Proof. Use an actual black closed-cell crossing in the configuration with a crossing. In the other configuration, closed-set planar duality gives an open white crossing in the complementary direction, relative to the quad. To see the applicable convention, map the quad to a rectangle and apply the separation theorem for a closed subset and its open complement. The white path can be chosen polygonal and with clearance from the black closed set. Both paths must visit the region where the configurations differ.

Trim each path at surrounding curves, using its first hit of the outer curve and its last preceding hit of the inner one. Its two parts lead to its two target sides. Outside the change, the two black parts cannot connect to each other: that would give the black crossing without the change. The same argument applies to the white parts, using their open clearance. Planar separation puts the four parts in alternating order. This proves the bulk implication.

Near one straight side, the parts leading to the other three sides give the three half-plane arms, with the middle one leading to the opposite side. The passage to disjoint site paths also respects the literal closed-cell convention. A convex cell intersects a straight side in a connected set; a same-colored shared-cell shortcut either removes a repeated portion or would already connect the target sides without the change. For white arms use relative open paths and pass through cell interiors and common sides. Black paths of zero width on the straight boundary are represented by the cells containing them, with the allowed margins. At a corner, at least one target arc of the crossing is at a fixed positive distance; following the corresponding part of a crossing supplies the stated one-arm test. Nothing in this argument changes if other prescribed regions are held fixed: only the paths in the agreement annulus are used. ◻

Proof. Cover the enlarged shell by finitely many squares of side \(c_1t\), where \(c_1\) is smaller than the required margins, and require a nucleus in each square. The Poisson empty-square formula and a union bound give (10). A point in the smaller shell then has a nucleus within \(C c_1t\). A nucleus beyond the enlargement cannot be its closest nucleus. The same comparison along a segment in a convex cell bounds the entire part of any cell meeting the smaller shell and hence bounds its star. This is also why even a very long cell belonging to a remote prescribed insertion cannot cross a screened shell.

Use dyadic shells, with spare shells for the buffers. On complete success an ordinary arm event with slightly changed radii bounds the necessary probe. Otherwise group outcomes by the outermost failed screen, at scale \(t\). Discard all demands up to scale \(Ct\) and keep the arm test beyond that scale, independently of the failed occupancy test. Quasi-multiplicativity and (7) give, with the convention that an inverted annulus costs one, \[\exp[-c(t/e)^2]A_e(Ct,R) \le C A_e(r,R)\,(t/r)^g\exp[-c(t/e)^2].\] The dyadic sum is bounded uniformly. Its tail with \(t\ge Mb_e\) goes to zero after division by \(A_e(r,R)\) when \(r\asymp b_e\). When \(r\) is fixed, the whole sum goes to zero. A failure at the outermost scale is covered by the same inequality using the power lower bound for \(A_e(r,R)\). This argument does not assert independence for events involving unrestricted Voronoi cells.

For a side probe, use bulk annuli up to the distance from the side and then half-annuli centered at the projection onto the side. Leave disjoint buffers where the two types meet. The assumed comparison of the half-plane cost with \(A_e\), followed by quasi-multiplicativity, gives the same calculation. We shall verify the comparison in every triangular orientation below. ◻

Proof. Let \(Z_e=\mathbb P^{\eta_e}(C_Q)\). Restrict the relevant geometry to a large fixed box surrounding \(Q\). The resulting error tends to zero exponentially as \(e\to0\) by localization. Partition this box into squares of side \(s\), and resample the Poisson process in one square at a time. Denote the resulting conditional crossing probability by \(Z_e^{(i)}\). Efron–Stein gives \[ \mathop{\mathrm{Var}}Z_e\le \frac12\sum_i\mathbb E(Z_e-Z_e^{(i)})^2+o_e(1). \tag{12}\] Here and throughout this proof \(e\to0\) first, with \(s>0\) fixed. Couple the colors outside the resampled square. Screens make the two tilings agree outside a constant enlargement, except on an event tending to zero. A difference then requires the arm tests of Lemma 4 starting at radius \(Cs\).

For squares a fixed distance from all vertices and marks, bulk and side tests, together with (9), contribute at most \(C s^{2+2\alpha}\) per square. If necessary decrease \(\alpha>0\) so that the side test has at least this exponent. There are \(O(s^{-2})\) such squares.

We spell out the corner summation. Suppose the distance to the nearest vertex or mark, capped at a small fixed constant, is comparable to \(u\gg s\). Probe from \(Cs\) to \(c_Q u\), using four arms followed, when necessary, by half-plane arms. Choose \(c_Q\) small enough that the probe meets at most one side, also at narrow polygon angles. Independently, with a buffer, require one arm from \(C_Q u\) to a fixed radius whenever the square is near a corner. The root-second-moment bound for the resulting necessary event is \[C(s/u)^{1+\alpha}u^{\beta_0}\] for some \(\beta_0>0\). There are \(O((u/s)^2)\) squares in this dyadic distance class. Their total contribution to (12) is at most \(C(s/u)^{2\alpha}u^{2\beta_0}\). The bounded number of squares within distance \(Cs\) of a corner contribute \(C s^{2\beta_0}\) by the one-arm test alone. It follows that \[\limsup_{e\to0}\mathop{\mathrm{Var}}Z_e \le C s^{2\alpha}+C s^{2\beta_0} +C\sum_{\substack{u\text{ dyadic}\\s\le u\le u_0}} (s/u)^{2\alpha}u^{2\beta_0} \longrightarrow0\] as \(s\downarrow0\). The dyadic sum tends to zero whether \(\alpha\) is smaller than, equal to, or larger than \(\beta_0\); in the equal case the extra factor is only \(\log(1/s)\). The supplied scalar Cardy theorem identifies \(\mathbb EZ_e\) and therefore proves the proposition. A finite union gives the last assertion. ◻

Proof. Include the buffered enlargements used below in the countable annulus family before extracting. The first assertion is the subsequence principle for convergence in probability, followed by a diagonal extraction over compact sets, rational tests, and the specified annuli. Occupancy gives the compact mesh assertion; Proposition 6 and arm concentration give the others. At fixed radii the annealed probabilities are bounded away from zero by RSW and quasi-multiplicativity. For \(r\le R/2\), absolute concentration therefore gives the relative comparisons. For \(r>R/2\), the arm event from \(r\) to \(2R\) is contained in the event from \(r\) to \(R\). Use this inclusion for the lower comparison and the bound \(1\) for the upper comparison; fixed-ratio RSW bounds the annealed probabilities above and below by positive constants. Nearby centers or radii are handled by inclusions with fixed-factor margins.

Here is the crossing-convention argument. A Schoenflies strip about each side of the limiting Jordan quad supplies inner and outer comparison quads. For a lower bound, trim in from the two non-target sides, also near their ends, and extend the target caps beyond the actual boundary. Keep positive clearance from the non-target sides. Approximate the comparison quad by a rational polygon. Any crossing of this polygon gives a crossing of the original test after trimming at its target sides. First convert the crossing to a path through cell interiors and common sides, using the clearance, and only then trim it. This order handles even a crossing initially supplied as a closed connected set.

The complementary color gives the opposite inequality by the same construction. As the strip widths decrease, both Cardy values tend to the desired value, by continuity of conformal maps under Jordan boundary convergence. For an approximating graph disk the comparison paths are trimmed on first meeting its boundary. Thus they use only fair interior values and reach neighbors of the terminal sides; prescribed exterior boundary colors are not used as parts of the crossing paths. Outer comparison tests similarly bound crossings originally expressed using closed cells. This proves the claimed sandwich for all the stated conventions.

For a fixed open track, use a finite chain of overlapping thin rectangles, together with transverse crossings in their overlaps. Their crossings concatenate, and such chains can also make circuits. Cardy or RSW gives a positive lower bound for each rectangle. Harris’ inequality [14] gives the product lower bound for requirements of one color; opposite-colored requirements are placed in disjoint tracks and use disjoint sites once the mesh is small. For the triangular model start with Smirnov’s theorem [23] for the usual lattice approximations and apply exactly the same sandwich. Virtual terminal edges incur only a vanishing mesh error before the positive-clearance trimming. ◻

Proof. Among the crossings of a prescribed first color, take the leftmost simple path. One way to make this choice precise is to add an external closing edge on the right. The union of all available crossing paths with that edge is a union of cycles sharing the edge and is two-connected. Its exterior envelope supplies the leftmost crossing. Equivalently, it has no bypass to its left between distinct path vertices, including the terminal incidences. Whether a proposed path is this leftmost path is determined by the colors on it and to its left; vertices strictly to its right remain independent fair bits.

If an ordered family exists, its first path may be replaced by this leftmost path while all remaining paths stay strictly to the right. Conditional on the leftmost path and the revealed left part, apply induction to the remaining graph. The conditional probability of its remaining \(k-1\) ordered crossings does not depend on their colors. Integrating proves independence of the last \(k-1\) colors. Global color reversal then removes dependence on the first one as well. Vertex deletions change neither the envelope construction nor the product law on the unrevealed vertices. ◻

Proof. We first compare a color sequence containing both colors with an alternating sequence. Choose two adjacent arms of opposite colors, trim them to proper crossings, and consider the gap between them. The disagreement interfaces in its triangulation contain a strand joining the two annular ends: the switch parity on the two ends, followed by pairing the interface ends, gives such a strand. Its visited triangles have no vertices on the other arms. Slitting along it and deleting the vertices of those triangles leaves a graph in the slit rectangle, possibly with further vertex deletions, containing the other arms. Each bank of the strand already supplies a monochromatic arm, of opposite colors on its two sides.

For precision, cut the actual disagreement curves at every contact with either annular boundary and sum over their maximal through components and their two color directions. Sum also over the at most \(k\) cyclic positions of the opposite-color gap; this fixes the residual ordered color list and costs only a factor depending on \(k\). A candidate strand’s occurrence is determined by the bits of its incident triangles. Conditional on those bits, the vertices left in the slit rectangle are still independent and fair. Lemma 8 replaces the ordered colors of the remaining arms by those completing an alternating family, without changing the conditional probability. The strand banks lie at the extremes relative to the remaining paths, so the replaced family and the banks really do give the alternating arm event.

The overcount in this sum has uniformly bounded moments. To see this, let \(N\) be the number of through strands after cutting at boundary contacts. In a smaller annulus, a bank of a wall gives a connected site walk: successive triangles connect their same-side vertices, with short spokes to the wall. The walk cannot cross another wall. After proper trimming it gives an arm in that gap. A gap is used by at most its two bordering strands, so \(N\) through strands imply at least a constant times \(N\) disjoint black arms in a buffered subannulus. If \(R/r\) is larger than a fixed constant, keep a fixed-ratio middle annulus. A white circuit there has probability at least \(c>0\) by Lemma 7; hence a black crossing has probability at most \(1-c\). BK on the fixed graph gives an exponential tail for the number of disjoint black crossings. Thus \(\limsup\mathbb EN^p\le C_p\) for every finite \(p\), uniformly in \(R/r\). Bounded ratios require only an enlarged constant.

Writing \(A_k^{\rm alt}\) for the alternating event in the same annulus, with the fixed end margins just used absorbed in its radii, put \(P_k^{\rm alt}=\mathbb P(A_k^{\rm alt})\). For each candidate strand, conditional color-order invariance replaces its residual-arm event by the completing alternating event with equal probability. Summing these replaced probabilities counts at most \(C_kN\) strands on configurations with an alternating family. Therefore \[\mathbb P(\text{specified mixed $k$ arms}) \le C_k\mathbb E[N\mathbf1_{A_k^{\rm alt}}].\] The moment bound and Hölder’s inequality give \[\limsup\mathbb P(\text{specified mixed $k$ arms}) \le C_p\bigl(\limsup P_k^{\rm alt}\bigr)^{1-1/p}.\] For \(k=4\), choose \(p\) large in the upper bound \(P_4^{\rm alt}\le C(r/R)^{1+\alpha}\). For \(k=6\), five of the alternating arms and the remaining arm have disjoint site witnesses. BKR for the fair product law and the five-arm and one-arm bounds give \(P_6^{\rm alt}\le C(r/R)^{2+\beta}\). Another sufficiently large \(p\) preserves an exponent strictly larger than two.

