Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model

Proof of Lemma 2.1 and Proposition 2.2. Fix a sufficiently large integer B and put w_k=B^kw. In the appropriate parity space, with plane centering in the even case, define

The lines \R v_k are uniformly nonzero and bounded by (2.7), and S_kv_k=v_{k+1} by (2.8). If F is supported in B_{w_k}, subtract \alpha v_k to cancel its response in the disk of radius 4w_{k+1}. Equations (2.3) and (2.7) give |\alpha|\le C\norm{F}_\infty. Applying (2.4) to this difference proves the quotient bound

Here X_k is the space of bounded functions of the required parity on B_{w_k}, with plane mean zero in the even case and with right-angle rotation invariance imposed when p=3; its norm is the sup norm. The same estimate holds after subtracting any multiple of v_k, so it is indeed an estimate for the quotient norm. Combining (2.3) and (2.4) at the fixed ratio B also bounds S_k uniformly.

Choose uniformly bounded projections onto \R v_k, using for example the response in a disk at one fixed large multiple of w_k and dividing by its nonzero response on v_k. In the resulting bounded splittings,

Enlarge B until q\le B^{b-p+\nu}<1. Writing S_{k-1}\cdots S_0F=a_kv_k+u_k gives

Consequently a_k converges to a linear functional a_\infty(F) and

Define \ell_b(F)=w^ba_\infty(F). Since the line is transported exactly, its coefficient is one on \mathcal O_b. Constants have already been removed. Dividing (2.14) by B^{kb} gives (2.11) at the radii w_k and gives (2.9)’s bound. The construction permits an arbitrary initial function on the first square; it need not initially be a function of the cutting contour.

To identify the functional independently of the chosen initial scale, first divide (2.14) by B^{kb} and test the resulting unscaled decomposition in a plus disk at a fixed sufficiently large multiple of w_k. Conditional averaging preserves that expectation. After multiplication by the disk scale to the power b, the error is O(B^{k(b-p+\nu)}) and tends to zero because p-\nu>b; the line term tends to \kappa_b \ell_b(F) by (2.6). This proves the limit in (2.9). It also proves that different initial scales and bounded splittings give the same functional. For an intermediate radius R, stop one scale earlier when necessary, choosing k so that B\le R/w_k<B^2. The final averaging ratio then lies in a fixed admissible compact interval. The input estimates are uniform on such an interval: otherwise a sequence of failing ratios would have a convergent subsequence, contrary to their stated converging-ratio version. One additional cut therefore proves both decay bounds and the disk limit for every sufficiently large radius.

For a cut enclosing F, its disk expectation agrees with that of F in every disk containing the cut, and its plane expectation also agrees. The disk-limit formula now proves P-invariance. A fixed shift of the disk center gives the same limit: apply the decay decomposition about the old center, whose remainder still has sup norm O(R^{-p+\nu}) in a square inside the shifted disk. The line term has the same calibrated one-point limit, by bulk one-point convergence with the insertion displaced by O(R^{-1}) in rescaled coordinates. This proves translation invariance; the disk symmetry proves rotation and reflection invariance.

The bounds in (2.12) at the center are (2.7). For a displaced site, first cut at a square centered there of radius comparable to R and contained inside the requested cut, and then use the tower property and contraction in sup norm. For a single centered bond, its one-point calibration has the same nonzero scalar limit as the four-bond average, by the separated reference field theorem for both bond orientations. The recurrence proving (2.7) uses (2.4) with p=2 in this case: at a fixed large ratio,

Iteration bounds the left side uniformly. Finally all four incident bonds have equal charge by square symmetry, and their average has charge one; thus every centered bond has charge one.□

Proof. Set n=|z| and \zeta=z/n. The plane separated energy limit gives (2.16). To extract the other two coefficients without a lattice collision estimate, fix a small \rho>0 and cut separately in the disjoint squares of radius \rho n about 0 and z. Put

and

Rounding the square radii leaves fixed buffers. The two conditional replacements preserve expectations in the plane and in every sufficiently large plus disk: condition on both contours and use independence of their interiors. They consequently preserve charges by Lemma 2.1. Proposition 2.2 gives \norm{\Phi_n}_\infty+\norm{\Psi_n}_\infty\le C, with both supports in B_{Cn}(0). Their scale-normalized charges are

and are bounded uniformly in n.

For either \Theta_n=\Phi_n or \Psi_n, let b be its leading dimension, and put \chi_n=\ell_b(\Theta_n)/n^b. Subtract from \Theta_n the field \chi_n n^bP_{Cn}\mathcal O_b. This subtraction has bounded norm and zero charge. Its expectation in a disk of radius 4Bn, with B fixed and large, is therefore O(B^{-p+\nu}), by (2.11). Calibration of the subtracted line yields

For a sequence of varying directions the same assertion follows from compact-separated convergence after passing to a convergent subsequence of \zeta. This is the quantitative separation of the two limits: first n\to\infty with B fixed, and then B\to\infty.

For the even product the limit of d_{Bn}(\Phi_n) is

Rescale this disk by B. The first expansion in (2.15) gives

Thus B d_{Bn}(\Phi_n) tends to zero in the iterated limit. The error in (2.20) also tends to zero, since the general even exponent p=2 is already larger than b=1. This proves (2.17).

For the odd product, plane expectation vanishes, and the separated limit is aa_\sigma\langle\varepsilon(\zeta)\sigma(0)\rangle^+ _{\mathrm{disk}(4B)}. Let m_\sigma=\langle\sigma(0)\rangle^+_{\mathrm{disk}(4)}. The second expansion in (2.15) gives

Since \kappa_d=a_\sigma m_\sigma, the odd instance of (2.20) gives \mu_n\to a/2 in this sign normalization. Squaring gives (2.18). All the continuum remainders just used are uniform for |\zeta|=1, which also proves the claimed uniformity in direction.□

Proof. The conditional-law assertions are the buffered and shrinking-patch comparison [19, Corollary 2.2]; the comparison across physical boundaries has the same common-pin convention. Apply (2.21) to the law in the inner square under an arbitrary cutting configuration and to its mixture induced by the reference plane law. Pointwise comparison of these laws, integrated against W\ge0, gives C^{-1}Z\le P_{s,z}W\le CZ. The inequality |fW|\le\norm{f}_\infty W gives the bound for M. The two mean identities are the tower property. Finally, global spin flip commutes with the reference kernel, which proves the parity assertion.□

Proof. Choose an intermediate annular contour separated from both squares by fixed relative collars. For any outer configuration \tau, the induced law on this contour has a density H_\tau relative to the reference plane law satisfying \langle H_\tau\rangle_0=1 and 0\le H_\tau\le C. This is the pointwise comparison used in [19, Proposition 4.1]. Hence PF(\tau)=\langle FH_\tau\rangle_0.

Apply the outer-to-inner version of finite soft-weight replacement [19, Lemma 3.3] to H_\tau/C. For any \eta_0>0, it supplies finitely many positive normalized weights U_1,\ldots,U_m and vectors c(\tau)\in\R^m with \sum_i|c_i(\tau)|\le C, such that, simultaneously for all F,

Because U_i\ge0 and \langle U_i\rangle_0=1, each functional F\mapsto\langle FU_i\rangle_0 has norm at most one. Cover the finite-dimensional coefficient ball \{c:\norm{c}_{\ell^1}\le C\} by finitely many sets of \ell^1-diameter at most \eta/2. From each occupied set choose one actual outer configuration \tau_k. For every \tau in that set, the two approximation errors and the coefficient difference give

Take \eta_0\le\eta/(4C). The number of sets depends on the tolerance and geometry, but not on the fine mesh. Taking the supremum over \tau proves (2.24). Finally, Minkowski’s inequality and \max_k|X_k|^q\le\sum_k|X_k|^q prove (2.25).□

Proof. A connected animal of n fine cells has a spanning-tree traversal of length at most 2n in fine-cell units. Divide this walk into consecutive pieces of length at most L. Each piece, together with its prescribed O(R)-thickening, lies in a bounded number of parent cells. There are at most 1+2n/L pieces, proving the bound for one carrier. A principal halo has fine diameter O(S) and, since L is large compared with S, its parent thickening has bounded size. Taking the union proves (3.15). The 1000HR contact rule makes the thickened union connected. Every cut and every tested bad center is within a fixed HR distance of one of the input carriers. Their input principal halos add only Sr\ll R. The selection test and fresh rounding seeds are within the same neighborhood. This proves the dependence claim, not just the assertion about spin coordinates. In a pure selected bad output the whole component lies in an O(Mr) square; its cuts and tests thus fit in a fixed parent halo. An unselected bad input uses an even smaller halo. Short singleton outputs fit in a fixed parent halo for the same reason. Choosing S to contain these fixed neighborhoods completes the proof.□

Proof. Fix one product F of expansion occurrences. Conditional on all selected cutting spins and all remaining exterior spins, the reference law factors over the selected interiors. The retained bad supports are outside these interiors. Therefore

This is the tower property with the factors P_Qe^B left outside the normalized operators. If an ordinary singleton cut is used, its interior has a collar from all other factors, including the selected normalizations and retained bad fields. Conditional averaging of its one occurrence consequently leaves the right-hand integral unchanged. The resulting integrand is precisely e^{B'}m(I). Thus each expansion term preserves its integral. Summing first for finitely many inputs and sufficiently small scalar amplitudes gives \int e^{H(\zeta)}\dd\mu_0=\int e^{B'}\mathcal F(\zeta)\dd\mu_0. By construction B'+\log\mathcal F=s'+H'. Lemma 3.3 justifies this logarithm, its exponentiation, and the extension to the whole small-entry ball, including finite source jets. Absolute finite-volume summability, proved in Proposition 3.6, permits truncation of an infinite slot list. This proves (3.16). Its denominator at zero sources is strictly positive, so division gives (3.17). No disorder expectation was taken.□

Proof. For any subset of labeled occurrences, its transformed moment factors as a function over the components of its induced contact graph. Indeed a selected cutting square cannot meet supports in two different components: their distance would then be at most 2HR plus their halos, much smaller than the contact threshold. Each normalized operator therefore acts on only one component and sends empty factors to 1. Ordinary singleton cuts likewise act on just that component. This proves algebraic factorization without asserting probabilistic independence of the spin functions.

