Proof. An even finite dual edge set determines a unique spin configuration equal to plus far away: the sign at a primal vertex is the parity of crossings of any path from that vertex to infinity. Evenness makes this parity independent of the path. Restricting the contour to \(E_V\) gives a bijection between the compatible spin configurations on \(V\) and \(\mathcal P_V\). Across a primal bond, the spin product is \(-1\) exactly when its crossing dual edge belongs to the contour. Thus \[\exp\left(\beta\sum_{\{x,y\}:\{x,y\}\cap V\ne\varnothing} J_{\{x,y\}}\sigma_x\sigma_y\right) =\exp\left(\beta\sum_{e\in E_V}J_{e^\bullet}\right) \exp\left(-2\beta\sum_{e\in P}J_{e^\bullet}\right).\] The first factor is constant over \(P\). Every exterior-only energy is constant as well. Normalization therefore gives (5), whose formula does not involve \(P_o\). ◻

Proof. For the invariant subspace use the displayed upper-limit estimates in [22], dividing the companion’s density-normalized operator by \(d^2\). The measured coefficient is \[\ell_s(f)=\frac H g\mathop{\mathrm{Cov}}_0(f,W_{Hs}),\] where \(W\) is the fixed exterior annular test there and \(g\ne0\) is its radial degree-one coefficient. The remainder in the transfer norm is \(C d^{-3}+C/(dH^2)\), and that in coefficient compatibility is \(C/(dH^2)\). Choose \(H\) large enough for all the prescribed ratios, then the replacement accuracies, and finally \(s_*\) to absorb the uniform upper-limit errors into \(C d^{-3}\). Center the companion’s fixed interior reference test under \(\mathbb E_0\) and divide it by its measured coefficient to obtain the centered bounded right inverse. Centering changes neither that coefficient nor the transfer estimates. The radial spectral gap requires only quarter turns, so reflection averaging in this application is unnecessary.

For completeness, the same method proves (13). Before the angular averaging in the proof of the companion’s radial gap, the polynomial expansion has the form \[(F,D_{-\log r}G)_{\mathrm{rad}} =\mathbb E\overline F\,\mathbb EG +r\,\overline{c(F)}c(G)+O_{F,G}(r^2).\] Both displayed coefficients are rotation scalars. Before smearing, the second factor is \(c(\prod_i\sigma(z_i))=S(z_1,\ldots,z_{2m})A_2(z)/A_0(z)\), where \[A_q(z)=\sum_{\substack{u_i\in\{-1,1\}\\\sum_i u_i=q}} \prod_{i<j}|z_i-z_j|^{u_i u_j/2}.\] All distances in this expression are rotation invariant. Thus both vanish on the even zero-quarter-turn-average subspace. A polynomial self-pairing there is the integral of \(r^a\) against a positive spectral measure and is \(O_F(r^2)\). Positive mass in any compact subinterval of \([0,2)\) would contradict this bound as \(r\downarrow0\). Polynomial density therefore places the whole spectrum of this subspace in \([2,\infty)\), giving the uniform radial bound \(r^2\|F\|_2\|G\|_2\).

Apply bounded replacement to the observable and to the buffered boundary density used in the proof of uniform transfer. Project the first proxy with \(I-\mathcal A\); this changes its norm and the replacement error by at most a factor two. The radial bound just proved then gives \(C d^{-2}\), uniformly over the boundary assignment. The same finite-list choice of accuracies and threshold makes the lattice inequality uniform in \(s\) and in the observable. This proves the lemma without a spatial symmetry assumption on the original even observable. ◻

Proof. Start with a bounded right inverse \(v_0\) of \(\ell_{s_*}\) and propagate it by \[m_j=\ell_{s_{j+1}}(P_{s_{j+1}}v_j),\qquad v_{j+1}=m_j^{-1}P_{s_{j+1}}v_j.\] Equations (11)–(12) show by induction that \(\|v_j\|_\infty\) stays bounded and \(m_j=d^{-1}(1+O(d^{-2}))\). Indeed, choose a fixed bound larger than four times the constant in (11), then choose \(d\) so large that the \(d^{-2}\) errors preserve it.

Write \(f_j=a_jv_j+k_j\), where \(\ell_{s_j}(k_j)=0\). Transfer has the triangular form \[a_{j+1}=m_ja_j+t_jk_j,\qquad k_{j+1}=K_jk_j, \qquad \|t_j\|+\|K_j\|\leq C d^{-3}.\] The last estimate follows by subtracting \(v_{j+1}\ell_{s_{j+1}}(P_{s_{j+1}}k_j)\) from the transferred kernel vector. Put \(M_{j,n}=\prod_{h=j}^{n-1}m_h\), with \(M_{j,j}=1\), and define \[\beta_j(a v_j+k) =a+\sum_{n=j}^{\infty} \frac{t_n K_{n-1}\cdots K_jk}{M_{j,n+1}}.\] For \(n=j\) the product of \(K\)’s is the identity. The series converges in operator norm because the ratio of its successive bounds is \(O(d^{-2})\); its correction to \(a\) has norm \(O(d^{-2})\|k\|_\infty\). Shifting the series proves exact compatibility with multiplier \(m_j\).

If \(\beta_j(f)=0\), the scalar at each later time is the negative of the remaining tail and has magnitude at most \(C d^{-2}\|k_n\|_\infty\). Hence the whole vector contracts at rate \((C d^{-3})^{n-j}\). Enlarge \(d\) so that this is bounded by \(d^{-w(n-j)}\). If \(\beta_j(f)\ne0\), division by \(M_{j,n}\) shows that its scalar converges to the nonzero value \(\beta_j(f)\) while its kernel part tends to zero. It therefore enters (15). Finally, for a vector in that cone, \[\frac{\|P_{ds}f\|_\infty}{|\ell_{ds}(P_{ds}f)|} \leq \frac{C+C d^{-2}M}{1-C d^{-2}M}.\] Choose \(M\) first, larger than the limiting bound, and then \(d\) large. This proves invariance of the cone. ◻

Proof. If a transferred \(B\) belonged to the stable hyperplane, then (14) and the Markov property, tested against a distant \(B_x\), would give \(\mathbb E_0BB_x=O(|x|^{-w})\). This contradicts (10), since \(w>2\). The cone assertion follows from Lemma 4.

We next show that, for one sufficiently large fixed \(T\), any centered unit-norm cone observable \(f_s\) satisfies \[ \mathbb E_0[f_s\,\tau_{Ts e_1}f_s]\geq c_T>0, \tag{17}\] uniformly for all sufficiently large \(s\); choose \(T\) divisible by \(8\), and \(\tau_x\) translates an observable by \(x\). The value \(c_T\) can be chosen of order \(T^{-2}\).

To see this, choose nonzero nonnegative smooth bumps \(\psi_1,\psi_2\) in two disjoint regions strictly inside the unit square. Let \(Y_s\) be the quarter-turn average of \(\Phi_s(\psi_1)\Phi_s(\psi_2)\), centered under \(\mathbb E_0\). The corresponding continuum spin-smear product, averaged under quarter turns and centered, has nonzero scalar coefficient. For two unsmeared pairs \(z_1,z_2\) and \(Te_1+y_1,Te_1+y_2\), the squared four-spin formula of the companion gives \[\mathop{\mathrm{Cov}}\bigl(\sigma(z_1)\sigma(z_2), \sigma(Te_1+y_1)\sigma(Te_1+y_2)\bigr) =\frac{|z_1-z_2|^{3/4}|y_1-y_2|^{3/4}}{4T^2} +o(T^{-2}).\] Here the neutral assignments within each pair give the product term; their cross logarithms are \(O(T^{-2})\) and cancel to first order after sign summation. The two assignments with charge \(2\) and \(-2\) on the respective pairs give the displayed strictly positive coefficient after taking the positive square root. The expansion is uniform on the separated bump supports, so it survives smearing and averaging.

Take a sufficiently large fixed \(A\) and put \(V_s=\chi_A(Y_s)-\mathbb E_0\chi_A(Y_s)\), where \(\chi_A(t)=\max\{-A,\min\{A,t\}\}\). The nonzero scalar coefficient survives for all sufficiently large \(A\): the moment bounds give ordinary \(L^2\) convergence of the truncated continuum test, and the radial coefficient is continuous in that norm. Its measured lattice coefficient is then bounded away from zero once \(H\) and the lattice threshold are large.

To see why \(A\) may be fixed before \(T\), put \(\varepsilon_A=\sup_s\mathbb E_0|Y_s-V_s|\), which tends to zero by the moment bounds. Buffered comparison gives \[\|P_{2s}(Y_s-V_s)\|_\infty\leq K\varepsilon_A,\qquad \|P_{2s}Y_s\|_\infty+\|P_{2s}V_s\|_\infty\leq C.\] The last constant is independent of \(A\), since \(\mathbb E_0|V_s|\leq2\mathbb E_0|Y_s|\). Apply the overall \(C/T\) transfer bound of Lemma 3 to both factors, now conditioned on these radius-\(2s\) collars, and integrate in the two disjoint radius-\(Ts/4\) squares. Expanding the difference of the two products yields \[\bigl|\mathbb E_0[Y_s\tau_{Ts e_1}Y_s] -\mathbb E_0[V_s\tau_{Ts e_1}V_s]\bigr| \leq CT^{-2}\varepsilon_A.\] Thus a fixed large \(A\) preserves the positive \(T^{-2}\) coefficient. For all large fixed \(T\), the covariance of \(V_s\) is at least \(cT^{-2}\) once the lattice threshold is large enough.

Now put \(a_s=\ell_s(f_s)/\ell_s(V_s)\). The cone condition and the uniform coefficient bounds give \(0<c\leq|a_s|\leq C\). The observable \(f_s-a_sV_s\) has measured coefficient zero. Transfer it and the other factor into the disjoint squares of radius \(Ts/4\) about their centers. Lemma 3 bounds the first factor by \(CT^{-3}\) and the other centered factor by \(CT^{-1}\). Thus \[\mathbb E_0[f_s\,\tau_{Ts e_1}f_s] =a_s^2\mathbb E_0[V_s\,\tau_{Ts e_1}V_s]+O(T^{-4}),\] which proves (17). In this argument choose the truncation, then \(T\), then a common measurement distance for the finite list \(d,T/4,T/8\), and finally the lattice threshold. The ratio \(T/8\) is used after transferring a truncation error to its radius-\(2s\) collar. Reapply Lemma 4 with these choices; its constants are uniform and its conclusions are unchanged. For this final choice, the covariance contradiction at the start of the proof again shows that \(P_sB\) is not stable, so it eventually enters the newly constructed cone.

