Background and relation to earlier work

Berezinskii’s spin-wave analysis predicted algebraic correlations for planar spins at low temperature [5]. Kosterlitz and Thouless described a transition driven by the unbinding of vortex pairs [12], and Kosterlitz developed a renormalization-group analysis of the critical behavior of the nearest-neighbor model [11]. For the Villain formulation, Janke records the critical prediction, up to a multiplicative constant, as \[|x|^{-1/4}(\log|x|)^{1/8} \left(1+O\left(\frac{\log\log|x|}{\log|x|}\right)\right);\] see [10]. Theorem 1 establishes the leading equivalent, with a relative \(o(1)\) error, for the ordinary cosine interaction and the free-box thermodynamic correlation in Equation (2).

Rigorous work established the phase picture through correlation bounds and dual representations. McBryan and Spencer proved power-law upper bounds for two-dimensional planar-spin correlations [15]. Fröhlich and Spencer proved a low-temperature algebraic regime for the plane rotator and Villain models through a multiscale Coulomb-gas analysis [9]. For the cosine XY model, van Engelenburg and Lis proved exponential decay below a threshold and the lower bound \(1/(8|x-y|)\) at and above that threshold in the limit over free boxes [20]. These results include sharp information about the phases, while their stated bounds do not determine the critical equivalent.

The dual height field provides another route to the transition. Aizenman, Harel, Peled, and Shapiro related height depinning to nonsummable spin correlations, including the Bessel height interaction dual to the cosine XY model [1]. Lammers identified an exact relation between the XY and dual-height correlation lengths [13], and proved a localization dichotomy with a universal positive lower bound on logarithmic height fluctuations in the delocalized phase [14]. These results constrain the transition through height information.

The lattice-Gaussian inequality of Regev and Stephens-Davidowitz [19] underlies the finite comparisons developed in [16] and passed to Bessel weights in [17]. In a related approach, Aizenman, Harel, Peled, and Shapiro extended sublattice monotonicity of height moment-generating functions to annealed Gaussian interactions; their application to the XY dual uses a representation credited there to Raoufi [1]. Bauerschmidt developed finite-range decompositions of Gaussian fields using functional calculus [2]. Bauerschmidt, Park, and Rodriguez obtained Gaussian scaling limits for discrete Gaussian heights at high temperature [3, 4]. Falco’s critical Coulomb-gas analysis gives a rigorous example in which a marginal recursion produces a power law with a multiplicative logarithm [8]. Its fractional electric-charge observable is different from the magnetic XY insertion considered here.

Theorem 1 sharpens the companion result \(C_{b_c}(r)=r^{-1/4+o(1)}\) of [17]. We use from that paper the critical height coefficient \(a(b_c)=8\pi\), finite Bessel comparisons, fixed-geometry and conditional height limits, and a four-face characteristic estimate. The analytic local maps come from [16], and the sector costs for fixed positive holes and annular kernel estimates from [18]. The annular argument is adapted to the pinned rim below. The cutoff initialization, quantitative coefficient estimate, comparison of different starting histories, and extraction of the amplitude are proved here.

The route to the amplitude

Fourier expansion of the spin interactions represents a correlation as a ratio of partition functions for integer-valued heights on the dual lattice. The numerator shifts the height increments by a connection whose circulation records the spin insertion. This magnetic insertion is not determined by convergence of height averages alone.

For an integer \(n\ge1\), let \[ M_n=\mathbb E^0_{n,b_c}\cos\theta_0, \tag{3}\] where \(\mathbb E^0_{n,b_c}\) denotes expectation for the measure obtained from Equation (1) on \(\Lambda_n\) by fixing all boundary angles at zero and integrating only the interior angles. The geometric argument proves two comparisons: \[\begin{aligned} \frac{M_{n'}}{M_n}&\longrightarrow d^{-1/8} &&\text{if }n,n'\to\infty\text{ and }n'/n\to d\in(0,\infty),\\ \frac{C_{b_c}(r)}{M_r^2}&\longrightarrow c_{\rm shape}\in(0,\infty) &&\text{as }r\to\infty. \end{aligned}\] Annular interfaces separate the microscopic insertions from an exterior with holes kept at fixed positive radius. The sector and annular-kernel estimates of [18] control these interfaces, and the common microscopic factors cancel in the comparisons. A pinned-rim argument relates the two-insertion calculation to the free-box thermodynamic correlation. The first comparison gives \(M_n=n^{-1/8+o(1)}\), which will make the cutoff loss negligible relative to the center observable, and later extends an asymptotic from geometric sizes to all integers. The second reduces the theorem to a center asymptotic.

To compute \(M_n\), we cut off differences of noisy cell averages, choosing the cell size according to the target square. For the squares used in the analytic comparison, the rough center bound makes this change negligible in the center partition ratio. After averaging to the associated starting scale, the bulk interaction is small. For the center insertion, every expansion term carries one labelled component that retains the source and its attached dependencies. The norm of the resulting labelled activity may grow polynomially with the cell size; a linear propagation estimate controls it.

Let \(L>1\) be the fixed dyadic block ratio and \(j\) the absolute scale index. After separating the Gaussian reference with height coefficient \(a=8\pi\), the smoothed bulk interaction at block side \(L^j\) retains \(t_j/2\) times the allocated gradient energy and a fundamental periodic term; the linear part of the remaining activity’s evolution contracts. Coordinates adapted to the pure Gaussian curve remove its drift and give a gradient coordinate \(u_j\) and a signed fundamental Fourier coordinate \(Z_j\). Their leading recursion is \[u_{j+1}-u_j\simeq-HZ_j^2,\qquad Z_{j+1}-Z_j\simeq-2(\log L)u_jZ_j,\qquad H>0.\] Physical torus variance and phase tests keep each finite history near the Gaussian curve. A phase lower bound derived from the four-face estimate of [17] prevents the fundamental direction from disappearing. Comparing histories through the same finite physical torus tests controls changes of starting scale without a rate for the height scaling limit. Together with the recursion, these constraints yield the critical balance \(HZ_j^2\sim2(\log L)u_j^2\), which gives harmonic decay of \(u_j\). On the controlled interior ranges of the size-dependent histories used in the comparison, the coordinate changes give \(t_j=u_j+O(j^{-2})\). The resulting estimate, uniform on each fixed such range, is \[t_j=\frac{1}{2(\log L)j} +O\left(\frac{\log j}{j^2}\right).\]

Let \(g_n\) be the minimum squared edge norm of a connection with the \(2\pi\) circulation of one insertion in the corresponding dual square. For \(n_0=L^N\) and \(n_1=L^{N+1}\), we use the same entry and a common stop \(j'=N-O(\log N)\), leaving a number of terminal blocks that grows as a small power of \(N\). We choose the field-independent factors removed at each step so that their complete contributions to the insertion ratio agree between sizes. The source term remaining at the stop is controlled relative to the source coefficient already extracted. Integration of the common phase then removes the linear periodic term, and a comparison of the terminal Gaussian fields leaves the gradient contribution with a summable error: \[\log\frac{M_{n_1}}{M_{n_0}} =-\frac{1-t_{j'}}{2a}\bigl(g_{n_1}-g_{n_0}\bigr)+\epsilon_N, \qquad \sum_N|\epsilon_N|<\infty.\] Here \(g_{L^N}=2\pi(\log L)N+c_N\) with \(c_N\) convergent. Substitution of the trajectory estimate gives the constant increment \(-(\log L)/8\) and the correction \(1/(16N)\). The unweighted Green differences \(c_{N+1}-c_N\) telescope, and summation by parts treats the differences weighted by \(1/N\); the other errors are absolutely summable. Summation therefore gives a positive finite center amplitude with logarithmic power \(1/16\). The first geometric comparison extends this equivalent to all integer sizes, and the bulk comparison squares the logarithmic power to \(1/8\).

Dual partition functions

Choose one orientation of each edge of a finite nearest-neighbor graph \(G=(V,E)\) and put \(\nabla_eh=h_y-h_x\) for \(e=(x,y)\). We call these height vertices and edges primary, to distinguish them from the auxiliary subdivisions in the comparison proof; primal refers to the spin lattice. The height takes values in \(2\pi\mathbb Z\). Its increment weight is \[ p_b(j)=e^{-b}I_j(b),\qquad \sum_{j\in\mathbb Z}p_b(j)e^{tj}=\exp\{b(\cosh t-1)\}. \tag{4}\] Here \(I_j\) is the modified Bessel function. Thus \(p_b\) is the law of the difference of two independent Poisson variables of mean \(b/2\). An integral connection is an antisymmetric edge function \(\tau:E\to2\pi\mathbb Z\). The weight in that sector is \[\prod_{e\in E}p_b\left(\frac{\nabla_eh+\tau_e}{2\pi}\right).\] A connection is flat where its circulation around every contractible cycle is zero. Its circulations around the remaining holes are its periods.

We allow nonnegative quadratic penalties in real linear observations, hard pins, and prescribed relative height data. Every observation used after a cut is supported in one component. A component with no absolute constraint or confining observation is summed modulo a common translation by \(2\pi\mathbb Z\); when its constant is confined, all its translation copies are summed. A hard pin fixes that constant instead. Relative translations between cut components are summed again on restoration. Write \(Z_G^\tau\) for a partition function with these conventions and \(Z_G^0\) for the same graph, cuts, and penalties with all affine data centered.

An integral gauge change replaces \(\tau\) by \(\tau+\nabla\psi\) and the height variable by \(h-\psi\), where \(\psi:V\to2\pi\mathbb Z\). This is an exact reindexing when every affine datum is changed with the height. A penalty \(\|Bh-y\|^2/2\) then has center \(y-B\psi\), and absolute data \(b^p\) on a set \(p\) become \(b^p-\psi|_p\). For relative data \(h_z-h_{z_0}=u_z\), the new value is \(u_z-\psi_z+\psi_{z_0}\). Thus a hard zero pin is preserved when \(\psi=0\) on it, and a constraint requiring all heights on a set to be equal is preserved when \(\psi\) is constant there.

The reference form on a retained primary graph is the unit-conductance form \[ \langle g,A_Gg\rangle=\sum_{\{x,y\}\in E}(g_x-g_y)^2. \tag{5}\] It converges under lattice scaling to \(\int|\nabla g|^2\). For the four nearest steps this is exactly the operator \(A_0\) in [16]. We write \(A\) when the graph is understood.

The critical height inputs

The next proposition uses absolute pin data: \(h_H=0\) fixes every primary height in \(H\), and \(b^p\) specifies a compatible absolute height at every primary vertex of \(p\). Its approximation by finite guard observations is understood in probability, at each prescribed tolerance, under the sampled law stated there. The finite list of smooth guard probes is fixed before the lattice limit and enlarged afterward to reduce the tolerance.

We also use the restoration consequence [17]. Suppose every endpoint of a restored bond, and the support of every restored form, lies in a core \(H\) separated from a guard \(p\). If \(r_\Delta\) is the conditional restoration factor divided by its value at zero guard data, then \(r_H\le r_\Delta\le1\). The guard laws in the two topologies transfer events whose probabilities tend to zero in either direction, also under a fixed real linear tilt. The finite-observation approximations above concern conditional Laplace transforms and \(r_H\); no additional limit for \(r_\Delta\) is asserted. In each use, the graph, pin regions, and observation cells are fixed before the lattice limit.

Separating the insertion from the exterior

For a graph with the boundary and constant conventions of Section 2, write \(\Phi_G^\tau=Z_G^\tau/Z_G^0\) for its bare sector ratio. The fixed-domain sector input evaluates this ratio on a free exterior. Let \(U\) be a bounded connected rectilinear domain with a simple outer boundary and finitely many disjoint simple holes, all with dyadic vertices, positive side lengths, and positive passages. Give the height graph free walls. If \(\vartheta_0\) is a smooth closed one-form near \(\overline U\) whose periods about the holes are prescribed multiples of \(2\pi\), set \[\mathcal M_U(\vartheta_0) =\inf_{g\in H^1(U)}\int_U|\vartheta_0+\nabla g|^2.\] Let \(\mathcal G_\nu(U)\) be its nearest-neighbor primary graph at mesh \(\nu^{-1}\). For integral lattice connections that are flat on the retained graph away from the holes and have these periods, the sector theorem of [18] states that \[ \Phi_{\mathcal G_\nu(U)}^\tau \longrightarrow \exp\{-\mathcal M_U(\vartheta_0)/(2a)\} \qquad(\nu\to\infty), \tag{9}\] where \(a=8\pi\). The limit holds along every coherent bounded-step rounding of this fixed geometry. The energy depends only on the periods. This statement concerns free height walls; a pinned exterior will be treated below.

To connect (9) to a microscopic source, use the annular construction of [18]. Fix a small \(\varepsilon>0\) and a rectilinear disk \(P\) with dyadic vertices such that \[ B(0,1-\varepsilon)\subset P\subset B(0,1+\varepsilon). \tag{10}\] Choose a sufficiently large power of four \(\lambda\), and an integer \(\varrho_0\) aligned so that all relevant array divisions lie between rows of dual vertices. Put \(\varrho_i=\varrho_0\lambda^i\). The \(i\)-th annulus has walls \(\varrho_{i-1}P\) and \(\varrho_iP\). Write \(Q_s=(-s,s)^2\) and \(A_\lambda=\lambda P\setminus P^\circ\). In the reference annulus \(A_\lambda\), the guard is \(\overline{Q_{2\sqrt\lambda}}\setminus Q_{\sqrt\lambda}\), and the inner and outer end regions are \(A_\lambda\cap\overline{Q_4}\) and \(A_\lambda\setminus Q_{\lambda/4}\), respectively. Scale these three regions by \(\varrho_{i-1}\), and call the guard \(\mathsf P_i\). It is a full separating band, and each end contains every endpoint in that annulus of a bond to be restored at its wall. These prescribed bands, with positive gaps, are the geometry used in the flat-pin estimate below.

Choose one vertex in \(\mathsf P_i\). A guard datum records every height difference from that vertex. In the annular quotient choose its height to be zero, so the datum specifies every guard height. The nearest-neighbor weight then factors between the two sides conditionally on that datum. Write \(\mu_i^\tau\) for the guard law in the cut annulus, with its one common constant quotiented out.

A core is a connected finite block filling the first hole; it may carry the source and has no guard data. At an interface in this construction, let \(T^\tau(u)\) be the restored partition sum divided by the cut partition sum at its incident guard data \(u\), and put \(d_T=T^0(0)\). The restored sum includes the relative integer translations of the joined components. The same definition applies to one joint interface joining several last annuli to a connected exterior, with no direct bonds between distinct incident annuli. Its data are those in the incident annuli, while all exterior heights are summed together inside that one kernel. For a vertex region \(H\), let \([H]\) denote the event that all its heights are equal, without prescribing their common value. In a centered cut annulus with guard \(\mathsf P_i\), put \[\overline D_i(H,\mathsf P_i) =\log\frac{\mathbb P([H]\mid[\mathsf P_i])}{\mathbb P([H])}.\] The upper kernel bound of [18], its Equation (8), holds for all these interfaces, including a core interface. For an annulus–annulus or joint exterior interface, its Equation (9) also gives the centered average bound: \[ 0<\frac{T^\tau(u)}{d_T}\le1, \qquad \int\frac{T^0}{d_T}\prod_i\,\mathrm d\mu_i^0 \ge \exp\left\{-\sum_i\overline D_i(H_i,\mathsf P_i)\right\}. \tag{11}\] The sum and product use the incident annuli and the appropriate end in each. The first bound is pointwise in the data, and the second is an average under independent centered cut annuli. The centered divisor \(d_T\) therefore works for the shifted kernel as well. The flat-pin estimate of [18] makes each interaction on the right arbitrarily small in the mesh limit by taking \(\lambda\) large at fixed \(P\).

