\section{Proofs of the site tools and of the bond corollary}\label{app:tools}

This appendix proves the statements of Section~\ref{sec:tools}, which are site adaptations of the
corresponding lemmas of~\cite[Section~2]{OAINU}, writing out every changed step, and then
Corollary~\ref{cor:bond}.

\subsection{The site tools}

\begin{proof}[Proof of Lemma~\ref{lem:crit-finite}]
An open self-avoiding path with $n+1$ sites from $x$ has probability at most $d^np^{n+1}$, so there is
no infinite cluster when $pd<1$. For the upper bound, the number of connected sets of size $n$
containing $x$ is at most $d^{2n}$~\cite[Lemma~2.2]{OAINU}. If $C_x=A$ then $\partial A$ is closed and
$|\partial A|\ge h_Vn$, so $\Pp_p(|C_x|<\infty)\le(1-p)+\sum_{n\ge1}d^{2n}(1-p)^{h_Vn}<1$ for $p$ near
$1$. As in~\cite[Lemma~2.2]{OAINU}, $|B(x,r+1)|\ge(1+h_V)|B(x,r)|$, so $G$ has exponential growth, and
with $p_c<1$ the site theorem~\cite[Theorem~1.1(2)]{Pei26b} gives no infinite cluster at $p_c$.
Countability gives all vertices at once, and a cluster in an induced subgraph lies inside a full-graph
cluster.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:tools}]
Harris's inequality~\cite{Har60} and the van den Berg--Kesten inequality~\cite{vdBK85}: the proofs
quoted in~\cite[Lemma~2.3]{OAINU} are for an arbitrary finite product of Bernoulli bits followed by
exhaustion, and apply verbatim to vertex bits. For the tree bound, on the event take a simple open path
$\pi$ from $x$ to $y$ in $D$ and a simple open path from $z$ to $x$, and let $w$ be its first vertex
on $\pi$. The three sets $\pi[x,w]$ (which contains $w$), $\pi(w,y]$ and the part of the $z$-path
strictly before $w$ are pairwise disjoint sets of open vertices. The second is empty if and only if
$w=y$, and otherwise it is an open path from a neighbour of $w$ to $y$; likewise for $z$. A union over
$w$ and the two neighbours, and the van den Berg--Kesten inequality, give
$\Pp\le\sum_wT(x,w)(I+\Adj T)(w,y)(I+\Adj T)(w,z)$, and~\eqref{eq:H} gives~\eqref{eq:tree}. Summing
over $y,z$ gives $\E_ps_x^2\le\kappa_p\chi_{\max}^3$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:threshold-order}]
The proof of~\cite[Lemma~2.4]{OAINU} applies verbatim. Under almost sure uniqueness at $p$, some
vertex has $\Pp_p(|C_x|=\infty)>0$, by countability. Harris's inequality with the event that the
finitely many sites of a fixed path from $x$ to $y$ are open makes this positive at every vertex, and
the finitely many types give $\inf_x\theta_x=\theta>0$. Uniqueness and Harris's inequality give
$T_p(x,y)\ge\Pp_p(x\lr\infty,\ y\lr\infty)\ge\theta^2$, so $\ip{\1_F}{T_p\1_F}\ge\theta^2|F|^2$ for
finite $F$, and $T_p$ is unbounded. The site kernel $T_p$ is entrywise increasing in $p$, so the
parameters with bounded $T_p$ form a downward interval, and $\ptwo\le p_u$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:sharpness}]
Finiteness of $\chi_x(p)$ for $p<p_c$ is site sharpness on quasi-transitive
graphs~\cite[Theorem~2]{AV08}, \cite[Corollary~8.3]{Pei26b}; the abstract of~\cite{AV08} states that
its results apply to site as well as bond percolation. Since $\chi_x(p)$ depends only on the orbit of
$x$, $\chi_{\max}(p)$ is a maximum over finitely many orbits and is finite. The second moment is Lemma~\ref{lem:tools}. For the lower comparison, along a path
$x=z_0,\dots,z_l=y$, $\chi_x\ge\E[|C_y|\1\{z_0,\dots,z_{l-1}\text{ open}\}]\ge p^l\chi_y$ by Harris's
inequality. Positive semidefiniteness: $\sum_{x,y}\bar f(x)T(x,y)f(y)=\E\sum_C|\sum_{x\in C}f(x)|^2$, the
sum over open clusters, which is consistent with $T(x,x)=\Pp(x\text{ open})$. The rest is as
in~\cite[Lemma~2.10]{OAINU}.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:russo}]
\emph{Finite domain.} In a finite induced $\Lambda$ the event $\{x\lr y\text{ in }\Lambda\}$ is
increasing in finitely many site bits, and finite-product differentiation gives
$\frac d{dp}\Pp=\sum_v\Pp(v\text{ pivotal})$. Pivotality of $v$ does not depend on $\omega_v$, so
$\Pp(v\text{ pivotal})=(1-p)^{-1}\Pp(v\text{ closed},\ v\text{ pivotal})$. A closed $v$ is pivotal if
and only if $x\nlr y$ and both $x,y$ lie in the cluster of $v$ after opening it, that is, $x,y\in
\widehat C^\Lambda_v$. This is~\eqref{eq:russo-a} in $\Lambda$, with each closed pivotal site counted
once.