Finally, an all-one-color \(k\)-arm event is a disjoint occurrence of the event of \(k-1\) such arms and the event of one such arm. Reimer’s inequality at density \(1/2\) bounds its probability by the intersection of the first event with the color-reversal of the second [18]. That intersection supplies a mixed \(k\)-arm family, since opposite colors are automatically site-disjoint. If color words are anchored, take the finite union over the cyclic position of the opposite-colored arm. The mixed-color estimate therefore proves the same bounds for monochromatic families. The bank and shadow argument used above is topological and also works after a homeomorphic change of coordinates with vanishing mesh. ◻

Proof. If a connected triangle piece had vanishing physical diameter but coordinate diameter bounded below, a Hausdorff subsequential limit of its preimage would be a nondegenerate continuum \(K\) on which \(f\) is constant. Uniform convergence of the maps gives this conclusion. Such a continuum cannot enter the open disk, where \(f\) is conformal and injective. A nondegenerate continuum in the boundary circle contains a nonconstant arc, contradicting the hypothesis. This proves the mesh assertion. Ordinary univalent distortion transfers the bulk estimates to balls in compact subsets of the coordinate disk.

In a patch where the indicated boundary interval projects injectively, crop the non-target sides of a coordinate test and trim its target arcs away from the ends. Its connecting subdisk has Jordan image. Approximate its interior sides by graph crosscuts and join them to the specified bank interval in order. The resulting cycles have Jordan boundary convergence in the physical plane. Apply Lemma 7, extending the target caps only after this cropping, and trim comparison paths at the graph boundary. They approach the specified bank from inside this subdisk, even if a different bank projects to the same physical neighborhood. Unused independent full-plane bits may be added for the comparison tests. Their trimmed witnesses use only the fair interior bits of the subdisk.

For the last assertion do this separately for the two complementary crossings. Their lower bounds approach the two complementary Cardy numbers; Hex duality supplies the squeeze. Thus the assertion invokes Cardy only for Jordan domains in the physical plane, not for a test on an unidentified covering surface or on the whole nonsimple projection. ◻

Proof. We first record the boundary one-arm estimate, which does not use trimming. In a chart of Lemma 10, a half-annulus crossing has Cardy’s limit. Its logarithmic image is a rectangle of aspect ratio \(\pi^{-1}\log(R/r)\). The rectangle cross ratio in Cardy’s formula gives a long-crossing bound \(C\exp(-\pi t/3)\) at aspect ratio \(t\), and hence \[ \limsup\mathbb P(\text{one free half-plane arm from $r$ to $R$}) \le C(r/R)^{1/3}. \tag{13}\] Margins give the same upper bound for graph cuts approximating the half-annulus. Only free interior vertices are used.

At either base endpoint of a trim semicircle, a visit within relative distance \(d\) by an arm continuing across the collar requires a one-arm event from scale \(d\) to a fixed scale. Equation (13) makes the combined probability arbitrarily small by first choosing \(d\) small. Away from these endpoint neighborhoods, cover the trim curve by \(O(1/d')\) balls of relative radius \(d'\ll d\). If two distinct arms approach in such a ball, their continuations in both directions give four disjoint site arms from scale \(Cd'\) to a fixed collar scale. Lemma 9 bounds the sum by \(C_d(d')^{c_0}\), which tends to zero. Distortion is uniform on these balls because their radii are small compared with their distance from the coordinate boundary. In a full annulus there are no base endpoints, so only this second argument is needed.

Define the bad-collar event to be exactly the union of the endpoint one-arm tests and the four-arm tests on this grid. These tests are all contained in a fixed enlargement of the collar; making \(d,d'\) smaller does not enlarge that region. Choose a grid of boxes finer than the resulting separation tolerance, and take the finite family of all ordered selections with the required separations. Properly trim the simple arms between the chosen curves. Any approach of a different arm to an endpoint box would have triggered the same four-arm test: that arm and the arm ending there both continue out of the wider collar in both directions. Thus the tenfold enlargements may be reserved for their own arms alone. The retained portions use sites only on the retained side of the trim, up to a vanishing mesh error absorbed inside these boxes. Proper crossings preserve their linear or cyclic order. ◻

Proof. Around each endpoint box choose two slightly larger boxes. In the annular gap between them require a circuit of the endpoint’s color, surrounding the endpoint box. The retained path reaches the small box from outside the larger one, so this circuit intersects it. Begin a corridor inside the circuit and require a crossing chain from there to the prescribed destination. This is the lasso construction: the circuit connects every possible retained witness to the corridor, without specifying where the witness first entered its endpoint box.

Route the corridor normally away from the trim until it is on the extension side with clearance, and then route it along a polygonal open track to its destination. The preserved order makes the tracks disjoint. Boxes can be made small compared with the track layout, or the tracks can taper near the boxes. Further lassos or transverse crossings may be placed at the destinations. Their rectangle chains have positive lower bounds by Lemma 7; boundary destinations use Lemma 10. Cardy’s values depend only on the normalized coordinate shapes, so these lower bounds are uniform over the stipulated layouts and do not depend on the physical distortion of a chart. All tracks have positive widths. No path is required to follow a prescribed curve exactly.

Let \(U_B,U_W\) be the sets of sites used by the black and white extension requirements. These sets are disjoint for fine meshes. Where an extension meets a possible retained arm, it lies inside that arm’s reserved enlargement; all arms of the other color avoid it. Consequently, after fixing the bits outside \(U_B\cup U_W\), the existence event \(H\) is increasing in black bits on \(U_B\) and in white bits on \(U_W\). The extension event has exactly the same coordinate monotonicities. Give the white coordinates the reversed order and apply Harris’ inequality to this product space. The extension event itself depends only on these coordinates and has probability at least \(c>0\). Thus the conditional probability of its intersection with \(H\) is at least \(c\) times the conditional probability of \(H\). Integration proves (14), also in the presence of restrictions outside these coordinates. A bounded finite number of choices of endpoint boxes is handled by summing, or by retaining a choice of maximal probability. At no point do we condition on an already selected arm or assert that its unexplored side has a fresh law. ◻

Proof. The one-arm assertion is (13). By Lemma 8, applied to a graph half-annulus viewed as a rectangle, it suffices for two arms to consider opposite colors. This use is valid after deleting all base vertices; it requires no law for them.

First suppose the two arms have separated trims at scales comparable to \(r\) and \(R\), with fixed-factor extension room. Lemma 12 extends their small ends to disjoint intervals on the base within distance \(Cr\) of each other, and their large ends to two prescribed separated arcs on a boundary at scale \(R\). There are only finitely many endpoint-box and color-order choices. We claim that the extended event has probability at most \(Cr/R\) in the limit.

Choose a half-disk at scale \(R\), with the two far arcs reserved for the two colors. Vary one marked point \(t\) on the base through an interval containing both near endpoints. Define \(E_t\) to be the black crossing from its far arc to the base target arc on the right of \(t\). The other marks stay in fixed separated positions at scale \(R\). Up to color reversal and reflection, the near endpoints are black then white from left to right, whereas the far black and white arcs are on the left and right respectively. When \(t=t_-\) lies to the left of the short near interval, the black connection gives \(E_{t_-}\). When \(t=t_+\) lies to its right, the white connection is a complementary crossing and excludes \(E_{t_+}\). These are nested events on the same fair bits, and \(t_+-t_-=O(r)\). Terminal-side Hex duality gives the same implication on the graph approximation. The ordinary Cardy limits of the two tests follow from Lemma 10. Their cross ratios stay in a compact subinterval of \((0,1)\), and Cardy’s formula is differentiable there, with derivative in the normalized moving mark bounded by a constant. Therefore \[\limsup\mathbb P(\text{extended two-arm event}) \le \lim_n\bigl[\mathbb P(E_{t_-})-\mathbb P(E_{t_+})\bigr] \le C r/R.\] Smooth boundary pieces flat on the relevant base interval avoid any corner issue in this comparison. The extension lower bound now proves the same estimate for the separated retained event.

It remains to bound the trimming exceptions without assuming the estimate being proved. Let \(F(k)\) be the supremum of the limit superior of the two-arm probability over the admitted half-annuli of ratio \(2^k\), allowing graph cuts converging to their nominal boundaries. Fixed-factor changes of radii alter \(k\) by a bounded integer. Choose the two end collars with disjoint buffers. Each bad collar has probability at most \(z\) by Lemma 11. On that failure keep the necessary two-arm event in the remaining half-annulus, deleting at most \(k_0\) end octaves, where \(k_0\) is absolute. The bad-collar test and the remaining arms use disjoint sites. Thus, for sufficiently large \(k\), \[ F(k)\le C_z2^{-k}+2z F(k-k_0). \tag{16}\] Intermediate graph cuts converging to dyadic semicircles justify the same recurrence when the original ends are not exact circles. Choose \(z\) so that \(2z2^{k_0}<1/2\), and then choose \(D\) large enough for the finitely many initial ratios and for \(C_z\). Induction in (16) gives \(F(k)\le D2^{-k}\). This proves the two-arm bound for all ratios, with the mesh limit always taken first.

Finally three disjoint paths contain disjoint witnesses for a two-arm event and a one-arm event. BKR and the first two estimates give the exponent \(1+1/3=4/3\). Color order in the rectangle makes this valid for every prescribed three-color list. Small changes of the base or graph boundary are handled with margins. For an upper bound permitting paths right up to a discretized physical boundary, one can extend the fair tiling slightly beyond that boundary and use the same Jordan sandwiches; no prescribed boundary colors are thereby counted as free arms. ◻

Proof. Only finitely many cells meet a given compact rectangle almost surely. Their number has moments bounded by a polynomial in \(e^{-1}\) as \(e\downarrow0\): localize in successively larger boxes using the occupancy screens, and combine their exponential failure tails with the ordinary Poisson count moments. At any fixed mesh the same argument gives finite moments. Conditional on this finite set, the crossing indicator is a nonconstant monotone Boolean function: it is zero when all cells are white and one when all are black. Some edge of a Boolean path from the all-white state to the all-black state is therefore pivotal. Every coloring has positive fair-color probability, proving strict positivity of the conditional mean and hence of the mean. The triangular assertion is immediate from its finite number of relevant cells.

For Poisson–Voronoi percolation, [27] estimates the derivative of the crossing probability of a \(2:1\) rectangle by \(R^2\alpha_{4,p}(R)\) at unit intensity. Its derivative is with respect to the color parameter, so Russo’s formula counts precisely the color pivotals used here. To include \(p=1/2\), which is an open endpoint in that statement, fix \(R\) and let \(p\downarrow1/2\). The annealed correlation length tends to infinity for the sufficiently small defining crossing constant used there. Localization and the integrable cell count justify continuity of the derivative and of the finite-scale arm probabilities. Rescaling now gives (17). The usual rectangle event agrees with our intersection-with-the-rectangle convention, apart from null general-position exceptions.

