\section{The vertex-transitive case}\label{app:cocycle}\label{app:vt}

For vertex-transitive graphs, Theorem~\ref{thm:b8} combines Lemma~\ref{lem:cocycle} with the site
tilted-threshold separation of Tran and Nguyen. This deduction is not needed for Theorem~\ref{thm:main},
since Theorem~\ref{thm:b8} covers vertex-transitive graphs too.

\begin{proposition}[{Tran--Nguyen~\cite[Proposition~5.5]{TN26} and Hutchcroft}]\label{prop:vt}
Let $G$ be an infinite, connected, locally finite, vertex-transitive graph whose automorphism group is
nonunimodular. For Bernoulli site percolation:
\begin{enumerate}[label=\textup{(\arabic*)},leftmargin=2.2em]
\item $\norm{T_p}_{2\to2}\le\chi_{p,\lambda}(v)$ for every $p\in[0,1]$ and $\lambda\in\R$;
\item $p_c<p_t=p_c(1/2)\le\ptwo\le p_u$, so in particular $p_c<\ptwo\le p_u$;
\item $\norm{T_{p_c}}_{2\to2}\le\chi_{p_c,1/2}(v)<\infty$, and $\sup_{0\le p<p_c}\norm{T_p}_{2\to2}=
\norm{T_{p_c}}_{2\to2}$.
\end{enumerate}
\end{proposition}

Here $\chi_{p,\lambda}(v)$ does not depend on $v$. The proof identifies the objects
of~\cite[Section~5]{TN26} with Hutchcroft's for the site kernel: the same modular function, the same
normalization in $\lambda$, and $p_{st}=p_c(1/2)=p_t$. Neither source states the combined inequality.

\paragraph{The objects of~\cite[Section~5]{TN26}.} The standing hypothesis of~\cite[Section~5]{TN26} is
that $G$ is infinite, connected, locally finite and vertex-transitive, with nonunimodular canonical
action, and $\Delta=\Delta_{\Aut(G)}$ is taken from~\cite[Section~2.1]{HutNU}; for transitive $\Gamma$
this is $\Delta(x,y)=|\Stab_yx|/|\Stab_xy|$. The fixed-root transport~\cite[(5.30)]{TN26} is the tilted
mass-transport principle~\cite[Proposition~2.2]{HutNU} with the random root replaced by $v$, which is
legitimate for transitive $\Gamma$. The site tilted susceptibility
$\chi_{p,\lambda}(v)=\sum_u\Pp_p(v\lr u)\Delta(v,u)^\lambda$ of~\cite[Auxiliary Lemma~5.A]{TN26} is
Hutchcroft's $X_{p,\lambda}(v)$ for the site kernel $\tau_p=T_p$, with the same exponent convention, and
$p_{sc}(\lambda)=\sup\{p:\chi_{p,\lambda}(v)<\infty\}$ is Hutchcroft's $p_c(\lambda)$ for $\tau_p$. The
conditional version $\widehat\chi_{p,\lambda}=\chi_{p,\lambda}/p$ of~\cite[(5.13)]{TN26} has the same
finiteness set for $p>0$. The $u=v$ term is $\Pp_p(v\lr v)=p$ in both. Finally, \cite{TN26} defines
$p_{st}=p_{sc}(1/2)$, while Hutchcroft defines $p_t=\sup\{p:X_{p,\lambda}<\infty\text{ for some }\lambda\}$;
for the site kernel these agree. Indeed $p_t\ge p_{st}$ is trivial; conversely, if
$\chi_{p,\lambda}<\infty$, then~\cite[(5.30)]{TN26} applied to $F(a,b,\omega)=\1\{a\lr b\text{ in }
\omega\}\Delta(a,b)^\lambda$, which vanishes unless $a$ and $b$ are open, and $\Delta(x,v)=\Delta(v,x)^{-1}$
give $\chi_{p,\lambda}(v)=\chi_{p,1-\lambda}(v)$, and Cauchy--Schwarz against $\Pp_p(v\lr\cdot)$ gives
$\chi_{p,1/2}\le(\chi_{p,\lambda}\chi_{p,1-\lambda})^{1/2}<\infty$, so $p\le p_{sc}(1/2)$, as in the
proof of~\cite[Proposition~5.5]{TN26}.

\paragraph{The inputs of~\cite[Proposition~5.5]{TN26}.} Its hypotheses, the estimates (5.23)--(5.26),
(5.34), (5.59)--(5.60) and (5.73) there, are each derived inside~\cite[Section~5]{TN26} from earlier
material of that section and from the external results cited there, including the tilted
mass-transport principle~\cite[Proposition~2.2]{HutNU} and site sharpness~\cite{AV08}. Only~(5.26) and~(5.34)
enter the inequality $p_{sc}<p_{sc}(\lambda)\le p_{st}$; its proof applies (5.34) with $1-\eps=3/4$ and
(5.26) with $\eta=1/2$, and both exponents match. We use~\cite[Lemma~5.2(2)--(3)]{TN26} as a cited
input; its proofs have not been checked line by line here.

\begin{proof}[Proof of Proposition~\ref{prop:vt}]
(1) $T_p$ is nonnegative and symmetric, and $\chi_{p,\lambda}(u)$ does not depend on $u$, by invariance
and transitivity; apply Lemma~\ref{lem:cocycle}.
(2) The hypothesis of~\cite[Section~5]{TN26} is the present one, so~\cite[Proposition~5.5]{TN26} gives
$p_c<p_{sc}(\lambda)\le p_{st}$ for $0<\lambda<1$; at $\lambda=1/2$, $p_c<p_{st}=p_t$. Since
$\chi_{p,\lambda}(v)$ is nondecreasing in $p$, every $p<p_{st}$ has $\chi_{p,1/2}<\infty$, hence
$\norm{T_p}<\infty$ by~(1), and $\ptwo\ge p_t$. The bound $\ptwo\le p_u$ is
Lemma~\ref{lem:threshold-order}.
(3) By~(2) there is $p'>p_c$ with $\chi_{p',1/2}<\infty$, hence $\chi_{p_c,1/2}<\infty$, and~(1) bounds
$\norm{T_{p_c}}$. The supremum identity is Lemma~\ref{lem:passage}.
\end{proof}