\section{Proof of Theorem~\ref{thm:main} from the two cases}\label{sec:assembly}

The proof rests on two theorems, one for each type of automorphism group. The first is the site
form of~\cite[Theorem~2.6]{OAINU}, proved in Sections~\ref{sec:goodness}--\ref{sec:critical}.

\begin{theorem}[Critical boundedness for a unimodular action]\label{thm:unimod}
Let $G$ be an infinite, connected, locally finite, simple, quasi-transitive graph with $h_V(G)>0$. If
$\Aut(G)$ is unimodular, then for Bernoulli site percolation $\norm{T_{p_c}}_{2\to2}<\infty$.
\end{theorem}

The second is proved in Section~\ref{sec:nonunimod}. Let $\Delta$ be Hutchcroft's modular function of
$\Gamma=\Aut(G)$~\cite[Section~2.1]{HutNU}, and for Bernoulli site percolation put
\[
\chi_{p,\lambda}(u)=\sum_xT_p(u,x)\Delta(u,x)^\lambda,\qquad
p_c(\lambda)=\sup\Bigl\{p:\sup_u\chi_{p,\lambda}(u)<\infty\Bigr\},
\]
and $p_t(G,\Gamma)=\sup\{p:\sup_u\chi_{p,\lambda}(u)<\infty\text{ for some }\lambda\in\R\}$.

\begin{theorem}\label{thm:b8}
Let $G$ be an infinite, connected, locally finite, quasi-transitive graph such that $\Gamma=\Aut(G)$
is nonunimodular. For Bernoulli site percolation:
\begin{enumerate}[label=\textup{(\arabic*)},leftmargin=2.2em]
\item $p_c<p_c(\lambda)\le p_c(1/2)=p_t(G,\Gamma)$ for every $0<\lambda<1$;
\item $\norm{T_p}_{2\to2}\le\sup_u\chi_{p,\lambda}(u)$ for every $p$ and $\lambda$; hence
$p_c<p_t(G,\Gamma)\le\ptwo\le p_u$;
\item $\norm{T_{p_c}}_{2\to2}\le\sup_u\chi_{p_c,1/2}(u)<\infty$, and $\sup_{p<p_c}\norm{T_p}_{2\to2}=
\norm{T_{p_c}}_{2\to2}$.
\end{enumerate}
Vertex-transitivity is neither assumed nor excluded.
\end{theorem}

For vertex-transitive $G$, Theorem~\ref{thm:b8} is the combination of~\cite[Proposition~5.5]{TN26}
with Hutchcroft's operator bound (Proposition~\ref{prop:vt}). The nonunimodular case uses no
nonamenability hypothesis, since a nonunimodular automorphism group forces nonamenability
(Section~\ref{sec:b8-setting}).

\begin{proof}[Proof of Theorem~\ref{thm:main}]
By Section~\ref{sec:conventions} we may assume that $G=G_s$ is simple. We split according to whether
$\Aut(G)$ is unimodular; the two cases are exhaustive.

\emph{$\Aut(G)$ unimodular.} Theorem~\ref{thm:unimod} gives $\norm{T_{p_c}}<\infty$. By
Lemma~\ref{lem:crit-finite}, $p_c<1$, so Lemma~\ref{lem:sprinkling} at $p=p_c$ with
$0<a<(d(1+d\norm{T_{p_c}}))^{-1}$ gives a bounded $T_{p_2}$ at $p_2:=p_c+(1-p_c)a\in(p_c,1)$; hence
$p_c<\ptwo$. Lemma~\ref{lem:threshold-order} gives $\ptwo\le p_u$, and Lemma~\ref{lem:passage} with
$p_*=p_c$ gives the supremum identity. This proves (1) and (2). For (3), choose $p_1\in(p_c,p_2)$ and
apply Proposition~\ref{prop:coupled}.

\emph{$\Aut(G)$ nonunimodular.} Theorem~\ref{thm:b8}(2)--(3) gives
$p_c<p_t(G,\Aut G)\le\ptwo\le p_u$, $\norm{T_{p_c}}<\infty$ and
$\sup_{p<p_c}\norm{T_p}=\norm{T_{p_c}}$, which are (1) and (2). For (3), pick $p_2\in(p_c,\ptwo)$. Then
$T_{p_2}$ is bounded, because the parameters with bounded $T_p$ form a downward interval by entrywise
domination~\cite[Lemma~2.4]{OAINU}; and $p_2<\ptwo\le p_u\le1$. Take $p_1\in(p_c,p_2)$ and apply
Proposition~\ref{prop:coupled}.
\end{proof}