\section{Nonunimodular automorphism groups}\label{sec:nonunimod}

This section proves Theorem~\ref{thm:b8}. Parts~(2) and~(3) follow from part~(1) by Hutchcroft's
operator bound, which is model-free (Section~\ref{sec:cocycle-main}). Part~(1), the strict inequality
$p_c<p_c(\lambda)$ for the site tilted susceptibility, is the quasi-transitive form of Tran and Nguyen's
\cite[Proposition~5.5]{TN26}. We obtain it by running~\cite[Section~5]{TN26} on a quasi-transitive graph.
The steps that use vertex-transitivity are replaced with the help of the devices of
Section~\ref{sec:b8-devices}; the full list of replacements is in Appendix~\ref{app:b8}. Every other
step of~\cite[Section~5]{TN26} is used as written, so this case inherits the correctness of that
unrefereed preprint.

\subsection{The operator bound}\label{sec:cocycle-main}

Let $\Gamma\le\Aut(G)$ be quasi-transitive, with Hutchcroft's modular function
$\Delta=\Delta_\Gamma$~\cite[Section~2.1]{HutNU}. For a nonnegative kernel $\tau$ and $\lambda\in\R$
put $\chi_\lambda(u)=\sum_x\tau(u,x)\Delta(u,x)^\lambda$ and $\chi^*_\lambda=\sup_u\chi_\lambda(u)$.

\begin{lemma}[{Hutchcroft~\cite[Lemma~7.1]{HutNU}, \cite[proof of Theorem~2.9]{Hut19}}]
\label{lem:cocycle}
Let $G$ be connected and locally finite and $\Gamma\le\Aut(G)$ quasi-transitive. For every nonnegative
kernel $\tau$, every $\lambda\in\R$, $n\ge1$ and $v\in V$, $(\tau^n)(v,v)\le(\chi^*_\lambda)^n$. If
$\tau$ is also symmetric, then $\norm\tau_{2\to2}\le\chi^*_\lambda$.
\end{lemma}

For $\tau=T_p$ this is the first step of the proof of~\cite[Theorem~2.9]{Hut19}, which obtains
$\sup_vT_p^n(v,v)\le\chi_{p,\lambda}^n$ from the proof of~\cite[Lemma~7.1]{HutNU} and then the norm bound
from~\cite[Proposition~2.2]{Hut19}.\footnote{\cite[Lemma~7.1]{HutNU} is written for transitive $\Gamma$
and $n=3$. The reference in~\cite{Hut19} to~\cite[Theorem~1.11]{HutNU} corresponds to Theorem~1.8 in
arXiv:1711.02590v3, which is stated for transitive $\Gamma$, although Sections~3--6 there are written for
quasi-transitive $\Gamma$.} The telescoping of~\cite[(7.1)--(7.2)]{HutNU} uses only the cocycle identity
$\Delta(u,v)\Delta(v,w)=\Delta(u,w)$, valid for quasi-transitive $\Gamma$~\cite[Lemma~2.3]{HutNU}, and
$\tau\ge0$, so it holds for every nonnegative kernel and every $n$. For symmetric $\tau$, the Schur test with weight $w=\Delta(o,\cdot)^\lambda$, for which
$\sum_y\tau(x,y)w(y)=w(x)\chi_\lambda(x)\le\chi^*_\lambda w(x)$ by the cocycle identity, gives the norm
bound directly. Applied to the site kernel $T_p$, Lemma~\ref{lem:cocycle} gives
$\norm{T_p}\le\sup_u\chi_{p,\lambda}(u)$.

For vertex-transitive $G$, Theorem~\ref{thm:b8} follows from this bound and Tran and Nguyen's
tilted-threshold separation~\cite[Proposition~5.5]{TN26}; Appendix~\ref{app:cocycle} records the
deduction (Proposition~\ref{prop:vt}). The proof below does not use it, since it covers vertex-transitive
graphs too.

