Proof. For \(d\in\mathcal L/2\mathcal L\) put \(z_d(s)=\sum_{u\in d+2\mathcal L}e^{-(u+s,T(u+s))/4}\). Coercivity on \(V\) makes these sums and their derivatives locally uniformly convergent for ambient shifts. The bijection \((u,v)\mapsto(u+v,u-v)\) gives \[w_T(x)w_T(y)=\sum_d z_d(x+y)z_d(x-y),\qquad \sum_d z_d(s)^2=w_T(0)w_T(s).\] Cauchy–Schwarz proves Equation (21). Taking logarithms and differentiating twice in \(y\) at zero gives \(D^2\log w_T(x)\ge D^2\log w_T(0)\). Direct differentiation of the sum gives \(D^2\log w_T(x)=-T+T\operatorname{Cov}_x(H)T\). Both covariances are supported on \(V\). Restricting the Hessian inequality to directions in \(V\) and inverting \(T_V\) there proves covariance minimality.

For \(x_T=T_V^{-1}P_VTx\) and \(u\in V\), completion of the square gives \[(u+x,T(u+x))=(u+x_T,T_V(u+x_T))+c_T(x),\qquad c_T(x)=(x,Tx)-(x_T,T_Vx_T)\ge0.\] For fixed \(x\) and \(T_t=T+tB\), \(B\ge0\), differentiation in the ambient space gives \[\frac{d}{dt}\log\frac{w_{T_t}(x)}{w_{T_t}(0)} =-\tfrac12\operatorname{tr}\bigl[B(\operatorname{Cov}_x(H)-\operatorname{Cov}_0(H) +\mathbb E_xH\otimes\mathbb E_xH)\bigr]\le0.\] This derivative keeps the ambient displacement fixed and includes the \(T_t\)-dependence of both \(x_{T_t}\) and the normal scalar \(c_{T_t}(x)\). A real tilt by \(\ell\) changes the effective displacement on \(V\) to \(x_T-T_V^{-1}\ell_V\). Its covariance therefore still dominates the centered covariance. For the centered law, integrating the Hessian of its logarithmic moment generating function on the segment from \(0\) to \(\ell\) proves the first inequality in Equation (22); symmetry makes the first derivative at zero vanish. Differentiating this logarithmic moment generating function with respect to \(T_t\) gives the same nonpositive expression with the tilted covariance and mean. Differentiating the resulting centered comparison twice at zero proves the variance comparison. The second derivative at \(s=0\) of \(\operatorname{Var}_{sY}(X)\ge\operatorname{Var}_0(X)\) proves Equation (23).

Let \(\mathcal L^*\) be the dual lattice in \(V\). Poisson summation on \(V\) gives \[w_T(x)=e^{-c_T(x)/2}C_{T_V}\sum_{q\in\mathcal L^*} e^{-2\pi^2(q,T_V^{-1}q)}e^{2\pi i(q,x_T)}\le w_T(0).\] Completion of the square now proves the upper bound in Equation (22). For the centered characteristic function, the same summation gives the strictly positive formula \[\mathbb Ee^{i(\ell,H)} =\frac{\displaystyle\sum_{q\in\mathcal L^*} e^{-\frac12(2\pi q-\ell_V,T_V^{-1}(2\pi q-\ell_V))}} {\displaystyle\sum_{q\in\mathcal L^*}e^{-2\pi^2(q,T_V^{-1}q)}}>0.\] It is a shifted-to-centered partition ratio on the dual lattice. Equation (21) there proves the assertion about its logarithmic Hessian.

A collection of centered equalities selects \(\mathcal L_E=\mathcal L\cap\bigcap_j\ker\ell_j\), a lattice in its span. The restricted precision is coercive, so the preceding argument applies directly to its conditional centered law. In particular the displayed dual sum proves strict characteristic positivity there. This law is also the dominated limit of the additional precisions \(t\sum_j\ell_j\otimes\ell_j\), \(t\uparrow\infty\), which transfers the centered comparisons with the original law.

For a free graph choose one representative \(h_{x_0}=0\) on each connected component. Its representative lattice has a span on which the massless graph precision is coercive. The same dual formula directly proves strict characteristic positivity on that centered law, and hence for its translation-invariant characters. To compare a graph with a cut graph, first add \(\eta\operatorname{Id}\). On a connected component choose representatives \(h_{x_0}=0\) and sum its common translation \(2\pi j\). Completion of the square shows that this sum, divided by \((2\pi\eta|B|)^{-1/2}\), tends to a constant independent of the representative and is uniformly bounded for \(0<\eta\le\eta_0\). The remaining representative weight is bounded by a constant times its massless weight and converges pointwise to it. Dominated convergence, also after any fixed real tilt annihilating constants, transfers the comparison inequalities. The same argument applies to bounded observations invariant under common \(2\pi\) translations, a fact used below for phases. Finally, for an equality event \(E\), differentiate its probability along \(T+tB\) before taking any infinite-penalty limit. Its logarithmic derivative is \(\frac12\operatorname{tr}B(\operatorname{Cov}(H)-\operatorname{Cov}(H\mid E))\ge0\) by the just-proved centered pin comparison; both means vanish by inversion symmetry. This proves the last assertion, and dominated limits give its free-graph versions whenever the event is invariant under the released translations. ◻

Proof. We give separately the cutting, folding, and distributional steps.

Cutting and the existence of \(a\). Cut a square of side \(M\) into \(m\)-squares and strips of width less than \(m\). In its affine observation \(x_1/M\), the squared cost of omitted seam and strip flows is at most \(C_J(m^{-1}+m/M)\): there are at most \(C_J(M^2/m+mM)\) affected edges and each coefficient is at most \(C_J/M\). On the remaining squares the observation is \(m/M\) times the cell observation. Cutting increases variance, and the cut components are independent. The triangle inequality in \(L^2\) consequently gives \[ \sqrt{a_M}\le\sqrt{a_m}+C_J\sqrt{k}(m^{-1}+m/M)^{1/2}. \tag{28}\] First let \(M\to\infty\), then take \(m\to\infty\) along a liminf subsequence. This proves convergence. Notice also that each \(a_m(k)\) is continuous for \(k>0\), by locally uniform convergence of its defining Gaussian sums, and Equation (28) is uniform when \(k\) ranges over a compact subset of \((0,\infty)\). In particular \(\sqrt{a(k)}\le\sqrt{a_m(k)}+C_J\sqrt{k/m}\). Square symmetries make the covariance matrix of the two affine tests scalar. Cutting a rectangle into small squares proves the affine upper bound \(a\int_B|\nabla g|^2\). For its reverse, let \(m_1(n),m_2(n)\) be the rectangle’s side counts and set \(V_n(p)=\operatorname{Var}_{B_n}(h,A_{B_n}(p\cdot x/n))\). Tile a square of side \(M\), divisible by both side counts, with copies of this rectangle. The normalized affine test \(p\cdot x/M\) restricts on each cell to \(n/M\) times the cell test, up to an irrelevant constant. There are \(M^2/(m_1(n)m_2(n))\) cells. Cutting and the seam-flow estimate, followed by \(M\to\infty\) at this fixed \(n\), give \[\sqrt a\,|p| \le\frac{n}{\sqrt{m_1(n)m_2(n)}}\sqrt{V_n(p)} +\frac{C_{J,k,p}}{\sqrt{\min(m_1(n),m_2(n))}}.\] Since \(m_1(n)m_2(n)/n^2\to|B|\), this proves \(\liminf_nV_n(p)\ge a|p|^2|B|\) along every admissible rectangle sequence. Together with the cutting upper bound it gives \[\lim_n\operatorname{Var}_{B_n}(h,A_{B_n}g)=a\int_B|\nabla g|^2\] for every affine \(g\) and every such sequence.

Cutting into cells of small fixed macroscopic diameter and replacing \(\nabla g\) by constants proves, for smooth or continuous piecewise smooth \(g\) with bounded piece derivatives and polygonal interfaces, \[ \limsup_n\operatorname{Var}(h,A_{B_n}g)\le a\int_B|\nabla g|^2. \tag{29}\] At a fixed partition the omitted edge count is \(O(n)\) and coefficients are \(O(n^{-1})\), so their cost tends to zero. In cell interiors the error cost is bounded by the \(L^2\) modulus of continuity of the gradient, which tends to zero as the partition is refined. The identical proof applies to partial gradient observations restricted to polygonal pieces.

Call a shift \(g\) saturated on \(B\) if, for every admissible sequence of discrete rectangles \(B_n\) converging to \(B\), \[\lim_{n\to\infty}\operatorname{Var}_{B_n}(h,A_{B_n}g(\cdot/n)) =a\int_B|\nabla g|^2.\] For a partial gradient observation use the same full-limit definition with the energy integral over its retained region.

This class is linear. Fix a finite list of saturated shifts and any admissible rectangle sequence. From any subsequence one can extract a further subsequence on which their covariance matrix converges, by the bare bounds. If \(C\) is that limiting matrix and \(E_{ij}=a\int_B\nabla g_i\cdot\nabla g_j\), then Equation (29), applied to every real linear combination, gives \(E-C\ge0\). Full convergence for each \(g_i\) gives \((E-C)_{ii}=0\) on this same subsequence. A positive-semidefinite matrix with zero diagonal has all its entries zero. Thus \(C=E\). Since this applies to every initial subsequence, the covariance matrices converge along the original sequence. In particular every finite linear combination is saturated.

Saturation on a larger rectangle passes to an enclosed rectangle. Partition the complement into finitely many rectangles, indexed together with the inner rectangle by \(i=0,\ldots,r\), and write \(E_i=\int_{B_i}|\nabla g|^2\). Vanishing seam-flow errors and cutting give \[V_{\mathrm{outer},n}\le\sum_{i=0}^rV_{i,n}+o(1).\] The outer variance converges to \(a\sum_iE_i\), while \(\limsup_nV_{i,n}\le aE_i\) for every piece. Consequently \[\liminf_nV_{0,n} \ge a\sum_iE_i-\sum_{i=1}^r\limsup_nV_{i,n} \ge aE_0.\] Together with the upper bound this is full convergence on the inner rectangle, on every admissible sequence. Strict enlargements suffice; compatible rounding of the partition changes only vanishing-cost flows.

The class is also closed under gradient-energy approximation. If \(g_j\) are saturated and \(\|\nabla(g_j-g)\|_{L^2(B)}\to0\), with convergence of the corresponding discrete difference energies, then the bare bound and the triangle inequality for standard deviations give \[\limsup_n\left| \sqrt{\operatorname{Var}(h,A_{B_n}g)}-\sqrt a\,\|\nabla g\|_{L^2(B)} \right| \le(\sqrt k+\sqrt a)\|\nabla(g-g_j)\|_{L^2(B)}.\] Letting \(j\to\infty\) proves full convergence for \(g\). The same argument shows that an observation changed by a flow of cost tending to zero retains its full variance limit. These statements apply to the fixed smooth and continuous piecewise smooth shifts used below, whose discrete energies converge by Riemann summation.

The fold. We justify the FKG correlation input (Fortuin et al. 1971) directly. A quadratic attractive height law has the lattice condition \(p(u\wedge v)p(u\vee v)\ge p(u)p(v)\), since it holds edge by edge for \(e^{-c(h_x-h_y)^2/(2k)}\). On a finite height box this condition implies association by induction in the number of vertices. Conditional laws in one fewer coordinate satisfy the same condition. Between two values of the last coordinate the likelihood ratio of their conditional laws is increasing. The induction hypothesis applied to that ratio proves that the conditional expectation of every increasing function increases. The conditional covariance decomposition and the one-variable identity \(\operatorname{Cov}(F(Z),G(Z))=\frac12\mathbb E[(F(Z)-F(Z'))(G(Z)-G(Z'))]\ge0\) complete the induction. Letting the box grow proves association in our proper finite Gaussian law. This argument also proves association after conditioning any set of vertices.

Suppose a rectangle and its discrete graph are invariant under reflection across a site axis or diagonal. Pin a site of the reflection line and condition its entire height trace. The conditional law remains invariant under reflection and associated, including interactions crossing the line. For a saturated shift odd about that line up to a constant, let \(X_+\) and \(X_-\) be its observations on edges internal to the two open sides. Then \(X_-\) is the reflected negative of \(X_+\) and their sum differs from the full observation by a vanishing-cost flow. If \(X_+\) can be changed by such a flow to have nonnegative off-line vertex coefficients, conditional association gives a nonpositive conditional covariance with \(X_-\). Their conditional means are negatives, so the unconditional covariance is also nonpositive. Write \(V_{\pm,n}=\operatorname{Var}(X_\pm)\) and \(E_\pm=\int_{B\cap\{\pm\text{ side}\}}|\nabla g|^2\). The vanishing-cost modifications give \(\operatorname{Cov}(X_+,X_-)\le o(1)\), and the full observation differs from \(X_++X_-\) by a vanishing-cost flow. Its full variance convergence therefore gives \[a(E_++E_-) =\lim_n V_{\mathrm{full},n} \le\liminf_n(V_{+,n}+V_{-,n}).\] More directly, \(V_{+,n}\ge V_{\mathrm{full},n}-V_{-,n}-o(1)\), so the separate upper bound gives \[\liminf_nV_{+,n} \ge a(E_++E_-)-\limsup_nV_{-,n}\ge aE_+.\] The same argument applies to the minus side. With \(\limsup_nV_{\pm,n}\le aE_\pm\), both partial variances converge along the full sequence. The folded-shift observation differs from its partial observation by a vanishing-cost flow, so it too has the required full variance limit.

Here are the required coefficient checks and error bounds. For an axis fold of \(g=x_1\), pair each step \((z_1,z_2)\) with \((-z_1,z_2)\). The coefficients of \(A_{B_+}g\) vanish away from the two normal-face strips, are nonpositive in the strip adjacent to the cut, and nonnegative at the outer normal face. Tangential boundaries preserve this cancellation. Move every cut-strip coefficient to the conditioned line along at most \(R_J+1\) nearest-neighbor edges. There are \(O(n)\) coefficients of size \(O(n^{-1})\), and the paths have bounded overlap, giving cost \(O(n^{-1})\). Thus \((x_1-t)_+\) is saturated after coherent rounding of the reflection line and restriction from an enclosing symmetric rectangle. Finite linear combinations and gradient approximation give every smooth function of either coordinate.

For the diagonal fold put \(Y=y+t\) and use an enclosing square \(p\le x,Y\le q\). Take \(g=F(x)-F(Y)\) with \(F'''<0\) and \(F'(p),F'(q)>0\) on a slightly larger interval. In the interior of \(x>Y\), \(A_Bg\ge0\): the symmetric second differences of \(F\) decrease with their center, and the step lists in the two coordinates coincide. At the outer face \(x=q\) the missing-step contribution has leading term a positive constant times \(F'(q)/n\); at \(Y=p\) it has leading term a positive constant times \(F'(p)/n\). The tangential first-order terms cancel in pairs. Uniform strict endpoint positivity absorbs the \(O(n^{-2})\) remainder. More precisely, in integer coordinates \(P\le i,j\le Q\), the exceptional sets can be taken as \(D=\{0<i-j\le2R_J\}\) and \(K=\{Q-i\le R_J,\ j-P\le R_J\}\). Outside \(D\) neither the lower \(i\)-face nor the upper \(j\)-face truncates a step; outside \(K\) the other two faces cannot both truncate. This proves the stated sign everywhere outside \(D\cup K\). Move all coefficients in \(D\cup K\) horizontally from \((i,j)\) to \((j,j)\) on the line. There are \(O(n)\) short paths of length at most \(2R_J\) and \(O(1)\) paths of length \(O(n)\), with bounded overlap. Their coefficients have size \(O(n^{-1})\). If the overlap is \(M\), their cost is at most \((M/c)\sum_v d(v)^2|P_v|=O(n^{-1})\). Define the opposite-side modification by reflected negative, so the line terms cancel exactly. The fold criterion proves saturation of \[ (F(x)-F(y+t))\mathbf 1_{\{x>y+t\}}. \tag{30}\] Every smooth \(F\) on the fixed interval is a difference of two functions with the strict properties just used: add a sufficiently large fixed cubic with negative third derivative, and then a sufficiently large positive linear function. Thus Equation (30) is saturated for arbitrary smooth \(F\).

For clarity, coherent rounding preserves these full limits. Choose \(t_n=r_n/n\) with \(r_n=\operatorname{round}(nt)\), and enclose the target rectangle in the exactly symmetric lattice square \(P_n\le i\le Q_n\), \(P_n-r_n\le j\le Q_n-r_n\). Its reflection is \((i,j)\mapsto(j+r_n,i-r_n)\), and its scaled endpoints can be chosen to converge. The odd seed \(F(x)-F(y+t_n)\) differs from the fixed saturated seed \(F(x)-F(y+t)\) by discrete gradient energy \(O(|t_n-t|^2)\). For the folded functions \(f_{t_n},f_t\) of Equation (30), the squared difference energy is at most \(C(|t_n-t|+n^{-1}+|t_n-t|^2)\). Indeed the strip between their diagonals, enlarged by \(R_J/n\), meets \(O(n^2|t_n-t|+n)\) edges, each carrying an \(O(n^{-1})\) difference increment; away from it, smooth derivative differences are \(O(|t_n-t|)\). Axis hinges satisfy the same estimate. Thus the fold and restriction limits transfer to each fixed target shift along every admissible rectangle sequence by the vanishing-cost perturbation bound.

To see that these folds suffice, extend a smooth target \(g\) slightly and choose a smooth family \(F_t\) of compact support in \(t\), on all relevant parameters, satisfying \(F_t'(x)=-g_{xy}(x,x-t)\). Differentiating the integral of Equation (30) gives \[\partial_x\partial_y\int (F_t(x)-F_t(y+t))\mathbf 1_{\{x>y+t\}}\,dt=-F_{x-y}'(x)=g_{xy}(x,y).\] The residual is a sum of two univariate functions. Riemann sums of this integral converge in gradient \(L^2\): translations of the bounded piecewise smooth gradients are continuous in \(L^2\), including across the moving diagonal. The bare bound then transfers saturation to \(g\).

Laplace transforms and smears. The lower bound in Equation (26) follows from saturation and Equation (22). For the upper bound, cut into cells of side comparable to a small fixed \(r\). For fixed \(r\), the discarded-flow observation \(E_n\) has bare variance tending to zero. Hölder with exponents \(p>1\) and \(p/(p-1)\) removes \(E_n\) at a cost tending to zero. Precision comparison bounds the remaining log transform by the sum of independent cell log transforms with argument \(p\). Each cell variable has bare variance \(O(r^2)\), so its subGaussian bound gives \[\log\mathbb Ee^{pX}=\tfrac12p^2\operatorname{Var}X+O_p(r^3),\] uniformly in the fine mesh. For example Taylor’s integral remainder is bounded by \(C_p\mathbb E[|X|^3e^{p|X|}]=O_p(r^3)\) using the Gaussian tail. There are \(O(r^{-2})\) cells. Apply Equation (29), let \(n\to\infty\), then \(r\downarrow0\), then \(p\downarrow1\). This proves the matching upper bound. Apply it to arbitrary linear combinations to obtain the joint law; the uniform exponential bounds give convergence of moments.

Finally nearest-neighbor Poincaré gives \(\|u-\bar u\|_2^2\le C_Bn^2(u,A_{B_n}u)\). Consequently the bare variance of a density error is bounded by its scaled \(L^2\) norm squared. Approximate \(f\) by finitely many nonconstant Neumann cosines of the rectangle, and use their smooth inverse-Laplacian shifts. For a Neumann shift, \(n^2A_{B_n}g\) converges in density \(L^2\) to \(-\Delta g\): interior errors vanish by Taylor expansion; boundary coefficients are \(O(n^{-2})\) because the normal derivative vanishes and tangential steps cancel. At corners both derivatives vanish. The \(O(n)\) boundary-layer sites have vanishing scaled area, so their bounded scaled errors disappear. Use the actual discrete rectangle endpoints in this approximation and then let them converge. Poincaré and bare domination complete the density approximation and prove Equation (27) and its joint version. ◻

Proof. Write \(g_m=(x_1-(m-1)/2)/m\), \(X=(h,A_Bg_m)\), and \(Y=(h,\rho-u_m)\), where \(\rho\) is the uniform right-face probability. The flow bound makes \(Y\) uniformly subGaussian. Reflection changes \(X\) to \(-X\) and exchanges right and left faces, so \[ \operatorname{Cov}(X,Y)\longrightarrow a/2. \tag{36}\] Here is a discrete verification of the normalization. The profile \(A_Bg_m\) is supported in fixed-width side layers. Except in boundedly many corner sites, its right-layer row sums are constant, its left-layer row sums are their negatives, and all other sums vanish, by pairing steps. Transport these coefficients to their respective faces along bounded paths; the cost is \(O(m^{-1})\). Corner discrepancies of size \(O(m^{-1})\) can be routed along \(O(m)\) paths, at cost \(O(m^{-1})\). Thus \(A_Bg_m-t_m(\rho_R-\rho_L)\) has vanishing flow cost. Pairing with \(g_m\) and using \((g_m,A_Bg_m)\to1\) gives \(t_m\to1\). This proves Equation (36).

Fix \(l\ne0\) and suppose along a subsequence \(w_B(lu_m)\to r>0\). Uniform subGaussian bounds on \(X,Y\) allow a further subsequence on which \[F_m(t,u)=\mathbb E\exp\{il(h,u_m)+itX+iuY\}\] converges locally uniformly with all derivatives on \(\mathbb C^2\). This follows from the uniform bound \(|F_m(t,u)|\le\mathbb Ee^{|\Im t||X|+|\Im u||Y|}\), Cauchy’s formula, and a diagonal Arzelà–Ascoli argument. The limit \(F\) is real and nonnegative on \(\mathbb R^2\) by Equation (31).

The Gaussian limit and moment convergence of \(X\) imply \(\operatorname{Var}(V_{XX})=2\kappa(X,X,X,X)\to0\). The fourth moments of \(Y\) are bounded, so \(\operatorname{Var}(V_{YY})\) is bounded. By Equation (34), \[\operatorname{Var}(V_{XY})=2\kappa(X,X,Y,Y) =\operatorname{Cov}(V_{XX},V_{YY})\longrightarrow0.\] Multiply \(V_{XX}\) or \(V_{XY}\) by the conditional characteristic function of \(D\) with phase \(l\bar D+tD_X+uD_Y\). Its modulus is at most one, so the concentration just proved and conditional independence permit replacement of \(V_{ij}\) by \(2C_{ij}\). Direct expansion in the two original copies gives \[\mathbb E[S_iS_j e^{i(l\bar D+tD_X+uD_Y)}] =-2(F_m F_{m,ij}-F_{m,i}F_{m,j}).\] Since \(\mathbb Ee^{i(l\bar D+tD_X+uD_Y)}=F_m^2\), it follows wherever \(F>0\) that \((\log F)_{tt}=-a\) and \((\log F)_{tu}=-a/2\). At \((0,0)\) reflection gives \(F_t=0\). Analytic continuation yields \[ F(t,u)=F(0,u)\exp\{-at^2/2-atu/2\}. \tag{37}\] Equation (24) ensures \(F(0,l)>0\), because the bare cost of \(lY\) is bounded.

Now join two horizontally adjacent \(m\)-squares to form a free rectangle, and use its centered neutral charge law. For a charge \(q\), let \(Q\) be its left total, the right total being \(-Q\). Use the following trial quadratic form: in each cell route the residual \(q-Q_B\rho_B\) with its minimizing internal flow, where \(\rho_B\) is its seam-face probability; route \(Q\) uniformly across the \(m\) nearest-neighbor seam edges; add the penalty \(Q^2\). The resulting form is \[k\sum_{B=L,R}(q_B-Q_B\rho_B,G_B(q_B-Q_B\rho_B))+D_mQ^2, \qquad D_m=1+k/(cm).\] It dominates the actual precision form \(k(q,Gq)\) by Equation (20). Conditional on \(Q\), the trial cells are independent shifted neutral lattice Gaussians, whose affine variances dominate the centered ones. By Equation (32) and saturation, their sum tends to \(2(k-a)/k^2\), exactly the actual variance of the global affine test \(x_1/m\) on the two-cell rectangle. Precision comparison gives the opposite upper bound for the trial total variance. Thus the variance, over \(Q\), of its conditional affine mean tends to zero.

The sector weights, after common normalization, are \(e^{-D_mQ^2/2}w_L(Q\rho_L)w_R(-Q\rho_R)\). Their normalizing sum is at most \(\sum_Q e^{-Q^2/2}\) and the \(Q=0\) term is one. The \(Q=l\) term stays positive by Equation (24). In that sector Equation (33) and Equation (37) give \(k(g,\mathbb E[q-l\rho])\to-al/2\) in the left cell. Reflection and charge reversal give the same residual pairing in the right cell. The two seam-face contributions to the global affine mean differ by only \(O(m^{-1})\). Its \(Q=l\) conditional mean therefore tends to \(-al/k\ne0\), whereas the \(Q=0\) conditional mean is zero. This contradicts the vanishing sector-mean variance. Thus \(w_B(lu_m)\to0\).

If Equation (35) failed, its positive lower bound would transfer to \(lu_m\) by Equation (24), which is impossible. Joint convergence with the Gaussian observations follows by Fourier polynomials on the circle and joint characteristic functions; tightness on the circle and the already proved Gaussian tightness make this convergence of the full joint law. Exponential bounds extend the joint assertion to moments of the Gaussian coordinates. ◻

Proof. Tile the \(Gm\)-square into \(G^2\) cells \(B_i\). The restriction of the large square’s face probability to \(B_i\) has total \(\xi_i\) and equals the sum of its exterior-face probabilities with coefficient \(1/(4G)\) per exterior face. Thus \(\xi\) is supported on boundary cells, \(\sum_i\xi_i=1\), and \(\sum_i\xi_i^2\le C/G\). For a neutral real profile \(r\), let \(Q_i=\sum_{B_i}r\) and choose a linear map from \(Q\) to an antisymmetric nearest-neighbor coarse flow \(F\) satisfying \(\operatorname{div}F=Q\) and zero exterior flux. The quadratic form \[ \begin{split} \mathcal T_m(r)={}&k\sum_i \left(r_i-\sum_eF_{ie}\rho_{i,e}, G_{B_i}\left(r_i-\sum_eF_{ie}\rho_{i,e}\right)\right)\\ &+\frac{k}{cm}\sum_{\langle i,j\rangle}F_{ij}^2+\sum_iQ_i^2 \end{split} \tag{39}\] is at least \(k(r,G_{B_{Gm}}r)\): use the internal minimizing flows and send each coarse flux uniformly across the \(m\) nearest-neighbor seam edges. The last term is an extra positive penalty. The internal and seam edges are disjoint, so their costs add. The form is positive definite on the neutral space because zero cost forces \(Q=0\) and then \(r=0\).

In the numerator of Equation (31) take \(r=q-\bar\rho_{B_{Gm}}\); in the denominator take \(r=q\). Precision monotonicity of a shifted-to-centered ratio shows that replacement by \(\mathcal T_m\) gives a lower bound for \(z_{Gm}\). Sum this new partition function by integer cell totals \(n_i=\sum_{B_i}q\). Then \(Q=n-\xi\) in the numerator and \(Q=n\) in the denominator. Divide both sums by the product of the cells’ centered neutral partition functions. Their internal factors become \(\prod_iw_{B_i}(s_i)\), where \[s_i=\sum_eH_{ie}\rho_{i,e},\qquad H_{ie}=\begin{cases} F_{ie},&\text{internal face},\\ 1/(4G),&\text{exterior face in the numerator},\\ 0,&\text{exterior face in the denominator}. \end{cases}\] In the denominator exterior \(H\) is zero. In either case \(\sum_eH_{ie}=n_i\). All these factors lie in \((0,1]\). The penalty \(\exp(-\frac12\sum_iQ_i^2)\) is summable over the finite-rank integer sector lattice, uniformly in \(m\). Thus dominated convergence is valid at every fixed \(G\). Each nonzero denominator sector contains a nonzero integer-total face profile and tends to zero by hypothesis. The zero sector equals one, so the normalized denominator tends to one.

Keep numerator sectors \(n=e_{i_0}\) with \(i_0\) at distance at least \(G/4\) from the boundary. Write \(Y_i=s_i-n_i\bar\rho_{B_i}\). The logarithmic Hessian bound for \(w\) gives \[ \prod_iw_{B_i}(s_i) \ge z_m\exp\left\{-\tfrac12\sum_i\operatorname{Var}_{B_i}(h,Y_i)\right\}. \tag{40}\] For the linear term at zero, inversion symmetry suffices. On the total-one hyperplane put \(\Psi_B(H)=\log w_B(\sum_eH_e\rho_{B,e})\), where \(\sum_eH_e=1\). Equation (31) makes this function smooth there, and square symmetry makes its differential at \(H_0=(1/4,1/4,1/4,1/4)\) invariant under the square group. Every tangent vector with zero coordinate sum has group average zero, so this differential vanishes on it. The face coefficients of each \(Y_i\) have zero sum, and \(n=e_{i_0}\) uses only the base profiles zero and \(\bar\rho_{B_i}\).

Put \(D_{ij}=(H_{i,+j}-H_{i,-j})/2\) and \(S_{ij}=H_{i,+j}+H_{i,-j}\). Reflection in each coordinate splits \(Y_i\) into two odd face differences and an even remainder. These three pieces have zero mutual covariance by symmetry. The odd differences have limiting variance \(a\) and zero cross covariance, as in Equation (36); the even remainder has a bounded bare variance times \(\sum_jS_{ij}^2\), by the face-to-strip flow estimate. Consequently, for fixed \(G\) and fixed face coefficients, \[ \limsup_m\sum_i\operatorname{Var}(h,Y_i) \le a\sum_{i,j}D_{ij}^2+C_{J,k}\sum_{i,j}S_{ij}^2. \tag{41}\]

We construct one coarse flow for each retained sector with bounds uniform in \(G,i_0\). In the continuum square of side \(G\), let \(x_0\) be the center of cell \(i_0\). There is a vector field with divergence \(\delta_{x_0}\) and constant outer normal flux \(1/(4G)\), of the form \[ V(x)=\frac{x-x_0}{2\pi|x-x_0|^2}+E(x),\qquad |E(x)|\le C/G,\quad |\nabla E(x)|\le C/G^2. \tag{42}\] For completeness, subtract the expanding field \((x-x_{\rm center})/(2G^2)\), which has divergence \(G^{-2}\) and the desired boundary flux. Solve the resulting zero-Neumann problem by even reflection to the torus of side \(2G\). After rescaling to a fixed torus, subtract cutoff logarithms around its finitely many reflected poles. Their mutual separations and distances from the original boundary are bounded below, uniformly for \(x_0/G\) in the central square. The remainder solves a smooth mean-zero Poisson equation with uniformly bounded derivatives. Its Fourier series has coefficients \(\widehat f(p)/|p|^2\); repeated integration by parts gives summability after any fixed number of differentiations. This proves the smooth bounds in Equation (42), including up to the reflected boundary. Adding back the expanding field proves the assertion.

Integrate \(V\) over the faces of each continuum unit cell, with outward sign, to define \(H\). It has divergence \(e_{i_0}\) and the prescribed exterior flux. The central faces have bounded fluxes. Off the central cell, \[|S_{ij}|\le C(1+|i-i_0|)^{-2}+C/G^2, \qquad D_{ij}=V_j(i)+O((1+|i-i_0|)^{-2}+G^{-2}).\] Hence \[ \sum_{i,j}S_{ij}^2\le C,\qquad \sum_{i,j}D_{ij}^2\le\frac1{2\pi}\log G+C. \tag{43}\] The coefficient follows by integrating the squared radial field: \(\int_1^{CG}(4\pi^2r^2)^{-1}2\pi r\,dr=(2\pi)^{-1}\log G+O(1)\). The comparison of the sum and integral has summable derivative error. The cross term with \(E\) is \(O(G^{-1}\sum_{r\le CG}1)=O(1)\), and \(\sum|E|^2=O(1)\). All constants are uniform for the retained cells. The internal part \(F\) has divergence \(e_{i_0}-\xi\). These vectors, for central \(i_0\), are linearly independent because their central coordinates are distinct and \(\xi\) is supported at the boundary. Prescribe the linear flow map on them by this construction and extend it to a basis of the neutral coarse space, using any path flow on the remaining basis vectors. Thus a single quadratic form (39) serves all retained sectors; separate nonlinear choices of precision have not been made.

For these sectors the penalty \(\|e_{i_0}-\xi\|_2^2\) is bounded by a common constant. The crossing cost tends to zero as \(m\to\infty\) for each fixed \(G\). Equations (40)–(43) therefore give at least \(c z_mG^{-a/(4\pi)}\) from each sector in the liminf of the ratio. There are at least \(c'G^2\) retained sectors. Division by the denominator, which tends to one, proves Equation (38), with one positive constant for all \(G\ge G_0\). ◻

Proof. If \(0<a<8\pi\), Theorem 6 gives the hypothesis of Lemma 7. Choose a fixed \(G\) so large that its right side exceeds two. The definition of liminf then gives \(z_{Gm}\ge\frac32z_m\) for all sufficiently large \(m\). Iteration along \(m,Gm,G^2m,\ldots\) contradicts \(0<z_m\le1\). The upper bound \(a\le k\) was proved in Theorem 5. ◻

Proof. Good phases at arbitrarily large scales. Some nonzero mean Fourier coefficient stays positive along a subsequence. Otherwise \(w_{B_m}(ju_m)\to0\) for all \(j\ne0\). Every fixed face profile of integer total \(j\) differs from \(ju_m\) by a test of variance tending to zero, by the face extension of Theorem 5 at \(a=0\). The inequality \(|\mathbb Ee^{iU}-\mathbb Ee^{iV}|\le\sqrt{\mathbb E(U-V)^2}\) would then give the hypothesis of Lemma 7 with \(a=0\), whose growth contradiction is impossible. Hence along a subsequence, extracted diagonally over \(j\), \[w_{B_m}(ju_m)\longrightarrow r_j=r_{-j}\ge0, \qquad r_0=1,\qquad r_l>0\quad\text{for some }l\ge1.\]

Use Equation (39) on a \(Gm\)-square, now with shift \(l u_{Gm}\) instead of the face probability. Cell totals are \(Q_i=n_i-l/G^2\) in numerator sectors, with \(\sum_i n_i=l\), and \(Q_i=n_i\) in denominator sectors. For each fixed \(G\) and sector, the crossing cost vanishes as \(m\to\infty\). The internal shifted profile differs from \(n_i u_m\) by a fixed linear combination of face and volume probabilities of total zero, so its coefficient tends to \(r_{n_i}\). The quadratic penalty gives summable domination of the sector sums, exactly as in Lemma 7. Therefore, with \(b(j)=r_je^{-j^2/2}\), \[ \liminf_m w_{B_{Gm}}(lu_{Gm}) \ge e^{l^2/(2G^2)}\frac{b^{*G^2}(l)}{b^{*G^2}(0)}. \tag{46}\] The ratio on the right tends to one as \(G\to\infty\). Here is a direct local limit argument. Normalize \(b\) to a probability \(p\). It is even, has exponential tails, positive mass at zero, and positive variance since \(p(l)>0\). Let \(d\mathbb Z\) be its support subgroup; then \(l\in d\mathbb Z\). Its characteristic function \(\psi\) has modulus less than one away from the finitely many points \(2\pi j/d\) in a period: equality in the triangle inequality, together with \(p(0)>0\), requires all support phases to equal one. Near each such point, \(\psi(t)=1-\frac12\sigma^2(t-t_0)^2+O(|t-t_0|^4)\). Fourier inversion of \(\psi(t)^N e^{-ilt}\), splitting into fixed small neighborhoods of these points and their complement, gives \(p^{*N}(l)=d(2\pi N\sigma^2)^{-1/2}(1+o(1))\), and the same formula at zero. The complement is exponentially small, and the neighborhood calculation uses \(u=\sqrt N(t-t_0)\) with a Gaussian dominating bound. This proves the ratio claim. Choose \(G\) first and then arbitrarily large members \(m\) of the subsequence in Equation (46). We obtain arbitrarily large scales at which \(w(lu)\) is arbitrarily close to one.

A joint phase bound. Fix \(0<d_0<\pi/10\). Tile a torus by \(m\)-squares, with averages \(\bar h_i\). For any \(\epsilon>0\) we can choose one of the preceding good scales so that, for every collection \(I\) of distinct cells, \[ \mathbb P\{\operatorname{dist}(l\bar h_i,2\pi\mathbb Z)\ge d_0\text{ for all }i\in I\} \le\epsilon^{|I|}, \tag{47}\] uniformly in the number of torus cells. We provide the comparison that makes this a joint, rather than single-cell, assertion. Add independent \(N(0,s^2)\) noises to the phases and wrap them modulo \(2\pi\). Let \(p_T(y)\) be the resulting joint density. Expanding the wrapped normal introduces integers \(j_i\) and the quadratic costs \(\sum_i(y_i+2\pi j_i-l\bar h_i)^2/(2s^2)\). Together with the height energy these are a lattice Gaussian on the height and \(j\) coordinates, shifted only in the \(j\) coordinates. Add a small centered height mass to make it positive definite. Increasing the height precision from the cut free cells to the torus therefore decreases \(p_T(y)/p_T(0)\) by Lemma 4. Let the mass decrease to zero using the translation-invariant clause of that lemma. This yields \[\frac{p_T(y)}{p_T(0)}\le\prod_{i\in I}\frac{p_i(y_i)}{p_i(0)}.\] The wrapped-noise supremum bounds \(p_T(0)\le(C/s)^{|I|}\). Fourier positivity in Equation (31) bounds \(p_i(0)\ge(2\pi)^{-1}\). At a good scale the free-cell phase is within any fixed small distance of zero with probability arbitrarily close to one: use \(\mathbb E\cos(l\bar h)=w(lu)\) and \(1-\cos x\ge1-\cos\delta\) when \(\operatorname{dist}(x,2\pi\mathbb Z)\ge\delta\). Thus, for the noisy free-cell phase, \[\mathbb P\{\operatorname{dist}(l\bar h+Z,2\pi\mathbb Z)\ge d_0/2\} \le C e^{-c d_0^2/s^2}+\delta_m, \qquad \delta_m\longrightarrow0\] along increasingly good scales. Integrate the density-ratio bound over the product of these bad sets. Conditional on original phases which are all bad at threshold \(d_0\), the independent noises retain badness at threshold \(d_0/2\) with probability at least \(\mathbb P(|Z|\le d_0/2)^{|I|}\). Hence the left side of Equation (47) is at most \[\left[ \frac{C}{s\mathbb P(|Z|\le d_0/2)} \bigl(e^{-c d_0^2/s^2}+\delta_m\bigr) \right]^{|I|}.\] Choose \(s\) small first, then a sufficiently good large scale. This proves Equation (47) with the stated uniformity.

Neighbor differences and bad animals. Declare a coarse cell bad if its phase is bad, or if its average differs from any axis-neighbor average by more than \(d_0/l\). For any list of \(q\) pairs of neighboring cells with mutually disjoint end cells, precision comparison after cutting into pair rectangles gives, for each choice of signs \(\eta_j\in\{-1,1\}\), \[\mathbb E\exp\left\{T\sum_{j=1}^q\eta_j (\bar h_{i_j}-\bar h_{i'_j})\right\} \le e^{q\delta_m(T)},\qquad \delta_m(T)\longrightarrow0.\] The free pair difference is a zero-total bounded density test, so the limit follows from Equation (27) at \(a=0\); the bound is uniform in the pair’s location and orientation. Chernoff and the \(2^q\) sign choices give a bound \([2e^{-Td_0/l+\delta_m(T)}]^q\) for simultaneous failures. Choose \(T\) large first and \(m\) large afterward. This exponential base can thus be made as small as desired, simultaneously with Equation (47).

For a prescribed sup-norm-connected set of \(j\) coarse sites, assign each bad site one of its five failure types (phase or one of four directed neighbors). One type occurs at least \(j/5\) times. For an edge type, a greedy extraction gives at least \(j/C_0\) mutually disjoint pairs, since each pair meets only a bounded number of pairs of the same type. For the phase type use Equation (47) directly. Union over the at most \(5^j\) assignments. Taking the preceding bases sufficiently small proves, with \(\eta>0\) arbitrarily small, \[ \mathbb P\{\text{every site of a prescribed sup-connected size-$j$ set is bad}\} \le\eta^j. \tag{48}\] The number of such sets containing a specified root is at most \(C_1^j\): a depth-first traversal of a spanning tree has at most \(2j\) steps, each with at most eight choices. We fix the scale so that \(C_1\eta\) is smaller than a sufficiently small absolute constant. Good axis-neighbors have the same nearest integer label to \(l\bar h/(2\pi)\), because the difference between their labels times \(2\pi\) is at most \(3d_0<2\pi\).

