Proof. For \(0<q<1\), apply Theorem 1. For \(1\le q<4\), let \(W\) be the wired-arc boundary edges and \(E'=E(G_{\delta_n})\setminus W\). For \(A'\subseteq E'\) and \(C\subseteq W\), the endpoints of every edge of \(W\) are already identified, so \(k_\xi(A'\cup C)=k_\xi(A')\). Summing their independent edge factors gives \[\sum_{C\subseteq W}p(q)^{|C|}(1-p(q))^{|W|-|C|}=1.\] Thus the marginal on \(E'\) is exactly the finite law used in (OpenAI 2026b, Theorem 1.1(i)), which removes \(W\) from sampling and declares its edges open for the same medial drawing. The wired/free arcs, orientation, marked approximation and curve metric agree, so that theorem proves the remaining range. These applications give convergence along the full mesh sequence in the stated cases. ◻

Proof. Contract the open conditioned edges and delete the closed ones. Components that no longer contain an unconditioned vertex contribute a fixed power of \(q\). All other components are counted in the quotient graph, giving (4). Adding an edge to a fixed state either leaves its component count unchanged or reduces it by one, so its opening-to-closing odds are \(v\) or \(v/q\). A single identification of two vertices similarly multiplies a state’s weight by \(1\) or \(q^{-1}\). Products of a fixed number of such factors, including the normalizing constants, give the last assertion. ◻

Proof. Thicken the open primal graph together with the fixed boundary cap trees on the sphere. If a planar ribbon graph has \(V'\) vertices, \(E'\) edges, and \(k'\) components, then its regular neighborhood has Euler characteristic \(V'-E'\) and also \(2k'-\ell\), because every component has genus zero. Thus \(\ell=2k'+E'-V'\). The cap trees change the expression only by a constant depending on the fixed completion, and their contractions make \(k'=k_\xi(A)\). Consequently \[\ell=2k_\xi(A)+|A|+c_\xi, \qquad d^\ell=d^{c_\xi}q^{k_\xi(A)}d^{|A|}.\] This is the odds form of the FK weight. A change of one pairing can merge or split only a bounded number of loops. Its weight ratio is therefore bounded above and below by fixed powers of \(d\); normalization preserves a bounded two-sided comparison. ◻

Proof. For a connected plane graph on the sphere, the dual subgraph on \(E\setminus A\) has as many components as the faces of the primal subgraph. Euler’s identity gives \(k^*(E\setminus A)=|A|-|V|+k(A)+1\). Multiplying out \(q^{k^*(E\setminus A)} \prod_{e\notin A}(q/v_e)\) shows that its ratio to \(q^{k(A)}\prod_{e\in A}v_e\) is constant in \(A\). Fixed cap trees can be included before applying this identity and then contracted; their complementary dual incidences give the stated boundary convention. Disconnected ambient graphs are handled componentwise or by adding fixed exterior trees. ◻

Proof. We first identify the squarefree coefficients of \(F_H\), then the signs of its logarithmic coefficients, and finally the convergence domain. The sign argument uses labeled copies of vertices: an auxiliary cancellation removes repeated types before the logarithm is taken.

Proof. The labeled construction above involves only types in \(\mathop{\mathrm{supp}}(m)\). After the factors in (13) are extracted, \(b_m\) is a finite sum of nonnegative contributions, one for each ordinary union \(D\). Apart from a nonnegative integer from the hard-edge sum and a positive factorial denominator, each contribution is \[v_D\prod_{D_0}\bigl[-Z(D_0;-u,1+v)\bigr].\] For a connected graph, differentiating \(Z\) with respect to a nonloop weight gives the partition function of its contraction, which is negative by (12); differentiating in a loop weight gives the partition function with that loop removed, also negative. Hence every factor \(-Z\) is nondecreasing in the edge weights. Adding an allowed induced edge merely adds nonnegative terms as well. This proves the first assertion, with parallel edges combined before applying it.

For the second assertion, fix the states of the original edges. Every component meeting \(S\) has factor one in (6). Identifying vertices in \(S\) only merges some of these components and leaves all other components unchanged. The product of factors is therefore unchanged for every edge state. The independent edge law is unchanged as well; edges that become loops can be retained. Taking expectations proves the polynomial identity, and taking its logarithm at zero proves the coefficient identity. ◻

Proof. Regard the common vertex as \(o\), remove it to obtain \(H\), and regard its incident nonloop edges as hub links. By (7) and Theorem 7, the part of \(\log Z_{H+o}\) depending on those links is \[ \sum_h\log(1+v_h) -\sum_{m\ne0}c_m\prod_h(1+v_h)^{-m_{i(h)}}. \tag{14}\] Every monomial in the second sum has nonnegative mixed derivative in the odds of two distinct links, including links at the same site. The first sum has mixed derivative zero. Absolute convergence on a larger polydisk justifies termwise differentiation, including at zero link odds by continuity. Thus \(\partial_{v_e}\partial_{v_f}\log Z\le0\) for distinct incident edges. For positive odds, \[ \partial_{v_e}\partial_{v_f}\log Z =\frac{\mathop{\mathrm{Cov}}(\mathbf 1_{\{e\text{ open}\}}, \mathbf 1_{\{f\text{ open}\}})}{v_ev_f}. \tag{15}\] Loops are independent of all other edge states and cause no exception.

For the plane statement, complementary dual edges have odds \(q/v_e\). Two primal edges on a common face become two dual edges at a common vertex. Complementing both opening indicators preserves their covariance, so the adjacent inequality in the dual gives the claimed cofacial inequality. Zero odds and contraction limits follow by continuity of finite partition functions after removal of their common leading factors. ◻

Proof. Add a distinct auxiliary edge \(f\) between \(a\) and \(o\), initially of odds zero. Its logarithmic partition derivative is \[ \left.\partial_{v_f}\log Z\right|_{v_f=0} =\mathbb P(a\leftrightarrow o)+q^{-1}\mathbb P(a\not\leftrightarrow o) =q^{-1}-(q^{-1}-1)\mathbb P(a\leftrightarrow o). \tag{17}\] The mixed derivative with respect to the strengthened edge is nonpositive by Lemma 9. Because \(q^{-1}-1>0\), the tested connection probability is nondecreasing. The cofacial argument is identical. Sending incident auxiliary odds to infinity proves the terminal-identification assertion.

For (16), induct on the number of edges, discarding loops. If no nonloop edge touches \(o\), both tested events are constant. Otherwise condition on an edge \(e\) incident to \(o\). The conditional laws are FK on the deletion and contraction minors, and their conditional covariance is nonnegative by induction. Moreover, each of the two conditional means is at least as large when \(e\) is open as when it is closed: vary its odds from zero to infinity in the connection monotonicity just proved. The covariance of these two conditional means is consequently nonnegative. The conditional covariance identity proves the induction step. Cases in which a tested vertex is contracted to \(o\) simply make its event constant one. ◻

Proof. Differentiate (7) with respect to \(s_a\): \[ s_a\partial_{s_a}\log Z_{H+o} =1+\sum_m m_a c_m x^m. \tag{23}\] This derivative has a second interpretation. Conditional on the original open graph, the component \(C_a\) has hub-link factor \(s_{C_a}-u\). Its contribution to the derivative is \[\frac{s_{C_a}}{s_{C_a}-u} =1+\frac{u}{s_{C_a}-u},\] while its conditional probability of having no hub link is \(q/(s_{C_a}-u)\). Averaging under the marginal original-edge law of the augmented graph and comparing with (23) proves (19). At \(s_i=1\) for every \(i\), the hub is isolated, so the right side equals one. Nonnegativity and the support assertion now follow from Theorem 7. For the \(H/S\) specialization, adjoin a fresh hub with links only at \(S\) and send their odds to infinity. Then \(x_i=0\) on \(S\) and \(x_i=1\) elsewhere, giving (20), including when \(a\in S\).

For (21), put \(x_i=\mathbf 1_{\{i\notin S\}}\) in (6). A component meeting \(S\) then contributes one instead of q. Hence \[\frac{F_H(\mathbf 1_{V\setminus S})}{F_H(\mathbf 1_V)} =\mathbb E_H q^{-k_S}.\] Take logarithms and use (8). For a singleton, \(k_{\{a\}}=1\) in every configuration, so \(K_H(\{a\})=\log t\). The final identity is the inclusion-exclusion identity for the two support-hitting events in the nonnegative sum. ◻

Proof. If \(a\in S\), both probabilities are one. Otherwise induct on the number of edges, ignoring independent loops. If no nonloop edge is incident to \(S\), both probabilities are zero. Choose a nonloop edge \(e\) with endpoint \(s\in S\). Write \(\mu=\mathbb P_H\), \(\nu=\mathbb P_{H/S}\), and let \(\alpha,\beta\) be their respective probabilities that \(e\) is open. To form \(H/S\), add auxiliary links from \(s\) to all other vertices of \(S\) and send their odds to infinity. Every such link is incident to \(e\). Lemma 9 therefore gives \[ \alpha\ge\beta. \tag{28}\]

For \(j=0,1\), let \(h_j\) and \(\widetilde h_j\) be the conditional hit probabilities under \(\mu\) and \(\nu\), respectively, given the state \(j\) of \(e\). Conditional FK laws are deletion and contraction minors. Identification of \(S\) commutes with both operations, with its image used after contraction. The induction hypothesis therefore gives \(h_j\ge\widetilde h_j\). If \(e\) becomes a loop in \(H/S\), its state is independent of the hit event and \(\widetilde h_1=\widetilde h_0\). Otherwise it is incident to the collapsed target terminal, and Lemma 10 gives \(\widetilde h_1\ge\widetilde h_0\). Thus \[\begin{align*} \mathbb P_H(a\leftrightarrow S) &=\alpha h_1+(1-\alpha)h_0\\ &\ge\alpha\widetilde h_1+(1-\alpha)\widetilde h_0\\ &\ge\beta\widetilde h_1+(1-\beta)\widetilde h_0 =\mathbb P_{H/S}(a\leftrightarrow S). \end{align*}\] If contraction brings \(a\) into the image of \(S\), the corresponding conditional probabilities are both one. Zero odds are obtained by continuity, or by deleting those edges at the start. ◻

Proof. Raise the auxiliary odds successively from zero to their prescribed values. The logarithm of (29) is the sum of the integrals of \(\partial_{v_f}\log Z\) for \(f\in X\) during these changes. Its derivative with respect to the odds of \(h\) is nonpositive by the cofacial assertion of Lemma 9. This proves the first statement; all embeddings are kept fixed, even when an edge has odds zero.

For the second, delete the edges of \(K\) incident to \(U\) in an exposed order. Start at the face sector incident to \(a\). Once one sector at a vertex of \(U\) is exposed, delete its incident edges successively around its cyclic fan, thereby enlarging the same face. An edge leading to another vertex of \(U\) exposes a sector there. A spanning tree of the connected set \(U\) ensures that this procedure reaches every vertex of \(U\). All original edges incident to \(U\) can therefore be deleted, and every edge at its deletion borders the expanding face. The edges of \(X\) remain on its boundary, because none touches \(U\) and deleting an edge never splits a face. If the deleted edge has the same face on both sides, that face remains available along both banks; otherwise its two incident faces merge. A loop can be removed without affecting any gain. At each step the first part of the lemma shows that the auxiliary-edge gain cannot decrease. Isolated vertices left in \(U\) contribute the same factors to numerator and denominator and can be discarded.

Finally send every auxiliary odds to infinity and divide the gain by their product. For a fixed original edge state, the tree \(X\) identifies all \(k_B\) components meeting \(B\), changing the cluster count by \(-(k_B-1)\). The limiting gain is consequently \(\mathbb E_K q^{1-k_B}\), and after erasure it is \(\mathbb E_{K-U}q^{1-k_B}\). This proves (30). An open component is connected in the underlying graph, so it satisfies the same geometric hypothesis. ◻

Proof. Our base law has \(A,C\) separately wired. First we compare an \(A\)–\(B\) path in \(H\) with a network hit. We then expose an extremal such witness and compare the uninspected \(C\) side with the separate \(C,D\) network. The two comparisons will give the two factors on the right of (31).

Proof. Draw successive links along the terminal intervals narrowly outside the rectangles, and a test edge between exposed representatives in the remaining main exterior face. This edge has two incident exterior sectors. An edge of the larger rectangle not belonging to the smaller one has a point outside the smaller closed rectangle. That point can be joined through the complement of the smaller rectangle to an exterior gap away from the wire intervals, crossing only edges to be removed. Perturb the route off vertices. Starting from the two test-edge faces, delete edges encountered on these routes by face expansion. At each deletion the edge being removed is cofacial with the test edge. The cofacial comparison therefore makes connection no larger after deletion. The cap links do not obstruct these routes: they lie along the common boundary intervals, and no added rectangle edge lies in their narrow outer caps. Take the wire limits after the finite-odds comparison. ◻

Proof. The \(A,C\) caps are disjoint. In their wired law, Lemma 12 still compares a hit of the as-yet unwired set \(B\) with the law that also wires its image, even if \(B\) has a shared endpoint with \(C\). In comparing the latter law with the \(A,B\) law, adding the \(C\) wire enlarges the \(B\) terminal when \(B\cap C\ne\varnothing\); otherwise it is the same exterior cofacial addition as in Lemma 14. If a hit in the \(A,C\) quotient uses \(C\) as a shortcut, its first visit to \(C\) already gives the desired connection; otherwise it contains an open \(A\)-to-\(B\) path in \(G\). Thus the initial event comparison in that proof also remains valid.

Use the spokes and access arcs from the hypotheses in the extremal search. If its path \(\gamma\) meets \(C\), the desired connection already holds. Otherwise the access arcs place every \(C\) incidence on the uninspected side, and every \(D\) incidence on the other side or on \(\gamma\). A shared endpoint in \(B\cap D\) or \(D\cap A\) is permitted on \(\gamma\); if it is not there, its assigned access sector places it on the \(D\) side. Thus the separator-wiring comparison and the conditional one-vertex factorization in the interlacing proof apply unchanged. This proves (35). ◻

Proof. Contract the boundary edges on the two tested sides. They are loops after wiring and do not affect other edges. Spherical planar duality at odds \(v=\sqrt q\) gives a dual graph consisting of the \(w\times h\) cell vertices, their ordinary nearest-neighbor edges, and one exterior hub with spokes to the bottom and top rows. Absence of the primal crossing is exactly an actual dual path from the bottom contacts to the top contacts through the cells, not a hop through that common hub. This is the disk path–cut alternative, obtained for example by tracing the boundary of the vertices reachable from the first primal electrode.

Split the exterior hub into a bottom hub and a top hub. The resulting network, after rotation, is \(R_{h+1,w-1}\) with its two full crossing sides wired separately; boundary edges that are loops can again be ignored. Let its connection probability be \(g'\). Joining the hubs changes the weight by \(q^{-1}\) on their separation and by one on their connection. Consequently \[ 1-g_x(w,h)=\frac{qg'}{1-(1-q)g'}. \tag{38}\] For \(w=h+1\), the split graph is isomorphic to the original network, so \(g'=g_x(w,h)\). The solution in \([0,1]\) is \[g_x(h+1,h)=\frac{1}{1+\sqrt q}.\]

Absorbing an internal vertical cut into a terminal does not decrease connection. After that contraction the part beyond the cut attaches at one terminal only and factors out of the connection probability. Thus decreasing width cannot decrease \(g_x\). Increasing height also cannot decrease it: first apply Lemma 16 to the old terminal intervals and then enlarge them along the new sides. Comparing with the just-established offset square proves both bounds, and rotations give the corresponding vertical bounds. ◻

Proof. Enlarge the left electrode to the entire indicated low interval. Its complementary dual boundary arcs are the lower arc and the arc going around above it, which includes the uncontracted upper portion of the left side and the top. Split the dual exterior hub along these two arcs. A bounded change of density relates this split law to the dual law of the primal problem: one hub join changes any weight by a factor between one and \(q^{-1}\).

Retain the cell grid, the spokes from its bottom row to the lower hub, and those from an available upper portion of its left column to the upper hub. Removing other spokes at either tested hub cannot increase its connection probability. By (39), it suffices to lower-bound the corresponding deterministic-contact network.

In the cell grid put a square of side \(m=\min(w-1,h-1)\) against the bottom-left corner. The upper half of its left side connects to its full right side with probability at least \(c/2\): reflect the square across its horizontal center line and apply (34) to its full left side. Interlace this chord with the full top–bottom square chord. This gives a lower bound \(c^2/2\) for connection from the bottom side to the upper half of the left side. Those contacts are available among the retained dual spokes when \(w\) is large; in particular they lie above the low primal electrode. Extend to the full cell grid using Lemma 16. Two applications of (39) and the bounded hub-splitting density therefore give a positive lower bound for the complementary dual crossing. The claimed primal upper bound follows. ◻

Proof. Choose \(\epsilon\) so that \(1-\epsilon\) exceeds both upper bounds in Lemmas 18 and 19. Restricting a high-probability corridor between two level transversals preserves its lower bound. We shall sum fourth powers of the spans of disjoint such passages. Except for passages of total cost \(O(R^4)\), the square and corner obstructions force a collection of passages in the other coordinate direction with strictly larger total fourth power. Applying the same argument in both directions bounds both sums by \(O(R^4)\). The initial vertical crossings then give the claimed bound on their number.

Level cuts and restriction. We use the following elementary fact about a Jordan quadrilateral. If every end-to-end path must meet a transverse coordinate line during its interior passage, in the situations where the ends are on strict opposite sides or every such path visits both strict sides, that line contains an opposite-wall transversal. To see this, cut the disk along all its finitely many interior line crosscuts. The adjacency graph of the remaining components is a tree: each crosscut splits one disk into two. The components incident to an open part of either end form a connected subtree, as follows by walking along that end. The forcing hypothesis makes these two subtrees disjoint. An edge on the tree path between them cannot terminate on an end and cannot cut off only a same-wall pocket. It therefore corresponds to an opposite-wall transversal. There are no unaccounted corner contacts here. Endpoints of an orthogonal level cut on a dual wall have half-integer transverse coordinate, whereas the tested line has integer coordinate. Parallel intermediate levels do not meet the ends at all.

If two transversals are put into the first and second terminal, respectively, connection does not decrease by terminal enlargement. The part between them then has its two full cut-vertex sets wired, and the discarded parts attach at just their respective single terminal. They factor out. A lattice edge crossing an integer-coordinate cut does so at a vertex of that cut; a primal path cannot pass through a cut endpoint on a dual wall. Edges along the cut are harmless loops after contraction. Thus every intervening pure subproblem has connection at least \(1-\epsilon\). A cut with no primal vertex would instead make the original passage impossible, so does not arise here.

The two variation sums. Let \(V_x\) be the maximum of the sum of fourth powers of the absolute level differences for subquadrilaterals between vertical transversals, chosen with disjoint longitudinal interiors in each corridor and summed over all corridors. Define \(V_y\) using horizontal transversals. The total number of candidate cuts is \(O(R^2)\): the disjoint simple dual walls have that total combinatorial length, and each unit wall edge has only a bounded number of integer-coordinate intersections. Consequently the maxima are finite. Interior horizontal levels near the two corridor ends give \[ V_y\ge m(H-2)^4. \tag{41}\] We will show that a large value of either variation forces a strictly larger value of the other, up to an \(O(R^4)\) error.

A slab core and its rectangle comparison. Consider a selected subquadrilateral between vertical cuts whose level difference is \(s>0\). Choose integers \(a<b\) strictly between its end levels. There are transversals at both levels. Among their ordered cuts choose consecutive ones of different types, and let \(J\) be the closed subquadrilateral between them, with ends \(e_a,e_b\). Its longitudinal order may be either \(a\) then \(b\) or the reverse. There is no further opposite-wall cut at either level inside \(J\). Every remaining component of either line through the interior of \(J\) is therefore a same-wall crosscut cutting off a side pocket away from the ends. The pockets are nested or disjoint. Remove their outermost members, replacing each wall excursion by its straight mouth. The remaining closed Jordan disk \(K\) has the same ends; its connected interior avoids both lines and hence lies in \(a<x<b\). Its boundary contacts with these lines are the ends and the separate mouth intervals. A removed pocket graph attaches only through its own mouth. Assign edges on mouths to the core.

On each longitudinal side of \(K\), choose an original-wall crossing from \(a\) to \(b\) with interior strictly in the slab, by taking a last contact before a passage to the opposite side. Call the lower and upper crossings \(\alpha,\beta\) in slab order and set \[\begin{align*} p_-&=\min y(\alpha),& P_-&=\max y(\alpha),\\ p_+&=\min y(\beta),& P_+&=\max y(\beta). \end{align*}\] Each crossing separates the infinite vertical slab into an upper and a lower component. The connected interior of \(K\) lies above \(\alpha\) and below \(\beta\) in this separation: it misses each crossing and meets its designated side near the corresponding wall. Points below the minimum of \(\alpha\) are in its lower component, and points above the maximum of \(\beta\) are in its upper component. It follows that \[ p_-\le p_+,\qquad P_-\le P_+, \qquad K\subset\{p_-\le y\le P_+\}. \tag{42}\] All contacts of \(K\) with either extreme vertical line lie between the heights of the corresponding endpoints of \(\alpha,\beta\). Moreover every path between \(e_a,e_b\) in \(K\) has minimum height at most \(p_+\) and maximum height at least \(P_-\). For an interior transversal path, the upward vertical ray from an interior maximum of \(\alpha\) starts on the lower side of that path and reaches its upper side; it must therefore intersect the path. The downward ray from an interior minimum of \(\beta\) gives the other bound. Use interior points and limits if an extremum is at an endpoint. Approximation inside the Jordan quadrilateral gives the same statement for a general continuous end-to-end path. It also holds for paths in \(J\): replace every excursion into a pocket by a path on its mouth, setting the horizontal coordinate to the mouth level and clipping the vertical coordinate to the mouth interval. Entry and exit are fixed, and this replacement cannot enlarge the transverse range of the path.

We claim that the pure crossing probability of \(J\) is at most that of the full lattice rectangle \[ [a,b]\times[\lfloor p_-\rfloor,\lceil P_+\rceil], \tag{43}\] whose two terminal intervals contain all the respective core contact vertices on \(x=a,b\). Condition first on the pocket edges. They induce noncrossing ties within each separate mouth, and no ties between mouths. On the core the remaining law is precisely FK with these ties and the two end wires. Draw the separate mouth links and end links narrowly outside their intervals and a terminal test edge in the main exterior. Every missing rectangle edge has a point outside the closed disk \(K\). Through its complement this point has access to an exterior gap not contained in \(K\), crossing only missing edges. No cap link seals such a gap: its supporting interval is contained in \(K\) on an extreme slab line, and the cap lies on the opposite side from the rectangle interior. Delete missing edges by face expansion from the test-edge faces. Cofacial Rayleigh compares the core below the full rectangle still carrying the separate ties. Now, and only now, enlarge the two tested terminals to absorb all their respective mouth contacts. Terminal monotonicity cannot decrease connection and subsumes every pocket tie. This proves the comparison conditionally on any pocket configuration and hence unconditionally. In particular we may further enlarge either terminal to its full rectangle side.

Figure 4 illustrates the separate-mouth convention.

Good slabs and their cost. Fix a small number \(\tau>0\), say \(\tau=1/100\). Choose \(\zeta>0\) sufficiently small compared with \(\tau^4\), and subsequently choose a fixed integer \(S_0\) sufficiently large compared with \(\zeta^{-1}\) and all lattice roundoff constants. Take a maximizing collection for \(V_x\). Its terms with \(s<S_0\) cost at most \(O(R^2S_0^4)\), which is \(O(R^4)\) with these fixed choices. Group the remaining spans by \(h\le s<2h\), where \(h=2^kS_0\). Round the two end levels inward to a common integer grid of spacing \(\lfloor\zeta h\rfloor\), choosing \(a,b\) strictly inside them. Each moves by at most \(\zeta h+1\), so \[b-a\ge s-O(\zeta h+1).\] Construct one \(J,K,\alpha,\beta\) as above for each chosen passage. Call it good if \[ (p_+-p_-)+(P_+-P_-) +|\beta(a)_y-\alpha(a)_y| +|\beta(b)_y-\alpha(b)_y|\le\zeta h. \tag{44}\]

Members of the maximizing collection may share an end cut. The chosen levels \(a,b\) lie strictly between each member’s own end levels, so its subquadrilateral \(J\) does not use such a common end cut. Their original side-wall subarcs have disjoint longitudinal interiors. Hence the constructed core interiors are disjoint; any remaining boundary contact only permits equality in the ordering below. For a fixed pair \(a,b\), the core disks are slab-ordered. Their bounding crossings cannot interleave: a slab crossing belonging to one core would then separate points of the connected interior of another. Minima, maxima, and the two endpoint heights of ordered disjoint crossings are ordered as well, with weak inequalities sufficient. Thus each of the four nonnegative gap sums in (44) telescopes to at most \(H\), and their total is at most \(4H\). The number of bad choices at fixed \(a,b\) is at most \(4H/(\zeta h)\). There are \(O_\zeta((R/h)^2)\) rounded pairs, so at this scale there are \(O_\zeta((R/h)^3)\) bad choices. Their contribution to \(V_x\) is \(O_\zeta(R^3h)\). Summing over the dyadic scales \(h\le R\) gives \(O(R^4)\).

Amplification of a good passage. For a good passage the height \(P_+-p_-\) is at least \(b-a-O(1)\). Otherwise the comparison rectangle (43) would be hard in the sense of (36), contradicting the connection probability at least \(1-\epsilon\) in \(J\). Subtracting the two extrema gaps in (44) gives the mandatory inner span \[ T:=P_--p_+\ge s-O(\zeta h+1). \tag{45}\] Choose integer levels a bounded distance inside \(p_+\) and \(P_-\). Every end-to-end path visits both strict sides of each of these levels, so the level-cut fact gives horizontal transversals. The span between two such low and high cuts is \(T-O(1)\).

If \(T\ge1.05s\), that one horizontal passage already has fourth power larger than \((1+\gamma)s^4\) for some fixed \(\gamma>0\), once \(S_0\) is large. Otherwise the height and width of (43) are within the aspect-ratio range of Lemma 19. Every point of each end interval must be at least \(s/16\), for definiteness, above the lower extreme and below the upper extreme, after increasing \(S_0\) and decreasing \(\zeta\) if necessary. Indeed if one endpoint interval approaches an extreme more closely, the endpoint gap bound in (44) places all contacts on that vertical side within \(s/16+O(\zeta h+1)\) of that extreme. The comparison box then has a corner electrode contained in its bottom or top eighth, and Lemma 19 contradicts high connection.

Starting from the longitudinal first end of \(J\), consider the earlier of the selected low and high transversals. The original end lies in the central height range just established, while that earlier cut is near the corresponding extreme. Choose an intermediate horizontal integer level between them at distance at least \(\tau s\) from the extreme cut. The two ends of this first subquadrilateral are on strict opposite sides of the intermediate level, so it contains a transversal there. The resulting extra horizontal passage precedes the low–high passage and has disjoint longitudinal interior from it. This works in either order of the low and high cuts. Its span is at least \(\tau s\), with bounded roundoff absorbed by the choices of constants.

The two contributions are therefore at least \[\bigl(1-O(\zeta+S_0^{-1})\bigr)^4s^4+\tau^4s^4 \ge(1+\gamma)s^4\] for a fixed \(\gamma>0\), by choosing \(\zeta\) and then \(S_0^{-1}\) small enough compared with \(\tau^4\). The constructed horizontal passages are contained in their respective original vertical subquadrilaterals, so all chosen contributions retain disjoint interiors.

