Proof. A finite even dual subgraph has even intersection parity with every closed primal walk. One way to see this is to reduce a closed walk modulo two to elementary primal faces; the intersection parity around such a face is the degree of the dual vertex it encloses. Starting with a plus spin at infinity, flip signs on crossing an occupied dual edge. Path independence follows from this parity statement. Finiteness fixes the plus sign in the unbounded region and gives uniqueness.

The exterior cell centers \(\mathbb Z^2\setminus\Omega\) form a connected nearest-neighbor graph. Indeed the complement of a Jordan polyomino has no bounded component, and a path in its exterior can be perturbed to cross cell sides away from vertices, giving a path of exterior cells. The Jordan condition excludes a diagonal pinch. A primal path using only exterior vertices crosses only dual edges outside \(E_\Omega\). Their occupied set is \(P_o\), so all exterior signs are fixed independently of \(P\).

Conversely, extend any assignment in \(\Omega\) by these fixed exterior spins. Its disagreement set outside \(E_\Omega\) is \(P_o\), and its complete dual disagreement graph is even. The interior part therefore belongs to \(\mathcal P_\Omega^\xi\). These two constructions are inverse.

For the signed-arc assertion, traverse the polygonal boundary. At a flat boundary vertex, the two successive exterior neighboring cells are separated by one exterior primal bond. At an unmarked vertex the crossing exterior dual edge is absent by the parity condition, so the two exterior signs agree. At a convex corner, the corresponding exterior path has two bonds; its crossing parity is again even at an unmarked vertex. At a reflex corner, the two boundary edges have the same adjacent exterior cell. At either mark the exterior parity is odd, and the sign changes. Reflex corners cannot support an exterior stub of the required kind. Thus there are exactly the two asserted constant, opposite arc signs.

Now fix an interior spin configuration and its contour \(P\). A nonconstant interaction term in (3) has a support \(S\) meeting \(E_\Omega\). Write \[X=S\cap E_\Omega,\qquad Y=S\setminus E_\Omega.\] The product of disagreement indicators is one exactly when \(X\subseteq P\) and \(Y\subseteq P_o\). Summing over all such supports gives precisely the double sum in (1). Terms supported entirely outside \(E_\Omega\) are constant in the interior spins. The nearest-neighbor energy is \(2\beta J\left|P\right|\) up to an exterior-dependent constant. This proves equality of the conditional laws.

Finally, choose an edge in a nonempty support of \(U\). Finite range leaves only finitely many possible other edges relative to it, hence only finitely many support types up to translation. Expanding the products of \(d_e\) gives finite local spin polynomials, invariant under spin flip and under the prescribed square symmetries. ◻

Source and the terminal symmetry. The first three assertions are Corollary 2.5, “Buffered and small-patch comparisons”, in Section 2, “Pointwise comparison through buffers”, of (OpenAI 2026). We record the symmetry needed below. For the positive joint marginal \(p(a,b)\), put \(f=\log p\). The source’s Lemmas 2.1 and 2.2, “Extreme rectangles in a ferromagnetic marginal” and “Pointwise ratios with arbitrary common pins”, bound \[\left|f(a,b)+f(a',b')-f(a,b')-f(a',b)\right| \le 4\operatorname{arctanh}q_0,\] where \(q_0\) is the zero-field FK connection probability after deleting the common pinned vertices and separately contracting the two terminal sets. Deletion retains the fields induced by the pinned spins in the original law; all these fields are then set to zero in the comparator. The right side is symmetric in the terminals. A dual circuit in the separating collar bounds \(q_0\) away from one; several separated dual circuits give its power decay in the small-patch case. Normalizing either family of conditional laws turns the oscillation bound into the displayed pointwise ratio bounds. Thus reversing which terminal is observed requires no new geometric estimate; this is also Remark 2.3 of the companion. ◻

Proof. First change the spins outside \(B_{2R}\), keeping all inner pins common. Proposition 3 across the outer collar costs a fixed factor. Next change the spins in \(B_{r/2}\), keeping the outer data common. Use the terminal-symmetric form of the same proposition across the collar from \(r/2\) to \(r\): the small terminal is now the source and \(A\) lies beyond its fixed-ratio collar. This costs another fixed factor. The product is independent of the distance between the collars. Finally mix the exterior data according to the full-plane law. Cross-domain comparison is legitimate because both collars are common lattice charts. ◻

Source and signed-measure interpretation. These are Proposition 4.2, “Transmission”, and Corollary 4.11, “Random and signed sources”, in Section 4, “Ordered transmission across a cut”, of (OpenAI 2026). To clarify the signed assertion, put \(m=\mathbb Eq_+=\mathbb Eq_-\). If \(m=0\), both sides vanish. Otherwise the source probability measures \(m^{-1}q_+\mu_S\) and \(m^{-1}q_-\mu_S\) are stochastically ordered by Definition 5. Couple them in that order, apply the deterministic-source transmission inequality, and multiply by \(m\). Its nonnegative magnetic difference permits averaging and gives (4). No division by a mesh-dependent lower bound on \(m\) occurs. ◻

Proof. The second line sum is an \(\mathcal F\)-measurable competitor for the orthogonal projection. Apply (5). This is also the Corollary 5.2, “Prediction from a nearby contour”, in Section 5 of (OpenAI 2026). ◻

Sources and the domination needed for smearing. The construction of smeared magnetization fields from spin correlations is developed in (Camia et al. 2015, sec. 3). We give the domination argument for the spatial test functions used here. The spin limit and its full-plane normalization follow from (Chelkak et al. 2015, Theorems 1.1–1.2, Remark 1.4, and Equation (1.6)). That paper uses a rotated lattice with nearest-neighbor spacing \(\sqrt2\) times its mesh parameter; its positive normalization constant is absorbed into \(a_\sigma\). The plus-boundary one-point convergence in a fixed square and the full-plane two-point asymptotic give (8) and (7) for large distances. Enlarging \(C\) covers the finitely many smaller distances.

For (9), put disjoint squares of radius a small fixed fraction of the nearest other insertion distance around the distinct insertion sites. Conditional on the spins outside these squares, the squares are independent. Boundary monotonicity bounds the absolute conditional mean at each insertion by the corresponding plus-square mean. A site appearing an even number of times contributes one; for a repeated site the corresponding factors on the right of (9) are one. Multiplication gives the bound, after harmless bounded-distance changes of the boxes.

For each fixed \(n\ge2\), the rescaled correlation kernel \[K_{\delta,n}(z_1,\ldots,z_n) =a_\sigma^{-n}\delta^{-n/8} \mathbb E_0\prod_{i=1}^n\sigma_{z_i/\delta}, \qquad z_i\in\delta\mathbb Z^2,\] is uniformly bounded in \(L^2\) on bounded products, with normalized lattice counting measure \(\delta^{2n}\). To see this, square the nearest-distance majorant in (9) and bound the product of nearest-neighbor maxima by \[|K_{\delta,n}(z_1,\ldots,z_n)|^2 \le C_n\sum_{\substack{f:\{1,\ldots,n\}\to\{1,\ldots,n\}\\ f(i)\ne i\text{ for all }i}} \prod_{i=1}^n(\delta+|z_i-z_{f(i)}|)^{-1/4}.\] There are finitely many directed choices \(f\). Each resulting graph consists of directed cycles with attached trees. Integrate tree leaves first. A two-cycle has a single singularity of power \(1/2\); on a longer cycle integrate one vertex by Cauchy–Schwarz and then the remaining path. Only one-variable singularities of power at most \(1/2<2\) occur. Their bounded-domain integrals and mesh-regularized lattice sums are uniformly bounded, including when lattice insertion sites coincide. This proves the \(L^2\) bound; its constants may depend on the fixed \(n\), with no assertion here about their growth in \(n\). The same integration argument with \(\delta=0\) gives the corresponding bound for the continuum correlation kernel. The first moment vanishes by spin symmetry, and the zeroth moment is one.

On a fixed bounded product, the set where any two insertion points are within distance \(\varepsilon\) has normalized counting measure \(O_n(\varepsilon^2+\delta^2)\). Cauchy–Schwarz therefore bounds its contribution to a smeared moment by \(C_n(\varepsilon+\delta)\), with the fixed test-function bounds absorbed in \(C_n\). Away from these diagonals the correlation convergence is uniform. Multiply by a continuous cutoff that vanishes when a pair is within distance \(\varepsilon\) and equals one when all pairwise distances are at least \(2\varepsilon\). The remaining continuum integrand is bounded and Riemann integrable: its discontinuities lie in the finite union of the coordinate lifts of the spatial test functions’ null discontinuity sets. Its lattice Riemann sums consequently converge. First take the mesh limit and then let \(\varepsilon\downarrow0\), using the preceding uniform bound. This proves the asserted joint smeared moments. Finally the covariance rule follows directly from (6) and \(|\psi(z)-\psi(w)|=|z-w|\,|\psi'(z)\psi'(w)|^{1/2}\) for Möbius maps, with complex conjugation included for an orientation-reversing map. ◻

Proof. The finite-bond argument below is a special case of the reflection-positivity method of Fröhlich et al. (1978). In a finite reflection-symmetric box, reflection through a bond line leaves the within-half weights paired. Each bond crossing the line has matrix \((e^{K_0st})_{s,t=\pm1}\), whose eigenvalues are \(2\cosh K_0\) and \(2\sinh K_0\), both positive. Tensoring its positive decomposition and summing the spins in the two halves writes \(\mathbb E[F\,\overline{F\circ\vartheta}]\) as a sum of nonnegative squares for every half-box function \(F\). For a reflection through vertices, condition on the spins on the line: the two remaining halves are independent with reflected laws and the same conclusion follows. Exhaust the plane by symmetric boxes, then use the moment convergence of Proposition 9 for polynomials in smeared fields whose supports lie strictly on one side. Approximation up to the line, when needed, follows in \(L^2\) from (7). ◻

Normalization and exact criticality. Use only the pure specialization of (Giuliani et al. 2012, Equations (1.8), (1.11), and Theorem 1.2). Their scaled energy is \(a^{-1}\sigma_x\sigma_{x+ae_j}\), and their superscript \(T\) denotes connected expectation. At two insertions their massless formula is \(1/(\pi^2|z-w|^2)\), independently of both orientation labels: the two permutations and the two nonzero sign choices each give \(|z-w|^{-2}\), with prefactor \(1/(4\pi^2)\). The remainder bound in their Equation (1.16), together with the following propagator estimate, is uniform when the two points remain in a fixed annulus of separations. This supplies uniformity in the direction of the displacement.

Their formulation permits a nonzero mass parameter tending to zero. To obtain the exactly critical lattice limit, fix a sequence \(a\to0\) and choose a subcritical \(\beta(a)\) so close to \(\beta_0\) that the finitely many spin covariances being tested differ from their critical values by \(o(a^2)\), and \(|\beta(a)-\beta_0|=o(a)\). Such a choice is possible by continuity of finite-spin expectations at the unique critical Gibbs state. The cited formula applies along this sequence; the error divided by \(a^2\) vanishes. This proves (10). Subtraction of the mean does not change a covariance. The finite-sum assertions then follow term by term, since each fixed displacement \(d_q\) is negligible compared with \(|x|\) and every leading coefficient has the same positive sign. ◻

Conversion from the published spin-arm estimate. Pointwise comparison reduces the event to a plus-boundary model in a fixed larger square, with a multiplicative constant depending only on the outer collar. Rescale so that \(R=1\). For a fixed \(u\), couple a subsequence of complete loop collections to their limit as in Theorem 13. Only loops of diameter at least \(1-u\), or a fixed smaller cutoff after relaxing radii, can contain the passages under consideration. There are finitely many such limit loops. Uniform parametrized convergence preserves four passages after replacing the two radii by \(2u\) and \(1/2\). Indeed choose the passages by last and first hits of the two circles, pass to convergent endpoint parameters, and use uniform continuity of the limiting parametrizations. An interval crossing the relaxed annulus cannot collapse to a point. Disjoint interval interiors remain distinct; injectivity of a limiting loop prevents two nontrivial intervals from having the same image. This also treats passages belonging to one loop and passages traversed in opposite directions.

We next pass from this limiting geometric event to strong alternating spin arms. Crop its through subarcs once more, to the radii \(4u\) and \(1/4\). The finitely many chosen compact subarcs of the simple, pairwise disjoint limit loops have disjoint thin neighborhoods after this crop. They are crosscuts of the annulus. Reading their sides in cyclic order changes the spin sign at each crosscut. If more than four through crosscuts occur, keep four consecutive ones and the adjacent alternating sides. One can follow each chosen side in its own thin neighborhood from the inner to the outer cropped circle.

Now use leftmost approximations for a plus side and rightmost approximations for a minus side, in the same spin configuration. By (Benoist and Hongler 2019, sec. 2.5), a leftmost loop has a strong plus path on its left, and a rightmost loop has a strong minus path on its right. The simultaneous circuit approximation and common-edge refinement in (Benoist and Hongler 2019, Theorem 6 and Remark 7) make both extremal approximations converge to the same selected loop, with its orientation. Their chosen side paths therefore lie in the disjoint neighborhoods and cross the cropped annulus for all sufficiently fine meshes. They supply four disjoint strong spin paths, of alternating colors. The cropping allows their endpoints to be moved by the bounded lattice distance needed to follow the colored side.

This is an eventual implication on the coupled limit event, so Fatou’s lemma bounds its probability by the lower limit of the four-strong-arm probabilities at the further relaxed radii. Wu’s Theorem 1.2 gives the interior alternating spin exponent \((16j^2-1)/24\), hence \(21/8\) for four arms. More precisely, (Wu 2018, Equation (5.4) and Corollary 5.4) gives the required mesh-first upper bound \(u^{21/8+o(1)}\) for these annular events, uniformly in a larger containing square. Combining the two relaxations yields (11); multiplying \(u\) by a fixed constant does not change its limiting exponent. Square and circular annuli contain one another after fixed-factor radius changes.

