Proof. For square collars this is [28]. Its proof uses the critical FK-Ising RSW estimate with arbitrary boundary wiring [16]. A finite collection of rectangular dual crossings supplies a separating circuit with probability bounded below. Repeated disjoint annular tests give \(q_0\leq C(w/R)^\theta\). In a clipped chart restore the missing lattice vertices and edges when bounding \(q_0\) from above, and then apply the same tests in the restored collar. Common fixed boundary spins have already been deleted. Polygonal collars use finitely many such rectangular tests. Conditioning on a farther contour represents each continuation as a mixture of the same local kernels. These observations give the stated extensions of the square version. ◻

Proof. For a contour configuration \(\tau\), let \(Z_1^{\rm out}(\tau)\) and \(Z_2^{\rm out}(\tau)\) be the two outer continuation weights, including any summation over remote data. Their ratio \(w(\tau)=Z_2^{\rm out}(\tau)/Z_1^{\rm out}(\tau)\) is independent of every inner datum. For a fixed configuration at \(W\), 2 bounds the oscillation of this ratio over \(\tau\); normalization of the contour marginal does not change its supremum-to-infimum ratio. Conditioning at \(W\) and mixing proves (16) for an arbitrary continuation. This argument also applies to a different outer domain, since the graph in the collar is unchanged.

Rescale \(w\) so that \(1\le w\le K\), and put \(d=\left\|\nu_1-\nu_2\right\|_{\mathop{\mathrm{TV}}}\). For an event \(A\), \[\left|\frac{\nu_1(w\mathbf 1_A)}{\nu_1(w)} -\frac{\nu_2(w\mathbf 1_A)}{\nu_2(w)}\right| \le Kd+|\nu_1(w)-\nu_2(w)|\le 2Kd.\] The bounds on the integrals follow from the layer-cake formula for nonnegative functions bounded by \(K\). Taking the supremum over \(A\) proves (17). Conditional on \(T\), the actual outer Gibbs kernel is the same for both inner laws, so applying that kernel cannot increase total variation. ◻

Proof. We compare the pinned law, the law with both pins and fields, and the law with fields alone. All error budgets in this proof may first be divided by the finite transport constants in 7.

Proof. We first prove the one-sided assertion. Smooth the observable by conditioning on a separating square contour at its end of the buffer. For the outer observable, a square just inside \(B_{R/2}\) can be used. The conditional mean is bounded by one, and replacing the observable by that mean preserves its pairings with all tests on the other side of the cut.

Break the contour at its corners and, in a boundary chart, at its intersections with the physical boundary. Choose intersections transversely and away from corners, or include a coincident corner in the same deletion. Delete short pieces in small disks at these points. The values in a deleted piece can be assigned arbitrarily in the far conditional law, at a cost tending to zero by 3. More explicitly, average the original bounded contour function over the deleted spins conditional on the retained ones. For every assignment of the retained spins, compare the far law for each deleted-spin assignment to a single prescribed one. The uniform pointwise ratio bound controls the resulting pairing error. Comparing that prescribed law back to the conditional mixture costs the same bound. Thus the new pairing is again in the original reference measure, with the conditionally averaged contour function. This operation does not assert a pointwise approximation of the observable. It also handles the vanishing displacements of intersection points in an approaching boundary chart.

The retained segments have positive distance from the physical edge and from the far test. Subdivide them into finitely many sufficiently short straight pieces, and delete much smaller gaps at the subdivision points by the same argument. Each piece now lies strictly inside its own square, and these squares have disjoint enlarged buffers. The buffers may be thin: for adjacent pieces of length \(\ell\) separated by a gap \(g\), one may take square half-sides just above \(\ell/2\) and outer walls at \(\ell/2+g/4\), with strictly nested band and target levels in between. The stopping theorem allows any such fixed positive clearances [28]; its constants need not be uniform as \(g/\ell\) shrinks. Choose the piece lengths after the original corner and physical-edge deletions, smaller than their positive geometric clearances. This also keeps squares on different sides disjoint. All these sizes and their number are fixed before the mesh limit. The observable, for the purpose of the desired pairings, has become a bounded function of spins in these finitely many source squares.

In each square reveal an interior square containing its source piece and apply the outward part of 5. Choose a separating target square contour \(T\) outside the stopping band, and a more distant constant-sign reference wall \(W\), both still inside the square’s buffer; see 1. The theorem gives finitely many deterministic regular signed barriers. Off an exceptional descriptor, adding the chosen barrier changes the reference target law by the prescribed small total-variation amount, for every realizable stopping transcript.

The actual continuation can include all other source squares, their transcripts, and the physical boundary. These data lie beyond \(W\). The law on \(T\) is therefore tilted by the common weight in 7, independently of the transcript and its replacement. The per-transcript TV estimate transports to this continuation. Its further law is the same given \(T\). The exceptional probability is still small, by the buffered comparison for the band marginal. Thus both required parts of the stopping theorem transport: the exceptional-event estimate uses the band density, and the conditional target estimate uses the common target-contour tilt.

Apply these comparisons successively to the finite family of source squares. Conditional on their transcripts the remaining law is Gibbs with the revealed pins. Each square’s unchanged outer collar permits the same conditional comparison, so the errors telescope. After all barriers have been inserted, they separate the initially observed spins from every far test. The law for the latter is the original reference law with just the selected deterministic signed barriers imposed; it no longer depends on the transcript behind them.

Let \(A_i\) be the good descriptor bins, with a joint index if several squares are used. The corresponding coefficient is \[c_i(\delta)=\mathbb E[F_\delta\mathbf 1_{A_i}], \qquad \sum_i|c_i(\delta)| \le\mathbb E[|F_\delta|\mathbf 1_{\cup_i A_i}]\le1.\] Here \(F_\delta\) denotes the bounded function after the initial conditioning and deletions. Its pairings already differ from the original ones by only the allocated error. Exceptional bins contribute at most their probabilities. We have therefore obtained a signed combination of deterministic pinned laws with coefficient mass at most one, uniformly against every bounded far test.

Apply 8 to every member of the finite barrier list. For a joint barrier choice, add the profiles in its disjoint squares. The field-only measure is exactly the original reference measure tilted by the single normalized weight for this sum profile. Normalization is global; no product of local normalizing constants is assumed. The coefficient mass does not change. Allocate the remaining errors among the finitely many choices and squares. This proves (20).

All source squares were placed on fixed-factor contours at the observable’s end of the original collar. Consequently their profiles remain within \(Cs\) at the inner end, or between \(cR\) and \(CR\) at the outer end, regardless of how small the deletion disks, strip widths, or approximation tolerances are chosen. They avoid the center. The deletion and subdivision choices put them strictly in the interior, also in boundary charts. This proves the support claims.

If a finite symmetry group preserves the reference law and the geometry, average the replacement over that group, with its appropriate character for an odd observable. Enlarge the finite profile list by the corresponding images. The sum of absolute coefficients cannot increase. In the plane this includes the right-angle lattice rotations and spin flip; free boundary also permits spin flip.

The two-sided finite-array assertion requires uniform integrability when a soft weight replaces one of the original bounded tests. We give its proof after establishing the required moment bounds in 10. The preceding one-sided construction is independent of those bounds. ◻

Proof. First consider one compact set. Buffered comparisons reduce bounds on nonnegative functions there to bounds in the plane, or to a fixed homogeneous square reference. The standard two-point estimate is \[\delta^{-1/4}\bigl|\mathbb E[\sigma_z\sigma_{z'}]\bigr| \le C(|z-z'|\vee\delta)^{-1/4}\] in the plane. Its singularity is integrable in two dimensions, so the second moments of the smeared fields are uniformly bounded. The buffered comparison gives the same conclusion in compact interior parts of the other reference geometries, and hence bounded first moments as well.

For a nonnegative profile \(h\), the GHS inequality [23], in the nonnegative-field form of [7], gives in a homogeneous plus or free reference, for \(t\ge0\), \[\log\mathbb Ee^{tX_{\delta,h}} \le t\mathbb EX_{\delta,h} +\tfrac12t^2\mathop{\mathrm{Var}}(X_{\delta,h}).\] The coefficients on the right are uniformly bounded. The negative exponential is bounded by the corresponding positive one using spin flip and monotonicity; minus boundary follows by spin flip. Plane and half-plane references follow by taking the homogeneous finite-volume limits. Write a signed profile as \(h=h_+-h_-\) and apply Cauchy–Schwarz to the two exponentials. Covering a compact support by finitely many buffered squares and applying Hölder, if necessary, proves (23) for every finite profile list. These are also the smeared-field exponential estimates included in 6; the argument explains why no sign restriction on \(h\) is needed.

We next justify moment convergence at diagonals. In a homogeneous plus or free reference, cluster coloring bounds the absolute spin product mean by the probability that every insertion connects to the boundary of a small square around it. Choose that square’s radius to be a fixed fraction of the minimum of its distance to another insertion and its distance to the physical boundary. The squares can be taken disjoint. Conditioning successively outside them and using the fully wired one-arm upper bound gives the product bound \[ \delta^{-b/8}\left|\mathbb E\prod_{i=1}^b\sigma_{z_i}\right| \le C_b\prod_{i=1}^b(d_i\vee\delta)^{-1/8}, \tag{25}\] where \(d_i\) is the indicated nearest-point or boundary distance. Repeated lattice sites are allowed by taking the trivial arm at radius \(\delta\). Fixed minus boundary is reduced to plus by spin flip.

On compact interior sets, the right side of (25) has uniformly integrable collision singularities. To see the relevant power counting, group close points in dyadic clusters. A cluster of \(q\ge2\) points costs at most singular degree \(q/8\), while its relative coordinates have dimension \(2(q-1)\). At every nontrivial cluster in the nested grouping, \(2(q-1)-q/8>0\). Summing the resulting geometric series, from the smallest clusters outwards, bounds the integral and makes the contribution of tuples within distance \(\varepsilon\) of a diagonal tend to zero uniformly in \(\delta\). The same counting bounds the lattice sums, with the final dyadic shell truncated at \(\delta\).

