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%

\subsection{Brydges\textendash Federbush Section 9 on slit domains}
\label{sec:7.6}\label{ssec:7.6}

In this subsection we bound the activities of Section~\ref{sec:7.5} on $\Gp$ by
\cite[Section~9]{BF80} and \cite[Section~7]{GM82}. On $\Gp$ the argument
differs from these references in the following points. There is a
one-body deficit, and Lemma~\ref{lem:7.31} moves its quadratic part into the covariance.
Lemmas~\ref{lem:7.34} and~\ref{lem:7.35} avoid both an extension of the wall field across $\Xi$ and a
Poincar\'e inequality across a slit.
Finally, the covariance rate is small and independent of $\tau$ (Proposition~\ref{prop:7.38}).
The proofs of this subsection are written out in full in Section~\ref{S4-S:S4.6}, with the rate conditions of Proposition~\ref{prop:7.38} checked one by one, together with Section~\ref{S4-S:S4.4} for Lemma~\ref{lem:7.31}, including Theorem~\ref{thm:7.29} and Corollary~\ref{cor:7.30} for $(C',G^\natural,\fc\cN)$. Section~\ref{S4-S:S4.7} does the same for Appendix~\ref{app:B}, and Section~\ref{S4-S:S4.8} for Lemmas~\ref{lem:7.39} and~\ref{lem:7.47} and Propositions~\ref{prop:7.long}, \ref{prop:7.51} and~\ref{prop:7.56}, the last with the encoding written out.

Throughout, $X$ is a polymer (a union of elements of $\tilde Y(h)$ in the sense of
Definition~\ref{def:7.23}), $|X|$ is its number of $\cU$-cells,
$\vartheta:=\psi+D$, where $D:=g-H$ is the field in the phases, and
$\tilde\delta:=P_X\vartheta$, where $P_X\vartheta:=\vartheta-\avg_\Omega\vartheta$
on each cell $\Omega\subseteq X$ and $P_X\vartheta:=0$ off $X$. We put
$C_u:=2+6/c_g$, so that $\#\Delta<C_u\mt^{-3}$ for each $\cU$-cell $\Delta$
by (R7), and $\|A\|_r:=\sup_x\sum_y|A(x,y)|e^{r\,d_{\Gp}(x,y)}$ for a matrix
$A$ on $V$ and $r\ge0$. Constants $C,C_n$ depend only on those of
Theorem~\ref{thm:6.5} and Proposition~\ref{prop:6.6}.

\paragraph{The deficit decomposition.}
By Proposition~\ref{prop:7.4} the inhomogeneity is confined to the term
$G_{2,3}$ of \eqref{eq:7.G2}, which by (S3) is
$\sum_{m\neq0}\sum_{x\in V}(\rho_1(m,x)-\rho^{\mathrm{ref}}(m))(\cos(m\bGM^{1/2}\vartheta(x))-1)$.
Let $G_{2,3}^{\Dep}$ be its part with $x\in\Dep$, and put
\begin{align}
\mu_{\Dep}(x)&:=\bGM\sum_{m\neq0}m^2\bigl(\rho_1(m,x)-\rho^{\mathrm{ref}}(m)\bigr)
\,1_{V\setminus\Dep}(x),\label{eq:7.6.mu}\\
G_{2,3}^{\natural}&:=\sum_{m\neq0}\sum_{x\in V\setminus\Dep}
\bigl(\rho_1(m,x)-\rho^{\mathrm{ref}}(m)\bigr)
\bigl(\cos(m\bGM^{1/2}\vartheta(x))-1+\tfrac12m^2\bGM\psi(x)^2\bigr).\notag
\end{align}
Then, identically in $\psi$,
\begin{equation}\label{eq:7.6.split}
G_{2,3}=G_{2,3}^{\Dep}-\tfrac12\langle\psi,\mu_{\Dep}\psi\rangle+G_{2,3}^{\natural}.
\end{equation}
We write $G_{2,3}^{\Dep}(Y)$, $G_{2,3}^\natural(Y)$ for the sums over $x\in Y$. Finally, let
$G^\natural:=G_1+G_{2,1}+G_{2,2}+G_{2,3}^{\Dep}+G_{2,3}^{\natural}$ and
$C':=(C_0^{-1}+\nu+\mu_{\Dep})^{-1}$, where the
operator $\mu_{\Dep}$ depends only on $\Gp$ (through $\rho_1$).
At a site with $D(x)=0$ we have $\partial_{\psi(x)}G_{2,3}
=-\mu_{\Dep}(x)\psi(x)+O(\theta'\mt^2\bGM|\psi(x)|^3)$, with
$\theta'\ge1-e^{-\theta_0}$ fixed, that is, with no power of $\bGM$, whereas
\cite[(9.813) to (9.819)]{BF80} need $\bGM^{1/2}$ per derivative of a
one-body coefficient of order $\mt^2/\bGM$. For this reason we move the quadratic part into
the covariance, as $\nu$ is, and $G_{2,3}^\natural$ is of fourth order
(type vi of \cite[(9.811)]{BF80}).

\begin{lemma}\label{lem:7.31}
Assume the standing hypothesis \eqref{eq:7.standing}, $\eps_\nu+\theta'\le1/7$,
with $\eps_\nu$ of Section~\ref{ssec:7.1} (the output form of (S5)) and $\theta'$ from \eqref{eq:7.thetaprime}. Then,
uniformly in $\psi$, $h$, $k$, $\Xi$ and $\Gp$, with $C_{\Dep}$ the constant $C$ of
Proposition~\ref{prop:7.5}(b):
(a) $|\mu_{\Dep}(x)|\le\theta'\mt^2$ for every $x$; for every union $Y$
of $\cU$-cells,
$-C_{\Dep}(e^{-\mu}+e^{-2\kappa_S})\,\cdep N_{\Dep}(Y)\le G_{2,3}^{\Dep}(Y)\le
C_{\Dep}\cdep N_{\Dep}(Y)$ and
\[
|G_{2,3}^\natural(Y)|\le\tfrac12\theta'\mt^2\textstyle\sum_{x\in Y\setminus\Dep}
\bigl(\vartheta(x)^2+\psi(x)^2\bigr).
\] (b) At a site $x\notin\Dep$ with $D(x)=0$, the $n$-th derivative in
$\psi(x)$ of the $x$-term of $G_{2,3}^\natural$ is at most
$C_n\theta'\mt^2\bGM|\psi(x)|^{4-n}$ for $0\le n\le4$ and at most
$C_n\theta'\mt^2\bGM^{n/2-1}$ for $n\ge4$. At $x\in\Dep$, the $n$-th derivative
($n\ge1$) of the $x$-term of $G_{2,3}^{\Dep}$ is at most
$C_n(\mt^2/\bGM)\bGM^{n/2}$. (c) $C'$ exists, is positive definite, and $C'\le\alpha_DC_0$ as
forms, with $\alpha_D:=(1-\eps_\nu-\theta')^{-1}\le 7/6$. (d) The bounds of Theorem~\ref{thm:6.12} hold for $C'$ and for every
interpolated covariance $C'_s$ of \eqref{eq:C.decomp}:
$C'_s(x,x)=C'(x,x)\le\alpha_DK_1M_Y$,
$|C'_s(x,y)|\le 2K_1''M_Ye^{-r_{\mathrm{cov}}d_{\Gp}(x,y)}$,
$\|C'_s\|_{r_{\mathrm{cov}}}\le 2K_3\mt^{-2}$ and
$\|C'_s\|_{2\to2}\le\alpha_D\mt^{-2}$. (e) As measures on $\mathbb R^V$,
\begin{equation}\label{eq:7.6.cov}
e^{G(\psi+g)}\,d\mu_C(\psi)=\fc\,e^{G^\natural}\,d\mu_{C'}(\psi),\qquad
\fc:=\Bigl(\frac{\det C'}{\det C}\Bigr)^{1/2},
\end{equation}
and $\fc$ does not depend on $h$, $k$ or $\Xi$. (f) The representation \eqref{eq:7.translated}, Corollary~\ref{cor:7.11}\,(d) and Proposition~\ref{prop:7.25} hold with $(C,G,\cN)$ replaced by
$(C',G^\natural,\fc\cN)$; Theorem~\ref{thm:7.29} and Corollary~\ref{cor:7.30} hold for the activities
and outer partition functions built with $(C',G^\natural,\fc\cN)$, with the same left sides
$Z(1,\Xi)$, $Z_r(R)$ and $W_{\Gp}(C)$; and $W_{\Gp}(C)$, every $Z_p$ and every $Z_r(R)$ are unchanged.
\end{lemma}

\begin{proof}
\emph{(a)} Arguing as in the proof of Proposition~\ref{prop:7.5}(a), with $m^2$
replaced by $|m|^n$, we obtain, for $x\notin\Dep$ and $n\ge2$,
\begin{equation}\label{eq:7.6.mom}
\bGM\sum_{m\neq0}|m|^n\,|\rho_1(m,x)-\rho^{\mathrm{ref}}(m)|\le C_n\theta'\mt^2,
\end{equation}
with $C_2=1$ once $C$ in \eqref{eq:7.thetaprime} is large, and at each $x$
the left side is at most $C_n\mt^2$. For $n=2$ this bounds $\mu_{\Dep}$.
In the same case, together with $|\cos y-1+\frac12z^2|\le\frac12y^2+\frac12z^2$, applied with
$y=m\bGM^{1/2}\vartheta(x)$ and $z=m\bGM^{1/2}\psi(x)$, it also bounds
$G_{2,3}^\natural$. The bound on $G_{2,3}^{\Dep}(Y)$ follows from Proposition~\ref{prop:7.5}(b) together with
Proposition~\ref{prop:6.18}.