Conclusion. Combining the good passages and the two error bounds gives \[V_y\ge(1+\gamma)(V_x-C_0R^4).\] The same construction with the two coordinate directions exchanged, and with the original longitudinal wall roles unchanged, gives \[V_x\ge(1+\gamma)(V_y-C_0R^4).\] Substitution bounds each variation by a constant multiple of \(R^4\). Together with (41), this proves (40). ◻

Proof. Suppose otherwise. Lemma 18 and positivity on each fixed finite graph give a sequence with \[ n\longrightarrow\infty,\qquad n\le W_n\le\Lambda n, \qquad g_x(W_n,n)\longrightarrow0. \tag{47}\] All limits in \(n\) below are along (47). Every macroscopic separation, relative interval width, and bulk clearance invoked in a convergence assertion is fixed and positive before taking that limit. These constants may depend on the finite number of chords to be used, but not on \(n\). A convergence assertion for a stated class of rectangles and intervals means that it holds for every sequence of integer choices satisfying those same fixed conditions. Equivalently, the convergence is uniform over that class: a failure of uniformity would select a violating sequence, to which the same subsequence contradiction applies.

Separated arches vanish. For each fixed \(\sigma>0\), consider ordinary rectangles with positive integer width and height at most \(n/8\), and two nonempty top intervals whose horizontal separation is at least \(\sigma n\). Their network probability tends to zero uniformly over these choices. Otherwise, after taking a subsequence, there is such a chord bounded below. Subdivide both intervals into a bounded number of subintervals of diameter at most \(\sigma n/10+O(1)\). By (34), a pair of these localized intervals still has a chord bounded below. Translate copies of its rectangle along the top of \(R_{W_n,n}\), starting at the left side, with integer displacement close to half the separation of the two localized centers. Consecutive chords interlace strictly; they retain their lower bounds by Lemma 16. Continue while the rectangles fit. A bounded number of applications of (35), depending only on \(\sigma,\Lambda\), joins the first endpoint to the last far endpoint with probability bounded below. The first lies within \(n/8\) of the left side and the last within \(n/8+n/16+O(1)\) of the right side.

A square of side \(\lfloor n/2\rfloor\) at the upper-left corner has a chord from its left side to the far half of its top side bounded below: reflection and subadditivity give the half-side crossing, then interlacing with an opposite full square crossing gives the stated chord. The analogous upper-right square gives the other end chord. They interlace the long top chord to the full left and right sides. This contradicts (47). Reflection gives the same vanishing statement for bottom arches. Translation gives it on the shifted grid as well.

Only aligned chords can survive. Fix \(0<c_-\le c_+\le1/8\) and \(0<\theta<1/2\). For integer square sides \(c_-n\le h\le c_+n\), a top electrode contained in \([0,(1/2-\theta)h]\) or in \([(1/2+\theta)h,h]\) connects to the full bottom side with probability tending uniformly to zero. If not, reflect that electrode and its chord across the vertical center line. Interlace the two top–bottom chords separately with the full left–right square crossing, whose probability is bounded below. This yields chords from the left top portion to the right side and from the right top portion to the left side. Interlacing them gives a separated same-top arch, contradicting the preceding paragraph. The corresponding off-center bottom chords also vanish.

Decompose each of the top and bottom sides into its two off-center portions and the central interval \([(1/2-\theta)h,(1/2+\theta)h]\), with endpoints rounded to lattice vertices. The full vertical crossing has a uniform lower bound. Subadditivity twice, and the vanishing just proved, therefore give a positive lower bound for the chord between the two central neighborhoods.

Now use an integer strip of height \(h\) and width \(\ell=4h+O(1)\), with the bound in \(O(1)\) fixed, \(h/n\) bounded above and below by fixed positive constants and both dimensions at most \(n/8\). Translated squares give bounded-below chords between aligned top and bottom patches of fixed positive relative width. Their centers may vary provided their distances from the vertical ends are at least \((1/2+\eta)h\) for a fixed \(\eta>0\).

In this strip, consider boundary electrodes lying entirely to the left and right, respectively, of a horizontal gap. Suppose that the gap contains centers \(c_1<c_2\) whose distances from its endpoints and from each other are at least \(\eta h\), and whose distances from the vertical strip ends are at least \((1/2+\eta)h\), for some fixed \(\eta>0\). The connection probability of those electrodes tends uniformly to zero. Choose the aligned patches at these centers with length \(\eta h/2+O(1)\) after lattice rounding, so that the two patches are disjoint and remain in the gap. Interlace a hypothetical bounded-below chord with the aligned vertical chord at the first center, joining that top patch to the right electrode, and with the aligned chord at the second center, joining that top patch to the left electrode. The resulting chords interlace to give a forbidden separated same-top arch.

Aligned short electrodes are crossed with probability tending to one. Fix \(\theta,\eta>0\) and equal aligned intervals on the top and bottom of this strip, each of length at least \(\theta h\) and contained between horizontal coordinates \((1/2+\eta)h\) and \(\ell-(1/2+\eta)h\). Contract their boundary edges. The complementary dual crossing is between the exterior arcs going around the left and right ends. Split the corresponding exterior hub into two hubs. This costs only a bounded density factor. Absorb all adjacent contact vertices into their respective hubs, which cannot decrease the network probability by terminal monotonicity. The result is an ordinary interior cell rectangle with two separately wired boundary electrodes whose horizontal coordinates are separated by the gap between the endpoints of the original intervals. This gap contains two centers satisfying the preceding fixed-clearance conditions, with constants depending only on \(\theta,\eta\). The preceding vanishing argument, on the shifted grid and with bounded lattice roundoff, applies to this network. Thus the complementary dual probability tends to zero, and the original aligned crossing tends to one. Smaller aligned subintervals of any fixed positive relative width are covered by the same argument.

Many alternating network crossings. The preceding steps have shown that every fixed aligned pair of short intervals in the indicated strip is crossed with probability tending to one. We next arrange finitely many such primal and dual crossings in a common law. Their simultaneous success will use a union bound; no correlation inequality between these events is needed.

For definiteness take \(h=\lfloor n/64\rfloor\) and strip width \(4h\). Choose an arbitrarily large but fixed finite ordered list of disjoint small interval labels in a central bulk part of the top side, and matching labels in the same order on the bottom. On the primal graph wire the odd intervals, each separately, by contracting their boundary edge runs. Leave no other primal wires. Round the runs so that each intervening even interval retains an uncontracted boundary-edge stretch of fixed positive relative width. Only interior labels with neighboring buffers will be used.

Every desired odd top–bottom network connection tends to one. This follows first in the pure two-electrode graph by the aligned estimate, then in the multiwire graph by adding the other disjoint interval caps cofacially with its test edge. The deletion order is the exposed boundary face expansion used in Lemma 16.

The dual consists of the cell grid and a common outer hub with stochastic spokes dual to all uncontracted boundary edges. Split this hub into separate hubs for the maximal complementary intervals. Since the number of intervals is fixed, the density change is bounded by a fixed power of \(q^{-1}\). We claim that each desired even top–bottom connection in this split network also tends to one, despite the finite spoke odds.

Retain first only the two spoke patches over smaller aligned intervals. Fix a deterministic bottom subinterval wire in that range. For any fixed integer \(k\), choose \(k\) disjoint smaller top patches aligned within it. Each is connected to the bottom wire with probability tending to one by the pure estimate and terminal enlargement. In the single rooted-multiset law based at the collapsed bottom electrode, all \(k\) patches are therefore hit with probability tending to one. Their union has multiplicity at least \(k\) on that event. Formula (39) gives lower limit at least \(1-(1+v)^{-k}\) for the finite top-spoke connection. Since \(k\) is arbitrary, this probability tends to one. Now keep this finite top-spoke graph as the base rooted at its hub. Apply the same argument to \(k\) disjoint deterministic bottom subintervals, using the conclusion just established for each of them. This makes the bottom spokes finite as well, still with connection tending to one. The order of limits is first \(n\to\infty\) for fixed \(k\), then \(k\to\infty\).

Extra spokes at the tested hubs cannot decrease connection. Spokes of each other hub occupy its own exposed boundary interval and can be removed by exterior face expansion from the test-edge faces; reversing this removal does not decrease the tested connection. Thus the desired even events tend to one in the full split dual network. The bounded hub-joining density transfers them back to the common original law. Their event definition still uses only the individual split-hub shortcuts, not the common hub as a path shortcut.

From networks to physical crossings. All these finitely many primal and dual successes coexist with probability tending to one. Draw shortcuts narrowly along their separate boundary intervals, including dual spokes through the appropriate boundary-edge midpoints. Complementary primal and dual states prevent crossings of opposite-color open paths. For topology only, place the rectangle and all narrow caps inside a slightly enlarged disk, and join each designated terminal to a distinct point on its boundary by outward spokes, preserving the order of the terminal intervals. For a dual event, the shortcuts here are those of its individual split hubs, although its edge state is now sampled in the common law.

For each network success, choose a simple witness in this enlarged disk, with the narrow cap shortcuts included as edges. Its only contacts with the enlarged boundary are its prescribed top and bottom endpoints, so it is a crosscut even if it uses another same-color cap along the way. Opposite-color witnesses are disjoint: their physical edges cannot cross, and their cap shortcuts and outward spokes occupy disjoint boundary intervals.

Fix a successful interior label whose two opposite-color neighbors also succeed. The target crosscut places the endpoints of one neighbor on one side and those of the other neighbor on the other side. Their witnesses cannot cross the target, so they lie on those respective sides and are disjoint from each other. The region between them meets the top and bottom boundary exactly between the neighboring labels. Every other same-color cap lies outside that region: its narrow connection to its own boundary interval is disjoint from the two neighbors. The target witness therefore cannot use another same-color wire or hub. Its segment from the last departure from its initial terminal to the first subsequent arrival at its final terminal is therefore an actual physical crossing between its labeled intervals. For a dual label, the crossing retains its initial and final stochastic boundary spokes and an ordinary cell path, with no intermediate hub jump. This proves simultaneous physical success for the slightly smaller range of interior labels that has all required buffers.

Uninspected high corridors. Select many odd interior labels spaced by at least four. For each selected \(i\), expose a physical dual crossing for label \(i-1\) from its left side, and one for label \(i+1\) from its right side. The search uses the physical cell graph with its available boundary half-extensions at that pair of intervals and fictitious always-open terminal stars confined to their outer caps. Seed the complementary-face search at the corresponding exterior side. Query an edge only from a reached face, and advance to the opposite face only when its physical bond in the path color is closed. On crossing success this exploration stays outside every blocking terminal path. Setting all unqueried edges to the nonpath state still leaves a path, by the plane path–cut alternative, so choose a simple witness measurably from the exposed edges. No edge with interior on its other side was queried. The ordered boundary spokes preserve this property at the endpoints. Stop each wall at the physical rectangle boundary-edge midpoint.

Let \(\mathcal R_i\) be the combined record of the two searches, including the queried edge states and the chosen walls. Let \(E_i\) be the event, determined by this record, that both searches succeed with disjoint walls in the proper order. The simultaneous physical crossings, including the odd target path separating its two neighbors, imply \(\mathbb P(E_i)\to1\).

Fix a value of \(\mathcal R_i\) in \(E_i\). Every fully contained primal edge between the walls, other than the deterministic boundary edges along the target wires, is uninspected, and every primal edge crossing a wall is closed. The top and bottom intervals contain exactly the two target primal wires: the wall endpoints lie in the flanking uncontracted intervals, and no exterior tie enters the corridor. Omit the boundary edges along the target wires, which are independent loops after contraction, and let \(G_i\) be the remaining contained-edge graph. Finite conditioning now gives exactly the FK law on \(G_i\) with the two target electrodes separately wired. Indeed the queried closed edges separate the corridor from the exterior across both walls, and the original boundary partition has no other class meeting the corridor; integrating the exterior therefore introduces no additional identification. Restoring the omitted loops with their independent ordinary law changes no crossing probability.

Let \(g_i\) be the pure corridor probability from Definition 20, measurable from \(\mathcal R_i\), and let \(T_i\) be the target’s actual physical crossing event in the original strip. On \(E_i\), such a path cannot cross either dual wall, so it is contained in the corridor. The preceding conditional law and the two-wire path equivalence give \(\mathbb P(T_i\mid\mathcal R_i)=g_i\) on \(E_i\). Since \(\mathbb P(T_i)\to1\), \[ \mathbb E\bigl[(1-g_i)\mathbf 1_{E_i}\bigr] =\mathbb P(E_i\cap T_i^c)\longrightarrow0. \tag{48}\] Markov’s inequality gives \(g_i\ge1-\epsilon\) on \(E_i\) with probability tending to one. Perform these searches on the same configuration for each of the finitely many selected indices. Equation (48) was applied separately to each search; no joint unexplored independence is required. With probability tending to one all the high-corridor conclusions hold together with the intervening physical crossings. In particular intervening odd primal crossings separate different pairs of dual walls, so the resulting corridors have disjoint closures.

The strip aspect ratio is fixed. Choose the finite number of selected labels larger than the constant allowed by Lemma 21. For large \(n\), any realization with all the preceding properties violates (40). Such realizations have probability tending to one, a contradiction. This proves (46). ◻

Proof. In the drawing with all links present, start from a face sector incident to \(e\). The dual adjacency graph of regions into which the links cut \(F\) is connected, since the original face \(F\) was connected. Remove a frontier auxiliary link to expand the reached \(e\)-face and repeat, also removing auxiliary links with that reached face on both sides. Eventually every link is removed. Reverse this order. At each addition, the link and \(e\) are cofacial in the graph including the new link. Cofacial Rayleigh implies that its log partition gain, integrated from zero to its specified odds, is nonincreasing in the odds of \(e\). Sum these log gains. The removal order may be chosen separately for each physical edge and each background graph.

For auxiliary odds \(\lambda_f\), divide the augmented partition function by \(\prod_{f\in T}\lambda_f\) and send every \(\lambda_f\) to infinity. Only the term with all auxiliary links open survives, and it is exactly the contracted-graph partition function. Retaining the same links and normalizing product after any physical minor handles links that become loops or redundant. This proves the limit statement as well. ◻

Proof. There are three steps. A component exploration bounds the number of physical crossings by a geometric tail. A covariance inequality turns that tail into the asserted upper bound on \(\Delta_H(S,W)\). Finally, an exact full-wire identity and cofacial comparison give the two bounds for an arbitrary partition \(X\).

Proof. Conditional component counting makes the unnormalized density factor \(Z_{X,F_W}/Z_X\). By Lemma 24 its ratio to the fixed factor \(Z_{F_W}/Z\) lies between \(e^{-\varepsilon_p}\) and one, where \(\varepsilon_p\to0\). Division by its average gives a normalized density between \(e^{-\varepsilon_p}\) and \(e^{\varepsilon_p}\), uniformly in \(X\). ◻

Proof. Relative to the \(X\)-law, the extra partition gain is the nonnegative random weight \[f(A)=q^{k_{H/(X,Y)}(A)-k_{H/X}(A)}, \qquad \mathbb E_{H/X}f=\frac{Z_{X,Y}}{Z_X}.\] On no actual crossing, explore all complete physical open components touching \(W\), without using ties as physical links. Their union \(U\) contains \(W\) and avoids \(S\). All edges from \(U\) to its complement are closed. Every \(X\) tie is contained in \(U\), and conditional component counting shows that the conditional mean of \(f\) is exactly the \(Y\)-partition gain in the ordinary graph \(H-U\).

This gain is at least the unconditional mean of \(f\) in the \(X\)-law. Draw all prospective cap links. With \(Y\) temporarily absent in describing faces, delete the \(X\) links in an order expanding the exterior face from the \(S\) cap. Then peel all physical edges touching \(U\) from this exposed face: start at \(W\), delete its inward fans, and continue along paths in each explored component to the vertices reached. All those components touch \(W\), and no \(S\) vertex is removed. At each deleted edge \(e\), the \(Y\) log partition gain cannot decrease. For a precise finite-odds comparison, peel the prospective \(Y\) links from the face sector incident to \(e\) through their cap, then add them in reverse order. Each newly added link is cofacial with \(e\) in the graph containing both. Cofacial Rayleigh makes its log gain nonincreasing with the odds of \(e\). Decrease those odds to zero and repeat. The order for \(Y\) may be chosen anew for each deleted edge; there is no claim that all nested cap links are simultaneously cofacial. Finally send auxiliary odds to infinity after dividing out their common leading product. During physical peeling retain the vertices of \(U\) temporarily. Since \(U\) avoids all \(Y\) endpoints, its isolated vertices contribute the same factor \(q^{|U|}\) to the ordinary gain’s numerator and denominator and then cancel. This proves the asserted comparison with \(H-U\).

Consequently \[\mathbb E_{H/X}[f\mid\text{no hit}]\ge\mathbb E_{H/X}f.\] Tilting by \(f\) does not decrease the no-hit probability, and hence does not increase the physical crossing probability. To compare the \(X\)-law with the blank law, use the same argument with the two ends exchanged and with the initial partition empty. ◻

Proof. The upper bound is the finite-odds cofacial cap comparison used at the end of Lemma 24, now with arbitrary noncrossing \(Y\) in place of a full wire. Peel the cap links from the main exterior and add them in reverse order, ordering the opposite cap as necessary for each comparison. Each pair is cofacial when Rayleigh is applied. Integrating and taking the normalized contraction limits gives the upper bound.

For the lower bound, use the no-hit component exploration from Lemma 26 in the \(X\)-law. On no actual hit its conditional \(Y\) gain is the blank \(Y\) gain in \(H-U\). By just the physical-edge peeling part of that proof, comparing this time against the empty base cap, that gain is at least \(Z_Y/Z\). Lemma 26 gives no-hit probability at least \(1-b_H\). Keeping only this nonnegative contribution to the mean of \(f\) yields \[\frac{Z_{X,Y}}{Z_X}=\mathbb E_{H/X}f \ge\mathbb P_{H/X}(\text{no hit})\frac{Z_Y}{Z} \ge(1-b_H)\frac{Z_Y}{Z}.\] This is the claimed inequality. ◻

Proof. Contract each level arc separately. Edges along a contracted arc are loops and contribute factors independent of the remaining configuration, so they may be omitted. The ordinary plane dual has precisely the interior cell graph just described, with a single exterior hub carrying the spokes from both flanks. Its odds are \(q/v=v\). Euler’s formula gives the FK dual law with these odds. The disk path–cut alternative identifies absence of the primal level-to-level connection with an open dual traversal from one flank to the other through the cells, without using the common hub as a shortcut.

Split the common hub into its two flank hubs and let \(p\) be their connection probability in the split law. Joining the hubs changes the component count by one on separation and by zero on connection. Consequently the joined law gives the traversal event probability \[\frac{p}{p+t(1-p)}=\frac{q p}{1-up}.\] This is \(1-Q_k(r,R)\). Solving for either probability proves (57); direct substitution proves that \(\mathcal H\) is involutive. This reasoning uses only the disk boundary order and the square-cell incidence structure, so applies to the cell-center passage as well. No Euclidean identification across a slit or between different sheets of the angular cover is made. ◻

Proof. All the comparisons here use auxiliary interval chains in narrow exterior caps and an exterior terminal test edge. When a rectangle is flush to a continuous exposed boundary stretch, every unwanted edge can be reached from a test-edge face across only unwanted edges. Successive face expansion therefore peels the unwanted edges, and cofacial Rayleigh gives the rectangle-to-disk comparison in the required direction. Interval caps can be chosen with no physical edge in their thin pockets, so they do not obstruct the peeling.

Proposition 22 first gives chords from a fixed positive-length patch to the opposite side of a fitting subrectangle. For two patches on adjacent sides, the two patch-to-opposite-side chords interlace. Lemma 14 therefore gives a positive lower bound for their connection. For two patches on the same side, first connect each toward the opposite side. Interlace each of these chords with a transverse rectangle crossing, choosing the right-hand endpoint for the left patch and the left-hand endpoint for the right patch. The resulting two chords again interlace and connect the original patches. When two input full-side electrodes meet at a corner, use Corollary 17; the two patches to be connected are disjoint and stay away from that corner. All rectangles have fixed aspect ratios and positive relative patch sizes. At a convex corner the same adjacent-side construction takes place in an inward square.

Here is the corresponding construction at a reflex corner. In local coordinates take \([-h,h]^2\) with its southwest quadrant removed. Let the two boundary patches lie on its negative horizontal and negative vertical rays, away from the corner. In the upper rectangle, connect the first patch to an interval in the upper portion of the right wall. In the right rectangle, connect the second patch to an interval in the right portion of the top wall. The adjacent-side construction just given bounds both probabilities below. Extend each rectangle cofacially to the clipped square. The four intervals occur in the cyclic order \[\text{negative horizontal ray},\quad \text{negative vertical ray},\quad \text{right wall},\quad \text{top wall}.\] The two chords consequently interlace. A further application of Lemma 14 connects the two original patches with a fixed positive lower bound.

Subdivide the boundary route into a fixed finite number of sufficiently short comparable pieces, putting disjoint ordered patches \(I_0,\ldots,I_N\) on their middle portions, with \(I_0\subset I\) and \(I_N\subset J\). Choose the subdivision so a two-piece span meets at most one corner and fits one of the preceding local constructions. Those constructions give \[ g(I_j,I_{j+2})\ge c_0>0. \tag{58}\] Interlace the chord from \(I_0\) to \(I_j\) with the chord from \(I_{j-1}\) to \(I_{j+1}\), beginning with \(j=2\). After a bounded number of applications this gives \[ g(I_0,I_N)\ge c_1>0. \tag{59}\] Enlarge the terminal intervals to \(I,J\) by adjacency. Finally, the face-expansion comparison in the first paragraph proves the stated extension property. ◻

Proof. For (i), use the convex-corner case of Lemma 31 along the outer polygon rim of \(C_k(r,R)\), starting and ending on flank patches at distances comparable to \(R\) from the center. When \(R/r\) is sufficiently large, the route stays a fixed fraction of \(R\) away from the inner rim. On each sheet its straight pieces have lengths comparable to \(R\), and the local inward rectangles and corner squares have sides a small fixed fraction of \(R\). Crossing an intervening quadrant seam does not change the local square-grid geometry. There are only finitely many pieces for fixed \(k\). The two banks are always treated as separate disk sides. The lemma therefore gives a deterministic-contact network lower bound between the two flank patches.

Replacing such a deterministic contact by its stochastic spokes loses at most the factor \[\eta=\frac{v}{1+v}>0.\] Indeed, with the other tested terminal fixed, the rooted formula of Lemma 11 writes the new isolated hub’s connection probability as \[\mathbb E_\mu\bigl[1-(1+v)^{-N}\bigr] \geq \eta\,\mu(N>0),\] where \(N\) counts rooted-multiset multiplicity on the contact patch. The right-hand hitting probability is the deterministic-contact network. Perform this comparison for the two hubs in turn. Adding their other spokes does not decrease their connection probability by adjacent Rayleigh. This proves (i), with a factor \(\eta^2\).

For (ii), contract the two circular layer cuts of the stated spoke-free passage and apply Lemma 30. The split dual has ordinary square-grid interiors and stochastic flank spokes. Its outer boundary route has the same local form as the one just used. Taking a cell dual shifts its rows and endpoints by bounded mesh distances; straight lengths and available inward depths remain comparable to \(m\), and successive corner rows still meet at right angles. For \(m\) sufficiently large the fixed clearances used above absorb all these shifts. The split dual flank-to-flank probability is therefore at least a positive constant \(c_4\). Its complementary radial probability is at most \(\mathcal H(c_4)<1\) by exact duality. Take \(c_1=1-\mathcal H(c_4)\). ◻

Proof. How the obstruction works. We first replace the partition law by a spreadable one and express it as an iid mixture of labeled pairs. Noncrossing leaves only private matched bonds and at most one repeated label on either side. The merger identities and pressure inequality leave two pure cases; in the private case the same pressure inequality makes the bond probability constant. The matched tests exclude that case, and the same-side tests exclude the repeated-label case.

Making the law spreadable. For each \(m\) there are finitely many partitions of \(2m\) labeled vertices. Color an increasing \(m\)-tuple of pair indices by a finite quantization of its vector of restriction probabilities. Iterated infinite Ramsey (Ramsey 1930, Theorem A) gives nested infinite sets \(J_j\) such that, for every \(m\leq j\), these vectors differ by at most \(2^{-j}\) between any two increasing \(m\)-tuples in \(J_j\). Restrict the law to the first \(j\) indices of \(J_j\) and take a projective subsequential limit, using compactness of each finite partition simplex. The resulting law is invariant under increasing injections of the pair indices. Every condition in the statement survives: it involves finitely many partition probabilities, the merger quotients have positive denominators, and increasing restrictions preserve the specified boundary order and the separation of \(U,V\). We may therefore assume this spreadability from now on.

Give each partition block an independent uniform label in \([0,1]\). There are countably many blocks, and their labels are almost surely distinct. Let \(X_i,Y_i\) label \(a_i,b_i\), and put \(Z_i=(X_i,Y_i)\). This sequence is spreadable on the compact space \([0,1]^2\). The spreadable form of the de Finetti representation is due to Ryll–Nardzewski (Ryll-Nardzewski 1957); we include the argument for this compact label space. For a bounded continuous real function \(f\) and two disjoint index blocks \(B,C\), each of length \(L\), spreadability gives \[ \mathbb E\left[\left(\frac1L\sum_{i\in B}f(Z_i) -\frac1L\sum_{i\in C}f(Z_i)\right)^2\right] =\frac2L\bigl(\mathbb Ef(Z_1)^2-\mathbb E[f(Z_1)f(Z_2)]\bigr). \tag{64}\] For functions \(f_1,\ldots,f_m\), average their product using one index in each of \(m\) successive length-\(L\) blocks. Its expectation equals \(\mathbb E\prod_j f_j(Z_j)\) by spreadability. Replacing every block average by the average of the same function over the first block changes this expectation by \(O(L^{-1/2})\), by (64) and a telescoping product estimate. The laws of the empirical measures of the first block have a subsequential weak limit on the compact space of probability measures on \([0,1]^2\). Calling the resulting random measure \(\nu\), we obtain \[\mathbb E\prod_{j=1}^m f_j(Z_j)=\mathbb E\prod_{j=1}^m\nu(f_j).\] Products of continuous functions determine the finite-dimensional laws. We can consequently realize the labeled pairs as iid with common law \(\nu\), conditionally on \(\nu\).

Noncrossing restricts the directing law. Almost every directing law gives a noncrossing partition with conditional probability one, simultaneously for all finite restrictions. Its first marginal has at most one atom: if two distinct labels \(x,y\) had positive mass, the event \(X_1=x,X_2=y,X_3=x,X_4=y\) would have positive probability and would cross. The second marginal has the same property. Denote their possible atom masses by \(\alpha,\beta\), putting the mass equal to zero for a diffuse marginal.

A diagonal tie \(X_i=Y_i\) at a label other than a common marginal atom has positive probability only if both marginals are diffuse. For example, if the first marginal has atom \(x\) of mass \(\alpha>0\) and such diagonal ties have mass \(\delta>0\), three independent pairs produce \(X_1=X_3=x\) and \(X_2=Y_2\ne x\) with probability \(\alpha^2\delta>0\). The two blocks interlace in the disk boundary order, which is impossible. The second-side argument is identical. Let \[D_0=\{\alpha=\beta=0\},\qquad d_0=\mathbb P(D_0).\] On \(D_0\), labels from different pairs are almost surely distinct, since each marginal is diffuse and different pairs are independent. Thus the only ties are private matched bonds, conditionally independent with a probability \(\gamma=\nu\{(x,x):x\in[0,1]\}\). Outside \(D_0\) every repetition uses a marginal atom, and no diffuse matched bonds occur. The two atoms, if both exist, need not be equal.

