Proof. The lattice-Gaussian inequality underlying these comparisons is due to Regev and Stephens-Davidowitz [16]; related extensions to annealed integer-valued Gaussian fields appear in [1]. The statements at fixed \(b\) are [14]. We recall the approximation which also gives the last assertion. Replace each primary edge by \(N\) bonds in series, with normalized lattice-Gaussian bond weight \((b/(2N))^{k^2}\). On compact sets of real \(t\) its transform is \[\frac{\sum_k(b/(2N))^{k^2}e^{tk}} {\sum_k(b/(2N))^{k^2}} =1+\frac bN(\cosh t-1)+O(N^{-2}).\] The full chain therefore converges to (9), including all fixed exponential moments. Matching endpoints, cycle constraints, period constraints, and the additional observations are positive quadratic forms or limits of such forms on the enlarged lattice. The finite lattice-Gaussian comparison inequalities apply there. Increasing \(b\) decreases the bond precision \(2\log(2N/b)\), proving the required monotonicities before passage to the limit. A spanning-forest parametrization and full-chain exponential moments dominate the sums on each fixed primary graph. This justifies the limit, including fixed polynomial insertions and real tilts. Confined constants are dominated by their observation penalties; unconfined constants are factored out. ◻

Proof. Release the centered endpoint, curl, period, and observation constraints in the full-chain construction of Lemma 2. The comparison bounds the transform by a product of independent whole-chain transforms. Their limiting logarithm is \(b\sum_e(\cosh(2\pi tw_e)-1)\), which is bounded by the right side of (14) when the arguments are bounded. The same bound holds uniformly in sufficiently fine subdivision. For a flow normalized by its Euclidean norm every coefficient has modulus at most one. Fixed exponential moments then give (13) and the normalized transform bound. This is the argument of [14]; its constants are uniform on \(I\). ◻

Proof. Expand \((N)_\ell\) as the sum over ordered choices of \(\ell\) Poisson units, recording their edges and signs. Removing those units leaves a partition sum with an integral edge connection and the same primary height variables, pins, and observation penalties. In particular the remaining Poisson differences obey the shifted curl and period constraints determined by that connection. Every removed unit contributes \(b/2\). Lemma 2 bounds each shifted sum by its centered zero-observation counterpart. Summing the \((2M)^\ell\) choices proves the assertion. Repeated choices of the same edge cause no difficulty: falling factorials are exactly the factors canceled by removing several units from its Poisson factorial. ◻

Proof. The parameter likelihood in the augmented variables is exactly \[ \exp\{-(b-b_0)M\}(b/b_0)^N. \tag{17}\] Expand the second factor in falling factorials. For real observations, Lemma 4 bounds the absolute sum of its positive-order terms, divided by \(Z_{b_0}^0\), by \[\sum_{\ell\ge1}\frac{|b-b_0|^\ell}{\ell!b_0^\ell}(b_0M)^\ell =e^{|b-b_0|M}-1.\] The scalar exponential in (17) has the same bound. For \(\eta=\eta_1+i\eta_2\), the absolute observation penalty is its real-\(\eta_1\) penalty times \(e^{\|\eta_2\|^2/2}\), uniformly bounded on \(K\). This proves (16). At zero observation it gives \(|Z_b^0/Z_{b_0}^0-1|<1/2\) after decreasing \(c\), hence nonvanishing. All sums are analytic on compact parameter sets by the same expansion. ◻

Proof. Let \(C_b\) be the covariance operator on mean-zero tests. For each oriented edge \(e\), removing one positive Poisson unit gives the shifted height weight \[\prod_{d\in E}p_b\bigl(\nabla_dh/(2\pi)-\mathbf 1_{\{d=e\}}\bigr).\] Let \(r_e\) be its partition sum divided by the centered sum, and \(v_e\) its mean of \(X\). Reversing \(h\) identifies the negative-removal law with its reflection, so its partition ratio is \(r_e\) and its mean is \(-v_e\). By Lemma 2, \(0<r_e\le1\) and each shifted variance is at least \(\mathop{\mathrm{Var}}_b(X)\).

Differentiating the normalized augmented law inserts the score \(N/b\); the deterministic term \(-|E|\) cancels. Since the centered mean of \(X\) is zero, \[\partial_b\mathop{\mathrm{Var}}_b(X)=b^{-1}\mathop{\mathrm{Cov}}_b(X^2,N).\] Removing a positive or negative unit separately now gives the exact identity \[ \partial_b\mathop{\mathrm{Var}}_b(X) =\sum_{e\in E}r_e\bigl(\mathop{\mathrm{Var}}_{e,+}(X)-\mathop{\mathrm{Var}}_b(X)+v_e^2\bigr) \ge\sum_{e\in E}r_ev_e^2. \tag{19}\] The same removal with the insertion \(X\) gives \[\mathbb E_b[(U_e-V_e)X]=br_ev_e.\] As \(\nabla_eh=2\pi(U_e-V_e)\), this is \[ \nabla_e(C_bf)=2\pi br_ev_e. \tag{20}\] Discrete integration by parts and Cauchy–Schwarz yield \[\begin{split} \mathop{\mathrm{Var}}_b(X)^2 &=\langle\nabla A_G^+f,\nabla C_bf\rangle^2\\ &\le G(f)\sum_e|\nabla_eC_bf|^2 \le4\pi^2b^2G(f)\sum_er_ev_e^2. \end{split}\] Together with (19), this proves the lower bound. For the upper bound, Lemma 4 gives \(\mathbb EN^2\le(bM)^2+bM\), while Lemma 3 gives \(\mathbb EX^4\le C_IG(f)^2\). Cauchy–Schwarz in the score identity proves \(|\partial_b\mathop{\mathrm{Var}}_b(X)|\le C_IMG(f)\), since \(M\ge1\) and \(b\in I\). The finite-volume analyticity established above justifies every differentiation. No bound on the inverse precision of a subdivided Gaussian chain is used. ◻

Proof. We first describe the parts of the finite expansion whose estimates have to be extended. The geometry and algebra are those of [15]; only their physical weights change with \(b\). The observation cells are partitioned into tiles of side comparable to \(R_0\). A finite colored sequence of isolation events deletes the remaining bonds between tiles. At an event with open patch \(o_i\) and cut patch \(c_i\), its proxy is \[ \mathcal B_i(b,\eta)=C_i(b) e^{-(E_{o_i}(\eta)-E_{c_i}(\eta))/2},\qquad C_i(b)=\frac{Z_{o_i,b}(0)}{Z_{c_i,b}(0)}. \tag{34}\] One writes the original factor as its proxied cut factor plus their difference. Selecting the difference marks that event. Later events within its fixed neighborhood are skipped. After the finite sequence, undeleted intertile bonds lie in the union of the marked neighborhoods. We call each connected part of this union a halo. Outside the halos, an isolated tile contributes either one or its normalized factor minus one. The latter choice is a selected tile.

The remaining Gaussian mismatch is expanded in links \(e^{b_\ell w_fw_{f'}}-1\), with \[ |b_\ell|\le Ce^{-c(R_0+d_\ell)}. \tag{35}\] Here \(d_\ell\) records the entire tagged path length, including winding on a torus. A link covers at most \(C(1+d_\ell/R_0)\) entry blocks. An individual pattern \(\pi\) consists of marked events, selected tiles, and links, with their complete occupied cover \(X(\pi)\). Write \(l(\pi)\) for the number of marks and selected tiles. Each of these physical items uses at most \(C_{R_0}\) observation cells and \(C_{R_0}s^2\) primary edges. Link-only cells carry no physical height factor. All identical factors off the affected halos cancel before the pattern is defined. Its denominator is a product of the affected local scalars \(C_i(b)\) and isolated-tile normalizers \(Z_{Q,b}(0)\). In particular, a centered partition function of an arbitrarily large halo is not a denominator of a defining pattern.

We keep every Gaussian energy in (34) fixed. Thus the finite algebra, the link coefficients, the occupied covers, and the copying identities remain valid with the actual parameter in every physical factor and scalar normalizer. It remains to bound and sum the resulting patterns.

Proof. The exact activity map and its uniform norm bounds are [15]; the triangular form is [15]. We recall the coordinate construction to specify the derivative bounds being used. Put \(\ell_j=\prod_{h\ge j}\lambda_h=1+O(e^{-cj})\) and \(\mathsf C_{h:j}=\mathsf C_{h-1}\cdots\mathsf C_j\). The balanced linear coordinates are \[T=t+\sum_{h\ge j}P_h^t\mathsf C_{h:j}\mathcal R, \qquad Z=\ell_jz+\sum_{h\ge j}\ell_{h+1}P_h^z\mathsf C_{h:j}\mathcal R.\] The series converge geometrically. At the chosen entry, initialize an exact Gaussian interaction \(\exp(d\sum_Be_B^0/2)\) and iterate the same map. Its balanced gradient is \(T_j^g(d)=d+O(d^2)\) and its remainder is \(\mathcal R_j^g(d)=O(d^2)\), uniformly on a common complex disk. Define \(u=(T_j^g)^{-1}(T)\) and \(Q=\mathcal R-\mathcal R_j^g(u)\), as in [15]. This gives (46); in particular its linear remainder rows are bounded, while derivatives of its quadratic scalar terms are \(O(|u|)\).

The parity estimates of [15] give leading scalar terms \(-H_jZ^2\) and \(-b_juZ\), with no nonconstant term on the Gaussian curve. Their remainders have the differentiated orders used in (47). Moreover [15] gives \(|H_j-H|+|b_j-b_*|\le C(1+j)^{-2}\). Rescaling by (45) yields the displayed map. Terms with a small coefficient multiplying \(Q\) are absorbed into a slightly larger contraction constant. The copying assertion and the separate torus contraction are part of the same local-map statement; no equality between plane and torus remainders is required. ◻

Proof. These are the terminal analytic functionals of [15]. Their analyticity does not require a scaling limit for the physical input. Indeed, on the fixed \(D_0^2\)-block grid the Gaussian and common-mean integral of the terminal subset sum is \(1+O(D_0^2\nu_j)\), uniformly on a fixed complex source disk. Choose the activity ball so that this error is less than \(1/2\). Both normalized numerator and denominator are then analytic and the denominator is nonzero.

Completing the Gaussian source \(wX_{n,s}\) before entry averaging gives the factor \(e^{v^0_{n,s}w^2/2}\) and translates the interaction by a microscopic cosine \(F(w)=w\gamma_{n,s}\cos(2\pi v_1/n)\). Writing \(\beta_{n,s}\) for the cell-average cosine factor, \[B_s\cos(2\pi v_1/n)=\beta_{n,s}\cos(2\pi x_{Q,1}/n),\qquad \gamma_{n,s}=\frac{2v^0_{n,s}}{\alpha\beta_{n,s}},\qquad \beta_{n,s}=1+O((s/n)^2).\] The linear gradient singleton contributes \(t_j\langle F,AF\rangle/2\) to the logarithm; mean integration removes the first harmonic at linear order. Since \(\langle\cos(2\pi v_1/n),A\cos(2\pi v_1/n)\rangle =2n^2\sin^2(\pi/n)\), this gives the first line and (51). The remainder costs its norm and all higher terms cost \(O(\nu_j^2)\).

For the phase insertion the common-mean integral instead selects the fundamental singleton. Its coefficient is \[p_{j,s}=\frac{D_0^2}{2} \exp\{\tau_{j,s}/2-aD_{{\rm tail},j}/2\}>0,\] where \(D_{{\rm tail},j}\) is the diagonal of the retained mean-free covariance and \(\tau_{j,s}\) is the variance, in height units, of the integrated high mean including the observation noise. Both are bounded on the fixed terminal geometry. At the same absolute stop the high covariances telescope to the same constant-mode continuation. The only entry-dependent contribution to \(\tau_{j,s}\) is the noise variance \((s/n)^2\). This proves the second assertion and the comparison of multipliers. Cauchy estimates on a smaller activity ball give all stated derivatives. ◻

Proof. The cell-flow proof of (29) also bounds the probability that a prescribed face difference exceeds \(s/2\) by \(Ce^{-cs^2}\), uniformly in the compact \(b\) interval. On its complement, \(1-\chi_s\) has that bound. Summing over faces and applying Hölder with the uniform reference moments gives the unnormalized loss for the phase and the first two moments of \(X_{n,s}\), as in [15]. The cutoff mass is consequently \(1-o(1)\) after normalization by the uncut law.

To differentiate a normalized expectation, insert the Poisson score \(N/b\). Lemma 4 bounds every fixed moment of \(N\) by a fixed power of \(n\), uniformly on the parameter interval. Apply Hölder to the product of this insertion, the observable, and the cutoff loss. The face tail retains a bound \(e^{-c's^2}\), while the other factors cost a power of \(n\). Differentiating the cutoff normalizer has the same estimate because it is bounded away from zero. This proves (52) for derivatives. Integrating the derivative bound proves the divided-difference version. ◻

Proof. The spatial mean of \(B_sh\) is exactly the microscopic mean. The independent mean noise has variance \((s/n)^2\), proving the phase identity. The profile \(B_s^*f_{n,s}\) projects onto the microscopic first cosine with coefficient \(\beta_{n,s}\) and has no first sine component, by reflection around arithmetic cell centers. The remaining Fourier modes have frequency-index distance at least \(cn/s\) from zero. Their total squared coefficient norm is at most \(Cn^{-2}(s/n)^2\), whereas \(A^+\) restricted to these modes has norm at most \(Cs^2\). Their reference cost is therefore \(O((s/n)^4)\). Translation invariance of the physical torus law makes distinct Fourier modes exactly orthogonal. Thus the variance is the first-cosine variance plus the alias variance, without a cross term. The reference bound controls the latter variance. Lemma 6, applied to the alias test, controls its derivative by its reference cost times \(|E|=2n^2\). The factor \(\beta_{n,s}^{-2}\) is uniformly bounded and independent of \(b\). Integration gives the divided-difference estimate. ◻

Proof. The family and its confinement through \(4\cdot2^k\) are [15]. We explain both the second-coordinate precision and the extension to longer critical histories.

