\section{Amplification of local endpoint maps}\label{sec:amplification}

This section proves the site form of~\cite[Proposition~4.3]{OAINU}: a heavy tail of the critical
cluster size yields local rules whose endpoint kernels are dispersive much faster than their success
probability decays or their cap grows. Throughout, $G$ is simple, infinite, connected, locally finite
and quasi-transitive with $h_V(G)>0$, $\Gamma=\Aut(G)$ is unimodular, and percolation has parameter
$p_c$, with $0<p_c<1$ by Lemma~\ref{lem:crit-finite}. Put
\[
c_0:=(1-p_c)^{d-1}\in(0,1).
\]
Suppose that
\begin{equation}\label{eq:heavy}
[\E_{p_c}s_o^\alpha]=\infty\qquad\text{for some }0<\alpha<\tfrac12,
\end{equation}
and choose $\beta>\alpha$ and $\eps>0$ so that, with $\tau=\frac12+\eps$, $\beta+\tau<1$, as
in~\cite[Section~4]{OAINU}. Goodness (Definition~\ref{def:good}) always uses this $\tau$, and enters only
through the inequality $\dpiv(B)\le KM^\tau$, in Lemma~\ref{lem:coding}.

\subsection{Local endpoint maps}\label{sec:maps}

\begin{lemma}[{Selection of an orbit; site form of~\cite[Lemma~4.1]{OAINU}}]\label{lem:scales}
There are a vertex orbit $O$, a constant $c>0$ and an unbounded sequence $\cN$ of dyadic integers such
that, for $n=2^j\in\cN$ and $\pi=[\Pp_{p_c}(n\le s_o<2n)]$,
\begin{equation}\label{eq:scales}
\begin{array}{@{}l@{\qquad}r@{}l@{\qquad}l@{}}
\textup{(a)}&\pi&{}\ge n^{-\beta},&\\[2pt]
\textup{(b)}&[\Pp_{p_c}(n/2\le s_o<2n)]&{}\le(1+2^\beta)\pi,&\\[2pt]
\textup{(c)}&\E_{p_c}\bigl[|C_x\cap O|\1\{n\le s_x<2n\}\bigr]&{}\ge cn\pi&(x\in O),\\[2pt]
\textup{(d)}&[\Pp_{p_c}(2^i\le s_o<2^{i+1})]&{}\le2^{\beta(j-i)}\pi&(0\le i\le j),\\[2pt]
\textup{(e)}&\E_{p_c}[s_x^2;\ 1\le s_x<2n]&{}\le C_On^2\pi&(x\in O),
\end{array}
\end{equation}
where~\textup{(e)} follows from~\textup{(d)}, with $C_O=4/(\sigma_O(1-2^{\beta-2}))$ and $\sigma_O$ the type
weight of $O$. The orbit and the sequence can be kept fixed throughout the amplification.
\end{lemma}

Bounds~(a)--(c) are those of~\cite[Lemma~4.1]{OAINU}, with $s_o=0$ when $o$ is closed. Bounds~(d)
and~(e) are new. The proof selects $\cN$ among the record indices $j$ of $j\mapsto2^{\beta j}\pi_j$,
where $\pi_j=[\Pp_{p_c}(2^j\le s_o<2^{j+1})]$, that is, $2^{\beta i}\pi_i<2^{\beta j}\pi_j$ for all
$i<j$; this gives~(d), whose case $i=j-1$ gives~(b). We call the $n\in\cN$ \emph{record scales}. The
second-moment bound~(e) at record scales is used only in Lemma~\ref{lem:second-moment}, to replace a
lower size window that the site argument loses. Both are proved in Appendix~\ref{app:amp-routine}.