Regard each nonempty connected subset of labels as a polymer with activity its transformed moment. Two polymers are incompatible if they overlap or a contact edge joins them. The hard-core polymer partition polynomial equals the transformed moment polynomial: every induced contact graph has a unique decomposition into connected components. Its logarithmic coefficient is the sum over partitions of the labels into connected blocks, with the hard-core Mayer coefficient of the quotient contact graph. For a finite graph Q that coefficient is

Its absolute value is at most \tau(Q). To see this directly, strictly order the edges and group each connected subgraph by its greedy minimum spanning tree. For a specified tree, an additional edge is permitted exactly when it is later than every edge on its tree path. Such edges are optional independently, so their alternating sum is zero or has absolute value one. The sum over trees gives the bound.

For every connected block choose an internal spanning tree, and for every edge of a quotient spanning tree choose one original contact edge between the corresponding blocks. Allowing all these choices can only enlarge the bound. Their union is a spanning tree of G with each edge marked internal or between blocks. Conversely these marks recover the blocks and all the chosen edges. There are at most 2^{k-1} markings per spanning tree. Every polymer activity has norm at most the product of its occurrence norms, since all its conditional operators, tilted or ordinary, are sup-norm contractions. This proves (3.19), including the vanishing for disconnected G. Labeling occurrences has avoided any difficulty with repetitions; returning to the exponential series gives the stated 1/k!.

We give the placement count because the indexed norms alone contain no summation. A rooted animal of n cells has a depth-first traversal of a spanning tree with at most 2n nearest or corner steps. Choosing a deterministic traversal for counting gives at most 8^{2n} possible encodings. Choices of an anchor and of a distinguished contact cell contribute polynomial factors in n, hence can be included in e^{a_{\rm an}n} after enlarging a_{\rm an}. If a parent occurrence has size n_1, a child occurrence of size n_2 touching its 1000HR neighborhood can be placed in at most

ways. One chooses a fine cell in the parent’s neighborhood, at cost CL^2(1+n_1), and then the child’s animal, its contact cell and anchor. Principal halos change only the constant in this estimate. A root whose parent anchor is a fixed cell costs at most CL^2 anchor choices and the same exponential animal count. The average assignment of anchors permits rooting at an input whose parent is the required output anchor; summing this choice costs at most k, harmless below. For a marked output, root instead at the occurrence containing its first mark.

By Lemma 3.1, an output weight v_{\rm out} costs

Choose L so that Cv_{\rm out}/L uses only a small fixed fraction of the smallest available input weight. The remaining weight pays the animal count and leaves positive slack b at each vertex. At a vertex with q children, the factors 1+n in (3.20) are paid by

For example, apply x^q/q!\le e^x to a fixed fraction of bn, and absorb the replacement of n by 1+n in C_b. Thus a rooted labeled tree contributes at most C_L^k\prod_a q_a! times the input weighted bounds. Ordering the children at every vertex introduces exactly \prod_a q_a! orders. After division by k!, these objects are counted by plane rooted trees, whose number at size k is at most 4^{k-1}. Factors k and 2^{k-1} are absorbed by increasing C_L. This proves the claimed geometric order bound. Only L^2 covering factors have entered, so C_L is polynomial in L for fixed weights.

The argument kept the norm of every occurrence as a separate nonnegative factor. The assigned output footprint depends only on the input indices, and the anchor assignment has at most k choices, with only a fixed additional multiplicity for parent rounding. We dominate these allowed assignments by their sum. Conditional kernels are bounded uniformly by one; we do not sum over badness patterns or possible cutting kernels. This gives deterministic nonnegative kernels independent of the bad amplitudes. One may therefore retain an arbitrary factor, or finitely many factors, while bounding all other good background factors by 2\delta on principal slots and by \delta e^{-wn} on animal slots. Principal input weights are fixed constant costs. A fixed number of distinguished occurrences adds only a fixed power of k, which a geometric series absorbs. For one retained factor the terms of order k\ge2 carry C_L^k\delta^{k-1}; their sum is O(C_L\delta) after enlarging C_L and taking \delta smaller. The same proof applies to finite rectangular arrays of input indices. Applying the multilinear bounds to each elementary tensor in (3.4), summing its products of norms, and taking the infimum proves the projective statements. For tensor outputs allocate the weights and counting slack separately to the finitely many factors. This explains why only finitely many sufficiently separated weights were fixed.

Finally, the background connected series converges whenever C_L\delta<1. With finitely many square-free labels, at most that many marked occurrences can appear; the background series still converges and the resulting jet is polynomial in the bounded marked coefficients. Exponentiation of the absolutely convergent connected series recovers the transformed moment series. For finite lists this identity first follows near zero from formal power series and extends through the small-entry ball by convergence. The same estimates permit passage to absolutely summable slot lists. Projection (3.2) costs at most two, and a saved scalar costs at most the uncentered function norm, so all the bounds survive these operations.□

Proof. A connected union of n' parent cells has diameter at most Cn'R. Run Lemma 3.3 with twice the required output weight, and retain one weight after the other has paid the ordinary norm. The remaining factor is at most e^{-cD/R}. Every coefficient considered contains a small entry, so its summed background bound is C_L\delta; the sum over orders at least two is smaller. A short singleton or pure bad output has a footprint of diameter CR and cannot be nonchart when m is large. A nonchart connected footprint crosses a distance cN in torus coordinates, or has a lift crossing that distance in the wrapping case, and therefore contains at least cm parent cells. There are at most Cm^2 anchors. The preceding estimate, applied before the scalar loses its footprint, gives the claimed total. Retained norm factors survive because the tree proof bounds every occurrence separately.□

Proof. Pure bad outputs are principal. Short singleton outputs are also principal, regardless of nearby bad fields. Every remaining output is therefore either of order at least two in small entries or comes from a singleton animal of size n\ge L/2. The first class is bounded by C_L\delta^2 with weight w by Lemma 3.3. For the second, the weight allocation in (3.15) leaves a fixed positive fraction of wn unused. Since n\ge L/2, this gives the factor e^{-cL} while retaining enough slack for the placement sum. Conditional averaging under any bad tilt cannot enlarge its sup norm. These two estimates give (3.22). Choose L to make the second coefficient at most 1/2, and then \delta to make C_L\delta\le1/2.□

Proof. On a torus use torus distances and connected torus animals. Make cuts only in injective square charts of radius comparable with HR; the scale is required to be smaller than a fixed sufficiently small multiple of the torus side. A wrapping carrier is retained as an animal. The counting and walk-covering arguments hold in charts, and identifying possible placements can only reduce their number. In a twisted torus the local reference specification includes the twist, with a local gauge choice in each chart. Only the background deviation from that reference enters the list. Center chart terms after identifying that local reference with the plane reference. A spin gauge change conjugates the conditional operators and multiplication of functions, preserves sup norms and threshold tests, and commutes with the geometric rules. Thus, using the same decision seeds, the complete local map is gauge covariant. The saved scalars of corresponding chart terms are identical because their reference expectations agree after the gauge identification. The possible constant spin reversal between two gauges acts trivially on the even background. In particular, these scalars cancel when comparing a torus and a twisted torus on matching chart footprints. The conditional identities and positivity argument are unchanged. The possible nonchart scalar discrepancy is bounded by Corollary 3.4.

For a box use an ambient grid extending past it and keep prescribed exterior spins as formal coordinates until they are evaluated. Forbid a selected or ordinary singleton cut if its rule neighborhood meets a fixed scale halo of the boundary. Retain its insertions instead; short entries keep the principal assignments already specified. When testing singleton clearance, include every retained bad insertion, including those retained because of the boundary rule. All cuts that remain lie entirely in the reference bulk, so their conditional kernels are exactly the reference kernels used above. Center using the formal plane convention before evaluating exterior spins, or use a fixed bounded projection off constants for boundary entries. No smallness of a boundary principal slot is needed. The termwise proof of (3.18) therefore applies in each finite box.

There are only finitely many possible anchors at a fixed step in a finite system: initially they lie in a bounded ambient grid, and each step moves an anchor only to a parent or a bounded neighboring parent. For each anchor the number of animals of size n is at most e^{a_{\rm an}n}. Since w>a_{\rm an}, the tail bound proves total absolute summability of all background animal slots. The same argument, with their fixed exponential weights and finitely many mark anchors, proves absolute summability for source jets at every fixed number of steps. Their fixed factors L^{qd} do not affect this conclusion. Truncating the animal lists and passing to the limit proves the full finite-volume partition identity and justifies all finite-volume expansions used in Proposition 3.2.

For completeness let K bound the input principal norms at a given step. A selected component producing a fixed principal output has at most C_L fine centers; only C_L such centers can contribute to the same output halo. Equation (3.10) bounds its centered logarithm by C_LK. Retained principal inputs obey the same bound simply by summation. The convergent small-entry series adds at most a constant C_L. Hence

Induction gives a finite bound at each fixed step, independent of the volume. This bound need not be uniform in the number of steps and is not a stochastic smallness assertion.

