\section{Good critical clusters}\label{sec:goodness}

The local endpoint maps of Section~\ref{sec:amplification} splice pieces cut from capped critical
clusters, and the inverse of each splice must locate its seam among the separating vertices of the
output cluster. This section proves the site form of~\cite[Proposition~3.2]{OAINU}: outside a
stretched-exponentially small event, every piece left after a deletion from a capped critical cluster
has few separating cut vertices relative to its number of contacts. Throughout, $G$ is simple,
infinite, connected, locally finite and quasi-transitive with $h_V(G)>0$, and $p=p_c$. No
unimodularity is used in this section.

\subsection{Cut vertices and good clusters}

For sites, an open edge between two open vertices cannot be closed independently, so the bridges
of~\cite[Definition~3.1]{OAINU} are replaced by cut vertices.

\begin{definition}[Site pivotal distance and good clusters]\label{def:good}
Let $H$ be a finite connected set of open vertices, with its induced graph. A \emph{cut vertex
separating $x$ from $y$} is a vertex $z\in H\setminus\{x,y\}$ such that $x$ and $y$ lie in different
components of $H\setminus\{z\}$. Let $\piv_H(x,y)$ be the number of such $z$, and put
$\dpiv(H)=\max_{x,y\in H}\piv_H(x,y)$ and $\rpiv(H,x)=\max_{y\in H}\piv_H(x,y)$.

Fix $0<\eps<1/2$ and put $\tau=1/2+\eps$. A finite open cluster $C$ with $|C|\le M$ is \emph{good for cap
$M$} if the following holds for every $A\subseteq C$: for every open component $B$ of $C\setminus A$,
let $K$ be the number of graph edges between $B$ and $A$; whenever $K\ge1$,
\[
\dpiv(B)\le K\,M^\tau,
\]
with the pivotal distance computed inside $B$.
\end{definition}

An open component is a connected component of the induced graph on $C\setminus A$. Since $C$ is open,
every edge between $B$ and $A$ joins two open vertices, so $K$ is the site reading of the number of
open edges from $B$ to $A$ in~\cite[Definition~3.1]{OAINU}. Call a vertex of $C\setminus B$ adjacent
to $B$ a \emph{contact} of $B$. Every contact lies in $A$, the set $\partial B\cap C$ is exactly the
set of contacts, every vertex of $\partial B\setminus C$ is closed, and with $k$ the number of
contacts, $k\le K\le dk$. If $C$ is connected and $A\ne\varnothing$, then $K\ge1$ for every component
$B$, and $C_x=\varnothing$ is good.

\begin{lemma}[Elementary facts]\label{lem:cut}
Let $H$ be finite and connected, and $x,y,w\in H$.
\begin{enumerate}[label=\textup{(\arabic*)},leftmargin=2.2em]
\item \textup{(Order.)} The cut vertices separating $x$ from $y$ lie on every $x$--$y$ path in $H$, and
they occur in the same order on every simple $x$--$y$ path. More generally, for $x$ and a nonempty set
$Y\subseteq H\setminus\{x\}$, the vertices $z\in H\setminus\{x\}$ that meet every path from $x$ to $Y$
($z\in Y$ allowed) occur in one order along every simple path from $x$ to a fixed $y\in Y$.
\item \textup{(Triangle inequality.)} $\piv_H(x,y)\le\piv_H(x,w)+\piv_H(w,y)+1$.
\item \textup{(Deletion.)} If $R\subseteq H$ and $x,y$ lie in one component $H''$ of $H\setminus R$,
then every cut vertex of $H$ separating $x$ from $y$ that is not in $R$ lies in $H''$ and separates $x$
from $y$ in $H''$. Hence $\piv_{H''}(x,y)\ge\piv_H(x,y)-|R|$.
\item \textup{(Separated indices.)} If $z_1,\dots,z_m$ are the cut vertices separating $x$ from $y$,
listed in the order of~\textup{(1)}, then $\piv_H(z_t,z_{t'})\ge|t-t'|-1$.
\item $\piv_H(x,y)\le|H|-2$ when $|H|\ge2$, so $\dpiv(H)\le|H|-1$.
\end{enumerate}
\end{lemma}

The proofs are short graph arguments (Appendix~\ref{app:cut}). The $+1$ in~(2) is absent for bridges,
since an edge separating $x$ from $y$ separates $x$ from $w$ or $w$ from $y$; it costs only constants
below.