The merger weights. Write \[ w(m)=t^{-m+\mathbf 1\{m>0\}},\qquad A_j(x)=t(1-ux)^j+(1-t)(1-x)^j, \qquad u=1-t^{-1}. \tag{65}\] If \(M\) is binomial with parameters \(j,x\), then \[ \mathbb Ew(M)=A_j(x),\qquad A_1(x)=1,\qquad (1-ux)^j\leq A_j(x)\leq t(1-ux)^j,\qquad A_j(x)\leq1. \tag{66}\] The first identity follows by separating \(M=0\) and summing \(t\mathbb E[t^{-M}]\) on \(M>0\); the bounds also follow directly from the formula, or from \(0<w(m)\leq1\).

Test \(k\) first-side labels and \(s\) second-side labels with exactly \(l\) overlapping pair indices, where \(1\leq l\leq k\) is fixed and \(s\to\infty\). Normalize the reweighting by \(t^{k+s-2}\). On \(D_0\), if \(m\) of the \(l\) matched bonds occur, the union has \(k+s-m\) blocks and \(C=\mathbf 1\{m>0\}\), so the normalized weight is \(w(m)\). The conditional denominator and numerator are respectively \[ A_l(\gamma),\qquad A_l(\gamma)-(1-\gamma)^l. \tag{67}\] On \(\{\beta=0<\alpha\}\) there are no cross-group ties, and the denominator is \(A_k(\alpha)\) while the numerator is zero. On \(\{\beta>0\}\) the normalized weight tends to zero in conditional expectation: impose the second-group merge first. Its normalized factor is \(w(M_s)\), where \(M_s\) counts occurrences of its atom; the remaining first-group merge has rank gain at most \(k-1\). Thus the whole normalized weight is at most \(w(M_s)\leq1\), and \(M_s\to\infty\) almost surely. Bounded convergence applies also after averaging over \(\nu\).

Equation (62), multiplied by its denominator, therefore yields, for every \(k\geq l\), \[ \mathbb E\bigl[\mathbf 1_{D_0}\{A_l(\gamma)-(1-\gamma)^l\}\bigr] =h\left(\mathbb E[\mathbf 1_{D_0}A_l(\gamma)] +\mathbb E[\mathbf 1_{\{\beta=0<\alpha\}}A_k(\alpha)]\right). \tag{68}\] Since \(A_k(\alpha)\to0\) on \(\alpha>0\), take \(l=1\), let \(k\to\infty\), and compare with \(k=1\). As \(A_1=1\) and \(h>0\), this proves \(\mathbb P(\beta=0<\alpha)=0\). Reversing the two sides proves \(\mathbb P(\alpha=0<\beta)=0\). No division by a limiting denominator has been used. Hence outside \(D_0\) both atom masses are positive.

Eliminating a mixed diffuse/atomic law. Choose \(D=\{a_1,\ldots,a_s\}\) and \(F=\{b_{s+1},\ldots,b_{2s}\}\). On \(D_0\) all selected labels are distinct. Outside \(D_0\) the number of repetitions of each marginal atom tends to infinity. Consequently each of \[\mathbb E[t^{k_D-s}],\qquad \mathbb E[t^{k_F-s}],\qquad \mathbb E[t^{k_{D\cup F}-2s}]\] tends to \(d_0\). For the last term, use \(k_{D\cup F}\leq k_D+k_F\); all three integrands are bounded by one. If \(0<d_0<1\), this gives \[K(D)+K(F)-K(D\cup F)\longrightarrow\log d_0<0,\] contrary to (61).

The purely diffuse case. If \(d_0=1\), put \(V=1-u\gamma\) and use the two disjoint vertex sets \(D=\{a_1,b_1\}\) and \(F=\{a_2,b_2\}\). Conditional independence of the private bonds gives \[\mathbb Et^{k_D}=t^2\mathbb EV,\qquad \mathbb Et^{k_{D\cup F}}=t^4\mathbb EV^2.\] Pressure positivity forces \((\mathbb EV)^2\geq\mathbb EV^2\), so \(\gamma\) is constant. The singleton matched test in (62) gives \(\gamma=h>0\). For groups of size \(s\) with all indices paired, (67) gives failure probability \[\frac{(1-\gamma)^s}{A_s(\gamma)} \leq\left(\frac{1-\gamma}{1-u\gamma}\right)^s\longrightarrow0.\] Their connection probabilities cannot all equal \(h<1\).

The positive-atom case. If \(d_0=0\), the first marginal has exactly one atom of mass \(\alpha>0\) almost surely. Two distinct first-side vertices connect with probability \(\mathbb E\alpha^2>0\). For ordered-separated groups of sizes \(s,k\), their atom counts are independent conditional on \(\nu\). There is at most one block shared by the groups. Their normalized merger weight is the product of their two single-group factors, and connection requires a positive atom count in both. Thus \[ G(a_U,a_V)= \frac{\mathbb E[(A_s(\alpha)-(1-\alpha)^s) (A_k(\alpha)-(1-\alpha)^k)]} {\mathbb E[A_s(\alpha)A_k(\alpha)]}. \tag{69}\] By (63) this lies in \([\mathbb E\alpha^2,1-c]\) for every \(s,k\).

If \(\operatorname{ess\,inf}\alpha=0\), fix \(k=1\). The probability laws weighted by \(A_s(\alpha)\) concentrate at zero. Indeed, for any \(0<\epsilon<1\), the positive-mass set \(\{\alpha\leq\epsilon/2\}\) contributes at least \(\mathbb P(\alpha\leq\epsilon/2)(1-u\epsilon/2)^s\) to the denominator, whereas \(\{\alpha\geq\epsilon\}\) contributes at most \(t(1-u\epsilon)^s\). Their ratio tends to zero. Since \(A_1=1\) and \(A_1-(1-\alpha)=\alpha\), the ratio in (69) is at most the expectation of \(\alpha\) under this weighted law and therefore tends to zero. This contradicts its positive lower bound. If instead \(\alpha\geq a>0\) almost surely, then uniformly \[\frac{(1-\alpha)^s}{A_s(\alpha)} \leq\left(\frac{1-a}{1-ua}\right)^s\longrightarrow0.\] Taking \(k=s\) in (69) makes the ratio tend to one, contradicting its strict upper bound. All possibilities for \(d_0\) have been excluded. ◻

Proof of Proposition 29. For the fixed \(k\) and orientation, the nonincreasing supremum in (56) has a limit \(M\geq0\). Lemma 32(i) and exact duality imply \(M\leq\mathcal H(c_0)<1\). Suppose \(M>0\). Choose sectors with \(R/r\to\infty\) and \(Q_k(r,R)\to M\), and put \[h=\mathcal H(M)\in(0,1).\] Fix \(n\). Inside each sector choose integer levels satisfying \[ r\ll s_1^-\ll s_1^+\ll s_2^-\ll\cdots \ll s_n^-\ll s_n^+\ll R, \tag{70}\] where every consecutive ratio tends to infinity along the chosen sequence. For example, divide the logarithmic interval from \(r\) to \(R\) into \(2n+1\) equal parts and round the resulting levels; rounding does not affect these ratios. In particular the interior levels tend to infinity.

Use the whole interior cell graph \(C_k(r,R)\) as a base. On the first flank put distinct hubs \(a_1,\ldots,a_n\), with \(a_i\) having precisely the odds-\(v\) spokes dual to flank edges between \(s_i^-\) and \(s_i^+\). Put analogous hubs \(b_1,\ldots,b_n\) on the second flank. There are no other spokes. Draw each fan in a narrow exterior strip along its own boundary interval. These strips are disjoint, so the network partition of the hubs is noncrossing in the order \(a_1,\ldots,a_n,b_n,\ldots,b_1\).

For a matched pair \(a_i,b_i\), retain just the cell subband between \(s_i^-\) and \(s_i^+\) and its two fans. Its connection probability is exactly \(\mathcal H(Q_k(s_i^-,s_i^+))\). Extending it to the full interior cell graph, and then adding the remaining separate fans, does not decrease this tested network probability. Here is the cofacial comparison. Draw the test edge joining the two hubs in the exterior of the disk. The complement of the closed subband has an inner and an outer portion. Each unwanted interior-grid edge has access through its portion to the corresponding inner or outer rim, and thence to a test-edge face, crossing only unwanted edges. The two retained fan strips occupy only their stated flank intervals; their pockets contain no unwanted interior edges and they do not close these complementary openings. Face expansion therefore deletes all unwanted edges in an order cofacial with the test edge. Cofacial Rayleigh gives the claimed extension comparison. Each other fan lies on a disjoint boundary stretch; peel its edges successively from the main exterior face and reverse this order to add the fan. The same cofacial comparison applies at each addition. All these are comparisons of finite-odds edges, so no unproved boundary-condition monotonicity is involved. Thus \[ \liminf\mathbb P(a_i\sim b_i)\geq h, \tag{71}\] because each subband ratio tends to infinity and the defining supremum for \(M\) bounds its \(Q_k\) from above by \(M+o(1)\).

Conversely, merge all first-flank hubs into one terminal and all second-flank hubs into another, then add every missing flank spoke. Each merger can be made by edges incident to the corresponding tested terminal, and the added spokes are incident there too. Adjacent Rayleigh shows that the network probability does not decrease at any step. The result is the full split dual network, with connection probability \(\mathcal H(Q_k(r,R))\to h\). More generally, for any nonempty \(U,V\subset\{1,\ldots,n\}\) with \(U\cap V\ne\varnothing\), choose \(i\in U\cap V\). Enlarge the matched terminals \(a_i,b_i\) to \(a_U,b_V\), and then to both full flank terminals. The two preceding comparisons squeeze the merger test to give \[ G(a_U,b_V)\longrightarrow h. \tag{72}\] Here \(G\) is exactly (60): the FK change of component count on identification is its rank-gain reweighting.

There is also a uniform strict upper bound for ordered-separated groups on the first flank. Suppose \(\max U<\min V\). In a gap between consecutive band intervals lying between these groups, choose an integer \(m\to\infty\) so that the two cell layers \(m+\tfrac12\) and \(2m-\tfrac12\) and the entire passage between them lie strictly inside the gap. This is possible by (70). Enlarge the inner tested terminal to the whole inner layer and the outer terminal to the whole outer layer. All vertices, hubs, and fans below the gap may be absorbed into the first terminal, and all those above it into the second. No fan crosses the gap. Every enlargement is by links incident to a tested terminal and does not decrease its connection probability. After these identifications the two exterior portions attach to the passage only at its respective terminal; their partition functions factor out. The remaining network is precisely the pure spoke-free cell-layer passage of Lemma 32(ii). Therefore \[ \mathbb P(a_i\sim a_j)\leq G(a_U,a_V)\leq1-c_1 \quad(i\in U,\ j\in V) \tag{73}\] once the original sector is sufficiently far along the sequence. The lower bound is the direct terminal-enlargement comparison in the original finite hub graph.

Finally Theorem 7, applied to that finite graph, gives (61) for every pair of disjoint hub sets. For fixed \(n\), pass to a subsequential limit of its finite partition law. Equations (72) and (73), noncrossing, and pressure positivity survive. Let \(n\) increase and take another projective subsequential limit on the two infinite lists. This is legitimate because each finite partition space is finite, and all constraints on a fixed restriction hold for all sufficiently large \(n\). The resulting law satisfies every hypothesis of Lemma 33, with \(c=c_1\), a contradiction. Hence \(M=0\).

For each fixed larger \(k\) the outer route has more, but still boundedly many, straight pieces and convex corners. The local comparisons, cell duality, radial band order, and gap contraction are unchanged on the noncyclic angular cover. Thus the same proof establishes the claimed range of \(k\), with constants allowed to depend on it. Rotations, reflections, and lattice translations preserve all graph incidences and odds. ◻

Proof. Keep the order of vertices on both cuts, and treat the cuts as impassable walls for the complementary search. Their tangential physical edges are not queried. They are unnecessary to detect an actual radial crossing: trim any such crossing after its last visit to \(A\) and before its first subsequent visit to \(C\). The disk path/cut alternative for these walls says that either an actual \(A\)–\(C\) path exists, or a complementary path joins the flanks, but not both. One way to verify this deterministic alternative is to contract each cut to its own terminal in the drawing and then trim the resulting terminal path as just described. No probability law is changed in using this topological test.

Split the complementary exterior into the two flank faces. Starting at one of them, explore its entire connected component using complementary edges, namely crossings of closed physical edges. The starting vertex is in an exterior flank collar: entering an adjacent interior face requires testing the intervening physical boundary edge and finding it closed. In particular, adjacent interior faces are not seeded for free. No passage out through \(A\) or \(C\) is allowed. One implementation keeps a fixed ordered queue of reached complementary faces and tests every previously untested physical edge incident to the next face. A closed edge adds its opposite face to the queue. An open edge does not. Stop on reaching the other flank, or on exhausting the queue. Thus the history is generated by successive tests of individual edge states.

If the other flank is reached, the complementary path blocks every actual \(A\)–\(C\) path. Otherwise set every uninspected physical edge temporarily closed. The reached complementary component does not enlarge: every edge leaving it was inspected and found open. There is still no complementary flank-to-flank crossing, so the disk path/cut alternative supplies an open radial crossing in this temporary configuration. All its physical edges were consequently inspected and found open. Choose a simple one by fixed deterministic tie breaks, and retain the portion after its last near-cut visit and before its first subsequent far-cut visit. This defines \(P\) and proves (i). Temporary closure is only a device for selecting the witness, not additional conditioning of the actual configuration.

Every reached face lies on the starting side of any open radial blocker. In particular no tested edge interior lies strictly on the opposite side of the chosen frontier: an edge is tested only from an incident reached face. The closed edges crossed during the search give a chain of face expansions from the starting flank. Deleting them exposes the appropriate side of \(P\) to that exterior face. These observations prove (ii). If \(P\) touches a flank, use separate exterior boundary slots, or equivalently slightly prolong that flank outward, to distinguish its two incident sides. Only the component accessible from the opposite flank is relevant; any further pieces separated by a boundary touch attach at \(P\) after contraction.

Orient the near-cut arc from the first flank to the second. Among simple actual radial paths, with internal visits to the cuts removed, consider the possible near endpoints. The search from the first flank has a leftmost such endpoint. To check this directly, let \(C_0\) be any such blocker. The exploration cannot reach a face strictly beyond \(C_0\). An inward edge with near endpoint strictly beyond the endpoint of \(C_0\) has its entire edge interior on the blocked side; neither incident complementary face is reached, so it cannot have been inspected. Hence the near endpoint of the selected inspected-open path cannot be farther along the arc than the endpoint of \(C_0\). The reversed search similarly gives a rightmost endpoint. If their arc coordinates are \(x_-\) and \(x_+\) in an arc of length \(\ell\), then \(x_-\le x_+\) and \[\max\{\ell-x_-,x_+\}\ge\ell/2.\] This gives (iii) and \(F_1\cup F_2=E\). For each search, its frontier and its own clearance test are functions of that search’s recorded leaf. An adaptive edge exploration with deterministic next-edge choices leaves the ordinary FK domain Markov law on untested edges, proving the final assertion. No joint history of the two searches is used. ◻

Proof. For the first estimate the explored band is \([a,s]\), its near cut is \(a\), and the targeted end is \(r\). For the second it is \([r,s]\), its near cut is \(s\), and the targeted end is \(R\). In either case write \(E\) for the actual crossing event in the explored band. In the first instance the opposite original end wire at \(R\) lies beyond the retained band. In the second, the opposite original end wire at \(r\) meets the measured path on the retained far cut. This is the only difference in the removal of end identifications below. Apply Lemma 35 from both flanks. On a favorable leaf \(\omega\), namely a successful search with its own clearance in that lemma, call the uninspected flank the right flank and its measured open path \(P\). Let \(G_\omega\) be the conditional graph obtained from the full \(H_k(r,R)\) by deleting the inspected closed edges, contracting the inspected open edges, and imposing the two original separate end wires. The remaining edges in this graph have exactly the FK law conditional on \(\omega\). Write \(T\) for the targeted end terminal in \(G_\omega\); all vertices of \(P\) already belong to one terminal there. Thus the probability of contact in the full network conditional on \(\omega\) is \(\mathbb P_{G_\omega}(P\leftrightarrow T)\).

Attach to the ordinary explored band a closed lattice rectangle running along the right flank across the missing end gap. Its transverse width is \[ w=\left\lfloor \frac{\min\{\text{target radius},\text{near radius}\}}8 \right\rfloor. \tag{78}\] The rectangle lies in the quadrant adjacent to that flank. Its opening on the near cut lies strictly to the right of the endpoint of \(P\): the clearance is at least half the whole near arc, whereas \(w\) is at most one eighth of its radius. Its aspect ratio is bounded in terms of \(L\); \(w\) is positive at the scales under consideration. Let \(H_*\) be the ordinary explored band together with this rectangle, and let \(B_0\) be the rectangle’s end interval on the targeted radial cut.

Define the reference graph \[R_\omega=H_* /(P,B_0).\] It has all ordinary edges of the explored band and attached rectangle, with the vertices of \(P\) and \(B_0\) wired separately and no other identifications. In particular it has none of the other measured deletions or contractions in \(G_\omega\). We claim \[ \mathbb P_{G_\omega}(P\leftrightarrow T) \ge \mathbb P_{R_\omega}(P\leftrightarrow B_0) =g_{H_*}(P,B_0). \tag{79}\]

We compare these two graphs through changes that cannot increase the tested probability in \(G_\omega\). For these comparisons, represent its fixed ties by auxiliary links of finite odds and take their contraction limits afterward. Shrinking the targeted end wire to \(B_0\) does not increase the tested connection probability by adjacency; its removed vertices remain ordinary vertices of the graph. The deleted closed edges expose \(P\) to the starting-flank exterior. Draw the terminal test edge through that accessible region and the exterior to an exterior representative of \(B_0\). Keep auxiliary cap chains narrow.

Every edge outside \(H_*\) can now be removed by cofacial face expansion. On the target side, the part of the end gap omitted by the rectangle opens through the remaining target boundary or through a flank; the restricted \(B_0\) cap seals neither access. On the far side, the omitted part opens through the far boundary and the flanks. Thus each unwanted edge has a midpoint accessible from a test-edge face across only unwanted edges. If the opposite full end wire is beyond the retained band, remove its auxiliary cap links first from that same exterior. For each finite comparison, cofacial Rayleigh says that the ensuing deletion does not increase the network connection probability. Fixed open ties can equivalently be assigned odds tending to infinity and then contracted, so this argument applies also in their presence.

Delete the vertices of \(P\) from \(H_*\) and take the component accessible from the rectangle. Define \(U_\omega\) by adjoining to this component its incident edges into \(P\) and the terminal vertex sets \(P,B_0\). Every edge interior of \(U_\omega\) outside \(P\) is uninspected and has its ordinary odds. Now contract \(P\); its own edges become loops and may be omitted. The searched side, and any pieces cut off from the rectangle by contacts of \(P\) with a flank, attach to \(U_\omega\) only at this contracted vertex. In the second application the retained opposite end wire also meets \(P\); remove its additional identifications first. This again does not increase the tested probability by adjacency at the \(P\) terminal.

Now compare the reduced conditional graph with \(R_\omega\). They have exactly the same ordinary graph on \(U_\omega\), with \(P\) and \(B_0\) as separate terminals. In either graph every remaining piece outside \(U_\omega\) attaches to it only at \(P\). An FK graph attached through one vertex contributes a partition-function factor independent of the other side’s states, so those pieces cancel from the terminal connection marginal. The marginal is therefore the one in \(R_\omega\). Its tested probability is therefore at most that in \(G_\omega\), proving (79).

It remains to bound its right-hand side uniformly in the history. Choose an interval \(J\) of fixed positive relative length on the left band flank, with radius comparable to the near radius and away from both radial cuts. In the first application retain temporarily only the rectangle and the subband \([a,4a]\), choosing \(J\) there. In the second retain the rectangle and \([\lfloor s/4\rfloor,s]\), again choosing \(J\) in its interior radial range. The assumed large ratio ensures that these subbands fit.

There is a boundary route from \(B_0\) along the right flank, around the truncation rim opposite the rectangle opening, and back along the left flank to \(J\). In the first case it uses the outer rim; in the second it uses the inner rim. Along the rectangle, its inward clearance is a fixed fraction of \(w\), and along the sector it is a fixed fraction of the near radius. Their ratio is bounded in terms of \(L\). Straight portions and corner neighborhoods therefore admit the hypotheses of Lemma 31, with constants depending only on \(L,k\). At the inner-rim corners use the clipped-square construction in that lemma. At split banks use their own sides of the abstract disk, never a neighborhood crossing the slit. It follows that the pure \(B_0\)–\(J\) connection probability in this truncated shape is at least \(c(q,L,k)>0\).

Extend to \(H_*\) cofacially. The further band pieces open to the exterior beyond the truncated rim; the two terminal caps are on \(B_0,J\) and do not obstruct this access. Consequently \[g_{H_*}(B_0,J)\ge c(q,L,k).\] Every graph route from the attached rectangle to \(J\) meets \(P\), because \(P\) is a radial separator and the rectangle opening is on its right. Add \(P\) to the \(J\) terminal by adjacency. Once \(P\) is contracted, all the remaining pieces on the \(J\) side attach only at that terminal and factor out. Wiring their \(J\) vertices to it does not change the other side’s marginal law. Therefore \[g_{H_*}(P,B_0) =g_{H_*}(P\cup J,B_0) \ge g_{H_*}(J,B_0)\ge c(q,L,k).\] Together with (79), this is the required lower bound given each favorable exploration history.

A successful network connection here supplies the indicated actual extension. First expand every contracted measured-open component along its already measured physical edges. The only remaining abstract jumps are in the two original end wires. Starting from the target terminal, trim its path after its last use of the target wire. If it reaches the opposite end wire or leaves the explored band past the far cut before reaching \(P\), its physical portion has already reached that cut. Otherwise it reaches the measured physical path \(P\), which continues to the far cut. Erasing loops gives an actual crossing, stopped at its first visit to the far cut. A target-wire jump only chooses the starting vertex of this physical path.

For the last probability step, let \(F_1,F_2\) be the favorable history events for the two flank explorations. They are both contained in \(E\), and Lemma 35 gives \(F_1\cup F_2=E\). Apply the conditional estimate separately to each exploration, not to a joint unexplored law. If \(E'\) is the extended actual-crossing event, then \[\mathbb P(E')\ge c\mathbb P(F_i)\quad(i=1,2),\qquad \mathbb P(E')\ge c\max_i\mathbb P(F_i)\ge\frac c2\mathbb P(E).\] This proves both asserted inequalities. In particular the second one is a fresh application under the original law, not an application under a law conditioned on the outcome of the first search. ◻

Proof of Proposition 34. Only the qualitative blank decay remains to be shown. Fix \(k\) and an orientation. If it fails, there is \(\epsilon>0\) and sectors with arbitrarily large \(R/r\) for which \(b_k(r,R)\ge\epsilon\).

Choose such a sector with \(R/r\) so large that it contains \(n\) mutually disjoint radial bands, each of ratio at least \(n\), and each of inner radius at least \(n\). The bands may be separated by additional radial gaps. This is possible for every \(n\): for example, choose the original ratio larger than \(n^{4n}\) and use successive geometrically spaced integer radii, increasing the ratio slightly to absorb roundoffs. Removing one closed band leaves an inner piece and an outer piece with no physical route between them. Conditional on all edges outside that band, each piece induces a noncrossing partition only on its own radial cut. There are no lateral wires and no through ties. By (74), the conditional crossing probability in that band is bounded by its own blank probability.

A full-sector crossing requires crossings of all the bands. Conditioning outside one band at a time therefore gives \[\epsilon\le b_k(r,R) \le \prod_{i=1}^n b_k(r_i,R_i).\] No independence is asserted here. At least one factor is at least \(\epsilon^{1/n}\). Selecting such a band for every \(n\) produces a new sequence, renamed \((r,R)\), with \[ r\longrightarrow\infty,\qquad R/r\longrightarrow\infty, \qquad b_k(r,R)\longrightarrow1. \tag{80}\]

Choose a fixed integer \(L\ge2\) sufficiently large that Proposition 29 gives \[ Q_k(x,y)\le q-\varepsilon_0 \qquad\text{whenever }y/x\ge L \tag{81}\] for some fixed \(\varepsilon_0>0\). Enlarge \(L\) once if necessary for integer cut conventions. Consider the actual crossing of the central band \[[a,s]=[2Lr,\lfloor R/(2L)\rfloor].\] Its probability tends to one in the blank law by (80), since a full crossing contains such a central crossing.

Insert the full inner end wire, using its collar \([r,Lr]\). Conditional on the exterior of this collar, the exterior gives only an opposite-cut cap partition. Lemma 24 and (81) bound the density and its reciprocal on that exterior by a constant \(C_0<\infty\) independent of \(r,R\). For example, the logarithmic variation is at most \[\log\left(1+\frac{(t-1)(q-\varepsilon_0)} {1-t(q-\varepsilon_0)}\right),\] whose denominator is positive. Now insert the full outer end wire, using \([\lfloor R/L\rfloor,R]\). The same estimate applies: the already inserted inner wire is part of the exterior and still induces only a noncrossing cap partition on the collar’s opposite cut. The central event is exterior to both collars. Its failure probability after both insertions is at most \(C_0^2\) times its blank failure probability, and hence tends to zero.

Apply the two estimates of Lemma 36 under this law with full separate end wires. Their constants are fixed, because \(q,L,k\) are fixed. They give \[\liminf_{R/r\to\infty} \mathbb P_{H_k(r,R)/(A_r,A_R)} (\text{actual $r$--$R$ crossing}) \ge \frac{c(q,L,k)^2}{4}>0\] along the sequence (80). With only these two separate full wires, terminal network connection is equivalent to existence of an actual path between the two end arcs: take a simple terminal path and remove its initial and final wire portions. The displayed probability is therefore \(Q_k(r,R)\), contradicting Proposition 29. This proves the blank decay, and the end-cap comparisons were established at the start. ◻

Proof. Choose an integer \(M\ge2\) so large that the supremum in Proposition 34 at ratio \(M\) is at most some fixed \(\theta<1\). Between \(r\) and \(R\) place the bands \[[(2M)^j r,\ M(2M)^j r],\] for all consecutive integers \(j\ge0\) for which they fit. Their number \(N\) is at least \(\log(R/r)/\log(2M)-1\), unless this lower bound is negative, in which case the eventual estimate follows by increasing \(C\). The intervening gaps make the edge sets disjoint. Conditioning outside each band induces only separate end caps, so the same iteration as above gives a crossing probability at most \(\theta^N\). This argument also applies with prescribed separate cap partitions at the two original ends, because neither partition joins the inner and outer components left by removing a band. Taking \[\eta=\frac{-\log\theta}{\log(2M)}\] and absorbing the one-band discrepancy into \(C\) proves the first two assertions. The last follows from (75). ◻

Proof. Full wiring decreases the number of components by \(k_D-1\), so \[ K_G(D)=\log t+\log Z_{G/F_D}-\log Z_G. \tag{90}\] Substitute this expression in each pressure term in (88). Wiring all of \(A\) subsumes either \(X_i\); wiring all of \(E\) subsumes either \(Y_j\); and wiring \(A\cup E\) subsumes both. The full-wire terms therefore cancel in the indicated alternating sum, leaving exactly the displayed mixed log ratio. This argument also works when the two rim images overlap.

