,P_m)$ of pairwise disjoint $r'$-connected site sets with $\dist(P_i,Z)\le R_*$ and $\sum_i|P_i|=u$, of the deterministic events $\{P_1\cup\dots\cup P_m\subseteq\Mm\}$, each of which has probability at most $q_m^u$ by Lemma~\ref{lem:5.11}. A family is fixed by the multiset of roots $z_i\in Z$ within $R_*$ of $P_i$, the sizes, marks $y_i\in P_i$ within $R_*$ of $z_i$, and the sets $P_i\ni y_i$. These sets are connected in a graph of degree less than $(2r'+1)^3$, and Lemma~\ref{lem:5.12} bounds their number. There are $2^{|Z|+m}$ values for the multiset of roots, $2^u$ for the sizes and $(2R_*+1)^3$ for each mark. Summing over $m\le u$, we bound the probability by $2^{|Z|}\sum_{u\ge u_0}\epsc^{\,u}\le2^{|Z|+1}\epsc^{u_0}$. \qed
\end{proof}

\begin{lemma}\label{lem:5.14}
Let $k\ge1$ and $j\in\{0,1,2,3\}$ be integers, put $c_T(k):=1+\log(2^{k+1}\kpl)$ and $c_T:=c_T(2)=2+\log96$, and let $\eta\in(0,1]$, $\eps\in(0,\frac12]$ and $F_*\ge1$. Let $E$ be an event with $E\subseteq\bigcup_{\Xi'}E_{\Xi'}$, the union over the plaquette sets $\Xi'$ of Lemma~\ref{lem:5.12} with $|\Xi'|\ge RT$, where $\PD(E_{\Xi'})\le2^{k|\Xi'|+j}\eps^{(|\Xi'|-(1-\eta)RT)/F_*}$. If $\eta|\log\eps|/F_*\ge c_T(k)$, then $\PD(E)\le2^{4|C|+j+2}e^{-RT}$, and every $X$ with $0\le X\le RT$ and $X<\eta RT$ off $E$ satisfies $\ED e^{\sigma X}\le e^{\sigma\eta RT+4|C|+4}$ for every $\sigma\in(0,1]$.
\end{lemma}

\begin{proof}
With $\lambda:=\log(2^{k+1}\kpl)$ and $\omega:=|\log\eps|/F_*$, Lemma~\ref{lem:5.12} bounds $\PD(E)$ by $2^{4|C|+j}$ times a geometric series in $a\ge RT$ of ratio $e^{-(\omega-\lambda)}\le e^{-1}$, and the hypothesis reads $\eta\omega\ge1+\lambda$. The moment bound then follows from $0\le X\le RT$, as we show in Section~\ref{S:proofs5}. \qed
\end{proof}

\begin{proposition}[Phase-1 output]\label{prop:5.15}
Assume $\epsc\le\frac12$ and $\etaT^{(1)}:=c_T\rbar^{2}\Cfree/|\log\epsc|\le1$. Then for every rectangle $C$ with $\min(R,T)\ge R_1(\beta)$ and every $\sigma\in(0,1]$, the two bounds of \eqref{eq:5.1} hold with $A^1$, $\etaT^{(1)}$ and $2^{4|C|+3}$ in place of $\AGp(C)$, $\etaT$ and $2^{4|C|+5}$.
\end{proposition}

\begin{proof}
We argue as in the proof of Theorem~\ref{thm:5.1}, without block families. On $E:=\{A^1\le(1-\etaT^{(1)})RT\}$, Proposition~\ref{prop:5.7} with $\mathcal Y^1$ and Lemma~\ref{lem:5.10} give $\sum N(P)\ge u_0(|\Xi'|):=(|\Xi'|-(1-\etaT^{(1)})RT)/(\rbar^{2}\Cfree)$ over the parts within $R_*$ of $Z(\Xi')$, where $\Xi':=\supp\Sigma^\circ$ is a function of $D$ by Lemma~\ref{lem:5.5}. For each deterministic $\Xi'$, Proposition~\ref{prop:5.13} bounds the probability of this event by $2^{2|\Xi'|+1}\epsc^{u_0(|\Xi'|)}$. Lemma~\ref{lem:5.14}, applied with $k=2$, $j=1$, $\eps=\epsc$, $F_*=\rbar^{2}\Cfree$ and $X:=RT-A^1\in[0,RT]$ (by Lemma~\ref{lem:5.3}), yields both bounds. \qed
\end{proof}

\begin{lemma}[Threshold $K_*$]\label{lem:5.16}
For $K>K_*=377.38$ there is a $\beta$-independent $\gamma_T(K)>0$ with
\begin{equation}\label{eq:5.9}
|\log\epsc|\ge\tau\Bigl(\frac{c_\delta(K)}{2K}-14.9662\Bigr)-O(\log\tau+\log r_0)\ge\gamma_T(K)\,\tau-O(\log\tau+\log r_0),
\end{equation}
and \eqref{eq:5.9} holds with $\eps''$ of \eqref{eq:5.epspp} in place of $\epsc$.
\end{lemma}

\begin{proof}
One mark costs $\delta^{1/2}=e^{-\tau c_\delta(K)/(2K)}$, and the root and part entropies of Proposition~\ref{prop:5.13} cost $e^{6\log R_*+O(\log r_0)}\le e^{14.9662\,\tau+O(\log\tau+\log r_0)}$, since $6\pi^2\vCb(0)<14.9662$. For the second inequality we use the lower bound on $c_\delta(K)$ of Lemma~\ref{lem:2.6} rather than its asymptotic form, and we take $\gamma_T(K):=(c_\delta^{\mathrm{sp}}(K)-1/(4a_*))/(2K)-14.9662$, which vanishes at $K=377.3635$ by the computation in Section~\ref{S:const5}. \qed
\end{proof}

\subsection{Phase 2: separators and leaves}
\label{ssec:5.4}

In Sections~\ref{ssec:5.4} to~\ref{ssec:5.6} we bound the Phase-2 cut bonds by the insulation around them. Since a bond can lie next to many cuts, the main tool is the Leaf Lemma below: $n$ cuts at one place and scale have $n+1$ pairwise disjoint sides, each of which crosses an annulus around that place and carries insulation of its own there.

\emph{Setting.} The cuts $j$, their times $t_j$, components $\KK_j$ and sides $U_j$ are those of Section~\ref{ssec:4.4}. We put $U_j':=\KK_j\setminus U_j$, $\mathrm{cut}_j:=\partial_{\Gp_{t_j}}U_j$ and $A_j:=|\mathrm{cut}_j|$. At a cut time the state is hull-closed without auxiliary material, so (S-face) and (S-plug) hold (Theorem~\ref{thm:4.rest}) and each cell at time $t_j$ lies in one block and has at least $c_1L_c^3$ sites (Theorem~\ref{thm:4.18}). A \emph{c-path} is a sequence of sites, consecutive ones joined by a $\Gp$-bond or lying in a common cell. A step of a c-path moves less than $2L_c$ in each coordinate. The \emph{box graph} is $\Lc$ with all coarse nearest-neighbour bonds (holes and frozen bonds ignored), with box-components and box paths, and $\partial(Y,W)$ denotes the set of box bonds from $Y$ to $W$. Bonds are \emph{square-adjacent} if they are edges of a common unit square. The distance of two bonds is the least $\ell^\infty$ distance of their ends, and a family of bond sets is \emph{$k_P$-linked} if the graph joining members with bonds at distance at most $k_P$ is connected.

