\section{Proofs for Section~\ref{sec:critical}}\label{app:critical}

Throughout, $G$ is simple, infinite, connected, locally finite and quasi-transitive with $h_V(G)>0$, and
$\Aut(G)$ is unimodular. Steps described as verbatim are those of~\cite{OAINU}, read with site bits; each
was checked for site percolation.

\subsection{Moments}\label{app:moments}

\begin{lemma}[{Maximal susceptibility in an intrinsic ball; site form of~\cite[Lemma~5.1]{OAINU}}]
\label{lem:maxpot}
For a deterministic domain $D$ put $f_D(v)=\E_p|C^D_v|$, which is zero off $D$. Let $B^D(v,M)$ be the
set of vertices reachable from $v$ by an open path in $D$ of length at most $M$, empty if $v$ is closed;
a maximum over the empty set is $0$. If $v\in D$ and $0<p<p_c$, then
\[
\E_p\max_{z\in B^D(v,M)}f_D(z)\le\Bigl(1+\frac{(1-p)M}p\Bigr)f_D(v).
\]
\end{lemma}

\begin{proof}
Take $D$ finite first. \emph{Resampling.} For $w\in D$, expose $B=C^D_w(\omega)$, and resample every
site of $\Nb[B]\cap D$ (resample $\{w\}$ if $B=\varnothing$). The event $\{C^D_w=B\}$ is measurable with
respect to these sites, so the resampled $\tilde\omega$ has product law independent of $B$, and
$\sum_w\Pp(Z\in B)=\E\max f_D$ as in~\cite[Lemma~5.1]{OAINU}, where $Z$ is the maximizer for
$\tilde\omega$. \emph{First boundary site.} Suppose $v\notin B$ and $Z\in B$. Along a simple
$\tilde\omega$-open path $v=z_0,\dots,z_l=Z$ with $l\le M$, the first vertex in $\Nb[B]$ is some
$y=z_i\in\partial B$ with $i\le l-1\le M-1$, because the path must cross $\partial B$ to enter $B$; this
$y$ is $\omega$-closed, and if $i\ge1$ then $z_0,\dots,z_{i-1}$ lie in $D\setminus\Nb[B]$, where
$\tilde\omega=\omega$. Let $N_w$ count the $y\in\partial B$ such that $y=v$, or $v$ is joined to $N(y)$
in $D\setminus\Nb[B]$ by an $\omega$-open path of length at most $M-2$; set $N_w=0$ if $v\in B$. Then
$\Pp(Z\in B)\le T^D(v,w)+\E N_w$. \emph{Opening $y$.} Open one counted $y$. In $\omega^y$, $v\lr w$, and
$y$ is a pivotal open site, since closing it returns $\omega$, where $v\nlr w$. All pivotal sites of
$v\lr w$, including the terminal $v$, lie in a fixed order on every simple open $v$--$w$ path. The prefix
$z_0,\dots,z_i=y$ has $i+1\le M$ vertices, so $y$ is among the first $M$ pivotal sites from $v$. For a
fixed output and distinguished site, closing it recovers the input, and comparing the two values of that
bit gives $\E N_w\le((1-p)/p)MT^D(v,w)$. \emph{Infinite domains and $M=0$.} Exhaustion is as
in~\cite[Lemma~5.1]{OAINU}; for $M=0$, $\E[\1\{v\text{ open}\}f_D(v)]\le f_D(v)$.
\end{proof}