\emph{(b)} If $D(x)=0$, then
$\vartheta(x)=\psi(x)$ and $f(\psi):=\cos(m\bGM^{1/2}\psi)-1+\frac12m^2\bGM\psi^2$
vanishes to fourth order with $|f^{(n)}|\le|m|^n\bGM^{n/2}$. At $x\in\Dep$,
$|\partial_\psi^n\cos(m\bGM^{1/2}\vartheta)|\le|m|^n\bGM^{n/2}$. We then sum with
\eqref{eq:7.6.mom}.

\emph{(c)} The Schur test gives
$\|\nu\|_{2\to2}\le\|\nu\|_{r_{\mathrm{cov}}}\le\eps_\nu\mt^2$, part (a) gives
$\|\mu_{\Dep}\|_{2\to2}\le\theta'\mt^2$, and
$C_0^{-1}=u^{-1}+\mt^2\ge\mt^2$ by Definition~\ref{def:7.1}. Hence
$C_0^{-1}+\nu+\mu_{\Dep}\ge(1-\eps_\nu-\theta')C_0^{-1}\ge\frac67C_0^{-1}>0$,
and (c) follows, since inversion reverses the order of positive definite forms.

\emph{(d)} Theorem~\ref{thm:6.12} applies to $C'=(C_0^{-1}+\nu')^{-1}$,
$\nu':=\nu+\mu_{\Dep}$, as
$2K_3\|\nu'\|_{r_{\mathrm{cov}}}\le2K_3(\eps_\nu+\theta')\mt^2\le\mt^2$,
where we used $K_3\le3.5$. The diagonal bound follows from (c). Block compressions keep the
diagonal and positivity and increase no other bound, which extends these bounds to the
interpolated covariances.

\emph{(e)} By \eqref{eq:7.6.split},
$e^{G(\psi+g)}=e^{G^\natural}e^{-\frac12\langle\psi,\mu_{\Dep}\psi\rangle}$. Since
$C^{-1}=C_0^{-1}+\nu$ by Proposition~\ref{prop:7.8}, we have
$e^{-\frac12\langle\psi,\mu_{\Dep}\psi\rangle}d\mu_C(\psi)
=(\det2\pi C)^{-1/2}e^{-\frac12\langle\psi,C'^{-1}\psi\rangle}d\psi=\fc\,d\mu_{C'}(\psi)$.
The operators $C_0$, $\nu$, $\mu_{\Dep}$ depend only on $\Gp$, $M_Y$ and $\mt$, so the factor $\fc$ does not depend on $h$, $k$ or $\Xi$, and since \eqref{eq:7.6.cov} is an identity of measures, it also preserves absolute integrability.

\emph{(f)} We insert \eqref{eq:7.6.cov} in Proposition~\ref{prop:7.8}. Those
proofs use only factorization over cells and sites, the margin of
Lemma~\ref{lem:C.4} ($q_M\le1/6$ for $C'$) and the quadratic bound on
$|E(X,s)|$, and, as $G-G^\natural=-\frac12\langle\psi,\mu_{\Dep}\psi\rangle$ does not depend on the wall field, \eqref{eq:7.6.cov} applies to each configuration of $Z(k,\Xi)$, $Z_p$ and $Z_r(R)$, so these are unchanged; the outer partition functions with $X\ne\emptyset$ are not the same for the two choices, and from here on only those built with $(C',G^\natural,\fc\cN)$ are used.
\end{proof}

Like the $\mathcal Z_p(X)$, the activities are now built with
$(C',G^\natural,\fc\cN)$. For a polymer $X$, $\cN(X)$ is the set of wall
configurations $N_X:=N|_{E(X)}$ of cell-constant labels $h_X$ with values in
$P\mathbb Z$ ($P=2\pi\bGM^{-1/2}$) whose walls lie in $E(X)$. At $k=0$ the
statements of this subsection and Lemma~\ref{lem:7.53} hold for the long
wall fields of Definition~\ref{def:7.27}, by Proposition~\ref{prop:7.long}.
We write $n_I(X)$ for the number of interfaces with both cells in $X$, and put $A_w:=2\Funit$.

\begin{lemma}[Wall entropy]\label{lem:7.32}
(a) $\Floc(Y)\ge A_w\sum_{I\subseteq Y}N_I^2$ for every union $Y$ of elements
of $\tilde Y(h)$, where $N_I$ is the constant value of $N$ on the interface
$I$ (Lemma~\ref{lem:7.6}(a)). (b) For every polymer $X$ and every $c_4>0$ with
$e^{-c_4A_w}\le\frac13$,
$\sum_{N_X\in\cN(X)}e^{-c_4\Floc(X)}\le(1+3e^{-c_4A_w})^{n_I(X)}$
$\le e^{c_4'|X|}$ with $c_4':=9C_ue^{-c_4A_w}/(c_gc_1(\mt L)^3)$.
\end{lemma}

\begin{proof}
We adapt the proof of \cite[Lemmas~9.2, 9.3]{BF80} to $\Gp$ as follows.
(1) The face sum of \cite[(9.117)]{BF80} becomes the interface sum. Indeed, walls
are supported on interfaces by Lemma~\ref{lem:7.6}(a), and by Theorem~\ref{thm:7.14}
with $\mathrm{good}(I)\ge c_g'L^2$, which holds by (R8) in fine units and Lemma~\ref{lem:4.17},
an interface $I\subseteq Y$ gives at least
$\frac1{18}\mt^2\frac L2c_g'L^2\cdot4\pi^2N_I^2/\bGM=2\Funit N_I^2$. (2) The
sum over $h$ becomes the injection $N_X\mapsto(N_I)_I\in\mathbb Z^{n_I(X)}$,
each interface giving $\sum_ne^{-c_4A_wn^2}\le1+3e^{-c_4A_w}$. (3) The cube
count becomes $n_I(X)\le3C_u|X|/(c_gc_1(\mt L)^3)$, since a cell has at least
$c_1L^3$ sites by (R5) and at most $6/c_g$ neighbours by (R8), and a $\cU$-cell has
fewer than $C_u\mt^{-3}$ sites. With $\log(1+3t)\le3t$ this gives the second inequality of (b),
and by Corollary~\ref{cor:7.11}(d) no perimeter term arises. For the value $c_4=c'/4$ of Table~\ref{tab:A.1}, $c_4'$
is doubly exponentially small in $\tau$, as
$A_w\ge(\pi^2c_g'/72)c_L^3(\mt\bGM)^{-1}$ by $\mt L\ge c_L/2$.
\end{proof}

In Debye units, for the Peierls factor $r_\Omega$ of \eqref{eq:7.rOmega},
we put $v_\Omega:=\mt^3|\Omega|$, $\hat\rho(m):=\mt^{-3}\rho^{\mathrm{ref}}(m)$,
$\bD:=\bGM\mt$, $\hat P:=2\pi\bD^{-1/2}$, and
\begin{equation}\label{eq:7.6.Rhat}
\hat R_v(x):=\frac{\exp\bigl\{v\sum_m\hat\rho(m)\bigl[e^{im\bD^{1/2}x}-1\bigr]\bigr\}}
{\sum_{n\in\mathbb Z}\exp\bigl\{-\frac12v(x-n\hat P)^2\bigr\}},
\end{equation}
so that $r_\Omega(A)=\hat R_{v_\Omega}(\mt^{-1/2}(A+k\bGM^{-1/2}\bar f_\Omega))$.
By (R5) in fine units and $c_L/2\le\mt L\le c_L$, we have
$v_\Omega\in[v_-,v_+]:=[c_1c_L^3/8,\,8c_L^3]$. Let
$\eta_*:=\sup_{0<s\le\pi}[1-2\bD\sum_m\hat\rho(m)(1-\cos ms)/s^2]
\le\bar\eta:=1-\frac4{\pi^2}\bigl(1-C_{\hat\rho}e^{-2\kappa_S}\bigr)+C_{\hat\rho}e^{-2\kappa_S}\le1-\frac4{\pi^2}+2C_{\hat\rho}e^{-2\kappa_S}$, where
$C_{\hat\rho}:=4C\ge\bD\sum_{|m|\ge2}|\hat\rho(m)|\,m^2e^{2\kappa_S(|m|-1)}$ with $C$ of \eqref{eq:6.S4c}
($\mt\ge\mD/2$); unlike $\eta_*$, the bound $\bar\eta$ does not depend on $\bD$.

\begin{lemma}\label{lem:7.33}
Fix $\eta_H\in(\bar\eta,1)$. There are $b_0>0$ and $c_7,c_7',c_7''$, depending only
on $v_\pm$, $\eta_H$, $C_{\hat\rho}$ and $\kappa_S$, such that for
$\bD\le b_0$, $v\in[v_-,v_+]$, all real $x$ and all integers $N\ge0$,
\[
|\partial_x^N\hat R_v(x)|\le c_7\,(c_7'\bD^{1/6})^Ne^{c_7''N\log N}
e^{\frac12\eta_Hvx^2}.
\]
Consequently, uniformly in $h$, with $\partial A_\Omega/\partial\psi(x)=1/|\Omega|$
($x\in\Omega$),
\[
|\partial_A^Nr_\Omega(A_\Omega)|\le c_7\,(c_7'\bD^{1/6}\mt^{-1/2})^N
e^{c_7''N\log N}\exp\Bigl(\tfrac12\eta_H\mt^2\sum_{x\in\Omega}\vartheta(x)^2\Bigr).
\]
\end{lemma}

