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\subsection{Constants of Subsections~\ref{ssec:7.0} to~\ref{ssec:7.5} and Appendix~\ref{app:C}}\label{S:constF}

\paragraph{Proposition~\ref{prop:7.5}.} Off $\Df$, (S4$'$) gives
$|\rho_1(\pm1,x)-\rho^{\mathrm{ref}}(1)|\le\rho^{\mathrm{ref}}(1)(1-e^{-\theta_0}+Ce^{-\mu})$, and
$\sum_{|m|\ge2}(|\rho_1(m,x)|+|\rho^{\mathrm{ref}}(m)|)m^2\le2Ce^{-2\kappa_S}m_D^2/\bGM$. Since
$2\bGM\rho^{\mathrm{ref}}(1)=\mt^2-\bGM\sum_{|m|\ge2}m^2\rho^{\mathrm{ref}}(m)\le(1+Ce^{-2\kappa_S})\mt^2$
and $m_D\le2\mt$, we obtain, after enlarging $C$,
\[
\sum_m|\rho_1(m,x)-\rho^{\mathrm{ref}}(m)|\,m^2\le\theta'\,\mt^2/\bGM\qquad(x\notin\Df),
\]
with $\theta'$ of \eqref{eq:7.thetaprime}. On $\Df$ we have $|\rho_1(\pm1,x)|\le\rho^{\mathrm{ref}}(1)(1+Ce^{-\mu})$ and, since $e^{-\Phi}\le1$,
$\rho_1(\pm1,x)-\rho^{\mathrm{ref}}(1)\le Ce^{-\mu}\rho^{\mathrm{ref}}(1)$. These bounds, the same tails and
$2\bGM\rho^{\mathrm{ref}}(1)\le2\mt^2$ give the constants of part (b). At $\theta_0=0.01$ we have
$1-e^{-\theta_0}=0.00995\ldots$, and hence the standing hypothesis \eqref{eq:7.standing} holds
once $\varepsilon_\nu+C(e^{-\mu}+e^{-2\kappa_S})\le0.1329$.

\paragraph{The partial fractions \eqref{eq:7.partialfrac} and Lemma~\ref{lem:7.18}.}
If $\lPV\le\frac14$, then $\varpi=(1-4\lPV^2)^{-1/2}\in[1,1.155]$, $c_+\le\varpi$,
$c_-\le1.26\,\lPV^2$, $\mt^2\le r_-\le1.08\,\mt^2$ and $r_+\ge0.932\,\MY^2$.
In part (b), with $\alpha=\mt/3$, the bound $\cosh x-1\le\frac{x^2}2\cosh x\le0.5281\,x^2$
for $x\le\frac13$ gives $6(\cosh\alpha-1)\le0.3521\,\mt^2$. With this $\alpha$, part (a) gives
$\|e^\rho R_{r_-}e^{-\rho}\|\le(r_--0.3521\mt^2)^{-1}\le1.544\,\mt^{-2}$ and
$\|e^\rho R_{r_+}e^{-\rho}\|\le(r_+-0.3521\mt^2)^{-1}\le1.1\,\MY^{-2}=1.1\,\lPV^2\mt^{-2}$, and hence
$\|e^\rho\mathcal Ge^{-\rho}\|\le(1.155\cdot1.544+1.26\cdot1.1\,\lPV^4)\mt^{-2}\le1.80\,\mt^{-2}$.
The matrix $e^\rho\nabla^*e^{-\rho}$ has entries $\pm e^{\rho(x)-\rho(b)}$ with
$|\rho(x)-\rho(b)|\le\frac13$, at most $6$ per row and $2$ per column. Hence its
norm is at most $\sqrt{12}\,e^{1/3}\le4.835$, and $1.80\cdot4.835\le8.71$.
In part (c), $\|[\Delta,\chi]\|\le6\Lambda_\chi$ and $\|\Delta\|\le12$, so
$\|[C_0^{-1},\chi]\|\le\MY^{-2}\cdot2\cdot12\cdot6\,\Lambda_\chi+6\Lambda_\chi=\Lambda_\chi(144\MY^{-2}+6)\le150\,\Lambda_\chi\MY^{-2}$, since $\MY\le1$.