\begin{lemma}[{A local map in a deleted domain; site form of~\cite[Lemma~5.2]{OAINU}}]\label{lem:mappot}
Fix a site map from Proposition~\ref{prop:efficient}, with critical success probability $u>0$, cap
$M\ge2$ and $H>1$, and run it at parameter $p$. On a map-dependent interval $p_0\le p<p_c$ with
$p_0\ge p_c/2$, $u_p\ge u/2$ and $\norm{K_p}\le e^{-(H-1)}$. For deterministic $A\subseteq V$ put
\[
F_A=f_{V\setminus\Nb[A]}/\chi\quad(\text{zero on }\Nb[A]),\qquad
h_A(v)=\sum_{a\in\Nb[A],\ \operatorname{dist}(v,a)\le M}\ \sum_{b\sim a}F_A(b).
\]
There are $Z$ and $C_{\mathrm{map}}$, uniform on the interval, with $2\le Z\le CM/u$ and, for every
$v\in O$, $(K_pF_A)(v)\le ZF_A(v)+C_{\mathrm{map}}h_A(v)$.
\end{lemma}

\begin{proof}
The continuity paragraph of~\cite[Lemma~5.2]{OAINU} is verbatim with site bits; it uses that success,
with output $Y\in C_v\cap O$ and $s_v\le M$, is determined by the read site bits, which follows from
Definition~\ref{def:map} since $\{s_v\le M,\ Y\in C_v\}$ is determined by the bits of $B(v,M)$, and that
the auxiliary randomness has a law not depending on $p$. On success at $p$ there is a simple open path
of length at most $M-1$ from $v$ to $Y$, because $Y\in C_v$ and $s_v\le M$. If the path avoids $\Nb[A]$,
then $Y\in B^{V\setminus\Nb[A]}(v,M)$, and Lemma~\ref{lem:maxpot} with $D=V\setminus\Nb[A]$ bounds the
contribution by $(1+(1-p)M/p)F_A(v)$; this case is empty if $v\in\Nb[A]$. If $Y\in\Nb[A]$, the
contribution is zero. Otherwise let $b$ be the first vertex after the last visit to $\Nb[A]$ and $a$ its
predecessor; then $a\in\partial A$, and the terminal segment from $b$ to $Y$ lies in $V\setminus\Nb[A]$
with length $l\le M$. The events that the segment is open and $|C^{V\setminus\Nb[A]}_Y|$ are increasing,