\begin{definition}[{Local endpoint map; site form of~\cite[Definition~4.2]{OAINU}}]\label{def:map}
A \emph{local endpoint map} on an orbit $O$ is a randomized rule at each $x\in O$ that reads site bits
in a deterministic finite ball about $x$ and uses auxiliary randomness independent of those bits. Its
conditional rules are covariant under $\Aut(G)$. The rule either fails or outputs $y\in C_x\cap O$. A
map with integer cap $M$ is required, on every successful input pattern, to have $s_x\le M$ and $C_x$
good for cap $M$. Write $u>0$ for its success probability and
\[
K(x,y)=\Pp(\text{output }y\mid\text{success at }x),\quad r=\norm K,\quad U=\log(1/u),\quad m=\log M,
\quad H=\log(1/r).
\]
All probabilities include the auxiliary randomness, and the norm is on counting-measure $\ell^2(O)$.
\end{definition}

The definition is read with two conventions, which hold for every map constructed below. (i) A
successful input pattern is a pattern of the read bits together with an auxiliary outcome on which the
rule succeeds, and the guarantees $y\in C_x\cap O$, $s_x\le M$ and goodness hold for every
configuration agreeing with the pattern on the read ball; they are properties of the pattern, since
$\{s_x\le M,\ C_x\text{ good},\ y\in C_x\}$ is determined by the bits within distance $M$ of $x$. (ii)
The rule is a fixed probability kernel from read-bit patterns to outcomes, realized with auxiliary
uniforms whose law does not depend on $p$. Invariance makes $u$ independent of $x\in O$; $K$ has row
sums one and, by Lemma~\ref{lem:MTP} restricted to $O\times O$, column sums one; so $0<r\le1$.

\begin{proposition}[{Efficient site endpoint maps; site form of~\cite[Proposition~4.3]{OAINU}}]
\label{prop:efficient}
Under~\eqref{eq:heavy}, for every $\eta>0$ there is a local endpoint map on the orbit $O$ of
Lemma~\ref{lem:scales}, with $M\ge2$ and $H>1$, such that
\[
\frac{U+m}{H}<\eta.
\]
Moreover the maps can be chosen so that $H\to\infty$. Their radii are finite, but need not be bounded
uniformly over the construction.
\end{proposition}

The proof follows~\cite[Section~4]{OAINU}: a starting map from a random walk with $H$ as large as
desired (Lemma~\ref{lem:initial}), an amplification step built from $k$ old steps by splicing
(Lemma~\ref{lem:step}), and an iteration whose parameter order makes $(U+m)/H$ decrease geometrically.
The site changes are in the splicing step, Sections~\ref{sec:switching}--\ref{sec:second-moment}; the
starting map, the step and the iteration are adapted in Appendix~\ref{app:amp-routine}.

\begin{lemma}[{Starting map; site form of~\cite[Lemma~4.4]{OAINU}}]\label{lem:initial}
For every $H_0$ there is a site local endpoint map on $O$ with $H>H_0$, with cap as large as
prescribed.
\end{lemma}

\subsection{The experiment and the site switching procedure}\label{sec:switching}

Fix an old map with parameters $(u,M,r)$, an integer $k\ge1$, and $n\in\cN$ with $kM\le n/2$. The
seeds are as in~\cite[Section~4.2]{OAINU}: old-map seeds, independent continuous priorities on the
edges of $G$ ordering contact lists, and a uniform $V_i\in[0,1]$ for each index $1\le i\le k$. Ties
have probability zero and are rejected.

For fixed $x,z\in O$, require in a distinguished site field $\omega$ that $A=C_x(\omega)$ satisfies
$n\le|A|<2n$, so that $x$ is $\omega$-open. Independently, start at $z$ and take $k$ old-map steps, with new
independent site fields and new seeds for each step. List a successful chain in reverse order,
$y_0,y_1,\dots,y_k=z$; let $E_i$ be the field used for the step from $y_i$, with output $y_{i-1}$, and
$C_i=C_{y_i}(E_i)$, which contains $y_i$ and $y_{i-1}$, has $|C_i|\le M$ and is good for cap $M$.
Require $y_0\in A$. The experiment has mass $1/n$ if all this holds \emph{and all rejection tests
below pass}, and zero otherwise.