Away from the diagonals, 6 supplies separated-point convergence, and the regular profile pieces give Riemann-sum convergence. Removing the small diagonal region now proves every joint polynomial moment limit. The exponential bound permits uniform approximation by Taylor polynomials: for example the tail after degree \(N\) is bounded in expectation by \(2^{-N}\mathbb Ee^{2|X_{\delta,h}|}\). This proves convergence of exponential expectations and their joint versions. Jensen’s inequality and the bounded first moments keep each normalizing denominator bounded away from zero. Higher exponential moments therefore give the asserted uniform integrability of normalized weights.

Finally, the buffered density-ratio bound implies \[\mathbb E[A\mid\hbox{spins in the second region}]\le K\mathbb EA\] for every nonnegative \(A\) in the first region: compare a fixed second-region assignment to its unconditional mixture. Multiply by the nonnegative normalized second-region weight and integrate. This proves (24). Arbitrarily large finite fields are allowed in the pointwise estimate, and the bound does not depend on the chosen profile. ◻

Completion of the proof of 9. For the finite families in (22), first choose the inner replacements with a small error budget. For each row \(\alpha\), the one-sided statement gives one vector \(c_{\alpha i}\) that works for all outer columns. The inner profile list is finite and independent of these rows. By 10, choose a finite \(Q\ge1\) such that the tails \[\mathbb E\bigl[U_{\delta,h_i} \mathbf 1_{\{U_{\delta,h_i}>Q\}}\bigr]\] are uniformly small for every profile in those lists and every sufficiently small mesh. Apply the outer one-sided replacement to every \(H_{\beta,\delta}\), choosing its error smaller by the factor \(Q\). It gives one vector \(d_{\beta j}\) that works simultaneously against all the bounded inner tests \((U_{\delta,h_i}\wedge Q)/Q\). Thus the coefficients factor by row and column, as required in (22). The unused fixed-factor buffer margin allows its inner test region to contain the whole support of the first replacement, rather than just the original \(B_s\).

The truncation error against an original bounded outer observable is bounded by the displayed tail. Against any positive normalized outer replacement it is bounded by a geometric constant times that tail, by (24). The sum of absolute outer coefficients is at most one. Thus this error estimate was available before the outer profile list was chosen; no bound on the eventual field strengths is needed. Removing the truncations and summing the allocated finite error budgets proves the two-sided assertion, including pairings with the constant one. ◻

Proof. We first prove a polynomial identity. On a set \(I\) of \(2m\) points, write \(D_+(\mu)\) and \(D_-(\mu)\) for the Vandermonde determinants on the points with the indicated signs, and put \[V_s(I)=\sum_{\sum_{i\in I}\mu_i=s}|D_+(\mu)D_-(\mu)|^2.\] Then \[ \sum_{\sum\mu_i=0}|D_+(\mu)D_-(\mu)|^2 \left|\sum_i\mu_i z_i\right|^2 =2V_2(I). \tag{32}\] Here and below the Vandermonde on an empty set is one.

For completeness, work in the exterior algebra of \(\mathbb C^{2m}\otimes\mathbb C^2\), with orthonormal site and color basis \(e_i\otimes e_\pm\). Define \[p_k^\pm=\sum_{i\in I}z_i^k e_i\otimes e_\pm, \qquad \Psi=\bigwedge_{k=0}^{m-1}(p_k^+\wedge p_k^-).\] Let \(Q\) project to the subspace with exactly one exterior factor at each site. Both \(\Psi\) and \(v=Q\Psi\) are invariant under simultaneous \(SU(2)\) transformations of the colors: each two-column wedge is multiplied by the determinant, which is one, and \(Q\) commutes with color transformations. In the basis indexed by one sign per site, \(v\) has only balanced coordinates, whose absolute values are \(|D_+(\mu)D_-(\mu)|\).

Let \(\tau_1,\tau_2,\tau_3\) be the Pauli matrices and let \(A_j=\sum_i z_i\tau_j^{(i)}\) act on this subspace. The Gram matrix \(\langle A_jv,A_kv\rangle\) is invariant under every real three-dimensional rotation, because \(v\) is \(SU(2)\) invariant. It is therefore a scalar multiple of the identity. For \(A_+=(A_1+iA_2)/2\) we obtain \[ \|A_3v\|^2=2\|A_+v\|^2. \tag{33}\] The left side is the left side of (32). On the full exterior algebra \(A_+\) is the derivation sending \(e_i\otimes e_-\) to \(z_i e_i\otimes e_+\) and sending a plus factor to zero; this derivation commutes with \(Q\). Applied to \(\Psi\), all terms vanish by repeated columns except the replacement \(p_{m-1}^-\mapsto p_m^+\). The remaining plus columns have degrees \(0,\ldots,m\), and the remaining minus columns degrees \(0,\ldots,m-2\). Thus the absolute values of its projected coordinates are exactly the Vandermonde products in \(V_2(I)\). Equation (33) proves (32). When \(m=0\) the identity is the trivial equality \(0=0\).

We now multiply the numerator on the left of (31) by \(W_0(X)\). In the resulting sum over two balanced assignments \(\mu,\nu\), partition the sites into the agreement set \(A=\{i:\mu_i=\nu_i\}\) and its complement \(B\). Both restrictions of \(\mu\) are balanced, since \[\sum\mu_i=\sum\nu_i=0 \quad\Longrightarrow\quad \sum_{i\in A}\mu_i=\sum_{i\in B}\mu_i=0.\] The factors joining \(A\) to \(B\) cancel in \(w_X(\mu)w_X(\nu)\). For either group \(I=A,B\), put \(q(I)=\prod_{i<j\in I}|z_i-z_j|^{-1}\). The remaining factor in that group is \[\prod_{i<j\in I}|z_i-z_j|^{\mu_i\mu_j} =q(I)|D_+(\mu)D_-(\mu)|^2.\] If \(M_I=\sum_{i\in I}\mu_i z_i\), the mixed terms in \(|M_A+M_B|^2\) sum to zero by reversal of all signs on one balanced group. Equation (32) therefore shows that the sum under consideration equals \[ 2\sum_{\substack{A\sqcup B=X\\|A|,|B|\text{ even}}}q(A)q(B) \bigl[V_2(A)V_0(B)+V_0(A)V_2(B)\bigr]. \tag{34}\] The same grouping applied to two charge-two assignments has charge two on \(A\) and charge zero on \(B\). Hence the first sum in brackets is \(W_2(X)^2\). Applied to charges two and minus two, it has charge zero on \(A\) and charge two on \(B\), so the second sum is \(W_2(X)W_{-2}(X)\). Sign reversal gives \(W_{-2}=W_2\). Thus (34) is \(4W_2(X)^2\); division by \(W_0(X)^2\) proves the assertion. ◻

Proof. We first record how pointwise expansions determine the spectral assertions. For \(v\) in the dense span of point states, \[\langle v,D_tv\rangle =\int_{[0,\infty)}t^q\,d\nu_v(q), \qquad \nu_v\geq0.\] Suppose its expansion consists of finitely many specified powers below \(p\), with an \(O_v(t^p)\) remainder. The limit at zero identifies the mass at zero. After removing that mass, positivity excludes spectral mass on each interval below the first remaining specified power: any positive mass on a compact subinterval there would decay more slowly than the expansion allows. Dividing by that first power and taking a limit identifies its atom. Repeating this argument identifies the subsequent atoms and excludes the intervening intervals, including the interval up to \(p\). Polarization identifies their matrix elements. If the excluded spectral projection vanishes on the dense point-state span, it vanishes on the entire Hilbert space. Consequently no uniformity of the constants in the point-state remainders is needed for this step.

Normalize the plane spin field temporarily so that its two-point function is \(|z-w|^{-1/4}\). The exact formula for an even list \(Z\) is \[\bigl(\mathbb ES_Z\bigr)^2=2^{-|Z|/2}W_0(Z).\] For lists \(X=(x_i)_{i=1}^k\) and \(Y=(y_j)_{j=1}^l\) of the same parity, the reflected, dilated matrix element \(F_{X,Y}(t)=\langle S_Y,D_tS_X\rangle\) consequently satisfies \[\begin{align*} F_{X,Y}(t)^2 &=2^{-(k+l)/2}\sum_s t^{s^2/4} \sum_{\substack{\sum_i\mu_i=s\\\sum_j\nu_j=-s}} w_X(\mu)w_Y(\nu) \prod_{i,j}|1-tx_i\overline y_j|^{\mu_i\nu_j/2}. \tag{35}\end{align*}\] Indeed the inner dilation contributes \(t^{(s^2-k)/4}\) to the spin correlation square and \(t^{k/4}\) from (29). Inversion of the outer list contributes powers of \(|y_j|\) that cancel its primary factors and the cross-distance powers. This leaves exactly (35). Each remaining cross factor has a convergent integer-power expansion near zero.

For odd lists the lowest charges are \(s=\pm1\). Their leading coefficients are equal and factor as \(W_1(X)W_1(Y)\). Taking the positive square root gives \[F_{X,Y}(t)=a(X)a(Y)t^{1/8}+O_{X,Y}(t^{9/8}),\] where \(a\) is nonzero, since \(W_1\) is positive. In the even sector the charge-zero term has leading coefficient \(m(X)^2m(Y)^2\), where \(m(X)=\mathbb ES_X>0\) for a nonempty even list. The coefficient linear in \(t\) from charge zero vanishes by sign reversal on one list. The charges \(s=\pm2\) therefore give \[ F_{X,Y}(t) =m(X)m(Y)+m(X)\frac{W_2(X)}{W_0(X)} m(Y)\frac{W_2(Y)}{W_0(Y)}\,t +O_{X,Y}(t^2). \tag{36}\] The order-one matrix has rank one and is nonzero already for a two-spin state. Empty states are handled directly: their pairings are the means, and they have no nonconstant coefficient.

