\section{The quasi-transitive form of Tran--Nguyen's argument}\label{app:b8}

This appendix proves Theorem~\ref{thm:b8}(1). We run~\cite[Section~5]{TN26} in the setting of
Section~\ref{sec:b8-setting}. Appendix~\ref{app:b8-devices} proves the devices of
Section~\ref{sec:b8-devices} whose proofs are deferred, Appendix~\ref{app:b8-list} lists the changes, and
Appendix~\ref{app:DT} proves Proposition~\ref{prop:DT}. We write TN(5.$\ast$), TN~Lemma~5.2, TN~Proposition~5.5
and so on for the equations and numbered statements of~\cite{TN26}; unprefixed numbers refer to this
paper. Every statement of~\cite[Section~5]{TN26} not mentioned below is used as written; every statement
that changes is listed in the order of~\cite{TN26}, either as ``as written except'' or with a replacement
proof. The common degree of~\cite{TN26} becomes the maximal degree $d$; every use of it is an upper bound
on neighbour counts, boundary sizes, connector costs or path counts. ``Root'' means a vertex $v$ with the
law $\widehat\Pp^{v,x}=\Pp_p(\,\cdot\mid v\text{ open},\ R_v=x)$, and
$\widehat\E^{m,n}_p(k;a)=\sup_{v,x}\widehat\E^{v,x}_p[(X^{m,n}_k(v))^a]$ keeps its definition~\tn{5.31} as a
supremum over all $v\in V$ and $x\in[0,1)$.

\subsection{Proofs of the devices}\label{app:b8-devices}

\begin{proof}[Proof of Lemma~\ref{lem:Q1}]
Let $y'$ be a sloped vertex at minimal distance from $y$, so $\operatorname{dist}(y,y')\le
R_{\mathrm{sl}}$. On a shortest path from $y$ to $y'$ every vertex before $y'$ is a plateau vertex, by
minimality, and a step from a plateau vertex keeps $h$ unchanged by (S3); so the path is level. Step from
$y'$ to a strictly lower neighbour, which lowers $h$ by at least $\delta_0$, and repeat from the new
vertex. After $\lceil t/\delta_0\rceil$ rounds, each of length at most $R_{\mathrm{sl}}+1$, the height has
dropped by at least $t$, and the last step is strict. The reversed statement is identical, with a
strictly higher neighbour of a sloped vertex.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:Q2}]
Apply Lemma~\ref{lem:Q1} with $t=(R^*+1)t_0$; this gives a nonincreasing path from $y$ to $y'$ with
$h(y')\le h(y)-(R^*+1)t_0$, of length at most $L(t)$. Append a shortest path from $y'$ to $O^*$, of
length at most $R^*$; along it, $h\le h(y')+R^*t_0\le h(y)-t_0$. The endpoint $q$ satisfies
$h(q)\ge h(y')-R^*t_0\ge h(y)-L(t)t_0-R^*t_0$. Take $M_*=L(t)+R^*$ and $H_*=(L(t)+R^*)t_0$, and
loop-erase if needed. The case (up) is the mirror image.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:Q5}]
All three quantities are orbit-invariant, so we may replace $v$ by a vertex $v'$ in its orbit with
$\operatorname{dist}(u,v')\le R^*$, and fix a path $\pi$ from $u$ to $v'$ of length at most $R^*$. Let
$c_\Delta=\min\{\Delta(u,v')^\lambda,1\}\ge e^{-|\lambda|t_0R^*}$. Part~(a) is Hutchcroft's inequality
$\chi(u)\ge\tau_p(u,v')\Delta^\lambda(u,v')\chi(v')$~\cite[proof of Proposition~3.1]{HutNU}, with
$\tau_p(u,v')\ge p^{R^*+1}$. For~(b) and~(c), let $f\ge0$ be increasing with $f(0)=0$. On $\{\pi\text{
open}\}$, $K_u=K_{v'}$ and $|K_u|_{u,\lambda}=\Delta(u,v')^\lambda|K_{v'}|_{v',\lambda}$, by the cocycle
identity. Under $\widehat\Pp^u$, a product measure on $V\setminus\{u\}$, Harris's inequality gives
$\widehat\E^u[f(|K_u|_{u,\lambda})]\ge p^{R^*}\widehat\E^u[f(\Delta(u,v')^\lambda|K_{v'}|_{v',\lambda})]$,
and Harris's inequality for the increasing event $\{u\text{ open}\}$ gives $\widehat\E^u[f(\cdot)]\ge
\E_p[f(\cdot)]$. Since $f(0)=0$ and $K_{v'}=\varnothing$ when $v'$ is closed,
$\E_p[f(\cdot)]=p\,\widehat\E^{v'}[f(\cdot)]$. Take $f(s)=s^a$ for~(b), where $\Delta^{\lambda a}\ge
c_\Delta$; and $f(s)=1-e^{-\theta s}$ for~(c), where concavity gives $1-e^{-\theta cs}\ge\min(c,1)(1-e^{-\theta s})$. So
$c_5=p^{R^*+1}c_\Delta$ works.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:5G}]
We follow~\cite{TN26}, with $z^\pm$ replaced as follows. For the first inequality, mark
$z\in K_v\cap L_{j-1}(v)$ and take the path of Lemma~\ref{lem:Q1} (up) from $z$ with $t=t_0$,
truncated at its first visit $w$ to $L_j(v)$ (S5); its sites lie in $L_{j-1,j}(v)$, at heights at most
$t_0(j+x)<0=h_v(v)$. For the second, with $z\ne v$, mark $z\in K_v\cap L_{j+1}(v)$ and take the path of
Lemma~\ref{lem:Q1} (down) from $z$, truncated at its first visit to $L_j(v)$; its heights are at most
$h_v(z)<0$, since $v$ is the unique peak. The case $z=v$ is possible only when $j=-1$, since $v\in
L_0(v)$. If $v$ is sloped, first step to a strictly lower neighbour and continue as above. If $v$ is a
plateau vertex, $P_v$ forces $K_v=\{v\}$, so $F_j=0$ for $j\le-1$ while $F_0>0$; this is why the case
$j=-1$ needs sloped $v$. The map opens the path and closes its neighbours outside $K_v$ and the path;
$v$ stays the unique peak and $w$ is counted by $X_j(v)$. For fixed $(v,x)$ the path depends only on
$z$, at most $(d+1)(L(t_0)+2)$ coordinates change, Lemma~\ref{lem:Q0} applies to each marked $z$, and the
map $z\mapsto w$ is at most $d^{L(t_0)+2}$-to-one. The proof of~\tn{5.67} uses~\tn{5.66} at
$j-1$, $j$ and $j+1$, so it inherits the stated ranges.
\end{proof}

\subsection{The change list}\label{app:b8-list}

\paragraph{Standing hypothesis.} Replace it by: $G$ infinite, connected, locally finite and
quasi-transitive, and $\Gamma=\Aut(G)$ nonunimodular; $\Delta$ is the quasi-transitive modular function
(S1).