Start with $S=A$ and no index selected. An index is \emph{eligible} if it exceeds every index already
selected. While $z\notin S$:
\begin{enumerate}[label=\textup{(\arabic*)},leftmargin=2.2em]
\item \emph{Selection.} Select the largest eligible $i$ with $C_i\cap\Nb[S]\ne\varnothing$.
\item \emph{Piece.} In the \emph{standard case} $y_i\notin\Nb[S]$, let $Y_i$ be the component of $y_i$
in the $E_i$-open subgraph induced by $V\setminus\Nb[S]$; let $W_i$ be the set of $E_i$-open vertices of
$\partial S$ adjacent to $Y_i$; and let the \emph{contact list} be the $K_i$ graph edges between $Y_i$
and $W_i$, ordered by priority. Choose the contact edge of rank $\lceil K_iV_i\rceil$, write it
$\{w_i,b_i\}$ with $w_i\in W_i$ and $b_i\in Y_i$, and call $w_i$ the \emph{seam}. In the
\emph{degenerate case} $y_i\in\partial S$, which is possible only for $i=k$ (Lemma~\ref{lem:splicing}),
put $w_k=z$ and $Y_k=\varnothing$; no uniform is used.
\item \emph{Junction rejection.} Put $R_i=N(w_i)\setminus(\Nb[S]\cup\Nb[Y_i])$, with
$\Nb[\varnothing]=\varnothing$. Reject, with mass zero, unless every vertex of $R_i$ is closed in the
current $\omega$.
\item \emph{Exchange.} Exchange the current $\omega$ and $E_i$ bits on the mask
$\mathrm{Mask}_i=\{w_i\}\cup(\Nb[Y_i]\setminus\Nb[S])$.
\item Set $S\leftarrow S\cup\{w_i\}\cup Y_i$, and call $P_i=\{w_i\}\cup Y_i$ the \emph{inserted piece}.
\end{enumerate}

Opening a closed seam $w$ attaches every currently open component adjacent to $w$, up to $d-1$
foreign branches; this is the junction problem. Steps~(3)--(4) force the new root cluster to be exactly
$S\cup P_i$: the mask makes $Y_i$ open and its outer boundary off $\Nb[S]$ closed, both from $E_i$, and
the rejection set $R_i$ covers every other neighbour of $w_i$ off $\Nb[S]$, whatever its $E_i$-state.
The rejection set must include $E_i$-closed neighbours of $w_i$ off $\Nb[S]\cup\Nb[Y_i]$ too, since they
are outside the mask and an $\omega$-open one would attach. The set $R_i$ is not exchanged at its own step; it
meets later masks only at the next seam, whose input bit the inverse restores by undoing that mask, so
the inverse has no ambiguity there.

\begin{lemma}[{Geometry of the site switching procedure; site form of~\cite[Lemma~4.5]{OAINU}}]
\label{lem:splicing}
Every step is well defined, with $K_i\ge1$ in the standard case. If the step's rejection test passes,
its effect on the distinguished root cluster is exactly $S\mapsto S\cup\{w_i\}\cup Y_i$, and $w_i$ is a
cut vertex of the new root cluster separating $S$ from $Y_i$. The selected indices increase. Let
$i_1<\dots<i_m$ be the selected indices and $S_j$ the set before the $j$-th selection, $S_1=A$. Then
every later piece avoids the closed neighbourhood of every earlier set: $P_{i_{j'}}\cap\Nb[S_j]=
\varnothing$ for $j'>j$, and $Y_{i_j}\cap\Nb[S_j]=\varnothing$. Consequently $w_{i_j}$ is the only vertex
of $C'\setminus S_j$ adjacent to $S_j$, where $C'=C_x(\omega')$ is the final root cluster, so
\[
S_j=\text{the component of }x\text{ in }C'\setminus\{w_{i_j}\};
\]
for $j<m$, and for $j=m$ in the standard case, $w_{i_j}$ separates $x$ from $z$ in $C'$. In the
degenerate case $w_{i_m}=z$. Finally
\begin{equation}\label{eq:support}
z\in C_x(\omega'),\qquad s_x(\omega')\le M'=2n+kM.
\end{equation}
\end{lemma}