\emph{Splitting.} Consider first $v\notin\{x,y\}$. Then $x\in C_a$ and $y\in C_b$ with $a,b\sim v$, so
$x,y$ are open, $C_x\ne C_y$ and $v\in\partial C_x\cap\partial C_y$. Conversely, every $v\in\partial
C_x\cap\partial C_y$ with $C_x\ne C_y$ nonempty is closed and pivotal. This gives $K^c$. The terms $v=x$
and $v=y$ give $R$; when $x=y$ they are one term, equal to $\Pp(x\text{ closed})=1-p$. For $x\ne y$, the
event $\{x\text{ closed},\ y\in\widehat C_x\}$ requires $y\in C_a$ for some $a\sim x$, so its probability
is at most $(\Adj T)(x,y)$; hence the bound on $R$.

\emph{Majorant.} $\1\{x,y\in\widehat C_v,\ x\nlr y\}\le(\delta_{xv}+\sum_{a\sim v}\1\{x\in C_a\})
(\delta_{yv}+\sum_{b\sim v}\1\{y\in C_b\})$, and when $x\in C_a$, $y\in C_b$ and $C_a\ne C_b$ the two
clusters are disjoint witnesses, so the van den Berg--Kesten inequality gives
$(1-p)T'(x,y)\le\sum_v(I+T\Adj)(x,v)(I+\Adj T)(v,y)$.

\emph{Exhaustion.} Fix $[p',p'']\subset(0,p_1]$ with $p_1<p_c$. For $\Lambda_n\uparrow V$, the indicator
$\1\{v\text{ closed},\ x,y\in\widehat C^{\Lambda_n}_v,\ x\nlr y\text{ in }\Lambda_n\}$ converges pointwise
to its full-graph version: finite open paths witness membership in $\widehat C_v$, and connection in $V$
is eventually witnessed in $\Lambda_n$. The summands are dominated by
$(I+T_{p_1}\Adj)(x,v)(I+\Adj T_{p_1})(v,y)$, which is summable in $v$ and $y$, with row sums at most
$(1+d\chi_{\max}(p_1))^2$. Dominated convergence passes the integrated identity to $V$.

\emph{Operator norm.} From here the proof of~\cite[Lemma~2.11]{OAINU} applies verbatim with the majorant
$(1-p_1)^{-1}(I+T_{p_1}\Adj)(I+\Adj T_{p_1})$ in place of the bond majorant: row differentiation in
$\ell^1$, finitely many row types, invariance, symmetry and the row and column bound.