On the triangular lattice, Kesten’s pivotal/four-arm method [15], together with arm separation and extension as in [16], yields the same estimate. We verify the exact boundary convention directly. For the lower bound, every site in a small fixed central patch can send four separated alternating arms from distinct neighbors to macroscopic landing intervals directed toward the four sides. Inner arm separation, followed by a finite prescription at a bounded lattice radius, connects to the neighbors without using the site itself. Outer landing rectangles and disjoint RSW tracks continue the arms across the four side arcs away from corners. The probability is at least \(c a_e\). A black setting makes the black crossing; a white setting makes an open white crossing with clearance, so this is pivotal for the stated closed-cell event. For the upper bound, use the straight-side three-arm estimate in every orientation established above. Write its exponent as \(\gamma>g\), choosing the triangular \(g<4/3\). For a relevant lattice site let \(u\) be its distance to the nearest rectangle corner, capped at a small constant and bounded below by \(b_e\), and let \(d\le u\) be its distance to the nearest side, also bounded below by \(b_e\). Lemma 4 and independent buffered probes give \[\mathbb P(\text{the site is pivotal}) \le C A_e(b_e,d)(d/u)^\gamma \le C a_e d^{-g}(d/u)^\gamma.\] When \(d\asymp u\) the half-annulus is omitted. The number of sites in a dyadic \((u,d)\) class is at most \(C m_eud\), including the bounded-width row just outside the rectangle whose hexagons meet it. Summing first over dyadic \(d\le u\) gives at most \(C m_ea_eu^{2-g}\); summing over \(u\) is bounded since \(g<2\). The bounded number of corner-scale sites is covered by the same estimate and the lower bound \(a_e\ge c b_e^g\). This proves the upper estimate with the exact closed-cell convention. Finally (7) gives \(m_ea_e\ge c e^{g-2}\to\infty\), proving the last assertion. ◻

Proof. Begin with the boundary cycle. Adding the successive triangles of the exploration band preserves two-connectivity: a triangle is attached along the edge through which it is entered, and any new corner has two neighbors in the graph already present. Adding the remaining edges between revealed vertices preserves this property. Boundaries of bounded faces of a two-connected plane graph are simple cycles. An interior chord would be an edge between revealed vertices and would already have split the face, proving the chord assertion.

The part of the visited band off the original boundary is connected. Two successive steps have an interior corner in common whenever they meet the boundary: otherwise the common edge would be an interior chord of the initial cycle. The initial and final switch triangles cause no exception to this connectivity. If a face had two separate intervals on the original boundary, a crosscut through that face between two of its boundary contacts would leave parts of this connected interior band on both sides. They could not be joined without crossing the crosscut or the face. This proves the interval assertion, including an isolated contact vertex by taking a small crosscut around that vertex.

Draw the exploration as a simple dividing arc in the triangulated disk. Its black neighbors are on one bank and its white neighbors on the other; an edge joining the two banks belongs to the visited band. Consequently a complementary face sees only one color on its boundary. For the target face at a stop, continue the dividing arc through that face to the target. The same separation gives one black and one white boundary arc, meeting at the stated edges. ◻

Proof. We use the multiple-crossing compactness argument of Aizenman and Burchard [5], spelling out its fixed-scale application to these paths and face boundaries. Cut a wall traversal between two graph cycles in a buffered annulus. Following the corners of the successive triangles on either bank gives a connected site path, its shadow. Crop the shadow farther from the two ends so that it cannot run around an end of the wall traversal. Between disjoint proper wall traversals the corresponding shadows lie in the intervening gaps. Thus many traversals supply many site-disjoint arms, with at most a fixed loss at the ends. A positive fraction of these arms have one color. The BK inequality [26], together with the positive probability of an opposite-color circuit in a fixed-ratio subannulus, bounds the probability of arbitrarily many such arms by a geometric tail. Increasing the annulus ratio if necessary makes the one-crossing bound strictly less than one.

In a boundary half-annulus, \(k\) disjoint proper wall traversals have at least \(k-1\) free shadows, in their internal gaps. Here and below free means that the arm uses only random interior sites. The original boundary sites can occur only in the two outside gaps. If neither bank of any traversal uses those sites, the two outside shadows are free as well, giving \(k+1\) free arms. These assertions can be read between intermediate graph crosscuts and then cropped, so that a large site star cannot join around a cut. They depend only on the plane triangulation. Lemma 13 and the same BK argument give the boundary estimates. For a face boundary the random pieces are monochromatic simple site paths. Its original-boundary pieces and switches are bounded in number and cause no additional oscillations. The free pieces obey the preceding arm bounds. Near each fixed switch use the fixed-point one-arm bound instead.

For clarity, fixed-annulus estimates suffice for tightness in the curve metric. Fix an excursion size \(r>0\). Cover first the switches, then the boundary, and then the remaining compact interior by finitely many smaller neighborhoods whose surrounding annuli have outer diameter less than \(r/10\). Every successive displacement of size \(r\) requires a traversal of one of these annuli. Their crossing counts therefore bound the number of such displacements. Do this for \(r=2^{-j}\). Choose bounds with exceptional probabilities summable in \(j\), and parametrize curves by assigning summably small time budgets to their successive steps at each scale. This gives a common modulus of continuity outside an arbitrarily small exceptional event. The same bounds apply simultaneously to face boundaries, since their crossings are among the site arms just counted. Finite initial mesh indices can be accommodated separately. The nondegeneration argument in Lemma 19 below supplies the finite numbers of macroscopic faces needed to list those boundaries jointly. ◻

Proof. We first identify the arms forced by a contact. All traversals below are properly trimmed in a buffered annulus, so that their shadows cannot join around an annular end.

Exploration contacts. A triple visit in the bulk, with positive excursions between visits, gives six wall traversals and hence six free arms. A double visit on the original boundary gives four traversals and three free arms. A double visit on a fixed interior segment gives four free arms. If a wall comes to a boundary bank and leaves it without either shadow using that bank, the two traversals have three free shadows. A shadow uses a bank site only when a crossed edge is incident with that site, up to one triangle, so this count is independent of the precise wall drawing. A second visit to a specified switch requires a free arm from a vanishing neighborhood of that point. A nonconstant interval of the wall on the boundary likewise gives such an arm at some point of a fixed countable dense set of boundary points.

The exterior of a face. A monochromatic face arc needs an additional planar observation. At a finite mesh let \(C\) be the simple site cycle bounding the face. Every site on its random part has an opposite-colored neighbor across an edge of the explored band. These neighbors belong to one connected opposite-colored bank, denoted by \(K\), which includes its original boundary arc and lies in the closed exterior of \(C\). Suppose four properly trimmed traversals of the random part of \(C\) cross a bulk annulus. Each traversal has an exterior-side incidence with a neighboring gap, and one gap accounts for at most two of these incidences. Thus at least two distinct gaps abut a traversal on its exterior side. Other pieces of \(C\) may still enter such a gap. We show that each selected gap contains an opposite-colored radial crossing.

Choose one of these gaps between two proper traversals \(P,Q\) of a wider annulus, with the gap on the exterior side of \(P\). Cut across this gap at two intermediate graph curves, leaving the ends of \(P,Q\) beyond the cuts. The resulting working quadrilateral has subarcs of \(P,Q\) as its lateral sides, and its interior avoids the whole of \(P\) and \(Q\); this construction allows radial backtracks. Write \(p_-,p_+\) for the ends of the contiguous face-cycle subarc \(P\), and join their opposite neighbors to them by short spokes beyond the working cuts. If \(K\) has no longitudinal crossing of the quadrilateral, rectangle separation supplies a transverse crosscut avoiding \(K\), from an interior point \(x\) of \(P\) to \(Q\), with interior disjoint from all of \(P\). Stop it at its first subsequent meeting \(y\) with the entire cycle \(C\). It starts on the exterior side of \(P\), so this prefix lies in the exterior. Other pieces of \(C\) may cause an earlier stop, but \(y\) is outside \(P\) because the original crosscut avoids \(P\).

The relevant ambient region is the whole exterior of \(C\), compactified by the point at infinity: it is a topological disk. The boundary points occur in the cyclic order \(p_-,x,p_+,y\). Our crosscut therefore separates the two endpoint spokes in this disk. The spokes avoid its working annulus and put their opposite-colored endpoints in different components. Those endpoints cannot be joined by \(K\) without crossing the crosscut, a contradiction. In particular a connection going around an end of the annulus does not escape this separation. Crop once more to remove the short spokes. The resulting opposite-colored crossings in the two selected gaps are disjoint; with the four monochromatic traversals they supply six free arms.

A self-return of the random face arc in the bulk forces exactly this configuration. Near the original boundary, four free traversals already exceed the three-arm requirement. If the face arc has the same color as the nearby original bank, an intermediate contact gives two arms along the arc and one opposite arm from \(K\). The opposite arm cannot use that original bank, so all three arms are free. The same count applies when the two ends of a macroscopic face arc approach each other as its shared boundary interval collapses; a singleton attachment is included. At a fixed switch or fixed patch endpoint, one free arm suffices. Contacts with an original bank of the other color need not be excluded: they will be retained as distinct boundary accesses.

Applying the arm estimates. For a fixed positive outer excursion scale, cover the compact interior by \(O(r^{-2})\) balls of radius \(r\). The six-arm bound in Lemma 9, whose exponent is greater than two, excludes both interior triple visits of the wall and self-returns of a face arc. Along a boundary interval there are only \(O(r^{-1})\) balls, and the free three-arm exponent \(4/3>1\) in Lemma 13 excludes the boundary configurations just listed. The four-arm exponent greater than one similarly excludes double visits on a fixed interior segment. The fixed-point one-arm estimate handles switches, patch endpoints, and the dense boundary tests.

All centers and outer radii are fixed before passing to the mesh limit. A contact at a random point with positive excursions is covered by an enlarged ball of a rational grid. Exhaust excursion sizes by a countable list and then decrease the grid scale. Thus none of these exclusions requires an arm estimate conditioned at a random limiting contact point. ◻

Proof. A face boundary consists of its random part and at most one interval of the initial boundary, by Lemma 16. The random part has no self-return, by Lemma 18. If there is an original boundary interval, it has the face’s color. The same lemma excludes an intermediate return to that interval or an additional visit to its endpoints. A nontrivial interval cannot have the entire free arc collapse, because its two endpoints are separated in the initial Jordan boundary parametrization; intervals under consideration contain no switch in their interior. Nor can a macroscopic free part have its attachment interval collapse. If no original interval is present, a nonconstant limiting cycle cannot fold into a doubly traced arc: that would be a distinct visit of its random part. After deleting pauses the limit is therefore Jordan.

These facts also give the claimed finiteness. If arbitrarily many disjoint faces had diameter at least \(r\), their inscribed radii would have a subsequence tending to zero. Curve tightness would give a nonconstant limiting boundary of such a subsequence. Its Jordan interior contains a disk; convergence in order forces the approximating face to contain a slightly smaller disk, a contradiction. The argument proves tightness of the number and excludes surplus macroscopic faces in any coupled subsequential limit.

The face boundaries lie within a vanishing distance of the band or the initial boundary. Jordan convergence prevents the band from entering a limiting face interior. Conversely a compact ball in an open component off the limiting trace, together with finitely many compact joining paths in that component, eventually avoids the discrete band. It is thus recovered by one of the discrete faces. This identifies the components. The side of a fixed point off the trace is preserved by the winding number of the wall completed along the initial boundary, so the color labels pass to the limit. Finally a nontrivial original boundary interval cannot be supplied entirely by the limiting free arc: this would contradict the fixed-point one-arm test on that interval. Hence true intervals of attachment, as well as their order, are recovered. ◻

Proof. Until the stopping query, every interior vertex of the separating path is unknown. Its bulk portion joins, through unknown vertices on the back side, the target edge. The chord-free initial boundary ensures a connected inner row at that edge: the triangles adjacent to consecutive boundary edges meet through their interior corner stars. Consequently the outgoing edge is incident to the target face. This argument is applied outside fixed endpoint neighborhoods; chords from the crosscut to the rim near its ends cannot affect it. Lemma 16 gives the two colored boundary arcs of that face.

