Critical Center Magnetization in the Planar XY Model

1 Introduction

For an integer R\geq1, let \Lambda_R=[-R,R]^2\cap\Z^2. The planar XY model assigns an angle \theta_x\in\R/(2\pi\Z) to each vertex. With free boundary conditions and inverse temperature b>0, its probability density is proportional to

relative to product uniform measure. Each undirected edge is counted once, and there are no exterior bonds. Write \E^{\mathrm{free}}_{R,b} for expectation under this law, and define

The existence of these limits and the fact that 0<b_c<\infty with m(b_c)=0 are among the inputs recalled in Section 2.

In \Lambda_n, now fix \theta_x=0 at every boundary vertex and retain every nearest-neighbor bond inside \Lambda_n, including those incident to its boundary. Let \E^0_{n,b} denote this law. Our observable is

It measures how strongly an aligned boundary influences the center at criticality. The power-law exponent alone does not determine its normalization: a slowly varying multiplier can remain invisible to fixed-ratio scaling limits.

The dependence on the three companion results is substantive. The height limit fixes the critical Gaussian coefficient, the local maps supply the analytic renormalization framework, and the spin-field results provide a rough positive lower bound and a ratio limit. We state the portions used below before beginning the proof. In particular, the critical magnetic contraction and the exact cancellation of its scalar factors are proved here; they are not asserted as consequences of a low-temperature magnetic-flow theorem.

1.1 Historical context

The critical behavior of two-dimensional systems with continuous symmetry differs from the conventional picture of ordering through spontaneous magnetization. Berezinskii’s analysis of the spin-wave regime and the vortex picture developed by Kosterlitz and Thouless identified the mechanism behind a phase with algebraic correlations [Ber71, KT73]. Kosterlitz’s renormalization analysis predicted the critical exponent and the essential singularity of the correlation length [Kos74]. The treatment of vortices and symmetry-breaking perturbations by José, Kadanoff, Kirkpatrick, and Nelson made the relation with Coulomb-gas and periodic-field descriptions particularly explicit [JKKN77]. These works supply the physical and methodological ancestry of the marginal recursion used here.

Rigorous correlation estimates established important parts of this picture without determining the exact critical normalization. McBryan and Spencer’s complex-rotation argument gives an algebraic upper bound [MS77]. Fröhlich and Spencer proved low-temperature algebraic lower bounds for the plane rotator and related models [FS81, Theorem C]. More recently, van Engelenburg and Lis proved the exponential–polynomial dichotomy with a polynomial lower bound also at the critical point [vEL23, Theorem 1]. Lammers identified the correlation-length relation between the XY model and its dual height model, relating the massless and delocalized phases [Lam23, Theorem 1]. These results concern the transition and quantitative correlation bounds; they do not determine the logarithmic factor in (1.2).

The relevance of logarithmic corrections has long been recognized in finite-size scaling. For example, Kenna and Irving discuss the renormalization-group prediction \chi_N\asymp N^{7/4}(\log N)^{1/8} for the susceptibility on a periodic square [KI95]. This is a prediction for a different observable, not a proof of the aligned-boundary one-point asymptotic. A square-root relation for the root-mean-square spatially averaged magnetization cannot by itself identify the center observable studied here.

Rigorous renormalization methods also give more precise information in related models. Dimock and Hurd control the ultraviolet-regularized sine-Gordon model at weak coupling with parameter \beta>8\pi [DH00]. Falco constructs marginal trajectories of order 1/j for the dilute Coulomb gas and proves multiplicative logarithms for suitable fractional-charge correlations [Fal12, Fal13]. Those probe observables and their logarithmic exponents are not interchangeable with magnetic observables of the nearest-neighbor cosine model. The common mathematical feature is that a coupling which decays as 1/j, when accumulated over scales, produces a power of a logarithm.

Our immediate inputs are the three companion manuscripts [Ope26a, Ope26b, Ope26c]. Their roles are different. The local maps and analytic norms come from [Ope26a]; the critical Bessel-height coefficient 8\pi and the required mixed-geometry height limits come from [Ope26b]. The spin-field construction in [Ope26c] uses the exact center magnetization as its normalization and leaves its asymptotic size unspecified. Theorem 1.1 identifies that size. The additional work is to enter the critical maps on finite histories, compare those histories without a rate in the height limit, and cancel the magnetic scalar factors exactly before estimating terminal errors. We do not use the public results for related models as substitutes for these companion inputs.

1.2 Proof strategy

The logarithmic factor in (1.2) reflects the marginal coordinate of the critical height recursion. The positive finite amplitude requires the finer, summable control described next.

Fourier duality expresses a_n as a ratio of two partition sums for heights in 2\pi\Z. The numerator has a circulation of 2\pi around the center, whereas the denominator has zero circulation. The proof has two principal difficulties. First, the observation density of the discrete height model cannot simply be assumed small in a Gaussian large-field norm. Second, entering the recursion in the presence of a circulation defect can create a large normalization factor. An estimate of that factor would be an unnecessary additional problem.

We resolve the first difficulty by working on overlapping finite intervals of scales. At a large target index J, introduce observations on cells of side comparable to J^3, and remove exceptionally large neighboring observation differences by a smooth cutoff. The cutoff error is exponentially small in J^3, whereas the finite physical boxes used in this history have logarithmic side of order J. The resulting interaction is small in the full analytic norm of the height maps. This construction is carried out in Sections 3 and 4.

Each history then runs far enough beyond its target interval to determine the critical branch of the recursion. Two observables of the original height law, the variance of a long-wave fluctuation and a characteristic coefficient of the mean height, compare the coordinates of different histories at a common absolute scale. Section 5 proves these comparisons without assuming a rate for the height scaling limit. Section 6 combines them with the marginal recursion. Its conclusion for the running gradient coefficient t_j, at block side L^j, is

where L is a fixed, sufficiently large power of two. Sparse changes of history have summable effects on this asymptotic.

To resolve the second difficulty, compare squares of side 2n and 2Ln in the dual height representation using the same history and the same stopping scale. Section 7 keeps the circulation insertion linear throughout the exact partition algebra. All scalar factors produced near the insertion are identical in the two squares. Their size is immaterial because they cancel in the ratio. The residual insertion contracts quickly enough that the rough lower bound on a_n makes its contribution negligible even relatively.

What remains is a terminal Gaussian calculation. The energy of the circulation produces the leading power n^{-1/8}; its coupling to (1.3) produces the logarithmic correction. Section 8 proves, for n=L^j/2,

for some \epsilon>0. Absolute summability, rather than an estimate of the initial magnetic normalization, yields the finite positive amplitude. The spin-field ratio limit then transfers the result from these geometric sizes to all integers.

2 Height conventions and local map inputs

The proof combines three inputs: the critical height coefficient from [Ope26b], the local integration maps of [Ope26a], and the identical-nest comparison of [Ope26c]. We first fix their common normalization. We then specify the analytic spaces in which the local maps operate. These inputs include finite-graph inequalities, scaling limits, and local activity maps. The entry construction for the critical XY model is supplied below.

2.1 Dual heights and finite comparisons

For a graph G consisting of retained nearest-neighbor edges, choose one orientation of each edge and write \partial v for the oriented gradient. Its unit-conductance Laplacian is A_G=\partial^*\partial, so that

Thus each undirected edge contributes once. Heights take values in 2\pi\Z, and an edge difference 2\pi k has weight

An integral connection l is an edge cochain with values in 2\pi\Z; it replaces \partial h by \partial h+l in the weights. Write Z_G^l for the resulting sum and F_G^l=Z_G^l/Z_G^0. Without observations or absolute pins, a constant translation by 2\pi\Z is divided out on each free component. If observations confine the constants, their translation copies are summed instead. Changing l by an integral gradient is an exact change of height variables.

We shall use the following finite-graph rules in exactly this normalization. A centered precision means an additional nonnegative quadratic form, including a limit enforcing centered linear equalities. A reference cost for a linear statistic X=\sum_e w_e\partial_eh is \mathcal R(w)=\sum_e w_e^2.

The finite Gaussian antecedent is the lattice inequality of Regev and Stephens-Davidowitz [RS17]. The required Bessel-weight comparisons are transferred in [Ope26b] through independent discrete Gaussian chains in series: replace a primary edge by N increments with normalized weights (b/(2N))^{k^2}, impose endpoint and period constraints, and then let N\to\infty at fixed graph. The physical sum along a chain tends to the law assigning mass p_b(k) to 2\pi k. Quadratic observations remain positive forms in the enlarged coordinates. This procedure also transfers the parity identities for two Gaussian replicas and the theta comparison used for complex observations. For example, the latter bounds the imaginary-to-real partition ratio at a real affine shift by the corresponding centered imaginary ratio; see [Ope26a, Equation (9.29)] and [Ope26b, Equation (5.26)]. The uniform bounds used here are the primary costs in (2.2), not inverses of the microscopic chain precision.

2.2 The critical Gaussian coefficient

For the free height square B_m=\{0,\ldots,m-1\}^2, put g_m(x)=x_1/m. The structural coefficient is the limit

The existence and the following identifications are inputs from [Ope26b].

These statements are [Ope26b, Sections 1.1 and 6, Theorems 2.4, 2.5, and 3.1]. Here and below all mesh limits precede variation of the fixed geometry. We also use the separated-pin Theorem 3.2 of [Ope26b] in the following precise form. In a centered high law \mu_m, with penalty \frac12\norm{B_mh}^2 in finitely many component-local box averages controlling the constants, let H,p be positively separated box regions. Then

satisfy D_m\ge0, 0<r_{H,m}\le1, and \E_{\mu_m}r_{H,m}=e^{-D_m}. The quantities have Gaussian limits with precision a^{-1}\int|\nabla v|^2+\norm{Bv}^2. For functions of random guard data, this means approximation in probability by a sufficiently long fixed list of smooth guard probes, whose joint law and function converge to the Gaussian counterparts. The probe list is enlarged after the mesh limit. Conditional Laplace transforms and weak laws have this approximation as well, including when the data are sampled under a fixed finite real linear tilt. The expectation identity just stated refers to the centered law.