\begin{definition}[Separators]\label{def:5.17}
Fix a cut $j$ and write $U:=U_j$, $U':=U_j'$. Let $U^\alpha$ be the box-components of $U$, and $\mathcal W(\alpha)$ the set of box-components $W$ of $\Lc\setminus U^\alpha$ with $W\cap U'\ne\emptyset$. Put $\Sep^*_j:=\bigcup_\alpha\bigcup_{W\in\mathcal W(\alpha)}\partial(U^\alpha,W)$, and call each $\partial(U^\alpha,W)$ a \emph{piece} of $\Sep^*_j$. Define $\Sep^{**}_j$ and its pieces in the same way with $U$ and $U'$ exchanged, and put $\Sep_j:=\Sep^*_j\cup\Sep^{**}_j$.
\end{definition}

\begin{lemma}[Separators]\label{lem:5.18}
For every cut $j$: (a) every bond of $\Sep^*_j$ joins a site of $U$ to a site outside $U$ and is a cut bond of $j$ or insulating at time $t_j$ (a bond of $\partial\XX_{t_j}$ or of $S_{t_j}$); the same holds for $\Sep^{**}_j$ with $U'$; (b) every piece is square-connected; (c) every piece contains a cut bond of $j$, distinct pieces of $\Sep^*_j$ are disjoint, and in particular $\Sep^*_j$ has at most $A_j$ pieces; the same holds for $\Sep^{**}_j$; (d) every cut bond of $j$ lies in a piece of $\Sep^*_j$ and in a piece of $\Sep^{**}_j$; (e) every box path from $U$ to $U'$ contains a bond of $\Sep^*_j$ and a bond of $\Sep^{**}_j$; (f) the pieces of $\Sep_j$ are $k_P$-linked.
\end{lemma}

\begin{proof}
Items (a) and (c) to (e) are elementary and are proved in Section~\ref{S:proofs5}. The separators are defined in the box graph, and the argument stays in the box graph except for (a) and the paths of (c) and (f).

(b) The set $\Pi:=\partial(U^\alpha,W)$ is a minimal set of box bonds separating a site $y\in U^\alpha$ from a site $x\in W$. It separates, because each box neighbour of $W$ outside $W$ lies in $U^\alpha$. It is minimal, because for $e=(u,v)\in\Pi$, $u\in U^\alpha$, $v\in W$, a box path inside $U^\alpha$ from $y$ to $u$, then $e$, then a box path inside $W$ from $v$ to $x$, meets $\Pi$ only in $e$. The unit squares generate the cycle space of the box graph over $\ZZ_2$ (Box topology (iv), Section~\ref{ssec:5.2}). We use Timár's lemma only in the box graph, since in the slit graph the unit squares need not generate the cycle space. By \cite[Lemma~1]{Ti}, for a minimal separating edge set $\Pi$ and a generating set $\mathcal C$ of the cycle space, each partition $(\Pi_1,\Pi_2)$ of $\Pi$ admits a cycle of $\mathcal C$ meeting both $\Pi_1$ and $\Pi_2$. Taking for $\mathcal C$ the unit squares, we obtain the square-connectedness of $\Pi$.

(f) A cell is $\Omega=Q\cap\hat\KK$ with $\hat\KK$ a component of $\Gp_{t_j}\cap\hat Q(Q)$. In particular, two sites of one cell are joined by a $\Gp$-path inside $\hat Q(Q)$, which by (G4) is a box of sides less than $k_P$, and this path lies in $\KK_j$ if the cell does. Bonds with all ends in $\hat Q(Q)$ are at distance at most $k_P-1$. A $\Gp_{t_j}$-path in $\KK_j$ enters or leaves $U^\alpha$ only through cut bonds, and such a bond with outer end $v$ lies in the piece $\partial(U^\alpha,W)$, where $W\ni v$ is a box-component of $\Lc\setminus U^\alpha$. Here $v\in\KK_j\setminus U=U'$, as a site of $U$ box-adjacent to $U^\alpha$ lies in $U^\alpha$.

We first link the pieces with the same $\alpha$ and different $W$. A $\Gp$-bond inside $U'$ joins box-adjacent sites outside $U^\alpha$ and therefore stays in one box-component of $\Lc\setminus U^\alpha$. For $W_1\ne W_2$ in $\mathcal W(\alpha)$ we take a c-path in $U'$ from $W_1\cap U'$ to $W_2\cap U'$, which exists because $U'$ is c-connected. Along it the box-component of $\Lc\setminus U^\alpha$ changes only at same-cell steps, between two sites of a cell $\Omega'\subseteq U'$ lying in components $W\ne\tilde W$. The $\Gp$-path inside $\hat Q$ joining them is a box path from $W$ to $\tilde W$ and hence meets $U^\alpha$. Its first entry crosses a cut bond of $\partial(U^\alpha,W)$ and its last exit one of $\partial(U^\alpha,\tilde W)$, since the path stays in $W$ before the entry and in $\tilde W$ after the exit. As both bonds lie in $\hat Q$, these two pieces contain bonds at distance at most $k_P$. Following the c-path, we conclude that all pieces $\partial(U^\alpha,W)$, $W\in\mathcal W(\alpha)$, are linked.

We now turn to pieces with different $\alpha$. A $\Gp$-bond inside $U$ stays in one $U^\alpha$. Thus, along a c-path in $U$, the box-component changes only at same-cell steps between sites of a cell $\Omega\subseteq U$ lying in $U^{\alpha_1}\ne U^{\alpha_2}$. The $\Gp$-path inside $\hat Q$ joining them leaves $U^{\alpha_1}$ and enters $U^{\alpha_2}$ through cut bonds of pieces of $\alpha_1$ and of $\alpha_2$, both in $\hat Q$. Each $\alpha$ has a piece, since the box-component of $\Lc\setminus U^\alpha$ containing a site of $U'$ lies in $\mathcal W(\alpha)$. Combining this with the first case, we find that all pieces of $\Sep^*_j$ are linked. The same holds for the pieces of $\Sep^{**}_j$, with $U$ and $U'$ exchanged, and by (d) a cut bond lies in a piece of each. Hence all pieces of $\Sep_j$ are linked. \qed
\end{proof}

For a site $x$ and $r>0$, $B_\infty(x,r)$ is the set of bonds with both ends within $\ell^\infty$ distance $r$ of $x$. Put $c_{\mathrm{sh}}:=c_1/216$, and let $\mathrm{side}_{\min}$ be the least coarse side of $\Lc$.

\begin{lemma}[Admissible cuts]\label{lem:5.22}\label{def:5.21}\label{lem:5.20}\label{lem:5.19}
For a site $x$ of $\Lc$ and $27L_c\le\rho\le\mathrm{side}_{\min}/24$, a cut $j$ is \emph{admissible at $(x,\rho)$} if (i) a cut bond of $j$ has both ends within distance $\rho/2+2$ of $x$; (ii) $U_j$ and $U_j'$ (at time $t_j$) each contain a site farther than $9\rho$ from $x$; (iii) $|\mathrm{cut}_j\cap B_\infty(x,9\rho)|\le c_{\mathrm{sh}}\rho L_c/2$. If $j$ is admissible at $(x,\rho)$, then $B_\infty(x,9\rho)$ contains at least $c_{\mathrm{sh}}\rho L_c$ bonds of each of $\Sep^*_j$ and $\Sep^{**}_j$.
\end{lemma}