\paragraph{$t_0$, $z^\pm$, $H_t$, $A^s_p$, \tn{5.1}--\tn{5.4} and left continuity.}
\begin{itemize}
\item Delete the choice of neighbours $z^\pm$ with $\log\Delta(z,z^\pm)=\pm t_0$; such neighbours need not
exist, for instance at pendant vertices.
\item For each vertex $v$ put $H_t(v)=\{z:h_v(z)\ge t\}$, $A^s_{p,v}(t)=\Pp_p(v\lr H_t(v)\text{ inside }
H_0(v))$ and $\widehat A^s_{p,v}=A^s_{p,v}/p$, and let $a(t)=\inf_v\widehat A^s_{p,v}(t)$, a minimum over
orbits.
\item \emph{Replacement for \tn{5.1}:}
\begin{equation}\label{eq:51}
\widehat A^s_{p,v}(t+s)\ge p\,\widehat A^s_{p,v}(t)\,a(s),\qquad\text{hence}\qquad a(t+s)\ge p\,a(t)a(s)
\qquad(s,t\ge0).
\end{equation}
Let $W=H_0(v)\setminus H_t(v)$, explore the open cluster $K$ of $v$ in $W$, and let $E_1$ be the event
that $K$ has a neighbour in $H_t(v)$. If $v\lr H_t(v)$ inside $H_0(v)$, the first vertex of a
witnessing path in $H_t(v)$ is preceded by a vertex of $K$, so $E_1$ contains this event. On $E_1$ choose
a neighbour $w\in H_t(v)$ of $K$ measurably from the exploration. By Lemma~\ref{lem:Q3} the labels of
$H_t(v)$ are untested. The event
\[
E_2(w)=\{w\text{ open},\ w\lr H_{t+s}(v)\text{ inside }H_t(v)\}
\]
depends only on them and contains $\{w\lr H_s(w)\text{ inside }H_0(w)\}$, because $h_v(w)\ge t$; so conditionally
$\Pp(E_2(w))\ge A^s_{p,w}(s)\ge pa(s)$. Finally $E_1\cap E_2(w)$ is contained in
$\{v\lr H_{t+s}(v)\text{ inside }H_0(v)\}$; divide by $p$. This replaces the one-layer buffer $t_0$ of~\tn{5.1}, and no site tested on the
first interface is used.
\item \emph{Replacement for \tn{5.2}--\tn{5.4}.} With $c_p=-\log p$ and $f(t)=-\log a(t)$, \eqref{eq:51} gives
$f(t+s)\le f(t)+f(s)+c_p$, so $b(n)=f(nt_0)+c_p$ is subadditive. By Fekete's lemma and monotonicity of
$a$, $\alpha^s_p=\lim_t-t^{-1}\log a(t)$ exists, $\alpha^s_p=\inf_nb(n)/(nt_0)$, and
$a(t)\le p^{-1}e^{t_0\alpha^s_p}e^{-\alpha^s_pt}$. The factor $e^{t_0\alpha^s_p}$ is finite: forcing a path
of Lemma~\ref{lem:Q1} (up) gives $a(t)\ge p^{L(t)}$ up to a constant, so
$\alpha^s_p\le|\log p|\limsup_tL(t)/t=|\log p|(R_{\mathrm{sl}}+1)/\delta_0<\infty$.
\item \emph{Uniformity in the root.} $\widehat A^s_{p,v}(t)\ge p^{M_*+1}\widehat A^s_{p,w}(t)$ for all
$v,w,t$: Lemma~\ref{lem:Q2} (up) from $v$, targeting the orbit of $w$, gives a path $\pi$ to $w'$ in that
orbit with all heights at least $h(v)$ and $h(w')>h(v)$, so $H_0(w')\subseteq H_0(v)$ and
$H_t(w')\subseteq H_t(v)$, and Harris's inequality for $\{\pi\text{ open}\}$ with the endpoint change of
Lemma~\ref{lem:Q5} gives the claim. Hence $\sup_v\widehat A^s_{p,v}(t)\le p^{-M_*-1}a(t)\le
p^{-M_*-2}e^{t_0\alpha^s_p}e^{-\alpha^s_pt}$, which is \tn{5.4} uniformly in the root, and
$-t^{-1}\log\widehat A^s_{p,v}(t)\to\alpha^s_p$ for every $v$.
\item \emph{Left continuity.} As written: $a(t)$ is a minimum of finitely many orbit functions
$\widehat A^s_{p,O}(t)$, each left-continuous and increasing in $p$, as in~\cite[Lemma~5.5]{HutNU}, so
$a(t)$ is left-continuous and increasing, and $-\log a(t)+c_p$ is left-continuous and decreasing. The
normalization remarks \tn{5.2}--\tn{5.3} are as written with $\widehat A^s_p$ replaced by $a$.
\end{itemize}

\paragraph{TN~Theorem~5.1 and the layer set-up.} TN~Theorem~5.1, $\alpha^s_{p_c}=1$, is neither
needed nor claimed. \tn{5.6} is as written (S5). Delete~\tn{5.7}, the implication $u\in L_k(v)\Rightarrow
u^+\in L_{k+1}(v)$; its uses are replaced below. The conditional laws and \tn{5.8}--\tn{5.9} are as written;
they are site BK and tree bounds and use no transitivity.

\paragraph{$p_c(\lambda)$, TN~Auxiliary Lemma~5.A, \tn{5.12}--\tn{5.13} and $p_{st}$.}
\begin{itemize}
\item $p_c(\lambda)=\sup\{p:\chi_{p,\lambda}(v)<\infty\}$ does not depend on $v$ by
Lemma~\ref{lem:Q5}(a), and $\sup_u\chi_{p,\lambda}(u)=\chi^*_{p,\lambda}$ is a maximum over orbits.
Positivity is as written.
\item $p_c(\lambda)<1$ for $0\le\lambda<1/2$: for $\lambda=0$ this is $p_c<1$, below; for $\lambda>0$ the
paragraph of~\cite{TN26} is as written except that the path and its endpoint $\gamma o$ come from
Lemma~\ref{lem:Q2} (up) with target the orbit of $o$, giving $\Delta(o,\gamma o)\ge e^{t_0}>1$ and a
path of length at most $M_*$; the translates argument, $\Delta(o,\gamma^no)=\Delta(o,\gamma o)^n$, is as
written.
\item TN~Auxiliary Lemma~5.A: ``Transitivity makes the supremum equal to $\chi_{p,\lambda}(v)$'' becomes ``the
supremum is at most $\chi^*_{p,\lambda}$'', so $\frac d{dp}\chi_{p,\lambda}(v)\le C(\chi^*_{p,\lambda})^2$;
and $Y_{\Lambda_R}(s)\le\chi_{s,\lambda}(v)$ becomes $Y_{\Lambda_R}(s)\le\chi^*_{s,\lambda}$. The
divergence at the threshold and \tn{5.12} are then as written for $\chi^*$, and Lemma~\ref{lem:Q5}(a)
transfers \tn{5.12} to every root: $\widehat\chi_{q-\eps,\lambda}(v)\ge c\eps^{-1}$ for every $v$.
\item \tn{5.13} is as written. The fixed-root symmetry $\widehat\chi_{p,\lambda}=\widehat\chi_{p,1-\lambda}$
becomes the averaged symmetry $\widehat\E_\mu\widehat\chi_{p,\lambda}(\rho)=\widehat\E_\mu
\widehat\chi_{p,1-\lambda}(\rho)$, which is Lemma~\ref{lem:Q4} with $F(a,b)=\1\{a\lr b\}\Delta(a,b)^\lambda$;
by Lemma~\ref{lem:Q5}(a), $p_c(\lambda)=p_c(1-\lambda)$. $p_{st}=p_c(1/2)$ as written.
\end{itemize}

