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\subsection{Constants of Sections~\ref{sec:2} and~\ref{sec:8}: the ordered list, the threshold $K_*$ and the conditions defining $\beta_0(K)$}\label{S:constA}

Here we give the values of the constants; the order in which they are fixed, and the reasons for it, are explained in Section~\ref{app:D}. The values were evaluated by computer in high-precision arithmetic, but the proofs need only the inequalities stated.

\subsubsection{The ordered list}\label{ssec:D.1}

Each constant is fixed after all constants above it, and the constants in items~1 to~14 are independent of $\beta$, $\tau$, $K$, the loop $C$ and $D$, except that $\bar r_0$ of item~1 is $r_0$ or $2r_0-1$ according to $\Grid(C)$, and that the depletion constant $\cdep$ of item~12 depends on $\tau$. The scales of item~11 depend on $\beta$ through $\tilde m$ alone, and the derived quantities $c_g'$ and $c_u$, which lie between two constants of this kind, depend on $\beta$ only through the size of $L_c$ ($c_g'$) and through its rounding ($c_u$). Constants built from $c_u$, such as $D_{\cU}$ of Appendix~\ref{app:A} and $K_0^B$ of Appendix~\ref{app:B}, likewise lie between two constants of this kind, and we treat them as constants. Illustrative values are given at $(c_g,c_s,r_0)=(1/4,1,1)$. We use coarse units for the regularization and fine units in Sections~\ref{sec:stage1} and~\ref{sec:stage2}.

