The lattice model and the main result

Fix \(q\in[1,4)\) and put \[ p_q=\frac{\sqrt q}{1+\sqrt q},\qquad \kappa=\frac{4\pi}{\arccos(-\sqrt q/2)},\qquad d=1+\frac{\kappa}{8},\qquad x=2-d. \tag{1}\] Thus \(4<\kappa\le6\) and \(1/4\le x<1/2\). Let \(D=(0,1)^2\), with marked corners \(a=(0,0)\) and \(b=(1,1)\). The counterclockwise boundary arc from \(a\) to \(b\), consisting of the bottom and right sides, is denoted by \(F\). The complementary arc is \(W\).

For \(n\ge1\), the graph has vertex set \(n^{-1}\mathbb Z^2\cap\overline D\) and all nearest-neighbor edges in \(\overline D\). Identify all vertices on \(W\), including \(a,b\), in a single block; all other vertices are distinct. Denote this partition by \(\xi\). Remove the boundary edges on \(W\) from the sampled set of edges, and call the remaining set \(E_n'\). The probability of a set of open edges \(A\subset E_n'\) is \[\mathbb P_n(A)=\frac{1}{Z_n} p_q^{|A|}(1-p_q)^{|E_n'|-|A|}q^{k_\xi(A)},\] where \(k_\xi(A)\) counts connected components after the prescribed identification, including isolated vertices. The removed edges on \(W\) are declared open for drawing the interface. Edges on \(F\) remain random.

We use the Dobrushin medial strand from \(a\) to \(b\). Medial vertices are midpoints of primal edges; medial edges join midpoints of consecutive sides of a lattice square. At an open primal edge, pair the four medial half-edges in the two turns around the adjacent dual faces. At a closed primal edge, use the turns around the primal endpoints. The boundary turns separate the primal wired arc from the complementary dual wired arc. Two distinguished terminal half-edges occur at the boundary changes. The pairing determines the strand and its order, including the order at a medial vertex visited twice.

Let \(e_{n,1},\ldots,e_{n,N_n}\) be the full medial-edge traversals of this strand, including boundary traversals but excluding the two terminal half-edges. Let \(z_{n,j}\) be the midpoint of \(e_{n,j}\). Define \[\Pi(u,v)=\bigl(\max(0,\min(1,u)),\max(0,\min(1,v))\bigr).\] Project the drawn strand by \(\Pi\) and join its terminal points to \(a,b\) by straight segments. The resulting oriented curve is \(\eta_n\). The terminal connectors lie within \(O(n^{-1})\) of the marked corners and carry no counting mass. For \(c>0\), set \[ \nu_{n,c}=c n^{-d}\sum_{j=1}^{N_n}\delta_{\Pi(z_{n,j})}. \tag{2}\] This convention counts traversals, not Euclidean length. In fact, a full medial edge cannot recur on this strand: after resolving the pairings, the strand is a component of a graph of degree two, with its two terminal ends. A medial vertex can nevertheless be visited twice. This distinction will matter when we replace counting by local hit indicators.

For continuous oriented curves \(\gamma_1,\gamma_2:[0,1]\to\overline D\), use the distance \[ d_{\mathrm{curv}}(\gamma_1,\gamma_2) =\inf_{\phi_1,\phi_2}\sup_{0\le t\le1} |\gamma_1(\phi_1(t))-\gamma_2(\phi_2(t))|, \tag{3}\] where the infimum is over increasing homeomorphisms of \([0,1]\). We identify curves at distance zero. This topology retains traversal order, unlike a topology on trace sets alone.

Let \(\eta\) be chordal \(\mathrm{SLE}_\kappa\) in \((D;a,b)\), oriented from \(a\) to \(b\). Its \(d\)-dimensional Minkowski-content measure is normalized by \[ \mu_{\eta,r} =r^{-x}\Pi_*\bigl[ \mathbf 1\{\operatorname{dist}(z,\operatorname{tr}\eta)<r\}\,\mathrm dA(z)\bigr], \qquad \mu_{\eta,r}\longrightarrow\mu_\eta \quad\text{in probability as }r\downarrow0. \tag{4}\] Here the area measure is on the whole plane, not only on \(D\). The convergence is weak convergence of finite measures on \(\overline D\). The existence of this whole-trace measure, including the stated projection convention, is justified in Lemma 28 from the SLE content and Green-function theorems.

The value of \(c(q)\) depends on the stated discrete and continuum units; its numerical evaluation is not needed. Every estimate below is for a fixed \(q\). We make no claim of uniformity as \(q\uparrow4\).

Context and inputs

Schramm introduced SLE to describe conformally invariant limits of planar lattice interfaces (Schramm 2000). Beffara proved that the SLE trace has dimension \(1+\kappa/8\) for \(\kappa\le8\) (Beffara 2008). Dimension predicts the exponent in (2); it does not identify a counting-measure limit. Lawler and Sheffield introduced a Doob–Meyer construction of natural parametrization (Lawler and Sheffield 2011); Lawler and Zhou established its existence throughout \(0<\kappa<8\) (Lawler and Zhou 2013). Lawler and Rezaei proved that the resulting natural measure agrees, up to normalization, with \(d\)-dimensional Minkowski content (Lawler and Rezaei 2015). We use their interior content theorem and Euclidean Green-function estimates (Lawler and Rezaei 2012), and supply the whole-square boundary argument in Section 8.

There are two particularly relevant precedents for passing from a lattice curve to its natural measure. For critical triangular-lattice site percolation, Garban, Pete, and Schramm constructed an interface measure measurable from the limiting SLE\(_6\) curve, with conformal covariance exponent \(7/4\) (Garban et al. 2013). Their normalization at mesh \(\eta\) is \(\eta^2/\alpha_2^\eta(\eta,1)\), where \(\alpha_2^\eta(\eta,1)\) is the alternating two-arm probability. Holden, Li, and Sun identified the resulting measure with Minkowski content and proved convergence in natural parametrization (Holden et al. 2022). The later sharp alternating-arm asymptotics of Du, Gao, Li, and Zhuang (Du et al. 2024) permit that normalization to be replaced by a constant times \(\eta^{7/4}\). These are triangular-site percolation results, not occupation results for square-lattice FK interfaces.

For square-lattice loop-erased random walk, Lawler and Viklund proved convergence to SLE\(_2\) when each step receives a deterministic constant times the \(5/4\) power of the mesh (Lawler and Viklund 2021). Their result uses model-specific Green-function estimates and is stated for bounded simply connected domains with analytic boundary. In the FK setting, the Ising case already has ordered-trace convergence to SLE\(_{16/3}\) (Chelkak et al. 2014). These precedents distinguish the geometric trace limit from the microscopic estimates required for natural measure convergence.

For the present FK model we take chordal ordered-trace convergence from the companion manuscript (OpenAI 2026a, Theorem chordal:trace). This is the trace input specified in Input 2. We separately identify the additional companion results used in the argument: buffered FK comparison and passage estimates, the whole-plane loop limit, and the square-row representation. They are cited at their stated scopes in Section 2; none supplies the occupation asymptotic proved here. The public crossing and arm-separation inputs are due to Duminil-Copin, Manolescu, and Tassion (Duminil-Copin et al. 2021), building on the FK-Ising separation arguments of Chelkak, Duminil-Copin, and Hongler (Chelkak et al. 2016). The electric normalization is fixed by the six-vertex height covariance (Duminil-Copin et al. 2026; OpenAI 2026b).

One parameter issue requires care. The angular-current proofs in the GFF companion are presented in a parameter interval different from the FK interval. In Section 4 we apply their algebraic arguments to the FK companion’s row representation and verify the required hypotheses there. We do not import an angular-spectrum theorem outside its stated range.

How the normalization is determined

There are two distinct obstacles. First, one must obtain information more precise than an arm exponent. Second, one must compare that information with visits to a single medial edge. The proof separates these tasks.

We orient loops and insert phases depending on their winding around marked lattice points. Summing the orientations gives scalar loop weights; for small charges these weights are positive. Regeneration across surrounding open circuits gives quantitative loss of dependence on the exterior and zero-free complex neighborhoods for the associated partition functions. These estimates are developed in Section 3.

The height potential associated with small electric charges is nearly harmonic, with a summable power error. The proof uses two directions of row slicing: the untwisted current controls the potential away from the line of insertions, and twisted rows control its differences near that line. Step-two lattice potential theory then gives a Coulomb correlation formula with a power remainder. Complex continuation extends it to charges that impose arm events, and a reflection symmetry removes the remaining linear ambiguity in the local normalization. Sections 4 and 5 contain this part of the argument.

To reach a single lattice edge, we condition two opposite-color arms to reach a large radius. Section 6 constructs a coupling that makes the exterior configurations identical with power accuracy. The successful separating band has exactly two medial strands crossing it, so its exterior continuation is independent of the microscopic starting configuration. This last property is stronger than separation of two chosen arms; it is what makes the coupling useful for counting a distinguished interface.

Section 7 compares two electric charges at a mesoscopic separation with charges at neighboring lattice points. The coupling applies even though the remaining electric factor is signed. A nonzero mesoscopic amplitude controls the relative error, and a summable comparison between scales \(n\) and \(N\in[n,2n]\) gives a genuine microscopic amplitude. A second comparison, with a disk probe, produces deterministic constants for replacing lattice visits by neighborhood area.

Finally, Section 8 performs this replacement in \(L^2\). Ordered-trace convergence identifies the disk probes, while SLE Minkowski content identifies their small-radius limit. Integrable boundary-strip estimates complete the passage to measures on the closed square.

The potentially reusable ingredients are the complex stability of radially tilted loop partition functions, the two-direction estimate for the electric height potential, and the separating-band coupling used to convert mesoscopic correlation amplitudes into microscopic normalization. Their roles are quantitative: they remove errors that an exponent-only argument cannot detect.

Trace and local FK estimates

This is precisely (OpenAI 2026a, Theorem “Dobrushin convergence for \(1\le q<4\),” label chordal:trace). We use it as the trace input, without deriving any assertion about occupation measures from trace convergence alone.

Here and below, a buffer has fixed positive relative width and at least a sufficiently large fixed number of mesh layers. A collar is a full annulus, or one of the fixed rectangular or straight-wall sector geometries explicitly described below. The graph and deterministic wire portions are fixed inside the collar; only the partition transmitted from the unobserved side varies. Only the number of prescribed deterministic through-wire blocks is bounded, and no additional identification jumps the collar. The exterior-induced partition on the unobserved side is arbitrary: its number of blocks is not bounded. Nor is any bound imposed on the number of random open components crossing the collar. The full annuli used for inward comparison have no prescribed through-wires.

