\section{Discussion}\label{sec:discussion}

\paragraph{Where nonamenability enters the unimodular case.} On an amenable quasi-transitive graph
$p_c=p_u$, so an argument that ran unchanged there would prove a false statement. In the unimodular
case nonamenability enters through $\rho=\norm P<1$ (Lemma~\ref{lem:walk}) and through $p_c<1$. The
decisive uses are those of $\rho<1$: the starting map of Lemma~\ref{lem:initial} needs $H>1$, that is,
an endpoint kernel of norm less than one; the norm loss $2t\rho^km_0^2$ of Proposition~\ref{prop:norm}
and the error $\rho^ka_3$ of Proposition~\ref{prop:decrease} need decay in $k$; the smoothing and escape
estimates of Lemmas~\ref{lem:smoothing}--\ref{lem:escapes} need $\rho^l\to0$; and the choice of $k$ in
the proof of Theorem~\ref{thm:unimod} uses $\log(1/\rho)>0$. On an amenable unimodular quasi-transitive
graph every invariant finite-range stochastic kernel on an orbit has norm one, by testing against
F\o lner sets with column sums one (Lemma~\ref{lem:MTP}), so $H=0$ for every local endpoint map and
Proposition~\ref{prop:efficient} fails there; the moment argument then has no input. None of the site changes, namely the Russo remainder, the degree-dependent
constants, the deleted domains $V\setminus\Nb[A]$, junction rejection, the pruning lemma, joint restoration
in the second moment, the induced corridors and the site avoidance event of Lemma~\ref{lem:escapes},
touches these uses; each changes only
graph-dependent constants or the deleted set.

\paragraph{Where nonunimodularity enters.} If $\Aut(G)$ is unimodular, then $\Delta\equiv1$,
$\chi_{p,\lambda}=\chi_p$ for every $\lambda$, and $p_t=p_c$, so the conclusion $p_c<p_t$ of
Theorem~\ref{thm:b8}(1) is false; for amenable graphs it must be, since $p_c=\ptwo=p_u$ there. The
argument of Section~\ref{sec:nonunimod} uses nonunimodularity in four places: $t_0>0$ and $\delta_0>0$,
for every connector and layer argument; the existence of sloped vertices, for Lemma~\ref{lem:Q6}; the
tilt in~\cite[Proposition~5.5]{TN26}, which compares the divergence of a tilted moment at $p_c(\lambda)$
with the upward decay $e^{-(t_0-\zeta)k}$ of~\cite[(5.34)]{TN26}; and the lower bound $\alpha^s_{p_c}\ge1$
of~\cite[(5.73)]{TN26} for the height exponent of~\eqref{eq:591}. This bound comes from the tilted symmetry $\widehat\chi_{p,1}=\widehat\chi_{p,0}$ and is
the source of the asymmetry between upward and downward crossings. The argument does not need $G$ to be
non-vertex-transitive: for vertex-transitive $G$ every vertex is sloped, $\mu$ is a point mass,
and the argument reduces to~\cite[Section~5]{TN26}.

\paragraph{The full nonuniqueness interval.} Theorem~\ref{thm:main}(3) gives one interval
$[p_1,p_2]$. H\"aggstr\"om, Peres and Schonmann prove that, for bond percolation on every infinite,
locally finite, connected, quasi-transitive graph, in the standard coupling almost surely
$N(p)=\infty$ for all $p\in(p_c,p_u)$~\cite[Theorem~1.3]{HPS99}, and they state that their results and
proofs may be adapted to site percolation~\cite[Section~1]{HPS99}. With that adaptation, which we have
not checked, Theorem~\ref{thm:main}(2) gives infinitely many infinite site clusters simultaneously for all
$p\in(p_c,p_u)$.

\paragraph{Scope.} The bond paper also deduces mean-field critical behaviour and exponential
connectivity below $\ptwo$ from the operator bound~\cite[Corollary~1.3]{OAINU}. We have not checked the
site forms of those deductions, and claim none of them.