\section{Critical boundedness for a unimodular action}\label{sec:critical}

This section proves Theorem~\ref{thm:unimod}. It follows~\cite[Sections~5--9]{OAINU}. The efficient
maps of Proposition~\ref{prop:efficient} give critical fractional moments, hence an almost linear
susceptibility bound and a subpower bound for the triangle trace; a deformation of $T_p$ by pivotal
corridors then contradicts unboundedness of $T_{p_c}$. Section~\ref{sec:moments} states the site
forms of the moment and trace estimates, whose proofs are routine adaptations given in
Appendix~\ref{app:critical}. The new ingredient is the site corridor, Sections~\ref{sec:corridors}
and~\ref{sec:legs}. Throughout, $G$ is simple, infinite, connected, locally finite and
quasi-transitive with $h_V(G)>0$, and $\Aut(G)$ is unimodular.

\subsection{Moments, susceptibility and the triangle trace}\label{sec:moments}

\begin{proposition}[{Critical fractional moments; site form of~\cite[Proposition~5.3]{OAINU}}]
\label{prop:moments}
For every $0<\alpha<1/2$, $\sup_v\E_{p_c}s_v^\alpha<\infty$.
\end{proposition}

The proof is by contradiction with~\eqref{eq:heavy}, and uses Proposition~\ref{prop:efficient}
only through four properties of its maps: success implies an output $y\in C_x\cap O$ with $s_x\le M$;
locality; covariance; and the efficiency inequality. A map is run at $p$ slightly below $p_c$, in the
deleted domain $V\setminus\Nb[A]$ of an independent size-biased cluster $A=C_x$. Its small norm sends
endpoint mass outside $\Nb[A]$, and comparing the susceptibility of that domain with $\chi$ bounds $\Psi(p)$
from below by $c_\gamma\chi^{-\gamma}$ for every $\gamma>0$, so that the inequality $(1-p)\chi'\ge
\chi^2\Psi(p)$ of Lemma~\ref{lem:russo} gives $\chi'\ge c\chi^{2-\gamma}$;
an exploration comparison with critical percolation then contradicts~\eqref{eq:heavy}. For sites,
removing $\Nb[A]$ rather than $A$ makes the exterior bits fresh, and the upper bound for the
deleted-domain potential costs a factor $d$, from the at most $d$ clusters adjacent to a closed vertex
of $\partial A$ (Appendix~\ref{app:moments}).

\begin{proposition}[{Almost linear susceptibility; site form of~\cite[Proposition~5.5]{OAINU}}]
\label{prop:susceptibility}
For every $\eta>0$ there is $C_\eta$ with $\chi_{\max}(p)\le C_\eta(p_c-p)^{-1-\eta}$ for
$0\le p<p_c$.
\end{proposition}

This is the decision-tree argument of~\cite[Proposition~5.5]{OAINU}, with the covariance bound
\cite[Lemma~5.4]{OAINU} for an arbitrary finite product law and an exploration by site queries, whose
revealment is at most $(d+2)C_an^{-a}$ by Proposition~\ref{prop:moments}.

Write $T=T_p$, $m_p=\norm{T_p}$ and $a_j=\tr T^j$.

\begin{proposition}[{Subpower triangle growth; site form of~\cite[Proposition~6.2]{OAINU}}]
\label{prop:trace}
For every $\eta>0$, $a_3(p)\le C_\eta(p_c-p)^{-\eta}$ for $p_c/2\le p<p_c$.
\end{proposition}

The proof (Appendix~\ref{app:trace}) differentiates $a_3$ by Lemma~\ref{lem:russo}. The contact term
is controlled, as in~\cite[Lemma~6.1]{OAINU}, by comparing a full and an exterior exploration score of
one cluster; for sites their difference is exactly $-p\,k_s(A,C_v)$, that is, $-p$ times the number of
common boundary vertices, all of which are closed. The remainder contributes $3\tr(T^2R_p)\le
C\,a_3$, by Lemma~\ref{lem:trace} and $a_2\le a_3/p$, and adds a bounded coefficient to the
differential inequality; and $a_3\ge p^3$ replaces $a_3\ge1$.