Apply (17) to \(f_s=P_sB/\|P_sB\|_\infty\). By simultaneous Markov integration in the two disjoint squares, \[\mathbb E_0[(P_sB)\,\tau_{Ts e_1}(P_sB)] =\mathbb E_0[B B_{Ts e_1}].\] The cone lower bound and (10) give the upper bound in (16). The inequality \(|\mathbb E_0[f_s\tau_{Ts e_1}f_s]|\leq1\) gives its lower bound. We first obtain these estimates along the geometric scales. For an intermediate radius, the tower property bounds its norm between the norms at the two adjacent scales, proving (16) for every large \(s\). ◻

Proof. We first work at the origin on the geometric scales and with quarter-turn-invariant centered functions. By Lemma 5, the vector \(E_j=P_{s_j}B\) lies eventually in the cone and has norm comparable to \(s_j^{-1}\). Since \(\beta_j=\ell_{s_j}+O(d^{-2})\) and \(d\) is large, \[c/s_j\leq|\beta_j(E_j)|\leq C/s_j.\] Define temporarily \(\widetilde Q(f)=\beta_j(f)/\beta_j(E_j)\) for a function supported in \(B_{s_j}\). Exact compatibility of both numerator and denominator makes this definition invariant under transfer to later geometric scales. This is also consistent when the same function is regarded as belonging to a later support space: the difference between it and its transfer vanishes after the next transfer, hence has zero \(\beta\) by compatibility. Its kernel is exactly the stable hyperplane, and \(|\widetilde Q(f)|\leq Cs_j\|f\|_\infty\).

Increase the fixed starting scale once so that the calibration holds on the whole grid. For a function supported in radius \(s\) above this threshold, use the next containing geometric scale \(r\), so that \(s\leq r\leq ds\). At that scale compare \(f\) with \(\widetilde Q(f)P_rB\), whose norm is at most \(C\|f\|_\infty\) by (16). Their difference is stable. Lemma 4, followed by norm contraction between adjacent scales, proves (21) with \(v=w\) and \(\widetilde Q\) in place of \(Q\). Radii \(R\) within a fixed factor of \(s\) are absorbed by increasing the constant.

For a general even function, apply this construction to its quarter-turn average and set \(\widetilde Q(f)=\widetilde Q(\mathcal A_s f)\). The complementary component contracts at rate \((Cd^{-2})^j\) by (13). Choose \(d\) large enough that this is at most \(d^{-uj}\). This proves the general remainder estimate with \(v=u\). Together with \(\|P_RB\|_\infty\leq C/R\), it proves (20).

For fixed \(f\), pair its remainder with \(B_x\) and integrate both factors in disjoint squares of radii proportional to \(|x|\). The remainder has norm \(O_f(|x|^{-u})\), and the transferred \(B_x\) has norm \(O(|x|^{-1})\). Since \(u>1\), their covariance is \(o(|x|^{-2})\). Equation (10) therefore gives \[|x|^2\mathbb E_0[fB_x]\longrightarrow\kappa\widetilde Q(f).\] This proves existence of (18) and identifies \(Q=\widetilde Q\). In particular the definition is independent of the geometric grid and its starting scale. Translation and square invariance follow from this intrinsic limit and the corresponding symmetries of \(\mu_0\); the limit is unchanged by a fixed displacement of its distant test. Equation (10) gives \(Q(b_e)=1\) for every edge. Finally the Markov property preserves the covariance with every sufficiently distant \(B_x\), so it preserves \(Q\) under transfer through any containing square. This also establishes the stated results about arbitrary centers. For the finitely many radii below the fixed threshold, transfer first to that threshold and enlarge the universal constants. Integrating independent auxiliary variables at the outset proves the asserted extension to them. ◻

Proof. Put \(r=|x-y|\) and integrate in disjoint squares of radius a small fixed fraction of \(r\). Write \(F,G\) for the transferred observables and \(U_x,U_y\) for the corresponding transfers of \(B_x,B_y\). Theorem 6 gives \[\|F\|_\infty+\|G\|_\infty\leq C_Dl^{-1},\qquad \|F-Q(f)U_x\|_\infty+\|G-Q(g)U_y\|_\infty \leq C_Dl^{-v},\] where \(v=u\), or \(v=w\) in the symmetric case. Notice that these estimates compare with the bond observable after transfer to a containing scale; the untransferred \(Q(f)B_x\) can have norm of order \(s\). It follows that \[ \bigl|\mathbb E_0fg-Q(f)Q(g)\mathbb E_0B_xB_y\bigr| \leq C_Dl^{-v-1}. \tag{24}\] Since \(v>1\), (10) proves (22). Its error is uniform: \(r=ls\geq l\), the energy remainder tends uniformly to zero outside growing disks, and \(|q_s(f)|+|q_s(g)|\leq C_D\).

Simultaneous integration in the disjoint squares also preserves the charge of their product. Indeed it preserves its correlation with every sufficiently distant energy test, so (18) applies. All transferred products fit in a square of radius \(Cr\). The charge bound, applied to the same product difference as in (24), therefore gives \[ \frac1s\bigl|Q(fg)-Q(f)Q(g)Q(B_xB_y)\bigr| \leq C_D\frac rs l^{-v-1}=C_Dl^{-v}. \tag{25}\]

It remains to prove \[ |Q(B_xB_y)|\leq Cr^{-w}. \tag{26}\] For distinct edges, Kramers–Wannier duality at the critical coupling gives \[ \mathbb E_0\prod_{i=1}^n b_{e_i} =(-1)^n\mathbb E_0^*\prod_{i=1}^n b_{e_i^*}. \tag{27}\] Here \(e_i^*\) is the crossing dual edge. To check the centering directly, differentiate the finite planar partition-function duality [18] in the distinct couplings. Its coupling-dependent prefactor is \(\prod_e(\sinh(2K_e))^{1/2}\), and \(\sinh(2K_e)\sinh(2K_e^*)=1\). At \(K_0\) the logarithmic prefactor derivative is \(\sqrt2\) and \(\mathrm dK_e^*/\mathrm dK_e=-1\); thus \(\partial_{K_e}-1/\sqrt2\) becomes \(-(\partial_{K_e^*}-1/\sqrt2)\). Distinct derivatives introduce no contact terms. Exhausting the plane and using uniqueness of its pure critical state proves (27).

Apply (27) to the three separated bond averages in the definition of \(Q(B_xB_y)\). After identifying the dual lattice with the original lattice, the distant dual average \(B'_z\) is centered and has charge one. Thus \(B'_z-B_z\) is centered and has charge zero. With \(x,y\) fixed, separately transfer the centered product \(B'_xB'_y-\mathbb E_0(B'_xB'_y)\) and this distant difference into squares of radii proportional to \(|z|\). Their norms are respectively \(O_{x,y}(|z|^{-1})\) and \(O(|z|^{-u})\), by Theorem 6. Their covariance is therefore \(O_{x,y}(|z|^{-u-1})=o(|z|^{-2})\). Replacing the distant dual average by \(B_z\) leaves the charge limit unchanged, and consequently \[Q(B_xB_y)=-Q(B'_xB'_y),\] where the primed average consists of the four dual edges surrounding the original site.

There are four identifications obtained by the translations \(h=(\pm\frac12,\pm\frac12)\). They send the dual average about \(x\) to the boundary-edge averages of the four unit plaquettes incident to \(x\). Let \(B'_{i,x}\), \(1\leq i\leq4\), denote the resulting observables, and put \[C_x=\frac14\sum_{i=1}^4B'_{i,x},\qquad D_{i,x}=B'_{i,x}-C_x.\] The observable \(C_x\) is quarter-turn invariant about \(x\), has charge one, and is supported in a fixed-radius square. Each \(D_{i,x}\) has charge zero, and their average vanishes. Hence \[\frac14\sum_iQ(B'_{i,x}B'_{i,y}) =Q(C_xC_y)+\frac14\sum_iQ(D_{i,x}D_{i,y}).\] The charge bound after separate transfers gives \[|Q(D_{i,x}D_{i,y})|\leq Cr^{1-2u},\qquad |Q(C_xC_y)-Q(B_xB_y)|\leq Cr^{-w}.\] For the first estimate both factors have zero charge and decay as \(r^{-u}\). For the second, expand the difference with one factor \(C-B\); it is quarter-turn invariant and has zero charge, while the other centered factor decays as \(r^{-1}\). The common-support charge bound supplies the factor \(Cr\) in each case. Averaging the negative duality identity now gives \(2|Q(B_xB_y)|\leq C(r^{-w}+r^{1-2u})\). Since \(2u-1>w\), this is (26).

Finally, its contribution in (25) is at most \[\frac{|Q(f)Q(g)|}{s}\,Cr^{-w} \leq C_Ds(ls)^{-w} =C_Ds^{1-w}l^{-w}\leq C_Dl^{-w}.\] This proves both charge bounds uniformly for \(s\geq1\). ◻

Proof. First integrate the auxiliary variables and put \[W(\sigma_{B_m})=\mathbb E_{\eta}e^{G(\sigma_{B_m},\eta)}, \qquad Z=\mathbb E_0W, \qquad h(\xi)=\frac{\mathbb E_0[W\mid\sigma_{\partial B_{2m}}=\xi]}{Z}.\] The function \(W\) is positive and even. The fixed-ratio buffered comparison of [22], applied to \(B_m\) and \(\partial B_{2m}\), gives a constant \(K\ge1\) such that \[K^{-1}\le h(\xi)\le K \quad\text{for every }\xi.\] Indeed, the conditional law on \(B_m\) has pointwise density between \(K^{-1}\) and \(K\) relative to its full-plane marginal: the latter is a mixture of the same conditional laws. Integrating the positive function \(W\) proves the displayed bounds, regardless of its size.