For the improved even estimate, set \(A_X=\sum_i\mu_i x_i\) and \(A_Y=\sum_j\nu_j y_j\) under their independent charge-zero weights. The linear cross exponent is \[-\tfrac12 t\operatorname{Re}(A_X\overline{A_Y}).\] The quadratic term in the logarithm has angular frequency two. Averaging rotation of \(X\) over the four right-angle rotations therefore leaves, from the square of the linear term, only \[\frac1{16}\mathbb E_0|A_X|^2\mathbb E_0|A_Y|^2t^2.\] The order-\(t\) correction to the \(s=\pm2\) cross factors has angular frequency one, and so supplies no invariant coefficient of order \(t^2\). Put \(a_X=W_2(X)/W_0(X)\) and similarly for \(Y\). Lemma 12 says that the invariant expansion of the correlation square, relative to its constant term, is \[1+2a_Xa_Yt+(a_Xa_Y)^2t^2+O(t^3).\] The order-\(t\) coefficient before averaging is already rotation invariant. We may therefore expand the square root first and then average: its invariant quadratic coefficient is \[\tfrac12(a_Xa_Y)^2 -\tfrac18(2a_Xa_Y)^2=0.\] Thus the remainder in (36) is \(O(t^3)\) after projecting one state to right-angle invariance. This proves all the plane spectral statements by the preceding positivity argument. It also identifies \(\Pi_0\) as the vacuum projection.

In the half-plane, the squares of the fixed/free spin formulas have the form \[ c_n\prod_i(2\operatorname{Im}z_i)^{-1/4} \sum_{\mu_i=\pm1}\chi_n(\mu) \prod_{i<j} \left|\frac{z_i-z_j}{z_i-\overline z_j}\right|^{\mu_i\mu_j/2}. \tag{37}\] For fixed plus boundary, \(c_n=2^{-n/2}\) and \(\chi_n=1\). For free boundary and even \(n\), \(c_n=(-1)^{n/2}2^{-n/2}\) and \(\chi_n=\prod_i\mu_i\). These are the fixed/free formulas in Proposition 6. Fixed minus follows by spin flip.

In a reflected pairing the inner height factors cancel the dilation factor, and the outer height factors cancel the primary factors exactly. At \(t=0\) the cross factors are one. Since \(c_{k+l}=c_kc_l\) for all list lengths in the fixed case and for even list lengths in the free case, the constant term is exactly the product of the separate correlation squares.

The linear cross term is odd under simultaneous reversal of the signs on either list. Internal weights are invariant under that reversal; so is \(\prod_i\mu_i\) when the list is even. The linear term therefore vanishes. The separate means are nonzero for every fixed-boundary point product, and for every even free-boundary point product. This follows from the positive one- and two-point functions and the ferromagnetic product inequalities, or directly from the formulas. Taking the square root, with the sign prescribed by fixed minus where necessary, gives \[F_{X,Y}(t)=m(X)m(Y)+O_{X,Y}(t^2).\] The spectral extraction argument again applies, proving the boundary statement. The zero state and linear combinations are included by direct pairing and polarization. ◻

Proof. For a fixed profile, Lemma 10 constructs the exponential by its polynomial series. Integrated spin products belong to the completion of the point-state span: remove small diagonals, approximate the remaining integrals by finite sums of point states, and then restore the diagonals. The integrable product-arm bounds used in Lemma 10 control this procedure in the squared Hilbert norm as well, since inner and reflected outer supports are separated. Exponential-series tails converge in that norm by the same moment bounds; absolute values are dominated by a nonnegative majorant profile and the nonnegative fixed-plus or free spin correlations. Fixed minus is reduced to plus by reversing the profile. This proves membership in \(\mathcal H\), not merely existence of the ordinary expectation.

Let \(U\) be such a normalized positive weight and let \(\mathcal JU\) be its reflected weight, including the Jacobian and primary factors in the reflected profile. Conformal covariance gives \(\mathbb E\mathcal JU=\mathbb EU=1\). On the lattice, the separated-support ratio estimate bounds the conditional mean of either normalized positive weight, given the spins supporting the other, by a fixed constant. Applying this estimate to fixed-profile approximations and taking the limit by Lemma 10 gives \[\|U\|_{\rm rad}^2=\mathbb E[U\mathcal JU]\leq C.\] Crucially, the lattice bound does not depend on the weights’ amplitudes. It therefore holds uniformly over all the finite dictionaries, including dictionaries obtained with smaller approximation errors. The triangle inequality proves the last assertion. ◻

Proof. Rescale the disk to radius four and apply the spin and scalar-energy one-point convergence in the reference input Proposition 6. Plane centering is the standard energy centering in this one-point statement. The fixed-boundary magnetization at the center is positive, and the plane-centered scalar-energy one-point function there is nonzero. Averaging the four incident orientations preserves its common scalar limit. These observations give (38); no rate of convergence is used. ◻

Proof of Proposition 11. Normalize \(\|F\|_\infty\leq1\). For any assignment \(\tau\) on the cutting contour at radius \(R\), condition first on a separating contour at a fixed smaller radius, for example \(R/2\). Relative to the critical plane law, its distribution induced by \(\tau\) has a density \(H_\tau\) with \[\mathbb EH_\tau=1,\qquad 0\leq H_\tau\leq C.\] The bound follows from the ratio estimate across the intervening fixed-modulus buffer and is uniform over every configuration \(\tau\), not just typical configurations. The same construction represents disk-plus expectation by a normalized density \(H_D\) on that contour, with the same kind of bound. The Markov property therefore identifies \[ \begin{split} P_RF(\tau)-m(F)&=\mathop{\mathrm{Cov}}(F,H_\tau),\\ d_R(F)&=\mathop{\mathrm{Cov}}(F,H_D). \end{split} \tag{39}\]

Apply the finite-matrix assertion (22) to these pairings, scaling the columns by their common sup-norm bound and adjoining a constant row and column to control the separate means. It gives one coefficient vector per row and per column, each with bounded total absolute mass. Lemma 10 passes each fixed dictionary to the continuum. The resulting inner and outer states have uniformly bounded norms by Lemma 14. Their supports lie at radial ends of radii \(C_1s\) and \(c_1R\), where \(C_1,c_1>0\) do not depend on replacement accuracy. After rescaling the radial ends, the dilation parameter is \[t=\frac{C_1s}{c_1R}.\] We choose \(\ell_0\) large enough that \(t<1\). Parity and, when needed, right-angle invariance are imposed by averaging the inner replacements. These projections commute with \(D_t\), so no symmetry of the outer density is required. Subtracting the separate means removes the vacuum term.

For sufficiently fine mesh, every such finite matrix is therefore, to arbitrarily small absolute error, a matrix of the form \[ t^d a_i b_j+E_{ij}, \qquad |a_i|,|b_j|\leq C, \qquad |E_{ij}|\leq Ct^p, \tag{40}\] by Proposition 13. Coefficients in the finite dictionaries may depend on the mesh; along an arbitrary sequence we pass to a subsequence on which they converge. Neither this argument nor (40) asserts convergence in Hilbert norm of the original lattice observables. In particular, the first inequality in the proposition follows at once from the disk column.

To obtain the second inequality we show that the disk column has \(|b_D|\) bounded away from zero. Adjoin a fixed inner calibration row: in the odd sector take a spin smeared against a nonnegative profile in a small interior box; in the even sector take a product of spins smeared in two small disjoint boxes, and average rotations when appropriate. Normalize these at the inner length scale. They are fixed states of bounded Hilbert norm. Lemma 10 and truncation allow them to be paired with the same outer replacements as the bounded row \(F\). Uniform control of the truncation tails against normalized positive outer weights follows from their separated-support density-ratio bound.

Their connected disk responses have a nonzero leading coefficient of order \(t^d\). For the odd calibration this is the positive bulk magnetization in a large fixed-plus disk. For the even calibration, the two-point case of (12), with \(q=|z-w|/|z-\overline w|\), gives the explicit ratio \[\frac{G_{2,\mathbb H}^{+}(z,w)}{G_{2,\mathbb C}(z,w)} =(1+q)^{1/2}(1-q^2)^{-1/8} =1+\frac q2+O(q^2).\] Here \(G_{2,\mathbb C}(z,w)=|z-w|^{-1/4}\) in the normalization of Proposition 6; the ratio is independent of the overall spin normalization. In a disk of radius \(H\) centered at zero, the same formula holds with \[q=q_H(z,w)=\frac{H|z-w|}{|H^2-z\overline w|} =\frac{|z-w|}{H}+O(H^{-3})\] for points in the two fixed boxes. Indeed a Möbius map to the half-plane gives this \(q_H\), and covariance cancels the primary factors between the disk and plane two-point functions. The coefficient of \(H^{-1}\) in their difference is therefore \(\tfrac12|z-w|^{3/4}>0\) in the displayed normalization. Smearing with the fixed positive profiles gives the connected calibration response \[ c_*t^d+o(t^d),\qquad c_*>0, \tag{41}\] as \(t\downarrow0\). This qualitative asymptotic is sufficient for detection; it does not require a lattice convergence rate. Rotation averaging preserves the positive leading coefficient.

Include this calibration row, the row \(F\), the disk column, and the column \(H_\tau\) in one finite matrix. For all sufficiently small \(t\), (41) and (40), with replacement error smaller than \(t^p\), give \(|a_{\rm cal}b_D|\geq c>0\). Since \(|a_{\rm cal}|\leq C\), this implies \(|b_D|\geq c/C\). Every other column has \(|b_j|\leq C\), whence \[|t^d a_Fb_\tau+E_{F,\tau}| \leq C|t^d a_Fb_D+E_{F,D}|+Ct^p.\] By (39) this is (27), first in the fine-mesh limsup sense.