Lemma~\ref{lem:splicing} is proved in Appendix~\ref{app:amp-switch}, by induction over the steps. Before
each step the current root set $S$ is the open cluster of $x$ in the current distinguished field. After a
passed rejection test and the exchange, the outer boundary of $S\cup\{w_i\}\cup Y_i$ lies in
$(\partial S\setminus\{w_i\})\cup(\partial Y_i\setminus\Nb[S])\cup R_i$, which is closed, so the new root
cluster is exactly $S\cup\{w_i\}\cup Y_i$. Maximality of the selected index keeps every later piece
outside the closed neighbourhood of every earlier set. The same induction shows that the rejection bits
are fresh: $R_i\subseteq V\setminus\Nb[S]$, while every earlier mask and rejection set lies in $\Nb[S]$,
so no $\omega$-bit of $V\setminus\Nb[S]$ has been read.

Thus $C'$ is a chain $A-w_{i_1}-Y_{i_1}-w_{i_2}-\cdots-w_{i_m}-Y_{i_m}$ in which each set is
adjacent to the earlier ones only through the next seam; $w_{i_{j+1}}$ may be adjacent to $w_{i_j}$ or
to vertices of $Y_{i_j}$, but to nothing in $S_j$.

\subsection{Inverse codes}\label{sec:coding}

For $c\in C'\setminus\{x\}$ let $X(c)$ be the component of $x$ in $C'\setminus\{c\}$ and
$\Hang(c)=C'\setminus X(c)$, the \emph{hanging set} at $c$. It contains $c$, and $c$ is its only vertex
adjacent to $X(c)$, so $c$ is determined by $\Hang(c)$.

\begin{lemma}[{Inverse codes and contact probabilities; site form of~\cite[Lemma~4.6]{OAINU}}]
\label{lem:coding}
Fix the output fields, $x$, $z$, the old-map seeds and the priorities. Let $C'=C_x(\omega')$, and let
$\Lambda=(\lambda_0,\lambda_1,\dots,\lambda_q)$ be the list with $\lambda_0=z$ followed by the cut
vertices of $C'$ separating $x$ from $z$, ordered from $z$ toward $x$. A valid switching input is
specified by its set of selected indices and positive gaps locating the selected seams in $\Lambda$.
Each fixed code determines at most one input bit tuple, without using the uniforms $V_i$. After
integrating those uniforms, the total mass of valid codes is at most
\begin{equation}\label{eq:W}
W^k,\qquad W=1+\sum_{l=1}^M\min\Bigl\{1,\frac{4dM^\tau}{l}\Bigr\}\le CM^\tau\log M\quad(M\ge2).
\end{equation}
\end{lemma}

The proof (Appendix~\ref{app:amp-switch}) reads the seams off the hanging sets. By
Lemma~\ref{lem:splicing}, $\Hang(w_{i_j})=C'\setminus S_j$, so the seams occur in $\Lambda$ in the order
$w_{i_m},\dots,w_{i_1}$, and from a code one recovers the sets $S_j$, the pieces and the masks, undoes
the exchanges in reverse order and replays the procedure. Goodness of the old cluster $C_{i_j}$, with
deleted set $C_{i_j}\cap\Nb[S_j]$, gives the gap bound $l_j\le3+K_{i_j}M^\tau$; this is the only place
where goodness enters. The prescribed seam is an endpoint of at most $\min\{d,K_{i_j}\}$ contact edges,
so the uniform $V_{i_j}$ lies in a set of measure at most $\min\{1,d/K_{i_j}\}\le\min\{1,4dM^\tau/l_j\}$,
and summing over codes gives~\eqref{eq:W}.

