% \providecommand{\bGM}{\beta_{\mathrm{GM}}} \providecommand{\mt}{\tilde m} \providecommand{\lPV}{\lambda_{\mathrm{PV}}} \providecommand{\Gp}{G'} \providecommand{\Dep}{\mathfrak{D}} \providecommand{\bD}{b_{D}} \providecommand{\qpin}{q_{\mathrm{pin}}} \providecommand{\Kcell}{K_{\mathrm{cell}}} \providecommand{\cU}{\mathcal{U}} \providecommand{\eps}{\varepsilon} \section{Brydges\textendash Federbush Section 9.8 bookkeeping on slit domains} \label{app:B} In this appendix we prove Lemma~\ref{lem:7.40} and Proposition~\ref{prop:7.41}, the bookkeeping of \cite[\S9.8, (9.81) to (9.824)]{BF80} on $\Gp$, again in condensed form (the complete proofs are in Section~\ref{S4-S:S4.7} of the supplement). In Debye units ($\bD:=\bGM\mt$) each field derivative carries $\bD^{1/2}$ and each coefficient $1/\bD$. A lone cell is of class A, and by the minimality of \cite[(E.11)]{GM82} it contains a link end or a Mayer cube, which gives it net weight $\ge\frac16$. Cells of $X_{\Sigma}$ are paid by $F_{1}^{\mathrm{loc}}$. We control factorials by pinning. Since $\Ord((1+r)^{3})$ cells lie within distance $r$ of a cell, a sum of $N$ distances grows like $N^{4/3}$, which outweighs $\log((N!)^{\qpin})$. \paragraph{Setting.} Cells are $\cU$-cells, grid cells are denoted $\Omega$, and $\mathrm{vol}_{D}(\Delta):=\mt^{3}\#\Delta\in[c_{u},C_{u}]$. Classes A, B and $X=Z\sqcup X_{\Sigma}\sqcup X_{o}$ are those of Section~\ref{sec:7.6} (seed cells in wall polymers count with $X_{\Sigma}$). On a class-A cell $\vartheta=\psi$ modulo $P\ZZ$ in each phase. In the seeded activity of Section~\ref{sec:7.6}, $X_{i}:=Y_{1}\cup\dots\cup Y_{i}$ and $\kappa(\bar y,s)=\kappa(n-1)\cdots\kappa(1)$ as in \cite[(E.9), (E.10)]{GM82} and Appendix~\ref{app:C}. When its brackets are expanded, $\kappa(i)$ is of one of five \emph{kinds}: K0 $=\partial_{s_{i}}E^{(i)}$, K1 $=\dot C\partial_{x}\partial_{y}$, K2 $=\dot C\partial_{x}(\partial_{y}E)$, K3 $=\dot C(\partial_{x}E)\partial_{y}$ and K4 $=\dot C(\partial_{x}E)(\partial_{y}E)$. Here $\partial_{x}:=\partial/\partial\psi(x)$, $x\in Y_{i+1}$ and $y\in X_{i}$. An \emph{$\mathcal{E}$-factor} is one Mayer term of $E(X,s)$ of the forms \cite[(E.5a) to (E.5d)]{GM82}, with its coefficient $\rho_{t}$ and its $t$ factors. Derivatives act on the factors to the right of the differentiation, as in \cite[(E.13) to (E.16)]{GM82}, which is the rule of \cite[p.~229]{BF80}. The $\psi$-dependent \emph{atoms} of $Q^{\natural}(X)$ are: (a1) $r_{\Omega}$ of \eqref{eq:7.rOmega}, for each grid cell $\Omega\subseteq X$; (a2) $e^{G_{2}(\Omega)}$, $G_{2}(\Omega):=G_{2,1}(\Omega)+G_{2,2}(\Omega)$; (a3) $e^{g_{z}^{\Dep}}$, $z\in X\cap\Dep$; (a4) $e^{g_{z}^{\natural}}$, $z\in X\setminus\Dep$; (a5) $e^{-F_{2}(X)}=e^{-\langle1_{X}\psi,F_{2}'\rangle}$. Here $g_{z}^{\Dep}$, $g_{z}^{\natural}$ are the $z$-terms of $G_{2,3}^{\Dep}$, $G_{2,3}^{\natural}$ and, pairing $m$ with $-m$ by (S3), \begin{equation}\label{eq:B.G2} G_{2,1}(\Omega)=\sum_{z\in\Omega}\sum_{m}\rhoref(m)f_{2}(m\bGM^{1/2}\tilde\delta(z)),\qquad G_{2,2}(\Omega)=\sum_{z\in\Omega}\sum_{m}\rhoref(m)v_{m}(\Omega)f_{1}(m\bGM^{1/2}\tilde\delta(z)), \end{equation} with $f_{1}(y):=e^{iy}-1-iy$, $f_{2}(y):=\cos y-1+\frac12y^{2}$ and $v_{m}(\Omega):=e^{im\bGM^{1/2}\mathrm{avg}_{\Omega}\vartheta}-1$. Thus $G_{2,1}$ is of fourth order in $\tilde\delta$. The factors of an $\mathcal{E}$-factor are $\eps(\xi)=e^{im\bGM^{1/2}(\vartheta(x)+h(x))}-1$ in (E.5a), $\tilde\eps(\xi):=\eps(\xi)-im\bGM^{1/2}\psi(x)$ in (E.5b) to (E.5d), and the \emph{$\psi$-factor} $im\bGM^{1/2}\psi(x)$ in (E.5c), (E.5d). \begin{lemma}[Debye units]\label{lem:B.1} Put $d\hat x:=\mt^{3}\times$ counting measure on $V$, $\hat\psi:=\mt^{-1/2}\psi$, $\delta/\delta\hat\psi(\hat x):=\mt^{-5/2}\partial/\partial\psi(x)$, $\hat C(\hat x,\hat y):=\mt^{-1}C'_{s}(x,y)$ (likewise for $\partial_{s_{i}}C'_{s}$), and $\hat c:=\mt^{-3}c$ for a coefficient $c(x)$ per site. Then (a) $\sum_{x,y}K(x,y)\partial_{\psi(x)}\partial_{\psi(y)}=\iint d\hat x\,d\hat y\,\mt^{-1}K(x,y)\frac{\delta}{\delta\hat\psi(\hat x)}\frac{\delta}{\delta\hat\psi(\hat y)}$; (b) if $F(\psi)=\sum_{z}c(z)f(\bGM^{1/2}\psi(z))$, then $\delta F/\delta\hat\psi(\hat x)=\hat c(\hat x)\,\bD^{1/2}f'(\bD^{1/2}\hat\psi(\hat x))$: each derivative of a function of $\bGM^{1/2}\psi$ produces exactly $\bD^{1/2}$; (c) $|\hat C(\hat x,\hat y)|\le(2K_{1}''/\lPV)e^{-\rcov d(x,y)}$, $\hat C(\hat x,\hat x)\le\alpha_{D}K_{1}/\lPV$ and $\sup_{\hat x}\int d\hat y\,|\hat C(\hat x,\hat y)|e^{\rcov d(x,y)}\le2K_{3}$; (d) for a cell $\Delta$, $k\ge2$ and $C_{k}\le Ck!(2\kappa_{S})^{-k}$: (M1) $\int_{\Delta}d\hat x\sum_{m}|\hat\rho^{\mathrm{ref}}(m)||m|^{k}\le C_{k}\mathrm{vol}_{D}(\Delta)/\bD$; (M2) $\int_{\Dep\cap\Delta}d\hat x\sum_{m}|\hat\rho_{1}-\hat\rho^{\mathrm{ref}}||m|^{k}\le C_{k}\cdep$; (M3) $\int_{\Delta\setminus\Dep}d\hat x\sum_{m}|\hat\rho_{1}-\hat\rho^{\mathrm{ref}}||m|^{k}\le C_{k}\theta'\mathrm{vol}_{D}(\Delta)/\bD$; (M4) for $t\ge2$, a set $\mathfrak a$ of cells and a cell $a_{*}\in\mathfrak a$, \[ \frac1{t!}\int_{\Xi_{t}(\mathfrak a)}d\hat x_{1}\cdots d\hat x_{t}\sum_{m_{1},\dots,m_{t}}|\hat\rho_{t}|\,e^{2\kappa_{S}\sum_{i}|m_{i}|}\le t\,K_{M4}C_{M4}^{\,t}\,\frac{4\,\mathrm{vol}_{D}(a_{*})}{\bD}\,e^{-\mu(t-1)}e^{-(1-\delta_{\mathrm{GM}})(A/\mt)\hat L(\mathfrak a)}, \] and hence, for all integers $k_{1},\dots,k_{t}\ge0$, the left side with $\prod_{i}|m_{i}|^{k_{i}}$ in place of $e^{2\kappa_{S}\sum_{i}|m_{i}|}$ is at most $\prod_{i}k_{i}!