\subsection{Induced corridors}\label{sec:corridors}

Fix $p_c/2\le p<p_c$, $q=1-p$ and $k\ge3$, and write $T_0=T_p$, $m_0=\norm{T_0}$, $\chi_0=\chi(p)$ and
$\chi_0^+=\max_x\E_ps_x$, as in~\cite[Section~7]{OAINU}. The deformation of~\cite{OAINU} penalizes a
connection whenever it must traverse an isolated open corridor, a path specified by the loop erasure of
a raw walk path whose edges are all pivotal. For sites the corridor is a set of pivotal vertices. A
chord between two of its vertices, or an open neighbour of one of them, would let a connection bypass
the vertices in between, so the corridor must be an induced path surrounded by closed vertices. We
obtain an induced path by shortcutting the loop erasure greedily.

\begin{definition}[{Site templates; replaces~\cite[Definition~7.1]{OAINU}}]\label{def:templates}
\leavevmode
\begin{itemize}
\item A \emph{raw path} $w=(w_0,\dots,w_k)$ is a $k$-step path of $P$, with base $u=w_0$, endpoint
$v=w_k$ and probability $\mu_u(w)=\prod_{i<k}P(w_i,w_{i+1})$, as in~\cite[Definition~7.1]{OAINU}.
\item Let $\lambda_w$ be its chronological loop erasure, with holding steps removed. Its \emph{greedy
shortcut} $\gamma_w=(g_0=u,g_1,\dots,g_\ell=v)$ is defined by $i_0=0$ and
$i_{t+1}=\max\{j>i_t:\lambda_w(j)\sim\lambda_w(i_t)\}$, $g_t=\lambda_w(i_t)$. Then $\gamma_w$ is an
\emph{induced} path: if $g_s\sim g_t$ with $t\ge s+2$, the maximality at step $s$ is contradicted. Its
vertices form a subset of the vertex set $\cW(w)$ of the raw path.
\item $\ell_w$ is the length of $\gamma_w$ and $I_w=\{g_1,\dots,g_{\ell-1}\}$ its interior, so
$u,v\notin I_w$. The raw path is retained as a \emph{template} if and only if $\ell_w\ge3$.
\item The \emph{blockers} are $B_w=N(I_w)\setminus\gamma_w$, with $b_w=|B_w|$. The weight is
$c(w)=\mu_u(w)\,p^{-|I_w|}q^{-b_w}$.
\item A template is \emph{eligible} in a domain $D$ if $\gamma_w\subseteq D$. A blocker outside $D$
counts as closed (absent).
\item For $x,y\in D$ connected in $D$, $N^D_{xy}$ is the sum of $c(w)$ over eligible templates such
that $x,y\notin I_w$, every vertex of $I_w$ is open and pivotal for $x\lr y$ in $D$ (closing it
destroys the connection), and every blocker is closed or absent. Put $N^D_{xy}=0$ on disconnection.
\item As in~\cite[Definition~7.1]{OAINU}, $W^D_t(x,y)=\1\{x\lr y\text{ in }D\}e^{-tN^D_{xy}}$,
$X^D_t(x)=\sum_{y\in D}W^D_t(x,y)$, $s^D_t(x)=\E_pX^D_t(x)$, $T_t(x,y)=\E_pW^V_t(x,y)$,
$\chi_t=[s^V_t(o)]$ and $m_t=\norm{T_t}$. Now $W^D_t(x,x)=\1\{x\text{ open}\}$.
\end{itemize}
\end{definition}

\begin{lemma}[Isolation]\label{lem:isolation}
Let a template be counted for $x\lr y$ in $D$. Then every simple open $x$--$y$ path in $D$ contains $u$,
then all of $I_w$ consecutively in corridor order, then $v$, or the same in reverse order.
\end{lemma}

\begin{proof}
Pivotality puts every interior vertex on the path. An open, present neighbour of $g_i$ outside $I_w$ is
$u$ (only if $i=1$) or $v$ (only if $i=\ell-1$): there is no chord, by inducedness, and blockers are
closed or absent. The path starts and ends outside $I_w$, so it enters $I_w$ through one of $u,v$ and
leaves through the other, and it cannot re-enter.
\end{proof}

\begin{lemma}[{Basic bounds and monotonicity; site form of~\cite[Lemma~7.2]{OAINU}}]\label{lem:basic}
There is a fixed $C$ with $c_k=e^{Ck}$ such that $\sum_{w:z\in I_w}c(w)\le c_k/k$ for every vertex
$z$. Moreover $N^D_{xy}\le c_k|C^D_x|$; $W^D_t(x,y)$ is increasing under site openings and domain
enlargement; $T_t$ is invariant and symmetric with $0\le T_t\le T_0$; and
$m_t\le\sup_xs^V_t(x)\le c_\sigma\chi_t$. Differentiability in $t$ holds as in~\cite{OAINU}.
\end{lemma}