Section 3.6 of [Ope26b] supplies the restoration bounds used in the entry argument. At zero guard data, a centered restoration ratio lies between its value in a free patch and its value with a separating guard pinned. If all restored bonds and forms are supported in H, pinning every primary height in H makes restoration a fixed scalar. The free-to-guard multiplicative gap is bounded by

No observation may cross the separating guard in this comparison.

2.3 Comparing center observables at different sizes

The information from [Ope26c] that we need is weaker than the asymptotic formula to be proved: it permits an arbitrary slowly varying factor. Its sequential form will eventually pass from geometric square sizes to all integers.

2.4 Covariances, blocks, and analytic norms

We now fix the reference \alpha=\sqrt{8\pi}. Its mean-free Gaussian field \psi has covariance A^+, the inverse of the primary unit-edge Laplacian on nonconstant modes; its common mean is independently uniform modulo \omega=2\pi/\alpha. On the plane we use only finite-support covariance formulas. For a dyadic integer u, set

The apparent singularities in this polynomial are removable. The argument 1-A/64 is the specialization of 1-Q/16 in [Ope26a, Equation (9.11)]: for nearest neighbors Q=A/4 and v_J^2=1/4, so A_0=Q/v_J^2=A. At an entry side m, and at subsequent sides u=mL^j, the covariances are

with C_{<m} integrated at entry. The first two inverses mean polynomial continuation at the zero eigenvalue; the last formula acts on nonconstant modes. No independent-variance factor is inserted. The same polynomials are used on free squares, with even reflection at their walls. Positivity, finite range, and exact local copying apply to the polynomial accumulated and shell covariances: their ranges are O(u) and O(Lu), respectively. The terminal covariance need not have finite range. These assertions and the derivative bounds follow from [Ope26a, Lemmas 8.1 and 4.1] with this argument. In particular positive-order shell and terminal estimates through the orders used below hold uniformly when u\le M/D_0, with fixed sufficiently large D_0; the plane shell diagonal is (2\pi)^{-1}\log L with an error that is a negative power of u.

Use the block geometry of [Ope26a, Lemma 4.3]: with p=32, an intermediate connectivity radius is r_0=4p+20 and the polymer connectivity radius is \rho=r_0+8. These fixed radii may be enlarged for the prescribed test collars. Shell covariance ranges are measured in the output blocks of side Lu, whereas the accumulated covariance C_{<u} has range O(u). The radius \rho is distinct from the tile parameter R used in the entry construction. Blocks have side u on the plane and torus. On a free square use nested translated grids, with possible end rectangles whose side lengths lie in [u,2u). A polymer X is a nonempty finite block set connected by steps of sup block-index distance at most \rho; |X| counts its blocks. Two polymers are compatible when their mutual block-index distance exceeds \rho. The coarse closure \bar X consists of the Lu-blocks meeting X. The geometry of [Ope26a, Lemma 4.3] supplies a fixed n_0 such that

if |X|\le n_0, the coarse index diameter of \bar X is at most one. The number of size-p polymers through a fixed block is at most C_\rho^p.

Let D_u(X) contain X and all blocks at index distance at most two, with the fixed difference stencils and whole observation cells included. Activities depend on the field in X and a fixed microscopic collar; D_u(X) is the larger testing domain. Our activities are locally analytic functions of real fields and are periodic under the common shift \psi\mapsto\psi+\omega. Field-even means K(X;-\psi)=K(X;\psi), a different symmetry from the half-period parity used below. For D=D_u(X), write E for the ambient nearest-neighbor edge set and use the half-edge energy and boundary term

Physical free-wall stencils are interpreted by even extension. Set

For a direction f define

The analytic point norm, at a real background \psi, is

The term with r=0 is |F(\psi)|. These real derivatives have locally convergent analytic extensions. The full derivative sum controls complex translations of test norm less than h_0 and obeys the product inequality.

The entry construction needs one additional, homogeneous quadratic regulator. Let B_s average over disjoint side-s observation cells. On an open union D of whole cells define

with free constants on every component. Here A_D retains only internal bonds; it differs at artificial boundaries from the half-edge allocation g_D. Choose constants 1<p_1<p_2 and 0<h_*<1/p_2, with the strict reserves required by the entry estimates. At scale u=mL^j, put

The Cameron norm \norm\zeta_{<u} belongs to the ambient accumulated covariance C_{<u}, including the pieces integrated before entry. Only its restriction to D enters the supremum. The variable e is the independent unit observation noise, included once. For a singular covariance we use its Gaussian Hilbert space, without inverting a null direction.

For a polymer activity K our norm is

The supremum includes all real fields. The activity weight \mathfrak A is unrelated to the Laplacian A. A fixed finite collection of stronger weights, slightly larger regulator coefficients, and analytic-radius reserves may be chosen throughout.

The key covariance bound underlying this input is A^{1/2}C_{<u}A^{1/2}\le I. Restricting to an open support only lowers the minimized energy in (2.9). It is therefore the homogeneous energy that is used in (2.10), also when measuring activities in a sector with a magnetic connection.

2.5 The exact partition map and its balanced coordinates

At a finite block scale define

including the empty collection with value one. Equivalently one sums over block subsets and multiplies the activities of their \rho-connected components. A compulsory insertion activity I uses the degree-one functional

It is the coefficient of \tau in \mathcal Z_u(K+\tau I;\psi), with \tau^2=0. An input polymer is called small when |X|\le n_0, and large when |X|>n_0, with the geometric threshold fixed above.

These are Proposition 4.8, Lemma 4.5, Corollary 4.7, and Proposition 8.3 of [Ope26a]. Before localization, Gaussian integration groups the union of the coarse closures into components connected at radius r_0. Different such components are beyond the shell covariance range. The subtracted pieces are assigned to coarse singletons with their original exclusion information retained. Only designated field-independent coefficients enter a_b. In particular, a translated quadratic polynomial may retain its nonzero value at zero field inside the activity. The same algebra applies in degree one in \tau; no smallness of the vector I is required to apply the derivative of the map at a small background. Coefficients copy from the plane whenever their predecessor neighborhoods fit inside the domain with the fixed clearance supplied by the covariance range.

For a full bulk block define the two singleton functions

Their coefficients in a general activity are denoted by t/2 and z: the singleton part is t e_B^0/2+zc_B^\alpha. Denote the remaining activity by \mathcal R. The linear map is triangular,

where \lambda_j=L^2e^{-\alpha^2\Gamma_u(0,0)/2}. At marginality the product \ell_j=\prod_{h\ge j}\lambda_h converges. Set C_{h:j}=C_{h-1}\cdots C_j for h>j and C_{j:j}=I. The balanced coordinates are

Both projection series have geometric tails. Their linear map is (T,Z,\mathcal R)\mapsto(T,Z,C_j\mathcal R).

The exact Gaussian curves of [Ope26a, Lemma 10.3] are obtained by initializing the interaction

without a lattice factor. For each fixed entry scale they have T_j^g(d)=d+O(d^2), Z_j^g(d)=0, and \mathcal R_j^g(d)=O(d^2), analytically on a disk independent of the step. In the following coordinate formulas u denotes the Gaussian scalar coordinate rather than a physical block side. Define

Thus (u,0,0) is an exact fixed-coordinate Gaussian trajectory. Split B=B_e+B_o into its even and odd parts under the half-period shift \psi\mapsto\psi+\pi/\alpha.

This proposition is a local analytic input, conditional on entry into and continued residence in its small tube. The uniformity needed when the entry scale varies, the absence of exit, and the quantitative rate in (2.17) are established in the bulk argument below.

Finally, we record the source estimates used with these local maps. For the one-center connection l of Lemma 2.1, let r minimize \norm{\partial r+l}^2 on the free dual square and put \eta_l=\partial r+l. If G_n is the inverse of the killed unit-edge Laplacian on the interior spin vertices, then orthogonal energy minimization gives

By [Ope26a, Lemmas 4.2 and 5.2],

On a source-free enlarged rectangle the connection is an exact real gradient; the assertion includes the reflected stencils at a free wall. Plane and finite-square connection gradients have the same local limit. The stronger finite-volume difference estimate needed for the magnetic calculation follows there by subtracting the common finite-range pieces and summing the positive-order tails.

All norm choices follow a fixed order: geometry and test collars; the observation-regulator reserves; \delta, then \kappa with the necessary integration reserves; the analytic radius h_0; a sufficiently large power-of-two block factor L; activity weights; and the small activity tube. For the XY entry construction, the observation-regulator choices must also satisfy the stronger reserve condition in Proposition 4.1. This condition is imposed before fixing L and the weights. The subsequent tile parameters and observation-cell size are chosen after these map parameters. Constants may depend on parameters already fixed, but never on the volume or the running scale unless explicitly indicated.

3 Observations, affine Gaussian shapes, and isolation

We prepare the critical Bessel height law for the local height maps. The preparation has two parts. First, an entire cutoff changes the finite-volume quantities under consideration by an exponentially small amount. Second, an exact isolation expansion expresses the cutoff interaction in terms of tile errors, restoration errors, and exponentially small Gaussian links. The affine Gaussian estimates below allow the same construction in the sector with one circulation. The next section proves the analytic norm estimates needed to enter the maps.

3.1 Observation cells and the cutoff

Set \alpha=\sqrt{8\pi}, and use the nearest-neighbor polynomial covariances and subset algebra specified in the inputs. Thus A is the unit-edge height Laplacian, and the covariance integrated below side m is

with polynomial continuation on constants and with the spectral-argument convention fixed there. There is no additional independent-variance factor. We use the same polynomials on free dual squares. The remaining reference field has a common constant uniform modulo 2\pi/\alpha. Statements about the plane concern local formulas; no probability law on absolute plane heights is required. The block collars, analytic tests, regulators, and compatibility rules are those of [Ope26a, Sections 9, 8, and 4]. The tile parameter R below is distinct from the fixed block connection radius \rho.

For a dyadic integer s, partition the microscopic graph into square observation cells of side s. Write B_s for cell averaging, and introduce independent standard Gaussian variables e_c, one per cell. The observations are

All cuts and clippings in this section retain whole observation cells. In particular a retained observation face consists of all the microscopic bonds joining the two cells on its sides.