\begin{proposition}[{Good critical clusters; site form of~\cite[Proposition~3.2]{OAINU}}]
\label{prop:good}
For each $\eps\in(0,1/2)$, uniformly in $x$ and for all sufficiently large $M$,
\[
\Pp_{p_c}(s_x\le M,\ C_x\text{ not good for cap }M)\le e^{-M^\eps}.
\]
The event that $s_x\le M$ and $C_x$ is good for cap $M$ is determined by the site bits in the
deterministic ball $B(x,M)$. The constants depend only on $d$, $p_c$ and $\eps$.
\end{proposition}

\subsection{Marked pivotal moments and the radius tail}

The probabilistic input is the site form of the marked-cluster estimate of~\cite[Lemmas~3.3
and~3.4]{OAINU}. In a deterministic domain $D\ni x$, mark vertices independently with probability
$h\in(0,1/2]$, independently of $\omega$, and let $E=E(D,x,h)$ be the event that $C_x^D$ contains a
mark; it requires $x$ to be open. A vertex is \emph{pivotal for $E$} if changing its state changes
$\1_E$, with the marks and all other bits fixed. Let $N_{\mathrm{piv}}$ be the number of open pivotal
vertices \emph{other than $x$}. On $E$ the root $x$ is always an open pivotal, since closing it
empties $C_x^D$; excluding it makes the recursion below a recursion of the same quantity. On $E$, an
open $v\ne x$ is pivotal if and only if every path in $C_x^D$ from $x$ to a mark meets $v$, so by
Lemma~\ref{lem:cut}(1), with $Y$ the set of marks, the open non-root pivotals are linearly ordered
from $x$ toward the marks. (If $x$ itself is marked there are no open non-root pivotals, since closing
any $v\ne x$ leaves $x$ marked, and the ordering is vacuous.)

\begin{lemma}[{Site marked pivotal moments; site form of~\cite[Lemma~3.3]{OAINU}}]\label{lem:marked}
There is a constant $C$, depending only on $d$ and $p_c$, such that, with
$A_h=C\sqrt{\log(1/h)/h}\ge1$, for every domain $D$, every $x\in D$, every integer $i\ge0$ and every
$r\ge0$,
\[
\E\Bigl[\binom{N_{\mathrm{piv}}}{i}\1_E\Bigr]\le A_h^i\,\Pp(E),\qquad
\Pp(E,\ N_{\mathrm{piv}}\ge r)\le2\Pp(E)e^{-r/(4A_h)}.
\]
\end{lemma}

The proof (Appendix~\ref{app:marked}) follows~\cite{OAINU}: an exploration score bounds the first
moment, through Hoeffding's lemma and the identity $\E[Z\1_E]=(1-p_c)\E[(N_{\mathrm{piv}}+\1_E)\1_E]$,
and an exact recursion over the first pivotal gives the higher binomial moments. For sites the
recursion removes the root cluster with the first pivotal $v$ closed, $W=C_x^{D\setminus\{v\}}$, and
continues in the exterior domain $(D\setminus\Nb_D[W])\cup\{v\}$ rooted at $v$; there is no separate
factor $p_c$, since the openness of $v$ is part of the continued event. Also $\Pp(E)\ge p_ch$ in place
of $h$, as $x$ must be open.

\begin{lemma}[{Site radius tail; site form of~\cite[Lemma~3.4]{OAINU}}]\label{lem:radius}
Fix $\eps\in(0,1/2)$ and $\kappa_0\in(0,1]$. There are $c>0$ and $M_0$, depending only on $d$,
$p_c$, $\eps$ and $\kappa_0$, such that, uniformly in $x$, for all $M\ge M_0$ and all real $\kappa$ with
$\kappa_0\le\kappa\le M^{1-\tau}$,
\[
\Pp_{p_c}\bigl(s_x\le M,\ \rpiv(C_x,x)\ge\kappa M^\tau/2\bigr)\le\exp\bigl(-c\kappa^2M^{2\eps}/\log M\bigr).
\]
\end{lemma}

For integers $1\le\kappa\le M^{1-\tau}$ this is the bound of~\cite[Lemma~3.4]{OAINU}, with the
bridge replaced by a cut vertex; it follows from Lemma~\ref{lem:marked} with a single mark at a witness $y$
(Appendix~\ref{app:marked}). The real range $\kappa\ge\kappa_0$, used with $\kappa_0=1/(2d)$ below,
changes only $M_0$ and $c$.