With probability tending to one as endpoint neighborhoods decrease, the portions of the original banks from the target up to, and a little beyond, the endpoints of \(L\) remain untouched. To reach one of these fixed neighborhoods, the band’s bank of the opposite color must supply a free arm out of it. The fixed-point one-arm bound therefore applies. Each target-face arc consists of an original bank followed by random sites of the same color. Away from the termination, its simplicity follows from Lemma 18. There is one additional check at the separator: an extra approach of a random arc to \(L\), with positive excursions of that arc before and after it, supplies two arms along the arc and an opposite arm through its neighbors in the band. All three arms lie on the front side of \(L\), up to the vanishing mesh error. Lemma 13, applied in this smooth crosscut patch, excludes such an approach. The same count excludes an additional visit to the termination. The fixed endpoint tests handle the two ends of \(L\).

The residual face is now described without identifying distinct boundary accesses. In the discrete target face the separator goes from one old bank to the other, and the queried vertex would be next to the endpoints of the outgoing edge. Cutting there, and removing that one triangle if necessary, separates two side polygons from the back. In the limit each side polygon has a boundary formed by a subinterval of its own old bank, its simple random arc, and its subarc of \(L\). This is a Jordan curve by the contact exclusions. An encounter with the other-colored bank does not create a repeated point of this side curve: that other bank is not part of its boundary traversal. The two side disks may touch each other in their closed exteriors. They remain joined to the back through the two open seams, not through the hit point. Figure 1 records the order of these pieces.

This decomposition proves kernel convergence: compact subsets of each side interior, the back, and the open seams persist; compact joining paths in the displayed union also persist. It specifies the limiting prime ends. On the interior of each piece of side boundary one uses its Jordan access, even when another bank touches its image. Near an attaching endpoint one joins the Jordan neighborhoods through the incident open seam. At the hit point the two neighborhoods are joined through the back. Thus the boundary traversal is continuous in exactly the order of the discrete cycles.

For the required uniform convergence of maps, use the admissible domain version of boundary convergence in [9]. Its hypotheses are as follows: the boundary is the ordered union of three individually simple arcs \(J_1,J_2,J_3\); \(J_3\) avoids the interiors of the other two arcs and is accessible from the exterior at each of its points; and the three corresponding boundary arcs converge in the uniform curve metric. The other two arcs are allowed to touch. Here choose \(J_3\) to be a closed arc around the target strictly inside the untouched round part of the boundary. The remaining boundary, divided at the hit point, gives \(J_1,J_2\). Their separate simplicity and their convergence were just proved; \(J_3\) has exterior accesses in the round disk and misses their interiors. The approximating Jordan graph disks meet the same conditions. A normalization point can be fixed in the back, and the two marked version gives convergence of the start and target prime ends. The cited corollaries therefore give uniform convergence of maps on the closed disk. Composing with the preceding uniformly convergent maps proves the physical statement. Equivalently, marks chosen in boundary order converge because a sequence of boundary subarcs of diameter tending to zero cannot have preimages of positive diameter: that would make a nonconstant limiting parameter arc constant under the limiting map. ◻

Proof. First fix a mark in the interior of \(L\). The event of a black connection from the black bank to the separator between that mark and the white end forces the hit to the corresponding side of the mark. The complementary white connection gives the reverse bound. This is the elementary Hex separation in the front disk: the two colored paths along the explored banks terminate at the neighbors of the first separator vertex about to be queried. The white path separates the black bank from separator accesses beyond that vertex toward the white end, and the black path gives the symmetric separation. The comparison is used with a small slack at the mark, so no identity for the individual terminal triangle is needed.

These two comparisons are within the scope of the scalar theorem. Trim each target arc slightly short of its endpoints, and crop a subdisk connecting it to its mate before extending the targets for a lower crossing bound. One target lies on a separately simple bank and the other lies in the interior separator. Their cropped test has Jordan image and stays off the other boundary accesses. The Jordan sandwich of Lemma 7 and the conformal-coordinate conclusion of Lemma 10 apply to it. In particular this is not an application of Cardy’s theorem to an unidentified covering surface. As the cropping and the slack decrease, the two bounds approach the complementary Cardy values. End stops are negligible: the endpoint is on an open monochromatic bank away from the switches, and an approach there carries a free arm of the opposite color out of its neighborhood. The limiting hit distribution is therefore Cardy’s hitting distribution and has no atoms. The same argument works for piecewise smooth indentations of \(L\).

To determine more than the hit marginal, attach finitely many closed Jordan bumps to disjoint subintervals of \(L\) on its front side. Their sides are simple piecewise smooth curves, and except for their feet the bumps are interior. Keep a specified interval of possible hits disjoint from all feet. Replace the relevant pieces of \(L\) by the bump sides. The preceding hitting argument determines the limiting probability of reaching the retained interval on this indented separator. On this event the exploration has avoided the bumps and the retained hit is also its original hit on \(L\).

The comparison can be made on the same graph and the same bits. Approximate the bump sides by simple graph paths in the front disk, joining the separator in order. These paths agree with the old separator away from shrinking neighborhoods of the junctions. Approach each junction from an interior portion of the bump side and trim at the original crosscut; the endpoint vertices may vary inside those shrinking neighborhoods. There is consequently no requirement to approximate a prescribed smooth arc through a prescribed microscopic vertex. These nested graph separators really hide the bump interiors from the original start side. Before querying a vertex on the new separator the exploration cannot query a hidden part of the old one. On the unchanged retained interval the stopping rules agree.

It follows that the indented Cardy value lies between the limiting probability of strict inclusion of the bumps in the side domains, with hit in the open retained interval, and the probability of inclusion in their closures, with hit in the closed interval. Here strict inclusion means inclusion with positive clearance from the side boundary, allowing the foot to continue through its open seam. Conversely strict inclusion and the retained hit force the indented test to pass, by the domain convergence of Lemma 20. Enlarge the bumps slightly by regular parallel or polygonal approximations. Inclusion of an enlarged closed bump implies strict inclusion of the original; the Cardy values converge as the enlargement decreases. Since hit endpoints have zero mass, these sandwiches determine the strict inclusion probabilities. The hit’s position relative to each foot specifies which side domain contains that bump.

These tests determine both open domains, including their joint law. Indeed a rational ball compactly contained in a side domain can be joined to its open seam by a polygonal finger of positive width whose closure stays in that domain. A countable collection of rational fingers and balls tests all such inclusions. Finite intersections are still bump tests: slightly enlarge overlapping bumps on the same side, fuse them, fill bounded holes, and fill pockets against the intervening seam intervals. Their containing side domain is Jordan, so these fillings are forced by inclusion. Under strict inclusion the enlargement can be taken with positive clearance. Bumps from opposite sides that intersect are incompatible; a filling that covers the retained hit likewise gives an empty event. Subdivide the retained hit intervals to distinguish these cases. Thus the preceding probabilities give a measure-determining class. The simple boundaries then recover the two ordered arcs from their open domains.

Finally compare this characterized law to the classical SLE law in a round disk. The Cardy hitting and hull-avoidance characterization is stated in [9]. The single-semicircle hulls have the side-boundary and first-contact properties just used; these may also be read from the ordinary triangular-lattice coupling in [9]. Here this last result is used solely to supply properties of the established \(\mathrm{SLE}_6\) law in the standard Jordan setting. For that law the indented test is precisely hitting the retained interval before the bumps, so it has the same Cardy value. The measure-determining argument identifies our kernel with that SLE kernel. ◻

Proof. Fix such a point \(w\) and time \(t\). Take a small open boundary arc around the target whose neighborhood is disjoint from the trace up to time \(t\). Allow connections through that arc to the exterior, and include the complementary closed boundary arc in the obstruction. The obstruction \(K_t\) is the image of a continuous curve obtained by traversing that closed arc and then the exploration. Its endpoint, the current tip, is visited only once. Disconnection of \(w\) from the target is now separation of \(w\) from infinity by \(K_t\).

We need a small local-connectivity observation with the terminal point excluded. For every \(\rho>0\) there is \(d>0\) such that any two points of \(K_t\) other than its endpoint at distance less than \(d\) can be joined on \(K_t\) by a path of diameter less than \(\rho\) contained in some strictly earlier obstruction. Otherwise take counterexample pairs converging to a common point and their curve parameters converging to limiting parameters. If both limiting parameters are terminal, the short terminal subpath joins the pair. If neither is terminal, join each point along its short parameter subpath to the common limiting value and concatenate; both subpaths lie strictly before the terminal time. The mixed case would give an earlier visit to the endpoint, contrary to the assumption. Uniform continuity proves the claimed diameter bound in each case, a contradiction.

If \(K_t\) separates \(w\), polygonal closed walks of nonzero winding number about \(w\) can be drawn arbitrarily close to \(K_t\). For example take a fine square grid in general position, form the union of closed squares meeting \(K_t\), and trace the boundary of the bounded complementary component containing \(w\). Choose the grid finer than a fixed fraction of \(\mathop{\mathrm{dist}}(w,K_t)\). Subdivide such a walk, replace its vertices by nearby points of \(K_t\) other than the endpoint, and join consecutive replacement points by the small paths just constructed. The resulting closed walk has the same winding number: each replacement remains in a small neighborhood avoiding \(w\), and can be joined to the original small segment there. It consists of finitely many paths, each lying strictly before time \(t\), and is therefore contained in \(K_{t'}\) for some \(t'<t\). Nonzero winding implies that \(K_{t'}\) already separated \(w\). This contradicts first disconnection at \(t\). ◻

Proof. Let \(\gamma\) be such a limit, in the initial disk coordinates. For a fixed sampling radius \(v\), couple its countably many sampled domains with an SLE trace \(\xi\) having their law; the latter can be sampled conditionally on the domains. A diagonal extraction over finite numbers of steps gives all the samples in their order on \(\gamma\). At every sample, the path passes through its tip, the preceding part of \(\gamma\) avoids the open target domain, and the following part lies in its closure. These are direct limits of the corresponding band inclusions, using the domain convergence above.

For fixed \(v\), the SLE capacity increments between samples are independent and identically distributed, strictly positive, and bounded above by \(Cv^2\). The upper bound is the elementary half-plane capacity bound for a hull contained in a half-disk of radius \(v\); positivity and independence follow from the nonconstant chordal SLE trace and its domain Markov property with the stated normalization. Their sums therefore diverge almost surely. SLE continuity at the start and transience to its target allow its time interval to be compactified, with the target as the final point.

Take a sequence \(v\downarrow0\). For each \(v\) retain the closed monotone correspondence between the positions of the samples on the parameter intervals of \(\gamma\) and \(\xi\), including both endpoint pairs. Take a subsequential Hausdorff limit of these correspondences, using tight parametrizations of the two curves. At the same time retain, at every rational SLE time \(s\), both the compact closure \(\overline{D_s}\) of the SLE target domain and its closed complement. Encoding both is necessary to pass the two band inclusions. The joint law of the SLE curve with these closed sets is independent of \(v\), so every subsequential limit retains the target domains of its SLE coordinate. No continuity of the map from traces to target domains is needed. The sample capacity gaps are at most \(Cv^2\); hence the limiting correspondence has full projection on the SLE parameter interval. Its paired points coincide in space. Tail sample pairs accumulate at the target, so the final endpoint causes no missing projection.