Spin-flip symmetry gives \(h(-\xi)=h(\xi)\), and the tower property gives \(\mathbb E_0h=1\). If \(t=\|G\|_\infty\), then both \(W\) and \(Z\) lie between \(e^{-t}\) and \(e^t\). Consequently \[\|h-1\|_\infty \le \min\{K+1,e^{2t}-1\} \le C_K\min\{1,t\}.\] Apply the even transfer estimate in Theorem 6 to the centered even function \(h-1\), supported in \(B_{2m}\). Since \(R\ge8m\), it gives \[\|P_R(h-1)\|_\infty \le C\frac mR\min\{1,t\}.\] The Ising Markov property, followed by conditional expectation, yields \[\frac{\mathcal Z_R^G(\tau)}{Z} =\mathbb E_0[h\mid\sigma_{\partial B_R}=\tau] =1+P_R(h-1)(\tau)\ge K^{-1}.\] The logarithm is \(K\)-Lipschitz on \([K^{-1},\infty)\). Its oscillation in the last display is therefore at most \(2K\|P_R(h-1)\|_\infty\), which proves (28). ◻

Proof. A parent whose \(F_i\) meets \(I_a\) is within \(6\) parent units of \(a\). Suppose a comparison edge has endpoint neighborhoods meeting two selected squares, centered at \(a\) and \(b\). An edge of length at most \(12\) cannot do this, since it would give \(\|a-b\|_\infty\le24<D\). For a longer edge, an endpoint \(i\) within \(6\) of a selected parent \(a\ne i\) has already lost the shorter comparison \(\{i,a\}\). Its martingale difference is therefore identically zero. For both endpoint contributions to survive, both endpoints would have to be selected; their common comparison coin makes this impossible. We may consequently declare the support of a normalizer using only those endpoint supports whose differences have not vanished. These declarations depend on coins and priorities, not on the functions being Taylor-expanded. Selected \(F_j\) themselves have radius less than one parent unit, so their separation is immediate. ◻

Proof. Since \(\mathbb E_eU_e=0\), every coin normalizer starts at degree two in the child potentials. The derivative of (31) at zero is conditional expectation. These facts give (32). Averaging the transfer in the full-plane reference law gives zero by the centering of each child; averaging \(A_j\) also preserves this centering. On a small ball, real conditional exponentials are uniformly bounded above and below. The ordinary power series for the exponential and logarithm consequently give the asserted bounds in the supremum norm. There are only finitely many terms in each stencil, independently of \(s\). ◻

Proof. Choose \(C\) large enough to contain the comparison, inactivity, and isolation ranges in parent units. Then choose \(L\) so large that old supports overhang a parent by less than \(1/10\) parent unit and \(100d_0/L<1/10\). Every selection or status check, local allocation, and conditional partition function then uses a fixed parent stencil. Old environment dependence adds only an overhang \(C/L\) in these units. The new priorities and thresholds are independent and located in the same stencil. Increasing \(C\) once, before fixing \(L\), covers all these supports. No integration requires environment variables beyond the terms that touch its finite square.

For any probability kernel \(P\) and real bounded \(H\), \(|\log Pe^H|\le\|H\|\). The same inequality and its Lipschitz version bound (31) and (33) by sums of the input norms. The factors \(p^{-1}\) and the finite numbers of terms depend only on the fixed step. Centering at most doubles this bound, proving (35). Spin flip commutes with every operation. The symmetric choices and allocation rules give the stated invariance in distribution. Exactness follows from the normalized decoration, (31), and (34). Pure conditional kernels in an embedded square depend only on its boundary assignment, so the same identities hold on a torus. We always subtract the same full-plane centering constants, rather than torus means. ◻

Proof. Use the full local comparison stars. If a parent’s complete decoration and allocation stencil leaves \(W\), set its \(F_j\) to zero in every \(U_e\) and retain its original child logs separately. Thus every fresh coin with a nontrivial likelihood is inside \(W\) and is sampled. Skip integrations whose stencils leave \(W\). Include retained terms in any interior integration they meet. Skip special integrations near the boundary and keep active ones separated from them as above. Every operation affecting the eroded chart then uses precisely its full stencil. All remaining discrepancies are supported in the removed boundary layer. The argument applies also to annular charts. ◻

Proof. First sample the entire output Gibbs field, including the new coins. Independently sample a uniform variable at every candidate parent center, and mark \(j\) when it is at most \(p_j=1-e^{-C_LS_j}\), with \(S_j\) the environmental norm sum in (36). These parameters depend only on the environment, so the marks have exactly the independence stated in the proposition, before any further conditioning.

Condition now on the new coins and the common exterior of the updated squares. The ratio of the old conditional density to the new one is bounded below by \(e^{-C_LS_j}\), where \(S_j\) is the local norm sum in (36). This follows from the real log bounds used in Proposition 11; for a special square it applies to the tilt from \(B_A\) to \(B_A+G_A\). If two laws satisfy \(\mu\ge(1-p)\nu\) and \(p>0\), sample from \(\nu\), retain that sample with probability \(1-p\), and otherwise sample from \([\mu-(1-p)\nu]/p\). The Bernoulli decision can be independent of the \(\nu\) sample; when \(p=0\) the laws agree and the sample is retained. Use this residual resampling in each marked updated square. Conditional factorization over the separated squares proves that the resampled field has the old conditional law. Unused centers keep their dummy marks and cause no change. Each update lies inside its radius-\(5s'\) square. Finally discard the conditionally decorated coins. Their decoration was normalized at each old field, so this recovers the input marginal. Taking environmental expectation in (36) gives the last bound. ◻

Proof. Only a finite neighborhood of the insertion has nonzero decorations or replacement terms. Every observable outside a larger buffered region has the same expectation against the normalized input and output densities, because the decorations and replacements preserve its unnormalized pairing and the partition function. Apply this to a pure bond tending to infinity. Theorem 6 identifies the limiting pairing with \(Q\), so the total lattice charge is unchanged. Dividing by the output scale \(Ls\) instead of \(s\) gives the factor \(L^{-1}\). Scalar centering constants cancel in \(D_f\). Analyticity near zero gives the coefficientwise assertion. ◻

Completion of Proposition 15. The last line follows by combining (48), (50), and (55): \[X'\le\bigl(\eta_L+CL^{2/P-1}+C_Lb\bigr)X +C_{L,b}X^2+C_L(\sqrt W+Y).\] Choose \(L\) large enough that the first two coefficients, and the coefficient \(CL^{2-w}\) in (47), leave a strict margin below one. This is possible because \(u>1\), \(w>2\), and \(P>2\). Choose \(b\) next to put flag-free stencils in the Taylor neighborhood and make \(C_Lb\) smaller than the remaining margin. Finally choose \(x_*\) so that \(C_{L,b}X\) is smaller than that margin for \(X\le x_*\). Increasing the resulting contraction factor to one common \(\rho<1\) proves all three assertions. ◻

Proof. On the unflagged stencils, the fifth-order Taylor remainder and Hölder’s inequality give \(O_L(X^5)\). On flagged stencils use the raw bound for the actual outputs and (49). Products of four fourth-order Taylor polynomials have degree at most \(16<P\), so the same bound controls every term there. Terms of input degree between five and sixteen have expectation \(O_L(X^5)\) when \(X\le1\). Products of separately taken expectations obey the identical estimates. ◻

Proof. The quadratic coefficient is \(C_2(i,j)=2\mathop{\mathrm{Cov}}(q_s(f_i),q_s(f_j))\). Its contribution is therefore exactly \(W\), because covariances vanish past the dependence range. Every remaining coefficient has three or four potential occurrences. Their supports fit in a ball of radius \(O(Hs)\). The charge bound in Theorem 6, the boundedness of pure expectation, and Hölder’s inequality bound each coefficient of degree \(d\) by \(C_H X^d\). There are finitely many tuples with first slot zero and diameter at most \(H\). ◻

Proof. Decompose \(f_i=\bar f_i+\widetilde f_i\), where \(\bar f_i=\mathbb Ef_i\). An occurrence of \(\widetilde f_i\) has size order one, and an occurrence of \(\bar f_i\) has size order two. Products of total size order at least five contribute \(O_H(z^5)\) by Hölder’s inequality. For the remaining coefficients connect two occurrences when their cell indices are within \(R_0\). A connected component containing at most four occurrences has bounded diameter. Moreover, a component consisting of a single centered occurrence has zero environmental expectation in every term of the covariance expansion. Thus, once \(H\) is large, a nonzero coefficient in the shell has precisely two distant groups: either a centered pair and one mean, or two centered pairs. Each has total size order four. This argument applies also to the products of means subtracted in a covariance.

Call the two groups \(A\) and \(B\). Let \(1+g_A\) and \(1+g_B\) be their separately normalized insertion densities, expanded formally in their own indeterminates. Every coefficient of \(g_A,g_B\) is pure centered. Define \[ T_A=q_s(g_A),\quad T_B=q_s(g_B),\quad h_{AB}=q_s(g_Ag_B),\quad c_{AB}=\mathbb E_0(g_Ag_B). \tag{63}\] Multiplying the two densities and normalizing again gives the exact formal identity \[ \mathcal T_{A\cup B} =\frac{T_A+T_B+h_{AB}}{1+c_{AB}}. \tag{64}\] If the anchor distance is \(l\), Proposition 7 gives, coefficient by coefficient, \[ h_{AB}=O(l^{-u}),\qquad c_{AB}=l^{-2}\bigl(\kappa T_AT_B+o(1)\bigr). \tag{65}\] The estimates carry the product of the norms of the local coefficient functions. All group diameters are bounded, so their constants are uniform in the slots and in \(s\).

The variance of \(T_A+T_B\) has no coefficient using both independent groups. Contributions with two cross factors \(h_{AB}\) or \(c_{AB}\) have shell sum bounded by \(C H^{2-2u}z^4=o_H(1)z^4\). For a single \(h_{AB}\) we use the stronger, symmetric part of Proposition 7. To justify its applicability, first replace the cutoff on total diameter by the cutoff on anchor distance and sum all bounded relative positions inside each group. Environmental factorization turns the required coefficient into a finite sum of bilinear charge pairings of deterministic functions. More explicitly, the two possible local sums, besides \(\bar f_x\), are \[\begin{align*} D_x&=\sum_{|a|\le R_0} \mathbb E\bigl[q_s(\widetilde f_x)\widetilde f_{x+a}\bigr],\\ M_x&=\sum_{|a|\le R_0} \mathbb E\bigl[\widetilde f_x\widetilde f_{x+a} -\mathbb E_0(\widetilde f_x\widetilde f_{x+a})\bigr]. \end{align*}\] The pair-plus-mean sector gives pairings \(q_s(D_x\bar f_y)\); the pair-plus-pair sector gives \(q_s(D_xM_y)\), with their exchanges and the fixed scalar factors from Taylor expansion. This list follows by taking the linear charge from one group and the term linear in that group in \(h_{AB}\); the other group supplies its mean or its quadratic normalized density. Terms leaving a centered singleton vanish as above. All displayed functions are pure centered, and \(D_x,M_x\) have norm \(O(z^2)\). Rotation of the offset sum preserves both its index set and the environment law, while charge is a rotation-invariant scalar. Thus each function is quarter-turn invariant about its anchor. The resulting bound is \(C l^{-w}z^4\), whose shell sum is \(O(H^{2-w})z^4=o_H(1)z^4\).