The first bound is proved as \cite[Lemma~9.7, (9.51)]{BF80}, with $\tilde l_D=1$,
$L^3\to v$, $\beta\to\bD$, $\rho_i\to\hat\rho(m)$, $\tau\to\hat P$, except for the following
points. (1) There are infinitely many charges. The Taylor expansions of the factors
I, II of \cite[(9.53)]{BF80} converge for $|\mathrm{Im}\,x|\le\bD^{-1/6}$ by
the tails of (S4$'$), as $e^{|m|\bD^{1/3}}e^{-2\kappa_S|m|}$ is summable.
(2) The constants are uniform in $v\in[v_-,v_+]\subset(0,\infty)$, not for a
fixed volume $L^3$. (3) We need an explicit exponent for
$\alpha_Dp'\eta_H<1$ in (L8-1). The inequalities $1-\cos s\ge2s^2/\pi^2$ on $[0,\pi]$ and
$2\bGM\rho^{\mathrm{ref}}(1)\ge(1-C_{\hat\rho}e^{-2\kappa_S})\mt^2$ bound $\eta_*$, and
each $\eta_H\in(\bar\eta,1)$ is admissible in \cite[(9.59)]{BF80}. The second
bound follows from the $\hat P$-periodicity, from
$A_\Omega+k\bGM^{-1/2}\bar f_\Omega=h_\Omega+\avg_\Omega\vartheta$ with
$h_\Omega\in P\mathbb Z$, and from
$|\Omega|(\avg_\Omega\vartheta)^2\le\sum_{x\in\Omega}\vartheta(x)^2$. With
$\gamma:=\eta_H\mt^2$, the second bound is the analogue on $\Gp$ of \cite[(7.9), (7.10)]{GM82}: each
derivative of $G_1$ contributes a factor $\bD^{1/6}$, times $e^{\frac12\gamma\sum_X\vartheta^2}$.

\paragraph{H\"older exponents and the hypotheses (L8-1) to (L8-4).}
We fix $\eta_H\in(\bar\eta,1)$, H\"older exponents $p_1,\dots,p_4>1$ with
$\sum_i1/p_i=1$ and $p_1$ even, and $p'>p_3$, $q'$ with $1/p'+1/q'=1/p_3$,
at the values given in Section~\ref{S:G.L8}. Let $C_W^f$, $\kbox=\rbox L$,
$N_{\mathrm{ov}}$ and $\Kflat_\Omega\supseteq\Omega$, with sites in the box
$\hat Q(Q)$ of fine side at most $\kbox$, be those of the cell Poincar\'e
inequality of Proposition~\ref{prop:6.10}(iv), and let
$C_P':=C_W^f\rbox^2N_{\mathrm{ov}}$, so that $C_P'L^2=C_W^f\kbox^2N_{\mathrm{ov}}$.
Hypothesis (L8-1) is the standing hypothesis \eqref{eq:7.standing} together with
$\hat\eta:=\alpha_Dp'\eta_H<1$, which implies $\alpha_Dp_3\eta_H<1$ (in \cite[(9.638)]{BF80} the constant $\hat\eta$
is denoted $\hat\beta<1$). (L8-2) requires
$C_P'(\mt L)^2\le\eta_H/8$, so that
$\theta_P:=(4\hat\eta/\eta_H)C_P'(\mt L)^2\le\frac12$ and
$\nu_0:=\max(\hat\eta,\theta_P)<1$. (L8-3) requires $\eps(L')\le 2(p'/q')^2$, with
$\eps(L')$ of Theorem~\ref{thm:7.19}, and (L8-4) states that each $\Kflat_\Omega$ has
$d_{\Gp}$-diameter less than $L'$. At the values fixed above, (L8-1) holds. Hypothesis (L8-2) holds by the choice of
$c_L$, since $\mt L\le c_L$, (L8-3) holds by Corollary~\ref{cor:7.21}, and (L8-4)
follows from (L8-2) by Proposition~\ref{prop:6.10}(v). A cell
$\Omega\subseteq Q$ is \emph{near} if the $\ell^\infty$-distance from the
sites of $\hat Q(Q)$ to $C$ is at most $\frac12$, and \emph{far} otherwise.
For a union $Y$ of elements, $E(\cN(Y))$ is the union of the $E(\Kflat_\Omega)$
over the near cells $\Omega\subseteq Y$.

\paragraph{The free-space solid angle.}
Let $\Omega_\Xi(y)\in(-2\pi,2\pi)$ be the signed solid angle of the flat
rectangle $\Xi$ seen from a point $y$ off its plane, increasing by $4\pi$
when $y$ crosses $\Xi$ in the direction in which $\xi_b=+1$. For a site $x$
of $\Gp$ (a dual site of $\mathbb Z^3$, never in the plane of $\Xi$) we put
$f^C(x):=\frac12\Omega_\Xi(x)$ and, for each dual bond $b=(x,x')$ of
$\mathbb Z^3$, $R^C(b):=f^C(x')-f^C(x)-j(b)$, where $j(b)=2\pi\xi_b$, which on the
bonds of $\Gp$ is the jump field of $\Xi$ (Section~\ref{sec:7.1}). The field
$f^C$ depends only on $C$, not on $\Gp$, unlike the Neumann solid angle
$f_{\Gp}$ of Lemma~\ref{lem:7.2}, and $R^C$ differs from $R_f$. Let
$S(D):=\sum_{b:\,d(b,C)\le D}|R^C(b)|^2$, the sum running over dual bonds of
$\mathbb Z^3$, and $D_{\cN}:=\sqrt3\,\kbox+1$.

\begin{lemma}\label{lem:7.34}
(a) Assume (L8-4). Let $Y$ be a union of elements of $\tilde Y(h)$ and
$\Omega\subseteq Y$ a cell. For every $\vartheta\in\mathbb R^V$, every
function $a$ on the sites of $\Kflat_\Omega$ and every $\eps\in(0,1]$,
\begin{gather*}
\|P_\Omega\vartheta\|^2\le(1+\eps)C_W^f\kbox^2\|\nabla\vartheta+w_Y\|^2_{E(\Kflat_\Omega)}
+(1+1/\eps)k^2\bGM^{-1}\delta_\Omega(a)^2,\\
\delta_\Omega(a):=(C_W^f)^{1/2}\kbox\|\nabla a-j\|_{E(\Kflat_\Omega)}+\|P_\Omega a\|_\Omega ,
\end{gather*}
where $w_Y:=\sum_{S_\beta\subseteq Y}w_\beta$. If $\Omega$ is far, there is an
$a$ with $\delta_\Omega(a)=0$. (b) For every dual bond $b$ of $\mathbb Z^3$, $d(b,C)\ge\frac12$ and
$|R^C(b)|\le 2\pi/d(b,C)$; at every dual site, $|f^C|<\pi$. (c) On every output $\Gp$,
$\sum_{\Omega\text{ near}}\|R^C\|^2_{E(\Kflat_\Omega)}\le N_{\mathrm{ov}}\|R^C\|^2_{E(\cN(V))}
\le N_{\mathrm{ov}}S(D_{\cN})$, and if $|C|\ge8\kbox$, then
\[
S(D_{\cN})\le C_{\mathrm{BS}}|C|\,(\log\kbox+C_{\mathrm{sh}}),\qquad C_{\mathrm{BS}}:=24\pi^3,\quad
C_{\mathrm{sh}}:=92 .
\]
\end{lemma}

\begin{proof}
Heuristically, on a box crossed by $C$ the trivial primitive leaves a flat sheet of
jumps, of energy $\asymp\kbox^2$, and with the solid angle only the
Biot\textendash Savart residual remains, of energy $\asymp\kbox\log\kbox$ per unit length
of $C$.

\emph{(a)} We first show that $w=w_Y$ on $E(\Kflat_\Omega)$. Let
$b'\in E(\Kflat_\Omega)$ be a wall bond. It belongs to one component $w_{\beta'}$ and has a wall
site $z$ as an endpoint. Fix $x\in\Omega$. Then $d_{\Gp}(x,z)<L'$ by (L8-4). Hence
each $\cU$-cell met by a $\Gp$-path of length $d_{\Gp}(x,z)$ from $x$ to $z$
lies in $\hat\Sigma$, consecutive ones being joined by a bond, and by Proposition~\ref{prop:7.9} the $\cU$-cell of
$x$ lies in $S_{\beta'}$. This $\cU$-cell also lies in the
element of $Y$ containing $\Omega$, and each $S_\beta$ lies in one element, in view of
Definition~\ref{def:7.23}. Hence $S_{\beta'}\subseteq Y$. Other bonds carry
$w=0=w_Y$. Next we define $a_\Omega:=h-k\bGM^{-1/2}a$ on the sites of $\Kflat_\Omega$.
Here, at $k=0$, for long wall fields, the label is replaced by the primitive of
Lemma~\ref{lem:7.prim} (Proposition~\ref{prop:7.long}), and at $k=1$, for the inside fields
of outer class in $\Glong$, by the primitive of Lemma~\ref{lem:7.insideprim}\,(b). As $h$ is constant on cells, $P_\Omega h=0$ and
$P_\Omega\vartheta=P_\Omega(\vartheta+a_\Omega)+k\bGM^{-1/2}P_\Omega a$. On
$E(\Kflat_\Omega)$, since $w=\nabla h-k\bGM^{-1/2}j$, we have
$\nabla(\vartheta+a_\Omega)=\nabla\vartheta+w_Y+k\bGM^{-1/2}(j-\nabla a)$. The
Poincar\'e inequality of Proposition~\ref{prop:6.10}(iv) for
$\vartheta+a_\Omega$ on $\Kflat_\Omega$ and the triangle inequality give
$\|P_\Omega\vartheta\|\le(C_W^f)^{1/2}\kbox\|\nabla\vartheta+w_Y\|_{E(\Kflat_\Omega)}
+k\bGM^{-1/2}\delta_\Omega(a)$. We then square, using
$(x+y)^2\le(1+\eps)x^2+(1+1/\eps)y^2$.

