Proof of Lemma 2 and Proposition 3. 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 \[v_k=w_k^bP_{w_k}\mathcal O_b, \qquad S_k=B^bP_{w_{k+1}}.\] The lines \(\mathbb Rv_k\) are uniformly nonzero and bounded by (14), and \(S_kv_k=v_{k+1}\) by (15). 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 (10) and (14) give \(|\alpha|\le C\left\lVert F\right\rVert_\infty\). Applying (11) to this difference proves the quotient bound \[ \left\lVert S_k:\,X_k/\mathbb Rv_k\longrightarrow X_{k+1}/\mathbb Rv_{k+1}\right\rVert \le CB^{b-p}. \tag{20}\] 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 (10) and (11) at the fixed ratio \(B\) also bounds \(S_k\) uniformly.

Choose uniformly bounded projections onto \(\mathbb Rv_k\), 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, \[S_k= \begin{pmatrix}1&\beta_k\\0&D_k\end{pmatrix},\qquad \left\lVert\beta_k\right\rVert\le C,\quad \left\lVert D_k\right\rVert\le CB^{b-p}=:q.\] Enlarge \(B\) until \(q\le B^{b-p+\nu}<1\). Writing \(S_{k-1}\cdots S_0F=a_kv_k+u_k\) gives \[\left\lVert u_k\right\rVert\le Cq^k\left\lVert F\right\rVert_\infty,\qquad a_{k+1}-a_k=\beta_ku_k.\] Consequently \(a_k\) converges to a linear functional \(a_\infty(F)\) and \[ |a_\infty(F)|\le C\left\lVert F\right\rVert_\infty, \quad \left\lVert S_{k-1}\cdots S_0F-a_\infty(F)v_k\right\rVert_\infty \le Cq^k\left\lVert F\right\rVert_\infty. \tag{21}\] 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 (21) by \(B^{kb}\) gives (18) at the radii \(w_k\) and gives (16)’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 (21) 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 (13). This proves the limit in (16). 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 (19) at the center are (14). 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 (14) uses (11) with \(p=2\) in this case: at a fixed large ratio, \[(Bw)\left\lVert P_{Bw}F\right\rVert_\infty \le C+CB^{-1}w\left\lVert P_wF\right\rVert_\infty.\] 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 (23). 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 \[E_0=P_{\rho n,0}e(0),\quad E_z=P_{\rho n,z}e(z),\quad S_0=P_{\rho n,0}\sigma_0,\] and \[\Phi_n=n^2(E_zE_0-\langle E_zE_0\rangle_0),\qquad \Psi_n=n^{1+d}E_zS_0.\] 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. Proposition 3 gives \(\left\lVert\Phi_n\right\rVert_\infty+\left\lVert\Psi_n\right\rVert_\infty\le C\), with both supports in \(B_{Cn}(0)\). Their scale-normalized charges are \[ \lambda_n=\frac{\ell(\Phi_n)}n=n\,\ell(e(z)e(0)),\qquad \mu_n=\frac{\ell_s(\Psi_n)}{n^d}=n\,\ell_s(e(z)\sigma_0), \tag{26}\] 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 (18). Calibration of the subtracted line yields \[ \limsup_{n\to\infty} \left|\chi_n-\kappa_b^{-1}B^b d_{Bn}(\Theta_n)\right| \le CB^{b-p+\nu}. \tag{27}\] 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 \[a^2\bigl(\langle\varepsilon(\zeta)\varepsilon(0)\rangle^+ _{\mathrm{disk}(4B)}-1\bigr).\] Rescale this disk by \(B\). The first expansion in (22) gives \[\langle\varepsilon(\zeta)\varepsilon(0)\rangle^+ _{\mathrm{disk}(4B)} =B^{-2}\langle\varepsilon(\zeta/B)\varepsilon(0)\rangle^+ _{\mathrm{disk}(4)} =1+O(B^{-2}).\] Thus \(B d_{Bn}(\Phi_n)\) tends to zero in the iterated limit. The error in (27) also tends to zero, since the general even exponent \(p=2\) is already larger than \(b=1\). This proves (24).

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 (22) gives \[\langle\varepsilon(\zeta)\sigma(0)\rangle^+ _{\mathrm{disk}(4B)} =B^{-1-d}\bigl(\tfrac12 Bm_\sigma+O(1)\bigr).\] Since \(\kappa_d=a_\sigma m_\sigma\), the odd instance of (27) gives \(\mu_n\to a/2\) in this sign normalization. Squaring gives (25). 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]; the comparison across physical boundaries has the same common-pin convention. Apply (28) 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 \(|fW|\le\left\lVert f\right\rVert_\infty W\) 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]. Hence \(PF(\tau)=\langle FH_\tau\rangle_0\).