Uniform second moments from coarse connections. Let the coarse torus side be \(M\). For any two prescribed coarse cells and \(1\le t\le M/200\), their radius-\(Ct\) neighborhoods are joined by a good nearest-neighbor path except on an event of probability \[ C e^{-c_0t}, \tag{49}\] where \(C,c_0>0\) do not depend on \(M\) or on the two cells. Here are the geometric and counting details. A missing good crossing of a coarse rectangle requires a sup-connected bad crossing in the perpendicular direction: take the boundary of the set reachable from the starting side by good nearest-neighbor paths and trace its separating boundary. Its length is at least the shorter rectangle side. For bounded aspect ratio, Equation (48) and the preceding rooted count bound its probability by \(Cs^2e^{-cs}\) when the width is comparable to \(s\). Four overlapping strip crossings produce a good circuit in a square annulus of radii \(s,2s\). Request such circuits around each prescribed cell for \(s=t,2t,4t,\ldots\) up to one scale \(s_*\) comparable to \(M/100\). Join consecutive circuits by a crossing of a bounded-aspect rectangle whose ends are inside the smaller circuit and outside the larger. Connect the two largest circuits by a bounded number of circuits and crossing rectangles along a chain of centers spaced by at most \(s_*/4\). All these rectangles have lifts to planar rectangles of bounded aspect, so the same separation argument applies even on the torus. A union bound sums at most \(C\sum_{r\ge0}(2^rt)^2e^{-c2^rt}\), plus a bounded number of \(M\)-scale errors, and proves Equation (49) after adjusting constants. For \(t\) comparable to \(M/200\) one uses the same construction with a single macroscopic scale.

Write \(A_i=\bar h_i\). On successful connection there are good end cells \(i',j'\) within distance \(Ct\) of the prescribed cells \(i,j\), and \(|A_{i'}-A_{j'}|\le2d_0/l\) by label agreement. Let \[B_t=2d_0/l+ \max_{\operatorname{dist}(i,i')\le Ct}|A_i-A_{i'}| +\max_{\operatorname{dist}(j,j')\le Ct}|A_j-A_{j'}|.\] Nearest-neighbor trial paths and the bare Gaussian estimate imply \(\mathbb EB_t^4\le C_m(1+t)^p\) for a fixed integer \(p\) independent of \(M\). Explicitly, connect any two vertices of cells at coarse distance at most \(Ct\) by a path of length at most \(C m(t+1)\). Its flow cost is at most \(C_Jm(t+1)\); averaging such paths bounds the average-difference variance by that number. Its fourth moment is at most a constant times the square of that variance, by the subGaussian tail. There are \(O((1+t)^2)\) possible end cells for each maximum, which proves a polynomial bound, for example with \(p=4\). The constants may depend on the now fixed \(m\).

Partition into the first successful integer value of \(t\). For \(t\ge2\), that event is contained in failure at \(t-1\), whose probability is at most \(Ce^{-c_0(t-1)}\). Cauchy–Schwarz therefore gives \[\mathbb E(A_i-A_j)^2\mathbf 1_{\{\text{first success at }t\}} \le C_m(1+t)^{p/2}e^{-c_0(t-1)/2}.\] The \(t=1\) term is bounded by the same moment bound without its exponential factor. Sum until the final macroscopic scale. On remaining failure, use a torus path of length \(O(mM)\) to bound the fourth moment of \(A_i-A_j\) by \(C_mM^2\) and multiply by the square root of its exponentially small failure probability. This term is uniformly bounded. Thus \(\sup_{M,i,j}\mathbb E(A_i-A_j)^2<\infty\) for large \(M\). Within a fixed cell, the point-minus-average fourth moment is uniformly bounded by paths of length \(O(m)\), so the same uniform second-moment bound holds for every pair of prescribed vertices. Finitely many smaller coarse tori can be absorbed into the constant.

Finally embed any desired \(Q_R\) and its \(J\)-interaction collar in a much larger torus with side divisible by \(m\). Choose a vertex \(y\) outside that collar and impose the centered equalities \(h_z=h_y\) at every vertex outside \(Q_R\). The conditional relative law on \(Q_R\) is exactly the zero-exterior law at temperature \(v^2k\); no interaction wraps across the embedding. Centered equality comparison in Lemma 4 bounds \(\operatorname{Var}(h_0-h_y)\) in this conditional law by its unconditioned torus variance. The latter has the bound just proved, independent of \(R\). Symmetry centers the conditional variable and proves Equation (45). ◻

Proof. At a fixed partition, Theorem 5 and Theorem 6 give joint convergence of the mean-free observations and of \(S\) modulo \(2\pi\), with the latter uniform and independent. Summing translations consequently integrates the constant against \(\,\mathrm dc/(2\pi)\). This assertion first holds for continuous compactly supported functions of the actual constant and a finite list of observations: periodize in the constant and use the stated joint convergence. To remove compact support, put \(\bar h_K=|K|^{-1}\sum_d|d|h_d\). Jensen’s Inequality gives \(\sum_d|d|h_d^2\ge |K|\bar h_K^2\), while \(S-\bar h_K\) is a mean-free smear of bounded bare variance. Summing the Gaussian translation tail and using Gaussian domination with a slightly smaller negative coefficient gives uniform integrability of every polynomial in the finitely many observations. The same argument allows a sufficiently small factor \(e^{\eta S^2}\). For an additional translation-invariant observation \(X_n\) of uniformly bounded bare variance, the one-dimensional Gaussian translation sum also gives the direct bound \[\sup_{h^0}\sum_{m\in\mathbb Z}W(h^0+2\pi m)<\infty\] at each fixed partition and strength. Its weighted polynomial and exponential tails are therefore controlled by the free subGaussian tails of \(X_n\). The band normalizer has a positive limit at these fixed parameters. This proves the asserted extension to such observations; the same bounds give equicontinuity in coefficients on compact sets.

For uniformity as the partition is refined, do the constant integral first in the continuum. The remaining penalty is \(q\) times a positive quadratic form in the mean-free Neumann field, for \(W^q\). It is the squared \(L^2(K)\) norm of its cell projection with its \(K\)-average subtracted. Let \(C_\ell\) be the covariance operator of this projected field. The nonzero Neumann eigenvalues of a square are comparable to \(m_1^2+m_2^2\). Thus its inverse has square-summable eigenvalues, and \[ \sup_\ell\|C_\ell\|_{\mathrm{HS}}<\infty. \tag{56}\] If \(t_j\) are the eigenvalues of \(\lambda C_\ell\), the determinant factor in \(\mathcal P(W^p)/\mathcal P(W)^p\) has logarithm \[\frac12\sum_j\{p\log(1+t_j)-\log(1+pt_j)\}.\] The linear terms cancel, and its absolute value is bounded by \(C_p\sum_jt_j^2\). The explicit one-dimensional constant integrals contribute bounded positive factors. This proves the bound without \(e^{\eta S^2}\). Neumann Poincaré’s Inequality and the control of the constant by \(\bar h_K\) give \[\|v\|_{L^2(F)}^2\le C_\lambda \left(a^{-1}\|\nabla v\|_2^2+ \lambda\sum_d|d|v_d^2\right)\] uniformly under refinement. The Gaussian covariance of \(S\) under the \(p\)-weighted measure is therefore uniformly bounded. Its one-dimensional Gaussian exponential integral proves (53) after decreasing \(\eta\). The fixed partition lattice limit already proved transfers this bound in the iterated-limsup sense asserted.

The cell projections converge strongly in \(L^2(K)\). The displayed coercivity and weak \(H^1\) compactness show that the corresponding minimizers of \(\frac12(v,Tv)-(f,v)\) converge to those for \(T_\lambda\). Their energies converge as well, by using smooth approximants as recovery sequences. This proves convergence of the inverse quadratic forms and hence of the finite Gaussian laws. Uniform exponential moments just established permit moment convergence. Since \(g\) vanishes near \(K\) and \(\partial F\), \(T_\lambda(ag)=-\Delta g\). This proves (55). Increasing \(\lambda\) decreases the continuum covariance, already finite at \(\lambda_0\), so also gives the final fourth-moment bound. ◻

Proof. Extend any variation on the unvisited vertices by zero on visited vertices. On a visited-to-unvisited nearest-neighbor edge the posterior energy pays at least its original conductance times the squared difference of this extension. Thus the nearest-neighbor energy of the zero extension is bounded by a fixed multiple of the posterior energy. The constant depends on \(J\), which contains the nearest-neighbor steps.

In a fixed-factor enlargement of a touched \(\ell\)-cell, stop a visited path when it has first traveled distance \(c\ell\) from its initial point in that cell. One coordinate projection spans \(c'\ell n\) levels. Its jumps are bounded by \(\max_{z\in J}|z|\); there are therefore visited zeros on a fixed positive fraction of these lattice levels. Take a comparable enlarged rectangle with \(N\asymp\ell n\) rows and columns. On each anchored row the one-dimensional telescoping estimate gives \[\sum_{x\text{ in row}}v_x^2 \le C N^2\sum_{e\text{ in row}}(\nabla_ev)^2.\] Let \(\bar v\) be the mean on the rectangle. Since the anchored rows contain a fixed fraction of its vertices, \[N^2\bar v^2\le C\sum_{x\text{ in anchored rows}}v_x^2 +C\sum_x(v_x-\bar v)^2.\] The ordinary rectangle Poincaré Inequality controls the second sum by \(CN^2\) times the gradient energy. Combining the two estimates and dividing counting area by \(n^2\) gives \[n^{-2}\sum_{x\text{ in cell}}v_x^2 \le C\ell^2\mathcal E_{\mathrm{post}}(v; \text{enlarged cell}).\] Enlargements have bounded overlap. Cauchy–Schwarz for \(\sum |d|b_dv_d\) and the variational formula for inverse quadratic forms prove (58). With one fixed \(r\)-sized anchored rectangle, the global Poincaré Inequality controls the mean and energy on the whole connected square, proving the \(V_r\) assertion. For an observation cell use the same proof at scale \(\delta\): its normalized density has squared \(L^2\) norm \(O(\delta^{-2})\), canceling the \(\delta^2\) in the local estimate. For exterior exploration, the filled exterior supplies visited zeros for the extension, and any path entering a touched interior cell supplies the required segment. No planarity of the \(J\)-bond graph is used. ◻

Proof. Given signless data, \(\mathbb E_\sigma S^2=Q_s\). Chebyshev’s Inequality gives \(\mathbb P_\sigma(|S|\le Ct)\ge1-4/C^2\) on the bin. For \(Y=(h,f)\), the Rademacher fourth-moment bound \(\mathbb E_\sigma Y^4\le3(\mathbb E_\sigma Y^2)^2\) and Paley–Zygmund give fixed positive probability that \(|Y|\ge c(\sum_CM_C(f)^2)^{1/2}\). Choose \(C\) so the two events intersect with a fixed positive probability. On each gradient orbit the number of translations satisfying \(|S|\le Ct\) is at most \(C'(1+t)\). A mean-zero \(Y\) is translation invariant and has the ordinary Gaussian upper tail by Lemma 4. This proves (61) and the first bound. For general \(f\) use \(Y=(h,f^0)+S\sum f\) on the same intersection, integrate the tail, and obtain (60). Vanishing bare norm is interpreted by continuity; then the corresponding mean-zero functional vanishes identically.

Let \(S_U\) be the signed complement mean. Lemma 11 and bare domination give \(\mathbb E(e^{zS_U}\mid\mathcal F_C)\le e^{V_rz^2/2}\). Condition on the signless complement and use Jensen: \(Q_U^p\le\mathbb E_\sigma|S_U|^{2p}\). Integrating the subGaussian tail yields \[\mathbb E(Q_U^p\mid\mathcal F_C)\le 2(2V_r)^pp!.\] Expansion of the exponential proves \(\mathbb E(e^{\delta Q_U}\mid\mathcal F_C) \le(1+2\delta V_r)/(1-2\delta V_r)\le3\). For large enough \(t_0(r)\) this implies \(\mathbb P(Q_U\le4t^2\mid\mathcal F_C)\ge1/2\). Since \(m_C^2\le4t^2\), this event gives \(Q_s\le8t^2\). Multiply by \(I(C)\mathbf 1_{m_C\le2t}\), apply conditional expectation, and sum using the least-root disintegration. The argument applies to extended nonnegative integrals, so does not presuppose the finiteness it proves. ◻

Proof. Choose a small fixed diameter threshold \(r\) and a finite family of separated probability box smears \(p_j\) such that every cluster of diameter less than \(r\) misses at least one box. If these small clusters contribute at least \(t^2/2\) to \(Q_s\), assign each one a missed box. For that box \(M_C(p_j-u)=-m_C\), so for some \(j\) its mean-zero squared-mass sum is at least \(ct^2\). Equation (61) bounds this part of the bin by \(C(1+t)e^{-ct^2}\), even without the damping factor.

Otherwise the macroscopic clusters give the pointwise majorant \[\mathbf 1_{t^2<Q_s\le4t^2} \le \sum_{C:\operatorname{diam}C\ge r} \frac{2m_C^2}{t^2}\mathbf 1_{m_C\le2t}\mathbf 1_{Q_s>t^2}.\] Expose each \(C\). Write \(S=\sigma_Cm_C+S_U\). If \(m_C\ge t/2\), the events \(|S_U|\le m_C/2\) and its complement give \[\mathbb E(e^{-\eta S^2}\mid\mathcal F_C) \le e^{-\eta m_C^2/4}+\mathbb P(|S_U|>m_C/2\mid\mathcal F_C) \le Ce^{-ct^2}.\] If \(m_C<t/2\), the condition \(Q_s>t^2\) implies \(Q_U>3t^2/4\), whose conditional probability has the same bound by (62). The remaining intrinsic sum \(\sum (m_C^2/t^2)\mathbf 1_{m_C\le2t}\) is finite after (63); indeed on its reinserted bin its pointwise value is bounded by a constant and the bin has mass \(O(1+t)\). Absorbing this factor proves (64).

To transfer moments, let \(H\) be a squared-mass root or a selected signed sum, and let \(A_t\) be a dyadic bin for \(Q_s\). Conditional Khintchine bounds followed by (60) give \(\mathcal P_n(|H|^{2p}\mathbf 1_{A_t})\le P_p(t)D^{2p}\), for a polynomial \(P_p\) and the appropriate bare-cost scale \(D\). Hölder’s Inequality with exponents \(4,4,2\) gives \[\begin{align*} \mathcal P_n(\widehat W|H|^p\mathbf 1_{A_t}) &\le \mathcal P_n(\widehat W^4e^{\eta S^2})^{1/4} \mathcal P_n(e^{-\eta S^2}\mathbf 1_{A_t})^{1/4} \mathcal P_n(|H|^{2p}\mathbf 1_{A_t})^{1/2}. \end{align*}\] The exponential factors cancel exactly in this application. Equations (53) and (64) make the dyadic sum finite and its tail uniformly small. The lowest bin is handled directly by (60).

For completeness, the bare variance of \(h_d-S\) is \(O(1+\log(1/\ell))\). Cut and compare to a fixed containing square using nearest-neighbor edges. In its discrete Neumann cosine basis the rescaled coefficients of a probability \(\ell\)-cell smear are bounded. Summing inverse eigenvalues up to frequency \(\ell^{-1}\) costs \(C\log(1/\ell)\). Above that frequency, Parseval and the bound \(\|d-u\|_2^2=O(\ell^{-2})\) give a bounded remainder. This proves (65). The same argument works with an independent auxiliary fair-sign sample: exchanging the old and new samples preserves the base measure, so the density moment estimate is unchanged. ◻

Proof. On \(Q_s\le t^2\), expand the difference of the two quadratic mass energies in the original and resampled signs. Orthogonality of Rademacher monomials bounds its squared base integral by \[ C\lambda^2\int\mathbf 1_{Q_s\le t^2} \sum_{C:\operatorname{diam}C\ge r}\sum_{C'\ne C} \left(\sum_d|d|M_C(d)M_{C'}(d)\right)^2\,\mathrm d\nu_n. \tag{67}\] Expose \(C\), retain the intrinsic condition \(m_C\le t\), and drop the complement truncation. The inner sum, after its fair signs are averaged, is the conditional second moment of \(\sum_d|d|M_C(d)h_d^U\). Equation (58) bounds it by \(C\ell^2\sum_d|d|M_C(d)^2\). Reinsert \(Q_s\le C(1+t^2)\) using Lemma 12, and then use (60). The result is \[ \mathcal P_n\left(\mathbf 1_{Q_s\le t^2} |\log W(\sigma')-\log W(\sigma)|^2\right) \le C_{\lambda,t,r}\ell^2D_\ell^2+o_n(1). \tag{68}\] The bin has finite base mass. Equation (53) gives uniform \(L^p\) bounds for both normalized densities there. Split where the logarithmic difference is larger than a fixed \(\delta\); on that set use Hölder and (68), and on its complement use \(|e^x-e^y|\le(e^\delta-1)(e^x+e^y)\). First take the two mesh limits, then \(\delta\downarrow0\). Lemma 13 removes the bin cutoff. This proves total variation convergence, also against functions with the moment bounds of that lemma.

Partition \(F\) into \(r\)-cells. A cluster of diameter below \(r\) meets only a bounded number of neighboring cells. For a bounded density \(f\), Cauchy–Schwarz over those cells bounds its squared mass contribution by \(C\|f\|_\infty^2r^4\) times the sum of the cell squared-mass contributions. There are \(O(r^{-2})\) cells. Equation (60) consequently gives, on a fixed bin, signed second moment at most \(C_t\|f\|_\infty^2r^2(1+\log(1/r))\). Higher fixed moments are bounded there. Interpolation, the Hölder transfer in Lemma 13, and its tail truncation prove the asserted vanishing in moments.

Resample signs first at a threshold smaller than \(r\). The fair second moment of the macroscopic signed contributions is their squared-mass sum, and all terms for nonnegative smears are nonnegative. Remove the still smaller clusters by the preceding bound and apply Lemma 10 to the full smears. This proves (66). Jensen in the signs gives \((\sum M_C(f)^2)^p\le\mathbb E_\sigma|\sum\sigma_CM_C(f)|^{2p}\); the same resampling and small-cluster argument bounds its limit by the Gaussian \(2p\)-moment. The covariance decreases with \(\lambda\) and is finite at \(\lambda_0\), giving the stated uniformity. ◻

Proof. Symmetry and bare domination give \(1\le\mu(e^L)\) and \(\mu(e^{L+tX})\le\exp((\varepsilon+|t|\sqrt V)^2/2)\). Convexity applied at \(t=\pm\varepsilon/\sqrt V\) proves the mean bound; if \(\varepsilon=0\), then \(L=0\) almost surely and the mean bound follows from centeredness. For \(V>0\) and \(z\ge0\), Chernoff with this same bound proves the displayed tail. The case \(V=0\) gives \(X=0\) almost surely and hence the non-tail estimates directly. If \(\varepsilon\le1\), then \[\mu((e^L-1)^2)\le e^{2\varepsilon^2}-1\le C\varepsilon^2.\] Cauchy–Schwarz, the bare fourth-moment bound, and the denominator at least one give the second-moment error \(CV\varepsilon\). Subtracting the squared mean gives the variance error. For \(\varepsilon\ge1\) the tail bound controls the two second moments by \(CV(1+\varepsilon)^2\), which proves (71). Conditional symmetry on \(E\) gives \(\mu(e^L\mid E)=\mu(\cosh L\mid E)\ge1\), proving (72).

For the final assertion let \(\phi(t)=\mathbb E\cos(tY)>0\). The minimal log Hessian of the positive characteristic function from Lemma 4, applied in the \(X\) direction at frequency \(tY\), gives \[\mathbb E[X^2\cos(tY)]\le \mathbb EX^2\,\phi(t) -\frac{(\mathbb E[X\sin(tY)])^2}{\phi(t)}.\] Subtract from the equality at \(t=0\) and integrate the weaker inequality against \(2\,\mathrm dt/(\pi t^2)\) on \((0,\infty)\). The identity \(|y|=(2/\pi)\int_0^\infty(1-\cos ty)t^{-2}\,\mathrm dt\) and Tonelli prove (73). The bias \(|Y|\) preserves inversion symmetry, so its mean for \(X\) is zero. ◻

Proof. Choose finite coverings by small observation squares \(B\) of an inner ring and by squares \(i\) of an outer ring, leaving the clearances shown in Figure 2. Use slightly overlapping enlarged squares so every path crossing the rings hits the coverings also after lattice rounding. The maximum scaled jump tends to zero, so a path from the starting set to the target must meet both coverings, in order.

Let \(C_0\) be the uniform bare variance bound for the remaining average in a cell reached by a path from afar, from Lemma 11. The cell radii can be chosen so small that their continuum prior variances under \(T_\lambda^{-1}\) are at least \(C_0+2c_0\) for some \(c_0>0\), uniformly over \(\lambda>0\). To see this directly, center a truncated logarithm at the cell and cut it off inside a fixed surrounding ball disjoint from the mass. Its mean over a radius-\(\delta\) cell is \(c\log(1/\delta)+O(1)\), while its gradient energy is \(c'\log(1/\delta)+O(1)\). The inverse variational principle therefore makes the cell-average variance grow at least as \(c''a\log(1/\delta)\). This trial pays no mass term. Fix the coverings at this point. Lemma 10 gives the same strict deficit \(c_0\) for fine \(\ell,n\) at each fixed \(\lambda\).

Write the average at \(B\) after exterior exploration as \(h_B=e_B+u_B\), its exposed and unexposed contributions. On omitting the linear tilt, the conditional form is obtained from the prior form by zero-pinning visited vertices and tightening the cable remainders. The partial average \(u_B\) is precisely the original full test after those zero pins. Thus its centered variance \(w_B\) is at most the full prior variance \(v_B\) by Lemma 4, and on a visit to \(B\) it is at most \(v_B-c_0\) by the local bound. Equations (71) and (75) show that restoring the tilt changes the conditional second moment, and the cross term with \(e_B\), by \(o(1)\) after averaging. The needed moments of \(e_B\) follow from Lemma 13. The prior identity \(\mathbb Eh_B^2=v_B\) consequently implies \[c_0\mathbb P_W(B\text{ reached})\le\mathbb E_We_B^2+o(1).\] Every cluster contributing to \(e_B\) is macroscopic, with a threshold fixed by its distance from the starting set. Sign resampling, including the same signless selection of exterior clusters, converts its second moment to the squared-mass sum: \[ c_0\mathbb P_W(B\text{ reached}) \le\mathbb E_W\sum_{C\text{ meeting the starting set}}M_C(B)^2+o(1). \tag{77}\]

A second deficit bounds this right side. For each \(x\in B\) with \(\mathbb E_W|h_x|>0\), bias the prior by \(|h_x|\) and explore only the cluster of \(x\). Its pre-exploration variance for \(h_i\) is at least \(v_i\) by (73). The bias is measurable at completed exploration and hence cancels from the conditional law of the complement. Its centered remaining variance is at most \(v_i\), and is at most \(v_i-c_0\) if the cluster visits \(i\). Its exposed average has absolute value \(M_{C(x)}(i)\). Apply total second moment in this biased law, multiply by \(\mathbb E_W|h_x|\) and the averaging weight of \(x\) in \(B\), and sum in \(x\). The terms without a visit or a tilt cancel in the comparison with \(v_i\). Clusters causing a tilt must reach the mass cells, and clusters contributing at \(i\) must reach \(i\); both are macroscopic because all three regions have positive separation. Their remaining errors are bounded by a constant times the expectation in (76), which tends to zero. A point with \(\mathbb E_W|h_x|=0\) contributes zero directly. The result is the exact weighted deficit \[ c_0\mathbb E_W\sum_{C:C\cap i\ne\varnothing}M_C(B) \le\mathbb E_W\sum_CM_C(B)M_C(i)^2+o(1). \tag{78}\] In particular no estimate on an unsigned sum over all microscopic clusters has been used.

The inverse forms \(T_\lambda^{-1}\) decrease as \(\lambda\to\infty\) to the form inverse with every function zero almost everywhere on \(K\). Indeed bounded energies give weak \(H^1\) compactness and \(\int_Kv^2\to0\); the variational principle gives both the limit and recovery by functions already zero there. This hard band separates the inner and outer regions, so its inverse has zero cross quadratic form for smears on opposite sides. Equation (66) therefore makes \(\mathbb E\sum_CM_C(B)M_C(i)\) tend to zero after (50). Truncate first at \(\max_CM_C(i)\le M\). On this event the right side of (78) is at most \(M\) times this vanishing cross sum. Its tail tends to zero uniformly as \(M\to\infty\): \[\sum_CM_C(B)M_C(i)^2\le\sqrt{Q_B}\,Q_i,\] and Lemma 14 gives arbitrarily high uniform moments after the first two limits. Thus \(\mathbb E\sum_{C\cap i\ne\varnothing}M_C(B)\to0\).

Every exterior-connected cluster meeting \(B\) must visit some outer covering cell \(i\). On \(\max_CM_C(B)\le M\) its contribution to (77) is at most \(M\) times the sum of the just estimated first-mass quantities. The remaining tail is again controlled by the uniform moments of \(Q_B\). Send \(M\to\infty\) after the three limits. Each \(B\) is reached with probability tending to zero, and their finite cover gives the assertion. The proof for a different starting region only replaces the filled exterior by that region; the path anchoring and the two separated coverings have exactly the same hypotheses. ◻

Proof. Consider \(X_n=(h,A_0g_n)\) in the free square with the band. On the success event of Proposition 16, the exposed contribution to \(X_n\) is zero. Its centered conditional variance is at most its massless zero-Dirichlet variance in \(D_n\): the centered posterior has zeros on the whole exterior, possibly further pins and tightened edges, and the remaining positive band precision. Drop the latter extra conditions using Lemma 4. The linear-tilt error vanishes by (71) and (75). On failure, Cauchy–Schwarz bounds \(\mathbb E_W[X_n^2\mathbf 1_{\mathrm{failure}}]\) by its fourth-moment root times the square root of the failure probability. Lemma 10 bounds this fourth moment uniformly after the first two limits and toward the last limit. Hence (55) implies \[ \liminf_{n\to\infty}\operatorname{Var}_D(h,A_0g_n) \ge a\int_D|\nabla g|^2. \tag{81}\] This is valid along every subsequence of lattice sizes: one uses that subsequence in the band argument before the other limits. Free-square cutting gives the opposite bound, with the same gradient energy, for the zero-Dirichlet law. Its logarithmic moment generating function has the covariance quadratic lower bound of Lemma 4. The matching upper bound follows by cutting into cells of relative side \(r\), Taylor-expanding their centered subGaussian logarithmic moment generating functions to second order, and summing the \(O(r^3)\) remainder over \(O(r^{-2})\) cells, as in Theorem 5. Discarded edge flows have vanishing bare cost before cutting. Thus it has the asserted Gaussian limit for compactly supported shifts.

To pass to density tests, use the deterministic inverse-form liminf and recovery argument given later in the proof of Lemma 21, with the zero condition imposed on every face of this square. That part of the proof uses only nearest-neighbor compactness, finite differences, and smooth recovery; it does not use the height transfer proved here. For the area-weighted samples \(f_n\) of a smooth density, let \(A_{0,D_n}\) denote the zero-exterior Dirichlet operator. Its inverse quadratic form therefore converges to \((f,(-\Delta_D)^{-1}f)\). Put \(v=(-\Delta_D)^{-1}f\) and choose \(g_j\in C_c^\infty(D)\) converging to \(v\) in \(H_0^1(D)\). The inverse variational identity gives, at each fixed \(j\), \[\lim_{n\to\infty} (f_n-A_{0,D_n}g_{j,n}, A_{0,D_n}^{-1}(f_n-A_{0,D_n}g_{j,n})) =\int_D|\nabla(v-g_j)|^2.\] Here the cross pairing and the sampled shift energy converge by finite differences; the inverse term is the deterministic limit just cited. Letting \(j\to\infty\) makes the bare error vanish, so Gaussian domination transfers the Laplace limits. This proves the full density-test statement and joint convergence by real linear combinations. In this argument the band constant was integrated, rather than fixed; the resulting Dirichlet law is consequently unprojected.

For a torus shift supported in a small square chart, impose the equalities that all heights outside that square have one common value. They can be imposed by diverging positive quadratic penalties. Modulo the additive lattice constant, the resulting law is the zero-Dirichlet square law. The observable annihilates constants, so comparison gives the same lower bound (81). Free-cell cutting gives the upper bound \(a\int|\nabla g|^2\) for every smooth torus shift. The limiting difference between this upper quadratic form and the actual covariance is positive semidefinite on each finite list, on every convergent subsequence. A vector at which it vanishes belongs to its null space. Each chart-supported shift has full variance convergence by the liminf and limsup bounds just proved. A finite partition of unity writes an arbitrary smooth torus shift as a finite sum of such shifts. On every covariance-convergent subsequence, all their defect diagonal entries therefore vanish simultaneously. Positive semidefiniteness annihilates their span, and the arbitrariness of the initial subsequence proves full variance convergence for the sum. For \(a=0\) the upper bound alone gives saturation. The same small-cell upper bound and Hessian lower bound give the Gaussian Laplace limit for these shifts.

Given mean-zero \(f\), solve \(-\Delta g=f\) on the torus with zero mean. On samples, \(A_0g_n\) differs from the area-weighted sample \(n^{-2}f_n\), with its sampled mean subtracted, by a density error tending to zero in scaled \(L^2\). This follows from the quadratic symbol of \(A_0\) and smooth finite differences. The discrete Poincaré Inequality and bare domination control the error in every exponential moment with a fixed argument. This proves (79) and all joint smooth projections.

Finally consider the Fourier coefficient at \(m\in\mathbb Z^2\) of \(H_n\). If \(m\) is zero modulo \(n\), mean subtraction makes the coefficient zero. Otherwise replace \(m\) by its representative \(\tilde m\) with coordinates of magnitude at most \(n/2\). The nearest-neighbor lower bound on the symbol gives its bare second moment at most \(Ck/|\tilde m|^2\le Ck\), uniformly in \(n,m\). Consequently \[\sup_n\mathbb E\|H_n\|_{H^{-2}(\mathbb T^2)}^2 \le Ck\sum_{m\in\mathbb Z^2}(1+|m|^2)^{-2}<\infty.\] Compact embedding into \(H^{-3}\) gives tightness; the smooth joint projections identify every limit and hence the full sequence. A nonzero smooth test identifies \(a\) uniquely. Restoring the normalization in Equation (18) proves (80). ◻

Proof. When \(a=0\), Proposition 9 gives the uniform zero-boundary point-variance bound. Conversely suppose the supremum on the right were finite, say at most \(C_0\). For any vertex \(x\) of any zero-boundary square, translate that square by \(-x\) and enlarge to a centered square. Relaxing the additional zero pins increases variance, so \(\mathbb Eh_x^2\le C_0\). Cauchy–Schwarz then gives \(\operatorname{Var}(h,f)\le C_0(\sum_x|f_x|)^2\) for every smear.

Choose a compactly supported logarithmic pole in an interior disk, smoothed at radius \(\delta\). More explicitly, let it equal \(\log(1/|z|)\) on \(2\delta\le|z|\le r_0\), smooth it radially to a constant at \(|z|\le\delta\), and multiply by a fixed cutoff supported strictly inside the square. Its gradient energy grows as \(2\pi\log(1/\delta)+O(1)\), while \(\int|\Delta g_\delta|\) stays bounded: the inner radial Laplacian has bounded total mass, and the outer cutoff is fixed. At each fixed \(\delta\), finite differences give a uniform bound on \(\sum_x|(A_0g_{\delta,n})_x|\) as \(n\to\infty\). Equation (81) would therefore bound \(a\int|\nabla g_\delta|^2\) by a constant independent of \(\delta\). This is impossible if \(a>0\). ◻

Proof. Finite precision comparison proves monotonicity and the bare bound \(a\le k\). The affine cutting estimate used to define \(a\) gives, for each fixed cell size \(m\), \[\sqrt{a(k)}\le\sqrt{a_m(k)}+C\sqrt{k/m},\] with harmless vanishing normalization errors if the finite affine energy is not exactly one. The constants are uniform for \(k\) in compact subintervals of \((0,\infty)\). Each finite-cell quantity is continuous in \(k\), by absolute Gaussian summation, and \(a_m(k)\to a(k)\). Taking the upper limit as \(k'\to k\) and then \(m\to\infty\) proves upper semicontinuity.

For strict growth let \(A_b\) be a free-cell Laplacian and let \(g\) be a nonconstant affine shift. Work modulo constants, put \(X=(h,A_bg)\), \(E_g=(g,A_bg)\), and \(U(k)=\operatorname{Var}X/E_g>0\). Differentiation of its finite Gaussian sum gives \[U'(k)=\frac{\operatorname{Cov}(X^2,(h,A_bh))}{2k^2E_g}.\] Diagonalize the positive form \(A_b\) on the mean-free space. The paired fourth-cumulant positivity of Lemma 4 bounds the numerator below by \(2(q,A_bq)\), where \(q=\operatorname{Cov}(h,X)\). Since \((q,A_bg)=\operatorname{Var}X\), Cauchy–Schwarz in the \(A_b\)-energy gives \((q,A_bq)\ge(\operatorname{Var}X)^2/E_g\). Thus \(U'\ge U^2/k^2\). Integrating the derivative of \(1/U\) and passing to growing affine cells proves (83). Its right side is positive since \(a(k_1)\le k_1\), and is strictly smaller than \(1/a(k_1)\), proving strict increase.

The gap in Corollary 8 is \(a\in\{0\}\cup[8\pi,\infty)\). Monotonicity makes a nonempty positive set an upper interval. At its infimum, upper semicontinuity and the gap force a positive value; \(a\le k\) implies \(k_c\ge8\pi\). Corollary 18 identifies this same infimum with the stated physical criterion and includes its endpoint. Nonemptiness and the exact endpoint value, differentiability above it, and the infinite endpoint slope are established in the later weak-height and endpoint arguments; none of those conclusions is assumed here. ◻

Proof. For smooth functions on a neighborhood of the closure, the discrete energies converge to \(\int|\nabla v|^2\): the symbol of \(A_0\) has quadratic part \(|p|^2\), internal Riemann sums converge, and the number of edges within a fixed lattice distance of the finitely many faces and corners is \(O(s)\), each with squared smooth difference \(O(s^{-2})\). Nearest-neighbor edges control the \(H^1\) norm of piecewise affine interpolants on each constituent square, modulo a componentwise constant. Conversely all fixed-range differences are bounded by a fixed multiple of nearest-neighbor energy on local face-connected patches: a retained jump can be replaced by a nearest-neighbor route of bounded lattice length in one of the finitely many local square-sector patterns. There are no opposite-only contacts. Thus bounded energies give weak \(H^1\) compactness on the squares, strong \(L^2\) compactness, and matching traces across retained faces. The difference quotients for each step pass weakly to their directional derivatives. Their sum and the lattice symmetries give the continuum liminf energy with its exact coefficient.

Here is the density statement with pins and corners. First truncate an \(H^1\) function to bounded magnitude; this decreases gradient energy and preserves every zero region. Around each polygon vertex use a cutoff which is zero within radius \(\delta^2\), one outside radius \(\delta\), and linear in \(\log r\) between. Its squared gradient integral is \(O(1/|\log\delta|)\). For a bounded function this removes all vertices at vanishing energy and \(L^2\) cost. Away from them, a pinned straight face is treated by the zero-trace \(H^1\) approximation on a half-rectangle, extending by zero into the pin before mollification. On a free face use even local extension and mollification. Interior patches are mollified directly. A finite partition of unity combines these approximations. They show that restrictions of smooth ambient functions are dense without pins, and functions smooth away from vertices and vanishing in a neighborhood of the pinned region are dense with pins. Sampling these functions gives recovery sequences. Observation-cell averages are continuous under strong \(L^2\) convergence. The same liminf and recovery proof applies to a square with zero trace on every face, using zero extension on each face; this is the deterministic \(H_0^1\) inverse-form assertion used in Theorem 17. Neumann Poincaré and cell masses, or a fixed positive-area hard pin, control constants. Minimizing the coercive forms minus a linear density source now proves inverse convergence by the liminf and recovery statements. The same reasoning, modulo constants, treats zero-total densities in an unpinned component.

For the height law, enclose the component in a larger free square. For a smooth ambient shift, separate its gradient observation into the retained pieces on the component and on the finitely many remaining polygonal pieces. The omitted seam flows have vanishing bare squared cost by their \(O(s)\) number and \(O(s^{-1})\) coefficients. Cutting increases covariances; the resulting independent component variances each have the continuum gradient upper bound, by further subdivision into squares. Their sum already attains that bound in the enclosing free square by Theorem 5. Write \(U_i\) for the retained pieces and \(V_{i,s}\) for their partial gradient-observation variances. For each component \(i\), subtract the limsup upper bounds for the other finitely many pieces from the full limiting enclosing-square variance. The cutting inequality gives \(\liminf_sV_{i,s}\ge a_h\int_{U_i}|\nabla g|^2\). Together with its upper bound this proves full variance convergence on that component, along every admissible mesh sequence. The density of ambient shifts just proved and bare error bounds give full variance convergence on its full \(H^1\) energy space. Cutting into small cells and the second-order logarithmic moment expansion used in Theorem 17 give the Gaussian Laplace limits. The form approximation transfers them to the stated density observations.

Finally consider a character with nonzero integer total charge on one component, including fixed mean-free observations if desired. Extend its source by zero to the enclosing square. The dual shifted-ratio precision comparison from Lemma 4 bounds its positive characteristic function in the cut component by that in the uncut square. This can first be done with a tiny positive mass, then passed to the translation-modulo law because its total is an integer. Approximation by bounded-density profiles is justified by the bare norm bounds. The enclosing character tends to zero by Theorem 6. The same argument with joint mean-free sources, or Fourier approximation of a bounded function of those sources, proves independence of the uniform fractional mean. Distinct free components were independent from the outset. ◻

Proof. It suffices first to use a smooth shift \(g\) vanishing on a neighborhood of the pin \(p\), but allowed to reach the outer free walls. Choose a subband of positive width between \(p\) and \(\operatorname{supp}g\), separated by positive gaps from both. It may follow level strips of sup distance to the whole pin region, clipped to the component. Select levels on a sufficiently fine fixed rectangular grid, then subdivide into mass cells. All clearances are fixed before the fine lattice limit. Where the pin has several components, use the union of these strips and retain those connected pieces through which a path to the support is possible.

Start with the component unpinned, put the band weight on it, and explore from the entire pin region. Lemma 21 supplies the free Gaussian and fractional-mean inputs to Lemma 10. Its Hilbert–Schmidt bound follows by cutting the domain into finitely many squares: their \(H^1\)-to-\(L^2\) inclusions have fourth-power summable singular values, which bounds the square of the Neumann inverse after the component constant is removed. The band controls that constant as before.

We verify the local hypotheses of the exploration lemmas at free walls and junctions. A fixed-factor enlargement of a touched small cell within the component is a connected patch of a bounded number of comparable rectangles, joined by straight faces of comparable length. Its Poincaré constant is \(C\ell^2\): apply the rectangle inequalities and bound the differences of their means across the common faces by the same gradient energy. Truncate a visiting path when it spans a comparable cell distance. Some coordinate then spans \(c\ell s\) levels. Bounded jumps and the bounded number of local rectangles put a positive fraction of anchored rows or columns into one of these rectangles. The one-dimensional estimate in Lemma 11 anchors its mean; the patch Poincaré Inequality anchors the remaining rectangles. At a corner use all adjacent sectors of the same component. The no-opposite-only convention ensures that a retained jump lies in a sector or crosses between sectors joined through retained faces. A filled starting pin region itself supplies the initial zero extension for variations. Thus the same local inequality (58) holds, and a fixed macroscopic path anchors the whole component. Small-cell mean-free bare variance still costs \(C(1+\log(1/\ell))\): compare first to a containing fixed square patch using its Neumann modes, and then use the component Poincaré Inequality to compare means.

The two coverings can be made of subdivisions reaching the free walls. A logarithmic trial restricted from the plane to a straight or sector patch has cell average and gradient energy proportional to \(\log(1/\delta)\), with positive geometric constants; it therefore gives the large pre-exposure variance required in Proposition 16. The anchored post-exposure bound is independent of the cell radius. For the size-bias step take the starting covering on the test side, with the other covering on the pin side; both are away from the mass. Every nonzero tilt or exposed contribution between them again requires a macroscopic cluster. All of Lemmas 12–15 and both deficits (77), (78) therefore apply with the same limit order. The hard-band inverse separates the sides, so reach of the support has vanishing probability.

The weak Laplacian source of \(g\) is the energy functional \(F_g(v)=\int_U\nabla g\cdot\nabla v\), including its free-face contributions. Its discrete observation \(X_s=(h,A_{0,U_s}g_s)\) is translation invariant and has bounded bare variance \[\operatorname{Var}_{\rm bare}X_s \le k_h(g_s,A_{0,U_s}g_s)=O(1).\] At each fixed band partition and strength, include \(X_s\) in the joint free-component limit from Lemma 21. The translation-sum bound in the proof of Lemma 10 and this bare bound give uniform integrability of every fixed power of \(X_s\) under the normalized band law. Since every mass-cell average of \(g\) is zero, the continuum projected band form on \(U\), \[\mathcal T_{\ell,\lambda}(v,w) =a_h^{-1}\int_U\nabla v\cdot\nabla w +\lambda\sum_d|d|v_dw_d ,\] satisfies \[\mathcal T_{\ell,\lambda}(a_hg,v)=F_g(v).\] Consequently, at every fixed \(\ell,\lambda\), \[\lim_{s\to\infty}\mathbb E_W X_s^2=a_h\int_U|\nabla g|^2,\qquad \lim_{s\to\infty}\mathbb E_W X_s^4 =3\left(a_h\int_U|\nabla g|^2\right)^2 .\] The fourth-moment limit is independent of \(\ell,\lambda\). On successful exploration the centered posterior has at least the desired pin, so its variance is at most the massless mixed variance. Equation (75) controls the tilt errors. On failure, Cauchy–Schwarz bounds the second moment by \((\mathbb E_WX_s^4)^{1/2}\mathbb P_W(\mathrm{failure})^{1/2}\). The displayed fourth-moment limit and vanishing reach make this term vanish in the prescribed order, with no finite-mesh uniformity in \(\lambda\) required. Run this comparison on any lattice subsequence and then take the stated band limits. It gives the liminf lower bound \(a_h\int|\nabla g|^2\) for the actual hard-pin variance. Free cutting gives the matching limsup, hence full variance convergence. Free cutting also gives the upper logarithmic moment bound, and minimal Hessian gives the matching lower logarithmic moment bound. The pinned density and form approximation of Lemma 21 extends these conclusions to the asserted tests.