\begin{proof}
Both sides of the cut cross about $\rho/(9L_c)$ disjoint shells around $x$, and in each shell the separator, which by Lemma~\ref{lem:5.18}(e) contains all box bonds between the sides it separates, is large by a box Cheeger inequality. We carry this out briefly here, and in full in Section~\ref{S:proofs5}. In a box $B=[n_1]\times[n_2]\times[n_3]\subset\ZZ^3$ (shell boxes need not be cubes), with $n:=\max_in_i$, each $A\subseteq B$ with $|A|\le|B|/2$ has at least $|A|/(2n)$ bonds to $B\setminus A$. To see this, we route all ordered pairs along coordinate-ordered paths. At most $n|B|$ of them use a given bond, and at least $|A|\,|B|/2$ join $A$ to $B\setminus A$. The \emph{shell boxes} form a tensor-product partition of $\Lc$ with sides in $[6L_c,12L_c)$, and the shell $\mathrm{Sh}_i$ is the union of the boxes at index distance $3i$, $3i+1$ or $3i+2$ from the box of $x$. A c-path moves less than $2L_c$ per step and cells lie in one block. It follows that each of the at least $\rho/(9L_c)$ disjoint, face-connected shells $\mathrm{Sh}_i$ with $\rho/(36L_c)+1\le i\le\rho/(4L_c)-1$ lies within $9\rho$ of $x$ and contains a whole cell of $U_j$ and one of $U_j'$, by (i) and (ii). In each shell, the Cheeger inequality in one shell box or in the union of two adjacent ones gives $(c_1/24)L_c^2$ bonds of $\Sep^*_j$ and as many of $\Sep^{**}_j$. Summing over the shells, we obtain $(\rho/(9L_c))(c_1/24)L_c^2=c_{\mathrm{sh}}\rho L_c$ bonds of each. \qed
\end{proof}

\begin{lemma}[Leaf Lemma]\label{lem:5.24}\label{lem:5.23}
Let $n\ge1$ and let $j_1,\dots,j_n$ be distinct cuts admissible at $(x,\rho)$. Let $I$ be the set of bonds $b\in B_\infty(x,9\rho)$ such that for some $i$, $b\in\Sep_{j_i}$, $b$ is not a cut bond of $j_i$, and $b$ is insulating at time $t_{j_i}$. Then $|I|\ge(n+1)\,c_{\mathrm{sh}}\rho L_c/4$.
\end{lemma}

\begin{proof}
The $2n$ sides $U_{j_i}$, $U_{j_i}'$ (at the respective times) are pairwise disjoint or nested (Lemma~\ref{lem:3.18}(b)) and distinct, since the component of a later cut lies in one side of an earlier cut or outside its component. Hence the minimal sides, the \emph{leaves}, are pairwise disjoint. We show that there are at least $n+1$ of them by adding the cuts in firing order. When a cut $j$ is added, its two sides are leaves, since an earlier side contains $\KK_j$ or is disjoint from it. An earlier leaf stops being one only if it contains $\KK_j$, while the earlier sides containing $\KK_j$ form a chain, with at most one minimal element. Thus each cut adds two leaves and removes at most one. For a leaf $L=U_j$ (resp.\ $U_j'$), Lemma~\ref{lem:5.22} gives at least $c_{\mathrm{sh}}\rho L_c$ bonds of $\Sep^*_j$ (resp.\ $\Sep^{**}_j$) in $B_\infty(x,9\rho)$, of which at most $c_{\mathrm{sh}}\rho L_c/2$ are cut bonds of $j$ by Lemma~\ref{lem:5.22}(iii), while Lemma~\ref{lem:5.18}(a) shows that the others are insulating at $t_j$ and have an end in $L$, so that they lie in $I$. Since a bond has two ends and the leaves are disjoint, a bond of $I$ is counted for at most two leaves. \qed
\end{proof}

\subsection{Phase 2: charges and domination}
\label{ssec:5.5}

A cut fires only on a fragmented split, and the scale cost of such a split exceeds its area. We spread the area of the cut over its thin clusters and, using the Leaf Lemma, charge each thin cluster to insulation of its own, at a rate $\epsD$ that is doubly exponentially small.

\emph{Scales.} For the scales $s=2^jL\ge\RNF$ of Definition~\ref{def:3.17}, which are lattice lengths, we put $\rho(s):=s/(128\rbar)$, a length in coarse units. Points of two sites at coarse distance $d$ are at lattice Euclidean distance at most $\sqrt3\,\rbar(d+1)$. Consequently, (U1) two cut bonds whose ends are pairwise within coarse distance $d$ lie in one $s$-cluster if $\sqrt3\,\rbar(d+1)\le s$, and (U2) cut bonds of different $s$-clusters have all their ends at coarse distance greater than $s/(\sqrt3\rbar)-1>73\rho(s)-1$. The \emph{charge boxes} at scale $s$ partition the coarse lattice into cubes $Q$ of side $\lceil\rho(s)\rceil$ with centres $x_Q$.

\emph{Thin clusters, good scales.} Fix a cut $j$ and a scale $s$. An $s$-cluster $c$ of $\mathrm{cut}_j$ is \emph{thin} if it contains a bond with an end $x_c$ such that $|c\cap B_\infty(x_c,10\rho(s))|\le c_{\mathrm{sh}}\rho(s)L_c/2$, and \emph{fat} otherwise. For a thin cluster we fix such a bond and end $x_c$. The scale $s$ is \emph{good} for $j$ if $\varphi_s\ge2$ and at least half of the $s$-clusters are thin. Let $\varphi_s^{\mathrm{thin}}$ be the number of thin $s$-clusters.

\begin{lemma}[Charges]\label{lem:5.25}
Assume $\RNF\ge\max\bigl(3456\rbar L_c,\ 2048\rbar L_c/(c_{\mathrm{sh}}\thetaNF\Funit)\bigr)$. For every cut $j$: (a) $J'_j:=\sum_{s\ \mathrm{good}}(\varphi_s-1)\ge J(U_j)/2$; (b) with $q_j(s,c):=A_j(\varphi_s-1)/(J'_j\varphi_s^{\mathrm{thin}})$ ($s$ good, $c$ a thin $s$-cluster), $\sum_{s\ \mathrm{good}}\sum_{c\ \mathrm{thin}}q_j(s,c)=A_j$ and $q_j(s,c)<4L_c^2/(\thetaNF\Funit)$; (c) for good $s$ and thin $c$, with $Q\ni x_c$ the charge box, $j$ is admissible at $(x_Q,\rho(s))$ and $x_{c'}\notin Q$ for every other thin $s$-cluster $c'$.
\end{lemma}

\begin{proof}
(a) A fat cluster has more than $c_{\mathrm{sh}}\rho L_c/2$ bonds, $\rho=\rho(s)$, and clusters are disjoint subsets of $\mathrm{cut}_j$. Consequently, at a scale with $\varphi_s\ge2$ that is not good, $\varphi_s-1<2\varphi_s^{\mathrm{fat}}<4A_j/(c_{\mathrm{sh}}\rho L_c)=512\rbar A_j/(c_{\mathrm{sh}}sL_c)$. Summing over dyadic $s\ge\RNF$ gives less than $1024\rbar A_j/(c_{\mathrm{sh}}\RNF L_c)\le J(U_j)/2$, by the assumption on $\RNF$ and the inequality $J(U_j)>\thetaNF\Funit A_j/L_c^2$, which holds because the split is fragmented.

(b) We have $\sum_cq_j(s,c)=A_j(\varphi_s-1)/J'_j$, and these sums add up to $A_j$ over good $s$. At a good scale $\varphi_s-1<\varphi_s\le2\varphi_s^{\mathrm{thin}}$, so $q_j(s,c)<2A_j/J'_j\le4A_j/J(U_j)<4L_c^2/(\thetaNF\Funit)$.