(2\kappa_{S})^{-k_{i}}$ times the right side. Here $\Xi_{t}(\mathfrak a)$ is the set of $(x_{1},\dots,x_{t})\in V^{t}$ such that $\mathfrak a$ is the set of the cells that contain some $x_{i}$, $A$ is the rate of (S2), $\hat L(\mathfrak a):=\mt L_{\Gp}(\mathfrak a)$ is the tree length of (S2) of the cells of $\mathfrak a$, each listed once, in Debye units, $C_{M4}:=(1-C)^{-1}$ per point and $K_{M4}:=e^{2\kappa_{S}}(1-C)^{-10}$, where $C$ is the parameter of (S2), fixed in $(0,\frac14]$ (item~(12) of Section~\ref{app:D}). The left side is the sum over all assignments of the $t$ points to the cells of $\mathfrak a$ that use every cell; it contains $\int_{a_{1}}\cdots\int_{a_{t}}$ for each $t$-tuple of cells with $\{a_{1},\dots,a_{t}\}=\mathfrak a$. \end{lemma} \begin{proof} Parts (a) and (b) follow from the change of variables, since $\mt^{-5/2}\bGM^{1/2}=\mt^{-3}\bD^{1/2}$. Part (c) is Theorem~\ref{thm:6.12} for $C'\le\alpha_{D}C_{0}$, which holds by Lemma~\ref{lem:7.31}, with $\MY/\mt=1/\lPV$ and entrywise domination of $C'_{s}$. Bound (M1) follows from $\bGM\sum_{m}m^{2}\rhoref(m)=\mt^{2}$ and (S4$'$). Bounds (M2) and (M3) follow from Propositions~\ref{prop:6.18}, \ref{prop:6.6} and \eqref{eq:7.6.mom}. We derive (M4) from (S2). As $e^{2\kappa_{S}\sum_{i}|m_{i}|}=e^{2\kappa_{S}}e^{2\kappa_{S}(\sum_{i}|m_{i}|-1)}$, the left side is $e^{2\kappa_{S}}\frac1{t!}\sum_{x\in\Xi_{t}(\mathfrak a)}\sum_{m}|\rho_{t}|e^{2\kappa_{S}(-1+\sum_{i}|m_{i}|)}$, since in Debye units $\int d\hat x\,|\hat\rho|=\sum_{x}|\rho|$. Each $x\in\Xi_{t}(\mathfrak a)$ has a coordinate in $a_{*}$. As $\rho_{t}$, the weight and $\Xi_{t}(\mathfrak a)$ are symmetric, the sum is at most $t$ times its part with $x_{1}\in a_{*}$. For $x\in\Xi_{t}(\mathfrak a)$, $L_{\Gp}(x_{1},\dots,x_{t})\ge L_{\Gp}(\mathfrak a)$. Indeed, in a tree for $x_{1},\dots,x_{t}$ we keep one marked vertex in each cell of $\mathfrak a$ and declare the others auxiliary. Hence the part is at most $e^{-(1-\delta_{\mathrm{GM}})AL_{\Gp}(\mathfrak a)}$ times the left side of \eqref{eq:6.S2} with $s=t$ and $a_{1}=a_{*}$, and (S2), $(1-C)^{-t-10}=(1-C)^{-10}C_{M4}^{\,t}$ and $|a_{*}|\mD^{2}/\bGM\le4\,\mathrm{vol}_{D}(a_{*})/\bD$, which holds as $\mD\le2\mt$, give (M4). Here the sum over the $|\mathfrak a|!\,S(t,|\mathfrak a|)$ assignments ($S$ as in the proof of Lemma~\ref{lem:A.7}) is a part of the sum over all ordered tuples of points in (S2), and the decay is read off the weight on that part, so the sum over these assignments is bounded by (M4) without a factorial. The form with $\prod_{i}|m_{i}|^{k_{i}}$ follows from $|m|^{k}\le k!(2\kappa_{S})^{-k}e^{2\kappa_{S}|m|}$. \qed \end{proof} Thus the expansion is as in \cite[\S9.8]{BF80}, with $\tilde l_{D}=1$, $\beta\to\bD$, on a fine lattice with a regularized covariance, and infinitely many charges only produce the $k!$ in (d). A factor $f$ has \emph{weight} $w$ if $|f|\le K\bD^{w}\cdot(\text{field monomial})\cdot(\text{numerical factors})$ for all positions in their cells and all fields, with $K$ uniform in $\bD$, $L'$ and $\tau$. Here a field monomial is a product of values $|\hat\psi(\hat x)|$ and averages $\mathrm{avg}_{\Omega}|\hat\psi|$, paid by the Wick bound $T_{1}$ (the Wick bound gives a constant $c_T$, independent of $\bD$, $L'$, $\tau$, per field factor, together with the factorials $\prod_jn_j!$ counted in Lemma~\ref{lem:B.9}). A numerical factor is $|\delta\hat D(\hat x)|:=|\hat D(\hat x)-\mathrm{avg}_{\Omega}\hat D|$ or $|\hat F_{2}'(\hat x)|$, with $\hat D:=\mt^{-1/2}D$, $\hat F_{2}':=\mt^{-5/2}F_{2}'$. The weight of a term is the sum of those of its factors. \begin{lemma}\label{lem:B.2} Expand every derivative of $\kappa(\bar y,s)$ by the Leibniz rule, as in \cite[(9.84), (9.85)]{BF80}. (a) $\kappa(\bar y,s)Q^{\natural}(X)$ is a finite sum of terms, each labelled by the kinds, the $\mathcal{E}$-factors, and a map $l$ that sends each derivative to one atom or one factor of an $\mathcal{E}$-factor which depends on the derivative's point and lies to its right; the term is $[\prod_{i}\partial_{s_{i}}C'(x_{i},y_{i}|s)]\prod_{a}\partial^{l^{-1}(a)}a\prod_{e}[\text{coefficient of }e\cdot\prod_{\phi\in e}\partial^{l^{-1}(\phi)}\phi]$. (b) The multiple derivatives are of the types: i) $\partial^{S}r_{\Omega}$; ii) from an observable (absent here); iii) $\partial^{S}e^{-F_{2}}=(-1)^{|S|}\prod_{j\in S}F_{2}'(x_{j})e^{-F_{2}}$; iv) $\partial^{S}\eps$, $\partial^{S}\tilde\eps$, $\partial^{S}(im\bGM^{1/2}\psi)$; v), vi) the Fa\`a di Bruno blocks $\rhoref(m)\partial^{B}[v_{m}f_{1}]$ and $\rhoref(m)\partial^{B}[f_{2}]$ of $\partial^{S}e^{G_{2}(\Omega)}$; vii), viii) the Fa\`a di Bruno blocks of $\partial^{S}e^{g_{z}^{\Dep}}$ and $\partial^{S}e^{g_{z}^{\natural}}$. (c) With $n_{\alpha}$ link ends, $m_{\alpha}$ factors of $\mathcal{E}$-factors and $w_{\alpha}$ ($=0$ here) factors of an observable in the cell $\Delta_{\alpha}$, the number of maps $l$ is at most $\prod_{\alpha}(m_{\alpha}+w_{\alpha}+5)^{n_{\alpha}}$, and for $c_{p}>0$, $n\ge1$ and integers $m,w\ge0$, $(m+w+5)^{n}\le C_{r}^{n+w}n!\,e^{c_{p}m}$ with $C_{r}:=4e^{5}\max(1,1/c_{p})$. \end{lemma} \begin{proof} The lemma extends \cite[(9.84) to (9.87), Lemma~9.10]{BF80} by two new types, vii) and viii). A bare derivative at $x\in\Delta_{\alpha}$ can hit $r_{\Omega(x)}$, $e^{G_{2}(\Omega(x))}$, $e^{g_{x}^{\Dep}}$ or $e^{g_{x}^{\natural}}$, $e^{-F_{2}}$, or one of the $m_{\alpha}+w_{\alpha}$ factors in $\Delta_{\alpha}$. The count therefore has $+5$ in place of the $+3$ of \cite[(9.86)]{BF80}. The undifferentiated exponentials recombine into $e^{G^{\natural}(X)}$ and $e^{-F_{2}}$, which stay in $T_{4}$ and $T_{2}$ of Proposition~\ref{prop:7.37}. We bound each $r_{\Omega}$ by Lemma~\ref{lem:7.33}, whose factor is included in $T_{3}$. The last inequality follows from $(m+w+5)^{n}\le2^{n}(m^{n}+(w+5)^{n})$, $m^{n}\le n!c_{p}^{-n}e^{c_{p}m}$ and $(w+5)^{n}\le n!e^{w+5}$. The factor $n!$ is controlled by the pinning of Lemma~\ref{lem:B.4}. Within a fixed $l$ the further expansions, namely Fa\`a di Bruno, the Leibniz rule inside $v_{m}f_{1}$, the two parts of $\partial\tilde\delta(z)/\partial\psi(x)$ and the charge sums of Lemma~\ref{lem:B.1}(d), cost at most $C^{n_{\alpha}}(n_{\alpha}!)^{2}$ per cell, and $e^{c_{p}\sum_{\alpha}m_{\alpha}}=\prod_{e}e^{c_{p}t_{e}}$ is a constant per $\mathcal{E}$-factor. \qed \end{proof} \begin{lemma}[Witness]\label{lem:B.3} (a) \emph{Link localization.