Source and normalization. These are Proposition 8.2, Identification of the tuned parameter, and Theorem 8.5, Physical critical point, in Section 8 of [18]. The definition of magnetization there first normalizes each quenched Gibbs law and then averages its infinite-volume plus magnetization. It is therefore the threshold used here. The bounds \(K_0/(1+\varepsilon)\le\beta_c(\varepsilon)\le K_0/(1-\varepsilon)\) imply \(\theta\to0\). The companion’s identification of \(h=0\) uses torus partition functions and a separate sharpness argument to identify the physical threshold. No inference from self-duality alone is needed. ◻

Source. This is Lemma 8.3, Local FK barriers, in Section 8 of [18], followed by (5). That lemma gives no rate for the exceptional environmental probability. Our uses fix all physical annular radii before taking the mesh limit. ◻

Source and parameter choices. The local construction and (12) are Proposition 3.3, The local scale map, of [18]. Proposition 4.1, One-step estimates, first makes the contraction coefficients small by increasing \(L\), then controls the Taylor remainder by decreasing the flag threshold, and finally restricts the input moments. Theorem 6.2, A small critical trajectory, gives \(X_k\le A\sqrt{v_k}\) and \(v_k\le(v_0^{-1}+ck)^{-1}\) for fixed positive constants \(A,c\) and \(v_0=O(\theta^2)\). Here \(v_k\) is a scalar variance coordinate of that construction, distinct from the edge odds \(v_e\). Proposition 2 identifies the resulting trajectory with \(h=0\). These proofs give the stated freedom in the parameter choices; they do not require summability of \(X_k\) over \(k\). ◻

Proof. We first prove the estimate for pinned rays by a four-point boundary correlation. Duality then turns the free-ray kernel into arbitrary products of dual spins; an FK screening argument makes the four-point estimate usable uniformly in the number of these insertions. Finally we treat a change of boundary condition by a fixed-scale arm comparison.

Proof. The spin Markov property factors the conditional law across the two caps into the shell kernel and positive factors on its two sides. If the shell diameter is \(a\), the conditional distributions on the inner cap, for any two remote conditions, have relative densities in \([e^{-a},e^a]\). Their mixtures with the conditional law of the inner patch have the same comparison. With free rays, spin reversal shows that the distribution of the patch modulo global sign depends only on the outer spin vector modulo sign. Thus the quotient kernel gives precisely this comparison for even functions, even when the remote spins were specified with absolute signs. Integrating independent auxiliary variables preserves all the bounds.

At a straight side, (15) now yields (26). It also bounds the oscillation of the logarithm of the expectation of any positive function by \(Cd^{-2}\). To retain the factor \(\|H\|_\infty\) when \(\|H\|_\infty\le1\), apply relative-density comparison to \(e^H-1\); its supremum norm is at most \(e\|H\|_\infty\), while all expectations of \(e^H\) are at least \(e^{-1}\). When \(\|H\|_\infty>1\), the direct projective bound is already at most \(Cd^{-2}\|H\|_\infty\).

At a corner or change point, divide the region into \(m\) successive shells of the fixed ratio in Lemma 6, with \(m\ge c\log d-C\). A diameter bound \(D<\infty\) gives a common positive fraction \(e^{-D}\) in every pair of normalized rows. Consequently conditional expectation contracts oscillation by at most \(q=1-e^{-D}<1\) through each shell. Composition gives a bound \(Cq^m\|H\|_\infty\), which is (26) for some \(\eta>0\). The quotient by sign is preserved at every free shell.

For (27) with \(\|H\|_\infty\le1\), apply the same contraction to \(e^H\) and then use the lower bound \(e^{-1}\). For \(\|H\|_\infty>1\), first pass through one shell. The resulting positive function \(h\) satisfies \(\mathop{\mathrm{osc}}\log h\le D\), independently of the size of \(H\). Multiplying \(h\) by a constant, arrange \(1\le h\le e^D\). The remaining shells contract its ordinary oscillation to at most \(q^{m-1}(e^D-1)\) and preserve its lower bound \(1\). Its logarithmic oscillation has the same upper bound. Absorbing the first shell into the constant gives \(Cd^{-\eta}\le Cd^{-\eta}\|H\|_\infty\). This argument uses no exponential moment assumption on \(\|H\|_\infty\). ◻

Proof. Allocate the initial bond perturbations as in [18], using only bonds present in the polygon. A cell receives the allocated terms \((K_e-K_0)\sigma_x\sigma_y\), followed by the specified centering. For fixed \(s_0\) there are boundedly many such terms per cell, and \(|K_e-K_0|=O(\theta)\); this proves the initial bound. Subtracting constants changes only the scalar partition factor.

The identities (32), (33), and (35) prove exactness at each step. They hold for every conditional pure probability kernel; the presence of the boundary merely reduces the set of spins integrated in a square. The allocation and status checks use the same finite stencils as in the plane. The choice of \(C\) in the local map already covers these stencils and the \(C/L\) overhang from the child scale. Restricting the pure integrations to polygon spins does not enlarge either radius. This proves locality. Choosing \(B\) with slack so that \(B>C_1+B/L\), where \(C_1\) bounds the complete parent determining radius, shows inductively that every input of a nonboundary output is also nonboundary. There the construction is precisely the plane construction. The pad similarly contains all possibly nonzero outputs.

Every determining neighborhood sees at most one local pin sign. Spin flip commutes with the conditional integrals, and with every algebraic operation in the decoration and the isolated ratio. Hence changing that sign conjugates the local potential by simultaneous flip of its spin and pin arguments. The corresponding norms agree; using the same thresholds and priorities therefore gives the same flags, isolation decisions, and local scalar subtraction. If no pin lies in a determining neighborhood, its potential is spin-flip even. The constants used for boundary centering are deterministic functionals of the local potential: dependence on the fixed pure polygon does not introduce dependence on remote random bonds.

In particular the \(+-\) potentials need not be centered for the pure \(+-\) expectation. Exactness only requires that the subtracted constants agree between the two constructions. Iterating the exact identities with those constants gives (36). ◻

Proof. There are four contributions to the recurrence. The linear transfer of boundary logs contracts by the summable side estimate; unflagged Taylor remainders are small because their input norms are at most \(2b_0\). On flagged stencils, a distant independent flag gives an extra moment factor. The remaining dependent logs form an isolated group, to which the positive-weight transfer estimate applies. Bulk inputs in each case are bounded by a constant times \(X_k\).

We first estimate the linearized regular map. Its plane formula from [18] remains valid before the specified output centering: \[p_{\rm sel}^{-1}A_j\sum_{i\in\mathcal C_j} \mathbb E_0^{++}[f_i\mid I_j^c].\] The derivative of each coin normalizer is zero, and the derivative of a conditional log integral is conditional expectation. For a boundary child the pure mean is zero, so its transferred supremum is bounded by its oscillation. The reference variables in its patch are integrated with their pure product law. The same bound for the other pin assignment follows by the local flip described in Lemma 10.

We claim that the sum of the transfer coefficients over boundary children in one determining stencil is \(o_L(1)\). Distances in the next paragraph are measured in child units. All these cells lie in a strip of fixed width \(B\) along a side. If a cell is within a fixed distance of a corner or a change, choose an inner sector shell containing its whole support and determining patch. The corner or change estimate of Lemma 6, composed to a radius proportional to \(L\), makes its transfer coefficient tend to zero as \(L\) grows. All those shells fit inside the parent integration square with room to condition onward.

For the remaining cells let \(t\) be their distance along the side to the nearest corner or change, capped at \(L\); in a neighborhood without a feature set \(t=L\). Once \(t\) exceeds a fixed support-dependent constant, a straight half-shell reaches a radius proportional to \(t\) without seeing a different boundary condition. Its coefficient is at most \(C't^{-2}\) by Lemma 6. On a free side the full determining stencil avoids pins, so the potential is even, as required by that lemma. There are \(O(t)\) boundary cells in each dyadic range \([t,2t]\), including \(O(L)\) in the last capped range. Consequently, for any sufficiently large fixed \(t_0\), \[ \sum_i a_{i,L} \le \sum_{t_i\le t_0}a_{i,L}+\frac{C''}{t_0}, \qquad \lim_{L\to\infty}a_{i,L}=0 \quad\text{for each member of the first, bounded list}. \tag{39}\] First choosing \(t_0\) and then \(L\) proves the claim. The same calculation holds for a fixed enlargement of the child patches or the boundary strip. Patches with no free spins cause no difficulty, since only their reference variables remain to be integrated.