\begin{enumerate}
\item[0.] \emph{Absolute constants.} Among them are $c_I=2^{2/3}$ and the Debye exponent $\pi^2v_{\mathrm{Cb}}(0)=2.49435508\ldots$, where $v_{\mathrm{Cb}}(0)=(-\Delta_{\mathbb Z^3})^{-1}(0,0)=\frac{\sqrt6}{192\pi^3}\Gamma(\frac1{24})\Gamma(\frac5{24})\Gamma(\frac7{24})\Gamma(\frac{11}{24})=0.25273101\ldots$ is half of Watson's integral \cite{Wat39} in the normalization of \cite{Zu11}, whose closed form is \cite[(3.6)]{Zu11}, and $\log(1/m_D)=\pi^2v_{\mathrm{Cb}}(0)\tau-\frac12\log(8\pi^2\tau)$. The rounding rule for the Debye exponent is in Section~\ref{ssec:D.4}. The other absolute constants are $C_{\mathrm{bl}}=384$ (Proposition~\ref{prop:6.10}), the window count $151^3$ (Definition~\ref{def:3.4}), $\alpha=\ln8$ and $8/31$ (Theorem~\ref{thm:7.52} and Lemma~\ref{lem:7.58}), $c_T=2+\log96$ and the threshold $14.9662>6\pi^2v_{\mathrm{Cb}}(0)=14.96613\ldots$ (Lemma~\ref{lem:2.6} and Section~\ref{sec:mincut}), and the absolute constants among those of Göpfert\textendash Mack, Brydges\textendash Federbush and Stage~1 quoted in Sections~\ref{sec:stage1}, \ref{sec:stage2} and Appendices~\ref{app:A} to~\ref{app:C}. The constants of these authors and of Stage~1 that depend on the parameters $C$, $\delta_{\mathrm{GM}}$, $\mu$, $\kappa_S$, $\lambda_{\mathrm{PV}}$ of Theorem~\ref{thm:6.5} are fixed together with these parameters, in items~12 and~13.
\item[1.] $r_0\ge1$, an integer, and $\bar r_0:=2r_0-1$, or $\bar r_0:=r_0$ if $\Grid(C)$ has no exceptional layer; they are used in Definition~\ref{def:3.4}, Lemma~\ref{lem:5.11} and $\eta_T$.
\item[2.] $c_g:=1/4$, the constant of the property (S-face) of Definition~\ref{def:3.14}, which the construction provides for all $c_g\le1-134\vartheta$ by Theorem~\ref{thm:4.rest}. The proofs hold under $0<c_g\le1/4$, so every $c_g\in(0,1/4]$ is admissible.
\item[3.] $\eta_w=1/10$, the hull-weight deficit of the frozen bonds (Definition~\ref{def:3.8}).
\item[4.] $c_\rho=1/12$, that is, $\rho_\lambda=L_c/12$ and $\lambda=12/L_c$, the equality case of Lemma~\ref{lem:4.8}.
\item[5.] Before $\chi$ and $c_L$ we fix $k_P/L_c=304$, so that the windows of Definition~\ref{def:3.4} have sides less than $151\cdot2L_c<k_P$, as in (G4), and we set $r_\square:=(k_P+3)/L_c\le305$.
\item[6.] $c_1:=\min(c_g/2,((1-\eta_w)c_Ic_\rho/2)^3)$, which equals $2.109\cdot10^{-4}$ for $c_g\ge4.22\cdot10^{-4}$. We set $c_{\mathrm{sh}}:=c_1/216$; Table~\ref{tab:S:B1.hull} gives their uses.
\item[7.] $c_s\in[3.3\cdot10^{-7},10]$ (illustration $c_s=1$), with $w_*:=(1+c_s)|\partial\Delta|$ (Definition~\ref{def:3.11}) and $\lambda_U:=2c_s/\ell$ (Definition~\ref{def:3.forced}); then $2c_s\ell^2>1$, as $\ell\ge308L_c\ge1232$ (used in Theorem~\ref{thm:4.42}).
\item[8.] The forcing threshold $\vartheta=3\cdot10^{-5}$ and the penalty $\mu_{\rm B}=25$ of Definition~\ref{def:3.forced}, and the thin-set constant $\varepsilon_{\rm th}=0.008$ of Section~\ref{ssec:4.3}. They satisfy $24.6<\mu_{\rm B}$ (Lemma~\ref{lem:4.forced}), $128\vartheta<\varepsilon_{\rm th}$, $2\varepsilon_{\rm th}+12\vartheta\le0.0164$ and $2/\sqrt{0.0164}>14$ (Lemma~\ref{lem:4.thin}), and $254.4\vartheta\le\varepsilon_{\rm th}$ (Theorem~\ref{thm:4.rest}); the arithmetic is in Section~\ref{S:B1.cells}.
\item[9.] The weak-link constant $\chi:=2.0\cdot10^{-8}$ and $\chi_{\rm N3}:=\chi/3=6.67\cdot10^{-9}$ of Definition~\ref{def:3.13}. Its value depends only on items~3, 5 and~8, item~3 through the factor $12/\eta_w$ of Proposition~\ref{prop:4.sealcount} in the expansion bound, and this value is constrained by Theorem~\ref{thm:4.rest}(c), which provides the expansion $h_\star=2.2\cdot10^{-11}/L_c$ on each component of $\Gp\cap\hat Q$ at rest. Since $3k_Ph_\star=2.006\cdot10^{-8}\ge\chi$ (Section~\ref{S:B1.cells}), (N3) holds with $\chi_{\rm N3}$ and (R6) with $\chi$.
\item[10.] \emph{Consequences, fixed with the constants above and independent of $L_c$ and $\ell$.} $C_1:=7+27\cdot24.6/\vartheta=2.214\cdot10^7$, $C_{\rm seal}:=726$, $\Cst:=C_1+2C_{\rm seal}=2.2141\cdot10^7$ and $C_{\mathrm{free}}:=\Cst(2+1/(2c_s)+1/3696)=5.536\cdot10^7$ (Propositions~\ref{prop:4.sealcount} and~\ref{prop:4.39}); $C_a:=4362+5808/c_s$, $C_b:=2+4/c_s$ and $C_c:=1+2/c_s$, equal to $10170$, $6$, $3$ (Lemma~\ref{lem:5.31}), and the response load $C_{\mathrm{resp}}:=375C_a+125C_c\Cst=8.307\cdot10^9$ of Section~\ref{ssec:5.6}; finally, $C_{\mathrm{rm}}=20$, $C_{\mathfrak B}:=30\Cst+170C_{\mathrm{resp}}+30=1.413\cdot10^{12}$ and $C_3:=C_{\mathrm{free}}+20(2\Cst+4C_{\mathrm{resp}}+1)=6.655\cdot10^{11}$ (Proposition~\ref{prop:5.36}, Theorem~\ref{thm:5.1}).
\item[11.] The constant $c_L$ is subject to $C_P'c_L^2\le(1-4/\pi^2)/8$ and $c_L\le\frac12$. The first condition holds if $2916r_\square^5C_{\mathrm{bl}}^2c_L^2/\chi^2\le(1-4/\pi^2)/8$, since $C_P'=C_W^fr_\square^2N_{\mathrm{ov}}\le2916r_\square^5C_{\mathrm{bl}}^2/\chi^2$, where $C_W^f=108C_{\mathrm{bl}}^2/\chi^2$ is the constant of Proposition~\ref{prop:6.10}(iv) and $N_{\mathrm{ov}}\le27r_\square^3$. At $r_\square=305$ this yields the bound $8.09\cdot10^{-12}\chi$, and the admissible choice is $c_L\le8.0\cdot10^{-12}\chi$, that is, $c_L=1.6\cdot10^{-19}$ at $\chi=2.0\cdot10^{-8}$. This choice implies $\tilde mL\le\frac12$ for Lemma~\ref{lem:4.19} and $c_L\le1.99\cdot10^{-4}\chi_{\rm N3}=1.33\cdot10^{-12}$ for Lemma~\ref{lem:7.47}. The scales of Definition~\ref{def:3.1} are then $L_c:=c_L\tilde m^{-1}/r_0$ rounded down to an even integer, $L:=r_0L_c$, $k_P:=304L_c$, $\ell_D\approx\tilde m^{-1}$, a multiple of $L$, and $\ell:=\ell_D/r_0\ge k_P+4L_c$. The derived quantities are $c_g'=c_g(1+1/L_c)^{-2}$ and $c_u:=r_0^3V_*\tilde m^3=((1-\eta_w)c_I\tilde mL/24)^3$, which lies in $[((1-\eta_w)c_Ic_L/48)^3,((1-\eta_w)c_Ic_L/24)^3]$ once $c_L\tilde m^{-1}\ge4r_0$ (Section~\ref{S:B1.scales}). Further derived quantities are $R_{\mathrm{HK}}=(1-\eta_w)c_Ic_\rho L/3$ and the constant $K_0:=(1-\eta_w)c_Ic_\rho c_L/6=0.01984\,c_L$, whose uses are listed in Section~\ref{S:E1.scales}. Since $c_L/2\le\tilde mL\le c_L$, we have $K_0\le\tilde mR_{\mathrm{HK}}\le2K_0$.
\item[12.] \emph{Stage~1 and Stage~2 constants.} These are the constants of Sections~\ref{sec:stage1} and~\ref{sec:stage2} and Appendices~\ref{app:A} to~\ref{app:C}. We first fix the parameters $C\in(0,\frac14]$ and $\delta_{\mathrm{GM}}\in(0,\frac12]$ of Theorem~\ref{thm:6.5}, where this $C$ is not the loop; the bound $\delta_{\mathrm{GM}}\le\frac12$ is used for (S5) on outputs and in Lemma~\ref{lem:B.4} and Proposition~\ref{prop:B.7}. The bound $C\le\frac14$ on this parameter implies $2C_{M4}=2/(1-C)\le\frac83$, hence $q:=2C_{M4}e^{c_p}/e\le0.9829$ since $c_p<\frac1{540}$. In Proposition~\ref{prop:B.7}, therefore, one factor $e^{-(t_e-1)}$ absorbs the per-$\mathcal E$-factor constants $t_e(2C_{M4}e^{c_p})^{t_e-1}$ up to the factor $\sup_{t\ge1}tq^{t-1}\le22$, which contributes one factor per cell; here $t_e$ is the factor of (M4), Lemma~\ref{lem:B.1}(d), which bounds the sum over all assignments of the points to their cells.

