% % % \subsection{Numerical checks for Sections~\ref{sec:2}, \ref{sec:stage1}, \ref{sec:stage2}, \ref{sec:8} and Appendix~\ref{app:A}}\label{app:E} The computations reported in this subsection illustrate statements of the paper and play no part in its proofs. Each number comes from a script of the archive released with the paper, and the file names of the scripts are given in brackets. The scripts use multiprecision arithmetic (30 to 60 significant digits) for the Bessel-function checks, exact rational arithmetic for the tree-graph integrals, and double precision for the lattice computations. \subsubsection{Bessel functions and the Hartman\textendash Watson law}\label{ssec:E.1} \emph{Complete monotonicity} (Lemma~\ref{lem:2.2}) [\texttt{hw\_tail.py}]. For $g(\lambda):=I_{\sqrt{2\lambda}}(\beta)/I_0(\beta)$ and the step $h=\beta^{-2}$, the forward differences satisfy $(-1)^k\Delta_h^kg(0)\ge0$ for all orders $k\le21$, at $\beta=2$, $10$ and $40$, as \cite[Theorem~1.3]{HW74} requires. \emph{The Villain coupling matched to the Wilson weight} [\texttt{dual\_weights.py}]. With $t_V(\beta):=1/(2\log(I_0(\beta)/I_1(\beta)))$, the Villain weight $e^{-n^2/(2t_V)}$ has the same ratio at $n=1$ as $I_n(\beta)/I_0(\beta)$. One finds $\beta-t_V(\beta)=0.6106$, $0.5650$, $0.5246$, $0.5112$, $0.5054$ at $\beta=2$, $5$, $10$, $20$, $40$, in line with $t_V=\beta-\frac12+O(1/\beta)$. At each of these $\beta$ the Wilson weight exceeds the matched Villain weight already at $n=2$, the first sign of its heavier tail. At the matched coupling the bound of \cite{GM82} is of order $e^{-2.494\beta}$, against the exponent $2.49436K\beta$ with $K>\Kst$ of \eqref{eq:1.sigma}: the constants of Theorem~\ref{thm:1.2} are far from sharp. \subsubsection{Villain monotonicity}\label{ssec:E.2} \emph{Monotonicity in each coupling} [\texttt{guth\_check.py}]. We test Lemma~\ref{lem:2.3}(b) on the complex of two adjacent unit cubes (eleven plaquettes), with $C$ the boundary of an outer face, using the dual representation \eqref{eq:2.dual} truncated at $\abs{k}\le40$. For 400 random choices of the eleven couplings in $[0.2,6]$, raising any one coupling by $30\%$ never lowered $\ex{\cos\theta_C}$ (0 decreases in 4400 trials). \subsubsection{The density of frozen plaquettes}\label{ssec:E.3} \emph{The exponent of $\delta$} (Lemma~\ref{lem:2.6}) [\texttt{delta\_table.py}]. Table~\ref{tab:E.1} compares $-\beta^{-1}\log$ of the bound of Lemma~\ref{lem:2.4}(iv), minimized over $\nu$ without restriction (golden-section search on $[\frac12\astar\beta,\frac32\astar\beta]$), with the lower bound $\cdelsp(K)-1/(4\astar)$ of Lemma~\ref{lem:2.6}. The minimizer is $\nu=\astar\beta$ to the displayed precision, and the finite-$\beta$ values approach $\cdelsp(K)-1/(4\astar)$ from below at the rate of the $O(\log\beta/\beta)$ term of \eqref{eq:2.cdelta}. The threshold values of Lemma~\ref{lem:2.6} [\texttt{kstar.py}] are recorded in Section~\ref{ssec:D.3}. \begin{table}[h] \centering \caption{The exponent $-\beta^{-1}\log\bigl(\min_\nu e^{\nu^2\eps/2}\sum_nI_{\sqrt{n^2+\nu^2}}(\beta)/I_0(\beta)\bigr)$ at $\eps=1/(K\beta)$.