\emph{The identity~\eqref{eq:russo-c}.} Summing~\eqref{eq:russo-a} over $y$, which the same domination
justifies, gives $(1-p)\chi'_x=\sum_v\E[\1\{v\text{ closed},\ x\in\widehat C_v\}(|\widehat C_v|-s_x)]$. If
$x\in C_a\subseteq\widehat C_v$, the $y\in\widehat C_v$ with $y\nlr x$ are those of
$\widehat C_v\setminus C_x$; if $x=v$ is closed, $s_x=0$. Lemma~\ref{lem:MTP} with the nonnegative mass
$m(x,v)=\E[\1\{v\text{ closed},\ x\in\widehat C_v\}(|\widehat C_v|-s_x)]$ moves the root to $v$:
$(1-p)\chi'=[\E\1\{o\text{ closed}\}\sum_{x\in\widehat C_o}(|\widehat C_o|-s_x)]=[\E\1\{o\text{
closed}\}(|\widehat C_o|^2-\sum_in_i^2)]$. Finally $(1+\sum_in_i)^2-\sum_in_i^2=1+2N_o+Q_o$.

\emph{The bounds~\eqref{eq:russo-d}.} $Q_o\ge0$, and $\E[\1\{o\text{ closed}\}N_o]\le\sum_{a\sim o}
\chi_a\le d\chi_{\max}$.
\end{proof}

For comparison with the bond quantity, $Q_o\le\sum_{b,c\sim o}s_bs_c\1\{C_b\ne C_c\}\le(d^2/4)Q_o$, since
each cluster $i$ is met by $m_i$ neighbours of $o$ and $m_im_j\le d^2/4$.

\begin{proof}[Proof of Lemma~\ref{lem:sprinkling}]
Take independent site fields $\omega_1\sim\Pp_p$ and $\omega_2\sim\Pp_a$; their union has law
$\Pp_{p+(1-p)a}$. On a simple union-open path $x=z_0,\dots,z_n=y$, let $s_1,\dots,s_j$ be its
$\omega_1$-closed vertices in order; they are $\omega_2$-open. The maximal $\omega_1$-open segments
between them are pairwise disjoint $\omega_1$-open witnesses. The segment before $s_1$ is empty
($x=s_1$) or joins $x$ to a neighbour of $s_1$, with kernel $I+T_p\Adj$. A segment between $s_r$ and
$s_{r+1}$ is empty ($s_r\sim s_{r+1}$) or joins a neighbour of $s_r$ to a neighbour of $s_{r+1}$, with
kernel $\Adj+\Adj T_p\Adj=\Adj(I+T_p\Adj)$. The final segment gives $I+\Adj T_p$. Independence of
$\omega_2$ at the distinct $s_r$ gives $a^j$, and the van den Berg--Kesten inequality in $\omega_1$ gives
the product; extra terms only increase the bound. For the norms, $\norm{I+T_p\Adj}\le1+d\norm{T_p}$ and
$\norm{\Adj(I+T_p\Adj)}\le d(1+d\norm{T_p})$, so the series converges when $ad(1+d\norm{T_p})<1$. If
$p_c<1$, a positive such $a$ exists at $p=p_c$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:passage}]
The proof of~\cite[Lemma~9.2]{OAINU} applies verbatim: finite-domain site connection probabilities are
polynomials in $p$, and connectivity is an increasing union of finite-domain events, so
$T_p(x,y)\uparrow T_{p_*}(x,y)$ as $p\uparrow p_*$, and the bilinear-form argument there is
model-free.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:coupled}]
We follow~\cite[Proposition~9.3]{OAINU}. Let $N_{12}$ be the number of $p_2$-clusters containing an
infinite $p_1$-cluster. Every automorphism-invariant event of the label field has probability zero or
one, by approximation with finite-coordinate events of vertex labels and an automorphism moving their
support off itself; so $N_{12}$ is almost surely a constant, which is positive since $p_1>p_c$.

\emph{$N_{12}\ne1$.} Put $\theta(x)=\Pp_{p_1}(|C_x|=\infty)$. Some $\theta(x)$ is positive, and
$\theta(x)\ge p_1^l\theta(y)$ along an open path of length $l$ by Harris's inequality, so
$\theta_*=\inf_x\theta(x)>0$ by the finitely many types. If $N_{12}=1$, Harris's inequality gives
$T_{p_2}(x,y)\ge\theta_*^2$, contradicting boundedness as in Lemma~\ref{lem:threshold-order}.