For (89), first wire \(A\) and \(E\) separately. Joining the two resulting terminals has partition gain \[p+t(1-p)=t(1-up).\] Insert this gain and (90) into the definition of \(\Delta_G\). The asserted bounds follow from \(0\le p\le1\). ◻

Proof. Corollary 37 and a union bound over the fixed number of components give \[b_H:=\mathbb P_H(\text{an actual path joins }A\text{ to }E) \le C_0(r/R)^{c_0}.\] Lemma 26 applied to the two full end wires gives \(p=g_H(A,E)\le b_H\). Choose \(L\) so that \(tp\le1/2\). By Lemma 24, \[0\le\Delta_H\le \log\left(1+\frac{(t-1)p}{1-tp}\right) \le C_1(r/R)^{c_0}.\] The one-sided identity, with images understood, is \[I_H(X,F_E)=\Delta_H-\Delta_{H/X}.\] Its nonnegative left side bounds \(\Delta_{H/X}\) by \(\Delta_H\); the argument with \(A,E\) interchanged bounds \(\Delta_{H/Y}\). Lemma 27 gives \[0\le I_H(X,Y)\le-\log(1-b_H)\le C_2(r/R)^{c_0}.\] Finally (88), with empty base caps, yields \[\Delta_{H/(X,Y)} =I_H(X,Y)-\Delta_H+\Delta_{H/X}+\Delta_{H/Y} \le C_3(r/R)^{c_0}.\] This proves both conclusions. The argument uses the accessible, separate disk arcs in its hypotheses, not just the fact that the underlying graph has a planar drawing. ◻

Proof. Fix one of the four quotients. Discard all cap classes absorbed into its hubs, obtaining clean residual caps \(X',Y'\) in \(H\). In \(H/(X',Y')\), set to zero the site variables of all vertices that will be absorbed into a hub. A cluster meeting one of these vertices has factor one in the defining product for \(F\). Identifying any of these zero-variable vertices with each other cannot alter a factor from a cluster avoiding them: if two such clusters were joined by the identification, each would already meet a zero-variable vertex and would already have factor one. The underlying independent edge states are unchanged. Loops and parallel-edge combinations preserve the same product law on the remaining states.

Thus the two polynomials with the indicated site variables clamped are identical. Their negative logarithms have identical coefficients on the retained sites. The hub-avoiding part of the quotient’s hit-both mass is therefore a submass of the clean pressure for \(H/(X',Y')\), bounded by Lemma 40. Every other multiset hits at least one of at most \(J\) final hub vertices. Equation (87) and a union bound give the remaining \(J\log t\). Apply (88) to the four quotients to bound the mixed interaction. ◻

Proof. In the multiset expansion proving Theorem 7, put \(\gamma_z=\prod_{e\ni z}(1+v_e)^{-1}\). The coefficient on \(m\) is the coefficient in the rescaled variables \(y_z=\gamma_zx_z\), multiplied by \(\prod_z\gamma_z^{m_z}\). The rescaled coefficient depends only on the induced weighted graph on \(\mathop{\mathrm{supp}}m\): its connected-structure expansion uses only ordinary edges between members of that support and the hard edges between copies of a site. It is nonnegative. The rescaled coefficients are therefore identical in the two graphs, whereas the incidence assumption makes every relevant \(\gamma_z\) at least as large in the second. This proves (94), including the case of a zero coefficient. ◻

Proof. For a fixed state on each side, absorb its edge weights and the component factors of clusters not touching the collar into positive weights \(a_X,b_Y\). Component counting then gives the combined partition function as \[Z_{\mathrm{comb}}=\sum_{X,Y}a_Xb_YZ_{H/(X,Y)}.\] Several states inducing the same cap may be combined into its weight. The assumed bound is exactly \[e^{-C}\frac{Z_{H/X}Z_{H/Y}}{Z_H} \le Z_{H/(X,Y)} \le e^C\frac{Z_{H/X}Z_{H/Y}}{Z_H}.\] Multiplication by \(a_Xb_Y\) and summation factor the right and left expressions into a product of a first-side sum and a second-side sum. This remains true when either side’s weights are changed, including when its cut is fully wired. Taking a mixed ratio for two choices on each side cancels the separate sums and leaves a factor between \(e^{-4C}\) and \(e^{4C}\). In particular the interaction of the two full cut wires in the combined graph is bounded. Equation (89) adds at most \(\log t\) to obtain the pressure bound. This proof also explains why a side graph attaching on a bank or bypassing the collar is excluded. ◻

Proof. We first bound the unnormalized mass of a fixed avoided-slit pattern. Take a small square annular collar compactly contained in \(V\), with fixed positive inner radius and a sufficiently large fixed ratio of outer to inner radius. Slit it along one coordinate ray. Consider supports that meet the inner ball, reach outside the outer cut, and avoid all sites of the ray across the band. Delete these forbidden ray sites and replace the band by the regular full-angle disk sector with its two banks distinct. Keep all allowed sites, including those in the inner and outer side graphs, and all their induced edges.

At an allowed collar site, a new bank edge can only replace an old edge to one of the deleted ray sites, with at most one replacement in each coordinate direction. At the rim transitions keep the original allowed edges. The regular collar is closed at its radial cuts and requires no additional outward continuations there; in particular an attached side-graph edge is not duplicated by a filler edge. Omit unnecessary edges to new forbidden sites outside the filled part. Hence no incident external product is increased. There is no old edge between the two banks after the ray sites have been removed, and the inner and outer graphs attach only at their respective cuts. Their cap partitions are noncrossing in the linearized cyclic order. Lemmas 42, 40, and 43 therefore bound the mass of this pattern by a constant depending on the fixed collar. Grid centers and radii can be rounded by bounded mesh steps. When the avoided ray has a positive physical clearance from the support, these roundings do not change the pattern for small meshes.

The space \(V\) is locally compact and second countable, so \(V^\dagger\) is compact metrizable; its Hausdorff hyperspace is compact as well. This proves the existence of the subsequential laws in the statement. We now pass the fixed-pattern bound to any such limit.

For a fixed collar compactly contained in \(V\), choose an open ball strictly inside its inner cut, and let \(\Sigma\) be its compact slit segment. For \(\varepsilon>0\), consider the hyperspace event \[\{K:K\text{ meets the inner ball and } \operatorname{dist}(K,\Sigma)>\varepsilon\}.\] It is open: meeting an open ball is an open Hausdorff condition, and the distance to a fixed compact set is continuous. If \(K_\delta\) belongs to this event, the inner hit comes from the original support, not from \(\infty\). That support is connected in the quotient graph, by Theorem 7, and it contains a marked site outside \(V\). Since the collar lies in clear bulk, with no quotient jumps or bypass through it, the support must cross the outer cut. On a fixed compact neighborhood of the collar, the compatible metric and the Euclidean metric induce the same uniformity. Moving slit points by a bounded number of mesh steps therefore moves them uniformly by \(o(1)\) in the compatible metric. The positive clearance from \(\Sigma\) makes the support avoid the rounded slit sites for all sufficiently small meshes. The fixed-pattern mass bound therefore applies. Dividing by the diverging total mass and using the open-set part of the Portmanteau theorem shows that this event has probability zero under the limit law. The adjoined point \(\infty\) has supplied compactness only; it has not been counted as an outer exit.

It suffices to use a countable collection of rationally specified collars, inner balls, and positive clearance margins. Strict hits and clearances persist under sufficiently small changes of these data. Every \(K_\delta\) meets the fixed compact \(C\), so every weak limit meets \(C\) almost surely. Suppose such a limit does not fill \(V\). The set \(K\cap V\) is closed relative to \(V\). Choose a point of \(V\setminus K\) sufficiently near a point of \(K\cap V\) and work inside a compact ball contained in \(V\). A Euclidean nearest point of \(K\cap V\) is attained there, and a sufficiently small complementary open disk tangent to \(K\) at that point has closure in \(V\). At least one coordinate direction points strictly into this disk. A short segment of that ray, away from its starting point, has positive distance from \(K\). Choose a large fixed-ratio collar about the tangency point with its slit on this segment and with a strict inner hit. Perturbing its data slightly to the countable rational family preserves the hit and clearance. This is one of the probability-zero events just excluded, a contradiction. ◻

Proof. If the pressure were unbounded along meshes tending to zero, Lemma 44 would force its normalized supports to fill \(V\). Place a small half-box collar at the midpoint of \(J\), inside \(W_0\cup J\), with outer radius less than \(d_0/2\) and a large fixed ratio between its two radii. This is a two-quadrant disk sector. Its lateral boundary lies on the free wall and has no attachments. The graph inside its inner cut and the remainder outside its outer cut induce separate end caps. Lemmas 40 and 43 bound the mass of connected supports meeting both cuts.

Choose a fixed tiny bulk patch strictly inside the inner cut. Density makes the normalized support meet this patch with probability tending to one. Connectedness to a remote marked set forces an exit from the outer cut, even if the other mark lies on the wall rim. This contradicts the bounded unnormalized cut mass.

It remains to justify uniformity when the mesh is bounded below. Every counted support meets the fixed compact set in the hypothesis, which contains only a uniformly bounded number of represented sites by the local counting property of the clear family. Equation (87) bounds the pressure by their number times \(\log t\), uniformly over the remote graphs and partitions. Thus an unbounded sequence would necessarily have a subsequence with mesh tending to zero, which was just excluded. ◻

Proof. The planar path/cut alternative gives the frontier implementation in the statement. Start in the left exterior face; entering an interior face requires querying its boundary edge and finding it closed, so interior faces adjacent to the flank are not seeded for free. Exhaust the complementary faces reached from there, querying an edge only when one of its incident faces is reached. If a primal crossing exists, this component cannot reach the opposite side. Its separating frontier contains a simple top-to-bottom path supported on measured open edges. Equivalently, setting every unqueried edge closed still leaves such a crossing. Fix deterministic tie breaks to select the extremal simple frontier. No reached face lies strictly on its right, and an edge with midpoint strictly on its right has both incident faces there. Hence every such edge is uninspected. Boundary contacts are handled by outward prolongations of the side incidences; the enlargement from \(B\) to \(L\) puts the strict right region in a single disk. Denote it by \(\Omega\). All measurement changes lie in \(B\). For a fixed successful leaf \(\omega\) and a fixed permitted remote cap choice, let \(G_\omega\) be the conditional untested graph, with measured closed edges deleted, measured open edges contracted, and that cap imposed. Let \(p_\omega\) be its single vertex containing the physical path \(P\).

Proof. Put \(X_0=Y_0=\varnothing\), \(X_1=X\), \(Y_1=Y\), and use the notation \(Z_{ij}\) from Lemma 39. We must bound above and below the ratio \[T=\frac{Z_{11}Z_{00}}{Z_{10}Z_{01}}=\exp(-I_H(X,Y)).\] A free rim will bound the three pressures with at least one blank cap, and hence bound \(T\) above. For the lower bound we expose a local primal or dual path, use its conditional pressure bound, and sum over the successful search leaves.

Proof. Realize each end wire by a narrow chain along its boundary interval, between successive tested slots. The dual slots strictly inside these chains have their own faces. All remaining exterior dual slots belong to one face: they are exactly the two longitudinal flank intervals, joined for weighting. Interior bonds are in diagonal or spoke duality cell by cell. The disk path/cut alternative identifies absence of the tested end connection with an actual opposite-color connection between the flanks, ignoring their common identification.

Let \(a\) be the connection probability when these two flank wires are separate. Joining them multiplies weights by \(t\) on separation and by one on connection, so the actual connection probability becomes \(a/(a+t(1-a))\). Equating this with \(1-p(Q)\) gives \(a=H(p(Q))\). Specifying the first and last eligible slots moves the four corner positions by only bounded mesh distance. ◻

Proof. Work first in the complete tested monochrome grid on an angular lift. Choose a regular sector \(D\) whose angular walls lie at least \(2\pi\) in either direction from the focal ray. Its radial band is deep: two comparable-radius patches, one on each angular wall, are separated from either radial end by a large multiplicative factor. Wire the patches separately and call the resulting terminals \(x',y'\). The fixed sector flank-chord construction, extended cofacially from a fixed-ratio subband, gives \(\mathbb P_D(x'\leftrightarrow y')\ge c_0>0\). Consequently the pressure mass hitting both terminals equals \(j(\mathbb P_D(x'\leftrightarrow y'))\ge j(c_0)\).

This paired mass can be kept away from both radial end rows \(E\). By the rooted hitting identity, the mass hitting \(x'\) and \(E\) is at most \[\frac{u}{q}\, \mathbb P_D(x'\leftrightarrow E\text{ fully wired}).\] Enlarge \(x'\) by all vertices in a central subband containing \(y'\). By adjacent-terminal comparison, this enlargement does not decrease the last probability. An outward connection must now cross one of two ordinary regular sector collars. They are networks in parallel. Conditional ties from the other piece either keep their two terminals separate or join them; joining upweights absence of an actual connection. Their pure radial probabilities therefore give a union upper bound. Proposition 29 makes this bound arbitrarily small by choosing the radial depths large. Fix the depths so that at least \(j(c_0)/2\) of the paired pressure mass avoids \(E\).

We next place a copy of this sector chart across the actual channel. Take as active its ordinary strict sites accessible from the focal ray, before a barrier is met, in a slightly larger radial band. Use radius-spanning barrier subarcs, trimmed from a last exit to a first reach. Between their angular lifts, each physical strict site occurs once. The active component does not reach the artificial angular walls. Every nearest-neighbor chart step from an internal active site either remains active or has an ordinary contact edge to its indicated hub, possibly a half-tile spoke. There is no missing incidence across the focal ray. Extra folds simply remove parts inaccessible from that ray.

Prewire into its indicated hub the active endpoint of every contact incidence at an obstructed chart step, including immediate ordinary contact incidences. Read these incidences from a band slightly larger than the template, so that the same prescription applies at its radial ends. The two absorbed sets stay apart because the focal strip has many-step width.

This enlargement has a bounded reversible cost. Hold one hub as the rooted vertex and remove the incident edges of the other hub to form the base graph in Lemma 11. For each remaining site \(i\), put \(s_i=\prod_{h\ni i,\,h\ni o}(1+v_h)\), the product over all links from \(i\) to the removed hub \(o\); parallel links are included separately. Its connection probability is \[\mathbb E_\mu\left[1-\prod_i s_i^{-m_i}\right].\] Whenever a multiset meets the set being prewired, at least one affected link has odds at least \(v\), and the displayed integrand is at least \(v/(1+v)\) times its full-wire limit. Off that set the integrands agree. Thus an entire hub enlargement costs at most \((1+v)/v\), independently of the number of absorbed sites. Apply this once for each hub.

Fill the nonactive chart sites and identify each filler site with the hub on its angular side of the focal strip. Sites on further sheets beyond the corresponding barrier have the same label. No required chart edge connects two different such labels: the untouched strip separates them throughout the enlarged radial band. Edges between retained strict sites, and edges from them to a prewired neighbor, are present at their ordinary odds. Every remaining required edge is a hub loop or an available contact incidence. In particular, if both sides of a missing chart step are retained across two cut half-tiles, precollaring has put both endpoints in the same hub. The missing edge is then a loop. Add or allocate loop copies as necessary; this does not change the enlarged network test.

Now split each hub into the distinct chart copies assigned to it, the appropriate wired artificial patch \(x'\) or \(y'\), and residual vertices carrying all other original edges. The complete chart \(D\) is present after splitting. Merging the pieces back does not decrease the terminal connection probability. At all internal chart sites the induced graph agrees with the benchmark, and the incident products are no larger; extra original hub edges have been assigned to residual pieces. Only radial end rows can retain unallocated outside incidences. By Lemma 42, coefficients supported on chart multisets avoiding those rows are at least the benchmark coefficients. Positive coverage therefore gives paired pressure mass at least \(j(c_0)/2\), hence a positive network connection probability. Reversing the two precollaring enlargements proves the assertion in the original graph. ◻

Proof. Buffered square events. In an ordinary clear box, horizontal actual crossings of a square, or a rectangle whose transverse dimension differs by a bounded number of steps, have probability bounded below uniformly in planar exterior data. Indeed a square crossing or its opposite cell crossing has probability at least \(1/2\). In the latter case duality, a rigid grid symmetry, and bulk annular density transfer that event to the desired color and direction in a larger clear buffer. The complementary cell crossing has exterior adjacent rows; trimming changes dimensions only by bounded steps. One may instead start with a self-dual box one step wider than its transverse height.

The same bounded-density transfer applies to a crossing with endpoint constraints on any fixed convex clipped-square support. Surround the whole support, within its positive clearance, by two simple ordinary rectilinear polygonal cuts. Their geometry is fixed and nondegenerate, although the relative collar width may be small. The proof of Proposition 47 uses connected clear regions, straight-wall pieces with blank ends, and bounded local path defects. Each of these constructions fits along these finitely many offset straight pieces, with constants depending on their widths and angles. Its pressure bounds therefore give the same reciprocal density comparison here. Rigidly moved copies may be compared through overlapping larger neighborhoods. Conditioning outside their buffers only changes the allowed planar caps. No physical hard crossing or sector screen is used for this transfer.

A route and its templates. Choose a simple polygonal path with axis-parallel segments in the tested monochrome grid directions, entering the tangent disks along axis rays making angle at most \(\pi/4\) with their inward normals. It can be chosen with a bounded number of segments, positive lower bounds on their lengths, and positive clearance off its endpoint approaches. To see uniformity, join the two disk centers within their cleared corridor, approximate by a sufficiently fine fixed pixel path, erase loops, and perturb its axis segments inside the clearance. Insert a small interior detour if necessary so both axial directions occur. A subsequential limit of cleared corridors and contact normals gives the same conclusion with strict sub-clearances. If the physical endpoints coincide, fix short rays in their separate accessible regions first, and join their interior endpoints; simplicity is in the disk topology.

Along this path place crossing templates, separated by small positive gaps between their support envelopes. Crop each crossing between two straight transverse caps. A finite cover of each cap and pigeonhole selection give tiny endpoint patches supporting a positive-probability crossing event. Their widths can be made arbitrarily small relative to the gap; their center displacements are then recorded. Slight cap shifts avoid vertices. If a cap crosses a bond, use the corresponding half-edge with its actual bond state and orient the witness toward the far cap, retaining its portion after its last near-cap hit and before its first subsequent far-cap hit. Bounded mesh rounding affects only these endpoint neighborhoods.

Here are explicit templates sufficient for the construction.

1. A straight \(+x\) advance of any prescribed fixed length is made from two identical small horizontal-square crossing templates, with one reflected to reverse its \(y\)-displacement. The transverse displacements of their patch centers cancel. Choose their widths just below half the desired advance and allocate equal small gaps at the ends. Longer advances are subdivided to remain inside the chosen clearance.

2. For a \(+x\)-to-\(+y\) turn, use the support \[[0,s]^2\cap\{x+y\le1.35s\}\] from its left edge to the oblique cap. Crop a horizontal crossing ending in the upper half, selected using reflection and transfer. Its endpoint has \(x+y\ge1.5s\), whereas its starting point has \(x+y\le s\), so it crosses the cap. Reflect and reverse traversal across that cap for the turn’s second half, with a small normal spacing. The second support is ahead of the vertical entry and the first is below the horizontal exit by a fixed fraction of \(s\). Much thinner incoming and outgoing straight templates consequently miss the nonadjacent half. Grid symmetries give the other turns.

3. For a contact with inward normal proportional to \((1,\tau)\), \(0\le\tau\le1\), use \[[0,s]^2\cap\{x+\tau y\ge0.65s\}\] from the oblique cap to its right edge. It is obtained by cropping a horizontal crossing starting in the lower half. Place the starting patch center a distance \(\epsilon\) inside the contact along the normal, with \[s^2/c'\ll\epsilon\ll s,\] where \(c'\) is the tangent radius. Tangent-disk exclusion then puts the whole template inside the clear approach and the wall in the opposite half-plane in a smaller contact annulus. Every exit advances a fixed fraction of \(s\) both axially and into the tangent half-plane. Much thinner subsequent straight templates are therefore safe. Reverse the construction at the other endpoint.

Choose bend and endpoint sizes first, sufficiently small relative to all route clearances and segment lengths. Their selected patch offsets and gap displacements are then fixed. Adjust total advances on two independent axis segments by the resulting small amounts, and realize the intervening lengths by the straight cancellation construction. Its widths are much smaller than adjacent bend or endpoint sizes. The final gaps are smaller still, and patch tolerances smaller than the gaps. Bounds on all segment lengths and widths give a finite bound on the number of pieces before these last choices. Thus the remaining advances can be fitted exactly, up to bounded mesh errors, without moving nonadjacent envelopes into one another. Near coincident endpoint closures the envelopes stay in the separate tangent approaches.

Conditioning and network concatenation. The disjoint clear buffers give a positive joint probability of all template crossings by successive conditioning. On their success, perform the complementary-face searches of Lemma 35 in the separate clipped envelopes, always from the same longitudinal side of the oriented route. Seed only the exterior flank face: an adjacent interior face is reached only after its intervening physical bond has been queried closed. Fictitious outward spokes confined to the endpoint patches specify search topology only; they add neither FK ties nor physical path edges. The complementary face search exhausts on success. Setting every unqueried edge closed still leaves a crossing, so a simple actual open blocker is supported on measured edges. No reached complementary face lies beyond it, and ordinary incidences into its unsearched side remain untouched.

At a gap, prolong the two blocker endpoints along their short terminal spokes and connect them inside the innermost gap neighborhood, solely for separation topology. Between a few gap widths and a small fixed fraction of the template length the blockers span the annulus in opposite half-planes. The focal ray on the unsearched side is in the untouched gap. An accessible fold terminates on the same blocker hub; it introduces no foreign boundary condition. Other terminal caps are outside the enlarged join annulus. At an endpoint the continuous wall and its short tangent access replace one of the blockers. Only its single wall label is locally accessible, and tangent-disk exclusion gives the same half-plane separation.

Lemma 50 therefore bounds every successive two-hub connection below in the same conditional FK graph. Hubs already connected cause no difficulty. Shared-root positive correlation gives \[\mathbb P(i\leftrightarrow k) \ge \mathbb P(i\leftrightarrow j)\mathbb P(j\leftrightarrow k).\] Iterating along the finite chain gives a positive conditional probability of connection of the original hubs. Averaging over successful leaves proves the lemma. This argument does not claim simultaneous physical bridges confined to the individual gaps. ◻

Proof. Let \(M\) be the supremum limit in the assertion. If \(M=1\), choose disks \(Q_n\) of diverging end-separation ratio with \(p(Q_n)\to1\), and then choose integer intermediate radii \(1+r_n\ll m_n\ll R_n\). Proposition 53 gives a subsequence and constants fixed on it. In the bounded alternative its opposite-color route uses boundedly many tiles, with bounded moves to the eligible flank slots. Conditional finite energy gives a positive lower bound for the companion flank connection. In the divergent alternative, Lemma 51 gives such a bound for the two flank hubs of the companion pure law. Either conclusion contradicts Lemma 49, since \(H(p(Q_n))\to0\). Thus \(M<1\). The lower bound here may depend on the extracted subsequence; no uniform localization constants over all disks were needed.

Suppose \(M>0\), and take \(p(Q)\to M\) at diverging end-separation ratios. Put \(h=H(M)\in(0,1)\). For any fixed list length, choose pairs of integer level crosscuts \[C_{s_i^-},\ C_{s_i^+}\] with all paired ratios, intervening gaps, and end clearances diverging successively. They can be ordered intrinsically: select the last desired cut, retain its piece containing \(I\), and repeat backwards. The intercut pieces are tiled disks of the same class, with full cuts as ends.

For each isolated intercut disk, let \(a_i,b_i\) be precisely the two companion flank electrodes prescribed by Lemma 49, including its eligible corner slots. Embed these same slots in their accessible fans of the companion graph of the full \(Q\), and wire each electrode separately there. The cuts alternate slots and contain a whole edge. Thus these are exactly the isolated companion tests, with no independent endpoint rounding; all electrodes lie on the original flanks away from the original end corners.

Each matched probability has lower limit at least \(h\). Indeed duality gives this for its isolated intercut disk, and cofacial extension preserves the lower bound. For this extension all unwanted graph is beyond the tile cuts and can be peeled through those end openings from the exterior faces of a terminal test edge. The wire chains hug their exposed boundary subintervals and seal no unwanted region. A corner gaining extra incidences upon extension is interpreted in its same accessible disk fan. The other disjoint interval chains can then be added by the same exterior cofacial argument.

Conversely merge all \(a_i\)’s, merge all \(b_i\)’s, and complete both flank intervals. Adjacent-terminal comparison shows that the test does not decrease; the final test is the companion probability of \(Q\), which tends to \(h\). Therefore every merged cross-list subgroup test containing a matched pair tends to \(h\).

For two ordered-separated nonempty index sets \(U,V\) on the first list, enlarge their terminals to the two complete cuts of an unused intervening diverging-ratio gap. No other local interval wire bridges that gap. Each exterior now attaches only at its own terminal and factors out, leaving the pure radial test of an intercut disk of this class. In the notation \(G\) of (60), terminal enlargement in the original finite hub graph and then this gap comparison give, for every \(i\in U,j\in V\), \[\mathbb P(a_i\sim a_j)\le G(a_U,a_V)\le M+o(1).\] Choose \(c_*=(1-M)/2>0\) before choosing the list length. For each fixed length the last upper bound is at most \(1-c_*\) sufficiently far along the disk sequence, simultaneously for its finitely many ordered-separated groups.

The induced partition has the reversed boundary orders of Lemma 33; pressure coverage gives its inequalities (61). For each fixed list length, take a subsequential limit in its finite partition simplex. The matched and cross-list subgroup tests converge to \(h\), and the same-list bounds above survive. Let the list length increase and take a projective subsequential limit, as in the proof of Proposition 29. Every fixed finite constraint holds eventually, and the merger denominators are positive. The resulting partition law satisfies all hypotheses of Lemma 33 with \(c=c_*\), a contradiction. Hence \(M=0\). ◻

Proof. We use the full-side skeleton rounding of Proposition 53 at divergent scale. A curve consisting of boundedly many line or circle pieces of total length \(O(\ell)\) meets \(O(\ell)\) cells. The same skeleton construction stays within \(O(1)\) tile steps and uses \(O(\ell)\) edges. Keep a grid-aligned central segment and follow each outward branch to its first original-wall incidence. Macroscopic clearance places that incidence in the intended terminal neighborhood, with its correct label. Loop erasure preserves the central segment and the separated branch tubes. Opposite accesses in coalescing terminal neighborhoods cannot meet in the interior, since that would give an excluded small opposite-flank cut.

There is a useful refinement when the final approach is the radial segment from the center \(x\) of an interior tangent ball to its contact \(z\). At every point \(y\) on this segment, \[ \mathop{\mathrm{dist}}(y,\partial D)=|y-z|. \tag{104}\] The upper bound uses \(z\) and the lower bound uses the tangent ball. Hence an earlier contact caused by rounding is only \(O(1)\) steps from \(z\). If \(\widetilde z\) is the rounded endpoint, center the terminal balls at \(\widetilde z\) and choose their normalized radii generically in slightly smaller fixed intervals, still separated by a fixed gap. The displacement \(|\widetilde z-z|=O(1)=o(\ell)\) preserves strict containment and separation. Reapplying Lemma 52 in these recentered balls gives wall continuations at the actual cut endpoint. We use precisely these endpoint-centered neighborhoods below.