and on the first $C^{V\setminus\Nb[A]}_b=C^{V\setminus\Nb[A]}_Y$; Harris's inequality gives
$f_{V\setminus\Nb[A]}(b)\ge p^{l+1}f_{V\setminus\Nb[A]}(Y)$, so $F_A(Y)\le p^{-M-1}F_A(b)$. Summing over
last exits with $\operatorname{dist}(v,a)\le M$ bounds this contribution by $p^{-M-1}h_A(v)$. Dividing by
$u_p\ge u/2$ gives $Z=\max(2,(2/u)(1+2M/p_c))\le CM/u$ and $C_{\mathrm{map}}=2(2/p_c)^{M+1}/u$.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:moments}]
Assume~\eqref{eq:heavy}, so that Proposition~\ref{prop:efficient} applies, and let $\Psi$ be as in
Lemma~\ref{lem:russo}, so $(1-p)\chi'\ge\chi^2\Psi(p)$.

\emph{Upper bound.} Expose $A=C_x$, $x\in O$, in an independent $p$-field and put
$\widehat\E[G(A)]=\E_p[(s_x/\chi)G(C_x)]$. For fixed $a$: if $a\in C_x$, all neighbours of $a$ lie in
$\Nb[C_x]$, where $F_A=0$. If $a\in\partial C_x$, then $a$ is closed and $C_x$ is one of its adjacent
clusters, say cluster $i$, with $n_i=s_x$. Conditionally on $C_x$ the bits outside $\Nb[C_x]$ are fresh;
for $b\sim a$ with $b\notin\Nb[C_x]$, in the same configuration $|C^{V\setminus\Nb[C_x]}_b|=s_b$, and if
$b$ is open its cluster is some $j\ne i$. Hence
\[
\begin{split}
\widehat\E\Bigl[\1\{a\in\Nb[A]\}\sum_{b\sim a}F_A(b)\Bigr]&=\chi^{-2}\E\Bigl[s_x\1\{a\in\partial C_x\}
\sum_{b\sim a,\ b\notin\Nb[C_x]}s_b\Bigr]\le\chi^{-2}\E\Bigl[\1\{a\text{ closed}\}n_i\,d\sum_{j\ne i}
n_j\Bigr]\\
&\le d\chi^{-2}\E[\1\{a\text{ closed}\}Q_a]\le dc_\sigma\Psi(p),
\end{split}
\]
the last step because $\E[\1\{a\text{ closed}\}Q_a]$ is constant on types with type average
$\chi^2\Psi$. Bounded degree gives $\widehat\E h_A(v)\le C'_{\mathrm{map}}\Psi(p)$ uniformly in $v$, and
iterating Lemma~\ref{lem:mappot} with $F_A(x)=0$ and stochasticity gives
$\widehat\E(K^j_pF_A)(x)\le C_{\mathrm{map}}Z^j\Psi(p)$, with a constant renamed.