Next come $\eta_H=0.62$, $(p_3,p',q',p_1,p_2,p_4)=(1.2,1.3,15.6,20,20,15)$, $c_\lambda=1/24$, $\theta_0=0.01$ and the link-sum rate $c_{10}$ with $0<c_{10}\le\frac79r_C$, where $r_C$ is evaluated at $\tilde mL=c_L/2$. We use this bound in Proposition~\ref{prop:B.7}, where, for $\delta_{\mathrm{GM}}\le\frac12$, the rate of (S2) in Debye units is at least $2r_C$ for all $\lambda_{\mathrm{PV}}\le\frac14$, by Proposition~\ref{prop:6.11}, and pinning uses $4c_p<\frac29r_C$ of this rate for every $\bar\delta$. The quantity $r_C=\min(\kappa_0/2,(2c_{\mathrm{gs}})^{1/2})$, as in Theorem~\ref{thm:6.12}, is computed with the constant $K_0$ of item~11, the value of $\tilde mR_{\mathrm{HK}}$ at $\tilde mL=c_L/2$, which is the worst case of $\tilde mL\in[c_L/2,c_L]$. It is therefore determined by items~0 to~11, and hence $c_{10}$ is fixed independently of $\beta$, $\lambda_{\mathrm{PV}}$ and $\bar\delta$.

We then fix $c''_{10}$ with $e^{c''_{10}}\ge16e\max(1,c_{c_{10}/2})$ (Lemma~\ref{lem:A.7}), and then the parameters $\mu$ and $\kappa_S$ of Theorem~\ref{thm:6.5}, both large: $\mu\ge1+2c''_{10}$, after $c_{10}$ and $c''_{10}$ (the factor $e^{-\mu(s-1)}$ of (S2) pays $e^{c''_{10}t}\le e^{2c''_{10}(t-1)}$, Proposition~\ref{prop:B.7}), and $\kappa_S$ so large that $(1+C_{\mathrm{ref}}e^{-2\kappa_S})\eta_H/8\le\frac17$ (Lemma~\ref{lem:C.4}), where $C_{\mathrm{ref}}:=8C$ with $C$ the constant of \eqref{eq:6.S4c} on $\Zthree$, so that $\bGM\sum_mm^2|\rho^{\mathrm{ref}}(m)|\le(1+C_{\mathrm{ref}}e^{-2\kappa_S})\mt^2$ (as $\mD\le2\mt$). This constant $C$ comes from the tail bound \eqref{eq:6.S4c}, that is, from (S2) at $s=1$ with $a_1=\{x\}$ and from \eqref{eq:6.lead}, whose right sides involve the parameter $C$ and $\mu$ but not $\kappa_S$. Hence the constant $C$ of the tail bound can be chosen independently of $\kappa_S$ and $\lambda_{\mathrm{PV}}$, and this last condition can be met. At $C=\frac14$, for instance, $c''_{10}=2200$, $\mu=4401$ and $\kappa_S\ge5.8$ are admissible; the binding condition on $\kappa_S$ is \eqref{eq:7.6.cl}.