Now let $\Omega\subseteq Q$ be far, and let $\Pi$ be the plane of $\Xi$. A dual bond
crossing $\Pi$ crosses it at the centre of a plaquette, and $j(b)\neq0$ iff
that plaquette lies in $\Xi$. The plaquettes crossed by bonds of $\hat Q(Q)$
form a rectangle. If this rectangle contained plaquettes in and outside $\Xi$, it would
contain two sharing an edge $e\in\partial\Xi=C$ at $\ell^\infty$-distance
$\frac12$ from a site of $\hat Q(Q)$, which is excluded for a far cell. Hence
$j=2\pi\sigma\nabla1_{\Pi^+}$ on $E(\hat Q(Q))$ for some $\sigma\in\{-1,0,1\}$,
where $\Pi^+$ is the set of sites on one side of $\Pi$. With this $\sigma$, the function
$a:=2\pi\sigma1_{\Pi^+}$ has $\nabla
a=j$ on $E(\Kflat_\Omega)$ and is constant on $\Omega$, as $Q$ lies on one side
of $\Pi$ by (A3) of Lemma~\ref{lem:3.5}. Thus $\delta_\Omega(a)=0$.

\emph{(b)} Primal edges are at distance at least $\frac12$ from dual bonds, which gives the first assertion.
Off $C$, $\nabla\Omega_\Xi$ extends continuously across $\Xi$ as the
Biot\textendash Savart field of $C$. Hence $|R^C(b)|\le\frac12\max_{y\in b}J(y)$,
where $J(y):=\oint_C|dl|\,|y-l|^{-2}$. Let $y=y'+h\mathbf e$ with $y'\in\Pi$ and $h\ge0$.
The length of $\{l\in C:|y'-l|\le t\}$ vanishes for $t<d(y',C)$ and is at
most $2\pi t$, since this set lies in the boundary of a convex subset of a disc of radius
$t$. Integration by parts and the substitution $t=h\tan\varphi$ give
$J(y)\le4\pi/d(y,C)$. Finally, $|\Omega_\Xi|<2\pi$, since the radial projection of
the rectangle from a point off $\Pi$ lies in an open hemisphere.

\emph{(c)} For a near cell $\Omega\subseteq Q$, the bonds of $\Kflat_\Omega$
are dual bonds of $\mathbb Z^3$ joining sites of $\hat Q(Q)$, a box of
diameter at most $\sqrt3\,\kbox$ within Euclidean distance $\frac{\sqrt3}2$ of
$C$, so $d(b,C)\le D_{\cN}$. Each $|R^C(b)|^2$ is a fixed nonnegative number
attached to a dual bond of $\mathbb Z^3$, independent of $\Gp$, and passing from $\mathbb Z^3$ to $\Gp$ can only delete
terms of free-space $\ell^2$ sums, since freezing a plaquette deletes a bond and a hole deletes sites and bonds.
The bounds are therefore free-space bounds, independent of the Neumann field and its enhancement at the slit
edges, and with the overlap
$N_{\mathrm{ov}}$ these facts give the first two inequalities. For the last one, we use the
number $\mathcal N(r)$ of dual bonds with $d(b,C)\le r$, which vanishes for
$r<\frac12$. The midpoints of these bonds lie within $r+\frac12$ of $C$ and form three
translates of $\mathbb Z^3$. Hence, with $r_*:=\frac12+\frac{\sqrt3}2$,
$\mathcal N(r)\le3|\{y:d(y,C)\le r+r_*\}|\le3\pi|C|(r+r_*)^2+16\pi(r+r_*)^3$.
By (b) and summation by parts,
$\sum_{d(b,C)\le D}d(b,C)^{-2}=\mathcal N(D)D^{-2}+2\int_{1/2}^D\mathcal N(r)r^{-3}dr$.
The term $3\pi|C|r^2$ gives $6\pi|C|\log D+\Ord(|C|)$, and the other terms give
$\Ord(|C|+D)$. The factor $(2\pi)^2$ of (b) gives $C_{\mathrm{BS}}=(2\pi)^2\cdot6\pi=24\pi^3$.
At $D=D_{\cN}\le2\kbox$ and $|C|\ge8\kbox$ the remaining constants add up to
$C_{\mathrm{sh}}=92$ (Section~\ref{S:G.34}).
\end{proof}

For a polymer $X$ put
\[
B_X(\vartheta):=\tfrac12\mt^2\sum_{x\in X}\vartheta(x)^2
+\frac{2\mt^2}{\eta_H}\sum_{x\in X}(P_X\vartheta)(x)^2,\qquad
T_3:=\Bigl[\int d\mu_{C'_s}(\psi)\,e^{p_3\eta_HB_X(\psi+D)}\Bigr]^{1/p_3}.
\]
With $\gamma=\eta_H\mt^2$ we have
$\frac12p_3\gamma\sum_X\vartheta^2+2p_3\mt^2\sum_X\tilde\delta^2=p_3\eta_HB_X(\vartheta)$.
Hence $T_3$ is the third factor of the H\"older split
(Proposition~\ref{prop:7.37}), and it is the analogue on $\Gp$ of \cite[(9.64)]{BF80}. The
near-cell factor is
\[
\fn'(Y):=2k^2\bGM^{-1}\Bigl[\theta_P\|R^C\|^2_{E(\cN(Y))}
+\frac{4\hat\eta\mt^2}{\eta_H}\sum_{\Omega\subseteq Y\text{ near}}\|P_\Omega f^C\|^2_\Omega\Bigr].
\]
We also use its cellwise version
$\fn''(Y):=2k^2\bGM^{-1}\sum_{\Omega\subseteq Y\text{ near}}\bigl[\frac{\theta_P}{N_{\rm ov}}\|R^C\|^2_{E(\Kflat_\Omega)}
+\frac{4\hat\eta\mt^2}{\eta_H}\|P_\Omega f^C\|^2_\Omega\bigr]$, which is additive over disjoint unions of cells
and satisfies $\fn''\le\fn'$, as each bond of $E(\cN(Y))$ lies in at most $N_{\rm ov}$ of the sets
$E(\Kflat_\Omega)$ (Theorem~\ref{thm:3.20}\,(R6)).

\begin{lemma}\label{lem:7.35}
Assume (L8-1) to (L8-4), and let $c_8$ and $\cnear$ be the constants of Section~\ref{S:G.35}. For every $h$, every polymer $X$ and every $s$,
\[
T_3\le\exp\Bigl\{\frac{c_8|X|+\fn'(X)}{p'}+\frac{\Floc(X)}{p_3}\Bigr\},
\]
and $\fn'=0$ for $k=0$. If
$\min(R,T)\ge R_1(\beta)=(2\rbox+5)L$, then $\fn'(V)\le\cnear|C|$, and uniformly for $\bGM\ge\bGMo$,
\[
\cnear\le0.1264\,\theta_PC_{\mathrm{BS}}+\frac{2\theta_PC_{\mathrm{BS}}C_{\mathrm{sh}}
+1024\pi^2\hat\eta(2\rbox+5)^2c_L^2/(\eta_Hc_1)}{\bGM}.
\]
\end{lemma}

\begin{proof}
\emph{Step 1 (local untranslation identity).} We absorb the shift $D$ by a translation, which turns
$\frac12\langle\psi,C_0^{-1}\psi\rangle+F_2+\Floc$ into one quadratic form of
$\psi+D$. In local form, this is the step of \cite[p.~226]{BF80} in which (7.8) to (7.10) are applied backwards.
Let $Y$ be a union of
elements, let $D_Y$, $w_Y$, $s_Y$ be the sums of $D_\beta$, $w_\beta$, $s_\beta$ over
$S_\beta\subseteq Y$, and let
$\mathcal Q_Y(\vartheta):=\mt^2\|\vartheta\|^2+\|\nabla\vartheta+w_Y\|^2+M_Y^{-2}\|\Delta\vartheta-\nabla^*w_Y\|^2$.
Then for all $\psi\in\mathbb R^V$
\begin{equation}\label{eq:7.6.untr}
\tfrac12\langle\psi,C_0^{-1}\psi\rangle+F_2(Y)+\Floc(Y)=\tfrac12\mathcal Q_Y(\psi+D_Y).
\end{equation}
Indeed, since $C_0^{-1}=\mt^2-\Delta+M_Y^{-2}\Delta^2$ and $-\Delta=\nabla^*\nabla$,
the right side is
$\frac12\langle\psi,C_0^{-1}\psi\rangle+\langle\psi,C_0^{-1}D_Y+s_Y\rangle+\frac12\mathcal Q_Y(D_Y)$
with $s_Y=M_Y^{-2}(-\Delta+M_Y^2)\nabla^*w_Y$. By Proposition~\ref{prop:7.9},
$D_\beta=-\chi_\beta C_0s_\beta$ and $(1-\chi_\beta)s_\beta=0$, so
$C_0^{-1}D_\beta+s_\beta=-[C_0^{-1},\chi_\beta]C_0s_\beta=F_{2,\beta}'$,
supported in $S_\beta$. Therefore the cross term is $F_2(Y)$. Since the functions $D_\beta$,
$\nabla D_\beta$, $w_\beta$, $\Delta D_\beta$, $\nabla^*w_\beta$ are supported in
$S_\beta$ or $E(S_\beta)$, terms of different $\beta$ have disjoint
supports, and
$\frac12\mathcal Q_Y(D_Y)=\sum_{S_\beta\subseteq Y}F_1(S_\beta)=\Floc(Y)$
by Proposition~\ref{prop:7.10} and Corollary~\ref{cor:7.11}.

