Proof. Draw noncrossing exterior primal trees realizing the designated identifications and thicken the resulting open primal graph. Its perimeter is the medial loop collection, apart from a fixed number of exterior components. Euler’s formula gives the configuration-dependent part of the perimeter count as \[\ell=2k_\xi(A)+|A|+\text{constant}.\] Since \(d^2=q\) and \(p_q/(1-p_q)=d\), the weights agree up to a constant independent of \(A\). The domain Markov property gives the conditional-law statement. On a torus the thickened graph has Euler characteristic \(|V|-|A|=2k-\ell-2g\), with \(g\le1\), so the additional factor is bounded. ◻

Proof. Buffered bulk crossing gives a conditional probability bounded below for an actual primal circuit around any fixed box in each of successively separated annuli. The probability of missing the first \(m\) such screens is at most \((1-c)^m\), regardless of the exterior partition. The same argument gives arbitrarily large actual dual circuits. These two colors of screens prevent a medial strand meeting the box from escaping to infinity. Taking the countable union over boxes proves finiteness.

A surgery at finitely many tile connections changes only finitely many loops at a fixed mesh. Relative to the unmodified plane law, its weight ratio is \(d^{\Delta\ell}\), bounded above and below by constants that may depend on this finite set of connections. Normalization defines its positive law. Equivalently, exhaust by planar domains; the same circuit argument makes the local completion independent of remote boundaries. Lemma 3 then gives the conditional law. ◻

Proof. First work on a finite graph and a finite remote edge set. Split each through wire transversally, preserving its connections on each side and retaining just one representative connection across the cut. The unknown data induce a partition \(\xi\) at their section. Include all common deterministic wires for every choice of \(\xi\).

For two ordered partitions, the likelihood ratio of the coarser law to the finer law is increasing: the rank of the additional identifications decreases as open components merge. Its marginal likelihood ratio on the remote set remains increasing, by the domain Markov property and conditional FKG. Consequently its maximum is attained at the all-open configuration and its minimum at the all-closed configuration. It is enough to compare those two events for the maximal and minimal partitions; every other partition lies between them.

An actual primal crosscut in every sector, or a primal circuit in the annulus, screens the remote graph with uniformly positive probability. Such screens can be discovered from the unknown side without examining edges strictly beyond them. In a sector, explore all actual dual paths from its unknown-facing arc and query edges incident to the reached dual vertices. If a primal wall-to-wall crossing exists, the search does not reach the remote arc. The elementary complementary crossing alternative then places a blocking primal path among the queried edges: otherwise, setting the unqueried primal edges closed would produce a dual crossing. The alternative follows by tracing the perimeter of the actual component reachable from the starting side; corner gap vertices belong to the arcs when their color permits. The annular version is the alternative between a primal circuit and a radial dual connection. Perform the searches in a subcollar leaving space before both sections. Reversing colors gives the dual searches as well.

Condition on a successful primal search under the minimal law. Each screen is already connected. Joining all screens and the bounded number of through-wire representatives requires only boundedly many additional block identifications. Each has an unnormalized weight ratio in \([q^{-1},1]\), hence changes expectations by a bounded density factor. The resulting partition dominates every partition transmitted from the unknown side. In particular it dominates the maximal law restricted at this same screen geometry, without requiring that law to contain the screen. The positive probability of the search proves the desired comparison of all-open probabilities.

For all-closed probabilities, make the actual dual searches under the maximal law. No actual primal path crosses these screens. Only the boundedly many through-wire representatives can still transmit ties; remove these ties at bounded cost. The resulting free cut is stochastically smaller on the remote graph than any other imposed cut partition, including the restriction of the minimal law. This proves the decreasing-event comparison. The extremal likelihood-ratio argument now gives the assertion for every remote event and every nonnegative remote function. Exhaustion proves the infinite-exterior version. For a rectangular bulk collar use the corresponding enclosing-circuit searches. ◻

Proof. Use a radius ratio such as \(256\) to leave two subcollars of the preceding type. Lemma 5, applied nearer the unknown side, gives a common marginal part of uniformly positive mass on the other subcollar. Couple those configurations identically on a fixed fraction of a reference marginal. In case (i), additionally impose a primal screen meeting and joining every through wire. In case (ii), impose a primal screen on the wired side and a dual screen on the other side. These requirements have uniformly positive conditional probability. On their occurrence the partition transmitted to the next section depends only on the shared configuration: every route through the unknown region must enter and leave through a primal screen already connected to its through wires, and the other sector transmits no primal connection.

For a nonnegative remote function, take the largest and smallest expectations over partitions at each successive section. Conditioning nests these intervals of possible expectations. The common part just constructed contracts their difference by a factor \(1-c\). Their ratio is bounded initially by Lemma 5; iterating therefore gives the stated relative estimate. A positive unknown-side tilt is a mixture over the same partitions. In particular, when propagating away from a gap, wire connections made or lost inside that gap belong to the unknown partition, which explains the need for both colors of screen in case (ii). ◻

Proof. First obtain one positive exponential rate. Disjoint simple radial strand subarcs split a slightly cropped annulus into ordered strips. Along each passage is an actual primal crossing path. Paths in distinct strips cannot join in the cropped graph, because open primal edges cannot cross the intervening medial separators. Choose the endpoint rim bands strictly inside the region traversed by the separators. Thus \(X\) passages force at least \(X/2-O(1)\) different crossing components in the actual cropped graph.

Explore complete actual components successively from one rim, ignoring all external identifications when deciding membership in a component. After a component is removed, its incident unexplored boundary edges are closed. The surviving FK law is dominated by full wiring at the original rims and walls; removing vertices can only decrease actual crossings. Even with this full wiring an actual dual blocker across the cropped annulus has probability at least \(c>0\). Conditional on every successful discovery, the chance of another crossing is therefore at most \(1-c\). The number of components has a geometric tail, proving (9) for one \(B>0\).

To obtain any fixed rate, divide the radial distance into \(J\) separated slices of thickness comparable to \(r/J\). Cover a middle rim in each slice by \(O(J)\) small boxes whose buffered annuli remain within the slice. Use wall-centered boxes near a straight wall and smaller bulk boxes elsewhere, counting the two sides of a wall separately. The overlap graph has a bounded coloring independent of \(J\). Every original passage crosses one small annulus in each slice, so \[JX\le\sum_i X_i,\] with \(O(J^2)\) counts having the first exponential bound. Conditional iteration within each color class, followed by Hölder across the bounded number of colors, gives a rate \(c'J\). Choose \(J\) arbitrarily large but fixed. Before these boxes fit in the mesh, a deterministic finite-size bound suffices. A finite covering by bulk, straight-wall, and corner buffers gives the macroscopic version: a strand segment of the designated diameter must leave one box of an inner cover whose outer buffer is much smaller than that diameter. ◻

Proof. The case \(b=1\) is Lemma 7. For \(0<b<1\), set \(D_i=\log(F_{i-1}/F_i)\ge0\). Jensen’s inequality and the exponential bound give, for any sufficiently small fixed \(\varepsilon>0\), \[\begin{align*} D_i &\le |\log b|\,\mathbb E_{\nu_{i-1}}X_i\\ &\le\varepsilon\log \mathbb E_{\nu_{i-1}} \exp\bigl(|\log b|X_i/\varepsilon\bigr)\\ &\le C_\varepsilon+ \varepsilon\sum_{i-k<l<i}D_l. \end{align*}\] In the last inequality, drop decreasing factors back to level \(i-k\). The remaining tilt is supported outside the buffer of \(X_i\), so the conditional passage bound applies. Choose \(\varepsilon(k-1)<1\); induction bounds every \(D_i\). Dropping the same bounded number of levels then proves every asserted exponential moment.

For \(b>1\), choose \(B>k\log b\) as large as desired and induct on \(j\) that \(\mathbb E_{\nu_j}e^{BX_i}\le C_*\) for \(i\ge j\). When \(i\ge j+k\), the tilt and the buffer are disjoint. Otherwise let \(j_0=i-k<j\). The normalizing increment from \(j_0\) to \(j\) is at least one, so \[\mathbb E_{\nu_j}e^{BX_i} \le\mathbb E_{\nu_{j_0}} \exp\left(BX_i+(\log b)\sum_{j_0<l\le j}X_l\right).\] Give each summand in the sum Hölder weight \(\log b/B\) and use the inductive bound at the earlier tilt \(j_0\). The remaining Hölder factor is an exponential moment of \(X_i\) at a larger fixed rate, whose buffer is disjoint from the tilt at \(j_0\). The right side is at most \(C C_*^{k\log b/B}\), which closes the induction for sufficiently large \(C_*\). Negative levels mean no tilt. The moment bounds give the ratio bounds using (10). The argument is unchanged after adjoining the indicated passage counts. ◻

Proof. Choose \(\eta>0\) so that \(\eta\log C_0<\varepsilon\) and split at \(X_n=\eta\log n\). On the first event the multiplicative factor lies between \(n^{-\varepsilon}\) and \(n^{\varepsilon}\). Cauchy–Schwarz and the exponential moment bound at an arbitrarily large fixed rate make both \(\mathbb E[Y_n;X_n>\eta\log n]\) and \(\mathbb E[Y_n C_0^{X_n};X_n>\eta\log n]\) smaller than any prescribed inverse power after accounting for the polynomial second moment. Choose that power larger than \(A+1\). Division by the stated lower bound for \(\mathbb E Y_n\) proves both estimates, enlarging constants for bounded \(n\). ◻

Proof. Enclose each segment in its own patch with a vacant bulk tile collar and room for a further disjoint buffer. Let \(X_s\) be a passage count through that collar. Compare the incremental loop change \(\Delta\ell_s\) of the slit surgery with \(\Delta\ell_s^\circ\) computed after closing the patch ports by the same arbitrary pairing before and after the surgery. Their difference is \(O(X_s)\). To see this, cut the segment ports. The background pairing is determined within the patch except for ends whose strand first traverses the collar. There are \(O(X_s)\) such ends, and changing their pairing changes either completed cycle count by at most their number, also when other surgeries have been made outside.

Put \(G_s=d^{\Delta\ell_s^\circ}\mathbf1_{\{X_s\le K\}}\). For fixed sufficiently large \(K\), simultaneous truncation has probability bounded below under the normalized law with all completed slits, by Lemma 7. On the truncated event the surgery weights and their local versions are comparable. Therefore \[Z_{\rm all}\asymp\mathbb E_0\prod_sG_s, \qquad Z_s\asymp\mathbb E_0G_s,\] where \(\mathbb E_0\) is the unmodified plane expectation. Apply Lemma 5 successively to the separated patch factors to obtain (11).

Translate an isolated segment by one tile step in its tangent direction. This is a symmetry of the homogeneous plane loop law, with duality interchanging the colors when needed. Its shifted local caps agree with the reversed caps except at boundedly many ports near tips and changes. Their remaining pairings differ on only boundedly many ends, which has bounded cycle-count cost. Applying such a lattice symmetry to the isolated slit and its cap data gives the last assertion. This is a comparison of partition functions; it does not assert bounded pointwise density for changing all pairings along a wall. ◻

Proof. For nest weights these claims, apart from the signs already noted, are Lemmas 5 and 8. Symmetries of the local loop weights and caps give the geometric and color symmetries. At a flat change reflection exchanges the two assignments relative to the gap color. Bending or erasing distant cap connections changes only the possible exterior partition.

We give the arm argument because it concerns actual connectivity. In a tile half-box, boundary regions separated by raw boundary-to-boundary arcs are precisely the actual color components touching boundary gaps. Closed interior loops cannot split these boundary classes. If the region at the marked gap does not reach the outer rim, some raw arc with both ends on the flat wall straddles the gap. One way to see this is to form the tree of regions separated by the disjoint arcs. The wall and rim walks both traverse the tree path between their common end regions. A wall visit to a region absent from the rim excursion is entered and left across the same separating edge within the wall portion. Conversely, a straddling arc cuts off the gap and its actual color paths. Thus absence of all straddling arcs inside radius \(r\) forces an arm to a comparable radius, and an arm to a larger fixed multiple excludes all such arcs.

Arm probabilities at constant multiples of a radius are comparable. Given the shorter arm, impose a same-color wall-to-wall screen in an intermediate annulus and a crossing from its inner side to the desired outer radius. The arm meets the screen and the added crossing meets it; FKG and the wall crossing estimates give a uniform extension cost. The construction is valid under arbitrary distant boundary data. Starting from a bounded-mesh event and iterating gives a polynomial lower bound. Conversely, actual opposite-color screens in separated annuli obstruct the arm and have uniformly positive conditional probability. Their iteration gives (15). Dual-color requirements use FKG for decreasing primal events. No wire identification has been used as an edge of an arm. ◻

Proof. Take a family of disk arcs closed under the arcs inside their disks, with no holes, and remove them from the inside outward. Removing two adjacent remaining endpoints creates a loop exactly when their exterior partners already pair them; otherwise reconnect those partners. The caps still pair successive high-low ports on each remaining interval, with arbitrary pairings of its unmatched ends. A high-low arc within an interval creates one loop. A low-high arc merely transports the pattern. Across a flip, equal high statuses transport an unmatched end without creating a loop. If two transported unmatched ends were already tied, the extra loop erases an entire remaining interval. There are boundedly many such exceptions; also allow the final bounded error for a no-flip circle. The unremoved boundary loops are at most the number of remaining arcs.

Include all sufficiently small arcs, together with arcs inside their disks. Any others contribute a macroscopic passage count \(X\) in boundary bands. Along a local straddling nest the starting parities alternate, because ports between consecutive starts pair internally on that side. The two relative weights in (22) multiply to \(f(g)\) after division by the corresponding cap weights \(d\) and \(1\). One unpaired factor is bounded. This proves the claimed \(f(g)^{N/2}\) comparison. Cutoff patches are disjoint with wider buffers, and a straddling arc omitted because it exits its patch is charged to \(X\). ◻

Proof of (17). Take a convex tile rectangle with both side lengths comparable to \(n\). Put a change in the middle of the bottom side and another in the middle of the top side. Give \(H\) drops \(s,t\) at these changes and no others. Since \(s+t=1\) and the total boundary turn is \(2\pi\), the factors (20) are periodic around the contour.

In the positive-sign expansion, an arc not separating the changes uses an interval containing neither change and has its ordinary cap weight. An arc enclosing one change has \(w(s)=w(t)=1\). Closed interior loops have weight \(d\). Lemma 13 and Lemma 7 therefore bound the spin partition function above by a constant times the positive cap partition \(Z_+^{\rm cap}\). The latter includes the common tile coefficients.

In the negative-sign expansion the change drops are \(-s,-t\) and each convex corner contributes \(1/2\). An arc containing neither change has an interval containing at most two corners and has positive weight for both high statuses. An arc separating the changes can use the interval containing the \(s\) mark and \(k\) corners; its weight is \(\sin(k\lambda/2)/\sin\lambda\). These weights are nonnegative and vanish exactly for \(k=0\), namely a straddling arc on the bottom side. Let \(E\) be the event that the \(s\)-gap vertex has an actual path of its color to another side of the rectangle. Every weight is strictly positive on \(E\), and the local straddles at \(t\) have weight \(d\). Relative to the cap law with the complementary wires, the spin partition is consequently at least \[ \mathbb E_- \left[\mathbf1_E d^{N_t} \prod_{v\text{ corner}} f(1/2)^{N_v/2} C^{-(1+X)}\right]. \tag{23}\] Here \(X\) counts macroscopic boundary passages. For instance, choose all small arcs below a fixed diameter fraction so their disks contain at most one feature. The cap elimination accounts for those arcs, and remaining factors on \(E\) lie in a fixed finite set of positive values.

Without \(C^{-(1+X)}\), the expectation in (23) is bounded below by a constant times \[\exp\left(P+D_*+\sum_v C_{i(v)}\right).\] Indeed the separated positive local factors compare conditionally by Lemma 5. After conditioning on their small patches, extend the local \(s\) arm through a fixed corridor avoiding those patches to another side, by the screens and crossing extensions in Lemma 11. The remaining law is FK with an induced partition. Lemma 9 loses only \(n^\varepsilon\) upon reinstating the passage factor, because all local first and second moments have polynomial cost by Lemma 8.

Choose odd horizontal tile length and even vertical tile length. The four corners then have the same relative gap-versus-wire type, since the left and right sides belong to opposite wire intervals. Compare the two tensor expansions and repeat on this very same rectangle with every wire polarity reversed. The ratios \(Z_+^{\rm cap}/Z_-^{\rm cap}\) cancel upon multiplication. One run has four type-\(0\) corners and the other four type-\(1\) corners; both have the same \(P,D_*\) up to bounded changes of local convention. Taking logarithms proves (17) with the error convention (16). ◻

Proof of (18). Take a tile cylinder whose spatial period has two positive coordinate components, each comparable to \(n\), with even sum. In a lift, its lower boundary repeats all the east steps followed by all the north steps. The upper boundary is a parallel translated copy at transverse distance of order \(Kn\log n\), where \(K\) will be a sufficiently large fixed constant. Each boundary has one convex and one concave corner. There are no wire flips and \(H\) is constant. Use positive caps for the reference laws and free periodic spin contraction, with no flux projection, for the tensor. The normal-angle lift is periodic.

Every same-boundary raw arc bounds a disk against the boundary interval it cuts off. Its lifted endpoint interval spans less than one period: otherwise its endpoints and those of a translate would interleave, forcing intersecting arcs. Its positive-expansion weights are ordinary cap weights, and its negative-expansion drops are \(0\) or \(\pm1/2\), so those weights are positive. A contractible interior loop has weight \(d\). An essential loop has total turn zero and hence weight \(2\); the zero-turn assertion follows by isotopy of a simple generator in the flat cylinder.

Reserve bottom and top bands of width \(O(n)\) containing all boundary features and their enlarged buffers. In the empty middle strip let \(k_0\) count essential closed medial loops wholly within that strip, and put \[ M=(2/d)^{k_0}. \tag{24}\] It depends only on middle-strip tiles and is at least one. Under either positive cap law, \[ \mathbb E M^p\le n^{O_p(K)}\qquad(p<\infty). \tag{25}\] To prove this, cover the middle strip by \(O(K\log n)\) buffered bulk boxes of size a small fraction of the period. Each essential loop supplies a passage, and conditional iteration of Lemma 7 with a bounded coloring proves the bound. Essential loops omitted from \(M\), macroscopic boundary arcs, and other macroscopic loops approaching features are charged to a count \(X\) in the enlarged boundary bands. This \(X\) has all fixed exponential moments uniformly, independently of \(K\).

Let \(T\) be the event of a raw arc joining the two boundaries. Then \(M=1\) on \(T\), and \[ \mathbb P_\pm(T)\le C n^{-cK}. \tag{26}\] Indeed, in each successive transverse band of width comparable to the period, actual crossings can form a noncontractible cycle of one color. Use a necklace of overlapping strip crossings with transversal joins; its buffers are injective on the cylinder. In the lift the connected crossings continue through every translate, and their projection contains a noncontractible cycle. FKG and crossing estimates give a uniform conditional success probability. Such a cycle blocks \(T\), so iteration through order \(K\log n\) bands proves (26).

The positive expansion has absolute partition value at most \[Z_+^{\rm cap}\,\mathbb E_+[M C^{1+X}].\] Off \(T\), the negative expansion is positive and its integrand relative to \(Z_-^{\rm cap}\) is at least \[M\prod_v f(g_v)^{N_v/2}C^{-(1+X)}, \qquad g_v\in\{1/2,-1/2\}.\] On \(T\), its absolute integrand is bounded by the same local factors times \(C^{1+X}\), without \(M\). These statements are Lemma 13; through arcs are macroscopic, so their bounded weights are included in \(X\). Hölder, the polynomial local tilt bounds, and (26) bound the possible signed through contribution by \(n^{-c'K+O(1)}\) relative to its caps.

Condition on the middle-strip tiles and use the separated feature patches to compare the positive main expectation, without its \(X\) factor, to \[(\mathbb E_-M) \exp\left(\sum_{v:g_v=1/2}C_{i(v)}+ \sum_{v:g_v=-1/2}L_{i(v)}\right).\] It is bounded below by an inverse power of \(n\) whose exponent is independent of \(K\), because \(M\ge1\). Choose \(K\) large enough that the through error is negligible. Now keep \(K\) fixed and apply Lemma 9, using (25), to absorb all \(C^{\pm X}\) factors at an arbitrarily small power loss.

Run the comparison for both choices of the two boundary wire colors, then reverse both colors on exactly the same cylinder, cutoffs and middle strip. Tensor equality cancels the two ratios of \(Z_\pm^{\rm cap}\mathbb E_\pm M\) when the inequalities are multiplied. The two runs together count twice each of \(C_0,C_1,L_0,L_1\). Taking logarithms and dividing by two proves (18). ◻

Proof of (19). Use a cylinder with straight horizontal boundaries, with even horizontal period comparable to \(n\), and lower boundary traversed eastward. Put the two flat changes on the bottom, separated by order \(n\), and no change on top. Place the finitely many bulk points \(z_j\) at mutually separated distances of order \(n\) in the bottom band. Their buffered cutoff patches are disjoint; make the band wide enough to leave clear corridors nearer the bottom. The height is again of order \(Kn\log n\).

Draw tile-edge strings from the \(s\)-gap to each \(z_j\) and multiply the spin crossing its string by \(\exp(-i\lambda a_j c)\), where \(c=+1\) denotes an arrow along the left normal of the oriented string step. Use raw drops \(s-1,t\) for \(H\) on the lower boundary, with total zero, and constant \(H\) on top. Oriented intersections give the following sign convention: in the positive expansion, a string of charge \(-a_j\) shifts a boundary disk arc’s cosine argument, in units of \(\lambda\), by \(+a_j\) for a launch in its interval and by \(-a_j\) for an endpoint in its disk. The boundary child lies to the arc’s right when the arc is followed from the earlier to the later port. Thus the local effective bottom drops are \(s,t\); in the negative expansion they are \(-s,-t\), with \(a=\sum_{z_j\text{ in disk}}a_j\) added for the endpoints inside.

For a boundary disk arc the negative-expansion numerators, over the common denominator \(\sin\lambda\), are \[ \begin{array}{c|l} \text{number of enclosed flips}&\text{numerators}\\ \hline 0&\sin((2-a)\lambda),\quad\sin((1+a)\lambda)\\ 1&\sin(a\lambda)\quad[s],\qquad \sin((2-a)\lambda)\quad[t]\\ 2&\sin((3-a)\lambda),\quad\sin(a\lambda). \end{array} \tag{27}\] The interval spans less than one period, so each change is counted at most once. Top boundary disks use the zero-flip row. Since \(0\le a\le1\) and \(0<\lambda\le\pi/3\), every entry is nonnegative. A vanishing entry always belongs to a child interval containing \(s\); this includes the additional zero \(\sin(3\lambda)=0\) at \(q=1\) in the two-flip row. Contractible closed loops in the negative expansion have relative weight \(\cos((1-a)\lambda)/\cos\lambda>0\); their interior is on the left for counterclockwise orientation. The two expansions therefore give the prescribed local weights \(u_\pm(a_j)\). Essential loops have positive weight \(2\cos(a\lambda)\), where \(a\) is the charge separated from the launches, and loops in the empty middle strip have weight \(2\). Keep \(M\) as in (24).

Let \(E\) be the event that the \(s\)-gap vertex connects by actual paths of its color to a noncontractible open cycle. Construct that cycle near the bottom, before the conditioned endpoint patches and middle strip. Conditionally on those patches and the middle, \[\mathbb P_-(E\mid\text{patches and middle})\gtrsim e^P.\] This follows from the local arm comparison, same-color FKG and crossing extension, and the periodic necklace construction used for (26). On \(E\) no child disk arc can cut off \(s\), since the noncontractible cycle cannot fit behind it, and there is no through arc.

Set \(U_\pm=\prod_j u_\pm(a_j)^{N_j}\) and let \(Y_\pm\) be the spin integrands relative to their respective positive cap laws. Local cap and loop counting gives \[\begin{align*} |Y_+|&\le M U_+ C^{1+X},\\ Y_-&\ge M U_- C^{-(1+X)}\quad\text{on }E,\\ Y_-&\ge0\quad\text{off }T. \end{align*}\] The local \(t\)-straddles in the negative expansion contribute \(d\) each, which may be discarded in this lower bound. A loop whose endpoint enclosure differs from a local nest has a macroscopic passage near a feature: a contractible disk containing \(z_j\) must separate it from the bottom. Thus it is charged to \(X\), with all cutoffs small compared with the feature separations. On the through event \(T\) one still has \[|Y_-|\le U_-d^{N_t}C^{1+X},\qquad M=1.\] All these local costs are polynomial and the through probability is \(O(n^{-cK})\), so choose \(K\) fixed and large to dominate the signed through error.

Conditioning and Lemma 5 compare the separated local factors to \(\prod_jF_\pm(a_j)\), the middle factor to \(\mathbb E_\pm M\), and the event \(E\) to its lower arm cost \(e^P\). Lemma 9 removes the passage errors at \(n^\varepsilon\) cost. Repeat with every wire designation reversed, using the same domain, strings, drops, cutoffs and \(M\). Equality of the two tensor expansions cancels both cap and middle-factor ratios upon multiplying the two comparisons. The resulting inequality is \[2P+2\sum_j\log\frac{F_-(a_j)}{F_+(a_j)} \le o(\log n),\] which gives (19). ◻

Proof. First prescribe the desired relative corner types for one designation. There remain at least four spare convex corners, one of which adjusts the parity of the sum of the absolute corner parities. If \(b_i\) denotes the integer coordinate parity of side \(i\), a corner between sides \(i-1\) and \(i\) has parity \(p_i=b_{i-1}+b_i\) modulo two. This cyclic system is solvable exactly when \(\sum_i p_i=0\) modulo two. Choose the side coordinates with those parities within boundedly many mesh units of the prescribed ones. For the fixed simple polygon, sufficiently small independent shifts preserve simplicity. Global wire reversal just exchanges the relative types.

Pair the balanced convex and concave sets and apply (18). If \(m\) is even, four convex terms remain; if \(m\) is odd, five convex terms and one nonpositive concave term remain. Since \(C_i\ge0\), Equation (17) implies \[C_i\le-\tfrac12(P+D_*)+o(\log n).\] Paying at most five such terms yields (28). With no corner factors, (17) alone gives \(P+D_*\le o(\log n)\), which is the corresponding flat budget. The endpoint-charge assertion is exactly (19). ◻

Proof. We first identify the plane state, including the effect of the zero-spin projection. Take periodic lists of time and space angles in a strict range, with the two direction ranges ordered and transversely separated in a half-circle. Use tori whose period sizes are even multiples of the list periods. The resulting rhombic square array is embedded and has bounded angles. Indeed the cyclic order at each vertex gives local injectivity, while averages of runs of either family have uniformly positive length and uniformly transverse directions. Thus index distances control Euclidean distances in both directions, giving properness and global embedding.