The replacement of diameter by anchor distance affects only strips of bounded thickness at the two ends of the shell. There are \(O(H)\) possible anchor displacements in these strips, and the unsymmetrized bound \(O(H^{-u})\) gives an error \(O(H^{1-u})z^4\). Thus the replacement and its reversal both have vanishing error as \(H\to\infty\).

Only the single-\(c_{AB}\) contribution remains. Up to size order four it is \[ -2\mathop{\mathrm{Cov}}\bigl(T_A+T_B,(T_A+T_B)c_{AB}\bigr). \tag{66}\] Its four occurrences must all be centered; a mean would increase the size order. Write \(U=T_{A,1}\) and \(V=T_{B,1}\). These are independent centered random scalars. At bidegree \((2,2)\), \[\begin{align*} -2\mathop{\mathrm{Cov}}(U+V,(U+V)\kappa l^{-2}UV) &=-2\kappa l^{-2}\mathbb E[(U+V)^2UV]\\ &=-4\kappa l^{-2}\mathbb EU^2\mathbb EV^2. \tag{67}\end{align*}\] The \(o(1)\) in (65) gives \(o_H(1)z^4\) after shell summation, since \(L\) is fixed and \(\sum_{H<|y|\le LH}|y|^{-2}\) is bounded.

We finally spell out the slot count. Put \(a_{ij}=\mathop{\mathrm{Cov}}(q_s(f_i),q_s(f_j))\). Suppose the four labeled slots split into the far pairs \(\{1,2\}\) and \(\{3,4\}\). The coefficient of \(t_1t_2\) in \(\mathbb EU^2\) is \(2a_{i_1i_2}\), even when the two cell indices coincide; the occurrences still have distinct indeterminates. The other pair contributes \(2a_{i_3i_4}\). Hence the polarized normalization contribution is \[-16\kappa |i_3-i_1|^{-2}a_{i_1i_2}a_{i_3i_4}.\] There are three partitions of four labeled slots into two unordered pairs. Their sums agree by permutation of the dummy slots, and a tuple in a sufficiently distant shell has at most one such far pairing. The prefactor from (60) is \(1/4!\), so the numerical coefficient is \(-16\cdot3/4!=-2\). For the pairing just displayed, anchoring \(i_1=0\) and writing \(y=i_3\) gives \[\sum_{i_2}a_{0i_2}\sum_{i_4}a_{y i_4}=W^2.\] Both sums are finite, and stationarity makes the second equal to \(W\). Changes of the cutoff caused by bounded shifts of anchors have the strip bounds already proved. A change of anchor in the kernel itself also has vanishing error: for bounded \(\delta\) and \(|y|\asymp H\), \(\bigl||y+\delta|^{-2}-|y|^{-2}\bigr|\le C|\delta|H^{-3}\), whose shell sum is \(O(H^{-1})\). This proves (62). ◻

Proof. First use the regular map and retain the child indeterminates. For finitely many labeled inputs its normalized total insertion charge obeys \[ \mathcal T_{\mathrm{out}}=L^{-1}\mathcal T_{\mathrm{in}} \tag{69}\] by Corollary 14. One can also see the normalization directly: the conditional coin likelihood integrates to one at every fixed old configuration, and each conditional integration preserves pairing with a test outside the affected region. Pairing with a distant pure bond identifies \(Q\) by Theorem 6. Centering the allocated output logs changes only a scalar, which disappears from the normalized insertion. The factor \(L^{-1}\) in (69) is exactly the change from \(Q/s\) to \(Q/(Ls)\).

This argument holds for every fixed realization of the fresh priorities. Taking environmental variance and then Taylor coefficients therefore preserves complete coefficients, including their mean subtractions. The variance factor \(L^{-2}\) cancels the \(L^2\) children per parent when sums are taken per unit area. If an allocation distinguishes positions within a parent, take the area average over complete parent periods. There is no assumption of invariance of individual allocated coefficients under child translations.

We must compare the two cutoffs. Every output Taylor coefficient uses children in a fixed stencil. Thus classifying a coefficient by parent diameter \(H\) rather than child diameter \(LH\) changes its classification only when the child diameter lies in a strip of width \(O_L(1)\) about \(LH\). Keep together the full covariance coefficient for each fixed tuple of output arguments, including all its mean subtractions, and only then expand its arguments in child slots. The parent cutoff is constant on precisely this grouping. More explicitly, write \(R_j\) for the regular output at parent \(j\), as a function of its child potentials. A Taylor contribution with \(m\) output arguments and child degree \(d\), where \(2\le m\le d\le4\), assigns its \(d\) occurrence slots to nonempty blocks \(B_1,\ldots,B_m\) and uses the random functions \[ G_a=D^{|B_a|}R_{j_a}(0) [f_{i_b}:b\in B_a],\qquad 1\le a\le m. \tag{70}\] The directions in this derivative are supported at their indicated child labels; repeated labels are allowed. Multilinear substitution gives the complete polarized covariance coefficient of \(G_1,\ldots,G_m\), with its Taylor factors and the single common weight \(\mathbf 1_{\{\mathop{\mathrm{diam}}(j_1,\ldots,j_m)\le H\}}\).

Discard size orders at least five, at cost \(O_H(z^5)\), and connect child slots when their enlarged stencils can share environment variables. A singleton centered child has zero expectation even conditional on the fresh priorities: the Taylor kernel depends on those independent priorities and takes the child function as a multilinear argument. A singleton mean costs size order two. Two components containing means only cannot create a lower-order term through random allocation. At the only relevant order their output is the linear map of Lemma 10, which is independent of priorities; its selection indicators are reference auxiliary variables, integrated by \(Q\). In particular its charges are deterministic when its inputs are deterministic. We are therefore left with two distant components of bounded diameter and size order two each.

Every output argument receiving a child slot lies in that slot’s bounded stencil. These components consequently partition the relevant output arguments into two distant groups as well: each block \(B_a\) in (70) lies wholly in one component. Apply (64) to the two finite lists of output arguments \(G_a\), with separate insertion indeterminates \(u_a\). The groups are independent in the environment, including their fresh priorities. For their separate charges the formal identity \[\mathop{\mathrm{Var}}\bigl(T_A(\mathbf u_A)+T_B(\mathbf u_B)\bigr) =\mathop{\mathrm{Var}}T_A(\mathbf u_A)+\mathop{\mathrm{Var}}T_B(\mathbf u_B)\] has zero coefficient involving variables from both groups, regardless of their separate means. Thus the unmixed contribution cancels for each fixed output tuple and each child-block partition, before any sum over output labels. Repeated child labels merely repeat derivative directions inside a group; no within-group independence is used. The remaining terms have a cross factor and hence an \(O_L(H^{-u})\) bound from Proposition 7. The strip therefore costs \(O_L(H^{1-u})z^4=o_H(1)z^4\). Keeping the covariance subtractions with each coefficient is essential here and is permitted because the cutoff depends only on the tuple of labels.

Finally apply Corollary 17. For each fixed output tuple, (58) is a sum of bounded multilinear expressions in at most four output potentials, including products of separately taken expectations. The charge bound on its fixed-\(H\) support makes the kernels uniformly bounded in \(s\). The corollary applies to the finite union of all determining stencils, including their status checks. It therefore replaces the regular Taylor output by the actual output and removes child degrees above four with total error \(O_H(X^5)=O_H(z^5)\). In particular the highest intermediate degree is \(4\cdot4=16<P=40\), so the finite moment assumption suffices. This proves (68). ◻

Proof. Combine Lemmas 19, 20, and 21. ◻

Proof. There are two bonds per lattice site. Periodicity of the allocation therefore gives \(\sum_e\alpha_{0,e}=2s_0^2\). Since \(Q(b_e)=1\), \[m_0=2s_0\mathbb E[K(h+\theta\xi)-K_0].\] Independence of distinct bond signs and \(\sum_i\alpha_{i,e}=1\) give the exact identity \[W_0=s_0^{-2}\sum_e\alpha_{0,e} \Bigl(\sum_i\alpha_{i,e}\Bigr)\mathop{\mathrm{Var}}(K(h+\theta\xi)) =2\mathop{\mathrm{Var}}(K(h+\theta\xi)).\] Taylor expansion of the smooth function \(K\) proves the claimed asymptotics and derivative. The finite number of bounded bond functions in a cell gives the bounds on \(X_0\) and \(Y_0\). Lemma 19 gives the assertion about \(v_{H,0}\). ◻

Proof. We first choose constants so that the moment bounds propagate whenever the mean remains between the walls \(m=\pm Dv\). In the estimates of Proposition 15, write \(C_1\) for a common constant and retain its contraction constant \(\rho<1\). Choose \(A\) large enough for the initial \(X_0/\sqrt{v_0}\) bound and so that \[ \rho A+2C_1< A. \tag{77}\] Choose \(D\) next, with \[ (L-1)D>C_1A^2+1. \tag{78}\] Choose \(T\) next so that, for every subsequently fixed \(H\), the initial ratios \(m_0/(Dv_{H,0})\) at \(h=-T\theta^2\) and \(h=T\theta^2\) are strictly below \(-1\) and above \(1\) for small \(\theta\). Lemma 23 permits this choice independently of \(H\): the leading denominator is \(2Da^2\theta^2\), whereas the leading numerator divided by \(\theta^2\) is an affine function of \(h/\theta^2\) with positive slope \(2s_0a\). Finally choose \(B\) sufficiently large that \[ \rho B+C_1(D+A^2)<B \tag{79}\] and \(Y_0\le Bv_{H,0}\) for \(|h|\le T\theta^2\) and sufficiently small \(\theta\), for every subsequently fixed \(H\). This last choice is possible from \(Y_0\le C_{s_0}(1+T)\theta^2\) and \(v_{H,0}/\theta^2\to2a^2\). The order so far is \(A,D,T,B\).

For data satisfying (75), Lemma 19 gives \[ W=v+O_H(v^{3/2}), \tag{80}\] where the implicit constant includes \(A^3\). Use Proposition 22 with \(z=\sqrt v\). Its constants \(A,B\) are now fixed. Choose \(H\) sufficiently large that the \(e_Hv^2\) error is smaller than one quarter of the negative leading term, using the positive limit \(2\pi\log L\) of the shell sum. Then restrict \(v\) to a sufficiently small interval, also within the smallness requirement of Proposition 15, so that (80) and the \(O_H(v^{5/2})\) remainder yield \[ -C_2v^2\le v'-v\le-cv^2, \qquad v'>0, \qquad v'/v=1+O(v). \tag{81}\] Here \(c,C_2>0\) are fixed and the smallness interval is uniform in the scale. Positivity follows, for example, by taking \(C_2v<1/2\).