\paragraph{Magnetization, \tn{5.14}--\tn{5.26} and TN~Auxiliary Lemmas~5.B--5.D.} This replaces the averaging;
every root-local step is as written.
\begin{itemize}
\item With $q=p_c(\lambda)$ and $0\le\lambda<1/2$, the temporary hypothesis
$\widehat\E^v_q[|K_v|^{(1+\eta)/2}_{v,\lambda}]<\infty$ is taken at one root; by Lemma~\ref{lem:Q5}(b) it
holds at every root, with finite supremum.
\item TN~Auxiliary Lemmas~5.B and~5.C, \tn{5.14}, the conditioning construction and \tn{5.15} are as written at a
fixed root. In TN~Auxiliary Lemma~5.D, delete the sentence asserting that transitivity makes the last
moment independent of $z$; $M_a=\sup_z\widehat\E^z[|K_z|^a_{z,\lambda}]$ is finite by
Lemma~\ref{lem:Q5}(b), and \tn{5.16} is as written, with $M_\Delta$ a maximum over finitely many edge
orbits. \tn{5.17} is as written at a fixed root.
\item \tn{5.17}$\to$\tn{5.18}: take $\widehat\E_\mu$, that is, set $v=\rho\sim\mu$, before splitting according
to $\Delta(\rho,u)\le1$ or $>1$, and apply Lemma~\ref{lem:Q4} in place of~\tn{5.30}, after integrating out the ghosts, to
\[
F(a,b)=\1\{a\lr b,\ \Delta(a,b)>1\}e^{-\theta|K_a|_{a,\lambda}}\Delta(a,b)^{\lambda(1+\delta)}.
\]
The transported sum is
\[
\widehat\E_\mu\sum_{u\in K_\rho}\1\{\Delta(\rho,u)<1\}e^{-\theta|K_u|_{u,\lambda}}\Delta(\rho,u)^{1-\lambda(1+\delta)},
\]
which is \tn{5.18} at $\rho$; the following
estimates are pointwise.
\item \tn{5.19}, averaged: $\widehat\E_\mu\widehat\Pp^\rho(G_{\rho,\theta}(K_\rho)\ge2,\ D^c_\rho)\le
C(p/(1-p))\theta^{1+\delta}\chi_\theta$ with $\chi_\theta=\widehat\E_\mu\widehat\chi_{p,\lambda,\theta}(\rho)$. \tn{5.20}--\tn{5.21}
are root-local; with $M_\theta=\widehat\E_\mu\widehat M_{p,\lambda,\theta}(\rho)$, $\widehat\E_\mu\widehat\Pp^\rho
(T_\rho=1)=\theta\partial_\theta M_\theta$ and $\widehat\E_\mu\widehat\Pp^\rho(D_\rho)\le(\max_v\widehat M_\theta(v))^2\le
c_5^{-2}M_\theta^2$ by Lemma~\ref{lem:Q5}(c), uniformly in $\theta>0$.
\item \tn{5.22}, averaged: $M_\theta\le CM_\theta^2+\theta\partial_\theta M_\theta+C(p/(1-p))\theta^{1+\delta}\partial_\theta M_\theta$, the averaged
form of~\cite[Lemma~4.3]{HutNU} with the site constants.
\item \tn{5.23}--\tn{5.25} are as written with $\varphi(\theta)=M_\theta$ at $p=q$: $\varphi'(0+)=\infty$ by \tn{5.12} at
every root; $\psi(t)\le-\log(1-t)$ since $X\ge1$ at every root; and \tn{5.25} uses the averaged moment,
finite by Lemma~\ref{lem:Q5}(b).
\item \tn{5.26}: the induction and contradiction are as written, and give
$\widehat\E_\mu\widehat\E^\rho_{p_c(\lambda)}[|K_\rho|^{(1+\eta)/2}_{\rho,\lambda}]=\infty$. Every orbit
has positive weight, so by Lemma~\ref{lem:Q5}(b),
\begin{equation}\label{eq:526}
\widehat\E^v_{p_c(\lambda)}[|K_v|^{(1+\eta)/2}_{v,\lambda}]=\infty\qquad\text{for every }v,\ 0\le\lambda<
1/2,\ \eta>0.
\end{equation}
\end{itemize}

\paragraph{TN~Auxiliary Lemmas~5.E and~5.F, $p_c<1$ and the root-peak denominator.} TN~Auxiliary Lemma~5.E and
the Reimer remarks are general and as written; \tn{5.27} is as written; delete \tn{5.28}, whose role is taken
by the surgery in the subcritical-peak item below. TN~Auxiliary Lemma~5.F is replaced by the site theorem
for nonunimodular groups~\cite[Theorem~1.2]{Pei26b}: if $\Aut(G)$ is nonunimodular and $p_c<1$, critical
site percolation has almost surely no infinite cluster; so at every $0<p\le p_c$ every open cluster is
finite almost surely. The bound $p_c<1$ is as written with root $o$: a nonunimodular $\Gamma$ forces
nonamenability (Section~\ref{sec:b8-setting}), and the count of connected sets needs only bounded degree.
In the root-peak denominator, $(1-p)^d$ becomes $(1-p)^{\deg(v)}\ge(1-p)^d$.

\paragraph{\tn{5.30} and the transports.} \tn{5.30} is replaced by Lemma~\ref{lem:Q4}. Every transport
of~\cite{TN26} is used only through Lemma~\ref{lem:Q4} with $\rho\sim\mu$: the susceptibility symmetry,
the large-$\Delta$ ghost half, the up/down identity \tn{5.49}, the peak-to-layer transports \tn{5.68}
and~\tn{5.113}, and the H\"older transport \tn{5.92}.

