%
%

\providecommand{\bGM}{\beta_{\mathrm{GM}}}
\providecommand{\mt}{\tilde m}
\providecommand{\MY}{M_Y}
\providecommand{\lPV}{\lambda_{\mathrm{PV}}}
\providecommand{\Floc}{F_1^{\mathrm{loc}}}
\providecommand{\UU}{\mathcal{U}}
\providecommand{\Df}{\mathfrak{D}}
\providecommand{\sat}{\operatorname{sat}}
\providecommand{\supp}{\operatorname{supp}}
\providecommand{\one}{\mathbf{1}}
\providecommand{\Z}{\mathbb{Z}}
\providecommand{\R}{\mathbb{R}}
\providecommand{\bC}{\mathbf{C}}
\providecommand{\cM}{\mathcal{M}}

\section{G\"opfert\textendash Mack Appendix E on slit domains}
\label{app:C}

In this appendix we give a condensed proof of Proposition~\ref{prop:7.25}. The complete proof, which comprises the Gaussian layer, the abstract expansion of \cite[Appendix~E]{GM82} and its verification on $G'$, is written out in Section~\ref{S4-S:S4.5}, together with complete proofs of Lemmas~\ref{lem:C.3} and~\ref{lem:C.5}. The expansion of \cite[Appendix~E]{GM82}
is an algebraic identity, built from Bernoulli interpolation of the couplings between consecutive
sets, the fundamental theorem of calculus and Gaussian integration by parts, and its only analytic
input is absolute convergence. On $G'$ absolute convergence needs the Poincaré inequality on cells,
as the Peierls identity of Proposition~\ref{prop:7.4} removes the mass in the cell-fluctuation directions
(Lemma~\ref{lem:C.4}). On block cubes this is the bound of \cite[(9.640) to (9.642)]{BF80}.

\paragraph{Setting.} We fix $h$. Everything below depends on it only through its walls and the integrand of Lemma~\ref{lem:7.24}, and therefore
holds equally for wall fields (Lemma~\ref{lem:7.wall}). The covariance $\bC$ is $C=(C_0^{-1}+\nu)^{-1}$ of Proposition~\ref{prop:7.8} or $C'$
of Lemma~\ref{lem:7.31}, positive definite on $V$ with $\bC\le\alpha C_0$, $\alpha=2$ for $C$ and
$\alpha=\alpha_D\le\frac76$ for $C'$. The integrand is $e^EQ_h$, $Q_h=e^Ge^{-\Floc(V)}e^{-F_2}$, with $G^\natural$
for $G$ when $\bC=C'$. We write $\partial_x:=\partial/\partial\psi(x)$. A \emph{sequence} is $\bar y=(Y_1,\dots,Y_m)$ of nonempty
admissible sets (Definition~\ref{def:7.23}), pairwise disjoint, with the first set $Y_1$
fixed before the expansion starts. Its union is $X(\bar y):=\bigcup_iY_i$.