The \(X\) estimate now gives \[X'\le\rho A\sqrt v+C_1\sqrt W+C_1Bv \le\bigl(\rho A+C_1+o(1)\bigr)\sqrt v \le A\sqrt{v'}.\] The last inequality follows from (77) and \(v'/v\to1\). Similarly, \[Y'\le\bigl(\rho B+C_1(D+A^2)\bigr)v\le Bv'\] by (79). At the upper wall, the mean estimate gives \[m'\ge LDv-C_1A^2v>Dv';\] at the lower wall it gives \(m'<-Dv'\). We decreased the smallness interval if necessary to use (78) and (81). Thus the walls point outward, while both norm bounds propagate up to and including the first exit step.

We now restrict \(\theta\) so that the initial interval \([-T\theta^2,T\theta^2]\) has all the properties fixed above and lies in the uniform smallness range just obtained. On this interval \(v_0=2a^2\theta^2+o(\theta^2)\) uniformly. The initial \(X\) bound is covered by \(A\), because changing \(h\) by \(O(\theta^2)\) does not change the leading \(O(\theta)\) bound on \(X_0\). Thus the complete parameter order is \(A,D,T,B,H\), followed by the small interval for \(\theta\).

All moment data are continuous in \(h\) at every finite number of steps. To check the only possible discontinuities, fix \(h\) and couple the constructions for nearby parameters with the same environment, priorities, and threshold variables. At a threshold comparison, the threshold is independent of the data being compared and has a continuous distribution. Equality therefore has probability zero. At finitely many steps only finitely many comparisons enter any specified local stencil, so the corresponding potentials are almost surely continuous at the fixed \(h\). The crude real-log bounds of Proposition 11 dominate them on every compact initial parameter interval by a finite deterministic bound at each fixed number of steps. Dominated convergence applies to \(X,Y,m,W\) and to the finite polynomial \(v_H\).

Continuity first supplies a closed initial subinterval \(I_0\) on which \(-1\le m_0/(Dv_0)\le1\), with its two endpoints mapped to \(-1\) and \(1\). Suppose \(I_k\) has been chosen so that all parameters in it satisfy (75) through time \(k\), and the endpoints lie on opposite time-\(k\) walls. The estimates just proved give positive \(v_{k+1}\) and the two norm bounds throughout \(I_k\). Its endpoints map strictly outside the opposite time-\((k+1)\) walls. The intermediate value theorem therefore gives a closed subinterval \(I_{k+1}\subset I_k\) on which \(-1\le m_{k+1}/(Dv_{k+1})\le1\), with endpoints on the two walls. For example, take a last crossing of the lower wall before a first subsequent crossing of the upper wall. The preceding-time bounds remain valid on this smaller interval. Compactness gives a parameter \(h\in\bigcap_{k\ge0}I_k\). No monotonicity of an iterated coordinate in \(h\) is needed.

For this parameter, (81) holds at every step. Taking reciprocals yields \[\frac1{v_{k+1}}\ge\frac1{v_k}+c,\] and proves (76). The other conclusions follow from (75) and \(v_0=O(\theta^2)\). ◻

Proof. Activate the finitely many relevant marks in deterministic order. For fixed \(\eta\) and exterior data, conditioning on all previous marks charges the first instability at \(x\) to \(p_{x,k-1}(\eta)\) times a pivotal possibility event with that mark omitted. If adding its square first permits two test values, choose a witnessing assignment and revert the new square, retaining all the other assigned values. The resulting pair differs only in that square. Every passage required outside it by the geometric implication is consequently possible using only the previous marks. We may enlarge these to all other marks in their respective ring neighborhoods.

Consider a retained family of rings with checks \(T_1,\ldots,T_m\). For fixed environment, condition off the first ring’s eroded chart \(W_{1,k}\) and on the marks whose squares do not meet its spin support. The other ring checks are measurable under this conditioning; the remaining spins have the specified conditional Gibbs law, and the remaining marks retain their independent law. Its factor is at most \(B^+_{k,T_1}(\eta)\). Remove each ring in turn. Pointwise in the original exterior data this proves \[ \sup_\zeta\bigl[p_{x,k-1}(\eta) \mathbb P_{k,\eta}^{\zeta}(T_1^+\cap\cdots\cap T_m^+)\bigr] \leq p_{x,k-1}(\eta)\prod_{\ell=1}^m B^+_{k,T_\ell}(\eta). \tag{91}\] The enlarged ring neighborhoods are disjoint and avoid the environment stencil of \(p_{x,k-1}\). All factors on the right are therefore functions of disjoint sets of the underlying independent environment variables. Taking their expectation, including that of the inserted mark, gives \[ \mathbb E\left[p_{x,k-1}\prod_{\ell=1}^m B^+_{k,T_\ell}\right] =\mathbb Ep_{x,k-1}\prod_{\ell=1}^m\mathbb EB^+_{k,T_\ell} \leq C_LX_{k-1}\prod_{\ell=1}^m(q_{i(\ell)}^0+\tau_{i(\ell)}). \tag{92}\] This is why conditional exterior suprema, and the omission of rings near the inserted square, are needed. A product of unconditional quenched check probabilities would not justify this equality.

Retain rings of ratio \(a\) with successive inner radii in ratio \(Ga\) in each of the three ranges in (85), illustrated in Figure 2. A fixed number of omissions handles their endpoints, recenterings, and transitions. By (84), the product in (92), apart from \(C_LX_{k-1}\), is at most \[ C_T(b/u)^{p_4}(u/v)^{p_h}(v/a_T)^{p_1}. \tag{93}\] If a range has too few scales for a retained ring, its bounded factor loss is included in \(C_T\).

It remains to sum over edit locations. Normalize \(a_T=1\) and suppose \(b<1\). The centers with dyadically specified truncated distances \(u,v\) occupy area at most \(C_Tuv\): near a vertex, the portion of side at distance \(O(v)\) has length \(O_T(v)\) and its \(u\)-neighborhood has that area; away from vertices use the finite total side length. The class \(u=v=1\) covers the remaining interior centers. Since the center grid has spacing comparable to \(b\), enlarging each class by one grid cell gives at most \(C_Tuv/b^2\) centers. Its contribution is bounded by \[\begin{align*} C_TX_{k-1}\frac{uv}{b^2}(b/u)^{p_4}(u/v)^{p_h}v^{p_1} &=C_TX_{k-1}(b/u)^{p_4-2}(u/v)^{p_h-1}v^{p_1}. \tag{94}\end{align*}\] Put \(d=\min\{p_4-2,p_h-1,p_1\}>0\). The three base ratios in the last expression have product \(b\); each dyadic class therefore contributes at most \(C_TX_{k-1}b^d\). There are \(O_T((1+\log(1/b))^2)\) classes. Absorb this factor into \(b^{d/2}\) and take \(c=d/2\). For \(b\geq1\), only \(O_T(1)\) grid centers have squares meeting the test support, and the crude pivotal bound one gives \(C_TX_{k-1}\). This proves (90), after restoring \(a_T\). Finally \(T^+-T\leq D_T\), and partial activation only reduces the permitted mark parameters. ◻

Proof. We prove both assertions by strong induction on dyadic ranges of \(r/s_k\). This induction includes all levels and all frames at once. At a terminal ratio, \(M_0s_k\leq r<M_0s_{k+1}\), only a bounded number of level-\(k\) potentials meet the free annular chart. Put \(S\) equal to the sum of their norms. Formula (86) implies, for every exterior assignment, \[\|\mu_{k,\eta,W_k}^{\zeta}-\mu_{0,W_k}^{\zeta}\|_{\mathop{\mathrm{TV}}} \leq \min\{1,2S\},\qquad \mathbb ES\leq C_{\rm boot}X_k.\] Together with the pure bound \(q_i^0\) this proves the terminal plain estimate. Since \(s_k/r\geq(M_0L)^{-1}\), increasing the fixed constant puts it in the form (95).

For a nonterminal plain check, perform one step in its free chart. Conditional on the remaining exterior, the new interior has its level-\((k+1)\) specification. Reversal and (88) bound its old probability by the new plain probability plus the probability of instability under marks from step \(k\). The new plain check has ratio \(r/s_{k+1}=(r/s_k)/L\) and is already covered by induction. Every ring used to bound the instability has inner radius at most \(r/2\) and is evaluated at level \(k+1\), with the single-level possibility law there. Its ratio to the level spacing is strictly smaller than \(r/s_k\), and it is monitored because the first retained radius was chosen above a sufficiently large multiple of \(s_{k+1}\). Its eroded collars remain in the region where the new specification is valid. Thus the inductive possibility bound supplies the hypothesis of Lemma 25. That lemma costs \(C X_k(s_{k+1}/r)^c\), which is a fixed multiple of \(X_k(s_k/r)^c\). Adding it to the inductive plain estimate gives (95).

Once the plain estimate at a given ratio has been proved, bound its possibility increase using marks from step \(k-1\). Its pivotal rings are now at level \(k\), with inner radii at most \(r/2\), so their possibility bounds belong to strictly smaller dyadic ranges of \(r/s_k\). Lemma 25 gives the last term in (96). This also proves the terminal possibility estimate. If no monitored ring fits between the required first radius and the cutoff, the test size and edit radius differ by a bounded factor; the crude bounded count of candidate marks gives the same estimate without a recursive call. This starts the induction.

To close it, the geometric spacing gives \[\sum_{j=k}^{J(r)}(s_j/r)^c \leq \frac{M_0^{-c}}{1-L^{-c}}.\] Choose \(C_{\rm boot}\) to cover the terminal comparison and the two one-level additions, and then choose \(\varepsilon_*\) so that this weighted sum and the additional single-level term are smaller than \(\min_i\tau_i\). In each inductive call only these fixed tolerances were used to obtain the pivotal power. This proves both estimates without any assumption about a configuration edited at several levels. ◻

Proof. Couple each successive level by the reversal in Proposition 13. After averaging over the environment, the probability that a test changes at step \(j\) is at most \[C_TX_j\min\{1,(\delta s_{j+1}/a_T^{\rm phys})^c\}.\] This follows from Lemma 25 and Proposition 26; if the test is no larger than the edit square, use the crude candidate count. Sum over steps and the fixed family. Equation (97) makes the sum vanish. At level \(K\) only a bounded number of potentials remain on the torus, so their expected total norm is \(O_M(X_K)\). The same density comparison as in the terminal bootstrap couples this level to the pure torus with expected total variation tending to zero. Gluing the successive couplings and using the union bound proves (98). Only the errors of individual reversals were summed; no probability estimate was imposed on the union of their marks.