\paragraph{\tn{5.31}--\tn{5.33}, TN~Lemma~5.2, the kernels and the first moments.}
\begin{itemize}
\item \tn{5.31}--\tn{5.33} are as written, with suprema over all $v\in V$ and $x\in[0,1)$.
\item In~\cite[Lemma~5.2]{TN26}, replace the hypothesis as above; clauses~1--3 are claimed and clause~4 is
dropped. The uniformity of clause~1 in the root and the offset holds for the supremum-type bounds, the
first-moment and probability estimates \tn{5.41}--\tn{5.62} and \tn{5.65}, and \tn{5.34}. It does not hold for the
peak lower bounds \tn{5.69}, \tn{5.70}, \tn{5.72}, \tn{5.86}, \tn{5.90} and \tn{5.91}, which hold at every sloped vertex,
uniformly in the offset and over the finitely many sloped orbits; at a plateau vertex $P_v$ forces
$K_v=\{v\}$. Theorem~\ref{thm:b8} never uses root-uniformity of the peak lower bounds.
\item The cluster kernel \tn{5.36}--\tn{5.38}, the two-level boundary kernel and the first-hit residual kernel
are as written; they hold on any countable graph.
\item \tn{5.39}, replaced: $\widehat\Pp_p(m+n)\ge p\,\widehat\Pp_p(m)\widehat\Pp_p(n)$ for $m,n\ge0$. Explore
the open cluster of $v$ in $L_{0,m-1}(v)$; by Lemma~\ref{lem:Q3}, a connection to $L_{m,\infty}(v)$ passes
through an untested neighbour $w\in L_m(v)$ of that cluster; $L_{0,\infty}(w)=L_{m,\infty}(v)$ by the
shift rule, with untested labels; continue from $w$ as in~\cite{TN26}. Then \tn{5.40} has $a_p(n)+c_p$
subadditive without the index shift, and the limit identification is as written.
\item \tn{5.41}, replaced. The first inequality, $a(t_0(n+1))\le\widehat\Pp^{v,x}(v\lr L_n(v)\text{ in }
L_{0,\infty}(v))$, is the inclusion argument of~\cite{TN26} at $v$ with $\widehat A^s_{p,v}\ge a$. The
second, $\widehat\Pp^{v,x}(v\lr L_n(v)\text{ in }L_{0,\infty}(v))\le p^{-L(t_0)}\sup_w\widehat
A^s_{p,w}(t_0n)$: replace $v^-$ by the endpoint $v'$ of a path $\pi$ of Lemma~\ref{lem:Q1} (down) from $v$
with $t=t_0$; relative to $v'$, $L_{0,\infty}(v)$ lies at raw height at least $h_{v'}(v)-t_0\ge0$, $\pi$
lies at heights at least $0$, and $L_n(v)$ lies above raw height $t_0n$; the Harris and endpoint-change
step is as written with the event $\{\pi\text{ open}\}$ and the factor $p^{|\pi|}$. Both inequalities are uniform
in $(v,x,n)$ by the uniformity in the root above.
\item \tn{5.42}, replaced. Let $u$ be the endpoint of a path of Lemma~\ref{lem:Q1} (down) from $v$ with
$t=(r+1)t_0$, of length at most $L((r+1)t_0)\le C_L(r+1)$. Then $v\in L_{r+1+i}(u)$ for some
$i\in[0,\lceil L((r+1)t_0)\rceil]$, the shift rule gives $L_{-r,\infty}(v)=L_{1+i,\infty}(u)\subseteq
L_{1,\infty}(u)$ and $L_n(v)=L_{n+r+1+i}(u)$, and the reversed path lies in $L_{0,r+1+i}(u)$. The rest is
as written with $p^{r+1}$ replaced by $p^{|\text{path}|}$, and gives
$\sup_{v,x}\widehat\Pp^{v,x}(v\lr L_n(v)\text{ in }L_{-r,\infty}(v))\le C_pe^{C_0(r+1)}e^{-t_0\alpha^s_pn}$
with $C_0=C_L|\log p|$.
\item The path decompositions \tn{5.43}--\tn{5.45} and the submultiplicativity of $U_p$ and $D_p$ are as
written.
\item \tn{5.46}, replaced: $\sup_{v,y}\widehat\E^{v,y}X^{-\infty,k+1}_{k+1}(v)\le B_p\inf_{w,x}\widehat
\E^{w,x}X^{-\infty,k}_k(w)$. The inclusion of~\cite{TN26} and \tn{5.43} give, at each fixed $v$,
$\sup_y\widehat\E^{v,y}X_{k+1}(v)\le B_p^{(0)}\inf_{y'}\widehat\E^{v,y'}X_k(v)$; the proof
in~\cite{TN26} uses transitivity to pass between roots. Across orbits, fix $(v,y)$ and $(w,x)$.
Lemma~\ref{lem:Q2} (up) from $w$, targeting the orbit of $v$, gives $v'$ in that orbit with $v'\in
L_j(w)$, $j\in[1,J_2]$, $J_2=\lceil H_*/t_0\rceil+1$, by a path with heights in $[h(w),h(w)+H_*]$, hence
inside $L_{0,J_2}(w)$. With $R'=R_{v'}$ induced by $R_w=x$, orbit invariance and the per-vertex inequality
give $\widehat\E^{v,y}X_{k+1}(v)\le B_p^{(0)}\widehat\E^{v',R'}X_k(v')$; first-visit
submultiplicativity, the first line of \tn{5.43}, gives $\widehat\E^{v',R'}X_k(v')\le U_p(j)\widehat
\E^{v',R'}X_{k-j}(v')$; and for $k\ge J_2$ Harris's inequality for the path and the endpoint change give
$\widehat\E^{w,x}X^{-\infty,k}_k(w)\ge p^{M_*+1}\widehat\E^{v',R'}X^{-\infty,k-j}_{k-j}(v')$, because
$L_{-\infty,k-j}(v')=L_{-\infty,k}(w)$, $L_{k-j}(v')=L_k(w)$ and the path lies in $L_{0,J_2}(w)\subseteq
L_{-\infty,k}(w)$. Chaining gives $B_p=B_p^{(0)}\max_{j\le J_2}U_p(j)p^{-M_*-1}$; the finitely many
$k<J_2$ have positive finite values and are absorbed.
\item \tn{5.47}, replaced: with $D^v_y(k)=\widehat\E^{v,y}X^{-\infty,0}_{-k}(v)$,
\begin{equation}\label{eq:547}
\sup_{v,y}D^v_y(k+1)\le B'_p\inf_{w',x'}D^{w'}_{x'}(k)\qquad(k\ge0).
\end{equation}
Offsets are treated as in~\cite{TN26}: layer sets and counts may be evaluated with any deterministic
offset at any vertex, since the configuration is independent of the offset field. Put $G_1=L(t_0)$ and
$J=G_1+3$.
\begin{itemize}
\item \emph{Step A (upward connector, in place of $w=v^+$).} Fix $(v,y)$, let $\pi$ be the path of
Lemma~\ref{lem:Q1} (up) from $v$ with $t=t_0$, and let $w$ be its endpoint; $\pi$ is nondecreasing,
$|\pi|\le G_1$ and $g:=h_v(w)\in[t_0,G_1t_0]$. For every offset $x$ at $w$: $L_{-\infty,0}(v;y)\subseteq
L_{-\infty,0}(w;x)$, since the upper boundaries are $t_0y<t_0\le g+t_0x$ in raw height from $v$;
$\pi\subseteq L_{-\infty,0}(w;x)$, since heights along $\pi$ are at most $h(w)$; and
$L_{-k-1}(v;y)\subseteq\bigcup_{j=1}^JL_{-k-j}(w;x)$, since for $z$ on the left $h_w(z)/t_0=h_v(z)/t_0-
g/t_0\in(-k-3-G_1,-k-2+y]\subseteq(-k-3-G_1,-k-1+x)$. For $u$ counted on the left, with connection event
$E_u$ in $L_{-\infty,0}(v;y)$, Harris's inequality gives $p^{|\pi|}\widehat\Pp^{v,y}(E_u)\le\widehat
\Pp^{v,y}(E_u\cap\{\pi\setminus\{v\}\text{ open}\})$; the right side forces $v$ and $w$ open, so it equals
$\widehat\Pp^{w,x}(E_u\cap\{\pi\setminus\{w\}\text{ open}\})$, the endpoint change of~\cite{TN26}; and on