} Let $1\le i\le n-1$ and let $Y\in\tilde Y(h)$ with $Y\subseteq Y_{i+1}$. Then $Y$ contains the point $x_{i}$ of the bare derivative of $\kappa(i)$ (kinds K1, K2), or a cube of an $\mathcal{E}$-factor of $\kappa(i)$. (b) Every lone cell is of class A, and $F_{2}'=0$ on it. (c) Let $\Delta\in X_{o}$ with $\Delta\subseteq Y_{i+1}$, $i\ge1$. Then (W1) the bare derivative of $\kappa(i)$ sits at $x_{i}\in\Delta$ and hits an atom of kind (a1) to (a4) of $\Delta$, or (W2) an $\mathcal{E}$-factor of $\kappa(i)$ has a cube equal to $\Delta$. \end{lemma} \begin{proof} (a) By \cite[(E.11)]{GM82}, $Y_{i+1}$ is the smallest union of elements of $\tilde Y(h)$ containing $\mathcal{O}-X_{i}$, where $\mathcal{O}$, the support of the term of $\kappa(i)$, consists of the cubes of its $\mathcal{E}$-factors and the cube of a bare $x_{i}$ if there is one. An element $Y\subseteq Y_{i+1}$ meeting no point of $\mathcal{O}-X_{i}$ would make $Y_{i+1}\setminus Y$ a smaller such union. As cells are $\cU$-cells, ``meets'' is ``contains''. (b) We have $\mathrm{supp}\,D_{\beta}$, $\mathrm{supp}\,F_{2,\beta}'\subseteq S_{\beta}$ by Proposition~\ref{prop:7.9}, and a lone cell lies in no wall polymer. (c) By (a), $\Delta$ contains $x_{i}$, or it contains a cube of an $\mathcal{E}$-factor of $\kappa(i)$, which is case (W2). In the first case the bare derivative hits a factor to its right that depends on $\psi(x_{i})$. An atom (a1) to (a4) at $x_{i}$ gives (W1). The atom (a5) and the derivatives of $e^{-F_{2}}$ give $F_{2}'(x_{i})=0$ by (b). A factor at $x_{i}$ of the $\mathcal{E}$-factor of the $y$-bracket of $\kappa(i)$ (kind K2) gives (W2). All other factors are functions of $\psi$ on $X_{i}$, which is disjoint from $Y_{i+1}$. \qed \end{proof} For each $\Delta\in X_{o}$ we fix one witness, (W1) if available and otherwise an $\mathcal{E}$-factor $e$ as in (W2). Let $W(e)$ denote the set of lone cells witnessed by $e$. Each lone cell lies in exactly one $Y_{i+1}$, and by (W2) of Lemma~\ref{lem:B.3}(c) a witness $e$ of a lone cell $\Delta\subseteq Y_{i+1}$ is an $\mathcal{E}$-factor of $\kappa(i)$. It follows that all cells of $W(e)$ lie in $Y_{i(e)+1}$, where $\kappa(i(e))$ is the link of $e$. Each $\kappa(i)$ has at most two $\mathcal{E}$-factors. The cells of $\Gp$ have polynomial growth: $\#\{\Delta:\mathrm{dist}(\Delta_{0},\Delta)\le r\}\le K_{g}(1+r)^{3}$ in Debye units, for the Euclidean distance between cells and $K_{g}:=(4\pi/3)(1+2C_{\cU})^{3}/c_{u}$, by (R7), $\mathrm{vol}_{D}\ge c_{u}$ and $\mt\le1$ (the sites of the cells lie in a ball of radius $r+2C_{\cU}+\sqrt3\mt/2\le(1+r)(1+2C_{\cU})$). Links have one end in $Y_{i+1}$ and one in $X_{i}$, and the sets $Y_{i+1}$ are pairwise disjoint. \begin{lemma}\label{lem:B.4} (a) Let $\Delta_{0}$ be a cell and $\Delta_{1},\dots,\Delta_{N}$ cells, each repeated at most twice, with $r_{k}:=\mathrm{dist}(\Delta_{0},\Delta_{k})$ (Euclidean, in Debye units). For $q\ge1$ and $c_{p}>0$, $(N!)^{q}\le c_{N}^{N}\exp(c_{p}\sum_{k}r_{k})$ with $c_{N}:=e^{c_{p}}\max(1,3q/(c_{p}a_{g}))^{3q}$, $a_{g}:=\frac34(2K_{g})^{-1/3}$. (b) Let $d:=\sum_{i}\mathrm{dist}(\Delta_{i}',\Delta_{i}'')$ be the total length of the links, $\hat L^{E}(e)$ the Euclidean Steiner length, in Debye units, of the cells of the $\mathcal{E}$-factor $e$ (the least Steiner length of points chosen in its cells), and $\hat L_{0}:=\sum_{e}\hat L^{E}(e)$. Pin each $\mathcal{E}$-factor $e$ at a set $\Pi(e)$ of at most four of its cells (chosen in Rule~D, before Theorem~\ref{thm:B.6}), and let $N_{\alpha}:=\#\{e:\Delta_{\alpha}\in\Pi(e)\}$, in either class. Then, with $c=c(q,c_{p},K_{g},C_{\cU})$, $\prod_{\alpha}(n_{\alpha}!)^{q}\le c^{\sum n_{\alpha}}e^{c_{p}d}$ and $\prod_{\alpha}(N_{\alpha}!)^{q}\le c^{\#\mathcal{E}}e^{4c_{p}(\hat L_{0}+2d)}$. \end{lemma} \begin{proof} For (a) we argue as in \cite[Lemma~9.11, (9.820)]{BF80}, with the growth $K_{g}(1+r)^{3}$ in place of the lattice count of \cite[Appendix~2]{BF80}. In increasing order the distances satisfy $r_{(k)}\ge(k/(2K_{g}))^{1/3}-1$. Summing these bounds, we obtain $\sum_{k}r_{k}\ge a_{g}N^{4/3}-N$. Moreover, $qN\log N-c_{p}a_{g}N^{4/3}\le3qN\log^{+}(3q/(c_{p}a_{g}))$. (b) We first treat the derivatives. Since $\Delta_{\alpha}$ lies in exactly one $Y_{i+1}$, at most one $x$-end lies in $\Delta_{\alpha}$. All other ends in $\Delta_{\alpha}$ are $y$-ends $y_{j}$ of links $j$ with $\Delta_{\alpha}\subseteq X_{j}$, whose partners $x_{j}$ lie in $Y_{j+1}$, disjoint for distinct $j$. We apply (a) to the $y$-ends, with $n!\le2^{n}(n-1)!$. For the $\mathcal{E}$-factors, let $e$ be pinned at $\Delta_{\alpha}$, with link $i=i(e)$. If $\Delta_{\alpha}\subseteq Y_{i+1}$, there are at most two such $e$. Otherwise $\Delta_{\alpha}\subseteq X_{i}$, and $e$ reaches $Y_{i+1}$. For K0 this holds since $\partial_{s_{i}}E^{(i)}$ keeps only Mayer terms with cubes in $X_{i}$ and in $Y_{i+1}$. The $x$-side $\mathcal{E}$-factor of K3, K4 does so through a factor at $x_{i}$, and the $y$-side one of K2, K4 through a factor at $y_{i}$, which is joined to $x_{i}$ by link $i$. Thus $\mathrm{dist}(\Delta_{\alpha},Y_{i+1})\le\hat L^{E}(e)+2C_{\cU}+d_{i}$, as $\hat L^{E}(e)$ is a Steiner length over points of the cubes of $e$ and cells have Euclidean diameter at most $C_{\cU}$ in Debye units. The partner is the cell of $Y_{i+1}$ nearest to $\Delta_{\alpha}$. Since distinct links give disjoint $Y$'s, a partner is used only by $\mathcal{E}$-factors of a single link, hence at most twice (see the paragraph after Lemma~\ref{lem:B.3}). We apply (a). Each $e$ appears at most $|\Pi(e)|\le4$ times and each link at most twice, and the factor $e^{8c_{p}C_{\cU}}$ is absorbed into $c$. \qed \end{proof} With $c_{p}$ of Proposition~\ref{prop:7.38} and $r_{C}=\rcov/\mt$, the link decay $e^{-r_{C}\mathrm{dist}(\Delta_{i}',\Delta_{i}'')}$ of Lemma~\ref{lem:B.1}(c) pays the factor $e^{9c_{p}d}$ and leaves $e^{-(\bar\delta+(r_{C}-\bar\delta)/2)\mathrm{dist}}$. In the proof of Proposition~\ref{prop:B.7}, the Mayer decay of (M4) absorbs the factor $e^{4c_{p}\hat L_{0}}$ together with the factors $e^{c_{10}L_{\eta^{A}}}$ of $\kappa'$. This uses $\hat L\ge3^{-1/2}\hat L^{E}$ and, on outputs, $(1-\delta_{\mathrm{GM}})A/\sqrt3\ge\frac12\kappa_{Y}\MY\ge2r_{C}\mt>4c_{p}\mt$ for $\delta_{\mathrm{GM}}\le\frac12$ (Proposition~\ref{prop:6.11}). Finally, $c^{\#\mathcal{E}}\le c^{2|X|}$ is a constant per cell, as a link has at most two $\mathcal{E}$-factors and there are $n-1<|X|$ links (each $Y_{i+1}$ contains a $\cU$-cell). In \cite[(9.821)]{BF80}, $N_{\alpha}$ counts distinguished factors $\eps$ in class-A cubes, but the proof in \cite[Appendix~2]{BF80} does not use the class, and (b) pins $\mathcal{E}$-factors at cells of either class, as required by Proposition~\ref{prop:B.10}. As in \cite[(9.822)]{BF80}, we multiply each term by $1=f_{q}\cdot\prod_{\alpha}(n_{\alpha}!)