Finally suppose a claimed lattice bound, with a slightly larger constant, failed at arbitrarily large \(s\) for the fixed ratio. Choose the failing functions and, for the supremum estimate, failing contour configurations along that sequence. Apply the finite-matrix argument just given to those rows and columns; the finite dictionaries are uniform over the chosen functions. Choose their accuracy below the fixed remainder scale \(t^p\) and then take the fine-mesh limit. This contradicts the limsup bound. It supplies a threshold uniform over \(F\) and \(\tau\). The same argument applies when the ratios approach the fixed ratio, since all geometric buffers retain their positive clearances. The constants used in the spectral and calibration estimates are uniform for all sufficiently small \(t\), hence for all sufficiently large fixed ratios, as asserted. ◻

Proof. First take a straight boundary and a fixed homogeneous reference half-plane law. As in (39), represent each cutting-contour assignment on a slightly inner contour by a bounded normalized density relative to that law. The physical boundary type is identical in both laws; only the artificial cut is changed. Lemma 9 replaces the resulting pairings by uniformly bounded radial states. Subtract the reference mean of \(F\), so that the vacuum projection disappears. The boundary case of Proposition 13 bounds every remaining pairing by \(C(s/R)^2\|F\|_\infty\). This is uniform over the arbitrary cut assignment. For fixed minus use spin flip; in the free case use the even projection on the inner state.

The radial origin may be moved to the nearby physical boundary point. The support then lies within a square of radius \((1+C_0)s\) about that point, while the outer test remains at a distance comparable to \(R\) if \(\ell_0\) is large enough. This changes only the fixed geometric constants. The limiting half-plane may be rotated after taking the continuum limit; covariance makes its radial estimate independent of orientation. The lattice buffers themselves may remain axis-parallel.

For locally graphical curved boundaries use the reference-chart construction in Section 3. Its capped reference domain agrees with the actual physical edge on the buffer under consideration, so the bounded-density representation remains valid. If the asserted thresholds did not exist for a fixed ratio, one could choose a sequence of failing examples with \(s\to\infty\), graph slope deviations tending to zero, and chart windows increasing without bound. After rescaling and passing to a subsequence, the tangent orientations converge and the charts converge locally to a half-plane. The contour replacement and moment convergence statements hold in these charts, with the fixed positive radial buffers described in Lemma 9. The same finite-matrix limsup argument as in Proposition 11 would then contradict the straight-boundary bound. This proves the existence of a finite window, a positive flatness threshold, and a finite mesh threshold. Compactness of the set of orientations makes these thresholds uniform in orientation. No rate for convergence of the graph or of a nearest-neighbor correlation is needed. ◻

Proof. Choose a maximal collection of mutually separated points of \(A\), with separation a fixed multiple of \(R\). Enlarged \(R\)-boxes about them cover \(U(A)+B_R\), so \(N_R(U(A))\) is bounded by a constant times their number. If there is only one such point this gives the constant term in (45). Otherwise, from each chosen point a path in \(A\) must leave a smaller concentric box. The \(r\)-neighborhood of the initial part of that path contains at least \(cRr\) sites. Choose these smaller boxes disjoint, including the \(r\)-thickening. Summing their contributions to \(|A+B_r|\) proves the other term. The argument uses only boxes of size comparable to \(R\) and applies unchanged in a torus chart. A coarse \(r\)-grid covering of \(A+B_{5R}\) costs at most \(C_L\) times a covering of \(A+B_r\), proving the second assertion. ◻

Proof. We prove the integral identity first, then bound the connected coefficients and sum them in the weighted norms. The same estimates will also control marked coefficients and the removed scalars.

Integral preservation. If an occurrence is an isolated vertex of a list’s overlap graph, its cutting square has a nearest-neighbor buffer from the supports of all other factors. Cutting squares of distinct isolated vertices likewise have such a buffer from one another. Conditional on their cutting contours and the remaining spins, the reference measure makes their interiors independent. Replacing their factors by conditional expectations therefore leaves the list’s integral unchanged. A long singleton is not altered. This proves exactness for every list, including lists with nonsingleton components; it does not require the components themselves to be independent in the unconditioned reference measure. Consequently the integral of \(\mathcal W_j(h)\) is the original partition integral. Exponentiating the scalar and nonscalar parts of \(\log\mathcal W_j(h)\) recovers \(\mathcal W_j(h)\) pointwise, giving the required partition identity.

Connected coefficients. Fix \(k\) labeled occurrences and their overlap graph \(G\). The transformed moment of any subset of these labels factors over the components of its induced graph. It follows either by the partition formula for a logarithm, or by the following polymer expansion, that the logarithmic coefficient vanishes unless \(G\) is connected. For connected \(G\) its sup norm is at most \[ 2^{k-1}\tau(G)\prod_{i=1}^k\|f_i\|_\infty, \tag{50}\] where \(\tau(G)\) is the number of spanning trees of \(G\).

The connected logarithm uses the hard-core Mayer expansion and the Penrose tree-graph bound [18]. We use the minimum-spanning-tree partition scheme [29]; the extra factor \(2^{k-1}\) will come from the internal/external edge markings below. To prove (50), use square-free variables for the \(k\) occurrences. Regard a connected subset of labels as a polymer. Its activity is the product of the original functions if its size is at least two, and the linearly transformed function if its size is one. Polymers are incompatible when their label sets overlap or an edge of \(G\) joins them. The hard-core polymer partition polynomial is exactly the transformed moment polynomial, because every induced graph has a unique decomposition into its components.

Its logarithm is a sum over partitions of the labels into connected blocks. For a fixed partition, the absolute Mayer coefficient is bounded by the spanning-tree sum of the quotient contact graph. For completeness, order that graph’s edges strictly and group its connected spanning subgraphs according to their greedy minimum spanning tree. For a given tree, the possible extra edges are optional independently: an edge may be added precisely when it is later than every edge on its tree path. Their alternating sum is zero or has absolute value one. Summing over trees proves the bound.

Choose a spanning tree within each connected block, and for every edge of a quotient tree choose an original contact edge. Allowing all these choices only increases the bound. Their union is a spanning tree of \(G\) whose edges are marked as internal to a block or between blocks. Conversely, the tree and its edge marks determine the blocks and these choices. There are at most \(2^{k-1}\) markings for each spanning tree of \(G\). Finally each polymer activity is bounded by the product of its input norms, since conditional averaging is a sup-norm contraction. This proves (50). Repeated input indices cause no change: their occurrences were labeled, and the original expansion still carries \(1/k!\).

Weighted summation. Subadditivity and 18 give \[ 2N_R(Q)\le 2\sum_iN_R(U(A_i)) \le Ck+\frac{C}{L}\sum_iN_r(A_i). \tag{51}\] Choose \(L\) so large that the last coefficient is, for example, less than one quarter. Thus a fixed fraction of each input exponential weight remains after paying the enhanced output weight.

Root a spanning tree at an input whose test neighborhood meets the output rooted block. Such a root carrier meets \(B_{CR}(x)\); covering that square by \(O(L^2)\) input rooted blocks costs \(C_L\). If a child touches its parent’s protective neighborhood, its carrier meets \(A_{\rm parent}+B_{5R}\). By 18, the sum of possible child terms costs at most \(C_L(1+N_r(A_{\rm parent}))\) times the child norm. For a vertex with \(b\) children, the unused exponential weight pays this factor to the power \(b\), using \[(1+t)^b e^{-ct}\le b!\,C_c^{\,b+1},\qquad t\ge0.\] Reserve fixed fractions of the weight for this inequality and for (51); they do not depend on \(k\).

The resulting tree bound is \(C_L^k\prod_i b_i!\) times the product of input norms. Summing rooted labeled trees and dividing by \(k!\) costs only another constant to the power \(k\): ordering the children identifies these objects with plane rooted trees, whose number on \(k\) vertices is the Catalan number \(\frac1k\binom{2k-2}{k-1}\le4^{k-1}\). Choices of root and the factor \(2^{k-1}\) are included in \(C_L^k\). Consequently the order-\(k\) contribution, including the doubled test weight, is bounded by \[ C_L^k\|h\|_r^k. \tag{52}\] The order-one rule is also bounded with this weight. For its short part the output test neighborhoods have bounded \(N_R\) and the root covering cost is \(C_L\); for its long part use (51) directly. Projection costs at most two. Summing (52) proves analyticity and (48), including the usual Lipschitz bounds on a smaller ball.

If a connected test carrier has diameter at least \(D\), its \(R\)-neighborhood has cardinality at least \(cDR\), hence \(N_R(Q)\ge cD/R\). The estimate with \(e^{2N_R(Q)}\), after retaining the ordinary output weight, leaves (49). In the torus the same argument applies to distances up to a fixed fraction of the side; a wrapping or non-chart connected carrier in particular crosses such a distance. For a singleton the same estimate is made on \(U(A)\) even when its actual carrier is smaller. Each input norm has remained a separate factor throughout the argument. A small factor in one of them therefore persists when the diameter restriction is imposed.

Marked coefficients and scalars. For marked coefficients, root a tree at an occurrence carrying a specified mark. Summation of this root is by (44), and the remaining argument is identical. For multiple labels, regard each input coefficient as carrying the product of its labels and retain only disjoint label sets. There are only finitely many allocations for any fixed total label set. The background occurrences are summed by the convergent series above; the marked occurrences give polynomials in the marked coefficients. Equivalently one can differentiate on a small source polydisc whose radius depends on their norm bound. This proves all the marked analytic and Lipschitz statements. In particular, terms in a one-mark derivative other than its zero-background linear term contain a background factor, which proves the second bound in (48) in marked norm. The removed scalar of a function has absolute value at most its sup norm. Before discarding its carrier, apply exactly the same estimate; this proves the asserted bounds for scalar increments and their background differences. No parity assumption was needed for these marked estimates.