\subsection{Pruning at a retained contact}

The bond proof keeps one open contact edge from the rest of the root cluster into a bad piece $B$;
then $B$ has one attachment, and its bridge tree embeds isometrically~\cite[Section~3]{OAINU}. For
sites the retained object is a contact vertex $a$, which may have several neighbours in $B$ and so
short-circuit the cut vertices of $B$: if $B$ is a path and $a$ is adjacent to both of its ends, then
$B\cup\{a\}$ is a cycle, with no cut vertices at all. We therefore also close all but one neighbour of
$a$ in $B$. The next lemma shows that the kept neighbour can be chosen so that the pivotal distance
drops by at most a factor $2d$.

\begin{lemma}[Pruning]\label{lem:pruning}
Let $B$ be a finite connected graph with $\dpiv(B)=\mathsf D$, and let $N\subseteq B$ be nonempty with
$|N|\le d$. Then there are $b^*\in N$ and $v\in B$ with the following property. Put
$R=N\setminus\{b^*\}$, and let $B''$ be the component of $b^*$ in $B\setminus R$. Then $v\in B''$ and
\[
\piv_{B''}(b^*,v)\ge\frac{\mathsf D}{2d}-d-\frac12.
\]
\end{lemma}

\begin{proof}
For $u\in B$ let $n(u)$ be a vertex of $N$ nearest to $u$ in the graph distance of $B$, ties broken by
a fixed order, and put $V_b=\{u\in B:n(u)=b\}$ for $b\in N$. These Voronoi cells partition $B$.

(i) \emph{Each $u\in V_b$ lies in the component of $b$ in $B\setminus(N\setminus\{b\})$.} If $u\in N$
then $n(u)=u$, so $u=b$. Otherwise take a geodesic in $B$ from $u$ to $b$, of length
$\operatorname{dist}(u,N)$. A vertex $q\ne b$ on it lying in $N$ would satisfy
$\operatorname{dist}(u,q)<\operatorname{dist}(u,b)$, which is impossible. So the geodesic avoids
$N\setminus\{b\}$.

(ii) \emph{Some cell contains a far vertex.} Put $L=\max_{b\in N}\max_{u\in V_b}\piv_B(b,u)$. Choose a
pair $(u_0,w_0)$ with $\piv_B(u_0,w_0)=\mathsf D$, and list its separating cut vertices
$z_1,\dots,z_{\mathsf D}$ in the order of Lemma~\ref{lem:cut}(1). If $z_t,z_{t'}\in V_b$, then by
Lemma~\ref{lem:cut}(4) and~(2),
\[
|t-t'|-1\le\piv_B(z_t,z_{t'})\le\piv_B(z_t,b)+\piv_B(b,z_{t'})+1\le2L+1.
\]
So the indices $t$ with $z_t\in V_b$ lie in an interval of length at most $2L+2$, and each cell
contains at most $2L+3$ of the $z_t$. The at most $d$ cells contain all $\mathsf D$ of them, so
$\mathsf D\le d(2L+3)$, that is, $L\ge\mathsf D/(2d)-3/2$.

(iii) \emph{Deleting $R$.} Choose $(b^*,v)$ with $v\in V_{b^*}$ and $\piv_B(b^*,v)=L$, by a fixed
order. By (i), $v\in B''$. By Lemma~\ref{lem:cut}(3) with $|R|\le d-1$,
$\piv_{B''}(b^*,v)\ge L-(d-1)\ge\mathsf D/(2d)-d-1/2$.
\end{proof}

The lemma is deterministic, so $(b^*,v,R)$ is a fixed function of $(B,N)$; it uses no randomness, no
independence and no group action.

\subsection{Proof of Proposition~\ref{prop:good}: the modified configuration}\label{sec:good-proof}