Apply the outer-to-inner version of finite soft-weight replacement [19] to \(H_\tau/C\). For any \(\eta_0>0\), it supplies finitely many positive normalized weights \(U_1,\ldots,U_m\) and vectors \(c(\tau)\in\mathbb R^m\) with \(\sum_i|c_i(\tau)|\le C\), such that, simultaneously for all \(F\), \[\left|PF(\tau)-\sum_{i=1}^m c_i(\tau) \langle FU_i\rangle_0\right| \le C\eta_0\left\lVert F\right\rVert_\infty.\] 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 \(\{c:\left\lVert c\right\rVert_{\ell^1}\le C\}\) 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 \[|PF(\tau)-PF(\tau_k)| \le (2C\eta_0+\eta/2)\left\lVert F\right\rVert_\infty.\] 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 (31). Finally, Minkowski’s inequality and \(\max_k|X_k|^q\le\sum_k|X_k|^q\) prove (32). ◻

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 (47). 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 \[ \int e^{B_{\rm ret}+\sum_{\mathcal C}B_{\mathcal C}}F\,\,\mathrm d\mu_0 =\int e^{B'}\left( \prod_{\mathcal C}P_{Q_{\mathcal C}}^{B_{\mathcal C}} \right)F\,\,\mathrm d\mu_0. \tag{50}\] 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 \(\int e^{H(\zeta)}\,\mathrm d\mu_0=\int e^{B'}\mathcal F(\zeta)\,\mathrm d\mu_0\). By construction \(B'+\log\mathcal F=s'+H'\). Lemma 9 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 12, permits truncation of an infinite slot list. This proves (48). Its denominator at zero sources is strictly positive, so division gives (49). 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 \[\sum_{F\subset E(Q):\,(V(Q),F)\text{ connected}}(-1)^{|F|}.\] 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 (51), 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 \[ C L^2(1+n_1)e^{a_{\rm an}n_2} \tag{52}\] 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 7, an output weight \(v_{\rm out}\) costs \[e^{v_{\rm out}n'}\le e^{Cv_{\rm out}k} \prod_{a=1}^ke^{Cv_{\rm out}n_a/L}.\] 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 (52) are paid by \[ (1+n)^q e^{-bn}\le q!\,C_b^{q+1},\qquad n\ge1. \tag{53}\] 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 (36), 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 (34) 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 9 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 9. For the second, the weight allocation in (47) 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 (54). 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 10.

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 (50) 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 8.

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 (42) 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 \[ K'\le C_L(1+K). \tag{55}\] 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 \(D\le4\), (59) 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 (58) on them, and bound every other good occurrence by its deterministic threshold or tail bound. The majorant is bounded by \[C s^5\sum_{k\ge5} k^C C_L^k\delta^{k-5}=O(s^5),\] 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 \[\mathop{\mathrm{cum}}(Z_1,\ldots,Z_k) =\sum_{\pi\in\mathcal P([k])}(-1)^{|\pi|-1}(|\pi|-1)! \bigotimes_{A\in\pi}\mathbb E\bigotimes_{a\in A}Z_a\] 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 \[D=\frac{P_{Cw}e^B}{\langle e^B\rangle_0},\qquad N=\frac{P_{Cw}(e^BF)}{\langle e^B\rangle_0}.\] Lemma 5 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 \(|N|\le C\left\lVert F\right\rVert_\infty\), uniformly in \(B\). Both functions are even and \(\langle D\rangle_0=1\). The even part of Proposition 3, followed by the tower property, gives \[P_RD=1+d_R,\qquad P_RN=\langle N\rangle_0+n_R, \qquad \left\lVert d_R\right\rVert_\infty\le Cw/R, \quad \left\lVert n_R\right\rVert_\infty\le C(w/R)\left\lVert F\right\rVert_\infty.\] Since \(P_RD\ge c\), subtracting \(\langle N\rangle_0\) from their ratio proves (61). The estimate remains uniform upon replacing \(B\) by \(tB\), \(0\le t\le1\). Integrate \[\frac{\,\mathrm d}{\,\mathrm dt}\log P_Re^{tB} =\frac{P_R(Be^{tB})}{P_Re^{tB}}\] in the quotient by constants and apply (61) with \(F=B\). This proves (62) without introducing \(e^{\left\lVert B\right\rVert}\) for the bound linear in \(\left\lVert B\right\rVert_\infty\). 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 \(C\left\lVert d_R\right\rVert_\infty\le Cw/R\). 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 \[\left\lVert X_i\mathbf 1_{\mathcal B_c}\right\rVert_p =\left\lVert X_i\right\rVert_p\mathbb P(\mathcal B_c)^{1/p} \le C B^2\delta^{-1}s^2e^{-vn_i}.\] 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 \(p\)th 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 (58) bounds its expectation by \((Bs)^p\). After taking the \(p\)th root and including attenuation, this part costs \[ C L^{-1+2/p}M^{C_p} Bs. \tag{68}\]