Finally, every output footprint includes the full input footprints, all cut contours, all selection and threshold neighborhoods, and the boundary-clearance tests. If these footprints avoid the boundary, the finite-volume and plane operations read exactly the same spin functions, edge variables and seeds, so they give exactly the same slot values. Iterate this statement along the ancestors of a slot. One may equivalently impose boundary clearance by a fixed multiple of each current scale; the accumulated clearances are bounded by C\sum_{a\le j}r_a\le C'r_j. This proves exact bulk agreement and completes the finite-volume construction.□

Proof. Expand each output slot into its pure bad part, its singleton part, and its absolutely convergent connected small-field series. The pure bad part has centered sup norm bounded by twice the sum of the norms of the bad principal inputs that generate it. Each such input carries its own bad indicator. All conditional moments in the other two parts are bounded by the product of the input sup norms, because the conditional laws, including the tilted laws, are normalized.

First consider terms without a bad part. On the event that all local bad tests relevant to a fixed coefficient fail, that coefficient is the ordinary Taylor coefficient. The difference between the two versions is bounded by their sum of absolute majorants and a union of their local bad events. For total degree D\le4, (4.4) bounds each such difference by O(s^p) with all retained index weights. The tree calculation above sums the coefficients and the possible test centers. A term containing a pure bad part is treated by expanding its finite sum of principal input norms and choosing one of its bad indicators. For products of at most four outputs the same inequality applies; each remaining factor either contributes an input norm or a convergent small-field series.

It remains to bound degrees at least five in those series. Retain five input factors, use (4.3) on them, and bound every other good occurrence by its deterministic threshold or tail bound. The majorant is bounded by

after decreasing \delta. The same argument bounds a high-degree term that has already supplied a bad part: retain enough input factors to reach degree five, or use the bad indicator when fewer factors are present. There are at most four unrestricted bad-only factors, so p>8\max q supplies all required Hölder powers. This proves the raw moment assertion.

For clarity, the projective estimate here is an estimate before any spin evaluation: a random elementary tensor has projective norm equal to the product of its factor norms, and the norm of its expectation is at most the expectation of that product. Applying the finite partition formula

with factors put in their original argument order proves the cumulant assertion and its error bound. Finally, disjoint dependence carriers use disjoint primitive disorder variables and seeds. Their joint moment generating polynomial factors, so its logarithm has no coefficient involving both families. This is the claimed connected-index restriction.□

Proof. Choose an intermediate square, of radius a fixed multiple of w, whose contour lies beyond the collar. Put

Lemma 2.4 bounds, pointwise, the reference law of the inner spins under any contour assignment above and below by fixed multiples of its plane law. Integration against e^B>0 therefore gives c\le D\le C and |N|\le C\norm F_\infty, uniformly in B. Both functions are even and \langle D\rangle_0=1. The even part of Proposition 2.2, followed by the tower property, gives

Since P_RD\ge c, subtracting \langle N\rangle_0 from their ratio proves (4.6). The estimate remains uniform upon replacing B by tB, 0\le t\le1. Integrate

in the quotient by constants and apply (4.6) with F=B. This proves (4.7) without introducing e^{\norm B} for the bound linear in \norm B_\infty. For the bound independent of that norm, subtract \log\langle e^B\rangle_0 and use \log P_RD=\log(1+d_R): since P_RD\ge c, its norm is at most C\norm{d_R}_\infty\le Cw/R. Taking the smaller of the two bounds proves the stated minimum.□

Proof. For the first inequality, use the absolute decomposition just proved. The nonlinear good terms cost C_L\delta Bs, and long singleton terms cost \lambda_LBs. A witness is independent of the short input it multiplies. Thus

Minkowski’s inequality and the finite witness sums give O(s^2) at fixed L. It remains to estimate the attenuated short sum.

Expand its pth power. Join two of the p input anchors when their distance is at most C M, choosing C so that different components have disjoint dependence carriers. Call the connected components bunches. For a term with one bunch, choose the first anchor in at most CL^2 ways. A spanning tree on the p labeled occurrences places each successive anchor in at most CM^2 ways. There are at most p^{p-2} such trees. Animal shapes and slot lengths are summed with the reserved weights. Hence the count is at most CL^2M^{C_p}, and (4.3) bounds its expectation by (Bs)^p. After taking the pth root and including attenuation, this part costs

For two or more bunches, their random variables are independent. Suppose a bunch contains k<p occurrences. Hölder within it uses only L^k norms. For 2\le k<p, interpolate by

for k=1 use \norm{X_i}_1\le\norm{X_i}_2. The product of all bunch estimates, after its pth root, is therefore

For the last inequality, a singleton already contributes one to p\theta. If there are no singletons and there are b\ge2 bunches, then \sum k\theta_k=2p(b-1)/(p-2)\ge1. All placement counts are polynomial in L: there are at most p bunch roots, each with CL^2 choices, followed by the preceding short-distance counts. For each of the finitely many bunch patterns, weighted Young’s inequality gives, for any fixed \varepsilon>0,

This also follows directly by maximizing the left side minus \varepsilon B as a function of B/B_2. Since 1/\theta\le p, the resulting constant remains polynomial in L. Sum over the finitely many patterns. Increase L to make (4.13) and (4.9) small; choose \varepsilon for the multi-bunch terms; then decrease \delta for the nonlinear terms. Allocating less than 1/8 in total to these coefficients of B proves (4.10).

For the second inequality, first use (4.5). The nonprincipal part of Th has input length at least L/2, and its L^2 norm is at most CL^Ce^{-cL} Bs. A principal output of Th has the form \pi P_{HR}A, where A is the sum of the short-rule inputs anchored in its parent cell and \pi denotes plane centering. All these inputs are supported in one inner square, separated by a fixed proportional buffer from the HR contour. The weighted placement bound and Minkowski give

The finite dictionary of Lemma 2.5 supplies N=N(\eta,L) actual contour assignments \sigma_1,\ldots,\sigma_N such that

This is a deterministic inequality and thus applies to the random A. Replace each tested value of Th by that of h' using (4.5). The L^2 norm of the maximum of N values is at most the square root of the sum of their second moments. By (4.12), after restoring the principal weight, this costs \sqrt N F_*s+O(s^2). The net error costs C_L\eta Bs. Choose L so that the exponential long-slot coefficient is smaller than the desired \eps_*/2; this is possible against the polynomial C_L^*. Then choose \eta with C_L\eta<\eps_*/2 and increase r_0 to the dictionary threshold. This proves (4.11). Notice the order of the argument: (4.12) is a tensor hypothesis, supplied later by the Taylor flow, whereas the conclusion is a probability norm of a sup norm. They are not the same assertion.□

Proof. For a fixed source input define the finite measure \dd\mu_i=Y_i^q\dd\Prob. Its mass is at most U^qe^{-vq n_i}, and (4.15) bounds the pth moment of every X_m in this measure. Consequently, for nonnegative integers a_m of total D\le p,

Indeed, apply Hölder with exponents p/a_m and p/(p-D), the last factor being the mass of \mu_i. The powers of U and of the source weight sum to exactly q. A bad indicator supplies the unused p-D powers as in (4.4).

For q possibly distinct source slots, Hölder first gives

Thus (4.18) retains decay in every source and background index. For fewer source occurrences include unweighted probability measures in this Hölder step. Apply this calculation to the tree expansion, retaining three background factors in a remainder. Additional good factors are bounded deterministically, and a changed bad decision is paid by the unused mixed moment powers. The same summable series as in Proposition 4.1 proves the asserted O(U^qs^3) bound. In L^q retain one background factor and use (4.18) with degree q. A bad flag costs O(Us^{p/q}). This proves (4.16), since p/q>1.

We give the high mixed estimate separately. Normalized tilted conditional moments and the deterministic good-field bounds imply a positive linear majorant

Here the coefficients are deterministic majorants, not the realized coefficients of the map. They include the factor L^d. Lemma 3.3, with the source vertex retained, proves that their weighted row sums are bounded by C L^d; reserving additional input length weight makes the part with n_i>M bounded by CL^Ce^{-cM}. These statements mean row sums after paying the requested output weight from the input weight; they also hold with all the fractional weights appearing below. In particular no factor \norm{\log P e^B} occurs in (4.19). To construct K_{ai}, sum the absolute tree majorants over the allowed carrier and anchor assignments, taking the prescribed enlarged union of input carriers for each output. Every realized routing is one of these boundedly many assignments. There is no sum over patterns of threshold outcomes or over the amplitudes of bad fields.

Fix output background slot b and source input i. Estimate \norm{\norm{h'_b}_\infty}_{L^p(\mu_i)} using the deterministic decomposition preceding Proposition 4.3. If n_i>M, crude Minkowski and (4.15) give

There is still a reserved exponential in n_i for its subsequent source sum.

Suppose now n_i\le M. Long background inputs and nonlinear good terms cost, respectively, \lambda_L B_s s and C_L\delta B_s s, times the common factor U^{q/p}e^{-vq n_i/p-vn_b}. For short background inputs with anchors within 10M of the source anchor, use (4.15). There are at most CM^2 anchors. Animal lengths and shapes cost a constant after their weight sums. The attenuated part of (4.8), estimated by Minkowski, therefore costs

For its witness part the witness carrier is disjoint from the union of the source and background input carriers. Exact independence gives

so this part gains an additional factor O(Bs/\delta) and is O(U^{q/p}s^2) at fixed constants. For short background anchors farther than 10M, the background input is independent of Y_i. Drop any witness indicator and use

Even a crude sum over these anchors has only a polynomial cost in L. Combining these estimates gives, for short i,

We now check the powers when summing sources. Put F_i=(\E Y_i^q\norm{h'_b}_\infty^p)^{1/p}. Minkowski in L^q, applied to (4.19), yields

This is an \ell^{p/q} norm with weights K_{ai}. Apply its triangle inequality to the three terms in braces in (4.21). Raising the input source weight to p/q gives (e^{-vq n_i/p})^{p/q}=e^{-vn_i}, exactly the weight needed in the positive source sum. The background output weight remains e^{-vn_b} after taking the power q/p. Hence the short-source coefficient of B_s s is at most

whereas the far-background term is a constant C_{L,B}s independent of B_s. For the long-source part use (4.20) and the reserved exponential in its row sum; its coefficient is at most CL^Ce^{-cM q/p}. First choose L large. Since dq/p<1 and M is a power of \log L, the first attenuation coefficient tends to zero, as do both long coefficients. Then choose \delta small. Their sum can be made less than 1/8. Witness errors have an extra positive power of s and remain o(s) after (4.22), whose use of the \ell^{p/q} triangle inequality is homogeneous of degree one in F_i. This proves (4.17).□

Proof. Put M=(\log L)^3. If an input length exceeds M, the unused exponential weight gives C L^C e^{-cM} after all linear placement sums. Charge projection adds only polynomial factors in the lengths. Consequently these long inputs, including the charge needed to compensate their removal, have norm at most \eta_L\|A\|, where \eta_L\to0 faster than every fixed negative power of L. It suffices to estimate the inputs of lengths at most M; constants denoted by C M^C include their weighted index sums.