Proof. Blank decay and the choice of a reference law. Separate cap partitions at the two ends give actual crossing probability at most \(b(Q)\), by Lemma 26. Suppose that blank decay fails, and choose a sequence with end-separation ratios tending to infinity and crossing probabilities at least \(\varepsilon_0>0\). Choose integer scales \[1+r\ll m\ll m'\ll R\] and the two families from Lemma 55. Its constants are fixed on one subsequence. Write \(F\) for the disk between \(C_0\) and \(C'_0\). Every original crossing traverses \(F\). Conditioning its blank exterior induces separate end caps, because the two exterior pieces are separated by \(F\). Thus no-hit comparison gives \(b(F)\ge\varepsilon_0\).

At a divergent-scale end, consider the collar between \(C_0,C_1\). The cut \(C_0\) has length \(O(\ell)\) and is at distance at least \(a\ell\) from \(C_1\). Cover \(C_0\) by \(N\) interval patches of diameter at most \(\alpha\ell\), where \(\alpha>0\) is a sufficiently small fixed constant and \(N\) is bounded independently of the mesh. By Theorem 54, the pure probability from each patch to the whole of \(C_1\) is less than \(q/2\). To apply that theorem with the patch as the end, include the unused parts of \(C_0\) in the free flanks; this changes neither the ordinary collar graph nor that pure test. Lemma 24 bounds its cross pressure by \(\log(2-q)\), since \(t=1/q\). Positive coverage consequently gives \[ \Delta(C_0,C_1)\le N\log(2-q). \tag{105}\] The one-sided identity and cofacial cap comparison now show that the full-\(C_0\) wire gain varies by bounded factors under every induced noncrossing cap at \(C_1\). Equivalently, inserting that wire changes the marginal beyond this collar by bounded reciprocal densities. The opposite end wire may already be present there: it only changes the cap induced at \(C_1\). At a bounded-scale end the cut has bounded rank, so finite component counting gives the same conclusion directly.

Insert the two end wires successively. Denote the resulting, fixed law on \(F\) by \(\mu\); its only ties are the two separate full end wires. For cuts \(A,B\) write \(E(A,B)\) for an actual traversal supported in the disk between those cuts. The original full traversal implies the central event, whose support is beyond both end collars. The marginal comparisons therefore give \[ \mu(E(C_2,C'_2))\ge c_0\varepsilon_0>0. \tag{106}\] At a bounded end use \(C_0\) in place of \(C_2\) in this notation. It remains to extend this actual central crossing to the two end wires.

Proof. The proof separates two estimates. When a continuation region has small area, an earlier level contains a short separating cut, so the uniform hard-passage bound makes the initial segment unlikely. The continuations beyond the midpoint are then summed by area. The two estimates use conditioning in opposite directions; neither requires independence of the branches.

We first record the disk decomposition used throughout the proof. Take an intermediate integer level. Its open intervals in the disk have disjoint interiors, and their closures are proper crosscuts; endpoints that coincide physically are separated in their boundary fans. Splitting a disk along these crosscuts produces a tree of disk regions, with an edge for each crosscut. Root this tree at the region incident to \(v\). The first crosscuts visible from that region will be denoted by \(w\). The side of \(w\) away from the root, including all its descendant regions, is a subdisk \(V_w\). These subdisks have disjoint interiors. Every terminal-level point is behind at least one such crosscut: otherwise a path from the root region to it would avoid the intermediate level. This also covers boundary contacts by using the closed disk and its split incidences.

For each \(w\), denote the other closed subdisk by \(V_w^-\). An actual successful path supplies both events \[\begin{aligned} E_w&=\{v\longleftrightarrow w\text{ actually in }V_w^-\},\\ F_w&=\{w\longleftrightarrow\{\varrho=s+\epsilon D\} \text{ actually in }V_w\} \end{aligned}\] for some visible \(w\). Indeed, use its first encounter with the relevant crosscut for the initial path and its last encounter for the remaining path. No edge is allocated to both subdisks, since the cuts follow tile edges; shared vertices on \(w\) belong to both end sets.

We prove (110) by induction on \(D\). If \(D\) is bounded, success requires at least one of the counted cells, so increasing \(C\) covers these cases. For the inductive step cut at \(s+\epsilon m\), where \(m=\lfloor D/2\rfloor\). The remaining distance \(D'=D-m\) is smaller than \(D\) and is at least \(\lfloor D/2\rfloor\). Write \(A_w\) for the cell count in \(V_w\) between the midpoint and terminal levels. Disjointness gives \[ \sum_w A_w\le A_v. \tag{113}\]

Here is the uniform estimate for the initial paths when the area is small. For a fixed \(w\), sum the lengths of the integer-level crosscuts in \(V_w^-\) at distances between \(D/8\) and \(D/4\) from the starting level. Each counted tile contributes a bounded total length to this sum: its radial range has length at most one, and it has only a bounded number of sides. Hence the sum is at most \(c_0 A_v\), for an absolute \(c_0\). There are a number of admissible levels proportional to \(D\) once \(D\) is large. At one of them the total crosscut length is at most \(c_2 A_v/D\).

One interval at this chosen level separates \(v\) from \(w\). To see this, split \(V_w^-\) along all the level intervals. The regions incident to each of the two connected end arcs form connected subtrees; those subtrees are disjoint because the arcs lie at strictly different radial levels on opposite sides of the chosen level. An edge between these subtrees is the required separating interval. Call its closure \(\sigma\). It has diameter at most \(c_2 A_v/D\), whereas its distance from \(w\) is at least \(c_3D\) for an absolute \(c_3>0\). Choose any point of \(\sigma\) as the center for the hard-passage test; then \(\sigma\) lies in a ball of radius \(c_2 A_v/D+O(1)\) and \(w\) lies outside a concentric ball of radius \(c_3D-O(1)\).

Condition first on all tile states in \(V_w\). These states induce only a noncrossing end partition on \(w\) in \(V_w^-\); the original partition is on \(v\). Next, for the purpose of bounding a traversal from \(\sigma\) to \(w\), condition on the side of \(\sigma\) containing \(v\). The remaining intercut disk has ordinary free longitudinal boundaries and at most joint planar end ties on \(\sigma,w\). Such ties are allowed by Theorem 56. A \(v\)-to-\(w\) path necessarily has an actual \(\sigma\)-to-\(w\) segment. The ratio in that theorem is bounded below by \[\frac{c_3D}{1+c_2 A_v/D}.\] Choose a fixed \(c_1>0\) sufficiently small, and subsequently a fixed \(D_0\) sufficiently large. If \(A_v<c_1D^2\) and \(D\ge D_0\), the uniform hard-passage estimate therefore gives, for every conditioning on \(V_w\), \[ \mathbb P(E_w\mid\text{states in }V_w)\le\frac1{10}. \tag{114}\] The additive mesh-scale term is controlled by the choice of \(D_0\).

There is a separate use of conditioning for \(F_w\). Fixing the states outside \(V_w\) induces a noncrossing partition only on \(w\), since \(w\) is its sole interface with the complementary subdisk. Applying the induction hypothesis and then averaging gives \(\mathbb P(F_w)\le C A_w/(D')^2\). Combining this with (114), which conditions in the opposite direction, yields \[\begin{align*} \mathbb P(\text{success}) &\le \sum_w\mathbb P(E_w\cap F_w) \le \frac1{10}\sum_w\mathbb P(F_w)\\ &\le \frac{C A_v}{D^2} \frac1{10}\left(\frac{D}{\lfloor D/2\rfloor}\right)^2. \end{align*}\] The final multiplier is less than one for all sufficiently large \(D\). The finitely many remaining \(D\) are covered by increasing \(C\). If \(A_v\ge c_1D^2\), the bound is immediate from \(C\ge1/c_1\). This proves the induction without requiring independence between different branches.

For (111), cut at an integer \(m\) nearest \(R/2\). In the same decomposition, each conditional initial event \(E_w\) has probability at most \(\eta(m/(1+s))\), where \(\eta(L)\to0\) is a uniform modulus supplied by Theorem 56. This applies directly to the ends \(v,w\), with the original end cap on \(v\) and the cap induced by conditioning on \(V_w\). The continuation slabs have total area at most \(c_4 R^2\) and remaining distance comparable to \(R\). Their bounds from (110) consequently sum to at most a constant depending on \(q\). Repeating the preceding conditional sum proves the outward limit.

For (112), use an integer \(m\) nearest \(\sqrt{(1+r)s}\). Then \(r+1\ll m\ll s\). Read the initial intercut disk in reverse, with \(w\subset\{\varrho=m\}\) as its small end and \(v\subset\{\varrho=s\}\) as its far end. Its conditional traversal chance is at most \(\eta(s/(1+m))=o(1)\). The disjoint continuation slabs have total area at most \(c_4 m^2\), and their radial distance \(m-r\) is comparable to \(m\). Their area bounds again sum to a bounded quantity. This proves the inward limit, irrespective of the number of exits. ◻

Proof. Select integer radii \[ a<b<\ell<c<h \tag{115}\] inside the available annulus, so that \(a\) is large in tile units and \(b/a,\ell/b,c/\ell,h/c\) all tend to infinity. Let \(U\) be the closed tiled component in \(a\le\varrho\le h\) accessible from the focal ray before crossing a barrier. Split its boundary contacts in their individual fans. A barrier diagonal leaves a half-tile. If it is a measured open primal edge, its endpoints already belong to the same hub, so replacing that boundary edge by the ordinary loop prescribed by the half-tile convention changes only a common partition-function factor. The opposite-color virtual slot is treated by that same convention.

We first justify the disk topology needed below. The two spanning barriers separate the focal ray from the opposite ray, preventing a wrap around the annulus. A physical spanning barrier joins the excluded inner and outer regions. Every additional excluded barrier portion is physically connected to one of those regions, by the exit assumption. Thus the excluded set is connected on the sphere; common hub labels alone would not establish this fact. The complement of its focal component remains connected: any other complementary component has boundary on this excluded set. There is therefore no hole in the focal component. With finitely many polygonal edges, splitting boundary pinches into their intrinsic fans gives a topological disk. Its interior is injectively embedded. Every nonradial boundary contact of \(U\) has one of the two specified labels. A strict primal site, including a strict radial-cut site, has one chart position and ordinary incidences. Any primal pinch on a contact barrier already belongs to its designated hub.

All connections of the unmeasured graph to fixed exterior data occur at the cuts \(\varrho=a,h\) or at \(x,y\). Let \(S\) consist of those radial-cut sites together with representatives of both hubs. The latter are included even if a hub is physically represented only outside a particular piece. An exterior condition is consequently represented by a planar partition \(X\) on \(S\).

Proof. For a bulk collar this is Proposition 47, with polygonal cuts and ordinary buffers. For sectors, retain every named bank class as a hub, including every partial bank interval. In each of the four cross pressure masses entering the alternating partition-function identity, discard terms hitting a final hub class. Positive coverage and the singleton identity charge at most \(J\log t\) for these terms, as in Lemma 41. All cap classes absorbed into a hub are discarded with it. The residual-cap condition in Definition 59 supplies the separate noncrossing caps on the coordinated cuts in each quotient. This is the cap order needed by the clean comparison; the bound on \(J\) alone would not supply it.

Each complete clean sector is a tiled disk, and its blank actual radial crossing probability is arbitrarily small at sufficiently large fixed radius ratio, by Theorem 56. Their finite disjoint union has the same property by a union bound. The separate-cap and no-hit comparisons, followed by the alternating pressure identity, therefore bound all four residual masses. This is precisely the sector density argument, applied to the coordinated disjoint charts; adding back the discarded hub terms leaves a uniform bound. The mixed partition-function bound says that the relative weight of an exterior state changes by a bounded factor when the opposite cap changes. Summing over exterior states and normalizing proves the stated marginal comparison. Reversing radial order makes the same proof apply inward.

If a chosen positive cap completion adds a fixed bounded number of further block identifications, each changes every configuration weight by either \(1\) or \(q^{-1}\), and these additions change normalized densities by a bounded factor. Any additional bank class instead retained in the collar must be included in \(J\) and satisfy the same attachment and residual-cap conditions. Thus the comparison does not cover an unbounded collection of partial bank classes or unlisted lateral attachments. ◻

Proof. We give the finite FK comparison and coupling argument. It also explains the restrictions on the geometry. Throughout a comparison, keep the entire common graph beyond the current section fixed, including its far boundary data. Collar marginals can depend on this remote graph; all constants below are uniform in that dependence.

Split each designated wire transversely at a section, retaining its connections separately on either side and one representative of the possible through connection. There are only boundedly many such representatives. The unknown graph then enters the common graph as a permitted partition \(\xi\) on that section, with all the common deterministic wire pieces included. Any bounded extra split or merger has weight ratio between \(\min(q,q^{-1})\) and \(\max(q,q^{-1})\) per operation. The bounded collar comparison therefore applies equally to this representation.

Use a sufficiently large fixed ratio, for example enlarge a factor \(256\) further if needed, to leave two buffered subcollars at every step. Fix a reference permitted partition. The initial comparison subcollar makes the marginal \(\mu_\xi\) on the second subcollar dominate \(K^{-1}\mu_*\) for every \(\xi\), with a constant \(K\) and the same reference marginal \(\mu_*\). This is a pointwise inequality of finite measures on the subcollar edge configurations.

Inside that second subcollar, in case (i) require a primal circuit, or a primal wall-to-wall screen meeting every named primal bank wire. In case (ii) require a primal screen in the wired sector and a dual screen in the other. Proposition 62 gives a reference probability at least \(p_*>0\) for this event \(G\). On \(G\), the partition transmitted to the next section is determined by the common subcollar configuration alone. Indeed every connection from the unknown side to the remote side in a screened primal component must meet a connected primal screen. All named bank wires there already meet that same screen. Connections through the unknown region can therefore make no further distinction among its remote contacts. In the unwired second sector the dual screen allows no actual primal crossing, and there is no bank or hub attachment to bypass it. The absence of other intersector contacts is used at exactly this point.

Any cluster of the common subcollar meeting the next section but avoiding the transmitting screen is therefore unable to acquire an extra connection through the unknown region. Components wholly on the exposed side contribute factors independent of all remote states and cancel when the conditional law is normalized. By the finite domain Markov identity, the entire conditional remote law thus depends only on this transmitted partition.

For completeness, screens can be discovered without querying edges strictly beyond them. Explore complementary-color components from the unknown-facing section. If the screen exists, the search stops short of the remote section; otherwise the disk path/cut alternative, or its annular version, would give a complementary crossing. The measured frontier consequently contains a blocking physical path or circuit. Perform this search in the buffered middle and reverse colors in the unwired sector. Thus the same transmitted-partition description applies either by revealing the whole fixed subcollar or by stopping at its measured screens.

To specify the transition precisely, reveal the tile states up to the next section and let \(\eta\) be the partition they induce there from the exposed side, together with the common deterministic wire pieces. The planar drawing and the absence of unlisted lateral attachments keep \(\eta\) in the permitted class. Conditional on the exposed states, the law of the fixed remote graph is the FK law with partition \(\eta\), by Lemma 3. Thus, for a remote observable \(f\), its conditional expectation is a function \(g(\eta)\) of that partition. On the screen event \(G\), the preceding separation argument shows that \(\eta\) is determined by the second subcollar configuration alone and is independent of the entering partition \(\xi\). Away from \(G\) it may also depend on the first subcollar and on \(\xi\).

The measure \[\lambda=K^{-1}\mathbf 1_G\mu_*\] is a common submeasure of all second-subcollar laws and has mass at least \(p_*/K\). On its support the just-defined map to \(\eta\) is common to all entering partitions. Its pushforward is therefore a common submeasure \(\nu\) of the laws \(L_\xi\) of the induced next partition, with the remote graph still held fixed. These laws are the transition kernels used below. Choose a fixed \(c>0\) not exceeding \(p_*/K\), and reduce \(\nu\) to mass \(c\) if necessary. Every such kernel has the form \[ L_\xi=c\widehat\nu+(1-c)\widetilde L_\xi, \tag{124}\] where \(\widehat\nu\) is the same probability measure for all \(\xi\). This is the common positive part of a coupling. No stochastic monotonicity of FK boundary conditions is required.

For a bounded nonnegative \(f\), make the relative-error deduction explicit. Number the sections \(0,1,\ldots,m\) toward the support of \(f\) and reserve the final bounded-comparison collar beyond section \(m\). Let \([a_j,b_j]\) be the range of conditional expectations of \(f\) over all permitted partitions at section \(j\), including the common remaining graph. Conditioning on the next section gives \[[a_j,b_j]\subseteq[a_{j+1},b_{j+1}],\qquad b_j-a_j\le(1-c)(b_{j+1}-a_{j+1}),\] where the second assertion follows from (124). The last comparison collar gives \(b_m\le C a_m\). If \(a_m=0\), then \(b_m=0\) and all expectations vanish. Otherwise \(a_0\ge a_m>0\) and \[\frac{b_0-a_0}{a_0} \le (1-c)^m\frac{b_m-a_m}{a_m} \le (C-1)(1-c)^m.\] It follows that either expectation is at most \([1+(C-1)(1-c)^m]\) times the other. This proves (123) for bounded \(f\). For a general nonnegative \(f\), apply those two inequalities to \(f\wedge M\) and let \(M\) increase by monotone convergence. No quotient of infinite expectations is used.

The same proof works inward, using the reversed collar comparison and the same separation by actual screens. A positive modification on the unknown side merely changes the mixture over its induced partitions: every member has the same common submeasure and lies in the same interval \([a_0,b_0]\). Integration over that mixture proves the assertion about tilts. In particular no fixed connection through a gap has to be retained when the gap belongs to the unknown region. In case (ii) this is why both colors of screen, rather than only the screen meeting the primal wires, are required. ◻

Proof. Fix one marker, write \(B\) for its opposite-color traversal, and condition on the edge states \(F\) outside its broad window. Other marker constraints and \(E\) are thereby decided. Trim any witness for \(A\) at two cuts lying just inside the window on its respective sides of the central marker. It gives actual paths from the two required sources to these cuts in disjoint side domains. A source can include a previously measured physical continuation through \(F\), but no search follows a wire identification as a path edge.

At each near cut choose two small, spaced opening intervals away from the flanks and corners. Orient each witness toward its target cut and retain the portion after its last source-set visit and before its first subsequent target-cut visit. Every resulting endpoint is a fixed positive distance to one side of at least one opening. Run a complementary search from that endpoint’s side, treating previously fixed far states as known data. The shielding argument given before Proposition 61 makes its measured frontier end on the same side of the selected opening. In particular that opening is on its unqueried side with a uniform margin. There are only a bounded number of choices of opening and orientation at each end. Let \(T_1,\ldots,T_N\) be the corresponding two-search success alternatives. They satisfy \[ A\subseteq\bigcup_{i=1}^N T_i \tag{126}\] for each fixed \(F\), with \(N\) independent of mesh and scale. On each successful leaf retain the simple actual frontier \(P\) connected physically back to its required source. Each \(T_i\) is a union of leaves of its own pair of searches, performed in the two disjoint side domains. The uninspected regions have their usual conditional FK laws on individual leaves; we do not condition those laws on the union in (126).

We verify the uniform tangent access needed to reconstruct between the two frontiers. From the empty central region enter a chosen opening, then move a small fixed radial distance into its side band and along a level transversal toward the searched flank. This route must meet \(P\): the source-to-cut crosscut separates the opening side from that flank. Start at a fixed clear point in the central region, and choose a clearing radius \(a\) smaller than its boundary distance, the opening separation, and all side-band margins. Move a center from that point along the route until its first distance-\(a\) approach to \(P\). The resulting disk is empty and tangent to \(P\). On entering the opening its endpoint is excluded by the angular margin; along the transverse portion it is excluded by the radial depth. Both cuts, all fixed far data, and the other flank stay at larger fixed distances. Any approachable point of the searched flank is screened by \(P\), or is itself a contact with \(P\). Hence there is no other locally accessible wall before the tangent contact. The unqueried incidences up to it are ordinary. Both endpoints of \(P\) lie outside a fixed enlargement of the contact neighborhood by the cut margins, so both branches of \(P\) continue out of it. Folds behind \(P\) do not affect this disk. Tile rounding is absorbed by decreasing the chosen fixed margins. This verifies all the measured-terminal hypotheses of Proposition 61, uniformly in the successful leaves. Perform the same construction at the other end.

In the clear center of the marker put a skinny tube. Two separated branches go to the tangent approaches just obtained; two further branches go to the two walls, with their attachment sites spaced on opposite tube sides. Use a wired contact or the free-wall seed as required by each wall’s tested-color designation. Seed rectangles are disjoint protected patches, exposed first. The radial branch detours remain in their respective radial neighborhoods, while the wall-to-wall portion remains wholly inside the marker. The four-terminal layout of Proposition 61 consequently has a uniform positive conditional probability \(c_0\) on every successful leaf. It both joins the two physically source-connected frontiers, preserving \(A\), and contains a same-color wall-to-wall screen inside the marker. The screen excludes \(B\) by planar path/cut separation.

Here is the relative, rather than merely absolute, estimate. Integration over the leaves of any fixed alternative gives \[ \mathbb P(A\cap B^c\mid F)\ge c_0\mathbb P(T_i\mid F) \qquad(1\le i\le N). \tag{127}\] Take the largest of these right sides and use (126). It follows that \[\mathbb P(A\cap B^c\mid F) \ge\frac{c_0}{N}\mathbb P(A\mid F),\qquad \mathbb P(A\cap B\mid F) \le\left(1-\frac{c_0}{N}\right)\mathbb P(A\mid F).\] Thus truncated success is used as an upper cover of the original arm event; it need not already imply that event. Averaging in \(F\) while retaining the exterior requirements and iterating over the separated windows proves (125) with \(c=c_0/N\).

For the two-arm event \(T\) in the statement, use the same opening and tangent constructions and join their two radial branches in the clear middle. No transverse screen is required. The two-success alternatives cover the separately required arms, and the identical bounded-union calculation gives the fixed relative probability. For extension of a single arm use one truncated search and one new outer terminal. The constants are uniform in the fixed rescaled geometry. Starting at a bounded number of mesh steps, a specified finite physical path has a positive conditional probability depending only on that bound and the fixed \(q\). Multiplying a fixed positive extension factor over \(O(\log(R/r))\) collars gives a polynomial lower bound. For an upper bound, separated fixed-ratio bands have conditional probability bounded below of an actual opposite-color screen by Proposition 62. An actual arm avoids every such screen. There are at least \(a\log(R/r)-b\) tests for fixed \(a>0\) and \(b<\infty\), so successive conditioning gives an upper bound \((1-c')^{a\log(R/r)-b}\le C(r/R)^{c''}\) for some \(c''>0\). These estimates require no independence of the successive bands. ◻

Proof. Write \(\mathcal D_{\mathrm{out}}\) for just the \(t\)-box and middle data. First remove the conditioning in the endpoint boxes one at a time. Each disjoint bulk-annular buffer separates that box from the entire support of \(\mathbf 1_{R_{\mathrm{arm}}}\). The inward or outward bounded comparison of Lemma 60, while holding all other data fixed, compares a specified box state with the mixture over its states. The number of boxes is fixed, so \[\mathbb P(R_{\mathrm{arm}}\mid\mathcal D) \ge c_0'\mathbb P(R_{\mathrm{arm}}\mid\mathcal D_{\mathrm{out}}).\] Next apply that lemma inward across the \(s\)-centered half-sector between \(c_1n\) and \(c_2n\). The remaining cylinder law and the reference law agree on its inner side, including the now unconditioned endpoint boxes. The \(t\) and middle states change only its permitted outer partition, with every bank class included in the named bound. Consequently \[\mathbb P(R_{\mathrm{arm}}\mid\mathcal D) \ge c\mathbb P_{\mathrm{ref}}(R_{\mathrm{arm}}) \ge c\mathbb P_{\mathrm{ref}}(A_{\mathrm{ref}})=c p_{\mathrm{ref}}.\] Both comparisons use the single joint observable \(\mathbf 1_{R_{\mathrm{arm}}}\). In the fixed-ratio gap about scale \(r\), these requirements force the two truncated successes used in the splice proof. The finitely many small separated endpoint patches can be avoided by disjoint interior bulk routes with fixed rescaled clearance; take the patch radii small before fixing these routes. Apply the two-arm splice and its bounded-alternative relative deduction. This constructs the extended physical arm with a fixed fraction of the probability of \(R_{\mathrm{arm}}\).

Search this extended-arm event afresh at the \(c_1n\) target cut, using its two stated openings. The one-sided truncated search gives a frontier with an untouched opening and tangent approach exactly as in Lemma 65. Use that opening’s cleared branch to attach the frontier to the joining tube, then use the stated corridor winding between its two tube positions. The winding layout of Proposition 61 supplies an actual noncontractible cycle in the same actual component as the arm, with a uniform conditional probability on each successful leaf. The truncated cover argument again turns this into a relative lower bound for the extended-arm event. Combining the two fixed losses gives the first inequality of (128).

Choose a fixed large spacing ratio for the intermediate markers, leaving their splice windows disjoint and separated from the two endpoint constructions. A maximal regularly spaced family has size proportional to \(\log(n/r)\) up to a bounded additive error. Apply (125) to the outer radial traversal appearing in \(R_{\mathrm{arm}}\), retaining its separated short gap arm, \(\mathcal D\), and all previously decided marker requirements in the exterior conditioning. The marker windows lie strictly inside this outer traversal domain. This proves (129) directly for \(R_{\mathrm{arm}}\); it does not require conditioning on the subsequent successful extension or cycle construction. ◻

Proof. An ordered crossing test gives one exponential rate. We first prove a geometric tail with one positive exponential rate. For each configuration, select a maximizing packing only to derive the following deterministic implications. No conditional law below fixes that choice of packing. Cover the relevant buffered annular slice by finitely many small bulk boxes and boxes flush with a straight bank. A subarc moving a fixed fraction of the slice width has a subarc between two separated tile-coordinate levels in one of these boxes. This follows by subdividing each coordinate into a fixed finite grid of levels and trimming at a last departure and a first arrival. Slight cropping keeps the levels inside the available buffer. Near a corner or a tip, make the selections in boxes away from the individual apex; an arc of the prescribed diameter must travel through one of them. There are only finitely many choices of box, coordinate, and pair of levels. The same selection applies to a diameter test: an arc of the prescribed diameter has a coordinate displacement bounded below by a fixed fraction of that diameter. Thus the argument below proves the first exponential rate for both crossing and diameter counts.

Suppose one choice receives many disjoint raw subarcs. They are ordered separators in the rectangular strip between its two levels. Small outward prolongations at lateral boundary contacts permit the usual disk order without adding any random tiles. On either side of a raw separator, its incident corner sites form an actual path: at each switch consecutive sites are either equal or joined by an actual open bond. One side has one color and the other side has the other color. These paths do not cross a raw separator. By retaining a fixed fraction of the separators and leaving unused separators between selections, we obtain disjoint actual crossings whose colors alternate in strip order. Trim them to be simple and to have no end-level contacts except their initial and terminal vertices. Lateral contacts are harmless. Their initial points retain their order along the first level.