For finitely many input terms the partition identity follows first as an identity of power series near zero. The convergent logarithmic series then defines its exponential throughout the common small ball, and analytic continuation gives the identity there. In a finite system the total absolute norm is finite, bounded by finitely many rooted norms: every nonconstant term meets a variable site. Truncating the list and using the proved convergence gives the identity for all interactions in the ball. This argument does not expand a thermodynamic partition function under the interacting measure.

For the preliminary finite-range transformation, take protective neighborhoods at the fixed radius \(r_0\) and make \(r_0\) larger than every microscopic support. The same exactness and tree proof apply. All covering costs are finite constants depending on this radius and the range; the enhanced output weight costs at most a fixed factor per input occurrence. A prescribed larger source carrier is left unchanged in the singleton rule, with its own protective neighborhood retained. The marked rooted estimates then have finite additional constants depending on that carrier. This proves the last assertion. ◻

Proof. 15 gives constants \(\kappa_d\ne0\) such that \[ R^d d_R(\mathcal O_d(z))\longrightarrow\kappa_d. \tag{57}\] Whenever the conditioning square is inside the disk, the Markov property and the tower property give \[ d_R(P_{s,z}\mathcal O_d(z))=d_R(\mathcal O_d(z)). \tag{58}\] This proves (56) for all sufficiently large radii, and gives the lower bound in (55) by taking a disk of radius comparable to \(r_j\). The difference of two expectations of a canonical representative is bounded by twice its sup norm.

For the upper bound, choose a fixed large ratio \(B\), independently of the later blocking ratio \(L\). At all sufficiently large \(s\), apply 11 to \(F=P_{s,z}\mathcal O_d(z)-m(\mathcal O_d)\) and to the outer radius \(Bs\). The relevant remainder exponent is \(p=3\) for the square-symmetric even field and \(p=1+1/8\) for the odd field. Equations (57)–(58) yield \[(Bs)^d\|P_{Bs,z}\mathcal O_d(z)-m(\mathcal O_d)\|_\infty \le C+C B^{d-p} s^d\|P_{s,z}\mathcal O_d(z)-m(\mathcal O_d)\|_\infty.\] Take \(B\) large enough that the last coefficient is below one. Iterating from radii in a fixed compact interval above the threshold proves a uniform bound for every large radius. Integer radii are handled by ratios tending to \(B\), which are allowed in 11. The initial compact interval costs only a finite constant. Passing from plane centering to \(\pi\) costs at most two. This bound was obtained with \(B\) fixed, so is independent of the later choice of \(L\). Finally \(N_{r_j}(B_{r_j}(z))\) is bounded above and below by absolute constants, giving the marked-norm assertion. ◻

Proof. We first prove the quotient estimate in the homogeneous plane space. Sum all short input functions at one anchor \(z\) into \(F_z\). It is square-symmetric and even, is supported in \(B_s(z)\) with \(s=CMr\), and obeys \[ \|F_z\|_\infty\le Cr^{-2}\|h\|_r. \tag{63}\] Indeed, every carrier contains its anchor. For any \(x\), all terms anchored at \(z\in B_r(x)\) contribute to the rooted sum in (43). Translation invariance therefore makes this sum at least \(|B_r|\) times the total weighted sum at a single anchor. This proves (63) without a further factor of \(M\).

Subtract \(\alpha r^{-2}v_j^{(1)}(z)\) from \(F_z\), choosing \(\alpha\) to make its disk response at radius \(R\) vanish. Equations (56) and 11 show \[|\alpha|\le CM\|h\|_r.\] The adjusted function remains supported in \(B_s(z)\) and has norm at most \(C(1+M)r^{-2}\|h\|_r\). All its short terms are mapped to the same carrier \(B_R(z)\) and are added there. Applying 11 with \((d,p)=(1,3)\) to this sum, and then projecting off constants, bounds the remaining function by \[C(CM/L)^3(1+M)r^{-2}\|h\|_r.\] There are \(O(R^2)\) anchors whose output squares meet an \(R\)-rooted block, and their output weights are bounded constants. The quotient norm of the short output is consequently at most \[ C L^2(1+M)(CM/L)^3\|h\|_r=o_L(1)\|h\|_r. \tag{64}\] Add the long-term bound (53). Since the distinguished line is exactly invariant, the same estimate applies after subtracting any multiple of the input line. Optimizing gives an operator bound \(o_L(1)\) on the quotient spaces, rather than merely a bound on a particular representative.

The norm bounds for \(\boldsymbol v_j\) give uniformly bounded projections onto these lines: choose a norm-bounded dual functional equal to one on \(\boldsymbol v_j\). Choose the functionals first on real lists and extend them complex-linearly. Their norms remain uniformly bounded, and the resulting projections commute with complex conjugation. The corresponding complement projections have uniform norm. In these splittings the lower-left entry vanishes by (60); the quotient estimate bounds the lower-right entry, and the bounded full linear map bounds the upper-right entry. Taking \(L\) larger if necessary proves (61).

For a singleton-source list sum its short terms at the marked anchor. The sum has norm at most \(\|u\|_{r,\bullet}\) and support in \(B_{CMr}\). The same response cancellation subtracts at most \(CM^d\|u\|_{r,\bullet}\) times \(v_j^{(d)}\). There is now no volume factor. After multiplying by \(L^d\), the quotient bound is \[ C L^d(1+M)^d(CM/L)^p\|u\|_{r,\bullet}, \qquad (d,p)=(1,2)\ \text{or}\ (1/8,1+1/8). \tag{65}\] Both expressions tend to zero with \(L\). The scaled long-term bound also tends to zero. The exact line multiplier is one by (60), proving (62) as before. Averaging a projection over the finite rotation group preserves its value on the invariant even line and keeps its norm bounded.

Finally let the input be a boundary family. A short carrier touching the edge has its anchor within \(CMr\) of it. 16, with support radius \(s=CMr\), gives for each such term \[ \|\pi P_{R,z}f_{A,z}\|_\infty \le C(CM/L)^2\|f_{A,z}\|_\infty. \tag{66}\] The use of an anchor just off the edge is allowed by that proposition; one can also recenter at a boundary point at distance at most \(s\), changing only fixed support and separation factors.

For a fixed output rooted square, the relevant anchors lie in a band of width \(CMr\) along an edge segment of length \(O(R)\). In a one-sheet graph chart this band is covered by at most \(CL(1+M)^C\) input \(r\)-blocks. Each block bounds the sum of the input norms anchored there, since an anchor belongs to its carrier. Summing (66) therefore costs at most \[CL(1+M)^C(CM/L)^2\|h\|_r=o_L(1)\|h\|_r.\] The long terms are bounded by (53). An output square of a short boundary term still touches the boundary, because the anchor is within \(CMr<R\) of it; a long unchanged carrier retains its contact. This proves the boundary assertion. ◻

Proof. We give the forward-stable, backward-thermal solution operator, since the spaces and linear maps depend on the scale. Sequence-space constructions for scale-dependent renormalization dynamics are also developed in [2]; the contraction needed here is proved explicitly below. For an imposed forcing \(f_j=(f_j^1,f_j^2)\), first solve the stable equation forward: \[w_j=D_{j-1}\cdots D_0w_0+ \sum_{i=0}^{j-1}D_{j-1}\cdots D_{i+1}f_i^2,\] where an empty product is the identity. Then solve the thermal equation backward: \[ t_j=-\sum_{i=j}^{\infty}L^{j-1-i}(B_iw_i+f_i^1). \tag{70}\] On the sequence space with norm \[\left\|(t,w)\right\|_{\gamma_0} =\sup_{j\ge0}\gamma_0^{-j}(|t_j|+\left\|w_j\right\|_{r_j}),\] these operators are bounded. Indeed the stable forcing sum costs at most \((\gamma_0-\rho)^{-1}\left\|f^2\right\|_{\gamma_0}\), and the backward sum costs at most \((L-\gamma_0)^{-1}(C_L\left\|w\right\|_{\gamma_0} +\left\|f^1\right\|_{\gamma_0})\). Any solution in this space must obey (70): iterate its thermal recurrence to time \(k\) and let \(k\to\infty\), using \(L^{j-k}t_k\to0\).

Substitute \(f_j=n_j(t_j,w_j)\) in this bounded linear solution operator. The uniform quadratic bounds imply \[\left\|n(t,w)\right\|_{\gamma_0} \le C_L\left\|(t,w)\right\|_{\gamma_0}^2,\] and its Lipschitz constant on a sequence ball of radius \(s\) is at most \(C_Ls\). A sufficiently small ball, with radius a fixed multiple of \(\left\|w_0\right\|_{r_0}\), is therefore invariant and contractive. The contraction theorem proves existence, uniqueness, and (69) in this weighted sequence ball. To obtain the stated uniqueness, any two trajectories in the ordinary \(\epsilon\)-ball with finite weighted norm obey the same linear solution formulas. Their difference is bounded in weighted norm by \(C\epsilon\) times itself, by the local Lipschitz bound on the remainders and the bounded solution operator. Choose \(\epsilon\) so that \(C\epsilon<1\); the constructed trajectory lies in this ball when \(w_0\) is sufficiently small.

The complex analytic contraction, or equivalently the analytic implicit function theorem on this sequence space, gives analytic dependence on \(w_0\). Evaluation at time zero gives \(\psi\). The tangent space of a graph over \(W_0\) has trivial intersection with the pure thermal line, proving transversality. ◻

Proof. Relative to the nearest-neighbor reference at \(K_c\), the microscopic background in the exponent is \[ (\beta J-K_c)\sum_{\{x,y\}:x\sim y}\sigma_x\sigma_y +\beta\lambda\sum_A V(A)\prod_{x\in A}\sigma_x. \tag{72}\] For the nearest-neighbor temperature displacement, assign half of each unordered bond coefficient to each endpoint, and use its two-site carrier. Thus its anchored function at \(z\) is exactly \(2(\beta J-K_c)e(z)\). Assign finite connected carriers and anchors to the remaining interaction shapes, distributing coefficients among equivalent anchors when needed. This preserves all lattice symmetries and spin-flip parity. These are lists of coefficients; no infinite-volume partition function is being expanded.