\emph{Step 2 (the Gaussian integral at fixed $Y$).} We show
\begin{equation}\label{eq:7.6.JY}
J_Y:=\int d\mu_{C_0}(\psi)\exp\bigl\{\hat\eta B_Y(\psi+D)-\Floc(Y)-F_2(Y)\bigr\}
\le\exp\{c_8|Y|+\fn''(Y)\}.
\end{equation}
Each $S_\beta$ lies in $Y$ or is disjoint from it, so $D=D_Y$ on $Y$. Let
$N:=\hat\eta\mt^21_Y+(4\hat\eta\mt^2/\eta_H)P_Y$, so that
$\hat\eta B_Y(\vartheta)=\frac12\langle\vartheta,N\vartheta\rangle$,
since $P_Y=1_YP_Y1_Y$ is an orthogonal projection. By \eqref{eq:7.6.untr} and $\vartheta=\psi+D_Y$, $J_Y$ is the ratio of
the integrals over $\mathbb R^V$ of
$\exp\{-\frac12\mathcal Q_Y(\vartheta)+\frac12\langle\vartheta,N\vartheta\rangle\}$
and of $e^{-\frac12\langle\psi,C_0^{-1}\psi\rangle}$, with quadratic part
$-\frac12\langle\vartheta,(C_0^{-1}-N)\vartheta\rangle$.
For positivity we use \cite[(9.637), (9.640) to (9.642)]{BF80}, with the
cell Poincar\'e inequality summed with overlap $N_{\mathrm{ov}}$, that is,
$\|P_Y\vartheta\|^2\le C_P'L^2\|\nabla\vartheta\|^2$, in place of the Neumann
Laplacian of the cubes in (9.642). We obtain $\langle\vartheta,N\vartheta\rangle\le
\hat\eta\mt^2\|\vartheta\|^2+\theta_P\|\nabla\vartheta\|^2\le
\nu_0\langle\vartheta,C_0^{-1}\vartheta\rangle$, the last step because $C_0^{-1}\ge\mt^2-\Delta$. Thus
$K:=C_0^{1/2}NC_0^{1/2}$ satisfies $0\le K\le\nu_0<1$, and hence
$J_Y=\det(1-K)^{-1/2}\exp\{\sup_\vartheta[-\frac12\mathcal Q_Y(\vartheta)
+\frac12\langle\vartheta,N\vartheta\rangle]\}$.

We first bound the supremum by Lemma~\ref{lem:7.34}, and only near cells leave a remainder. For a far cell we apply Lemma~\ref{lem:7.34}(a) with
$\delta_\Omega(a)=0$ and $\eps\to0$, and for a near cell with $\eps=1$, $a=f^C$
and $\delta_\Omega(f^C)^2\le2C_W^f\kbox^2\|R^C\|^2_{E(\Kflat_\Omega)}+2\|P_\Omega f^C\|^2_\Omega$.
Summing over the cells of $Y$ with overlap $N_{\mathrm{ov}}$ and multiplying
by $2\hat\eta\mt^2/\eta_H$ gives, as
$(2\hat\eta\mt^2/\eta_H)\cdot2C_W^f\kbox^2N_{\mathrm{ov}}=\theta_P$,
$\frac12\langle\vartheta,N\vartheta\rangle\le\frac12\hat\eta\mt^2\|\vartheta\|^2_Y
+\theta_P\|\nabla\vartheta+w_Y\|^2+\fn''(Y)$, the near-cell remainders adding up exactly to $\fn''(Y)$. Since $\mathcal Q_Y(\vartheta)\ge\mt^2\|\vartheta\|^2+\|\nabla\vartheta+w_Y\|^2$,
the exponent is at most
$-\frac12(1-\hat\eta)\mt^2\|\vartheta\|^2-(\frac12-\theta_P)\|\nabla\vartheta+w_Y\|^2+\fn''(Y)\le\fn''(Y)$.

For the determinant, we have $-\frac12\log\det(1-K)\le\operatorname{tr}K/(2(1-\nu_0))$
\cite[Appendix~3]{BF80}, and $\operatorname{tr}K=\operatorname{tr}(NC_0)\le
\hat\eta(1+4/\eta_H)\mt^2K_1M_Y\#Y\le2(1-\nu_0)c_8|Y|$ by $P_Y\le1_Y$,
Theorem~\ref{thm:6.12}, $M_Y=\mt/\lPV$ and $\#Y<C_u\mt^{-3}|Y|$.

\emph{Step 3 (assembly).} We follow \cite[(9.65) to (9.626)]{BF80}, adapted to $\Gp$
as follows. (1) We apply the convexity step \cite[(9.67) to (9.611)]{BF80},
fact S3 of Table~\ref{tab:C.1}, to the block compressions
$C'_\epsilon=\sum_J1_{Y_J}C'1_{Y_J}$ of \eqref{eq:C.decomp},
$C'_s=\sum_\epsilon\lambda_\epsilon(s)C'_\epsilon$. The factors in this step are finite, because
for $\frac12\langle\vartheta,N_1\vartheta\rangle=p_3\eta_HB_X(\vartheta)$ we have
$\alpha_DN_1\le N$ (as $p_3<p'$), $N$ commutes with each $1_{Y_J}$, and
$C'_\epsilon\le\alpha_DC_{0\epsilon}$, so that
$\|N_1^{1/2}C'_\epsilon N_1^{1/2}\|\le\nu_0<1$ by the positivity step.
(2) In the monotonicity step (9.612) to (9.617) we use Lemma~\ref{lem:7.31}(c), $C'_\epsilon\le\alpha_DC_{0\epsilon}$ with
$C_{0\epsilon}:=\sum_J1_{Y_J}C_01_{Y_J}$, instead of (9.618) to (9.622). (3) After H\"older's inequality (9.625)
with exponents $p'/p_3$, $q'/p_3$, both integrals factor over the sets
$Y_J$, unions of elements (9.626), and the first factor is at most
$\exp\{(c_8|X|+\fn''(X))/p'\}\le\exp\{(c_8|X|+\fn'(X))/p'\}$ by \eqref{eq:7.6.JY}, the additivity of $\fn''$ and $\fn''\le\fn'$. (4) Corollary~\ref{cor:7.20}
with $t_C=1$ and (L8-3) replaces (9.632):
$\bigl(\int d\mu_{C_0}e^{(q'/p')F_2(Y_j)}\bigr)^{1/q'}e^{\Floc(Y_j)/p'}\le e^{\Floc(Y_j)/p_3}$,
as $1/p_3-1/p'=1/q'$. For $k=0$, $\fn'=0$, since $\fn'$ carries the factor $k^2$.

\emph{Step 4 (the perimeter bound).} Lemma~\ref{lem:7.34}(c) and $k\le1$ show that
the first part of $\fn'(V)$ is at most
$2\bGM^{-1}\theta_PC_{\mathrm{BS}}|C|(\log\kbox+C_{\mathrm{sh}})$, as
$|C|=2(R+T)\ge4R_1(\beta)\ge8\kbox$. By Lemma~\ref{lem:7.34}(b),
$\|P_\Omega f^C\|^2_\Omega\le\pi^2|\Omega|\le8\pi^2L^3$. It remains to count the near
cells. A near cell $\Omega\subseteq Q$ lies in a block $Q\subseteq\hat Q(Q)$
all of whose sites are within $\ell^\infty$-distance $\kbox+\frac12$ of $C$.
The $\ell^\infty$-neighbourhood of width $s$ of $C$ has volume at most
$(2s)^2(|C|+8s)$, and for $s=\kbox+2$ and $L\ge4$, $2s\le(2\rbox+5)L$ and
$8s\le(8\rbox+20)L$. Blocks are disjoint with at least $L^3$ sites, and by (R5) a
block contains fewer than $8/c_1$ cells. Hence, as
$|C|\ge4R_1(\beta)=(8\rbox+20)L$,
$N_{\mathrm{near}}\le\frac8{c_1}(2\rbox+5)^2(|C|+(8\rbox+20)L)/L
\le\frac{16}{c_1}(2\rbox+5)^2|C|/L$, and the second part of $\fn'(V)$ is at most
$2\bGM^{-1}\frac{4\hat\eta\mt^2}{\eta_H}8\pi^2L^3N_{\mathrm{near}}
\le\frac{1024\pi^2\hat\eta}{\eta_Hc_1}(2\rbox+5)^2c_L^2\frac{|C|}{\bGM}$, using
$\mt L\le c_L$. Adding the two parts gives $\fn'(V)\le\cnear|C|$. The bound on $\cnear$,
uniform in $\bGM\ge\bGMo$, then follows (Section~\ref{S:G.35}) from
$\log\kbox\le\log(\rbox c_L)+\log(1/\mt)$ and the value of $\mt$.
\end{proof}

By Proposition~\ref{prop:7.8}, $E(\psi)=E'(\psi+g)+\frac12\langle\psi,\nu\psi\rangle$,
where $E'$ collects the Mayer terms of order $s\ge2$ of
Theorem~\ref{thm:6.5}. We write $E(X)$ for the part of $E$ localized in $X$,
as in Section~\ref{sec:7.5}. Both $G_{2,1}$ and $G_{2,2}$ are real, by (S3). Let
$c_\lambda:=\frac14(1-1/p_3)$, $c'_\lambda:=C(e^{-\mu}+\eps_\nu)$,
$c_E:=c'_\lambda+\theta'$,
$G_2^\natural:=G_{2,1}+G_{2,2}+G_{2,3}^{\Dep}+G_{2,3}^\natural$, and assume
\begin{equation}\label{eq:7.6.cl}
2\alpha_Dp_4c_E\le\tfrac12,\qquad
\frac{p_4\alpha_Dc_E^2}{1-2\alpha_Dp_4c_E}+c_E\le c_\lambda .
\end{equation}
For the values of Section~\ref{S:G.36}, \eqref{eq:7.6.cl} follows from
$c_E\le1/70$, which is the case for $\theta_0=0.01$ and $\mu$, $\kappa_S$, $\bGM$
large.