\subsection{The quasi-transitive setting}\label{sec:b8-setting}

From now on $G$ is infinite, connected, locally finite and quasi-transitive, $\Gamma=\Aut(G)$ is
nonunimodular, and $d$ is the maximal degree; every use of the common degree in~\cite[Section~5]{TN26}
is an upper bound and becomes $d$. A nonunimodular quasi-transitive automorphism group forces
nonamenability, since every quasi-transitive group of automorphisms of an amenable graph is
unimodular~\cite{SW90,Sal92}, \cite[Section~2.2]{Pei26b}; so $h_V(G)>0$, and $p_c<1$
(Appendix~\ref{app:b8-list}, item on $p_c<1$).

\begin{enumerate}[label=\textup{(S\arabic*)},leftmargin=2.6em]
\item \emph{Modular function.} $\Delta$ is diagonally $\Gamma$-invariant, satisfies
$\Delta(u,v)\Delta(v,w)=\Delta(u,w)$, and $\Delta(x,\cdot)$ is harmonic for simple random
walk~\cite[Lemma~2.3]{HutNU}; $\Gamma$ is unimodular if and only if $\Delta\equiv1$. Write
$h_v(z)=\log\Delta(v,z)$ for the raw height of $z$ relative to $v$, and $h(z)=h_o(z)$ for a fixed $o$, so
that $h_v(z)=h(z)-h(v)$. (The companion paper normalizes heights by $t_0$; here they are raw.)
\item \emph{The constants $t_0$ and $\delta_0$.} $t_0=\max_{x\sim y}\log\Delta(x,y)>0$ and
$\delta_0=\min\{|\log\Delta(x,y)|:x\sim y,\ \Delta(x,y)\ne1\}>0$; both are well defined because there are
finitely many edge orbits and $\Delta\not\equiv1$, that is, $\Gamma$ is nonunimodular. An edge changes $h$
by at most $t_0$.
\item \emph{Sloped and plateau vertices.} A vertex $y$ is \emph{sloped} if some neighbour $z$ has
$h(z)<h(y)$, and a \emph{plateau} vertex otherwise. By harmonicity a plateau vertex has all its
neighbours at its own height, and a sloped vertex also has a neighbour of strictly larger height.
Slopedness is $\Gamma$-invariant. Sloped vertices exist, since some edge has $\Delta\ne1$ and both its
endpoints are sloped. By quasi-transitivity $R_{\mathrm{sl}}:=\sup_y\operatorname{dist}(y,\text{sloped
set})<\infty$, and $R^*:=\max_{O}\sup_y\operatorname{dist}(y,O)<\infty$ over the orbits $O$.
\item \emph{Orbit weights.} $\mu$ is the measure of~\cite[Section~2.1]{HutNU} on the finite set of
orbits, and $\mu_{\min}=\min_O\mu(O)>0$.
\item \emph{Layers.} As in~\cite[(5.6)]{TN26}, an independent uniform offset $R_o\in[0,1)$ is
transported equivariantly by $R_v=R_o-h_o(v)/t_0\bmod1$, and
$L_j(v)=\{x:j+R_v-1<h_v(x)/t_0\le j+R_v\}$, $L_{m,n}(v)=\bigcup_{m\le j\le n}L_j(v)$. The exact shift
rule $u\in L_k(v)\Rightarrow L_j(u)=L_{j+k}(v)$ holds by the cocycle identity, and the joint law of the
configuration and the offset field is $\Gamma$-invariant. A path cannot skip a layer, and a path along
which $h$ is weakly monotone, going from $L_a(v)$ to height beyond $L_b(v)$, visits $L_b(v)$.
\end{enumerate}