Part (i) is (Duminil-Copin et al. 2021, Theorem 1.2 and Propositions 6.2, 6.3, and 6.5). The last three propositions apply to fixed color words, including the two words needed here. Parts (ii)–(v) are respectively the lemmas “Bounded densities across a collar,” “Relative comparison,” “Exponential passage moments,” and “Local tilt bounds” in (OpenAI 2026a, Section “Tiles, positive laws, and localization”); their source labels are prelim:collar, prelim:relative-collar, prelim:passages, and prelim:tilts. The compact-uniform assertion in Part (v) follows from those proofs by bounding \(|\log b|\) on the compact set. We require Part (iii) only in full annuli. In particular, these inputs do not assert comparison across arbitrary irregular walls or under complex-valued tilts.

The separation argument with high probability used in Section 6 is the actual-cluster construction of (Chelkak et al. 2016, Lemma 5.9), whose original statement concerns FK-Ising. For our range of \(q\), its conditional crossing steps are supplied by (Duminil-Copin et al. 2021, Theorem 1.2), as in the proof of that paper’s Proposition 6.2. Section 6 specifies the extra buffers and the arm continuations needed for the coupling; we do not identify ordinary separation up to constants with that stronger conclusion.

The free and wired plane laws coincide, are invariant under lattice symmetries, and are exchanged by primal–dual symmetry at these self-dual weights; see (Duminil-Copin et al. 2018, Theorem 1.1) and its square-lattice specialization in (OpenAI 2026a, Section “Crossing estimates and finite plane loops”). Buffered circuits of both colors show that every plane medial strand is almost surely a finite loop. A translation by one tile may interchange the two colors, but preserves this loop law.

Plane loops and angular rows

This is the separately cited Theorem “Nested full-plane convergence for \(q<4\),” label loop:plane, in (OpenAI 2026a, Section “From squares to the free plane law and the sphere”). Its statement concerns exactly the plane law used here. It is stronger than Input 2, and is used for the finite-resolution nesting limits in Sections 3 and 7. Passage moments from Input 3 provide the required uniform integrability when nesting functionals are unbounded.

Orienting a completed contractible loop and assigning total turning weights \(e^{i\lambda}\) and \(e^{-i\lambda}\) recovers its unoriented weight \(m\). Summing the local orientations gives the homogeneous six-vertex weights \[a=b=1,\qquad c=2\cos(\lambda/2),\qquad \Delta=\frac{2-c^2}{2}=-\cos\lambda.\] The induced height changes by one across an arrow. The plane arrow law is the zero-slope law obtained by first taking the longitudinal limit of balanced even-width tori and then letting the width tend to infinity. Here balance means zero total arrow flux across a transverse section. The identification of finite local arrow and phase words with the FK loop expansion is part of the plane-kernel construction in (OpenAI 2026a, Section “Row algebra and the infinite-plane state”). It agrees with the balanced-plane law in (Duminil-Copin et al. 2026, Theorem 2.2 and Equation (18)).

These are the plane-kernel, angle-generator, and fixed-band continuation lemmas, labels bulk:plane-kernels, bulk:boost, and bulk:edge-band, together with Equation bulk:inner-factorization, in (OpenAI 2026a, Section “Angular transfer and its two leading modes”). The argument in (OpenAI 2026b, sec. 4) for the current cocycle, finite-inner spectrum, spectral-width tails, and variogram is stated there for \(\pi/3<\lambda<\pi\). Those stronger conclusions are not included in Input 5. In Section 4 we transfer the needed arguments to the representation above and prove the conclusions at our parameter.

The covariance normalization

Only a separated-increment covariance, rather than a pointwise Gaussian limit for the height, is needed to determine the electric constant. Let \(h_\delta\) be the height of the preceding balanced plane law on mesh \(\delta\), with an arbitrary additive constant, and put \(G(u,v)=-(2\pi)^{-1}\log|u-v|\).

Indeed, \(c=2\cos(\lambda/2)\in[\sqrt3,2)\), so the covariance part of (Duminil-Copin et al. 2026, Definition 2.7 and Theorem 2.8) applies directly to this physical law and unit-height convention. Its multiplier is \(1/\arcsin(c/2)=2/(\pi-\lambda)\). The same normalization is recorded in the main theorem of (OpenAI 2026b). In Section 5 we use Input 6 only for four distinct collinear points. Thus this use needs neither the smaller-\(c\) part of that companion’s GFF theorem nor an extension of the DKLM theorem beyond its stated range.

Passage bounds at a circuit and under a tilt

The FK estimates used here are the bounded-density collar lemma, exponential passage moments, and local tilt bounds in (OpenAI 2026a, sec. 2, “Comparison through a collar” and “Passages and positive local tilts”); their precise scope is recorded in Input 3. We first explain the extension to an irregular wired circuit, where a bulk-buffer assertion alone would not suffice.

A newly counted enclosing loop whose maximum radius crosses a specified cutoff has a passage in one of a fixed number of buffers at that scale: it either goes around \(u\) in the band or leaves the band. The same observation charges every enclosing loop whose status changes when a subannulus is resampled. Loops which do not enclose \(u\) need not be charged. Thus, for passage counts \(X_j\) on \(l\) geometrically spaced bands, finite coloring of their buffers and conditional iteration give \[ \mathbb E\exp\left(t\sum_{j=1}^l X_j\right) \le \exp(C_t(l+1)),\qquad t<\infty. \tag{9}\] The bound is conditional on data beyond the buffered cuts.

At the outer wall of a primal-wired circuit, count large enclosing loops using their dual sides. Those sides have free boundary at the wall. In a cropped annular or rectangular passage region, the ordered medial separators force distinct actual dual crossing components: two neighboring sides may use one component, so a factor two and bounded end effects are allowed. The crop ends lie strictly inside the region traversed by the separators. An inward switch arc beside an open circuit edge still has an actual interior dual gap on its other side; it cannot use a fictitious path through the exterior.

Explore complete actual dual components from one end of the crop, including unsuccessful components. After a component is removed, its incident unexplored boundary edges are closed in the explored color. The remaining law is dominated by filling the cropped box and wiring its cut boundary. The original wall is free in this color, and conditioning farther away can identify cut-boundary vertices only. Filling the missing vertices and, if necessary, wiring the entire boundary can only increase actual crossing probabilities. A blocker in the filled box has uniformly positive probability by the crossing bounds of Input 3, including the wall version of (Duminil-Copin et al. 2021, Theorem 1.2). Successive component discoveries therefore give a geometric tail. To obtain an arbitrary fixed exponential rate, subdivide the passage distance into \(J\) separated slices, cover each by buffered small boxes, and apply the same exploration to the boxes intersected with the circuit interior. Every original passage crosses a small buffer in each slice. If \(X\) is the original count and \(X_i\) the \(O(J^2)\) small-buffer counts, then \(JX\le\sum_iX_i\). Their buffers have a coloring with \(k\) colors independent of \(J\). Conditional iteration in each color class, followed by Hölder across classes, therefore bounds \(\mathbb Ee^{cJX/k}\) using the common first exponential rate \(c\) of the \(X_i\). The prefactor may depend on \(J\), which is harmless: choose \(J\) large and fixed to obtain any prescribed rate. The finitely many scales where these boxes do not fit are covered by a deterministic finite-size bound. This proves (9) also in the outer circuit collar. The argument uses the specified wired wall, not an arbitrary partition on a jagged boundary.

Now first count nests only up to a radius a fixed factor inside the inscribed radius of the circuit. The local tilt bounds compare successive cutoff masses and give every fixed passage moment at the last weighted level. They are uniform for \(v\in V\): in their proof, the Hölder exponents can be chosen using \(\sup_{v\in V}|\log v|\), on either side of \(1\). Moving the cutoff back a bounded number of levels leaves the passage buffers outside the tilt. Hölder’s inequality then restores those levels. The preceding wall estimate restores the remaining boundedly many outer levels at bounded cost. Negative powers have the same comparison, by Jensen’s inequality followed, when needed, by this cutoff argument. Bounded densities across a full bulk collar now compare the interior cutoff mass to that for any other comparable state. This proves (6), first for comparable sizes and then by multiplying comparisons across scales.

These statements remain uniform if counting begins at a lower maximum-radius threshold \(h\). Start the cutoff induction with no weight below \(h\). In particular, under a tilt supported below the inner end of \(l\) bands, their joint passage moments still obey (9); the innermost boundedly many buffers use the local tilt bounds, and all others are separated from its support. Adding the weights in those \(l\) bands changes the density, including its normalization, by a factor whose fixed moments cost \(\exp(C(l+1))\). Hence (9) holds under the full positive tilt up to the outer end of those bands as well. These are the real estimates needed for regeneration.

Stopping circuits and exponential waiting times

Fix \(K\) sufficiently large and use levels \(r_i=r_0K^i\), \(i\ge0\), with \(r_0\) large enough for all crossing constructions. Each trial band has radii comparable to \(r_i\) and \(3r_i\) and has full collars on both sides. The value of \(K\) leaves these enlarged trial regions disjoint and leaves an unused inward collar after a starting circuit. Assign a state the level just below its inscribed radius; states at one level have sizes within a fixed factor. A plane cutoff is assigned in the same way, and its first trial lies well inside that cutoff.

Inspect trials inward until finding a surrounding primal-open circuit, choosing the outermost circuit in that band. The usual outer exploration is a stopping exploration: no edge strictly inside the selected circuit need be queried. Intersecting candidate circuits can be merged by taking their outer boundary. Failed tests concern previous, disjoint bands. At level zero use a terminal symbol with continuation mass one. Conditional on a stopped circuit, the ordinary continuation is its wired FK law. If \(D\) counts the weighted enclosing loops outside that circuit, then \[ F(\gamma;v)=\mathbb E_\gamma\!\left[v^D F(\gamma_1;v)\right], \tag{10}\] where \(\gamma_1\) denotes the stopped state, including the terminal case. For a plane start, \(D\) also respects its original maximum-radius cutoff. The assignment of circuit-side arcs makes this factorization exact. It follows that the stopped process remains a regeneration process under the tilt.

Let \(J\) be the number of levels crossed in one transition. Assigning the loops counted by \(D\) to their maximum-radius bands gives, under any of the positive tilts under discussion, \[ \mathbb E_{\gamma,v}[e^{tD};J=j]\le e^{C_t(j+1)},\qquad t<\infty. \tag{11}\] We next prove \[ \mathbb P_{\gamma,v}(J>l)\le Ce^{-cl}. \tag{12}\] This needs more than a crossing bound after arbitrary neighboring edges have been exposed, because such an exposure need not leave a favorable conditional law under the tilt.