^{q}(N_{\alpha}!)^{q}$, $f_{q}:=\prod_{\alpha}(n_{\alpha}!)^{-q}(N_{\alpha}!)^{-q}$, and bound the second factor by (b). We state the bounds below in the variables of Lemma~\ref{lem:B.1}, at fixed positions, for $k$ derivatives at points of the factor's cell (for site factors, at the site). Here $\hat\delta:=\mt^{-1/2}\tilde\delta$, $\hat\psi_{\Omega}:=\mathrm{avg}_{\Omega}\hat\psi$, and $v_{\Omega}\in[v_{-},v_{+}]$ is the Debye volume of a grid cell. The chain-rule factors $\delta(\hat\psi_{\Omega})/\delta\hat\psi(\hat x)=1_{\Omega}(\hat x)/v_{\Omega}$ and $\delta\hat\delta(\hat z)/\delta\hat\psi(\hat x)=\hat\delta(\hat x-\hat z)-1_{\Omega}(\hat x)/v_{\Omega}$, $\hat\delta(\hat x-\hat z):=\mt^{-3}1[x=z]$, are not shown. They are absorbed when the line end is integrated, or bounded by $1/v_{-}$. In class B, $|\hat\delta|\le|\hat\delta\hat\psi|+|\delta\hat D|$. A row of Table~\ref{tab:B.5} that carries an equation number of \cite{BF80} restates that bound in Debye units, with the charges and volumes shown, and the other rows are new. \begin{table}[tbp] \centering\footnotesize \begin{tabular}{p{0.30\textwidth}p{0.08\textwidth}p{0.40\textwidth}p{0.10\textwidth}} \hline\noalign{\smallskip} factor (\cite{BF80}) & $k$ & bound & weight\\ \noalign{\smallskip}\hline\noalign{\smallskip} v) $v_{m}(\Omega)$, any class (9.814) & $\ge1$; $0$ & $(|m|\bD^{1/2}/v_{-})^{k}$; $2$ & $k/2$; $0$\\ $v_{m}(\Omega)$, class A & $0$ & $|m|\bD^{1/2}|\hat\psi_{\Omega}|$ & $1/2$\\ vi) $f_{1}(m\bD^{1/2}\hat\delta)$ (9.815) & $0$; $1$; $\ge2$ & $\frac12m^{2}\bD|\hat\delta|^{2}$; $m^{2}\bD|\hat\delta|$; $(|m|\bD^{1/2})^{k}$ & $1$; $1$; $k/2$\\ $f_{1}$, bounded form & $1$ & $2|m|\bD^{1/2}$ & $1/2$\\ vi) $f_{2}(m\bD^{1/2}\hat\delta)$ & $\le4$; $\ge4$ & $m^{4}\bD^{2}|\hat\delta|^{4-k}$; $(|m|\bD^{1/2})^{k}$ & $2$; $k/2$\\ $f_{2}$, bounded form & $2$; $3$ & $2m^{2}\bD$; $|m|^{3}\bD^{3/2}$ & $1$; $3/2$\\ iv) $\tilde\eps$, class A (9.816) & $0$; $1$; $\ge2$ & $\frac12m^{2}\bD|\hat\psi|^{2}$; $m^{2}\bD|\hat\psi|$; $(|m|\bD^{1/2})^{k}$ & $1$; $1$; $k/2$\\ $\tilde\eps$, class B (9.818) & $0$; $1$ & $2+|m|\bD^{1/2}|\hat\psi|$; $2|m|\bD^{1/2}$ & $0$; $1/2$\\ $\psi$-factor, any class (9.816) & $0$; $1$ & $|m|\bD^{1/2}|\hat\psi|$; $|m|\bD^{1/2}$ (zero for $k\ge2$) & $1/2$; $1/2$\\ $\eps$, any class (9.817) & $\ge1$ & $(|m|\bD^{1/2})^{k}$ & $k/2$\\ $\eps$ undistinguished; distinguished, class A (9.819) & $0$ & $2$; $|m|\bD^{1/2}|\hat\psi|$ & $0$; $1/2$\\ vii) $\cos(m\bD^{1/2}\hat\vartheta)-1$ & $\ge1$ & $(|m|\bD^{1/2})^{k}$ & $k/2$\\ viii) $f_{2}(m\bD^{1/2}\hat\psi)$, class A & $\le4$; $\ge4$ & $m^{4}\bD^{2}|\hat\psi|^{4-k}$; $(|m|\bD^{1/2})^{k}$ & $2$; $k/2$\\ viii), class B & $1$; $2$; $\ge3$ & $|m|\bD^{1/2}+m^{2}\bD|\hat\psi|$; $2m^{2}\bD$; $(|m|\bD^{1/2})^{k}$ & $\frac12$; $1$; $k/2$\\ i) $r_{\Omega}$ (Lemma~\ref{lem:7.33}) & $\ge0$ & $c_{7}(c_{7}'\bD^{1/6}/v_{-})^{k}e^{c_{7}''k\log k}e^{\frac12\eta_{H}\mt^{2}\sum_{\Omega}\vartheta^{2}}$ & $k/6$\\ iii) $\hat F_{2}'$, class B only & & numerical factor & $0$\\ \noalign{\smallskip}\hline \end{tabular} \caption{Bounds per type (Lemma~\ref{lem:B.5}(a)).}\label{tab:B.5} \end{table} \begin{lemma}\label{lem:B.5} (a) With $C$ numerical and $|m|$ the charge of the factor, the factors satisfy the bounds of Table~\ref{tab:B.5}. (b) \emph{Vertex weights.} A vertex is a block $B$ on a $G_{2}$-, vii)- or viii)-atom with its coefficient, an $r_{\Omega}$ with all its derivatives, an $\mathcal{E}$-factor with its coefficient and factors, or one factor $\hat F_{2}'$. The coefficient weight is $-1$ for one-body, off-$\Dep$ and Mayer coefficients ((M1), (M3), (M4)) and $-\frac16$ for deficit coefficients on $\Dep$ ((M2) with $\cdep\le\bD^{-1/6}$ for $\tau\ge\tau_{0}$). For a block with $k=|B|\ge1$: a $G_{2,1}$-block has weight $\ge1$ in class A, $\ge0$ in class B if $k\ge2$, and for $k=1$ in class B it is a term of weight $0$ equal to $\hat\delta(\hat x)$ at the derivative point plus a term of weight $-\frac12$ with no field or numerical factor; a $G_{2,2}$-block has weight $\ge\frac12$ in class A and $\ge-\frac12$ in class B ($\ge0$ if $k\ge2$), with no numerical factor; a vii)-block has weight $\ge\frac13$; a viii)-block has weight $\ge1$ in class A and $\ge-\frac12$ in class B ($\ge0$ if $k\ge2$); $r_{\Omega}$ with $k$ derivatives has weight $k/6$; $\hat F_{2}'$ has weight $0$. So in class A every non-$\mathcal{E}$ vertex has weight $\ge0$, and $\ge\frac16$ if it carries a derivative; in class B every non-$\mathcal{E}$ vertex has weight $\ge-\frac12$, and is negative only for single-derivative blocks. \end{lemma} \begin{proof} (a) By Lemma~\ref{lem:B.1}(b) each derivative of a function of $\bGM^{1/2}\psi=\bD^{1/2}\hat\psi$ brings $\bD^{1/2}$, and each derivative of $\hat\psi_{\Omega}$ brings $1/v_{\Omega}$. This gives the rows of \cite{BF80}. For the new rows we use the following facts. In class A, $v_{m}(\Omega)=e^{im\bGM^{1/2}\mathrm{avg}_{\Omega}\psi}-1$, since $h\in P\ZZ$, and $|e^{it}-1|\le|t|$. Next, $|f_{1}'|=|e^{iy}-1|\le\min(|y|,2)$. Further, $f_{2}^{(j)}(0)=0$ for $j\le3$ and $|f_{2}^{(4)}|\le1$, hence $|f_{2}^{(k)}(y)|\le|y|^{4-k}/(4-k)!