We use the notation of~\cite[Section~5]{TN26}:
\begin{itemize}
\item $K_v=C_v$ is the open cluster of $v$, and $|K_v|_{v,\lambda}=\sum_{u\in K_v}\Delta(v,u)^\lambda$ for
$\lambda\in\R$;
\item $\widehat\Pp^v_p=\Pp_p(\,\cdot\mid v\text{ open})$, and $\widehat\Pp^{v,x}_p$ also conditions on $R_v=x$;
\item $X^{m,n}_j(v)=|\{x\in L_j(v):v\lr x\text{ in }L_{m,n}(v)\}|$;
\item a vertex $z$ is the \emph{peak} of a set $A\ni z$ if $h_z(u)<0$ for all $u\in A\setminus\{z\}$; $P_z$ is
the event that $z$ is the peak of its open cluster, and $P_v(s)$ the event that $v$ is the peak of the set
joined to it inside $L_{-s,\infty}(v)$;
\item $Q_{v;k,\ell,r}=\{v\nlr L_{\ell-1}(v)\text{ in }L_{-k-r-2,\infty}(v)\}$;
\item $\widehat\chi_{p,\lambda}=\chi_{p,\lambda}/p$ is the conditional tilted susceptibility
of~\cite[(5.13)]{TN26}.
\end{itemize}
The supremum $\widehat\E^{m,n}_p(j;a)$ of~\cite[(5.31)]{TN26} is used only in Appendix~\ref{app:b8} and is
defined there.

The two obstructions to running~\cite[Section~5]{TN26} on $G$ are these. First, the argument there
moves between layers along neighbours $z^\pm$ with $\log\Delta(z,z^\pm)=\pm t_0$; on a quasi-transitive
graph such neighbours need not exist, for instance at pendant vertices. Second, several estimates are
used uniformly in the root, and transports are applied at a fixed root; on a quasi-transitive graph
the lower bounds for peak events fail at plateau vertices, where $P_v$ forces $K_v=\{v\}$, so that
$\widehat\Pp(v\lr L_{-k}(v),\ P_v)=0$ for $k\ge1$.

\subsection{Devices}\label{sec:b8-devices}

\begin{lemma}[{Finite modifications~\cite[(F.1)]{Pei26b}}]\label{lem:Q0}
Let $0<p<1$, let $B\subseteq V$ be finite, and let $\Phi$ be a measurable map of configurations with
$\Phi(\omega)=\omega$ off $B$. Then $\Pp_p(A)\ge\min(p,1-p)^{|B|}\Pp_p(\Phi(\omega)\in A)$ for every event $A$.
The same holds with independent extra randomness, such as the offset field, that $\Phi$ does not change.
\end{lemma}

The first device replaces the neighbours $z^\pm$ by monotone paths of bounded length.

\begin{lemma}[{Monotone paths with a length bound; \cite[Lemma~F.1]{Pei26b} with lengths}]
\label{lem:Q1}
For every vertex $y$ and every $t>0$ there is a path from $y$ to a vertex $z$, of length at most
$L(t):=(\lceil t/\delta_0\rceil+1)(R_{\mathrm{sl}}+1)$, such that $h(z)\le h(y)-t$, $h$ is nonincreasing
along the path, and $h$ strictly decreases at its last step. The same holds with all inequalities
between heights reversed.
\end{lemma}

The proof (Appendix~\ref{app:b8-devices}) walks along a shortest path to a nearest sloped vertex, at
distance at most $R_{\mathrm{sl}}$; that path is level, since a step from a plateau vertex keeps $h$
unchanged. It then steps to a strictly lower neighbour, which lowers $h$ by at least $\delta_0$, and
repeats $\lceil t/\delta_0\rceil$ times.