Two consequences of [18] control how the averaged loss passes through a whole nest. For any \(\xi\in(0,1/64)\), one can first take \(\varepsilon\) small, then \(\lambda\) large, and finally the fixed aligned threshold \(\varrho_0\) large, so that every interface between successive annuli of these radii satisfies \[ \int\left(1-\frac{T_i^\tau}{d_{T_i}}\right) \,\mathrm d(\mu_i^\tau\otimes\mu_{i+1}^\tau)\le\xi. \tag{12}\] This holds in the centered sector and in either unit-period sector, when restoration introduces no source. It follows from the exact sector identity (10) in that proposition and (9); it is the normalized estimate (11) there. All late annuli obey the same bound because they refine the same fixed geometry. Fix one such core, containing the source in a unit sector, and fix its connection once on the increasing family. In a nest of \(J\) annuli, the law of its last guard then has density at most \(2\) with respect to \(\mu_J^\tau\), for all sufficiently large \(J\). This holds in all three sectors and for identical lattice translates. The only lower assumption on the first core kernel is its strict positivity; no uniform positive lower bound on that kernel is required. This density bound compares the actual last-guard law, after all inner blocks have been joined, with the cut law used in the averaged estimate.

It remains to attach the whole nests to their exterior. Let \(\mathcal N_J\) be the bare sector ratio of one whole free nest of depth \(J\), with its core and a unit source. Integral gauge changes and reversal of all heights show that the same \(\mathcal N_J\) is obtained from any lattice translate and either sign. This is the common factor that will cancel in both comparisons. Suppose \(k\) such nests, with disjoint closures, attach to a connected exterior. The graph of cut blocks, with an edge for each interface, is a tree rooted at the exterior; there are no other block adjacencies, in particular no direct bonds between distinct incident last annuli. Write \(\mu_{J,i}^\tau\) for the cut guard law in the \(i\)-th incident last annulus, with its restricted connection. In the following ratio, “joined” means the exterior after joining only the \(k\) last annuli; the holes then have the inner radii of those annuli. Define \[ \mathfrak q^\tau_{\rm ext} =\frac{\Phi_{\rm joined}^\tau} {\Phi_{\rm ext}^\tau\prod_{i=1}^k\Phi_{{\rm last},i}^\tau}. \tag{13}\] Releasing those interfaces gives \(0<\mathfrak q^\tau_{\rm ext}\le1\). Exact summation of the interface kernel, which is Equation (10) of [18], gives \[ \int\frac{T^\tau_{\rm ext}}{d_{\rm ext}} \prod_i\,\mathrm d\mu_{J,i}^\tau =\left(\int\frac{T^0_{\rm ext}}{d_{\rm ext}} \prod_i\,\mathrm d\mu_{J,i}^0\right)\mathfrak q^\tau_{\rm ext}. \tag{14}\] Thus a sufficient attachment condition, along the lattice sequence in question, is \[ \sum_i\overline D_i(H_i,\mathsf P_i)\le\delta+o(1), \qquad \log\mathfrak q^\tau_{\rm ext}\ge-\delta-o(1), \tag{15}\] with \(\delta>0\) arbitrarily small. Indeed, (11) and (14) bound the shifted average loss by \(1-e^{-2\delta-o(1)}\), and the centered one by \(1-e^{-\delta-o(1)}\). This estimate is averaged over the independent cut-annulus laws, without any uniform lower bound at individual data.

The lower thermodynamic comparison below retains a hard pin in the exterior. We therefore check that the same reasoning applies when that pin fixes the exterior constant and is kept in both the restored and cut graphs. The entire hard constraint remains in the exterior partition sum and joint kernel; for the rim below, this includes every equality \(h_x=0\) on the rim. Accordingly, \(d_{\rm ext}=T_{\rm ext}^0(0)\) is evaluated with that pin retained. Prescribed relative guard data are affine equalities, so Proposition 3, with the independent increments on cut bonds, still gives \(0<T^\tau/d_T\le1\). To prove the centered bound in (11), impose the flat end event \([H_i]\) in each incident annulus and choose its representative to be zero on that end. The restored-bond factor now depends on the relative integer translations and the exterior heights, and is independent of the guard data. Adding these end constraints decreases the data-to-zero ratio, giving the product of the annular flat-pin ratios in (11). This proof uses no unpinned exterior law. For multiplication, place the exterior, with its full pin constraint, at the root of the tree of cut blocks. Its configurations are summed subject to every retained pin equality. Once its additive constant is fixed, the remaining block constants are in bijection with the integer differences along the tree edges. Conditioning on the full annular guards separates their two sides, so the tree multiplication of [18] and the identity (14) follow with this pin retained.

To finish the attachment calculation, let \(e_\tau\) be the expectation of \(T^\tau_{\rm ext}/d_{\rm ext}\) under the product of the propagated last-guard laws of the whole nests, and define \(e_0\) in the same way. Their densities relative to the independent cut laws are at most \(2^k\), so (15) implies \(1-e_\tau,1-e_0\le 2^k(1-e^{-2\delta-o(1)})\). Exact multiplication and cancellation of \(d_{\rm ext}\) give \[ \Phi_{\rm full}^\tau =\mathcal N_J^k\Phi_{\rm ext}^\tau\frac{e_\tau}{e_0}, \qquad \log\Phi_{\rm full}^\tau =k\log\mathcal N_J+\log\Phi_{\rm ext}^\tau +O_k(\delta)+o(1). \tag{16}\] Here \(\delta\) is small at fixed \(k\), and \(J\to\infty\) with the core, \(P,\lambda,\varrho_0\) fixed. The first identity is exact; the second is the attachment consequence we shall use.

A pinned comparison for the thermodynamic correlation

We first place the desired thermodynamic correlation between two finite height ratios. For a large dyadic \(R_{\rm out}\ge8\), let \[\mathsf F_{R_{\rm out}} =\overline{Q_{7R_{\rm out}/8}}\setminus Q_{3R_{\rm out}/4}.\] The region \(\mathsf F_{R_{\rm out}}\) is a positive-width rim, expressible as an aligned union of positive-area boxes with dyadic vertices. In lattice units let \(G_{R_{\rm out},r}\) be the dual primary array in \(rQ_{R_{\rm out}}\), with coherent rounding, and let \(\mathsf F_{R_{\rm out},r}\) be the filled primary boxes representing \(r\mathsf F_{R_{\rm out}}\). Put \[\Phi_{R_{\rm out},r}^{\rm fr}=\Phi_{G_{R_{\rm out},r}}^\tau, \qquad \Phi_{R_{\rm out},r}^{\rm pin} =\Phi_{G_{R_{\rm out},r},\,h_{\mathsf F_{R_{\rm out},r}}=0}^\tau,\] where the sources are \(0,re_1\), with periods \(+2\pi,-2\pi\).

There is one connection suitable for both comparisons. In the primal lattice take the actual nearest-neighbor path \[(0,0),(1,0),\ldots,(r,0)\] and give its edges the unit current directed from one source to the other. The dual connection is supported on the dual bonds crossing these path edges. They lie inside \(rQ_{R_{\rm out}/2}\), for all large \(r\), so the connection vanishes on every bond meeting the rim or lying outside it. Use this connection in every larger free spin box.

For a box \(\Lambda_R\) containing \(rQ_{R_{\rm out}}\) in its interior, release the matching equalities for all bonds crossing the boundary of \(G_{R_{\rm out},r}\), retaining their independent Bessel increments as in the finite comparison. The resulting inner height component has a free wall; the connection on the exterior component is zero and its ratio cancels. Proposition 3 therefore gives the upper bound by \(\Phi_{R_{\rm out},r}^{\rm fr}\). For the lower bound add the common hard pin \(h_{\mathsf F_{R_{\rm out},r}}=0\) before making any cut. Added precision decreases the sector ratio. This filled rim separates the interior from the entire exterior. Because the connection is supported on the specified path inside the rim, the partition factor for all exterior variables, including the original fixed exterior face, is identical in the numerator and denominator. It cancels. The same cancellation of the outer collar occurs in \(G_{R_{\rm out},r}\) with this rim pinned, so the remaining ratio is exactly \(\Phi_{R_{\rm out},r}^{\rm pin}\). Hence, for every sufficiently large \(R\) at each fixed \(R_{\rm out},r\), \[\Phi_{R_{\rm out},r}^{\rm pin}\le \mathbb E^{\rm free}_{R,b_c}\cos(\theta_0-\theta_{re_1}) \le \Phi_{R_{\rm out},r}^{\rm fr}.\] Taking \(R\to\infty\) at this point proves \[ \Phi_{R_{\rm out},r}^{\rm pin}\le C_{b_c}(r) \le \Phi_{R_{\rm out},r}^{\rm fr}. \tag{17}\] The finite comparisons just used are the affine and matching-constraint comparisons of [17]; the exterior cancellation is why the location of the connection matters.

Figure 1 shows the separation used for the pinned sector estimate. In the later annular application, its two holes are those left in the exterior when the two insertion nests are removed from the full pinned comparator \(\Phi_{R_{\rm out},r}^{\rm pin}\). Its retained arc \(\Gamma\) is the portion of the primal source path between the two hole walls. The guard separates the strip bonds from the artificial side boundary of the patch, including at the hole exits; the rim lies beyond the patch.

The energies outside the nests

Let \(D=(-1,1)^2\) be the reference square, and write its Dirichlet Green kernel as \[G_D(z,w)=\frac1{2\pi}\log\frac1{|z-w|}+R_D(z,w).\] The regular part \(R_D\) is smooth across the diagonal on compact subsets of \(D\times D\). This Dirichlet kernel is the stream function for the free-height variational problem: rotating the gradient of \(2\pi G_D\) gives the unit-period trial field tangent to the outer wall. The exterior-energy calculation of [18] gives, for one circular hole of radius \(t\) at the center, \[ \mathcal M_{\rm cen}(t) =2\pi\log(1/t)+4\pi^2R_D(0,0)+O(t). \tag{23}\] More generally, for separated unit charges \(\sigma_i\) at \(z_i\), that lemma gives the constant term \(4\pi^2[\sum_iR_D(z_i,z_i)+2\sum_{i<j}\sigma_i\sigma_jG_D(z_i,z_j)]\) and the leading term \(2\pi k\log(1/t)\). The error is \(O(t)\) at each fixed separated configuration. It follows by integrating the Green energy on the small circles; the regular harmonic part has zero total normal derivative there, so no additional logarithmic error is created.

For the pair \(0,e_1\) in \(Q_{R_{\rm out}}=R_{\rm out}D\), put \[\mathfrak h_{R_{\rm out}} =R_{Q_{R_{\rm out}}}(0,0)+R_{Q_{R_{\rm out}}}(e_1,e_1) -2G_{Q_{R_{\rm out}}}(0,e_1).\] Green scaling gives \(R_{Q_{R_{\rm out}}}(z,w)=(2\pi)^{-1}\log R_{\rm out} +R_D(z/R_{\rm out},w/R_{\rm out})\). The singular logarithm between \(0\) and \(e_1\) is zero, so \[\mathfrak h_{R_{\rm out}} =R_D(0,0)+R_D(e_1/R_{\rm out},e_1/R_{\rm out}) -2R_D(0,e_1/R_{\rm out}) \longrightarrow0.\] The \(\log R_{\rm out}\) terms have canceled by neutrality. The same exterior formula therefore reads \[ \mathcal M_{{\rm pair},R_{\rm out}}(t) =4\pi\log(1/t)+4\pi^2\mathfrak h_{R_{\rm out}} +O_{R_{\rm out}}(t). \tag{24}\] In both (23) and (24), replacing circular holes by \(tP\) changes the right side by \(O(\varepsilon)\): restrict admissible forms between the exteriors of the circles of radii \(t(1-\varepsilon)\) and \(t(1+\varepsilon)\), and use their logarithmic terms. The constant in this error is independent of small \(t\). The same restriction argument on a circular annulus proves, for either unit period, \[ \mathcal M_{\lambda P\setminus P^\circ}(\pm1) =2\pi\log\lambda+O(\varepsilon), \tag{25}\] with an error independent of large fixed \(\lambda\). This is also the annular energy estimate in the proof of [18].

For the pinned lower bound we exhibit a trial attaining the same pair asymptotics. Identify the plane with \(\mathbb C\), and let \[\Theta(z)=\arg z-\arg(z-1).\] Use the branch on \(\mathbb C\setminus[0,1]\) that tends to zero at infinity. Its gradient has periods \(+2\pi,-2\pi\), and \(\Theta(z)=O(|z|^{-1})\), \(\nabla\Theta(z)=O(|z|^{-2})\) at infinity. Rotation identifies \(\nabla\Theta\), up to orientation, with the gradient of \(\log|z|-\log|z-1|\). Green’s identity on the plane outside the two radius-\(t\) disks gives \[ \int_{\mathbb R^2\setminus(B(0,t)\cup B(e_1,t))}|\nabla\Theta|^2 =4\pi\log(1/t)+O(t). \tag{26}\] Indeed each small circle contributes \(2\pi\log(1/t)+O(t)\). The cross logarithm is \(\log|e_1|=0\), and the boundary term at infinity vanishes by the displayed decay. Harmonicity of each regular part removes a logarithmically amplified error, as in the preceding Green calculation.

Choose a smooth cutoff \(\chi_{R_{\rm out}}\) equal to one on \(Q_{R_{\rm out}/2}\), zero outside \(Q_{2R_{\rm out}/3}\), and satisfying \(|\nabla\chi_{R_{\rm out}}|\le C/R_{\rm out}\). The cut \([0,1]\) lies where \(\chi_{R_{\rm out}}=1\). Thus \(\vartheta_{R_{\rm out}}=d(\chi_{R_{\rm out}}\Theta)\) is a well-defined smooth closed one-form off the holes: its two branches have the same differential across that cut. It has the required periods and a lift identically zero near \(\mathsf F_{R_{\rm out}}\). The energy of both the discarded tail and the cutoff correction is \(O(R_{\rm out}^{-2})\), since on the transition region \(|\Theta\nabla\chi_{R_{\rm out}}|+|\nabla\Theta| \le C R_{\rm out}^{-2}\) and its area is \(O(R_{\rm out}^2)\). The polygon inclusions (10) and (26) consequently give \[ \int_{Q_{R_{\rm out}}\setminus(tP\cup(e_1+tP))}|\vartheta_{R_{\rm out}}|^2 \le4\pi\log(1/t)+C\bigl(t+\varepsilon+R_{\rm out}^{-2}\bigr). \tag{27}\] The constants here are uniform for large \(R_{\rm out}\) and sufficiently small \(t,\varepsilon\).

We record carefully how these energies apply when the last radius moves with the lattice. Suppose \(t_\nu\to t>0\). Around each limiting source, choose fixed dyadic stair holes lying just inside the disk of radius \((1-3\varepsilon)t\) and just outside the disk of radius \((1+3\varepsilon)t\). Small further perturbations of their dyadic centers and sides give strict eventual inclusions of the rounded moving holes \(t_\nu P\). For the pair, choose exit sides whose interiors meet the fixed source-to-source path transversely. All these holes contain the source, remain separated, and leave positive passages. Cut comparison brackets the three sector ratios, using restrictions of the one connection chosen before the cuts. For the pinned ratios keep exactly the same rim in both brackets; the rim is connected, so the integral potential relating two flat representatives with the same periods and zero connection near the rim is constant there and can be normalized to zero. Apply (9) to the fixed free exteriors and Lemma 6 to the fixed pinned inner bracket. The changes of radius cost \(O(\varepsilon)\) in the energy formulas. This is the moving-hole argument of [18], with the pin retained in the lower comparison. Bounded rounding of the outer arrays and filled rim boxes is included in the fixed mixed limits. In particular, every sector limit just invoked has holes of positive radius.