For each of the first \(l\) trials, designate its central band as a region to be resampled by the ordinary FK conditional law. All counts used below have buffers in that trial’s enlarged collar. A trial is called typical if the old passage counts which can charge changed enclosing loops are at most \(H\), and, for every boundary partition which changes outside its collar, the ordinary conditional redraw has probability at least \(b\) of producing a surrounding circuit and at most \(H\) new charging passages. Here \(b>0\) and \(H<\infty\) will be fixed independently of \(l\) and of the lattice scale. The second condition depends only on collar edges outside the resampled band, so remains true after other trial bands are changed.

We justify that typicality fails with arbitrarily small ordinary conditional probability. Expose the collar up to its far cuts, and let \(Y\) count its actual components joining the band-facing cuts to the far cuts, on both radial sides, plus one. Only these components can acquire new identifications when the far partition is changed. For fixed collar edges, any two redraw kernels thus have density ratio at most \(q^{C Y}\). Complete-component exploration in cropped subcollars, as above, gives a uniform exponential tail for \(Y\). First fix a large bound \(Y_0\). Given \(Y\le Y_0\), the worst conditional probability, over the induced far partitions, of more than \(H\) new passages is at most \(q^{CY_0}\) times that probability under the actual partition, or its conditional mixture. The latter probability has arbitrarily small expectation as \(H\to\infty\), by the ordinary conditional passage estimate. Markov’s inequality then makes the collar-edge event on which this worst probability exceeds a fixed tolerance arbitrarily rare. Success without the passage restriction has conditional probability at least \(2b>0\) by RSW. Choose the tolerance below \(b\), and then \(H\) large. The old passage test is handled by the same moment bound.

All these estimates hold conditional on data outside the enlarged trial collar, uniformly in its induced partition. For any prescribed set of trials their failures therefore have probability at most a chosen small constant to the size of that set, by conditional iteration. A binomial sum proves, at any desired exponential rate, that fewer than \(l/3\) typical trials is unlikely under ordinary FK. To transfer this statement to the fully tilted law, first retain only the tilt below the inner end of these \(l\) trials, omitting boundedly many end trials for the buffers. The same ordinary conditional argument still applies: the retained weight counts loops wholly inside that lower cutoff, so is measurable outside each enlarged trial collar. Conditional on the exterior of a collar, it is constant, and the conditional law in the collar is ordinary FK. The remaining change of density has second moment at most \(e^{C(l+1)}\), by (9) and the real mass comparisons. Cauchy–Schwarz, choosing the preceding ordinary rate larger than \(C\), gives \[ \mathbb P_{\gamma,v}(|T|<l/3)\le Ce^{-cl}, \tag{13}\] where \(T\) is the set of typical trials. This proof applies equally when a lower radius threshold is present.

For completeness, the insertion argument converting (13) into (12) is as follows. Write \(A_l\) for failure at all \(l\) trials, and for a fixed subset \(S\) of the trials put \(A_S=A_l\cap\{S\subset T\}\). Redraw the bands in \(S\) successively with their ordinary FK heat baths, retaining only outcomes with success and at most \(H\) new passages. Each retention has probability at least \(b\), even after the preceding redraws. At one accepted redraw, at most \(CH\) old or new enclosing loops change their weight. Consequently the ratio of input to output tilt densities is at most \(M=\exp(CH\sup_{v\in V}|\log v|)\) per redraw. Ordinary heat baths preserve the ordinary law, including their successive composition. If \(E_S\) is the event with exact success set \(S\) among these \(l\) trials, integration of the accepted transitions therefore gives \[\mathbb P_{\gamma,v}(E_S) \ge (b/M)^{|S|}\mathbb P_{\gamma,v}(A_S).\] The events \(E_S\) are disjoint. With \(\delta=b/M\), summing over \(S\) yields the useful integrated inequality \[ 1\ge \mathbb E_{\gamma,v}\left[\mathbf 1_{A_l}(1+\delta)^{|T|}\right]. \tag{14}\] The set \(T\) stays inside the expectation; no deterministic choice of typical trials is required. Equations (13) and (14) prove (12).

Combining the tail with (11) gives a joint exponential moment: there exist \(a,t>0\) such that \[ \sup_{\gamma,v}\mathbb E_{\gamma,v}e^{aJ+tD}<\infty. \tag{15}\] Indeed fix an exponent \(T_0>0\) in (11) and apply Hölder on \(\{J=j\}\): \[\mathbb E[e^{tD};J=j] \le \mathbb E[e^{T_0D};J=j]^{t/T_0} \mathbb P(J=j)^{1-t/T_0}.\] For \(t/T_0\) small, the resulting exponent in \(j\) is negative; then choose \(a>0\) smaller than its absolute value.

Coupling states at different levels

At a common level, the next-band transitions of any two states have a common part of uniformly positive mass. To see this, use bounded densities through the unused collar to minorize both ordinary band marginals by a fixed fraction of one reference marginal, and restrict to its successful circuits. Given the next screen, its continuation mass compares to any reference mass at that level by (6). The number of newly charged loops can be restricted to \(D\le H_1\) at arbitrarily small discarded ordinary probability, by the fixed-band passage estimate. On this restriction \(v^D\) is bounded above and below uniformly. Subtract the two discarded parts from the reference marginal and then take its common minimum. The normalizations also compare to the same continuation mass, so the common part retains a fixed mass \(p_0>0\) under the tilted laws. In particular a jump of one level has probability bounded below.

For starts at the same or neighboring levels, advance whichever state has the higher remaining level; when the levels agree, attempt coupling on this common part. Use a fixed mass strictly smaller than the available overlap. After a failed attempt the residual transition moments are bounded by the original moments divided by \(1-p_0\). This avoids conditioning estimates depending on an overlap which might itself be close to one.

Here is the quantitative reason this procedure also works for asynchronous levels. If their current difference is \(d\ge1\) and the higher state jumps by \(J\), the new difference is \(|d-J|\). For sufficiently small \(a>0\), (15) gives an exponential drift outside a fixed finite set: \[\mathbb Ee^{a|d-J|} \le e^{ad}\mathbb E[e^{-aJ};J\le d] +\mathbb E[e^{a(J-d)};J>d] \le \theta e^{ad},\qquad d\ge d_0,\] with \(\theta<1\). The last tail term is exponentially small relative to \(e^{ad}\), while \(J\ge1\) bounds the first coefficient below one. From \(\{0,\ldots,d_0\}\), finitely many one-level jumps bring the levels together with uniformly positive probability, followed by the common-part coupling. The exponential drift bounds return times, and the failed-attempt residual moments remain bounded. Geometrically summing these returns proves an exponential moment for the number of transitions until meeting. More explicitly, the kernel of one unsuccessful excursion from this finite set has mass strictly below one at discount one and a bounded exponential moment; the same is true at a sufficiently small positive time exponent. Summing its powers proves the assertion.

Iterating (15) and reducing its exponents then bounds total distance and charged-loop cost up to meeting. If \(L\) denotes the largest number of levels descended by either chain, and \(D_1^*,D_2^*\) their respective costs before the common state, \[ \mathbb E\exp\{c(L+D_1^*+D_2^*)\}\le C. \tag{16}\] One may require each chain to take one transition before permitting meeting; the bound is unchanged. The terminal symbol provides the same construction at the finitely many bottom levels.

If the initial size is \(R\), reaching a radius comparable to \(s\) without meeting requires order \(\log(R/s)\) descended levels. Equation (16), with a smaller exponent to cover a final overshoot, bounds this probability by \(C(s/R)^c\). On meeting, both chains have exactly the same surrounding circuit and the same conditional tilted continuation. Their configurations can therefore be made identical inside it. This proves the inward coupling assertion in Proposition 7. Terminal exhaustion, though a valid meeting for partition-function identities, is counted as failure in this spatial coupling; the coupling attempts here are stopped before the radii drop below \(s\).

Complex stability and removal of the inner cutoff

Fix a real \(v\in V\). For a state \(\gamma\) at level \(i\) define \[G_\gamma(w) =\frac{F(\gamma;ve^w)}{F(\gamma;v)} =\mathbb E_{\gamma,v}e^{wN_\gamma}.\] At finite scale these are holomorphic functions. We prove by induction on the level that they do not vanish and that, for states at the same or neighboring levels, \[ |\log G_\gamma(w)-\log G_{\gamma'}(w)|\le B|w|. \tag{17}\] All logarithms start at zero at \(w=0\). The terminal levels have uniformly bounded size and give the initial assertion directly.

To compare two new starts, use the preceding coupling, requiring one transition in each chain so that their meeting state lies below the highest level being proved. Regeneration gives \[G_{\gamma_j}(w)= \mathbb E[e^{wD_j^*}G_{\gamma_*}(w)],\qquad j=1,2,\] with the same random meeting state \(\gamma_*\). Divide both sides by \(G_\sigma(w)\) for any reference state \(\sigma\) at the preceding level. Chaining the inductive bounds through the lower levels gives \[Q(w)=\frac{G_{\gamma_*}(w)}{G_\sigma(w)},\qquad |\log Q(w)|\le B|w|(L+2).\] By (16), for each fixed \(B\) and all sufficiently small \(|w|\), both divided expectations lie in \(\{z:|z-1|<1/4\}\). Their difference has the sharper bound \[\begin{align*} \left|\mathbb E[(e^{wD_1^*}-e^{wD_2^*})Q(w)]\right| &\le |w|\mathbb E\left[(D_1^*+D_2^*) e^{|w|(D_1^*+D_2^*)+B|w|(L+2)}\right]\\ &\le C_0|w|. \end{align*}\] Here \(C_0\) is independent of \(B\), provided \(B|w|\) is smaller than a fixed fraction of the exponent in (16). The logarithm is uniformly Lipschitz on the displayed disk. Thus choose \(B\) larger than the resulting fixed multiple of \(C_0\), and only then choose \(\delta>0\) so that the disk and exponent conditions hold for \(|w|<\delta\). This closes both the nonvanishing induction and (17). Telescoping levels proves (7). The real comparison to a terminal state bounds \(|\log F(\gamma;v)|\) by \(C\log(2+R)\), and another telescoping sum proves the asserted logarithmic bound in the complex neighborhood.

It remains to compare full and thresholded masses. Put \(h=\epsilon R\) and let \(l\) be the number of levels between \(R\) and \(h\). In each regeneration path stop at its first screen whose level is at most \(J\) levels above \(h\), where \(1\ll J\le l/2\). Call this stop typical if its level is between one and \(J+O(1)\) levels above \(h\). The probability of an atypical stop is \(Ce^{-cJ}\) under both the full and thresholded tilts: sum the jump-tail bound over its possible origins, each level being visited at most once. On a typical path every loop already charged encloses the stopping screen or lies outside it, and hence has maximum radius above \(h\). The two weighting rules have therefore agreed up to this stop.