For the mean, collect all short slots with anchor c into one even function supported in a square of radius C M r. Homogeneity and square symmetry make this grouped function square symmetric about c. Its charge is zero, up to the discarded long-input charge. Subtract that charge using a translated copy of v_j before applying the reference estimate. The square-symmetric charge-zero exponent is 3-\nu, so one child contribution at the parent contour costs C M^C L^{-3+\nu}. There are L^2 child anchors. Thus the short part is bounded by C M^C L^{-1+\nu}\|m\|, plus \eta_L\|m\| for the compensation. The grouped function is formed before taking its sup norm; the triangle inequality and the weighted slot sum bound that norm by the input array norm. This is where the square symmetry of the law is used.

For the covariance, connectivity and the short-length restriction put the two anchors within C M r of one another. Separate pairs whose anchors go to different parent cells from pairs going to the same parent. In the first case, an anchor lies in a strip of width C M r about the boundary of a parent cell. There are C L M such fine cells per parent. Each centered factor costs at most C M/L under averaging; after the weighted sums over slot shapes and the other anchor, this contribution is at most

This bound applies also to pairs inside one parent for which the first anchor lies in that strip.

For the remaining pairs work first with one elementary projective tensor F\otimes G. If c is its first anchor and z_c the center site of c, replace both factors by their charges times the same unit-charge field:

Their supports and the original supports are contained in a square of radius C M r. After subtracting the charge, the even improved estimate and the tower property bound a replaced-factor error at the parent contour by C M^C L^{-2+\nu} times the original factor norm. The other factor has its usual C M^C/L bound. Summing L^2 first anchors gives C M^C L^{-1+\nu}\|K\| for terms containing a replacement error.

It remains to sum the term in which both factors were replaced. With the first anchor fixed, the transported elementary pair is the same for every choice of its second slot or anchor. Its scalar coefficient is therefore the corresponding sum in (5.2), and vanishes on the kernel of g_r. For an anchor outside the strip, every short connected second anchor goes to the same parent, so this cancellation is complete. Restriction to short inputs leaves only the discarded long-input charge, already bounded by \eta_L. Strip anchors were estimated in (5.6). Homogeneity is needed here to make the zero coordinate condition hold at every first anchor; it is not a statement about an arbitrary nonhomogeneous covariance. Taking infima over projective decompositions proves the asserted tensor bound without any factor depending on the dimension of the function spaces.

Finally, a connected short k-tuple has all its anchors within C_k M r of its first anchor. The first anchor has C L^2 possible positions in a rooted parent cell, and each of the k centered factors costs C M/L. Weighted sums bound the result by C M^C L^{2-k}\|A\| for k=3,4. Together with the long part and the preceding estimates this tends to zero. The intertwining assertions follow from (5.5).□

Proof. For a short source slot the ordinary odd estimate gives C((n+S)/L)^d before source scaling. Multiplication by L^d and summation with the unused exponential weight give a constant independent of L; one sums (n+S)^d against that weight, rather than replacing it by M^d. Long slots are again bounded by C L^C e^{-cM}.

For each short slot individually, subtract its normalized charge times the representative at its input scale. Both functions fit in a square of radius C(n+S)r containing y. The charge-zero odd estimate, whose exponent is 1+d-\nu, followed by source scaling, bounds the difference between the transported slot and its charged representative by C(n+S)^C L^{-1+\nu} times its norm. Summing over slots is legitimate by the exponential weights. Long slots admit the same subtraction with a beyond-polynomially small bound, since charge extraction has only a polynomial size cost. This gives a componentwise decomposition into a bounded rank-one map and a remainder of arbitrarily small norm.

Apply that decomposition separately to each factor of an elementary projective q-tensor. The all-line term is exactly Q_{j,q,y}(U)w_{j+1,y}^{\otimes q}. Every other term contains a small remainder and at most q-1 bounded factors. The sum of its 2^q-1 choices is small for fixed q. Infimizing over decompositions proves the result in the projective norm. In particular no norm comparison involving the number of spin configurations is used.

A connected array with an additional background factor has no L^2 volume sum: its root is the fixed mark. Connectivity bounds the remaining anchor sums by powers of the slot lengths. At least one even factor supplies C M^C/L, whereas the scaled marked factors stay bounded. This proves contraction in the mixed sector. All these are componentwise deterministic inequalities, so applying the center-coordinate projection to each random realization and then taking its L^p norm gives the last assertion. The buffer H keeps every point of a cell uniformly away from its cutting contour, making the constants independent of the position of y in its cell.□

Proof. Linear insertions propagate by nested averaging. Thus the coefficient of t_i at scale r is P_{Hr,c(i)}e(i) in the principal slot of its current cell, giving K_j^*=V_j exactly, and the same statement for the spin gives w_{j,y}. We explain separately why the mean profile stays bounded despite the expanding mean direction.

For one microscopic variable, write H_{i,k}^{(j)} for its t_i^k coefficient in the plane-centered background log list at scale j. All its anchors follow the same chain of parent cells. The fixed-degree carrier estimate of Lemma 3.1 bounds the number of occupied slot types and their lengths by constants independent of j. Increasing L makes these slots short for linear transport. Constants in the present formal calculation may depend on L and r_0. We claim

The first bound is the reference energy estimate. At a step from radius s to Ls, every freshly generated quadratic term costs C\|H_{i,1}\|^2, hence at most C_L(Ls)^{-2}. To bound the transported old quadratic terms, condition on one contour outside all earlier cuts. The exact identity, with saved constants restored, represents the original factor 1+t_i e(i). Its quadratic coefficient is zero. Its first scalar coefficient is zero because e(i) is plane centered. Equating quadratic coefficients therefore gives, modulo constants,

Here each H denotes the sum of its list entries before this common outer average. The linear rule puts these old short entries in the same parent principal slot; cancellation in (6.6) thus occurs before taking its norm. The right side is bounded by C s^{-2}. Combining old and fresh terms proves the second bound at the next scale. At the first scale the bound is included by the allowed constant depending on L,r_0. This reasoning also covers earlier saved scalar coefficients: they are reinstated for the exact identity and disappear upon plane centering. Their footprints obey the same carrier bound.

Only squares of individual signs contribute to [\lambda^2]\E h_j. There are at most C r_j^2 microscopic sites which can contribute to a fixed output anchor. Summing the second estimate in (6.5) proves the bound on m_j^*. The finitely many fixed-degree slot lengths permit the same conclusion with the indexed exponential weights.

For completeness, the remaining profile recurrences follow by the moment–cumulant partition formula. The lowest nonzero fourth-cumulant coefficient has four linear background letters, so Y_j^* transports homogeneously by T_j^{(4)}. At degree four the third cumulant has, besides its homogeneous term, only a quadratic output letter and two linear output letters. Its input partitions use either Y_j^* or two copies of V_j. These terms are bounded by the multilinear estimates of Proposition 4.1. The strict contractions of Proposition 5.1 therefore bound Y_j^* and W_j^*. In (6.2), the constant marked tensor is deterministic. Its first nonzero covariance with a background uses one linear marked correction and one linear background letter. The recurrence is the contracting connected marked/background transport, with bounded forcing from the constant marked tensor and V_j; hence Z_{j,q,y}^* is bounded by Proposition 5.2. The degree-two coefficient of \cum(u_{j,y}^{\otimes q},h_j,h_j) is zero: at that degree the marked argument is deterministic, and a joint cumulant with a deterministic argument vanishes.

Finally, in the degree-four covariance increment the linearly transported covariance has identical g coordinate. The remaining partitions use W_j^*,Y_j^*,V_j\otimes V_j, or m_j^*\otimes V_j, all bounded. In the degree-two marked-charge increment the transported marked tensor likewise has unchanged charge; the remaining terms use m_j^*,V_j and Z_{j,q,y}^*, while the connected two-background profile just discussed vanishes. This proves the asserted bounds on the increments.□

Proof. Fix the coefficient and place an outer contour beyond every cut and carrier which occurs in it. For each fixed spin assignment on that contour, apply Proposition 3.2 to the system inside it. The termwise conditional-expectation proof of that proposition leaves these contour spins fixed: each inner cut is an application of the tower property within this conditional reference measure. Thus the conditional partition expression, as a function of the outer spin assignment, agrees before and after every step. Its source derivative gives the analogous identity with the single odd insertion. Let C_j be the saved scalar formal series, so that the restored weight is e^{C_j}\exp H_*^{(j)}. Taking the reference plane average proves that e^{C_j}\langle\exp H_*^{(j)}\rangle_0 is independent of j. Taking the disk-response limit defining \ell or \ell_s proves the corresponding invariance of each charge numerator multiplied by e^{C_j}. The saved scalar with one odd source is zero, so the marked identity has this same scalar factor. Every operation involves a finite coefficient and finitely many cuts; no interchange with a convergent infinite formal expansion is involved. Dividing cancels e^{C_j} and proves the claim.□

Proof. We first prove (6.10). The coordinate \beta_J is the coefficient of degree four in the variance per unit microscopic area of \ell(H_*^{(J)}). To see this directly, restrict to a large union of cells, sum its linear charges, divide its variance by the number of microscopic sites, and let the window grow. At fixed J the coefficient arrays have bounded range, so the boundary contribution vanishes. Formula (5.2) is precisely the resulting anchored covariance sum. At degree four, averaging the independent signs leaves only one-site terms and unordered two-site terms. Terms supported on disjoint noninteracting insertions cancel in the variance subtraction. The surviving pair separations are at most C_Lr_J, by the fixed-degree carrier bound.