Further, we require $\varepsilon_\nu+\theta'\le1/7$ and $c_E\le1/70$. The depletion constant $\cdep:=A_0^D+A_1^D\log(C'\tau/\theta_0)$ of Proposition~\ref{prop:6.18} depends on $\tau$, but all its occurrences are in lower bounds on $\tau$ (Theorem~\ref{thm:7.42}). Next we fix $K_{\mathrm{col}}=2>27\cdot2.4944/(4\pi^2)=1.706$ and $\varepsilon_0=0.003125$ (Corollary~\ref{cor:7.21}), and $0<\delta_2<\bar\delta<r_C$ and $c'_{\mathrm{rate}}=\frac12(r_C-\bar\delta)$ (Proposition~\ref{prop:7.38}). Last come the pure number $c'_0:=\frac12(1-1/p_3-c_\lambda)=1/16$, a lower bound for the fraction $c'$ of Proposition~\ref{prop:7.37}, then $\varepsilon_E=c'_0/8=1/128$, $c_w=c'/16$ and $\alpha=\ln8$.
\item[13.] $\lambda_{\mathrm{PV}}\le\min\bigl(\frac14,\lambda_0\bigr)$, chosen after $c_L$ and after $\mu$, $\kappa_S$ of item~12, where $\lambda_0=\lambda_0(\kappa_S,\mu,C,\delta_{\mathrm{GM}})$ of Theorem~\ref{thm:6.5} comes from the Debye condition in its proof. Here $M_Y$ is the smallest fixed point of \eqref{eq:6.mass} and $\tilde m:=\lambda_{\mathrm{PV}}M_Y$, as in Section~\ref{ssec:6.2}. The constants above are the same for every $\lambda_{\mathrm{PV}}$ allowed by the a priori bound $\lambda_{\mathrm{PV}}\le\frac14$, whereas those of Göpfert\textendash Mack, Brydges\textendash Federbush and Stage~1 that depend on $(\lambda_{\mathrm{PV}},\kappa_S,\mu,C,\delta_{\mathrm{GM}})$, among them the threshold $\bGMo$ of Theorem~\ref{thm:6.5}, are fixed here, after $\lambda_{\mathrm{PV}}$.
\item[14.] \emph{The cut-rule constants.} Those of Definition~\ref{def:3.17} are $\theta_{\mathrm{NF}}:=c'_0c_g/(90(c_g+4))=4.08\cdot10^{-5}$, $c_3^{\mathrm{NF}}:=c'_0c_g^2/(64(c_g+4))=1.44\cdot10^{-5}$ and $\kappa_{\mathrm{NF}}:=4(c_g+4)/c_g^2=272$ at $c_g=1/4$. They belong to the regularization but come after item~12, because they are built from the pure number $c'_0$, and the constants above are independent of $\theta_{\mathrm{NF}}$ and $c_3^{\mathrm{NF}}$.
\item[15.] $K>K_*=377.38$, which implies $c_\delta^{\mathrm{sp}}(K)/(2K)>14.9662$ (see Section~\ref{ssec:D.3}). Proposition~\ref{prop:5.15} and Theorem~\ref{thm:5.1} use this threshold.
\item[16.] $\beta_0(K)$, determined by the conditions of Section~\ref{ssec:D.5}.
\end{enumerate}

\noindent Among the scales determined by these constants are $\tilde m^{-1}$, with $\tilde m\in[m_D/2,2m_D]$, and $L_c$, $L$, $k_P$, $\ell$ as in item~11. These constants further determine $V_*=(1-\eta_w)^3\rho_\lambda^3/2$, $V_{\mathcal U}=\tilde m^{-3}$, $R_*=(J+1)w'+6L_c=O((\ell/c_s)\log(\ell/c_s))$ of Theorem~\ref{thm:4.42} together with $r'=2R_*+2$, and $R_{\mathrm{resp}}=10\ell$ together with $R'=R_*+2R_{\mathrm{resp}}$. The remaining scales are $L'=2(\beta_{\mathrm{GM}}/a)\tilde m^{-1}$, $R_1(\beta)=(2r_\square+5)L$, $F_{\mathrm{unit}}=\sigma_FL^2=(\pi^2c_g'/18)(\tilde mL)^3/(\tilde m\beta_{\mathrm{GM}})$ of Definition~\ref{def:3.17}, $R_{\mathrm{NF}}=Le^{c_3^{\mathrm{NF}}F_{\mathrm{unit}}}$, and the box $\Lamo$ of side $M_\Lambda'\in\{M_\Lambda,M_\Lambda+1\}$, where $M_\Lambda=\max(3\max(R,T),r_0(\ell+3))$.