Write \[Q_h(\sigma)=F_{\ge h}(\sigma;v)/F(\sigma;v)\] for the continuation quotient. Normalize by its value \(q_h\) at any reference screen at the threshold level. Real comparisons, including their thresholded version, give on typical stops \[ e^{-C(J+1)}\le Q_h(\sigma)/q_h\le e^{C(J+1)}. \tag{18}\] Couple the two full-tilt paths from the original comparable states. Their stopped screens agree except with probability \(Ce^{-cl}\), after changing \(c\) to account for \(J\le l/2\). On typical stopped paths regeneration gives the exact identity \[I_\gamma:=\mathbb E_{\gamma,v} [\mathbf 1_{\mathrm{typ}}Q_h(\sigma)] =\frac{F_{\ge h}(\gamma;v)}{F(\gamma;v)} \mathbb P_{\gamma,v,\ge h}(\mathrm{typ}),\] where the probability on the right uses the thresholded tilt. The difference of the two \(I_\gamma\), divided by \(q_h\), is at most \(Ce^{C(J+1)-cl}\). The lower bound in (18) converts this into a relative error \(Ce^{C'(J+1)-cl}\).

The omitted atypical paths also have small relative weight. Indeed their fraction of the full mass is their probability under the full tilt, and their fraction of the thresholded mass is their probability under the thresholded tilt. Both are \(Ce^{-cJ}\). The factorization is used only on typical paths, where the already charged counts agree; an overshoot below \(h\) does not require such a factorization. Taking \(J\) a sufficiently small positive fraction of \(l\) proves (8) and completes the proof of Proposition 7.

Several marked points

Fix \(p\ge2\) labeled tile vertices \(\mathbf u=(u_1,\ldots,u_p)\). For \(\varnothing\ne I\subsetneq\{1,\ldots,p\}\) let \(N_I\) be the number of plane loops enclosing exactly the points with labels in \(I\). Assign relative weights \(\mathbf v=(v_I)\), with \(v_\varnothing=v_{\{1,\ldots,p\}}=1\), and define \[C(\mathbf v;\mathbf u) =\mathbb E\prod_{\varnothing\ne I\subsetneq\{1,\ldots,p\}} v_I^{N_I}.\] This is well defined even for complex weights. Every separating loop intersects a fixed finite collection of tile paths joining the points; only finitely many switch strands use those paths. Let \(b_0\in(1,2)\) be a fixed continuity cutoff for the limiting loop law, and write \[F_n(v)=\mathbb Ev^{\#\{\ell:u\text{ inside }\ell, R_u(\ell)\le b_0n\}}.\] Translation and duality make this function independent of the choice of tile vertex \(u\). Rounding and color conventions are transported by these symmetries.

Rows, currents, and their spectral measure

We first specify the row coordinates and the exchange identity. Put \(v_0=\pi/2\). A strip angle \(z\), with \(|\Re z|<v_0\), corresponds to the half-plane argument \(w=e^{iz}\); conversely \(z=-i\operatorname{Log}w\), using the logarithm on \(\Re w>0\). The argument of the local matrix in a homogeneous row is \[u(z)=\rho(v_0-z),\qquad R(u)=\begin{pmatrix} \sin(\lambda-u)/\sin\lambda&0&0&0\\ 0&\sin u/\sin\lambda&1&0\\ 0&1&\sin u/\sin\lambda&0\\ 0&0&0&\sin(\lambda-u)/\sin\lambda \end{pmatrix}.\] On an even spatial circle of size \(N\), with quantum spin space \((\mathbb C^2)^{\otimes N}\), define the monodromy and marked row by \[\mathcal M_{N,c}(z)=R_{c0}(u(z))\cdots R_{c,N-1}(u(z)),\qquad T_N^D(z)=\operatorname{tr}_c\bigl(D_c\mathcal M_{N,c}(z)\bigr).\] The index \(c\) denotes a two-dimensional auxiliary space. Let \(\Omega_N\) be the positive unit Perron vector of the untwisted rows on the zero-total-spin sector, and let \(\Lambda_N(z)\) be its eigenvalue for real strip angles, analytically continued using that common eigenvector. The marked rows themselves act on the full quantum spin space; in particular a local matrix \(O\) need not preserve the zero-total-spin sector. For real strip angles, divide by \(\Lambda_N(z)\) and take plane word limits. Denote the resulting row and its bounded holomorphic continuation by \(\widehat A^D(z)\). We use \(A^D(w)=\widehat A^D(-i\operatorname{Log}w)\) below. A finite row word means a vector \[\widehat A^{D_r}(z_r)\cdots\widehat A^{D_1}(z_1)\Omega, \qquad r<\infty,\] initially with real \(z_j\in(-v_0,v_0)\); the empty word is the vacuum \(\Omega\).

The local Yang–Baxter identity, and its row consequence, are \[\begin{align*} R_{12}(u-u')R_{13}(u)R_{23}(u') &=R_{23}(u')R_{13}(u)R_{12}(u-u'),\\ \mathcal R_{12}(z,z')\mathcal M_{N,1}(z)\mathcal M_{N,2}(z') &=\mathcal M_{N,2}(z')\mathcal M_{N,1}(z) \mathcal R_{12}(z,z'),\tag{25}\\ \mathcal R_{12}(z,z')&=R_{12}\bigl(\rho(z'-z)\bigr). \end{align*}\] These are the finite identities in the Row Algebra and the Infinite-Plane State subsection of (OpenAI 2026a). In particular, if \(\mathcal R\) is invertible and \[\mathcal R(D\otimes E)\mathcal R^{-1} =\sum_\ell D_\ell\otimes E_\ell,\] then auxiliary tracing, followed by the plane limit, gives the marked exchange rule \[ \widehat A^D(z)\widehat A^E(z') =\sum_\ell\widehat A^{E_\ell}(z')\widehat A^{D_\ell}(z). \tag{26}\] There are at most sixteen terms after expansion in tensor matrix units. The normalization cancels on the two sides. Analyticity extends this identity wherever its intertwiner is invertible.

Set \[Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad X=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad K(\zeta)=\operatorname{diag}(\zeta,\zeta^{-1}), \qquad |\zeta|=1,\] and \[P_\zeta(w)=A^{K(\zeta)}(w),\qquad P(w)=A^I(w),\qquad J(w)=A^Z(w).\] The Hilbert space completion of the row words is denoted by \(\mathcal H\), and \(\Omega\) has norm one. The rows are bounded and holomorphic for \(\Re w>0\), with \(\|A^D(w)\|\le2\|D\|\). Their reflection rule is \[ \widehat A^D(z)^* =\widehat A^{X\overline D X}(-\overline z),\qquad A^D(w)^*=A^{X\overline D X}(\overline w). \tag{27}\] This is Equation bulk:adjoint of (OpenAI 2026a), continued holomorphically from real strip angles. In particular, \(J(w)^*=-J(\overline w)\). The \(P_\zeta\) rows with the same twist commute and are normal; \(P_\zeta(1)\) is self-adjoint. Every \(P_\zeta(w)\) is a contraction in the closed right half-plane away from zero. At its two boundary arguments the untwisted rows are the spatial shift and its inverse: \[ P(i)=S,\qquad P(-i)=S^{-1},\qquad A^D(i)=SD_0,\qquad A^D(-i)=(XD^tX)_0S^{-1}. \tag{28}\] Here \(D_0\) acts at one site and \(S\Omega=\Omega\). Dilation of the argument is implemented by a strongly continuous unitary group fixing \(\Omega\). Also \(P(w)\Omega=\Omega\).

These facts follow from the Plane Kernels, Angle Generator, and Scalar Continuation in a Fixed Band lemmas in (OpenAI 2026a, Section “Angular transfer and its two leading modes”). We record the analytic estimates in the latter proof which will also justify differentiation at an endpoint. For one radius \(r_\lambda>0\), independent of all sufficiently large even \(N\), the two quotients \[ T_N^D(z)T_N^I(z)^{-1},\qquad T_N^I(z)^{-1}T_N^D(z) \tag{29}\] are holomorphic and bounded by \(C_\lambda\|D\|\) on the discs \(|z-v_0|<r_\lambda\) and \(|z+v_0|<r_\lambda\). This is Equation bulk:division and the crossing argument in that proof. On these endpoint discs \(T_N^I(z)\) is invertible, so \(\Lambda_N(z)\) is nonzero there. We use nonvanishing of its complex continuation only on these discs. In particular, the marked vacuum vectors have uniform holomorphic bounds on these discs.

For clarity, the dense set used to extend these vectors consists of finite words whose real angles lie in \((-v_0/3,v_0/3)\). It is dense because orthogonality to such words extends, one argument at a time, by holomorphy. Exchanging an endpoint row through any one such word uses Equation (26). Its finitely many intertwiners stay invertible on a common disc about that endpoint: the word angles are separated from the endpoint by the middle third of the strip. The continuation radius is independent of the word, although its vector bound may depend on the word. This is the dense-set continuation step in the proof of the same Scalar Continuation lemma. The denominators of those intertwiners do not depend on \(D\). Thus the scalar continuation discs, and the constants in the low-band arguments below, are uniform for the unit diagonal marks \(K(\zeta)\).

We will need the following concrete consequence for a fixed local spin matrix \(O\). For the finite vacuum \(\Omega_N\) just specified, there are some \(r_O>0\) and \(C_O<\infty\), \[ \sup_N\sup_{|z-v_0|<r_O} \left\|\frac{T_N^I(z)}{\Lambda_N(z)}O\Omega_N\right\| \le C_O. \tag{30}\] Only sufficiently large circles containing the support of \(O\) are included. To obtain this bound, translate the fixed support into an interval, expand \(O\) in tensor matrix units, and express each resulting vector by a finite product of positive endpoint rows, using Equation (28). Translation commutes with the unmarked row. Move the row at \(z\) to the vacuum through those endpoint rows. The intertwiners are now near \(R(0)\), the invertible swap, so their coefficients are bounded on a sufficiently small disc. The remaining endpoint rows are bounded shifts with local marks, and the marked vacuum vector is bounded by Equation (29). This proves Equation (30) without a bound depending on \(N\). Cauchy’s formula consequently bounds the \(k\)th vector Taylor coefficient by \(C_O r_O^{-k}\) after decreasing \(r_O\). Plane word convergence and these bounds pass endpoint derivatives and their Gram pairings to the plane.