An admitted finite observation penalty can now be integrated against these joint massless limits. In a pinned component bare Gaussian bounds give uniform integrability for polynomial and fixed exponential insertions. In free components use the independent uniform fractional means and their translation sums. The common lower bound on the quadratic form of the released constants gives a uniform finite-dimensional Gaussian theta bound on these sums, as in Lemma 10. On a fixed compact coefficient and center set the limiting normalizers are continuous and bounded away from zero. The same tail bounds give equicontinuity of the integrals there. This proves the last assertion and its stated compact-set uniformity. ◻

Proof. For the local cell form, Poincaré on each permitted rectangle and the upper and lower area bounds give the uniform norm equivalence \[ c\left(\|v\|_2^2+\|\nabla v\|_2^2\right) \le\mathcal T_U(v,v) \le C\left(\|v\|_2^2+\|\nabla v\|_2^2\right). \tag{91}\] The discrete version follows from the same cell inequality, nearest-neighbor domination, and the bounded routes for retained \(J\)-bonds. The proof applies at any fixed positive gradient coefficient. More quantitatively, if \(a'\in I=[a_-,a_+]\Subset(0,\infty)\), then \[\min\{1,a_h/a_+\}\mathcal T_{U,a_h} \le\mathcal T_{U,a'}\le \max\{1,a_h/a_-\}\mathcal T_{U,a_h}.\] Thus the coercivity, cutoff absorption, and layer-contraction constants below can be chosen uniformly on \(I\), in the continuum and on the discrete graphs. In the local estimates below, \(\mathcal T_U\) denotes the chosen local cell form, with its coefficient held fixed while topologies are compared. Each vector in \(V_p\) minimizes energy subject to its values on \(p\); it is therefore weakly harmonic for its specified form off \(p\).

We give the local estimates behind compactness and locality. Let \(v\) be harmonic away from a source region. Test its equation by \(\chi^2v\), with a Lipschitz cutoff supported away from the source. Expanding the gradient term and applying \(2|ab|\le\varepsilon a^2+\varepsilon^{-1}b^2\) absorbs half the gradient term and bounds the rest by the \(L^2\) mass on the cutoff layer. The average penalty in a cell wholly inside the constant part of the cutoff is nonnegative. Cells meeting its transition are contained in a bounded enlargement of the layer, and their average cross terms are bounded by that layer’s \(L^2\) mass. Thus \[\mathcal T_U(v,v;\text{inner patch}) \le C\|v\|_{L^2(\text{enlarged patch})}^2\] whenever the inner patch is separated from sources by a fixed positive margin. Here local energy includes entire cells, and enlarging by finitely many cell layers absorbs the average terms. This proof uses no integration boundary term, so is valid up to free walls. Discrete differences obey the same expansion, with cutoff differences \(O(s^{-1})\) and a fixed enlargement by the interaction range.

Using instead a cutoff which retains \(v\) near its source and removes its distant tail, harmonic minimality and (91) imply \[\mathcal T_U(v,v;\operatorname{dist}\ge r+c_1) \le C\mathcal T_U(v,v;r\le\operatorname{dist}\le r+c_1).\] Equivalently each extra layer reduces the remaining tail by a factor at most \(C/(1+C)<1\). Iteration proves exponential energy and \(L^2\) decay. A source in one observation cell obeys the same bound outside that cell. Distances may be measured in the retained face-cell graph. For estimates on changing topologies they may also be measured in the uncut inclusion-region graph: a retained path can only become longer on removing bonds, while a distance cutoff in the inclusion graph still has uniformly bounded differences on retained bonds. Polynomial ball volumes control the number of layer cells. These cutoff estimates use only the local cell and route bounds and the graph distance, so they hold on a finite periodic identification with the quotient inclusion distance as well.

To obtain a trace estimate, cut a bounded patch into its rectangular mass cells. The inclusion from broken cell \(H^1\) into \(L^2\) has singular values bounded by \(Cj^{-1/2}\), with the appropriate cell-count factor, by the Neumann cosine eigenvalues on each rectangle and the uniform side-length bounds. The same bound holds uniformly on discrete rectangles: the nearest-edge cosine eigenvalues control all frequencies up to the finite lattice cutoff. Apply this inclusion on a first patch, the harmonic interior bound on a smaller patch, a second such inclusion, and then another interior bound. The singular-value product inequality \(s_{2j-1}(AB)\le s_j(A)s_j(B)\) gives \[ s_j(\text{restriction of harmonic extension}) \le Cj^{-1}. \tag{92}\] This is Hilbert–Schmidt, with uniformly vanishing squared singular-value tails. Leave an additional distance margin before these two restrictions; the exponential operator bound already proved then multiplies the Hilbert–Schmidt bound. Projection onto \(V_H\) is unchanged when a function is multiplied by a cutoff equal to one near \(H\), since their difference vanishes on \(H\). The estimates therefore apply to the cross projections. For a fixed finite observation penalty on a fixed domain, Poincaré and control of the released constants still give coercivity with fixed constants. If a pin gap is smaller than a cell, or an observation crosses a cutoff patch, retain the finitely many involved averages as extra coordinates \(F(v)\in\mathbb R^m\) in the interior estimate: \[\|v\|_{H^1(\text{inner})} \le C\bigl(\|v\|_{L^2(\text{outer})}+|F(v)|\bigr).\] The same estimate holds discretely. Each augmented inclusion \(H^1\oplus\mathbb R^m\to L^2\oplus\mathbb R^m\) has \(s_{j+m}\le Cj^{-1/2}\). The two restrictions therefore give \(s_{2j+O(m)}\le Cj^{-1}\), which is the same square-summable tail after a fixed index shift. This proves compactness and (92) for the modified fixed form. Its constants may depend on that penalty; no exponential uniformity as a band rank or strength increases is asserted.

The intersection \(V_p\cap V_H\) is zero. In fact choose a smooth cutoff equal to one on \(p\) and zero near \(H\). It decomposes any \(H^1\) test into a function vanishing on \(p\) and a function vanishing on \(H\). A vector in both orthogonal complements is orthogonal to every test, hence zero. Compactness of \(C\) then gives \(\|C\|<1\): equality would have an attaining unit vector whose two projections are both isometries, placing it in the intersection. For the local cell masses with the stated fixed positive pin clearance, the cutoff can be chosen with uniformly bounded multiplier norm \(\|\chi v\|_{\mathcal E_U}\le M\|v\|_{\mathcal E_U}\), by the uniform local coercivity. For a unit \(v\in V_p\), \(v-\chi v\) vanishes on \(p\) and \(\chi v\) vanishes on \(H\), so \[1=\langle v,\chi v\rangle_{\mathcal E_U} =\langle(\operatorname{Id}-\Pi_H)v,\chi v\rangle_{\mathcal E_U} \le M\|(\operatorname{Id}-\Pi_H)v\|_{\mathcal E_U}.\] Thus \(\|C\|^2\le1-M^{-2}\) uniformly. The determinant formula proved below and the Hilbert–Schmidt bound then give \[D_U^{\mathrm{cg}}(H,p) \le\frac{\|C\|_{\mathrm{HS}}^2}{2(1-\|C\|^2)} \le\frac{M^2}{2}\|C\|_{\mathrm{HS}}^2,\] which supplies the uniform determinant prefactor in the local cell estimates. For a fixed nonlocal penalty the preceding pointwise gap and penalty-dependent constants suffice. For finite lists of pin observations the Gaussian Schur formula gives \(-\frac12\log\det(\operatorname{Id}-C^*C)\). Exhausting the observation subspaces by smooth density probes gives (88), since \(C^*C\) is trace class and has norm below one. Densities in \(p\) are total for \(V_p\): orthogonality to all their representers means the tested energy function is zero a.e. on \(p\). The analogous statement holds for \(H\). This totality and Schur-formula exhaustion are in the specified energy space, so the same argument gives the displayed determinant for every modified fixed form in the statement.

We spell out why lattice refinement loses no trace. Lemma 21 gives convergence of the Gram matrices of every fixed finite list of density representers and of their pinned projections. A sequence of unit-energy discrete vectors harmonic off \(p\) has weakly convergent interpolants; testing on the dense shifts from that lemma shows that its weak limit is harmonic off \(p\). Conversely, projecting a fixed representer onto \(V_p\) converges strongly in energy, by convergence of its variational minimum and norm. If such a bounded sequence has weak limit zero, compact \(L^2\) convergence and the interior estimate make its energy near \(H\) tend to zero. For a fixed observation penalty the finitely many averages converge too, so cause no exception. The cutoff observation just used then gives norm convergence to zero of its projection onto \(V_H\). It follows by contradiction that residual cross projections outside a growing finite probe list have uniformly small operator norm in the lattice limit. Equation (92) makes their Hilbert–Schmidt norms uniformly small as well: first retain finitely many singular directions and use the operator bound there, then sum the uniform \(j^{-2}\) tail. Thus the cross products converge in trace norm and are eventually bounded in norm by a fixed number less than one. The Fredholm logarithm, given by \(-\log\det(\operatorname{Id}-K)=\sum_{m\ge1}\operatorname{tr}(K^m)/m\), converges. This proves the full discrete Gaussian determinant limit. The exponential Hilbert–Schmidt estimate proves the claimed decay of the determinant when the separation is large; for bounded separation its fixed geometry bound suffices.

It remains to verify the local cell statements about \(\mathsf E_{U,a'}\). Minimizing (85) gives \(v_y=T_{U,a'}^{-1}B^*y\) and \(E_U(y;a')=y^T(\operatorname{Id}-BT_{U,a'}^{-1}B^*)y\). The source-tail estimate applied to \(y=e_i\) gives exponential off-diagonal entries. Restoring bonds adds a positive form, so the change of the minimum energy is positive semidefinite. If the added bonds are at a core, use as a competitor the old minimizer cut to zero in a neighborhood of that core. Its energy excess over the old minimum is exactly the squared norm of the correction in the old form, since the old minimizer is stationary. The correction is supported near the core and has energy bounded by the tail estimate. There is no new cross-bond energy because it vanishes on both ends of every restored bond. This bounds diagonal changes by \(Ce^{-c\operatorname{dist}(i,\mathrm{core})}\); positive-form Cauchy–Schwarz bounds their entries by the geometric mean of diagonal bounds. Let \(D_{\rm full}\) and \(D_{\rm patch}\) be the full and open-patch core changes, extending the latter by zero. If the patch retains the whole component and all its ambient bonds, they agree. Otherwise the core estimate bounds their difference by \(Ce^{-c(d_i+d_j)}\), where \(d_i\) is the distance from index \(i\) to the core. When both indices lie in the patch, cutting its boundary is a direct sum because the mass is cell-local. The same difference is then a difference of two boundary changes and is bounded by \(Ce^{-c(b_i+b_j)}\), where \(b_i\) is the distance to that boundary. The guard gives \(d_i+b_i\ge r-O(1)\), so the minimum of the two bounds has a factor \(e^{-cr}\). If an index is outside the patch, the patch entry is zero and that index has core distance at least \(r-O(1)\), giving the same factor from the core bound. Independently, the full and patch energy matrices have the off-diagonal spatial decay just proved. Taking a geometric mean with that bound and weakening \(c\) proves (89) for every entry. Retaining another fraction of the core bound permits summation over cores with polynomial cell counts.

Without hard pins a componentwise constant \(y\) has zero minimum energy; symmetry therefore gives zero row sums for the matrices and their differences. A symmetric zero-row-sum matrix \(F\) obeys \[y^TFy=-\frac12\sum_{i,j}F_{ij}(y_i-y_j)^2.\] For each pair choose a shortest nearest-cell path in the inclusion region. Its length times the sum of squared differences along it bounds \((y_i-y_j)^2\). Pairs of length \(l\) using a given edge have both endpoints in its \(l\)-ball, whose volume is polynomial. Summing the exponential entry bound over \(l\) proves (90); no separate flow-congestion assertion is needed. At junctions bounded microscopic jumps can be shadowed by face-cell paths with bounded excess length, precisely because opposite-only contacts are excluded. For hard-wall changes the constant need not lie in the null space, so one uses the proven magnitude bounds directly. ◻

Proof. It suffices to treat one component meeting both regions. Place a band \(K\) between them with positive clearances, and replace the unit-cell mass by \(W_{\ell,\lambda}\). Denote the resulting interaction by \(D_W\). For a pin set \(Q\in\{\varnothing,H,p,H\cup p\}\), let \(Z_{M,Q}\) be its partition sum with mass penalty \(M\), where \(M=U\) denotes the unit-cell mass and \(M=W\) the band mass. Direct cancellation gives the exact identity \[ D_U-D_W= \log\frac{Z_{U,H\cup p}}{Z_{W,H\cup p}} +\log\frac{Z_{U,\varnothing}}{Z_{W,\varnothing}} -\log\frac{Z_{U,H}}{Z_{W,H}} -\log\frac{Z_{U,p}}{Z_{W,p}}. \tag{94}\] Each ratio is a change between two fixed finite-rank confining penalties on the same massless domain. For \(Q\ne\varnothing\), it is the ratio of their expectations in the normalized massless mixed law. For \(Q=\varnothing\), use the sigma-finite per-translation-orbit normalization. Its common massless factor cancels inside this ratio. Proposition 22 and Lemma 21 show that, at fixed band partition and strength, every ratio has precisely its Gaussian limit. No normalization belonging to two different pin sets has been compared. By Proposition 23, the corresponding Gaussian identity can be written with the full continuum determinants. Consequently \[ D_U-D_W\longrightarrow D_U^{\mathrm{cg}}-D_W^{\mathrm{cg}} \quad\text{at each fixed }\ell,\lambda. \tag{95}\]

To remove the band, explore all clusters meeting \(p\) under its unpinned band law \(\mu\). On the success event \(\mathcal S\) that \(H\) is untouched, the centered posterior \(\mu_{0,\mathcal F}\) has zero pins throughout the visited set, tightened outgoing cable remainders, and the restricted centered band mass. It is obtained from \(\mu(\,\cdot\mid h_p=0)\) by further zero pins and increased precision. Its probability of the untouched pin \(h_H=0\) is therefore at least \(\mu(h_H=0\mid h_p=0)\). The actual posterior has the linear tilt from the exposed cell averages, with bare variance at most \(\varepsilon^2\) in (74). The event \(h_H=0\) is inversion symmetric, so (72) gives the relative bound \[\mu(h_H=0\mid\mathcal F) \ge e^{-\varepsilon^2/2}\mu(h_H=0\mid h_p=0) \quad\text{on }\mathcal S.\] Averaging, for every \(\delta>0\), \[ e^{-D_W} \ge\mathbb E_W[\mathbf 1_{\mathcal S}e^{-\varepsilon^2/2}] \ge e^{-\delta/2} \mathbb P_W(\mathcal S\cap\{\varepsilon^2\le\delta\}). \tag{96}\] The mixed-domain verification in Proposition 22 permits Proposition 16 and (75) here. First the tilt vanishes at each fixed \(\lambda\); then the reach probability vanishes as \(\lambda\) increases. Since \(D_W\ge0\), sending \(\delta\downarrow0\) proves that \(D_W\to0\) in the three prescribed limits. This is a relative bound, independent of how small either microscopic pin probability is.

The same conclusion holds for \(D_W^{\mathrm{cg}}\). A direct way to avoid separate divergent determinants for an infinitely strong local mass is to run the proof on real Gaussian lattices with coefficient \(a_h\). Their translation integral is Lebesgue measure, all free and mixed form limits follow from Lemma 21, and the cable-sign, path, tail, and tilt proofs are unchanged. Replace each pin probability by its joint density at zero. Its centered density increases under added precision. The tilt ratio at zero obeys (72), since the conditional centered Gaussian on that pin still has a symmetric linear tilt. To justify averaging posterior densities, first average probabilities of small boxes around zero on \(\mathcal S\), divide by their volumes, and use Fatou and the continuous Gaussian densities. This proves (96) for densities. Proposition 23 passes their full-pin interactions to the continuum at every fixed band partition. Thus \(D_W^{\mathrm{cg}}\to0\) as well. Combining these two vanishing statements with (95) proves (93): take a lattice limsup at fixed \(\ell,\lambda\), then refine the band, and finally increase its strength. ◻

Proof. Let \(X_s\) be one real smear, \(b=h_p\), and put \[m_s(b)=\mathbb E(X_s\mid b),\qquad v_{p,s}=\operatorname{Var}(X_s\mid h_p=0).\] Denote its full and pinned Gaussian limiting variances by \(v\) and \(v_p\). Conditioning on \(b\) gives an affine Gaussian-lattice law with the same quadratic part as the centered \(p=0\) law, including when the affine shift has components normal to the remaining lattice span. Minimal covariance in Lemma 4, also after a further real tilt \(tX_s\), gives \[ \operatorname{Var}(X_s\mid b;\text{ tilt }tX_s)\ge v_{p,s}. \tag{101}\] Let \(Z_s^{(N)}\) be the first \(N\) pin smears and let \(L_s^{(N)}\) be their linear combination using the limiting Gaussian regression coefficients for \(X\). Redundant directions may be removed, or the inverse Gram matrix taken on its range. Conditional expectation and variance decomposition give the exact bound \[\begin{align*} \mathbb E|m_s-L_s^{(N)}|^2 &=\mathbb E|X_s-L_s^{(N)}|^2 -\mathbb E\operatorname{Var}(X_s\mid b)\\ &\le\mathbb E|X_s-L_s^{(N)}|^2-v_{p,s}. \tag{102}\end{align*}\] The full and pinned Gaussian limits are given by Proposition 22. If \(f\) is the energy representer of \(X\) and \(P_N\) projects onto the first \(N\) pin representers, the limiting right side is \(\|(\Pi_p-P_N)f\|_{\mathcal E_U}^2\). It tends to zero by totality of the pin density probes. Thus \(m_s\) is approximated in \(L^2\) by finite Gaussian linear predictions, and its limiting law is Gaussian of variance \(v-v_p\).

Integrating (101) twice in \(t\) yields \[ Y_s(b):=\mathbb E(e^{X_s}\mid b) \ge\exp\{m_s(b)+v_{p,s}/2\}. \tag{103}\] Both sides have limiting expectation \(e^{v/2}\). For the right side this follows from the just proved prediction convergence and uniform exponential integrability: Jensen gives \(\mathbb Ee^{q m_s}\le\mathbb Ee^{qX_s}\), whose bare Gaussian bound is uniform for each fixed \(q\). For the left side it is the full Laplace limit. Their nonnegative difference therefore tends to zero in \(L^1\). This proves the conditional exponential assertion for \(X\), and for every real linear combination of a fixed finite list. To obtain conditional weak laws, take a countable dense set of those linear combinations. From any subsequence one can extract a subsequence on which their probability errors tend to zero almost surely. Conditional exponential moment bounds make the finite-dimensional laws tight, and their Laplace transforms identify the Gaussian conditional law. Approximation by bounded continuous functions then gives the asserted conditional weak convergence, including conditional expectations of fixed smooth quadratic penalties.

For completeness, describe the Gaussian pin ratio explicitly. Let \(s_j\) be the singular values of \(C=\Pi_H|_{V_p}\) from Proposition 23. In canonical coordinates the pin observations are standard Gaussian \(\xi_j\) on \(p\), and their correlated standard coordinates on \(H\), say \(\eta_j\), satisfy \[\eta_j\mid\xi_j\ \sim N(s_j\xi_j,1-s_j^2).\] The ratio of the conditional density at zero to its centered conditional value is consequently \[ r_H^{\mathrm{cg}} =\exp\left\{-\frac12\sum_j \frac{s_j^2}{1-s_j^2}\xi_j^2\right\}. \tag{104}\] This is the limit of finite-probe density ratios, rather than a separate density for a full continuum pin. The exponent is finite almost surely because \[\sum_j\frac{s_j^2}{1-s_j^2} \le\frac{\|C\|_{\mathrm{HS}}^2}{1-\|C\|^2}<\infty.\] It is therefore strictly positive, and Gaussian integration gives \[ \mathbb Er_H^{\mathrm{cg}} =\prod_j\sqrt{1-s_j^2} =e^{-D_U^{\mathrm{cg}}(H,p)}. \tag{105}\]

At finite lattice spacing, prescribing \(b\) identifies an affine copy of the zero-\(p\) lattice. Apply the shifted-to-centered ratio comparison while increasing the quadratic precision on \(H\). It shows that every normalized conditional penalty ratio lies at most one and decreases when that penalty is increased. A sufficiently large multiple of the full coordinate-square penalty on \(H\) dominates any fixed smooth positive penalty there. Taking this multiple to infinity yields \(0<r_H\le1\). Integrating its definition immediately gives (98).

Choose increasing finite smooth penalties \(Q_N\) exhausting the pin on \(H\): successively include a total list of density probes and increase their positive strengths. Define \[R_{N,s}(b)= \frac{\mathbb E(Q_N\mid h_p=b)}{\mathbb E(Q_N\mid h_p=0)}.\] The preceding ordering gives \(r_{H,s}\le R_{N,s}\le1\). For fixed \(N\), conditional weak convergence and the hard-\(p=0\) limit identify \(R_{N,s}\) with its Gaussian counterpart. Moreover \[ \mathbb E(R_{N,s}-r_{H,s}) =\mathbb ER_{N,s}-e^{-D_{U,s}}. \tag{106}\] The full interaction limit (93) and the cross-projection trace approximation in Proposition 23 make the right side tend to zero as \(N\to\infty\) after \(s\to\infty\). Indeed the corresponding Gaussian expectations decrease to (105), by finite Gaussian conditioning and the convergent determinant. Markov’s Inequality in (106) proves the claimed approximation in probability for the full ratio. In particular its strictly positive Gaussian limit implies \[ \lim_{\varepsilon\downarrow0}\limsup_{s\to\infty} \mathbb P(r_{H,s}\le\varepsilon)=0. \tag{107}\]

A tilt from the admitted fixed observation list has a uniformly bounded \(L^q\) density for every fixed finite \(q\): its normalizing denominator is at least one and the uniform bare covariance bound controls the numerator’s moments. The bound is uniform when its coefficients range over a fixed compact set. Hölder therefore transfers every convergence-in-probability statement just proved to these tilted laws. For a convergent sequence of coefficients in that compact set, the difference of the corresponding tests has bare variance tending to zero. Gaussian domination and uniform exponential integrability make the centered conditional numerators and denominators differ in \(L^1\) by \(o(1)\). Their joint convergence and the strictly positive Gaussian denominator give convergence of the tilted ratios in probability. A subsequence argument therefore gives the stated compact-coefficient uniformity. The same argument applies to fixed finite lists of data and admitted penalties. For a fixed real linear tilt \(T\), conditional exponential moments under the tilted law are \[\mathbb E_T(e^X\mid h_p)= \frac{\mathbb E(e^{X+T}\mid h_p)}{\mathbb E(e^T\mid h_p)}.\] Apply the joint convergence already proved to numerator and denominator; the Gaussian denominator is strictly positive. This proves the tilted conditional assertion. The centered mean identity was used before this change of measure in Equation (106); neither that identity nor the centered density bound in Equation (100) is asserted unchanged after tilting.

Finally, the restored-bond penalty is supported inside \(H\), so the same shifted-ratio comparison puts \(r_\Delta\) between \(r_H\) and one. Bayes’ formula on the \(p\)-marginal gives \[\frac{\,\mathrm d\nu_p}{\,\mathrm d\mu_p}(b) =\frac{\mathbb E_\mu(Q_\Delta\mid b)}{\mathbb E_\mu Q_\Delta} =\frac{r_\Delta(b)}{\mathbb E_\mu r_\Delta}.\] The denominator lies in \([e^{-D_U},1]\), proving (100). For an event \(A_s\) about \(p\)-data, the upper density bound transfers \(\mu_p(A_s)\to0\) to \(\nu_p(A_s)\to0\), since \(D_U\) has a finite limit. Conversely, for every \(\varepsilon>0\), \[\mu_p(A_s)\le\mu_p(r_H\le\varepsilon) +\varepsilon^{-1}\nu_p(A_s).\] Use (107) after taking the lattice limit. This proves the reverse centered transfer. If the pin data are then evaluated under an admitted observation tilt \(T\), the tilted pin marginal in either topology has density \[g_T(b)=\frac{\mathbb E(e^T\mid h_p=b)}{\mathbb Ee^T}\] relative to that topology’s centered pin marginal. Conditional Jensen and the same bare bound give a uniform \(L^q\) bound for \(g_T\). Thus one first transfers a centered pin-data event by (100) and (107), and then applies Hölder separately in each topology. This completes the conditional interfaces without changing the centered density comparison. ◻

Proof. The square of the Dirichlet sum is a polynomial in \(\cos z\) of degree \(t-1\), so \(Q_t\) has degree \(p(t-1)\). For \(t\) integral, \[Q_{2t}(\cos z)=Q_t(\cos z)\cos^{2p}(tz/2)\le Q_t(\cos z).\] Iteration proves positivity of every difference in (108). Since \(Q_t(1)=1\), division by \(\lambda\) is removable, and multiplication by \(1-\gamma\lambda\) leaves degree at most \(p(S-1)\). The spectrum of \(\mathcal L\) lies in \([0,8]\), where \(1-\gamma\lambda\ge1/2\). A polynomial of degree \(d\) in a nearest-neighbor operator has range at most \(d\). This proves positivity and the stated, slightly enlarged, range. Telescoping proves the decomposition, including the empty shell sum when \(m_*=1\) because \(Q_1=1\). Its continuation at zero is positive since \[Q_t(1-\lambda/16)=1-\frac{t^2-1}{3}\lambda+O(\lambda^2).\]

Write a shell as the sum of its binary pieces \(H_t\), \(t=m,2m,\ldots,S/2\). On the Fourier square the nearest-neighbor symbol is \(\lambda(\xi)=4\sum_{i=1}^2\sin^2(\xi_i/2)\asymp |\xi|^2\). For \(0\le\lambda\le8\), the identity just used and \(|\sin u|\le\min(1,|u|)\) give \[ 0\le H_t(\lambda)\le C\min\{t^2,\lambda^{-1}\min(1,(t\sqrt\lambda)^{-2p})\}. \tag{112}\] Indeed the difference factor \(1-\cos^{2p}(tz/2)\) is at most \(Ct^2z^2\), whereas \(Q_t(\cos z)\le C\min(1,(tz)^{-2p})\); here \(z\asymp\sqrt\lambda\). A difference of order \(r\) has Fourier multiplier at most \(C_r|\xi|^r\). Integrating (112) in polar coordinates, split at \(|\xi|=t^{-1}\), gives \[\int |\xi|^r H_t(\lambda(\xi))\,\frac{\,\mathrm d\xi}{(2\pi)^2}\le C_rt^{-r},\qquad0\le r\le12.\] The corresponding normalized Fourier sums obey the same inequality whenever \(t\le M/D_0\): divide the frequency square into annuli of width \(M^{-1}\) and use that their number of points is bounded by a constant times their area times \(M^2\), with the zero-frequency contribution \(O(t^2/M^2)\). Taking suprema instead of integrals gives \(\sup_\xi |\xi|^{2r}H_t\le C_rt^{2-2r}\). For the full shell and \(r=1\), use directly \(\lambda\Gamma_m\le C\); for \(r>1\) the binary bounds form a geometric series. For \(r=0\) their operator bounds sum to \(CS^2\). Their kernels vanish beyond distance \(2pt+O(r)\), so only \(t\ge c\operatorname{dist}(x,y)\) can contribute. Counting these dyadic scales proves the logarithmic bound and summing \(t^{-r}\) proves the positive-order bounds.

Even reflection of the cell coordinates across \(-1/2\) and \(M-1/2\) identifies the free operator with the even subspace of the \(2M\) torus operator. Its kernel is the sum of the four reflected images in two dimensions. Since the range is less than a fixed small fraction of \(M\), at most a fixed number of periodic images enter the preceding estimates. The reflected difference conventions are the ones in those estimates. Orthogonal restriction, with the reflection multiplicity in the counting norm, also proves the operator estimates.

For completeness, the lower bound is not inferred from positivity of image terms. Choose a small fixed \(a>0\). Whenever both coordinates of the Fourier frequency are between \(a/t\) and \(4a/t\), the binary multiplier is at least \(c_at^2\). This follows by the expansion at zero of \(Q_t-Q_{2t}\), uniformly on the indicated compact annulus after decreasing \(a\). In the plane the annulus has area at least \(c_at^{-2}\). On the torus it has at least \(c_a(M/t)^2\) points for \(D_0\) sufficiently large. In the free cosine basis the factors at cell coordinate \(i\) are \(\cos(\pi u(i+1/2)/M)\). Pair each index \(u\) in a subinterval \([aM/t,2aM/t]\) with \(2u\). The elementary inequality \[\cos^2\theta+\cos^2(2\theta)\ge7/16\] ensures a uniform lower bound on each pair, and each frequency has bounded pairing multiplicity. Taking the product of the two one-dimensional sums proves \(H_t(x,x)\ge c\) also at a corner. There are \(\log_2L\) binary pieces, proving the diagonal assertion.

For the terminal remainder use \(\lambda^{-1}Q_{m_*}\le C\lambda^{-1}\min\{1,(m_*\sqrt\lambda)^{-2p}\}\). A positive-order diagonal integral is finite at zero and bounded by \(C_rm_*^{-2r}\); the discrete sum has the same bound uniformly for \(M\ge2m_*\): there are at most \(C(M/m_*)^2\) modes in the disk \(|\xi|\le m_*^{-1}\), and the remaining annuli form a convergent power sum. The operator bounds follow from the same multiplier with \(\lambda\ge cM^{-2}\). Reflection gives the free version. For \(M=1\) there are no nonconstant modes. No estimate on the value of the terminal constant mode is needed. ◻

Proof. Use the decomposition of Lemma 26 with the factors \((1-\gamma\lambda)/(1+s)\) omitted. Odd reflection gives its Dirichlet version. Sum binary pieces up to a scale comparable to \(M\) and keep the smoothed positive-spectrum remainder. A piece of scale \(t\) has differences bounded by \(C_rt^{-r}\) and vanishes when \(t<c|x-y|\), apart from reflected images whose distances are no smaller than the direct distance. Summing proves (113) and (114); nonnegativity follows independently from the maximum principle. The terminal remainder has differences \(O(M^{-r})\) and value \(O(1)\), since the smallest Dirichlet eigenvalue is at least \(cM^{-2}\).

For the local lattice limit, all fixed-scale pieces eventually coincide with the plane pieces by finite range. The tail of a derivative of positive order is bounded by \(C\sum_{t\ge T}t^{-r}\le C_rT^{-r}\). Taking the volume limit and then \(T\to\infty\) proves convergence. For the continuum limit, the discrete sine modes converge to the continuum sine modes: on every fixed mode \(n^2\lambda_n\) converges to its continuum eigenvalue. If \(f,g\) are smooth and compactly supported in the interior, repeated summation by parts against those sine modes bounds their discrete coefficients by \(C_q(1+|k|)^{-q}\), uniformly in \(n\), for every fixed \(q\). The eigenvalue inequality \(n^2\lambda_n(k)\ge c|k|^2\) then permits dominated summation, identifying the distributional limit of the kernels with \((-\Delta_D)^{-1}\). Equation (114), multiplied by \(n\) for first differences on compact subsets away from the diagonal, gives equicontinuity; (113) gives boundedness there. Distributional identification therefore upgrades to local uniform convergence. Rotating an odd-reflected potential gradient makes its normal component odd and its tangential component even. Its curl vanishes away from source plaquettes; path integration there supplies an even-reflected local lift with the asserted difference estimates. ◻

Proof. Choose a spanning tree of the fine block graph of \(X\) and traverse it in both directions. Its traversal has fewer than \(2|X|\) steps, each of spatial length at most \(C Rm\). Break it after each accumulated length \(S/4\). A resulting piece has diameter at most \(S/2\) and meets a bounded number of coarse rectangles because every coarse side is at least \(S\). This proves the first inequality in (115); the second follows from nesting. Successive fine blocks in the traversal meet coarse blocks at index distance at most \(1+CR/L\), which is less than \(r_0\) for large \(L\). Take \(n_0>4C_0\) and then \(L>4C_0R\). The half-size assertion follows. Increase \(L\) further so \(CRn_0m<S/2\). A coordinate interval of this length cannot meet three consecutive coarse intervals, proving the diameter assertion.

A canonical depth-first traversal records a rooted animal by a word of length at most \(2n\) in a fixed alphabet of allowed fine steps, giving the first count. To encode a family, join its \(j\) fine components by a tree using pairs of blocks whose containing coarse blocks have distance at most \(r_0\). Each joining displacement has length at most \(CL\) in fine block indices and hence has at most \(CL^2\) choices. Traverse fine component trees and the joining tree together. A word specifying step types, component boundaries, and traversal returns has at most \(C^n\) choices; the long steps are traversed at most \(2(j-1)\) times. The first fine root has at most \(CL^2\) positions in the specified coarse block. This gives (116), with one unused factor. All estimates use only the lower and upper bounds \(m,2m\) on side lengths and therefore include end blocks. In the finite-volume decomposition above, whenever a shell is used its upper scale is at most \(m_*<M/D_0\) with \(D_0\) larger than the fixed neighborhood radii, so every small neighborhood used here unwraps on a torus. ◻

Proof. We first record the elementary estimates underlying the claim. If \(B\) is any rectangle of side lengths in \([m,2m]\) and \(a\) is a first difference based on its stencil neighborhood, averaging along coordinate paths gives \[\begin{align*} m\sum_{v\in F}|a(v)|^2&\le C\sum_{v\in B}|a(v)|^2+Cm^4\max_B|\nabla a|^2,\tag{122}\\ m^2\max_B|a|^2&\le C\sum_{v\in B}|a(v)|^2+Cm^4\max_B|\nabla a|^2, \tag{123}\end{align*}\] where \(F\) is one face, including an adjacent stencil face. To verify these inequalities, compare \(a(v)\) with \(a(w)\) along a path of at most \(4m+4\) unit steps; its difference is at most \(Cm\max|\nabla a|\). Square the comparison and average over all \(w\in B\). For (122), sum over its at most \(2m\) face sites and multiply by \(m\). All reflected versions follow by applying the same comparison to the even extension.

Put \(D'=D_S(Y)\) and \(E=D'\setminus D\). The collars and nesting imply \(D\subset D'\). Counting at most \(CL^2\) fine blocks in each coarse block gives \[\begin{align*} T_m(D)&\le CL^{-2}T_S(D'),\tag{124}\\ P_m(D)&\le L^{-1}P_S(D')+Cg_E+CL^{-2}T_S(D'). \end{align*}\] For the second inequality an old boundary face that survives on \(\partial D'\) keeps the same edge increments with coefficient \(m/S\). Every other old face abuts a fine block in \(E\). Apply (122) to that block, basing a crossing increment at its outside endpoint. Each block is charged on at most four faces. The first term is bounded by \(Cg_E\), and the second sums as in the first line. The same argument for two sets at the same scale charges disappearing faces to the added blocks; their \(g\) and \(T\) increments absorb \(\delta\) times this cost when \(\delta C\le1\). This proves monotonicity, including a nonconnected added region.

The pure-fluctuation estimate is uniform in \(L\). The one-dimensional inequality \[|u(i)|\le\frac1{q}\sum_{j=1}^{q}|u(j)|+\sum_{j=1}^{q-1}|u(j+1)-u(j)|\] on an interval of length \(q\asymp m\), applied in each coordinate and followed by Cauchy–Schwarz, yields the two-dimensional discrete Sobolev bound \[\max_B |u|^2\le C\sum_{a\in\{0,1\}^2}m^{2|a|-2}\sum_{v\in B^+}|\nabla^a u(v)|^2.\] Here \(B^+\) adds only the fixed difference stencils and has bounded overlap as \(B\) varies. Apply it to \(u=m\nabla\zeta\) and \(u=m^2\nabla^2\zeta\). It bounds \(U_m(X,\zeta)\) by a fixed multiple of the squared Euclidean norm of the stacked Gaussian vector \[Z=(m^{j-1}\nabla^j\zeta(v):1\le j\le4,\ v\in D^+).\] The operator norm of its covariance is at most \(C\) by (111); its trace is at most \(C|X|\) because each diagonal entry is \(O(m^{-2})\) and \(|D^+|\le C m^2|X|\). If its eigenvalues are \(\lambda_i\), then \[\mathbb Ee^{a|Z|^2}=\prod_i(1-2a\lambda_i)^{-1/2} \le\exp\{2a\textstyle\sum_i\lambda_i\},\qquad 2a\max_i\lambda_i\le1/2.\] Thus \(\mathbb Ee^{aU_m(X,\zeta)}\le e^{Ca|X|}\) for all sufficiently small \(a\). Its exponential cost can be made less than \(2^{c|X|}\) with any fixed \(c>0\). The terminal positive-difference bounds prove (121) by this same calculation.

It remains essential to treat the bulk cross term without a squared triangle inequality. For an edge \(e=\{v,w\}\) set \(a_H(e)=(\mathbf 1_H(v)+\mathbf 1_H(w))/2\), and put \[B_H(\phi,\zeta)=\sum_ea_H(e)\partial\phi(e)\partial\zeta(e),\qquad g_H(\phi+\zeta)=g_H(\phi)+2B_H(\phi,\zeta)+g_H(\zeta).\] The gradient operator bound gives \(\operatorname{Var}B_E\le Cg_E(\phi)\). Summation by parts on \(D'\), orienting crossing edges from \(v\in D'\) to \(w\notin D'\) and setting \(h_e=\phi(v)-\phi(w)\), gives the exact identity \[ \partial^*a_{D'}\partial\phi =\mathbf 1_{D'}\mathcal L\phi-\frac12\sum_{e\in\partial D'}h_e(\delta_v+\delta_w). \tag{125}\] For the first term, \(\|\Gamma_m\|\le CS^2\) and the pointwise second-difference bound show \[\operatorname{Var}\langle \mathbf 1_{D'}\mathcal L\phi,\zeta\rangle \le CS^2\sum_{D'}|\mathcal L\phi|^2\le CT_S(D',\phi).\] For the boundary term, the absolute covariance row sum over all crossing endpoints is at most \(CS\). Indeed finite range leaves only a bounded number of nearby coarse faces, each of length \(O(S)\), and on one face \[\sum_{j=0}^{CS}\left(1+\log^+\frac{S}{m+j}\right)\le CS;\] the estimate follows either by integration of \(\log(S/t)\) or by summing dyadic intervals. Irregular sides have length at most \(2S\) and reflected images have bounded multiplicity. The matrix row-sum inequality therefore bounds that variance by \(CS\sum_{\partial D'}h_e^2=CP_S(D')\). Since \(B_D=B_{D'}-B_E\), we have proved \[ \operatorname{Var}B_D\le C\{g_E+P_S(D')+T_S(D')\}. \tag{126}\]

Use the squared triangle inequality only on \(P_m,T_m\). Cauchy–Schwarz, followed by the exact Gaussian moment of \(B_D\), bounds \(\mathbb Ee^{\kappa U_m(\phi+\zeta)}\) by \[(\mathbb Ee^{4\kappa U_m(\zeta)})^{1/2} \exp\{\kappa g_D+2\kappa\delta P_m+2\kappa T_m+4\kappa^2\operatorname{Var}B_D\}.\] Subtract the desired exponent \(\kappa(g_{D'}+\delta P_S+T_S)\). By (124) and (126) its coefficients are bounded above by \[\kappa(-1+2C\delta+4C\kappa)\,g_E,\quad \kappa(2\delta/L+4C\kappa-\delta)\,P_S,\quad \kappa(C(1+\delta)L^{-2}+4C\kappa-1)\,T_S.\] Choose \(\delta\) so \(2C\delta<1/4\), then \(\kappa\) so all \(8C\kappa\) terms fit within a quarter of the corresponding reserves, then \(L\) large. The same choices with reserve cover \(2\kappa\). Decrease \(\kappa\) further to make the fluctuation prefactor at most \(2^{|X|}\). This proves (120) with coefficient exactly \(\kappa\) on the bulk energy.