Multiplication by \(A_j/p_{\rm sel}\) costs at most \(p_{\rm sel}^{-1}\), which is independent of \(L\), and output centering costs at most a factor two. Minkowski’s inequality and (39) therefore bound the boundary-child part of the linearized output by \(o_L(1)Z_k\). All bulk inputs can be bounded crudely, giving \(C_LX_k\). Only finitely many fixed shell ratios are needed once \(L\) is fixed; we subsequently choose \(s_0\) large enough for all their pure estimates.

It remains to control the actual map in the \(P\)th moment. Set \(X=Z_k+X_k\). Every child norm in an enlarged determining stencil has root moment at most \(X\). On a stencil without flags, every child norm is at most \(2b_0\). Conditional integrations are probability kernels, so the regular map has uniform Taylor bounds after \(L\) is fixed. Writing \(S\) for the sum of its child norms, its remainder is bounded by \(C_LS^2\le C_Lb_0S\). Its root-moment cost is thus \(C_Lb_0X\). The real map, without any smallness restriction, has the crude bound \(C_LS\) by \(|\log\mathbb E_0e^H|\le\left\lVert H\right\rVert_\infty\) and the analogous Lipschitz bound for differences of such logarithms.

Small flag probability alone would not improve this last estimate in the \(P\)th moment. We use the location of the flags, as in [18]. Enlarge the determining stencil to include all isolation checks and all child neighborhoods of radius \(100D_0L+2C\) about a contributing child. If a flag \(h\) in this enlarged stencil is farther than \(2C\) child units from a charged child \(i\), their environmental dependence neighborhoods are disjoint. Its fresh threshold is independent too, and Markov’s inequality gives \(\mathbb P(h\text{ is a flag})\le b_0^{-P}X^P\). Hence \[ \mathbb E\!\left[\left\lVert f_i\right\rVert_\infty^P {\bf1}_{\{h\text{ is a flag}\}}\right] =\mathbb E\left\lVert f_i\right\rVert_\infty^P\, \mathbb P(h\text{ is a flag}) \le C_{b_0}X^{2P}. \tag{40}\] There are only finitely many charges for fixed \(L\), so these contributions cost \(C_{L,b_0}X^2\) in root moment.

If no such independent flag is available for a contributing term, every relevant flag lies within \(2C\) of that term. These flags form one bounded child-radius group. The enlarged stencil contains the whole isolation neighborhood, so either they are an isolated group of the construction, or a farther flag is present and (40) applies. In the isolated case, all logs possibly dependent on the triggering flags are included in \(G_A\). For every probability kernel \(P_0\), \[ \left|\log\frac{P_0e^{B_A+G_A}}{P_0e^{B_A}} -\log P_0e^{G_A}\right| \le2\left\lVert B_A\right\rVert_\infty. \tag{41}\] Each child in \(B_A\) is outside the concentration neighborhood and is independent of an actual triggering flag. Its occurrence can again be charged using (40). The same applies to every term omitted from the concentrated contribution.

After this estimate, the only remaining large term to bound is the centered form of \[\log\mathbb E_0^{++}[e^{G_A}\mid I_A^c].\] Corollary 7 applies to this positive weight for any size of \(\left\lVert G_A\right\rVert_\infty\). A group containing a boundary child has bounded child diameter. There are boundedly many possible flag patterns at each anchor and boundedly many logs in each concentration neighborhood, with bounds independent of \(L\). Thus its logs all lie in a fixed enlargement of the boundary strip, and the shell coefficient at its anchor multiplies the sum of their norms. Summing over all possible boundary anchors gives the same arbitrarily small sum as in (39). We may sum before taking the root moment by Minkowski’s inequality; neither a supremum over anchors nor independence among their events is needed. Groups containing only bulk inputs contribute at most \(C_LX_k\) by the crude bound. The linearized terms restricted to flagged stencils, if subtracted in comparing the two maps, obey the identical dichotomy: either charge them to an independent flag or use the linear shell transfer for the bounded local group.

Combining these estimates gives \[Z_{k+1}\le \bigl(o_L(1)+C_Lb_0\bigr)Z_k+C_L'X_k +C_{L,b_0}(Z_k+X_k)^2.\] Choose \(L\) so the first term leaves a strict contraction margin, then choose \(b_0\) within that margin, and finally choose \(s_0\) for the finite list of pure shell estimates. These choices are compatible with the parameter freedoms in Proposition 4. This proves (38).

For completeness, take a deterministic scalar majorant for all the finite polygon runs, starting from their uniform \(O_{s_0}(\theta)\) bound and iterating the right side of (38). Choose a small invariant interval so that the quadratic term has Lipschitz constant less than half the remaining contraction margin. Proposition 4 makes \(\sup_kX_k\) as small as required by decreasing \(\theta\); the initial majorant is small by the same choice. The majorant therefore stays in that interval. Since \(X_k\to0\), its limsup \(z\) satisfies \(z\le q_*z+C_{L,b_0}z^2\), which forces \(z=0\) in the chosen interval. This proves uniform smallness and decay. ◻

Proof of Proposition 9. All determining stencils used in Lemma 11 lie in ordinary injectively projected charts. The physical-edge signs within each such stencil have their usual iid law. In particular, disjoint dependence neighborhoods used in (40) are genuinely independent. There is no requirement of independence between different sheets: the entire enlarged stencil, including its overhang, was placed in one ordinary chart before the terminal scale was selected.

By (30), the number of possibly nonzero terminal potentials, including those in the pad, is bounded for this fixed polygon. This bound may depend on its geometry and on \(\alpha\), but not on the mesh. Proposition 4 and Lemma 11 therefore give \[S_\delta=\sum_i\max_{\eta\in\{++, +-\}} \left\lVert f_i^{(N),\eta}\right\rVert_\infty \longrightarrow0 \quad\text{in }L^1\text{ of the environment and construction choices}.\] Applying (36) yields \[\left|\log\frac{\mathcal Z_J^\eta}{\mathcal Z_0^\eta} -a_\delta\right|\le S_\delta, \qquad \eta\in\{++, +-\}.\] The common scalar cancels from the difference. Since \(\tanh\) is Lipschitz, (29) proves the comparison with an error bounded by \(S_\delta\). The partition ratios themselves do not depend on the construction choices. Averaging the bound over those choices thus gives convergence in environmental probability alone.

The local boundary shapes are the straight side, the two orthogonal corner angles, and a single fixed/free change. Their constants, and all scale-map parameters, were chosen independently of the polygon; only the positive terminal physical scale depended on it. This proves the asserted uniform disorder threshold. On the dual grid, duality replaces \(v_e\) by \(2/v_e\), hence \(\theta\xi_e\) by \(-\theta\xi_e\). This has the same environmental law, and the same argument applies. ◻

Proof. Fix the internal switches. The two possible boundary pairings acquire numbers of additional loops differing by one under separate primal closure. The noncrossing outcome acquires the extra factor \(\sqrt2\). The local edge factors are unchanged. Thus the separate-closure law has density proportional to \(\mathbf 1_{E_{\mathrm P}}+\sqrt2\mathbf 1_{E_{\mathrm D}}\) relative to the open-cap law. Normalization proves (42) and the density bounds. Conditioning this same two-valued density on the full revealed history proves the conditional identity. In particular, no Markov property of a putative limiting curve is being used. This is the finite cap calculation of [17]; it does not require constant edge odds. ◻

Proof. For a planar polygon this is the four-mark observable theorem of Chelkak–Smirnov [6], with boundary pairings and deterministic closures converted as in Lemma 12. We explain why its proof applies to \(P\). This is necessary because the projection of \(P\) need not be injective.

The finite observable. The construction combines the two Dobrushin observables obtained by joining \(c\) to \(d\), or \(a\) to \(d\), with a deterministic exterior arc. Its local s-holomorphic identities and its relation to the two pairing weights are identities on tiles. The primitive of its square is single-valued on the disk and is constant on each of the four boundary arcs. In cyclic order its normalized boundary values are \(0,1,\kappa_\delta,\kappa_\delta\). If \(p_\delta=\mathbb P_{\mathrm{open}}(E_{\mathrm P})\), the exact relation is \[ \kappa_\delta= \left(\frac{t_\delta^2+\sqrt2t_\delta} {t_\delta^2+\sqrt2t_\delta+1}\right)^2, \qquad t_\delta=\frac{1-p_\delta}{p_\delta}. \tag{45}\] This extends continuously to \(p_\delta=0,1\) and is one-to-one. It is the normalization in [6], with their pairing probability equal to \(1-p_\delta\).

The phases can be computed in the frame supplied by the projection. A simple closed curve bounding a disk in this flat surface has total turning \(2\pi\) by Gauss–Bonnet. Closing an observable path by a boundary arc therefore gives the same phase relations as in the plane. The collar accommodates the deterministic closing arcs. There is no additional period for the primitive or ambiguity in the half-differential on a simply connected polygon.