The second steers a path into a prescribed orbit while controlling its heights; it replaces the uses of
transitivity that move a root into a given position.

\begin{lemma}[Orbit-targeted connectors]\label{lem:Q2}
There are constants $M_*$ and $H_*$ such that for every vertex $y$ and every orbit $O^*$:
\begin{itemize}
\item \textup{(down)} there is a path $y=y_0,\dots,y_m=q$ with $m\le M_*$, $q\in O^*$, $h(y_j)\le h(y)$
for all $j$, and $h(y)-H_*\le h(q)\le h(y)-t_0$;
\item \textup{(up)} there is a path with the mirrored properties: $q\in O^*$, $h(y_j)\ge h(y)$ for all
$j$, and $h(y)+t_0\le h(q)\le h(y)+H_*$.
\end{itemize}
\end{lemma}

The proof (Appendix~\ref{app:b8-devices}) applies Lemma~\ref{lem:Q1} with $t=(R^*+1)t_0$ and appends a
shortest path to $O^*$, of length at most $R^*$, along which $h$ changes by at most $R^*t_0$; so
$M_*=L((R^*+1)t_0)+R^*$ and $H_*=M_*t_0$ work.

The third replaces the one-layer buffer of~\cite[(5.1)]{TN26}: an exploration below a layer leaves the
layer above untested.

\begin{lemma}[First crossing]\label{lem:Q3}
Let $W\subseteq V$ be deterministic once the offset field is fixed, let $v\in W$, and explore the open
cluster $K$ of $v$ in the induced graph on $W$, testing boundary sites one at a time, as in the cluster
kernel~\cite[(5.36)]{TN26}.
\begin{enumerate}[label=\textup{(\arabic*)},leftmargin=2.2em]
\item No site of $V\setminus W$ is tested, and conditionally on the exploration the labels of
$V\setminus W$ are i.i.d.\ uniform, by~\cite[(5.36)--(5.38)]{TN26} with $Q_{\mathrm{old}}=\varnothing$.
\item If $W=L_{-\infty,k}(v)$, every neighbour of $K$ outside $W$ lies in $L_{k+1}(v)$ and is adjacent to
$K\cap L_k(v)$.
\item If $W=L_{-\ell,\infty}(v)$, every neighbour of $K$ outside $W$ lies in $L_{-\ell-1}(v)$ and is
adjacent to $K\cap L_{-\ell}(v)$.
\end{enumerate}
\end{lemma}

\begin{proof}
The exploration queries only sites of $W$. For~(2)--(3), an edge changes $h$ by at most $t_0$, and the
layers have thickness exactly $t_0$.
\end{proof}

The fourth replaces the fixed-root tilted transport~\cite[(5.30)]{TN26} by an average over a random
root.

\begin{lemma}[Random-root conditioned tilted mass transport]\label{lem:Q4}
Let $F(x,y,\omega,R)\ge0$ be measurable and diagonally $\Gamma$-invariant, jointly with the
configuration $\omega$ and the equivariant offset field $R$, and suppose that $F$ vanishes unless both
$x$ and $y$ are open. Let $\rho\sim\mu$ be independent, and let $\widehat\E_\mu$ denote expectation over
$\rho$, the offset field, and $\omega$ under $\widehat\Pp^\rho$. Then
\begin{equation}\label{eq:Q4}
\widehat\E_\mu\sum_xF(\rho,x)=\widehat\E_\mu\sum_xF(x,\rho)\Delta(\rho,x).
\end{equation}
Moreover, $\sup_vQ(v)\le\mu_{\min}^{-1}\E_\mu Q(\rho)$ for every orbit-invariant $Q:V\to[0,\infty]$.
\end{lemma}

\begin{proof}
Multiply both sides of~\eqref{eq:Q4} by $p$. Each summand forces $\rho$ open, so the identity becomes
$\E\sum_xF(\rho,x)=\E\sum_xF(x,\rho)\Delta(\rho,x)$, which is the tilted mass-transport
principle~\cite[Proposition~2.2]{HutNU} applied to the diagonally invariant function
$(x,y)\mapsto\E F(x,y,\omega,R)$. This is the argument of~\cite{TN26} for~(5.30) there, with the fixed
root replaced by $\rho$. For the last assertion, a maximizing orbit has weight at least $\mu_{\min}$.
\end{proof}