SLE has no constant time interval. A value of the \(\gamma\) parameter cannot therefore be paired with two distinct SLE times. Any interval missing from the first projection has its two endpoints paired with the same SLE time \(t\). If the correspondence does not already identify the two curves up to pauses, there is consequently a nonconstant excursion of \(\gamma\) that leaves and returns to \(\xi(t)\) inside such a missing interval. Every point \(w\) on this excursion satisfies \[ w\in\bigcap_{s<t}\overline{D_s}, \qquad w\notin\bigcup_{s>t}D_s. \tag{19}\] It suffices here to use rational \(s\). For the first inclusion, apply the suffix condition at a sample between \(s\) and \(t\); for the second, use the prefix condition at a sample between \(t\) and \(s\). Monotonicity of the target domains and the retained closed sets pass these conditions to the limit.

If \(\xi(t)\) is on the initial boundary, the excursion violates the absence of a boundary double visit of \(\gamma\). If it is interior and also visited by SLE at another time, the full projection of the correspondence supplies a third visit of \(\gamma\) in addition to the two excursion endpoints. This violates the absence of interior triple points. We are left with a uniquely visited interior tip.

A point satisfying (19) that is off the SLE trace and off the initial boundary would have disconnection time exactly \(t\). Indeed it has a neighborhood free of trace; while a connection to the target exists, a compact connecting path remains open for a short later time. The two inclusions say respectively that disconnection has not happened earlier and that it has happened immediately afterward. Lemma 22 excludes this possibility at a uniquely visited interior tip. The hidden excursion must therefore lie entirely on the SLE trace or the initial boundary.

Take a small interior neighborhood of \(\xi(t)\). Every other point of the hidden excursion there that lies on the SLE trace has another visit supplied by the correspondence, and is a double point of \(\gamma\). A nonconstant excursion leaving \(\xi(t)\) must cross one of a fixed countable collection of line segments forming arbitrarily fine rational grids inside this neighborhood. That crossing would be an interior double point on a fixed segment, excluded by Lemma 18. Thus the excursion is constant. No missing nonconstant interval remains, and the ordered curve is SLE. ◻

Proof. First work at a fixed mesh, with a virtual black bank whose actual color is white. List the edges from that bank to revealed vertices off the bank, including the switch edges. Order them along the bank and then within successive interior vertex stars. Thus several entries may use one bank vertex. Label each entry by the color of its off-bank endpoint. The white entries are exactly the places where the virtual wall crosses a bank-incident edge. Their order along the wall is the bank/star order: the short spokes joining them to the bank lie on one side of the simple wall and cannot cross.

Between two consecutive white entries the list has the form \[W_i,B_1,\ldots,B_k,W_{i+1}.\] Write \(E_i\) for the virtual excursion between those white entries and \(I_i\) for the intervening bank interval, retaining the star order if the interval is a single vertex. Apart from its end triangles, \(E_i\) is an actual interface after restoration. We describe the returning chain that completes it.

Consecutive entries bound a common face, possibly a triangle, because there are no rim chords and every incident face has a single bank interval or vertex by Lemma 16. For \(k>0\), each middle wedge \(B_j,B_{j+1}\) is a triangle or an unknown face whose boundary is black before restoration. Restoring the bank to white creates exactly two switches there. Join them directly through the triangle, or by exploring that unknown face with its actual two-switch boundary conditions. The end wedges \(W_i,B_1\) and \(B_k,W_{i+1}\) have mixed colors before restoration, so they cannot be unknown monochromatic faces; they are band triangles. After restoration they join the returning chain to the surviving ends of \(E_i\). Consequently a nonempty run completes that same excursion to one real loop. An empty run through only bank triangles has no macroscopic contribution. White entries separate different runs, so this operation cannot concatenate distinct excursions. Unused interfaces lie in the other unknown faces, including residual faces of these closing explorations. This is a finite incidence identity; it asserts no independence of the virtual excursions.

We next show that the identity passes to the limit. Work in the initial disk coordinates of this step. A sequence of excursions of diameter bounded below has, by curve tightness, a limiting pair of endpoints and an ordered limiting arc. Its interior cannot acquire a contact with the virtual bank: during the discrete excursion neither bank shadow uses that bank, so such a contact between positive excursions gives three free half-plane arms as in Lemma 18. Conversely a compact subarc that avoids the bank in the limit eventually avoids it discretely. There is no boundary interval of the limiting wall, no boundary double visit, and no return at a switch. Hence positive-size excursions have distinct endpoints and match in their order; no positive excursion can collapse to a spike at one boundary point.

Every limiting contact with the open bank is approximated by an actual recorded occurrence, not just the contacts selected by a diameter cutoff. Otherwise the approaching and departing wall segments in an occurrence-free neighborhood would give the same three-arm event. Together with the initial and final bank points, this recovers the full order of the contacts.

For a matched excursion, the intervening bank interval is incident with one limiting face of positive size. Every compact subinterval of that interval has an adjacent strip avoiding the band; its limiting face has a Jordan boundary by Lemma 19. Its two attachments converge to the excursion endpoints, so its actual two-switch marks also converge. Proposition 15 supplies the closing exploration. Any other incident faces involved in that run have attachments collapsing to the endpoint neighborhoods. Their diameters tend to zero: a positive diameter face of the virtual bank’s color with a collapsing attachment interval, including a singleton, is excluded by Lemma 18. Thus their pieces cause no error in the curve metric when the loop is pasted.

The same argument rules out a surplus macroscopic loop completed from vanishing excursions at this step. If \(E_i\) shrinks, its endpoints coalesce, and continuity of the inverse Jordan-bank parametrization makes \(I_i\) shrink as well. Every completion face for this run is attached to \(I_i\); all its band triangles are within a vanishing mesh distance of that interval. The diameter of the completed loop is therefore bounded by the diameter of \(E_i\), the diameter of \(I_i\), twice the largest completion-face diameter, and a vanishing mesh error. The collapsing-attachment exclusion makes that largest diameter tend to zero. This also excludes an accumulation of many small completion pieces into a macroscopic loop. An other-colored face whose boundary merely touches the bank in the limit, without genuine cycle edges on it, receives no new switch assignment. True intervals of attachment have been distinguished by Lemma 19. This distinction prevents an incorrect extra reconnection at a limiting contact.

Actual two-switch explorations leave monochromatic faces again. Their unknown sites are fresh conditional on all queried colors by the product-law argument preceding Lemma 16. Select and match faces, for example, using interior points from a fixed dense countable set that avoid the traces. Along every deterministic converging sequence of selected Jordan faces, Proposition 15 applies. The bounded-integration argument used for stopped kernels therefore gives joint convergence of each finite set of operations. Uniform continuity of the Jordan coordinate maps transfers all these statements back to physical coordinates. ◻

Proof. Lemma 24, iterated, proves convergence of every fixed finite number of recursive operations. It remains to justify that finitely many operations discover all loops above a fixed scale with arbitrarily high probability. We use a property of the already identified continuum SLE construction, checking its precise scope.

Fix a conformal map from the unit disk onto the initial Jordan domain, extended as a homeomorphism of closures. In these initial disk coordinates choose the exploration order and boundary endpoints as in [8]: priority is by maximal coordinate span, with its specified dense-point tie breaking; monochromatic domains use the maximal-span endpoint rule, and two-color excursion domains use their two prescribed switches. Start with that construction’s monochromatic disk and virtual first exploration and transport the recursion by the fixed map. For this continuum recursion the number of steps until every remaining domain has coordinate diameter below a fixed \(r>0\) is almost surely finite. Here is the exact consequence of [8] that proves this assertion. Theorem 5 there gives finite-step coupling with the reorganized triangular construction, and its proof, equation (20), proves tightness of the stronger triangular stopping count until all remaining domains have coordinate span below a prescribed cutoff. The final part of that proof transfers this estimate from the original construction to the reorganized one by its maximal-span comparison.

To pass that statement to the continuum, suppose that after \(n\) steps a continuum remaining domain has diameter greater than \(r\). Choose a compact connected subset of its interior with diameter greater than \(r-z\), by joining two interior points with a compact path. This subset has positive distance from the finitely many explored boundaries. Under the finite-step coupling it stays in a corresponding triangular remaining domain for all sufficiently small lattice meshes. Hence that triangular domain has diameter greater than \(r-z\), and its coordinate span exceeds \((r-z)/\sqrt{2}\). Taking the mesh limit in the tight stopping-count estimate, and then \(n\to\infty\), proves the continuum assertion, using a smaller cutoff to obtain the strict diameter inequality. This use of the triangular theorem establishes a fact about the known continuum SLE recursion; it supplies no convergence assertion for the present tessellations.

Uniform continuity of the fixed initial conformal map converts this coordinate exhaustion into exhaustion at every positive physical diameter. We may now couple finitely many steps of our construction to those continuum choices. On each matched boundary choose the nearest ordered discrete endpoints. The choice depends only on the coupled past, so it leaves the unknown discrete colors with their correct conditional product law. This avoids imposing an unwarranted continuity assumption on a maximizing endpoint rule. At each step Lemma 19 matches the remaining macroscopic faces with no surplus. Induction gives that property after every fixed finite number of operations.

Fix an error probability and first choose, using continuum exhaustion, a deterministic number of operations after which all continuum remaining faces are smaller than the desired cutoff with the stated high probability. If an excursion already drawn still needs its matching face exploration to complete a macroscopic loop, perform those additional operations as well. For each positive excursion cutoff there are only finitely many such mates; include all of them and then decrease this auxiliary cutoff. The no-surplus assertion of Lemma 24 ensures that vanishing excursions at any of these finitely many steps cannot produce a missing macroscopic loop. All other loops lie within the remaining small faces. Finite-step convergence now gives the required loop matching, and then the error probability and auxiliary cutoffs may decrease. The incidence rule retains orientation and multiplicity throughout. Finally, on deterministic geometry different replicas have independent colors, so their finite recursive laws are independent products; the same limiting and cutoff arguments give their joint convergence. ◻

Proof. A crosscut between opposite sides separates the other pair. Paths running arbitrarily close to its banks give the asserted color connections; they can be taken with clearance on their respective sides. The two types of opposite crosscut cannot coexist, because they would intersect.

In their absence every crosscut has both ends on one side or on consecutive sides. On the boundary circle the endpoint pairing is noncrossing. A same-side pair cuts off a pocket containing no mark. A pair on consecutive sides cuts off the corresponding corner. Remove the pockets and, for each corner that is cut off, its outermost corner pocket. The remaining region is connected and incident to each of the four side intervals between the removed pockets. This can also be proved by induction: remove an innermost paired interval first, then restore it as a pocket along one boundary interval of the remaining region. Interior closed curves only remove interior pockets. Consequently the region left between the corner pockets meets all four sides. A path through it joins either pair of sides, and its color decides which crossing occurs. The bank of an outermost corner cut facing this region has that color. All paths used here lie in the quad; no identification of different boundary points or external connection is made. ◻

Proof. We give both the geometric interpolation and the arm test for a single local change. Choose \(u>0\) smaller than the separations of all unrelated sides and squares, and put balls of radius \(Cu\) around the polygon vertices, marks, and square corners. Outside these balls divide each straight edge into \(O(s^{-1})\) intervals of length comparable to \(s\), where \(s\ll u\). For a displacement much smaller than \(s\), perform the motion one interval at a time, using a piecewise linear cutoff supported on a slightly longer interval. Leave the portions within distance \(2u\) of the vertices for the corner moves, and taper the edge moves between distances \(2u\) and \(3u\); motions in balls of radius \(4u\) then complete the interpolation. There are boundedly many corner moves and \(O(s^{-1})\) edge moves. A polygonal collar identifies the vicinity of each edge with a straight strip. Interpolating its sufficiently small piecewise linear ambient displacements preserves the Jordan property; it also gives the familiar quad sandwiches by shifting target and non-target sides in their respective directions. Squares are treated by the same construction. Each intermediate prescribed region remains a connected disk, and the part common to the two regions in a local comparison is connected. This last property can be ensured by taking the motions to be graphs in their collars and retaining the unchanged inner square. All constructions are fixed before \(e\to0\).