For small connected \(X\), its fine collar contains at most a fixed number of fine blocks and lies inside a fixed number of coarse blocks. Equation (123) at side \(S\) gives \[g_D+P_m(D)+T_m(D)\le CL^{-2}\{g_{D'}+T_S(D')\}.\] Now a squared triangle inequality is harmless for every term: the deterministic coefficient is \(O(L^{-2})\). The pure fluctuation bound proves the claimed half-exponent estimate for large \(L\). Finally, separated collars make the involved Gaussian subvectors independent by finite range; sums of their energies are the energy of the union, and hence their product estimates follow. For same-scale regrouping only disjoint collars and monotonicity are needed, not Gaussian independence. ◻

Proof. Constant invariance permits replacing \(\phi\) on the fine collar by \(\phi-\phi(v_0)\). Equation (123) on the enclosing coarse blocks and paths of length at most \(C m\) imply \[\begin{align*} \|\phi-\phi(v_0)\|_{m,D_m(X)}&\le CL^{-1}\sqrt{U_S(Y,\phi)},\tag{130}\\ \|f-f(v_0)\|_{m,D_m(X)}&\le C/L,\qquad \|f\|_{S,D_S(Y)}\le1. \end{align*}\] The paths and their stencils lie in the coarse collar. For the second differences use \(m^2|\nabla^2\phi|\le CL^{-2}\sqrt{T_S}\), which is stronger than needed. The constants depend on \(R,n_0\), already fixed before \(L\).

We spell out why the derivative sum and arbitrarily large real fields are compatible with Taylor expansion. At a real interpolated field \(t(\phi-\phi(v_0))\), convolution and the small-polymer part of Lemma 29 bound the input analytic derivative sum by \[C u\exp\{(\kappa/2)t^2U_S(Y,\phi)\}.\] After an output derivative, only the nonconstant part of its direction enters \(H\). All these directions together use at most \(Ch_0/L<h_0/2\) of the fine analytic radius. The remaining radius bounds the fixed number \(r\) of Taylor derivatives by \(C_rh_0^{-r}\). Taylor’s formula with its integral remainder and (130) therefore bounds the output derivative sum of an order-\(r\) remainder by \[ C_ruL^{-r}(1+\sqrt{U_S(Y,\phi)})^r e^{\kappa U_S(Y,\phi)/2}\le C_{r,\kappa}uL^{-r}W_S^\kappa(Y,\phi). \tag{131}\] The last inequality is the finite supremum of \((1+\sqrt t)^re^{-\kappa t/2}\) over \(t\ge0\). This argument justifies the derivative sums by their nonnegative absolute majorants. Evenness removes the linear Taylor term. Thus constant subtraction gives order two in the even case and order one otherwise.

For quadratic localization, Taylor expansion to order two leaves the order-three bound (131). For each \(v\in B\) and \(\epsilon\), replace the field modulo constants by the affine function with coordinate slopes \(\epsilon_i\nabla_{\epsilon_i e_i}\phi(v)\). The error in fine test norm is at most \(CL^{-2}\sqrt{U_S}\), by summing second differences along coordinate paths. On output directions its norm is at most \(CL^{-2}\). The difference between the two quadratic forms contains one such error and one factor of size \(CL^{-1}(1+\sqrt{U_S})\), or two errors. Applying the same radius allocation and exponent slack gives \(CuL^{-3}W_S^\kappa\). Every full fine block has \(m^2\) sites; averaging those affine quadratic forms over \(v,\epsilon\) and then over \(B\in X\) gives exactly (128). At zero field the derivative bound and \(\|\ell_i\|_{m,D_m(X)}\le C\) prove \(|c_X|+\|D^X\|\le Cu\).

Finally define \(D_B=\sum_{X\ni B}D^X/|X|\). Translation covariance makes it independent of \(B\); reflection changes the sign of each off-diagonal entry, and a quarter-turn interchanges the diagonal entries. Hence \(D_B=d\operatorname{Id}\). Direct substitution, retaining both signs of each lattice direction, yields \[\frac1{2m^2}\sum_{v\in B}\operatorname{ave}_{\epsilon}p^T(d\operatorname{Id})p =\frac d4\sum_{v\in B}\sum_{e=\pm e_i}(\nabla_e\phi(v))^2 =\frac d2 e_B(\phi).\] The rooted animal count in Lemma 28 bounds \(|d|\) by \(C\varepsilon\sum_{n\le n_0}(C_R/A)^n\le C\varepsilon\). Quadratic localization was asserted only for full bulk blocks; no replacement of an end-block volume by \(m^2\) has been made. ◻

Proof. Set \(w(v)=\Gamma_m(v,v_0)\), \(w_0=w(v_0)\). The binary positive-order kernel estimates and \(\operatorname{diam}D_m(X)\le Cm\) imply \[\|w-w_0\|_{m,D_m(X)}\le C,\] uniformly in \(L\). The Gaussian shift identity, with \(\tau=\operatorname{sgn}(q)\), is \[ \mathbb EF_q(\phi+\zeta) =e^{a^2w_0/2}\mathbb E\left[e^{-ia\tau\zeta(v_0)} F_q(\phi+\zeta+ia\tau w)\right]. \tag{134}\] The shift \(ia\tau w_0\) is constant and contributes \(e^{-a|q|\alpha w_0}\). Its nonconstant remainder has fine norm at most \(Ca\). A coarse direction is its anchor value, of absolute value at most one, plus a nonconstant part of fine norm at most \(C/L\). The first contributes \(|q|\alpha\) per derivative; summing it gives \(e^{h_0|q|\alpha}\). Choose \(h_0>4Ca\) and \(L\) so large that the imaginary remainder and all residual directions consume less than \(h_0\). The fine derivative sum then bounds their full contribution by the input norm at the real field \(\phi+\zeta\). Lemma 29 gives (133) times \(CuW_S^{\kappa/2}\).

For rigor in (134), use the Gaussian eigen-coordinates with positive eigenvalues. Their exact real Cameron–Martin identity is \(\mathbb EF_q(\phi+\zeta)=\mathbb E[e^{-z\zeta(v_0)-z^2w_0/2}F_q(\phi+\zeta+zw)]\) for real \(z\). Both sides continue analytically in the scalar shift in a neighborhood containing the segment \([0,ia\tau]\): the constant character handles its constant part and the stated radius bound handles its residual part. A slightly larger Gaussian regulator parameter bounds the integrands and their differentiated series along this compact segment. Dominated continuation proves the identity. Null covariance directions do not enter \(w\) and cause no change.

Take \(a=1\). The diagonal lower bound \(w_0\ge c\log L\) allows first choosing \(L\) so \(w_0\ge2h_0+2\) and then \(\alpha_0\) so \[(\alpha|q|-1/2)w_0-h_0\alpha|q|\ge4\log L+c'\alpha(|q|-1)\log L.\] The resulting geometric series over \(|q|\ge1\) proves (132). Fourier projection and inversion are valid for values and each derivative because periodic analyticity gives exponentially decaying Fourier coefficients on every compact real-field neighborhood. The obtained summable output-norm majorant permits summing the full analytic derivative series as well. ◻

Proof. A small support contains at most \(n_0\) blocks, with boundedly many shapes at its anchor. A coarse rectangle has at most \(CL^2\) fine anchors; its fixed-width boundary strip has at most \(CL\); a fixed-width neighborhood of a point has at most \(C\). End rectangles have side between \(S\) and \(2S\), leaving these counts unchanged. Since \(|\bar X|\le|X|\), the activity weight ratio is at most one. Multiply the counts by \(L^{-3},L^{-2},L^{-1}\), respectively. The last assertion follows directly from convolution, the regulator bound, and the finite number \(C_{L,A}\) of small supports and assigned pieces. It invokes no bulk cancellation across a physical wall. ◻

Proof. We give the algebra before bounding it. For a configuration of compatible fine polymers take the union of their coarse closures and its \(r_0\)-connected components. Fine fields belonging to different such components have mutual distance greater than the shell range: the support collar uses only a fixed number of lattice steps and \(r_0>4p+10\). Their Gaussian subvectors are independent. For each temporary component \(Y\) sum the expectations of products of all compatible fine families with closure union exactly \(Y\); call this sum \(\widetilde K(Y)\). Independence proves that the integrated partition functional is precisely the temporary-radius partition expansion in these \(\widetilde K\). Its linear term is \(\sum_{\bar X=Y}\mathbb EK(X)\).

Apply the fixed designated localization to each small linear summand, using Lemma 30 for the ordinary choice. Collect pieces assigned to a coarse block \(b\in Y\) as \(F_{Y,b}\). Write identically \[\widetilde K(Y)=\left(\widetilde K(Y)-\sum_{b\in Y}F_{Y,b}\right)+\sum_{b\in Y}F_{Y,b}.\] Expand this choice independently in every temporary configuration. A residual retains support \(Y\); a chosen piece receives support \(\{b\}\), while its original exclusion information \(Y\) is retained. Group the union of these new supports into its \(R\)-connected components. This factorizes even with the retained exclusions. Indeed a support that shrinks came from a small linear input and has index diameter at most one. If two original supports violate a temporary exclusion, their new supports have distance at most \(r_0+2<R\). They therefore belong to the same final component. There is no exclusion constraint between distinct final components. Within a component it is retained in the defining finite sum. Denote the resulting final-radius activities by \(H(Y)\).

Let \(a_b\) be the sum of the designated field-independent coefficients assigned at \(b\). Assume \(|a_b|<1/2\) and put \(d_b=(1+a_b)^{-1}-1\). Multiply the partition expansion with activities \(H\) by \(\prod_b(1+d_b)\). View a chosen \(d_b\) as a constant decoration that may overlap an occupied old block. Group the occupied union again by radius \(R\). Since the scalar multiplier is exactly \(\prod_b(1+a_b)^{-1}\), this defines \(K'\) and proves (136). At first order \(d_b=-a_b\), so precisely the designated constants cancel. A translated gradient polynomial can have a nonzero value at zero field; if that value is not designated for extraction, the algebra keeps it in \(K'\). This observation will matter for defects.

We next justify the norm bounds, including uniformity of the sums. Initially use the stronger output weight \(4A\). Expand each singleton input into its part measured by the activity norm and its separately measured polynomial parts. After summing the absolute bounds of any separately retained regular components, an ordinary input has point norm at most \(\varepsilon A^{-|X|}W_m^\kappa\). After summing \(\nu\), a separately measured polynomial singleton has point norm at most \(C_{\rm pol}\sum_\nu|t_{B,\nu}|W_m^\kappa\le(C_A\varepsilon)A^{-1}W_m^\kappa\), with \(C_A=AC_{\rm pol}\). For \(P=e_B/2\), the uniform local bound follows from \(e_B=g_B\le U_m\) and the bounds \(C\sqrt{U_m}\) and \(C\) on its first and second derivatives in unit test directions; the regulator absorbs these terms. Compatible fine polymers have disjoint collars. The product inequality and Lemma 29 give, for a family of total size \(n\) and \(j\) components, of which \(j_{\rm pol}\) are separately measured polynomial singletons after summing their indices, a bound \[\varepsilon^j C_A^{j_{\rm pol}}(2/A)^n W_S^\kappa(Y).\] For each large component \(X_i\), (115) gives \(|\bar X_i|\le |X_i|/2\), whereas every component obeys \(|\bar X_i|\le|X_i|\). Hence multiplication by \((4A)^{|Y|}\) costs at most a product with factor \(8^{|X_i|}A^{-|X_i|/2}\) on a large component and \(8^{|X_i|}\) on a small one. Use (116). For fixed \(L\) the sum over the size of each large component is bounded by \[\sum_{n>n_0}(C A^{-1/2})^n,\] and tends to zero as \(A\to\infty\). The sum over each small component has at most \(n_0\) terms and a finite cost depending on the geometry, with the additional fixed \(C_A\) cost only for separately measured singletons. Large components retain their original weight gain. The long-jump factor \((CL^2)^{2j}\) is a fixed factor per component. Thus all families with \(j\ge2\) sum to at most \[\sum_{j\ge2}(C_{L,A}\varepsilon)^j\le C_{L,A}\varepsilon^2\] when \(\varepsilon<(2C_{L,A})^{-1}\). The \(j=1\) terms are bounded linearly, and the large ones can be made an arbitrarily small linear multiple by increasing \(A\). Bounding the rooted sum over all output supports is stronger than the required supremum over one output support, so this argument proves the claimed norm estimate without a volume factor.

For the ordinary localization, \(|c_X|+\|D^X\|\le Cu\) and the normalized formula (128) give the required complete singleton bound: its value is at most \(Cu(1+U_S)\), and its first two derivatives in unit coarse directions are at most \(Cu\sqrt{U_S}\) and \(Cu\). The regulator absorbs these terms. The finite small-support count then verifies the stated rooted assignment and scalar bound; for other designated maps use that bound directly. In the relocation estimate keep each \(F_{Y,b}\) as one atom carrying its original label \(Y\), with support \(Y\) for its subtraction and \(\{b\}\) for its assignment. Its fine summands remain combined. A small \(Y\) has index diameter at most one, so its possible labels at a fixed \(b\) and the weight change between these two supports have a fixed \(C_{L,A}\) cost; regulator monotonicity handles the support change. Distinct temporary objects have distance greater than \(r_0\), and shrinking a small label by at most one block leaves their assigned collars disjoint. Count a final connected family by the previous tree of fine objects with one added coarse vertex for each such atom. An added vertex has only a fixed number of label choices, and its tree attachments have at most \(CL^2\) fine positions or a fixed number of coarse positions. Marking attachment positions costs only an exponential factor in the old fine size and the number of added atoms. The rooted assignment bound supplies \(C_{L,A}\varepsilon\) per added vertex. Thus the same exponential family estimate applies, with large objects retaining their original weight gain. The bound at weight \(4A\) remains available after relocation, and terms involving at least two objects are bounded by the same geometric series starting at degree two. The subtraction and designated singleton insertion are complex-linear, so all other changes so far are included in (137).

For clarity, the last decoration sum can be bounded directly without discarding overlaps. Fix a final support \(Y\) and its old occupied set \(V\subset Y\). The old components are determined by \(V\). Decorations are compulsory on \(Y\setminus V\) and optional on \(V\). Put \(d_* =\sup_b|d_b|\le C_{L,A}\varepsilon\). Monotonicity of the regulator and disjointness of old collars imply that the absolute sum for this \(V\) is at most its old activity coefficient times \[d_*^{|Y\setminus V|}(1+d_*)^{|V|}W_S^\kappa(Y).\] The old weight \(4A\), compared with the final weight \(A\), supplies \(4^{-|V|}\); optional overlaps cost at most \(2^{|V|}\). The holes supply \((Ad_*)^{|Y\setminus V|}\). Thus the subset sum is bounded by \[\sum_{V\subset Y}2^{-|V|}(Ad_*)^{|Y\setminus V|} =(1/2+Ad_*)^{|Y|}\le(5/8)^{|Y|}\] when \(Ad_*\le1/8\), leaving an exponential reserve. A term with an old activity and a hole contains at least one factor \(\varepsilon\) from each. If there are no holes, the nonempty-overlap sum is bounded by \(|Y|d_*(1+d_*)^{|Y|}\); the exponential reserve absorbs \(|Y|\). Pure decorations on a support of at least two blocks are degree at least two; on a singleton, \(d_b=-a_b+O(a_b^2)\). These bounds give the stated quadratic estimate at the last step.

The designated maps are bounded complex-linear maps of the measured input vector. All remaining operations are sums of bounded multilinear maps and the absolutely convergent scalar inverse series for \(1+a_b\). Their absolute bounds on a slightly larger input ball imply convergence of the complex Taylor series and its termwise derivatives on the smaller ball. Symmetry and periodicity are preserved by the explicit operations. Locality is preserved because every field used by an output object is inside its output support and fixed collar; its numerical coefficients can depend on predecessor data only in a fixed larger neighborhood. Iteration expands that neighborhood by the geometric sum of earlier scale lengths, bounded by a constant times the present scale. Choosing \(D_0\) larger than this constant ensures exact plane/torus agreement on unwrapped neighborhoods through the stopping scale. Finally, under the hypothesis on the terminal activity grid, \(M/m_*\le D_0L\) bounds the number of terminal blocks. When the terminal covariance also satisfies (121), that estimate controls their final Gaussian integrations. Under these terminal hypotheses, if every terminal activity norm tends to zero, then its finite subset expansion tends to one after the designated scalar factors have been removed. ◻

Proof. Perimeter edges contribute constants because both endpoint angles are zero. Contracting the perimeter in the sphere and discarding the final redundant edge leaves \(M^2\) faces, indexed by the cells. Each surviving primal edge separates two cells, including the edges ending at the fused pin, so the dual has precisely the free square adjacency.

For a divergence-free current \(u\), the rotated edge increments \(\mathcal R^{-1}u\) have zero sum around every elementary dual plaquette: this sum is the primal divergence at the corresponding interior vertex. The free dual grid is simply connected. Integrating these increments from a base cell gives a stream \(h\), unique modulo constants. Integral increments give integral \(h\) after fixing its value at the base cell. Conversely rotated gradients have zero primal divergence. This proves both identities, including their integral assertion. In particular, there are no additional harmonic currents in this calculation. One can also transpose this statement: the integer curl map on primal edge cochains is onto the integer cell charges subject to its single total constraint, and its kernel consists of integer primal gradients with the pin fixed. Surjectivity follows by routing charges along a dual tree; exactness of the kernel follows by integrating along primal paths. These integral statements apply to the multigraph, including parallel edges.

The Fourier coefficient at \(u\in\mathbb Z\) of \(\sum_m e^{-b(t+2\pi m)^2/2}\) is a positive constant, independent of \(u\), times \(e^{-u^2/(2b)}\). Integrating each interior angle in the product of these Fourier series enforces \(d^*u=-l\). The source need not have total zero: its compensating divergence occurs at \(\mathfrak p\). An integer solution \(u^0\) exists by routing each signed source along a path to the pin. Since \(d^*J=-l\), the first part of the lemma gives \(u^0-J=\mathcal R d'a\). All integral solutions are exactly \(J+\mathcal R d'(a+h)\), \(h\) integral modulo constants. Discrete summation by parts gives \[(J,\mathcal R d'f)=0,\qquad \lVert J\rVert^2=(l,G_nl),\] because a rotated dual gradient has zero interior divergence and the primal potential \(G_nl\) vanishes at the pin. Consequently its energy splits orthogonally. Setting \(\psi=(h+a)/\sqrt b\) gives exactly the comb constraint \(\psi+\vartheta\in\omega\mathbb Z\) and proves (142). Gaussian determinants and the constant Jacobian of the quotient by common height translations cancel between numerator and denominator. Finally \(\mathcal R^{-1}dG_nl\) is co-closed on the free dual graph, and \(\eta-d'\vartheta=\mathcal R^{-1}u^0/\sqrt b\); these observations give (141). ◻

Proof. The circulation of \(\mathcal R^{-1}J\) around a dual plaquette is a signed primal divergence of \(J\) and therefore vanishes off \(l\). Integrating around paths in a source-free rectangle gives the lift; the congruence in (141) makes its values agree with \(\vartheta\) up to a single phase offset and integer multiples of \(\omega\). A rectangle clipped to the square remains simply connected. At its boundary the odd extension of the Dirichlet potential makes the rotated normal increment odd and the tangential increment even, giving the free-wall extension of this lift. The derivative estimates and local convergence now follow by applying Lemma 27 to \(J=-\sum_i\varsigma_i dG_n(\cdot,x_i)\). In the local limit all fixed finite range pieces agree with their plane versions, and the tails of each positive-order derivative are summable. A different common phase offset does not affect either the gradients or neutral projections. ◻

Proof. For each residual real field \(\phi\) condition independent \(N(0,d_0)\) variables \(\xi_v\) to satisfy \(\xi_v+\phi_v+\vartheta_v\in\omega\mathbb Z\). Their conditional probabilities are their Gaussian densities on the translated lattice, divided by the corresponding lattice sum. The independent convolution of the normalized comb at \(v\) is \[ w_v=w(\phi_v+\vartheta_v),\qquad w(t)=\sum_{q\in\mathbb Z}e^{-q^2\alpha^2d_0/2}e^{iq\alpha t}. \tag{149}\] This is an equality of a Gaussian convolution with a positive atomic measure, so it does not require a smooth microscopic sine potential.

Assign to \(v\) half of each incident oriented squared gradient of \(\phi+\xi\) plus \(\eta\) and call the resulting local expression \(E_v\). Then \(\sum_vE_v=\lVert \partial(\phi+\xi)+\eta\rVert^2\). Expanding the product of \[ 1+F_v=w_v e^{sE_v/2} \tag{150}\] into occupied site subsets, and taking conditional expectations, gives activities on their radius-\(R\) components. The conditional variables remain independent. Energies involve only nearest neighbors, so components at distance greater than \(R\) use disjoint variable sets and factor exactly. Extract the initial singleton constants by the exact decoration operation of Proposition 33, using \[ c_v^0=\frac s2\mathbb Ee_v^0(\xi), \tag{151}\] where this expectation is unconditioned and \(e_v^0\) has the same energy allocation. The extracted factors \(1+c_v^0\) are identical in the two partition integrals even at a wall.

Here are uniform estimates that justify these operations and the claimed norm. The analytic derivative sums of \(w-1\), \(w^{-1}-1\), and of the conditional first and second moment errors relative to \(N(0,d_0)\) are \(O(e^{-cb})\). Indeed a nonzero Fourier mode and its analytic derivatives are bounded by \[C(1+|q|\alpha)^a e^{-q^2\alpha^2d_0/2+h_1|q|\alpha},\] where \(a\) is fixed for polynomial insertions and \(h_1>h_0\) is a fixed radius reserve. Their sum is \(O(e^{-cb})\) uniformly for small \(s\). The conditional normalization is bounded away from zero, and its inverse has the same estimate by its convergent geometric series.

For the remaining derivative sums, fix all constrained lattice indices. Then \(\phi_v+\xi_v=\omega h_v-\vartheta_v\), so \(E_v\) is independent of \(\phi\) in that summand. A derivative consequently falls on Gaussian densities or conditional normalizations. The absolute derivative sum of a density at \(\xi\) is bounded by that density times \[\exp\{h_1|\xi|/d_0+h_1^2/(2d_0)\}.\] This follows by expanding the exponential generating function of the Hermite derivatives and replacing its coefficients by their absolute values. Uniformly over translated lattices of mesh \(\omega\le1\), their Gaussian sums, multiplied by \(\omega\), are bounded when one inserts this factor and \(e^{\epsilon\xi^2}\) for any sufficiently small fixed \(\epsilon>0\). To see the uniformity, cover the line by its mesh intervals, compare the Gaussian envelope on each interval with its integral over that interval and its two neighbors, and sum; the bounded number of intervals around the maximum contributes a bounded amount.

Now expand \(F_v\) as \((w_v-1)+w_v(e^{sE_v/2}-1)\). Each occupied site costs \(C(z+|s|)\), using \(|e^y-1|\le |y|e^{|y|}\). Each Gaussian variable appears in a bounded number of local energies. The preceding square exponential bounds absorb its factors, while the remaining field polynomials and their small quadratic exponent are absorbed by \(W_1^\kappa(X,\phi)\). Since \(|s|\) can be made arbitrarily small, this works for any subsequently fixed positive regulator exponent. The result before decorations is \[|\mathcal K_0(X)|_{1,X,\phi} \le [C(z+|s|)]^{|X|}W_1^\kappa(X,\phi).\] The same argument with \(|e^y-1-y|\le\tfrac12|y|^2e^{|y|}\) bounds a singleton Taylor remainder by \(C s^2W_1^\kappa\). Conditional first and second moments then give \[\mathbb E_{\mathrm{cond}} E_v =e_v(\phi)+\mathbb Ee_v^0(\xi)+O(z)W_1^\kappa\] in the untwisted bulk norm. Its scalar term is removed by (151), leaving exactly \(s e_B/2\) at first order. All multi-factor and decoration errors are quadratic by the explicitly convergent expansion in Proposition 33. Starting that expansion at a larger fixed site weight absorbs its decoration cost and proves (148). The fixed bound \(C_\eta\) adds only a constant depending on \(C_\eta\) per touched site to these estimates. Finally, when \(\eta=\partial\lambda\) and \(\lambda\equiv\vartheta\pmod\omega\), both the constrained Gaussian lattice sum and its energy become their untwisted versions at \(\phi+\lambda\). The extracted constants were chosen independently of the twist. This proves the last two assertions as exact identities, not asymptotic estimates. ◻

Proof. Start with \(t_0=s\) and Proposition 36. On small inputs from \(\mathcal P_j\) use neutral quadratic localization and charged contraction. For the singleton \(t_je_B/2\), assign its entire Gaussian expectation, including its scalar part and its unchanged gradient part, to its coarse singleton. The scalar part is bounded by the positive-derivative estimates in Lemma 26. The maps commute with translations by full blocks and with square rotations and reflections. Lemma 30 therefore makes the sum of the assigned Hessians a scalar matrix, constant over the full coarse block; the sum of its nonconstant pieces is exactly \(t_{j+1}e_{B'}/2\). Consequently the exact map has \[\begin{align*} t_{j+1}&=t_j+\sigma_j(\mathcal P_j),& |\sigma_j(\mathcal P_j)|&\le C\lVert \mathcal P_j\rVert,\tag{153}\\ \lVert \mathcal P_{j+1}\rVert&\le \theta\lVert \mathcal P_j\rVert +C_{L,A_U}(|t_j|+\lVert \mathcal P_j\rVert)^2 . \tag{154}\end{align*}\] Here \(\theta>0\) is as small as desired. Indeed there are \(O(L^2)\) small fine inputs per output, and their neutral and charged remainders are \(O(L^{-3})\) and \(O(L^{-4})\), respectively. Large inputs have an arbitrarily small weight gain. The separately assigned gradient input has zero subtracted linear remainder, and all remaining operations cost the quadratic term in (154) by Proposition 33.

Choose \(\theta<\rho<1\), let \(\varepsilon=C_1z\), and take fixed \(T,C_1\) large. For \(s\in[-T\varepsilon,T\varepsilon]\), as long as \(|t_j|\le T\varepsilon\rho^j\), induction in (154) gives \(\lVert \mathcal P_j\rVert\le2\varepsilon\rho^j\). At \(j=0\) this follows from \(C(s^2+z)\le2\varepsilon\) for sufficiently small \(z\). For the induction step, choose \(z\) so small that \[2\theta\varepsilon\rho^j+ C_{L,A_U}(T+2)^2\varepsilon^2\rho^{2j} \le2\varepsilon\rho^{j+1}.\] At either boundary \(t_j=\pm T\varepsilon\rho^j\) of its envelope, (153) gives \[\pm t_{j+1}\ge(T-2C)\varepsilon\rho^j >T\varepsilon\rho^{j+1},\] provided \(T(1-\rho)>2C\). Thus a boundary hit is followed by a strict exit on the same side.

The finite prefix of the map is continuous in \(s\) while these bounds hold. This assertion requires only continuity at each fixed support and each fixed derivative, rather than continuity in a stronger completed norm. For a fixed support there are finitely many fine supports in a step; the Gaussian variables can be coupled by rescaling as \(s\) varies. The analytic norm and regulator bounds, with their parameter reserve, dominate the relevant expectations, and dominated convergence applies inductively, also at converging real backgrounds and directions. The Hessian coefficient is a finite local sum and is therefore continuous. The bound permits computing one additional step at an envelope boundary.

Classify a parameter by the side of its first eventual exit. These two classes are disjoint and relatively open: a strict exit persists by finite-prefix continuity, while an equality at a first hit forces a strict exit one step later and has the same property. They contain the two respective endpoints of the interval. They cannot cover a connected interval. A parameter in their complement remains in every envelope, which proves (152). Translating all plane blocks conjugates exactly the local maps and their scalar coefficients, so the same chosen parameter works for every translated hierarchy. ◻

Proof. Use the nested cuts of Lemma 28: a free end block has side length between \(m\) and \(2m\), and every retained cut lies at least \(m\) from the ends. No divisibility of \(M\) is needed. Use the stopping convention of Lemma 26: for \(M>D_0\), stop at the largest nonnegative power \(m_*\) of \(L\) strictly below \(M/D_0\); for \(M\le D_0\), put \(m_*=1\) and integrate no shells. In all cases \(M/m_*\le D_0L\), so the number of terminal blocks is bounded by a constant depending on \(D_0,L\), and \(m_*\to\infty\) along every integer \(M\to\infty\). Whenever \(m_*=1\), the only activity level is the initialized level, and the asserted norm bound is its \(j=0\) bound. Bounds on physical ratios in these finitely many boxes are obtained directly from their positive exact finite partition sums.

Call a support bulk-good when its sites lie at distance greater than \(Hm\) from free walls. On the torus require that it fit in coordinate arcs leaving gaps greater than \(Hm\) in both periods. All other supports are wall supports. Localize small bulk-good inputs using exactly the plane pieces, even if the shell being integrated could itself reach a wall from that input. For small wall inputs subtract only the neutral constant of the full activity, with no gradient localization. The resulting maps remain exact by Proposition 33.

For a bulk-good output every contributing fine support, the one-step neighborhood of every assigned piece, and each singleton normalization lies in the output neighborhood with the clearance used in its definition. The source support of a shell path lies within \(2pS\). Taking \(H\) larger than these range and assignment constants makes these fine data bulk-good and makes their kernels identical to the plane kernels. Connectivity and exclusions also agree. Induction proves the exact copy assertion. For a torus, the gaps provide a common lift of each connected neighborhood; block-period translations preserve the plane prescriptions, so the same assertion holds across a chosen cut.

Small wall supports in a fixed coarse output can lie only in a strip of bounded fine width next to a physical wall. There are \(O(L)\) such placements. Each activity is even under field negation, and constant localization therefore leaves \(O(L^{-2})\) by Lemma 30; its charged part has the stronger charged gain. There are no small wrapping inputs before the stopping scale on the torus. Large inputs gain by weight. Good linear inputs that can feed a wall output are bounded in full by \(C_{L,A_U}u_j\), including their assigned gradient pieces; no cancellation of these pieces on a wall output is asserted. The nonlinear estimate in Proposition 33 gives \[ w_{j+1}\le\theta w_j+C_{L,A_U}u_j+ C_{L,A_U}(u_j+w_j)^2. \tag{156}\] Initialization gives \(w_0=O(z)\) because \(s=O(z)\). Substituting (152) and iterating the geometric convolution in (156), with \(z\) small, proves (155) for any fixed \(\rho_1>\max(\rho,\theta)\) sufficiently below one. At the terminal scale only boundedly many blocks remain. Every nonempty term of their finite polymer partition has at least one activity whose norm tends to zero. Lemma 29 and the terminal part of Lemma 26 bound its Gaussian expectation uniformly. The empty term equals one, proving the last assertion. ◻

Proof. First increase the lower bound on \(b\) so that Corollary 75, applied with \(J=J_{\mathrm{nn}}\) and \(k=\beta/v_{\mathrm{nn}}^2=4\pi^2b\), gives \(a(k)>0\). Thus the hypothesis of Theorem 17 holds at the bare height temperature \(\beta=\pi^2b\), independently of the Villain calculation below. The physical roughness criterion in Corollary 18 places this input in the rough regime; we may further increase the lower bound to lie strictly above its finite threshold.

Let \(f\) be a smooth real mean-zero test on the unit torus. Set \[f_M(v)=2\pi\sqrt b\,M^{-2} \left(f(v/M)-M^{-2}\sum_w f(w/M)\right).\] Then \((f_M,\psi)\) is precisely the test of the centered field \(\sigma=2\pi\sqrt b\,\psi\). Gaussian translation, after the initial convolution (or first against a bounded smooth comb regularization), gives the Laplace prefactor \[ \exp\{(f_M,Gf_M)/(2(1+s))\} \tag{158}\] and translates the entire interaction and comb by \(\chi=Gf_M/(1+s)\). Every exact partition identity with its fixed subtractions is an identity as a function of background field. Therefore the same scalar factors may be used, and the source ratio is the ratio of terminal averages with backgrounds \(\phi+\chi\) and \(\phi\).

Fourier series and the rapid decay of the coefficients of \(f\) show that \(M|\nabla\chi|\) and \(M^2|\nabla^2\chi|\) are uniformly bounded. For example the discrete inverse multiplier divided by \(M^2\) tends on each nonzero fixed mode \(r\) to \((4\pi^2|r|^2)^{-1}\), and repeated summation by parts bounds the sampled source coefficients by \(C_a(1+|r|)^{-a}\) uniformly up to the lattice cutoff. Multiplication by one or two scaled differences leaves an absolutely summable series. The terminal regulator evaluated at \(\phi+\chi\) thus has bounded expectation, using its positive-order Gaussian bounds and a small parameter reserve. The decaying terminal activities in Proposition 38 make both terminal averages tend to one. All manipulations with combs are legitimate by the initial positive Gaussian lattice sums and the domination in Proposition 36.

The same Fourier calculation in (158) gives \[\log\mathbb Ee^{H_N(f)}\longrightarrow \frac{2\pi^2b}{1+s}(f,(-\Delta_\mathbb T)^{-1}f).\] Thus its covariance coefficient is \(4\pi^2b/(1+s)\). The coefficient in Equation (6) is \(\beta_{\mathrm{eff}}/v_{\mathrm{nn}}^2=4\beta_{\mathrm{eff}}\) by (144); uniqueness of the torus coefficient in Theorem 17 gives (157), using the positivity established before this calculation. This comparison uses no inference from smooth height convergence to a spin observable; those observables are treated through (145) next. ◻

Proof. Use the initial activities in Proposition 36. For small shift-good inputs use exactly the pieces of untwisted localization, evaluated at their common lift. Designate for extraction exactly the same scalar coefficients as in the untwisted map. In particular the value of a translated gradient polynomial at zero background is retained inside that polynomial; it is not added to the designated scalar. For a small defect input use only the neutral constant localization in the original, untranslated field variable. These prescriptions are permitted exact splittings in Proposition 33.

The translated-copy assertion follows by induction. If an output is shift-good, every fine input, assigned piece, and singleton normalization entering it fits in its enlarged rectangle. Clearance provides a common source-free lift throughout that rectangle. The induction hypothesis then translates all products and Gaussian convolutions by the same field, and the identical pieces and designated constants preserve the equality. A defect output requires no lift on its entire support. In particular a gradient piece from a good input is translated on the fine block where it was assigned; moving its support label to a coarse singleton does not evaluate a singular lift at a source core. All activities remain periodic under a uniform background shift by \(\omega\).

The estimates for translations can be stated directly in the regulators. On the collar of a good input, (147) bounds \(m|\nabla\lambda|\) and \(m^2|\nabla^2\lambda|\) by a constant. For \(k\le2\) this is uniform without separation. Under (159), put \(d=\epsilon M\). At each collar site \(v\), at most one mark has distance strictly less than \(d/2\) from \(v\); every other mark has distance at least \(d/2\). The collar is at distance at least \((H-C)m\) from every mark. Hence, pointwise at \(v\), \[m^{r+1}|\nabla^r\eta(v)| \le C_r b^{-1/2}\left[ \left(\frac{m}{1+(H-C)m}\right)^{r+1} +k\left(\frac{m}{d}\right)^{r+1}\right], \qquad r=0,1.\] Since \(m\le cd/[L(k+1)]\), the right side is bounded independently of \(k\). At the initial scale \(m=1\), the same division into one nearest mark and the remaining marks, with the nearest contribution bounded by one without a clearance assumption, gives the fixed bound on \(\sup|\eta|\) required by initialization. Using \(\partial\lambda=\eta\) and summing the pointwise bounds over the collar gives \[U_m(X,\phi+\lambda)\le2U_m(X,\phi)+C|X|.\] The weight reserve \(A_U/A_D\) absorbs \(e^{C\kappa|X|}\) and the regulator exponent doubles. Common values of the lift cause no loss since the norm is a supremum over real backgrounds; only its derivatives enter the regulator. Assigned gradient pieces on a good fine block are bounded on their coarse singleton by the same estimate: \(m\eta\) is bounded there, and the coarse field gradients and analytic test gradients have their prescribed bounds. Hence all linear forcing from good inputs is bounded by \(C(u_j+w_j)\), including pieces that reach a defect output.

Small defect inputs within a specified coarse output have only \(O(1)\) possible fine placements per nearby mark, because they have bounded fine diameter and their enlarged hull reaches a mark. In the regime under consideration at most two marks are locally relevant; counting multiplicity covers a repeated pair. Field-negation symmetry can fail for a twisted input. Its neutral constant subtraction nevertheless gains \(L^{-1}\) by Lemma 30; \(O(1)\) placements make this a contraction. The charged contraction of Lemma 31 and the large-polymer weight gain treat the other linear terms. Products, relocations, and decorations have the quadratic bound of Proposition 33, now using its allowed exponent \(2\kappa\). This proves the first inequality of (160). Initialization uses \(\sup|\eta|\le C\) from Lemma 35. Iterating with (152) and (155), and decreasing \(z\) if needed, proves the second inequality.

For fixed \(k,\epsilon\), failure of (159) occurs only once \(m\) is a fixed positive fraction of \(M\), up to a factor depending on \(k,\epsilon,L\). For sufficiently large \(M\), let \(j_e\) be the last activity index in the decaying regime. It satisfies \(j_e\ge\log_L M-C_{k,\epsilon}\), so the entering activity norms are at most \(C_{k,\epsilon}zM^{\log_L\rho_2}=o(1)\) at the already fixed \(b\). There are only boundedly many later scales and boundedly many blocks at their start. Crude versions of the same finite sums and Gaussian estimates have constants and positive smallness radii allowed to depend on \(k,\epsilon\). The entering \(o(1)\) eventually lies inside each such radius, and every remaining map has no term independent of the activities, so finitely many maps send \(o(1)\) to \(o(1)\). On good inputs the scaled lift derivatives are bounded by a \(k\)-dependent constant, and all weight losses occur on boundedly many blocks. The terminal Gaussian regulator estimate proves the stated convergence of the terminal average. Here \(k\) is fixed before \(M\to\infty\); no increase of \(b_0\) with that fixed order is needed. ◻

Proof. Let \(a_{j,B'}^{\rm tw}\) and \(a_{j,B'}^{\rm u}\) be the designated constants of one exact map. Their differences can only come from small defect inputs. On good inputs they were designated to be identical, including when a translated gradient piece has a nonzero value at zero. There are boundedly many relevant fine supports near each mark, and their norms in the decaying regime give \[ \sum_{B'}|a_{j,B'}^{\rm tw}-a_{j,B'}^{\rm u}| \le Ckz\rho_2^j. \tag{162}\] All individual constants are real and uniformly small. Thus \(1+a_{j,B'}>0\) and \(|\log(1+a)-\log(1+a')|\le2|a-a'|\) for the values occurring here. The initial constants agree exactly. The exact partition identities therefore give \[ \log\mathcal R_{n,b}(l) =\sum_{j,B'}\log\frac{1+a_{j,B'}^{\rm tw}} {1+a_{j,B'}^{\rm u}} +\log\frac{T^{\rm tw}_{n,b}}{T^{\rm u}_{n,b}}, \tag{163}\] where the two terminal factors tend to one. The logarithmic tail in the sum is bounded by \(Ckz\rho_2^J\) beyond depth \(J\) in the decaying regime. The bounded number of late steps for a separated configuration contributes \(o(1)\). For \(k\le2\) there are no such late steps, and the terminal factors stay bounded above and below uniformly for large \(n\). Finitely many smaller boxes can be included by their positive exact partition sums. This proves the uniform ratio bounds.

For a single interior mark, translate all nested cuts by its lattice location. This changes neither the physical ratio nor the tuned scalar parameter. At every fixed depth the scalar differences in (163) are calculated in a bounded neighborhood of that mark, in fixed relative block positions. Eventually the kernels there equal the plane kernels. Lemma 35 makes their increments tend to those of the isolated plane source. Their phases modulo \(\omega\) tend to the corresponding isolated phases up to a common phase offset. Such an offset translates the background argument; neutral constant projection is invariant under this translation, and the designated good constants were already fixed. Thus it does not alter a scalar difference. Negating the source and the field also leaves these constants unchanged. Continuity of each fixed local map in its finitely many phases and increments follows by the Gaussian lattice-sum domination in Proposition 36 and the finite-depth dominated-convergence argument in Proposition 37. Each fixed-depth log increment therefore tends to a value independent of source location and sign. The summable tail (162) defines a finite real sum \(\log c(b)\), with \(|\log c(b)|\le Cz\).

To treat \(k\) separated marks, use one hierarchy for their joint problem and for each of the \(k\) corresponding single-mark problems. At any fixed depth their scalar-difference neighborhoods are disjoint for large \(n\). Near one mark the contributions of the other marks to the connection tend to zero by (147); their remaining common phase offsets have no effect on the neutral constants. The local calculations consequently approach those of its single-mark problem. The position of a mark relative to all cuts through a fixed depth has only finitely many possibilities. Passing over these possibilities gives uniformity without selecting a subsequence. The physical single-mark ratio has the limit \(c(b)\) regardless of the hierarchy, as just proved using translated cuts. More explicitly, for a single unit source \(l=\varsigma\delta_x\), let \(S_J^{\mathcal H}\) be the scalar log sum through depth \(J\) in a hierarchy \(\mathcal H\), continuing its exact single-mark maps gives \[\left|\log\mathcal R_{n,b}(l)-S_J^{\mathcal H}\right| \le Cz\rho_2^J+o_n(1)\] uniformly in the cuts under consideration. The full physical ratio is exactly independent of \(\mathcal H\); the finite-depth sums need not agree. Its limit from the mark-anchored hierarchy therefore places every residue-class limit of \(S_J^{\mathcal H}\) within \(Cz\rho_2^J\) of \(\log c(b)\). Finally let the fixed depth tend to infinity in (162). The terminal and late-step errors vanish, yielding (161) and its uniformity. ◻

Proof. Insert (161) into (145). The exact identity \[(l,G_nl)=\sum_iG_n(x_i,x_i)+ 2\sum_{i<j}\varsigma_i\varsigma_jG_n(x_i,x_j)\] holds for a list of unit sources, including repeated sites when the list is retained with multiplicity. Its self-energies cancel the full diagonal multipliers, leaving the off-diagonal expression. The factors \(A_{\mathrm V}^k\) cancel \(c(b)^k\), and the off-diagonal convergence in Lemma 27 proves (164). In particular no finite spatial part of the Dirichlet diagonal has been discarded.