Let \(\Phi_{{\rm cen},\nu}^{\rm ext}(t_\nu)\) denote the free exterior in the reference square \(D\), with one moving polygonal hole at the center. The preceding argument gives \[ \limsup_{\nu\to\infty}\left| \log\Phi_{{\rm cen},\nu}^{\rm ext}(t_\nu)+ \frac{2\pi\log(1/t)+4\pi^2R_D(0,0)}{2a}\right| \le C(\varepsilon+t). \tag{28}\] For the pair exterior in \(Q_{R_{\rm out}}\), write \(\Phi_{{\rm pair},R_{\rm out},\nu}^{\rm ext,fr}(t_\nu)\) for the free-wall ratio and \(\Phi_{{\rm pair},R_{\rm out},\nu}^{\rm ext,pin}(t_\nu)\) for the ratio with \(\mathsf F_{R_{\rm out}}\) pinned. Releasing that pin bounds the latter by the former. Combining (24), (27), and Lemma 6 shows the following useful uniform statement. Given \(\delta>0\), first choose \(R_{\rm out}\) sufficiently large, then choose a bound \(\eta>0\) and \(\varepsilon>0\) sufficiently small. For either boundary choice \(\star\in\{\mathrm{fr},\mathrm{pin}\}\) and every sequence \(t_\nu\to t\in(0,\eta]\), \[ \limsup_{\nu\to\infty}\left| \log\Phi_{{\rm pair},R_{\rm out},\nu}^{\mathrm{ext},\star}(t_\nu) +\frac{4\pi\log(1/t)}{2a}\right|\le\delta. \tag{29}\] Here one uses \(\mathfrak h_{R_{\rm out}}\to0\) and \(R_{\rm out}^{-2}\to0\) first, then the \(O_{R_{\rm out}}(t)\) error. The bound follows for the pinned ratio from its lower trial bound and its free upper bound; it asserts no separate pinned sector theorem. It is uniform over the stated positive values of \(t\) because the energy errors are bounded by \(C_{R_{\rm out}}\eta+C\varepsilon\) and \(C(\eta+\varepsilon+R_{\rm out}^{-2})\). For each such value the fixed stair brackets are chosen after its limit has been extracted.

Cancellation of the microscopic costs

The averaged interaction and its subset targets

From now on \(p=p_{b_c}\), \(\alpha=\sqrt a\), and \(\omega=2\pi/\alpha\). Let \(G\) be a finite retained graph whose primary vertices are partitioned into whole square cells of side \(s\); the estimates below concern the specified tile and box topologies. Let \(B_s\) take the average in every retained cell. Introduce independent standard Gaussian noises \(e\) and observations \[y=B_sh+e.\] For prescribed \(y\), let \(Z_G^\tau(y)\) include the penalty \(\exp(-\|B_sh-y\|^2/2)\), with every retained observation row. The corresponding unpinned real Gaussian energy is \[ E_G^\tau(y)=\min_u \left\{a^{-1}\lVert \nabla u+\tau\rVert_G^2+\lVert B_su-y\rVert^2\right\}. \tag{31}\] On a cut graph the minimum varies every free component constant. At fixed \(y\), the observation penalty confines those constants in the height sum as well; all component translation copies are summed.

We use the reference representation on the connected full graphs: a compatible torus in the centered sector, or an aligned free square in its centered or center-unit sector. On such a graph choose \(v\) so that \(\eta=\nabla v+\tau\) is co-closed, including at a free boundary. Orthogonality to gradients gives \[ E_G^\tau(y)=a^{-1}\lVert \eta\rVert_G^2+E_G^0(y-B_sv). \tag{32}\] Let \(N_{\rm c}\) be the number of observation cells and \(\mathcal Y_G=\mathbb R^{N_{\rm c}}/(2\pi\mathbb Z\mathbf1)\). Integrals on this quotient use ambient Lebesgue measure on a fundamental domain. Write \(Z_G^\tau\) without an observation argument for the bare partition sum, modulo one common height translation. Summing all height translation copies in \(Z_G^\tau(y)\) and integrating their Gaussian penalties gives exactly \[\int_{\mathcal Y_G}Z_G^\tau(y)\,\,\mathrm dy =(2\pi)^{N_{\rm c}/2}Z_G^\tau .\]

Let \(\phi_0\) have covariance \(A^+\) on the mean-free modes, let \(\theta\) be independent and uniform on \([0,\omega)\), and let \(e\) be independent standard observation noise. Put \[Y_0=B_s\alpha(\phi_0+\theta\mathbf1)+e,\qquad Y_\tau=B_sv+Y_0 .\] Their quotient densities are \[\frac{\,\mathrm d\mu_G^0}{\,\mathrm dy} =\kappa_Ge^{-E_G^0(y)/2},\qquad \frac{\,\mathrm d\mu_G^\tau}{\,\mathrm dy} =\kappa_G e^{-\{E_G^\tau(y)-a^{-1}\|\eta\|_G^2\}/2}.\] The same \(\kappa_G\) includes the common-mean Jacobian and cancels. Consequently the exact reference identity is \[ \frac{Z_G^\tau}{Z_G^0} =e^{-\|\eta\|_G^2/(2a)} \frac{\mathbb E\!\left[e^{E_G^\tau(Y_\tau)/2}Z_G^\tau(Y_\tau)\right]} {\mathbb E\!\left[e^{E_G^0(Y_0)/2}Z_G^0(Y_0)\right]} . \tag{33}\] The same calculation applies when either observation integrand is multiplied by a function on \(\mathcal Y_G\) for which the integrals are finite, including the cutoff defined below. This reference identity uses the connected full graph and its co-closed \(v\). The cut-graph sums used later instead factor through their confined component observations.

Initialization will be used on compatible tori, in local prescriptions on the plane, and on aligned free height squares. The free height grid for \(M_n\) has \(2n\) primary dual vertices on a side and one circulation \(2\pi\) at its center. Choose a straight primal current path from the center to a side along observation-cell divisions, and support \(\tau\) on the dual bonds crossing that path. Use the same cut direction and grid origins for every square. In particular the center is the junction of four observation cells. Let \(v_n\) be real vertex values such that \[\eta_n=\nabla v_n+\tau,\qquad \langle \eta_n,\nabla f\rangle=0\quad\hbox{for every real }f.\] The latter identity includes the free boundary and characterizes the minimum-energy connection. Write \(\eta_\infty\) and \(v_\infty\) for the corresponding centered plane connection and branch. Plane expressions in this section are local prescriptions; there is no infinite-plane sector partition integral.

Here is the exact Gaussian split used in the construction. The dyadic ratio \(L\) will be fixed in the parameter choice below. With the four nearest steps, the notation of [16] has \(Q=A/4\). For an integral \(u\ge1\), put \[P_u(\cos t)= \left(\frac{\sin(ut/2)}{u\sin(t/2)}\right)^{64}, \qquad \mathsf P_u=P_u(1-A/64).\] The removable values are understood. The split of [16], with no independent first piece, is \[ \begin{split} C_{<u}&=A^{-1}(1-\mathsf P_u),\\ \Gamma_u&=A^{-1}(\mathsf P_u-\mathsf P_{Lu}),\\ C_{\ge u}&=A^+\mathsf P_u\quad\hbox{on nonconstant modes}. \end{split} \tag{34}\] The first two quotients use their polynomial continuations at zero. They are positive, have ranges \(O(u)\) and \(O(Lu)\), respectively, and telescope on nonconstant modes. Their continued constant modes may be integrated along with the other modes: the final common mean is uniform modulo \(\omega\), so adding an independent Gaussian constant does not change a periodic integral. On a free square these operators are the even restrictions of the operators on the doubled torus.

Orient the retained faces between observation cells. If a face \(f=(i,j)\) is retained, the chosen connection is constant on its microscopic crossing edges; call that value \(\tau_f\). For the observations of (31), set \[ \begin{split} w_f&=y_j-y_i+\tau_f,\\ \chi_s(z)&= \frac{\displaystyle\int_{-s}^{s}\exp\{-(z-t)^2/2\}\,\mathrm dt} {\displaystyle\int_{-s}^{s}\exp\{-t^2/2\}\,\mathrm dt},\\ \widehat Z_G^\tau(y)&= Z_G^\tau(y)\prod_{f\in\mathcal F(G)}\chi_s(w_f). \end{split} \tag{35}\] Here \(\mathcal F(G)\) is the set of retained cell faces. A cut removes its cutoff factor along with its microscopic bonds. The function \(\chi_s\) is entire and even; on the real line it lies in \((0,1]\) and equals one at zero. For \(s\ge1\), Gaussian tails give \[ \begin{split} 1-\chi_s(x)&\le Ce^{-cs^2}\quad (|x|\le s/2),\\ \chi_s(x)&\le C\exp\{-\tfrac12(|x|-s)^2\} \quad (|x|\ge s). \end{split} \tag{36}\]

Fix the connection and collar radii in the block calculus of [16]. At side \(u\), a polymer \(X\) is a finite collection of blocks connected within that fixed block distance; \(|X|\) denotes its block cardinality. Let \(D_u(X)\) consist of the sites in the blocks of \(X\) and in all grid blocks at index distance at most two from \(X\), as in [16]. Difference stencils based there remain in a fixed microscopic stencil neighborhood, denoted \(D_u(X)^+\) when needed. At the aligned scales used here, block sides are multiples of \(s\), and the finite grids and clipped end blocks are whole \(s\)-cell arrays. An open energy \(E_{D_u(X)}\) uses the corresponding union of whole observation cells, all primary bonds with both endpoints in that union (including intercell bonds), and every one of their observation rows; each resulting component constant is free to minimize. Compatible polymers have separated collars. For a finite grid \(\mathcal B_u\), the subset functional is \[ \mathcal Z_u(K;\phi)= \sum_{V\subset\mathcal B_u} \prod_{X\in\mathop{\mathrm{Comp}}(V)}K(X,\phi), \tag{37}\] where components use the fixed connection radius and the empty product is one.

At the compatible entry side \(m\) selected in (59), define for a graph \(\mathcal G\) in one of the connected full classes just listed the interaction after entry averaging by \[ \begin{split} \mathcal H_{\mathcal G,m}^\tau(\psi) &=\mathbb E_{\zeta_{<m},e} \left[ e^{E_{\mathcal G}^\tau(y)/2} \frac{\widehat Z_{\mathcal G}^\tau(y)} {Z_{\mathcal G}^0(0)} \right]_{\,y=B_s(v+\alpha(\psi+\zeta_{<m}))+e},\\ &\hspace{18mm}\mathop{\mathrm{Cov}}(\zeta_{<m})=C_{<m}. \end{split} \tag{38}\] For the centered sector \(v=0\); for a square unit sector \(v=v_n\). The noises and the Gaussian field are independent. Subsequent integration of \(\mathcal H_{\mathcal G,m}^\tau\) and \(\mathcal H_{\mathcal G,m}^0\) against the remaining covariance \(C_{\ge m}\) and the uniform common mean gives the two expectations in the cutoff version of (33), divided by the same \(Z_{\mathcal G}^0(0)\). Their ratio therefore represents the cutoff physical sector ratio after multiplication by \(\exp\{-\|\eta\|^2/(2a)\}\), with \(\eta=\eta_n\) in the square unit sector. The remaining Gaussian covariance is the same in both sectors.

The construction has an exact algebraic target before any norm is chosen. In the centered calculation it will produce a positive field-independent scalar \(c_{\mathcal G,m}\) and ordinary component activities \(K_{\mathcal G,m}^0\) satisfying \[ \mathcal H_{\mathcal G,m}^0(\psi) =c_{\mathcal G,m}\mathcal Z_m(K_{\mathcal G,m}^0;\psi). \tag{39}\] In the center-unit calculation it will use the same scalar and produce unlabelled component activities \(K_{\mathcal G,m}^\tau\) together with one labelled component \(J_{\mathcal G,m}\), so that \[ \mathcal H_{\mathcal G,m}^\tau(\psi) =c_{\mathcal G,m}[x]\, \mathcal Z_m(K_{\mathcal G,m}^\tau+xJ_{\mathcal G,m};\psi). \tag{40}\] Here \([x]\) selects the coefficient of \(x\). The component counted by \(J\) contains the compulsory source marker and every dependency attached to it; \(K^\tau\) counts the other components. The common scalar cancels between the two sectors of this square. Theorem 12 gives the parameter range, makes the ordinary activities small in the norm defined next, and permits polynomial growth of the labelled norm. It also identifies the activity prescriptions that copy exactly between aligned geometries.

To reach these identities with those bounds, we first obtain a large-field reserve from the cutoff and isolate the single circulation cost in the Gaussian energy. The finite expansion then records each component’s complete occupied cover. Its real estimate passes from signed differences to positive halo topologies, cuts those topologies into bounded pieces, and pays the resulting local Gaussian costs. An analytic strip estimate upgrades that real bound to the derivative norm before the connected components are summed.

The activity norm and the local map

The local calculus below uses the exact partition map of [16]. In a related analytic construction, Bauerschmidt, Park, and Rodriguez begin their high-temperature discrete Gaussian analysis with smoothing, then use the successive finite-range Gaussian integrations described by Brydges [6] and a localization of the noncontracting terms inspired by Brydges and Slade [7]; see [3].

The analytic point norm uses all real directional derivatives at a fixed radius \(h_0\): \[ \begin{split} \lVert f\rVert_{u,D} &=\max_{x\in D}\bigl\{|f(x)|,u|\nabla f(x)|, u^2|\nabla^2f(x)|\bigr\},\\ |F|_{u,X,\phi} &=\sum_{k\ge0}\frac{h_0^k}{k!} \sup_{\lVert f_i\rVert_{u,D_u(X)}\le1} |D^kF(\phi)[f_1,\ldots,f_k]|. \end{split} \tag{41}\] All fixed difference stencils are included in the padding. The ordinary regulator \(W_u^\kappa(X,\phi)\) is the exponential regulator in [16]: it controls the gradient energy on \(D_u(X)\), its boundary trace, and scaled suprema of second differences. At a free side the field is extended evenly. We use exactly that regulator, rather than a bound restricted to bounded fields.

Use the accumulated covariance \(C_{<u}\) of (34). Above a common absolute side, this polynomial split is independent of the chosen entry side. Finite-range decompositions by functional calculus have a general antecedent in [2]; the exact polynomials and boundary conventions here are those of [16]. For now regard \(p_1,p_2,h_*\) as parameters satisfying \(1<p_1<p_2\) and \(0<h_*<1/p_2\). They will be fixed with the strict reserve supplied by (57). With \(D=D_u(X)\), put \[ V_u(X,\phi)=\frac12\sup_{\zeta,e} \left\{h_* E_D^0\bigl(B_s\alpha(\phi+\zeta)+e\bigr) -p_1^{-1}\bigl(\|\zeta\|_{C_{<u}}^2+\|e\|^2\bigr)\right\}, \tag{42}\] where \(\|\zeta\|_{C_{<u}}\) is the Cameron norm on the Gaussian Hilbert space, including when the covariance is singular. The full norm is \[ \lVert F\rVert_{u,A_w,V} =\sup_{X,\phi} A_w^{|X|}\frac{|F|_{u,X,\phi}} {W_u^\kappa(X,\phi)e^{V_u(X,\phi)}}. \tag{43}\] When \(u=L^j\), we also write \(V_j=V_{L^j}\), with \(j\) the absolute block index. The block weight \(A_w\) is distinct from the Laplacian \(A\).