(c) Take for $x_Q$ a central site of $Q\cap\Lc$ (nonempty, as it contains $x_c$; for instance the centre of $Q$ moved coordinatewise into $\Lc$), so that $x_Q\in Q$ and $|x_c-x_Q|_\infty\le\lceil\rho\rceil/2\le\rho/2+1$, $\rho=\rho(s)$. The numerical inequalities used in (c), and the range $27L_c\le\rho\le\mathrm{side}_{\min}/24$ of Lemma~\ref{lem:5.22}, are established in Section~\ref{S:const5}. (i) The chosen bond of $c$ has both ends within $\rho/2+2$ of $x_Q$. (ii) Since $\varphi_s\ge2$, there is a cut bond in another $s$-cluster, with one end in $U_j$ and one in $U_j'$. By (U2) its ends are farther than $9\rho$ from $x_Q$. (iii) $B_\infty(x_Q,9\rho)\subseteq B_\infty(x_c,10\rho)$, and by (U1) all cut bonds of $j$ in $B_\infty(x_c,10\rho)$ lie in $c$, which is thin. Finally, if $x_c,x_{c'}\in Q$, the two chosen bonds lie in one cluster by (U1). \qed
\end{proof}

\emph{Classes.} For a bond set $Y$, $\Bfr(Y)$ is the set of blocks containing an end of a bond of $Y$. The \emph{classes} $\mathcal J$ are the connected components of the graph on cuts with $j\sim j'$ if $\Bfr(\Sep_j)\cap\Bfr(\Sep_{j'})\ne\emptyset$. We put $\mathfrak X(\mathcal J):=\bigcup_{j\in\mathcal J}\Sep_j$, $\Bfr(\mathcal J):=\Bfr(\mathfrak X(\mathcal J))$ and $A(\mathcal J):=\sum_{j\in\mathcal J}A_j$. Let $\Scut$ be the set of all bonds frozen by the cut rule and $S_1^{(2)}$ the set of bonds frozen by seal steps in Phase 2, and put $\mathrm{ins}^1:=\partial\XX^1$. The \emph{payable bonds} form $\Bpay:=\mathrm{ins}^1\cup\Scut\cup S_1^{(2)}\cup\partial\XX_{\mathrm{final}}\cup\bigcup_\Delta\partial\Delta$, the last union over the cells removed in Phase 2. We denote by $m_q$ the number of marks of a block $q$, and by $h(\Bfr)$ the number of sites in the blocks of $\Bfr$ that are hull material at some time of Phase 2. By Proposition~\ref{prop:4.4} a bond is frozen at most once, so each cut bond belongs to exactly one cut.

\begin{lemma}[Classes]\label{lem:5.26}
(a) Distinct classes have disjoint $\mathfrak X$'s and disjoint $\Bfr$'s. (b) The cut bonds of every $j\in\mathcal J$ lie in $\mathfrak X(\mathcal J)$, and every cut bond lying in $\mathfrak X(\mathcal J)$ belongs to a cut of $\mathcal J$. Moreover, $|\mathfrak X(\mathcal J)\cap\Scut|=A(\mathcal J)$. (c) $\Bfr(\mathcal J)$ is a union of at most $2A(\mathcal J)$ sets, each $*$-connected in the block lattice ($26$-adjacency), which are linked at block distance at most $k_b:=\lceil k_P/L_c\rceil+2$. (d) Every bond of $\mathfrak X(\mathcal J)\setminus\Bpay$ is, at some time of Phase 2, a hole face of a site $y\notin\XX^1$ that is hull material at that time and lies in a block of $\Bfr(\mathcal J)$. So $|\mathfrak X(\mathcal J)\setminus\Bpay|\le6\,h(\Bfr(\mathcal J))\le12L_c\sum_{q\in\Bfr(\mathcal J)}m_q$.
\end{lemma}

\begin{proof}
(a) A common bond or block relates two cuts.

(b) The first claim follows from Lemma~\ref{lem:5.18}(d). For the second, a cut bond of $j''$ in $\Sep_j$ lies in $\Sep_{j''}$, so $j''\sim j$. The equality in (b) follows from the two claims.

(c) In view of Lemma~\ref{lem:5.18}(c), the set $\Sep_j$ has at most $2A_j$ pieces, and Lemma~\ref{lem:5.18}(b), (f) shows that they are square-connected, hence with $*$-connected blocks, and $k_P$-linked. Bonds at distance $\le k_P$ have blocks at block distance $\le k_P/L_c+1\le k_b$, and related cuts share a block, which gives the linking asserted in (c).

(d) Let $b\in\Sep_j\setminus\Bpay$, $j\in\mathcal J$. Cut bonds and bonds frozen at $t_j$ lie in $\Scut\cup S_1^{(2)}\subseteq\Bpay$, since by Theorem~\ref{thm:4.30} the bonds frozen in Phase 1 are converted. It follows by Lemma~\ref{lem:5.18}(a) that $b=(y,u)$ is a hole face at $t_j$, with $y\in\XX_{t_j}$ and $u\in\KK_j\subseteq\Gp_{t_j}$. If $y\in\XX^1$, then $b\in\partial\XX^1=\mathrm{ins}^1$, as $u\in\Gp_{t_j}\subseteq\Gp_{\XX^1}$, where the inclusion holds because $\Gp$ only shrinks. If $y$ lies in a cell $\Delta$ removed in Phase 2 before $t_j$, then $b\in\partial\Delta$, as $u\notin\Delta\subseteq\XX_{t_j}$. Otherwise $y\notin\XX^1\cup R_{t_j}\supseteq\Mm\cup\Uf^1\cup R_{t_j}=B_{t_j}$, where $R_t$ is as in Section~\ref{ssec:4.4} and we use that Phase 2 fills only by removal. Hence, in this case, $y$ is hull material at $t_j$, also if it lies in a cell removed after $t_j$. Its block contains an end of $b$ and hence lies in $\Bfr(\mathcal J)$. Since a site carries six bonds, this gives the first inequality in (d). For the last bound, let $q\not\subseteq\Uf^1$ be a block. If its removal cell is never removed, the hull material $(\XX_t\setminus B_t)\cap q=(\XX_t\cap q)\setminus\Mm$ only grows in $t$. If the cell is removed at time $t_\Delta$, all hull material of $q$ before $t_\Delta$ is hull material just before $t_\Delta$, because removal cells are unions of blocks by Lemma~\ref{lem:3.3}(c). Corollary~\ref{cor:4.10} at the final time, or just before the removal, gives fewer than $2L_cm_q$ sites per block. \qed
\end{proof}

We call a hole face of (d) \emph{transient} if its other end is absorbed later. There can be $L_c$ times as many transient faces as local sources. We absorb them into $\epsD$ and attach them to the contour only through blocks.

\begin{proposition}\label{prop:5.27}\label{eq:5.10}
Under the assumption on $\RNF$ of Lemma~\ref{lem:5.25}, for every set $\mathcal J$ of cuts, $A(\mathcal J)\le\epsD|\mathfrak X(\mathcal J)|$ with $\epsD:=3.2768\cdot10^{7}\rbar L_c/(c_{\mathrm{sh}}\thetaNF\Funit\RNF)$, and if $\mathcal J$ is a class and $\epsD\le\frac12$,
\begin{equation}\label{eq:5.11}
A(\mathcal J)\le2\epsD\Bigl(|\mathfrak X(\mathcal J)\cap\Bpay\setminus\Scut|+12L_c\sum_{q\in\Bfr(\mathcal J)}m_q\Bigr).
\end{equation}
\end{proposition}

\begin{proof}
We assign the charge $q_j(s,c)$ of Lemma~\ref{lem:5.25} to the pair $(Q,s)$ with $Q\ni x_c$. Fix $(Q,s)$ and let $j_1,\dots,j_n$ be the cuts of $\mathcal J$ with a charge at $(Q,s)$. By Lemma~\ref{lem:5.25}(b), (c), each of them has exactly one charge there, of size less than $4L_c^2/(\thetaNF\Funit)$, and is admissible at $(x_Q,\rho(s))$. The Leaf Lemma gives $I_Q\subseteq B_\infty(x_Q,9\rho(s))\cap\mathfrak X(\mathcal J)$ with $|I_Q|\ge(n+1)c_{\mathrm{sh}}\rho(s)L_c/4$. Hence the charge at $(Q,s)$ is at most $4nL_c^2/(\thetaNF\Funit)\le16L_c\,|\mathfrak X(\mathcal J)\cap B_\infty(x_Q,9\rho(s))|/(c_{\mathrm{sh}}\thetaNF\Funit\rho(s))$. A bond lies in $B_\infty(x_Q,9\rho(s))$ for at most $20^3$ charge boxes $Q$ of one scale, and $\sum_s1/\rho(s)\le256\rbar/\RNF$ over dyadic $s\ge\RNF$. Summing over $(Q,s)$, we obtain $A(\mathcal J)\le20^3\cdot16\cdot256\,\rbar L_c|\mathfrak X(\mathcal J)|/(c_{\mathrm{sh}}\thetaNF\Funit\RNF)=\epsD|\mathfrak X(\mathcal J)|$. For \eqref{eq:5.11}, we write $\mathfrak X(\mathcal J)$ as the disjoint union of $\mathfrak X\cap\Scut$, $\mathfrak X\cap\Bpay\setminus\Scut$ and $\mathfrak X\setminus\Bpay$. The first has $A(\mathcal J)$ elements and the last is bounded by Lemma~\ref{lem:5.26}(b), (d). Thus $A\le\epsD(A+R)$ with $R$ the bracket of \eqref{eq:5.11}, and $A\le\epsD R/(1-\epsD)\le2\epsD R$. \qed
\end{proof}