The original current sums have positive coefficients, so every signed expectation is nonnegative. The maximum principle gives \(G_n(x,y)\ge0\). For two unit sources, Proposition 41 bounds the remaining ratio without separation. The logarithmic estimate in Lemma 27 then bounds the worst sign choice by \(C\exp\{G_n(x,y)/K\}\) and hence by the right-hand side of (165), with \(Q=C_0/K\). Increasing the fixed lower bound on \(b\) makes \(Q<2\), since \(s=O(e^{-cb})\) and \(K/b\to1\). For repeated opposite sources this also bounds \(A_{\mathrm V}^2e^{G_n(x,x)/K}\), so the diagonal case is included. The arbitrary-side construction of Proposition 38, together with the finite-residue comparison in Proposition 41, establishes the assertion about the full sequence. ◻

Proof. Let \(U\) be an independent uniform angle and rotate every spin, including all boundary spins, by \(U\). The resulting boundary angle is common and uniform. Conditional on that angle, subtracting it gives precisely the original fixed-boundary measure. Put \(X=\operatorname{Re}(e^{iU}W)\). The Lee–Yang Theorems for finite XY and Villain graphs, including this boundary convention and nonnegative vertex weights, assert that \(\mathbb Ee^{zX}\) has only imaginary zeros (Newman and Wu 2019, Theorems 1 and 3, Remark 5). For the square used here, the theorem can be applied before boundary fusion: attach a single new vertex of observable weight zero to every original boundary vertex, and let these added ferromagnetic couplings tend to infinity. The approximating graphs are simple, and their limit is exactly the common uniformly rotated boundary law in Remark 5. Thus no multigraph convention is needed after fusion. The variable \(X\) is symmetric and bounded, so it satisfies the exponential square-integrability hypothesis of Newman and Wu (2019, Proposition 13). That proposition gives \[\mathbb Ee^{zX}=e^{Bz^2}\prod_j(1+z^2/y_j^2),\qquad B\geq0,\quad \sum_jy_j^{-2}<\infty,\qquad B+\sum_jy_j^{-2}=\tfrac12\mathbb EX^2.\] For real \(t\), the inequality \(1+u\leq e^u\) therefore implies \[ \mathbb Ee^{tX}\leq \exp\left(\tfrac12t^2\mathbb EX^2\right) =\exp(t^2v/4). \tag{168}\] The equality uses uniform rotation, which gives \(\mathbb EX=0\) and \(\mathbb EX^2=v/2\). The same statement applies to \(Y=\operatorname{Im}(e^{iU}W)\), since its law equals that of \(X\). When \(v>0\), Chernoff’s Inequality and \(|e^{iU}W|^2=X^2+Y^2\) give \[\mathbb P\{|W|>r\}\leq4\exp(-r^2/(2v)),\qquad r>0.\] Integrating \(p r^{p-1}\) times this bound proves (166). If \(v=0\), then \(W=0\) almost surely and the conclusion is immediate. The rotation has not changed \(|W|\). In particular, the estimate concerns the original fixed-boundary field, even though the scalar Lee–Yang Theorem was applied after rotation.

For positivity, orient the edges arbitrarily and expand each edge kernel in its absolutely convergent Fourier series. For XY, its coefficients are \[I_k(b)=\sum_{r=0}^{\infty} \frac{(b/2)^{2r+|k|}}{r!(r+|k|)!}\geq0.\] They are strictly positive when \(b>0\); at \(b=0\), \(I_0(0)=1\) and \(I_k(0)=0\) for \(k\ne0\). For Villain they are a common positive constant times \(\exp(-k^2/(2b))\). These formulas also apply edge by edge when the parameters vary. Integration of the interior angles restricts the integer edge currents to the divergence prescribed by the inserted integer charges. The fixed boundary angles contribute the factor one and impose no further constraint there. Every surviving summand is nonnegative. The denominator is positive, proving (167); repeated vertices merely add their charges. ◻

Proof. Fix a compact set \(E\Subset D\) containing all supports, and choose \(0<\rho<1\) so that the closed \(2\rho\)-neighborhood of \(E\) is contained in \(D\). On that neighborhood the fundamental-solution decomposition is \[G_D(u,v)=\frac1{2\pi}\log\frac1{|u-v|}+R_D(u,v),\] where \(R_D\) is smooth, in particular bounded, also across \(u=v\). Write \(q=(2\pi K)^{-1}<2\). For \(a,b\leq\rho\), let \(C_{a,b}(x,y)\) denote the covariance of the two circle averages of \(\Phi_D/\sqrt K\). We first prove the uniform estimate \[ C_{a,b}(x,y) =q\log\frac1{\max\{|x-y|,a,b\}}+O_{E,K}(1). \tag{170}\] This includes coincident centers and equal or unequal radii. Indeed, if \(r=|x-y|\) and \(M=\max\{r,a,b\}\), the double circle average of the logarithm of the distance is \[L_{a,b}(r)=\frac1{2\pi}\int_0^{2\pi} \log\max\{a,|r+be^{it}|\}\,\mathrm dt.\] This follows by first averaging over the circle of radius \(a\) and using \(\frac1{2\pi}\int\log|z-ae^{it}|\,\mathrm dt=\log\max\{|z|,a\}\). Pointwise comparison gives \(L_{a,b}(r)\leq\log(2M)\). If \(a=M\), its integrand is at least \(\log M\). Otherwise its average is at least the average of \(\log|r+be^{it}|\), which is \(\log\max\{r,b\}=\log M\). Thus \(\log M\leq L_{a,b}(r)\leq\log M+\log2\). The bounded remainder gives (170). The logarithmic integrals in this calculation are finite, so it also justifies the covariance of every circle average used here.

Gaussian integration, with the full diagonal variance subtracted from the characteristic exponent, gives for arbitrary radii \(\varepsilon_1,\ldots,\varepsilon_m\leq\rho\) \[ \mathbb E\prod_{i=1}^m V_{\varepsilon_i}^{\sigma_i}(x_i) =\exp\left\{-\sum_{i<j}\sigma_i\sigma_j C_{\varepsilon_i,\varepsilon_j}(x_i,x_j)\right\}. \tag{171}\] Here the pointwise superscript means complex conjugation for sign \(-1\). For distinct centers and \(m\geq2\), set \[d_i=\min\{\rho,\min_{j\ne i}|x_i-x_j|\},\qquad r_i=\max\{\varepsilon_i,d_i/4\}.\] The enlarged circles remain in \(D\). For \(i\ne j\), both \(d_i/4\) and \(d_j/4\) are at most \(|x_i-x_j|/4\), whence \[\max\{|x_i-x_j|,r_i,r_j\} =\max\{|x_i-x_j|,\varepsilon_i,\varepsilon_j\}.\] Equation (170) shows that replacing the radii by \(r_i\) changes each off-diagonal covariance by a bounded amount. The covariance matrix \(C^r\) of the enlarged circle averages is positive semidefinite. Therefore \[-\sum_{i<j}\sigma_i\sigma_j C^r_{ij} =\frac12\sum_i C^r_{ii} -\frac12\sum_{i,j}\sigma_i\sigma_j C^r_{ij} \leq\frac12\sum_i C^r_{ii}.\] Since \(C^r_{ii}\leq q\log(4/d_i)+C_{E,K}\), we have the domination \[ 0<\mathbb E\prod_i V_{\varepsilon_i}^{\sigma_i}(x_i) \leq C_{E,K,m}\prod_i d_i^{-q/2}. \tag{172}\] For \(m=1\), the expectation is exactly one. The constants in (172) are independent of every radius, which is essential when two different regularizations are compared.

Here is an explicit proof that the dominating function is integrable. Put \(a=q/2<1\). Each factor satisfies \[d_i^{-a}\leq\rho^{-a}+\sum_{j\ne i}|x_i-x_j|^{-a}.\] Expanding the product gives finitely many directed graphs on \(\{1,\ldots,m\}\), each vertex having at most one outgoing edge and no self-edge. An edge \(i\to j\) contributes \(|x_i-x_j|^{-a}\). Every component is a tree or a tree attached to a single directed cycle. Integrate vertices with no incoming edge first. Such a vertex has at most one remaining incident edge, and \[\sup_{y\in E}\int_E |x-y|^{-s}\,\mathrm dx<\infty,\qquad 0\leq s<2,\] by polar integration over a fixed disk containing all differences \(E-E\). This removes all attached trees at a uniform cost. A two-vertex directed cycle contributes \(|x-y|^{-2a}\), which is integrable because \(2a=q<2\). On a cycle with at least three vertices, integrate one vertex using \[\int_E|x-y|^{-a}|x-z|^{-a}\,\mathrm dx \leq\left(\int_E|x-y|^{-2a}\,\mathrm dx\right)^{1/2} \left(\int_E|x-z|^{-2a}\,\mathrm dx\right)^{1/2}\leq C_{E,a}.\] The remaining graph is a path and can be integrated from its ends. Isolated vertices contribute \(|E|\). This proves integrability of every term of the expansion, including all simultaneous collision configurations.

At distinct centers, the covariances in (171) tend to \(G_D(x_i,x_j)/K\) as all radii vanish, without any constraint on their relative rates. Dominated convergence now gives convergence of all integrated mixed kernels. In particular, the second mixed kernel with radii \(\varepsilon,\delta\) shows that \(V_\varepsilon(f)\) is Cauchy in \(L^2\). Uniform bounds on the even absolute moments follow by applying (172) with \(m\) positive and \(m\) negative signs and bounding the test functions by their suprema. The \(L^2\) limit and these bounds imply convergence in every finite \(L^p\): for a given \(p\), use a uniform even moment of order strictly larger than \(p\) and uniform integrability. Hölder’s Inequality then passes every mixed product to the limit and proves (169).

For completeness, the limits form a random distribution. Embed the square in a larger flat torus and let \(e_\nu\), \(\nu\in\mathbb Z^2\), be its normalized Fourier basis. For \(\chi\in C_c^\infty(D)\), the second-kernel bound gives \[\sup_{\varepsilon,\nu}\mathbb E|V_\varepsilon(\chi e_\nu)|^2<\infty.\] Each coefficient is Cauchy in \(L^2\) and \(\sum_{\nu\in\mathbb Z^2}(1+|\nu|^2)^{-2}<\infty\). Dominated convergence in this series proves that \(\chi V_\varepsilon\) is Cauchy in \(L^2(\Omega;H^{-2})\). On a countable nested family of cutoffs these limits are compatible, because multiplication by a fixed smooth cutoff is continuous on \(H^{-2}\). They define \(V_{K,D}\in H^{-2}_{\mathrm{loc}}(D)\) almost surely, with the claimed local convergence. In particular this construction uses the circle averages and the complete Dirichlet variance specified in the statement, including its finite spatial part. ◻

Proof. All signed products of the \(U_n\) have nonnegative expectations by Lemma 43, since their deterministic multipliers are positive. We first record consequences of (174). If \(B\) is a square of side \(d\leq1\) in a fixed compact interior set, then \[ \mathbb E|S_n^+(1_B)|^2\leq C_E(d+n^{-1})^{4-Q}. \tag{175}\] To verify this including the lattice diagonal, fix \(x\in B\cap D_n\). The number of lattice points \(y\) with \(j/n\leq|x-y|<(j+1)/n\) is at most \(C(j+1)\). Writing \(h=n^{-1}\), summation up to \(C(d/h+1)\) gives \[h^2\sum_{y\in B\cap D_n}(|x-y|+h)^{-Q} \leq C h^{2-Q}\sum_{j\leq C(d/h+1)}(j+1)^{1-Q} \leq C(d+h)^{2-Q}.\] There are at most \(C(d/h+1)^2\) possible \(x\), which proves (175). The same summation on any fixed compact set gives a uniform second-moment bound for its indicator smear. For arbitrary complex weights supported there, positivity and the two-point bound imply \[ \mathbb E|S_n^+(f)|^2\leq C_E\|f\|_\infty^2. \tag{176}\]

Lemma 43 applies to \(S_n^+(w)\) for every nonnegative weight function \(w\), since \(n^{-2}a_n(x)w(x)\geq0\). Decompose a complex \(f\) as \[f=(\operatorname{Re}f)_+-(\operatorname{Re}f)_- +i(\operatorname{Im}f)_+-i(\operatorname{Im}f)_-.\] Each of the four weights is nonnegative and at most \(\|f\|_\infty\). Minkowski’s Inequality, (166), and (176) give \[ \|S_n^+(f)\|_{L^p}\leq C_E\sqrt p\,\|f\|_\infty, \qquad p\geq2, \tag{177}\] uniformly in \(n\). This estimate applies to bounded weights, so indicators used in the collision argument cause no smoothness issue.

Fix \(m\geq2\), signs \(\sigma_i\), and test functions \(f_i\) supported in \(E\Subset D\). Expand the mixed moment as a lattice sum. For a specified pair of indices \(i,j\), consider the part where \(|x_i-x_j|<d\). A grid of squares of side \(d\), enlarged by a fixed factor, provides at most \(C_Ed^{-2}\) squares \(B\) containing every such pair. For \(d\) small these enlarged squares lie in a fixed larger compact \(E'\Subset D\). Nonnegativity of every signed correlation and \(|f_i|\leq\|f_i\|_\infty\) show that the absolute value of this part is at most a fixed test-function factor times \[\sum_B\mathbb E\left[ S_n^{\sigma_i}(1_B)S_n^{\sigma_j}(1_B) \prod_{r\ne i,j}S_n^{\sigma_r}(1_{E'})\right].\] The expectations in this display are nonnegative; their absolute values are bounded by the expectations of the products of absolute values. Hölder with \(m\) equal exponents and Lemma 43 give for each square \[\left|\mathbb E\left[ S_n^{\sigma_i}(1_B)S_n^{\sigma_j}(1_B) \prod_{r\ne i,j}S_n^{\sigma_r}(1_{E'})\right]\right| \leq C_{E',m}\mathbb E|S_n^+(1_B)|^2 \leq C_{E',m}(d+n^{-1})^{4-Q}.\] Indeed, each of the two box factors has \(L^m\) norm at most a constant times its \(L^2\) norm, and all other factors have uniformly bounded \(L^m\) norm. Consequently the contribution from all collision pairs obeys \[ \limsup_{n\to\infty} |\hbox{collision contribution at distance }d| \leq C_{E,m,f_1,\ldots,f_m}d^{2-Q}\longrightarrow0. \tag{178}\] This includes repeated lattice sites and every combination of signs; it does not assume a correlation inequality for absolute values of individual spins.

On configurations with mutual distances at least \(d\), use (173). Its uniformity and boundedness show that the lattice sums converge to the corresponding integrals of the continuous separated-point kernel. One can use a smooth cutoff which vanishes when a pair is within \(d\) and is one when all pairs are at least \(2d\) apart; its Riemann sums converge directly. The omitted lattice contribution tends to zero by (178). The omitted continuum contribution tends to zero by the integrable domination in Lemma 44. Letting \(d\downarrow0\) proves convergence of the mixed moment to (169). For \(m=1\), uniformity in (173) on the support and ordinary Riemann summation give the assertion without a collision step.

These moment limits identify the joint laws. For any finite list \(f_1,\ldots,f_k\), each real linear combination \(Z_n\) of the real and imaginary parts of the \(S_n^+(f_j)\) satisfies \(\|Z_n\|_{L^p}\leq C\sqrt p\) by (177). Thus, for every \(t<\infty\), \[\sup_n\mathbb Ee^{t|Z_n|}<\infty;\] this follows by expanding the exponential, since \((C t\sqrt p)^p/p!\) is summable in integer \(p\). Every even moment of the corresponding linear combination \(Z\) of the \(V_{K,D}(f_j)\) is the limit of the lattice moments just proved. The same moment bound and the same exponential-series argument give \(\mathbb Ee^{t|Z|}<\infty\). Its characteristic function is therefore determined by its moments, by absolute convergence of the exponential series. The finite-dimensional lattice laws are tight by their second moments. Along any weakly convergent subsequence, the uniform moment bounds make every polynomial uniformly integrable; hence its limit has the continuum mixed moments. Every real projection of that limit has the characteristic function of \(Z\). The Cramér–Wold Criterion identifies the vector law with that of \((V_{K,D}(f_1),\ldots,V_{K,D}(f_k))\). All subsequential limits agree, giving full-sequence joint convergence of both real and imaginary parts.

Finally choose a larger flat torus containing \(D\), and a nested sequence of smooth cutoffs \(\chi_j\) exhausting \(D\), with \(\chi_{j+1}=1\) on a neighborhood of \(\operatorname{supp}\chi_j\). Equation (176) gives, uniformly in \(n\) and the Fourier index \(\nu\), \[\mathbb E|S_n^+(\chi_j e_\nu)|^2\leq C_j.\] Therefore \[ \sup_n\mathbb E\|\chi_j S_n^+\|_{H^{-2}}^2 \leq C_j\sum_{\nu\in\mathbb Z^2}(1+|\nu|^2)^{-2}<\infty. \tag{179}\] The embedding \(H^{-2}\hookrightarrow H^{-3}\) on the torus is compact: truncate the Fourier series and use that the ratio of the two squared weights is \((1+|\nu|^2)^{-1}\to0\). For any \(\eta>0\), choose bounds \(R_j\) so that the sum of the Markov bounds \(C'_j/R_j^2\) from (179) is below \(\eta\). The simultaneous constraints \(\|\chi_j T\|_{H^{-2}}\leq R_j\) have relatively compact closure in \(H^{-3}_{\mathrm{loc}}(D)\), by this compact embedding and a diagonal subsequence argument. Compatibility of cutoff limits follows from continuity of multiplication by the cutoffs, so the resulting local limits are distributions. This proves tightness in \(H^{-3}_{\mathrm{loc}}(D)\). The already identified test-function laws identify every subsequential limit in that space: a countable dense family of smooth compactly supported tests separates distributions and generates its Borel sigma-field. Hence the full sequence converges to \(V_{K,D}\) there. ◻

Proof. The exact microscopic measure, without an added-noise or small-activity replacement, was initialized in Proposition 36. Propositions 37 and 41 give the same \(s(b)\) and \(c(b)\) for all domains, locations, and signed unit observables. Corollary 42 verifies both hypotheses of Theorem 45, and \(s=O(e^{-cb})\) permits the fixed lower bound on \(b\) required there. Applying that theorem proves the field and moment conclusions. Proposition 39 supplies the effective-coefficient identity with the height law, with the same \(s\). ◻

Proof. On the central interval, \(1-\cos(x/\sqrt b)\ge c x^2/b\). On a fixed slightly larger interval the same inequality holds with a smaller \(c>0\). Outside that interval the Gaussian tail of \(h_b\) gives the bound \(h_b(x)e^{b(\cos(x/\sqrt b)-1)}\le C e^{-c x^2}\), after choosing \(\upsilon\) small. This proves the bound globally on the real line. Directly from the integral defining \(h_b\), \(|h_b(x+iy)|\le e^{y^2/(2\upsilon^2)}h_b(x)\); also \(b|\cos(x/\sqrt b)|(\cosh(y/\sqrt b)-1) \le C y^2 e^{|y|/\sqrt b}\). Hence \(|f_b(x+iy)|\le C e^{(1/2-c)x^2+C y^2 e^{|y|/\sqrt b}}\).

To obtain the factor \(b^{-1}\), on \(|x|\le\sqrt b\) write \(f_b-1\) as \((h_b-1)e^{z^2/2+b(\cos(z/\sqrt b)-1)}\) plus \(e^{z^2/2+b(\cos(z/\sqrt b)-1)}-1\). The first term uses the exponentially small cutoff error, with the same imaginary bound. Taylor’s formula with integral remainder bounds the exponent in the second by \(Cb^{-1}(|x|+|y|)^4e^{|y|/\sqrt b}\). The inequality \(|e^u-1|\le |u|\max(1,e^{\Re u})\) and the previously proved real Gaussian margin absorb the \(x\) polynomial into a fixed increase of its square coefficient; the \(y\) polynomial is absorbed into \(C e^{C y^2e^{|y|/\sqrt b}}\). On \(|x|>\sqrt b\), the global bound and \(b\le C e^{c x^2/2}\) give the same result. These increases leave \(\kappa_0<1/2\). Finally differentiation under the positive real integral shows \[(\log h_b)''(x)=-\upsilon^{-2} +\upsilon^{-4}\operatorname{Var}_x(t)\ge-\upsilon^{-2}.\] Adding the second derivative \(1-\cos(x/\sqrt b)\ge0\) of the other exponent proves the Hessian bound. ◻

Proof. Apply Equation (181) on each edge, with \(x=\sqrt b\,d\theta\). For a branch integer \(m\), put \(w=\mathcal R^{-1}\sqrt b(d\theta+2\pi m)\). Its divergence is \(\partial^*w=\mathfrak p q\) for an integral mean-zero \(q\). The incidence matrix of a connected graph maps integral edges onto integral mean-zero vertex arrays: a rooted spanning tree gives a solution by assigning to each parent edge the sum of charges below it. Thus every such \(q\) occurs. Two integral branch solutions with the same \(q\) differ by an integral primal gradient after crossing. To verify this last claim, integrate their increments along primal paths; zero sums around dual plaquettes make the result path independent, and fixing its value at the fused vertex makes it integral and unique. Summing over these gradients unwraps the angular fundamental domain to the complete transverse real space. Its Jacobian is constant, independent of \(q\) and of the source. Orthogonal decomposition gives \(w=X+V_q\), with \(X\in\operatorname{ran}P_T\); its Gaussian square splits into \(\|X\|^2+\|V_q\|^2\).

In the angular insertion the continuous lift gives \(-i(X,\eta)\). Changing that lift by the gradient part of \(2\pi m\) contributes \(-2\pi(dG_nl,m)=2\pi(q,a)\) modulo \(2\pi\), because the selected total current is integral. This is exactly the second factor in Equation (184). Orthogonality and the definition of \(G_n\) give the last two assertions. All sums converge on a finite graph by the real Gaussian branch bound in Lemma 48. ◻

Proof. At first take \(s>0\), finitely many Fourier indices, and multiply every compensating quadratic by a number \(\lambda<1\). Integration then uses the ordinary complex Gaussian density whose real precision on \(w,p\) is positive. Normalize its Gaussian integral by its determinant, continuously from real positive diagonal precision. The mixed off-diagonal entries are purely imaginary, so this prescription is unambiguous. On restoring the full compensation its formal exponent becomes \(-\|w\|^2/2-i(w,p)\). Integrating a fixed neutral Fourier charge by Gaussian completion gives real mean \(V_q\) and covariance \(P_T\) for \(w\); the remaining charge factor is \(e^{-\|V_q\|^2/2}\). The matrix calculation is also obtained by first integrating \(p\) at \(\lambda<1\) and letting \(\lambda\uparrow1\); the covariance tends to the stated positive semidefinite matrix. The determinant ratio is independent of \(q\).

For the offset identity put \(w'=w-i d_c\eta/\mu\) in the full compensated exponent. Expansion gives exactly \[-\tfrac12\|w'\|^2-i(w',p)-i(w',\eta) +\tfrac12(s+d_c^2/\mu)\|\eta\|^2.\] Every additional linear term involving \(p\) vanishes since \((p,\eta)=0\). The vertex shift supplies \(e^{i\mathfrak p(q,\vartheta)}\). The contour translation in \(w\) is allowed at every fixed regularization by the strict real Gaussian margin of Equation (182); for an entire integrand the vertical sides of a translating rectangle have integrals tending to zero.

For completeness the uniform domination needed here, and in subsequent local calculations, is proved in Lemma 60 and Proposition 62: at fixed depth it retains \(e^{-c b\sum q_v^2}\) after any fixed deterministic contour shifts and permits \(\lambda\uparrow1\) and Fourier truncation removal. The use of that estimate is noncircular: its proof starts with the same strict, finite regularizations and only uses their finite Gaussian completion identities. It proves domination before asserting the present limiting identities. At \(s=0\), Equation (185) forces \(s_{12}=0\); take a limit from the permitted region. Source-independent determinants cancel in every partition ratio. Thus no positive joint probability law for \((w,\psi)\) is being assumed at a degenerate endpoint. ◻

Proof. Each \(P_d\) is a nonnegative polynomial at most one on the spectrum, of degree \(p_0(d-1)\). Thus \(Q_r-Q_{2r}=Q_r(1-P_{2r})\ge0\) and its zero at \(\lambda=0\) makes Equation (189) a polynomial of degree \(O(r)\). This proves positivity and range. On the Fourier square \(\lambda\asymp|\xi|^2\), and the product over dyadic \(d\) gives \[|\widehat\Gamma_r(\xi)|\le C\min\{r^2,\ |\xi|^{-2} e^{-c(\log^+(r|\xi|))^2}\}.\] Indeed for \(d|\xi|\ge1\) the factor is bounded by \(C(d|\xi|)^{-2p_0}\), whose logarithms sum to a negative quadratic in \(\log(r|\xi|)\); for \(r|\xi|\le1\) use \(1-P_{2r}\le Cr^2|\xi|^2\). Multiplication by \(|\xi|^j\) and integration in two dimensions reduces the pointwise estimate to \(r^{-j}\int_1^\infty t^{j-1}e^{-c(\log t)^2}\,dt\), bounded by \(C e^{C_0j^2}r^{-j}\) by completing the square. Taking a supremum instead of an integral proves the square-root operator estimate. Discrete Fourier sums give the same bounds on finite periodic extensions. Reflection has only a fixed number of images within range at scales below the domain; the terminal Fourier sum gives the remaining cases. For \(j\le8+\lceil\log(2+r)\rceil\), the stencil extension is still \(O(r)\).

On a small annular frequency band \(a/r\le |\xi|\le4a/r\), the earlier product is bounded below since its relative errors sum as \(\sum_{d\le r}d^2|\xi|^2\le C a^2\); the last difference is bounded below by \(c_a r^2\). Each binary piece consequently contributes a positive constant to its diagonal. For reflected cosine eigenfunctions, pair a frequency index \(u\) with \(2u\) inside a slightly wider band: at least one of their squared cosines is bounded below at every site. Apply this in both coordinates. The terminal clearance ensures enough frequencies per band. Summing the binary scales proves the diagonal assertion.

The scalar pieces telescope to \(G\) and \(G_n\), respectively. Applying gradients yields \(P_L+P_T=I\); on a torus the harmonic addition supplies the missing two directions. Finally \(E_{\ge S}\) is the identity minus a finite sum of polynomial kernels of total range \(CS\). ◻

Proof. Use the test convention in Equation (193), and put \[W(t)=\sup_{j\in\mathbb N_0}t^j e^{-K_0j^2}.\] Choose a periodic lattice box of side \(T\asymp S\) containing the fine region and its collar. A cutoff \(\chi\) equals one there, is supported in the prescribed coarse neighborhood, and satisfies \[|\nabla^k\chi|\le C^{k+1}(k!)^3S^{-k}.\] Such a cutoff is obtained by sampling a fixed compact Gevrey bump after rescaling by \(S\). For scalar fields expand \(f=\chi(\psi-\psi(v_0))\), and for edge fields expand \(f=S\chi w\). Write \(c_\ell(f)\) for normalized discrete Fourier coefficients and \(r_\ell=1+|\ell|\), using representatives with \(|\ell_i|\le T/2\).

The cutoff preserves the derivative allowance at the cost of an exponential factor in the order. Indeed, a Leibniz term with \(k\) derivatives on the cutoff has, relative to \(e^{K_0j^2}\), the factor \[(k!)^3 e^{K_0(j-k)^2-K_0j^2} \le (k!)^3e^{-K_0k^2}\le C_{K_0}.\] Summing the Leibniz terms, including harmless lattice shifts, gives \(S^j|\nabla^jf|\le C^{j+1}e^{K_0j^2}\) for \(0\le j\le j(S)\) on normalized coarse tests. Summation by parts in a coordinate with maximal \(|\ell_i|\) gives \[|c_\ell(f)|\le C C_1^j r_\ell^{-j}e^{K_0j^2}, \qquad 0\le j\le j(S).\] For sufficiently small fixed \(c>0\), \[A_\ell(z)=W(cr_\ell)c_\ell(f)\] is therefore a bounded coarse atom. Scalar coefficients annihilate constants because of the anchor subtraction. The finite derivative budget suffices: a maximizing index for \(W(t)\) can be chosen at most \(J\) whenever \(\log t\le K_0(2J+1)\); this holds for \(t=cr_\ell\), \(|\ell|\le CT\), and \(J=j(S)\).

Let \(e_\ell\) denote a Fourier mode. On the fine region the scalar mode \(e_\ell-e_\ell(v_0)\) has fine test norm at most \[C L^{-1}r_\ell W(Cr_\ell/L).\] Subtracting its discrete affine interpolant gives the bound \[C L^{-2}r_\ell^2W(Cr_\ell/L).\] At orders zero and one this is discrete Taylor’s formula; at orders at least two the two frequency factors are already present. For edges the Fourier mode carries the factor \(S^{-1}\), and the corresponding polynomial powers of \(r_\ell\) are zero and one. The cutoff equals one at every site needed for the anchor and its difference stencils, so these are exact identities for the specified restrictions and affine errors.

For every fixed positive integer \(N\), shifting the integer in the supremum defining \(W\) gives \[W(t)\ge e^{-K_0N^2}t^N W(te^{-2K_0N}).\] If \(L\ge(C/c)e^{2K_0N}\), then \[\frac{W(Cr/L)}{W(cr)}\le e^{K_0N^2}(cr)^{-N}.\] Choose \(N>P+2\), where \(P\le2\) is the polynomial power above. Thus \[\sum_{\ell\in\mathbb Z^2} r_\ell^P\frac{W(Cr_\ell/L)}{W(cr_\ell)}\le C.\] Normalize the fine modes by their displayed bounds and absorb their remaining factors into the numerical coefficients. This proves the two asserted coefficient sums. Real and imaginary parts may be separated at a fixed additional cost. Near a wall perform the expansion on the reflected extension and average the modes over the finitely many nearby reflections, with their prescribed scalar or oriented-edge signs. Choose \(\chi=1\) also on the required image regions. The identities and normalized bounds persist. The periodic boxes under consideration unwrap with the stated clearance, so the same construction applies there. ◻

Proof. Write \(\mathcal D_r=\nabla\Gamma_r^{1/2}\). The multiplier bounds in Equations (189) and (190) imply \[|d_r(\xi)|\le C\min\{r|\xi|, e^{-c(\log^+(r|\xi|))^2}\}.\] Here \(|\xi|\) may equivalently be replaced by the square root of the lattice Laplacian symbol. On nonconstant terminal modes the same bound holds because the terminal scale is comparable to the box side. The white resolution, Equation (191), or Fourier Cauchy–Schwarz gives, for \(F=\sum_r\mathcal D_rg_r\), \[\|F\|_2^2\le C\sum_r\|g_r\|_2^2.\] Set \(R_u=T_ug-Q_uF\). The term from scale \(r\) has multiplier \((1-Q_u)d_r\) for \(r\ge u\), and \(-Q_ud_r\) for \(r<u\). For every \(q\ge0\), \[\|(u\nabla)^qR_{u,r}\|_{2\to2} \le Ce^{C(q+1)^2}\min(r/u,u/r).\] For \(r<u\), put \(t=u|\xi|\). The multiplier is bounded by \(C(r/u)t^{q+1}e^{-c(\log^+t)^2}\). For \(r\ge u\), put \(t=r|\xi|\) and \(\rho=u/r\). Use \(|1-Q_u|\le C\min\{1,(\rho t)^2\}\) and \[(\rho t)^q\min\{1,(\rho t)^2\} \le\rho^{q+1}t^{q+1}.\] The supremum of a fixed power of \(t\) times the logarithmic Gaussian decay costs at most \(Ce^{C(q+1)^2}\). Young’s Inequality on binary scale indices consequently gives \[\sum_u\|(u\nabla)^qR_u\|_2^2 \le Ce^{C(q+1)^2}\sum_r\|g_r\|_2^2.\] For \(q\ge1\) the same estimate holds directly for \(T_ug\), because only \(r\ge u\) occurs and the derivative gives the summable gain \((u/r)^q\).

The scaled lattice Sobolev estimate in two dimensions is \[\sup_{|y-x|\le Au}|H(y)|^2 \le C\sum_{k=0}^2u^{2k-2} \sum_{|z-x|\le Bu}|\nabla^kH(z)|^2.\] To obtain it, apply a cutoff on the larger box and use Fourier Cauchy–Schwarz with weight \((1+u^2|\xi|^2)^2\); the inverse-weight sum is \(O(u^{-2})\). Summing the displayed estimate over \(x\) and then over \(u\) proves the non-tangential bounds for \(R_u\) and the positive-order tails. At order zero the remaining term satisfies \[\sup_{|y-x|\le Au}|Q_uF(y)|\le C M(|F|)(x),\] where \(M\) is the lattice averaging maximal operator: \(Q_u\) has range \(O(u)\) and kernel size \(Cu^{-2}\). The square bound for \(M\) follows from the cube covering inequality and layer-cake integration. These arguments prove the asserted bounds. Their constants grow at most as \(Ce^{C(q+1)^2}\); summing with \(e^{-K_0q^2}\) is legitimate for sufficiently large fixed \(K_0\).

Reflection reduces the finite wall problem to periodic extensions with the required signs and a fixed volume multiplicity. The same Fourier and local estimates hold there, counting norms per period. The last remainder has the stated box-scale estimates. Harmonic constants, where present, have zero approximation error since \(Q_u1=1\), zero positive differences, and a square bound from the white resolution. This also covers the finite direct sums in the statement. A grouped shell is obtained by restricting the binary tail indices and by taking its corresponding coordinate direct sum, which does not enlarge these estimates. ◻

Proof. Let \(\mathcal M g\) be the derivative-weighted maximal function in Lemma 53. A scalar atom born at size \(h\) may subtract the future scalar field’s anchor value. A lattice path bounds the zeroth order by \(Ch\) times its gradient maximum; the higher test orders are controlled by the corresponding terms in \(\mathcal M\). An edge atom obeys the same bound because its test normalization contains the extra factor \(h\). Thus either atom acts on future fields with absolute value at most \[Ch\inf_{x\in B}\mathcal M g(x),\] where \(B\) is its comparable birth box and the fixed aperture is large enough to contain the required collar. For each atom Abel summation writes a weighted future sum exactly as \[\sum_{r\ge h}a_rF_r =a_hT_h+\sum_{r>h}(a_r-a_{r^-})T_r.\] The sum of coefficient magnitudes is uniformly bounded by the assumed magnitude and total variation, so the preceding atom bound persists. All tails here start at a size at least the atom birth.

For the family of birth boxes put \[\rho(x)=\sum_\alpha h_\alpha^{-1}1_{B_\alpha}(x).\] Bounded overlap and geometric summation over dyadic sizes give \[\|\rho\|_\infty\le C,\qquad \rho(x)^2\le C\sum_\alpha h_\alpha^{-2}1_{B_\alpha}(x), \qquad \|\rho\|_2^2\le Cn_a.\] For the middle inequality compare both sides to the inverse square of the smallest occupied size at \(x\). Since \(|B_\alpha|\asymp h_\alpha^2\), duality and Lemma 53 give \[\begin{split} |\langle I,g\rangle| &\le C\sum_\alpha h_\alpha \inf_{B_\alpha}\mathcal M g \le C\sum_x\rho(x)\mathcal M g(x)\\ &\le C\sqrt{n_a} \left(\sum_r\|g_r\|_2^2\right)^{1/2}. \end{split}\] This proves the source bound on all stated components. Apply Lemma 53 once more, now to the scalar-coordinate source itself, to obtain the output square bound. The proof uses the finite extensions and terminal estimates already included in that lemma, so it applies to all the stated geometries. ◻

Proof. First omit the output coefficients and write \(f=\sum_\alpha f_\alpha\) for the full output. Keep the allowed atom-dependent input coefficients. Testing a binary covariance derivative kernel at \(r\ge\max(h_\alpha,u)\) against its atom gives \(Ch_\alpha r^{-2-j}\) for \(j=0,1\). For scalar atoms subtract a kernel constant before testing; for edge atoms the second covariance derivative and edge normalization give exactly the same bound. The high derivative bounds in Equation (190) are absorbed by the test weights, using \(K_0\) as fixed above. The kernel vanishes unless its range reaches the atom box. Summing over binary pieces, and using the uniform bound on the input coefficients, yields \[ |\nabla^jf_\alpha^{\ge u}(x)| \le\frac{Ch_\alpha} {(h_\alpha+u+\operatorname{dist}(x,B_\alpha))^{2+j}}, \qquad j=0,1. \tag{196}\] The terminal piece satisfies the same estimate at box scale.

Let \(Q\) have side \(u\ge1\). Call an atom near if \(h_\alpha\le C_0u\) and its birth box is within \(C_0u\) of \(Q\); all others are far. Choose \(C_0\) greater than the range and cone constants. The number of near atoms at size \(h\) is at most \(C(u/h)^2\), and their total number is at most \(Cu^2\). Lemma 54, applied just to this subfamily and then to the full output operator, gives \[\frac1{|Q|}\sum_Q|f_{\rm near}|\le C.\] Equation (196) bounds their tail from \(u\) pointwise on \(Q\) by \[\sum_{\alpha\ {\rm near}} \frac{Ch_\alpha}{(h_\alpha+u)^2} \le C\sum_{\substack{h\le C_0u\\h\ {\rm dyadic}}}h^{-1} \le C.\] On \(Q\) the full output of each far atom equals its tail from \(u\): its birth size is larger, or finite range excludes all smaller pieces. The variation of all far outputs across \(Q\) is at most \[C\sum_\alpha \frac{u h_\alpha} {(u+h_\alpha+\operatorname{dist}(Q,B_\alpha))^3} \le C\sum_{h\ {\rm dyadic}}\frac{u}{h(u+h)}\le C.\] For the middle bound, bounded overlap at size \(h\) compares the spatial sum with \(h^{-2}\) times the integral of the kernel; that integral is \(O((u+h)^{-1})\). Small sizes contribute \(C\sum_{h\le u}h^{-1}\) and large ones \(C\sum_{h>u}u h^{-2}\), both bounded. It follows that \[\frac1{|Q|}\sum_Q|f-f_Q|\le C, \qquad |f^{\ge u}(y)-f_Q|\le C\quad(y\in Q).\] In particular \(f\) has uniformly bounded BMO norm and, with a fixed enlargement of \(Q\) for the cone, \[T_*I:=\sup_u\sup_{|y-x|\le Au}|f^{\ge u}(y)| \le C M(|f|)+C.\] The same kernel estimates bound unit-scale neighbor differences. Thus the piecewise-constant extension has bounded ordinary cube BMO norm, including cubes smaller than a lattice cell.

Here is the needed exponential BMO estimate, in the form of the John–Nirenberg Inequality (John and Nirenberg 1961). In a cube \(Q\), stop on maximal dyadic subcubes where the average of \(|f-f_Q|\) exceeds twice the uniform BMO bound. Their total volume is at most \(|Q|/2\). Their averages differ from \(f_Q\) by a fixed multiple of that bound, because their parents did not stop. Repeat inside each selected cube with its own average. After \(k\) generations the selected union has volume at most \(2^{-k}|Q|\); outside it, \(|f-f_Q|\le C(k+1)\). Consequently \[|\{x\in Q:|f(x)-f_Q|>t\}|\le C|Q|e^{-ct}.\] This argument applies componentwise to real and imaginary parts.

Select maximal averaging cubes of \(|f|\) above a sufficiently large fixed level. By Lemma 54 and the square maximal inequality their total volume is \(O(n_a)\). The averages on these cubes are uniformly bounded, since their parents are below the chosen level. Outside their union \(|f|\) is below that level. Applying the preceding exponential estimate on the selected cubes therefore gives \[|\{x:|f(x)|>t\}|\le Cn_a e^{-ct},\qquad t\ge C.\] Below this fixed level use \(\|f\|_2^2\le Cn_a\) instead. Layer-cake integration gives \[\sum_x|f(x)|^p\le n_a(Cp)^p,\qquad p\ge2.\] For completeness the \(L^p\) maximal bounds used here need no endpoint hypothesis: the cube covering inequality applied to \(|f|1_{\{|f|>t/2\}}\) bounds \(|\{M|f|>t\}|\) by \(Ct^{-1}\sum_{|f|>t/2}|f|\); integration gives a constant uniformly bounded in operator norm for \(p\ge2\).