Expand each \(R\) into its two adjacent-side pairings with oriented turn weight \(e^{i\rho W/2}\). The pairing at the corner between the two positive steps, and at its opposite corner, has coefficient \(a(u)\); the other pairing has coefficient \(b(u)\). For example, the mixed entry follows from \(a(u)e^{iu}+b(u)e^{-i(\lambda-u)}=1\) and its conjugate. In a single-pairing entry the two turns cancel. A contractible loop therefore has weight \(d\). An essential simple oriented torus loop has total turn zero. All essential loops are parallel up to orientation. Their number \(k\) is even, by the even intersection parities with both period cycles and primitivity of their common homology class. The allowed orientation count after projection onto zero spin on either chosen quantum section is \[ \chi=2^k\quad\text{if the intersection is zero},\qquad \chi=\binom{k}{k/2}\quad\text{otherwise}. \tag{40}\] Since \(d<2\), there is \(c(d)>0\) such that \(c(d)d^k\le\chi\le2^k\) for every even \(k\).

Replacing \(\chi\) by \(d^k\) gives the canonical isoradial FK law, up to the bounded torus genus factor of Lemma 3. To check the parameter, the relative open weight at a primal corner between the positive steps is \(\sqrt q\,b/a\), and at the other primal corner it is \(\sqrt q\,a/b\). If \(\gamma\) is the angle at the primal corner, the odds divided by \(\sqrt q\) are \[\frac{\sin(\rho\gamma)}{\sin(\rho(\pi-\gamma))}.\] This is the canonical weight, with the complementary-angle convention of (Duminil-Copin et al. 2018, Theorem 1.1). That theorem applies to these fixed doubly periodic bounded-angle isoradial grids. It gives uniqueness and buffered crossing estimates, uniformly over the boundary partition on the outer buffer. The bulk passage-count argument of Lemma 7 consequently applies on each such grid. No estimate as an angle degenerates is used.

Here is why the essential-orientation tilt does not change the local limit, even for unbounded aspect ratios. Let \(r\le R\) denote the shorter and longer period sizes. Cover the torus by \(O(R/r)\) buffered boxes of size a small fraction of \(r\), with a bounded number of disjoint-buffer colors, so that every essential loop crosses one of them. Include \(O(R/r)\) translates, by medium periods, of a target annulus about the query. Let \(Y\) be the sum of the passage counts in this cover. For each fixed \(c>0\), conditional passage moments, iteration within each color, and Hölder’s inequality give \[ \mathbb E e^{cY}\le e^{C_cR/r},\qquad k\le Y. \tag{41}\] The density of the tilted law is bounded above by \(C e^{c_0Y}\) before normalization and its normalizing denominator is bounded below. Consequently \(\mathbb E_{\rm tilt}e^Y\le C e^{CR/r}\), and Jensen’s inequality gives \(\mathbb E_{\rm tilt}Y\le CR/r\). Translation invariance then bounds the expected count at the target annulus uniformly.

Condition on the exterior of its inner box. The number of exterior excursions from that boundary to distances comparable to \(r\) before returning is tight. If it is at most \(K\), changing the filling can alter only \(O(K)\) essential loops: every affected loop must leave the injective chart along such an excursion. The common homology class can change only when no essential loop remains unaffected. Thus the ratios of the genus and orientation tilt over all fillings are bounded in terms of \(K,q\). For the binomial case this also follows directly by comparing central binomial coefficients with indices differing by \(O(K)\); a switch between the two cases in (40) can occur only when there are \(O(K)\) essential loops. The conditional inner law is therefore comparable with FK under the induced partition. Successive buffered primal and dual circuits show that an arm or medial strand from a fixed window reaches toric distances with probability tending to zero under the tilted law as well.

Now condition outside a large fixed test box containing the query. With probability tending to one, no exterior actual cluster or medial path incident to this box reaches toric distances. On that event changing its filling leaves both the essential factor and the genus factor exactly unchanged. Its conditional law is ordinary FK. Letting the test box grow, comparison of increasing cylinder events with free and wired laws and uniqueness identifies the local plane law.

The same conclusion holds with finitely many marked matrices \(D_i\). Given the unoriented pairing, the absolute marked orientation ratio is at most \(\prod_i2\|D_i\|\): only marked loops are broken, each newly cut arc has at most two orientations, unaffected contractible loops retain weight \(d\), and any residual flux constraint has no more orientations than the central count in (40). When all marked strands stay in a fixed window, they do not encounter the distant flux projection. The absence of escaping strands therefore proves marked local convergence too.

Sending the time length to infinity first gives the Perron-normalized coefficient at fixed spatial circumference. Sending the circumference to infinity, and diagonalizing the two limits, gives the stated plane coefficient. Any finite word can be included in a periodic list; unmarked filler rows disappear from its normalized longitudinal vacuum coefficient.

There is also a transverse quantization. Take a circle in the time index, transfer in decreasing space index, and let the auxiliary space spin flow in increasing time. The same difference relation makes the untwisted transverse transfers commute and gives common Perron data. The seam marks of the original word become one tensor-product quantum insertion. Taking cylinder limits in this order shows that its plane coefficient is unchanged when the strict transverse angles are changed. Rotating both families and then returning the transverse angles to \(v\) proves common time-angle translation invariance. Continuity follows from the finite-box FKG sandwich and uniqueness as a common finite periodic medium varies; sufficiently large fixed circuits at the limiting medium control long marked strands, and all remaining local orientation factors are continuous.

Finite-volume positivity now gives the limiting Gram form. To prove boundedness before completing, let \(h\) be a formal word vector and set \(T=(A^D)^*A^D\), with the adjoint understood first as the reflected formal row in (38). The marked expansion bounds \(\|T^jh\|^2\le C_h(2\|D\|)^{4j}\). Iterated Cauchy–Schwarz applied to the positive form \(\langle h,Th\rangle\) gives \[\|A^Dh\|^2 \le \|h\|^{2-2^{1-m}} \|T^{2^{m-1}}h\|^{2^{1-m}}.\] Letting \(m\to\infty\) gives (39); the same inequality shows that null vectors remain null. Continuity of kernels, and words with rational angles and matrix entries, give separability. ◻

Proof. For each word \(h\), the function \(f_h(t)=\langle h,V_th\rangle\) is continuous and exponentially convex on a sufficiently small interval: every permitted Hankel matrix is a Gram matrix by (43). The interval Bernstein–Widder theorem, in the form of (Buescu and Paixão 2021, Theorem 3.5 and Corollary 3.6), represents \(f_h\) as a Laplace transform of a positive measure with finite integrals throughout that interval. In particular it is analytic near zero. Inner products of finite differences of \(V_th\) are the corresponding differences of \(f_h(s+t)\). Their limits show that all Hilbert-space derivatives \(h^{(k)}\) exist and satisfy \[ \|h^{(k)}\|^2=f_h^{(2k)}(0) \le C_h(2k)!r_h^{-2k}. \tag{45}\] The Taylor series recovers the vector curve: this follows by comparing its inner products and its squared norm using (43).

On the span of all these derivative vectors, define \(B_0h^{(k)}=h^{(k+1)}\). The cross versions of (43) make this well defined and symmetric. More explicitly, if a linear combination of derivatives vanishes, its proposed derivative has zero inner product with every word by differentiating the cross identity; density makes that derivative zero. The same identity gives symmetry between derivative vectors. The domain is dense and invariant. Bound (45), also with \(k\) replaced by \(k+j\), implies that every vector in this domain is analytic for this same operator: for some \(t>0\) its series \(\sum_{j\ge0}t^j\|B_0^jh^{(k)}\|/j!\) converges. Nelson’s theorem (Neeb 2011, Definition 4.7 and Theorem 4.8) therefore makes \(B_0\) essentially self-adjoint. Denote its closure by \(B\).

Finite differences of ordinary words approximate \(h^{(k)}\) both in norm and after applying \(B\), by the Taylor series just obtained. Ordinary words are thus a graph core. The vector series gives \(V_th=e^{tB}h\) for small real \(t\) on each word.

For fixed \(D\), the row family is strongly continuous, by the Gram continuity and uniform bound. The word identity \(A^D(x+t)V_th=V_t(A^D(x)h)\) yields, weakly on the graph core, \[\frac{d}{dx}\langle g,A^D(x)h\rangle =\langle Bg,A^D(x)h\rangle-\langle g,A^D(x)Bh\rangle.\] Integrate this equality and approximate \(g,h\) in graph norm to extend it to \(\operatorname{Dom}B\). The extension in Equation (44) then satisfies the Cauchy–Riemann equations between domain vectors, with continuous partial derivatives. Its uniform bound \(2\|D\|\) extends weak holomorphicity from that dense set to all vectors, hence gives bounded operator holomorphicity throughout the strip. ◻

Proof. We first obtain a uniform finite-volume vacuum estimate. Temporarily write \(T_D(u)=T_N^D(v-u/\rho)\). For \(u\) in a fixed sufficiently small complex disc about zero, both \[ T_D(u)T_I(u)^{-1},\qquad T_I(u)^{-1}T_D(u) \tag{47}\] are holomorphic with norms at most \(C\|D\|\), uniformly for all sufficiently large \(N\).

To prove the first assertion, multiply by the transfer \(\widetilde T_I(-u)\) having reversed auxiliary order. Fold its auxiliary index \(c'\) by transposition. The product is the two-auxiliary partial trace of \((D\otimes I)\prod_jG_j\), in increasing site order, where \[G_j=R_{cj}(u)R_{c'j}(-u)^{t_{c'}}.\] The line spanned by \(|++\rangle+|--\rangle\) is invariant in the auxiliary space, and on this line \(G_j\) is \(r(u)I\), where \(r(u)=1-b(u)^2\), by inversion. At \(u=0\) the range of \(G_j\) is entirely in that line tensored with the site, by the swap identity. In the common triangular decomposition, \(G_j/r(u)\) has identity on this line, a complementary diagonal block of norm at most \(C|u|\), and a bounded off-diagonal block. A product of these matrices has uniformly bounded norm, since the off-diagonal block is bounded by a geometric series. Its partial trace without \(D\) differs from the identity by at most \(C(C|u|)^N\): the off-diagonal block has zero partial trace, the line block gives the identity, and the remaining diagonal product has the stated norm. This partial trace is invertible on one fixed disc, proving the first bound in (47). Reversing the product in \(G_j\) gives the same argument with the invariant covector, proving the second. Crossing gives the corresponding bounds near \(u=\lambda\).

The common vacuum is an eigenvector of \(T_I(u)\) by analyticity, and its eigenvalue is nonzero in these discs. Hence \(A_N^D(z)\Omega_N\) has a uniform vector bound in discs of a fixed radius about \(z=\pm v\). Pairing with each fixed limiting word, normal-family compactness gives analytic subsequential limits; their values on the interior fix them uniquely. The vector norm bound identifies these limits with a bounded holomorphic \(\mathcal H\)-valued function. Uniform derivative bounds also give strong convergence at the real endpoint to the swap expression (42), initially in the enlarged space.

Now commute this endpoint row through a word with all angles in \((-v/3,v/3)\). Each exchange uses an intertwiner whose inverse is analytic on one fixed disc about the endpoint, independent of the word: the middle-third angle restriction keeps its parameter uniformly away from inversion zeros. The exchange expresses the result as a finite sum of bounded interior rows applied to the continued vacuum vectors. It follows that the endpoint preserves \(\mathcal H\) and that \(A^D(z)h\) continues holomorphically on this same disc for every middle-third word \(h\). Its bound may depend on the word, because the number of exchanges does. This does not alter the radius. Strong convergence of interior rows to the endpoint holds first on this dense set and then on all of \(\mathcal H\), by the uniform interior operator bound.

Choose a countable dense set of middle-third words \(h_j\), and a finite equivalent measure in the spectral representation of \(P_\zeta\). If \(r\) is the common continuation radius, write the continued vector as \(F_j(z)=\sum_{k\ge0}a_{jk}(z-v)^k\) at the right endpoint. For \(r'<R<r\), Cauchy’s estimate gives \[\|a_{jk}\|_{L^2}\le M_{j,R}R^{-k},\qquad \sum_{k\ge0}(r')^k\|a_{jk}\|_{L^2}<\infty.\] Since the measure is finite, the corresponding pointwise series of fiber norms is summable almost everywhere. Intersect these full-measure sets over the countable core and a countable family of smaller radii. We obtain pointwise holomorphic vector series on the same disc for every \(j\). The left endpoint is identical. On its interior portion the vector equals \(f(z)h_j\). Analyticity consequently makes the continued fiber proportional to \(h_j\) everywhere. At almost every spectral point at least one vector in the countable dense set is nonzero; ratios using any such vector agree on the interior and hence throughout the disc. This constructs the scalar continuation with one common radius.

The swap formulas show that \(P_\zeta(v)\) and \(P_\zeta(-v)\) are unitary inverses. Strong boundary convergence and (44) give unitary horizontal limits at \(\pm v+ib\). For countably many \(b\) within the continuation discs, the scalar value there has modulus one almost everywhere. Continuity gives modulus one on each full vertical diameter. Schwarz reflection across that diameter now implies \(f(z)\overline{f(2v-\overline z)}=1\), and the analogous identity at \(-v\), throughout the symmetric disc. Thus there are no zeros there. Rational imaginary translations preserve null sets and cover both boundary lines by discs of the same radius. This gives the asserted fixed bands.

The bounded-function maximum principle on the strip gives \(|f|\le1\). Finally each nonzero analytic \(f\) has isolated zeros, so Fubini’s theorem shows that on a fixed vertical orbit almost every argument has no zero for almost every spectral function. The covariance of the spectral measure class transfers this null-set assertion to any prescribed point of that orbit. ◻

Proof. Draw the strands smoothly and disjointly, with transverse path crossings. At a marked crossing use the tangent-angle lift rotated by less than \(\pi/2\) to the orthogonal crossing direction; this agrees with the lattice convention at a tile edge. Sliding the source on its strand while its tangent changes by \(b\) changes the sum of the turns of the two departing pieces by \(-2b\). The winding factor is consequently multiplied by \(e^{-i\rho b}\), which cancels the change of the marker gauge. All other spin constraints and prefix factors correspond directly.

A generic isotopy also creates or removes a pair of crossings at a tangency. Terms marked elsewhere are unchanged because the two new diagonal factors cancel. The two terms marked at the new crossings cancel one another. To check the sign, orient a test tangent \(\beta\) of the strand approximately in the path direction near tangency, so crossings \(1,2\) occur in that order both along the path and along this test direction. With the source at crossing \(2\), let \(c_1\) be the signed spin at crossing \(1\); the strand there travels opposite to \(\beta\), away from the source. The orthogonalized lifts satisfy \[ (\alpha_2-\beta_2)-(\alpha_1-\beta_1)=-c_1\pi. \tag{65}\] For example, when the first path tangent is clockwise from \(\beta\), the offsets are \(-\pi/2,+\pi/2\) and \(c_1=-1\); the other case reverses the signs. The ratio of the marker gauge times the source-piece winding factors is therefore \(e^{-ic_1\lambda}\). The extra prefix factor is \(Q^{c_1}\), and \[e^{-ic_1\lambda}Q^{c_1}=-1.\] The outside constraints and factors of the two terms are identical. This proves the cancellation.

A generic isotopy has only isolated tangencies of this kind, besides transverse slides, and can be chosen to keep all fixed insertions disjoint. This proves the primitive identity. For a pure diagonal string, only the first cancellation is needed; equivalently it is the tile ice-conservation identity. ◻

Proof. By Lemma 19, for each pairing we may choose a simple path crossing minimally the boundary arcs separating \(p\) from \(z\), followed, if necessary, by the nested closed loops. Such paths exist in the disk regions cut out by the disjoint arcs and loops. A sole outward source on a closed loop is inconsistent with the remaining arrow constraints, so it gives zero.

For a parent-to-child crossing, the boundary-ordered arc travels along the positive normal of the path. Let \(\beta\) be its tangent lift, continued from \(\theta_i\). The correct lift relation is the exact identity \[ \beta-\alpha=\pi/2, \tag{67}\] without a multiple of \(2\pi\). To verify this, close a simple curve by taking the boundary from \(p\) to \(i\), the arc to the crossing, and the reversed path back to \(p\). Its positive full turn, with the left right-angle turns at the port and crossing and the short turn at \(p\), is \(\beta-\alpha+3\pi/2\). It equals \(2\pi\), proving (67).

For an outward source the sum of strand turns is \(\theta_i+\theta_j-\pi-2\beta\). Multiplying its winding weight by the two outward boundary factors and by \(e^{i\rho\alpha}\) gives \[\frac1{2\sin\lambda} e^{-i\lambda(h_i+h_j)} e^{i\rho(\alpha-\beta-\pi/2)} =c_*a_e.\] Summing the two orientations of an unmarked arc gives \(w_e\). If a prefix string crosses it, multiplication by \(K(Q)\) changes that sum to \(k_e\). These are precisely the tree weights in (66).

For a pure string, the same calculation holds on every separating boundary arc. A counterclockwise enclosing loop has intersection factor \(Q^{-1}\), so the sum of its two orientations is \(e^{i\lambda}Q^{-1}+e^{-i\lambda}Q=-2\). Crossing each such medial loop switches the adjacent color, which proves the final parity assertion. ◻

Proof. There are two matters to check: the compared paths must have common endpoints, and rerouting the other charges must contribute the same phase to every possible source position.

Let \(a,b\) be the two positive-time unit steps, namely \(X,Y\) in the diagonal slicing. An increment in a negative direction is exactly the negative of the corresponding forward increment ending at \(z\). This rewriting is local, even if previous cuts have been made. Choose a point on the common prefix in the intact neighborhood, express each increment as a difference of two complete primitives from that point, cancel their common initial marked terms, and apply Lemma 19 inside the neighborhood. Thus no earlier path or cut is altered.

Use the following common endpoints for the high stencil:

• For directions \(a,b\), start at \(z\) and append two after-twist steps in the other direction to end at \(z+2a+2b\).

• For directions \(-a,-b\), start at \(z-2a-2b\), prepend two before-twist steps in the other direction, and end at \(z\).

• For directions \(a,-b\), use the trajectory starting at \(z-2b\) with steps \(b,b,a,a\).

The case \(-a,b\) is the same with \(a,b\) interchanged. Backward differencing changes the relative sign exactly as required to obtain the coefficient for the two positive directions.

Approach the common start by the same prefix, with a final positive-direction segment. A simple replacement can first bend around the stencil and approach from the required side, even when the original prefix arrived from later times. In a box with spare room, choose lattice corridors preserving the entrance and reserve a clear rectangle for the last positive steps. All marker lifts now differ by the short angle between \(a\) and \(b\) in their common time frame; no extra \(2\pi\) turn is introduced. The low case is the same construction with one increment. The explicit prefix charge is \(Q^\epsilon\), with \(\epsilon=1\) or \(-1\) according to the reflected type.

We next consolidate diagonal chains in the clear band. Keep as common background every chain except the explicit prefix steps between the common start and the marker. In particular, the charge arriving at the common start is included in this background. Using real charges modulo \(2\pi\), route it to carry a total phase \(\zeta\) on a common positive-time seam before the start, and a phase \(\xi\) from the start through the comparison block and to the future seam. These are the net oriented charges through ordinary sections, and \[\zeta/\xi=Q^\epsilon.\] All endpoints are preserved. Joining and funneling the chains can be done outside the sandwich cuts but inside the larger vacant margins, including transverse lattice displacements. Choose the cuts through level vertices of the positive-time seam with many positive steps on each side. This construction does not require a monotone path between the original remote cut positions.

Inside the comparison rectangle, first route the \(\xi\) background along one common skirt on the higher-space-coordinate side of both positive-step paths, avoiding every possible marker. In \(X,Y\) coordinates it runs around the northwest side of the small positive rectangle between the common endpoints. Tile ice conservation then moves this skirt onto either compared path, always on the same, left spin side of a broken source edge. At a marker it therefore multiplies \(L_-\) or \(L_+\) on a fixed side by the diagonal twist. This multiplication is one common scalar.

It remains to justify that the change from the original background routing to this common skirt has a common phase. Their difference is a closed tile-edge chain in the larger slab, hence a weighted sum of face boundaries. Tile ice conservation cancels every face contribution except at a source edge, whose signed spin jump is fixed and equal to two relative to its positive step. All prospective source edges lie in the same winding region of this common closed-chain difference. Indeed other original chains avoid the intact neighborhood; the chosen prefix approaches the common start from the past without entering the positive rectangle; the replacement before the start follows that same final incoming line with net charge \(\zeta\); and the common skirt and outgoing seam stay outside the rectangle. No edge of the routing difference separates the interiors of two possible marker edges. Extend the coincident clear incoming and outgoing parts by many lattice edges if needed. The face coefficient seen by the source is consequently identical in every stencil term. Summing conservation gives one common phase. There are no contributions from other defects, since all changes remain inside the clear band and its margins. The same reasoning can be phrased as a sequence of path slides consistently kept on one side of the entire marker rectangle. Every phase remains unitary.

We have now obtained exactly the before- and after-twist operators in (31), with their required relative phase, the compensating even pairs described above, and one common bounded scalar. The two exterior states are independent of which term is used. Theorem 15 gives the claimed bound.

For later repeated cuts, note also how these states glue. A consolidated chain at a diagonal cut ends at a shared level vertex. Reflection conjugates its charge to the inverse in the mirrored direction, preserving the spin convention. Joining after reversing path direction therefore continues with the same charge and creates no endpoint at the cut. At preliminary tile-axis cuts, route the chains transversely over the line through its level vertices, with no segment lying on the line inside the spare margin; the same argument applies. Reflections preserve the high or low type at every other intact stencil point. This proves the reflected assertion and the compatibility needed for successive extractions. ◻

Proof. Attach a thin exterior annulus containing the local caps, with the four unmatched ports continued to its outer boundary. Each cap cuts off a sliver at the intervening gap. The remaining annular regions connect the wire-color visits of each interval and lead to the four outer intervals. Consequently precisely the wired classes remain accessible from that boundary. Two noncrossing open strands cut a disk into three boundary regions, with one further region for each boundary loop. The other tree regions give exactly the loops in (71). This also gives the opposite-pair alternative.

These alternatives concern actual connections. Indeed the switch arcs are perimeters separating the two thickened diagonal graphs. Visits of one color in a single boundary region belong to the same actual component; reinstating internal loops cannot divide that boundary class. There are only two designated intervals of each color. The two possible pairings of four ports agree or disagree with the deterministic exterior pairing, adding two or one loops, respectively, as asserted. Finally, the path rule of Lemma 20 gives (73)–(74); all ordinary internal-loop weights cancel. Separating the term marked on \(e\) gives (75). ◻

Proof. An unpassed \(L=3\) edge kills every term of \(H\), so only its last edge can mark. Once the first \(b\)- or \(d_0\)-leg edge has been passed, \(k_e=0\) kills all later marked terms. At node \(0\), \[ D=F,\qquad H/P_0= \begin{cases}F,&c_0=\mathfrak A,\\-y^3F,&c_0=\mathfrak B. \end{cases} \tag{85}\] The first \(b\)-leg coefficient \(a_e/P_0\) is \(-y^2\) or \(y^{-1}\), according as the parent color is \(\mathfrak A\) or \(\mathfrak B\); the corresponding \(d_0\)-leg coefficients are \(-y^4\) or \(y\). This gives (81) at the near branch or shared node.

On the middle chain, \(a_e/P_0=y\) in type \(X\) and \(-y^2\) in type \(Y\). If the first chain edge has \(k_e=0\), its weight is \(w_e=1\) and only that chain mark contributes. Otherwise \(F=d\); the chain terms and the target mark contribute \((2-d^2)\) times the chain amplitude. Indeed the second chain edge already has \(k=0,w=1\), or, when \(m=1\), the target edge has the same amplitude. The identities \[ d+(2-d^2)y=yG,\qquad -dy^3-(2-d^2)y^2=G \tag{86}\] give the far-branch values; the first-\(k=0\) cases give the same values.

For (82), consider first the earlier port of an arc, and let \(r=1\) for a high port and \(r=-1\) otherwise. With “before” and “after” referring to boundary-gap visits, \[ \Re(a_e/A_R)=\cos((m_e-r\Delta)\lambda) =c_{\rm after}k_e-c_{\rm before}w_e. \tag{87}\] At the later port, using its own high status, \[ -\Re(a_e/A_R)=-\cos((m_e+r\Delta)\lambda) =c_{\rm after}w_e-c_{\rm before}k_e. \tag{88}\] These product-to-sum identities and (75) prove the invariant, including across a corner in the same interval.

For \(R=B,C\), direct substitution gives \[ \Re(\Gamma/A_R)=c_R,\qquad d\Gamma(\text{wire color})+\Gamma(\text{other color})=-A_R. \tag{89}\] Thus \(H^\circ\) is unchanged on \(L=0\) edges in \(B,C\). It is unchanged on \(c\)-leg edges as well: those have \(w=1,k=-d\), and \(a=A_C\) for a high first port, whose parent color is \(\mathfrak B\), and \(a=A_B\) otherwise. The table bounds all other parts of the tree: an unpassed \(p\)-leg kills terms; the middle chain kills a complete prefix in at most two steps; and an \(L=0\) tail has the two-step rule (80). The \(c\)-leg is the only place where the uncorrected variables can grow, and the correction is constant there. This proves the uniform bound for \(H^\circ\), including atypical trees.

At a node incident to a \(c\)-leg edge, the \(B,C\) line invariants, anchored by (81), intersect at \(P_0GT\) when \(d>1\). The four line directions after division by \(P_0G\) are \[ 1,\quad -y^{-1},\quad y^2,\quad -y \qquad\text{on }A,B,C,D, \tag{90}\] as follows from \(iA_R/(P_0G)\) up to nonzero real factors. At \(d=1\) the same identity follows by continuity for each fixed finite tree. The descendants beyond a \(c\)-leg edge lie in its \(B,C\) boundary disk; its \(L=0\) side branches therefore preserve this value.