Partition that level into a fixed number of intervals of length at most \(\eta\) times the level separation. One interval contains a consecutive block of at least a fixed fraction of the alternating crossings, up to a fixed additive loss. We choose the fixed \(\eta>0\) sufficiently small below. Thus a large passage count forces a long ordered alternation of proper actual crossings starting in one such interval.

We now test this implication without conditioning on the existence of the entire alternation. For the first color, perform complementary path/cut searches from both lateral sides of the strip, looking for proper actual crossings from the chosen interval to the other level. For these searches only, ignore bonds running along the two end levels and split the incidences at an end-level site into separate leaves in their fan order. Give the target leaves auxiliary outward spokes. A simple path in this split drawing is exactly a physical crossing with no additional end contacts. No switch state or probabilistic boundary identification has been changed.

Seed each search at its prescribed exterior flank face. An interior face is entered only from an already reached face, across a bond that has been queried and found closed to the tested color. The search therefore cannot inspect strictly beyond a witnessing crossing. Equivalently, after a successful search, closing every still uninspected tested-color edge leaves a crossing by the planar path/cut alternative, so one may select a simple blocker made of measured edges. An existing proper crossing shields all edges strictly on its other side. Consequently, when a long ordered alternation exists, the two searches find disjoint measured blockers, with all but a bounded number of the intervening witnesses still between them.

Condition on the two search leaves. The disk between these blockers is uninspected in its interior. For the opposite color, its longitudinal flanks are free, and its possible inherited identifications occur only at the two end cuts. To see this at tile level, each cutter follows actual open diagonals of the blocker color. The opposite-color half-tile slots along it are dead or dangling slots, not entrances to an exterior wire. Splitting a lateral contact into its fan incidences does not introduce a longitudinal connection. In the conditional opposite-color graph, all bonds crossing a cutter are closed; outside connections therefore meet the intercut disk only at its end cuts. Unqueried exterior states can be integrated out after this observation.

The near end of this disk remains within the chosen interval, whereas the other end is at the original positive level separation. Theorem 56, including its allowed end-cap identifications, gives \[ \mathbb P(\text{an opposite-color proper traversal} \mid\text{the two search leaves})\le p_0<1 \tag{131}\] once \(\eta\) is sufficiently small and the mesh is sufficiently fine. The bound is uniform over the irregular measured blockers.

Repeat the two lateral searches inside the current disk, now testing the opposite color, and then alternate colors. At each stage the first and last witnesses of the tested color shield the remaining middle witnesses; discarding at most two witnesses at each end preserves the claim. New blockers cannot use dead longitudinal slots or the original lateral wires. The searches stop when the required disjoint pair is absent. A long initial alternation forces linearly many successful rounds. Except for the first round, success requires the event in (131). Iterated conditioning on the search leaves, not conditioning on the desired alternation, therefore gives \[\mathbb P(X\ge m\mid\text{exterior states}) \le C e^{-\beta m}\] for some fixed \(\beta>0\). Taking the finite union over the initial geometric choices proves (130) at one rate \(B_0>0\).

Proof. For \(b=1\), this is Proposition 69.

Proof. There are \(O(\log n)\) dyadic levels, so (133) proves the polynomial bounds and the fixed-factor cutoff assertion. The bounded reciprocal marginal-density comparisons in Lemmas 60 and 63 give the exterior assertion. Finally \(t\mapsto\log\mathbb EW^t\) is convex. Its values on a slightly larger compact interval are \(O(\log n)\), by the bounds just proved; its secant slopes on the smaller interval are therefore \(O(\log n)\). ◻

Proof. Put \(L=\log C_0\) and choose \(\eta>0\) with \(\eta L<\varepsilon/2\); the case \(L=0\) is immediate. On \(X_n\le\eta\log n\), either multiplier lies between \(n^{-\varepsilon/2}\) and \(n^{\varepsilon/2}\). For every fixed \(t>0\), Cauchy–Schwarz gives \[\mathbb E[Y_ne^{LX_n};X_n>\eta\log n] \le (\mathbb EY_n^2)^{1/2} \bigl(n^{-t\eta}\mathbb Ee^{(2L+t)X_n}\bigr)^{1/2} \le C_t n^{A/2-t\eta/2}.\] The same bound, with \(L=0\), controls the omitted \(\mathbb E[Y_n;X_n>\eta\log n]\). Choose \(t\) so large that both errors, after division by \(\mathbb EY_n\), are smaller than any prescribed inverse power. Retaining the truncated event proves the lower bounds, and adding the tail proves the upper bounds. Enlarge constants for bounded \(n\). ◻

Proof. Work in the cut drawing without using exterior caps or wire identifications in its connectivity test; the sampling law retains its prescribed boundary partition. Within each tile, the corner sites on a given side of a switch strand are equal or joined by an actual open diagonal. Concatenating these local incidences identifies boundary-touching regions of the drawing with the actual color components incident to the boundary gaps. This identification uses only physical edges. The disjoint raw boundary-to-boundary arcs have a tree as their region-adjacency graph. A closed raw loop has no wall port and encloses no boundary gap; it therefore does not split a boundary-touching class. Apparent bank contacts in an unsmoothed drawing retain their distinct fan incidences on the cut surface and do not join such classes.

Let \(S\) be the set of tree vertices visited by the outer-rim walk. It is a connected subtree; neither that walk nor the wall walk is assumed to be a simple path. If the region of the marked gap is absent from \(S\), take the first edge separating it from \(S\). An arc with an endpoint on the rim has both adjacent regions visited by the rim walk, so this separating edge cannot have such an endpoint. Both ends of its raw arc are on the wall. The wall walk enters and leaves the component containing the gap across that edge, hence the two ends lie on opposite sides of the gap. This is a straddling raw arc. Conversely, a wall-to-wall raw arc straddling the gap separates it from a sufficiently distant rim, and an actual path of the gap color cannot cross it. The region argument is unchanged by interior closed loops or mixed wire designations, since no exterior identification has been used. Applying it in two nested buffered half-boxes, with fixed margins to absorb the tile diameter, proves (137).

For a fixed-factor extension use the single-arm version of Lemma 65. Its finitely many truncated-search success events \(T_1,\ldots,T_m\) cover the shorter-arm event. On each successful leaf, Proposition 61 and Lemma 58 extend the measured source-connected frontier through the remaining window with probability at least \(c_0\). Thus, for the permitted states \(F\) fixed away from that window, \[\mathbb P(\mathcal A_{Cn}\mid F) \ge c_0\max_i\mathbb P(T_i\mid F) \ge \frac{c_0}{m}\mathbb P(\mathcal A_n\mid F).\] This is a relative event estimate, not an assertion under arbitrary conditioning on the full states of an incoming arm. Iterating from a bounded-mesh event of positive finite-energy probability gives the polynomial lower bound. In the other direction, Proposition 62 provides opposite-color actual wall-to-wall screens in separated annuli, each with uniformly positive conditional probability. Each screen blocks the arm. Conditional iteration over \(O(\log n)\) annuli gives the strict polynomial upper bound. The bounded collar comparisons in Lemmas 60 and 63 make these estimates uniform under the specified exterior changes. None of these paths traverses a wire identification. The assertions for \(D_*\) follow from Corollary 71; its upper bound is immediate from \(d<1\) and \(N_n\ge0\). ◻

Proof. Fix \(b=b_1\). At a dyadic scale \(r_i\) well between the mesh and the cutoff, choose a fixed large integer \(k\), larger than all buffer ranges in Proposition 70. Let \(N_{\rm in}\) count straddles wholly inside \(r_{i-k}\), and let \(N_{\rm out}\) count straddles whose entire raw traces lie outside \(r_{i+k}\) and inside the large cutoff. In particular, outside refers to the trace, not to the region enclosed by the arc. These counts have disjoint supports. Put \[M=N_n-N_{\rm in}-N_{\rm out}\ge0,\qquad \,\mathrm dQ_i= \frac{b^{N_{\rm in}+N_{\rm out}}} {\mathbb Eb^{N_{\rm in}+N_{\rm out}}}\,\,\mathrm d\mathbb P.\] Every omitted straddle counted by \(M\) supplies a passage in one of a fixed number of shell bands between \(r_{i-k}\) and \(r_{i+k}\), allowing fixed buffer shifts. Indeed it is neither confined below the lower cutoff nor confined above the upper one; its radial travel or its crossing between opposite rays has substantial diameter near one of those levels. Distinct straddles give disjoint subarcs. The small-box construction used for (132) therefore bounds \(M\) by a fixed sum of shell passage counts.

These counts have all fixed exponential moments under \(Q_i\). For a count near the inner edge, condition on states outside an intermediate radius below the support of the outer tilt. The outer factor is then constant, and the conditional law is the inner tilt with a compatible exterior partition. Its edge moment is bounded by Proposition 70. For a count near the outer edge, reverse this conditioning and use the inward version of that proposition. Middle buffers miss both supports and use Proposition 69 directly. Hölder’s inequality over the fixed number of counts gives \[ \mathbb E_{Q_i}e^{tM}\le C_t\qquad(t\ge0), \tag{140}\] uniformly in \(i,n\) and the exterior states. This argument uses conditional estimates, not independence of the two tilts.

Between the two tilt supports, choose two separated regular subbands. Proposition 62 constructs proper actual wall-to-wall paths of opposite colors in these bands. Their conditional probabilities are bounded below uniformly after fixing states away from their supports; the two tilt factors are measurable there. Applying the construction successively gives an event \(E_i\) with \[Q_i(E_i)\ge p>0 .\] On \(E_i\), at least one raw straddle lies between the two paths. Indeed raw strands cannot cross either actual path. Along each flank, the end vertices of the two paths have opposite colors, so the wall segment between them contains an odd number of raw wall ports. In the intervening disk, pairings of ports on a single flank remove them in pairs. Hence at least one raw arc joins the two flanks. It is trapped between the blockers and is a localized straddle.

The full tilt \(\nu\) by \(b^{N_n}\) is obtained from \(Q_i\) by the density \(b^M/\mathbb E_{Q_i}b^M\). Cauchy–Schwarz yields \[Q_i(E_i)^2 \le \mathbb E_{Q_i}[b^M\mathbf 1_{E_i}]\, \mathbb E_{Q_i}[b^{-M}\mathbf 1_{E_i}],\] and consequently \[ \nu(E_i)\ge \frac{p^2} {\mathbb E_{Q_i}b^M\,\mathbb E_{Q_i}b^{-M}} \ge c_b>0 . \tag{141}\] The last inequality follows from (140); because \(M\ge0\), one of the two denominator factors is at most one. Thus the same argument works for \(b<1\) and \(b>1\).

Choose \(c_1\log n-O(1)\) scales whose localization bands are disjoint. Their forced straddles are distinct, so, pointwise, \(N_n\ge\sum_i\mathbf 1_{E_i}\). Taking expectation under the single full tilt and using (141) gives \[\mathbb E_{\nu}N_n\ge c_2\log n-O(1).\] For the finite graph, \(\psi(t)=\log\mathbb Ee^{tN_n}\) is differentiable and convex, with \(\psi'(\log b_1)=\mathbb E_{\nu}N_n\). The supporting-line inequality between \(\log b_1\) and \(\log b_2\) proves (138).

Finally choose any fixed \(0<b_1<d\) and put \(b_2=d\). Since the summand for \(N_n=0\) has weight one, \[\mathbb Eb_1^{N_n}\ge\mathbb P(N_n=0) \ge\mathbb P(\mathcal A_{Cn}).\] Lemma 73 compares the final probability with \(\mathbb P(\mathcal A_n)\) by a fixed factor. Combining this with (138) proves (139). ◻

Proof. For \(0<x<\lambda\), the two strip Poisson kernels are \[P_0(x,t)=\frac{\sin(\pi x/\lambda)} {2\lambda\{\cosh(\pi t/\lambda)-\cos(\pi x/\lambda)\}},\qquad P_\lambda(x,t)=\frac{\sin(\pi x/\lambda)} {2\lambda\{\cosh(\pi t/\lambda)+\cos(\pi x/\lambda)\}}.\] Their masses are \(1-x/\lambda\) and \(x/\lambda\), respectively. Their sum has a uniformly bounded first absolute moment, and exponentially decaying tails. Thus \[U(x+\mathrm iy)=\int_\mathbb R \{P_0(x,y-t)+P_\lambda(x,y-t)\}h(t)\,\mathrm dt\] is harmonic and \(U(x+\mathrm iy)-|y|\) is bounded, since \(h(t)-|t|\) is bounded. A harmonic conjugate on the simply connected strip defines \(F=\exp(U+\mathrm iV)\). Fix its constant phase by \(F(\lambda/2)>0\). This also proves uniqueness.

The boundary data are real analytic. For example, near any point of the imaginary edge choose the analytic branch of \(\tfrac12\log r(u)\) that is real on that edge. The difference of its real part and \(U\) vanishes there, so odd harmonic reflection extends that difference across the edge. This gives a local holomorphic extension of \(F\), still nonvanishing. The other edge is identical. Symmetries of the Poisson integral and the phase at \(\lambda/2\) give the first two identities in (145). On the imaginary edge, \(F(\mathrm iy)F(-\mathrm iy)=|F(\mathrm iy)|^2=r(\mathrm iy)\), so analytic continuation gives the third identity.

The only zeros of \(r\) closest to the imaginary axis are at \(u=\pm\lambda\); all its zeros are real. Hence \(r(u)/F(-u)\) extends \(F\) without zeros to \(-\lambda<\Re u<0\). Reflection about \(\lambda/2\) gives the extension across the other edge and (146). Along the real interval \([0,\lambda]\), \(F\) is real, nonzero, and positive at the midpoint, hence positive throughout. At zero the third identity gives \(F(0)^2=r(0)=1\); the endpoint values follow. The growth assertion on the extended substrips follows from \(F(-u)=r(u)/F(u)\) and symmetry, because \(\log|r(u)|-2|\Im u|\) is bounded at the two imaginary infinities on every bounded real interval. On a compact remaining rectangle the extended function is continuous and nonzero. ◻

Proof. Let \(J=\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)\). Direct multiplication gives \[\begin{align*} R_{12}(x-y)R_{13}(x)R_{23}(y) &=R_{23}(y)R_{13}(x)R_{12}(x-y),\tag{157}\\ R(u)R(-u)&=r(u)I,\tag{158}\\ J_pR(u)^{t_p}J_p&=R(\lambda-u), \tag{159}\end{align*}\] where \(t_p\) is transpose only in the auxiliary spin. For completeness, the nonconstant entries in the one-minus-spin sector of (157) reduce to \[\begin{align*} b(x-y)+a(x)b(y)&=b(x)a(y),\\ b(y)+a(x)b(x-y)&=b(x)a(x-y),\\ a(x)+b(x-y)b(y)&=a(x-y)a(y). \end{align*}\] These are sine addition identities. The all-equal-spin sectors are scalar, and spin reversal gives the remaining sector. Equations (158) and (159) follow from their displayed \(2\)-by-\(2\) blocks.

Slide the intertwiner \(R_{12}(x-y)\) across the two auxiliary rows using (157). Their boundary tensor \(K(\zeta)\otimes K(\zeta)\) commutes with the intertwiner because \(R\) preserves total spin. For generic arguments the intertwiner is invertible, and cyclicity of the auxiliary trace proves row commutativity. Analytic continuation removes the genericity restriction.

On taking an adjoint, the order of the site factors reverses. Auxiliary partial transpose reverses that order once more: its matrix entries belonging to distinct physical sites commute. Crossing then gives, for any \(D\), \[T_k^D(u)^*=T_k^{J\bar D J}(\lambda-\bar u).\] For a unitary diagonal twist \(J\overline{K(\zeta)}J=K(\zeta)\). The symmetries of \(F\) prove the second identity in (155). Commutativity with the adjoints makes each row normal and gives simultaneous orthogonal diagonalization.

On \(u=\mathrm iy\), the real-coefficient formula for \(R\) and inversion show that \(R(\mathrm iy)/\sqrt{r(\mathrm iy)}\) is unitary. A product of these matrices on the auxiliary and physical spins is unitary too. Since \(|F(\mathrm iy)|=\sqrt{r(\mathrm iy)}\) and auxiliary dimension is two, its partial trace with \(D\) has norm at most \(2\|D\|\). On \(u=\lambda+\mathrm iy\), crossing converts the traced row into a reversed product at \(-\mathrm iy\), with the transpose and spin flip moved to \(D\). The same bound follows. For each fixed \(k\), the growth of \(F\) and the explicit trigonometric entries of \(R\) show that the normalized matrix entries are bounded throughout the strip. The bounded-strip maximum principle, applied to every unit-vector matrix element, gives (156). ◻

Proof. In the one-minus-spin sector, in the order \((++-),(+-+),(-++)\), direct multiplication gives \[ G(u)= \begin{pmatrix} bc&b&-b\\ a&ac&0\\ c&1&-b^2 \end{pmatrix}. \tag{163}\] Because \(ac=1-b^2=r\), the vector \((0,1,1)\) is an eigenvector with eigenvalue \(r\). Spin reversal gives the other physical-spin fiber of the invariant auxiliary line. On the quotient, using \(a-c=-db\), the mixed block in the unrescaled basis is \[b\begin{pmatrix}c&1\\-d&-b\end{pmatrix}.\] Rescaling its middle auxiliary vector by \(\sqrt d\) produces the first mixed block of (161); spin reversal negates that middle vector and produces the other block. The two unmixed states have value \(-ab\). This proves the complete quotient formula.

On \(\Re u=-\lambda/2\) one has \[\bar c=-b,\qquad |a|^2=|b|^2+d,\qquad |b|=|c|.\] Each mixed block of \(\mathcal L\) has adjoint-square \(|a|^2I\); the unmixed entries have the same modulus. Since \(r=ac\), \(Y=(b/r)\mathcal L\) is unitary on that boundary line.

On \(\Re u=\lambda/2\), take the partial transpose of \(\mathcal L\) in the rescaled auxiliary basis and, if desired, exchange \(e_+,e_-\). Its two mixed blocks have diagonal entries \(-a,-b\) and opposite real off-diagonal entries \(\sqrt d,-\sqrt d\); the two unmixed entries equal \(c\). Here \[a=\bar b,\qquad |c|^2=|b|^2+d,\] so the transformed matrix is unitary times \(|c|\). Again the factor \(b/r\) makes its norm one.

Let \(Q_k(u)=Y_1(u)\cdots Y_k(u)\). On the left boundary \(\|Q_k\|=1\). On the right boundary its auxiliary partial transpose is the reversed product of the individually transposed factors, because entries at distinct physical sites commute. That product is unitary. Undoing a partial transpose on a three-dimensional auxiliary space costs at most the fixed factor three in operator norm: writing unit vectors as \(\sum_{i=1}^3e_i\otimes x_i\) and \(\sum_{j=1}^3e_j\otimes y_j\), bound each transposed block by the original operator norm and use \(\sum_i\|x_i\|\sum_j\|y_j\|\le3\). Thus \(\|Q_k\|\le3\) on the right boundary.

There are no zeros of \(r\) in the closed fusion strip \(|\Re u|\le\lambda/2\). Each quotient factor contains \(b(u)\), so \(Q_k\) has a zero of order at least \(k\) at zero. The map \[B(u)=\tan\frac{\pi u}{2\lambda}\] has its only zero in the strip at zero, has modulus one on its boundary lines, and maps its interior into the unit disk. Hence \(Q_k/B^k\) has a removable singularity at zero and no other singularities there. Its entries are bounded at both imaginary infinities: \(b\mathcal L/r\) is bounded there and \(B\) tends to a number of modulus one. The strip maximum principle applied to its scalar matrix elements gives (162). ◻

Proof. The product of the two auxiliary row traces can be regrouped as the two-auxiliary trace of \(\prod_jG_j(u)\). Indeed the factors \(R_{1i}\) and \(R_{2j}\) commute for \(i\ne j\), so no same-site order is changed. The scalar denominator is \(r(u)^k\) by (146).

The twist \(K(\zeta)\otimes K(\zeta)\) preserves the invariant line and acts on it as the identity. In a basis beginning with that line, both this twist and every \(G_j\) are block upper triangular. The trace therefore equals the invariant-line contribution \(I\) plus the trace of the quotient product against a bounded auxiliary matrix. The off-diagonal block contributes exactly zero to the trace, regardless of its size. Lemma 78 proves (164).

For the marker use \[Q=D\otimes K(\zeta) +e^{-\mathrm i\tau\lambda}K(\xi)\otimes D.\] If \(D=|-\rangle\langle+|\) and \(\tau=\epsilon\), then \[Qs_0=(\zeta^{-1}+e^{-\mathrm i\epsilon\lambda}\xi^{-1})|--\rangle=0.\] If \(D=|+\rangle\langle-|\) and \(\tau=-\epsilon\), then \[Qs_0=(\zeta+e^{\mathrm i\epsilon\lambda}\xi)|++\rangle=0.\] Both cancellations use (149). Thus \(Q\) kills the invariant line. Its product with the block triangular fusion product again has zero contribution from the off-diagonal block, and now its invariant-line contribution is zero as well. Taking its quotient trace and extracting \(e^{\mathrm i\tau\rho z}\) proves (165).

On the right edge \(z=v+\mathrm iy\), one has \(z-\pi=-\bar z\). Consequently (155) turns (164) into approximate unitarity. On the left edge apply the same identity at \(z+\pi\). The errors converge to zero uniformly for \(y\) in a fixed compact interval. To pass into the strip, truncate at \(|\Im z|=Y\). On the two vertical sides the norm is at most \(1+o_k(1)\); on the horizontal sides the uniform bound is two. The harmonic measure of the horizontal sides tends uniformly to zero on any fixed compact subset as \(Y\to\infty\). The maximum principle for unit-vector matrix elements, first with \(k\to\infty\) and then \(Y\to\infty\), proves the claimed limiting contraction. The edge estimates include the real endpoints. ◻

Proof. The row bound (156) gives a normal family in \(\mathcal S\). The lower bound at \(1\) prevents its limit from being identically zero. The asymptotic contraction (154) gives \(|f|\le1\) throughout the strip: its boundary version, together with the uniform row bound, can first be applied on a finite rectangle, and the harmonic measure of the two horizontal sides tends to zero as the rectangle grows. Equivalently, one can use the interior contraction already obtained from this argument. Equation (166) passes to the limit.

The inversion estimate (164) makes \(f_k\) zero-free on every compact subset of each spectral band about a strip edge, for all sufficiently large \(k\). Under \(u=\rho(v-z)\), the band \(|\Re u|<\lambda/2\) about the right edge is \(0<\Re z<\pi\); the corresponding band about the left edge is \(-\pi<\Re z<0\). These two bands cover the strict strip except for its center line. Hurwitz’s theorem therefore excludes zeros of \(f\) off that center line, which maps to the positive real axis in the \(w\)-plane.

We give the additional argument needed at an edge; mere convergence on the strict strip would not suffice. Fix a disk \(D\) centered at \(z_0=v+\mathrm iy_0\), with closure compactly contained in the right-edge spectral band. In this paragraph write \(f_k(z)\) for \(f_k(e^{\mathrm iz})\). On \(D\), \[q_k(z):=f_k(z)f_k(z-\pi)\longrightarrow1\] uniformly. For all sufficiently large \(k\), choose the analytic square root of \(q_k\) near \(1\), and set \[g_k(z)=\frac{f_k(z)}{\sqrt{q_k(z)}}, \qquad U_k(z)=-\log|g_k(z)|.\] Both eigenvalue factors are nonzero on \(D\). On its vertical diameter the adjoint identity gives \(q_k(z)=|f_k(z)|^2\), so that \(|g_k(z)|=1\). Consequently \(U_k\) is harmonic on \(D\) and odd under reflection in that diameter. On the physical half of \(D\), the row bound and \(q_k\to1\) give \(U_k\ge-C\). At a fixed point strictly inside that half-disk, \(U_k\) is bounded above as well: the limiting eigenvalue has no zero there by the preceding paragraph.

For clarity, the elementary boundary estimate used here is as follows. For a harmonic function \(U\) odd across a diameter of a disk, its Poisson formula is an integral over one semicircle with kernel \[K(z,t)=P(z,t)-P(z,t^{\rm ref}),\] where \(P\) is the disk Poisson kernel and \(t^{\rm ref}\) is the reflected boundary point. This kernel is nonnegative when \(z\) and \(t\) are on the same side of the diameter. On any strictly smaller half-disk, \[ 0\le K(z,t)\le C K(z_*,t) \tag{167}\] for a fixed interior radial point \(z_*\) on that side. The assertion follows directly from the Poisson formula: the difference has the product of the two signed distances to the diameter in its numerator, and its remaining denominators are bounded above and below on the smaller disk. If \(U\ge-C\) on the physical half-disk, the negative part of its semicircle data has a bounded contribution. The formula at \(z_*\) therefore bounds the positive contribution there, and (167) bounds it at all points of the smaller half-disk. Reflection supplies the other half.

Applying this observation on an intermediate disk inside \(D\) bounds \(|U_k|\) on a smaller closed disk. Thus \(f_k\) and \(f_k^{-1}\) are uniformly bounded there. Normal families extend the already chosen interior limit across the edge, and the limit has modulus one on the diameter. This extension is unique, so taking further subsequences does not change it. The left edge is treated with the shifted inversion identity. These two arguments cover every nonzero point of the imaginary axis. ◻

Proof. Multiply \(f\) by its sign at \(1\) and put \(a=|f(1)|\). For a nonconstant function, Schwarz–Pick in the right half-plane gives, for \(r>0\), \[\left|\frac{\sigma f(r)-a}{1-a\sigma f(r)}\right| \le t:=\frac{|r-1|}{r+1}.\] Writing the fraction on the left as \(h\), one obtains \[|1-\sigma f(r)| = (1-a)\left|\frac{1-h}{1+ah}\right| \le (1-a)\frac{1+t}{1-t} \le s\max(r,r^{-1}).\] Constant functions satisfy the same conclusion directly. Choose \(c_0\) sufficiently small. This estimate excludes zeros on the positive real axis even on an annulus with inner radius \(c_0^{-1}s/4\) and outer radius \(4c_0/s\). Lemma 80 excludes zeros elsewhere in its right half. On that half-annulus, therefore, \[U(w):=-\log|f(w)|\] is a nonnegative harmonic function, vanishing on the imaginary-axis boundary segments and extending oddly through them. On the positive real axis the same estimate gives \[ U(r)\le Cs\max(r,r^{-1}). \tag{169}\]

Fix \(r\in[c_0^{-1}s,c_0/s]\) and rescale by \(r\). A bounded Harnack chain inside \(\{1/4<|w|<4,\ \Re w>0\}\) controls \(U\) on the unit semicircle away from its endpoints by \(CU(r)\). Near either endpoint, use a disk centered on the imaginary axis, of fixed radius after rescaling. Its interior reference point is controlled by the same Harnack chain; the reflected Poisson estimate (167), now with nonnegative data on the physical half, controls the smaller half-disk up to the diameter. Consequently \[ 0\le U(w)\le Cs\max(r,r^{-1}) \quad (|w|=r,\ \Re w\ge0). \tag{170}\] The right side is bounded by a fixed constant on the asserted annulus, proving its uniform inverse bound.

For \(r\) in a fixed neighborhood of \(1\), the bound is \(Cs\). After odd reflection it bounds \(|U|\) in fixed neighborhoods of the two unit semicircle endpoints as well. Harmonic gradient estimates therefore give \(|\nabla U|\le Cs\) in a neighborhood of the whole unit semicircle. Choose a branch of \(\log(\sigma f)\) in a thin simply connected neighborhood of that semicircle, with value \(\log a\) at \(1\). The Cauchy–Riemann equations identify the modulus of its derivative with \(|\nabla U|\). Integration along the semicircle gives \(\log(\sigma f)=O(s)\), and hence the second assertion in (168).