Fix two sites i\ne k and abbreviate

Their invariant energy charge is exactly

The numerator uses \ell(e(i))=\ell(e(k))=1 and \ell(1)=0; the denominator uses the centering of each energy. Through degree three,

Here O(t^4) denotes only discarded formal monomials of total degree at least four. Consequently the degree-four variance contribution after t_i=\lambda\eta_i, t_k=\lambda\eta_k is

The square of the quadratic mixed term supplies A_{ik}^2; its mean is zero. The two linear/cubic crosses each supply -2C_{ik}. Pure quadratic terms, if present in a different log representation, are deterministic after the sign substitution and disappear in the variance.

We justify replacing the linear charge by this invariant charge with a bounded total error. Write A_i=[t_i]H_*, B_i=[t_i^2]H_* and D_{ik}=[t_it_k]H_* at scale r=r_J. The bounds above give

All columns are plane centered. In particular the denominator in (6.7) has zero first-degree term. At mixed degree two, \exp H_*-H_* contributes A_iA_k, whose energy charge is O(r^{-1}). Thus the linear charge has first-degree coefficients exactly one and mixed second-degree coefficient A_{ik}+O(r^{-1}). At t_i^2t_k, the numerator corrections beyond the linear log charge are the charges of

The denominator correction is its quadratic coefficient multiplied by a first-degree numerator. That coefficient is bounded by C(r^{-2}+(hr)^{-1}) (in fact the plane-centering cancellation may improve some of its terms). Multiplying the displayed sup-norm products by the charge cost Cr gives the same bound. Therefore the linear and invariant coefficients at t_i^2t_k and t_it_k^2 differ by

Since |A_{ik}|\le C/h by Lemma 2.3, squaring the second-degree coefficient adds at most C((hr)^{-1}+r^{-2}). The first-degree/cubic crosses obey the same bound. This proves that the pair contribution differs from (6.14) by at most C((hr)^{-1}+r^{-2}).

For a single site the invariant charge is exactly t_i. The pure cubic charge discrepancy consists of the products A_iB_i, A_i^3/6, and the quadratic denominator term times the linear numerator, and is O(r^{-2}). Hence its degree-four variance contribution is O(r^{-2}) per site. No bound on the third log column itself is being assumed: its linear charge is determined by the invariant identity and these lower-column corrections.

The pair errors have a bounded sum per unit area, since in two dimensions

Indeed the shell n\le |z|<n+1 has at most C(n+1) lattice points; the first summand contributes C/r per shell and the second has at most C r^2 terms. The one-site contribution is bounded as well. One may use the disk |z|\le C_Lr for the pair sum: any noninteracting pair in this range has zero linear variance contribution and satisfies the same comparison, and beyond this range the linear coefficient is zero. Thus

The half accounts for unordered pairs. If the fixed-scale list is only grid-translation invariant, the intermediate expression averages its linear pair contribution over the first site’s residue class in a grid cell; the invariant expression is fully translation invariant and the uniform error bound gives exactly (6.17).

Lemma 2.3 gives, uniformly in direction, \langle e(z)e(0)\rangle_0=a^2|z|^{-2}+o(|z|^{-2}) and \ell(e(z)e(0))^2=o(|z|^{-2}). The lattice sum satisfies

For example compare each unit lattice square away from the origin with its radial integral; the gradient error is O(|z|^{-3}), which is summable outside any fixed disk. For an o(|z|^{-2}) remainder, split at a fixed large radius, make its coefficient arbitrarily small beyond that radius, and use the same logarithmic bound. Its sum is consequently o(\log R). Applying these facts in (6.17) yields \beta_J=-4\pi a^2\log r_J+o(\log r_J). Telescoping (6.3) subtracts only the bounded initial coefficient \beta_0, proving (6.10).

The marked increment

For one energy site i and mark y, the invariant normalized spin charge is exactly

The denominator is one because the energy is centered. Write F_0=[1]f, F_1=[t_i]f, and retain the background columns A_i,B_i above. At first degree the difference between \ell_s(f) and \mathcal S is \ell_s(F_0A_i), which is O(r^{-1}) by (6.9). At second degree the numerator corrections are the charges of

Their sup norms, multiplied by the odd charge cost Cr^d, are bounded by C((hr)^{-1}+r^{-2}). The denominator correction has the same bound. It follows that the linear marked charge is

Its qth power has degree-two coefficient

since |S_i(y)|\le C/h. Sign averaging eliminates distinct-site quadratic monomials, so only this one-site calculation is needed. The marked coefficient has range at most C_Lr; sum (6.19) over those sites and use (6.16). Uniformly in y,

The finitely many coincident or adjacent sites contribute only a bounded amount. The uniform OPE asymptotic \ell_s(\sigma_y e(i))^2\sim a^2/(4|i-y|^2) and the preceding lattice summation give (6.11) after subtracting \alpha_{0,q}(y). Finally \log r_J=J\log L+\log r_0 gives both Cesàro assertions.□

Proof. For q=1 the invariant coefficient of t_i^2 in (6.18) is exactly zero, rather than merely smaller than its logarithmic scale. Thus (6.20) gives |\alpha_{J,1}(y)|\le C. The initial coefficient is uniformly bounded, and (6.4) telescopes. This stronger conclusion will control the first marked moment; a zero Cesàro mean alone would not suffice.□

Proof. We first derive the estimates under the corresponding bounds on an input prefix, and then choose the constants so that the estimates reproduce those bounds. This separation is useful because no sign condition is available for an individual b_j.

The finite Taylor rule in Proposition 4.1 determines the cumulant recursion with remainder O(G_j^{5/2}). The required algebra can be described explicitly. Assign a separate letter to every factor in a Taylor monomial, even when two letters have the same slot index. If I_1,\ldots,I_k are the groups belonging to the k output factors, the cumulant of their products is

The join condition says precisely that the partition connects the output groups. Formula (7.8) follows by expanding moments into cumulants and applying the defining alternating sum for the output cumulant. The coefficient of a partition not connecting the groups is zero. It therefore applies also to correlated input slots.

A singleton block costs m=O(G), a pair costs K=O(G), and blocks of sizes three and four cost W,Y=O(G^2), where G=G_j. For covariance the nonlinear degree patterns with at most four letters are (1,2),(2,1),(1,3),(2,2),(3,1). The three-letter patterns retain W and the connecting mK partitions; the four-letter patterns retain Y and the connecting KK partitions. For the third cumulant the only further pattern has four letters and retains Y and KK. For the fourth cumulant there is only homogeneous linear transport to the stated accuracy. The mean retains its linear term and the quadratic Taylor coefficient applied to K; its other terms have size O(G^2). In particular the recursions have the form

Here Q_j is the quadratic Taylor coefficient contracted against a covariance; \mathcal B_j and \mathcal F_j denote the sums of exactly the connected partitions just listed, including their Taylor coefficients. They are bounded linear maps on the displayed direct sums of tensor spaces. The indexed estimates in Proposition 4.1 justify all slot sums.

The charge of linear covariance transport is unchanged by Proposition 5.1. Equations (7.3) and (7.10) therefore first give the coarse estimate g_{j+1}-g_j=O(G_j^2). On the kernel of that charge the linear operator contracts. Subtracting g_{j+1}V_{j+1}, and using the exact transport of V_j, gives (7.4) by convolution with this contraction. For clarity, every such convolution uses the elementary estimate

for the finitely many positive exponents \alpha used here, where \kappa<\rho<1. To verify it, use G_i/G_j\le(1+B_0\bar g)^{j-i} and decrease D so that \kappa(1+B_0\bar g)^\alpha<\rho.

Subtract next the recursion for g_jm_j^* from (7.9). The stable mean component is controlled forward by (7.13). Its forcing is O(G_j^2+G_j\rho^j). The thermal component is controlled backward: if its error is z_j, then

Summing this equality from N backwards gives |z_j|\le CG_j(G_j+\rho^j+\rho^{N-j}), after enlarging \rho<1. This proves (7.5). The fourth and third cumulant recursions, in that order, have contracting homogeneous operators. Subtract their formal-profile recursions, use g_{j+1}^2-g_j^2=O(G_j^3), and apply (7.13). The resulting errors are those in (7.6). An unmatched initial profile contributes only the geometric term. All comparisons at scale zero can likewise be paid by that term.

The definition of b_j is the charge of the nonlinear term in (7.10) evaluated on the formal profiles. Inserting (7.4)–(7.6) therefore gives (7.7). The terminal term in that formula is the cost of using only a finite thermal prefix.

We now show why this refined estimate reproduces the asserted covariance range. The coarse one-step bound first makes the tested g_{N+1} positive and comparable to G_{N+1}, so reciprocals are legitimate. Taking reciprocals in (7.7) and summing gives, for 1\le k\le N+1,

Indeed g_j/G_j is comparable to one on the input prefix, the two geometric errors have bounded sums, and the reciprocal Taylor remainder is O(G_j), bounded by O(s_j). Proposition 6.3 says that R_k=\sum_{j<k}(b_j+B_0)=o(k). Thus for each \eta>0 there is C_\eta<\infty such that |R_k|\le\eta k+C_\eta for every k. Since s_j\le\sqrt{\bar g},

First choosing \eta small and then D small makes this strictly smaller than the amount allowed by G_k/2<g_k<2G_k. It also proves that

uniformly over all kept prefixes. Only the averaged coefficient was used; individual coefficients may have either sign.