\subsection{The load and its mean}\label{sec:load}

All comparisons are made on finite bit spaces, by conditioning on the bits outside a deterministic ball
about $x$ that contains every query, mask and rejection set of every successful experiment at $(x,z)$,
as in~\cite[Section~4.3]{OAINU} (Appendix~\ref{app:amp-switch}). At fixed seeds an exchange swaps bits
between equally distributed i.i.d.\ fields on a common mask, and rejection changes no bit; so the number
of open bits is preserved, and each successful input tuple and its output have the same product
probability. With the injective inverse at a fixed code (Lemma~\ref{lem:coding}), define, for a single
reference i.i.d.\ distinguished site field $\omega'$, $L_x(z,\omega')$ as $1/n$ times the sum over codes of
the valid-code indicators, integrated over reference i.i.d.\ auxiliary output fields and all seeds,
including the $V_i$. This is the density of the switched mass at $z$ relative to the law of $\omega'$,
and by~\eqref{eq:W}, $0\le L_x(z,\omega')\le W^k/n$. It vanishes unless~\eqref{eq:support} holds, and it
is a covariant function of $\omega'$ in the ball of radius $M'$ about $x$ (Appendix~\ref{app:amp-switch}).
Put $L=\sum_{z\in O}L_x(z,\omega')$ and $b_0=\E L$. As in~\cite{OAINU}, the reference product law is not
a claim about the law of the actual output of a switch.

\begin{lemma}[{Mean load with junction rejection; site form of~\cite[Lemma~4.7]{OAINU}}]
\label{lem:mean}
Let $F(x,z)=\E L_x(z,\omega')$, $B_n(x,y)=\Pp_{p_c}(y\in C_x,\ n\le s_x<2n)$ and
$F_0(x,z)=(u^k/n)\sum_{y\in O}B_n(x,y)K^k(z,y)$, the bond formula of~\cite[Lemma~4.7]{OAINU}. Then,
entrywise and for the row sum $b_0$ of $F$, which is constant on $O$,
\[
c_0^kF_0\le F\le F_0,\qquad cc_0^k\pi u^k\le b_0\le C\pi u^k,\qquad\norm{F/b_0}\le c_0^{-k}r^k.
\]
\end{lemma}

Dropping the rejection tests gives the bond formula, so $F\le F_0$. The lower bound $F\ge c_0^kF_0$ is
new: conditionally on the root cluster $A$, the old chain with its fields and the seeds, and on the event
that the earlier tests pass, the rejection sets of the selected steps are determined, lie outside
$\Nb[A]$ and outside all earlier masks, and have at most $d-1$ vertices each, so all tests pass with conditional probability at least $(1-p_c)^{(d-1)k}=c_0^k$. The
row-sum and norm bounds then follow as in~\cite[Lemma~4.7]{OAINU}. The proof is in
Appendix~\ref{app:amp-switch}.

\subsection{The second moment and joint restoration}\label{sec:second-moment}

The second-moment bound of~\cite[Lemma~4.8]{OAINU} undoes two switching inverses at once. For two
inverses whose inserted pieces are far apart, it uses that removing two disjoint hanging components
from the root cluster leaves a connected set containing $x$. For bonds the first seams are bridges and
this holds. For sites it fails: if $c^1$ and $c^2$ lie far apart on a cycle of $C'$ through $x$, each
carrying its pieces, removing both hanging sets cuts the cycle. We record the two seams and their
neighbourhoods in the remaining set, which costs a polynomial factor, and replay the switching from a
set rather than a connected cluster.

\begin{lemma}[{Second moment; site form of~\cite[Lemma~4.8]{OAINU}}]\label{lem:second-moment}
There is a fixed polynomial $\cQ(k,M)\ge1$, for example a large graph-dependent multiple of
$1+\nu+\nu^2+d^2\nu^3+4^d\nu^4$ with $\nu=kM$, such that
\begin{equation}\label{eq:second-moment}
\E L^2\le \cQ(k,M)\,\pi u^{2k}\qquad\text{whenever }kM\le n/2\text{ and }n\ge(W/u)^k.
\end{equation}
\end{lemma}

\paragraph{Outline of the proof of Lemma~\ref{lem:second-moment}.} Expand $L^2$ with a common reference field $\omega'$ and two independent families of auxiliary output
fields and seeds; a summand is a pair of valid coded inverses with endpoints $z_1,z_2$. For a switching
inverse put $B=\Hang(w_{i_1})=C'\setminus A=P_{i_1}\cup\dots\cup P_{i_m}$ (Lemma~\ref{lem:splicing}); it
contains its endpoint and has $|B|\le \nu$. There are three classes.