that event $u$ is joined to $w$ inside $L_{-\infty,0}(w;x)$ with $u\in L_{-k-j}(w;x)$ for some $j\in[1,J]$.
Summing over $u$ and applying the last-visit decomposition at $L_{-k}(w;x)$, the second line of \tn{5.44},
\begin{equation}\label{eq:stepA}
D^v_y(k+1)\le p^{-G_1}\Bigl(\sum_{j=1}^JD_p(j)\Bigr)D^w_x(k)\qquad\text{for every }x\in[0,1).
\end{equation}
\item \emph{Step B (cross-orbit step, in place of the transitivity that makes $D^w_x(k)$ independent of
$w$).} Fix $(w',x')$. Lemma~\ref{lem:Q2} (down) from $w'$, targeting the orbit of $w$, gives a path
$\sigma$ to $w''$ in that orbit with $|\sigma|\le M_*$, heights along $\sigma$ at most $h(w')$, and
$h(w'')\in[h(w')-H_*,h(w')-t_0]$. With $R''$ the offset at $w''$ induced by $x'$, $w''\in L_{-j'}(w';x')$
for some $j'\in[1,J_2]$, and $L_i(w'';R'')=L_{i-j'}(w';x')$. Also $\sigma\subseteq L_{-\infty,0}(w';x')$,
and on $\{\sigma\text{ open}\}$ every vertex of $L_{-k}(w';x')=L_{-k+j'}(w'';R'')$ joined to $w''$ inside
$L_{-\infty,j'}(w'';R'')=L_{-\infty,0}(w';x')$ is counted by $X^{-\infty,0}_{-k}(w')$. Domain
monotonicity, Harris's inequality and the endpoint change give, for $k\ge J_2$,
$D^{w'}_{x'}(k)\ge p^{M_*+1}D^{w''}_{R''}(k-j')$, and the last-visit decomposition gives
$D^{w''}_{R''}(k)\le D^{w''}_{R''}(k-j')D_p(j')$. Hence
$D^{w''}_{R''}(k)\le p^{-M_*-1}D_p(j')D^{w'}_{x'}(k)$ for $k\ge J_2$.
\item \emph{Step C (combination).} As $w''$ lies in the orbit of $w$ and the joint law of the
configuration and the equivariant offsets is $\Gamma$-invariant, $D^w_x(k)=D^{w''}_x(k)$ for every $x$.
Apply~\eqref{eq:stepA} with $x=R''$, then Step~B: for $k\ge J_2$, $D^v_y(k+1)\le B'_pD^{w'}_{x'}(k)$ with
$B'_p=p^{-G_1-M_*-1}(\max_{j'\le J_2}D_p(j'))\sum_{j=1}^JD_p(j)$, finite for $p\in I^s_h$.
\item \emph{Small $k$.} For $0\le k<J_2$, $\sup D(k+1)\le D_p(k+1)<\infty$, and $\inf_{w',x'}
D^{w'}_{x'}(k)\ge p^{L((k+2)t_0)}>0$ by forcing a path of Lemma~\ref{lem:Q1} (down) from $w'$ of depth
$(k+2)t_0$, truncated at its first visit to $L_{-k}(w')$. These cases are absorbed into $B'_p$.
\end{itemize}
Every downstream use of \tn{5.47} needs the full supremum-infimum form~\eqref{eq:547}: the chain after
\tn{5.47}, the pointwise lower bound in the second line of \tn{5.51}, the pointwise lower bounds for $k<0$ in
\tn{5.54}, and the summation for \tn{5.60} at every root.
\item \tn{5.48}--\tn{5.49} and the sentence that the quantities do not depend on $v$, replaced. Define
$\bar U_p(n)=\widehat\E_\mu\int_0^1\widehat\E^{\rho,x}X^{-\infty,n}_n(\rho)\,dx$ and $\bar D_p(n)$
likewise. Lemma~\ref{lem:Q4} applied to the transport of~\cite{TN26} gives
$\bar U_p(n)\asymp e^{-t_0n}\bar D_p(n)$, which is \tn{5.49} with $\rho\sim\mu$. Then
$\inf_{v,x}\le\text{average}\le\sup_{v,x}$, and \tn{5.46}--\eqref{eq:547} put every fixed-$(v,x)$ moment
within constants and a one-layer shift of the averages.
\item One-layer comparison $U_p(k)\le C_UU_p(k+1)$, replaced: mark $z$, take the path of
Lemma~\ref{lem:Q1} (up) from $z$ with $t=t_0$, truncated at its first visit $w$ to $L_{k+1}(v)$, and
force it open; the image is counted by $X^{-\infty,k+1}_{k+1}(v)$. Lemma~\ref{lem:Q0} is applied per mark,
and $z\mapsto w$ is at most $d^{L(t_0)+1}$-to-one.
\item One-layer comparison for $D_p$, replaced; \cite{TN26} obtains it from the maximal-decrement map and
\tn{5.47}. For every $(v,x)$ and $k\ge0$, $D^v_x(k)\le C_{Dp}D^v_x(k+1)$ with
$C_{Dp}=d^{L(t_0)+1}\min(p,1-p)^{-(L(t_0)+1)}$, hence $D_p(k)\le C_{Dp}D_p(k+1)$. Mark $z\in L_{-k}(v)$
joined to $v$ inside $L_{-\infty,0}(v)$, and let $\pi_z$ be the path of Lemma~\ref{lem:Q1} (down) from $z$
with $t=t_0$, truncated at its first visit $w$ to $L_{-k-1}(v)$; it is nonincreasing, so
$\pi_z\subseteq L_{-k-1,-k}(v)\subseteq L_{-\infty,0}(v)$ for $k\ge0$. For $k\ge1$ its heights are
negative, so $v\notin\pi_z\setminus\{z\}$; for $k=0$ it may start at $z=v$, which is open. The map opens
$\pi_z$, changes at most $L(t_0)+1$ coordinates, does not change the root's state, and maps
$\{z\text{ joined to }v\text{ in }L_{-\infty,0}(v)\}$ into $\{w\text{ joined to }v\text{ in }
L_{-\infty,0}(v)\}$. Lemma~\ref{lem:Q0} is applied per mark, and $z\mapsto w$ is at most
$d^{L(t_0)+1}$-to-one since $\operatorname{dist}(z,w)\le L(t_0)$. The other direction,
$D_p(k+1)\le D_p(k)D_p(1)$, is last-visit submultiplicativity, as written.
\item \tn{5.50}, rechecked. First chain: $D^{\sup}(k)\le B'_p\bar D(k-1)$ by~\eqref{eq:547}, then $\le
C_pe^{t_0(k-1)}\bar U(k-1)$ by the averaged \tn{5.49}, then $\le C_pe^{t_0(k-1)}U^{\sup}(k-1)\le
C_pC_U^2e^{t_0k}U^{\sup}(k+1)$ by the $U$ comparison twice. Reverse chain: $U^{\sup}(k)\le B_p\bar U(k-1)$
by \tn{5.46}, then $\le C_pe^{-t_0(k-1)}\bar D(k-1)$, then $\le C_pe^{-t_0(k-1)}D^{\sup}(k-1)\le
C_pC_{Dp}^2e^{-t_0k}D^{\sup}(k+1)$ by the $D$ comparison twice. Both lines of \tn{5.50} hold with a new
$C_p$, and the deductions after \tn{5.50} are as written.
\item \tn{5.50}--\tn{5.62} are as written; they manipulate suprema and per-$(v,x)$ inequalities. At
\tn{5.56}--\tn{5.58} the comparison becomes $\sup_w\widehat A^s_{p,w}(t_0k)\ge p^{L(t_0)}\sup_{v,x}
\widehat\Pp^{v,x}(\cdots)$, by the second inequality of \tn{5.41}. The lower inequality of \tn{5.41} at fixed
$(v,x)$ gives $\widehat\Pp^{v,x}(\cdots)\ge a(t_0(k+1))\ge p^{M_*+1}\sup_w\widehat A^s_{p,w}(t_0(k+1))$ by
the uniformity in the root, so the lower bound in \tn{5.59} holds at every $(v,x)$, and \tn{5.59}--\tn{5.60}
follow as written. Where~\cite{TN26} uses $\widehat\chi_{p,1/2}<\infty$ to place $p$ in $I^s_h$, it needs
$\sup_v\widehat\chi_{p,1/2}(v)<\infty$, a maximum over orbits finite by Lemma~\ref{lem:Q5}(a); the lower
bounds of \tn{5.51} used there hold at every $(v,x)$ by \tn{5.46}--\eqref{eq:547}. Site sharpness on
quasi-transitive graphs~\cite{AV08} identifies $\widehat\chi_{p,0}(v)<\infty$ with $p<p_c$ at every
root.
\end{itemize}