The following precise features of the cited calculus will be used. They are analytic statements about activity functions and do not assume that the functions came from a discrete Gaussian height model.

On a regular block \(B\) of side \(u\), the two retained bulk singletons are \[ e_B^0(\phi)=\frac12\sum_{x\in B}\sum_{e=\pm e_1,\pm e_2} |\nabla_e\phi(x)|^2,\qquad c_B^\alpha(\phi)=u^{-2}\sum_{x\in B}\cos(\alpha\phi_x). \tag{45}\] Summing \(e_B^0\) over the blocks gives the reference energy. The weak map of [16] retains \(t e_B^0/2+z c_B^\alpha\); Section 5 states its normal coordinates and the estimates used here.

Reference and cutoff estimates

The source background controls the deterministic cost of translating the reference field. The reflected covariance bounds and the prescribed-observation cutoff bound then provide the reserves used at entry.

Localizing the energy in face variables

The Gaussian proxy in the isolation expansion must still be local when its face values carry the unit circulation. A comparison matrix written only in the values \(y_i\) does not expose this point. We therefore extend the observation minimum to independent real face data.

The entry family in this subsection consists of the free retained graphs produced by the finite tile isolation and box cuts below, together with their current patch restrictions. Every present observation face is wholly retained or wholly cut, and every present observation row is retained. Natural clipped walls and lifted periodic charts are allowed as in that construction. A current patch inherits all earlier cuts. Temporary hard primary guards belong only to the finite comparisons and are not members of this free-energy family.

Regard the primary sites in each cell as a separate set, joining them only across retained faces. If \(w\in\mathbb R^{\mathcal F(G)}\), write \(\nabla_G^w\xi\) for the ordinary difference inside a cell and for \(\xi_j-\xi_i+w_f\) on a microscopic edge crossing the oriented face \(f=(i,j)\). Define \[\mathcal Q_G(w)=\min_\xi \left\{a^{-1}\|\nabla_G^w\xi\|^2+\sum_i|(B_s\xi)_i|^2\right\}.\] For physical \(w_f=y_j-y_i+\tau_f\), the substitution \(\xi|_i=u|_i-y_i\) proves \(\mathcal Q_G(w)=E_G^\tau(y)\). If all four faces incident to a cell vertex \(z\) are retained, let \(\sigma_z(w)\) be their oriented sum around \(z\). There is no such variable at a junction with a cut face. Denote the set of full junctions by \(\mathcal V_4(G)\).

The separated term in (51) is deterministic in every Gaussian integration here. Indeed, if \(y=y^0+B_sv_n\), then \[ \sigma_z\bigl(\nabla_{\rm c}y^0+\nabla_{\rm c}B_sv_n+\tau\bigr) =\sigma_z(\tau), \tag{56}\] where \((\nabla_{\rm c}y)_f=y_j-y_i\). It is zero or the fixed number \(2\pi\). Tile deletions preserve the source junction, so this term cancels exactly in their before-minus-after proxy differences. An auxiliary seam used for a bound can remove the junction, at a cost \(O(\log(2+s))\) once at the source. In (56) neither the high Gaussian field nor the observation noise is present on the right. The term therefore has zero Hessian in their coordinates and creates no Gaussian determinant. Imaginary field translations have gradient face data as well, so the same observation applies to them.

The initialization statement

Combining (47) with (54), and decreasing a fixed \(c_*>0\) if necessary, gives the large-field reserve \[ e^{E_G^\tau(y)/2}\frac{\widehat Z_G^\tau(y)}{Z_G^0(0)} \le \exp\left\{\frac{1-c_*}{2}E_G^\tau(y) +C\,1_{\{\text{unit junction is full in }G\}}\log(2+s)\right\}. \tag{57}\] Indeed the upper bound for \(E_G^\tau\) bounds a fixed positive multiple of that energy by the face-square sum plus the displayed unit term. We may and do take \(0<c_*<1\). This is the reserve that replaces the bare discrete-Gaussian large-field estimate in the entry proof of [16].

We specify the parameter order. First fix the interaction, connection, and collar radii of the local calculus, and the ordinary regulator parameters with the exponent reserves needed for Hölder. Choose \[ 1-c_*<h_1<h_*<1/p_2,\qquad 1<p_1<p_2, \tag{58}\] leaving room to increase \(h_*\) and the ordinary regulator exponent slightly. These choices are possible because \(c_*>0\), and they are made before \(R\) or \(s\). Fix the analytic derivative radius and imaginary strip widths needed for the marginal fundamental frequency \(\alpha=\sqrt{8\pi}\), then a sufficiently large fixed dyadic \(L\), then sufficiently large block weights \(A_w\), and finally a sufficiently small activity ball, in the order of [16]. We may fix a finite number of such weights and regulators at once. In particular we keep an ordinary unmarked weight stronger than the weight used after translating activities by a source lift.

Next fix a dyadic coloring period \(q\) as in the finite isolation expansion of [16]. Choose \(q\) larger than a fixed multiple of the physical proximity, core, open-proxy, guard, testing, and skip radii in units of \(R\) cells. These dimensionless radii are fixed before \(q\) and describe only the local isolation comparisons. With \(q\) fixed, choose a large dyadic \(R\) and put \(P=R^5\). The period \(q\) does not increase with \(R\) or \(s\). The entry side is \[ m=qRs=L^{j_0}. \tag{59}\] The spatial quantities have the following distinct roles:

After \(R,P,q\) are fixed we choose a finite large-field threshold \(U_{\rm big}\), and only then take \(s\) sufficiently large among the dyadic values satisfying (59). Such values are arbitrarily large.

The occupied support in the statement records all evaluated coordinates and local rule data, together with the complete paths that tag a link’s winding or wall dependence. We use its norm padding \(D_m(X)\) as a sufficient region for the entry copy tests. For a torus, say that \(D_m(X)\) unwraps when its sites, bonds, fixed stencils, and occupied path incidences admit one injective lift to the aligned plane block grid. A square copy test requires that this region, including the fixed stencil neighborhood, miss the wall. For a square-to-plane unit copy, identify the cell and block grids by their common center, straight cut ray and direction, origins, and additive branch convention, and require the local cell and block types on \(D_m(X)\) to match under that identification. The construction below checks these conditions summand by summand. Its auxiliary \(P\)-boxes are containers used for estimates and do not add inputs to an activity.

The assertions concern arbitrarily late starts, with constants uniform for all sufficiently large admissible \(s\). They make no uniform claim in temperature and assume no rate for a height scaling limit. In particular the smallness can be prescribed in advance by increasing the fixed \(R\), while the label is only required to obey (60).

The finite expansion and the occupied cover

We prove Theorem 12 by adapting the finite expansion and pattern estimates in [16]. The theorem there is stated for quadratic discrete Gaussian heights; we use its model-independent algebra, Gaussian estimates, and counting, and prove the needed Bessel estimates with the cutoff. The strengthened-law entry of [17] is not an entry theorem for the present law. Its proofs of the bounded-data and analytic estimates will be used with the inputs verified below.

In observation-cell units, start with the square grid of side \(R\). Remove the square of side \(R/2\) centered at each vertex from its incident squares and make each removed square a tile. The trimmed squares and these vertex tiles partition the cells. At most three tiles meet at a junction, and tiles meeting there share a face segment. At a free wall clip complete aligned arrays, terminate the tiles at the wall, and use the same rules on every complete neighborhood. All pieces remain unions of whole cells and belong to the rectilinear geometries of Proposition 4. The four observation cells meeting the source lie with clearance inside a vertex tile. No tile deletion cuts through their unit junction.

Use the finite colors and bounded proximity groups of [16]. For clarity, the colors specify the tile type and its square-symmetry orbit modulo \(q\) in units of \(R\). The period is greater than the fixed physical proximity, core, open-proxy, guard, testing, and skip radii of the isolation step. Each proximity group has a bounded number of members, and groups of one color have disjoint enlarged physical neighborhoods used by the isolation comparisons. Order the colors. At an event isolate its group of tiles by deleting all still-retained bonds to other tiles, and delete the corresponding cutoff faces. Its core, open patch, guard, and larger guard-testing patch have the fixed multiples of \(R\) and separations in the cited construction. The same coloring and grouping apply to clipped wall groups.

Each open patch is a restriction of the current graph and therefore keeps its earlier cuts and retained observation rows. Measured by center sup-distance from a group member in cell units, the cited construction takes core tiles at distance at most \(R\), open-patch tiles at distance at most \(5R\), and guard tiles in that patch starting at distance at least \(3R\). After the fixed stencil inflation of (53), the core and changed artificial boundary still have separation \(cR\), including for clipped or disconnected groups. Thus every remaining Gaussian deletion-versus-patch difference estimated below is an instance of that four-graph estimate.

If \(o_i,c_i\) are the current open patch before and after event \(i\), set \[ \mathcal B_i^\tau(y)=C_i \exp\{-\tfrac12(E_{o_i}^\tau(y)-E_{c_i}^\tau(y))\}, \qquad C_i=\frac{Z_{o_i}^0(0)}{Z_{c_i}^0(0)}. \tag{64}\] Here \(C_i\) is the centered open-patch scalar; only the Gaussian shape depends on the sector. Replace the old weight by \(\mathcal B_i^\tau\) times the cut weight plus their difference. Selecting the difference marks the event; after a mark skip every later event within the fixed skip radius of its group. The radius is larger than the physical core, open-proxy, guard, and testing collars of the isolation rules. A later considered patch then misses each earlier marked core and guard. Its proxy does not depend on the choice of either signed term inside that earlier difference. This is a finite binary identity at the level of the integrand. After the last color, all undeleted cross-tile bonds lie in fixed halos of marks.

Let the superscript \(0{\rm m}\) denote the path with no marks. Dividing the expansion by \[\prod_i\mathcal B_i^{\tau,0{\rm m}}(y) \prod_b Z_b^0(0)e^{-E_b^\tau(y)/2} =\left(\prod_i C_i^{0{\rm m}}\prod_b Z_b^0(0)\right) e^{-E_{\rm asm}^\tau(y)/2}\] defines the assembled Gaussian energy \(E_{\rm asm}^\tau\). The index \(b\) runs over the isolated tiles on that path. The scalar on the right is positive and is exactly the same in the centered and unit calculations. At a real zero centered observation every retained cutoff equals one, so these scalar normalizers need no cutoff correction. In a marked halo retain the whole signed height factor and every affected proxy. Outside the halos expand each isolated factor as one plus \[T_Q^\tau(y)-1,\qquad T_Q^\tau(y)= \frac{\widehat Z_Q^\tau(y)e^{E_Q^\tau(y)/2}} {Z_Q^0(0)} .\] Selecting this second term selects an additional tile. In every term of the unit calculation insert one formal compulsory marker \(I_\star\) at the source. If the source tile is isolated, leave its normalized tile factor unexpanded and attach it to this marker. If it lies in a marked halo, the marker carries the factor one and that halo retains its entire signed factor and its ordinary marked items. The marker is adjacent to them and hence labels their component. In particular, an arbitrarily long chain of attached halos keeps the union of their individual covers; it is not hidden inside the fixed primitive cover of \(I_\star\). Every term has exactly one marker. Identical proxies and final tile factors off the affected halos cancel exactly, before any estimates are made.

The remaining Gaussian mismatch is \((E_{\mathcal G}^\tau-E_{\rm asm}^\tau)/2\). Telescope it over the deletions in the no-mark path. The \(D_s\sigma_z^2\) terms of (51) telescope exactly, since a full junction has the same \(D_s\) in all graphs in which it persists. Lemma 11 therefore expresses the mismatch as a quadratic sum in face values whose coefficients, after summing deletion positions, obey \[C e^{-cR-c\,d(f,f')}.\] The retained core-distance decay in (53) justifies this summation with a constant independent of volume. Enumerate its diagonal and off-diagonal summands as \(b_\ell w_fw_{f'}\), with all symmetry factors included in \(b_\ell\), and expand \[ e^{(E_{\mathcal G}^\tau-E_{\rm asm}^\tau)/2} =\prod_\ell(1+\ell_\ell), \qquad \ell_\ell=e^{b_\ell w_fw_{f'}}-1. \tag{65}\] If \(d_\ell\) is the full tagged length of the link, then \(|b_\ell|\le Ce^{-c(R+d_\ell)}\). The elementary inequality \(|e^z-1|\le |z|e^{|z|}\), followed by \(2|w_fw_{f'}|\le w_f^2+w_{f'}^2\), gives a factor \(Ce^{-c'(R+d_\ell)}\) times a quadratic exponential with coefficients \(Ce^{-c'(R+d_\ell)}\) on its incident faces. For example choose those coefficients proportional to \(|b_\ell|^{1/2}\), which pays the polynomial prefactor while retaining half the exponential gain. Such costs may also be charged on its connecting path. The sum of all reserved coefficients at one face is \(Ce^{-c'R}\), since there are only polynomially many endpoints and path descriptions at a given length. At a fixed finite real observation the product in (65) is absolutely convergent; the integrated absolute convergence will follow from the estimates below.

We retain the following geometric information in a link. On a torus, periodize the columns of the massive observation inverse and retain each displacement image separately. This is an exact periodization because the columns have the exponential decay proved above and the local forcing and proxy rules are periodic. The link’s path keeps its entire winding. In a free square use the plane coefficients between retained faces, with coordinate paths inside the square, and put their difference from the square matrix into additional links. Give each additional link a path that also reaches a nearest wall. The cut-distance bound in (52), together with the direct face-distance and \(R\)-gains, bounds this coefficient by \(Ce^{-c(R+d_\ell)}\) with the wall path included in \(d_\ell\). The fixed \(O(R)\) proxy ranges are absorbed by decreasing \(c\). Average coordinate-order choices and split ties to preserve symmetry. Each path has linear length, so its cover uses \(O(1+d_\ell/R)\) entry blocks; no bounding rectangle is filled. This decomposition makes interior copying exact, since every term that differs at a wall has a tag reaching that wall.

We now specify the occupied supports in the subset functional. For an ordinary mark or selected tile \(I\), first choose one fixed physical envelope containing its tile halo and every core, open-proxy, guard, testing, and seam-comparison piece used for any allowed translate. The envelope has \(O(R^2)\) observation cells per item, with a constant that may depend on the fixed \(q\); every later seam comparison is clipped to this envelope. Some cells in the envelope serve only the estimates; occupying them is a fixed conservative choice made before the later box translate is optimized. Add all fixed field and Gaussian-averaging collars of the local calculus. For a link include its entire chosen connecting path and every winding or wall-reaching tag, with the same collars. All these sets are enlarged to whole observation cells. Let \(\operatorname{cov}_m(I)\) be all \(m\)-blocks met by this full set. The collars include the finite range of \(C_{<m}\) where needed for the averaging dependency. They are fixed once the map parameters and \(q\) are fixed. A halo cover has bounded entry-block size, and a link cover has size \(O(1+d_\ell/R)\). The Gaussian-averaging collars can overlap even for events of one color; the full-cover adjacency below merges every such dependency.