In observation-cell units use the tile geometry of [Ope26a, Theorem 9.1 and Subsection 9.2]. Start with squares of side R, remove the square of side R/2 centered at each grid vertex from the incident squares, and make each removed square a vertex tile. The remaining parts of the original squares and the vertex tiles partition the cells. Here R is a large dyadic integer, P=R^5, q is a fixed dyadic period parameter large enough for all fixed collars, and

The block factor L is a sufficiently large power of two. We use square and torus sides compatible with the grids, at least sP, and divisible by sP. Free-wall tiles are the allowed box clips of this construction; their cuts lie between cell layers.

Choose the common square-symmetry origins at grid vertices, which are elementary-square centers in site coordinates. In a pierced square put the source at the central grid vertex and take the integral connection l along a grid ray. Thus l vanishes inside each observation cell and has a constant value l_f on the microscopic edges crossing an oriented observation face f=(c,c'). The source lies in the central vertex tile, even though four observation cells meet there. Tori in this proof are unpierced and carry ordinary single-valued heights.

On every observation face retained in the graph insert

The function \chi_T is entire. For real arguments it lies in (0,1), and 1-\chi_T(d)\le C e^{-cT^2} for |d|\le T/2. Its role is to give the observation law a quadratic large-field reserve on the finite scale intervals used later.

For the pierced square, the input lower bound a_n\ge c n^{-1/2} converts (3.5) into a relative error bound. Its use here requires no information about the eventual normalizing amplitude; the bound comes from [Ope26c, Equations (18)–(19)] and the fixed-spin duality of its Section 2. In particular a polynomial loss when dividing by the magnetic partition ratio is harmless when s is chosen on the finite histories below.

3.2 The affine observation identity and local lifts

Let G be a graph obtained by the cell cuts just described. Write \mathcal F(G) for its retained observation faces. Free constants are allowed on every connected component. Define the Gaussian shape energy by

Write E_G=E_G^0. Let Z_G^l(y) be the Bessel height sum with observation penalty \exp(-|B_sh-y|^2/2) and with all component constants summed. No cutoff occurs in this definition. More explicitly, apart from the common normalization of the observation noise, it is

Here E(G) denotes the set of microscopic edges, each counted once; the observation penalty makes the sum over each component constant finite. When subsequently integrating observations, one divides out common translations, or uses one period for each free constant.

On the full connected graph, let r minimize the unobserved affine energy, and put \eta_l=\partial r+l. The minimizer is determined up to a constant and satisfies

For v=r+u, orthogonality gives |\partial v+l|^2=|\eta_l|^2+|\partial u|^2. Gaussian integration and completion of the observation square consequently give the affine Gaussian observation law

The prime omits constants. The constant of \psi is uniform modulo 2\pi/\alpha, and e has the independent standard Gaussian law already specified. This calculation may equivalently be made by first summing height translations and then integrating y, with the common translation divided out.

For clarity, define the interaction on every cut graph by

On the full graph suppress G. Relative to the unpierced Gaussian integral, evaluating J^l under (3.8) introduces the scalar

To see that this is the complete scalar, the Gaussian integral with fixed observations equals a common determinant factor times e^{-E^l(y)/2}. Translating by r makes its determinant exactly the unpierced determinant and leaves the constant energy |\eta_l|^2/\alpha^2. Multiplication by (3.9) cancels the observation shape. Thus no additional factor involving a chosen local harmonic lift is present.

We record the lift estimates that determine how the eventual activities copy between the plane and a square. They are consequences of [Ope26a, Lemmas 4.2 and 5.2, and the duality in Lemma 5.1], with the source period rescaled to 2\pi in height units.

Extend the chosen integral ray connection l to the plane. Let \eta_\infty be the divergence-free infinite-plane connection obtained as the local limit of the square minimizing connections, equivalently the rotated infinite-plane Green gradient normalized to circulation 2\pi at the source and zero circulation elsewhere. Its energy grows logarithmically with the radius; no finite total plane energy is assumed. Since \eta_\infty-l has zero circulation around every lattice plaquette, path integration defines a single-valued real height r_\infty by

Use the same ray convention for r and r_\infty, and align their additive constants at one fixed height vertex adjacent to the mark, chosen consistently with the grids.

For later copying, a support is called shift-good when its spanning axis rectangle, enlarged by Hm for a sufficiently large fixed H, misses the mark. On such a support the connection gauges away and the ordinary calculation is evaluated at \psi+\lambda. At an interior support near the mark, the pierced-plane calculation is evaluated at \psi+\gamma_M. The spanning rectangle in this definition will include the numerical dependencies of links, as specified below. Thus these statements apply to the entire calculation on a support, including an insertion, and not just to the fields explicitly occurring in a factor.

3.3 A real large-observation bound

The cutoff yields a uniform strict saving in face differences. We first prove the estimate for the height sum without the cutoff. It is useful on every open patch, including a clipped patch carrying the connection.

3.4 Localization of the affine shape energy

The Gaussian part of the isolation expansion compares energies on different cut graphs. We need these comparisons to be localized in observation-cell distance with constants independent of s. A nonzero circulation creates a logarithmic energy near a cell junction. The next lemma isolates that energy in an exactly local term; the remaining matrix has uniform exponential bounds.

For an observation junction z at which all four faces are retained, let \kappa_z(d) be their oriented circulation. No circulation term is assigned to a junction whose retained sectors have tree adjacency. For data d_f=y_{c'}-y_c+l_f, all \kappa_z(d) vanish except possibly at the source, where its absolute value is 2\pi if the full star is retained.

The same proof supplies the extra estimate used in isolation. Compare a true deletion with its calculation on an open patch having clearance of order R from artificial boundaries. For entries near the deletion, the graph-comparison argument has a common box of radius cR. For entries far from it, each deletion change is already exponentially small by (3.18). Combining these bounds with separation decay gives, for the error \Delta_i at event i with deletion region D_i,

All local pieces, including the logarithmic circulation terms, cancel exactly in this difference: the true and patch calculations have the same retained faces and affine data near the deletion. A physical free wall is the same wall on both calculations. Enlarging the fixed collars allows the estimate for the bounded groups of simultaneous events used next. This is the precise reason that links have an e^{-cR} factor with no additional \log s loss.

3.5 Exact isolation and links with numerical dependencies

We now apply the finite covariant isolation construction of [Ope26a, Subsection 9.2]. We describe both the scalar normalization and the support labels, because exact copying depends on them.

Color tile centers by their tile type and the square-symmetry orbit of their coordinates modulo q, in units of R. In one color group nearby centers into proximity components, with a fixed large proximity radius. Take q greater than sixteen times that radius. A square-symmetry orbit has at most eight residue labels. If a group contained two distinct centers with the same label, choose a shortest proximity path between any such pair. Its interior labels are distinct, so the first repeated label occurs within eight steps. Its displacement is therefore less than q; since its endpoints have the same residue, they must be the same center, a contradiction. Thus each label occurs at most once and the group has at most eight centers. Distinct groups of the same color have disjoint enlarged neighborhoods. Order the finite list of colors, and process groups within a color simultaneously. An event isolates every tile in its group, including from the other members of the group, deleting a bond only when it is still present.

As in the cited construction, a group’s core contains its tiles and their immediate neighbors, its open patch extends a fixed number of tile units farther, and its guard separates core from exterior. One may use distances 1, 5, and 3 for core, patch, and start of guard, and distances 12 and 10 for an enlarged patch and guard when testing pins. Enlarge these fixed constants, and first the proximity radius, whenever the fixed collars demand it. All distances in this paragraph are in units of R observation cells.

Let o and c denote the current open patch before and after one such deletion. Its proxy is

This is [Ope26a, Equation (9.6)] with affine shape energies. Its constants are always centered and have no cutoff. Delete the physical face cutoff together with that face’s microscopic bonds; thus a cutoff is present exactly when its face is retained.

At each considered event write the original local weight as B_i^l times its cut weight plus their difference. Selecting the difference marks that event. After a mark skip subsequent events near that group, with the fixed skip distance enlarged along with the collars; distance 30 in the preceding geometry suffices before such enlargements. Entire groups are skipped together. The resulting binary expansion is an exact finite identity. A later considered patch does not meet an earlier marked core or guard, so its proxy is independent of the signed choice within the earlier difference. The fixed number of batches and the skip rule imply that every remaining cross-tile bond lies in a fixed tile halo of a mark. All departures from the path with no marks are confined to these halos.

Let B_i^{l,(0)} be the proxies on the path with no marked events. Divide the expansion by

where b runs through the isolated tiles. This is the normalization of [Ope26a, Equation (9.7)], again with centered no-cutoff constants and affine energies. A marked halo retains its full signed height sum and every affected proxy. Outside these halos expand an unused tile factor as

In the magnetic calculation there is one exception: if the tile containing the source remains unused, retain its whole factor rather than expanding it around 1. It will be the compulsory insertion. If the source belongs to a marked halo, that halo supplies the compulsory factor instead. Exactly one connected component of every resulting pattern contains the source.

Write E_0^l for the energy appearing in (3.26), so that \mathcal B^l(y) is a constant times e^{-E_0^l(y)/2}. Combining it with (3.9) leaves the Gaussian factor

Telescope the energy changes along the no-mark path. For each event, the true change and the open-patch change have identical local parts; their remaining difference is (3.24). Summing over events gives a matrix M on face arrays with

Before summing, the event localization in (3.24) remains available. It makes the sum over event positions uniformly convergent. This argument also treats the true free-wall clips, since both paths use the same physical wall.