\paragraph{\tn{5.73}.} As written, except that the tilted symmetry $\widehat\chi_{p,1}=\widehat\chi_{p,0}$
becomes the averaged symmetry together with Lemma~\ref{lem:Q5}(a): $\widehat\chi_{p,1}(v)<\infty$ for
every $v$ when $p<p_c$. Log-convexity in the tilt is pointwise. So $\alpha^s_{p_c}\ge1$, with $\alpha^s$
as above.

\paragraph{Subcritical peaks, \tn{5.63}--\tn{5.72}.}
\begin{itemize}
\item \tn{5.63}--\tn{5.65}, replaced by a first-crossing surgery. Fix $p<p_c$ and $(v,x)$. Explore the open
cluster $K$ of $v$ in $W=L_{-\infty,k}(v)$, which is finite almost surely; let $N=|K\cap L_k(v)|$ and
$H'_k=\{K\text{ has a neighbour in }L_{k+1}(v)\}$. By Lemma~\ref{lem:Q3}(2) and the first-entry argument,
$\{v\lr L_{k+1}(v)\}\subseteq H'_k\subseteq\{v\lr L_k(v)\}$, so by \tn{5.59},
$\widehat\Pp^{v,x}(H'_k)\asymp_pe^{-t_0\beta^s_pk}$ uniformly in $(v,x)$. Since $N\le
X^{-\infty,k}_k(v)$, \tn{5.51} gives $\widehat\E^{v,x}N\le C_pe^{-t_0\beta^s_pk}$, hence
$\widehat\E^{v,x}[N\mid H'_k]\le C'_p$; this replaces \tn{5.63}. On $H'_k$ choose a neighbour $w\in
L_{k+1}(v)$ of $K$ measurably; by Lemma~\ref{lem:Q3}, $w$ and all neighbours of $K$ outside $W$ are
untested, at most $dN$ of them, all in $L_{k+1}(v)$. Prescribe $w$ open, and prescribe closed every other
untested neighbour of $K$ and every untested neighbour of $w$; a tested neighbour of $w$ is in $K$, if
tested open, or closed. So at most $d(N+1)$ sites are prescribed closed, and by \tn{5.36} the conditional
probability is at least $p(1-p)^{d(N+1)}$. On the prescribed event the full cluster is $K\cup\{w\}$, and
since $h_v(w)>t_0(k+R_v)\ge\max_Wh_v$, $w$ is its unique peak. \tn{5.64} is as written with $Z_k$ replaced
by $K\cap L_k$: conditional Jensen gives $c'_p>0$ and $\widehat\Pp^{v,x}(\mathrm{peak}(K_v)\in L_{k+1}(v))\ge
c'_p\widehat\Pp^{v,x}(H'_k)\ge ce^{-t_0\beta^s_pk}$. The upper bound is as written, which gives \tn{5.65}
uniformly in $(v,x)$ at every vertex.
\item TN~Auxiliary Lemma~5.G and \tn{5.67} are replaced by Lemma~\ref{lem:5G}.
\item \tn{5.68}, replaced. Lemma~\ref{lem:Q4} applied to $T_k(a,b)=\1\{P_a,\ b\in K_a\cap L_{-(k+1)}(a)\}$
gives $p\,\widehat\E_\mu\int_0^1F_{-(k+1)}(\rho,y)\,dy=\widehat\E_\mu\E\sum_a\1\{a=\mathrm{peak}(K_\rho),\
a\in L_{k+1}(\rho)\}\Delta(\rho,a)$, with weight $\Delta(\rho,a)\asymp e^{t_0k}$, and \tn{5.65} at every
vertex gives that the average $\sum_O\mu(O)\int F_{-(k+1)}(O,y)\,dy$ is $\asymp_pe^{-t_0(\beta^s_p-1)k}$.
\item \tn{5.69}, replaced. The upper bound holds at every $v$, with $\int F\le\mu_{\min}^{-1}$ times the
average. At a sloped $v_0$, Lemma~\ref{lem:Q6} and Lemma~\ref{lem:5G} iterated $J_6$ times give
$\int F_{-(k+1)}(v_0,y)\,dy\ge c\max_O\int F_{-(k+1)}(O,y)\,dy\ge ce^{-t_0(\beta-1)k}$, and \tn{5.67} at $v_0$
replaces the offset average by any fixed $x$; the case $k=0$ is as written, using $(1-p)^d\le F_0\le
\widehat\chi_{p,0}$. So \tn{5.69} holds at every sloped vertex, uniformly in the offset and over the
finitely many sloped orbits:
\[
\widehat\E^{v_0,x}[X^{-\infty,\infty}_{-k};\ P_{v_0}]\asymp_pe^{-t_0(\beta^s_p-1)k}.
\]
\item \tn{5.70}--\tn{5.72} are as written at a sloped $v_0$, and \tn{5.61} holds at every vertex; so \tn{5.72} holds
at every sloped vertex, uniformly in $x,k,\ell$. This is the site form of~\cite[Lemma~5.17]{HutNU}, which
asserts the existence of a suitable vertex.
\end{itemize}

\paragraph{Reimer, the surgery map, \tn{5.76} and critical peak survival, \tn{5.74}--\tn{5.91}.}
\begin{itemize}
\item \tn{5.74}--\tn{5.75} are as written, with $M=(d+1)s$.
\item \tn{5.76}--\tn{5.83} are replaced by Proposition~\ref{prop:DT}, proved in Section~\ref{app:DT}.
\item Denominator positivity is needed only at a sloped $v_0$: a path of Lemma~\ref{lem:Q1} (down) from
$v_0$ with first step strict, as for $\sigma$ in Lemma~\ref{lem:Q6}, of depth at least $(\ell+1)t_0$,
together with its closed exterior, gives $\widehat\Pp(N_\ell\cap P_{v_0}(S))\ge p^L(1-p)^{d(L+1)}>0$.
\item \tn{5.84} is as written, except that the uniformity in the root becomes a maximum over the finitely
many orbits of sequences tending to zero; each tends to zero since clusters at $p\le p_c$ are finite,
hence of bounded height. The one-layer offset enlargement is as written.
\item \tn{5.85}--\tn{5.91} are as written at a sloped $v_0$: \tn{5.86} is \tn{5.72} at $v_0$ at level $p_-$;
\tn{5.87}--\tn{5.89} are conditional on $A^-_k$ at $v_0$; and the residual-kernel bound uses \tn{5.84}, a supremum
over all roots. The conclusion is~\eqref{eq:591}.
\end{itemize}

\paragraph{TN~Proposition~5.3.} Every offset-averaged quantity in \tn{5.93}, \tn{5.94}--\tn{5.95}
and \tn{5.106} becomes the average over $\rho\sim\mu$ and the offset, $\widehat\E_\mu\int dx\,
\widehat\E^{\rho,x}$, and ``the root is fixed and the offset is averaged'' becomes ``the root
$\rho\sim\mu$ and the offset are averaged''. \tn{5.92}--\tn{5.93} are as written with Lemma~\ref{lem:Q4}; the
H\"older step \tn{5.94}--\tn{5.95} is as written under the joint average; and the up/down recursions
\tn{5.96}--\tn{5.105} are as written, per fixed $(v,x)$, with each entry's cluster bounded by the supremum over
all roots.

\paragraph{TN~Proposition~5.4.}
\begin{itemize}
\item \tn{5.107}, replaced: $\widehat\E^{m,n}_p(k;a)\le\mu_{\min}^{-1}\sum_{j=-1}^1\widehat\E_\mu\int dy\,
\widehat\E^{\rho,y}[(X^{m-1,n+1}_{k+j}(\rho))^a]$, by the per-vertex offset inclusion of~\cite{TN26} and
the last assertion of Lemma~\ref{lem:Q4}. \tn{5.108} is as written with $C_p$ replaced by
$C_p\mu_{\min}^{-1}$.
\item \tn{5.109}, replaced: $\widehat\E^{m+r,n+r}_p(k+r;a)\le C_{sh,0}e^{C_{sh}|r|}\widehat
\E^{m,n}_p(k;a)$. Take $\gamma$ to be the path of Lemma~\ref{lem:Q1} from $v$ (up if $r>0$, down if
$r<0$) with $t=|r|t_0$, truncated at its first visit to $L_r(v)$; its length is at most
$L(|r|t_0)\le(|r|t_0/\delta_0+2)(R_{\mathrm{sl}}+1)$, and it lies in $L_{0,r}(v)$, respectively
$L_{r,0}(v)$, hence in $L_{m+r,n+r}(v)$ under the stated hypotheses. The Harris and endpoint-change
argument \tn{5.110}--\tn{5.111} is as written with $p^{|r|}$ replaced by $p^{|\gamma|}$; here
$C_{sh}=|\log p|(R_{\mathrm{sl}}+1)t_0/\delta_0$ and $C_{sh,0}=p^{-2(R_{\mathrm{sl}}+1)}$.
\item \tn{5.112} is proved in Section~\ref{app:DT}, through Proposition~\ref{prop:DT} and \tn{5.113}--\tn{5.117}
at a sloped $v_0$.
\item \tn{5.118}--\tn{5.125} are as written, with the constant prefactors $C_{sh,0}$ and $\mu_{\min}^{-1}$,
which enter finitely many times, and the choices of $\delta$ in \tn{5.119} and \tn{5.122} with the new $C_{sh}$.
The order of choice of constants, after $\eps$ and $\zeta$ and before $v,x,k$, is unaffected, since every
new constant depends only on $(G,p)$. So \tn{5.34} holds as stated, uniformly in the root and offset.
\end{itemize}

\paragraph{TN~Proposition~5.5.} The statement is as written with the hypotheses
\eqref{eq:526}, \tn{5.34}, \tn{5.59}--\tn{5.60} and \tn{5.73}, all proved above. In the proof, \tn{5.126} is
\eqref{eq:526} at a fixed root $v$, and \tn{5.127} is as written at that $v$, using \tn{5.34}, which is a
supremum over roots; this gives $p_c<p_c(\lambda)$ for $0<\lambda<1/2$. The fixed-root symmetry is
replaced by the averaged symmetry and Lemma~\ref{lem:Q5}(a); Cauchy--Schwarz is pointwise; and the strict
lower bound at $\lambda=1/2$ is as written. The conclusion is $p_c<p_c(\lambda)\le p_{st}=p_c(1/2)$ for
$0<\lambda<1$. The final paragraph of the proof in~\cite{TN26}, which proves
$\alpha^s_{p_c}=\beta^s_{p_c}=1$, is not needed.