\emph{$N_{12}$ is not a finite $n\ge2$.} By countability there are fixed $x,y$ which, with positive
probability, lie in infinite $p_1$-clusters in distinct $p_2$-clusters. Replace the labels of the
vertices of a fixed finite path from $x$ to $y$, endpoints included, by independent uniforms on
$[0,p_1]$. The modified law is absolutely continuous with respect to the original, and the modification
only opens sites at both parameters. Opening finitely many sites, each adjacent to at most $d$ old
clusters, joins only finitely many old clusters into any new cluster, so every infinite new
$p_1$-cluster contains an infinite old one; every old counted $p_2$-cluster lies in a counted new one;
and old $p_2$-clusters only merge. The two distinguished counted clusters merge, so the new count is at
most $n-1$, contradicting absolute continuity. Hence $N_{12}=\infty$ almost surely.

\emph{Simultaneity.} On this event choose an infinite $p_1$-cluster in each of infinitely many distinct
counted $p_2$-clusters. For every $p\in[p_1,p_2]$ their containing $p$-clusters are infinite and
distinct, because their $p_2$-clusters are distinct.
\end{proof}

\subsection{Bond percolation: proof of Corollary~\ref{cor:bond}}\label{app:bond}

\begin{proof}[Proof of Corollary~\ref{cor:bond}]
Let $L=L(G)$ be the line graph: its vertices are the edges of $G$, loops and parallel copies
included, and two distinct edges are adjacent when they share an endpoint. Let $d$ be the maximal
number of edges at a vertex of $G$. Then $L$ is infinite, connected and locally finite, and
automorphisms of $G$ act on $L$ with finitely many orbits, because every edge contains a vertex from
a finite set of orbit representatives, each of which lies in at most $d$ edges. For a finite nonempty
$F\subseteq V(L)$, let $W$ be the set of endpoints of edges in $F$. Each vertex $y\in\partial W$ is
joined by an edge $g$ to some $w\in W$; the edge $g$ is not in $F$, since its endpoint $y$ is not in
$W$, and $g$ shares $w$ with an edge of $F$. Distinct $y$ give distinct $g$, so
$|\partial_LF|\ge|\partial W|\ge h_V(G)|W|\ge(h_V(G)/d)|F|$, as $|F|\le d|W|$. Hence $h_V(L)>0$.

Bond percolation on $G$ with edge labels is site percolation on $L$ with the same labels. The open
clusters of $L$ are the edge sets of the bond clusters that contain an edge, and a bond cluster is
infinite if and only if its edge set is, by local finiteness. Hence the numbers of infinite clusters
agree for every configuration, so $p_c$ and $p_u$ agree, and~(3) for bond percolation on $G$ is~(3)
of Theorem~\ref{thm:main} for $L$. For the two-point kernels, $x\lr y$ in bond percolation on $G$
means $x=y$, or some open edge $e\ni x$ and some open edge $f\ni y$ lie in one open cluster of $L$.
With the incidence kernel $B(x,e)=\1\{x\in e\}$ this gives, entrywise,
\[
T_p^{\mathrm{bond}}\le I+B\,T_p^{\mathrm{site},L}B^*,\qquad\norm{B}^2=\norm{BB^*}\le2d,
\]
since $\sum_y(BB^*)(x,y)=\sum_{e\ni x}|e|\le2d$. Theorem~\ref{thm:main}(1) for $L$ therefore bounds
$T^{\mathrm{bond}}_{p_c}$, and~(2) for $L$ gives $\ptwo^{\mathrm{bond}}(G)\ge\ptwo^{\mathrm{site}}(L)>
p_c$. The inequality $\ptwo^{\mathrm{bond}}\le p_u^{\mathrm{bond}}$ is~\cite[Lemma~2.4]{OAINU}, and the
supremum identity is~\cite[Lemma~9.2]{OAINU}.
\end{proof}