The untwisted commuting rows have spectral functions \(f\) satisfying \[ |f(w)|\le1\quad(\Re w>0),\qquad f(\overline w)=\overline{f(w)},\qquad |f(it)|=1\quad(t\ne0). \tag{31}\] They continue through the imaginary axis away from zero. Their zeros \(a_j\) lie in a fixed cone \(\Re a_j\ge c_0|a_j|\); the same cone and continuation statements hold for a fixed twist. The resulting inner factorization has the form \[ f(w)=\gamma e^{-a w-b/w} \prod_j\frac{w-a_j}{w+\overline{a_j}}\,\gamma_j, \qquad a,b\ge0,\quad |\gamma|=|\gamma_j|=1, \tag{32}\] with the usual summability condition on the zeros. This is precisely the factorization established in the Inner Factorization and Low Spectral Bands subsection of (OpenAI 2026a).

Let \(\mathcal D\) be the closed subspace generated by words whose marks are diagonal. It is invariant under \(P,J\) and their adjoints, and contains finite height-phase insertions, by Equation (28). All untwisted spectral statements in the next proposition are on this subspace.

The proposition supplies two different estimates for the electric potential. Dilation decay first forces the spectral functions to be finite products, and the current cocycle then supplies their measure. Its width tail makes modes containing both very small and very large zeros rare; this will give the extra power in the off-axis Laplacian estimate. The degree moment has a separate role: it makes the logarithmic height variogram integrable, giving the growth bound needed for potential inversion in Section 5.

Electric insertions and the estimate away from their line

Fix \(p\) distinct tile vertices \(u_1,\ldots,u_p\) on one tile axis, in a disk of radius \(Mn\), where \(M\) and \(p\) are fixed and \(n\ge2\). For real charges \(t=(t_1,\ldots,t_p)\) with \(\sum_jt_j=0\), define \[ U_t=\exp\left(i\sum_jt_jh(u_j)\right),\qquad C(t;u)=\mathbb EU_t. \tag{44}\] Neutrality makes the expression independent of the additive height constant. Its loop expansion gives relative weight \[ v_I(t)=\frac{\cos(\lambda+\sum_{j\in I}t_j)}{\cos\lambda} \tag{45}\] to a loop enclosing exactly the point set \(I\). Indeed a diagonal height string contributes \(e^{it}\) when a counterclockwise loop encloses charge \(t\), while its two winding weights before the insertion are \(e^{i\lambda}\) and \(e^{-i\lambda}\). Summing their orientations gives Equation (45). The same identity follows for endpoint strings from Equation (28); ice conservation permits rerouting the strings. These finite-word identities use the plane-kernel identification in Input 5. Spin reversal also gives \(C(t;u)=C(-t;u)\).

Choose \(b>0\) sufficiently small in terms of \(p\) and \(\lambda\), and assume \(\max_j|t_j|\le b\). All factors in either winding expansion are then positive. The number of charged loops has mean \(O(1+\log n)\) by the buffered passage estimates: assign loops to geometric scales about the marked points until scale \(Mn\), and use a bounded number of buffers at each scale. Since \(\log v_I(t)\ge-Cb\), Jensen’s inequality gives \[ C(t;u)\ge c n^{-Cb}. \tag{46}\] The constants do not require a lower bound on the separation of distinct marked vertices.

Fix a tile vertex \(z_0\) and define the complex-valued potential \[ H(z)=\frac{\mathbb E[(h(z)-h(z_0))U_t]}{C(t;u)}. \tag{47}\] Changing \(z_0\) adds a constant. Its unit increments are bounded uniformly in \(n\): condition on the unoriented loops in the positive tilt of Equation (45). A marked spin lies on one loop, and its orientation average is a ratio of two orientation sums with denominator bounded away from zero for these charges. The numerator is uniformly bounded. Proposition 10 also gives, for each fixed insertion experiment, \[ |H(z)-H(z')|\le C_{t,u}\sqrt{1+\log(2+|z-z'|)}. \tag{48}\] This is Cauchy–Schwarz in the positive physical six-vertex law, using \(|U_t|=1\) and the logarithmic variance.

For a function on tile vertices, write \[\Delta_2H(z)=H(z+2e_1)+H(z-2e_1) +H(z+2e_2)+H(z-2e_2)-4H(z).\] The factor two removes the signs of the slowly damped row modes.

Twisted rows and finite differences near the axis

The preceding proof uses a vacuum half-plane and thus loses its damping at the insertion line. A different slicing supplies bounds there: an interval between insertions contains a constant diagonal twist, and a step-two height increment is small on its low bands.

For a fixed unit twist define \[J_\zeta(w)=A^{ZK(\zeta)}(w),\qquad H_\zeta=-\log|P_\zeta(1)|,\qquad Q_\zeta(u)=\mathbf 1_{[0,u]}(H_\zeta).\] Split \(Q_\zeta(u)\) further into the two signs of \(P_\zeta(1)\). For each fixed argument \(w\), the joint spectral projection of \(\{f:f(w)=0\}\) is zero, by the Scalar Continuation in a Fixed Band lemma, label bulk:edge-band, in (OpenAI 2026a). In particular this holds at \(w=1\), so \(H_\zeta\) is a densely defined nonnegative self-adjoint operator. The same zero budget as in Equation (51), now including the possible exponential factors in Equation (32), shows that on \(Q_\zeta(u)\) both \(P_\zeta(w)\) and its inverse are uniformly bounded for \[ A u<|w|<a/u,\qquad \Re w\ge0, \tag{54}\] and on the unit semicircle \[ P_\zeta(w)Q_\zeta^\sigma(u) =\sigma Q_\zeta^\sigma(u)+O(u),\qquad \sigma\in\{+1,-1\}. \tag{55}\] The constants \(A,a\) and the errors are uniform over unit twists.

The power in Equation (59) makes its right-hand side summable on each of the four coordinate-parity cosets. Together with the sublogarithmic growth in Equation (48), this is the input needed to recover the potential from its discrete Laplacian.

The potential kernel and local charge profiles

Set \(A=1/[2\pi(\pi-\lambda)]\). The covariance calculation below identifies this constant in the potential-kernel argument.

We first derive (60) for real charges with an unspecified coefficient in place of \(A\); the continuation and the identification of that coefficient will follow below. Choose \(b\) small enough that all subset weights in both winding conventions are strictly positive. The inward coupling of Proposition 7, in its multipoint form in Proposition 8, shows that the unit increments of \(H_{\mathbf t}\) near \(u_j\) approach a profile depending only on \(t_j\). Indeed, inside a common surrounding stopping circuit a marked spin belongs to a loop inside that circuit. Its orientation average depends only on whether that loop encloses \(u_j\), and hence only on the single charge \(t_j\). No remote charge enters this average. Comparing configurations at separation scales \(a\) and \(2a\) gives a power error; summing these errors over successive doublings constructs a unique limiting increment profile. Neutrality can always be maintained by splitting the remote compensating charge into finitely many small charges. The same coupling compares two different remote configurations, including different numbers of compensators: pad them with zero charges to obtain a common finite number of marked points. Thus the limit is independent of that choice.

Let \(\phi_t(y)\) be the step-two Laplacian of this limiting potential, with its charge at the origin. Translation invariance, including an odd tile translation with the corresponding color exchange, identifies these profiles for every possible location of the charge. They also agree when the remote points are placed on the other tile axis: the conditional loop law and the orientation average inside the matched stopping circuit are identical. This observation will allow us to compare all four parity classes of the step-two Laplacian.

The profiles have a summable power tail. To see this at a point \(y\), compare the limiting profile to an experiment whose separation is \((1+|y|)^B\). First choose \(B\) so large that the inward coupling error is smaller than \((1+|y|)^{-2-c_2}\) for some \(c_2>0\). Then decrease \(b\) so that the factor \(n^{Cb}\) in (59), at this value of \(n\), does not exhaust the power gain of that estimate. It follows that \[ |\phi_t(y)|\le C(1+|y|)^{-2-c_2}. \tag{62}\] All constants here can be chosen uniformly for the small charges under consideration, since their positive weights lie in a fixed neighborhood of one.

Put \(g_{\mathbf t}(z)=\Delta_2H_{\mathbf t}(z)\). The local convergence just established gives the following useful quantitative form. After decreasing \(b\) and \(\xi\), there is \(c_3>0\) such that \[ \sum_{z\in\mathbb Z^2} \left|g_{\mathbf t}(z)- \sum_j\phi_{t_j}(z-u_j)\right| \log(n+|z|+2)=O(n^{-c_3}). \tag{63}\] Here and below the tile coordinate lattice is identified with \(\mathbb Z^2\). For completeness, take disjoint balls of radius \(n^\varepsilon\) around the \(u_j\). Within these balls the coupling has a fixed power gain, with a possible loss \(n^{C\xi}\) from the ratio between the configuration scale and its least separation. Choose \(\xi\) and then \(\varepsilon\) small enough that summing over \(O(n^{2\varepsilon})\) sites, and multiplying by a logarithm, still leaves a power gain. Outside the balls, (59) and (62) give a bound of order \(n^{Cb-\varepsilon c_1}\), with a possibly smaller exponent than \(c_1\) and an extra logarithm. Decrease \(b\) once more to make this a negative power. The same estimates on the unbounded part of the lattice include the displayed logarithmic weight. This proves (63).

We now apply ordinary lattice potential theory, keeping its parity issue explicit. For each \(a\in\{0,1\}^2\) let \(\Lambda_a=a+2\mathbb Z^2\). Let \(K\) be the potential kernel on \(2\mathbb Z^2\), normalized by \[\Delta_2K=\delta_0,\qquad K(y)=k\log|y|+k'+O(|y|^{-1})\quad (|y|\longrightarrow\infty), \qquad k\ne0.\] The kernel normalization, asymptotic and increment estimates follow from (Lawler and Limic 2010, Proposition 4.4.2, Theorem 4.4.4 and Corollary 4.4.5), after rescaling each parity coset to the integer lattice and adjusting the generator normalization. The same kernel is used on each translated lattice. By (59), the convolution \[U_a(z)=\sum_{w\in\Lambda_a}K(z-w)g_{\mathbf t}(w), \qquad z\in\Lambda_a,\] is absolutely convergent, has growth \(O_n(1+\log(2+|z|))\), and has Laplacian \(g_{\mathbf t}\). On the other hand, the exact variogram bound in Proposition 10 and Cauchy–Schwarz give \[|H_{\mathbf t}(z)-H_{\mathbf t}(z_0)| \le |C(\mathbf t;\mathbf u)|^{-1} \bigl(\mathbb E|h(z)-h(z_0)|^2\bigr)^{1/2} =O_n\!\left(1+\sqrt{\log(2+|z-z_0|)}\right).\] Thus \(H_{\mathbf t}-U_a\) is a sublinearly growing harmonic function on \(\Lambda_a\), and the lattice Liouville theorem makes it constant; this is the sublinear-growth consequence of the gradient estimate (Lawler and Limic 2010, Theorem 6.3.8(a)), with the same rescaling. The tail bound also gives \[U_a(z)= \left(\sum_{w\in\Lambda_a}g_{\mathbf t}(w)\right) (k\log|z|+k')+o(1).\] The sublogarithmic growth of \(H_{\mathbf t}\) therefore forces \[ \sum_{w\in\Lambda_a}g_{\mathbf t}(w)=0 \quad\text{for every }a. \tag{64}\] In particular \(U_a(z)\to0\) at infinity. The unit increments of \(H_{\mathbf t}\) tend to zero along the off-axis normals, as proved in Section 4. Evaluating neighboring parity classes along those normals shows that their four additive constants agree. Consequently the convolution formula determines all point differences of \(H_{\mathbf t}\), not only differences within one parity class.