In the proof of that proposition, let \(Y_j=|Z_j|\) for the spliced critical histories. Equations (135)–(137) of [15] give \[u_j\asymp Y_j\asymp j^{-1},\qquad u_j=\sqrt{H/b_*}\,Y_j+O(j^{-2}),\qquad u_j=(b_*j)^{-1}+O(\log(2+j)/j^2).\] The overlap bounds in [15] transfer both coordinates between histories with \(O(j^{-2})\) error on the indicated fractional ranges. The phase formula [15], together with positive physical characteristic coefficients and the exponentially small cutoff loss, selects \(Z_j>0\) there. Rescaling by (45) gives (54). The plane remainder bound is part of the same proposition and its odd cone estimate. For a larger torus, its separate contraction is driven by the copied scalar sequence. Contracting from an earlier fractional range suppresses its initial contribution exponentially and sums an \(O(j^{-2})\) forcing, giving the asserted bound at every intermediate stage.

For the last assertion, the physical input to critical confinement is [15]. It gives the critical torus Gaussian first-cosine limit and factorization with the mean phase. For any fixed tolerance, first \(P_0\) and then \(s\) can be chosen so that the variance and mixed phase discrepancies are below that tolerance for every compatible \(n\ge sP_0\). No rate of convergence in \(n\) is asserted or needed. The mixed phase functional has a nonzero multiplier on \(z\), as shown in [15]. The variance multiplier in Proposition 9 is nonzero as well.

Suppose there were a first exit from a tube of radius \(\rho\). Before that exit both plane and torus remainder contractions give \[\|\mathcal R_j\|+\|\mathcal R^{\rm tor}_{j|j}\| \le C\theta^{j-j_0}\rho_{\rm ent}+C\rho^2,\] where \(\rho_{\rm ent}\) is the entry norm. At the candidate exit the bounded map leaves all states inside a fixed multiple of the tube, so the terminal observables are still defined. The variance and mixed phase identities then bound the two singleton coefficients by \(C(\rho_{\rm ent}+\rho^2+\delta_{\rm phys}+\delta_{\rm cut})\). Here \(\delta_{\rm phys}\) is the uniform critical discrepancy just described. The number of initial indices with \(n_j<sP_0\) is at most \(\max(0,\lceil\log_L(R_0^4/(D_0q))\rceil)\), independent of \(s\); their amplification is absorbed by the small entry norm. Finally, uniformly for \(j\le J_{\max}(s)\), \[\delta_{\rm cut}\le Cn_j^Ce^{-cs^2} \le \exp\{O(J_{\max}(s))-cs^2\}\longrightarrow0.\] Choose \(\rho\) small, then the entry and physical discrepancies small relative to \(\rho\), and finally \(s\) large. The preceding bound excludes the first exit. Separate torus contractions give the same conclusion at all earlier stages of each valid stopped torus. This is the finite-history argument of [15] with precisely the larger range permitted by the cutoff estimate. ◻

Proof. We omit the history superscript until we compare two entries. The entry analyticity in Proposition 7 gives an initial derivative bound \(Cs_k^2\). Propagation in the chosen tube bounds all plane derivatives by \(Cs_k^2e^{\gamma j}\); contraction of a torus remainder driven by the same scalars gives the weaker common bound \(Cs_k^2e^{2\gamma j}\). These estimates are used only to remove the initial remainder contribution.

On a slightly larger fixed fractional interval than that in the statement, Proposition 12 gives \[W_j=\frac{(1,1)}j+O\!\left(\frac{\log j}{j^2}\right),\qquad \|Q_j\|+\|\mathcal R_{j|h}^{\mathrm{tor}}\|=O(j^{-2}).\] Here \(h\ge j\) is any valid stop within the history’s horizon. Differentiating the map of Proposition 8 therefore gives \[ V_{j+1}=A_jV_j+B_j\partial_b Q_j, \qquad A_j=I+\frac{\mathsf M}{j} +O\!\left(\frac{\log j}{j^2}\right), \qquad \|B_j\|\le\frac Cj, \qquad \mathsf M=\begin{pmatrix}0&-2\\-1&-1\end{pmatrix}. \tag{57}\] For a fixed torus stop \(h\), put \[r_{j|h}=\|\partial_bQ_j\| +\|\partial_b\mathcal R_{j|h}^{\mathrm{tor}}\|.\] The analytic remainder estimates, the absence of linear forcing from the scalar inputs into the unbalanced torus remainder, and the coordinate changes in Proposition 8 imply \[ r_{j+1|h}\le\theta_1r_{j|h}+\frac Cj|V_j|. \tag{58}\] Terms with a small coefficient multiplying either remainder derivative have been included in \(\theta_1\). The constants are independent of \(h\).

We give the backward estimate that makes this contraction useful. Fix \(j\) in the asserted interval and take \(l=\lfloor j/2\rfloor\). For large \(k\), the interval \([l,j]\) lies inside the slightly enlarged fractional range on which the preceding bounds hold. Define \[S_v=\max_{l\le i\le j}r^{j-i}|V_i|, \qquad S_q=\max_{l\le i\le j}r^{j-i}r_{i|h}.\] Iterating (58), and using \(i\asymp j\) on this interval, gives \[ S_q\le\frac CjS_v+Cr^{j-l}r_{l|h}. \tag{59}\] For large \(j\), the matrices \(A_i\) are invertible and \(\|A_i^{-1}\|\le1+C/j\) in a fixed equivalent scalar norm. Solving (57) backward from \(j\), multiplying at index \(i\) by \(r^{j-i}\), and summing the geometric factors \(r^{v-i}\) over intermediate indices \(i\le v<j\), gives \[ S_v\le C|V_j|+\frac CjS_q. \tag{60}\] Absorbing \(CS_q/j^2\) between these inequalities yields \[r_{j|h}\le\frac Cj|V_j|+Cr^{j-l}r_{l|h}.\] The crude initial derivative estimate and the choice of \(\gamma\) make the last term exponentially small on these fractional intervals. Consequently \[ r_{j|h}\le\frac Cj|V_j|+e^{-cj}. \tag{61}\] The same proof without the torus term gives the plane assertion.

We next use the physical meaning of the two scalar coordinates. The height part of the first-cosine observation has variance bounded below by a positive constant on the critical tori and has bounded reference cost. Lemma 6 thus gives a fixed positive lower bound for its temperature derivative. Lemma 10 allows us to use the cutoff readout with an exponentially small error. Differentiating Proposition 9, using (61) and the coordinate changes, bounds this response by \(C|V_j|+e^{-cj}\). In particular \[ |V_j|\ge c>0. \tag{62}\] This lower bound is the step that will exclude a purely decaying solution of the linearized recurrence.

To compare two entries, define the normalized diagnostic vector \[ \mathcal{O}_{j,s}(b)= \begin{pmatrix} \bigl(\widehat\mathop{\mathrm{Var}}_b X_{n_j,s} -\|f_{n_j,s}\|_2^2\bigr)/\beta_{n_j,s}^2\\[2pt] e^{(s/n_j)^2/2}\widehat\mathbb E_b e^{i\bar\eta} \end{pmatrix}. \tag{63}\] The subtracted term is the independent observation-noise variance in the uncut law. Lemma 11 compares this vector with the same physical pair \((\mathop{\mathrm{Var}}_b^{\mathrm{tor}}H_{n_j}, \mathbb E_b^{\mathrm{tor}}e^{i\bar h})\) for either entry. At zero activity, its linear scalar coefficient in the \(W_j,Q_j\) coordinates is \[\operatorname{diag}\left( \frac{\kappa_{n_j,s}}{b_*\beta_{n_j,s}^2}, \frac{e^{(s/n_j)^2/2}p_{j,s}\ell_j^{-1}}{\sqrt{b_*H}}\right).\] Here \(\ell_j=1+O(e^{-cj})\) is the balancing factor in the proof of Proposition 8. The linear remainder rows are controlled by (61). Its constant background is independent of \(b\). Differentiating and using (61) gives \[ D_jV_j+O(|V_j|/j)+O(e^{-cj}), \qquad D_j=\operatorname{diag}\left( \frac{a\mathfrak g}{b_*}, \frac{p_j}{\sqrt{b_*H}}\right). \tag{64}\] The numbers \(p_j\) and their inverses are uniformly bounded and positive. At a common absolute stop this diagonal matrix may be chosen identically for both entries: the difference of the actual multipliers is exponentially small and is included in the error. The variance comparison error is bounded by \(Cn_j^2(s/n_j)^4\); the phase comparison is exact after removal of the mean noise, apart from the cutoff error. All these errors are exponentially small when the entries being compared have \(s\asymp j\). Since both readouts describe the same finite physical torus, (64) and (62) give \[ |V_j^{(k)}-V_j^{(k')}| \le \frac Cj\bigl(|V_j^{(k)}|+|V_j^{(k')}|\bigr) \tag{65}\] whenever the index lies in the fixed interior ranges for both histories. In particular their norms are comparable. Only two physical scalar readouts were compared; no subtraction of remainders belonging to different activity norms is involved.

Define a single sequence by taking \(V_j=V_j^{(k)}\) for \(2^k\le j<2^{k+1}\). At an ordinary step, (57) and (61) show that this sequence obeys \[V_{j+1}=\left(I+\frac{\mathsf M}{j}+E_j\right)V_j, \qquad \|E_j\|\le C\frac{\log j}{j^2}.\] Indeed an error vector of norm at most \(\eta|V_j|\) can be written as a matrix of norm at most \(\eta\) applied to \(V_j\). The nonvanishing bound permits this representation also for the exponentially small errors. At a change of entry there is, by (65), an additional matrix error of norm \(C/j\).

The matrix \(\mathsf M\) has eigenvectors \(q_u=(1,-1/2)\) and \(q_s=(1,1)\), with eigenvalues \(1\) and \(-2\), respectively. Let \((U_k,S_k)\) be the coordinates of \(V_{2^k}\) in this basis. Telescoping products over a dyadic interval gives \[ \begin{pmatrix}U_{k+1}\\S_{k+1}\end{pmatrix} =\left\{ \begin{pmatrix}2&0\\0&1/4\end{pmatrix}+\mathsf E_k \right\} \begin{pmatrix}U_k\\S_k\end{pmatrix}, \qquad \|\mathsf E_k\|\le Ck2^{-k}. \tag{66}\] For completeness, the unperturbed growing product is exactly \(\prod_{j=n}^{2n-1}(1+1/j)=2\), and the decaying product is \(\prod_{j=n}^{2n-1}(1-2/j)=1/4+O(n^{-1})\). All partial products within one dyadic interval and their inverses are bounded. Summing \(C\log j/j^2\) over the interval, together with its one \(C/j\) change of entry, proves the stated bound on \(\mathsf E_k\).

For all sufficiently large \(k\), the cone \(\{|S_k|\le|U_k|\}\) is invariant under (66). If a sequence never enters this cone, then \(|U_k|<|S_k|\) at every sufficiently large index, and (66) gives \(|S_{k+1}|\le\tfrac12|S_k|\). Both coordinates would tend to zero, contrary to (62). Thus the sequence enters the cone. Thereafter \[U_{k+1}=2U_k(1+\eta_k),\qquad |\eta_k|\le Ck2^{-k}.\] The factors \(1+\eta_k\) are positive on the tail and their logarithms are absolutely summable. It follows that \(U_k/2^k\) converges to a finite nonzero number \(\mathcal P\). The second row of (66), divided by its first, gives \[\frac{|S_{k+1}|}{|U_{k+1}|} \le \rho\frac{|S_k|}{|U_k|}+Ck2^{-k} \quad\text{for some }\rho<1,\] so this ratio tends to zero. The same product estimates on partial dyadic intervals show that \(V_j/j\to\mathcal Pq_u\) at every intermediate index. Applying (65) then gives the uniform assertion for either history on its interior range. Finally the first component of (64) has positive physical response. Its leading term would be negative and of order \(j\) if \(\mathcal P<0\). Hence \(\mathcal P>0\). ◻

Proof. The extended critical confinement in Proposition 12 keeps all these states in the chosen small tube. Put \(i_0=\lceil J/100\rceil\). At every \(i\ge i_0\), compare the history with a canonical history \(\mathcal H^{(k)}\) for which \(2^k\le i<2^{k+1}\). Both are exact descriptions of the same physical torus of side \(n_i\). The first choice in (55) ensures \(s/n_i\le e^{-ci}\) throughout this range. The observation sampling errors and multiplier differences are consequently exponentially small. The cutoff error is uniformly at most \[C\exp\{CK(s)-cs^2\},\] and is also exponentially small in \(i\), since \(K(s)=o(s^2)\). The canonical history has the bounds of Proposition 12.

Write \(d_i=|t_i|+|z_i|\), and let \(R_i\) be the maximum of \(\|\mathcal R_i\|\) and the remainder norms at stage \(i\) of all the separate tori with stops in \([i,K(s)]\). The two nonzero scalar multipliers in Proposition 9 give \[ d_i\le\frac Ci+CR_i+Cd_i^2. \tag{68}\] In writing this inequality we absorbed remainder squares and mixed terms using the uniform small tube. Remainder contraction, with copied scalar forcing also on each torus, gives \[ R_{i+1}\le\theta_1R_i+Cd_i^2. \tag{69}\] The family of tori in the maximum only decreases when \(i\) increases, so the same inequality holds for the maximum itself.

Absorb \(Cd_i^2\) into the left side of (68). Then \(d_i\le C/i+CR_i\), and, since \(R_i\) is uniformly small, (69) becomes \[R_{i+1}\le\theta_2R_i+\frac C{i^2} \quad\text{for a fixed }\theta_2<1.\] Iteration from \(i_0\) shows that \(R_i\le C\theta_2^{i-i_0}+C/i^2\). For \(i\ge J/2\) the first term is at most \(C/i^2\), uniformly for all later \(i\), when \(s\) is large. Thus \(R_i=O(i^{-2})\) and \(d_i=O(i^{-1})\).