The additive constant in the comparison of \(T_*I\) with \(M(|f|)\) is not summed over the whole plane. On \(\{T_*I\le C_1\}\) use \((T_*I)^p\le C_1^{p-2}(T_*I)^2\) and the separate square bound in Lemma 54. On its complement choose \(C_1\) large enough that \(T_*I\le C M(|f|)\) and use the \(L^p\) maximal inequality. This proves \(\sum_x(T_*I)^p\le n_a(Cp)^p\). Finally Abel summation at each output point bounds the requested weighted output by \(C T_*I\), proving Equation (195).

In finite reflected or periodic geometries, perform the argument on the prescribed extensions and count volume per period. Bounded overlap implies that the number of atoms per period is at most a constant times its volume; the square estimate therefore bounds the whole-period average by a fixed constant. Choose the stopping level above it. All stopping cubes then have the stated parent bound, and the same proof gives the required exponential tail with prefactor \(Cn_a\). The final box-scale remainder is included in Equation (196); harmonic constants, if present, have no variation and obey their square bound. Thus neither the finite endpoint nor reflection changes any constant in the conclusion. ◻

Proof. Initially each selected vertex factor appears once at its vertex, and each selected edge factor of a specified type appears once and contains its two endpoints. Convolution takes closures, a residual keeps the full temporary support, and products take unions. The only shrinking operation is coefficient detachment in Definition 56(iii), where the old Gaussian root, including its atoms, has ended. A new polynomial begins with its own support and atoms. Scalar decorations introduce no field, and choosing a summand or one contour derivative does not copy a microscopic factor. This proves (a) inductively within each root. A tilt factor and an \(F_b\) factor may use the same physical edge; the no-repetition assertion is separate for each type, as required below.

For (d), fix a Taylor change at scale \(S\). Its child is a single input, so all live leaves in that child receive the same relative factor. For any other leaf with a different factor, follow the branches to their first product ancestor. If that product is formed at this regrouping, they were distinct temporary objects and their original supports have distance greater than \(r_0S\). If it occurs later, the two unshrunk ancestors are distinct compatible fine components, or distinct temporary objects at that later step. Their distance is therefore greater than the corresponding fixed radius times a scale at least \(S\). There is no case of differently changed live branches inside the same temporary Taylor input: that input was linear. By (a), later closure cannot have hidden an earlier nearby leaf. Choose \(r_0\) larger than the range constant in Equation (191). Its white tail then has the asserted zero cross blocks. Unchanged leaves count as a group with relative multiplier one.

Here is the collar induction needed for (b). If a newly emitted atom uses \(c_0\) birth-block widths and all inherited dependence uses \(c_*\) old-block widths, the new dependence is contained in \(c_0+c_*/L+c_{\rm st}\) current-block widths, with a fixed stencil allowance \(c_{\rm st}\). Fix \(c_*=2(c_0+c_{\rm st})+1\); the required induction inequality then holds for every \(L\ge2\). Let \(c_{\rm ker}\) bound the relevant kernel range in current block units. The small-closure conclusion of Lemma 28 makes a support relocation at most one current block, so fix \(c_{\rm move}=1\) for that conclusion. Choose the temporary radius \(r_0>2c_*+c_{\rm ker}+4\) and the compatibility radius \(R>r_0+2c_{\rm move}+4\), increasing them if necessary for earlier kernel ranges. The small-support threshold in Lemma 28 uses these fixed radii, and \(L\) is then taken large enough for its small-closure conclusion. The resulting fine small-support diameter is fixed before this last choice of \(L\), so the Fourier cutoff uses a fixed number of coarse blocks. These separations are the locality hypotheses used in the factorization proof of Proposition 33, with the present block collars in place of its smaller microscopic collar. Before compatible inputs meet, all their already integrated shells then have zero cross covariance, including their atom dependence. The current shell at a multiple-input convolution is shared from the outset. Before products of temporary objects, use their stronger temporary separation. Gaussian characteristic functions factor when these cross blocks vanish; Fourier uniqueness proves factorization for the finite complex Gaussian measures of the strict regularization. The limiting factorization follows after the endpoint bounds below. This proves the common-stack induction. When a finite-grid good piece is copied from the plane, only equality of the restricted covariances is needed; the chosen square roots need not be identical.

For (c), a selected Taylor term emits only its fixed number of direction atoms at the upper birth scale, and only a single-input temporary object emits them. Distinct batches in one final history are separated at that birth scale by the same first-product-ancestor argument. In particular an upper block may contain \(O(L^2)\) fine inputs, but its multiple-input convolution emits no new Taylor atoms: the old atoms retain their old birth scales. A tensor is a normalized vertex/orientation sum; a selected summand contains only its two linear factors, not a pair at every vertex. The same observation applies to a localized quadratic assignment. Products of these primitives have fine compatibility or temporary separation. At a fixed scale there are only finitely many such types. If a coefficient calculation ends, its older atoms end with it and are not stacked on its newly assigned polynomial. Repeated opaque occurrences of that coefficient add no atoms to the new root; if expanded, each copy is checked on its own completed stack. At the microscopic scale the list of primitive types and polynomial degrees is fixed. This proves the bounded overlap without an \(L^2\) loss. Probes have the same ancestry and a fixed number per node. Reflections, periodic distance, and the terminal harmonic piece do not alter any of these support arguments. ◻

Proof. Restrict the white kernels to the distinct edges under consideration and write \(A_r\) for their diagonal attenuations. The row covariance is \(\sum_rA_rE_rA_r\). If the history stops early, add the missing positive tail with the last attenuation. Remove changes in reverse chronological order. After all later changes have been removed, a change at \(S\) leaves a tail \(P Q E_{\ge S}Q P\), where \(P\) records all earlier factors and \(Q\) the new relative factors. By Lemma 57(d), the restricted tail is block diagonal between groups on which \(Q\) differs. On each block \(Q=qI\) with \(0\le q\le1\), whence \(QE_{\ge S}Q\preceq E_{\ge S}\). Congruence by \(P\) preserves this order, even when \(P\) varies within a block. Removing every change leaves \(\sum_rE_r=I\). This proves the \(R_w\) and \(R_F\) bounds. Replacing \(D_rD_r^*\) by \(E_r\) proves the \(R_p\) bound. Finally the same attenuation in both fields gives \(R_wJ=R_p\) row by row. Deterministic probe and twist backgrounds do not change these linear maps. ◻

Proof. Equation (200) follows by substituting the linear maps of Lemma 58 into the quadratic tilts. Those contraction bounds imply \(M=I+O(|s_\bullet|)\) and \(H=J+O(|s_\bullet|)\), while \(0\preceq S_0\preceq sI\). The Schur complement is therefore positive and close to \(I\). The perturbation of the untilted inverse covariance has rank at most a fixed multiple of the number of tilt leaves. The untilted covariance has bounded operator norm, including as \(s\downarrow0\); its product with the precision perturbation is small in operator norm. Thus the determinant ratio is a product of at most \(CN_h\) numbers of modulus in a fixed neighborhood of one. This proves its bound independently of the number of unused Gaussian coordinates.

For \(S_0>0\), integration in \(y\) or direct block inversion gives covariance \[\begin{pmatrix}V&-iC'\\-i(C')^t&D\end{pmatrix}.\] Its characteristic function with a \(y\) source proves the asserted mean and exponential factor. Positivity of \(V\) is the Schur-complement identity \(V^{-1}=M+HS_0^{-1}H^t\); limits prove the positive semidefinite case.

Let \(A=I-R_p^t\Theta R_p\), \(\Delta=C'-J\). Using \(R_wJ=R_p\) gives \[\begin{align*} H^t\Delta &= (I-S_0D)-(d_cI-s_{12}R_p^t\Theta R_p)\\ &= -A(sD+s_{12}I). \end{align*}\] For nonsingular \(A\) and \(s>0\) the Cameron square satisfies \[\Delta^tV^{-1}\Delta =\Delta^tM\Delta+ (sD+s_{12}I)^t A(sD+s_{12}I)/s\preceq CI,\] since \(s_{12}^2/s\le d_*^{-1}\). The same inequality holds on the supported space by approximation when \(A\) is singular. At \(s=0\) one has \(s_{12}=0\) and \(H^t\Delta=0\), which is exactly the required range condition for the possibly singular \(V\).

Translate the real Gaussian variable by \(i\Delta I\). Its density changes by a phase of modulus one and the factor \(\exp(\|\Delta I\|_{V^{-1}}^2/2)\le e^{C\|I\|^2}\). The remaining evaluation shift is \(iJI\). The entire-function translation is justified first with positive covariance and strict compensation by Equation (182), then by the same Gaussian majorant on the supported limiting space. Linear \(x\)-source factors can be left in the integrand during this translation; their extra deterministic factor is bounded by Cauchy–Schwarz using their source norm. No unrestricted imaginary Gaussian is substituted into \(F_b\). ◻

Proof. The fixed-radius atom contours and the bound \(|d_A|\le C_{\rm tag}\) give \(\prod_A|e^{t_A d_A}|\le e^{CN_h}\) within this endpoint integral. Physical scalar phases have modulus one. The decorated microscopic edge offsets are uniformly bounded. These facts use exactly the admissible decoration rules and do not move any early factor into the structural coefficient.

On a shell acting on current phase charges \(k\), the original source is \(\Gamma_r^{1/2}k\) in the \(y\) coordinates. It is real because all previous charge attenuations were real. Write \(Y=Y_o+Y_a\); the preceding source estimates give \(\|Y_a\|^2\le CN_h\), and every linear \(x\) source has the same bound. Lemma 59 reduces the imaginary evaluations at \(F_b\) leaves to the attenuated outputs of \(JI\), \(I=\Im Y_a\), together with bounded deterministic offsets. By Lemma 55, their values \(v_e\) obey \(\sum_e|v_e|^p\le N_h(Cp)^p\), \(p\ge2\). Consequently \[\begin{align*} \sum_e |v_e|^2 e^{|v_e|/\sqrt b} &\le N_h\sum_{k\ge0} \frac{C^{k+2}(k+2)^{k+2}}{k!\,b^{k/2}} \le CN_h \tag{205}\end{align*}\] for all sufficiently large fixed \(b\). The ratio \((k+2)^{k+2}/k!\) is bounded by an exponential in \(k\) times a fixed polynomial, so the displayed series converges. Bounded offsets add at most \(C\) per leaf. Thus the imaginary growth term in Equation (182) costs \(e^{CN_h}\).

Choose \(\kappa_1\) strictly between \(\kappa_0\) and \(1/2\). Small parameter size and Lemma 58 give a fixed \(\nu>0\) such that \[ 2\kappa_1R_F\{V+C'D^{-1}(C')^t\}R_F^t =2\kappa_1R_FM^{-1}R_F^t\preceq(1-\nu)I. \tag{206}\] For a real Gaussian \(\xi\) of covariance \(\Sigma=R_FVR_F^t\), completing the square in \(\mathbb Ee^{\kappa_1\|\xi+R_FC'Y_o\|^2}\) gives determinant \(\det(I-2\kappa_1\Sigma)^{-1/2}\), at most \(e^{Cn_F}\). Its shifted-mean exponent is \(\kappa_1\langle R_FC'Y_o,(I-2\kappa_1\Sigma)^{-1}R_FC'Y_o\rangle\). Equation (206), or its Schur complement, bounds this by \((1-\nu)(Y_o,DY_o)/2\) after decreasing \(\nu\) if necessary. It therefore leaves a fixed part of the charge exponential \(e^{-(Y_o,DY_o)/2}\) unspent.

Means from \(Y_a\) and deterministic offsets have square \(O(N_h)\) on the evaluation edges. The difference \(\kappa_1-\kappa_0\) absorbs them with \(e^{CN_h}\), using \(2|uv|\le\delta u^2+\delta^{-1}v^2\). The same inequality absorbs cross terms in \(-\Re(Y,DY)/2\), and linear real Gaussian sources, with an arbitrarily small additional fraction of the retained charge square. Their source squares are \(O(N_h)\); Gaussian integration or a small Hölder enlargement gives \(e^{CN_h}\). Since \(D\) is uniformly positive, the remaining suppression is \(e^{-c\|Y_o\|^2}\). Each branch factor contributes the separate \(b^{-1}\) in Equation (182). Polynomial derivatives use their fixed circles and add only a bounded number of source atoms. This proves Equation (204) in strict regularization, with all constants independent of that regularization. ◻

Proof. Fix one selected live root, rather than the sum of its alternatives or the forest of its separately completed coefficient dependencies. At a charge-carrying node let \(H=h_m(k;X)\) be the incoming test budget. At a merger budgets add subadditively by test restriction and the triangle inequality. Reblocking without localization is nonincreasing by Equation (194). At a neutral Taylor node its real attenuation \(\rho\in[0,1]\) gives \[H_{\rm out}\le\rho(C/L)H,\qquad H-H_{\rm out}\ge(1-C/L)H.\] Thus a fixed multiple of this decrease pays Equation (202). At termination the remaining budget is consumed, which also permits bounded final test directions.

For a charged small linear node put \(Q=\sum k\ne0\), \(q=|Q|\), \(\ell=\log L\), and let \(\Gamma\) be its shell. Choose an anchor \(v\) in its support. Lemma 51, summing its positive-order binary bounds, shows that \(\Gamma(\cdot,v)-\Gamma(v,v)\) on the local support and collar is a fixed multiple of a fine test. Consequently \[d:=\Gamma(v,v)\ge c_0\ell,\qquad |(k,\Gamma e_v)-Qd|\le C_1H.\] Fix \(\epsilon=c_0/(2C_1)\) when \(C_1>0\). If \(H\le\epsilon q\ell\), positive-semidefinite Cauchy–Schwarz gives \[ (k,\Gamma k)\ge\frac{|(k,\Gamma e_v)|^2}{d} \ge\frac{q^2d}{4}\ge c q^2\ell. \tag{208}\] If \(C_1=0\) this conclusion holds without a case split. Otherwise, for \(H>\epsilon q\ell\), Equation (194) and choices \(C/L\le1/2\), \(\ell\ge4/\epsilon\) give \[ H-H_{\rm out}\ge(1-C/L)H-q\ge H/4 \ge(\epsilon/4)q\ell. \tag{209}\] At all other nodes the first inequality in Equation (194) is retained; thus no negative budget decrease is hidden in this dichotomy.

The sum of child budgets minus the parent budget is a nonnegative potential loss at every node. Summing these losses over this live root, including the final retained budget, gives at most its original leaf budget \[ H_{\rm leaf}\le C\mathfrak p\sum_v|q_v|. \tag{210}\] Only neutral total charges are multiplied by a Taylor attenuation. Mergers add totals, so every nonzero sector total remains in \(\mathfrak p\mathbb Z\), even though individual entries inside a neutral branch may cease to be integral multiples. Hence \(q\ge\mathfrak p\) at every recorded charged node. A fixed potential multiplier pays both neutral probes and \(4\ell\) at each node of Equation (209), once \(b\) is large.

The alternative using shell energy requires a separate account. At the shell of a recorded charged step, its temporary branch was a single input. Every other live charge group in the eventual combined live root is either another temporarily separated object at this step or first joins at a later product ancestor. Lemma 57(a),(d), with the scalar shell range included in the radius choice, gives zero cross-covariance between that charged branch and every other such group on this shell. Thus its scalar energy splits off exactly, with a nonnegative remaining square. Two selected branches at that shell are mutually separated as well. Different recursion levels use disjoint shell coordinates. A later Taylor attenuation changes future fields only, so it does not alter a payment on an already integrated shell. Writing \(\mathcal E\) for the sum of the selected energies and \(E_0\) for the first-piece square, we have \[ E_0=\gamma\mathfrak p^2\sum_vq_v^2, \qquad E_0+\mathcal E\le\|Y_o\|^2. \tag{211}\] This is a separation statement about generated single-input histories; positivity alone would not allow one to split arbitrary charge squares.

Let \(c\) be the constant in Equation (204). Choose a fixed multiplier \(D_0'\) large enough for all potential payments and the specified final normalized tests. Since \(|q_v|\le q_v^2\) for nonzero integers, increasing \(b\) yields \[D_0'H_{\rm leaf}\le(c/4)E_0.\] Equations (208) and \(q\ge\mathfrak p\) similarly give \(4\ell N_E\le(c/4)\mathcal E\), where \(N_E\) counts the energy-paid steps. These fixed allocations use disjoint portions of Equation (211); a fixed positive fraction of \(E_0\) and of the full source square remains. It supplies the microscopic Gaussian in Equation (207). Detached scalar calculations have their own completed integrals and are absent from this live source square. Reusing an evaluated scalar multiplies its established bound once per opaque occurrence; it does not reinsert any charge leaf or shell reserve. If the occurrences are expanded, apply this proof separately to each completed root and multiply the resulting bounds. No joint overlap or charge-square assertion across those roots is used. ◻

Proof. For a specified representation write each selected term in the form \[c_h(\alpha)\,\ell_h(\alpha) \left(\prod_{\nu=1}^{m_h}a_{h,\nu}\right) \mathcal I_h(z;\mathbf d_h,\alpha).\] Here \(h\) is its unique live root, or the empty root with integral one. The variables \(\alpha\) range over the compact Taylor and contour parameter spaces with their positive product measure. The structural coefficient \(c_h\) contains the displayed parameter powers, finite averaging coefficients, Fourier/Taylor direction weights, and contour kernel factors. The factor \(\ell_h\) is only the explicitly extracted late numerical product defined above, and equals one when there are no such events. The \(a_{h,\nu}\) are opaque occurrences of previously completed coefficients. The admissible early record \(\mathbf d_h\) and the generated phase-probe factors paid by Lemma 61 stay inside \(\mathcal I_h\) and are not part of \(c_h\) or \(\ell_h\).

Give the live root the weight \[ W(h)=C_*^{N_h}\,b^{-n_F} \exp\{-c_*b\textstyle\sum_vq_v^2\}\,L^{-4n_{\rm ch}}, \tag{212}\] with \(W=1\) for the empty root. All counts and charges in this display belong only to \(h\). If \(a\) has a chosen completed representation, let \(\mathsf m(a)\) be the bound supplied by this same rule at its earlier evaluation, on the source domain where the size will be used. A size at one fixed source point may instead use the exact modulus \(|a|\) there. For a uniform estimate or a later Cauchy formula on a source polydisc, \(\mathsf m(a)\) must be a completed size uniform on that polydisc, or another proved bound for the supremum of \(|a|\) there; a modulus at one point does not suffice. Equivalently one may differentiate \(a\) in its separately completed root and use that derivative’s completed size. These choices also apply when completed numbers are combined before forming a new primitive. In particular each source-independent component of the full unmarked tensor may use \(\mathsf m(t_{ab,j})=|t_{ab,j}|\le |t_j|_\infty\) after its evaluated increments have been summed, preserving their cancellation. This choice asserts no derivative bound in the real reference parameters; only their continuity is used in (v). The acyclic coefficient dependence makes this a finite-depth recursive definition. Set \[\mathsf s_X(\mathcal R)=\sum_h\int |c_h(\alpha)|\,|\ell_h(\alpha)| \left(\prod_{\nu=1}^{m_h}\mathsf m(a_{h,\nu})\right) W(h)\,d\alpha.\] Choose \(C_*\) sufficiently large and \(c_*>0\) sufficiently small from the bounds in Lemmas 60 and 61, which are uniform over the admissible early records and sufficiently large \(L\). Induction over coefficient evaluations then gives \(|a|\le\mathsf m(a)\) and makes this size a positive majorant of every endpoint forest. No arbitrary numerical factor can be hidden in the endpoint integral because its admissible records have the specified generation rules and bounds.

The primitive bounds in (i) follow directly: polynomial contours contribute a fixed number of atoms, and Equation (198) has two explicit powers of \(s_\bullet\) and only a bounded tilt strength in its exponential. A full tensor has normalized vertex averages: each selected summand has two normalized atoms and coefficient \(O(|t|)\).

When compatible live roots are concatenated, their live leaf lists and actual event lists are disjoint. The common-stack construction joins them into one live root, while their structural and late numerical factors multiply and their lists of opaque coefficient occurrences concatenate. At the final endpoint apply Equation (207) once to the concatenated live root, whose geometric properties were proved in Lemma 57. Thus no fresh \(C^{N_h}\) is charged each time an old root is convolved. Each opaque numerical coefficient is already bounded at its own endpoint. In particular an inverse-normalizer term \(a_B^m\) has size \(\mathsf m(a_B)^m\), and its absolute series is at most \(\mathsf m(a_B)/(1-\mathsf m(a_B))\) when \(\mathsf m(a_B)<1\). If these occurrences are expanded, they are \(m\) separately completed roots and their endpoint weights multiply; no common atom-overlap estimate is applied to them. This proves the product rule. The purely spatial enumeration in Proposition 33 uses precisely these multiplicative sizes: nonlinear inputs carry at least two factors, large supports use the polymer weight gain, and the inverse scalar decorations use their absolutely convergent small scalar series. It does not require a pointwise norm at arbitrary remaining fields. This proves (ii).

For (iii) Taylor’s integral formula of order \(a\) contributes \(a\) direction arguments and a fixed number of contour events. Apply the exact \(CL^{-1}\) directional decomposition of Lemma 52 to each argument before using Cauchy’s formula. This gives \(CL^{-a}\) in the positive coefficient sum. The evenness assumption makes the linear term vanish when \(a=2\) or \(3\). In the bulk, the quadratic Taylor term has two arguments; replacing one by its affine approximation costs \(CL^{-2}\) and the other \(CL^{-1}\), so the two-factor difference has \(CL^{-3}\). The affine polynomial itself is the assigned tensor prescription of Equation (197). Its normalized coefficient is a terminated history with bounded probes, hence costs \(Cu\) by Lemma 61. Scalar extraction is the same statement with zero Taylor arguments. The charged gain is exactly the recorded factor in Equation (212). Multi-input terms receive no such Taylor or charged gain, consistently with both the grammar and the quadratic counting rule.

For (iv), apply \(T_\Lambda\) only to an untagged live root in the displayed representation, and as the identity to an empty root, using the same source domain and completed coefficient bounds. The transport rule changes \(\mathbf d_h\) inside \(\mathcal I_h\) and leaves \(c_h,\ell_h\), every \(\mathsf m(a_{h,\nu})\), and the event list defining \(W(h)\) unchanged. Its own-birth bound proves that the new record is admissible. The equality of representation sizes in (iv) is therefore termwise in their defining nonnegative sums, before any support relocation; multiplying by \(A_D^{|X|}\) gives the displayed comparison with the \(A_U\) norm. In particular no factor per old event is charged to a support weight. The endpoint bound for the transported integrand is available because Lemma 60 was proved uniformly over these records. This is a comparison of the same chosen representations, not merely a comparison of their endpoint values. At finite strict regularization the factor-by-factor substitution is an exact representation of the translated activity. The uniform majorants used below pass this identity to the limiting evaluated expressions on the bounded background domains used here.

The derivative claims in (iv) follow from the derivative version of Lemma 60 and the final-test allowance in Lemma 61. A Cauchy derivative in a remaining field direction acts on \(\mathcal I_h\) with every opaque scalar occurrence fixed. A derivative in a source parameter on which an opaque scalar depends instead uses its uniform polydisc bound in the preceding definition, or differentiates that scalar within its separately completed root. Thus cutting a root never discards a source-parameter derivative. The summable later-probe calculation proves that a tagged root stays in the admissible class under further defect operations. At late birth sizes only the separate factor \(\ell_h\) changes; its bounded number of events gives a fixed-configuration factor per live root. It does not change an earlier event weight or an opaque completed coefficient size. Thus the inverse-temperature threshold remains independent of the fixed correlation order.

Finally fix a finite calculation depth and a finite geometry. All its support choices and discrete Fourier direction decompositions are finite; compact Taylor and contour parameters can be fixed independently of a small parameter variation. Only the microscopic integral charge indices remain unbounded. Before removing regularizations, all identities are ordinary integrated identities of entire functions with strict real-density integrability: the \(x\) growth is below its Gaussian threshold by Equation (182), the real \(y\) precision is positive with strict compensation, and the mixed terms are imaginary on real coordinates except for controlled lower-degree sources. Lemma 59 proves the integrated formulas and Lemmas 60–61 give bounds uniform as compensation becomes full. At fixed geometry the completion before the Cameron estimate is also continuous: its real covariance \(V\) may become singular, but positive Gaussian measures converge to the measure on its supported space, and its complex shifts are bounded for fixed contour choices. The retained real-charge Gaussian square dominates the remaining fixed-background costs, which grow only exponentially linearly in the charge magnitudes. Thus dominated convergence removes charge truncation and strict compensation at this fixed depth. The same majorants hold uniformly on each smaller source polydisc, so integrating Cauchy formulas proves holomorphy and convergence of source derivatives. It also passes real parameter continuity to the limit. Scalar division is continuous whenever its normalizer stays away from zero, as ensured in the small-activity map. Plane calculations at finite depth use an auxiliary large torus whose preterminal covariance entries agree on all required supports, so the same reasoning applies there. This proves (v) and completes the limiting justification promised in Proposition 50. ◻

Proof. Use the temporary-component, relocation, and constant-decoration identities of Proposition 33. They are algebraic identities for local activities, used with the enlarged fixed separation radii proved in Lemma 57. Their hypotheses here are finite shell range, the retained temporary exclusions, one-step locality of assigned pieces, and multiplicative bounds for compatible inputs. Finite range follows from Lemma 51; the history construction supplies the other hypotheses, including the fact that evaluated coefficients are scalar factors and do not retain their extinct microscopic leaves.

For completeness, the relevant counts are as follows. The maximum fine-block count \(n_0\) of a small component and the connectivity radius are fixed before \(L\). The number of small animals containing a specified fine block is therefore a constant. A coarse block has \(O(L^2)\) fine blocks, giving at most \(CL^2\) small linear inputs per coarse output. Their neutral and charged contributions have total operator bound \[ C L^2(L^{-3}+L^{-4})\le C/L. \tag{217}\] In the neutral bulk quadratic remainder, replace each argument by its affine approximation. The two-factor difference is the sum of one term with the first argument replaced and one with the second replaced. The error direction has size \(C/L^2\) and the other direction size \(C/L\), so each term has size \(C/L^3\). The first-order term vanishes because the full neutral projection is even under \((w,\psi)\mapsto(-w,-\psi)\). This explains the gain used in (217) without assigning a Taylor gain to a nonlinear convolution.

For a large input, the closure estimate leaves an exponential weight gain at least \(A^{-|X|/2}\), after the fixed per-input history cost. Taking \(A\) large after \(L\) makes the sum of its linear contributions as small as prescribed. To check nonlinear terms, root a compatible family at a specified output block. If it has \(q\) fine components and a total of \(v\) fine blocks, a depth-first traversal of the fine animals and a tree connecting their closures gives at most \[C^v(CL^2)^{2q}\] choices. The long jumps of that tree have range \(CL\) in fine-block units. For large components the weight gain sums these choices; the total size of the small components is at most \(qn_0\). Thus all terms with \(q\ge2\) sum to \(C_{L,A}(|t_j|+r_j)^2\). Use a temporary output weight \(4A\) before the constant-decoration step. If \(a_B\) is the assigned constant, then \(|a_B|\le C_{L,A}(|t_j|+r_j)\) and \[d_B=(1+a_B)^{-1}-1=-a_B+O(a_B^2).\] Overlapping a decoration with an old occupied site costs at most two choices, paid by the extra weight. A decoration at a hole supplies a factor \(|d_B|\). The subset sum converges when \(A\sup_B|d_B|<1/8\). Every term containing an old component and a decoration is quadratic; pure decorations are quadratic apart from the displayed cancelling linear singleton. Relocated multiple-object terms are also quadratic, because their original temporary exclusions are retained. These counts prove the stated nonlinear bound.

The entire expectation of a tensor singleton is assigned to the coarse singleton. Its nonconstant part is exactly the same quadratic form; its covariance contraction is a bounded constant. At any specified fine block only a bounded number of small supports contribute quadratic coefficients, each bounded by \(Cr_j\). Averaging over orientation signs and anchors and summing all fine blocks in a full coarse block preserves translation and square symmetries. Hence the sum is precisely \(\mathcal E_B(t_{j+1}-t_j)\), with coefficient bound \(Cr_j\). There is no factor \(L^2\) in this coefficient: the sum over fine sites is the edge sum already present in (213). Complex conjugation combined with \(\psi\mapsto-\psi\) makes the three extracted parameters real. This proves (214) and (215).

At the first independent piece, the linear compensation factors give the stated tensor and their designated covariance constants. The remaining single primitives cost \(C/b\), \(\sum_{q\ne0}e^{-cbq^2}\), or \(C|t_0|^2\). Each edge has two endpoints and only finitely many choices of primitive type. The same animal and decoration counts therefore give (216).

All these identities may first be performed with a finite Fourier cutoff and strict covariance regularization. At a fixed depth there are finitely many geometric choices for each coefficient. The charge sums retain Gaussian domination; the contour and Taylor parameters range over compact sets. The completed finite-dimensional integrals are continuous, including singular real Gaussian covariances, and the scalar denominators stay away from zero. The domination in Proposition 62 consequently permits removing the regularizations and taking parameter limits in each of these exact identities. It also permits fixed smooth-source Cauchy derivatives. This proves the asserted continuity without a positivity assumption on the mixed Gaussian pair.

For the torus take \(F\) larger than the fixed support, kernel, and assignment clearances. Every small input through the indicated scale has an unambiguous lift to the plane, with its required neighborhood; a winding input is large. The local tensor assignments are therefore identical to the plane assignments. Large winding inputs obey the large-support weight estimate. The transverse harmonic directions are added at the terminal integration, as prescribed in Lemma 51, and do not change any earlier local map. ◻

Proof. Translate the full fields before undoing the compensation. The longitudinal vortex argument becomes \(V_q+d_c\lambda g\): the \(\tau\)-terms cancel because \(s-d_ca=0\). The linear source on \(w\) is \((\mu\lambda+iB_c\tau)g\), the source on \(p\) is \(id_c\lambda g\), and the direct translation factor is \[\exp\{-\mu\lambda^2/2-iB_c\lambda\tau +(s+\mu a^2)\tau^2/2\}.\] The constant scalar mode enforces \(\sum q_v=0\). At each such charge configuration the \(p\)-source gives the further factor \(\exp\{-d_c\lambda V_g-d_c^2\lambda^2/2\}\). Since \((X,g)=0\), the linear \(w\)-source is evaluated at \(V_g+d_c\lambda\). Combining these factors gives the exact ratio \[\begin{align*} \frac{Z(\lambda,\tau)}{Z(0,0)} &=\exp\!\left\{\frac{-\mu-d_c^2+2\mu d_c}{2}\lambda^2 +iB_c(d_c-1)\lambda\tau +\frac{s+\mu a^2}{2}\tau^2\right\}\\[-2pt] &\quad\times\mathbb E_\nu\left[ e^{((\mu-d_c)\lambda+iB_c\tau)V_g} \frac{f(w+d_c\lambda g)}{f(w)}\right]. \tag{222}\end{align*}\] The identity first holds in the strict regularizations and then passes to the prescribed finite-graph limit by Proposition 50. Real positivity of \(f\) makes the displayed logarithmic derivatives well defined. Differentiating (222) proves all three Hessian identities; in particular the direct mixed term is \(iB_c(d_c-1)=iB_cs_{12}\).

The branch estimate in Lemma 48 gives \((\log f_b)''\ge-C_f\). As \(\sum_e g_e^2=1\), \[\partial_g^2\log f(w) =\sum_e g_e^2(\log f_b)''(w_e)\ge-C_f.\] Rearranging (220), and bounding its deterministic terms, gives \(\operatorname{Var}_\nu(O)\le C\). Because \(B_c\) stays bounded away from zero, (219) yields \(|\operatorname{Cov}_\nu(O,V_g)|\ge(|s_{12}|-C_2\delta)_+.\) Cauchy–Schwarz for the positive measure \(\nu\) now gives (221). ◻

Proof. First choose \(d_*\) small using the universal variance constant in Lemma 64. That constant requires only \(|H_{\lambda\lambda}|\le1\) and a fixed small parameter neighborhood; it does not require a decay estimate or a choice of \(T\). Fix the history geometry and a sufficiently small \(\theta>0\). Choose \(L\) large enough for the bulk, wall, and defect small-input contraction counts in Propositions 63 and 66. Next choose the defect weight \(A_D\) sufficiently large for the large-input contraction and decoration estimates, and then choose the untwisted weight \(A_U\) sufficiently large, with \(A_U/A_D\) large enough for the support and assignment counts accompanying the same-representation transport comparison in Proposition 66. In the bulk recurrence and in this shooting argument set \(A=A_U\). Both weights are now fixed and will not be enlarged after a survivor is chosen. The estimates requiring these choices depend only on the history geometry and the uniform one- or two-source bounds, also valid in the separated many-source decaying regime; they do not depend on a correlation order or on the eventual surviving parameters. Fix \(\rho\in(\theta,1)\), and fix \(\rho<\rho_1<\rho_2<1\) for the forced recurrences, together with the diagnostic torus factor \(F\). All recurrence, translation, and terminal constants below are evaluated at these fixed choices. Choose \(C_*\) to dominate the microscopic \(b^{-1}\) constant. Choose \(T\) large afterwards, and finally choose the lower bound on \(b\) large enough for the bulk shooting estimates and the wall and defect smallness conditions with these same weights.

Let \(h=T\varepsilon\), \(\delta_j=h\rho^j\), and use the parameter domain \[ P_h=\{(s_{11},s_{12},s): |s_{11}|,|s_{12}|\le h, \ d_*s_{12}^2\le s\le h\}. \tag{224}\] It is a closed topological ball when \(d_*h<1\). Its lower face is the graph \(s=d_*s_{12}^2\). If \(|t_i|_\infty\le\delta_i\) through step \(j-1\), induction in (215) gives \(r_j\le2\varepsilon\rho^j\): it suffices that \[C_{L,A}(T+2)^2\varepsilon\le2(\rho-\theta),\] and (216) gives the initial inequality by increasing \(b\). At a coordinate equality \(|t_{j,\alpha}|=\delta_j\), the increment estimate gives \[|t_{j+1,\alpha}|\ge\delta_j-2C\varepsilon\rho^j>\delta_{j+1}\] on the same signed side, provided \(T(1-\rho)>2C\). This strict crossing will control boundary contacts of the exit map.

We next exclude an exit through the positive \(22\)-face for a parameter on the lower face of \(P_h\). Suppose the trajectory has stayed inside through step \(j-1\), and at step \(j\) has \(t_{22,j}\ge\delta_j\). The increment bound implies \(|t_j|_\infty\le M_0\delta_j\), with fixed \(M_0\), and the remainder bound still holds at step \(j\). There is no such event at \(j=0\), since \(d_*h^2<h\).

Perform the calculation on the auxiliary torus of side \(FL^j\). After removing its designated constants there are at most \(C_F\) terminal blocks. The terminal expression is its tensor contribution plus an error whose smooth-source derivatives of order at most two are bounded by \(C_F(r_j+|t_j|^2)\). Indeed a single tensor expectation has its unmodified external-field quadratic and a field-independent covariance constant; every other linear input is a remainder, and every multi-input term has at least two tensor/remainder factors. The terminal history estimate bounds their integrals and Cauchy derivatives on fixed small circles in \((\lambda,\tau)\). The normalized zero-source partition is \(1+O_F(|t_j|+r_j)\), so logarithmic differentiation has the same error bound. The chosen sine mode obeys all the required box-scale test bounds uniformly in \(j\). On the fixed complex \((\lambda,\tau)\)-polydisc, its scalar background, including the constant value on each terminal block, contributes at most \(\exp\{C h_{L^j}(k;X)\}\) for the current charge array \(k\) on a terminal support \(X\). This cost is paid by the final normalized-test allowance in Lemma 61; the test class defining \(h_m\) includes its zeroth-order bound and does not quotient out constants. The finitely many terminal test events only change \(C_F\), which is fixed before \(T\). The uniform majorants on smaller source polydiscs in Proposition 62(iv)–(v) justify the Cauchy derivatives also at the degenerate endpoint \(s=0\). Consequently \[\begin{align*} H_{\lambda\lambda}&=t_{11,j} +O_F(r_j+|t_j|^2),\\ H_{\lambda\tau}/i&=t_{12,j}+a t_{11,j} +O_F(r_j+|t_j|^2),\\ H_{\tau\tau}&=t_{22,j}-2a t_{12,j}-a^2t_{11,j} +O_F(r_j+|t_j|^2). \tag{225}\end{align*}\] All constants here are independent of \(T\).

For small \(h\), the first line ensures \(|H_{\lambda\lambda}|\le1\); the second gives \(|H_{\lambda\tau}|\le C\delta_j\). Equations (218) and (221) therefore imply, on the lower face, \[ H_{\tau\tau}\le d_*s_{12}^2 +C d_*^2s_{12}^4 -c_1(|s_{12}|-C_2\delta_j)_+^2 \le C_3\delta_j^2. \tag{226}\] For the last inequality use \((x-y)_+^2\ge x^2/2-y^2\), choose \(d_*<c_1/4\), and then make \(h\) small enough to absorb the fourth-order term. On the other hand, \(|a|\le Ch\) and the last line of (225) imply \[H_{\tau\tau}\ge\delta_j-C h\delta_j -C_F\bigl(2\delta_j/T+M_0^2\delta_j^2\bigr) \ge\tfrac12\delta_j.\] Choose \(T\) sufficiently large, then \(h\) sufficiently small. This contradicts (226). These choices are not circular: the variance bound preceded \(d_*\), and the terminal constant \(C_F\) was fixed before \(T\).

Here is the topological conclusion, including boundary intersections. Put \(x_j=t_j/\delta_j\), and interpolate consecutive \(x_j\)’s linearly in a time variable. If no infinite survivor exists, every parameter has a first contact with the boundary of the cube \(Q=[-1,1]^3\). Denote that point by \(E(s_\bullet)\). This defines a continuous map \(P_h\to\partial Q\), as follows. Put \(D_j=\{s_\bullet:x_i(s_\bullet)\in Q\text{ for }i<j\}\). Finite-recursion continuity on these relative domains makes the \(D_j\) nested compact sets. If their intersection is empty, then \(D_N=\varnothing\) for some finite \(N\). Let \(n\) be the first index at which \(x_n\notin Q\). Strict crossing implies that every \(x_i\), \(i<n-1\), is in the interior; the first-contact point lies on the final segment, possibly at its initial endpoint. Along any convergent parameter sequence, pass to a subsequence with the same \(n\) and with convergent first-contact interpolation fractions. Closedness of \(D_n\) and relative continuity give convergence of both segment endpoints. Their limiting interpolated point lies on \(\partial Q\). There can be no earlier boundary contact: an equality at an earlier integer time would force the next endpoint outside \(Q\), contrary to membership in \(D_n\). If the final segment starts on the boundary, strict outward crossing makes that initial point its unique boundary point; otherwise it starts in the interior and again has a unique boundary point. This also covers a final endpoint exactly on the boundary. Thus every subsequential limit is the first-contact point at the limiting parameter, proving continuity including at corners.

On the top and on either vertical face of \(P_h\), the initial point already belongs to \(\partial Q\), so \(E(s_\bullet)=s_\bullet/h\). The lower-face exclusion just proved implies that its first-contact image avoids the relative interior of the top cube face. Indeed a contact there would require the next discrete endpoint to have \(t_{22,j}\ge\delta_j\), while every earlier endpoint was retained. At the intersections with a vertical face the initial image lies on that vertical face; it is never in the relative interior of the top face.

The boundary map \(E|_{\partial P_h}\) has degree one. For example, the point \((0,0,1)\) has exactly the preimage \((0,0,h)\), in the relative interior of the top parameter face; in a neighborhood there the boundary map is the orientation-preserving map \((s_{11},s_{12})\mapsto(s_{11}/h,s_{12}/h)\). There are no preimages on the lower or vertical faces. But any map from the ball \(P_h\) to the sphere \(\partial Q\) restricts to a null-homotopic boundary map, by contracting the ball to a point, and has boundary degree zero. This contradiction proves the existence of a survivor. Fix one such parameter choice for each \(b\). ◻

Proof. Let \(c_{\rm loc}\) bound, in current coarse-block units, every assignment, normalization-provenance, and collar neighborhood used by one local map. This includes inherited coefficient provenance: its earlier radii sum geometrically in current units, as in the proof of Proposition 33. Let \(C_X\) bound the diameter of a small input in fine-block units. The fixed geometric choices include \(H>c_{\rm loc}\) and \((H-c_{\rm loc})L>H+C_X\), as well as the kernel clearances. Thus a good output’s enlarged hull contains all the support and scalar-normalizer provenance needed to compute it, with a clearance margin.

An untwisted support is bulk-good if its required neighborhood has distance greater than \(Hm\) from the physical walls. On a torus require coordinate gaps of that size so the neighborhood unwraps. The remaining supports are wall supports. Copy the bulk localization on small good inputs. On small wall inputs subtract only their neutral constant; the full untwisted activity is even. To determine a good output, every contributing fine support and every assigned piece lies in its coarse sites or within the fixed one-step assignment neighborhood. The clearance ensures that all these supports are good and that the shell covariance agrees with the bulk covariance there. Connectivity tests, temporary exclusions, and singleton normalizations then agree as well. This proves equality with the bulk on good outputs by induction, including the normalization step.