\subsection{Proof of Proposition~\ref{prop:DT}}\label{app:DT}

Fix $p,k,\ell,r$ and $(v,x)$, and write $J$ for the residual coordinates. Let $O^*$ be an orbit attaining
$\max_O\sup_y\widehat\E^{O,y}[X^{-k-r,0}_{-k}(O);\ Q_{O;k,\ell+1,r+a_*}]$; the maximum exists because there
are finitely many orbits, the quantity depends only on the orbit by $\Gamma$-invariance of the joint law
of the configuration and the equivariant offsets, and the maximum equals the left side of the
proposition.

\emph{Step 1: exploration}, as in~\cite{TN26} except for the retained event. Expose the open cluster $A$ of
$v$ in $L_{-\ell,\infty}(v)$ with its tested external boundary $\Theta$ in that graph. Retain only
outcomes $(A,\Theta)$ on which $v$ is the peak of $A$ and $A$ has a neighbour in $L_{-\ell-1}(v)$, that is,
on which $N_\ell(v)$ holds; call them admissible. Every site of $L_{-\infty,-\ell-1}(v)$ is untested
(Lemma~\ref{lem:Q3}). Select by a fixed enumeration an edge $\{u,q_1\}$ with $u\in A$ and
$q_1\in L_{-\ell-1}(v)$; since an edge changes height by at most $t_0$, $u\in L_{-\ell}(v)$.

\emph{Step 2: connector}, in place of the maximal-decrement neighbours of~\tn{5.77}. Lemma~\ref{lem:Q2}
(down) with target $O^*$ gives a path $\gamma'=(q_1,\dots,q_m=q)$ with $m\le M_*$ and $q\in O^*$, along
which $h(q_j)\le h(q_1)$, with $h(q_1)-H_*\le h(q)\le h(q_1)-t_0$. Choose $\gamma'$ deterministically from
$(q_1,O^*)$ and loop-erase it. Every site of $\gamma'$ has height at most $h(q_1)$, so
$\gamma'\subset L_{-\infty,-\ell-1}(v)$, and $\gamma'$ is untested and disjoint from $A\cup\Theta$. Since
$h(q)\le h(q_1)-t_0$ and layers have thickness $t_0$, $q\in L_{-\ell-a}(v)$ for some integer
$a\in[2,a_*]$ with $a_*:=2+\lceil H_*/t_0\rceil$, and $L_j(q)=L_{j-\ell-a}(v)$ for every $j$. Put
$\gamma=(u,q_1,\dots,q)$; its non-root sites lie strictly below the exploration domain, which is the
hypothesis of the map \tn{5.75}.

\emph{Step 3: the residual events}, as in~\cite{TN26} with the new indices. Put $\Theta^*=L_{-S,\infty}(v)
\setminus(A\cup\Theta)$. For $w\in L_{-k-1,-k+1}(q)$, let $E^{A,\Theta}_{q,w}$ be the event that $q$ is open,
that $q$ is joined to $w$ in $L_{-S,-\ell-1}(v)\setminus(A\cup\Theta)$, and that the open cluster $C_q$ of
$q$ in $\Theta^*$ does not meet $H^+(v):=\{z:h_v(z)\ge0\}\setminus\{v\}$. The event $B^{A,\Theta}$ is
defined as in~\cite{TN26}: an open path leaves $A$ through an untested site below $L_{-\ell}(v)$, uses no
site of $\Theta$, and returns to $H^+(v)$ inside $\Theta^*$. The slab used for $w$ is $L_{-k-1,-k+1}(q)$,
and the $q$--$w$ path domain is $L_{-k-r-1,1}(q)=L_{-k-r-1-\ell-a,1-\ell-a}(v)$; since $a\ge2$ its top
index is at most $-\ell-1$, and since $a\le a_*$ its bottom index is at least $-S$. So the domain lies in
$L_{-S,-\ell-1}(v)$, strictly below the explored region.

\emph{Step 4: the Reimer step}, \tn{5.78}--\tn{5.79}, as written. TN~Auxiliary Lemma~5.E and its finite exhaustion
use no transitivity; the disjoint-witness argument uses only that a $B$-path meeting $C_q$ would join $q$
to $H^+(v)$ inside $\Theta^*$, and that $\partial_{\Theta^*}C_q$ is closed. All clusters are finite at
$p\le p_c$, by definition for $p<p_c$ and by~\cite[Theorem~1.2]{Pei26b} at $p=p_c$. Hence
$\widehat\Pp(E^{A,\Theta}_{q,w}\cap B^{A,\Theta}\mid A,\Theta)\le\widehat\Pp(E^{A,\Theta}_{q,w}\mid A,\Theta)
\widehat\Pp(B^{A,\Theta}\mid A,\Theta)$, and \tn{5.80} gives $\Pp(R\mid B^c)\ge\Pp(R)$ for
$R=E^{A,\Theta}_{q,w}$.

\emph{Step 5: identification of $B^c$}, as written. Given an admissible $(A,\Theta)$, the event $P_v(S)$
is exactly $B^c$: the set joined to $v$ inside $L_{-S,\infty}(v)$ is $A$ together with whatever is reached
by leaving $A$; every neighbour of $A$ inside $L_{-\ell,\infty}(v)$ is in $\Theta$ and closed, so leaving
$A$ means entering an untested site below $L_{-\ell}(v)$; hence $v$ stays the peak if and only if no such
excursion reaches $H^+(v)$.

\emph{Step 6: the surgery}, \tn{5.75} with $\gamma$ from Step~2. The map $\Phi_\gamma$ opens
$q_1,\dots,q_{m-1}$ ($q$ is already open on $R$), closes every site adjacent to $\gamma\setminus\{u\}$ outside
$A\cup\Theta\cup C_q\cup\gamma$, and changes nothing else. Only coordinates in a set $S_\gamma\subset J$
with $|S_\gamma|\le(d+1)M_*$ change, so $\widehat\Pp(E\mid A,\Theta)\le2^{(d+1)M_*}[p(1-p)]^{-(d+1)M_*}
\widehat\Pp(\Phi_\gamma(E)\mid A,\Theta)$. Preservation is as in~\cite{TN26}, with
$\gamma\setminus\{u\}\subset L_{-\infty,-\ell-1}(v)$ disjoint from $H^+(v)$: opening $\gamma$ only attaches
$A$ to $C_q\cup(\gamma\setminus A)$, and $C_q\cap H^+(v)=\varnothing$ on $R$; every other new exit from
$\gamma$ is closed; and a $B$-path in the image avoiding $\gamma$ was a $B$-path before, because the map
only closes sites off $\gamma$. So $\Phi_\gamma$ maps $R\cap B^c$ into $B^c\cap\{u\text{ joined to }w\text{
through }\gamma\}$. On the image, $A\cup\gamma\cup(\text{the }q\text{--}w\text{ path})$ is an open connected
set in $L_{-S,0}(v)$, since $A\subset L_{-\ell,0}(v)$ ($v$ is its peak) and $\gamma$ and the $q$--$w$ path
lie in $L_{-S,-\ell-1}(v)$. A site of $L_{-k+j}(q)$, $j\in\{-1,0,1\}$, lies in $L_{-k-\ell-(a-j)}(v)$ with
$a-j\in[1,a_*+1]$. Summing over $w$ gives, for every admissible $(A,\Theta)$,
\begin{equation}\label{eq:DT1}
\sum_{i=1}^{a_*+1}\widehat\E^{v,x}\bigl[X^{-S,0}_{-k-\ell-i}(v)\1_{B^c}\bigm|A,\Theta\bigr]\ge
c_{d,p,M_*}\,\widehat\Pp(B^c\mid A,\Theta)\,p\,\E^{q;A,\Theta}_{p,J}[Z_q;\ E_q],
\end{equation}
with $Z_q=\sum_{j=-1}^1X^{-k-r-1,1}_{-k+j}(q)$ counted in the residual domain, $E_q=\{q\nlr H^+(v)\text{
in }\Theta^*\}$, and $\E^{q;A,\Theta}_{p,J}$ the residual product law on $J$ with $q$ conditioned open.
The factor $\widehat\Pp(B^c\mid A,\Theta)$ comes from \tn{5.80} summed over $w$, and~\eqref{eq:DT1} is \tn{5.81}
multiplied by it, with the new index ranges.