Repeat the readout comparison with these improved bounds. Each nonlinear and remainder error is now \(O(i^{-2})\). Consequently the two unbalanced scalar coordinates agree with those of the canonical history to \(O(i^{-2})\). Each coordinate change to \(W,Q\) differs from its linear part by \(O(i^{-2})\), by Proposition 8. The canonical critical estimates therefore give (67). ◻

Proof. Entry analyticity and propagation in the small tube first give, through stage \(J\), the crude bound \[ \text{distance between the states at \(b_c-\varepsilon\) and \(b_c\)} \le C\varepsilon s^2e^{2\gamma J}. \tag{72}\] This statement uses each history’s own norm. The torus bound follows by contracting its remainder with the same scalar inputs. Since \(\varepsilon\asymp s^{-3}\) and \(2\gamma C_*<1\), the right side decays as a negative power of \(s\); in particular it is smaller than every fixed negative power of \(J\). Induction with a fixed margin to the boundary of the tube justifies the propagation up to \(J\).

Denote the scalar secants by \(\widetilde V_i\), with denominator \(-\varepsilon\). On \([J/2,J]\), Lemma 14 and (72) allow us to integrate the analytic derivatives of the map along the segment between its two input states. This gives the same form of scalar and remainder inequalities as (57) and (58). The extra coefficients from (72) are exponentially small in \(J\), and hence fit inside their displayed errors. The crude bound for any initial remainder secant is \(Cs^2e^{2\gamma J}\). The weighted estimates (59)– (60), now on a fixed-fraction interval ending at \(J\), therefore give \[ \widetilde r_J\le\frac CJ|\widetilde V_J|+e^{-cJ}. \tag{73}\] Indeed the homogeneous contribution is bounded by a constant times \(r^{c_1J}s^2e^{2\gamma J}\). The choices of \(C_*\) and \(\gamma\) make its exponent strictly negative; taking, for example, an interval of length \(J/4\) within \([J/2,J]\) suffices. The estimate is uniform for the torus remainders behind later stops.

To identify this secant, choose a canonical history with \(s_k\asymp J\) and \(J\) in its fixed interior range. In that history we can compare the secant directly with its derivative. For all stages through \(J\), analytic continuation in the parameter is valid and bounded on a disk of radius \[\rho_J=cJ^{-2}e^{-2\gamma J}\] about \(b_c\). To see this, the entry displacement is bounded by \(Cs_k^2\rho_J\), and its amplification through \(J\) is bounded by \(Ce^{\gamma J}\); the resulting displacement tends to zero. The same argument and contraction control the separate torus states. Cauchy’s estimate bounds second parameter derivatives of the stopped states by \(C\rho_J^{-2}\). Since \(\log\varepsilon=-3J/C_*+O(1)\), \[ \varepsilon\rho_J^{-2} \le CJ^4\exp\{(-3/C_*+4\gamma)J\} \longrightarrow0. \tag{74}\] Thus the canonical scalar secant is \(V_J^{(k)}+o(1)\), with analogous absolute control of its remainder secants. This is the only use of a second temperature derivative; it is taken in the history with entry side comparable to \(J\).

Compare the two histories through their physical readout secants. Lemma 11 and Lemma 10 apply uniformly on the real parameter segment \([b_c-\varepsilon,b_c]\), so their first-derivative bounds apply also to the divided differences. Taylor expansion of \(\mathcal{O}_{J,s}\) from (63) on the segment between each history’s two temperature states, followed by (73), gives \[ D_J\widetilde V_J =D_JV_J^{(k)} +O\!\left( \frac{|\widetilde V_J|+|V_J^{(k)}|}{J}\right)+o(1), \tag{75}\] with the same uniformly invertible diagonal matrix as in (64). The multipliers and observation errors are exponentially small because the first condition in (55) also gives \(n_J^2(s/n_J)^4\le e^{-cJ}\).

Proposition 13 gives \(|V_J^{(k)}|=O(J)\). Absorbing \(C|\widetilde V_J|/J\) in (75) shows \(|\widetilde V_J|=O(J)\), then \(\widetilde V_J-V_J^{(k)}=O(1)+o(1)\). Proposition 13 and (73) prove all the assertions. ◻

Proof. Set \(\Delta W_i=W_i(b_c-\varepsilon)-W_i(b_c)\) and define \(\Delta Q_i\) similarly. We first prove that, for a sufficiently small fixed \(\eta>0\), uniformly on \(J\le i\le\eta/\delta\), \[ \Delta W_i=-\varepsilon i\bigl(\mathcal Pq_u+O(\eta^2)+o(1)\bigr), \qquad \|\Delta Q_i\|\le C\varepsilon. \tag{80}\] Here the constant in \(O(\eta^2)\) is independent of sufficiently small \(\eta\), and \(o(1)\to0\) as \(\varepsilon\to0\). The critical history has the estimates of Lemma 14 through every fixed multiple of \(\delta^{-1}\), by (70).

Bootstrap the estimate \(|\Delta W_i|\le C_1\varepsilon i\). The differentiated remainder bounds in Proposition 8 give \[\|\Delta Q_{i+1}\| \le\theta_1\|\Delta Q_i\| +C_{C_1}\bigl(i^{-1}\varepsilon i+(\varepsilon i)^2\bigr).\] The second term is \(O_{C_1}(\varepsilon)\) on the indicated range. Lemma 15 and contraction therefore give \(\|\Delta Q_i\|\le C_{C_1}\varepsilon\). Subtract the two scalar recurrences and linearize their leading quadratic terms about the critical trajectory. The result is \[ \Delta W_{i+1} =\left(I+\frac{\mathsf M}{i}\right)\Delta W_i+e_i, \qquad |e_i|\le C_{C_1}\left( \frac{\varepsilon\log i}{i}+\frac\varepsilon i+\varepsilon^2i^2\right). \tag{81}\] The first error uses the \(O(\log i/i^2)\) critical correction and the differentiated cubic terms. The second uses \(\|\Delta Q_i\|=O(\varepsilon)\) and the \(O(i^{-1})\) scalar remainder row. The last is the quadratic term in \(\Delta W_i\).

In the eigenbasis of \(\mathsf M\), the unperturbed products from \(h\) to \(i\) have norm at most \(Ci/h\), and send \(q_u\) to \((i/h)q_u\) exactly. Variation of constants in (81) and Lemma 15 therefore gives, after division by \(\varepsilon i\), a total error bounded by \[o(1)+C_{C_1}\sum_{h\ge J}\frac{\log h+1}{h^2} +C_{C_1}\varepsilon\sum_{h\le i}h \le o(1)+C_{C_1}\left(\frac{\log J}{J}+\eta^2\right).\] Choose \(C_1\) larger than the leading bound, then choose \(\eta\) small. The improved inequality closes the bootstrap. The trajectory stays within the chosen tube throughout this argument, by the same bound and the critical estimates. This proves (80). The unbalanced remainder on every torus behind a stop of order \(\delta^{-1}\) satisfies its corresponding bounds by the same contraction with copied scalars.

For the continuous-scale limit, linearly interpolate \(x_j/\delta,y_j/\delta\) at times \(w=\delta j\). On every compact subinterval of \((0,\eta]\), (80) and the critical estimates bound these interpolants uniformly and give \(Q_j=O(\delta^2)\). The map in Proposition 8 then gives bounded slopes and, uniformly on such a subinterval, the limiting equations \[ X'=-Y^2,\qquad Y'=-XY. \tag{82}\] Indeed the scalar map error is \(O(\delta^3)\) there, while a step has length \(\delta\) and the scalar coordinates were divided by \(\delta\). Its contribution to the slope tends to zero. Compactness and integration of the discrete equations therefore show that every subsequential limit solves (82).

Applying (80) with arbitrarily small \(\eta\) determines its behavior at the left endpoint: \[ (X(w),Y(w)) =\frac{(1,1)}w -w\bigl(\mathcal Pq_u+O(w^2)\bigr) \qquad(w\downarrow0). \tag{83}\] The equations preserve \(X^2-Y^2\), whose value is \(-3\mathcal P\) by (83). Thus \(X'=-(X^2+3\mathcal P)\). Its pole at zero and the positive sign of \(Y\) near zero give uniquely \[X(w)=k_0\cot(k_0w),\qquad Y(w)=k_0\csc(k_0w).\] Every subsequential limit is the same, so convergence holds on compact subintervals of \((0,\eta]\).

We extend it to each compact subinterval of \((0,T)\). Starting from any fixed time in \((0,\eta)\), choose a bound strictly larger than the supremum of the displayed solution on the target compact interval. While the discrete scalars divided by \(\delta\) remain within this bound, remainder contraction gives \(\|Q_j\|=O(\delta^2)\), so the scalar error in the Euler approximation tends uniformly to zero. The quadratic vector field is Lipschitz on this bounded set. The discrete Gronwall inequality then gives uniform convergence up to the end of the interval and excludes a first exit from the larger bound. This proves (76), including the analogous torus remainder bounds behind these stops.

It remains to control the final part of the trajectory, where the scaled solution diverges but the unscaled activity is still small. Fix a time \(w_0\in(T/2,T)\). Starting at \(j_{\mathrm{start}}=\lfloor w_0/\delta\rfloor\), let \(j_e\) be the first index with \(-x_{j_e}\ge d\), provided such an index exists. We prove its existence and the estimates in (78) using a cone on which both scalar coordinates have a definite direction of motion.

Take a large fixed \(M>k_0\) and let \(w_M\in(T/2,T)\) be the outgoing time at which \(Y(w_M)=M\). At an index with \(\delta j\to w_M\), compact convergence gives \[y_j=(M+o(1))\delta, \qquad -x_j/y_j=1+O(M^{-2})+o(1).\] The remainder estimate at this index can be made \[ \|Q_j\|+\|\mathcal R_{j|h}^{\mathrm{tor}}\| \le K M^2\delta^2 \tag{84}\] with \(K\) independent of large fixed \(M\). Here \(h\) can be any later stop up to a fixed sufficiently large multiple of \(\delta^{-1}\). To verify the independence, use the fixed scaled interval on which the limiting outgoing \(Y\) increases from \(M/2\) to \(M\). Throughout it \(|W|\le CM\delta\) for all sufficiently small \(\delta\), with an absolute \(C\), and its number of discrete steps tends to infinity. Contraction bounds the forced remainder by \(CM^2\delta^2\) and removes the contribution from the earlier state. The threshold for \(\delta\) may depend on \(M\), as is appropriate when \(M\) is fixed before taking \(\delta\to0\).

For sufficiently small coordinates consider \[ y>0,\qquad \frac12\le q:=-x/y\le2, \qquad \|Q\|+\|\mathcal R^{\mathrm{tor}}\|\le K_1y^2. \tag{85}\] Choose \(K_1\) sufficiently large for the remainder contraction. In this cone the scalar map gives \[\begin{align*} y_+&=y\{1+qy+O_{K_1}(y^2+j^{-2}y)\}, \tag{86}\\ q_+-q&=y(1-q^2)+O_{K_1}(y^2+j^{-2}y). \tag{87}\end{align*}\] At each endpoint of \([1/2,2]\) the leading term in (87) points inward. It also points uniformly inward on fixed small neighborhoods of the endpoints. Away from those neighborhoods a sufficiently small step cannot cross either endpoint. Thus the ratio condition is preserved for small \(y\) and large \(j\). Equation (86) gives constants \(0<c<C\) such that \[y+cy^2\le y_+\le y+Cy^2.\] Remainder contraction preserves the last condition in (85): its forcing is at most \(C'y^2\), so choose \(K_1>C'/(1-\theta_1)\), and use \(y_+\ge y\). This applies to the plane remainder and separately to every stopped torus under consideration.

For large fixed \(M\), (84) places the trajectory in this cone. Choose \(d\) small enough that the cone estimates hold until the first crossing of \(-x=d\), including that crossing. Before it, \(y\le2d\), and reciprocal summation gives \[\frac1{y_i}-\frac1{y_{i+1}} =\frac{y_{i+1}-y_i}{y_iy_{i+1}}\ge c'>0.\] It follows that the exit occurs within \(C/(M\delta)\) further steps: otherwise the last inequality would force \(1/y_i<0\). At the crossing, the change of \(x\) is \(O(y^2)=O(d^2)\). The cone and remainder bounds therefore give the last three assertions in (78).

Compact convergence on \([w_0,T-\zeta]\), for any fixed \(\zeta>0\), shows that no such exit occurs there when \(\delta\) is small. Hence \(\liminf\delta j_e\ge T\). The preceding upper bound gives \[\limsup_{\delta\downarrow0}\delta j_e\le w_M+C/M.\] Since \(w_M\uparrow T\) as \(M\to\infty\), this proves \(\delta j_e\to T\). Throughout this argument all unscaled coordinates remain small; no activity map is used beyond its domain. The fixed horizon of stopped tori can be taken larger than \((T+1)/\delta\), and the exit lies before it for small \(\delta\).

Finally apply the variance readout of Proposition 9. At a stop with \(\delta j\to w\in(0,T)\), the scalar activity is \(O(\delta)\) and the remainder is \(O(\delta^2)\), so the variance converges to \(a\mathfrak g\). At the exit, the coordinate change gives \[t_{j_e}=-\frac d{b_*}+O(d^2),\qquad \|\mathcal R_{j_e|j_e}^{\mathrm{tor}}\|+\nu_{j_e}^2=O(d^2).\] Here \(\nu_{j_e}\) is the full stopped activity size. The readout multiplier tends to \(a\mathfrak g\), and therefore \[\mathop{\mathrm{Var}}_{b,n_{j_e}}^{\mathrm{tor}}H_{n_{j_e}} =a\mathfrak g-\frac{a\mathfrak g}{b_*}d +O(d^2)+o_{\varepsilon\downarrow0}(1).\] To pass from the cutoff observation to \(H_n\), remove the independent noise and use the sampling comparison and cutoff bound. Their errors tend to zero both before and at the exit, since \(\log n_j=O(\delta^{-1})=o(s^2)\) and \(s/n_j\to0\). Taking \(d\) sufficiently small gives (79), for example with \(c_d=a\mathfrak g d/(2b_*)\) after decreasing its allowed range. ◻

Proof. For positive rational \(q\), set \(m_k(q)=\lfloor L^{q/\delta_k}\rfloor\). A diagonal subsequence gives a limit of \(a_{m_k(q)}(b_k)\) for every such \(q\). If \(q<q'\), then \(m_k(q)/m_k(q')\to0\) and \(m_k(q)\to\infty\). Equation (89) orders the two limits. Denote these rational limits by \(\alpha(q)\) and define their right-limit regularization by \[\mathcal A(w)=\sup_{q>w,\ q\in\mathbb Q}\alpha(q).\] This is nonincreasing and right continuous. Values at rational jump points need not be preserved; only continuity points will be used.