Apply one preliminary version of 19, with protective neighborhoods at radius \(r_0\) and every microscopic singleton averaged on its cutting square \(B_{r_0}\). The finite range makes the same connected-list estimates convergent on a fixed neighborhood of \((\beta_0,0)\); their constants may depend on \(r_0,V\). This gives an analytic initial list \(h_0(\beta,\lambda)\in\mathcal H_0\), equal to zero at \((\beta_0,0)\). The preliminary map is local, has the same reserved exponential weight as the subsequent maps, and preserves the finite-volume integral, with its scalar prefactor retained.

The normalization of its thermal derivative is explicit. Because each unoriented bond occurs twice in the sum of incident bonds, \[\sum_z e(z)=\frac12\sum_{\{x,y\}:x\sim y} \sigma_x\sigma_y.\] Thus the preliminary derivative with respect to \(\beta\), at \((\beta_0,0)\), has the term \(2J P_{r_0,z}e(z)\) at every anchor \(z\). By the definition of the homogeneous thermal list in 20, \[ \partial_\beta h_0(\beta_0,0) =2Jr_0\boldsymbol v_0. \tag{73}\] This equality is in the quotient by constants and therefore also holds for the canonical representatives.

Write \(h_0(\beta,\lambda)=t_0(\beta,\lambda)\boldsymbol v_0 +w_0(\beta,\lambda)\). The scalar analytic equation \[t_0(\beta,\lambda)-\psi(w_0(\beta,\lambda))=0\] has \(\beta\)-derivative \(2Jr_0\ne0\) at \((\beta_0,0)\), since \(\partial_\beta w_0(\beta_0,0)=0\). The implicit function theorem produces \(\beta_c(\lambda)\). All maps commute with complex conjugation, so the branch and trajectory are real for real \(\lambda\). Shrinking the interval makes \(\beta_c\) positive. Along the branch, \(\left\|w_0\right\|_{r_0}=O(|\lambda|)\), so 22 gives (71). The argument applies to both signs of \(\lambda\) and uses no sign assumption on \(V\). ◻

Proof. A compact \(C^2\) Jordan boundary has uniform one-sheet graph charts whose slope variation is arbitrarily small on sufficiently short physical windows. Choose \(\eta_D\) so that every enlarged local window required by 21 is such a chart up to the stated terminal scale. The normalized chart constants are fixed universally; only the permitted physical radius depends on \(D\). For small \(a\) the preliminary scale also lies in these windows.

Before preliminary blocking, the finite-domain prescription differs from the restriction of (72) only within the fixed interaction range of the edge. A crossing interaction with fixed exterior spins remains a function of its available spins, with an ambient carrier containing the original interaction set. Deleting crossing interactions gives the corresponding free prescription. Enlarging carriers by a fixed microscopic amount accommodates any symmetry-preserving anchor decomposition. The rooted norm of these discrepancies is \(O(|\lambda|)\), because \(\beta_c(\lambda)-\beta_0=O(|\lambda|)\). The local preliminary map therefore gives \[ \left\|b_0^{a,b}\right\|_{r_0}\le C_0|\lambda|. \tag{75}\] The bound uses local density of interaction shapes and the fixed preliminary scale, not the total length of the boundary.

We verify the locality needed at subsequent steps. For an intact short-singleton output, its cutting square and all nearest-neighbor edges required for conditional averaging are present in \(\Lambda_a\). The conditional kernel is exactly the plane kernel at that square. For an intact connected output of order at least two, its carrier contains all the protective neighborhoods of its inputs, and hence every reference neighborhood used in its coefficient. All these inputs and kernels also agree with their plane counterparts. A long unchanged singleton uses no outside kernel. Thus coefficients agree on intact output carriers. Any omission or discrepancy in comparing the two maps has a carrier touching the edge; the same is true of an output involving a boundary-family input.

The canonical representative subtracts the independent uniform-spin mean on the available variables. On an intact carrier that set of variables is identical in the plane and the domain, so this subtraction also agrees exactly. At the edge the discarded scalar keeps its generation carrier for any source bookkeeping. Subtracting constants therefore neither produces an interior discrepancy nor transports an edge discrepancy into an interior carrier.

Let \(\mathcal R_j^{a,b}\) denote the finite-domain map. The preceding locality and the absolute coefficient estimates of 19 give the bulk leakage bound \[ \left\|\mathcal R_j^{a,b}(\operatorname{Res}_{a,j}h_j) -\operatorname{Res}_{a,j+1}\mathcal R_j(h_j)\right\|_{r_{j+1}} \le C_L\left\|h_j\right\|_{r_j}. \tag{76}\] No boundary contraction is asserted for this difference. In contrast, the linear image of the boundary family itself has norm at most \(\rho\left\|b_j^{a,b}\right\|_{r_j}\) by 21. The nonlinear terms involving that family cost at most \(C_L(\left\|h_j\right\|_{r_j}+\left\|b_j^{a,b}\right\|_{r_j}) \left\|b_j^{a,b}\right\|_{r_j}\). Consequently, writing \(q_j=\left\|b_j^{a,b}\right\|_{r_j}\), we have \[ q_{j+1}\le\rho q_j+C_L\left\|h_j\right\|_{r_j} +C_L(\left\|h_j\right\|_{r_j}+q_j)q_j. \tag{77}\] Evenness in the free case is preserved by every operation. In the fixed cases the unrestricted boundary estimate in 21 applies instead.

Choose \(\alpha\) with \(\rho<\alpha<\gamma\). On a sufficiently small common ball the last term in (77) is at most \((\alpha-\rho)q_j\). Choose \(B\ge C_0\) large enough that \(C_LC\le B(\gamma-\alpha)\), where \(C\) is the constant in (71). Since \(\gamma_0^j\le\gamma^j\), induction then gives \(q_j\le B|\lambda|\gamma^j\). Decrease \(\lambda_0\) so this bound and (71) stay in the analytic balls and in the preceding smallness condition. Every constant in this choice is independent of \(D\); the domain affects only \(\eta_D\) and the threshold for the mesh. This closes the induction and proves (74). ◻

Proof. Choose \(\vartheta\) sufficiently small, depending on the already fixed map parameters, that every retained carrier with its norm neighborhood and every short-term cutting window under consideration lies in an isometric torus chart. The periodic microscopic interactions are their usual images; only \(N\) larger than the interaction range and the preliminary scale is relevant here.

Compare one periodic map applied to \(\Pi_Nh_j\) with \(\Pi_N\mathcal R_j(h_j)\). A connected list for which the entire union of protective and test neighborhoods lies in a sufficiently small max-ball lifts uniquely, up to an overall period. Its input coefficients and contacts agree with those in the plane. The local conditional kernels on all singleton cutting squares agree as well, and canonical uniform-spin subtraction uses identical variables. The resulting coefficients therefore agree whenever neither an input nor the output is removed by the diameter truncation. This argument uses equality of conditional kernels in a chart, not equality of plane and torus marginal laws.

Every exception has extent at least a fixed multiple of \(N\): it contains an omitted input or output carrier, or its connected union of test neighborhoods leaves every such small chart, including by winding around the torus. In the plane or torus where that list is being estimated, the reserved exponential weight in 19 therefore supplies \(\exp(-cN/r_{j+1})\). Rooting near a norm block costs only the usual local covering constant. A long unchanged singleton is compared directly with its truncated counterpart and has the same tail bound. Summing all exceptional lists, using the small analytic ball to sum higher powers of the background, gives \[ \left\|\mathcal R_j^N(\Pi_Nh_j) -\Pi_N\mathcal R_j(h_j)\right\|_{r_{j+1}} \le C_L\left\|h_j\right\|_{r_j}\exp(-cN/r_{j+1}). \tag{81}\] The retained small carriers and their neighborhoods have comparable norm weights in the two spaces by the chart choice. The preliminary finite-range map satisfies the same comparison, giving \[ \left\|H_0^N-\Pi_Nh_0\right\|_{r_0} \le C_0|\lambda|\exp(-cN/r_0). \tag{82}\]

Here is the summation that preserves a spatial exponential after propagation. Set \(E_j=\left\|H_j^N-\Pi_Nh_j\right\|_{r_j}\) and enlarge the uniform local Lipschitz constant to \(K\ge1\). As long as the compared backgrounds remain in the analytic balls, (81) and (71) imply \[E_{j+1}\le K E_j+C_1|\lambda|\gamma^j \exp(-cN/r_{j+1}).\] For a terminal index \(j\), put \(x=N/r_j\ge\vartheta^{-1}\) and \(q=K/\gamma\). Iteration, including (82), gives \[ \frac{E_j}{|\lambda|\gamma^j} \le C_0q^j e^{-cxL^j} +\frac{C_1}{\gamma}\sum_{m=0}^{j-1}q^m e^{-cxL^m}. \tag{83}\] At \(\lambda=0\) the discrepancy is identically zero, so the displayed division is needed only when \(\lambda\ne0\). For \(0<c'<c\), \[\sum_{m\ge0}q^m e^{-cxL^m} \le e^{-c'x} \sum_{m\ge0}q^m \exp\bigl(-(cL^m-c')/\vartheta\bigr) \le C_{L,K,\gamma,\vartheta}e^{-c'x}.\] The series converges because its exponential decay in \(L^m\) dominates \(q^m\). The initial term in (83) is its \(m=j\) term and has the same bound. This proves (80), after renaming \(c'\). Together with (78) it gives (79). Decreasing the common \(\lambda_0\) makes these bounds stay strictly inside all analytic balls, and closes the finite-scale induction uniformly in \(N\).