\begin{lemma}\label{lem:7.36}
Assume $\eps_\nu+\theta'\le1/7$ and \eqref{eq:7.6.cl}. For every $h$, every polymer
$X$ and every $s$,
\[
T_4:=\Bigl[\int d\mu_{C'_s}\bigl|e^{E(X)}e^{G_2^\natural(X)}e^{-2\mt^2\sum_X\tilde\delta^2}\bigr|^{p_4}\Bigr]^{1/p_4}
\le\exp\bigl\{c_9|X|+C_{\Dep}\cdep N_{\Dep}(X)+c_\lambda\Floc(X)\bigr\},
\]
with $c_9:=2\alpha_Dc_EK_1C_u/\lPV$.
\end{lemma}

\begin{proof}
The proof of \cite[Lemma~9.9, (9.72) to (9.718)]{BF80} carries over to $\Gp$ with the following changes.
(1) In (9.72), the bracket of $G_{2,1}$ lies in $[0,\frac12y^2]$, where
$y=m\bGM^{1/2}\tilde\delta$, and that of $G_{2,2}$ is at most $y^2$, so by (S4$'$)
$G_{2,1}(X)+G_{2,2}(X)\le\frac32(1+Ce^{-2\kappa_S})\mt^2\sum_X\tilde\delta^2\le2\mt^2\sum_X\tilde\delta^2$.
(2) By Lemma~\ref{lem:7.31}(a), $e^{G_{2,3}^{\Dep}(X)}\le e^{C_{\Dep}\cdep N_{\Dep}(X)}$
and $|G_{2,3}^\natural(X)|\le\frac12\theta'\mt^2\sum_X(\vartheta^2+\psi^2)$. Thus
$c'_\lambda$ of (9.712) becomes $c_E$. (3) On $\Gp$ the bounds (9.73) to (9.712),
$|E(X)|\le\frac12c'_\lambda\mt^2\sum_X(\vartheta^2+\psi^2)$, remain valid.
Indeed, $|\eps_k|\le|m|\bGM^{1/2}|\vartheta|$ as in (9.78), since the phase at $\psi+g$
is $m\bGM^{1/2}(\vartheta+h)$ with $h\in P\mathbb Z$, and (9.711) follows from (S2) of
Theorem~\ref{thm:6.5}, which is uniform in $\Gp$, by the Schur test
\cite[Lemma~9.6]{BF80} together with the bound $\|\nu\|\le\eps_\nu\mt^2$ of (S5). (4) In (9.716) to
(9.718), the bound $C'_s\le\alpha_D\mt^{-2}$ of Lemma~\ref{lem:7.31}(d) takes the place of the lower
bound on $C_s^{-1}$. With it, $K_4:=2p_4c_E\mt^2C_s'^{1/2}1_XC_s'^{1/2}$ has
$\|K_4\|\le2\alpha_Dp_4c_E\le\frac12$, and
$-\frac12\log\det(1-K_4)\le\operatorname{tr}K_4\le p_4c_9|X|$, the last inequality
by Theorem~\ref{thm:6.12}. Condition \eqref{eq:7.6.cl} is (9.718), and the bound
$\frac12\mt^2\|1_XD\|^2\le\Floc(X)$ of Corollary~\ref{cor:7.11} replaces
$\frac12\int f^2\le F_1$.
\end{proof}

Let $\kappa'$ be a term of the expansion of an activity, with the parameters
of \cite[(7.3) to (7.5)]{GM82} referring to $\cU$-cells: as in Lemma~\ref{lem:A.7}, the cubes
$(a_1,\dots,a_t)$ of a Mayer factor are replaced by its number $t$ of points and
its set of $\cU$-cells. We put
$\cP'_G:=e^{\fn'(V)/p'}$ and
\[
T_1:=\Bigl[\int d\mu_{C'_s}\prod_i|\psi(x_i)|^{p_1}\Bigr]^{1/p_1},\qquad
T_2:=\Bigl[\int d\mu_{C'_s}e^{-p_2F_2(X)}\Bigr]^{1/p_2}.
\]

\begin{proposition}[H\"older split]\label{prop:7.37}
Assume (L8-1) to (L8-4), \eqref{eq:7.6.cl} and
\begin{equation}\label{eq:7.6.coll}
p_2\eps(L')\le\tfrac12(1-1/p_3-c_\lambda).
\end{equation}
Fix $h$, $X$, $s$ and a term $\kappa'$.
(a) There is a kernel $k((x_i))\ge0$ such that
\begin{multline*}
\int d\mu_{C'_s}\bigl|e^{E(X,s)}\kappa'e^{G^\natural(X)}e^{-\Floc(X)}e^{-F_2(X)}\bigr|_0\\
\le\int_{x_i\in X}k((x_i))\int d\mu_{C'_s}e^{E}e^{G_2^\natural}e^{-F_2}
\prod_i|\psi(x_i)|\,e^{\frac12\gamma\sum_X\vartheta^2}e^{-\Floc(X)} .
\end{multline*} (b) The right side of (a) is at most
$\int k((x_i))\,T_1T_2\,e^{-\Floc(X)}\,T_3T_4$, with $T_3$ as in
Lemma~\ref{lem:7.35} and $T_4$ as in Lemma~\ref{lem:7.36}. (c) $T_1\le c_T^{\sum_jn_j}\prod_jn_j!$, where $n_j$ is the number of
points $x_i$ in the $j$-th $\cU$-cell and $c_T$ depends only on $p_1$, $r_C$,
$\alpha_DK_1/\lPV$, $K_1''/\lPV$, $c_u$ and $C_{\cU}$; and
$T_2\le e^{p_2\eps(L')\Floc(X)}$. (d) We have
\[
T_2\,e^{-\Floc(X)}\,T_3\,T_4\le\cP'_G\,e^{c_{13}|X|+C_{\Dep}\cdep N_{\Dep}(X)}
e^{-c'\Floc(X)},
\]
with $c_{13}:=c_8/p'+c_9$ and
$c':=1-1/p_3-c_\lambda-p_2\eps(L')\ge c_0':=\frac12(1-1/p_3-c_\lambda)$; at the
values of Section~\ref{S:G.36}, $c_0'=1/16$. For $k=0$, $\cP'_G=1$.
\end{proposition}

\begin{proof}
\emph{(a)} is the analogue on $\Gp$ of \cite[(7.9)]{GM82}, that is, of
\cite[(9.22), (9.23), (9.81) to (9.823)]{BF80}, with the
differentiated factors of Appendix~\ref{app:B}, the rates of
Proposition~\ref{prop:7.38} and Lemma~\ref{lem:7.33} for the derivatives of
$G_1$. By (S3), $E$ and $G_2^\natural$ are real for real $\psi$.
\emph{(b)} follows as in \cite[(7.11)]{GM82}, that is, \cite[(9.25)]{BF80}, since
$(e^{\frac12\gamma\sum_X\vartheta^2}e^{2\mt^2\sum_X\tilde\delta^2})^{p_3}=e^{p_3\eta_HB_X(\vartheta)}$.
\emph{(c)} For $T_1$ we use \cite[(9.31) to (9.33)]{BF80}, with Lemma~\ref{lem:7.31}(d)
in place of the Neumann series of (9.32) and (R7) for the number of
$\cU$-cells within a given distance. The bound on $T_2$ follows from Corollary~\ref{cor:7.20} with
$t_C=\alpha_D\le2$. \emph{(d)} This is \cite[Lemma~13]{GM82} on $\Gp$. By (c) and Lemmas~\ref{lem:7.35},
\ref{lem:7.36}, the exponents of $\Floc(X)$ add up to
$-1+p_2\eps(L')+1/p_3+c_\lambda=-c'$, and \eqref{eq:7.6.coll} gives
$c'\ge c_0'$. Finally, $e^{\fn'(X)/p'}\le\cP'_G$, as $\fn'$ is monotone in $X$.
\end{proof}

\medskip\noindent\emph{Remark.} In these bounds the perimeter is carried by the factor $\cP'_G$ alone.
Far cells contribute nothing, and $E_{\Gp}$ of Corollary~\ref{cor:7.11}\,(d) drops out
of the bounds, since the factor $e^{-k^2E_{\Gp}/2\bGM}$ of Proposition~\ref{prop:7.3} cancels
against $\Floc(V)=F_1+\frac12k^2\bGM^{-1}E_{\Gp}$ and every later use of $\Floc$
is a lower bound. Hence Theorem~\ref{thm:7.61} contains no factor $\log(1+\min(R,T))$,
and $\cP'_G\le e^{2\cnear(R+T)/p'}$.
\medskip

We measure distances between $\cU$-cells in Debye units,
$\dist(A,B):=\mt\,d_{\Gp}(A,B)$. The covariance rate in these units,
$r_C:=r_{\mathrm{cov}}/\mt$, does not depend on $\tau$, by Theorem~\ref{thm:6.12}.
The rate conditions \cite[(9.123), (9.124)]{BF80} require a rate of at least
$(1-\delta_1)/\tilde l_D\ge\frac12$, whereas on $\Gp$ the rate $r_C$ is small. In (9.95) we take the
exponent with the minus sign that (9.124) has.

\begin{proposition}\label{prop:7.38}
Let $0<\delta_2<\bar\delta<r_C$ and put $c'_{\mathrm{rate}}:=\frac12(r_C-\bar\delta)$.
Every step of \cite[Section~9]{BF80} that uses the rate conditions
\cite[(9.123), (9.124)]{BF80}, the covariance bounds \cite[(9.32), (9.34)]{BF80}
or \cite[(9.95)]{BF80} holds on $\Gp$ with the bounds of
Lemma~\ref{lem:7.31}(d) for $C'_s$, with (S2) of Theorem~\ref{thm:6.5} and with any
$\tau$-independent margin, in particular $c'_{\mathrm{rate}}$, in place of a lower bound on the
rates. The pinning factor of
\cite[Lemmas 9.10, 9.11]{BF80} is used with the margin
$c_p:=(r_C-\bar\delta)/18$.
\end{proposition}

\begin{proof}
The argument uses the rate conditions through margins, and the changes are as follows.

