Proof. Theorem 1 of van Engelenburg and Lis [14] is stated for free-boundary XY spins with density \(\exp\{\frac b2\sum_{\{x,y\}} (\sigma_x\overline{\sigma_y}+ \overline{\sigma_x}\sigma_y)\}\). Writing \(\sigma_x=e^{i\theta_x}\) gives exactly (1). Their infinite-volume correlation is the free finite-graph exhaustion, hence agrees with (2); reflection of all angles removes its imaginary part. Their theorem gives exponential decay below a threshold and the lower bound \(1/(8|x-y|)\) at and above it. These alternatives have positive and zero horizontal mass, respectively, so their threshold is (3). Specialize to \(x=0,y=ne_1\). ◻

Proof. Expand each factor \(e^{b\cos(\theta_x-\theta_y)}\) in its Fourier series. Integrating the spins imposes zero divergence on the integer edge current in the denominator. Inserting \(e^{i(\theta_0-\theta_x)}\) changes its divergence to the two unit sources. The insertion with the opposite sign has the same sum by current reversal, so taking the real part makes no change. The factors \(e^b\) cancel between the two partition sums.

Rotate currents onto dual edges. In a planar rectangle, every divergence-free integral current is a difference of integer face heights with exterior height zero. This follows by summing the rotated current along dual paths: path independence is exactly the zero-divergence condition at the enclosed primal vertices. Subtract a fixed unit current on a path between the sources in the numerator. The remaining current is divergence free and gives the same height variables. The subtracted current is the claimed connection. Changing its path is an integral change of height variables. Equivalently, the numerator has multivalued heights with the stated monodromies and single-valued branches on simply connected charts. Releasing identifications across a cut gives the asserted cut version. ◻

Proof. Replace every primary edge by \(N\) bonds in series, each with normalized integer increment weight \[q_N^{j^2},\qquad q_N=b/(2N).\] For all sufficiently large \(N\) this is a genuine lattice Gaussian. For every fixed real \(t\), uniformly on compact sets of \(t\), \[\frac{\sum_jq_N^{j^2}e^{tj}}{\sum_jq_N^{j^2}} =1+\frac bN(\cosh t-1)+O_t(N^{-2}).\] The sum of the \(N\) increments therefore converges to \(p_b\), with convergence of exponential moments at every fixed real argument. Matching chain endpoints and imposing the cycle and period constraints gives a quadratic lattice law on the enlarged graph. The numerator is an affine lattice parallel to the denominator lattice.

All partition ratios just listed converge at fixed primary graph. For domination, use a spanning forest to express vertex differences as sums of chain increments; the remaining normalized edge weights are bounded by one. The forest variables have uniformly bounded exponential moments. If a constant is confined, sum its translations against the confining Gaussian. This also dominates fixed real tilts and polynomial factors. Normalizers of the independent chain laws agree in the two sectors.

On the enlarged lattice, the covariance and partition comparisons are precisely BKTref@st:finite-gaussian [12]

. They hold on a proper affine subspace by restricting to its parallel span and completing the square in the remaining fixed coordinates. For clarity, with quadratic precision \(T+tQ\), \(Q\ge0\), differentiation of the log shifted-to-centered ratio gives \[-\frac12\mathop{\mathrm{tr}}Q\bigl( \mathop{\mathrm{Cov}}_{\mathrm{shift}}-\mathop{\mathrm{Cov}}_0 +m_{\mathrm{shift}}\otimes m_{\mathrm{shift}}\bigr)\le0.\] After a fixed real tilt, completing the square gives the same formula with the tilted covariance and mean. This proves the tilted version of (ii), not merely its untilted version. Infinite centered penalties implement the required equalities. Poisson summation on the enlarged lattice proves (iii). Taking the fixed-graph limit proves the stated rules. Vanishing auxiliary masses justify released translation modes; alternatively one can use integer gradient coordinates. ◻

Proof. Keep independent full-chain increments as variables and release their endpoint, curl, and period constraints. Lemma 4 bounds the original centered transform by this product transform. For one subdivision bond and bounded \(t\), pairing opposite increments in its moment generating function gives \[\log\frac{\sum_jq_N^{j^2}e^{2\pi tj}} {\sum_jq_N^{j^2}} \le C_{b,R}t^2/N\qquad (|t|\le R).\] Summing over \(N\) bonds gives \(C_{b,R}t^2\) for a whole chain. For a flow normalized by \(\sqrt{\mathcal R(w)}\), every coefficient has absolute value at most one. Independence therefore bounds its log transform at a fixed real argument by a constant. This proves the normalized exponential and moment bounds, and rescaling proves (4). The same inequalities on the original law follow by the fixed-graph limit. In this argument a centered primary observation is a real linear function of the enlarged coordinates, so its squared penalty is an additional positive form. Cycle, period, and endpoint conditions are likewise limits of positive equality penalties. They can all be removed before applying the independent whole-chain bound. No uniform estimate on the inverse microscopic Gaussian precision is asserted or used. ◻

Proof. We give the changes to the cutting and folding proof of BKTref@st:free-rectangle-limit [12]

; the observation Laplacian here is the primary \(A\), not the subdivided quadratic precision.

Cut an \(M\)-square into \(m\)-squares and strips of width less than \(m\). In its affine observation, the seam and strip flows have reference cost \(O(m^{-1}+m/M)\). Cutting increases covariance, and the retained cell observations become independent. Consequently \[ \sqrt{a_M}\le\sqrt{a_m} +C_b(m^{-1}+m/M)^{1/2}. \tag{9}\] First let \(M\to\infty\), and then let \(m\) follow a liminf subsequence. This proves existence and finiteness of \(a\). Square symmetry makes the limiting covariance of the two affine observations scalar.

Cutting rectangles into squares gives the upper variance bound \(a\int_B|\nabla g|^2\) for affine \(g\). For the reverse bound, at each fixed discrete rectangle tile an arbitrarily large square by copies of that rectangle, choosing its side divisible by both rectangle side counts. Apply the same seam estimate, first sending that square side to infinity. If \(m_1(n),m_2(n)\) are the rectangle side counts and \(V_n(p)\) is its variance for \(g=p\cdot x/n\), this gives \[\sqrt a\,|p| \le\frac{n}{\sqrt{m_1(n)m_2(n)}}\sqrt{V_n(p)} +\frac{C_{b,p}}{\sqrt{\min(m_1(n),m_2(n))}}.\] Thus every affine shift attains the upper bound along every admissible rectangle sequence.

Cut into cells of small fixed macroscopic diameter and approximate each gradient there by a constant. Seams have vanishing reference cost, and the interior error is controlled by the squared \(L^2\)-modulus of continuity. This proves \[ \limsup_n\mathop{\mathrm{Var}}\langle h,A_{B_n}g\rangle \le a\int_B|\nabla g|^2 \tag{10}\] for smooth or continuous piecewise smooth \(g\), and for partial gradient observations restricted to polygonal pieces. Call equality in this upper bound saturation. Saturated shifts form a linear space: on a convergent covariance subsequence the upper-bound defect is positive semidefinite, and its diagonal vanishes for saturated shifts, so every entry vanishes. Saturation passes to an enclosed rectangle by cutting its complement into rectangles and comparing the sum of their upper bounds with the saturated outer variance. It also passes through gradient-energy approximation, by Lemma 5.

To obtain nonaffine saturated shifts, condition on the whole trace of a site-axis or diagonal reflection line and pin one of its vertices. The subdivided quadratic graph is attractive: its density satisfies the lattice condition edge by edge. Its conditioned law is associated, and association passes to the fixed-primary-graph limit. If a saturated seed is odd about the line, the two partial tests on its open sides are reflected negatives. If the off-line coefficients on one side are nonnegative after a vanishing-cost transport to the line, conditional association gives nonpositive conditional cross covariance. The two conditional means are negatives, so the unconditional cross covariance is nonpositive as well, up to the vanishing transport error. The saturated sum and the separate upper bounds (10) force each partial test to saturate. The corresponding folded shift differs from that partial test only along a vanishing-cost seam.