\paragraph{Interpolation.} For pairwise disjoint admissible $Y_1,\dots,Y_n$ with union
$X$, let $Y_{n+1}:=V\setminus X$, $\chi_i:=\one_{Y_i}$ and, for $s\in[0,1]^n$,
\begin{equation}\label{eq:C.E3}
p(x,y\mid s):=\sum_{1\le i<j\le n+1}s_i\cdots s_{j-1}\bigl[\chi_i(x)\chi_j(y)+\chi_i(y)\chi_j(x)\bigr]
+\sum_{i\le n+1}\chi_i(x)\chi_i(y),
\end{equation}
and $\bC(s)(x,y):=p(x,y\mid s)\bC(x,y)$, as in \cite[(E.2), (E.3)]{GM82}, with $\chi_{n+1}=\one_{V\setminus X}$. For $x\in Y_a$, $y\in Y_b$, $a\le b$,
$p(x,y\mid s)=\prod_{a\le k<b}s_k$, and hence $\partial_{s_k}p\in[0,1]$ vanishes unless $x$, $y$ lie on different
sides of the cut between $\bigcup_{i\le k}Y_i$ and $\bigcup_{i>k}Y_i$. For $\varepsilon\in\{0,1\}^n$ let $\pi(\varepsilon)$ be
the partition of $\{1,\dots,n+1\}$ into the maximal runs of consecutive indices joined by
the $k$ with $\varepsilon_k=1$, $Y_J:=\bigcup_{i\in J}Y_i$ (admissible), and
$\lambda_\varepsilon(s):=\prod_ks_k^{\varepsilon_k}(1-s_k)^{1-\varepsilon_k}$. Then
\begin{equation}\label{eq:C.decomp}
\bC(s)=\sum_\varepsilon\lambda_\varepsilon(s)\,\bC_\varepsilon,\qquad\bC_\varepsilon:=\sum_{J\in\pi(\varepsilon)}\one_{Y_J}\bC\one_{Y_J},
\end{equation}
since $\sum_\varepsilon\lambda_\varepsilon(s)\bC_\varepsilon(x,y)$ is $\bC(x,y)$ times the probability, for independent
Bernoulli$(s_k)$ variables $\varepsilon_k$, that $a$ and $b$ lie in one run. Hence
$\lambda_{\min}(\bC)\le\bC(s)\le\|\bC\|\le\alpha\mt^{-2}$, $\bC(s)(x,x)=\bC(x,x)$, $|\bC(s)(x,y)|\le|\bC(x,y)|$ (so
each entrywise monotone norm of $\bC(s)$ is at most that of $\bC$), and
$\|\cM^{1/2}\bC(s)\cM^{1/2}\|\le\|\cM^{1/2}\bC\cM^{1/2}\|$ for each symmetric $\cM\ge0$ commuting with all
$\chi_i$. For a sequence $\bar y$ of length $n$, we write $\bC^{\bar y}(s,s_n)$ for \eqref{eq:C.E3} built from $\bar y$
($s=(s_1,\dots,s_{n-1})$), $d\mu'^{\,\bar y}_{ss_n}$ for its Gaussian measure and $d\mu^{\bar y}_s$ for the measure at
$s_n=0$.

\paragraph{The weights of $E$.} By (S1) and \cite[(E.5a) to (E.5d)]{GM82}, $E$ is a sum
$\sum_\tau e_\tau$ of terms of two kinds: for $t\ge3$ particles,
$e_\tau=\frac1{t!}\rho_t(\xi_1,\dots,\xi_t)\,\varepsilon_k(\xi_1)\cdots\varepsilon_k(\xi_t)$, and for $t=2$ the three terms $\frac12\rho_2\tilde\varepsilon_k(\xi_1)\tilde\varepsilon_k(\xi_2)$, $\frac12\rho_2\tilde\varepsilon_k(\xi_1)\,im_2\bGM^{1/2}\psi(x_2)$ and
$\frac12\rho_2\,im_1\bGM^{1/2}\psi(x_1)\tilde\varepsilon_k(\xi_2)$. Here $x_i\in V$, $m_i\ne0$,
$\varepsilon_k(\xi)=e^{im(\bGM^{1/2}(\psi+g)(x)+kf(x))}-1$ and $\tilde\varepsilon_k(\xi):=\varepsilon_k(\xi)-im\bGM^{1/2}\psi(x)$. With
$\supp\tau:=\{x_1,\dots,x_t\}$, $I(\tau)$ is the set of $k\le n$ such that $\supp\tau$ meets both
$\bigcup_{i\le k}Y_i$ and $\bigcup_{i>k}Y_i$. We put $w_s(\tau):=\prod_{k\in I(\tau)}s_k$, $E(Y,s):=\sum_{\supp\tau\subseteq Y}w_s(\tau)e_\tau$ and
$E(Y):=E(Y,1)$, and $E^{\bar y}(Y,(s,s_n))$ denotes $E(Y,\cdot)$ built from $\bar y$. These functions agree with $E(X,s)$ of \cite[p.~89]{GM82} and
$E(\Lambda,ss_n)$ of \cite[p.~90, (v)]{GM82}, with sites for the unit cubes $a_i$.

\paragraph{Absolute convergence.} Let $P_V$ be the cell-fluctuation projection,
$(P_V\varphi)|_\Omega:=\varphi|_\Omega-\operatorname{avg}_\Omega\varphi$, and $\|\nu\|_0:=\sup_x\sum_y|\nu(x,y)|$. Summing the fine Poincaré inequality on
cells, given by Theorem~\ref{thm:3.20}\,(R6) and Proposition~\ref{prop:6.10}\,(iv), with the overlap
$N_{\mathrm{ov}}$, we obtain $\|P_V\varphi\|^2\le C_P'L^2\|\nabla\varphi\|^2$, $C_P':=C_W^f(k_\square/L)^2N_{\mathrm{ov}}$, and by (L8-2) of
Subsection~\ref{ssec:7.6}, $C_P'(\mt L)^2\le\eta_H/8<\frac17$. The constants $c_V$, $b_V$, $C_\alpha$ below
serve only to prove convergence and may depend on $G'$.