Consider an edge move supported in a ball of radius \(Cs\). If it changes a crossing, take a black crossing from the configuration where the crossing occurs and an open white complementary crossing from the other configuration. Outside the changed neighborhood their paths run in the common tiling and coloring. For a quad-side move, their portions going to the other three sides give three disjoint free arms from scale \(Cs\) to scale \(cu\). They lie in a half-plane translated by \(O(s)\) from the original supporting line. Trimming at first and last hits of auxiliary curves preserves the order of the arms. The paths stay in the common region of the two domains; using the enlarged half-plane therefore does not require the intermediate boundary itself to be straight.

At an edge of a black prescribed square, the two arms of the white complementary path are free. At least one of the two black branches towards the target sides is also free until scale \(cu\). Otherwise both target sides would connect, before entering the changed neighborhood, to the common black prescribed region. That region is connected also in the comparison without a crossing, so it would already join the target sides there, a contradiction. The white branches cannot connect to one another in the common part so as to avoid the changed region, since they would then block the black crossing in the other comparison. After proper trimming we thus have three disjoint free arms in the exterior half-plane of the square edge. A white prescribed square is handled with the colors interchanged. The open white connections through a white prescribed region exist with clearance, because its forced cells form a connected union by side adjacency.

Here and below the “cells meeting” convention is compatible with that connectivity. Follow a path inside the connected prescribed region through the cells it meets. Consecutive cells share a side, or meet at a tiling vertex where the three cells are pairwise side-adjacent. The resulting side-adjacency chain connects all cells meeting the region. A path in those cells can be moved into cell interiors and relative interiors of shared sides. On an unchanged straight boundary piece, convexity of each cell and the shortcut argument for crossing and complementary crossing remove possible reuse of a cell by two candidate arms. The finitely many bends introduced by a fixed interpolation require one further observation. A shared cell there, despite the arms continuing to a fixed outer scale, would give a free arm from a vanishing neighborhood of that bend to that scale. Its probability tends to zero by the one-arm bound. Thus the graph-arm extraction is valid with vanishing error, including for the exact closed-cell crossing convention.

At a corner move, a changed crossing requires at least one free arm from scale \(Cu\) to a fixed positive scale. At a quad vertex or mark this is a branch of the crossing towards a target arc away from that vertex. At a prescribed square it is a branch of the color opposite to that square; this branch cannot use the prescribed region. Since all other prescribed regions and unrelated boundary pieces are at fixed positive distances, the arm is free throughout the annulus used for this estimate.

A union bound and (20) give, for displacement small enough in terms of \(u,s\), \[ \limsup_{e\to0}\mathbb P(\hbox{some compared indicator differs}) \le C u^\gamma+C s^{-1}(s/u)^\beta. \tag{21}\] Constants may depend on the fixed tests and their separations. The vanishing errors at the finitely many interpolation vertices have already disappeared in the mesh limit. Given an error tolerance, first choose \(u\) so that the corner term is small, then choose \(s\) so that the edge term is small, and finally restrict the displacement. Equivalently, in the displayed bound the limits are taken as \(e\to0\), then \(s\downarrow0\) at fixed \(u\), and then \(u\downarrow0\). This proves the lemma. ◻

Proof. Approximate the outer boundary and hole boundaries in disjoint thin strips by simple site cycles, in boundary order, using the graph approximation construction of Section 2. Hole rims are free vertices. The graph outside these cycles, with each hole triangulated by its fan, is a finite planar triangulation. The terminal neighbors are exactly the sites on the corresponding target arc; the terminals create no connection between other boundary vertices.

Shadow an original closed-cell crossing by a site path with vanishing error. Before it enters a black forced square, it crosses the rim of the enlarged middle square through free black sites; crop there and replace its subsequent passage through that square by the black fan. An original black path stays away from every shrunken white hole. The quad collar slack allows the shadow to be clipped at the middle target arcs and keeps it clear of the middle non-target sides. This produces the claimed terminal path. These statements only use the small mesh of cells on the fixed compact region and the strictly positive slack, so hold for all sufficiently fine configurations on the given crossing event.

Choose an outermost black terminal path on one side. More precisely, add an external edge joining the terminals, take the union of all simple black terminal paths together with that edge, and take the appropriate exterior boundary cycle of this union. Each constituent path with the external edge is a cycle, and these cycles share that edge; their union is two-connected. Its exterior boundary therefore supplies a simple terminal path \(P\). No black path can leave \(P\) into the exterior side and return to a distinct vertex of \(P\): taking its first departure and first return would produce a terminal path farther on that side, contrary to the boundary construction. We call this the no-bypass property. It is the same elementary outermost-path construction used in Lemma 8.

A simple \(P\) visits each fan vertex at most once and never visits a white fan vertex. Remove the fan edges and terminal edges. The remaining segments join distinct rim or terminal sets and consist entirely of free black sites. There are at most \(J+1\) of them. Many traversals of a fixed annulus by one segment yield equally many disjoint free black arms, after proper trimming. The one-color disjoint-occurrence bound and the one-arm bound control these numbers. The successive-excursion compactness argument in Lemma 17 consequently applies to these segments, also at their endpoints. Notice that the segments may be selected using all the colors; the bound is on the existence of the disjoint arms and does not assume freshness conditional on the selected path.

We give the additional argument that rules out a self-touch in a limit. Suppose first that the contact lies in the bulk, at positive distance from all rims. Two visits with a nonconstant intervening excursion produce four proper traversals of a sufficiently small annulus, with a fixed positive outer radius. Between the traversals, on the side forbidden by the no-bypass property, there are at least two disjoint radial gaps. Indeed, in cyclic order the four directed traversals have distinct sectors on their forbidden sides; at most two traversals can border the same such sector. Each selected gap contains a white arm. To justify the last assertion without assuming smooth graph cuts, choose two intermediate graph cycles, with wide radial margins. Within the gap each cycle has a transverse vertex path between its two path banks. One way to see this is to label just the vertices of that cycle black and apply Hex in the gap: an avoiding radial white path would cross the separating cycle, which is impossible. Use an innermost such cut at the inner cycle, and then one at the outer cycle. They enclose a quadrilateral whose side banks are portions of \(P\). A black crossing between these banks would leave into the forbidden side and return to \(P\); trim at its first return to obtain a bypass. Hex therefore gives a white crossing between the transverse cuts. With the cuts chosen away from the trimmed ends, the departure points on \(P\) are internal and both adjacent path edges are still present, as required for the no-bypass argument. Trim the white crossing to the smaller annulus. Doing this in the two disjoint gaps yields four black and two white site-disjoint arms. All vertices involved are free; no fan edge or rim has entered this annulus.

By Lemma 9, the probability of such a six-arm event in any of \(O(r^{-2})\) balls of radius \(r\) is \(O(r^{c})\), when its outer radius is fixed. It tends to zero. At a point of a straight rim or outer boundary, a self-touch gives at least three free traversals in the corresponding half-plane. There are three even if one of the two visits is an endpoint of the segment. Covering that side by \(O(r^{-1})\) balls and applying Lemma 13 again gives a vanishing bound. At the finitely many polygon corners, a single free arm reaching from vanishing scale to a fixed scale already has probability tending to zero. The endpoints of a segment lie in distinct, positively separated rim or terminal sets, so they cannot coincide with each other. These exclusions at countably many outer scales rule out every nontrivial repeated point of a limiting segment. ◻

Proof. It suffices to consider a subsequence on which the asserted mismatch has positive limiting probability. Include its indicator, the crossing vectors, and the at most \(J+1\) segments of Lemma 28 in a joint subsequential representation. Pad the segment list when necessary and include its finite combinatorial itinerary through the hubs as a coordinate in a finite discrete space. In the representation this itinerary stabilizes almost surely, though its limit may be random. Tightness gives convergence of the segment curves; Lemma 28 makes each nonempty limit a simple arc. On the represented mismatch event their ordinary critical crossing vectors agree in the limit, coordinate by coordinate.

For each limiting segment, choose its two end neighborhoods as follows. An end at a black hub lies strictly inside the final enlarged black square; an end at an outer terminal lies in the collar on the appropriate side of the final target cap. The segment stays a positive distance from the final white squares, because these are strictly smaller than the middle white holes. Its non-target boundary approaches have room inside the final non-target collar. Near the two ends, cut the simple arc at points still within the prescribed end neighborhoods. The planar Schoenflies theorem gives a rectangular neighborhood of the intervening simple arc, which can be chosen as a thin tube around it. Choose a polygon quad in this tube whose target end cuts are inside those neighborhoods and whose lateral sides leave room around the arc. A sufficiently close rational polygon approximation preserves these properties. Uniform convergence in curve order implies that the corresponding free segment in the first model crosses this fixed quad for all sufficiently large indices.

The witnesses can depend on the limiting configuration, but are selected from a countable list. On the representation, convergence of a Boolean coordinate means eventual equality of that coordinate. Thus, simultaneously for every member of the list, a witnessed black crossing in the first sequence is also present in the second for all sufficiently large indices. There are only finitely many witnesses on the event under consideration. Their crossings in the second sequence avoid its final white squares: the tube clearance eventually exceeds every cell diameter in the fixed region, so none of the cells in a tube can be recolored white by a distant square. If a crossing enters an enlarged black square, that only helps it. At a hub, the two adjoining witnesses have endpoints in the open square, so the forced black cells connect them there with clearance. At the outer ends, join in their prescribed order and clip the resulting path at the final target caps. This yields a black crossing of the final test, contradicting the represented mismatch. The tubes need not be disjoint: their only required joins occur inside their specified black squares, and any additional intersection of black witnesses is harmless.

For several tests apply this argument to each positive indicator; for an indicator equal to zero use the same implication in the reverse coupling when deriving equality of laws. For replicas keep a separate segment list and crossing vector in each replica. The arm bounds apply separately, and a finite union bound proves the assertion without any assumption that the replicas have independent geometries. ◻

Proof. For unmodified tests the assertion on a deterministic good sequence was proved in Subsection 4.1. On any subsequence, compactness of the countable Boolean crossing space permits a representation in which these ordinary test indicators converge coordinatewise. Couple two candidate subsequential limits through their common unmodified crossing vector. Lemma 29 transfers every positive prescribed test from either sequence to the other’s arbitrarily small favorable perturbation. By Lemma 27, replacing that perturbation by the specified test costs a probability tending to zero with its size. For precision, let \(X_e^{a,i}\) be test \(i\) in sequence \(a\in\{1,2\}\), and let \(X_e^{a,i,+v}\) denote a favorable perturbation of size \(v>0\). The transfer implication in both directions gives \[\limsup_{e\to0}\mathbb P(X_e^1\ne X_e^2) \le \sum_i\sum_{a=1}^2\limsup_{e\to0} \mathbb P(X_e^{a,i,+v}\ne X_e^{a,i}).\] The right side tends to zero as \(v\downarrow0\) by continuity. Thus the resulting Boolean vectors agree with probability tending to one under the coupling, so their subsequential laws coincide. This proves existence and commonality of the limit, including different prescriptions in different tests on the same colors.