(i) \emph{Tree factor} $e^{\bar\delta d}$, $d=\sum_i\dist(\Delta'_i,\Delta''_i)$.
Link $i$ decays at rate $r_C$ by Lemma~\ref{lem:7.31}(d), and pinning brings a factor
$e^{9c_pd}$. The rest $r_C-\bar\delta-9c_p=c'_{\mathrm{rate}}$ is left for the sum over the
positions of $\Delta''_i$, as $d_{\Gp}$ dominates the Euclidean distance and
by (R7) at most $\frac{4\pi}{3}(r+2C_{\cU})^3/c_u$ $\cU$-cells lie within
Debye distance $r$ of a given one.

(ii) \emph{Mayer factors.}
The factors $e^{\bar\delta L_0}$ and $e^{rO_0}$ of \cite[(7.3), (7.4)]{GM82} are absorbed by
the tree decay of (S2) at the Stage~1 rate
$\sqrt3\kappa_YM_Y\ge4\sqrt3\,r_{\mathrm{cov}}>\bar\delta\mt$
of Proposition~\ref{prop:6.11} and by $e^{-\mu(s-1)}$ of (S2). The details are in the \emph{Tree weights} step of the proof of
Proposition~\ref{prop:B.7}.

(iii) \emph{Wick's theorem.} The rate
$\frac12$ is needed only to obtain (9.32) by a Neumann series, whereas
Lemma~\ref{lem:7.31}(d) gives the pointwise bound directly. The
contraction count holds for all rates, with constants depending on $r_C$,
$K_1''/\lPV$, $c_u$, $C_{\cU}$.

(iv) \emph{Clustering.} Since each application has $W\subseteq Z$ or $W\supseteq Z$, we do not use the factor
$e^{-(1-2\delta_1)\dist(X_1,W)}$ of \cite[(9.122)]{BF80}.

(v) We work in the finite box, and the only role of \cite[(A4.25),
(A4.26)]{BF80} is in the infinite-volume limit of
\cite[Lemma~A4.1]{BF80}.
\end{proof}

\paragraph{Seed configurations and the seeded activity.}
A pair $(U,k)$ is \emph{admissible} if $U=V$ and $k\in\{0,1\}$, or
$U=X_0^c:=V\setminus X_0$ with $X_0\subsetneq V$ a union of $\cU$-cells and
$k=0$. A \emph{seed configuration} in $(U,k)$ is a pair $\sigma=(Z,N_Z)$ with
$Z\subseteq U$ a nonempty, possibly disconnected union of $\cU$-cells,
$Z\supseteq X_1$ if $k=1$, and $N_Z$ a wall field on $E(Z)$ whose wall polymers have their closures in $Z$ (so that $Z$ is a union of elements of $N_Z$); for label configurations, $N_Z\in\cN(Z)$. For $X\supseteq Z$,
$\cN_\sigma(X)$ is the set of $N_X\in\cN(X)$ with $N_X|_{E(Z)}=N_Z$ for which
$Z$ is a union of elements of $N_X$. The \emph{seeded activity}
$\cK^{(Z)}_{N_X}(X)$ is the activity of Proposition~\ref{prop:7.25} at $N_X$
with first set $Y_1=Z$, built with $(C',G^\natural)$:
$\cK^{(Z)}_{N_X}(X)=\sum_{\bar y}\int_{[0,1]^{n-1}}ds\int
d\mu_{C'_s}(\psi)\,e^{E(X,s)}\kappa(\bar y,s)\,Q^\natural_{N_X}(X)$, the sum
over $\bar y=(Z,Y_2,\dots,Y_n)$ with the $Y_j$ ($j\ge2$) disjoint unions of
elements and $\bigcup_jY_j=X$, and
$Q^\natural_{N_X}(X):=e^{G^\natural(X)}e^{-\Floc(X)}e^{-F_2(X)}$. We write
$X=Z\sqcup X_\Sigma\sqcup X_o$, where $X_\Sigma$ consists of the non-seed
$\cU$-cells in wall polymers and $X_o$ of the other non-seed $\cU$-cells (the
\emph{lone} cells). A cell is of \emph{class B} if it is one of the $\cU$-cells of
$\bigcup_\beta S_\beta$, and of \emph{class A} otherwise. Since
$\operatorname{supp}D\cup\operatorname{supp}F_2'\subseteq\bigcup_\beta S_\beta$ by Proposition~\ref{prop:7.9}\,(b), (e),
we have $D=F_2'=0$ on every class-A cell.
As in \cite[p.~214]{BF80}, $|\cdot|_0$ is the absolute value taken inside the
sums and integrals that result when all differentiations in $\kappa'$ are
performed.

\begin{lemma}\label{lem:7.39}
Let $(U,k)$ be admissible, $\sigma=(Z,N_Z)$ a seed configuration in $(U,k)$,
$c_1'>0$, $c_4'$ as in Lemma~\ref{lem:7.32} with $c_4=c_1'$, $e^{a''}\ge4C_\varphi$, where
$C_\varphi<\infty$ is the link sum of Lemma~\ref{lem:A.7}, and
$c_2^L:=\log(2eD_{\cU})+4+c_4'+a''+\alpha$. Then
\[
\sum_{\substack{X\supseteq Z\\ X\subseteq U}}\ \sum_{N_X\in\cN_\sigma(X)}
|\cK^{(Z)}_{N_X}(X)|\,e^{\alpha|X|}
\le\sup_{(\cdot)}e^{c_1'\Floc(X,N_X)+c_2^L|X|+\delta_2d}\int d\mu_{C'_s}
\bigl|e^{E(X,s)}\kappa'Q^\natural_{N_X}(X)\bigr|_0 ,
\]
where the supremum runs over $n$, $T$, $\bar y$, the elementary terms,
$s\in[0,1]^{n-1}$ and $N_X\in\cN_\sigma(X)$; $\kappa'$ is the elementary term
without its tree weight $q(T,s)$ (Lemma~\ref{lem:A.1}) and with every Mayer
factor $\mathcal E$ replaced by
$\sum_{\eta^A}e^{c_{10}L_{\eta^A}}e^{c''_{10}t}\mathcal E_{\eta^A}$
(\cite[(9.18), (9.18a), (9.19)]{BF80}); and $d:=\sum_i\dist(\Delta'_i,\Delta''_i)$, the sum over the links $i$ with a covariance line (types (2) to (5), Lemma~\ref{lem:A.1}).
The constant $c_2^L$ depends only on $D_{\cU}$, $c_4'$, $C_\varphi$, $\alpha$,
not on $L'$, and it is bounded uniformly in $\bGM$ and $\tau$ ($c_4'$ depends on them through $A_w$, but it is doubly exponentially small by Lemma~\ref{lem:7.32}). The bound also holds for the sum restricted by
$X\cap W\neq\emptyset$ if $W\subseteq Z$ or $W\supseteq Z$.
\end{lemma}

This lemma is the version on $\Gp$ of \cite[Lemmas 9.1, 9.4]{BF80}, in the distance-zero form, for
seeds that may depend on the configuration, be disconnected, omit $X_1$ when
$k=0$, and lie in $X_0^c$, and we prove it in Proposition~\ref{prop:A.8}. For the last assertion of the lemma, note that every term of the sum then
satisfies the restriction. In the supremum we take
together the elementary terms with the same link data in the sense of Lemma~\ref{lem:A.7},
as in the definition of $\kappa'$ above and in Proposition~\ref{prop:A.8}; that is, $\kappa'$ contains, inside
$|\cdot|_0$, the sums over the assignments of the points of each
$\mathcal E$-factor to the cells of its set of cells, over their positions and
over their charges (and, for $t=2$, over the forms).