\subsubsection{The threshold $K_*$ of Lemma~\ref{lem:2.6}}\label{ssec:D.3}

Lemma~\ref{lem:2.6} gives, with $a_*$ the solution of $a=K\log(2a)$,
\begin{equation}\label{eq:D.csp}
c_\delta(K)=\frac{|\log\delta|}{\beta}\ge c_\delta^{\mathrm{sp}}(K)-\frac1{4a_*}-O\Big(\frac{\log\beta}\beta\Big),\qquad c_\delta^{\mathrm{sp}}(K):=a_*\Big(\tfrac12\log(2a_*)-1\Big)+1 .
\end{equation}
The function $K\mapsto c_\delta^{\mathrm{sp}}(K)/(2K)$ is increasing. Indeed $da_*/dK=a_*\log(2a_*)/(a_*-K)$ and $a_*-K=K(\log(2a_*)-1)$, which yield
\[
\frac{dc_\delta^{\mathrm{sp}}}{dK}=\Big(\tfrac12\log(2a_*)-\tfrac12\Big)\frac{da_*}{dK}=\tfrac12\log^2(2a_*),
\]
and $K\cdot\frac12\log^2(2a_*)-c_\delta^{\mathrm{sp}}(K)=K\log(2a_*)-1>0$. Hence $\frac{d}{dK}\frac{c_\delta^{\mathrm{sp}}(K)}{K}>0$. The threshold of Lemma~\ref{lem:2.6} and \eqref{eq:2.threshold} is $14.9662$, that is, $6\pi^2v_{\mathrm{Cb}}(0)=14.96613\ldots$ rounded up. The equation $c_\delta^{\mathrm{sp}}(K)/(2K)=14.9662$ has the unique root $K=377.3635$, and $K_*:=377.38$ lies above it. Hence $c_\delta^{\mathrm{sp}}(K)/(2K)>14.9662$ for all $K\ge K_*$. Numerically, at $K=K_*$,
\[
\frac{c_\delta^{\mathrm{sp}}(K_*)}{2K_*}=14.96639,\qquad a_*=3321.43,\qquad \frac1{4a_*}=7.5\cdot10^{-5}.
\]

\emph{The computation behind \eqref{eq:2.cdelta}} (proof of Lemma~\ref{lem:2.6}). We put $\nu:=\astar\beta$ in Lemma~\ref{lem:2.4}(iv). Then
$\nu^2\eps/2=\astar^2\beta/(2K)$. Let $h(\mu):=(\beta/2)^\mu/\Gamma(\mu+1)$ and
$\mu_n:=(n^2+\nu^2)^{1/2}\ge\nu$. From the series
$I_\mu(\beta)=\sum_{k\ge0}(\beta/2)^{\mu+2k}/(k!\,\Gamma(\mu+k+1))$ and
$\Gamma(\mu+k+1)=\Gamma(\mu+1)\prod_{i=1}^k(\mu+i)\ge\Gamma(\mu+1)(\mu+1)^k$,
we get $I_\mu(\beta)\le h(\mu)\sum_{k\ge0}(\beta^2/(4(\mu+1)))^k/k!=h(\mu)e^{\beta^2/(4(\mu+1))}\le h(\mu)e^{\beta/(4\astar)}$ for
$\mu\ge\nu$. Since $\log\Gamma$ is convex, its difference quotients increase.
Thus for $\mu\ge\nu$
\[
\log\Gamma(\mu+1)-\log\Gamma(\nu+1)\ge(\mu-\nu)\bigl(\log\Gamma(\nu+1)-\log\Gamma(\nu)\bigr)=(\mu-\nu)\log\nu ,
\]
hence $h(\mu)\le h(\nu)(\beta/(2\nu))^{\mu-\nu}\le h(\nu)2^{-(\mu-\nu)}$, because
$\nu=\astar\beta\ge\beta$. As $\mu_n-\nu\ge0$ for $\abs{n}\le\nu$ and
$\mu_n-\nu\ge\abs{n}-\nu$ otherwise,
\[
\sum_{n\in\ZZ}I_{\mu_n}(\beta)\le e^{\beta/(4\astar)}h(\nu)\Bigl(2\nu+1+2\sum_{m>\nu}2^{-(m-\nu)}\Bigr)\le(2\astar\beta+5)\,e^{\beta/(4\astar)}h(\nu).
\]
Stirling's inequality $\Gamma(x+1)\ge(x/e)^x$ gives
$\log h(\nu)\le-\astar\beta(\log2\astar-1)$. Finally, for $\beta\ge\pi^{-2}$,
using $\cos\theta\ge1-\theta^2/2$ on $[0,\beta^{-1/2}]\subseteq[0,\pi]$, we obtain
$I_0(\beta)=\frac1\pi\int_0^\pi e^{\beta\cos\theta}d\theta\ge\frac{1}{\pi\sqrt\beta}e^{\beta-1/2}$.
Collecting these bounds, we obtain
\[
\log\delta\le-\beta\Bigl[\astar(\log2\astar-1)-\frac{\astar^2}{2K}+1-\frac{1}{4\astar}\Bigr]+\log(2\astar\beta+5)+\log(\pi\sqrt\beta)+\frac12 ,
\]
and $\astar^2/(2K)=\frac12\astar\log2\astar$ by the definition of $\astar$, which
gives \eqref{eq:2.cdelta}, with exactly the function $c_\delta^{\mathrm{sp}}(K)$ and the
correction $1/(4a_*)$ of \eqref{eq:D.csp}.