Let \[b_a(t)=\sum_{y\in\Lambda_a}\phi_t(y).\] Equation (63), followed by (64) and \(n\to\infty\), implies \[\sum_j b_{a-u_j}(t_j)=0\] in every separated neutral experiment; the subscripts are taken modulo two. In a two-charge experiment, changing one point by one tile along the axis shows that \(b_a(t)\) is independent of the corresponding coordinate parity. Repeating the experiment on the other axis proves independence of both parities. Denote the common sum by \(b_*(t)\). Two charges give \(b_*(-t)=-b_*(t)\), and three give \(b_*(s+t)=b_*(s)+b_*(t)\) whenever these charges are small. The uniform bound from (62) excludes discontinuous additive functions. Hence \(b_*(t)=b_1t\) for a constant \(b_1\).

Evaluate the convolution formula at \(u_i\). The contribution of its own profile is \[g_0(t_i)=\sum_{y\in2\mathbb Z^2}K(-y)\phi_{t_i}(y),\] which depends only on \(t_i\). Every other profile contributes \[b_*(t_j)\bigl(k\log|u_i-u_j|+k'\bigr) +O(n^{-c}),\] after possibly decreasing \(c\). To justify the power error, split the profile sum at a small power of \(|u_i-u_j|\); use the kernel asymptotic and its increment bound on the inner part, and (62) on the remaining part. The error from replacing \(g_{\mathbf t}\) by its profiles is already controlled by (63). Neither this argument nor the self contribution depends on the point parity.

For \(0\le l\le1\) differentiation of the characteristic function gives the exact identity \[\frac{\,\mathrm d}{\,\mathrm dl}\log C(l\mathbf t;\mathbf u) =i\sum_i t_iH_{l\mathbf t}(u_i).\] The common additive constant disappears by neutrality. The cross terms integrate to \(2A\sum_{i<j}t_it_j\log|u_i-u_j|\) with \(2A=ikb_1\). The self terms integrate separately at each point. The terms involving \(k'\) are also single-point terms, since \(\sum_{i\ne j}t_it_j=-\sum_i t_i^2\). Absorb them into \(f(t_i)\), taking \(f(0)=0\). Uniformity of the preceding estimates for \(l\in[0,1]\) proves (60) for small real charges.

Analytic continuation and the electric constant

The complex estimates of Section 3 turn this local calculation into a formula at finite charges. First they determine the scaling exponent of each positive nesting weight, without yet determining its amplitude. On thin complex neighborhoods of compact subintervals of \(I_\lambda\), the functions \[\frac{\log F_n(v(t))}{\log n}\] form a locally bounded holomorphic family; logarithms are continued from real positive weights. Extract any convergent subsequence. For small neutral systems, the all-positive comparison in Proposition 8 bounds the normalized correlation above and below. Equation (60), with separations of order \(n\), therefore shows that its limiting exponent \(g\) satisfies \[\sum_j g(t_j)=-A\sum_jt_j^2.\] The two- and three-charge identities imply \(g(t)=-At^2+Bt\) near zero. Holomorphic uniqueness extends this identity along \(I_\lambda\). Since \[v(t)=v(-2\lambda-t),\] the same symmetry holds for \(g\), and it forces \(B=-2\lambda A\). Every subsequential exponent is thus the same. We have proved, locally uniformly also in those complex neighborhoods, \[ F_n(v(t))=\exp\{\beta(t)\log n+o(\log n)\}, \qquad \beta(t)=-A(t^2+2\lambda t). \tag{65}\]

Divide \(C(\mathbf t;\mathbf u)\) by the cross-distance product in (61). Neutrality gives \(\sum_j\beta(t_j)=-A\sum_jt_j^2\), so Proposition 8 and (65) bound this quotient by \(n^{o(1)}\) on compact complex neighborhoods with macroscopic separations. If the least separation is only \(n^{1-\xi}\), the bound is \(n^{o(1)+C\xi}\). These are bounds for the quotient itself; no nonvanishing claim about the correlation is needed here.

Here is the quantitative continuation argument. Fix a reference configuration with macroscopic separations and fixed parities. Compare its quotients at sizes \(n\) and \(N\), uniformly for \(n\le N\le2n\). Equation (60) makes their difference \(O(n^{-c})\) on a small real neighborhood of zero. Apply the two-constants theorem in disks slit along a compact real segment, successively in the \(p-1\) independent charge variables. A negative power on the segment and an \(n^{o(1)}\) bound on the other boundary give a negative power on a smaller complex neighborhood. A finite chain of overlapping disks carries this estimate along any prescribed compact path in \(\mathcal D_p^+\). The resulting exponent may be smaller, but remains positive. Summing the comparison over dyadic scales, and using its uniformity for every \(N\in[n,2n]\), gives a holomorphic limit \(M_p\) through all integers with a power error.

Next subtract the reference quotient from the quotient of any allowed configuration. Their difference has the same small-charge power bound by (60), while its bound along the continuation path is \(n^{o(1)+C\xi}\). Choose \(\xi\) sufficiently small after fixing that path. The same finite sequence of two-constants estimates still leaves a negative power. This proves the geometry-uniform statement in (61). The opposite winding convention proves the assertion on \(\mathcal D_p^-\). The initial amplitude is therefore real analytic. To obtain the same conclusion for \(f\), compare the logarithmic amplitudes of the neutral triples \((s,t,-s-t)\) and pairs \((s+t,-s-t)\). Their difference is the analytic function \(f(s)+f(t)-f(s+t)\). Averaging this identity in \(s\) over a fixed small interval first shows that the continuous function \(f\) is continuously differentiable; differentiation at \(s=0\) then expresses \(f'(t)\) as a constant minus an analytic function. Hence \(f\) is real analytic.

It remains to identify \(A\). Take four distinct collinear continuum points \(a,b,c,d\) and the neutral charges \((-s,s,-t,t)\) at their lattice approximations. The mixed derivative at zero of \(\log C\) is minus the covariance of the two height increments. In (60) it kills all single-point terms and yields \[-2A\log\frac{|a-c|\,|b-d|}{|a-d|\,|b-c|}\] for that covariance in the limit. Differentiation of the limit is legitimate by the locally uniform analytic convergence just proved. Input 6, with \(G(z,w)=-(2\pi)^{-1}\log|z-w|\) and \(c=2\cos(\lambda/2)\) in the six-vertex parametrization, gives \[ 2A=\frac{\sigma(2\cos(\lambda/2))^2}{2\pi} =\frac{1}{\pi(\pi-\lambda)}, \qquad A=\frac{1}{2\pi(\pi-\lambda)}. \tag{66}\] This is the separated-endpoint covariance assertion of (OpenAI 2026b, Theorem 1.1, Part (iii)); at the present parameters it lies in the established interval \(\sqrt3\le c<2\) of (Duminil-Copin et al. 2026, Theorem 2.8). Only the covariance statement at that scope is used here. This completes the proof of Proposition 14.

Pure nesting amplitudes and nonzero special charges

We next specify the charges used to detect microscopic arms. A single group consists of one charge \(\alpha=\pi/2-\lambda\) and at least two small negative charges whose sum is \(-\alpha\). Choose sufficiently many compensators that each lies, with its negative, in the interval of Proposition 15. The number of charges is now fixed. They need not all lie in the small-system neighborhood used in the proof of (59) for that number of points.

Arm estimates needed for microscopic comparison

Let \(\pi_1(R)\) be the plane probability that a point is connected in its own color to distance \(R\), with a fixed bounded microscopic inner scale. Let \(\pi_2(r,R)\) be the plane probability of a primal arm and a dual arm across the concentric square annulus with radii \(r,R\). All connections here are actual lattice paths. Bounded changes of these radii are harmless by the arm estimates in Input 3.

Three starting configurations

Fix a center and a target cut of radius \(R\). We consider the following starts, contained well inside that cut.

1. A tile side \(e\). Its two end sites have opposite colors, and the associated full medial segment crosses that side. The event \(S_R(e)\) requires an actual path of each color from its end site to the target cut.

2. Two sites \(u,v\) of opposite colors, with \(|u-v|\asymp r\) and both within distance \(O(r)\) of the center. The event \(S_R(u,v)\) requires their respective color paths to reach the target cut.

3. A round whole-tile disk of radius \(r\). The event \(S_R(r)\) requires both colors to cross the intervening annulus, starting at gaps of the disk boundary.

We sometimes write \(S_R\) when the start is specified separately. Every path in this definition is restricted to the region between the start and its first arrival at the target cut.

These definitions have useful descriptions in the strand picture. For a point start, its color region reaches the target cut if and only if no completed loop inside that cut surrounds the point. Indeed, a region that does not reach the cut has a boundary loop separating it from the cut; conversely, an actual color path cannot cross a loop. Thus in cases (i) and (ii), \(S_R\) says that no completed loop surrounds either starting site. In case (iii), \(S_R(r)\) is equivalent to the existence of a strand subarc joining the two rims. To see the latter assertion, cut all strands at both rims. An arc with both ends on the same rim cuts off a region incident only to that rim. If there are no through-arcs, at most one color can have a region incident to both rims. If there are through-arcs, the regions between consecutive through-arcs give paths of both colors between the rims. The rounded drawing makes this a statement about disjoint arcs, even when the straight medial drawing has a double visit.

Write \(\pi_1(r)\) for the plane one-arm probability to radius \(r\), and \(\pi_2(r,R)\) for the plane alternating two-arm probability across a square annulus. Their estimates from Lemmas 17 and 18 are available below. A fixed change of a radius, or of the shape of a rim between the round and square choices, changes the following comparisons by a constant only.

The proof constructs an annular band which has positive probability and transmits exactly one region of each color. We first explain its geometry and the gluing that preserves an arbitrary event on its central edges. We then show that a conditioned exploration has enough opportunities to use this band.