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 \[\left\lVert X_i\right\rVert_k\le\left\lVert X_i\right\rVert_2^{\theta_k} \left\lVert X_i\right\rVert_p^{1-\theta_k}, \qquad \theta_k=\frac{2(p-k)}{k(p-2)};\] for \(k=1\) use \(\left\lVert X_i\right\rVert_1\le\left\lVert X_i\right\rVert_2\). The product of all bunch estimates, after its \(p\)th root, is therefore \[s B^{1-\theta}B_2^\theta \exp\{-v\textstyle\sum n_i/p\},\qquad \theta\ge1/p.\] 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 \(\varepsilon>0\), \[C L^C B^{1-\theta}B_2^\theta \le\varepsilon B+C_{p,\varepsilon}L^{C_p}B_2.\] 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 (68) and (64) 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 (65).

For the second inequality, first use (60). 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 \[\left\lVert\left\lVert A\right\rVert_\infty\right\rVert_2\le C_L Bs.\] The finite dictionary of Lemma 6 supplies \(N=N(\eta,L)\) actual contour assignments \(\sigma_1,\ldots,\sigma_N\) such that \[\left\lVert\pi P_{HR}A\right\rVert_\infty \le \max_{\nu\le N}|\pi P_{HR}A(\sigma_\nu)| +C\eta\left\lVert A\right\rVert_\infty.\] This is a deterministic inequality and thus applies to the random \(A\). Replace each tested value of \(Th\) by that of \(h'\) using (60). The \(L^2\) norm of the maximum of \(N\) values is at most the square root of the sum of their second moments. By (67), 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 \(\varepsilon_*/2\); this is possible against the polynomial \(C_L^*\). Then choose \(\eta\) with \(C_L\eta<\varepsilon_*/2\) and increase \(r_0\) to the dictionary threshold. This proves (66). Notice the order of the argument: (67) 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 \(\,\mathrm d\mu_i=Y_i^q\,\mathrm d\mathbb P\). Its mass is at most \(U^qe^{-vq n_i}\), and (70) bounds the \(p\)th moment of every \(X_m\) in this measure. Consequently, for nonnegative integers \(a_m\) of total \(D\le p\), \[ \int\prod_m X_m^{a_m}\,\,\mathrm d\mu_i \le U^q(B_s s)^D e^{-vq n_i-v\sum_m a_mn_m}. \tag{73}\] 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 (59).

For \(q\) possibly distinct source slots, Hölder first gives \[\mathbb E\left[\prod_{\ell=1}^qY_{i_\ell}\prod_mX_m^{a_m}\right] \le\prod_{\ell=1}^q \left(\int\prod_mX_m^{a_m}\,\,\mathrm d\mu_{i_\ell}\right)^{1/q}.\] Thus (73) 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 13 proves the asserted \(O(U^qs^3)\) bound. In \(L^q\) retain one background factor and use (73) with degree \(q\). A bad flag costs \(O(Us^{p/q})\). This proves (71), 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 \[ Y'_a\le\sum_i K_{ai}Y_i,\qquad K_{ai}\ge0. \tag{74}\] Here the coefficients are deterministic majorants, not the realized coefficients of the map. They include the factor \(L^d\). Lemma 9, 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 \(\left\lVert\log P e^B\right\rVert\) occurs in (74). 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 \(\left\lVert\left\lVert h'_b\right\rVert_\infty\right\rVert_{L^p(\mu_i)}\) using the deterministic decomposition preceding Proposition 15. If \(n_i>M\), crude Minkowski and (70) give \[ \left\lVert\left\lVert h'_b\right\rVert_\infty\right\rVert_{L^p(\mu_i)} \le C_L U^{q/p}B_s s\,e^{-vq n_i/p-vn_b}. \tag{75}\] 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 (70). There are at most \(CM^2\) anchors. Animal lengths and shapes cost a constant after their weight sums. The attenuated part of (63), estimated by Minkowski, therefore costs \[C M^C L^{-1} B_s s\, U^{q/p}e^{-vq n_i/p-vn_b}.\] For its witness part the witness carrier is disjoint from the union of the source and background input carriers. Exact independence gives \[\big(\mathbb EY_i^q X_m^p\mathbf 1_{\mathcal B_c}\big)^{1/p} =\big(\mathbb EY_i^qX_m^p\big)^{1/p}\mathbb P(\mathcal B_c)^{1/p},\] 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 \[\big(\mathbb EY_i^qX_m^p\big)^{1/p} = (\mathbb EY_i^q)^{1/p}(\mathbb EX_m^p)^{1/p} \le U^{q/p} Bs\,e^{-vq n_i/p-vn_m}.\] Even a crude sum over these anchors has only a polynomial cost in \(L\). Combining these estimates gives, for short \(i\), \[ \left\lVert\left\lVert h'_b\right\rVert_\infty\right\rVert_{L^p(\mu_i)} \le U^{q/p}e^{-vq n_i/p-vn_b} \{a_LB_s s+C_LBs+o(s)\}, \quad a_L=C M^C/L+\lambda_L+C_L\delta. \tag{76}\]