For \(q_-<w<q_+\), a sequence as in the statement eventually lies between \(m_k(q_-)\) and \(m_k(q_+)\). Applying (89) in these two directions bounds the limsup and liminf of \(\sqrt{a_{m_k}(b_k)}\) by the values at \(q_-\) and \(q_+\). Letting these rationals tend to the continuity point \(w\) proves (90).

Centered covariance increases with \(b\), as follows either from the chain approximation or from the finite comparisons of Section 2. Thus \(a_m(b_k)\le a_m(b_c)\). The critical height coefficient is \(a\) by [14], and \(a_m(b_c)\to a\) by [14]. Consequently \(0\le\mathcal A\le a\). ◻

Proof. We give the moving-temperature version of the cutting and folding argument in [14]. Its one infinite-size affine comparison is replaced below by a finite macroscopic enclosure.

First consider an affine shift \(g(x)=p\cdot x\). Tile most of \(U\) by squares of a small fixed macroscopic side, leaving strips of arbitrarily small total area. The coefficient of every retained square tends to \(a_*\) by (90). Releasing the cuts, using independence of the resulting laws, and bounding the strip flows by reference cost give \[ \limsup_k\mathop{\mathrm{Var}}(h,A_{U_k}g_k)\le a_*|U|\,|p|^2. \tag{92}\] The strip bound is sent to zero after \(k\to\infty\).

To prove the reverse inequality, place \(U\) strictly inside a fixed square \(S\) and partition \(S\setminus U\) into finitely many rectangles of positive width. Use the actual \(U_k\) endpoints for the complementary cuts; their rescaled lengths converge to the prescribed positive lengths. The full affine observation on \(S_k\) is the sum of the affine observations on these pieces and \(U_k\), apart from seam flows of reference cost \(o(1)\). Its variance tends to \(a_*|S|\,|p|^2\). Cutting increases covariance; each piece has the upper bound (92). The sum of those upper bounds is exactly \(a_*|S|\,|p|^2\). Hence the bound must be attained on each piece. Otherwise a subsequence with a fixed deficit on one piece, together with the upper bounds on the others, would contradict the limiting variance on \(S_k\). This proves affine saturation on \(U\). All enclosing lengths are asymptotic to fixed positive multiples of \(n_k\).

For a smooth or continuous piecewise smooth \(g\), approximate its gradient on small fixed macroscopic rectangles by constant vectors. The seams have vanishing reference cost; the interior error is bounded by its squared \(L^2\) gradient error. Letting the rectangles shrink after the mesh limit gives \[ \limsup_k\mathop{\mathrm{Var}}(h,A_{U_k}g_k)\le a_*\int_U|\nabla g|^2. \tag{93}\] The same estimate holds for partial gradient observations restricted to polygonal pieces. Explicitly, for a piece \(H\) the partial test is \[J_{H,k}(g)=\sum_{\substack{\{x,y\}\in E(U_k)\\x,y\in H_k}} (g_k(y)-g_k(x))(h_y-h_x),\] where \(H_k=\{x\in U_k:x/n_k\in H\}\), and its energy bound is \(a_*\int_H|\nabla g|^2\). Edges crossing a fixed cut contribute a vanishing reference-cost error. Call equality saturation. On a convergent joint covariance subsequence, the difference between the quadratic form on the right of (93) and the limiting covariance is positive semidefinite. Its null space is linear. Saturation therefore passes to linear combinations, to \(L^2\)-gradient limits, and to restrictions obtained by cutting an enclosing rectangle and comparing the sum of the energy bounds.

For completeness, the finite reflection step used to generate nonaffine saturated shifts has no temperature-limit hypothesis. Work first in a rectangle symmetric about the reflection line. Choose the representative with one height on the line fixed at zero, and let \(\mathcal T\) be the sigma-field generated by its full trace. For an odd saturated shift, denote by \(Y_+\) and \(Y_-\) the partial tests on the two open sides after transporting exceptional coefficients to the line in a reflection-symmetric manner. Suppose that the off-line coefficients of \(Y_+\) are nonnegative; those of \(Y_-\) are then nonpositive. The subdivided quadratic law is conditionally associated, and this property passes to the primary graph. Consequently \(\mathop{\mathrm{Cov}}(Y_+,Y_-\mid\mathcal T)\le0\). Reflection fixes the prescribed trace and exchanges the two sides, while oddness changes the sign of the test. Thus the conditional means satisfy the exact identity \[\mathbb E(Y_-\mid\mathcal T)=-\mathbb E(Y_+\mid\mathcal T).\] The total covariance is therefore \[\mathop{\mathrm{Cov}}(Y_+,Y_-) =\mathbb E\mathop{\mathrm{Cov}}(Y_+,Y_-\mid\mathcal T) -\mathop{\mathrm{Var}}\bigl(\mathbb E(Y_+\mid\mathcal T)\bigr)\le0.\] The transport has vanishing reference cost, so its contribution to covariances tends to zero by the uniform second-moment bound. The saturated sum and the separate energy bounds (93) then force both original partial tests to saturate. An odd shift vanishes on its reflection line, so the weak gradient of \(g\mathbf 1_H\) is \(\mathbf 1_H\nabla g\). Its lattice gradient test differs from \(J_{H,k}(g)\) only by the stated vanishing-cost seam. Thus the folded shift itself saturates.

Here are the shifts and transports from [14]. For a hinge at \(t\), use a fixed enclosing rectangle symmetric about \(x_1=t\) and the odd affine seed \(g=x_1-t\). Transport the negative coefficients adjacent to that vertical cut to the cut. There are \(O(n_k)\) of them, each of size \(O(n_k^{-1})\), on bounded-length paths of bounded overlap. Their total squared cost is \(O(n_k^{-1})\). This yields the hinges \((x_1-t)_+\); restriction gives their saturation on the target rectangle. Linear combinations and energy approximation then give every smooth function of \(x_1\), and likewise of \(x_2\). For a diagonal cut write \(Y=x_2+t\) and start with \(F(x_1)-F(Y)\) on a square symmetric under interchanging \(x_1,Y\). Initially impose \(F'''<0\) and positive derivatives at the two end faces. The interior coefficients in \(x_1>Y\) are nonnegative because the centered second differences of \(F\) decrease with their center; the two outer-face coefficients have positive leading terms. Exceptional diagonal-strip coefficients again have total transport cost \(O(n_k^{-1})\). We obtain saturation of \[ (F(x_1)-F(x_2+t))\mathbf 1_{\{x_1>x_2+t\}}. \tag{94}\] Every smooth \(F\) is a difference of two functions with those strict properties, by adding a sufficiently large cubic of negative third derivative and then a sufficiently large positive linear function. Thus (94) holds for arbitrary smooth \(F\). Rounding a reflection line changes the squared gradient cost by \(O(n_k^{-1})\).

Given smooth \(g\), choose a bounded interval \(I\) containing all \(x_1-x_2\) in the target rectangle. Choose a smooth family \(F_t\), compactly supported in \(I\) after slightly enlarging \(I\), with \(F_t'(x)=-g_{12}(x,x-t)\) whenever \((x,x-t)\) lies in that rectangle. Integrate (94) over \(t\in I\). Indeed, differentiating first in \(x_1\) and then in \(x_2\) gives the distributional identity \[\partial_{12}\int_I (F_t(x_1)-F_t(x_2+t))\mathbf 1_{\{x_1>x_2+t\}}\,dt =-F'_{x_1-x_2}(x_1)=g_{12}(x_1,x_2).\] The difference from \(g\) is a sum of one-variable functions. Riemann sums converge in gradient energy, including across the moving reflection lines. Linearity and energy approximation therefore prove saturation for every smooth \(g\).

The finite centered comparison gives \(\log\mathbb Ee^X\ge\mathop{\mathrm{Var}}(X)/2\), providing the Laplace lower bound. For the upper bound, cut into cells of fixed macroscopic side \(r\). A retained smooth-gradient test \(X\) on one cell has reference-cost square root \(O(r)\). The normalized exponential moment estimate gives, for each fixed \(p>1\), \[\log\mathbb Ee^{pX}=\tfrac12p^2\mathop{\mathrm{Var}}(X)+O_p(r^3), \qquad \mathbb E(|X|^3e^{p|X|})=O_p(r^3),\] with constants uniform in \(k\). Hölder’s inequality and the same reference bound remove the seam errors. There are \(O(r^{-2})\) cells. First send \(k\to\infty\), then \(r\downarrow0\), and finally \(p\downarrow1\). This proves (91). Apply it to real linear combinations to obtain joint Gaussian limits; the exponential bounds supply convergence of moments.

For densities, approximate by finite sums of Neumann eigenfunctions and use their smooth inverse-Laplacian shifts. The discrete Neumann form approximation and Poincaré inequality bound the reference cost of the error by its squared scaled \(L^2\) norm. Using the actual rectangle endpoints before passing to their limits handles the rounding. Finally move a face probability to a parallel strip of width \(r\). Parallel normal flows cost \(O(r+n_k^{-1})\); the strip is a density smear. Let \(k\to\infty\) and then \(r\downarrow0\). ◻

Proof. We describe the soft joining and replica argument of [14], keeping its lattice and geometric limits separate. Start with a square of side \(m=m_k\) on the chosen logarithmic scale. Let \(u_m\) be its volume probability, \(\rho_m\) its right-face probability, and set \[X=(h,A_{B_m}(x_1/m)),\qquad Y=(h,\rho_m-u_m).\] Lemma 18 gives joint Gaussian limits, with \(\mathop{\mathrm{Var}}X\to a_*\) and \(\mathop{\mathrm{Cov}}(X,Y)\to a_*/2\). Suppose along a subsequence that \(\phi(lu_m)\) is bounded away from zero for a nonzero integer \(l\). Uniform exponential bounds provide a further subsequence on which \[F_m(t,v)=\mathbb Ee^{i(h,lu_m)+itX+ivY}\] converges locally with all derivatives. On real arguments the limit \(F\) is strictly positive by (95).

The replica identity is a finite quadratic-lattice identity. Let \(F_{m,N}\) denote the preceding transform with each primary edge replaced by \(N\) Gaussian chain bonds and the observations evaluated at the primary endpoints. For two independent fields under this subdivided law put \(S=h_1+h_2\), \(D=h_1-h_2\). Choose both representatives to vanish at the same reference vertex. At every vertex of the subdivided graph, \(S\equiv D\) modulo \(4\pi\mathbb Z\). Conditional on this full common parity class, \(S\) and \(D\) are independent and identically distributed. If \(V_{ij}\) is the conditional second moment of \(S_i,S_j\), for linear observations indexed by \(i,j\), then \[ \mathbb EV_{ij}=2\mathop{\mathrm{Cov}}(h_i,h_j),\qquad \mathop{\mathrm{Cov}}(V_{ij},V_{rs})=2\kappa(h_i,h_j,h_r,h_s), \tag{97}\] where all moments in this identity are in the subdivided law and \(\kappa\) is its fourth joint cumulant. This follows by expanding \(\mathbb ES_iS_jD_rD_s\) in the original two copies. Differentiating \(F_{m,N}\) in its observation variables gives \[\mathbb ES_iS_j e^{i(l(h_1-h_2,u_m)+tX(D)+vY(D))} =-2(F_{m,N}F_{m,N,ij}-F_{m,N,i}F_{m,N,j}).\] In the conditional expectation on the left, replace \(V_{ij}\) by its mean. The error is at most \(\sqrt{\mathop{\mathrm{Var}}V_{ij}}\). First take the chain limit on each fixed primary graph. Then the Gaussian moment limits in Lemma 18 make the needed fourth cumulants zero. For \(ij=XX,XY\) this yields \[(\log F)_{tt}=-a_*,\qquad (\log F)_{tv}=-a_*/2.\] Reflection gives \(F_t(0,0)=0\). Consequently \[ F(t,v)=F(0,v)\exp\{-a_*t^2/2-a_*tv/2\}. \tag{98}\]

Softly join two horizontal squares as follows. Keep independent seam increments with the Bessel law and put \(H_2=(h_1,\rho_R)-(h_2,\rho_L)-\bar\eta\), where \(\bar\eta\) is their mean with the joining orientation. Penalize this mismatch by \(e^{-H_2^2/2}\). Sum relative cell translations; quotient by their common translation. Adding exact seam matching recovers the joined rectangle. The soft variance of the sum of the internal affine tests is bounded above by twice the free square variance and below by its variance in the joined rectangle. The joined full affine test differs from this sum by a seam flow of squared cost \(O(m^{-1})\). Lemma 18 therefore makes the soft variance tend to \(2a_*\).

The comparisons just used with summed translations can be obtained from ordinary finite comparisons without reweighting the original free cells. First confine each cell mean by an auxiliary precision \(\zeta\sum_i\bar h_i^2\). For each representative mean \(\mu\), its translation sum is \(\sum_{q\in\mathbb Z}e^{-\zeta(\mu+2\pi q)^2/2}\). After division by its Gaussian integral scalar, this tends uniformly to one as \(\zeta\downarrow0\). After adding the soft penalty, divide out the common-translation scalar; the relative-translation Gaussian sum is uniformly bounded in its representative-dependent shift. The finite precision comparisons at \(\zeta>0\) therefore pass to \(\zeta=0\) by dominated convergence, using the uniform internal moments. This justifies both variance comparisons with the original free-cell product law.