Compactness and boundary values. The bounds on the primitive use its boundary values, the local boundary modification by half-rhombi, and the discrete maximum principle. They do not require a global injective coordinate. Interior estimates for the observable and its primitive apply in ordinary projection charts. A finite chart cover of each compact subset consequently gives subsequential convergence to a bounded harmonic primitive \(H\) whose complex differential has the corresponding holomorphic square root.

The constant data and one-sided normal conditions pass to the limit near open straight sides by the same boundary estimates. In each such chart, comparison with distant boundary data first stops at the boundary of a small chart neighborhood; the comparison function there is bounded. Its contribution near the side is then suppressed by ordinary planar discrete harmonic measure. Corners have ordinary wedge charts. Their isolated values do not affect the bounded harmonic Dirichlet problem. The marked points lie in straight portions, so their two one-sided conditions are handled in disjoint neighborhoods in the same way.

Identification of the remaining constant. The sign condition needed here is concrete. Denote the two auxiliary primitives by \(H_\delta^\bullet\) and \(H_\delta^\circ\), on the black and white graphs of the observable construction, respectively. The black primitive is subharmonic and the white one superharmonic; these color names follow [6]. On the arc \(cd\), \(H_\delta^\bullet\) at an inward neighbor is at least its boundary value \(\kappa_\delta\). On \(da\), \(H_\delta^\circ\) has the reversed inequality. These follow from the observable’s boundary tangent condition and the half-face modification in [6].

To see that these conditions survive in the required sense, suppose that a limiting harmonic primitive satisfies \(H<\kappa\) on a small interior contour \(\mathcal C\) whose endpoints lie in a straight subinterval of \(cd\). Enclose the region between this contour and the subinterval, entirely within one chart. Let \(V_\delta\) be the discrete harmonic measure of a smaller subinterval strictly between the endpoints. It vanishes on the contour. Discrete Green’s formula and the inward inequality give \[ \sum_{u\in\mathcal C_\delta} V_\delta(u_{\rm in}) \bigl(H_\delta^\bullet(u)-\kappa_\delta\bigr)w_u\ge0, \tag{46}\] where \(u_{\rm in}\) is the inward neighbor and \(w_u>0\) is the local conductance; multiple boundary incidences are summed separately. On a fixed middle portion, \(V_\delta(u_{\rm in})\ge c_1\delta\) and there are order \(\delta^{-1}\) incidences. Compact convergence thus bounds that portion of the sum above by a fixed negative number \(-c\). At distance \(r\) from an endpoint, the local harmonic-measure bound is \(V_\delta(u_{\rm in})\le C\delta r^{\eta-1}\) for some \(\eta>0\). Summing over the endpoint portions of length \(\mu\), and using the bounded primitive, gives a contribution at most \(C\mu^\eta\). The remaining compact portion has nonpositive contribution for small mesh. Thus (46) would imply \(0\le-c+C\mu^\eta\), a contradiction for small \(\mu\). The dual argument excludes \(H>\kappa\) near a subinterval of \(da\). This is exactly the local comparison in [6]; all its objects lie in one chart.

Pull the limiting boundary-value problem to a conformal disk. For \(0<\kappa<1\) its solution is the imaginary part of a conformal map to a horizontal strip of height one with a slit at height \(\kappa\), sending \(a\) to the lower left end, \(b\) to the right end, and \(c\) to the upper left end. If \(d\) mapped to the lower slit bank rather than the tip, one would have \(H<\kappa\) near a part of \(cd\); the upper bank would give \(H>\kappa\) near a part of \(da\). The preceding contradiction excludes both. It also excludes \(\kappa=0,1\) when the four marks are distinct. Since the polygon boundary is Jordan in the surface, every nonempty boundary interval contains a straight subinterval on which the argument can be applied.

Consequently \(d\) maps to the slit tip. In the normalized half-plane quad \((\mathbb H;0,\chi_P,1,\infty)\) the strip parameter is \(\kappa=1-\chi_P\). Solving (45) for the pairing probability gives (43). This identifies every subsequential limit.

Boundary-layer conventions. If the chosen medial graph differs from a primal polygon approximation by an incident layer, place an easier and a harder orthogonal quad in the collar, with strict margins. The easier quad extends beyond the two free sides and ends inside the two wired sides. Restrict the target and test to their shared grid edges. The target’s unmatched pieces attach only at the two electrodes and cannot join the opposite exterior zones. Their induced partition is therefore dominated by separate wiring of the easier electrodes. Every target crossing contains a shared-edge test crossing. This gives the upper comparison by FK monotonicity. Interchanging the roles with the harder quad gives the lower comparison. These are intrinsic disk statements on the same lifted grid, as in [17]. Let the mesh tend to zero with margins fixed, then shrink the margins. Marked Jordan convergence in the collared disk gives convergence of the two moduli to \(\chi_P\), completing the proof. ◻

Proof. Combine Proposition 9 with Proposition 13, then invert the strictly increasing continuous map \(\mathcal G\) in (42). ◻

Proof. We recall the finite construction of [17]. Make one tile for every edge, including boundary edges. Glue the four triangles directed toward an interior square center. For each boundary edge, keep its exterior triangle as a separate tab with its own exterior dual vertex. Different exterior tabs are not glued along their radial sides; primal boundary vertices retain their interior incidences. Flatten the tabs and add a thinner Jordan collar. The displacement is \(O(\delta_n)\), and tracing the tabs in the original boundary order proves (i).

Draw the two possible switches as disjoint local arcs. Draw boundary caps and the wires behind them in the collar; unmatched ports lead to the marked outer points. There are finitely many local choices, all made deterministically. The switches follow the prescribed medial pairings. Where a wired boundary edge is bypassed by a cap, either switch state gives the same continuation of the inside ports, up to a microscopic detour and possibly a microscopic loop. The edge still retains its random state in the FK law. On the free side, the cap around a primal vertex gives the turn beside the dual boundary wire. Match the ordered pieces of the original and drawn strands. Each matched pair lies in a bounded number of neighboring tiles; the resulting increasing parametrizations prove (ii).

The tile Euler identity gives the FK component weight under each deterministic closure. Revealing an encountered switch therefore does exactly the usual contraction or deletion and leaves the other edge odds unchanged. This proves (iii), including a stop inside a tile if that entire encountered tile is revealed.

On an open primal turn the bank uses the actual revealed edge; on the other turn it stays at a primal corner. Along a cap it stays at a vertex or follows vertices of the same wire interval. These pieces give the asserted short physical accesses. The dual statement is identical. The embedding of every completion follows from the local noncrossing pairings. Projection back to the original grid moves points by \(O(\delta_n)\) and duplicates no edge, although separate boundary copies of a vertex can project to one physical vertex. This proves (iv). None of the construction uses equality of edge odds. ◻

Proof. For each fixed polygon, Corollary 14 gives convergence in environmental probability. For each fixed physical annulus, its inner radius in lattice units tends to infinity, so Lemma 3 gives goodness with probability tending to one. Enumerate the tests and annuli. Choose the \(j\)th mesh of the further subsequence so that the probability of an error larger than \(1/j\) in any of the first \(j\) polygon tests, or failure of goodness for any of the first \(j\) annuli, is at most \(2^{-j}\). The Borel–Cantelli lemma proves the simultaneous assertions. Independence between different meshes is not used. The dual assertions follow in the same way, since the critical odds duality preserves the law of the environment. ◻

Proof. Fix an integer \(m\). Once \(r/R\) is sufficiently small, an arm from radius \(r\) to radius \(R\) must cross \(m\) separated intermediate square annuli of the fixed barrier ratio. The square annuli may have fixed positive gaps between them and the two original rims. Choose their centers from a sufficiently fine finite net of the bounded set \(K\), and their radii from the dense countable collection. The gaps allow these choices with enough slack that the assertion holds for every center \(z\in K\). Only finitely many fixed annuli are involved for this choice of \(r,R,m\).

By Lemma 16, they are all good for sufficiently large \(n\). Condition on all edges outside one intermediate annulus. The induced partition of its boundary is dominated by the partition wiring its two rims together. Its conditional crossing probability is therefore at most \(1-c_0\). Successive conditioning over the separated annuli gives the bound \((1-c_0)^m\). Letting \(m\) tend to infinity proves (47). The finite net depends on \(r\); this causes no problem because \(r\) precedes the mesh limit.