\paragraph{Theorem~\ref{thm:7.19}.} The constant is $C_\varepsilon:=3.8\cdot10^9$. With
$d(A',V_B)\ge L'/2-2$, $\Lambda_\chi=4/L'$, $\MY^{-4}=\lPV^4\mt^{-4}$ and $\mt\le1$,
Lemma~\ref{lem:7.18}\,(b), (c) give
$\|F'_{2,\beta}\|^2\le(150\cdot\frac4{L'}\MY^{-2})^2(8.71\,\mt^{-2})^2e^{4/3}e^{-\mt L'/3}\|w_\beta\|^2
\le1.71\cdot10^6\cdot16\,e^{4/3}\,\mt^2(\mt L')^{-2}\lPV^4\mt^{-8}e^{-\mt L'/3}\|w_\beta\|^2$.
Theorem~\ref{thm:7.14} and $c_g'\ge\frac9{16}c_g$ give
\[
\sum_{\beta:S_\beta\subseteq X}\|w_\beta\|^2\le18\,\Floc(X)/(c_g'\mt^2L/2)\le32\,\Floc(X)/(c_g\mt^2L/2).
\]
The product of the numerical factors is $1.71\cdot10^6\cdot16\cdot32\,e^{4/3}\le3.4\cdot10^9\le C_\varepsilon$.

\paragraph{Corollary~\ref{cor:7.21}.} The Hölder exponents and the constant of
Subsection~\ref{ssec:7.6} are $p_2=20$, $p_3=1.2$, $p'=1.3$, $q'=15.6$ and
$c_\lambda=\frac1{24}$, so
$\varepsilon_0=\min\{(1-\frac1{1.2}-\frac1{24})/40,\,2(1.3/15.6)^2\}=\min\{0.003125,\,0.0139\}=0.003125$.
Dropping $(\mt L')^{-2}\le1$ in \eqref{eq:7.epsL}, we obtain $\varepsilon(L')\le\varepsilon_0$
as soon as
\[
\mt L'\ge27\log(1/\mt)+3\log\bigl(C_\varepsilon\lPV^4/(c_g\varepsilon_0\,\mt L/2)\bigr).
\]
The second term is bounded, since $c_L/2\le\mt L\le c_L$ by Definition~\ref{def:3.1}.
For the first, Definition~\ref{def:3.1} and Section~\ref{S:G.35} give $\log(1/\mt)\le2.4944\,\tau=(2.4944/4\pi^2)\bGM$
for $\tau\ge\tau_0$, so the right side is at most $1.706\,\bGM+O(1)$, while
$\mt L'=\Kcol\bGM=2\bGM$.

\paragraph{Proposition~\ref{prop:7.22}.} The constant of part (c) is
\[
\Phi_\UU:=\frac{\pi c_gc_u}{24}\,\frac{(\mt L)^3}{(\mt\bGM)(\mt L')^3(11/10)^3}.
\]
Dividing (a) by (b) gives the share $\frac{\pi c_g'c_u}{12}(\mt L)^3/((\mt\bGM)(\mt L')^3(1+\eta_{\mathrm{col}})^3)$,
which is at least $\Phi_\UU$ because $c_g'\ge\frac9{16}c_g\ge\frac12c_g$ and
$\eta_{\mathrm{col}}\le\frac1{10}$ for $\bGM\ge\bGMo$. The latter holds since $L$ and $D_E$ are $O(\mt^{-1})$,
while $L'=2\bGM\mt^{-1}$. With $\mt L\ge c_L/2$ and $\mt L'=2\bGM$,
$\Phi_\UU\ge\pi c_gc_uc_L^3/(24\cdot64\cdot1.331\,\mt\bGM^4)$, and
$\mt\bGM^4\to0$ because $\log(1/\mt)=\pi^2\vCb(0)\tau+O(\log\tau)$ and $\bGM=4\pi^2\tau$.