No microscopic inclusion of arbitrary resolved passages in the strong-arm event was used. Such an inclusion would be false as a justification at an individual alternating face, where a colored side of the fixed resolution may have only weak adjacency. Complete oriented convergence and the extremal side paths are the steps that remove this convention difference. ◻

Proof. Theorem 1 of (Duminil-Copin et al. 2011) gives critical \(q=2\) FK rectangle crossing bounds uniform over boundary wirings. Its Section 2.1 gives the domain Markov property, association, planar duality, and Edwards–Sokal coloring. A finite chain of overlapping rectangle crossings, including overlap-square crossings, constructs an FK connection through a prescribed corridor. The analogous finite construction gives an FK circuit in a band. All aspect ratios and the number of rectangles are fixed. A dual circuit in a disjoint outer collar isolates the constructed cluster from the boundary. Uniform conditional RSW permits this dual event and the inner open tests to be imposed in sequence. The isolated cluster then receives either prescribed spin color with probability \(1/2\). For several opposite-color constructions keep their isolating collars disjoint; their clusters can be colored independently. Same-color connections may merge, and spin FKG combines the finitely many same-color tests. This proves positivity in the reference free or plane charts.

Mixed-sign spin pins are not arbitrary FK wirings: identifying the two notions would invalidate this argument. Instead, Proposition 3 transfers the reference lower bound through each unpinned collar. Reveal separated constructions successively; the remaining collars stay free and the lower bounds remain valid. For a single annular circuit one can first transfer the individual rectangular tests and then use pure-spin FKG. Consequently the argument does not require the entire annulus to be a square box avoiding all interior pins.

For the power bound choose \(m\ge c\log(R/r)-C\) annular bands of a fixed aspect ratio with separated test collars. Each has a conditional probability at least \(p>0\) of a strong spin circuit of either specified color. A circuit of the opposite color blocks a monochromatic arm. A strong spin circuit of either color also blocks a contour passage, since disagreement edges cannot cross its equal-spin bonds. Conditional iteration therefore gives \(C(1-p)^m\le C'(r/R)^\rho\), with a union bound for two possible arm colors. This is a spin and contour deduction from FK RSW, coloring, and buffered comparison; no FK arm exponent is being identified with a spin arm exponent. Pins touching a corridor’s terminals without a buffer are outside the assertion. ◻

Proof. Let \(Q:L^2(\nu_D)\to L^2(\nu_{F_*})\) be conditional expectation from \(D\) to \(F_*\). Its integral kernel is the density of the joint law of \((\sigma_{F_*},\sigma_D)\) relative to the product of its marginals. The buffered comparison bounds this density by a fixed constant, hence \(\|Q\|_{\mathrm{HS}}\leq K_1\). Choose \(m+1\) equally spaced nested enlargements \(F_0\subset\cdots\subset F_m=F_*\). Regarding all conditional expectations as orthogonal projections in \(L^2(\nu)\) gives \[\sum_{j=1}^m\|(P_{F_j}-P_{F_{j-1}})Q\|_{\mathrm{HS}}^2 \leq\|Q\|_{\mathrm{HS}}^2\leq K_1^2.\] Choose \(m\) so that \(KK_1m^{-1/2}\leq t\). At least one summand is at most \(K_1^2/m\). For its endpoints \(F_-,C\), nesting implies, for every intermediate \(F\), \[\|(P_C-P_F)h\|_2 \leq\|(P_C-P_{F_-})Q\|_{\mathrm{op}}\|h\|_2\leq t.\] We used \(P_Fh=P_FQh\), which holds since \(F\subset F_*\). This proves the simultaneous, rather than likelihood-dependent, assertion. ◻

Proof. Put \(a=P_Ch\) and \(z=(I-T_M)a\). Then \(\mathbb E_\nu z=0\), \(|z|\leq2K\), and, for every increasing \(g\) of the core spins, \[ \mathbb E_\nu[zg]=\mathbb E[\mathop{\mathrm{Cov}}_\nu(a,g\mid Y)]\geq0. \tag{22}\] To justify the last inequality, conditioning on \(Y=y\) changes the Ising weight by \(\exp(Ms^{-7/8}\sum_x y_x\sigma_x)\) times a constant. The conditional law remains ferromagnetic, with external fields of arbitrary signs, so conditional FKG applies (Fortuin et al. 1971, Proposition 1 and pp. 98–99). The function \(z\) itself need not be increasing. Positivity of \(I-T_M\) and its contraction bound also give \[ \langle z,a\rangle =\langle a,(I-T_M)a\rangle\geq\|z\|_2^2. \tag{23}\]

Suppose \(\|z\|_2\geq b\). Choose the window tolerance less than \(b^2/(4K)\). For every \(F\) in that window, \[\langle P_Fz,h\rangle=\langle z,P_Fh\rangle \geq b^2-2Kt\geq b^2/2.\] Subdivide an inner fixed portion of the window into \(n\) equal radial steps. Denote the successive spin sets by \(F_0',\ldots,F_n'\) and choose a separating square contour, or the two contours of a band, \(J_i\) strictly between \(F_i'\) and \(F_{i+1}'\). Its distance from \(F_i'\) is comparable to \(w=s/n\), with a constant depending on the already fixed window. Define \(m_i=s^{-7/8}\sum_{x\in J_i}\sigma_x\). The signed source \(P_{F_i'}z\) is order-positive: an increasing \(F_i'\)-measurable test has the same pairing with it as with \(z\). Subdivide \(J_i\) into straight pieces of a sufficiently small fixed multiple of \(w\). Their enlarged charts are pin-free, they have bounded overlap, and the target layer \(D\) is at distance comparable to \(s\). These are exactly the admissible cuts of Proposition 6. For its fixed exponent \(\eta>0\), that proposition yields \[ \langle P_{F_i'}z,m_i\rangle\geq c_b n^{-7/8+\eta}. \tag{24}\] The constant does not deteriorate further as \(n\) increases.

The observation predicts each \(m_i\) with arbitrarily small error when \(n\) is fixed. To see this quantitatively, average it over parallel nearby lines, including the corresponding short endpoint corrections, in a strip of width \(\theta s\) lying inside \(C\). Call the average \(\bar m_i\). All these lines have the same length and buffer bounds, with displacement at most \(C\theta s\); a square contour is treated side by side, including its short endpoint pieces. For fixed \(\theta\), Proposition 7 is uniform over the averaging lines: otherwise choose a maximizing line at each mesh, obtaining a sequence of pairs that still satisfies that proposition with \(w/s=\theta\). Minkowski’s inequality therefore gives, even though the number of averaging lines grows with \(s\), \[\limsup_{s\to\infty}\|m_i-\bar m_i\|_2\leq C\theta^{3/8}.\] The average is a linear combination of the signals \(s^{-7/8}\sigma_x\), with coefficients of size \(O((\theta s)^{-1})\) on \(O(\theta s^2)\) sites. Replacing each signal by \(Y_x/M\) gives a \(Y\)-measurable estimator of \(\bar m_i\) with noise variance at most \(C/(M^2\theta)\). Consequently \[\limsup_{s\to\infty}\|m_i-\mathbb E[m_i\mid Y]\|_2 \leq C\theta^{3/8}+C/(M\sqrt\theta).\] Since \(\langle z,m_i\rangle=\mathbb E[\mathop{\mathrm{Cov}}(a,m_i\mid Y)]\), first choosing \(\theta\) small and then \(M\) large makes this pairing smaller in absolute value than \(\tfrac12c_b n^{-7/8+\eta}\).

Set \(d_i=(P_{F_{i+1}'}-P_{F_i'})z\). The variable \(m_i\) is measurable on \(F_{i+1}'\), whereas \(d_i\) is orthogonal to all \(F_i'\)-measurable functions. Therefore \[|\langle d_i,m_i-P_{F_i'}m_i\rangle| =|\langle z,m_i\rangle-\langle P_{F_i'}z,m_i\rangle| \geq\tfrac12c_b n^{-7/8+\eta}.\] Corollary 8, with a nearby contour in \(F_i'\) and with \(n\) fixed before \(s\to\infty\), bounds its second factor by \(Cn^{-3/8}\). Hence \(\|d_i\|_2\geq c_b'n^{-1/2+\eta}\) for all sufficiently large \(s\). Orthogonality of the increments gives the contradiction \[ 4K^2\geq\|z\|_2^2\geq\sum_{i=0}^{n-1}\|d_i\|_2^2 \geq c_b''n^{2\eta}. \tag{25}\] Choose \(n\) large enough before choosing \(\theta,M,s\). Thus the complete order is \(b\), the window tolerance and width, \(n\), \(\theta\), \(M\), and finally the mesh threshold. Only finitely many contours and windows occur after \(n\) has been fixed, so their thresholds can be combined. ◻

Proof. Use Lemma 18, then (26), and finally the finite-mode construction. To pass from the reference law \(\nu\) to a full-plane pairing, choose the buffered reference chart inside the specified enlargement, disjoint from the paired supports, and condition the plane law on its exterior spins. The conditional marginal on \(D\) has a likelihood \(h\) in (18), and its core likelihood is \(P_Ch\). The paired observable is measurable outside that chart, so integrating the bound against its signed values gives the same correlation estimate under \(\mu_0\). For the finite-mode error, every such core likelihood is bounded by \(K\); its \(L^1(\nu)\) error therefore costs at most that fixed factor. The replacing function is the same for every exterior configuration. For finitely many required supports the menu is their finite product, and errors are divided by their number. The bounded functions of the finite-dimensional sums are equicontinuous; tightness of those sums and compactness on bounded coordinate sets give the subsequential assertion by a diagonal argument. These replacements and their smeared coordinates are now evaluated under \(\mu_0\). Their limiting field law is therefore the full-plane spin-field law: Proposition 9 gives joint moment convergence for the fixed polygonal cell indicators. The prescribed rounding errors lie in vanishing neighborhoods of their finitely many edges. The uniform \(L^2\) bounds on fixed-order rescaled correlation kernels proved there, and the vanishing lattice counting measure of these neighborhoods, make the errors tend to zero by Cauchy–Schwarz. The finite menu of spatial smearings is fixed before the mesh limit. The unsquared version of the ordering argument (30) supplies exponential integrability of each fixed smeared sum and hence moment determinacy. Averaging over the finite symmetry group preserves all bounds. Applying the result to one side and then the other costs at most the sum of the two errors. ◻

Proof. Set \(w_i=(\prod_{j\ne i}|z_i-z_j|)^{-1}\) and \(C_z=\prod_{i<j}|z_i-z_j|^{1/2}\). For the positive subset \(I=\{i:u_i=1\}\), direct multiplication gives \[ W_z(u)=C_z\,|\Delta(z_I)|^2\prod_{i\in I}w_i, \tag{40}\] where \(\Delta\) is the Vandermonde determinant. Let \(Z_k\) be the sum of the determinant weights on the right, without \(C_z\), over \(|I|=k\); set \(Z_0=1\). For the positive measure \(\sum_iw_i\delta_{z_i}\) on this finite set of complex points, let \(p_j\) be the monic orthogonal polynomial of degree \(j\), and let \(h_j=\sum_iw_i|p_j(z_i)|^2\). Cauchy–Binet applied to the evaluation matrix gives \[ Z_k=h_0\cdots h_{k-1},\qquad 1\leq k\leq2m. \tag{41}\] The weight in (40) without \(C_z\) is also \(\prod_{i\in I,j\notin I}|z_i-z_j|^{-1}\). It is unchanged under complementation. Thus \(Z_{2m-k}=Z_k\) and \[ h_mh_{m-1}=Z_{m+1}/Z_{m-1}=1. \tag{42}\]

Consider the probability distribution on \(m\)-subsets proportional to these weights. Its inclusion probabilities are the minors of the orthogonal projection \(P\) onto polynomials of degree at most \(m-1\), viewed on the finite weighted point set. This is another direct Cauchy–Binet identity: summing squared minors containing a prescribed set leaves the corresponding principal minor of \(P\). Let \(M_z\) denote multiplication by \(z\) on that space. Expanding the absolute-square variance with the one- and two-point inclusion probabilities gives \[\begin{align*} \mathop{\mathrm{Var}}\left(\sum_{i\in I}z_i\right) &=\operatorname{tr}(PM_z^*M_z) -\operatorname{tr}(PM_z^*PM_z)\\ &=\|(I-P)M_zP\|_{\mathrm{HS}}^2 =h_m/h_{m-1}. \end{align*}\] The last equality uses the orthonormal polynomial basis: multiplication by \(z\) sends its first \(m-1\) vectors into \(\operatorname{ran}P\), and the last vector has complement equal to \(p_m/\sqrt{h_{m-1}}\). This argument uses complex absolute-square variance; no assumption that the points lie on a real line is present.

Complementation shows that \(\mathbb E\sum_{i\in I}z_i=\tfrac12\sum_i z_i\). Since \(\sum_i u_i z_i=2\sum_{i\in I}z_i-\sum_i z_i\), we obtain \[\frac{A_0^d}{A_0}=4\frac{h_m}{h_{m-1}}=4h_m^2, \qquad \frac{A_2}{A_0}=\frac{Z_{m+1}}{Z_m}=h_m,\] which proves (39). If \(|z_i|\leq1\), the minimizing property of monic orthogonal polynomials, applied to \(zp_{j-1}\), gives \(h_j\leq h_{j-1}\). Together with (42) this implies \(h_m\leq1\). Ratios \(Z_{m+j}/Z_m=h_m\cdots h_{m+j-1}\) are at most one, and complementation treats negative charges. ◻

Proof. We compute first with an even number \(n=2m\) of inner insertions \(z_i\) and \(n'=2m'\) reflected outer insertions \(y_j\). Both insertion sets are inside the unit disk and have distinct points. After including the dilation and reflection weights, the squared spin formula in Proposition 9 becomes \[ 2^{-(n+n')/2}\sum_q r^{q^2/4} \sum_{\substack{\sum u_i=q\\\sum v_j=-q}} W_z(u)W_y(v) \prod_{i,j}|1-rz_i\bar y_j|^{u_i v_j/2}. \tag{44}\] For example, the power of \(r\) is the primary factor \(r^{n/4}\) in the squared expression times the internal-distance factor \(r^{(q^2-n)/4}\). All powers of \(|y_j|\) from inversion cancel. This verifies the normalization of the radial degree.