\begin{lemma}[Growth of $\ln Q_h$]\label{lem:C.3}
Let $\vartheta:=\psi+D$. For every admissible $Y$, every sequence, every $s$ and real $\psi$:
(a) $E(Y,s)$ is real, and $E(Y,s)=B_{Y,s}(\psi)+\frac12\langle\psi,\nu^Y_s\psi\rangle$ with $|B_{Y,s}(\psi)|\le b_V$ and
$\nu^Y_s(x_1,x_2):=\one_Y(x_1)\one_Y(x_2)w_s(x_1,x_2)\nu(x_1,x_2)$, where $w_s(x_1,x_2):=\prod_{a\le k<b}s_k$ for $x_1\in Y_a$,
$x_2\in Y_b$, $a\le b$; moreover $\|\nu^Y_s\|_{2\to2}\le\|\nu\|_0\le\varepsilon_\nu\mt^2$, and $|\partial^\alpha E(Y,s)|\le C_\alpha(1+\|\psi\|)$
for $|\alpha|\ge1$; (b) $G_{2,1}(Y)\le\frac12(1+C_{\mathrm{ref}}e^{-2\kappa_S})\mt^2\|\one_YP_V\vartheta\|^2$; (c) $G_1(Y)\le c_V$,
$|G_{2,2}(Y)|\le c_V(1+\|\psi\|_1)$ and $|G_{2,3}(Y)|\le c_V$; for the terms $G^\Df_{2,3}$, $G^\natural_{2,3}$ of $G^\natural$
(Lemma~\ref{lem:7.31}), $G^\Df_{2,3}(Y)\le c_V$ and $G^\natural_{2,3}(Y)\le c_V+\frac12\theta'\mt^2\|\psi\|^2$; (d) $\Floc(Y)\ge0$ does not
depend on $\psi$, $F_2(Y)$ is linear in $\psi$, and $|\partial^\alpha\ln Q_h(Y)|\le C_\alpha(1+\|\psi\|)$ for $|\alpha|\ge1$.
\end{lemma}

\begin{proof}[Sketch]
(a) By (S3), the terms pair into complex conjugates. With $\varepsilon_k=\tilde\varepsilon_k+im\bGM^{1/2}\psi$ the terms with
$t=2$ add up to $\frac12\rho_2\varepsilon_k\varepsilon_k$ plus a part with charge sum $\frac12\nu$. Using (S2$^+$), we bound $B_{Y,s}$ by
$\sum_{t\ge2}\frac{2^t}{t!}\sum|\rho_t|$, and its $\psi$-derivatives by the same sums with the weights $(\sum_i|m_i|)^n$, and the Schur test with $0\le w_s\le1$ gives the norm bound. (b) Here
$\tilde\delta=P_V\vartheta$, and by (S3) $G_{2,1}$ is a sum of terms $\rho^{\mathrm{ref}}(m)[\cos y-1+\frac12y^2]$ of modulus at most $\frac12|\rho^{\mathrm{ref}}(m)|y^2$, since $0\le\cos y-1+\frac12y^2\le\frac12y^2$, with
$\bGM\sum_mm^2|\rho^{\mathrm{ref}}(m)|\le(1+C_{\mathrm{ref}}e^{-2\kappa_S})\mt^2$. Part (c) is explicit on the finite set $V$, with
Proposition~\ref{prop:7.5} off $\Df$. Part (d) follows from Corollary~\ref{cor:7.11}, Proposition~\ref{prop:7.9}, and the fact that $\ln r_\Omega$ is
smooth and periodic in $A_\Omega$.
\end{proof}