We now check the powers when summing sources. Put \(F_i=(\mathbb EY_i^q\left\lVert h'_b\right\rVert_\infty^p)^{1/p}\). Minkowski in \(L^q\), applied to (74), yields \[ (\mathbb E(Y'_a)^q\left\lVert h'_b\right\rVert_\infty^p)^{1/p} \le \left(\sum_i K_{ai}F_i^{p/q}\right)^{q/p}. \tag{77}\] This is an \(\ell^{p/q}\) norm with weights \(K_{ai}\). Apply its triangle inequality to the three terms in braces in (76). 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 \[(CL^d)^{q/p}\big(CM^C/L+\lambda_L+C_L\delta\big),\] whereas the far-background term is a constant \(C_{L,B}s\) independent of \(B_s\). For the long-source part use (75) 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 (77), whose use of the \(\ell^{p/q}\) triangle inequality is homogeneous of degree one in \(F_i\). This proves (72). ◻

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 \[ C M^C L^{-1}\|K\|. \tag{83}\] 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: \[F\longmapsto q_r(F)\,rP_{Hr,c}e(z_c),\qquad G\longmapsto q_r(G)\,rP_{Hr,c}e(z_c).\] 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 (79), 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 (83). 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 (82). ◻

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 7 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 \[ \|H_{i,1}^{(j)}\|\le Cr_j^{-1},\qquad \|H_{i,2}^{(j)}\|\le Cr_j^{-2}. \tag{89}\] 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, \[ P H_{i,2}=-\tfrac12 P(H_{i,1}^2). \tag{90}\] 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 (90) 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 \([\lambda^2]\mathbb Eh_j\). There are at most \(C r_j^2\) microscopic sites which can contribute to a fixed output anchor. Summing the second estimate in (89) 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 13. The strict contractions of Proposition 17 therefore bound \(Y_j^*\) and \(W_j^*\). In (86), 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 18. The degree-two coefficient of \(\mathop{\mathrm{cum}}(u_{j,y}^{\otimes q},h_j,h_j)\) 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 8 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 (94). 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 (79) 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 \[C_{ik}=\langle e(i)e(k)\rangle_0,\qquad A_{ik}=\ell(e(i)e(k)).\] Their invariant energy charge is exactly \[ \mathcal E= \frac{t_i+t_k+A_{ik}t_it_k}{1+C_{ik}t_it_k}. \tag{96}\] 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, \[ \mathcal E=t_i+t_k+A_{ik}t_it_k -C_{ik}(t_i^2t_k+t_it_k^2)+O(t^4). \tag{97}\] 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 \[ A_{ik}^2-4C_{ik}. \tag{98}\] 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 \[\|A_i\|\le C/r,\quad \|B_i\|\le C/r^2,\quad \|D_{ik}\|\le C/(hr).\] All columns are plane centered. In particular the denominator in (91) 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 \[A_iD_{ik},\qquad B_iA_k,\qquad \tfrac12 A_i^2A_k.\] 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 \[ O\bigl(r^{-2}+(hr)^{-1}\bigr). \tag{99}\] Since \(|A_{ik}|\le C/h\) by Lemma 4, 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 (98) 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 \[ \sum_{1\le |z|\le C_Lr} \left(\frac1{r(1+|z|)}+\frac1{r^2}\right)\le C_L. \tag{100}\] 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 \[ \beta_J=\frac12\sum_{0<|z|\le C_Lr_J} \left\{\ell(e(z)e(0))^2-4\langle e(z)e(0)\rangle_0\right\}+O(1). \tag{101}\] 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 (101).

Lemma 4 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 \[\sum_{0<|z|\le R}|z|^{-2}=2\pi\log R+O(1).\] 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 (101) yields \(\beta_J=-4\pi a^2\log r_J+o(\log r_J)\). Telescoping (87) subtracts only the bounded initial coefficient \(\beta_0\), proving (94).