\emph{Step 7: removing the residual conditioning}, as in~\cite{TN26} with the new parameters. Fix $y\in
[0,1)$. The half-open layer definition gives $L^y_{-k}(q)\subseteq L^{R_q}_{-k-1,-k+1}(q)$ and
$L^y_{-k-r,0}(q)\subseteq L^{R_q}_{-k-r-1,1}(q)$, which is the first part of \tn{5.82}, so
$X^{-k-r,0}_{-k}(q;y)\le Z_q$ pointwise. For the events: first, $\Theta^*\subseteq
L^{R_q}_{-S+\ell+a,\infty}(q)=L^{R_q}_{-k-r-1-(a_*-a),\infty}(q)\subseteq L^y_{-k-(r+a_*)-2,\infty}(q)$.
Second, every $z\in H^+(v)$ has $h_q(z)=h_v(z)-h_v(q)>t_0(\ell+a-1)\ge t_0(\ell+1)$, using $a\ge2$; a path
from $q$, at normalized height $0$, to normalized height above $\ell+1\ge\ell+y$ moves in steps of at most
$1$, and its first vertex above $\ell+y-1$, which is at least $1>0$, lies in $L^y_\ell(q)$. So every path
from $q$ to $H^+(v)$ meets $L^y_\ell(q)=L^y_{(\ell+1)-1}(q)$. The nonconnection in
$Q^y_{q;k,\ell+1,r+a_*}$ holds in $L^y_{-k-r-a_*-2,\infty}(q)\supseteq\Theta^*$, hence in $\Theta^*$, and
$Q^y_{q;k,\ell+1,r+a_*}\subseteq E_q$. So $X^{-k-r,0}_{-k}(q;y)\1_{Q^y_{q;k,\ell+1,r+a_*}}\le Z_q\1_{E_q}$,
which is \tn{5.83} with the new parameters. The variable $Z_q\1_{E_q}$ uses only $J$-coordinates, since the
slab of $Z_q$ lies in $L_{-S,-\ell-1}(v)$ and $E_q$ is defined in $\Theta^*\subseteq J$. Extending the
residual law by fresh Bernoulli($p$) variables on $A\cup\Theta$, as in~\cite{TN26}, gives
$\E^{q;A,\Theta}_{p,J}[Z_q;\ E_q]\ge\widehat\E^{q,y}[X^{-k-r,0}_{-k}(q);\ Q_{q;k,\ell+1,r+a_*}]$ for every
$y$. In place of the transitivity used at this point in~\cite{TN26}: $q\in O^*$, so the supremum over $y$
of the right side equals the left side of the proposition, by the choice of $O^*$.

\emph{Step 8: averaging}, as written. Sum~\eqref{eq:DT1} against $\widehat\Pp^{v,x}(A,\Theta)$ over
admissible $(A,\Theta)$. By Step~5, $\sum_{(A,\Theta)}\widehat\Pp(A,\Theta)\widehat\Pp(B^c\mid A,\Theta)=
\widehat\Pp^{v,x}(N_\ell(v)\cap P_v(S))$, since $N_\ell(v)\cap P_v(S)$ forces $v$ to be the peak of $A$,
which lies in the set joined inside $L_{-S,\infty}(v)$. This gives
$\sum_i\widehat\E^{v,x}[X^{-S,0}_{-k-\ell-i}(v);\ P_v(S)]\ge c'\widehat\Pp^{v,x}(N_\ell(v)\cap P_v(S))\cdot
\mathrm{LHS}$. Divide by $\widehat\Pp^{v,x}(P_v(S))$, which is at least $(1-p)^d>0$ because it contains the
isolated-root event, and rearrange; this proves the proposition with $C_p=1/c'$.
\qed

\paragraph{Downstream: \tn{5.113}--\tn{5.117}, hence \tn{5.112}.}
\begin{itemize}
\item \emph{\tn{5.113}, with a random root.} For $i\in[1,a_*+1]$ put $K=k+\ell$ and
$F_i(a,b)=\1\{b\in L_{-K-i}(a),\ a\lr b\text{ in }L_{-S,0}(a),\ P_a(S)\}$. It is diagonally invariant with
the equivariant offsets and vanishes unless $a$ and $b$ are open, so Lemma~\ref{lem:Q4} gives
\[
p\sum_O\mu(O)\int_0^1\sum_i\widehat\E^{O,y}[X^{-S,0}_{-K-i}(O);\ P_O(S)]\,dy=\E_\mu\E\sum_i\sum_a
F_i(a,\rho)\Delta(\rho,a).
\] If $F_i(a,v)=1$, then $a\in L_{K+i}(v)$ and $v$ is joined to $a$ in
$L_{K+i-S,K+i}(v)=L_{-(r+a_*+1-i),K+i}(v)$, with lower depth at most $r+a_*$; there is at most one such
$a$, as in~\cite{TN26}, and $\Delta(v,a)\le e^{t_0(K+i+1)}$. The quasi-transitive \tn{5.42}, uniform in
$(v,x)$, bounds the right side by $Ce^{-t_0(\alpha-1)K+C_0r}$. Every orbit has weight at least
$\mu_{\min}$, so for every vertex $v_0$, $\int\sum_i\widehat\E^{v_0,y}[X^{-S,0}_{-K-i}(v_0);\
P_{v_0}(S)]\,dy\le(p\mu_{\min})^{-1}Ce^{-t_0(\alpha-1)(k+\ell)+C_0r}$, and some offset $x_{v_0}$ attains at
most this value.
\item \emph{\tn{5.114}.} Take $v_0$ sloped, so that~\eqref{eq:591} holds for every offset. At
$(v_0,x_{v_0})$ the denominator of Proposition~\ref{prop:DT} is at least $\widehat\Pp^{v_0,x}(v_0\lr
L_{-\ell-1}(v_0),\ P_{v_0})$: a path from $v_0$ to $L_{-\ell-1}(v_0)$, stopped at its first entry, stays in
$L_{-\ell-1,\infty}(v_0)$ and witnesses $N_\ell(v_0)$, and $P_{v_0}$ implies $P_{v_0}(S)$. By~\eqref{eq:591}
this is at least $c_\xi e^{-t_0(\alpha-1+\xi)(\ell+1)}$, and the numerator, divided by
$\widehat\Pp(P_{v_0}(S))\ge(1-p)^d$, is at most $Ce^{-t_0(\alpha-1)(k+\ell)+C_0r}$. Hence, for every
$\xi>0$, $\sup_{v,x}\widehat\E^{v,x}_{p_c}[X^{-k-r,0}_{-k}(v);\ Q_{v;k,\ell+1,r+a_*}]\le C_\xi
e^{C_0r-t_0(\alpha-1)k+t_0\xi\ell}$; the $(\alpha-1)\ell$ terms cancel, as in~\cite{TN26}.
\item \emph{\tn{5.115}--\tn{5.117}.} The complement of $Q_{v;k,\ell+1,r+a_*}$ is $\{v\lr L_\ell(v)\text{ in }
L_{-k-r-a_*-2,\infty}(v)\}$. The quasi-transitive \tn{5.42} and $\alpha^s_{p_c}\ge1$ give
$\sup\widehat\Pp(Q^c)\le Ce^{C_0(k+r)-t_0\ell}$, the factor $e^{C_0a_*}$ being absorbed. \tn{5.116}--\tn{5.117}
are then as written with $\ell=\max\{2,\lceil t_0^{-1}\log N+C_1(k+r)\rceil\}$, where $C_1>C_0/t_0$ is the
constant of~\cite{TN26}; the event is now $Q_{v;k,\ell+1,r+a_*}$, and the shift $\ell\to\ell+1$ is
absorbed into constants. This proves \tn{5.112}.
\end{itemize}