Poisson summation of the relative translations expresses the soft characteristic transform as a positive sum of integer sectors, with weights \(e^{-q^2/2}\chi_m(q)\); here \(0<\chi_m(q)\le1\) is the independent seam characteristic. Explicitly, \(\chi_m(q)=\exp\{mb(\cos(2\pi q/m)-1)\}\), so \(\chi_m(q)\to1\) for fixed \(q\), uniformly in the temperatures here. To specify the sector functions, write \(F_m^{(q)}(t,v)=\mathbb Ee^{iq(h,u_m)+itX+ivY}\). If \(\rho_R,\rho_L\) are the two vertical face probabilities, reflection and charge reversal give \[\psi_q(t)=\phi(q\rho_R+tA(x_1/m)) \phi(-q\rho_L+tA(x_1/m)) =F_m^{(q)}(t,q)^2.\] Each sector satisfies \((\log\psi_q)''\ge-2\mathop{\mathrm{Var}}X\). The logarithmic Hessian of a positive sum is the mean of the sector logarithmic Hessians plus the variance of the sector logarithmic first derivatives. Thus the soft variance is at most twice the free variance minus this latter, nonnegative variance. Differentiation under the sector sum is justified uniformly: with \(V=2\mathop{\mathrm{Var}}X\), \[\psi_q| (\log\psi_q)'|^2 \le2V(-\psi_q\log\psi_q)\le2V/e,\] and raw second derivatives are bounded by \(V\). The Gaussian weights are summable.

At \(t=0\) the normalizing constant is \(Z_m=\sum_qe^{-q^2/2}\chi_m(q)\psi_q(0)\), and \(1\le Z_m\le\sum_qe^{-q^2/2}\) because the zero sector is one. Its normalized weight \(p_0\) is therefore bounded below and its logarithmic derivative is zero. The sector \(q=l\) also has normalized weight \(p_l\) bounded below: (95) moves \(lu_m\) to a seam face at bounded cost. By reflection, charge reversal and (98), its logarithmic derivative tends to \(-a_*l\), up to orientation. If \(D_q\) denotes the sector logarithmic derivative, their contribution bounds the variance of derivatives below by \(p_0p_l(D_l-D_0)^2\). Its liminf is strictly positive, contradicting convergence of the soft variance to \(2a_*\). Hence \(\phi(lu_m)\to0\). Equation (95) then proves (96) for square profiles of bounded cost.

For a fixed rectangle, enclose it in a fixed-multiple square. Regard its mean phase as a phase on the square. The difference from the square mean has bounded reference cost, so its square coefficient tends to zero. Cutting the complement decreases characteristic coefficients and leaves the rectangle phase on one free component; hence the rectangle coefficient also tends to zero. Bounded-cost phase transfers inside the rectangle give the stated generality. Mixed real characteristic functions with the mean-free observations now vanish whenever \(l\ne0\), and for \(l=0\) they have the Gaussian limit. Fourier approximation on the circle proves joint independence. Uniform exponential moments permit passage with any fixed real tilt and polynomial insertion in the mean-free observations. ◻

Proof. Write \(n=n_k\) and \(b=b_k\) in this proof, and index primary vertices by \(x\in nU\cap\mathbb Z^2\), with the permitted boundary rounding. For a continuum density \(f\), put \(f_n(x)=n^{-2}f(x/n)\); equivalent cell quadratures give the same limits. All discrete pairings use ordinary sums: \((h,f_n)=\sum_xh_xf_n(x)\), whereas \(g_n(x)=g(x/n)\). Probability averages, including cell averages, use weights equal to the reciprocal of their full primary counts. The unit-edge form is denoted by \(A_{U,n}\); its energy tends to \(\mathcal E_U\). All unspecified finite-graph constants below are uniform for \(b\in J\). We first prove the assertion for \[X_{g,n}=(h,A_{U,n}g_n),\] where \(g\) is smooth on the rectangle, may meet its free walls, and vanishes in a neighborhood of \(p\). The final approximation will use the form grounded on \(p\) to reach arbitrary densities.

Fix a closed neighborhood \(V\) of \(\mathop{\mathrm{supp}}g\), separated from \(p\), so that the primary support of \(A_{U,n}g_n\) lies in \(V\) for all sufficiently large \(k\). Choose a positive-width band \(K\) separating \(p\) from \(V\), with positive gaps on both sides. Such a band can be chosen as a finite union of rectangular strips, clipped at the free walls. Partition it into rectangular cells \(d\) of side comparable to \(\ell\), with uniformly bounded aspect ratios. If \(h_d\) is the probability average in \(d\), set \[ W_{\ell,\lambda}(h) =\exp\left\{-\frac\lambda2\sum_d|d|h_d^2\right\}, \qquad \widehat W_{n,\ell,\lambda} =\frac{W_{\ell,\lambda}}{\mathcal P_n(W_{\ell,\lambda})}. \tag{100}\] Here \(\mathcal P_n\) is the sigma-finite free measure obtained by normalizing the law modulo common \(2\pi\mathbb Z\)-translations and then summing all translations. The band confines its constant, so the denominator in (100) is finite and positive. We write \(\mathbb E_W\) for integration against \(\widehat W\,d\mathcal P_n\). Throughout the proof the order is \[ k\to\infty,\qquad \ell\downarrow0, \qquad \lambda\to\infty. \tag{101}\] In particular, every band partition and strength is fixed before the moving-parameter mesh limit.

We will prove that these successive band limits give variance \(a_*\mathcal E_U(g,g)\) for \(X_{g,n}\), and that this value is no larger than \(\liminf_k\mathop{\mathrm{Var}}_pX_{g,n}\). The free variance upper bound will then give equality. The comparison is the main task. The finite exploration argument below controls the linear tilts introduced by the band and proves that exploration from \(p\) reaches \(V\) with vanishing probability. When it does not reach \(V\), the exposed test is zero and the centered posterior variance is at most the hard-pin variance; the bounded band fourth moment controls the contribution when it does reach \(V\). We finish by squeezing Laplace transforms and using the grounded form to approximate density smears.

Band limits. At a fixed partition and strength, the assumed joint free limits turn translation summation into integration of the constant against \(dc/(2\pi)\). This passage also holds with every fixed polynomial insertion. Indeed, if \(S\) is the probability average over \(U\) and \(\bar h_K=|K|^{-1}\sum_d|d|h_d\), then \[W_{\ell,\lambda}\le e^{-\lambda|K|\bar h_K^2/2}, \qquad S-\bar h_K\text{ has bounded primary reference cost}.\] The reference-cost moment and fixed-linear-exponential bounds, uniform on \(J\), control this difference; the displayed Gaussian factor controls the sum over translations. They give uniform integrability of the normalizing constants and polynomial insertions. The limiting law is therefore the Gaussian with precision \(a_*^{-1}\mathcal E_U+\lambda\sum_d|d|v_d^2\).

The subsequent cell refinement is a statement about these Gaussian forms. Integrating their constant first leaves the covariance \(C_\ell\) of the cell projection of the mean-free Neumann field, after subtracting its \(K\)-average. The inverse Neumann Laplacian on a rectangle is Hilbert–Schmidt, so \(\sup_\ell\|C_\ell\|_{\mathrm{HS}}<\infty\). For \(q\ge1\), if \(t_j\) are the eigenvalues of \(\lambda C_\ell\), the determinant factor in the normalized \(q\)-th weight moment satisfies \[\frac12\sum_j\{q\log(1+t_j)-\log(1+qt_j)\} \le C_q\sum_jt_j^2.\] The constant integral is a bounded positive factor at fixed \(\lambda\). Poincaré’s inequality, with the constant controlled by the band, also bounds every moment of \(S\) in the resulting Gaussian law. Consequently, for every fixed \(q\ge1\), \(r\ge0\), and \(\lambda>0\), \[ \limsup_{\ell\downarrow0}\limsup_{k\to\infty} \mathcal P_n\bigl(\widehat W^q(1+|S|)^r\bigr)<\infty. \tag{102}\] Strong convergence of cell projections and the ordinary form approximation give the band covariance \[ T_\lambda^{-1},\qquad T_\lambda=-\Delta_U/a_*+\lambda\mathbf 1_K. \tag{103}\] These are the ingredients of the band-density argument in [14], now deduced from the assumed sequence limits. Since \(g\) vanishes near \(K\), \(T_\lambda(a_*g)=-\Delta_Ug\) in the weak Neumann sense. Thus the limiting variance of \(X_{g,n}\) is \(a_*\mathcal E_U(g,g)\). Its limiting fourth moment is bounded uniformly for \(\lambda\ge\lambda_0>0\), by Gaussian covariance monotonicity.

Finite cable estimates. Metric-graph Gaussian sign clusters are developed in [13]; the Brownian-bridge extension of integer-restricted Gaussian fields is described in [1]. We use this construction on the finite Gaussian-chain approximants. We record the finite inputs used to remove the band and impose the primary pin. Replace each primary edge by \(N\) lattice-Gaussian bonds with increment weight \(q_N^{j^2}\), where \(q_N=b/(2N)\), and attach the corresponding independent Brownian bridges conditional on their endpoint heights. Let \(\mathcal P_{n,N}\) be the resulting unweighted sigma-finite cable measure, and let \(\mathbb E_{W,N}\) and \(\mathbb P_{W,N}\) denote its normalized band law; write \(\widehat W_N=W/\mathcal P_{n,N}(W)\). A cable bond is open when its bridge does not hit zero. Under the unweighted cable measure, conditional on the magnitudes and the open graph, the nonzero clusters have independent fair signs. Exploration stops a bridge at its first zero or at its next vertex. The centered unvisited law has additional zero pins and increased precision on the stopped segments. These finite disintegration and comparison facts, together with the estimates below, are those of [14]. All cluster quantities in the following argument refer to these finite-\(N\) measures. Subdivision is removed by a \(\limsup_{N\to\infty}\) at each fixed primary graph, before taking \(k\to\infty\). Primary observables and their band normalizers have actual fixed-graph limits, equal to those of the Bessel law. No limit of the cable clusters is required. A remainder denoted by \(o_k(1)\) in a cable estimate means that its absolute value has \(\limsup_{k\to\infty}\limsup_{N\to\infty}\) equal to zero at the fixed auxiliary parameters then under consideration.

The dependence on \(b\) is controlled by the whole-chain estimate \[ \log\frac{\sum_{j\in\mathbb Z}q_N^{j^2}e^{2\pi tj}} {\sum_{j\in\mathbb Z}q_N^{j^2}} \le \frac{C_{J,R}t^2}{N},\qquad |t|\le R, \tag{104}\] for all sufficiently large \(N\), uniformly on \(J\). This is [14] with its parameter dependence made explicit. Grouping opposite integers proves this directly: the numerator increment is bounded by a constant times \(t^2\sum_{j\ge1}q_N^{j^2}j^2e^{2\pi Rj}=O_{J,R}(t^2/N)\). A primary flow extended through all \(N\) bonds therefore has a uniform moment cost. Additional zero pins and increased precision decrease this centered cost. No bare resistance of the subdivided network is used.

For clarity, write \(\mathcal R(f)\) for the minimum squared unit-edge flow cost of a mean-free primary test \(f\). After exposing a cluster of scaled diameter at least \(r>0\), the full-component unvisited mean has grounded reference cost at most \(V_r\). If \(\mathcal D\) is the collection of band cells touched by an exposed path and \(\ell<r/C\), the remaining averages satisfy \[ \mathcal R_{\rm post} \left(\sum_{d\in\mathcal D}|d|c_dh_d^U\right) \le C\ell^2\sum_{d\in\mathcal D}|d|c_d^2. \tag{105}\] Here the superscript \(U\) denotes the unvisited contribution, extended by zero on visited primary vertices. The same bound holds for exploration from a filled region. In particular, the remaining probability average in a cell reached by a path from a fixed multiple of its diameter away has uniformly bounded centered posterior moments. These statements follow from path Poincaré on the unit-edge network: a spanning path anchors a positive fraction of the rows or columns in a fixed enlargement of a touched cell, giving an \(\ell^2\) factor; the enlargements have bounded overlap. Duality gives (105), and (104) transfers it to the cable posterior. The local estimate is [14].

We also need the finite estimates that control the sigma-finite constant before band weighting. Write its sign disintegration as \(\nu_{n,N}(d\omega)\mathcal R_\omega(d\sigma)\), and put \[M_C(f)=\sum_{x\in C}|h_x|f_x,\qquad m_C=M_C(u),\qquad Q=\sum_Cm_C^2,\] where \(u\) is the probability weight on all primary vertices of \(U\). If \(u_d\) is the probability weight in a cell \(d\), we abbreviate \(M_C(u_d)\) to \(M_C(d)\). A cluster is counted once if it contains a primary vertex; internal cable vertices contribute no mass. For \(t\ge1\) and every integer \(q\ge1\), the bin bounds are \[\begin{align*} \nu_{n,N}(Q\le4t^2)&\le C(1+t),\\ \int_{Q\le4t^2}\left(\sum_CM_C(f)^2\right)^q\,d\nu_{n,N} &\le C_q(1+t) \left(t\left|\sum_xf_x\right| +\mathcal R\left(f-u\sum_xf_x\right)^{1/2}\right)^{2q}. \tag{106}\end{align*}\] For any nonnegative cost \(I(C)\) determined by a completed cluster, and all sufficiently large \(t=t(r)\), reinsertion gives \[ \int\sum_{\mathop{\mathrm{diam}}C\ge r}I(C)\mathbf 1_{\{m_C\le2t\}}\,d\nu_{n,N} \le2\int\mathbf 1_{\{Q\le8t^2\}} \sum_{\mathop{\mathrm{diam}}C\ge r}I(C)\mathbf 1_{\{m_C\le2t\}}\,d\nu_{n,N}. \tag{107}\] Finally, for every \(j>0\), some \(M<\infty\) satisfies \[ \mathcal P_{n,N}\left((1+|S|)^{-M} \mathbf 1_{\{t^2<Q\le4t^2\}}\right) \le C_j(1+t)^{-j}. \tag{108}\] The constants in these estimates depend only on fixed geometry, the displayed moment orders, \(r\) when present, and \(J\). Indeed, the proof of (106) uses fair-sign Khintchine and Paley–Zygmund bounds, at most \(C(1+t)\) translations with \(|S|\le Ct\), and the primary reference moments. After a macroscopic cluster is exposed, path anchoring bounds every moment of the complementary squared-mass sum; its probability of being at most \(4t^2\) is at least \(1/2\), proving (107). Polynomial damping follows by separating clusters below a fixed diameter from the macroscopic ones: a small cluster misses one of finitely many fixed boxes and is controlled by the corresponding mean-free test; a large cluster anchors the complementary mean. These are finite estimates, proved in [14]; their only parameter-dependent inputs are the uniform primary moment bounds and (104). All cable statements are made first at a fixed primary graph and then passed through its subdivision limit. Thus they require no convergence rate for \(N\) uniform in \(k\).