\emph{Lower bound identity.} Fresh exterior exposure gives
$\E[s_xf_{V\setminus\Nb[C_x]}(v)]=\E[s_xs_v\1\{v\notin\Nb[C_x]\}]=\E(s_xs_v;\ C_x\ne C_v)$, since when
both are open, $v\notin\Nb[C_x]$ if and only if $C_v\ne C_x$. So
$\widehat\E(K^j_pF_A)(x)=\chi^{-2}\sum_{v\in O}K^j_p(x,v)\E_p(s_xs_v;\ C_x\ne C_v)$, as
in~\cite[Proposition~5.3]{OAINU}.

\emph{Diagonal loss.} Harris's inequality and Lemma~\ref{lem:sharpness} give the lower bound $c_0$
without distinctness, and the diagonal-loss identity of~\cite[Proposition~5.3]{OAINU} holds verbatim (if
$x$ is closed, $s_x=0$), with $\E s_x^2\le\kappa_p\chi_{\max}^3$ (Lemma~\ref{lem:tools}).

\emph{Conclusion.} The choice of $j$, the resulting lower bound $\Psi(p)\ge c_\gamma\chi^{-\gamma}$, the
inequality $\chi'\ge c\chi^{2-\gamma}$ (now from the lower bound in~\eqref{eq:russo-d}) and its
integration are verbatim. The transcript comparison is verbatim with site exploration, which makes at
most $(d+1)N$ site queries, so $\E_pL^b_N\le e^{(d+1)C_b}$. The final summation is verbatim.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:susceptibility}]
As in~\cite[Proposition~5.5]{OAINU}, with these changes. The algorithm reveals the marks, then explores
each marked cluster by site queries: the marked vertex, then the neighbours of each discovered open
vertex. A non-marked vertex $z$ is queried only if some $y\in\Nb[z]$ is open with $C_y$ containing a mark,
so its revealment is at most $1/n+(d+1)\sup_y\E_p\min(1,s_y/n)\le(d+2)C_an^{-a}$, uniformly, by
Proposition~\ref{prop:moments}, the restriction coupling and monotonicity in $p\le p_c$. Increasing site
functions have resampling influence $2p(1-p)\Pp(\text{pivotal})$, and $F'(p)=\sum_z\Pp(z\text{ pivotal})$
by finite-product differentiation. The rest, including the covariance bound~\cite[Lemma~5.4]{OAINU}, is
verbatim.
\end{proof}

\subsection{The triangle trace}\label{app:trace}

Write $T=T_p$, $m_p=\norm{T_p}$ and $a_j=\tr T^j$. For a nonnegative invariant function $F$ of an
ordered pair of distinct \emph{nonempty} open clusters put
$\cD(F)=[\E_p\1\{o\text{ open}\}s_o^{-1}\sum_{B\ne C_o}F(C_o,B)]$, the sum over open clusters $B$, and
$k_J(A,B)=\sum_{x\in A,y\in B}J(x,y)$.