Here the coefficient verification can be made on unit edges. For the axis seed \(g=x_1\), the only negative coefficients of the right-side partial observation occur next to the cut. Move them to the conditioned line along bounded-length paths. There are \(O(n)\) coefficients of size \(O(n^{-1})\), with bounded overlap, so the cost is \(O(n^{-1})\). The hinges \((x_1-t)_+\), and then all smooth univariate functions, are therefore saturated. For a diagonal seed, put \(Y=y+t\) and take \(g=F(x)-F(Y)\) on a symmetric enclosing square \(p\le x,Y\le q\). Initially assume \(F'''<0\) and \(F'(p),F'(q)>0\). In \(x>Y\), away from the diagonal strip, the interior coefficient is nonnegative because the symmetric second differences of \(F\) decrease with their center. The coefficients on \(x=q\) and \(Y=p\) have positive leading terms \(F'(q)/n\) and \(F'(p)/n\); their \(O(n^{-2})\) remainders are harmless. Move the exceptional diagonal-strip coefficients to the line. Their total path cost is again \(O(n^{-1})\). The fold criterion proves saturation of \[ (F(x)-F(y+t))\,1_{\{x>y+t\}}. \tag{11}\] Every smooth \(F\) is a difference of two functions with the strict properties just used: add a sufficiently large cubic with negative third derivative and then a sufficiently large positive linear function. Hence (11) holds for arbitrary \(F\). Coherently round a reflection line by \(O(n^{-1})\). The changed fold has squared gradient cost \(O(n^{-1})\): only an \(O(n^{-1})\)-width strip changes its side, and away from that strip the smooth gradient changes by \(O(n^{-1})\). Restriction from the enclosing symmetric rectangle consequently gives the claim on every admissible target rectangle sequence.

Finally choose a smooth family \(F_t\), compactly supported in the relevant \(t\)-range, with \(F_t'(x)=-g_{xy}(x,x-t)\). The mixed derivative of the integral of (11) is \(g_{xy}\). Its difference from \(g\) is therefore a sum of univariate functions. Riemann sums converge in gradient \(L^2\), including across the moving diagonals. Linearity and energy approximation prove saturation for every smooth \(g\).

The lower Laplace bound now follows from Lemma 4. For the upper bound, cut into cells of side comparable to a fixed \(r\). Remove seam errors by Holder’s inequality and the reference-cost exponential bound. A cell test has reference-cost square root \(O(r)\); hence, for fixed \(p>1\), \[\log\mathbb Ee^{pX} =\tfrac12p^2\mathop{\mathrm{Var}}X+O_p(r^3).\] The remainder follows from \(\mathbb E[|X|^3e^{p|X|}]=O_p(r^3)\), which uses normalized exponential moments, not a bare Gaussian upper tail. There are \(O(r^{-2})\) cells. First take the microscopic limit, then \(r\downarrow0\), then \(p\downarrow1\). This proves (7); applying it to real linear combinations proves the joint assertion and moment convergence.

Nearest-neighbor Poincare bounds the reference cost of a density error by a constant times its scaled \(L^2\)-norm squared. Approximate \(f\) by finite sums of Neumann cosines, use their smooth inverse-Laplacian shifts, and then remove the approximation. For such a shift \(n^2A_{B_n}g\) converges in density \(L^2\) to \(-\Delta g\); the boundary coefficients are \(O(n^{-2})\) and occupy \(O(n)\) sites. Using the actual rectangle endpoints before letting them converge handles arbitrary admissible sequences. This proves (8). A face probability can be moved to a strip of width \(r\) at reference cost \(O(r+n^{-1})\), by parallel normal flows. Strip approximation proves the face extension. ◻

Proof. We first isolate the replacement for the quadratic charge-space joining used in BKTref@st:fractional-mean [12]

and BKTref@st:critical-face-ratio [12]

. Tile a rectangle by free cells, keeping their lattice translations, and independently assign every seam edge a difference of law \(2\pi p_b\). Let \(\mathcal V\) be the neutral Euclidean space of cell totals. Choose a linear flow right inverse \(P\) on the neighboring-cell graph, so \(\operatorname{div}P(d)=d\) for \(d\in\mathcal V\). Define \(H\in\mathcal V\) by \[ \langle H,d\rangle =\sum_{\langle i,j\rangle}P(d)_{ij} \left(\langle h_i,\rho_{ij}\rangle-\langle h_j,\rho_{ji}\rangle -\overline\eta_{ij}\right). \tag{13}\] Here \(\rho_{ij}\) is the uniform seam-face probability and \(\overline\eta_{ij}\) is the mean of the independent seam differences, with consistent orientation. Translating the cells by constants \(c_i\) changes \(H\) by the orthogonal projection of \(c\) onto \(\mathcal V\). Weight this law by \(e^{-|H|^2/2}\), summing relative translations and taking the common translation modulo \(2\pi\). Adding exact seam matching makes \(H=0\) and recovers the joined rectangle. The soft law thus has larger internal variances and smaller invariant characteristic coefficients than the joined law. These comparisons with translation-summed laws can be justified without biasing the initial free-cell laws. First add the auxiliary precision \(\eta\sum_i\overline h_i^2\), where \(\overline h_i\) is the cell mean. For a fixed cell shape with representative mean \(\mu\), its translation sum is \(\sum_{k\in\mathbb Z}e^{-\eta(\mu+2\pi k)^2/2}\). After division by its Gaussian integral scalar, this tends uniformly in \(\mu\) to one and is uniformly bounded, by Poisson summation. Thus the original internal law tends to the product of the free-cell laws, also in moments of invariant tests. After adding \(|H|^2\), divide out the common translation scalar. The relative translation sum is a periodized Gaussian on the projected lattice, uniformly bounded in its shape-dependent shift. Dominated convergence again applies with the internal moments. Apply precision comparison at \(\eta>0\) and then let \(\eta\downarrow0\). In particular the soft variance of the sum of two internal affine tests is at most their independent free-cell variance.

For a phase whose cell restrictions are \(s_i\), of totals \(\xi_i\), Poisson summation in the relative translations gives, up to a common scalar, the numerator \[ \sum_{\substack{n_i\in\mathbb Z\\\sum n_i=\sum\xi_i}} e^{-|d|^2/2} \prod_i w_i\left(s_i+\sum_{j\sim i}P(d)_{ij}\rho_{ij}\right) \chi_m(P(d)),\qquad d=n-\xi. \tag{14}\] The denominator uses \(s=0\). The signs of \(P\) are chosen so the displayed cell totals are \(n_i\). Indeed the reciprocal vectors of the projected translation lattice are exactly the neutral integer vectors; the phase shifts them to \(n-\xi\). Fourier transforming the Gaussian soft weight supplies \(e^{-|d|^2/2}\); the remaining factors are the cell and independent-seam characteristic functions. The latter factor \(\chi_m\) is positive and at most one. For fixed cell fluxes it tends to one, since each seam average has variance \(O(m^{-1})\). For two cells one may use a scalar mismatch penalty, giving the same formula with integer sectors \(l\) and weights \(e^{-l^2/2}\). All these calculations can first be performed on the subdivided quadratic graph and then passed to the fixed primary graph.

We prove the uniform phase. Let \(\rho\) be the right-face probability, and set \[X=\langle h,A(x_1/m)\rangle,\qquad Y=\langle h,\rho-u_m\rangle.\] On the unit square graph \(X\) is exactly the right-face average minus the left-face average. Theorem 6 gives joint Gaussian limits, \(\mathop{\mathrm{Var}}X\to a\), and, by reflection, \(\mathop{\mathrm{Cov}}(X,Y)\to a/2\). Suppose, for some nonzero integer \(l\), that \(w_m(lu_m)\) stays bounded away from zero on a subsequence. The reference-cost exponential bounds give a further subsequence on which \[F_m(t,v)=\mathbb Ee^{il\langle h,u_m\rangle+itX+ivY}\] converges locally with all derivatives to an entire function \(F\). On real arguments it is positive: this follows from (5) and the assumed positive value at the origin.