At a fixed coarse wall output, a small wall input must lie in a strip of bounded fine-block width along a side. There are \(O(L)\) such positions and only a fixed number of small animals at each position. Corner positions are already included in this count. Thus even neutral wall remainders cost \(CL\cdot L^{-2}\), and charged ones cost \(CL\cdot L^{-4}\). Small torus-winding supports are absent in the regime where the torus clearance is used. Large inputs contract by the weight as before. A good input contributing to a bad output is bounded crudely by \(Cu_j\), including any assigned tensor polynomial; it is not required to cancel there. The product and decoration counts in the proof of Proposition 63 give the quadratic term. The initial size is \(w_0\le C\varepsilon\), which proves the first recursion.

For a spin source array \(l=\sum_{r=1}^k\varsigma_r\delta_{x_r}\), \(\varsigma_r\in\{1,-1\}\), use the exact connection of Proposition 50. Its offsets are \[\eta=-\mathcal R^{-1}dG_nl/\sqrt b, \qquad \vartheta=a_l/\sqrt b, \qquad (w,p,\psi)\mapsto (w-id_c\eta/\mu,p+\eta,\psi+\vartheta).\] The symbol \(a_l\) here denotes the connection lift, not the scalar \(a=s/d_c\) in the diagnostic. A support is shift-good if its rectangular hull enlarged by \(Hm\) misses every marked source. On that hull, clipped to the physical square when necessary, the connection has a common lift modulo its period; the reflected extensions obey the same identity on the stencil neighborhoods. Let \(\mathcal R_j^{\rm un}(X)\) be the chosen untwisted representation; each selected live root is untagged. On a good support represent the full twisted activity by \(T_\Lambda\mathcal R_j^{\rm un}(X)\), where \(\Lambda=(-i d_c\eta/\mu,\Lambda_\psi)\) and \(\partial\Lambda_\psi=\eta\) on its enlarged hull.

The first independent-piece integration, temporary grouping, and relocation of the full linear \(U_e\) expectation construct \(K_0\) before good and defect outputs are classified. The initial field collar and retained exclusion provenance lie within a fixed number of unit blocks of each output support, covered by the chosen \(H\). On an initial good output the exact microscopic substitution with the common lift commutes with this finite algebra. The designated contraction scalars are source independent, while the whole shifted polynomial remains in the activity. Thus \(K_0^{\rm tw}(X,z)=K_0^{\rm un}(X,z+\Lambda)\) there, and we choose its representation afresh as \(T_\Lambda\mathcal R_0^{\rm un}(X)\). Initial defect outputs retain their intrinsic tagged representations, including shifted \(U_e\) polynomials. The no-return induction below begins at these classified \(K_0\) outputs and applies only to subsequent \(m\to Lm\) maps.

For localization on small good inputs copy the untwisted pieces and retain their untwisted designated constants. In particular no background value of a translated tensor is reassigned as a scalar constant. On small defect inputs subtract only the neutral constant.

Every support and normalization neighborhood contributing to a good output fits within its enlarged hull with a clearance margin. More explicitly, if \(Y\) is good at \(S=Lm\), every contributing fine predecessor \(X\), including a predecessor of a scalar normalizer, obeys \[\operatorname{dist}(x_i,\operatorname{Hull}X) >(H-c_{\rm loc})S-C_Xm>Hm\] for each mark \(x_i\); residual predecessors have the stronger bound because their closures lie in \(Y\). Thus all fine inputs are good and admit the same lift. Using the exact fine activity identities with this common lift, substitution commutes with the convolution, product, and assignment identities. This proves the good-output identity inductively, including every designated scalar normalizer and its provenance. After proving that identity at each still-good output \(Y\), choose its representation afresh as \(T_{\Lambda_Y}\mathcal R_{j+1}^{\rm un}(Y)\). Thus the transport is applied to an untagged root; the previously assembled tagged representation of that good output is discarded. This is a choice of a representation of the same exact activity.

At a defect output retain, in each selected term, its concatenated live root with its existing tags and its separately completed coefficient occurrences. No lift is chosen on that output and no further \(T_\Lambda\) is applied to that root. An assigned piece originating at a good fine block is transported only at that fine block, before relocation, even if its target coarse singleton is a defect. Its opaque untwisted coefficient is unchanged. The following no-return property justifies this rule. If a fine support \(X\) is defective, some mark has distance at most \(Hm\) from its hull. A residual output contains \(\bar X\), so it is defective at \(S=Lm\). A singleton receiving a small defect coefficient belongs to \(\bar X\) and meets \(X\); its distance from that mark is at most \((H+C_X)m\). Any output whose assignment or normalization neighborhood uses that singleton is within a further \(c_{\rm loc}S\). The chosen inequality \((H-c_{\rm loc})L>H+C_X\) makes such an output defective as well. This includes a defect-specific scalar inverse decoration: it is an opaque number supported at its singleton, and every later use retains that support or the stated normalization provenance. Defect localization emits only a neutral constant, so it cannot create a relocated live polynomial elsewhere. A good-origin polynomial first landing in a defect then follows the residual rule. Consequently no tagged defect ancestry, including its scalar-normalizer provenance, re-enters a good output.

The derivative bounds for the connection give bounded scaled positive derivatives at every point of a good fine collar for one or two sources. In the separated \(k\)-source regime, at each folded collar point at most one source is within \(\epsilon_0n/3\); the identity of this source may vary with the point. The others contribute at most \(Ckm/(\epsilon_0n)\) in the scaled bound. The own-birth estimate after Equation (203) applies to every old atom, since its birth size \(h\le m\) obeys the same early inequality. Choosing \(c\) small therefore puts the transport of each good input in the admissible early class.

Apply the explicit comparator in Proposition 62(iv) to that input before relocation: \[A_D^{|X|}\mathsf s_X(T_\Lambda\mathcal R_j^{\rm un}) =A_D^{|X|}\mathsf s_X(\mathcal R_j^{\rm un}) \le (A_D/A_U)^{|X|}\|\mathcal R_j^{\rm un}\|_{A_U}.\] Its structural coefficient, every opaque completed coefficient size, and the live event weight are identical on the two sides. All bounded early contour translations remain inside the endpoint integrand. The fixed ratio \(A_U/A_D\) is used only in the subsequent spatial enumeration of support and assignment choices. The product bounds apply to concatenated live roots and multiply the sizes of completed scalar occurrences separately. This gives good-input forcing \(C(u_j+w_j)\), including assigned polynomials on their original good fine blocks, without a cost for every old event at each scale.

A small defect input has bounded fine diameter and an enlarged hull meeting a mark. There are only \(O(1)\) such fine supports per mark in a coarse output. In the indicated separated regime the number of relevant marks in this local calculation is bounded; for at most two sources it is bounded at every scale. Consequently the neutral and charged linear costs are \(C(L^{-1}+L^{-4})\). Large defect inputs contract by the weight. The product counts give the second recursion; the microscopic offset bounds give \(v_0\le C\varepsilon\).

To solve the recurrences, take the initial sizes sufficiently small that each quadratic term can absorb a fixed small multiple of the unknown wall or defect size into a contraction coefficient strictly less than \(\rho\). Iterating the resulting linear inequalities and summing their geometric convolutions gives the displayed \(C\varepsilon\rho_1^j\) and \(C\varepsilon\rho_2^j\) bounds. Equivalently, substitution of these bounds into the two recurrences proves them inductively after choosing their constants and then increasing \(b\).

For a fixed separated configuration, the last scale in the decaying regime is proportional to \(n\). There are only \(O_{k,\epsilon_0}(1)\) remaining scales, on grids with \(O_{k,\epsilon_0}(1)\) blocks. The entering activities tend to zero. The finite remaining maps and terminal integrals have finite bounds by the history calculus with the actual source offsets; their constants may depend on \(k,\epsilon_0,b\). Each new term has at least one entering activity, so finitely many applications of these bounds preserve \(o(1)\). In each live root, the number of atoms whose own birth sizes are late is bounded by \(C_{k,\epsilon_0,L}\), as proved after Equation (203). Only their explicitly recorded late numerical product may change by a fixed-configuration factor. Early decorations stay in the uniform endpoint class, and opaque completed coefficient occurrences are not transported. Thus neither the original charge and \(F_b\) weights nor the microscopic lower bound on \(b\) is increased with \(k\). ◻

Proof. After extracting the designated constants at each step, the terminal normalized integrals in both problems tend to one. There are boundedly many terminal blocks, and Proposition 66 and the terminal history estimate bound every nonempty activity contribution by \(o(1)\). The same reasoning includes the bounded number of late steps for a fixed separated configuration.

Write \(1+a_{j,B}^{\mathrm{tw}}\) and \(1+a_{j,B}^{\mathrm{un}}\) for the extracted factors. They are real: on real background fields complex conjugation combined with \(w\mapsto-w\) preserves the compensated integral and its local rules, including the real connection offsets. Their distance from one is small in the decaying regime, so they are positive. The microscopic designated constants are identical in the two problems. At subsequent scales a difference arises only from a small defect input. The bounded number of such inputs per source, the extracted-coefficient estimate, and Proposition 66 imply \[ \sum_B\left|\log(1+a_{j,B}^{\mathrm{tw}}) -\log(1+a_{j,B}^{\mathrm{un}})\right| \le Ck\varepsilon\rho_2^j. \tag{231}\] This follows from \(|\log(1+x)-\log(1+y)|\le2|x-y|\) for the small real coefficients in question. In a separated configuration the bounded late contributions are \(o(1)\). For two sources the estimate holds throughout. The terminal history bound then gives a uniform ratio bound for all sufficiently large grids. The finitely many remaining grid sizes are covered by enlarging that bound; their positive partition functions are finite and their source locations form finite sets.

For a single signed mark remaining macroscopically interior, translate the nested cuts by its lattice position. This translation does not change the bulk parameters or the exact physical ratio. At any fixed depth all constants that differ between the two problems are computed within a fixed lattice neighborhood of the mark. In this neighborhood, as the distance to the boundary tends to infinity, the grids and kernels are eventually the plane ones. The connection increments converge to the isolated plane connection. Its phase values modulo the period converge up to one common phase. A common phase translates the activity argument in the scalar field; neutral constant extraction is invariant under that translation, and the good-input lift rule preserves the invariance at every step. The finite-depth continuity of the history integrals therefore proves convergence of each local constant difference. Simultaneous negation of the fields and the connection proves that these limits are unchanged by reversing the source sign. They are also independent of the macroscopic source position, because the fixed-depth limiting grid and increments are the same.

The sum of these isolated differences is absolutely convergent by (231). Define its exponential to be \(c_b\). It is finite and strictly positive. The terminal ratio tends to one, so it is also the limit of the exact single-source physical remaining ratio. In particular its value is independent of cuts, although cuts were translated to compute that limit.

For several separated sources, compare with the individual single-source problems on the same hierarchy. At each fixed depth the neighborhoods supporting differing constants around distinct marks are disjoint for large \(n\). The other sources’ connection increments tend to zero in each neighborhood; their phase contribution is a common phase to leading order and thus does not affect neutral constants. Finite-depth continuity makes the local differences converge to the corresponding single-source differences. A mark has only finitely many positions modulo the cuts at a fixed depth, so this comparison is uniform in those positions. First sum through a fixed depth, then use (231) to send that depth to infinity. The result is \(k\log c_b\), proving (230) with no subsequence or location-dependent adjustment. ◻

Proof. By Proposition 65, \(s+d_c^2/\mu=1+O(b^{-1})>0\). This proves the stated asymptotic for the coefficient \(K_{\mathrm{XY}}(b)\) and, on increasing \(b_0\), the required lower bound. Let \(K=K_{\mathrm{XY}}(b)\), \(A=c_b^{-1}\), and write \(Z_n(x)=A e^{G_n(x,x)/(2K)}e^{i\theta_x}\). For signed factors let \(Z_n(x)^{[+]}=Z_n(x)\) and \(Z_n(x)^{[-]}=\overline{Z_n(x)}\). Equation (229) gives the exact cancellation \[ \mathbb E\prod_{r=1}^k Z_n(x_r)^{[\varsigma_r]} =A^k R_n\!\left(\sum_r\varsigma_r\delta_{x_r}\right) \exp\left\{-\frac1K\sum_{r<t} \varsigma_r\varsigma_tG_n(x_r,x_t)\right\}. \tag{233}\] In particular every self-energy, including its finite position-dependent Dirichlet part, cancels. By Proposition 67 and the lattice Green convergence in Lemma 27, at separated convergent interior points this tends to \[\exp\left\{-\frac1K\sum_{r<t} \varsigma_r\varsigma_tG_D(z_r,z_t)\right\}.\] These are precisely the separated signed moments of the full-variance Wick exponential. Convergence along every convergent separated configuration sequence is uniform on separated compact configuration sets: otherwise compactness would supply a convergent sequence on which the error stayed bounded away from zero.

We check the remaining hypotheses of Theorem 45. The two-source ratio bound and the Green estimates give, uniformly in all lattice points of a fixed interior compact set, \[ 0\le\mathbb E[Z_n(x)\overline{Z_n(y)}] \le C(|x-y|+n^{-1})^{-Q},\qquad Q<2. \tag{234}\] One may take \(Q=C_G/K<2\), where \(C_G\) is the constant in the discrete logarithmic Green bound, by increasing the lower bound on \(b\) once. Repeated locations are included: their diagonal Green factor is bounded by the same estimate. The same-sign two-point expression is bounded above by a constant since \(G_n\ge0\).

All signed-spin expectations are nonnegative. Indeed the Fourier coefficients of \(e^{b\cos\theta}\) are nonnegative: expand \(\exp((b/2)e^{i\theta})\exp((b/2)e^{-i\theta})\) and group equal powers. Haar integration at interior vertices selects currents with the required integer divergence. Every surviving current has nonnegative weight, and the pinned boundary supplies the remaining divergence. The positive deterministic multipliers in \(Z_n\) preserve this property.

Finally fuse all fixed boundary vertices into one pin vertex and give that vertex an independent uniform angle, rotating every spin by the same angle. The resulting measure is the finite free ferromagnetic plane-rotator measure, with nonnegative edge couplings, and this rotation preserves the modulus of each nonnegative weighted complex smear. The Lee–Yang estimate in Lemma 43 therefore applies to its real and imaginary projections and gives the absolute-smear moment bounds needed in Theorem 45. Its public theorem hypotheses are finite graph, nonnegative couplings, and nonnegative smear weights; all are satisfied here. General complex test functions are handled by splitting real and imaginary parts into nonnegative parts, as in that theorem.

Thus the hypotheses of the transfer theorem are all available: separated signed multipoint limits for every fixed order, the unrestricted integrable two-point bound (234), positivity of every signed moment, and the rotated Lee–Yang moment bound. That theorem controls collisions, proves moment determinacy, and gives local negative-Sobolev tightness. It yields the claimed full complex law and joint smeared convergence. The large fixed \(b_0\) was chosen in the bulk and the at-most-two-source estimates; Proposition 66 supplies every higher fixed order at this same \(b\). The arbitrary-size hierarchy and the cut-independent amplitude prove the assertion for all integers \(n\), rather than only a geometric subsequence. ◻

Proof. For integer scale ratios the elementary identity \[P_{Lt}(\cos z)=P_t(\cos z) \left(\frac{\sin(Ltz/2)}{L\sin(tz/2)}\right)^{64} \le P_t(\cos z)\] holds, with values at removable singularities defined by continuity. The degree of \(P_t\) is \(32(t-1)\), and \(P_t(1-q/16)=1-(t^2-1)q/3+O(q^2)\). Thus the quotients by \(Q\) are polynomials with nonnegative spectral values, also at zero. A polynomial of degree \(d\) in \(Q\) has range at most \(d\rho\) in sup distance, proving the range bounds. Moreover, \(0\le1-\gamma Q\le\operatorname{Id}\) and \(0\le\mathsf P_t\le\operatorname{Id}\) on the spectrum. Multiplication by \(A_0^{1/2}\) on both sides of the ultraviolet multiplier proves Equation (241); both sides of its equality vanish on the constant mode.

For a long-wave shell \(\Gamma_m\), split it into its binary pieces. In \(|\xi|\le c/\rho\), the angle \(z=\arccos(1-Q(\xi)/16)\) satisfies \((m/d_\rho)z\asymp m|\xi|\), and the binary covariance multiplier is bounded by \[C\min\{m^2,|\xi|^{-2}(m|\xi|)^{-64}\}.\] Outside this region it is bounded by \(C\rho^2(\rho/m)^{64}\). Multiplying by the required nearest-difference symbols and integrating gives, for a binary piece at side \(m\), \(|\nabla^r\Gamma|\le C_rm^{-r}\) for every required \(r\le12\). The high-frequency integral has an extra factor \(\rho^{66}m^{-64}\); increasing \(B\) absorbs this factor after each of the finitely many rescalings. Spectral supremum estimates give \(\|\Gamma_m\|\le C(Lm)^2\) and \(\|\nabla^r\Gamma_m^{1/2}\|^2\le C_rm^{-2(r-1)}\) for \(1\le r\le6\). For the terminal remainder use the same low- and high-frequency multiplier bounds, with \(\mathsf P_{m_*/d_\rho}\) in place of the shell difference. Positive-order spectral suprema and the normalized torus Fourier sum split at \(|\xi|=m_*^{-1}\) give the operator and diagonal bounds at scale \(m_*\) whenever \(n\ge2m_*\). In the low region the radial annulus count gives \(C_rm_*^{-2r}\) for the diagonal of \(r\) differences, \(1\le r\le6\); in the high region the same factor \(\rho^{66}m_*^{-64}\) is absorbed by the choice of \(B\). These constants do not require an upper bound for \(n/m_*\). Summing the binary pointwise estimates and using their ranges gives the logarithmic kernel bound and the positive-order mixed-distance bounds. A fixed low-frequency annulus in each binary piece gives its strictly positive diagonal lower bound. The Fourier sums have enough frequencies in those annuli by the choice of \(D_0\). These are precisely the kernel hypotheses used in Lemmas 29 and 31.

For convergence of these shells, put \(\eta=m\xi\) and divide the integral at \(|\eta|=(m/\rho)^\varepsilon\), with a fixed sufficiently small \(\varepsilon>0\). The tail just estimated is a negative power of \(m/\rho\), even after eight scaled differences. In the inner region, \[\frac{m}{2d_\rho}\arccos(1-Q(\eta/m)/16) =u|\eta|\{1+O((\rho|\eta|/m)^2)\}.\] Taylor expansion of the sine ratios and of the difference multipliers therefore has a power-small integrated error. All quotients are regular at zero, so no additional infrared truncation is required. On the diagonal the continuum integral is \[G_L(0)=\frac1{2\pi}\int_0^\infty (P_u(r)-P_u(Lr))\frac{\,\mathrm dr}{r} =\frac{\log L}{2\pi}.\] The last identity follows by integrating from \(\delta\) to \(T\), changing variables in the second term, and sending \(\delta\downarrow0\) and \(T\uparrow\infty\), using \(P_u(0)=1\) and its decay. This proves Equation (243) with geometric summability.

The multiplier for \(D_{\rm uv}/v^2\) tends to one almost everywhere as \(\rho\to\infty\). Outside \(|\xi|<c/\rho\) it is uniformly bounded. Inside that disk, the bound \(C\min\{m_0^2/\rho^2,1/(\rho^2|\xi|^2)\}\) has integral at most \(C\rho^{-2}(1+\log m_0)\). Dominated convergence outside, followed by this vanishing inside contribution, proves Equation (244). Range makes the ultraviolet entries identical on all sufficiently large tori.

For fixed \(J\), let \(d\) be the largest power of two not exceeding its sup range, and use \(m/d\) in the polynomial indices. Square symmetry gives \(Q(\xi)=v^2|\xi|^2+O_J(|\xi|^4)\); inclusion of the nearest neighbors implies that zero is its only Fourier zero. Choose \(\gamma\sup Q<1\) and repeat the shell and terminal arguments with the fixed range in place of \(\rho\). The polynomial positivity, range, and ultraviolet energy bound follow directly as before. This proves the fixed-\(J\) assertions for the first family.

We give the estimates for the entry family with its literal, unrescaled polynomial indices. On the Fourier square, \(Q(\xi)\asymp_J|\xi|^2\). For \(z=\arccos(1-Q(\xi)/16)\) one has exactly \(\sin(z/2)=\sqrt{Q(\xi)/32}\). The elementary sine bound and the binary identity give \[\begin{split} 0\le\mathsf P_t(\xi)&\le C\min\{1,(t\sqrt{Q(\xi)})^{-64}\},\\ D_t(\xi):=\mathsf P_t-\mathsf P_{2t} &=\mathsf P_t\{1-\cos^{64}(tz/2)\}\\ &\le C\min\{t^2Q(\xi),\,\min(1,(t\sqrt{Q(\xi)})^{-64})\}. \end{split}\] Consequently the binary covariance \(\Delta_t=v^2D_t/Q\) is bounded by \(C_J\min\{t^2,|\xi|^{-2}\min(1,(t|\xi|)^{-64})\}\). It is a polynomial of degree \(O(t)\), so its range is \(C_Jt\). Multiplication by the nearest-axis difference symbols and radial integration, split at \(|\xi|=t^{-1}\), prove \(|\nabla_x^a\nabla_y^c\Delta_t(x,y)|\le C_{J,r}t^{-r}\) for \(0\le r=|a|+|c|\le12\). Spectral suprema similarly give \(\|\nabla^r\Delta_t^{1/2}\|^2\le C_{J,r}t^{2-2r}\) for \(0\le r\le6\). Summing over the binary sides \(t=M,2M,\ldots,LM/2\) proves the operator bound at order zero and at orders \(2\le r\le6\); at order one use directly \(|\xi|^2v^2(\mathsf P_M-\mathsf P_{LM})/Q\le C_J\). At positive pointwise order, a piece can contribute only when \(t\ge c_J\operatorname{dist}(x,y)\), up to the fixed difference stencil. The geometric sum of \(t^{-r}\) then gives \(C_{J,r}(M+\operatorname{dist}(x,y))^{-r}\), and the same counting at order zero gives the displayed logarithm. On a fixed annulus \(c_J/t\le|\xi|\le C_J/t\) with its constants sufficiently small, \(\mathsf P_t\) is bounded below and \(1-\cos^{64}(tz/2)\ge c_Jt^2Q\) for all large \(t\). Its integral gives a positive constant to each binary diagonal. This proves the lower bound. Choosing \(D_0\) larger than twice the polynomial range constant, enlarged for the fixed difference stencils, makes each retained torus kernel the periodization of its plane kernel with at most one contributing image, and makes its diagonal identical to the plane diagonal. Thus all these bounds hold on the retained tori as stated.

For the terminal covariance, its nonconstant multiplier is bounded by \[C_J|\xi|^{-2}\min\{1,(M_*|\xi|)^{-64}\}.\] Its spectral supremum after \(2r\) differences proves the positive-order operator bound. Splitting the normalized torus Fourier sum at \(|\xi|=M_*^{-1}\) proves the positive-order diagonal bound: the number of modes in every radial dyadic annulus is bounded by its area times \(Cn^2\), with the same bound for the innermost disk when \(n\ge2M_*\). For \(1\le r\le6\) the resulting sum is at most \(C_{J,r}M_*^{-2r}\). This argument has no dependence on an upper bound for \(n/M_*\). At order zero the annuli between the smallest nonzero frequency and \(M_*^{-1}\) each contribute at most \(C_J\), and there are \(O(1+\log(n/M_*))\) of them; the higher-frequency sum is bounded. This proves the stated terminal value-diagonal bound. Polynomial division at zero proves positivity, range, and the asserted energy inequality for \(C^{\rm en}_{<M}\).

For the stated rate of continuum convergence put \(\eta=M\xi\) and use \[Q(\eta/M)=v^2|\eta|^2/M^2+O_J(|\eta|^4/M^4),\qquad \frac M2z(\eta/M)=\upsilon_{\rm en}|\eta| \{1+O_J(|\eta|^2/M^2)\}.\] The rescaled multiplier is \(v^2(\mathsf P_M-\mathsf P_{LM})/(M^2Q)\). On \(|\eta|\le M^{1/100}\), write each sine ratio as \([z/(2\sin(z/2))]^{64}[\sin(tz/2)/(tz/2)]^{64}\). Taylor’s formula, the preceding two expansions, and the bounds \(|(\sin r/r)^{64}|\le C(1+r)^{-64}\) and \(|((\sin r/r)^{64})'|\le C(1+r)^{-64}\) for \(r\ge1\) give an error bounded by \(C_{J,L}M^{-2}(1+|\eta|)^{-60}\) after division by \(|\eta|^2\). At zero the difference of the two sine ratios vanishes quadratically, so this bound extends across the removable quotient. The product of \(r\) scaled difference symbols differs from the corresponding derivative symbol by at most \(C_rM^{-1}|\eta|^{r+1}\) on this region. These errors are integrable through \(r=8\). On the complementary region both covariance multipliers, times a symbol of order \(r\), have integral bounded by \[C_{J,L}\int_{M^{1/100}}^\infty t^{r-65}\,\,\mathrm dt \le C_{J,L}M^{-(64-r)/100};\] the same estimate bounds the part outside the rescaled Fourier square for the continuum kernel. For \(r\le8\) this is \(O(M^{-1/2})\). Fourier inversion proves the stated uniform scaled-difference estimate. Its diagonal case and \(G_{L,\upsilon_{\rm en}}(0)=\log L/(2\pi)\) give \(|\delta_j^{\rm en}|\le C_{J,L}(mL^j)^{-1/2}\). The mean-value bound for \(e^{-a'\delta/2}\), uniformly on \(I\), now proves the displayed summable estimate. The fixed parameter \(\upsilon_{\rm en}\) can leave every compact interval when \(J\) varies, which is why this second argument is not a spread-out estimate. ◻

Proof. Write the first Gaussian field as \(\eta+\zeta\), where \(\operatorname{Cov}(\eta)=\gamma v^2\operatorname{Id}\) and \(\operatorname{Cov}(\zeta)=C_{\rm uv}\). Conditioning each \(\eta_x\) on the comb gives the periodic density \[w(t)=\sum_{q\in\mathbb Z}e^{-q^2a'\gamma v^2/2}e^{iq\alpha t}.\] Expand in occupied site factors \(w(\phi_x+\zeta_x)-1\) and in occupied oriented-edge factors \[e^{sE_\ell(\phi+\zeta+\eta)/2}-1, \qquad E_{(x,y)}(\psi) =\frac{(\psi_{x+y}-\psi_x)^2}{2|J|v^2}.\] Assign an edge to the block of its base vertex and group all occupied factors by the enlarged block connectivity. Gaussian range and the retained collars give exact factorization between separated components. This is the initial instance of the algebra in Proposition 33.

Here are estimates justifying the expansion and its derivatives. An occupied site costs \(\delta_0\) in the analytic point norm; its Fourier derivatives introduce powers of \(\alpha|q|\) and factors \(e^{h_0\alpha|q|}\), absorbed by the gap \(\gamma-\gamma'\). An occupied edge costs \(C|s|\), using \(|e^{sE/2}-1|\le C|s|(1+E)e^{cE}\) for a fixed arbitrarily small \(c\). For conditional derivatives one differentiates the shifted Gaussian density while keeping the constrained lattice value fixed. At the at most twice as many touched endpoints as occupied edges, the resulting weights are bounded by \[C\exp\{h_0|\eta_x|/(\gamma v^2)+h_0^2/(2\gamma v^2)\}.\] The conditional normalizer and its inverse have the same estimates, since \(w=1+O(\delta_0)\) uniformly in its argument. These bounds are uniform also for the first two conditional moments, with polynomial factors absorbed by the same exponential slack.

The field energy from a selected set of edges is bounded by a constant times the regulator gradient energy on their collars. Indeed, resolve each \(y\) into an axis path of length at most \(C|y|\) and apply Cauchy–Schwarz along the path. Summing its translates and then its coefficient \(1/(|J|v^2)\) uses \(\sum_{y\in J}|y|^2=4|J|v^2\), and gives a bounded constant. For the \(\zeta\) energy, put \(c=(|J|v^2)^{-1}\) and let \(\mathsf{B}\) have one row per oriented interaction edge, with \((\mathsf{B}\psi)_\ell=\sqrt{c/2}\,\nabla_\ell\psi\). Both edge orientations are included, so \(\mathsf{B}^*\mathsf{B}=A_0\) and \(E_\ell(\psi)=|(\mathsf{B}\psi)_\ell|^2\). For a selected edge set \(S\), let \(\mathsf{B}_S\) be the row restriction and \(T_S=\mathsf{B}_SC_{\rm uv}\mathsf{B}_S^*\). Equation (241) implies \[\|\mathsf{B}C_{\rm uv}^{1/2}\|^2 =\|C_{\rm uv}^{1/2}A_0C_{\rm uv}^{1/2}\|\le1, \qquad 0\le T_S\le\operatorname{Id},\qquad \operatorname{tr}T_S\le|S|.\] Gaussian diagonalization therefore gives, for \(0\le\tau<1/2\), \[ \mathbb E\exp\left\{\tau\sum_{\ell\in S}E_\ell(\zeta)\right\} =\det(\operatorname{Id}-2\tau T_S)^{-1/2} \le\exp\left\{\frac{\tau|S|}{1-2\tau}\right\}. \tag{247}\] Here \(-\log(1-x)\le x/(1-x)\) bounds each eigenvalue term. Polynomial occupation factors are absorbed by \[\prod_{\ell\in S}(1+E_\ell) \le C_\varepsilon^{|S|} \exp\left\{\varepsilon\sum_{\ell\in S}E_\ell\right\},\] with a fixed reserve below the exponent \(1/2\). Use \(E_\ell(\phi+\zeta+\eta) \le3E_\ell(\phi)+3E_\ell(\zeta)+3E_\ell(\eta)\). The background part is placed directly in the output regulator by the deterministic path estimate; only the selected edge energies of \(\zeta\) are integrated using Equation (247). In particular Lemma 29 is not applied to \(C_{\rm uv}\). For the independent field, each touched vertex is incident to at most \(2|J|\) oriented edges, and hence \[\sum_{\ell\in S}E_\ell(\eta) \le\frac2{v^2}\sum_{x\text{ touched}}\eta_x^2.\] There are at most \(2|S|\) touched vertices. Shifted-lattice Gaussian sums with the displayed derivative weights and this sufficiently small quadratic growth are bounded by \(C\) per touched vertex, including their normalizers. Combining these facts bounds a term with \(r\) occupations by the product of its site and edge costs, times \(C^rW_{m_0}^\kappa\).

On one edge, replacing its conditional expectation by the unconditioned linear term costs \(C\rho^C(s^2+|s|\delta_0)\). The unconditioned expectation is exactly \(sE_\ell(\phi)/2\) plus a constant. Extract the sum of these constants by the constant-decoration identity of Proposition 33; its nonconstant linear part is \(s e_B^0/2\). A single site’s fundamental Fourier terms, after the \(\zeta\) integration, have coefficient \(2e^{-a'D_{\rm uv}/2}\). Summing sites in \(B\) gives \(z_0c_B^\alpha\). All its higher harmonics cost at least \(C\delta_0^2\) in the compact-\(a'\) regime.

For a prescribed connected support the local choices for \(r\) occupations are bounded by \((Cm_0^2|J|)^{Cr}\), by encoding the counts in each block and the sites and edges for each occupation. At least one occupation is required for every occupied block. Since \(m_0\) is polynomial in \(\rho\), this pays any fixed prescribed initial activity weight. Summing the absolutely convergent expansion with at least two occupations, and the single-occupation errors just identified, proves Equation (246). Differentiating the edge factors proves the asserted derivative bound by the identical occupation sum with one power of \(|s|\) removed. The weaker bounds follow by putting all site occupations, including the fundamental one, in the remainder. For fixed \(J\) all counting factors are fixed, so the same proof gives its asserted version. Finally, since \(\gamma'>1/2\) and \(D_{\rm uv}=v^2(1+o(1))\), the exponent in \(\delta_0^2/z_0\) is strictly negative for large \(\rho\); polynomial factors do not affect the stated \(o(z_0)\) conclusion. ◻

Proof. The preceding neutral and fundamental subtractions give \(CL^{-3}\) in point norm before support counting. There are \(O(L^2)\) small fine supports above a prescribed coarse block, with weight ratios bounded independently of \(L,A\). Thus these subtracted linear terms have norm at most \(C/L\). Large supports have the closure-size gain of Lemma 28, which makes their linear contribution arbitrarily small by increasing \(A\) after \(L\). The extracted scalar projections are bounded by \(C_{L,A}\). This proves the triangular linear part and \(\|C_j\|\le\theta\).

Apply Proposition 33 with the enlarged collars. Temporary connectivity contains the covariance range, so convolution factorizes; all exclusions on an original support are retained when a localized piece is relocated to a singleton. Only small supports shrink, and their coarse diameter is at most one; hence separated final components carry no remaining mutual exclusion. Constant decorations are removed by their convergent inverse series. The counting and product estimates in that proposition bound every term with at least two inputs by \(C_{L,A}(|t|+|z|+\|R\|)^2\). They also converge absolutely for complex activities in the same real-field norm. Bounded multilinear sums and the constant inverse series prove analyticity and the differentiated bounds, without an appeal to analytic dependence on the period.

If \(a'>k_0\) stays in a compact interval, the fundamental charged estimate before counting is \(CL^{-a'/(4\pi)}\). Its product with the \(O(L^2)\) small-support count is uniformly small once \(L\) is chosen for the positive gap, so fundamental localization is unnecessary. For \(\alpha\ge2\sqrt{k_0}\) use a contour shift of the fixed size \(2\sqrt{k_0}\). Its gain is \(\exp\{-[2\sqrt{k_0}\alpha|q|-2k_0]\Gamma_m(0,0)\}\). The coefficient is already at least \(2k_0\) for \(|q|=1\) and increases linearly with \(\alpha|q|\). Taking \(L\) large pays the output constant derivative factors uniformly and leaves at least \(L^{-4}\) in point norm. This proves both no-cosine variants.

A small output support depends on predecessor data within a fixed number of its blocks: convolution uses a fixed enlargement, relocations originate only from small supports, and normalization adds singleton decorations. Iterating these dependencies contributes a geometric sum of smaller physical radii, bounded by a fixed multiple of the current scale. With \(D_0\) larger than this multiple, the needed covariance entries and activities unwrap identically to the plane. Large torus supports retain the closure-size contraction and need not be identified with plane supports. Applying the same norm estimates inductively proves the torus assertion, and applying their differentiated forms proves its derivative assertion. At fixed depth, absolute expansions or dominated Gaussian integrals with regulator reserve give continuity in all parameters of every fixed local coefficient. Uniform geometric tails then give continuity of the scalar projection series used below. ◻

Proof. The linear changes in Equations (251) and (252), together with their inverses, are uniformly bounded on the product space. Apply them to the analytic expansion in Proposition 71, keep the quadratic terms containing only \(T,Z\), and place every term with a factor \(R\) in the displayed error. The quadratic terms in \(R_+\) have the same bound by the inverse coordinate change. Translation by \(\pi/\alpha\) commutes with convolution, reblocking, and localization, preserves energy, and changes the sign of the fundamental cosine. The projection series commute with this involution as well. Its even and odd scalar parities exclude \(TZ\) from the first equation and \(T^2,Z^2\) from the second. Cauchy estimates on a smaller fixed ball give all claimed differentiated bounds. ◻

Proof. We first justify the limiting calculation, and only then calculate its coefficients. A quadratic coefficient in the balanced map is the quadratic image of its two singleton inputs followed by the linear projection series. Truncating those series at depth \(r\) costs at most \(C\theta^r\) (or \(C(\theta/\lambda)^r\)). At any fixed depth there are finitely many connected block configurations: quadratic images of pure singleton inputs have bounded size, and each step has bounded range in the rescaled block coordinates. Each coefficient is thus a finite volume-normalized sum of Gaussian polynomial/character expectations and their Taylor projections on constant or affine backgrounds. Lemma 69 gives convergence of every covariance and derivative entering these expectations. The rescaled difference in direction \(y\), divided by \(|y|\), converges to its directional derivative; the weighted sum of its squares converges to \(\int_B|\nabla\phi|^2\). The frame prescription has the same limit even before its Hessian is scalar, because two frame derivatives transformed back to the axes recover the entire usual gradient. Spatial Riemann sums and Gaussian moment formulas therefore give convergence at finite depth. Uniform geometric tails permit sending \(r\to\infty\) afterwards. This proves existence of the limiting quadratic map. Its kernels are related by dilation, so its coefficients are independent of scale, and \(\ell_j=\lambda=1\) at marginality.

To identify the coefficients, start at side one with the exponential interaction of \(t e_B^0/2+zc_B^\alpha\) and iterate \(r\) shells, with \(M=L^r\). At order one its remainder is zero, and its order-two initial remainder is bounded independently of \(r\). At order two the marginal increments add, while \(R^{(2)}_{j+1}=C_jR^{(2)}_j+O(1)\) keeps the remainder coefficients uniformly bounded. Thus the final scalar quadratic coefficients are \(r\) times the unknown coefficients in Equation (254), up to bounded errors. The unbalanced coordinates have the same leading terms because their difference from the balanced coordinates is a bounded projection of \(R^{(2)}\). This argument is a finite-order calculation in the exact block algebra; no infinite-volume interacting measure is assumed to exist for nonzero formal \(t,z\).

Two diagnostics read these scalar terms. The first is the fundamental cosine coefficient of the connected log density per output block on a constant background. The second is its neutral Hessian in \(g_1\) on the affine background \(\theta+g\cdot x/M\), after averaging \(\theta\) over one period. Here connected log density means only its Taylor polynomial through degree two, computed in a large box and then per output block with boundary terms removed. At fixed \(r\) all its kernels have bounded range. The linear singleton terms read \(z\) and \(t\) exactly; the linear remainder terms are bounded by the activity norm. The degree-two connected terms of the unchanged first-order singleton inputs are also bounded in output block units. An affine derivative may be anchored in its cluster because of the neutral phase average, and the resulting fixed powers of its diameter are summable against its exponential support weight. Therefore each diagnostic differs from its output scalar coefficient by \(O(1)\), uniformly in \(r\). Extracted constants disappear from both diagnostics. These facts justify reading the secular coefficients from the original Gaussian integral rather than from its block representation.

That integral uses the covariance \(G_M\) in Equation (242), with \(G_M(0)=\log M/(2\pi)\). In the \(t^2\) contribution to the neutral gradient Hessian, a background gradient can occur only in a linear gradient fluctuation. Its covariance integrates to a derivative integral of \(G_M\), hence to zero. Boundary terms vanish on taking the per-block limit because this kernel has finite range. There is no \(t^2\) secular term.

For the mixed term, Gaussian differentiation gives \[\begin{align*} &\operatorname{Cov}\left(\tfrac12|\nabla\zeta(x)|^2, \cos(\alpha(\theta+\zeta(y)))\right)\\ &\hspace{20mm}=-\frac{\alpha^2}{2}|\nabla G_M(x-y)|^2 e^{-\alpha^2G_M(0)/2}\cos(\alpha\theta). \end{align*}\] The output block area therefore makes the mixed diagnostic \[ -\frac{\alpha^2}{2}M^2e^{-\alpha^2G_M(0)/2} \int_{\mathbb R^2}|\nabla G_M(x)|^2\,\mathrm dx. \tag{255}\] Parseval and radial integration show \[\int|\nabla G_M|^2 =\frac1{2\pi}\int_0^\infty (P_u(r)-P_u(Mr))^2\frac{\,\mathrm dr}{r} =\frac{\log M}{2\pi}+O(1).\] For example, subtract the indicator of \([M^{-1},1]\) in this last integral; near each endpoint its error is integrable uniformly by \(P_u(r)=1+O(r^2)\) and \(P_u(r)=O(r^{-64})\). At \(\alpha^2=8\pi\), \(M^2e^{-\alpha^2G_M(0)/2}=1\). Dividing Equation (255) by \(r=\log M/\log L\) gives \[ b=\frac{\alpha^2}{4\pi}\log L=2\log L>0. \tag{256}\]

For the \(z^2\) term, averaging the phase in the connected product of two cosines gives \[\frac12e^{-\alpha^2G_M(0)} (e^{\alpha^2G_M(x-y)}-1) \cos\big(\alpha g\cdot(x-y)/M\big).\] The second-cumulant factor \(1/2\) and two derivatives in \(g_1\), with output area \(M^2\) cancelling the affine scaling, give \[ -\frac{\alpha^2}{4}e^{-\alpha^2G_M(0)} \int_{\mathbb R^2}x_1^2(e^{\alpha^2G_M(x)}-1)\,\mathrm dx. \tag{257}\] We verify its logarithmic asymptotics, including its sign. Put \[F_u(x)=\int_{\mathbb R^2} \frac{1-\cos(\eta\cdot x)}{|\eta|^2} P_u(\eta)\frac{\,\mathrm d\eta}{(2\pi)^2}.\] Radial integration gives \(F_u(x)=(2\pi)^{-1}\log|x|+d_u+o(1)\) as \(|x|\to\infty\), uniformly for \(u\) in a compact interval. One direct proof replaces \(P_u\) by the indicator of a unit radial disk: their difference is integrable in \(\,\mathrm dr/r\) because it is \(O(r^2)\) at zero and decays at infinity. In the disk term the circle character average at argument \(t\) is \(O(t^{-1/2})\). This bound follows by excluding arcs of size \(t^{-1/2}\) about its two stationary angles and integrating by parts on the remaining arcs. Its radial integral therefore has a convergent constant after subtraction of \(\log|x|\). The same proof yields \(F_u(x)=(2\pi)^{-1}\log(1+|x|)+O(1)\) globally.