\emph{Rerooting.} $\cD(k_JF)=[\E_p\sum_vJ(o,v)F(C_o,C_v)\1\{o,v\text{ open},\ C_o\ne C_v\}]$ and
$\cD(F(A,B))=\cD(F(B,A))$, by transport from an open $x$ to each $y\in C_x$, as
in~\cite[Section~6]{OAINU}.

\emph{Contacts.} For distinct open clusters $A,B$, $k_s(A,B)\le k_{\Adj^2}(A,B)\le d^2k_s(A,B)$. Each
common boundary vertex $z$ gives a path $x\sim z\sim y$; conversely the middle vertex of such a path with
$x\in A$ and $y\in B$ is not in $A\cup B$, since distinct open clusters are not adjacent, so it lies in
$\partial A\cap\partial B$, and each $z$ carries at most $d^2$ such paths.

\begin{lemma}[{Contacts of two large clusters; site form of~\cite[Lemma~6.1]{OAINU}}]\label{lem:contacts}
Fix $0<\alpha<1/2$ and $\eps_1>0$ with $r=1/2+\eps_1-2\alpha<0$. Uniformly for $p_c/2\le p<p_c$ and
$L\ge1$, $\cD(k_s^2\1\{\min(|A|,|B|)\ge L\})\le CL^r$.
\end{lemma}

\begin{proof}
Fix $x,v$ and $H_n=\{x,v\text{ open},\ C_x\ne C_v,\ n\le\min(s_x,s_v)<2n\}$. The van den Berg--Kesten
inequality, monotone coupling and Proposition~\ref{prop:moments} give $\Pp(H_n)\le Cn^{-2\alpha}$.
\emph{Full score.} Explore $C_v$ by site queries: query $v$, then every unqueried neighbour of each
discovered open vertex, with cap $N=\lceil2(d+1)n\rceil$. When the exploration completes, the queried set
is exactly $\Nb[C_v]$, and $S_{\mathrm{full}}=\sum_{\text{queried}}(\omega_z-p)$ satisfies
$\Pp(|S_{\mathrm{full}}|>t)\le2e^{-2t^2/N}$. \emph{Exterior score.} Expose $A=C_x$ with $\partial A$, that
is, the bits of $\Nb[A]$. If $v\notin\Nb[A]$, explore the cluster of $v$ in $V\setminus\Nb[A]$ with the
same rule and cap; the queried bits are fresh, so the same tail bound holds. If $x$ is closed, set
$S_{\mathrm{ext}}=0$; this case never enters the next step, since $H_n\subseteq\{x\text{ open}\}$.
\emph{Difference.} On $H_n\cap\{s_v\le s_x\}$, $v$ is open and $C_v\ne C_x$, so $v\notin\Nb[A]$ and
$C_v\cap\Nb[A]=\varnothing$. Both explorations finish, since $|\Nb[C_v]|\le(d+1)s_v<N$, with queried sets
$\Nb[C_v]$ and $C_v\cup(\partial C_v\setminus\partial A)$; note $\partial C_v\cap A=\varnothing$ because the
clusters are not adjacent. The difference is $\partial A\cap\partial C_v$, which is all closed, so
exactly $S_{\mathrm{full}}-S_{\mathrm{ext}}=-p\,k_s(A,C_v)$. \emph{Conclusion.} Since $k_s\le|\partial C_v|
\le ds_v<2dn$ on this event, the rest of~\cite[Lemma~6.1]{OAINU} is verbatim and gives
$\E[k_s\1_{H_n}]\le Cn^r$. Finally $\cD(k_s^2\1\{\cdot\})\le\cD(k_{\Adj^2}k_s\1\{\cdot\})$; apply the
rerooting with $J=\Adj^2$. There are at most $d^2$ vertices $v$ with $\Adj^2(o,v)>0$, each with weight at
most $d$. The dyadic bins are as in~\cite{OAINU}.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:trace}]
\emph{Russo for the trace.} By Lemma~\ref{lem:russo} and cyclicity, $a_3'=3\tr(T^2T')$ almost everywhere.
With $J=T^2$, Lemma~\ref{lem:russo} and the rerooting give
\[
(1-p)a_3'=3\cD(k_sk_J)+3\tr(JR_p).
\]
\emph{Terminal terms.} $\tr(JR_p)\le(1-p)a_2+\tr(T^2\Adj T)+\tr(T^3\Adj)=(1-p)a_2+2\tr(\Adj T^3)$. By
Lemma~\ref{lem:trace} with $A=T^3$ (positive semidefinite, invariant and bounded) and $B=\Adj$
(self-adjoint, invariant and bounded), $\tr(\Adj T^3)\le\norm\Adj\tr T^3\le da_3$. Also
$a_3\ge[\sum_yT(o,y)T(y,y)T(y,o)]=pa_2$, so $a_2\le a_3/p$. Hence
$0\le\tr(JR_p)\le((1-p)/p+2d)a_3\le Ca_3$.