Here is the replica calculation responsible for its form. For two independent copies of a centered subdivided quadratic height, put \(S=h_1+h_2\), \(D=h_1-h_2\). Conditioned on their common parity class modulo \(4\pi\mathbb Z\), they are independent and identically distributed. Writing \(V_{ij}=\mathbb E(S_iS_j\mid\text{parity})\), we have \[ \mathbb EV_{ij}=2\mathop{\mathrm{Cov}}(h_i,h_j),\qquad \mathop{\mathrm{Cov}}(V_{ij},V_{rs})=2\kappa(h_i,h_j,h_r,h_s). \tag{15}\] To verify the second identity, put \(C_{ij}=\mathbb Eh_ih_j\) and \(M_{ijrs}=\mathbb Eh_ih_jh_rh_s\). Conditional independence and expansion in the two original copies give \[\mathbb EV_{ij}V_{rs} =\mathbb ES_iS_jD_rD_s =2M_{ijrs}+2C_{ij}C_{rs} -2C_{ir}C_{js}-2C_{is}C_{jr}.\] Subtracting \(\mathbb EV_{ij}\mathbb EV_{rs}=4C_{ij}C_{rs}\) gives exactly twice the fourth joint cumulant, without symmetrizing the indices. In particular \(\mathop{\mathrm{Var}}(V_{XX})=2\kappa(X,X,X,X)\) and \(\mathop{\mathrm{Var}}(V_{XY})=2\kappa(X,X,Y,Y)\). After first taking the fixed-graph chain limit and then the Gaussian limit, both vanish.

For completeness, write \(F_{m,N}\) for the transform before the chain limit, and put \(\theta(h)=l\langle h,u_m\rangle+tX+vY\). Inversion symmetry makes \(F_{m,N}\) real on real arguments. For the observation indices \(i,j\in\{X,Y\}\), differentiating in their respective phase variables gives \[\mathbb ES_iS_j e^{i\theta(D)} =-2\bigl(F_{m,N}F_{m,N,ij} -F_{m,N,i}F_{m,N,j}\bigr).\] By conditional independence the left side equals \(\mathbb E[V_{ij}\mathbb E(e^{i\theta(D)}\mid\text{parity})]\). Replacing \(V_{ij}\) by \(2C_{ij}\) has absolute error at most \(\sqrt{\mathop{\mathrm{Var}}(V_{ij})}\), and \(\mathbb Ee^{i\theta(D)}=F_{m,N}^2\). Fixed-graph moment convergence passes this bound through \(N\to\infty\); Gaussian moment convergence then removes the error as \(m\to\infty\). Thus \[-2FF_{tt}+2F_t^2=2aF^2,\qquad -2FF_{tv}+2F_tF_v=aF^2.\] Since \(F>0\) on real arguments, division by \(-2F^2\) yields \[(\log F)_{tt}=-a,\qquad(\log F)_{tv}=-a/2.\] Reflection gives \(F_t(0,0)=0\); hence \[ F(t,v)=F(0,v)\exp(-at^2/2-atv/2). \tag{16}\]

Now softly join two horizontal \(m\)-squares and test the sum of their internal affine observations. Its variance is at most twice the free-cell variance, and at least its variance in the fully joined rectangle. The joined rectangle’s full affine test differs from this sum by the seam test, whose reference cost is \(O(m^{-1})\): there are \(m\) seam coefficients of size \(m^{-1}\). Its limiting variance is consequently \(2a\), so the soft-law variance also tends to \(2a\).

In (14), every sector function has logarithmic second derivative at least minus twice the free-cell variance. The logarithmic second derivative of a positive sum is the weighted mean of these derivatives plus the variance of the sector logarithmic first derivatives. Thus the negative second derivative of the normalized sum, the soft-law variance, is at most twice the free variance minus that nonnegative sector variance. Differentiation of the sector sum is justified by the following summable bounds. Write \(\psi_l(t)\) for the product of the two cell characteristic coefficients in sector \(l\), and \(V=2\mathop{\mathrm{Var}}(X)\) for the independent-cell variance. Then \(0<\psi_l\le1\) and \((\log\psi_l)''\ge-V\), so the derivative estimate underlying (5) gives \[\psi_l|(\log\psi_l)'|^2 \le 2V(-\psi_l\log\psi_l)\le 2V/e.\] Multiplication by \(e^{-l^2/2}\chi_m(l)\le e^{-l^2/2}\) gives a summable bound, uniform in \(m\), on the logarithmic-derivative square terms. The raw second derivatives are bounded by the independent-cell second moment \(V\). The analogous first-derivative bound follows by Cauchy–Schwarz. Hence termwise differentiation and the mixture-Hessian identity are valid even for sectors whose coefficients become small. The sector zero has weight one before normalization. The sector \(l\) stays bounded away from zero: transfer from \(lu_m\) to the seam face costs a bounded amount, and \(\chi_m\to1\). By (16) at \(v=l\), reflection and charge reversal give its logarithmic first derivative tending to \(-al\), up to the irrelevant common orientation. The zero sector derivative is zero. If \(a>0\), this contradicts the vanishing sector variance. Therefore \(w_m(lu_m)\to0\). Equation (5) proves (12); Fourier approximation on the circle and the Gaussian limits prove the joint uniform-phase assertion.

It remains to obtain the sharp numerical gap. Let \(\overline\rho_m\) be one quarter of the sum of the four face probabilities and put \(z_m=w_m(\overline\rho_m)>0\). Tile a \(Gm\)-square into \(G^2\) cells, with \(G\) fixed. In the outer-face phase, the cell totals \(\xi_i\) are supported at the boundary, sum to one, and satisfy \(\sum_i\xi_i^2\le C/G\). The denominator of (14) tends to one: its zero sector is one; every other sector has a cell of nonzero integer total and tends to zero by (12); Gaussian sector weights permit dominated convergence at fixed \(G\).

Retain numerator sectors \(n=e_{i_0}\), where \(i_0\) is at least \(G/4\) from the boundary. The radial-flow construction of BKTref@st:radial-cost [12]

supplies outward face fluxes \(H_{ie}\) with divergence \(e_{i_0}\), constant outer flux \(1/(4G)\), and, on putting \[D_{ij}=(H_{i,+j}-H_{i,-j})/2,\qquad S_{ij}=H_{i,+j}+H_{i,-j},\] the bounds \[ \sum_{i,j}D_{ij}^2\le\frac1{2\pi}\log G+C,\qquad \sum_{i,j}S_{ij}^2\le C. \tag{17}\] For clarity, its continuum field is \[V(x)=\frac{x-x_0}{2\pi|x-x_0|^2}+E(x),\qquad |E|\le C/G,\quad|\nabla E|\le C/G^2.\] The correction fixes the constant outer normal flux; it is obtained by the zero-Neumann problem after even reflection. Integrating \(V\) across unit-cell faces gives \(H\). The squared radial field integrates to \((2\pi)^{-1}\log G+O(1)\); face-sum derivative errors are summable and give the second bound in (17). These constants are uniform for the retained central cells. The internal flows have divergences \(e_{i_0}-\xi\). These vectors are linearly independent, since their central coordinates are distinct and \(\xi\) is supported on boundary cells. Prescribe \(P\) on them by these flows and extend linearly. Thus one soft precision works for all retained sectors.