\emph{Class~1} (at least one inverse makes no switch) is verbatim from~\cite[Lemma~4.8]{OAINU} and
contributes at most $C\pi u^{2k}$. \emph{Class~2} (both switch, and $B^1,B^2$ are nested or at graph
distance at most $2$) is a counting bound: hanging sets are nested or disjoint, so, given the first
inverse, the second endpoint takes at most $\nu+\nu^2+d^2\nu^3$ values, each of integrated mass at most
$W^k/n$, and the class contributes at most $C(\nu+\nu^2+d^2\nu^3)\pi u^{2k}$. Both are proved in
Appendix~\ref{app:amp-second}. The site change is in Class~3.

\emph{Class 3: both switch, and $\operatorname{dist}(B^1,B^2)\ge3$, that is, $\Nb[B^1]\cap\Nb[B^2]=
\varnothing$.}

\emph{Locality of each inverse.} Every vertex that inverse $a$ exchanges, $\mathrm{Mask}^a:=
\bigcup_j\mathrm{Mask}^a_{i_j}$, every rejection set, every contact set $W^a_j$ and every contact edge
lies in $\Nb[B^a]$, because $w^a_j,Y^a_j\subseteq B^a$. So the two mask collections are disjoint,
including their seams and rejection sets.

\emph{Joint restoration.} Let $\omega^a$, with restored auxiliary family $E^a$, be the input of inverse $a$
alone, so that $C_x(\omega^a)=A^a=C'\setminus B^a$, $n\le|A^a|<2n$, and all of $\partial A^a$ is
$\omega^a$-closed. Write $c^a=w^a_{i_1}$ for the first seam of inverse $a$, so that $B^a=\Hang(c^a)$ and
$c^a\in\partial A^a$. Undo both inverses (the masks are disjoint, so the order is irrelevant), and call
the result $\omega_{00}$; it equals $\omega^1$ off $\mathrm{Mask}^2$ and $\omega^2$ off $\mathrm{Mask}^1$. Put
$A_{00}:=C'\setminus(B^1\cup B^2)=A^1\setminus B^2$, so $|A_{00}|<2n$. The set $A_{00}$ need not be
connected. What survives is the following.

\begin{enumerate}[label=\textup{(J\arabic*)},leftmargin=2.6em]
\item \emph{$A_{00}$ is a union of $\omega_{00}$-clusters; $A_x:=C_x(\omega_{00})$ is its component containing
$x$, and every other component is adjacent to $c^1$ or $c^2$.} We have $\mathrm{Mask}^a\cap A^a=
\varnothing$, since each $\mathrm{Mask}^a_{i_j}$ misses $S^a_j\supseteq A^a$; so $A_{00}$ is
$\omega_{00}$-open, as $\omega_{00}=\omega'$ there. A vertex of $\partial A_{00}$ inside $C'$ is $c^1$ or $c^2$, the only
vertex of $B^a$ adjacent to $X(c^a)\supseteq A_{00}$; and $c^a\in\partial A^a$ is $\omega^a$-closed and not in
$\mathrm{Mask}^b$ for $b\ne a$, so it is $\omega_{00}$-closed. A vertex $y\in\partial A_{00}\setminus C'$ is
in $\partial A^1\cap\partial A^2$; if $y\in\mathrm{Mask}^a$, then $y\notin\mathrm{Mask}^b$ and
$\omega_{00}(y)=\omega^a(y)$ is closed; if $y$ is in neither mask, $\omega_{00}(y)=\omega'(y)$ is closed. So $\partial A_{00}$
is $\omega_{00}$-closed. A component of $A_{00}$ not containing $x$ has a neighbour in $C'\setminus A_{00}=
B^1\cup B^2$, as $C'$ is connected, hence is adjacent to $c^1$ or $c^2$.
\item \emph{$c^a\in\partial A_x$.} Since $\Hang(c^1)\cap \Hang(c^2)=\varnothing$, $c^1\notin \Hang(c^2)$, so some
$x$--$c^1$ path $\gamma$ in $C'$ avoids $c^2$; it stays in $X(c^2)$, and before its first visit to $c^1$
it stays in $X(c^1)$. So the part of $\gamma$ before $c^1$ is a connected subset of
$X(c^1)\cap X(c^2)=A_{00}$ containing $x$; it lies in $A_x$, and its last vertex is adjacent to $c^1$.
Likewise for $c^2$.
\item \emph{Recovering $A_{00}$ with bounded extra data.} By (J1),
$A_{00}=A_x\cup\bigcup_{y\in N^1\cup N^2}C_y(\omega_{00})$ with $N^a:=N(c^a)\cap A_{00}$. Record the extra
datum $\xi=(c^1,c^2,N^1,N^2)$. The seam $c^a$ lies in the first selected old cluster of chain $a$, hence
in its trajectory union $T^a$, with $|T^a|\le \nu$, which is computable from the restored chain, and
$N^a\subseteq N(c^a)$ has at most $2^d$ values. So, given the restored tuple, $\xi$ takes at most
$4^d\nu^2$ values, and $(\omega_{00},\xi)$ determines $A_{00}$.
\end{enumerate}