The gap walk along \(B\) goes from the end of the \(b\)-leg to the end of the \(c\)-leg. Its \(L=0\) excursions preserve \(H^\circ\), so only type-\(X\) middle nodes need checking. Similarly only type-\(Y\) nodes need checking for \(C\). At node \(0\), \[ H^\circ/(P_0G)=F\quad(c_0=\mathfrak A),\qquad H^\circ/(P_0G)=yF\quad(c_0=\mathfrak B). \tag{91}\] After the first zero \(k\), the value stays at the far-branch value. If the first \(k\) is nonzero, the first jump is \(-y^{-1}\) in type \(X\) with \(c_0=\mathfrak A\), and \(-y^2\) in type \(Y\) with \(c_0=\mathfrak B\). Thus node \(1\) already has value \(T\) in normalized units. These are precisely the alternatives in the statement. ◻

Proof. The table and the two-step cancellation bound \(|H_V|+|D_V|\) by \(C\exp(C\ell)\), where \(\ell\) is the number of \(c\)-leg edges passed on the root-to-\(V\) path. A passed \(c\)-leg arc has a child disk containing \(z\). It cannot be wholly inside a box at \(c\) of radius a small fixed multiple of \(\min(1,|z-c|)\), because the flat wall makes such a child disk local. Fixed buffers and the dyadic bands between that radius and macroscopic scale count these passages. The conditional passage estimates of Lemmas 7 and 8, grouped into disjoint buffer colors, give the asserted moments. Bounded open-cap reweighting allows their use under a deterministic-cap FK law. If two nearby interior vertices have different boundary-tree regions, a raw boundary arc travels from their vicinity to the wall. Bulk RSW circuits make this probability uniformly small; Hölder’s inequality and the moment bound give equicontinuity, including between lattice parities.

For the boundary claims, work on separated scales in the relevant wall sector. In disjoint buffers impose two actual wall-to-wall crosscuts, of opposite colors, the first inside the second. Up-to-wall RSW gives a uniform conditional lower bound. A raw medial separator lies between the two thickened paths and joins the two banks. One direct verification is to trace the perimeters: on either bank the gaps at the two crosscut ends have opposite colors, so an odd number of ports lies between them; the paths prohibit radial escape, and at least one of these ports must be paired to the other bank. Such an arc separates the deeper box from the exterior. The same argument works in a corner sector. Iteration gives probability tending to one at diverging scale ratio. It produces the claimed local raw arch at an ordinary point and the appropriate terminal-leg separator at each mark. ◻

Proof. We first cancel a positively weighted residual at a fixed interior \(z\). Replace the marked primitive by the full \(K(Q)\) string from \(p\) to \(z\). If \(N\) is the number of internal closed loops surrounding \(z\), their weight ratio is \(-2/d\). Let \(\sigma\) take opposite signs on the two colors. Nesting parity gives \((-1)^N=\sigma(\operatorname{col}(z))\sigma(\operatorname{col}(V))\). Equality of the two winding expansions implies \[ \mathbb E_{\rm open} [\sigma(\operatorname{col}(V))D_V(2/d)^N]=o(1). \tag{93}\] Here is the required absolute estimate in the wrong expansion. Relative to the complementary cap partition it is at most \[ K_*^{1+X_*} \prod_{\nu=p,b,d_0}\mathbf1_{\{N_\nu=0\}}\, d^{N_c}\prod_{v\text{ a corner}}f(g_v)^{N_v/2}, \qquad g_v\in\{-\tfrac12,\tfrac12\}. \tag{94}\] The counts use disjoint small macroscopic cutoffs. The variable \(X_*\) counts macroscopic passages and has every fixed exponential moment; its constants may depend on \(z\). Away from singularities, ordinary small arcs have the complementary zero-drop cap weights. At \(b,d_0,c\) the effective wrong drops are \(-s,-s,-t\), with weights \(0,0,d\). At \(p\), use a locally forward boundary interval. Its arc meets the exiting string with net intersection \(-1\); after changing the winding convention the effective drop is \(s+1\), again giving weight zero. This depends on the net intersection, not the placement of the string. Corner alternation and cap elimination give the remaining factors in (94). Internal loops around \(z\) have absolute weight \(2|\cos(2\lambda)|\le d\), and each nonlocal arc costs at most a bounded factor per passage.

Without the \(K_*^{1+X_*}\) factor, the logarithm of the expectation is bounded above by \[ 3P+D_*+\sum_{v\text{ convex}}C_{i(v)} +\sum_{v\text{ concave}}L_{i(v)}+O(1). \tag{95}\] Its second moment is polynomial in \(n\). Hölder with exponent slightly above one absorbs \(X_*\) at an arbitrarily small loss in the coefficient of \(\log n\), by Lemma 9. The balanced-corner budget (28) is strictly negative by (15). Our global color choice makes the complementary cap partition at most a constant times the correct open-cap partition. This proves (93).

We next remove the positive weight, retaining relative error estimates. Let \(\mu\) be the correct deterministic-cap FK law and \(R_*=d^{\ell_{\rm open}-\ell_{\rm det}}\); open expectation is \(\mathbb E_\mu[R_*\cdot]/\mathbb E_\mu R_*\). For the interval \(R\) containing \(u\), put \[ B_*=R_*\sigma(\operatorname{col}(V)) d^{-g_R(\operatorname{col}(V))}D_V, \qquad T_*=d^{g_R(\operatorname{col}(V))}(2/d)^N. \tag{96}\] Thus \(\mathbb E_\mu[B_*T_*]=o(1)\) at every fixed \(z\). Choose scales in the regular sector at \(u\) with \[ |z-u|\ll r_1\ll r_2\ll r_3\ll1, \tag{97}\] all ratios diverging as \(z\to u\), and take the mesh limit first.

Except on an event of vanishing probability, \(B_*\) is determined outside \(r_2\). To see this, use actual screens to ensure that no raw strand from the \(r_2\) rim reaches \(r_3\). Every changing boundary arc then lies in one small part of \(R\), so has \(L=0\). All \(L>0\) data, the marked subtree and \(R_*\) are fixed. Regard arcs touching the connected patch as affected and arcs wholly outside it as stable. Stable separators either separate the entire patch from \(p\) or do not; noncrossing puts the affected on-path arcs after them. Off-path factors cancel, while the on-path \(k/w\) suffix telescopes by (80). The color factor in \(B_*\) cancels that suffix exactly. Set \(B_{\rm out}=B_*\) on this exterior good event and zero otherwise.

Truncate \(T_*\) to \(T_{\rm in}\) by counting only enclosing loops wholly inside \(r_1\) and reconstructing the tree color from their parity and the fixed lattice color of \(z\). This is exact unless a raw path runs from \(O(|z-u|)\) to the \(r_1\) scale: every enclosing loop meets the segment from \(z\) to its nearest wall point. To measure both truncation errors, let \(N_0\) count the enclosing loops wholly inside the box of radius \(\operatorname{dist}(z,\partial\Omega)/8\) at \(z\), and put \(U_z=(2/d)^{N_0}\). Work under the positive tilt \(\nu_{U_z}\propto U_z\mu\). The rare events remain rare under this tilt by conditional RSW outside its support. The variables \(B_*\), \(T_*/U_z\), and \(T_{\rm in}/U_z\) have uniformly bounded fixed moments as \(z\to u\). For \(B_*\), every relevant \(c\)-leg arc gives a macroscopic passage away from \(u\); an arc wholly local at \(u\) would have \(L=0\). Every extra loop in \(T_*\) gives a passage in a band at distance comparable to \(\operatorname{dist}(z,\partial\Omega)\), by the segment observation and its escape from the cutoff. Finitely many bulk and sector buffers cover this band uniformly. Near a corner, use a corner buffer and small straight-bank buffers, or only the latter if the corner is farther away than this scale. The tilt-edge passage estimates apply in each buffer. Hölder therefore makes the product truncation errors \(o(\mathbb E_\mu U_z)\), and \[ \mathbb E_\mu T_{\rm in}\asymp\mathbb E_\mu U_z\ge1. \tag{98}\]

The inward relative comparison of Lemma 6 between \(r_1\) and \(r_2\), applied to this positive inner factor and the absolute parts of \(B_{\rm out}\), gives \[ \frac{\mathbb E_\mu[B_{\rm out}T_{\rm in}] -\mathbb E_\mu B_{\rm out}\,\mathbb E_\mu T_{\rm in}} {\mathbb E_\mu U_z}=o(1). \tag{99}\] First use (93) at fixed \(z\), then let \(z\to u\). Equations (98)–(99) show that \(\mathbb E_\mu B_*\) vanishes in this order of limits. Finally apply the same separation without a tilt to \(B_*\) times any bounded function of the tree color, reconstructing that color from the inner-loop parity. Its expectation factors up to \(o(1)\) into \(\mathbb E_\mu B_{\rm out}\) and a bounded inner expectation. Taking the bounded color factor that reverses the extra factors in (96), and dividing by \(\mathbb E_\mu R_*\), proves (92). ◻

Proof. Fixed up-to-wall RSW corridors give both actual crossings positive probability under deterministic caps, and hence under open caps; thus \(0<r<1\). Shielding and the moment bounds give the interior limits \(0,U_0,yV_0\) at \(p,b,d_0\): near \(p\) an unpassed \(L=3\) edge kills \(H\), and near \(b,d_0\) the first \(k=0\) edge fixes it at its marked value. At every other boundary point the line invariant, shielding, and Lemma 25 put \(h\) on the corresponding line of direction (90), with offsets fixed by these values. In a conformal half-plane coordinate, whose boundary extension is homeomorphic, the line-normal harmonic component vanishes continuously on each unmarked interval. Harmonic Schwarz reflection, followed by extension of its harmonic conjugate, extends \(h\) holomorphically across that interval.

On \(B,C\) the traces lie on the actual segments \([U_0,T_0]\) and \([T_0,yV_0]\), respectively. Indeed \(H^\circ\) has, for each connection event, only the two values in Lemma 23; the segment direction is fixed, so averaging keeps it between the averaged endpoints. Its correction has zero limiting expectation at every fixed boundary point. As that point approaches \(c\), shielding gives a \(c\)-leg separator enclosing it and nearby interior points. The uniform bound for \(H^\circ\) therefore makes the boundary trace tend to \(T_0\).

There is also continuity at \(c\) from the interior. In a half-plane coordinate \(\zeta\) with \(c=0\), the flat wall and the moment bounds give polynomial growth of \(h-T_0\). Its two boundary directions are \(-y^{-1}\) on the negative and \(y^2\) on the positive real axis. Consequently \[ e^{-2i\lambda}\zeta^{3\rho-1}(h(\zeta)-T_0) \tag{102}\] has real boundary values on both sides, with the branch in the upper half-plane. Schwarz reflection makes it holomorphic on a punctured disk; polynomial growth makes its singularity at worst a pole. Its leading Laurent power, after undoing the multiplier, must have positive exponent because the two boundary traces tend to zero. Thus \(h\to T_0\) throughout the interior approach as well.

The target polygon is counterclockwise and simple. For \(d>1\) its vertex \(T_0\) is reflex: in the basis \(1,y\) it lies strictly inside the triangle with vertices \(0,U_0,yV_0\). When \(d=1\), it lies on the third side of that triangle. For any point off the four boundary lines, straighten each continuous boundary trace in its own line. Its winding number is that of the target polygon. The argument principle therefore gives one preimage of every generic interior point and none of every generic exterior point. The open mapping theorem puts the open image inside the polygon, and then the traces on \(A,D\) also lie on their actual segments. The segment control on \(B,C\) already established is needed at the reflex corner. No interior point lies on the trace, and the preimage count proves conformal bijectivity.

The four target angles are \(\rho\pi,\rho\pi,(2-3\rho)\pi,\rho\pi\). The Schwarz–Christoffel formula therefore has derivative proportional to \(\zeta^{\rho-1}(\zeta-\chi)^{\rho-1}(\zeta-1)^{1-3\rho}\); see, for example, (Pommerenke 1992). The \(B\) and \(C\) side vectors are \(-(1-r)y^{-1}\) and \(ry^2\), so their lengths are \(1-r\) and \(r\). Their ratio gives (101). At \(d=1\) the straight marked point has exponent zero, as required. Lemma 22 reweights the two open-cap alternatives by \(d\) and \(d^2\), giving the last assertion. ◻

Proof. Between consecutive endpoint scales use corridor annuli with radii \(Kr_j\) and \(r_{j+1}/K\), for a large fixed geometric constant \(K\). Each winding loop wholly in corridor \(j\) has ratio \(v(\sum_{i\le j}a_i)\ge1\), because the total charge magnitude is \(2\). Let \(M_j\) be the product of these ratios and \(M=\prod_jM_j\). Besides the \(U\) nests and these corridor loops, all nontrivial loops are counted by a variable \(X\) having every fixed exponential moment uniformly under \(\nu_U\), with constants independent also of sufficiently large \(L\). Indeed a loop excluding \(z_*\) but surrounding an endpoint outside its local cutoff has a passage at that endpoint scale, since it separates the endpoint from \(z_*\). A loop winding about \(z_*\) and enclosing some but not all endpoints either lies in its corridor or passes a bordering scale. Fixed box and shell covers, disjoint from other endpoint buffers, and the tilt-edge passage estimates give the claim. Therefore, for a fixed \(C\), \[ |Y/U|\le MC^X,\qquad Y/U\ge MC^{-X}\quad\text{off }\mathcal B, \tag{111}\] where \(\mathcal B\) is the event that some loop excluding \(z_*\) encloses at least two endpoints. Off \(\mathcal B\), all remaining ratios belong to a fixed finite positive set. Moreover \[ \mathbb E_{\nu_U}M^b\le C_bL^{C_b}\qquad(b>0\text{ fixed}). \tag{112}\] To prove this, use \(O(\log L)\) dyadic passage bands per corridor, group disjoint buffers by colors, and apply Hölder; those buffers avoid the supports of the endpoint tilts.

If a bad loop has smallest enclosed endpoint index \(j\), it crosses corridor \(j\) radially, and hence excludes every winding loop counted by \(M_j\). Write \(\mathcal B_j\) for this event. Successive actual circuit blockers give \(\mathbb P(\mathcal B_j\mid U\text{ patches},(M_i)_{i\ne j}) \le CL^{-\alpha}\), uniformly. On \(\mathcal B_j\) one has \(M_j=1\). First apply this conditional estimate to \(\prod_{i\ne j}M_i\), then Hölder with exponent \(1+\delta\) close to one to absorb \(C^X\), interpolating against the polynomial moment bound (112). For some \(\alpha'>0\) independent of \(L\), \[ \mathbb E_{\nu_U}[MC^X\mathbf1_{\mathcal B}] \le C'L^{-\alpha'}\mathbb E_{\nu_U}M. \tag{113}\] The same interpolation, using arbitrary fixed exponential rates for \(X\), gives for every \(\varepsilon>0\) \[ \mathbb E_{\nu_U}[MC^{-X}] \ge c_\varepsilon L^{-\varepsilon}\mathbb E_{\nu_U}M. \tag{114}\] Equivalently, Jensen under the \(M\) tilt and the exponential bounds give an arbitrarily small coefficient of \(\log L\) in its mean of \(X\). Take \(\varepsilon<\alpha'\) and then \(L\) large. Equations (111)–(114), and \(M\ge1\), prove the positive lower bound in (110). Its moment bound follows from (111), (112), and Hölder. ◻

Proof. Under \(\nu_U\), bulk RSW circuits in the empty scales between the wall and the first endpoints make the probability that a raw open arc reaches the splitting scales tend to zero. On its complement the phase comparison above is exact. Lemma 27 and (115) make the error negligible even on the exceptional event, by Hölder.

Choose intermediate radii \(1\ll R_1\ll R_2\ll R_0\). The ideal tree integrand, including its open-to-deterministic reweighting, is readable inside \(R_1\) when its boundary arcs stay there; set it to zero otherwise. Similarly \(Y/U\) is readable outside \(R_2\) unless a raw path joins that scale to the first endpoint bands: new loops at smaller scales have trivial ratios. All product truncation errors vanish under \(\nu_U\) by the moment estimates and RSW. The inward relative comparison across the intervening bulk annuli then factors their expectation up to \(o(1)\). Applied to absolute parts, that comparison also replaces the inner tilted expectation by the unweighted one. The exterior coefficient is the fixed phase times \(\mathbb E_{\nu_U}(Y/U)\) and the bounded tree/cap normalizations. Interior-exterior independence completes (116); Lemma 27 bounds the coefficient away from zero. ◻

Proof. We first work at one probe pair with ideal real cuts, before a small generic perturbation. Divide a bounded vertical interval containing all data, with empty end margins, into finitely many distinctly labeled cells, including the two marker cells (128). Horizontal folds act on a word of cell labels. Splitting a word of length \(m\) after position \(j\) makes the two words \[ WW^{\rm rev}\quad\text{and}\quad V^{\rm rev}V, \tag{133}\] where \(W\) is the prefix of length \(j\) and \(V\) the suffix of length \(m-j\). Their lengths are \(2j\) and \(2(m-j)\).

Under the size-biased weights \(j/m\) and \(1-j/m\), the vector of label frequencies is a bounded martingale. The expected terminal label frequency multiplied by the initial word length equals the expected terminal label count under the ordinary half-weights. Choose \(j\) within a bounded distance of \(m/2\) to maximize the conditional frequency variance. These frequencies converge almost surely to a pure label. Here are the details. If the frequency vector \(\pi\) stays away from the vertices of its simplex, then some admissible near-middle prefix has \[ \|\operatorname{counts}_{\le j}-j\pi\|\ge c>0. \tag{134}\] Indeed two consecutive choices differ by a single-label vector minus \(\pi\); they cannot both have small norm. The mixed length-two case is immediate. The conditional frequency variance is then at least \(c'/m^2\).

Along the size-biased chain, \(\sum m^{-2}=\infty\) almost surely. Lengths change by bounded amounts; the probabilities of its symmetric increments are \(1/2+O(1/m)\). Both the absolute conditional drift and conditional variance of \(\log m\) are \(O(m^{-2})\). If the sum were finite, localization and martingale convergence would make \(\log m\) converge to a finite limit. Then \(m\) would be bounded, contradicting the same assumed summability. A bounded frequency martingale cannot accumulate the divergent conditional variance in (134) while tending to a mixed limit. Length one, if reached, is already pure. This proves purity and hence arbitrarily small expected impurity at a sufficiently large finite depth.

Outer-end folds retaining the whole word and its reflection can also make its length arbitrarily large; the empty child has zero size weight. In a branch, concentrate on its majority label if that label is a marker. For any prescribed finite run depth, only a proportion bounded by that depth times the minority frequency, plus the end contribution, fails to lie deep in a pure run of the majority. Their expected count can therefore be made arbitrarily small. Adjacent equal-label cells in the folded word alternate orientation. In a pure run of marker cell \(i\), the vertical projection to the original cell is a triangular wave of period \(2H_i\). It sends every marker ordinate \(b'\) to \(y_i\) and \(b'+H_i\) to \(y_{*,i}\), by (128).

We next choose vertical cuts to align the marker abscissas with the primary gaps. All copies of a majority marker start from the same original abscissa \(x_0\). Its farther clear side up to a neighboring vertical wall has width at least a fixed fraction of \(G_*\); suppose this is the left side, reflecting the following construction otherwise. Take \(H_i\) much smaller than this width. Cut just to the right of the current rightmost copy \(a_{\max}\) and retain the left child, including all marker columns there. The discarded child’s marker count is zero, so this loses no expected majority count. The left tail below \(x_0\) still projects identically. If the cut is \(c'\), the new rightmost copy and right-tail projection are \[ a'_{\max}=2c'-x_0,\qquad s'\longmapsto a'_{\max}+x_0-s'. \tag{135}\] Keep the central projection in the wall-free abscissa interval. The first cut is close to \(x_0\) on the right; subsequent distances \(c'-a_{\max}>0\) can lie in the ample leftward clearance sampled by the old right tail.

Drift the copy abscissas modulo \(2H_i\) toward \(A_*=X_*-H_i\). Lift their distribution initially to mean \(x_0\) and variance zero. At each step, with lifted mean \(\mu\), choose \(c'\equiv\mu+v\pmod{H_i}\); reflected lifts are \(2(\mu+v)-a\). The new mean is \(\mu+v\) and the variance increases by exactly \(v^2\). Choose \(Nv\) to realize the required residue displacement to \(A_*\) in the rightward direction, with bounded positive total displacement, and then choose \(N\) large. The first cut fits its small right clearance and the later cuts their larger available clearance. Chebyshev’s inequality makes all but an arbitrarily small fraction of copies have residue within a small fraction of the primary gap width of \(A_*\). At the last step, \[ a_{\max}+x_0\equiv2A_*\pmod{2H_i}. \tag{136}\]

For such a successful copy \((a,b')\), consider its slope-\(+1\) line. Choose the pure-run depth larger than the entire folded horizontal support span divided by the cell width; this finite bound is known from the abscissa schedule before the word depth is chosen. Throughout the horizontal support of all data, the line then projects vertically into the clear marker band, so meets no horizontal wall or other feature height. In the central portion its abscissas avoid vertical walls. In the left tail, every projected vertical wall lies at \(X_*\) modulo \(2H_i\), and hence at horizontal offset \(H_i\) from \(a\) up to the small residue error. The same is true in the right tail by (135)–(136). The triangular-wave identity therefore sends the ordinate at each such encounter into the primary gap at \(y_{*,i}\). The line has a wall-free slab of positive thickness. By designing first the finite abscissa schedule and its required run depths for the fixed \(h\), then the word depth, the expected non-targeted or unsuccessful marker counts are arbitrarily small for each of the two original markers.

Finally perturb every cut generically by a sufficiently small fixed amount. The finite successful inventory and strict clearances persist. Distinct stencil copies acquire distinct time projections: for copies with different reflection histories, their difference depends nontrivially on the last cut where the histories differ; distinct original markers with identical histories were separated by the generic-coordinate assumption. Keep cuts away from endpoints, marked features, tips, and coincident runs along solids. At every transverse solid intersection insert a secondary gap at its preimage on the original wall. There are finitely many such preimages, away from marks and corners after the perturbation. Thus all intended axis cuts become vacant.

Each template works on a neighborhood of its initial probe pair. A finite cover of a compact support gives finitely many templates; take the union of all their secondary gaps, then choose \(\sigma\) sufficiently small. Additional removed portions only improve the clearances. Round cuts to vertex lines. These roundings and the \(O(1)\)-mesh shifts of the polygon preserve all fixed margins for fine mesh. Apply the finite median-extraction procedure above, with a further arbitrarily small residual count. This proves the lemma. ◻

Proof. Choose the expected extraction deficit \(\varepsilon\) in Lemma 30 small relative to \(c_0\) in (132). The angular costs contribute at most \[ -(1+\rho)(1-\varepsilon)\log n -(1-\rho)(1-\varepsilon)\log n =(-2+2\varepsilon)\log n. \tag{138}\] The baseline (131) cancels the precise normalization in (107). The remaining feature cost is at most \(-c_0\log n+o(\log n)\) by (132). There are finitely many nodes, so all Hölder allowances may be chosen with total smaller than, say, \(c_0/4\). Taking \(2\varepsilon<c_0/4\) leaves a strictly negative extra exponent. This proves (137). It uses finite templates fixed before the mesh, not any uniformity as their gaps shrink. ◻

Proof. Take a simultaneous locally convergent subsequence \(h_n\to h\), \(J_n\to J\). The ideal compactness statements do not use holomorphicity. On a product lattice coset for the step-two stencil, multiply (137) by a compactly supported smooth test function \(\varphi(z,u)\) in an eligible product neighborhood and sum with normalized mesh weights. Twice summing by parts and dividing by \((2/n)^2\) gives \[ \int \mathcal H_g(z,u) (\partial_1+i\partial_2)_z\partial_{j,u}\varphi(z,u) \,d^2z\,d^2u=o(1), \tag{139}\] where the integral denotes the corresponding lattice Riemann sum before passage to a limit. The difference-to-derivative error vanishes because \(\mathcal H_g\) is locally bounded at fixed gaps by the absolute moment bounds. Every coset gives the same conclusion.

First let the mesh tend to zero. Proposition 29, with its positive multiplier bounded above and below, transfers the zero limit from the secondary gaps to the primary-gapped wall as \(\sigma\to0\). Apply it again as the primary widths \(h\to0\) to reach the complete wall. Finally use Proposition 28, whose scalar is bounded away from zero, and then let \(R_0\to\infty\). The ideal functions have compact convergence, so their lattice sums are ordinary Riemann sums. We obtain \[ \int h(z)J(u)(\partial_1+i\partial_2)_z \partial_{j,u}\varphi(z,u)\,d^2z\,d^2u=0. \tag{140}\] No convergence of either comparison scalar is required: for each fixed geometric stage its inverse is bounded, and that stage’s error tends to zero in the stated order.

Fix an interior \(z\) off the finitely many feature projection lines. There is a generic exterior probe box eligible for this \(z\) on which some distributional derivative \(\partial_jJ\) is nonzero. Otherwise both derivatives would vanish off the finitely many excluded lines in the connected strip; continuity would make \(J\) constant on the whole strip, contradicting its different limiting values at \(p,d_0\). Shrink the \(z\) neighborhood and this probe box to retain all strict margins. Product tests in (140), with a probe test pairing nontrivially with \(\partial_jJ\), give \((\partial_1+i\partial_2)h=0\) distributionally near \(z\). Thus \(h\) is holomorphic there. Its continuity extends holomorphicity across the finitely many exceptional straight lines by Morera’s theorem, splitting triangles along those lines and passing to the limit. Proposition 26 now applies. ◻

Proof. Make one tile for each edge of the polygon graph, including boundary edges. Glue the four triangles directed toward each interior square center. The exterior triangle of a boundary edge is a separate tab, with its own outer dual vertex; do not glue different exterior tabs along their radial sides. Each primal boundary vertex retains all its interior incidences. Flatten the exterior tabs toward their bases, to less than half their original height, and add a still thinner Jordan collar. The tabs are then disjoint except at their prescribed vertices. All changes lie within \(O(\delta_n)\) of the original boundary, and tracing the tabs and collar along the original parametrization proves (i).

Draw the switches and boundary caps as disjoint arcs. A local wire can be drawn behind its cap in the collar. The unmatched ports continue to the marked outer-boundary points. The finite choices inside each tile and the collar can be made deterministically. At a wired boundary edge, capping its outer dual vertex joins the inside ports in the same way for either switch state. Declaring that edge open for the drawing merely substitutes a cap-sized detour and possibly deletes a microscopic loop. On the free side, the cap around a primal vertex is exactly the turn beside the dual wired arc. Inside the polygon, glued half-segments give the specified medial edges and their quarter-turn pairings. Match corresponding ordered pieces in the two drawings. Each substituted piece lies in boundedly many neighboring tiles, so simultaneous parametrizations have distance \(O(\delta_n)\). Zero-length pieces are handled by limits of increasing reparametrizations. This also covers a midpoint visited with both prescribed pairings and proves (ii).

