%
\providecommand{\Nlong}{\mathcal N^{\mathrm{long}}}
\providecommand{\Kflat}{\mathfrak K^{\flat}}
\providecommand{\kbox}{k_{\square}}
\providecommand{\cN}{\mathcal N}
\providecommand{\Floc}{F_1^{\mathrm{loc}}}
\providecommand{\Funit}{F_{\mathrm{unit}}}
\subsection{Full proofs for Subsections~\ref{sec:7.7} to~\ref{sec:7.9}}\label{S:H2}

\paragraph{Proof of Proposition~\ref{prop:7.long}.}
The bounds of Subsection~\ref{ssec:7.6} and Appendices~\ref{app:A},
\ref{app:B} are stated for the configurations $\cN(X)$, $\cN_\sigma(X)$ produced by
labels. We go through the uses of the label in their proofs and show that
each of them survives for long wall fields. To this end we fix $k=0$ and a configuration $N_X\in\Nlong(X)$ (or
$\Nlong_\sigma(X)$), with the integrand of Lemma~\ref{lem:7.wall}.

(i) \emph{Sums over configurations.} They enter only through the following.
Lemma~\ref{lem:7.32}\,(b) maps $N_X$ to its interface values
$(N_I)_{I\subseteq X}\in\mathbb Z^{n_I(X)}$ and drops the realizability constraints. A wall
field is constant on interfaces and vanishes off them. Hence the map is
injective on $\Nlong(X)$, and the bound holds there as well. In the proof of
Proposition~\ref{prop:A.9}, step (2) splits by the first set
$\sat_{N_X}(S_0)$ through Lemma~\ref{lem:7.26}, which holds in wall form
by Lemma~\ref{lem:7.wall}\,(b), and the seed configuration it produces is long
by Lemma~\ref{lem:7.45}\,(a). Step (3) of that proof uses Lemma~\ref{lem:5.12} to count wall polymers with their walls
through their wall interfaces and values at the Euclidean link scale $g$.
This is again a count of integer functions on interfaces, combined
with $\Floc(\mathcal S)\ge2\Funit\sum_IN_I^2$ from Lemma~\ref{lem:7.32}\,(a). The sum over
link data and sets in Propositions~\ref{prop:A.5} and~\ref{prop:A.8} does not
involve the configuration beyond these.

(ii) \emph{Bounds at a fixed configuration.} Each is a bound on a Gaussian
integral whose integrand is a function of $\psi$, $D$, $w$ and $F_2'$, that
is, of $\psi$ and $N_X$, by Lemma~\ref{lem:7.wall}. The label enters these proofs
only in the following places.
\begin{itemize}
\item Appendix~\ref{app:B} uses the label in the phases of class-A cells, through
$e^{im\bGM^{1/2}h(x)}=1$. Under Lemma~\ref{lem:7.wall} the phase is
$e^{im\bGM^{1/2}\vartheta(x)}$ by definition. The same holds for the cell function
$\hat R_v$ of Lemma~\ref{lem:7.33}, evaluated at $\operatorname{avg}_\Omega(\psi+D)$, and
for Lemma~\ref{lem:7.43}, where $D=0$ on the wall-free cell $\Delta$.
\item Theorem~\ref{thm:7.14} enters through Lemma~\ref{lem:7.32}\,(a), and it holds for
every wall field by Lemma~\ref{lem:7.wall}\,(b).
\item In Step~1 of Lemma~\ref{lem:7.35}, the identity \eqref{eq:7.6.untr} is stated
in terms of $D_Y$, $w_Y$ and $s_Y$ for unions of elements and involves no
extension of $h$. At $k=0$ the near-cell term vanishes.
\item The functions $G^\natural$ of Lemma~\ref{lem:7.31} are functions of
$\vartheta=\psi+D$ and $\psi$, bounded pointwise.
\item Lemma~\ref{lem:7.34}\,(a) is used in Step~2 of Lemma~\ref{lem:7.35}. Its proof
first shows that $w=w_Y$ on $E(\Kflat_\Omega)$, using only (L8-4) and
Proposition~\ref{prop:7.9}, and then puts $a_\Omega:=h$ on $\Kflat_\Omega$, using
$P_\Omega h=0$ and $\nabla h=w$ on $E(\Kflat_\Omega)$. For a long configuration, Lemma~\ref{lem:7.45}\,(a) shows that the
field $N_X\one_{E(Y)}$ on the union $Y$ of elements is long, and Lemma~\ref{lem:7.prim} gives $h':\Kflat_\Omega\to P\mathbb Z$,
constant on $\Omega$, with $\nabla h'=w_Y$ on $E(\Kflat_\Omega)$. We apply
Proposition~\ref{prop:6.10}\,(iv) to $\vartheta+h'$. Since $\nabla(\vartheta+h')=\nabla\vartheta+w_Y$ on
$E(\Kflat_\Omega)$ and $P_\Omega(\vartheta+h')=P_\Omega\vartheta$, we obtain
$\|P_\Omega\vartheta\|^2\le C_W^f\kbox^2\|\nabla\vartheta+w_Y\|^2_{E(\Kflat_\Omega)}$, which is (a)
at $k=0$. Step~2 of Lemma~\ref{lem:7.35} then goes through without change.
\end{itemize}
Absolute convergence at a fixed configuration (Proposition~\ref{prop:7.25})
follows as in Section~\ref{S:F2}, the estimates used there being
pointwise in $(\psi,D)$. Lemma~\ref{lem:7.34}\,(a) is the only place where the
bounds use that the configuration is long: for a wall field with a nonzero
period on a short cycle of $\Kflat_\Omega$, or on a same-cell closure through
$\Omega$, no primitive of $w_Y$ constant on $\Omega$ exists. With (i) and (ii), the
proofs of Propositions~\ref{prop:7.37}, \ref{prop:7.41}, Lemmas~\ref{lem:7.39},
\ref{lem:7.40} and Proposition~\ref{prop:A.9} hold with the same constants, and
so does Lemma~\ref{lem:7.53}. \qed