Angular frequency refers here to the terms \(e^{ij\theta}\) produced by rotating one whole insertion set, \(z_i\mapsto e^{i\theta}z_i\). Divide (44) by its constant term, and abbreviate the charged sums of the \(y\) variables by \(B_q,B_0^d\). The \(q=\pm2\) terms have first-order coefficient \[p=2(A_2/A_0)(B_2/B_0).\] In the charge-zero term the first-order cross factor is proportional to \(\operatorname{Re}[(\sum_i u_i z_i) \overline{(\sum_jv_j y_j)}]\) and vanishes after summing either sign family. To find the angularly constant second-order coefficient, use \[\log\prod_{i,j}|1-rz_i\bar y_j|^{u_iv_j/2} =-\tfrac r2\operatorname{Re}(X\bar Y) -\tfrac{r^2}{4}\operatorname{Re}(X_2\overline{Y_2})+O(r^3),\] where \(X=\sum u_i z_i\), \(X_2=\sum u_i z_i^2\), and similarly for \(Y,Y_2\). Squaring the linear term in the exponential contributes \(r^2|X|^2|Y|^2/16\) in angular frequency zero. Its normalized sum is \[q_0=\frac{A_0^dB_0^d}{16A_0B_0}.\] Every other second-order term has angular frequency one or two: frequency one comes from the first cross correction in the \(q=\pm2\) sector, and frequency two from the remaining charge-zero terms. Charges of absolute value at least four begin at \(r^4\).

The correlation itself is the positive square root. Its scalar second-order coefficient is \(q_0/2-p^2/8\). By Lemma 20, \(q_0=p^2/4\), so this coefficient is zero. Averaging one insertion set over quarter turns kills the nonzero angular frequencies just listed. The first-order coefficient in the unsquared correlation is \[ \bigl(S(z_1,\ldots,z_n)A_2/A_0\bigr) \bigl(S(y_1,\ldots,y_{n'})B_2/B_0\bigr). \tag{45}\] It factors across the gap. An empty insertion set contributes only the constant state.

These computations survive smearing. For a fixed number of insertions and \(|r|\) sufficiently small, expand the real-analytic factors into their absolutely convergent power series in \(r\) and the angular variables. The ratios of charged sums are bounded by Lemma 20; the cross factors and their first three derivatives are bounded uniformly when \(|z_i|,|y_j|\leq1\) and \(|r|\leq r_0<1\). The remaining integrable dominating function is a constant times the product of the within-set spin correlations. Thus every even quarter-turn invariant smeared polynomial has the bilinear expansion \[ (F,D_{-\log r}G)_{\mathrm{rad}} =\mathbb E\overline F\mathbb EG+r\overline{c(F)}c(G)+O_{F,G}(r^3), \tag{46}\] with the linear functional \(c\) given by (45) and linear extension.

We explain why the polynomial-dependent remainder in this calculation implies the uniform bound in the proposition. In a self-pairing the spectral functional calculus (Bühler and Salamon 2016, Theorem 6.4.1 and Section 6.5, especially Theorem 6.5.4) represents the left side of (46) as \(\int_{[0,\infty)}r^d\,d\mu_F(d)\) with \(\mu_F\) positive. Letting \(r\downarrow0\) identifies its mass at zero as \(|\mathbb EF|^2\). After subtracting it, divide by \(r\). Positivity excludes mass on \((0,1)\) and identifies the mass at \(1\) as \(|c(F)|^2\). After subtracting this atom as well, the remainder is positive and \(O_F(r^3)\), which excludes mass on \((1,3)\): any positive mass in a compact interval with upper endpoint below \(3\) would violate that bound. Density of the polynomial tests excludes these spectral intervals on the entire stated subspace.

Polarization and the two factorizations in (46) show that the projections at \(0\) and \(1\) have rank one, provided the latter is nonzero. It is nonzero already for a two-spin test smeared in two small disjoint regions: choose nonzero nonnegative smooth compactly supported bumps \(\psi_1,\psi_2\) in those regions and use \(\Phi(\psi_1)\Phi(\psi_2)\). The coefficient is strictly positive since \(S(z_1,z_2)A_2/A_0>0\) for distinct points; square-group averaging preserves this positive scalar coefficient. Choose a unit real vector in that eigenspace and denote the corresponding coefficient by \(\ell\). Then \(|\ell(F)|\leq\|F\|_{\mathrm{rad}}\leq\|F\|_2\). On the remaining spectral subspace, the norm of \(D_{-\log r}\) is at most \(r^3\). Cauchy–Schwarz proves (43). Finally (37) and polynomial density give the asserted extension. ◻

Proof of Theorem 16. Choose \(L\) large enough that the term \(C/L\) in (54) is less than \(a/4\) and the fixed support enlargements are disjoint. Next choose an integer \(H>L\) large enough that all terms \(CL/H^2\) are less than \(a/4\), and that the lower bound needed in (51) holds. Choose every replacement accuracy small compared with the now fixed amplification factors \(H,L^2,HL^2\). In each replacement construction, choose the window, its finite subdivision, the observation strength, and the coarse partition before taking the mesh small. Finally choose \(s_0\) so that the uniform upper limits (54) and (56) are within their remaining error margins, and so that the norm and right-inverse bounds hold for every \(s\geq s_0\). Homogeneity gives the estimates for arbitrary \(f\).

All square contours and cell boundaries used here are specified by integer rounding. They remain in the same fixed buffered charts; their rescaled locations converge to the prescribed contours and cells. Changes of cell indicators are confined to shrinking neighborhoods of their finitely many polygonal edges; their normalized lattice counting measure tends to zero. The uniform \(L^2\) bounds on fixed-order rescaled correlation kernels and Cauchy–Schwarz therefore make the resulting smearing errors tend to zero. Spatial test functions and their finite menus are fixed before the mesh limit. Only these specified geometric roundings and the convergent cell-normalization scalars depend on the mesh, not arbitrary varying Riemann-integrable spatial functions. The separate line-continuity estimate controls the auxiliary square contours. Consequently changing an auxiliary contour by a bounded number of lattice layers affects none of the limiting estimates or their uniform thresholds. The conditioning convention in (14), however, is fixed throughout, so the Markov identity (55) is exact at every scale. ◻

Proof. Since both \(f_x\) and its conditional expectation lie in \([-a,a]\), the bracket in (58) lies in \([0,2]\). Its conditional integral equals one. For a configuration \(\sigma\), the incoming mass of an \(x\)-refresh applied to \(\mu_t\) is \[\mu_t(\sigma)\left[1+ \frac{f_x(\sigma)-P_{t,x}f_x(\sigma)}{2a}\right].\] Indeed, summing the old interior configuration first leaves its exterior marginal, which is then multiplied by the refresh conditional law. Subtract the outgoing mass \(\mu_t(\sigma)\) and multiply by \(2ac\). The result is \(c\mu_t(\sigma)(f_x-P_{t,x}f_x)\). Summing over \(x\) proves the adjoint identity, including its sign and normalization. The finite-state forward equation has a unique solution, so the reversed family is the law of the stated time-inhomogeneous chain. The zero case is immediate. ◻

Proof. Choose a connected covering list \(\mathcal C\) of \(B_{Ls}\) by \(c_L\ge2\) squares of radius \(s\), all contained in \(B_{Ls}\), including the root square. Such a list can be obtained from centers \((as,bs)\), \(-L+1\le a,b\le L-1\). It has size depending only on \(L\), and its union is exactly \(B_{Ls}\).

Expand the numerator in (63) in its exponential series. Expand the reciprocal denominator as \[\frac1{P^0e^H} =\sum_{j\ge0}\bigl(-P^0(e^H-1)\bigr)^j.\] The absolute scalar series is bounded by \[\|f\|_\infty\frac{e^h}{2-e^h},\qquad h=\|H\|_\infty<\log2.\] Its coefficient of degree \(p\) is bounded by \(K_0K_1^p\|f\|_\infty\) for fixed numerical \(K_0,K_1\). Each degree-\(p\) coefficient is a sum of products of pure conditional expectations with precisely \(p\) inserted translates of coefficients of \(V\). It depends only on the spins in the covering list \(\mathcal C\) and the inserted lists. Give it the concatenated list consisting of \(\mathcal C\), followed by all inserted lists, retaining repeated boxes. An inserted list meets the integrated square and hence is connected to \(\mathcal C\). If its input lengths are \(m_1,\ldots,m_p\), its output length is exactly \[ n=c_L+\sum_{i=1}^p m_i. \tag{67}\] Center each output coefficient with the fixed linear projection. This costs a factor of at most two in its sup norm. The placement bound (62) supplies a factor \(C_Lm_i\) per insertion; the factors \(s^2\) have canceled with the translation normalization. Taking the square-group average of this rule makes it equivariant, without increasing a sum of absolute norms. Spin flip commutes with every pure conditional integration.

Here is a convergent majorant, including the dependence on list size. Let \(a_m(t)\) be the sum of nonnegative coefficient majorants on lists of length \(m\), and put \[X_b(t)=\sum_{m\ge1}m e^{b(m-1)}a_m(t),\qquad Y_b(t)=\sum_{m\ge1}m(m-1)e^{b(m-1)}a_m(t).\] The majorant includes the absolute values of the initial coefficients and of both fixed forcing terms \(-f\) and \(P^0f\). By (67), for \(d/2\le b\le d\) the contribution of terms with \(p\ge1\) insertions to its \(X_b\) derivative is at most \[ K\|f\|_\infty K^p \left[c_L X_b^p+p(X_b+Y_b)X_b^{p-1}\right], \tag{68}\] where \(K=K_{L,d}\). To verify this bound, the exponential weight of an output is the product of the input weights times \(e^{b(c_L-1+p)}\), absorbed in \(K^{p+1}\). Its additional size factor \(n\) is the sum of \(c_L\) and the \(m_i\). Choosing the latter term at the \(i\)-th insertion replaces one factor \(\sum m e^{b(m-1)}a_m=X_b\) by \(\sum m^2 e^{b(m-1)}a_m=X_b+Y_b\). This proves (68), including the unbounded size factor.

Set \(b(t)=3d/4-dt/8\), reserving a final exponent gap \(b(1)-d/2=d/8\). As long as \(KX_{b(t)}\le1/2\), summing the geometric series and its derivative in (68) gives \[ D^+X_{b(t)}(t) \le -\frac d8Y_{b(t)}(t) +C_{L,d}\|f\|_\infty\bigl(1+Y_{b(t)}(t)\bigr). \tag{69}\] Furthermore \[X_{3d/4}(0)\le \left(\sup_{m\ge1}m e^{-d(m-1)/4}\right)\|V\|_{s,d}.\] Choose the input radius so that \(C_{L,d}\|f\|_\infty\le d/16\) and the resulting bound \(X_{b(t)}(t)\le X_{3d/4}(0)+C_{L,d}\|f\|_\infty t\) is strictly below \(1/(2K)\). A first-exit argument then gives this bound through time one. In particular it gives (65).

For completeness, the same estimates construct the solution rather than only bounding a hypothetical one. First retain only output lists of length at most \(M\). By (67), every nonlinear coefficient equation at length \(n\) uses only coefficients of lengths strictly smaller than \(n\); only the fixed forcing occurs at length one or at the fixed cover length. The truncated equations therefore have successive explicit integral solutions. Their coefficients are polynomials in the finitely many input coefficients of lengths at most \(M\). Truncations agree on all smaller lengths. The corresponding nonnegative triangular majorants obey (69) and increase to a majorant for all coefficients. Their tails satisfy, uniformly on the complex input ball, \[\sup_{0\le t\le1}\sum_{m>M}e^{(d/2)(m-1)}a_m(t) \le \frac{C\|V\|_{s,d}}{M+1}.\] Thus the truncations converge uniformly into \(C([0,1],\mathcal A_{s,d/2})\). They are holomorphic polynomials in finite-dimensional projections of the input, so their locally uniform limit is analytic. The reserved exponent gap also gives uniform convergence of the derivatives in \(\mathcal A_{s,d/2}\): the vector field is bounded in \(\mathcal A_{s,5d/8}\) by \(C\|f\|_\infty[1+\sum_{p\ge1}(KX_{5d/8})^p]\), so its tail above length \(M\), measured at exponent \(d/2\), is at most \(C\|f\|_\infty e^{-dM/8}\). Absolute convergence permits termwise conditional integration and time differentiation. This proves (64); uniqueness for this labeled expansion follows successively in each length. This construction is also valid for real data and commutes with symmetry averaging.

Integrating the two fixed forcing terms gives \((V-f)+P^0f\) at time one. Every remaining term contains one factor \(f\) and at least one coefficient of \(V(t)\). Its \(\mathcal A_{s,d/2}\) norm is bounded by \(C\|f\|_\infty\sum_{p\ge1}(KX_{d/2}(t))^p\), which is \(O(\|V\|_{s,d}^2)\). Integration proves (66). The same analytic bound on a complex ball, followed by the Cauchy estimates on a smaller ball, gives the two derivative estimates. All constants used above are independent of \(s\). ◻

Proof. Choose a spanning tree of the connected box graph using its vertex labels, and traverse its doubled edges starting at the root. The resulting walk has \(2(m-1)\) steps, each of sup-norm length at most \(2s\). Put \(q=\lfloor(L-1)/2\rfloor\), and retain the centers at times \(0,q,2q,\ldots\), through the last such time in the walk. Every old center occurs in the walk and is at distance at most \(2s(q-1)\) from the preceding retained center. Its radius-\(s\) square is therefore contained in the radius-\(Ls\) square at that retained center. Consecutive retained squares overlap. Removing repetitions leaves a connected covering list, containing the root, of length \(M\) with \[M-1\le\left\lfloor\frac{2(m-1)}q\right\rfloor \le\frac{m-1}{2}.\] In the final inequality we used \(q\ge4\). If the original union fits in \(B_{Ls}\), choose that single square instead. The same inequalities hold. The procedure uses only the labeled graph, so it can be chosen square-equivariantly; alternatively one can average its conjugates under the square group. Such averaging preserves all norm bounds.