Band-weighted clusters and tilt errors. We now combine those finite estimates with (102). This is the point where the free limits enter the exploration argument. If \(F=(\sum_CM_C(f)^2)^{1/2}\), or if \(F\) is a signed partial sum selected using signless data, and \(A_t\) is a dyadic bin of \(Q\), then \[\begin{align*} \mathcal P_{n,N}(\widehat W_N|F|^q\mathbf 1_{A_t}) &\le \mathcal P_{n,N}(\widehat W_N^4(1+|S|)^M)^{1/4} \mathcal P_{n,N}((1+|S|)^{-M}\mathbf 1_{A_t})^{1/4}\\ &\hspace{35mm}\times \mathcal P_{n,N}(|F|^{2q}\mathbf 1_{A_t})^{1/2}. \end{align*}\] The bin moments grow polynomially in \(t\), whereas (108) has arbitrarily high polynomial decay. Choosing its exponent after \(q\) therefore bounds the sum and its tail. In particular, with \(D_\ell^2=1+\log(1/\ell)\), \[ \mathbb E_{W,N}\left(\sum_CM_C(d)^2\right)^q +\mathbb E_{W,N}|d_*|^{2q} \le C_{q,\lambda}D_\ell^{2q}+o_k(1). \tag{109}\] Here \(d_*\) is any signed selection whose nonnegative coefficients are bounded by the full cell average. The logarithm is the ordinary reference cost of a probability cell average minus \(S\). Fixed observation boxes have a fixed constant in its place. The remainder is taken at fixed \(\ell,\lambda\); the constants remain valid in the subsequent cell refinement.

Replacing the signs of all clusters of diameter at least \(r\) by independent fair signs changes the band law by vanishing total variation in the first two limits of (101). Here is the quantitative estimate ensuring this assertion for our sequence. On \(Q\le t^2\), expand the difference of the two band energies in the old and new signs. Fair-sign orthogonality, followed by exposure of one cluster, (105), and (107), gives \[ \mathcal P_{n,N}\left(\mathbf 1_{\{Q\le t^2\}} |\log W(\sigma')-\log W(\sigma)|^2\right) \le C_{\lambda,t,r}\ell^2D_\ell^2+o_k(1). \tag{110}\] The intermediate conditional second moment is bounded by \(C\ell^2\sum_d|d|M_C(d)^2\), which explains the factor \(\ell^2\). On this finite-mass bin, split at a fixed tolerance for the logarithmic difference, use Hölder and (102) on the exceptional set, and use \(|e^x-e^y|\le(e^\eta-1)(e^x+e^y)\) when \(|x-y|\le\eta\). Then send \(\eta\) to zero. The weighted moments remove the bin cutoff, also against fixed polynomial observables. Exchanging the old and new sign samples preserves the unweighted measure and \(Q\), so the two weighted tails obey the same bound. This supplies the sequence version of the argument in [14].

Clusters of diameter less than \(r\) contribute a second moment at most \(C_t r^2(1+\log(1/r))\) on a fixed bin to a bounded density smear: partition into \(r\)-cells and use that such a cluster meets only boundedly many neighboring cells. Higher bin moments and the weighted tail bound extend the vanishing to every fixed moment. It follows, by fair-sign resampling, the band Gaussian limit, and then removal of smaller clusters, that for nonnegative fixed density smears \(f_1,f_2\), \[ \limsup_{\ell\downarrow0}\limsup_{k\to\infty}\limsup_{N\to\infty} \mathbb E_{W,N}\sum_{\mathop{\mathrm{diam}}C\ge r}M_C(f_1)M_C(f_2) \le (f_1,T_\lambda^{-1}f_2). \tag{111}\] If \(Q_f=\sum_{\mathop{\mathrm{diam}}C\ge r}M_C(f)^2\), fair-sign Jensen also bounds the limiting \(q\)-th moment of \(Q_f\) by the Gaussian \(2q\)-th moment of its smear. These bounds are uniform for \(\lambda\ge\lambda_0\), since the Gaussian covariance decreases with \(\lambda\). Uniformity toward large strength is thus obtained after the mesh and cell limits.

During exploration write \(h_d=d_*+h_d^U\). Relative to its centered posterior with the restricted band precision, the unvisited law has linear tilt \[L=-\lambda\sum_d|d|d_*h_d^U,\qquad \zeta^2=C\lambda^2\ell^2\sum_d|d|d_*^2.\] The latter quantity bounds its reference cost by (105). For one completed macroscopic cluster define \(\zeta_C^2\) by replacing \(d_*\) with \(M_C(d)\), and put \(E_2=\sum_{\mathop{\mathrm{diam}}C\ge r}\zeta_C^2\). Weighted Jensen and (109) give \[ \mathbb E_{W,N}\zeta^{2q}+\mathbb E_{W,N}E_2^q \le C_{q,\lambda,r}\ell^{2q}D_\ell^{2q}+o_k(1). \tag{112}\] When \(\zeta\le1\), the centered posterior \(\mu\) satisfies \[1\le\mu(e^L),\qquad \mu(e^{2L})\le e^{C\zeta^2},\qquad \mu((e^L-1)^2)\le C'\zeta^2.\] Thus fixed-cost posterior means and second moments change by \(O(\zeta)\), using their fourth moments. Large tilts are removed by polynomial truncation. In the size-biased exploration used below, the precise bound for fixed boxes \(B,i\) is \[ \sum_{\mathop{\mathrm{diam}}C\ge r}M_C(B)\mathbf 1_{\{\zeta_C>1\}} (1+h_i^2+M_C(i)^2) \le \sqrt{Q_BE_2}\,(1+h_i^2+Q_i). \tag{113}\] For \(\zeta_C\le1\), the corresponding errors are bounded by \(\sqrt{Q_B}(1+\sqrt{Q_i})(\sqrt{E_2}+E_2)\). Hölder, the weighted moments, and (112) make both expectations vanish at fixed \(\lambda\) after the mesh and cell limits. For exploration from a filled region the same conclusion follows directly by discarding \(\zeta>1\) using (112) and the actual band moments. The finite tilt estimates used here are [14]; the argument above identifies the sequence limits of all their moment inputs.

Vanishing reach. It remains to show that exploration from \(p\) is unlikely to cross the band and reach \(V\). Choose finite small-cell coverings \(i\) and \(B\) on the starting and target sides, respectively, with fixed positive gaps from the band, the starting region, and the target region. They cover separator strips, so any crossing cluster meets a cell on each side. The radius \(\rho\) of these cells is fixed before the band refinement. Choose the macroscopic threshold \(r>0\) below the fixed separations that a cluster joining the starting region, either comparison strip, the band, or the target must span. Every cluster contributing to the cross terms below then has diameter at least \(r\), and the previous moment and tilt estimates apply with this one threshold. Path anchoring bounds the remaining centered variance of an average in a reached cell by a constant \(C_0\) independent of \(\rho\). On the other hand, a logarithmic trial function supported away from the band gives the continuum band variance lower bound \[c a_*\log(1/\rho)-C.\] The trial pays no band mass, so this bound is uniform in \(\lambda\). At a free wall it is restricted to the rectangle, which is permitted for the Neumann form. Since \(a_*>0\), choose \(\rho\) so small that the prior variance exceeds \(C_0+2c_0\) for a fixed \(c_0>0\). The band limits then give a gap at least \(c_0\) in the discrete laws, for each fixed strength and sufficiently fine partition and mesh. This is the only step that requires a positive height coefficient; it uses \(a_*>0\) in the limiting band law. The finite estimates require only \(b\in J\), with no assumption on \(a(b_k)\).

There are two useful consequences of this gap. Write a target average as exposed plus unvisited height. The centered posterior variance is at most the prior variance and loses at least \(c_0\) if the cell is reached. Restoring its tilt has the vanishing errors just proved. Fair-sign resampling of the exposed term gives \[ c_0\mathbb P_{W,N}(B\text{ reached}) \le\mathbb E_{W,N}\sum_{C:C\cap p\ne\varnothing}M_C(B)^2+o(1). \tag{114}\] Here and below \(o(1)\) vanishes in the first two limits at fixed \(\lambda\). A second comparison starts by biasing with \(|h_x|\), for \(x\in B\), and exposing its cluster. The finite size-bias inequality \[\mathbb E[X^2|Y|]\ge\mathbb EX^2\,\mathbb E|Y|\] holds for real linear tests under centered band precision. It follows from the minimal characteristic logarithmic Hessian in the finite comparison rules: integrate \(\mathbb E[X^2\cos(tY)]\le\mathbb EX^2\,\mathbb E\cos(tY)\) against \(2\,dt/(\pi t^2)\) after subtracting the identity at zero. Consequently the pre-exposure second moment at \(i\) is at least its unbiased variance. The completed-cluster bias cancels from the conditional complement law. Apply the same variance deficit, multiply by \(\mathbb E|h_x|\), and average \(x\) over \(B\). This yields \[ c_0\mathbb E_{W,N}\sum_{C:C\cap i\ne\varnothing}M_C(B) \le\mathbb E_{W,N}\sum_C M_C(B)M_C(i)^2+o(1). \tag{115}\] Only clusters spanning the fixed separation contribute to the exposed or tilted terms. They have a fixed positive diameter, so (113) justifies their summed errors. Equations (114) and (115) are the two finite deficit comparisons in [14].

As \(\lambda\to\infty\), the inverse forms \(T_\lambda^{-1}\) decrease to the inverse with a hard zero band on \(K\). This follows from weak energy compactness and recovery by functions zero on \(K\). The band separates \(B\) from \(i\), so \((B,T_\lambda^{-1}i)\to0\). By (111), the limiting expectation of \(\sum_CM_C(B)M_C(i)\) therefore vanishes. On \(\max_CM_C(i)\le R\), the right side of (115) is at most \(R\) times this cross sum. Its remaining tail tends to zero as \(R\to\infty\), uniformly after the first two limits and toward large \(\lambda\), because \[\sum_CM_C(B)M_C(i)^2\le\sqrt{Q_B}\,Q_i\] and the \(Q\)’s have arbitrary fixed moments uniformly in that order of limits. Thus the first-mass sum on the left of (115) vanishes. Every cluster joining \(p\) to \(B\) visits some cell \(i\). Truncating \(\max_CM_C(B)\) in (114) and applying these first-mass bounds, with the same moment control of its tail, proves \[ \lim_{\lambda\to\infty}\limsup_{\ell\downarrow0} \limsup_{k\to\infty}\limsup_{N\to\infty} \mathbb P_{W,N}\bigl(\text{exploration from }p \text{ reaches }V\bigr)=0. \tag{116}\]

Pinned tests and the grounded approximation. On a finite subdivided graph, explore every cluster meeting the primary pin region. If it does not reach \(V\), the exposed part of \(X_{g,n}\) is zero. The centered posterior has the pin \(p\), possibly further zero pins and increased segment precision, and a nonnegative restricted band form. Its variance is at most the variance of \(X_{g,n}\) under the primary hard-pin law on that subdivided graph, by finite precision comparison. At fixed primary graph, this latter variance converges as \(N\to\infty\) to the Bessel hard-pin variance. The averaged tilt errors vanish as above. On failure, Cauchy–Schwarz bounds the actual second-moment contribution by the fourth-moment root times the square root of the failure probability. The band fourth moment is uniformly bounded in the successive limits, whereas (116) makes that probability vanish. Hence \[\liminf_{k\to\infty}\mathop{\mathrm{Var}}_p X_{g,n} \ge a_*\mathcal E_U(g,g).\] The free variance bound gives the opposite limsup. For every real \(t\), the finite lower bound for the logarithmic moment generating function and the upper bound obtained by releasing the pin give \[\frac{t^2}{2}\mathop{\mathrm{Var}}_p X_{g,n} \le \log\mathbb E_p e^{tX_{g,n}} \le \log\mathbb E_{\rm free}e^{tX_{g,n}}.\] The free release is permitted because \(A_{U,n}g_n\) has total zero. The two endpoints converge to \(a_*t^2\mathcal E_U(g,g)/2\). Applying this to finite linear combinations gives the joint Gaussian Laplace limit for these tests; uniform primary exponential moments give their polynomial moment limits. Weak free-face terms may be approximated first by bounded density smears in primary reference cost, with the gap from \(p\) and \(K\) preserved.

Finally let \(f\) be any bounded piecewise continuous density, and set \(v=G_pf\). Smooth shifts \(g_j\) vanishing near \(p\), and allowed to meet the free walls, are dense in \(\mathcal H_p\) in energy. Choose \(\mathcal E_U(v-g_j,v-g_j)\to0\). The ordinary mixed-form approximation on these rectangular geometries, as stated in [14], gives \[ \lim_{k\to\infty} \mathcal R_{p,n}\bigl(f_n-A_{U,n}g_{j,n}\bigr) =\mathcal E_U(v-g_j,v-g_j), \tag{117}\] where \(\mathcal R_{p,n}\) is the squared dual norm of the unit-edge form grounded on \(p\). To see the precise form of this approximation, let \(A_{p,n}\) be the restriction to unpinned coordinates and put \(v_n=A_{p,n}^{-1}f_n\). The left side before its limit is \[(f_n,A_{p,n}^{-1}f_n)-2(f_n,g_{j,n}) +(g_{j,n},A_{U,n}g_{j,n}).\] Mixed inverse-form convergence, quadrature, and smooth energy consistency give respectively \(\mathcal E_U(v,v)\), \(\mathcal E_U(v,g_j)\), and \(\mathcal E_U(g_j,g_j)\), proving (117). Values of a test on pinned vertices are immaterial in this norm. Equivalently, a flow representing its unpinned coefficients may discharge into the pinned vertices. This is why (117) imposes no zero-total condition on \(f\).