The weight bound combines $p^{-|I_w|}q^{-b_w}\le((2/p_c)(1-p_c)^{-d})^k$ with the fact that the raw paths
through $z$ have total walk probability at most $k+1$. Monotonicity follows from
Lemma~\ref{lem:isolation}: a template counted after an enlargement contains $u$, $I_w$ and $v$ on every
old simple open path, so it was eligible, open and pivotal before. The proof is in
Appendix~\ref{app:basic}.

\begin{lemma}[{Corridor calibration; site form of~\cite[Lemma~7.3]{OAINU}}]\label{lem:calibration}
Let $Q_w=\{I_w\text{ open},\ B_w\text{ closed}\}$ and $D_w=V\setminus(I_w\cup B_w)=(V\setminus\Nb[I_w])
\cup\{u,v\}$. Then $Q_w$ is independent of the percolation in $D_w$ and $c(w)\Pp_p(Q_w)=\mu_u(w)$. For
$x,y\notin I_w$, on $Q_w$ the template is counted for $x\lr y$ in $V$ if and only if
$x\in C_u^{D_w}$, $y\in C_v^{D_w}$ and $C_u^{D_w}\ne C_v^{D_w}$, or the same with $u,v$ interchanged.
\end{lemma}

\begin{proof}
\emph{Independence and calibration.} $Q_w$ is determined by the bits of $I_w\cup B_w$, which is disjoint
from $D_w$, and $\Pp(Q_w)=p^{|I_w|}q^{b_w}$.
\emph{($\Leftarrow$)} On $Q_w$ the open set is $\mathrm{open}(D_w)\cup I_w$, and $I_w$ is joined to the
rest of the open set only through $u$ and $v$, by inducedness and the closed blockers. So $x\lr y$ via
$u$, the corridor and $v$, and there is no $D_w$-bypass; every interior vertex is pivotal.
\emph{($\Rightarrow$)} By Lemma~\ref{lem:isolation}, the path splits into an open $x$--$u$ path and an
open $v$--$y$ path. These avoid $I_w$, which is traversed once, and $B_w$, which is closed, so they lie
in $D_w$. If $C_u^{D_w}=C_v^{D_w}$, there is a bypass, contradicting pivotality.
\emph{Terminals.} If $x\in B_w$, then $x$ is closed on $Q_w$ and lies outside $D_w$, and both sides are
false.
\end{proof}

\begin{proposition}[{Small change of operator norm; site form of~\cite[Proposition~7.4]{OAINU}}]
\label{prop:norm}
For every $t\ge0$, $0\le T_0-T_t\le2t\,T_0P^kT_0$ entrywise, and $m_t\ge m_0-2t\rho^km_0^2$.
\end{proposition}

\begin{proof}
``Counted'' implies $Q_w$. Lemma~\ref{lem:calibration}, independence, and the van den Berg--Kesten
inequality for the two distinct exterior clusters in $D_w$, with $T^{D_w}\le T_0$, bound the term of
$w$ by $\mu_u(w)(T_0(x,u)T_0(v,y)+T_0(x,v)T_0(u,y))$. The rest is as
in~\cite[Proposition~7.4]{OAINU}.
\end{proof}

\subsection{Two legs and the decrease of the susceptibility}\label{sec:legs}

The deformation decreases $\chi_t$ at a rate bounded below by the deformed susceptibilities at the two
corridor endpoints, in the domain with the corridor deleted. Two estimates of~\cite[Section~7]{OAINU}
carry over with site constants; their proofs are in Appendix~\ref{app:deformation}.