For the last assertions, fill the missing ordinary edges, identify copies of the same physical vertex when necessary, and wire the rims. FK monotonicity for \(q=2\) dominates the original law on all shared edges. Every original actual arm gives a radial path in this completion. The absence of other wire attachments is exactly what ensures domination by this boundary partition. Boundedly many lattice-step changes of the rims are absorbed in the fixed gaps. ◻

Proof. The missing-color argument is the one used in the Macroscopic avoidability Lemma of [17]; we verify its probabilistic input in the present setting. An avoidable component cannot access the relative interiors of both designated prime boundary arcs strictly between the rims. To see this in the finite slit disk, join interior points on the two short access arcs by a path in the open annular component. Removing loops and shortening the accesses gives a crosscut between the two designated arcs, still between buffered rims. In the prime-end disk its endpoints lie on opposite boundary arcs between tip and target, so the crosscut separates those two ends. Coincident physical points on opposite banks are distinct prime accesses in this argument. An incidence at the current tip tile can be moved a bounded number of lattice steps along its bank and is absorbed by the rim buffers. Choose the color whose designated arc is absent. A future strand traversal has a bank of that color, and the bank shadows an actual radial path in a slightly cropped annulus. A known open edge or a wire incidence of that color in the middle would give an \(O(\delta_n)\) access to its designated arc. This is excluded by the choice of component.

To handle all components with this missing color simultaneously, keep the union of their uncertain edges between buffered grid rims and condition on its complement. Apply FK comparison to this joint conditional law on the union, rather than to separate component marginals. Away from those rims, all induced attachments are ordinary physical-vertex incidences: any other attachment would have the forbidden access to the designated arc. Fill to the full grid and wire the rims. Lemma 17 bounds the probability of any of their actual arms by a single full-grid arm probability. Sum over the two possible missing colors. The fixed buffers absorb the error of a tile, including the last local arc at the stop. ◻

Proof. Fix a point away from the limiting marked points. Choose a small outer ball, with a fixed buffer, which meets at most one boundary condition type. Such a ball exists by uniform marked Jordan convergence: away from the changes, the two types lie on disjoint compact boundary arcs. If the ball is interior, choose either color; otherwise choose the color with no designated wire in that ball. For a four-change law, use the deterministic separate closure of this color. Its density relative to the open-cap law, and the reciprocal density, are bounded by constants depending only on \(q\).

Consider \(m\) disjoint radial traversals of an annulus inside this ball, and crop its inner and outer rims by fixed factors. The territorial argument of [17] gives at least \(m/2-O(1)\) distinct actual crossing components of the chosen color in the cropped graph. Here is the needed form of that argument. The simply drawn traversal arcs are disjoint radial separators in a wider annulus. On the indicated side of each separator, a path in the color territory shadows an ordinary edge path: in a tile it moves to the appropriate corner, and passage to another corner uses the actual edge prescribed by the switch. The error is bounded by a fixed number of lattice steps. Keep short territorial tails out to the wider rims before cropping. Paths assigned to different intervening strips cannot meet in the cropped graph without crossing a separator. At most two assignments belong to a single strip. No cap joins different vertices of this color in the buffered ball, because it contains no wire of that color. Tabs and any separate boundary copies are retained while making this planar argument.

Condition on the graph outside the crop, allowing arbitrary rim partitions. Explore the actual components meeting its inner rim one at a time; component here means connected by ordinary open edges, ignoring artificial rim identifications. After a crossing component has been completely explored and removed, all edges from it to the remaining interior have been revealed closed. Thus its removal creates no remote interior wiring. The only arbitrary identifications left in the conditional law are still at the rim bands. Deleted closed edges weaken the full-grid comparison, and identification of copies of the same physical vertex in that comparison can only strengthen it.

Choose the fixed inner radius sufficiently small that the full-grid arm probability in a slightly narrower crop is at most \(1/2\) for all sufficiently large \(n\), by Lemma 17. At every completed component exploration the conditional chance that any crossing component remains is therefore at most \(1/2\). Equivalently, after exploring noncrossing components until the next crossing is found, the same bound applies again. If \(K_n\) is the number of actual crossing components, induction gives \[\mathbb P(K_n\ge k)\le 2^{-k},\qquad k\ge1,\] with an immaterial adjustment of the first factor. This proves a geometric tail for the number of strand traversals at these fixed scales. The argument uses whole-component exploration, not independence of distinct components.

Now fix a displacement accuracy \(\rho\). Surround the marked points by disjoint neighborhoods of diameter much smaller than \(\rho\). For every other possible curve point choose an inner ball and an outer ball as above, with outer diameter much smaller than \(\rho\). The inner balls cover the remaining compact set, and a finite subcover suffices. Every displacement of size \(\rho\) has a subpassage outside the mark neighborhoods which enters one of these inner balls and leaves its outer ball. Consequently an excessive number of displacements forces excessive traversals of at least one annulus in this finite family. The geometric tails just proved bound \(N_\rho\) in probability.

Apply this bound at a sequence of accuracies decreasing to zero, with summable failure probabilities. Subdivision at the successive displacement times gives the usual Arzelà–Ascoli compactness criterion for curves modulo increasing reparametrization. Finitely many meshes discarded by the limsup can be accommodated separately. Prefixes have no more displacements than their full curves. Uniform continuity of the initial charts transfers the conclusion to the common disk. ◻

Proof. We first describe the deterministic obstruction to a good approximation, and then bound its probability using actual stopping rules. On the indicated range, the half-plane hull lies in a fixed bounded region: the standard Loewner height and horizontal-range bounds use only \(T\) and \(R\). Thus the disk prefix stays outside a fixed cap at \(1\). The inverse map is close to the identity near infinity, so the crosscuts used below also stay outside a possibly smaller fixed target cap.

For each \(t\) and sufficiently small \(y\), the length-area estimate for \(\Psi^{-1}\circ g_{n,t}^{-1}\) supplies a radius \(y'\in[y,\sqrt y]\) whose upper semicircle about \(W_n(t)\) has an image crosscut \(C\) of length at most \(c/\sqrt{|\log y|}\). Indeed, integration of the squared image lengths with logarithmic measure is bounded by a constant times the area of \(\mathbb D\); see also [19]. The crosscut contains \(w=H_n(t,y')\) and separates the tip access from the target. The hyperbolic geodesic from \(w\) to that tip is the image of the vertical segment from \(W_n(t)+iy'\) to \(W_n(t)\) and contains \(H_n(t,y)\). By the diameter form of the Gehring–Hayman theorem [19], used in the same way in [14], if \(w\) can be joined to the tip in the slit disk by a path of diameter \(a\), this geodesic has diameter at most a universal constant times \(a\). The assertion extends to prime-end endpoints by taking limits.

Fix a small crosscut cutoff \(r\). Decrease \(y\) until the above crosscuts have diameter less than \(r\). A small perturbation of their endpoints, retaining \(w\), gives regular endpoints with distinct physical positions and diameter less than \(2r\). The prime boundary maps are continuous for each finite simple drawing; we may avoid the slit root and tip in this perturbation. If both endpoints lie on the original disk circle, the small circle cap cut off by \(C\) has diameter \(O(r)\) and avoids the target. Since \(C\) separates tip from target, the tip is in this cap. It can be joined to \(w\) there; an existing slit disjoint from the crosscut does not change this separation.

Otherwise let \(s\) be the latest prefix time at an endpoint of \(C\). If \(\eta_n[s,t]\) stays in a small ball about \(\eta_n(s)\), following \(C\) to that endpoint and then the prime bank to the tip gives a small-diameter path from \(w\) to the tip. Hence failure of the claimed tip accuracy implies a fixed \(b>0\), depending on \(\varepsilon\) but independent of \(r,y,n\), such that \[ \eta_n[s,t]\not\subset B(\eta_n(s),b). \tag{50}\] At time \(s\), the crosscut joins the tip either to an earlier slit point or to the original circle. In the first case it and the intervening prefix form a Jordan loop. In the second case use the prefix from its beginning and the circle arc avoiding the target to close the loop. The portion of the disk cut off by this loop is on the non-target side. The tail relevant to (50) lies on that side: it is disjoint from \(C\), and its final tip access is separated from target by \(C\). This is the small gate whose probability we must control; Figure 2 illustrates the case of two slit endpoints.

Subdivide the curve successively into pieces of diameter less than \(b/100\), by ending a piece when it attains a chosen still smaller fixed diameter. Subdivide each of the two original circle arcs into pieces with the same diameter bound. The number of curve pieces is stochastically bounded by Proposition 19. Every gate just constructed has its two endpoints in a pair \((J,K)\), where \(K\) is a later curve piece and \(J\) is an earlier curve piece or a circle piece. They cannot belong to the same curve piece: that would make the loop too small to contain the excursion in (50).