In each cell, expand the logarithm of its characteristic coefficient around the symmetric four-face profile of the same total, either zero or \(\overline\rho_m\). The linear term vanishes by square symmetry. The logarithmic Hessian lower bound therefore gives a product lower bound \[z_m\exp\left\{-\frac12\sum_i\mathop{\mathrm{Var}}\langle h,Y_i\rangle\right\},\] where \(Y_i\) is the face-profile difference. Coordinate reflections split \(Y_i\) into two mutually orthogonal odd face differences and an even remainder. The odd differences have limiting variance \(a\); the even remainder has reference cost bounded by \(C\sum_jS_{ij}^2\). Consequently, as also calculated in BKTref@st:face-energy-bound [12]

, \[\limsup_m\sum_i\mathop{\mathrm{Var}}\langle h,Y_i\rangle \le a\sum_{i,j}D_{ij}^2+C_b\sum_{i,j}S_{ij}^2.\] The soft penalty is bounded below uniformly on these sectors, since \(|e_{i_0}-\xi|^2\) is bounded, and the seam factor tends to one at fixed \(G\). There are at least \(cG^2\) retained sectors. Characteristic monotonicity from the soft law to the joined law yields \[ \liminf_{m\to\infty}\frac{z_{Gm}}{z_m} \ge c_bG^{\,2-a/(4\pi)}, \tag{18}\] with \(c_b>0\) independent of sufficiently large fixed \(G\). If \(0<a<8\pi\), choose one integer \(G\) making the right side larger than two. Then \(z_{Gm}\ge(3/2)z_m\) for every sufficiently large \(m\). Iterating along \(m,Gm,G^2m,\ldots\) contradicts \(0<z_m\le1\). This proves the gap. ◻

Proof. In a sufficiently large free spin box, put disjoint circular dyadic annuli of radii \(r,2r\) around each source, through radii of order \(n\). On the fixed annulus \(1<|z|<2\), choose a smooth tangential divergence-free form \(p\), compactly supported there and approximating \(d\arg z\) in \(L^2\). A radial cutoff provides such forms. Discretize \(p\) at scale \(r\) by taking differences of a sampled stream function on faces. Its pairing with twisted differences is deterministic, by circulation, and tends to \[D=\int p\cdot\nabla\arg z.\] Use the opposite sign at the negative source.

Insert in the twisted numerator the real exponential of \(\lambda\) times the sum of these deterministic pairings. Release endpoint constraints between a grid of small rectangular cells at every annular scale, retaining independent seam increments; release all remaining exterior constraints as well. Lemma 4(ii), with its real tilt, bounds this normalized tilted numerator by the resulting product of transforms. Each retained cell is simply connected and avoids the sources, so its integral connection disappears by an integer gauge change. Here the tilt is on the affine edge differences, the same coordinates as the edge weights. If the connection on a chart is \(\alpha=\nabla\kappa\), the reindexing \(h'=h+\kappa\) replaces \(\nabla h+\alpha\) by \(\nabla h'\) in both the weight and the tilt. It therefore introduces no residual real-tilt scalar.

Take cells of side comparable to \(tr\), lying within the annulus, with \(t>0\) small and fixed. There are only \(O_t(r)\) leftover nonzero edge coefficients, each \(O(r^{-1})\); their independent-chain log moment is \(o_r(1)\). On the squares, approximate the coefficients by constant gradients. Theorem 6, Holder’s inequality, and the reference-cost bound on the coefficient error give log transform at most \[\frac{\lambda^2a}{2}\int|p|^2+o_r(1)+o_t(1).\] The errors are taken first at fixed \(t,\lambda,p\); rounding changes only vanishing reference-cost terms.

There are \(\log n/\log2+O(1)\) annuli per source. At any fixed \(n\) all cuts are internal to a sufficiently large box, so we may first take its prescribed free-box limit. Averaging the convergent estimates along the increasing annular scales gives \[\limsup_n\frac{\log C_b(n)}{\log n} \le\frac{2}{\log2} \left(-\lambda D+\frac{\lambda^2a}{2}\int|p|^2\right),\] after \(t\downarrow0\). Now let \(p\to d\arg z\); both \(D\) and \(\int|p|^2\) tend to \(2\pi\log2\). For \(a>0\), the optimal choice \(\lambda=1/a\) yields \(-2\pi/a\). The factor two is from the two sources. For \(a=0\), each fixed \(\lambda>0\) is allowed, giving \(-4\pi\lambda\); letting this fixed parameter be arbitrarily large proves the stated interpretation. ◻

Proof. At a fixed band partition, the free limits and the uniform fractional mean make translation summation converge to integration of the constant against \(\,\mathrm dc/(2\pi)\). To justify unbounded insertions, let \(\bar h_K=|K|^{-1}\sum_d|d|h_d\). The band energy is at least \(\lambda|K|\bar h_K^2/2\), while \(S-\bar h_K\) is a mean-free test of bounded primary reference cost. Lemma 5 supplies all its moments and fixed linear exponential moments. Summing the Gaussian translation tail gives uniform integrability for every fixed polynomial insertion. This also proves convergence of the normalizing constants and of the weighted moments at each fixed partition.

After that limit, integrate the constant first. On the mean-free Neumann field the remaining quadratic penalty is the squared \(L^2(K)\) norm of its cell projection after subtracting its \(K\)-average. Its covariance \(C_\ell\) satisfies \(\sup_\ell\lVert C_\ell\rVert_{\mathrm{HS}}<\infty\): the square’s inverse Neumann eigenvalues are square summable; cutting a permitted domain into finitely many rectangles gives the same bound by the compact energy inclusions of BKTref@an:forms [12]

. If \(t_j\) are the eigenvalues of \(\lambda C_\ell\), the logarithm of the determinant ratio in the normalized \(p\)-th weight moment is \[\frac12\sum_j\{p\log(1+t_j)-\log(1+pt_j)\} \le C_p\sum_jt_j^2.\] The constant integrals are bounded positive factors. Poincare’s inequality and control of the constant by the band give, uniformly under cell refinement, \[\lVert v\rVert_{L^2(U)}^2\le C_\lambda \left(a^{-1}\mathcal E_U(v)+\lambda\sum_d|d|v_d^2\right).\] Thus every fixed moment of \(S\) under the continuum \(p\)-weighted law is bounded. This proves (23) in its stated iterated-limsup sense. It does not assert a lattice quadratic-exponential moment.

Strong convergence of the cell projections in \(L^2(K)\), weak energy compactness, and smooth recovery sequences give convergence of the inverse forms to \(T_\lambda^{-1}\), exactly as in BKTref@an:band-density [12]

. These prove the finite Gaussian limits. If \(g\) vanishes near \(K\), then in the weak Neumann sense \(T_\lambda(ag)=-\Delta_Ug\); hence its test variance is \(a\mathcal E_U(g)\). Gaussian covariance decreases with \(\lambda\), which proves the fourth-moment assertion. ◻

Proof. Extend variations on unvisited primary vertices by zero on the visited primary set. Apply the ordinary unit-edge path-Poincare argument of BKTref@an:path-poincare [12]

: in a fixed-factor enlargement of a touched cell, stop the primary path after it spans a comparable cell distance. One coordinate spans a positive fraction of the lattice levels, giving anchored rows or columns. One-dimensional telescoping on those rows, followed by rectangle Poincare, bounds the cell’s scaled \(L^2\) norm by \(C\ell^2\) times its local unit-edge energy. The enlargements have bounded overlap. Duality with flows gives (25). A macroscopic anchored rectangle and global Poincare control the full-component mean.

At a free wall or a permitted junction, use the bounded union of comparable rectangles within the same component. Their face Poincare inequalities control differences of rectangle means. The no-opposite-only convention ensures these rectangles are joined through faces; one anchored rectangle anchors the union. A filled starting region provides the initial zeros. For a normalized cell average its squared \(L^2\) density is \(O(\ell^{-2})\), canceling the local \(\ell^2\).

These are inequalities for the primary reference network. Extend the optimizing flows through chains and use (24), imposing the posterior constraints and increased precisions by Lemma 4. One may insert the removed internal vertices at zero, ground them, and tighten the first surviving segment; the resulting centered quadratic law is precisely the unvisited posterior. The flow identity involves only the unvisited test, since all inserted and visited coordinates in this comparison are zero. This transfers the inequalities to centered posterior moments, uniformly in \(N\). ◻

Proof. Given signless data, \(\mathbb E_\sigma S^2=Q\). On \(Q\le4t^2\), Chebyshev gives high probability of \(|S|\le Ct\). For \(Y=(h,f)\), the Rademacher fourth-moment bound and Paley–Zygmund give a fixed positive probability of \[|Y|\ge c\left(\sum_CM_C(f)^2\right)^{1/2}.\] Choose \(C\) so the two events intersect with a fixed positive probability. On every translation orbit, \(|S|\le Ct\) allows at most \(C'(1+t)\) translations. A mean-free \(Y\) is translation invariant and has moments of every order bounded by Lemma 5. This gives arbitrary polynomial bin tails, and integration gives (26). For nonzero total charge, use \(Y=(h,f-u\sum f)+S\sum f\) on the same intersection.