\emph{Square bound and the small part.} The square bound of~\cite[Proposition~6.2]{OAINU} is verbatim,
since distinct clusters are disjoint witnesses. For the small-$A$ part, $k_s\le k_{\Adj^2}$; reroot with
$J'=\Adj^2$, expose $\Nb[A]$ with $A=C_o$, and use $v\notin\Nb[A]$, so that the exterior cluster is fresh.
This gives $\cD(k_sk_J\1\{|A|<L\})\le d^2c_\sigma a_3[\E s_o\1\{s_o<L\}]\le CL^{1-\alpha}a_3$. The large
part is Cauchy--Schwarz with Lemma~\ref{lem:contacts}, as in~\cite{OAINU}.

\emph{Differential inequality.} $a_3\ge p^3\ge(p_c/2)^3$ replaces $a_3\ge1$. So
$b=\sqrt{a_3}\ge(p_c/2)^{3/2}$ and $b'\le CL^{1-\alpha}b+Cb+CL^{1/4+\eps_1/2-\alpha}m_p^{3/2}$; the added
term $Cb$ has a bounded, hence integrable, coefficient. The exponent choices and the integration are
verbatim, using Proposition~\ref{prop:susceptibility}.
\end{proof}

\subsection{Deformation and escapes}\label{app:deformation}

\begin{proof}[Proof of Lemma~\ref{lem:basic}]\label{app:basic}
\emph{Weight bound.} $p^{-|I|}q^{-b}\le((2/p_c)(1-p_c)^{-d})^k$, and $\sum_uP(w_i=z)=
\sum_uP^i(u,z)=1$ for each step $i$, so $\sum_u\sum_{w\ni z}\mu_u(w)\le k+1$; choose $C$ accordingly.
\emph{Penalty bound.} Charge each counted template to an interior vertex of one fixed simple open path.
\emph{Monotonicity.} For an enlargement $(D',\omega')\ge(D,\omega)$ of a pair connected before, take a
template counted after. Every old simple open path is a new one, so it contains $I_w$ and, by
Lemma~\ref{lem:isolation}, $u$ and $v$. Hence the template was eligible in $D$, and open and pivotal
before. Blockers closed or absent after were closed or absent before, and the condition
$x,y\notin I_w$ is deterministic. \emph{Exhaustion.} The rest is as in~\cite[Lemma~7.2]{OAINU}, using
$\E|C_x|^2<\infty$ (Lemmas~\ref{lem:tools} and~\ref{lem:sharpness}): once an exhaustion contains
$C^D_x$, all counted templates agree, since their blockers lie in $\Nb[C^D_x]$ and are closed or absent
both before and after they enter.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:deletion}]
\emph{Reduction to a visit.} Suppose that $W^D_t(v,y)>W^{D\setminus A'}_t(v,y)$. If the connection is
destroyed, every old simple path visits $A'$. Otherwise a template is newly counted after deletion. It is
eligible in $D$, its interior is open, and the terminal condition is unchanged. Either an interior vertex
was not pivotal before, in which case an old simple path avoiding it visits $A'$; or a blocker
$\beta\in A'$ was open before, in which case $\beta\sim z$ for some interior $z\notin A'$, and every
surviving simple path, being an old one, visits $z\in\Nb[A']$. So the whole loss is at most the
indicator that some old simple open $v$--$y$ path in $D$ visits $\Nb[A']$. \emph{Splitting.} Split at the
visited $z$: a prefix containing $z$, and a suffix that is empty ($z=y$) or runs from $N(z)$ to $y$; the
two are disjoint witnesses, and the van den Berg--Kesten inequality gives
$\E(W^D-W^{D\setminus A'})\le\sum_{z\in\Nb[A']}T_0(v,z)(I+\Adj T_0)(z,y)$. \emph{Summing} over $y$,
$\sum_y(I+\Adj T_0)(z,y)\le1+d\chi_0^+\le(1/p+d)\chi_0^+$, since $\chi_0^+\ge p$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:correlation}]
\emph{Exposure.} Expose $A=C^D_u$ with its closed boundary, that is, the bits of $\Nb[A]\cap D$. Then
$X^D_t(u)$ is determined: counted templates have interior and terminals in $A$ and blockers in
$\Nb[A]$, and a blocker in $A$ would be open, so the template would not be counted. The bits of
$D\setminus\Nb[A]$ are fresh. \emph{Deleting $\Nb[A]$ changes nothing outside.} For $v\notin\Nb[A]$, which