Telescoping the numerical dependence. An exponentially small coefficient computed from the whole graph is not yet an exactly local coefficient. We resolve it into local calculations before assigning supports. For each event and pair of argument faces, take open cell boxes containing the event patch and the two faces. The initial radius is a sufficiently large fixed multiple of R plus such a multiple of the largest separation among the event and argument faces. Include a fixed R-size halo for every numerical tile rule. Increase the radius one cell at a time. Let \Delta_i^{[D]}(f,g) denote the calculation clipped at radius D, with the same grids and the same true free walls if one is met. Write the exact coefficient as the first clipped calculation plus successive differences

On finite squares the sequence becomes exact when the clip contains the graph. On the plane it converges by the solution comparison in Lemma 3.4. Every summand depends on a finite box and has exponentially small size in its radius, after weakening constants. The first summand has the same bound by (3.24), because its initial radius is controlled by R and the argument distances. Subsequent differences have it by comparison on the common interior box. In particular the coefficients retain both the e^{-cR} gain and summable decay in the added radius.

On a torus perform the same telescoping only up to a sufficiently small fixed fraction of the period, and then use the exact residual as a final term. Give that term a winding label. If the initial radius already reaches the threshold, no initial clipped calculation is needed; the exact term has exponential decay in the period length because R and the argument distances then control that length. Otherwise the last residual has the same decay by graph comparison. Include actual winding paths in the support of either kind of terminal term.

Index the bilinear summands resulting from this telescoping by \nu, and write x_\nu(y) for one-half its coefficient times d_fd_g, with the chosen symmetric matrix convention. Define the link \ell_\nu(y)=e^{x_\nu(y)}-1. The exact version of [Ope26a, Equation (9.10)] is now

At every fixed finite-volume field the coefficient series and this product converge absolutely. Their integrated absolute estimates are proved in the entry section.

Support labels and copying. A link whose numerical box has radius D' in cell units is covered by a path, or a fixed number of paths, of total length O(R+D'). These paths contain its argument faces and span its numerical dependence. Choose paths on the underlying uncut grid; deleted faces do not obstruct a support label. Extend them to the coordinate extrema of the dependence box so that their spanning rectangle contains that box. When a numerical calculation reaches a free wall, include a path to the wall in its support. Equivalently one may always extend the paths to all four sides of the dependence box. Include the symmetric image choices to retain square covariance. On a torus retain winding lengths and the winding paths specified above. The number of m-blocks met by such a cover is

The support is the blocks on the paths and their prescribed fixed collars; the spanning rectangle is used to check the full region on which its numerical coefficient depends.

These rules make copying exact. First telescope on the same cell grids in the plane and in a box. Any coefficient whose dependence box and required clearance avoid the physical wall is then the identical local calculation in both geometries. A coefficient that does depend on the wall is flagged by a support reaching the wall. On a torus, a small unwrapped neighborhood contained in a sufficiently small fixed fraction of the side sees precisely the plane terms, since all final residual terms carry winding supports. In a simply connected patch avoiding the mark, gauge change turns the face arguments into ordinary differences; the numerical rules are unchanged. Near the mark the same construction copies the pierced-plane calculation with the translation (3.11). This explains why the spanning rectangle, including its dependence box, was required in the definition of shift-good supports.

There are only polynomially many face, event, origin, path, and radius labels of a given length above an anchor block. The coefficient decay therefore pays the exponential support weights associated with (3.32) when R is large. More concretely,

Assign to each link a small positive face-square cost using these inequalities. Summing over every radius and label gives an O(e^{-cR}) total coefficient at each face. The link fields themselves use only their two argument faces. No value of a branch potential along a long covering path is estimated or inserted. The enlarged labels are used also for the unmarked calculation and leave the subsequent local maps unchanged.

Finally cover every selected tile and marked event by its required halo, and every link by the just specified paths. Average the resulting patterns over e and the Gaussian modes below m in (3.8). The covariance (3.1) is positive and has range O(m) by [Ope26a, Lemma 8.1]; its shell continuation has the same properties at subsequent scales. Choose the fixed connection radius \rho large enough to include this range and the activity collars. Distinct connected components of a pattern then have independent Gaussian integrations, and their numerical choices and field factors separate. Grouping components gives the compatible subset algebra of [Ope26a, Proposition 4.8]. All original exclusions are retained, including separation from the component containing the source. In the pierced sector exactly that component is the compulsory insertion. We have obtained the exact algebraic representation and its copying rules; the remaining task is to bound its background and insertion activities in the analytic norm.

4 Analytic entry with one magnetic insertion

The preparation in Section 3 expresses the observed height interaction as an exact sum of local patterns. We now estimate that sum in the full analytic norm of (2.11). The distinction between an ordinary pattern and a pattern meeting the magnetic source is important: ordinary activities will be arbitrarily small, whereas the single activity carrying the source will have a polynomial bound in the observation scale. No smallness of that insertion is required.

We use the notation of Section 3. In particular, s is the observation-cell side, R is the tile parameter, P=R^5, m=qRs, and J_G^l is the cutoff interaction associated with the affine observation energy E_G^l. The polynomial covariance below m, its continuation on zero modes, and the analytic norm are those fixed in Section 2. The common constant of the reference field is uniform modulo 2\pi/\alpha.

The regulator must be chosen to accommodate the large-observation saving. By (3.14) and (3.19), there is a fixed h_b\in(0,1) such that

The constants here are independent of R,P,s. Fix h_{\rm ent} with h_b<h_{\rm ent}<1. An entry-admissible reserve will mean a choice

Such choices exist by taking p_1,p_2 sufficiently close to 1. We make this choice, with room for any fixed finite collection of stronger regulator regimes, before choosing L or activity weights. The proof below keeps all actual quadratic charges below h_{\rm ent}.

In a pierced square, the compulsory cover is the set of entry blocks covering the mark, a fixed CP-cell neighborhood of it, and the prescribed collars. Here C is a sufficiently large fixed geometric constant. The cover is declared occupied whether or not an isolation error occurs there, and is also called the token. Its size can depend on R but not on s or on the square side.

We prove the proposition by adapting the real-pattern and analytic estimates of [Ope26b, Section 5, Lemma 5.4], together with the normalization and connected pattern arguments of [Ope26a, Section 9]. The point of the proof is to check the hypotheses of those arguments for the unstrengthened XY law with cutoffs and for its affine magnetic sector. The Gaussian coefficient in every comparison below is \alpha^2=8\pi.

4.1 Bounded observations and conditional restoration

An isolation error has two positive terms: the restoration kernel before a cut, and the restoration kernel after the cut multiplied by its Gaussian proxy. Denote these kernels by K_o and K_c, and the proxy by B_i. After conditioning on the guard heights and prescribing the observations, its relative absolute error is

The kernels in this formula include the face cutoffs retained in their respective graphs. A cutoff common to both terms cancels from the ratio. In particular, inserting these cutoffs does not change the conditional sampling of guard heights at fixed observations.

First suppose that the observations range over a fixed compact set and that the patch has no magnetic source. On this set \chi_{\sqrt{s}}(d_f) tends uniformly to 1, with positive values; its logarithm therefore contributes a uniformly vanishing error to (4.5). Other positive real factors only decrease when multiplied by cutoffs. For a selected isolated tile, whose factor is its normalized interaction minus 1, the same compact convergence applies to that difference.

The remaining convergence inputs are precisely the mixed-smear, separated-pin, and conditional-pin limits of [Ope26b], with the observation mass retained. Core pins and surrounding guards have fixed positive clearance in tile units. All cuts follow whole observation cells, and free-wall patches are clipped with the same actual wall. These are the mixed polygonal geometries of the cited limits. The observation mass controls the constant on each component. Compactness of observation data is understood modulo component periods, using the joint components of all faces appearing in a restoration test.

For clarity, the conditional-pin input is convergence in probability under the relevant guard laws, not convergence uniformly at arbitrary guard values. In the proof of [Ope26b, Equation (5.20)], conditional terms are integrated under either of the two restoration topologies. The restoration comparisons of [Ope26b, Lemma 3.9] give exactly these changes of sampling law. The local-limit inputs required there are smear limits with observation mass, separated hard-pin and conditional-pin limits, their versions with fixed observation tilts, and the corresponding finite-geometry Gaussian statements. For the present unstrengthened law these are supplied directly by [Ope26b, Theorem 3.1 and the following restoration estimates]; the additional averaged-bond precision used in that paper’s strengthened-law entry lemma is absent here.

These statements are uniform on the compact observation sets just specified. One way to see the needed uniformity is to let the observation data themselves converge in such a compact set. Sampling tilts then vary continuously, with uniform integrability supplied by the observation exponential-moment bounds. For each finite list of conditional Laplace terms, differences at converging arguments tend to zero in probability: their derivatives are bounded by conditional moments on a fixed larger compact set. The finite Gaussian terms and proxies converge in the same way, and the limiting positive normalizers stay bounded away from zero. Compactness now gives the claim.

Here is the quantitative estimate supplied by these limits. Let \mathcal E_{\rm box} be the sum of squared affine face differences on the underlying portion of a P-box in the padded pattern. It may include faces from a patch whose difference is being tested. Recenter one observation in each joint component into [-\pi,\pi] by a height period. Paths within the box then give

Thus a fixed bound on \mathcal E_{\rm box} gives a compact observation set. Call an error deep if its core, open patch, and testing guard lie inside the bounding box with the prescribed positive clearances. At fixed R,P, there are finitely many box geometries, positive topologies, and choices of deep errors, and their number j_{\rm deep} is at most a fixed polynomial N_P in P.

For a deep error i, write \mathcal D_i=\log(K_o/(B_iK_c)). Decompose this logarithm into its centered zero-guard value, the change of centered restoration normalization with guard data, and the two conditional observation Laplace terms minus the proxy energy change. The first term is O(e^{-cR}) after refinement. The second lies between \log r_{H_i} and zero, where r_{H_i} is the conditional full-core pin ratio on a neighborhood H_i containing the restored bonds. These are the restoration and pin-interaction estimates just cited.