A band with exactly two through-arcs

Landing extensions and gluing

We use a strengthened form of arm separation which retains the crossing clusters. For a fixed-ratio bulk annulus and every \(\varepsilon>0\), the probability in the plane FK law that some crossing arms admit no separated landing extensions in their respective original color clusters is at most \(\varepsilon\), when the separation parameter is chosen small enough. This is precisely the event \(B_n\) preceding Lemma 5.9 of (Chelkak et al. 2016). The replacement arms there are required to belong to the same respective primal or dual clusters. The proof uses the quadrilateral crossing bounds, including at the exposed jagged sides, the small- and large-extremal-length bounds, and an upper bound on the number of crossing components. These are all available for \(1\le q<4\) by (Duminil-Copin et al. 2021, Theorem 1.2 and Section 6); indeed its proof of Proposition 6.2 invokes this argument. Thus we use the proof at its stated probabilistic inputs, rather than an Ising-specific identity.

We need two geometric features of these extensions. Their endpoints stay away from the corners of the square cut, and the extension boxes can be much smaller than the separation between different landings. In the proof of (Chelkak et al. 2016, Lemma 5.9), separation occurs at the scale denoted there by \(\delta^3 n\), whereas extensions are constructed at smaller scales \(\delta^k n\), \(k\ge4\). Only finitely many such scales are needed for a specified error tolerance. The arms consequently protrude a positive relative distance beyond their landing cut, inside disjoint boxes having a fixed-factor margin between them. All constants may depend on the error tolerance. Bounded lattice rounding is harmless once the trial scale exceeds a corresponding fixed minimum.

Suppose a past has been exposed up to a square cut of radius \(l\), and the conditional, unconditioned-FK probability that the specified start reaches a larger cut of radius \(8l\) is at least \(h>0\). Here and below the factors \(8\) and \(16\) may be increased by fixed amounts to accommodate collars. Apply the preceding landing statement on an annulus strictly between those two cuts. Its failure event can be assigned ordinary plane probability less than \(h/(2C)\), where \(C\) is the collar-comparison constant. Its conditional probability given the past is then less than \(h/2\). Thus, with conditional probability at least \(h/2\), the two start-connected clusters have separated outer extensions before radius \(16l\). The crossing portions are taken after their last visits to the annulus’s inner rim. Replacing those portions within their same clusters preserves connection to the start. A finite cover of the landing rim by small boxes allows the landing locations to be restricted to one pair of boxes, at a cost depending on \(h\) only.

There is a corresponding construction facing inward from the remaining annulus toward \(R\). Arms beginning at a scale comparable to \(K l\), where \(K\) is a sufficiently large fixed number, can have separated inward protrusions and can be required to reach a little beyond the target cut. Their probability is bounded below by \(c\pi_2(l,R)\). To check the relative cost of the landing failure, crop the remaining arms through a fixed-ratio buffer before the annulus where the landing test is performed. Collar comparison and quasimultiplicativity bound the joint probability of that failure and the remaining arms by \(C\varepsilon\pi_2(l,R)\). Choose \(\varepsilon\) small, or use the equivalent well-separated-arm comparison. This also shows that the construction is uniform under conditioning on data separated from it by the specified collars.

The two sets of protrusions can be joined through the free space between them. Group their locations on a sufficiently fine but fixed grid along each cut. For every resulting pair of location choices, there are disjoint corridors joining the primal tabs and the dual tabs, with widths bounded below by a positive multiple of \(l\). The constants can be small functions of \(h\); they do not change the total number of geometric trial scales. In a protrusion box, require a transverse crossing at a level between the landing cut and the protrusion’s outermost required level. It must meet the protruding arm. Overlapping longitudinal and transverse crossings then carry that contact through its corridor. Only one color is involved in each corridor and its overlap boxes. The conditional crossing estimates and FKG in that color give a positive fixed gluing probability. This is the tube construction of (Chelkak et al. 2016, Proof of Proposition 5.4).

The distinction between the central band and its extensions is essential here. An arbitrary event \(E\) may fix every edge of the central band, so no gluing edge is allowed to lie there. The protrusions extend outside that band; their overlap boxes and the connecting corridors lie strictly outside it. Within an overlap box the only requirement inherited from the extension is a connection in that box’s color, which is increasing in its color edges. Thus FKG applies after \(E\) is fixed. The auxiliary extensions are integrated out when we take a measure on the central-band configurations. They are not additional band data that must agree in the eventual coupling.

Frequent opportunities under the conditioned law

We now prove Proposition 20. Denote the common ambient law, including its permitted exterior reweighting, by \(\mathbb P_0\), and write \[Q_a=\mathbb P_0(\,\cdot\mid S_R(a))\] for a particular start \(a\). Until the target scale, all conditional edge laws before the \(S_R(a)\) conditioning are ordinary FK laws with induced partitions, or mixtures of such laws. Consequently the collar and crossing comparisons just established apply to \(\mathbb P_0\) as well.

Choose a fixed large ratio \(K\) leaving room for the past-extension annulus, the reference band with its two collars, and the outer-extension annulus. Trial radii are \(l_i=K^i l_0\), where \(l_0\) is a fixed large multiple of \(s\), increased to a fixed minimum mesh scale when necessary. Stop the trials before a fixed fraction of \(R\). The number \(k\) of available trials satisfies \[k\ge \frac{\log(R/s)}{\log K}-C.\] When the right side is bounded, enlarging the constant in (71) suffices. We may therefore assume there are many trials.

Let \(\mathcal F_i\) contain all edges inside the radius-\(l_i\) cut, together with the identities of the start-connected color regions there. Call trial \(i\) good for start \(a\) if \[ \mathbb P_0(S_{8l_i}(a)\mid\mathcal F_i)\ge h. \tag{72}\] The threshold \(h>0\) is chosen after \(K\). The preceding separation and gluing constructions then use narrower corridors if necessary; they require no increase of \(K\) depending on \(h\).

Under \(Q_a\), with probability at least \(1-Ce^{-c k}\), at least \(3k/4\) of the trials are good. Here is the conditioning calculation. The event \(S_R(a)\) implies \(S_{8l_i}(a)\) at every trial. For a specified collection \(J\) of bad trials, expose success to \(8l_i\) before passing to the next selected trial. Since badness is \(\mathcal F_i\)-measurable, the tower property and the negation of (72) give a factor at most \(h\) for each \(i\in J\). Moreover, by (70) and quasimultiplicativity, \[\frac{\mathbb P_0(S_R(a))}{\mathbb P_0(S_{l_0}(a))} \ge C^{-1}e^{-C_0 k}.\] The microscopic or one-arm factors in (70) cancel in this ratio. Summing over collections of at least \(k/4\) bad trials gives \[ Q_a\{\text{at least }k/4\text{ bad trials}\} \le C e^{C_0 k}2^k h^{k/4}. \tag{73}\] Choose \(h\) so small that the right side decreases exponentially in \(k\). This choice is uniform over all the permitted starts and exterior laws.

A common band measure and loss of the past

At a good trial, place a scaled copy of the reference band between the past-extension and outer-extension annuli. Its central configuration has a subprobability measure \(\nu_l\) defined as follows: take the plane FK law, restrict to the successful reference-band construction of Lemma 21, including the auxiliary protrusions, and then retain only the central-band edges. There is a fixed \(v_0>0\) such that \(\nu_l(\text{all band data})\ge v_0\).

For every event \(E\) of central-band data and every good past, \[ \mathbb P_0\{\text{band data}\in E,\ S_R(a) \mid\mathcal F_i\} \ge c_h\,\nu_l(E)\,\pi_2(l,R). \tag{74}\] Indeed, the good past supplies the separated start-connected extensions with probability bounded below in terms of \(h\). Require the reference-band event with its protrusions, and separated arms continuing toward \(R\). Before gluing, these observations are separated by full collars. Bounded-density comparison can therefore be applied successively from the two sides of the central band. It preserves the factor \(\nu_l(E)\) for an arbitrary, possibly nonmonotone, set \(E\), and gives the factor \(\pi_2(l,R)\) for the remaining arms. There is still unobserved space for the two-color tube construction. Its same-color overlap requirements leave \(E\) fixed, as explained above, and complete both paths from the start to the target. This proves (74).

To make the averaging explicit, condition beyond a collar containing all these constructions, and let \(\xi\) denote the resulting exterior state. Given the already fixed past, write \(p_\xi\) for the conditional probability of reaching \(8l\) and \(\mu\) for the conditional distribution of \(\xi\). Goodness means \(\int p_\xi\,\mu(\,\mathrm d\xi)\ge h\); it does not assert \(p_\xi\ge h\) for each \(\xi\). Uniform collar comparison makes the landing-error probability at most \(\varepsilon\) for every \(\xi\), with \(\varepsilon<h/2\). In the preceding construction, keep the probability of the start-connected extensions as a factor instead of replacing it by a constant. The finitely many landing-location choices all have the same lower gluing bound. The conditional lower bound is therefore \[c\nu_l(E)\pi_2(l,R)(p_\xi-\varepsilon)_+.\] Its integral is at least \(c\nu_l(E)\pi_2(l,R)(h-\varepsilon)\), as required in (74). This proves the assertion for the exterior mixture without imposing pointwise goodness. No identification jumps an annulus used in these comparisons.

In the reverse direction, any successful continuation from the past must have two arms in the remaining annulus. Cropping through a collar and using (70) gives \[ \mathbb P_0(S_R(a)\mid\mathcal F_i)\le C\pi_2(l,R). \tag{75}\] Dividing (74) by (75) shows that, under \(Q_a\), the band marginal after every good past dominates \(c\nu_l\). Hence two good pasts, even for different starts, have a uniformly positive common part on identical central-band data.

It remains to check that coupling this band is enough. Cut its strands at both rims. There are two through-arcs. All remaining arcs have both ends on the same rim. Whatever the configuration inside the inner rim, completing its pairings joins the two through-arcs to one another: they are the only two strand ends left after the same-rim arcs and the interior pairings are followed. Interior loops can change in number, but their loop weight is then a factor depending solely on that interior completion. The effective pairing seen on the outer rim is fixed by the band.

The same fact can be expressed in FK terms. Exactly one primal region and one dual region transmit through the band. An exterior connection that enters the past must do so through the transmitted region of its color, whose trace on the outer rim is already determined by the band. Thus the induced exterior primal partition is independent of the earlier past. On \(S_R(a)\), the starting regions have made contact with these two transmitted regions. Conditional on that contact, the remaining constraint is only that the transmitted regions continue to the target cut. Paths that cross the band repeatedly use the same transmitted region and impose no additional constraint. The domain Markov property now shows that the conditional law on and outside the outer rim, given the common band and successful continuation, is identical for the two pasts. A common exterior weight or an ambient wall farther out changes this common law in the same way in both cases.