Appendix~\ref{app:amp-second} completes Class~3. Run virtually from the \emph{set} $S_0=A_{00}$ with the
field $\omega_{00}$, the switching procedure reproduces both original runs step by step, because, by
distance $3$, a vertex of $\Nb[B^1]$ is neither in $B^2$ nor adjacent to it. So the joint
undo, together with the datum $\xi$ of~(J3), is injective, and the map to $(\omega_{00},E^1,E^2)$ is at most
$4^d\nu^2$-to-one. By~(J2) each restored trajectory hits $\Nb[C_x(\omega_{00})]$, and the trajectory-hit
bound of~\cite[Lemma~4.8]{OAINU}, with the record-scale bound~(e) of Lemma~\ref{lem:scales} in place of the
lost lower size window, bounds Class~3 by $C4^d\nu^4\pi u^{2k}$. Summing the three classes
gives~\eqref{eq:second-moment} with $\cQ(k,M)=C(1+\nu+\nu^2+d^2\nu^3+4^d\nu^4)$.

The extra factor $4^d\nu^4$ is polynomial in $\nu=kM$, hence $O_M(\log(k+1))$ after taking logarithms, and
it is absorbed exactly as the bond term $\nu^3$ is.

\subsection{One amplification step}

\begin{lemma}[{One amplification step; site form of~\cite[Lemma~4.9]{OAINU}}]\label{lem:step}
Fix an old map. For $n\in\cN$ tending to infinity and $k=O(\log n)$ with $kM\le n/2$ and
$n\ge(W/u)^k$, there is, for all sufficiently large such $n$, a new site local endpoint map with
\[
u'\ge c'c_0^{2k}\pi/\cQ(k,M),\qquad r'\le2c_0^{-k}r^k,\qquad M'=2n+kM.
\]
The threshold for $n$ may depend on the fixed old map.
\end{lemma}

Lemma~\ref{lem:step} truncates $L$ at a multiple of $\cQ(k,M)u^kc_0^{-k}$ using
Lemmas~\ref{lem:mean} and~\ref{lem:second-moment}, discards root clusters that are not good for cap
$M'$, which costs relative mass $\cQ(k,M)c_0^{-2k}n^\beta e^{-(M')^\eps}\to0$ because $c_0^{-2k}=n^{O(1)}$
for $k=O(\log n)$, and outputs $z$ with probability $L_x(z,\omega')$ divided by the truncation level. The
junction cost enters as $c_0^{2k}$ in $u'$ and $c_0^{-k}$ in $r'$; both are linear in $k$ in the
exponent, and in the iteration of Proposition~\ref{prop:efficient} they are absorbed by keeping
$t=U+(\tau+\xi)m$ and $H$ above fixed multiples of $\log(1/c_0)$. Neither touches $\beta+\tau<1$. The proofs
of Lemma~\ref{lem:step} and Proposition~\ref{prop:efficient} are in Appendix~\ref{app:amp-routine}.