For each ordered pair use the following revealing stopping rule during piece \(K\): stop the first time in a finite deterministic schedule that there exists a crosscut of diameter at most \(2r\) from the current tip to an earlier point of \(J\). Only non-root slit endpoints and circle-interior endpoints are admitted. Such a schedule can always detect the witness above. At fixed mesh there are finitely many complete switch configurations. For each configuration exhibiting a witness choose one witness time in an exploration-compatible clock, and take the union of those finitely many times. Although the selections used the full configurations, their union is a deterministic schedule. The detection test at each scheduled time depends only on the current prefix. It is therefore a stopping rule, and it occurs no later than the witness time for the pair in question. If a time is inside a local piece, only its already encountered switch is revealed. A witnessing crosscut at detection may also be selected deterministically from the current history.

Let \(C_1\) be this first detected crosscut for the witness pair. We claim that its trapped region and that of \(C\) agree outside \(B(\eta_n(s),b/8)\), after taking \(r\) sufficiently small. To prove this, compare the two closed loops modulo two. Their long common prefix segments cancel. The remaining prefix subarcs lie in \(J\) and \(K\), and the other remaining arcs are \(C\) and \(C_1\); in the circle case there is also a subarc of the one circle piece \(J\). The two pieces have small diameter and are joined by a small crosscut. Thus this entire symmetric difference is confined to the displayed ball. The winding numbers modulo two of the loops coincide outside it. Since each loop is Jordan, its winding parity is exactly the indicator of its trapped region there. This proves the claim without any bound on the length or shape of the intervening common prefix.

The far excursion in (50) consequently visits the region cut off by \(C_1\). At detection, its tip endpoint has access to the target side directly across \(C_1\) within a ball of radius \(O(r)\). An excursion reaching the far trapped region either starts into that region at the detecting tip, or enters it later through \(C_1\), since the other sides of its boundary belong to the past or the original boundary. Both entrances occur inside the inner ball. The excursion therefore produces a radial crossing, on the trapped side, from radius \(4r\) to a fixed radius, say \(b/3\), about the detection tip. Outside the inner ball, the trapped region is bounded only by the past and the original boundary. Its annular components therefore cannot connect to the target-side region without entering the inner ball. A route from tip to target is available on the other side of the gate and avoids all these components, so they are avoidable. The argument allows repeated future crossings of the fictitious crosscut; every such entrance still occurs inside the inner rim.

By the disk version of Lemma 18, the probability of this event for each ordered pair tends to zero as \(r\downarrow0\) after the mesh limit. To finish, first use Proposition 19 to bound the number of pieces with arbitrarily high probability. The resulting number of pairs is finite and independent of \(r\). Next choose \(r\) to make the sum of their crossing bounds small, and finally choose \(y\) small enough for the crosscut construction. This proves (49) with the asserted order of limits. ◻

Proof. At first entrance, the cap has not previously been visited. It gives a route from the tip access to the target inside a ball of radius \(O(r)\). Stop at the entering local piece and allow a one-step buffer. A subsequent displacement of diameter \(b\) forces a crossing from an enlarged inner rim to a fixed positive distance, and every relevant annular component is avoidable because of that tip-to-target route. Apply Lemma 18 in disk coordinates. This is also the proof of the corresponding target-cap Corollary in [17]. Initial inverse-chart continuity gives the analogous assertion in physical coordinates. ◻

Proof. Choose a disk \(O\) containing all physical domains with a fixed margin. For a small ball away from \(B(z_n,r_0/2)\), the probability that Brownian motion from \(z_n\) hits that ball before leaving \(O\) tends uniformly to zero with its radius. For example, the logarithmic harmonic function in an enclosing annulus gives a bound of order \(1/|\log r|\). Killing earlier on leaving \(U_n\) only decreases this probability.

In the normalized unit disk, a connected set of diameter at least \(a>0\) has hitting probability from the origin bounded below by a positive number depending only on \(a\). This is the crosscut form of the Beurling estimate: a connected set joining two separated small sectors is either met before exit or separates a boundary interval of uniformly positive harmonic measure from the origin. The estimate also applies to connected sets close to the circle; the set itself, rather than its boundary projection, lies in the disk. Conformal invariance therefore implies that a small physical path away from the basepoint has small diameter in the disk chart. Near the basepoint the same conclusion follows from interior distortion on the fixed base disk. The maps from the disk to \(R_n\) form a compact family on the closed disk because the four angles remain separated. This proves the asserted modulus for \(u_n\).

Every electrode attachment and every known open bank edge has an access to \(u_n=0\) or \(u_n=1\) within \(O(\delta_n)\), by Lemma 15. A short possible connection from \(B_n(\beta)\) to such an attachment would, after adding this access, have chart oscillation at least \(\beta\) and physical diameter tending to zero. Choose the physical distance so that its modulus is smaller than \(\beta/2\), and then absorb the tile error by taking \(n\) large. This also shows that the two electrodes have not already met along the revealed bank under the stated localization. ◻

Proof. Couple the original law and the law with the edges in \(S\) closed monotonically. Explore, in the stronger configuration, the actual clusters from the deletion endpoints, revealing their incident edges in both configurations. FK partition monotonicity remains valid at each step. Once these clusters are enclosed by their closed edge cuts, the conditional laws outside agree and may be coupled identically. If no explored cluster reaches an electrode, the crossing decision is the same. Known connections along a bank count as electrode access in this argument.

For the second assertion choose an outer radius \(R_\beta>0\) using Lemma 23. Every component of possible edges in a ball of radius \(R_\beta\) that meets the deleted central incidences has no accessible wire or known open attachment there. All its remaining boundary identifications arise at the rim of the ball. Project this graph to the physical grid, fill missing edges, identify any copies of a physical vertex, and wire the rim. These operations increase the induced FK partition and dominate its actual arm event. Cropping the annulus by a few tile widths accommodates the drawing and the possible initial test step. Lemma 17 bounds the resulting probability of an arm from radius \(r\) to \(R_\beta\) by a quantity tending to zero in the asserted order. A union bound over the fixed \(m\) balls completes the proof. ◻

Proof. Let \(I\) be one bin and let \(J\) be a short interval adjacent to the top or bottom of \(R_n\). By Cauchy–Schwarz, \[ \int_I \operatorname{length} \bigl(F_n(\{x\}\times J)\bigr)\,dx \leq (|I||J|)^{1/2} \left(\int_{I\times J}|F_n'(w)|^2\,dA(w)\right)^{1/2}. \tag{54}\] The last integral is at most the area of the common physical enclosing disk. By making the height fraction sufficiently small, the set of levels with tail length exceeding \(r\) occupies an arbitrarily small fraction of \(I\).

We can simultaneously separate the endpoints. On the compact portions of the horizontal rectangle sides containing the bins, harmonic measure from the basepoint is uniformly comparable to abscissa length. An endpoint in a small physical ball contributes at most the Brownian probability of reaching that ball before leaving the enclosing disk. This tends uniformly to zero as the ball shrinks, by the logarithmic estimate used in Lemma 23. Choose the endpoints successively, excluding fixed small balls around the endpoints already chosen. There are only finitely many choices, and these exclusions together with the length exclusions occupy less than the bin. A positive separation can therefore be retained before taking a subsequence.

Choose the top and bottom levels independently, then join them by an interpolation inside the middle of the same bin. Interior convergence of \(F_n\) gives convergence of the retained cores after a further subsequence. Different bins are disjoint, so the resulting fibers are disjoint and ordered. Endpoints can be chosen away from the finitely many tile corners and exceptional prime accesses. ◻

Proof. First consider increasingly fine finite samples of \(\partial U\). The normalized projections from the unit disk to the punctured domains form a bounded normal family. Their derivative at \(0\) is bounded below by the radius of the common base disk: that disk lifts to the cover, and Schwarz’s lemma gives the derivative comparison. Any subsequential limit is therefore nonconstant. Hurwitz’s theorem shows that it omits each point of the dense boundary sample. Its image is consequently contained in \(U\), the component at \(z\).

Conversely, every simply connected domain compactly contained in \(U\) and containing \(z\) lifts into all the covers under consideration once \(n\) is large. The same derivative comparison bounds the projection derivative below by the conformal radius of that domain. Exhausting \(U\) and then applying Schwarz’s lemma to the limit map into \(U\) shows equality with the conformal radius of \(U\). Thus the limit is precisely the normalized Riemann map onto \(U\). This proves the asserted approximation on compact sets and along their access paths. The order is to fix the desired compact accuracy, choose a finite puncture list, and only then let \(n\) tend to infinity.

For a fixed list with distinct limiting punctures, the charts of the covers converge under continuation in physical coordinates. One direct verification is to move the finitely many punctures by diffeomorphisms supported in disjoint small neighborhoods, fixing the base disk and the outer boundary, with dilatations tending to one. Their lifts, in normalized disk coordinates, converge to the identity. On every compact set away from the punctures this identifies the limiting cover charts and preserves continuation along any fixed finite collection of paths.