After a macroscopic cluster is exposed, let \(S_U\) be the complement signed mean. Lemma 12 gives every conditional moment of \(S_U\), uniformly. Jensen in its remaining fair signs yields \[Q_U^p\le\mathbb E_\sigma|S_U|^{2p}.\] Thus \(\mathbb P(Q_U\le4t^2\mid\mathcal F_C)\ge1/2\) for sufficiently large \(t\). If \(m_C\le2t\), this event implies \(Q\le8t^2\). Multiply by \(I(C)1_{\{m_C\le2t\}}\), condition, and sum by the least-primary-root disintegration. This proves (27), also for initially extended nonnegative integrals.

For (28), choose a small fixed diameter threshold \(r\) and finitely many separated probability box smears \(p_j\) so every cluster of diameter below \(r\) misses one box. If small clusters contribute at least \(t^2/2\) to \(Q\), assign each a missed box. Since \(M_C(p_j-u)=-m_C\) there, one mean-free squared-mass sum is at least \(ct^2\). Its polynomial bin tail, of an arbitrarily high order, controls this part.

Otherwise the macroscopic clusters supply the majorant \[1_{\{t^2<Q\le4t^2\}} \le\sum_{\mathop{\mathrm{diam}}C\ge r} \frac{2m_C^2}{t^2}1_{\{m_C\le2t\}}1_{\{Q>t^2\}}.\] Expose \(C\), and write \(S=\sigma_Cm_C+S_U\). If \(m_C\ge t/2\), separate \(|S_U|\le m_C/2\) from its complement. Polynomial damping handles the first event; arbitrarily high conditional moments of \(S_U\) handle the second. If \(m_C<t/2\), the condition \(Q>t^2\) forces \(Q_U>3t^2/4\), again giving arbitrary polynomial decay. Precisely, retain the intrinsic cost \[I_t(C)=\frac{2m_C^2}{t^2} 1_{\{\mathop{\mathrm{diam}}C\ge r,\ m_C\le2t\}}.\] After dropping the original upper-bin restriction, the conditional expectation of \((1+|\sigma_Cm_C+S_U|)^{-M}1_{\{m_C^2+Q_U>t^2\}}\) is at most \(C_K(1+t)^{-K}\), uniformly on this intrinsic cutoff, by the two cases just proved. This calculation is under the unweighted base disintegration. Reinsertion gives \[\int\sum_C I_t(C)\,\mathrm d\nu_n \le2\int1_{\{Q\le8t^2\}}\sum_CI_t(C)\,\mathrm d\nu_n \le32\,\nu_n(Q\le8t^2)=O(1+t).\] Choose \(K>j+1\). No weighted conditioning or count of microscopic clusters enters this estimate. ◻

Proof. Let \(F\) be a squared-mass root or such a signed sum and let \(A_t\) be a dyadic bin of \(Q\). Khintchine’s inequality and (26) give \(\mathcal P_n(|F|^{2p}1_{A_t})\le P_p(t)D^{2p}\), where \(P_p\) is a polynomial and \(D\) the relevant reference cost scale. Holder with exponents \(4,4,2\) gives \[\begin{align*} \mathcal P_n(\widehat W|F|^p1_{A_t}) &\le \mathcal P_n\bigl(\widehat W^4(1+|S|)^M\bigr)^{1/4}\\ &\quad{}\times \mathcal P_n\bigl((1+|S|)^{-M}1_{A_t}\bigr)^{1/4} \mathcal P_n(|F|^{2p}1_{A_t})^{1/2}. \end{align*}\] The two polynomial weights cancel exactly. Use Lemma 11 and choose the decay order in (28) after \(p\), so that the dyadic sum and its tail converge uniformly. The lowest bin is bounded directly by (26).

The primary reference cost of \(h_d-S\) is \(O(1+\log(1/\ell))\). On a containing rectangular patch, sum the inverse Neumann cosine eigenvalues up to frequency \(\ell^{-1}\); this costs \(O(\log(1/\ell))\). Above that frequency Parseval and the squared density bound \(O(\ell^{-2})\) give a bounded remainder. Component Poincare compares the patch and component means. This proves (29). An auxiliary independent sign sample has the same estimates, by exchanging the old and new samples in the base measure. ◻

Proof. On \(Q\le t^2\), expand the difference of the two quadratic band energies in the original and new signs. Orthogonality of Rademacher monomials bounds its squared base integral by \[C\lambda^2\int1_{\{Q\le t^2\}} \sum_{\mathop{\mathrm{diam}}C\ge r}\sum_{C'\ne C} \left(\sum_d|d|M_C(d)M_{C'}(d)\right)^2\,\mathrm d\nu_n.\] Expose \(C\), retain the intrinsic condition \(m_C\le t\), and drop the complement bin restriction. Averaging the complement signs turns the inner sum into the conditional second moment of \(\sum_d|d|M_C(d)h_d^U\). Lemma 12 bounds it by \(C\ell^2\sum_d|d|M_C(d)^2\). Reinsert a bin using (27), and apply (26). Thus \[\mathcal P_n\left(1_{\{Q\le t^2\}} |\log W(\sigma')-\log W(\sigma)|^2\right) \le C_{\lambda,t,r}\ell^2D_\ell^2+o_n(1).\] The bin has finite base mass. The normalized densities have uniform \(L^p\) bounds there. Split according to whether the logarithmic difference exceeds a fixed \(\delta>0\), use Holder on that exceptional set and \(|e^x-e^y|\le(e^\delta-1)(e^x+e^y)\) on its complement. Then let \(\delta\downarrow0\). Lemma 14 removes the bin cutoff, including against observables with the stated high moments. For this last step, use the enlarged base space carrying the old and new independent macroscopic sign samples, with the small-cluster signs shared. Swapping the two samples preserves the base measure and the signless variable \(Q\). Thus the old and new normalized densities have identical \(Q\)-tails, and for every fixed \(p\), \[\int_{\{Q>T\}}|\widehat W(\sigma)-\widehat W(\sigma')| \le2T^{-p}\mathbb E_WQ^p.\] The right side is uniformly small in the first two limits. For auxiliary-sign observables the unweighted bin moment estimate is unchanged. For each density use its own component mean in both damping factors; swapping the samples gives the bound for the other density. Hence the weighted Holder estimate controls those observables under either density.

Partition \(U\) into \(r\)-cells. A cluster of diameter below \(r\) meets only boundedly many neighboring cells. For a bounded density, Cauchy–Schwarz and (26) bound its signed second moment on a fixed bin by \(C_t r^2(1+\log(1/r))\). Higher fixed moments are bounded there. Interpolation, followed by the weighted tail transfer, gives their vanishing.

Resample first at a threshold below \(r\), remove the smaller clusters, and use Lemma 11. Fair-sign second moments are the cluster cross sums; nonnegative smears make their summands nonnegative. This proves (30). Jensen in the fair signs similarly bounds the \(p\)-th moment of a squared-mass sum by the Gaussian \(2p\)-th smear moment. The latter is uniformly bounded for \(\lambda\ge\lambda_0\), since the Gaussian covariance decreases with \(\lambda\). ◻

Proof. Choose finite small-cell coverings \(i\) on the starting side and \(B\) on the target side, away from the band and with positive clearances. Enlarge overlapping cells slightly so every crossing path meets both coverings after rounding; see Figure 1. Concretely, first choose two separator strips, clipped to the free component, with a fixed clearance \(\eta>0\) from the band and the filled starting and target regions. Then cover them by cells of radius \(\delta\ll\eta\). A reaching path contains, in a fixed-factor enlargement of the touched cell, a segment spanning \(c\delta\). The path-Poincare constant on this rescaled patch depends only on the fixed finite face and corner geometry, not on \(\delta\); at a wall use the adjacent face-connected rectangular pieces of that same component. The primary path bound gives a uniform upper bound \(C_0\) for the centered remaining average in a reached cell. A logarithmic trial supported in a fixed ball away from the band gives continuum prior variance at least \(c\,a\log(1/\delta)-C\) for a cell of radius \(\delta\). Cut the trial off at a radius \(\rho<\eta/4\) fixed before \(\delta\). It pays no band mass. At a free wall or corner restrict it to the component; this is an admissible Neumann energy function. Its cell average is bounded below by \(c\log(\rho/\delta)-C\), and its energy is at most \(C\log(\rho/\delta)+C\), uniformly over the bounded cell shapes. Thus shrinking the cells increases the prior lower bound without increasing the path-anchoring constant. Choose the covering cells so small that the prior variance exceeds \(C_0+2c_0\), for some \(c_0>0\), uniformly in \(\lambda\). At each fixed \(\lambda\), band convergence gives the strict gap \(c_0\) on fine meshes.