\begin{lemma}[{Deletion estimate; site form of~\cite[Lemma~7.6]{OAINU}}]\label{lem:deletion}
For $A'\subseteq D$ and every $v$,
$s^{D\setminus A'}_t(v)\ge s^D_t(v)-C_{\mathrm{del}}(p)\,\chi_0^+\sum_{z\in\Nb[A']}T_0(v,z)$.
\end{lemma}

\begin{lemma}[{Two distinct exterior clusters; site form of~\cite[Lemma~7.7]{OAINU}}]
\label{lem:correlation}
There is $C'$, depending only on $G$ and uniform in $D$, $p\in[p_c/2,p_c)$, $k$ and $t$, such that for
every domain $D$ and $u,v\in D$,
\[
\E_p\bigl(X^D_t(u)X^D_t(v);\ C^D_u\ne C^D_v\bigr)\ge s^D_t(u)s^D_t(v)-C'(\chi_0^+)^2T_0^3(u,v).
\]
\end{lemma}

In both, the site change is that a template whose interior meets a deleted set can be newly counted
after the deletion, either because an interior vertex was not pivotal before or because a blocker in the
deleted set was open before; in the second case every surviving path visits $\Nb[A']$, which is why the
deleted neighbourhood appears. Lemma~\ref{lem:correlation} uses Lemma~\ref{lem:deletion} with
$A'=\Nb[C^D_u]\cap D$, (H1) and the tree bound~\eqref{eq:tree}, with $C'=C_{\mathrm{del}}c_{\mathrm{nb}}
\kappa_p$ at $p=p_c/2$.

The two-leg decomposition is where the induced corridor changes the bookkeeping.

\begin{lemma}[{Two legs; site form of~\cite[Lemma~7.8]{OAINU}}]\label{lem:legs}
On $Q_w$, if $x\in C_u^{D_w}$, $y\in C_v^{D_w}$ and $C_u^{D_w}\ne C_v^{D_w}$, then
\[
N^V_{xy}\le2c_k+N^{D_w}_{xu}+N^{D_w}_{vy}.
\]
\end{lemma}

\begin{proof}
\emph{Templates near the corridor.} Templates $w'$ with $I_{w'}\cap(I_w\cup\{u,v\})\ne\varnothing$ have
total weight at most $(k+1)c_k/k\le2c_k$, by Lemma~\ref{lem:basic}. This class is new for sites: it
contains templates whose interior passes through $u$ and whose endpoint lies in $I_w$. Such templates
are counted in $V$ but are eligible in no leg.

\emph{The remaining templates lie in one leg.} Take a counted $w'$ with $I_{w'}$ disjoint from
$I_w\cup\{u,v\}$. Then $I_{w'}$ is a connected set of open vertices on the $x$--$y$ path outside
$I_w\cup\{u,v\}$, so it lies in one exterior cluster, say $C_u^{D_w}$. Its endpoints $u',v'$ are open
and adjacent to $I_{w'}$. If $v'\in I_w$, its open neighbour in $I_{w'}$ would be $u$ or $v$, which is
excluded. So $u',v'\notin I_w\cup B_w$, and $\gamma_{w'}$ is eligible in $D_w$.

\emph{It is counted for the leg.} We have $x,u\notin I_{w'}$. Pivotality for $x\lr u$ in $D_w$ holds: a
bypass followed by the corridor and a $v$--$y$ path, which lies in the other cluster and hence avoids
$I_{w'}$, would bypass in $V$. Blockers closed in $V$ are closed or absent in $D_w$. So $w'$ is counted
in $N^{D_w}_{xu}$ with the same weight. The case of the cluster of $v$ is symmetric.
\end{proof}

Let $\E_{\mathrm{path}}$ denote averaging over the type-weighted root $u=o$ and a length-$k$ $P$-path
from $u$, with endpoint $v$, as in~\cite[Section~7]{OAINU}, and put $a_3=\tr(T_0^3)$.

\begin{proposition}[{Decrease of the averaged susceptibility; site form
of~\cite[Proposition~7.5]{OAINU}}]\label{prop:decrease}
For all $t\ge0$,
\[
-\chi_t'\ge e^{-2tc_k}\Bigl(\E_{\mathrm{path}}\bigl[\1\{\ell_w\ge3\}s^{D_w}_t(u)s^{D_w}_t(v)\bigr]
-C'(\chi_0^+)^2\rho^ka_3\Bigr).
\]
\end{proposition}

\begin{proof}
This is the proof of~\cite[Proposition~7.5]{OAINU}, with Lemma~\ref{lem:calibration} for the
calibration, Lemma~\ref{lem:legs} in place of the bond two-leg bound, which turns the factor $e^{-tc_k}$
into $e^{-2tc_k}$, Lemma~\ref{lem:correlation}, and $\E_{\mathrm{path}}T_0^3(u,v)=\tr(P^kT_0^3)\le
\rho^ka_3$.
\end{proof}