\subsection{Phase 2: responses}
\label{ssec:5.6}

Cuts change $\Gp$ only on their cut bonds, which cross block faces and carry the hull weight $2-\eta_w$. The hull, removal and seal steps therefore respond to cuts as they respond to marks, and we bound the responses locally by the marks, the cut bonds and the bonds of $\mathrm{ins}^1=\partial\XX^1$. Since each frozen bond is a grid bond and supports no hull material (Corollary~\ref{cor:4.10}), a response creates no new source, so the bound on the responses need not be iterated. Section~\ref{S:const5} collects the constants of this subsection and the next, with the arithmetic behind them. Let $m(Z)$ be the number of marks in $Z$, and let $\ncut(Z)$ and $n_{\mathrm{seal}}(Z)$ be the numbers of cut bonds and of bonds frozen by seal steps in Phase~2 (converted ones included) with both ends in $Z$. For a removal cell $\Delta_0$, $\mathcal N(\Delta_0)$ is the set of the at most $27$ removal cells within index distance $1$ of $\Delta_0$, with union $U(\Delta_0)$, and $\mathcal N_2(\Delta_0)$ is the set of the at most $125$ removal cells within index distance $2$ of $\Delta_0$, with union $U_2(\Delta_0)$.

\emph{Response units.} These are (1) the bonds of $S_1^{(2)}$, (2) the bonds of $\partial\XX_{\mathrm{final}}\setminus\mathrm{ins}^1$, that is, the hole faces created in Phase 2 and present at the end, and (3) the pairs $(\Delta,b)$ with $\Delta$ a cell removed in Phase 2 and $b\in\partial\Delta$, and transient hole faces are not units. Each unit is placed at its bond, and $\mathrm{units}(Z)$ denotes the number of units with an end in $Z$. We assign each unit to the removal cell of the first of its ends in $\Lc$, in a fixed order (a bond of a unit of type (3) may leave $\Lc$). Each bond of $\Bpay\setminus\Scut$ lies in $\mathrm{ins}^1$ or is the bond of a unit.

\begin{lemma}[Units]\label{lem:5.31}
For every removal cell $\Delta_0$, $\mathrm{units}(\Delta_0)\le C_a\,m(U_2(\Delta_0))+C_b\,\ncut(U_2(\Delta_0))+C_c\,|\mathrm{ins}^1\cap E(U_2(\Delta_0))|$, with $C_a:=4362+5808/c_s$, $C_b:=2+4/c_s$ and $C_c:=1+2/c_s$.
\end{lemma}

\begin{proof}
We bound the units near $\Delta_0$ by the seal count or, through Lemma~\ref{lem:4.21}, by comparison with the Phase-1 output. Let $U:=U(\Delta_0)$, $m:=m(U_2(\Delta_0))$, $n:=\ncut(U)$ and $i:=|\mathrm{ins}^1\cap E(U)|$. A seal bond with an end in a block $q$ lies on one of the six faces of $q$, and by Proposition~\ref{prop:4.sealcount} the face shared with $q'$ carries at most $121(m_q+m_{q'})$ of them. Summing over the blocks of $\Delta_0$, resp.\ of $U$, we find that at most $1452\,m$ units of type (1) have an end in $\Delta_0$ and that $n_{\mathrm{seal}}(U)\le1452\,m$. A unit $(\Delta,b)$ with an end in $\Delta_0$ has $\Delta\in\mathcal N(\Delta_0)$, and by Lemma~\ref{lem:4.21}(b) the units of type (3) of the cells of $\mathcal N(\Delta_0)$ number less than $u_3:=c_s^{-1}(i+2n_{\mathrm{seal}}(U)+2n)$. We turn to the units of type (2). For $1\le r\le L_c$ the boxes $N_r:=\{x:\dist(x,\Delta_0)\le r\}$ lie in $U$ and have disjoint outer layers. A final hole face with an end in $\Delta_0$ lies in $E(N_r)$ and has exactly one end in $\XX_\infty$, so it is counted in $\mathcal A_2(E(N_r);\infty)$, which \eqref{eq:4.I5} bounds by $\mathrm{Ins}_{N_r}(\XX^1\cup R_\infty)+\mathrm{cut}_{N_r}$. A bond of $E(N_r)$ with exactly one end in $\XX^1\cup R_\infty$ is a bond of $\mathrm{ins}^1$ or the bond of a unit of type (3) of a cell of $\mathcal N(\Delta_0)$, and a bond of $\bar S_\infty\cap E(N_r)$ with no end in $\XX^1$ was frozen in Phase~2. Hence $\mathrm{Ins}_{N_r}(\XX^1\cup R_\infty)\le i+u_3+2n_{\mathrm{seal}}(U)+2n$. The sites counted in $\mathrm{cut}_{N_r}$ are hull material in the outer layer of $N_r$, each with at most three bonds leaving $N_r$, and the blocks of $U$ hold fewer than $2L_cm$ sites of hull material (Corollary~\ref{cor:4.10}). Consequently, $\mathrm{cut}_{N_r}\le6m$ for some $r$. Adding these bounds, we obtain $\mathrm{units}(\Delta_0)\le1452m+2u_3+i+2n_{\mathrm{seal}}(U)+2n+6m$. Inserting $n_{\mathrm{seal}}(U)\le1452\,m$ here and in the bound on the units of type (3) gives the claim. \qed
\end{proof}

\begin{lemma}\label{lem:5.32}
There are weights $\varpi(u,z)\ge0$, for $u$ a response unit and $z$ a \emph{source} (a mark, a cut bond or a bond of $\mathrm{ins}^1$), with $\varpi(u,z)=0$ if $\dist(u,z)>\Rresp:=10\ell$ and $\sum_z\varpi(u,z)=1$ for every $u$, the weights being functions of $D$, such that every mark receives $\sum_u\varpi(u,z)\le125C_a$, every cut bond at most $125C_b$, and every bond of $\mathrm{ins}^1$ at most $125C_c$.
\end{lemma}

\begin{proof}
Since Lemma~\ref{lem:5.31} bounds the units of a cell by the sources around it, we spread each unit over those sources. For a unit $u$ assigned to $\Delta_0$ put $\varpi(u,z):=c_z/D(\Delta_0)$ for the sources $z$ of $U_2(\Delta_0)$, that is, the marks in $U_2(\Delta_0)$, the cut bonds and the bonds of $\mathrm{ins}^1$ in $E(U_2(\Delta_0))$. Here $c_z=C_a$, $C_b$ or $C_c$ according to the kind of $z$, and $D(\Delta_0)$ is the right side of Lemma~\ref{lem:5.31}. Then $\sum_z\varpi(u,z)=1$, and $D(\Delta_0)\ge\mathrm{units}(\Delta_0)>0$ if a unit is assigned to $\Delta_0$. A source lies in $U_2(\Delta)$ for at most $125$ cells $\Delta$ and therefore receives at most $125c_z$. A unit assigned to $\Delta_0$ has an end in $\Delta_0$, a source of $U_2(\Delta_0)$ lies in a cell at index distance at most $2$, and cells have sides less than $3\ell$. Hence $\dist(u,z)\le9\ell+1\le\Rresp$. \qed
\end{proof}

The constant $\Cresp:=375C_a+125C_c\Cst$ is fixed before $K$. A part $P$ owns the bonds of $\partial Y_P$, which by Theorem~\ref{thm:3.21}(a) are at most $\Cst N(P)$ bonds of $\mathrm{ins}^1$. The marks and the $\mathrm{ins}^1$ bonds of $P$ therefore receive at most $(125C_a+125C_c\Cst)N(P)\le\Cresp N(P)$, and a cut bond receives at most $125C_b\le\Cresp$.