Relabeling a function does not change it. The inequality \(e^{d(M-1)}\le e^{(d/2)(m-1)}\), summed over lists, proves the first bound. For \(m\ge2\), comparison with the original weight gives the extra factor \(e^{-d(m-1)/2}\le e^{-d/2}\), proving (71). Finally, the factor \(L^2\) cancels the change from \(s^{-2}\) to \((Ls)^{-2}\) in the translation normalization, which proves (70) term by term. ◻

Proof. Take a bounded single-square right inverse \(w_0\). Recursively define \[a_k=\ell_{k+1}T_{s_k}w_k,\qquad w_{k+1}=a_k^{-1}T_{s_k}w_k.\] The vectors remain single-square by (72). If \(\|w_k\|\le M\), Equation (74) gives \[|a_k-L|\le aM,\qquad \|w_{k+1}\|\le\frac{C_1L+aM}{L-aM}.\] Choose a fixed \(M\) larger than the initial right-inverse bound and \(2C_1+1\), and choose the parameters to make the displayed ratio at most \(M\). Induction gives \(\sup_k\|w_k\|\le M\) and \(a_k\ge L/2\ge4\). We have used real vectors, so the scalar multipliers are positive; their complexifications use the same fixed multipliers.

Every \(V_k\) has the unique decomposition \(V_k=\alpha_kw_k+u_k\), with \(\alpha_k=\ell_kV_k\) and \(u_k\in\ker\ell_k\). These coordinates have norms uniformly equivalent to \(\|V_k\|\). For \(u\in\ker\ell_k\), its single-square part has \(\ell_{s_k}\pi_{s_k}u=0\). Consequently (74) and (71) imply \[\|T_{s_k}u\|\le(a+\kappa)\|u\|.\] Set \[b_ku=\ell_{k+1}T_{s_k}u,\qquad D_ku=T_{s_k}u-w_{k+1}b_ku.\] Thus \(D_k:\ker\ell_k\to\ker\ell_{k+1}\). The uniform functional and vector bounds give \(\|b_k\|+\|D_k\|\le C(a+\kappa)\). Choose \(a\) and then \(\kappa\) so that \(\|D_k\|\le1/8\). The coordinate form of the linear recurrence is exactly triangular: \[ \alpha_{k+1}=a_k\alpha_k+b_ku_k, \qquad u_{k+1}=D_ku_k. \tag{76}\] There is no term from \(\alpha_k\) into the second equation, by the definition of \(w_{k+1}\).

We record the inverse of the inhomogeneous recurrence. Given \(u_0\) and forcing terms \(g_k\in\mathbb C\), \(v_k\in\ker\ell_{k+1}\) with finite norm \(\sup_k2^k\max\{|g_k|,\|v_k\|\}\), its only decaying solution is \[\begin{align*} u_k&=D_{k-1}\cdots D_0u_0 +\sum_{j=0}^{k-1}D_{k-1}\cdots D_{j+1}v_j, \tag{77}\\ \alpha_k&=-\sum_{j=k}^{\infty} \frac{b_ju_j+g_j}{a_k a_{k+1}\cdots a_j}. \tag{78}\end{align*}\] An empty product in these expressions is the identity. The first formula is ordinary forward substitution. Iterating the scalar equation to time \(n\), dividing by \(a_k\cdots a_n\), and letting \(n\to\infty\) gives the second; the terminal term vanishes for a decaying sequence. Conversely, absolutely convergent sums in (77)–(78) satisfy both equations. In the norm \[\|(\alpha,u)\|_*= \sup_{k\ge0}2^k\max\{|\alpha_k|,\|u_k\|\},\] they are a bounded linear function of \(u_0\) and the forcing norm \(\sup_k2^k\max\{|g_k|,\|v_k\|\}\). For example the forward convolution is bounded by the geometric sum \(2\sum_{j\ge0}(2/8)^j\), and the backward convolution by \(\sum_{j\ge0}4^{-j-1}2^{-j}\). These constants are uniform in \(k\) and in the lattice scales.

For the nonlinear recurrence use the forcing \[g_k=\ell_{k+1}R_{s_k}(\alpha_kw_k+u_k),\qquad v_k=(I-w_{k+1}\ell_{k+1})R_{s_k}(\alpha_kw_k+u_k).\] Equation (73) implies that on \(\|(\alpha,u)\|_*\le\rho\) this forcing has norm at most \(C\rho^2\) and Lipschitz constant at most \(C\rho\). To see the latter explicitly, at time \(k\) both arguments have norm at most \(C\rho2^{-k}\), and their difference has norm at most \(C2^{-k}\|(\alpha,u)-(\widetilde\alpha,\widetilde u)\|_*\). Multiplication by \(2^k\) leaves a factor at most \(C\rho\).

Insert this forcing into the two sums. Their bounded linear inverse is a contraction on a ball of radius \(C\|u_0\|\) when \(u_0\) is small enough. It produces the asserted unique sequence. Iteration of the contraction also proves analytic dependence on complex \(u_0\): the iterates are analytic and converge uniformly on smaller balls. Set \(h(u_0)=\alpha_0\). The construction respects conjugation, so \(h\) is real analytic on real data.

Finally the derivative at zero is given by the same sums with the quadratic forcing deleted. It parametrizes exactly all decaying solutions of (76). Uniform equivalence of coordinates proves the assertion about the tangent space and the trajectory estimate. ◻

Proof. Let \(G(V)=\ell_0V-h((I-w_0\ell_0)V)\). The graph in Lemma 25 is exactly \(G(V)=0\). We prove that the derivative of \(G(V_0(\beta,\lambda))\) in \(\beta\) at \((\beta_0,0)\) is nonzero.

The temperature derivative of (79) is the single-square vector \(Z_0=Js_0^2E\), modulo constants. Its successive linear images remain single-square and are \[ Z_k=Js_k^2\mathbb E_0[E\mid\sigma_{\partial B_{s_k}}],\qquad k\ge1, \tag{82}\] modulo constants. For the induction step, the pure Markov property gives \[\mathbb E_0\!\left[\mathbb E_0[E\mid\sigma_{\partial B_{s_k}}] \mid\sigma_{\partial B_{s_{k+1}}}\right] =\mathbb E_0[E\mid\sigma_{\partial B_{s_{k+1}}}].\] Although the two boundary sigma-fields are not nested, this identity follows by first conditioning on the full exterior of \(B_{s_k}^\circ\). The larger boundary is measurable there, and the conditional expectation of \(E\) given that exterior is the inner-boundary conditional expectation. Multiplication by \(L^2\) at each step proves (82).

Put \(y_k=(2s_k,0)\) and \(E_{y_k}=E\circ\tau_{y_k}\). This bounded observable is exterior to \(B_{s_k}^\circ\), so another use of the pure Markov property yields \[ \mathop{\mathrm{Cov}}_0(Z_k,E_{y_k}) =Js_k^2\mathop{\mathrm{Cov}}_0(E,E_{y_k}). \tag{83}\] Proposition 11 says that the right-hand side converges to a strictly positive number. Its square symmetrization is precisely what rules out cancellation between bond orientations. On the other hand \[|\mathop{\mathrm{Cov}}_0(Z_k,E_{y_k})| \le 2\|E\|_\infty\left\|Z_k\right\|_{\mathrm q}.\] Thus the quotient norms of \(Z_k\) do not tend to zero. If \(Z_0\) were tangent to \(G=0\), Lemma 25 would imply \(\|Z_k\|\le C2^{-k}\), a contradiction. This proves the required nonzero derivative.

The scalar analytic implicit-function theorem applied to \(G(V_0(\beta,\lambda))=0\) now gives a unique local real analytic branch \(\beta=\beta_c(\lambda)\) through \((\beta_0,0)\). All data are real, so the branch is real on a two-sided real interval. Shrink the interval so that \(\beta_c(\lambda)>\beta_0/2>0\) and the initial point lies in the small graph neighborhood. The analytic sequence constructed in Lemma 25 vanishes at \(\lambda=0\), hence its sequence norm is at most \(C_U|\lambda|\) on a smaller interval. Shrink it once more to obtain (80). Equation (81) is then Lemma 23. Every ingredient in this construction is a bulk object fixed before a domain or exterior configuration is chosen. ◻

Proof. A connected list of \(m\) radius-\(s\) squares has sup-norm diameter at most \(2sm\). Set \(M=\lfloor N/(8s)\rfloor\), changing fixed constants if needed. Every list of length less than \(M\) then has diameter less than \(N/2\). Such a translated support cannot intersect two distinct periodic copies of \(B_{Ls}^\circ\): the distance between these copies is at least \(N-2Ls\ge4N/5\), greater than that diameter. Accordingly, for the potential truncated to these lists, the plane conditional Hamiltonian with periodic exterior data is the same as the torus conditional Hamiltonian, up to terms constant in the updated spins. This is an identity of local Hamiltonians, not an assertion that plane and torus Gibbs measures are equal.

For the discarded lists, the weighted estimate gives \[ \sum_{|\ell|\ge M}|\ell|\|v_\ell(t)\|_\infty \le C r_s e^{-cN/s}. \tag{85}\] The placement count of (62) therefore bounds the sup norm of their plane conditional Hamiltonian by \(Cr_s e^{-cN/s}\). The same bound holds for the torus conditional Hamiltonian. To see the latter, enumerate every torus translation that can depend on the update spins by choosing one of its declared boxes meeting a periodic copy of the update square. The number of distinct translations is at most \(C_Lms^2\); counting a translation more than once only increases the upper bound. This proves the torus version of the same placement estimate, including supports with repeated periodic images.

If two real conditional Hamiltonians differ by at most \(h\) after subtracting constants, their normalized conditional densities have ratio between \(e^{-2h}\) and \(e^{2h}\). Thus their expectations of a bounded function differ by at most \(C\|f\|_\infty h\) for \(h\le1\). Apply this observation to truncation of each Hamiltonian. The exact identity for short lists and (85) yield, uniformly in the entire spin configuration, \[ \left|\left.(P_{Ls}^{V(t)}f)_x\right|_{\mathrm{periodic}} -P_{t,x}^{N}f_x\right| \le C r_s e^{-cN/s}. \tag{86}\] We used \(\|f\|_\infty\le r_s\le C\) to weaken the bound and simplify notation. For a long list, periodic evaluation of the plane conditional function can involve an image of an updated spin in what was the plane exterior. This causes no problem: the comparison is pointwise, and the entire dependence in question is included in the discarded tail.

Differentiate the finite Gibbs density and use Equation (64) with periodic evaluation. Replacing each plane conditional term by its exact torus counterpart changes the logarithmic derivative by at most \(Cr_s(N/s)^2e^{-cN/s}\), before centering. The torus counterpart has zero \(\mu_t\)-mean. Subtracting the mean of the error costs at most another factor two. Multiplying by \(\mu_t\) proves (84). ◻

Proof. For the duration of this proof put \(K=K_N\). At each scale retain the deterministic intermediate potential \(V_k(t)\) supplied by Lemma 23, with \(f_k=\pi_{s_k}V_k\) fixed. The endpoint Gibbs law is exactly the starting law for the next scale: \[\mu_{s_k,N}^{V_k(1)}=\mu_{s_{k+1},N}^{V_{k+1}},\] by the density identity (70), also after periodic evaluation.

First bound the final perturbation. Maximality of \(K\) gives \(N/s_{K+1}<10L\), whereas \(K\to\infty\) with \(N\). Hence \[ \|\mathcal H_{s_{K+1},N}(V_{K+1})\|_\infty \le (N/s_{K+1})^2\|V_{K+1}\|_{s_{K+1},d} \le C_L\epsilon2^{-K-1}=o(1). \tag{88}\] The normalized density relative to \(\mu_{0,N}\) lies between the exponentials of plus and minus twice this bound. Its total variation distance from \(\mu_{0,N}\) therefore tends to zero. Couple a sample from this endpoint law and a pure sample so that they agree outside an event of that probability.

For each site and each scale, now independently generate a Poisson process on \([0,1]\) of rate \[r_{k,x}=2s_k^{-2}\|f_k\|_\infty.\] These rates are deterministic; they do not involve the current spins, the exterior data of a refresh, or its time within the step. If \(f_k=0\) we take no clocks at that scale. Generate all these processes independently of the coupled endpoint and pure samples, and take \(M_{k,x}=1\) exactly when its process has at least one point. Then \[\mathbb P(M_{k,x}=1)=1-e^{-r_{k,x}} \le2s_k^{-2}\|f_k\|_\infty \le2\epsilon2^{-k}s_k^{-2},\] and all asserted independence properties already hold. Additional randomness used to choose refreshed spins can likewise be generated in advance; no independence assertion about those spin values is needed.

Run scales in the order \(K,K-1,\ldots,0\). At scale \(k\), traverse the deterministic measure path backwards in time. At a clock of site \(x\), refresh \(B_{Ls_k}^\circ(x)\) with the law in (58), using the exact torus conditional of \(\mu_{s_k,N}^{V_k(t)}\) and the fixed function \((f_k)_x\). The normalization and nonnegativity proved in Lemma 22 apply at every time, irrespective of whether the evolving sample has the instantaneous Gibbs law. There are finitely many Poisson clocks on a finite torus, so this constructs a finite-state time-inhomogeneous Markov chain. It changes spins only in marked squares.

Lemma 27, with \(r_s=C\epsilon2^{-k}\) from (81), bounds the error of its desired reverse probability equation by \[C\epsilon2^{-k}(N/s_k)^2e^{-cN/s_k}.\] These errors add in total variation. Explicitly, for a prescribed probability path \(p_t\) and a Markov evolution with generator \(\mathcal L_t\), the difference satisfies the inhomogeneous forward equation with forcing \(\dot p_t-\mathcal L_t^*p_t\). Variation of constants expresses the difference as the initial difference propagated by a Markov kernel, plus the integral of the propagated forcing. Markov kernels contract total variation of signed measures, so its norm is bounded by the initial norm plus the integrated forcing norm. Apply this identity on each unit-time reverse step and use the exact equality of consecutive endpoint laws. The output law differs from \(\mu_{\lambda,N}\) by at most \[ C\sum_{k=0}^{K_N}\epsilon2^{-k}(N/s_k)^2e^{-cN/s_k}. \tag{89}\] This quantity tends to zero. For each fixed \(k\), \(N/s_k\to\infty\). Moreover \(r^2e^{-cr}\) is bounded for \(r\ge0\), so the summands, extended by zero for \(k>K_N\), are dominated by a constant times the summable sequence \(\epsilon2^{-k}\). Dominated convergence proves the assertion without requiring the largest-scale error to vanish at a fixed radius ratio.