The tile/cap Euler identity gives exactly the conditional FK law asserted in (iii). On a queried primal-open tile the adjacent primal corners are joined by its actual edge; on a queried closed turn the bank stays at one primal corner. Along a cap it stays at one vertex or uses neighbors already belonging to that wire interval. Thus the bank is a known chain attached to the appropriate original interval, and every queried primal-open edge accesses this bank within one tile. The same statement holds for dual edges. Designated outer vertices access their arcs through the collar. These are physical accesses, not additional identifications. They prove (iv), also when the last begun tile is revealed in full. Projecting back to the usual primal or dual lattice moves points by \(O(\delta_n)\) and duplicates no edge; separate virtual copies of a vertex may become one physical vertex. ◻

Proof. The substitutions \(u^{-1}\) in \(I_C\) and \(u=1-(1-\chi)t\) in \(I_B\), followed by the Euler transformation of the beta integral, give \[I_B(\chi)=\left(\frac{1-\chi}{\chi}\right)^h I_C(1-\chi).\] Both \(I_C(0)\) and \(I_C(1-)\) are finite and strictly positive: at the possible singularity \(u=1\), the latter integrand has exponent \(-2\rho>-1\), while the integrand at infinity has exponent \(-1-\rho<-1\). Substitution into (141) proves both claims, with \(c_q=I_C(0)/I_C(1-)\). ◻

Proof. Away from the normalizing point, a connected set in a ball of radius \(r\) is hit before domain exit by Brownian motion from that point with probability tending uniformly to zero as \(r\downarrow0\). Comparison with a fixed enclosing Euclidean annulus gives this bound. In the normalized disk, a connected set of a fixed positive diameter has hitting probability bounded below; this follows, for example, by a crosscut between separated angular sectors and the Beurling estimate. The rectangle maps of the disk form a compact family because the four marks remain separated. Thus a small physical continuum cannot have a macroscopic rectangle-coordinate oscillation. Near the basepoint use interior distortion instead. Endpoint accesses follow by interior approximation.

By Lemma 35, every known primal-open edge and fixed primal wire attachment accesses \(u_n=0\) or \(1\) within \(O(\delta_n)\). A short possible connection from the central band to one of them would contradict the just-proved modulus. Small buffers in the band absorb individual tile errors. In particular the two opposite designated intervals have not already met through the revealed bank under the stated nondegeneracy. ◻

Proof. Couple the deleted and original laws monotonically. Explore the actual open clusters from all deletion endpoints in the stronger configuration, revealing the corresponding edges in both laws. FK partition monotonicity persists at every step. After the clusters are enclosed by their closed cuts, the conditional laws outside agree and can be coupled identically. If no explored cluster reaches a wire interval, the crossing decision is unchanged. This proves the first assertion, including contact with a wire along a known bank.

Choose the fixed outer radius small enough for Lemma 38. Every component of possible edges inside this ball that meets the central-band set has no accessible wire or known primal-open attachment. Its uncertain boundary connections can occur only at the ball rim. Project to the physical lattice, fill missing edges, and wire that rim. Splitting physical vertex copies only weakens the original induced partition. The full-lattice wired-rim law therefore dominates every relevant actual arm simultaneously. Cropping the ball slightly absorbs tile errors. Its one-arm probability from radius \(r\) to the fixed outer radius tends to zero: buffered dual circuits from RSW block such an arm. A finite union bound gives the last assertion. ◻

Proof. The extra actual primal arm cannot make a lateral connection to either known primal leg inside the wider annulus, even in the all-unknown-primal-open completion, by Lemma 38. On the side containing the extra arm it is bracketed by the two known primal legs. A planar crossing alternative in each intervening strip supplies a dual radial separator. The original dual separator on the other side remains. After cropping at both rims, these three primal and three dual arms are disjoint and alternate.

For precision, thicken actual paths to their tile color territories. An open transversal between two primal territories would give an actual lateral bridge, with an error of at most one tile. Its absence forces a dual territorial radial crossing in the intervening topological rectangle, which shadows a dual edge path in the cropped annulus. If the no-bridge test is made after every unknown primal edge is opened, every dual separator so obtained consists of known closed tiles. Thus the four old legs and the separators used in this argument are actual original edges; no auxiliary boundary condition manufactures them.

Apply (144) in the original full-sample law to each cell of the deterministic grid, then sum over \(O(\ell^{-2})\) cells. The center may have been selected by the past because the sum was taken over the entire deterministic grid. Bounded cap densities control the averaged conditional probabilities under either history law. Markov’s inequality now makes the conditional flagged-cell cost small outside a set of histories of small probability. This is an annealed-to-conditional argument, with \(R_1\) and all comparison constants fixed before \(\ell\downarrow0\). ◻

Proof. We give the construction and its scale order. Retain only prefix excursions reaching depth \(r\) from the limiting Jordan boundary, including each complete segment between successive visits to depth \(h_b\ll r\) and the last incomplete segment if present. Their number is tight, uniformly in small \(h_b\), because each visits one of a fixed finite family of bulk annuli at depth \(r/2\). First bound this number, then choose \(h_b\) small. Join their boundary-near endpoints to the exterior of a slightly larger polygon through protected balls of radius \(O(h_b)\). Lemma 39 controls central attachments in these finitely many balls. No boundary arm exponent for the rough Jordan curve is needed.

Break the retained excursions into blocks of diameter at most \(\epsilon\ll h_b\), each contained with margin in a bulk square \(S\) of side \(10\epsilon\). Successive displacement thresholds and the bulk passage estimate make the number of blocks tight. After fixing a block-count cutoff, snap its endpoints to a much finer deterministic grid and protect them by balls of radius \(e\ll\epsilon\). The actual known primal bank gives a path between the true endpoints in \(S\); loop erasure gives a simple path \(\pi\), possibly trivial. Take \[\ell\ll R_1\ll e,\] so all flag annuli are bulk annuli. Declare forbidden every closed \(\ell\)-cell within \(3\ell\) of a candidate, except flagged cells and cells in enlarged endpoint protections.

There is a connector between the protected endpoints avoiding all forbidden cells. Otherwise planar cell separation gives a polygonal circuit through a chain of touching forbidden cells and possibly the exterior of \(S\), separating the two endpoints. It can be chosen away from their protection balls and within a cell-sized error of the cell chain. Choose a candidate witness in each cell. Two successive witnesses can be joined avoiding \(\pi\) inside a disk of radius \(O(R_1)\). If they could not, a proper crosscut of the simple path \(\pi\) would separate them in a circle of radius between \(4R_1\) and \(6R_1\). Its full-path endpoints lie outside because the cell is away from endpoint protections. The crosscut passes within \(O(\ell)\) of the witness segment. The two witnesses have possible arms to that circle, on opposite sides of \(\pi\), and cannot connect locally to \(\pi\) by Lemma 38. The two known radial legs of \(\pi\) and the two possible arms consequently interlace. In each intervening strip the optimistic no-bridge test supplies a known dual radial separator, as in the proof of Lemma 40. The cell is therefore flagged, a contradiction.

Push the separating circuit through the witness points and these avoiding joins. Its displacement is \(O(R_1)\), smaller than its distance from the endpoints, so its winding distinction is preserved. Along the exterior of \(S\) avoidance is automatic because \(\pi\) has an interior margin. We have obtained a separating circuit disjoint from \(\pi\), which is impossible. Overlapping endpoint protections or a trivial \(\pi\) are handled by a connector within their union.

Select one polygonal connector for each allowed finite grid pattern. Their union and the protected exterior attachments form an obstacle \(K\) attached to the complement of the larger polygon. The connectors need not form one simple curve. They stay close to the retained prefix, and every omitted prefix piece is boundary-near; conversely the entire prefix is approached by this obstacle or the outer wall. Within distance \(\ell/2\) of \(K\), candidates occur only in flagged or protected error regions. The basepoint component of the complement is simply connected, because each obstacle component is attached to the exterior. It is a fixed polygonal domain, possibly with slits and pinches. An interior radial approximation in its Riemann map, followed by a polygonal perturbation, gives a Jordan domain \(H\) whose boundary is within \(\ell/10\) of the obstacle or outer wall. The outer wall was chosen much farther than \(\ell\) from the original graph. Hence the asserted clearance holds. Every choice is from a finite list at fixed scales and count cutoffs.

Normalize maps of \(H\) and \(U_n\) at the same basepoint, and transfer marks by finite-grid approximations to their disk angles. On a Hausdorff subsequence of the original boundary and prefix, every compact path in the limiting basepoint component eventually belongs to both constructions, while every limiting boundary point is approached by complements of both. Thus their pointed kernels agree, and their normalized interior charts have the same limit. This is an interior statement; no locally connected boundary for the limiting slit domain is asserted. The fixed basepoint margin and separation of the marks persist throughout. ◻

Proof of Theorem 37. It suffices to prove an upper bound for the primal connection under separate deterministic primal closure. Interchanging the colors and using the other deterministic closure gives the complementary upper bound; (142) then yields the open-cap limit.

Use Lemma 41, and write \(u\) and \(v\) for the rectangle abscissas in \(U_n\) and \(H\). Test on the Jordan subquad of \(H\) between \(v=a_*\) and \(v=1-a_*\), with these sides separately wired. At fixed accuracy there are finitely many such quads; approximate each on the original color grid by the polygon construction of Proposition 33. Their limiting crossing probabilities differ from \(g(\chi_{U_n})\) by arbitrarily small trimming, chart-angle, and kernel errors, uniformly under the nondegeneracy assumptions.

We justify comparison on shared edges. Take a fine finite grid of abscissa bins in the central part of \((0,1)\), with width much smaller than both \(\beta\) and \(a_*\). In each bin choose a nearby vertical fiber for each chart. Its central part is a compact internal curve and is matched in the other chart by kernel convergence. Its two remaining tails can be confined to balls of vanishing radius, with matched centers. Here is the required length–area argument. Remove small fractions of rectangle height at top and bottom. For each fixed fraction, inverse maps and derivatives converge on the compact retained rectangles. Let that fraction decrease slowly. For a bin \(I\) and a deleted height interval \(J\), Cauchy–Schwarz gives \[\int_I\!\operatorname{length}\bigl(\Phi(\{x\}\times J)\bigr)\,dx \le (|I||J|)^{1/2} \left(\int_{I\times J}|\Phi'(z)|^2\,dA(z)\right)^{1/2},\] where \(\Phi\) is the inverse rectangle map. The last integral is bounded by the common physical area bound. Choose levels in each bin controlling the tails of both charts. Their initial endpoints match on the retained compact pieces, so matched small endpoint balls contain all four tails. This argument does not require uniform local connectedness of the explored boundaries. Tile errors are absorbed by decreasing the fractions slowly enough. Lemma 39 controls the conditional attachment cost of these finitely many endpoint balls.

Delete still unknown edges touching a slightly narrower band, initially \(3\beta\le u\le1-3\beta\), if their endpoints fail containment in \(H\) with a sufficiently small fixed fraction of the clearance \(\ell\), or if their two abscissas differ by more than \(a_*/10\). These deletions have negligible conditional crossing cost. To see this, suppose an endpoint of such an edge reaches an electrode by an actual path, allowing the initial test step. Paths meeting a flagged cell, protection ball, or fiber-endpoint ball while centrally located already have negligible total cost. For every other path, trace it inward from the electrode after its last entry into the broader band \(2\beta\le u\le1-2\beta\). It crosses a comparison \(u\)-fiber on its main internal piece, giving an anchor in \(H\) with nearly equal abscissas. It cannot subsequently leave \(H\), since its central vertices are candidates and the wall has clearance from them. Each later crossing of either chart’s comparison fiber is again on its main piece. If \(v\) drifted by more than a few bin widths from \(u\), it would cross a \(v\)-fiber in a nonmatching \(u\)-bin, a contradiction. The endpoint therefore has neither proposed defect. Vertices without a possible electrode connection can be discarded at no crossing cost. Smaller nested bands and clearance constants accommodate complete edge segments and physical-grid projection.

Now keep the surviving uncertain edges in \(4\beta\le u\le1-4\beta\) that also belong to the trimmed test graph. A nonshared original piece can attach to these only on a designated side of the test quad: clearance excludes its free horizontal sides, and the abscissa buffers exclude an attachment from a noncentral piece. No nonshared piece, even with history wires, connects the two designated sides. Such a traversal must pass through the middle shared strip by chart matching, while all identifications lie in the two separate groups near \(u=0,1\). After matching physical vertices, the induced original partition is thus dominated by the test law’s separate wiring. Every modified original actual crossing contains a shared crossing between the test sides, by taking its traversal from the first exterior group to the second. Domain Markov and partition monotonicity give the desired test upper bound.

For completeness, the construction has the following quantifiers. If convergence in probability failed with fixed error and probability, choose first \(a_*\) small, then \(\beta\) and fiber precision smaller. Choose obstacle accuracies tending to zero, with, at each stage, a deep-excursion cutoff, boundary protection radius \(h_b\), block sizes and count cutoffs, endpoint protections \(e\), the intermediate radius \(R_1\), and finally the flag scale \(\ell\). The conditional protection costs and the averaged flag cost (145) are made small in that order. For each fixed stage choose the lattice mesh small enough for every geometric margin and each of the finitely many Jordan test limits. Select failing histories outside all exceptional sets, and pass to a common kernel subsequence. The fiber argument and additional deletions along this subsequence have uniform conditional error by Lemma 39, and contradict the failure. All six-arm summations took place under the original law before this deterministic selection of histories. This proves (143). ◻

Proof. Insert on the original free side a small separate primal interval whose endpoints in the initial coordinate approximate \(x,x+s\). Use the four-open-change experiment. Until the specified stop its cross-ratio is \[\chi_t=\frac{g_t(x+s)-g_t(x)}{g_t(x+s)-W_t}.\] The real Loewner solutions remain separated from the driver and from one another. At each fixed \(s>0\) the four marks, including \(\infty\), therefore stay nondegenerate. A physical point corresponding to \(iM\) for sufficiently large fixed \(M\) has a fixed unaffected disk; it supplies the basepoint in Theorem 37. The trajectory cannot end at either inserted mark within this localization.

Let \(Z_t\) be the conditional pairing probability. It is an exact bounded martingale. At any of the allowed stopping rules, Theorem 37 gives \(Z_t-f(\chi_t)\to0\) in probability, and hence in mean, for fixed \(s\). Divide the martingale identity by \(f(s/(x+s))\) only at this fixed \(s\). Lemma 36 and analytic ODE estimates near \(x\) give, uniformly in the stopped range, \[ \frac{f(\chi_t)}{f(s/(x+s))} =\left(\frac{xg_t'(x)}{g_t(x)-W_t}\right)^h+o_s(1), \tag{149}\] after the mesh limit. The expression on the right is bounded above and away from zero independently of small \(s\): \(g_t(x)-W_t\) is bounded below, \(g_t(x)\) is bounded above, and \[g_t'(x)=\exp\left(-\int_0^t\frac{2\,du}{(g_u(x)-W_u)^2}\right)\] has the same bounds.

We next compare the laws, after replacing the normalized conditional probabilities by the bounded expressions in (149). Separate deterministic four-change closure adds just the small wire interval to the Dobrushin law. Couple monotonically and explore clusters from that interval in the stronger law. The probability that its actual cluster reaches a fixed small outer neighborhood tends to zero as the interval shrinks. Inside the intervening annulus there are no wire attachments away from the rims, since the old primal wire is a positive distance away. Filling to the full-lattice wired-rim law dominates the arm, which is blocked by the buffered RSW circuits. Outside this neighborhood the configurations can consequently be coupled identically. The open-four and deterministic-four laws also have vanishing total variation difference: their density depends only on the pairing, and the exceptional pairing has probability \(g(s/(x+s))\to0\). The stopped strands stay outside a fixed neighborhood of the insertion point, by the separated real and nearby complex solutions of (146). The drawing and caps were chosen identical off that neighborhood.

Thus bounded stopped-history functions of \(M\) transfer with vanishing error, first as the mesh tends to zero and then as \(s\downarrow0\). We never divide a coupling error by \(f(s/(x+s))\). The same stopping rules are applied in the coupled experiments, so the conclusion includes mesh-dependent rules and yields (148). ◻

Proof. We first work inside the preceding localization. Set \[X_t(x)=g_t(x),\qquad A_t(x)=x^h g_t'(x)^h, \qquad M_t(x)=A_t(x)(X_t(x)-W_t)^{-h}.\] The processes \(X,A\) are uniformly bounded and uniformly Lipschitz in time there, with \(A\) bounded away from zero. Take \(2R+1<x_1<x_2\). Their real solutions stay separated: their gap solves \[\partial_t(X_t(x_2)-X_t(x_1)) =-\frac{2(X_t(x_2)-X_t(x_1))} {(X_t(x_1)-W_t)(X_t(x_2)-W_t)}.\] It consequently has a positive lower bound on the stopped range.

Here is the two-test convexity argument in detail. At a stopping time \(\sigma\), freeze \(X,A\) and write \(F_j(w)=A_\sigma(x_j)(X_\sigma(x_j)-w)^{-h}\). Both derivatives are positive, and \[\frac{F_1'(w)}{F_2'(w)} =\frac{A_\sigma(x_1)}{A_\sigma(x_2)} \left(\frac{X_\sigma(x_2)-w}{X_\sigma(x_1)-w}\right)^{h+1}\] has derivative uniformly bounded below by a positive constant on \([-R,R]\). Thus for \(a_\sigma=F_1'(W_\sigma)/F_2'(W_\sigma)\), a bounded predictable quantity, integration of the derivative difference on either side of \(W_\sigma\) gives \[ F_1(w)-F_1(W_\sigma) -a_\sigma\bigl(F_2(w)-F_2(W_\sigma)\bigr) \ge c(w-W_\sigma)^2. \tag{151}\] The change of the frozen \(X,A\) between \(\sigma\) and \(\tau\) costs at most \(C(\tau-\sigma)\). Applying (148) to these two increments, with the predictable coefficient incorporated into the test, yields \[ \mathbb E[(W_\tau-W_\sigma)^2] \le C\mathbb E[\tau-\sigma]+o(1) \tag{152}\] for every sequence of allowed stopping rules. Boundedness and (152) give the stopping-time tightness criterion in the Skorokhod \(J_1\) topology. Since every path is continuous, all subsequential limits are continuous, and the convergence may be realized uniformly on compact stopped time intervals.

At inner localization levels chosen to be continuity levels of the hitting operation, ODE continuity and (148) make each \(M(x)\) a martingale in the limit. Such levels can increase to any outer localization. The identity \[W_t=X_t(x)-\left(A_t(x)/M_t(x)\right)^{1/h}\] and the positive bounds on \(A,M\) show that \(W\) is a continuous semimartingale. Write its decomposition as a local martingale plus a continuous finite-variation process \(B^{\mathrm{fv}}\), and let \([W]\) be its quadratic variation. With \(Z=X-W\), \[dX=2Z^{-1}\,dt,\qquad d\log A=-2hZ^{-2}\,dt.\] Itô’s formula gives the finite-variation part of \(dM/M\) as \[hZ^{-1}\,dB^{\mathrm{fv}}-4hZ^{-2}\,dt +\frac{h(h+1)}2Z^{-2}\,d[W].\] Since \(M\) is a martingale, multiplication by \(Z^2/h\) yields the identity of signed measures \[ (X-W)\,dB^{\mathrm{fv}}-4\,dt +\frac{1+h}{2}\,d[W]=0. \tag{153}\] Subtract the identities for \(x_1,x_2\). Their positive separation forces \(dB^{\mathrm{fv}}=0\), and then \(d[W]=8(1+h)^{-1}dt\). Lévy’s characterization identifies the localized law as Brownian with the rate in (150). The trigonometric identity follows from \(\arccos(-d/2)=\pi-\lambda\).

Finally, the probability of exiting a large outer driver bound before \(T\) is bounded in the limit by the Brownian probability of hitting inner bounds approaching it. These probabilities tend to zero as the bounds increase. Remove localization to obtain full compact-time driving convergence. Every subsequential law has been identified, so the convergence holds along the full mesh sequence. ◻

Proof. The rounded medial arcs divide tiles into primal and dual territories. A path within a territory shadows a path of its color on the physical lattice: within a tile it passes to the corresponding corner, and passage between opposite corners uses precisely the edge supplied by that switch. The shadowing has an error bounded by one tile. Actual paths stay inside their territories.

Take disjoint simple radial sublegs, perturbing both annular radii within buffers to regular values. Thin parallel crosscuts on their two sides give ordered territorial crossings of opposite colors at each separator. Compress successive runs of equal color in the cyclic list. Each medial separator contributes a color transition, so at least \(m\) alternating crossings remain. Shadow them by lattice walks in the central crop. Near the wider rims keep territorial tails until they can be spliced to the walks; thus no positive mesh-width clearance of the thin territorial paths at the rims is needed. Opposite-color paths are disjoint. Same-color members of the alternating list lie in different strips separated by an opposite-color territorial path and cannot meet. Disjoint tails from the inner wider rim preserve order at first hits of the outer cropped rim; taking last passages to those hits gives the claimed actual arms. The same strip ordering, after ignoring the other color’s assignments, proves the last assertion. ◻

Proof. Bulk passage counts in a fixed finite family of buffered annuli are tight by Section 2. We first show that a macroscopic displacement confined to a shrinking neighborhood of the original boundary has vanishing probability. In fixed plane Schoenflies coordinates for \(D\), such a displacement crosses longitudinally one of finitely many boundary subarcs of positive length, after shortening the subpassage to avoid the two change points. On each subarc the boundary type is homogeneous. Its nonwired-color bank is an actual path using that color’s edges.

Surround the subarc by a thin Jordan ribbon and place the permitted path in a still thinner band with strict margins. Fill to the full lattice of the bank color and wire the ribbon’s outer boundary. This dominates the actual crossing: there is no nearby designated wire of this color, and all exterior attachments to the shared graph occur on its ordinary sides. Longitudinal ribbon crossings become unlikely as the ribbon narrows. One concrete proof chooses any number \(N\) of ordered intermediate positions on the subarc, each with its own disjoint fixed neighborhood. At each position two across-ribbon sections can be confined to arbitrarily small physical balls. A longitudinal crossing must then traverse a Euclidean annulus of arbitrarily large logarithmic aspect ratio inside that subribbon; its extremal length tends to infinity. Choose transverse blocker quads with bounded transverse extremal distance and with outer margins beyond the permitted band. The strong quad RSW theorem (Duminil-Copin et al. 2021, Theorem 1.2) gives an actual opposite-color transverse blocker with conditional probability at least \(c>0\), under arbitrary exterior partitions. Approximate the fixed Jordan corridors by polygons with strict margins before taking the mesh limit. Successive conditioning over the disjoint quads bounds the crossing probability by \((1-c)^N\). Let \(N\) grow and then narrow the ribbon. Every geometric scale here was fixed before the mesh limit.

At a fixed displacement accuracy, outside these unlikely boundary travels, every excess collection of successive displacements forces many passages through a finite annulus cover of the compact part away from the ribbon. The passage estimates therefore bound the number of successive displacements at each accuracy. Subdivide curves at these displacement times and apply the usual Arzelà–Ascoli argument modulo reparametrization. This proves tightness and also rules out a nonconstant interval of a limit confined to the original boundary.

Three genuine visits to an interior point, with positive intervening excursions, produce six disjoint medial legs to some fixed positive outer radius. Lemma 44 turns them into six alternating actual arms across a cropped annulus. Cover the possible contact points by a deterministic grid of mesh \(\ell\). The bulk six-arm estimate and the \(O(\ell^{-2})\) grid count give a bound \(O(\ell^{c_0})\). First take the lattice mesh limit, then \(\ell\downarrow0\), and finally a countable union over interior compact sets and rational excursion radii. This excludes precisely three visits separated by nonconstant excursions; constant waiting intervals are irrelevant to the curve topology. ◻

Proof. An avoidable component cannot access both designated primal and dual prime boundary arcs strictly between the rims. A path joining such accesses contains a crosscut separating the tip from the target. The future simple strand traverses one component between rims; choose a color whose designated boundary is absent there. Its bank supplies an actual arm of unqueried edges in a cropped annulus. A known-open edge or wire jump of that color would give a short physical access to its designated boundary there, contrary to the choice of component.

To bound all such components at once, keep their random edges of this color within a cropped grid annulus. Unqueried tiles are uncut; an adjacency of incident edges, or attachment to a known edge or wire vertex, has a corresponding \(O(\delta_n)\) path in \(U_n\) until it reaches a designated wire. Thus away from the cropped rims the induced partition contains only ordinary physical-vertex attachments. Delete known edges of the opposite state, fill the rest to the full lattice, and wire both rims. FK monotonicity dominates all the relevant actual arms simultaneously by one full-lattice arm event. Buffered dual circuits give its vanishing bound. Sum over the two possible missing colors. The argument uses the conditional FK law at actual revealing stops; an optional last piece of length \(O(\delta_n)\) is absorbed by the buffers. ◻

Proof. At the first entrance the previously untouched cap gives a route from the tip to the target inside a small physical ball. Stop just before entrance, or at the entering piece and use its bank with a one-step buffer. Outside this inner ball every relevant annular component is avoidable. Apply Lemma 46. The diameters of the initial conformal caps shrink uniformly because the initial inverse disk maps converge uniformly up to the boundary. ◻

Proof. The double-point and boundary-hit dimension theorems of Miller–Wu (Miller and Wu 2017, Theorem 1.1 and Corollary 1.7) give, respectively, \[1+\frac\kappa8-\frac6\kappa \quad\hbox{and}\quad 2-\frac8\kappa.\] Their difference is \((\kappa-4)^2/(8\kappa)>0\), so the trace has an interior double point almost surely. Let \(E_T\) mean that both visits occur strictly before \(T\). Scaling makes \(\mathbb P(E_T)\) independent of \(T>0\), while \(\bigcup_{m\ge1}E_m\) has probability one. Thus \(\mathbb P(E_T)=1\) for every \(T>0\). At a deterministic time \(a\), the domain Markov property restarts ordinary SLE with additive capacity clock. An interior double point of the restarted trace maps under \(g_a^{-1}\) to an interior point of the original remaining domain. Apply this observation to all rational open intervals. ◻

Proof. Use the continuous Brownian SLE trace (Rohde and Schramm 2005); here \(4<\kappa\le6\). Suppose that the claimed excursion occurs. By a countable choice, there is a small closed ball \(J\) with a larger buffered neighborhood disjoint from the SLE trace and original boundary, approached deeply inside a swallowed component at finite limiting capacity. Fix bounds for this capacity and the driver, and a high basepoint corresponding to \(iM\) in the initial half-plane coordinate. Take the first discrete approach to an intermediate buffer around \(J\), before the alleged hit. Along a subsequence of these approach times the prefix keeps a fixed spare distance from \(J\), whereas its kernel viewed from the basepoint excludes \(J\). This follows from driving convergence and equality of the accessible Loewner domains; the whole buffered ball, being disjoint from the trace, lies in one excluded component.