For fixed \(j\), (80) tends to zero as \(N\to\infty\). The omitted plane carriers also have vanishing weighted tails at that fixed scale. Identifying the remaining local carriers with their unique lifts proves the claimed coefficient convergence. Finally the preliminary and subsequent maps preserve real coefficients and the exact finite-volume identity of 19. Exponentiating the real residual background gives a strictly positive weight, completing the last assertion. ◻

Proof. Use the bounded splittings from 21, with the distinguished vectors from 20. For a normalized one-source coefficient the zero-background matrix is \[A_j= \begin{pmatrix}1&B_j\\0&D_j\end{pmatrix}, \qquad \left\|B_j\right\|\leq C_L, \qquad \left\|D_j\right\|\leq\rho<1.\] The derivative on the critical plane trajectory differs from \(A_j\) by an operator \(E_j\) with \[\left\|E_j\right\|\leq C_L\left\|h_j\right\|_{r_j} \leq C|\lambda|\gamma^j.\] Products of consecutive \(A_j\) are uniformly bounded: their lower-right blocks contract geometrically and their upper-right blocks are bounded by \(C_L\sum_{s\geq0}\rho^s\). Expanding a perturbed product by the locations of its factors \(E_j\) therefore bounds it by a constant times \(\prod_j(1+C\left\|E_j\right\|)\). This product is finite, uniformly for the coupling in a sufficiently small complex disc. Thus the one-source trajectory is bounded.

Write its coordinates as \(\alpha_j,w_j\). The stable recurrence gives \[\left\|w_j\right\| \leq \rho^j\left\|w_0\right\| +C|\lambda|\sum_{i<j}\rho^{j-1-i}\gamma^i.\] This tends to zero and is summable in \(j\); it is bounded by \(C\theta^j\) for any fixed \(\max(\rho,\gamma)<\theta<1\), after enlarging \(C\). Moreover \[|\alpha_{j+1}-\alpha_j| \leq C_L\left\|w_j\right\|+C|\lambda|\gamma^j.\] Hence \(\alpha_j\) converges, and subtracting its limit times \(v_j^{(d_\ell)}\) proves (87). The preliminary source map and every later derivative are analytic in the coupling. The bounds just given are uniform on a smaller complex disc, so the limiting scalar is analytic there. Reality of all maps on real inputs gives real values on the real axis. Spin-flip symmetry makes every plane spin singleton scalar contribution identically zero.

Translation covariance makes the limit independent of the marked site. Choose the energy splitting equivariantly under the square symmetries. Rotation about the mark carries the four incident bond sources into one another and fixes the scalar energy line, so their limit scalars agree. At zero coupling the spin source is exactly on its distinguished line after the preliminary step. The average of the four bond sources is \(e(z)\), so this average likewise starts exactly on the energy line. The normalized line multiplier is one. Linearity of a one-source derivative and equality of the four scalars give both asserted intercepts. Continuity now gives simultaneous nonvanishing after one decrease of the coupling interval. All choices were made on the plane background and for the two field types, without reference to a domain or a collection of other marks. ◻

Proof. Let \(\Pi_F\) denote restriction to intact carriers in a domain, or the small-diameter periodic projection used in 25. At each scale use the trial coefficients \[W_j^{\{\ell\}}=\Pi_F u_j^\ell, \qquad W_j^S=0\quad(|S|\geq2), \qquad \widetilde c_j^S=0\] for the scalar labels retained in (91). Let \(\mathcal I\) be this collection of scalar label sets: all sets with at least two labels, and every spin singleton. On source states use the finite product norm \[\|(U,c)\|_{j,\mathrm{src}} :=\sum_{\varnothing\ne S}\|U^S\|_{r_j,\bullet} +\sum_{S\in\mathcal I}|c^S|.\] Write \(\mathsf U_j=(U_j,c_j)\) for the exact state and \(\mathsf W_j=(W_j,\widetilde c_j)\) for the trial state. Let \(\Phi_j^F\) be the source step (85) with the actual finite-system background \(H_j^F\) held fixed. Thus no background error is included among these source variables. Define the error and the one-step trial defect by \[\mathfrak e_j=\mathsf U_j-\mathsf W_j, \qquad \mathfrak d_j=\Phi_j^F(\mathsf W_j)-\mathsf W_{j+1}.\] Energy singleton scalars are omitted from these states, consistently with (86).

The defect is bounded by \[ \|\mathfrak d_j\|_{j+1,\mathrm{src}} \leq\varepsilon_j\exp(-c\Delta_*/r_{j+1}), \qquad \sup_j\varepsilon_j<\infty, \qquad \varepsilon_j\longrightarrow0. \tag{92}\] We verify both factors; a fixed small stopping fraction alone would not make the exponential in (92) tend to zero.

First, the list/log map has connected generating carriers. A coefficient joining two distinct marks must span their separation. A discrepancy between an interior singleton calculation and its plane calculation must reach the physical edge; a term containing a boundary-family input likewise connects its mark to the edge. In the torus comparison a projection exception must have a nonlocal or winding generating neighborhood. These are precisely diameter conditions covered by the doubled-weight and marked estimates in 19. They give the exponential factor in (92), including before scalar projection. They also cover long unchanged singleton carriers.

For the vanishing factor, terms containing a background input have the extra bound \(C|\lambda|\gamma^j\), by [flow:branch,flow:boundary]. The corresponding torus bounds follow from 25; in particular its background discrepancy has the stronger norm bound \[C|\lambda|\gamma^j\exp(-c'N/r_j),\] which supplies a spatial gain even when that discrepancy is used as one input without reestablishing its generating history.

For terms with no background input, decompose every trial singleton by 26 into its ideal term and remainder. A term containing at least one remainder has a factor tending to zero in marked norm. This factor is retained simultaneously with the spatial exponential: the latter is paid from the reserved carrier weights, whereas the former is one of the multilinear input norms. A term containing only ideal singletons cannot be exceptional or join distinct marks. Each such singleton and its protective neighborhood has radius \(O(r_{j+1})\) about its own mark. The stopping fraction makes these neighborhoods pairwise disjoint and interior, so the logarithm of their factorizing list weight has no multiple-label coefficient. Their singleton transfers are exactly the plane ones.

There is no missing spin scalar in this comparison. In the plane it is zero by parity. On intact interior carriers the local graph and the available-spin representative projection are the same in the finite system and on the plane. An interior ideal spin therefore produces no scalar discrepancy. Every remaining spin singleton discrepancy reaches the edge or involves the torus discrepancy or a remainder, and is covered by the preceding estimates. Multiple-label scalar coefficients have the same connected generating carriers as their nonscalar counterparts, so the same proof applies to them. Energy singleton scalars are deliberately absent from the error state by (86). For example, after decreasing \(c\) one may take a bounded multiple of \(|\lambda|\gamma^j+\sum_\ell\left\|\eta_j^\ell\right\|_{r_j,\bullet}\) as \(\varepsilon_j\). This proves (92).

The preliminary defect obeys \(C\exp(-c\Delta_*/r_0)\), with a fixed prefactor, by the preliminary local-list estimates. The microscopic sources are interior and separated. On a unit neighborhood of the bounded trial jets, the source maps have a scale-independent Lipschitz constant \(K\geq1\). It includes all the finite multipliers \(L^{d(S)}\) for the retained scalars. Consequently \(\|\mathfrak e_l\|_{l,\mathrm{src}}\) at any \(l\leq k\), as long as the state stays in this neighborhood, is bounded by \[ C K^l e^{-c\Delta_*/r_0} +C\sum_{i<l}K^{l-i-1}\varepsilon_i e^{-c\Delta_*/r_{i+1}}. \tag{93}\] Put \(q_* =\inf \Delta_*/r_k>0\). At \(l=k\), reindexing the sum by \(h=k-i-1\) bounds it by \[C\sum_{h=0}^{k-1}K^h\varepsilon_{k-h-1}e^{-c q_*L^h}.\] The majorant \(C\sup_i\varepsilon_i\,K^h e^{-c q_*L^h}\) is summable. For each fixed \(h\) the corresponding \(\varepsilon_{k-h-1}\) tends to zero, so dominated convergence gives an \(o(1)\) bound. The same bound is uniform over \(l\leq k\): for \(l\) large use the same tail majorant and the smallness of the late \(\varepsilon_i\), while the finitely many early \(l\) have \(\Delta_*/r_l\to\infty\). The preliminary term also tends to zero uniformly, since geometric growth of \(\Delta_*/r_0\) dominates \(K^k\). Thus the estimates bootstrap inside the unit neighborhood. Together with (87), they prove all three conclusions. No rate of convergence for that remainder was needed. ◻

Proof. We first establish an unscaled marked contraction. Split a short source function into its spin-flip even and odd parts. Combining the response and remainder bounds in 11 bounds the norm of its conditional average modulo constants by \(C(CM/L)^{1/8}\) times its sup norm; the odd sector gives the slower decay. Here there is neither a volume multiplier nor the multiplier \(L^d\) used for a scaling field. The doubled output weight on its fixed-size carrier costs only a constant. Long source carriers have the exponentially small tail from 19. The total marked norm sums the list directly, so these estimates give \[ \left\|D\mathcal R_j(0)u\right\|_{r_{j+1},\bullet} \leq q_0\left\|u\right\|_{r_j,\bullet}, \qquad q_0<1, \tag{96}\] after including this condition when choosing the large block ratio. Canonical subtraction of constants costs at most a factor two, already included here. No parity restriction remains on \(u\). The estimate holds on tori through the chart scales in 25, since short cutting squares have the same local critical continuation and long carriers are handled by their norm tail.