It remains to fix the constants in the bounds used above. First choose the covariance-kernel allowance larger than its initial and line contributions. Contraction reproduces this allowance, with its nonlinear O(G^2) loss left for the final smallness choice. Choose X so large that, under this covariance bound,

Choose the remaining mean allowance from this tube, the initial mean, and stable contraction. Next choose the fourth-cumulant allowance and then the third-cumulant allowance, using (7.12) and (7.11). These choices furnish a fixed bound F_* on the actual next principal raw second tensor. Indeed, at each principal slot i,

after enlarging F_* by the fixed factor e^v if necessary. This is precisely the tensor input required by the moment reproduction estimate; no output probability norm has been assumed to obtain it.

For the moment bounds in Proposition 4.3, choose B_2 larger than the fixed multiple of F_* and the initial second moment bound, and B larger than the initial high-moment bound and C_L^*B_2, with strict room in both inequalities. The dictionary error was chosen to have \eps_* C_L^* sufficiently small, so these choices can also satisfy \eps_*B\ll B_2. Finally decrease D to pay all nonlinear remainders, the variation of G_j, and all fixed constants in the profile comparisons. The tail bound is reproduced by the exact map. This order closes the input-prefix induction and its tested next step. In particular, the covariance improvement uses only the previously kept prefix, not an as yet unproved next thermal bound.□

Proof. At every fixed scale the expected list coordinates are continuous in x_0. Indeed the map has finite random footprints; each fresh threshold has a continuous distribution, so equality with that threshold has probability zero. Almost-sure continuity away from those equalities, together with the finite-step moment bounds and summable coefficient bounds, gives dominated convergence. This applies uniformly on each compact parameter interval used below.

Choose the initial interval I_0 so that its endpoints have x_0=-XG_0 and x_0=XG_0. Suppose I_j is a closed interval all of whose parameters are kept through j, and whose endpoints have x_j=-XG_j and x_j=XG_j. By (7.17), their next coordinates lie strictly below -XG_{j+1} and above XG_{j+1}, respectively. Continuity supplies a subinterval I_{j+1}\subset I_j on which |x_{j+1}|\le XG_{j+1} and whose endpoints attain both bounds. For an explicit choice, take the first hitting of the upper bound and the last preceding hitting of the lower bound. Between those points neither bound can be crossed. The nested nonempty compact intervals have a common parameter, giving the asserted trajectory. Finally, apply the finite-prefix profile bounds to arbitrarily long prefixes of this trajectory, with j fixed, and let N tend to infinity.□

Proof. In (7.14), R_j=o(j) and \sum_{i<j}s_i=o(j) for each fixed D>0, because s_i\to0. Thus g_j^{-1}=B_0j+o(j). The coarse increment estimate and \sum_jG_j^2\le C\bar g give the total-variation bound.□

Proof. Put a_j=A_{j,q}^{1/q}=\norm{\alpha_j}_{L^q} and k_j=\norm{u_j-\alpha_jw_{j,y}}_{v,q}. The source-line projection is bounded in the weighted norm. Its complement contracts under the scaled linear transport, and Proposition 4.4 bounds the nonlinear difference by CUs_j when U=C_*a_j. Since linear transport preserves normalized spin charge,

where the first inequality is the reverse triangle inequality in L^q. In particular a_{j+1}>0 and a_{j+1}/a_j=1+O(s_j) once disorder is small. This proves positivity before any normalization by the next moment is used.

To close (8.2), reserve a fixed part of C_*a_j for the line and the rest for the kernel. Choose C_* large enough for the bounded line norm and the initial list. The strict contraction in the second inequality of (8.5) then reproduces the kernel allowance relative to a_{j+1}, after decreasing D. The mixed output estimate in Proposition 4.4 is

Choose B_s after the background constants so that its first two terms leave strict room below B_s s_j, and then decrease D so that the remaining room pays the changes from U to C_*a_{j+1} and from s_j to s_{j+1}. Initially the source is deterministic, so the mixed bound follows directly from the background moment bound. This closes the simultaneous induction for (8.2)–(8.3).

At zero background the first step gives exactly the representative w_{1,y}. Its perturbation is O(a_0s_0), hence k_1\le Ca_1s_1. Divide the kernel recursion in (8.5) by a_{j+1}s_{j+1}. The ratios a_j/a_{j+1}=1+O(s_j) and s_j/s_{j+1}=1+O(G_j) make its contraction coefficient strictly smaller than one. Iteration proves (8.4).□

Proof. The mixed estimates give initially \norm{C_{k,j}}\le CA_js_j^k for k=1,2. Apply the source Taylor rule through two background degrees, counting the displayed background arguments among those degrees. For C_2 the only retained term is homogeneous linear transport: a nonlinear term in either background output has at least three background letters, and a nonlinear term in the transported source tensor also adds a third letter. The degree-zero transport of the grouped source tensor is deterministic and linear on that tensor. Thus no independent S_jK_j term is present in this cumulant recursion. By Proposition 5.2,

The initial grouped source tensor is deterministic, so C_{2,0}=0. Contraction now proves (8.7). More explicitly, for each of the finitely many exponents b below,

Indeed (8.5) gives A_i/A_j\le(1+C\sqrt{\bar g})^{j-i}, and s_i/s_j\le(1+B_0\bar g)^{(j-i)/2}; their product can be absorbed in the strict contraction by choosing D small.

The corresponding expansion of C_1 has its homogeneous term and terms with two background letters. To see their structure without any assumption on the disorder law, set U=u^{\otimes q} in the moment-cumulant identity. For example,

with the indicated permutations of indices. The source-linear, background-quadratic term and the source-background term covaried with one background are obtained from this identity by subtracting their products of expectations. Their only possible leading inhomogeneous contribution is a bounded bilinear operation on S,K; all the C_2 and C_1m terms are O(A_js_j^3) at this stage. Consequently

where \mathcal H_j is exactly the sum of these degree-two inhomogeneous Taylor terms. Since C_{1,0}=0, contraction first gives C_{1,j}=O(A_jG_j).

In the expansion of S_{j+1}, the degree-one terms now use C_{1,j} and S_jm_j, and the degree-two terms use S_jK_j; all other terms cost O(A_js_j^3). Since m_j,K_j=O(G_j), taking product spin charge improves (8.5) to

To identify its coefficient, expand u_j=\alpha_jw_{j,y}+(u_j-\alpha_jw_{j,y}) in the q tensor factors. Every term except the all-line term contains a kernel factor. Hölder, (8.2), and (8.4) bound its expectation by CA_js_j, proving (8.9).

The formal profile satisfies

Subtract this equation multiplied by A_{j+1}g_{j+1} from (8.13). The replacements of S_j and K_j by their line parts cost O(A_jG_j(s_j+\rho^j)), by (8.9) and (7.4). The change of scalar factors costs O(A_jG_j^2), by (8.14) and g_{j+1}-g_j=O(G_j^2). Contraction and (8.12) prove (8.8). Finally insert the three profile comparisons and (7.5), with its terminal term removed, into the Taylor formula for the spin charge of S_{j+1}. Its leading coefficient is exactly c_{j,q}(y) by definition. The accumulated error is that in (8.10). Scale zero has bounded coefficients and errors of order A_0G_0, which the geometric term allows.□

Proof. The error in (8.10) has summable absolute values, because

All factors are positive and uniformly close to one for small D. Taking logarithms therefore gives

The error is uniform in y. Let \mu_q=\binom q2 B_0/8. Proposition 6.3 gives \sum_{j<J}(c_{j,q}(y)-\mu_q)=o(J) uniformly in y. Abel summation then gives \sum_{1\le j<J}(c_{j,q}(y)-\mu_q)/j=o(\log J): for each \eta>0, the partial sums are at most \eta j+C_\eta, and the summation-by-parts remainder is at most \eta\log J+O_\eta(1). The coefficients are bounded, so replacing 1/(B_0j) by g_j=(1+o(1))/(B_0j) costs another o(\log J). This proves (8.15) from (8.17).

For the bound uniform in D, write c_j=c_{j,2}(y) and \mu=B_0/8. Uniform Cesaro convergence gives |\sum_{i<k}(c_i-\mu)|\le\eta k+C_\eta. Since G_j decreases, summation by parts implies

By (7.16), replacing G_j by g_j costs at most \norm{c}_{\infty}\sup_j|g_j/G_j-1|\sum_{j<J}G_j. Choose \eta and then the disorder bound so that the total coefficient of the logarithm is less than 1/4. The sums of the Taylor errors and of g_j^2 are uniformly bounded (in fact they tend to zero with D). Applying the logarithmic estimate proves (8.16).□

Proof. Set U_j^*=C(1+B_0j\bar g)^{1/8}. Equations (8.2), (8.3), and (8.16), with q=2, give all one-source absolute estimates with A_j in the previous proof replaced by U_j^*. To retain the full background smallness in this passage, write Y=\norm{u_{j,i}} and H=\prod_m\norm{h_{j,m}}^{a_m}, with b=\sum_m a_m\le p. Cauchy–Schwarz gives

The first factor uses the mixed q=2 measure and the second the background moment bound. Unused powers in both measures pay any large-field indicator in the Taylor remainder as in Proposition 4.4. Apply the same connected-cumulant recursions to the single grouped source u_j. They yield

The change of its scalar coordinate is bounded absolutely by |A_{j+1,1}-A_{j,1}|\le CU_j^*G_j. When comparing the C_1 recursion with its formal profile, use this absolute difference bound in place of (8.14). Together with |A_{j,1}|\le U_j^* it bounds the change of A_{j,1}g_j by CU_j^*G_j^2. The same contracting convolution proves

Consequently, with a_j=A_{j,1},

Corollary 6.4 gives a fixed bound on every partial sum of c_{j,1}(y), uniformly in y. Abel summation and Corollary 7.3 therefore imply

Also \sum_jc_{j,1}(y)^2g_j^2\le C\bar g. Thus the products P_J=\prod_{j<J}(1+c_{j,1}(y)g_j) are positive and satisfy \sup_{J,y}|P_J-1|\le C\bar g. The inhomogeneous errors have total cost

The exponent 11/8>1 is what makes the first sum integrable on the scale j\asymp\bar g^{-1}. Variation of constants in (8.20), with a_0=1, gives

It follows that \sup_{J,y}|a_J-1|\le C\sqrt{\bar g}, which proves both assertions.□

Proof. A newly generated coefficient containing both labels is bilinear in the two singleton lists. Apply the connected coefficient bounds of Lemma 3.3, retaining the output carrier containing both marks. At zero background degree a pair of principal slots does not merge: their input carriers, averaging squares, and contact neighborhoods are disjoint, also in output units, by our choice of F. Their transformed moment therefore factors and contributes zero to the joint logarithmic coefficient. At the first step the two input sources are principal. At later steps a zero-background merge must therefore contain a remainder from (9.6); Hölder’s inequality in L^4 gives a factor s_j in its L^2 cost.