\subsection{Counting and the proof of Theorem~\ref{thm:5.1}}
\label{ssec:5.7}

In this subsection $(\XX,S)$ is the Phase-2 output, with $\Ffr$ and the minimizer $\Sigma^\circ$ of Lemma~\ref{lem:5.5} computed for it, and we write $\Xi':=\supp\Sigma^\circ$ and $a:=a_\Sigma=|\Xi'|\ge RT$. Let $R':=R_*+2\Rresp$ and
\begin{equation}\label{eq:5.epspp}
\eps'':=8e\,(2R'+4L_c+1)^3\,(2r'+1)^3\,q_m .
\end{equation}
The hole set only grows, as Proposition~\ref{prop:4.4} shows, and hence $\XX^1\subseteq\XX$.

As pieces we take the sets $Y_P$, $P\in\cP$, of Theorem~\ref{thm:3.21}(a), the sets $Y_\Delta:=\Delta\setminus\XX^1$ for the cells $\Delta$ removed in Phase 2, which lie in $\XX$, and the sets $Y_{\Delta'}:=(\XX\setminus\XX^1)\cap\Delta'$ for the removal cells $\Delta'$ not removed in Phase 2. These sets partition $\XX$. A piece is touched if $\Xi'$ meets its $F(\cdot)$, and a part $P$ is touched if $Y_P$ is. The cell of a touched $Y_\Delta$ or $Y_{\Delta'}$ is a \emph{touched cell}. Let $u_{\mathrm{touch}}$ be the number of response units that have an end in a touched cell or are of type (1) with a coarse face containing a plaquette of $\Xi'$ (each unit counted once), $c_{\mathrm{touch}}$ the number of bonds of $S\cap\Scut$ whose coarse face contains a plaquette of $\Xi'$, and $i_{\mathrm{touch}}$ the number of bonds of $\mathrm{ins}^1$ with an end in a touched cell.

\begin{proposition}\label{prop:5.34}
$\#(\Xi'\cap\Ffr)\le\rbar^{2}\bigl[\Cfree\sum_{P\ \mathrm{touched}}N(P)+20(u_{\mathrm{touch}}+c_{\mathrm{touch}}+i_{\mathrm{touch}})+(26L_c/\ell)\sum_{\Delta'\ \mathrm{touched}}m(\Delta')\bigr]$, the last sum over the touched cells $\Delta'$.
\end{proposition}

\begin{proof}
We apply Proposition~\ref{prop:5.7} to the pieces defined above. Roughly speaking, the removal cells cut the Phase-2 hole material into pieces of diameter less than $9\ell$, each bounded by Phase-1 insulation or by response units and entering with its volume divided by $\ell$. A touched part costs at most $\Cfree N(P)$, as in Lemma~\ref{lem:5.10}. A touched $Y_\Delta$ costs $2|\partial Y_\Delta|+|Y_\Delta|/\ell$, with $|Y_\Delta|/\ell<27\ell^2\le4.5|\partial\Delta|$. A bond of $\partial Y_\Delta$ either lies in $\partial\Delta$, in which case it is the bond of a unit $(\Delta,b)$ of type (3), or joins $\XX^1\cap\Delta$ to $\Delta\setminus\XX^1$, in which case it is a bond of $\mathrm{ins}^1$ with both ends in $\Delta$.

A touched $Y:=Y_{\Delta'}$ costs $2|\partial Y|+|Y|/\ell$. A bond $(y,v)$ of $\partial Y$, $y\in Y$, is of one of the following four types. (a) $v\in\Gp$, and then the bond is a final hole face with no end in $\XX^1$, the bond of a unit of type (2). (b) $v\in\XX^1$, so the bond lies in $\mathrm{ins}^1$. (c) $v$ lies in a cell $\Delta$ removed in Phase 2, so $(\Delta,(y,v))$ is a unit of type (3). (d) $v\in(\XX\setminus\XX^1)\cap\Delta''$ for another cell $\Delta''$ not removed in Phase 2. For a bond of type (d), we walk from $y$ along its coordinate line into $\Delta'$. If the walk leaves $Y$ inside $\Delta'$, it leaves through a bond from $Y$ to $\Delta'\setminus Y\subseteq\Gp\cup\XX^1$, of type (a) or (b) with both ends in $\Delta'$, and distinct walks end at distinct bonds. Otherwise the walk crosses $\Delta'$ inside $Y$, through at least $\ell$ sites, and we charge $1/\ell$ to each of them. Since a site lies on three lines, each with at most two type-(d) bonds at its ends, it receives at most $6/\ell$. Hence $|\partial Y|$ is at most the number of bonds of types (a) to (c) with an end in $Y$, plus those of types (a), (b) inside $\Delta'$, plus $6|Y|/\ell$. Phase 2 fills only by removal, so $Y$ is hull material and Corollary~\ref{cor:4.10} at the final time gives $|Y|<2L_cm(\Delta')$.

Since the Phase-1 output has no frozen bond, a touched bond of $S$ lies in $S_1^{(2)}$ or in $S\cap\Scut$. In the first case it is a unit of type (1), counted in $u_{\mathrm{touch}}$, and in the second it is counted in $c_{\mathrm{touch}}$. Each unit or bond of $\mathrm{ins}^1$ charged above has an end in the touched cell charging it, so it is counted in $u_{\mathrm{touch}}$ or $i_{\mathrm{touch}}$, and the coefficients add up to at most $20$. \qed
\end{proof}

\emph{The attached family.} Let $\mathcal T$ be the set of classes with a cut bond counted in $c_{\mathrm{touch}}$, and $\mathcal F$ the smallest set of classes containing $\mathcal T$ with the closure property: if $\varpi(u,z)>0$ for a cut bond $z$ of a class $\mathcal J'$, where (i) $u$ is a unit counted in $u_{\mathrm{touch}}$, or (ii) the bond of the unit $u$ lies in $\mathfrak X(\mathcal J)$ for some $\mathcal J\in\mathcal F$, then $\mathcal J'\in\mathcal F$. We put $\Bfr_{\mathcal F}:=\bigsqcup_{\mathcal J\in\mathcal F}\Bfr(\mathcal J)$ and $A_{\mathcal F}:=\sum_{\mathcal J\in\mathcal F}A(\mathcal J)$. Let $Z_{\mathcal F}$ be the set of sites of the blocks of $\Bfr_{\mathcal F}^{+1}$ (the blocks at block distance at most $1$ from $\Bfr_{\mathcal F}$) and $\Qatt$ the set of parts $P$ with $\dist(P,Z(\Xi')\cup Z_{\mathcal F})\le R'$, and put $N_{\mathcal Q}:=\sum_{P\in\Qatt}N(P)$. In Lemma~\ref{lem:5.35} and Proposition~\ref{prop:5.36} we assume the condition
\begin{equation}\label{eq:5.16}
8\epsD(1+8L_c)(C_\star'+3\Cresp+6)(48\Cresp+2)\le1,
\end{equation}
which holds for $\beta\ge\beta_0(K)$, as $\epsD$ is doubly exponentially small.