We spell out why these scalar arguments are uniform for the growing matrices. If any compressed diagonal norm assertion failed, choose an offending eigenvalue, a point in the indicated compact set, and a subsequence of site numbers. The points have a convergent subsequence. Lemma 80 supplies a scalar limit, including local convergence through an edge when necessary; (168) then contradicts the failure. For the sign assertion one first fixes the sign subsequence. The same argument rules out zeros of the finite compressed eigenvalues on the compact half-annulus for all sufficiently large \(k\), so that their inverses are defined there. No bound on the number of eigenvalues is used.

Finally, scalar continuation across an imaginary boundary interval with modulus one is given by Schwarz reflection, \(f(w)=1/\overline{f(-\overline w)}\) on its other side. The just-proved inverse bounds therefore control this continuation near either ray, as well as its inverse. The edge-continuation argument and the same subsequence contradiction give the corresponding local bounds for the finite compressed matrices. One may decrease \(c_0\) once to keep all neighborhoods inside the wider annulus used above. ◻

Proof. The row bound (156), including the scalar factor in \(M\), gives \[\|m_k(w)\|\le C|w|^{\tau\rho}\] on the physical right half-plane. Lemma 81 gives the same limsup bound, up to a constant, for (172). Thus the scalar matrix elements of both expressions are normal families on their respective half-annuli. It remains to check that their limits glue through both imaginary rays, including local boundedness there.

Use \(z\) coordinates near \(z=v+\mathrm iy\). Compressing the fusion identity (165) gives \[ m_k(z)p_{\zeta,k}(z-\pi) +p_{\xi,k}(z)m_k(z-\pi)=o(1) \tag{173}\] uniformly on compact neighborhoods, in operator norm. Here \(m_k(z)\) abbreviates the continued compression at argument \(z\), and the same convention applies to \(p\). The error tends to zero for each fixed \(s\) and compact neighborhood. The inversion identity also gives \[p_{\xi,k}(z)=p_{\xi,k}(z-\pi)^{-1}+o(1),\] and the analogous relation for the other twist. All these inverses are locally bounded by Lemma 81. In a small neighborhood of the edge, one of \(z,z-\pi\) is in the physical closed strip. Solving (173) for the other marker shows that both continued marker expressions are locally bounded. Solving for \(m_k(z)\) in particular gives \[m_k(z)= -p_{\xi,k}(z-\pi)^{-1}m_k(z-\pi) p_{\zeta,k}(z-\pi)^{-1}+o(1).\] Since \(e^{\mathrm i(z-\pi)}=-e^{\mathrm iz}\), this is exactly the compatibility of the physical and external expressions across the upper imaginary ray. Across the lower ray use the same identity with \(z+\pi\) in place of \(z\) and solve in the opposite direction. Again one argument is physical, and inversion produces precisely (172).

Montel’s theorem and a diagonal subsequence over compact sets now give the asserted holomorphic function on the full annulus. This uses scalar matrix elements only: the two twist spaces may have different eigenbases and dimensions tending to infinity. There is no assertion that limits of individual matrix factors can be multiplied. Their finite-dimensional identities and uniform norm bounds are used before taking subsequences. Bound (171) follows on each half-annulus from its defining expression, and on the rays by continuation. ◻

Proof. Expand the scalar limit in Lemma 82 as \(m(w)=\sum_{j\in\mathbb Z}b_jw^j\). Cauchy’s estimate on circles just inside the two annular radii, with a fixed proportional margin, gives \[ |b_j|\le C(C's)^{|j-\tau\rho|}. \tag{176}\] For example, use the outer circle when \(j>\tau\rho\) and the inner circle when \(j<\tau\rho\). Shrinking \(s_0\) makes \(C's_0<1/2\). Because \(0<\rho<1/2\), summing these bounds on the unit circle yields \[\begin{align*} |m(w)|&\le Cs^\rho,\tag{177}\\ \frac{m(w)-m(-w)}2 &=b_\tau w^\tau+O(s^{1+\rho}), &|b_\tau|&\le Cs^{1-\rho}. \tag{178}\end{align*}\] Indeed \(j=0\) has exponent \(\rho\), the nearest nonzero exponent is \(1-\rho\), and among odd indices other than \(\tau\) the smallest exponent is \(1+\rho\). The remaining geometric tails are uniform.

By the arbitrary-unit-vector subsequence argument in Lemma 82, (177) gives \[ \limsup_{k\to\infty} \sup_{|w|=1,\ \Re w\ge0}\|m_k(w)\|\le Cs^\rho. \tag{179}\] The supremum causes no difficulty: choose offending points, pass to a convergent subsequence on the compact semicircle, and use the local uniform convergence, including at its endpoints.

Further split each low spectral projection according to the sign of \(P(0)\), and first work on one pair of sign blocks, with signs \(\sigma_\xi,\sigma_\zeta\in\{1,-1\}\). Lemma 81 gives, in limsup uniformly on the physical unit semicircle, \[p_{\xi,k}=\sigma_\xi I+O(s),\qquad p_{\zeta,k}=\sigma_\zeta I+O(s),\] and the same estimates for their inverses. Combining (172) and (179), every scalar subsequential limit satisfies \[ m(-w)=-\sigma_\xi\sigma_\zeta m(w)+O(s^{1+\rho}). \tag{180}\] The compressed anticommutator is exactly \[E_\xi N(z)E_\zeta =m_k(z)p_{\zeta,k}(z)+p_{\xi,k}(z)m_k(z).\] If the signs disagree, its leading terms cancel and it is \(O(s^{1+\rho})\) in limsup. If both signs equal \(\sigma\), (180) and (178) instead give, in each tested scalar subsequential limit, \[ E_\xi N(x)E_\zeta =2\sigma b_\tau e^{\mathrm i\tau x}+O(s^{1+\rho}). \tag{181}\] This equation denotes the corresponding scalar matrix element, not an identification of spaces of different dimensions. Its leading term has size \(O(s^{1-\rho})\) and cancels between \(x\) and \(y\) after multiplication by \(e^{\mathrm i\tau(x-y)}\). Applying the same argument to arbitrary sequences of test vectors and of \(x,y\) proves (174)–(175). Summing the four sign blocks changes the constant only.

Finally, an even product of at most a fixed number of twist rows equals \(I+O(s)\) on its sign block, because every factor equals \(\sigma I+O(s)\). Multiplying the compressed anticommutator by such products changes it by at most \(Cs\|m_k\|\) in limsup, hence by \(Cs^{1+\rho}\). This reasoning is separate for the two terms, so their additional lists need not agree. All these additional rows commute with the spectral projections on their own side. This proves the last assertion. ◻

Proof of Theorem 76. The row contraction is (154). It remains to insert the two outer products in the low-band estimates. For either twist write \[h_k=-\log|P(0)|,\] with value \(+\infty\) on the kernel. The spectral projections of \(h_k\) commute with every row of their twist. If a band has positive lower endpoint \(t\), the at least \(n\) zero-argument factors in its outer product give the bound \(e^{-nt}\) on that band. The other factors have asymptotic norm at most one. Since their list is finite and fixed while \(k\to\infty\), the full product has limsup norm at most \(e^{-nt}\) there. On a band without a positive lower endpoint its limsup norm is at most one by contraction of all the factors.

For \(n\) large enough, choose \(J\) so that \(2^J/n\le s_0<2^{J+1}/n\), and split each side into \[I_0=(-\infty,1/n],\qquad I_i=(2^{i-1}/n,2^i/n]\quad(1\le i\le J), \qquad I_\infty=(2^J/n,+\infty].\] Set \(D_0=1\) and \(D_i=e^{-2^{i-1}}\) for \(i\ge1\). For a pair of finite bands \(I_i,I_j\), apply Lemma 83 with \[s=\frac{\max(2^i,2^j)}n.\] Both band projections are further restrictions of the appropriate low projections. Their outer products contribute \(D_iD_j\). For either exponent \(a\in\{1-\rho,1+\rho\}\), the triangle inequality therefore bounds the sum of these blocks by \[ C n^{-a}\sum_{i,j=0}^{J} D_iD_j\max(2^i,2^j)^a \le C_a n^{-a}. \tag{182}\] The infinite double sum converges, since the \(D_i\) decrease exponentially in \(2^i\). In particular there is no factor counting the number of bands.

The remainder has \(h_k>2^J/n>s_0/2\) and therefore contributes \(e^{-ns_0/2}\) on its side. Group all blocks with a remainder on the left into a single compression, and then all remaining blocks with a remainder on the right into another. The unrestricted row bounds control \(N\) and its two-term difference; contraction controls the outer product on the other side. Thus these two groups together contribute at most \(Ce^{-ns_0/2}\) in limsup. Additional bounded row lists cause only a fixed constant, and their low-band contribution was already covered by Lemma 83.

Taking \(a=1-\rho\) in (182) proves the first asserted sandwich bound. Taking \(a=1+\rho\) proves the second, with its phase \(e^{\mathrm i\tau(x-y)}\) and with independently chosen additional even row lists. The bounded range of smaller \(n\) follows directly from contraction of the outer products and the uniform marker bound, after enlarging the constant. At every stage the band decomposition is finite before taking the site-number limsup; only the resulting scalar estimates are then summed and bounded uniformly in \(n\). ◻

Proof. Multiply the Yang–Baxter identity on the two site spaces by their exchange \(P_{12}\). Since this exchange interchanges the site labels, the result is (187); invertibility holds at \(x-y=\lambda/2\) by positivity of \(C\).

We first record the adjoint calculation for arbitrary \(D\). Let \(J=\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)\) reverse the two basis states of the single auxiliary spin, and put \(D^\sharp=J\overline D J\). Taking the ordinary adjoint reverses the site order. Auxiliary transpose reverses it back, because the entries belonging to distinct site spaces commute. Applying \(J_pR(u)^{t_p}J_p=R(\lambda-u)\) gives a forward row with arguments \(\lambda-\overline{u_j(U)}\) and boundary matrix \(D^\sharp\). These arguments are the exchanged stagger at central argument \(\lambda-\overline U\). Applying (187) on each complete pair gives \[ G_N^{-1}T_N^D(U)^*G_N =T_N^{D^\sharp}(\lambda-\overline U). \tag{190}\] Equation (183) gives the same transformation for \(\Phi\). For a unitary diagonal twist \(K(\zeta)^\sharp=K(\zeta)\), proving (188). The usual two-auxiliary Yang–Baxter calculation applies with these fixed site offsets: the difference of the two row arguments is the same at every site, and \(K(\zeta)\otimes K(\zeta)\) commutes with the auxiliary intertwiner. Taking the auxiliary traces proves commutativity. The adjoint belongs to the same commuting family, so the family consists of commuting normal matrices in the \(G_N\) inner product.

For the bound, take \(\Re U=0\) and write \(x=U+\delta_*\), \(y=U-\delta_*\). Then \(\overline x=-y\) and \(\overline y=-x\). For \(E=R_{p1}(x)R_{p2}(y)\), taking the transpose of the exchange identity and using \(R(t)R(-t)=r(t)\mathop{\mathrm{id}}\) gives \[ E^{*C}E=r(x)r(y)\mathop{\mathrm{id}}=|\Phi(U)|^2\mathop{\mathrm{id}}. \tag{191}\] The last equality follows by multiplying the four factors in \(|F(x)F(y)|^2\) and using (183). The normalized pair is consequently unitary for the metric \(\mathop{\mathrm{id}}_p\otimes C\). Embedded in the full space, it is unitary for \(\mathop{\mathrm{id}}_p\otimes G_N\), since all other factors of \(G_N\) act on disjoint sites. The product over pairs has norm one. Auxiliary trace with \(D\) costs at most \(2\|D\|\), proving the bound on \(\Re U=0\). Equation (190) proves it on \(\Re U=\lambda\), because \(\|D^\sharp\|=\|D\|\).

For fixed \(N\), each normalized matrix entry is bounded as \(|\Im U|\to\infty\): each \(R\) factor grows at most as \(\exp(|\Im U|)\), while \(\log|F(U\pm\delta_*)|=|\Im U|+O(1)\), also on the continued portions by (183). The strip maximum principle applied to matrix elements between \(G_N\)-unit vectors proves (189) throughout the strip. ◻

Proof. Applying (187) to each auxiliary row separately, and commuting factors acting on disjoint tensor factors before and after each application, exchanges the arguments in a pair of fused site matrices. The invariant line is in the auxiliary spaces, so this identity passes to the quotient. The scalar product \(r(x)r(y)\) is unchanged by exchange. Hence, for \(x-y=\lambda/2\), \[ C_{12}Y_1(x)Y_2(y)C_{12}^{-1}=Y_1(y)Y_2(x). \tag{196}\]

We give the reflected inversion identities needed at the two edges. Let \(Z(u)=Y(u)^{t_{\rm aux}}\), where the transpose is in the rescaled three-dimensional auxiliary basis. Direct multiplication of (193) gives the meromorphic identities \[ Y(u)^TY(-\lambda-u)=\mathop{\mathrm{id}}, \qquad Z(u)^TZ(\lambda-u)=\mathop{\mathrm{id}}. \tag{197}\] Here \(T\) transposes both auxiliary and site indices. For completeness, if \(w=-\lambda-u\), then \(b(w)=-c(u)\), \(c(w)=-b(u)\), and \(a(u)a(w)=d-b(u)c(u)\). A mixed block of \(\mathcal L(u)^T\mathcal L(w)\) therefore equals \((d-bc)\mathop{\mathrm{id}}\); the product of the scalar factors \(b/(ac)\) is \(1/(a(u)a(w))\). The unmixed entries of \(Y(u)\) and \(Y(w)\) are \(-b/c\) and \(-c/b\). This proves the first identity. For \(w=\lambda-u\), one has \(a(w)=b(u)\), \(b(w)=a(u)\), and \(c(u)c(w)=d+a(u)b(u)\). After a fixed reordering, the mixed blocks of \(\mathcal L^{t_{\rm aux}}\) have diagonal entries \((-a,-b)\) and opposite off-diagonal entries \(\pm\sqrt d\). Their transpose products equal \((d+ab)\mathop{\mathrm{id}}\), canceled by the scalar product \(1/(c(u)c(w))\); the unmixed entries are \(b/a\) and \(a/b\). This proves the second identity.

Write \(Q(U)=Y_1(U+\delta_*)Y_2(U-\delta_*)\) and \(Q_s(U)=C Q(U)C^{-1}\). If \(\Re U=-\lambda/2\), the first reflection in (197) interchanges the two arguments after conjugation. Thus \[Q(U)^*=Q_s(U)^{-1}=C Q(U)^{-1}C^{-1}, \qquad Q(U)^{*C}Q(U)=\mathop{\mathrm{id}}.\] If \(\Re U=\lambda/2\), put \(H(U)=Q(U)^{t_{\rm aux}}=Z_2(U-\delta_*)Z_1(U+\delta_*)\). The order reversal is legitimate because entries on the two distinct site spaces commute. Transposing the exchange identity gives \(C H(U)C^{-1}=Z_2(U+\delta_*)Z_1(U-\delta_*)\). The second reflection identity now gives \(H(U)^*=(C H(U)C^{-1})^{-1}\), hence \(H(U)^{*C}H(U)=\mathop{\mathrm{id}}\).

The complete quotient product is therefore unitary in the metric \(\mathop{\mathrm{id}}_{\mathbb C^3}\otimes G_N\) on the left edge. On the right edge its auxiliary transpose, a reversed product of these unitary pairs, is unitary. Undoing a transpose in a three-dimensional tensor factor costs at most three in operator norm. Indeed, its block-matrix expansion and Cauchy–Schwarz give this bound; conjugation by \(G_N^{1/2}\) commutes with auxiliary transpose, so the same bound holds for the present norm. The strip maximum principle yields a uniform bound in between.

There are no poles from \(r(u_j(U))\) on this closed strip, since \(|\Re u_j(U)|\le3\lambda/4<\lambda\). Also \(Y(u)\) has a scalar factor \(b(u)\) and hence vanishes at \(u=0\). The complete product consequently has a zero of order at least \(N/2\) at each of \(U=\delta_*\) and \(U=-\delta_*\). The two functions in (194) have precisely these zeros, modulus one on the strip edges, and modulus less than one inside. Dividing the product by \(B_+(U)^{N/2}B_-(U)^{N/2}\) removes these zeros. For fixed \(N\) the quotient remains bounded at the two imaginary ends of the strip, as follows directly from the trigonometric formulas. Applying the maximum principle again proves (195). ◻

Proof. The three changes from the homogeneous argument are the \(G_N\) metric, the damping basepoint \(e^{\mathrm i\gamma_0}\), and the two-zero fusion error in (195). We give the analytic argument to track these changes explicitly. All diagonalizations and projections in this proof are orthogonal for \(G_N\); no comparison with the original Euclidean norm is used. Suppress the superscript \(\mathrm{stag}\).

Map the open \(z\) strip to the right half-plane by \(w=e^{\mathrm iz}\), using the branch with \(-v<\Re z<v\). In a common eigenbasis of a fixed twist, let \(f_N(w)\) be any eigenvalue. It is holomorphic and uniformly bounded. The adjoint identity gives \(f_N(\overline w)=\overline{f_N(w)}\) and hence \(f_N(1)\in\mathbb R\). For a fixed small \(s>0\) restrict to eigenvalues satisfying \(|f_N(e^{\mathrm i\gamma_0})|\ge e^{-s}\). Every sequence has a subsequence converging on compact subsets to a nonzero holomorphic function \(f\) with \(|f|\le1\). Nonzeroness follows at \(e^{\mathrm i\gamma_0}\), and the bound follows from the edge unitarity and strip bound proved above. Schwarz–Pick between \(e^{\mathrm i\gamma_0}\) and \(1\) gives, with \(\alpha=\tan(\gamma_0/2)<1\), \[ |f(1)|\ge\frac{e^{-s}-\alpha}{1-\alpha e^{-s}} \ge1-C_\lambda s. \tag{205}\] The first inequality is the elementary extremal modulus bound for a disk of pseudohyperbolic radius \(\alpha\) centered at \(f(e^{\mathrm i\gamma_0})\); take \(s\) small enough that \(e^{-s}>\alpha\). Thus a large eigenvalue at the physical angle is also close in modulus to a real sign at the center.

The remaining continuation argument uses the same analytic facts as Lemmas 80–83. We verify the correspondence before applying their proofs:

1. The fixed-twist rows are commuting normal holomorphic matrices in the \(G_N\) inner product, with adjoint reflection (188) and the uniform strip bound (189).

2. The two-row and marker inversion errors in (200)–(201) tend uniformly to zero on compact subsets of the same edge bands \(0<\Re z<\pi\) and \(-\pi<\Re z<0\) used in the homogeneous proof. The two zeros in the fused error change its rate of convergence, not these bands or the resulting identities.

3. The marker has the same growth factor \(|w|^{\tau\rho}\), and (205) bounds the limiting row modulus at \(w=1\) below by \(1-C_\lambda s\).

These are precisely the inputs to the scalar continuation and Laurent estimates. All their finite matrix elements can be taken between \(G_N\)-unit vectors: the proofs use only the indicated intrinsic norms, not a bounded comparison between \(G_N\) and the Euclidean metric.

Here are their consequences with the present spectral cutoff. Define \[E_{\eta,N}(s)= \mathbf 1_{\{|P_{N,\eta}(\gamma_0)|\ge e^{-s}\}},\qquad \eta\in\{\xi,\zeta\}.\] By the adjoint identity, the moduli at \(\gamma_0\) and \(-\gamma_0\) agree on every common eigenvector. The scalar limits on this low subspace are zero-free off the positive real \(w\) axis by inversion and Hurwitz’s theorem. Dividing by the analytic square root of the two-row product gives modulus one on each strip edge; the reflected half-disk estimate from Lemma 80 bounds the continued eigenvalues and their inverses there. Thus the scalar limits extend across the two imaginary rays.

For \(\sigma=\mathop{\mathrm{sgn}}f(1)\), Schwarz–Pick based at \(1\) gives \[ |f(r)-\sigma|\le C_\lambda s\max(r,r^{-1}),\qquad r>0. \tag{206}\] Consequently the half-annulus proof, with \(C_\lambda s\) in place of \(s\), gives bounded inverses on \[ c_0^{-1}s<|w|<c_0/s,\qquad \Re w\ge0, \tag{207}\] and the sign estimate \[ f(w)=\sigma+O_\lambda(s),\qquad |w|=1,\quad\Re w\ge0. \tag{208}\] The constants absorb this fixed rescaling of \(s\). An offending-eigenvalue subsequence would contradict these scalar estimates, so they hold in limsup for the diagonal compressions \(p_{\eta,N}=P_{N,\eta}|_{\operatorname{ran}E_{\eta,N}(s)}\) as well. For sufficiently small fixed \(s\) and all large \(N\), their center values are nonzero by (205); hence we may refine each projection according to the sign of \(P_{N,\eta}(0)\). Further spectral restrictions preserve every estimate.

Compress the marker to \(m_N=E_{\xi,N}(s)M_NE_{\zeta,N}(s)\). On the left half of the annulus its continuation is represented, at \(-w\) with \(\Re w>0\), by \[ -p_{\xi,N}(w)^{-1}m_N(w)p_{\zeta,N}(w)^{-1}. \tag{209}\] The marker inversion identity glues this expression to the physical one across both rays, exactly as in Lemma 82. The inverse bounds control its errors. For every sequence of pairs of \(G_N\)-unit test vectors, Montel’s theorem therefore gives a scalar subsequential limit on the full annulus with \(|m(w)|\le C_\lambda|w|^{\tau\rho}\). No products of limiting operators on varying spaces are used.

Writing \(m(w)=\sum_{j\in\mathbb Z}\beta_jw^j\), Cauchy’s estimates give \(|\beta_j|\le C_\lambda(C_\lambda s)^{|j-\tau\rho|}\) and hence \[ |m(w)|\le C_\lambda s^\rho,\qquad \frac{m(w)-m(-w)}2=\beta_\tau w^\tau +O_\lambda(s^{1+\rho}),\qquad |\beta_\tau|\le C_\lambda s^{1-\rho} \quad (|w|=1). \tag{210}\] On equal-sign blocks the marker sum is \(2\sigma\beta_\tau e^{\mathrm i\tau x}+O_\lambda(s^{1+\rho})\); on opposite-sign blocks its two terms cancel to order \(O_\lambda(s^{1+\rho})\). The arbitrary-unit-vector subsequence argument therefore yields, uniformly for real \(x,y\in[-v,v]\), \[\begin{align*} \limsup_N\|E_{\xi,N}(s)\mathcal N_N(x)E_{\zeta,N}(s)\|_{G_N} &\le C_\lambda s^{1-\rho},\tag{211}\\ \limsup_N\|E_{\xi,N}(s) (\mathcal N_N(x)-e^{\mathrm i\tau(x-y)}\mathcal N_N(y)) E_{\zeta,N}(s)\|_{G_N} &\le C_\lambda s^{1+\rho}. \tag{212}\end{align*}\] There are only four sign blocks. On each one, an additional bounded even row product is \(\mathop{\mathrm{id}}+O_\lambda(s)\) by (208). Its error times the marker bound is \(O_\lambda(s^{1+\rho})\). This proves the same estimates with the allowed additional lists, even when the two terms use different lists.

Finally decompose each side spectrally according to \(h_{\eta,N}=-\log|P_{N,\eta}(\gamma_0)|\), assigning value \(+\infty\) at a zero. Use the band \((-\infty,n^{-1}]\), successive dyadic bands up to a fixed sufficiently small \(s_0\), and the remainder. On a band with positive lower endpoint \(t\), each damping row at either physical angle has norm at most \(e^{-t}\), since the two moduli agree. Commutativity and (202) show that the whole exterior product has limsup norm at most \(e^{-nt}\) on that band; on the first band use \(1\). For any pair of small bands use (211) or (212) with \(s\) the larger upper endpoint. The double dyadic sum is bounded by \(C_\lambda n^{-\beta}\) for \(\beta=1-\rho\) or \(1+\rho\), respectively: after writing each upper endpoint as \(2^j/n\), its summands are bounded by \(C_\lambda n^{-\beta}2^{\beta\max(j,k)}e^{-c(2^j+2^k)}\), with the first band interpreted as having no exponential cost. Terms involving a remainder are exponentially small in \(n\), by the uniform strip bound and damping at least \(e^{-ns_0}\). This proves both assertions. ◻

Proof. We verify the collapse directly in components. Spin labels take values \(\pm1\). The collapsed vertices have entries \[R(0)_{p,d;q,c}=\mathbf 1_{p=c}\mathbf 1_{d=q},\qquad R(\lambda)_{p,d;q,c}=\mathbf 1_{p=-d}\mathbf 1_{q=-c}.\] At the square vertex the equality \(a(\lambda/2)=b(\lambda/2)\) gives \[ (R_*)_{-c_0,d_0;-d_1,c_1} =C_{d_0,d_1;c_0,c_1}. \tag{216}\] Each identity follows by listing the six nonzero entries of \(R\). In the auxiliary trace use indices \(p_0,p_1,\ldots,p_N=p_0\) so that the site \(j\) factor is \((R(u_j))_{p_{j-1},d_j;p_j,c_j}\). For the positive step the even sites force \(p_{2j-1}=c_{2j}\) and \(p_{2j}=d_{2j}\), giving, with cyclic indices, \[(\mathsf T_N(+))_{d;c} =\prod_{j=1}^{N/2} (R_*)_{d_{2j-2},d_{2j-1};c_{2j},c_{2j-1}} =(G_N)_{S^{-1}d;c}.\] The last equality uses that \(R_*\) commutes with exchange of its two spin factors. For the negative step the odd sites force \(p_{2j-2}=-d_{2j-1}\) and \(p_{2j-1}=-c_{2j-1}\); hence \[(\mathsf T_N(-))_{d;c} =\prod_{j=1}^{N/2} (R_*)_{-c_{2j-1},d_{2j};-d_{2j+1},c_{2j}} =(G_N)_{Sd;c},\] by (216). All auxiliary labels are fixed in these sums, so neither collapse introduces an additional cycle factor. This proves (213).

For a general auxiliary insertion, leave the trace indices separate and multiply by \(D_{p_N,p_0}\). At the positive step the last collapsed vertex has \(p_N=d_N\), so this factor applies \(D\) to the last output spin of the untwisted row. At the negative step the first collapsed vertex has \(p_0=-d_1\). Comparing with an untwisted output spin \(\widetilde d_1\) gives the factor \(D_{-\widetilde d_1,-d_1}=(JD^TJ)_{d_1,\widetilde d_1}\). These are the two identities in (214).

The product defining \(G_N\) is invariant under translation by two sites, so \([G_N,S^2]=0\). Consequently \[(SG_N)^{*G_N}=S^{-1}G_N,\qquad SG_NS^{-1}G_N=S^{-1}G_NSG_N.\] This proves mutual adjointness, commutativity, normality, and equality of the two norms.