Finally maximally couple the output with a sample from the exact desired law. Its probability of disagreement is bounded by (89). This final correction can be sampled conditionally on the already constructed output and leaves the joint law of the pure input and all marks unchanged. On the complement of the endpoint-coupling failure and this final disagreement, the target sample is obtained from the pure sample by precisely the described marked-square refreshes. Both exceptional probabilities tend to zero, which proves the proposition. ◻

Proof. Choose \(a_T\) smaller than the separation of distinct nonincident sides, the lengths of all sides, and the distances between the disjoint terminal sets, with additional fixed reductions depending on the finitely many vertex angles. Then a ball below this scale meets no test wall, just one straight wall, or walls incident to one nearby vertex. This choice is essential for a thin polygon: distance to a vertex alone need not exclude a second, nonincident side. Capping at \(a_T\) removes that difficulty. Enlargement and recentering cost only fixed factors for this test.

First work in a neighborhood meeting no test wall. Cut the common exterior contours at their intersections with the enlarged edit square, making an arbitrarily small displacement of the cut to avoid vertices. Follow every exterior end to its first return to the square or to the local outer cutoff. An exterior connection which reaches that cutoff supplies two long ends; short returns can be contracted for this argument. The number of long ends is even. If it is zero, the edit is screened within the cutoff. If it is two, the interior filling, together with the short exterior returns, must connect these same two ends in either configuration. It can add small closed curves but cannot change their through connection or the two side colors seen outside the cutoff. Since there is no terminal in this neighborhood, these small curves cannot create a terminal visit. A changed passage count therefore requires at least four long contour portions. This argument is about cut parameter intervals, and allows all four portions to come from one loop.

More explicitly, the filling together with short exterior returns is a degree-two network whose only free ends are the long ends. With two such ends it has one connecting path and terminal-free cycles. Restrict each resulting contour to the permitted test region, breaking its parameter interval wherever it exits that region. On each remaining component record the successive inner and outer terminal visits in parameter order, cyclically on a closed component, collapsing consecutive visits of the same type. The maximal number of disjoint passages is the number of changes of terminal type within these components. A confinement-wall exit ends a component; visits on opposite sides of such an exit are not paired. Replacing the terminal-free connecting path in the present wall-free neighborhood preserves these sequences and their component endpoints. This also covers repeated visits by one contour and small terminal-free loops.

For a color-crossing test the same conclusion also follows by juxtaposing a successful path and the blocking path of the opposite color in the configuration where the test fails. The triangulation gives the exact crossing–blocking alternative. Both paths must meet the edit square. Outside it they supply four alternating color continuations, whose intervening contours give four long portions. Cutting a band along a radial path gives the identical argument for a circuit and its radial blocker.

Next suppose that only one straight wall is present. There are two different wall roles for passage tests. At a confinement wall, a passage disrupted by the edit has a part on either side of the disrupted portion leading back to the local cutoff in the permitted half-plane; its terminal walls lie beyond that cutoff. These are the two required portions. At a terminal wall, stop every passage at its first terminal hit. A portion which hits this terminal before entering the edit neighborhood is unchanged. Among the remaining portions arriving from the opposite terminal, a terminal hit can disappear only if the new filling pairs its long end with another long end on the permitted side. Pairing with an end on the other side still crosses the terminal wall and preserves a hit. If there is only one long end on the permitted side, neither pairing can remove the hit. Consequently a changed terminal visit supplies two long portions in the permitted half-plane. A hairpin tip moved across a terminal is covered by this case: its two local arms supply the required half-plane portions. In a confined test only one arm may continue all the way to the opposite terminal, so the passage count need not change by two.

For a crossing and its blocker, at most the color arm directed towards this single wall can terminate before the local cutoff. The other three occur in alternating order on the permitted side. The two transitions between them give two disjoint contour portions in that half-plane. The circuit argument is the same after the radial cut. In each case start the half-plane rings at a fixed multiple of the distance to the wall and stop before the next vertex scale; the contours in question stay on one consistent permitted side. If the test permits either side, taking the union of the two checks costs at most a factor two per retained ring.

Above the vertex scale we retain only one contour portion. For a crossing and blocker, two adjacent incident walls can terminate two of the color continuations, but a transition between the remaining colors persists towards the separated terminal. For a passage test, an affected passage still has a portion towards its other terminal, at distance at least \(a_T\). Equivalently, if no contour exits the local cutoff, every change is confined to components which cannot connect the separated terminal sets or alter a separating circuit. Thus a pivotal change forces one passage up to \(a_T\).

At every step the portions are cropped at the appropriate first or last hits. No positive-length interval is used twice, even when a contour returns repeatedly to a wall. The fixed-factor omissions accommodate the chosen cuts, recentering, and the transition between the three geometric regimes. An \(O(\delta)\) displacement of a wall is absorbed by increasing the starting square and, for the half-plane checks, translating its line outwards by \(O(\delta)\). ◻

Proof. First consider a rectangle of fixed shape, with an unexamined buffer around it. Contour crosscuts between its horizontal sides, confined between two vertical sides, are linearly ordered from left to right. After cropping their ends, each crosscut has two colored side routes. One obtains these routes by following the open spin regions next to the interpolated contour and then following vertices in the triangulation. The two routes associated with one crosscut have opposite signs. Reading the routes in their planar order and retaining successive sign changes produces an alternating list whose length is at least a fixed fraction of the number of crosscuts. Opposite signs cannot meet. Routes of the same sign separated by an opposite route are disjoint as crossings. The fixed end crop and side margins absorb the interpolation distance. Thus it suffices to bound the number of ordered alternating spin crossings of a fixed rectangle.

Explore the leftmost crossing of a prescribed sign, if one exists. The exploration examines only the sites on its left and the crossing itself. For example, reveal the vertices reachable from the examining side before a crossing of that sign blocks the exploration; their outer envelope gives the same leftmost path. An alternative path lying farther left somewhere is detectable on this examined side by splicing at its first and last common vertices. Consequently the transcript leaves the spins to the right with the ordinary Ising conditional law, and the only newly exposed spins facing that region on the separating path have the prescribed sign. Repeat for the opposite sign in the remaining right region. Previously examined sites lie behind the most recent spanning path.

There is a uniform probability \(c_0>0\), conditional on each successful transcript, that no further opposite crossing can occur. We construct the blocking event in a deterministic rectangle, rather than asking RSW for a crossing ending on the random exposed path. For one fixed choice of proportions, take the counting slab to be \([-1,1]\times[-2,2]\), and let \(\Gamma\) be the exposed simple color-\(s\) crossing from its bottom to its top. Use the central rectangle \([-2,2]\times[-1,1]\) for a horizontal test, with end gates on \(x=-2\) and \(x=2\), and surround it by the larger rectangle \([-4,4]\times[-3/2,3/2]\). Fixed changes of these proportions cover the original rectangle shapes.

In the larger rectangle, fill the side to the left of \(\Gamma\) by color \(s\), leave its right side free, and put color \(-s\) on the outer boundary. The side is determined by the bottom-to-top crosscut in the counting slab; its restriction to the central levels suffices, even when it has bulges or crosses a horizontal level several times. Nearest-neighbor conditional independence makes the filled left side invisible from the right except through the color-\(s\) path. All earlier revealed sites are screened by that path. No sites to the right of the counting slab have been examined, so the lateral extension retains a free region past the right gate. Thus, for increasing tests when \(s=+\), the actual remaining law dominates this right-side law; putting the outer pins at minus only lowers it. For \(s=-\) the order is reversed.

The filled comparison law in turn dominates, in the color-\(s\) order, the ordinary law in the same larger rectangle with all boundary spins \(-s\) and no interior pins. In this last, deterministic law, a color-\(s\) horizontal crossing of the central test has probability at least \(c_0\) by Proposition 15; its gates and test collars are a positive distance from the outer pins. Such a crossing must meet \(\Gamma\), since its gates lie on opposite sides of the counting slab. Its portion after the last meeting with \(\Gamma\) connects the exposed path to the right gate. It therefore blocks another color-\(-s\) vertical crossing confined to the slab. This suffix uses only the unexplored side, so the two monotone comparisons prove the asserted conditional stopping probability. The central vertical crop leaves clearance from the endpoints of \(\Gamma\); lattice rounding is absorbed by the fixed margins.

The chance of continuing the alternating exploration after \(m\) successful crossings is therefore at most \((1-c_0)^{m-1}\), with a factor two for the initial color. This gives uniformly bounded moments of all orders for the number of contour crosscuts in a fixed cropped rectangle.

Cover a fixed enlargement of \(K\) by finitely many squares of diameter much smaller than \(a\). Every departure by distance \(a\) from one such square contains a passage through one of finitely many buffered rectangles; choose the first side through which an appropriate enlargement is left. Disjoint parameter intervals yield distinct passages in this finite collection. Their total number thus has bounded expectation, indeed bounded moments.

Cut the contours at the boundary of an enlarged chart containing the \(20a\)-neighborhood of \(K\), and retain only components that meet \(K\) and have diameter at least \(10a\). Subdivide each such parameter interval successively when its distance from the last subdivision point first reaches \(a\). For a closed component choose a root without reference to the later center or direction and subdivide cyclically. Retain every union of five consecutive subdivision intervals whose middle interval meets \(K\). The interval containing a qualifying visit is therefore the middle of one listed union. That union lies within distance less than \(10a\) of the visit, after an absolute reduction of the subdivision distance if needed. It therefore lies between the two distant visits in the statement. The buffer from \(K\) to the cutting boundary ensures that these intervals exist on both sides of every qualifying visit. Endpoint remainders and cyclic intervals through a root are included. The rectangle count bounds the number of retained intervals, including separate traversals of the same geometric region. This constructs the required list without subdividing any part of a contour far from the fixed chart.

Complete-resolution convergence from Theorem 13, with slightly relaxed rectangles, preserves every fixed finite collection of these macroscopic passages. Fatou’s lemma transfers the expectation bound to the limit. Alternatively perform the same successive subdivision on the limiting loops: uniform continuity makes the list finite on each loop, and the relaxed passage counts bound its expected length. The construction lists no microscopic loops unrelated to the excursions under consideration. ◻

Proof. Use a larger round disk with plus boundary as a reference and average the center over the lattice translates of a small disk in its interior. Translation invariance of the plane law and pure buffered density comparison bound the plane probability at a specified center by a constant times this averaged reference probability. Off-lattice centers are covered by an \(O(\delta)\) relaxation of the distances. This step uses only lattice translations. Complete loop convergence with relaxed distances then bounds its upper limit by the analogous continuum event averaged over area. Indeed every witnessing subarc has a fixed positive diameter, so it belongs to the macroscopic part of the loop ensemble; subsequential uniform parametrized convergence preserves its closed near-visit and half-plane conditions after the relaxation. The reference disk and the averaging disk are rotationally invariant. We may therefore average also over the direction in this continuum bound. For directions or centers varying with the mesh, take convergent subsequences and relax the geometric inequalities before applying loop convergence. This gives the same uniform bound and does not assert rotational invariance of the lattice.

Use Lemma 31 with a subdivision length smaller than one tenth of the fixed excursion length. If the event occurs, one member \(C\) of its list contains the near visit and lies entirely between the distant visits. This list is independent of \(z\) and \(\theta\). Write \(n_\theta\) and \(t_\theta\) for the upward normal and the unit tangent to the line, and put \[m_C(\theta)=\min_{w\in C}n_\theta\cdot w, \quad C_{\theta,h}=\{w\in C:n_\theta\cdot w\leq m_C(\theta)+h\}.\] For almost every \(\theta\) the support line of the compact convex hull of \(C\) has a unique contact point. One elementary justification is to use the support function of that convex hull: it is Lipschitz on the circle and differentiable almost everywhere, and at a differentiability direction the two extreme tangent coordinates of the supporting face coincide. Compactness then gives \[ \operatorname{diam}\{t_\theta\cdot w:w\in C_{\theta,h}\} \longrightarrow0\qquad(h\downarrow0) \tag{96}\] at each such direction.

If \(C\) supplies a near visit, its height above its minimum at that visit is at most \(C_Ku\). The normal coordinate of \(z\) lies in an interval of length \(C_Ku\) about \(m_C(\theta)\), and its tangent coordinate lies within \(Cu\) of the projection of \(C_{\theta,C_Ku}\). The area of the possible centers is therefore bounded by \[C_Ku\left(\operatorname{diam} \{t_\theta\cdot w:w\in C_{\theta,C_Ku}\}+Cu\right).\] By (96) this is \(o_K(u)\) for almost every direction and is at most \(C_Ku\) times a fixed chart diameter. The expected number of arcs in the list is finite. Dominated convergence first over directions and then over that list proves the averaged \(o_K(u)\) bound. The preceding buffered comparison and rotational averaging yield (95). ◻

Proof. Normalize \(R=1\) and write \(u=r/R\). Select two confined passages. If there are fewer than four unconstrained passages from radius \(u\) to radius \(Ku\), they belong to the same loop and their inner ends are joined inside radius \(O(Ku)\). To verify the statement including multiplicities, first suppose that the selected passages belong to different loops. The complementary arc on each loop supplies a second passage to \(Ku\), giving four in total. On a single loop the two complementary arcs either both join an inner end to an outer end, again giving four, or join inner to inner and outer to outer. In the latter case an excursion of the inner connecting arc to \(Ku\) supplies two further passages. These may share their outer hit, as allowed by our convention, and have disjoint interiors. Thus that excursion is excluded.

Concatenate the two confined passages and their inner connecting arc. This subarc has a near visit to \(z\) and goes a fixed positive distance on both sides, all above the original line lowered by \(CKu\). Its probability is \(o_K(u)\) by Lemma 32. In the complementary case there are four passages from \(u\) to \(Ku\), and the original confined passages still give two confined passages from \(C_0Ku\) to \(1\). Here \(C_0\) is a fixed buffer constant. Condition on the spins outside radius \(C_0Ku\). Pure buffered comparison bounds the conditional four-passage probability by \(CK^{-2-c}\), uniformly in those outside spins, by (94).

Let \(p(u)\) be the mesh upper limit of the plane confined-passage probability, with the uniform translation and direction convention just described. We have proved, for every sufficiently large fixed \(K\) and all sufficiently small \(u\), \[ p(u)\leq o_K(u)+CK^{-2-c}p(C_0Ku). \tag{98}\] All constants associated with buffers and relaxed squares are fixed before \(K\) is enlarged. Square and Euclidean radii differ by fixed factors and give the same recursion after changing \(C_0\).