Proof. For \(q=1\) the invariant coefficient of \(t_i^2\) in (102) is exactly zero, rather than merely smaller than its logarithmic scale. Thus (104) gives \(|\alpha_{J,1}(y)|\le C\). The initial coefficient is uniformly bounded, and (88) telescopes. This stronger conclusion will control the first marked moment; a zero Cesàro mean alone would not suffice. ◻

Proof. Put \(a_j=A_{j,q}^{1/q}=\left\lVert\alpha_j\right\rVert_{L^q}\) and \(k_j=\left\lVert u_j-\alpha_jw_{j,y}\right\rVert_{v,q}\). The source-line projection is bounded in the weighted norm. Its complement contracts under the scaled linear transport, and Proposition 16 bounds the nonlinear difference by \(CUs_j\) when \(U=C_*a_j\). Since linear transport preserves normalized spin charge, \[ |a_{j+1}-a_j|\le C C_*a_js_j, \qquad k_{j+1}\le\kappa k_j+C C_*a_js_j, \tag{127}\] 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 (124), 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 (127) then reproduces the kernel allowance relative to \(a_{j+1}\), after decreasing \(D\). The mixed output estimate in Proposition 16 is \[U^{q/p}\bigl(B_s s_j/8+C_{L,B}s_j+o(s_j)\bigr).\] 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 (124)–(125).

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 (127) 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 (126). ◻

Proof. The mixed estimates give initially \(\left\lVert C_{k,j}\right\rVert\le CA_js_j^k\) 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 18, \[ C_{2,j+1}=\mathcal L_{2,j}C_{2,j}+O(A_js_j^3), \qquad \left\lVert\mathcal L_{2,j}\right\rVert\le\kappa. \tag{133}\] The initial grouped source tensor is deterministic, so \(C_{2,0}=0\). Contraction now proves (129). More explicitly, for each of the finitely many exponents \(b\) below, \[ \sum_{i<j}\kappa^{j-1-i}A_i s_i^b\le C_b A_j s_j^b. \tag{134}\] Indeed (127) 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, \[\mathbb E[U\otimes h\otimes h] =C_2+S\otimes K+S\otimes m\otimes m +C_1\otimes m+m\otimes C_1,\] 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 \[ C_{1,j+1}=\mathcal L_{1,j}C_{1,j} +\mathcal H_j(S_j,K_j)+O(A_js_j^3), \qquad\left\lVert\mathcal L_{1,j}\right\rVert\le\kappa, \tag{135}\] 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 (127) to \[ A_{j+1}/A_j=1+O(G_j). \tag{136}\] 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, (124), and (126) bound its expectation by \(CA_js_j\), proving (131).

The formal profile satisfies \[Z_{j+1,q,y}^*=\mathcal L_{1,j}Z_{j,q,y}^* +\mathcal H_j(w_{j,y}^{\otimes q},V_j).\] Subtract this equation multiplied by \(A_{j+1}g_{j+1}\) from (135). The replacements of \(S_j\) and \(K_j\) by their line parts cost \(O(A_jG_j(s_j+\rho^j))\), by (131) and (108). The change of scalar factors costs \(O(A_jG_j^2)\), by (136) and \(g_{j+1}-g_j=O(G_j^2)\). Contraction and (134) prove (130). Finally insert the three profile comparisons and (109), 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 (132). Scale zero has bounded coefficients and errors of order \(A_0G_0\), which the geometric term allows. ◻

Proof. The error in (132) has summable absolute values, because \[\sum_{j\ge0}G_j(s_j+\rho^j)<\infty,\qquad \sum_{j\ge0}g_j^2<\infty.\] All factors are positive and uniformly close to one for small \(D\). Taking logarithms therefore gives \[ \log A_{J,q}=\sum_{j<J}c_{j,q}(y)g_j+O_D(1). \tag{139}\] The error is uniform in \(y\). Let \(\mu_q=\binom q2 B_0/8\). Proposition 21 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 (137) from (139).

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 \[ \sum_{j<J}c_jG_j \le(\mu+\eta)\sum_{j<J}G_j+C_\eta\bar g \le\left(\frac18+\frac\eta{B_0}\right) \log(1+B_0J\bar g)+C_\eta. \tag{140}\] By (120), replacing \(G_j\) by \(g_j\) costs at most \(\left\lVert c\right\rVert_{\infty}\sup_j|g_j/G_j-1|\sum_{j<J}G_j\). 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 (138). ◻