\begin{lemma}[Margin]\label{lem:C.4}
Assume the standing hypothesis \eqref{eq:7.standing}, and let $\kappa_S$ be so large that
$(1+C_{\mathrm{ref}}e^{-2\kappa_S})\eta_H/8\le\frac17$, so that (L8-2) gives $(1+C_{\mathrm{ref}}e^{-2\kappa_S})C_P'(\mt L)^2\le\frac17$. Put
$\cM:=(1+C_{\mathrm{ref}}e^{-2\kappa_S})\mt^2P_V+(\varepsilon_\nu+\theta')\mt^2$ for $\bC=C'$ and
$\cM:=(1+C_{\mathrm{ref}}e^{-2\kappa_S})\mt^2P_V+\varepsilon_\nu\mt^2$ for $\bC=C$. Then $\cM\ge0$ is symmetric, commutes with $\one_\Omega$ for every cell $\Omega$,
$\cM\ge(1+C_{\mathrm{ref}}e^{-2\kappa_S})\mt^2P_V+\|\nu\|_0+\theta'\mt^2\one_{[\bC=C']}$, and
\begin{equation}\label{eq:C.margin}
\langle\varphi,\cM\varphi\rangle\le q_M\langle\varphi,\bC^{-1}\varphi\rangle,\qquad q_M\le\tfrac16\ (\bC=C'),\quad q_M\le\tfrac27\ (\bC=C).
\end{equation}
Moreover $\|\cM^{1/2}\bC(s)\cM^{1/2}\|\le q_M$ for every sequence of admissible sets and every $s$,
and the same holds for $\one_U\bC\one_U$ on $\R^U$ whenever $U$ is a union of cells.
\end{lemma}

\begin{proof}
We control the cell fluctuations by the gradient through the Poincaré
inequality on cells, and the constant part by the mass. Both are dominated by
$C_0^{-1}=\MY^{-2}\Delta^2-\Delta+\mt^2\ge\mt^2-\Delta$, and $C_0^{-1}\le\alpha\bC^{-1}$. The projection $P_V$ is block diagonal over cells,
and thus $\cM$ commutes with each $\one_\Omega$. Moreover, $\|\nu\|_0\le\varepsilon_\nu\mt^2$ by (S5). By the Poincaré inequality
$\|P_V\varphi\|^2\le C_P'L^2\|\nabla\varphi\|^2$,
\[
\langle\varphi,\cM\varphi\rangle\le(1+C_{\mathrm{ref}}e^{-2\kappa_S})C_P'(\mt L)^2\|\nabla\varphi\|^2+(\varepsilon_\nu+\theta')\mt^2\|\varphi\|^2
\le\tfrac17\langle\varphi,C_0^{-1}\varphi\rangle,
\]
which is at most $\frac\alpha7\langle\varphi,\bC^{-1}\varphi\rangle$, with $\alpha=\alpha_D\le\frac76$ for $C'$ and $\alpha=2$ for $C$ (where $\theta'$ is
absent). We now turn to the last statement, for which any $q_M<1$ suffices.
Inequality \eqref{eq:C.margin} says that $\|\cM^{1/2}\bC\cM^{1/2}\|\le q_M$. Each $Y_i$ and $V\setminus X$ is a union of cells, and hence
$\cM$ commutes with each $\chi_i$ and \eqref{eq:C.decomp} applies. Since $\one_U$ commutes with $\cM$, we also get
$\|\cM^{1/2}\one_U\bC\one_U\cM^{1/2}\|\le q_M$, and \eqref{eq:C.decomp} applies on $\R^U$.
\end{proof}

\begin{lemma}\label{lem:C.5}
Call $F:\R^V\to\mathbb C$ \emph{admissible} if $F$ is $C^2$ and, for $|\alpha|\le2$,
$|\partial^\alpha F(\psi)|\le K_F(1+\|\psi\|)^{d_F}\exp(\frac12\langle\psi,\cM\psi\rangle+k_F\|\psi\|_1)$. Under the hypotheses of
Lemma~\ref{lem:C.4}: (a) for every admissible $Y$ and $s'$, and every finite composition $\mathfrak L$
of operators $\partial_x$ and of multiplications by sub-series, with weights in $[0,1]$, of the term expansions of $E(\cdot,s')$, of $\partial_{s_n}E(\cdot,s')$ and of their $\psi$-derivatives (all lie in one polynomial class), the function
$e^{E(Y,s')}\mathfrak LQ_h(Y)$ is admissible, uniformly in $s'$; (b) if $t\mapsto\bC(t)$ is one of the
covariances \eqref{eq:C.E3} with one $s$-variable moving and $F$ is admissible, then
$\int d\mu_{\bC(t)}|F|$ is bounded uniformly in $t$ and in the other variables, $t\mapsto\int d\mu_{\bC(t)}F$ is
$C^1$ on $[0,1]$, and
\begin{equation}\label{eq:C.E14}
\partial_t\int d\mu_{\bC(t)}F=\frac12\sum_{x,y\in V}\partial_t\bC(t)(x,y)\int d\mu_{\bC(t)}\,\partial_x\partial_yF ;
\end{equation}
(c) if $F$ also depends on $t$, is $C^1$ in $t$, and $F$ and $\partial_tF$ are admissible uniformly in $t$,
then $\partial_t\int d\mu_{\bC(t)}F=\int d\mu_{\bC(t)}\partial_tF$ plus the right side of \eqref{eq:C.E14}.
\end{lemma}