The fifth compares quantities at different roots, so that finiteness or divergence at one root
transfers to all.

\begin{lemma}[Orbit comparability]\label{lem:Q5}
There is $c_5=c_5(G,p,\lambda)>0$ such that for all vertices $u,v$:
\begin{enumerate}[label=\textup{(\alph*)},leftmargin=2.2em]
\item $\chi_{p,\lambda}(u)\ge c_5\chi_{p,\lambda}(v)$;
\item $\widehat\E^u[|K_u|^a_{u,\lambda}]\ge c_5\widehat\E^v[|K_v|^a_{v,\lambda}]$ for $0<a\le1$;
\item $\widehat M_{p,\lambda,\theta}(u)\ge c_5\widehat M_{p,\lambda,\theta}(v)$ for all $\theta>0$, where
$\widehat M_{p,\lambda,\theta}(v)=\widehat\E^v[1-e^{-\theta|K_v|_{v,\lambda}}]$ is the tilted magnetization.
\end{enumerate}
In particular, finiteness or divergence of each quantity does not depend on the root, and its supremum
over roots is a maximum over orbits.
\end{lemma}

All three quantities are orbit-invariant. The proof (Appendix~\ref{app:b8-devices}) moves the second root
into the orbit of $v$ at distance at most $R^*$ from $u$ and uses a path $\pi$ of length at most $R^*$
between them. Part~(a) is Hutchcroft's inequality $\chi(u)\ge T_p(u,v')\Delta^\lambda(u,v')\chi(v')$~\cite[proof
of Proposition~3.1]{HutNU}. For~(b) and~(c), on $\{\pi\text{ open}\}$ the two clusters coincide and the
cocycle identity relates their tilted sizes, and Harris's inequality gives the comparison;
$c_5=p^{R^*+1}e^{-|\lambda|t_0R^*}$ works, since $\min\{\Delta(u,v')^\lambda,1\}\ge e^{-|\lambda|t_0R^*}$.

The sixth transfers peak estimates to a sloped vertex. Put $F_j(v,x)=\widehat\E^{v,x}[X^{-\infty,\infty}_j
(v);\ P_v]$, as in~\cite[Auxiliary Lemma~5.G]{TN26}.

\begin{lemma}[Peak transfer]\label{lem:Q6}
Let $v_0$ be a sloped vertex and $0<p\le p_c$. There are $C_6$ and $J_6$ such that for every vertex $u$
and every $k\ge1$,
\[
\int_0^1F_{-k}(u,x)\,dx\le C_6\sum_{j=0}^{J_6}\int_0^1F_{-k-j}(v_0,y)\,dy.
\]
\end{lemma}

\begin{proof}
\emph{The connector $\sigma$.} Since $v_0$ is sloped, there is $\sigma=(v_0=\sigma_0,\sigma_1,\dots,
\sigma_s)$: first a step to a strictly lower neighbour, then the path of Lemma~\ref{lem:Q1} (down) with
$t=(2R^*+1)t_0$. So $h(\sigma_i)<h(v_0)$ for $i\ge1$, $h(\sigma_s)\le h(v_0)-t$, and $s\le1+L(t)$.

\emph{The path $\pi_u$.} For a vertex $u$, choose $g\in\Gamma$ with $\operatorname{dist}(u,g\sigma_s)\le
R^*$, and let $\pi_u$ be a shortest path from $u$ to $g\sigma_s$ followed by the reversal of $g\sigma$; its
endpoint is $w=gv_0$, in the orbit of $v_0$. Along the first segment $h\le h(u)+R^*t_0$;
$h(g\sigma_s)\ge h(u)-R^*t_0$, so $h(w)\ge h(u)-R^*t_0+t>h(u)+R^*t_0$; and along the reversed $g\sigma$
every vertex except $w$ lies strictly below $h(w)$. Hence $w$ is the unique highest vertex of $\pi_u$,
$0<h(w)-h(u)\le H_6:=(R^*+s)t_0$, and $|\pi_u|\le M_6:=R^*+s$.