$ for $k\le4$, $|f_{2}^{(k)}|\le1$ for $k\ge4$, $|f_{2}^{(2)}|=|1-\cos y|\le2$ and $|f_{2}^{(3)}|=|\sin y|\le1$. For vii), $|\partial_{t}^{k}e^{imt}|=|m|^{k}$. In class A the factor viii) is $f_{2}(m\bD^{1/2}\hat\psi(\hat z))$, and in class B its first derivative is $-m\bD^{1/2}\sin(\cdot)+m^{2}\bD\hat\psi$ and its second $m^{2}\bD(1-\cos(\cdot))$. Row i) is Lemma~\ref{lem:7.33} with $\partial A_{\Omega}/\partial\psi(x)=\mt^{3}/v_{\Omega}$. For iii), $F_{2}=\int d\hat x\,\hat\psi\hat F_{2}'$ and $\mathrm{supp}\,F_{2}'\subseteq\bigcup S_{\beta}$. (b) The coefficient weights are those of (M1) to (M4). Moreover, $\cdep=\Ord(\log\tau)$, while $\bD^{-1/6}=e^{\frac16\pi^2\vCb(0)\tau(1+o(1))}$. For $G_{2,1}$ in class A the weight is $-1+2$ for $k\le4$ and $-1+k/2$ for $k\ge4$, while in class B with $k=2,3$ the bounded forms give $-1+1$ and $-1+\frac32$. In class B with $k=1$, as $\sum_{z\in\Omega}\tilde\delta(z)=0$ and $f_{2}'(y)=y-\sin y$, we have $\partial_{x}G_{2,1}(\Omega)=\mt^{2}\tilde\delta(x)-\sum_{m}\rhoref(m)m\bGM^{1/2}[\sin y_{x}-\mathrm{avg}_{\Omega}\sin y]$, whose first term is $(\bD\sum_{m}m^{2}\hat\rho^{\mathrm{ref}}(m))\hat\delta(\hat x)=\hat\delta(\hat x)$ (weight $-1+\frac12+\frac12$) and whose second has weight $-1+\frac12$. For $G_{2,2}$ we split $k=k_{1}+k_{2}$ between $v_{m}$ and $f_{1}$. In class A the weight is $-1+\frac12+\max(1,k_{2}/2)$ if $k_{1}=0$, else $-1+k_{1}/2+\max(1,k_{2}/2)$. In class B we distinguish three cases. If $k_{2}=0$, then $\sum_{z\in\Omega}f_{1}(y_{z})=\sum_{z\in\Omega}(e^{iy_{z}}-1)$ as $\sum_{z}y_{z}=0$, which is bounded by $2\#\Omega$ with no numerical factor, and hence the weight is $-1+k_{1}/2\ge-\frac12$. If $k_{2}=1$, the bounded form gives $-1+k_{1}/2+\frac12$. If $k_{2}\ge2$, the weight is $\ge-1+k_{1}/2+k_{2}/2\ge0$. For a vii)-block the weight is $-\frac16+k/2$. For a viii)-block it is $-1+2$ in class A, and $-1+\frac12$, $-1+1$, $-1+k/2$ in class B. \qed \end{proof} A factor of an $\mathcal{E}$-factor is \emph{gaining} if its weight is at least $\frac12$. This is the case for all differentiated factors, in either class, for all distinguished $\eps$'s (class A), for all $\tilde\eps$'s and $\psi$-factors in class A, and for all $\psi$-factors in class B. Each factor has weight $\ge0$. We select distinguished factors as in \cite[p.~232]{BF80}, with the attribution to cells fixed by Rule D. \emph{Rule D.} Let $e$ be an $\mathcal{E}$-factor of type (E.5a), $t\ge3$. (D-a) For each $\Delta\in W(e)$ in which $e$ has no differentiated factor, we distinguish one undifferentiated $\eps$ of $e$ in $\Delta$, which exists since $e$ has a cube equal to $\Delta$. (D-b) While $e$ has fewer than three gaining factors, we distinguish another undifferentiated $\eps$ of $e$ in a class-A cell, if there is one. All other undifferentiated $\eps$'s are bounded by $2$. For types (E.5b) to (E.5d) no choice is made. The pinning set $\Pi(e)$ consists of the cells of the factors distinguished in (D-b) (at most three) for (E.5a), or of the cells of $e$ (at most two) for (E.5b) to (E.5d), together with the charged cell defined below, if there is one. Hence $|\Pi(e)|\le4$. Let $P(e)$ be the weight of $e$ under Rule D. An $\mathcal{E}$-factor gives $\frac16$ to each cell of $W(e)$. Its remainder $\varrho(e):=P(e)-|W(e)|/6$ is assigned to one chosen class-B cube of $e$ (its \emph{charged cell}) if $e$ has a class-B cube, and to any cell of $e$ otherwise. We call $e$ \emph{charged} if $\varrho(e)<0$. Each non-$\mathcal{E}$ vertex is assigned to its own cell. \begin{theorem}[Net weight per lone cell]\label{thm:B.6} In every term, after this distribution: (i) every cell $\Delta\in X_{o}$ receives weight at least $\frac16$; (ii) every class-A cell receives weight at least $0$; (iii) every class-B cell receives weight at least $-N_{\Delta}^{\mathrm{ch}}-\frac12n_{\Delta}^{(1)}$, where $N_{\Delta}^{\mathrm{ch}}$ is the number of $\mathcal{E}$-factors charged to $\Delta$ and $n_{\Delta}^{(1)}\le n_{\Delta}$ the number of single-derivative blocks in $\Delta$. The value $\frac16$ is attained, both by a lone cell whose only vertex of positive weight is $r_{\Omega}$ with one derivative, and by an $\mathcal{E}$-factor of type (E.5a) with $t=3$ whose three factors are undifferentiated $\eps$'s in three cells of $W(e)$. \end{theorem} \begin{proof} We first treat the $\mathcal{E}$ ledger. If all cubes of $e$ are of class A, then $P(e)\ge|W(e)|/6$ and $P(e)\ge0$. In general $P(e)\ge-1+|W(e)|/2$, and hence $\varrho(e)\ge-1$. Indeed, suppose first that all cubes are of class A. For (E.5a), (D-b) stops only when $e$ has $g(e)\ge3$ gaining factors, and the cells of $W(e)$ contain distinct gaining factors by (D-a), hence $g(e)\ge\max(3,|W(e)|)$ and $P(e)\ge-1+g(e)/2\ge g(e)/6\ge|W(e)|/6$. For (E.5b), $P(e)\ge-1+1+1$, and for (E.5c), (E.5d), $P(e)\ge-1+1+\frac12$, both $\ge\frac26$. In general the cells of $W(e)$ are lone, hence of class A by Lemma~\ref{lem:B.3}(b), each contains a gaining factor of $e$, and all other factors have weight $\ge0$. (ii) A class-A cell receives its non-$\mathcal{E}$ vertices, which have weight $\ge0$ by Lemma~\ref{lem:B.5}(b), credits $\frac16$, and remainders $\varrho(e)\ge0$ of $\mathcal{E}$-factors with no class-B cube. (i) Let $\Delta\in X_{o}$, with its witness from Lemma~\ref{lem:B.3}(c). In case (W1) a derivative hits an atom of $\Delta$. If this atom is $r_{\Omega}$, its vertex has weight $\ge\frac16$, and otherwise the Fa\`a di Bruno block containing the derivative has weight $\ge\frac13$, since the cell is of class A. In case (W2), $\Delta\in W(e)$ receives $\frac16$ from $e$. All other items received are $\ge0$, as in (ii). (iii) Class-B non-$\mathcal{E}$ vertices have weight $\ge-\frac12$, with negative weight only for single-derivative blocks. Charged remainders are $\ge-1$, and uncharged ones are $\ge0$. As for attainment, the first example