The level-zero quenched probabilities depend only on the original bond environment \(\omega\). Averaging the bounds over the auxiliary algorithm choices therefore leaves exactly the expectation in (98). Finally, choosing \(K\) by the first scale above a fixed multiple of \(\delta^{-1}\) keeps the physical period between two fixed bounds whose ratio is \(L\). ◻

Proof. Choose the terminal spacing \(s_K\) comparable to \(r\) with a sufficiently small fixed proportionality constant depending on this geometry. Run the local construction through level \(K\). The sum of the removed boundary thicknesses is \(O(s_K)\), so this choice leaves all prescribed buffers free. At each reversal use the conditional-supremum version of (91). For example, a measurable upper bound for the error is the sum over inserted centers of their mark parameters times the products of local ring suprema, with the factor one when only the crude count is used. These functions depend on local environment, and bound the error for every exterior assignment before any mixture of those assignments is taken. Summing them over steps and the finite family gives an envelope with expectation at most \[C_{\mathcal T}\sum_{j=0}^{K-1}X_j \min\{1,(s_{j+1}/r)^c\}.\] At the terminal scale add \(C\min\{1,\sum_j\|f_j^{(K)}\|\}\) over potentials touching the remaining window. Their number is bounded, so this contribution has expectation \(O_{\mathcal T}(X_K)\). Conditional on its exterior, deleting those potentials gives the pure conditional law; integrating the exterior produces the stated mixture. Formula (97), with effective mesh \(r^{-1}\), proves the vanishing expectation.

If every pure conditional law gives an event probability at least \(c_*>0\), the perturbed probability is at least \(c_*-\mathcal E_r(\eta)\) for every initial exterior. Markov’s inequality then transfers any lower bound less than \(c_*\). In particular this applies to the fixed compatible systems of buffered colored corridors and circuits in [22]. To state the result using only the original bond environment, average the envelope over the algorithm choices. The worst level-zero conditional probability itself depends only on the bonds in the window and its incident layer; hence its good-environment event is local. All estimates are uniform in the center, whereas their convergence rate may depend on the fixed geometry. ◻

Proof. For a specified center, pack separated rings of ratio \(a\) from a fixed multiple of \(r\) to a fixed fraction of \(R\). At every fixed physical \(r>0\), their inner lattice radii eventually exceed \(M_0s_0\). Condition off their free charts as in (91) and use the plain bound of Proposition 26. Their environment neighborhoods are disjoint, so (92), without an inserted mark, yields \[\mathbb E_{\rm env}\mu_\omega(F_4(z;r,R))\leq C(r/R)^{p_4}\] in the mesh upper limit, with \(p_4>2\). The bound already includes the supremum over exterior data of the ambient domain. For fixed \(r\), cover \(K_0\) by an \(r\)-grid of \(O_{K_0}(r^{-2})\) centers. Enlarge the inner squares and reduce the outer radius by fixed factors, so four passages about any center imply a grid check. A union bound gives \(C_{K_0,R}r^{p_4-2}\) after the mesh limit, which tends to zero. The finite test family is enlarged only after its mesh limit has been taken; lattice rounding is absorbed by the same fixed crops. ◻

Proof. We record the finite-volume form of Kramers–Wannier duality [18], including its homology convention; the four-sector matrix also appears in [4]. For \(b\in(\mathbb Z/2\mathbb Z)^2\), let \[W_b(K)=\sum_{\substack{A\subset E(\mathbb T_N)\text{ even}\\{}[A]=b}} \prod_{e\in A}\tanh K_e.\] The high-temperature expansion gives \[Z_\alpha(K)=2^{N^2}\prod_e\cosh K_e \sum_b(-1)^{\alpha\cdot b}W_b(K).\] The low-temperature expansion of the dual spin system with holonomy \(\beta\) has domain walls of winding class \((\beta_2,\beta_1)\). Each admissible wall configuration comes from two spin configurations, so \[Z_\beta(K^*)=2e^{\sum_eK_{e^*}^*} W_{(\beta_2,\beta_1)}(K).\] Substitution gives the sign \((-1)^{\alpha_1\beta_2+\alpha_2\beta_1}\) in (104). Its prefactor follows from \(|E(\mathbb T_N)|=2N^2\) and \(e^{-K_{e^*}^*}\cosh K_e=\sqrt{\sinh(2K_e)/2}\). At \(K_e=K_0\), the dual couplings equal \(K_0\) and \(a(K)=1\); hence \(\mathsf HZ(K_0)=Z(K_0)\). The displayed matrix satisfies \(\mathsf H^2=I\).

For the ratio, expand each edge factor as \[e^{K_e\tau_e^\alpha\sigma_x\sigma_y} =e^{-K_e}\left(1+v_e\, \mathbf 1_{\{\sigma_x=\tau_e^\alpha\sigma_y\}}\right).\] On a chosen open subgraph the spin constraints are consistent exactly when every open cycle has zero evaluation under \(\alpha\). In that case there are \(2^{k(A)}\) compatible spin assignments. Dividing the resulting sum by its untwisted version proves (105). ◻

Proof. Choose a fixed large integer \(M\) and torus periods \(N_k=Ms_k\), so that every complete support and environment-dependence neighborhood of Proposition 11, through scale \(k\), embeds in the torus. The primitive environment variables in each such neighborhood have their plane distribution, so each resulting potential has its plane law when expressed in local spin coordinates. Apply the scale construction with iid torus disorder, simultaneously in the four spin holonomies. Let \(\zeta\) denote the independent algorithmic choices used in this run. In every contractible neighborhood a flat twist has a spin trivialization; changes of trivialization on overlaps are constant spin flips. Every interaction is even, and the auxiliary variables are unchanged by spin flip. The local identities therefore hold in each holonomy, with the same scalar factors removed by centering. In particular, for a common positive factor \(c_k(\omega,\zeta)\) the resulting partition functions satisfy \[e^{-S_k}Z_\alpha(K_0) \le c_k(\omega,\zeta)^{-1}Z_\alpha(K(h_*,\omega)) \le e^{S_k}Z_\alpha(K_0),\] where \(S_k=S_k(\omega,\zeta)\) can be the largest, over the four holonomies, of the sums of norms of the remaining potentials. There are only \(M^2\) cells at the last scale. The trajectory bound \(X_k\to0\) gives \(\mathbb ES_k\le C M^2 X_k\to0\). Consequently \[ \bigl\|\widehat Z(K(h_*,\omega))- \widehat Z(K_0)\bigr\|_2\longrightarrow0 \quad\text{in environmental probability}. \tag{106}\] The partition vector itself does not depend on the algorithmic choices, so this conclusion also holds after retaining only the original bond environment.

The dual odds satisfy \(v_{e^*}^*=2/v_e\). Thus duality sends \((h,\xi_e)\) to \((-h,-\xi_e)\), with the signs transported along the crossing-edge bijection. This transformation preserves their iid law. By Lemma 30, the normalized duality transform is the orthogonal matrix \(\mathsf H\), and its pure target is fixed. Therefore (106) holds with \(-h_*\) as well.

Take \(\alpha=10\), and let \(A_\alpha\) be the increasing event that some open cycle has odd evaluation under \(\alpha\). Its pure probability \(q_N^0\) satisfies \[ c\le q_N^0\le1-c \tag{107}\] uniformly for large \(N\). For the lower bound, a finite chain of pure FK rectangle crossings in a torus band constructs an open cycle winding once horizontally. For the upper bound, the corresponding construction of a closed-dual cycle winding once vertically prevents such an open cycle. The constructions use finitely many embedded rectangles of fixed aspect ratios and the uniform pure FK bounds recalled in [22]. They can be imposed successively with arbitrary induced boundary wiring; for a closed-dual crossing use the worst, fully wired primal boundary.

All ratios in (105) belong to \([0,1]\), so the untwisted coordinate of \(\widehat Z\) is at least \(1/2\). Hence (106) implies, on the same original signs, \[q_{N_k,\omega}(h_*)-q_{N_k}^0\longrightarrow0, \qquad q_{N_k,\omega}(-h_*)-q_{N_k}^0\longrightarrow0\] in probability, where \(q_{N,\omega}(h)= \phi_{\mathbb T_N,h,\omega}(A_\alpha)\). The two convergence statements hold simultaneously by a union bound. Querying every edge in (102) and using (103) gives \(q'_{N,\omega}(h)\ge q_{N,\omega}(h)(1-q_{N,\omega}(h))\). If \(h_*\ne0\), monotonicity and (107) keep this product bounded below on the interval between \(-h_*\) and \(h_*\), with probability tending to one. Integration forces an endpoint difference bounded below by a positive constant, whereas both endpoints approach the same \(q_{N_k}^0\). This contradiction proves \(h_*=0\). ◻

Proof. Choose two separated buffered subbands of an annulus of fixed sufficiently large ratio \(a\). The pure probability of simultaneous plus and minus spin circuits, one in each subband, is bounded below by a constant \(2c_0>0\), uniformly over pins outside the buffers; this is precisely [22]. Proposition 28 transfers this lower bound to the random spin law, with probability tending to one over the local environment and uniformly over the boundary pins. Apply it to Edwards–Sokal spins with the wired boundary cluster pinned plus. An ordinary open FK path has a constant spin. Such a path cannot cross both separating circuits of opposite colors. The common triangulation used for spin circuits causes no exception: a nearest-neighbor lattice path can cross a circuit only through a vertex of that circuit. The ordinary radial FK crossing probability is thus at most \(1-c_0\).

Defining goodness by this wired crossing probability makes it a function of the original local bonds alone. Its failure probability tends to zero after integrating out all algorithmic choices. FK monotonicity gives the assertion for \(h\le0\) and for weaker wiring.