For exterior exploration write \(h_B=e_B+u_B\), exposed plus unexposed. The centered posterior is obtained by zero-pinning visited vertices and increasing cable precision. Thus its variance for \(u_B\) is at most the full prior variance \(v_B\), and at most \(v_B-c_0\) on a visit to \(B\). Restoring the tilt changes the averaged second moment and cross term by \(o(1)\), by the good- and bad-tilt estimates above. The identity \(\mathbb E_Wh_B^2=v_B\) therefore gives \(c_0\mathbb P_W(B\text{ reached})\le\mathbb E_We_B^2+o(1)\). All contributing clusters are macroscopic, with a threshold fixed by the separation from the starting set. Sign resampling, with that same signless selection, yields \[ c_0\mathbb P_W(B\text{ reached}) \le\mathbb E_W\sum_{C\text{ meeting the starting set}}M_C(B)^2+o(1). \tag{35}\]

For the second deficit, bias the prior by \(|h_x|\), where \(x\in B\), and expose only its cluster. The pre-exposure second moment at \(i\) is at least \(v_i\) by (34). The bias is measurable at completed exploration, so cancels from the conditional complement law. Its centered remaining variance is at most \(v_i\), and at most \(v_i-c_0\) when the cluster visits \(i\). Multiply the total second-moment comparison by \(\mathbb E|h_x|\) and the primary averaging weight of \(x\), then sum over \(x\in B\). This produces precisely the weight \(M_C(B)\), not a count of microscopic clusters. Only clusters reaching the band or \(i\) cause a nonzero tilt or exposed term; the separating gaps make all of them macroscopic. The estimates preceding the proposition give \[ c_0\mathbb E_W\sum_{C:C\cap i\ne\varnothing}M_C(B) \le\mathbb E_W\sum_C M_C(B)M_C(i)^2+o(1). \tag{36}\] Points with \(\mathbb E|h_x|=0\) contribute zero.

As \(\lambda\to\infty\), the inverse forms \(T_\lambda^{-1}\) decrease to the inverse with every energy function zero on \(K\). Weak energy compactness and recovery by such functions prove this statement directly. The hard band separates the sides, so its cross covariance is zero. Equation (30) therefore makes \(\mathbb E\sum_CM_C(B)M_C(i)\) vanish in the prescribed order. Truncate first at \(\max_CM_C(i)\le M\); on this event the right side of (36) is at most \(M\) times that cross sum. Its remaining tail vanishes uniformly as \(M\to\infty\), since \[\sum_CM_C(B)M_C(i)^2\le\sqrt{Q_B}\,Q_i\] and Lemma 15 supplies arbitrarily high moments uniform toward large \(\lambda\). Consequently the first-mass sum on the left of (36) vanishes. Every starting-set cluster meeting \(B\) visits one of the finitely many cells \(i\). Truncate \(\max_CM_C(B)\) in (35) and use these first-mass bounds; its tail is controlled by the moments of \(Q_B\). This proves vanishing reach.

All exploration calculations were on finite subdivided Gaussian graphs. Their primary bounds are uniform in large \(N\), or are passed first at a fixed primary graph. Thus no continuum-time cable realization of the limiting Bessel law is needed. ◻

Proof of Theorem 9. On a component meeting \(p\), start with a smooth shift \(g\) vanishing near the entire pin but allowed to meet the free walls. Choose a separating band between \(p\) and its support, with fixed gaps. The band may be a union of strips clipped to the component. Use the ordinary mixed-domain geometry from BKTref@an:mixed-scaling [12]

; Lemma 12 already verifies its reference inequalities at faces and junctions.

For the weak Neumann Laplacian test of \(g\), the band variance is \(a\mathcal E_U(g)\). Explore from every primary vertex of \(p\). On success, the exposed test is zero and the centered posterior has at least the desired primary pin, possibly further zero pins, tighter chains, and restricted band precision. Its variance is therefore at most that under the desired massless hard-\(p\) law. The tilt errors vanish as above. On failure the actual second moment is bounded by its fourth-moment root times the square root of the failure probability. Lemma 11 bounds that fourth moment uniformly after the first two limits and toward large \(\lambda\). Proposition 16 therefore gives \[\liminf_{n\to\infty}\mathop{\mathrm{Var}}_p(h,A g_n) \ge a\mathcal E_U(g).\] Run the argument on an arbitrary primary subsequence. The free variance upper bound gives the opposite limsup. The free logarithmic transform upper bound and the centered-variance lower bound under real tilt then give the matching Gaussian Laplace limit.

Weak free-face terms may first be approximated by bounded density smears in primary reference cost, preserving their gap from \(p\) and the band. The real mixed inverse-form approximation and density of smooth shifts vanishing near \(p\), from BKTref@an:forms [12]

, extend the result to all asserted tests. Lemma 5 controls the errors independently of subdivision. It supplies uniform integrability for every fixed linear exponential and polynomial insertion.

Unpinned components were handled before Lemma 11. Finally integrate a fixed finite-rank quadratic penalty against these massless limits. On pinned components the preceding moment bounds suffice. On unpinned components sum translations and use the independent uniform fractional mean, with the confining penalty controlling the constants exactly as in the band-density proof. This proves the theorem, including its limit order and its statements for rounded geometry sequences. ◻

Proof of the remaining assertions of Theorem 10. Let \(X_n\) be a fixed real smear, \(m_n(b^p)=\mathbb E(X_n\mid h_p=b^p)\), and \(v_{p,n}=\mathop{\mathrm{Var}}(X_n\mid h_p=0)\). At prescribed primary data, the subdivided law is an affine Gaussian lattice law with the same quadratic part as at zero data. Lemma 4, including an additional real tilt, and then its fixed-graph limit give \[ \mathop{\mathrm{Var}}(X_n\mid h_p=b^p;\text{ tilt }tX_n)\ge v_{p,n}. \tag{40}\]

Choose the first \(J\) smooth density probes on \(p\), and let \(L_n^{(J)}\) be their linear combination with the limiting Gaussian regression coefficients. Remove redundant directions or invert on their Gram-matrix range. Conditional variance decomposition gives the exact bound \[\begin{align*} \mathbb E|m_n-L_n^{(J)}|^2 &=\mathbb E|X_n-L_n^{(J)}|^2 -\mathbb E\mathop{\mathrm{Var}}(X_n\mid h_p)\\ &\le \mathbb E|X_n-L_n^{(J)}|^2-v_{p,n}. \tag{41}\end{align*}\] Theorem 9 identifies both limiting variances. As \(J\to\infty\), totality of the pin probes makes the right side tend to zero: in the Gaussian energy space it is the squared difference between the full pin projection and the \(J\)-probe projection of the representer of \(X\). Thus the conditional mean is approximated in \(L^2\) by the finite Gaussian linear predictions.

Integrating (40) twice in \(t\) gives \[Y_n:=\mathbb E(e^{X_n}\mid h_p) \ge\exp\{m_n+v_{p,n}/2\}.\] Both sides have limiting expectation \(e^{v/2}\), where \(v\) is the full limiting variance. For the right side, use (41) and uniform exponential integrability: \(\mathbb Ee^{q m_n}\le\mathbb Ee^{qX_n}\) by Jensen for every fixed real \(q\). Their nonnegative difference tends to zero in \(L^1\), proving the conditional Laplace assertion. Apply the same argument to a countable dense set of linear combinations of any fixed test list. Subsequence extraction and conditional exponential moment bounds give tightness and identify all conditional finite-dimensional weak limits. This also proves conditional convergence of fixed smooth quadratic penalties.

