%
\providecommand{\Floc}{F_1^{\mathrm{loc}}}
\providecommand{\UU}{\mathcal U}
\providecommand{\one}{\mathbf 1}
\providecommand{\bGM}{\beta_{\mathrm{GM}}}
\providecommand{\mt}{\tilde m}
\providecommand{\MY}{M_Y}
\subsection{Full proofs for Subsection~\ref{ssec:7.5}}\label{S:F2}

\paragraph{Proof of Lemma~\ref{lem:7.wall}.}
The integrand of Proposition~\ref{prop:7.8} depends on the configuration only
through the field $\psi+g+k\bGM^{-1/2}f$, read modulo $P\Z$ at each site and in
each cell average, and through the walls. We write $\vartheta:=\psi+D$.

(a) Let $h\in\mathcal H$ and $k\in\{0,1\}$. By \eqref{eq:7.shift},
$g+k\bGM^{-1/2}f=h+D$. Since $h$ takes values in $P\Z=2\pi\bGM^{-1/2}\Z$,
$e^{im\bGM^{1/2}h(x)}=1$, so each phase $e^{im(\bGM^{1/2}(\psi+g)(x)+kf(x))}$ of
$E'$, $\varepsilon_k$, $G_{2,2}$ and $G_{2,3}$ equals $e^{im\bGM^{1/2}\vartheta(x)}$. In
$G_1=\sum_\Omega\ln r_\Omega(A_\Omega)$ the function $r_\Omega$ of \eqref{eq:7.rOmega}
depends on $A_\Omega$ only through $A_\Omega+k\bGM^{-1/2}\bar f_\Omega$ and is $P$-periodic
in it. This quantity is $\operatorname{avg}_\Omega\vartheta+h|_\Omega$, as $h$ is constant
on $\Omega$. Hence $r_\Omega(A_\Omega)$ is $r_\Omega$ at $k=0$ evaluated at
$\operatorname{avg}_\Omega\vartheta$. Likewise
$\tilde\delta=(\psi+g+k\bGM^{-1/2}f)-\operatorname{avg}_\Omega(\psi+g+k\bGM^{-1/2}f)=\vartheta-\operatorname{avg}_\Omega\vartheta$
on $\Omega$. The terms added in $G^\natural$ (Lemma~\ref{lem:7.31}) are functions of
$\vartheta$ and $\psi$ through the same phases. The term $F_1(S_\beta)$ is given by
\eqref{eq:7.F1Sbeta}, and $F_2(Y)=\langle\one_Y\psi,F_2'\rangle$ with
$F_2'=\sum_\beta F'_{2,\beta}$ and $F'_{2,\beta}=-[C_0^{-1},\chi_\beta]C_0s_\beta$
as in Proposition~\ref{prop:7.9}\,(e), both functions of $w$ alone. Hence the two
definitions of $Q_N(Y)$ and of $E(Y,s)$ agree.

(b) After the replacement, the action is a sum of three kinds of terms.
The cell terms are $\ln r_\Omega(\operatorname{avg}_\Omega\vartheta)$. The site terms are the
terms of $G_{2,1}$, $G_{2,2}$, $G_{2,3}$, of $G^\natural$ and of $E$, functions of
$\psi$ and $\vartheta$ at the sites of their support (the $E$-terms with weights
$w_s(\tau)$). The wall-component terms $F_1(S_\beta)$ and
$\langle\psi,F'_{2,\beta}\rangle$ are supported in $S_\beta$
by Proposition~\ref{prop:7.9}\,(e). On a union $Y$ of wall components and of
$\UU$-cells outside $\hat\Sigma$, Proposition~\ref{prop:7.9}\,(f) shows that $D|_Y=\sum_{S_\beta\subseteq Y}D_\beta|_Y$ depends on $N$
only through $N|_{E(Y)}$. The proof of Lemma~\ref{lem:7.24} needs no more than
these facts. An admissible set contains
each cell and each wall component entirely or not at all, which gives the
factorization and the locality for every wall field. Definition~\ref{def:7.23}
and the proof of Lemma~\ref{lem:7.26} use only the wall sites, the wall
components and Proposition~\ref{prop:7.9}\,(a), all of which are defined from
$N$. On $U\ne V$ we take closures in $U$, which is consistent because the
wall polymers of $N$ have their closures in $U$.

Proposition~\ref{prop:7.25} is
proved in Appendix~\ref{app:C} from the factorization, the admissibility of the
sets of a sequence and absolute convergence. Absolute convergence rests on
Lemma~\ref{lem:C.3}, whose bounds are pointwise in $(\psi,\vartheta)$, and on the Poincar\'e inequality on cells
of Lemma~\ref{lem:C.4}, which does not involve the configuration. Here Lemma~\ref{lem:C.3}\,(b) is a bound
on $G_{2,1}$ by $\|\one_YP_V\vartheta\|^2$.

In
Theorem~\ref{thm:7.14}, the bound \eqref{eq:7.H6p} of
Corollary~\ref{cor:7.11}\,(c) with $g-H=D$ and $\nabla g+k\bGM^{-1/2}R_f=w+\nabla D$ reads
$\Floc(Y)\ge\frac12\mt^2\|D\|^2_Y+\frac12\|w+\nabla D\|^2_{E(Y)}$, which follows from
\eqref{eq:7.F1Sbeta} by dropping the term with $\MY^{-2}$, the supports of
different $\beta$ being disjoint. In the same theorem, the tube argument uses only that $w$ vanishes
on the bonds inside cells and is constant on interfaces
(Lemma~\ref{lem:7.13}). The proof of Proposition~\ref{prop:7.5} bounds
$\cos(mu)-1$ at each site, for an arbitrary real number $u$. Here
$u=\bGM^{1/2}(\varphi+k\bGM^{-1/2}f-h)(x)$, which after the translation
$\varphi=\psi+g$ equals $\bGM^{1/2}\vartheta(x)$. \qed