is forced when $v$ is open and $C_v\ne A$, $C^D_v\cap\Nb[A]=\varnothing$, and weighted connections from
$v$ in $D$ and in $D\setminus\Nb[A]$ agree exactly: eligibility and pivotality are cluster-internal, and a
blocker of a template inside $C_v$ cannot lie in $A$, since it would be open and adjacent to $C_v$, so it
is closed or absent in both domains. Including $v\in\Nb[A]$ by the zero convention,
$\E(X(u)X(v);\ C_u\ne C_v)=\E[X^D_t(u)s^{D\setminus\Nb[A]}_t(v)]$. \emph{Error.} Apply
Lemma~\ref{lem:deletion} with $A'=\Nb[A]\cap D$, and note $\Nb[A']\subseteq\{z:\operatorname{dist}(z,A)\le
2\}$. By (H1), $\sum_{z\in\Nb[A']}T_0(v,z)\le c_{\mathrm{nb}}(p)\sum_{a\in A}T_0(v,a)$, and by
Lemma~\ref{lem:tools}, $\E(|A|\1\{a\in A\})\le\kappa_p\chi_0^+T_0^2(u,a)$. So
$C'=C_{\mathrm{del}}c_{\mathrm{nb}}\kappa_p$, evaluated at $p=p_c/2$, works.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:endpoint}]
Let $F=\{\pi_0,\dots,\pi_{l-1}\text{ open}\}$, so $\Pp(F)=p^l$. Templates counted for $b\lr y$ whose
interior meets the path have total weight at most $(l+1)c_k/k$. For the others, interior pivotality for
$z\lr y$ holds by concatenation, and $z\notin I_{w'}$ because $z$ lies on the path; blockers and
eligibility are unchanged. Harris's inequality for $\1_F$ and $W^D_t(z,y)$, both increasing by
Lemma~\ref{lem:basic}, finishes the proof.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:smoothing}]
The proof of~\cite[Lemma~8.2]{OAINU} applies verbatim with $T_0(b,b)=p$ and $\tr T_0=p$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:escapes}]
Use the labelled experiment of~\cite[Lemma~8.1]{OAINU}. Replace the event that the escape avoids every
raw-path vertex other than the start by the \emph{site avoidance event}: every escape vertex other than
the start is at graph distance at least $2$ from every raw-path vertex other than the start. Since
$I_w\subseteq\cW(w)\setminus\{u,v\}$, this keeps the escape inside $D_w=V\setminus(\Nb[I_w]\setminus
\{u,v\})$. The event refers only to the vertex set of the raw path and the escape, so rooted reversal
transfers its bound from $u$ to $v$ unchanged.
\begin{itemize}
\item \emph{Deterministic part.} $\operatorname{dist}(g_{-i},g_j)=i+j\ge2$ for $i,j\ge1$ along the
geodesic segments, so the negative half never violates avoidance against the positive half.
\item \emph{Near-collision bounds.} Replace each point collision by membership in $\Nb[x]$, which costs a
factor at most $d+1$: a middle time $r$ near a negative-segment vertex $g_{-i}$, $i\ge1$, has probability
at most $(d+1)\rho^r$, and zero for $r<J+i-1$; a tail time $l$ near a positive-segment vertex has
probability at most $(d+1)\rho^l$, and zero for $l<J$; a tail--middle collision has
$\sum_xP^l(g_{-J},x)((I+\Adj)P^r)(x,g_J)\le(1+d)\rho^{l+r}$, and zero for $l+r<2J-1$. So
$E^s_J\le(d+1)c_*^{-1}(2(J+1)\sum_{r\ge J-1}\rho^r+\sum_{n\ge2J-1}(n+1)\rho^n)\to0$.
\item \emph{Suffix collisions.} $\norm{(I+\Adj)\nu H_{J,i}}\le(1+d)\sqrt{c_\sigma}c_*^{-1}\rho^L$, so
$F^s_J=(d+1)F_J$, in the notation of~\cite[Lemma~8.1]{OAINU}.
\item \emph{$\operatorname{dist}(u,v)\ge3$} is unchanged, and any $u$--$v$ path has length at least
$\operatorname{dist}(u,v)\ge3$, so $\ell_w\ge3$.
\item \emph{Potential test.} Over $S=\Nb[\Nb[I_w]]$, with $|S|\le(1+d+d^2)(k+1)$, using
Lemma~\ref{lem:smoothing}.
\item \emph{Constants.} The order of choice ($J$, then $L_0$, then $k_0$), the domination and
$b_*=\vartheta_0/(2D_J)$ are as in~\cite[Lemma~8.1]{OAINU}.\qedhere
\end{itemize}
\end{proof}