is (W1) with Lemma~\ref{lem:B.5}(b). In the second, $P(e)=-1+\frac32=3\cdot\frac16$, and the remainder is zero. \qed \end{proof} Each field factor of an $\mathcal{E}$-factor lies in a cell of $\Pi(e)$ or is the single (D-a) factor of a lone cell, and a non-$\mathcal{E}$ vertex carries at most four field factors per derivative. Hence the number $n_{j}$ of field factors in the cell $\Delta_{\alpha}$ satisfies $n_{j}\le1+4(n_{\alpha}+N_{\alpha})$. \begin{proposition}[Per-cell form of (9.824)]\label{prop:B.7} Fix the constants of Section~\ref{sec:7.6}, with $c_{10}\le\frac79r_{C}$ in Lemma~\ref{lem:7.39}, $\qpin:=q_{0}+2$ with $q_{0}:=8+c_{7}''$, and $c_{p}$ of Proposition~\ref{prop:7.38}. For every term $\kappa'$ of the activity of a polymer $X$ with seed $Z$ (kinds and link-end cells fixed, and for each $\mathcal{E}$-factor $e$ its number $t_{e}$ of points and the set $\mathfrak a(e)$ of cells containing them; $\kappa'$ contains, inside $|\cdot|_{0}$, the sum over all configurations of the points of $e$ with this set of cells, that is, over the assignments of the $t_{e}$ points to the cells of $\mathfrak a(e)$ that use every cell, over their positions in these cells and over their charges, and over the forms \cite[(E.5b) to (E.5d)]{GM82} if $t_{e}=2$), every $h_{X}$ and every $s$, \begin{align*} \int d\mu_{C'_{s}}\bigl|e^{E(X,s)}\kappa'Q^{\natural}(X)\bigr|_{0}&\le\mathcal{P}'_{G}\,e^{c_{13}|X|+C_{\Dep}\cdep N_{\Dep}(X)}\,e^{-c'F_{1}^{\mathrm{loc}}(X)+\frac{c'}{16}F_{1}^{\mathrm{loc}}(X\setminus Z)+\frac{c'}{32}F_{1}^{\mathrm{loc}}(Z)}\\ &\qquad\times\prod_{i}\bigl[K_\ell\,e^{-(\bar\delta+(r_{C}-\bar\delta)/2)\mathrm{dist}(\Delta_{i}',\Delta_{i}'')}\bigr]\prod_{e}\mathfrak{m}_{e}\prod_{\Delta\subseteq X}\Psi_{\Delta}, \end{align*} where $\prod_i$ runs over the links with a covariance line (types (2) to (5), Lemma~\ref{lem:A.1}), $K_\ell:=2K_{1}''(1+v_{+}/v_{-})^{2}/\lPV$ is the line constant, $\mathfrak{m}_{e}:=e^{-(\mu-1-2c''_{10})(t_{e}-1)}e^{-r_{C}\hat L^{E}(e)}$ is the leftover Mayer weight, $\hat L^{E}(e)$ the Euclidean Steiner length of the cells of $\mathfrak a(e)$ (Lemma~\ref{lem:B.4}(b)) ($\mathfrak{m}_{e}\le1$ for $\mu\ge1+2c''_{10}$), and $\Psi_{\Delta}\le K_{0}^{B}$ for every class-A cell, $\Psi_{\Delta}\le K_{0}^{B}\bD^{1/6}$ for $\Delta\in X_{o}$, and $\Psi_{\Delta}\le K_{0}^{B}e^{c_{B}(\bD)}$ for every class-B cell, with $c_{B}$ as in Proposition~\ref{prop:B.10}. The constant $K_{0}^{B}$ depends only on $\lPV,C,\mu,\kappa_{S},\delta_{\mathrm{GM}},\qpin,c_{p},c'$, the constants of Theorem~\ref{thm:6.12}, Lemma~\ref{lem:7.33} and the Wick bound, and the geometry constants $c_{u},C_{u},C_{\cU},D_{\cU},v_{\pm}$; it does not depend on $\bD$, $L'$ or $\tau$, and it contains neither $K_\ell$ nor $e^{c_{13}}$. \end{proposition} We give the proof after Lemma~\ref{lem:B.9}. The numerical factors $\delta\hat D=\hat D-\mathrm{avg}_{\Omega}\hat D$ and $\hat F_{2}'$ sit at line ends integrated freely over their cells. For a term we put $\mathcal{N}:=\prod_{j}\int_{\Delta_{j}}|N_{j}(\hat x)|\,d\hat x$, the product over its numerical factors $N_{j}$, and $n_{\Delta}^{N}\le n_{\Delta}$ denotes the number of them in the cell $\Delta$. \begin{lemma}[Numerical factors]\label{lem:B.8} All numerical factors lie in class-B cells. For every union $Y$ of elements of $\tilde Y(h)$ and every $c>0$, the numerical factors located in $Y$ satisfy \[ \prod_{j:\,\Delta_{j}\subseteq Y}\int_{\Delta_{j}}|N_{j}|\,d\hat x\le\prod_{\Delta\subseteq Y}\Bigl[\Bigl(\frac{C_{u}}{2c}\Bigr)^{n_{\Delta}^{N}/2}(n_{\Delta}^{N}!)^{1/2}\Bigr]e^{3cF_{1}^{\mathrm{loc}}(Y)}. \] In particular, with $3c=c'/16$ on $X\setminus Z$ and $3c=c'/32$ on $Z$, $\mathcal{N}\le\prod_{\Delta}[\cdots]\,e^{\frac{c'}{16}F_{1}^{\mathrm{loc}}(X\setminus Z)+\frac{c'}{32}F_{1}^{\mathrm{loc}}(Z)}$. \end{lemma} \begin{proof} First, $\mathrm{supp}\,D_{\beta}$, $\mathrm{supp}\,F_{2,\beta}'\subseteq S_{\beta}$ by Proposition~\ref{prop:7.9}(b), (e). Hence all numerical factors lie in class-B cells, and $S_{\beta}$ lies in one element of $\tilde Y(h)$. By the Cauchy\textendash Schwarz inequality on each cell and $\|\delta\hat D\|_{L^{2}(\Delta)}\le\|\hat D\|_{L^{2}(\Delta)}$, the left side is at most $\prod_{\Delta}(C_{u}a_{\Delta})^{n_{\Delta}^{N}/2}$ with $a_{\Delta}:=\|\hat D\|^{2}_{L^{2}(\Delta)}+\|\hat F_{2}'\|^{2}_{L^{2}(\Delta)}$, and $a^{n/2}\le(2c)^{-n/2}(n!)^{1/2}e^{ca}$. Finally $\sum_{\Delta\subseteq Y}a_{\Delta}\le(2+\eps(L'))F_{1}^{\mathrm{loc}}(Y)\le3F_{1}^{\mathrm{loc}}(Y)$ by Corollary~\ref{cor:7.11}, Theorem~\ref{thm:7.19} and Corollary~\ref{cor:7.21}. \qed \end{proof} \begin{lemma}\label{lem:B.9} All factorials produced in a term are at most $\prod_{\alpha}C^{n_{\alpha}+N_{\alpha}}(n_{\alpha}!)^{q_{0}}(N_{\alpha}!)^{q_{0}}$ with $q_{0}:=8+c_{7}''$. Consequently the pinning factor $f_{\qpin}$ with $\qpin=q_{0}+2$ absorbs them and leaves $\prod_{\alpha}(n_{\alpha}!)^{-2}(N_{\alpha}!)^{-2}$. \end{lemma} \begin{proof} We count the factorials per cell, up to factors $C^{n_{\alpha}+N_{\alpha}}$. The Wick bound $T_{1}\le c_{T}^{\sum n_{j}}\prod_{j}n_{j}!$ with $n_{j}\le1+4(n_{\alpha}+N_{\alpha})$ gives $(n_{\alpha}!)^{4}(N_{\alpha}!)^{4}$, the expansions inside the types and Lemma~\ref{lem:B.2}(c) give $(n_{\alpha}!)^{3}$, Lemma~\ref{lem:7.33} gives $(k!)^{c_{7}''}$, since $k\log k\le\log k!+k$, and Lemma~\ref{lem:B.8} gives $(n_{\alpha}!)^{1/2}$. The total exponent is $7.5+c_{7}''\le q_{0}$. \qed \end{proof} \begin{proof}[Proof of Proposition~\ref{prop:B.7}] We follow the argument of \cite[(9.88) to (9.823)]{BF80} on $\Gp$, make it quantitative, and add the bound on the tree weights of $\kappa'$. \emph{Assignments.