\begin{proof}[Sketch]
(a) We collect the exponents of Lemma~\ref{lem:C.3} and use the lower bound on $\cM$ of
Lemma~\ref{lem:C.4}. (b) By Lemma~\ref{lem:C.4} and \eqref{eq:C.decomp}, $\bC(t)^{-1}\ge(\cM+2\epsilon)/q'$ with
$q':=q_M+2\epsilon\|\bC\|<1$ for small $\epsilon>0$. Therefore $|F|g_t$, where $g_t$ is the Gaussian density, is integrable
uniformly in $t$. Since $\partial_tg_t=\frac12\sum_{x,y}\partial_t\bC(t)(x,y)\partial_x\partial_yg_t$, two integrations by parts give
\eqref{eq:C.E14}, the step ``By partial integration'' of \cite[p.~91]{GM82}. (c) This follows from the product rule
under the same domination.
\end{proof}

\paragraph{The operators $\kappa$.} We fix $\bar y$ of length $n$ with $X:=X(\bar y)\subsetneq V$ and write
$E:=E^{\bar y}(V,(s,s_n))=\sum_\tau w(\tau)e_\tau$, $E_x:=\partial_xE$. As $e^{-E}\partial_x\partial_y(e^EF)=(\partial_x+E_x)(\partial_y+E_y)F$ is
symmetric in $x,y$, and $\partial_{s_n}\bC^{\bar y}(x,y)=0$ unless $x,y$ lie on different sides of $X$,
Lemma~\ref{lem:C.5}\,(c) yields
$\partial_{s_n}\int d\mu'^{\,\bar y}_{ss_n}e^EF=\int d\mu'^{\,\bar y}_{ss_n}e^E\tilde\kappa(n)F$ for $F$ not depending on $s_n$ with $e^EF$
admissible, where
\begin{equation}\label{eq:C.E16}
\tilde\kappa(n):=\partial_{s_n}E+\sum_{x\in V\setminus X}\sum_{y\in X}\partial_{s_n}\bC^{\bar y}(x,y)(\partial_x+E_x)(\partial_y+E_y),
\end{equation}
as in \cite[(E.15), (E.16)]{GM82}. As GM note on p.~92, the factor $\frac12$ of (E.16) cancels by symmetry. Inserting $E=\sum_\tau w(\tau)e_\tau$, we write $\tilde\kappa(n)=\sum_\theta\mathfrak t_\theta$ with elementary terms of
five types: (1) $(\partial_{s_n}w(\tau))e_\tau$; (2) $\partial_{s_n}\bC(x,y)\partial_x\partial_y$; (3) $\partial_{s_n}\bC(x,y)w(\tau)(\partial_x\partial_ye_\tau)$; (4)
$\partial_{s_n}\bC(x,y)w(\tau)(\partial_ye_\tau)\partial_x$, respectively $(\partial_xe_\tau)\partial_y$; (5)
$\partial_{s_n}\bC(x,y)w(\tau)w(\tau')(\partial_xe_\tau)(\partial_ye_{\tau'})$. The support $S(\theta)$ is $\supp\tau$ in type (1), and $\{x,y\}$
with the supports of the $\tau$'s involved otherwise. If $\mathfrak t_\theta\ne0$, then $S(\theta)\not\subseteq X$, since in type (1)
$n\in I(\tau)$ and otherwise $x\in V\setminus X$. We put $Y_{n+1}(\theta):=\sat_h(S(\theta)\setminus X)$ and
\begin{equation}\label{eq:C.E20}
\kappa(n)[Y']:=\sum_{\theta:\,Y_{n+1}(\theta)=Y'}\mathfrak t_\theta,\qquad\tilde\kappa(n)=\sum_{Y'}\kappa(n)[Y'],
\end{equation}
as in \cite[(E.11), (E.20)]{GM82}. With $\bar y':=(\bar y,Y')$ and $X':=X\cup Y'$, each $\theta$ in $\kappa(n)[Y']$ has
$S(\theta)\subseteq X'$, coefficients depending on $\psi|_{X'}$ only, derivatives at $x\in Y'$, $y\in X$, and
$E$-terms of $E^{\bar y'}(X',(s,s_n))$ with the same weights. Thus $\kappa(n)[Y']$ depends on $s_1,\dots,s_n$
only. Moreover, $\partial_{s_n}\bC^{\bar y}(x,y)=s_a\cdots s_{n-1}\bC(x,y)=\partial_{s_n}\bC^{\bar y'}(x,y)$ for $x\in Y'$,
$y\in Y_a$. For $\bar y=(Y_1,\dots,Y_m)$ the ordered product $\kappa(\bar y,s):=\kappa(m-1)\circ\cdots\circ\kappa(1)$, with
$\kappa(i):=\kappa(i)[Y_{i+1}]$ built from $(Y_1,\dots,Y_i)$ and each operator acting on everything to its
right, is $\kappa(\bar y,s)$ of \cite[(E.9), (E.10)]{GM82}.

