\section{Correspondence with the bond argument}\label{app:table}

Tables~\ref{tab:corr1} and~\ref{tab:corr2} list each numbered statement of~\cite[Sections~2--9]{OAINU}, its
site form here, and the change. ``As stated'' means that the statement of~\cite{OAINU} is used without
change; ``verbatim'' means that its proof carries over with site bits. We use as stated only the graph,
operator and product-measure statements of~\cite{OAINU}, namely the weighted mass transport, the
invariant trace, the walk with a spectral gap, and the decision-tree covariance
bound~\cite[Lemmas~2.7, 2.8, 2.9 and~5.4]{OAINU}, and we check their hypotheses at each use. We have
checked for site percolation every step of the proofs of~\cite{OAINU} that we cite as carrying over.

\begin{table}[ht]
\centering\small
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.16\textwidth}>{\raggedright\arraybackslash}p{0.24\textwidth}>{\raggedright\arraybackslash}p{0.54\textwidth}@{}}
\toprule
\cite{OAINU} & Here & Change for sites\\
\midrule
Lemma 2.1 & Section~\ref{sec:conventions} & simplification to $G_s$ instead of subdivision\\
Lemma 2.2 & Lemma~\ref{lem:crit-finite} & Peierls with the closed vertex boundary; \cite[Theorem~1.1(2)]{Pei26b}\\
Lemma 2.3 & Lemma~\ref{lem:tools} & vertex bits; tree bound with $\kappa_p$\\
Lemma 2.4 & Lemma~\ref{lem:threshold-order} & verbatim\\
Proposition 2.5 & Theorem~\ref{thm:b8} & replaced: \cite[Theorem~2.9]{Hut19} is a bond theorem\\
Theorem 2.6 & Theorem~\ref{thm:unimod} & site proof, Sections~\ref{sec:goodness}--\ref{sec:critical}\\
Lemmas 2.7--2.9 & Lemmas~\ref{lem:MTP}--\ref{lem:walk} & as stated\\
Lemma 2.10 & Lemma~\ref{lem:sharpness} & site sharpness~\cite{AV08}; $T_p(x,x)=p$\\
Lemma 2.11 & Lemma~\ref{lem:russo} & closed pivotal sites; remainder $R_p$\\
Lemma 2.12 & Lemma~\ref{lem:sprinkling} & kernels $I+T_p\Adj$ and $\Adj(I+T_p\Adj)$\\
Definition 3.1 & Definition~\ref{def:good} & cut vertices in place of bridges\\
Proposition 3.2 & Proposition~\ref{prop:good} & pruning (Lemma~\ref{lem:pruning}); sequential reconstruction\\
Lemma 3.3 & Lemma~\ref{lem:marked} & root excluded; exterior $(D\setminus\Nb[W])\cup\{v\}$; $\Pp(E)\ge p_ch$\\
Lemma 3.4 & Lemma~\ref{lem:radius} & real range $\kappa\ge\kappa_0$\\
--- & Lemma~\ref{lem:pruning} & new\\
Lemma 4.1 & Lemma~\ref{lem:scales} & new bounds (d)--(e): bins and second moment at record scales\\
Definition 4.2 & Definition~\ref{def:map} & site bits; conventions (i)--(ii)\\
Proposition 4.3 & Proposition~\ref{prop:efficient} & junction costs absorbed in the iteration\\
Lemma 4.4 & Lemma~\ref{lem:initial} & visited vertices, including $x$, open; coin $p_c^{\ell+1-e}$\\
Lemma 4.5 & Lemma~\ref{lem:splicing} & junction rejection; seams are cut vertices\\
Lemma 4.6 & Lemma~\ref{lem:coding} & hanging sets; gap bound $l\le3+KM^\tau$\\
Lemma 4.7 & Lemma~\ref{lem:mean} & factor $c_0^k$\\
Lemma 4.8 & Lemma~\ref{lem:second-moment} & joint restoration at a site obstruction (the jointly restored root set can be disconnected); extra factor $4^d\nu^2$\\
Lemma 4.9 & Lemma~\ref{lem:step} & factors $c_0^{2k}$ and $c_0^{-k}$\\
\bottomrule
\end{tabular}
\caption{Sections~2--4 of~\cite{OAINU}.}\label{tab:corr1}
\end{table}

\begin{table}[ht]
\centering\small
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.16\textwidth}>{\raggedright\arraybackslash}p{0.24\textwidth}>{\raggedright\arraybackslash}p{0.54\textwidth}@{}}
\toprule
\cite{OAINU} & Here & Change for sites\\
\midrule
Lemma 5.1 & Lemma~\ref{lem:maxpot} & resampling of $\Nb[B]$; first boundary site\\
Lemma 5.2 & Lemma~\ref{lem:mappot} & deleted domain $V\setminus\Nb[A]$\\
Proposition 5.3 & Proposition~\ref{prop:moments} & factor $d$ in the upper bound\\
Lemma 5.4 & --- & as stated\\
Proposition 5.5 & Proposition~\ref{prop:susceptibility} & revealment $(d+2)C_an^{-a}$\\
Lemma 6.1 & Lemma~\ref{lem:contacts} & score difference $-p\,k_s(A,C_v)$\\
Proposition 6.2 & Proposition~\ref{prop:trace} & remainder at most $Ca_3$; $a_3\ge p^3$\\
Definition 7.1 & Definition~\ref{def:templates} & greedy shortcut; vertex blockers\\
--- & Lemma~\ref{lem:isolation} & new\\
Lemma 7.2 & Lemma~\ref{lem:basic} & monotonicity through Lemma~\ref{lem:isolation}\\
Lemma 7.3 & Lemma~\ref{lem:calibration} & $D_w=V\setminus(I_w\cup B_w)$\\
Proposition 7.4 & Proposition~\ref{prop:norm} & verbatim after calibration\\
Proposition 7.5 & Proposition~\ref{prop:decrease} & factor $e^{-2tc_k}$\\
Lemma 7.6 & Lemma~\ref{lem:deletion} & visits of $\Nb[A']$; constant $C_{\mathrm{del}}$\\
Lemma 7.7 & Lemma~\ref{lem:correlation} & $C'=C_{\mathrm{del}}c_{\mathrm{nb}}\kappa_p$\\
Lemma 7.8 & Lemma~\ref{lem:legs} & new template class of weight $2c_k$\\
Lemma 7.9 & Lemma~\ref{lem:endpoint} & factor $e^{-t(l+1)c_k}$\\
Lemma 8.1 & Lemma~\ref{lem:escapes} & site avoidance; factors $d+1$\\
Lemma 8.2 & Lemma~\ref{lem:smoothing} & $T_0(b,b)=p$\\
Claim 9.1 & Appendix~\ref{app:conclusion}, step (3) & $c_f=e^{-(L_f+1)}p_0^{L_f}$\\
Lemma 9.2 & Lemma~\ref{lem:passage} & verbatim\\
Proposition 9.3 & Proposition~\ref{prop:coupled} & vertex labels; at most $d$ clusters joined per site\\
\bottomrule
\end{tabular}
\caption{Sections~5--9 of~\cite{OAINU}.}\label{tab:corr2}
\end{table}