The continuity assertion follows directly from Lemma 27. For mesh-dependent centers or radii, bracket each moving square between two fixed concentric squares with radii arbitrarily close to its limiting radius. Force the inner or outer square according to the monotone direction of its color: enlarging black helps a black crossing, while enlarging white hinders it. With a corresponding quad sandwich if needed, the specified moving test lies between two fixed tests. Continuity makes their disagreement probability vanish. This proves the moving-parameter statement without requiring estimates at mesh-dependent positive scales.

On a deterministic tiling independent color replicas have product laws. Apply the preceding convergence to the finite vector in each replica; their joint law therefore converges to the corresponding product. For Poisson geometry, Lemma 7 says that from every vanishing-mesh sequence one can extract a subsequence along which the environments are almost surely good deterministic sequences. On this subsequence, each conditional atom probability for our finite Boolean vector converges almost surely to its just identified deterministic value. There are finitely many atoms, so the conditional total-variation distance tends to zero almost surely. The subsequence criterion proves convergence in probability along the original sequence. The argument applies to the product color measure on the same fixed environment and hence retains the shared geometry in the replica assertion. Averaging gives the corresponding annealed limits. ◻

Proof. We first establish a pointwise geometric bound and then integrate it. Put \(W(\mathbf w)=\prod_{i,r\in I_i}\|w_{i,r}\|_{\mathop{\mathrm{TV}}}\); the assertion is immediate if one of these factors is zero. For fixed geometry and unselected fair bits, the signed integral in (28) vanishes if \(F\) is independent of any one selected history, because its weight has mass zero. Thus a necessary condition for a nonzero integral is the following: at every selected coordinate there are settings of the inserted histories for which changing that coordinate changes at least one tested crossing. The settings witnessing this condition may differ from coordinate to coordinate. We use only the resulting arm tests in regions containing no inserted site, where all the colors are the original fair bits. In particular, no enumeration of all global history assignments is required.

Our geometric objective is one four-arm factor \(a_e\) for each inserted site, with the remaining cost expressed as a product of integrable singularities in intersite distances and distances to polygon vertices and marks. The merging forest will select the distances and the disjoint annuli that supply these factors.

Proof. Choose \(p>1\) with \(gp<2\). For fixed \(k\), there are only finitely many ordinary forests on the \(k\) sites and the \(J\) fixed roots. The pointwise estimate (35), multiplied by \((m_eq_e)^k\), is bounded by a constant times the sum over all such forests having at most one root per component of \(\prod_f(|x_{i(f)}-x_{j(f)}|\vee b)^{-g}\), with a root coordinate interpreted as fixed. Raising this finite sum to the power \(p\) costs only a constant depending on \(k\). Leaf integration, now with exponent \(gp<2\) and with distances bounded by a fixed constant containing \(B\) and all roots, bounds each forest integral uniformly in \(e\). The lattice estimate is the same dyadic calculation as (36). This proves (38).

Hölder’s inequality bounds the expression before the limits in (39) by \(C_k^{1/p}\sigma_e^{\otimes k}(\mathcal B_\rho)^{1-1/p}\). For area measure this volume tends to zero as \(\rho\downarrow0\). For lattice volume its mesh limit is the same: the finitely many polygon edges and the pairwise diagonals have zero area in the relevant product space, and their \(\rho\)-neighborhoods can be bounded by finite unions of boxes with arbitrarily small total volume as \(\rho\) decreases. This proves the ordered-limit assertion and includes vertices, marks, and box edges.

It remains to justify removal of \(\Gamma_e\). Adding a fixed number of nuclei can only improve its occupancy requirements. Thus the difference between truncated and untruncated kernels is bounded uniformly in the locations by \(C_k\exp(-c/e^2)\). The normalization \((m_eq_e)^k\) grows only polynomially in \(1/e\), by the lower arm bound in Lemma 3. The normalized difference therefore converges uniformly to zero, also in \(L^p\). ◻

Proof. Choose deterministic graph trim cycles a fixed number of doublings inside the two end collars, leaving a factor of room on both sides. Use the quenched alternating-arm lower bound on a longer annulus. A bad trim means that two distinct proper arms, or an additional visit of another arm, approach within relative distance \(\delta\) of a candidate endpoint. Both relevant path pieces continue to the two sides of the collar. Lemma 11 bounds the probability of such an event in the collar by a quantity tending to zero with \(\delta\): cover the trim curve by \(O(\delta^{-1})\) balls, and use the any-color four-arm bound of Lemma 9 in each ball.

This absolute collar estimate yields a relative error here. Retain the necessary alternating arms in the rest of the original annulus, deleting only a fixed number of end scales and a buffer at the bad collar. Its color coordinates are disjoint from those of the collar test. Multiplying the two bounds and using quasi-multiplicativity gives at most \(o_\delta(1)A_e(l,L)\), uniformly in the intervening ratio. Choose \(\delta\) so that the two bad ends together cost less than half the arm lower bound. The retained proper arms then end in four boxes from a fixed finite list; only the appropriate arm enters the reserved enlargement of each box. Loop erasure and first-contact trimming keep retained paths on their designated side of the trim, up to the vanishing mesh margin.

For each choice of endpoint boxes, apply Lemma 12. A lasso surrounds each endpoint box, a corridor starts inside that lasso, and disjoint corridors route in cyclic order to the target patches. A cyclic offset is allowed. These routes use only the fixed-ratio end spaces, so their widths and crossing costs do not depend on \(L/l\). Their overlaps with retained paths occur solely in the reserved color-specific enlargements. Conditioning on coordinates outside the connector supports, Harris’ inequality in those supports gives a positive factor times the retained arm event. The event being extended is the existence of suitable paths, not a conditioning on a previously inspected choice of paths. Taking a maximum over the finite endpoint choices proves (44) and the asserted support and monotonicity properties. ◻

Proof of Lemma 33. Apply Lemma 35 between scales comparable to \(h\) and a fixed macroscopic radius about the square center. At the small end continue the four separated tracks into four patches lying strictly inside \(B_\infty(x,h)\). At the large end continue the black tracks across the target sides of the rectangle and the white tracks across the other two sides, away from its corners. These end extensions have uniformly positive conditional probability on typical geometry. Forcing the square black joins its two black tracks and gives the tested crossing. Forcing it white joins its white tracks and gives a complementary white crossing with clearance from black. Thus the square changes the crossing. The event has conditional probability at least \(cA_e(h,1)\) on typical geometry. Averaging, and taking the mesh sufficiently small for the fixed \(h\), proves (42). All tracks can be chosen with fixed positive widths for centers in a small central neighborhood, giving the final uniform assertion. ◻

Proof. Leave a core annulus strictly between the end regions, of ratio \(L/l\) which will be chosen large. Guarded alternating arms through the sleeve cost at least \(c(l/L)^g\), where \(g<2\), by Lemma 35 and the power lower bound. Exclude in the core four alternating arms together with a fifth disjoint arm of either color. The quenched upper bound for this event on typical geometry is at most \(C(l/L)^2\). The end margins change these bounds only by fixed constants. Choosing \(L/l\) large and then fixing it leaves positive probability, uniformly for small meshes. The exclusion uses only core coordinates and therefore preserves the endpoint monotonicities.

We verify that the exclusion produces the claimed cycle. Trim the four arms to proper simple traversals of the core. They divide it into four topological rectangles, each with an opposite-colored arm as each of its two lateral boundaries. In such a gap, let \(\mathcal B\) be the black cluster reached from its black lateral boundary, using only vertices of the gap. If \(\mathcal B\) is adjacent to the white cluster of the other lateral boundary, there is a transverse path that is first black and then white, with a single color switch. Otherwise let \(N(\mathcal B)\) be the external vertex boundary of \(\mathcal B\) in the gap. Every vertex of \(N(\mathcal B)\) is white, and every transverse path meets it. Temporarily color \(N(\mathcal B)\) white and its complement black. Hex duality in the gap then makes \(N(\mathcal B)\) contain a proper white path between the two annular ends, disjoint from the white lateral boundary and from the four original arms. This is the excluded fifth arm. Hence the single-switch transverse path exists in every gap. Orient it from its black to its white side, stop at its first white-side contact, and start at the last preceding black-side contact. Erase loops within each color segment. Its interior then avoids both original side arms.

Join the two contacts on each original arm by the intervening subpath of that arm. The four transverse paths lie in disjoint gaps, and these four joining subpaths lie on disjoint arms. Their union is a simple enclosing cycle; if two contacts on an arm agree, the joining subpath is a single vertex. Its only color changes are the four switches on the transverse paths. It therefore has the required four runs. Each original arm meets its corresponding run, and its portions on either side supply the connections to the two guarded ends. ◻

Proof. Hex duality in the disk bounded by the inner necklace says that precisely one of the two black runs being joined or the two white runs being joined holds. Connections include the vertices of the runs. The four alternating arms transmit whichever join holds to the corresponding runs of the outer necklace. The opposite join at the outer necklace is impossible by planar separation. Thus the bit is preserved.

For the last assertion cut the graph at the outer cycle. Each run is already connected along the cycle, so the only undetermined same-color connection among boundary vertices is the join of the two black runs, or equivalently the alternative white join. An exterior path can therefore be read from this bit and the fixed exterior graph. The cycle is separated from every tested boundary. Site paths and cell paths agree at this cut: an inner site cannot connect to the exterior without meeting a cycle site, and the cycle cells and their stars are localized by the screens. Outside that cut the original closed-cell convention is left unchanged. A quad disjoint from the inner disk cannot use it. The same argument permits arbitrary fixed histories at the other insertion regions, which lie outside the trial. ◻

Proof. Condition first on all Poisson geometries, keeping only realizations in \(T_J\) and in the geometry part of \(E\). In each source environment retain four arms from the inner beginning of its probe to the inner trim curve, and four arms from its outer end down to the outer trim curve. Stop at first contact from the retained side, with the graph margins prescribed above. Since the trim is not bad, their endpoint boxes can be chosen from a fixed finite list with matching cyclic order and the reserved avoidance conditions. For each choice let \(\mathcal R\) be the event that such retained paths exist, together with failure of all trials other than \(J\). Delete the color part of disqualification, failure of trial \(J\), and all arm requirements between its two trims. The events \(\mathcal R\) for the finitely many box and order choices cover the left-hand event of (51). Their definition concerns existence of retained paths; we do not reveal or condition on a selected path.

In the four sleeves, two in each source environment, impose the events of Lemma 36, with cardinal endpoint patches of matching alternating colors. Sleeve coordinates are disjoint from the retained path coordinates and from the other trial supports. Thus these four events have joint conditional probability at least \(c_0^4\), and intersecting them with \(\mathcal R\) costs at most that fixed factor.

We describe precisely the additional connector supports. In each extension space use disjoint polygonal tracks from the retained endpoint boxes to the corresponding sleeve patches, first leaving the trim in its normal direction and then respecting annular cyclic order. At each end start a track inside the box surrounded by the guarded lasso. In the seam space use four fixed separated tracks, of the same respective colors for every inside–outside combination, and require rectangle-chain connections in each of its four sewn geometries. Every rectangle and lasso used by a connector lies in these tracks or in a reserved endpoint enlargement. Include their cell stars in the supports.

Let \(V_{\rm B}\) and \(V_{\rm W}\) be, respectively, the original source-bit coordinates used by black and white connector requirements, taking the union over the four sewn combinations. The positive widths between opposite-color tracks and the mesh screens ensure \(V_{\rm B}\cap V_{\rm W}=\varnothing\); a source site cannot belong to two opposite track supports, even if it is used in several sewn geometries. On \(V=V_{\rm B}\cup V_{\rm W}\) put the product order that favors black on \(V_{\rm B}\) and white on \(V_{\rm W}\). Every connector event is increasing in this order. Its conditional probability, with geometries fixed, has a uniform positive lower bound from the crossing conditions in \(T_J\). Harris’ inequality therefore bounds the probability of all connector events by a positive constant from below, since their number is fixed. These events depend only on the coordinates in \(V\).