Proof. Set \(U_j^*=C(1+B_0j\bar g)^{1/8}\). Equations (124), (125), and (138), 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 \(Y=\left\lVert u_{j,i}\right\rVert\) and \(H=\prod_m\left\lVert h_{j,m}\right\rVert^{a_m}\), with \(b=\sum_m a_m\le p\). Cauchy–Schwarz gives \[\mathbb E[YH]\le\bigl(\mathbb E[Y^2H]\bigr)^{1/2} \bigl(\mathbb E[H]\bigr)^{1/2} \le C U_j^*s_j^b e^{-vn_i-v\sum_m a_mn_m}.\] 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 16. Apply the same connected-cumulant recursions to the single grouped source \(u_j\). They yield \[C_{2,j}=O(U_j^*s_j^3),\quad C_{1,j}=O(U_j^*G_j),\quad S_j=A_{j,1}w_{j,y}+O(U_j^*s_j).\] 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 (136). 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 \[C_{1,j}=A_{j,1}g_jZ_{j,1,y}^* +O\bigl(U_j^*G_j(s_j+\rho^j)\bigr).\] Consequently, with \(a_j=A_{j,1}\), \[ a_{j+1}=(1+c_{j,1}(y)g_j)a_j+e_j, \qquad |e_j|\le CU_j^*G_j(s_j+\rho^j). \tag{142}\]

Corollary 22 gives a fixed bound on every partial sum of \(c_{j,1}(y)\), uniformly in \(y\). Abel summation and Corollary 25 therefore imply \[\sup_J\left|\sum_{j<J}c_{j,1}(y)g_j\right| \le C\left(g_0+\sum_{j\ge0}|g_{j+1}-g_j|\right) \le C\bar g.\] 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 \[\begin{align*} \sum_{j\ge0}|e_j| &\le C\bar g^{3/2}\sum_{j\ge0} (1+B_0j\bar g)^{-11/8} +C\bar g\sum_{j\ge0}\rho^j (1+B_0j\bar g)^{-7/8}\\ &\le C\sqrt{\bar g}. \end{align*}\] The exponent \(11/8>1\) is what makes the first sum integrable on the scale \(j\asymp\bar g^{-1}\). Variation of constants in (142), with \(a_0=1\), gives \[a_J=P_J\left(1+\sum_{j<J}e_j/P_{j+1}\right).\] 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 9, 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 (148); 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 \(X_a=\left\lVert u_{j,a,i_a}\right\rVert\) and \(H=\left\lVert h_m\right\rVert\). The mixed bound of Proposition 16, for source power four, and Hölder interpolation with respect to the measure \(X_a^4\,\,\mathrm d\mathbb P\) give \[\mathbb E(X_a^4H^2) \le \bigl(\mathbb EX_a^4H^p\bigr)^{2/p} \bigl(\mathbb EX_a^4\bigr)^{1-2/p} \le C U_a^4s_j^2, \qquad U_a=C A_{j,4}(y_a)^{1/4}.\] It follows that \[\left\lVert X_1X_2H\right\rVert_{L^2}^2 \le\bigl(\mathbb EX_1^4H^2\bigr)^{1/2} \bigl(\mathbb EX_2^4H^2\bigr)^{1/2} \le C U_1^2U_2^2s_j^2.\] 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 \(B_m=\{\left\lVert h_m\right\rVert>\delta\}\); a random threshold in \([\delta,2\delta]\) can only reduce this event. The same mixed bound gives \[\mathbb E(X_a^4\mathbf 1_{B_m}) \le\delta^{-p}\mathbb E(X_a^4\left\lVert h_m\right\rVert^p) \le C U_a^4s_j^p, \qquad \left\lVert X_1X_2\mathbf 1_{B_m}\right\rVert_{L^2} \le C U_1U_2s_j^{p/2}.\] 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 9 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 \[ C A_{j,4}(y_1)^{1/4}A_{j,4}(y_2)^{1/4}s_j \exp(-c\ell/r_{j+1}). \tag{152}\] 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 (152) 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 \[\sum_{m\ge0}K^m\exp(-cF L^m)<\infty.\] 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 (151), 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 \[U_{J,1}U_{J,2}-Z_Jw_{J,y_1}w_{J,y_2} =R_1U_{J,2}+ \widetilde\alpha_{J,y_1}w_{J,y_1}R_2\] and (148) 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 (157), (150), and (158) give the first assertion, including a positive lower bound. For the second, the triangle inequality and its reverse in \(L^2\) give \[\left|\left\lVert r_J^{2d}C_{*,\omega}(y_1,y_2)\right\rVert_{L^2} -q_J\left\lVert Z_J\right\rVert_{L^2}\right| \le O(J^{1/8-\eta+o(1)}).\] 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 \[ r_{J_n}\le n/(\log n)^2. \tag{161}\] 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 28, 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 \[ C K^{J_n}\exp(-c'n/r_{J_n}) \le \exp\bigl(C\log n-c''(\log n)^2\bigr). \tag{162}\] 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 (162) 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 \(M<\infty\), \[\left\lVert\langle\sigma_y\rangle^+_{n,*,\omega}\right\rVert_{L^{400}} \le r_{J_n}^{-d}\left(C(1+J_n)^M +C K^{J_n}e^{-c'n/r_{J_n}}\right) =n^{-d+o(1)}.\] Raising this inequality to the power \(400\) proves (160). 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 \[ |A|-|V|+k(A)=k(A^*)-1+r(A). \tag{166}\] 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 (165).