In the limiting Gaussian problem, let u_{i,c} be the cut harmonic prediction from the guard data. Set \varepsilon_i=\vartheta_i u_{i,c}, where \vartheta_i equals one near H_i and vanishes before the guard. The norm \norm{\cdot}_{\mathcal T} is the continuum gradient-plus-unit-cell-observation energy norm on the cut patch, with gradient coefficient \alpha^{-2}, corresponding to (3.20). The guarded localization estimate of [Ope26a, Equation (9.23)] gives, for every r\ge2,

Here \tau may be either restoration topology, and the expectation is over its Gaussian guard law with fixed observation tilt y. The constants and polynomial degree are independent of r. Indeed the centered prediction is \mathsf A g in the energy Hilbert space, where g has identity covariance and \norm{\mathsf A}_{\rm HS}\le CP^C e^{-cR}. Diagonalization and Gaussian moments bound its L^r energy norm by C\sqrt r\,\norm{\mathsf A}_{\rm HS}. Restoring bonds decreases the guard covariance. The observation tilt adds a deterministic prediction bounded by CP^Ce^{-cR}\norm y_2, proving (4.6) under both laws.

Subtracting \varepsilon_i from a cut minimizer makes it admissible for restoration without changing the guard data. Orthogonal projection in the energy space gives

and bounds the linear conditional-Laplace discrepancy by C\norm y_2\norm{\varepsilon_i}_{\mathcal T}. The conditional quadratic discrepancy from the open-patch proxy is at most Ce^{-cR}\norm y_2^2, by the same guarded localization. If restoration joins free faces, first prescribe finitely many smooth guard probes: the cut and restored minimizers belong to nested admissible energy spaces, and the correction makes the cut minimizer trace-matching. Increasing these finite probe lists gives the assertion. Thus the calculation uses finite-energy predictions, not pointwise values of a rough continuum trace.

Since \delta_i\le\min(1,|\mathcal D_i|), these estimates give Gaussian moment bounds of the form

The microscopic conditional approximation transfers these bounded moments under either topology. For this transfer use the normalized conditional restoration ratio: the conditional expectation of the positive restoration factor in the centered cut law, divided by its value at zero guard. This ratio lies between r_{H_i} and 1, and the limiting Gaussian r_{H_i} is strictly positive. Exceptional sets therefore have vanishing probability under both tilted laws, and there \delta_i\le1 suffices. The cutoff ratios add only uniformly vanishing logarithms on the compact observation set.

Let \mu_{\tau,y} be a positive topology’s conditional guard law in the box at fixed observations, and let \mathcal I_{\rm deep} be its retained deep errors. Hölder’s inequality, without an independence assumption on these errors, now yields

For j_{\rm deep}\ge1, use moment order \max(2,j_{\rm deep}). The polynomial loss is at most

Maximizing its energy-dependent part after multiplication by e^{-R^{-12}\mathcal E_{\rm box}} costs e^{O(j_{\rm deep}\log R)}, since j_{\rm deep}\le N_P=\operatorname{poly}(P) and P=R^5. Large R absorbs this loss in the exponential rarity. For j_{\rm deep}=0 the left-hand side is one. The remaining quadratic allowance is precisely

This transfer needs no convergence rate uniform in R. Fix R,P and the real-data threshold first, and prove the Gaussian version of (4.7) with a stronger exponential constant. Compactness and the finite list of geometries make the refinement errors, including exceptional-set probabilities, smaller than any prescribed positive number. Choose this number below the finite minimum of the remaining slack for 1\le j_{\rm deep}\le N_P, then take s sufficiently large. The prediction constants and exponential gain have already been fixed; only the refinement threshold depends on this transfer tolerance. For an isolated selected tile the normalized Gaussian interaction is one, so the same compact convergence makes its error from one at most e^{-cR}. These are the event and tile gains used below in the real pattern estimate.

If a simply connected observation box misses the source, the ray connection is removed by an integral gauge constant on each observation cell. The preceding argument then applies without change, with d_f^2 in the bounding face cost. Enlarged guards and counting halos can be increased by fixed factors. At an actual free wall, including a corner, the guards are clipped with that wall; no periodic replacement of the boundary condition is used.

4.2 Centered constants and cuts into bounded boxes

The estimates must retain the centered normalization even in the magnetic calculation. In each pattern telescope only the true centered partition ratios in its affected halos. These are Z_G^0(0) without cutoffs. The restoration and pin-interaction estimates of [Ope26b] therefore give the same bounded exponential cost per marked event as in [Ope26b, Equations (5.16) and (5.17)]. A fixed distinguished halo may have an extra e^{C_R} cost. Nothing is telescoped through an unaffected portion of the volume.

After this accounting, each positive topology has the real field factor

multiplied by a conditional expectation of products of (4.5) and by the small energy-error factors. Here \mathcal F(G) is the set of observation faces retained in that topology. The double telescoping of normalization constants in [Ope26a, Lemma 9.3] now uses the affine matrices of Section 3. Their local curl terms cancel exactly: the two paths compare the same deletions with the same nearby affine data and connection. At artificial boundaries the two paths are clipped identically in the unchanged collar. Only the exponentially small coefficients on face squares remain.

To estimate a topology, cut it into observation boxes of side P. The shifted comparison [Ope26a, Equation (9.20)] applies with the connection and with prescribed guard values by Proposition 2.2. Its guard probabilities still use centered partition sums. Drop the cutoffs on the new box cuts. The conditional kernels at deep errors, and hence their cutoff ratios, are unchanged. The Gaussian localization estimate gives, after averaging translates of the cuts, the logarithmic cost

where \mathcal E is a face-square cost confined to the original padded support. Indeed each face is within the fixed tile-size neighborhood of a seam for a proportion O(R/P) of the translates; the exponentially decaying matrix tails have the same bound after summation. A local curl term can add C\log(2s) only in the box containing the magnetic junction. No other elementary cell junction has nonzero circulation.

The same averaging retains a fixed positive proportion, for example one half, of the deep errors while satisfying (4.10), after increasing its constant. One can first separate the common upper charge for the magnetic curl and then apply the ordinary averaging argument. This is the only exceptional seam charge associated with the source.

4.3 The compulsory cover and large observations

In the pierced calculation use the compulsory cover defined before Proposition 4.1. Choose its constant C large enough that every bounding box belonging to a component disjoint from this cover can be tested away from the source. There is exactly one token in a pattern.

In the token component use the good-box estimates outside the fixed number of boxes meeting the source. In those exceptional boxes take only \delta_i\le1. They contain at most \operatorname{poly}(P) events, and the missing gains can therefore be paid by a constant depending on R. We next give a bound that applies both to these boxes and to boxes with large observations.

The cutoff large-field estimate and (3.19) give the bound used in choosing h_b above:

To check the strict inequality h_b<1, combine (3.13) with the Gaussian decay of the cutoff on the remaining faces. The logarithm of the right-hand side of (4.9) is at most

Since (3.19) bounds E_G^l by a fixed multiple of that face sum plus the displayed curl charge, a fixed fraction of E_G^l/2 is saved. This fraction is independent of s,R,P. The initial choice of h_b can be made positive by enlarging it while preserving h_b<1. For a selected isolated-tile difference the same argument allows the sum of this bound and 1.

It remains to express these affine costs in the homogeneous energy used in the norm. Write

taking r=0 in the ordinary sector, and let D be the underlying padded support. Restrictions of a trial field for E_D(x) show that the sum of the cut homogeneous observation energies is at most E_D(x). A quadratic inequality in (3.7), followed by (3.19), bounds the affine part from B_sr by O(\operatorname{poly}(P)) per broad box, and by O_R(\log(2s)) for the token calculation. Here the affine face differences of B_sr are uniformly bounded by the source estimates of Section 3. Choose the quadratic-inequality parameter so that its increase in h_b is less than one quarter of h_{\rm ent}-h_b. This is a fixed choice made before R; its deterministic cost only changes the constants depending on R.

Small face costs are handled by the two-cell Poincare estimate. Their deterministic affine parts cost their coefficient times the number of faces. In particular, (4.10) adds O((R/P)R^2) per affected halo, and (4.8) adds O(R^{-12}P^2) per occupied bounding box. Link costs have their exponentially small summable coefficients. Both large-observation thresholds and the regulators are evaluated on real data. If a box is declared large because \mathcal E_{\rm box}>U_{\rm big}^2, insert

with fixed sufficiently small \epsilon_1>0. Choose it so that its increase in the energy coefficient uses less than another quarter of h_{\rm ent}-h_b. Its positive part uses only a further small increase of the energy coefficient and O(\operatorname{poly}(P)) per such box. Its negative part will pay the lost good-box gains.

The remaining seam, good-box, and link coefficients tend to zero as R increases. Choose R so that their sum uses less than the remaining half of h_{\rm ent}-h_b. The actual energy coefficient h_{\rm used} therefore satisfies

with the regulator parameters already fixed in (4.1). This ordering is uniform for large s and applies in each of the finitely many reserve regimes.

4.4 Gaussian averaging and the real pattern bound

We now average the observation noise and the reference modes below m. This step uses the contraction of the homogeneous observation energy and the localized determinant estimate in [Ope26a, Lemma 9.2]. It is important to apply the trace estimate to the actual quadratic costs before replacing them by their common energy majorant. A broad box has only O(P^2) observation coordinates, so its contribution has rank O(P^2). This remains true for a box meeting the magnetic source. Every Gaussian adjacent-cell difference has bounded variance: use the two-cell flow from Section 3, now in the ordinary Gaussian field. Hence the remaining trace estimate is exactly the localized trace estimate of [Ope26b, Equation (5.24)]. Links retain their exponential path costs.

After fixing norm parameters and then large R,P, choose the finite U_{\rm big} so that the negative term in (4.12) pays the broad-box trace, all polynomial box costs, and all lost item gains. Only after this choice do we invoke the compact-data estimates and increase s. The identities P=R^5 give

so the seam and good-box allowances leave strict room for the exponential gains.

For a pattern \pi, let j_1 count its isolation errors and j_2 its selected tiles. Let d_\ell denote the cell-distance parameter of a link, including its numerical dependence radius. If X is the covered block support and F_\pi its averaged field factor, the resulting real estimate has the form

For a pattern with the token, multiply the right-hand side by C_Rs^{C_R}. The fixed number of exceptional boxes and the O_R(\log(2s)) curl charge are precisely the sources of this factor. There is only one such factor. The sum over choices of small- and large-observation boxes costs a fixed exponential per ordinary item, or a fixed C_R in the token component.