Consider now a fiber tail. Its projection lies in a ball of radius \(r\) away from the basepoint. Brownian motion on the cover, projected to \(O\), gives the same upper bound for hitting this tail as for hitting its physical ball before exit from \(O\). The connected-set estimate in the unit disk used in Lemma 23 thus makes its diameter small in the cover’s disk coordinate. The disk-to-rectangle maps form a compact family on the closure, so its rectangle diameter is small as well. Apply this observation to successively shorter tails approaching the endpoint. The lift has a continuous endpoint in the closed uniformizing disk. That endpoint cannot be interior: its projection tends to a puncture. It is therefore an ideal boundary point of the cover.

On the retained core, both charts are close to the limiting chart of \(U\). Taking the omitted height fractions small locates the top and bottom endpoints near the corresponding horizontal sides, away from their corners. The lifted fiber is consequently a crosscut joining those sides. Its entire \(v_n\)-range lies in its bin with a small enlargement. These crosscuts are disjoint because \(\iota_n\) is injective, and their order is the order seen on the common compact cores.

Each crosscut separates the whole cover disk into two components. The two components of \(U_n\) cut by its original fiber lift to the corresponding sides: each is connected, and the side is determined on an interior core. A point between consecutive fibers is therefore trapped between their lifted crosscuts. Both its \(u_n\) and \(v_n\) coordinates lie between the adjacent bins, up to the chosen errors. This proves (55). The same separation argument outside the two extreme fibers gives the one-sided bounds. In particular, the conclusion is global in the lifted domain; compact convergence alone would not give it. ◻

Proof. Start from a central candidate and follow a possible connection toward an electrode until its first exit from \(B_n(\beta)\). Lemma 23 excludes a wire jump before this exit. The path crosses the anchor fiber on that side. Up to this crossing it lies in the band where the puncture-ball edges were deleted, so it stays away from the punctures. Its crossing of the anchor cannot be in a deleted tail. Following the retained anchor core and then one of its fixed compact access paths joins the candidate to the basepoint while still avoiding the smaller puncture balls. Physical accesses at vertices can be shifted slightly into \(U_n\), retaining this clearance; tile errors are smaller than every fixed margin.

We show that all points accessible in this way have bounded sheet indices. Trim \(U_n\) by mutually disjoint disks centered at \(P_n\), with slightly smaller positive radii than the second deletion radii, and let \(T_n\) be the component containing \(z_n\). The preceding paths put all the central candidates in \(T_n\), or in its boundary accesses. The trimming circles can be chosen in general position, so that they meet the finite polygonal slit drawing transversely in finitely many points.

The prime boundary of \(T_n\) consists of inherited portions of the prime boundary of \(U_n\), in their original cyclic order, interleaved with arcs of the trimming circles. To verify this description, note that each puncture disk reaches the complement of \(U_n\). A whole circle surrounding its center cannot lie in the simply connected domain \(U_n\). Each portion of a trimming circle inside \(U_n\) is therefore a crosscut in the prime-end disk: its two physical endpoints may coincide on opposite slit banks, but its endpoint accesses are distinct. The crosscuts from the disjoint trimming circles are disjoint, and successive crosscut separation gives the asserted boundary description of each remaining component. The two banks of the slit are counted separately in this description.

The inserted circle arcs have uniformly bounded total length: on each circle their interiors are disjoint. The inherited boundary has a uniform bound on the number of successive oscillations of every fixed positive size. Indeed, it consists of the original boundary traversal and two traversals of the prefix; the original boundaries converge uniformly and the prefixes belong to a compact family of curves. The same oscillation bound holds for the trimmed boundary. More explicitly, subdivide the original parametrization into a bounded number of pieces of diameter less than a prescribed small fraction of the oscillation scale. Inherited arcs occur in that order. Inserted arcs whose endpoints lie in one such piece can make a substantial additional excursion only if their length exceeds a fixed fraction of the scale. Their number is bounded by the total circle length. Arcs crossing subdivision points account for only a bounded number of further changes.

It remains to turn this boundary control into control of an access path. Join a point of \(T_n\) and \(z_n\) by straight segments to their first boundary hits, and connect the two resulting prime accesses along the trimmed boundary. The straight segments have bounded length. The intervening boundary path has the oscillation bound just established. Polygonal prime-boundary paths admit arbitrarily close interior approximations with those accesses, including on the two sides of a slit. Thus one obtains an interior path from \(z_n\) to the point with uniformly bounded oscillation count at a fixed scale smaller than its clearance from every puncture. An additional arbitrarily short access segment handles a boundary incidence.

Cover the common compact physical region away from the punctures by finitely many ordinary chart disks, smaller than this clearance. Subdivide the access path whenever its diameter reaches a still smaller fixed scale. There are boundedly many pieces, and each lies in one of these ordinary disks. Its lift is obtained by a bounded number of continuations among a finite list of charts, so it lies in a compact part of the universal cover. The same compact chart bound works for the varying punctures by Lemma 26. Only one controlled access path is needed for each point: any other path in \(U_n\) reaches the same lift because \(\iota_n\) is single-valued. This proves the lemma. ◻

Proof. A compact collared polygon away from the punctures is covered by finitely many ordinary projection charts. Choose rational rectangles inside these charts, with overlaps large enough for the required continuations. Their gluings have identity projection transitions. Subdividing further, the polygon and a collar can be specified by finitely many rational rectangles or squares, their sheet labels, and the incidence identifications on overlaps. There are countably many finite descriptions of this kind, including when projections of different sheets overlap.

These finite chart continuations stay a positive distance from the limiting punctures. As the punctures move, the same projected paths and neighborhoods avoid them, and their lift identifications agree for all large \(n\), by the finite-list continuity in Lemma 26. Thus both the physical polygon and its square-grid rounding have fixed chart and incidence descriptions. Copies of the same physical edge have the same disorder, exactly as in Proposition 9; they are distinct FK edges when they lie on different sheets. The simultaneous fixed-test limit on the chosen environment, followed by Proposition 13, gives the assertion. ◻

Proof. Take the central range for shared edges inside \(B_n(5\beta)\), allowing fixed smaller margins at its boundary. Since \(\beta<a_*/20\), this range extends well past both vertical test cuts. The chart errors in (55) and the polygonization error are chosen much smaller than these margins.

There is no attachment from a shared edge to a nonshared original piece through a horizontal test side. Indeed, every possible central attachment belongs to the compact candidate set, which lies on the inner side of those walls with positive clearance. A noncentral piece cannot enter there either: the global strip trapping and its one-sided bounds put such pieces beyond the corresponding vertical cut. Consequently a nonshared original piece can attach to the shared graph only along one of the two test electrodes.

Nor can a nonshared piece connect the two electrodes to each other. Between small neighborhoods of the vertical cuts there is a middle strip in which every candidate edge is shared. A possible path from the left exterior group to the right must traverse that strip, by (55). All identifications from known banks and original wires belong to the separate left and right groups: Lemma 23 excludes a central access to either wire and excludes an already known connection between them.

These statements remain valid when the drawing is matched to physical lattice vertices. Each uncertain switch supplies only one random edge, by Lemma 15. For a primal-open switch, the primal bank follows the revealed ordinary edge, attaching both endpoint vertex regions to that bank. For a primal-closed switch, the encountered turn cuts off only its visited primal corner, where the bank stays at the same vertex. The opposite, unvisited corner retains a connected small neighborhood of its incidences. If that other turn is visited later, the same rule then attaches its corner to the bank. Thus every vertex whose required identifications have been separated by the explored strand is already bank-attached and is excluded from the central band by Lemma 23.

The boundary construction retains all interior incidences of a primal boundary vertex. Exterior dual tabs are separate vertices of the drawn graph. Apart from the prescribed electrode-wire identifications already handled above, distinct tabs require no identification with one another, and the comparison does not merge their different lifts. If two such copies occupy the same ordinary vertex in one lifted test chart, their merger in the test only strengthens the local partition. Interchanging the two colors gives the same switch argument for a dual test. Consequently all identifications required among the remaining unbanked incidences have representatives in a common small vertex neighborhood. Away from the puncture balls these representatives have a common local lift, identical to that of the ordinary grid incidences. No additional central identification can therefore join different sheets or connect the two exterior groups around the shared strip.