\emph{The map.} Fix $u$ and $x$. Let $z\in K_u\cap L_{-k}(u)$. On $E_z=\{u\text{ open},\ P_u,\
z\in K_u\cap L^x_{-k}(u)\}$, let $\Phi$ open every site of $\pi_u$ and close every site of
$N(\pi_u)\setminus(K_u\cup\pi_u)$. The set $B=\pi_u\cup N(\pi_u)$ has at most $(d+1)(M_6+1)$ sites. After
$\Phi$, the open cluster of $w$ is exactly $K_u\cup\pi_u$: every open neighbour of $\pi_u$ lies in
$K_u\cup\pi_u$, and neighbours of $K_u$ off $\pi_u$ keep their states and were closed, being boundary
sites of $K_u$.

\emph{The peak.} The unique peak of the image cluster is $w$: $K_u$ lies at heights at most
$h(u)<h(w)$, and $\pi_u\setminus\{w\}$ lies strictly below $h(w)$. So $\Phi(E_z)\subseteq E'_z:=\{w\text{
open},\ P_w,\ z\in K_w\cap L_{-k-j}(w)\}$, where $j\in[0,J_6]$ is defined by $u\in L_{-j}(w)$; the
exact shift rule gives $L_{-k}(u)=L_{-k-j}(w)$, and $J_6:=\lceil H_6/t_0\rceil+1$.

\emph{The bound.} Lemma~\ref{lem:Q0} gives $\Pp(E_z)\le\min(p,1-p)^{-|B|}\Pp(E'_z)$. Sum over $z$ and
divide by $p$ on both sides; both events force their root open. Integrate over $x$: the induced offset at
$w$ is $R_w=x-h_u(w)/t_0=x+h_w(u)/t_0\bmod1$, a translation, which preserves Lebesgue measure. Orbit
invariance gives $F(w,\cdot)=F(v_0,\cdot)$ with the equivariant offsets. This proves the claim with
$C_6=\min(p,1-p)^{-(d+1)(M_6+1)}$.
\end{proof}

At a sloped vertex the one-layer comparison of peak moments of~\cite[Auxiliary Lemma~5.G]{TN26} holds in
its full range; at a plateau vertex it can fail at $j=-1$.

\begin{lemma}[{Quasi-transitive form of~\cite[Auxiliary Lemma~5.G]{TN26}}]\label{lem:5G}
At a sloped vertex $v$, for every $j\le-1$, $F_{j-1}(v,x)\le CF_j(v,x)$ and $F_{j+1}(v,x)\le CF_j(v,x)$.
At an arbitrary vertex $v$, the first inequality holds for $j\le-1$ and the second for $j\le-2$.
Consequently~\cite[(5.67)]{TN26} holds at sloped $v$ for $j\le-1$, and at every $v$ for $j\le-2$.
\end{lemma}

The proof (Appendix~\ref{app:b8-devices}) follows~\cite{TN26}, with the neighbours $z^\pm$ replaced by
paths of Lemma~\ref{lem:Q1} truncated at their first visit to the target layer; each path is opened, its
neighbours outside $K_v$ and the path are closed, and Lemma~\ref{lem:Q0} is applied to each marked vertex.
At a plateau vertex $P_v$ forces $K_v=\{v\}$, so $F_j=0$ for $j\le-1$ while $F_0>0$; this is why the case
$j=-1$ needs a sloped vertex.