Uniform reference-cost bounds make the difference between the two smears in (117) tend to zero in every fixed moment, first as \(k\to\infty\) and then \(j\to\infty\). Their fixed linear exponential bounds give the same passage for Laplace transforms, by Cauchy–Schwarz applied to \(e^{t(h,f_n)}-e^{tX_{g_j,n}}\). The limiting variance is \(a_*\mathcal E_U(v,v)=a_*\int_UfG_pf\). Apply the argument to every finite linear combination of the \(f_i\)’s and use polarization to obtain (99), with the asserted joint Laplace and moment convergence. ◻

Proof. The free and pinned assertions have been proved. For the torus upper bound, cut into small fixed rectangles, restrict the gradient test to each piece, and use (93). The omitted seams and the gradient approximations have uniformly vanishing reference cost. Therefore the limsup in (118) is at most its right side. This already proves the assertion when \(a_*=0\).

For \(a_*>0\), first suppose that \(g\) is supported in an unwrapped rectangle strictly inside one torus chart. Add primary zero pins on a collar of fixed positive width surrounding its support inside that chart. Centered precision comparison decreases variance. The collar separates the interior law from the torus exterior, and Lemma 20 gives the interior variance limit. Indeed the torus test equals the grounded interior test there, and \((-\Delta_D)^{-1}(-\Delta g)=g\) because \(g\) vanishes near the collar. Its variance is therefore \(a_*\int|\nabla g|^2\), matching the upper bound. For general \(g\), use a finite smooth partition of unity to write \(g\) as a sum of shifts supported in such charts. On each joint covariance subsequence the upper-bound defect is a positive semidefinite quadratic form. Since its diagonal vanishes on each summand, its cross terms with those summands vanish as well. It therefore vanishes on their sum.

For the cosine density \(f(x)=\cos(2\pi x_1)\), its inverse torus Laplacian is \(g=f/(4\pi^2)\) and \(\int|\nabla g|^2=1/(8\pi^2)\). The discrete eigenvalue converges to \(4\pi^2\) after multiplication by \(n_k^2\). The normalizing correction has vanishing reference cost, which proves the last assertion. ◻

Proof. If \(a_m(b_k)\) failed to tend to zero uniformly on the stated set, there would be offending \(m_k\) and a subsequence with \(\delta_k\log_Lm_k\to w\in I\). This contradicts (90). Square symmetry gives \(\mathop{\mathrm{Var}}X_{m,p}=a_m(b_k)|p|^2\), uniformly small for \(p\in K\). The reference exponential bound at \(\pm2\lambda\) supplies uniform integrability of \(e^{\lambda X_{m,p}}\). Since \(X_{m,p}\to0\) in probability uniformly over these choices, the expectations tend to one uniformly. Translation does not change the law. The extra side factors change \(\delta_k\log_Lm\) by \(o(1)\); compactness gives the last statement. ◻

Proof. We use the finite soft joining formula [14]. Tile \(B_{Gm}\) by \(G^2\) free \(m\)-squares. The outer-face phase has cell totals \(\xi_i\), supported on boundary cells, with \(\sum_i\xi_i=1\) and \(\sum_i\xi_i^2\le C/G\). On the space of neutral cell totals choose a flow right inverse \(P\) on the neighboring-cell graph, and let \(\rho_{ij}\) be the probability on the face of cell \(i\) adjoining cell \(j\). Define a vector \(\mathsf H\) in that neutral Euclidean space by \[(\mathsf H,d)=\sum_{\langle i,j\rangle}P(d)_{ij} \bigl((h_i,\rho_{ij})-(h_j,\rho_{ji}) -\bar\eta_{ij}\bigr),\] where \(\bar\eta_{ij}\) is the mean of the independent Bessel seam increments with the same orientation. Translating cells by constants changes \(\mathsf H\) by the neutral projection of those constants. Penalize by \(e^{-|\mathsf H|^2/2}\) and sum relative translations, quotienting by their common translation. The auxiliary-confinement argument in the proof of Lemma 19 justifies the finite precision comparisons for these sums. Poisson summation gives the positive numerator \[ \sum_{\substack{q_i\in\mathbb Z\\\sum_iq_i=1}} e^{-|q-\xi|^2/2}\, \prod_i\phi_{m,b}\left(s_i+ \sum_{j\sim i}P(q-\xi)_{ij}\rho_{ij}\right) \chi_m(P(q-\xi)), \tag{123}\] up to a common scalar. Here \(s_i\) is the restriction of the outer phase to cell \(i\), \(\rho_{ij}\) is its seam-face probability, and \(\chi_m\) is the characteristic function of the independent seam averages. The denominator is the corresponding sum with \(s=0\) and total zero. This follows by Fourier transforming the Gaussian penalty on the neutral space. The reciprocal of the projected cell-translation lattice is the neutral integer lattice. Exact seam matching adds centered precision, so the joined coefficient is at least this soft coefficient.

For fixed \(G\), the denominator tends to one along the sequence in (121). Its zero sector is one. Every nonzero sector contains a nonzero integer cell total and tends to zero by Lemma 19; the Gaussian weights permit dominated convergence. Each fixed-flux seam characteristic tends to one because the seam averages have variance \(O(m^{-1})\).

Retain the sectors \(q=e_{i_0}\) with \(i_0\) at distance at least \(G/4\) from the boundary. The radial flow of [14] has outward face fluxes \(H_{i,\pm j}\), constant outer flux \(1/(4G)\), and cell divergence \(e_{i_0}\). Set \[D_{ij}=\tfrac12(H_{i,+j}-H_{i,-j}),\qquad S_{ij}=H_{i,+j}+H_{i,-j}.\] The unit-cell radial field, corrected to have constant boundary flux, gives \[ \sum_{i,j}D_{ij}^2\le\frac1{2\pi}\log G+C, \qquad \sum_{i,j}S_{ij}^2\le C. \tag{124}\] One may obtain it by integrating \((x-x_0)/(2\pi|x-x_0|^2)+E(x)\) across cell faces, where the Neumann boundary correction has \(|E|\le C/G\) and \(|\nabla E|\le C/G^2\). The radial energy is \((2\pi)^{-1}\log G+O(1)\), and the face-sum errors are summable. The constants are uniform for the retained central cells. Prescribe \(P(e_{i_0}-\xi)\) to be these flows for all retained \(i_0\) and extend linearly. This is possible because the vectors \(e_{i_0}-\xi\) are linearly independent, their central coordinates being distinct.

In each cell expand the logarithm of its characteristic coefficient about the symmetric four-face profile of the same total. Square symmetry makes the linear term vanish on every zero-total face perturbation. The logarithmic Hessian bound thus gives the product lower bound \[ z_m(b)\exp\left\{-\tfrac12\sum_i\mathop{\mathrm{Var}}(h,Y_i)\right\}, \tag{125}\] where \(Y_i\) is the difference from that symmetric profile. Since \(q_i=S_{i1}+S_{i2}\), its decomposition is explicitly \[\begin{align*} Y_i={}&\sum_{j=1}^2D_{ij}(\rho_{i,+j}-\rho_{i,-j})\\ &+\frac{S_{i1}-S_{i2}}4 (\rho_{i,+1}+\rho_{i,-1}-\rho_{i,+2}-\rho_{i,-2}). \end{align*}\] Square reflections make these three components mutually orthogonal. The two odd face differences have variance exactly \(a_m(b)\); the even component has reference cost at most \(C(S_{i1}^2+S_{i2}^2)\). Lemma 18 and the reference bound give \[\limsup_k\sum_i\mathop{\mathrm{Var}}(h,Y_i) \le a_*\sum_{i,j}D_{ij}^2+C\sum_{i,j}S_{ij}^2.\] The Gaussian sector weight is bounded below because \(|e_{i_0}-\xi|\) is bounded. There are at least \(c_0G^2\) such sectors. Combining these facts with (124) proves (121). The additive energy constant and the even component constant are uniform, since \(a_*\le a\) and the reference bounds are uniform. Therefore its prefactor \(c\) does not depend on \(G\) or on the positive continuity value.

For (122), keep \(G\) fixed and retain just one central sector. Bound every variance in (125) by its reference cost. The soft denominator is at most the fixed finite-dimensional Gaussian lattice sum, since all its characteristic factors are at most one. The selected seam factor has a uniform positive lower bound: a mean over \(m\) independent Bessel increments at fixed flux \(q\) has characteristic function \(\exp\{bm(\cos(2\pi q/m)-1)\}\), which is bounded below uniformly in large \(m\) and in the compact temperature interval. There are only finitely many seams. These bounds prove (122). Crucially, (125) retains the factor \(z_m(b)\) regardless of how small it is. ◻

Proof. At \(b_c\), [14], the fixed-temperature version of (121), has coefficient \(a=8\pi\). Taking \(c_1\) to be the smaller of \(1\) and half the prefactor in that liminf bound, we obtain a constant independent of sufficiently large fixed \(r\) such that for \(G=L^r\), \[z_{Gm}(b_c)\ge c_1z_m(b_c)\] for all \(m\) beyond a threshold depending on \(r\). Fix \(r\) and iterate on each residue class of \(j\) modulo \(r\). The finitely many initial coefficients are strictly positive. Hence \[\liminf_{j\to\infty}j^{-1}\log z_{D_0L^j}(b_c) \ge r^{-1}\log c_1.\] Now let \(r\to\infty\) and use \(z_m\le1\) to obtain (126). Characteristic coefficients decrease with \(b\) by the chain precision comparison, giving the last claim. The lower bound is therefore uniform for all \(b\le b_c\). ◻

Proof. First we exclude a continuity value in \((0,a)\). If such a value exists, monotonicity and continuity give a compact interval \([u,v]\subset(0,\infty)\) of positive length and \(\eta>0\) such that all its continuity values lie in \([\eta,a-\eta]\). Choose a fixed integer stride \(r\) so large that, for \(G=L^r\), \(cG^{2-(a-\eta)/(4\pi)}>2\), where \(c\) is from Lemma 23.

On one residue class modulo \(r\) take consecutive indices \(j\) for which \(\delta_k j\) runs across \([u,v]\), and define on their cells of width \(\delta_kr\) the step function \[F_k(t)=\log\frac{z_{D_0L^{j+r}}(b_k)}{z_{D_0L^j}(b_k)}.\] Choose the endpoints of this union of cells to converge to \(u,v\). At almost every \(t\in(u,v)\) the profile is continuous. The starting cell size has logarithmic scale tending to \(t\), so (121) gives \(\liminf_k F_k(t)\ge\log2\). Equation (122) gives the common lower bound \(F_k\ge\log c(G)\). Extend by zero outside the chosen cells in a fixed enclosing interval. Fatou’s lemma therefore gives \[ \liminf_k\int F_k(t)\,dt\ge(v-u)\log2>0. \tag{128}\] On the other hand, the integral telescopes exactly: \[\int F_k(t)\,dt =\delta_kr\bigl[\log z_{D_0L^{j_+(k)}}(b_k) -\log z_{D_0L^{j_-(k)}}(b_k)\bigr].\] Since \(z\le1\), Lemma 24 bounds its limsup by \(\limsup_k\delta_kr\,o(j_-(k))=0\); here \(\delta_kj_-(k)\to u>0\). This contradicts (128). Thus every continuity value belongs to \(\{0,a\}\).

For \(0<w<T\), Proposition 16 gives the torus cosine variance limit \(a\mathfrak g\), whereas Proposition 21 gives \(\mathcal A(w)\mathfrak g\) at continuity points. Thus \(\mathcal A(w)=a\) at all such points. Suppose a continuity point \(w_0>T\) also had value \(a\). Monotonicity and the bound \(\mathcal A\le a\) would make the profile identically \(a\) on a neighborhood of \(T\), so \(T\) would be a continuity point. The moving exit sizes \(n_{j_e}=D_0L^{j_e}\) in Proposition 16 have \(\delta_kj_e\to T\). Proposition 21 would give cosine variance tending to \(a\mathfrak g\) there, contradicting the strictly smaller exit limsup. Every continuity point above \(T\) therefore has value zero. Finally a monotone function whose values on the continuity points of an open interval are a single constant is that constant everywhere on the interval. This proves (127). ◻

Proof. The Ginibre inequalities [8] imply positive correlation for cosine characters and monotonicity of two-point correlations under increasing ferromagnetic edge couplings. In particular, finite free correlations are bounded above by their free-exhaustion limits; see also [18]. These limits are translation invariant: translating any finite graph gives another finite graph in the same monotone exhaustion.

We first record supermultiplicativity. For three vertices \(u,v,z\), Ginibre’s inequality gives \[\left\langle \cos(\theta_u-\theta_v)\cos(\theta_z-\theta_v) \right\rangle \ge C_{G,b}(u,v)C_{G,b}(z,v).\] The expectation of \(\sin(\theta_u-\theta_v)\sin(\theta_z-\theta_v)\) is nonnegative. To see this, condition on \(\theta_v\), rotate it to zero, and then condition on all cosine coordinates. Writing a sine coordinate as its absolute value times its sign, the remaining signs have a ferromagnetic zero-field Ising distribution: the sign coupling on an edge \(\{x,y\}\) is \(b|\sin\theta_x\sin\theta_y|\ge0\). The sign at \(v\), whose sine is zero, is irrelevant. Their two-point sign expectations are therefore nonnegative. The cosine addition formula now proves \[C_{G,b}(u,z)\ge C_{G,b}(u,v)C_{G,b}(v,z).\] After free exhaustion this becomes \(C_b(x+y)\ge C_b(x)C_b(y)\). For every positive integer \(n\), \(C_b(jne_1)\ge C_b(ne_1)^j\). Taking \(j\to\infty\) in the definition of the axis mass yields \(C_b(ne_1)\le e^{-m(b)n}\). If \(x=(p,q)\) and \(|p|\ge|q|\), reflection symmetry and supermultiplicativity give \[C_b(x)^2=C_b(p,q)C_b(p,-q)\le C_b(2p,0) \le e^{-2m(b)|p|}.\] Interchanging coordinates treats the other case and proves (131) with the mass specified in the statement of the main theorem.