\paragraph{Proof of Proposition~\ref{prop:7.25}.} We follow the proof of \cite[Lemma~E.1,
Proposition~E.2, (E.2) to (E.24)]{GM82}, an induction in $n$. Each step is the fundamental
theorem of calculus \cite[(E.17)]{GM82} in a new variable $s_{n+1}$, started from the
measure and the $E$ of the previous step by the consistency and decoupling relations
\cite[(E.21a), (E.21b)]{GM82}, which hold on $G'$ by \eqref{eq:C.E3} and the definition of
$w_s(\tau)$. Lemma~\ref{lem:C.5} justifies each exchange of derivatives and integrals. The following changes are
needed on $G'$.

(1) \emph{Atoms are sites}, whereas the atoms of \cite[p.~89]{GM82} are unit cubes.
Since each $Y_j$ is admissible, $w_s(\tau)$ depends only on which $Y_j$ contain the points.

(2) \emph{One finite matrix with mass at every site}, and $\chi_{n+1}=\one_{V\setminus X}$, so that the region
outside $\Lambda$ of \cite[p.~92]{GM82} does not occur. Termination occurs at $m\le e_h+1$, since each $Y_j$, $j\ge2$,
contains an element outside $Y_1$ (\cite[p.~93]{GM82}).

(3) \emph{First sets may depend on $h$
and be disconnected}. In \cite[p.~90, (iii)]{GM82} the first set is a fixed union of unit cubes. Here the factorization $Q_h=Q_h(Y_1)Q_h(V\setminus Y_1)$ of Lemma~\ref{lem:7.24} suffices for the
step $n=1$, and the admissibility of each $\bigcup_{j\le i}Y_j$ for the selection rule; every comparison is made at one value of $h$. Type (2) terms
reach the element of $x$ at any distance at which $\bC(x,y)\ne0$, and therefore the sets $X(\bar y)$ are
disconnected in general.

(4) \emph{On $U=X_0^c$ the covariance is $\one_U\bC\one_U$}: marginals of
Gaussian measures are compressions, and hence $\mathcal K_{\bar y}$ involves the covariance only through the block $p(x,y\mid s)\bC(x,y)$,
$x,y\in X(\bar y)$. Lemma~\ref{lem:C.4} covers $\one_U\bC\one_U$.

(5) \emph{The selection rule: $S(\theta)$ contains
the bare point $x$} of each covariance term, so that all derivatives of a term of
$\kappa(n)[Y']$ act in $X\cup Y'$. By contrast, \cite[(E.11)]{GM82} is stated for the supports of the
$E$-terms, and for the term without $E$-terms \cite[p.~91; (E.19), p.~93]{GM82} take $Y_{n+1}\ni x$.

(6) \emph{``$\chi C\chi$'' is an exact block-sum decomposition}: the convex combination of
\cite[p.~88]{GM82} is \eqref{eq:C.decomp}, with compressions to admissible blocks. The bounds on $\bC(s)$ in Lemmas~\ref{lem:C.4}, \ref{lem:C.5} and in Section~\ref{sec:stage2}
(Table~\ref{tab:C.1}) follow from this decomposition. The integrals $\int d\mu_\bC e^{E(Y)}Q_h(Y)$ are positive, since $E(Y)$, $G$, $\Floc$, $F_2$ are
real by Lemma~\ref{lem:C.3}, and they are finite by Lemma~\ref{lem:C.5}\,(b).