For the compulsory source item, let the compulsory region be the observation-cell square within a fixed large multiple of \(P\) of the center. Begin with this region and the local source-tile dependency when that tile factor is attached. Apply the same full inflation, including every padded dependency of that local factor and any tag assigned to it, and call the resulting set of entry blocks \(\mathfrak C_{\star,m}\). Choose the multiplier of \(P\) large enough that, for any item outside the compatibility neighborhood of this cover, every allowed \(P\)-box containing part of its physical region, together with its fixed testing enlargement, misses the source and uses no stencil near it. This is possible because such a box and enlargement reach only \(O(P)\) cells from the item, whereas the initial square may use a larger fixed multiple of \(P\). Its size is at most \(C_R\), because \(m=qRs\) and \(P,R,q\) are fixed as \(s\) increases. It is connected and contains the center.

Declare two items adjacent whenever their full covers are within the fixed connection distance of the subset calculus. Its initial choice includes the Gaussian range and the binary expansion’s exclusion rules: \(C_{<m}\) has range less than \(32m\) for the printed polynomial, and the exclusions have diameter \(O(Rs)=O(m/q)\). Thus one bound chosen before \(L\) works for every later \(R,s\). This distance is fixed independently of the size of the compulsory cover; the large cover occupies more blocks and does not enlarge the connection radius used by subsequent maps. For a component \(\pi\) without the compulsory item, define its occupied support to be the full union \[ X(\pi)=\bigcup_{I\in\pi}\operatorname{cov}_m(I). \tag{66}\] For a component \(\pi_\star\) containing the compulsory item, define its occupied support to be exactly \[ X(\pi_\star)=\mathfrak C_{\star,m} \ \cup\! \bigcup_{\substack{I\in\pi_\star\\I\ne I_\star}} \operatorname{cov}_m(I). \tag{67}\] In particular, a summand of \(J(X)\) is assigned only to \(X=X(\pi_\star)\). The full set \(\mathfrak C_{\star,m}\) is occupied even where the source factor is constant in the field. Every padded dependency and every tag of an attached item is occupied as well. We never replace this set by a root block or by a smaller subcover. Items that meet any part of this compatibility neighborhood belong to \(\pi_\star\), even if their factors would otherwise be independent. Thus their ordinary patterns have not been discarded; they are summed in the labelled component.

Separated components now have separated field and noise coordinates and separated Gaussian subvectors below \(m\). They also have no exclusion constraint between them: every such constraint was included in the adjacency just defined. Within a component, skip and validity constraints are functions of its recorded items and local rule data. Potentially incompatible exterior items need not lie in \(D_m(X)\); when selected, their exclusion is enforced by subset compatibility rather than consulted as an input to this component’s activity. Averaging a pattern in (38) therefore factors over these components. The sum over components without the compulsory item defines the ordinary activity; the sum over the component \(\pi_\star\) defines \(J\). All constraints within a component remain in its defining sum. This gives the subset algebra (39) and (40), once the absolute convergence below justifies the exchanges with Gaussian integration. The common scalar is the positive scalar of the no-mark divisor divided by \(Z_{\mathcal G}^0(0)\). No independence of the random guards or of the binary marks has been assumed.

In the unit calculation, define \(K^\tau(X)\) by its ordinary pattern sum whenever \(X\) can coexist with \(\mathfrak C_{\star,m}\). Values on supports that cannot coexist do not enter \([x]\mathcal Z_m(K^\tau+xJ)\), by (67). Set those values to zero, except that when the rectangle spanned by \(D_m(X)\) misses the source, set them to the translated centered prescription. This convention is useful for later local maps and does not alter the exact identity.

We check the sufficient copy region at the level of each summand assigned to \(X\). Every evaluated field or noise coordinate and every local isolation or proxy stencil lies in the full cover and hence in \(D_m(X)\). The grid phase and color rule is common under the compatible aligned identification. The cover also contains the whole averaging collar for \(C_{<m}\), whose range is less than \(32m\). It occupies every projected path site, while the summand retains the complete lifted winding and tag data. Compatible aligned block identifications preserve these local data and the within-component constraints. The optimized \(P\)-grid, its energy classification, and empty portions of its boxes are used only in the proof of the bound; they are not data in the defining activity.

There is one qualification for the numerical link coefficients. An untagged coefficient is the fixed universal plane table indexed by its faces and displacement; the definition of that table may use the plane inverse. A torus image carries its complete winding path. A square-minus-plane coefficient may depend on the whole square, but its additional link carries a complete path to the wall. Consequently a summand with such an ambient obstruction cannot be assigned to an \(X\) whose \(D_m(X)\) unwraps, or respectively misses the wall. All contributing rules and coefficients in that case are the common plane ones. This proves the centered copy statements.

In a source-free rectangle spanning \(D_m(X)\), an integral gauge removes \(\tau\). It turns every retained factor in (35) into the ordinary neighbor cutoff and changes the field argument to the whole lifted translate \(\psi+\lambda_X\), proving (62). Near a wall the two sectors use the same finite-square centered coefficient, including its possible dependence on remote parts of that square. Under the centered aligned identification in the theorem, when \(D_m(X)\) misses the wall, the affine energies, cutoff, and source factor are the same plane-unit rules. Changing \(v_\infty\) to \(v_n\) translates their argument by (63). The full compulsory cover and every attached cover remain occupied under these identifications. The artificially filled entries above are either zero or the same translated centered prescription. Thus the identities hold for the complete component sums with \(D_m(X)\) as a sufficient entry copy region. The later causal neighborhood in Section 6 tracks the subsequent maps separately.

Bounds for the pattern sums

Fix a pattern \(\pi\). Let \(j_1\) be its number of binary marks, \(j_2\) its number of additional selected tiles, and \(d_\ell\) the tagged lengths of its links. Let \(X\) be the full occupied cover just defined. Write \(F_\pi(\psi)\) for its signed factor after averaging the noise and the Gaussian pieces below \(m\), with the common scalar removed. Put \[\mathcal C_\pi(s)= \begin{cases} 1,&\pi\text{ has no compulsory item},\\ C_Rs^{C_R},&\pi\text{ has the compulsory item}. \end{cases}\] The constants \(C_R\) in this notation may increase during the proof, but remain independent of \(s\) and the volume.

Analytic coordinates and a coefficient estimate

We state precisely the analytic part of the companion recursion used here. Within one history omit its superscript \(k\). Write \(\Phi_j\) for the complete normalized unbalanced plane weak map on the additive coordinates above. It includes the scalar inverse decorations in the partition algebra of [16], and \(\Phi_j(0)=0\). At the fixed reference \(a=8\pi\), its linearization is \[ (t,z,\mathcal R)\longmapsto \bigl(t+P_j^t\mathcal R,\ \lambda_jz+P_j^z\mathcal R, \ \mathsf C_j\mathcal R\bigr), \qquad \|\mathsf C_j\|\le\vartheta_0<1. \tag{93}\] The projections are bounded uniformly in \(j,k\); the omitted part is analytic with norm at most \(C(|t|+|z|+\|\mathcal R\|)^2\), with its Taylor derivative bounds. These statements are [16], applied with the regulator of [16]. They concern activities at a fixed frequency. For a compatible torus the singleton coefficients copy the plane coefficients up to its stop, while its own remainder satisfies the same contraction and analytic bounds. No equality between the plane and torus remainders is part of this assertion.

For our polynomial shells, \(\lambda_j=L^2 e^{-a\Gamma_{L^j}(0,0)/2}\). The continuum value is one. The fixed nearest-neighbor estimate in [16], with the independent microscopic piece omitted as in [16], gives \[ |\lambda_j-1|\le CL^{-c_1j},\qquad \ell_j:=\prod_{h\ge j}\lambda_h=1+O(L^{-c_1j}) \tag{94}\] after increasing the first absolute scale. Here and below an exponent may be decreased without changing notation. Put \(\mathsf C_{h:j}=\mathsf C_{h-1}\cdots\mathsf C_j\) for \(h>j\) and \(\mathsf C_{j:j}=I\). The balanced coordinates are exactly \[ T=t+\sum_{h\ge j}P_h^t\mathsf C_{h:j}\mathcal R, \qquad Z=\ell_jz+\sum_{h\ge j}\ell_{h+1}P_h^z\mathsf C_{h:j}\mathcal R. \tag{95}\] The series have geometric operator tails. These are [16] at the marginal eigenvalue one. In particular the small linear errors in \(\lambda_j\) have already been balanced; they are not included in a nonlinear activity error. The formulas for the maps and the shells above a common absolute side are independent of the entry side. The entry-dependent regulator is used only to bound these formulas.

We use one neutral terminal response both for an exact Gaussian trajectory and for the physical histories below. Fix a sufficiently large dyadic \(D_0\), exceeding all causal padding radii, and set \[ n_j=D_0L^j. \tag{96}\] For each trajectory used below, a torus stopped at side \(L^j\) is initialized separately at its entry side on the compatible grid of side \(n_j\). It has \(D_0^2\) terminal blocks, and the fixed buffer makes its singleton coefficients copy the plane coefficients through that stop. This construction uses only the local map. The physical cutoff tori introduced below will also have to satisfy \(n_j\ge s_kP\).

On this torus let \(A\) be the unit-conductance Laplacian, and let \(\mathbb E_j^{\rm ter}\) integrate a mean-free Gaussian \(\psi\) with covariance \(C_{\ge L^j}=A^+\mathsf P_{L^j}\) and an independent constant \(\theta\), uniform modulo \(\omega\); write \(\phi=\psi+\theta\). Let \(q_n(x)=\cos(2\pi x_1/n)\), with the origin fixed by the compatible grid, and put \(F_{n,\gamma}(w)=w\gamma q_n\). Here \(\gamma\) ranges over a fixed compact subinterval of \((0,\infty)\). Since \(n=n_j=D_0L^j\), the field norms of (41) satisfy \(\sup_{\gamma,X}\|\gamma q_{n_j}\|_{L^j,D_{L^j}(X)}\le C\). Choose a fixed complex source disk \(|w|<w_{\rm ter}\) with \(w_{\rm ter}C<h_0\) and a fixed strict margin. All energy pairings involving complex sources below are the complex bilinear extensions of the real lattice pairing. Thus \(\langle F_{n,\gamma}(w),AF_{n,\gamma}(w)\rangle =w^2\gamma^2\langle q_n,Aq_n\rangle\).

At this fixed grid, and for each entry regulator separately, let \(K\) range over complex local \(\omega\)-periodic analytic activities whose field dependence has the support collars in Proposition 7. Write its unbalanced coordinates as \((t,z,\mathcal R^{\rm tor})\), and use the open complex ball \(\nu=|t|+|z|+\|\mathcal R^{\rm tor}\|<\rho_{\rm ter}\), with the current torus norm for that entry. All activity variations below are taken within one such space. The finite subset sum and the source Taylor series controlled by \(h_0\) make \(\mathbb E_j^{\rm ter}\mathcal Z_{L^j}(K;\phi+F_{n_j,\gamma}(w))\) jointly analytic in the activity coordinates and \(w\). Its absolute expansion on the fixed \(D_0^2\)-block grid bounds its difference from one by \(CD_0^2\nu\), uniformly on the source disk. Choose \(CD_0^2\rho_{\rm ter}<1/2\). Both the shifted and unshifted integrals are then nonzero, so define \[ \mathcal W_j(K;F)= \log\frac{\mathbb E_j^{\rm ter}\mathcal Z_{L^j}(K;\phi+F)} {\mathbb E_j^{\rm ter}\mathcal Z_{L^j}(K;\phi)}. \tag{97}\] The common phase has the same normalized law in both integrals, and the logarithm takes the branch equal to zero at \(F=0\). On fixed smaller concentric activity and source domains, joint Cauchy bounds control every fixed input and source derivative uniformly in the entry and stop. The tubes below lie in such an inner activity ball; its convexity keeps the segments between actual and pure states inside the larger analytic domain.

For the half-period action \((\mathcal S K)(X,\phi)=K(X,\phi+\omega/2)\), applied to every constituent together, the activity ball is invariant. Changing the common phase in both integrals gives \(\mathcal W_j(\mathcal S K;F)=\mathcal W_j(K;F)\). On the inner domains the terminal expansion gives, with two source derivatives, \[ \begin{split} \mathcal W_j(K;F_{n_j,\gamma}(w)) &=\frac t2\langle F_{n_j,\gamma}(w),AF_{n_j,\gamma}(w)\rangle +O(\|\mathcal R^{\rm tor}\|+\nu^2),\\ \left.\partial_w^2\mathcal W_j(K;F_{n_j,\gamma}(w))\right|_{w=0} &=\kappa_{n_j,\gamma}t+O(\|\mathcal R^{\rm tor}\|+\nu^2),\\ \kappa_{n,\gamma} &=\gamma^2\langle q_n,Aq_n\rangle =2\gamma^2n^2\sin^2(\pi/n)\asymp1. \end{split} \tag{98}\] In particular this reads the actual unbalanced gradient \(t\). To obtain the expansion, retain the terms linear in \(K\) in the terminal subset sum. The centered Gaussian removes the cross term in the gradient energy, and the common phase removes the fundamental singleton even after the shift. The linear remainder costs its norm; the remaining terms cost \(O(\nu^2)\) by the finite block count and the regulator reserve. This is the calculation in [16], here for the specified first cosine.

For each chosen entry side there is also an exact Gaussian trajectory. At side \(m\), for every nonempty polymer \(X\) connected in the subset convention, define the local occupied-component input by \[g_B(d,\phi)=e^{d e_B^0(\phi)/2}-1,\qquad K_m^g(X,\phi;d)=\prod_{B\in X}g_B(d,\phi).\] For every occupied set \(V\), the product over its components is \(\prod_{B\in V}g_B\). Hence on each finite grid \(\mathcal Z_m(K_m^g;\phi)=\prod_B(1+g_B) =\exp\{d\sum_Be_B^0(\phi)/2\}\). Use the same local input on the plane and compatible tori. With the fixed activity weights, a common sufficiently small \(d\)-disk makes this an allowed analytic input: its linear term is the singleton \(d e_B^0/2\), and its remaining singleton and multi-block terms have norm \(O(d^2)\). Every factor is invariant under common field translations, so the map preserves the vanishing of all charged coordinates. Denote the balanced gradient, unbalanced gradient, and unbalanced remainder at scale \(j\) by \(T_j^g(d)\), \(t_j^g(d)\), and \(\mathcal R_j^g(d)\). On a fixed complex disk, uniformly in all sufficiently large entry sides and all later steps, \[ T_j^g(d)=d+O(d^2),\qquad Z_j^g(d)=0, \qquad \|\mathcal R_j^g(d)\|\le C|d|^2. \tag{99}\] The bounds are analytic, the functions \(T_j^g\) have inverses on a common smaller disk, and the step carries the point with parameter \(d\) to the point with that same parameter. The fixed-entry curve is [16]; we record why its estimates are uniform as the entry side varies. For entry side \(m\) and each stop \(j\), let \(K^{g,\rm tor}_{j|j}(d)\) be the terminal activity of the separately initialized pure torus. Apply the exact local-map identities to \(\phi+F\), where \(F=F_{n_j,\gamma}(w)\). The field-independent extraction factors cancel in (97). The shells from \(m\) to \(L^j\) and the terminal covariance telescope to \(C_{\ge m}\) on mean-free modes, and \(\sum_B e_B^0(\phi)=\langle \phi,A\phi\rangle\) on the torus. Gaussian completion thus gives, while the terminal state lies in the readout ball, the exact identity \[ \mathcal W_j(K^{g,\rm tor}_{j|j}(d);F) =\frac d2\langle A^{1/2}F, (I-dA^{1/2}C_{\ge m}A^{1/2})^{-1}A^{1/2}F\rangle. \tag{100}\] The gradient energy and the mean-free shift are insensitive to the common constant. On mean-free modes \(A^{1/2}C_{\ge m}A^{1/2}=\mathsf P_m\) lies between zero and \(I\). For \(|d|<\delta<1\) the Gaussian determinant is nonzero and cancels in the normalized ratio, while the resolvent has norm at most \((1-\delta)^{-1}\). The second source derivative of (100) is consequently \(O(|d|)\), uniformly in the entry and stop. These pure tori are available at every stop after their entry; they do not use the physical cutoff sewing.