By 19, the derivative at a residual background differs by at most \(C_L\left\|H_i^N\right\|_{r_i}\) from this derivative. The torus bound in 25 and one decrease of \(|\lambda|\) therefore give a uniform contraction factor \(q_1<1\) at every permitted step. Denote the scalar-increment functional for this unscaled derivative by \(A_i(H_i^N)\). The same proposition gives \[ \left\|A_i(H_i^N)\right\|\leq C_L, \qquad \left\|A_i(H_i^N)-A_i(0)\right\| \leq C_L\left\|H_i^N\right\|_{r_i}. \tag{97}\]

Suppose first that a bounded test \(F\) is inserted at scale \(j\) with support in a square of radius \(D r_j\). Subtract its uniform-spin mean \(c\) and use that square as its marked carrier. The initial marked norm is at most a constant \(C_D\). This carrier may initially be long; the long-carrier part of (96) applies to it. Thus \(D\) changes the initial bound, not the contraction factor or the coupling interval. Propagate this unscaled derivative through any terminal index \(s\) with \(r_s\) below the torus chart threshold. Let \(w_i\) be its nonscalar list at scale \(i\). The exact first-derivative version of the partition identity reads \[ {\mathbb E}_{\nu_j^N}F =c+\sum_{i=j}^{s-1} A_i(H_i^N)w_i +{\mathbb E}_{\nu_s^N}w_s, \qquad \left\|w_i\right\|_{r_i,\bullet}\leq C_D q_1^{i-j}. \tag{98}\] The final expectation is bounded by its marked norm, using positivity.

Compare this calculation with the same one on the zero-background trajectory, starting from the same \(F\), and denote its lists by \(w_i^0\). The operator difference is bounded by \(C|\lambda|\gamma^i\). Using a common contraction factor \(q_1\) in both trajectories gives \[\left\|w_i-w_i^0\right\|_{r_i,\bullet} \leq C_D|\lambda|\gamma^j(i-j)q_1^{i-j-1} \quad(i>j).\] Equations (97) and (98) can now be summed geometrically. The difference of their scalar sums is at most \(C_D|\lambda|\gamma^j\); the two terminal expectations contribute at most \(2C_Dq_1^{s-j}\). Thus \[ \left|{\mathbb E}_{\nu_j^N}F-{\mathbb E}_{0,N}F\right| \leq C_D\bigl(|\lambda|\gamma^j+q_1^{s-j}\bigr). \tag{99}\] Taking the last permitted \(s\) makes \(s-j\to\infty\) whenever \(N/r_j\to\infty\). This proves the second assertion, uniformly over the stated tests.

For the first assertion, fix a bounded observable \(F\) of finitely many lattice sites, and choose a fixed connected carrier containing them. Insert it at the microscopic scale. If this carrier does not fit a preliminary cutting square, use the unchanged long-singleton rule, with its protective neighborhood. Its fixed size only changes the finite constants in the preliminary marked bound. Thereafter the unscaled contraction and scalar estimate give a summable tail \(C_F\sum_{i\geq m}q_1^i\), uniformly in every sufficiently large torus on which these scales are available.

For any fixed finite set of scales, the background coefficients converge to their plane counterparts by 25. The same is true of the local derivative and each scalar increment. Indeed, lists contained in a common chart agree with the plane list rule; nonchart lists or omitted long inputs have a diameter comparable to \(N\) and vanish by the marked diameter estimates. The weighted tails permit the passage from finite lists to the complete coefficient. Thus the first \(m\) scalar increments have a limit as \(N\to\infty\). The sum of subsequent increments and the final nonscalar expectation is uniformly negligible as \(m\to\infty\), by the contraction just proved. Applying the telescope from the preliminary scale proves that \({\mathbb E}_{\lambda,N}F\) converges.

Taking \(F\) to be cylinder indicators gives consistent limiting finite-dimensional probabilities, hence the asserted thermodynamic state. This construction does not assume uniqueness of other Gibbs states or a choice of a symmetry-broken phase. Its coupling restriction comes from a uniform one-mark operator bound and is independent of \(F\) and of the number of sites on which \(F\) depends. ◻

Proof. Fix separated physical insertion points and a small positive stopping radius below all their pairwise separations. Choose \(k=k(a)\) so that \(a r_k\) stays between two positive constants below that radius. At zero background, singleton transport is exactly nested conditional averaging. Its nonscalar list is therefore \(u_k^\ell=r_k^{d_\ell}\pi P_{r_k,z_\ell}X_\ell\), also for an individual directed bond. By [src:scalar,rg:fields] these lists have bounded marked norm. Hence there is a constant \(c_{a,\ell}\) such that \[f_{a,\ell}=r_k^{d_\ell}P_{r_k,z_\ell}X_\ell-c_{a,\ell}\] has uniformly bounded sup norm. The constants vanish for spins and are immaterial for centered energies. The cutting squares are disjoint and their union lies in one fixed physical square \(B_B\). These functions are identical in the plane and in every sufficiently large disk \(D_T\) containing \(B_B\): each conditional kernel is determined inside its own cutting square.

The conditional Markov identity for every subproduct identifies the original scaled centered correlation with \((a r_k)^{-d(\{1,\ldots,q\})}\) times the corresponding centered product of the \(f_{a,\ell}\), in either system. Let \(p_a\) and \(p_{a,T}\) denote these original nearest-neighbor correlations in the plane and in \(D_T\) with plus boundary. Conditioning on a common surrounding square and applying 2 with repeated critical RSW in the annuli between \(B_B\) and the distant wall gives a total-variation error \(\omega(T)\to0\) on their common support, uniformly for sufficiently small \(a\). The uniform sup norms control all subset moments. Actual-state centering is a finite polynomial in those moments, so \[\limsup_{a\downarrow0}|p_a-p_{a,T}|\le C\omega(T).\] For every fixed \(T\), bounded-domain convergence in 6 gives \(p_{a,T}\to\mathcal C_{D_T}^{+}\). Consequently \[\limsup_{a,b\downarrow0}|p_a-p_b|\le2C\omega(T).\] Sending \(T\) to infinity proves the full mesh limit and identifies it with \(\lim_{T\to\infty}\mathcal C_{D_T}^{+}\). The same comparison proves independence of the bond directions. The order is a fixed positive stopping radius, then the mesh limit at each fixed \(T\), and finally \(T\to\infty\). Only bounded-domain mixed convergence and the preceding uniform conditional-field bounds have been used. ◻

Proof of 1. Fix the allowed potential once and for all. 23 gives the critical inverse-temperature branch with the specified intercept, positive on a small real coupling interval. The reference perturbation has real coefficients for either sign of the coupling; none of the background estimates requires a sign condition on the finite-range potential. 26 gives the common nonzero normalizations with their unit intercepts. Choose the interval once so that the background, boundary, torus, nonvanishing, and unscaled-contraction bounds used above all hold. These choices do not involve a domain, insertion count, or insertion configuration. Constants in a fixed finite source jet may depend on those data, as the theorem permits.

For any fixed allowed bounded domain, boundary type, and separated configuration, 27 and the bounded-domain evaluation prove the requested full-sequence limit. The microscopic energy sources may have either specified direction. Their plane scalars coincide, and the critical symmetric-average comparison has exactly the common nearest-neighbor energy limit. Equation (86) centers every source in its actual finite-domain state. For a fixed negative boundary, the same argument uses its physical prescription throughout; no boundary-dependent normalization has been introduced.

For the full-plane assertion, the first part of 28 establishes the required periodic-box limit at each fixed mesh. Consider an arbitrary sequence \(a\to0\) and choose its stopping scales by (88). On tori with \(N/r_k\to\infty\), 27 still reduces the final source jet to the bounded ideal singleton fields, with vanishing remaining scalar and multiple-source coefficients. Products of any subset of these ideal fields have uniformly bounded sup norms and support in a square of radius \(D r_k\), for a fixed \(D\) depending on the configuration. The second part of 28 therefore replaces their residual tilted expectations by critical torus expectations. This step does not assert smallness of a total background on an expanding torus.

The latter local critical expectations can in turn be replaced by plane expectations as \(N/r_k\to\infty\). Surround their support by a planar chart. The pointwise ratio estimate and repeated critical RSW in the intervening annuli make the effect of arbitrary spins at the distant chart wall tend to zero, uniformly for the bounded tests. Both the torus and plane laws are mixtures of those continuations. The cutting squares defining the ideal fields are identical local critical squares in both systems. Their simultaneous conditional expectation identity now recovers the original microscopic critical spins and symmetric energies in the plane. 29 finishes the same comparison there. The separately cited plane spin formula identifies its nonzero two-spin amplitude.

To make the order of limits explicit, at each fixed mesh all unscaled moments appearing in the centered product converge as \(N\to\infty\), by 28. The centered moment is a finite polynomial in these moments. Choose \(N\) large enough that its error after the fixed mesh normalization is smaller than any prescribed sequence tending to zero, and also so that \(N/r_k\to\infty\). The joint comparison just proved applies to these chosen boxes. Since the original mesh sequence was arbitrary, this proves the stated iterated, full-sequence plane limit. Its two-spin coefficient is nonzero: it is \(Z_\sigma(\lambda)^2\) times the positive nearest-neighbor plane amplitude multiplying \(|x-y|^{-1/4}\). In particular the selected temperature branch has the massless power-law behavior required here; that critical behavior was a conclusion, not an input to its construction.

Finally let \(\phi:D\to D'\) be as in the theorem. The nearest-neighbor spin field and the energy field centered relative to the plane background have their usual primary covariance, by 6. If \(m_D^b(y)\) denotes the domain one-point function of that energy field, it has the same factor \(|\phi'(y)|\) as the field itself. Expand a domain-centered mixed correlation as \[\sum_{T\subseteq\{1,\ldots,m\}} (-1)^{m-|T|} \prod_{k\notin T}m_D^b(y_k) \left\langle \prod_{i=1}^n\sigma(x_i) \prod_{k\in T}\epsilon(y_k) \right\rangle_D^b.\] Every term has the common conformal factor \(\prod_i|\phi'(x_i)|^{1/8}\prod_k|\phi'(y_k)|\). Thus the actual-domain-centered nearest-neighbor comparison functions have precisely the covariance stated in the theorem. The constants \(Z_\sigma^nZ_\epsilon^m\) do not depend on the domain and preserve this identity. Homogeneous fixed and free boundary types are transported unchanged. This completes every bulk correlation and covariance assertion; no claim concerning coincident or boundary insertions, or interface convergence, is required or used. ◻