The ratio of the two unnormalized weights is \[\begin{align*} \frac{2^{k(A)}\prod_{e\in A}v_e} {2^{k(A^*)}\prod_{e\notin A}(2/v_e)} &=\Big(\prod_{e\in E}v_e\Big) 2^{k(A)-k(A^*)-|E|+|A|}\\ &=\Big(\prod_{e\in E}v_e\Big) 2^{|V|-|E|-1}\,2^{r(A)}, \end{align*}\] where the last equality uses (166). 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 12 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 7. 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 9 and Corollary 10. Summing the possible nonchart footprints, including the neighborhoods used by the decisions, bounds the uncancelled scalar increment at this step by \[C_L\delta m^2e^{-cm}.\] 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 \[ \frac{Z^\chi(K)}{Z^0(K)} =\frac{Z^\chi(K^{\mathrm{ref}})}{Z^0(K^{\mathrm{ref}})} e^{c_J^\chi-c_J^0} \frac{\langle e^{H_J^\chi}\rangle_{J,\mathrm{ref},\chi}} {\langle e^{H_J^0}\rangle_{J,\mathrm{ref},0}}. \tag{170}\] 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 \(|\log\langle e^H\rangle|\le\left\lVert H\right\rVert_\infty\) now give \[ \left|\log\frac{Z^\chi(K)/Z^0(K)} {Z^\chi(K^{\mathrm{ref}})/Z^0(K^{\mathrm{ref}})} \right| \le P_J+C\delta Q^2+C_L\delta Q^2e^{-c'Q}, \qquad P_J\xrightarrow{\mathbb P}0. \tag{171}\] 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 (167) and (168), 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 (169). ◻

Proof. For a fixed environment abbreviate its FK law to \(\phi\), and set \(X_e=\mathbf 1_{\{e\in A\}}\) and \(c_e(\omega)=\mathop{\mathrm{Cov}}_{\phi_{t,\omega}}(\mathbf 1_{\mathcal A},X_e)\ge0\). Differentiating the finite sum in (163) gives \[ f_\ell'(t)=\sum_e\mathbb E[\lambda_e(t,\omega)c_e(\omega)], \qquad \lambda_e=\frac{\,\mathrm d}{\,\mathrm dt}\log v_e. \tag{173}\] Since \(\,\mathrm dK_e/\,\mathrm dt=\tfrac12\tanh(2K_e)\), 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 \[X_{e_i}=\mathbf 1_{\{U_i\ge1-p_i\}},\qquad p_i=\phi_{t,\omega}(X_{e_i}=1\mid X_{e_1},\ldots,X_{e_{i-1}}).\] 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 \[p(P)=\phi(X_{e_i}=1\mid P),\qquad a_b(P)=\phi(\mathcal A\mid P,X_{e_i}=b),\quad b=0,1.\] 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 \[\begin{align*} I_{U_i} &=\mathbb E\,\phi\bigl(a_1(P)-a_0(P)\bigr)\\ &\le\frac{1}{\eta(1-\eta)} \mathbb E\,\phi\bigl(\mathop{\mathrm{Cov}}_\phi(\mathbf 1_{\mathcal A},X_{e_i}\mid P)\bigr) \le C\mathbb Ec_{e_i}(\omega). \tag{174}\end{align*}\] For the last inequality, total covariance gives an additional term \[\mathop{\mathrm{Cov}}_\phi\bigl(\phi(\mathcal A\mid P), \phi(X_{e_i}=1\mid P)\bigr)\ge0.\] 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 \[ \phi_+(\mathcal A)-\phi_-(\mathcal A) =\frac{(R-1)\mathop{\mathrm{Cov}}_{\phi_-}(\mathbf 1_{\mathcal A},X_e)} {1+(R-1)\phi_-(X_e)}. \tag{175}\] 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 \(\mathbb Ec_e(\omega)\), since both low- and high-bond covariances are nonnegative. Thus \(I_{V_e}\le C\mathbb Ec_e(\omega)\) as well.