For ordinary analytic terms of positive background degree, retain one background factor before bounding the others by the small thresholds and the deterministic tail bounds. To spell out why the retained factor is small despite dependence, suppress the slot weights and write X_a=\norm{u_{j,a,i_a}} and H=\norm{h_m}. The mixed bound of Proposition 4.4, for source power four, and Hölder interpolation with respect to the measure X_a^4\,\dd\Prob give

It follows that

The slot weights interpolate in the same way.

There is a separate estimate for changes caused by a large-field decision. The actual map and its ordinary analytic comparison agree unless one of the finitely many local badness tests in that term succeeds. For a tested principal slot m, let B_m=\{\norm{h_m}>\delta\}; a random threshold in [\delta,2\delta] can only reduce this event. The same mixed bound gives

Normalized tilted moments, and their joint logarithmic derivatives, are bounded by a fixed multiple of X_1X_2; their bounds contain no exponential of a large principal norm. Thus the last display also bounds the decision-dependent terms. The number and positions of tested centers cost only the polynomial carrier factors already paid by the tree weights. Summing the tests and the analytic coefficients as in Lemma 3.3 gives a fresh joint coefficient of cost at most C U_1U_2s_j, since p/2>1. All additional small factors and placements are paid by the thresholds and exponential slack. This separates the smallness supplied by a retained analytic factor from the smallness supplied by the rare decision event.

An animal containing both marks has length at least c\ell/r_{j+1}. Retain one further fixed positive exponential weight on the output before summing it or saving a scalar. The fresh cost is therefore at most

The finitely many initial-scale adjustments obey this bound with a fixed constant. Once generated, a joint coefficient is transported linearly in that coefficient. The deterministic source bound, with normalized tilts and the fixed source rescaling, bounds its total weighted sup cost by a constant K per step, uniformly in the sizes of the principal background fields. Scalars obey the same bound when their rescaling is included. Thus their accumulated cost is bounded by the sum of (9.10) multiplied by K^{J-j-1}, over 0\le j<J.

For completeness, this sum is uniform over the whole stopping window. For j\ge J/2, s_j\le C_Ds_J and A_{j,4}(y_a)^{1/4}\le C_{D,\eta}J^{3/16+\eta} for every \eta>0. Writing m=J-j-1, the remaining numerical series is bounded by

For j<J/2, the same exponential contains a factor at most \exp(-cF L^{J/2-1}), which pays K^J and any polynomial bound on the earlier marked moments. This proves (9.9), with arbitrarily small fixed power losses in place of its o(1).

Finally set R_a=U_{J,a}-\widetilde\alpha_{J,y_a}w_{J,y_a}. The identity

and (9.6) give the same L^2 bound by Hölder with fourth moments. No independence is used in this last estimate.□

Proof. For each fixed separation, free ferromagnetic correlations increase as the volume increases and remain in [0,1]. Their limit exists, and bounded convergence passes both disorder moments to the limit. Here J and r_J are fixed during this passage. The first line of (9.15), (9.8), and (9.16) give the first assertion, including a positive lower bound. For the second, the triangle inequality and its reverse in L^2 give

The main term is J^{1/8+o(1)}, so squaring proves the second assertion. All finite-volume identities hold for each choice of seeds, whereas the original correlation depends only on the bonds. Averaging over the seeds thus leaves the asserted disorder moments unchanged.□

Proof. Under the Edwards–Sokal coupling [12], the wired arm probability equals the spin expectation with the corresponding plus wall. Start a single source at y and let J_n be the largest integer satisfying

Thus r_{J_n}\asymp n/(\log n)^2, J_n=O(\log n), and the principal halo remains far from the wall for every j\le J_n.

Use the finite-box blocking rule, which forbids cuts near the wall and leaves those interactions in the lists. Every singleton term whose full footprint stays in the bulk equals its plane counterpart. By the weighted L^{400} source estimate and Proposition 8.3, the sum of the sup costs of these bulk terms at the final scale has L^{400} norm bounded by a fixed power of 1+J_n. Indeed A_{j,400}^{1/400}=j^{399/16+o(1)}, and summing the exponential slot weights only changes its constant. A local odd scalar is zero: before the boundary spins are evaluated, the bulk kernels and the plane centering are invariant under simultaneous spin flip.

It remains to estimate all terms that can see the wall, including saved scalar increments. Every such source footprint contains the mark and reaches the boundary region. At scale j its carrier length is therefore at least c n/r_j. The deterministic weighted source transport costs a fixed factor K per step, even with arbitrary principal background sizes: selected averages are normalized, retained large terms do not enlarge a source norm, and the small-field coefficient sums are bounded. This is the source estimate of Section 4 applied without averaging over the randomness. Keep an additional exponential carrier weight. At each possible scale of first boundary contact it supplies \exp(-c'n/r_j). Sum these contributions, retaining the source rescaling up to J_n; changing K absorbs both that rescaling and the sum over the possible first-contact steps. The total deterministic cost at the stopping scale is at most

One can equivalently subtract the plane and finite-box constructions: the first coefficient where they differ must have a footprint meeting the wall, and each later difference is transported with the same weighted estimate. This verifies the bound for all nonbulk coefficients without assuming that their laws agree with plane laws. For an odd scalar the carrier weight is used before its support is forgotten, so (9.20) includes all nonzero scalar contributions. Initial fixed constants merely change C.

The exact normalized one-source identity now expresses the plus magnetization, multiplied by r_{J_n}^d, as the terminal expectation of the scaled source plus its saved scalar. Its absolute value is bounded by the total sup costs just estimated, irrespective of the terminal background. Hence, for some fixed M<\infty,

Raising this inequality to the power 400 proves (9.18). Translating the box changes none of the constants because the source estimates are uniform in its mark. Moving the wall by a bounded number of lattice layers changes the lower bound on the boundary-reaching carrier length only by a fixed factor, proving the remaining conventions.□

Proof. The cycle space of (V,A) has dimension |A|-|V|+k(A). A cellular boundary is supported on A precisely when its face coefficients are constant on each complementary dual component. The space of such boundaries therefore has dimension k(A^*)-1: its face coefficients have k(A^*) degrees of freedom, and adding the constant two-chain does not change the boundary. Consequently

Apply the same identity on the dual cellulation and add the two identities. The torus Euler formula |V|-|E|+|F|=0 gives r(A)+r(A^*)=2. Primal and complementary dual cycles have zero intersection, since a primal edge and its transverse dual cannot both be present. This inclusion, together with the dimension identity, proves (10.3).

The ratio of the two unnormalized weights is

where the last equality uses (10.4). The first factor is independent of A, and 1\le2^{r(A)}\le4. Normalization proves the claimed comparison.□

Proof. Apply the exact finite-volume map of Proposition 3.6 to the untwisted model and to the \chi-twisted model, using the corresponding critical reference partition function in each case and the same decision seeds. A twist is removed on a contractible chart by a spin gauge transformation. Every background function is even, so the ambiguity of a simultaneous sign change on the chart does not alter it. Conditional reference averaging, the local decisions, and plane centering in this gauge therefore give identical chart coefficients and identical saved scalar increments in the two runs.

Here is why the errors outside charts remain uniform in J. Choose Q larger than the fixed halos and chart margins in the construction. A principal output uses only a short parent footprint, by Lemma 3.1. That footprint lifts to the plane at every scale up to J, and its environment and seeds have the planar law. A nonchart footprint has length at least cm when the torus has side m at the scale carrying that footprint. Changing between input and output cells changes this constant by the fixed factor L only. It cannot satisfy the short-footprint rule for a principal output. This also proves that a seam crossing a short footprint causes no exception: it is removed by the local gauge, whereas a genuinely nonchart input remains in the tail.

Keep the scalar supports until applying the enhanced tree weights in Lemma 3.3 and Corollary 3.4. Summing the possible nonchart footprints, including the neighborhoods used by the decisions, bounds the uncancelled scalar increment at this step by

There are O(m^2) possible roots; the additional length weight pays e^{-cm} after the animal sum. The side lengths m decrease geometrically and remain at least Q, so their total is bounded by C_L\delta Q^2e^{-c'Q}, independently of J. The final tail functions have total supremum norm at most C\delta Q^2. These estimates use the finite-volume weighted bounds before forgetting supports, and apply also to normalized conditional operations on nonchart terms.