Before a first exit from a ball of radius \(\rho\), the bounded map gives every candidate plane and stopped-torus state full size at most \(C\rho\). Choose \(\rho\) so that these states lie inside the fixed inner readout ball. Remainder contraction bounds both the plane and stopped-torus remainders by \(C(|d|^2+\rho^2)\). In this pure case the fundamental coordinate vanishes. Comparing the second derivative in (100) with (98), whose multiplier is bounded below and whose scalar is the plane \(t_j^g\) by copying, gives \[|t_j^g|\le C\bigl(|d|+\|\mathcal R^{g,\rm tor}_{j|j}\| +( |t_j^g|+\|\mathcal R^{g,\rm tor}_{j|j}\|)^2\bigr) \le C(|d|+\rho^2).\] The candidate full-size bound makes the squared term \(O(\rho^2)\). The constants are independent of the entry and stage. Choose \(\rho\) small and then one disk radius \(\delta>0\) so that these bounds exclude every first exit for \(|d|\le\delta\). Every finite composition is consequently analytic and uniformly bounded on that common disk. The triangular linearization gives \(\mathcal R_j^g(0)=(\mathcal R_j^g)'(0)=0\) and \(T_j^g(0)=0,(T_j^g)'(0)=1\). Cauchy estimates on a smaller common disk give (99) and its derivative bounds; one further shrink gives the common inverse disk. This is the no-exit mechanism in [17], using here only the neutral contraction at the marginal parameter.

Define, for this history, \[ u_j=(T_j^g)^{-1}(T_j),\qquad Q_j=\mathcal R_j-\mathcal R_j^g(u_j),\qquad Q_j=Q_{e,j}+Q_{o,j}, \tag{101}\] where \(Q_e,Q_o\) are the even and odd parts under the common shift \(\phi\mapsto\phi+\pi/\alpha\). The pure Gaussian points in these coordinates are \((u,0,0)\) with \(u\) unchanged by the step. Write \(Y_j=|Z_j|\).

For the precision in Proposition 14 we need a rate for these two map coefficients. The next estimate is a refinement of the finite-depth argument in the proof of the cited proposition.

Physical constraints from stopping tori

For history \(k\), the valid physical stops on the compatible cell and block grids are \[ \mathcal J_k=\{j\in\mathbb Z:j_0(k)\le j\le4\cdot2^k,\ n_j\ge s_kP\}. \tag{111}\] For every \(j\in\mathcal J_k\), initialize the exact cutoff torus of side \(n_j\) separately at \(m_k\) and run it to side \(L^j\). At a stage \(j_0(k)\le i\le j\), its scalar coordinates are the plane values \(t_i^{(k)},z_i^{(k)}\), by copying their padded neighborhoods. Write \(\mathcal R_{i|j}^{\rm tor,(k)}\) for its own remainder. Its norm is the torus version of \(\|\cdot\|_{i,k}\) with the same entry regulator. We suppress \(k\) while working within one history. No torus is continued from a different size. When \(n_j<s_kP\) only plane steps are needed. There are at most \(\max\{0,\lceil\log_L(R^4/(D_0q))\rceil\}\) of them. The bounded map amplifies the entry norm \(Ce^{-cR}\) by only \(R^{O(1)}\) on these steps, so increasing \(R\) controls them. These are the stopping conventions and the initial estimate in [16].

Let \(\mathbb E_n^{\rm tor}\) denote the original periodic Bessel height law on \((\mathbb Z/n\mathbb Z)^2\), with its constant modulo \(2\pi\). Put \(f(x)=\cos(2\pi x_1)\) on the unit torus and \[ \mathfrak g=\langle f,(-\Delta_{\mathbb T^2})^{-1}f\rangle>0,\qquad x_n=\langle h,n^{-2}f(\cdot/n)\rangle,\qquad X_{n,s}=\langle y,f_{n,s}\rangle, \tag{112}\] where \(f_{n,s}(Q)=(s/n)^2 f(x_Q/n)\). Here \(x_Q\) is the arithmetic center of the primary sites of \(Q\), in the same coordinates as \(q_n\): for \(Q=v_Q+\{0,\ldots,s-1\}^2\), \(x_Q=v_Q+\tfrac{s-1}{2}(1,1)\). Both tests have zero total. Also write \(\chi_{n,s}=e^{i\bar y}\), where \(\bar y\) is the mean of the cell observations, and write \(e^{i\bar h}\) for the uniform height character.

For \(j\in\mathcal J_k\), let \(\widehat\mathbb E_{j,k}\) be the normalized cutoff expectation on the torus (96) in history \(k\). The cutoff bound of Section 4, summed over the faces of these tori, changes bounded phase observables and the fixed smooth Laplace diagnostics by at most \[ \delta_k\le \exp(-c_5 4^k),\qquad j\le4\cdot2^k, \tag{115}\] for large \(k\). Indeed their log volume is \(O(2^k)\), while \(s_k^2\asymp4^k\). The same statement for the fixed source derivatives follows from the tilted estimate and its moment bound.

We now identify the physical Laplace correction with the neutral response (97). Let \(K_{j|j}^{\rm tor}\) be the terminal activity at a valid stop, with remainder \(\mathcal R_{j|j}^{\rm tor}\), and set \(\nu_j=|t_j|+|z_j|+\|\mathcal R_{j|j}^{\rm tor}\|\). On zero-total cell vectors the full reference observation covariance is \(\Sigma^0_{n,s}=I+aB_sA^+B_s^*\), and \(v^0_{n,s}=\langle f_{n,s},\Sigma^0_{n,s}f_{n,s}\rangle\) is the reference variance of \(X_{n,s}\). Complete the source \(wX_{n,s}\) before the entry averaging. It contributes \(e^{v^0_{n,s}w^2/2}\) and translates the observation argument of the entire interaction by \(w\Sigma^0_{n,s}f_{n,s}\). Cell translation and reflection symmetry make this a first cell cosine. The corresponding microscopic first cosine has a nonzero cell average, so choose \(F_{n,s}(w)\) with \(\alpha B_sF_{n,s}(w)=w\Sigma^0_{n,s}f_{n,s}\). The finite cosine sum shows that \(F_{n,s}(w)=w\gamma_{n,s}q_n\), with \(\gamma_{n,s}\) in a fixed compact subinterval of \((0,\infty)\) when \(n\ge sP\), after \(P\) and the entry threshold are large. Choose the interval in (98) to contain these amplitudes. The exact map identities then give, uniformly on a fixed source disk, \[ \begin{split} \log\widehat\mathbb E_{j,k} e^{wX_{n_j,s_k}}-\tfrac12v^0_{n_j,s_k}w^2 &=\mathcal W_j(K_{j|j}^{\rm tor};F_{n_j,s_k}(w))\\ &=\tfrac12 t_j\langle F_{n_j,s_k}(w),AF_{n_j,s_k}(w)\rangle +O(\|\mathcal R_{j|j}^{\rm tor}\|+\nu_j^2). \end{split} \tag{116}\] The first equality uses the same extraction scalars in numerator and denominator, so they cancel. The estimate includes two source derivatives. The uniform mean integral removes the fundamental term at linear order, and \(\langle F_{n,s}(w),AF_{n,s}(w)\rangle/w^2\to a\mathfrak g\) as \(s/n\to0\). This is the physical use of (98); the finite cosine estimates, including their errors, are derived in the overlap comparison below.

There is also a mixed phase diagnostic; write \(s=s_k\) in this paragraph. On the same activity ball and source disk, define the normalized phase functional \[ \mathcal C_j(K;F)= \frac{\mathbb E_j^{\rm ter} [e^{i\alpha\theta}\mathcal Z_{L^j}(K;\phi+F)]} {\mathbb E_j^{\rm ter}\mathcal Z_{L^j}(K;\phi)}. \tag{117}\] The inserted factor has modulus one, so the same terminal bounds give joint analyticity and uniform Cauchy bounds on the fixed inner domains. Changing the common phase by \(\omega/2\) gives \(\mathcal C_j(\mathcal S K;F)=-\mathcal C_j(K;F)\); the denominator is invariant. Let \(\tau_{j,s}\) be the variance of the integrated high mean in height units, including the mean observation noise. These means are independent of the retained mean and of the mean-free source. The factor \(e^{\tau_{j,s}/2}\) and its inverse are bounded. For the actual \(K=K_{j|j}^{\rm tor}\), the completion in (116) identifies the full physical phase functional, after removal of the Gaussian prefactor, with \(e^{\tau_{j,s}/2}\mathcal C_j(K;F_{n_j,s}(w))\). This formula defines its analytic, odd extension to the activity ball. Subtracting the value at \(w=0\) gives \[ \begin{split} &e^{-v^0_{n_j,s}w^2/2}\widehat\mathbb E_{j,k} [\chi_{n_j,s}e^{wX_{n_j,s}}] -\widehat\mathbb E_{j,k}\chi_{n_j,s}\\ &\qquad=e^{\tau_{j,s}/2} [\mathcal C_j(K;F_{n_j,s}(w))-\mathcal C_j(K;0)]\\ &\qquad=z_jM_{j,s}(w)+O(\|\mathcal R_{j|j}^{\rm tor}\|+\nu_j^2). \end{split} \tag{118}\] The second equality is the terminal expansion, where \[ M_{j,s}(w)=p_{j,s}\operatorname{ave}_x (e^{-i\alpha F_{n_j,s}(w,x)}-1), \qquad p_{j,s}=\frac{D_0^2}{2} e^{\tau_{j,s}/2-aD_{{\rm tail},j}/2}. \tag{119}\] Here \(D_{{\rm tail},j}\) is the diagonal of the remaining mean-free field covariance. The average in (119) is over the microscopic torus sites. This is [16]. To see the sign of its high-mean factor, averaging the interaction over an independent Gaussian constant multiplies its first Fourier coefficient by \(e^{-\tau_{j,s}/2}\). The physical mean insertion must undo that averaging, giving \(e^{\tau_{j,s}/2}\). Both \(p_{j,s}\) and its inverse are bounded. For a fixed small real \(w\ne0\), the average in (119) stays strictly negative when \(s/n_j\) is small, so \(|M_{j,s}(w)|\) is bounded below. In particular both scalar directions have nonzero diagnostic multipliers.

The odd signal and comparison of histories

The phase diagnostic also gives a lower bound, on the absolute scale, which prevents an arbitrarily late entry from losing the odd direction. Let \(B_n=\{0,\ldots,n-1\}^2\). For \(e\in\{\pm e_1,\pm e_2\}\), let \(\rho_{n,e}\) be the uniform probability on its face with outward normal \(e\), including the \(n\) vertices of that face. Put \[\begin{gathered} \bar\rho_n=\frac14\sum_e\rho_{n,e},\qquad \rho_n^{\rm vol}=n^{-2}\mathbf 1_{B_n},\\ z_n^{\rm face}=\mathbb E_{B_n}^{h,\rm free}e^{i\langle h,\bar\rho_n\rangle}, \quad z_n^{\rm vol}=\mathbb E_{B_n}^{h,\rm free}e^{i\langle h,\rho_n^{\rm vol}\rangle}. \end{gathered}\] where \(\mathbb E_{B_n}^{h,\rm free}\) is the free Bessel height law at \(b_c\), modulo its common \(2\pi\) translation. Both profiles have total one, and these centered characteristic coefficients are positive. The critical face estimate [17], using \(a=8\pi\), says that for every fixed sufficiently large \(G=L^r\), \[\liminf_{n\to\infty}z_{Gn}^{\rm face}/z_n^{\rm face}\ge c_6>0,\] where \(c_6\) is independent of \(r\). Replace it by a smaller number in \((0,1)\). For fixed \(r\), iterate the eventual lower bound \(c_6/2\) on the \(r\) residue classes of the absolute index in \(n_j=D_0L^j\). It follows that \[\limsup_{j\to\infty}\frac{-\log z_{n_j}^{\rm face}}{j} \le \frac{\log(2/c_6)}{r}.\] Letting \(r\to\infty\) proves \(-\log z_{n_j}^{\rm face}=o(j)\). The difference between the face and uniform profiles has bounded primary reference cost. Indeed, for one face route each site’s mass \(n^{-2}\) to its projection on that face along a row or column. Each of \(O(n^2)\) edges carries at most \(1/n\), so the squared flow energy is \(O(1)\); averaging the four flows gives the same bound for \(\bar\rho_n-\rho_n^{\rm vol}\). The phase-transfer inequality [17] therefore gives \[\bigl|\sqrt{-\log z_n^{\rm face}} -\sqrt{-\log z_n^{\rm vol}}\bigr|\le C,\] and hence \(-\log z_{n_j}^{\rm vol}=o(j)\). Restoring the periodic bonds increases this centered characteristic by [17]. We have proved \[ \mathbb E_{n_j}^{\rm tor} e^{i\bar h}\ge e^{-o(j)}. \tag{121}\] This is a statement about one physical torus law at each absolute \(j\). The noise factor in Lemma 17 is bounded below, and the cutoff error (115) is negligible compared with (121) on any fixed-fraction interval. Thus \(|\widehat\mathbb E_{j,k}\chi_{n_j,s_k}|\ge e^{-o(j)}\) uniformly there.

We next compare two histories through the same physical torus at a common absolute index. The following statement quantifies the overlap of their useful windows.

The recurrence with dyadic changes of history

We finish the proof of Proposition 14. At absolute index \(j\) choose \((u_j,Y_j)\) from \(\mathcal H^{(k)}\) when \(2^k\le j<2^{k+1}\). These spliced scalars are positive in the second coordinate and tend to zero. At a step that stays in one history, Lemmas 15, 16, and 19 give \[\begin{align*} u_{j+1}-u_j &=-Y_j^2\{H+O(|u_j|+Y_j+(1+j)^{-2})\}, \tag{131}\\ \frac{Y_{j+1}}{Y_j} &=1-b_*u_j+O(u_j^2+Y_j^2+|u_j|(1+j)^{-2}). \tag{132}\end{align*}\] At a crossing \(j+1=2^k\), first take the old history’s ordinary step and then replace it at the same absolute \(j+1\) by the new history. Lemma 20 adds at most \(C Y_j^2\) to (131), and at most \(C(|u_j|+Y_j)\) to the ratio in (132). Here the slow step ratio and the overlap ratio make the old and new \(Y\)’s comparable. These bounds allow the \(CY_j\) term at a switch before \(u_j\) and \(Y_j\) are known to be comparable.