\begin{lemma}\label{lem:7.40}
Insert Proposition~\ref{prop:7.37} and the Wick bound into the right side of
Lemma~\ref{lem:7.39}. In the resulting bound, the factor $e^{c_2'|X|}$ of
\cite[(9.823)]{BF80}, with $e^{c_2^L|X|}$ and $e^{c_{13}|X|}$, is at most
$\prod_{\Delta\subseteq X}e^{c_o}\cdot\prod_{\Delta\text{ class B}}e^{c_B(\bD)}
\cdot\prod_{\Delta\cap\Dep\neq\emptyset}e^{C_{\Dep}\cdep}$, the products running over
$\cU$-cells, and the near-cell factor is pulled out globally as $\cP'_G$.
Here $c_o$ does not depend on $L'$ and is bounded uniformly in $\bD$ and $\tau$, and, with the leftover pinning
exponent $q_1=2$ of Lemma~\ref{lem:B.9},
$c_B(\bD)\le2(C_{\mathcal E}^{1/2}+C_G^{1/2})\bD^{-1/2}$, where $C_{\mathcal E}$ and
$C_G$, the constants of Proposition~\ref{prop:B.10}, are independent of $L'$. Every
class-B cell lies in $X_\Sigma$ or in the seed, and every factor on a class-A cell
carries in addition a nonnegative power of $\bD$. The resulting bound depends on the scale $L'$ only
through the set of cells in $X_\Sigma$, through $\eps(L')$, which is decreasing, and
through the monotone hypotheses (L8-3), (L8-4) and \eqref{eq:7.6.coll}.
\end{lemma}

\begin{proposition}\label{prop:7.41}
Insert Proposition~\ref{prop:7.37} and the Wick bound into
Lemma~\ref{lem:7.39} with $c_1'=c'/4$, and distribute $\Floc$ by the
fractions of Table~\ref{tab:A.1}. Then for $\bGM\ge\bGMo$, in the resulting bound,
every non-seed $\cU$-cell $\Delta$ of $X$ carries a factor at most
$\eps_\Delta$, where
\begin{align*}
\eps_\Delta&:=\Kcell\,e^{\alpha+C_{\Dep}\cdep}(\bD/\lPV)^{1/6}\quad(\Delta\subseteq X_o),\\
\eps_\Delta&:=\Kcell\,e^{\alpha+C_{\Dep}\cdep+c_B(\bD)-(c'/8)\Phi_{\cU}/(1+D_{\cU})}\quad(\Delta\subseteq X_\Sigma),
\end{align*}
and every seed cell carries at most $\Kcell e^{\alpha+C_{\Dep}\cdep}$, times
$e^{c_B(\bD)}$ if it is of class B. Here
$\Kcell:=K_Be^{(c_2^L+c_{13})C_u}\max(1,\lPV^{1/6})$ with $K_B$ independent
of $\bD$, $L'$ and $\tau$, $\Phi_{\cU}$ is the share of
Proposition~\ref{prop:7.22}, and the exponent $p_0=1/6$ is attained.
\end{proposition}

Lemma~\ref{lem:7.40} and Proposition~\ref{prop:7.41} are a quantitative version on $\Gp$ of \cite[Section~9.8,
(9.81) to (9.824)]{BF80} and are proved in
Appendix~\ref{app:B}. With $p_0:=1/6$ we put
\begin{equation}\label{eq:7.6.c16D}
c_{16}^D:=2\Kcell\,e^{\alpha+C_{\Dep}\cdep}\max\Bigl\{(\bD/\lPV)^{p_0},\;
e^{c_B(\bD)-(c'/8)\Phi_{\cU}/(1+D_{\cU})}\Bigr\}.
\end{equation}

\begin{theorem}[Removal of the logarithm]\label{thm:7.42}
Fix $\alpha$. There is $\tau_0$ such that $c_{16}^D\le1$ for $\tau\ge\tau_0$,
and $c_{16}^D\to0$ as $\tau\to\infty$. Consequently the depletion factor
$e^{C_{\Dep}\cdep N_{\Dep}(X)}$, and with it the $\log\tau$ in $\cdep$, is
absorbed by the per-cell smallness of Proposition~\ref{prop:7.41}.
\end{theorem}

\begin{proof}
The first branch of \eqref{eq:7.6.c16D} is at most $1$ if
$\alpha\le p_0\log(\lPV/\bD)-C_{\Dep}\cdep-\log(2\Kcell)$. This holds for
$\tau\ge\tau_0$, because the per-cell gains are exponentially small in $\tau$ and the depletion exponent is
$O(\log\tau)$: we have $\log(1/\bD)\ge\pi^2\vCb(0)\tau-\frac32\log\tau-C$
as $\mt\le2m_D$, $\cdep=O(\log\tau)$ by Proposition~\ref{prop:6.18}, and
$\log\Kcell$ is affine in $\alpha$. In the second branch, Proposition~\ref{prop:7.22} with
$\mt L\ge c_L/2$ and $\mt L'=2\bGM$ gives
$(c'/8)\Phi_{\cU}/(1+D_{\cU})\ge c\,\bD^{-1}\bGM^{-3}$, which outweighs the bound
$c_B(\bD)\le C\bD^{-1/2}$ of Lemma~\ref{lem:7.40}. A depleted cell that is lone
or in $X_\Sigma$ has its factor $e^{C_{\Dep}\cdep}$ inside $\eps_\Delta\le\frac12$
(Section~\ref{S:G.42}).
\end{proof}

Let $\tilde\cK(\Delta)$ be the activity of the single element $\Delta$ at
$k=0$ (first set $\Delta$, $n=1$), with no wall on $\Delta$.

\begin{lemma}\label{lem:7.43}
For $\tau\ge\tau_0$ and every wall-free $\cU$-cell $\Delta$, $\tilde\cK(\Delta)\ge\frac12$.
\end{lemma}

\begin{proof}
A single $\cU$-cell that is an element lies in no $S_\beta$, so $D=0$ on
$\Delta$ and $\Floc(\Delta)=F_2(\Delta)=0$. The label is constant on $\Delta$
and, at $k=0$, contributes only a multiple of $2\pi$ to each phase. Hence
$\vartheta=\psi$ on $\Delta$,
$\tilde\cK(\Delta)=\int d\mu_{C'_\Delta}e^{E(\Delta)+G^\natural(\Delta)}$ with
$C'_\Delta$ the $\Delta\times\Delta$ block of $C'$, and the exponent is real by
(S3). Positivity at $k=0$ allows us to use Jensen's inequality, which gives
\[
\tilde\cK(\Delta)\ge\exp\Bigl(-\mathbb E|E(\Delta)|-\mathbb E|G_1(\Delta)|
-\mathbb E|G_{2,1}(\Delta)+G_{2,2}(\Delta)|-\mathbb E\bigl(G_{2,3}^{\Dep}(\Delta)\bigr)_-
-\mathbb E|G_{2,3}^\natural(\Delta)|\Bigr),
\]
expectations under $d\mu_{C'_\Delta}$. A supremum bound does not suffice here, since
$\sup_\psi(-G_{2,3}^{\Dep}(\Delta))$ is only bounded by
$C(e^{-\mu}+e^{-2\kappa_S})\cdep$, which grows like $\log\tau$. Let $x_b:=\alpha_DK_1\bD/\lPV$. Then
$\bGM C'(x,x)\le x_b$ by Lemma~\ref{lem:7.31}(d) and $M_Y=\mt/\lPV$. Moreover,
$\#\Delta<C_u\mt^{-3}$, and $\Delta$ contains at most $C_u/v_-$ cells. As we will show below, each
expectation is at most $C(\bD^{1/2}+\cdep\bD)$, with $C$ depending only on the
constants fixed before $\tau_0$. Since $\cdep\bD=O(\log\tau)e^{-2.494\tau}\to0$,
the sum is then at most $\ln2$ for $\tau\ge\tau_0$.

We begin with $G_{2,3}^{\Dep}$. At $x\in\Dep$ the $x$-term can be negative only because of the
excess $\rho_1(m,x)>\rho^{\mathrm{ref}}(m)$, which is at most
$Ce^{-\mu}\rho^{\mathrm{ref}}(1)$ for $m=\pm1$ by Proposition~\ref{prop:6.6} and
$e^{-\Phi}\le1$, and is bounded by the $|m|\ge2$ moment bound otherwise. With
$1-\cos u\le\frac12u^2$, $\mathbb E\psi(x)^2=C'(x,x)$ and
Proposition~\ref{prop:6.18},
$\mathbb E(G_{2,3}^{\Dep}(\Delta))_-\le C(e^{-\mu}+e^{-2\kappa_S})\mt^2\,\alpha_DK_1M_Y\,\#(\Dep\cap\Delta)
\le C(e^{-\mu}+e^{-2\kappa_S})\alpha_DK_1\cdep\bD/\lPV$.

The other four terms are moment bounds with $\bGM C'(x,x)\le x_b$.
First, $\mathbb E|E(\Delta)|\le C\#\Delta(m_D^2/\bGM)x_b^{3/2}\le C\bD^{1/2}$, as
each term of $E(\Delta)$ has at least three factors $\eps$ or
$\tilde\eps:=\eps-im\bGM^{1/2}\psi$ and the weight
$e^{2\kappa_S(\sum|m_i|-1)}$ of (S2) absorbs the charges.
Second, $\mathbb E|G_1(\Delta)|\le C\bD$, as
$\log\hat R_v(x)=O(v\bD x^4+e^{-\pi^2v/\bD})$ for $|x|\le\bD^{-1/2}$ by Taylor's
formula and the periodization, with a Gaussian tail beyond.
Third, $\mathbb E|G_{2,1}+G_{2,2}|\le C(\bD+\bD^{1/2})$, as the brackets are of
fourth and third order and
$\mathrm{Var}\,\tilde\delta(x)\le2\alpha_D(K_1/\lPV+1/v_-)\mt$. Finally,
$\mathbb E|G_{2,3}^\natural(\Delta)|\le3C_4\theta'C_u(\alpha_DK_1)^2\bD/\lPV^2$
by Lemma~\ref{lem:7.31}(b) with $n=0$ and $\mathbb E\psi^4=3C'(x,x)^2$.
\end{proof}

The bound $\tilde\cK(\Delta)\ge\frac12$ replaces \cite[(A4.16) to (A4.21)]{BF80},
which use translation invariance, whereas here the positivity at $k=0$ of Lemma~\ref{lem:7.44}
and the diagonal bound of Theorem~\ref{thm:6.12} suffice.