For the pin ratio, precision comparison at prescribed \(b^p\) shows that every normalized conditional positive-penalty ratio is at most one and decreases when the penalty is increased. Letting a full coordinate pin on \(H\) tend to infinity yields \(0<r_{H,n}\le1\). Its definition gives exactly \(\mathbb Er_{H,n}=e^{-D_n}\).

Let \(Q_J\) be increasing finite smooth quadratic penalties exhausting the pin on \(H\), and put \[R_{J,n}(b^p)= \frac{\mathbb E(Q_J\mid h_p=b^p)}{\mathbb E(Q_J\mid h_p=0)}.\] Then \(r_{H,n}\le R_{J,n}\le1\). For fixed \(J\), the conditional weak convergence and the centered hard-\(p\) limit identify \(R_{J,n}\) with its Gaussian counterpart. Moreover, \[ \mathbb E(R_{J,n}-r_{H,n})=\mathbb ER_{J,n}-e^{-D_n}. \tag{42}\] The interaction limit (39) and the finite-probe determinant approximation in BKTref@an:gaussian-localization [12]

make the right side tend to zero as \(J\to\infty\) after \(n\to\infty\).

More explicitly, let \(s_j\) be the singular values of the cross projection between the Gaussian pin spaces of \(p\) and \(H\). Positive separation gives \(\sup_js_j<1\) and \(\sum_js_j^2<\infty\). In canonical standard Gaussian pin coordinates \(\xi_j\), \[r_H^{\mathrm g} =\exp\left\{-\frac12\sum_j \frac{s_j^2}{1-s_j^2}\xi_j^2\right\}>0 \quad\hbox{almost surely},\qquad \mathbb Er_H^{\mathrm g}=\prod_j\sqrt{1-s_j^2}=e^{-D^{\mathrm g}}.\] Finite probes exhaust these convergent expressions. Markov’s inequality in (42) therefore proves the asserted finite-observation approximation of the full ratio. In particular \[ \lim_{\eta\downarrow0}\limsup_{n\to\infty} \mu_n(r_{H,n}\le\eta)=0. \tag{43}\]

Finally a fixed finite linear tilt has uniformly bounded \(L^q\) density for every fixed finite \(q\): its denominator is at least one, and the mixed Laplace bounds control its numerator’s moments. Holder transfers the preceding convergence statements to the tilted law. Here \(r_{H,n}\), \(D_n\), and the conditional functions being sampled keep their centered definitions; in particular we do not claim \(\mathbb E_{\mu_{n,L}}r_{H,n}=e^{-D_n}\). If a conditional transform under the tilted baseline is needed, use \[\mathbb E_{\mu_{n,L}}(e^X\mid h_p) =\frac{\mathbb E_{\mu_n}(e^{X+L}\mid h_p)} {\mathbb E_{\mu_n}(e^L\mid h_p)}.\] The denominator has a strictly positive Gaussian limit, so the established joint conditional convergence applies. This completes the theorem. ◻

Proof. For each restored edge \(e=(x,y)\), enlarge the cut law by an independent normalized chain of increments \(\xi_{e,1},\ldots,\xi_{e,N}\in2\pi\mathbb Z\) in height units, and put \(J_e=\sum_k\xi_{e,k}\). Restoring the edge imposes \(J_e=h_y-h_x\). Conditional on primary heights, its factor is the chain-convolution probability \(p_{b,N}((h_y-h_x)/(2\pi))\), tending to the Bessel weight. Thus the primary factor need not itself be quadratic. On the expanded lattice, matching is the limit of the positive quadratic penalties \(\exp[-t(J_e-h_y+h_x)^2/2]\).

Write \(Z_{b^p}\) and \(Z_0\) for the expanded cut sums at the two guard data, and use a superscript \(\Delta\) for restored matching and any further restored positive forms. The guard fibers are affine and centered copies of the same expanded lattice, and \[r_\Delta(b^p)= \frac{Z_{b^p}^{\Delta}/Z_{b^p}} {Z_0^{\Delta}/Z_0}.\] Precision comparison gives \(r_\Delta\le1\); adding the full primary \(H\)-pin can only decrease this ratio. With both endpoints pinned, every new chain matches \(J_e=0\). Its integrated factor is the scalar \(p_{b,N}(0)>0\), independent of guard data. Hence \(Z_{b^p}^{\Delta+H}=\kappa_N Z_{b^p}^{H}\) and \(Z_0^{\Delta+H}=\kappa_N Z_0^{H}\), with the same \(\kappa_N\). The resulting normalized ratio is exactly \(r_{H,n}\), proving its lower bound. All independent-chain normalizers were identical in these sums, and both they and \(\kappa_N\) cancel. Internal chain variables therefore need not be pinned. Take matching penalties to infinity and then the fixed-primary-graph subdivision limit.

Bayes’ formula gives the middle identity in (44). Its denominator lies in \([e^{-D_n},1]\), proving the last two bounds. The interaction has a finite limit, so the upper density bound transfers \(\mu_{n,p}(A_n)\to0\) to \(\nu_{n,p}(A_n)\to0\). Conversely, for \(\eta>0\), \[\mu_{n,p}(A_n) \le\mu_{n,p}(r_{H,n}\le\eta) +\eta^{-1}\nu_{n,p}(A_n).\] Use (43), sending \(\eta\downarrow0\) after the lattice limit. Fixed linear tilts transfer by Holder and the bounds for both positive and negative fixed exponential moments. All these statements concern the finite-observation sense at typical data specified above. ◻

Proof. We separate the centered normalization, the random guards, and the local gain.

Proof of Proposition 18. Choose \(M\), then arbitrarily large fixed \(R\), and then sufficiently large fixed \(m_*\) as in Proposition 19. Combining (46), (57), and (58) gives, uniformly in sufficiently distant outer boundaries, \[\log \mathbb E_{\mu_{L,b}}\cos(\theta_0-\theta_{ne_1}) \ge -\frac{2\pi}{a}\log n -C_M(1+\log n)\eta_R-O_{m_*,R,M,b}(1).\] For each fixed \(n\), first take the prescribed free-box limit \(L\to\infty\). Divide by \(\log n\), and then take the liminf as \(n\to\infty\). All hole and insertion constants were fixed before this limit and disappear. Finally let \(R\to\infty\), obtaining the claimed lower exponent. Since \(0<C_b(n)\le1\), this polynomial lower bound also implies \(m(b)=0\). ◻

Proof. For fixed \(t\), the additional energy is a finite-rank quadratic tilt of finitely many subcell averages. Theorems 9 and 10 give its limit on each fixed pin set. Integration against the nonnegative quadratic weight is justified by its boundedness and the confining observation masses; fixed linear tilts are handled by the exponential moment bounds. Free component constants are treated by the joint uniform-phase limit before summing their confining translations.

The full pin interaction also transforms by finite-dimensional quantities. If \(W\) is the added quadratic weight, and subscripts indicate centered conditioning on the designated pins, then \[ D_W(H,p)-D(H,p) =\log\mathbb E_{H,p}W+\log\mathbb EW -\log\mathbb E_HW-\log\mathbb E_pW . \tag{63}\] Each expectation has a strictly positive Gaussian limit. This proves, at fixed \(t\), convergence to the pin interaction for the tilted continuum Gaussian precision, not merely convergence of unconditioned smears.

In observation-cell units, let \(P_t\) denote averages over squares of side \(1/t\), and \(B\) averages over the unit observation cells. The continuum form obtained in this first limit is \[ T_t(f)=a^{-1}\int|\nabla f|^2 +\delta\sum_{\langle i,j\rangle} \bigl((P_tf)_i-(P_tf)_j\bigr)^2 +\lVert Bf\rVert_2^2 . \tag{64}\] It converges in inverse forms to \[ T(f)=a'^{-1}\int|\nabla f|^2+\lVert Bf\rVert_2^2, \tag{65}\] also with any of the specified zero-region pins. Here is the form argument, including its normalization. If \(\zeta=1/t\), the averaged-gradient term is a sum of \(\zeta^2|D_\zeta P_tf|^2\). Bounded \(T_t\)-energy gives weak \(H^1\) and strong \(L^2\) compactness on each fixed component. The averaged difference quotients converge weakly to the directional derivatives on its interior, giving the lower bound with coefficient \(\delta\). Conversely, smooth functions give the matching upper bound by Riemann sums. The averaged-gradient term is uniformly bounded by a constant times continuum gradient energy, by the two-cell Poincare inequality. Energy approximation therefore extends recovery to all admissible functions, using corner cutoffs and density with zero neighborhoods of the pins, as in BKTref@an:forms [12]

. No component is glued through a point. This proves the claimed inverse-form convergence and, in particular, convergence of all finite covariance and pinned inverse matrices.