Nevertheless \(J\) is still internal in every discrete simple slit disk. Normalize its inverse disk map \(\Phi_n\) at the basepoint and write \(z_n=\Phi_n^{-1}(\operatorname{center}J)\). Kernel convergence forces \(|z_n|\to1\). Put \(\epsilon_n=1-|z_n|\), and take circular disk crosscuts centered at \(z_n/|z_n|\), with radii between \(2\epsilon_n\) and \(\sqrt{\epsilon_n}\). If \(L_n(t)\) is their image length, Cauchy–Schwarz and the area formula give \[ \int_{2\epsilon_n}^{\sqrt{\epsilon_n}} L_n(t)^2\,\frac{dt}{t} \le2\pi\operatorname{area}(U_n). \tag{154}\] The right side is uniformly bounded and the logarithmic integration range diverges, so one image crosscut has diameter tending to zero. Its disk cap contains \(z_n\), and its endpoints access the slit boundary. Since that boundary stays a fixed distance from \(J\), the crosscut separates all of \(J\) from the basepoint for sufficiently large \(n\).

The target is on the basepoint side. Indeed the driver and capacity bounds give a fixed route high above the half-plane hulls from the basepoint to the target, with positive physical separation from the prefix. A small gate touching the prefix misses this route. A gate with both endpoints on the original outer wall cannot separate two fixed interior balls when its diameter tends to zero, by uniform Jordan convergence. Adding a boundary-to-interior simple slit disjoint from that gate does not change its separation. Gates with an endpoint at the initial slit attachment are included among gates touching the prefix.

We must use a stopping rule, not a gate selected from the future excursion. Fix a small \(\ell>0\) and stop at the first discrete step at which there exists a crosscut of diameter at most \(\ell\) separating \(J\) from both basepoint and target, while the prefix still keeps the chosen spare distance from \(J\) and the basepoint is safe. The preceding construction shows that this stop occurs before the offending buffered hit for fine meshes. Choices can be fixed measurably from each finite history. At first occurrence a witnessing gate has an endpoint on the latest microscopic step, including its attachment point. Otherwise it was a crosscut of the previous slit domain with the same separation: adding a disjoint simple slit cannot change that separation. This contradicts first occurrence. The tip therefore reaches the gate and its target-side bank within a ball of radius \(\ell+O(\delta_n)\).

Choose a fixed outer radius smaller than the spare distance to \(J\). In the annulus from \(4\ell\) to this radius about the new gate endpoint, components on the trapped side are avoidable: a tip-to-target route crosses the gate inside the inner ball and stays on its other side. Any later hit of \(J\) requires a radial traversal on the trapped side, since every return through the gate is inside the inner rim. Lemma 46, applied at this first-gate stop, makes its conditional probability vanish as \(\ell\downarrow0\) after the mesh limit. There is no sum over possible gate locations. This contradicts the alleged excursion. ◻

Proof. By Proposition 43 and Lemma 45, take a joint subsequence realized with compact driving convergence and uniform physical curve convergence up to reparametrization. Write \(Y(t)\) for the physical Brownian SLE trace, and \(\gamma(s)\) for the full limiting physical curve in an arbitrary forward parametrization. Before its first target visit, let \(c(s)\) be the half-plane capacity of the hull of its initial segment. Kernel convergence makes \(c\) continuous and nondecreasing and identifies it as the limit of the corresponding discrete capacities. These mapped segments are bounded locally away from the target. As the target is approached, \(c(s)\to\infty\): a driver bounded on every finite capacity interval has its finite-time hulls in bounded portions of the initial half-plane.

The domain accessible toward the target after time \(s\) is the Loewner remaining domain of \(Y\) at time \(c(s)\), by kernel convergence and the Loewner ODE. On every interval where \(c\) changes, the newly added trace segments of \(\gamma\) and \(Y\) have a point in common: a boundary point of the smaller shared remaining domain inside the older domain belongs to both increments. Continuity, with shrinking such intervals, implies \[\gamma(s)=Y(c(s))\] at every time outside the interiors of plateaus of \(c\), including their active endpoints. A nonconstant plateau is therefore an extra excursion based at \(Y(t)\), with \(t\) its constant capacity, contained in the old closed hull or original boundary.

Lemma 49 excludes motion inside an open swallowed component. Hence an extra plateau can move only on \(Y([0,t])\) or the original boundary. By Lemma 45, it has no varying subinterval entirely on that boundary. If the plateau were nonconstant, choose a compact varying subinterval in its interior whose image avoids both the boundary and its basepoint. Every image point has a unique old SLE time: two old visits together with this genuine extra visit would give three interior visits separated by nonconstant excursions, forbidden by Lemma 45. This old-time inverse is continuous. Indeed any sequence of old times has a convergent subsequence in \([0,t]\), and trace continuity and uniqueness identify its limit. Composing with the extra subcurve yields a continuous nonconstant real function, whose image contains a nonempty capacity interval. Lemma 48 provides a point in that interval with two old visits, a contradiction. Thus every plateau is constant. The same reasoning at capacity zero excludes an initial extra motion.

Corollary 47 excludes further motion after the first limit target visit. The capacity change of time therefore identifies the entire ordered curve with \(Y\), up to the permitted increasing reparametrizations and constant waiting intervals. Finally Lemma 35 transfers this conclusion from the rounded simple drawing to the specified medial strand at distance \(O(\delta_n)\), including repeated midpoint visits with fixed pairings. All subsequential limits agree, proving the claimed full-sequence convergence. ◻

Proof. At a cut, every matching connection is either the consumed switch pair or a local cap on one of its sides. Consequently the factors \(q^{\ell/2}\) for unstarted loops, and the factor for eventually closing the current strand, have no remaining dependence involving both sides of the cut. Their product therefore factors over the components. A two-change component has one strand between its ports; closing these ports outside gives its marginal cap law. The deterministic triangles implement exactly those identifications. No random switch is hidden in such a triangle.

Until a switch into a homogeneous pocket, every traversed pair belongs to the loop through the originally opened cap. At a pocket switch that loop remains on the other side of the actual open diagonal. The new cap is on a loop not yet traced in the retained pocket. This also proves the last assertion. For finitely many targets keep the cut-off components, including their boundary states, and process active branches in a fixed priority order. The factorization gives the claimed conditional independence before taking any limit. ◻

Proof. We explain the reduction to a simple square-lattice quad, so that the up-to-boundary crossing theorem applies with its exact hypotheses. For the color \(X\), retain the edge-connected component around the basepoint of square faces whose centers have \(\ell^\infty\) distance more than \(k\delta\) from the planar complement; a fixed \(k=20\) suffices. The failing faces are edge-connected, as one sees by rasterizing a thickened path in the connected complement, including slit banks. Other components of passing faces are assigned to the exterior. Both face families are then edge-connected. Alternating interior and exterior faces at a self-touch vertex would separate one of these families, so the retained polygon has a simple boundary. It lies in unqueried territory and eventually contains every compact chart core.

An incident failing face on its perimeter can be followed, through \(O(k)\) adjacent failing faces, to the first wall contact. Thus every perimeter point has a short exterior access to the actual wall. The chart modulus implies that the perimeter approaches the unit circle uniformly and that its separated accesses have the correct cyclic order. The winding about the basepoint fixes that order globally.

Approximate the two longitudinal sides of the desired corridor by simple \(X\)-lattice crosscuts inside this polygon. First approximate their compact interior portions with positive clearance; then append terminal paths in disjoint chart neighborhoods, stopping at the first perimeter contact. Grid shadowing and loop erasure preserve the chart margins, since each short connected error is chart-small. Together with the appropriate perimeter intervals these crosscuts bound a simple lattice quad. The transverse margins and angular order bound its extremal distance by a constant depending only on the buffered corridor.

This last bound can be made directly in discrete extremal length. Thicken vertex weights over balls of a few mesh steps, with density scaled by \(\delta^{-1}\). Each member of a fixed transverse family of longitudinal chart paths shadows an edge path in the quad, cropped to the two terminal sides. Grouping successive connected pieces of displacement \(\delta\) charges only boundedly many nearby vertices. The shadowed length is therefore bounded by a constant times its density integral; the squared area integral is bounded by a constant times the sum of squared vertex weights. The fixed chart family gives the asserted extremal-distance bound.

Theorem 1.2 of Duminil-Copin et al. (2021) now gives a positive crossing probability for the simple quad at \(p_q\), for \(1\le q<4\), with arbitrary boundary partition. Equivalently, use the free law as the lower comparison. Do not query edges running along its terminal perimeter sides. From a reached vertex, an outward failing-face chain gives \(O(k)\) available \(X\)-edges toward the intended wall. The extension uses only those unqueried terminal-side edges and edges outside the quad. Finite energy gives a uniform positive conditional cost. Stop at the first wall contact; if an opposite-color wall blocks the next step, the current vertex is already the \(X\)-corner of its triangle. This proves the crossing assertion, including its endpoint convention. Circuits and interior connections with strict margins follow by ordinary box-crossing and gluing.

For the collar statement, find the same-color screen nearest the unknown-facing side of a deterministic buffered subcollar. Paths may be restricted to the strip between the prescribed wall intervals, pruned at terminal returns, and augmented by ordered spokes to common endpoints. Minimizing the region on the unknown side selects a simple outermost crossing. Any competing path entering strictly nearer that side has an open excursion that can replace part of the selected path and decrease this region. Hence the selection tests only on and on that side of the screen. Its other side retains an FK law with a boundary partition.

Actual paths of the screen’s color must attach to it in order to cross. An opposite-color path cannot pass it except through the bounded through-wire blocks; the added segment in a wall triangle creates no other bypass. Splitting or merging those blocks changes a weight by a bounded factor. The primal and dual screening searches used in the collar proof of Section 2 thus apply without change to arbitrary-event densities. All chart margins are fixed before the mesh limit, as required. ◻

Proof. Such cycles provide edge-disjoint vertical passages in a fixed-height middle horizontal band. By bounded degree and Menger’s theorem, their number is bounded by a fixed multiple of the maximal number of vertex-disjoint passages. For primal paths one may add random switches on the initial wired boundary edges, which does not change the interior marginal under consideration. For dual paths use the rectangle of interior faces. Thus it suffices to work in an ordinary fixed-aspect color-lattice rectangle, with its induced positive FK law.

Prune bottom-to-top paths so they do not return to their terminal rows. Select the leftmost open path, then the leftmost path to its right that is vertex-disjoint from it, and repeat. To justify the information claim, attach noncrossing spokes just below and above the terminal rows to common endpoints, pad the rectangle sides, and minimize left area. Another eligible path entering strictly left gives an excursion with distinct endpoints on the augmented path and permits replacement in either traversal direction, decreasing left area. End excursions through a spoke use only terminal contacts. The selected path is therefore determined by its own edges and the edges on its left. If \(m\) vertex-disjoint passages exist, at least \(m-1\) remain to its right, and the argument iterates.

After a selection \(p\), there is a uniformly positive conditional probability of blocking another disjoint passage. If \(p\) comes within ten mesh steps of the right wall on some horizontal half-row, force an opposite-color connection along that row from outside the right wall to first adjacency to \(p\); its finite-energy cost is bounded. If \(p\) already blocks, no forcing is necessary. Otherwise follow \(p\) on its right through opposite-lattice face centers, joining consecutive ones inside the right wedges at its vertices. Erase loops and trim terminal returns to obtain a simple path \(p_*\) between the extreme interior half-rows. Those rows are single intervals on the right side because \(p\) was pruned. With them and the rightmost interior column, \(p_*\) bounds a simple opposite-lattice quad of fresh edges.

Horizontal rows give a bounded-extremal-distance family across this quad: there are order-height disjoint rows, each of length at most order-width. Lemma 52, or directly Duminil-Copin et al. (2021, Theorem 1.2), supplies an opposite-open crossing with uniformly positive probability. Extend it one step beyond the right boundary at finite-energy cost. At its other endpoint, append a segment in the incident square to a vertex of \(p\). This screen blocks all actual vertical paths on the right avoiding vertices of \(p\). Its edges are fresh under the leftmost-path conditioning. Consequently the successive selections have a geometric tail; blocker tests need not themselves be exposed when continuing the search. In particular, if \(N\) is their number, then for constants depending only on \(q\) and the fixed rectangle shape, \[ \mathbb P(N\ge m)\le(1-c)^{m-1},\qquad \sup_{\delta}\mathbb E[e^{aN}]<\infty \quad\text{for }0<a<-\log(1-c). \tag{158}\] This proves the first assertion, and also gives the corresponding bound for a strictly bulk rectangle.

A generation begins in a homogeneous \(C\)-pocket, whose present wall is its reference wall. Read the \(C\)-arc from \(O\) to \(W\) in counterclockwise order for primal and clockwise order for dual. It consists of a prefix of the remaining reference interval followed by a suffix of actual open edges revealed during this generation. A triangle step shortens this path; a fresh diagonal extends it or returns to an earlier vertex. At a same-color switch retaining part of the prefix, the next gap is precisely its first retained reference edge. The description is thus preserved, even when the prefix becomes empty. These switches take place on the reference wall.

Start a new generation at the first homogeneous pocket whose retained arc has no reference edge; every opposite-color switch has this property. Its wall consists entirely of actual diagonals discovered since the previous start. Consumed edges are not reintroduced by known triangles. The successive new reference cycles are consequently edge-disjoint, separately for each color. They surround the basepoint ball present up to the fixed horizon. If a wall walk has repeated vertices, its winding extracts a surrounding cycle. The first assertion bounds the number of generations. For finitely many branches sum the bounds, allowing an ancestral generation to be counted along each of its descendants. ◻

Proof. We supply the modifications needed to apply the moving-domain comparison of Section 11.

Proof. Use two ordered test points \(x\) in Proposition 54. The relative-convexity inequality (151) and the inversion argument of Proposition 43 identify the stopped chordal driver and its bracket. They also give stopped second-moment estimates. On a sufficiently short segment started with \(\theta\in[e,2\pi-e]\), the chordal and radial clock rates and coordinate changes are bounded above and below. Thus crossing a fixed interior angular interval in vanishing radial time has probability tending to zero, uniformly at causal starting stops. A macroscopic angular increment started near an endpoint must cross such an interior interval. Aldous’ criterion and the vanishing within-update oscillations give continuous tightness of \(\theta\); in particular its numbers of fixed-size alternating swings are tight.

After the tightness of \(W,O\) proved below, localize on rational start ranges and inner continuity levels to pass the re-centered identities to a joint subsequential limit, retaining tests of the past. The ordinary chordal-to-radial target change, before any loss of the force access, gives (160). This is precisely the stopped coordinate change in Sheffield (2009, Proposition 3.12); no continuation through collisions is inferred from that proposition. ◻

Proof. Work in the disk chart at the starting stop. The radial Loewner ODE confines the updates during time \(u\) to a strip of depth \(C\sqrt u\) from its boundary. A driver advance of \(r\) forces a positive angular traversal through a buffered part of the corridor from angle \(W_0+r/5\) to \(W_0+r/2\).

Here is the boundary-coordinate justification of this implication. Stop at the first \(|W-W_0|=r\), so the lift cannot wrap around. If the advancing growth never passes, for example, \(W_0+2r/3\) in the starting chart, a neighborhood of radius comparable to \(r\) at \(W_0+5r/6\) remains untouched. Its prepoint image stays \(O(\sqrt u)\) from its original angle. First check the ODE at depth a large constant times \(\sqrt u\). Near the untouched arc the map reflects analytically; radial monotonicity bounds the log-radius harmonic part by the absolute original log radius. In the stated neighborhood this is \(O(r)\), so the interior gradient estimate bounds the angular derivative. The boundary estimate follows by reflection. The untouched prime access is distinct from the tip, and the small boundary relocations also miss it. The driver therefore cannot pass its lifted image, contrary to the stipulated advance. This proves the required physical traversal in the starting chart. All within-step and strand-surrogate errors vanish for fixed \(r,u\).

Place order \(r/\sqrt u\) disjoint buffered bars across that angular corridor. Each has width comparable to \(\sqrt u\) and extends to a larger fixed multiple of the allowed depth. By Lemma 52, each has a uniformly positive conditional chance of an actual dual-open path from the dual corner of a triangle on the primal wall to its inner end. Complete this path across that triangle to the midpoint of its primal base.

One successful bar prevents the required advance before a dual reset. For completeness, follow the actual traversed switch pairs and add a microscopic connector at each primal restart. At that restart the connector passes from the encountered pair through the neighborhood of the open primal diagonal and hitting vertex to the new cap’s earlier port, approaching along its near base endpoint. It stays in the retained sectors and avoids all actual dual paths. A positive angular passage within the shallow strip would have to cross the completed bar positively. It cannot cross its actual dual part. Traversal of the cap pair in the fixed triangle crosses the added segment only negatively, because primal is on the left. If this triangle itself is selected for the restart, the connector reaches the earlier port without crossing the segment. Every other reset connector stays beside its own primal vertices and avoids that fixed segment. An excluded triangle is never used again. Thus the positive crossing is impossible. This argument is in the prime closure of the stop domain, where all short accessed moves are uniformly chart-small.

While \(\theta<r/16\) there is no dual reset. Multiplying the conditional failure bounds for the disjoint bars proves (161). Also, a negative motion of order \(r\) is impossible in this zone: \(O\) is nondecreasing, including its primal resets, and \(W=O-\theta\). The reflected argument proves the other endpoint statement. ◻

Proof. For a smooth function constant near both endpoints, the localized interior identities glue to the martingale problem for \(\mathcal A\). To admit a smooth function with \(f'(0)=f'(2\pi)=0\), approximate it by such cutoffs with uniformly bounded second derivatives. Its cotangent term is bounded near the ends, since \(|f'(\theta)|=O(\min(\theta,2\pi-\theta))\). The endpoint time is zero, so dominated convergence gives the limiting martingale identities and, by applying the same argument to \(f^2\), their brackets.

In particular \(X=\cos(\theta/2)\) solves \[ dX_t=\frac{\sqrt\kappa}{2}\sqrt{1-X_t^2}\,dB_t -\frac{3\kappa-8}{8}X_t\,dt, \qquad -1\le X_t\le1. \tag{165}\] The diffusion coefficient is one-half Hölder and the drift is Lipschitz; the one-dimensional square-root uniqueness criterion gives pathwise uniqueness. Hence this equation fixes the instantaneous reflected law of \(\theta\); a sticky alternative is also excluded by zero endpoint time.

Near either endpoint the scaled distance \(R\) has Bessel dimension \[ \mathfrak d=3-\frac8\kappa\in(1,2). \tag{166}\] Its drift differs from \((\mathfrak d-1)/(2R)\) by a bounded smooth term of order \(R\). Stopped Girsanov, equivalently in the squared coordinate where the excess drift divided by the diffusion coefficient extends boundedly, transfers the usual Bessel path properties. Thus \(\theta\) is locally Hölder of every exponent below \(1/2\), and each endpoint set has Hausdorff dimension at most \[ d_0=1-\mathfrak d/2<1/2. \tag{167}\] The needed Bessel Hölder statement is, for example, Hutzenthaler et al. (2014, Theorem 1.2 and Corollary 2.14). The zero-set statement follows from the inverse-local-time description in Revuz and Yor (1999, Exercise XI.1.25) and the stable-range dimension in Bertoin (1996, Theorem III.15). Only a bounded stopped perturbation at one endpoint is being used here.

Write the possible remaining term as the signed measure \[d\nu=dO-\cot(\theta/2)\,dt.\] It is supported on the two endpoint sets and has the sign specified after (162). We give the time-block argument explicitly, so that a Hausdorff-dimension bound, rather than an unstated Minkowski bound, suffices. Choose \(\eta>0\) so small that \[p=\tfrac12-\eta>d_0, \qquad u_m=2^{-m},\qquad r_m=u_m^p.\] By the Hölder property, every sufficiently fine dyadic block of length \(u_m\) meeting an endpoint set has its end-gap at most \(r_m/32\) throughout. Indeed choose a Hölder exponent strictly between \(p\) and \(1/2\). Apply Lemma 56 with harmless fixed buffer changes to the finitely many blocks at level \(m\), taking the mesh limit first. The probability that some level-\(m\) block stays in the stated endpoint zone and has one-signed \(O\)-variation exceeding \(Cr_m\) is at most \[ C(1+T/u_m)\exp(-c u_m^{-\eta}). \tag{168}\] This is summable in \(m\). The estimate for \(O\) follows from that for \(W\) because their difference is the end-gap, of size at most \(r_m/32\) on such a block. Borel–Cantelli gives, almost surely, the variation bound \(Cu_m^p\) on every sufficiently fine endpoint-meeting block, separately for the two endpoints and on each fixed horizon.

On an endpoint neighborhood, \(dO\) is one-signed, and so is the cotangent time measure. Its restriction to the endpoint set is zero, whereas the restriction of \(dO\) there is precisely \(d\nu\). Consequently \(|\nu|(I)\le\operatorname{Var}_I(O)\) for every small interval in that neighborhood. Cover the endpoint set by intervals of variable lengths whose sum of \(p\)th powers is arbitrarily small, using (167). Replace each by a bounded number of comparable dyadic intervals that meet the set. The preceding variation bound yields arbitrarily small \(|\nu|\)-mass. Thus \(\nu=0\), proving (164).

Finally, a Bessel process of dimension greater than one is a semimartingale with integrable inverse-distance drift and no local-time term in its ordinary coordinate. Undoing (165) in the endpoint neighborhoods, and using the interior identities elsewhere, gives \(d\theta=-\sqrt\kappa\,dB+((\kappa-4)/2)\cot(\theta/2)dt\). Together with (164) this is (160) globally. There is therefore no unspecified boundary pushing rate. ◻

Proof. Use the sum of currently accrued target clocks as a common parameter \(v\), freezing each target at its prescribed horizon. Every clock \(t_i(v)\) is nondecreasing and \(1\)-Lipschitz. Several run together while their targets share an active component; separate components are processed in serial priority order. A flat clock has a constant state. All tightness bounds and the interior identities hold between adapted common-clock stops, including tests of the whole past. Work in other components during an interrupted segment is conditionally independent of its unqueried switches by Lemma 51; neither the schedule nor the stop gives advance information.

Extract limits jointly with the clocks. The preceding endpoint arguments can be applied in each marginal’s own radial clock, using tightness and inverse-clock extraction across flat stretches. Thus the martingale part \(M_i\) of \(W_i\) is a continuous local martingale relative to the common past, with \[ d\langle M_i\rangle=\kappa\,dt_i. \tag{169}\] The Loewner equations determine its target kernels and normalized maps.

While targets \(z_i,z_j\) remain in one component, put \(\xi=\Phi_i(z_j)\). Radial Loewner evolution and the conformal-radius change give \[ dt_j=P^2\,dt_i,\qquad d\log|\xi|=P\,dt_i,\qquad P=\frac{1-|\xi|^2}{|e^{iW_i}-\xi|^2}. \tag{170}\] Their marked boundary angles correspond by the normalized disk Möbius map. On a compact interval with \(|\xi|<1\) these formulas are regular and pass to the limit. The targets are still in the same kernel exactly while they retain an interior connecting path: kernel convergence preserves paths with positive clearance. Thus they have the same growth up to kernel separation.

Suppose now their limiting kernels have separated but their discrete components have not yet split. Along such a portion \(|\xi|\to1\) by monotonicity. On a subinterval where both limiting angle gaps stay in the interior, the two marked prepoints remain separated in both charts. The disk Möbius formula forces one of \(e^{iW_i},e^{iO_i}\) to be within \(C(1-|\xi|)\) of \(\xi\) in the prelimit. Since they are separated in the \(i\)-chart, the choice cannot change on a connected subinterval. If it is the tip, \(P\to\infty\); if it is the forcepoint, \(P\to0\). Equation (170) then forces one limiting clock to be flat. This remains true along subsequences of exact discrete split times. After an exact split, serial growth gives disjoint time interiors for the two clocks. There are only boundedly many changes of component choice for a finite target list and finite horizons. Endpoint angle sets have zero mass for the corresponding own clock by Proposition 57. The covariance inequality for continuous local martingales and (169) therefore show that \[ d\langle M_i,M_j\rangle=0 \quad\hbox{after their kernel separation}. \tag{171}\] Frozen targets are omitted from this comparison.

We spell out why these characteristics determine independent continuations, without asserting that the clocks are independent. In each still-accessible active group choose its first indexed target as representative. Until a separation or freeze, the representative’s motion determines all other states and clocks by (170). Separation and freeze times are intrinsic stopping stages: mutual accessibility is read from the radial ODE up to first approach to unit modulus. While two other members remain regular with the representative, they cannot lose their shared access before either is lost from that representative. A nonrepresentative’s horizon is likewise read by its clock correspondence. At a multiple separation, take the continuous state limits in all own clocks; the subgroups are exactly those retaining internal mutual paths in their kernels. Serial speeds of unrelated groups have no effect on these stages.

For each target accumulate its martingale coordinate only over the suffix during which it is an active representative, keeping it flat otherwise. More precisely, with the predictable choice at the stage endpoints, set \[N_i(v)=\int_0^v \mathbf 1_{\{i\text{ is an active representative at }s\}}\,dM_i(s).\] Once it becomes representative it remains representative of its descendant group until it freezes. These representative martingales \(N_i\) are pairwise orthogonal by (171). Their terminal brackets can be finite, so make the completion explicit. Let \(T_i\) be the prescribed horizons and take the deterministic common terminal parameter \(V=\sum_iT_i\), extending all stopped states constantly if necessary. Divide each \(N_i\) by \(\sqrt\kappa\), and enlarge the common filtration after \(V\) with independent Brownian motions \(\widehat B_i\), independent also of the entire original terminal sigma-field. Set \[\widehat N_i(V+s)=N_i(V)/\sqrt\kappa+\widehat B_i(s),\qquad s\ge0,\] and use \(\widehat N_i(v)=N_i(v)/\sqrt\kappa\) for \(v\le V\). The extended processes are continuous local martingales in this enlarged common filtration, remain pairwise orthogonal, and have infinite terminal brackets. Knight’s orthogonal-martingale time-change theorem now gives independent standard Brownian motions in their respective bracket clocks (Revuz and Yor 1999, Theorem V.1.9); see also Meyer (1971, Theorem 2). Restricting to each original terminal bracket supplies precisely its original representative martingale. The auxiliary continuation changes neither the past filtration nor any stage of the exploration.

For the conditional assertion, let \(\mathcal G_v\) be the common causal filtration and let \(S\) be a separation or freeze stage. Restart the active representative martingales at \(S\), stop at the next common stage, and pad constantly to a deterministic horizon. After localizing to bounded brackets, these restarted processes remain orthogonal martingales under \(\mathbb P(\,\cdot\mid E)\) for every \(E\in\mathcal G_S\) of positive probability, with the same bracket identities. Completing them as above and applying Knight’s theorem under this measure gives the same product Wiener law for every \(E\). Thus the continuation noises are independent of \(\mathcal G_S\). A continuing representative uses its original martingale increments, and a newly selected one starts its representative suffix. No conditioning on terminal bracket values or future separation data is involved.