For completeness, put \(q(u)=p(u)/u\), \(d=C_0K\), and \(\vartheta=CC_0K^{-1-c}\). Fix \(K\) so large that \(\vartheta<1\). Then \(q(u)\leq e_K(u)+\vartheta q(du)\), where \(e_K(u)\to0\). Iterate until \(d^mu\) first reaches a fixed small interval bounded away from zero. The terminal value is bounded because \(p\leq1\), and \[q(u)\leq\sum_{j=0}^{m-1}\vartheta^j e_K(d^ju) +\vartheta^m q(d^mu).\] This first gives boundedness of \(q\) near zero. Splitting the sum at a fixed index, letting \(u\downarrow0\), and then sending that index to infinity gives \(q(u)\to0\). This proves the plane assertion. A final fixed buffered density comparison proves the same assertion in the other pure bulk charts. ◻

Proof. Condition on the original pure spins outside \(W_j\) and on all marks outside \(\mathcal J_j\). For each fixed value of the remaining local marks, \(T_j^+\) is simply an event of the pure spins on \(A_j\). Corollary 4 gives the pointwise density bound \[ \frac{d\mu_{0,N}(\sigma_{A_j}\mid\sigma_{W_j^c})} {d\mu_{0,N}(\sigma_{A_j})}\leq D. \tag{100}\] For clarity, the two parts of this comparison have fixed aspect: the outer data are separated from \(A_j\) by the collar outside \(S_{R_j}\), and the inner data by the collar inside \(S_{r_j}\). For the second comparison interchange the terminal sets in the symmetric density-odds bound and regard the already fixed outer data as common pins. The two losses are independent of \(R_j/r_j\). No pins occur in the intervening free collars.

For rotated squares this comparison uses a fixed patch cover, as in (OpenAI 2026, Corollary 2.6), without rotating the lattice. Write \(S_t=z+e^{i\theta}[-t,t]^2\). Partition \(A=S_R\setminus S_r\) into a bounded number of sets, each contained in an axis-aligned square of radius \(cR\), with \(c>0\) fixed and small. Each patch is separated from the changed data outside \(S_{2R}\) by a distance comparable to \(R\), uniformly in \(\theta\). Factor the probability of the configuration on \(A\) into successive patch conditionals. Earlier patches and all inner data are common pins, allowed by Proposition 3, so the bounded product costs a fixed factor. To change the inner data, partition \(S_{r/2}\) into a bounded number of patches of diameter \(cr\) and change them one at a time. Each is separated from the whole observed annulus by distance comparable to \(r\). Terminal symmetry gives the same bound, with the remaining inner patches and the outer data common. These two finite products are uniform in \(R/r\) and \(\theta\). Finally mix the reference exterior data to obtain (100). Fixed positive patch clearances absorb lattice rounding.

The marks in \(\mathcal J_j\) are independent of the conditioning and of the original spins. Integrating the pointwise bound over their unchanged distribution gives \[\mathbb P(T_j^+\mid\sigma_{W_j^c},M_{\mathcal J_j^c}) \leq D\mathbb P(T_j^+).\] All other \(T_i^+\) are measurable with respect to this conditioning. Remove the factors one at a time to obtain (99). This proof uses conditional laws of \(\sigma^0\) only. In particular, it neither defines nor assumes a conditional mixing estimate for an edited spin law. ◻

Proof. Use physical lengths and rescale \(a_T\) to one. Write \(b=\delta b_k^{\mathrm{lat}}/a_T\), and similarly rescale \(u,v\). Discard centers whose edit square does not meet the finite spin support of \(T\): their insertion cannot affect the test. For \(b\leq1\), divide the insertion centers into dyadic classes with truncated distances comparable to \(u,v\), where \(b\leq u\leq v\leq1\). The area of one such class is at most \(C_Tuv\). Indeed, when \(v<1\) the relevant portion of an edge has length \(O_T(v)\) near a vertex, and its \(u\)-neighborhood has area \(O_T(uv)\). When \(v\) is comparable to one, the total finite edge length is \(O_T(1)\). Centers far from the edges belong to the capped class \(u=v=1\). Lower truncation at \(b\) allows the same estimate after enlarging by one lattice cell, so there are at most \(C_Tuv/\delta_T^2\) centers, where \(\delta_T=\delta/a_T\) is the rescaled mesh.

The marking probability is bounded by \(C_L\epsilon2^{-k}\delta_T^2/b^2\). Multiplication by the number of centers and (103) gives exactly \[\begin{align*} &C_T\epsilon2^{-k}\frac{uv}{b^2} (b/u)^{p_4}(u/v)^{p_h}v^{p_1}\\ &\hspace{15mm}= C_T\epsilon2^{-k} (b/u)^{p_4-2}(u/v)^{p_h-1}v^{p_1}. \tag{105}\end{align*}\] This equality is where both location-count thresholds enter. Let \(m=\min\{p_4-2,p_h-1,p_1\}>0\). The three base ratios in the last line belong to \((0,1]\) and their product is \(b\). Every term is therefore at most \(C_T\epsilon2^{-k}b^m\). There are \(O_T((1+\log(1/b))^2)\) dyadic pairs. Absorbing this factor into \(b^{m/2}\) proves (104), with \(c_*=m/2\).

If \(b\geq1\), the centers whose squares can affect the test occupy area at most \(C_Tb^2\). The crude bound one on pivotality, combined with the marking probability, gives \(C_T\epsilon2^{-k}\). The same crude count covers \(b\) within a fixed factor of one. Omitted transition rings at smaller scales have already been absorbed into \(C_T\) in (103). The computation uses only upper bounds on individual mark probabilities; partial activation can therefore be arbitrary in space. For a fixed annular shape the geometric constants scale with its inner radius and are independent of its location or orientation. ◻

Proof. There are finitely many mark variables on the torus. Activate one level at a time, increasing its Bernoulli parameters continuously from zero to their prescribed values. Conditional on the pure spins, the expectation of an increasing event of these marks is a polynomial in their parameters. Differentiating in a parameter gives the probability that this index is pivotal when its own mark is omitted. Integration is the usual finite-product interpolation identity and requires no monotonicity in the spins. For any collection of checks, even a continuum of geometric frames, these expectations are uniformly Lipschitz in the finitely many parameters: the sum of their absolute parameter changes bounds the change of every event probability. Thus their supremum is continuous as well.

Before any marks are activated, the estimates hold with strict slack, by the choice of \(s_0\). At each stage monitor only checks with inner radius at least \(C_{\rm mon}\) times the current radius. When the cutoff increases, some checks leave this collection. They are not needed again: the rings used for a later insertion start at a larger fixed multiple of that insertion’s radius. Conversely, any check still monitored at a given stage has been monitored throughout all earlier stages.

Suppose there were a first failure of (106). Up to that stage all ring possibility estimates used to control an insertion satisfy (106); by continuity the same weak bounds hold at the candidate first failure. Each pivotal insertion for any monitored check therefore satisfies (103). Apply Lemma 35 to that check, summing all previous levels and the activated fraction of the current level. Since these checks are translates, rotations, and dilates of only three fixed shapes, their total probability increase is at most \[C_{\mathrm{boot}}\epsilon\sum_{k\geq0}2^{-k} \leq2C_{\mathrm{boot}}\epsilon.\] Choose \(\epsilon_*\) so that this is less than \(\frac12\min_i\tau_i\). The first failure cannot occur. The same estimate, being uniform over the frames, rules out a failure occurring only as a supremum. This proves the bootstrap on each finite torus, before taking any limit.

The order of the choices is now explicit: fix the collars and gap factor, choose \(A\) and the tolerances, enlarge \(s_0\), and then choose \(\epsilon_*\). The finitely many universal shapes determine \(C_{\mathrm{boot}}\). No later polygonal test enters this choice. The enlargement of \(s_0\) is made before the critical trajectory of Theorem 26 is constructed. ◻

Proof of Theorem 29. Fix one of the tests \(T\) in the statement. During increasing activation of the marks, apply the finite-product interpolation identity to \(D_T\). Its initial value is zero. If a new mark first makes \(D_T=1\), choose two permitted assignments with different test values and revert the newly permitted square in a witnessing assignment. Before its addition all assignments gave one value. We obtain two configurations which differ only inside that square and change the test. Lemma 30 applies, and its required outside checks are possibility events using the older marks. Proposition 36 controls these checks whenever the insertion is smaller than the test. For larger insertions use the crude overlapping-square count in Lemma 35. We conclude that \[ \mathbb P(D_T=1)\leq C_T\epsilon \sum_{k\geq0}2^{-k} \min\left\{1, \left(\frac{\delta(Ls_k+O(1))}{a_T}\right)^{c_*}\right\}. \tag{107}\] Absent torus levels can be set to zero; the displayed infinite sum is a convenient upper bound. For every fixed \(k\), its factor in braces tends to zero with \(\delta\). Every summand is dominated by \(2^{-k}\), whose sum is finite. Dominated convergence therefore proves \(\mathbb P(D_T=1)\to0\). Equivalently first make the level tail small, and then refine the mesh for the finitely many remaining levels. This argument keeps \(\epsilon\) fixed; it does not require a new smallness condition depending on \(C_T\).

A union bound proves (93) for the fixed finite family. Under Proposition 28, the perturbed configuration belongs to \(\mathcal C(\sigma^0,M)\) except with probability \(o(1)\). On the complement of that exceptional event and all \(D_T\), the entire test vector agrees with the pure vector. Its total-variation distance from the pure test-vector law tends to zero. The joint positivity and four-passage conclusions follow respectively from Proposition 15 and Proposition 14. ◻

Proof. We construct one factor on a tree \(T\), and then give different constructions disjoint sets of bits. For each complete spin pattern \(t\in\{-1,+1\}^{V(T)}\), reserve two private bits on every edge of \(T\), with the same coupling \(a\). Conditional on all the physical spins, these bits are independent in the unperturbed representation. For the two bits on \(e=uv\), set \(D_e=b_{e,1}-b_{e,2}\). Their conditional moments are \[ \mathbb E[D_e\mid\sigma]=0,\qquad q_e(\sigma_u,\sigma_v) :=\mathbb E[D_e^2\mid\sigma] =2\operatorname{sech}^2\bigl(a(\sigma_u+\sigma_v)\bigr)>0. \tag{113}\] At a vertex \(v\) of this private copy of \(T\), multiply its vertex factor by \[ 1+c_{v,t}\,\mathbf 1_{\{\sigma_v=t_v\}} \prod_{\substack{e\in E(T)\\e\ni v}}D_e. \tag{114}\] An empty product is one. In the expansion of the product over vertices, a chosen subset \(S\subseteq V(T)\) contributes one copy of \(D_e\) for each end of \(e\) in \(S\). If \(S\) is nonempty and proper, the connectedness of \(T\) gives an edge with exactly one end in \(S\). Its centered difference makes this term vanish upon summing the private bits. The only surviving terms are therefore the empty selection and the full selection. Their sum is \[ 1+\mathbf 1_{\{\sigma_{V(T)}=t\}} \left(\prod_{v\in V(T)}c_{v,t}\right) \left(\prod_{uv\in E(T)}q_{uv}(t_u,t_v)\right). \tag{115}\] The same formula holds for a singleton tree, with an empty edge product.

For a target factor \(F\) on a subset of \(V(T)\), put \[\varepsilon_t=F(t)-1,\qquad Q_t=\prod_{uv\in E(T)}q_{uv}(t_u,t_v)>0.\] Choose coefficients whose product is \(\varepsilon_t/Q_t\). For example, if \(m=|V(T)|\), take absolute values \(\lvert\varepsilon_t/Q_t\rvert^{1/m}\) at every vertex, place the sign of \(\varepsilon_t\) at one prescribed vertex, and take all coefficients zero when \(\varepsilon_t=0\). Every coefficient tends to zero as \(F\to1\), for either sign of \(\varepsilon_t\). Equation (115) then realizes \(1+\varepsilon_t\mathbf 1_{\{\sigma_{V(T)}=t\}}\). Use distinct private bits for different patterns. Conditional independence allows their contributions to be multiplied, and exactly one pattern is active at a given physical spin configuration. Hence \[\prod_t\left(1+\varepsilon_t \mathbf 1_{\{\sigma_{V(T)}=t\}}\right)=F(\sigma).\]

There are finitely many trees, finitely many patterns on each, and only finitely many translated trees using a given bond. First count these uses, and choose \(M\) large enough to assign all their pairs of private bits disjointly. Unused bits retain their original factor. This allocation can be made translation covariant; its number is independent of the size of the torus. Choose \(a\) only after this finite allocation, using (111). The product of all the factors (114) at \(v\), times \(W_v^0\), defines \(W_v\). Each \(D_e\) has absolute value at most two. Since a vertex belongs to only finitely many assigned trees, all correction factors are positive when their coefficients are small enough. Their tables then converge uniformly to one, proving (112). Summing all private channels, with the physical spins fixed, proves the claimed exact identity. ◻

Proof. At \(\lambda=0\), put \(H=a\sum_{j\ne i}b_j\). Then \[F_v^0(b)=2\cosh\left(a\sum_jb_j\right),\qquad d_i\log F_v^0=\log\cosh(H+a)-\log\cosh(H-a).\] Because \(-1<\tanh u<1\) for finite \(u\), \[ -2a<d_i\log F_v^0<2a. \tag{117}\] The increments of \(\log P_v^{-,0}\) and \(\log P_v^{+,0}\) are exactly \(-2a\) and \(2a\). For two distinct bits the mixed difference of \(\log F_v^0\), writing \(H\) now for the contribution from all the remaining bits, is \[\log\cosh(H+2a)+\log\cosh(H-2a)-2\log\cosh H>0,\] by strict convexity. Finally, the log odds of the free spin are \(2a\sum_jb_j\), with increment \(4a>0\) in each bit. There are only finitely many entries in these tables. Their strictly positive minimum gaps persist under the uniform positive perturbation in (112). This proves all the assertions. ◻

Proof. Let \(\mathcal B(V)\) consist of all bits on bonds with at least one endpoint in \(V\). Condition in each law on all other bits. After summing every free physical spin, the conditional weight on \(\mathcal B(V)\) is a product of vertex factors, restricted to the conditioned values where necessary. A vertex in \(V\) has every incident bit in \(\mathcal B(V)\). Its factor is the identical attractive function \(F_v\) in the two laws. Its contribution to the conditional log odds of a bit is therefore ordered whenever the two configurations of varying bits are ordered.