The retained-and-sleeve event has the same monotonicities on \(V\). For retained paths the only overlaps are their reserved, color-specific endpoint enlargements. For sleeves the only overlaps are their guarded end patches; the five-arm exclusion giving the necklace lies strictly in the core and uses no coordinate of \(V\). Requirements that paths avoid other boxes are restrictions on allowed path locations, so favoring their own colors preserves the existence of the required paths. Failures of other trials use no coordinate of \(V\), or of these sleeves, because the enlarged trial supports are disjoint. They need not be monotone.

Now condition on all color coordinates outside \(V\), apply Harris in \(V\) to the retained-and-sleeve event and the intersection of the connector events, and integrate back. This costs another fixed positive factor. All joining requirements are realized simultaneously, including those in geometries that combine different sources. Lassos ensure actual connections to retained paths, while the reserved tracks ensure disjointness, including between arms of the same color. In the two original environments the resulting arms complete \(E\). In all four combinations they give \(U_J\), with the common necklaces supplied by the sleeves. Other trials still fail. Hence the probability for each retained choice is at most a constant times the right-hand probability in (51). Summing over the bounded finite choices and then integrating the geometries proves the claim. ◻

Proof of Proposition 34. Choose a desired relative error \(z>0\). First fix \(M\) so that the cutoff error in (46) is sufficiently small. Next choose \(l_*/R\) so that (49) is sufficiently small. Fix the sleeve and extension ratios, and the trim tolerances so that the disqualification term \(\rho\) is sufficiently small. These choices fix \(C_*\). Choose \(n\) so large that \(C_*C(M,R)/n\) is sufficiently small, and place that many buffered trials below the protection annulus. Only now take \(h\) sufficiently small to lie below every trial; for each such fixed \(h\), let \(e\downarrow0\). All local constructions are therefore at positive scales when their geometry conditions are imposed.

On configurations with a successful protected seam, the adaptive measure-preserving involution and the signed identity following Lemma 37 identify the two product integrals. The remaining portions are bounded in total variation. A nonzero signed contribution has the necessary probes at every insertion point; the independent probes at the other points contribute \(C P_e\). At the first point the discarded portions are bounded by (46), (49), and (52), together with the vanishing fixed-scale geometry errors. The weights contribute only their fixed total-variation bounds. Thus the limsup of the ratio in (43) is at most \(Cz\); since \(z\) is arbitrary, it is zero.

All constants are uniform on the fixed compact separated location range. Local layouts are translated copies of the same finite lists. In the Poisson model Mecke points at other insertions lie outside their buffered supports, and the screens make those annotations irrelevant there. In the triangular model translations are by lattice sites. This proves the stated uniformity and completes the comparison. ◻

Proof. Fix a compact range of insertion locations separated from one another and from all polygon boundaries. Write \(K_{e,s}(x_1,\ldots,x_k)\) for the all-site kernel and \(K_{e,h}(x_1,\ldots,x_k)\) for the kernel with all insertions replaced by squares of radius \(h\). Proposition 34, applied successively at the \(k\) locations, gives \[ (m_e q_e)^k K_{e,s}(\boldsymbol x) =c_e^k\frac{K_{e,h}(\boldsymbol x)}{D_e(h)^k} +o_h(1), \tag{54}\] uniformly on this range, in the ordered sense \(\lim_{h\downarrow0}\limsup_{e\downarrow0}|o_h(1)|=0\). Here and below a kernel includes its specified signed weights. To check the factors in this iteration, divide the single replacement estimate by \(D_e(h)\ge c_0 a_{e,h}\). Each already replaced location contributes at most \(S_e/D_e(h)\le C a_e/a_{e,h}\), while its square kernel costs \(C a_{e,h}\). Thus every accumulated error is \(o_h(a_e^k)\) before normalization. The bound \(m_e q_e a_e\le C\) proves (54), including when \(c_e\) is small.

For fixed sufficiently small \(h\), Proposition 30 gives common critical limits for \(K_{e,h}\) and \(D_e(h)\). The limiting denominator is positive: the square lower bound and the four-arm lower power bound give \(D_e(h)\ge c_h>0\) at this fixed \(h\). These limits hold also for convergent insertion locations and are continuous on the separated range. Consequently the convergence is uniform there. Indeed, a failure of uniform convergence would give a sequence of locations with a convergent subsequence, contradicting the assertion for convergent locations. Since \(\sigma_e\) converges to Lebesgue measure, integrals against a fixed continuous cutoff therefore converge too. This applies to the triangular counting measure as well as to the Poisson location integral.

Apply this observation first with \(k=1\), \(F=\mathbf1_{C_{Q_0}}\), and weight \(\delta_1-\delta_0\). Its full normalized first coefficient is \[ m_e q_e\int K_{e,s}(x)\,d\sigma_e(x) =q_e\mathbb EN_e(Q_0)=1. \tag{55}\] One may first use the localized box from Section 5; its omission contributes \(o(1)\) also to this coefficient. By Lemma 32, for every \(z>0\) there is a continuous cutoff \(0\le\chi\le1\), supported on a compact range away from the polygon boundary and the box boundary, such that the integral in (55) changes by at most \(z+o(1)\) when multiplied by \(\chi\). Nonnegativity is used here.

Let \(J_\chi(h)\) be the common limit of \[\int\chi(x)\frac{K_{e,h}(x)}{D_e(h)}\,d\sigma_e(x).\] The separate insertion upper bound and the square lower bound imply \(0\le J_\chi(h)\le C_\chi\), uniformly for sufficiently small \(h\). For any subsequential limit \(c_*\) of \(c_e\), \[ |1-c_*J_\chi(h)|\le z+o_h(1). \tag{56}\] Choose \(z<1/4\) and then \(h\) small. This proves that every such \(c_*\) is bounded below by a positive constant. If \(c_+\) and \(c_-\) are two subsequential limits, possibly from different models, the same cutoff and the same \(J_\chi(h)\) work for both. Since all \(c_e\) have a common upper bound \(C\), (56) also gives \(J_\chi(h)\ge(1-z-o_h(1))/C\). Subtracting the two instances yields \[|c_+-c_-|\le \frac{2C(z+o_h(1))}{1-z-o_h(1)}.\] First let \(h\downarrow0\) and then \(z\downarrow0\). All subsequential limits agree, proving \(c_e\to c>0\).

For a general fixed coefficient, restrict its location integral to a compact separated range, losing at most an arbitrarily small amount by Lemma 32. Equation (54) and the common critical square limits compare any two subsequential limits of the restricted coefficient. Their difference tends to zero as \(h\downarrow0\). Removing the cutoff proves uniqueness of the full coefficient limit across both models. Boundedness follows from Proposition 31, so these limits exist. The factor \(1/k!\) in the expansion and the finite sum over replica assignments do not affect this argument. ◻

Proof. The first expression is the normalized first coefficient of \(\mathbf1_{C_{Q_*}}\) with weight \(\delta_1-\delta_0\). Lemma 39 gives a common finite limit \(B\). For positivity restrict insertion locations to a small open patch near the center of \(Q_*\). The square lower bound of Lemma 33 gives \(K_{e,h}(x)\ge c_1 a_{e,h}\) uniformly on a smaller patch, whereas \(D_e(h)\le C_1a_{e,h}\). Choose a nonzero nonnegative cutoff in that patch. Equation (54) with \(k=1\) and \(c_e\to c>0\) shows that this part of the limiting first coefficient is strictly positive. All omitted parts are nonnegative. Hence \(B>0\). Taking reciprocals gives (57); \(q_e\to0\) by Lemma 14. ◻

Proof. First use the auxiliary scale \(q_e\). Expand the expectation of any bounded function of the specified indicators as in Section 5. The degree-zero term has its common critical limit by Proposition 30, and every positive-degree coefficient has a common limit by Lemma 39. The terms with repeated replica labels at one spatial site tend to zero. The majorant in Proposition 31 is summable uniformly on every bounded parameter range, so the whole expectation has a common limit.

At fixed degree the history weights are continuous on their finite labeled history space, including when times coincide or change order. The kernel integrals are multilinear in these weights. They therefore have continuous limiting coefficients, and local uniform summability makes their sum continuous. The same argument applies to convergent parameter vectors.

At the prescribed scale, the parameter \(\lambda\) corresponds to \(t=\lambda/(B+o(1))\) at the auxiliary scale, by Lemma 40. The conclusion for convergent vectors proves the assertion in the required normalization. ◻

Proof of Theorem 2. Finiteness and positivity of the expectations follow from Lemma 14; their inverse scales vanish by Lemma 40.

At fixed mesh, only finitely many cells meet any particular quad almost surely. Its crossing indicator is monotone in the parameter, and the closed mark rule gives, at every finite deterministic \(t\), \[ \{\tau_e(Q)\le t\}=\{C_Q\text{ occurs at }p_e(t)\} \quad\text{almost surely}. \tag{58}\] Thus Proposition 41 supplies common limits of the joint distribution functions of every finite set of thresholds. These limits are continuous at finite parameter vectors.

For completeness, compactification introduces no ambiguity. Take any weakly convergent subsequence of threshold laws on \(\mathcal K\), which exists by compactness. At finite continuity points of a limiting coordinate marginal, (58) identifies its distribution function with the continuous limiting crossing function. Squeezing between such continuity points excludes a finite atom and gives the same identity at every finite point. Apply this to each coordinate of a finite projection. Its joint distribution function then agrees with the one supplied by Proposition 41. Finite-endpoint rectangles determine measures on \(\overline\mathbb R^k\) as well: their increasing or decreasing limits recover events involving either endpoint at infinity. All finite projections are therefore uniquely determined. Since the index set is countable, they determine the law on \(\mathcal K\).

In particular, the triangular laws have a full weak limit, which we denote by \(\nu_\triangle\). The annealed one-replica Voronoi laws have the same limit. Apply the preceding argument also to two replicas. For the triangular model their joint law is a product at every mesh, so its limit is \(\nu_\triangle\otimes\nu_\triangle\). Proposition 41 shows that two color replicas on a common Poisson geometry have this same limit.

Let \(f\) be a real continuous function on \(\mathcal K\), and put \[Z_{e,f}=\mathbb E[f(\tau_e^V)\mid\eta_e], \qquad m_f=\int f\,d\nu_\triangle.\] Conditional independence of the two color replicas gives \[\mathbb EZ_{e,f}\longrightarrow m_f, \qquad \mathbb EZ_{e,f}^{2} =\mathbb E[f(\tau_e^{V,1})f(\tau_e^{V,2})] \longrightarrow m_f^2.\] Hence \(Z_{e,f}\to m_f\) in \(L^2\) over geometry.

The family \(\mathcal F=\{f:\|f\|_\infty\le1,\ \mathop{\mathrm{Lip}}_{d_{\mathcal K}}(f)\le1\}\) is uniformly bounded and equicontinuous on compact \(\mathcal K\). For any \(\zeta>0\) it has a finite uniform \(\zeta\)-net \(f_1,\ldots,f_m\). For every sampled geometry, \[d_{\mathrm{BL}}\bigl(\mathop{\mathrm{Law}}(\tau_e^V\mid\eta_e),\nu_\triangle\bigr) \le 2\zeta+\max_{1\le i\le m}|Z_{e,f_i}-m_{f_i}|.\] The maximum tends to zero in probability. Letting \(\zeta\downarrow0\) proves (6). All arguments apply along arbitrary vanishing-mesh sequences, completing the proof. ◻