The term $O(\log\beta/\beta)$ is absorbed in the term $O(\log\tau+\log r_0)$ of Lemma~\ref{lem:5.16}. The margins $c_\delta^{\mathrm{sp}}(K)/(2K)-14.9662$ that appear in $\eta_T$ are:
\[
\begin{array}{c|cccc}
K & 377.38 & 10^3 & 3\cdot10^3 & 10^4\\ \hline
c_\delta^{\mathrm{sp}}(K)/(2K) & 14.9664 & 19.521 & 25.289 & 32.372\\
\text{margin} & 1.9\cdot10^{-4} & 4.55 & 10.32 & 17.41
\end{array}
\]

\emph{Numerical values.} To more digits, $a_*(K_*)=3321.434$ and $c_\delta^{\mathrm{sp}}(K_*)/(2K_*)=14.966392$; moreover $1/(8K_*a_*)=9.97\cdot10^{-8}$, and the root is $377.363524$, or $377.363533$ with the correction $-1/(4a_*)$. The correction $1/(4a_*)$ in \eqref{eq:D.csp} moves $c_\delta/(2K)$ by $10^{-7}$ and the root by less than $10^{-4}$. The inequalities $\Gamma(\mu+1)(\mu+1)^k/\Gamma(\mu+k+1)\le1$ and $I_\mu(\beta)\le h(\mu)e^{\beta^2/(4(\mu+1))}$ proved in the series step hold numerically on a grid of $\mu\in[0,10^4]$, $k\le59$, $\beta\in[0.1,100]$, and at $K=K_*$ and $\beta=0.2,1,3$ the bound on $\log\delta$ before the last two steps of the computation above lies below the right side of \eqref{eq:2.cdelta}. The entropy exponent of Lemma~\ref{lem:5.16} is $6\pi^2v_{\mathrm{Cb}}(0)<6\cdot2.4943551=14.96613\ldots<14.9662$, so the threshold is computed without the rounding of the Debye exponent up to $2.4944$ used elsewhere; with the rounded exponent the threshold is $14.9664$, whose root $377.3807$ exceeds $K_*$.

\subsubsection{The string tension and the rounding rule}\label{ssec:D.4}

The slit-domain tension of Theorem~\ref{thm:7.61} is $\sigma_V=\varepsilon_E\sigma_F=(c_g'\tilde mL/9216)\tilde m/\tau$, with $c_g'=c_g(1+1/L_c)^{-2}$ and with $\tilde mL>c_L-2r_0\tilde m$ by Section~\ref{ssec:3.1}. For $L_c\ge35$, one of the conditions of Section~\ref{ssec:D.5}(e), we have $c_L\tilde m^{-1}\ge35r_0$, and hence $c_g'\tilde mL\ge(36/35)^{-2}(1-2/35)\,c_gc_L\ge0.89\,c_gc_L\ge(9/16)c_gc_L$. In \eqref{eq:8.sigma}, $18432\cdot16/9=32768=2^{15}$ exactly and $(8\pi^2)^{1/2}/2^{16}=1.3559\cdot10^{-4}$. We take $c_g=1/4$ and $c_L=8.0\cdot10^{-12}\cdot2.0\cdot10^{-8}=1.6\cdot10^{-19}$, as in items~9 and~11, although any smaller admissible $c_L$ also works; at these values \eqref{eq:8.sigma} gives $\sigma(\beta)\ge5.4\cdot10^{-24}(K\beta)^{-1/2}e^{-2.49436K\beta}$.