If a favorable cut translate was selected according to the real argument, dominate that selection by the sum over all translates. The number of choices gives at most a polynomial-in-R factor per error, absorbed in the exponential item gain. No optimized-cut comparison is needed for a pattern with no error event; fixed token costs are already permitted. In particular, this step never introduces a determinant whose rank grows with the microscopic cell size s.

The same regulator argument applies on free squares. Restricting to an open support lowers the homogeneous minimum energy. The positive-order covariance estimates for shells and for the terminal field follow from the binary-polynomial kernels of [Ope26a], with the convention of Section 2 and even reflection at free walls. The independent-normal factor from other covariance splits is absent here. Enlarge supports only by aligned whole cells. This proves the energy contraction, localized trace, and finite-range additivity needed in the free-square estimates.

The strict reserve also allows real translations of the background. If f has scaled analytic test norm at most 1, replacing \psi by \psi+xf in (4.13) costs at most

while leaving the final regulator evaluated at \psi. This is the quadratic completion with reserve in [Ope26a, Equation (9.28)].

4.5 Complex directions and analytic derivatives

The real bound alone does not imply the full analytic norm. We obtain a separate estimate on a fixed complex strip and then interpolate. Let z be a real observation direction. On the extended Gaussian chain lattice, an affine connection and prescribed real observations become a real linear tilt and a scalar after choosing coordinates on the underlying centered lattice. The theta parity identity of [Ope26a, Equation (9.29)] therefore gives

The fixed-graph Gaussian-chain limit preserves this inequality. The right-hand side is the ordinary centered imaginary ratio for the topology, without cutoffs. Its characteristic factor is positive, and centered pinning increases that factor.

Discard the rarity of the errors, expand into positive topologies, and apply (4.15) to each whole integrated topology. Use the argument of [Ope26b] from its complex-comparison Equation (5.26) through its strip estimate (5.27): pin whole observation-cell strips of width R/4 along translated P-box walls. The mixed limits on the resulting fixed pinned boxes give their Gaussian imaginary ratios uniformly on each fixed compact set of imaginary directions. The limiting ratios are strictly positive. The additional imaginary Gaussian wall cost is

on a strip of fixed height H_{\rm strip}. We distinguish this strip height from the geometric clearance H in Proposition 4.1. The homogeneous imaginary quadratics cancel against the same proxy quadratics as in the ordinary argument; connection offsets are real.

The cutoff factors need a simpler estimate. For real a,v, their defining Gaussian integral gives

If the field direction has bounded scaled analytic test norm, the imaginary part of an adjacent observation difference is O_{H_{\rm strip}}(s/m). A halo has O(R^2) cells, so the sum of these extra logarithmic costs is O_{H_{\rm strip}}(1) per halo. It does not require a new wall estimate. The real parts of the positive-topology estimates still have the bounds already proved: on compact-data boxes use the same comparisons with rarity discarded, and on large-data boxes retain their Gaussian completion and payments. For a selected tile discard the subtraction from 1 in this rough bound. The token component may again pay its single factor C_Rs^{C_R}.

Thus, for real x,t with |t|\le H_{\rm strip} and a direction f of scaled point norm at most 1,

A token multiplies this by C_Rs^{C_R}. The term o(R) includes the wall cost: per halo (4.16) is O(R^3/P)=O(R^{-2}). Absolute Gaussian integration and analyticity follow from the broad real estimate with reserve and this strip bound.

Choose r_{\rm an}>4eh_0, where h_0 is the analytic radius, and then H_{\rm strip}>4r_{\rm an}, before choosing R. Apply the three-lines theorem on both half-strips to

where the constant D_{\rm damp}>C_{H_{\rm strip}} is used only in this interpolation. The real boundary has (4.13) and (4.14); the outer boundary has (4.18). On |z|\le r_{\rm an} the harmonic weight of the real boundary is at least 3/4. The exponential gains therefore survive, with a smaller positive constant. Removing the damping costs e^{C|X|}, which is absorbed by the item and link gains because

in the ordinary case, with an additional fixed C_R for a token. Finally Cauchy’s estimate followed by real polarization bounds the order-r analytic derivative contribution by the same pattern bound times

The series in r converges. We have proved (4.13) in the full analytic point norm, with weakened but still positive exponential gains.

4.6 Summation, exact copying, and translated norms

All bounding boxes above use only the underlying portions of the original padded activity supports. They are devices for estimating a pattern, not new activity domains. A cut crossing a topology may charge underlying faces there, so the estimate also controls differences on an error patch. All good-error patches tested after cutting are deep in their bounding boxes. Choose cut translates on the tile grid and imaginary pin strips on the R/4 grid; these choices preserve the tile-junction geometry. A group meeting a free wall uses its clipped events and patches, with the plane ordering restricted to present tiles. Its size and counting bounds, including at corners, are the same bounds used above.

The connected-pattern summation following [Ope26a, Equation (9.34)] now applies. Exponential item gains pay the activity weights, halo covers, and polynomial labels. A link with dependence radius D' has a cover of O(1+D'/R) blocks and decay exponential in R+D', as proved in Section 3; its gain therefore also pays its cover and radius labels. On a torus the last telescoping term uses a winding label with the same bound. The finite-range Gaussian averaging factorizes on compatible components with their original exclusions. Every ordinary connected component contains an item or a link and hence retains a small factor. By increasing R, their sum gives (4.2).

The component containing the compulsory cover has the bound (4.4); all other components are ordinary background activities. No additional optional token is summed. This gives exactly the coefficient partition function (4.3). The normalization constants were centered throughout, so the extracted scalar is the ordinary one on the same square. The absolute estimates justify every exchange of pattern summation and Gaussian integration. In particular these are convergent exact representations, not identities of formal expansions.

Plane objects in this construction are defined on finite supports. The tile and link rules can respect all block translations and square symmetries, using symmetric spanning paths when necessary. Padding an event cover includes the numerical rules in a sufficiently large fixed multiple of its tile-size halo. Link covers also record their numerical dependence. Consequently a support with its stated clearance has exactly the same local calculation in the plane and in the finite graph.

For the magnetic copying statement, recall that the harmonic energy scalar has already been removed in (3.9). The remaining localized preparation, including all links, depends on a harmonic solution only through the arguments of its functions. The polynomial reference covariances themselves copy with clearance. Replacing r_\infty by r therefore changes an interior argument precisely by \gamma_M from (3.11). On a shift-good support an integral gauge removes the connection and gives the ordinary activity at the corresponding local lift.

There is one harmless convention needed for background activities at entry. A support incompatible with the fixed compulsory token footprint cannot appear in the coefficient partition function. Define the background on such a support to be its prescribed ordinary translate if it is shift-good, and zero otherwise. This leaves the represented partition function unchanged and preserves the pierced-plane copying rule. It also permits choosing H from the fixed map geometry, independently of P. Use identical positions in the initial hierarchy in every comparison, and require the inner supports of the token construction to fit with clearance. The square applications have M/(sP)\longrightarrow\infty, so this requirement is satisfied.

Lastly we check the norm convention for lifted backgrounds, rather than merely their pointwise copying. On a shift-good support and its collars the lift satisfies

The ordinary regulator can use a doubled exponent, at a cost C^{|X|} on support X, as in [Ope26a, Proof of Proposition 5.7]. For the variational regulator, increase its parameter h_* slightly. The optimized quadratic is bounded by local gradient energy and increases at least proportionally with h_*; applying the quadratic inequality to the translation again costs only C^{|X|}. A stronger untwisted activity weight pays both losses. These changes use the finitely many reserve regimes fixed at the start. Each downstream contraction may be performed in its own fixed regime, with the same regulator at successive steps; no reserve is consumed anew at every iteration.

Choose h_b,h_{\rm ent} from the uniform preparation estimates, then the analytic translation radii and entry-admissible regulator reserves before L, take L sufficiently large for the desired local-map contractions, and then take the activity weights sufficiently large. Once these downstream parameters are fixed, choose R,P, then U_{\rm big}, and finally s_0 as above. This order proves all assertions of Proposition 4.1 simultaneously. □

5 Finite critical histories and physical readouts

The entry construction provides a small interaction at a large scale, but its cutoff is useful only on a finite range of larger volumes. We therefore work with a family of finite histories. This section proves that these histories stay close to the marginal Gaussian reference, and obtains two physical observables that compare histories with different entry scales. Section 6 will use those comparisons to determine the gradient coefficient with a summable error.

Throughout this section a=\alpha^2=8\pi, and L is the fixed block factor chosen after the geometric and analytic norm parameters. An absolute index k refers to the physical side L^k, rather than to the number of steps since entry. This distinction matters when two histories have different entry scales.

5.1 The family of histories and the local maps

Fix the parameters of Proposition 4.1, including R, P=R^5 and q, independently of the history length. For a large integer J, choose a dyadic observation side s=s_J such that

Such a choice is possible because q,R,L are powers of two: among any fixed number of successive dyadic choices of s, one has the required congruence of exponents. In particular i(J)=O_L(\log J). Starting from the unpierced plane activity of Proposition 4.1, run the local maps through absolute index 8J, stopping provisionally if the activity leaves the small ball of the map theorem. We call this the J-history. A superscript (J) distinguishes its coordinates when necessary. Figure 1 displays the overlap of two such histories and the intervals that will be retained when they are joined in Section 6.

We use the singleton and remainder coordinates from Proposition 2.7. The unbalanced coordinates are (t,z,\mathcal R): the first two multiply the gradient singleton e_B^0/2 and the fundamental singleton c_B^\alpha, respectively. The balanced gradient and fundamental coordinates are (T,Z). Parametrizing the exact Gaussian curves gives the coordinate u, and subtracting the corresponding Gaussian remainder gives B=B_e+B_o, where the subscripts denote the even and odd parts under translation by \pi/\alpha. In these coordinates an exact Gaussian trajectory is (u,0,0,0) with u unchanged by the map. The curve depends on its entry index; the reference covariance kernels at a specified absolute index do not.

5.2 Periodic physical diagnostics

We now identify observables which test the two nearly marginal coordinates. Fix a sufficiently large dyadic D_0 and use tori of side

The constant D_0 exceeds the causal clearance of the local maps. For each admissible stop, the torus has its own exact recursion from the same entry scale as the plane history. Its singleton coefficients agree with the plane ones. Its remainder need not agree with the plane remainder and will be estimated separately.