Condition now on all nonshared original edges. Their induced partition on the shared graph identifies vertices only within one test electrode or within the other. It is therefore dominated by the partition that wires each test electrode separately. The weights on every shared edge are exactly those in \(Q_n\). Adding the remaining test edges and applying FK monotonicity gives a stochastic upper comparison on the shared graph. Finally, every actual original crossing contains a shared-edge crossing between the two test electrodes: take its traversal from the left exterior group to the right, and use the absence of any nonshared bypass. This event inclusion and the conditional stochastic comparison prove (56) after averaging the conditioned edges. The proof is unchanged for the dual grid and its actual odds. ◻

Proof of Proposition 22. Suppose that (51) fails with a fixed positive error and a fixed positive probability along a subsequence. Tightness of the prefixes lets us restrict to a compact family of curves while losing less than this probability. In particular, their numbers of successive displacements of every fixed size have deterministic bounds on that family. Select a deterministic failing history at each remaining mesh. The initial boundaries already converge, so we may take the convergent-prefix, basepoint, angle, and kernel subsequences used above.

It is enough to contradict failure of an upper crossing bound in one of the two separate closures. Indeed, if \(p_n\) is the conditional open-cap pairing probability, those crossing probabilities are exactly \(T(p_n)\) and \(T(1-p_n)\), respectively, by Lemma 12. The continuous increasing map \(T\) has a continuous inverse on \([0,1]\). The two upper bounds by \(g(\chi_{U_n})=T(f(\chi_{U_n}))\) and \(g(1-\chi_{U_n})=T(1-f(\chi_{U_n}))\) therefore force \(p_n-f(\chi_{U_n})\to0\).

Fix an error tolerance. Choose \(a_*\) small, then \(\beta<a_*/20\) and the fiber-bin precision \(\lambda\) much smaller than \(\beta\). Choose the tail radius so that the four anchor-tail deletions have small total cost and the lifted tails have sufficiently small chart diameter. Lemma 25 gives a fixed finite number of fibers and their limiting endpoints. Choose a fixed finite puncture sample fine enough for Lemma 26. Only after that choice, take the puncture-ball radii small enough for both the required core clearances and a small total deletion cost. Lemma 27 then gives one compact part of the cover containing the surviving central candidates. Choose the horizontal inset of the test smaller than the clearance of this compact set, and finally choose the rational polygon with strictly smaller approximation errors.

All these data are fixed before taking the mesh limit. The deletion bound and Lemmas 28 and 29 give \[\limsup_{n\to\infty} \left[ \phi_n^{\mathrm{sep}}(\text{original crossing}\mid\mathcal F_n) -g(\chi_{U_n}) \right] \leq \text{deletion error}+\text{modulus error}.\] The modulus error tends to zero as the vertical trim \(a_*\), the horizontal inset, and the polygonization error tend to zero. The cover rectangle was given the same four-angle data as \(U_n\), and all rectangle heights remain in a compact nondegenerate range, so continuity of the pure four-mark function justifies this comparison uniformly in the limiting height. The deletions have the independently small costs proved above. The right side can therefore be made arbitrarily small. Applying the same argument to the dual proves the two upper bounds and contradicts the selected failure.

Although the geometry was selected after fixing an environment and a failing history subsequence, the polygon used at the last step is one fixed member of the previously simultaneous countable stock. No estimate is applied to a boundary that continues to change with the mesh, and no uniform rate over all polygons is needed. This completes the proof for arbitrary stopping rules satisfying the stated localization and tightness assumptions. ◻

Proof. Fix \(s>0\), initially without taking a limit in \(s\). Insert a separate primal wired interval on the formerly free side, with lattice endpoints whose initial half-plane coordinates tend to \(x,x+s\). In the open-cap four-change law, explore the strand starting at the original initial mark \(a_n\), whose half-plane coordinate is \(0\). By Lemma 15, the domain, initial charts, and drawings may be chosen the same away from this insertion. Regard the Dobrushin stopping rules and \(F_n\) as functions of revealed histories. Apply the same rules on the common tree of prefix histories, extend them as stopping rules outside that tree by stopping at its first exit, and extend \(F_n\) boundedly there. These conventions retain the order of the two stops and their localization; below we show that the relevant stopped prefixes stay in the common tree. Thus the same symbols will denote well-defined stopping times and history tests under open caps.

We check the localization required by Proposition 22. Real Loewner trajectories from \(x\) and \(x+s\) remain separated from the driver, and their gap solves \[ \partial_t\bigl(g_{n,t}(x+s)-g_{n,t}(x)\bigr) =-\frac{2\bigl(g_{n,t}(x+s)-g_{n,t}(x)\bigr)} {(g_{n,t}(x+s)-W_n(t))(g_{n,t}(x)-W_n(t))}. \tag{60}\] For fixed \(s\), this gap is bounded below by a positive constant. The same ODE is regular in a complex neighborhood of the interval, uniformly on the stopped range. The prefix therefore stays out of a fixed neighborhood of the insertion, and cannot terminate at either inserted mark before the stop. A point initially corresponding to \(iY\), with \(Y\) sufficiently large, has a fixed disk untouched by all these hulls. It supplies the basepoint of Proposition 22. The four marks, including tip and infinity, are uniformly separated in the normalized charts for each fixed \(s\). Initial marked Jordan convergence transfers these statements to the physical drawings. The history tightness hypothesis follows from Proposition 19 for the four fixed marks.

Let \(Z_{n,t}^{s}\) be the conditional probability of the rare primal pairing under the open-cap law. It is an exact bounded martingale in the revealed-switch filtration. At each of the allowed stopping times, Proposition 22 gives \[ Z_{n,t}^{s}-f(\chi_{n,t}^{s})\longrightarrow0 \quad\hbox{in probability and in mean},\qquad \chi_{n,t}^{s}= \frac{g_{n,t}(x+s)-g_{n,t}(x)} {g_{n,t}(x+s)-W_n(t)}. \tag{61}\] The vanishing initial errors in the lattice mark positions are included here; they do not affect the fixed-\(s\) limit. Optional sampling and the boundedness of \(F_n\) thus give the approximate martingale identity for \(f(\chi_{n,t}^{s})\).

Divide this identity by \(f(s/(x+s))\) while \(s\) is fixed. The small-interval behavior in Proposition 13, together with analytic ODE bounds near \(x\), gives \[ \frac{f(\chi_{n,t}^{s})}{f(s/(x+s))} =M_{n,t}(x)+o_s(1)+o_n(1). \tag{62}\] Here \(o_n(1)\) is taken first for fixed \(s\), and \(o_s(1)\) is uniform over the stopped paths. More explicitly, \[\frac{\chi_{n,t}^{s}}{s/(x+s)} =\frac{xg'_{n,t}(x)}{g_{n,t}(x)-W_n(t)}+O(s),\] with the ratio bounded above and away from zero; applying \(f(u)\sim c\sqrt u\) proves the displayed replacement. By (58), the expression after replacement is bounded independently of small \(s\).

We can now compare laws using only bounded functions of the stopped histories. First compare the deterministic separate primal closure with the Dobrushin law. The former adds just the short wired interval, so FK monotonicity gives a coupling. Explore its actual open clusters in the stronger law. Unless one reaches a fixed outer neighborhood of the insertion, the configurations outside that neighborhood can be coupled identically: the explored clusters are enclosed by closed cuts, and the conditional outside laws then agree. The insertion has physical diameter tending to zero as \(s\downarrow0\), by the uniform continuity of the initial inverse maps. The old primal wire stays a positive distance away. The intervening annulus therefore has no primal wire attachments except at its rims, and Lemma 17 shows that the probability of reaching the outer neighborhood tends to zero, first in the mesh limit and then as \(s\downarrow0\).

Second, compare this deterministic closure to open caps. Lemma 12 gives a likelihood depending only on the pairing, with two fixed positive values. If the exceptional pairing has probability \(p\) under the separate-closure law, normalizing these two weights shows that the total variation difference is at most \(C p\). By Proposition 22 at the empty prefix and Lemma 12, for fixed \(s\) that separate-closure pairing probability tends to \(g(s/(x+s))\), which tends to zero as \(s\downarrow0\).

The stopped prefixes avoid the spare neighborhood of the insertion established above. On the successful coupling event their drawings, revealed histories, Loewner maps, and stopping rules therefore agree. The extension convention also makes the stopped tests agree on these common histories. Combining the two comparisons with (62) yields \[\limsup_{n\to\infty} \left|\mathbb E_n\!\left[ F_n(M_{n,\upsilon_n}(x)-M_{n,\sigma_n}(x))\right]\right| \le e(s),\qquad \lim_{s\downarrow0}e(s)=0.\] This proves (59). Notice that the coupling errors were applied after replacement by bounded tests; they were never divided by the small number \(f(s/(x+s))\). ◻

Proof. We first prove tightness under the fixed localization \(T,R\). Set \(h=1/2\) and, for \(x>2R+1\), write \[X_{n,t}(x)=g_{n,t}(x),\qquad A_{n,t}(x)=x^h g'_{n,t}(x)^h,\qquad M_{n,t}(x)=A_{n,t}(x)(X_{n,t}(x)-W_n(t))^{-h}.\] The real-flow bounds show that \(X_n,A_n\) are uniformly bounded and Lipschitz in time, with \(A_n\) bounded away from zero. Choose \(2R+1<x_1<x_2\). Equation (60) also gives a positive lower bound on \(X_{n,t}(x_2)-X_{n,t}(x_1)\) throughout the stopped range.