We also need convergence of the infinite-dimensional interaction determinant. Form convergence alone would not give it. We verify the uniform compactness used in BKTref@an:gaussian-localization [12]

for the additional averaged term. For an energy-harmonic function off a pin or source, test its equation against \(\chi^2f\), where \(\chi\) is a smooth cutoff in a separating margin. On a subcell replace \((P_t(\chi^2f))_i\) by \(\chi_i^2(P_tf)_i\). The error satisfies \[ \left|(P_t(\chi^2f))_i-\chi_i^2(P_tf)_i\right| \leq C\bigl(t^{-1}|\chi_i|+t^{-2}\bigr) (P_tf^2)_i^{1/2}. \tag{66}\] Pairing this error with each incident averaged bond and applying Cauchy–Schwarz absorbs the term with \(|\chi_i|\) into a fixed part of the positive weighted bond energy. The remaining \(t^{-2}\) terms cost local \(L^2\) mass: use \[|(P_tf)_i-(P_tf)_j| \leq(P_tf^2)_i^{1/2}+(P_tf^2)_j^{1/2}\] and the bounded number of incident bonds. The differences of the cutoff values themselves are controlled by the discrete product inequality. All bounds are uniform for large \(t\).

The unit-cell observation term may cross an intermediate cutoff. On the fixed geometry its forcing consists of finitely many bounded cell-average coefficients. Carry these coefficients as finite-dimensional coordinates through the successive restriction maps. Their bounds follow from the original energy bound. The remaining terms are local on an enlarged margin for large \(t\). We consequently obtain the same nested gradient-from-\(L^2\) estimates as for the form with unit-cell mass alone, with this fixed finite-rank enlargement.

Two broken-domain \(H^1\)-to-\(L^2\) compact embeddings, each with singular numbers \(O(j^{-1/2})\), separated by these harmonic interior estimates, give a uniform \(O(j^{-1})\) singular-value tail for the cross projections between the separated pin regions. A fixed finite-rank enlargement changes only finitely many singular directions. Bounded-energy sequences harmonic off one pin have weak limits harmonic off that pin for \(T\): test their equations against smooth functions, whose averaged differences converge strongly. For a weakly null sequence, compact \(L^2\) convergence and the interior estimate give strongly vanishing energy near the other pin; the observation averages vanish as well.

For completeness, identify all the varying energy spaces with the same broken \(H^1\) space \(V\). The forms \(T_t\) are uniformly equivalent there, since the original gradient and observation terms are a lower bound and the averaged-gradient term has a uniform upper bound. Let \(V_{0,H}\) be the fixed subspace of functions zero on \(H\). Its \(T_t\)-orthogonal complement \(U_H^t\) is the source projection space. If \(\phi\) is a smooth density supported in \(H\), orthogonality of \(v_t\in U_H^t\) to the \(T_t\)-Riesz representative of \(\phi\) is exactly \(\int v_t\phi=0\), independently of \(t\).

Take \(T_t\)-unit vectors in \(U_H^t\) orthogonal to increasing finite lists of these probes. Every weak limit vanishes on \(H\) and is \(T\)-harmonic off \(H\), by the smooth-test passage above. It lies both in \(V_{0,H}\) and its \(T\)-orthogonal complement, so is zero. The finite observation forcing coordinates are the actual averages \(Bv_t\); strong \(L^2\) convergence makes them zero too. For a cutoff \(\chi_p\) equal to one near the opposite pin, the local estimate gives \(T_t(\chi_pv_t)\to0\). Moreover \(\lVert P_p^tv_t\rVert_{T_t}\leq\lVert \chi_pv_t\rVert_{T_t}\), since \(v_t-\chi_pv_t\) vanishes on \(p\). This proves uniform finite-probe approximation of the cross projection. The same argument applies in reverse, and the finite-probe Gram matrices converge by inverse-form convergence. Combined with the uniform square-summable singular tails, these facts give convergence of the cross determinants. The limiting cross projection has norm strictly smaller than one, as in BKTref@an:gaussian-localization [12]

; operator convergence preserves a norm gap. In particular the logarithmic determinant defining \(D(H,p)\) converges.

The conditional conclusions now follow by the regression and minimal-Hessian argument used in Theorem 10, with the covariance and determinant just obtained. Conditional Laplace transforms are first approximated using finitely many smooth guard probes. The full pin ratio follows by increasing finite core penalties and using the centered identity \(\mathbb Er_H=e^{-D(H,p)}\). For use in both neighboring restoration topologies, take a common finite probe list. The bound \(r_H\leq r_\Delta\leq1\) from Lemma 17, and strict positivity of the Gaussian limiting \(r_H\), transfer negligible exceptional sets in both directions. Fixed real tilts preserve convergence for these same centered-defined quantities by the exponential moment bounds; the centered expectation identity is not asserted under tilted sampling.

Every geometry and probe list is fixed before refinement. Compact-data uniformity follows by taking convergent sequences of observation vectors and using the same exponential integrability. Operationally, given an error tolerance one first chooses a sufficiently large \(t\), then a sufficiently large \(s_0\) depending on \(t\). No single refinement threshold uniform over all \(t\) is required. ◻

Proof. On the subdivided quadratic lattice, complete the square in the joint height and observation variables. Centered theta domination bounds the shifted sum by the centered sum times the exponential of minus half its real minimum energy. Discard the nonnegative microscopic chain energy in that minimum, retaining the added average energy and \(\lVert B_sh-y\rVert_2^2\). We show that this retained energy dominates \(c_*E_G(y)\) for every real trial height.

Write \(u_i=(B_{s_0}h)_i\). On the union of two adjacent \(t\)-by-\(t\) subcell squares, the ordinary grid Poincare inequality gives \[ |\overline u_{\rm left}-\overline u_{\rm right}|^2 \leq \frac{2}{t^2}\sum_i|u_i-\overline u|^2 \leq C\sum_{\langle i,j\rangle}(u_i-u_j)^2. \tag{68}\] The constant does not depend on \(t\). Apply this to each same-component observation face and sum with bounded overlap. The observation residuals then show that \[ \sum_{\langle x,x'\rangle\ {\rm in}\ G}(y_x-y_{x'})^2 \leq C_\delta\left\{ \delta\sum_{\langle i,j\rangle}(u_i-u_j)^2 +\lVert B_sh-y\rVert_2^2\right\}. \tag{69}\] Only retained faces within a component occur.

Conversely, \(E_G(y)\) is at most a constant times the left side of (69). One explicit trial interpolates the values \(y_x\) at scale \(s\), with central plateaus and a piecewise affine subdivision joining cell centers, face midpoints, and vertices. At a midpoint or vertex use the local average within each incident face-connected sector. Both its gradient energy and its observation error are bounded by the nearby face-difference sum. There is no bond requirement between opposite-only sectors. These local estimates are uniform under the permitted cuts, and agree under lattice refinement. They prove the required comparison of minima.

The constants do not use the microscopic chain precision or the value of \(\widetilde b\). Pass to the fixed-graph Skellam limit using Lemma 4. If necessary decrease \(c_*\) so that \(c_*\leq1\). This gives (67) uniformly in the stated parameters. ◻

Proof. We give the entry estimates, especially the places where the physical law differs from the quadratic source. The choices are made in this order: the downstream norm and regulator parameters and imaginary strip widths; \(q,R,P\); a finite large-field threshold; \(t\); \(s_0\) depending on \(t\); and finally the interval \(I\).