\subsection{Endpoint escapes and the proof of Theorem~\ref{thm:unimod}}\label{sec:conclusion}

The last ingredients bound the deformed row sums from below near the corridor ends. Their proofs are
in Appendix~\ref{app:deformation}.

\begin{lemma}[{Comparison along a path; site form of~\cite[Lemma~7.9]{OAINU}}]\label{lem:endpoint}
If a simple path in $D$ joins $b$ to $z$ and has length $l$, then
$s^D_t(b)\ge e^{-t(l+1)c_k}p^ls^D_t(z)$.
\end{lemma}

\begin{lemma}[{Smoothing; site form of~\cite[Lemma~8.2]{OAINU}}]\label{lem:smoothing}
$(P^lT_0)(a,b)\le\sqrt{c_\sigma p}\,\rho^l\le\sqrt{c_\sigma}\,\rho^l$ for all vertices $a,b$ and $l\ge0$.
\end{lemma}

\begin{lemma}[{Two endpoint escapes; site form of~\cite[Lemma~8.1]{OAINU}}]\label{lem:escapes}
For every $\tau>0$ there are $J,C_1,k_0,b_*>0$, depending only on $G$ and $\tau$, such that for every
$p_c/2\le p<p_c$, $k\ge k_0$ and $R\ge1$, a set of raw paths of $\E_{\mathrm{path}}$-probability at least
$b_*$ has $\ell_w\ge3$ and admits, from each of $u$ and $v$, a simple path in $D_w$ of length at most
$j=J+\lceil C_1\log((k+1)R)\rceil$ to a vertex $z$ with
\[
\sum_{b\in\Nb[\Nb[I_w]]}T_0(z,b)\le\tau/R.
\]
\end{lemma}

For Lemma~\ref{lem:escapes} the escape must stay in $D_w=V\setminus(\Nb[I_w]\setminus\{u,v\})$, so the
avoidance event of~\cite[Section~8]{OAINU} is replaced by a site avoidance event: every escape vertex
other than the start is at graph distance at least $2$ from every raw-path vertex other than the start.
Each point collision in the bond estimates becomes membership in a neighbourhood $\Nb[x]$, at a cost of a
factor at most $d+1$, and the remaining choices of constants are those of~\cite{OAINU}.

The proof of Theorem~\ref{thm:unimod} is that of~\cite[Theorem~2.6]{OAINU} in~\cite[Section~9]{OAINU},
with the site lemmas above; it is given in Appendix~\ref{app:conclusion}. Suppose that $T_{p_c}$ is
unbounded, and let $\delta=p_c-p$. Sprinkling (Lemma~\ref{lem:sprinkling}) forces $m_0\ge c\delta^{-1}$,
and Propositions~\ref{prop:susceptibility} and~\ref{prop:trace} make $R=\chi_0^+/m_0$ and $a_3$ subpower
in $\delta$. A row comparison from Lemma~\ref{lem:endpoint}, the escapes of Lemma~\ref{lem:escapes} and
the deletion estimate keep $s^{D_w}_t(u)s^{D_w}_t(v)$ of order $\chi_t^2$ on a positive proportion of raw
paths, up to a factor polynomial in $kR$. For a corridor length $k=o(\log(1/\delta))$ chosen from
$\rho<1$, Proposition~\ref{prop:norm} keeps $m_t\ge m_0/2$ up to time $t_*=(kR)^{C_2+2}/m_0$, while
Proposition~\ref{prop:decrease} gives $-\chi_t'\ge c(kR)^{-C_2}\chi_t^2$; integrating, $\chi_{t_*}$ would
be smaller than the bound $m_{t_*}\le c_\sigma\chi_{t_*}$ allows. The site changes in this deduction are
the constant $C_{\mathrm{del}}$ in the choice of $\tau$, the factor $e^{-2tc_k}$ of
Proposition~\ref{prop:decrease}, the lower bound $a_3\ge(p_c/2)^3$ in place of $a_3\ge1$, and the constant
$1/d^2$ in the sprinkling bound.