At a revealing stopping time \(\sigma_n\), freeze these two quantities and define, for \(-R\le w\le R\), \[F_j(w)=A_{n,\sigma_n}(x_j) (X_{n,\sigma_n}(x_j)-w)^{-h},\qquad j=1,2.\] Both derivatives are positive. Their ratio is \[r(w):=\frac{F'_1(w)}{F'_2(w)} =\frac{A_{n,\sigma_n}(x_1)}{A_{n,\sigma_n}(x_2)} \left(\frac{X_{n,\sigma_n}(x_2)-w} {X_{n,\sigma_n}(x_1)-w}\right)^{h+1}.\] The separation of the two real solutions and their uniform bounds imply \(r'(w)\ge c>0\). Also \(F'_2(w)\) is bounded above and away from zero. Put \(a_n=r(W_n(\sigma_n))\). Integrating \(F'_2(w)(r(w)-a_n)\) on either side of \(W_n(\sigma_n)\) gives \[ F_1(w)-F_1(W_n(\sigma_n)) -a_n\bigl(F_2(w)-F_2(W_n(\sigma_n))\bigr) \ge c_1|w-W_n(\sigma_n)|^2. \tag{63}\] The predictable coefficient \(a_n\) is uniformly bounded.

Between \(\sigma_n\) and another allowed stop \(\upsilon_n\), changing the frozen \(X_n,A_n\) costs at most \(C(\upsilon_n-\sigma_n)\). Apply Lemma 30 to the two increments, using the bounded \(\mathcal F_{n,\sigma_n}\)-measurable coefficient \(a_n\) as a test for the second. Equation (63) then yields \[ \mathbb E_n|W_n(\upsilon_n)-W_n(\sigma_n)|^2 \le C\mathbb E_n(\upsilon_n-\sigma_n)+o(1). \tag{64}\] This holds for every sequence of allowed stopping times. In particular, for the driver stopped at \(\tau_n\), it implies the stopping-time tightness criterion of Aldous [1] in the Skorokhod \(J_1\) topology. The drivers are bounded on this range. Every prelimit driver is continuous, and the continuous paths form a closed subspace for \(J_1\) convergence. Thus all limits are continuous, and convergence along a subsequence may be realized uniformly on compact stopped intervals.

We identify any such limit inside an inner driver bound, choosing inner levels which are continuity levels for the stopping operation. These levels can be increased to the outer bound. Continuity of the Loewner ODE under uniform driver convergence gives limits \(X_t(x),A_t(x)\) and \[ M_t(x)=A_t(x)(X_t(x)-W_t)^{-h}. \tag{65}\] Use bounded continuous functions of finitely many earlier driver values as tests in (59), and then a monotone-class argument. It follows that each \(M(x)\) is a martingale in the limiting filtration. This argument applies at the localized stops as well. The bounds above and below persist.

Before applying Itô’s formula, note that \[W_t=X_t(x)-\left(\frac{A_t(x)}{M_t(x)}\right)^{1/h}.\] The processes \(X,A\) are continuous of finite variation, \(M\) is a continuous martingale, and the function in this identity is smooth on the range of their positive bounds. Hence \(W\) is a continuous semimartingale. Write its decomposition as a continuous local martingale plus a continuous finite-variation process \(B^{\rm fv}\), and denote its quadratic variation by \([W]\).

Put \(Z_t(x)=X_t(x)-W_t\). The Loewner equations give \[dX_t(x)=2Z_t(x)^{-1}\,dt,\qquad d\log A_t(x)=-2hZ_t(x)^{-2}\,dt.\] The finite-variation part of \(dM_t(x)/M_t(x)\) is consequently \[hZ_t(x)^{-1}\,dB_t^{\rm fv} -4hZ_t(x)^{-2}\,dt +\frac{h(h+1)}{2}Z_t(x)^{-2}\,d[W]_t.\] It vanishes because \(M(x)\) is a martingale. Multiplication by \(Z_t(x)^2/h\) gives an identity of signed measures, \[ Z_t(x)\,dB_t^{\rm fv}-4\,dt +\frac{1+h}{2}\,d[W]_t=0. \tag{66}\] Subtract this identity for \(x_1\) and \(x_2\). Their strictly positive separation forces \(dB^{\rm fv}=0\). Substitution back into (66) yields \[d[W]_t=\frac{8}{1+h}\,dt=\frac{16}{3}\,dt.\] Lévy’s characterization therefore identifies the stopped limit as Brownian motion with variance rate \(16/3\). If the initial half-plane picture was reflected, reflecting back negates the driver and leaves this Brownian law unchanged.

It remains to remove the outer driver bound. For any fixed capacity horizon, the limiting upper bound on the probability that \(W_n\) exits \([-R,R]\) is bounded by the Brownian probability of reaching inner levels tending to \(R\). This follows by applying the stopped convergence at continuity levels below \(R\); exiting the outer interval entails first reaching each such inner level. These Brownian probabilities tend to zero as \(R\to\infty\). Thus the full drivers are tight on every compact capacity interval, and each subsequential law has the Brownian identification just proved. This proves the proposition. ◻

Proof of Theorem 1. Fix the environment and subsequence of Lemma 16. By Proposition 31, the drivers converge uniformly on compact time intervals in law to \(W=\sqrt{16/3}\,B\). For every fixed height \(y>0\), continuity of the inverse Loewner equation gives convergence of \[H_n(t,y)=\Psi^{-1}\!\left(g_{n,t}^{-1}(W_n(t)+iy)\right)\] uniformly in \(t\) on each compact capacity interval to the analogous \(H(t,y)\) for \(W\). A large driver bound has arbitrarily high probability on that interval, so Proposition 20 applies after localization.

For completeness, the uniform conclusion at height zero follows without assuming uniform boundary convergence of the inverse maps. For two small positive heights, the triangle inequality and (49) show that \(H_n(\cdot,y)\) and \(H_n(\cdot,y')\) are close in the uniform metric, in probability, after the mesh limit. Passing at these fixed heights to the limiting driver proves that \(H(\cdot,y)\) is Cauchy in probability in the space of continuous disk-valued functions on the time interval as \(y\downarrow0\). Its limit is continuous. The SLE trace theorem [20] identifies its pointwise values with the SLE trace, first at rational times and then at all times by continuity. The same approximation argument now gives \[ \eta_n\ \Longrightarrow\ \eta \quad\hbox{uniformly on every compact capacity interval}, \tag{67}\] where \(\eta\) is chordal \(\mathrm{SLE}_{16/3}\) in \(\mathbb D\) from \(-1\) to \(1\). In particular, a macroscopic excursion cannot be hidden in a sequence of capacity intervals of vanishing length: the uniform tip estimate already controls all times in those intervals.

To include the terminal part of the curve, first fix a small cap about \(1\). Chordal SLE tends to its target almost surely [20]; hence with probability as close to one as desired its tail after a sufficiently large deterministic time lies in a smaller cap. By (67), the discrete curve has entered the original cap by that time with the same high probability. Lemma 21 bounds the diameter of its subsequent tail. Letting the cap shrink therefore makes both tails uniformly small. Parametrize the two curves by the common capacity clock until the chosen time and then by any increasing parametrizations of their tails. This proves convergence in the full uniform oriented-curve metric modulo increasing reparametrization.

The initial inverse disk charts converge uniformly on their closures by marked Jordan convergence. Composing with them transfers the curve convergence to the original physical embedding. Finally, the drawing comparison gives \(d(\gamma_n,\widehat\gamma_n)=O(\delta_n)\) in that same oriented metric. We have proved the asserted weak convergence of quenched laws for every environment in the full-probability event of Lemma 16.

Return now to the entire prescribed mesh sequence. The oriented-curve space, with constant waiting intervals identified, is separable, and its weak probability-law topology is metrized by the bounded-Lipschitz distance for the bounded curve metric. Each finite quenched law is measurable in its bonds. From every subsequence, Lemma 16 selects a further deterministic subsequence along which this distance from the SLE law tends to zero almost surely. The subsequence criterion for convergence in probability proves convergence in environmental probability along the full sequence. All choices requiring small disorder were made before choosing the domain and its approximation, as in the polygon comparison. This completes the proof. ◻