The matrix \(C\) preserves the number of positive spins, has positive diagonal entries, and has a positive exchange entry on each unlike pair. Thus a step of \(SG_N\) can hold every complete pair or can exchange any chosen subset of its unlike pairs, followed by the cyclic shift. Over \(N\) steps, holding all pairs gives the identity permutation. Holding everywhere except for one chosen pair exchange gives any prescribed adjacent transposition: the shifts before and after the exchange move that complete pair to any desired cyclic bond. All the corresponding weights are positive. Adjacent transpositions connect all configurations in a fixed-spin sector, so \(SG_N\) is irreducible there. The same argument works with \(S^{-1}G_N\).

There is also a one-step self-loop in every sector. For a sector with at most \(N/2\) positive spins, place them on odd sites and put negative spins on every even site. Exchange each unlike complete pair before applying \(S\); the original configuration is recovered. For a sector with more than \(N/2\) positive spins, reverse all spins in this construction. For \(S^{-1}G_N\), reverse the roles of the two parities. These self-loops and irreducibility prove primitivity.

Perron–Frobenius gives a unique positive ray in each sector and strict modulus inequality for its remaining eigenvalues. Normality says that the largest Perron eigenvalue over the sectors equals \(\Lambda_N\). The operator \(S^2\) commutes with \(SG_N\) and preserves positivity, so uniqueness of the positive ray makes \(S^2\Omega\) a positive multiple of \(\Omega\). Since \(S^2\) has finite order, that multiple equals one. It follows that \(S^{-1}G_N\Omega=S^{-2}(SG_N)\Omega =\Lambda_N\Omega\). Taking \(G_N\) adjoints and then canceling the invertible rightmost \(G_N\) gives the last identity in (215). Finally, \(F\) is positive on the real interval under consideration; (183) gives \(F(0)=F(\lambda)=1\). Thus \(\Phi(\lambda/4)=\Phi(3\lambda/4)=F(\lambda/2)\), as claimed. ◻

Proof. Theorem 1 of (Duminil-Copin et al. 2022), with its parameters \[\zeta=\lambda,\qquad \theta=\pi/2,\qquad r=\frac1{2\cos(\lambda/2)},\] has exactly the paired weights \((a_*,a_*,1)\). Its disordered-regime formula is the first expression in (218): the integral over the real line is even and its numerator at \(\theta=\pi/2\) contains \(\sinh(\lambda t)\). The theorem supplies the iterated cylinder pressure; choosing the second period sufficiently large for each first period gives the stated sequence. Restricting also the second period to even integers does not change the cylinder Perron limit.

Here is the identification with \(F\), including its normalization. The harmonic function \[H(u)=\log|F(u)/a(u)|\] on \(0<\Re u<\lambda\) has zero data on the left edge and right-edge data \[g(y)=\log\left|\frac{\sin(\lambda+\mathrm iy)}{\sin(\mathrm iy)}\right| =\int_0^\lambda \frac{\sin(2\alpha)}{\cosh(2y)-\cos(2\alpha)}\,\,\mathrm d\alpha.\] These statements follow from \(F(u)F(-u)=r(u)\), the reflection symmetry of \(F\), and \(|a(\mathrm iy)|^2=r(\mathrm iy)\). There is only a logarithmic singularity at the right-edge origin; the imaginary growth is bounded after subtracting \(\log|a|\). Exhaustion by rectangles therefore gives the strip Poisson solution. For the Fourier transform convention \(\widehat h(t)=\int_{\mathbb R}e^{-\mathrm ity}h(y)\,\,\mathrm dy\), residue integration on a rectangle of imaginary height \(\pi\) gives \[\int_{\mathbb R}e^{-\mathrm ity} \frac{\sin(2\alpha)}{\cosh(2y)-\cos(2\alpha)}\,\,\mathrm dy =\frac{\pi\sinh((\pi/2-\alpha)t)}{\sinh(\pi t/2)}.\] The poles are at \(\mathrm i\alpha\) and \(\mathrm i(\pi-\alpha)\); the value at \(t=0\) is obtained by continuity. The multiplier from right-edge data to the midpoint is \(1/(2\cosh(\lambda t/2))\). Integrating in \(\alpha\) thus gives \[H(\lambda/2)=\frac14\int_{\mathbb R} \frac{\cosh(\pi t/2)-\cosh((\pi/2-\lambda)t)} {t\sinh(\pi t/2)\cosh(\lambda t/2)}\,\,\mathrm dt.\] Use the difference-of-cosines identity, replace \(t\) by \(2t\), and use evenness. This is the integral in (218). Since both \(F(\lambda/2)\) and \(a_*\) are positive, the asserted equality follows. ◻

Proof. This proof compares finite row norms with the periodic pressure; it does not use a plane limit. For even \(N,M\), the physical indexed torus has partition function \[ Z_{N,M}=\mathop{\mathrm{Tr}}\bigl((T_N^+T_N^-)^{M/2}\bigr)=Z_{M,N}. \tag{220}\] Every second indexed tile is a physical square. On that parity, lifted indices map to ordinary square coordinates by \[(j,t)\longmapsto\left(\frac{j+t-1}{2},\frac{j-t-1}{2}\right),\] up to the choice of origin. The other indexed tiles identify duplicate spins with weight one. Each collapsed time edge is incident to an uncollapsed neighboring site, so no component consists only of duplicates and no extra cycle factor remains. The indexed periods become \[\left(\frac N2,\frac N2\right),\qquad \left(\frac M2,-\frac M2\right).\] Their determinant has absolute value \(NM/2\), the number of physical vertices. Interchanging \(N,M\) reflects this ordinary torus. Since the physical weights have \(a=b\), that reflection proves the second equality in (220), with all flux sectors retained.

The two rows commute and are each other’s \(G\)-adjoints. If their common eigenvalues are \(\alpha_j,\overline{\alpha_j}\), then \[Z_{N,M}^{1/(NM)}= \left(\sum_j|\alpha_j|^M\right)^{1/(NM)}.\] The unnormalized finite-dimensional \(\ell^p\) norm decreases with \(p\). Thus this expression decreases with even \(M\), and by symmetry with even \(N\). Moreover, \[\lim_{M\to\infty}Z_{N,M}^{1/(NM)}=\Lambda_N^{1/N}.\]

Fix an ordinary periodic \(k\)-by-\(l\) rectangle. Among its seam spin assignments, choose one carrying at least \(2^{-2(k+l)}\) of its partition function. Repeating this same assignment glues independent interiors of such rectangles. If both indexed periods are multiples of \(2\operatorname{lcm}(k,l)\), their two physical period vectors lie in \(k\mathbb Z\times l\mathbb Z\). The quotient then contains exactly \(NM/(2kl)\) ordinary rectangles, whose opposite seam assignments already agree. Consequently, along this cofinal family, \[Z_{N,M}^{1/(NM)} \ge \left(2^{-2(k+l)}Z^{\mathrm{ord}}_{k,l}\right)^{1/(2kl)}.\] For fixed even \(N_0,M_0\), choose such \(N\ge N_0\), \(M\ge M_0\). Separate monotonicity bounds \(Z_{N_0,M_0}^{1/(N_0M_0)}\) below by the same right side. Letting \(M_0\to\infty\), and then letting \(k,l\) grow along the sequence in Lemma 88, gives \(\Lambda_{N_0}^{1/N_0}\ge F(\lambda/2)^{1/2}\) for every even \(N_0\). The asymptotic contraction in (202) of the normalized staggered twist row, applied to the untwisted physical step, gives the opposite inequality in the limit for the ratio in (219). Thus that ratio tends to one, not merely its \(N\)-th root.

The number of rows in each sandwich term is fixed while \(N\to\infty\). Its products are bounded in the \(G\) norm and the change of divisor is \(1+o(1)\) per row. This remains true when the compared terms have different fixed row counts. Their difference therefore changes by an error tending to zero. ◻

Proof. We first work exclusively in the plane. The screen and collar estimates used here are Propositions 62 and 64. Only full annuli are needed; they have no bank classes, and the surrounding primal circuit determines the boundary partition transmitted toward the query. The bounded collar densities and actual circuit screens of Propositions 47, 62, and 64 give the following uniform statement. For any fixed finite inner query, its conditional law differs by a quantity tending to zero as the surrounding buffer grows, uniformly over planar outside partitions. Indeed couple a common positive part on an intermediate annulus. With a probability bounded below, independent of the two outside partitions, this common part contains an actual primal circuit. Conditioning toward the inside, the circuit induces the same wired boundary there. Iterating in separated collars contracts the range of inner probabilities by a fixed factor, starting from bounded density. Alternating primal and dual circuit screens also imply that no raw strand meeting a fixed query escapes all large buffers. This proves existence, uniqueness, self-duality, and lattice symmetry of the planar local limit.

It is important not to apply a planar boundary estimate directly to the exterior of a disk in a torus. Instead cut a fundamental chart and remove \(O(N+M)\) seam edges and vertex identifications. The remaining core is an ordinary plane graph. For any fixed core edge configuration, restoring the seam changes its component count by at most \(C(N+M)\). Summing over seam edge states changes its weight relative to the free planar-core law by a factor between \(\exp(-C(N+M))\) and \(\exp(C(N+M))\), also after normalization. All constants depend only on the fixed model parameter.

In the winding expansion an oriented switch loop of total turn \(W\) has factor \(e^{\mathrm i\rho W/2}\), with a common factor \(a_*\) per physical tile; the local orientation sum is \(R(\lambda/2)\), as verified in Lemma 92 below. A simple contractible loop has turn \(\pm2\pi\), while a simple essential loop has turn zero, by isotopy to a straight essential cycle on the flat torus. Summing the two orientations therefore gives loop weights \(2\cos(\pi\rho)=d\) and \(2\), respectively. To compare with torus FK, thicken the open primal graph. If its total genus is \(g\), then \(g\in\{0,1\}\), and its number of perimeter components is \[|A|-|V|+2k(A)-2g.\] Thus the FK-to-loop discrepancy, apart from fixed factors, is the bounded genus factor \(q^{-g}\) and the essential-loop tilt \((2/d)^{\ell_{\mathrm{ess}}}\). Each essential strand has length at least a constant times \(\min(N,M)\); disjointness gives \[\ell_{\mathrm{ess}}\le C\frac{NM}{\min(N,M)}\le C(N+M).\] Consequently the complete unmarked spin-loop core law satisfies the same pointwise density comparison with the free planar-core law: \[ e^{-C(N+M)}\le \frac{\,\mathrm d\mu_{N,M}^{\mathrm{tor,core}}} {\,\mathrm d\mu_{N,M}^{\mathrm{pl,core}}} \le e^{C(N+M)}. \tag{221}\]

We next obtain a stronger concentration bound under the planar reference law. Let \(h\) be a bounded local switch statistic and write \(\mu(h)\) for its plane mean. For a tolerance \(\eta>0\), choose a fixed buffer size \(L\) so that every conditional mean of an interior translate of \(h\), given the states outside its buffer, is within \(\eta\) of \(\mu(h)\). This uses only planar outside states. Color the translates into a fixed number \(O(L^2)\) of classes having disjoint buffers. In one such class, conditioning on any earlier values of \(h\) is a mixture of full outside configurations for the next buffer. Its conditional mean is therefore still within \(\eta\) of \(\mu(h)\). The elementary bounded-variable exponential estimate, iterated over that class, gives an exponential tail in the number of its translates. Taking the union over the fixed color classes yields \[ \mu_{N,M}^{\mathrm{pl,core}} \left(\left|\overline h-\mu(h)\right|>3\eta\right) \le C_\eta e^{-c_\eta NM} \tag{222}\] for the average over translates outside the seam buffer. The omitted strip contains \(O(L(N+M))\) translates. Its fraction tends to zero whenever both periods grow. By (221), the same deviation probability under the torus law is bounded by \(C_\eta\exp(C(N+M)-c_\eta NM)\), which tends to zero without an aspect-ratio restriction. Translation invariance and boundedness then give convergence of each unmarked local expectation.

Finally fix finitely many spin insertions. First take a large fixed buffer containing them. The planar alternating-screen event in that buffer has probability tending to one with its radius, and by the unmarked local convergence the same is true on the torus. On that event all raw strands meeting insertions are local. Summing their orientations makes the insertion ratio a local function of the switches. For \(m\) inserted edge matrices, at most \(m\) raw loop components meet insertions, and breaking them at the marked edges leaves at most \(2m\) oriented pieces. Summing their orientations bounds the modified numerator by a constant depending on the fixed matrices and \(m\). Their unmarked divisors are products of at most \(m\) factors, each equal to \(d\) or \(2\), and are at least \(d^m>0\). Thus the ratio is bounded uniformly in the torus periods. This also treats off-diagonal insertions, including those whose total spin change makes their contraction zero. The complement of the screening event therefore has arbitrarily small contribution. This proves the assertion for marked ratios and for both slicings.

Fixed displacements of periods and finitely many changed positive time steps in a balanced period alter only bounded-width seams; the same proof applies. A long-time cylinder limit followed by a circumference limit is realized by choosing each time period large enough to approximate its finite-circumference vacuum coefficient. The comparison above is applied along this joint sequence, never with one period fixed. ◻

Proof. At fixed \(N\), primitivity identifies the sector-averaged low coefficient, or each of the two sector-averaged high coefficients separately, with the long balanced-cylinder vacuum coefficient of its physical diagram. Cauchy–Schwarz in \(G_N\), followed by Cauchy–Schwarz in the finite sector average, gives \[|\mathcal C_N^a| \le\|B_{\mathrm{out},N}\mathcal D_N^a B_{\mathrm{in},N}\|_{G_N} \sqrt{R_{\mathrm{out},N}R_{\mathrm{in},N}}.\] At either physical angle the strip divisor is \(F(\lambda/2)^{N/2}\). Thus each row in (225) is its normalized staggered row multiplied by \(F(\lambda/2)^{N/2}/\Lambda_N\), which tends to one by Proposition 89. Each row list is fixed while \(N\) grows. Proposition [physical:finite-staggered] therefore gives the operator limsup \(C_{\lambda,L}n^{-1+\rho}\) or \(C_{\lambda,L}n^{-1-\rho}\). This use of the divisor comparison is asymptotic, including when the two high terms have different row counts.

For each \(N\), choose one sufficiently large balanced time period that approximates this finite set of vacuum coefficients: both scalar terms in the high case and both reflected coefficients. Proposition 90 applies along the resulting joint sequence, since (214) puts every row twist or marker on one physical edge. The row counts and all other insertion supports are fixed during this limit, so the diagrams have finitely supported insertions and both periods tend to infinity. A twist on each of finitely many such seam edges is covered by that proposition; no insertion extends around a growing period. It identifies the limits in the statement. In particular the reflected limits are finite, and they are nonnegative as limits of the norm squares in (223); this does not assert positivity of individual winding summands. Taking the limit in the preceding Cauchy–Schwarz inequality proves the two bounds.

For an ordinary tile-axis cut, the untwisted homogeneous row at \(u=\lambda/2\) is self-adjoint by (155). Within each spin sector its matrix has a positive diagonal and positive transitions for adjacent unlike-spin exchanges, obtained by two auxiliary exchanges; these connect the sector, so the row is primitive. Reflection matches the spin coordinates directly. Using its own Perron divisor and the ordinary Hilbert norm gives the same reflected finite identity and sector-averaged Cauchy–Schwarz inequality. The fixed-diagram plane limit then gives the last assertion. ◻

Proof. At a tile, the oriented tensor vanishes unless there are two incoming arrows. For adjacent incoming arrows one pairing is allowed and its two turns cancel. For opposite incoming arrows the two pairing contributions sum to \(2\cos(\lambda/2)\). Multiplication by \(a_*=1/(2\cos(\lambda/2))\) therefore gives the stated real six-vertex tensor for both signs.

Close a boundary arc against its forward boundary interval, using the normal endpoint tangents. Its turn is \(\theta_j-\theta_i-\pi\). Multiplying its two boundary factors by its plus winding weight and summing its two orientations gives \(w(h_i-h_j)\); the lifted normal terms cancel. Reversing the winding sign gives exactly the replacement \(-h_i-\theta_i/\pi\). Ports inside the child interval pair among themselves. Its endpoint high statuses are consequently opposite for an even number of flips and equal for an odd number. This proves the drop rules, and substitution of \(\Delta_0\) gives (233). The string assertion is the same orientation sum with its extra crossing factors. ◻

Proof. Remove arcs from descendants outward, so the next two endpoints are adjacent among the remaining ports. Joining them makes one completed loop exactly when their exterior partners already pair them; otherwise it reconnects those partners. Within a designated interval, a high–low pair closes one cap loop and a low–high pair transports the same local pattern. Across a flip the equal high statuses transport an unmatched end without closing a loop. If two such transported ends were already tied, the exceptional loop erases an entire remaining designated interval. There are only boundedly many whole-interval erasures. Other cap statuses are inherited unchanged. At the end, each remaining arc accounts for at most a bounded number of unremoved boundary loops. This proves the aggregate error. Changing boundedly many extra ties changes the loop count by only a bounded amount. No sign or positive corner-cost assertion is used. ◻

Proof. Fix \(0<\varepsilon<1/2\) and put \(r=n^{1-\varepsilon}\). Use a straight cylinder with even circumference comparable to \(n\) and height \(Kn\log n\), where \(K\) will be fixed sufficiently large. Place two bottom flips, named \(s\) and \(t\), a distance of order \(n\) apart, and no top flips. The same letters denote their effective drops from (231). All bulk endpoints \(z_j\) are grouped near the \(s\)-gap: their heights, pairwise separations, and distances from that gap are comparable to \(r\). Give them disjoint cutoff boxes and larger buffers of small fixed fractions of \(r\), leaving clear corridors around them. The \(t\)-cutoff is a small fixed fraction of \(n\).

Choose the two positive cap completions as a paired choice under global color reversal, including any residual planar ties. Thus reversing every designation exchanges the two completed switch laws exactly; it does not change the cylinder or its tile weights.

Draw strings of charges \(-a_j\) from the \(s\)-gap to \(z_j\). Use raw bottom drops \(s-1,t\), whose sum is zero, and constant raw base on top. An oriented intersection shifts a correct disk drop by \(+a_j\) if its interval contains the launch and by \(-a_j\) if its disk contains the endpoint. Thus the local effective drops are \(s,t\). In the wrong expansion they are \(-s,-t\), with the enclosed endpoint charge added. Figure 8 records the cylinder and the separate arm requirements used below.

The comparison must keep the local positive weights on the same events throughout the proof. In particular, the possibly small factor from raw straddles at the distant flip will multiply both the positive main event and the exceptional event carrying a negative term. A saving only for their unweighted probabilities would not suffice.

Proof. Attach a thin annulus carrying the caps and continue the four open ports to its outer boundary. Along each interval, the annulus joins precisely the visits of its designated color to the outer boundary. The other intervening gaps are cut off by the caps. Therefore a raw region remains accessible from the outer boundary exactly when it is wired. An unwired region acquires only the adjacent capped slivers; it neither merges with another raw region nor becomes accessible. After attachment there are two noncrossing open strands. These leave three regions touching the outer boundary, and each additional closed loop adds exactly one region not touching that boundary. This proves (249).

The raw arcs are the perimeters separating the thickened graphs of the two colors. Boundary visits of one color in the same raw region are connected in that graph: trace within its thickening, then retract to edges. Internal closed perimeters only enclose components with no boundary visits, so restoring them cannot split this boundary class. There are exactly two noncrossing pairings of the four open ports. They give, respectively, the actual opposite connection of color \(\mathfrak A\) and of color \(\mathfrak B\); both cannot occur because their endpoints alternate. Against the exterior pairing that wires \(A,C\) separately, the first pairing forms one loop and the second forms two. This proves the connection and completion assertions.

Lemma 120 gives \(w_e\) on an unpassed arc, \(k_e\) on a passed arc, and \(c_*a_e\) on the marked arc of the primitive. For the primitive, an internal loop retains weight \(d\) unless it is marked, in which case its contribution is zero; its factors therefore cancel against the open-cap weight. For the full diagonal string, each enclosing internal loop instead contributes the ratio \(-2/d\). Removing those ratios leaves \(D_V\), while removing \(c_*\) from the primitive and dividing the local-cap factors by \(d^{\mathcal I}\) gives (251)–(252) and (253). In the difference \(H_v-H_u\), every mark before \(e\) has the same factors on both paths. The only new term is the mark on \(e\), proving (254), including when some \(w_e\) vanishes. ◻

Proof. We use the actual crossings and buffered routing of Propositions 22, 34, and 61, in the screening form of Proposition 62. The precise consequence needed is as follows: in a fixed-ratio bulk annulus there are two nested actual circuits of opposite colors with conditional probability at least \(\epsilon>0\); in a fixed-ratio wall sector there are two successive actual crosscuts of opposite colors between its banks with the same type of lower bound. The bound is uniform over the permitted exterior data of Definition 59; in these fixed wall charts every designated bank class, including a partial interval, is among the bounded named classes. Use disjoint buffers for the two colors, and condition successively; no positive association is used. The constants may depend on the fixed polygon and \(d\). All estimates obtained under deterministic completion transfer to the open-cap law by its bounded density ratio.

Two such wall crosscuts force a raw arc joining the banks between them. To see this, trace their thickened perimeters. The gaps adjacent to the two crosscut ends have opposite colors on each bank, so there is an odd number of ports between those ends. The actual paths prevent a raw strand in this intervening strip from escaping through either crosscut. If no raw strand joined the banks, all the ports on either bank would be paired on that bank, contradicting oddness. Thus at least one joining raw arc separates the deeper neighborhood from the outside. The same argument applies in the finitely many convex or concave corner sectors of the polygon. In the bulk, the nested circuits prevent a raw boundary arc from passing from the inside to the outside.

Under deterministic caps, for \(K\) disjoint scale buffers, conditioning bounds the probability of no successful screen by \((1-\epsilon)^K\). More generally the probability of fewer than \(m_0\) successful buffers is at most \[ \sum_{j=0}^{m_0-1}\binom Kj(1-\epsilon)^{K-j}. \tag{274}\] Indeed, for each proposed set of at most \(m_0-1\) successes, require failure on its complement and condition on those disjoint buffers one at a time. The sum tends to zero for fixed \(m_0\) as \(K\to\infty\). For the open-cap law, multiply these upper bounds by its fixed density-comparison constant.

If two nearby interior vertices belong to different boundary-tree regions, a raw boundary arc intersects a short lattice path between them and continues to the wall. Surround that path by disjoint bulk buffers out to a fixed fraction of its distance from the wall. The preceding bound makes this event uniformly unlikely as their separation tends to zero. The bound on \(H\) in (264) then gives equicontinuity of its expectation. The same argument allows vertices of different lattice parities. Piecewise interpolation, followed by Arzelà–Ascoli on an exhaustion of interior compact sets, gives the asserted compactness.

At an ordinary boundary point, a successful local wall screen traps the deeper neighborhood in a disk whose boundary visits belong only to its interval. Every raw region there has a visit on that interval. Taking arbitrarily many buffers between \(|z-u|\) and a fixed small radius proves (272). At a mark, the forced raw arc has the corresponding one-mark descendant interval, except at \(p\), where its descendant interval contains \(b,c,d_0\) and the local neighborhood is on the root side. Hence the asserted edges are passed at \(b,c,d_0\) and unpassed at \(p\). Buffers between a mesh-scale neighborhood of each mark and a fixed radius show that each terminal leg is nonempty with probability tending to one. Finally, take buffers between \(|z-c|\) and a fixed radius avoiding the other marks. Their successful raw arcs are distinct nested \(c\)-leg edges, all passed before reaching \(V(z)\). Equation (274) proves (273), with the mesh limit taken first. ◻

Proof. We use the same tile pairings in both cut conventions. A raw arc or port interaction is affected if it uses a newly missing port; the gapped diagram joins the filled raw paths which end there. For the source case set \(S=UT_f\), and for the diagonal case set \(S=U_zUV_{\rm wt}T_f\). Put \(T=T_g/T_f\) and \[ \nu_g=\frac{ST\mu_g}{\mathbb E_{\mu_g}(ST)},\qquad \nu_f=\frac{S\mu_f}{\mathbb E_{\mu_f}S}. \tag{304}\] All factors of \(S\) are supported away from the gaps being filled. Our goal is to separate the normalized diagram into a positive local multiplier near the gaps and a common signed factor away from them. The separation is exact on a screening event. Uniform moments under both tilted laws will control its complement before the relative collar comparison factors the two expectations.

Proof. Fix sufficiently large \(L\). For every sufficiently large fixed \(R_0\), (302) bounds \(|a_n^I(R_0,L)|\) away from zero for all sufficiently large \(n\); the mesh threshold may depend on \(R_0\). Restrict to this mesh tail. By the exact factorization (301), the ratio in the theorem is \(\mathcal R_f(z)/a_n^I(R_0,L)\) for our fixed choice of \(\sigma\), and its negative for the other choice. The parity identity (284) has already converted each enclosing-loop factor \((-2/d)\) into the signed tree factor and the positive magnitude \(2/d\). It remains to show that \(\mathcal R_f(z)\) vanishes and then use the nonzero exterior factor.

Use \(\mathcal R_g\) from (286), with the unmarked exterior profile \(H^{\rm diag}\) in (275) and the charge routes of Section 9.3. The two distant charge lists each have total magnitude one and satisfy (277). Besides their endpoint factor \(U\), the exterior normalization has the flat-flip factor \[ V_{\rm wt}=w(1/4)^{\sum_{\nu=1}^4N^{\rm ext}_\nu}, \tag{313}\] where the four local straddle counts use disjoint buffered patches. Open gaps of width comparable to \(h\) at every true corner, away from all these patches and from the string corridors. The remaining solids are separated straight fragments. For the gapped configuration use the tip factor \(T_g\) and normalize its vacuum tensor by \[ Z_g^{\rm cap}\mathbb E_{\mu_g}[T_g S],\qquad S=U_z U V_{\rm wt}. \tag{314}\] All supports of \(S\) remain away from the shrinking corner gaps.

Proof. Let \(R\) be the unmarked boundary interval containing \(u\) and put \(g_R=1/2\) on its wire color and \(g_R=-1/2\) on the other color. Under \(\mu_{\rm in}\) write the numerator integrand of (312) as \(B_*T_*\), where \[ B_*=R_*\sigma(\operatorname{col}(V)) d^{-g_R(\operatorname{col}(V))}D_V, \qquad T_*=d^{g_R(\operatorname{col}(V))}(2/d)^{N(z)}. \tag{317}\] The first factor is bounded uniformly, by Lemma 96. Choose scales \[ |z-u|\ll r_1\ll r_2\ll r_3\ll1, \tag{318}\] with all consecutive ratios diverging as \(z\to u\), and take the mesh limit first at every fixed choice of \(z\) and these scales.

Actual screens in the regular sector at \(u\) make \(B_*\) measurable outside the \(r_2\) patch, except on an event whose probability tends to zero in the scale ratios. Use as the good event the absence of a raw traversal of the annulus from its \(r_2\) rim to its \(r_3\) rim; this event is determined by the annular diagram, hence by the configuration outside \(r_2\). Here is the finite invariance needed for measurability. For every compatible filling inside \(r_2\) on this exterior event, every boundary arc affected by changing that filling lies in one small portion of \(R\) and contains no mark in its child interval. All marked-subtree data and the open-to-completed pairing factor \(R_*\) remain fixed. Stable exterior separators put the whole connected inner patch on one side or the other. Noncrossing therefore puts all affected on-path edges after the stable prefix. Their off-path factors cancel the local caps; their on-path suffix telescopes by the ordinary-tail rule (260). The sign and color factor in \(B_*\) cancel that suffix. Define the exterior-readable truncation \(B_{\rm out}\) to equal \(B_*\) on this good event and zero otherwise.