The pathwise uniqueness of (165), followed by (164), determines the angles and maps through the next intrinsic stage. A newly assigned representative uses its assigned independent noise; a continuing one uses the continuation of the same Brownian solution. A change caused solely by a representative’s horizon gives a fresh continuation at that stopping stage. Induction over the finite target groups identifies the usual common-branch, independent-after-separation construction.

The resulting equations and angular convention are the target-invariant radial tree of Sheffield (2009, Propositions 3.13–3.14 and Section 4.2): a one-sided force weight \(\kappa-6\), with the side reversed in an enclosed component after a full turn. Our sign convention is \(\theta=O-W\); exchanging the sign of its Brownian motion translates to the opposite angular convention. Increasing the finite horizons proves the assertion at all finite clocks. ◻

Proof. Passage counts and compactness. Consider a vertical band of fixed positive width, with its vertical sides interior to the square and its height equal to that of the square. Each disjoint medial passage between its sides has an actual dual path beside it: consecutive adjacent dual vertices are identical or joined by an open dual diagonal, including at wall caps. Crop to a slightly narrower band. The medial separators continue beyond both cropped sides and are disjoint from dual-open edges, so they divide the band into strips. Each accompanying dual path lies in the strip on its separator’s dual side, with at most two assignments to a strip. Paths in different such strips cannot communicate in the actual dual graph there. Hence many passages force many distinct actual dual components crossing the cropped band.

The graph of internal dual faces is a full lattice rectangle with arbitrary induced boundary data; it has no dual crossing of the initial deterministic primal wall. The disjoint-passage proof of Lemma 53 bounds its crossing-component count. Even with all boundary vertices wired, an actual left-right crossing is bounded away from probability one by an up-to-boundary primal blocker. End that blocker just beyond the extreme dual rows by bounded finite-energy extensions to neighboring internal primal vertices. The rectangle’s dual boundary edges themselves are random, so the crossing theorem applies. The same argument holds for horizontal bands.

A finite family of trimmed bands at each fixed scale bounds both the number of loops with diameter above that scale and the number of their disjoint subarcs with a given positive displacement. These bounds yield compactness modulo reparametrization. To make the extraction explicit, subdivide each positive loop by successive first-displacement times into arcs of diameter at most \(2^{-j+O(1)}\), retaining marks from previous subdivisions. The number of marks at each fixed \(j\) is tight. Pass to subsequences on which their positions and cyclic orders converge. Assign nested parameter intervals with maximal length tending to zero and positive lengths at every fixed subdivision, splitting an interval further in two when necessary. The corresponding polygonal curves approximate uniformly. Diagonalize over diameter cutoffs tending to zero, retaining multiplicities of matched elements. This proves collection tightness.

Proof. The stopped chordal equivalence is only valid while its force access is available. To continue after the original target is cut off, use the branch that retains this access, as prescribed by the tree construction. A bounded finite chordal-capacity portion has bounded hull and driver. An open set of targets mapped to sufficiently large heights remains accessible above it, with tip and force angles strictly separated on that portion. The countable dense set supplies such a target. Its branch records the default continuation even if it has already separated from \(z\). Target invariance gives consistency when the target or the finite portion is enlarged. This realizes all finite portions and hence the complete strand; it does not extend Proposition 3.12 of Sheffield (2009) beyond its stopping condition.

All these curves are continuous. Over a finite trace time the remaining domains have locally connected boundaries, by the locally-connected-continuum boundary theorem applied to the continuous trace and the starting boundary. Their prime-access conformal images can thus be regarded as continuous physical curves.

For ordinary chordal SLE, Miller and Wu (2017, Theorem 1.1 and Corollary 1.7) give double points in the interior when \(4<\kappa<8\). Indeed their double-point dimension exceeds the boundary-hit dimension by \((\kappa-4)^2/(8\kappa)>0\). Both visits occur at finite times. Scaling makes the probability of such a double visit before time \(t\) independent of \(t>0\); since its union over finite times has probability one, this probability is one for every \(t>0\). The domain Markov property then gives an interior double visit in every rational open capacity interval. Conformal maps preserve an interior double visit. Every open traversal interval of the canonical loop contains an interior-time portion either of its full-turn excursion or of its finite default continuation. On a sufficiently short localized portion its law is ordinary chordal SLE. The preceding arbitrary-short-time statement applies before any positive localization time, proving the assertion. ◻

Proof. The loop through the newly opened cap lies inside its homogeneous starting pocket. At the cap, the side toward the base, of the starting color, is exterior to this loop; the other side is interior. During the episode the actual color banks preserve connections to these sides. If a bank hits its same-color wall, the retained homogeneous target pocket is connected to the cap along that bank. The traversed loop is outside the pocket: its past pairs have been removed, the new open diagonal is a barrier, and the continuation to the old force side stays in the main component. The target can reach that pocket’s wall color without crossing the loop. Comparing this color with the starting color proves that the target is inside the loop precisely at a polarity change. The same argument works with separate boundary occurrences.

Every other unstarted candidate surrounding the target stays intact in the retained component. Actual diagonals and walls cannot cross it, and a loop wholly inside a simply connected pocket has its interior there too. Its deterministic pairs in any wall triangles are also intact, because cuts by other loops never traverse them. If the loop entered at a new cap surrounds the target, it is outermost among the remaining candidates: its access to the homogeneous wall at that cap crosses none of them. Thus successful episodes enumerate the surrounding loops, with possible unsuccessful same-color episodes between them.

Before such a loop’s episode, growth remains outside its enclosed region; during the episode it follows that loop up to step-sized errors. A ball centered at the target inside its region therefore survives until the switch. If its radius is \(r>0\), then \(\operatorname{crad}(U_t,z)\ge r\) up to that time, bounding the radial clock by \(\log(\operatorname{crad}(U_0,z)/r)\). Conversely, Koebe’s bound on a fixed horizon gives a positive-radius ball in the newly retained pocket, and its surrounding successful loop lies outside that ball, up to vanishing step errors. ◻

Proof. Near \(0\) the entire dual wall arc and both end sides are in a tiny boundary neighborhood of the stopping chart. Until a primal reset, the growing dual bank remains actually connected to that arc; stop also before a fixed interior-angle threshold, so no opposite-end switch has yet occurred. In successive buffered dyadic chart half-annuli around this neighborhood, place actual primal transversal paths from primal wall to primal wall. Lemma 52 gives a uniformly positive conditional success chance at each separated scale, after the mesh limit at the chosen cutoffs. A successful path confines the connected dual bank, and hence the slit pieces, inside its small wall sector.

A hull confined to such a sector has a small new angular footprint: its basepoint-side maps converge to the identity at the origin and, by reflection, on each closed boundary arc away from that sector. The trapped dual arc remains in its short footprint. Therefore it cannot produce a fixed positive angular advance. Letting the number of half-annuli grow proves the first assertion. Exchange colors and reverse directions at \(2\pi\).

A true polarity change approaches the specified opposite endpoint, up to vanishing update errors. Uniform angle convergence and induction therefore preclude a premature macroscopic limit for \(b_k^n\). Conversely, after the limiting first endpoint hit, instantaneous reflection produces a positive excursion within any prescribed additional time almost surely. Truncate the clocks and choose an exit size so that failure has arbitrarily small probability. Start at the first completed-state approach within a small tolerance after the previous true polarity change. The just-proved estimate excludes an excursion of this size before the corresponding reset. Let the tolerance tend to zero. An earlier matching reset already suffices. Thus no endpoint hit is skipped, and \(b_k^n\to b_k\).

For the final assertion, stop at successive approaches to the starting endpoint after a fixed-size advance. A successful episode that makes such an approach must then advance macroscopically without the corresponding same-color reset, since that reset would end the episode. The first assertion bounds each such event. The number of fixed-size swing tests is tight by angular tightness, so truncation and iteration give the stated retreat estimate. ◻

Proof. Fix a rational \(s\in(a_k,b_k)\). In any open subinterval of \((a_k,s)\) choose \(t<t'\) strictly inside it. The target kernel strictly shrinks between these times. There is a new boundary point inside the earlier open kernel; it belongs to the continuous canonical trace over \([t,t']\). Every small ball about this point with closure in the earlier kernel must encounter new prelimit growth between the corresponding times. Otherwise the ball stays clear of new boundary. It also meets a compact internal patch of the later kernel. Connectedness then puts that whole ball in the retained prelimit component, extending the limiting kernel through its alleged boundary point, a contradiction. These prelimit boundary pieces are slit updates tracing the successful episode, up to vanishing physical step errors.

Apply this test simultaneously to any finite ordered list of such intervals. Completed-update stops can be moved slightly inside them because capacities and angular within-step errors vanish. We obtain points of the discrete winding-loop traversal, in chronological order, converging to canonical trace points in those intervals.

Now take any finite list of finite chordal intervals on the default continuation after \(s\). Choose, realization by realization, a target \(w\in\mathcal Z\) in the current kernel whose chart image is safely above the entire bounded portion in question. It was still accessible with \(z\) up to \(s\). Their time correspondence there is regular and passes to the mesh; they cannot have split earlier because the limiting kernel contains internal connecting paths with clearance. The branch toward \(w\) follows the default continuation with tip and force angles separated by a positive margin on the selected portion, by Lemma 60. Therefore its discrete branch has no reset there and continues the same original winding loop, in order, after the common prefix. The new-boundary test applies to these intervals in the \(w\)-clock. The target \(w\) need not have a successful episode for this loop. Its post hoc choice costs no uniformity assumption: all targets, clocks, and regular sharing correspondences from the countable set have already been included in the coupling.

By continuity, shrinking the testing intervals and increasing the finite continuation range shows that every finite increasing list of canonical traversal points has representatives in order on \(L_z^k\). On a countable dense set of canonical parameter values, take a single nondecreasing cyclic choice of representative parameters by compactness and a diagonal subsequence. Omitted endpoint values are included by continuity. The order uses at most one turn: the discrete episode and its no-reset continuation traverse a simple closed loop from its released cap without passing that cap a second time. Two separated canonical visits cannot collapse to one representative parameter, because every intervening open traversal interval has actual variation.

If \(\widetilde L_z^k=\widetilde L_w^j\) but the two lattice-limit elements were distinct, each would contain the same canonical interior double visit from Lemma 60. This would give more than two total limiting visits at that point, contrary to Lemma 59(ii). Thus equal canonical labels have equal lattice-limit labels, with multiplicity understood as stipulated above. ◻

Proof. Only the forward implication remains. On the event that the discrete labels are shared, the two branches cannot split before entering this common loop: before its episode the loop and its interior remain intact across every cut by other loops. Choose clock horizons large enough to include both successful switches except on an arbitrarily small event.

While they share a domain, write \(\xi=\Phi_z(w)\). For a fixed small \(e>0\), stop at first approach to \(|\xi|\ge1-e\), if it occurs before the relevant winding switches. For fine mesh this can be a completed state also satisfying \(|\xi|\le1-e/2\). An exact split cannot precede this threshold inside the comparison range: the ODE and clock change are regular for \(|\xi|\le1-e/2\), and elementary capacities vanish there.

We first show that an unstarted loop surrounding both targets at this stop has probability tending to zero as \(e\downarrow0\). Place separated dyadic chart half-annuli around the boundary direction of \(\xi\), between radius a multiple of \(e\) and a fixed small radius. Choose their buffered wall intervals to avoid the two marked accesses and sides; deleting the bounded number of affected scales leaves a number tending to infinity. Each bar separates a sector containing the radial access from \(w\) to the wall from the sector containing \(z\) and a remote wall portion.

Use an actual-color transversal path in each bar, choosing the color to match one wall end. At that end it reaches an actual wall vertex. At the other it does so as well if the wall has the same color. If that wall has the other color, complete the path from its specified-color triangle corner to the base by the segment of Lemma 52. After trimming at first wall contacts this is a simple crosscut in the prime closure. Its added segment meets only the one local cap pair, once.

An unstarted simple loop inside the domain surrounding both targets must cross this screen at least twice: it must meet both of the separated sectors, and its interior contains both targets. It cannot cross the actual open path, and cannot traverse that one cap pair twice. Thus one successful screen precludes the event, even with mixed wall colors. The conditional success probability of each fixed-aspect buffered bar is bounded below by Lemma 52. Iteration, with the mesh limit first, proves the asserted small probability.

We next obtain a shared interior precursor for a shared successful loop. If the common loop is processed before the near-separation threshold is needed, its episode visibly passes through interior angles, comparable in the two charts while \(|\xi|\le1-e/2\). If the loop is still current at the threshold, its \(z\)-angle cannot be arbitrarily close to that episode’s starting endpoint and later succeed without a same-color reset, by Lemma 62. Otherwise, its progression from the start to the current state has already passed a fixed interior margin. The \(w\)-angle at that shared state has a comparable interior margin, with constants allowed to depend on \(e\). The unstarted case has just been made unlikely. Thus, except on an arbitrarily small event, the two successful episodes have a common state before kernel separation, with both angles interior by fixed positive margins.

From this precursor to the respective winding switches, neither angle returns arbitrarily near its episode’s starting endpoint, by the successful-episode retreat estimate in Lemma 62. Choose in order the clock cutoff, the near-separation threshold, the interior margins, and the retreat tolerance, making their error probabilities successively small. With strict buffers a subsequential precursor remains in a common kernel with \(|\xi|\le1-e/2\), lies after both previous polarity changes, and lies in the interiors of both limiting successful excursions. Their endpoint histories and starting excursion time agree along the common branch by the regular chart correspondence. They share the successful front to this interior state and use the same default continuation to the same force access for its rest. Therefore their canonical loop elements coincide. Let all the preceding errors tend to zero. Countability of the labels proves the simultaneous assertion. ◻

Proof. The two-winding-disk property of Lemma 59(iii) provides a disk of winding one for every positive limiting loop with an interior point. It contains a target from \(\mathcal Z\), so the loop is one of its exhausted winding labels. For a labeled loop, Lemma 64 makes its dense-target winding indicators identical to those of its canonical partner: winding one at a target is equivalent to equality with one of that target’s successive surrounding-loop labels, in both collections.

Suppose an interior trace point of the lattice-limit loop were outside its canonical partner trace. A small ball about it avoids that canonical trace and has constant canonical winding. The two-winding-disk property supplies disks of both lattice winding statuses in the same ball, each containing a dense target. This contradicts the indicator equality. Hence the entire interior lattice trace lies on its partner. The reverse trace inclusion follows from ordered containment. If the canonical partner is internal, the connected lattice trace containing it cannot reach the wall without adding an interior point outside that partner. If the lattice trace is internal, so is the contained partner. Thus in either wall-away case the full traces are equal.

It remains to exclude extra traversal of this trace. Invert the nondecreasing cyclic representative selection in Lemma 63 by a generalized inverse. Because every canonical open traversal interval has actual variation, this inverse is continuous, nondecreasing, and of degree one. The traces agree at all its non-plateau-interior times. A plateau would insert an extra portion of the lattice traversal anchored at one canonical point.

Assume such a portion varies. By trace equality it lies on the canonical trace. At a visited value different from its anchor there can be only one canonical parameter: two canonical visits plus this inserted visit would violate the total-visit bound of Lemma 59(ii). On a varying subarc away from the anchor, the assignment to that unique canonical parameter is continuous, by compactness. It therefore sweeps a nontrivial canonical traversal interval all of whose values have unique canonical visits. This is impossible by the interior-double-visit property of Lemma 60. Hence every inserted portion is constant. Equality modulo reparametrization follows, including the preserved multiplicity of collection elements. ◻

Proof. We first give the uniform comparison estimates that allow an upgrade from a local limit. For a fixed compact disk \(K\), the probability that any discrete loop visits \(K\) and reaches a distant surrounding circle tends to zero as that circle tends to infinity, uniformly for sufficiently small meshes and for larger initial wired squares. Actual monochromatic circuits in buffered annuli block such a loop; conditional RSW gives a fixed success chance at each separated scale. Use a slightly enlarged \(K\) and strict radial margins when passing to limits.

The free infinite-volume law and the homogeneous-square law can be coupled identically on a growing intermediate disk with probability tending to one as the square grows. Bracket the exterior partition by the minimal and maximal partitions, use monotone FK coupling, and expose actual boundary-attachment clusters in the maximal sample. Dual RSW screens make their probability of reaching the deep interior tend to zero. Once exterior influence is separated, the remaining conditional laws agree and one uses identical switches. Combining this with the preceding escape estimate makes the entire loops visiting \(K\) coincide, since those loops stay in the intermediate disk. This argument uses the specified free plane law and FK domain Markov coupling; it assumes no interface limit.

In a fixed-square subsequential coupling, Proposition 65 identifies all wall-away macroscopic loop elements, with their traversals. Its remaining possible discrepancies involve the wall. The escape estimate makes it unlikely that a lattice-limit loop both hits \(K\) and reaches that wall. The same holds for a canonical CLE loop, because that loop is contained in its lattice-limit partner. Thus, when the square is large, the square CLE and the free plane lattice law have arbitrarily close matching configurations on loops visiting \(K\), in the successive mesh and square limits.

The large-domain local limit of nested CLE is established in Miller et al. (2015, Appendix A, Theorem A.1). Its comparison uses conformal identifications of the relevant random-region loop configurations, with maps approaching the identity on compact sets; it does not assert equality at unchanged Euclidean coordinates. The work cited here is The conformal loop ensemble nesting field, arXiv:1401.0218, whose Appendix A contains that theorem. For \(4<\kappa<8\) this construction agrees with the whole-plane branching construction by Gwynne et al. (2021, Remark 2.6), and its law is invariant under inversion, hence under Möbius maps, by their Theorem 1.1. Centered squares and disks give the same local limit: normalize their conformal radii, and the comparison maps tend to the identity on compact subsets. All these parameter restrictions hold for our \(4<\kappa\le6\).

To pass from this local law to the required spherical topology, fix \(\epsilon>0\) and choose \(K\) so large that its exterior has spherical diameter less than \(\epsilon\). Loops entirely outside \(K\) are then irrelevant at that matching accuracy. The uniform escape bounds control the loops that visit \(K\) but leave a much larger disk. They hold for square CLEs uniformly as the squares grow, by their subsequential square-law couplings and the containment argument above. Consequently the laws of the square CLEs are Cauchy for spherical matchings: compare two large squares on a large fixed intermediate disk to the same free-plane lattice law, let the mesh tend to zero, and use the escape tails and the exterior diameter bound.

Any resulting spherical limit has the established local whole-plane CLE law. That local law, together with these tails, determines the full spherical collection: at each accuracy all relevant loops are seen in a bounded window except on an arbitrarily small event. Conversely, the same comparisons force every subsequence of the free-plane lattice laws to this limit. Hence convergence holds for the full mesh sequence. The target labels ranged over all \(k\ge1\) and the clock cutoffs increased without bound, so no nested generations have been discarded. This proves Theorem 1(ii) for \(1\le q<4\). ◻

Proof. Bulk RSW in a sequence of buffered annuli gives a polynomial bound for an actual primal or dual arm between their extreme rims: each annulus has a uniformly positive conditional probability of a blocking circuit of the opposite color.

We also need the standard elementary count consequence of this argument. The number of distinct actual clusters of one color crossing a fixed-aspect annulus has a geometric tail. Explore complete actual components from the inner rim, revealing their closed exiting edges. Remove every explored component, including noncrossing ones, until the next crossing component is found. At this stopping time, extend the surviving configuration by closed edges on the deleted graph. Its FK law is dominated by the law in the original annulus with all vertices on both original rims wired together. Removed bulk components do not wire surviving bulk vertices. A dual circuit in a buffered middle annulus of this auxiliary residual configuration blocks another actual primal crossing, and has probability bounded below even under the maximal primal boundary condition. Repeating the exploration bounds the chance of each additional crossing by a fixed number below one. Apply the same argument with the colors exchanged.

Each loop is the unique exterior contour of its interior-side actual cluster, primal or dual. Thus distinct loops give distinct colored clusters. Cover \(B(0,M)\) by \(O((1+M/a)^2)\) balls of radius \(a/100\). A loop of diameter at least \(a\) meeting one of these balls has an interior-side bank crossing an annulus about its center whose two radii are fixed multiples of \(a\). Different global clusters give different actual components in that annulus. The preceding geometric tail and Minkowski’s inequality prove (177).

For (178), a bank of an escaping loop gives an actual arm from scale \(M\) to scale \(R\). For (179), a bank of the indicated loop gives an arm from scale \(r+C\delta\) about \(z\) to a fixed fraction of \(a\). The polynomial arm bound proves both assertions. ◻

Proof. This is the construction preceding the metric conclusion, with its parameters retained. In the notation of (Duminil-Copin, Kozlowski, Krachun, et al. 2026, Definition 3.8 and Section 3.6, pp. 49–50), choose \(\eta=N^{-b}\) and \(t=N^{-b^2}\), for \(b>0\) sufficiently small. On an event of probability \(1-O(N^{-c})\), the selected nail clusters survive the whole extrema-resampling and track-exchange chain. The same construction preserves the bijection of non-trivial loops and their reduced words at each step. This covers both colors of contour.

Initially, the nail grid spacing is \(t^{1/2}N\), nail diameters are \(O(tN)\), and the nail window has size comparable to \(t^{1/4}N\). The terminal nail positions differ by these errors from their images under \(M_{\pi/2,\alpha}\). The independent FK argument in (Duminil-Copin, Kozlowski, Krachun, et al. 2026, sec. 4.2, proof of Theorem 1.7) proves \(M_{\pi/2,\alpha}=\mathrm{id}\). We may therefore use the already constructed coupling with corresponding nails at the same limiting positions.

For clarity, these scales can be arranged for every small physical mesh \(\delta\). Set \(a=b^2\) and choose \(N=2\lfloor\tfrac12\delta^{-1/(1-3a/8)}\rfloor\), so \(N\) is even as required by the track construction. Then the physical grid spacing, nail diameter, and nail-window size are respectively of orders \[N^{-a/8},\qquad N^{-5a/8},\qquad N^{a/8}.\] The homogeneous observation window has scale \(\delta N\), which also diverges. Center the nail window by translating each endpoint sample by a period vector of its own homogeneous lattice nearest the same centering vector. The two chosen translations differ by \(O(\delta)\); this only adds a vanishing error to corresponding nail positions. The endpoint restrictions have exactly the homogeneous laws, by (Duminil-Copin, Kozlowski, Krachun, et al. 2026, Proposition 2.11). After centering, both the nail window and the homogeneous observation windows exhaust the plane. The probability \(O(N^{-c})\) tends to zero. This proves the stated extraction; no conclusion about curve-set convergence is used to recover the bijection. ◻

Proof of Theorem 67. The rectangular model on \(\mathbb L(\alpha)\) is invariant under reflection \(S_{\alpha/2}\) about \(e^{i\alpha/2}\mathbb R\) (Duminil-Copin, Kozlowski, Krachun, et al. 2026, Remark 1.8). Its fair-sign height correlations inherit that symmetry. If the chosen endpoint rounding is not reflection equivariant, the reflected representatives differ by \(O(\delta)\), which is harmless by (181). Apply (184) at the original and reflected tuples. The square model is consequently asymptotically invariant under \(S_{\alpha/2}\), uniformly on every compact subset of \(\mathcal D_k\). Composing with its exact horizontal-reflection symmetry yields invariance under rotation through \(\alpha\). Square symmetries cover the endpoints and the other half of the circle. This proves the assertion for each fixed rotation.

The estimate (181) upgrades this to uniformity in the rotation angle. Indeed, the orthogonal images of \(K\) form a compact subset of \(\mathcal D_k\), and a finite angular net approximates every rotation with endpoint displacement at most \(C_K\) times its angular mesh. First use convergence at the finitely many net angles, and then let the angular mesh decrease. Reflections follow by composition with an exact square symmetry. This argument requires no estimate uniform in a degenerating isoradial angle.

Finally, any translation differs from a period translation of the mesh-\(\delta\) square lattice by a vector of length \(O(\delta)\). Exact period invariance and the location-uniform estimate (181) handle this residual displacement, uniformly over all translation vectors. Every plane isometry is a translation followed by an orthogonal map. The fixed rotation and dilation identifying \(\mathbb L(\pi/2)\) with \(\mathbb Z^2\) merely change the mesh convention and compact set. These observations prove (174) for the original square lattice. ◻

Proof. Lemma 68 identifies the discrete gradient law with the isotropic six-vertex law at \((\mathbf a,\mathbf b,\mathbf c) =(1,1,2)\). Theorem 67 gives precisely the compact-\(\mathcal D_k\), all-isometries symmetry statement used as Theorem 4.1 in (Duminil-Copin, Kozlowski, Lammers, et al. 2026), at this parameter. We may therefore apply its isotropic Theorem 2.8, using the remaining spectral, regularity, and free-energy results cited and proved there. The passage from correlations to smooth field tests is part of its Section 8. No anisotropic extension or general-parameter loop-functional assertion is needed here.

At \(\mathbf c=2\) one has \(\Delta=-1\), so the variance multiplier in that theorem is \(2/\arccos(-1)=2/\pi\). A fixed rotation and dilation of the lattice does not change the neutral whole-plane Green quadratic form: the dilation adds a constant to the logarithmic kernel and each test has integral zero. For every real \(\rho\in C_c^\infty(\mathbb R^2)\) with \(\int\rho=0\), the convergence therefore gives \[\mathbb E e^{i\langle h_\delta,\rho\rangle} \longrightarrow \exp\left\{-\frac1\pi \iint \rho(x)G(x,y)\rho(y)\,dx\,dy\right\}.\] Apply the same assertion to every real linear combination of a finite family of neutral tests to obtain joint convergence. This is exactly Proposition 73. ◻

Proof. It suffices, by linear decomposition, to use differences of two smooth unit masses. Expand a product of \(m\) such pairings, integrating the two endpoints of each occurrence separately. For fixed sampled endpoints \(a_i,b_i\), put \[d_i(L)=\mathbf1_{\{a_i\in\operatorname{int}L\}} -\mathbf1_{\{b_i\in\operatorname{int}L\}}.\] Conditional on the loops, averaging the fair signs gives exactly \[ \mathbb E_{\epsilon}\prod_{i=1}^m \sum_L\epsilon_Ld_i(L) =\sum_{f:\{1,\ldots,m\}\to\{\text{loops}\}} \prod_i d_i(f(i)) \prod_L\mathbf1_{\{|f^{-1}(L)|\text{ is even}\}}. \tag{188}\] In a nonzero term every used loop separates the endpoints in each slot assigned to it and is used in at least two slots. If \(d_*\) is the smallest distance between the \(2m\) sampled endpoints, truncated at \(1\), such a loop has diameter at least \(d_*\). Notice that even repeated copies of the same smeared charge have separately sampled endpoints; coincidences have zero integration measure.