For completeness, this also rules out percolation at \(h=0\). Pack concentric annuli with a fixed large geometric separation, so that their bond sets, including incident layers, are disjoint. Conditional on all edges off one annulus, its induced FK boundary partition is dominated by full wiring. The probability of an arm crossing \(m\) such annuli is therefore bounded, at fixed environment, by the product of their fully wired crossing bounds. Choose the first radius large enough that every bad-annulus probability is at most \(1/2\). Independence of the local bond sets bounds the environmental mean of this product by \((1-c_0/2)^m\). Letting \(m\to\infty\) in the wired infinite-volume limit proves (108). ◻

Proof. We work at \(x=0\); the estimates are uniform over translates. For every vertex \(z\in B_{N+1}\), pack \(m_N\ge c\log N\) separated annuli centered at \(z\), with inner radii between \(N^{1/4}\) and a sufficiently small multiple of \(N^{1/2}\). Their full annular graphs lie in \(B_{N^{1/2}}(z)\), and their bond neighborhoods are disjoint. Let \(\eta_N\to0\) be an upper bound for their bad-environment probabilities, uniformly over all these radii and centers, provided by Lemma 32. Independence for the annuli around one fixed \(z\) gives \[\mathbb P\bigl(\text{at least }m_N/2\text{ annuli at }z \text{ are bad}\bigr) \le 2^{m_N}\eta_N^{m_N/2}.\] Let \(\mathcal G_N\) be the event that fewer than half are bad at every \(z\in B_{N+1}\). A union bound, without any independence assertion between distinct centers, gives \[ \mathbb P(\mathcal G_N^c) \le C N^2 2^{m_N}\eta_N^{m_N/2} =o(N^{-A})\qquad\text{for every }A>0. \tag{110}\] Indeed \(m_N\) is bounded above and below by constant multiples of \(\log N\), while \(\log(1/\eta_N)\to\infty\).

On \(\mathcal G_N\), multiplication of the good-annulus bounds gives, for some fixed \(\alpha>0\), \[ \phi_{B_{2N},h,\omega}^{\mathrm w} \bigl(z\leftrightarrow\partial B_{N^{1/2}}(z) \text{ inside }B_{N^{1/2}}(z)\bigr) \le C N^{-\alpha}, \qquad z\in B_{N+1},\quad h\le0. \tag{111}\] For large \(N\) all these small boxes lie within \(B_{2N}\). Induced boundary conditions are harmless because the annular bounds already use full wiring. The event \(\mathcal G_N\) works simultaneously for every \(h\le0\) by monotonicity.

Fix such an environment and write \[q_\omega(h)=\phi_{B_{2N},h,\omega}^{\mathrm w} (0\leftrightarrow\partial B_N\text{ inside }B_N).\] Choose an integer \(\ell\) uniformly between \(N/4\) and \(N/2\), and explore all ordinary open clusters in \(B_N\) that meet the layer \(\partial B_\ell\). This determines the event defining \(q_\omega\): every path from \(0\) to \(\partial B_N\) meets that layer. The exploration uses only ordinary edges in \(B_N\), although its law is the marginal of the wired measure on \(B_{2N}\). This is the layer exploration of [10].

An edge is queried only if an endpoint lies on the chosen layer or has an ordinary open path to it. For a fixed endpoint \(z\), the probability, over \(\ell\), that the layer is within distance \(N^{1/2}\) of \(z\) is at most \(C N^{-1/2}\). Otherwise such a path forces an arm of length \(N^{1/2}\) at \(z\). Thus (111) bounds the maximum averaged revealment by \[\max_e\delta_e\le C\bigl(N^{-1/2}+N^{-\alpha}\bigr) \le C N^{-d},\qquad d=\min\{1/2,\alpha\}>0.\] Edges outside \(B_N\) have revealment zero. Their covariances are nonnegative, so retaining them in the derivative sum only strengthens the bound obtained from (102). Equations (102)–(103) yield, for all \(h\le0\), \[ q'_\omega(h)\ge cN^d q_\omega(h)(1-q_\omega(h)). \tag{112}\] Moreover (111) at \(z=0\) gives \(q_\omega(0)\le C N^{-\alpha}\). For sufficiently large \(N\), integrating the derivative of \(\log(q_\omega/(1-q_\omega))\) over \([-b,0]\) gives \[\frac{q_\omega(-b)}{1-q_\omega(-b)} \le \frac{q_\omega(0)}{1-q_\omega(0)}e^{-cbN^d} \le e^{-cbN^d}.\] Combining this estimate with (110) proves (109). In a larger graph, condition outside \(B_{2N}(x)\) and dominate the induced boundary partition by full wiring. This proves the final assertion, including infinite-volume limits. ◻

Proof. Lemma 32 and monotonicity give \(m_\theta(h)=0\) for \(h\le0\). Fix \(h=b>0\). Planar FK duality replaces the edge odds by \(2/v_e\), so the dual environment has the law of the model at \(-b\). In finite domains the dual boundary condition is dominated by full wiring. Taking limits, the dual of the infinite-volume wired primal law therefore satisfies the local annealed arm bounds of Lemma 33.

A closed-dual circuit surrounding \(B_M\) whose farthest distance from the origin lies in \([R,2R)\) has diameter at least \(cR\). It consequently contains an ordinary dual arm of length \(c'R\) from some dual vertex in \(B_{2R}\), with absolute positive constants \(c,c'\). There are \(O(R^2)\) possible vertices. Using (109) with any \(A>2\) and summing over dyadic \(R\ge c''M\) gives \[ \mathbb E\phi_{b,\omega}^{\mathrm w} \bigl(\text{a closed-dual circuit surrounds }B_M\bigr) \le C_{b,A}\sum_{j\ge0}(2^jM)^{2-A} \longrightarrow0. \tag{114}\] Here the arm bound is applied in a box of radius comparable with \(R\), with a further collar of comparable width; its validity for arbitrary exterior boundary conditions was part of Lemma 33.

If every open cluster meeting \(B_M\) were finite, their union would be finite. It contains \(B_M\), and its lattice edge boundary consists of closed edges. Connecting the union through the box, if necessary, does not change its exterior boundary. Its outer dual boundary therefore contains a closed-dual circuit surrounding \(B_M\). The complement of the event in (114) thus forces some vertex of \(B_M\) to belong to an infinite open cluster. For large \(M\) this has positive joint probability. The joint environment–wired FK law is translation invariant, whence \[0<\mathbb E\phi_{b,\omega}^{\mathrm w}(B_M\leftrightarrow\infty) \le |B_M|\,m_\theta(b).\] This proves positivity for every \(b>0\).

Finally, the left side of (113), viewed as a function of \(\beta\ge0\), is continuous and strictly increasing from zero to infinity. Let \(\beta_*\) be its unique solution, and set \[v_\pm=e^{2\beta_*(1\pm\epsilon)}-1, \qquad \theta_\epsilon=\frac12\log\frac{v_+}{v_-}.\] Then \(v_\pm=\sqrt2e^{\pm\theta_\epsilon}\). Comparison with the constant couplings gives \(K_0/(1+\epsilon)\le\beta_*\le K_0/(1-\epsilon)\), so \(\theta_\epsilon\to0\) as \(\epsilon\to0\). The model at \(\beta_*\) therefore falls within the result just proved and has zero plus magnetization. The same holds below \(\beta_*\) by monotonicity. For \(\beta>\beta_*\) both log odds increase, and \[b:=\min_{s\in\{-1,1\}} \log\frac{e^{2\beta(1+s\epsilon)}-1} {e^{2\beta_*(1+s\epsilon)}-1}>0.\] The bonds at \(\beta\) dominate those of (100) with parameters \((b,\theta_\epsilon)\) in the same sign environment. Their plus magnetization is consequently positive. This proves \(\beta_*=\beta_c(\epsilon)\) with the normalization and plus boundary convention of Theorem 2. ◻

Proof. On \(\{\varepsilon\le a/2\}\), the second conditioning probability is at least \(a/2\). Subtracting the two conditional probability fractions gives \[\bigl\|p(v\in\cdot\mid e=1) -q_\omega(v\in\cdot\mid e=1)\bigr\|_{\mathop{\mathrm{TV}}} \le \frac{4\varepsilon(\omega)}a.\] Couple these two vectors maximally and glue this coupling to the two order couplings. Extend the outer vectors to samples with laws \(\lambda\) and \(\mu_\omega\) by their conditional laws given \(U\) and \(W\). The outer vectors are ordered unless the two middle vectors disagree. On \(\{\varepsilon>a/2\}\) take any coupling with the same outer marginals. Markov’s inequality bounds this exceptional environmental probability by \(2\mathbb E_{\mathrm{env}}\varepsilon/a\), proving (115). All sets are finite. The coupling matrices and subsequent conditional probabilities can be selected measurably; for instance, fix an enumeration of feasible basic solutions of the finite coupling constraints and choose the first available one. ◻

Proof. Put the corridors, their reveal regions, and the finitely many strips inside a torus chart with a fixed positive external buffer. Choose a torus period allowed by Proposition 27. In this chart use the same bond environment as the target; complete the torus with independent bonds. Write \(E_h\) for the event that both cutting corridors have their prescribed lengthwise colored crossings, and write \(V\) for the vector of ordinary plus crossings of the strips on the torus. The pure buffered crossing bounds give \[p_n(E_h)\ge a_h>0, \qquad p_n=\mathcal L_0(\mathbf 1_{E_h},V),\] for all sufficiently fine meshes. Here \(a_h\) may depend on the fixed geometry. With \(q_{n,\omega}\) denoting the corresponding quenched torus vector law, Proposition 27 gives \[ \mathbb E_{\mathrm{env}}\|q_{n,\omega}-p_n\|_{\mathop{\mathrm{TV}}}\longrightarrow0. \tag{117}\] At this stage the environmental expectation includes the independently added torus bonds.

We verify the two order comparisons needed for Lemma 35. In either torus, conditional on \(E_h\), reveal the prescribed crossing nearest the exterior side of each corridor. The stopped reveal of [22] leaves only pins of the prescribed color facing the unrevealed center. The transcript already determines crossing existence, so no additional conditioning on \(E_h\) remains in that center. Disjoint reveal regions allow both explorations.

First compare the pure Dobrushin law in \(A_h^-\) with the conditioned pure torus. The revealed plus path is an upper plus pin against a lower free spin. It screens the plus arc of \(A_h^-\); every accessible remaining boundary incidence is a lower minus pin against an upper free spin. Meetings of a cut and boundary give upper plus against lower minus. The minus reveal is outside \(A_h^-\). Ferromagnetic boundary comparison therefore puts the pure-domain strip vector below \(p_n(V\in\cdot\mid E_h)\).

Next compare the conditioned random torus, as the lower law, with the target, as the upper law. They have identical couplings on every bond in their common free region. The minus path screens the target minus arc. The remaining boundary incidences, with upper entry first, are exactly \[(+,\mathrm{free}),\qquad(\mathrm{free},-),\qquad(+,-).\] Each comparison follows by conditioning a free spin to its extremal value and using the nearest-neighbor ferromagnetic order. This also explains why no comparison between different random bond strengths is needed. The source’s geometric incidence check, including the corridor caps, shows that these are all incidences encountered by the common components containing the strips. A nearest-neighbor edge cannot cross the interior of a separating triangulated diagonal. Virtual plus values outside \(D_n\) give the same order for portions of a strip that exit and reenter the target. Thus \(q_{n,\omega}(V\in\cdot\mid E_h)\) is below the target strip vector.