To see precisely how these errors enter the desired ratio, let c_J^\chi,H_J^\chi denote the accumulated scalar and terminal background for a twist, and use superscript 0 for the untwisted run. Exact partition preservation gives

The brackets are normalized terminal reference expectations in the respective exact partition identities. Chart scalar increments cancel in c_J^\chi-c_J^0, leaving the nonchart bound above. At the final scale there are only O(Q^2) principal slots. Their laws agree in charts with the planar laws, so the high-moment bounds along the tuned flow imply that their total supremum norm converges to zero in probability as J\to\infty. Denote this total, in the two runs together, by P_J. Exact partition preservation and |\log\langle e^H\rangle|\le\norm{H}_\infty now give

The reference ratios here are on the same microscopic torus; exact blocking has not replaced them by a limiting torus ratio. First fix Q, then choose \delta and the permitted disorder so that the deterministic errors are smaller than a fixed tolerance depending on a_0, and finally let J grow. By (10.5) and (10.6), on an event of probability tending to one all three tuned character probabilities lie in a common compact subinterval of (0,1). Their values always lie in [0,1], so averaging proves (10.7).□

Proof. For a fixed environment abbreviate its FK law to \phi, and set X_e=\one_{\{e\in A\}} and c_e(\omega)=\Cov_{\phi_{t,\omega}}(\one_{\mathcal A},X_e)\ge0. Differentiating the finite sum in (10.1) gives

Since \dd K_e/\dd t=\tfrac12\tanh(2K_e), the coefficients \lambda_e=\tanh(2K_e)e^{2K_e}/(e^{2K_e}-1) have uniform positive lower and upper bounds.

Represent each environment bit by a uniform variable V_e, with \xi_e=1 when V_e\ge1/2. Order the FK edges arbitrarily as e_1,\ldots,e_M and use further independent uniforms U_i to set

The conditional FK laws are increasing both in the prescribed past and in the couplings. Induction in i therefore makes the entire sample increasing in all the U_i and V_e. Let F be the Boolean indicator of \mathcal A in this product representation. For one uniform coordinate z, its influence I_z is the probability, over all other coordinates, that the section of F in z has values zero and one on sets of positive Lebesgue measure.

We first bound an FK-uniform influence. Fix the environment and let P=(X_{e_1},\ldots,X_{e_{i-1}}) be the past. Write

The future uniforms couple the two branches monotonically. As both intervals for X_{e_i} have length at least \eta, the section in U_i is nonconstant exactly when the two resulting Boolean outputs differ. Averaging those future uniforms gives a_1(P)-a_0(P). Consequently

For the last inequality, total covariance gives an additional term

Both functions are increasing in P, and the marginal distribution of P is positively associated. This proves the indicated sign.

For the environment coordinate V_e, condition on every other bond. Let \phi_- and \phi_+ be the FK laws with its low and high value, and put R=v_e^+/v_e^-\ge1. Changing this bond multiplies the weight by R^{X_e}, so direct normalization gives

Monotonicity of the common-uniform coupling makes the left side the conditional influence of V_e. The ratio R is uniformly bounded. After averaging the other bonds, the low-bond covariance is at most twice \E c_e(\omega), since both low- and high-bond covariances are nonnegative. Thus I_{V_e}\le C\E c_e(\omega) as well.

Translation invariance of the averaged law and of \mathcal A makes \E c_e(\omega) constant within each of the two edge orientations. Each orientation has \ell^2 edges. Equation (10.11) therefore implies

There is no symmetry requirement on the chosen ordering or the sequential sampler: only the averaged covariances were compared by translation. There are N=2M=4\ell^2 product coordinates. The BKKKL inequality [3], in the null-set-invariant formulation [15, Theorem 2.3 and Equation (2.3)], gives

Combine this with (10.14) to obtain (10.10).□

Proof. The dual transformation v_{e^*}^*=2/v_e is equivalent to \sinh(2K_e)\sinh(2K_{e^*}^*)=1. Thus, in the coordinates (1.7), it sends (t,D,\xi) to (-t,D,-\xi) on the dual lattice. At t=0 the environment law is invariant. Write a_i=\E\phi_{\mathbb T_\ell,0,\omega}(r(A)=i). Lemma 10.1 gives a_0/4\le a_2\le4a_0. In particular, a_1+a_2\ge1/5 and a_0+a_1\ge1/5.

There are three nonzero characters of \mathbb F_2^2. A one-dimensional homology space is detected by two of them, whereas a two-dimensional space is detected by all three. Therefore

It follows that at t=0 at least one character has probability bounded below by an absolute positive constant, and at least one has probability bounded away from one. The two characters need not be the same.

Apply Lemma 10.3 to each \mathcal H_\chi along \ell_J=Qr_J. On (0,1) it states

If t_*>0, integrate from 0 to t_* and use the upper bound in (10.7). Every character probability at zero would tend to zero. If t_*<0, integrate from t_* to zero and use the lower bound in (10.7); every character probability at zero would tend to one. Both conclusions contradict the preceding absolute bounds. Hence t_*=0.□

Proof. Put m=\lfloor n^{1/2}\rfloor and, for z\in\Lambda_{2n}, define

Use the wall convention of Proposition 9.3; shifting the wall one lattice step does not affect its uniform estimate. Let \mathcal G_n=\{q_z\le n^{-1/50}\text{ for every }z\in\Lambda_{2n}\}. Markov’s inequality, the 400th moment in Proposition 9.3, and a union bound give

All boxes use the same global environment; independence of these box events is unnecessary. On \mathcal G_n, the same inequalities hold throughout t\in[t',0] by monotonicity in t.

Fix such an environment, and consider

Choose k uniformly from \{1,\ldots,n\}, independently of the FK configuration. Explore all open components in \Lambda_n meeting its square shell \partial\Lambda_k, revealing all edges of the induced graph on \Lambda_n incident to the vertices reached. Retain their actual connectivity; the shell vertices are not identified with one another. Every path witnessing \mathcal A_n meets the chosen shell, and its whole open component is explored. Thus this algorithm determines \mathcal A_n.

For an edge e=xy within \Lambda_n, the chance that the chosen shell lies within distance m+2 of x or y is at most Cm/n. Otherwise, querying e requires that one endpoint is already joined to the shell by a revealed open path, and hence has an open arm of radius m. The endpoint box is contained in \Lambda_{4n}. Conditional on all edges outside that box, the domain Markov property gives an induced boundary partition, dominated by full wiring. Its conditional arm probability is consequently at most q_x or q_y, uniformly for t\in[t',0]. The revealment \delta_e(t) therefore satisfies

Edges outside \Lambda_n are never queried.

The monotone-measure OSSS inequality [11, Theorem 1.1], averaged over the independent choice of shell, gives

The shell construction is the exploration underlying [11, Lemma 3.2]; here its revealment is bounded using the quenched local arms on \mathcal G_n. Differentiating the FK weights as in (10.11) and using their positive lower derivative bound yields

The event that all four edges incident to zero are closed has probability at least \eta^4, by sequential conditioning and finite energy. Hence f_\omega(0)\le1-\eta^4. Integration of the preceding logit inequality gives

After averaging, (10.16) contributes at most Cn^{-6}. Finally, the restriction of the infinite wired law to this cylinder event is dominated by the wired law in \Lambda_{4n}, by domain comparison. This proves (10.15).□

Proof. At zero, Proposition 9.3 implies \E\phi^{\mathrm w}_{\Lambda_n,0,\omega} (0\longleftrightarrow\partial\Lambda_n)\to0 by Hölder’s inequality. Equations (10.2) and monotonicity give zero magnetization for t\le0.

Fix s>0 small enough that -s lies in the parameter window. Planar free/wired duality sends the infinite free FK law at (s,D,\xi) to the infinite wired dual law at (-s,D,-\xi). This identity follows first from the finite planar bond expansion and then by taking free and wired limits; see also the boundary conventions in [6, Section 2.5]. The dual environment still has the same independent binary law. Failure of a primal crossing of a fixed-aspect rectangle of diameter comparable to n entails a dual crossing in the transverse direction, and hence a dual arm of length at least cn from a vertex in that rectangle. There are O(n^2) possible vertices. Lemma 10.5, translated to each such vertex and applied at -s, therefore bounds the averaged failure probability by Cn^{-4}.

For each n=2^kn_0, use the four rectangular sides of \Lambda_{2n}\setminus\Lambda_n, each of width n and length 4n. Their longitudinal crossings intersect in the corner squares and contain an open circuit surrounding \Lambda_n. Require also a left–right crossing of [n/2,5n]\times[-n/4,n/4]. Its initial point lies inside \Lambda_n and its endpoint lies outside \Lambda_{4n}, so it must intersect both this circuit and the circuit at scale 2n. Rounding rectangle endpoints by a lattice step is harmless; one may choose n_0 divisible by four. All aspect ratios are fixed and there are five requested crossings per scale. The probability that any of them fails is at most

which is smaller than one for large n_0. On the complementary event the joined circuits form an infinite open connected set. Thus the joint law obtained by first sampling the environment and then its infinite free FK configuration has positive probability of an infinite cluster. This joint law is translation stationary. If the origin had zero probability of belonging to an infinite cluster, every vertex would have probability zero, and countable subadditivity would exclude an infinite cluster anywhere. Hence the origin has positive connection probability. Wired domination and (10.2) give positive averaged plus magnetization at s. Monotonicity extends this conclusion to every t>0.□

Proof of Theorem 1.1. Fix \eps\in(0,\eps_0) and use D=D_*(\eps)>0. By (10.19) and Proposition 10.4, the tuned free infinite-volume correlation of Proposition 9.2 is C_{\eps,\omega}(r) at the physical critical inverse temperature. That proposition gives, at its stopping scale J=\log r/\log L+O(1),

The factors r_J^{4d} cancel in the ratio of the second moment to the square of the first. Thus

which proves the asserted limit along all integer r\to\infty. The disorder remains fixed throughout this final limit. The thermodynamic limits and the separate Gibbs normalizations required in the definition of R_\eps are those already established in Proposition 9.2.□