Throughout, and in particular in Section~\ref{ssec:8.1}, we round as follows. The Debye exponent is $\pi^2v_{\mathrm{Cb}}(0)=2.49435508\ldots<2.4943551$. In a lower bound on $\tilde m$, and hence on a tension, the exponent is rounded up, to $2.49436$, and prefactors are rounded down, to $1.35\cdot10^{-4}$. In an upper bound on a scale the exponent is rounded up to $2.4944$: since $\tilde m\ge m_D/2$, $\log(1/\tilde m)\le2.4943551\tau-\frac12\log(8\pi^2\tau)+\log2\le2.4944\tau$ for all $\tau\ge1$. We use this form for $K_{\mathrm{col}}>27\cdot2.4944/(4\pi^2)$, in the logarithmic step of the proof of Lemma~\ref{lem:6.19}, and in the bound $\log(1/\tilde m)/\beta_{\mathrm{GM}}\le2.4944/(4\pi^2)=0.063184<0.0632$ of Remark~\ref{rem:7.62}. The bounds that determine $K_*$ use the exponent without rounding. There the entropy $6\log R_*$ of a root and a part in Lemma~\ref{lem:5.16}, with $\ell\approx\tilde m^{-1}/r_0$ and with $R_*$, $r'$ and $R'$ of Section~\ref{sec:mincut}, is bounded with $\pi^2v_{\mathrm{Cb}}(0)<2.4943551$, as in Sections~\ref{ssec:D.3} and~\ref{S:const5} (for the threshold with the rounded exponent, see the end of Section~\ref{ssec:D.3}). Where the exponent serves as a lower bound on $\log(1/\tilde m)$, it is rounded down: the hypothesis of Lemma~\ref{lem:6.19} has $2.494\tau$, because it is used to bound $\tilde m\le2m_D\le17.8\,\tau^{1/2}e^{-2.494\tau}$ from above.