At an endpoint outside \(V\), the table in the statement and (116) order the two contributions to those log odds, regardless of the values of all other incident bits. This last qualification allows both the separately conditioned bits and the other varying bits at that endpoint to differ. Adding the contributions of the two endpoints orders the full single-bit conditional log odds.

Run single-bit heat-bath chains in the two finite state spaces, updating the same coordinate and using the same uniform random number at each step. The ordered conditional probabilities preserve the coordinatewise order. Start from any ordered pair, and take a subsequential stationary limit of the coupled chains. Strict positivity makes each marginal chain irreducible and gives its prescribed Gibbs measure as its unique stationary law. We have thus obtained an ordered coupling of the bits.

Conditional on all bits, the free physical spins are independent, with the kernels \(k_v\). At the vertices of \(V\) these kernels are the same in both laws and are increasing by Lemma 38. Another family of common uniform random numbers gives the ordered physical spins. Finally, the argument worked for every pair of exterior bit configurations. Integrating these couplings against the two exterior distributions proves the mixture assertion. ◻

Proof. Fix a deterministic query order to resolve all choices. A transcript lists queried vertices \(v_1,\ldots,v_m\) and their values \(s_1,\ldots,s_m\). Any configuration agreeing with these values causes the same successive choices and the same stopping decision, by induction on the number of queries. Thus the transcript event is the cylinder \(\{\sigma_{v_i}=s_i:1\le i\le m\}\). Conditioning on a transcript-measurable event adds nothing once a compatible transcript has been specified. Independent exploration randomness, if used, can first be conditioned on and then averaged out.

For the annular construction, query the outer graph boundary. Keep a queue of all revealed vertices of color \(-s\), and query every previously unqueried triangulation-neighbor of a queued vertex before removing that vertex from the queue. Newly discovered vertices of color \(-s\) join the queue; vertices of color \(s\) do not. Declare failure just before a query would enter the protected inner disk. Declare success when the queue has been exhausted. This is a finite adaptive procedure of the kind just considered.

On success let \(V\) be the unqueried nearest-neighbor component containing the inner disk, or the union of such components if the chosen inner set has more than one component. An unqueried nearest neighbor of \(V\) belongs to \(V\). A queried neighbor cannot have color \(-s\), since every neighbor of such a vertex was queried before its queue entry was removed. Hence every boundary incidence of \(V\) meets a revealed \(s\) vertex. If a color-\(s\) circuit separates the protected disk from the outer boundary, every queued vertex has a color-\(-s\) path to the outer boundary and so lies outside the circuit. A further queried neighbor cannot lie strictly inside: the edge to it would cross the circuit without meeting a circuit vertex. Planarity forbids this, and expansion stops at vertices of the circuit because they have color \(s\). The positive separation of the narrower band from the protected disk proves the asserted implication from a circuit to success.

For a topological rectangle, seed the exploration on the inspecting side and expand color \(-s\) within the rectangle. A color-\(-s\) path reaching the opposite side certifies failure. Otherwise the boundary of the reached color-\(-s\) set facing that side gives a color-\(s\) crossing between the two end sides. To check this finite planar assertion directly, join the reached components by a thin deterministic strip immediately outside the inspecting side. Trace the boundary of a small polygonal neighborhood of this connected set on the side facing the opposite boundary. Since the reached set does not meet that boundary, this trace has a component running between the two end sides. Every graph vertex encountered just beyond the reached set was queried and has color \(s\), because otherwise it would have been added to the queue. As the trace passes through a triangular face, consecutive such vertices coincide or are joined by the remaining edge of that face. They consequently give a walk of color \(s\) between the end sides. Erasing closed subwalks gives a simple crossing. Conversely a crossing between the end sides blocks any opposite-color path between the inspecting and opposite sides, since the two planar paths would have to share a vertex. Boundary arcs may be separated at their corners by one lattice step; the prescribed buffered gates allow this harmless choice of finite rectangle convention. Thus the existence of the desired crossing is determined by the transcript, and on success no queries pass strictly beyond its facing boundary. In applying this construction, the two end gates lie past the comparison region, so a nearest-neighbor path cannot go around a gate within that region.

Finally, a nearest-neighbor bond cannot cross a simple path or circuit of the triangulation in the interior of an edge. If its ends lie on opposite sides, it must meet a path vertex. This proves the incidence assertion even when the separating path uses diagonal steps. ◻

Proof. For \(u\le v\) and \(s\le t\), define \[\Delta(u,v;s,t)=f(u,s)+f(v,t)-f(u,t)-f(v,s)\ge0.\] Writing \(D=f(a,b)+f(a',b')-f(a,b')-f(a',b)\), expansion gives the exact identity \[\begin{align*} \Lambda-D={}&\Delta(i_-,a';j_-,b) +\Delta(a,i_+;b',j_+)\\ &+\Delta(a',i_+;j_-,b') +\Delta(i_-,a;b,j_+). \end{align*}\] Every term is nonnegative; no order between \(a,a'\), or between \(b,b'\), is required. Interchanging \(b,b'\) proves the other bound.

If \(q_b,q_{b'}\) are the two conditional masses on \(I\), (119) says that \(\log(q_b/q_{b'})\) has oscillation at most \(\Lambda\). Its density ratio has mean one under \(q_{b'}\), so its minimum is at most one and its maximum is at least one. Both bounds in the statement follow. They persist after taking arbitrary mixtures in either law. ◻

Proof. All sets in this proof are chosen in physical coordinates before the mesh tends to zero. We identify a lattice vertex with its rescaled position when using \(Q\) and \(\widetilde Q\). Let \(I\) contain all bits incident to the vertices of \(Q\). Choose two surrounding square annular bands \(A_1,A_2\), with \(A_1\) inside \(A_2\), and then a square cut outside \(A_2\), all within \(\widetilde Q\). Leave positive distances between successive objects. The inner protected disk for an exploration in \(A_1\) contains the endpoints of every bit in \(I\); the inner protected disk for an exploration in \(A_2\) contains all of \(A_1\). Choose slightly wider exploration bands around \(A_1,A_2\), still disjoint and separated from the cut. A circuit in the original, narrower band then implies exploration success by Lemma 40.

More precisely, take a lattice square vertex set \(S\) containing both exploration bands, with its boundary separated from them, and let \(J\) be all bit channels on the nearest-neighbor bonds with one endpoint in \(S\) and the other outside. We require \(I\) to lie strictly inside this cut. Let \(\mathcal B_{\mathrm{int}}(S)\) be the bits on bonds with both endpoints in \(S\). Every physical vertex in \(S\) is free. Once \(J=j\) is fixed, the conditional mass of the inside spins and bits is \[K_S(\sigma_S,b_{\mathcal B_{\mathrm{int}}(S)}\mid j) =\frac1{Z_S(j)}\prod_{v\in S}W_v(\sigma_v,b_{\mathcal B(v)}), \qquad b_J=j.\] Every bit incident to a vertex of \(S\) is either an interior bit or belongs to \(J\); all remaining vertex factors lie outside \(S\). This proves the factorization and shows that \(K_S\) is identical in the domain and torus laws. In particular, their inside-bit marginals can differ only through their mixing distribution on \(j\). The representation is the same for all the laws by Proposition 37; exterior pins in the spin model are retained exactly by Lemma 2.

Let \(\widehat\mu\) denote the joint torus law of spins and bits, and write \(\widehat\mu_j=\widehat\mu(\,\cdot\mid J=j)\). For a sign \(s\), let \(C_r^s\) be the fixed spin test of a color-\(s\) circuit in \(A_r\), and let \(E_r^s\) be success of the exploration with that color in its wider band. These are different events. The geometric implication is \[ C_r^s\subseteq E_r^s. \tag{121}\] Only the fixed tests \(C_r^s\) are used as inputs to bulk stability. By Theorem 29 and the pure simultaneous signed-circuit positivity of Proposition 15, for all sufficiently fine meshes \[ \widehat\mu(C_1^+\cap C_2^-) \ge c,\qquad \widehat\mu(C_1^-\cap C_2^+)\ge c \tag{122}\] for a fixed \(c>0\). We may take \(c\le1\).

We first show that, for every cut configuration \(j\), \[ \widehat\mu_j(C_1^+)\ge c,\qquad \widehat\mu_j(C_1^-)\ge c. \tag{123}\] Condition the unconditioned torus experiment on \(E_2^-\), and then on a successful complete exploration transcript. All physical spins of \(A_1\) remain free. Let \(V\) be the unqueried nearest-neighbor component containing the protected inner disk, and hence all of \(A_1\). It is a component of the common free set in the revealed experiment and in \(\widehat\mu_j\). Every neighbor outside \(V\) is a revealed minus spin in the first experiment and a free spin in the second. The queried outer boundary of the exploration separates \(V\) from the cut, so no bit of \(J\) touches \(V\). The minus-pin/free case of Proposition 39 orders the conditional experiment below \(\widehat\mu_j\) on the spins of \(A_1\). Arbitrary further exterior bit conditions are allowed by that proposition. Average over the successful transcripts. Since \(C_1^+\) is increasing, this gives \[\begin{align*} \widehat\mu_j(C_1^+) &\ge \widehat\mu(C_1^+\mid E_2^-)\\ &=\frac{\widehat\mu(C_1^+\cap E_2^-)} {\widehat\mu(E_2^-)} \ge \widehat\mu(C_1^+\cap C_2^-)\ge c. \end{align*}\] The denominator is positive by (122). Interchanging the colors and applying the comparison to the decreasing event \(C_1^-\) proves the second inequality. This step uses no monotonicity in the cut configuration \(j\).

Now take the two extreme cut configurations \(j_-,j_+\), and denote the corresponding laws by \(\widehat\mu_-\) and \(\widehat\mu_+\). Under \(\widehat\mu_-\), the event \(E_1^+\) has probability at least \(c\), by (123) and (121). Conditional on any successful transcript, the component containing all endpoints of \(I\) has facing plus pins. Compare it to \(\widehat\mu_+\) before any reveal. The plus-pin/free case of Proposition 39 makes the former law larger on the bits \(I\). The cut bits are outside the comparison component in both experiments. The increasing event \(\{I=i_+\}\) therefore satisfies \[ \widehat\mu_-(I=i_+)\ge \widehat\mu_-(E_1^+)\widehat\mu_+(I=i_+) \ge c\,\widehat\mu_+(I=i_+). \tag{124}\] The opposite construction, starting from \(\widehat\mu_+\) and revealing a minus shield, gives \[ \widehat\mu_+(I=i_-)\ge c\,\widehat\mu_-(I=i_-). \tag{125}\] These inequalities compare possibly very small probabilities; they do not require a lower bound for either all-equal configuration on \(I\).

The joint torus bit marginal \(p(I,J)\) is positive and MTP\(_2\), by Lemma 38 and marginalization as proved above. Its extreme ratio satisfies \[e^{\Lambda} =\frac{\widehat\mu_+(I=i_+)\widehat\mu_-(I=i_-)} {\widehat\mu_-(I=i_+)\widehat\mu_+(I=i_-)} \le c^{-2},\] by (124) and (125). Apply Lemma 41. It bounds the density ratio of every two mixtures of the laws of \(I\) given \(J\) between \(c^2\) and \(c^{-2}\). One mixture is the torus law, and the other is the domain law, by the exact cut factorization.

Finally, conditional on \(I\), the physical spins in \(Q\) have the same product of Bernoulli kernels with plus probabilities \(k_v(I)\) in both laws. Integrating the pointwise bit-density inequalities against this positive kernel gives (120), with \(C=c^{-2}\). Summation gives the event version. Every geometric separation was fixed before taking the fine-mesh threshold, so the constant is independent of the domain and its exterior pins. ◻

Proof. Place all sets in a torus chart with a fixed positive external buffer. Let \(E_h\) be the event of simultaneous lengthwise minus and plus crossings of \(C_h^-\) and \(C_h^+\). Proposition 15 gives \(\mu_{0,N}(E_h)\geq q_h>0\) for small mesh. By Theorem 29, \(\mu_{\lambda,N}(E_h)\geq q_h/2\) eventually, and the joint laws of \[\bigl(\mathbf 1_{E_h},V(S_1),\ldots,V(S_m)\bigr)\] in the pure and perturbed tori have total variation distance \(\varepsilon_n\to0\). Here each \(V(S_i)\) is an ordinary bulk crossing test in the torus, without virtual filling. For any two laws \(p,q\) on a finite space, an event \(E\) with \(p(E),q(E)\geq q_*>0\), and a random vector \(V\) on that space, subtraction of the two conditional fractions gives \[ \bigl\|p(V\in\cdot\mid E)-q(V\in\cdot\mid E)\bigr\|_{\mathop{\mathrm{TV}}} \leq \frac{2\|p-q\|_{\mathop{\mathrm{TV}}}}{q_*}. \tag{129}\] Thus the two middle vectors conditioned on \(E_h\) can be coupled to agree with probability \(1-o(1)\). The laws \(p,q\) in this application are the joint indicator laws just displayed; no total variation estimate on the underlying spin fields is used.

In each conditioned torus reveal the crossing closest to the radially exterior side of each corridor. Use the stopped reveal of Lemma 40. The transcript examines the exterior side up to the crossing and leaves only pins of its prescribed color facing the center. The crossing and the event of its being the first such crossing are determined by this transcript. After both transcripts have been conditioned upon, no additional crossing-existence condition remains on the unrevealed center. The two scopes are disjoint, so the second reveal does not enter the core left by the first.