Only stops with N_k\ge sP are used. Before this condition holds there are at most O(1+\log R) plane steps. Their bounded map amplification is R^{O(1)}, which is absorbed by the exponentially small entry norm when R is increased. All stopping tori are compatible with the observation and tile grids.

Let f be the first real horizontal cosine on the unit torus, and put

Here f_N is sampled at observation-cell centers and includes the cell-area weights, and \bar y is the uniform mean of the observations. Heights are taken modulo a common translation by 2\pi, so e^{i\bar h} is well defined. The expectation in the next lemma is for the periodic Bessel height law before inserting the cutoff, together with its independent observation noises.

We record the map interpretation of these tests. A source wX_N translates the remaining field by a smooth function F satisfying B_s\alpha F equal to the reference observation covariance applied to the source. It can be chosen as a microscopic cosine: translation symmetry makes the observation-cell function a cosine, and cell averaging of the corresponding microscopic cosine has a multiplier bounded away from zero. Gaussian completion gives

for a fixed c>0, with uniform scaled derivatives in the point norm. One can obtain this error by periodic inverse-energy approximation of the sampled density and the Taylor expansion of the first discrete eigenvalue.

At a stop with D_0^2 blocks, the analytic terminal expansion of [Ope26a, Equations (10.3) and (10.4)] gives

Here \nu_k=|t_k|+|z_k|+\norm{\mathcal R_k^{\rm tor}}, and \E_a denotes the reference Gaussian observation law. For one fixed sufficiently small nonzero real w, both displayed linear multipliers are bounded away from zero. Explicitly the second multiplier is

The tail diagonal D_{{\rm tail},k} is bounded, and the last average tends to a strictly negative number for the fixed w. The number \tau_k is the variance, in observation units, of the integrated Gaussian means. It is bounded because the zero-mode continuation at side u contributes O(u^2/N_k^2). The compensating factor e^{\tau_k/2} is necessary: insertion of the retained uniform mean gives the physical mean-phase insertion times e^{-\tau_k/2}. This prescription is exact for the cutoff interaction, which is periodic in the common mean. Before tilt and subtraction, the same formula with spatial factor one is a real positive multiplier bounded above and below.

5.3 No exit and the nonexponential odd sector

The preceding lemma gives convergence to the Gaussian fixed point. To determine the approach rate we must also rule out exponential decay in the odd direction. The relevant lower bound is a statement about the original periodic model at absolute side N_k; it therefore does not depend on the entry history.

5.4 Readouts common to different entry scales

The estimates obtained so far do not yet identify a single trajectory: each J-history uses different observation cells and different cutoffs. We now compare their coordinates using observables of the same original periodic model.

Let \widetilde X_{N_k} be the ordinary microscopic sampled cosine, without observation averaging or noise, and define

Both quantities depend only on the absolute index k.

We have proved three facts about every long interior window of a finite history: its state tends to zero, its odd coordinate is positive and only subexponentially small, and its two scalar coordinates can be read from physical observables independent of entry. These are the inputs for the trajectory argument that follows.

6 Joining finite histories and determining the critical trajectory

The bulk estimates provide a small trajectory on each finite scale window. The histories have different entry scales, so they need not agree as activity families on their overlaps. Their physical readouts nevertheless determine the two marginal coordinates accurately enough to compare them. We first use those comparisons to form one infinite sequence, and then determine its leading coefficient. The coefficient of the gradient activity obtained at the end of this section is the one needed in the magnetic terminal calculation.

6.1 An abstract lemma for sparse changes of history

An ordinary evolution step increases the scale index by one. A change of history takes place at a fixed scale index and does not count as an additional evolution step. The following notation keeps these two operations distinct. We write (u_j,Z_j) for the state used to start the ordinary step at index j, and (\widetilde u_{j+1},\widetilde Z_{j+1}) for the output of that step. At an index where no change of history occurs, the tilded state is (u_{j+1},Z_{j+1}). At a change of history, it is the state immediately before the change.

6.2 Application to the bulk histories

For a dyadic integer J, attach a superscript (J) to the coordinates of the history whose entry index is i(J) and whose last index is 8J. The entry estimate gives i(J)=O(\log J). Thus, for every fixed c>0, the interval [cJ,8J] lies after entry when J is large. The estimates below are uniform on such intervals, with constants allowed to depend on c.

The argument uses uniform convergence on positive-fraction windows and the common physical readouts with their stated nonlinear errors. The additional errors from entry-scale smoothing and cutoff removal are exponentially small on those windows. No rate of convergence in the height scaling limit was used to obtain the coefficient 1/(2\log L).

7 The magnetic recursion in two squares

The bulk trajectory controls activities far from the boundary and the source. We now extend that control to the free walls and the magnetic mark. The purpose of the extension is to compare two squares using exactly the same scalar normalization at the mark. The insertion at entry can have a polynomially large norm; its residual part will contract, while its extracted scalar will cancel in the comparison.

Fix a large integer j, put M=L^j, and consider the free dual squares of sides Q=M and Q=LM, with their common center at the mark. Choose the J-history of Proposition 6.3, where J is a power of two and J\le j<2J. Its entry scale is m=L^i=qRs, with s\asymp_L J^3 and i=O(\log J). In both squares stop at the same scale

Thus j^{1/10}\le D<Lj^{1/10} and M/(Ps)\to\infty. In particular the entry footprint and every fixed number of blocks about the mark fit strictly inside both squares at all scales used.

We use the blocks, compatibility relation, analytic activity norms, and covariance shells of the preceding sections. Write \mathscr Z_k(K;\phi) for the subset partition functional at scale L^k: it sums products of the activities K(X;\phi) over compatible collections of nonempty connected supports, including the empty collection. Thus \mathscr Z_k is the functional \mathcal Z_{L^k} of Section 2, indexed here by its absolute scale. For another activity family I, define

Each term on the right has exactly one I factor; the original exclusions between that factor and the background factors are retained. The entry representation of Proposition 4.1 has this form in the pierced square: in its notation, \mathscr I_i(K,I;\phi)=\mathcal Z_m^{(1)}(K,I;\phi).

For every support X at scale L^k, let \operatorname{Rect}_k(X) be its axis-parallel spanning rectangle, and let \operatorname{Rect}_k^H(X) be that rectangle enlarged by HL^k in each coordinate direction. In a square, call X bulk-good if \operatorname{Rect}_k^H(X) lies strictly inside the square, and a wall support otherwise. Independently, call X shift-good if \operatorname{Rect}_k^H(X) misses the mark, and a defect support otherwise. These predicates apply to supports of every size; they specify both the full wall and defect norms below and the terminal support decomposition.

The constant H is fixed before L, independently of the entry box parameter P. It is large enough for the testing and assignment radii and for the geometric sum of predecessor-dependence radii used in copying. The same definition at an output scale uses H coarse blocks. It does not require a bulk-good fine input to be clear of its entire shell, whose range can be of order L fine blocks. The entry convention for unused background supports inside the compulsory footprint allows H to remain independent of P.

Figure 2 shows the geometry of the common stop. The local computations determining the marked scalars take place in the same interior region; the wall contributions remain in the terminal activities.

Let K^0_{k,Q} and K^l_{k,Q} denote the ordinary and pierced background activities. On a shift-good support the local lift \lambda_Q satisfies \partial\lambda_Q=\eta_l/\alpha and is defined up to a constant multiple of the field period. On interior supports meeting the mark, the comparison with the pierced plane calculation instead uses the smooth field

The lift estimates and entry copying established earlier apply in these field units.

We prove the proposition in four steps. The localizations and placement estimates are those of the exact partition map in [Ope26a, Section 4]. The wall and lifted-copy constructions follow the method of its Propositions 5.5 and 5.7. Their low-temperature conclusions are not inputs here: the marginal charge estimate, critical forcing, and insertion contraction needed for the present proposition are established below.

7.1 The wall estimate at the marginal frequency

A small support means one in the small linear-input class of the partition map. Its fine-block size is bounded by the fixed threshold n_0, and its coarse closure has index diameter at most one. In an ordinary square use the plane localization, including the fundamental harmonic localization, on small bulk-good inputs. On small wall inputs extract only the neutral constant. Since a small support has bounded diameter, the latter inputs lie within a fixed number of fine-block layers of a wall. A small bulk-good input still uses the plane prescription if its shell convolution can see the wall; its contribution to a wall output is treated as forcing. In contrast, the H coarse-block clearance of a bulk-good output ensures that every predecessor needed for copying lies in the interior with its required shell and numerical dependence.

Ordinary activities are even in the field. After neutral constant subtraction, the localization lemma of [Ope26a, Lemma 4.5] gives a factor CL^{-2} for a small wall input. There are O_H(L) fine placements at a prescribed coarse block along a wall, and only O_H(1) at a corner. It remains to check the charged part at \alpha^2=8\pi.

Let \Gamma be the shell covariance from side m' to side Lm', and let x lie on a small wall support. The binary kernel estimates, with even reflection at a free wall, give

Both constants are independent of L after the geometric radii are fixed. For the diagonal estimate, the direct image supplies the plane value. An image contribution vanishes at binary scales smaller than a fixed multiple of its separation, by finite range. At scales larger than a sufficiently large fixed multiple of that separation it is nonnegative: the positive plane binary diagonal and the derivative estimate give this assertion. Only boundedly many intermediate binary scales can contribute negatively, and each is bounded. At a corner the same argument applies to the finite set of reflected images. Other walls are beyond the shell range in the applications where this description is used. Positive-order binary estimates summed over the shell prove the second bound in (7.9).

Apply the charge-contour estimate of [Ope26a, Equations (4.26) and (8.14)], with shift amplitude a=\alpha. For nonzero integer charge q, its exponential gain is bounded by

Using (7.9), the fundamental charges have gain CL^{-2}, and the higher charges form a summable series with stronger powers of L^{-1}. The analytic radius h_0 is chosen before L to allow the fixed nonconstant imaginary shift. The observation regulator is admissible in this estimate by [Ope26a, Lemma 9.2] and its free-square version verified at entry. Thus the charged wall terms have the same CL^{-2} bound required for the placement count.