\subsubsection{The conditions defining $\beta_0(K)$}\label{ssec:D.5}

We fix the constants of items~0 to~15. Each condition below is a lower bound on $\tau=K\beta$ that remains true for all larger $\tau$. Indeed, in each condition the left side is a fixed function of $\tau$ that tends to its limit monotonically for large $\tau$ and is built from powers of $\tilde m$, of $\tau$, of $\log\tau$ and of the doubly exponential scale $R_{\mathrm{NF}}$, and hence the set of admissible $\tau$ contains a half-line. Then $\beta_0(K)$ is $\tau_0/K$ for the largest of the thresholds $\tau_0$.
\begin{enumerate}
\item[(a)] \emph{Section~\ref{sec:mincut}.} The conditions used in Proposition~\ref{prop:5.13}, Lemma~\ref{lem:5.14}, Proposition~\ref{prop:5.15}, Lemma~\ref{lem:5.25}, Proposition~\ref{prop:5.27}, Lemma~\ref{lem:5.35}, Proposition~\ref{prop:5.36}, Lemmas~\ref{lem:5.37}, \ref{lem:5.38} and Theorem~\ref{thm:5.1} are
\begin{gather*}
\varepsilon_c\le\tfrac12,\qquad\eta_T^{(1)}\le1,\qquad\varepsilon''\le\tfrac12,\qquad\eta_T\le1,\qquad 2^{29}\cdot26e\cdot\varepsilon''^{1/(3C_{\mathfrak B})}\le\tfrac12,\\
2\big(2\lceil R'/L_c\rceil+5\big)^3\,\varepsilon''^{\,1/(48\varepsilon_D(1+8L_c)(\Cst+3C_{\mathrm{resp}}+6))}\le\tfrac12,\\
8\varepsilon_D(1+8L_c)(\Cst+3C_{\mathrm{resp}}+6)(48C_{\mathrm{resp}}+2)\le1\quad\text{(that is, \eqref{eq:5.16})},\\
R_{\mathrm{NF}}\ge\max\big(3456\bar r_0L_c,\ 2048\bar r_0L_c/(c_{\mathrm{sh}}\theta_{\mathrm{NF}}F_{\mathrm{unit}})\big).
\end{gather*}
Here $\varepsilon_D=3.2768\cdot10^7\bar r_0L_c/(c_{\mathrm{sh}}\theta_{\mathrm{NF}}F_{\mathrm{unit}}R_{\mathrm{NF}})$ is doubly exponentially small. The factor $2^{29}$ contains the factor $2^b$ for the sizes of the components in Lemma~\ref{lem:5.38}. The binding condition is the first of \eqref{eq:5.17}, $|\log\varepsilon''|\ge75.16\,C_{\mathfrak B}\approx1.1\cdot10^{14}$, while $\eta_T\le1$ needs about $1.7\cdot10^{13}$ and $\eta_T^{(1)}\le1$ needs about $3.6\cdot10^8$; all three values are computed in Section~\ref{S:const5}.
\item[(b)] \emph{Section~\ref{sec:stage1}.} The conditions are $\tilde m\le K_0(2C_H)^{-1/3}$; $M_Y\le m_W$ and $\tilde m\le\min(1/68,m_W/2)$ for Theorem~\ref{thm:6.12}, with $m_W=\min(1,c_{\mathrm{ex}}/2)$ of Proposition~\ref{prop:6.11}; $M_Y\le\frac15$, for Lemma~\ref{lem:F.7} on $\Lambda^\circ$ in the proof of Theorem~\ref{thm:6.16}; $m_D/2\le\tilde m\le2m_D$; the coupling condition of Lemma~\ref{lem:6.19}, $\tau\ge2$ and $2.494\tau\ge\ln(C^{\mathrm{dep}}_*/\theta_0)+2.5\ln\tau$, whose exponent is rounded as in Section~\ref{ssec:D.4}; and the threshold $\beta_{\mathrm{GM}}\ge\bGMo(\lambda_{\mathrm{PV}},\kappa_S,\mu,C,\delta_{\mathrm{GM}})$ of Theorem~\ref{thm:6.5}.
\item[(c)] \emph{Section~\ref{sec:stage2} and Appendices~\ref{app:A} to~\ref{app:C}.} The conditions are the slack in $\varepsilon_\nu+\theta'\le1/7$ and $c_E\le1/70$, which needs the terms $\beta_{\mathrm{GM}}M_Y$ and $e^{-\mu}$, $e^{-2\kappa_S}$ to be small; $\varepsilon(L')\le\varepsilon_0$ of Corollary~\ref{cor:7.21}; ($\beta1$) and ($\beta3$) of \eqref{eq:A.beta}; the cell-diameter conditions of Lemma~\ref{lem:7.47} and Proposition~\ref{prop:6.10}, of the form $2+C(k_\square/\chi)\log k_\square<L'$; the conditions of Proposition~\ref{prop:7.41} and Theorem~\ref{thm:7.42}, the latter a condition $0.4157\,\tau\ge C_{\mathfrak D}A_1^D\log(C'\tau/\theta_0)+\alpha+O(\log\tau)$; $c_{16}e^{c_2}\le1/8$ of Lemma~\ref{lem:A.11}; the conditions of Lemma~\ref{lem:7.53}; $\Theta_4\le\frac14$ of Proposition~\ref{prop:7.57}; and $F_{\mathrm{unit}}\ge(8C'/c')(1+\ln\beta_{\mathrm{GM}})$ of Proposition~\ref{prop:7.56}.
\item[(d)] \emph{Section~\ref{sec:8}.} The condition is $\eta_T+\eta_S\le\frac12$. Both deficits are explicit upper bounds that are eventually decreasing in $\tau$: $\eta_T=O_K(1/\tau)$, because, by Lemma~\ref{lem:5.16}, $|\log\varepsilon''|\ge\gamma_T(K)\tau-O(\log\tau+\log r_0)$ with $\gamma_T(K)>0$ for $K>K_*$, and $\eta_S=O(\tilde m\beta_{\mathrm{GM}}\log\tau)$.
\end{enumerate}
\begin{enumerate}
\item[(e)] \emph{Rounding of $L_c$ in Section~\ref{sec:3}.} The conditions are $L_c\ge4$, which implies $c_g'\ge(16/25)c_g\ge(9/16)c_g$ for Lemma~\ref{lem:4.17}; $L_c\ge35$, used for $\lambda=12/L_c\le1$ in Proposition~\ref{prop:4.39}, in Lemma~\ref{lem:7.47} and for $c_g'\tilde mL\ge(9/16)c_gc_L$ in Section~\ref{ssec:D.4} and Section~\ref{ssec:8.1}; and $c_L\tilde m^{-1}\ge4r_0$, which gives $\tilde mL\ge c_L/2$ (the bounds on $c_u$ and on $C_{\mathcal U}$, Lemma~\ref{lem:4.19}).
\item[(f)] \emph{Section~\ref{sec:2}.} The conditions are $\beta\ge\pi^{-2}$, for Lemma~\ref{lem:2.6}; \eqref{eq:2.threshold}, that is, $\beta\ge\beta_1(K)$; and $\delta<1$, which follows from \eqref{eq:2.cdelta} for all large $\beta$ and is used in Step~\eqref{eq:8.1}, with $\delta'=\delta$ in Proposition~\ref{prop:2.9}.
\end{enumerate}
In addition, the range of Remark~\ref{rem:8.3}, where Phase~2 is empty, is nonempty once $\sqrt3(r_0(\ell+3)+1)\le R_{\mathrm{NF}}$. This condition belongs to that remark rather than to Theorem~\ref{thm:1.2}, and it holds for all large $\tau$ because $r_0\ell\approx\tilde m^{-1}$ is exponential and $R_{\mathrm{NF}}$ is doubly exponential in $\tau$.

All these thresholds are finite but very large (Section~\ref{app:D}).