\subsection{Proof of Theorem~\ref{thm:unimod}}\label{app:conclusion}

We follow the proof of~\cite[Theorem~2.6]{OAINU} in~\cite[Section~9]{OAINU}, with these changes, in its
order.
\begin{enumerate}[label=\textup{(\arabic*)},leftmargin=2.2em]
\item \emph{Setup.} Suppose that $T_{p_c}$ is unbounded. Put $p_0=p_c/2$ and, for $p_0\le p<p_c$,
$\delta=p_c-p$, $S=\log(1/\delta)$, $m_0=\norm{T_p}$, $R=\chi_0^+/m_0\ge1$ and $a_3=\tr(T_p^3)$; here
$R\ge1$ since $m_0\le\chi_0^+$.
\item \emph{Lower bound on $m_0$.} Sprinkling (Lemma~\ref{lem:sprinkling}) with $a=\delta/(1-p)$ would
bound $T_{p_c}$ if $ad(1+dm_0)<1$. Hence $m_0\ge(1-p)/(d^2\delta)-1/d\ge c\delta^{-1}$ for small
$\delta$. With Propositions~\ref{prop:susceptibility} and~\ref{prop:trace}, and
$(p_c/2)^3\le a_3\le C_\eta\delta^{-\eta}$, this gives $\log R=o(S)$ and $|\log a_3|=o(S)$.
\item \emph{Row comparison.} $s^V_t(z)\ge c_f\chi_t$ whenever $tc_k\le1$, with $c_f=e^{-(L_f+1)}p_0^{L_f}$,
by Lemma~\ref{lem:endpoint} along a path of length at most $L_f$ to an orbit maximizing $s^V_t$, where
every vertex is within distance $L_f$ of every orbit.
\item \emph{Fixing $\tau$.} Choose $0<\tau<c_f/(4c_\sigma C_{\mathrm{del}}(p_0))$ and the constants of
Lemma~\ref{lem:escapes} for this $\tau$. Then $p^{2j}\ge c(kR)^{-C_2}$ for $k\ge k_0$ and $R\ge1$, as
in~\cite[Section~9]{OAINU}. With $\lambda=-\log\rho>0$ choose $K_*$ with $\lambda K_*>C_2+4$, and put
$A_\delta=\log(2R)+\log(2+a_3)+\log S$, $k=\lceil K_*A_\delta\rceil$, $B_*=(kR)^{C_2+2}$ and
$t_*=B_*/m_0$. Then $k\to\infty$, $k=o(S)$, and $B_*\rho^k=o(1)$, $R^2a_3\rho^k=o((kR)^{-C_2})$ and
$t_*jc_k=o(1)$, as in~\cite[Section~9]{OAINU}; the last estimate also gives $t_*(j+1)c_k=o(1)$ and
$t_*c_k=o(1)$.
\item \emph{Retained norm.} Proposition~\ref{prop:norm} and $B_*\rho^k=o(1)$ give $m_t\ge m_0/2$ and
$\chi_t\ge m_0/(2c_\sigma)$ on $0\le t\le t_*$, using $m_t\le c_\sigma\chi_t$ (Lemma~\ref{lem:basic}).
\item \emph{Positive term.} For a raw path selected by Lemma~\ref{lem:escapes}, with escape endpoint $z$,
Lemma~\ref{lem:deletion} with $A'=I_w\cup B_w$ (so $\Nb[A']\subseteq\Nb[\Nb[I_w]]$), the escape bound,
the row comparison and the retained norm give
\[
s^{D_w}_t(z)\ge c_f\chi_t-C_{\mathrm{del}}\chi_0^+\tau/R=c_f\chi_t-C_{\mathrm{del}}\tau m_0\ge
(c_f-2c_\sigma C_{\mathrm{del}}\tau)\chi_t\ge(c_f/2)\chi_t.
\]
Lemma~\ref{lem:endpoint} along the two escapes gives $s^{D_w}_t(u)s^{D_w}_t(v)\ge e^{-2t(j+1)c_k}p^{2j}
(c_f\chi_t/2)^2$, and since such paths have $\E_{\mathrm{path}}$-probability at least $b_*$,
$\E_{\mathrm{path}}[\1\{\ell_w\ge3\}s^{D_w}_t(u)s^{D_w}_t(v)]\ge c_1(kR)^{-C_2}\chi_t^2$ on $[0,t_*]$.
\item \emph{Error and contradiction.} The error in Proposition~\ref{prop:decrease} is
$C'(\chi_0^+)^2\rho^ka_3=C'R^2m_0^2\rho^ka_3\le4C'c_\sigma^2R^2\rho^ka_3\chi_t^2=o((kR)^{-C_2}\chi_t^2)$,
and the prefactor $e^{-2tc_k}$ is bounded below on $[0,t_*]$. So $-\chi_t'\ge c_2(kR)^{-C_2}\chi_t^2$ on
$[0,t_*]$. Integrating, $1/\chi_{t_*}\ge c_2(kR)^{-C_2}t_*=c_2(kR)^2/m_0$, while the retained norm gives
$1/\chi_{t_*}\le2c_\sigma/m_0$. These are incompatible as $kR\to\infty$. Hence $T_{p_c}$ is bounded.
\qed
\end{enumerate}