For precision, the free comparison components and all their boundary incidences are as follows. In the first comparison, the lower law is the pure law in \(A_h\), and the upper law is the conditioned pure torus. Take the components of the common free region on the center side of its revealed plus path. This path is an upper plus pin against lower free spins. All accessible portions of \(\partial A_h\) are lower minus pins against upper free spins: the plus boundary is screened by the revealed path, whose caps continue outside \(A_h\). At a meeting of a path and a boundary, the pair is upper plus against lower minus. The minus reveal is outside \(A_h\) by (127). These checks give a joint comparison on all spins used by the strips, and therefore \[V_A\leq V_0^{E_h}.\]

In the second comparison the lower law is the conditioned perturbed torus and the upper law is the target. Take the common free components on the center side of the revealed minus path. On this path the lower pin is minus and the upper vertex is free. The target minus arc is screened by this path and its caps. Every remaining accessible target boundary neighbor is an upper plus pin against a lower free vertex. Where the two cuts meet, the pair is upper plus against lower minus. The plus reveal lies outside the target. The comparisons are precisely \[(\text{upper},\text{lower})=(+,\text{free}),\quad (\text{free},-),\quad (+,-).\] No same-sign pinned pair, and no monotonicity of an arbitrary pinned perturbed factor, is being assumed. A diagonal segment of a separating triangulated path cannot be crossed by the interior of a nearest-neighbor edge. Thus each nearest-neighbor incidence leaving a comparison component actually meets one of the path or arc pins just enumerated. This remains true when a strip exits and reenters \(D_n\). Condition on all other auxiliary bits as in Proposition 39; the strict inequalities are uniform in those bits and hence in the unprescribed spin pattern farther behind the arcs. The resulting spin comparison and the extremal virtual values outside \(D_n\) give \[V_\lambda^{E_h}\leq V_n^+.\]

Finally glue the two order couplings with the equality coupling supplied by (129). This is elementary disintegration on finite spaces: first sample each common middle vector, then sample the remaining variables from the relevant conditional laws. Extend the outer vectors to their complete configurations in the same way. This proves (128). In particular the conclusion is about a finite vector of tests, not a total variation comparison of spin configurations. Interchanging plus and minus and exchanging the enlarged and eroded arcs proves the other assertion. ◻

Proof. Consider first a plus strip. Its boundary contacts lie on a compact subarc of \(I_+\) separated from the marks. For large \(n\), the same is true of its contacts with \(\partial D_n\). Every exterior vertex sharing a nearest-neighbor bond with the domain at those contacts is already plus, by Lemma 2. Fill the remaining exterior vertices of the incident square faces with plus for the purpose of drawing only.

Inspect a square face of the primal lattice, or equivalently the four cells incident to one dual vertex. The Jordan condition allows one, two adjacent, or three interior cells at a boundary vertex; two opposite interior cells alone would give a diagonal pinch. With one interior cell, its two nearest exterior neighbors are already plus. Only the opposite exterior vertex might have a different actual spin, and filling it leaves every disagreement edge in \(E_{\Omega_n}\) unchanged. The resulting zero set joins the two occupied domain edges when there are two. With two adjacent or three interior cells, every exterior cell of the face is a nearest exterior neighbor, so its spin already has the filling value. In each case the interpolated zero segments have exactly the partial contour’s connections: its unique connection at degree two, and the prescribed NE/SW pairing at degree four. Faces wholly inside the domain have the same property without filling.

Redraw the target portions in all faces meeting the strip, together with one adjacent lattice layer, as the corresponding zero segments. A monochromatic edge of the triangulation is disjoint from that zero set. Both the original drawing and each replacement meet a face side at the same disagreement-edge midpoint. The replacements therefore glue to one another and to the unchanged portions without changing a local connection. Matching the successive face crossings in their original order gives the displayed \(O(\delta_n)\) curve distance.

The minus construction is identical with the colors exchanged. The boundary-contact sets of the fixed plus and minus strip lists have positive separation from each other and from the marks. Therefore, for sufficiently fine mesh, no face receives conflicting filling instructions. In the bulk both instructions use the actual spins. The replacements can consequently be made simultaneously, leaving the marked neighborhoods and half-stubs untouched. This is a modification of the drawing, not an interpolation of the actual full-plane exterior configuration. In particular all farther exterior spins and every interaction across a narrow fjord remain in the Gibbs conditional law. ◻

Proof. Work in the closed disk using a fixed Schoenflies map. If \(P(L)\cap M(U)\) were nonempty, connect a point of that intersection to an interior point of \(I_+\) through \(P(L)\) and to an interior point of \(I_-\) through \(M(U)\). The two side domains are Jordan domains. Small endpoint arcs and interior polygonal paths, with loops removed, therefore give a simple crosscut between these opposite boundary arcs, contained in their union except for its two boundary endpoints. This crosscut is disjoint from \(K\) by (131). Its endpoints alternate with \(a,b\) around the circle, so it separates \(a\) from \(b\) in the closed disk. This contradicts connectedness of \(K\). Hence \(P(L)\cap M(U)=\varnothing\). It follows that \(P(L)\subset P(U)\cup U\). No interior point of \(U\) can lie in the open set \(P(L)\): every neighborhood of such a point meets \(M(U)\). Thus \(P(L)\subset P(U)\).

Let \(X,Y\) be the ordinary planar areas of the two plus sides after this fixed disk map. They are bounded random variables, \(X\leq Y\), and their distributions agree when the trace laws agree. Thus \(\mathbb E(Y-X)=0\) and \(X=Y\) almost surely. Proper inclusion of these particular side domains would give strictly different areas. Indeed a point of \(P(U)\setminus P(L)\) either belongs to \(M(L)\), or lies on \(L\); in the latter case its open neighborhood in \(P(U)\) meets \(M(L)\). In either case the difference contains a nonempty open subset of \(M(L)\), of positive area. Consequently \(P(L)=P(U)\) and their interior boundaries give the same chord. This reasoning does not require the chords to have zero area. Finally (131) puts \(K\) inside that chord. The inverse of a simple parametrization sends \(K\) to a connected subset of \([0,1]\) containing both endpoints, which is all of \([0,1]\). ◻

Proof. The inverse coordinate \(\Gamma^{-1}\) is continuous on the compact trace. Extend it continuously, with values in \([0,1]\), to a compact ambient neighborhood and call the extension \(p\). Setting \(u_n=p\circ g_n\) gives (133) by uniform continuity: a point close to \(\Gamma(v)\) has \(p\)-value close to \(v\), and hence is close to \(\Gamma(p(\cdot))\). The endpoints have the asserted coordinates.

Here is the matching when drops vanish. The running maximum \[m_n(t)=\max_{s\leq t}u_n(s)\] is continuous, nondecreasing, and satisfies \(0\leq m_n(t)-u_n(t)\leq B_n\). Its endpoint values tend to \(0,1\). For all large \(n\) let \[v_n(t)=\frac{m_n(t)-m_n(0)}{m_n(1)-m_n(0)},\qquad h_n(t)=(1-\kappa_n)v_n(t)+\kappa_n t,\] where \(0<\kappa_n\downarrow0\). Then \(h_n\) is an increasing homeomorphism of \([0,1]\) and \(\|h_n-u_n\|_\infty\to0\). If \(\omega_\Gamma\) denotes the modulus of continuity of \(\Gamma\), then \[d_{\rm curv}(g_n,\Gamma) \leq e_n+\omega_\Gamma(\|h_n-u_n\|_\infty)\longrightarrow0.\] This uses exactly the allowed matching in (2), with the identity on \(g_n\) and \(h_n\) on \(\Gamma\).

For the converse, the continuous function \(u_n(s)-u_n(t)\) attains its maximum on \(0\leq s\leq t\leq1\). If \(B_n\geq\eta>0\), a maximizing pair has \(s<t\) and \(u_n(s)-u_n(t)\geq\eta\). Set \(c_n=(u_n(s)+u_n(t))/2\in[\eta/2,1-\eta/2]\). When the endpoint coordinate errors are less than \(\eta/4\), each of the intervals \([0,s]\), \([s,t]\), and \([t,1]\) has endpoints on opposite sides of \(c_n\), at coordinate distance at least \(\eta/4\). Each therefore contains a visit to \(c_n\). Injectivity and compactness imply \[q_\eta=\min_{|v-w|\geq\eta/4}|\Gamma(v)-\Gamma(w)|>0, \qquad d_\eta=\mathop{\mathrm{dist}}(\Gamma([\eta/2,1-\eta/2]),\partial D)>0.\] Choose \(R_\eta<\min(q_\eta,d_\eta)/4\). Once \(e_n<R_\eta\), the four anchor points \(g_n(0),g_n(s),g_n(t),g_n(1)\) are outside the radius-\(R_\eta\) disk about \(\Gamma(c_n)\), whereas each of the three level visits is within \(e_n\) of its center. In each time interval crop the incoming and outgoing portions at the last outer and first inner hits, and at the last inner and first outer hits. For every \(e_n<r<R_\eta\) this supplies two annular passages. The six parameter interiors are disjoint, and simplicity gives disjoint spatial interiors. They are thus six passages in the stated convention, and in particular at least four. Fixed-factor cropping gives the same assertion for square annuli.

We give the measurability and order of limits for a random \(\Gamma\). Take a measurable continuous representative, for example the image of a disk SLE representative in the coupling above. We need not choose a random extension \(p\). Write \(\Delta_n=d_H(\operatorname{tr}g_n,\operatorname{tr}\Gamma)\), \(z_n=2\Delta_n+1/n\), and define instead \[ u_n(t)= \frac{\int_0^1 v\,(z_n-|g_n(t)-\Gamma(v)|)_+\,dv} {\int_0^1 (z_n-|g_n(t)-\Gamma(v)|)_+\,dv}. \tag{134}\] The denominator is positive since every \(g_n(t)\) is within \(\Delta_n\) of the trace, and continuity gives a parameter interval with positive weight. This coordinate is continuous in \(t\) and jointly measurable. Let \[\iota_\Gamma(r)=\sup\{|v-w|:|\Gamma(v)-\Gamma(w)|\leq r\}.\] It tends to zero as \(r\downarrow0\). All parameters receiving positive weight in (134) have diameter at most \(\iota_\Gamma(2z_n)\), and their weighted average lies between their extreme values. Consequently \[e_n\leq z_n+ \omega_\Gamma(\iota_\Gamma(2z_n))\longrightarrow0\] in probability; the endpoint coordinate errors also tend to zero. These assertions follow first almost surely along any subsequence on which \(\Delta_n\) and the endpoint errors converge almost surely, and hence hold in probability for the full sequence.

Fix \(0<\eta<1\) and an exceptional probability \(\zeta>0\). Since \(q_\eta,d_\eta\) are strictly positive random variables, choose deterministic \(q,d>0\) such that \[\mathbb P(q_\eta<q\ \text{or}\ d_\eta<d)<\zeta.\] The random centers then lie in the deterministic compact set \(F_d=\{z\in\overline D:\mathop{\mathrm{dist}}(z,\partial D)\geq d\}\), and one deterministic \(R<\min(q,d)/8\) works. For each fixed \(0<r<R\), the preceding six-passage argument, outside an event of probability \(\zeta+o(1)\), bounds the probability of a drop of size \(\eta\) by \(\mathbb P(\mathcal B_n(F_d;r,R))\). First let \(n\to\infty\), then \(r\downarrow0\) using (132), and finally \(\zeta\downarrow0\). Thus \(B_n\to0\) in probability. To deduce curve distance convergence, first choose a deterministic coordinate tolerance on which \(\omega_\Gamma\) is small outside probability \(\zeta\), then use the running-maximum bound and let \(n\to\infty\). Finally remove \(\zeta\). This localizes the continuity modulus as well as the injectivity and boundary-distance moduli. All annulus radii and compact charts are deterministic before the mesh limit is taken. ◻

Proof of Theorem 1. For the fixed finite-range real potential \(U\), Lemma 2 converts the exact contour weight into a finite-range, even, square-symmetric spin perturbation, with all interactions meeting the interior retained. Theorem 16 and Theorem 26 give an analytic bulk choice \(\beta_c(J,U,\lambda)\) near \(\beta_0=\log(1+\sqrt2)/(2J)\), positive for sufficiently small real \(\lambda\) of either sign and equal to \(\beta_0\) at zero. Shrink this interval once to satisfy the independent-edit stability threshold and the finite-table strict-incidence margins. This gives a number \(\lambda_0(J,U)>0\) before any choice of \(D,a,b\), approximation, exterior contours, corridor, or strip. The universal fixed-shape bootstrap in Theorem 29 is what permits all later fixed geometries to use this same interval.

Fix any \(|\lambda|<\lambda_0\), any bounded Jordan marked domain, any sequence of meshes and uniformly boundary-parametrized Jordan lattice domains in the theorem, and any admissible sequence \(P_{o,n}\). On each subsequence pass further so that the choice of plus boundary arc is fixed. The exact conditional-law reduction permits the target to be placed in a buffered torus chart while retaining all its exterior pins. Finite range permits any continuation beyond that chart without changing the conditional law. Proposition 43, the finite strip construction, and the slow diagonal produce a joint limit of every further Hausdorff-convergent subsequence with two comparison chords of the same \(\mathop{\mathrm{SLE}}_3\) law. Their forbidden-side inclusions and Lemma 45 identify the target trace with that chord. Thus the target traces converge in law in Hausdorff distance to chordal \(\mathop{\mathrm{SLE}}_3\).

Lift the subsequential trace coupling to the target paths using their conditional distributions given their traces. At each mesh the spin space is finite, so this disintegration is immediate and preserves every target passage probability. No tightness of these path variables is needed to perform the lift. The limiting chord already has a measurable continuous representative from its SLE marginal. The interior density comparison and bulk four-passage estimate give (132), uniformly in the permitted exterior configurations. Lemma 46 therefore gives convergence in probability in the oriented curve metric in this representation, and hence convergence in distribution in that metric. Since every subsequence has a further subsequence with this same limit, the full sequence converges. This argument also gives tightness in the requested curve topology; it did not assume it for the perturbed paths.

The conclusion concerns the prescribed \(\gamma_n\). The local redrawings used for its barriers preserve traversal order and have \(O(\delta_n)\) curve distance from it; they were used only to identify its trace limit. The half-stubs remained part of the original curve, with endpoints tending to \(a,b\), and the orientation is always from the incoming midpoint to the outgoing midpoint. The unit-lattice potential \(U\) was fixed throughout; only coordinates were multiplied by \(\delta_n\). No sign condition on \(U\) or \(\lambda\), boundary regularity beyond the stipulated Jordan parametrizations, or restriction on the additional exterior contours was imposed. The value of \(\beta_c\) and its interval of validity are consequently independent of every domain and boundary choice required in the theorem. ◻