For the charge-two estimate, fix an axis and abbreviate \(s_x=\sin\theta_x\), \(c_x=\cos\theta_x\). Rotation invariance gives \[C_{G,b}(u,v)=2\langle s_us_v\rangle_{G,b},\qquad C^{(2)}_{G,b}(u,v)=8\langle c_uc_vs_us_v\rangle_{G,b}.\] The same conditional sign argument gives \(\langle s_us_v\mid(c_x)_{x\in G}\rangle\ge0\). Since \(c_uc_v\le1\), the second identity is at most \(8\langle s_us_v\rangle_{G,b}=4C_{G,b}(u,v)\). Edge monotonicity finishes (132). ◻

Proof. Condition on the undirected paired geometry. Choose one reference orientation for each loop and write its random orientation as \(\sigma_\gamma\in\{-1,1\}\). Equation (135) gives \[F_n=\sum_\gamma\sigma_\gamma F_n(\gamma),\qquad H_n=\sum_\gamma\sigma_\gamma H_n(\gamma),\] where \(F_n(\gamma)=2\pi\sum_zf_{n,z}W_\gamma^{\rm ref}(z)\), and similarly for \(H_n(\gamma)\). The conditional means vanish, and the conditional covariance is \(\sum_\gamma F_n(\gamma)H_n(\gamma)\). If one summand is nonzero, the loop has nonzero winding around at least one face in each support. Each such face lies in the horizontal range of the loop. Consequently this loop has horizontal diameter at least \(dn-O(1)\).

Let \(N=\sum_eN_e\) be the total number of traversals, and let \(\mathcal B_n\) be the event that some loop has horizontal diameter at least \(dn/2\). A loop of length \(|\gamma|\) has winding number at most \(|\gamma|\) in absolute value. Hence \[\left|\sum_\gamma F_n(\gamma)H_n(\gamma)\right| \le (2\pi M)^2\sum_\gamma|\gamma|^2\,\mathbf1_{\mathcal B_n} \le (2\pi M)^2N^2\mathbf1_{\mathcal B_n}.\] There are \(O(n^4)\) ordered pairs of vertices in \(G_n\). A loop counted by \(\mathcal B_n\) visits a pair separated by at least \(dn/2\). Using (136) and Lemma 26, we obtain \[\mathbf P_{G_n,b}(\mathcal B_n) \le C n^4 e^{-c n m(b)}.\] On \(N\le n^4\), the preceding conditional-covariance bound is therefore at most \(C n^{12}e^{-c n m(b)}\) after expectation.

It remains to bound the weighted tail of \(N\). Before the zero-divergence conditioning, \(N\) is Poisson with mean \(\mu=b|E(G_n)|\le C_I n^2\). The probability of that conditioning event is at least the probability of the zero configuration, namely \(e^{-\mu}\). Thus \[\mathbf E_{G_n,b}[N^2\mathbf1_{\{N>n^4\}}] \le e^{\mu}\, \mathbb E_{\operatorname{Pois}(\mu)}[N^2\mathbf1_{\{N>n^4\}}] \le Ce^{-c n^4\log n}.\] For the last estimate one may use the Poisson exponential moment with parameter \(\log n\): its logarithm is \(\mu(n-1)=O(n^3)\), whereas the tail factor is \(e^{-n^4\log n}\); the polynomial insertion changes only a polynomial prefactor. Pairings do not change the marginal law of \(N\). This proves (137). ◻

Proof. If the assertion failed, a further subsequence and numbers \(0<w<T\), \(\eta>0\) would satisfy \[\delta_k\log_L\frac1{m(b_k)}\le w-2\eta.\] Take \(n_k=\lfloor L^{w/\delta_k}\rfloor\). Then, for all large \(k\), \[n_km(b_k)\ge L^{\eta/\delta_k} \ge n_k^{\eta/(2w)}.\] The coefficients of \(F_{n_k}\) and \(H_{n_k}\) have uniformly bounded absolute sums. Lemma 27 makes their covariance tend to zero, because an exponential in a positive power of \(n_k\) dominates the polynomial in (137). This contradicts (138). ◻

Proof. Choose \(T<u<v<w'\), and use radii \[\mathcal R_k=\{4^j:j\in\mathbb Z,\quad u\le\delta_k\log_L(4^j)\le v\}.\] Their number \(N_k\) obeys \[ N_k=\frac{v-u}{\delta_k\log_L4}+O(1),\qquad \frac{N_k}{\log r_k}\longrightarrow q:=\frac{v-u}{w'\log4}>0. \tag{141}\] For \(R\in\mathcal R_k\), the annuli \(R<|z|<2R\) about the source at \(0\) are disjoint. Their maximal radius is \(o(r_k)\); therefore they avoid the other source \(x\), uniformly over the displacements in the proposition. Figure 2 compares this annular geometry with the pinned geometry used for the lower bound.

Choose a nonnegative, nonzero smooth radial function \(\chi\) supported in \((1,2)\), and let \[p(z)=\chi(|z|)\nabla\arg z, \qquad D=\int_{\mathbb R^2}p(z)\cdot\nabla\arg z\,dz>0.\] This is a smooth divergence-free form, compactly supported in the unit annulus. Choose its radial stream function \(\psi\), constant on each complementary component, with \(\nabla^\perp\psi=p\). At scale \(R\), sample \(\psi_R(z)=\psi(z/R)\) on the faces of the primary height graph and take its rotated discrete difference. Denote the resulting edge form by \(\eta_R\), choosing its sign to agree with the positive source. Thus, on an edge with midpoint \(z_e\), its corresponding coordinate is \(R^{-1}p(z_e/R)+O(R^{-2})\). It has zero discrete divergence, \[ \max_e|\eta_R(e)|\le C/R,\qquad \sum_e|\eta_R(e)|^2\le C. \tag{142}\] For any affine height difference \(\nabla h+\alpha\) representing the unit spin twist at \(0,x\), discrete summation by parts gives \[ \sum_e\eta_R(e)(\nabla h+\alpha)_e=D_R, \qquad D_R\longrightarrow D>0. \tag{143}\] Indeed the gradient part pairs to zero. The remaining pairing is the difference of the stream-function values at the two source faces, multiplied by their circulations \(\pm2\pi\). The stream function is constant near each source, so there is no dependence on the height configuration or on the path chosen for \(\alpha\). The convergence in (143) also follows directly by Riemann sums; its error is uniform here because the second source lies outside every annulus. Fix \(d_{\rm circ}>0\) so that \(D_R\ge d_{\rm circ}\) for all sufficiently large radii.

We make the partition-ratio estimate at fixed \(k,x\) in a finite free spin box large enough to contain all these annuli in its interior. Write \(Z_\alpha\) and \(Z_0\) for the twisted and centered height sums, both with exterior face height zero. Free-boundary duality gives \(C_{G,b_k}(0,x)=Z_\alpha/Z_0\). For any fixed \(\lambda>0\), insertion of the deterministic tilt from (143) gives the exact identity \[ e^{\lambda\sum_{R\in\mathcal R_k}D_R}C_{G,b_k}(0,x) =\frac{Z_\alpha\left( e^{\lambda\sum_R(\eta_R,\nabla h+\alpha)}\right)}{Z_0}. \tag{144}\] Here \(Z_\alpha(F)\) means the unnormalized twisted sum with insertion \(F\). We will bound the right side independently of the outer spin box.

Fix a small relative cell size \(t>0\). Cover the support of each \(\eta_R\) by square primary cells of side \(m_R=\lfloor tR\rfloor\), with disjoint vertex sets, leaving individual edges between cells as seam increments. The support of \(\chi\) has positive distance from the annular boundary. Thus for sufficiently small fixed \(t\), every cell meeting this support lies inside the annulus, avoids both sources, and is simply connected. There are \(O(t^{-2})\) such cells at each radius. Edges not in a retained cell and carrying a nonzero tilt number \(O_t(R)\).

Release the endpoint constraints between these cells and the seam increments, and release all remaining constraints outside the cells. The tilted affine-to-centered comparison [14] bounds the normalized expression in (144) by the product of the cell transforms and the independent whole-edge seam transforms. Untilted factors on the rest of the graph have affine-to-centered ratio at most one. On a retained cell the connection is an integral gradient, \(\alpha=\nabla\kappa\); reindexing \(h'=h+\kappa\) removes it from both the weight and the statistic being tilted. Accordingly the factor for that cell is the centered free transform \[\mathbb E_{Q,b_k}^{\rm free} \exp\{\lambda(\eta_R,\nabla h)_Q\}.\] In particular, this gauge change introduces no additional real exponential factor. Released constant modes are taken modulo common translations; the edge-difference tests annihilate every such constant. Equivalently, one can retain vanishing quadratic penalties on these constants in both sectors and cancel their identical factors before removing them. No outer-box normalizer is present in the resulting bound.

We now estimate the factors for one annulus. The total squared coefficient cost of its seam increments is \(O_t(R^{-1})\) by (142). The Bessel increment transform is \(\exp\{b_k(\cosh(2\pi s)-1)\}\), so their combined logarithmic cost is \(O_{t,\lambda}(R^{-1})\), uniformly in \(k\). On each retained square replace \(\eta_R\) by its value at a chosen point of the square. Smoothness of \(p\) makes the total squared cost of the coefficient error over all cells \(O(t^2)+o_R(1)\). The bounded-flow estimate (14) and Hölder’s inequality with exponent two therefore give \[\begin{align*} &\sum_Q\log\mathbb E_{Q,b_k}^{\rm free} e^{\lambda(\eta_R,\nabla h)_Q} \\[-2pt] &\qquad\le \frac12\sum_Q\log\mathbb E_{Q,b_k}^{\rm free} e^{2\lambda(\bar\eta_{R,Q},\nabla h)_Q} +C_\lambda t^2+o_{t,\lambda,R}(1), \tag{145}\end{align*}\] where \(\bar\eta_{R,Q}\) is the constant coefficient on \(Q\). The error coefficients multiplied by the fixed tilt are uniformly bounded, as required by (14). That estimate bounds each error transform by its squared coefficient cost. Summing the costs gives the total \(C_\lambda t^2\) in (145).

Each constant test in the sum is a linear combination of the two square affine observations defining \(a_{m_R}(b_k)\). Its coefficient vector has size at most \(C_\lambda m_R/R\le C_\lambda t\). Translations of the square do not change its law. Moreover, \[\delta_k\log_Lm_R =\delta_k\log_LR+\delta_k\log_Lt+o(1).\] Thus these scales lie in a slightly enlarged compact subinterval of \((T,w')\) for all large \(k\). By Corollary 22, the transforms of these affine tests tend uniformly to one, uniformly also in their bounded coefficient vectors. Since the number of cells is uniformly bounded once \(t\) is fixed, their total logarithmic cost is \(o_k(1)\) uniformly over \(R\in\mathcal R_k\). The reference estimates and \(\min\mathcal R_k\to\infty\) also make all seam and discretization errors uniform. We obtain, for every fixed \(\lambda,t\), the per-annulus bound \[ \log(\text{product of its transforms}) \le C_\lambda t^2+o_k(1). \tag{146}\]

Choose \(\lambda\) first, and then \(t\) so small that \(C_\lambda t^2\le\lambda d_{\rm circ}/4\). For all sufficiently large \(k\), the uniform error in (146) is at most \(\lambda d_{\rm circ}/4\). Equations (144) and (141) imply \[C_{G,b_k}(0,x) \le\exp\{-\lambda d_{\rm circ}N_k/2\}.\] This estimate holds uniformly in every sufficiently large outer spin box, at the fixed values of \(b_k,r_k,x\). We may therefore first take the prescribed free-box thermodynamic limit. The resulting estimate for \(C_{b_k}(x)\) is uniform in the indicated range of \(x\). Finally choose the fixed \(\lambda\) large enough that \(\lambda d_{\rm circ}q/4>K\). Equation (141) then gives (140). ◻

Proof. At fixed \(b,r\), take the free-box thermodynamic limit in (147). For a target \(se_1\), apply the inequality successively in translates of \(H_r\) centered at the boundary point reached in the previous step. Each step moves the center by at most \(r\) in the supremum norm. Hence at least \(\lfloor s/r\rfloor-1\) iterations are possible before the target can belong to the next translated square. Translation invariance and \(C_b\le1\) give \[C_b(se_1)\le S_{r,b}^{\lfloor s/r\rfloor-1}.\] Taking \(-s^{-1}\log\) and then \(s\to\infty\) proves the claim. ◻

Proof. Fix \(w'>T\), and set \(r_k=\lfloor L^{w'/\delta_k}\rfloor\). Proposition 29, with any fixed \(K>1\), and finite-to-infinite monotonicity imply \[S_{r_k,b_k}\le\sum_{z\in\partial H_{r_k}}C_{b_k}(z) \le8r_k\,r_k^{-K}<\frac12\] for all sufficiently large \(k\). Thus Lemma 30 yields \(m(b_k)\ge(\log2)/r_k\). Consequently \[\limsup_k\delta_k\log_L\frac1{m(b_k)} \le\lim_k\delta_k\log_L\frac{r_k}{\log2}=w'.\] Letting \(w'\downarrow T\) proves (148). ◻

Proof of Theorem 1. Start with any sequence \(b_k<b_c\) tending to \(b_c\). The construction in Sections 4 and 5 admits a subsequence with the profile (129). Its transition point is \[T=\frac{\pi}{\sqrt{3\mathcal P}},\] where \(\mathcal P\in(0,\infty)\) is independent of the extracted sequence. Propositions 28 and 31 prove (130) on that subsequence. If the asserted limit failed for the original parameter approach, there would be a sequence on which its distance from \(T\) stayed bounded below; applying the same extraction to that sequence would give a contradiction. Therefore the logarithmic-scale limit holds as \(b\uparrow b_c\) without extraction. Converting from base \(L\) to natural logarithms gives \[\lim_{b\uparrow b_c}\sqrt{b_c-b}\log\frac1{m(b)} =(\log L)\,T =\frac{\pi\log L}{\sqrt{3\mathcal P}} =:A_{\rm XY}\in(0,\infty).\] All spin correlations used above are limits of the prescribed free spin boxes before their separation tends to infinity. Thus the mass in this identity is exactly the mass in Theorem 1. ◻