\subsection{The surgery inequality at a sloped vertex}\label{sec:DT}

The decisive step of the critical analysis in~\cite[Section~5]{TN26} is the surgery
inequality~\cite[(5.76)]{TN26}, the site analogue of~\cite[Lemma~6.4]{HutNU}. It bounds a restricted
first moment, uniformly over roots, by a ratio of moments at one root. Its proof in~\cite{TN26} uses
transitivity in four places: the maximal-decrement neighbours used to leave the explored region, the
independence of a supremum from the root of the residual cluster, the fixed-root transport in the
downstream step~(5.113), and a denominator bound uniform in the root. We replace the first by
Lemmas~\ref{lem:Q3} and~\ref{lem:Q2}, the second by steering the residual root into a maximizing orbit,
the third by Lemma~\ref{lem:Q4}, and the fourth by evaluating at one sloped vertex.

\begin{proposition}[{Quasi-transitive form of~\cite[(5.76)]{TN26}}]\label{prop:DT}
There are integers $a_*\ge2$ and $C_p<\infty$, depending only on $G$ and $p$, such that the following
holds for every $0<p\le p_c$, every $k,r\ge0$ and every $\ell\ge2$. Put $S=k+\ell+r+a_*+1$ and let
$N_\ell(v)$ be the event that the open cluster $A$ of $v$ in the induced graph on $L_{-\ell,\infty}(v)$
has a neighbour in $L_{-\ell-1}(v)$. For every $(v,x)$ with
$\widehat\Pp^{v,x}(N_\ell(v)\cap P_v(S))>0$,
\[
\sup_{v',x'}\widehat\E^{v',x'}\bigl[X^{-k-r,0}_{-k}(v');\ Q_{v';k,\ell+1,r+a_*}\bigr]
\le C_p\,\frac{\sum_{i=1}^{a_*+1}\widehat\E^{v,x}\bigl[X^{-S,0}_{-k-\ell-i}(v)\bigm|P_v(S)\bigr]}
{\widehat\Pp^{v,x}\bigl(N_\ell(v)\bigm|P_v(S)\bigr)}.
\]
\end{proposition}

Compared with~\cite[(5.76)]{TN26}, the left side has the slack parameters $(\ell+1,r+a_*)$, the right
side sums $i\in[1,a_*+1]$ at depth $S$, $N_\ell(v)$ replaces $\{v\lr L_{-\ell}(v)\}$, and the infimum over
$(v,x)$ is restricted to positive denominators. The first three are bounded shifts, absorbed downstream;
the last is harmless, because the downstream use needs one $(v,x)$. These are the quasi-transitive
features of~\cite[(6.19)]{HutNU}.

\paragraph{The two new steps.} The proof (Appendix~\ref{app:DT}) follows~\cite{TN26} with the new indices,
except for two steps. Let $O^*$ be an orbit attaining the left side, which is a maximum over the finitely
many orbits. First, the proof explores the open cluster $A$ of $v$ in $L_{-\ell,\infty}(v)$ with its tested
boundary, keeps the outcomes on which $v$ is the peak of $A$ and $A$ has a neighbour $q_1\in
L_{-\ell-1}(v)$, and descends from $q_1$ along a connector of Lemma~\ref{lem:Q2} (down) to a vertex $q\in
O^*$. Every site of the connector lies at height at most $h(q_1)$, hence in the untested region
$L_{-\infty,-\ell-1}(v)$ (Lemma~\ref{lem:Q3}), and $q\in L_{-\ell-a}(v)$ for some $a\in[2,a_*]$; this
replaces the maximal-decrement neighbours of~\cite[(5.77)]{TN26}. Second, the surgery opens the connector
and closes its neighbours outside $A$, the tested boundary, the residual cluster $C_q$ of $q$ and the
connector, changing at most $(d+1)M_*$ coordinates. As the connector lies strictly below the explored
region, opening it attaches $A$ only to $C_q$, which on the event considered does not reach the heights at
least $h(v)$, so $v$ stays the peak. Because $q\in O^*$, the residual moment at $q$ dominates the moment
$\widehat\E^{q,y}[X^{-k-r,0}_{-k}(q);\ Q_{q;k,\ell+1,r+a_*}]$ for every offset $y$, and the supremum of these
over $y$ is the left side itself; this replaces the transitivity that makes the residual
supremum independent of the root in~\cite{TN26}.