Fix $\eps\in(0,1/2)$; constants depend only on $d$, $p_c$ and $\eps$. Take $M$ so large that
$M^\tau\ge4d^2$. Let $\mathrm{Bad}=\{s_x\le M,\ C_x\text{ not good for cap }M\}$; on $\mathrm{Bad}$, $x$ is
open. Choose a witnessing pair $(A,B)$ by a fixed finite ordering: $A\subseteq C_x$, $B$ an open
component of $C_x\setminus A$ with $K\ge1$ graph edges to $A$, and $\mathsf D:=\dpiv(B)>KM^\tau$. Since
$\mathsf D\le|B|-1<M$ (Lemma~\ref{lem:cut}(5)), $K<M^{1-\tau}$. The proof closes a set $J$ of open
vertices of $C_x$, chosen by fixed rules, and shows that the resulting configuration $\omega'$ has a
pivotal radius of order $\mathsf D$; a Peierls-type count then compares with Lemma~\ref{lem:radius}.
\begin{itemize}
\item \emph{Case 1, $x\in B$.} $J$ is the set of all contacts of $B$. In $\omega'$ every vertex of
$\partial B$ is closed, so $C'_x=B$. A diameter pair of $B$ and Lemma~\ref{lem:cut}(2) give
$\rpiv(C'_x,x)\ge(\mathsf D-1)/2$.
\item \emph{Case 2, $x\notin B$.} Take a simple open path $\gamma$ in $C_x$ from $x$ to $B$, by a fixed
ordering. Its vertex just before its first vertex in $B$ is a contact, so $\gamma$ meets the set of
contacts; let $a$ be the \emph{first} contact on $\gamma$ ($a=x$ is possible). Put $N=N(a)\cap B$, which
is nonempty with $|N|\le d$, and apply Lemma~\ref{lem:pruning} to $(B,N)$, obtaining $b^*$, $v$,
$R=N\setminus\{b^*\}$ and $B''$. Let $J$ consist of all contacts other than $a$, together with $R$.
\end{itemize}
In Case~2 the following hold (Appendix~\ref{app:good-proof}).
\begin{enumerate}[label=\textup{(\alph*)},leftmargin=2.2em]
\item\label{good:a} $\gamma$ from $x$ to $a$ is open in $\omega'$.
\item\label{good:b} In $\omega'$, the only open vertex outside $B''$ with a neighbour in $B''$ is $a$, and the
only neighbour of $a$ in $B''$ is $b^*$.
\item\label{good:c} Hence every cut vertex of $B''$ separating $b^*$ from $v$, and $b^*$ itself, separate
$x$ from $v$ in $C'_x$, and Lemma~\ref{lem:pruning} gives $\rpiv(C'_x,x)\ge\mathsf D/(2d)-d+1/2$.
\end{enumerate}
Closing only shrinks clusters, and $\mathsf D>KM^\tau\ge4d^2$, so in both cases $\omega'$ lies in the
radius-tail event
\[
O_K:=\{s_x\le M,\ \rpiv(C_x,x)\ge\kappa M^\tau/2\},\qquad\kappa=K/(2d),\quad1/(2d)\le\kappa\le M^{1-\tau}.
\]
Appendix~\ref{app:good-proof} completes the proof by counting. Let $\mathrm{Bad}_K$ be the part of
$\mathrm{Bad}$ whose witness has count $K$. The configuration $\omega$ is $\omega'$ with at most $K+d$
vertices of $C_x$ reopened, and these can be listed one at a time as neighbours of the growing root
cluster, so each $\omega'$ has at most $(K+d+1)(dM)^{K+d}$ preimages in $\mathrm{Bad}_K$. With
$\vartheta=\max\{1,p_c/(1-p_c)\}$, a pointwise comparison on the finite product of the bits in $B(x,M)$
gives $\Pp(\mathrm{Bad}_K)\le(K+d+1)(dM)^{K+d}\vartheta^{K+d}\,\Pp(O_K)$. Lemma~\ref{lem:radius} with
$\kappa_0=1/(2d)$ and a sum over the fewer than $M$ values of $K$ give $\Pp(\mathrm{Bad})\le e^{-M^\eps}$
for all sufficiently large $M$, uniformly in $x$, and a breadth-first exploration of $C_x$ stopped after
more than $M$ open vertices gives locality.

Compared with~\cite{OAINU}, Case~1 closes the $k\le K$ contact vertices in place of the open contact
edges. Case~2 replaces retention of the first contact edge entering $B$ by retention of the
first contact vertex $a$ on $\gamma$, closure of $a$'s other neighbours in $B$, and the choice of $b^*$
by Lemma~\ref{lem:pruning}. This loses the factor $2d$ in the radius, absorbed by the real range of
Lemma~\ref{lem:radius}, and adds at most $d-1$ changed vertices, absorbed in the summation. The
sequential listing of the reopened vertices replaces the observation of~\cite[Section~3]{OAINU} that
every changed edge is incident to $B\subseteq C'_x$: after pruning, $B\setminus B''$ and some contacts can
leave $C'_x$.