\begin{lemma}[Attached family]\label{lem:5.35}
(a) the units counted in $u_{\mathrm{touch}}$ and the units whose bond lies in $\bigcup_{\mathcal F}\mathfrak X(\mathcal J)$ number at most $\Cresp(N_{\mathcal Q}+A_{\mathcal F})$ each, so at most $2\Cresp(N_{\mathcal Q}+A_{\mathcal F})$ together; (b) $A_{\mathcal F}\le8\epsD(1+8L_c)(C_\star'+3\Cresp+6)\,N_{\mathcal Q}\le N_{\mathcal Q}/(48\Cresp+2)$; (c) $c_{\mathrm{touch}}\le A_{\mathcal F}$, $i_{\mathrm{touch}}\le C_\star'N_{\mathcal Q}$ and $|\Bfr_{\mathcal F}|\le C_{\Bfr}N_{\mathcal Q}$ ($C_{\Bfr}$ of Section~\ref{S:const5}); (d) $\Bfr_{\mathcal F}$ has $m\le2A_{\mathcal F}$ $*$-components.
\end{lemma}

\begin{proof}
(a) A unit of the first family has an end within $3\ell$ of $Z(\Xi')$, since a touched cell meets $Z(\Xi')$ and has sides less than $3\ell$, and since a unit of type (1) whose face meets $\Xi'$ has both ends in $Z(\Xi')$. A unit of the second family has an end in $Z_{\mathcal F}$. Thus, by Lemma~\ref{lem:5.32}, a source $z$ with $\varpi(u,z)>0$ lies within $\Rresp+3\ell+1$ of $Z(\Xi')\cup Z_{\mathcal F}$. If such a source is a mark, it is a mark of a part of $\Qatt$. If it is a cut bond, it belongs to a class of $\mathcal F$ by the closure property, and there are at most $A_{\mathcal F}$ such bonds, by Lemma~\ref{lem:5.26}(b). If it is a bond of $\mathrm{ins}^1=\bigsqcup_P\partial Y_P$, its owner $P$ has a site of $Y_P\subseteq P^{+R_*}$ within $\Rresp+3\ell+2$ of $Z(\Xi')\cup Z_{\mathcal F}$ and so lies in $\Qatt$. As each unit has total weight at least $1$, the loads stated after Lemma~\ref{lem:5.32} imply that each family has at most $\Cresp N_{\mathcal Q}+125C_bA_{\mathcal F}\le\Cresp(N_{\mathcal Q}+A_{\mathcal F})$ elements.

(b) We sum \eqref{eq:5.11} over $\mathcal J\in\mathcal F$. The bonds of $\mathfrak X(\mathcal J)\cap\Bpay\setminus\Scut$ are bonds of $\mathrm{ins}^1$ or of units. A bond of $\mathrm{ins}^1$ lies in $\partial Y_P$ for one part $P$, which has an end of it in $Y_P\subseteq P^{+R_*}$, so $P$ lies within $R_*+1$ of $Z_{\mathcal F}$ and belongs to $\Qatt$. A part owns at most $C_\star'N(P)$ such bonds, by Theorem~\ref{thm:3.21}(a). The units are bounded by (a), and the marks in blocks of $\Bfr_{\mathcal F}$ belong to parts of $\Qatt$. Therefore
\[
A_{\mathcal F}\le2\epsD\bigl[C_\star'N_{\mathcal Q}+2\Cresp(N_{\mathcal Q}+A_{\mathcal F})+12L_cN_{\mathcal Q}\bigr].
\]
The coefficient of $A_{\mathcal F}$ on the right is $4\epsD\Cresp\le\frac12$ by \eqref{eq:5.16}. Hence $A_{\mathcal F}\le4\epsD(C_\star'+2\Cresp+12L_c)N_{\mathcal Q}$, which gives the first inequality. The second inequality is \eqref{eq:5.16}.

(c) Lemma~\ref{lem:5.26}(b) implies that a touched cut bond belongs to a cut of a class in $\mathcal T$, and thus $c_{\mathrm{touch}}\le A_{\mathcal F}$. A bond counted in $i_{\mathrm{touch}}$ has an end within $3\ell$ of $Z(\Xi')$, so its owner lies within $R_*+3\ell+1\le R'$ of $Z(\Xi')$, in $\Qatt$, and Theorem~\ref{thm:3.21}(a) gives $i_{\mathrm{touch}}\le C_\star'N_{\mathcal Q}$. For $|\Bfr_{\mathcal F}|$, we charge each block of $\Bfr(\mathcal J)$ to a bond of $\mathfrak X(\mathcal J)$ with an end in it if that bond lies in $\Bpay$, with at most two blocks per bond. Otherwise, by Lemma~\ref{lem:5.26}(d), the block is that of a site $y$ of hull material or one of the six face neighbours of the block of that site, and we charge it to the first mark of the block of $y$, which exists by Corollary~\ref{cor:4.10}. In this way each mark receives at most seven blocks. Counting as in (a) and (b) gives $|\Bfr_{\mathcal F}|\le C_{\Bfr}N_{\mathcal Q}$.

(d) By Lemma~\ref{lem:5.26}(c), $\Bfr(\mathcal J)$ is a union of at most $2A(\mathcal J)$ $*$-connected sets. \qed
\end{proof}

\begin{proposition}\label{prop:5.36}
$\#(\Xi'\cap\Ffr)\le\rbar^{2}C_3N_{\mathcal Q}$, $|\Bfr_{\mathcal F}|\le C_{\Bfr}N_{\mathcal Q}$ and $m\le M_mN_{\mathcal Q}$, with $C_3$, $C_{\Bfr}$ free of $\beta$, $\tau$, $K$, $C$, $D$ and $M_m=O(\epsD L_c\Cresp)$ (Section~\ref{S:const5}).
\end{proposition}

\begin{proof}
We insert Lemma~\ref{lem:5.35} into Proposition~\ref{prop:5.34}. Touched parts lie in $\Qatt$ by Lemma~\ref{lem:5.10}. By Lemma~\ref{lem:5.35}(c) and the proof of (a), $u_{\mathrm{touch}}+c_{\mathrm{touch}}+i_{\mathrm{touch}}\le(\Cresp+C_\star')N_{\mathcal Q}+(\Cresp+1)A_{\mathcal F}$. The marks of the touched cells belong to parts of $\Qatt$, so $\sum_{\Delta'}m(\Delta')\le N_{\mathcal Q}$, and $26L_c/\ell\le26/308$. Together with $A_{\mathcal F}\le N_{\mathcal Q}/(48\Cresp+2)$ from Lemma~\ref{lem:5.35}(b), these estimates give the first bound, and the other two follow from Lemma~\ref{lem:5.35}(c), (b), (d). \qed
\end{proof}

\begin{lemma}\label{lem:5.37}
Assume $\eps''\le\frac12$. For a fixed set $Z$ of sites, a fixed set $\Bfr'$ of blocks and $u_0>0$, $\PD\bigl(\sum_{P\in\cP:\ \dist(P,Z\cup\bigcup\Bfr')\le R'}N(P)\ge u_0\bigr)\le2^{|Z|+|\Bfr'|+1}(\eps'')^{u_0}$.
\end{lemma}

\begin{proof}
We argue as in the proof of Proposition~\ref{prop:5.13}, with roots in $Z\cup\Bfr'$, which give at most $2^{|Z|+|\Bfr'|+m}$ multisets of roots, and with the mark $y_i$ within $R'$ of its root. As block sides are less than $2L_c$, there are $(2R'+4L_c+1)^3$ choices for each mark. \qed
\end{proof}