We give the barrier argument with these switch errors. Sample the tail at fixed lengths \(K\). At most one switch occurs in a sampled step when the initial index is sufficiently large. For fixed \(K\), the slow ratios give \(Y_i/Y_j=1+o(1)\) throughout the step, and its change in \(u\) is \(O_K(Y_j^2)\). Choose \(K\) larger than a fixed multiple of the constant in the switch error. Summing (131), then going sufficiently far into the tail, gives, with \(+\) now denoting one sampled step, \[ -C_2 K Y^2\le u_+-u\le -C_1 K Y^2 \tag{133}\] for fixed \(C_1,C_2>0\). The sampled \(u\)’s decrease to zero, so they are positive. Summing the ratio equation and absorbing the possible \(C|u|\) switch error into its coefficient gives \[ \frac{Y_+}{Y}=1-\widetilde b u+\mathcal E, \qquad C_3K\le\widetilde b\le C_4K,\qquad |\mathcal E|\le C_5Y+C_KY^2. \tag{134}\] The constants \(C_1,\ldots,C_5\) can be fixed for all sufficiently large \(K\); the tail threshold and \(C_K\) may depend on \(K\). To justify this form, use \(u_i=u+O_K(Y^2)\) inside the step and expand the product of its \(K\) ratios. The coefficient errors \(O(|u_i|(1+i)^{-2})\) sum to \(O_K(uj^{-2}+Y^2j^{-2})\), since the sampled \(u\) is positive and \(|u_i|\le u+C_KY^2\). Absorb the first part into \(\widetilde b\) and the second into \(\mathcal E\). The remaining ordinary errors and products are \(O_K(u^2+uY+Y^2)=O_K(u^2+Y^2)\); their \(u^2\) part changes \(\widetilde b\) by \(O_K(u)\). The switch term splits into a part bounded by \(Cu\), absorbed there as well, and a remainder bounded by \(C_5Y+C_KY^2\). Thus \(\widetilde b=Kb_*+O(1)+O_K(u+j^{-2})\). Taking \(K\) large and then the tail sufficiently late leaves the displayed positive coefficient interval.

Choose \(c_->0\) small. In the region \(0<u\le c_-Y\), (133)–(134) imply \[(c_-Y_+-u_+)-(c_-Y-u) \ge \{C_1K-C_4Kc_-^2-C_5c_--C_Kc_-Y\}Y^2.\] Choose \(c_-\) so the quadratic loss is small, take \(K\) large for the fixed switch loss, and then take the tail large for the last loss. The right side is a positive multiple of \(KY^2\). Thus a nonnegative lower gap becomes strictly positive and keeps increasing as long as it is in this region; the same inequality keeps it in the region. This contradicts \(u,Y\to0\). Similarly, in the region \(u\ge c_+Y\), \[(u_+-c_+Y_+)-(u-c_+Y) \ge \{-C_2K+C_3Kc_+^2-C_5c_+-C_Kc_+Y\}Y^2.\] Choose \(c_+\) sufficiently large, then \(K\) and the tail as above. The upper gap also increases strictly, again contradicting convergence. It follows that \(c_-Y<u<c_+Y\) on the sampled tail. The bounds within a sampled step give \(u_j\asymp Y_j\) at every sufficiently large index.

Increase the fixed \(K\) once more if needed so that \(C_3Kc_->2C_5\). Equations (134) and the two barriers then give \(Y-Y_+\asymp K Y^2\) on the sampled steps. Their ratios tend to one, so taking reciprocals gives \(1/Y_+-1/Y\asymp K\). Summing over the sampled indices and filling the bounded gaps proves \[ u_j\asymp Y_j\asymp j^{-1}. \tag{135}\]

It remains to read the constant with the required error. Put \(c_0=\sqrt{H/b_*}\) and \(I_j=u_j^2-c_0^2Y_j^2\). At an ordinary step, (131)–(135) give \[u_{j+1}-u_j=-HY_j^2+O(j^{-3}),\qquad Y_{j+1}-Y_j=-b_*u_jY_j+O(j^{-3}).\] In \(I_{j+1}-I_j\), the leading terms \(-2Hu_jY_j^2+2c_0^2b_*u_jY_j^2\) cancel, and all remaining terms are \(O(j^{-4})\). At a switch the extra changes in \(u,Y\) are \(O(j^{-2})\), so the corresponding bound is \(O(j^{-3})\). Since \(I_j\to0\), summing its increments to infinity gives \[|I_j|\le C\sum_{i\ge j}i^{-4} +C\sum_{2^k\ge j}(2^k)^{-3} \le Cj^{-3}.\] Together with the positive lower bound for \(u_j+c_0Y_j\) in (135), this yields \[ u_j=c_0Y_j+O(j^{-2}). \tag{136}\] At ordinary steps the reciprocal form of (132) is now \[\frac1{Y_{j+1}}-\frac1{Y_j}=b_*c_0+O(j^{-1}).\] At each switch there is an additional \(O(1)\) error. There are only \(O(\log j)\) switches up to index \(j\); the ordinary errors also sum to \(O(\log j)\). Thus \(1/Y_j=b_*c_0j+O(\log(2+j))\), and \[ u_j=\frac1{b_*j} +O\!\left(\frac{\log(2+j)}{j^2}\right). \tag{137}\]

For an arbitrary history \(\mathcal H^{(k)}\) at an interior index, compare it with the spliced history at that same \(j\). Both indices are interior in their extended ranges. The ratio assertion in Lemma 20 and (135) first give \(Y_j^{(k)}\asymp j^{-1}\); its gradient bound then changes (137) by only \(O(j^{-2})\). Finally, (99), (126), and Lemma 19 give \[\begin{align*} t_j^{(k)}&=u_j^{(k)}+O(j^{-2}),\qquad z_j^{(k)}=O(j^{-1}),\\ \|\mathcal R_j^{(k)}\|_{j,k} &\le C\{(u_j^{(k)})^2+(Y_j^{(k)})^2+|u_j^{(k)}|Y_j^{(k)}\} =O(j^{-2}). \end{align*}\] Substituting \(b_*=2\log L\) proves Proposition 14. The scale in that conclusion is the absolute \(j\); no single uniformly valid microscopic start and no quantitative height scaling limit have been assumed.

The reference energy.

The unit field is \(2\pi\) times the rotated gradient of the killed primal Green function \(G_n(\,\cdot\,,0)\). The stream identity gives \[ g_n=4\pi^2\langle G_n(\,\cdot\,,0), A_n^{\rm kill}G_n(\,\cdot\,,0)\rangle =4\pi^2G_n(0,0). \tag{141}\] For the sequence \(n=L^N\), choose one sufficiently large fixed dyadic \(D_G\), distinct from the growing \(D_N\), and put \(v=n/D_G\). For late \(N\), \(v\) is a dyadic integer. Write \(\mathsf P_t^{\rm kill}=P_t(1-A_n^{\rm kill}/64)\) for the polynomials of (34). Since \(\mathsf P_1^{\rm kill}=I\), their exact telescope is \[ (A_n^{\rm kill})^{-1} =\sum_{\substack{t=1,2,4,\ldots\\t<v}} (A_n^{\rm kill})^{-1} (\mathsf P_t^{\rm kill}-\mathsf P_{2t}^{\rm kill}) +(A_n^{\rm kill})^{-1}\mathsf P_v^{\rm kill}. \tag{142}\] There is no independent microscopic piece in this identity. The quotients use their polynomial continuations when compared with the plane operators.

Choose \(D_G\) larger than the fixed range constant. Finite range then makes every center diagonal in the sum in (142) equal to its plane diagonal \[H_t(0,0),\qquad H_t=A^{-1}(\mathsf P_t-\mathsf P_{2t})\] for the plane nearest-neighbor Laplacian. The fixed-ratio Fourier estimate used in the proof of Lemma 16 applies at ratio \(2\). To check the omitted microscopic factor in [16], its nearest-neighbor kernel contains \(1-\gamma Q\), with \(Q=A/4\). The difference from \(H_t\) is \(\frac{\gamma}{4}(\mathsf P_t-\mathsf P_{2t})\); its plane diagonal is \(O(t^{-2})\), by \(\widehat{\mathsf P_t}(\xi)\le C(1+t|\xi|)^{-64}\). This correction and the Fourier power error are summable over dyadic \(t\). The continuum diagonal is \((\log 2)/(2\pi)\): if \(P(r)\) is the limiting radial cutoff, then \[\int_0^\infty\bigl(P(r)-P(2r)\bigr)\,\frac{\,\mathrm dr}{r}=\log 2\] because \(P(0)=1\) and \(P\) decays at infinity, and the angular factor is \(1/(2\pi)\). Thus, including the finitely many early sides, \[H_t(0,0)=\frac{\log 2}{2\pi}+\varepsilon_t,\qquad \sum_{t=1,2,4,\ldots}|\varepsilon_t|<\infty.\] The sum of the center diagonals in (142) is therefore \((2\pi)^{-1}\log v\) plus a convergent constant and \(o(1)\).

The last center diagonal in (142) has a limit for this fixed \(D_G\). For the normalized positive sine modes \(e_{k,n}\) and their Dirichlet eigenvalues \(\lambda_{k,n}\), \[|e_{k,n}(0)|^2\le Cn^{-2},\qquad \lambda_{k,n}\ge c|k|^2n^{-2},\qquad \mathsf P_v^{\rm kill}(k) \le C\min\{1,(|k|/D_G)^{-64}\}.\] Its summands are consequently bounded by \(C|k|^{-2}\min\{1,(|k|/D_G)^{-64}\}\), a summable function of the positive mode indices when \(D_G\) is fixed. Each fixed mode converges with its Dirichlet normalization, so dominated convergence gives the limit. Since \(\log v=\log n-\log D_G\), \[G_{L^N}(0,0)=\frac{\log L}{2\pi}N+c+o(1).\] Together with (141), this proves (140). No rate is needed for its convergent remainder.

The paired local computation

Take the entry side \(m=L^{j_0}=qRs\) supplied by Theorem 12 for the epoch containing \(N\). Thus \(s\asymp N\) and \(j_0=O(\log N)\). Use this same choice in the two squares, with grids centered on the same unit. At absolute index \(j\), put \(m_j=L^j\). All steps have \(j_0\le j<j'\), so \[m_{j+1}\le m'=\frac{n_0}{D_N}.\] The centered and unit sectors within a square have the same Gaussian covariance; their difference is in the activities and in the factor \(e^{-g_n/(2a)}\) in (33).

Fix the finite set of regulator and weight reserves allowed by Theorem 12. Use a stronger norm for the untwisted activities than for their translated unit copies, and leave the weakest block weight, denoted \(A_w\), large enough for the rooted sums below. These choices are fixed before \(N\). In a recurrence, \(\|\cdot\|_j\) denotes the chosen norm for that class of activities at side \(m_j\), with the same choice at successive indices. Define the size of the common plane bulk state by \[U_j=|t_j|+|z_j|+\|\mathcal R_j\|_j.\] Lemma 18 keeps \(U_j\) uniformly small throughout the history, and Proposition 14 gives \(U_j=O(N^{-1})\) on every fixed-fraction interval ending at \(j'\). Such an interval is always taken with a slightly longer interval behind it in the same history.

We now specify the exact algebra whose scalars will cancel. For the free dual square \(\mathcal G_n\), the entry theorem gives \[\mathcal H_{\mathcal G_n,m}^0(\phi) =c_{\mathcal G_n,m}\mathcal Z_m(K_m^{0,n};\phi),\qquad \mathcal H_{\mathcal G_n,m}^\tau(\phi) =c_{\mathcal G_n,m}[x]\mathcal Z_m(K_m^{\tau,n}+xJ_m^n;\phi).\] Here \(K_m^{\tau,n}\) sums unit-sector components without the compulsory source marker, while \(J_m^n\) sums the component containing that marker. Thus \([x]\) selects exactly one labelled component. The positive entry scalar is common to the two sectors of this square; it need not be common to the two square sizes.

Use the exact unbalanced partition map and work in \(\mathbb R[x]/(x^2)\). Let \(\mathcal B_{m_{j+1}}^n\) be the output block grid in square \(n\), and let \(\zeta_j^n\) have the shell covariance \(\Gamma_{m_j}\) there. The unit step has the exact form \[ \begin{split} &\mathbb E_{\zeta_j^n} \mathcal Z_{m_j}(K_j^{\tau,n}+xJ_j^n;\phi+\zeta_j^n)\\ &\quad = \prod_{B\in\mathcal B_{m_{j+1}}^n} (1+a_{jB}^{\tau,n}+x\dot a_{jB}^{\tau,n})\, \mathcal Z_{m_{j+1}}(K_{j+1}^{\tau,n}+xJ_{j+1}^n;\phi) \pmod{x^2}. \end{split} \tag{143}\] Use \(a_{jB}^{0,n}\) for the corresponding centered scalar. Each \(a\) is the complete sum of designated field-independent pieces assigned to \(B\); \(\dot a\) is its coefficient in the label direction. Only the unlabelled \(a\) must be small, since \[(1+a+x\dot a)^{-1} =(1+a)^{-1}-x\dot a(1+a)^{-2}\pmod{x^2}.\] The nonlinear map is a convergent sum of bounded multilinear maps of degree at least two, so \[\|D\mathcal N(K)J\| \le C\|K\|\,\|J\|\] at a small unlabelled input, for every finite-norm direction \(J\). No small-ball condition on \(xJ\) at a fixed nonzero \(x\) is used.

There are two copying operations. Within a square, an activity away from the source is the untwisted activity at a local lift. We keep its translated nonconstant pieces inside the activity and designate the untwisted constant; this removes scalar differences away from the source. Between the two squares, central unit activities are the same plane-unit prescription at different smooth backgrounds. Designating the neutral constant before that background is inserted makes the remaining scalar independent of the square size. The central activities themselves retain the background difference.

For the precise support conditions, include in the causal padding \(D_{m_j}(X)\), its fixed stencils, and an additional collar of fixed positive width in current-block units. Call a support wall-free when its required sites and causal padding stay a fixed number \(H_0\) of current blocks from the square boundary. Call it shift-good when its enclosing rectangle with that padding misses the unit; a support failing this condition is a defect. The full norm domain of a shift-good support is then at distance at least \(c m_j\) from the unit. Choose \(H_0\) once to include the shell range, assignment neighborhoods, and the geometric sum of all earlier causal padding. Occupied support, norm padding, and causal dependence have distinct roles here: every obstruction tag remains occupied in the support; the norm padding contains the evaluated coordinates; the causal neighborhood contains the predecessor data used by the local operations. The entry copy conditions are those of Theorem 12. These classifications are used in the proofs of [16], whose stated results concern a decaying Villain flow. We derive the needed estimates for the critical height history.

Fix a constant \(H_A\) large enough for the assignments described below. In the common centered grid define the geometric set \[ \mathcal A_j= \{\,m_{j+1}\text{-blocks meeting the }H_A m_{j+1} \text{-neighborhood of the unit}\,\}. \tag{144}\] This is a set of possible scalar destinations, not the set where a numerical difference happens to be nonzero. Its cardinality is bounded by a fixed constant. For late \(N\), it lies in both squares because \(m_{j+1}\le m'\ll n_0\).

Let \(w_j\) be the maximum over the two squares of the full untwisted norm restricted to supports that are not wall-free, and let \(d_j\) be the corresponding maximum of the full unlabelled unit norm on defects. Write \(\|J_j\|_j=\max_{n\in\{n_0,n_1\}}\|J_j^n\|_j\). Each norm here is the fixed choice for its class.

Choose the entry and bulk tube small enough that the coefficient in (148) is at most a fixed \(\theta'<1\). The recurrences then keep the actual unlabelled maps in the required small ball. The polynomial entry bound gives \[ \|J_{j'}\|_{j'}\le C_Rs^{C_R}(\theta')^{j'-j_0} \le e^{-cN}. \tag{154}\] The constant \(c>0\) may decrease between occurrences. Iterating (145) and (146) from an earlier fixed-fraction interval, with the quadratic small factors absorbed in the contraction, gives \[ U_{j'}+w_{j'}+d_{j'}\le C/N. \tag{155}\] The same bound holds for the plane centered-unit defect prescriptions. More generally, Lemma 18 and these geometric convolutions show that \(w_j+d_j=o(1)\) uniformly on every fixed-fraction interior interval. Earlier portions of the history contribute exponentially little.