For each endpoint and each size octave, the number of loops of that size which wind around it has moments of every order bounded uniformly in the mesh. Indeed such a loop supplies a passage in one of a bounded number of buffered annuli of comparable size. Ordering their medial passages produces distinct actual bank components, with bounded multiplicity. Successive actual-component exploration and bulk RSW give a geometric tail for this count. Between \(d_*\) and the diameter of the fixed support there are \(O(1+|\log d_*|)\) octaves. For larger loops, separation of two support points forces a visit to a fixed bounded slit joining them. Reaching the octave of radius \(R\) then requires an actual arm from that bounded region to scale \(R\), whose probability decays as a power of \(R\). Cauchy–Schwarz and higher passage-count moments make the sum of these outer octaves finite. Consequently the absolute value of the integrand in (188), after expectation, is bounded by \[C_m\bigl(1+|\log(d_*\vee\delta)|\bigr)^m.\] The same reasoning bounds any fixed higher absolute moment of the geometric majorant. Logarithmic singularities are locally integrable in the independently sampled endpoints. This proves uniform integrability of every fixed moment. Proposition 73 and uniform integrability now give joint moment convergence.

If a drawing modification changes a contributing indicator at one of a fixed collection of distinct sampled endpoints, a loop of positive diameter enters an \(O(\delta)\) neighborhood of that endpoint. The bulk one-arm estimate makes this event negligible; the preceding higher moment bounds allow its removal under the integrals. ◻

Proof. For a vertical cut through a primal column, a bisected tile contributes the seam operator \[I+|\mathrm{Bell}\rangle\langle\mathrm{Bell}|, \qquad \mathrm{Bell}=e_1\otimes e_1+e_2\otimes e_2,\] which is positive. The horizontal case is identical. A diagonal cut along tile edges contracts equal indices, and is positive as well. These statements give a positive Gram form for arbitrary finite network replacements strictly on one side of the cut. Each switch intertwines the transports on its reference-directed pairs, so simultaneous conjugation preserves all contractions; reflection therefore implements it unitarily.

One may first use large reflection-periodic tori, with two reflection seams. For a diagonal seam take periods along and perpendicular to the diagonal. The all-loop torus measure differs from periodic FK by a bounded genus factor. A fixed patch is, with probability tending to one, screened by contractible actual circuits of both colors. On that event the genus data are exterior to the patch. Bulk uniqueness and the circuit tails identify every local subsequential limit with the plane FK state. Equivalently, finite closure of all strands meeting the patch gives its plane domain Markov law directly. Independent binary indices pass to this limit. The finite-volume positive forms therefore give the asserted plane form. ◻

Proof. Scale the outer radius to a fixed small constant. Choose mirror boxes of radius \(s\) at horizontal distance one, centered at \(x\) and \(y\). In the box at \(x\), take a positive unit mass in a disk of radius \(\varepsilon s\) at \(x+(s/2,0)\) and a negative unit mass near \(x-(s/2,0)\); use their reflected placement at \(y\). Fix smooth mass profiles and connecting slits inside the boxes. Their height differences are denoted \(h_x,h_y\).

For a traceless complex \(2\times2\) matrix \(V\), insert \(\exp(taV)\) along each successive directed flux piece, where \(a\) is its contribution to this smeared dipole. Equivalently, if \(\rho_x\) is the dipole density, choose a compactly supported one-form \(\alpha_x\) in its box with \(d\alpha_x=\rho_x\,dx\wedge dy\), and integrate \(\alpha_x\) along the directed pieces. Such a form exists because \(\int\rho_x=0\). Let \(A(V)\) be the second-order coefficient as a vector in the reflected space. It is an equivariant quadratic map on the complexified conjugation representation of \(SU(2)\). Its symmetric square decomposes into a scalar representation and one irreducible spin-two representation. Set \[E=e_1e_2^*,\qquad F=E^*,\qquad H=\operatorname{diag}(1,-1).\] The tensor \(E\otimes E\) lies entirely in spin two, whereas the spin-two projection of \(H\otimes H\) is nonzero. Unitary equivariance and irreducibility therefore give, with an absolute constant, \[ \|A(E)\|^2 \le C\left\|A(H)-\frac12\mathbb E(h_x^2)\mathbf1\right\|^2 =\frac C4\operatorname{Cov}(h_x^2,h_y^2). \tag{190}\] Here \(\mathbf1\) is the vacuum vector. This does not assert that the entire scalar sector consists of multiples of \(\mathbf1\): projection onto spin two annihilates \(\mathbf1\), which is all the inequality requires. For diagonal \(H\), the second coefficient is \(h_x^2/2\), giving the last equality with the reflected orientation convention.

By Lemma 74, the covariance in (190) converges to its Gaussian value. The covariance of the two neutral charges is \(O(s^2)\), since the logarithmic kernel is smooth between the boxes and each charge has total mass zero and first absolute moment \(O(s)\). The Gaussian identity \(\operatorname{Cov}(X^2,Y^2)=2\operatorname{Cov}(X,Y)^2\) gives \[ \limsup_{\delta\downarrow0}\|A(E)\|^2\le C_\varepsilon s^4. \tag{191}\]

We obtain a matching lower bound from two passages at each box. For a loop visiting both boxes, compress each maximal run at one box before its next visit to the other into one factor. Nilpotence gives alternating factors \[I+ta_1E,\ I+ub_1F,\ I+ta_2E,\ I+ub_2F,\ldots.\] Close an excursion by an arc through the omitted other box, where the relevant one-form vanishes. The closing arc can be chosen disjoint from the excursion, giving a simple curve. Its flux is an averaged winding difference, so \(|a_i|,|b_i| \le1\). The fluxes are integers if the excursions avoid the endpoint supports. Let \(M\) be the total number of alternating periods, over all loops visiting both boxes. There are \(2M\) disjoint connecting passages. The coefficient \(K\) of \(t^2u^2\) in the product of normalized traces satisfies \[ |K|\le C M^4,\qquad K=0\quad\hbox{if }M<2. \tag{192}\] Loops visiting only one box contribute \(1\) and can be omitted.

If \(M=2\) and the endpoint supports are missed, then \(K\ge0\). For a single two-period loop its only possible nonzero contribution is \(a_1a_2b_1b_2/2\). The full winding difference belongs to \(\{-1,0,1\}\), so nonzero integer fluxes satisfy \(a_1=-a_2\) and \(b_1=-b_2\); the coefficient is \(1/2\). For two one-period loops the coefficient is \(a_1b_1a_2b_2/4\). If nonzero, both simple loops separate the two endpoint pairs. Their interiors are disjoint or nested, so their choices of the contained endpoints either agree in both boxes or are complementary in both. Hence the products \(a_1b_1\) and \(a_2b_2\) have the same sign, and this coefficient is \(1/4\).

For completeness, the errors in this sign calculation must be relative to two four-arm probabilities. Let \(\pi_4=\pi_4^\delta(c_1s,c_2)\), for fixed suitable constants. Bulk component searches give \[\begin{align*} \mathbb E\bigl[M^4\mathbf1_{\{M\ge3\}}\bigr] &\le o_s(1)\pi_4^2,\tag{193}\\ \mathbb P(M=2,\text{ an endpoint support is hit}) &\le C\varepsilon^\alpha\pi_4^2, \tag{194}\end{align*}\] for some \(\alpha>0\), with the mesh taken sufficiently small at each fixed \(s,\varepsilon\). Here is the exploration underlying these estimates. Around either box the \(2M\) radial separators, cropped with end margins, give \(2M\) alternating bank arms and at least \(M\) distinct actual primal crossing components of a cropped annulus. Conversely, two distinct actual primal crossing components force two intervening dual separators, hence four alternating arms in a slightly narrower annulus. Thus the first two successful discoveries already cost \(C\pi_4\). Expose each complete actual component, including all its exiting edges revealed closed, and do not count wire identifications as open paths. The remaining graph is dominated by the original annular lattice with maximal identifications at its original rims. In separated dyadic rings, a buffered dual circuit blocks every actual radial crossing with uniformly positive conditional probability. Each further discovered crossing therefore costs at most \(Cs^\alpha\). Iterating this bound gives a summable tail even after multiplication by \(M^4\). The two box annuli have disjoint bulk collars, so their first-four-arm costs multiply up to a constant; this proves (193). An endpoint hit additionally requires a bank arm from radius \(\varepsilon s\) to a fixed small multiple of \(s\), in a buffered region inside the hole of its corresponding four-arm annulus. Both four-arm costs are retained. The same conditional blocking estimate proves (194). In particular an absolute \(o(1)\) bound would not suffice in either estimate.

Finally there is a compatible positive event of probability at least \(c\pi_4^2\), with \(c\) independent of sufficiently small \(\varepsilon\). Prescribe primal east/west and dual north/south landings at each box annulus, with buffered extensions. Attach the two primal inner landings to small actual circuits around the positive and negative endpoint disks, respectively, at radii \(\eta s\), where the small constant \(\eta>0\) is fixed before choosing \(\varepsilon\ll\eta\). At the large scale join the inward primal arms directly between the boxes, and the outward pair around their exterior. Join the upper dual arms to each other and the lower dual arms to each other between these routes. At each inner rim join its two dual arms in a central corridor between the terminals. Together these dual connections form a closed oval separating the two primal groups. Retain one inner-to-outer landing pairing of comparable probability at the first annulus and its mirror at the second; the gluing works for either pairing. One of the two pairing classes has at least half the separated-arm probability, so no additional theorem concerning a prescribed pairing is needed.

All these joins can be made in finitely many disjoint tubes of the requested colors, outside the core annuli except for their landing extensions. Use transverse crossings in overlap rectangles to join the extensions to the tubes. Condition on the complement of one color’s tubes and apply FKG and buffered RSW there, then do the other color. Since only that color’s extensions enter its tubes, the requirements are increasing in the appropriate conditional model. The two core annulus probabilities multiply by the bulk collar comparison. The resulting positive and negative terminal circuits belong to two distinct actual primal clusters, the first connecting the two positive terminals and the second the two negative ones.

There are at least two distinct closed interfaces separating these two connected primal groups: color parity makes their number even, and at least one is necessary. Each such interface separates both endpoint pairs and misses their support disks. On the part of this event with \(M=2\), the two-loop coefficient is \(1/4\). Subtracting (193) and (194), first choose \(\varepsilon>0\) small and fixed, and then \(s\) small. Positivity on all the remaining \(M=2\) configurations and (192) show \[\|A(E)\|^2=\mathbb E K\ge c'\pi_4^2.\] Combining this with (191) proves \(\limsup_{\delta\downarrow0}\pi_4\le Cs^2\). Quasi-multiplicativity and fixed-factor rim changes give (189). Bulk collars give its domain version. ◻

Proof. Fix an interior compact set and an outer displacement cutoff \(R>0\). A point with two genuinely distinct visits, each having displacement at least \(R\), gives four medial passages across every sufficiently small surrounding annulus. With rim margins, their actual banks give four alternating arms. A grid cover at scale \(r\) has \(O(r^{-2})\) cells, each with limiting probability at most \(C(r/R)^2\). Its expected number of required cells is therefore bounded uniformly in \(r\), after the mesh limit. For every \(\eta>0\) the expected \(\eta\)-dimensional covering content is \(O(r^\eta)\); along dyadic scales it tends to zero almost surely. Countable compact sets, displacement cutoffs, and positive \(\eta\) prove the claim. ◻

Proof. An exterior connection of either color can enter only through the same-color wall arc, already connected by actual open edges. Hence it creates no additional interior partition. For a closed wall no \(C'\) vertex is wired through that wall, giving the stated free model (equivalently, apply planar duality). There are exactly two transition ports in the mixed experiment. The loop through one must leave the interior through the other; its one independent fair index is fixed by the condition at the first port. ◻

Proof. Order the medial radial separators and crop their actual \(C'\) bank paths with margins at both ends. Different strips cannot communicate across a separator, and each strip is charged at most twice. Thus many passages force a proportional number of distinct actual crossing components of the cropped graph. Reveal these components successively from one rim, ignoring partition jumps. A deleted component does not wire surviving interior sites through its former neighbors. Since the wall is free for this color, the remaining graph is dominated by the full annular lattice with maximal ties only at its original rims. A buffered circuit of the other color blocks actual crossing, with conditional probability bounded below by bulk RSW. Iteration gives a geometric tail for the component count and hence all its polynomial moments. Applying these blockers in separated dyadic rings proves the power bound for one arm. Missing edges or sites only lower actual connectivity; the same comparison applies to a product on disjoint subgraphs. Enlarge inner radii by a fixed mesh multiple whenever needed. ◻

Proof. A loop winding around \(z\) and of diameter in \([r,2r]\) lies within \(O(r)\) of \(z\) and supplies a macroscopic passage in one of a bounded number of annuli of radius comparable with \(r\). For a homogeneous wall use its free bank color. In a fixed mixed polygon, choose a subpassage away from the transition points by a fixed fraction of \(r\). The two wall arcs separate linearly away from a transition, so the chosen patch avoids one of them and has a free bank color. Lemma 79 gives bounded moments per octave. At mesh scale a deterministic local bound suffices. The same free bank argument works for arbitrary homogeneous wall shapes and for the stated products.

Expand (195) and average the signs. A loop which survives must be assigned to at least two of the distinct points and therefore has diameter at least \(d_*\). The remaining offset factors have absolute value at most one. Minkowski’s inequality over \(O(1+|\log(d_*\vee\delta)|)\) octaves gives (196) and the asserted higher moment majorants. Integrating the logarithmic singularities gives uniform integrability. Uniform second moments for compactly supported Fourier tests give tightness in a sufficiently weak local negative Sobolev space, hence tightness as distributions.

To move one point a small distance while the tuple stays distinct, any changed contributing loop indicator must belong to a loop of diameter bounded below which approaches that point. In the mixed case a changed offset requires the distinguished strand to approach it. The bulk one-arm estimate makes these events uniformly rare; the higher count moments control their contribution. This proves equicontinuity, allowing the \(O(\delta)\) ambiguity of the drawing. The same annular passage counts give tight macroscopic curve counts and tightness modulo reparametrization by the displacement-arc subdivision argument of Section 15. ◻

Proof. First note the needed uniform norm bound. For a wall factor let \(W_j\) denote its switch constraints only, and let \(p_j\) be their isolated vacuum probability. Allowed lattice translations and reflections preserve \(p_j\) exactly. The arbitrary-event comparison across vacant bulk collars gives \[ c\le\mathbb E\prod_j\frac{\mathbf1_{W_j}}{p_j}\le C. \tag{198}\] The constants may depend on the fixed gaps and rectangles. A single marked-port projection contributes exactly \(1/2\) to the expectation of the positive wall constraints. Bounded charge or index factors can be omitted in absolute value. The same upper bound applies to separated subfamilies, their lattice translates, and their reflected copies. In particular the reflected norms needed below are bounded uniformly in the mesh. No lower bound on an individual, possibly very rare, wall probability \(p_j\) is asserted.

On the algebraic reflected space let \(S_\delta\) translate an insertion deeper into the half-plane by an admissible lattice period in the normal direction. Reflection and translation invariance make this operator symmetric. Cauchy–Schwarz gives \[\|S_\delta^n v\|^2 \le \|v\|\,\|S_\delta^{2n}v\|.\] At fixed mesh every finite insertion is bounded, including its normalizing constants, so the norms on the right are bounded as the displacement varies. Iterating this inequality proves \(\|S_\delta^n v\|\le\|v\|\). It also shows that null vectors remain null. Thus \(S_\delta\) extends to a self-adjoint contraction on the reflected Hilbert space. Use \(T_\delta=S_\delta^2\) to obtain a positive self-adjoint contraction. Write \(p_\delta=O(\delta)\) for its physical translation step.

Reserve a small fixed part of the vacant slab. After absorbing the rest into the positions of the two base vectors, the correlations have the spectral interpolation \[ F_\delta(z)= \left\langle a_\delta, T_\delta^{N_\delta+z/p_\delta}b_\delta\right\rangle, \qquad N_\delta p_\delta\longrightarrow a>0. \tag{199}\] The vector norms are bounded by (198). Spectral calculus therefore makes these a bounded holomorphic family on every closed half-plane strictly inside \(\operatorname{Re}z>-a\). A bounded integer rounding of the exponent changes a correlation by \(o(1)\) locally there: for \(n\) of order \(\delta^{-1}\) and bounded real \(\theta\), \[\sup_{0\le t\le1}|t^{n+\theta}-t^n|=O(n^{-1}).\] Thus rounding a physical shift by a bounded number of admissible normal translation periods gives the same limit. Take a diagonal subsequence on a countable array of rational shifts and apply normal-family compactness to (199). Every resulting limit is holomorphic on a neighborhood of the entire nonnegative real ray, including zero. The assertion for one cut only uses the other translations as fixed parameters. ◻

Proof. Put \(Y_\delta=X_\delta(f)-\mu(f)\). If an allowed reflection line separates this cell from all the others with a strict projection margin, reflection Cauchy–Schwarz proves the claim immediately. Indeed the squared norm of \(Y_\delta\) tends to zero: the two reflected Laplacian tests have disjoint supports and their whole-plane Gaussian covariance is \[\sigma^2\int\nabla f\cdot\nabla(f\circ\theta)=0.\] Proposition 73 consequently factors their characteristic functions. Reflection conjugations and orientation signs do not change this conclusion. The other reflected vector has bounded norm by (198).

For general cell placements choose the original cells from a finite rectangular grid with positive gaps, and call the row and column of \(Y_\delta\) its central row and column. The following finite displacement procedure creates the required separating line.

1. Translate all rows above the central row upward, and all rows below it downward, each as a single group.

2. Translate the columns to the right of the central column rightward, and those to the left leftward, by amounts larger than the preceding vertical spread.

3. The central cell is now isolated in its \(x+y\) projection. Partition all other cells into the groups with larger and smaller \(x+y\) projections. Move these groups far in the directions \((1,1)\) and \((-1,-1)\), respectively, until the former are above and to the right of all remaining cells and the latter below and to their left.

4. Move the upper group farther right and the lower group farther downward. All other cells then have larger \(x-y\) projection than the central cell.

Each movement begins with a vacant projection slab and increases its width. Individual rectangular collars stay vacant: each translated group moves rigidly, and the two groups at a cut only become farther apart. Each named directional movement is a separate one-parameter translation. Empty groups need no movement. The amounts are chosen successively sufficiently large, with each threshold allowed to depend on all earlier choices.

In the final configuration a diagonal reflection line separates \(Y_\delta\), so every limiting correlation is zero. Apply Lemma 82 to the last displacement. Analytic uniqueness extends the zero correlation from all sufficiently large values to zero. Continue in reverse order. At each stage the earlier parameters are still sufficiently large for the relevant vacant slab, and the later ones have been returned to zero. A countable set of rational choices suffices for a common subsequence. The triangular dependence of the thresholds therefore causes no loss of parameters. Eventually the correlation at the original placements is zero. Every subsequence has the same conclusion, proving the \(o(1)\) assertion. ◻

Proof. Consider one gap of width \(\eta\) at a homogeneous wall point. The switches still to be fixed lie in a ball of radius \(O(\eta)\), and are to be open in the wall color \(C\). First omit the marked-index condition. In the opposite color \(C'\), the gapped model is the larger of the two models. Couple by exploring all actual \(C'\) components incident to the fixing edges in this larger sample, starting from both endpoints and revealing also all exiting edges. Use the ordered FK conditional probabilities for common queried edges. Explore actual open edges only, without following boundary partition identifications.

The only differing fixed data lie on the fixing edges. On the event that the discovered components stay inside a fixed small homogeneous neighborhood, their outside edges are closed in both samples. This screens the changes, so the unqueried graphs have identical induced laws and may be sampled identically. This is also a direct whole-plane exploration with only specified edge states; it requires no boundary wire jump. The probability that the exploration leaves the neighborhood is at most \(C(\eta/R)^\beta\) by Lemma 79, where \(R>0\) is its fixed radius.

Every discrepant switch is open for \(C'\) in the larger sample, so both its \(C'\) vertices belong to the discovered components. A loop or strand touching a discrepancy in either model has its actual \(C'\) bank in those components. On the screening event it therefore cannot reach any charge support or the marked port, all outside the neighborhood. All remote loops and their binary indices can be matched exactly. Restoring the marked-index condition changes the exceptional probability by at most a factor two, since its conditional probability is exactly \(1/2\). Repeat at the finitely many gaps and let their widths tend to zero after the mesh limit. ◻

Proof of Theorem 81. Take a subsequential distributional limit \(h\), using Lemma 80. The preceding gap construction, Lemmas 83 and 84 show that, in a neighborhood of every tuple of interior positions, the joint Laplacian tests have the Gaussian law with covariance \[ \mathbb E\bigl[L(f)L(g)\bigr] =\sigma^2\int_V\nabla f\cdot\nabla g, \qquad L(f)=\langle h,-\Delta f\rangle. \tag{201}\] For coincident positions all the tests are combined in their single vacant cell before applying the characteristic-function argument. This detail includes the diagonals of \(V^m\) for every \(m\).

More explicitly, the integrated moment bounds give moment distributions of every order for \(h\), and hence for its Laplacian. On product neighborhoods of every position tuple they coincide with the Gaussian moment distributions just computed. Tensor test functions, partitions of unity, and their density in smooth compactly supported tests show equality on all of \(V^m\). This excludes any unseen distribution supported on a coincidence diagonal. Every linear combination of finitely many \(L(f)\) has the Gaussian moments; Gaussian moment determinacy identifies its law. Thus \(L\) is a centered Gaussian isonormal field on \(C_c^\infty(V)\) with (201).

Extend \(L\) by mean-square density to \(H_0^1(V)\). For a smooth compactly supported density \(g\), let \(v_g\in H_0^1(V)\) solve \(-\Delta v_g=g\) weakly, and set \[h^0(g)=L(v_g).\] Then \(h^0\) has the law of \(\sigma\Phi_V\). For \(f\in C_c^\infty(V)\), \(v_{-\Delta f}=f\), so \(\langle h-h^0,-\Delta f\rangle=0\). Consequently \[ H=h-h^0-u \tag{202}\] is distributionally harmonic, and therefore represented almost surely by a smooth harmonic function. No independence between \(H\) and \(h^0\) has been assumed or is needed.

It remains to show that \(H=0\). We use moment kernels, not pointwise values of the limiting random distribution. Let \(x,y\) approach distinct boundary points which are neither corners nor transitions. The uncentered second kernel of \(h-u\) tends to zero. To see this before passing to the limit, use (195). In the mixed case the wrong \(0/1\) side at a point near its prescribed wall requires the marked strand to cross a short segment from that point to the wall. It must then leave a fixed local patch, since both marked ports are elsewhere. Likewise, a loop containing both \(x\) and \(y\) has diameter bounded below; a segment from \(x\) to the nearby wall must cross it, so the loop approaches that wall point while leaving the patch. The free bank one-arm estimate makes both events rare. The uniform higher count moments in Lemma 80 absorb their contributions. Finally the deterministic harmonic function \(u\) tends to the prescribed boundary value. The homogeneous case has no marked term and is the same argument. Thus \[ K_{h-u}(x,y)\longrightarrow0 \quad\text{at distinct ordinary boundary points}. \tag{203}\] Kernel convergence and equicontinuity on interior compact sets justify taking the mesh limit before this boundary limit.

Let \(\psi:\mathbb D\to V\) be conformal. Polygon corner behavior implies that \(\psi\) and its inverse extend Hölder continuously to the boundary. In particular, for some \(a,C>0\), \[|\psi(z)-\psi(w)|\ge C^{-1}|z-w|^a.\] If \(r,s\ge1/2\), then \(|re^{i\theta}-se^{i\varphi}|\ge c|e^{i\theta}-e^{i\varphi}|\). The logarithmic second-kernel bound consequently gives an integrable majorant \[ C\left(1+ \bigl|\log|e^{i\theta}-e^{i\varphi}|\bigr|\right)^2 \tag{204}\] uniformly over radii approaching one. The same type of bound holds for the Dirichlet Green kernel, whose value tends to zero at two distinct boundary points.

Choose nonnegative smooth functions \(\rho_t\) of integral one, each supported on a compact radial interval in \((0,1)\) whose two endpoints tend to one. For every integer \(n\), consider the smooth distribution test which, for an ordinary function \(v\), is \[B_{n,t}(v)=\int_0^1\rho_t(r) \frac1{2\pi}\int_0^{2\pi} v(\psi(re^{i\theta}))e^{-in\theta}\,d\theta\,dr.\] For distributions this is the pullback test; its density in \(V\) includes the conformal change-of-variables Jacobian. It is smooth and compactly supported for each fixed \(t\). Equations (203) and (204), followed by dominated convergence in the two independent angles and radii, give \[B_{n,t}(h-u)\longrightarrow0\quad\text{in }L^2.\] The Dirichlet kernel gives the same conclusion for \(B_{n,t}(h^0)\). Hence \(B_{n,t}(H)\to0\) in \(L^2\).

The harmonic function \(H\circ\psi\) has Fourier expansion with \(n\)-th circle coefficient \(c_n r^{|n|}\). Therefore \[B_{n,t}(H)=c_n\int_0^1\rho_t(r)r^{|n|}\,dr \longrightarrow c_n.\] All coefficients, including \(c_0\), vanish almost surely. Thus \(H=0\), and (197) follows for every subsequential limit. Uniform integrability gives convergence of all integrated moments. The equicontinuity of the kernels at distinct interior tuples then upgrades their distributional convergence to pointwise convergence there, proving the last assertion of the theorem. ◻

Proof. Draw each cluster as a small regular neighborhood of its actual graph. Its boundary follows the paired medial pieces within \(O(\delta)\). Every perimeter behind its external perimeter lies in a bounded face of that graph. Such a face is enclosed by a simple cycle. Equivalently, the bank of the external perimeter is a closed graph walk; decomposing this walk into simple cycles shows that every wound point with the specified clearance belongs to a maximal cycle interior.

Two maximal cycle interiors cannot overlap without one containing the other. Otherwise their boundaries meet at two or more vertices, their union is contained in one biconnected block, and the simple outer boundary of that block encloses both. This contradicts maximality. Shared vertices or edges cause no difficulty, since the open interiors remain disjoint.

We justify the assertion about revealing information. Fix a possible list of cycles and expose its wall edges and everything outside its open disks. Any connection between vertices of one cycle that uses its interior already has a connection along that cycle. More strongly, adding arbitrary interior edges cannot join two distinct exterior biconnected blocks. After removal of an exterior cut vertex, the attachments belonging to any one wall cycle remain connected along the rest of that cycle. Thus an interior addition cannot repair an exterior cut vertex. The exterior block containing a maximal cycle already has all its vertices in that cycle’s closed disk: otherwise the simple external boundary of the block would be a larger cycle. A new cycle that both enters and leaves a listed disk would have two distinct wall attachments and would consequently belong to its wall block. It cannot enlarge that block outside the disk. The same argument applies if several listed disks are traversed, since they would then have to belong to the same exterior block. Nor can internal changes alter an external cluster perimeter. Consequently the event specifying the proposed list and its exterior is measurable from the revealed edges. Each wall is actually connected, so the domain Markov property gives the asserted product of homogeneous laws.