Let p_k be the full ordinary plane activity norm, and let w_k be the full norm restricted to ordinary wall supports as defined above, without a small-size restriction. Large linear inputs have arbitrarily strong weight contraction after the activity weight is chosen sufficiently large. The analytic nonlinear bound of the partition map and the preceding small-input estimates give

Here C_Lp_k includes plane inputs and assigned pieces that reach a wall output. Their field evaluations occur on their original fine blocks with the prescribed collars, so no wall cancellation is needed to bound them. All norm regimes, including their regulator reserves, are fixed once; the wall contraction does not consume another reserve at each step.

7.2 Defect localization and exact copying

On a small shift-good input in the pierced square, use exactly the localization pieces of its ordinary counterpart, evaluated at the lifted field. Designate exactly the same scalar for extraction. In particular a translated quadratic may have a nonzero value at zero field, but that value remains part of the quadratic piece; it is not extracted as an additional scalar.

For a small defect input F_Q, let

be its neutral projection. In the plane extract \E(F_\infty)_0(\zeta), where \zeta is the current shell. In the square extract instead

The neutral projection is invariant under constant field shifts, so the alignment constant in \gamma_Q does not affect this definition. Every small defect input is interior: its bounded fine-block diameter and its failure to be shift-good place it within O_H(L^k) of the mark, while L^k\le v/L=M/(DL) for every input to a step before the stop.

The changed subtraction has the same contraction as ordinary neutral constant subtraction. Indeed the latter gives CL^{-1} without evenness. On a small fine support at side m'=L^k, the nonconstant part of \gamma_Q has fine test norm at most Cm'/Q. By the analytic derivative bound with its fixed regulator reserve,

The last inequality holds uniformly through the chosen stop for large j. Charged small defect inputs have gain CL^{-2} by the same contour argument, now with the interior shell estimates. At a prescribed output there are only O_H(1) small fine defect placements. Consequently their total residual norm is at most CL^{-1} times their input norm.

Translations of shift-good inputs use the norm reserves fixed at entry. On the relevant fine collars, m'|\nabla\lambda_Q| and (m')^2|\nabla^2\lambda_Q| are bounded. The quadratic inequality therefore permits a fixed increase in the ordinary regulator exponent, and a fixed increase in the coefficient of the variational observation regulator. Each costs at most C^{|X|} on a support X, absorbed by the reserved activity weight. A localized piece that is assigned to a coarse singleton is still evaluated on the fine block where its lift is defined. This applies to both gradient pieces and fundamental harmonics; a deterministic real phase does not enlarge the analytic norm of the latter. Thus pieces reaching defect outputs have controlled norm without evaluating a lift at the singular core.

Let d_k be the full defect norm, in this fixed reserved regime. The same multilinear partition-map estimates as for the wall, with the large-input weight gain in the defect norm, give

The forcing term includes the translated ordinary pieces just described. This proves the analytic estimate needed near the mark. We next verify the numerical equality that will make its scalar normalization disappear between the two squares.

Suppose an output support has the stated clearance. Every fine input, assigned singleton piece, and scalar normalization entering that output then fits in its enlarged neighborhood. On a shift-good output they share a common lift. Products and Gaussian convolution commute with this deterministic translation, and the identical localizations preserve the translated-copy identity. For an interior defect output use instead

The shell kernels agree there by finite range, and (7.11) gives the exact equality

It follows inductively that both activities and designated coefficients copy the corresponding plane calculation.

This induction includes numerical, as well as field, dependence. In the exact map, predecessor dependence from relocation and singleton normalizers extends by at most a fixed multiple of the current scale. Over earlier steps these distances sum to a geometric series. The link construction at entry records any numerical dependence reaching a wall in its support, so the same argument starts at entry. Enlarging H by a fixed amount pays these causal radii. A support that still reaches a wall cannot become a small marked input before the stop: the distance from the mark to the wall, in stop blocks, tends to infinity. Large inputs do not supply scalar extractions. Thus neither a long entry link nor a large insertion support can introduce a hidden wall dependence into the local constants at the mark.

The only differences between pierced and ordinary designated constants arise from small defect inputs, so they occur at O_H(1) blocks per step. At each of these blocks both the pierced and ordinary constants are the same plane constants in the two squares. All other factors cancel in the pierced-to-ordinary product within each square. This proves (7.5) exactly.

7.3 Scalar extraction with one insertion

For clarity, we record the scalar algebra before estimating the insertion. Write one exact background map as

Here the a_{k,b} are the designated coefficients of small linear inputs. Extend the same exact splitting algebra to coefficients in \R[\varepsilon]/(\varepsilon^2). Its degree-zero terms use the background localization; its degree-one terms use the marked localization and allocation specified below. Write

Thus \mathcal T_k(K) is the linear map in the insertion slot of this extended construction. It need not be the derivative of the unmarked map \mathcal F_k, since the two slots have different localization prescriptions. Put

The coefficient of \varepsilon in the exact identity is

This is differentiation of the partition identity with the stated degree-dependent splittings. No finite nonzero perturbation by the possibly large I is required.

If an earlier step has already produced a scalar \sigma, linearity applied to \sigma\mathscr Z_k(K)+\mathscr I_k(K,I) gives

In particular the earlier scalar is kept outside the later map. It is not put back into the activity as \sigma K. This elementary identity is important because \sigma need not be small, whereas the operator controlling I_+ only sees the small background K.

We also specify how the mark is preserved in the exact algebra. Use (7.11) for small insertion inputs, and assign their extracted constant to the coarse block containing the designated fine mark block. This assignment lies in their coarse closure, as required by the partition map. A residual input keeps its closure label, including the mark. In the scalar normalization step the decoration at block b is

Only a small marked input can give a'_b\ne0, and its designated block is the mark block. Hence a differentiated decoration is also supported at the mark. Every term in a differentiated product contains either an insertion input or such a decoration; its final connected support therefore contains the mark. Original exclusions remain attached to relocated pieces, exactly as in [Ope26a, Proposition 4.8]. The differentiation does not turn a compatible collection into an unrestricted product.

At zero background, the complete residual insertion map after scalar extraction is the subtracted small linear map together with the unchanged large linear map. The small-input bound just proved is C/L, with boundedly many marked placements. By increasing the large-support weight after fixing L, its remaining linear contribution is as small as desired. The multilinear bounds for the exact partition map give \norm{\mathcal T_k(K)-\mathcal T_k(0)}\le C_L\norm{K} on the fixed small background ball, and hence

where \delta_A tends to zero as the weight is increased. For this degree-one map the background and insertion may have different designated localizations: the exact splitting algebra is linear in each input, and its multilinear estimates use the same bounds on these designated pieces. Thus (7.18) applies to the prescribed marked splitting as well. It imposes no smallness assumption on \norm{I} itself.

Choose L so that C/L<\tfrac13L^{-2/3}, the weight so that \delta_A<\tfrac13L^{-2/3}, and then the background tube so that C_L\norm{K}<\tfrac13L^{-2/3} at every step. Iteration and the entry insertion bound give (7.8).

Finally every summand in \beta_k depends only on a small marked input and the background normalizer at its assigned block. The copying argument in the preceding subsection makes both its numerator and denominator identical in the two squares. This also holds for the insertion activities by induction through (7.16). At entry the compulsory footprint fits with clearance in both squares; a support too small to contain it has insertion coefficient zero in both. Starting with \sigma_i=0, Equation (7.17) therefore gives \sigma_M=\sigma_{LM}. This equality does not require a bound or a sign for the accumulated scalar.

7.4 Completion of the background bounds

8 Terminal integration and the normalization constant

We now compare two squares using the same finite history. The comparison removes all scalar contributions from the microscopic magnetic core. What remains is a deterministic energy difference and a small terminal partition function. The accuracy of the latter comparison, rather than an estimate on the core amplitude, will give a convergent normalization.

Let M=L^j, n=M/2, and use the history with J\leq j<2J as in Proposition 7.1. For Q\in\{M,LM\}, the subscript Q will indicate the free dual square of side Q. Both calculations stop at the scale in (7.1):

In particular,

There are O_L(D^2) terminal blocks in either square. All constants below may depend on the parameters fixed before choosing the history, including L, but not on j.

Write \E_{*,Q} for expectation under the remaining reference Gaussian field in the square of side Q, together with its common constant, which is uniform modulo 2\pi/\alpha. Recall that \alpha^2=8\pi. For \varepsilon\in\{0,l\}, define

Here \mathscr Z_h is the compatible-support partition polynomial, and \mathscr I_h is its part with exactly one marked activity, as in Proposition 7.1. In particular, \mathcal J_Q does not include the separately extracted scalar \sigma_Q.

8.1 Uniform terminal integration

The fact that D grows is useful: it separates the magnetic core from the wall at the terminal scale. It also requires a terminal estimate whose constants do not depend on the total number of blocks.

8.2 Cancellation of the magnetic scalar

Let \widehat a_{Q/2} denote the ratio with the observation cutoffs used for this history. This notation depends on the chosen history; only the two squares under simultaneous comparison use it below. Set

The common preparation scalar cancels between the two sectors. The exact identities (7.4) and (7.6) therefore give

Both \rho_j and \sigma_j are exactly the same for Q=M and Q=LM. The equality of \sigma_j is algebraic; its sign has not been assumed.

The two scalar logarithms in (8.8) cancel when the square is enlarged. It remains to compute the difference of the terminal background terms to a summable accuracy.

8.3 The first-order magnetic difference

Define the first-order term

and its magnetic double difference

The word “double” here refers simply to taking first a difference between the magnetic and ordinary sectors and then a difference between the two square sizes.

8.4 The deterministic energy increment

We next compute the constant in the leading energy increment. Its normalization is important: the Laplacian here counts each undirected unit edge once.

8.5 A summable ratio and the full limit

All quantities involving the microscopic magnetic core have now cancelled. The universal logarithmic correction comes from the running gradient coefficient in Lemma 8.3.