Since \(G_M(x)=G_M(0)-F_u(x)+F_u(x/M)\), writing \(c_u=-d_u\) gives \[ G_M(x)=\frac1{2\pi}\log\frac M{|x|}+c_u+o(1) \quad(1\ll|x|\ll M), \qquad G_M(x)\le\frac1{2\pi}\log^+\frac M{1+|x|}+C. \tag{258}\] For completeness the finite range and logarithmic error also have an elementary radial representation. The function \(\sin(u|\eta|)/(u|\eta|)\) is the characteristic function of the planar projection of the uniform probability measure on the sphere of radius \(u\) in \(\mathbb R^3\): rotate \(\eta\) to an axis and integrate the uniform axial coordinate on \([-u,u]\). Thus \(P_u\) is the Fourier transform of its \(64\)-fold convolution, a radial probability density \(q_u\) supported in the disk of radius \(R_u=64u\). It is bounded and smooth to every order used here, as follows from the integrability of \(|\eta|^rP_u(\eta)\) for these orders. The fundamental-solution identity \(\Delta\log|x|=2\pi\delta_0\) gives \[F_u(x)=\frac1{2\pi}\int q_u(y) \{\log|x-y|-\log|y|\}\,\mathrm dy.\] Both sides vanish at zero and have Laplacian \(q_u\); the radial harmonic difference is therefore zero. Angular averaging of the logarithm is \(\log\max\{|x|,|y|\}\), by its mean-value identity inside or outside the corresponding circle. Consequently \(F_u(x)=(2\pi)^{-1}\log|x|-c_u\) exactly for \(|x|\ge R_u\), where \(c_u=(2\pi)^{-1}\int q_u(y)\log|y|\,\mathrm dy\). The scaling \(q_u(y)=u^{-2}q_1(y/u)\) also proves \(c_u=c_1+(\log u)/(2\pi)\). The formula for \(G_M\) now shows \(G_M(x)=0\) for \(|x|\ge MR_u\), confirming the claimed range without inferring pointwise positivity from spectral positivity. At \(\alpha^2=8\pi\), the integrand in Equation (257) has prefactor \(M^{-4}\). On intermediate logarithmic shells its radial part is \(e^{8\pi c_u}\,\mathrm d|x|/|x|\), and \(\int_0^{2\pi}\cos^2\vartheta\,\mathrm d\vartheta=\pi\). The regions \(|x|\le R_u\) and the \(-1\) term contribute \(O(1)\), uniformly on compact \(u\) intervals. On \(R_u\le|x|\le MR_u\) the exact formula for \(F_u\) makes the exponential part of the radial integral \[\pi e^{8\pi c_u}\int_{R_u}^{MR_u} e^{8\pi F_u(r/M)}\frac{\,\mathrm dr}{r} =\pi e^{8\pi c_u}\log M+O(1).\] Indeed \(F_u(t)=O(t^2)\) near zero, and after \(t=r/M\) the integral of \((e^{8\pi F_u(t)}-1)/t\) is bounded on \([0,R_u]\). It follows that \[ H=\frac{\alpha^2\pi}{4}e^{8\pi c_u}\log L>0. \tag{259}\] The continuity and boundedness of \(c_u\) on the compact \(u\) interval make this positivity uniform. The finite-depth convergence proved at the start, first uniformly in small \(\mu\) and then in the summable lattice error, now proves all discrete assertions of the proposition. This computes both signs directly in the compensated positive-gradient convention of Equation (236).

For the entry family, the fixed-depth Gaussian expectations at the start of the proof converge by its order-eight scaled-difference bound, and the same geometric projection tails are uniform once its own diagonal errors are small. The localizations and normalized singleton inputs are unchanged: in particular \(c_B^\alpha=M^{-2}\sum_{x\in B}\cos(\alpha\phi_x)\) converges to the unit-block cosine integral in both families. The limiting calculation therefore uses \(G_{M,\upsilon_{\rm en}}\) in the two diagnostics. Equations (256) and (259) give precisely \(b_{\rm en}\) and \(H_{\rm en}\) displayed above. The identity \(c_u=c_1+(\log u)/(2\pi)\) gives \(e^{8\pi(c_{\upsilon_{\rm en}}-c_{u_{\rm wh}})}=d^4\). No coordinate factor changes this number: the balanced factor \(\ell_j\) tends to one at criticality in either continuum limit. ◻

Proof. Use the no-cosine map of Proposition 71. In (i) take \(L\) for the gap \(a'-k_0\) and then \(\rho\) large; in (ii) use its uniform high-frequency contraction. Balance \(t\) by Equation (251), now with no \(Z\) coordinate. The recursion is \[T_{j+1}=T_j+O((|T_j|+\|R_j\|)^2),\qquad R_{j+1}=C_jR_j+O((|T_j|+\|R_j\|)^2).\] Proposition 70, after allowing exponent slack, supplies \(\|R_0\|\le Cx_*\) and \(T_0=s+O(x_*)\) on \(|s|\le C'x_*\). The constant in the last \(O(x_*)\) can be made independent of any fixed \(C'\): the dependence on \(s\) outside its linear term is at most \(C(s^2+|s|\delta_0)\) and hence is \(o(x_*)\) on every such interval.

Choose \(\theta<\theta_3<1\) and a fixed large envelope constant \(D\). While \[ |T_j|\le Dx_*\theta_3^j \tag{260}\] holds, the remainder recursion and geometric summation imply \(\|R_j\|\le D_Rx_*\theta_3^j\) for a fixed \(D_R\): its forced contribution is bounded by \(Cx_*^2\sum_{h<j}\theta^{j-1-h}\theta_3^{2h}\), which is at most \(C'x_*^2\theta_3^j\). This estimate also propagates torus remainders through their stopping scales. Choose \(C'\) so that the two endpoints \(s=\pm C'x_*\) start outside the two opposite sides of Equation (260).

If equality first occurs at scale \(j\), then \(|T_{j+1}-T_j|\le Cx_*^2\theta_3^{2j}\), whereas the envelope decreases by \(D(1-\theta_3)x_*\theta_3^j\). For small enough \(x_*\) the next iterate exits strictly on the same side. A strict exit persists under a sufficiently small change of \(s\) by finite-time continuity. Thus the sets of parameters eventually exiting on the positive and on the negative side are disjoint relatively open sets; when equality occurs its strict next exit gives the same openness. Both sets are nonempty. If there were no survivor, they would disconnect the shooting interval, which is impossible. The maps can be evaluated for the one extra step at an exit because the whole enlarged envelope lies in their fixed small domain. A survivor obeys Equation (260) and the remainder estimate at all scales, proving decay of every activity.

It remains to identify the shot with the physical coefficient. For a smooth mean-zero continuum test \(f\), let \(f_n\) be its sampled lattice source including the area \(n^{-2}\) and discrete mean subtraction. Completing the Gaussian square in the original comb integral gives the prefactor \(\exp\{(f_n,A_0^{-1}f_n)/2\}\) for the \(\phi\) field and the ratio of the full interaction integrals at the translated background \(F_n=A_0^{-1}f_n\) and at zero. This translation includes the constraint itself before ultraviolet integration. Therefore all field-independent constants extracted at every scale cancel exactly in the ratio. At \(m_*\) there are boundedly many blocks. The terminal field obeys the regulator estimates with reserve; \(F_n\) and its first two scaled differences are uniformly bounded by discrete Fourier summation of the smooth sampled source. The activity decay thus makes each of the two terminal interaction averages tend to one, by the finite block expansion and dominated Gaussian integration. Finally, \[(f_n,A_0^{-1}f_n)\longrightarrow (f,(-\Delta_{\mathbb T})^{-1}f)_{L^2(\mathbb T^2)}\] by the quadratic symbol at zero and the rapid Fourier decay of \(f\). Since \(\sigma=\sqrt{a'}\phi\), the height Laplace limit has coefficient \(a'\). We next prove positivity of the structural coefficient at this input using only its unconditional free-rectangle upper bound, before using the positive-coefficient transfer theorem.

Fix a nonzero smooth real mean-zero \(f\) and let \(g=(-\Delta_{\mathbb T^2})^{-1}f\), with zero mean. Write \(g_n\) for its sample and \(\operatorname{Var}_{\mathbb T_n}\) for variance in the physical height law on the torus of side \(n\). For a fixed integer \(q\ge2\), partition a fundamental square into \(q^2\) axis rectangles by rounding the cuts \(in/q\), \(0\le i\le q\), and delete every interaction edge joining different rectangles or crossing the fundamental-square seam. For all sufficiently large \(n\), these are free rectangles of side comparable to \(n/q\). Define \[X_n=(\sigma,A_0g_n),\qquad X_n^{\rm cut}=\sum_{B\text{ in the partition}} (\sigma|_B,A_Bg_n|_B).\] There are at most \(C_{J,q}n\) deleted edges, and periodic smoothness gives \(|\nabla_e g_n|\le C_{J,g}/n\) on all of them, including the wraparound edges. The difference \(X_n-X_n^{\rm cut}\) is represented by these edge flows with weights \(c_e\nabla_e g_n\), so its flow cost in Equation (20) is at most \(C_{J,q,g}/n\). Lemma 4 therefore gives \[ \operatorname{Var}_{\mathbb T_n}(X_n-X_n^{\rm cut})\le C_{J,q,g}k/n. \tag{261}\] Each summand in \(X_n^{\rm cut}\) annihilates the constant on its rectangle. Deleting the crossing precisions and releasing the rectangle constants are thus permissible in the free-graph comparison of Lemma 4. Under the cut law the rectangle representatives are independent, and hence \[\operatorname{Var}_{\mathbb T_n}(X_n^{\rm cut}) \le\sum_B\operatorname{Var}_{B,k}(\sigma,A_Bg_n|_B).\] For this fixed finite partition, apply the unconditional Equation (29) along \(n=L^N\to\infty\) to each sequence of rectangles. Their rounded endpoints converge to the indicated continuum endpoints. The \(L^2\) triangle inequality and Equation (261) give \[\sqrt{\operatorname{Var}_{\mathbb T_n}X_n} \le\left\{\sum_B\operatorname{Var}_{B,k}(\sigma,A_Bg_n|_B)\right\}^{1/2} +C_{J,q,g}\sqrt{k/n},\] and therefore \[ \limsup_{N\to\infty}\operatorname{Var}_{\mathbb T_{L^N}}(\sigma,A_0g_{L^N}) \le a(k)\sum_B\int_B|\nabla g|^2 =a(k)(f,(-\Delta_{\mathbb T^2})^{-1}f). \tag{262}\] No positivity assumption on \(a(k)\) enters this comparison, and no limit in \(q\) is required.

To replace the gradient observation by the density source used in the shot, put \(e_n=f_n-A_0g_n\). The quadratic symbol and smooth finite differences imply \(n^{-2}\sum_x|n^2e_n(x)|^2\to0\), with \(\sum_xe_n(x)=0\). The torus Poincaré bound from the nearest-neighbor edges gives \[\operatorname{Var}_{\mathbb T_n}(\sigma,e_n) \le k(e_n,A_0^+e_n) \le C_Jk n^2\sum_x|e_n(x)|^2\longrightarrow0.\] Thus Equation (262) holds with \(f_n\) in place of \(A_0g_n\). Bare Gaussian domination for every fixed real multiple of \(f_n\) gives uniform exponential moments; the shot’s Laplace limit consequently gives convergence of its second moments. Its limiting variance is \(a'(f,(-\Delta_{\mathbb T^2})^{-1}f)\). The energy of the chosen nonzero \(f\) is positive, so Equation (262) implies \(a(k)\ge a'>0\). Theorem 17 now applies on the positive branch and identifies the same torus Laplace limit with coefficient \(a(k)\). Uniqueness on this one nonzero test proves \(a(k)=a'\), as claimed. ◻

Proof. For each sufficiently large \(a'\), Proposition 74 gives an input \(k(a')\) with \(a(k(a'))=a'\) and \(|k(a')-a'|\le C_Je^{-c_Ja'}\), after absorbing the factor \(a'\) into a smaller exponent. This already proves nonemptiness and finite onset by the equivalence of positive coefficient and physical roughness in Corollary 18. To cover every high input \(k\), choose \(\delta_k=e^{-c_Jk/2}\) and shoot at \(a'_- =k-\delta_k\) and \(a'_+=k+\delta_k\). For large \(k\) the shot errors are less than \(\delta_k/2\), so \(k(a'_- )<k<k(a'_+)\). The monotonicity in Proposition 19 gives \(k-\delta_k\le a(k)\le k+\delta_k\). This proves the first assertion of Equation (263). Substitute \(k=\beta/v^2\) and \(\beta_{\mathrm{eff}}=v^2a(k)\) to obtain the second. ◻

Proof. Corollary 8 and Corollary 18 give \(k_c(J_\rho)=\beta_{\mathrm c}(J_\rho)/v_{J_\rho}^2\ge8\pi\). For every fixed \(a'>8\pi\), regime (i) of Proposition 74 gives a rough input \(k(a',\rho)=a'+o(1)\). Hence \(\limsup_{\rho\to\infty}k_c(J_\rho)\le a'\). Letting this fixed \(a'\) decrease to \(8\pi\) proves the result. The block factor may depend on \(a'-8\pi\) in this argument; the limit is taken in \(\rho\) before decreasing \(a'\), so that dependence does not affect the squeeze. ◻

Proof. Write \(E_D(y)=\langle y,My\rangle\), and let \(L\) map the Gaussian Hilbert coordinates \(\xi\) of \((\zeta,e)\) to \(B_s\alpha\zeta+e\) on \(D\). The fundamental bound is \[ \|M^{1/2}L\|\leq1. \tag{278}\] For the uncut full observation covariance it is the defining inverse covariance identity. Restriction to an open set only lowers the minimized energy: use the actual restricted field as trial function. Equivalently, the field portion follows from \(A_0^{1/2}C_{<u}A_0^{1/2}\leq I\), and the observation minimization also permits its actual noise. The same proof includes any future Gaussian pieces. Constants are killed by the energy, including on the torus.

Put \(H=h_*M\) and \(Q=L^*HL\). Equation (278) and \(p_1h_*<1\) imply \[ 2V_j=\langle f,Hf\rangle +\langle L^*Hf,(p_1^{-1}I-Q)^{-1}L^*Hf\rangle, \qquad f=B_s\alpha\psi. \tag{279}\] This proves finiteness, positivity, and the first bound in Equation (277). The second follows by trying \(\alpha\psi\) with zero noise in the minimization defining \(E_D\). Increasing \(D\) increases the minimized energy; taking the same full Gaussian Hilbert coordinates proves monotonicity of the supremum. For separated compatible sets the accumulated covariance has finite range inside their fixed collars; their coordinates and energies split, proving additivity.

For composition let \(z\) be standard coordinates for the next shell and \(Tz=B_s\alpha\zeta_{\Gamma_u}\). Completing its Gaussian square gives the identity \[\begin{align*} \left(\mathbb E_z e^{p_1V_j(f+Tz)}\right)^{1/p_1} &=\det(I-p_1T^*H_jT)^{-1/(2p_1)} \exp\left\{\frac12\sup_z [2V_j(f+Tz)-p_1^{-1}\|z\|^2]\right\}, \tag{280}\end{align*}\] where \(H_j\) is the quadratic matrix in Equation (279). The supremum on the right is exactly the supremum defining the next \(V\), since it combines the old coordinates and the shell coordinates with the same Cameron penalty. Enlarging the support only increases it. Applying Equation (278) to both coordinate sets shows that the determinant has a strict stability margin depending only on Equation (275).

For a positive matrix \(F\) with \(\|F\|\leq1-\delta\), \(-\log\det(I-F)\leq\delta^{-1}\operatorname{tr}F\). The trace needed here is at most a constant times \[\mathbb EE_D(B_s\alpha\zeta_{\Gamma_u}) \leq\mathbb E\langle \zeta_{\Gamma_u},A_{0,D}\zeta_{\Gamma_u}\rangle \leq C|X|.\] The last inequality is precisely the positive-order difference bound for the shell kernels; the factors \(u^{-2}\) cancel the number of sites in each block. Its constant is independent of the microscopic cell size and of sufficiently large \(L\). The terminal covariance satisfies the same bound. No determinant over all microscopic height variables is introduced by this argument.

Apply Hölder with exponents \(p_1\) and \(p_1/(p_1-1)\) to the two regulators. Choose \(\kappa\) small enough for the latter power and for the analytic reserve required by Lemma 29. Its proof then gives \(W_{Lu}^{\kappa/2}\) for a small polymer after the shell \(u\to Lu\), while Equation (280) transports the factor \(e^V\) without increasing its coefficient. Since \(V_j(t\psi)=t^2V_j(\psi)\) for \(0\leq t\leq1\) and constants do not affect it, all Taylor interpolation arguments in Lemma 30 keep this factor bounded by its output value. The remaining polynomial losses are absorbed by the half-exponent reserve in \(W\). Imaginary charge shifts use only bounded nonconstant directions, whose internal energy is \(O(|X|)\); their costs follow from the same reserve and completion. Thus the linear powers of \(L^{-1}\) are unchanged. The extra fixed \(C^{|X|}\) is paid by the same large-component weight gain, choosing \(L\) and \(A\) as in the toolkit. Product, decoration, and regrouping estimates use monotonicity and additivity just proved, so their quadratic nonlinear bounds also hold. ◻

Proof. Write \(\pi\) for the pattern together with a choice of its positive topology. After all common factors have canceled, let \(\mathcal A\) be the affected halo tiles and let \(\mathcal I_\pi,\mathcal I_0\) be the remaining proxy events on this path and the no-mark path. The set \(\mathcal I_\pi\) includes every surviving unmarked replacement and the factor \(B_i\) on every marked cut choice. Write \(\Delta E_i=E_{o_i}-E_{c_i}\), so that \(B_i=C_i e^{-\Delta E_i/2}\). The notation \(Z_G,E_G\) includes the product and sum over all components of \(G\), respectively. Define \[\mathcal Q_G^{\rm all}(y) =e^{E_G(y)/2}\frac{Z_G(y;k)}{Z_G(0;k)} \mathbb E_{G,y;k}\prod_{i\ {\rm marked}}\delta_i.\] Apart from the separately selected tile factors and links, this positive topology’s contribution to the majorant after division by Equation (270) is exactly \[\frac{\prod_{i\in\mathcal I_\pi}B_i^\pi(y)} {\prod_{i\in\mathcal I_0}B_i^0(y)} \frac{Z_G(y;k)\mathbb E_{G,y;k}\prod_{i\ {\rm marked}}\delta_i} {\prod_{b\in\mathcal A}[Z_b(0;k_h)e^{-E_b(y)/2}]} =\mathsf C_{\pi,G}(k)e^{\mathsf Q_{\pi,G}(y)/2} \mathcal Q_G^{\rm all}(y),\] where \[\begin{split} \mathsf C_{\pi,G}(k) &=\frac{Z_G(0;k)\prod_{i\in\mathcal I_\pi}C_i^\pi} {\prod_{i\in\mathcal I_0}C_i^0 \prod_{b\in\mathcal A}Z_b(0;k_h)},\\ \mathsf Q_{\pi,G} &=\sum_{b\in\mathcal A}E_b-E_G -\sum_{i\in\mathcal I_\pi}\Delta E_i^\pi +\sum_{i\in\mathcal I_0}\Delta E_i^0. \end{split}\] Thus every external proxy factor is in \(\mathsf C_{\pi,G}\) and \(\mathsf Q_{\pi,G}\); the defining \(B_i\) inside each \(\delta_i\) remains there. The two telescopes below bound these two prefactors.

At \(y=0\), a true restoration ratio for a positive bond form supported on a core is sandwiched between its open-patch ratio and its ratio with the outer guard pinned. To see the direction, interpolate the bond precision: the derivative of the logarithm of the centered partition ratio is the negative centered expectation of the added quadratic form. Added exterior precision or centered pins decrease this expectation by Lemma 4. The guard-pinned ratio is independent of the exterior. The logarithmic gap between these two ratios is bounded by \(D(H,p)\), where \(H\) is the union of core tiles and \(p\) is the guard: increasing core precision as far as a full core pin can only increase the guard pin probability further. By Proposition 24 and the localization estimate, this gap is bounded and at \(k_h\) differs from its exponentially small Gaussian counterpart by an arbitrarily small amount for large \(s\). Fixed-patch continuity permits Equation (269) to keep its constants at \(k_h\) when \(k\) is nearby.

To count only local errors, take the union of tiles within distance \(S_0\) of all marks, with \(S_0\) larger than all skip and dependence radii, and telescope on its enlargement through distance \(2S_0\). Perform separately the ordinary sequence ending at \(G\) and the sequence ending at the topology with no marks. Count every event touching this enlarged union, including events clipped at its artificial boundary; there are \(O(j_1)\) of them. At a clipped event use the open-patch proxy clipped in exactly the same way on both paths. The two paths are identical near the artificial boundary, so these proxy factors cancel in their quotient. Their final tile integrals also cancel off the affected halo. The remaining true/proxy ratios are bounded by the centered comparison just proved, one per event. The only \(k\) versus \(k_h\) isolated-tile constants are those in the affected halo; they cost a bounded factor per tile. This gives \(\mathsf C_{\pi,G}(k)\le e^{Cj_1}\) and excludes any extensive normalization loss.

Apply the identical double telescoping to the Gaussian energies. Proposition 23 bounds each local-versus-true error matrix by an exponentially small prefactor, with decay in both cell separation and distance from that event’s core. Each matrix has zero row sums on its unpinned components. Express it in pair differences and assign shortest face-cell paths inside the padded union. The absence of opposite-only junctions makes these paths available. A path of length \(d\) using a fixed edge has endpoints within its \(d\)-ball; polynomial ball growth and exponential decay bound the total charge on that edge. Summing event locations retains \(|\mathsf Q_{\pi,G}(y)|/2\le Ce^{-cR}\mathcal E(y)\), with a weaker \(c\). On a small torus the same argument is performed on the torus itself; the number of halo tiles is still \(O(j_1)\), and image paths are charged with their full lengths. This proves Equation (282). ◻

Proof. Translate each connected underlying component by a common height period so that one observed cell value lies in \([-\pi,\pi]\). The ratios in question are exactly invariant under this simultaneous translation of heights and observations. Paths in a box bound every remaining \(|y_x|\) by a polynomial in \(P\) times \(1+\sqrt{\mathcal E_{\rm box}(y)}\): a spanning tree in each connected clipped component has at most \(CP^2\) cells and gives such paths without any convexity assumption. Thus the stipulated gradient bound leaves a compact set of observation vectors. There are finitely many box geometries, positive topologies, mark choices, and translations at fixed \(R,P\).

In the Gaussian limit the centered-normalized partition factor \(e^{E_G(y)/2}Z_G(y;k_h)/Z_G(0;k_h)\) tends to one. For a deep mark, decompose the log ratio in Equation (281) into its centered \(y=b=0\) value, the change in its conditional centered normalization as the guard changes from zero to \(b\), and the conditional log Laplace change due to \(y\), minus its comparison-energy change. The first term is \(O(e^{-cR})\) by Propositions 24 and 23. The second is between \(\log r_H(b)\) and zero by Proposition 25, with \(H\) containing the restored bonds.

For the quantitative limiting bounds let \(u_c\) be the cut Gaussian harmonic prediction of the guard data, and let \(\chi_H\) equal one near \(H\) with support disjoint from the guard. The localization lemma gives, for every \(r\geq2\), \[ \big\|\|\chi_Hu_c\|_T\big\|_{L^r} \leq C\sqrt r\,\operatorname{poly}(P) e^{-cR}\bigl(1+\sqrt{\mathcal E_{\rm box}(y)}\bigr). \tag{286}\] Here \(T\) is the continuum gradient-plus-unit-cell-observation form. The bound holds when the data have either topology’s centered law: restoring bonds decreases their covariance. In the energy Hilbert space with norm \(\|\cdot\|_T\), write the centered random prediction as \(\mathsf A g\), where \(g\) denotes standard Gaussian energy coordinates. The localized prediction map satisfies \(\|\mathsf A\|_{\mathrm{HS}}\leq C\operatorname{poly}(P)e^{-cR}\) by Proposition 23. This normalization includes the coefficient \(1/a_h\) in \(T\); \(g\) has identity covariance in these energy coordinates. Diagonalize \(\mathsf A\mathsf A^*\), with eigenvalues \(\lambda_i\geq0\). Minkowski’s Inequality in \(L^{r/2}\) and the standard real Gaussian moment bound give \[\big\|\|\mathsf A g\|_T\big\|_{L^r}^{\,2} =\Big\|\sum_i\lambda_i g_i^2\Big\|_{L^{r/2}} \leq\sum_i\lambda_i\|g_i\|_{L^r}^{\,2} \leq Cr\,\operatorname{tr}(\mathsf A\mathsf A^*).\] Prove this first for finite projections; the Hilbert–Schmidt bound and monotone convergence give the same estimate for the full prediction. Covariance ordering only decreases the trace for restored data. A tilt by \(y\) adds a deterministic prediction \(\mu_y\). The localized operator bound and the period recentering already used on the box give \[\|\mu_y\|_T\leq C\operatorname{poly}(P)e^{-cR}\|y\|_2 \leq C\operatorname{poly}(P)e^{-cR} \bigl(1+\sqrt{\mathcal E_{\rm box}(y)}\bigr),\] enlarging the fixed polynomial if necessary. The triangle inequality proves Equation (286) with the displayed \(\sqrt r\) dependence. Its constant and polynomial degree are independent of \(r\), so the subsequent choice \(r=O(j_{\rm deep})\) has only polynomial moment losses.

The negative log Gaussian full-core pin ratio is at most \(\|\chi_Hu_c\|_T^2/2\): this cutoff corrects the conditional mean to zero on \(H\) without changing its guard values. The linear-in-data part of the conditional Laplace discrepancy is bounded by \(C\|y\|\|\chi_Hu_c\|_T\). To verify this also when restoring bonds glues two free faces, first prescribe a finite list of smooth guard probes. The cut and restored predictions minimize the same cut energy on nested admissible spaces. Subtracting \(\chi_Hu_c\) from the cut minimizer leaves all guard probes unchanged and makes it admissible in the restored, trace-matching subspace. Orthogonal projection then bounds the difference of minimizers by \(\|\chi_Hu_c\|_T\). The conditional quadratic parts differ from the open-patch comparison in Equation (269) by at most \(Ce^{-cR}\|y\|^2\), again by guarded localization. All these arguments use finite-energy predictions, so they never prescribe pointwise values of a continuum rough Gaussian trace.

To transfer these estimates to the microscopic guards, first work under the centered high laws and use finite smooth guard lists and finite smooth core penalties. Proposition 25 gives the joint conditional convergence for each topology in its own centered law. Increasing the probe lists and the core penalty approximates the full pin ratio there: the centered identity \(\mathbb Er_H=e^{-D(H,p)}\) and Proposition 24 close the approximation error, and the Gaussian ratio is strictly positive almost surely.

For a cut centered law \(\mu\) and a restored centered law \(\nu\) with identical exterior, let \(A_s\) be any failure event about their common guard data. Equations (100) and (107) give \[\nu_p(A_s)\le e^{D_U(H,p)}\mu_p(A_s),\qquad \mu_p(A_s)\le\mu_p(r_H\le\varepsilon) +\varepsilon^{-1}\nu_p(A_s).\] The interactions have finite limits. Taking \(s\to\infty\) and then \(\varepsilon\downarrow0\) transfers vanishing failure probabilities in both directions. Thus both topologies’ conditional approximations hold jointly when their guard data are sampled under either centered law.

Only after this centered transfer, change the sampling law separately in each topology \(\tau\). For a fixed observation vector \(y\), \[\frac{\,\mathrm dP_{\tau,y;k_h}}{\,\mathrm dP_{\tau,0;k_h}}(h) =\frac{e^{(B_sh,y)}}{\mathbb E_{\tau,0;k_h}e^{(B_sh,y)}} .\] At fixed \(R,P,U\) these are tilts by a fixed finite list of cell observations with bounded bare covariance and coefficients in a compact set. The displayed densities have a common \(L^q\) bound for every fixed finite \(q\), by the density moment estimate in Proposition 25. Hölder in each topology therefore transfers the centered failure bounds to its own tilted law. There are only finitely many topologies and required moment orders at these fixed parameters, so the choices are common to all of them. On exceptional data use \(\delta_i\leq1\). Consequently bounded moments of \(\delta_i\) have the Gaussian upper bounds derived from Equation (286); no uniform claim for arbitrary prescribed microscopic guards is needed.

Use \(|(x-1)/(x+1)|\leq|\log x|\) for \(x>0\). Hölder over the \(j_{\rm deep}\leq\operatorname{poly}(P)\) retained marks gives their product a factor \(e^{-cRj_{\rm deep}}\), with polynomial losses in \(P\), the moment order, and \(1+\mathcal E_{\rm box}\). For clarity, if a per-mark bound has the form \(e^{-c_0R}P^Cj^C(1+\mathcal E)^C\), maximize \((1+\mathcal E)^{Cj}e^{-R^{-12}\mathcal E}\) to obtain a factor \([C(1+jR^{12})^C]^j\). Its logarithm is \(O(j\log R)\) because \(j\leq\operatorname{poly}(P)\), and is absorbed by \(c_0Rj/2\). This proves Equation (285) in the limiting Gaussian calculation with slack.

The same argument works along every convergent sequence in the compact set of translated observation vectors, using uniform exponential integrability of the finite probes. A subsequence contradiction therefore makes the bound uniform there for sufficiently large \(s\). All finitely many box calculations are then continuous in \(k,a'\); their strict slack gives a common neighborhood. This last continuity is on fixed finite microscopic graphs. It invokes no scaling theorem away from \(k_h\). Equation (271) is handled by the free mixed-domain convergence in exactly its own finite tile: its normalized ratio tends to one uniformly on the same compact sets, so its error can be made smaller than \(e^{-cR}\). ◻

Proof. We give the replica argument, since Gaussian convergence of \(X_n\) alone would not imply this assertion. First remove the independent observation noise; its nonconstant test variance tends to zero, and its mean characteristic is a bounded deterministic multiplier. Take two independent pinned lattice fields \(h^{(1)},h^{(2)}\). The sum and difference are conditionally independent, with the same centered law, when their common coordinatewise parity class is fixed. This is the elementary change of variables \((h^{(1)},h^{(2)})\mapsto(h^{(1)}+h^{(2)},h^{(1)}-h^{(2)})\): the quadratic energy splits and the two fields belong to the same coset of twice the height lattice.

Let \(V\) be the conditional second moment of the cosine test of the sum field. Expansion in the independent original copies gives \[\mathbb EV=2\operatorname{Var}X_n,\qquad \operatorname{Var}V=2\bigl(\mathbb EX_n^4-3(\mathbb EX_n^2)^2\bigr).\] The last expression tends to zero by Gaussian fourth-moment convergence. Multiplication by a bounded character of the difference field therefore replaces \(V\) by its mean with error tending to zero. Define \[F_n(t)=\mathbb Ee^{i\bar h+itX_n}.\] The corresponding product expectation of the squared sum test and the difference character is \(-2(F_n F_n''-(F_n')^2)\). More explicitly, for complex \(t\), \[-2(F_nF_n''-(F_n')^2) =2\operatorname{Var}(X_n)F_n^2+ \mathbb E[(V-2\operatorname{Var}(X_n)) e^{i\bar D+itX_n(D)}].\] If \(\mathbb Ee^{tX_n}\le e^{B_0t^2/2}\) for real \(t\), the last expectation has modulus at most \(\sqrt{2\kappa_4(X_n)}e^{2B_0(\operatorname{Im}t)^2}\) by Cauchy–Schwarz. Consequently every locally uniform subsequential limit \(F\) satisfies \[ FF''-(F')^2=-a_hg F^2 . \tag{302}\] Bare domination supplies locally uniform bounds on the entire functions and all their fixed derivatives. The real transforms are nonnegative by the Gaussian-lattice Fourier representation. If \(F\) is not identically zero, solve \((\log F)''=-a_hg\) on any positive interval. Analytic continuation then makes \(F\) an exponential quadratic everywhere; in particular it has no isolated zero. Reflection or translation by half a cosine wavelength changes \(X_n\) to \(-X_n\) without changing \(\bar h\), so \(F'(0)=0\). Thus \(F(t)=F(0)e^{-a_hgt^2/2}\). The identically zero limit has the same factorization. Analytic continuation to \(t=-iw\) gives the stated Laplace version. Every subsequence has only limits of this form, which proves the assertion for the original sequence. ◻

Proof. At the true reference both diagnostics tend to zero: Equation (300) uses the structural Laplace limit, and the mixed diagnostic uses Lemma 81. Their discrepancies can be made uniformly small for \(n\ge sP\) by choosing \(P\) large first and then \(s\) sufficiently large. The block-profile error and the noise variance are \(O((s/n)^2)\le O(P^{-2})\), which explains this order. Before the first candidate exit, the separate torus recursion gives \[\|\mathcal R_j^{\rm tor}\| \le C\theta^j\varepsilon+C\rho^2\] with the same earlier plane singleton inputs. At the candidate exit the new activities are still in a fixed multiple of the tube by the bounded map estimates. The nonzero two diagnostic multipliers therefore give \(|t_j|+|z_j|\le C(\delta_{\rm diag}+\varepsilon+\rho^2)\). Choose \(\rho\) first, with \(C\rho\ll1\), and then \(\delta_{\rm diag}+\varepsilon\ll\rho\). This excludes the exit. The initial plane steps corresponding to sizes below \(sP\) were controlled just after Equation (298).

For convergence let \(q\) be the limsup of the plane activity size. Contraction bounds the limsup of its remainder, and separately that of the stopped torus remainder, by \(Cq^2\). Taking limsups in the two diagnostics gives \(q\le C'q^2\). The tube is chosen smaller than \(1/C'\), so \(q=0\).

For the final assertion, right continuity gives \(a(k)\downarrow8\pi\) as \(k\downarrow k_c\). Fix a slightly larger input \(k_1>k_c\). Finite-volume Gaussian comparisons sandwich the mean-free Laplace functions for \(k_c\le k\le k_1\) between those at the two endpoints. Their limiting Gaussian coefficients approach the same value as \(k_1\downarrow k_c\). Entire-function bounds turn this into uniform smallness of the requisite second and fourth moment discrepancies. The replica calculation in Lemma 81 then gives the mixed discrepancy uniformly as well: otherwise a violating sequence has an entire subsequential limit obeying (302), which is impossible. Choose the auxiliary bracket endpoint \(k_1\) sufficiently close to \(k_c\) first, then a common large-scale cutoff for the two fixed endpoint laws and entry scales with \(sP\) above that cutoff. Finally shrink the actual input neighborhood to \([k_c,k_c+\delta]\subseteq[k_c,k_1]\) as required by entry. The preceding first-exit argument now holds throughout this right neighborhood. For each fixed input its limiting discrepancy is zero, so the individual limsup argument still proves convergence. ◻

Proof. At side \(m\) initialize with the exponential of \(d\sum_B e_B^0/2\), with no height-lattice factor. Expand it by the same occupied-block algebra. The singleton linear term is \(d e_B^0/2\) and the remainder is \(O(d^2)\) in any prescribed fixed activity weight, since the small exponential energy is absorbed by the regulator reserve. We check that this assertion is uniform in the starting scale. A direction of scaled point norm one has \(e_B^0(f)\le C_J\): there are \(O(m^2)\) sites in a block and each fixed-range difference is \(O_J(m^{-1})\). Polarization gives \(|De_B^0(\phi)[f]|\le C_J\sqrt{U_m}\) and \(|D^2e_B^0[f,g]|\le C_J\); all higher derivatives vanish. The analytic derivative sum of a sufficiently small energy exponential is therefore absorbed by a fixed part of \(W_m^\kappa\), uniformly for \(m\ge m_{\min}\). The occupied-block expansion with the fixed weight \(A\) has bounds \((C_A|d|)^{|X|}\) and, after removing its linear singleton, \(C_A|d|^2\). The optional factor \(e^V\ge1\) only improves this initialization. Common-shift invariance keeps every charged coordinate zero.

Direct Gaussian completion bounds all terminal logarithmic ratios under the smooth shift \(F\) by \(C|d|\), uniformly in the stopping scale. Indeed the tail covariance contracts the kinetic energy, and the relevant inverse \((I-dC^{1/2}A_0C^{1/2})^{-1}\) is uniformly bounded on a fixed complex disk; determinants cancel from the ratio. The kinetic contraction is the form inequality \(0\le C^{1/2}A_0C^{1/2}\le I\) for the remaining covariance, so its resolvent norm is at most \((1-|d|)^{-1}\). The selected kinetic energy has shell and terminal expectation at most \(C|X|\), uniformly in the starting scale, by the positive-order kernel estimates and Equation (278). Thus the terminal analytic estimates use the same regulator reserve at every such scale.

Take diagnostic tori of side \(n=DmL^j\) with the fixed \(D\) chosen large enough for locality. The first Fourier mode used in the gradient diagnostic has energy bounded above and below uniformly in \(m,j\). Indeed its symbol converges to the continuum symbol for \(m\ge m_{\min}\), and its \(s\)-cell averaging multiplier is bounded away from zero when \(s/n\le1/D\) (here \(s\le m\)). The independent observation-noise contribution is uniformly bounded as well. No lower restriction \(n\ge sP\) is needed for this Gaussian initialization, because there are no entry sewing operations. The gradient diagnostic consequently gives, before a first exit from a fixed small tube, a bound \(C|d|+C\|\mathcal R_j^{\rm tor}\|+C\rho^2\) on its scalar coordinate, with constants independent of \(m,s,j\). Contraction bounds the remainder by \(C|d|^2+C\rho^2\) there. Choose the tube radius first and then the common \(d_0\) sufficiently small. The same first-exit argument works for complex \(d\), using absolute values of the analytic diagnostic, and excludes an exit on that disk. More quantitatively, on each parameter disk \(|d|\le\delta\le d_0\) take the tube radius \(\rho=M\delta\), with \(M\) fixed larger than the diagnostic constant. For sufficiently small common \(d_0\) the same estimates exclude exit from that tube. This gives a uniform \(O(\delta)\) trajectory bound, and contraction improves the remainder to \(O(\delta^2)\). Applying the conclusion with \(\delta=|d|\) gives the stated pointwise bounds. The derivative at \(d=0\) is exactly the unchanged gradient singleton, so the uniform analytic bounds give \(T_j^g=d+O(d^2)\), with a uniform derivative bound for the error. Taking \(d_0\) smaller makes \(|(T_j^g)'(d)-1|<1/2\). The quantitative analytic inverse theorem gives the common inverse disk. Exactness of the Gaussian integrations shows that the trajectory with initial parameter \(d\) maps to the trajectory with that same parameter at the next step. This proves the last assertion. Finally \((\mathcal R_j^g)'(0)=0\), so the contracted linear remainder map is unchanged by (303). ◻

Proof. Differentiate the absolutely convergent pinned lattice sum: \[\partial_k\operatorname{Var}(h,f_0) =\frac1{2k^2}\operatorname{Cov}\bigl((h,f_0)^2,(h,A_0h)\bigr).\] Express the positive form \(A_0\) as a sum of squares of real linear forms \(\ell_r\). The paired fourth-cumulant inequality in Lemma 4 gives \[\operatorname{Cov}\bigl((h,f_0)^2,\ell_r(h)^2\bigr) \ge 2\operatorname{Cov}\bigl((h,f_0),\ell_r(h)\bigr)^2.\] Summing gives the first inequality in (304). On the quotient by constants, Cauchy–Schwarz gives \((f_0,Cf_0)^2\le(f_0,A_0^+f_0)(Cf_0,A_0Cf_0)\); pinning selects a representative and changes neither side.

Take \(f_0=B_s^*f_n\). Its bare Green form tends to \(g\). The observation-noise variance is independent of \(k\) and tends to zero, while the height variance tends to \(a(k)g\). Division by \(g\) proves (305). The neighborhood version follows from a positive lower variance bound and continuity of \(k\) and of the fixed bare denominator. ◻

Proof. Use the gapped scheme without \(Z\) and the coordinates of Lemma 83. Analyticity, exact vanishing on \(B=0\), and the diagonal balanced linearization give \[\begin{align*} \|B_+\|&\le[\theta+C(|u|+\|B\|)]\|B\|,\tag{307}\\ |u_+-u|&\le C(|u|+\|B\|)\|B\|. \tag{308}\end{align*}\] On a smaller tube the first factor is at most \(\theta'<1\). These inequalities keep nearby entry data in the tube, make \(B\) decay geometrically, and give a uniformly convergent limit \(u_\infty(k,a')\). For fixed \(a'\) it is analytic in \(k\) on a complex neighborhood. The convergence of its \(k\) derivatives follows by Cauchy’s formula on a smaller disk.