} \label{tab:E.1}\label{tab:D.2} \begin{tabular}{rrrrrr} \hline $K$ & $\astar$ & $\cdelsp(K)-\frac{1}{4\astar}$ & $\beta=10$ & $\beta=20$ & $\beta=40$\\ \hline 5 & 17.886 & 15.090 & 14.948 & 15.002 & 15.037\\ 10 & 44.998 & 57.236 & 57.105 & 57.153 & 57.186\\ 15 & 75.199 & 114.295 & 114.170 & 114.215 & 114.247\\ 20 & 107.393 & 181.935 & 181.814 & 181.857 & 181.887\\ \hline \end{tabular} \end{table} \emph{The Debye exponent} [\texttt{watson.py}]. The closed form of $\vCb(0)=(-\Delta_{\Zthree})^{-1}(0,0)$ given in item~0 of Section~\ref{ssec:D.1} and the integral $\int_0^\infty e^{-6t}I_0(2t)^3\,dt$ agree. Both give $\vCb(0)=0.2527310098587$, hence $\pi^2\vCb(0)=2.4943550872$ and $(8\pi^2)^{1/2}/2^{16}=1.35586\cdot10^{-4}$, the numbers used in \eqref{eq:8.sigma}. \subsubsection{The free-space field of the near-loop estimate}\label{ssec:E.4} \emph{The solid-angle primitive} (Lemma~\ref{lem:7.34}) [\texttt{lemmaN\_rectangle.py}, \texttt{slit\_nearcell.py}]. For the square loop $C=\partial([0,64]^2\times\{0\})$ and the flat surface $\Xi$ it bounds, we evaluate the free-space solid angle $f^C=\frac12\Omega_\Xi$ on the dual lattice of a box with margin $24$ around $C$ ($602{,}112$ dual sites). Its largest modulus is $3.0974<\pi$. The residual $R^C$ satisfies the bound $\abs{R^C(b)}\le2\pi/d(b,C)$ of Lemma~\ref{lem:7.34}(b), with exact segment-to-segment distances, on all $1{,}031{,}184$ dual bonds within distance $21$ of $C$. Over these bonds the largest ratio $\abs{R^C(b)}\,d(b,C)/(2\pi)$ is $0.3415$. The bound $\abs{R^C(b)}\le\frac12\sum_ed(b,e)^{-2}$, summed over the unit edges $e$ of $C$, holds on the same bonds with largest ratio $0.8645$. In a near box of side $32$ centred on a side of $C$, we build eighteen slit geometries (cracks parallel and perpendicular to $\Xi$, holes, windows, frozen pieces of $\Xi$ and combs of frozen sheets) by deleting dual bonds and sites. On the eighteen slit geometries and the free box this bound holds with largest ratio $0.8303$, and on each slit geometry the energy of $f^C$ on the remaining bonds, as well as the minimum of $\sum_b(\nabla a(b)-j(b))^2$ over fields $a$ on the slit graph, is at most the energy of $f^C$ on the free box. Since freezing only removes terms from sums of fixed free-space quantities, slit edges do not enter the near-loop estimate. \subsubsection{Further checks}\label{ssec:E.5} \emph{The tree-graph $s$-integral} (Lemmas~\ref{lem:A.2} and~\ref{lem:A.3}) [\texttt{s\_integral/tree\_s\_integral\_check.py}]. In exact rational arithmetic, with the $s$-integrals evaluated as exact polynomial integrals, we find for $n=2,\dots,7$ that $\sum_Tw(T)=n^{n-2}/(n-1)!$, that for random rational $y\ge0$ the left side of \eqref{eq:A.vol} equals the volume of the polytope $\{0\le u_m\le y_m+u_{m-1},\ 1\le m\le n-1\}$ ($u_0:=0$) and is at most $e^{y_1+\dots+y_{n-1}}$, that Lemma~\ref{lem:A.3} holds for random $y_1$ and $b_i\in[0,0.9]$, and that $w(T)=1/(n-2)!$ for the tree with $T(2)=1$ and $T(j)=2$ for $j\ge3$. \emph{Corner sites} (Remark~\ref{rem:6.7}) [\texttt{corner\_pair.py}, \texttt{corner\_pair\_hp.py}]. With $v(x)=\int_0^\infty e^{-6t}\prod_\mu I_{x_\mu}(2t)\,dt$ (30 digits), we obtain $v(1,1,0)=0.0551914336877$ and $v(1,1,1)=0.0435783543977$. The corner image sum is $0.4673457$, which gives $\Phi=9.22503\,\tau$, and $E_{\mathrm{pair}}=0.52927