Apply Lemma 35 with \(a=a_h\) and (117). It gives (116), extending the strip vectors to the complete spin samples with their conditional laws. Finally average over the extra torus bonds while retaining the original environment. The conditional pure marginal is unchanged, and the target marginal never used the extra bonds. Interchanging the signs proves the other assertion. ◻

Source and scope. The first assertions are [22]. Uniform boundary parametrization gives uniform convergence of the normalized conformal maps on the closed disk, including the two marks. Mapping one disk \(\mathrm{SLE}_3\) by these maps proves the displaced continuum convergence. For fixed \(h\), only pure convergence in an ordinary polygonal approximation is used.

The finite family is obtained by covering favorable simple comparison chords with open neighborhoods described by rational polygonal strips, transverse gates, and strict crossing margins. Compact middle subarcs are in \(D\); the two short terminal pieces tend to the marks. A countable such cover has a finite subfamily carrying arbitrarily large probability. Its entire list is fixed before taking the lattice limit; the sampled pure chord may select a strip only after the joint vector coupling has been made. Following the signed side of the lattice interface supplies the crossing with \(O(\delta_n)\) drawing error.

The last assertion is the winding and boundary-order argument of [22]. A signed crossing in one of these strips blocks a target contour from the corresponding side. To include the open signed boundary arc, erase loops from the crossing and close it outside \(D_n\). In the fixed \(\Psi\) coordinates, take radial connectors from its gates to a circle \(\rho=\rho_*>1\) and the circular arc on the signed side, where \(\rho_*\) is independent of \(h,n\). At fixed \(h\), the added path has positive clearance from \(\overline D\) and hence lies outside \(D_n\) for large \(n\). Include this threshold in the diagonal choice. The target marks are on the zero-winding side of the closed barrier. As the strip widths and terminal errors vanish, the gate centers tend to the marks, while the fixed-radius closure stays a positive distance beyond every compact subarc of the signed boundary. The winding exclusions therefore hold on every compact neighborhood of the forbidden side or boundary arc, giving (118). The argument uses only the deterministic geometry and noncrossing of opposite signed paths with the target contour; it applies to the couplings in Proposition 36. ◻

Proof. This is [22]. Here is the argument. A point of \(P(L)\cap M(U)\) could be connected inside this union to both signed boundary arcs. Removing loops gives a crosscut between those arcs that avoids \(K\). Its endpoints alternate with \(a,b\), so it separates \(a\) from \(b\), contradicting the connectedness of \(K\). Hence \(P(L)\subset P(U)\cup U\). No point of \(U\) can belong to the open set \(P(L)\), since every neighborhood of an interior point of \(U\) meets \(M(U)\). Thus \(P(L)\subset P(U)\).

Use a fixed Schoenflies homeomorphism to the disk and take the planar areas of the images of these two plus sides. They are bounded, ordered random variables with the same distribution, so their difference vanishes almost surely. If the inclusion of plus sides were proper, take \(z\in P(U)\setminus P(L)\). It either lies in \(M(L)\), or lies on \(L\). In the latter case a neighborhood of \(z\) inside \(P(U)\) meets \(M(L)\), by the two-sidedness of a Jordan chord. In either case the difference contains a nonempty open subset of \(M(L)\), and hence has positive area. The plus sides are therefore equal, and so are their interior boundary chords. By (118), \(K\) lies on this chord. The inverse chord parametrization maps it to a connected subset of \([0,1]\) containing both endpoints, which is all of \([0,1]\). ◻

Source and interpretation. This is [22], with its passage convention of disjoint parameter intervals. Extend the inverse parametrization of \(G\) continuously to an ambient neighborhood and compose it with \(g_n\) to obtain \(u_n\). When \(B_n\to0\), the running maximum of \(u_n\), normalized to endpoint values \(0,1\) and perturbed by a vanishing strictly increasing linear function, is an increasing homeomorphism uniformly close to \(u_n\). Uniform continuity of \(G\) gives the asserted metric convergence. The other implication is the companion’s deterministic passage argument. In the random statement, compact exhaustion of \(D\) and deterministic lower bounds on \(R_\eta\) on events of increasing probability reduce its random centers and radii to (119). Six passages imply four, so the latter condition suffices. No bound on target path lengths is assumed. ◻

Completion of the proof of Theorem 2. Fix \(\epsilon\) in the interval already chosen in Sections 7 and 8. Begin with any subsequence of the given approximations, and pass further so that the designation of \(I_+\) is fixed. We will construct, along this subsequence, deterministic comparison laws \(R_n\) and couplings such that \[ \mathcal L(L_n\mid\omega)=R_n,\qquad \mathcal L(\Gamma_n\mid\omega)=Q_{n,\omega},\qquad \mathbb E\,\min\{1,d_{\mathrm{curv}}(\Gamma_n,L_n)\}\longrightarrow0, \tag{120}\] where \(R_n\Longrightarrow\mathrm{SLE}_3(D;a,b)\) in the curve metric.

Choose \(\eta_j\downarrow0\) and the displacements and finite strip families of Lemma 37. For each fixed \(j\), apply Proposition 36 to the two entire families. Choose increasing deterministic thresholds \(n_j\) so that for all \(n\ge n_j\) the geometric placements are valid, the pure comparison laws have their required approximation bounds, and each strip-transfer failure probability is at most \(\eta_j\). The threshold may absorb the reciprocal of \(a_{h_j}\) from the conditioning estimate. Put \[j(n)=\max\{j\le n:n_j\le n\},\] with any definition before \(n_1\). This deterministic index tends to infinity, as slowly as necessary. Let \(L_n\) and \(U_n\) be the two pure comparison chords at displacement \(h_{j(n)}\), and denote the law of \(L_n\) by \(R_n\). Their deterministic marginal laws converge to the same \(\mathrm{SLE}_3(D;a,b)\) law.

For each \(n\), glue the two strip-transfer couplings over the common pair \((\omega,\Sigma_{n,\omega})\). Equivalently, disintegrate each comparison sample conditional on that pair and then sample both conditional kernels. Each comparison marginal conditional on \(\omega\) is still its prescribed pure law. With probability tending to one, the target has both signed barriers selected from the finite lists by its two comparison chords.

All target traces lie eventually in one fixed compact planar set. Indeed uniform boundary convergence bounds \(\partial D_n\), and the bounded components it encloses are bounded by the same large disk. The space of nonempty compact subsets of this disk is compact in Hausdorff distance. The ambient curve space is complete and separable: rational polygonal paths are dense, and from a Cauchy sequence one can select a subsequence with successive curve distances less than \(2^{-j}\). Recursively choose parametrizations of its representatives with successive uniform distances less than \(2^{1-j}\). They converge uniformly to a continuous path, which represents the limit of the original Cauchy sequence. Thus this space is Polish and the usual weak-convergence and representation theorems apply. The comparison chord laws are tight in this space. Along any further subsequence, pass again to a joint limit \((K,L,U)\) of the target trace and comparison chords. The limit \(K\) is connected, contains \(a,b\), and lies in \(\overline D\); connectedness and endpoint inclusion pass under Hausdorff convergence, and points at positive distance outside \(\overline D\) are eventually outside \(D_n\) by the boundary winding argument. Lemma 37 gives (118). Since \(L\) and \(U\) have the same simple chord law, Lemma 38 gives \[K=L=U\qquad\text{almost surely as traces}.\]

Represent this joint convergence with almost-sure convergence of the trace and comparison variables, and lift it to the target paths by their original conditional laws given those variables. At each mesh the target configuration space is finite, so this lift is immediate. It preserves every target passage probability and requires no prior tightness of the target paths in the curve metric. The marginal estimate Corollary 29 is precisely (119) for these target curves, including after this lift. Lemma 39 therefore gives \(d_{\mathrm{curv}}(\Gamma_n,L)\to0\) in probability in the representation. The represented \(L_n\) already converges to \(L\) in that metric. The triangle inequality yields \(d_{\mathrm{curv}}(\Gamma_n,L_n)\to0\) in probability. This statement depends only on the joint law of the pair at each \(n\), so it holds in the original couplings as well. Since every further subsequence admits this argument, it holds along the whole chosen subsequence. Boundedness then proves the expectation assertion in (120).

Let \(\pi_{n,\omega}\) be the conditional joint law of \((\Gamma_n,L_n)\) in that coupling, and write \(\overline d=\min\{1,d_{\mathrm{curv}}\}\). Every function \(F\) in the defining class for \(d_{\mathrm{BL}}\) satisfies \[\left|\int F\,\mathrm dQ_{n,\omega}-\int F\,\mathrm dR_n\right| \le\int \overline d(\gamma,\ell)\, \mathrm d\pi_{n,\omega}(\gamma,\ell).\] The right side is independent of \(F\), so taking its supremum and then environmental expectation gives \[ \mathbb E_{\mathrm{env}}d_{\mathrm{BL}}(Q_{n,\omega},R_n) \le \mathbb E\,\overline d(\Gamma_n,L_n)\longrightarrow0. \tag{121}\] This step uses the conditional pure marginal in (120); its annealed counterpart alone would not give (121). The deterministic laws \(R_n\) converge in the same separable curve space to the required SLE law, and hence their bounded-Lipschitz distances to that law tend to zero. The triangle inequality and Markov’s inequality prove, for every \(\eta>0\), \[\mathbb P_{\mathrm{env}}\!\left\{ d_{\mathrm{BL}}\bigl(Q_{n,\omega},\mathrm{SLE}_3(D;a,b)\bigr) >\eta\right\}\longrightarrow0\] on the chosen subsequence. If convergence failed on the original sequence, a subsequence with the displayed probability bounded below would contradict the construction just given. This proves the full sequence assertion.

All couplings used only finitely many bond variables at each mesh; integrating out added torus bonds and algorithmic variables preserves the conditional laws given the original infinite environment. The bounded-Lipschitz distances are measurable, since \(Q_{n,\omega}\) takes only finitely many values. Finally restore the prescribed half-stubs and local drawing. Their matched displacement is \(O(\delta_n)\) and their orientation remains incoming to outgoing. Every displacement, strip, and comparison above was made in the original Euclidean embedding, and the interval of allowed \(\epsilon\) was fixed before choosing any target geometry. ◻