} For each $\mathcal{E}$-factor $e$ we fix its form and a surjection $\pi_{e}:\{1,\dots,t_{e}\}\to\mathfrak a(e)$, the cell of each of its points. For the terms with these assignments we carry out all the steps below except the bound on the tree weights and the collection of the constants, with the maps $l$, Rule~D, the pinning sets $\Pi(e)$, the witnesses and the weights $P(e)$ of these terms. In the integration over the positions we leave out the points of the $\mathcal{E}$-factors. The bounds so obtained are the same for every assignment, because the numbers $m_{\alpha}$, $n_{\alpha}$, $N_{\alpha}$, $n_{j}$ appear only in Lemmas~\ref{lem:B.2}(c), \ref{lem:B.4}(b), \ref{lem:B.9} and in the Wick bound, whose right sides are uniform, and $\hat L^{E}(e)$ depends only on $\mathfrak a(e)$. The positions and charges of the points of $e$ appear only in $|\hat\rho_{t_{e}}|$ and $\prod_{i}|m_{i}|^{k_{i}}$, where $k_{i}$ is the number of derivatives on the $i$th factor. By $|m_{i}|^{k_{i}}\le k_{i}!(2\kappa_{S})^{-k_{i}}e^{2\kappa_{S}|m_{i}|}$, the factorials $k_{i}!$ are among those of Lemma~\ref{lem:B.9}, namely those of the charge sums in the proof of Lemma~\ref{lem:B.2}, and the weight $e^{2\kappa_{S}\sum_{i}|m_{i}|}$ does not depend on the assignment. We therefore bound the sum over the assignments, the positions and the charges of the points of $e$ at once by (M4) with $\mathfrak a=\mathfrak a(e)$. The sum over the forms costs at most a factor $3$. \emph{Count.} The sum over the maps $l$ and the further expansions is at most the number of terms, which Lemma~\ref{lem:B.2}(c) bounds, times the largest term, the ``Sup'' of \cite[(9.88)]{BF80}. \emph{Pinning.} We insert $f_{\qpin}$ and bound $\prod_{\alpha}(n_{\alpha}!)^{\qpin}(N_{\alpha}!)^{\qpin}$ by Lemma~\ref{lem:B.4}(b), with the resulting factors absorbed as explained after that lemma. This leaves $e^{-(\bar\delta+(r_{C}-\bar\delta)/2)\mathrm{dist}}$ per link. \emph{Tree weights.} In the term $\kappa'$ of Lemma~\ref{lem:7.39} an $\mathcal{E}$-factor $e$ appears as $\sum_{\eta^{A}}e^{c_{10}L_{\eta^{A}}}e^{c''_{10}t_{e}}\mathcal{E}_{\eta^{A}}$, where $L_{\eta^{A}}$ is the length \cite[(A1.3)]{BF80}, measured with $\mathrm{dist}$ and with auxiliary points in $V$, of an augmented tree $\eta^{A}$ whose marked points lie in the cells of $\mathfrak a(e)$, one in each, as in Lemma~\ref{lem:A.7}, and $\mathcal{E}=\sum_{\eta^{A}}\mathcal{E}_{\eta^{A}}$. Since only $\sum_{\eta^{A}}\mathcal{E}_{\eta^{A}}=\mathcal{E}$ is used, we may put $\mathcal{E}_{\eta^{A}_{e}}:=\mathcal{E}$ for one such tree $\eta^{A}_{e}$ of least length (a minimizer exists, since $V$ is finite and an auxiliary vertex of degree at most $2$ can be removed without increasing the length) and $\mathcal{E}_{\eta^{A}}:=0$ otherwise. Then $e$ carries the number $e^{c_{10}L^{d}(e)}e^{c''_{10}t_{e}}$, $L^{d}(e):=L_{\eta^{A}_{e}}$, which depends only on $\mathfrak a(e)$, that is, on the link datum. By Lemma~\ref{lem:A.7} and the \emph{Assignments} step, no sum over positions at a fixed $\eta^{A}$ occurs (unlike \cite[p.~214]{BF80}, where the positions are summed at a fixed $\eta^{A}$ with the constant weights $b_{\eta^{A}}$). If $L_{\Gp}(\mathfrak a(e))=\infty$, every configuration with the cells $\mathfrak a(e)$ has points in two components of $\Gp$, so $\rho_{t_{e}}$ vanishes on it by Proposition~\ref{prop:6.3}(d), and $\mathcal{E}=0$. Such a factor is absent, as noted before Lemma~\ref{lem:A.7}, so no product $0\cdot\infty$ arises, and in what follows $L_{\Gp}(\mathfrak a(e))<\infty$. Since $\mathrm{dist}=\mt d_{\Gp}$ and $L_{\Gp}$ is computed with $3^{-1/2}d_{\Gp}$, we have $L^{d}(e)=\sqrt3\,\mt L_{\Gp}(\mathfrak a(e))=\sqrt3\,\hat L(\mathfrak a(e))$. Thus the Mayer decay of (M4) is $e^{-\rho_{S}L^{d}(e)}$ with $\rho_{S}:=(1-\delta_{\mathrm{GM}})A/(\sqrt3\,\mt)\ge2r_{C}$, by the chain of inequalities after Lemma~\ref{lem:B.4}. Also, $L^{d}(e)\ge\hat L^{E}(e)$, since both lengths are minima over points of the cells of $e$ and $d_{\Gp}$ dominates the Euclidean distance. With the pinning factor $e^{4c_{p}\hat L^{E}(e)}$ of Lemma~\ref{lem:B.4}(b) we obtain \[ e^{c_{10}L^{d}(e)}e^{4c_{p}\hat L^{E}(e)}e^{-\rho_{S}L^{d}(e)}\le e^{-(\rho_{S}-c_{10}-4c_{p})\hat L^{E}(e)}\le e^{-r_{C}\hat L^{E}(e)}, \] because $\rho_{S}-c_{10}\ge0$ and $\rho_{S}-c_{10}-4c_{p}\ge2r_{C}-\frac79r_{C}-\frac29(r_{C}-\bar\delta)=r_{C}+\frac29\bar\delta$. The bound uses $L^{d}$, not $\hat L^{E}$, because on a slit graph $d_{\Gp}$ is not bounded by a multiple of the Euclidean distance, and hence $e^{-r_{C}\hat L^{E}(e)}$ alone cannot absorb $e^{c_{10}L^{d}(e)}$. Finally, $t_{e}\ge2$ for each $\mathcal{E}$-factor, by \cite[(E.5a) to (E.5d)]{GM82} and Appendix~\ref{app:C}. Hence $e^{c''_{10}t_{e}}\le e^{2c''_{10}(t_{e}-1)}$, which cancels against the factor $e^{-2c''_{10}(t_{e}-1)}$ of $e^{-\mu(t_{e}-1)}$ and leaves $e^{-(\mu-2c''_{10})(t_{e}-1)}$. \emph{H\"older and positions.} At fixed positions we bound $T_{1}$ by the Wick bound, $T_{2}e^{-F_{1}}T_{3}T_{4}$ by Proposition~\ref{prop:7.37}(d), and the factors $r_{\Omega}$ by Lemma~\ref{lem:B.5}. As $0\le\partial_{s_{i}}p(x,y|s)\le1$ in \cite[(E.3)]{GM82}, $|\partial_{s_{i}}\hat C|\le\mt^{-1}|C'|$. We bound each line by its supremum over its pair of end cells, using Lemma~\ref{lem:B.1}(c), and integrate its ends against a vertex point, a cell average, an $r_{\Omega}$-derivative (cost at most $v_{+}/v_{-}$) or a numerical factor (kept in $\mathcal{N}$). The points of each $\mathcal{E}$-factor $e$, summed over all assignments at once, are integrated against its coefficient by (M4) with $\mathfrak a=\mathfrak a(e)$, and the other internal points by (M1) to (M3) of Lemma~\ref{lem:B.1}(d). This gives at most $\prod_{i}(2K_{1}''/\lPV)e^{-r_{C}\mathrm{dist}(\Delta_{i}',\Delta_{i}'')}\prod_{\text{vertices}}[\text{mass}\cdot\bD^{w}](1+v_{+}/v_{-})^{2(n-1)}\mathcal{N}$. \emph{Numerical factors and factorials.} We bound these by Lemmas~\ref{lem:B.8} and~\ref{lem:B.9}. \emph{Weights.} We use Theorem~\ref{thm:B.6} and $\bD\le1$. A cell with weight $\ge\frac16$ gets at most $\bD^{1/6}$, and one with weight $\ge0$ at most $1$. A class-B cell gets $\bD^{-N_{\Delta}^{\mathrm{ch}}-n_{\Delta}^{(1)}/2}$ with $(N_{\alpha}!)