Translation invariance of the averaged law and of \(\mathcal A\) makes \(\mathbb Ec_e(\omega)\) constant within each of the two edge orientations. Each orientation has \(\ell^2\) edges. Equation (173) therefore implies \[ \max_z I_z\le C\ell^{-2}f_\ell'(t). \tag{176}\] 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], gives \[\max_z I_z\ge c f_\ell(t)(1-f_\ell(t))\frac{\log N}{N}.\] Combine this with (176) to obtain (172). ◻

Proof. The dual transformation \(v_{e^*}^*=2/v_e\) is equivalent to \(\sinh(2K_e)\sinh(2K_{e^*}^*)=1\). Thus, in the coordinates (7), it sends \((t,D,\xi)\) to \((-t,D,-\xi)\) on the dual lattice. At \(t=0\) the environment law is invariant. Write \(a_i=\mathbb E\phi_{\mathbb T_\ell,0,\omega}(r(A)=i)\). Lemma 33 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 \[\begin{align*} \sum_{\chi\ne0}\mathbb E\phi_{0,\omega}(\mathcal H_\chi) &=2a_1+3a_2,\\ \sum_{\chi\ne0}\mathbb E\phi_{0,\omega}(\mathcal H_\chi^c) &=3a_0+a_1. \end{align*}\] 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 35 to each \(\mathcal H_\chi\) along \(\ell_J=Qr_J\). On \((0,1)\) it states \[\frac{\,\mathrm d}{\,\mathrm dt}\log\frac{f_{\ell_J}(t)}{1-f_{\ell_J}(t)} \ge c\log\ell_J.\] If \(t_*>0\), integrate from \(0\) to \(t_*\) and use the upper bound in (169). Every character probability at zero would tend to zero. If \(t_*<0\), integrate from \(t_*\) to zero and use the lower bound in (169); 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 \[q_z(\omega)=\phi^{\mathrm w}_{z+\Lambda_m,0,\omega} (z\longleftrightarrow z+\partial\Lambda_m).\] Use the wall convention of Proposition 32; 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 \(400\)th moment in Proposition 32, and a union bound give \[\begin{align*} \mathbb P(\mathcal G_n^c) &\le Cn^2 n^{400/50}m^{-400/8+o(1)} =n^{-15+o(1)}\le Cn^{-6}. \tag{178}\end{align*}\] 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 \[f_\omega(t)=\phi^{\mathrm w}_{\Lambda_{4n},t,\omega}(\mathcal A_n), \qquad \mathcal A_n=\{0\longleftrightarrow\partial\Lambda_n \text{ by a path in }\Lambda_n\}.\] 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 \[ \max_e\delta_e(t) \le C(n^{-1/2}+n^{-1/50})\le Cn^{-1/50}. \tag{179}\] Edges outside \(\Lambda_n\) are never queried.

The monotone-measure OSSS inequality [11], averaged over the independent choice of shell, gives \[f_\omega(t)(1-f_\omega(t)) \le\sum_e\delta_e(t) \mathop{\mathrm{Cov}}_{\phi^{\mathrm w}_{\Lambda_{4n},t,\omega}} (\mathbf 1_{\mathcal A_n},X_e).\] The shell construction is the exploration underlying [11]; here its revealment is bounded using the quenched local arms on \(\mathcal G_n\). Differentiating the FK weights as in (173) and using their positive lower derivative bound yields \[f_\omega'(t)\ge c n^{1/50}f_\omega(t)(1-f_\omega(t)).\] 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 \[f_\omega(t')\le C\exp\{-c|t'|n^{1/50}\}.\] After averaging, (178) 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 (177). ◻

Proof. At zero, Proposition 32 implies \(\mathbb E\phi^{\mathrm w}_{\Lambda_n,0,\omega} (0\longleftrightarrow\partial\Lambda_n)\to0\) by Hölder’s inequality. Equations (164) 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]. 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 37, 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 \[C\sum_{k\ge0}(2^kn_0)^{-4},\] 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 (164) give positive averaged plus magnetization at \(s\). Monotonicity extends this conclusion to every \(t>0\). ◻

Proof of Theorem 1. Fix \(\varepsilon\in(0,\varepsilon_0)\) and use \(D=D_*(\varepsilon)>0\). By (181) and Proposition 36, the tuned free infinite-volume correlation of Proposition 31 is \(C_{\varepsilon,\omega}(r)\) at the physical critical inverse temperature. That proposition gives, at its stopping scale \(J=\log r/\log L+O(1)\), \[r_J^{2d}\mathbb EC_{\varepsilon,\omega}(r)\asymp1, \qquad r_J^{4d}\mathbb EC_{\varepsilon,\omega}(r)^2=J^{1/4+o(1)}.\] The factors \(r_J^{4d}\) cancel in the ratio of the second moment to the square of the first. Thus \[\log R_\varepsilon(r)=(1/4+o(1))\log J+O(1), \qquad \frac{\log J}{\log\log r}\longrightarrow1,\] 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 \(R_\varepsilon\) are those already established in Proposition 31. ◻