For the marked path, close each side using its wall arc and the appropriate half-sides of the terminal gates. The same-color bank and wall form a closed walk of actual open edges. The path lies outside every cycle made from this walk, so its exploration queries no switch strictly inside such a cycle. The preceding argument therefore applies on both sides. Finally, a further contour leaving a pocket and winding a point inside it would have to surround its wall cycle. This is impossible because that cycle belongs to an actual cluster connected to the original wall. ◻

Proof. On graphs of the opposite color, the smaller laws are independent free random-cluster laws. Their product is stochastically dominated by the restriction of the larger laws: reveal the remaining edges and use the induced boundary partitions, which dominate free boundary conditions. Couple these opposite-color configurations by inclusion.

Consider a smaller outer contour that is relevant to the test points. Its cluster is included in a larger-model cluster, but the latter may itself be surrounded by a different cluster. Select the outermost larger contour containing the filled hull of the smaller cluster. One must make this selection even when the two clusters do not merge. Using nested regular neighborhoods of included graphs, or the cycle decomposition in Lemma 85, shows that this selected contour contains all the wound test points with clearance. Every relevant smaller maximal cycle lies in a maximal cycle behind the selected larger contour. In the surrounding case, the smaller cycle lies in a bounded face of the outer cluster and is therefore enclosed by one of its maximal cycles.

Retain only the two sets of exterior and pocket data from this coupling. Now resample their interiors using the conditional laws of Lemma 85. Their wall color has reversed, so the same free opposite-color comparison applies at the next stage. Iterate. This is a sequence of couplings of conditional marginals; it does not claim monotonicity of nesting depth in the original edgewise coupling. At every fixed number of stages it gives inclusion at the same numerical nesting levels. A smaller contour’s group of test points can merge with another such group, but cannot split. There is no extra contour of that level between a selected outer contour and a pocket behind it: the pocket wall is part of that very contour’s cluster.

To translate this construction into moments, expand each height as the sum of signs of its surrounding contours and average the signs. A term survives exactly when every assigned contour receives an even number of the indices \(1,\ldots,m\). Encode that term by the vector of nesting levels \((\ell_1,\ldots,\ell_m)\) assigned at the respective points. The same vector is admissible in the larger model: mergers add even multiplicities and hence preserve their parity. Distinct vectors remain distinct assignments. This is an injection between the terms counted by the two sign-averaged moments.

For a surviving term, each assigned contour winds at least two distinct test points and has diameter bounded below by their separation. All its ancestors have that lower bound as well. Truncate the number of such contours and ancestors. The free-bank arm bound makes the event that any of these contours passes within \(O(\delta)\) of a test point negligible. On its complement all the required pocket containments hold. Lemma 79 and the higher count moments in Lemma 80 remove the exceptional event and then the truncation by uniform integrability. This proves (205). ◻

Proof. The first two assertions follow from Proposition 76 and Corollary 77, together with the one-arm estimate. For the two-status assertion use the attached-circuit argument of Section 15, with both banks retained. Here is the only modification required at \(q=4\). Enlarge the first crossing band a little laterally, and give a label only to an actual component containing a crossing of the original core. After a component has been searched and removed, comparison with full wiring on the enlarged band’s boundary is still valid. A buffered opposite-color transversal extending past both lateral ends of the core blocks any further core crossing. Bulk RSW gives this blocker a uniformly positive probability. The number of labels is therefore tight, and each label represents an actual connection in the revealed past. The leftmost crossings from a label’s bottom sources in successive core slabs can now be used exactly as in the attached-circuit argument; the fresh-side circuits lie in interior balls. For a distinguished path, close it along the remote wall when reading the two statuses. The argument also works with exposed bulk data outside the test patch. Only buffered bulk RSW is used (Duminil-Copin et al. 2018).

For the boundary assertion, fix a deterministic boundary point away from the marks. A macroscopic passage approaching it supplies an actual arm of the bank color that is not wired there. In local annuli this arm is bounded by the free-bank comparison. Along a primal wired arc the dual bank uses internal square faces and has no dual exit through that arc. Along the complementary arc the primal bank uses actual sampled edges and has no primal ties. Exterior dual identifications occur only beside the complementary arc. Turns can be followed by adjacent actual bank edges within \(O(\delta)\). Any other induced ties in a buffered annulus away from the other arc attach the surviving graph only at its rims. Thus the same arm bound applies in the specified Jordan approximations.

A nonconstant boundary subtraversal would cover an arc and hit a point of a fixed countable dense subset away from the marks, by a macroscopic passage. The preceding bound excludes this. For tightness, at each displacement scale use a finite cover of subpassage tests away from the two marks. Every displacement contains a smaller displacement a fixed fraction of that scale away from both marks. Marked uniform Jordan convergence and separation of the two boundary arcs allow its annulus to be cropped in a neighborhood missing at least one arc. The same passage estimates apply there. Interior no-spur and multiple-visit tests always retain positive clearance from the original boundary. ◻

Proof. Every point of \(T\cap V\) is approached by \(E\), by the two-status property. Points of \(T\cap\partial V\) are approached by the exterior of \(\overline V\). Hence \[ \overline E=\mathbb C\setminus U. \tag{206}\] Compact subsets of \(U\) persist in the approximating interiors by winding stability. Conversely, every excluded point is arbitrarily close to persistent exterior points, by (206). Thus the component containing a fixed point of \(U\) is exactly its pointed kernel. Kernels of simply connected domains are simply connected.

Let \(M\subset T\cap V\) be the distinct multiple-visit set, after constant pauses have been erased. Given \(\eta>0\), cover all of \(M\) by a countable collection of open balls \(D_i\), centered on \(M\), with \[\sum_i\operatorname{rad}(D_i)^\beta<\eta, \qquad \sup_i\operatorname{rad}(D_i)<\eta.\] This is possible because \(\dim_{\mathrm H}M\leq0\). Set \(G=E\cup\bigcup_iD_i\).

We first show that any \(x\in E\cap V\) can be joined to the exterior through \(G\), after adding at most one arbitrarily small ball centered on \(\partial V\). If the closure of the path component of \(x\) in \(G\) reaches \(\partial V\), such a ball connects it to the Jordan exterior. Otherwise that component has a compact frontier \(S\subset V\setminus G\) separating \(x\) from infinity. The set \(S\) meets \(T\) only at unique-preimage points; its other points lie in \(U\).

There are polygonal closed walks, arbitrarily close to \(S\), of nonzero winding about \(x\). To see this, grid the bounded component of \(\mathbb C\setminus S\) containing \(x\), retain its edge-adjacent squares reachable from \(x\) without meeting \(S\), and trace their external perimeter. We can lift these walks, with an error tending uniformly to zero, into the closed disks bounded by \(L_n\). Near a point of \(S\cap U\), a fixed interior ball persists. Near a point \(z\in S\cap T\), all boundary parameters approaching \(z\) tend to its unique preimage under \(L\). Nearby points of the approximating closed disk can therefore be connected near \(z\): first go by a closest segment to the boundary, and then use the short boundary arc between the resulting nearby parameters. The closest segments stay in the closed disk. The errors are uniform along the walk; otherwise compactness would give one of the two local situations just considered and contradict failure of uniformity. The lifted closed walk still winds about \(x\), whereas \(x\) lies outside the approximating disk. This contradicts contractibility of that disk and proves the escape assertion.

Choose a large square \(Q\) containing \(\overline V\) in its interior. Take finite increasingly fine nets of \(\overline E\cap Q\) whose vertices belong to \(E\). Join each net vertex to an exterior collar of \(Q\) using the escape assertion. The finitely many additional boundary-centered balls can have arbitrarily small total \(\beta\)-content. Select a finite polygonal tree joining these paths to one root on \(\partial Q\), otherwise lying in the open allowed region; gridding the paths and deleting cycles gives such a tree with axis-parallel edges. Remove a sufficiently thin polygonal neighborhood of this tree, attached to \(\partial Q\). Using narrow rectangular links and square junctions leaves one simple rectilinear polygon \(B\). Keep an open part of the collar outside \(\overline V\) in \(B\).

As the nets become finer, the removed tree approaches every point of \(\overline E\cap Q\). Any part of it entering \(U\) lies in the protecting balls, whose radii tend to zero and whose centers lie on \(T\) or \(\partial V\). Hence compact subsets of \(U\) persist, while the pointed kernels cannot extend through an excluded point. This proves the first two assertions.

For the finite selection, fix the tree and its closed thin neighborhood. It is compact and lies in the union of \(E\) and the countable protecting balls. A finite subcover uses compact subsets of \(E\) and finitely many of those balls, with a positive margin. The compact subsets of \(E\) are disjoint from the closed disks bounded by \(L_n\) for all large \(n\). Thus only the finitely many selected balls can contain cuts affecting the discrete pockets. Their total content is no larger than that of the full cover. This is why the countable cover is used before selecting the tree. A further \(O(\delta)\) adjustment makes the comparison polygon a regular wall polygon on the appropriate color lattice, without losing the protecting margins. ◻

Proof. Write \(\beta>0\) for the power in the free-bank arm bound. Fix a comparison polygon \(B\) and the finite protecting balls supplied by Lemma 88. In every pocket delete opposite-color edges outside the lattice approximation of \(B\). Equivalently, intersect its set of elementary wall-color cells with the corresponding cell set of \(B\), retaining only opposite-color edges between surviving adjacent cell centers. All changes are in the protecting balls.

The original opposite-color laws are product free laws. Couple them with the modified laws by exploring, in the larger law, the actual clusters incident to the deleted edges. Once those clusters are exposed, all discrepancies are shielded and the unqueried graphs have the same conditional law. The argument is the same as the gap-closing coupling in Section 18. If the ball radii are \(r_i\) and \(e>0\) is fixed, the probability that a discrepancy reaches farther than \(e\) from their centers is at most \[ C\sum_i(r_i/e)^\beta. \tag{207}\] Indeed every such discrepancy supplies an actual opposite-color arm in the original product free law, dominated by the full-lattice comparison. Contours through changed switches have banks in these discovered clusters and obey the same locality bound. Choose \(e\) smaller than the tuple’s separation and its clearance from \(T\cup\partial V\). Every sign-averaged residual moment term uses a contour of at least the pair-separation diameter. The macroscopic count moments and Hölder’s inequality therefore make the moment error tend to zero with the total protecting content in (207).

We verify the laws and offsets of the retained regions. Each side-connected component of the retained wall-color cells is again a simple polygon disk. Every globally missing cell has a path of missing cells to the exterior: it is outside at least one of the two simple grid polygons being intersected. Thus a retained component has no hole. Its boundary cannot pinch at a vertex with two opposite retained cells in that component and two opposite missing cells. The retained pair would be connected by a side-adjacent path, while the missing pair is connected through the exterior; these two disjoint connections between alternating sectors contradict planar separation. Apply this argument separately to each old pocket. The retained free opposite-color graph hence factors into independent homogeneous disk laws.

Each retained component also reaches an old pocket wall. Start from one of its cells and take a side-adjacent path in \(B\) to the part of \(B\) outside \(\overline V\). The first exit from the old pocket crosses an old wall edge; the path before that exit remains in the same retained component. The component therefore has an actual wall-color connection to the old wall. No exterior contour can surround it and add an unrecorded height jump. Its residual offset is zero. Finally, a connected compact neighborhood of any test point began in one old pocket for large \(n\): the bank walk tracking \(L_n\) winds around that compact, so Lemma 85 places it in a pocket.

For the upper bound, apply Lemma 86 from the retained disks into \(B\), and then apply Theorem 81. For the lower bound, make no deletions. Choose one regular polygon disk compactly inside each involved component \(\Omega\), containing its test points, and compare their product laws into the original pockets. Such inner disks can exhaust each simply connected \(\Omega\) in the pointed-kernel sense.

Dirichlet Green functions converge along simply connected pointed kernels. Between points whose limiting kernels are different, they tend to zero; this follows by conformal mapping convergence, or by the vanishing probability of crossing the separating bottleneck before exit. Therefore the Wick kernels of both the inner polygons and the protected outer polygons converge to those of independent Dirichlet fields in the components \(\Omega\). Take the mesh limit first at fixed polygons and protectors; next improve the inner and outer approximations and let their total protecting content vanish. The two bounds coincide. Already assigned ancestor signs, or the marked path’s side constants, multiply lower-order residual moments and are restored afterwards. They do not enter the moment comparison. ◻

Proof. For a test function \(f\), the independent sum on the right exists in \(L^2\), since Green-function monotonicity gives \[ \sum_{\Omega\in\mathcal O} \iint_{\Omega^2}|f(z)|G_\Omega(z,w)|f(w)|\,dz\,dw \leq \iint_{V^2}|f(z)|G_V(z,w)|f(w)|\,dz\,dw. \tag{209}\] The same estimate, with a fixed negative Sobolev norm, defines the sum as a random distribution.

Use a representation on which the extracted geometric data converge. A fixed distinct tuple of deterministic interior points is almost surely eventually covered, with positive limiting clearance, by the relevant winding components. Its offsets converge. Conditional on the discrete exterior pocket data, including the actual cycle walls, its residual moments converge by Proposition 89. That proposition applies along every realized convergent data sequence; it requires no regularity of the unexplored cycle walls beyond the proved pocket properties.

To pass to conditioning only on the stated limiting data, multiply the discrete moment identities by a bounded continuous function of those data. The necessary uniform integrability follows from Lemma 80. At level \(k\), terms already assigned to ancestors have offsets bounded by \(k\). The remaining sign-averaged terms require shared contours of at least the relevant pair-separation diameter, and their counts have all polynomial moments under the initial law. The marked offset is bounded by one. Thus, for each \(m\), a bound of the form \[C_{m,k}\bigl(1+|\log d_*|\bigr)^m, \qquad d_* = \min\bigl\{1,|z_i-z_j|:i\ne j\bigr\},\] and its higher-moment versions apply also before coverage by the limiting components has stabilized. Such bounds are integrable on tuples in compact subsets of \(V\), including near diagonals.

Smear the identities against tensor products of smooth test densities. Fubini and uniform integrability pass every joint moment to the limit, with every bounded continuous geometric multiplier retained. The almost-everywhere coverage removes the remaining set from these integrals. Higher even moments give uniform integrability of the smeared variables themselves. A countable determining family of test densities then yields the conditional moment identities for distributions. For each fixed conditional geometry, the Gaussian law with the bounded offset on the right of (208) is moment determining; equivalently, apply the one-dimensional Gaussian moment criterion to every rational linear combination of finitely many tests. Hence the moment identities identify the conditional law. Because the tests range over the entire initial domain, this argument also excludes a residual distribution supported on the area-zero complement of the components. ◻

Proof. In the homogeneous case, write \(\Phi=h/\sigma\) and \(m=v/\sigma\). Theorem 81 says that \(\Phi\) is a zero-boundary GFF. At the first level \(v\in\{-1,1\}\), so \(m\in\{-2\lambda_0,2\lambda_0\}\). In the marked case put \[X=\frac{h-1/2}{\sigma}=\Phi+u, \qquad m=\frac{v-1/2}{\sigma}-u,\] where \(\Phi\) is a zero-boundary GFF and \(u\) is the bounded harmonic function with boundary data \(-\lambda_0,\lambda_0\) on the respective arcs. Here \(|m|\leq2\lambda_0\) and \(m+u\in\{-\lambda_0,\lambda_0\}\) on each component.

In either case Proposition 90 gives the full conditional Dirichlet law of \(\Phi-m\) in \(V\setminus A\). This law depends on the exterior data only through \(A\). Conditioning down to \((A,m)\) proves locality. Moreover \(m\) is an ordinary bounded function, integrated over \(V\setminus A\), and the distributional identity has no additional part supported on \(A\). This is thinness. The uniqueness of a local set’s harmonic distribution from the field and set is also part of the local-set formalism (Aru et al. 2020, Lemma 2.3(1)); no extra choice of component labels is being added at this step.

For completeness, \(A\) is connected and contains the entire Jordan boundary. Its complementary components are simply connected by Lemma 88. If \(A\) were disconnected, split it into two nonempty disjoint compact sets, putting the connected \(\partial V\) in the first. A fine polygonal boundary around part of the second, at positive distance from the first and from \(\partial V\), would be a closed walk in a component of \(V\setminus A\) with nonzero winding about a point of \(A\). This contradicts that component’s simple connectivity. Thus \(A\) is a nontrivial connected compact set. It has no isolated points, and no component disjoint from \(\partial V\). These facts verify the topological condition in the definition of a bounded-type thin local set. Conformal transport to a disk preserves all these assertions. ◻

Proof. Apply the boundary-data uniqueness theorem with \(a=b=\lambda_0\). The initial domain, after conformal transport, has one complementary component, which is not a singleton. The boundary function has two changes and takes values in the closed allowed interval. Its interior range meets \((-\lambda_0,\lambda_0)\) by the strong maximum principle. The height-gap condition holds with equality. In every component of \(V\setminus A\), the total conditional mean is constant \(-\lambda_0\) or \(\lambda_0\), so the stated boundary criterion holds on every boundary component. Finally Lemma 91 supplies thinness, boundedness, and the required topology. All hypotheses of Aru et al. (2020, Theorem 3.21) are therefore met.

The completed Schramm–Sheffield level line coupled to the same field satisfies these same conditions. Uniqueness identifies the limiting closed set, together with its harmonic labels, with that level line plus the original boundary. In particular its interior trace is the simple chordal \(\mathrm{SLE}_4\) arc, which has no hits on the interiors of the two boundary arcs.

This closed-set statement identifies the original ordered curve as well. Any extra boundary point off the identified compact arc has a neighborhood disjoint from that arc; reaching it would force a nonconstant boundary subtraversal, excluded by Lemma 87. On a simple arc, any genuine backtracking repeats a nondegenerate subarc. Such a subarc has positive Hausdorff dimension, contradicting the dimension-zero multiple-visit bound. After suppressing constant pauses, the curve therefore traverses the arc once, in the prescribed endpoint order. ◻

Proof. The first-level set \(A\) in Lemma 91 has harmonic values \(\pm2\lambda_0\). Ordinary two-valued-set uniqueness identifies its coupling with the field as the labelled non-nested \(\mathrm{CLE}_4\) gasket.

We must still identify the discrete contour elements, since one such element could a priori have several winding components. Every positive contour has an interior-status disk, by the two-status property and the exclusion of boundary-only traversal. Thus it winds at least one hole of the identified gasket. Select two distinct holes by the first points they contain in a fixed countable dense set. Conditional on the gasket \(A\), the two unit offsets in our construction satisfy \[ \mathbb E[v(\Omega_1)v(\Omega_2)\mid A] =\mathbb P\{\Omega_1\text{ and }\Omega_2 \text{ belong to one contour}\mid A\}. \tag{210}\] Indeed the signs were independent and fair given the entire unlabelled contour geometry; different contours contribute zero covariance, whereas the same contour contributes one. In the unique two-valued-set coupling, different \(\mathrm{CLE}_4\) holes have independent fair labels, so the left side of (210) is zero. Countability excludes all grouping of distinct holes.

Consequently each positive contour winds exactly one simple CLE hole. Its full interior trace equals that hole’s Jordan boundary: the two-status property puts every trace point on the boundary of its winding region, and winding stability puts that boundary in the trace. Additional boundary pieces are excluded as in Proposition 92. The limiting winding is one. Any extra covering or backtracking repeats a nondegenerate subarc and violates the dimension-zero bound. Thus the original contour is exactly the corresponding simple-loop element, not just a set with the same filled interior.

Proceed conditionally through the hierarchy. Given the geometry and labels through an identified level, the residual fields in its holes are independent zero-boundary GFFs by Proposition 90. The next-level decomposition, with the parent constant subtracted, makes its missing set in each hole a thin local set with harmonic values \(\pm2\lambda_0\). Child winding components lie in the parent by discrete winding stability at points with clearance. They are simply connected and cover almost every point, by the arguments already given. Every positive child has an interior-status disk and is supported on the closure of its winding regions by the two-status property. Thus the same topology and thinness verification applies inside each parent hole.

Conditional uniqueness, including measurability from the respective residual field, gives independent fresh CLEs in the parent holes. Equation (210) and the trace argument again identify every individual child and its sign. Its parent can be read at any interior disk point off the traces. Induction identifies all finite generations. Finally every positive-diameter contour and all its ancestors occur in the retained finite-cutoff hierarchies, whose counts are tight. Sending the cutoff to zero identifies the full collection in the stated matching topology. ◻

Proof. Use the expanding-square coupling of Section 15. Its endpoint ingredients are bulk estimates: monotone boundary coupling and exploration of actual boundary connections couple the infinite-volume free switches with the square switches deep inside a large square, except when a boundary connection reaches the inner region. Actual dual circuits on successive annuli block that event, uniformly even under the maximal boundary law. Actual-color annular blockers likewise make a loop from a fixed disk reaching the coupling boundary unlikely. The finite-square limit is given by Proposition 93.

Nested simple CLEs in domains of conformal radius tending to infinity converge locally to nested whole-plane CLE; their spherical whole-plane law is Möbius invariant (Kemppainen and Werner 2016, Theorem 1, Proposition 4, and Section 3.2). Its range \(8/3<\kappa\leq4\) includes the present endpoint, and centered expanding squares meet its conformal-radius condition.

To pass from the local comparison to the spherical metric, first fix a spherical diameter cutoff. Loops lying entirely outside a sufficiently large fixed disk are below that cutoff. With high probability, every other loop above the cutoff remains inside the switch-coupling region, by the preceding escape bound. The number of such loops is tight, and the finite-square matching identifies them bijectively with their traversal order. Take the mesh limit, then enlarge the square and the fixed disk, and finally send the diameter cutoff to zero. This is the required global matching statement, including all nesting levels. ◻

Proof. Use the two-change disk drawing from Lemma 35: place tiles on every polygon-graph edge, give boundary edges distinct exterior tabs, and put local caps along the wired primal and complementary dual arcs. Flattening the tabs and separating the drawing in a thin collar changes the ordered curve by only \(O(\delta_n)\) in uniform distance. Boundary edges on the wired primal arc may be fixed open, while edges on the free arc retain their sampled law.

The primal bank of the distinguished path, closed along the primal wall, is a walk of actual open edges in the wired cluster. On the other side, the actual internal dual bank, its exits to exterior copies across the free arc, and the local complementary-wire links give the dual closing walk. All artificial dual links and copies stay within \(O(\delta_n)\) of that free arc. No dual exit through the primal wired arc is used. At fixed bulk points away from limiting traces these walks therefore have stable complementary winding statuses \(1\) and \(0\). Lemma 87 gives tightness, the two-status property, and the restrictions on multiple visits and boundary traversal for these drawings.

Construct a regular comparison polygon \(V^+\) close to the limiting marked domain \(V\) with the following two strict separations: the original primal wired arc, including its endpoints, lies outside \(\overline{V^+}\); and the comparison dual wall lies outside \(\overline V\). All margins are fixed before taking the mesh limit. One construction uses plane Schoenflies coordinates for \(\partial V\). Move the primal portion slightly inward, extending it a little beyond both marks, then rise outside the unit circle near the two ends and put both comparison transitions there. Keep the whole comparison dual portion outside the unit circle. Approximate the resulting simple curve by a simple rectilinear polygon, preserving all strict margins, and place the transitions inside straight sides. Its actual primal-open and dual-open wall approximations are regular experiments. Reverse the colors to construct \(V^-\): the original complementary arc is outside \(\overline{V^-}\), and the comparison primal wall is outside \(\overline V\). Both comparisons approach the same marked Jordan domain as the offsets vanish.

Couple the original law with that in \(V^+\) so that every original primal open edge in the common region is open in the comparison. To check the boundary partition needed for this ordering, restrict the original configuration to the common edges. Any connection departing from the comparison disk can induce ties only at its actual primal wall vertices: its dual wall and transitions lie outside the entire original closure, while the original wired vertices lie outside the comparison. The comparison connects all those primal wall vertices by fixed open wall edges. Its induced partition therefore dominates the original one, and random-cluster monotonicity gives the coupling. The cut is along a primal lattice wall, so no unaligned edge crossing is being treated as a wall vertex.

Every piece inside \(V^+\) of the original primal closing bank walk belongs to the comparison primal wall cluster: follow its original connection to the outer wire and take its first exit from \(V^+\). Suppose a test point has original primal winding status but comparison dual status, with positive clearance. On the comparison dual side it can be joined to the comparison dual wall without crossing the comparison primal wall cluster, and hence without crossing the original closing walk inside \(V^+\). From that wall, which is outside the entire original polygon, it can escape to infinity through the original exterior. This contradicts the original winding. Thus original primal status implies comparison primal status.

For \(V^-\) apply exactly the dual ordering. Original dual identifications and artificial bank links lie beside the complementary arc, outside \(V^-\). In the common region the dual walk uses only internal dual steps, and departing connections induce ties only at the aligned comparison dual wall vertices. An originally wired primal edge admits no dual crossing step. Original dual status therefore implies comparison dual status. The free-bank arm bound removes near-hits and tile ambiguities at any prescribed finite tuple of bulk points.

Let \(I_n(z)\) be the original primal-side indicator, and let \(I_n^-(z),I_n^+(z)\) denote those in the two comparisons. The separate monotone couplings give, for every fixed finite tuple \(\boldsymbol z\), the stochastic bounds \[ (I_n^-(z))_{z\in\boldsymbol z} \ \preceq\ (I_n(z))_{z\in\boldsymbol z} \ \preceq\ (I_n^+(z))_{z\in\boldsymbol z}, \tag{211}\] up to errors vanishing with the mesh. At fixed comparison polygons the limiting laws are the regular chordal \(\mathrm{SLE}_4\) laws by Proposition 92. Now let the offsets vanish. Conformal maps and their boundary extensions converge under marked uniform Jordan convergence, using uniform local connectivity of these boundary parametrizations. The two comparison SLE laws therefore have the same limit. Deterministic interior points avoid SLE, so their side-status arrays converge as well. Equation (211) identifies every finite-dimensional status law of an original subsequential limit.

On a fixed countable dense set, the statuses determine the interior trace: an interior trace point has both statuses in every neighborhood, whereas a point off the trace has a small disk of constant status. This follows from Lemma 87. Thus the limiting interior trace is the SLE trace. No boundary traversal and the dimension-zero multiple-visit bound identify the complete ordered curve exactly as in Proposition 92. Every tight subsequence has the same law, proving the full-sequence assertion. ◻