We can therefore complete the coupling as follows. At each trial, if both pasts are good, use the common submeasure \(c\nu_l\) to couple the central-band configurations with probability at least \(p>0\). On success, fill the interior gaps according to their respective conditional laws and sample the exterior from the common conditional law just described. On failure, use the residual band marginals and fill the data up to the next trial cut with their correct conditional laws. No edges beyond that next cut are inspected merely to decide whether a coupling attempt failed. This is a coupling of the successive conditional kernels, so each complete marginal remains \(Q_a\) throughout.

For completeness, this iteration does not require the trials to be independent. By (73) and a union bound over the two marginals, except with probability \(Ce^{-c k}\) there are at least \(k/2\) jointly good trials. Their goodness is known from the two pasts before the corresponding attempt. At each such trial, conditional on the entire coupled past and on all earlier failures, the overlap probability is at least \(p\). Iterating at these predictable trial times bounds the probability of \(k/2\) failed good attempts by \((1-p)^{k/2}\). The total failure probability is consequently at most \(Ce^{-c'k}\le C(s/R)^c\). On success the band and its exterior agree, and the two transmitted color regions have been reached by both starts. This proves Proposition 20.

For a tile-side start, the common through-strand passes through the associated medial segment: otherwise that segment would lie on a completed inner loop surrounding one of its two end sites, contrary to \(S_R(e)\). For a disk start, the common through-strand meets the disk, since every disk-to-target strand must pass through the band. Thus an exterior rule selecting the distinguished Dobrushin strand gives the same verdict for a microscopic segment and for a disk after successful coupling. This is the form used in the occupation comparison.

A signed factor with positive local conditioning

Fix the two neutral charge groups of Proposition 16. Each group has a core charge \(\alpha=\pi/2-\lambda\) and sufficiently many small negative compensating charges. Place the compensators at distinct points \(u_j(n)\) on the axis by fixed, color-preserving lattice roundings \(u_j(n)=n\zeta_j+O(1)\), where the \(\zeta_j\) are distinct fixed locations away from the origin. The two cores lie within distance \(r=o(n)\) of the origin, in opposite colors. The first core is held fixed. Write \(e\) for the neighboring pair corresponding to a medial segment, and write \(r\) for the separated pair, with the appropriate odd lattice separation. All rounding conventions preserve these colors.

Let \[B_* = \sum_{j\text{ remote}}\beta(t_j),\qquad \beta(t)=-A(t^2+2\lambda t),\qquad A=\frac{1}{2\pi(\pi-\lambda)}.\] Choose disjoint small patches of radius a fixed multiple of \(n\) around the remote points. Reweight only the loops wholly contained in these patches, using their single-point weights \(v(t_j)=\cos(\lambda+t_j)/\cos\lambda\). These weights are positive. Let \(Z_*(n)\) be the mass of this local reweighting and let \(\mathbb P_n^*\) be its normalized probability law. The cutoff comparison and the pure-nest asymptotic give \[ Z_*(n)\asymp\prod_{j\text{ remote}}F_n(v(t_j)) \asymp n^{B_*}. \tag{76}\] Constants are uniform over the finitely many rounding classes used below. We use the positive-winding expansion of the electric correlation. Its remaining loop factor, denoted by \(Y_r\), need not be nonnegative.

Let \(R\) be larger than the core separation by a fixed factor and smaller than a sufficiently small multiple of \(n\). The event \(S_R(r)\) means that both core colors reach the tile rim of radius \(R\). The scalar weight of a loop enclosing just one core and no remote point is zero, because \(\cos(\lambda+\alpha)=0\). A completed loop inside the \(R\)-disk enclosing both opposite-color cores contains another loop separating them. Thus any nonzero configuration has \(S_R(r)\). Conversely, factoring out its indicator merely restricts the original loop product. Consequently the full electric correlation satisfies \[ C_r(n)=Z_*(n)\mathbb P_n^*(S_R(r)) \mathbb E_n^*[Y_r\mid S_R(r)]. \tag{77}\] The same identity holds for the neighboring pair \(e\).

We record two uniform bounds used throughout this section. For every fixed \(j\ge1\), \[\begin{align*} \mathbb E_n^*[|Y_r|^j\mid S_R(r)]&\le C_j, \tag{78}\\ \frac{\mathbb P_n^*(S_R(r))}{\mathbb P(S_R(r))} &=1+O((R/n)^{c_0}). \tag{79}\end{align*}\] Here \(\mathbb P\) denotes the ordinary plane FK law; the same statements hold with \(r\) replaced by \(e\). To justify the first bound, discard configurations on which \(Y_r=0\). Every non-unit factor remaining on the other configurations belongs to a macroscopic loop. A separating loop must enclose a remote point, and an ancestor enclosing both cores contains such a separating loop. A loop charged only at remote points but not included in the local reweighting has left its remote patch. Each such loop therefore has a passage through one of finitely many buffered regions of size comparable to \(n\). These buffers may be chosen away from the support of \(S_R(r)\). Even a loop with an arbitrarily large excursion has such a passage. Conditional passage moments and the local-tilt estimate of Input 3 control every fixed moment of the product. Cutting the arm event back by a fixed factor leaves collars before the passage buffers, so the estimates cost its ordinary probability, not its reciprocal. This gives (78) after normalization. The relative collar comparison gives (79), because the nonnegative local reweighting is supported at distance comparable to \(n\).

The neighboring-pair amplitude

Set \(x_1=-\beta(\alpha)\). The parameter identities are \[ 2x_1+2A\alpha^2=x, \qquad x=\frac{\pi-2\lambda}{2(\pi-\lambda)}. \tag{80}\] For a sufficiently small fixed \(\xi>0\), choose an odd separation \(r\asymp n^{1-\xi}\). Proposition 16 gives \[ \begin{split} C_r(n) &=M_* n^{B_*-x}r^{2A\alpha^2}(1+O(n^{-c_2}))\\ &=M_* n^{B_*}r^{-2x_1}(r/n)^x(1+O(n^{-c_2})), \qquad M_*\ne0. \end{split} \tag{81}\] The amplitude depends on the fixed remote geometry, charges, and rounding convention, but not on \(n\) or \(r\) in this regime. Indeed the full Coulomb formula uses the actual pairwise distances; replacing the remote distances from either core by their limiting values changes its product by \(O(r/n)+O(n^{-1})\). These errors can be included in the power remainder. We will not use this equivalence with a fixed \(M_*\) when \(r/n\) is held positive.

Calibration by disk probes

A disk does not carry a prescribed electric charge. We therefore introduce a diagnostic loop factor which agrees with the neighboring pair’s factor after successful outward coupling. Its purpose is only to determine deterministic normalization constants.

Let \(s=\epsilon n<R\). Under the same remote local reweighting, let \(S_R(s)\) be the disk two-arm event. If exactly one global loop touches the disk and leaves the \(R\)-cut, assign that loop the factor it would receive as a separator of the two cores, with its actual remote enclosures. Give its ancestors the factors corresponding to enclosing both cores, and give other loops no core charge. Divide out the local remote weights already included in \(\mathbb P_n^*\). Call the resulting factor \(Y_s\). In every other case set \(Y_s=0\). In particular the distinguished loop receives value zero when it encloses no remote point. Define the deterministic quantity \[ W_s(n)=Z_*(n)\mathbb P_n^*(S_R(s)) \mathbb E_n^*[Y_s\mid S_R(s)]. \tag{87}\] The moment bound (78) also holds for \(Y_s\): every charged loop in a nonzero configuration spans between the disk and a remote patch, or leaves a remote patch.

In the coupling of the disk start and the edge start, the factors \(Y_s\) and \(Y_e\) coincide on success. The unique band through-loop touches the disk, and no second loop can both touch the disk and leave \(R\). Replacing its inside portion preserves its remote enclosures; the same is true for its ancestors. Thus \[ \left|\mathbb E_n^*[Y_s\mid S_R(s)]- \mathbb E_n^*[Y_e\mid S_R(e)]\right| \le C(s/R)^{c_6}. \tag{88}\]

Segment probes and their replacement by disks

Let \(\mathcal E_n\) be the deterministic set of full medial segments available in the discrete square, including the boundary collar of Section 1. A segment here is an edge of the drawn medial graph, before applying \(\Pi\). Write \(z_e\) for its projected midpoint and \(I_e\) for the indicator that it belongs to the distinguished strand. The pairing at every medial vertex splits that vertex into two vertices of degree two. In the resulting finite graph, the component containing a distinguished terminal half-edge is a path, with the other distinguished half-edge as its other end. It cannot repeat an edge: a repeated edge would put that component on a cycle, which cannot contain its degree-one terminal vertices. Consequently the counting multiplicity in (2) is exactly \(I_e\), even when the unsplit medial vertex is visited twice. Distinct collar edges are still counted separately if their projections coincide.

There is a constant \(A_0>0\), determined by the density of medial midpoints in the stated lattice units, such that \[ A_0n^{-2}\sum_{e\in\mathcal E_n}h(z_e) \longrightarrow \int_D h(z)\,\mathrm dA(z) \qquad\bigl(h\in C(\overline D)\bigr). \tag{93}\] Indeed the bulk midpoints form a periodic set of fixed density, and there are only \(O(n)\) additional boundary probes. Define \[ X_n(f)=A_0n^{-2}\sum_{e\in\mathcal E_n}f(z_e)n^x I_e =A_0n^{-d}\sum_{e\in\mathcal E_n}f(z_e)I_e. \tag{94}\]

Use the sequence of radii \(\epsilon\downarrow0\) supplied by Proposition 26, together with its constants \(L_\epsilon\asymp\epsilon^{-x}\). The physical scale of a unit tile is \(a_0/n\). For a probe \(e\) in the bulk, let \(B_e^\epsilon\) be the whole-tile disk of radius \(s=\epsilon n\) used in that proposition, translated to the probe. Its physical radius is \(a_0\epsilon\), up to \(O(n^{-1})\). Let \(J_e^\epsilon\) indicate that the distinguished strand enters this disk; the occurrence of an unrelated loop does not count. For \(f\in C_c(D)\), put \[ X_{n,\epsilon}(f) =A_0n^{-2}\sum_{e\in\mathcal E_n}f(z_e)L_\epsilon J_e^\epsilon. \tag{95}\] For fixed \(\epsilon\) sufficiently small, every disk occurring in this sum is inside \(D\) once \(n\) is sufficiently large.

Minkowski content in the closed square

We record the whole-domain consequence of the SLE content theorems, including the boundary estimate needed to match the convention of Equation (4).

Identification of the joint limit