^{-2}(n_{\alpha}!)^{-2}$ from Lemma~\ref{lem:B.9} and $N_{\Delta}^{\mathrm{ch}}\le N_{\alpha}$, since a charged $\mathcal{E}$-factor is pinned at its charged cell. Proposition~\ref{prop:B.10} bounds the product by $e^{c_{B}(\bD)}$. \emph{Constants.} The line constants $(2K_{1}''/\lPV)(1+v_{+}/v_{-})^{2}=K_\ell$ stay in $\prod_iK_\ell$, and $e^{c_{13}|X|}$, the factor of Proposition~\ref{prop:7.37}, stays as it is. Each remaining constant is per cell or per $\mathcal{E}$-factor. Recall that each $\mathcal{E}$-factor that occurs has its cells in one component of $\Gp$. The per-$\mathcal{E}$-factor constants that grow with $t_{e}$ are $2$ for each undistinguished $\eps$, from Rule~D, $C_{M4}$ for each point and the factor $t_{e}$ of (M4), and $e^{c_{p}}$ for each factor, from Lemma~\ref{lem:B.2}(c). Their product is at most $t_{e}(2C_{M4}e^{c_{p}})^{t_{e}}$. As $C\le\frac14$, we have $2C_{M4}\le\frac83$. Moreover, $c_{p}1$. \end{proposition} \begin{proof} By Theorem~\ref{thm:B.6}(iii) and the bound for a class-B cell in the proof of Proposition~\ref{prop:B.7}, the cell carries at most $(C_{\mathcal{E}}/\bD)^{N^{\mathrm{ch}}}(C_{G}\bD^{-1/2})^{n^{(1)}}(N_{\alpha}!)^{-q_{1}}(n_{\alpha}!)^{-q_{1}}$ with $N^{\mathrm{ch}}\le N_{\alpha}$, $n^{(1)}\le n_{\alpha}$. We may assume $C_{\mathcal{E}}/\bD,C_{G}\bD^{-1/2}\ge1$ and raise the exponents to $N_{\alpha}$, $n_{\alpha}$. For $q,a\ge1$, $\sup_{N}a^{N}/(N!)^{q}\le(\sum_{N}a^{N/q}/N!)^{q}=e^{qa^{1/q}}$. For the last assertion, we use $\Phi_{\cU}\asymp\bD^{-1}\bGM^{-3}$, which follows from Proposition~\ref{prop:7.22} as $\mt L'=K_{\mathrm{col}}\bGM$, and $\bD=e^{-\pi^2\vCb(0)\tau(1+o(1))}$. If $q_{1}>1$, then $c_{B}/\Phi_{\cU}\le C\bD^{1-1/q_{1}}\bGM^{3}\to0$. If $q_{1}=1$, the bound is of order $\bD^{-1}$ and the ratio grows like $\bGM^{3}$. As Lemma~\ref{lem:B.4} holds for all $q$, the choice $\qpin=q_{0}+2$ is admissible. \qed \end{proof} The order $\bD^{-1/q_{1}}$ cannot be improved within the bounds of Lemma~\ref{lem:B.5}: a class-B cell can contain an $\mathcal{E}$-factor all of whose factors are undifferentiated class-B $\eps$'s, of weight exactly $-1$, since $\eps=e^{im\bGM^{1/2}(\psi+D+h)}-1$ with $D\neq0$ has no small bound. \begin{remark}[One or two derivatives on $G_{2}$ in class-A cells]\label{rem:B.11} On \cite[p.~231]{BF80}, the bounds \cite[(9.814), (9.815)]{BF80} read $(c\beta^{1/2})^{n}\cdot2$ for the $n$th derivative of a term from v), and $c\beta(|\delta|^{2-n}+|g-h|^{2-n})$ if $n<2$, $(c\beta^{1/2})^{n}$ if $n\ge2$, for either term from vi). With these bounds a $G_{2}$-block with one or two derivatives in a class-A cell carries no net power of $\bD$. Indeed, a $G_{2,2}$-block whose single derivative falls on the factor vi) has weight $-1+0+1=0$, and a $G_{2,1}$-block with one or two derivatives has weight $-1+1=0$. A lone cell whose only witness is such a block then receives no power of $\bD$. The class-A bounds of Lemma~\ref{lem:B.5}(a) supply this power. In a class-A cell $|v_{m}(\Omega)|\le|m|\bGM^{1/2}|\mathrm{avg}_{\Omega}\psi|$ as $h\in P\ZZ$, and by charge symmetry $G_{2,1}$ is of fourth order in $\tilde\delta$. By Lemma~\ref{lem:B.5}(b), both give weight at least $\frac12$. Without charge symmetry the third-order form of the second term of vi) already gives $-1+\frac32=\frac12$. \end{remark} \begin{proof}[Proof of Lemma~\ref{lem:7.40}] We insert Proposition~\ref{prop:B.7} into Lemma~\ref{lem:7.39}. With $e^{c_{2}^{L}|X|}$ and the factor $e^{c_{13}|X|}$ of Proposition~\ref{prop:B.7}, its factors are distributed cell by cell as stated, with $c_{o}:=c_{2}^{L}+c_{13}+\log K_{0}^{B}+\log\max(1,K_\ell)$ (link $i$ is mapped injectively to a cell of $Y_{i+1}$, so $\prod_iK_\ell\le\prod_\Delta\max(1,K_\ell)$) and $c_{B}$ of Proposition~\ref{prop:B.10} with $q_{1}=2$. Since a class-B cell meets $\bigcup_{\beta}S_{\beta}$, it lies in a wall polymer, in $X_{\Sigma}$ or in the seed. Factors on class-A cells carry in addition $\bD^{w}$ with $w\ge0$, by Theorem~\ref{thm:B.6}(ii). Their constants are independent of $L'$, and so are $C_{\mathcal{E}}$ and $C_{G}$, which are built from the same quantities (see Section~\ref{S:G.B11}). Indeed, no class-A cell carries $D$, $F_{2}'$ or a quantity depending on $\hat\Sigma$, by Proposition~\ref{prop:7.9}(b), (e). This proves the assertion of the lemma on $L'$, with $\eps(L')$ coming from Lemma~\ref{lem:B.8}. \qed \end{proof} \begin{proof}[Proof of Proposition~\ref{prop:7.41}] We insert Proposition~\ref{prop:B.7} into Lemma~\ref{lem:7.39} with $c_{1}'=c'/4$. Of $F_{1}^{\mathrm{loc}}$ we use $c'/16$ on $X\setminus Z$ for the numerical factors of Lemma~\ref{lem:B.8} and $c'/8$ there for the wall polymers, and $c'/32$ on $Z$ for the numerical factors. The share $c'/8$ gives each cell of a wall polymer of $X\setminus Z$ the factor $e^{-(c'/8)\Phi_{\cU}/(1+D_{\cU})}$, since $F_{1}(S_{\beta})\ge\Phi_{\cU}\#S_{\beta}$ by Proposition~\ref{prop:7.22}(c), a closure adds at most $D_{\cU}$ cells per cell, and $F_{1}^{\mathrm{loc}}(\mathcal{S})\ge\sum_{S_{\beta}\subseteq\mathcal{S}}F_{1}(S_{\beta})$ by the definition of $F_{1}^{\mathrm{loc}}$ in Corollary~\ref{cor:7.11}. The link decay of Proposition~\ref{prop:B.7} absorbs $e^{\delta_{2}d}$, as $\delta_{2}<\bar\delta$. The Mayer cubes and the per-$\mathcal{E}$-factor constants growing with $t_{e}$ are absorbed inside Proposition~\ref{prop:B.7}, and the leftover Mayer weights of that proposition satisfy $\mathfrak{m}_{e}\le1$, since $\mu\ge1+2c''_{10}$ by item~(12) of the list in Section~\ref{app:D}. With $K_{B}:=K_{0}^{B}\max(1,K_\ell)$, independent of $\bD$, $L'$, $\tau$, the per-cell factors and the line constants (one per link, and the links inject into the cells) give $\eps_{\Delta}$ (Section~\ref{S:G.B11}). The factor $e^{C_{\Dep}\cdep}$ is present only on cells meeting $\Dep$, and the rest of the factor on $X\setminus Z$ is at most $1$. The exponent $p_{0}=\frac16$ is attained by Theorem~\ref{thm:B.6}. \qed \end{proof}