Downstream, Proposition~\ref{prop:DT} is
applied at a sloped vertex $v_0$, where, for every $\zeta>0$ and some $c_\zeta>0$, the critical
peak-survival bound
\begin{equation}\label{eq:591}
\inf_x\widehat\Pp^{v_0,x}_{p_c}\bigl(v_0\lr L_{-k}(v_0),\ P_{v_0}\bigr)\ge c_\zeta
e^{-t_0(\alpha^s_{p_c}-1+\zeta)k}
\end{equation}
holds by the argument of~\cite[(5.85)--(5.91)]{TN26} at $v_0$, whose subcritical input is supplied at
sloped vertices by Lemmas~\ref{lem:Q6} and~\ref{lem:5G}. Here $\alpha^s_p$ is the \emph{height
exponent}: the exponential decay rate in $t$ of the probability that an open root is joined to the
vertices at height at least $t$ above it inside the vertices at height at least its own
(Appendix~\ref{app:b8-list}).

\subsection{Proof of Theorem~\ref{thm:b8}}\label{sec:b8-proof}

\emph{Part~(1).} Appendix~\ref{app:b8} runs~\cite[Section~5]{TN26} in the present setting. With the
devices above, it proves the inputs of~\cite[Proposition~5.5]{TN26} on $G$: the divergence of tilted
moments at $p_c(\lambda)$ at every root, the critical fractional estimate~\cite[(5.34)]{TN26} uniformly
in the root and offset, the first-moment identities~(5.59)--(5.60), and the lower bound~(5.73) there.
The proof of~\cite[Proposition~5.5]{TN26} then gives $p_c<p_c(\lambda)$ for $0<\lambda<1/2$, and, as
there, at $\lambda=1/2$. In place of the fixed-root symmetry used there, the averaged symmetry of
Lemma~\ref{lem:Q4} together with Lemma~\ref{lem:Q5}(a) gives $p_c(\lambda)=p_c(1-\lambda)$, and
Cauchy--Schwarz gives $p_c(\lambda)\le p_c(1/2)$; so $p_c<p_c(\lambda)\le p_c(1/2)$ for $0<\lambda<1$.
Finally $p_t(G,\Gamma)=p_c(1/2)$: if $\sup_u\chi_{p,\lambda}(u)<\infty$ for some $\lambda$, the averaged
symmetry gives finiteness of $\chi_{p,1-\lambda}$ at $\rho$, hence at every root by
Lemma~\ref{lem:Q5}(a), and Cauchy--Schwarz bounds $\chi_{p,1/2}(u)$ by
$(\chi_{p,\lambda}(u)\chi_{p,1-\lambda}(u))^{1/2}$; the reverse inequality is trivial.

\emph{Part~(2).} The kernel $T_p$ is nonnegative and symmetric, so Lemma~\ref{lem:cocycle} gives
$\norm{T_p}\le\sup_u\chi_{p,\lambda}(u)$. For $p<p_t$, $\chi_{p,1/2}$ is finite at every root and its
supremum is a maximum over orbits (Lemma~\ref{lem:Q5}(a)), so $T_p$ is bounded and $\ptwo\ge p_t$. The
bound $\ptwo\le p_u$ is Lemma~\ref{lem:threshold-order}.

\emph{Part~(3).} By part~(1) there is $p'>p_c$ with $\sup_u\chi_{p',1/2}(u)<\infty$, and monotonicity in
$p$ gives $\sup_u\chi_{p_c,1/2}(u)<\infty$; part~(2) bounds $\norm{T_{p_c}}$. The supremum identity is
Lemma~\ref{lem:passage}.
\qed