The stopped identity and terminal integration

At the stop, write \[C_{*,n}=A_n^+\mathsf P_{m'}^n,\qquad \mathsf P_t^n=P_t(1-A_n/64),\qquad \mathop{\mathrm{Cov}}(\zeta_{*,n})=C_{*,n}\] where \(A_n\) is the free-square Laplacian and \(A_n^+\) is its inverse on the nonconstant modes. This is the terminal covariance in (34). The remaining common phase is uniform modulo \(\omega\). For \(\sigma\in\{0,\tau\}\), define the three terminal quantities \[ \begin{split} T_n^\sigma &=\frac1\omega\int_0^\omega \mathbb E_{\zeta_{*,n}} \mathcal Z_{m'}(K_{j'}^{\sigma,n};\zeta_{*,n}+t)\,\mathrm dt,\\ \dot T_n^\tau &=[x]\frac1\omega\int_0^\omega \mathbb E_{\zeta_{*,n}} \mathcal Z_{m'}(K_{j'}^{\tau,n}+xJ_{j'}^n; \zeta_{*,n}+t)\,\mathrm dt. \end{split} \tag{156}\] Thus \(T_n^\tau\) contains no compulsory component, whereas \(\dot T_n^\tau\) contains one surviving labelled component.

Let \(\widehat M_n\) denote the insertion ratio with the observation cutoff included in both physical sectors. Iterating (143), integrating the terminal field, and taking the coefficient of \(x\) give the exact stopped identity \[ \widehat M_n=e^{-g_n/(2a)}D_{\rm ex} \frac{c_{\rm ex}T_n^\tau+\dot T_n^\tau}{T_n^0}. \tag{157}\] The common entry scalar cancels within this square. Because the actual local maps remain in their small ball, the outside cancellation and complete-scalar copying in Lemma 22 allow us to write the remaining expressions exactly as \[ D_{\rm ex}= \prod_{\substack{j_0\le j<j'\\B\in\mathcal A_j}} \frac{1+a_{jB}^{\tau,n}}{1+a_{jB}^{0,n}}, \qquad c_{\rm ex}= \sum_{\substack{j_0\le j<j'\\B\in\mathcal A_j}} \frac{\dot a_{jB}^{\tau,n}}{1+a_{jB}^{\tau,n}}. \tag{158}\] Both are independent of \(n\in\{n_0,n_1\}\), although they may depend on the pair. Each factor uses the complete scalar assigned to its block. All its unlabelled denominators are positive, and \(D_{\rm ex}>0\). We next estimate the terminal quantities in (157) and then justify discarding its last term relatively.

The first estimate is uniform as the number of terminal blocks grows. It uses the same observation regulator as the local norms.

Apply this lemma with \(\nu=C/N\) from (155) and \(v\le e^{-cN}\) from (154). Since \(D_N^2/N=o(1)\), \[ T_n^0,T_n^\tau=1+O(D_N^2/N),\qquad |\dot T_n^\tau|\le e^{-cN}. \tag{164}\]

We need a lower bound on \(c_{\rm ex}\) only to discard the last term of (157) relatively. It follows from the physical observable and requires no lower bound on the label at entry. The cutoff loss in (48) and \(M_n=n^{-1/8+o(1)}\) from Proposition 5 give, for the two current sizes, \[ \widehat M_n=M_n(1+O(e^{-cN^2})). \tag{165}\] Indeed the unit loss is absolute relative to the uncut centered partition. Dividing it by \(M_n=e^{-O(N)}\) still leaves the displayed bound, and the centered loss has the same bound.

Set \(Q_n=\widehat M_n e^{g_n/(2a)}\). The same rough center asymptotic, (165), and (140) imply \[Q_n=e^{o(N)}>0.\] The scalar estimate in Lemma 22 also gives \[ |\log D_{\rm ex}| \le C\sum_{j=j_0}^{j'-1}(U_j+w_j+d_j)=o(N). \tag{166}\] For the last equality, there are only \(O(1)\) blocks in \(\mathcal A_j\) at each index. The summands are bounded in the small ball and tend uniformly to zero on every fixed-fraction interior interval. For fixed \(\delta>0\), indices below \(\delta N\) cost \(C\delta N+O(\log N)\); the rest cost \(o(N)\). Let \(N\to\infty\) and then \(\delta\downarrow0\).

Rearranging (157), \[c_{\rm ex}T_n^\tau =Q_n T_n^0/D_{\rm ex}-\dot T_n^\tau.\] The first term on the right is positive and at least \(e^{-o(N)}\) by (164) and (166); the second is exponentially small. Hence \(c_{\rm ex}>0\) for late \(N\), \(c_{\rm ex}\ge e^{-o(N)}\), and \(\dot T_n^\tau/(c_{\rm ex}T_n^\tau)=O(e^{-cN})\). Taking logarithms in the two sizes now cancels both \(D_{\rm ex}\) and \(c_{\rm ex}\) exactly. With (165), this gives \[ \log\frac{M_{n_1}}{M_{n_0}} =-\frac{g_{n_1}-g_{n_0}}{2a} +\left[\log(T_n^\tau/T_n^0)\right]_{n_0}^{n_1} +O(e^{-cN}), \tag{167}\] where \([F_n]_{n_0}^{n_1}=F_{n_1}-F_{n_0}\).

The terminal energy and the other supports

By Lemma 24, each terminal logarithm equals its single-activity sum with error \(O(D_N^4/N^2)\). We will prove \[ \left[\log(T_n^\tau/T_n^0)\right]_{n_0}^{n_1} =\frac{t_{j'}}{2a}(g_{n_1}-g_{n_0}) +O\left(\frac{D_N^4}{N^2}+\frac{1}{ND_N}\right). \tag{168}\] The same \(t_{j'}\) occurs in both computations.

In a single-activity expectation, integration of the common phase replaces an activity \(F\) by \[F_0(\phi)=\frac1\omega\int_0^\omega F(\phi+t)\,\mathrm dt.\] This neutral projection is invariant under every real common constant and has the same norm bound [16]. In particular the fundamental singleton has zero expectation, including after a nonconstant lift.

For a single-activity support \(X\), let \(k=|X|\) count its full occupied blocks, including all obstruction tags, and put \(D_X=D_{m'}(X)\). Choose a fixed small \(c>0\) and use the following disjoint partition: \[ \begin{split} \mathfrak G_n&=\{X:\ X\text{ is wall-free and shift-good}\},\\ \mathfrak W_n&=\{X:\ k\le cD_N,\ X\text{ is not wall-free}\},\\ \mathfrak C_n&=\{X:\ k\le cD_N,\ X\text{ is wall-free and a defect}\},\\ \mathfrak L_n&=\{X:\ k>cD_N,\ X\notin\mathfrak G_n\}. \end{split} \tag{169}\] The first class allows every size. These four classes cover every support exactly once. This terminal size split is separate from the fixed local-map threshold \(n_{\rm sm}\). A support in \(\mathfrak W_n\) is automatically shift-good when \(c\) is small: if its padded rectangle also met the unit, its occupied extrema in a wall-normal coordinate would span a fixed multiple of \(n\) minus \(O(m')\). Fixed-distance connectivity would then force \(k\ge c_0n/m'\), contradicting \(k\le cD_N\). This argument uses no occupied unit block in an unlabelled defect.

Let \(\mathcal I_n=\{B:\{B\}\in\mathfrak G_n\}\) be the singleton blocks in the first class. On such a block, the centered and unit activities are the plane activities at \(\phi\) and \(\phi+\lambda_n\). The fundamental singleton vanishes under the phase projection, and the energy singleton contributes exactly \[\frac{t_{j'}}2 \mathbb E\{e_B^0(\zeta_{*,n}+\lambda_n)-e_B^0(\zeta_{*,n})\} =\frac{t_{j'}}2e_B^0(\lambda_n).\] The cross term vanishes because \(\zeta_{*,n}\) is centered. On all supports in \(\mathfrak G_n\), the plane remainder has norm \(O(N^{-2})\). Its single-activity sum is bounded by the full rooted estimate \(O(D_N^2/N^2)\), with the weight reserve for the lift. This is an upper bound for the remainder on \(\mathfrak G_n\); the other three classes are treated below.

To identify the energy left by these singletons, define an allocation on every terminal block, including the omitted ones, directly from the connection: \[ E_{n,B}=\frac1{2a} \sum_{\substack{x\in B,\ e\in\{\pm e_1,\pm e_2\}\\ \{x,x+e\}\text{ internal to the square}}} \eta_n(x,e)^2. \tag{170}\] Each internal unoriented edge is counted at its two endpoints, so \[\sum_B E_{n,B}=g_n/a,\qquad E_{n,B}=e_B^0(\lambda_n)\quad(B\in\mathcal I_n)\] by (45). Thus the missing central allocations are defined without evaluating a lift at the unit.

For late \(N\), the complement of \(\mathcal I_n\) is a fixed-width strip of terminal blocks at the walls together with the same central singleton set in the two centered grids. The wall-strip energy in each square is \(O(m'/n)=O(D_N^{-1})\) by (46). On the identical central set, the difference of the two endpoint energy densities is bounded by \[C\left(\frac{1}{n_0(1+|x|)}+\frac1{n_0^2}\right).\] Write the difference of squares using the plane unit and (46) to obtain this bound. Its sum over \(|x|\le Cm'\), with the same internal neighbor stencils, is \(O(m'/n_0)=O(D_N^{-1})\). Consequently \[ \left[\sum_{B\in\mathcal I_n}e_B^0(\lambda_n)\right]_{n_0}^{n_1} =\frac{g_{n_1}-g_{n_0}}a+O(D_N^{-1}). \tag{171}\] Only this two-size difference is asserted; neither omitted central energy is required to be small by itself. Since \(t_{j'}=O(N^{-1})\), the allocation error contributes \(O((ND_N)^{-1})\) to (168).

For \(X\in\mathfrak W_n\), its sites remain a fixed fraction of \(n\) from the unit. The first two difference bounds in (46), and paths of \(O(k)\) blocks after one anchor is removed, give \[\|\lambda_n\|_{m',D_X}\le C(1+k)/D_N.\] The neutral projection of the untwisted wall activity is even under field negation. Gaussian symmetry therefore makes the first derivative at zero of \(\mathbb EF_0(\zeta_{*,n}+r\lambda_n)\) vanish. Taylor’s formula, the full analytic derivative norm, and the moment reserve in Lemma 24 bound its translated expectation difference by \[\frac{(1+k)^2 C^k}{N A_w^k D_N^2}.\] There are \(O_L(D_N)\) roots near a wall. The rooted shape sum is finite after the fixed weight choice, so the total contribution of \(\mathfrak W_n\) is \(O((ND_N)^{-1})\).

For \(X\in\mathfrak C_n\), an individual activity can have norm \(O(N^{-1})\) and need not be even. A size-\(k\) defect has an occupied block within \(O(k+1)\) block distance of the unit: its padded rectangle reaches the unit and its block graph is connected within a fixed distance. Choose \(c\) small enough that every such full padded set, including its stencils, lies in a common central fraction of the smaller square. These supports are then the same in the two grids. Their unit activities are the common plane prescription at \(\phi+\delta_n\) by (147). Modulo a common constant, \[\|\delta_n\|_{m',D_X}\le C(1+k)/D_N.\]

We compare the terminal fields on this full padded set. Choose a binary side \(v=2^b m'\asymp c_2n_0\), with fixed \(c_2>0\) small enough that the polynomial range through \(v\), added to the set and all its stencils, remains inside both squares. On nonconstant modes the exact split is \[A_n^+\mathsf P_{m'}^n =\sum_{\substack{t=m',\,2m',\,\ldots\\t<v}} A_n^{-1}(\mathsf P_t^n-\mathsf P_{2t}^n) +A_n^+\mathsf P_v^n.\] The quotients in the sum use their polynomial continuations at zero. Independent Gaussian constants may be added to realize these continuations because \(F_0\) is constant-invariant. Finite range makes the restrictions of every polynomial piece to the full padded set identical covariance matrices in the two squares. For this expectation couple those restrictions identically. The unmatched tails \(\zeta_{{\rm long},n}\) satisfy, for every fixed \(p<\infty\) and \(1\le i\le4\), \[\sup_x\|\nabla^i\zeta_{{\rm long},n}(x)\|_{L^p} \le C_{p,i}n_0^{-i}.\] This is the positive-order multiplier estimate in the proof of Lemma 23, now at \(v\asymp n_0\). The same local discrete Sobolev estimate applied to \(m'\nabla\zeta_{{\rm long},n}\) and \((m')^2\nabla^2\zeta_{{\rm long},n}\), using two further differences, gives their block maxima with bounds \(C_p/D_N\) and \(C_p/D_N^2\). Paths in an enclosing rectangle of \(O((1+k)^2)\) blocks control values relative to one anchor. Hence the two fields with their backgrounds, after subtracting their respective anchor values, obey \[ \left\|\,\|\xi_1-\xi_0\|_{m',D_X}\,\right\|_{L^p} \le C_p(1+k)^C/D_N. \tag{172}\] Here \(\xi_i\) is the restriction of \(\zeta_{*,n_i}+\delta_{n_i}\) in the coupling, with its anchor removed. The identical short fields cancel on the padded set.

The point norms at these two endpoints have the same local regulator forms. To see this precisely, let \(M_{D_X}\) be the positive matrix representing the open energy \(E_{D_X}^0\) on its observation coordinates, and let \(L_{D_X}\) be the joint Hilbert map from the old Gaussian and noise coordinates to \(B_s\alpha\zeta+e\) on those coordinates. The internal edges and whole observation cells are identical on the two padded sets, even when those sets have holes or several components. Thus \(M_{D_X}\) is identical. The restriction of \(C_{<m'}=A^{-1}(I-\mathsf P_{m'})\) is also identical by its polynomial finite range and the wall clearance; the noise law is fixed. Thus \(L_{D_X}L_{D_X}^*\) agrees. The optimized formula [16] depends on the Hilbert map only through this covariance, as its convergent inverse series shows. Hence \(V_{m',h_{\rm b}}\) agrees, and the ordinary regulator has the same edges and stencils as well. This equality is for the local high-mode regulator. The terminal covariances are the different covariances just coupled.

Remove the anchor pointwise from the neutral activity and apply the mean value theorem on the segment from \(\xi_0\) to \(\xi_1\). Use (172) in the exponent conjugate to \(p_{\rm seg}\). The identical local forms and the endpoint moment (160), with the deterministic backgrounds \(\delta_n\), bound the regulator at each point of this segment. For this support the expectation difference is at most \[\frac{(1+k)^C C^k}{ND_NA_w^k}.\] The same comparison applies to the centered activity, with zero background. The location bound for a size-\(k\) defect gives only polynomially many possible roots. The rooted shape and weight sum absorbs these factors, so \(\mathfrak C_n\) contributes \(O((ND_N)^{-1})\) in total.

Finally the supports in \(\mathfrak L_n\) have \(k>cD_N\). The absolute rooted bound with the remaining block weight gives \[C D_N^2N^{-1}e^{-c'D_N}=O((ND_N)^{-1}).\] This includes the large supports reaching both a wall and the unit. Large supports in \(\mathfrak G_n\) were already included in its plane remainder estimate.

The partition has now accounted for every single activity: the plane energy and the vanishing fundamental on \(\mathcal I_n\), the plane remainder on \(\mathfrak G_n\), and all terms in \(\mathfrak W_n,\mathfrak C_n,\mathfrak L_n\). Families with at least two constituents contribute the separate \(O(D_N^4/N^2)\) error from (161). Combining these estimates with (171) proves (168). Substitution in (167) proves (139), absorbing the exponentially small error. The reference-energy calculation proved (140), completing Proposition 21.