For Lemma~\ref{lem:5.38} we set $d_0:=\lceil R'/L_c\rceil+2$. The block distance is the $\ell^\infty$ distance of block indices, and $q(x)$ is the block of a site $x$. Since block sides are at least $L_c$, sites at distance $r$ lie in blocks at block distance at most $r/L_c+1$. A block set $\Bfr'$ is \emph{anchored to} a plaquette set $\Xi'$ if the graph on $\{o\}\cup\{*\text{-components of }\Bfr'\}$, in which $o$ is joined to a component $C$ when the component $C$ has a block within block distance $d_0$ of $q(x)$ for some $x\in Z(\Xi')$, and two components are joined when they have blocks within block distance $d_0$, is connected.

\begin{lemma}\label{lem:5.38}
(a) $\Bfr_{\mathcal F}$ is anchored to $\Xi'=\supp\Sigma^\circ$. (b) For every plaquette set $\Xi'$ with $|\Xi'|=a$, the number of block sets anchored to $\Xi'$ with $b$ blocks and $m$ $*$-components is at most $2^{2a+2b+m}(2d_0+1)^{3m}(26e)^b$.
\end{lemma}

\begin{proof}
(a) We build $\mathcal F$ by the closure, starting from $\mathcal F_0:=\mathcal T$ and letting $\mathcal F_{t+1}$ be $\mathcal F_t$ with the classes forced by $\mathcal F_t$. A class of $\mathcal T$ has a touched cut bond, whose ends lie in $Z(\Xi')$ and whose blocks lie in its $\Bfr$. A class $\mathcal J'\in\mathcal F_{t+1}\setminus\mathcal F_t$ has a cut bond $z$ with $\varpi(u,z)>0$, hence $\dist(u,z)\le\Rresp$, for a unit $u$ of type (i) or (ii) relative to $\mathcal F_t$. In case (i), $u$ has an end within $3\ell$ of $Z(\Xi')$, so the block of an end of $z$ is within block distance $(\Rresp+3\ell+1)/L_c+1\le d_0$ of $q(x)$ for some $x\in Z(\Xi')$. In case (ii), the block of an end of $z$ is within block distance $(\Rresp+1)/L_c+2\le d_0$ of $\Bfr(\mathcal J)$, $\mathcal J\in\mathcal F_t$, as $u$ has an end in a block of $\Bfr(\mathcal J)$. In both cases this block lies in $\Bfr(\mathcal J')$, as $z\in\mathfrak X(\mathcal J')$, and $\Bfr(\mathcal J')$ is a union of $*$-connected sets linked at block distance $k_b\le d_0$ by Lemma~\ref{lem:5.26}(c), each of these sets lying in one component of $\Bfr_{\mathcal F}$. By induction on $t$, each component meeting $\Bfr(\mathcal J)$, $\mathcal J\in\mathcal F_t$, is connected to $o$, and each component of $\Bfr_{\mathcal F}$ meets some $\Bfr(\mathcal J)$.

(b) Along a breadth-first spanning tree from $o$, built with fixed orders, the block set is fixed by the following data, which Section~\ref{S:proofs5} lists in more detail:
\begin{itemize}
\item the component sizes, with $2^b$ choices;
\item the anchors of the children of $o$ in $Z(\Xi')$, where $|Z(\Xi')|\le2a$, and of each component among its blocks, with $2^{2a+b+m}$ choices in all;
\item the root positions relative to the anchors, with $(2d_0+1)^{3m}$ choices;
\item the components, for which Lemma~\ref{lem:5.12} with $D_{\max}=26$ gives $(26e)^b$ choices. \qed
\end{itemize}
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:5.1}]
The conditions of item~(a) of Section~\ref{ssec:D.5} include \eqref{eq:5.16} and
\begin{equation}\label{eq:5.17}
2^{29}\cdot26e\cdot(\eps'')^{1/(3C_{\Bfr})}\le\tfrac12,\qquad2(2d_0+1)^3(\eps'')^{1/(3M_m)}\le\tfrac12 .
\end{equation}

On $E:=\{\AGp(C)\le(1-\etaT)RT\}$, as $a=\AGp(C)+\#(\Xi'\cap\Ffr)$, Proposition~\ref{prop:5.36} gives $N_{\mathcal Q}\ge\frac13(u_1(a)+b/C_{\Bfr}+m/M_m)$ with $b:=|\Bfr_{\mathcal F}|$ and $u_1(a):=(a-(1-\etaT)RT)/(\rbar^{2}C_3)$.

We first reduce the estimate to deterministic pairs. The minimizer $\Sigma^\circ$ and the block family $\Bfr_{\mathcal F}$ are functions of $D$, defined through fixed orders. Indeed, this holds for the output $(\XX,S)$ by Theorem~\ref{thm:3.20}(R1), for the minimizer by Lemma~\ref{lem:5.5}, for the classes, for the weights $\varpi$ by Lemma~\ref{lem:5.32}, and for the smallest closed family $\mathcal F$. Thus, on $E$, the pair $(\supp\Sigma^\circ,\Bfr_{\mathcal F})$ takes one deterministic value $(\Xi',\Bfr')$, with $\Xi'$ a plaquette set as in Lemma~\ref{lem:5.12} and with $\Bfr'$ anchored to $\Xi'$ by Lemma~\ref{lem:5.38}(a). Hence $E\subseteq\bigcup_{(\Xi',\Bfr')}E(\Xi',\Bfr')$, the union over all such deterministic pairs, where $E(\Xi',\Bfr')$ is the event that the parts $P$ within $R'$ of $Z(\Xi')\cup\bigcup\Bfr'^{+1}$ have $\sum N(P)\ge\frac13(u_1(|\Xi'|)+|\Bfr'|/C_{\Bfr}+m(\Bfr')/M_m)$. In $E(\Xi',\Bfr')$ the random partition appears only as a family of disjoint $r'$-connected mark sets, by Theorem~\ref{thm:3.21}, and Lemma~\ref{lem:5.37} sums over all such families.

It remains to sum over these pairs. Lemma~\ref{lem:5.37}, applied with $Z=Z(\Xi')$, where $|Z|\le2a$, and with the block set $\Bfr'^{+1}$ of at most $27b$ blocks, gives $\PD(E(\Xi',\Bfr'))\le2^{2a+27b+1}(\eps'')^{(u_1+b/C_{\Bfr}+m/M_m)/3}$. Summing over the anchored $\Bfr'$ with Lemma~\ref{lem:5.38}(b), we obtain the factors $(2^{29}\cdot26e\,(\eps'')^{1/(3C_{\Bfr})})^b$ and $(2(2d_0+1)^3(\eps'')^{1/(3M_m)})^m$, each at most $2^{-b}$, resp.\ $2^{-m}$, by \eqref{eq:5.17}. The union over $\Bfr'$ therefore has probability at most $2^{4a+3}(\eps'')^{u_1(a)/3}$. Finally, Lemma~\ref{lem:5.14} with $k=4$, $j=3$, $\eps=\eps''$, $F_*=3\rbar^{2}C_3$ and $X:=RT-\AGp(C)$, for which $\etaT|\log\eps''|/F_*=c_T''\ge c_T(4)$ and, by Lemma~\ref{lem:5.3}, $0\le X\le RT$, gives both bounds of \eqref{eq:5.1}.

\emph{The order of $\etaT$.} By Lemma~\ref{lem:5.16} with $\eps''$, $|\log\eps''|\ge\gamma_T(K)\tau/2$ for $\tau\ge\tau_0(K)$, where $\tau_0(K)$ is chosen so that the error term of \eqref{eq:5.9} is at most $\gamma_T(K)\tau/2$ (it grows logarithmically in $\tau$). This gives the first bound, and the second follows from $C_3=O((C_\star'+\Cresp)(1+1/c_s))$ (Section~\ref{S:const5}) and $\gamma_T(K)\ge1.4\cdot10^{-6}\log^2K$ for $K>K_*$ (Section~\ref{S4-S:S4.2}). \qed
\end{proof}