\paragraph{Structural facts of the expansion (Appendix~\ref{app:C}).} The bounds on
the activities in Subsections~\ref{ssec:7.6} and~\ref{ssec:7.7} and in Appendices~\ref{app:A}
and~\ref{app:B} use the expansion of Proposition~\ref{prop:7.25} only through
the facts of Table~\ref{tab:C.1}, and none of them uses translation invariance, an
infinite lattice, or the shape of the atoms.

\begin{table}[ht]
\centering\small
\begin{tabular}{p{0.04\textwidth}p{0.5\textwidth}p{0.18\textwidth}p{0.17\textwidth}}
\hline
& fact & used in & proof\\
\hline
S1 & $\bC_s(x,x)=\bC(x,x)$, $|\bC_s(x,y)|\le|\bC(x,y)|$, weighted norms and $\|\bC_s\|_{2\to2}$ bounded by those of $\bC$ & Lemma~\ref{lem:7.31}, App.~\ref{app:B} & Prop.~\ref{prop:7.25}, change~6\\
S2 & $\bC_s=\sum_\varepsilon\lambda_\varepsilon\bC_\varepsilon$, $\bC_\varepsilon$ block diagonal with admissible blocks & Lemma~\ref{lem:7.35} & Prop.~\ref{prop:7.25}, change~6\\
S3 & $\log\int d\mu_\bC\,e^{\frac12\langle\psi+D,M(\psi+D)\rangle}$ is convex in $\bC$ for $M\ge0$ & Lemma~\ref{lem:7.35} & \cite[(9.67) to (9.611)]{BF80}; Lemma~\ref{lem:7.35}, Step~3\\
S4 & the integrand of $\mathcal K_{\bar y}$ depends on $\psi|_{X(\bar y)}$ only; covariance derivatives have both ends in $X(\bar y)$ & Lemma~\ref{lem:7.39}, Thm.~\ref{thm:7.29}(b), (c) & Prop.~\ref{prop:7.25}\\
S5 & each $Y_{i+1}$ is a union of elements, each meeting $S(\theta_i)\setminus X_i$, the part of the support of its link outside $X_i$ & Lemma~\ref{lem:7.39} (App.~\ref{app:A}) & selection rule \eqref{eq:C.E20}\\
S6 & link $i$ carries $\prod_{k=T(i+1)}^{i-1}s_k$ (covariance link from $Y_{T(i+1)}$) or $\prod_{k=j_{\min}}^{i-1}s_k$ ($E$-term link); all other $s$-dependence is through weights in $[0,1]$ and through $\bC_s$ & Lemma~\ref{lem:7.39} (App.~\ref{app:A}) & \eqref{eq:C.E3}, \eqref{eq:C.E16}\\
S7 & five types of terms per link & App.~\ref{app:A} & \eqref{eq:C.E16}\\
S8 & link ends $x\in Y_{i+1}$, $y\in Y_{T(i+1)}$, counted in $\UU$-cells & App.~\ref{app:A} & selection rule \eqref{eq:C.E20}\\
S9 & sequences of disjoint admissible sets; finitely many & Lemma~\ref{lem:7.39}, Prop.~\ref{prop:7.51} & Prop.~\ref{prop:7.25}, change~2\\
S10 & $Z(Y)>0$; one-set activities are positive & Lemmas~\ref{lem:7.43}, \ref{lem:7.44}, Prop.~\ref{prop:7.57} & Prop.~\ref{prop:7.25}\\
S11 & the first set may depend on $h$ & Thm.~\ref{thm:7.29}, Lemma~\ref{lem:7.53} & Prop.~\ref{prop:7.25}, change~3\\
S12 & the first set may be disconnected & Thm.~\ref{thm:7.29}(c), Lemma~\ref{lem:7.53} & Prop.~\ref{prop:7.25}, change~3\\
S13 & expansion on $U=X_0^c$ with $\one_U\bC\one_U$; activities independent of $U$ & Thm.~\ref{thm:7.29}(b) & Prop.~\ref{prop:7.25}, change~4\\
S14 & absolute convergence of every term & all & Lemmas~\ref{lem:C.4}, \ref{lem:C.5}\\
\hline
\end{tabular}
\caption{Structural facts of the expansion used in Section~\ref{sec:stage2}.}
\label{tab:C.1}
\end{table}